LMFDB - Embedded newform 117.5.j.b.109.6

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			109.6
Root			$-0.0868646 + 0.0868646i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.6

$q$-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 $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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	−0.717881	−	0.696165i	$-0.754888\pi$
$2$	−0.0868646	+	0.0868646i	−0.0217161	+	0.0217161i	0.696165	+	0.717881i	$0.254888\pi$
$3$		0			0
$3$		0			0
$4$			15.9849i			0.999057i
$5$	14.6130	−	14.6130i	0.584519	−	0.584519i	−0.351623	−	0.936142i	$-0.614370\pi$
$5$	14.6130	−	14.6130i	0.584519	−	0.584519i	0.936142	+	0.351623i	$0.114370\pi$
$6$		0			0
$7$	21.4788	+	21.4788i	0.438342	+	0.438342i	0.891454	−	0.453112i	$-0.149686\pi$
$7$	21.4788	+	21.4788i	0.438342	+	0.438342i	−0.453112	+	0.891454i	$0.649686\pi$
$8$	−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	0.611747	−	0.791053i	$-0.290467\pi$
$11$	−21.6960	−	21.6960i	−0.179306	−	0.179306i	−0.791053	+	0.611747i	$0.790467\pi$
$12$		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	$0.143534\pi$
$17$			251.891i			0.871596i	−0.900045	+	0.435798i	$0.856466\pi$
$18$		0			0
$19$	133.428	−	133.428i	0.369608	−	0.369608i	−0.497726	−	0.867334i	$-0.665832\pi$
$19$	133.428	−	133.428i	0.369608	−	0.369608i	0.867334	+	0.497726i	$0.165832\pi$
$20$	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	$0.421510\pi$
$23$			1026.00i			1.93950i	−0.244091	+	0.969752i	$0.578490\pi$
$24$		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	−0.494723	−	0.869051i	$-0.664731\pi$
$31$	359.729	−	359.729i	0.374327	−	0.374327i	0.869051	+	0.494723i	$0.164731\pi$
$32$	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	0.994719	−	0.102634i	$-0.0327272\pi$
$37$	1221.26	+	1221.26i	0.892085	+	0.892085i	−0.102634	+	0.994719i	$0.532727\pi$
$38$			23.1804i			0.0160529i
$39$		0			0
$40$	−81.2001			−0.0507500
$41$	−381.553	+	381.553i	−0.226980	+	0.226980i	−0.811430	−	0.584450i	$-0.801310\pi$
$41$	−381.553	+	381.553i	−0.226980	+	0.226980i	0.584450	+	0.811430i	$0.301310\pi$
$42$		0			0
$43$			356.437i			0.192773i	0.995344	+	0.0963863i	$0.0307284\pi$
$43$			356.437i			0.192773i	−0.995344	+	0.0963863i	$0.969272\pi$
$44$	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	0.0894638	−	0.995990i	$-0.471485\pi$
$47$	−2002.52	−	2002.52i	−0.906526	−	0.906526i	−0.995990	+	0.0894638i	$0.971485\pi$
$48$		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	0.739650	−	0.672991i	$-0.234991\pi$
$59$	232.041	+	232.041i	0.0666592	+	0.0666592i	−0.672991	+	0.739650i	$0.734991\pi$
$60$		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	0.00411288	−	0.999992i	$-0.498691\pi$
$67$	4507.42	−	4507.42i	1.00410	−	1.00410i	0.999992	−	0.00411288i	$-0.00130917\pi$
$68$	−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	−0.138262	−	0.990396i	$-0.544152\pi$
$71$	4295.61	−	4295.61i	0.852134	−	0.852134i	0.990396	+	0.138262i	$0.0441516\pi$
$72$		0			0
$73$	−6223.95	−	6223.95i	−1.16794	−	1.16794i	−0.982692	−	0.185248i	$-0.940691\pi$
$73$	−6223.95	−	6223.95i	−1.16794	−	1.16794i	−0.185248	−	0.982692i	$-0.559309\pi$
$74$	−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	0.365804	−	0.930692i	$-0.380794\pi$
$83$	8931.56	−	8931.56i	1.29650	−	1.29650i	0.930692	−	0.365804i	$-0.119206\pi$
$84$		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	0.252111	−	0.967698i	$-0.418875\pi$
$89$	−5668.17	−	5668.17i	−0.715588	−	0.715588i	−0.967698	+	0.252111i	$0.918875\pi$
$90$		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	−0.886797	−	0.462159i	$-0.847075\pi$
$97$	−3995.41	+	3995.41i	−0.424637	+	0.424637i	0.462159	+	0.886797i	$0.347075\pi$
$98$	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	$0.324049\pi$
$101$			17363.6i			1.70215i	−0.525043	+	0.851076i	$0.675951\pi$
$102$		0			0
$103$			6664.68i			0.628210i	0.949388	+	0.314105i	$0.101704\pi$
$103$			6664.68i			0.628210i	−0.949388	+	0.314105i	$0.898296\pi$
$104$	−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	−0.693712	−	0.720253i	$-0.744026\pi$
$109$	315.337	−	315.337i	0.0265413	−	0.0265413i	0.720253	+	0.693712i	$0.244026\pi$
$110$	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	$-0.784561\pi$
$127$		−	20203.7i		−	1.25263i	0.779568	−	0.626317i	$-0.215439\pi$
$128$	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	0.998470	−	0.0552888i	$-0.0176079\pi$
$137$	17702.6	+	17702.6i	0.943182	+	0.943182i	−0.0552888	+	0.998470i	$0.517608\pi$
$138$		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	−0.734852	−	0.678228i	$-0.762748\pi$
$149$	−1257.11	+	1257.11i	−0.0566241	+	0.0566241i	0.678228	+	0.734852i	$0.262748\pi$
$150$		0			0
$151$	23761.6	+	23761.6i	1.04213	+	1.04213i	0.999073	+	0.0430569i	$0.0137097\pi$
$151$	23761.6	+	23761.6i	1.04213	+	1.04213i	0.0430569	+	0.999073i	$0.486290\pi$
$152$	−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	0.648802	−	0.760957i	$-0.275270\pi$
$163$	−2979.83	−	2979.83i	−0.112154	−	0.112154i	−0.760957	+	0.648802i	$0.775270\pi$
$164$	−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	−0.993086	−	0.117388i	$-0.962548\pi$
$167$	−30970.0	−	30970.0i	−1.11047	−	1.11047i	−0.117388	−	0.993086i	$-0.537452\pi$
$168$		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	$-0.884686\pi$
$173$		−	21213.4i		−	0.708792i	0.935095	−	0.354396i	$-0.115314\pi$
$174$		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	$-0.739067\pi$
$179$		−	46842.2i		−	1.46194i	0.682407	−	0.730972i	$-0.260933\pi$
$180$		0			0
$181$		−	35937.8i		−	1.09697i	−0.836161	−	0.548485i	$-0.815205\pi$
$181$		−	35937.8i		−	1.09697i	0.836161	−	0.548485i	$-0.184795\pi$
$182$	−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	0.137935	−	0.990441i	$-0.455954\pi$
$193$	−31755.0	−	31755.0i	−0.852506	−	0.852506i	−0.990441	+	0.137935i	$0.955954\pi$
$194$		−	694.120i		−	0.0184430i
$195$		0			0
$196$	23630.9			0.615132
$197$	−23870.6	+	23870.6i	−0.615080	+	0.615080i	−0.944265	−	0.329186i	$-0.893226\pi$
$197$	−23870.6	+	23870.6i	−0.615080	+	0.615080i	0.329186	+	0.944265i	$0.393226\pi$
$198$		0			0
$199$		−	44314.1i		−	1.11901i	−0.828826	−	0.559507i	$-0.810991\pi$
$199$		−	44314.1i		−	1.11901i	0.828826	−	0.559507i	$-0.189009\pi$
$200$	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	−0.151721	+	0.988423i	$0.548482\pi$
$223$	−56698.2	+	56698.2i	−1.14014	+	1.14014i	−0.988423	+	0.151721i	$0.951518\pi$
$224$	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	−0.282361	−	0.959308i	$-0.591118\pi$
$227$	34882.4	−	34882.4i	0.676947	−	0.676947i	0.959308	+	0.282361i	$0.0911177\pi$
$228$		0			0
$229$	−22893.1	−	22893.1i	−0.436550	−	0.436550i	0.454299	−	0.890849i	$-0.349890\pi$
$229$	−22893.1	−	22893.1i	−0.436550	−	0.436550i	−0.890849	+	0.454299i	$0.849890\pi$
$230$	−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	$-0.969903\pi$
$233$		−	10250.9i		−	0.188822i	0.995533	−	0.0944108i	$-0.0300967\pi$
$234$		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	−0.351863	+	0.936051i	$0.614452\pi$
$239$	−73567.0	+	73567.0i	−1.28791	+	1.28791i	−0.936051	+	0.351863i	$0.885548\pi$
$240$		0			0
$241$	51482.8	+	51482.8i	0.886397	+	0.886397i	0.994175	−	0.107778i	$-0.0343736\pi$
$241$	51482.8	+	51482.8i	0.886397	+	0.886397i	−0.107778	+	0.994175i	$0.534374\pi$
$242$	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	$-0.952618\pi$
$251$		−	18687.0i		−	0.296614i	0.988941	−	0.148307i	$-0.0473823\pi$
$252$		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	$-0.824366\pi$
$257$		−	69245.4i		−	1.04839i	0.851597	−	0.524197i	$-0.175634\pi$
$258$		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	−0.998286	−	0.0585225i	$-0.981361\pi$
$271$	−77613.1	−	77613.1i	−1.05681	−	1.05681i	−0.0585225	−	0.998286i	$-0.518639\pi$
$272$		−	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	$0.216594\pi$
$277$			96547.1i			1.25829i	−0.777289	+	0.629144i	$0.783406\pi$
$278$	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	0.975059	−	0.221945i	$-0.0712404\pi$
$281$	59466.7	+	59466.7i	0.753115	+	0.753115i	−0.221945	+	0.975059i	$0.571240\pi$
$282$		0			0
$283$			49885.9i			0.622880i	0.950266	+	0.311440i	$0.100811\pi$
$283$			49885.9i			0.622880i	−0.950266	+	0.311440i	$0.899189\pi$
$284$	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	0.923994	+	0.382407i	$0.124905\pi$
$293$	112153.	+	112153.i	1.30640	+	1.30640i	0.382407	+	0.923994i	$0.375095\pi$
$294$		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	0.390557	−	0.920579i	$-0.372282\pi$
$307$	−49954.0	−	49954.0i	−0.530022	−	0.530022i	−0.920579	+	0.390557i	$0.872282\pi$
$308$	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	$-0.915253\pi$
$311$		−	50895.9i		−	0.526213i	0.964767	−	0.263107i	$-0.0847471\pi$
$312$		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	−0.160346	+	0.987061i	$0.551261\pi$
$317$	−115302.	+	115302.i	−1.14741	+	1.14741i	−0.987061	+	0.160346i	$0.948739\pi$
$318$		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	−0.204627	−	0.978840i	$-0.565598\pi$
$331$	84823.6	−	84823.6i	0.774213	−	0.774213i	0.978840	+	0.204627i	$0.0655980\pi$
$332$	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	$0.0503053\pi$
$337$			35747.4i			0.314764i	−0.987538	+	0.157382i	$0.949695\pi$
$338$	−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	0.864019	−	0.503459i	$-0.167940\pi$
$349$	43916.5	+	43916.5i	0.360559	+	0.360559i	−0.503459	+	0.864019i	$0.667940\pi$
$350$		−	738.544i		−	0.00602893i
$351$		0			0
$352$	−2891.13			−0.0233336
$353$	35106.8	−	35106.8i	0.281736	−	0.281736i	−0.552065	−	0.833801i	$-0.686160\pi$
$353$	35106.8	−	35106.8i	0.281736	−	0.281736i	0.833801	+	0.552065i	$0.186160\pi$
$354$		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	0.167631	−	0.985850i	$-0.446388\pi$
$359$	−105453.	−	105453.i	−0.818219	−	0.818219i	−0.985850	+	0.167631i	$0.946388\pi$
$360$		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	−0.703795	−	0.710403i	$-0.748513\pi$
$379$	949.057	−	949.057i	0.00660715	−	0.00660715i	0.710403	+	0.703795i	$0.248513\pi$
$380$	62334.3			0.431678
$381$		0			0
$382$	2176.83	−	2176.83i	0.0149175	−	0.0149175i
$383$	126901.	−	126901.i	0.865105	−	0.865105i	−0.126821	−	0.991926i	$-0.540477\pi$
$383$	126901.	−	126901.i	0.865105	−	0.865105i	0.991926	+	0.126821i	$0.0404774\pi$
$384$		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	$0.0137847\pi$
$389$			13102.1i			0.0865849i	−0.999062	+	0.0432925i	$0.986215\pi$
$390$		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	1.00000		0.000446211i	$-0.000142033\pi$
$397$	157539.	+	157539.i	0.999554	+	0.999554i	−0.000446211		1.00000i	$0.500142\pi$
$398$	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	0.513833	−	0.857890i	$-0.328225\pi$
$401$	−55324.7	−	55324.7i	−0.344057	−	0.344057i	−0.857890	+	0.513833i	$0.828225\pi$
$402$		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	−0.979475	−	0.201566i	$-0.935397\pi$
$409$	−130129.	+	130129.i	−0.777909	+	0.777909i	0.201566	+	0.979475i	$0.435397\pi$
$410$	−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	−0.271987	−	0.962301i	$-0.587681\pi$
$421$	122352.	−	122352.i	0.690314	−	0.690314i	0.962301	+	0.271987i	$0.0876808\pi$
$422$	−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	−0.791716	−	0.610890i	$-0.790812\pi$
$431$	−33590.4	+	33590.4i	−0.180826	+	0.180826i	0.610890	+	0.791716i	$0.290812\pi$
$432$		0			0
$433$		−	81942.2i		−	0.437051i	−0.975831	−	0.218525i	$-0.929875\pi$
$433$		−	81942.2i		−	0.437051i	0.975831	−	0.218525i	$-0.0701246\pi$
$434$	−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	$-0.934907\pi$
$439$		−	78273.4i		−	0.406149i	0.979163	−	0.203074i	$-0.0650933\pi$
$440$	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	0.998733	−	0.0503165i	$-0.0160230\pi$
$449$	191202.	+	191202.i	0.948417	+	0.948417i	−0.0503165	+	0.998733i	$0.516023\pi$
$450$		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	−0.470685	−	0.882301i	$-0.655993\pi$
$457$	85965.6	−	85965.6i	0.411616	−	0.411616i	0.882301	+	0.470685i	$0.155993\pi$
$458$	3977.20			0.0189604
$459$		0			0
$460$	−239660.	+	239660.i	−1.13261	+	1.13261i
$461$	231988.	−	231988.i	1.09160	−	1.09160i	0.0962424	−	0.995358i	$-0.469318\pi$
$461$	231988.	−	231988.i	1.09160	−	1.09160i	0.995358	−	0.0962424i	$-0.0306824\pi$
$462$		0			0
$463$	19082.7	+	19082.7i	0.0890180	+	0.0890180i	0.750214	−	0.661196i	$-0.229951\pi$
$463$	19082.7	+	19082.7i	0.0890180	+	0.0890180i	−0.661196	+	0.750214i	$0.729951\pi$
$464$	−122294.			−0.568025
$465$		0			0
$466$	890.443	+	890.443i	0.00410048	+	0.00410048i
$467$		−	122781.i		−	0.562987i	−0.959563	−	0.281493i	$-0.909170\pi$
$467$		−	122781.i		−	0.562987i	0.959563	−	0.281493i	$-0.0908298\pi$
$468$		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	0.465011	−	0.885305i	$-0.346051\pi$
$479$	−96432.8	−	96432.8i	−0.420295	−	0.420295i	−0.885305	+	0.465011i	$0.846051\pi$
$480$		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	−0.985023	−	0.172425i	$-0.944840\pi$
$487$	−192723.	+	192723.i	−0.812598	+	0.812598i	0.172425	+	0.985023i	$0.444840\pi$
$488$	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	$-0.938850\pi$
$491$		−	92057.9i		−	0.381855i	0.981604	−	0.190927i	$-0.0611495\pi$
$492$		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	−0.832886	−	0.553445i	$-0.813313\pi$
$499$	−69581.0	+	69581.0i	−0.279441	+	0.279441i	0.553445	+	0.832886i	$0.313313\pi$
$500$	−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	−0.0560201	+	0.998430i	$0.517841\pi$
$509$	−273188.	+	273188.i	−1.05445	+	1.05445i	−0.998430	+	0.0560201i	$0.982159\pi$
$510$		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	0.664601	−	0.747198i	$-0.268601\pi$
$541$	−24174.5	−	24174.5i	−0.0825968	−	0.0825968i	−0.747198	+	0.664601i	$0.768601\pi$
$542$	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	−0.998902	−	0.0468427i	$-0.985084\pi$
$557$	−324441.	−	324441.i	−1.04575	−	1.04575i	−0.0468427	−	0.998902i	$-0.514916\pi$
$558$		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	$0.0647291\pi$
$563$			128026.i			0.403908i	−0.979395	+	0.201954i	$0.935271\pi$
$564$		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	$0.187047\pi$
$569$			358977.i			1.10877i	−0.832259	+	0.554386i	$0.812953\pi$
$570$		0			0
$571$		−	183795.i		−	0.563718i	−0.959456	−	0.281859i	$-0.909049\pi$
$571$		−	183795.i		−	0.563718i	0.959456	−	0.281859i	$-0.0909510\pi$
$572$	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	−0.787654	−	0.616118i	$-0.788704\pi$
$577$	−57109.1	+	57109.1i	−0.171535	+	0.171535i	0.616118	+	0.787654i	$0.288704\pi$
$578$	−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	−0.330905	−	0.943664i	$-0.607354\pi$
$587$	211138.	−	211138.i	0.612759	−	0.612759i	0.943664	+	0.330905i	$0.107354\pi$
$588$		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	0.942572	−	0.334004i	$-0.108400\pi$
$593$	214002.	+	214002.i	0.608567	+	0.608567i	−0.334004	+	0.942572i	$0.608400\pi$
$594$		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	−0.0467552	−	0.998906i	$-0.514888\pi$
$613$	357789.	−	357789.i	0.952151	−	0.952151i	0.998906	+	0.0467552i	$0.0148881\pi$
$614$	8678.47			0.0230201
$615$		0			0
$616$	−2589.45	+	2589.45i	−0.00682410	+	0.00682410i
$617$	402876.	−	402876.i	1.05828	−	1.05828i	0.0600878	−	0.998193i	$-0.480862\pi$
$617$	402876.	−	402876.i	1.05828	−	1.05828i	0.998193	−	0.0600878i	$-0.0191381\pi$
$618$		0			0
$619$	−169086.	−	169086.i	−0.441293	−	0.441293i	0.451153	−	0.892446i	$-0.351013\pi$
$619$	−169086.	−	169086.i	−0.441293	−	0.441293i	−0.892446	+	0.451153i	$0.851013\pi$
$620$	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	0.996259	+	0.0864220i	$0.0275433\pi$
$631$	431081.	+	431081.i	1.08268	+	1.08268i	0.0864220	+	0.996259i	$0.472457\pi$
$632$	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	$0.142412\pi$
$641$			355513.i			0.865245i	−0.901575	+	0.432623i	$0.857588\pi$
$642$		0			0
$643$	17508.7	−	17508.7i	0.0423479	−	0.0423479i	−0.685616	−	0.727964i	$-0.740467\pi$
$643$	17508.7	−	17508.7i	0.0423479	−	0.0423479i	0.727964	+	0.685616i	$0.240467\pi$
$644$	−352262.	−	352262.i	−0.849365	−	0.849365i
$645$		0			0
$646$	−5838.94			−0.0139917
$647$		−	617081.i		−	1.47412i	−0.675826	−	0.737061i	$-0.736213\pi$
$647$		−	617081.i		−	1.47412i	0.675826	−	0.737061i	$-0.263787\pi$
$648$		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	0.638711	−	0.769447i	$-0.279468\pi$
$661$	−57121.2	−	57121.2i	−0.130736	−	0.130736i	−0.769447	+	0.638711i	$0.779468\pi$
$662$			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	$-0.770392\pi$
$673$		−	598216.i		−	1.32077i	0.750926	−	0.660387i	$-0.229608\pi$
$674$	−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	0.388050	−	0.921638i	$-0.373149\pi$
$683$	−248913.	−	248913.i	−0.533589	−	0.533589i	−0.921638	+	0.388050i	$0.873149\pi$
$684$		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	−0.929849	−	0.367943i	$-0.880062\pi$
$691$	−268299.	+	268299.i	−0.561906	+	0.561906i	0.367943	+	0.929849i	$0.380062\pi$
$692$	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	$0.184788\pi$
$701$			539033.i			1.09693i	−0.836173	+	0.548465i	$0.815212\pi$
$702$		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	−0.996608	−	0.0822903i	$-0.973777\pi$
$709$	−542342.	−	542342.i	−1.07890	−	1.07890i	−0.0822903	−	0.996608i	$-0.526223\pi$
$710$	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	$0.250994\pi$
$719$			733372.i			1.41862i	−0.704896	+	0.709311i	$0.749006\pi$
$720$		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	$0.0216280\pi$
$727$			71768.0i			0.135788i	−0.997693	+	0.0678941i	$0.978372\pi$
$728$	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	−0.593982	−	0.804478i	$-0.702445\pi$
$733$	113097.	−	113097.i	0.210496	−	0.210496i	0.804478	+	0.593982i	$0.202445\pi$
$734$	−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	0.806487	−	0.591251i	$-0.201366\pi$
$739$	117545.	+	117545.i	0.215236	+	0.215236i	−0.591251	+	0.806487i	$0.701366\pi$
$740$			570543.i			1.04190i
$741$		0			0
$742$	6414.13			0.0116501
$743$	507699.	−	507699.i	0.919663	−	0.919663i	−0.0773413	−	0.997005i	$-0.524643\pi$
$743$	507699.	−	507699.i	0.919663	−	0.919663i	0.997005	+	0.0773413i	$0.0246431\pi$
$744$		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	$0.126599\pi$
$751$			436896.i			0.774636i	−0.921946	+	0.387318i	$0.873401\pi$
$752$	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	0.759583	−	0.650410i	$-0.225403\pi$
$761$	63224.5	+	63224.5i	0.109173	+	0.109173i	−0.650410	+	0.759583i	$0.725403\pi$
$762$		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	−0.884481	−	0.466577i	$-0.845487\pi$
$769$	−247132.	+	247132.i	−0.417904	+	0.417904i	0.466577	+	0.884481i	$0.345487\pi$
$770$	2366.09			0.00399070
$771$		0			0
$772$	507601.	−	507601.i	0.851702	−	0.851702i
$773$	−370482.	+	370482.i	−0.620024	+	0.620024i	−0.945537	−	0.325513i	$-0.894463\pi$
$773$	−370482.	+	370482.i	−0.620024	+	0.620024i	0.325513	+	0.945537i	$0.394463\pi$
$774$		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	0.558677	−	0.829385i	$-0.311309\pi$
$787$	−167668.	−	167668.i	−0.270708	−	0.270708i	−0.829385	+	0.558677i	$0.811309\pi$
$788$	−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	$0.0287022\pi$
$797$			114399.i			0.180097i	−0.995937	+	0.0900483i	$0.971298\pi$
$798$		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	−0.533280	−	0.845939i	$-0.679041\pi$
$811$	205643.	−	205643.i	0.312660	−	0.312660i	0.845939	+	0.533280i	$0.179041\pi$
$812$	−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	−0.0220168	−	0.999758i	$-0.507009\pi$
$821$	659037.	−	659037.i	0.977741	−	0.977741i	0.999758	+	0.0220168i	$0.00700872\pi$
$822$		0			0
$823$			1.08710e6i			1.60498i	0.596664	+	0.802491i	$0.296493\pi$
$823$			1.08710e6i			1.60498i	−0.596664	+	0.802491i	$0.703507\pi$
$824$	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	0.337088	−	0.941473i	$-0.390558\pi$
$827$	−413357.	−	413357.i	−0.604385	−	0.604385i	−0.941473	+	0.337088i	$0.890558\pi$
$828$		0			0
$829$			857909.i			1.24834i	0.781290	+	0.624169i	$0.214562\pi$
$829$			857909.i			1.24834i	−0.781290	+	0.624169i	$0.785438\pi$
$830$	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	0.528404	−	0.848993i	$-0.322791\pi$
$839$	−225669.	−	225669.i	−0.320588	−	0.320588i	−0.848993	+	0.528404i	$0.822791\pi$
$840$		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	0.999522	−	0.0309037i	$-0.00983851\pi$
$853$	704776.	+	704776.i	0.968619	+	0.968619i	−0.0309037	+	0.999522i	$0.509839\pi$
$854$	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	$-0.759574\pi$
$857$		−	1.00696e6i		−	1.37105i	0.728051	−	0.685523i	$-0.240426\pi$
$858$		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	−0.0242565	−	0.999706i	$-0.507722\pi$
$863$	726484.	−	726484.i	0.975449	−	0.975449i	0.999706	+	0.0242565i	$0.00772184\pi$
$864$		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	−0.861441	−	0.507857i	$-0.830438\pi$
$877$	−271952.	+	271952.i	−0.353584	+	0.353584i	0.507857	+	0.861441i	$0.330438\pi$
$878$	6799.18	+	6799.18i	0.00881998	+	0.00881998i
$879$		0			0
$880$	161867.			0.209022
$881$			252034.i			0.324719i	0.986732	+	0.162360i	$0.0519104\pi$
$881$			252034.i			0.324719i	−0.986732	+	0.162360i	$0.948090\pi$
$882$		0			0
$883$		−	927130.i		−	1.18910i	−0.804058	−	0.594551i	$-0.797330\pi$
$883$		−	927130.i		−	1.18910i	0.804058	−	0.594551i	$-0.202670\pi$
$884$	−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	$0.0454536\pi$
$907$			234146.i			0.284624i	−0.989822	+	0.142312i	$0.954546\pi$
$908$	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	−0.706122	−	0.708090i	$-0.749557\pi$
$929$	1698.00	−	1698.00i	0.00196746	−	0.00196746i	0.708090	+	0.706122i	$0.249557\pi$
$930$		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	−0.0466000	−	0.998914i	$-0.514839\pi$
$941$	843256.	−	843256.i	0.952314	−	0.952314i	0.998914	+	0.0466000i	$0.0148386\pi$
$942$		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	0.854046	−	0.520198i	$-0.174142\pi$
$947$	299398.	+	299398.i	0.333848	+	0.333848i	−0.520198	+	0.854046i	$0.674142\pi$
$948$		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	$-0.826533\pi$
$953$		−	941607.i		−	1.03677i	0.855146	−	0.518387i	$-0.173467\pi$
$954$		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	−0.808992	−	0.587819i	$-0.799987\pi$
$967$	−206817.	+	206817.i	−0.221173	+	0.221173i	0.587819	+	0.808992i	$0.299987\pi$
$968$	−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	−0.647651	−	0.761937i	$-0.724248\pi$
$977$	109090.	−	109090.i	0.114287	−	0.114287i	0.761937	+	0.647651i	$0.224248\pi$
$978$		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	0.819862	−	0.572561i	$-0.194050\pi$
$983$	238965.	+	238965.i	0.247302	+	0.247302i	−0.572561	+	0.819862i	$0.694050\pi$
$984$		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.109.6		20
3.2	odd	2		39.5.g.a.31.5	✓	20
13.8	odd	4	inner	117.5.j.b.73.6		20
39.8	even	4		39.5.g.a.34.5	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.5	✓	20	3.2	odd	2
39.5.g.a.34.5	yes	20	39.8	even	4
117.5.j.b.73.6		20	13.8	odd	4	inner
117.5.j.b.109.6		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
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

Related objects

Downloads

Learn more