LMFDB - Embedded newform 117.5.j.b.109.5

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			109.5
Root			$-0.755803 + 0.755803i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.755803 + 0.755803i) q^{2} +14.8575i q^{4} +(-25.9238 + 25.9238i) q^{5} +(55.8718 + 55.8718i) q^{7} +(-23.3222 - 23.3222i) q^{8} -39.1866i q^{10} +(-44.0036 - 44.0036i) q^{11} +(3.89259 - 168.955i) q^{13} -84.4562 q^{14} -202.466 q^{16} -77.6827i q^{17} +(-246.651 + 246.651i) q^{19} +(-385.164 - 385.164i) q^{20} +66.5162 q^{22} +513.460i q^{23} -719.088i q^{25} +(124.755 + 130.639i) q^{26} +(-830.116 + 830.116i) q^{28} +197.709 q^{29} +(108.512 - 108.512i) q^{31} +(526.180 - 526.180i) q^{32} +(58.7128 + 58.7128i) q^{34} -2896.82 q^{35} +(-739.885 - 739.885i) q^{37} -372.839i q^{38} +1209.20 q^{40} +(-1253.54 + 1253.54i) q^{41} -1757.44i q^{43} +(653.785 - 653.785i) q^{44} +(-388.075 - 388.075i) q^{46} +(-2396.04 - 2396.04i) q^{47} +3842.31i q^{49} +(543.489 + 543.489i) q^{50} +(2510.26 + 57.8343i) q^{52} +5503.35 q^{53} +2281.48 q^{55} -2606.11i q^{56} +(-149.429 + 149.429i) q^{58} +(4609.57 + 4609.57i) q^{59} -3179.03 q^{61} +164.028i q^{62} -2444.08i q^{64} +(4279.05 + 4480.87i) q^{65} +(-2737.08 + 2737.08i) q^{67} +1154.17 q^{68} +(2189.43 - 2189.43i) q^{70} +(-5327.99 + 5327.99i) q^{71} +(6132.89 + 6132.89i) q^{73} +1118.42 q^{74} +(-3664.62 - 3664.62i) q^{76} -4917.12i q^{77} -5972.92 q^{79} +(5248.70 - 5248.70i) q^{80} -1894.85i q^{82} +(-723.125 + 723.125i) q^{83} +(2013.83 + 2013.83i) q^{85} +(1328.28 + 1328.28i) q^{86} +2052.53i q^{88} +(5048.32 + 5048.32i) q^{89} +(9657.31 - 9222.34i) q^{91} -7628.75 q^{92} +3621.87 q^{94} -12788.3i q^{95} +(1216.16 - 1216.16i) q^{97} +(-2904.03 - 2904.03i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.755803	+	0.755803i	−0.188951	+	0.188951i	−0.795242	−	0.606292i	$-0.792656\pi$
$2$	−0.755803	+	0.755803i	−0.188951	+	0.188951i	0.606292	+	0.795242i	$0.292656\pi$
$3$		0			0
$3$		0			0
$4$			14.8575i			0.928595i
$5$	−25.9238	+	25.9238i	−1.03695	+	1.03695i	−0.0376619	+	0.999291i	$0.511991\pi$
$5$	−25.9238	+	25.9238i	−1.03695	+	1.03695i	−0.999291	+	0.0376619i	$0.988009\pi$
$6$		0			0
$7$	55.8718	+	55.8718i	1.14024	+	1.14024i	0.988406	+	0.151835i	$0.0485182\pi$
$7$	55.8718	+	55.8718i	1.14024	+	1.14024i	0.151835	+	0.988406i	$0.451482\pi$
$8$	−23.3222	−	23.3222i	−0.364410	−	0.364410i
$9$		0			0
$10$		−	39.1866i		−	0.391866i
$11$	−44.0036	−	44.0036i	−0.363666	−	0.363666i	0.501494	−	0.865161i	$-0.332784\pi$
$11$	−44.0036	−	44.0036i	−0.363666	−	0.363666i	−0.865161	+	0.501494i	$0.832784\pi$
$12$		0			0
$13$	3.89259	−	168.955i	0.0230331	−	0.999735i
$13$	3.89259	−	168.955i	0.0230331	−	0.999735i
$14$	−84.4562			−0.430899
$15$		0			0
$16$	−202.466			−0.790884
$17$		−	77.6827i		−	0.268798i	−0.990927	−	0.134399i	$-0.957090\pi$
$17$		−	77.6827i		−	0.268798i	0.990927	−	0.134399i	$-0.0429104\pi$
$18$		0			0
$19$	−246.651	+	246.651i	−0.683244	+	0.683244i	−0.960730	−	0.277486i	$-0.910499\pi$
$19$	−246.651	+	246.651i	−0.683244	+	0.683244i	0.277486	+	0.960730i	$0.410499\pi$
$20$	−385.164	−	385.164i	−0.962909	−	0.962909i
$21$		0			0
$22$	66.5162			0.137430
$23$			513.460i			0.970624i	0.874341	+	0.485312i	$0.161294\pi$
$23$			513.460i			0.970624i	−0.874341	+	0.485312i	$0.838706\pi$
$24$		0			0
$25$		−	719.088i		−	1.15054i
$26$	124.755	+	130.639i	0.184549	+	0.193253i
$27$		0			0
$28$	−830.116	+	830.116i	−1.05882	+	1.05882i
$29$	197.709			0.235088			0.117544	−	0.993068i	$-0.462498\pi$
$29$	197.709			0.235088			0.117544	+	0.993068i	$0.462498\pi$
$30$		0			0
$31$	108.512	−	108.512i	0.112916	−	0.112916i	−0.648391	−	0.761307i	$-0.724558\pi$
$31$	108.512	−	108.512i	0.112916	−	0.112916i	0.761307	+	0.648391i	$0.224558\pi$
$32$	526.180	−	526.180i	0.513848	−	0.513848i
$33$		0			0
$34$	58.7128	+	58.7128i	0.0507896	+	0.0507896i
$35$	−2896.82			−2.36475
$36$		0			0
$37$	−739.885	−	739.885i	−0.540457	−	0.540457i	0.383206	−	0.923663i	$-0.374820\pi$
$37$	−739.885	−	739.885i	−0.540457	−	0.540457i	−0.923663	+	0.383206i	$0.874820\pi$
$38$		−	372.839i		−	0.258199i
$39$		0			0
$40$	1209.20			0.755751
$41$	−1253.54	+	1253.54i	−0.745709	+	0.745709i	−0.973670	−	0.227961i	$-0.926794\pi$
$41$	−1253.54	+	1253.54i	−0.745709	+	0.745709i	0.227961	+	0.973670i	$0.426794\pi$
$42$		0			0
$43$		−	1757.44i		−	0.950480i	−0.879856	−	0.475240i	$-0.842361\pi$
$43$		−	1757.44i		−	0.950480i	0.879856	−	0.475240i	$-0.157639\pi$
$44$	653.785	−	653.785i	0.337699	−	0.337699i
$45$		0			0
$46$	−388.075	−	388.075i	−0.183400	−	0.183400i
$47$	−2396.04	−	2396.04i	−1.08467	−	1.08467i	−0.996067	−	0.0886054i	$-0.971759\pi$
$47$	−2396.04	−	2396.04i	−1.08467	−	1.08467i	−0.0886054	−	0.996067i	$-0.528241\pi$
$48$		0			0
$49$			3842.31i			1.60030i
$50$	543.489	+	543.489i	0.217396	+	0.217396i
$51$		0			0
$52$	2510.26	+	57.8343i	0.928349	+	0.0213884i
$53$	5503.35			1.95918			0.979592	−	0.200997i	$-0.0644180\pi$
$53$	5503.35			1.95918			0.979592	+	0.200997i	$0.0644180\pi$
$54$		0			0
$55$	2281.48			0.754210
$56$		−	2606.11i		−	0.831030i
$57$		0			0
$58$	−149.429	+	149.429i	−0.0444201	+	0.0444201i
$59$	4609.57	+	4609.57i	1.32421	+	1.32421i	0.910333	+	0.413876i	$0.135825\pi$
$59$	4609.57	+	4609.57i	1.32421	+	1.32421i	0.413876	+	0.910333i	$0.364175\pi$
$60$		0			0
$61$	−3179.03			−0.854349			−0.427175	−	0.904169i	$-0.640491\pi$
$61$	−3179.03			−0.854349			−0.427175	+	0.904169i	$0.640491\pi$
$62$			164.028i			0.0426711i
$63$		0			0
$64$		−	2444.08i		−	0.596700i
$65$	4279.05	+	4480.87i	1.01279	+	1.06056i
$66$		0			0
$67$	−2737.08	+	2737.08i	−0.609729	+	0.609729i	−0.942875	−	0.333146i	$-0.891890\pi$
$67$	−2737.08	+	2737.08i	−0.609729	+	0.609729i	0.333146	+	0.942875i	$0.391890\pi$
$68$	1154.17			0.249605
$69$		0			0
$70$	2189.43	−	2189.43i	0.446822	−	0.446822i
$71$	−5327.99	+	5327.99i	−1.05693	+	1.05693i	−0.0586519	+	0.998278i	$0.518680\pi$
$71$	−5327.99	+	5327.99i	−1.05693	+	1.05693i	−0.998278	+	0.0586519i	$0.981320\pi$
$72$		0			0
$73$	6132.89	+	6132.89i	1.15085	+	1.15085i	0.986382	+	0.164470i	$0.0525915\pi$
$73$	6132.89	+	6132.89i	1.15085	+	1.15085i	0.164470	+	0.986382i	$0.447408\pi$
$74$	1118.42			0.204240
$75$		0			0
$76$	−3664.62	−	3664.62i	−0.634457	−	0.634457i
$77$		−	4917.12i		−	0.829335i
$78$		0			0
$79$	−5972.92			−0.957046			−0.478523	−	0.878075i	$-0.658828\pi$
$79$	−5972.92			−0.957046			−0.478523	+	0.878075i	$0.658828\pi$
$80$	5248.70	−	5248.70i	0.820109	−	0.820109i
$81$		0			0
$82$		−	1894.85i		−	0.281805i
$83$	−723.125	+	723.125i	−0.104968	+	0.104968i	−0.757640	−	0.652672i	$-0.773648\pi$
$83$	−723.125	+	723.125i	−0.104968	+	0.104968i	0.652672	+	0.757640i	$0.273648\pi$
$84$		0			0
$85$	2013.83	+	2013.83i	0.278731	+	0.278731i
$86$	1328.28	+	1328.28i	0.179594	+	0.179594i
$87$		0			0
$88$			2052.53i			0.265047i
$89$	5048.32	+	5048.32i	0.637334	+	0.637334i	0.949897	−	0.312563i	$-0.101188\pi$
$89$	5048.32	+	5048.32i	0.637334	+	0.637334i	−0.312563	+	0.949897i	$0.601188\pi$
$90$		0			0
$91$	9657.31	−	9222.34i	1.16620	−	1.11367i
$92$	−7628.75			−0.901317
$93$		0			0
$94$	3621.87			0.409899
$95$		−	12788.3i		−	1.41698i
$96$		0			0
$97$	1216.16	−	1216.16i	0.129255	−	0.129255i	−0.639520	−	0.768775i	$-0.720867\pi$
$97$	1216.16	−	1216.16i	0.129255	−	0.129255i	0.768775	+	0.639520i	$0.220867\pi$
$98$	−2904.03	−	2904.03i	−0.302378	−	0.302378i
$99$		0			0
$100$	10683.9			1.06839
$101$			1635.21i			0.160299i	0.996783	+	0.0801497i	$0.0255398\pi$
$101$			1635.21i			0.160299i	−0.996783	+	0.0801497i	$0.974460\pi$
$102$		0			0
$103$			16777.5i			1.58144i	0.612177	+	0.790721i	$0.290294\pi$
$103$			16777.5i			1.58144i	−0.612177	+	0.790721i	$0.709706\pi$
$104$	−4031.19	+	3849.63i	−0.372707	+	0.355920i
$105$		0			0
$106$	−4159.45	+	4159.45i	−0.370189	+	0.370189i
$107$	−2250.96			−0.196608			−0.0983039	−	0.995156i	$-0.531342\pi$
$107$	−2250.96			−0.196608			−0.0983039	+	0.995156i	$0.531342\pi$
$108$		0			0
$109$	−4156.99	+	4156.99i	−0.349886	+	0.349886i	−0.860067	−	0.510181i	$-0.829578\pi$
$109$	−4156.99	+	4156.99i	−0.349886	+	0.349886i	0.510181	+	0.860067i	$0.329578\pi$
$110$	−1724.35	+	1724.35i	−0.142509	+	0.142509i
$111$		0			0
$112$	−11312.2	−	11312.2i	−0.901798	−	0.901798i
$113$	−20903.0			−1.63701			−0.818507	−	0.574497i	$-0.805198\pi$
$113$	−20903.0			−1.63701			−0.818507	+	0.574497i	$0.805198\pi$
$114$		0			0
$115$	−13310.8	−	13310.8i	−1.00649	−	1.00649i
$116$			2937.47i			0.218302i
$117$		0			0
$118$	−6967.86			−0.500421
$119$	4340.27	−	4340.27i	0.306495	−	0.306495i
$120$		0			0
$121$		−	10768.4i		−	0.735493i
$122$	2402.72	−	2402.72i	0.161430	−	0.161430i
$123$		0			0
$124$	1612.22	+	1612.22i	0.104853	+	0.104853i
$125$	2439.12	+	2439.12i	0.156104	+	0.156104i
$126$		0			0
$127$			8903.28i			0.552005i	0.961157	+	0.276002i	$0.0890098\pi$
$127$			8903.28i			0.552005i	−0.961157	+	0.276002i	$0.910990\pi$
$128$	10266.1	+	10266.1i	0.626595	+	0.626595i
$129$		0			0
$130$	−6620.78	−	152.538i	−0.391762	−	0.00902589i
$131$	−19604.9			−1.14241			−0.571204	−	0.820808i	$-0.693524\pi$
$131$	−19604.9			−1.14241			−0.571204	+	0.820808i	$0.693524\pi$
$132$		0			0
$133$	−27561.7			−1.55812
$134$		−	4137.38i		−	0.230418i
$135$		0			0
$136$	−1811.73	+	1811.73i	−0.0979527	+	0.0979527i
$137$	14793.4	+	14793.4i	0.788183	+	0.788183i	0.981196	−	0.193013i	$-0.0618259\pi$
$137$	14793.4	+	14793.4i	0.788183	+	0.788183i	−0.193013	+	0.981196i	$0.561826\pi$
$138$		0			0
$139$	29205.3			1.51158			0.755790	−	0.654814i	$-0.227253\pi$
$139$	29205.3			1.51158			0.755790	+	0.654814i	$0.227253\pi$
$140$		−	43039.6i		−	2.19590i
$141$		0			0
$142$		−	8053.82i		−	0.399416i
$143$	−7605.93	+	7263.35i	−0.371946	+	0.355194i
$144$		0			0
$145$	−5125.38	+	5125.38i	−0.243775	+	0.243775i
$146$	−9270.52			−0.434909
$147$		0			0
$148$	10992.9	−	10992.9i	0.501866	−	0.501866i
$149$	−4521.47	+	4521.47i	−0.203661	+	0.203661i	−0.801566	−	0.597906i	$-0.796000\pi$
$149$	−4521.47	+	4521.47i	−0.203661	+	0.203661i	0.597906	+	0.801566i	$0.296000\pi$
$150$		0			0
$151$	6088.39	+	6088.39i	0.267023	+	0.267023i	0.827899	−	0.560877i	$-0.189536\pi$
$151$	6088.39	+	6088.39i	0.267023	+	0.267023i	−0.560877	+	0.827899i	$0.689536\pi$
$152$	11504.9			0.497961
$153$		0			0
$154$	3716.38	+	3716.38i	0.156703	+	0.156703i
$155$			5626.10i			0.234177i
$156$		0			0
$157$	−12700.8			−0.515267			−0.257634	−	0.966243i	$-0.582943\pi$
$157$	−12700.8			−0.515267			−0.257634	+	0.966243i	$0.582943\pi$
$158$	4514.36	−	4514.36i	0.180835	−	0.180835i
$159$		0			0
$160$			27281.2i			1.06567i
$161$	−28687.9	+	28687.9i	−1.10675	+	1.10675i
$162$		0			0
$163$	18752.1	+	18752.1i	0.705787	+	0.705787i	0.965646	−	0.259859i	$-0.0836762\pi$
$163$	18752.1	+	18752.1i	0.705787	+	0.705787i	−0.259859	+	0.965646i	$0.583676\pi$
$164$	−18624.5	−	18624.5i	−0.692462	−	0.692462i
$165$		0			0
$166$		−	1093.08i		−	0.0396676i
$167$	11683.9	+	11683.9i	0.418942	+	0.418942i	0.884839	−	0.465897i	$-0.154268\pi$
$167$	11683.9	+	11683.9i	0.418942	+	0.418942i	−0.465897	+	0.884839i	$0.654268\pi$
$168$		0			0
$169$	−28530.7	−	1315.35i	−0.998939	−	0.0460540i
$170$	−3044.12			−0.105333
$171$		0			0
$172$	26111.2			0.882611
$173$		−	30610.1i		−	1.02276i	−0.859356	−	0.511378i	$-0.829135\pi$
$173$		−	30610.1i		−	1.02276i	0.859356	−	0.511378i	$-0.170865\pi$
$174$		0			0
$175$	40176.7	−	40176.7i	1.31189	−	1.31189i
$176$	8909.26	+	8909.26i	0.287618	+	0.287618i
$177$		0			0
$178$	−7631.08			−0.240850
$179$		−	3696.07i		−	0.115354i	−0.998335	−	0.0576772i	$-0.981631\pi$
$179$		−	3696.07i		−	0.115354i	0.998335	−	0.0576772i	$-0.0183694\pi$
$180$		0			0
$181$		−	9870.81i		−	0.301298i	−0.988587	−	0.150649i	$-0.951864\pi$
$181$		−	9870.81i		−	0.301298i	0.988587	−	0.150649i	$-0.0481363\pi$
$182$	−328.754	+	14269.3i	−0.00992494	+	0.430785i
$183$		0			0
$184$	11975.0	−	11975.0i	0.353705	−	0.353705i
$185$	38361.3			1.12086
$186$		0			0
$187$	−3418.32	+	3418.32i	−0.0977529	+	0.0977529i
$188$	35599.2	−	35599.2i	1.00722	−	1.00722i
$189$		0			0
$190$	9665.42	+	9665.42i	0.267740	+	0.267740i
$191$	28809.3			0.789708			0.394854	−	0.918744i	$-0.370795\pi$
$191$	28809.3			0.789708			0.394854	+	0.918744i	$0.370795\pi$
$192$		0			0
$193$	15092.0	+	15092.0i	0.405166	+	0.405166i	0.880049	−	0.474883i	$-0.157510\pi$
$193$	15092.0	+	15092.0i	0.405166	+	0.405166i	−0.474883	+	0.880049i	$0.657510\pi$
$194$			1838.36i			0.0488457i
$195$		0			0
$196$	−57087.3			−1.48603
$197$	−14144.9	+	14144.9i	−0.364474	+	0.364474i	−0.865457	−	0.500983i	$-0.832972\pi$
$197$	−14144.9	+	14144.9i	−0.364474	+	0.364474i	0.500983	+	0.865457i	$0.332972\pi$
$198$		0			0
$199$			11143.9i			0.281403i	0.990052	+	0.140702i	$0.0449359\pi$
$199$			11143.9i			0.281403i	−0.990052	+	0.140702i	$0.955064\pi$
$200$	−16770.7	+	16770.7i	−0.419268	+	0.419268i
$201$		0			0
$202$	−1235.90	−	1235.90i	−0.0302887	−	0.0302887i
$203$	11046.4	+	11046.4i	0.268057	+	0.268057i
$204$		0			0
$205$		−	64992.9i		−	1.54653i
$206$	−12680.5	−	12680.5i	−0.298815	−	0.298815i
$207$		0			0
$208$	−788.119	+	34207.7i	−0.0182165	+	0.790674i
$209$	21707.1			0.496946
$210$		0			0
$211$	−7090.45			−0.159261			−0.0796304	−	0.996824i	$-0.525374\pi$
$211$	−7090.45			−0.159261			−0.0796304	+	0.996824i	$0.525374\pi$
$212$			81766.1i			1.81929i
$213$		0			0
$214$	1701.29	−	1701.29i	0.0371492	−	0.0371492i
$215$	45559.5	+	45559.5i	0.985602	+	0.985602i
$216$		0			0
$217$	12125.5			0.257503
$218$		−	6283.74i		−	0.132222i
$219$		0			0
$220$			33897.2i			0.700355i
$221$	−13124.9	−	302.387i	−0.268727	−	0.00619126i
$222$		0			0
$223$	−61073.2	+	61073.2i	−1.22812	+	1.22812i	−0.263447	+	0.964674i	$0.584859\pi$
$223$	−61073.2	+	61073.2i	−1.22812	+	1.22812i	−0.964674	+	0.263447i	$0.915141\pi$
$224$	58797.3			1.17182
$225$		0			0
$226$	15798.6	−	15798.6i	0.309315	−	0.309315i
$227$	47906.1	−	47906.1i	0.929692	−	0.929692i	−0.0679936	−	0.997686i	$-0.521660\pi$
$227$	47906.1	−	47906.1i	0.929692	−	0.929692i	0.997686	+	0.0679936i	$0.0216597\pi$
$228$		0			0
$229$	−32915.8	−	32915.8i	−0.627673	−	0.627673i	0.319809	−	0.947482i	$-0.396381\pi$
$229$	−32915.8	−	32915.8i	−0.627673	−	0.627673i	−0.947482	+	0.319809i	$0.896381\pi$
$230$	20120.8			0.380355
$231$		0			0
$232$	−4611.02	−	4611.02i	−0.0856684	−	0.0856684i
$233$		−	93191.8i		−	1.71659i	−0.513160	−	0.858293i	$-0.671525\pi$
$233$		−	93191.8i		−	1.71659i	0.513160	−	0.858293i	$-0.328475\pi$
$234$		0			0
$235$	124229.			2.24951
$236$	−68486.8	+	68486.8i	−1.22965	+	1.22965i
$237$		0			0
$238$			6560.78i			0.115825i
$239$	7463.00	−	7463.00i	0.130652	−	0.130652i	−0.638757	−	0.769409i	$-0.720551\pi$
$239$	7463.00	−	7463.00i	0.130652	−	0.130652i	0.769409	+	0.638757i	$0.220551\pi$
$240$		0			0
$241$	4266.86	+	4266.86i	0.0734640	+	0.0734640i	0.742884	−	0.669420i	$-0.233457\pi$
$241$	4266.86	+	4266.86i	0.0734640	+	0.0734640i	−0.669420	+	0.742884i	$0.733457\pi$
$242$	8138.76	+	8138.76i	0.138972	+	0.138972i
$243$		0			0
$244$		−	47232.5i		−	0.793344i
$245$	−99607.4	−	99607.4i	−1.65943	−	1.65943i
$246$		0			0
$247$	40712.8	+	42633.1i	0.667325	+	0.698800i
$248$	−5061.49			−0.0822953
$249$		0			0
$250$	−3686.99			−0.0589918
$251$		−	59125.2i		−	0.938480i	−0.883071	−	0.469240i	$-0.844528\pi$
$251$		−	59125.2i		−	0.938480i	0.883071	−	0.469240i	$-0.155472\pi$
$252$		0			0
$253$	22594.1	−	22594.1i	0.352983	−	0.352983i
$254$	−6729.13	−	6729.13i	−0.104302	−	0.104302i
$255$		0			0
$256$	23587.0			0.359909
$257$			50094.3i			0.758442i	0.925306	+	0.379221i	$0.123808\pi$
$257$			50094.3i			0.758442i	−0.925306	+	0.379221i	$0.876192\pi$
$258$		0			0
$259$		−	82677.5i		−	1.23250i
$260$	−66574.7	+	63576.1i	−0.984832	+	0.940475i
$261$		0			0
$262$	14817.4	−	14817.4i	0.215859	−	0.215859i
$263$	−14909.8			−0.215556			−0.107778	−	0.994175i	$-0.534374\pi$
$263$	−14909.8			−0.215556			−0.107778	+	0.994175i	$0.534374\pi$
$264$		0			0
$265$	−142668.	+	142668.i	−2.03158	+	2.03158i
$266$	20831.2	−	20831.2i	0.294409	−	0.294409i
$267$		0			0
$268$	−40666.2	−	40666.2i	−0.566192	−	0.566192i
$269$	−55704.4			−0.769812			−0.384906	−	0.922956i	$-0.625766\pi$
$269$	−55704.4			−0.769812			−0.384906	+	0.922956i	$0.625766\pi$
$270$		0			0
$271$	−4600.69	−	4600.69i	−0.0626447	−	0.0626447i	0.675090	−	0.737735i	$-0.264105\pi$
$271$	−4600.69	−	4600.69i	−0.0626447	−	0.0626447i	−0.737735	+	0.675090i	$0.764105\pi$
$272$			15728.1i			0.212588i
$273$		0			0
$274$	−22361.8			−0.297856
$275$	−31642.5	+	31642.5i	−0.418413	+	0.418413i
$276$		0			0
$277$		−	76710.8i		−	0.999762i	−0.866094	−	0.499881i	$-0.833377\pi$
$277$		−	76710.8i		−	0.999762i	0.866094	−	0.499881i	$-0.166623\pi$
$278$	−22073.4	+	22073.4i	−0.285614	+	0.285614i
$279$		0			0
$280$	67560.3	+	67560.3i	0.861738	+	0.861738i
$281$	25395.9	+	25395.9i	0.321626	+	0.321626i	0.849391	−	0.527765i	$-0.176970\pi$
$281$	25395.9	+	25395.9i	0.321626	+	0.321626i	−0.527765	+	0.849391i	$0.676970\pi$
$282$		0			0
$283$			41249.2i			0.515042i	0.966273	+	0.257521i	$0.0829056\pi$
$283$			41249.2i			0.515042i	−0.966273	+	0.257521i	$0.917094\pi$
$284$	−79160.7	−	79160.7i	−0.981460	−	0.981460i
$285$		0			0
$286$	258.921	−	11238.3i	0.00316544	−	0.137394i
$287$	−140075.			−1.70058
$288$		0			0
$289$	77486.4			0.927748
$290$		−	7747.55i		−	0.0921231i
$291$		0			0
$292$	−91119.6	+	91119.6i	−1.06868	+	1.06868i
$293$	22812.4	+	22812.4i	0.265727	+	0.265727i	0.827376	−	0.561649i	$-0.189833\pi$
$293$	22812.4	+	22812.4i	0.265727	+	0.265727i	−0.561649	+	0.827376i	$0.689833\pi$
$294$		0			0
$295$	−238995.			−2.74628
$296$			34511.5i			0.393895i
$297$		0			0
$298$		−	6834.69i		−	0.0769638i
$299$	86751.8	+	1998.69i	0.970367	+	0.0223565i
$300$		0			0
$301$	98191.1	−	98191.1i	1.08378	−	1.08378i
$302$	−9203.25			−0.100908
$303$		0			0
$304$	49938.5	−	49938.5i	0.540367	−	0.540367i
$305$	82412.6	−	82412.6i	0.885919	−	0.885919i
$306$		0			0
$307$	75497.8	+	75497.8i	0.801047	+	0.801047i	0.983259	−	0.182212i	$-0.0583259\pi$
$307$	75497.8	+	75497.8i	0.801047	+	0.801047i	−0.182212	+	0.983259i	$0.558326\pi$
$308$	73056.3			0.770116
$309$		0			0
$310$	−4252.23	−	4252.23i	−0.0442479	−	0.0442479i
$311$			53069.8i			0.548690i	0.961631	+	0.274345i	$0.0884610\pi$
$311$			53069.8i			0.548690i	−0.961631	+	0.274345i	$0.911539\pi$
$312$		0			0
$313$	−46540.0			−0.475048			−0.237524	−	0.971382i	$-0.576336\pi$
$313$	−46540.0			−0.475048			−0.237524	+	0.971382i	$0.576336\pi$
$314$	9599.33	−	9599.33i	0.0973602	−	0.0973602i
$315$		0			0
$316$		−	88742.8i		−	0.888708i
$317$	−32779.9	+	32779.9i	−0.326203	+	0.326203i	−0.851141	−	0.524937i	$-0.824089\pi$
$317$	−32779.9	+	32779.9i	−0.326203	+	0.326203i	0.524937	+	0.851141i	$0.324089\pi$
$318$		0			0
$319$	−8699.92	−	8699.92i	−0.0854937	−	0.0854937i
$320$	63360.0	+	63360.0i	0.618750	+	0.618750i
$321$		0			0
$322$		−	43364.9i		−	0.418241i
$323$	19160.5	+	19160.5i	0.183655	+	0.183655i
$324$		0			0
$325$	−121494.	−	2799.12i	−1.15024	−	0.0265005i
$326$	−28345.7			−0.266718
$327$		0			0
$328$	58470.5			0.543487
$329$		−	267742.i		−	2.47357i
$330$		0			0
$331$	−10224.0	+	10224.0i	−0.0933177	+	0.0933177i	−0.752225	−	0.658907i	$-0.771019\pi$
$331$	−10224.0	+	10224.0i	−0.0933177	+	0.0933177i	0.658907	+	0.752225i	$0.271019\pi$
$332$	−10743.8	−	10743.8i	−0.0974729	−	0.0974729i
$333$		0			0
$334$	−17661.4			−0.158319
$335$		−	141911.i		−	1.26452i
$336$		0			0
$337$		−	83180.9i		−	0.732426i	−0.930531	−	0.366213i	$-0.880654\pi$
$337$		−	83180.9i		−	0.732426i	0.930531	−	0.366213i	$-0.119346\pi$
$338$	22557.7	−	20569.5i	0.197452	−	0.180048i
$339$		0			0
$340$	−29920.5	+	29920.5i	−0.258828	+	0.258828i
$341$	−9549.87			−0.0821275
$342$		0			0
$343$	−80528.9	+	80528.9i	−0.684484	+	0.684484i
$344$	−40987.3	+	40987.3i	−0.346364	+	0.346364i
$345$		0			0
$346$	23135.2	+	23135.2i	0.193251	+	0.193251i
$347$	113467.			0.942343			0.471172	−	0.882041i	$-0.343831\pi$
$347$	113467.			0.942343			0.471172	+	0.882041i	$0.343831\pi$
$348$		0			0
$349$	85102.7	+	85102.7i	0.698703	+	0.698703i	0.964131	−	0.265428i	$-0.0855132\pi$
$349$	85102.7	+	85102.7i	0.698703	+	0.698703i	−0.265428	+	0.964131i	$0.585513\pi$
$350$			60731.4i			0.495767i
$351$		0			0
$352$	−46307.7			−0.373739
$353$	−143909.	+	143909.i	−1.15488	+	1.15488i	−0.169320	+	0.985561i	$0.554157\pi$
$353$	−143909.	+	143909.i	−1.15488	+	1.15488i	−0.985561	+	0.169320i	$0.945843\pi$
$354$		0			0
$355$		−	276243.i		−	2.19197i
$356$	−75005.6	+	75005.6i	−0.591825	+	0.591825i
$357$		0			0
$358$	2793.50	+	2793.50i	0.0217963	+	0.0217963i
$359$	61789.1	+	61789.1i	0.479427	+	0.479427i	0.904948	−	0.425521i	$-0.139909\pi$
$359$	61789.1	+	61789.1i	0.479427	+	0.479427i	−0.425521	+	0.904948i	$0.639909\pi$
$360$		0			0
$361$			8647.58i			0.0663560i
$362$	7460.39	+	7460.39i	0.0569305	+	0.0569305i
$363$		0			0
$364$	137021.	+	143484.i	1.03415	+	1.08293i
$365$	−317976.			−2.38676
$366$		0			0
$367$	83785.3			0.622065			0.311032	−	0.950399i	$-0.399325\pi$
$367$	83785.3			0.622065			0.311032	+	0.950399i	$0.399325\pi$
$368$		−	103958.i		−	0.767651i
$369$		0			0
$370$	−28993.6	+	28993.6i	−0.211787	+	0.211787i
$371$	307482.	+	307482.i	2.23394	+	2.23394i
$372$		0			0
$373$	204445.			1.46946			0.734731	−	0.678358i	$-0.237308\pi$
$373$	204445.			1.46946			0.734731	+	0.678358i	$0.237308\pi$
$374$		−	5167.16i		−	0.0369410i
$375$		0			0
$376$			111762.i			0.790530i
$377$	769.602	−	33404.0i	0.00541481	−	0.235026i
$378$		0			0
$379$	34520.1	−	34520.1i	0.240322	−	0.240322i	−0.576661	−	0.816983i	$-0.695645\pi$
$379$	34520.1	−	34520.1i	0.240322	−	0.240322i	0.816983	+	0.576661i	$0.195645\pi$
$380$	190002.			1.31580
$381$		0			0
$382$	−21774.2	+	21774.2i	−0.149216	+	0.149216i
$383$	32143.1	−	32143.1i	0.219124	−	0.219124i	−0.589005	−	0.808129i	$-0.700480\pi$
$383$	32143.1	−	32143.1i	0.219124	−	0.219124i	0.808129	+	0.589005i	$0.200480\pi$
$384$		0			0
$385$	127471.	+	127471.i	0.859981	+	0.859981i
$386$	−22813.2			−0.153113
$387$		0			0
$388$	18069.1	+	18069.1i	0.120026	+	0.120026i
$389$		−	88729.2i		−	0.586364i	−0.956057	−	0.293182i	$-0.905286\pi$
$389$		−	88729.2i		−	0.586364i	0.956057	−	0.293182i	$-0.0947142\pi$
$390$		0			0
$391$	39887.0			0.260902
$392$	89611.3	−	89611.3i	0.583164	−	0.583164i
$393$		0			0
$394$		−	21381.5i		−	0.137735i
$395$	154841.	−	154841.i	0.992411	−	0.992411i
$396$		0			0
$397$	98911.6	+	98911.6i	0.627576	+	0.627576i	0.947457	−	0.319882i	$-0.103643\pi$
$397$	98911.6	+	98911.6i	0.627576	+	0.627576i	−0.319882	+	0.947457i	$0.603643\pi$
$398$	−8422.57	−	8422.57i	−0.0531714	−	0.0531714i
$399$		0			0
$400$			145591.i			0.909944i
$401$	8767.07	+	8767.07i	0.0545213	+	0.0545213i	0.733842	−	0.679320i	$-0.237725\pi$
$401$	8767.07	+	8767.07i	0.0545213	+	0.0545213i	−0.679320	+	0.733842i	$0.737725\pi$
$402$		0			0
$403$	−17911.3	−	18756.1i	−0.110285	−	0.115487i
$404$	−24295.2			−0.148853
$405$		0			0
$406$	−16697.8			−0.101299
$407$			65115.3i			0.393092i
$408$		0			0
$409$	90488.4	−	90488.4i	0.540937	−	0.540937i	−0.382867	−	0.923804i	$-0.625063\pi$
$409$	90488.4	−	90488.4i	0.540937	−	0.540937i	0.923804	+	0.382867i	$0.125063\pi$
$410$	49121.9	+	49121.9i	0.292218	+	0.292218i
$411$		0			0
$412$	−249272.			−1.46852
$413$			515090.i			3.01984i
$414$		0			0
$415$		−	37492.3i		−	0.217694i
$416$	−86852.7	−	90949.1i	−0.501876	−	0.525547i
$417$		0			0
$418$	−16406.3	+	16406.3i	−0.0938983	+	0.0938983i
$419$	255942.			1.45785			0.728926	−	0.684593i	$-0.240020\pi$
$419$	255942.			1.45785			0.728926	+	0.684593i	$0.240020\pi$
$420$		0			0
$421$	82355.8	−	82355.8i	0.464654	−	0.464654i	−0.435523	−	0.900177i	$-0.643437\pi$
$421$	82355.8	−	82355.8i	0.464654	−	0.464654i	0.900177	+	0.435523i	$0.143437\pi$
$422$	5358.99	−	5358.99i	0.0300925	−	0.0300925i
$423$		0			0
$424$	−128350.	−	128350.i	−0.713946	−	0.713946i
$425$	−55860.7			−0.309263
$426$		0			0
$427$	−177618.	−	177618.i	−0.974164	−	0.974164i
$428$		−	33443.7i		−	0.182569i
$429$		0			0
$430$	−68868.0			−0.372461
$431$	180532.	−	180532.i	0.971850	−	0.971850i	−0.0277649	−	0.999614i	$-0.508839\pi$
$431$	180532.	−	180532.i	0.971850	−	0.971850i	0.999614	+	0.0277649i	$0.00883898\pi$
$432$		0			0
$433$			249267.i			1.32950i	0.747064	+	0.664752i	$0.231463\pi$
$433$			249267.i			1.32950i	−0.747064	+	0.664752i	$0.768537\pi$
$434$	−9164.53	+	9164.53i	−0.0486554	+	0.0486554i
$435$		0			0
$436$	−61762.6	−	61762.6i	−0.324902	−	0.324902i
$437$	−126645.	−	126645.i	−0.663173	−	0.663173i
$438$		0			0
$439$		−	218037.i		−	1.13136i	−0.824625	−	0.565680i	$-0.808614\pi$
$439$		−	218037.i		−	1.13136i	0.824625	−	0.565680i	$-0.191386\pi$
$440$	−53209.3	−	53209.3i	−0.274841	−	0.274841i
$441$		0			0
$442$	10148.4	−	9691.29i	0.0519460	−	0.0496063i
$443$	−129384.			−0.659286			−0.329643	−	0.944106i	$-0.606928\pi$
$443$	−129384.			−0.659286			−0.329643	+	0.944106i	$0.606928\pi$
$444$		0			0
$445$	−261744.			−1.32177
$446$		−	92318.7i		−	0.464109i
$447$		0			0
$448$	136555.	−	136555.i	0.680382	−	0.680382i
$449$	−128000.	−	128000.i	−0.634920	−	0.634920i	0.314378	−	0.949298i	$-0.398204\pi$
$449$	−128000.	−	128000.i	−0.634920	−	0.634920i	−0.949298	+	0.314378i	$0.898204\pi$
$450$		0			0
$451$	110320.			0.542379
$452$		−	310567.i		−	1.52012i
$453$		0			0
$454$			72415.2i			0.351332i
$455$	−11276.1	+	489433.i	−0.0544676	+	2.36412i
$456$		0			0
$457$	−162330.	+	162330.i	−0.777258	+	0.777258i	−0.979364	−	0.202105i	$-0.935222\pi$
$457$	−162330.	+	162330.i	−0.777258	+	0.777258i	0.202105	+	0.979364i	$0.435222\pi$
$458$	49755.7			0.237199
$459$		0			0
$460$	197766.	−	197766.i	0.934623	−	0.934623i
$461$	78988.5	−	78988.5i	0.371674	−	0.371674i	−0.496413	−	0.868087i	$-0.665350\pi$
$461$	78988.5	−	78988.5i	0.371674	−	0.371674i	0.868087	+	0.496413i	$0.165350\pi$
$462$		0			0
$463$	3183.80	+	3183.80i	0.0148520	+	0.0148520i	0.714494	−	0.699642i	$-0.246657\pi$
$463$	3183.80	+	3183.80i	0.0148520	+	0.0148520i	−0.699642	+	0.714494i	$0.746657\pi$
$464$	−40029.5			−0.185928
$465$		0			0
$466$	70434.6	+	70434.6i	0.324350	+	0.324350i
$467$		−	178606.i		−	0.818959i	−0.912319	−	0.409479i	$-0.865710\pi$
$467$		−	178606.i		−	0.818959i	0.912319	−	0.409479i	$-0.134290\pi$
$468$		0			0
$469$	−305851.			−1.39048
$470$	−93892.7	+	93892.7i	−0.425046	+	0.425046i
$471$		0			0
$472$		−	215011.i		−	0.965110i
$473$	−77333.6	+	77333.6i	−0.345658	+	0.345658i
$474$		0			0
$475$	177364.	+	177364.i	0.786100	+	0.786100i
$476$	64485.7	+	64485.7i	0.284609	+	0.284609i
$477$		0			0
$478$			11281.1i			0.0493738i
$479$	127290.	+	127290.i	0.554785	+	0.554785i	0.927818	−	0.373033i	$-0.121682\pi$
$479$	127290.	+	127290.i	0.554785	+	0.554785i	−0.373033	+	0.927818i	$0.621682\pi$
$480$		0			0
$481$	−127888.	+	122127.i	−0.552762	+	0.527865i
$482$	−6449.82			−0.0277622
$483$		0			0
$484$	159991.			0.682976
$485$			63055.0i			0.268063i
$486$		0			0
$487$	−19975.0	+	19975.0i	−0.0842227	+	0.0842227i	−0.747963	−	0.663740i	$-0.768968\pi$
$487$	−19975.0	+	19975.0i	−0.0842227	+	0.0842227i	0.663740	+	0.747963i	$0.268968\pi$
$488$	74142.1	+	74142.1i	0.311333	+	0.311333i
$489$		0			0
$490$	150567.			0.627102
$491$		−	119626.i		−	0.496205i	−0.968734	−	0.248103i	$-0.920193\pi$
$491$		−	119626.i		−	0.496205i	0.968734	−	0.248103i	$-0.0798070\pi$
$492$		0			0
$493$		−	15358.6i		−	0.0631913i
$494$	−62993.1	−	1451.31i	−0.258130	−	0.00594712i
$495$		0			0
$496$	−21970.1	+	21970.1i	−0.0893034	+	0.0893034i
$497$	−595368.			−2.41031
$498$		0			0
$499$	42223.1	−	42223.1i	0.169570	−	0.169570i	−0.617220	−	0.786790i	$-0.711741\pi$
$499$	42223.1	−	42223.1i	0.169570	−	0.169570i	0.786790	+	0.617220i	$0.211741\pi$
$500$	−36239.2	+	36239.2i	−0.144957	+	0.144957i
$501$		0			0
$502$	44687.0	+	44687.0i	0.177327	+	0.177327i
$503$	−325582.			−1.28684			−0.643419	−	0.765514i	$-0.722485\pi$
$503$	−325582.			−1.28684			−0.643419	+	0.765514i	$0.722485\pi$
$504$		0			0
$505$	−42391.0	−	42391.0i	−0.166223	−	0.166223i
$506$			34153.4i			0.133393i
$507$		0			0
$508$	−132281.			−0.512589
$509$	−19696.3	+	19696.3i	−0.0760238	+	0.0760238i	−0.744096	−	0.668072i	$-0.767120\pi$
$509$	−19696.3	+	19696.3i	−0.0760238	+	0.0760238i	0.668072	+	0.744096i	$0.267120\pi$
$510$		0			0
$511$			685311.i			2.62450i
$512$	−182085.	+	182085.i	−0.694600	+	0.694600i
$513$		0			0
$514$	−37861.5	−	37861.5i	−0.143308	−	0.143308i
$515$	−434937.	−	434937.i	−1.63988	−	1.63988i
$516$		0			0
$517$			210869.i			0.788918i
$518$	62487.9	+	62487.9i	0.232882	+	0.232882i
$519$		0			0
$520$	4706.93	−	204301.i	0.0174073	−	0.755551i
$521$	−88391.6			−0.325638			−0.162819	−	0.986656i	$-0.552059\pi$
$521$	−88391.6			−0.325638			−0.162819	+	0.986656i	$0.552059\pi$
$522$		0			0
$523$	170185.			0.622184			0.311092	−	0.950380i	$-0.399305\pi$
$523$	170185.			0.622184			0.311092	+	0.950380i	$0.399305\pi$
$524$		−	291280.i		−	1.06084i
$525$		0			0
$526$	11268.9	−	11268.9i	0.0407296	−	0.0407296i
$527$	−8429.52	−	8429.52i	−0.0303516	−	0.0303516i
$528$		0			0
$529$	16199.6			0.0578886
$530$		−	215658.i		−	0.767738i
$531$		0			0
$532$		−	409498.i		−	1.44687i
$533$	206912.	+	216671.i	0.728335	+	0.762687i
$534$		0			0
$535$	58353.5	−	58353.5i	0.203873	−	0.203873i
$536$	127669.			0.444383
$537$		0			0
$538$	42101.6	−	42101.6i	0.145457	−	0.145457i
$539$	169076.	−	169076.i	0.581975	−	0.581975i
$540$		0			0
$541$	−335088.	−	335088.i	−1.14489	−	1.14489i	−0.987543	−	0.157348i	$-0.949706\pi$
$541$	−335088.	−	335088.i	−1.14489	−	1.14489i	−0.157348	−	0.987543i	$-0.550294\pi$
$542$	6954.43			0.0236735
$543$		0			0
$544$	−40875.1	−	40875.1i	−0.138121	−	0.138121i
$545$		−	215530.i		−	0.725630i
$546$		0			0
$547$	−55273.4			−0.184732			−0.0923658	−	0.995725i	$-0.529443\pi$
$547$	−55273.4			−0.184732			−0.0923658	+	0.995725i	$0.529443\pi$
$548$	−219793.	+	219793.i	−0.731903	+	0.731903i
$549$		0			0
$550$		−	47831.0i		−	0.158119i
$551$	−48765.2	+	48765.2i	−0.160623	+	0.160623i
$552$		0			0
$553$	−333718.	−	333718.i	−1.09126	−	1.09126i
$554$	57978.3	+	57978.3i	0.188906	+	0.188906i
$555$		0			0
$556$			433918.i			1.40365i
$557$	261902.	+	261902.i	0.844167	+	0.844167i	0.989398	−	0.145231i	$-0.0463926\pi$
$557$	261902.	+	261902.i	0.844167	+	0.844167i	−0.145231	+	0.989398i	$0.546393\pi$
$558$		0			0
$559$	−296928.	−	6840.99i	−0.950227	−	0.0218925i
$560$	586508.			1.87024
$561$		0			0
$562$	−38388.6			−0.121543
$563$			451899.i			1.42569i	0.701323	+	0.712844i	$0.252593\pi$
$563$			451899.i			1.42569i	−0.701323	+	0.712844i	$0.747407\pi$
$564$		0			0
$565$	541886.	−	541886.i	1.69751	−	1.69751i
$566$	−31176.3	−	31176.3i	−0.0973175	−	0.0973175i
$567$		0			0
$568$	248521.			0.770311
$569$		−	444187.i		−	1.37196i	−0.727621	−	0.685979i	$-0.759374\pi$
$569$		−	444187.i		−	1.37196i	0.727621	−	0.685979i	$-0.240626\pi$
$570$		0			0
$571$			388114.i			1.19038i	0.803584	+	0.595192i	$0.202924\pi$
$571$			388114.i			1.19038i	−0.803584	+	0.595192i	$0.797076\pi$
$572$	−107915.	−	113005.i	−0.329831	−	0.345388i
$573$		0			0
$574$	105869.	−	105869.i	0.321325	−	0.321325i
$575$	369223.			1.11674
$576$		0			0
$577$	−399739.	+	399739.i	−1.20067	+	1.20067i	−0.226711	+	0.973962i	$0.572797\pi$
$577$	−399739.	+	399739.i	−1.20067	+	1.20067i	−0.973962	+	0.226711i	$0.927203\pi$
$578$	−58564.5	+	58564.5i	−0.175299	+	0.175299i
$579$		0			0
$580$	−76150.4	−	76150.4i	−0.226369	−	0.226369i
$581$	−80804.6			−0.239378
$582$		0			0
$583$	−242167.	−	242167.i	−0.712489	−	0.712489i
$584$		−	286065.i		−	0.838764i
$585$		0			0
$586$	−34483.4			−0.100419
$587$	88922.4	−	88922.4i	0.258069	−	0.258069i	−0.566200	−	0.824268i	$-0.691587\pi$
$587$	88922.4	−	88922.4i	0.258069	−	0.258069i	0.824268	+	0.566200i	$0.191587\pi$
$588$		0			0
$589$			53529.3i			0.154298i
$590$	180634.	−	180634.i	0.518913	−	0.518913i
$591$		0			0
$592$	149802.	+	149802.i	0.427439	+	0.427439i
$593$	221413.	+	221413.i	0.629641	+	0.629641i	0.947978	−	0.318337i	$-0.103124\pi$
$593$	221413.	+	221413.i	0.629641	+	0.629641i	−0.318337	+	0.947978i	$0.603124\pi$
$594$		0			0
$595$			225033.i			0.635641i
$596$	−67177.9	−	67177.9i	−0.189118	−	0.189118i
$597$		0			0
$598$	−67077.9	+	64056.7i	−0.187576	+	0.179127i
$599$	400910.			1.11736			0.558681	−	0.829383i	$-0.311308\pi$
$599$	400910.			1.11736			0.558681	+	0.829383i	$0.311308\pi$
$600$		0			0
$601$	214969.			0.595149			0.297575	−	0.954699i	$-0.403822\pi$
$601$	214969.			0.595149			0.297575	+	0.954699i	$0.403822\pi$
$602$			148426.i			0.409561i
$603$		0			0
$604$	−90458.3	+	90458.3i	−0.247956	+	0.247956i
$605$	279157.	+	279157.i	0.762672	+	0.762672i
$606$		0			0
$607$	597701.			1.62221			0.811104	−	0.584901i	$-0.198867\pi$
$607$	597701.			1.62221			0.811104	+	0.584901i	$0.198867\pi$
$608$			259566.i			0.702167i
$609$		0			0
$610$			124576.i			0.334790i
$611$	−414150.	+	395497.i	−1.10937	+	1.05940i
$612$		0			0
$613$	−160710.	+	160710.i	−0.427684	+	0.427684i	−0.887839	−	0.460155i	$-0.847794\pi$
$613$	−160710.	+	160710.i	−0.427684	+	0.427684i	0.460155	+	0.887839i	$0.347794\pi$
$614$	−114123.			−0.302717
$615$		0			0
$616$	−114678.	+	114678.i	−0.302218	+	0.302218i
$617$	−59850.2	+	59850.2i	−0.157216	+	0.157216i	−0.781332	−	0.624116i	$-0.785459\pi$
$617$	−59850.2	+	59850.2i	−0.157216	+	0.157216i	0.624116	+	0.781332i	$0.285459\pi$
$618$		0			0
$619$	377131.	+	377131.i	0.984264	+	0.984264i	0.999878	−	0.0156145i	$-0.00497045\pi$
$619$	377131.	+	377131.i	0.984264	+	0.984264i	−0.0156145	+	0.999878i	$0.504970\pi$
$620$	−83589.9			−0.217456
$621$		0			0
$622$	−40110.4	−	40110.4i	−0.103675	−	0.103675i
$623$			564118.i			1.45343i
$624$		0			0
$625$	322968.			0.826797
$626$	35175.1	−	35175.1i	0.0897607	−	0.0897607i
$627$		0			0
$628$		−	188703.i		−	0.478475i
$629$	−57476.3	+	57476.3i	−0.145274	+	0.145274i
$630$		0			0
$631$	−51852.3	−	51852.3i	−0.130230	−	0.130230i	0.638988	−	0.769217i	$-0.279354\pi$
$631$	−51852.3	−	51852.3i	−0.130230	−	0.130230i	−0.769217	+	0.638988i	$0.779354\pi$
$632$	139302.	+	139302.i	0.348757	+	0.348757i
$633$		0			0
$634$		−	49550.2i		−	0.123273i
$635$	−230807.	−	230807.i	−0.572403	−	0.572403i
$636$		0			0
$637$	649179.	+	14956.6i	1.59987	+	0.0368598i
$638$	13150.9			0.0323082
$639$		0			0
$640$	−532274.			−1.29950
$641$		−	174399.i		−	0.424450i	−0.977221	−	0.212225i	$-0.931929\pi$
$641$		−	174399.i		−	0.424450i	0.977221	−	0.212225i	$-0.0680710\pi$
$642$		0			0
$643$	−46308.4	+	46308.4i	−0.112005	+	0.112005i	−0.760888	−	0.648883i	$-0.775236\pi$
$643$	−46308.4	+	46308.4i	−0.112005	+	0.112005i	0.648883	+	0.760888i	$0.275236\pi$
$644$	−426232.	−	426232.i	−1.02772	−	1.02772i
$645$		0			0
$646$	−28963.2			−0.0694034
$647$			158276.i			0.378099i	0.981968	+	0.189050i	$0.0605407\pi$
$647$			158276.i			0.378099i	−0.981968	+	0.189050i	$0.939459\pi$
$648$		0			0
$649$		−	405676.i		−	0.963141i
$650$	93940.9	−	89709.7i	0.222345	−	0.212331i
$651$		0			0
$652$	−278609.	+	278609.i	−0.655390	+	0.655390i
$653$	−251177.			−0.589052			−0.294526	−	0.955643i	$-0.595162\pi$
$653$	−251177.			−0.589052			−0.294526	+	0.955643i	$0.595162\pi$
$654$		0			0
$655$	508233.	−	508233.i	1.18462	−	1.18462i
$656$	253799.	−	253799.i	0.589769	−	0.589769i
$657$		0			0
$658$	202360.	+	202360.i	0.467384	+	0.467384i
$659$	514739.			1.18527			0.592634	−	0.805472i	$-0.298088\pi$
$659$	514739.			1.18527			0.592634	+	0.805472i	$0.298088\pi$
$660$		0			0
$661$	−99888.1	−	99888.1i	−0.228618	−	0.228618i	0.583497	−	0.812115i	$-0.301684\pi$
$661$	−99888.1	−	99888.1i	−0.228618	−	0.228618i	−0.812115	+	0.583497i	$0.801684\pi$
$662$		−	15454.6i		−	0.0352649i
$663$		0			0
$664$	33729.8			0.0765028
$665$	714503.	−	714503.i	1.61570	−	1.61570i
$666$		0			0
$667$			101516.i			0.228182i
$668$	−173593.	+	173593.i	−0.389028	+	0.389028i
$669$		0			0
$670$	107257.	+	107257.i	0.238932	+	0.238932i
$671$	139889.	+	139889.i	0.310698	+	0.310698i
$672$		0			0
$673$		−	852495.i		−	1.88218i	−0.338153	−	0.941091i	$-0.609802\pi$
$673$		−	852495.i		−	1.88218i	0.338153	−	0.941091i	$-0.390198\pi$
$674$	62868.4	+	62868.4i	0.138392	+	0.138392i
$675$		0			0
$676$	19542.8	−	423895.i	0.0427655	−	0.927610i
$677$	90015.3			0.196399			0.0981994	−	0.995167i	$-0.468692\pi$
$677$	90015.3			0.196399			0.0981994	+	0.995167i	$0.468692\pi$
$678$		0			0
$679$	135898.			0.294764
$680$		−	93934.0i		−	0.203144i
$681$		0			0
$682$	7217.82	−	7217.82i	0.0155181	−	0.0155181i
$683$	−231496.	−	231496.i	−0.496252	−	0.496252i	0.414017	−	0.910269i	$-0.364125\pi$
$683$	−231496.	−	231496.i	−0.496252	−	0.496252i	−0.910269	+	0.414017i	$0.864125\pi$
$684$		0			0
$685$	−767003.			−1.63462
$686$		−	121728.i		−	0.258668i
$687$		0			0
$688$			355822.i			0.751719i
$689$	21422.3	−	929819.i	0.0451261	−	1.95866i
$690$		0			0
$691$	−34118.8	+	34118.8i	−0.0714558	+	0.0714558i	−0.741931	−	0.670476i	$-0.766090\pi$
$691$	−34118.8	+	34118.8i	−0.0714558	+	0.0714558i	0.670476	+	0.741931i	$0.266090\pi$
$692$	454790.			0.949726
$693$		0			0
$694$	−85758.5	+	85758.5i	−0.178057	+	0.178057i
$695$	−757111.	+	757111.i	−1.56744	+	1.56744i
$696$		0			0
$697$	97378.1	+	97378.1i	0.200445	+	0.200445i
$698$	−128642.			−0.264041
$699$		0			0
$700$	596927.	+	596927.i	1.21822	+	1.21822i
$701$			413503.i			0.841479i	0.907182	+	0.420739i	$0.138229\pi$
$701$			413503.i			0.841479i	−0.907182	+	0.420739i	$0.861771\pi$
$702$		0			0
$703$	364987.			0.738527
$704$	−107549.	+	107549.i	−0.217000	+	0.217000i
$705$		0			0
$706$		−	217533.i		−	0.436432i
$707$	−91362.3	+	91362.3i	−0.182780	+	0.182780i
$708$		0			0
$709$	−256055.	−	256055.i	−0.509379	−	0.509379i	0.404957	−	0.914336i	$-0.367287\pi$
$709$	−256055.	−	256055.i	−0.509379	−	0.509379i	−0.914336	+	0.404957i	$0.867287\pi$
$710$	208786.	+	208786.i	0.414175	+	0.414175i
$711$		0			0
$712$		−	235476.i		−	0.464501i
$713$	55716.7	+	55716.7i	0.109599	+	0.109599i
$714$		0			0
$715$	8880.89	−	385469.i	0.0173718	−	0.754010i
$716$	54914.5			0.107118
$717$		0			0
$718$	−93400.8			−0.181176
$719$		−	115639.i		−	0.223691i	−0.993726	−	0.111845i	$-0.964324\pi$
$719$		−	115639.i		−	0.223691i	0.993726	−	0.111845i	$-0.0356761\pi$
$720$		0			0
$721$	−937390.	+	937390.i	−1.80322	+	1.80322i
$722$	−6535.87	−	6535.87i	−0.0125380	−	0.0125380i
$723$		0			0
$724$	146656.			0.279784
$725$		−	142170.i		−	0.270479i
$726$		0			0
$727$			754304.i			1.42718i	0.700565	+	0.713588i	$0.252931\pi$
$727$			754304.i			1.42718i	−0.700565	+	0.713588i	$0.747069\pi$
$728$	−440316.	−	10144.5i	−0.830809	−	0.0191412i
$729$		0			0
$730$	240327.	−	240327.i	0.450980	−	0.450980i
$731$	−136522.			−0.255487
$732$		0			0
$733$	−286004.	+	286004.i	−0.532310	+	0.532310i	−0.921259	−	0.388949i	$-0.872838\pi$
$733$	−286004.	+	286004.i	−0.532310	+	0.532310i	0.388949	+	0.921259i	$0.372838\pi$
$734$	−63325.2	+	63325.2i	−0.117540	+	0.117540i
$735$		0			0
$736$	270173.	+	270173.i	0.498753	+	0.498753i
$737$	240883.			0.443476
$738$		0			0
$739$	−491973.	−	491973.i	−0.900850	−	0.900850i	0.0946597	−	0.995510i	$-0.469824\pi$
$739$	−491973.	−	491973.i	−0.900850	−	0.900850i	−0.995510	+	0.0946597i	$0.969824\pi$
$740$			569954.i			1.04082i
$741$		0			0
$742$	−464792.			−0.844210
$743$	30681.5	−	30681.5i	0.0555776	−	0.0555776i	−0.678772	−	0.734349i	$-0.737487\pi$
$743$	30681.5	−	30681.5i	0.0555776	−	0.0555776i	0.734349	+	0.678772i	$0.237487\pi$
$744$		0			0
$745$		−	234428.i		−	0.422373i
$746$	−154520.	+	154520.i	−0.277656	+	0.277656i
$747$		0			0
$748$	−50787.8	−	50787.8i	−0.0907728	−	0.0907728i
$749$	−125765.	−	125765.i	−0.224180	−	0.224180i
$750$		0			0
$751$		−	100355.i		−	0.177934i	−0.996035	−	0.0889669i	$-0.971643\pi$
$751$		−	100355.i		−	0.177934i	0.996035	−	0.0889669i	$-0.0283565\pi$
$752$	485118.	+	485118.i	0.857850	+	0.857850i
$753$		0			0
$754$	24665.2	+	25828.5i	0.0433852	+	0.0454315i
$755$	−315668.			−0.553780
$756$		0			0
$757$	307678.			0.536915			0.268457	−	0.963292i	$-0.413486\pi$
$757$	307678.			0.536915			0.268457	+	0.963292i	$0.413486\pi$
$758$			52180.8i			0.0908181i
$759$		0			0
$760$	−298251.	+	298251.i	−0.516362	+	0.516362i
$761$	−42256.7	−	42256.7i	−0.0729670	−	0.0729670i	0.669681	−	0.742648i	$-0.266431\pi$
$761$	−42256.7	−	42256.7i	−0.0729670	−	0.0729670i	−0.742648	+	0.669681i	$0.766431\pi$
$762$		0			0
$763$	−464518.			−0.797908
$764$			428035.i			0.733319i
$765$		0			0
$766$			48587.7i			0.0828074i
$767$	796754.	−	760868.i	1.35436	−	1.29336i
$768$		0			0
$769$	257148.	−	257148.i	0.434841	−	0.434841i	−0.455431	−	0.890271i	$-0.650515\pi$
$769$	257148.	−	257148.i	0.434841	−	0.434841i	0.890271	+	0.455431i	$0.150515\pi$
$770$	−192685.			−0.324988
$771$		0			0
$772$	−224230.	+	224230.i	−0.376235	+	0.376235i
$773$	−503211.	+	503211.i	−0.842154	+	0.842154i	−0.989139	−	0.146985i	$-0.953043\pi$
$773$	−503211.	+	503211.i	−0.842154	+	0.842154i	0.146985	+	0.989139i	$0.453043\pi$
$774$		0			0
$775$	−78029.8	−	78029.8i	−0.129914	−	0.129914i
$776$	−56727.1			−0.0942035
$777$		0			0
$778$	67061.8	+	67061.8i	0.110794	+	0.110794i
$779$		−	618372.i		−	1.01900i
$780$		0			0
$781$	468902.			0.768740
$782$	−30146.7	+	30146.7i	−0.0492977	+	0.0492977i
$783$		0			0
$784$		−	777939.i		−	1.26565i
$785$	329254.	−	329254.i	0.534308	−	0.534308i
$786$		0			0
$787$	−170363.	−	170363.i	−0.275060	−	0.275060i	0.556073	−	0.831133i	$-0.312307\pi$
$787$	−170363.	−	170363.i	−0.275060	−	0.275060i	−0.831133	+	0.556073i	$0.812307\pi$
$788$	−210158.	−	210158.i	−0.338449	−	0.338449i
$789$		0			0
$790$			234059.i			0.375034i
$791$	−1.16789e6	−	1.16789e6i	−1.86659	−	1.86659i
$792$		0			0
$793$	−12374.7	+	537114.i	−0.0196783	+	0.854122i
$794$	−149515.			−0.237162
$795$		0			0
$796$	−165570.			−0.261310
$797$			1.00145e6i			1.57657i	0.615312	+	0.788284i	$0.289030\pi$
$797$			1.00145e6i			1.57657i	−0.615312	+	0.788284i	$0.710970\pi$
$798$		0			0
$799$	−186131.	+	186131.i	−0.291558	+	0.291558i
$800$	−378370.	−	378370.i	−0.591203	−	0.591203i
$801$		0			0
$802$	−13252.4			−0.0206037
$803$		−	539739.i		−	0.837053i
$804$		0			0
$805$		−	1.48740e6i		−	2.29528i
$806$	27713.3	+	638.494i	0.0426598	+	0.000982849i
$807$		0			0
$808$	38136.8	−	38136.8i	0.0584146	−	0.0584146i
$809$	−483971.			−0.739473			−0.369736	−	0.929137i	$-0.620552\pi$
$809$	−483971.			−0.739473			−0.369736	+	0.929137i	$0.620552\pi$
$810$		0			0
$811$	573330.	−	573330.i	0.871692	−	0.871692i	−0.120965	−	0.992657i	$-0.538599\pi$
$811$	573330.	−	573330.i	0.871692	−	0.871692i	0.992657	+	0.120965i	$0.0385989\pi$
$812$	−164122.	+	164122.i	−0.248917	+	0.248917i
$813$		0			0
$814$	−49214.4	−	49214.4i	−0.0742751	−	0.0742751i
$815$	−972250.			−1.46374
$816$		0			0
$817$	433474.	+	433474.i	0.649409	+	0.649409i
$818$			136783.i			0.204421i
$819$		0			0
$820$	965634.			1.43610
$821$	−393694.	+	393694.i	−0.584081	+	0.584081i	−0.936022	−	0.351941i	$-0.885522\pi$
$821$	−393694.	+	393694.i	−0.584081	+	0.584081i	0.351941	+	0.936022i	$0.385522\pi$
$822$		0			0
$823$			704258.i			1.03976i	0.854240	+	0.519879i	$0.174023\pi$
$823$			704258.i			1.03976i	−0.854240	+	0.519879i	$0.825977\pi$
$824$	391289.	−	391289.i	0.576292	−	0.576292i
$825$		0			0
$826$	−389307.	−	389307.i	−0.570600	−	0.570600i
$827$	222716.	+	222716.i	0.325642	+	0.325642i	0.850927	−	0.525285i	$-0.176041\pi$
$827$	222716.	+	222716.i	0.325642	+	0.325642i	−0.525285	+	0.850927i	$0.676041\pi$
$828$		0			0
$829$			511137.i			0.743753i	0.928282	+	0.371876i	$0.121285\pi$
$829$			511137.i			0.743753i	−0.928282	+	0.371876i	$0.878715\pi$
$830$	28336.8	+	28336.8i	0.0411334	+	0.0411334i
$831$		0			0
$832$	−412941.	−	9513.83i	−0.596542	−	0.0137439i
$833$	298481.			0.430157
$834$		0			0
$835$	−605781.			−0.868846
$836$			322513.i			0.461461i
$837$		0			0
$838$	−193442.	+	193442.i	−0.275462	+	0.275462i
$839$	−265771.	−	265771.i	−0.377558	−	0.377558i	0.492663	−	0.870220i	$-0.336024\pi$
$839$	−265771.	−	265771.i	−0.377558	−	0.377558i	−0.870220	+	0.492663i	$0.836024\pi$
$840$		0			0
$841$	−668192.			−0.944734
$842$			124490.i			0.175594i
$843$		0			0
$844$		−	105347.i		−	0.147889i
$845$	773723.	−	705526.i	1.08361	−	0.988096i
$846$		0			0
$847$	601648.	−	601648.i	0.838640	−	0.838640i
$848$	−1.11424e6			−1.54949
$849$		0			0
$850$	42219.7	−	42219.7i	0.0584356	−	0.0584356i
$851$	379902.	−	379902.i	0.524581	−	0.524581i
$852$		0			0
$853$	−73021.9	−	73021.9i	−0.100359	−	0.100359i	0.655145	−	0.755503i	$-0.272608\pi$
$853$	−73021.9	−	73021.9i	−0.100359	−	0.100359i	−0.755503	+	0.655145i	$0.772608\pi$
$854$	268489.			0.368138
$855$		0			0
$856$	52497.5	+	52497.5i	0.0716458	+	0.0716458i
$857$		−	361781.i		−	0.492588i	−0.969195	−	0.246294i	$-0.920787\pi$
$857$		−	361781.i		−	0.492588i	0.969195	−	0.246294i	$-0.0792128\pi$
$858$		0			0
$859$	441391.			0.598188			0.299094	−	0.954224i	$-0.403316\pi$
$859$	441391.			0.598188			0.299094	+	0.954224i	$0.403316\pi$
$860$	−676901.	+	676901.i	−0.915225	+	0.915225i
$861$		0			0
$862$			272893.i			0.367264i
$863$	995991.	−	995991.i	1.33732	−	1.33732i	0.438665	−	0.898651i	$-0.355452\pi$
$863$	995991.	−	995991.i	1.33732	−	1.33732i	0.898651	−	0.438665i	$-0.144548\pi$
$864$		0			0
$865$	793529.	+	793529.i	1.06055	+	1.06055i
$866$	−188397.	−	188397.i	−0.251211	−	0.251211i
$867$		0			0
$868$			180156.i			0.239116i
$869$	262830.	+	262830.i	0.348045	+	0.348045i
$870$		0			0
$871$	451789.	+	473097.i	0.595524	+	0.623612i
$872$	193901.			0.255004
$873$		0			0
$874$	191438.			0.250614
$875$			272556.i			0.355991i
$876$		0			0
$877$	182081.	−	182081.i	0.236736	−	0.236736i	−0.578761	−	0.815497i	$-0.696464\pi$
$877$	182081.	−	182081.i	0.236736	−	0.236736i	0.815497	+	0.578761i	$0.196464\pi$
$878$	164793.	+	164793.i	0.213771	+	0.213771i
$879$		0			0
$880$	−461924.			−0.596492
$881$			1.36681e6i			1.76099i	0.474054	+	0.880496i	$0.342790\pi$
$881$			1.36681e6i			1.76099i	−0.474054	+	0.880496i	$0.657210\pi$
$882$		0			0
$883$			564414.i			0.723896i	0.932198	+	0.361948i	$0.117888\pi$
$883$			564414.i			0.723896i	−0.932198	+	0.361948i	$0.882112\pi$
$884$	4492.72	−	195003.i	0.00574917	−	0.249538i
$885$		0			0
$886$	97789.0	−	97789.0i	0.124573	−	0.124573i
$887$	110785.			0.140811			0.0704053	−	0.997518i	$-0.477571\pi$
$887$	110785.			0.140811			0.0704053	+	0.997518i	$0.477571\pi$
$888$		0			0
$889$	−497442.	+	497442.i	−0.629418	+	0.629418i
$890$	197827.	−	197827.i	0.249750	−	0.249750i
$891$		0			0
$892$	−907397.	−	907397.i	−1.14043	−	1.14043i
$893$	1.18197e6			1.48219
$894$		0			0
$895$	95816.3	+	95816.3i	0.119617	+	0.119617i
$896$			1.14717e6i			1.42894i
$897$		0			0
$898$	193486.			0.239937
$899$	21453.9	−	21453.9i	0.0265452	−	0.0265452i
$900$		0			0
$901$		−	427515.i		−	0.526625i
$902$	−83380.5	+	83380.5i	−0.102483	+	0.102483i
$903$		0			0
$904$	487505.	+	487505.i	0.596544	+	0.596544i
$905$	255889.	+	255889.i	0.312431	+	0.312431i
$906$		0			0
$907$		−	966035.i		−	1.17430i	−0.809479	−	0.587149i	$-0.800250\pi$
$907$		−	966035.i		−	1.17430i	0.809479	−	0.587149i	$-0.199750\pi$
$908$	711766.	+	711766.i	0.863308	+	0.863308i
$909$		0			0
$910$	−361392.	−	378437.i	−0.436411	−	0.456995i
$911$	1.12436e6			1.35478			0.677392	−	0.735622i	$-0.263110\pi$
$911$	1.12436e6			1.35478			0.677392	+	0.735622i	$0.263110\pi$
$912$		0			0
$913$	63640.3			0.0763467
$914$		−	245379.i		−	0.293727i
$915$		0			0
$916$	489047.	−	489047.i	0.582854	−	0.582854i
$917$	−1.09536e6	−	1.09536e6i	−1.30262	−	1.30262i
$918$		0			0
$919$	−71843.6			−0.0850662			−0.0425331	−	0.999095i	$-0.513543\pi$
$919$	−71843.6			−0.0850662			−0.0425331	+	0.999095i	$0.513543\pi$
$920$			620877.i			0.733550i
$921$		0			0
$922$			119400.i			0.140456i
$923$	879451.	+	920930.i	1.03231	+	1.08099i
$924$		0			0
$925$	−532043.	+	532043.i	−0.621818	+	0.621818i
$926$	−4812.66			−0.00561259
$927$		0			0
$928$	104031.	−	104031.i	0.120800	−	0.120800i
$929$	767314.	−	767314.i	0.889082	−	0.889082i	−0.105353	−	0.994435i	$-0.533597\pi$
$929$	767314.	−	767314.i	0.889082	−	0.889082i	0.994435	+	0.105353i	$0.0335974\pi$
$930$		0			0
$931$	−947711.	−	947711.i	−1.09339	−	1.09339i
$932$	1.38460e6			1.59401
$933$		0			0
$934$	134991.	+	134991.i	0.154743	+	0.154743i
$935$		−	177232.i		−	0.202730i
$936$		0			0
$937$	−52590.1			−0.0598997			−0.0299499	−	0.999551i	$-0.509535\pi$
$937$	−52590.1			−0.0598997			−0.0299499	+	0.999551i	$0.509535\pi$
$938$	231163.	−	231163.i	0.262732	−	0.262732i
$939$		0			0
$940$			1.84574e6i			2.08888i
$941$	50182.4	−	50182.4i	0.0566725	−	0.0566725i	−0.678203	−	0.734875i	$-0.737241\pi$
$941$	50182.4	−	50182.4i	0.0566725	−	0.0566725i	0.734875	+	0.678203i	$0.237241\pi$
$942$		0			0
$943$	−643641.	−	643641.i	−0.723803	−	0.723803i
$944$	−933283.	−	933283.i	−1.04730	−	1.04730i
$945$		0			0
$946$		−	116898.i		−	0.130625i
$947$	−661892.	−	661892.i	−0.738053	−	0.738053i	0.234148	−	0.972201i	$-0.424770\pi$
$947$	−661892.	−	661892.i	−0.738053	−	0.738053i	−0.972201	+	0.234148i	$0.924770\pi$
$948$		0			0
$949$	1.06006e6	−	1.01231e6i	1.17705	−	1.12404i
$950$	−268104.			−0.297068
$951$		0			0
$952$	−202449.			−0.223379
$953$			1.38652e6i			1.52665i	0.646015	+	0.763325i	$0.276434\pi$
$953$			1.38652e6i			1.52665i	−0.646015	+	0.763325i	$0.723566\pi$
$954$		0			0
$955$	−746848.	+	746848.i	−0.818889	+	0.818889i
$956$	110882.	+	110882.i	0.121323	+	0.121323i
$957$		0			0
$958$	−192413.			−0.209654
$959$			1.65307e6i			1.79744i
$960$		0			0
$961$			899971.i			0.974500i
$962$	4353.54	−	188962.i	0.00470427	−	0.204185i
$963$		0			0
$964$	−63395.0	+	63395.0i	−0.0682183	+	0.0682183i
$965$	−782485.			−0.840275
$966$		0			0
$967$	975365.	−	975365.i	1.04307	−	1.04307i	0.0440424	−	0.999030i	$-0.485976\pi$
$967$	975365.	−	975365.i	1.04307	−	1.04307i	0.999030	−	0.0440424i	$-0.0140237\pi$
$968$	−251142.	+	251142.i	−0.268021	+	0.268021i
$969$		0			0
$970$	−47657.2	−	47657.2i	−0.0506506	−	0.0506506i
$971$	−850441.			−0.901999			−0.450999	−	0.892524i	$-0.648932\pi$
$971$	−850441.			−0.901999			−0.450999	+	0.892524i	$0.648932\pi$
$972$		0			0
$973$	1.63175e6	+	1.63175e6i	1.72357	+	1.72357i
$974$		−	30194.4i		−	0.0318279i
$975$		0			0
$976$	643647.			0.675691
$977$	418372.	−	418372.i	0.438302	−	0.438302i	−0.453138	−	0.891440i	$-0.649696\pi$
$977$	418372.	−	418372.i	0.438302	−	0.438302i	0.891440	+	0.453138i	$0.149696\pi$
$978$		0			0
$979$		−	444289.i		−	0.463554i
$980$	1.47992e6	−	1.47992e6i	1.54094	−	1.54094i
$981$		0			0
$982$	90413.4	+	90413.4i	0.0937584	+	0.0937584i
$983$	84309.7	+	84309.7i	0.0872511	+	0.0872511i	0.749385	−	0.662134i	$-0.230349\pi$
$983$	84309.7	+	84309.7i	0.0872511	+	0.0872511i	−0.662134	+	0.749385i	$0.730349\pi$
$984$		0			0
$985$		−	733377.i		−	0.755884i
$986$	11608.1	+	11608.1i	0.0119400	+	0.0119400i
$987$		0			0
$988$	−633422.	+	604892.i	−0.648902	+	0.619675i
$989$	902374.			0.922559
$990$		0			0
$991$	−1.44435e6			−1.47070			−0.735351	−	0.677686i	$-0.762983\pi$
$991$	−1.44435e6			−1.47070			−0.735351	+	0.677686i	$0.762983\pi$
$992$		−	114194.i		−	0.116043i
$993$		0			0
$994$	449981.	−	449981.i	0.455430	−	0.455430i
$995$	−288891.	−	288891.i	−0.291802	−	0.291802i
$996$		0			0
$997$	1.78714e6			1.79791			0.898954	−	0.438043i	$-0.144328\pi$
$997$	1.78714e6			1.79791			0.898954	+	0.438043i	$0.144328\pi$
$998$			63824.8i			0.0640809i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.109.5		20
3.2	odd	2		39.5.g.a.31.6	✓	20
13.8	odd	4	inner	117.5.j.b.73.5		20
39.8	even	4		39.5.g.a.34.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.6	✓	20	3.2	odd	2
39.5.g.a.34.6	yes	20	39.8	even	4
117.5.j.b.73.5		20	13.8	odd	4	inner
117.5.j.b.109.5		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)

\(f(q)\)	\(=\)	\(q+(-0.755803 + 0.755803i) q^{2} +14.8575i q^{4} +(-25.9238 + 25.9238i) q^{5} +(55.8718 + 55.8718i) q^{7} +(-23.3222 - 23.3222i) q^{8} -39.1866i q^{10} +(-44.0036 - 44.0036i) q^{11} +(3.89259 - 168.955i) q^{13} -84.4562 q^{14} -202.466 q^{16} -77.6827i q^{17} +(-246.651 + 246.651i) q^{19} +(-385.164 - 385.164i) q^{20} +66.5162 q^{22} +513.460i q^{23} -719.088i q^{25} +(124.755 + 130.639i) q^{26} +(-830.116 + 830.116i) q^{28} +197.709 q^{29} +(108.512 - 108.512i) q^{31} +(526.180 - 526.180i) q^{32} +(58.7128 + 58.7128i) q^{34} -2896.82 q^{35} +(-739.885 - 739.885i) q^{37} -372.839i q^{38} +1209.20 q^{40} +(-1253.54 + 1253.54i) q^{41} -1757.44i q^{43} +(653.785 - 653.785i) q^{44} +(-388.075 - 388.075i) q^{46} +(-2396.04 - 2396.04i) q^{47} +3842.31i q^{49} +(543.489 + 543.489i) q^{50} +(2510.26 + 57.8343i) q^{52} +5503.35 q^{53} +2281.48 q^{55} -2606.11i q^{56} +(-149.429 + 149.429i) q^{58} +(4609.57 + 4609.57i) q^{59} -3179.03 q^{61} +164.028i q^{62} -2444.08i q^{64} +(4279.05 + 4480.87i) q^{65} +(-2737.08 + 2737.08i) q^{67} +1154.17 q^{68} +(2189.43 - 2189.43i) q^{70} +(-5327.99 + 5327.99i) q^{71} +(6132.89 + 6132.89i) q^{73} +1118.42 q^{74} +(-3664.62 - 3664.62i) q^{76} -4917.12i q^{77} -5972.92 q^{79} +(5248.70 - 5248.70i) q^{80} -1894.85i q^{82} +(-723.125 + 723.125i) q^{83} +(2013.83 + 2013.83i) q^{85} +(1328.28 + 1328.28i) q^{86} +2052.53i q^{88} +(5048.32 + 5048.32i) q^{89} +(9657.31 - 9222.34i) q^{91} -7628.75 q^{92} +3621.87 q^{94} -12788.3i q^{95} +(1216.16 - 1216.16i) q^{97} +(-2904.03 - 2904.03i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 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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