LMFDB - Embedded newform 117.5.j.b.73.6

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			73.6
Root			$-0.0868646 - 0.0868646i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.5.j.b.109.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.0868646 - 0.0868646i) q^{2} -15.9849i q^{4} +(14.6130 + 14.6130i) q^{5} +(21.4788 - 21.4788i) q^{7} +(-2.77836 + 2.77836i) q^{8} -2.53870i q^{10} +(-21.6960 + 21.6960i) q^{11} +(134.520 - 102.301i) q^{13} -3.73149 q^{14} -255.276 q^{16} -251.891i q^{17} +(133.428 + 133.428i) q^{19} +(233.587 - 233.587i) q^{20} +3.76923 q^{22} -1026.00i q^{23} -197.922i q^{25} +(-20.5713 - 2.79870i) q^{26} +(-343.336 - 343.336i) q^{28} +479.065 q^{29} +(359.729 + 359.729i) q^{31} +(66.6281 + 66.6281i) q^{32} +(-21.8804 + 21.8804i) q^{34} +627.737 q^{35} +(1221.26 - 1221.26i) q^{37} -23.1804i q^{38} -81.2001 q^{40} +(-381.553 - 381.553i) q^{41} -356.437i q^{43} +(346.809 + 346.809i) q^{44} +(-89.1229 + 89.1229i) q^{46} +(-2002.52 + 2002.52i) q^{47} +1478.33i q^{49} +(-17.1924 + 17.1924i) q^{50} +(-1635.27 - 2150.29i) q^{52} -1718.92 q^{53} -634.086 q^{55} +119.351i q^{56} +(-41.6137 - 41.6137i) q^{58} +(232.041 - 232.041i) q^{59} -1458.96 q^{61} -62.4953i q^{62} +4072.84i q^{64} +(3460.65 + 470.817i) q^{65} +(4507.42 + 4507.42i) q^{67} -4026.46 q^{68} +(-54.5281 - 54.5281i) q^{70} +(4295.61 + 4295.61i) q^{71} +(-6223.95 + 6223.95i) q^{73} -212.169 q^{74} +(2132.84 - 2132.84i) q^{76} +932.007i q^{77} -321.183 q^{79} +(-3730.34 - 3730.34i) q^{80} +66.2868i q^{82} +(8931.56 + 8931.56i) q^{83} +(3680.88 - 3680.88i) q^{85} +(-30.9617 + 30.9617i) q^{86} -120.558i q^{88} +(-5668.17 + 5668.17i) q^{89} +(692.027 - 5086.61i) q^{91} -16400.5 q^{92} +347.896 q^{94} +3899.57i q^{95} +(-3995.41 - 3995.41i) q^{97} +(128.414 - 128.414i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0868646	−	0.0868646i	−0.0217161	−	0.0217161i	0.696165	−	0.717881i	$-0.254888\pi$
$2$	−0.0868646	−	0.0868646i	−0.0217161	−	0.0217161i	−0.717881	+	0.696165i	$0.754888\pi$
$3$		0			0
$3$		0			0
$4$		−	15.9849i		−	0.999057i
$5$	14.6130	+	14.6130i	0.584519	+	0.584519i	0.936142	−	0.351623i	$-0.114370\pi$
$5$	14.6130	+	14.6130i	0.584519	+	0.584519i	−0.351623	+	0.936142i	$0.614370\pi$
$6$		0			0
$7$	21.4788	−	21.4788i	0.438342	−	0.438342i	−0.453112	−	0.891454i	$-0.649686\pi$
$7$	21.4788	−	21.4788i	0.438342	−	0.438342i	0.891454	+	0.453112i	$0.149686\pi$
$8$	−2.77836	+	2.77836i	−0.0434118	+	0.0434118i
$9$		0			0
$10$		−	2.53870i		−	0.0253870i
$11$	−21.6960	+	21.6960i	−0.179306	+	0.179306i	−0.791053	−	0.611747i	$-0.790467\pi$
$11$	−21.6960	+	21.6960i	−0.179306	+	0.179306i	0.611747	+	0.791053i	$0.290467\pi$
$12$		0			0
$13$	134.520	−	102.301i	0.795975	−	0.605329i
$13$	134.520	−	102.301i	0.795975	−	0.605329i
$14$	−3.73149			−0.0190382
$15$		0			0
$16$	−255.276			−0.997171
$17$		−	251.891i		−	0.871596i	−0.900045	−	0.435798i	$-0.856466\pi$
$17$		−	251.891i		−	0.871596i	0.900045	−	0.435798i	$-0.143534\pi$
$18$		0			0
$19$	133.428	+	133.428i	0.369608	+	0.369608i	0.867334	−	0.497726i	$-0.165832\pi$
$19$	133.428	+	133.428i	0.369608	+	0.369608i	−0.497726	+	0.867334i	$0.665832\pi$
$20$	233.587	−	233.587i	0.583968	−	0.583968i
$21$		0			0
$22$	3.76923			0.00778766
$23$		−	1026.00i		−	1.93950i	−0.244091	−	0.969752i	$-0.578490\pi$
$23$		−	1026.00i		−	1.93950i	0.244091	−	0.969752i	$-0.421510\pi$
$24$		0			0
$25$		−	197.922i		−	0.316675i
$26$	−20.5713	−	2.79870i	−0.0304309	−	0.00414009i
$27$		0			0
$28$	−343.336	−	343.336i	−0.437929	−	0.437929i
$29$	479.065			0.569637			0.284818	−	0.958581i	$-0.408067\pi$
$29$	479.065			0.569637			0.284818	+	0.958581i	$0.408067\pi$
$30$		0			0
$31$	359.729	+	359.729i	0.374327	+	0.374327i	0.869051	−	0.494723i	$-0.164731\pi$
$31$	359.729	+	359.729i	0.374327	+	0.374327i	−0.494723	+	0.869051i	$0.664731\pi$
$32$	66.6281	+	66.6281i	0.0650665	+	0.0650665i
$33$		0			0
$34$	−21.8804	+	21.8804i	−0.0189277	+	0.0189277i
$35$	627.737			0.512439
$36$		0			0
$37$	1221.26	−	1221.26i	0.892085	−	0.892085i	−0.102634	−	0.994719i	$-0.532727\pi$
$37$	1221.26	−	1221.26i	0.892085	−	0.892085i	0.994719	+	0.102634i	$0.0327272\pi$
$38$		−	23.1804i		−	0.0160529i
$39$		0			0
$40$	−81.2001			−0.0507500
$41$	−381.553	−	381.553i	−0.226980	−	0.226980i	0.584450	−	0.811430i	$-0.301310\pi$
$41$	−381.553	−	381.553i	−0.226980	−	0.226980i	−0.811430	+	0.584450i	$0.801310\pi$
$42$		0			0
$43$		−	356.437i		−	0.192773i	−0.995344	−	0.0963863i	$-0.969272\pi$
$43$		−	356.437i		−	0.192773i	0.995344	−	0.0963863i	$-0.0307284\pi$
$44$	346.809	+	346.809i	0.179137	+	0.179137i
$45$		0			0
$46$	−89.1229	+	89.1229i	−0.0421186	+	0.0421186i
$47$	−2002.52	+	2002.52i	−0.906526	+	0.906526i	−0.995990	−	0.0894638i	$-0.971485\pi$
$47$	−2002.52	+	2002.52i	−0.906526	+	0.906526i	0.0894638	+	0.995990i	$0.471485\pi$
$48$		0			0
$49$			1478.33i			0.615712i
$50$	−17.1924	+	17.1924i	−0.00687696	+	0.00687696i
$51$		0			0
$52$	−1635.27	−	2150.29i	−0.604758	−	0.795224i
$53$	−1718.92			−0.611933			−0.305967	−	0.952042i	$-0.598980\pi$
$53$	−1718.92			−0.611933			−0.305967	+	0.952042i	$0.598980\pi$
$54$		0			0
$55$	−634.086			−0.209615
$56$			119.351i			0.0380585i
$57$		0			0
$58$	−41.6137	−	41.6137i	−0.0123703	−	0.0123703i
$59$	232.041	−	232.041i	0.0666592	−	0.0666592i	−0.672991	−	0.739650i	$-0.734991\pi$
$59$	232.041	−	232.041i	0.0666592	−	0.0666592i	0.739650	+	0.672991i	$0.234991\pi$
$60$		0			0
$61$	−1458.96			−0.392088			−0.196044	−	0.980595i	$-0.562810\pi$
$61$	−1458.96			−0.392088			−0.196044	+	0.980595i	$0.562810\pi$
$62$		−	62.4953i		−	0.0162579i
$63$		0			0
$64$			4072.84i			0.994345i
$65$	3460.65	+	470.817i	0.819089	+	0.111436i
$66$		0			0
$67$	4507.42	+	4507.42i	1.00410	+	1.00410i	0.999992	+	0.00411288i	$0.00130917\pi$
$67$	4507.42	+	4507.42i	1.00410	+	1.00410i	0.00411288	+	0.999992i	$0.498691\pi$
$68$	−4026.46			−0.870774
$69$		0			0
$70$	−54.5281	−	54.5281i	−0.0111282	−	0.0111282i
$71$	4295.61	+	4295.61i	0.852134	+	0.852134i	0.990396	−	0.138262i	$-0.0441516\pi$
$71$	4295.61	+	4295.61i	0.852134	+	0.852134i	−0.138262	+	0.990396i	$0.544152\pi$
$72$		0			0
$73$	−6223.95	+	6223.95i	−1.16794	+	1.16794i	−0.185248	+	0.982692i	$0.559309\pi$
$73$	−6223.95	+	6223.95i	−1.16794	+	1.16794i	−0.982692	+	0.185248i	$0.940691\pi$
$74$	−212.169			−0.0387453
$75$		0			0
$76$	2132.84	−	2132.84i	0.369259	−	0.369259i
$77$			932.007i			0.157195i
$78$		0			0
$79$	−321.183			−0.0514634			−0.0257317	−	0.999669i	$-0.508192\pi$
$79$	−321.183			−0.0514634			−0.0257317	+	0.999669i	$0.508192\pi$
$80$	−3730.34	−	3730.34i	−0.582866	−	0.582866i
$81$		0			0
$82$			66.2868i			0.00985824i
$83$	8931.56	+	8931.56i	1.29650	+	1.29650i	0.930692	+	0.365804i	$0.119206\pi$
$83$	8931.56	+	8931.56i	1.29650	+	1.29650i	0.365804	+	0.930692i	$0.380794\pi$
$84$		0			0
$85$	3680.88	−	3680.88i	0.509464	−	0.509464i
$86$	−30.9617	+	30.9617i	−0.00418628	+	0.00418628i
$87$		0			0
$88$		−	120.558i		−	0.0155680i
$89$	−5668.17	+	5668.17i	−0.715588	+	0.715588i	−0.967698	−	0.252111i	$-0.918875\pi$
$89$	−5668.17	+	5668.17i	−0.715588	+	0.715588i	0.252111	+	0.967698i	$0.418875\pi$
$90$		0			0
$91$	692.027	−	5086.61i	0.0835681	−	0.614251i
$92$	−16400.5			−1.93768
$93$		0			0
$94$	347.896			0.0393725
$95$			3899.57i			0.432086i
$96$		0			0
$97$	−3995.41	−	3995.41i	−0.424637	−	0.424637i	0.462159	−	0.886797i	$-0.347075\pi$
$97$	−3995.41	−	3995.41i	−0.424637	−	0.424637i	−0.886797	+	0.462159i	$0.847075\pi$
$98$	128.414	−	128.414i	0.0133709	−	0.0133709i
$99$		0			0
$100$	−3163.77			−0.316377
$101$		−	17363.6i		−	1.70215i	−0.525043	−	0.851076i	$-0.675951\pi$
$101$		−	17363.6i		−	1.70215i	0.525043	−	0.851076i	$-0.324049\pi$
$102$		0			0
$103$		−	6664.68i		−	0.628210i	−0.949388	−	0.314105i	$-0.898296\pi$
$103$		−	6664.68i		−	0.628210i	0.949388	−	0.314105i	$-0.101704\pi$
$104$	−89.5162	+	657.971i	−0.00827628	+	0.0608332i
$105$		0			0
$106$	149.313	+	149.313i	0.0132888	+	0.0132888i
$107$	4317.96			0.377147			0.188574	−	0.982059i	$-0.439614\pi$
$107$	4317.96			0.377147			0.188574	+	0.982059i	$0.439614\pi$
$108$		0			0
$109$	315.337	+	315.337i	0.0265413	+	0.0265413i	0.720253	−	0.693712i	$-0.244026\pi$
$109$	315.337	+	315.337i	0.0265413	+	0.0265413i	−0.693712	+	0.720253i	$0.744026\pi$
$110$	55.0796	+	55.0796i	0.00455204	+	0.00455204i
$111$		0			0
$112$	−5483.01	+	5483.01i	−0.437102	+	0.437102i
$113$	22445.2			1.75779			0.878896	−	0.477014i	$-0.158281\pi$
$113$	22445.2			1.75779			0.878896	+	0.477014i	$0.158281\pi$
$114$		0			0
$115$	14992.9	−	14992.9i	1.13368	−	1.13368i
$116$		−	7657.80i		−	0.569099i
$117$		0			0
$118$	−40.3122			−0.00289516
$119$	−5410.31	−	5410.31i	−0.382057	−	0.382057i
$120$		0			0
$121$			13699.6i			0.935699i
$122$	126.732	+	126.732i	0.00851464	+	0.00851464i
$123$		0			0
$124$	5750.23	−	5750.23i	0.373974	−	0.373974i
$125$	12025.3	−	12025.3i	0.769622	−	0.769622i
$126$		0			0
$127$			20203.7i			1.25263i	0.779568	+	0.626317i	$0.215439\pi$
$127$			20203.7i			1.25263i	−0.779568	+	0.626317i	$0.784561\pi$
$128$	1419.84	−	1419.84i	0.0866599	−	0.0866599i
$129$		0			0
$130$	−259.711	−	341.505i	−0.0153675	−	0.0202074i
$131$	−11605.8			−0.676291			−0.338145	−	0.941094i	$-0.609800\pi$
$131$	−11605.8			−0.676291			−0.338145	+	0.941094i	$0.609800\pi$
$132$		0			0
$133$	5731.76			0.324029
$134$		−	783.071i		−	0.0436106i
$135$		0			0
$136$	699.843	+	699.843i	0.0378375	+	0.0378375i
$137$	17702.6	−	17702.6i	0.943182	−	0.943182i	−0.0552888	−	0.998470i	$-0.517608\pi$
$137$	17702.6	−	17702.6i	0.943182	−	0.943182i	0.998470	+	0.0552888i	$0.0176079\pi$
$138$		0			0
$139$	−23995.5			−1.24194			−0.620969	−	0.783835i	$-0.713261\pi$
$139$	−23995.5			−1.24194			−0.620969	+	0.783835i	$0.713261\pi$
$140$		−	10034.3i		−	0.511955i
$141$		0			0
$142$		−	746.272i		−	0.0370101i
$143$	−699.027	+	5138.06i	−0.0341839	+	0.251262i
$144$		0			0
$145$	7000.56	+	7000.56i	0.332963	+	0.332963i
$146$	1081.28			0.0507263
$147$		0			0
$148$	−19521.8	−	19521.8i	−0.891243	−	0.891243i
$149$	−1257.11	−	1257.11i	−0.0566241	−	0.0566241i	0.678228	−	0.734852i	$-0.262748\pi$
$149$	−1257.11	−	1257.11i	−0.0566241	−	0.0566241i	−0.734852	+	0.678228i	$0.762748\pi$
$150$		0			0
$151$	23761.6	−	23761.6i	1.04213	−	1.04213i	0.0430569	−	0.999073i	$-0.486290\pi$
$151$	23761.6	−	23761.6i	1.04213	−	1.04213i	0.999073	−	0.0430569i	$-0.0137097\pi$
$152$	−741.423			−0.0320907
$153$		0			0
$154$	80.9584	−	80.9584i	0.00341366	−	0.00341366i
$155$			10513.4i			0.437603i
$156$		0			0
$157$	3164.10			0.128366			0.0641831	−	0.997938i	$-0.479556\pi$
$157$	3164.10			0.128366			0.0641831	+	0.997938i	$0.479556\pi$
$158$	27.8994	+	27.8994i	0.00111759	+	0.00111759i
$159$		0			0
$160$			1947.27i			0.0760652i
$161$	−22037.2	−	22037.2i	−0.850167	−	0.850167i
$162$		0			0
$163$	−2979.83	+	2979.83i	−0.112154	+	0.112154i	−0.760957	−	0.648802i	$-0.775270\pi$
$163$	−2979.83	+	2979.83i	−0.112154	+	0.112154i	0.648802	+	0.760957i	$0.275270\pi$
$164$	−6099.08	+	6099.08i	−0.226765	+	0.226765i
$165$		0			0
$166$		−	1551.67i		−	0.0563098i
$167$	−30970.0	+	30970.0i	−1.11047	+	1.11047i	−0.117388	+	0.993086i	$0.537452\pi$
$167$	−30970.0	+	30970.0i	−1.11047	+	1.11047i	−0.993086	+	0.117388i	$0.962548\pi$
$168$		0			0
$169$	7630.15	−	27522.9i	0.267153	−	0.963654i
$170$	−639.476			−0.0221272
$171$		0			0
$172$	−5697.61			−0.192591
$173$			21213.4i			0.708792i	0.935095	+	0.354396i	$0.115314\pi$
$173$			21213.4i			0.708792i	−0.935095	+	0.354396i	$0.884686\pi$
$174$		0			0
$175$	−4251.12	−	4251.12i	−0.138812	−	0.138812i
$176$	5538.47	−	5538.47i	0.178799	−	0.178799i
$177$		0			0
$178$	984.727			0.0310796
$179$			46842.2i			1.46194i	0.682407	+	0.730972i	$0.260933\pi$
$179$			46842.2i			1.46194i	−0.682407	+	0.730972i	$0.739067\pi$
$180$		0			0
$181$			35937.8i			1.09697i	0.836161	+	0.548485i	$0.184795\pi$
$181$			35937.8i			1.09697i	−0.836161	+	0.548485i	$0.815205\pi$
$182$	−501.959	+	381.734i	−0.0151539	+	0.0115244i
$183$		0			0
$184$	2850.59	+	2850.59i	0.0841974	+	0.0841974i
$185$	35692.6			1.04288
$186$		0			0
$187$	5465.03	+	5465.03i	0.156282	+	0.156282i
$188$	32010.0	+	32010.0i	0.905671	+	0.905671i
$189$		0			0
$190$	338.735	−	338.735i	0.00938323	−	0.00938323i
$191$	−25060.0			−0.686933			−0.343466	−	0.939165i	$-0.611601\pi$
$191$	−25060.0			−0.686933			−0.343466	+	0.939165i	$0.611601\pi$
$192$		0			0
$193$	−31755.0	+	31755.0i	−0.852506	+	0.852506i	−0.990441	−	0.137935i	$-0.955954\pi$
$193$	−31755.0	+	31755.0i	−0.852506	+	0.852506i	0.137935	+	0.990441i	$0.455954\pi$
$194$			694.120i			0.0184430i
$195$		0			0
$196$	23630.9			0.615132
$197$	−23870.6	−	23870.6i	−0.615080	−	0.615080i	0.329186	−	0.944265i	$-0.393226\pi$
$197$	−23870.6	−	23870.6i	−0.615080	−	0.615080i	−0.944265	+	0.329186i	$0.893226\pi$
$198$		0			0
$199$			44314.1i			1.11901i	0.828826	+	0.559507i	$0.189009\pi$
$199$			44314.1i			1.11901i	−0.828826	+	0.559507i	$0.810991\pi$
$200$	549.898	+	549.898i	0.0137474	+	0.0137474i
$201$		0			0
$202$	−1508.29	+	1508.29i	−0.0369642	+	0.0369642i
$203$	10289.7	−	10289.7i	0.249696	−	0.249696i
$204$		0			0
$205$		−	11151.2i		−	0.265348i
$206$	−578.924	+	578.924i	−0.0136423	+	0.0136423i
$207$		0			0
$208$	−34339.7	+	26114.9i	−0.793724	+	0.603617i
$209$	−5789.73			−0.132546
$210$		0			0
$211$	41706.9			0.936791			0.468396	−	0.883519i	$-0.344832\pi$
$211$	41706.9			0.936791			0.468396	+	0.883519i	$0.344832\pi$
$212$			27476.8i			0.611356i
$213$		0			0
$214$	−375.078	−	375.078i	−0.00819019	−	0.00819019i
$215$	5208.60	−	5208.60i	0.112679	−	0.112679i
$216$		0			0
$217$	15453.0			0.328167
$218$		−	54.7833i		−	0.00115275i
$219$		0			0
$220$			10135.8i			0.209418i
$221$	−25768.6	−	33884.4i	−0.527603	−	0.693769i
$222$		0			0
$223$	−56698.2	−	56698.2i	−1.14014	−	1.14014i	−0.988423	−	0.151721i	$-0.951518\pi$
$223$	−56698.2	−	56698.2i	−1.14014	−	1.14014i	−0.151721	−	0.988423i	$-0.548482\pi$
$224$	2862.18			0.0570428
$225$		0			0
$226$	−1949.70	−	1949.70i	−0.0381725	−	0.0381725i
$227$	34882.4	+	34882.4i	0.676947	+	0.676947i	0.959308	−	0.282361i	$-0.0911177\pi$
$227$	34882.4	+	34882.4i	0.676947	+	0.676947i	−0.282361	+	0.959308i	$0.591118\pi$
$228$		0			0
$229$	−22893.1	+	22893.1i	−0.436550	+	0.436550i	−0.890849	−	0.454299i	$-0.849890\pi$
$229$	−22893.1	+	22893.1i	−0.436550	+	0.436550i	0.454299	+	0.890849i	$0.349890\pi$
$230$	−2604.70			−0.0492382
$231$		0			0
$232$	−1331.01	+	1331.01i	−0.0247290	+	0.0247290i
$233$			10250.9i			0.188822i	0.995533	+	0.0944108i	$0.0300967\pi$
$233$			10250.9i			0.188822i	−0.995533	+	0.0944108i	$0.969903\pi$
$234$		0			0
$235$	−58525.4			−1.05976
$236$	−3709.15	−	3709.15i	−0.0665963	−	0.0665963i
$237$		0			0
$238$			939.929i			0.0165936i
$239$	−73567.0	−	73567.0i	−1.28791	−	1.28791i	−0.936051	−	0.351863i	$-0.885548\pi$
$239$	−73567.0	−	73567.0i	−1.28791	−	1.28791i	−0.351863	−	0.936051i	$-0.614452\pi$
$240$		0			0
$241$	51482.8	−	51482.8i	0.886397	−	0.886397i	−0.107778	−	0.994175i	$-0.534374\pi$
$241$	51482.8	−	51482.8i	0.886397	−	0.886397i	0.994175	+	0.107778i	$0.0343736\pi$
$242$	1190.01	−	1190.01i	0.0203198	−	0.0203198i
$243$		0			0
$244$			23321.3i			0.391718i
$245$	−21602.7	+	21602.7i	−0.359895	+	0.359895i
$246$		0			0
$247$	31598.6	+	4298.95i	0.517933	+	0.0704642i
$248$	−1998.91			−0.0325004
$249$		0			0
$250$	−2089.15			−0.0334264
$251$			18687.0i			0.296614i	0.988941	+	0.148307i	$0.0473823\pi$
$251$			18687.0i			0.296614i	−0.988941	+	0.148307i	$0.952618\pi$
$252$		0			0
$253$	22260.1	+	22260.1i	0.347765	+	0.347765i
$254$	1754.99	−	1754.99i	0.0272024	−	0.0272024i
$255$		0			0
$256$	64918.8			0.990582
$257$			69245.4i			1.04839i	0.851597	+	0.524197i	$0.175634\pi$
$257$			69245.4i			1.04839i	−0.851597	+	0.524197i	$0.824366\pi$
$258$		0			0
$259$		−	52462.5i		−	0.782077i
$260$	7525.97	−	55318.2i	0.111331	−	0.818316i
$261$		0			0
$262$	1008.14	+	1008.14i	0.0146864	+	0.0146864i
$263$	87200.1			1.26068			0.630341	−	0.776318i	$-0.282915\pi$
$263$	87200.1			1.26068			0.630341	+	0.776318i	$0.282915\pi$
$264$		0			0
$265$	−25118.5	−	25118.5i	−0.357686	−	0.357686i
$266$	−497.887	−	497.887i	−0.00703667	−	0.00703667i
$267$		0			0
$268$	72050.8	−	72050.8i	1.00316	−	1.00316i
$269$	−19288.6			−0.266561			−0.133280	−	0.991078i	$-0.542551\pi$
$269$	−19288.6			−0.266561			−0.133280	+	0.991078i	$0.542551\pi$
$270$		0			0
$271$	−77613.1	+	77613.1i	−1.05681	+	1.05681i	−0.0585225	+	0.998286i	$0.518639\pi$
$271$	−77613.1	+	77613.1i	−1.05681	+	1.05681i	−0.998286	+	0.0585225i	$0.981361\pi$
$272$			64301.7i			0.869130i
$273$		0			0
$274$	−3075.45			−0.0409645
$275$	4294.12	+	4294.12i	0.0567817	+	0.0567817i
$276$		0			0
$277$		−	96547.1i		−	1.25829i	−0.777289	−	0.629144i	$-0.783406\pi$
$277$		−	96547.1i		−	1.25829i	0.777289	−	0.629144i	$-0.216594\pi$
$278$	2084.36	+	2084.36i	0.0269701	+	0.0269701i
$279$		0			0
$280$	−1744.08	+	1744.08i	−0.0222459	+	0.0222459i
$281$	59466.7	−	59466.7i	0.753115	−	0.753115i	−0.221945	−	0.975059i	$-0.571240\pi$
$281$	59466.7	−	59466.7i	0.753115	−	0.753115i	0.975059	+	0.221945i	$0.0712404\pi$
$282$		0			0
$283$		−	49885.9i		−	0.622880i	−0.950266	−	0.311440i	$-0.899189\pi$
$283$		−	49885.9i		−	0.622880i	0.950266	−	0.311440i	$-0.100811\pi$
$284$	68664.9	−	68664.9i	0.851330	−	0.851330i
$285$		0			0
$286$	507.036	−	385.595i	0.00619879	−	0.00471410i
$287$	−16390.6			−0.198989
$288$		0			0
$289$	20071.8			0.240321
$290$		−	1216.20i		−	0.0144614i
$291$		0			0
$292$	99489.3	+	99489.3i	1.16684	+	1.16684i
$293$	112153.	−	112153.i	1.30640	−	1.30640i	0.382407	−	0.923994i	$-0.375095\pi$
$293$	112153.	−	112153.i	1.30640	−	1.30640i	0.923994	−	0.382407i	$-0.124905\pi$
$294$		0			0
$295$	6781.61			0.0779271
$296$			6786.21i			0.0774540i
$297$		0			0
$298$			218.397i			0.00245931i
$299$	−104960.	−	138017.i	−1.17404	−	1.54380i
$300$		0			0
$301$	−7655.82	−	7655.82i	−0.0845004	−	0.0845004i
$302$	−4128.08			−0.0452621
$303$		0			0
$304$	−34061.1	−	34061.1i	−0.368562	−	0.368562i
$305$	−21319.7	−	21319.7i	−0.229183	−	0.229183i
$306$		0			0
$307$	−49954.0	+	49954.0i	−0.530022	+	0.530022i	−0.920579	−	0.390557i	$-0.872282\pi$
$307$	−49954.0	+	49954.0i	−0.530022	+	0.530022i	0.390557	+	0.920579i	$0.372282\pi$
$308$	14898.0			0.157046
$309$		0			0
$310$	913.243	−	913.243i	0.00950304	−	0.00950304i
$311$			50895.9i			0.526213i	0.964767	+	0.263107i	$0.0847471\pi$
$311$			50895.9i			0.526213i	−0.964767	+	0.263107i	$0.915253\pi$
$312$		0			0
$313$	−140281.			−1.43189			−0.715947	−	0.698154i	$-0.754005\pi$
$313$	−140281.			−1.43189			−0.715947	+	0.698154i	$0.754005\pi$
$314$	−274.848	−	274.848i	−0.00278762	−	0.00278762i
$315$		0			0
$316$			5134.08i			0.0514149i
$317$	−115302.	−	115302.i	−1.14741	−	1.14741i	−0.987061	−	0.160346i	$-0.948739\pi$
$317$	−115302.	−	115302.i	−1.14741	−	1.14741i	−0.160346	−	0.987061i	$-0.551261\pi$
$318$		0			0
$319$	−10393.8	+	10393.8i	−0.102139	+	0.102139i
$320$	−59516.3	+	59516.3i	−0.581214	+	0.581214i
$321$		0			0
$322$			3828.50i			0.0369247i
$323$	33609.4	−	33609.4i	0.322149	−	0.322149i
$324$		0			0
$325$	−20247.6	−	26624.4i	−0.191693	−	0.252066i
$326$	517.684			0.00487112
$327$		0			0
$328$	2120.18			0.0197072
$329$			86023.2i			0.794737i
$330$		0			0
$331$	84823.6	+	84823.6i	0.774213	+	0.774213i	0.978840	−	0.204627i	$-0.0655980\pi$
$331$	84823.6	+	84823.6i	0.774213	+	0.774213i	−0.204627	+	0.978840i	$0.565598\pi$
$332$	142770.	−	142770.i	1.29527	−	1.29527i
$333$		0			0
$334$	5380.40			0.0482304
$335$			131734.i			1.17384i
$336$		0			0
$337$		−	35747.4i		−	0.314764i	−0.987538	−	0.157382i	$-0.949695\pi$
$337$		−	35747.4i		−	0.314764i	0.987538	−	0.157382i	$-0.0503053\pi$
$338$	−3053.56	+	1727.98i	−0.0267284	+	0.0151253i
$339$		0			0
$340$	−58838.5	−	58838.5i	−0.508984	−	0.508984i
$341$	−15609.3			−0.134238
$342$		0			0
$343$	83323.1	+	83323.1i	0.708235	+	0.708235i
$344$	990.308	+	990.308i	0.00836861	+	0.00836861i
$345$		0			0
$346$	1842.70	−	1842.70i	0.0153922	−	0.0153922i
$347$	−60713.6			−0.504228			−0.252114	−	0.967697i	$-0.581126\pi$
$347$	−60713.6			−0.504228			−0.252114	+	0.967697i	$0.581126\pi$
$348$		0			0
$349$	43916.5	−	43916.5i	0.360559	−	0.360559i	−0.503459	−	0.864019i	$-0.667940\pi$
$349$	43916.5	−	43916.5i	0.360559	−	0.360559i	0.864019	+	0.503459i	$0.167940\pi$
$350$			738.544i			0.00602893i
$351$		0			0
$352$	−2891.13			−0.0233336
$353$	35106.8	+	35106.8i	0.281736	+	0.281736i	0.833801	−	0.552065i	$-0.186160\pi$
$353$	35106.8	+	35106.8i	0.281736	+	0.281736i	−0.552065	+	0.833801i	$0.686160\pi$
$354$		0			0
$355$			125543.i			0.996177i
$356$	90605.2	+	90605.2i	0.714913	+	0.714913i
$357$		0			0
$358$	4068.92	−	4068.92i	0.0317478	−	0.0317478i
$359$	−105453.	+	105453.i	−0.818219	+	0.818219i	−0.985850	−	0.167631i	$-0.946388\pi$
$359$	−105453.	+	105453.i	−0.818219	+	0.818219i	0.167631	+	0.985850i	$0.446388\pi$
$360$		0			0
$361$		−	94714.7i		−	0.726780i
$362$	3121.72	−	3121.72i	0.0238219	−	0.0238219i
$363$		0			0
$364$	−81309.0	−	11062.0i	−0.613672	−	0.0834893i
$365$	−181901.			−1.36537
$366$		0			0
$367$	23389.1			0.173653			0.0868263	−	0.996223i	$-0.472327\pi$
$367$	23389.1			0.173653			0.0868263	+	0.996223i	$0.472327\pi$
$368$			261913.i			1.93402i
$369$		0			0
$370$	−3100.42	−	3100.42i	−0.0226473	−	0.0226473i
$371$	−36920.3	+	36920.3i	−0.268236	+	0.268236i
$372$		0			0
$373$	175048.			1.25817			0.629084	−	0.777337i	$-0.283430\pi$
$373$	175048.			1.25817			0.629084	+	0.777337i	$0.283430\pi$
$374$		−	949.435i		−	0.00678769i
$375$		0			0
$376$		−	11127.4i		−	0.0787079i
$377$	64443.7	−	49008.6i	0.453417	−	0.344818i
$378$		0			0
$379$	949.057	+	949.057i	0.00660715	+	0.00660715i	0.710403	−	0.703795i	$-0.248513\pi$
$379$	949.057	+	949.057i	0.00660715	+	0.00660715i	−0.703795	+	0.710403i	$0.748513\pi$
$380$	62334.3			0.431678
$381$		0			0
$382$	2176.83	+	2176.83i	0.0149175	+	0.0149175i
$383$	126901.	+	126901.i	0.865105	+	0.865105i	0.991926	−	0.126821i	$-0.0404774\pi$
$383$	126901.	+	126901.i	0.865105	+	0.865105i	−0.126821	+	0.991926i	$0.540477\pi$
$384$		0			0
$385$	−13619.4	+	13619.4i	−0.0918832	+	0.0918832i
$386$	5516.77			0.0370263
$387$		0			0
$388$	−63866.3	+	63866.3i	−0.424237	+	0.424237i
$389$		−	13102.1i		−	0.0865849i	−0.999062	−	0.0432925i	$-0.986215\pi$
$389$		−	13102.1i		−	0.0865849i	0.999062	−	0.0432925i	$-0.0137847\pi$
$390$		0			0
$391$	−258440.			−1.69046
$392$	−4107.31	−	4107.31i	−0.0267292	−	0.0267292i
$393$		0			0
$394$			4147.02i			0.0267143i
$395$	−4693.44	−	4693.44i	−0.0300813	−	0.0300813i
$396$		0			0
$397$	157539.	−	157539.i	0.999554	−	0.999554i	−0.000446211	−	1.00000i	$-0.500142\pi$
$397$	157539.	−	157539.i	0.999554	−	0.999554i	1.00000		0.000446211i	$0.000142033\pi$
$398$	3849.32	−	3849.32i	0.0243007	−	0.0243007i
$399$		0			0
$400$			50524.7i			0.315779i
$401$	−55324.7	+	55324.7i	−0.344057	+	0.344057i	−0.857890	−	0.513833i	$-0.828225\pi$
$401$	−55324.7	+	55324.7i	−0.344057	+	0.344057i	0.513833	+	0.857890i	$0.328225\pi$
$402$		0			0
$403$	85191.1	+	11590.1i	0.524546	+	0.0713639i
$404$	−277556.			−1.70055
$405$		0			0
$406$	−1787.62			−0.0108449
$407$			52993.1i			0.319912i
$408$		0			0
$409$	−130129.	−	130129.i	−0.777909	−	0.777909i	0.201566	−	0.979475i	$-0.435397\pi$
$409$	−130129.	−	130129.i	−0.777909	−	0.777909i	−0.979475	+	0.201566i	$0.935397\pi$
$410$	−968.647	+	968.647i	−0.00576233	+	0.00576233i
$411$		0			0
$412$	−106534.			−0.627617
$413$		−	9967.89i		−	0.0584391i
$414$		0			0
$415$			261033.i			1.51565i
$416$	15778.9	+	2146.70i	0.0911780	+	0.0124047i
$417$		0			0
$418$	502.922	+	502.922i	0.00287838	+	0.00287838i
$419$	78745.1			0.448534			0.224267	−	0.974528i	$-0.428001\pi$
$419$	78745.1			0.448534			0.224267	+	0.974528i	$0.428001\pi$
$420$		0			0
$421$	122352.	+	122352.i	0.690314	+	0.690314i	0.962301	−	0.271987i	$-0.0876808\pi$
$421$	122352.	+	122352.i	0.690314	+	0.690314i	−0.271987	+	0.962301i	$0.587681\pi$
$422$	−3622.85	−	3622.85i	−0.0203435	−	0.0203435i
$423$		0			0
$424$	4775.77	−	4775.77i	0.0265651	−	0.0265651i
$425$	−49854.8			−0.276013
$426$		0			0
$427$	−31336.6	+	31336.6i	−0.171869	+	0.171869i
$428$		−	69022.2i		−	0.376792i
$429$		0			0
$430$	−904.885			−0.00489392
$431$	−33590.4	−	33590.4i	−0.180826	−	0.180826i	0.610890	−	0.791716i	$-0.290812\pi$
$431$	−33590.4	−	33590.4i	−0.180826	−	0.180826i	−0.791716	+	0.610890i	$0.790812\pi$
$432$		0			0
$433$			81942.2i			0.437051i	0.975831	+	0.218525i	$0.0701246\pi$
$433$			81942.2i			0.437051i	−0.975831	+	0.218525i	$0.929875\pi$
$434$	−1342.32	−	1342.32i	−0.00712652	−	0.00712652i
$435$		0			0
$436$	5040.64	−	5040.64i	0.0265163	−	0.0265163i
$437$	136897.	−	136897.i	0.716856	−	0.716856i
$438$		0			0
$439$			78273.4i			0.406149i	0.979163	+	0.203074i	$0.0650933\pi$
$439$			78273.4i			0.406149i	−0.979163	+	0.203074i	$0.934907\pi$
$440$	1761.72	−	1761.72i	0.00909978	−	0.00909978i
$441$		0			0
$442$	−704.968	+	5181.73i	−0.00360849	+	0.0265235i
$443$	−169637.			−0.864395			−0.432198	−	0.901779i	$-0.642262\pi$
$443$	−169637.			−0.864395			−0.432198	+	0.901779i	$0.642262\pi$
$444$		0			0
$445$	−165658.			−0.836549
$446$			9850.14i			0.0495191i
$447$		0			0
$448$	87479.6	+	87479.6i	0.435864	+	0.435864i
$449$	191202.	−	191202.i	0.948417	−	0.948417i	−0.0503165	−	0.998733i	$-0.516023\pi$
$449$	191202.	−	191202.i	0.948417	−	0.948417i	0.998733	+	0.0503165i	$0.0160230\pi$
$450$		0			0
$451$	16556.3			0.0813975
$452$		−	358785.i		−	1.75613i
$453$		0			0
$454$		−	6060.09i		−	0.0294014i
$455$	84443.1	−	64217.9i	0.407888	−	0.310194i
$456$		0			0
$457$	85965.6	+	85965.6i	0.411616	+	0.411616i	0.882301	−	0.470685i	$-0.155993\pi$
$457$	85965.6	+	85965.6i	0.411616	+	0.411616i	−0.470685	+	0.882301i	$0.655993\pi$
$458$	3977.20			0.0189604
$459$		0			0
$460$	−239660.	−	239660.i	−1.13261	−	1.13261i
$461$	231988.	+	231988.i	1.09160	+	1.09160i	0.995358	+	0.0962424i	$0.0306824\pi$
$461$	231988.	+	231988.i	1.09160	+	1.09160i	0.0962424	+	0.995358i	$0.469318\pi$
$462$		0			0
$463$	19082.7	−	19082.7i	0.0890180	−	0.0890180i	−0.661196	−	0.750214i	$-0.729951\pi$
$463$	19082.7	−	19082.7i	0.0890180	−	0.0890180i	0.750214	+	0.661196i	$0.229951\pi$
$464$	−122294.			−0.568025
$465$		0			0
$466$	890.443	−	890.443i	0.00410048	−	0.00410048i
$467$			122781.i			0.562987i	0.959563	+	0.281493i	$0.0908298\pi$
$467$			122781.i			0.562987i	−0.959563	+	0.281493i	$0.909170\pi$
$468$		0			0
$469$	193628.			0.880283
$470$	5083.79	+	5083.79i	0.0230140	+	0.0230140i
$471$		0			0
$472$			1289.38i			0.00578759i
$473$	7733.25	+	7733.25i	0.0345653	+	0.0345653i
$474$		0			0
$475$	26408.4	−	26408.4i	0.117046	−	0.117046i
$476$	−86483.4	+	86483.4i	−0.381697	+	0.381697i
$477$		0			0
$478$			12780.7i			0.0559371i
$479$	−96432.8	+	96432.8i	−0.420295	+	0.420295i	−0.885305	−	0.465011i	$-0.846051\pi$
$479$	−96432.8	+	96432.8i	−0.420295	+	0.420295i	0.465011	+	0.885305i	$0.346051\pi$
$480$		0			0
$481$	39348.1	−	289220.i	0.170072	−	1.25008i
$482$	−8944.07			−0.0384983
$483$		0			0
$484$	218986.			0.934816
$485$		−	116770.i		−	0.496417i
$486$		0			0
$487$	−192723.	−	192723.i	−0.812598	−	0.812598i	0.172425	−	0.985023i	$-0.444840\pi$
$487$	−192723.	−	192723.i	−0.812598	−	0.812598i	−0.985023	+	0.172425i	$0.944840\pi$
$488$	4053.51	−	4053.51i	0.0170212	−	0.0170212i
$489$		0			0
$490$	3753.02			0.0156311
$491$			92057.9i			0.381855i	0.981604	+	0.190927i	$0.0611495\pi$
$491$			92057.9i			0.381855i	−0.981604	+	0.190927i	$0.938850\pi$
$492$		0			0
$493$		−	120672.i		−	0.496493i
$494$	−2371.37	−	3118.22i	−0.00971730	−	0.0127777i
$495$		0			0
$496$	−91830.0	−	91830.0i	−0.373268	−	0.373268i
$497$	184529.			0.747052
$498$		0			0
$499$	−69581.0	−	69581.0i	−0.279441	−	0.279441i	0.553445	−	0.832886i	$-0.313313\pi$
$499$	−69581.0	−	69581.0i	−0.279441	−	0.279441i	−0.832886	+	0.553445i	$0.813313\pi$
$500$	−192224.	−	192224.i	−0.768896	−	0.768896i
$501$		0			0
$502$	1623.24	−	1623.24i	0.00644131	−	0.00644131i
$503$	340240.			1.34477			0.672387	−	0.740200i	$-0.265269\pi$
$503$	340240.			1.34477			0.672387	+	0.740200i	$0.265269\pi$
$504$		0			0
$505$	253735.	−	253735.i	0.994940	−	0.994940i
$506$		−	3867.22i		−	0.0151042i
$507$		0			0
$508$	322955.			1.25145
$509$	−273188.	−	273188.i	−1.05445	−	1.05445i	−0.998430	−	0.0560201i	$-0.982159\pi$
$509$	−273188.	−	273188.i	−1.05445	−	1.05445i	−0.0560201	−	0.998430i	$-0.517841\pi$
$510$		0			0
$511$			267366.i			1.02391i
$512$	−28356.5	−	28356.5i	−0.108171	−	0.108171i
$513$		0			0
$514$	6014.97	−	6014.97i	0.0227671	−	0.0227671i
$515$	97390.8	−	97390.8i	0.367201	−	0.367201i
$516$		0			0
$517$		−	86893.2i		−	0.325091i
$518$	−4557.13	+	4557.13i	−0.0169837	+	0.0169837i
$519$		0			0
$520$	−10923.0	+	8306.82i	−0.0403958	+	0.0307205i
$521$	413731.			1.52420			0.762100	−	0.647459i	$-0.224168\pi$
$521$	413731.			1.52420			0.762100	+	0.647459i	$0.224168\pi$
$522$		0			0
$523$	331600.			1.21230			0.606151	−	0.795350i	$-0.292713\pi$
$523$	331600.			1.21230			0.606151	+	0.795350i	$0.292713\pi$
$524$			185518.i			0.675653i
$525$		0			0
$526$	−7574.60	−	7574.60i	−0.0273772	−	0.0273772i
$527$	90612.4	−	90612.4i	0.326262	−	0.326262i
$528$		0			0
$529$	−772831.			−2.76168
$530$			4363.82i			0.0155351i
$531$		0			0
$532$		−	91621.6i		−	0.323724i
$533$	−90359.5	−	12293.3i	−0.318067	−	0.0432727i
$534$		0			0
$535$	63098.2	+	63098.2i	0.220450	+	0.220450i
$536$	−25046.5			−0.0871800
$537$		0			0
$538$	1675.50	+	1675.50i	0.00578867	+	0.00578867i
$539$	−32073.7	−	32073.7i	−0.110401	−	0.110401i
$540$		0			0
$541$	−24174.5	+	24174.5i	−0.0825968	+	0.0825968i	−0.747198	−	0.664601i	$-0.768601\pi$
$541$	−24174.5	+	24174.5i	−0.0825968	+	0.0825968i	0.664601	+	0.747198i	$0.268601\pi$
$542$	13483.7			0.0458996
$543$		0			0
$544$	16783.0	−	16783.0i	0.0567117	−	0.0567117i
$545$			9216.03i			0.0310278i
$546$		0			0
$547$	−114112.			−0.381380			−0.190690	−	0.981650i	$-0.561073\pi$
$547$	−114112.			−0.381380			−0.190690	+	0.981650i	$0.561073\pi$
$548$	−282974.	−	282974.i	−0.942292	−	0.942292i
$549$		0			0
$550$		−	746.013i		−	0.00246616i
$551$	63920.8	+	63920.8i	0.210542	+	0.210542i
$552$		0			0
$553$	−6898.62	+	6898.62i	−0.0225586	+	0.0225586i
$554$	−8386.52	+	8386.52i	−0.0273251	+	0.0273251i
$555$		0			0
$556$			383566.i			1.24077i
$557$	−324441.	+	324441.i	−1.04575	+	1.04575i	−0.0468427	+	0.998902i	$0.514916\pi$
$557$	−324441.	+	324441.i	−1.04575	+	1.04575i	−0.998902	+	0.0468427i	$0.985084\pi$
$558$		0			0
$559$	−36463.7	−	47947.8i	−0.116691	−	0.153442i
$560$	−160246.			−0.510989
$561$		0			0
$562$	−10331.1			−0.0327095
$563$		−	128026.i		−	0.403908i	−0.979395	−	0.201954i	$-0.935271\pi$
$563$		−	128026.i		−	0.403908i	0.979395	−	0.201954i	$-0.0647291\pi$
$564$		0			0
$565$	327992.	+	327992.i	1.02746	+	1.02746i
$566$	−4333.31	+	4333.31i	−0.0135266	+	0.0135266i
$567$		0			0
$568$	−23869.4			−0.0739853
$569$		−	358977.i		−	1.10877i	−0.832259	−	0.554386i	$-0.812953\pi$
$569$		−	358977.i		−	1.10877i	0.832259	−	0.554386i	$-0.187047\pi$
$570$		0			0
$571$			183795.i			0.563718i	0.959456	+	0.281859i	$0.0909510\pi$
$571$			183795.i			0.563718i	−0.959456	+	0.281859i	$0.909049\pi$
$572$	82131.4	+	11173.9i	0.251025	+	0.0341517i
$573$		0			0
$574$	1423.76	+	1423.76i	0.00432128	+	0.00432128i
$575$	−203068.			−0.614193
$576$		0			0
$577$	−57109.1	−	57109.1i	−0.171535	−	0.171535i	0.616118	−	0.787654i	$-0.288704\pi$
$577$	−57109.1	−	57109.1i	−0.171535	−	0.171535i	−0.787654	+	0.616118i	$0.788704\pi$
$578$	−1743.53	−	1743.53i	−0.00521884	−	0.00521884i
$579$		0			0
$580$	111903.	−	111903.i	0.332649	−	0.332649i
$581$	383678.			1.13662
$582$		0			0
$583$	37293.7	−	37293.7i	0.109723	−	0.109723i
$584$		−	34584.7i		−	0.101405i
$585$		0			0
$586$	−19484.3			−0.0567400
$587$	211138.	+	211138.i	0.612759	+	0.612759i	0.943664	−	0.330905i	$-0.107354\pi$
$587$	211138.	+	211138.i	0.612759	+	0.612759i	−0.330905	+	0.943664i	$0.607354\pi$
$588$		0			0
$589$			95996.0i			0.276709i
$590$	−589.081	−	589.081i	−0.00169228	−	0.00169228i
$591$		0			0
$592$	−311759.	+	311759.i	−0.889561	+	0.889561i
$593$	214002.	−	214002.i	0.608567	−	0.608567i	−0.334004	−	0.942572i	$-0.608400\pi$
$593$	214002.	−	214002.i	0.608567	−	0.608567i	0.942572	+	0.334004i	$0.108400\pi$
$594$		0			0
$595$		−	158122.i		−	0.446639i
$596$	−20094.8	+	20094.8i	−0.0565706	+	0.0565706i
$597$		0			0
$598$	−2871.46	+	21106.1i	−0.00802973	+	0.0590209i
$599$	−85941.2			−0.239523			−0.119762	−	0.992803i	$-0.538213\pi$
$599$	−85941.2			−0.239523			−0.119762	+	0.992803i	$0.538213\pi$
$600$		0			0
$601$	−326765.			−0.904663			−0.452332	−	0.891850i	$-0.649408\pi$
$601$	−326765.			−0.904663			−0.452332	+	0.891850i	$0.649408\pi$
$602$			1330.04i			0.00367004i
$603$		0			0
$604$	−379827.	−	379827.i	−1.04115	−	1.04115i
$605$	−200191.	+	200191.i	−0.546934	+	0.546934i
$606$		0			0
$607$	69228.5			0.187892			0.0939459	−	0.995577i	$-0.470052\pi$
$607$	69228.5			0.187892			0.0939459	+	0.995577i	$0.470052\pi$
$608$			17780.2i			0.0480982i
$609$		0			0
$610$			3703.86i			0.00995393i
$611$	−64519.4	+	474237.i	−0.172825	+	1.27032i
$612$		0			0
$613$	357789.	+	357789.i	0.952151	+	0.952151i	0.998906	−	0.0467552i	$-0.0148881\pi$
$613$	357789.	+	357789.i	0.952151	+	0.952151i	−0.0467552	+	0.998906i	$0.514888\pi$
$614$	8678.47			0.0230201
$615$		0			0
$616$	−2589.45	−	2589.45i	−0.00682410	−	0.00682410i
$617$	402876.	+	402876.i	1.05828	+	1.05828i	0.998193	+	0.0600878i	$0.0191381\pi$
$617$	402876.	+	402876.i	1.05828	+	1.05828i	0.0600878	+	0.998193i	$0.480862\pi$
$618$		0			0
$619$	−169086.	+	169086.i	−0.441293	+	0.441293i	−0.892446	−	0.451153i	$-0.851013\pi$
$619$	−169086.	+	169086.i	−0.441293	+	0.441293i	0.451153	+	0.892446i	$0.351013\pi$
$620$	168056.			0.437190
$621$		0			0
$622$	4421.05	−	4421.05i	0.0114273	−	0.0114273i
$623$			243491.i			0.627345i
$624$		0			0
$625$	227751.			0.583042
$626$	12185.5	+	12185.5i	0.0310952	+	0.0310952i
$627$		0			0
$628$		−	50577.9i		−	0.128245i
$629$	−307626.	−	307626.i	−0.777537	−	0.777537i
$630$		0			0
$631$	431081.	−	431081.i	1.08268	−	1.08268i	0.0864220	−	0.996259i	$-0.472457\pi$
$631$	431081.	−	431081.i	1.08268	−	1.08268i	0.996259	−	0.0864220i	$-0.0275433\pi$
$632$	892.361	−	892.361i	0.00223412	−	0.00223412i
$633$		0			0
$634$			20031.3i			0.0498345i
$635$	−295237.	+	295237.i	−0.732189	+	0.732189i
$636$		0			0
$637$	151234.	+	198864.i	0.372709	+	0.490092i
$638$	1805.70			0.00443614
$639$		0			0
$640$	41496.0			0.101309
$641$		−	355513.i		−	0.865245i	−0.901575	−	0.432623i	$-0.857588\pi$
$641$		−	355513.i		−	0.865245i	0.901575	−	0.432623i	$-0.142412\pi$
$642$		0			0
$643$	17508.7	+	17508.7i	0.0423479	+	0.0423479i	0.727964	−	0.685616i	$-0.240467\pi$
$643$	17508.7	+	17508.7i	0.0423479	+	0.0423479i	−0.685616	+	0.727964i	$0.740467\pi$
$644$	−352262.	+	352262.i	−0.849365	+	0.849365i
$645$		0			0
$646$	−5838.94			−0.0139917
$647$			617081.i			1.47412i	0.675826	+	0.737061i	$0.263787\pi$
$647$			617081.i			1.47412i	−0.675826	+	0.737061i	$0.736213\pi$
$648$		0			0
$649$			10068.7i			0.0239048i
$650$	−553.925	+	4071.51i	−0.00131106	+	0.00963672i
$651$		0			0
$652$	47632.3	+	47632.3i	0.112049	+	0.112049i
$653$	−164123.			−0.384895			−0.192447	−	0.981307i	$-0.561642\pi$
$653$	−164123.			−0.384895			−0.192447	+	0.981307i	$0.561642\pi$
$654$		0			0
$655$	−169596.	−	169596.i	−0.395305	−	0.395305i
$656$	97401.2	+	97401.2i	0.226337	+	0.226337i
$657$		0			0
$658$	7472.37	−	7472.37i	0.0172586	−	0.0172586i
$659$	−401563.			−0.924662			−0.462331	−	0.886707i	$-0.652987\pi$
$659$	−401563.			−0.924662			−0.462331	+	0.886707i	$0.652987\pi$
$660$		0			0
$661$	−57121.2	+	57121.2i	−0.130736	+	0.130736i	−0.769447	−	0.638711i	$-0.779468\pi$
$661$	−57121.2	+	57121.2i	−0.130736	+	0.130736i	0.638711	+	0.769447i	$0.279468\pi$
$662$		−	14736.3i		−	0.0336259i
$663$		0			0
$664$	−49630.1			−0.112566
$665$	83758.0	+	83758.0i	0.189401	+	0.189401i
$666$		0			0
$667$		−	491519.i		−	1.10481i
$668$	495053.	+	495053.i	1.10943	+	1.10943i
$669$		0			0
$670$	11443.0	−	11443.0i	0.0254912	−	0.0254912i
$671$	31653.6	−	31653.6i	0.0703036	−	0.0703036i
$672$		0			0
$673$			598216.i			1.32077i	0.750926	+	0.660387i	$0.229608\pi$
$673$			598216.i			1.32077i	−0.750926	+	0.660387i	$0.770392\pi$
$674$	−3105.18	+	3105.18i	−0.00683545	+	0.00683545i
$675$		0			0
$676$	−439951.	−	121967.i	−0.962745	−	0.266901i
$677$	−249942.			−0.545332			−0.272666	−	0.962109i	$-0.587905\pi$
$677$	−249942.			−0.545332			−0.272666	+	0.962109i	$0.587905\pi$
$678$		0			0
$679$	−171633.			−0.372273
$680$			20453.6i			0.0442335i
$681$		0			0
$682$	1355.90	+	1355.90i	0.00291513	+	0.00291513i
$683$	−248913.	+	248913.i	−0.533589	+	0.533589i	−0.921638	−	0.388050i	$-0.873149\pi$
$683$	−248913.	+	248913.i	−0.533589	+	0.533589i	0.388050	+	0.921638i	$0.373149\pi$
$684$		0			0
$685$	517375.			1.10262
$686$		−	14475.7i		−	0.0307603i
$687$		0			0
$688$			90989.7i			0.192227i
$689$	−231229.	+	175847.i	−0.487083	+	0.370421i
$690$		0			0
$691$	−268299.	−	268299.i	−0.561906	−	0.561906i	0.367943	−	0.929849i	$-0.380062\pi$
$691$	−268299.	−	268299.i	−0.561906	−	0.561906i	−0.929849	+	0.367943i	$0.880062\pi$
$692$	339095.			0.708124
$693$		0			0
$694$	5273.86	+	5273.86i	0.0109499	+	0.0109499i
$695$	−350645.	−	350645.i	−0.725936	−	0.725936i
$696$		0			0
$697$	−96109.7	+	96109.7i	−0.197834	+	0.197834i
$698$	−7629.58			−0.0156599
$699$		0			0
$700$	−67953.8	+	67953.8i	−0.138681	+	0.138681i
$701$		−	539033.i		−	1.09693i	−0.836173	−	0.548465i	$-0.815212\pi$
$701$		−	539033.i		−	1.09693i	0.836173	−	0.548465i	$-0.184788\pi$
$702$		0			0
$703$	325903.			0.659443
$704$	−88364.3	−	88364.3i	−0.178292	−	0.178292i
$705$		0			0
$706$		−	6099.08i		−	0.0122364i
$707$	−372950.	−	372950.i	−0.746125	−	0.746125i
$708$		0			0
$709$	−542342.	+	542342.i	−1.07890	+	1.07890i	−0.0822903	+	0.996608i	$0.526223\pi$
$709$	−542342.	+	542342.i	−1.07890	+	1.07890i	−0.996608	+	0.0822903i	$0.973777\pi$
$710$	10905.3	−	10905.3i	0.0216331	−	0.0216331i
$711$		0			0
$712$		−	31496.4i		−	0.0621299i
$713$	369081.	−	369081.i	0.726010	−	0.726010i
$714$		0			0
$715$	−85297.2	+	64867.4i	−0.166849	+	0.126886i
$716$	748768.			1.46057
$717$		0			0
$718$	18320.2			0.0355371
$719$		−	733372.i		−	1.41862i	−0.704896	−	0.709311i	$-0.749006\pi$
$719$		−	733372.i		−	1.41862i	0.704896	−	0.709311i	$-0.250994\pi$
$720$		0			0
$721$	−143149.	−	143149.i	−0.275371	−	0.275371i
$722$	−8227.35	+	8227.35i	−0.0157829	+	0.0157829i
$723$		0			0
$724$	574463.			1.09593
$725$		−	94817.4i		−	0.180390i
$726$		0			0
$727$		−	71768.0i		−	0.135788i	−0.997693	−	0.0678941i	$-0.978372\pi$
$727$		−	71768.0i		−	0.135788i	0.997693	−	0.0678941i	$-0.0216280\pi$
$728$	12209.7	+	16055.1i	0.0230379	+	0.0302936i
$729$		0			0
$730$	15800.7	+	15800.7i	0.0296505	+	0.0296505i
$731$	−89783.2			−0.168020
$732$		0			0
$733$	113097.	+	113097.i	0.210496	+	0.210496i	0.804478	−	0.593982i	$-0.202445\pi$
$733$	113097.	+	113097.i	0.210496	+	0.210496i	−0.593982	+	0.804478i	$0.702445\pi$
$734$	−2031.68	−	2031.68i	−0.00377106	−	0.00377106i
$735$		0			0
$736$	68360.3	−	68360.3i	0.126197	−	0.126197i
$737$	−195586.			−0.360084
$738$		0			0
$739$	117545.	−	117545.i	0.215236	−	0.215236i	−0.591251	−	0.806487i	$-0.701366\pi$
$739$	117545.	−	117545.i	0.215236	−	0.215236i	0.806487	+	0.591251i	$0.201366\pi$
$740$		−	570543.i		−	1.04190i
$741$		0			0
$742$	6414.13			0.0116501
$743$	507699.	+	507699.i	0.919663	+	0.919663i	0.997005	−	0.0773413i	$-0.0246431\pi$
$743$	507699.	+	507699.i	0.919663	+	0.919663i	−0.0773413	+	0.997005i	$0.524643\pi$
$744$		0			0
$745$		−	36740.2i		−	0.0661957i
$746$	−15205.4	−	15205.4i	−0.0273226	−	0.0273226i
$747$		0			0
$748$	87358.1	−	87358.1i	0.156135	−	0.156135i
$749$	92744.5	−	92744.5i	0.165320	−	0.165320i
$750$		0			0
$751$		−	436896.i		−	0.774636i	−0.921946	−	0.387318i	$-0.873401\pi$
$751$		−	436896.i		−	0.774636i	0.921946	−	0.387318i	$-0.126599\pi$
$752$	511194.	−	511194.i	0.903962	−	0.903962i
$753$		0			0
$754$	−9854.98	−	1340.76i	−0.0173346	−	0.00235835i
$755$	694455.			1.21829
$756$		0			0
$757$	−435716.			−0.760348			−0.380174	−	0.924915i	$-0.624136\pi$
$757$	−435716.			−0.760348			−0.380174	+	0.924915i	$0.624136\pi$
$758$		−	164.879i		−	0.000286964i
$759$		0			0
$760$	−10834.4	−	10834.4i	−0.0187576	−	0.0187576i
$761$	63224.5	−	63224.5i	0.109173	−	0.109173i	−0.650410	−	0.759583i	$-0.725403\pi$
$761$	63224.5	−	63224.5i	0.109173	−	0.109173i	0.759583	+	0.650410i	$0.225403\pi$
$762$		0			0
$763$	13546.1			0.0232683
$764$			400582.i			0.686285i
$765$		0			0
$766$		−	22046.5i		−	0.0375735i
$767$	7476.15	−	54952.0i	0.0127083	−	0.0934098i
$768$		0			0
$769$	−247132.	−	247132.i	−0.417904	−	0.417904i	0.466577	−	0.884481i	$-0.345487\pi$
$769$	−247132.	−	247132.i	−0.417904	−	0.417904i	−0.884481	+	0.466577i	$0.845487\pi$
$770$	2366.09			0.00399070
$771$		0			0
$772$	507601.	+	507601.i	0.851702	+	0.851702i
$773$	−370482.	−	370482.i	−0.620024	−	0.620024i	0.325513	−	0.945537i	$-0.394463\pi$
$773$	−370482.	−	370482.i	−0.620024	−	0.620024i	−0.945537	+	0.325513i	$0.894463\pi$
$774$		0			0
$775$	71198.2	−	71198.2i	0.118540	−	0.118540i
$776$	22201.4			0.0368685
$777$		0			0
$778$	−1138.11	+	1138.11i	−0.00188029	+	0.00188029i
$779$		−	101820.i		−	0.167787i
$780$		0			0
$781$	−186395.			−0.305585
$782$	22449.3	+	22449.3i	0.0367104	+	0.0367104i
$783$		0			0
$784$		−	377381.i		−	0.613971i
$785$	46236.9	+	46236.9i	0.0750325	+	0.0750325i
$786$		0			0
$787$	−167668.	+	167668.i	−0.270708	+	0.270708i	−0.829385	−	0.558677i	$-0.811309\pi$
$787$	−167668.	+	167668.i	−0.270708	+	0.270708i	0.558677	+	0.829385i	$0.311309\pi$
$788$	−381570.	+	381570.i	−0.614499	+	0.614499i
$789$		0			0
$790$			815.387i			0.00130650i
$791$	482096.	−	482096.i	0.770514	−	0.770514i
$792$		0			0
$793$	−196259.	+	149252.i	−0.312092	+	0.237342i
$794$	−27369.1			−0.0434129
$795$		0			0
$796$	708356.			1.11796
$797$		−	114399.i		−	0.180097i	−0.995937	−	0.0900483i	$-0.971298\pi$
$797$		−	114399.i		−	0.180097i	0.995937	−	0.0900483i	$-0.0287022\pi$
$798$		0			0
$799$	504416.	+	504416.i	0.790125	+	0.790125i
$800$	13187.2	−	13187.2i	0.0206050	−	0.0206050i
$801$		0			0
$802$	9611.52			0.0149432
$803$		−	270070.i		−	0.418837i
$804$		0			0
$805$		−	644057.i		−	0.993877i
$806$	−6393.31	−	8406.86i	−0.00984138	−	0.0129409i
$807$		0			0
$808$	48242.4	+	48242.4i	0.0738935	+	0.0738935i
$809$	−1.23834e6			−1.89209			−0.946046	−	0.324032i	$-0.894961\pi$
$809$	−1.23834e6			−1.89209			−0.946046	+	0.324032i	$0.894961\pi$
$810$		0			0
$811$	205643.	+	205643.i	0.312660	+	0.312660i	0.845939	−	0.533280i	$-0.179041\pi$
$811$	205643.	+	205643.i	0.312660	+	0.312660i	−0.533280	+	0.845939i	$0.679041\pi$
$812$	−164480.	−	164480.i	−0.249460	−	0.249460i
$813$		0			0
$814$	4603.22	−	4603.22i	0.00694725	−	0.00694725i
$815$	−87088.4			−0.131113
$816$		0			0
$817$	47558.8	−	47558.8i	0.0712503	−	0.0712503i
$818$			22607.3i			0.0337864i
$819$		0			0
$820$	−178251.			−0.265097
$821$	659037.	+	659037.i	0.977741	+	0.977741i	0.999758	−	0.0220168i	$-0.00700872\pi$
$821$	659037.	+	659037.i	0.977741	+	0.977741i	−0.0220168	+	0.999758i	$0.507009\pi$
$822$		0			0
$823$		−	1.08710e6i		−	1.60498i	−0.596664	−	0.802491i	$-0.703507\pi$
$823$		−	1.08710e6i		−	1.60498i	0.596664	−	0.802491i	$-0.296493\pi$
$824$	18516.8	+	18516.8i	0.0272717	+	0.0272717i
$825$		0			0
$826$	−865.857	+	865.857i	−0.00126907	+	0.00126907i
$827$	−413357.	+	413357.i	−0.604385	+	0.604385i	−0.941473	−	0.337088i	$-0.890558\pi$
$827$	−413357.	+	413357.i	−0.604385	+	0.604385i	0.337088	+	0.941473i	$0.390558\pi$
$828$		0			0
$829$		−	857909.i		−	1.24834i	−0.781290	−	0.624169i	$-0.785438\pi$
$829$		−	857909.i		−	1.24834i	0.781290	−	0.624169i	$-0.214562\pi$
$830$	22674.6	−	22674.6i	0.0329141	−	0.0329141i
$831$		0			0
$832$	416654.	+	547877.i	0.601906	+	0.791474i
$833$	372377.			0.536652
$834$		0			0
$835$	−905128.			−1.29819
$836$			92548.3i			0.132421i
$837$		0			0
$838$	−6840.16	−	6840.16i	−0.00974043	−	0.00974043i
$839$	−225669.	+	225669.i	−0.320588	+	0.320588i	−0.848993	−	0.528404i	$-0.822791\pi$
$839$	−225669.	+	225669.i	−0.320588	+	0.320588i	0.528404	+	0.848993i	$0.322791\pi$
$840$		0			0
$841$	−477778.			−0.675514
$842$		−	21256.1i		−	0.0299819i
$843$		0			0
$844$		−	666681.i		−	0.935908i
$845$	513691.	−	290693.i	0.719430	−	0.407118i
$846$		0			0
$847$	294250.	+	294250.i	0.410156	+	0.410156i
$848$	438799.			0.610202
$849$		0			0
$850$	4330.62	+	4330.62i	0.00599393	+	0.00599393i
$851$	−1.25301e6	−	1.25301e6i	−1.73020	−	1.73020i
$852$		0			0
$853$	704776.	−	704776.i	0.968619	−	0.968619i	−0.0309037	−	0.999522i	$-0.509839\pi$
$853$	704776.	−	704776.i	0.968619	−	0.968619i	0.999522	+	0.0309037i	$0.00983851\pi$
$854$	5444.09			0.00746465
$855$		0			0
$856$	−11996.8	+	11996.8i	−0.0163726	+	0.0163726i
$857$			1.00696e6i			1.37105i	0.728051	+	0.685523i	$0.240426\pi$
$857$			1.00696e6i			1.37105i	−0.728051	+	0.685523i	$0.759574\pi$
$858$		0			0
$859$	−1.07340e6			−1.45470			−0.727352	−	0.686264i	$-0.759249\pi$
$859$	−1.07340e6			−1.45470			−0.727352	+	0.686264i	$0.759249\pi$
$860$	−83259.0	−	83259.0i	−0.112573	−	0.112573i
$861$		0			0
$862$			5835.63i			0.00785367i
$863$	726484.	+	726484.i	0.975449	+	0.975449i	0.999706	−	0.0242565i	$-0.00772184\pi$
$863$	726484.	+	726484.i	0.975449	+	0.975449i	−0.0242565	+	0.999706i	$0.507722\pi$
$864$		0			0
$865$	−309992.	+	309992.i	−0.414303	+	0.414303i
$866$	7117.87	−	7117.87i	0.00949105	−	0.00949105i
$867$		0			0
$868$		−	247016.i		−	0.327857i
$869$	6968.39	−	6968.39i	0.00922769	−	0.00922769i
$870$		0			0
$871$	1.06745e6	+	145225.i	1.40706	+	0.191428i
$872$	−1752.24			−0.00230441
$873$		0			0
$874$	−23783.0			−0.0311347
$875$		−	516579.i		−	0.674715i
$876$		0			0
$877$	−271952.	−	271952.i	−0.353584	−	0.353584i	0.507857	−	0.861441i	$-0.330438\pi$
$877$	−271952.	−	271952.i	−0.353584	−	0.353584i	−0.861441	+	0.507857i	$0.830438\pi$
$878$	6799.18	−	6799.18i	0.00881998	−	0.00881998i
$879$		0			0
$880$	161867.			0.209022
$881$		−	252034.i		−	0.324719i	−0.986732	−	0.162360i	$-0.948090\pi$
$881$		−	252034.i		−	0.324719i	0.986732	−	0.162360i	$-0.0519104\pi$
$882$		0			0
$883$			927130.i			1.18910i	0.804058	+	0.594551i	$0.202670\pi$
$883$			927130.i			1.18910i	−0.804058	+	0.594551i	$0.797330\pi$
$884$	−541638.	+	411909.i	−0.693114	+	0.527105i
$885$		0			0
$886$	14735.4	+	14735.4i	0.0187713	+	0.0187713i
$887$	−642869.			−0.817100			−0.408550	−	0.912736i	$-0.633965\pi$
$887$	−642869.			−0.817100			−0.408550	+	0.912736i	$0.633965\pi$
$888$		0			0
$889$	433951.	+	433951.i	0.549083	+	0.549083i
$890$	14389.8	+	14389.8i	0.0181666	+	0.0181666i
$891$		0			0
$892$	−906316.	+	906316.i	−1.13907	+	1.13907i
$893$	−534385.			−0.670118
$894$		0			0
$895$	−684503.	+	684503.i	−0.854534	+	0.854534i
$896$		−	60992.6i		−	0.0759734i
$897$		0			0
$898$	−33217.3			−0.0411919
$899$	172333.	+	172333.i	0.213231	+	0.213231i
$900$		0			0
$901$			432981.i			0.533358i
$902$	−1438.16	−	1438.16i	−0.00176764	−	0.00176764i
$903$		0			0
$904$	−62360.9	+	62360.9i	−0.0763089	+	0.0763089i
$905$	−525158.	+	525158.i	−0.641200	+	0.641200i
$906$		0			0
$907$		−	234146.i		−	0.284624i	−0.989822	−	0.142312i	$-0.954546\pi$
$907$		−	234146.i		−	0.284624i	0.989822	−	0.142312i	$-0.0454536\pi$
$908$	557592.	−	557592.i	0.676308	−	0.676308i
$909$		0			0
$910$	−12913.4	−	1756.85i	−0.0155940	−	0.00212154i
$911$	1.19549e6			1.44049			0.720244	−	0.693721i	$-0.244030\pi$
$911$	1.19549e6			1.44049			0.720244	+	0.693721i	$0.244030\pi$
$912$		0			0
$913$	−387558.			−0.464939
$914$		−	14934.7i		−	0.0178774i
$915$		0			0
$916$	365945.	+	365945.i	0.436139	+	0.436139i
$917$	−249279.	+	249279.i	−0.296447	+	0.296447i
$918$		0			0
$919$	716149.			0.847954			0.423977	−	0.905673i	$-0.360634\pi$
$919$	716149.			0.847954			0.423977	+	0.905673i	$0.360634\pi$
$920$			83311.1i			0.0984299i
$921$		0			0
$922$		−	40303.1i		−	0.0474107i
$923$	1.01729e6	+	138401.i	1.19410	+	0.162456i
$924$		0			0
$925$	−241715.	−	241715.i	−0.282501	−	0.282501i
$926$	−3315.22			−0.00386626
$927$		0			0
$928$	31919.2	+	31919.2i	0.0370643	+	0.0370643i
$929$	1698.00	+	1698.00i	0.00196746	+	0.00196746i	0.708090	−	0.706122i	$-0.249557\pi$
$929$	1698.00	+	1698.00i	0.00196746	+	0.00196746i	−0.706122	+	0.708090i	$0.749557\pi$
$930$		0			0
$931$	−197251.	+	197251.i	−0.227572	+	0.227572i
$932$	163860.			0.188644
$933$		0			0
$934$	10665.3	−	10665.3i	0.0122259	−	0.0122259i
$935$			159721.i			0.182700i
$936$		0			0
$937$	1.49808e6			1.70630			0.853152	−	0.521663i	$-0.174688\pi$
$937$	1.49808e6			1.70630			0.853152	+	0.521663i	$0.174688\pi$
$938$	−16819.4	−	16819.4i	−0.0191163	−	0.0191163i
$939$		0			0
$940$			935524.i			1.05876i
$941$	843256.	+	843256.i	0.952314	+	0.952314i	0.998914	−	0.0466000i	$-0.0148386\pi$
$941$	843256.	+	843256.i	0.952314	+	0.952314i	−0.0466000	+	0.998914i	$0.514839\pi$
$942$		0			0
$943$	−391472.	+	391472.i	−0.440228	+	0.440228i
$944$	−59234.4	+	59234.4i	−0.0664706	+	0.0664706i
$945$		0			0
$946$		−	1343.49i		−	0.00150125i
$947$	299398.	−	299398.i	0.333848	−	0.333848i	−0.520198	−	0.854046i	$-0.674142\pi$
$947$	299398.	−	299398.i	0.333848	−	0.333848i	0.854046	+	0.520198i	$0.174142\pi$
$948$		0			0
$949$	−200530.	+	1.47396e6i	−0.222663	+	1.63664i
$950$	−4587.91			−0.00508356
$951$		0			0
$952$	30063.5			0.0331716
$953$			941607.i			1.03677i	0.855146	+	0.518387i	$0.173467\pi$
$953$			941607.i			1.03677i	−0.855146	+	0.518387i	$0.826533\pi$
$954$		0			0
$955$	−366201.	−	366201.i	−0.401525	−	0.401525i
$956$	−1.17596e6	+	1.17596e6i	−1.28670	+	1.28670i
$957$		0			0
$958$	16753.2			0.0182544
$959$		−	760459.i		−	0.826873i
$960$		0			0
$961$		−	664712.i		−	0.719758i
$962$	−28540.9	+	21705.0i	−0.0308403	+	0.0234537i
$963$		0			0
$964$	−822948.	−	822948.i	−0.885561	−	0.885561i
$965$	−928070.			−0.996612
$966$		0			0
$967$	−206817.	−	206817.i	−0.221173	−	0.221173i	0.587819	−	0.808992i	$-0.299987\pi$
$967$	−206817.	−	206817.i	−0.221173	−	0.221173i	−0.808992	+	0.587819i	$0.799987\pi$
$968$	−38062.3	−	38062.3i	−0.0406204	−	0.0406204i
$969$		0			0
$970$	−10143.2	+	10143.2i	−0.0107803	+	0.0107803i
$971$	−1.54657e6			−1.64033			−0.820165	−	0.572128i	$-0.806118\pi$
$971$	−1.54657e6			−1.64033			−0.820165	+	0.572128i	$0.806118\pi$
$972$		0			0
$973$	−515393.	+	515393.i	−0.544394	+	0.544394i
$974$			33481.6i			0.0352930i
$975$		0			0
$976$	372437.			0.390979
$977$	109090.	+	109090.i	0.114287	+	0.114287i	0.761937	−	0.647651i	$-0.224248\pi$
$977$	109090.	+	109090.i	0.114287	+	0.114287i	−0.647651	+	0.761937i	$0.724248\pi$
$978$		0			0
$979$		−	245953.i		−	0.256618i
$980$	345318.	+	345318.i	0.359556	+	0.359556i
$981$		0			0
$982$	7996.57	−	7996.57i	0.00829241	−	0.00829241i
$983$	238965.	−	238965.i	0.247302	−	0.247302i	−0.572561	−	0.819862i	$-0.694050\pi$
$983$	238965.	−	238965.i	0.247302	−	0.247302i	0.819862	+	0.572561i	$0.194050\pi$
$984$		0			0
$985$		−	697642.i		−	0.719051i
$986$	−10482.1	+	10482.1i	−0.0107819	+	0.0107819i
$987$		0			0
$988$	68718.3	−	505100.i	0.0703977	−	0.517445i
$989$	−365703.			−0.373883
$990$		0			0
$991$	810923.			0.825719			0.412859	−	0.910795i	$-0.364530\pi$
$991$	810923.			0.825719			0.412859	+	0.910795i	$0.364530\pi$
$992$			47936.1i			0.0487123i
$993$		0			0
$994$	−16029.0	−	16029.0i	−0.0162231	−	0.0162231i
$995$	−647560.	+	647560.i	−0.654085	+	0.654085i
$996$		0			0
$997$	17086.1			0.0171890			0.00859452	−	0.999963i	$-0.497264\pi$
$997$	17086.1			0.0171890			0.00859452	+	0.999963i	$0.497264\pi$
$998$			12088.3i			0.0121368i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.73.6		20
3.2	odd	2		39.5.g.a.34.5	yes	20
13.5	odd	4	inner	117.5.j.b.109.6		20
39.5	even	4		39.5.g.a.31.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.5	✓	20	39.5	even	4
39.5.g.a.34.5	yes	20	3.2	odd	2
117.5.j.b.73.6		20	1.1	even	1	trivial
117.5.j.b.109.6		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)

\(f(q)\)	\(=\)	\(q+(-0.0868646 - 0.0868646i) q^{2} -15.9849i q^{4} +(14.6130 + 14.6130i) q^{5} +(21.4788 - 21.4788i) q^{7} +(-2.77836 + 2.77836i) q^{8} -2.53870i q^{10} +(-21.6960 + 21.6960i) q^{11} +(134.520 - 102.301i) q^{13} -3.73149 q^{14} -255.276 q^{16} -251.891i q^{17} +(133.428 + 133.428i) q^{19} +(233.587 - 233.587i) q^{20} +3.76923 q^{22} -1026.00i q^{23} -197.922i q^{25} +(-20.5713 - 2.79870i) q^{26} +(-343.336 - 343.336i) q^{28} +479.065 q^{29} +(359.729 + 359.729i) q^{31} +(66.6281 + 66.6281i) q^{32} +(-21.8804 + 21.8804i) q^{34} +627.737 q^{35} +(1221.26 - 1221.26i) q^{37} -23.1804i q^{38} -81.2001 q^{40} +(-381.553 - 381.553i) q^{41} -356.437i q^{43} +(346.809 + 346.809i) q^{44} +(-89.1229 + 89.1229i) q^{46} +(-2002.52 + 2002.52i) q^{47} +1478.33i q^{49} +(-17.1924 + 17.1924i) q^{50} +(-1635.27 - 2150.29i) q^{52} -1718.92 q^{53} -634.086 q^{55} +119.351i q^{56} +(-41.6137 - 41.6137i) q^{58} +(232.041 - 232.041i) q^{59} -1458.96 q^{61} -62.4953i q^{62} +4072.84i q^{64} +(3460.65 + 470.817i) q^{65} +(4507.42 + 4507.42i) q^{67} -4026.46 q^{68} +(-54.5281 - 54.5281i) q^{70} +(4295.61 + 4295.61i) q^{71} +(-6223.95 + 6223.95i) q^{73} -212.169 q^{74} +(2132.84 - 2132.84i) q^{76} +932.007i q^{77} -321.183 q^{79} +(-3730.34 - 3730.34i) q^{80} +66.2868i q^{82} +(8931.56 + 8931.56i) q^{83} +(3680.88 - 3680.88i) q^{85} +(-30.9617 + 30.9617i) q^{86} -120.558i q^{88} +(-5668.17 + 5668.17i) q^{89} +(692.027 - 5086.61i) q^{91} -16400.5 q^{92} +347.896 q^{94} +3899.57i q^{95} +(-3995.41 - 3995.41i) q^{97} +(128.414 - 128.414i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 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

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