LMFDB - Embedded newform 117.5.j.b.73.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,5,Mod(73,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("117.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	117.j (of order $4$, degree $2$, 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:	$12.0942856808$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 $ x^20 + 5446x^16 - 1452x^15 + 106320x^13 + 8376897x^12 - 1643220x^11 + 1054152x^10 + 232107192x^9 + 3337033804x^8 + 3105930144x^7 + 2297014848x^6 + 21034904256x^5 + 158658808624x^4 + 235144077504x^3 + 177846480000x^2 + 27353289600*x + 2103506496
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.7
Root			$1.99524 + 1.99524i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.5.j.b.109.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.99524 + 1.99524i) q^{2} -8.03802i q^{4} +(-3.82538 - 3.82538i) q^{5} +(-42.4180 + 42.4180i) q^{7} +(47.9617 - 47.9617i) q^{8} -15.2651i q^{10} +(152.629 - 152.629i) q^{11} +(102.350 - 134.483i) q^{13} -169.268 q^{14} +62.7818 q^{16} -195.803i q^{17} +(-418.159 - 418.159i) q^{19} +(-30.7485 + 30.7485i) q^{20} +609.065 q^{22} +96.2247i q^{23} -595.733i q^{25} +(472.537 - 64.1132i) q^{26} +(340.957 + 340.957i) q^{28} +814.896 q^{29} +(574.293 + 574.293i) q^{31} +(-642.122 - 642.122i) q^{32} +(390.675 - 390.675i) q^{34} +324.530 q^{35} +(-1272.11 + 1272.11i) q^{37} -1668.66i q^{38} -366.943 q^{40} +(1849.48 + 1849.48i) q^{41} +712.995i q^{43} +(-1226.84 - 1226.84i) q^{44} +(-191.991 + 191.991i) q^{46} +(-213.922 + 213.922i) q^{47} -1197.58i q^{49} +(1188.63 - 1188.63i) q^{50} +(-1080.97 - 822.688i) q^{52} -3263.61 q^{53} -1167.73 q^{55} +4068.88i q^{56} +(1625.91 + 1625.91i) q^{58} +(33.4862 - 33.4862i) q^{59} +1396.14 q^{61} +2291.70i q^{62} -3566.88i q^{64} +(-905.972 + 122.921i) q^{65} +(-89.7582 - 89.7582i) q^{67} -1573.87 q^{68} +(647.515 + 647.515i) q^{70} +(-1632.34 - 1632.34i) q^{71} +(3204.47 - 3204.47i) q^{73} -5076.32 q^{74} +(-3361.17 + 3361.17i) q^{76} +12948.5i q^{77} -6995.98 q^{79} +(-240.164 - 240.164i) q^{80} +7380.34i q^{82} +(-161.980 - 161.980i) q^{83} +(-749.022 + 749.022i) q^{85} +(-1422.60 + 1422.60i) q^{86} -14640.7i q^{88} +(10192.7 - 10192.7i) q^{89} +(1363.02 + 10046.0i) q^{91} +773.456 q^{92} -853.652 q^{94} +3199.23i q^{95} +(9961.57 + 9961.57i) q^{97} +(2389.45 - 2389.45i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) $ 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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 $

$2$	1.99524	+	1.99524i	0.498810	+	0.498810i	0.911068	−	0.412257i	$-0.135259\pi$
$2$	1.99524	+	1.99524i	0.498810	+	0.498810i	−0.412257	+	0.911068i	$0.635259\pi$
$3$		0			0
$3$		0			0
$4$		−	8.03802i		−	0.502376i
$5$	−3.82538	−	3.82538i	−0.153015	−	0.153015i	0.626448	−	0.779463i	$-0.284508\pi$
$5$	−3.82538	−	3.82538i	−0.153015	−	0.153015i	−0.779463	+	0.626448i	$0.784508\pi$
$6$		0			0
$7$	−42.4180	+	42.4180i	−0.865674	+	0.865674i	−0.991990	−	0.126316i	$-0.959685\pi$
$7$	−42.4180	+	42.4180i	−0.865674	+	0.865674i	0.126316	+	0.991990i	$0.459685\pi$
$8$	47.9617	−	47.9617i	0.749401	−	0.749401i
$9$		0			0
$10$		−	15.2651i		−	0.152651i
$11$	152.629	−	152.629i	1.26140	−	1.26140i	0.310985	−	0.950415i	$-0.399341\pi$
$11$	152.629	−	152.629i	1.26140	−	1.26140i	0.950415	−	0.310985i	$-0.100659\pi$
$12$		0			0
$13$	102.350	−	134.483i	0.605619	−	0.795755i
$13$	102.350	−	134.483i	0.605619	−	0.795755i
$14$	−169.268			−0.863614
$15$		0			0
$16$	62.7818			0.245242
$17$		−	195.803i		−	0.677521i	−0.940873	−	0.338760i	$-0.889992\pi$
$17$		−	195.803i		−	0.677521i	0.940873	−	0.338760i	$-0.110008\pi$
$18$		0			0
$19$	−418.159	−	418.159i	−1.15833	−	1.15833i	−0.984834	−	0.173501i	$-0.944492\pi$
$19$	−418.159	−	418.159i	−1.15833	−	1.15833i	−0.173501	−	0.984834i	$-0.555508\pi$
$20$	−30.7485	+	30.7485i	−0.0768712	+	0.0768712i
$21$		0			0
$22$	609.065			1.25840
$23$			96.2247i			0.181899i	0.995855	+	0.0909496i	$0.0289902\pi$
$23$			96.2247i			0.181899i	−0.995855	+	0.0909496i	$0.971010\pi$
$24$		0			0
$25$		−	595.733i		−	0.953173i
$26$	472.537	−	64.1132i	0.699020	−	0.0948420i
$27$		0			0
$28$	340.957	+	340.957i	0.434894	+	0.434894i
$29$	814.896			0.968961			0.484480	−	0.874802i	$-0.339009\pi$
$29$	814.896			0.968961			0.484480	+	0.874802i	$0.339009\pi$
$30$		0			0
$31$	574.293	+	574.293i	0.597599	+	0.597599i	0.939673	−	0.342074i	$-0.111129\pi$
$31$	574.293	+	574.293i	0.597599	+	0.597599i	−0.342074	+	0.939673i	$0.611129\pi$
$32$	−642.122	−	642.122i	−0.627072	−	0.627072i
$33$		0			0
$34$	390.675	−	390.675i	0.337954	−	0.337954i
$35$	324.530			0.264922
$36$		0			0
$37$	−1272.11	+	1272.11i	−0.929223	+	0.929223i	−0.997656	−	0.0684329i	$-0.978200\pi$
$37$	−1272.11	+	1272.11i	−0.929223	+	0.929223i	0.0684329	+	0.997656i	$0.478200\pi$
$38$		−	1668.66i		−	1.15558i
$39$		0			0
$40$	−366.943			−0.229339
$41$	1849.48	+	1849.48i	1.10023	+	1.10023i	0.994383	+	0.105846i	$0.0337551\pi$
$41$	1849.48	+	1849.48i	1.10023	+	1.10023i	0.105846	+	0.994383i	$0.466245\pi$
$42$		0			0
$43$			712.995i			0.385611i	0.981237	+	0.192806i	$0.0617587\pi$
$43$			712.995i			0.385611i	−0.981237	+	0.192806i	$0.938241\pi$
$44$	−1226.84	−	1226.84i	−0.633698	−	0.633698i
$45$		0			0
$46$	−191.991	+	191.991i	−0.0907332	+	0.0907332i
$47$	−213.922	+	213.922i	−0.0968411	+	0.0968411i	−0.753868	−	0.657026i	$-0.771814\pi$
$47$	−213.922	+	213.922i	−0.0968411	+	0.0968411i	0.657026	+	0.753868i	$0.271814\pi$
$48$		0			0
$49$		−	1197.58i		−	0.498782i
$50$	1188.63	−	1188.63i	0.475452	−	0.475452i
$51$		0			0
$52$	−1080.97	−	822.688i	−0.399769	−	0.304249i
$53$	−3263.61			−1.16184			−0.580920	−	0.813960i	$-0.697307\pi$
$53$	−3263.61			−1.16184			−0.580920	+	0.813960i	$0.697307\pi$
$54$		0			0
$55$	−1167.73			−0.386026
$56$			4068.88i			1.29747i
$57$		0			0
$58$	1625.91	+	1625.91i	0.483328	+	0.483328i
$59$	33.4862	−	33.4862i	0.00961970	−	0.00961970i	−0.702281	−	0.711900i	$-0.747835\pi$
$59$	33.4862	−	33.4862i	0.00961970	−	0.00961970i	0.711900	+	0.702281i	$0.247835\pi$
$60$		0			0
$61$	1396.14			0.375206			0.187603	−	0.982245i	$-0.439928\pi$
$61$	1396.14			0.375206			0.187603	+	0.982245i	$0.439928\pi$
$62$			2291.70i			0.596177i
$63$		0			0
$64$		−	3566.88i		−	0.870821i
$65$	−905.972	+	122.921i	−0.214431	+	0.0290937i
$66$		0			0
$67$	−89.7582	−	89.7582i	−0.0199951	−	0.0199951i	0.697039	−	0.717034i	$-0.254501\pi$
$67$	−89.7582	−	89.7582i	−0.0199951	−	0.0199951i	−0.717034	+	0.697039i	$0.754501\pi$
$68$	−1573.87			−0.340370
$69$		0			0
$70$	647.515	+	647.515i	0.132146	+	0.132146i
$71$	−1632.34	−	1632.34i	−0.323812	−	0.323812i	0.526416	−	0.850227i	$-0.323536\pi$
$71$	−1632.34	−	1632.34i	−0.323812	−	0.323812i	−0.850227	+	0.526416i	$0.823536\pi$
$72$		0			0
$73$	3204.47	−	3204.47i	0.601327	−	0.601327i	−0.339338	−	0.940665i	$-0.610203\pi$
$73$	3204.47	−	3204.47i	0.601327	−	0.601327i	0.940665	+	0.339338i	$0.110203\pi$
$74$	−5076.32			−0.927012
$75$		0			0
$76$	−3361.17	+	3361.17i	−0.581920	+	0.581920i
$77$			12948.5i			2.18392i
$78$		0			0
$79$	−6995.98			−1.12097			−0.560486	−	0.828164i	$-0.689386\pi$
$79$	−6995.98			−1.12097			−0.560486	+	0.828164i	$0.689386\pi$
$80$	−240.164	−	240.164i	−0.0375257	−	0.0375257i
$81$		0			0
$82$			7380.34i			1.09761i
$83$	−161.980	−	161.980i	−0.0235129	−	0.0235129i	0.695253	−	0.718765i	$-0.255293\pi$
$83$	−161.980	−	161.980i	−0.0235129	−	0.0235129i	−0.718765	+	0.695253i	$0.755293\pi$
$84$		0			0
$85$	−749.022	+	749.022i	−0.103671	+	0.103671i
$86$	−1422.60	+	1422.60i	−0.192347	+	0.192347i
$87$		0			0
$88$		−	14640.7i		−	1.89059i
$89$	10192.7	−	10192.7i	1.28680	−	1.28680i	0.350078	−	0.936720i	$-0.386155\pi$
$89$	10192.7	−	10192.7i	1.28680	−	1.28680i	0.936720	−	0.350078i	$-0.113845\pi$
$90$		0			0
$91$	1363.02	+	10046.0i	0.164596	+	1.21313i
$92$	773.456			0.0913819
$93$		0			0
$94$	−853.652			−0.0966107
$95$			3199.23i			0.354485i
$96$		0			0
$97$	9961.57	+	9961.57i	1.05873	+	1.05873i	0.998164	+	0.0605636i	$0.0192898\pi$
$97$	9961.57	+	9961.57i	1.05873	+	1.05873i	0.0605636	+	0.998164i	$0.480710\pi$
$98$	2389.45	−	2389.45i	0.248798	−	0.248798i
$99$		0			0
$100$	−4788.52			−0.478852
$101$		−	923.540i		−	0.0905343i	−0.998975	−	0.0452671i	$-0.985586\pi$
$101$		−	923.540i		−	0.0905343i	0.998975	−	0.0452671i	$-0.0144139\pi$
$102$		0			0
$103$			11272.8i			1.06257i	0.847192	+	0.531286i	$0.178291\pi$
$103$			11272.8i			1.06257i	−0.847192	+	0.531286i	$0.821709\pi$
$104$	−1541.15	−	11358.9i	−0.142488	−	1.05019i
$105$		0			0
$106$	−6511.69	−	6511.69i	−0.579538	−	0.579538i
$107$	6063.17			0.529580			0.264790	−	0.964306i	$-0.414697\pi$
$107$	6063.17			0.529580			0.264790	+	0.964306i	$0.414697\pi$
$108$		0			0
$109$	5709.40	+	5709.40i	0.480549	+	0.480549i	0.905307	−	0.424758i	$-0.139641\pi$
$109$	5709.40	+	5709.40i	0.480549	+	0.480549i	−0.424758	+	0.905307i	$0.639641\pi$
$110$	−2329.90	−	2329.90i	−0.192554	−	0.192554i
$111$		0			0
$112$	−2663.08	+	2663.08i	−0.212299	+	0.212299i
$113$	−9695.94			−0.759335			−0.379667	−	0.925123i	$-0.623962\pi$
$113$	−9695.94			−0.759335			−0.379667	+	0.925123i	$0.623962\pi$
$114$		0			0
$115$	368.096	−	368.096i	0.0278333	−	0.0278333i
$116$		−	6550.15i		−	0.486783i
$117$		0			0
$118$	133.626			0.00959681
$119$	8305.59	+	8305.59i	0.586512	+	0.586512i
$120$		0			0
$121$		−	31950.5i		−	2.18226i
$122$	2785.64	+	2785.64i	0.187157	+	0.187157i
$123$		0			0
$124$	4616.18	−	4616.18i	0.300220	−	0.300220i
$125$	−4669.76	+	4669.76i	−0.298865	+	0.298865i
$126$		0			0
$127$		−	12871.5i		−	0.798036i	−0.916943	−	0.399018i	$-0.869351\pi$
$127$		−	12871.5i		−	0.798036i	0.916943	−	0.399018i	$-0.130649\pi$
$128$	−3157.15	+	3157.15i	−0.192697	+	0.192697i
$129$		0			0
$130$	−2052.89	−	1562.38i	−0.121473	−	0.0924483i
$131$	19409.4			1.13102			0.565508	−	0.824743i	$-0.308680\pi$
$131$	19409.4			1.13102			0.565508	+	0.824743i	$0.308680\pi$
$132$		0			0
$133$	35474.9			2.00548
$134$		−	358.179i		−	0.0199476i
$135$		0			0
$136$	−9391.06	−	9391.06i	−0.507735	−	0.507735i
$137$	8904.59	−	8904.59i	0.474431	−	0.474431i	−0.428914	−	0.903345i	$-0.641104\pi$
$137$	8904.59	−	8904.59i	0.474431	−	0.474431i	0.903345	+	0.428914i	$0.141104\pi$
$138$		0			0
$139$	6138.32			0.317702			0.158851	−	0.987303i	$-0.449221\pi$
$139$	6138.32			0.317702			0.158851	+	0.987303i	$0.449221\pi$
$140$		−	2608.58i		−	0.133091i
$141$		0			0
$142$		−	6513.81i		−	0.323041i
$143$	−4904.45	−	36147.5i	−0.239838	−	1.76769i
$144$		0			0
$145$	−3117.28	−	3117.28i	−0.148266	−	0.148266i
$146$	12787.4			0.599896
$147$		0			0
$148$	10225.2	+	10225.2i	0.466820	+	0.466820i
$149$	17824.9	+	17824.9i	0.802887	+	0.802887i	0.983546	−	0.180659i	$-0.0578231\pi$
$149$	17824.9	+	17824.9i	0.802887	+	0.802887i	−0.180659	+	0.983546i	$0.557823\pi$
$150$		0			0
$151$	17720.1	−	17720.1i	0.777162	−	0.777162i	−0.202185	−	0.979347i	$-0.564804\pi$
$151$	17720.1	−	17720.1i	0.777162	−	0.777162i	0.979347	+	0.202185i	$0.0648043\pi$
$152$	−40111.2			−1.73611
$153$		0			0
$154$	−25835.3	+	25835.3i	−1.08936	+	1.08936i
$155$		−	4393.77i		−	0.182883i
$156$		0			0
$157$	−43199.0			−1.75256			−0.876282	−	0.481798i	$-0.839984\pi$
$157$	−43199.0			−1.75256			−0.876282	+	0.481798i	$0.839984\pi$
$158$	−13958.7	−	13958.7i	−0.559152	−	0.559152i
$159$		0			0
$160$			4912.72i			0.191903i
$161$	−4081.66	−	4081.66i	−0.157465	−	0.157465i
$162$		0			0
$163$	−29545.3	+	29545.3i	−1.11202	+	1.11202i	−0.119144	+	0.992877i	$0.538015\pi$
$163$	−29545.3	+	29545.3i	−1.11202	+	1.11202i	−0.992877	+	0.119144i	$0.961985\pi$
$164$	14866.2	−	14866.2i	0.552729	−	0.552729i
$165$		0			0
$166$		−	646.380i		−	0.0234569i
$167$	−6360.18	+	6360.18i	−0.228054	+	0.228054i	−0.811879	−	0.583826i	$-0.801555\pi$
$167$	−6360.18	+	6360.18i	−0.228054	+	0.228054i	0.583826	+	0.811879i	$0.301555\pi$
$168$		0			0
$169$	−7610.14	−	27528.5i	−0.266452	−	0.963848i
$170$	−2988.96			−0.103424
$171$		0			0
$172$	5731.07			0.193722
$173$			19966.6i			0.667132i	0.942727	+	0.333566i	$0.108252\pi$
$173$			19966.6i			0.667132i	−0.942727	+	0.333566i	$0.891748\pi$
$174$		0			0
$175$	25269.8	+	25269.8i	0.825137	+	0.825137i
$176$	9582.35	−	9582.35i	0.309348	−	0.309348i
$177$		0			0
$178$	40673.9			1.28374
$179$			21028.1i			0.656287i	0.944628	+	0.328144i	$0.106423\pi$
$179$			21028.1i			0.656287i	−0.944628	+	0.328144i	$0.893577\pi$
$180$		0			0
$181$			38363.4i			1.17101i	0.810669	+	0.585504i	$0.199104\pi$
$181$			38363.4i			1.17101i	−0.810669	+	0.585504i	$0.800896\pi$
$182$	−17324.5	+	22763.7i	−0.523021	+	0.687225i
$183$		0			0
$184$	4615.10	+	4615.10i	0.136315	+	0.136315i
$185$	9732.57			0.284370
$186$		0			0
$187$	−29885.4	−	29885.4i	−0.854624	−	0.854624i
$188$	1719.51	+	1719.51i	0.0486507	+	0.0486507i
$189$		0			0
$190$	−6383.24	+	6383.24i	−0.176821	+	0.176821i
$191$	50333.2			1.37971			0.689855	−	0.723947i	$-0.257674\pi$
$191$	50333.2			1.37971			0.689855	+	0.723947i	$0.257674\pi$
$192$		0			0
$193$	12651.4	−	12651.4i	0.339643	−	0.339643i	−0.516590	−	0.856233i	$-0.672799\pi$
$193$	12651.4	−	12651.4i	0.339643	−	0.339643i	0.856233	+	0.516590i	$0.172799\pi$
$194$			39751.5i			1.05621i
$195$		0			0
$196$	−9626.15			−0.250577
$197$	29465.8	+	29465.8i	0.759251	+	0.759251i	0.976186	−	0.216935i	$-0.0696059\pi$
$197$	29465.8	+	29465.8i	0.759251	+	0.759251i	−0.216935	+	0.976186i	$0.569606\pi$
$198$		0			0
$199$			51188.8i			1.29261i	0.763077	+	0.646307i	$0.223687\pi$
$199$			51188.8i			1.29261i	−0.763077	+	0.646307i	$0.776313\pi$
$200$	−28572.3	−	28572.3i	−0.714309	−	0.714309i
$201$		0			0
$202$	1842.69	−	1842.69i	0.0451594	−	0.0451594i
$203$	−34566.3	+	34566.3i	−0.838804	+	0.838804i
$204$		0			0
$205$		−	14150.0i		−	0.336703i
$206$	−22492.0	+	22492.0i	−0.530022	+	0.530022i
$207$		0			0
$208$	6425.69	−	8443.06i	0.148523	−	0.195152i
$209$	−127647.			−2.92225
$210$		0			0
$211$	−11413.1			−0.256353			−0.128176	−	0.991751i	$-0.540912\pi$
$211$	−11413.1			−0.256353			−0.128176	+	0.991751i	$0.540912\pi$
$212$			26233.0i			0.583681i
$213$		0			0
$214$	12097.5	+	12097.5i	0.264160	+	0.264160i
$215$	2727.48	−	2727.48i	0.0590043	−	0.0590043i
$216$		0			0
$217$	−48720.7			−1.03465
$218$			22783.3i			0.479406i
$219$		0			0
$220$			9386.24i			0.193931i
$221$	−26332.2	−	20040.4i	−0.539140	−	0.410319i
$222$		0			0
$223$	27797.2	+	27797.2i	0.558974	+	0.558974i	0.929015	−	0.370041i	$-0.120657\pi$
$223$	27797.2	+	27797.2i	0.558974	+	0.558974i	−0.370041	+	0.929015i	$0.620657\pi$
$224$	54475.1			1.08568
$225$		0			0
$226$	−19345.7	−	19345.7i	−0.378764	−	0.378764i
$227$	23277.0	+	23277.0i	0.451727	+	0.451727i	0.895927	−	0.444201i	$-0.146512\pi$
$227$	23277.0	+	23277.0i	0.451727	+	0.451727i	−0.444201	+	0.895927i	$0.646512\pi$
$228$		0			0
$229$	28256.2	−	28256.2i	0.538818	−	0.538818i	−0.384363	−	0.923182i	$-0.625579\pi$
$229$	28256.2	−	28256.2i	0.538818	−	0.538818i	0.923182	+	0.384363i	$0.125579\pi$
$230$	1468.88			0.0277671
$231$		0			0
$232$	39083.8	−	39083.8i	0.726140	−	0.726140i
$233$		−	28942.7i		−	0.533123i	−0.963818	−	0.266561i	$-0.914112\pi$
$233$		−	28942.7i		−	0.533123i	0.963818	−	0.266561i	$-0.0858875\pi$
$234$		0			0
$235$	1636.66			0.0296363
$236$	−269.163	−	269.163i	−0.00483271	−	0.00483271i
$237$		0			0
$238$			33143.3i			0.585116i
$239$	−46271.6	−	46271.6i	−0.810064	−	0.810064i	0.174580	−	0.984643i	$-0.444143\pi$
$239$	−46271.6	−	46271.6i	−0.810064	−	0.810064i	−0.984643	+	0.174580i	$0.944143\pi$
$240$		0			0
$241$	36808.4	−	36808.4i	0.633742	−	0.633742i	−0.315263	−	0.949004i	$-0.602093\pi$
$241$	36808.4	−	36808.4i	0.633742	−	0.633742i	0.949004	+	0.315263i	$0.102093\pi$
$242$	63748.9	−	63748.9i	1.08853	−	1.08853i
$243$		0			0
$244$		−	11222.2i		−	0.188495i
$245$	−4581.18	+	4581.18i	−0.0763212	+	0.0763212i
$246$		0			0
$247$	−99033.5	+	13436.7i	−1.62326	+	0.220242i
$248$	55088.1			0.895682
$249$		0			0
$250$	−18634.6			−0.298154
$251$			51658.9i			0.819969i	0.912092	+	0.409984i	$0.134466\pi$
$251$			51658.9i			0.819969i	−0.912092	+	0.409984i	$0.865534\pi$
$252$		0			0
$253$	14686.7	+	14686.7i	0.229448	+	0.229448i
$254$	25681.8	−	25681.8i	0.398069	−	0.398069i
$255$		0			0
$256$	−69668.7			−1.06306
$257$		−	120472.i		−	1.82397i	−0.410221	−	0.911986i	$-0.634548\pi$
$257$		−	120472.i		−	1.82397i	0.410221	−	0.911986i	$-0.365452\pi$
$258$		0			0
$259$		−	107920.i		−	1.60881i
$260$	988.042	+	7282.23i	0.0146160	+	0.107725i
$261$		0			0
$262$	38726.4	+	38726.4i	0.564163	+	0.564163i
$263$	−33541.8			−0.484926			−0.242463	−	0.970161i	$-0.577955\pi$
$263$	−33541.8			−0.484926			−0.242463	+	0.970161i	$0.577955\pi$
$264$		0			0
$265$	12484.5	+	12484.5i	0.177779	+	0.177779i
$266$	70781.1	+	70781.1i	1.00035	+	1.00035i
$267$		0			0
$268$	−721.478	+	721.478i	−0.0100451	+	0.0100451i
$269$	60299.3			0.833313			0.416656	−	0.909064i	$-0.363202\pi$
$269$	60299.3			0.833313			0.416656	+	0.909064i	$0.363202\pi$
$270$		0			0
$271$	−16273.7	+	16273.7i	−0.221588	+	0.221588i	−0.809167	−	0.587579i	$-0.800081\pi$
$271$	−16273.7	+	16273.7i	−0.221588	+	0.221588i	0.587579	+	0.809167i	$0.300081\pi$
$272$		−	12292.9i		−	0.166156i
$273$		0			0
$274$	35533.6			0.473302
$275$	−90926.4	−	90926.4i	−1.20233	−	1.20233i
$276$		0			0
$277$		−	22320.4i		−	0.290899i	−0.989366	−	0.145450i	$-0.953537\pi$
$277$		−	22320.4i		−	0.290899i	0.989366	−	0.145450i	$-0.0464629\pi$
$278$	12247.4	+	12247.4i	0.158473	+	0.158473i
$279$		0			0
$280$	15565.0	−	15565.0i	0.198533	−	0.198533i
$281$	102684.	−	102684.i	1.30044	−	1.30044i	0.372343	−	0.928095i	$-0.378554\pi$
$281$	102684.	−	102684.i	1.30044	−	1.30044i	0.928095	−	0.372343i	$-0.121446\pi$
$282$		0			0
$283$			108180.i			1.35075i	0.737473	+	0.675377i	$0.236019\pi$
$283$			108180.i			1.35075i	−0.737473	+	0.675377i	$0.763981\pi$
$284$	−13120.7	+	13120.7i	−0.162675	+	0.162675i
$285$		0			0
$286$	62337.5	−	81908.6i	0.762110	−	1.00138i
$287$	−156903.			−1.90488
$288$		0			0
$289$	45182.0			0.540966
$290$		−	12439.5i		−	0.147913i
$291$		0			0
$292$	−25757.6	−	25757.6i	−0.302093	−	0.302093i
$293$	74282.2	−	74282.2i	0.865266	−	0.865266i	−0.126678	−	0.991944i	$-0.540431\pi$
$293$	74282.2	−	74282.2i	0.865266	−	0.865266i	0.991944	+	0.126678i	$0.0404314\pi$
$294$		0			0
$295$	−256.195			−0.00294392
$296$			122025.i			1.39272i
$297$		0			0
$298$			71129.9i			0.800976i
$299$	12940.5	+	9848.55i	0.144747	+	0.110162i
$300$		0			0
$301$	−30243.8	−	30243.8i	−0.333814	−	0.333814i
$302$	70711.6			0.775313
$303$		0			0
$304$	−26252.8	−	26252.8i	−0.284072	−	0.284072i
$305$	−5340.77	−	5340.77i	−0.0574122	−	0.0574122i
$306$		0			0
$307$	−41201.1	+	41201.1i	−0.437151	+	0.437151i	−0.891052	−	0.453901i	$-0.850032\pi$
$307$	−41201.1	+	41201.1i	−0.437151	+	0.437151i	0.453901	+	0.891052i	$0.350032\pi$
$308$	104080.			1.09715
$309$		0			0
$310$	8766.64	−	8766.64i	0.0912241	−	0.0912241i
$311$			102377.i			1.05848i	0.848472	+	0.529240i	$0.177523\pi$
$311$			102377.i			1.05848i	−0.848472	+	0.529240i	$0.822477\pi$
$312$		0			0
$313$	91169.7			0.930598			0.465299	−	0.885154i	$-0.345947\pi$
$313$	91169.7			0.930598			0.465299	+	0.885154i	$0.345947\pi$
$314$	−86192.3	−	86192.3i	−0.874197	−	0.874197i
$315$		0			0
$316$			56233.9i			0.563150i
$317$	46694.5	+	46694.5i	0.464673	+	0.464673i	0.900184	−	0.435511i	$-0.143432\pi$
$317$	46694.5	+	46694.5i	0.464673	+	0.464673i	−0.435511	+	0.900184i	$0.643432\pi$
$318$		0			0
$319$	124377.	−	124377.i	1.22225	−	1.22225i
$320$	−13644.7	+	13644.7i	−0.133249	+	0.133249i
$321$		0			0
$322$		−	16287.8i		−	0.157091i
$323$	−81876.9	+	81876.9i	−0.784796	+	0.784796i
$324$		0			0
$325$	−80115.7	−	60973.0i	−0.758492	−	0.577259i
$326$	−117900.			−1.10938
$327$		0			0
$328$	177409.			1.64902
$329$		−	18148.3i		−	0.167666i
$330$		0			0
$331$	−41823.3	−	41823.3i	−0.381735	−	0.381735i	0.489992	−	0.871727i	$-0.337000\pi$
$331$	−41823.3	−	41823.3i	−0.381735	−	0.381735i	−0.871727	+	0.489992i	$0.837000\pi$
$332$	−1302.00	+	1302.00i	−0.0118123	+	0.0118123i
$333$		0			0
$334$	−25380.2			−0.227511
$335$			686.718i			0.00611912i
$336$		0			0
$337$			97325.0i			0.856968i	0.903549	+	0.428484i	$0.140952\pi$
$337$			97325.0i			0.856968i	−0.903549	+	0.428484i	$0.859048\pi$
$338$	39741.9	−	70110.0i	0.347868	−	0.613687i
$339$		0			0
$340$	6020.66	+	6020.66i	0.0520818	+	0.0520818i
$341$	175308.			1.50762
$342$		0			0
$343$	−51046.8	−	51046.8i	−0.433891	−	0.433891i
$344$	34196.4	+	34196.4i	0.288977	+	0.288977i
$345$		0			0
$346$	−39838.2	+	39838.2i	−0.332773	+	0.332773i
$347$	−114507.			−0.950981			−0.475490	−	0.879721i	$-0.657729\pi$
$347$	−114507.			−0.950981			−0.475490	+	0.879721i	$0.657729\pi$
$348$		0			0
$349$	−28924.8	+	28924.8i	−0.237476	+	0.237476i	−0.815804	−	0.578328i	$-0.803705\pi$
$349$	−28924.8	+	28924.8i	−0.237476	+	0.237476i	0.578328	+	0.815804i	$0.303705\pi$
$350$			100839.i			0.823174i
$351$		0			0
$352$	−196013.			−1.58198
$353$	−56172.5	−	56172.5i	−0.450790	−	0.450790i	0.444826	−	0.895617i	$-0.353265\pi$
$353$	−56172.5	−	56172.5i	−0.450790	−	0.450790i	−0.895617	+	0.444826i	$0.853265\pi$
$354$		0			0
$355$			12488.6i			0.0990962i
$356$	−81929.4	−	81929.4i	−0.646457	−	0.646457i
$357$		0			0
$358$	−41956.1	+	41956.1i	−0.327363	+	0.327363i
$359$	−136938.	+	136938.i	−1.06252	+	1.06252i	−0.0646065	+	0.997911i	$0.520579\pi$
$359$	−136938.	+	136938.i	−1.06252	+	1.06252i	−0.997911	+	0.0646065i	$0.979421\pi$
$360$		0			0
$361$			219393.i			1.68348i
$362$	−76544.3	+	76544.3i	−0.584111	+	0.584111i
$363$		0			0
$364$	80749.6	−	10956.0i	0.609449	−	0.0826892i
$365$	−24516.6			−0.184024
$366$		0			0
$367$	−40440.2			−0.300249			−0.150124	−	0.988667i	$-0.547967\pi$
$367$	−40440.2			−0.300249			−0.150124	+	0.988667i	$0.547967\pi$
$368$			6041.16i			0.0446092i
$369$		0			0
$370$	19418.8	+	19418.8i	0.141847	+	0.141847i
$371$	138436.	−	138436.i	1.00578	−	1.00578i
$372$		0			0
$373$	−198442.			−1.42632			−0.713158	−	0.701003i	$-0.752736\pi$
$373$	−198442.			−1.42632			−0.713158	+	0.701003i	$0.752736\pi$
$374$		−	119257.i		−	0.852591i
$375$		0			0
$376$			20520.1i			0.145146i
$377$	83404.2	−	109589.i	0.586821	−	0.771055i
$378$		0			0
$379$	−33116.9	−	33116.9i	−0.230554	−	0.230554i	0.582370	−	0.812924i	$-0.302125\pi$
$379$	−33116.9	−	33116.9i	−0.230554	−	0.230554i	−0.812924	+	0.582370i	$0.802125\pi$
$380$	25715.5			0.178085
$381$		0			0
$382$	100427.	+	100427.i	0.688214	+	0.688214i
$383$	205260.	+	205260.i	1.39929	+	1.39929i	0.802101	+	0.597189i	$0.203716\pi$
$383$	205260.	+	205260.i	1.39929	+	1.39929i	0.597189	+	0.802101i	$0.296284\pi$
$384$		0			0
$385$	49532.8	−	49532.8i	0.334173	−	0.334173i
$386$	50485.0			0.338835
$387$		0			0
$388$	80071.3	−	80071.3i	0.531880	−	0.531880i
$389$		−	93857.4i		−	0.620254i	−0.950695	−	0.310127i	$-0.899628\pi$
$389$		−	93857.4i		−	0.620254i	0.950695	−	0.310127i	$-0.100372\pi$
$390$		0			0
$391$	18841.1			0.123240
$392$	−57437.8	−	57437.8i	−0.373788	−	0.373788i
$393$		0			0
$394$			117583.i			0.757445i
$395$	26762.3	+	26762.3i	0.171526	+	0.171526i
$396$		0			0
$397$	125978.	−	125978.i	0.799305	−	0.799305i	−0.183681	−	0.982986i	$-0.558801\pi$
$397$	125978.	−	125978.i	0.799305	−	0.799305i	0.982986	+	0.183681i	$0.0588015\pi$
$398$	−102134.	+	102134.i	−0.644769	+	0.644769i
$399$		0			0
$400$		−	37401.2i		−	0.233758i
$401$	−71867.1	+	71867.1i	−0.446932	+	0.446932i	−0.894333	−	0.447401i	$-0.852350\pi$
$401$	−71867.1	+	71867.1i	−0.446932	+	0.446932i	0.447401	+	0.894333i	$0.352350\pi$
$402$		0			0
$403$	136011.	−	18453.8i	0.837459	−	0.113625i
$404$	−7423.44			−0.0454823
$405$		0			0
$406$	−137936.			−0.836808
$407$			388322.i			2.34424i
$408$		0			0
$409$	−9218.59	−	9218.59i	−0.0551084	−	0.0551084i	0.679015	−	0.734124i	$-0.262407\pi$
$409$	−9218.59	−	9218.59i	−0.0551084	−	0.0551084i	−0.734124	+	0.679015i	$0.762407\pi$
$410$	28232.6	−	28232.6i	0.167951	−	0.167951i
$411$		0			0
$412$	90611.3			0.533812
$413$			2840.83i			0.0166550i
$414$		0			0
$415$			1239.27i			0.00719565i
$416$	−152075.	+	20633.3i	−0.878762	+	0.119229i
$417$		0			0
$418$	−254686.	−	254686.i	−1.45765	−	1.45765i
$419$	−327438.			−1.86510			−0.932548	−	0.361046i	$-0.882420\pi$
$419$	−327438.			−1.86510			−0.932548	+	0.361046i	$0.882420\pi$
$420$		0			0
$421$	−148423.	−	148423.i	−0.837407	−	0.837407i	0.151110	−	0.988517i	$-0.451715\pi$
$421$	−148423.	−	148423.i	−0.837407	−	0.837407i	−0.988517	+	0.151110i	$0.951715\pi$
$422$	−22771.8	−	22771.8i	−0.127871	−	0.127871i
$423$		0			0
$424$	−156528.	+	156528.i	−0.870685	+	0.870685i
$425$	−116647.			−0.645794
$426$		0			0
$427$	−59221.6	+	59221.6i	−0.324806	+	0.324806i
$428$		−	48735.9i		−	0.266049i
$429$		0			0
$430$	10883.9			0.0588639
$431$	111444.	+	111444.i	0.599933	+	0.599933i	0.940295	−	0.340362i	$-0.110549\pi$
$431$	111444.	+	111444.i	0.599933	+	0.599933i	−0.340362	+	0.940295i	$0.610549\pi$
$432$		0			0
$433$			27383.4i			0.146054i	0.997330	+	0.0730268i	$0.0232659\pi$
$433$			27383.4i			0.146054i	−0.997330	+	0.0730268i	$0.976734\pi$
$434$	−97209.6	−	97209.6i	−0.516095	−	0.516095i
$435$		0			0
$436$	45892.3	−	45892.3i	0.241417	−	0.241417i
$437$	40237.2	−	40237.2i	0.210700	−	0.210700i
$438$		0			0
$439$			31599.3i			0.163964i	0.996634	+	0.0819819i	$0.0261250\pi$
$439$			31599.3i			0.163964i	−0.996634	+	0.0819819i	$0.973875\pi$
$440$	−56006.3	+	56006.3i	−0.289289	+	0.289289i
$441$		0			0
$442$	−12553.6	−	92524.4i	−0.0642574	−	0.473600i
$443$	24320.0			0.123924			0.0619620	−	0.998079i	$-0.480264\pi$
$443$	24320.0			0.123924			0.0619620	+	0.998079i	$0.480264\pi$
$444$		0			0
$445$	−77982.1			−0.393799
$446$			110924.i			0.557644i
$447$		0			0
$448$	151300.	+	151300.i	0.753847	+	0.753847i
$449$	214520.	−	214520.i	1.06408	−	1.06408i	0.0662794	−	0.997801i	$-0.478887\pi$
$449$	214520.	−	214520.i	1.06408	−	1.06408i	0.997801	−	0.0662794i	$-0.0211129\pi$
$450$		0			0
$451$	564571.			2.77566
$452$			77936.2i			0.381472i
$453$		0			0
$454$			92886.6i			0.450652i
$455$	33215.5	−	43643.6i	0.160442	−	0.210813i
$456$		0			0
$457$	84912.2	+	84912.2i	0.406572	+	0.406572i	0.880541	−	0.473969i	$-0.157179\pi$
$457$	84912.2	+	84912.2i	0.406572	+	0.406572i	−0.473969	+	0.880541i	$0.657179\pi$
$458$	112756.			0.537536
$459$		0			0
$460$	−2958.76	−	2958.76i	−0.0139828	−	0.0139828i
$461$	−106364.	−	106364.i	−0.500487	−	0.500487i	0.411102	−	0.911589i	$-0.365144\pi$
$461$	−106364.	−	106364.i	−0.500487	−	0.500487i	−0.911589	+	0.411102i	$0.865144\pi$
$462$		0			0
$463$	155075.	−	155075.i	0.723404	−	0.723404i	−0.245893	−	0.969297i	$-0.579081\pi$
$463$	155075.	−	155075.i	0.723404	−	0.723404i	0.969297	+	0.245893i	$0.0790811\pi$
$464$	51160.7			0.237629
$465$		0			0
$466$	57747.7	−	57747.7i	0.265927	−	0.265927i
$467$		−	39999.2i		−	0.183408i	−0.995786	−	0.0917039i	$-0.970769\pi$
$467$		−	39999.2i		−	0.183408i	0.995786	−	0.0917039i	$-0.0292313\pi$
$468$		0			0
$469$	7614.73			0.0346185
$470$	3265.54	+	3265.54i	0.0147829	+	0.0147829i
$471$		0			0
$472$		−	3212.11i		−	0.0144180i
$473$	108824.	+	108824.i	0.486410	+	0.486410i
$474$		0			0
$475$	−249111.	+	249111.i	−1.10409	+	1.10409i
$476$	66760.6	−	66760.6i	0.294650	−	0.294650i
$477$		0			0
$478$		−	184646.i		−	0.808136i
$479$	252164.	−	252164.i	1.09904	−	1.09904i	0.104515	−	0.994523i	$-0.466671\pi$
$479$	252164.	−	252164.i	1.09904	−	1.09904i	0.994523	−	0.104515i	$-0.0333290\pi$
$480$		0			0
$481$	40876.7	+	301276.i	0.176679	+	1.30219i
$482$	146883.			0.632234
$483$		0			0
$484$	−256818.			−1.09632
$485$		−	76213.5i		−	0.324003i
$486$		0			0
$487$	49095.0	+	49095.0i	0.207004	+	0.207004i	0.802993	−	0.595989i	$-0.203240\pi$
$487$	49095.0	+	49095.0i	0.207004	+	0.207004i	−0.595989	+	0.802993i	$0.703240\pi$
$488$	66961.3	−	66961.3i	0.281180	−	0.281180i
$489$		0			0
$490$	−18281.1			−0.0761397
$491$			179257.i			0.743555i	0.928322	+	0.371778i	$0.121252\pi$
$491$			179257.i			0.743555i	−0.928322	+	0.371778i	$0.878748\pi$
$492$		0			0
$493$		−	159559.i		−	0.656491i
$494$	−224405.	−	170786.i	−0.919558	−	0.699840i
$495$		0			0
$496$	36055.1	+	36055.1i	0.146556	+	0.146556i
$497$	138481.			0.560631
$498$		0			0
$499$	−215997.	−	215997.i	−0.867455	−	0.867455i	0.124735	−	0.992190i	$-0.460192\pi$
$499$	−215997.	−	215997.i	−0.867455	−	0.867455i	−0.992190	+	0.124735i	$0.960192\pi$
$500$	37535.7	+	37535.7i	0.150143	+	0.150143i
$501$		0			0
$502$	−103072.	+	103072.i	−0.409009	+	0.409009i
$503$	32799.4			0.129637			0.0648186	−	0.997897i	$-0.479353\pi$
$503$	32799.4			0.129637			0.0648186	+	0.997897i	$0.479353\pi$
$504$		0			0
$505$	−3532.89	+	3532.89i	−0.0138531	+	0.0138531i
$506$			58607.1i			0.228902i
$507$		0			0
$508$	−103462.			−0.400915
$509$	58817.8	+	58817.8i	0.227025	+	0.227025i	0.811448	−	0.584424i	$-0.198679\pi$
$509$	58817.8	+	58817.8i	0.227025	+	0.227025i	−0.584424	+	0.811448i	$0.698679\pi$
$510$		0			0
$511$			271855.i			1.04111i
$512$	−88491.5	−	88491.5i	−0.337568	−	0.337568i
$513$		0			0
$514$	240370.	−	240370.i	0.909816	−	0.909816i
$515$	43122.9	−	43122.9i	0.162590	−	0.162590i
$516$		0			0
$517$			65301.6i			0.244311i
$518$	215327.	−	215327.i	0.802490	−	0.802490i
$519$		0			0
$520$	−37556.4	+	49347.4i	−0.138892	+	0.182498i
$521$	−287522.			−1.05924			−0.529622	−	0.848234i	$-0.677666\pi$
$521$	−287522.			−1.05924			−0.529622	+	0.848234i	$0.677666\pi$
$522$		0			0
$523$	32066.9			0.117234			0.0586170	−	0.998281i	$-0.481331\pi$
$523$	32066.9			0.117234			0.0586170	+	0.998281i	$0.481331\pi$
$524$		−	156013.i		−	0.568196i
$525$		0			0
$526$	−66924.1	−	66924.1i	−0.241886	−	0.241886i
$527$	112448.	−	112448.i	0.404886	−	0.404886i
$528$		0			0
$529$	270582.			0.966913
$530$			49819.4i			0.177356i
$531$		0			0
$532$		−	285148.i		−	1.00751i
$533$	438017.	−	59429.6i	1.54183	−	0.209194i
$534$		0			0
$535$	−23193.9	−	23193.9i	−0.0810338	−	0.0810338i
$536$	−8609.90			−0.0299688
$537$		0			0
$538$	120312.	+	120312.i	0.415665	+	0.415665i
$539$	−182785.	−	182785.i	−0.629164	−	0.629164i
$540$		0			0
$541$	118440.	−	118440.i	0.404674	−	0.404674i	−0.475202	−	0.879877i	$-0.657625\pi$
$541$	118440.	−	118440.i	0.404674	−	0.404674i	0.879877	+	0.475202i	$0.157625\pi$
$542$	−64939.8			−0.221061
$543$		0			0
$544$	−125730.	+	125730.i	−0.424854	+	0.424854i
$545$		−	43681.2i		−	0.147063i
$546$		0			0
$547$	−401196.			−1.34086			−0.670428	−	0.741975i	$-0.733889\pi$
$547$	−401196.			−1.34086			−0.670428	+	0.741975i	$0.733889\pi$
$548$	−71575.3	−	71575.3i	−0.238343	−	0.238343i
$549$		0			0
$550$		−	362840.i		−	1.19947i
$551$	−340756.	−	340756.i	−1.12238	−	1.12238i
$552$		0			0
$553$	296756.	−	296756.i	0.970396	−	0.970396i
$554$	44534.6	−	44534.6i	0.145104	−	0.145104i
$555$		0			0
$556$		−	49339.9i		−	0.159606i
$557$	6502.04	−	6502.04i	0.0209575	−	0.0209575i	−0.696550	−	0.717508i	$-0.745283\pi$
$557$	6502.04	−	6502.04i	0.0209575	−	0.0209575i	0.717508	+	0.696550i	$0.245283\pi$
$558$		0			0
$559$	95885.4	+	72974.7i	0.306852	+	0.233533i
$560$	20374.6			0.0649700
$561$		0			0
$562$	409758.			1.29734
$563$			495416.i			1.56298i	0.623918	+	0.781490i	$0.285540\pi$
$563$			495416.i			1.56298i	−0.623918	+	0.781490i	$0.714460\pi$
$564$		0			0
$565$	37090.6	+	37090.6i	0.116190	+	0.116190i
$566$	−215846.	+	215846.i	−0.673770	+	0.673770i
$567$		0			0
$568$	−156579.			−0.485330
$569$			486731.i			1.50336i	0.659525	+	0.751682i	$0.270757\pi$
$569$			486731.i			1.50336i	−0.659525	+	0.751682i	$0.729243\pi$
$570$		0			0
$571$		−	32584.1i		−	0.0999387i	−0.998751	−	0.0499694i	$-0.984088\pi$
$571$		−	32584.1i		−	0.0999387i	0.998751	−	0.0499694i	$-0.0159124\pi$
$572$	−290555.	+	39422.1i	−0.888047	+	0.120489i
$573$		0			0
$574$	−313059.	−	313059.i	−0.950173	−	0.950173i
$575$	57324.2			0.173381
$576$		0			0
$577$	98903.1	+	98903.1i	0.297070	+	0.297070i	0.839865	−	0.542795i	$-0.182634\pi$
$577$	98903.1	+	98903.1i	0.297070	+	0.297070i	−0.542795	+	0.839865i	$0.682634\pi$
$578$	90149.0	+	90149.0i	0.269839	+	0.269839i
$579$		0			0
$580$	−25056.8	+	25056.8i	−0.0744851	+	0.0744851i
$581$	13741.8			0.0407090
$582$		0			0
$583$	−498123.	+	498123.i	−1.46555	+	1.46555i
$584$		−	307384.i		−	0.901270i
$585$		0			0
$586$	296422.			0.863207
$587$	192901.	+	192901.i	0.559833	+	0.559833i	0.929260	−	0.369427i	$-0.120446\pi$
$587$	192901.	+	192901.i	0.559833	+	0.559833i	−0.369427	+	0.929260i	$0.620446\pi$
$588$		0			0
$589$		−	480291.i		−	1.38444i
$590$	−511.170	−	511.170i	−0.00146846	−	0.00146846i
$591$		0			0
$592$	−79865.1	+	79865.1i	−0.227884	+	0.227884i
$593$	23881.7	−	23881.7i	0.0679133	−	0.0679133i	−0.672334	−	0.740248i	$-0.734708\pi$
$593$	23881.7	−	23881.7i	0.0679133	−	0.0679133i	0.740248	+	0.672334i	$0.234708\pi$
$594$		0			0
$595$		−	63544.1i		−	0.179490i
$596$	143277.	−	143277.i	0.403351	−	0.403351i
$597$		0			0
$598$	6169.27	+	45469.8i	0.0172517	+	0.127151i
$599$	−328881.			−0.916610			−0.458305	−	0.888795i	$-0.651543\pi$
$599$	−328881.			−0.916610			−0.458305	+	0.888795i	$0.651543\pi$
$600$		0			0
$601$	289076.			0.800319			0.400159	−	0.916446i	$-0.368955\pi$
$601$	289076.			0.800319			0.400159	+	0.916446i	$0.368955\pi$
$602$		−	120688.i		−	0.333019i
$603$		0			0
$604$	−142434.	−	142434.i	−0.390428	−	0.390428i
$605$	−122223.	+	122223.i	−0.333919	+	0.333919i
$606$		0			0
$607$	−27331.5			−0.0741797			−0.0370899	−	0.999312i	$-0.511809\pi$
$607$	−27331.5			−0.0741797			−0.0370899	+	0.999312i	$0.511809\pi$
$608$			537018.i			1.45272i
$609$		0			0
$610$		−	21312.2i		−	0.0572756i
$611$	6873.97	+	50663.6i	0.0184130	+	0.135711i
$612$		0			0
$613$	−91286.9	−	91286.9i	−0.242934	−	0.242934i	0.575129	−	0.818063i	$-0.304952\pi$
$613$	−91286.9	−	91286.9i	−0.242934	−	0.242934i	−0.818063	+	0.575129i	$0.804952\pi$
$614$	−164412.			−0.436111
$615$		0			0
$616$	621030.	+	621030.i	1.63663	+	1.63663i
$617$	−308320.	−	308320.i	−0.809901	−	0.809901i	0.174718	−	0.984619i	$-0.444099\pi$
$617$	−308320.	−	308320.i	−0.809901	−	0.809901i	−0.984619	+	0.174718i	$0.944099\pi$
$618$		0			0
$619$	−133583.	+	133583.i	−0.348634	+	0.348634i	−0.859601	−	0.510967i	$-0.829288\pi$
$619$	−133583.	+	133583.i	−0.348634	+	0.348634i	0.510967	+	0.859601i	$0.329288\pi$
$620$	−35317.2			−0.0918763
$621$		0			0
$622$	−204268.	+	204268.i	−0.527981	+	0.527981i
$623$			864711.i			2.22790i
$624$		0			0
$625$	−336606.			−0.861711
$626$	181906.	+	181906.i	0.464192	+	0.464192i
$627$		0			0
$628$			347234.i			0.880447i
$629$	249083.	+	249083.i	0.629568	+	0.629568i
$630$		0			0
$631$	−170966.	+	170966.i	−0.429389	+	0.429389i	−0.888420	−	0.459031i	$-0.848197\pi$
$631$	−170966.	+	170966.i	−0.429389	+	0.429389i	0.459031	+	0.888420i	$0.348197\pi$
$632$	−335539.	+	335539.i	−0.840057	+	0.840057i
$633$		0			0
$634$			186334.i			0.463567i
$635$	−49238.5	+	49238.5i	−0.122112	+	0.122112i
$636$		0			0
$637$	−161053.	−	122571.i	−0.396909	−	0.302072i
$638$	496324.			1.21934
$639$		0			0
$640$	24154.6			0.0589711
$641$		−	533276.i		−	1.29788i	−0.760838	−	0.648942i	$-0.775212\pi$
$641$		−	533276.i		−	1.29788i	0.760838	−	0.648942i	$-0.224788\pi$
$642$		0			0
$643$	165350.	+	165350.i	0.399929	+	0.399929i	0.878208	−	0.478279i	$-0.158739\pi$
$643$	165350.	+	165350.i	0.399929	+	0.399929i	−0.478279	+	0.878208i	$0.658739\pi$
$644$	−32808.5	+	32808.5i	−0.0791069	+	0.0791069i
$645$		0			0
$646$	−326729.			−0.782928
$647$		−	422965.i		−	1.01041i	−0.863000	−	0.505203i	$-0.831417\pi$
$647$		−	422965.i		−	1.01041i	0.863000	−	0.505203i	$-0.168583\pi$
$648$		0			0
$649$		−	10221.9i		−	0.0242686i
$650$	−38194.4	−	281506.i	−0.0904008	−	0.666287i
$651$		0			0
$652$	237486.	+	237486.i	0.558653	+	0.558653i
$653$	12686.1			0.0297510			0.0148755	−	0.999889i	$-0.495265\pi$
$653$	12686.1			0.0297510			0.0148755	+	0.999889i	$0.495265\pi$
$654$		0			0
$655$	−74248.2	−	74248.2i	−0.173063	−	0.173063i
$656$	116114.	+	116114.i	0.269822	+	0.269822i
$657$		0			0
$658$	36210.2	−	36210.2i	0.0836333	−	0.0836333i
$659$	−504025.			−1.16060			−0.580298	−	0.814404i	$-0.697064\pi$
$659$	−504025.			−1.16060			−0.580298	+	0.814404i	$0.697064\pi$
$660$		0			0
$661$	110698.	−	110698.i	0.253359	−	0.253359i	−0.568987	−	0.822346i	$-0.692665\pi$
$661$	110698.	−	110698.i	0.253359	−	0.253359i	0.822346	+	0.568987i	$0.192665\pi$
$662$		−	166895.i		−	0.380827i
$663$		0			0
$664$	−15537.7			−0.0352412
$665$	−135705.	−	135705.i	−0.306869	−	0.306869i
$666$		0			0
$667$			78413.1i			0.176253i
$668$	51123.3	+	51123.3i	0.114569	+	0.114569i
$669$		0			0
$670$	−1370.17	+	1370.17i	−0.00305228	+	0.00305228i
$671$	213092.	−	213092.i	0.473285	−	0.473285i
$672$		0			0
$673$			629694.i			1.39027i	0.718879	+	0.695136i	$0.244656\pi$
$673$			629694.i			1.39027i	−0.718879	+	0.695136i	$0.755344\pi$
$674$	−194187.	+	194187.i	−0.427465	+	0.427465i
$675$		0			0
$676$	−221274.	+	61170.5i	−0.484215	+	0.133859i
$677$	758327.			1.65455			0.827273	−	0.561800i	$-0.189891\pi$
$677$	758327.			1.65455			0.827273	+	0.561800i	$0.189891\pi$
$678$		0			0
$679$	−845100.			−1.83303
$680$			71848.7i			0.155382i
$681$		0			0
$682$	349781.	+	349781.i	0.752018	+	0.752018i
$683$	−49915.7	+	49915.7i	−0.107003	+	0.107003i	−0.758581	−	0.651578i	$-0.774107\pi$
$683$	−49915.7	+	49915.7i	−0.107003	+	0.107003i	0.651578	+	0.758581i	$0.274107\pi$
$684$		0			0
$685$	−68126.8			−0.145190
$686$		−	203702.i		−	0.432859i
$687$		0			0
$688$			44763.1i			0.0945679i
$689$	−334029.	+	438899.i	−0.703632	+	0.924541i
$690$		0			0
$691$	154157.	+	154157.i	0.322855	+	0.322855i	0.849861	−	0.527006i	$-0.176686\pi$
$691$	154157.	+	154157.i	0.322855	+	0.322855i	−0.527006	+	0.849861i	$0.676686\pi$
$692$	160492.			0.335152
$693$		0			0
$694$	−228468.	−	228468.i	−0.474359	−	0.474359i
$695$	−23481.4	−	23481.4i	−0.0486132	−	0.0486132i
$696$		0			0
$697$	362135.	−	362135.i	0.745427	−	0.745427i
$698$	−115424.			−0.236911
$699$		0			0
$700$	203119.	−	203119.i	0.414529	−	0.414529i
$701$			529356.i			1.07724i	0.842549	+	0.538620i	$0.181054\pi$
$701$			529356.i			1.07724i	−0.842549	+	0.538620i	$0.818946\pi$
$702$		0			0
$703$	1.06388e6			2.15270
$704$	−544411.	−	544411.i	−1.09845	−	1.09845i
$705$		0			0
$706$		−	224156.i		−	0.449718i
$707$	39174.7	+	39174.7i	0.0783731	+	0.0783731i
$708$		0			0
$709$	−305608.	+	305608.i	−0.607957	+	0.607957i	−0.942412	−	0.334455i	$-0.891448\pi$
$709$	−305608.	+	305608.i	−0.607957	+	0.607957i	0.334455	+	0.942412i	$0.391448\pi$
$710$	−24917.8	+	24917.8i	−0.0494302	+	0.0494302i
$711$		0			0
$712$		−	977721.i		−	1.92866i
$713$	−55261.1	+	55261.1i	−0.108703	+	0.108703i
$714$		0			0
$715$	−119517.	+	157039.i	−0.233785	+	0.307182i
$716$	169024.			0.329703
$717$		0			0
$718$	−546450.			−1.05999
$719$		−	559312.i		−	1.08192i	−0.841047	−	0.540962i	$-0.818060\pi$
$719$		−	559312.i		−	1.08192i	0.841047	−	0.540962i	$-0.181940\pi$
$720$		0			0
$721$	−478171.	−	478171.i	−0.919842	−	0.919842i
$722$	−437741.	+	437741.i	−0.839737	+	0.839737i
$723$		0			0
$724$	308366.			0.588287
$725$		−	485460.i		−	0.923587i
$726$		0			0
$727$		−	39892.6i		−	0.0754785i	−0.999288	−	0.0377392i	$-0.987984\pi$
$727$		−	39892.6i		−	0.0754785i	0.999288	−	0.0377392i	$-0.0120156\pi$
$728$	547193.	+	416448.i	1.03247	+	0.785774i
$729$		0			0
$730$	−48916.6	−	48916.6i	−0.0917932	−	0.0917932i
$731$	139607.			0.261259
$732$		0			0
$733$	−218804.	−	218804.i	−0.407237	−	0.407237i	0.473537	−	0.880774i	$-0.342977\pi$
$733$	−218804.	−	218804.i	−0.407237	−	0.407237i	−0.880774	+	0.473537i	$0.842977\pi$
$734$	−80687.9	−	80687.9i	−0.149767	−	0.149767i
$735$		0			0
$736$	61788.0	−	61788.0i	0.114064	−	0.114064i
$737$	−27399.5			−0.0504437
$738$		0			0
$739$	516317.	−	516317.i	0.945425	−	0.945425i	−0.0531607	−	0.998586i	$-0.516930\pi$
$739$	516317.	−	516317.i	0.945425	−	0.945425i	0.998586	+	0.0531607i	$0.0169296\pi$
$740$		−	78230.6i		−	0.142861i
$741$		0			0
$742$	552426.			1.00338
$743$	664866.	+	664866.i	1.20436	+	1.20436i	0.972827	+	0.231534i	$0.0743745\pi$
$743$	664866.	+	664866.i	1.20436	+	1.20436i	0.231534	+	0.972827i	$0.425625\pi$
$744$		0			0
$745$		−	136374.i		−	0.245708i
$746$	−395940.	−	395940.i	−0.711461	−	0.711461i
$747$		0			0
$748$	−240219.	+	240219.i	−0.429343	+	0.429343i
$749$	−257187.	+	257187.i	−0.458444	+	0.458444i
$750$		0			0
$751$		−	1.02624e6i		−	1.81956i	−0.415087	−	0.909782i	$-0.636249\pi$
$751$		−	1.02624e6i		−	1.81956i	0.415087	−	0.909782i	$-0.363751\pi$
$752$	−13430.4	+	13430.4i	−0.0237495	+	0.0237495i
$753$		0			0
$754$	385069.	−	52245.6i	0.677323	−	0.0918982i
$755$	−135572.			−0.237835
$756$		0			0
$757$	727963.			1.27033			0.635167	−	0.772375i	$-0.280931\pi$
$757$	727963.			1.27033			0.635167	+	0.772375i	$0.280931\pi$
$758$		−	132153.i		−	0.230005i
$759$		0			0
$760$	153440.	+	153440.i	0.265652	+	0.265652i
$761$	−121468.	+	121468.i	−0.209746	+	0.209746i	−0.804160	−	0.594413i	$-0.797384\pi$
$761$	−121468.	+	121468.i	−0.209746	+	0.209746i	0.594413	+	0.804160i	$0.297384\pi$
$762$		0			0
$763$	−484363.			−0.831997
$764$		−	404579.i		−	0.693134i
$765$		0			0
$766$			819088.i			1.39596i
$767$	−1076.01	−	7930.60i	−0.00182906	−	0.0134808i
$768$		0			0
$769$	78988.3	+	78988.3i	0.133570	+	0.133570i	0.770731	−	0.637161i	$-0.219891\pi$
$769$	78988.3	+	78988.3i	0.133570	+	0.133570i	−0.637161	+	0.770731i	$0.719891\pi$
$770$	197660.			0.333378
$771$		0			0
$772$	−101692.	−	101692.i	−0.170628	−	0.170628i
$773$	167987.	+	167987.i	0.281136	+	0.281136i	0.833562	−	0.552426i	$-0.186298\pi$
$773$	167987.	+	167987.i	0.281136	+	0.281136i	−0.552426	+	0.833562i	$0.686298\pi$
$774$		0			0
$775$	342125.	−	342125.i	0.569615	−	0.569615i
$776$	955547.			1.58682
$777$		0			0
$778$	187268.	−	187268.i	0.309389	−	0.309389i
$779$		−	1.54676e6i		−	2.54887i
$780$		0			0
$781$	−498285.			−0.816912
$782$	37592.6	+	37592.6i	0.0614736	+	0.0614736i
$783$		0			0
$784$		−	75186.1i		−	0.122322i
$785$	165252.	+	165252.i	0.268169	+	0.268169i
$786$		0			0
$787$	−515211.	+	515211.i	−0.831833	+	0.831833i	−0.987767	−	0.155935i	$-0.950161\pi$
$787$	−515211.	+	515211.i	−0.831833	+	0.831833i	0.155935	+	0.987767i	$0.450161\pi$
$788$	236847.	−	236847.i	0.381430	−	0.381430i
$789$		0			0
$790$			106794.i			0.171117i
$791$	411283.	−	411283.i	0.657336	−	0.657336i
$792$		0			0
$793$	142894.	−	187757.i	0.227232	−	0.298572i
$794$	502711.			0.797403
$795$		0			0
$796$	411457.			0.649379
$797$		−	1.07630e6i		−	1.69441i	−0.531268	−	0.847204i	$-0.678284\pi$
$797$		−	1.07630e6i		−	1.69441i	0.531268	−	0.847204i	$-0.321716\pi$
$798$		0			0
$799$	41886.7	+	41886.7i	0.0656118	+	0.0656118i
$800$	−382533.	+	382533.i	−0.597708	+	0.597708i
$801$		0			0
$802$	−286784.			−0.445868
$803$		−	978193.i		−	1.51703i
$804$		0			0
$805$			31227.8i			0.0481892i
$806$	308194.	+	234555.i	0.474411	+	0.361056i
$807$		0			0
$808$	−44294.5	−	44294.5i	−0.0678465	−	0.0678465i
$809$	−1.19149e6			−1.82051			−0.910254	−	0.414051i	$-0.864114\pi$
$809$	−1.19149e6			−1.82051			−0.910254	+	0.414051i	$0.864114\pi$
$810$		0			0
$811$	34214.2	+	34214.2i	0.0520193	+	0.0520193i	0.732638	−	0.680619i	$-0.238289\pi$
$811$	34214.2	+	34214.2i	0.0520193	+	0.0520193i	−0.680619	+	0.732638i	$0.738289\pi$
$812$	277844.	+	277844.i	0.421395	+	0.421395i
$813$		0			0
$814$	−774795.	+	774795.i	−1.16933	+	1.16933i
$815$	226044.			0.340312
$816$		0			0
$817$	298145.	−	298145.i	0.446667	−	0.446667i
$818$		−	36786.6i		−	0.0549773i
$819$		0			0
$820$	−113738.			−0.169152
$821$	−223368.	−	223368.i	−0.331386	−	0.331386i	0.521727	−	0.853113i	$-0.325288\pi$
$821$	−223368.	−	223368.i	−0.331386	−	0.331386i	−0.853113	+	0.521727i	$0.825288\pi$
$822$		0			0
$823$		−	88095.9i		−	0.130064i	−0.997883	−	0.0650318i	$-0.979285\pi$
$823$		−	88095.9i		−	0.130064i	0.997883	−	0.0650318i	$-0.0207149\pi$
$824$	540664.	+	540664.i	0.796293	+	0.796293i
$825$		0			0
$826$	−5668.15	+	5668.15i	−0.00830771	+	0.00830771i
$827$	245737.	−	245737.i	0.359302	−	0.359302i	−0.504254	−	0.863556i	$-0.668232\pi$
$827$	245737.	−	245737.i	0.359302	−	0.359302i	0.863556	+	0.504254i	$0.168232\pi$
$828$		0			0
$829$			662724.i			0.964326i	0.876082	+	0.482163i	$0.160149\pi$
$829$			662724.i			0.964326i	−0.876082	+	0.482163i	$0.839851\pi$
$830$	−2472.65	+	2472.65i	−0.00358927	+	0.00358927i
$831$		0			0
$832$	−479684.	−	365069.i	−0.692961	−	0.527386i
$833$	−234490.			−0.337935
$834$		0			0
$835$	48660.2			0.0697913
$836$			1.02603e6i			1.46807i
$837$		0			0
$838$	−653318.	−	653318.i	−0.930329	−	0.930329i
$839$	−177547.	+	177547.i	−0.252226	+	0.252226i	−0.821883	−	0.569657i	$-0.807076\pi$
$839$	−177547.	+	177547.i	−0.252226	+	0.252226i	0.569657	+	0.821883i	$0.307076\pi$
$840$		0			0
$841$	−43225.7			−0.0611154
$842$		−	592279.i		−	0.835415i
$843$		0			0
$844$			91738.6i			0.128786i
$845$	−76195.1	+	134418.i	−0.106712	+	0.188255i
$846$		0			0
$847$	1.35527e6	+	1.35527e6i	1.88912	+	1.88912i
$848$	−204895.			−0.284932
$849$		0			0
$850$	−232738.	−	232738.i	−0.322129	−	0.322129i
$851$	−122408.	−	122408.i	−0.169025	−	0.169025i
$852$		0			0
$853$	898.292	−	898.292i	0.00123458	−	0.00123458i	−0.706489	−	0.707724i	$-0.749722\pi$
$853$	898.292	−	898.292i	0.00123458	−	0.00123458i	0.707724	+	0.706489i	$0.249722\pi$
$854$	−236323.			−0.324033
$855$		0			0
$856$	290800.	−	290800.i	0.396868	−	0.396868i
$857$		−	284851.i		−	0.387843i	−0.981017	−	0.193921i	$-0.937879\pi$
$857$		−	284851.i		−	0.387843i	0.981017	−	0.193921i	$-0.0621206\pi$
$858$		0			0
$859$	−302470.			−0.409917			−0.204959	−	0.978771i	$-0.565706\pi$
$859$	−302470.			−0.409917			−0.204959	+	0.978771i	$0.565706\pi$
$860$	−21923.5	−	21923.5i	−0.0296424	−	0.0296424i
$861$		0			0
$862$			444716.i			0.598506i
$863$	577769.	+	577769.i	0.775769	+	0.775769i	0.979108	−	0.203340i	$-0.0651795\pi$
$863$	577769.	+	577769.i	0.775769	+	0.775769i	−0.203340	+	0.979108i	$0.565180\pi$
$864$		0			0
$865$	76379.8	−	76379.8i	0.102081	−	0.102081i
$866$	−54636.6	+	54636.6i	−0.0728530	+	0.0728530i
$867$		0			0
$868$			391618.i			0.519785i
$869$	−1.06779e6	+	1.06779e6i	−1.41399	+	1.41399i
$870$		0			0
$871$	−21257.6	+	2884.21i	−0.0280207	+	0.00380181i
$872$	547665.			0.720248
$873$		0			0
$874$	160566.			0.210199
$875$		−	396164.i		−	0.517439i
$876$		0			0
$877$	−457143.	−	457143.i	−0.594365	−	0.594365i	0.344443	−	0.938807i	$-0.388068\pi$
$877$	−457143.	−	457143.i	−0.594365	−	0.594365i	−0.938807	+	0.344443i	$0.888068\pi$
$878$	−63048.2	+	63048.2i	−0.0817868	+	0.0817868i
$879$		0			0
$880$	−73312.2			−0.0946697
$881$			438514.i			0.564979i	0.959270	+	0.282489i	$0.0911602\pi$
$881$			438514.i			0.564979i	−0.959270	+	0.282489i	$0.908840\pi$
$882$		0			0
$883$		−	215411.i		−	0.276278i	−0.990413	−	0.138139i	$-0.955888\pi$
$883$		−	215411.i		−	0.276278i	0.990413	−	0.138139i	$-0.0441121\pi$
$884$	−161085.	+	211658.i	−0.206135	+	0.270851i
$885$		0			0
$886$	48524.2	+	48524.2i	0.0618146	+	0.0618146i
$887$	−883724.			−1.12323			−0.561616	−	0.827398i	$-0.689820\pi$
$887$	−883724.			−1.12323			−0.561616	+	0.827398i	$0.689820\pi$
$888$		0			0
$889$	545985.	+	545985.i	0.690839	+	0.690839i
$890$	−155593.	−	155593.i	−0.196431	−	0.196431i
$891$		0			0
$892$	223435.	−	223435.i	0.280815	−	0.280815i
$893$	178907.			0.224349
$894$		0			0
$895$	80440.4	−	80440.4i	0.100422	−	0.100422i
$896$		−	267840.i		−	0.333626i
$897$		0			0
$898$	856037.			1.06155
$899$	467989.	+	467989.i	0.579050	+	0.579050i
$900$		0			0
$901$			639026.i			0.787171i
$902$	1.12646e6	+	1.12646e6i	1.38453	+	1.38453i
$903$		0			0
$904$	−465034.	+	465034.i	−0.569046	+	0.569046i
$905$	146755.	−	146755.i	0.179182	−	0.179182i
$906$		0			0
$907$		−	628760.i		−	0.764311i	−0.924098	−	0.382156i	$-0.875182\pi$
$907$		−	628760.i		−	0.764311i	0.924098	−	0.382156i	$-0.124818\pi$
$908$	187101.	−	187101.i	0.226937	−	0.226937i
$909$		0			0
$910$	153352.	−	20806.7i	0.185186	−	0.0251258i
$911$	−471997.			−0.568725			−0.284363	−	0.958717i	$-0.591782\pi$
$911$	−471997.			−0.568725			−0.284363	+	0.958717i	$0.591782\pi$
$912$		0			0
$913$	−49445.9			−0.0593183
$914$			338841.i			0.405605i
$915$		0			0
$916$	−227124.	−	227124.i	−0.270690	−	0.270690i
$917$	−823307.	+	823307.i	−0.979091	+	0.979091i
$918$		0			0
$919$	−738465.			−0.874377			−0.437189	−	0.899370i	$-0.644026\pi$
$919$	−738465.			−0.874377			−0.437189	+	0.899370i	$0.644026\pi$
$920$		−	35309.0i		−	0.0417166i
$921$		0			0
$922$		−	424444.i		−	0.499296i
$923$	−386589.	+	52451.9i	−0.453781	+	0.0615684i
$924$		0			0
$925$	757836.	+	757836.i	0.885710	+	0.885710i
$926$	618826.			0.721683
$927$		0			0
$928$	−523262.	−	523262.i	−0.607608	−	0.607608i
$929$	−945453.	−	945453.i	−1.09549	−	1.09549i	−0.994931	−	0.100560i	$-0.967937\pi$
$929$	−945453.	−	945453.i	−1.09549	−	1.09549i	−0.100560	−	0.994931i	$-0.532063\pi$
$930$		0			0
$931$	−500777.	+	500777.i	−0.577757	+	0.577757i
$932$	−232642.			−0.267828
$933$		0			0
$934$	79808.1	−	79808.1i	0.0914857	−	0.0914857i
$935$			228646.i			0.261541i
$936$		0			0
$937$	−759558.			−0.865130			−0.432565	−	0.901603i	$-0.642391\pi$
$937$	−759558.			−0.865130			−0.432565	+	0.901603i	$0.642391\pi$
$938$	15193.2	+	15193.2i	0.0172681	+	0.0172681i
$939$		0			0
$940$		−	13155.5i		−	0.0148886i
$941$	603301.	+	603301.i	0.681326	+	0.681326i	0.960299	−	0.278973i	$-0.0899940\pi$
$941$	603301.	+	603301.i	0.681326	+	0.681326i	−0.278973	+	0.960299i	$0.589994\pi$
$942$		0			0
$943$	−177966.	+	177966.i	−0.200131	+	0.200131i
$944$	2102.32	−	2102.32i	0.00235915	−	0.00235915i
$945$		0			0
$946$			434260.i			0.485253i
$947$	−706138.	+	706138.i	−0.787389	+	0.787389i	−0.981066	−	0.193676i	$-0.937959\pi$
$947$	−706138.	+	706138.i	−0.787389	+	0.787389i	0.193676	+	0.981066i	$0.437959\pi$
$948$		0			0
$949$	−102969.	−	758922.i	−0.114334	−	0.842684i
$950$	−994073.			−1.10147
$951$		0			0
$952$	796700.			0.879065
$953$		−	633625.i		−	0.697664i	−0.937185	−	0.348832i	$-0.886578\pi$
$953$		−	633625.i		−	0.697664i	0.937185	−	0.348832i	$-0.113422\pi$
$954$		0			0
$955$	−192544.	−	192544.i	−0.211117	−	0.211117i
$956$	−371932.	+	371932.i	−0.406957	+	0.406957i
$957$		0			0
$958$	1.00626e6			1.09642
$959$			755430.i			0.821405i
$960$		0			0
$961$		−	263897.i		−	0.285751i
$962$	−519559.	+	682676.i	−0.561416	+	0.737674i
$963$		0			0
$964$	−295866.	−	295866.i	−0.318377	−	0.318377i
$965$	−96792.4			−0.103941
$966$		0			0
$967$	664366.	+	664366.i	0.710484	+	0.710484i	0.966636	−	0.256152i	$-0.0824549\pi$
$967$	664366.	+	664366.i	0.710484	+	0.710484i	−0.256152	+	0.966636i	$0.582455\pi$
$968$	−1.53240e6	−	1.53240e6i	−1.63539	−	1.63539i
$969$		0			0
$970$	152064.	−	152064.i	0.161616	−	0.161616i
$971$	72281.4			0.0766634			0.0383317	−	0.999265i	$-0.487796\pi$
$971$	72281.4			0.0766634			0.0383317	+	0.999265i	$0.487796\pi$
$972$		0			0
$973$	−260375.	+	260375.i	−0.275026	+	0.275026i
$974$			195913.i			0.206512i
$975$		0			0
$976$	87652.3			0.0920161
$977$	943495.	+	943495.i	0.988440	+	0.988440i	0.999934	−	0.0114940i	$-0.00365872\pi$
$977$	943495.	+	943495.i	0.988440	+	0.988440i	−0.0114940	+	0.999934i	$0.503659\pi$
$978$		0			0
$979$		−	3.11142e6i		−	3.24634i
$980$	36823.7	+	36823.7i	0.0383420	+	0.0383420i
$981$		0			0
$982$	−357661.	+	357661.i	−0.370893	+	0.370893i
$983$	718661.	−	718661.i	0.743733	−	0.743733i	−0.229562	−	0.973294i	$-0.573729\pi$
$983$	718661.	−	718661.i	0.743733	−	0.743733i	0.973294	+	0.229562i	$0.0737292\pi$
$984$		0			0
$985$		−	225435.i		−	0.232354i
$986$	318360.	−	318360.i	0.327464	−	0.327464i
$987$		0			0
$988$	108005.	+	796033.i	0.110644	+	0.815487i
$989$	−68607.7			−0.0701424
$990$		0			0
$991$	−972544.			−0.990289			−0.495144	−	0.868811i	$-0.664885\pi$
$991$	−972544.			−0.990289			−0.495144	+	0.868811i	$0.664885\pi$
$992$		−	737531.i		−	0.749475i
$993$		0			0
$994$	276303.	+	276303.i	0.279648	+	0.279648i
$995$	195817.	−	195817.i	0.197789	−	0.197789i
$996$		0			0
$997$	−1.40966e6			−1.41816			−0.709080	−	0.705128i	$-0.750889\pi$
$997$	−1.40966e6			−1.41816			−0.709080	+	0.705128i	$0.750889\pi$
$998$		−	861933.i		−	0.865391i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.73.7		20
3.2	odd	2		39.5.g.a.34.4	yes	20
13.5	odd	4	inner	117.5.j.b.109.7		20
39.5	even	4		39.5.g.a.31.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.4	✓	20	39.5	even	4
39.5.g.a.34.4	yes	20	3.2	odd	2
117.5.j.b.73.7		20	1.1	even	1	trivial
117.5.j.b.109.7		20	13.5	odd	4	inner

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.99524 + 1.99524i) q^{2} -8.03802i q^{4} +(-3.82538 - 3.82538i) q^{5} +(-42.4180 + 42.4180i) q^{7} +(47.9617 - 47.9617i) q^{8} -15.2651i q^{10} +(152.629 - 152.629i) q^{11} +(102.350 - 134.483i) q^{13} -169.268 q^{14} +62.7818 q^{16} -195.803i q^{17} +(-418.159 - 418.159i) q^{19} +(-30.7485 + 30.7485i) q^{20} +609.065 q^{22} +96.2247i q^{23} -595.733i q^{25} +(472.537 - 64.1132i) q^{26} +(340.957 + 340.957i) q^{28} +814.896 q^{29} +(574.293 + 574.293i) q^{31} +(-642.122 - 642.122i) q^{32} +(390.675 - 390.675i) q^{34} +324.530 q^{35} +(-1272.11 + 1272.11i) q^{37} -1668.66i q^{38} -366.943 q^{40} +(1849.48 + 1849.48i) q^{41} +712.995i q^{43} +(-1226.84 - 1226.84i) q^{44} +(-191.991 + 191.991i) q^{46} +(-213.922 + 213.922i) q^{47} -1197.58i q^{49} +(1188.63 - 1188.63i) q^{50} +(-1080.97 - 822.688i) q^{52} -3263.61 q^{53} -1167.73 q^{55} +4068.88i q^{56} +(1625.91 + 1625.91i) q^{58} +(33.4862 - 33.4862i) q^{59} +1396.14 q^{61} +2291.70i q^{62} -3566.88i q^{64} +(-905.972 + 122.921i) q^{65} +(-89.7582 - 89.7582i) q^{67} -1573.87 q^{68} +(647.515 + 647.515i) q^{70} +(-1632.34 - 1632.34i) q^{71} +(3204.47 - 3204.47i) q^{73} -5076.32 q^{74} +(-3361.17 + 3361.17i) q^{76} +12948.5i q^{77} -6995.98 q^{79} +(-240.164 - 240.164i) q^{80} +7380.34i q^{82} +(-161.980 - 161.980i) q^{83} +(-749.022 + 749.022i) q^{85} +(-1422.60 + 1422.60i) q^{86} -14640.7i q^{88} +(10192.7 - 10192.7i) q^{89} +(1363.02 + 10046.0i) q^{91} +773.456 q^{92} -853.652 q^{94} +3199.23i q^{95} +(9961.57 + 9961.57i) q^{97} +(2389.45 - 2389.45i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 20 * q - 24 * q^5 - 24 * q^7 - 372 * q^11 - 224 * q^13 - 480 * q^14 - 2328 * q^16 - 840 * q^19 - 228 * q^20 + 3536 * q^22 + 828 * q^26 - 1984 * q^28 + 5064 * q^29 + 1712 * q^31 + 7260 * q^32 + 8040 * q^34 - 5976 * q^35 - 868 * q^37 + 6336 * q^40 + 6000 * q^41 - 20868 * q^44 - 1992 * q^46 - 1692 * q^47 - 8952 * q^50 - 12416 * q^52 + 18816 * q^53 - 3280 * q^55 + 1256 * q^58 + 6588 * q^59 - 5512 * q^61 - 33048 * q^65 - 4704 * q^67 + 28296 * q^68 - 31096 * q^70 - 9756 * q^71 + 13692 * q^73 + 28296 * q^74 - 17072 * q^76 - 5056 * q^79 - 42228 * q^80 - 24708 * q^83 - 4752 * q^85 + 77424 * q^86 - 36000 * q^89 + 19104 * q^91 - 12912 * q^92 + 112832 * q^94 + 50012 * q^97 - 11544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more