LMFDB - Embedded newform 23.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,4,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	23.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:	$1.35704393013$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.334189.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} - 5x + 4 $ x^4 - 2x^3 - 16x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.362907$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.24143 q^{2} -4.41777 q^{3} +9.98977 q^{4} +7.80430 q^{5} -18.7377 q^{6} -27.0572 q^{7} +8.43948 q^{8} -7.48328 q^{9} +O(q^{10})$	\(q+4.24143 q^{2} -4.41777 q^{3} +9.98977 q^{4} +7.80430 q^{5} -18.7377 q^{6} -27.0572 q^{7} +8.43948 q^{8} -7.48328 q^{9} +33.1014 q^{10} +35.2507 q^{11} -44.1325 q^{12} +58.1812 q^{13} -114.761 q^{14} -34.4776 q^{15} -44.1226 q^{16} +98.3726 q^{17} -31.7398 q^{18} -35.3289 q^{19} +77.9632 q^{20} +119.533 q^{21} +149.514 q^{22} -23.0000 q^{23} -37.2837 q^{24} -64.0928 q^{25} +246.772 q^{26} +152.339 q^{27} -270.295 q^{28} -235.531 q^{29} -146.235 q^{30} -55.0241 q^{31} -254.659 q^{32} -155.730 q^{33} +417.241 q^{34} -211.163 q^{35} -74.7562 q^{36} +401.458 q^{37} -149.845 q^{38} -257.031 q^{39} +65.8643 q^{40} -59.2600 q^{41} +506.990 q^{42} +11.2341 q^{43} +352.147 q^{44} -58.4018 q^{45} -97.5530 q^{46} -103.224 q^{47} +194.924 q^{48} +389.093 q^{49} -271.846 q^{50} -434.588 q^{51} +581.216 q^{52} -351.594 q^{53} +646.137 q^{54} +275.107 q^{55} -228.349 q^{56} +156.075 q^{57} -998.990 q^{58} -547.016 q^{59} -344.424 q^{60} +478.070 q^{61} -233.381 q^{62} +202.477 q^{63} -727.139 q^{64} +454.063 q^{65} -660.517 q^{66} +14.3681 q^{67} +982.720 q^{68} +101.609 q^{69} -895.633 q^{70} +843.177 q^{71} -63.1550 q^{72} +118.935 q^{73} +1702.76 q^{74} +283.148 q^{75} -352.927 q^{76} -953.786 q^{77} -1090.18 q^{78} -388.400 q^{79} -344.347 q^{80} -470.952 q^{81} -251.348 q^{82} -62.9800 q^{83} +1194.10 q^{84} +767.730 q^{85} +47.6488 q^{86} +1040.52 q^{87} +297.498 q^{88} +678.372 q^{89} -247.707 q^{90} -1574.22 q^{91} -229.765 q^{92} +243.084 q^{93} -437.816 q^{94} -275.717 q^{95} +1125.03 q^{96} +421.192 q^{97} +1650.31 q^{98} -263.791 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * 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 $

$2$	4.24143	1.49957	0.749787	−	0.661679i	$-0.230156\pi$
$2$	4.24143	1.49957	0.749787	+	0.661679i	$0.230156\pi$
$3$	−4.41777	−0.850201	−0.425100	−	0.905146i	$-0.639761\pi$
$3$	−4.41777	−0.850201	−0.425100	+	0.905146i	$0.639761\pi$
$4$	9.98977	1.24872
$5$	7.80430	0.698038	0.349019	−	0.937116i	$-0.386515\pi$
$5$	7.80430	0.698038	0.349019	+	0.937116i	$0.386515\pi$
$6$	−18.7377	−1.27494
$7$	−27.0572	−1.46095	−0.730476	−	0.682938i	$-0.760702\pi$
$7$	−27.0572	−1.46095	−0.730476	+	0.682938i	$0.760702\pi$
$8$	8.43948	0.372976
$9$	−7.48328	−0.277158
$10$	33.1014	1.04676
$11$	35.2507	0.966226	0.483113	−	0.875558i	$-0.339506\pi$
$11$	35.2507	0.966226	0.483113	+	0.875558i	$0.339506\pi$
$12$	−44.1325	−1.06166
$13$	58.1812	1.24127	0.620637	−	0.784098i	$-0.286874\pi$
$13$	58.1812	1.24127	0.620637	+	0.784098i	$0.286874\pi$
$14$	−114.761	−2.19081
$15$	−34.4776	−0.593473
$16$	−44.1226	−0.689416
$17$	98.3726	1.40346	0.701731	−	0.712442i	$-0.252411\pi$
$17$	98.3726	1.40346	0.701731	+	0.712442i	$0.252411\pi$
$18$	−31.7398	−0.415619
$19$	−35.3289	−0.426579	−0.213289	−	0.976989i	$-0.568418\pi$
$19$	−35.3289	−0.426579	−0.213289	+	0.976989i	$0.568418\pi$
$20$	77.9632	0.871655
$21$	119.533	1.24210
$22$	149.514	1.44893
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−37.2837	−0.317104
$25$	−64.0928	−0.512743
$26$	246.772	1.86138
$27$	152.339	1.08584
$28$	−270.295	−1.82432
$29$	−235.531	−1.50817	−0.754087	−	0.656774i	$-0.771920\pi$
$29$	−235.531	−1.50817	−0.754087	+	0.656774i	$0.771920\pi$
$30$	−146.235	−0.889956
$31$	−55.0241	−0.318794	−0.159397	−	0.987215i	$-0.550955\pi$
$31$	−55.0241	−0.318794	−0.159397	+	0.987215i	$0.550955\pi$
$32$	−254.659	−1.40681
$33$	−155.730	−0.821486
$34$	417.241	2.10460
$35$	−211.163	−1.01980
$36$	−74.7562	−0.346094
$37$	401.458	1.78377	0.891883	−	0.452266i	$-0.149384\pi$
$37$	401.458	1.78377	0.891883	+	0.452266i	$0.149384\pi$
$38$	−149.845	−0.639687
$39$	−257.031	−1.05533
$40$	65.8643	0.260351
$41$	−59.2600	−0.225728	−0.112864	−	0.993610i	$-0.536002\pi$
$41$	−59.2600	−0.225728	−0.112864	+	0.993610i	$0.536002\pi$
$42$	506.990	1.86263
$43$	11.2341	0.0398416	0.0199208	−	0.999802i	$-0.493659\pi$
$43$	11.2341	0.0398416	0.0199208	+	0.999802i	$0.493659\pi$
$44$	352.147	1.20655
$45$	−58.4018	−0.193467
$46$	−97.5530	−0.312683
$47$	−103.224	−0.320355	−0.160178	−	0.987088i	$-0.551207\pi$
$47$	−103.224	−0.320355	−0.160178	+	0.987088i	$0.551207\pi$
$48$	194.924	0.586142
$49$	389.093	1.13438
$50$	−271.846	−0.768895
$51$	−434.588	−1.19323
$52$	581.216	1.55000
$53$	−351.594	−0.911229	−0.455614	−	0.890177i	$-0.650580\pi$
$53$	−351.594	−0.911229	−0.455614	+	0.890177i	$0.650580\pi$
$54$	646.137	1.62830
$55$	275.107	0.674463
$56$	−228.349	−0.544900
$57$	156.075	0.362678
$58$	−998.990	−2.26162
$59$	−547.016	−1.20704	−0.603520	−	0.797347i	$-0.706236\pi$
$59$	−547.016	−1.20704	−0.603520	+	0.797347i	$0.706236\pi$
$60$	−344.424	−0.741082
$61$	478.070	1.00345	0.501726	−	0.865026i	$-0.332698\pi$
$61$	478.070	1.00345	0.501726	+	0.865026i	$0.332698\pi$
$62$	−233.381	−0.478056
$63$	202.477	0.404915
$64$	−727.139	−1.42019
$65$	454.063	0.866456
$66$	−660.517	−1.23188
$67$	14.3681	0.0261992	0.0130996	−	0.999914i	$-0.495830\pi$
$67$	14.3681	0.0261992	0.0130996	+	0.999914i	$0.495830\pi$
$68$	982.720	1.75253
$69$	101.609	0.177279
$70$	−895.633	−1.52927
$71$	843.177	1.40939	0.704695	−	0.709510i	$-0.251084\pi$
$71$	843.177	1.40939	0.704695	+	0.709510i	$0.251084\pi$
$72$	−63.1550	−0.103373
$73$	118.935	0.190689	0.0953445	−	0.995444i	$-0.469605\pi$
$73$	118.935	0.190689	0.0953445	+	0.995444i	$0.469605\pi$
$74$	1702.76	2.67489
$75$	283.148	0.435934
$76$	−352.927	−0.532678
$77$	−953.786	−1.41161
$78$	−1090.18	−1.58255
$79$	−388.400	−0.553145	−0.276572	−	0.960993i	$-0.589199\pi$
$79$	−388.400	−0.553145	−0.276572	+	0.960993i	$0.589199\pi$
$80$	−344.347	−0.481239
$81$	−470.952	−0.646025
$82$	−251.348	−0.338496
$83$	−62.9800	−0.0832886	−0.0416443	−	0.999132i	$-0.513260\pi$
$83$	−62.9800	−0.0832886	−0.0416443	+	0.999132i	$0.513260\pi$
$84$	1194.10	1.55104
$85$	767.730	0.979671
$86$	47.6488	0.0597454
$87$	1040.52	1.28225
$88$	297.498	0.360379
$89$	678.372	0.807947	0.403974	−	0.914771i	$-0.367629\pi$
$89$	678.372	0.807947	0.403974	+	0.914771i	$0.367629\pi$
$90$	−247.707	−0.290118
$91$	−1574.22	−1.81344
$92$	−229.765	−0.260376
$93$	243.084	0.271039
$94$	−437.816	−0.480397
$95$	−275.717	−0.297768
$96$	1125.03	1.19607
$97$	421.192	0.440882	0.220441	−	0.975400i	$-0.429250\pi$
$97$	421.192	0.440882	0.220441	+	0.975400i	$0.429250\pi$
$98$	1650.31	1.70109
$99$	−263.791	−0.267798
$100$	−640.273	−0.640273
$101$	−1.65795	−0.00163339	−0.000816695	−	1.00000i	$-0.500260\pi$
$101$	−1.65795	−0.00163339	−0.000816695		1.00000i	$0.500260\pi$
$102$	−1843.28	−1.78933
$103$	1057.63	1.01176	0.505879	−	0.862604i	$-0.331168\pi$
$103$	1057.63	1.01176	0.505879	+	0.862604i	$0.331168\pi$
$104$	491.019	0.462965
$105$	932.869	0.867035
$106$	−1491.26	−1.36645
$107$	−155.626	−0.140607	−0.0703036	−	0.997526i	$-0.522397\pi$
$107$	−155.626	−0.140607	−0.0703036	+	0.997526i	$0.522397\pi$
$108$	1521.83	1.35591
$109$	−1815.84	−1.59565	−0.797826	−	0.602888i	$-0.794017\pi$
$109$	−1815.84	−1.59565	−0.797826	+	0.602888i	$0.794017\pi$
$110$	1166.85	1.01141
$111$	−1773.55	−1.51656
$112$	1193.84	1.00720
$113$	485.235	0.403956	0.201978	−	0.979390i	$-0.435263\pi$
$113$	485.235	0.403956	0.201978	+	0.979390i	$0.435263\pi$
$114$	661.982	0.543862
$115$	−179.499	−0.145551
$116$	−2352.90	−1.88329
$117$	−435.386	−0.344029
$118$	−2320.13	−1.81005
$119$	−2661.69	−2.05039
$120$	−290.974	−0.221351
$121$	−88.3874	−0.0664068
$122$	2027.70	1.50475
$123$	261.797	0.191914
$124$	−549.678	−0.398085
$125$	−1475.74	−1.05595
$126$	858.792	0.607200
$127$	2468.34	1.72464	0.862322	−	0.506360i	$-0.169009\pi$
$127$	2468.34	1.72464	0.862322	+	0.506360i	$0.169009\pi$
$128$	−1046.84	−0.722879
$129$	−49.6298	−0.0338734
$130$	1925.88	1.29931
$131$	1071.51	0.714647	0.357323	−	0.933981i	$-0.383689\pi$
$131$	1071.51	0.714647	0.357323	+	0.933981i	$0.383689\pi$
$132$	−1555.70	−1.02581
$133$	955.901	0.623212
$134$	60.9414	0.0392876
$135$	1188.90	0.757959
$136$	830.214	0.523458
$137$	241.497	0.150602	0.0753009	−	0.997161i	$-0.476008\pi$
$137$	241.497	0.150602	0.0753009	+	0.997161i	$0.476008\pi$
$138$	430.967	0.265843
$139$	2659.19	1.62266	0.811329	−	0.584590i	$-0.198745\pi$
$139$	2659.19	1.62266	0.811329	+	0.584590i	$0.198745\pi$
$140$	−2109.47	−1.27345
$141$	456.018	0.272366
$142$	3576.28	2.11349
$143$	2050.93	1.19935
$144$	330.182	0.191078
$145$	−1838.16	−1.05276
$146$	504.456	0.285952
$147$	−1718.93	−0.964453
$148$	4010.48	2.22743
$149$	−182.051	−0.100095	−0.0500477	−	0.998747i	$-0.515937\pi$
$149$	−182.051	−0.100095	−0.0500477	+	0.998747i	$0.515937\pi$
$150$	1200.95	0.653716
$151$	−3377.13	−1.82005	−0.910023	−	0.414557i	$-0.863937\pi$
$151$	−3377.13	−1.82005	−0.910023	+	0.414557i	$0.863937\pi$
$152$	−298.157	−0.159104
$153$	−736.150	−0.388981
$154$	−4045.42	−2.11681
$155$	−429.425	−0.222531
$156$	−2567.68	−1.31781
$157$	−2843.19	−1.44530	−0.722648	−	0.691216i	$-0.757075\pi$
$157$	−2843.19	−1.44530	−0.722648	+	0.691216i	$0.757075\pi$
$158$	−1647.37	−0.829481
$159$	1553.26	0.774728
$160$	−1987.44	−0.982005
$161$	622.316	0.304630
$162$	−1997.51	−0.968762
$163$	2135.85	1.02633	0.513167	−	0.858289i	$-0.328472\pi$
$163$	2135.85	1.02633	0.513167	+	0.858289i	$0.328472\pi$
$164$	−591.994	−0.281872
$165$	−1215.36	−0.573429
$166$	−267.126	−0.124897
$167$	796.566	0.369103	0.184551	−	0.982823i	$-0.440917\pi$
$167$	796.566	0.369103	0.184551	+	0.982823i	$0.440917\pi$
$168$	1008.79	0.463275
$169$	1188.05	0.540759
$170$	3256.28	1.46909
$171$	264.376	0.118230
$172$	112.226	0.0497510
$173$	−4456.05	−1.95831	−0.979154	−	0.203121i	$-0.934892\pi$
$173$	−4456.05	−1.95831	−0.979154	+	0.203121i	$0.934892\pi$
$174$	4413.31	1.92283
$175$	1734.17	0.749093
$176$	−1555.35	−0.666132
$177$	2416.59	1.02623
$178$	2877.27	1.21158
$179$	−693.459	−0.289562	−0.144781	−	0.989464i	$-0.546248\pi$
$179$	−693.459	−0.289562	−0.144781	+	0.989464i	$0.546248\pi$
$180$	−583.420	−0.241587
$181$	−974.743	−0.400288	−0.200144	−	0.979767i	$-0.564141\pi$
$181$	−974.743	−0.400288	−0.200144	+	0.979767i	$0.564141\pi$
$182$	−6676.95	−2.71939
$183$	−2112.01	−0.853136
$184$	−194.108	−0.0777709
$185$	3133.10	1.24514
$186$	1031.03	0.406443
$187$	3467.70	1.35606
$188$	−1031.18	−0.400035
$189$	−4121.88	−1.58636
$190$	−1169.44	−0.446526
$191$	−775.067	−0.293623	−0.146811	−	0.989165i	$-0.546901\pi$
$191$	−775.067	−0.293623	−0.146811	+	0.989165i	$0.546901\pi$
$192$	3212.34	1.20745
$193$	2021.09	0.753789	0.376895	−	0.926256i	$-0.376992\pi$
$193$	2021.09	0.753789	0.376895	+	0.926256i	$0.376992\pi$
$194$	1786.46	0.661136
$195$	−2005.95	−0.736662
$196$	3886.95	1.41653
$197$	566.907	0.205028	0.102514	−	0.994732i	$-0.467311\pi$
$197$	566.907	0.205028	0.102514	+	0.994732i	$0.467311\pi$
$198$	−1118.85	−0.401582
$199$	−1724.07	−0.614150	−0.307075	−	0.951685i	$-0.599350\pi$
$199$	−1724.07	−0.614150	−0.307075	+	0.951685i	$0.599350\pi$
$200$	−540.910	−0.191241
$201$	−63.4751	−0.0222746
$202$	−7.03210	−0.00244939
$203$	6372.82	2.20337
$204$	−4341.43	−1.49001
$205$	−462.483	−0.157567
$206$	4485.86	1.51721
$207$	172.115	0.0577915
$208$	−2567.11	−0.855754
$209$	−1245.37	−0.412172
$210$	3956.70	1.30018
$211$	2757.94	0.899832	0.449916	−	0.893071i	$-0.351454\pi$
$211$	2757.94	0.899832	0.449916	+	0.893071i	$0.351454\pi$
$212$	−3512.34	−1.13787
$213$	−3724.97	−1.19827
$214$	−660.079	−0.210851
$215$	87.6745	0.0278109
$216$	1285.66	0.404993
$217$	1488.80	0.465743
$218$	−7701.77	−2.39280
$219$	−525.428	−0.162124
$220$	2748.26	0.842216
$221$	5723.43	1.74208
$222$	−7522.40	−2.27419
$223$	−4257.81	−1.27858	−0.639292	−	0.768964i	$-0.720773\pi$
$223$	−4257.81	−1.27858	−0.639292	+	0.768964i	$0.720773\pi$
$224$	6890.37	2.05528
$225$	479.624	0.142111
$226$	2058.09	0.605762
$227$	4320.03	1.26313	0.631564	−	0.775324i	$-0.282413\pi$
$227$	4320.03	1.26313	0.631564	+	0.775324i	$0.282413\pi$
$228$	1559.15	0.452883
$229$	1057.27	0.305094	0.152547	−	0.988296i	$-0.451253\pi$
$229$	1057.27	0.305094	0.152547	+	0.988296i	$0.451253\pi$
$230$	−761.333	−0.218264
$231$	4213.61	1.20015
$232$	−1987.76	−0.562513
$233$	−3194.98	−0.898327	−0.449163	−	0.893450i	$-0.648278\pi$
$233$	−3194.98	−0.898327	−0.449163	+	0.893450i	$0.648278\pi$
$234$	−1846.66	−0.515897
$235$	−805.588	−0.223620
$236$	−5464.56	−1.50726
$237$	1715.86	0.470284
$238$	−11289.4	−3.07471
$239$	131.327	0.0355433	0.0177717	−	0.999842i	$-0.494343\pi$
$239$	131.327	0.0355433	0.0177717	+	0.999842i	$0.494343\pi$
$240$	1521.25	0.409150
$241$	2429.45	0.649356	0.324678	−	0.945825i	$-0.394744\pi$
$241$	2429.45	0.649356	0.324678	+	0.945825i	$0.394744\pi$
$242$	−374.890	−0.0995819
$243$	−2032.60	−0.536590
$244$	4775.81	1.25303
$245$	3036.60	0.791842
$246$	1110.40	0.287790
$247$	−2055.47	−0.529501
$248$	−464.375	−0.118903
$249$	278.232	0.0708121
$250$	−6259.25	−1.58348
$251$	689.940	0.173501	0.0867503	−	0.996230i	$-0.472352\pi$
$251$	689.940	0.173501	0.0867503	+	0.996230i	$0.472352\pi$
$252$	2022.70	0.505626
$253$	−810.766	−0.201472
$254$	10469.3	2.58623
$255$	−3391.66	−0.832917
$256$	1377.01	0.336184
$257$	−7466.72	−1.81230	−0.906151	−	0.422955i	$-0.860993\pi$
$257$	−7466.72	−1.81230	−0.906151	+	0.422955i	$0.860993\pi$
$258$	−210.502	−0.0507956
$259$	−10862.3	−2.60600
$260$	4535.99	1.08196
$261$	1762.55	0.418003
$262$	4544.76	1.07167
$263$	−1688.09	−0.395786	−0.197893	−	0.980224i	$-0.563410\pi$
$263$	−1688.09	−0.395786	−0.197893	+	0.980224i	$0.563410\pi$
$264$	−1314.28	−0.306395
$265$	−2743.95	−0.636072
$266$	4054.39	0.934552
$267$	−2996.89	−0.686917
$268$	143.534	0.0327155
$269$	−3665.13	−0.830731	−0.415365	−	0.909655i	$-0.636346\pi$
$269$	−3665.13	−0.830731	−0.415365	+	0.909655i	$0.636346\pi$
$270$	5042.65	1.13661
$271$	289.699	0.0649371	0.0324686	−	0.999473i	$-0.489663\pi$
$271$	289.699	0.0649371	0.0324686	+	0.999473i	$0.489663\pi$
$272$	−4340.46	−0.967570
$273$	6954.55	1.54179
$274$	1024.29	0.225839
$275$	−2259.32	−0.495425
$276$	1015.05	0.221372
$277$	0.383938	8.32802e−5 0	4.16401e−5	−	1.00000i	$-0.499987\pi$
$277$	0.383938	8.32802e−5 0	4.16401e−5		1.00000i	$0.499987\pi$
$278$	11278.8	2.43330
$279$	411.761	0.0883565
$280$	−1782.10	−0.380361
$281$	−3507.39	−0.744602	−0.372301	−	0.928112i	$-0.621431\pi$
$281$	−3507.39	−0.744602	−0.372301	+	0.928112i	$0.621431\pi$
$282$	1934.17	0.408434
$283$	4231.65	0.888854	0.444427	−	0.895815i	$-0.353407\pi$
$283$	4231.65	0.888854	0.444427	+	0.895815i	$0.353407\pi$
$284$	8423.15	1.75994
$285$	1218.06	0.253163
$286$	8698.87	1.79851
$287$	1603.41	0.329778
$288$	1905.69	0.389908
$289$	4764.17	0.969707
$290$	−7796.42	−1.57870
$291$	−1860.73	−0.374839
$292$	1188.13	0.238118
$293$	−1066.58	−0.212663	−0.106331	−	0.994331i	$-0.533910\pi$
$293$	−1066.58	−0.212663	−0.106331	+	0.994331i	$0.533910\pi$
$294$	−7290.71	−1.44627
$295$	−4269.08	−0.842561
$296$	3388.10	0.665302
$297$	5370.07	1.04917
$298$	−772.159	−0.150100
$299$	−1338.17	−0.258823
$300$	2828.58	0.544360
$301$	−303.964	−0.0582067
$302$	−14323.9	−2.72929
$303$	7.32446	0.00138871
$304$	1558.80	0.294090
$305$	3731.00	0.700448
$306$	−3122.33	−0.583306
$307$	−2744.52	−0.510222	−0.255111	−	0.966912i	$-0.582112\pi$
$307$	−2744.52	−0.510222	−0.255111	+	0.966912i	$0.582112\pi$
$308$	−9528.10	−1.76271
$309$	−4672.36	−0.860198
$310$	−1821.38	−0.333701
$311$	2719.23	0.495798	0.247899	−	0.968786i	$-0.420260\pi$
$311$	2719.23	0.495798	0.247899	+	0.968786i	$0.420260\pi$
$312$	−2169.21	−0.393613
$313$	5919.04	1.06890	0.534448	−	0.845202i	$-0.320520\pi$
$313$	5919.04	1.06890	0.534448	+	0.845202i	$0.320520\pi$
$314$	−12059.2	−2.16733
$315$	1580.19	0.282646
$316$	−3880.03	−0.690724
$317$	1327.55	0.235213	0.117606	−	0.993060i	$-0.462478\pi$
$317$	1327.55	0.235213	0.117606	+	0.993060i	$0.462478\pi$
$318$	6588.06	1.16176
$319$	−8302.64	−1.45724
$320$	−5674.82	−0.991350
$321$	687.522	0.119544
$322$	2639.51	0.456815
$323$	−3475.39	−0.598688
$324$	−4704.70	−0.806705
$325$	−3729.00	−0.636454
$326$	9059.06	1.53906
$327$	8021.98	1.35663
$328$	−500.124	−0.0841912
$329$	2792.94	0.468024
$330$	−5154.88	−0.859899
$331$	3236.92	0.537514	0.268757	−	0.963208i	$-0.413387\pi$
$331$	3236.92	0.537514	0.268757	+	0.963208i	$0.413387\pi$
$332$	−629.156	−0.104004
$333$	−3004.22	−0.494386
$334$	3378.58	0.553497
$335$	112.133	0.0182880
$336$	−5274.10	−0.856326
$337$	12263.3	1.98226	0.991132	−	0.132878i	$-0.0424220\pi$
$337$	12263.3	1.98226	0.991132	+	0.132878i	$0.0424220\pi$
$338$	5039.03	0.810908
$339$	−2143.66	−0.343444
$340$	7669.44	1.22334
$341$	−1939.64	−0.308028
$342$	1121.33	0.177294
$343$	−1247.15	−0.196326
$344$	94.8102	0.0148600
$345$	792.986	0.123748
$346$	−18900.0	−2.93663
$347$	−10377.8	−1.60551	−0.802753	−	0.596311i	$-0.796632\pi$
$347$	−10377.8	−1.60551	−0.802753	+	0.596311i	$0.796632\pi$
$348$	10394.6	1.60117
$349$	4837.57	0.741975	0.370987	−	0.928638i	$-0.379019\pi$
$349$	4837.57	0.741975	0.370987	+	0.928638i	$0.379019\pi$
$350$	7355.39	1.12332
$351$	8863.28	1.34783
$352$	−8976.92	−1.35929
$353$	6761.01	1.01941	0.509706	−	0.860349i	$-0.329754\pi$
$353$	6761.01	1.01941	0.509706	+	0.860349i	$0.329754\pi$
$354$	10249.8	1.53890
$355$	6580.41	0.983809
$356$	6776.78	1.00890
$357$	11758.7	1.74325
$358$	−2941.26	−0.434219
$359$	4539.67	0.667394	0.333697	−	0.942680i	$-0.391704\pi$
$359$	4539.67	0.667394	0.333697	+	0.942680i	$0.391704\pi$
$360$	−492.881	−0.0721586
$361$	−5610.87	−0.818030
$362$	−4134.31	−0.600261
$363$	390.476	0.0564591
$364$	−15726.1	−2.26448
$365$	928.206	0.133108
$366$	−8957.93	−1.27934
$367$	−9564.79	−1.36043	−0.680215	−	0.733012i	$-0.738114\pi$
$367$	−9564.79	−1.36043	−0.680215	+	0.733012i	$0.738114\pi$
$368$	1014.82	0.143753
$369$	443.459	0.0625625
$370$	13288.9	1.86717
$371$	9513.15	1.33126
$372$	2428.35	0.338453
$373$	−5028.99	−0.698100	−0.349050	−	0.937104i	$-0.613496\pi$
$373$	−5028.99	−0.698100	−0.349050	+	0.937104i	$0.613496\pi$
$374$	14708.0	2.03352
$375$	6519.48	0.897772
$376$	−871.153	−0.119485
$377$	−13703.5	−1.87206
$378$	−17482.7	−2.37887
$379$	560.190	0.0759235	0.0379618	−	0.999279i	$-0.487913\pi$
$379$	560.190	0.0759235	0.0379618	+	0.999279i	$0.487913\pi$
$380$	−2754.35	−0.371830
$381$	−10904.6	−1.46629
$382$	−3287.40	−0.440309
$383$	5273.52	0.703563	0.351781	−	0.936082i	$-0.385576\pi$
$383$	5273.52	0.703563	0.351781	+	0.936082i	$0.385576\pi$
$384$	4624.70	0.614592
$385$	−7443.64	−0.985358
$386$	8572.33	1.13036
$387$	−84.0681	−0.0110424
$388$	4207.61	0.550539
$389$	−9371.31	−1.22145	−0.610725	−	0.791843i	$-0.709122\pi$
$389$	−9371.31	−1.22145	−0.610725	+	0.791843i	$0.709122\pi$
$390$	−8508.10	−1.10468
$391$	−2262.57	−0.292642
$392$	3283.74	0.423097
$393$	−4733.71	−0.607593
$394$	2404.50	0.307454
$395$	−3031.19	−0.386116
$396$	−2635.21	−0.334405
$397$	6200.86	0.783909	0.391955	−	0.919985i	$-0.371799\pi$
$397$	6200.86	0.783909	0.391955	+	0.919985i	$0.371799\pi$
$398$	−7312.52	−0.920963
$399$	−4222.95	−0.529855
$400$	2827.95	0.353493
$401$	2600.97	0.323906	0.161953	−	0.986798i	$-0.448221\pi$
$401$	2600.97	0.323906	0.161953	+	0.986798i	$0.448221\pi$
$402$	−269.225	−0.0334024
$403$	−3201.37	−0.395711
$404$	−16.5626	−0.00203965
$405$	−3675.45	−0.450950
$406$	27029.9	3.30412
$407$	14151.7	1.72352
$408$	−3667.70	−0.445044
$409$	6260.14	0.756831	0.378416	−	0.925636i	$-0.376469\pi$
$409$	6260.14	0.756831	0.378416	+	0.925636i	$0.376469\pi$
$410$	−1961.59	−0.236283
$411$	−1066.88	−0.128042
$412$	10565.5	1.26340
$413$	14800.7	1.76343
$414$	730.016	0.0866626
$415$	−491.515	−0.0581387
$416$	−14816.4	−1.74623
$417$	−11747.7	−1.37959
$418$	−5282.15	−0.618082
$419$	−6352.30	−0.740645	−0.370322	−	0.928903i	$-0.620753\pi$
$419$	−6352.30	−0.740645	−0.370322	+	0.928903i	$0.620753\pi$
$420$	9319.15	1.08269
$421$	3405.54	0.394242	0.197121	−	0.980379i	$-0.436841\pi$
$421$	3405.54	0.394242	0.197121	+	0.980379i	$0.436841\pi$
$422$	11697.6	1.34936
$423$	772.450	0.0887892
$424$	−2967.27	−0.339866
$425$	−6304.98	−0.719615
$426$	−15799.2	−1.79689
$427$	−12935.2	−1.46600
$428$	−1554.67	−0.175579
$429$	−9060.53	−1.01969
$430$	371.866	0.0417046
$431$	−17077.6	−1.90858	−0.954290	−	0.298881i	$-0.903386\pi$
$431$	−17077.6	−1.90858	−0.954290	+	0.298881i	$0.903386\pi$
$432$	−6721.61	−0.748597
$433$	15884.5	1.76296	0.881480	−	0.472222i	$-0.156548\pi$
$433$	15884.5	1.76296	0.881480	+	0.472222i	$0.156548\pi$
$434$	6314.65	0.698417
$435$	8120.56	0.895060
$436$	−18139.8	−1.99253
$437$	812.564	0.0889479
$438$	−2228.57	−0.243117
$439$	−5977.96	−0.649914	−0.324957	−	0.945729i	$-0.605350\pi$
$439$	−5977.96	−0.649914	−0.324957	+	0.945729i	$0.605350\pi$
$440$	2321.76	0.251558
$441$	−2911.69	−0.314404
$442$	24275.6	2.61238
$443$	−1747.71	−0.187440	−0.0937202	−	0.995599i	$-0.529876\pi$
$443$	−1747.71	−0.187440	−0.0937202	+	0.995599i	$0.529876\pi$
$444$	−17717.4	−1.89376
$445$	5294.22	0.563978
$446$	−18059.2	−1.91733
$447$	804.261	0.0851012
$448$	19674.4	2.07484
$449$	10320.1	1.08472	0.542358	−	0.840148i	$-0.317532\pi$
$449$	10320.1	1.08472	0.542358	+	0.840148i	$0.317532\pi$
$450$	2034.30	0.213106
$451$	−2088.96	−0.218105
$452$	4847.39	0.504429
$453$	14919.4	1.54741
$454$	18323.1	1.89415
$455$	−12285.7	−1.26585
$456$	1317.19	0.135270
$457$	−11934.9	−1.22164	−0.610821	−	0.791769i	$-0.709161\pi$
$457$	−11934.9	−1.22164	−0.610821	+	0.791769i	$0.709161\pi$
$458$	4484.35	0.457510
$459$	14986.0	1.52394
$460$	−1793.15	−0.181753
$461$	−5673.34	−0.573175	−0.286588	−	0.958054i	$-0.592521\pi$
$461$	−5673.34	−0.573175	−0.286588	+	0.958054i	$0.592521\pi$
$462$	17871.8	1.79972
$463$	−7559.91	−0.758831	−0.379416	−	0.925226i	$-0.623875\pi$
$463$	−7559.91	−0.758831	−0.379416	+	0.925226i	$0.623875\pi$
$464$	10392.3	1.03976
$465$	1897.10	0.189196
$466$	−13551.3	−1.34711
$467$	1203.47	0.119250	0.0596252	−	0.998221i	$-0.481009\pi$
$467$	1203.47	0.119250	0.0596252	+	0.998221i	$0.481009\pi$
$468$	−4349.40	−0.429597
$469$	−388.761	−0.0382758
$470$	−3416.85	−0.335335
$471$	12560.6	1.22879
$472$	−4616.53	−0.450197
$473$	396.011	0.0384960
$474$	7277.73	0.705226
$475$	2264.33	0.218725
$476$	−26589.7	−2.56037
$477$	2631.07	0.252555
$478$	557.016	0.0532999
$479$	7928.98	0.756335	0.378167	−	0.925737i	$-0.376554\pi$
$479$	7928.98	0.756335	0.378167	+	0.925737i	$0.376554\pi$
$480$	8780.05	0.834901
$481$	23357.3	2.21414
$482$	10304.4	0.973757
$483$	−2749.25	−0.258996
$484$	−882.970	−0.0829236
$485$	3287.11	0.307753
$486$	−8621.15	−0.804657
$487$	748.445	0.0696412	0.0348206	−	0.999394i	$-0.488914\pi$
$487$	748.445	0.0696412	0.0348206	+	0.999394i	$0.488914\pi$
$488$	4034.66	0.374264
$489$	−9435.70	−0.872591
$490$	12879.5	1.18743
$491$	−2201.18	−0.202318	−0.101159	−	0.994870i	$-0.532255\pi$
$491$	−2201.18	−0.202318	−0.101159	+	0.994870i	$0.532255\pi$
$492$	2615.30	0.239648
$493$	−23169.8	−2.11667
$494$	−8718.16	−0.794026
$495$	−2058.70	−0.186933
$496$	2427.81	0.219782
$497$	−22814.0	−2.05905
$498$	1180.10	0.106188
$499$	−12268.4	−1.10062	−0.550308	−	0.834961i	$-0.685490\pi$
$499$	−12268.4	−1.10062	−0.550308	+	0.834961i	$0.685490\pi$
$500$	−14742.3	−1.31859
$501$	−3519.05	−0.313811
$502$	2926.34	0.260177
$503$	10161.0	0.900711	0.450356	−	0.892849i	$-0.351297\pi$
$503$	10161.0	0.900711	0.450356	+	0.892849i	$0.351297\pi$
$504$	1708.80	0.151024
$505$	−12.9392	−0.00114017
$506$	−3438.81	−0.302122
$507$	−5248.52	−0.459754
$508$	24658.2	2.15360
$509$	18108.1	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$509$	18108.1	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$510$	−14385.5	−1.24902
$511$	−3218.05	−0.278588
$512$	14215.2	1.22701
$513$	−5381.98	−0.463197
$514$	−31669.6	−2.71768
$515$	8254.04	0.706246
$516$	−495.791	−0.0422984
$517$	−3638.70	−0.309536
$518$	−46071.9	−3.90788
$519$	19685.8	1.66495
$520$	3832.06	0.323167
$521$	−10987.7	−0.923950	−0.461975	−	0.886893i	$-0.652859\pi$
$521$	−10987.7	−0.923950	−0.461975	+	0.886893i	$0.652859\pi$
$522$	7475.72	0.626826
$523$	20489.8	1.71311	0.856553	−	0.516059i	$-0.172601\pi$
$523$	20489.8	1.71311	0.856553	+	0.516059i	$0.172601\pi$
$524$	10704.2	0.892394
$525$	−7661.19	−0.636879
$526$	−7159.91	−0.593511
$527$	−5412.87	−0.447416
$528$	6871.21	0.566346
$529$	529.000	0.0434783
$530$	−11638.3	−0.953838
$531$	4093.47	0.334542
$532$	9549.23	0.778217
$533$	−3447.82	−0.280190
$534$	−12711.1	−1.03008
$535$	−1214.55	−0.0981491
$536$	121.259	0.00977166
$537$	3063.54	0.246186
$538$	−15545.4	−1.24574
$539$	13715.8	1.09607
$540$	11876.9	0.946479
$541$	1261.91	0.100284	0.0501420	−	0.998742i	$-0.484033\pi$
$541$	1261.91	0.100284	0.0501420	+	0.998742i	$0.484033\pi$
$542$	1228.74	0.0973780
$543$	4306.20	0.340325
$544$	−25051.5	−1.97440
$545$	−14171.4	−1.11383
$546$	29497.3	2.31203
$547$	−5337.23	−0.417191	−0.208596	−	0.978002i	$-0.566889\pi$
$547$	−5337.23	−0.417191	−0.208596	+	0.978002i	$0.566889\pi$
$548$	2412.50	0.188060
$549$	−3577.53	−0.278115
$550$	−9582.75	−0.742927
$551$	8321.05	0.643355
$552$	857.526	0.0661209
$553$	10509.0	0.808118
$554$	1.62845	0.000124885 0
$555$	−13841.3	−1.05862
$556$	26564.7	2.02625
$557$	−10891.2	−0.828503	−0.414251	−	0.910163i	$-0.635957\pi$
$557$	−10891.2	−0.828503	−0.414251	+	0.910163i	$0.635957\pi$
$558$	1746.46	0.132497
$559$	653.615	0.0494543
$560$	9317.06	0.703067
$561$	−15319.5	−1.15293
$562$	−14876.4	−1.11659
$563$	19620.8	1.46877	0.734385	−	0.678733i	$-0.237471\pi$
$563$	19620.8	1.46877	0.734385	+	0.678733i	$0.237471\pi$
$564$	4555.52	0.340110
$565$	3786.92	0.281977
$566$	17948.3	1.33290
$567$	12742.7	0.943812
$568$	7115.98	0.525669
$569$	25207.8	1.85723	0.928617	−	0.371041i	$-0.120999\pi$
$569$	25207.8	1.85723	0.928617	+	0.371041i	$0.120999\pi$
$570$	5166.31	0.379636
$571$	18889.0	1.38438	0.692190	−	0.721715i	$-0.256646\pi$
$571$	18889.0	1.38438	0.692190	+	0.721715i	$0.256646\pi$
$572$	20488.3	1.49765
$573$	3424.07	0.249638
$574$	6800.77	0.494527
$575$	1474.14	0.106914
$576$	5441.38	0.393619
$577$	−8214.44	−0.592672	−0.296336	−	0.955084i	$-0.595765\pi$
$577$	−8214.44	−0.592672	−0.296336	+	0.955084i	$0.595765\pi$
$578$	20206.9	1.45415
$579$	−8928.72	−0.640872
$580$	−18362.8	−1.31461
$581$	1704.06	0.121681
$582$	−7892.17	−0.562098
$583$	−12393.9	−0.880453
$584$	1003.75	0.0711224
$585$	−3397.88	−0.240146
$586$	−4523.82	−0.318903
$587$	3022.88	0.212551	0.106276	−	0.994337i	$-0.466107\pi$
$587$	3022.88	0.212551	0.106276	+	0.994337i	$0.466107\pi$
$588$	−17171.7	−1.20433
$589$	1943.94	0.135991
$590$	−18107.0	−1.26348
$591$	−2504.47	−0.174315
$592$	−17713.4	−1.22976
$593$	−16386.9	−1.13479	−0.567394	−	0.823446i	$-0.692048\pi$
$593$	−16386.9	−1.13479	−0.567394	+	0.823446i	$0.692048\pi$
$594$	22776.8	1.57331
$595$	−20772.6	−1.43125
$596$	−1818.65	−0.124991
$597$	7616.53	0.522151
$598$	−5675.75	−0.388125
$599$	−4768.20	−0.325247	−0.162624	−	0.986688i	$-0.551996\pi$
$599$	−4768.20	−0.325247	−0.162624	+	0.986688i	$0.551996\pi$
$600$	2389.62	0.162593
$601$	1849.74	0.125545	0.0627725	−	0.998028i	$-0.480006\pi$
$601$	1849.74	0.125545	0.0627725	+	0.998028i	$0.480006\pi$
$602$	−1289.24	−0.0872852
$603$	−107.521	−0.00726132
$604$	−33736.8	−2.27273
$605$	−689.802	−0.0463545
$606$	31.0662	0.00208247
$607$	18350.2	1.22704	0.613519	−	0.789680i	$-0.289754\pi$
$607$	18350.2	1.22704	0.613519	+	0.789680i	$0.289754\pi$
$608$	8996.82	0.600114
$609$	−28153.7	−1.87331
$610$	15824.8	1.05037
$611$	−6005.67	−0.397649
$612$	−7353.96	−0.485729
$613$	13252.1	0.873160	0.436580	−	0.899666i	$-0.356190\pi$
$613$	13252.1	0.873160	0.436580	+	0.899666i	$0.356190\pi$
$614$	−11640.7	−0.765116
$615$	2043.15	0.133964
$616$	−8049.46	−0.526497
$617$	−5203.51	−0.339523	−0.169761	−	0.985485i	$-0.554300\pi$
$617$	−5203.51	−0.339523	−0.169761	+	0.985485i	$0.554300\pi$
$618$	−19817.5	−1.28993
$619$	−3974.35	−0.258066	−0.129033	−	0.991640i	$-0.541187\pi$
$619$	−3974.35	−0.258066	−0.129033	+	0.991640i	$0.541187\pi$
$620$	−4289.86	−0.277879
$621$	−3503.80	−0.226414
$622$	11533.4	0.743486
$623$	−18354.9	−1.18037
$624$	11340.9	0.727563
$625$	−3505.50	−0.224352
$626$	25105.2	1.60289
$627$	5501.75	0.350429
$628$	−28402.8	−1.80477
$629$	39492.5	2.50345
$630$	6702.27	0.423849
$631$	14357.8	0.905827	0.452913	−	0.891555i	$-0.350385\pi$
$631$	14357.8	0.905827	0.452913	+	0.891555i	$0.350385\pi$
$632$	−3277.90	−0.206310
$633$	−12184.0	−0.765038
$634$	5630.70	0.352719
$635$	19263.7	1.20387
$636$	15516.7	0.967419
$637$	22637.9	1.40808
$638$	−35215.1	−2.18523
$639$	−6309.73	−0.390625
$640$	−8169.86	−0.504597
$641$	2587.81	0.159458	0.0797288	−	0.996817i	$-0.474595\pi$
$641$	2587.81	0.159458	0.0797288	+	0.996817i	$0.474595\pi$
$642$	2916.08	0.179265
$643$	−18495.1	−1.13433	−0.567167	−	0.823603i	$-0.691960\pi$
$643$	−18495.1	−1.13433	−0.567167	+	0.823603i	$0.691960\pi$
$644$	6216.79	0.380398
$645$	−387.326	−0.0236449
$646$	−14740.7	−0.897776
$647$	5705.85	0.346708	0.173354	−	0.984860i	$-0.444540\pi$
$647$	5705.85	0.346708	0.173354	+	0.984860i	$0.444540\pi$
$648$	−3974.59	−0.240952
$649$	−19282.7	−1.16627
$650$	−15816.3	−0.954409
$651$	−6577.18	−0.395975
$652$	21336.6	1.28161
$653$	17240.3	1.03318	0.516588	−	0.856234i	$-0.327202\pi$
$653$	17240.3	1.03318	0.516588	+	0.856234i	$0.327202\pi$
$654$	34024.7	2.03436
$655$	8362.43	0.498851
$656$	2614.71	0.155621
$657$	−890.024	−0.0528511
$658$	11846.1	0.701837
$659$	1656.26	0.0979041	0.0489520	−	0.998801i	$-0.484412\pi$
$659$	1656.26	0.0979041	0.0489520	+	0.998801i	$0.484412\pi$
$660$	−12141.2	−0.716053
$661$	−14138.6	−0.831965	−0.415983	−	0.909373i	$-0.636562\pi$
$661$	−14138.6	−0.831965	−0.415983	+	0.909373i	$0.636562\pi$
$662$	13729.2	0.806042
$663$	−25284.8	−1.48112
$664$	−531.519	−0.0310647
$665$	7460.14	0.435025
$666$	−12742.2	−0.741368
$667$	5417.22	0.314476
$668$	7957.51	0.460906
$669$	18810.1	1.08705
$670$	475.606	0.0274242
$671$	16852.3	0.969562
$672$	−30440.1	−1.74740
$673$	19008.2	1.08873	0.544363	−	0.838850i	$-0.316772\pi$
$673$	19008.2	1.08873	0.544363	+	0.838850i	$0.316772\pi$
$674$	52013.9	2.97255
$675$	−9763.86	−0.556757
$676$	11868.3	0.675257
$677$	−2957.92	−0.167920	−0.0839602	−	0.996469i	$-0.526757\pi$
$677$	−2957.92	−0.167920	−0.0839602	+	0.996469i	$0.526757\pi$
$678$	−9092.19	−0.515020
$679$	−11396.3	−0.644108
$680$	6479.24	0.365394
$681$	−19084.9	−1.07391
$682$	−8226.86	−0.461910
$683$	−23345.5	−1.30789	−0.653947	−	0.756540i	$-0.726888\pi$
$683$	−23345.5	−1.30789	−0.653947	+	0.756540i	$0.726888\pi$
$684$	2641.05	0.147636
$685$	1884.71	0.105126
$686$	−5289.70	−0.294405
$687$	−4670.78	−0.259391
$688$	−495.679	−0.0274674
$689$	−20456.1	−1.13108
$690$	3363.40	0.185569
$691$	−11901.8	−0.655234	−0.327617	−	0.944811i	$-0.606246\pi$
$691$	−11901.8	−0.655234	−0.327617	+	0.944811i	$0.606246\pi$
$692$	−44514.9	−2.44538
$693$	7137.45	0.391240
$694$	−44016.9	−2.40758
$695$	20753.1	1.13268
$696$	8781.48	0.478249
$697$	−5829.56	−0.316801
$698$	20518.2	1.11265
$699$	14114.7	0.763758
$700$	17324.0	0.935408
$701$	−12803.0	−0.689817	−0.344909	−	0.938636i	$-0.612090\pi$
$701$	−12803.0	−0.689817	−0.344909	+	0.938636i	$0.612090\pi$
$702$	37593.0	2.02116
$703$	−14183.1	−0.760917
$704$	−25632.2	−1.37223
$705$	3558.91	0.190122
$706$	28676.4	1.52868
$707$	44.8596	0.00238631
$708$	24141.2	1.28147
$709$	8087.63	0.428403	0.214201	−	0.976790i	$-0.431285\pi$
$709$	8087.63	0.428403	0.214201	+	0.976790i	$0.431285\pi$
$710$	27910.4	1.47529
$711$	2906.51	0.153309
$712$	5725.11	0.301345
$713$	1265.56	0.0664732
$714$	49873.9	2.61412
$715$	16006.1	0.837193
$716$	−6927.49	−0.361582
$717$	−580.174	−0.0302190
$718$	19254.7	1.00081
$719$	−34338.9	−1.78112	−0.890560	−	0.454867i	$-0.849687\pi$
$719$	−34338.9	−1.78112	−0.890560	+	0.454867i	$0.849687\pi$
$720$	2576.84	0.133379
$721$	−28616.4	−1.47813
$722$	−23798.1	−1.22670
$723$	−10732.8	−0.552083
$724$	−9737.46	−0.499848
$725$	15095.9	0.773305
$726$	1656.18	0.0846646
$727$	2109.84	0.107633	0.0538167	−	0.998551i	$-0.482861\pi$
$727$	2109.84	0.107633	0.0538167	+	0.998551i	$0.482861\pi$
$728$	−13285.6	−0.676370
$729$	21695.3	1.10223
$730$	3936.93	0.199606
$731$	1105.13	0.0559162
$732$	−21098.4	−1.06533
$733$	15133.9	0.762599	0.381299	−	0.924452i	$-0.375477\pi$
$733$	15133.9	0.762599	0.381299	+	0.924452i	$0.375477\pi$
$734$	−40568.5	−2.04007
$735$	−13415.0	−0.673225
$736$	5857.16	0.293339
$737$	506.486	0.0253143
$738$	1880.90	0.0938171
$739$	−29057.4	−1.44641	−0.723204	−	0.690635i	$-0.757331\pi$
$739$	−29057.4	−1.44641	−0.723204	+	0.690635i	$0.757331\pi$
$740$	31299.0	1.55483
$741$	9080.62	0.450182
$742$	40349.4	1.99633
$743$	−3088.06	−0.152476	−0.0762382	−	0.997090i	$-0.524291\pi$
$743$	−3088.06	−0.152476	−0.0762382	+	0.997090i	$0.524291\pi$
$744$	2051.50	0.101091
$745$	−1420.78	−0.0698704
$746$	−21330.1	−1.04685
$747$	471.297	0.0230841
$748$	34641.6	1.69334
$749$	4210.81	0.205420
$750$	27651.9	1.34627
$751$	−11655.2	−0.566317	−0.283159	−	0.959073i	$-0.591382\pi$
$751$	−11655.2	−0.566317	−0.283159	+	0.959073i	$0.591382\pi$
$752$	4554.50	0.220858
$753$	−3048.00	−0.147510
$754$	−58122.4	−2.80729
$755$	−26356.2	−1.27046
$756$	−41176.6	−1.98092
$757$	−38979.8	−1.87152	−0.935762	−	0.352633i	$-0.885287\pi$
$757$	−38979.8	−1.87152	−0.935762	+	0.352633i	$0.885287\pi$
$758$	2376.01	0.113853
$759$	3581.78	0.171292
$760$	−2326.91	−0.111060
$761$	8875.64	0.422788	0.211394	−	0.977401i	$-0.432200\pi$
$761$	8875.64	0.422788	0.211394	+	0.977401i	$0.432200\pi$
$762$	−46251.0	−2.19882
$763$	49131.6	2.33117
$764$	−7742.75	−0.366653
$765$	−5745.13	−0.271524
$766$	22367.3	1.05504
$767$	−31826.0	−1.49827
$768$	−6083.32	−0.285824
$769$	27567.7	1.29274	0.646370	−	0.763024i	$-0.276286\pi$
$769$	27567.7	1.29274	0.646370	+	0.763024i	$0.276286\pi$
$770$	−31571.7	−1.47762
$771$	32986.3	1.54082
$772$	20190.2	0.941273
$773$	−5450.84	−0.253626	−0.126813	−	0.991927i	$-0.540475\pi$
$773$	−5450.84	−0.253626	−0.126813	+	0.991927i	$0.540475\pi$
$774$	−356.569	−0.0165589
$775$	3526.65	0.163459
$776$	3554.64	0.164438
$777$	47987.4	2.21562
$778$	−39747.8	−1.83165
$779$	2093.59	0.0962909
$780$	−20039.0	−0.919885
$781$	29722.6	1.36179
$782$	−9596.54	−0.438839
$783$	−35880.7	−1.63764
$784$	−17167.8	−0.782062
$785$	−22189.1	−1.00887
$786$	−20077.7	−0.911131
$787$	20985.1	0.950494	0.475247	−	0.879852i	$-0.342359\pi$
$787$	20985.1	0.950494	0.475247	+	0.879852i	$0.342359\pi$
$788$	5663.28	0.256023
$789$	7457.58	0.336498
$790$	−12856.6	−0.579010
$791$	−13129.1	−0.590161
$792$	−2226.26	−0.0998821
$793$	27814.7	1.24556
$794$	26300.5	1.17553
$795$	12122.1	0.540789
$796$	−17223.0	−0.766902
$797$	−30421.8	−1.35206	−0.676032	−	0.736872i	$-0.736302\pi$
$797$	−30421.8	−1.35206	−0.676032	+	0.736872i	$0.736302\pi$
$798$	−17911.4	−0.794557
$799$	−10154.4	−0.449607
$800$	16321.8	0.721330
$801$	−5076.45	−0.223929
$802$	11031.9	0.485721
$803$	4192.55	0.184249
$804$	−634.102	−0.0278147
$805$	4856.74	0.212643
$806$	−13578.4	−0.593398
$807$	16191.7	0.706288
$808$	−13.9923	−0.000609215 0
$809$	−12231.4	−0.531560	−0.265780	−	0.964034i	$-0.585629\pi$
$809$	−12231.4	−0.531560	−0.265780	+	0.964034i	$0.585629\pi$
$810$	−15589.2	−0.676233
$811$	6733.52	0.291548	0.145774	−	0.989318i	$-0.453433\pi$
$811$	6733.52	0.291548	0.145774	+	0.989318i	$0.453433\pi$
$812$	63663.0	2.75140
$813$	−1279.82	−0.0552096
$814$	60023.5	2.58455
$815$	16668.8	0.716421
$816$	19175.2	0.822629
$817$	−396.889	−0.0169956
$818$	26552.0	1.13492
$819$	11780.3	0.502610
$820$	−4620.10	−0.196757
$821$	−659.373	−0.0280296	−0.0140148	−	0.999902i	$-0.504461\pi$
$821$	−659.373	−0.0280296	−0.0140148	+	0.999902i	$0.504461\pi$
$822$	−4525.09	−0.192008
$823$	37882.3	1.60449	0.802245	−	0.596995i	$-0.203639\pi$
$823$	37882.3	1.60449	0.802245	+	0.596995i	$0.203639\pi$
$824$	8925.82	0.377361
$825$	9981.16	0.421211
$826$	62776.3	2.64439
$827$	−36305.0	−1.52654	−0.763271	−	0.646079i	$-0.776408\pi$
$827$	−36305.0	−1.52654	−0.763271	+	0.646079i	$0.776408\pi$
$828$	1719.39	0.0721655
$829$	35900.3	1.50406	0.752032	−	0.659127i	$-0.229074\pi$
$829$	35900.3	1.50406	0.752032	+	0.659127i	$0.229074\pi$
$830$	−2084.73	−0.0871832
$831$	−1.69615	−7.08049e−5 0
$832$	−42305.8	−1.76285
$833$	38276.1	1.59206
$834$	−49827.1	−2.06879
$835$	6216.64	0.257648
$836$	−12440.9	−0.514688
$837$	−8382.34	−0.346160
$838$	−26942.9	−1.11065
$839$	−40349.3	−1.66033	−0.830163	−	0.557521i	$-0.811753\pi$
$839$	−40349.3	−1.66033	−0.830163	+	0.557521i	$0.811753\pi$
$840$	7872.93	0.323383
$841$	31086.0	1.27459
$842$	14444.4	0.591194
$843$	15494.8	0.633062
$844$	27551.2	1.12364
$845$	9271.88	0.377470
$846$	3276.30	0.133146
$847$	2391.52	0.0970172
$848$	15513.3	0.628216
$849$	−18694.5	−0.755704
$850$	−26742.2	−1.07912
$851$	−9233.54	−0.371941
$852$	−37211.6	−1.49630
$853$	19569.0	0.785498	0.392749	−	0.919646i	$-0.371524\pi$
$853$	19569.0	0.785498	0.392749	+	0.919646i	$0.371524\pi$
$854$	−54864.0	−2.19837
$855$	2063.27	0.0825290
$856$	−1313.41	−0.0524431
$857$	24551.4	0.978598	0.489299	−	0.872116i	$-0.337253\pi$
$857$	24551.4	0.978598	0.489299	+	0.872116i	$0.337253\pi$
$858$	−38429.7	−1.52910
$859$	13459.2	0.534599	0.267300	−	0.963613i	$-0.413869\pi$
$859$	13459.2	0.534599	0.267300	+	0.963613i	$0.413869\pi$
$860$	875.849	0.0347281
$861$	−7083.51	−0.280378
$862$	−72433.5	−2.86206
$863$	−42260.5	−1.66694	−0.833468	−	0.552568i	$-0.813648\pi$
$863$	−42260.5	−1.66694	−0.833468	+	0.552568i	$0.813648\pi$
$864$	−38794.6	−1.52757
$865$	−34776.4	−1.36697
$866$	67373.2	2.64369
$867$	−21047.0	−0.824446
$868$	14872.8	0.581584
$869$	−13691.4	−0.534463
$870$	34442.8	1.34221
$871$	835.954	0.0325203
$872$	−15324.8	−0.595140
$873$	−3151.90	−0.122194
$874$	3446.44	0.133384
$875$	39929.4	1.54270
$876$	−5248.91	−0.202448
$877$	37306.1	1.43642	0.718208	−	0.695829i	$-0.244963\pi$
$877$	37306.1	1.43642	0.718208	+	0.695829i	$0.244963\pi$
$878$	−25355.1	−0.974595
$879$	4711.90	0.180806
$880$	−12138.5	−0.464986
$881$	−6563.30	−0.250991	−0.125496	−	0.992094i	$-0.540052\pi$
$881$	−6563.30	−0.250991	−0.125496	+	0.992094i	$0.540052\pi$
$882$	−12349.7	−0.471471
$883$	−27904.7	−1.06350	−0.531749	−	0.846902i	$-0.678465\pi$
$883$	−27904.7	−1.06350	−0.531749	+	0.846902i	$0.678465\pi$
$884$	57175.8	2.17537
$885$	18859.8	0.716346
$886$	−7412.79	−0.281081
$887$	4358.15	0.164974	0.0824872	−	0.996592i	$-0.473714\pi$
$887$	4358.15	0.164974	0.0824872	+	0.996592i	$0.473714\pi$
$888$	−14967.9	−0.565640
$889$	−66786.4	−2.51962
$890$	22455.1	0.845727
$891$	−16601.4	−0.624206
$892$	−42534.6	−1.59660
$893$	3646.77	0.136657
$894$	3411.22	0.127616
$895$	−5411.96	−0.202125
$896$	28324.6	1.05609
$897$	5911.72	0.220052
$898$	43772.2	1.62661
$899$	12959.9	0.480797
$900$	4791.34	0.177457
$901$	−34587.2	−1.27888
$902$	−8860.18	−0.327064
$903$	1342.85	0.0494874
$904$	4095.13	0.150666
$905$	−7607.19	−0.279416
$906$	63279.7	2.32045
$907$	−52106.6	−1.90758	−0.953788	−	0.300481i	$-0.902853\pi$
$907$	−52106.6	−1.90758	−0.953788	+	0.300481i	$0.902853\pi$
$908$	43156.1	1.57730
$909$	12.4069	0.000452708 0
$910$	−52109.0	−1.89824
$911$	34691.3	1.26166	0.630832	−	0.775920i	$-0.282714\pi$
$911$	34691.3	1.26166	0.630832	+	0.775920i	$0.282714\pi$
$912$	−6886.44	−0.250036
$913$	−2220.09	−0.0804757
$914$	−50621.1	−1.83194
$915$	−16482.7	−0.595522
$916$	10561.9	0.380977
$917$	−28992.2	−1.04406
$918$	63562.2	2.28526
$919$	5838.08	0.209555	0.104777	−	0.994496i	$-0.466587\pi$
$919$	5838.08	0.209555	0.104777	+	0.994496i	$0.466587\pi$
$920$	−1514.88	−0.0542870
$921$	12124.7	0.433791
$922$	−24063.1	−0.859519
$923$	49057.0	1.74944
$924$	42093.0	1.49866
$925$	−25730.6	−0.914613
$926$	−32064.9	−1.13792
$927$	−7914.52	−0.280417
$928$	59980.2	2.12171
$929$	−31619.5	−1.11669	−0.558344	−	0.829610i	$-0.688563\pi$
$929$	−31619.5	−1.11669	−0.558344	+	0.829610i	$0.688563\pi$
$930$	8046.44	0.283713
$931$	−13746.2	−0.483903
$932$	−31917.1	−1.12176
$933$	−12012.9	−0.421528
$934$	5104.44	0.178825
$935$	27063.0	0.946583
$936$	−3674.43	−0.128315
$937$	30060.8	1.04807	0.524036	−	0.851696i	$-0.324426\pi$
$937$	30060.8	1.04807	0.524036	+	0.851696i	$0.324426\pi$
$938$	−1648.91	−0.0573973
$939$	−26149.0	−0.908776
$940$	−8047.64	−0.279239
$941$	−45847.9	−1.58831	−0.794155	−	0.607715i	$-0.792086\pi$
$941$	−45847.9	−1.58831	−0.794155	+	0.607715i	$0.792086\pi$
$942$	53274.9	1.84266
$943$	1362.98	0.0470676
$944$	24135.8	0.832154
$945$	−32168.4	−1.10734
$946$	1679.65	0.0577276
$947$	16769.4	0.575431	0.287716	−	0.957716i	$-0.407104\pi$
$947$	16769.4	0.575431	0.287716	+	0.957716i	$0.407104\pi$
$948$	17141.1	0.587254
$949$	6919.78	0.236697
$950$	9604.00	0.327995
$951$	−5864.80	−0.199978
$952$	−22463.3	−0.764747
$953$	−10747.1	−0.365303	−0.182652	−	0.983178i	$-0.558468\pi$
$953$	−10747.1	−0.365303	−0.182652	+	0.983178i	$0.558468\pi$
$954$	11159.5	0.378724
$955$	−6048.86	−0.204960
$956$	1311.93	0.0443837
$957$	36679.2	1.23894
$958$	33630.2	1.13418
$959$	−6534.23	−0.220022
$960$	25070.1	0.842846
$961$	−26763.3	−0.898370
$962$	99068.5	3.32027
$963$	1164.59	0.0389704
$964$	24269.7	0.810864
$965$	15773.2	0.526174
$966$	−11660.8	−0.388384
$967$	−12138.2	−0.403659	−0.201829	−	0.979421i	$-0.564689\pi$
$967$	−12138.2	−0.403659	−0.201829	+	0.979421i	$0.564689\pi$
$968$	−745.944	−0.0247681
$969$	15353.5	0.509005
$970$	13942.1	0.461498
$971$	12928.1	0.427273	0.213636	−	0.976913i	$-0.431469\pi$
$971$	12928.1	0.427273	0.213636	+	0.976913i	$0.431469\pi$
$972$	−20305.2	−0.670052
$973$	−71950.3	−2.37063
$974$	3174.48	0.104432
$975$	16473.9	0.541114
$976$	−21093.7	−0.691797
$977$	14735.6	0.482532	0.241266	−	0.970459i	$-0.422437\pi$
$977$	14735.6	0.482532	0.241266	+	0.970459i	$0.422437\pi$
$978$	−40020.9	−1.30851
$979$	23913.1	0.780660
$980$	30334.9	0.988790
$981$	13588.4	0.442248
$982$	−9336.17	−0.303390
$983$	29251.2	0.949104	0.474552	−	0.880228i	$-0.342610\pi$
$983$	29251.2	0.949104	0.474552	+	0.880228i	$0.342610\pi$
$984$	2209.43	0.0715795
$985$	4424.32	0.143117
$986$	−98273.3	−3.17410
$987$	−12338.6	−0.397914
$988$	−20533.7	−0.661199
$989$	−258.385	−0.00830755
$990$	−8731.86	−0.280320
$991$	47986.6	1.53819	0.769094	−	0.639136i	$-0.220708\pi$
$991$	47986.6	1.53819	0.769094	+	0.639136i	$0.220708\pi$
$992$	14012.4	0.448482
$993$	−14300.0	−0.456995
$994$	−96764.2	−3.08770
$995$	−13455.1	−0.428700
$996$	2779.47	0.0884246
$997$	32463.8	1.03123	0.515617	−	0.856819i	$-0.327563\pi$
$997$	32463.8	1.03123	0.515617	+	0.856819i	$0.327563\pi$
$998$	−52035.5	−1.65046
$999$	61157.9	1.93689

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.4.a.b.1.4	✓	4
3.2	odd	2		207.4.a.e.1.1		4
4.3	odd	2		368.4.a.l.1.4		4
5.2	odd	4		575.4.b.g.24.7		8
5.3	odd	4		575.4.b.g.24.2		8
5.4	even	2		575.4.a.i.1.1		4
7.6	odd	2		1127.4.a.c.1.4		4
8.3	odd	2		1472.4.a.bf.1.1		4
8.5	even	2		1472.4.a.y.1.4		4
23.22	odd	2		529.4.a.g.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.b.1.4	✓	4	1.1	even	1	trivial
207.4.a.e.1.1		4	3.2	odd	2
368.4.a.l.1.4		4	4.3	odd	2
529.4.a.g.1.4		4	23.22	odd	2
575.4.a.i.1.1		4	5.4	even	2
575.4.b.g.24.2		8	5.3	odd	4
575.4.b.g.24.7		8	5.2	odd	4
1127.4.a.c.1.4		4	7.6	odd	2
1472.4.a.y.1.4		4	8.5	even	2
1472.4.a.bf.1.1		4	8.3	odd	2

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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q+4.24143 q^{2} -4.41777 q^{3} +9.98977 q^{4} +7.80430 q^{5} -18.7377 q^{6} -27.0572 q^{7} +8.43948 q^{8} -7.48328 q^{9} +O(q^{10})\)	\(q+4.24143 q^{2} -4.41777 q^{3} +9.98977 q^{4} +7.80430 q^{5} -18.7377 q^{6} -27.0572 q^{7} +8.43948 q^{8} -7.48328 q^{9} +33.1014 q^{10} +35.2507 q^{11} -44.1325 q^{12} +58.1812 q^{13} -114.761 q^{14} -34.4776 q^{15} -44.1226 q^{16} +98.3726 q^{17} -31.7398 q^{18} -35.3289 q^{19} +77.9632 q^{20} +119.533 q^{21} +149.514 q^{22} -23.0000 q^{23} -37.2837 q^{24} -64.0928 q^{25} +246.772 q^{26} +152.339 q^{27} -270.295 q^{28} -235.531 q^{29} -146.235 q^{30} -55.0241 q^{31} -254.659 q^{32} -155.730 q^{33} +417.241 q^{34} -211.163 q^{35} -74.7562 q^{36} +401.458 q^{37} -149.845 q^{38} -257.031 q^{39} +65.8643 q^{40} -59.2600 q^{41} +506.990 q^{42} +11.2341 q^{43} +352.147 q^{44} -58.4018 q^{45} -97.5530 q^{46} -103.224 q^{47} +194.924 q^{48} +389.093 q^{49} -271.846 q^{50} -434.588 q^{51} +581.216 q^{52} -351.594 q^{53} +646.137 q^{54} +275.107 q^{55} -228.349 q^{56} +156.075 q^{57} -998.990 q^{58} -547.016 q^{59} -344.424 q^{60} +478.070 q^{61} -233.381 q^{62} +202.477 q^{63} -727.139 q^{64} +454.063 q^{65} -660.517 q^{66} +14.3681 q^{67} +982.720 q^{68} +101.609 q^{69} -895.633 q^{70} +843.177 q^{71} -63.1550 q^{72} +118.935 q^{73} +1702.76 q^{74} +283.148 q^{75} -352.927 q^{76} -953.786 q^{77} -1090.18 q^{78} -388.400 q^{79} -344.347 q^{80} -470.952 q^{81} -251.348 q^{82} -62.9800 q^{83} +1194.10 q^{84} +767.730 q^{85} +47.6488 q^{86} +1040.52 q^{87} +297.498 q^{88} +678.372 q^{89} -247.707 q^{90} -1574.22 q^{91} -229.765 q^{92} +243.084 q^{93} -437.816 q^{94} -275.717 q^{95} +1125.03 q^{96} +421.192 q^{97} +1650.31 q^{98} -263.791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9	\( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 7 * q^3 + 20 * q^4 + 14 * q^5 - 17 * q^6 + 16 * q^7 - 63 * q^8 - 33 * q^9 - 70 * q^10 + 8 * q^11 - 67 * q^12 + 111 * q^13 - 144 * q^14 + 10 * q^15 + 64 * q^16 + 98 * q^17 + 49 * q^18 + 96 * q^19 + 140 * q^20 + 180 * q^21 + 220 * q^22 - 92 * q^23 - 188 * q^24 + 184 * q^25 - 229 * q^26 - 155 * q^27 + 282 * q^28 + 21 * q^29 - 406 * q^30 - 193 * q^31 - 432 * q^32 - 418 * q^33 + 666 * q^34 - 752 * q^35 - 629 * q^36 + 170 * q^37 + 748 * q^38 - 291 * q^39 - 26 * q^40 - 125 * q^41 + 640 * q^42 + 2 * q^43 + 830 * q^44 + 168 * q^45 - 46 * q^46 - 677 * q^47 + 551 * q^48 + 1220 * q^49 + 414 * q^50 - 340 * q^51 + 2247 * q^52 - 230 * q^53 + 641 * q^54 - 972 * q^55 - 2174 * q^56 + 1322 * q^57 - 1835 * q^58 - 1140 * q^59 - 804 * q^60 + 754 * q^61 + 443 * q^62 - 1092 * q^63 - 805 * q^64 + 1318 * q^65 - 398 * q^66 + 488 * q^67 + 284 * q^68 - 161 * q^69 - 3820 * q^70 - 401 * q^71 + 1503 * q^72 + 1509 * q^73 + 1366 * q^74 + 1401 * q^75 - 3832 * q^76 + 736 * q^77 - 1907 * q^78 - 838 * q^79 + 2846 * q^80 - 932 * q^81 - 949 * q^82 + 142 * q^83 + 2614 * q^84 + 112 * q^85 + 918 * q^86 + 2223 * q^87 - 404 * q^88 + 2342 * q^89 + 1784 * q^90 + 292 * q^91 - 460 * q^92 - 509 * q^93 + 1567 * q^94 - 956 * q^95 + 799 * q^96 + 1062 * q^97 + 2478 * q^98 - 1498 * q^99

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