LMFDB - Embedded newform 242.5.b.e.241.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,5,Mod(241,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("242.241");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	242.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$25.0155310663$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} + \cdots + 1499670491 $ x^16 - 4x^15 + 138x^14 - 428x^13 + 7783x^12 - 18620x^11 + 235604x^10 - 425164x^9 + 4199998x^8 - 5446172x^7 + 45313276x^6 - 38408204x^5 + 290282498x^4 - 132888724x^3 + 1016893202x^2 - 153602264*x + 1499670491
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16}\cdot 11^{10} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.16
Root			$0.809017 - 3.04259i$ of defining polynomial
Character	$\chi$	$=$	242.241
Dual form			242.5.b.e.241.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +16.1949 q^{3} -8.00000 q^{4} +27.1348 q^{5} +45.8061i q^{6} +45.4207i q^{7} -22.6274i q^{8} +181.275 q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +16.1949 q^{3} -8.00000 q^{4} +27.1348 q^{5} +45.8061i q^{6} +45.4207i q^{7} -22.6274i q^{8} +181.275 q^{9} +76.7489i q^{10} -129.559 q^{12} +156.647i q^{13} -128.469 q^{14} +439.446 q^{15} +64.0000 q^{16} -326.026i q^{17} +512.722i q^{18} +106.816i q^{19} -217.079 q^{20} +735.583i q^{21} -467.991 q^{23} -366.449i q^{24} +111.299 q^{25} -443.066 q^{26} +1623.93 q^{27} -363.366i q^{28} +177.059i q^{29} +1242.94i q^{30} +272.947 q^{31} +181.019i q^{32} +922.141 q^{34} +1232.48i q^{35} -1450.20 q^{36} +2139.43 q^{37} -302.121 q^{38} +2536.89i q^{39} -613.991i q^{40} -1299.98i q^{41} -2080.54 q^{42} -2337.65i q^{43} +4918.85 q^{45} -1323.68i q^{46} -2630.43 q^{47} +1036.47 q^{48} +337.959 q^{49} +314.802i q^{50} -5279.96i q^{51} -1253.18i q^{52} +1630.63 q^{53} +4593.18i q^{54} +1027.75 q^{56} +1729.87i q^{57} -500.800 q^{58} -4401.10 q^{59} -3515.57 q^{60} +4750.63i q^{61} +772.012i q^{62} +8233.62i q^{63} -512.000 q^{64} +4250.60i q^{65} -1184.26 q^{67} +2608.21i q^{68} -7579.06 q^{69} -3485.99 q^{70} +4314.87 q^{71} -4101.77i q^{72} +1891.62i q^{73} +6051.21i q^{74} +1802.48 q^{75} -854.528i q^{76} -7175.40 q^{78} -1270.31i q^{79} +1736.63 q^{80} +11616.2 q^{81} +3676.91 q^{82} -2317.11i q^{83} -5884.67i q^{84} -8846.67i q^{85} +6611.88 q^{86} +2867.46i q^{87} -13826.8 q^{89} +13912.6i q^{90} -7115.04 q^{91} +3743.93 q^{92} +4420.35 q^{93} -7439.99i q^{94} +2898.43i q^{95} +2931.59i q^{96} +1902.97 q^{97} +955.893i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9}+O(q^{10}) $ 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9	$ 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9} - 384 q^{12} + 384 q^{14} + 1352 q^{15} + 1024 q^{16} - 480 q^{20} - 2424 q^{23} + 5692 q^{25} + 4344 q^{27} - 428 q^{31} - 3904 q^{34} + 320 q^{36} - 228 q^{37} + 1440 q^{38} - 5568 q^{42} + 12152 q^{45} - 9228 q^{47} + 3072 q^{48} - 11956 q^{49} + 12468 q^{53} - 3072 q^{56} + 2944 q^{58} - 22320 q^{59} - 10816 q^{60} - 8192 q^{64} + 21524 q^{67} - 24704 q^{69} + 16960 q^{70} + 8868 q^{71} + 19716 q^{75} - 39424 q^{78} + 3840 q^{80} + 1136 q^{81} - 20992 q^{82} - 35616 q^{86} + 46596 q^{89} - 35600 q^{91} + 19392 q^{92} + 14832 q^{93} + 69448 q^{97}+O(q^{100}) $ 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9 - 384 * q^12 + 384 * q^14 + 1352 * q^15 + 1024 * q^16 - 480 * q^20 - 2424 * q^23 + 5692 * q^25 + 4344 * q^27 - 428 * q^31 - 3904 * q^34 + 320 * q^36 - 228 * q^37 + 1440 * q^38 - 5568 * q^42 + 12152 * q^45 - 9228 * q^47 + 3072 * q^48 - 11956 * q^49 + 12468 * q^53 - 3072 * q^56 + 2944 * q^58 - 22320 * q^59 - 10816 * q^60 - 8192 * q^64 + 21524 * q^67 - 24704 * q^69 + 16960 * q^70 + 8868 * q^71 + 19716 * q^75 - 39424 * q^78 + 3840 * q^80 + 1136 * q^81 - 20992 * q^82 - 35616 * q^86 + 46596 * q^89 - 35600 * q^91 + 19392 * q^92 + 14832 * q^93 + 69448 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/242\mathbb{Z}\right)^\times$.

$n$	$123$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	16.1949	1.79943	0.899716	−	0.436475i	$-0.143773\pi$
$3$	16.1949	1.79943	0.899716	+	0.436475i	$0.143773\pi$
$4$	−8.00000	−0.500000
$5$	27.1348	1.08539	0.542697	−	0.839929i	$-0.317403\pi$
$5$	27.1348	1.08539	0.542697	+	0.839929i	$0.317403\pi$
$6$	45.8061i	1.27239i
$7$	45.4207i	0.926953i	0.886109	+	0.463477i	$0.153398\pi$
$7$	45.4207i	0.926953i	−0.886109	+	0.463477i	$0.846602\pi$
$8$	− 22.6274i	− 0.353553i
$9$	181.275	2.23796
$10$	76.7489i	0.767489i
$11$	0	0
$11$	0	0
$12$	−129.559	−0.899716
$13$	156.647i	0.926908i	0.886121	+	0.463454i	$0.153390\pi$
$13$	156.647i	0.926908i	−0.886121	+	0.463454i	$0.846610\pi$
$14$	−128.469	−0.655455
$15$	439.446	1.95309
$16$	64.0000	0.250000
$17$	− 326.026i	− 1.12812i	−0.825734	−	0.564059i	$-0.809239\pi$
$17$	− 326.026i	− 1.12812i	0.825734	−	0.564059i	$-0.190761\pi$
$18$	512.722i	1.58247i
$19$	106.816i	0.295889i	0.988996	+	0.147945i	$0.0472657\pi$
$19$	106.816i	0.295889i	−0.988996	+	0.147945i	$0.952734\pi$
$20$	−217.079	−0.542697
$21$	735.583i	1.66799i
$22$	0	0
$23$	−467.991	−0.884671	−0.442335	−	0.896850i	$-0.645850\pi$
$23$	−467.991	−0.884671	−0.442335	+	0.896850i	$0.645850\pi$
$24$	− 366.449i	− 0.636195i
$25$	111.299	0.178079
$26$	−443.066	−0.655423
$27$	1623.93	2.22762
$28$	− 363.366i	− 0.463477i
$29$	177.059i	0.210534i	0.994444	+	0.105267i	$0.0335698\pi$
$29$	177.059i	0.210534i	−0.994444	+	0.105267i	$0.966430\pi$
$30$	1242.94i	1.38104i
$31$	272.947	0.284024	0.142012	−	0.989865i	$-0.454643\pi$
$31$	272.947	0.284024	0.142012	+	0.989865i	$0.454643\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	922.141	0.797700
$35$	1232.48i	1.00611i
$36$	−1450.20	−1.11898
$37$	2139.43	1.56277	0.781383	−	0.624052i	$-0.214515\pi$
$37$	2139.43	1.56277	0.781383	+	0.624052i	$0.214515\pi$
$38$	−302.121	−0.209225
$39$	2536.89i	1.66791i
$40$	− 613.991i	− 0.383744i
$41$	− 1299.98i	− 0.773339i	−0.922218	−	0.386670i	$-0.873625\pi$
$41$	− 1299.98i	− 0.773339i	0.922218	−	0.386670i	$-0.126375\pi$
$42$	−2080.54	−1.17945
$43$	− 2337.65i	− 1.26428i	−0.774855	−	0.632139i	$-0.782177\pi$
$43$	− 2337.65i	− 1.26428i	0.774855	−	0.632139i	$-0.217823\pi$
$44$	0	0
$45$	4918.85	2.42906
$46$	− 1323.68i	− 0.625557i
$47$	−2630.43	−1.19078	−0.595390	−	0.803437i	$-0.703002\pi$
$47$	−2630.43	−1.19078	−0.595390	+	0.803437i	$0.703002\pi$
$48$	1036.47	0.449858
$49$	337.959	0.140758
$50$	314.802i	0.125921i
$51$	− 5279.96i	− 2.02997i
$52$	− 1253.18i	− 0.463454i
$53$	1630.63	0.580502	0.290251	−	0.956951i	$-0.406261\pi$
$53$	1630.63	0.580502	0.290251	+	0.956951i	$0.406261\pi$
$54$	4593.18i	1.57517i
$55$	0	0
$56$	1027.75	0.327727
$57$	1729.87i	0.532432i
$58$	−500.800	−0.148870
$59$	−4401.10	−1.26432	−0.632160	−	0.774838i	$-0.717831\pi$
$59$	−4401.10	−1.26432	−0.632160	+	0.774838i	$0.717831\pi$
$60$	−3515.57	−0.976546
$61$	4750.63i	1.27671i	0.769743	+	0.638354i	$0.220385\pi$
$61$	4750.63i	1.27671i	−0.769743	+	0.638354i	$0.779615\pi$
$62$	772.012i	0.200836i
$63$	8233.62i	2.07448i
$64$	−512.000	−0.125000
$65$	4250.60i	1.00606i
$66$	0	0
$67$	−1184.26	−0.263813	−0.131907	−	0.991262i	$-0.542110\pi$
$67$	−1184.26	−0.263813	−0.131907	+	0.991262i	$0.542110\pi$
$68$	2608.21i	0.564059i
$69$	−7579.06	−1.59191
$70$	−3485.99	−0.711426
$71$	4314.87	0.855956	0.427978	−	0.903789i	$-0.359226\pi$
$71$	4314.87	0.855956	0.427978	+	0.903789i	$0.359226\pi$
$72$	− 4101.77i	− 0.791237i
$73$	1891.62i	0.354967i	0.984124	+	0.177484i	$0.0567956\pi$
$73$	1891.62i	0.354967i	−0.984124	+	0.177484i	$0.943204\pi$
$74$	6051.21i	1.10504i
$75$	1802.48	0.320441
$76$	− 854.528i	− 0.147945i
$77$	0	0
$78$	−7175.40	−1.17939
$79$	− 1270.31i	− 0.203543i	−0.994808	−	0.101771i	$-0.967549\pi$
$79$	− 1270.31i	− 0.203543i	0.994808	−	0.101771i	$-0.0324510\pi$
$80$	1736.63	0.271348
$81$	11616.2	1.77049
$82$	3676.91	0.546833
$83$	− 2317.11i	− 0.336349i	−0.985757	−	0.168175i	$-0.946213\pi$
$83$	− 2317.11i	− 0.336349i	0.985757	−	0.168175i	$-0.0537872\pi$
$84$	− 5884.67i	− 0.833995i
$85$	− 8846.67i	− 1.22445i
$86$	6611.88	0.893980
$87$	2867.46i	0.378843i
$88$	0	0
$89$	−13826.8	−1.74558	−0.872792	−	0.488092i	$-0.837693\pi$
$89$	−13826.8	−1.74558	−0.872792	+	0.488092i	$0.837693\pi$
$90$	13912.6i	1.71761i
$91$	−7115.04	−0.859200
$92$	3743.93	0.442335
$93$	4420.35	0.511083
$94$	− 7439.99i	− 0.842009i
$95$	2898.43i	0.321156i
$96$	2931.59i	0.318098i
$97$	1902.97	0.202250	0.101125	−	0.994874i	$-0.467756\pi$
$97$	1902.97	0.202250	0.101125	+	0.994874i	$0.467756\pi$
$98$	955.893i	0.0995307i
$99$	0	0
$100$	−890.394	−0.0890394
$101$	− 11991.7i	− 1.17554i	−0.809029	−	0.587769i	$-0.800007\pi$
$101$	− 11991.7i	− 1.17554i	0.809029	−	0.587769i	$-0.199993\pi$
$102$	14934.0	1.43541
$103$	−2440.14	−0.230007	−0.115004	−	0.993365i	$-0.536688\pi$
$103$	−2440.14	−0.230007	−0.115004	+	0.993365i	$0.536688\pi$
$104$	3544.53	0.327711
$105$	19959.9i	1.81042i
$106$	4612.12i	0.410477i
$107$	13103.4i	1.14450i	0.820080	+	0.572249i	$0.193929\pi$
$107$	13103.4i	1.14450i	−0.820080	+	0.572249i	$0.806071\pi$
$108$	−12991.5	−1.11381
$109$	− 934.389i	− 0.0786457i	−0.999227	−	0.0393228i	$-0.987480\pi$
$109$	− 934.389i	− 0.0786457i	0.999227	−	0.0393228i	$-0.0125201\pi$
$110$	0	0
$111$	34647.8	2.81209
$112$	2906.93i	0.231738i
$113$	6923.26	0.542193	0.271096	−	0.962552i	$-0.412614\pi$
$113$	6923.26	0.542193	0.271096	+	0.962552i	$0.412614\pi$
$114$	−4892.82	−0.376487
$115$	−12698.9	−0.960216
$116$	− 1416.48i	− 0.105267i
$117$	28396.2i	2.07438i
$118$	− 12448.2i	− 0.894009i
$119$	14808.3	1.04571
$120$	− 9943.52i	− 0.690522i
$121$	0	0
$122$	−13436.8	−0.902770
$123$	− 21053.1i	− 1.39157i
$124$	−2183.58	−0.142012
$125$	−13939.2	−0.892108
$126$	−23288.2	−1.46688
$127$	− 13850.4i	− 0.858728i	−0.903131	−	0.429364i	$-0.858738\pi$
$127$	− 13850.4i	− 0.858728i	0.903131	−	0.429364i	$-0.141262\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	− 37858.0i	− 2.27498i
$130$	−12022.5	−0.711392
$131$	− 23230.6i	− 1.35368i	−0.736128	−	0.676842i	$-0.763348\pi$
$131$	− 23230.6i	− 1.35368i	0.736128	−	0.676842i	$-0.236652\pi$
$132$	0	0
$133$	−4851.66	−0.274275
$134$	− 3349.58i	− 0.186544i
$135$	44065.2	2.41784
$136$	−7377.13	−0.398850
$137$	925.378	0.0493035	0.0246518	−	0.999696i	$-0.492152\pi$
$137$	925.378	0.0493035	0.0246518	+	0.999696i	$0.492152\pi$
$138$	− 21436.8i	− 1.12565i
$139$	− 35049.7i	− 1.81407i	−0.421051	−	0.907037i	$-0.638339\pi$
$139$	− 35049.7i	− 1.81407i	0.421051	−	0.907037i	$-0.361661\pi$
$140$	− 9859.87i	− 0.503054i
$141$	−42599.6	−2.14273
$142$	12204.3i	0.605252i
$143$	0	0
$144$	11601.6	0.559489
$145$	4804.48i	0.228513i
$146$	−5350.31	−0.251000
$147$	5473.21	0.253284
$148$	−17115.4	−0.781383
$149$	− 10504.3i	− 0.473147i	−0.971614	−	0.236574i	$-0.923976\pi$
$149$	− 10504.3i	− 0.473147i	0.971614	−	0.236574i	$-0.0760244\pi$
$150$	5098.18i	0.226586i
$151$	− 22237.5i	− 0.975286i	−0.873043	−	0.487643i	$-0.837857\pi$
$151$	− 22237.5i	− 0.975286i	0.873043	−	0.487643i	$-0.162143\pi$
$152$	2416.97	0.104613
$153$	− 59100.2i	− 2.52468i
$154$	0	0
$155$	7406.38	0.308278
$156$	− 20295.1i	− 0.833954i
$157$	−7325.16	−0.297179	−0.148589	−	0.988899i	$-0.547473\pi$
$157$	−7325.16	−0.297179	−0.148589	+	0.988899i	$0.547473\pi$
$158$	3592.98	0.143926
$159$	26407.9	1.04457
$160$	4911.93i	0.191872i
$161$	− 21256.5i	− 0.820048i
$162$	32855.6i	1.25193i
$163$	42452.8	1.59783	0.798917	−	0.601442i	$-0.205407\pi$
$163$	42452.8	1.59783	0.798917	+	0.601442i	$0.205407\pi$
$164$	10399.9i	0.386670i
$165$	0	0
$166$	6553.77	0.237835
$167$	− 14289.9i	− 0.512384i	−0.966626	−	0.256192i	$-0.917532\pi$
$167$	− 14289.9i	− 0.512384i	0.966626	−	0.256192i	$-0.0824679\pi$
$168$	16644.4	0.589723
$169$	4022.58	0.140842
$170$	25022.1	0.865818
$171$	19363.0i	0.662187i
$172$	18701.2i	0.632139i
$173$	− 28302.6i	− 0.945660i	−0.881154	−	0.472830i	$-0.843233\pi$
$173$	− 28302.6i	− 0.945660i	0.881154	−	0.472830i	$-0.156767\pi$
$174$	−8110.40	−0.267882
$175$	5055.29i	0.165071i
$176$	0	0
$177$	−71275.3	−2.27506
$178$	− 39108.0i	− 1.23431i
$179$	12260.6	0.382655	0.191328	−	0.981526i	$-0.438721\pi$
$179$	12260.6	0.382655	0.191328	+	0.981526i	$0.438721\pi$
$180$	−39350.8	−1.21453
$181$	−26057.2	−0.795372	−0.397686	−	0.917522i	$-0.630187\pi$
$181$	−26057.2	−0.795372	−0.397686	+	0.917522i	$0.630187\pi$
$182$	− 20124.4i	− 0.607546i
$183$	76936.0i	2.29735i
$184$	10589.4i	0.312778i
$185$	58053.0	1.69622
$186$	12502.6i	0.361390i
$187$	0	0
$188$	21043.5	0.595390
$189$	73760.3i	2.06490i
$190$	−8198.01	−0.227092
$191$	−11897.3	−0.326123	−0.163061	−	0.986616i	$-0.552137\pi$
$191$	−11897.3	−0.326123	−0.163061	+	0.986616i	$0.552137\pi$
$192$	−8291.78	−0.224929
$193$	18611.2i	0.499643i	0.968292	+	0.249821i	$0.0803719\pi$
$193$	18611.2i	0.499643i	−0.968292	+	0.249821i	$0.919628\pi$
$194$	5382.40i	0.143012i
$195$	68838.0i	1.81034i
$196$	−2703.67	−0.0703788
$197$	51063.1i	1.31575i	0.753125	+	0.657877i	$0.228545\pi$
$197$	51063.1i	1.31575i	−0.753125	+	0.657877i	$0.771455\pi$
$198$	0	0
$199$	−32814.0	−0.828616	−0.414308	−	0.910137i	$-0.635976\pi$
$199$	−32814.0	−0.828616	−0.414308	+	0.910137i	$0.635976\pi$
$200$	− 2518.41i	− 0.0629603i
$201$	−19178.9	−0.474714
$202$	33917.5	0.831230
$203$	−8042.17	−0.195156
$204$	42239.7i	1.01499i
$205$	− 35274.8i	− 0.839377i
$206$	− 6901.77i	− 0.162640i
$207$	−84834.8	−1.97985
$208$	10025.4i	0.231727i
$209$	0	0
$210$	−56455.2	−1.28016
$211$	65113.4i	1.46253i	0.682092	+	0.731266i	$0.261070\pi$
$211$	65113.4i	1.46253i	−0.682092	+	0.731266i	$0.738930\pi$
$212$	−13045.0	−0.290251
$213$	69878.9	1.54023
$214$	−37061.9	−0.809283
$215$	− 63431.8i	− 1.37224i
$216$	− 36745.4i	− 0.787583i
$217$	12397.5i	0.263277i
$218$	2642.85	0.0556109
$219$	30634.6i	0.638739i
$220$	0	0
$221$	51071.2	1.04566
$222$	97998.7i	1.98845i
$223$	18846.8	0.378990	0.189495	−	0.981882i	$-0.439315\pi$
$223$	18846.8	0.378990	0.189495	+	0.981882i	$0.439315\pi$
$224$	−8222.03	−0.163864
$225$	20175.7	0.398532
$226$	19581.9i	0.383388i
$227$	− 60295.2i	− 1.17012i	−0.810990	−	0.585061i	$-0.801071\pi$
$227$	− 60295.2i	− 1.17012i	0.810990	−	0.585061i	$-0.198929\pi$
$228$	− 13839.0i	− 0.266216i
$229$	82692.6	1.57687	0.788435	−	0.615119i	$-0.210892\pi$
$229$	82692.6	1.57687	0.788435	+	0.615119i	$0.210892\pi$
$230$	− 35917.8i	− 0.678975i
$231$	0	0
$232$	4006.40	0.0744352
$233$	− 40580.0i	− 0.747481i	−0.927533	−	0.373741i	$-0.878075\pi$
$233$	− 40580.0i	− 0.747481i	0.927533	−	0.373741i	$-0.121925\pi$
$234$	−80316.5	−1.46681
$235$	−71376.4	−1.29246
$236$	35208.8	0.632160
$237$	− 20572.5i	− 0.366261i
$238$	41884.3i	0.739431i
$239$	77291.9i	1.35313i	0.736385	+	0.676563i	$0.236531\pi$
$239$	77291.9i	1.35313i	−0.736385	+	0.676563i	$0.763469\pi$
$240$	28124.5	0.488273
$241$	81225.5i	1.39849i	0.714884	+	0.699243i	$0.246479\pi$
$241$	81225.5i	1.39849i	−0.714884	+	0.699243i	$0.753521\pi$
$242$	0	0
$243$	56584.6	0.958265
$244$	− 38005.1i	− 0.638354i
$245$	9170.47	0.152777
$246$	59547.1	0.983990
$247$	−16732.4	−0.274262
$248$	− 6176.09i	− 0.100418i
$249$	− 37525.3i	− 0.605237i
$250$	− 39426.0i	− 0.630816i
$251$	−69668.6	−1.10583	−0.552916	−	0.833237i	$-0.686485\pi$
$251$	−69668.6	−1.10583	−0.552916	+	0.833237i	$0.686485\pi$
$252$	− 65868.9i	− 1.03724i
$253$	0	0
$254$	39174.9	0.607213
$255$	− 143271.i	− 2.20332i
$256$	4096.00	0.0625000
$257$	−68603.4	−1.03867	−0.519337	−	0.854570i	$-0.673821\pi$
$257$	−68603.4	−1.03867	−0.519337	+	0.854570i	$0.673821\pi$
$258$	107079.	1.60866
$259$	97174.3i	1.44861i
$260$	− 34004.8i	− 0.503030i
$261$	32096.4i	0.471167i
$262$	65706.0	0.957199
$263$	− 29173.5i	− 0.421772i	−0.977511	−	0.210886i	$-0.932365\pi$
$263$	− 29173.5i	− 0.421772i	0.977511	−	0.210886i	$-0.0676349\pi$
$264$	0	0
$265$	44246.9	0.630073
$266$	− 13722.6i	− 0.193942i
$267$	−223923.	−3.14106
$268$	9474.05	0.131907
$269$	−8485.13	−0.117261	−0.0586305	−	0.998280i	$-0.518673\pi$
$269$	−8485.13	−0.117261	−0.0586305	+	0.998280i	$0.518673\pi$
$270$	124635.i	1.70967i
$271$	79870.0i	1.08754i	0.839235	+	0.543770i	$0.183003\pi$
$271$	79870.0i	1.08754i	−0.839235	+	0.543770i	$0.816997\pi$
$272$	− 20865.7i	− 0.282030i
$273$	−115227.	−1.54607
$274$	2617.36i	0.0348629i
$275$	0	0
$276$	60632.5	0.795953
$277$	48644.7i	0.633981i	0.948429	+	0.316990i	$0.102672\pi$
$277$	48644.7i	0.633981i	−0.948429	+	0.316990i	$0.897328\pi$
$278$	99135.6	1.28274
$279$	49478.4	0.635634
$280$	27887.9	0.355713
$281$	− 6387.19i	− 0.0808905i	−0.999182	−	0.0404452i	$-0.987122\pi$
$281$	− 6387.19i	− 0.0808905i	0.999182	−	0.0404452i	$-0.0128776\pi$
$282$	− 120490.i	− 1.51514i
$283$	− 127053.i	− 1.58639i	−0.608966	−	0.793196i	$-0.708416\pi$
$283$	− 127053.i	− 1.58639i	0.608966	−	0.793196i	$-0.291584\pi$
$284$	−34519.0	−0.427978
$285$	46939.8i	0.577899i
$286$	0	0
$287$	59046.2	0.716849
$288$	32814.2i	0.395619i
$289$	−22772.1	−0.272651
$290$	−13589.1	−0.161583
$291$	30818.4	0.363935
$292$	− 15133.0i	− 0.177484i
$293$	− 89042.3i	− 1.03720i	−0.855018	−	0.518598i	$-0.826454\pi$
$293$	− 89042.3i	− 1.03720i	0.855018	−	0.518598i	$-0.173546\pi$
$294$	15480.6i	0.179099i
$295$	−119423.	−1.37228
$296$	− 48409.7i	− 0.552521i
$297$	0	0
$298$	29710.8	0.334566
$299$	− 73309.6i	− 0.820008i
$300$	−14419.8	−0.160220
$301$	106178.	1.17193
$302$	62897.1	0.689631
$303$	− 194204.i	− 2.11530i
$304$	6836.22i	0.0739723i
$305$	128908.i	1.38573i
$306$	167161.	1.78522
$307$	88760.8i	0.941769i	0.882195	+	0.470885i	$0.156065\pi$
$307$	88760.8i	0.941769i	−0.882195	+	0.470885i	$0.843935\pi$
$308$	0	0
$309$	−39517.9	−0.413882
$310$	20948.4i	0.217986i
$311$	−128939.	−1.33310	−0.666551	−	0.745460i	$-0.732230\pi$
$311$	−128939.	−1.33310	−0.666551	+	0.745460i	$0.732230\pi$
$312$	57403.2	0.589695
$313$	−93944.6	−0.958922	−0.479461	−	0.877563i	$-0.659168\pi$
$313$	−93944.6	−0.958922	−0.479461	+	0.877563i	$0.659168\pi$
$314$	− 20718.7i	− 0.210137i
$315$	223418.i	2.25163i
$316$	10162.5i	0.101771i
$317$	3760.37	0.0374207	0.0187104	−	0.999825i	$-0.494044\pi$
$317$	3760.37	0.0374207	0.0187104	+	0.999825i	$0.494044\pi$
$318$	74692.7i	0.738625i
$319$	0	0
$320$	−13893.0	−0.135674
$321$	212208.i	2.05945i
$322$	60122.4	0.579862
$323$	34824.8	0.333798
$324$	−92929.7	−0.885247
$325$	17434.7i	0.165063i
$326$	120075.i	1.12984i
$327$	− 15132.3i	− 0.141518i
$328$	−29415.3	−0.273417
$329$	− 119476.i	− 1.10380i
$330$	0	0
$331$	−98750.2	−0.901326	−0.450663	−	0.892694i	$-0.648812\pi$
$331$	−98750.2	−0.901326	−0.450663	+	0.892694i	$0.648812\pi$
$332$	18536.9i	0.168175i
$333$	387824.	3.49740
$334$	40417.9	0.362310
$335$	−32134.6	−0.286341
$336$	47077.3i	0.416997i
$337$	120905.i	1.06460i	0.846556	+	0.532299i	$0.178672\pi$
$337$	120905.i	1.06460i	−0.846556	+	0.532299i	$0.821328\pi$
$338$	11377.6i	0.0995901i
$339$	112121.	0.975639
$340$	70773.3i	0.612226i
$341$	0	0
$342$	−54766.9	−0.468237
$343$	124405.i	1.05743i
$344$	−52895.0	−0.446990
$345$	−205657.	−1.72784
$346$	80052.0	0.668682
$347$	95654.8i	0.794416i	0.917729	+	0.397208i	$0.130021\pi$
$347$	95654.8i	0.794416i	−0.917729	+	0.397208i	$0.869979\pi$
$348$	− 22939.7i	− 0.189421i
$349$	121063.i	0.993945i	0.867766	+	0.496972i	$0.165555\pi$
$349$	121063.i	0.993945i	−0.867766	+	0.496972i	$0.834445\pi$
$350$	−14298.5	−0.116723
$351$	254385.i	2.06480i
$352$	0	0
$353$	−170898.	−1.37148	−0.685739	−	0.727848i	$-0.740521\pi$
$353$	−170898.	−1.37148	−0.685739	+	0.727848i	$0.740521\pi$
$354$	− 201597.i	− 1.60871i
$355$	117083.	0.929049
$356$	110614.	0.872792
$357$	239819.	1.88169
$358$	34678.4i	0.270578i
$359$	− 147200.i	− 1.14214i	−0.820902	−	0.571070i	$-0.806529\pi$
$359$	− 147200.i	− 1.14214i	0.820902	−	0.571070i	$-0.193471\pi$
$360$	− 111301.i	− 0.858804i
$361$	118911.	0.912450
$362$	− 73700.8i	− 0.562413i
$363$	0	0
$364$	56920.3	0.429600
$365$	51328.8i	0.385279i
$366$	−217608.	−1.62447
$367$	−90295.4	−0.670399	−0.335200	−	0.942147i	$-0.608804\pi$
$367$	−90295.4	−0.670399	−0.335200	+	0.942147i	$0.608804\pi$
$368$	−29951.4	−0.221168
$369$	− 235654.i	− 1.73070i
$370$	164199.i	1.19941i
$371$	74064.3i	0.538098i
$372$	−35362.8	−0.255541
$373$	− 225814.i	− 1.62305i	−0.584316	−	0.811526i	$-0.698637\pi$
$373$	− 225814.i	− 1.62305i	0.584316	−	0.811526i	$-0.301363\pi$
$374$	0	0
$375$	−225744.	−1.60529
$376$	59519.9i	0.421004i
$377$	−27735.9	−0.195146
$378$	−208626.	−1.46010
$379$	−222225.	−1.54708	−0.773542	−	0.633745i	$-0.781517\pi$
$379$	−222225.	−1.54708	−0.773542	+	0.633745i	$0.781517\pi$
$380$	− 23187.5i	− 0.160578i
$381$	− 224306.i	− 1.54522i
$382$	− 33650.6i	− 0.230604i
$383$	−81378.9	−0.554772	−0.277386	−	0.960759i	$-0.589468\pi$
$383$	−81378.9	−0.554772	−0.277386	+	0.960759i	$0.589468\pi$
$384$	− 23452.7i	− 0.159049i
$385$	0	0
$386$	−52640.4	−0.353301
$387$	− 423757.i	− 2.82940i
$388$	−15223.7	−0.101125
$389$	182748.	1.20769	0.603843	−	0.797103i	$-0.293635\pi$
$389$	182748.	1.20769	0.603843	+	0.797103i	$0.293635\pi$
$390$	−194703.	−1.28010
$391$	152577.i	0.998013i
$392$	− 7647.14i	− 0.0497654i
$393$	− 376217.i	− 2.43586i
$394$	−144428.	−0.930379
$395$	− 34469.6i	− 0.220924i
$396$	0	0
$397$	272916.	1.73160	0.865799	−	0.500391i	$-0.166811\pi$
$397$	272916.	1.73160	0.865799	+	0.500391i	$0.166811\pi$
$398$	− 92812.1i	− 0.585920i
$399$	−78572.1	−0.493540
$400$	7123.15	0.0445197
$401$	261811.	1.62817	0.814085	−	0.580746i	$-0.197239\pi$
$401$	261811.	1.62817	0.814085	+	0.580746i	$0.197239\pi$
$402$	− 54246.1i	− 0.335673i
$403$	42756.5i	0.263264i
$404$	95933.2i	0.587769i
$405$	315204.	1.92168
$406$	− 22746.7i	− 0.137996i
$407$	0	0
$408$	−119472.	−0.717704
$409$	215084.i	1.28576i	0.765966	+	0.642881i	$0.222261\pi$
$409$	215084.i	1.28576i	−0.765966	+	0.642881i	$0.777739\pi$
$410$	99772.3	0.593529
$411$	14986.4	0.0887184
$412$	19521.2	0.115004
$413$	− 199901.i	− 1.17197i
$414$	− 239949.i	− 1.39997i
$415$	− 62874.4i	− 0.365071i
$416$	−28356.2	−0.163856
$417$	− 567627.i	− 3.26430i
$418$	0	0
$419$	6203.42	0.0353348	0.0176674	−	0.999844i	$-0.494376\pi$
$419$	6203.42	0.0353348	0.0176674	+	0.999844i	$0.494376\pi$
$420$	− 159679.i	− 0.905212i
$421$	239940.	1.35375	0.676874	−	0.736099i	$-0.263334\pi$
$421$	239940.	1.35375	0.676874	+	0.736099i	$0.263334\pi$
$422$	−184169.	−1.03417
$423$	−476831.	−2.66491
$424$	− 36896.9i	− 0.205238i
$425$	− 36286.5i	− 0.200894i
$426$	197647.i	1.08911i
$427$	−215777.	−1.18345
$428$	− 104827.i	− 0.572249i
$429$	0	0
$430$	179412.	0.970320
$431$	169562.i	0.912795i	0.889776	+	0.456398i	$0.150861\pi$
$431$	169562.i	0.912795i	−0.889776	+	0.456398i	$0.849139\pi$
$432$	103932.	0.556905
$433$	−205388.	−1.09546	−0.547732	−	0.836654i	$-0.684509\pi$
$433$	−205388.	−1.09546	−0.547732	+	0.836654i	$0.684509\pi$
$434$	−35065.3	−0.186165
$435$	77808.0i	0.411193i
$436$	7475.11i	0.0393228i
$437$	− 49988.9i	− 0.261764i
$438$	−86647.7	−0.451657
$439$	155401.i	0.806350i	0.915123	+	0.403175i	$0.132093\pi$
$439$	155401.i	0.806350i	−0.915123	+	0.403175i	$0.867907\pi$
$440$	0	0
$441$	61263.4	0.315010
$442$	144451.i	0.739394i
$443$	146155.	0.744744	0.372372	−	0.928084i	$-0.378545\pi$
$443$	146155.	0.744744	0.372372	+	0.928084i	$0.378545\pi$
$444$	−277182.	−1.40605
$445$	−375187.	−1.89465
$446$	53306.8i	0.267987i
$447$	− 170117.i	− 0.851396i
$448$	− 23255.4i	− 0.115869i
$449$	−6081.04	−0.0301637	−0.0150819	−	0.999886i	$-0.504801\pi$
$449$	−6081.04	−0.0301637	−0.0150819	+	0.999886i	$0.504801\pi$
$450$	57065.5i	0.281805i
$451$	0	0
$452$	−55386.1	−0.271096
$453$	− 360134.i	− 1.75496i
$454$	170541.	0.827401
$455$	−193065.	−0.932570
$456$	39142.6	0.188243
$457$	342258.i	1.63878i	0.573235	+	0.819391i	$0.305688\pi$
$457$	342258.i	1.63878i	−0.573235	+	0.819391i	$0.694312\pi$
$458$	233890.i	1.11501i
$459$	− 529445.i	− 2.51302i
$460$	101591.	0.480108
$461$	394901.i	1.85818i	0.369860	+	0.929088i	$0.379406\pi$
$461$	394901.i	1.85818i	−0.369860	+	0.929088i	$0.620594\pi$
$462$	0	0
$463$	87290.1	0.407196	0.203598	−	0.979055i	$-0.434736\pi$
$463$	87290.1	0.407196	0.203598	+	0.979055i	$0.434736\pi$
$464$	11331.8i	0.0526336i
$465$	119946.	0.554726
$466$	114778.	0.528549
$467$	104456.	0.478960	0.239480	−	0.970901i	$-0.423023\pi$
$467$	104456.	0.478960	0.239480	+	0.970901i	$0.423023\pi$
$468$	− 227169.i	− 1.03719i
$469$	− 53789.8i	− 0.244542i
$470$	− 201883.i	− 0.913911i
$471$	−118630.	−0.534753
$472$	99585.5i	0.447005i
$473$	0	0
$474$	58187.9	0.258986
$475$	11888.5i	0.0526915i
$476$	−118467.	−0.522856
$477$	295592.	1.29914
$478$	−218614.	−0.956804
$479$	27956.3i	0.121845i	0.998142	+	0.0609226i	$0.0194043\pi$
$479$	27956.3i	0.121845i	−0.998142	+	0.0609226i	$0.980596\pi$
$480$	79548.2i	0.345261i
$481$	335136.i	1.44854i
$482$	−229740.	−0.988879
$483$	− 344246.i	− 1.47562i
$484$	0	0
$485$	51636.7	0.219520
$486$	160045.i	0.677596i
$487$	−281193.	−1.18562	−0.592811	−	0.805342i	$-0.701982\pi$
$487$	−281193.	−1.18562	−0.592811	+	0.805342i	$0.701982\pi$
$488$	107495.	0.451385
$489$	687519.	2.87519
$490$	25938.0i	0.108030i
$491$	367827.i	1.52574i	0.646552	+	0.762870i	$0.276210\pi$
$491$	367827.i	1.52574i	−0.646552	+	0.762870i	$0.723790\pi$
$492$	168425.i	0.695786i
$493$	57726.0	0.237508
$494$	− 47326.5i	− 0.193932i
$495$	0	0
$496$	17468.6	0.0710061
$497$	195985.i	0.793431i
$498$	106138.	0.427967
$499$	147512.	0.592416	0.296208	−	0.955124i	$-0.404278\pi$
$499$	147512.	0.592416	0.296208	+	0.955124i	$0.404278\pi$
$500$	111513.	0.446054
$501$	− 231423.i	− 0.922000i
$502$	− 197052.i	− 0.781942i
$503$	339342.i	1.34122i	0.741808	+	0.670612i	$0.233968\pi$
$503$	339342.i	1.34122i	−0.741808	+	0.670612i	$0.766032\pi$
$504$	186305.	0.733440
$505$	− 325392.i	− 1.27592i
$506$	0	0
$507$	65145.2	0.253435
$508$	110803.i	0.429364i
$509$	70836.1	0.273413	0.136706	−	0.990612i	$-0.456348\pi$
$509$	70836.1	0.273413	0.136706	+	0.990612i	$0.456348\pi$
$510$	405231.	1.55798
$511$	−85918.7	−0.329038
$512$	11585.2i	0.0441942i
$513$	173462.i	0.659128i
$514$	− 194040.i	− 0.734453i
$515$	−66212.9	−0.249648
$516$	302864.i	1.13749i
$517$	0	0
$518$	−274850.	−1.02432
$519$	− 458358.i	− 1.70165i
$520$	96180.1	0.355696
$521$	312685.	1.15194	0.575972	−	0.817470i	$-0.304624\pi$
$521$	312685.	1.15194	0.575972	+	0.817470i	$0.304624\pi$
$522$	−90782.2	−0.333165
$523$	255802.i	0.935192i	0.883942	+	0.467596i	$0.154880\pi$
$523$	255802.i	0.935192i	−0.883942	+	0.467596i	$0.845120\pi$
$524$	185845.i	0.676842i
$525$	81869.8i	0.297033i
$526$	82515.2	0.298238
$527$	− 88988.0i	− 0.320413i
$528$	0	0
$529$	−60825.6	−0.217358
$530$	125149.i	0.445529i
$531$	−797807.	−2.82949
$532$	38813.3	0.137138
$533$	203639.	0.716814
$534$	− 633350.i	− 2.22107i
$535$	355558.i	1.24223i
$536$	26796.7i	0.0932720i
$537$	198560.	0.688562
$538$	− 23999.6i	− 0.0829161i
$539$	0	0
$540$	−352522.	−1.20892
$541$	− 192337.i	− 0.657155i	−0.944477	−	0.328577i	$-0.893431\pi$
$541$	− 192337.i	− 0.657155i	0.944477	−	0.328577i	$-0.106569\pi$
$542$	−225906.	−0.769006
$543$	−421993.	−1.43122
$544$	59017.0	0.199425
$545$	− 25354.5i	− 0.0853615i
$546$	− 325912.i	− 1.09324i
$547$	112208.i	0.375017i	0.982263	+	0.187509i	$0.0600412\pi$
$547$	112208.i	0.375017i	−0.982263	+	0.187509i	$0.939959\pi$
$548$	−7403.02	−0.0246518
$549$	861169.i	2.85722i
$550$	0	0
$551$	−18912.8	−0.0622948
$552$	171495.i	0.562823i
$553$	57698.3	0.188674
$554$	−137588.	−0.448292
$555$	940162.	3.05223
$556$	280398.i	0.907037i
$557$	29232.1i	0.0942213i	0.998890	+	0.0471107i	$0.0150013\pi$
$557$	29232.1i	0.0942213i	−0.998890	+	0.0471107i	$0.984999\pi$
$558$	139946.i	0.449461i
$559$	366187.	1.17187
$560$	78878.9i	0.251527i
$561$	0	0
$562$	18065.7	0.0571982
$563$	− 267361.i	− 0.843494i	−0.906714	−	0.421747i	$-0.861417\pi$
$563$	− 267361.i	− 0.843494i	0.906714	−	0.421747i	$-0.138583\pi$
$564$	340797.	1.07136
$565$	187862.	0.588493
$566$	359359.	1.12175
$567$	527617.i	1.64117i
$568$	− 97634.4i	− 0.302626i
$569$	− 313405.i	− 0.968013i	−0.875065	−	0.484006i	$-0.839181\pi$
$569$	− 313405.i	− 0.968013i	0.875065	−	0.484006i	$-0.160819\pi$
$570$	−132766.	−0.408636
$571$	467241.i	1.43308i	0.697549	+	0.716538i	$0.254274\pi$
$571$	467241.i	1.43308i	−0.697549	+	0.716538i	$0.745726\pi$
$572$	0	0
$573$	−192675.	−0.586836
$574$	167008.i	0.506889i
$575$	−52087.0	−0.157541
$576$	−92812.5	−0.279745
$577$	−179158.	−0.538128	−0.269064	−	0.963122i	$-0.586714\pi$
$577$	−179158.	−0.538128	−0.269064	+	0.963122i	$0.586714\pi$
$578$	− 64409.1i	− 0.192793i
$579$	301406.i	0.899073i
$580$	− 38435.8i	− 0.114256i
$581$	105245.	0.311780
$582$	87167.5i	0.257341i
$583$	0	0
$584$	42802.5	0.125500
$585$	770526.i	2.25152i
$586$	251850.	0.733409
$587$	272980.	0.792236	0.396118	−	0.918200i	$-0.370357\pi$
$587$	272980.	0.792236	0.396118	+	0.918200i	$0.370357\pi$
$588$	−43785.7	−0.126642
$589$	29155.1i	0.0840397i
$590$	− 337779.i	− 0.970352i
$591$	826962.i	2.36761i
$592$	136923.	0.390692
$593$	− 11945.3i	− 0.0339694i	−0.999856	−	0.0169847i	$-0.994593\pi$
$593$	− 11945.3i	− 0.0339694i	0.999856	−	0.0169847i	$-0.00540665\pi$
$594$	0	0
$595$	401822.	1.13501
$596$	84034.7i	0.236574i
$597$	−531420.	−1.49104
$598$	207351.	0.579833
$599$	−285085.	−0.794550	−0.397275	−	0.917700i	$-0.630044\pi$
$599$	−285085.	−0.794550	−0.397275	+	0.917700i	$0.630044\pi$
$600$	− 40785.4i	− 0.113293i
$601$	− 439209.i	− 1.21597i	−0.793950	−	0.607984i	$-0.791979\pi$
$601$	− 439209.i	− 1.21597i	0.793950	−	0.607984i	$-0.208021\pi$
$602$	300316.i	0.828678i
$603$	−214676.	−0.590402
$604$	177900.i	0.487643i
$605$	0	0
$606$	549291.	1.49574
$607$	406708.i	1.10384i	0.833898	+	0.551919i	$0.186104\pi$
$607$	406708.i	1.10384i	−0.833898	+	0.551919i	$0.813896\pi$
$608$	−19335.8	−0.0523063
$609$	−130242.	−0.351169
$610$	−364606.	−0.979860
$611$	− 412051.i	− 1.10374i
$612$	472802.i	1.26234i
$613$	− 561402.i	− 1.49401i	−0.664819	−	0.747004i	$-0.731492\pi$
$613$	− 561402.i	− 1.49401i	0.664819	−	0.747004i	$-0.268508\pi$
$614$	−251053.	−0.665931
$615$	− 571272.i	− 1.51040i
$616$	0	0
$617$	−361868.	−0.950559	−0.475280	−	0.879835i	$-0.657653\pi$
$617$	−361868.	−0.950559	−0.475280	+	0.879835i	$0.657653\pi$
$618$	− 111773.i	− 0.292659i
$619$	−98980.1	−0.258325	−0.129163	−	0.991623i	$-0.541229\pi$
$619$	−98980.1	−0.258325	−0.129163	+	0.991623i	$0.541229\pi$
$620$	−59251.1	−0.154139
$621$	−759987.	−1.97071
$622$	− 364694.i	− 0.942645i
$623$	− 628022.i	− 1.61808i
$624$	162361.i	0.416977i
$625$	−447799.	−1.14637
$626$	− 265716.i	− 0.678060i
$627$	0	0
$628$	58601.3	0.148589
$629$	− 697509.i	− 1.76298i
$630$	−631921.	−1.59214
$631$	4305.14	0.0108126	0.00540628	−	0.999985i	$-0.498279\pi$
$631$	4305.14	0.0108126	0.00540628	+	0.999985i	$0.498279\pi$
$632$	−28743.8	−0.0719632
$633$	1.05450e6i	2.63173i
$634$	10635.9i	0.0264604i
$635$	− 375829.i	− 0.932058i
$636$	−211263.	−0.522287
$637$	52940.4i	0.130469i
$638$	0	0
$639$	782176.	1.91559
$640$	− 39295.4i	− 0.0959361i
$641$	−378590.	−0.921412	−0.460706	−	0.887553i	$-0.652404\pi$
$641$	−378590.	−0.921412	−0.460706	+	0.887553i	$0.652404\pi$
$642$	−600214.	−1.45625
$643$	−243073.	−0.587915	−0.293957	−	0.955818i	$-0.594972\pi$
$643$	−243073.	−0.587915	−0.293957	+	0.955818i	$0.594972\pi$
$644$	170052.i	0.410024i
$645$	− 1.02727e6i	− 2.46925i
$646$	98499.4i	0.236031i
$647$	211457.	0.505142	0.252571	−	0.967578i	$-0.418724\pi$
$647$	211457.	0.505142	0.252571	+	0.967578i	$0.418724\pi$
$648$	− 262845.i	− 0.625964i
$649$	0	0
$650$	−49312.9	−0.116717
$651$	200776.i	0.473750i
$652$	−339623.	−0.798917
$653$	239834.	0.562451	0.281225	−	0.959642i	$-0.409259\pi$
$653$	239834.	0.562451	0.281225	+	0.959642i	$0.409259\pi$
$654$	42800.7	0.100068
$655$	− 630358.i	− 1.46928i
$656$	− 83198.9i	− 0.193335i
$657$	342902.i	0.794401i
$658$	337930.	0.780503
$659$	474194.i	1.09191i	0.837816	+	0.545953i	$0.183832\pi$
$659$	474194.i	1.09191i	−0.837816	+	0.545953i	$0.816168\pi$
$660$	0	0
$661$	808699.	1.85090	0.925452	−	0.378865i	$-0.123686\pi$
$661$	808699.	1.85090	0.925452	+	0.378865i	$0.123686\pi$
$662$	− 279308.i	− 0.637334i
$663$	827092.	1.88160
$664$	−52430.2	−0.118917
$665$	−131649.	−0.297697
$666$	1.09693e6i	2.47304i
$667$	− 82862.2i	− 0.186254i
$668$	114319.i	0.256192i
$669$	305222.	0.681967
$670$	− 90890.4i	− 0.202474i
$671$	0	0
$672$	−133155.	−0.294862
$673$	682366.i	1.50656i	0.657699	+	0.753281i	$0.271530\pi$
$673$	682366.i	1.50656i	−0.657699	+	0.753281i	$0.728470\pi$
$674$	−341972.	−0.752785
$675$	180743.	0.396692
$676$	−32180.6	−0.0704209
$677$	814971.i	1.77814i	0.457776	+	0.889068i	$0.348646\pi$
$677$	814971.i	1.77814i	−0.457776	+	0.889068i	$0.651354\pi$
$678$	317127.i	0.689881i
$679$	86434.1i	0.187476i
$680$	−200177.	−0.432909
$681$	− 976474.i	− 2.10555i
$682$	0	0
$683$	303153.	0.649861	0.324930	−	0.945738i	$-0.394659\pi$
$683$	303153.	0.649861	0.324930	+	0.945738i	$0.394659\pi$
$684$	− 154904.i	− 0.331094i
$685$	25110.0	0.0535137
$686$	−351872.	−0.747715
$687$	1.33920e6	2.83747
$688$	− 149610.i	− 0.316070i
$689$	255434.i	0.538072i
$690$	− 581685.i	− 1.22177i
$691$	−122009.	−0.255525	−0.127763	−	0.991805i	$-0.540780\pi$
$691$	−122009.	−0.255525	−0.127763	+	0.991805i	$0.540780\pi$
$692$	226421.i	0.472830i
$693$	0	0
$694$	−270553.	−0.561737
$695$	− 951069.i	− 1.96898i
$696$	64883.2	0.133941
$697$	−423829.	−0.872418
$698$	−342419.	−0.702825
$699$	− 657189.i	− 1.34504i
$700$	− 40442.3i	− 0.0825353i
$701$	− 506988.i	− 1.03172i	−0.856673	−	0.515860i	$-0.827473\pi$
$701$	− 506988.i	− 1.03172i	0.856673	−	0.515860i	$-0.172527\pi$
$702$	−719510.	−1.46003
$703$	228525.i	0.462405i
$704$	0	0
$705$	−1.15593e6	−2.32570
$706$	− 483374.i	− 0.969781i
$707$	544669.	1.08967
$708$	570202.	1.13753
$709$	639601.	1.27238	0.636190	−	0.771533i	$-0.280510\pi$
$709$	639601.	1.27238	0.636190	+	0.771533i	$0.280510\pi$
$710$	331162.i	0.656937i
$711$	− 230275.i	− 0.455520i
$712$	312864.i	0.617157i
$713$	−127737.	−0.251268
$714$	678312.i	1.33056i
$715$	0	0
$716$	−98085.2	−0.191328
$717$	1.25173e6i	2.43486i
$718$	416345.	0.807614
$719$	54516.4	0.105455	0.0527277	−	0.998609i	$-0.483208\pi$
$719$	54516.4	0.105455	0.0527277	+	0.998609i	$0.483208\pi$
$720$	314807.	0.607266
$721$	− 110833.i	− 0.213206i
$722$	336332.i	0.645199i
$723$	1.31544e6i	2.51648i
$724$	208457.	0.397686
$725$	19706.6i	0.0374917i
$726$	0	0
$727$	−671062.	−1.26968	−0.634839	−	0.772644i	$-0.718934\pi$
$727$	−671062.	−1.26968	−0.634839	+	0.772644i	$0.718934\pi$
$728$	160995.i	0.303773i
$729$	−24532.0	−0.0461614
$730$	−145180.	−0.272433
$731$	−762136.	−1.42626
$732$	− 615488.i	− 1.14868i
$733$	882149.i	1.64185i	0.571035	+	0.820926i	$0.306542\pi$
$733$	882149.i	1.64185i	−0.571035	+	0.820926i	$0.693458\pi$
$734$	− 255394.i	− 0.474044i
$735$	148515.	0.274913
$736$	− 84715.4i	− 0.156389i
$737$	0	0
$738$	666530.	1.22379
$739$	− 87595.7i	− 0.160396i	−0.996779	−	0.0801981i	$-0.974445\pi$
$739$	− 87595.7i	− 0.160396i	0.996779	−	0.0801981i	$-0.0255553\pi$
$740$	−464424.	−0.848108
$741$	−270980.	−0.493516
$742$	−209486.	−0.380493
$743$	− 1.03199e6i	− 1.86938i	−0.355459	−	0.934692i	$-0.615676\pi$
$743$	− 1.03199e6i	− 1.86938i	0.355459	−	0.934692i	$-0.384324\pi$
$744$	− 100021.i	− 0.180695i
$745$	− 285033.i	− 0.513551i
$746$	638698.	1.14767
$747$	− 420033.i	− 0.752735i
$748$	0	0
$749$	−595164.	−1.06090
$750$	− 638499.i	− 1.13511i
$751$	647165.	1.14745	0.573727	−	0.819047i	$-0.305497\pi$
$751$	647165.	1.14745	0.573727	+	0.819047i	$0.305497\pi$
$752$	−168348.	−0.297695
$753$	−1.12827e6	−1.98987
$754$	− 78449.0i	− 0.137989i
$755$	− 603411.i	− 1.05857i
$756$	− 590082.i	− 1.03245i
$757$	1.07768e6	1.88061	0.940304	−	0.340337i	$-0.110541\pi$
$757$	1.07768e6	1.88061	0.940304	+	0.340337i	$0.110541\pi$
$758$	− 628547.i	− 1.09395i
$759$	0	0
$760$	65584.1	0.113546
$761$	136593.i	0.235862i	0.993022	+	0.117931i	$0.0376262\pi$
$761$	136593.i	0.235862i	−0.993022	+	0.117931i	$0.962374\pi$
$762$	634434.	1.09264
$763$	42440.6	0.0729009
$764$	95178.3	0.163061
$765$	− 1.60368e6i	− 2.74027i
$766$	− 230174.i	− 0.392283i
$767$	− 689421.i	− 1.17191i
$768$	66334.3	0.112465
$769$	− 24012.2i	− 0.0406050i	−0.999794	−	0.0203025i	$-0.993537\pi$
$769$	− 24012.2i	− 0.0406050i	0.999794	−	0.0203025i	$-0.00646292\pi$
$770$	0	0
$771$	−1.11102e6	−1.86902
$772$	− 148889.i	− 0.249821i
$773$	−75679.0	−0.126653	−0.0633266	−	0.997993i	$-0.520171\pi$
$773$	−75679.0	−0.126653	−0.0633266	+	0.997993i	$0.520171\pi$
$774$	1.19856e6	2.00069
$775$	30378.8	0.0505787
$776$	− 43059.2i	− 0.0715061i
$777$	1.57373e6i	2.60668i
$778$	516890.i	0.853963i
$779$	138859.	0.228823
$780$	− 550704.i	− 0.905168i
$781$	0	0
$782$	−431554.	−0.705702
$783$	287533.i	0.468991i
$784$	21629.4	0.0351894
$785$	−198767.	−0.322556
$786$	1.06410e6	1.72242
$787$	493819.i	0.797294i	0.917104	+	0.398647i	$0.130520\pi$
$787$	493819.i	0.797294i	−0.917104	+	0.398647i	$0.869480\pi$
$788$	− 408505.i	− 0.657877i
$789$	− 472462.i	− 0.758950i
$790$	97494.8	0.156217
$791$	314459.i	0.502587i
$792$	0	0
$793$	−744175.	−1.18339
$794$	771922.i	1.22443i
$795$	716573.	1.13377
$796$	262512.	0.414308
$797$	150592.	0.237074	0.118537	−	0.992950i	$-0.462180\pi$
$797$	150592.	0.237074	0.118537	+	0.992950i	$0.462180\pi$
$798$	− 222235.i	− 0.348985i
$799$	857590.i	1.34334i
$800$	20147.3i	0.0314802i
$801$	−2.50644e6	−3.90654
$802$	740514.i	1.15129i
$803$	0	0
$804$	153431.	0.237357
$805$	− 576791.i	− 0.890075i
$806$	−120934.	−0.186156
$807$	−137416.	−0.211003
$808$	−271340.	−0.415615
$809$	445235.i	0.680287i	0.940374	+	0.340144i	$0.110476\pi$
$809$	445235.i	0.680287i	−0.940374	+	0.340144i	$0.889524\pi$
$810$	891532.i	1.35883i
$811$	− 677570.i	− 1.03018i	−0.857137	−	0.515089i	$-0.827759\pi$
$811$	− 677570.i	− 1.03018i	0.857137	−	0.515089i	$-0.172241\pi$
$812$	64337.3	0.0975778
$813$	1.29349e6i	1.95695i
$814$	0	0
$815$	1.15195e6	1.73428
$816$	− 337917.i	− 0.507493i
$817$	249699.	0.374086
$818$	−608348.	−0.909171
$819$	−1.28977e6	−1.92285
$820$	282199.i	0.419689i
$821$	873586.i	1.29604i	0.761622	+	0.648021i	$0.224403\pi$
$821$	873586.i	1.29604i	−0.761622	+	0.648021i	$0.775597\pi$
$822$	42387.9i	0.0627334i
$823$	−124202.	−0.183370	−0.0916850	−	0.995788i	$-0.529225\pi$
$823$	−124202.	−0.183370	−0.0916850	+	0.995788i	$0.529225\pi$
$824$	55214.2i	0.0813198i
$825$	0	0
$826$	565405.	0.828705
$827$	− 72676.1i	− 0.106263i	−0.998588	−	0.0531313i	$-0.983080\pi$
$827$	− 72676.1i	− 0.106263i	0.998588	−	0.0531313i	$-0.0169202\pi$
$828$	678678.	0.989927
$829$	−365134.	−0.531304	−0.265652	−	0.964069i	$-0.585587\pi$
$829$	−365134.	−0.531304	−0.265652	+	0.964069i	$0.585587\pi$
$830$	177836.	0.258144
$831$	787796.i	1.14081i
$832$	− 80203.5i	− 0.115863i
$833$	− 110184.i	− 0.158791i
$834$	1.60549e6	2.30821
$835$	− 387753.i	− 0.556138i
$836$	0	0
$837$	443249.	0.632698
$838$	17545.9i	0.0249855i
$839$	−53598.8	−0.0761432	−0.0380716	−	0.999275i	$-0.512121\pi$
$839$	−53598.8	−0.0761432	−0.0380716	+	0.999275i	$0.512121\pi$
$840$	451642.	0.640082
$841$	675931.	0.955675
$842$	678652.i	0.957245i
$843$	− 103440.i	− 0.145557i
$844$	− 520907.i	− 0.731266i
$845$	109152.	0.152869
$846$	− 1.34868e6i	− 1.88438i
$847$	0	0
$848$	104360.	0.145125
$849$	− 2.05760e6i	− 2.85461i
$850$	102634.	0.142053
$851$	−1.00123e6	−1.38253
$852$	−559031.	−0.770117
$853$	− 305696.i	− 0.420138i	−0.977687	−	0.210069i	$-0.932631\pi$
$853$	− 305696.i	− 0.420138i	0.977687	−	0.210069i	$-0.0673688\pi$
$854$	− 610310.i	− 0.836825i
$855$	525412.i	0.718733i
$856$	296495.	0.404641
$857$	− 344594.i	− 0.469187i	−0.972094	−	0.234594i	$-0.924624\pi$
$857$	− 344594.i	− 0.469187i	0.972094	−	0.234594i	$-0.0753759\pi$
$858$	0	0
$859$	−267027.	−0.361884	−0.180942	−	0.983494i	$-0.557915\pi$
$859$	−267027.	−0.361884	−0.180942	+	0.983494i	$0.557915\pi$
$860$	507454.i	0.686120i
$861$	956246.	1.28992
$862$	−479593.	−0.645444
$863$	1.27833e6	1.71641	0.858207	−	0.513305i	$-0.171579\pi$
$863$	1.27833e6	1.71641	0.858207	+	0.513305i	$0.171579\pi$
$864$	293964.i	0.393791i
$865$	− 767988.i	− 1.02641i
$866$	− 580924.i	− 0.774610i
$867$	−368791.	−0.490617
$868$	− 99179.7i	− 0.131639i
$869$	0	0
$870$	−220074.	−0.290757
$871$	− 185511.i	− 0.244530i
$872$	−21142.8	−0.0278054
$873$	344959.	0.452626
$874$	141390.	0.185095
$875$	− 633128.i	− 0.826942i
$876$	− 245077.i	− 0.319370i
$877$	122610.i	0.159414i	0.996818	+	0.0797072i	$0.0253985\pi$
$877$	122610.i	0.159414i	−0.996818	+	0.0797072i	$0.974601\pi$
$878$	−439539.	−0.570175
$879$	− 1.44203e6i	− 1.86637i
$880$	0	0
$881$	870902.	1.12206	0.561032	−	0.827794i	$-0.310405\pi$
$881$	870902.	1.12206	0.561032	+	0.827794i	$0.310405\pi$
$882$	173279.i	0.222745i
$883$	−773237.	−0.991724	−0.495862	−	0.868401i	$-0.665148\pi$
$883$	−773237.	−0.991724	−0.495862	+	0.868401i	$0.665148\pi$
$884$	−408569.	−0.522831
$885$	−1.93404e6	−2.46933
$886$	413390.i	0.526614i
$887$	− 362911.i	− 0.461268i	−0.973041	−	0.230634i	$-0.925920\pi$
$887$	− 362911.i	− 0.461268i	0.973041	−	0.230634i	$-0.0740800\pi$
$888$	− 783990.i	− 0.994225i
$889$	629096.	0.796001
$890$	− 1.06119e6i	− 1.33972i
$891$	0	0
$892$	−150774.	−0.189495
$893$	− 280972.i	− 0.352339i
$894$	481162.	0.602028
$895$	332691.	0.415331
$896$	65776.2	0.0819319
$897$	− 1.18724e6i	− 1.47555i
$898$	− 17199.8i	− 0.0213290i
$899$	48327.9i	0.0597969i
$900$	−161406.	−0.199266
$901$	− 531628.i	− 0.654875i
$902$	0	0
$903$	1.71954e6	2.10880
$904$	− 156656.i	− 0.191694i
$905$	−707057.	−0.863291
$906$	1.01861e6	1.24094
$907$	−1.07813e6	−1.31056	−0.655282	−	0.755385i	$-0.727450\pi$
$907$	−1.07813e6	−1.31056	−0.655282	+	0.755385i	$0.727450\pi$
$908$	482361.i	0.585061i
$909$	− 2.17378e6i	− 2.63080i
$910$	− 546071.i	− 0.659427i
$911$	−561906.	−0.677060	−0.338530	−	0.940956i	$-0.609930\pi$
$911$	−561906.	−0.677060	−0.338530	+	0.940956i	$0.609930\pi$
$912$	110712.i	0.133108i
$913$	0	0
$914$	−968051.	−1.15879
$915$	2.08765e6i	2.49353i
$916$	−661541.	−0.788435
$917$	1.05515e6	1.25480
$918$	1.49750e6	1.77697
$919$	− 308378.i	− 0.365134i	−0.983193	−	0.182567i	$-0.941559\pi$
$919$	− 308378.i	− 0.365134i	0.983193	−	0.182567i	$-0.0584407\pi$
$920$	287342.i	0.339487i
$921$	1.43747e6i	1.69465i
$922$	−1.11695e6	−1.31393
$923$	675914.i	0.793392i
$924$	0	0
$925$	238116.	0.278295
$926$	246894.i	0.287931i
$927$	−442336.	−0.514746
$928$	−32051.2	−0.0372176
$929$	−408120.	−0.472886	−0.236443	−	0.971645i	$-0.575982\pi$
$929$	−408120.	−0.472886	−0.236443	+	0.971645i	$0.575982\pi$
$930$	339257.i	0.392250i
$931$	36099.4i	0.0416487i
$932$	324640.i	0.373741i
$933$	−2.08815e6	−2.39883
$934$	295446.i	0.338676i
$935$	0	0
$936$	642532.	0.733404
$937$	− 1.10225e6i	− 1.25545i	−0.778433	−	0.627727i	$-0.783985\pi$
$937$	− 1.10225e6i	− 1.25545i	0.778433	−	0.627727i	$-0.216015\pi$
$938$	152140.	0.172918
$939$	−1.52142e6	−1.72552
$940$	571011.	0.646232
$941$	− 1.54042e6i	− 1.73965i	−0.493363	−	0.869823i	$-0.664233\pi$
$941$	− 1.54042e6i	− 1.73965i	0.493363	−	0.869823i	$-0.335767\pi$
$942$	− 335537.i	− 0.378128i
$943$	608380.i	0.684151i
$944$	−281670.	−0.316080
$945$	2.00147e6i	2.24123i
$946$	0	0
$947$	−1.56708e6	−1.74739	−0.873697	−	0.486470i	$-0.838284\pi$
$947$	−1.56708e6	−1.74739	−0.873697	+	0.486470i	$0.838284\pi$
$948$	164580.i	0.183131i
$949$	−296317.	−0.329022
$950$	−33625.8	−0.0372586
$951$	60898.8	0.0673360
$952$	− 335074.i	− 0.369715i
$953$	43847.4i	0.0482790i	0.999709	+	0.0241395i	$0.00768459\pi$
$953$	43847.4i	0.0482790i	−0.999709	+	0.0241395i	$0.992315\pi$
$954$	836059.i	0.918629i
$955$	−322831.	−0.353971
$956$	− 618335.i	− 0.676563i
$957$	0	0
$958$	−79072.3	−0.0861576
$959$	42031.3i	0.0457021i
$960$	−224996.	−0.244136
$961$	−849021.	−0.919330
$962$	−947907.	−1.02427
$963$	2.37531e6i	2.56134i
$964$	− 649804.i	− 0.699243i
$965$	505011.i	0.542309i
$966$	973676.	1.04342
$967$	− 1.00897e6i	− 1.07901i	−0.841982	−	0.539506i	$-0.818611\pi$
$967$	− 1.00897e6i	− 1.07901i	0.841982	−	0.539506i	$-0.181389\pi$
$968$	0	0
$969$	563984.	0.600647
$970$	146051.i	0.155224i
$971$	328298.	0.348201	0.174100	−	0.984728i	$-0.444298\pi$
$971$	328298.	0.348201	0.174100	+	0.984728i	$0.444298\pi$
$972$	−452677.	−0.479132
$973$	1.59198e6	1.68156
$974$	− 795333.i	− 0.838361i
$975$	282354.i	0.297019i
$976$	304041.i	0.319177i
$977$	1.01240e6	1.06063	0.530316	−	0.847800i	$-0.322073\pi$
$977$	1.01240e6	1.06063	0.530316	+	0.847800i	$0.322073\pi$
$978$	1.94460e6i	2.03307i
$979$	0	0
$980$	−73363.7	−0.0763887
$981$	− 169381.i	− 0.176006i
$982$	−1.04037e6	−1.07886
$983$	1.04166e6	1.07800	0.538999	−	0.842306i	$-0.318802\pi$
$983$	1.04166e6	1.07800	0.538999	+	0.842306i	$0.318802\pi$
$984$	−476377.	−0.491995
$985$	1.38559e6i	1.42811i
$986$	163274.i	0.167943i
$987$	− 1.93490e6i	− 1.98621i
$988$	133860.	0.137131
$989$	1.09400e6i	1.11847i
$990$	0	0
$991$	−1.89936e6	−1.93401	−0.967006	−	0.254755i	$-0.918005\pi$
$991$	−1.89936e6	−1.93401	−0.967006	+	0.254755i	$0.918005\pi$
$992$	49408.8i	0.0502089i
$993$	−1.59925e6	−1.62187
$994$	−554328.	−0.561040
$995$	−890403.	−0.899375
$996$	300203.i	0.302619i
$997$	601955.i	0.605583i	0.953057	+	0.302791i	$0.0979186\pi$
$997$	601955.i	0.605583i	−0.953057	+	0.302791i	$0.902081\pi$
$998$	417227.i	0.418901i
$999$	3.47429e6	3.48125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.5.b.e.241.16		16
11.3	even	5		22.5.d.a.13.3	✓	16
11.7	odd	10		22.5.d.a.17.3	yes	16
11.10	odd	2	inner	242.5.b.e.241.8		16
33.14	odd	10		198.5.j.a.145.1		16
33.29	even	10		198.5.j.a.127.1		16
44.3	odd	10		176.5.n.c.145.4		16
44.7	even	10		176.5.n.c.17.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.d.a.13.3	✓	16	11.3	even	5
22.5.d.a.17.3	yes	16	11.7	odd	10
176.5.n.c.17.4		16	44.7	even	10
176.5.n.c.145.4		16	44.3	odd	10
198.5.j.a.127.1		16	33.29	even	10
198.5.j.a.145.1		16	33.14	odd	10
242.5.b.e.241.8		16	11.10	odd	2	inner
242.5.b.e.241.16		16	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:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	242.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +16.1949 q^{3} -8.00000 q^{4} +27.1348 q^{5} +45.8061i q^{6} +45.4207i q^{7} -22.6274i q^{8} +181.275 q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +16.1949 q^{3} -8.00000 q^{4} +27.1348 q^{5} +45.8061i q^{6} +45.4207i q^{7} -22.6274i q^{8} +181.275 q^{9} +76.7489i q^{10} -129.559 q^{12} +156.647i q^{13} -128.469 q^{14} +439.446 q^{15} +64.0000 q^{16} -326.026i q^{17} +512.722i q^{18} +106.816i q^{19} -217.079 q^{20} +735.583i q^{21} -467.991 q^{23} -366.449i q^{24} +111.299 q^{25} -443.066 q^{26} +1623.93 q^{27} -363.366i q^{28} +177.059i q^{29} +1242.94i q^{30} +272.947 q^{31} +181.019i q^{32} +922.141 q^{34} +1232.48i q^{35} -1450.20 q^{36} +2139.43 q^{37} -302.121 q^{38} +2536.89i q^{39} -613.991i q^{40} -1299.98i q^{41} -2080.54 q^{42} -2337.65i q^{43} +4918.85 q^{45} -1323.68i q^{46} -2630.43 q^{47} +1036.47 q^{48} +337.959 q^{49} +314.802i q^{50} -5279.96i q^{51} -1253.18i q^{52} +1630.63 q^{53} +4593.18i q^{54} +1027.75 q^{56} +1729.87i q^{57} -500.800 q^{58} -4401.10 q^{59} -3515.57 q^{60} +4750.63i q^{61} +772.012i q^{62} +8233.62i q^{63} -512.000 q^{64} +4250.60i q^{65} -1184.26 q^{67} +2608.21i q^{68} -7579.06 q^{69} -3485.99 q^{70} +4314.87 q^{71} -4101.77i q^{72} +1891.62i q^{73} +6051.21i q^{74} +1802.48 q^{75} -854.528i q^{76} -7175.40 q^{78} -1270.31i q^{79} +1736.63 q^{80} +11616.2 q^{81} +3676.91 q^{82} -2317.11i q^{83} -5884.67i q^{84} -8846.67i q^{85} +6611.88 q^{86} +2867.46i q^{87} -13826.8 q^{89} +13912.6i q^{90} -7115.04 q^{91} +3743.93 q^{92} +4420.35 q^{93} -7439.99i q^{94} +2898.43i q^{95} +2931.59i q^{96} +1902.97 q^{97} +955.893i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9}+O(q^{10}) \) 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9	\( 16 q + 48 q^{3} - 128 q^{4} + 60 q^{5} - 40 q^{9} - 384 q^{12} + 384 q^{14} + 1352 q^{15} + 1024 q^{16} - 480 q^{20} - 2424 q^{23} + 5692 q^{25} + 4344 q^{27} - 428 q^{31} - 3904 q^{34} + 320 q^{36} - 228 q^{37} + 1440 q^{38} - 5568 q^{42} + 12152 q^{45} - 9228 q^{47} + 3072 q^{48} - 11956 q^{49} + 12468 q^{53} - 3072 q^{56} + 2944 q^{58} - 22320 q^{59} - 10816 q^{60} - 8192 q^{64} + 21524 q^{67} - 24704 q^{69} + 16960 q^{70} + 8868 q^{71} + 19716 q^{75} - 39424 q^{78} + 3840 q^{80} + 1136 q^{81} - 20992 q^{82} - 35616 q^{86} + 46596 q^{89} - 35600 q^{91} + 19392 q^{92} + 14832 q^{93} + 69448 q^{97}+O(q^{100}) \) 16 * q + 48 * q^3 - 128 * q^4 + 60 * q^5 - 40 * q^9 - 384 * q^12 + 384 * q^14 + 1352 * q^15 + 1024 * q^16 - 480 * q^20 - 2424 * q^23 + 5692 * q^25 + 4344 * q^27 - 428 * q^31 - 3904 * q^34 + 320 * q^36 - 228 * q^37 + 1440 * q^38 - 5568 * q^42 + 12152 * q^45 - 9228 * q^47 + 3072 * q^48 - 11956 * q^49 + 12468 * q^53 - 3072 * q^56 + 2944 * q^58 - 22320 * q^59 - 10816 * q^60 - 8192 * q^64 + 21524 * q^67 - 24704 * q^69 + 16960 * q^70 + 8868 * q^71 + 19716 * q^75 - 39424 * q^78 + 3840 * q^80 + 1136 * q^81 - 20992 * q^82 - 35616 * q^86 + 46596 * q^89 - 35600 * q^91 + 19392 * q^92 + 14832 * q^93 + 69448 * q^97

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