LMFDB - Embedded newform 1045.4.a.d.1.3

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:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-4.73877 q^{2} +1.86006 q^{3} +14.4559 q^{4} +5.00000 q^{5} -8.81437 q^{6} +1.31072 q^{7} -30.5930 q^{8} -23.5402 q^{9} +O(q^{10})$	\(q-4.73877 q^{2} +1.86006 q^{3} +14.4559 q^{4} +5.00000 q^{5} -8.81437 q^{6} +1.31072 q^{7} -30.5930 q^{8} -23.5402 q^{9} -23.6938 q^{10} -11.0000 q^{11} +26.8888 q^{12} +70.6588 q^{13} -6.21117 q^{14} +9.30028 q^{15} +29.3258 q^{16} -75.7764 q^{17} +111.551 q^{18} -19.0000 q^{19} +72.2795 q^{20} +2.43800 q^{21} +52.1264 q^{22} +40.8312 q^{23} -56.9047 q^{24} +25.0000 q^{25} -334.835 q^{26} -94.0076 q^{27} +18.9476 q^{28} +38.5425 q^{29} -44.0718 q^{30} +243.671 q^{31} +105.776 q^{32} -20.4606 q^{33} +359.087 q^{34} +6.55358 q^{35} -340.295 q^{36} +194.735 q^{37} +90.0365 q^{38} +131.429 q^{39} -152.965 q^{40} -166.705 q^{41} -11.5531 q^{42} -423.484 q^{43} -159.015 q^{44} -117.701 q^{45} -193.489 q^{46} -556.208 q^{47} +54.5476 q^{48} -341.282 q^{49} -118.469 q^{50} -140.948 q^{51} +1021.44 q^{52} +88.0153 q^{53} +445.480 q^{54} -55.0000 q^{55} -40.0987 q^{56} -35.3411 q^{57} -182.644 q^{58} -311.398 q^{59} +134.444 q^{60} +106.589 q^{61} -1154.70 q^{62} -30.8545 q^{63} -735.853 q^{64} +353.294 q^{65} +96.9581 q^{66} +93.3695 q^{67} -1095.42 q^{68} +75.9483 q^{69} -31.0559 q^{70} -681.720 q^{71} +720.164 q^{72} +18.3756 q^{73} -922.804 q^{74} +46.5014 q^{75} -274.662 q^{76} -14.4179 q^{77} -622.813 q^{78} +658.842 q^{79} +146.629 q^{80} +460.726 q^{81} +789.974 q^{82} +1347.15 q^{83} +35.2435 q^{84} -378.882 q^{85} +2006.79 q^{86} +71.6911 q^{87} +336.523 q^{88} +1028.35 q^{89} +557.757 q^{90} +92.6136 q^{91} +590.251 q^{92} +453.241 q^{93} +2635.74 q^{94} -95.0000 q^{95} +196.749 q^{96} -1429.87 q^{97} +1617.26 q^{98} +258.942 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * 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 $

$2$	−4.73877	−1.67541	−0.837703	−	0.546126i	$-0.816102\pi$
$2$	−4.73877	−1.67541	−0.837703	+	0.546126i	$0.816102\pi$
$3$	1.86006	0.357968	0.178984	−	0.983852i	$-0.442719\pi$
$3$	1.86006	0.357968	0.178984	+	0.983852i	$0.442719\pi$
$4$	14.4559	1.80699
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−8.81437	−0.599742
$7$	1.31072	0.0707720	0.0353860	−	0.999374i	$-0.488734\pi$
$7$	1.31072	0.0707720	0.0353860	+	0.999374i	$0.488734\pi$
$8$	−30.5930	−1.35203
$9$	−23.5402	−0.871859
$10$	−23.6938	−0.749265
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	26.8888	0.646843
$13$	70.6588	1.50748	0.753739	−	0.657174i	$-0.228248\pi$
$13$	70.6588	1.50748	0.753739	+	0.657174i	$0.228248\pi$
$14$	−6.21117	−0.118572
$15$	9.30028	0.160088
$16$	29.3258	0.458215
$17$	−75.7764	−1.08109	−0.540544	−	0.841316i	$-0.681781\pi$
$17$	−75.7764	−1.08109	−0.540544	+	0.841316i	$0.681781\pi$
$18$	111.551	1.46072
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	72.2795	0.808109
$21$	2.43800	0.0253341
$22$	52.1264	0.505154
$23$	40.8312	0.370169	0.185085	−	0.982723i	$-0.440744\pi$
$23$	40.8312	0.370169	0.185085	+	0.982723i	$0.440744\pi$
$24$	−56.9047	−0.483984
$25$	25.0000	0.200000
$26$	−334.835	−2.52564
$27$	−94.0076	−0.670066
$28$	18.9476	0.127884
$29$	38.5425	0.246798	0.123399	−	0.992357i	$-0.460620\pi$
$29$	38.5425	0.246798	0.123399	+	0.992357i	$0.460620\pi$
$30$	−44.0718	−0.268213
$31$	243.671	1.41176	0.705879	−	0.708332i	$-0.250552\pi$
$31$	243.671	1.41176	0.705879	+	0.708332i	$0.250552\pi$
$32$	105.776	0.584335
$33$	−20.4606	−0.107931
$34$	359.087	1.81126
$35$	6.55358	0.0316502
$36$	−340.295	−1.57544
$37$	194.735	0.865250	0.432625	−	0.901574i	$-0.357587\pi$
$37$	194.735	0.865250	0.432625	+	0.901574i	$0.357587\pi$
$38$	90.0365	0.384365
$39$	131.429	0.539629
$40$	−152.965	−0.604647
$41$	−166.705	−0.634998	−0.317499	−	0.948259i	$-0.602843\pi$
$41$	−166.705	−0.634998	−0.317499	+	0.948259i	$0.602843\pi$
$42$	−11.5531	−0.0424449
$43$	−423.484	−1.50188	−0.750939	−	0.660372i	$-0.770399\pi$
$43$	−423.484	−1.50188	−0.750939	+	0.660372i	$0.770399\pi$
$44$	−159.015	−0.544827
$45$	−117.701	−0.389907
$46$	−193.489	−0.620184
$47$	−556.208	−1.72620	−0.863098	−	0.505037i	$-0.831479\pi$
$47$	−556.208	−1.72620	−0.863098	+	0.505037i	$0.831479\pi$
$48$	54.5476	0.164026
$49$	−341.282	−0.994991
$50$	−118.469	−0.335081
$51$	−140.948	−0.386995
$52$	1021.44	2.72399
$53$	88.0153	0.228110	0.114055	−	0.993474i	$-0.463616\pi$
$53$	88.0153	0.228110	0.114055	+	0.993474i	$0.463616\pi$
$54$	445.480	1.12263
$55$	−55.0000	−0.134840
$56$	−40.0987	−0.0956859
$57$	−35.3411	−0.0821235
$58$	−182.644	−0.413488
$59$	−311.398	−0.687127	−0.343564	−	0.939129i	$-0.611634\pi$
$59$	−311.398	−0.687127	−0.343564	+	0.939129i	$0.611634\pi$
$60$	134.444	0.289277
$61$	106.589	0.223726	0.111863	−	0.993724i	$-0.464318\pi$
$61$	106.589	0.223726	0.111863	+	0.993724i	$0.464318\pi$
$62$	−1154.70	−2.36527
$63$	−30.8545	−0.0617032
$64$	−735.853	−1.43721
$65$	353.294	0.674165
$66$	96.9581	0.180829
$67$	93.3695	0.170252	0.0851261	−	0.996370i	$-0.472871\pi$
$67$	93.3695	0.170252	0.0851261	+	0.996370i	$0.472871\pi$
$68$	−1095.42	−1.95351
$69$	75.9483	0.132509
$70$	−31.0559	−0.0530269
$71$	−681.720	−1.13951	−0.569755	−	0.821814i	$-0.692962\pi$
$71$	−681.720	−1.13951	−0.569755	+	0.821814i	$0.692962\pi$
$72$	720.164	1.17878
$73$	18.3756	0.0294617	0.0147308	−	0.999891i	$-0.495311\pi$
$73$	18.3756	0.0294617	0.0147308	+	0.999891i	$0.495311\pi$
$74$	−922.804	−1.44965
$75$	46.5014	0.0715936
$76$	−274.662	−0.414551
$77$	−14.4179	−0.0213386
$78$	−622.813	−0.904098
$79$	658.842	0.938298	0.469149	−	0.883119i	$-0.344561\pi$
$79$	658.842	0.938298	0.469149	+	0.883119i	$0.344561\pi$
$80$	146.629	0.204920
$81$	460.726	0.631997
$82$	789.974	1.06388
$83$	1347.15	1.78156	0.890779	−	0.454437i	$-0.150160\pi$
$83$	1347.15	1.78156	0.890779	+	0.454437i	$0.150160\pi$
$84$	35.2435	0.0457784
$85$	−378.882	−0.483477
$86$	2006.79	2.51625
$87$	71.6911	0.0883459
$88$	336.523	0.407653
$89$	1028.35	1.22477	0.612386	−	0.790559i	$-0.290210\pi$
$89$	1028.35	1.22477	0.612386	+	0.790559i	$0.290210\pi$
$90$	557.757	0.653253
$91$	92.6136	0.106687
$92$	590.251	0.668891
$93$	453.241	0.505364
$94$	2635.74	2.89208
$95$	−95.0000	−0.102598
$96$	196.749	0.209173
$97$	−1429.87	−1.49671	−0.748355	−	0.663298i	$-0.769156\pi$
$97$	−1429.87	−1.49671	−0.748355	+	0.663298i	$0.769156\pi$
$98$	1617.26	1.66701
$99$	258.942	0.262875
$100$	361.397	0.361397
$101$	570.691	0.562237	0.281118	−	0.959673i	$-0.409295\pi$
$101$	570.691	0.562237	0.281118	+	0.959673i	$0.409295\pi$
$102$	667.921	0.648373
$103$	127.512	0.121982	0.0609909	−	0.998138i	$-0.480574\pi$
$103$	127.512	0.121982	0.0609909	+	0.998138i	$0.480574\pi$
$104$	−2161.66	−2.03816
$105$	12.1900	0.0113298
$106$	−417.084	−0.382177
$107$	−978.154	−0.883754	−0.441877	−	0.897076i	$-0.645687\pi$
$107$	−978.154	−0.883754	−0.441877	+	0.897076i	$0.645687\pi$
$108$	−1358.96	−1.21080
$109$	123.688	0.108689	0.0543446	−	0.998522i	$-0.482693\pi$
$109$	123.688	0.108689	0.0543446	+	0.998522i	$0.482693\pi$
$110$	260.632	0.225912
$111$	362.218	0.309732
$112$	38.4377	0.0324288
$113$	−468.731	−0.390217	−0.195108	−	0.980782i	$-0.562506\pi$
$113$	−468.731	−0.390217	−0.195108	+	0.980782i	$0.562506\pi$
$114$	167.473	0.137590
$115$	204.156	0.165545
$116$	557.166	0.445962
$117$	−1663.32	−1.31431
$118$	1475.64	1.15122
$119$	−99.3213	−0.0765107
$120$	−284.523	−0.216444
$121$	121.000	0.0909091
$122$	−505.099	−0.374832
$123$	−310.080	−0.227309
$124$	3522.48	2.55103
$125$	125.000	0.0894427
$126$	146.212	0.103378
$127$	−1749.03	−1.22205	−0.611027	−	0.791609i	$-0.709244\pi$
$127$	−1749.03	−1.22205	−0.611027	+	0.791609i	$0.709244\pi$
$128$	2640.83	1.82358
$129$	−787.704	−0.537624
$130$	−1674.18	−1.12950
$131$	681.833	0.454748	0.227374	−	0.973808i	$-0.426986\pi$
$131$	681.833	0.454748	0.227374	+	0.973808i	$0.426986\pi$
$132$	−295.777	−0.195031
$133$	−24.9036	−0.0162362
$134$	−442.456	−0.285242
$135$	−470.038	−0.299662
$136$	2318.23	1.46166
$137$	−3188.28	−1.98827	−0.994134	−	0.108153i	$-0.965506\pi$
$137$	−3188.28	−1.98827	−0.994134	+	0.108153i	$0.965506\pi$
$138$	−359.901	−0.222006
$139$	2431.82	1.48392	0.741959	−	0.670445i	$-0.233897\pi$
$139$	2431.82	1.48392	0.741959	+	0.670445i	$0.233897\pi$
$140$	94.7378	0.0571915
$141$	−1034.58	−0.617923
$142$	3230.51	1.90914
$143$	−777.247	−0.454522
$144$	−690.334	−0.399499
$145$	192.712	0.110372
$146$	−87.0777	−0.0493603
$147$	−634.804	−0.356175
$148$	2815.07	1.56350
$149$	354.998	0.195185	0.0975926	−	0.995226i	$-0.468886\pi$
$149$	354.998	0.195185	0.0975926	+	0.995226i	$0.468886\pi$
$150$	−220.359	−0.119948
$151$	875.625	0.471903	0.235952	−	0.971765i	$-0.424179\pi$
$151$	875.625	0.471903	0.235952	+	0.971765i	$0.424179\pi$
$152$	581.267	0.310177
$153$	1783.79	0.942555
$154$	68.3229	0.0357508
$155$	1218.35	0.631358
$156$	1899.93	0.975103
$157$	−1195.12	−0.607523	−0.303762	−	0.952748i	$-0.598243\pi$
$157$	−1195.12	−0.607523	−0.303762	+	0.952748i	$0.598243\pi$
$158$	−3122.10	−1.57203
$159$	163.713	0.0816561
$160$	528.880	0.261323
$161$	53.5181	0.0261976
$162$	−2183.27	−1.05885
$163$	7.62502	0.00366404	0.00183202	−	0.999998i	$-0.499417\pi$
$163$	7.62502	0.00366404	0.00183202	+	0.999998i	$0.499417\pi$
$164$	−2409.87	−1.14743
$165$	−102.303	−0.0482684
$166$	−6383.84	−2.98483
$167$	−2879.05	−1.33406	−0.667028	−	0.745033i	$-0.732434\pi$
$167$	−2879.05	−1.33406	−0.667028	+	0.745033i	$0.732434\pi$
$168$	−74.5858	−0.0342525
$169$	2795.66	1.27249
$170$	1795.43	0.810020
$171$	447.264	0.200018
$172$	−6121.84	−2.71387
$173$	−3011.06	−1.32327	−0.661637	−	0.749824i	$-0.730138\pi$
$173$	−3011.06	−1.32327	−0.661637	+	0.749824i	$0.730138\pi$
$174$	−339.727	−0.148015
$175$	32.7679	0.0141544
$176$	−322.583	−0.138157
$177$	−579.217	−0.245970
$178$	−4873.10	−2.05199
$179$	−2827.01	−1.18045	−0.590225	−	0.807239i	$-0.700961\pi$
$179$	−2827.01	−1.18045	−0.590225	+	0.807239i	$0.700961\pi$
$180$	−1701.47	−0.704557
$181$	−574.401	−0.235883	−0.117942	−	0.993021i	$-0.537630\pi$
$181$	−574.401	−0.235883	−0.117942	+	0.993021i	$0.537630\pi$
$182$	−438.874	−0.178744
$183$	198.261	0.0800868
$184$	−1249.15	−0.500480
$185$	973.675	0.386951
$186$	−2147.80	−0.846691
$187$	833.541	0.325960
$188$	−8040.48	−3.11921
$189$	−123.217	−0.0474219
$190$	450.183	0.171893
$191$	−612.907	−0.232191	−0.116095	−	0.993238i	$-0.537038\pi$
$191$	−612.907	−0.232191	−0.116095	+	0.993238i	$0.537038\pi$
$192$	−1368.73	−0.514476
$193$	2967.99	1.10694	0.553472	−	0.832868i	$-0.313303\pi$
$193$	2967.99	1.10694	0.553472	+	0.832868i	$0.313303\pi$
$194$	6775.80	2.50760
$195$	657.147	0.241329
$196$	−4933.54	−1.79794
$197$	−776.809	−0.280941	−0.140470	−	0.990085i	$-0.544862\pi$
$197$	−776.809	−0.280941	−0.140470	+	0.990085i	$0.544862\pi$
$198$	−1227.07	−0.440423
$199$	−2532.37	−0.902084	−0.451042	−	0.892503i	$-0.648948\pi$
$199$	−2532.37	−0.902084	−0.451042	+	0.892503i	$0.648948\pi$
$200$	−764.824	−0.270406
$201$	173.672	0.0609448
$202$	−2704.37	−0.941975
$203$	50.5182	0.0174664
$204$	−2037.54	−0.699294
$205$	−833.523	−0.283980
$206$	−604.250	−0.204369
$207$	−961.174	−0.322735
$208$	2072.12	0.690749
$209$	209.000	0.0691714
$210$	−57.7656	−0.0189819
$211$	−4277.21	−1.39552	−0.697761	−	0.716330i	$-0.745820\pi$
$211$	−4277.21	−1.39552	−0.697761	+	0.716330i	$0.745820\pi$
$212$	1272.34	0.412192
$213$	−1268.04	−0.407908
$214$	4635.24	1.48065
$215$	−2117.42	−0.671660
$216$	2875.97	0.905950
$217$	319.383	0.0999130
$218$	−586.126	−0.182099
$219$	34.1797	0.0105463
$220$	−795.074	−0.243654
$221$	−5354.27	−1.62972
$222$	−1716.47	−0.518927
$223$	−3615.34	−1.08566	−0.542828	−	0.839844i	$-0.682646\pi$
$223$	−3615.34	−1.08566	−0.542828	+	0.839844i	$0.682646\pi$
$224$	138.642	0.0413545
$225$	−588.505	−0.174372
$226$	2221.21	0.653772
$227$	4739.10	1.38566	0.692831	−	0.721100i	$-0.256363\pi$
$227$	4739.10	1.38566	0.692831	+	0.721100i	$0.256363\pi$
$228$	−510.887	−0.148396
$229$	449.996	0.129854	0.0649270	−	0.997890i	$-0.479319\pi$
$229$	449.996	0.129854	0.0649270	+	0.997890i	$0.479319\pi$
$230$	−967.447	−0.277355
$231$	−26.8180	−0.00763852
$232$	−1179.13	−0.333679
$233$	−4166.46	−1.17148	−0.585738	−	0.810500i	$-0.699195\pi$
$233$	−4166.46	−1.17148	−0.585738	+	0.810500i	$0.699195\pi$
$234$	7882.09	2.20200
$235$	−2781.04	−0.771978
$236$	−4501.53	−1.24163
$237$	1225.48	0.335881
$238$	470.660	0.128186
$239$	−3217.04	−0.870682	−0.435341	−	0.900266i	$-0.643372\pi$
$239$	−3217.04	−0.870682	−0.435341	+	0.900266i	$0.643372\pi$
$240$	272.738	0.0733548
$241$	3012.00	0.805061	0.402531	−	0.915406i	$-0.368131\pi$
$241$	3012.00	0.805061	0.402531	+	0.915406i	$0.368131\pi$
$242$	−573.391	−0.152310
$243$	3395.18	0.896300
$244$	1540.84	0.404270
$245$	−1706.41	−0.444974
$246$	1469.40	0.380835
$247$	−1342.52	−0.345839
$248$	−7454.61	−1.90874
$249$	2505.78	0.637741
$250$	−592.346	−0.149853
$251$	−1597.43	−0.401709	−0.200854	−	0.979621i	$-0.564372\pi$
$251$	−1597.43	−0.401709	−0.200854	+	0.979621i	$0.564372\pi$
$252$	−446.029	−0.111497
$253$	−449.143	−0.111610
$254$	8288.22	2.04744
$255$	−704.742	−0.173069
$256$	−6627.44	−1.61803
$257$	−5343.11	−1.29686	−0.648432	−	0.761273i	$-0.724575\pi$
$257$	−5343.11	−1.29686	−0.648432	+	0.761273i	$0.724575\pi$
$258$	3732.75	0.900739
$259$	255.242	0.0612354
$260$	5107.18	1.21821
$261$	−907.297	−0.215173
$262$	−3231.04	−0.761888
$263$	1559.71	0.365687	0.182844	−	0.983142i	$-0.441470\pi$
$263$	1559.71	0.365687	0.182844	+	0.983142i	$0.441470\pi$
$264$	625.951	0.145927
$265$	440.077	0.102014
$266$	118.012	0.0272022
$267$	1912.79	0.438429
$268$	1349.74	0.307643
$269$	−3898.57	−0.883644	−0.441822	−	0.897103i	$-0.645668\pi$
$269$	−3898.57	−0.883644	−0.441822	+	0.897103i	$0.645668\pi$
$270$	2227.40	0.502056
$271$	1512.03	0.338927	0.169464	−	0.985536i	$-0.445796\pi$
$271$	1512.03	0.338927	0.169464	+	0.985536i	$0.445796\pi$
$272$	−2222.20	−0.495370
$273$	172.266	0.0381906
$274$	15108.5	3.33116
$275$	−275.000	−0.0603023
$276$	1097.90	0.239441
$277$	3713.74	0.805549	0.402774	−	0.915299i	$-0.368046\pi$
$277$	3713.74	0.805549	0.402774	+	0.915299i	$0.368046\pi$
$278$	−11523.8	−2.48617
$279$	−5736.05	−1.23085
$280$	−200.493	−0.0427920
$281$	122.027	0.0259058	0.0129529	−	0.999916i	$-0.495877\pi$
$281$	122.027	0.0259058	0.0129529	+	0.999916i	$0.495877\pi$
$282$	4902.62	1.03527
$283$	−8171.60	−1.71643	−0.858217	−	0.513287i	$-0.828428\pi$
$283$	−8171.60	−1.71643	−0.858217	+	0.513287i	$0.828428\pi$
$284$	−9854.87	−2.05908
$285$	−176.705	−0.0367267
$286$	3683.19	0.761509
$287$	−218.502	−0.0449400
$288$	−2489.99	−0.509458
$289$	829.064	0.168749
$290$	−913.218	−0.184917
$291$	−2659.63	−0.535774
$292$	265.636	0.0532369
$293$	337.461	0.0672856	0.0336428	−	0.999434i	$-0.489289\pi$
$293$	337.461	0.0672856	0.0336428	+	0.999434i	$0.489289\pi$
$294$	3008.19	0.596738
$295$	−1556.99	−0.307293
$296$	−5957.52	−1.16984
$297$	1034.08	0.202032
$298$	−1682.25	−0.327014
$299$	2885.08	0.558022
$300$	672.219	0.129369
$301$	−555.067	−0.106291
$302$	−4149.38	−0.790630
$303$	1061.52	0.201263
$304$	−557.189	−0.105122
$305$	532.944	0.100053
$306$	−8452.97	−1.57916
$307$	−1911.48	−0.355354	−0.177677	−	0.984089i	$-0.556858\pi$
$307$	−1911.48	−0.355354	−0.177677	+	0.984089i	$0.556858\pi$
$308$	−208.423	−0.0385585
$309$	237.180	0.0436656
$310$	−5773.49	−1.05778
$311$	−7720.72	−1.40772	−0.703861	−	0.710337i	$-0.748542\pi$
$311$	−7720.72	−1.40772	−0.703861	+	0.710337i	$0.748542\pi$
$312$	−4020.81	−0.729595
$313$	−8184.29	−1.47797	−0.738984	−	0.673723i	$-0.764694\pi$
$313$	−8184.29	−1.47797	−0.738984	+	0.673723i	$0.764694\pi$
$314$	5663.40	1.01785
$315$	−154.272	−0.0275945
$316$	9524.16	1.69549
$317$	−8507.35	−1.50732	−0.753660	−	0.657265i	$-0.771713\pi$
$317$	−8507.35	−1.50732	−0.753660	+	0.657265i	$0.771713\pi$
$318$	−775.799	−0.136807
$319$	−423.967	−0.0744125
$320$	−3679.27	−0.642741
$321$	−1819.42	−0.316356
$322$	−253.610	−0.0438916
$323$	1439.75	0.248018
$324$	6660.20	1.14201
$325$	1766.47	0.301496
$326$	−36.1332	−0.00613875
$327$	230.066	0.0389072
$328$	5099.99	0.858536
$329$	−729.030	−0.122166
$330$	484.790	0.0808692
$331$	−1254.97	−0.208396	−0.104198	−	0.994557i	$-0.533228\pi$
$331$	−1254.97	−0.208396	−0.104198	+	0.994557i	$0.533228\pi$
$332$	19474.3	3.21925
$333$	−4584.10	−0.754376
$334$	13643.1	2.23509
$335$	466.847	0.0761391
$336$	71.4963	0.0116085
$337$	10172.0	1.64423	0.822114	−	0.569322i	$-0.192794\pi$
$337$	10172.0	1.64423	0.822114	+	0.569322i	$0.192794\pi$
$338$	−13248.0	−2.13194
$339$	−871.866	−0.139685
$340$	−5477.08	−0.873636
$341$	−2680.38	−0.425661
$342$	−2119.48	−0.335112
$343$	−896.899	−0.141189
$344$	12955.6	2.03058
$345$	379.741	0.0592597
$346$	14268.7	2.21702
$347$	−5582.74	−0.863681	−0.431841	−	0.901950i	$-0.642136\pi$
$347$	−5582.74	−0.863681	−0.431841	+	0.901950i	$0.642136\pi$
$348$	1036.36	0.159640
$349$	8377.99	1.28500	0.642498	−	0.766287i	$-0.277898\pi$
$349$	8377.99	1.28500	0.642498	+	0.766287i	$0.277898\pi$
$350$	−155.279	−0.0237144
$351$	−6642.46	−1.01011
$352$	−1163.54	−0.176184
$353$	5335.49	0.804474	0.402237	−	0.915536i	$-0.368233\pi$
$353$	5335.49	0.804474	0.402237	+	0.915536i	$0.368233\pi$
$354$	2744.77	0.412099
$355$	−3408.60	−0.509605
$356$	14865.7	2.21315
$357$	−184.743	−0.0273884
$358$	13396.5	1.97773
$359$	6269.99	0.921776	0.460888	−	0.887458i	$-0.347531\pi$
$359$	6269.99	0.921776	0.460888	+	0.887458i	$0.347531\pi$
$360$	3600.82	0.527167
$361$	361.000	0.0526316
$362$	2721.95	0.395201
$363$	225.067	0.0325425
$364$	1338.81	0.192782
$365$	91.8780	0.0131757
$366$	−939.513	−0.134178
$367$	761.765	0.108348	0.0541741	−	0.998532i	$-0.482747\pi$
$367$	761.765	0.108348	0.0541741	+	0.998532i	$0.482747\pi$
$368$	1197.41	0.169617
$369$	3924.26	0.553628
$370$	−4614.02	−0.648301
$371$	115.363	0.0161438
$372$	6552.00	0.913187
$373$	−6072.68	−0.842979	−0.421490	−	0.906833i	$-0.638493\pi$
$373$	−6072.68	−0.842979	−0.421490	+	0.906833i	$0.638493\pi$
$374$	−3949.95	−0.546115
$375$	232.507	0.0320176
$376$	17016.0	2.33387
$377$	2723.36	0.372043
$378$	583.897	0.0794509
$379$	7758.52	1.05153	0.525763	−	0.850631i	$-0.323780\pi$
$379$	7758.52	1.05153	0.525763	+	0.850631i	$0.323780\pi$
$380$	−1373.31	−0.185393
$381$	−3253.29	−0.437457
$382$	2904.42	0.389013
$383$	−7928.04	−1.05771	−0.528856	−	0.848711i	$-0.677379\pi$
$383$	−7928.04	−1.05771	−0.528856	+	0.848711i	$0.677379\pi$
$384$	4912.09	0.652784
$385$	−72.0893	−0.00954289
$386$	−14064.6	−1.85458
$387$	9968.90	1.30943
$388$	−20670.0	−2.70454
$389$	−10444.4	−1.36132	−0.680660	−	0.732599i	$-0.738307\pi$
$389$	−10444.4	−1.36132	−0.680660	+	0.732599i	$0.738307\pi$
$390$	−3114.06	−0.404325
$391$	−3094.04	−0.400185
$392$	10440.8	1.34526
$393$	1268.25	0.162785
$394$	3681.12	0.470690
$395$	3294.21	0.419620
$396$	3743.24	0.475012
$397$	−4542.51	−0.574262	−0.287131	−	0.957891i	$-0.592702\pi$
$397$	−4542.51	−0.574262	−0.287131	+	0.957891i	$0.592702\pi$
$398$	12000.3	1.51136
$399$	−46.3221	−0.00581204
$400$	733.144	0.0916430
$401$	−1231.72	−0.153390	−0.0766950	−	0.997055i	$-0.524437\pi$
$401$	−1231.72	−0.153390	−0.0766950	+	0.997055i	$0.524437\pi$
$402$	−822.993	−0.102107
$403$	17217.5	2.12820
$404$	8249.85	1.01595
$405$	2303.63	0.282638
$406$	−239.394	−0.0292633
$407$	−2142.09	−0.260883
$408$	4312.03	0.523229
$409$	−4092.07	−0.494718	−0.247359	−	0.968924i	$-0.579563\pi$
$409$	−4092.07	−0.494718	−0.247359	+	0.968924i	$0.579563\pi$
$410$	3949.87	0.475781
$411$	−5930.37	−0.711736
$412$	1843.30	0.220420
$413$	−408.154	−0.0486293
$414$	4554.78	0.540713
$415$	6735.77	0.796737
$416$	7474.00	0.880872
$417$	4523.33	0.531195
$418$	−990.402	−0.115890
$419$	−13191.5	−1.53806	−0.769032	−	0.639210i	$-0.779261\pi$
$419$	−13191.5	−1.53806	−0.769032	+	0.639210i	$0.779261\pi$
$420$	176.218	0.0204727
$421$	8163.72	0.945072	0.472536	−	0.881311i	$-0.343339\pi$
$421$	8163.72	0.945072	0.472536	+	0.881311i	$0.343339\pi$
$422$	20268.7	2.33807
$423$	13093.2	1.50500
$424$	−2692.65	−0.308412
$425$	−1894.41	−0.216217
$426$	6008.93	0.683412
$427$	139.708	0.0158336
$428$	−14140.1	−1.59693
$429$	−1445.72	−0.162704
$430$	10034.0	1.12530
$431$	−4074.77	−0.455394	−0.227697	−	0.973732i	$-0.573120\pi$
$431$	−4074.77	−0.455394	−0.227697	+	0.973732i	$0.573120\pi$
$432$	−2756.84	−0.307034
$433$	−10706.3	−1.18825	−0.594126	−	0.804372i	$-0.702502\pi$
$433$	−10706.3	−1.18825	−0.594126	+	0.804372i	$0.702502\pi$
$434$	−1513.48	−0.167395
$435$	358.456	0.0395095
$436$	1788.01	0.196400
$437$	−775.792	−0.0849226
$438$	−161.969	−0.0176694
$439$	−14451.4	−1.57113	−0.785565	−	0.618779i	$-0.787628\pi$
$439$	−14451.4	−1.57113	−0.785565	+	0.618779i	$0.787628\pi$
$440$	1682.61	0.182308
$441$	8033.84	0.867492
$442$	25372.6	2.73044
$443$	8976.92	0.962768	0.481384	−	0.876510i	$-0.340134\pi$
$443$	8976.92	0.962768	0.481384	+	0.876510i	$0.340134\pi$
$444$	5236.19	0.559681
$445$	5141.74	0.547735
$446$	17132.3	1.81891
$447$	660.316	0.0698700
$448$	−964.494	−0.101714
$449$	14548.6	1.52915	0.764577	−	0.644533i	$-0.222948\pi$
$449$	14548.6	1.52915	0.764577	+	0.644533i	$0.222948\pi$
$450$	2788.79	0.292144
$451$	1833.75	0.191459
$452$	−6775.93	−0.705117
$453$	1628.71	0.168926
$454$	−22457.5	−2.32155
$455$	463.068	0.0477120
$456$	1081.19	0.111034
$457$	−11685.6	−1.19613	−0.598064	−	0.801448i	$-0.704063\pi$
$457$	−11685.6	−1.19613	−0.598064	+	0.801448i	$0.704063\pi$
$458$	−2132.43	−0.217558
$459$	7123.56	0.724399
$460$	2951.26	0.299137
$461$	−13163.6	−1.32991	−0.664954	−	0.746884i	$-0.731549\pi$
$461$	−13163.6	−1.32991	−0.664954	+	0.746884i	$0.731549\pi$
$462$	127.084	0.0127976
$463$	4010.15	0.402522	0.201261	−	0.979538i	$-0.435496\pi$
$463$	4010.15	0.402522	0.201261	+	0.979538i	$0.435496\pi$
$464$	1130.29	0.113087
$465$	2266.20	0.226006
$466$	19743.9	1.96270
$467$	18788.8	1.86176	0.930879	−	0.365329i	$-0.119043\pi$
$467$	18788.8	1.86176	0.930879	+	0.365329i	$0.119043\pi$
$468$	−24044.8	−2.37494
$469$	122.381	0.0120491
$470$	13178.7	1.29338
$471$	−2222.99	−0.217474
$472$	9526.58	0.929017
$473$	4658.33	0.452833
$474$	−5807.28	−0.562737
$475$	−475.000	−0.0458831
$476$	−1435.78	−0.138254
$477$	−2071.90	−0.198880
$478$	15244.8	1.45875
$479$	12618.3	1.20364	0.601821	−	0.798631i	$-0.294442\pi$
$479$	12618.3	1.20364	0.601821	+	0.798631i	$0.294442\pi$
$480$	983.746	0.0935451
$481$	13759.7	1.30435
$482$	−14273.1	−1.34881
$483$	99.5466	0.00937790
$484$	1749.16	0.164272
$485$	−7149.33	−0.669349
$486$	−16089.0	−1.50167
$487$	−4475.50	−0.416436	−0.208218	−	0.978082i	$-0.566766\pi$
$487$	−4475.50	−0.416436	−0.208218	+	0.978082i	$0.566766\pi$
$488$	−3260.87	−0.302485
$489$	14.1830	0.00131161
$490$	8086.28	0.745512
$491$	−5328.82	−0.489789	−0.244894	−	0.969550i	$-0.578753\pi$
$491$	−5328.82	−0.489789	−0.244894	+	0.969550i	$0.578753\pi$
$492$	−4482.49	−0.410744
$493$	−2920.61	−0.266811
$494$	6361.87	0.579421
$495$	1294.71	0.117561
$496$	7145.82	0.646889
$497$	−893.541	−0.0806454
$498$	−11874.3	−1.06847
$499$	10798.7	0.968771	0.484386	−	0.874855i	$-0.339043\pi$
$499$	10798.7	0.968771	0.484386	+	0.874855i	$0.339043\pi$
$500$	1806.99	0.161622
$501$	−5355.19	−0.477549
$502$	7569.85	0.673026
$503$	−21295.4	−1.88770	−0.943852	−	0.330370i	$-0.892827\pi$
$503$	−21295.4	−1.88770	−0.943852	+	0.330370i	$0.892827\pi$
$504$	943.931	0.0834246
$505$	2853.46	0.251440
$506$	2128.38	0.186992
$507$	5200.09	0.455511
$508$	−25283.7	−2.20824
$509$	6769.82	0.589522	0.294761	−	0.955571i	$-0.404760\pi$
$509$	6769.82	0.589522	0.294761	+	0.955571i	$0.404760\pi$
$510$	3339.61	0.289961
$511$	24.0852	0.00208506
$512$	10279.3	0.887272
$513$	1786.14	0.153724
$514$	25319.7	2.17277
$515$	637.560	0.0545520
$516$	−11387.0	−0.971479
$517$	6118.28	0.520468
$518$	−1209.53	−0.102594
$519$	−5600.74	−0.473690
$520$	−10808.3	−0.911492
$521$	−2040.13	−0.171554	−0.0857771	−	0.996314i	$-0.527337\pi$
$521$	−2040.13	−0.171554	−0.0857771	+	0.996314i	$0.527337\pi$
$522$	4299.47	0.360503
$523$	9782.44	0.817890	0.408945	−	0.912559i	$-0.365897\pi$
$523$	9782.44	0.817890	0.408945	+	0.912559i	$0.365897\pi$
$524$	9856.50	0.821724
$525$	60.9501	0.00506682
$526$	−7391.09	−0.612675
$527$	−18464.5	−1.52623
$528$	−600.023	−0.0494558
$529$	−10499.8	−0.862975
$530$	−2085.42	−0.170915
$531$	7330.36	0.599078
$532$	−360.004	−0.0293386
$533$	−11779.2	−0.957245
$534$	−9064.24	−0.734547
$535$	−4890.77	−0.395227
$536$	−2856.45	−0.230186
$537$	−5258.39	−0.422563
$538$	18474.4	1.48046
$539$	3754.10	0.300001
$540$	−6794.82	−0.541486
$541$	13915.7	1.10588	0.552941	−	0.833220i	$-0.313505\pi$
$541$	13915.7	1.10588	0.552941	+	0.833220i	$0.313505\pi$
$542$	−7165.15	−0.567841
$543$	−1068.42	−0.0844387
$544$	−8015.32	−0.631717
$545$	618.438	0.0486073
$546$	−816.330	−0.0639848
$547$	23962.7	1.87307	0.936535	−	0.350574i	$-0.114013\pi$
$547$	23962.7	1.87307	0.936535	+	0.350574i	$0.114013\pi$
$548$	−46089.4	−3.59278
$549$	−2509.12	−0.195058
$550$	1303.16	0.101031
$551$	−732.307	−0.0566194
$552$	−2323.48	−0.179156
$553$	863.555	0.0664052
$554$	−17598.5	−1.34962
$555$	1811.09	0.138516
$556$	35154.2	2.68142
$557$	14963.0	1.13824	0.569122	−	0.822253i	$-0.307283\pi$
$557$	14963.0	1.13824	0.569122	+	0.822253i	$0.307283\pi$
$558$	27181.8	2.06218
$559$	−29922.9	−2.26405
$560$	192.189	0.0145026
$561$	1550.43	0.116683
$562$	−578.258	−0.0434027
$563$	−20369.1	−1.52479	−0.762395	−	0.647112i	$-0.775977\pi$
$563$	−20369.1	−1.52479	−0.762395	+	0.647112i	$0.775977\pi$
$564$	−14955.7	−1.11658
$565$	−2343.66	−0.174510
$566$	38723.3	2.87573
$567$	603.880	0.0447277
$568$	20855.8	1.54065
$569$	3670.46	0.270428	0.135214	−	0.990816i	$-0.456828\pi$
$569$	3670.46	0.270428	0.135214	+	0.990816i	$0.456828\pi$
$570$	837.365	0.0615322
$571$	−4260.27	−0.312236	−0.156118	−	0.987738i	$-0.549898\pi$
$571$	−4260.27	−0.312236	−0.156118	+	0.987738i	$0.549898\pi$
$572$	−11235.8	−0.821315
$573$	−1140.04	−0.0831168
$574$	1035.43	0.0752928
$575$	1020.78	0.0740338
$576$	17322.1	1.25305
$577$	−4021.76	−0.290170	−0.145085	−	0.989419i	$-0.546346\pi$
$577$	−4021.76	−0.290170	−0.145085	+	0.989419i	$0.546346\pi$
$578$	−3928.74	−0.282723
$579$	5520.62	0.396251
$580$	2785.83	0.199440
$581$	1765.73	0.126084
$582$	12603.4	0.897640
$583$	−968.168	−0.0687778
$584$	−562.165	−0.0398331
$585$	−8316.61	−0.587777
$586$	−1599.15	−0.112731
$587$	−18448.3	−1.29718	−0.648589	−	0.761139i	$-0.724641\pi$
$587$	−18448.3	−1.29718	−0.648589	+	0.761139i	$0.724641\pi$
$588$	−9176.66	−0.643604
$589$	−4629.74	−0.323880
$590$	7378.20	0.514840
$591$	−1444.91	−0.100568
$592$	5710.75	0.396470
$593$	−17516.8	−1.21303	−0.606517	−	0.795071i	$-0.707434\pi$
$593$	−17516.8	−1.21303	−0.606517	+	0.795071i	$0.707434\pi$
$594$	−4900.28	−0.338486
$595$	−496.607	−0.0342166
$596$	5131.82	0.352697
$597$	−4710.34	−0.322917
$598$	−13671.7	−0.934914
$599$	−4435.94	−0.302583	−0.151292	−	0.988489i	$-0.548343\pi$
$599$	−4435.94	−0.302583	−0.151292	+	0.988489i	$0.548343\pi$
$600$	−1422.62	−0.0967968
$601$	−1023.98	−0.0694991	−0.0347496	−	0.999396i	$-0.511063\pi$
$601$	−1023.98	−0.0694991	−0.0347496	+	0.999396i	$0.511063\pi$
$602$	2630.33	0.178080
$603$	−2197.93	−0.148436
$604$	12657.9	0.852723
$605$	605.000	0.0406558
$606$	−5030.28	−0.337197
$607$	17864.4	1.19455	0.597275	−	0.802036i	$-0.296250\pi$
$607$	17864.4	1.19455	0.597275	+	0.802036i	$0.296250\pi$
$608$	−2009.74	−0.134056
$609$	93.9667	0.00625242
$610$	−2525.50	−0.167630
$611$	−39300.9	−2.60220
$612$	25786.3	1.70319
$613$	1926.30	0.126921	0.0634603	−	0.997984i	$-0.479786\pi$
$613$	1926.30	0.126921	0.0634603	+	0.997984i	$0.479786\pi$
$614$	9058.04	0.595363
$615$	−1550.40	−0.101656
$616$	441.085	0.0288504
$617$	−5099.70	−0.332749	−0.166375	−	0.986063i	$-0.553206\pi$
$617$	−5099.70	−0.332749	−0.166375	+	0.986063i	$0.553206\pi$
$618$	−1123.94	−0.0731576
$619$	−13110.2	−0.851283	−0.425642	−	0.904892i	$-0.639952\pi$
$619$	−13110.2	−0.851283	−0.425642	+	0.904892i	$0.639952\pi$
$620$	17612.4	1.14086
$621$	−3838.44	−0.248038
$622$	36586.7	2.35851
$623$	1347.87	0.0866796
$624$	3854.26	0.247266
$625$	625.000	0.0400000
$626$	38783.5	2.47620
$627$	388.752	0.0247612
$628$	−17276.6	−1.09779
$629$	−14756.3	−0.935410
$630$	731.061	0.0462320
$631$	−12754.4	−0.804669	−0.402335	−	0.915493i	$-0.631801\pi$
$631$	−12754.4	−0.804669	−0.402335	+	0.915493i	$0.631801\pi$
$632$	−20156.0	−1.26861
$633$	−7955.85	−0.499552
$634$	40314.3	2.52537
$635$	−8745.13	−0.546520
$636$	2366.62	0.147551
$637$	−24114.6	−1.49993
$638$	2009.08	0.124671
$639$	16047.8	0.993493
$640$	13204.1	0.815531
$641$	12372.3	0.762369	0.381184	−	0.924499i	$-0.375516\pi$
$641$	12372.3	0.762369	0.381184	+	0.924499i	$0.375516\pi$
$642$	8621.81	0.530024
$643$	9869.74	0.605326	0.302663	−	0.953098i	$-0.402124\pi$
$643$	9869.74	0.605326	0.302663	+	0.953098i	$0.402124\pi$
$644$	773.651	0.0473387
$645$	−3938.52	−0.240433
$646$	−6822.65	−0.415532
$647$	−22549.8	−1.37021	−0.685103	−	0.728446i	$-0.740243\pi$
$647$	−22549.8	−1.37021	−0.685103	+	0.728446i	$0.740243\pi$
$648$	−14095.0	−0.854480
$649$	3425.37	0.207177
$650$	−8370.88	−0.505128
$651$	594.070	0.0357656
$652$	110.227	0.00662087
$653$	20055.0	1.20186	0.600929	−	0.799302i	$-0.294797\pi$
$653$	20055.0	1.20186	0.600929	+	0.799302i	$0.294797\pi$
$654$	−1090.23	−0.0651854
$655$	3409.16	0.203369
$656$	−4888.74	−0.290965
$657$	−432.565	−0.0256864
$658$	3454.70	0.204678
$659$	26515.3	1.56736	0.783680	−	0.621164i	$-0.213340\pi$
$659$	26515.3	1.56736	0.783680	+	0.621164i	$0.213340\pi$
$660$	−1478.88	−0.0872204
$661$	−14491.9	−0.852751	−0.426375	−	0.904546i	$-0.640210\pi$
$661$	−14491.9	−0.852751	−0.426375	+	0.904546i	$0.640210\pi$
$662$	5946.99	0.349148
$663$	−9959.24	−0.583386
$664$	−41213.4	−2.40872
$665$	−124.518	−0.00726105
$666$	21723.0	1.26389
$667$	1573.73	0.0913572
$668$	−41619.2	−2.41062
$669$	−6724.74	−0.388630
$670$	−2212.28	−0.127564
$671$	−1172.48	−0.0674560
$672$	257.882	0.0148036
$673$	−11467.1	−0.656798	−0.328399	−	0.944539i	$-0.606509\pi$
$673$	−11467.1	−0.656798	−0.328399	+	0.944539i	$0.606509\pi$
$674$	−48202.8	−2.75475
$675$	−2350.19	−0.134013
$676$	40413.8	2.29938
$677$	4013.69	0.227856	0.113928	−	0.993489i	$-0.463657\pi$
$677$	4013.69	0.227856	0.113928	+	0.993489i	$0.463657\pi$
$678$	4131.57	0.234029
$679$	−1874.15	−0.105925
$680$	11591.1	0.653676
$681$	8815.00	0.496023
$682$	12701.7	0.713156
$683$	−18599.4	−1.04200	−0.521000	−	0.853557i	$-0.674441\pi$
$683$	−18599.4	−1.04200	−0.521000	+	0.853557i	$0.674441\pi$
$684$	6465.60	0.361430
$685$	−15941.4	−0.889181
$686$	4250.19	0.236550
$687$	837.018	0.0464836
$688$	−12419.0	−0.688183
$689$	6219.05	0.343871
$690$	−1799.51	−0.0992841
$691$	7378.21	0.406195	0.203097	−	0.979159i	$-0.434899\pi$
$691$	7378.21	0.406195	0.203097	+	0.979159i	$0.434899\pi$
$692$	−43527.5	−2.39114
$693$	339.399	0.0186042
$694$	26455.3	1.44702
$695$	12159.1	0.663628
$696$	−2193.25	−0.119446
$697$	12632.3	0.686488
$698$	−39701.4	−2.15289
$699$	−7749.85	−0.419351
$700$	473.689	0.0255768
$701$	−15695.9	−0.845686	−0.422843	−	0.906203i	$-0.638968\pi$
$701$	−15695.9	−0.845686	−0.422843	+	0.906203i	$0.638968\pi$
$702$	31477.1	1.69234
$703$	−3699.97	−0.198502
$704$	8094.39	0.433336
$705$	−5172.89	−0.276343
$706$	−25283.6	−1.34782
$707$	748.014	0.0397906
$708$	−8373.10	−0.444464
$709$	19051.2	1.00914	0.504572	−	0.863370i	$-0.331650\pi$
$709$	19051.2	1.00914	0.504572	+	0.863370i	$0.331650\pi$
$710$	16152.6	0.853795
$711$	−15509.3	−0.818064
$712$	−31460.2	−1.65593
$713$	9949.36	0.522589
$714$	875.455	0.0458867
$715$	−3886.23	−0.203268
$716$	−40866.9	−2.13306
$717$	−5983.87	−0.311676
$718$	−29712.0	−1.54435
$719$	25508.5	1.32310	0.661548	−	0.749903i	$-0.269900\pi$
$719$	25508.5	1.32310	0.661548	+	0.749903i	$0.269900\pi$
$720$	−3451.67	−0.178661
$721$	167.132	0.00863290
$722$	−1710.69	−0.0881793
$723$	5602.48	0.288186
$724$	−8303.48	−0.426238
$725$	963.561	0.0493597
$726$	−1066.54	−0.0545220
$727$	29515.1	1.50571	0.752857	−	0.658184i	$-0.228675\pi$
$727$	29515.1	1.50571	0.752857	+	0.658184i	$0.228675\pi$
$728$	−2833.32	−0.144244
$729$	−6124.37	−0.311150
$730$	−435.388	−0.0220746
$731$	32090.1	1.62366
$732$	2866.04	0.144716
$733$	13045.7	0.657370	0.328685	−	0.944440i	$-0.393395\pi$
$733$	13045.7	0.657370	0.328685	+	0.944440i	$0.393395\pi$
$734$	−3609.82	−0.181527
$735$	−3174.02	−0.159286
$736$	4318.96	0.216303
$737$	−1027.06	−0.0513330
$738$	−18596.1	−0.927552
$739$	12758.6	0.635090	0.317545	−	0.948243i	$-0.397141\pi$
$739$	12758.6	0.635090	0.317545	+	0.948243i	$0.397141\pi$
$740$	14075.3	0.699216
$741$	−2497.16	−0.123799
$742$	−546.678	−0.0270474
$743$	−31522.0	−1.55643	−0.778216	−	0.627996i	$-0.783875\pi$
$743$	−31522.0	−1.55643	−0.778216	+	0.627996i	$0.783875\pi$
$744$	−13866.0	−0.683268
$745$	1774.99	0.0872894
$746$	28777.0	1.41233
$747$	−31712.2	−1.55327
$748$	12049.6	0.589005
$749$	−1282.08	−0.0625450
$750$	−1101.80	−0.0536425
$751$	−16377.7	−0.795782	−0.397891	−	0.917433i	$-0.630258\pi$
$751$	−16377.7	−0.795782	−0.397891	+	0.917433i	$0.630258\pi$
$752$	−16311.2	−0.790969
$753$	−2971.31	−0.143799
$754$	−12905.4	−0.623324
$755$	4378.13	0.211041
$756$	−1781.22	−0.0856907
$757$	26305.0	1.26297	0.631487	−	0.775387i	$-0.282445\pi$
$757$	26305.0	1.26297	0.631487	+	0.775387i	$0.282445\pi$
$758$	−36765.8	−1.76173
$759$	−835.431	−0.0399529
$760$	2906.33	0.138715
$761$	−18726.2	−0.892015	−0.446008	−	0.895029i	$-0.647155\pi$
$761$	−18726.2	−0.892015	−0.446008	+	0.895029i	$0.647155\pi$
$762$	15416.6	0.732917
$763$	162.119	0.00769215
$764$	−8860.12	−0.419565
$765$	8918.96	0.421524
$766$	37569.1	1.77210
$767$	−22003.0	−1.03583
$768$	−12327.4	−0.579202
$769$	−13525.2	−0.634242	−0.317121	−	0.948385i	$-0.602716\pi$
$769$	−13525.2	−0.634242	−0.317121	+	0.948385i	$0.602716\pi$
$770$	341.614	0.0159882
$771$	−9938.48	−0.464236
$772$	42904.9	2.00023
$773$	30608.9	1.42423	0.712113	−	0.702065i	$-0.247738\pi$
$773$	30608.9	1.42423	0.712113	+	0.702065i	$0.247738\pi$
$774$	−47240.3	−2.19382
$775$	6091.76	0.282352
$776$	43743.9	2.02360
$777$	474.765	0.0219203
$778$	49493.7	2.28077
$779$	3167.39	0.145678
$780$	9499.64	0.436079
$781$	7498.92	0.343575
$782$	14661.9	0.670473
$783$	−3623.28	−0.165371
$784$	−10008.4	−0.455920
$785$	−5975.61	−0.271693
$786$	−6009.92	−0.272731
$787$	−1141.82	−0.0517170	−0.0258585	−	0.999666i	$-0.508232\pi$
$787$	−1141.82	−0.0517170	−0.0258585	+	0.999666i	$0.508232\pi$
$788$	−11229.5	−0.507657
$789$	2901.15	0.130904
$790$	−15610.5	−0.703034
$791$	−614.373	−0.0276164
$792$	−7921.81	−0.355416
$793$	7531.44	0.337263
$794$	21525.9	0.962123
$795$	818.567	0.0365177
$796$	−36607.6	−1.63005
$797$	−11300.2	−0.502227	−0.251113	−	0.967958i	$-0.580797\pi$
$797$	−11300.2	−0.502227	−0.251113	+	0.967958i	$0.580797\pi$
$798$	219.509	0.00973753
$799$	42147.4	1.86617
$800$	2644.40	0.116867
$801$	−24207.5	−1.06783
$802$	5836.85	0.256991
$803$	−202.132	−0.00888303
$804$	2510.59	0.110127
$805$	267.590	0.0117159
$806$	−81589.5	−3.56559
$807$	−7251.57	−0.316316
$808$	−17459.1	−0.760161
$809$	−32635.3	−1.41829	−0.709144	−	0.705064i	$-0.750918\pi$
$809$	−32635.3	−1.41829	−0.709144	+	0.705064i	$0.750918\pi$
$810$	−10916.4	−0.473533
$811$	24565.6	1.06364	0.531821	−	0.846857i	$-0.321508\pi$
$811$	24565.6	1.06364	0.531821	+	0.846857i	$0.321508\pi$
$812$	730.286	0.0315616
$813$	2812.46	0.121325
$814$	10150.8	0.437084
$815$	38.1251	0.00163861
$816$	−4133.42	−0.177327
$817$	8046.20	0.344554
$818$	19391.3	0.828854
$819$	−2180.14	−0.0930162
$820$	−12049.3	−0.513147
$821$	−3078.29	−0.130856	−0.0654282	−	0.997857i	$-0.520841\pi$
$821$	−3078.29	−0.130856	−0.0654282	+	0.997857i	$0.520841\pi$
$822$	28102.6	1.19245
$823$	8995.91	0.381018	0.190509	−	0.981685i	$-0.438986\pi$
$823$	8995.91	0.381018	0.190509	+	0.981685i	$0.438986\pi$
$824$	−3900.97	−0.164923
$825$	−511.515	−0.0215863
$826$	1934.14	0.0814739
$827$	−2580.29	−0.108495	−0.0542476	−	0.998528i	$-0.517276\pi$
$827$	−2580.29	−0.108495	−0.0542476	+	0.998528i	$0.517276\pi$
$828$	−13894.6	−0.583178
$829$	23520.8	0.985419	0.492710	−	0.870194i	$-0.336007\pi$
$829$	23520.8	0.985419	0.492710	+	0.870194i	$0.336007\pi$
$830$	−31919.2	−1.33486
$831$	6907.77	0.288361
$832$	−51994.5	−2.16657
$833$	25861.1	1.07567
$834$	−21435.0	−0.889968
$835$	−14395.2	−0.596608
$836$	3021.28	0.124992
$837$	−22906.9	−0.945971
$838$	62511.6	2.57688
$839$	−5449.28	−0.224231	−0.112116	−	0.993695i	$-0.535763\pi$
$839$	−5449.28	−0.224231	−0.112116	+	0.993695i	$0.535763\pi$
$840$	−372.929	−0.0153182
$841$	−22903.5	−0.939091
$842$	−38685.9	−1.58338
$843$	226.977	0.00927345
$844$	−61830.9	−2.52169
$845$	13978.3	0.569075
$846$	−62045.7	−2.52149
$847$	158.597	0.00643382
$848$	2581.12	0.104523
$849$	−15199.6	−0.614429
$850$	8977.17	0.362252
$851$	7951.26	0.320289
$852$	−18330.6	−0.737085
$853$	24877.8	0.998592	0.499296	−	0.866432i	$-0.333592\pi$
$853$	24877.8	0.998592	0.499296	+	0.866432i	$0.333592\pi$
$854$	−662.042	−0.0265276
$855$	2236.32	0.0894508
$856$	29924.6	1.19486
$857$	33490.4	1.33490	0.667451	−	0.744654i	$-0.267385\pi$
$857$	33490.4	1.33490	0.667451	+	0.744654i	$0.267385\pi$
$858$	6850.94	0.272596
$859$	7060.32	0.280437	0.140218	−	0.990121i	$-0.455220\pi$
$859$	7060.32	0.280437	0.140218	+	0.990121i	$0.455220\pi$
$860$	−30609.2	−1.21368
$861$	−406.427	−0.0160871
$862$	19309.4	0.762971
$863$	16600.4	0.654789	0.327395	−	0.944888i	$-0.393829\pi$
$863$	16600.4	0.654789	0.327395	+	0.944888i	$0.393829\pi$
$864$	−9943.74	−0.391543
$865$	−15055.3	−0.591786
$866$	50734.8	1.99081
$867$	1542.11	0.0604068
$868$	4616.96	0.180541
$869$	−7247.27	−0.282908
$870$	−1698.64	−0.0661945
$871$	6597.37	0.256651
$872$	−3783.97	−0.146951
$873$	33659.3	1.30492
$874$	3676.30	0.142280
$875$	163.839	0.00633004
$876$	494.098	0.0190571
$877$	−3296.96	−0.126945	−0.0634724	−	0.997984i	$-0.520217\pi$
$877$	−3296.96	−0.126945	−0.0634724	+	0.997984i	$0.520217\pi$
$878$	68481.7	2.63228
$879$	627.696	0.0240861
$880$	−1612.92	−0.0617857
$881$	32212.8	1.23187	0.615936	−	0.787797i	$-0.288778\pi$
$881$	32212.8	1.23187	0.615936	+	0.787797i	$0.288778\pi$
$882$	−38070.5	−1.45340
$883$	15207.3	0.579576	0.289788	−	0.957091i	$-0.406415\pi$
$883$	15207.3	0.579576	0.289788	+	0.957091i	$0.406415\pi$
$884$	−77400.8	−2.94487
$885$	−2896.08	−0.110001
$886$	−42539.5	−1.61303
$887$	35650.5	1.34952	0.674761	−	0.738037i	$-0.264247\pi$
$887$	35650.5	1.34952	0.674761	+	0.738037i	$0.264247\pi$
$888$	−11081.3	−0.418767
$889$	−2292.48	−0.0864872
$890$	−24365.5	−0.917678
$891$	−5067.98	−0.190554
$892$	−52263.0	−1.96177
$893$	10567.9	0.396016
$894$	−3129.08	−0.117061
$895$	−14135.0	−0.527913
$896$	3461.37	0.129058
$897$	5366.41	0.199754
$898$	−68942.3	−2.56195
$899$	9391.66	0.348420
$900$	−8507.36	−0.315088
$901$	−6669.48	−0.246607
$902$	−8689.72	−0.320772
$903$	−1032.46	−0.0380487
$904$	14339.9	0.527585
$905$	−2872.01	−0.105490
$906$	−7718.08	−0.283020
$907$	−27460.5	−1.00530	−0.502651	−	0.864489i	$-0.667642\pi$
$907$	−27460.5	−1.00530	−0.502651	+	0.864489i	$0.667642\pi$
$908$	68508.0	2.50387
$909$	−13434.2	−0.490191
$910$	−2194.37	−0.0799370
$911$	4350.73	0.158229	0.0791143	−	0.996866i	$-0.474791\pi$
$911$	4350.73	0.158229	0.0791143	+	0.996866i	$0.474791\pi$
$912$	−1036.40	−0.0376302
$913$	−14818.7	−0.537160
$914$	55375.5	2.00400
$915$	991.306	0.0358159
$916$	6505.10	0.234645
$917$	893.688	0.0321834
$918$	−33756.9	−1.21366
$919$	−32301.4	−1.15944	−0.579720	−	0.814816i	$-0.696838\pi$
$919$	−32301.4	−1.15944	−0.579720	+	0.814816i	$0.696838\pi$
$920$	−6245.74	−0.223822
$921$	−3555.46	−0.127205
$922$	62379.0	2.22814
$923$	−48169.5	−1.71779
$924$	−387.679	−0.0138027
$925$	4868.38	0.173050
$926$	−19003.2	−0.674387
$927$	−3001.66	−0.106351
$928$	4076.86	0.144213
$929$	41033.7	1.44916	0.724581	−	0.689190i	$-0.242033\pi$
$929$	41033.7	1.44916	0.724581	+	0.689190i	$0.242033\pi$
$930$	−10739.0	−0.378652
$931$	6484.36	0.228267
$932$	−60229.9	−2.11684
$933$	−14361.0	−0.503920
$934$	−89035.6	−3.11920
$935$	4167.70	0.145774
$936$	50885.9	1.77699
$937$	14459.8	0.504144	0.252072	−	0.967709i	$-0.418888\pi$
$937$	14459.8	0.504144	0.252072	+	0.967709i	$0.418888\pi$
$938$	−579.934	−0.0201871
$939$	−15223.2	−0.529065
$940$	−40202.4	−1.39495
$941$	34535.1	1.19640	0.598200	−	0.801347i	$-0.295883\pi$
$941$	34535.1	1.19640	0.598200	+	0.801347i	$0.295883\pi$
$942$	10534.2	0.364357
$943$	−6806.75	−0.235056
$944$	−9131.97	−0.314852
$945$	−616.086	−0.0212077
$946$	−22074.7	−0.758679
$947$	40768.9	1.39895	0.699477	−	0.714655i	$-0.253416\pi$
$947$	40768.9	1.39895	0.699477	+	0.714655i	$0.253416\pi$
$948$	17715.5	0.606932
$949$	1298.40	0.0444128
$950$	2250.91	0.0768729
$951$	−15824.1	−0.539572
$952$	3038.53	0.103445
$953$	16090.7	0.546936	0.273468	−	0.961881i	$-0.411829\pi$
$953$	16090.7	0.546936	0.273468	+	0.961881i	$0.411829\pi$
$954$	9818.23	0.333204
$955$	−3064.53	−0.103839
$956$	−46505.2	−1.57331
$957$	−788.602	−0.0266373
$958$	−59795.2	−2.01659
$959$	−4178.92	−0.140714
$960$	−6843.64	−0.230081
$961$	29584.3	0.993062
$962$	−65204.2	−2.18531
$963$	23025.9	0.770509
$964$	43541.1	1.45474
$965$	14839.9	0.495041
$966$	−471.728	−0.0157118
$967$	−40214.3	−1.33734	−0.668669	−	0.743560i	$-0.733136\pi$
$967$	−40214.3	−1.33734	−0.668669	+	0.743560i	$0.733136\pi$
$968$	−3701.75	−0.122912
$969$	2678.02	0.0887826
$970$	33879.0	1.12143
$971$	−3117.35	−0.103028	−0.0515142	−	0.998672i	$-0.516405\pi$
$971$	−3117.35	−0.103028	−0.0515142	+	0.998672i	$0.516405\pi$
$972$	49080.4	1.61960
$973$	3187.43	0.105020
$974$	21208.4	0.697700
$975$	3285.73	0.107926
$976$	3125.80	0.102515
$977$	−48359.1	−1.58357	−0.791783	−	0.610802i	$-0.790847\pi$
$977$	−48359.1	−1.58357	−0.791783	+	0.610802i	$0.790847\pi$
$978$	−67.2098	−0.00219748
$979$	−11311.8	−0.369283
$980$	−24667.7	−0.804062
$981$	−2911.63	−0.0947616
$982$	25252.0	0.820596
$983$	−38687.5	−1.25528	−0.627640	−	0.778504i	$-0.715979\pi$
$983$	−38687.5	−1.25528	−0.627640	+	0.778504i	$0.715979\pi$
$984$	9486.27	0.307329
$985$	−3884.05	−0.125641
$986$	13840.1	0.447016
$987$	−1356.04	−0.0437316
$988$	−19407.3	−0.624927
$989$	−17291.4	−0.555949
$990$	−6135.33	−0.196963
$991$	−13011.2	−0.417069	−0.208534	−	0.978015i	$-0.566869\pi$
$991$	−13011.2	−0.417069	−0.208534	+	0.978015i	$0.566869\pi$
$992$	25774.5	0.824940
$993$	−2334.31	−0.0745992
$994$	4234.28	0.135114
$995$	−12661.8	−0.403424
$996$	36223.3	1.15239
$997$	42961.7	1.36471	0.682353	−	0.731023i	$-0.260957\pi$
$997$	42961.7	1.36471	0.682353	+	0.731023i	$0.260957\pi$
$998$	−51172.6	−1.62309
$999$	−18306.6	−0.579774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.d.1.3	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.d.1.3	✓	22	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)

\(f(q)\)	\(=\)	\(q-4.73877 q^{2} +1.86006 q^{3} +14.4559 q^{4} +5.00000 q^{5} -8.81437 q^{6} +1.31072 q^{7} -30.5930 q^{8} -23.5402 q^{9} +O(q^{10})\)	\(q-4.73877 q^{2} +1.86006 q^{3} +14.4559 q^{4} +5.00000 q^{5} -8.81437 q^{6} +1.31072 q^{7} -30.5930 q^{8} -23.5402 q^{9} -23.6938 q^{10} -11.0000 q^{11} +26.8888 q^{12} +70.6588 q^{13} -6.21117 q^{14} +9.30028 q^{15} +29.3258 q^{16} -75.7764 q^{17} +111.551 q^{18} -19.0000 q^{19} +72.2795 q^{20} +2.43800 q^{21} +52.1264 q^{22} +40.8312 q^{23} -56.9047 q^{24} +25.0000 q^{25} -334.835 q^{26} -94.0076 q^{27} +18.9476 q^{28} +38.5425 q^{29} -44.0718 q^{30} +243.671 q^{31} +105.776 q^{32} -20.4606 q^{33} +359.087 q^{34} +6.55358 q^{35} -340.295 q^{36} +194.735 q^{37} +90.0365 q^{38} +131.429 q^{39} -152.965 q^{40} -166.705 q^{41} -11.5531 q^{42} -423.484 q^{43} -159.015 q^{44} -117.701 q^{45} -193.489 q^{46} -556.208 q^{47} +54.5476 q^{48} -341.282 q^{49} -118.469 q^{50} -140.948 q^{51} +1021.44 q^{52} +88.0153 q^{53} +445.480 q^{54} -55.0000 q^{55} -40.0987 q^{56} -35.3411 q^{57} -182.644 q^{58} -311.398 q^{59} +134.444 q^{60} +106.589 q^{61} -1154.70 q^{62} -30.8545 q^{63} -735.853 q^{64} +353.294 q^{65} +96.9581 q^{66} +93.3695 q^{67} -1095.42 q^{68} +75.9483 q^{69} -31.0559 q^{70} -681.720 q^{71} +720.164 q^{72} +18.3756 q^{73} -922.804 q^{74} +46.5014 q^{75} -274.662 q^{76} -14.4179 q^{77} -622.813 q^{78} +658.842 q^{79} +146.629 q^{80} +460.726 q^{81} +789.974 q^{82} +1347.15 q^{83} +35.2435 q^{84} -378.882 q^{85} +2006.79 q^{86} +71.6911 q^{87} +336.523 q^{88} +1028.35 q^{89} +557.757 q^{90} +92.6136 q^{91} +590.251 q^{92} +453.241 q^{93} +2635.74 q^{94} -95.0000 q^{95} +196.749 q^{96} -1429.87 q^{97} +1617.26 q^{98} +258.942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * q^99

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