LMFDB - Embedded newform 1045.4.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.41218$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41218 q^{2} -9.99772 q^{3} +3.64299 q^{4} +5.00000 q^{5} +34.1140 q^{6} -3.40924 q^{7} +14.8669 q^{8} +72.9544 q^{9} +O(q^{10})$	\(q-3.41218 q^{2} -9.99772 q^{3} +3.64299 q^{4} +5.00000 q^{5} +34.1140 q^{6} -3.40924 q^{7} +14.8669 q^{8} +72.9544 q^{9} -17.0609 q^{10} +11.0000 q^{11} -36.4216 q^{12} -6.49940 q^{13} +11.6329 q^{14} -49.9886 q^{15} -79.8725 q^{16} -39.4206 q^{17} -248.934 q^{18} +19.0000 q^{19} +18.2149 q^{20} +34.0846 q^{21} -37.5340 q^{22} -88.1922 q^{23} -148.635 q^{24} +25.0000 q^{25} +22.1771 q^{26} -459.439 q^{27} -12.4198 q^{28} +42.7789 q^{29} +170.570 q^{30} -128.680 q^{31} +153.604 q^{32} -109.975 q^{33} +134.510 q^{34} -17.0462 q^{35} +265.772 q^{36} +152.245 q^{37} -64.8315 q^{38} +64.9792 q^{39} +74.3346 q^{40} +257.243 q^{41} -116.303 q^{42} +46.2259 q^{43} +40.0729 q^{44} +364.772 q^{45} +300.928 q^{46} -499.380 q^{47} +798.543 q^{48} -331.377 q^{49} -85.3046 q^{50} +394.116 q^{51} -23.6772 q^{52} +305.432 q^{53} +1567.69 q^{54} +55.0000 q^{55} -50.6849 q^{56} -189.957 q^{57} -145.969 q^{58} +535.298 q^{59} -182.108 q^{60} +167.249 q^{61} +439.079 q^{62} -248.719 q^{63} +114.855 q^{64} -32.4970 q^{65} +375.254 q^{66} -223.592 q^{67} -143.609 q^{68} +881.721 q^{69} +58.1647 q^{70} +906.666 q^{71} +1084.61 q^{72} -215.861 q^{73} -519.488 q^{74} -249.943 q^{75} +69.2167 q^{76} -37.5016 q^{77} -221.721 q^{78} -215.529 q^{79} -399.363 q^{80} +2623.57 q^{81} -877.760 q^{82} -1.41222 q^{83} +124.170 q^{84} -197.103 q^{85} -157.731 q^{86} -427.691 q^{87} +163.536 q^{88} +229.956 q^{89} -1244.67 q^{90} +22.1580 q^{91} -321.283 q^{92} +1286.50 q^{93} +1703.98 q^{94} +95.0000 q^{95} -1535.69 q^{96} -517.635 q^{97} +1130.72 q^{98} +802.498 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	−3.41218	−1.20639	−0.603194	−	0.797594i	$-0.706106\pi$
$2$	−3.41218	−1.20639	−0.603194	+	0.797594i	$0.706106\pi$
$3$	−9.99772	−1.92406	−0.962031	−	0.272941i	$-0.912004\pi$
$3$	−9.99772	−1.92406	−0.962031	+	0.272941i	$0.912004\pi$
$4$	3.64299	0.455373
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	34.1140	2.32117
$7$	−3.40924	−0.184082	−0.0920408	−	0.995755i	$-0.529339\pi$
$7$	−3.40924	−0.184082	−0.0920408	+	0.995755i	$0.529339\pi$
$8$	14.8669	0.657031
$9$	72.9544	2.70201
$10$	−17.0609	−0.539513
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−36.4216	−0.876166
$13$	−6.49940	−0.138662	−0.0693311	−	0.997594i	$-0.522086\pi$
$13$	−6.49940	−0.138662	−0.0693311	+	0.997594i	$0.522086\pi$
$14$	11.6329	0.222074
$15$	−49.9886	−0.860467
$16$	−79.8725	−1.24801
$17$	−39.4206	−0.562406	−0.281203	−	0.959648i	$-0.590733\pi$
$17$	−39.4206	−0.562406	−0.281203	+	0.959648i	$0.590733\pi$
$18$	−248.934	−3.25968
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	18.2149	0.203649
$21$	34.0846	0.354184
$22$	−37.5340	−0.363740
$23$	−88.1922	−0.799537	−0.399769	−	0.916616i	$-0.630909\pi$
$23$	−88.1922	−0.799537	−0.399769	+	0.916616i	$0.630909\pi$
$24$	−148.635	−1.26417
$25$	25.0000	0.200000
$26$	22.1771	0.167281
$27$	−459.439	−3.27478
$28$	−12.4198	−0.0838259
$29$	42.7789	0.273925	0.136963	−	0.990576i	$-0.456266\pi$
$29$	42.7789	0.273925	0.136963	+	0.990576i	$0.456266\pi$
$30$	170.570	1.03806
$31$	−128.680	−0.745534	−0.372767	−	0.927925i	$-0.621591\pi$
$31$	−128.680	−0.745534	−0.372767	+	0.927925i	$0.621591\pi$
$32$	153.604	0.848552
$33$	−109.975	−0.580126
$34$	134.510	0.678480
$35$	−17.0462	−0.0823238
$36$	265.772	1.23042
$37$	152.245	0.676458	0.338229	−	0.941064i	$-0.390172\pi$
$37$	152.245	0.676458	0.338229	+	0.941064i	$0.390172\pi$
$38$	−64.8315	−0.276765
$39$	64.9792	0.266795
$40$	74.3346	0.293833
$41$	257.243	0.979869	0.489934	−	0.871759i	$-0.337021\pi$
$41$	257.243	0.979869	0.489934	+	0.871759i	$0.337021\pi$
$42$	−116.303	−0.427284
$43$	46.2259	0.163939	0.0819697	−	0.996635i	$-0.473879\pi$
$43$	46.2259	0.163939	0.0819697	+	0.996635i	$0.473879\pi$
$44$	40.0729	0.137300
$45$	364.772	1.20838
$46$	300.928	0.964552
$47$	−499.380	−1.54983	−0.774916	−	0.632065i	$-0.782208\pi$
$47$	−499.380	−1.54983	−0.774916	+	0.632065i	$0.782208\pi$
$48$	798.543	2.40125
$49$	−331.377	−0.966114
$50$	−85.3046	−0.241278
$51$	394.116	1.08210
$52$	−23.6772	−0.0631431
$53$	305.432	0.791590	0.395795	−	0.918339i	$-0.370469\pi$
$53$	305.432	0.791590	0.395795	+	0.918339i	$0.370469\pi$
$54$	1567.69	3.95066
$55$	55.0000	0.134840
$56$	−50.6849	−0.120947
$57$	−189.957	−0.441410
$58$	−145.969	−0.330460
$59$	535.298	1.18118	0.590592	−	0.806970i	$-0.298894\pi$
$59$	535.298	1.18118	0.590592	+	0.806970i	$0.298894\pi$
$60$	−182.108	−0.391834
$61$	167.249	0.351051	0.175525	−	0.984475i	$-0.443838\pi$
$61$	167.249	0.351051	0.175525	+	0.984475i	$0.443838\pi$
$62$	439.079	0.899404
$63$	−248.719	−0.497391
$64$	114.855	0.224325
$65$	−32.4970	−0.0620116
$66$	375.254	0.699858
$67$	−223.592	−0.407704	−0.203852	−	0.979002i	$-0.565346\pi$
$67$	−223.592	−0.407704	−0.203852	+	0.979002i	$0.565346\pi$
$68$	−143.609	−0.256105
$69$	881.721	1.53836
$70$	58.1647	0.0993145
$71$	906.666	1.51551	0.757757	−	0.652537i	$-0.226295\pi$
$71$	906.666	1.51551	0.757757	+	0.652537i	$0.226295\pi$
$72$	1084.61	1.77531
$73$	−215.861	−0.346091	−0.173045	−	0.984914i	$-0.555361\pi$
$73$	−215.861	−0.346091	−0.173045	+	0.984914i	$0.555361\pi$
$74$	−519.488	−0.816071
$75$	−249.943	−0.384812
$76$	69.2167	0.104470
$77$	−37.5016	−0.0555027
$78$	−221.721	−0.321858
$79$	−215.529	−0.306948	−0.153474	−	0.988153i	$-0.549046\pi$
$79$	−215.529	−0.306948	−0.153474	+	0.988153i	$0.549046\pi$
$80$	−399.363	−0.558126
$81$	2623.57	3.59886
$82$	−877.760	−1.18210
$83$	−1.41222	−0.00186760	−0.000933800	−	1.00000i	$-0.500297\pi$
$83$	−1.41222	−0.00186760	−0.000933800		1.00000i	$0.500297\pi$
$84$	124.170	0.161286
$85$	−197.103	−0.251515
$86$	−157.731	−0.197774
$87$	−427.691	−0.527049
$88$	163.536	0.198102
$89$	229.956	0.273879	0.136940	−	0.990579i	$-0.456273\pi$
$89$	229.956	0.273879	0.136940	+	0.990579i	$0.456273\pi$
$90$	−1244.67	−1.45777
$91$	22.1580	0.0255252
$92$	−321.283	−0.364088
$93$	1286.50	1.43445
$94$	1703.98	1.86970
$95$	95.0000	0.102598
$96$	−1535.69	−1.63267
$97$	−517.635	−0.541834	−0.270917	−	0.962603i	$-0.587327\pi$
$97$	−517.635	−0.541834	−0.270917	+	0.962603i	$0.587327\pi$
$98$	1130.72	1.16551
$99$	802.498	0.814688
$100$	91.0747	0.0910747
$101$	−1176.48	−1.15905	−0.579527	−	0.814953i	$-0.696763\pi$
$101$	−1176.48	−1.15905	−0.579527	+	0.814953i	$0.696763\pi$
$102$	−1344.80	−1.30544
$103$	−1608.76	−1.53899	−0.769493	−	0.638655i	$-0.779491\pi$
$103$	−1608.76	−1.53899	−0.769493	+	0.638655i	$0.779491\pi$
$104$	−96.6261	−0.0911054
$105$	170.423	0.158396
$106$	−1042.19	−0.954965
$107$	321.501	0.290474	0.145237	−	0.989397i	$-0.453606\pi$
$107$	321.501	0.290474	0.145237	+	0.989397i	$0.453606\pi$
$108$	−1673.73	−1.49125
$109$	890.448	0.782472	0.391236	−	0.920290i	$-0.372048\pi$
$109$	890.448	0.782472	0.391236	+	0.920290i	$0.372048\pi$
$110$	−187.670	−0.162669
$111$	−1522.10	−1.30155
$112$	272.305	0.229735
$113$	967.886	0.805761	0.402881	−	0.915252i	$-0.368009\pi$
$113$	967.886	0.805761	0.402881	+	0.915252i	$0.368009\pi$
$114$	648.167	0.532512
$115$	−440.961	−0.357564
$116$	155.843	0.124738
$117$	−474.160	−0.374667
$118$	−1826.53	−1.42497
$119$	134.394	0.103529
$120$	−743.176	−0.565353
$121$	121.000	0.0909091
$122$	−570.686	−0.423504
$123$	−2571.84	−1.88533
$124$	−468.778	−0.339496
$125$	125.000	0.0894427
$126$	848.674	0.600047
$127$	2155.72	1.50622	0.753109	−	0.657896i	$-0.228553\pi$
$127$	2155.72	1.50622	0.753109	+	0.657896i	$0.228553\pi$
$128$	−1620.74	−1.11918
$129$	−462.154	−0.315429
$130$	110.886	0.0748101
$131$	328.200	0.218893	0.109447	−	0.993993i	$-0.465092\pi$
$131$	328.200	0.218893	0.109447	+	0.993993i	$0.465092\pi$
$132$	−400.637	−0.264174
$133$	−64.7756	−0.0422312
$134$	762.938	0.491849
$135$	−2297.19	−1.46453
$136$	−586.063	−0.369518
$137$	1712.49	1.06794	0.533969	−	0.845504i	$-0.320700\pi$
$137$	1712.49	1.06794	0.533969	+	0.845504i	$0.320700\pi$
$138$	−3008.59	−1.85586
$139$	−1172.17	−0.715269	−0.357634	−	0.933862i	$-0.616417\pi$
$139$	−1172.17	−0.715269	−0.357634	+	0.933862i	$0.616417\pi$
$140$	−62.0991	−0.0374881
$141$	4992.66	2.98197
$142$	−3093.71	−1.82830
$143$	−71.4934	−0.0418082
$144$	−5827.05	−3.37214
$145$	213.894	0.122503
$146$	736.557	0.417520
$147$	3313.01	1.85886
$148$	554.627	0.308041
$149$	3118.28	1.71449	0.857247	−	0.514906i	$-0.172173\pi$
$149$	3118.28	1.71449	0.857247	+	0.514906i	$0.172173\pi$
$150$	852.851	0.464233
$151$	1678.94	0.904834	0.452417	−	0.891806i	$-0.350562\pi$
$151$	1678.94	0.904834	0.452417	+	0.891806i	$0.350562\pi$
$152$	282.472	0.150733
$153$	−2875.90	−1.51963
$154$	127.962	0.0669578
$155$	−643.398	−0.333413
$156$	236.718	0.121491
$157$	2700.85	1.37294	0.686470	−	0.727158i	$-0.259159\pi$
$157$	2700.85	1.37294	0.686470	+	0.727158i	$0.259159\pi$
$158$	735.425	0.370299
$159$	−3053.62	−1.52307
$160$	768.021	0.379484
$161$	300.668	0.147180
$162$	−8952.10	−4.34163
$163$	−1734.78	−0.833610	−0.416805	−	0.908996i	$-0.636850\pi$
$163$	−1734.78	−0.833610	−0.416805	+	0.908996i	$0.636850\pi$
$164$	937.133	0.446206
$165$	−549.874	−0.259440
$166$	4.81874	0.00225305
$167$	−2153.78	−0.997991	−0.498995	−	0.866605i	$-0.666298\pi$
$167$	−2153.78	−0.997991	−0.498995	+	0.866605i	$0.666298\pi$
$168$	506.733	0.232710
$169$	−2154.76	−0.980773
$170$	672.551	0.303425
$171$	1386.13	0.619884
$172$	168.400	0.0746536
$173$	−3665.16	−1.61073	−0.805366	−	0.592778i	$-0.798031\pi$
$173$	−3665.16	−1.61073	−0.805366	+	0.592778i	$0.798031\pi$
$174$	1459.36	0.635826
$175$	−85.2310	−0.0368163
$176$	−878.598	−0.376289
$177$	−5351.76	−2.27267
$178$	−784.650	−0.330405
$179$	−2513.72	−1.04963	−0.524816	−	0.851216i	$-0.675866\pi$
$179$	−2513.72	−1.04963	−0.524816	+	0.851216i	$0.675866\pi$
$180$	1328.86	0.550263
$181$	−1407.75	−0.578104	−0.289052	−	0.957313i	$-0.593340\pi$
$181$	−1407.75	−0.578104	−0.289052	+	0.957313i	$0.593340\pi$
$182$	−75.6072	−0.0307933
$183$	−1672.11	−0.675443
$184$	−1311.15	−0.525321
$185$	761.226	0.302521
$186$	−4389.78	−1.73051
$187$	−433.626	−0.169572
$188$	−1819.23	−0.705752
$189$	1566.34	0.602827
$190$	−324.157	−0.123773
$191$	−1781.72	−0.674976	−0.337488	−	0.941330i	$-0.609577\pi$
$191$	−1781.72	−0.674976	−0.337488	+	0.941330i	$0.609577\pi$
$192$	−1148.28	−0.431616
$193$	2783.10	1.03799	0.518995	−	0.854777i	$-0.326306\pi$
$193$	2783.10	1.03799	0.518995	+	0.854777i	$0.326306\pi$
$194$	1766.27	0.653662
$195$	324.896	0.119314
$196$	−1207.20	−0.439943
$197$	−4229.34	−1.52958	−0.764792	−	0.644277i	$-0.777158\pi$
$197$	−4229.34	−1.52958	−0.764792	+	0.644277i	$0.777158\pi$
$198$	−2738.27	−0.982830
$199$	−175.168	−0.0623985	−0.0311993	−	0.999513i	$-0.509933\pi$
$199$	−175.168	−0.0623985	−0.0311993	+	0.999513i	$0.509933\pi$
$200$	371.673	0.131406
$201$	2235.41	0.784447
$202$	4014.38	1.39827
$203$	−145.843	−0.0504246
$204$	1435.76	0.492761
$205$	1286.22	0.438211
$206$	5489.37	1.85662
$207$	−6434.01	−2.16036
$208$	519.124	0.173052
$209$	209.000	0.0691714
$210$	−581.515	−0.191087
$211$	1404.06	0.458101	0.229050	−	0.973415i	$-0.426438\pi$
$211$	1404.06	0.458101	0.229050	+	0.973415i	$0.426438\pi$
$212$	1112.68	0.360469
$213$	−9064.59	−2.91594
$214$	−1097.02	−0.350424
$215$	231.130	0.0733159
$216$	−6830.44	−2.15163
$217$	438.700	0.137239
$218$	−3038.37	−0.943965
$219$	2158.12	0.665900
$220$	200.364	0.0614025
$221$	256.210	0.0779844
$222$	5193.70	1.57017
$223$	5660.34	1.69975	0.849876	−	0.526983i	$-0.176677\pi$
$223$	5660.34	1.69975	0.849876	+	0.526983i	$0.176677\pi$
$224$	−523.674	−0.156203
$225$	1823.86	0.540403
$226$	−3302.60	−0.972061
$227$	1038.03	0.303508	0.151754	−	0.988418i	$-0.451508\pi$
$227$	1038.03	0.303508	0.151754	+	0.988418i	$0.451508\pi$
$228$	−692.010	−0.201006
$229$	−3830.77	−1.10543	−0.552717	−	0.833369i	$-0.686409\pi$
$229$	−3830.77	−1.10543	−0.552717	+	0.833369i	$0.686409\pi$
$230$	1504.64	0.431361
$231$	374.931	0.106791
$232$	635.990	0.179978
$233$	−349.957	−0.0983967	−0.0491983	−	0.998789i	$-0.515667\pi$
$233$	−349.957	−0.0983967	−0.0491983	+	0.998789i	$0.515667\pi$
$234$	1617.92	0.451994
$235$	−2496.90	−0.693106
$236$	1950.08	0.537880
$237$	2154.80	0.590588
$238$	−458.578	−0.124896
$239$	−2513.17	−0.680181	−0.340091	−	0.940393i	$-0.610458\pi$
$239$	−2513.17	−0.680181	−0.340091	+	0.940393i	$0.610458\pi$
$240$	3992.72	1.07387
$241$	5428.40	1.45093	0.725464	−	0.688260i	$-0.241625\pi$
$241$	5428.40	1.45093	0.725464	+	0.688260i	$0.241625\pi$
$242$	−412.874	−0.109672
$243$	−13824.9	−3.64965
$244$	609.288	0.159859
$245$	−1656.89	−0.432059
$246$	8775.60	2.27444
$247$	−123.489	−0.0318113
$248$	−1913.07	−0.489839
$249$	14.1189	0.00359338
$250$	−426.523	−0.107903
$251$	5565.76	1.39963	0.699817	−	0.714323i	$-0.253265\pi$
$251$	5565.76	1.39963	0.699817	+	0.714323i	$0.253265\pi$
$252$	−906.080	−0.226499
$253$	−970.115	−0.241069
$254$	−7355.73	−1.81708
$255$	1970.58	0.483931
$256$	4611.42	1.12583
$257$	1278.09	0.310214	0.155107	−	0.987898i	$-0.450428\pi$
$257$	1278.09	0.310214	0.155107	+	0.987898i	$0.450428\pi$
$258$	1576.95	0.380530
$259$	−519.040	−0.124524
$260$	−118.386	−0.0282384
$261$	3120.90	0.740150
$262$	−1119.88	−0.264070
$263$	−913.712	−0.214228	−0.107114	−	0.994247i	$-0.534161\pi$
$263$	−913.712	−0.214228	−0.107114	+	0.994247i	$0.534161\pi$
$264$	−1634.99	−0.381161
$265$	1527.16	0.354010
$266$	221.026	0.0509473
$267$	−2299.03	−0.526960
$268$	−814.544	−0.185657
$269$	5024.64	1.13888	0.569438	−	0.822034i	$-0.307161\pi$
$269$	5024.64	1.13888	0.569438	+	0.822034i	$0.307161\pi$
$270$	7838.44	1.76679
$271$	540.948	0.121255	0.0606277	−	0.998160i	$-0.480690\pi$
$271$	540.948	0.121255	0.0606277	+	0.998160i	$0.480690\pi$
$272$	3148.62	0.701887
$273$	−221.530	−0.0491120
$274$	−5843.32	−1.28835
$275$	275.000	0.0603023
$276$	3212.10	0.700527
$277$	6076.84	1.31813	0.659065	−	0.752086i	$-0.270952\pi$
$277$	6076.84	1.31813	0.659065	+	0.752086i	$0.270952\pi$
$278$	3999.67	0.862892
$279$	−9387.74	−2.01444
$280$	−253.425	−0.0540893
$281$	−5589.51	−1.18663	−0.593313	−	0.804971i	$-0.702181\pi$
$281$	−5589.51	−1.18663	−0.593313	+	0.804971i	$0.702181\pi$
$282$	−17035.9	−3.59742
$283$	1774.03	0.372633	0.186316	−	0.982490i	$-0.440345\pi$
$283$	1774.03	0.372633	0.186316	+	0.982490i	$0.440345\pi$
$284$	3302.97	0.690124
$285$	−949.783	−0.197405
$286$	243.948	0.0504370
$287$	−877.003	−0.180376
$288$	11206.1	2.29280
$289$	−3359.02	−0.683700
$290$	−729.846	−0.147786
$291$	5175.17	1.04252
$292$	−786.379	−0.157601
$293$	−4678.24	−0.932785	−0.466392	−	0.884578i	$-0.654447\pi$
$293$	−4678.24	−0.932785	−0.466392	+	0.884578i	$0.654447\pi$
$294$	−11304.6	−2.24251
$295$	2676.49	0.528242
$296$	2263.42	0.444454
$297$	−5053.83	−0.987383
$298$	−10640.1	−2.06835
$299$	573.197	0.110866
$300$	−910.539	−0.175233
$301$	−157.595	−0.0301782
$302$	−5728.84	−1.09158
$303$	11762.2	2.23009
$304$	−1517.58	−0.286313
$305$	836.247	0.156995
$306$	9813.11	1.83326
$307$	7956.78	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$307$	7956.78	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$308$	−136.618	−0.0252745
$309$	16083.9	2.96111
$310$	2195.39	0.402226
$311$	−1323.09	−0.241241	−0.120620	−	0.992699i	$-0.538488\pi$
$311$	−1323.09	−0.241241	−0.120620	+	0.992699i	$0.538488\pi$
$312$	966.040	0.175292
$313$	−5074.95	−0.916463	−0.458232	−	0.888833i	$-0.651517\pi$
$313$	−5074.95	−0.916463	−0.458232	+	0.888833i	$0.651517\pi$
$314$	−9215.81	−1.65630
$315$	−1243.59	−0.222440
$316$	−785.170	−0.139776
$317$	−5000.74	−0.886023	−0.443012	−	0.896516i	$-0.646090\pi$
$317$	−5000.74	−0.886023	−0.443012	+	0.896516i	$0.646090\pi$
$318$	10419.5	1.83741
$319$	470.567	0.0825916
$320$	574.273	0.100321
$321$	−3214.28	−0.558890
$322$	−1025.94	−0.177556
$323$	−748.991	−0.129025
$324$	9557.63	1.63883
$325$	−162.485	−0.0277324
$326$	5919.38	1.00566
$327$	−8902.45	−1.50552
$328$	3824.41	0.643804
$329$	1702.51	0.285295
$330$	1876.27	0.312986
$331$	−7826.20	−1.29960	−0.649799	−	0.760106i	$-0.725147\pi$
$331$	−7826.20	−1.29960	−0.649799	+	0.760106i	$0.725147\pi$
$332$	−5.14468	−0.000850455 0
$333$	11106.9	1.82780
$334$	7349.09	1.20396
$335$	−1117.96	−0.182331
$336$	−2722.42	−0.442025
$337$	1511.36	0.244300	0.122150	−	0.992512i	$-0.461021\pi$
$337$	1511.36	0.244300	0.122150	+	0.992512i	$0.461021\pi$
$338$	7352.43	1.18319
$339$	−9676.65	−1.55033
$340$	−718.043	−0.114533
$341$	−1415.48	−0.224787
$342$	−4729.74	−0.747821
$343$	2299.11	0.361925
$344$	687.238	0.107713
$345$	4408.61	0.687975
$346$	12506.2	1.94317
$347$	4150.88	0.642163	0.321082	−	0.947052i	$-0.395954\pi$
$347$	4150.88	0.642163	0.321082	+	0.947052i	$0.395954\pi$
$348$	−1558.07	−0.240004
$349$	9344.40	1.43322	0.716610	−	0.697474i	$-0.245693\pi$
$349$	9344.40	1.43322	0.716610	+	0.697474i	$0.245693\pi$
$350$	290.824	0.0444148
$351$	2986.08	0.454088
$352$	1689.65	0.255848
$353$	−449.766	−0.0678148	−0.0339074	−	0.999425i	$-0.510795\pi$
$353$	−449.766	−0.0678148	−0.0339074	+	0.999425i	$0.510795\pi$
$354$	18261.2	2.74172
$355$	4533.33	0.677758
$356$	837.725	0.124717
$357$	−1343.64	−0.199195
$358$	8577.27	1.26626
$359$	−1753.36	−0.257769	−0.128884	−	0.991660i	$-0.541140\pi$
$359$	−1753.36	−0.257769	−0.128884	+	0.991660i	$0.541140\pi$
$360$	5423.03	0.793942
$361$	361.000	0.0526316
$362$	4803.48	0.697418
$363$	−1209.72	−0.174915
$364$	80.7213	0.0116235
$365$	−1079.31	−0.154777
$366$	5705.55	0.814847
$367$	−767.933	−0.109226	−0.0546128	−	0.998508i	$-0.517392\pi$
$367$	−767.933	−0.109226	−0.0546128	+	0.998508i	$0.517392\pi$
$368$	7044.14	0.997829
$369$	18767.0	2.64762
$370$	−2597.44	−0.364958
$371$	−1041.29	−0.145717
$372$	4686.71	0.653212
$373$	−11081.0	−1.53821	−0.769103	−	0.639125i	$-0.779297\pi$
$373$	−11081.0	−1.53821	−0.769103	+	0.639125i	$0.779297\pi$
$374$	1479.61	0.204569
$375$	−1249.71	−0.172093
$376$	−7424.24	−1.01829
$377$	−278.037	−0.0379831
$378$	−5344.63	−0.727243
$379$	−8323.03	−1.12803	−0.564017	−	0.825763i	$-0.690745\pi$
$379$	−8323.03	−1.12803	−0.564017	+	0.825763i	$0.690745\pi$
$380$	346.084	0.0467203
$381$	−21552.3	−2.89806
$382$	6079.54	0.814283
$383$	−13017.9	−1.73678	−0.868389	−	0.495884i	$-0.834844\pi$
$383$	−13017.9	−1.73678	−0.868389	+	0.495884i	$0.834844\pi$
$384$	16203.7	2.15336
$385$	−187.508	−0.0248216
$386$	−9496.44	−1.25222
$387$	3372.38	0.442966
$388$	−1885.74	−0.246737
$389$	−5090.82	−0.663534	−0.331767	−	0.943361i	$-0.607645\pi$
$389$	−5090.82	−0.663534	−0.331767	+	0.943361i	$0.607645\pi$
$390$	−1108.60	−0.143939
$391$	3476.59	0.449664
$392$	−4926.56	−0.634767
$393$	−3281.25	−0.421164
$394$	14431.3	1.84527
$395$	−1077.65	−0.137271
$396$	2923.49	0.370987
$397$	3748.54	0.473888	0.236944	−	0.971523i	$-0.423854\pi$
$397$	3748.54	0.473888	0.236944	+	0.971523i	$0.423854\pi$
$398$	597.704	0.0752769
$399$	647.608	0.0812555
$400$	−1996.81	−0.249602
$401$	2176.45	0.271039	0.135520	−	0.990775i	$-0.456730\pi$
$401$	2176.45	0.271039	0.135520	+	0.990775i	$0.456730\pi$
$402$	−7627.64	−0.946348
$403$	836.341	0.103377
$404$	−4285.91	−0.527802
$405$	13117.9	1.60946
$406$	497.644	0.0608317
$407$	1674.70	0.203960
$408$	5859.29	0.710976
$409$	−2331.72	−0.281897	−0.140949	−	0.990017i	$-0.545015\pi$
$409$	−2331.72	−0.281897	−0.140949	+	0.990017i	$0.545015\pi$
$410$	−4388.80	−0.528652
$411$	−17121.0	−2.05478
$412$	−5860.68	−0.700814
$413$	−1824.96	−0.217434
$414$	21954.0	2.60623
$415$	−7.06108	−0.000835216 0
$416$	−998.336	−0.117662
$417$	11719.0	1.37622
$418$	−713.146	−0.0834476
$419$	−9085.76	−1.05935	−0.529676	−	0.848200i	$-0.677686\pi$
$419$	−9085.76	−1.05935	−0.529676	+	0.848200i	$0.677686\pi$
$420$	620.849	0.0721294
$421$	10852.2	1.25630	0.628151	−	0.778092i	$-0.283812\pi$
$421$	10852.2	1.25630	0.628151	+	0.778092i	$0.283812\pi$
$422$	−4790.90	−0.552647
$423$	−36431.9	−4.18766
$424$	4540.83	0.520099
$425$	−985.515	−0.112481
$426$	30930.0	3.51776
$427$	−570.194	−0.0646220
$428$	1171.22	0.132274
$429$	714.771	0.0804416
$430$	−788.657	−0.0884474
$431$	−9662.44	−1.07987	−0.539934	−	0.841707i	$-0.681551\pi$
$431$	−9662.44	−1.07987	−0.539934	+	0.841707i	$0.681551\pi$
$432$	36696.5	4.08695
$433$	−1214.50	−0.134793	−0.0673964	−	0.997726i	$-0.521469\pi$
$433$	−1214.50	−0.134793	−0.0673964	+	0.997726i	$0.521469\pi$
$434$	−1496.92	−0.165564
$435$	−2138.45	−0.235704
$436$	3243.89	0.356317
$437$	−1675.65	−0.183426
$438$	−7363.89	−0.803334
$439$	−13044.4	−1.41817	−0.709086	−	0.705122i	$-0.750892\pi$
$439$	−13044.4	−1.41817	−0.709086	+	0.705122i	$0.750892\pi$
$440$	817.681	0.0885941
$441$	−24175.4	−2.61045
$442$	−874.236	−0.0940795
$443$	−149.103	−0.0159911	−0.00799557	−	0.999968i	$-0.502545\pi$
$443$	−149.103	−0.0159911	−0.00799557	+	0.999968i	$0.502545\pi$
$444$	−5545.01	−0.592690
$445$	1149.78	0.122482
$446$	−19314.1	−2.05056
$447$	−31175.7	−3.29879
$448$	−391.567	−0.0412942
$449$	3959.77	0.416199	0.208099	−	0.978108i	$-0.433272\pi$
$449$	3959.77	0.416199	0.208099	+	0.978108i	$0.433272\pi$
$450$	−6223.34	−0.651936
$451$	2829.67	0.295442
$452$	3526.00	0.366922
$453$	−16785.5	−1.74096
$454$	−3541.93	−0.366148
$455$	110.790	0.0114152
$456$	−2824.07	−0.290020
$457$	18516.7	1.89535	0.947674	−	0.319239i	$-0.103427\pi$
$457$	18516.7	1.89535	0.947674	+	0.319239i	$0.103427\pi$
$458$	13071.3	1.33358
$459$	18111.3	1.84175
$460$	−1606.42	−0.162825
$461$	−13049.0	−1.31834	−0.659169	−	0.751995i	$-0.729092\pi$
$461$	−13049.0	−1.31834	−0.659169	+	0.751995i	$0.729092\pi$
$462$	−1279.33	−0.128831
$463$	4230.44	0.424633	0.212316	−	0.977201i	$-0.431899\pi$
$463$	4230.44	0.424633	0.212316	+	0.977201i	$0.431899\pi$
$464$	−3416.86	−0.341861
$465$	6432.52	0.641507
$466$	1194.12	0.118705
$467$	−16723.8	−1.65715	−0.828573	−	0.559881i	$-0.810847\pi$
$467$	−16723.8	−1.65715	−0.828573	+	0.559881i	$0.810847\pi$
$468$	−1727.36	−0.170613
$469$	762.280	0.0750508
$470$	8519.88	0.836155
$471$	−27002.4	−2.64162
$472$	7958.23	0.776075
$473$	508.485	0.0494296
$474$	−7352.57	−0.712478
$475$	475.000	0.0458831
$476$	489.596	0.0471441
$477$	22282.6	2.13889
$478$	8575.39	0.820563
$479$	9467.10	0.903054	0.451527	−	0.892257i	$-0.350879\pi$
$479$	9467.10	0.903054	0.451527	+	0.892257i	$0.350879\pi$
$480$	−7678.46	−0.730150
$481$	−989.502	−0.0937992
$482$	−18522.7	−1.75038
$483$	−3006.00	−0.283184
$484$	440.801	0.0413976
$485$	−2588.18	−0.242315
$486$	47173.0	4.40290
$487$	11828.6	1.10063	0.550313	−	0.834958i	$-0.314508\pi$
$487$	11828.6	1.10063	0.550313	+	0.834958i	$0.314508\pi$
$488$	2486.48	0.230651
$489$	17343.8	1.60392
$490$	5653.59	0.521231
$491$	−16651.1	−1.53045	−0.765226	−	0.643761i	$-0.777373\pi$
$491$	−16651.1	−1.53045	−0.765226	+	0.643761i	$0.777373\pi$
$492$	−9369.19	−0.858528
$493$	−1686.37	−0.154057
$494$	421.366	0.0383768
$495$	4012.49	0.364339
$496$	10278.0	0.930433
$497$	−3091.04	−0.278978
$498$	−48.1764	−0.00433501
$499$	6886.53	0.617802	0.308901	−	0.951094i	$-0.400039\pi$
$499$	6886.53	0.617802	0.308901	+	0.951094i	$0.400039\pi$
$500$	455.373	0.0407298
$501$	21532.9	1.92020
$502$	−18991.4	−1.68850
$503$	−3369.15	−0.298654	−0.149327	−	0.988788i	$-0.547711\pi$
$503$	−3369.15	−0.298654	−0.149327	+	0.988788i	$0.547711\pi$
$504$	−3697.68	−0.326801
$505$	−5882.42	−0.518345
$506$	3310.21	0.290823
$507$	21542.7	1.88707
$508$	7853.28	0.685891
$509$	−716.163	−0.0623642	−0.0311821	−	0.999514i	$-0.509927\pi$
$509$	−716.163	−0.0623642	−0.0311821	+	0.999514i	$0.509927\pi$
$510$	−6723.98	−0.583809
$511$	735.922	0.0637090
$512$	−2769.09	−0.239019
$513$	−8729.34	−0.751286
$514$	−4361.08	−0.374239
$515$	−8043.79	−0.688256
$516$	−1683.62	−0.143638
$517$	−5493.18	−0.467292
$518$	1771.06	0.150224
$519$	36643.2	3.09915
$520$	−483.130	−0.0407436
$521$	21932.7	1.84432	0.922159	−	0.386811i	$-0.126424\pi$
$521$	21932.7	1.84432	0.922159	+	0.386811i	$0.126424\pi$
$522$	−10649.1	−0.892908
$523$	−15869.0	−1.32677	−0.663387	−	0.748276i	$-0.730882\pi$
$523$	−15869.0	−1.32677	−0.663387	+	0.748276i	$0.730882\pi$
$524$	1195.63	0.0996781
$525$	852.115	0.0708369
$526$	3117.75	0.258442
$527$	5072.63	0.419293
$528$	8783.97	0.724003
$529$	−4389.13	−0.360741
$530$	−5210.94	−0.427073
$531$	39052.3	3.19158
$532$	−235.976	−0.0192310
$533$	−1671.93	−0.135871
$534$	7844.71	0.635719
$535$	1607.51	0.129904
$536$	−3324.13	−0.267874
$537$	25131.5	2.01956
$538$	−17145.0	−1.37393
$539$	−3645.15	−0.291294
$540$	−8368.65	−0.666906
$541$	1217.25	0.0967353	0.0483676	−	0.998830i	$-0.484598\pi$
$541$	1217.25	0.0967353	0.0483676	+	0.998830i	$0.484598\pi$
$542$	−1845.81	−0.146281
$543$	14074.2	1.11231
$544$	−6055.17	−0.477230
$545$	4452.24	0.349932
$546$	755.899	0.0592482
$547$	−15655.4	−1.22372	−0.611861	−	0.790965i	$-0.709579\pi$
$547$	−15655.4	−1.22372	−0.611861	+	0.790965i	$0.709579\pi$
$548$	6238.56	0.486311
$549$	12201.6	0.948544
$550$	−938.350	−0.0727480
$551$	812.798	0.0628428
$552$	13108.5	1.01075
$553$	734.790	0.0565036
$554$	−20735.3	−1.59018
$555$	−7610.52	−0.582070
$556$	−4270.21	−0.325714
$557$	−13712.0	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$557$	−13712.0	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$558$	32032.7	2.43020
$559$	−300.441	−0.0227322
$560$	1361.52	0.102741
$561$	4335.28	0.326266
$562$	19072.4	1.43153
$563$	−19995.8	−1.49684	−0.748420	−	0.663225i	$-0.769187\pi$
$563$	−19995.8	−1.49684	−0.748420	+	0.663225i	$0.769187\pi$
$564$	18188.2	1.35791
$565$	4839.43	0.360347
$566$	−6053.31	−0.449540
$567$	−8944.38	−0.662484
$568$	13479.3	0.995740
$569$	3824.27	0.281760	0.140880	−	0.990027i	$-0.455007\pi$
$569$	3824.27	0.281760	0.140880	+	0.990027i	$0.455007\pi$
$570$	3240.83	0.238147
$571$	−24392.8	−1.78775	−0.893876	−	0.448314i	$-0.852025\pi$
$571$	−24392.8	−1.78775	−0.893876	+	0.448314i	$0.852025\pi$
$572$	−260.449	−0.0190384
$573$	17813.1	1.29870
$574$	2992.49	0.217603
$575$	−2204.81	−0.159907
$576$	8379.14	0.606130
$577$	9588.87	0.691837	0.345918	−	0.938265i	$-0.387567\pi$
$577$	9588.87	0.691837	0.345918	+	0.938265i	$0.387567\pi$
$578$	11461.6	0.824808
$579$	−27824.7	−1.99716
$580$	779.214	0.0557847
$581$	4.81458	0.000343791 0
$582$	−17658.6	−1.25769
$583$	3359.75	0.238673
$584$	−3209.19	−0.227393
$585$	−2370.80	−0.167556
$586$	15963.0	1.12530
$587$	−3829.92	−0.269298	−0.134649	−	0.990893i	$-0.542991\pi$
$587$	−3829.92	−0.269298	−0.134649	+	0.990893i	$0.542991\pi$
$588$	12069.3	0.846477
$589$	−2444.91	−0.171037
$590$	−9132.67	−0.637265
$591$	42283.7	2.94301
$592$	−12160.2	−0.844226
$593$	−22150.1	−1.53389	−0.766944	−	0.641714i	$-0.778224\pi$
$593$	−22150.1	−1.53389	−0.766944	+	0.641714i	$0.778224\pi$
$594$	17244.6	1.19117
$595$	671.971	0.0462994
$596$	11359.9	0.780735
$597$	1751.28	0.120059
$598$	−1955.85	−0.133747
$599$	5236.82	0.357213	0.178607	−	0.983921i	$-0.442841\pi$
$599$	5236.82	0.357213	0.178607	+	0.983921i	$0.442841\pi$
$600$	−3715.88	−0.252834
$601$	−4937.22	−0.335097	−0.167548	−	0.985864i	$-0.553585\pi$
$601$	−4937.22	−0.335097	−0.167548	+	0.985864i	$0.553585\pi$
$602$	537.744	0.0364067
$603$	−16312.0	−1.10162
$604$	6116.35	0.412038
$605$	605.000	0.0406558
$606$	−40134.6	−2.69036
$607$	1747.55	0.116855	0.0584274	−	0.998292i	$-0.481391\pi$
$607$	1747.55	0.116855	0.0584274	+	0.998292i	$0.481391\pi$
$608$	2918.48	0.194671
$609$	1458.10	0.0970201
$610$	−2853.43	−0.189397
$611$	3245.67	0.214903
$612$	−10476.9	−0.691998
$613$	3552.41	0.234063	0.117031	−	0.993128i	$-0.462662\pi$
$613$	3552.41	0.234063	0.117031	+	0.993128i	$0.462662\pi$
$614$	−27150.0	−1.78450
$615$	−12859.2	−0.843144
$616$	−557.534	−0.0364670
$617$	16316.2	1.06461	0.532307	−	0.846551i	$-0.321325\pi$
$617$	16316.2	1.06461	0.532307	+	0.846551i	$0.321325\pi$
$618$	−54881.2	−3.57224
$619$	−24509.5	−1.59147	−0.795734	−	0.605647i	$-0.792915\pi$
$619$	−24509.5	−1.59147	−0.795734	+	0.605647i	$0.792915\pi$
$620$	−2343.89	−0.151827
$621$	40518.9	2.61831
$622$	4514.64	0.291030
$623$	−783.973	−0.0504161
$624$	−5190.05	−0.332962
$625$	625.000	0.0400000
$626$	17316.6	1.10561
$627$	−2089.52	−0.133090
$628$	9839.18	0.625200
$629$	−6001.59	−0.380444
$630$	4243.37	0.268349
$631$	−19405.7	−1.22429	−0.612146	−	0.790745i	$-0.709693\pi$
$631$	−19405.7	−1.22429	−0.612146	+	0.790745i	$0.709693\pi$
$632$	−3204.26	−0.201675
$633$	−14037.4	−0.881414
$634$	17063.4	1.06889
$635$	10778.6	0.673601
$636$	−11124.3	−0.693565
$637$	2153.75	0.133964
$638$	−1605.66	−0.0996376
$639$	66145.2	4.09494
$640$	−8103.69	−0.500510
$641$	−19064.1	−1.17471	−0.587354	−	0.809330i	$-0.699830\pi$
$641$	−19064.1	−1.17471	−0.587354	+	0.809330i	$0.699830\pi$
$642$	10967.7	0.674238
$643$	−6246.68	−0.383118	−0.191559	−	0.981481i	$-0.561354\pi$
$643$	−6246.68	−0.383118	−0.191559	+	0.981481i	$0.561354\pi$
$644$	1095.33	0.0670219
$645$	−2310.77	−0.141064
$646$	2555.69	0.155654
$647$	8977.07	0.545479	0.272739	−	0.962088i	$-0.412070\pi$
$647$	8977.07	0.545479	0.272739	+	0.962088i	$0.412070\pi$
$648$	39004.4	2.36457
$649$	5888.28	0.356140
$650$	554.428	0.0334561
$651$	−4386.00	−0.264057
$652$	−6319.78	−0.379604
$653$	−24936.4	−1.49439	−0.747195	−	0.664605i	$-0.768600\pi$
$653$	−24936.4	−1.49439	−0.747195	+	0.664605i	$0.768600\pi$
$654$	30376.8	1.81625
$655$	1641.00	0.0978919
$656$	−20546.7	−1.22288
$657$	−15748.0	−0.935142
$658$	−5809.26	−0.344177
$659$	−14809.7	−0.875422	−0.437711	−	0.899116i	$-0.644211\pi$
$659$	−14809.7	−0.875422	−0.437711	+	0.899116i	$0.644211\pi$
$660$	−2003.19	−0.118142
$661$	−5340.06	−0.314228	−0.157114	−	0.987581i	$-0.550219\pi$
$661$	−5340.06	−0.314228	−0.157114	+	0.987581i	$0.550219\pi$
$662$	26704.4	1.56782
$663$	−2561.52	−0.150047
$664$	−20.9953	−0.00122707
$665$	−323.878	−0.0188864
$666$	−37898.9	−2.20504
$667$	−3772.76	−0.219013
$668$	−7846.19	−0.454458
$669$	−56590.5	−3.27043
$670$	3814.69	0.219962
$671$	1839.74	0.105846
$672$	5235.54	0.300544
$673$	5883.89	0.337009	0.168505	−	0.985701i	$-0.446106\pi$
$673$	5883.89	0.337009	0.168505	+	0.985701i	$0.446106\pi$
$674$	−5157.03	−0.294720
$675$	−11486.0	−0.654956
$676$	−7849.75	−0.446618
$677$	6818.57	0.387089	0.193544	−	0.981092i	$-0.438002\pi$
$677$	6818.57	0.387089	0.193544	+	0.981092i	$0.438002\pi$
$678$	33018.5	1.87031
$679$	1764.74	0.0997417
$680$	−2930.31	−0.165254
$681$	−10377.9	−0.583967
$682$	4829.86	0.271180
$683$	−17108.0	−0.958446	−0.479223	−	0.877693i	$-0.659081\pi$
$683$	−17108.0	−0.958446	−0.479223	+	0.877693i	$0.659081\pi$
$684$	5049.66	0.282279
$685$	8562.43	0.477597
$686$	−7844.99	−0.436623
$687$	38299.0	2.12692
$688$	−3692.18	−0.204598
$689$	−1985.12	−0.109764
$690$	−15043.0	−0.829965
$691$	−4774.26	−0.262838	−0.131419	−	0.991327i	$-0.541953\pi$
$691$	−4774.26	−0.262838	−0.131419	+	0.991327i	$0.541953\pi$
$692$	−13352.1	−0.733485
$693$	−2735.91	−0.149969
$694$	−14163.5	−0.774698
$695$	−5860.86	−0.319878
$696$	−6358.45	−0.346288
$697$	−10140.7	−0.551084
$698$	−31884.8	−1.72902
$699$	3498.77	0.189321
$700$	−310.495	−0.0167652
$701$	−2934.54	−0.158111	−0.0790556	−	0.996870i	$-0.525190\pi$
$701$	−2934.54	−0.158111	−0.0790556	+	0.996870i	$0.525190\pi$
$702$	−10189.0	−0.547807
$703$	2892.66	0.155190
$704$	1263.40	0.0676366
$705$	24963.3	1.33358
$706$	1534.68	0.0818110
$707$	4010.91	0.213361
$708$	−19496.4	−1.03491
$709$	−3546.07	−0.187836	−0.0939179	−	0.995580i	$-0.529939\pi$
$709$	−3546.07	−0.187836	−0.0939179	+	0.995580i	$0.529939\pi$
$710$	−15468.5	−0.817640
$711$	−15723.8	−0.829379
$712$	3418.73	0.179947
$713$	11348.5	0.596082
$714$	4584.73	0.240307
$715$	−357.467	−0.0186972
$716$	−9157.45	−0.477975
$717$	25126.0	1.30871
$718$	5982.79	0.310969
$719$	−31872.2	−1.65317	−0.826587	−	0.562810i	$-0.809720\pi$
$719$	−31872.2	−1.65317	−0.826587	+	0.562810i	$0.809720\pi$
$720$	−29135.2	−1.50806
$721$	5484.64	0.283299
$722$	−1231.80	−0.0634941
$723$	−54271.6	−2.79168
$724$	−5128.40	−0.263253
$725$	1069.47	0.0547851
$726$	4127.80	0.211015
$727$	35074.0	1.78930	0.894651	−	0.446765i	$-0.147424\pi$
$727$	35074.0	1.78930	0.894651	+	0.446765i	$0.147424\pi$
$728$	329.421	0.0167708
$729$	67380.8	3.42330
$730$	3682.79	0.186721
$731$	−1822.25	−0.0922004
$732$	−6091.49	−0.307579
$733$	−8673.65	−0.437065	−0.218533	−	0.975830i	$-0.570127\pi$
$733$	−8673.65	−0.437065	−0.218533	+	0.975830i	$0.570127\pi$
$734$	2620.33	0.131769
$735$	16565.1	0.831309
$736$	−13546.7	−0.678449
$737$	−2459.52	−0.122927
$738$	−64036.4	−3.19406
$739$	−23606.3	−1.17507	−0.587533	−	0.809200i	$-0.699901\pi$
$739$	−23606.3	−1.17507	−0.587533	+	0.809200i	$0.699901\pi$
$740$	2773.14	0.137760
$741$	1234.60	0.0612069
$742$	3553.07	0.175792
$743$	3532.49	0.174421	0.0872103	−	0.996190i	$-0.472205\pi$
$743$	3532.49	0.174421	0.0872103	+	0.996190i	$0.472205\pi$
$744$	19126.3	0.942481
$745$	15591.4	0.766745
$746$	37810.3	1.85567
$747$	−103.027	−0.00504628
$748$	−1579.70	−0.0772184
$749$	−1096.07	−0.0534709
$750$	4264.25	0.207611
$751$	2401.09	0.116667	0.0583335	−	0.998297i	$-0.481421\pi$
$751$	2401.09	0.116667	0.0583335	+	0.998297i	$0.481421\pi$
$752$	39886.8	1.93420
$753$	−55644.9	−2.69298
$754$	948.713	0.0458224
$755$	8394.69	0.404654
$756$	5706.14	0.274511
$757$	16528.4	0.793571	0.396786	−	0.917911i	$-0.370126\pi$
$757$	16528.4	0.793571	0.396786	+	0.917911i	$0.370126\pi$
$758$	28399.7	1.36085
$759$	9698.93	0.463833
$760$	1412.36	0.0674100
$761$	1947.40	0.0927637	0.0463819	−	0.998924i	$-0.485231\pi$
$761$	1947.40	0.0927637	0.0463819	+	0.998924i	$0.485231\pi$
$762$	73540.5	3.49618
$763$	−3035.75	−0.144039
$764$	−6490.77	−0.307366
$765$	−14379.5	−0.679598
$766$	44419.6	2.09523
$767$	−3479.12	−0.163786
$768$	−46103.7	−2.16618
$769$	−1220.34	−0.0572256	−0.0286128	−	0.999591i	$-0.509109\pi$
$769$	−1220.34	−0.0572256	−0.0286128	+	0.999591i	$0.509109\pi$
$770$	639.812	0.0299444
$771$	−12778.0	−0.596872
$772$	10138.8	0.472673
$773$	11092.9	0.516149	0.258074	−	0.966125i	$-0.416912\pi$
$773$	11092.9	0.516149	0.258074	+	0.966125i	$0.416912\pi$
$774$	−11507.2	−0.534389
$775$	−3216.99	−0.149107
$776$	−7695.64	−0.356002
$777$	5189.22	0.239591
$778$	17370.8	0.800480
$779$	4887.62	0.224797
$780$	1183.59	0.0543325
$781$	9973.32	0.456944
$782$	−11862.8	−0.542470
$783$	−19654.3	−0.897045
$784$	26467.9	1.20572
$785$	13504.3	0.613998
$786$	11196.2	0.508087
$787$	7074.76	0.320442	0.160221	−	0.987081i	$-0.448779\pi$
$787$	7074.76	0.320442	0.160221	+	0.987081i	$0.448779\pi$
$788$	−15407.4	−0.696532
$789$	9135.03	0.412187
$790$	3677.12	0.165603
$791$	−3299.75	−0.148326
$792$	11930.7	0.535275
$793$	−1087.02	−0.0486775
$794$	−12790.7	−0.571693
$795$	−15268.1	−0.681137
$796$	−638.133	−0.0284146
$797$	−13460.8	−0.598250	−0.299125	−	0.954214i	$-0.596695\pi$
$797$	−13460.8	−0.598250	−0.299125	+	0.954214i	$0.596695\pi$
$798$	−2209.76	−0.0980257
$799$	19685.9	0.871634
$800$	3840.11	0.169710
$801$	16776.3	0.740025
$802$	−7426.45	−0.326979
$803$	−2374.47	−0.104350
$804$	8143.58	0.357216
$805$	1503.34	0.0658209
$806$	−2853.75	−0.124713
$807$	−50234.9	−2.19127
$808$	−17490.7	−0.761535
$809$	−33272.3	−1.44597	−0.722985	−	0.690863i	$-0.757231\pi$
$809$	−33272.3	−1.44597	−0.722985	+	0.690863i	$0.757231\pi$
$810$	−44760.5	−1.94163
$811$	−45228.5	−1.95831	−0.979154	−	0.203120i	$-0.934892\pi$
$811$	−45228.5	−1.95831	−0.979154	+	0.203120i	$0.934892\pi$
$812$	−531.306	−0.0229620
$813$	−5408.24	−0.233303
$814$	−5714.37	−0.246055
$815$	−8673.90	−0.372802
$816$	−31479.0	−1.35047
$817$	878.293	0.0376103
$818$	7956.24	0.340078
$819$	1616.52	0.0689694
$820$	4685.67	0.199549
$821$	−13563.8	−0.576590	−0.288295	−	0.957542i	$-0.593088\pi$
$821$	−13563.8	−0.576590	−0.288295	+	0.957542i	$0.593088\pi$
$822$	58419.8	2.47886
$823$	−5036.99	−0.213339	−0.106670	−	0.994295i	$-0.534019\pi$
$823$	−5036.99	−0.213339	−0.106670	+	0.994295i	$0.534019\pi$
$824$	−23917.3	−1.01116
$825$	−2749.37	−0.116025
$826$	6227.09	0.262310
$827$	36255.2	1.52445	0.762223	−	0.647314i	$-0.224108\pi$
$827$	36255.2	1.52445	0.762223	+	0.647314i	$0.224108\pi$
$828$	−23439.0	−0.983770
$829$	11529.3	0.483027	0.241513	−	0.970397i	$-0.422356\pi$
$829$	11529.3	0.483027	0.241513	+	0.970397i	$0.422356\pi$
$830$	24.0937	0.00100759
$831$	−60754.5	−2.53616
$832$	−746.486	−0.0311055
$833$	13063.1	0.543348
$834$	−39987.5	−1.66026
$835$	−10768.9	−0.446315
$836$	761.384	0.0314988
$837$	59120.4	2.44146
$838$	31002.3	1.27799
$839$	22796.7	0.938058	0.469029	−	0.883183i	$-0.344604\pi$
$839$	22796.7	0.938058	0.469029	+	0.883183i	$0.344604\pi$
$840$	2533.67	0.104071
$841$	−22559.0	−0.924965
$842$	−37029.6	−1.51559
$843$	55882.3	2.28314
$844$	5114.96	0.208607
$845$	−10773.8	−0.438615
$846$	124312.	5.05195
$847$	−412.518	−0.0167347
$848$	−24395.6	−0.987911
$849$	−17736.2	−0.716968
$850$	3362.76	0.135696
$851$	−13426.8	−0.540853
$852$	−33022.2	−1.32784
$853$	2578.64	0.103506	0.0517532	−	0.998660i	$-0.483519\pi$
$853$	2578.64	0.103506	0.0517532	+	0.998660i	$0.483519\pi$
$854$	1945.60	0.0779593
$855$	6930.66	0.277221
$856$	4779.73	0.190850
$857$	−35195.7	−1.40287	−0.701437	−	0.712731i	$-0.747458\pi$
$857$	−35195.7	−1.40287	−0.701437	+	0.712731i	$0.747458\pi$
$858$	−2438.93	−0.0970439
$859$	−30234.3	−1.20091	−0.600455	−	0.799659i	$-0.705014\pi$
$859$	−30234.3	−1.20091	−0.600455	+	0.799659i	$0.705014\pi$
$860$	842.002	0.0333861
$861$	8768.03	0.347054
$862$	32970.0	1.30274
$863$	−48744.5	−1.92269	−0.961346	−	0.275343i	$-0.911209\pi$
$863$	−48744.5	−1.92269	−0.961346	+	0.275343i	$0.911209\pi$
$864$	−70571.7	−2.77882
$865$	−18325.8	−0.720341
$866$	4144.10	0.162612
$867$	33582.5	1.31548
$868$	1598.18	0.0624950
$869$	−2370.82	−0.0925484
$870$	7296.80	0.284350
$871$	1453.22	0.0565331
$872$	13238.2	0.514109
$873$	−37763.7	−1.46404
$874$	5717.63	0.221283
$875$	−426.155	−0.0164648
$876$	7862.00	0.303233
$877$	26686.0	1.02751	0.513753	−	0.857938i	$-0.328255\pi$
$877$	26686.0	1.02751	0.513753	+	0.857938i	$0.328255\pi$
$878$	44510.0	1.71087
$879$	46771.8	1.79474
$880$	−4392.99	−0.168281
$881$	6125.56	0.234251	0.117126	−	0.993117i	$-0.462632\pi$
$881$	6125.56	0.234251	0.117126	+	0.993117i	$0.462632\pi$
$882$	82490.9	3.14922
$883$	30564.7	1.16487	0.582437	−	0.812876i	$-0.302099\pi$
$883$	30564.7	1.16487	0.582437	+	0.812876i	$0.302099\pi$
$884$	933.370	0.0355120
$885$	−26758.8	−1.01637
$886$	508.765	0.0192915
$887$	9292.82	0.351773	0.175886	−	0.984410i	$-0.443721\pi$
$887$	9292.82	0.351773	0.175886	+	0.984410i	$0.443721\pi$
$888$	−22629.0	−0.855157
$889$	−7349.38	−0.277267
$890$	−3923.25	−0.147761
$891$	28859.3	1.08510
$892$	20620.5	0.774021
$893$	−9488.22	−0.355556
$894$	106377.	3.97962
$895$	−12568.6	−0.469410
$896$	5525.49	0.206020
$897$	−5730.66	−0.213312
$898$	−13511.5	−0.502098
$899$	−5504.77	−0.204221
$900$	6644.29	0.246085
$901$	−12040.3	−0.445195
$902$	−9655.36	−0.356417
$903$	1575.59	0.0580647
$904$	14389.5	0.529411
$905$	−7038.73	−0.258536
$906$	57275.3	2.10027
$907$	−5274.24	−0.193085	−0.0965427	−	0.995329i	$-0.530778\pi$
$907$	−5274.24	−0.193085	−0.0965427	+	0.995329i	$0.530778\pi$
$908$	3781.52	0.138209
$909$	−85829.6	−3.13178
$910$	−378.036	−0.0137712
$911$	31601.0	1.14927	0.574637	−	0.818409i	$-0.305143\pi$
$911$	31601.0	1.14927	0.574637	+	0.818409i	$0.305143\pi$
$912$	15172.3	0.550883
$913$	−15.5344	−0.000563102 0
$914$	−63182.3	−2.28653
$915$	−8360.56	−0.302067
$916$	−13955.4	−0.503385
$917$	−1118.91	−0.0402942
$918$	−61799.2	−2.22187
$919$	22242.0	0.798365	0.399182	−	0.916872i	$-0.369294\pi$
$919$	22242.0	0.798365	0.399182	+	0.916872i	$0.369294\pi$
$920$	−6555.74	−0.234931
$921$	−79549.6	−2.84609
$922$	44525.6	1.59043
$923$	−5892.78	−0.210144
$924$	1365.87	0.0486296
$925$	3806.13	0.135292
$926$	−14435.0	−0.512272
$927$	−117366.	−4.15836
$928$	6571.02	0.232440
$929$	−21961.3	−0.775593	−0.387796	−	0.921745i	$-0.626764\pi$
$929$	−21961.3	−0.775593	−0.387796	+	0.921745i	$0.626764\pi$
$930$	−21948.9	−0.773907
$931$	−6296.16	−0.221642
$932$	−1274.89	−0.0448072
$933$	13227.9	0.464162
$934$	57064.8	1.99916
$935$	−2168.13	−0.0758348
$936$	−7049.29	−0.246168
$937$	54802.8	1.91070	0.955352	−	0.295471i	$-0.0954767\pi$
$937$	54802.8	1.91070	0.955352	+	0.295471i	$0.0954767\pi$
$938$	−2601.04	−0.0905404
$939$	50737.9	1.76333
$940$	−9096.17	−0.315622
$941$	−1690.15	−0.0585520	−0.0292760	−	0.999571i	$-0.509320\pi$
$941$	−1690.15	−0.0585520	−0.0292760	+	0.999571i	$0.509320\pi$
$942$	92137.0	3.18682
$943$	−22686.8	−0.783441
$944$	−42755.6	−1.47413
$945$	7831.68	0.269592
$946$	−1735.04	−0.0596313
$947$	27588.0	0.946663	0.473332	−	0.880884i	$-0.343051\pi$
$947$	27588.0	0.946663	0.473332	+	0.880884i	$0.343051\pi$
$948$	7849.91	0.268938
$949$	1402.97	0.0479897
$950$	−1620.79	−0.0553529
$951$	49996.0	1.70476
$952$	1998.03	0.0680215
$953$	5521.68	0.187686	0.0938430	−	0.995587i	$-0.470085\pi$
$953$	5521.68	0.187686	0.0938430	+	0.995587i	$0.470085\pi$
$954$	−76032.2	−2.58033
$955$	−8908.58	−0.301858
$956$	−9155.44	−0.309737
$957$	−4704.60	−0.158911
$958$	−32303.5	−1.08943
$959$	−5838.28	−0.196588
$960$	−5741.42	−0.193024
$961$	−13232.5	−0.444179
$962$	3376.36	0.113158
$963$	23454.9	0.784864
$964$	19775.6	0.660714
$965$	13915.5	0.464203
$966$	10257.0	0.341629
$967$	−46921.9	−1.56040	−0.780199	−	0.625531i	$-0.784882\pi$
$967$	−46921.9	−1.56040	−0.780199	+	0.625531i	$0.784882\pi$
$968$	1798.90	0.0597301
$969$	7488.20	0.248252
$970$	8831.33	0.292327
$971$	−45144.3	−1.49202	−0.746009	−	0.665936i	$-0.768032\pi$
$971$	−45144.3	−1.49202	−0.746009	+	0.665936i	$0.768032\pi$
$972$	−50363.8	−1.66196
$973$	3996.22	0.131668
$974$	−40361.4	−1.32778
$975$	1624.48	0.0533589
$976$	−13358.6	−0.438114
$977$	5548.53	0.181692	0.0908461	−	0.995865i	$-0.471043\pi$
$977$	5548.53	0.181692	0.0908461	+	0.995865i	$0.471043\pi$
$978$	−59180.3	−1.93495
$979$	2529.51	0.0825776
$980$	−6036.01	−0.196748
$981$	64962.1	2.11425
$982$	56816.5	1.84632
$983$	6193.74	0.200966	0.100483	−	0.994939i	$-0.467961\pi$
$983$	6193.74	0.200966	0.100483	+	0.994939i	$0.467961\pi$
$984$	−38235.4	−1.23872
$985$	−21146.7	−0.684051
$986$	5754.19	0.185853
$987$	−17021.2	−0.548926
$988$	−449.867	−0.0144860
$989$	−4076.77	−0.131076
$990$	−13691.3	−0.439535
$991$	−21698.6	−0.695538	−0.347769	−	0.937580i	$-0.613061\pi$
$991$	−21698.6	−0.695538	−0.347769	+	0.937580i	$0.613061\pi$
$992$	−19765.8	−0.632624
$993$	78244.1	2.50050
$994$	10547.2	0.336556
$995$	−875.838	−0.0279055
$996$	51.4351	0.00163633
$997$	−52385.5	−1.66406	−0.832029	−	0.554732i	$-0.812821\pi$
$997$	−52385.5	−1.66406	−0.832029	+	0.554732i	$0.812821\pi$
$998$	−23498.1	−0.745310
$999$	−69947.3	−2.21525

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.7	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-3.41218 q^{2} -9.99772 q^{3} +3.64299 q^{4} +5.00000 q^{5} +34.1140 q^{6} -3.40924 q^{7} +14.8669 q^{8} +72.9544 q^{9} +O(q^{10})\)	\(q-3.41218 q^{2} -9.99772 q^{3} +3.64299 q^{4} +5.00000 q^{5} +34.1140 q^{6} -3.40924 q^{7} +14.8669 q^{8} +72.9544 q^{9} -17.0609 q^{10} +11.0000 q^{11} -36.4216 q^{12} -6.49940 q^{13} +11.6329 q^{14} -49.9886 q^{15} -79.8725 q^{16} -39.4206 q^{17} -248.934 q^{18} +19.0000 q^{19} +18.2149 q^{20} +34.0846 q^{21} -37.5340 q^{22} -88.1922 q^{23} -148.635 q^{24} +25.0000 q^{25} +22.1771 q^{26} -459.439 q^{27} -12.4198 q^{28} +42.7789 q^{29} +170.570 q^{30} -128.680 q^{31} +153.604 q^{32} -109.975 q^{33} +134.510 q^{34} -17.0462 q^{35} +265.772 q^{36} +152.245 q^{37} -64.8315 q^{38} +64.9792 q^{39} +74.3346 q^{40} +257.243 q^{41} -116.303 q^{42} +46.2259 q^{43} +40.0729 q^{44} +364.772 q^{45} +300.928 q^{46} -499.380 q^{47} +798.543 q^{48} -331.377 q^{49} -85.3046 q^{50} +394.116 q^{51} -23.6772 q^{52} +305.432 q^{53} +1567.69 q^{54} +55.0000 q^{55} -50.6849 q^{56} -189.957 q^{57} -145.969 q^{58} +535.298 q^{59} -182.108 q^{60} +167.249 q^{61} +439.079 q^{62} -248.719 q^{63} +114.855 q^{64} -32.4970 q^{65} +375.254 q^{66} -223.592 q^{67} -143.609 q^{68} +881.721 q^{69} +58.1647 q^{70} +906.666 q^{71} +1084.61 q^{72} -215.861 q^{73} -519.488 q^{74} -249.943 q^{75} +69.2167 q^{76} -37.5016 q^{77} -221.721 q^{78} -215.529 q^{79} -399.363 q^{80} +2623.57 q^{81} -877.760 q^{82} -1.41222 q^{83} +124.170 q^{84} -197.103 q^{85} -157.731 q^{86} -427.691 q^{87} +163.536 q^{88} +229.956 q^{89} -1244.67 q^{90} +22.1580 q^{91} -321.283 q^{92} +1286.50 q^{93} +1703.98 q^{94} +95.0000 q^{95} -1535.69 q^{96} -517.635 q^{97} +1130.72 q^{98} +802.498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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