LMFDB - Embedded newform 1045.4.a.b.1.10

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			1.10
Root			$-0.259637$ 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)

$f(q)$	$=$	$q-1.25964 q^{2} +7.52739 q^{3} -6.41331 q^{4} +5.00000 q^{5} -9.48178 q^{6} -31.9473 q^{7} +18.1555 q^{8} +29.6616 q^{9} +O(q^{10})$	\(q-1.25964 q^{2} +7.52739 q^{3} -6.41331 q^{4} +5.00000 q^{5} -9.48178 q^{6} -31.9473 q^{7} +18.1555 q^{8} +29.6616 q^{9} -6.29819 q^{10} +11.0000 q^{11} -48.2755 q^{12} +67.9220 q^{13} +40.2420 q^{14} +37.6369 q^{15} +28.4371 q^{16} -77.4913 q^{17} -37.3628 q^{18} +19.0000 q^{19} -32.0666 q^{20} -240.480 q^{21} -13.8560 q^{22} -156.165 q^{23} +136.664 q^{24} +25.0000 q^{25} -85.5571 q^{26} +20.0348 q^{27} +204.888 q^{28} -18.1652 q^{29} -47.4089 q^{30} +116.499 q^{31} -181.065 q^{32} +82.8013 q^{33} +97.6109 q^{34} -159.736 q^{35} -190.229 q^{36} +357.585 q^{37} -23.9331 q^{38} +511.276 q^{39} +90.7777 q^{40} +63.0971 q^{41} +302.917 q^{42} -154.374 q^{43} -70.5465 q^{44} +148.308 q^{45} +196.711 q^{46} -599.793 q^{47} +214.057 q^{48} +677.629 q^{49} -31.4909 q^{50} -583.307 q^{51} -435.605 q^{52} -442.919 q^{53} -25.2366 q^{54} +55.0000 q^{55} -580.021 q^{56} +143.020 q^{57} +22.8816 q^{58} +256.143 q^{59} -241.378 q^{60} -389.147 q^{61} -146.746 q^{62} -947.607 q^{63} +0.579130 q^{64} +339.610 q^{65} -104.300 q^{66} +229.074 q^{67} +496.976 q^{68} -1175.51 q^{69} +201.210 q^{70} -175.317 q^{71} +538.522 q^{72} -757.114 q^{73} -450.427 q^{74} +188.185 q^{75} -121.853 q^{76} -351.420 q^{77} -644.022 q^{78} -589.435 q^{79} +142.186 q^{80} -650.053 q^{81} -79.4795 q^{82} -1222.01 q^{83} +1542.27 q^{84} -387.456 q^{85} +194.455 q^{86} -136.737 q^{87} +199.711 q^{88} +1250.38 q^{89} -186.814 q^{90} -2169.92 q^{91} +1001.54 q^{92} +876.930 q^{93} +755.521 q^{94} +95.0000 q^{95} -1362.95 q^{96} -1613.18 q^{97} -853.567 q^{98} +326.278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.25964	−0.445349	−0.222675	−	0.974893i	$-0.571479\pi$
$2$	−1.25964	−0.445349	−0.222675	+	0.974893i	$0.571479\pi$
$3$	7.52739	1.44865	0.724323	−	0.689460i	$-0.242152\pi$
$3$	7.52739	1.44865	0.724323	+	0.689460i	$0.242152\pi$
$4$	−6.41331	−0.801664
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−9.48178	−0.645153
$7$	−31.9473	−1.72499	−0.862496	−	0.506064i	$-0.831100\pi$
$7$	−31.9473	−1.72499	−0.862496	+	0.506064i	$0.831100\pi$
$8$	18.1555	0.802369
$9$	29.6616	1.09858
$10$	−6.29819	−0.199166
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−48.2755	−1.16133
$13$	67.9220	1.44909	0.724545	−	0.689227i	$-0.242050\pi$
$13$	67.9220	1.44909	0.724545	+	0.689227i	$0.242050\pi$
$14$	40.2420	0.768223
$15$	37.6369	0.647855
$16$	28.4371	0.444330
$17$	−77.4913	−1.10555	−0.552776	−	0.833330i	$-0.686432\pi$
$17$	−77.4913	−1.10555	−0.552776	+	0.833330i	$0.686432\pi$
$18$	−37.3628	−0.489250
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−32.0666	−0.358515
$21$	−240.480	−2.49890
$22$	−13.8560	−0.134278
$23$	−156.165	−1.41577	−0.707884	−	0.706329i	$-0.750350\pi$
$23$	−156.165	−1.41577	−0.707884	+	0.706329i	$0.750350\pi$
$24$	136.664	1.16235
$25$	25.0000	0.200000
$26$	−85.5571	−0.645351
$27$	20.0348	0.142804
$28$	204.888	1.38286
$29$	−18.1652	−0.116317	−0.0581585	−	0.998307i	$-0.518523\pi$
$29$	−18.1652	−0.116317	−0.0581585	+	0.998307i	$0.518523\pi$
$30$	−47.4089	−0.288521
$31$	116.499	0.674960	0.337480	−	0.941333i	$-0.390425\pi$
$31$	116.499	0.674960	0.337480	+	0.941333i	$0.390425\pi$
$32$	−181.065	−1.00025
$33$	82.8013	0.436783
$34$	97.6109	0.492357
$35$	−159.736	−0.771440
$36$	−190.229	−0.880690
$37$	357.585	1.58883	0.794413	−	0.607378i	$-0.207779\pi$
$37$	357.585	1.58883	0.794413	+	0.607378i	$0.207779\pi$
$38$	−23.9331	−0.102170
$39$	511.276	2.09922
$40$	90.7777	0.358831
$41$	63.0971	0.240344	0.120172	−	0.992753i	$-0.461655\pi$
$41$	63.0971	0.240344	0.120172	+	0.992753i	$0.461655\pi$
$42$	302.917	1.11288
$43$	−154.374	−0.547484	−0.273742	−	0.961803i	$-0.588261\pi$
$43$	−154.374	−0.547484	−0.273742	+	0.961803i	$0.588261\pi$
$44$	−70.5465	−0.241711
$45$	148.308	0.491299
$46$	196.711	0.630511
$47$	−599.793	−1.86146	−0.930732	−	0.365703i	$-0.880829\pi$
$47$	−599.793	−1.86146	−0.930732	+	0.365703i	$0.880829\pi$
$48$	214.057	0.643677
$49$	677.629	1.97560
$50$	−31.4909	−0.0890698
$51$	−583.307	−1.60155
$52$	−435.605	−1.16168
$53$	−442.919	−1.14792	−0.573958	−	0.818885i	$-0.694593\pi$
$53$	−442.919	−1.14792	−0.573958	+	0.818885i	$0.694593\pi$
$54$	−25.2366	−0.0635976
$55$	55.0000	0.134840
$56$	−580.021	−1.38408
$57$	143.020	0.332342
$58$	22.8816	0.0518017
$59$	256.143	0.565202	0.282601	−	0.959238i	$-0.408803\pi$
$59$	256.143	0.565202	0.282601	+	0.959238i	$0.408803\pi$
$60$	−241.378	−0.519362
$61$	−389.147	−0.816807	−0.408403	−	0.912802i	$-0.633914\pi$
$61$	−389.147	−0.816807	−0.408403	+	0.912802i	$0.633914\pi$
$62$	−146.746	−0.300593
$63$	−947.607	−1.89504
$64$	0.579130	0.00113111
$65$	339.610	0.648053
$66$	−104.300	−0.194521
$67$	229.074	0.417698	0.208849	−	0.977948i	$-0.433028\pi$
$67$	229.074	0.417698	0.208849	+	0.977948i	$0.433028\pi$
$68$	496.976	0.886282
$69$	−1175.51	−2.05095
$70$	201.210	0.343560
$71$	−175.317	−0.293047	−0.146523	−	0.989207i	$-0.546808\pi$
$71$	−175.317	−0.293047	−0.146523	+	0.989207i	$0.546808\pi$
$72$	538.522	0.881465
$73$	−757.114	−1.21388	−0.606942	−	0.794746i	$-0.707604\pi$
$73$	−757.114	−1.21388	−0.606942	+	0.794746i	$0.707604\pi$
$74$	−450.427	−0.707582
$75$	188.185	0.289729
$76$	−121.853	−0.183914
$77$	−351.420	−0.520104
$78$	−644.022	−0.934886
$79$	−589.435	−0.839451	−0.419725	−	0.907651i	$-0.637874\pi$
$79$	−589.435	−0.839451	−0.419725	+	0.907651i	$0.637874\pi$
$80$	142.186	0.198710
$81$	−650.053	−0.891705
$82$	−79.4795	−0.107037
$83$	−1222.01	−1.61606	−0.808032	−	0.589139i	$-0.799467\pi$
$83$	−1222.01	−1.61606	−0.808032	+	0.589139i	$0.799467\pi$
$84$	1542.27	2.00328
$85$	−387.456	−0.494418
$86$	194.455	0.243822
$87$	−136.737	−0.168502
$88$	199.711	0.241923
$89$	1250.38	1.48922	0.744608	−	0.667502i	$-0.232637\pi$
$89$	1250.38	1.48922	0.744608	+	0.667502i	$0.232637\pi$
$90$	−186.814	−0.218799
$91$	−2169.92	−2.49967
$92$	1001.54	1.13497
$93$	876.930	0.977778
$94$	755.521	0.829001
$95$	95.0000	0.102598
$96$	−1362.95	−1.44901
$97$	−1613.18	−1.68859	−0.844296	−	0.535877i	$-0.819981\pi$
$97$	−1613.18	−1.68859	−0.844296	+	0.535877i	$0.819981\pi$
$98$	−853.567	−0.879830
$99$	326.278	0.331234
$100$	−160.333	−0.160333
$101$	−24.3497	−0.0239889	−0.0119945	−	0.999928i	$-0.503818\pi$
$101$	−24.3497	−0.0239889	−0.0119945	+	0.999928i	$0.503818\pi$
$102$	734.755	0.713251
$103$	−491.240	−0.469935	−0.234968	−	0.972003i	$-0.575498\pi$
$103$	−491.240	−0.469935	−0.234968	+	0.972003i	$0.575498\pi$
$104$	1233.16	1.16271
$105$	−1202.40	−1.11754
$106$	557.917	0.511224
$107$	68.6565	0.0620306	0.0310153	−	0.999519i	$-0.490126\pi$
$107$	68.6565	0.0620306	0.0310153	+	0.999519i	$0.490126\pi$
$108$	−128.490	−0.114481
$109$	687.646	0.604262	0.302131	−	0.953266i	$-0.402302\pi$
$109$	687.646	0.604262	0.302131	+	0.953266i	$0.402302\pi$
$110$	−69.2800	−0.0600509
$111$	2691.68	2.30165
$112$	−908.488	−0.766465
$113$	−763.056	−0.635241	−0.317621	−	0.948218i	$-0.602884\pi$
$113$	−763.056	−0.635241	−0.317621	+	0.948218i	$0.602884\pi$
$114$	−180.154	−0.148008
$115$	−780.825	−0.633151
$116$	116.499	0.0932472
$117$	2014.68	1.59194
$118$	−322.647	−0.251712
$119$	2475.64	1.90707
$120$	683.319	0.519819
$121$	121.000	0.0909091
$122$	490.184	0.363764
$123$	474.956	0.348174
$124$	−747.142	−0.541091
$125$	125.000	0.0894427
$126$	1193.64	0.843953
$127$	1486.04	1.03830	0.519151	−	0.854682i	$-0.326248\pi$
$127$	1486.04	1.03830	0.519151	+	0.854682i	$0.326248\pi$
$128$	1447.79	0.999748
$129$	−1162.03	−0.793111
$130$	−427.786	−0.288610
$131$	−2583.23	−1.72288	−0.861441	−	0.507857i	$-0.830438\pi$
$131$	−2583.23	−1.72288	−0.861441	+	0.507857i	$0.830438\pi$
$132$	−531.031	−0.350154
$133$	−606.998	−0.395740
$134$	−288.550	−0.186022
$135$	100.174	0.0638639
$136$	−1406.90	−0.887061
$137$	−1265.07	−0.788924	−0.394462	−	0.918912i	$-0.629069\pi$
$137$	−1265.07	−0.788924	−0.394462	+	0.918912i	$0.629069\pi$
$138$	1480.72	0.913387
$139$	−2195.80	−1.33990	−0.669948	−	0.742408i	$-0.733683\pi$
$139$	−2195.80	−1.33990	−0.669948	+	0.742408i	$0.733683\pi$
$140$	1024.44	0.618436
$141$	−4514.87	−2.69660
$142$	220.836	0.130508
$143$	747.142	0.436917
$144$	843.490	0.488131
$145$	−90.8260	−0.0520185
$146$	953.690	0.540602
$147$	5100.78	2.86194
$148$	−2293.30	−1.27370
$149$	−2255.64	−1.24020	−0.620099	−	0.784524i	$-0.712907\pi$
$149$	−2255.64	−1.24020	−0.620099	+	0.784524i	$0.712907\pi$
$150$	−237.045	−0.129031
$151$	−2131.97	−1.14899	−0.574494	−	0.818509i	$-0.694801\pi$
$151$	−2131.97	−1.14899	−0.574494	+	0.818509i	$0.694801\pi$
$152$	344.955	0.184076
$153$	−2298.51	−1.21453
$154$	442.662	0.231628
$155$	582.493	0.301851
$156$	−3278.97	−1.68287
$157$	−429.527	−0.218344	−0.109172	−	0.994023i	$-0.534820\pi$
$157$	−429.527	−0.218344	−0.109172	+	0.994023i	$0.534820\pi$
$158$	742.474	0.373848
$159$	−3334.02	−1.66293
$160$	−905.324	−0.447326
$161$	4989.05	2.44219
$162$	818.831	0.397120
$163$	3915.62	1.88156	0.940782	−	0.339013i	$-0.110093\pi$
$163$	3915.62	1.88156	0.940782	+	0.339013i	$0.110093\pi$
$164$	−404.661	−0.192675
$165$	414.006	0.195335
$166$	1539.29	0.719712
$167$	−863.598	−0.400163	−0.200082	−	0.979779i	$-0.564121\pi$
$167$	−863.598	−0.400163	−0.200082	+	0.979779i	$0.564121\pi$
$168$	−4366.04	−2.00504
$169$	2416.40	1.09986
$170$	488.054	0.220189
$171$	563.570	0.252031
$172$	990.049	0.438898
$173$	671.032	0.294899	0.147450	−	0.989070i	$-0.452894\pi$
$173$	671.032	0.294899	0.147450	+	0.989070i	$0.452894\pi$
$174$	172.238	0.0750423
$175$	−798.682	−0.344998
$176$	312.808	0.133970
$177$	1928.09	0.818778
$178$	−1575.03	−0.663221
$179$	−1209.04	−0.504848	−0.252424	−	0.967617i	$-0.581228\pi$
$179$	−1209.04	−0.504848	−0.252424	+	0.967617i	$0.581228\pi$
$180$	−951.146	−0.393857
$181$	2220.50	0.911870	0.455935	−	0.890013i	$-0.349305\pi$
$181$	2220.50	0.911870	0.455935	+	0.890013i	$0.349305\pi$
$182$	2733.32	1.11323
$183$	−2929.26	−1.18326
$184$	−2835.26	−1.13597
$185$	1787.92	0.710544
$186$	−1104.61	−0.435453
$187$	−852.404	−0.333337
$188$	3846.66	1.49227
$189$	−640.059	−0.246336
$190$	−119.666	−0.0456918
$191$	−941.777	−0.356778	−0.178389	−	0.983960i	$-0.557089\pi$
$191$	−941.777	−0.356778	−0.178389	+	0.983960i	$0.557089\pi$
$192$	4.35934	0.00163858
$193$	2589.62	0.965829	0.482915	−	0.875667i	$-0.339578\pi$
$193$	2589.62	0.965829	0.482915	+	0.875667i	$0.339578\pi$
$194$	2032.02	0.752013
$195$	2556.38	0.938800
$196$	−4345.85	−1.58376
$197$	76.5299	0.0276778	0.0138389	−	0.999904i	$-0.495595\pi$
$197$	76.5299	0.0276778	0.0138389	+	0.999904i	$0.495595\pi$
$198$	−410.991	−0.147515
$199$	2778.99	0.989937	0.494968	−	0.868911i	$-0.335180\pi$
$199$	2778.99	0.989937	0.494968	+	0.868911i	$0.335180\pi$
$200$	453.889	0.160474
$201$	1724.33	0.605097
$202$	30.6718	0.0106835
$203$	580.329	0.200646
$204$	3740.93	1.28391
$205$	315.485	0.107485
$206$	618.784	0.209285
$207$	−4632.10	−1.55533
$208$	1931.51	0.643874
$209$	209.000	0.0691714
$210$	1514.59	0.497697
$211$	−5372.89	−1.75301	−0.876505	−	0.481392i	$-0.840131\pi$
$211$	−5372.89	−1.75301	−0.876505	+	0.481392i	$0.840131\pi$
$212$	2840.58	0.920244
$213$	−1319.68	−0.424521
$214$	−86.4823	−0.0276253
$215$	−771.870	−0.244842
$216$	363.743	0.114582
$217$	−3721.81	−1.16430
$218$	−866.185	−0.269108
$219$	−5699.10	−1.75849
$220$	−352.732	−0.108096
$221$	−5263.36	−1.60205
$222$	−3390.54	−1.02504
$223$	5849.29	1.75649	0.878246	−	0.478209i	$-0.158714\pi$
$223$	5849.29	1.75649	0.878246	+	0.478209i	$0.158714\pi$
$224$	5784.53	1.72542
$225$	741.540	0.219715
$226$	961.174	0.282904
$227$	−2159.63	−0.631452	−0.315726	−	0.948850i	$-0.602248\pi$
$227$	−2159.63	−0.631452	−0.315726	+	0.948850i	$0.602248\pi$
$228$	−917.235	−0.266427
$229$	−2305.18	−0.665198	−0.332599	−	0.943068i	$-0.607926\pi$
$229$	−2305.18	−0.665198	−0.332599	+	0.943068i	$0.607926\pi$
$230$	983.556	0.281973
$231$	−2645.28	−0.753448
$232$	−329.799	−0.0933292
$233$	−118.509	−0.0333210	−0.0166605	−	0.999861i	$-0.505303\pi$
$233$	−118.509	−0.0333210	−0.0166605	+	0.999861i	$0.505303\pi$
$234$	−2537.76	−0.708968
$235$	−2998.96	−0.832472
$236$	−1642.72	−0.453102
$237$	−4436.91	−1.21607
$238$	−3118.40	−0.849311
$239$	−706.961	−0.191337	−0.0956684	−	0.995413i	$-0.530499\pi$
$239$	−706.961	−0.191337	−0.0956684	+	0.995413i	$0.530499\pi$
$240$	1070.29	0.287861
$241$	−2400.89	−0.641722	−0.320861	−	0.947126i	$-0.603972\pi$
$241$	−2400.89	−0.641722	−0.320861	+	0.947126i	$0.603972\pi$
$242$	−152.416	−0.0404863
$243$	−5434.14	−1.43457
$244$	2495.72	0.654805
$245$	3388.15	0.883513
$246$	−598.273	−0.155059
$247$	1290.52	0.332444
$248$	2115.09	0.541567
$249$	−9198.56	−2.34110
$250$	−157.455	−0.0398332
$251$	2419.38	0.608406	0.304203	−	0.952607i	$-0.401610\pi$
$251$	2419.38	0.608406	0.304203	+	0.952607i	$0.401610\pi$
$252$	6077.30	1.51918
$253$	−1717.82	−0.426870
$254$	−1871.87	−0.462407
$255$	−2916.53	−0.716237
$256$	−1828.32	−0.446368
$257$	−5098.31	−1.23745	−0.618724	−	0.785608i	$-0.712350\pi$
$257$	−5098.31	−1.23745	−0.618724	+	0.785608i	$0.712350\pi$
$258$	1463.74	0.353211
$259$	−11423.9	−2.74071
$260$	−2178.03	−0.519521
$261$	−538.809	−0.127783
$262$	3253.93	0.767284
$263$	−201.947	−0.0473481	−0.0236741	−	0.999720i	$-0.507536\pi$
$263$	−201.947	−0.0473481	−0.0236741	+	0.999720i	$0.507536\pi$
$264$	1503.30	0.350462
$265$	−2214.59	−0.513364
$266$	764.598	0.176243
$267$	9412.11	2.15735
$268$	−1469.12	−0.334854
$269$	3514.10	0.796501	0.398250	−	0.917277i	$-0.369617\pi$
$269$	3514.10	0.796501	0.398250	+	0.917277i	$0.369617\pi$
$270$	−126.183	−0.0284417
$271$	5862.94	1.31420	0.657100	−	0.753803i	$-0.271783\pi$
$271$	5862.94	1.31420	0.657100	+	0.753803i	$0.271783\pi$
$272$	−2203.63	−0.491230
$273$	−16333.9	−3.62114
$274$	1593.53	0.351347
$275$	275.000	0.0603023
$276$	7538.95	1.64417
$277$	102.734	0.0222841	0.0111420	−	0.999938i	$-0.496453\pi$
$277$	102.734	0.0222841	0.0111420	+	0.999938i	$0.496453\pi$
$278$	2765.91	0.596721
$279$	3455.53	0.741496
$280$	−2900.10	−0.618980
$281$	8029.34	1.70459	0.852296	−	0.523060i	$-0.175210\pi$
$281$	8029.34	1.70459	0.852296	+	0.523060i	$0.175210\pi$
$282$	5687.10	1.20093
$283$	−4791.83	−1.00652	−0.503259	−	0.864136i	$-0.667866\pi$
$283$	−4791.83	−1.00652	−0.503259	+	0.864136i	$0.667866\pi$
$284$	1124.36	0.234925
$285$	715.102	0.148628
$286$	−941.128	−0.194581
$287$	−2015.78	−0.414592
$288$	−5370.67	−1.09885
$289$	1091.89	0.222246
$290$	114.408	0.0231664
$291$	−12143.0	−2.44617
$292$	4855.61	0.973128
$293$	−4377.93	−0.872905	−0.436453	−	0.899727i	$-0.643765\pi$
$293$	−4377.93	−0.872905	−0.436453	+	0.899727i	$0.643765\pi$
$294$	−6425.13	−1.27456
$295$	1280.71	0.252766
$296$	6492.14	1.27482
$297$	220.383	0.0430570
$298$	2841.29	0.552321
$299$	−10607.0	−2.05158
$300$	−1206.89	−0.232266
$301$	4931.83	0.944405
$302$	2685.51	0.511700
$303$	−183.289	−0.0347515
$304$	540.305	0.101936
$305$	−1945.74	−0.365287
$306$	2895.29	0.540892
$307$	−7176.13	−1.33408	−0.667042	−	0.745021i	$-0.732440\pi$
$307$	−7176.13	−1.33408	−0.667042	+	0.745021i	$0.732440\pi$
$308$	2253.77	0.416949
$309$	−3697.75	−0.680770
$310$	−733.730	−0.134429
$311$	−9273.13	−1.69077	−0.845387	−	0.534154i	$-0.820630\pi$
$311$	−9273.13	−1.69077	−0.845387	+	0.534154i	$0.820630\pi$
$312$	9282.49	1.68435
$313$	7609.07	1.37409	0.687044	−	0.726615i	$-0.258908\pi$
$313$	7609.07	1.37409	0.687044	+	0.726615i	$0.258908\pi$
$314$	541.048	0.0972391
$315$	−4738.04	−0.847486
$316$	3780.23	0.672958
$317$	8173.52	1.44817	0.724086	−	0.689710i	$-0.242262\pi$
$317$	8173.52	1.44817	0.724086	+	0.689710i	$0.242262\pi$
$318$	4199.66	0.740582
$319$	−199.817	−0.0350709
$320$	2.89565	0.000505850 0
$321$	516.805	0.0898605
$322$	−6284.39	−1.08763
$323$	−1472.33	−0.253631
$324$	4168.99	0.714848
$325$	1698.05	0.289818
$326$	−4932.26	−0.837952
$327$	5176.18	0.875362
$328$	1145.56	0.192845
$329$	19161.8	3.21101
$330$	−521.498	−0.0869925
$331$	−8112.01	−1.34706	−0.673530	−	0.739160i	$-0.735223\pi$
$331$	−8112.01	−1.34706	−0.673530	+	0.739160i	$0.735223\pi$
$332$	7837.15	1.29554
$333$	10606.5	1.74545
$334$	1087.82	0.178212
$335$	1145.37	0.186800
$336$	−6838.55	−1.11034
$337$	3855.89	0.623275	0.311638	−	0.950201i	$-0.399122\pi$
$337$	3855.89	0.623275	0.311638	+	0.950201i	$0.399122\pi$
$338$	−3043.79	−0.489824
$339$	−5743.82	−0.920240
$340$	2484.88	0.396357
$341$	1281.48	0.203508
$342$	−709.894	−0.112242
$343$	−10690.5	−1.68289
$344$	−2802.74	−0.439284
$345$	−5877.57	−0.917212
$346$	−845.256	−0.131333
$347$	−87.6239	−0.0135559	−0.00677795	−	0.999977i	$-0.502158\pi$
$347$	−87.6239	−0.0135559	−0.00677795	+	0.999977i	$0.502158\pi$
$348$	876.934	0.135082
$349$	5596.76	0.858417	0.429209	−	0.903205i	$-0.358793\pi$
$349$	5596.76	0.858417	0.429209	+	0.903205i	$0.358793\pi$
$350$	1006.05	0.153645
$351$	1360.81	0.206936
$352$	−1991.71	−0.301587
$353$	−5935.57	−0.894953	−0.447477	−	0.894296i	$-0.647677\pi$
$353$	−5935.57	−0.894953	−0.447477	+	0.894296i	$0.647677\pi$
$354$	−2428.69	−0.364642
$355$	−876.585	−0.131054
$356$	−8019.09	−1.19385
$357$	18635.1	2.76267
$358$	1522.95	0.224834
$359$	−626.506	−0.0921050	−0.0460525	−	0.998939i	$-0.514664\pi$
$359$	−626.506	−0.0921050	−0.0460525	+	0.998939i	$0.514664\pi$
$360$	2692.61	0.394203
$361$	361.000	0.0526316
$362$	−2797.02	−0.406100
$363$	910.814	0.131695
$364$	13916.4	2.00390
$365$	−3785.57	−0.542865
$366$	3689.81	0.526966
$367$	−3207.67	−0.456237	−0.228119	−	0.973633i	$-0.573257\pi$
$367$	−3207.67	−0.456237	−0.228119	+	0.973633i	$0.573257\pi$
$368$	−4440.88	−0.629068
$369$	1871.56	0.264037
$370$	−2252.13	−0.316440
$371$	14150.1	1.98015
$372$	−5624.03	−0.783850
$373$	3694.86	0.512902	0.256451	−	0.966557i	$-0.417447\pi$
$373$	3694.86	0.512902	0.256451	+	0.966557i	$0.417447\pi$
$374$	1073.72	0.148451
$375$	940.924	0.129571
$376$	−10889.6	−1.49358
$377$	−1233.82	−0.168554
$378$	806.242	0.109705
$379$	10271.8	1.39215	0.696077	−	0.717967i	$-0.254927\pi$
$379$	10271.8	1.39215	0.696077	+	0.717967i	$0.254927\pi$
$380$	−609.265	−0.0822490
$381$	11186.0	1.50413
$382$	1186.30	0.158891
$383$	−6220.43	−0.829893	−0.414947	−	0.909846i	$-0.636200\pi$
$383$	−6220.43	−0.829893	−0.414947	+	0.909846i	$0.636200\pi$
$384$	10898.1	1.44828
$385$	−1757.10	−0.232598
$386$	−3261.98	−0.430131
$387$	−4578.98	−0.601454
$388$	10345.8	1.35368
$389$	11544.6	1.50471	0.752356	−	0.658757i	$-0.228917\pi$
$389$	11544.6	1.50471	0.752356	+	0.658757i	$0.228917\pi$
$390$	−3220.11	−0.418094
$391$	12101.4	1.56521
$392$	12302.7	1.58516
$393$	−19445.0	−2.49585
$394$	−96.3999	−0.0123263
$395$	−2947.17	−0.375414
$396$	−2092.52	−0.265538
$397$	9872.42	1.24807	0.624033	−	0.781398i	$-0.285493\pi$
$397$	9872.42	1.24807	0.624033	+	0.781398i	$0.285493\pi$
$398$	−3500.52	−0.440867
$399$	−4569.11	−0.573288
$400$	710.928	0.0888660
$401$	14589.6	1.81688	0.908442	−	0.418010i	$-0.137272\pi$
$401$	14589.6	1.81688	0.908442	+	0.418010i	$0.137272\pi$
$402$	−2172.03	−0.269480
$403$	7912.82	0.978078
$404$	156.162	0.0192311
$405$	−3250.26	−0.398783
$406$	−731.004	−0.0893574
$407$	3933.43	0.479049
$408$	−10590.3	−1.28504
$409$	−11291.8	−1.36514	−0.682571	−	0.730820i	$-0.739138\pi$
$409$	−11291.8	−1.36514	−0.682571	+	0.730820i	$0.739138\pi$
$410$	−397.397	−0.0478684
$411$	−9522.71	−1.14287
$412$	3150.48	0.376730
$413$	−8183.06	−0.974969
$414$	5834.77	0.692665
$415$	−6110.06	−0.722725
$416$	−12298.3	−1.44946
$417$	−16528.7	−1.94104
$418$	−263.264	−0.0308054
$419$	7816.24	0.911332	0.455666	−	0.890151i	$-0.349401\pi$
$419$	7816.24	0.911332	0.455666	+	0.890151i	$0.349401\pi$
$420$	7711.36	0.895895
$421$	−3872.66	−0.448318	−0.224159	−	0.974553i	$-0.571964\pi$
$421$	−3872.66	−0.448318	−0.224159	+	0.974553i	$0.571964\pi$
$422$	6767.90	0.780702
$423$	−17790.8	−2.04496
$424$	−8041.43	−0.921053
$425$	−1937.28	−0.221110
$426$	1662.32	0.189060
$427$	12432.2	1.40898
$428$	−440.316	−0.0497278
$429$	5624.03	0.632939
$430$	972.276	0.109040
$431$	8658.75	0.967697	0.483848	−	0.875152i	$-0.339239\pi$
$431$	8658.75	0.967697	0.483848	+	0.875152i	$0.339239\pi$
$432$	569.733	0.0634520
$433$	13287.3	1.47470	0.737350	−	0.675511i	$-0.236077\pi$
$433$	13287.3	1.47470	0.737350	+	0.675511i	$0.236077\pi$
$434$	4688.13	0.518520
$435$	−683.683	−0.0753565
$436$	−4410.09	−0.484415
$437$	−2967.14	−0.324799
$438$	7178.79	0.783142
$439$	−14329.0	−1.55782	−0.778911	−	0.627135i	$-0.784228\pi$
$439$	−14329.0	−1.55782	−0.778911	+	0.627135i	$0.784228\pi$
$440$	998.555	0.108191
$441$	20099.6	2.17034
$442$	6629.93	0.713470
$443$	−8226.35	−0.882270	−0.441135	−	0.897441i	$-0.645424\pi$
$443$	−8226.35	−0.882270	−0.441135	+	0.897441i	$0.645424\pi$
$444$	−17262.6	−1.84515
$445$	6251.91	0.665997
$446$	−7367.99	−0.782252
$447$	−16979.1	−1.79661
$448$	−18.5016	−0.00195116
$449$	5830.98	0.612876	0.306438	−	0.951891i	$-0.400863\pi$
$449$	5830.98	0.612876	0.306438	+	0.951891i	$0.400863\pi$
$450$	−934.071	−0.0978501
$451$	694.068	0.0724665
$452$	4893.72	0.509250
$453$	−16048.2	−1.66448
$454$	2720.35	0.281217
$455$	−10849.6	−1.11789
$456$	2596.61	0.266661
$457$	−5297.88	−0.542285	−0.271143	−	0.962539i	$-0.587401\pi$
$457$	−5297.88	−0.542285	−0.271143	+	0.962539i	$0.587401\pi$
$458$	2903.69	0.296245
$459$	−1552.52	−0.157877
$460$	5007.68	0.507574
$461$	−1709.65	−0.172725	−0.0863624	−	0.996264i	$-0.527524\pi$
$461$	−1709.65	−0.172725	−0.0863624	+	0.996264i	$0.527524\pi$
$462$	3332.09	0.335547
$463$	19902.0	1.99767	0.998836	−	0.0482335i	$-0.0153592\pi$
$463$	19902.0	1.99767	0.998836	+	0.0482335i	$0.0153592\pi$
$464$	−516.566	−0.0516831
$465$	4384.65	0.437276
$466$	149.278	0.0148395
$467$	−3507.73	−0.347577	−0.173788	−	0.984783i	$-0.555601\pi$
$467$	−3507.73	−0.347577	−0.173788	+	0.984783i	$0.555601\pi$
$468$	−12920.7	−1.27620
$469$	−7318.28	−0.720526
$470$	3777.61	0.370740
$471$	−3233.21	−0.316303
$472$	4650.41	0.453501
$473$	−1698.11	−0.165073
$474$	5588.89	0.541574
$475$	475.000	0.0458831
$476$	−15877.0	−1.52883
$477$	−13137.7	−1.26108
$478$	890.514	0.0852117
$479$	16805.5	1.60305	0.801526	−	0.597960i	$-0.204022\pi$
$479$	16805.5	1.60305	0.801526	+	0.597960i	$0.204022\pi$
$480$	−6814.73	−0.648017
$481$	24287.9	2.30235
$482$	3024.25	0.285790
$483$	37554.5	3.53787
$484$	−776.011	−0.0728786
$485$	−8065.89	−0.755161
$486$	6845.05	0.638884
$487$	−4319.89	−0.401957	−0.200979	−	0.979596i	$-0.564412\pi$
$487$	−4319.89	−0.401957	−0.200979	+	0.979596i	$0.564412\pi$
$488$	−7065.18	−0.655381
$489$	29474.4	2.72572
$490$	−4267.84	−0.393472
$491$	−542.067	−0.0498231	−0.0249116	−	0.999690i	$-0.507930\pi$
$491$	−542.067	−0.0498231	−0.0249116	+	0.999690i	$0.507930\pi$
$492$	−3046.04	−0.279118
$493$	1407.64	0.128595
$494$	−1625.59	−0.148054
$495$	1631.39	0.148132
$496$	3312.88	0.299905
$497$	5600.90	0.505503
$498$	11586.8	1.04261
$499$	1002.36	0.0899232	0.0449616	−	0.998989i	$-0.485683\pi$
$499$	1002.36	0.0899232	0.0449616	+	0.998989i	$0.485683\pi$
$500$	−801.664	−0.0717030
$501$	−6500.64	−0.579695
$502$	−3047.54	−0.270953
$503$	1422.95	0.126136	0.0630680	−	0.998009i	$-0.479912\pi$
$503$	1422.95	0.126136	0.0630680	+	0.998009i	$0.479912\pi$
$504$	−17204.3	−1.52052
$505$	−121.748	−0.0107282
$506$	2163.82	0.190106
$507$	18189.2	1.59332
$508$	−9530.42	−0.832370
$509$	−22415.5	−1.95196	−0.975981	−	0.217854i	$-0.930094\pi$
$509$	−22415.5	−1.95196	−0.975981	+	0.217854i	$0.930094\pi$
$510$	3673.78	0.318975
$511$	24187.8	2.09394
$512$	−9279.29	−0.800958
$513$	380.662	0.0327615
$514$	6422.03	0.551096
$515$	−2456.20	−0.210161
$516$	7452.48	0.635809
$517$	−6597.72	−0.561252
$518$	14389.9	1.22057
$519$	5051.12	0.427205
$520$	6165.81	0.519978
$521$	−16329.0	−1.37310	−0.686551	−	0.727082i	$-0.740876\pi$
$521$	−16329.0	−1.37310	−0.686551	+	0.727082i	$0.740876\pi$
$522$	678.704	0.0569081
$523$	5385.26	0.450250	0.225125	−	0.974330i	$-0.427721\pi$
$523$	5385.26	0.450250	0.225125	+	0.974330i	$0.427721\pi$
$524$	16567.1	1.38117
$525$	−6011.99	−0.499781
$526$	254.379	0.0210864
$527$	−9027.62	−0.746203
$528$	2354.63	0.194076
$529$	12220.5	1.00440
$530$	2789.59	0.228626
$531$	7597.60	0.620918
$532$	3892.87	0.317251
$533$	4285.68	0.348281
$534$	−11855.8	−0.960772
$535$	343.283	0.0277409
$536$	4158.96	0.335148
$537$	−9100.90	−0.731346
$538$	−4426.50	−0.354721
$539$	7453.92	0.595664
$540$	−642.449	−0.0511974
$541$	18495.8	1.46986	0.734931	−	0.678142i	$-0.237215\pi$
$541$	18495.8	1.46986	0.734931	+	0.678142i	$0.237215\pi$
$542$	−7385.18	−0.585278
$543$	16714.6	1.32098
$544$	14030.9	1.10583
$545$	3438.23	0.270234
$546$	20574.8	1.61267
$547$	−19440.9	−1.51962	−0.759811	−	0.650144i	$-0.774708\pi$
$547$	−19440.9	−1.51962	−0.759811	+	0.650144i	$0.774708\pi$
$548$	8113.32	0.632452
$549$	−11542.7	−0.897325
$550$	−346.400	−0.0268556
$551$	−345.139	−0.0266849
$552$	−21342.1	−1.64562
$553$	18830.8	1.44805
$554$	−129.407	−0.00992418
$555$	13458.4	1.02933
$556$	14082.4	1.07415
$557$	4545.26	0.345761	0.172880	−	0.984943i	$-0.444693\pi$
$557$	4545.26	0.345761	0.172880	+	0.984943i	$0.444693\pi$
$558$	−4352.72	−0.330224
$559$	−10485.4	−0.793354
$560$	−4542.44	−0.342774
$561$	−6416.38	−0.482887
$562$	−10114.1	−0.759138
$563$	9419.23	0.705103	0.352552	−	0.935792i	$-0.385314\pi$
$563$	9419.23	0.705103	0.352552	+	0.935792i	$0.385314\pi$
$564$	28955.3	2.16177
$565$	−3815.28	−0.284088
$566$	6035.97	0.448252
$567$	20767.4	1.53818
$568$	−3182.98	−0.235132
$569$	14275.8	1.05179	0.525897	−	0.850548i	$-0.323730\pi$
$569$	14275.8	1.05179	0.525897	+	0.850548i	$0.323730\pi$
$570$	−900.769	−0.0661913
$571$	15872.4	1.16329	0.581647	−	0.813441i	$-0.302409\pi$
$571$	15872.4	1.16329	0.581647	+	0.813441i	$0.302409\pi$
$572$	−4791.66	−0.350261
$573$	−7089.12	−0.516845
$574$	2539.15	0.184638
$575$	−3904.13	−0.283154
$576$	17.1779	0.00124262
$577$	−12192.7	−0.879704	−0.439852	−	0.898070i	$-0.644969\pi$
$577$	−12192.7	−0.879704	−0.439852	+	0.898070i	$0.644969\pi$
$578$	−1375.39	−0.0989770
$579$	19493.1	1.39915
$580$	582.496	0.0417014
$581$	39040.0	2.78770
$582$	15295.8	1.08940
$583$	−4872.11	−0.346110
$584$	−13745.8	−0.973984
$585$	10073.4	0.711937
$586$	5514.60	0.388748
$587$	18151.6	1.27631	0.638156	−	0.769907i	$-0.279697\pi$
$587$	18151.6	1.27631	0.638156	+	0.769907i	$0.279697\pi$
$588$	−32712.9	−2.29432
$589$	2213.47	0.154846
$590$	−1613.23	−0.112569
$591$	576.070	0.0400954
$592$	10168.7	0.705962
$593$	15404.5	1.06676	0.533380	−	0.845876i	$-0.320922\pi$
$593$	15404.5	1.06676	0.533380	+	0.845876i	$0.320922\pi$
$594$	−277.603	−0.0191754
$595$	12378.2	0.852867
$596$	14466.1	0.994222
$597$	20918.5	1.43407
$598$	13361.0	0.913667
$599$	22290.5	1.52047	0.760237	−	0.649645i	$-0.225083\pi$
$599$	22290.5	1.52047	0.760237	+	0.649645i	$0.225083\pi$
$600$	3416.60	0.232470
$601$	−6328.64	−0.429535	−0.214768	−	0.976665i	$-0.568899\pi$
$601$	−6328.64	−0.429535	−0.214768	+	0.976665i	$0.568899\pi$
$602$	−6212.32	−0.420590
$603$	6794.69	0.458874
$604$	13673.0	0.921102
$605$	605.000	0.0406558
$606$	230.878	0.0154765
$607$	−10340.7	−0.691457	−0.345728	−	0.938335i	$-0.612368\pi$
$607$	−10340.7	−0.691457	−0.345728	+	0.938335i	$0.612368\pi$
$608$	−3440.23	−0.229473
$609$	4368.36	0.290665
$610$	2450.92	0.162680
$611$	−40739.2	−2.69743
$612$	14741.1	0.973649
$613$	−11683.3	−0.769792	−0.384896	−	0.922960i	$-0.625763\pi$
$613$	−11683.3	−0.769792	−0.384896	+	0.922960i	$0.625763\pi$
$614$	9039.32	0.594133
$615$	2374.78	0.155708
$616$	−6380.23	−0.417316
$617$	−9286.83	−0.605954	−0.302977	−	0.952998i	$-0.597981\pi$
$617$	−9286.83	−0.605954	−0.302977	+	0.952998i	$0.597981\pi$
$618$	4657.83	0.303180
$619$	23926.3	1.55360	0.776800	−	0.629747i	$-0.216841\pi$
$619$	23926.3	1.55360	0.776800	+	0.629747i	$0.216841\pi$
$620$	−3735.71	−0.241983
$621$	−3128.74	−0.202177
$622$	11680.8	0.752985
$623$	−39946.3	−2.56888
$624$	14539.2	0.932746
$625$	625.000	0.0400000
$626$	−9584.66	−0.611949
$627$	1573.22	0.100205
$628$	2754.69	0.175038
$629$	−27709.7	−1.75653
$630$	5968.21	0.377427
$631$	−4840.63	−0.305392	−0.152696	−	0.988273i	$-0.548796\pi$
$631$	−4840.63	−0.305392	−0.152696	+	0.988273i	$0.548796\pi$
$632$	−10701.5	−0.673549
$633$	−40443.9	−2.53949
$634$	−10295.7	−0.644942
$635$	7430.18	0.464343
$636$	21382.1	1.33311
$637$	46026.0	2.86282
$638$	251.697	0.0156188
$639$	−5200.18	−0.321934
$640$	7238.95	0.447101
$641$	9602.17	0.591674	0.295837	−	0.955238i	$-0.404401\pi$
$641$	9602.17	0.591674	0.295837	+	0.955238i	$0.404401\pi$
$642$	−650.986	−0.0400193
$643$	11515.8	0.706284	0.353142	−	0.935570i	$-0.385113\pi$
$643$	11515.8	0.706284	0.353142	+	0.935570i	$0.385113\pi$
$644$	−31996.3	−1.95781
$645$	−5810.17	−0.354690
$646$	1854.61	0.112954
$647$	−28834.1	−1.75206	−0.876031	−	0.482255i	$-0.839818\pi$
$647$	−28834.1	−1.75206	−0.876031	+	0.482255i	$0.839818\pi$
$648$	−11802.1	−0.715477
$649$	2817.57	0.170415
$650$	−2138.93	−0.129070
$651$	−28015.5	−1.68666
$652$	−25112.1	−1.50838
$653$	−27178.0	−1.62873	−0.814363	−	0.580355i	$-0.802914\pi$
$653$	−27178.0	−1.62873	−0.814363	+	0.580355i	$0.802914\pi$
$654$	−6520.11	−0.389842
$655$	−12916.1	−0.770497
$656$	1794.30	0.106792
$657$	−22457.2	−1.33355
$658$	−24136.9	−1.43002
$659$	12884.4	0.761617	0.380808	−	0.924654i	$-0.375646\pi$
$659$	12884.4	0.761617	0.380808	+	0.924654i	$0.375646\pi$
$660$	−2655.15	−0.156593
$661$	−5418.49	−0.318842	−0.159421	−	0.987211i	$-0.550963\pi$
$661$	−5418.49	−0.318842	−0.159421	+	0.987211i	$0.550963\pi$
$662$	10218.2	0.599911
$663$	−39619.4	−2.32080
$664$	−22186.3	−1.29668
$665$	−3034.99	−0.176980
$666$	−13360.4	−0.777333
$667$	2836.77	0.164678
$668$	5538.53	0.320796
$669$	44029.9	2.54454
$670$	−1442.75	−0.0831914
$671$	−4280.62	−0.246276
$672$	43542.4	2.49953
$673$	−17973.1	−1.02944	−0.514719	−	0.857359i	$-0.672104\pi$
$673$	−17973.1	−1.02944	−0.514719	+	0.857359i	$0.672104\pi$
$674$	−4857.03	−0.277575
$675$	500.871	0.0285608
$676$	−15497.1	−0.881722
$677$	−24334.4	−1.38146	−0.690730	−	0.723112i	$-0.742711\pi$
$677$	−24334.4	−1.38146	−0.690730	+	0.723112i	$0.742711\pi$
$678$	7235.13	0.409828
$679$	51536.7	2.91281
$680$	−7034.48	−0.396706
$681$	−16256.4	−0.914751
$682$	−1614.21	−0.0906321
$683$	18812.2	1.05392	0.526962	−	0.849889i	$-0.323331\pi$
$683$	18812.2	1.05392	0.526962	+	0.849889i	$0.323331\pi$
$684$	−3614.35	−0.202044
$685$	−6325.37	−0.352818
$686$	13466.1	0.749475
$687$	−17352.0	−0.963637
$688$	−4389.95	−0.243264
$689$	−30083.9	−1.66344
$690$	7403.61	0.408479
$691$	1210.13	0.0666216	0.0333108	−	0.999445i	$-0.489395\pi$
$691$	1210.13	0.0666216	0.0333108	+	0.999445i	$0.489395\pi$
$692$	−4303.54	−0.236410
$693$	−10423.7	−0.571375
$694$	110.374	0.00603711
$695$	−10979.0	−0.599219
$696$	−2482.53	−0.135201
$697$	−4889.47	−0.265713
$698$	−7049.88	−0.382295
$699$	−892.064	−0.0482703
$700$	5122.20	0.276573
$701$	9786.34	0.527282	0.263641	−	0.964621i	$-0.415077\pi$
$701$	9786.34	0.527282	0.263641	+	0.964621i	$0.415077\pi$
$702$	−1714.12	−0.0921587
$703$	6794.11	0.364501
$704$	6.37043	0.000341044 0
$705$	−22574.4	−1.20596
$706$	7476.67	0.398567
$707$	777.906	0.0413807
$708$	−12365.4	−0.656385
$709$	−3651.45	−0.193418	−0.0967089	−	0.995313i	$-0.530832\pi$
$709$	−3651.45	−0.193418	−0.0967089	+	0.995313i	$0.530832\pi$
$710$	1104.18	0.0583650
$711$	−17483.6	−0.922201
$712$	22701.4	1.19490
$713$	−18193.0	−0.955586
$714$	−23473.4	−1.23035
$715$	3735.71	0.195395
$716$	7753.94	0.404719
$717$	−5321.57	−0.277179
$718$	789.170	0.0410189
$719$	−24718.3	−1.28211	−0.641056	−	0.767494i	$-0.721503\pi$
$719$	−24718.3	−1.28211	−0.641056	+	0.767494i	$0.721503\pi$
$720$	4217.45	0.218299
$721$	15693.8	0.810634
$722$	−454.729	−0.0234394
$723$	−18072.4	−0.929628
$724$	−14240.8	−0.731013
$725$	−454.130	−0.0232634
$726$	−1147.30	−0.0586503
$727$	9396.23	0.479349	0.239675	−	0.970853i	$-0.422959\pi$
$727$	9396.23	0.479349	0.239675	+	0.970853i	$0.422959\pi$
$728$	−39396.2	−2.00566
$729$	−23353.5	−1.18648
$730$	4768.45	0.241765
$731$	11962.6	0.605272
$732$	18786.3	0.948581
$733$	−5569.82	−0.280663	−0.140331	−	0.990105i	$-0.544817\pi$
$733$	−5569.82	−0.280663	−0.140331	+	0.990105i	$0.544817\pi$
$734$	4040.50	0.203185
$735$	25503.9	1.27990
$736$	28276.0	1.41612
$737$	2519.81	0.125941
$738$	−2357.49	−0.117588
$739$	−3583.62	−0.178384	−0.0891919	−	0.996014i	$-0.528428\pi$
$739$	−3583.62	−0.178384	−0.0891919	+	0.996014i	$0.528428\pi$
$740$	−11466.5	−0.569618
$741$	9714.24	0.481594
$742$	−17823.9	−0.881856
$743$	−1937.44	−0.0956633	−0.0478316	−	0.998855i	$-0.515231\pi$
$743$	−1937.44	−0.0956633	−0.0478316	+	0.998855i	$0.515231\pi$
$744$	15921.1	0.784540
$745$	−11278.2	−0.554633
$746$	−4654.18	−0.228420
$747$	−36246.8	−1.77537
$748$	5466.73	0.267224
$749$	−2193.39	−0.107002
$750$	−1185.22	−0.0577043
$751$	17176.8	0.834608	0.417304	−	0.908767i	$-0.362975\pi$
$751$	17176.8	0.834608	0.417304	+	0.908767i	$0.362975\pi$
$752$	−17056.4	−0.827104
$753$	18211.6	0.881365
$754$	1554.16	0.0750653
$755$	−10659.8	−0.513843
$756$	4104.90	0.197478
$757$	14439.6	0.693287	0.346643	−	0.937997i	$-0.387321\pi$
$757$	14439.6	0.693287	0.346643	+	0.937997i	$0.387321\pi$
$758$	−12938.7	−0.619994
$759$	−12930.7	−0.618384
$760$	1724.78	0.0823214
$761$	7773.40	0.370283	0.185142	−	0.982712i	$-0.440726\pi$
$761$	7773.40	0.370283	0.185142	+	0.982712i	$0.440726\pi$
$762$	−14090.3	−0.669864
$763$	−21968.4	−1.04235
$764$	6039.91	0.286016
$765$	−11492.6	−0.543157
$766$	7835.49	0.369592
$767$	17397.7	0.819029
$768$	−13762.5	−0.646629
$769$	6777.95	0.317840	0.158920	−	0.987291i	$-0.449199\pi$
$769$	6777.95	0.317840	0.158920	+	0.987291i	$0.449199\pi$
$770$	2213.31	0.103587
$771$	−38377.0	−1.79262
$772$	−16608.1	−0.774271
$773$	−5366.37	−0.249696	−0.124848	−	0.992176i	$-0.539844\pi$
$773$	−5366.37	−0.249696	−0.124848	+	0.992176i	$0.539844\pi$
$774$	5767.85	0.267857
$775$	2912.46	0.134992
$776$	−29288.1	−1.35487
$777$	−85991.8	−3.97032
$778$	−14542.0	−0.670122
$779$	1198.84	0.0551387
$780$	−16394.9	−0.752603
$781$	−1928.49	−0.0883569
$782$	−15243.4	−0.697063
$783$	−363.937	−0.0166105
$784$	19269.8	0.877816
$785$	−2147.63	−0.0976462
$786$	24493.6	1.11152
$787$	−22829.9	−1.03405	−0.517026	−	0.855970i	$-0.672961\pi$
$787$	−22829.9	−1.03405	−0.517026	+	0.855970i	$0.672961\pi$
$788$	−490.810	−0.0221883
$789$	−1520.13	−0.0685907
$790$	3712.37	0.167190
$791$	24377.6	1.09579
$792$	5923.75	0.265772
$793$	−26431.7	−1.18363
$794$	−12435.7	−0.555825
$795$	−16670.1	−0.743683
$796$	−17822.5	−0.793597
$797$	13842.1	0.615199	0.307599	−	0.951516i	$-0.400474\pi$
$797$	13842.1	0.615199	0.307599	+	0.951516i	$0.400474\pi$
$798$	5755.43	0.255313
$799$	46478.7	2.05795
$800$	−4526.62	−0.200050
$801$	37088.3	1.63602
$802$	−18377.6	−0.809148
$803$	−8328.26	−0.366000
$804$	−11058.6	−0.485085
$805$	24945.2	1.09218
$806$	−9967.28	−0.435586
$807$	26452.0	1.15385
$808$	−442.082	−0.0192480
$809$	−33075.3	−1.43741	−0.718706	−	0.695314i	$-0.755265\pi$
$809$	−33075.3	−1.43741	−0.718706	+	0.695314i	$0.755265\pi$
$810$	4094.15	0.177597
$811$	1853.08	0.0802347	0.0401173	−	0.999195i	$-0.487227\pi$
$811$	1853.08	0.0802347	0.0401173	+	0.999195i	$0.487227\pi$
$812$	−3721.83	−0.160851
$813$	44132.6	1.90381
$814$	−4954.69	−0.213344
$815$	19578.1	0.841461
$816$	−16587.6	−0.711619
$817$	−2933.11	−0.125601
$818$	14223.6	0.607964
$819$	−64363.4	−2.74608
$820$	−2023.31	−0.0861670
$821$	29001.3	1.23283	0.616415	−	0.787422i	$-0.288584\pi$
$821$	29001.3	1.23283	0.616415	+	0.787422i	$0.288584\pi$
$822$	11995.2	0.508977
$823$	−2509.58	−0.106292	−0.0531461	−	0.998587i	$-0.516925\pi$
$823$	−2509.58	−0.106292	−0.0531461	+	0.998587i	$0.516925\pi$
$824$	−8918.73	−0.377062
$825$	2070.03	0.0873567
$826$	10307.7	0.434201
$827$	22871.1	0.961674	0.480837	−	0.876810i	$-0.340333\pi$
$827$	22871.1	0.961674	0.480837	+	0.876810i	$0.340333\pi$
$828$	29707.1	1.24685
$829$	−42739.8	−1.79061	−0.895305	−	0.445454i	$-0.853042\pi$
$829$	−42739.8	−1.79061	−0.895305	+	0.445454i	$0.853042\pi$
$830$	7696.46	0.321865
$831$	773.318	0.0322817
$832$	39.3357	0.00163909
$833$	−52510.3	−2.18412
$834$	20820.1	0.864438
$835$	−4317.99	−0.178958
$836$	−1340.38	−0.0554523
$837$	2334.03	0.0963869
$838$	−9845.62	−0.405861
$839$	−24711.2	−1.01684	−0.508419	−	0.861110i	$-0.669770\pi$
$839$	−24711.2	−1.01684	−0.508419	+	0.861110i	$0.669770\pi$
$840$	−21830.2	−0.896683
$841$	−24059.0	−0.986470
$842$	4878.15	0.199658
$843$	60440.0	2.46935
$844$	34458.1	1.40533
$845$	12082.0	0.491874
$846$	22410.0	0.910722
$847$	−3865.62	−0.156817
$848$	−12595.3	−0.510054
$849$	−36070.0	−1.45809
$850$	2440.27	0.0984713
$851$	−55842.2	−2.24941
$852$	8463.52	0.340323
$853$	9106.85	0.365548	0.182774	−	0.983155i	$-0.441492\pi$
$853$	9106.85	0.365548	0.182774	+	0.983155i	$0.441492\pi$
$854$	−15660.1	−0.627490
$855$	2817.85	0.112712
$856$	1246.50	0.0497715
$857$	4830.17	0.192527	0.0962635	−	0.995356i	$-0.469311\pi$
$857$	4830.17	0.192527	0.0962635	+	0.995356i	$0.469311\pi$
$858$	−7084.24	−0.281879
$859$	−32182.0	−1.27827	−0.639137	−	0.769093i	$-0.720708\pi$
$859$	−32182.0	−1.27827	−0.639137	+	0.769093i	$0.720708\pi$
$860$	4950.24	0.196281
$861$	−15173.6	−0.600597
$862$	−10906.9	−0.430963
$863$	32873.1	1.29666	0.648328	−	0.761361i	$-0.275469\pi$
$863$	32873.1	1.29666	0.648328	+	0.761361i	$0.275469\pi$
$864$	−3627.60	−0.142840
$865$	3355.16	0.131883
$866$	−16737.1	−0.656756
$867$	8219.11	0.321956
$868$	23869.2	0.933378
$869$	−6483.78	−0.253104
$870$	861.192	0.0335599
$871$	15559.1	0.605283
$872$	12484.6	0.484841
$873$	−47849.4	−1.85505
$874$	3737.51	0.144649
$875$	−3993.41	−0.154288
$876$	36550.1	1.40972
$877$	−39189.5	−1.50893	−0.754467	−	0.656338i	$-0.772105\pi$
$877$	−39189.5	−1.50893	−0.754467	+	0.656338i	$0.772105\pi$
$878$	18049.3	0.693774
$879$	−32954.4	−1.26453
$880$	1564.04	0.0599134
$881$	9867.32	0.377342	0.188671	−	0.982040i	$-0.439582\pi$
$881$	9867.32	0.377342	0.188671	+	0.982040i	$0.439582\pi$
$882$	−25318.2	−0.966561
$883$	−224.088	−0.00854039	−0.00427020	−	0.999991i	$-0.501359\pi$
$883$	−224.088	−0.00854039	−0.00427020	+	0.999991i	$0.501359\pi$
$884$	33755.6	1.28430
$885$	9640.43	0.366169
$886$	10362.2	0.392918
$887$	−7918.21	−0.299738	−0.149869	−	0.988706i	$-0.547885\pi$
$887$	−7918.21	−0.299738	−0.149869	+	0.988706i	$0.547885\pi$
$888$	48868.9	1.84677
$889$	−47474.8	−1.79106
$890$	−7875.13	−0.296601
$891$	−7150.58	−0.268859
$892$	−37513.4	−1.40812
$893$	−11396.1	−0.427049
$894$	21387.5	0.800118
$895$	−6045.19	−0.225775
$896$	−46252.9	−1.72456
$897$	−79843.4	−2.97201
$898$	−7344.92	−0.272944
$899$	−2116.22	−0.0785093
$900$	−4755.73	−0.176138
$901$	34322.3	1.26908
$902$	−874.274	−0.0322729
$903$	37123.8	1.36811
$904$	−13853.7	−0.509698
$905$	11102.5	0.407801
$906$	20214.8	0.741273
$907$	−1134.52	−0.0415338	−0.0207669	−	0.999784i	$-0.506611\pi$
$907$	−1134.52	−0.0415338	−0.0207669	+	0.999784i	$0.506611\pi$
$908$	13850.4	0.506213
$909$	−722.250	−0.0263537
$910$	13666.6	0.497850
$911$	16304.1	0.592953	0.296476	−	0.955040i	$-0.404188\pi$
$911$	16304.1	0.592953	0.296476	+	0.955040i	$0.404188\pi$
$912$	4067.09	0.147670
$913$	−13442.1	−0.487261
$914$	6673.40	0.241506
$915$	−14646.3	−0.529172
$916$	14783.8	0.533265
$917$	82527.1	2.97196
$918$	1955.62	0.0703105
$919$	41847.4	1.50209	0.751045	−	0.660252i	$-0.229550\pi$
$919$	41847.4	1.50209	0.751045	+	0.660252i	$0.229550\pi$
$920$	−14176.3	−0.508021
$921$	−54017.5	−1.93262
$922$	2153.53	0.0769228
$923$	−11907.9	−0.424651
$924$	16965.0	0.604012
$925$	8939.61	0.317765
$926$	−25069.2	−0.889661
$927$	−14571.0	−0.516260
$928$	3289.08	0.116346
$929$	−3783.21	−0.133609	−0.0668047	−	0.997766i	$-0.521280\pi$
$929$	−3783.21	−0.133609	−0.0668047	+	0.997766i	$0.521280\pi$
$930$	−5523.07	−0.194740
$931$	12875.0	0.453233
$932$	760.036	0.0267122
$933$	−69802.4	−2.44933
$934$	4418.47	0.154793
$935$	−4262.02	−0.149073
$936$	36577.5	1.27732
$937$	1622.54	0.0565699	0.0282849	−	0.999600i	$-0.490995\pi$
$937$	1622.54	0.0565699	0.0282849	+	0.999600i	$0.490995\pi$
$938$	9218.38	0.320886
$939$	57276.4	1.99057
$940$	19233.3	0.667363
$941$	38389.6	1.32993	0.664965	−	0.746874i	$-0.268446\pi$
$941$	38389.6	1.32993	0.664965	+	0.746874i	$0.268446\pi$
$942$	4072.68	0.140865
$943$	−9853.56	−0.340272
$944$	7283.95	0.251136
$945$	−3200.29	−0.110165
$946$	2139.01	0.0735149
$947$	−20721.7	−0.711051	−0.355526	−	0.934667i	$-0.615698\pi$
$947$	−20721.7	−0.711051	−0.355526	+	0.934667i	$0.615698\pi$
$948$	28455.3	0.974878
$949$	−51424.8	−1.75903
$950$	−598.328	−0.0204340
$951$	61525.3	2.09789
$952$	44946.5	1.53017
$953$	−10259.1	−0.348715	−0.174358	−	0.984682i	$-0.555785\pi$
$953$	−10259.1	−0.348715	−0.174358	+	0.984682i	$0.555785\pi$
$954$	16548.7	0.561619
$955$	−4708.88	−0.159556
$956$	4533.96	0.153388
$957$	−1504.10	−0.0508053
$958$	−21168.8	−0.713918
$959$	40415.7	1.36089
$960$	21.7967	0.000732797 0
$961$	−16219.1	−0.544429
$962$	−30593.9	−1.02535
$963$	2036.46	0.0681455
$964$	15397.7	0.514445
$965$	12948.1	0.431932
$966$	−47305.1	−1.57559
$967$	46006.0	1.52994	0.764970	−	0.644066i	$-0.222754\pi$
$967$	46006.0	1.52994	0.764970	+	0.644066i	$0.222754\pi$
$968$	2196.82	0.0729427
$969$	−11082.8	−0.367422
$970$	10160.1	0.336310
$971$	35834.8	1.18434	0.592170	−	0.805813i	$-0.298271\pi$
$971$	35834.8	1.18434	0.592170	+	0.805813i	$0.298271\pi$
$972$	34850.9	1.15004
$973$	70149.9	2.31131
$974$	5441.50	0.179011
$975$	12781.9	0.419844
$976$	−11066.2	−0.362932
$977$	52970.1	1.73456	0.867280	−	0.497821i	$-0.165866\pi$
$977$	52970.1	1.73456	0.867280	+	0.497821i	$0.165866\pi$
$978$	−37127.0	−1.21390
$979$	13754.2	0.449015
$980$	−21729.2	−0.708281
$981$	20396.7	0.663829
$982$	682.808	0.0221887
$983$	−15020.5	−0.487365	−0.243682	−	0.969855i	$-0.578355\pi$
$983$	−15020.5	−0.487365	−0.243682	+	0.969855i	$0.578355\pi$
$984$	8623.09	0.279364
$985$	382.649	0.0123779
$986$	−1773.12	−0.0572694
$987$	144238.	4.65162
$988$	−8276.50	−0.266509
$989$	24107.8	0.775110
$990$	−2054.96	−0.0659705
$991$	−40843.1	−1.30921	−0.654603	−	0.755972i	$-0.727164\pi$
$991$	−40843.1	−1.30921	−0.654603	+	0.755972i	$0.727164\pi$
$992$	−21093.8	−0.675129
$993$	−61062.3	−1.95141
$994$	−7055.11	−0.225125
$995$	13894.9	0.442713
$996$	58993.3	1.87678
$997$	−53503.0	−1.69956	−0.849778	−	0.527141i	$-0.823264\pi$
$997$	−53503.0	−1.69956	−0.849778	+	0.527141i	$0.823264\pi$
$998$	−1262.61	−0.0400472
$999$	7164.15	0.226890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-1.25964 q^{2} +7.52739 q^{3} -6.41331 q^{4} +5.00000 q^{5} -9.48178 q^{6} -31.9473 q^{7} +18.1555 q^{8} +29.6616 q^{9} +O(q^{10})\)	\(q-1.25964 q^{2} +7.52739 q^{3} -6.41331 q^{4} +5.00000 q^{5} -9.48178 q^{6} -31.9473 q^{7} +18.1555 q^{8} +29.6616 q^{9} -6.29819 q^{10} +11.0000 q^{11} -48.2755 q^{12} +67.9220 q^{13} +40.2420 q^{14} +37.6369 q^{15} +28.4371 q^{16} -77.4913 q^{17} -37.3628 q^{18} +19.0000 q^{19} -32.0666 q^{20} -240.480 q^{21} -13.8560 q^{22} -156.165 q^{23} +136.664 q^{24} +25.0000 q^{25} -85.5571 q^{26} +20.0348 q^{27} +204.888 q^{28} -18.1652 q^{29} -47.4089 q^{30} +116.499 q^{31} -181.065 q^{32} +82.8013 q^{33} +97.6109 q^{34} -159.736 q^{35} -190.229 q^{36} +357.585 q^{37} -23.9331 q^{38} +511.276 q^{39} +90.7777 q^{40} +63.0971 q^{41} +302.917 q^{42} -154.374 q^{43} -70.5465 q^{44} +148.308 q^{45} +196.711 q^{46} -599.793 q^{47} +214.057 q^{48} +677.629 q^{49} -31.4909 q^{50} -583.307 q^{51} -435.605 q^{52} -442.919 q^{53} -25.2366 q^{54} +55.0000 q^{55} -580.021 q^{56} +143.020 q^{57} +22.8816 q^{58} +256.143 q^{59} -241.378 q^{60} -389.147 q^{61} -146.746 q^{62} -947.607 q^{63} +0.579130 q^{64} +339.610 q^{65} -104.300 q^{66} +229.074 q^{67} +496.976 q^{68} -1175.51 q^{69} +201.210 q^{70} -175.317 q^{71} +538.522 q^{72} -757.114 q^{73} -450.427 q^{74} +188.185 q^{75} -121.853 q^{76} -351.420 q^{77} -644.022 q^{78} -589.435 q^{79} +142.186 q^{80} -650.053 q^{81} -79.4795 q^{82} -1222.01 q^{83} +1542.27 q^{84} -387.456 q^{85} +194.455 q^{86} -136.737 q^{87} +199.711 q^{88} +1250.38 q^{89} -186.814 q^{90} -2169.92 q^{91} +1001.54 q^{92} +876.930 q^{93} +755.521 q^{94} +95.0000 q^{95} -1362.95 q^{96} -1613.18 q^{97} -853.567 q^{98} +326.278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 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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