LMFDB - Embedded newform 117.5.j.b.109.10

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			109.10
Root			$5.10926 - 5.10926i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.5.j.b.73.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(5.10926 - 5.10926i) q^{2} -36.2092i q^{4} +(-3.64937 + 3.64937i) q^{5} +(-17.1355 - 17.1355i) q^{7} +(-103.254 - 103.254i) q^{8} +37.2912i q^{10} +(-114.571 - 114.571i) q^{11} +(28.1152 - 166.645i) q^{13} -175.100 q^{14} -475.757 q^{16} +378.636i q^{17} +(374.813 - 374.813i) q^{19} +(132.141 + 132.141i) q^{20} -1170.74 q^{22} +249.703i q^{23} +598.364i q^{25} +(-707.785 - 995.081i) q^{26} +(-620.464 + 620.464i) q^{28} +460.768 q^{29} +(1349.74 - 1349.74i) q^{31} +(-778.703 + 778.703i) q^{32} +(1934.55 + 1934.55i) q^{34} +125.068 q^{35} +(-644.748 - 644.748i) q^{37} -3830.04i q^{38} +753.623 q^{40} +(103.315 - 103.315i) q^{41} +1797.34i q^{43} +(-4148.51 + 4148.51i) q^{44} +(1275.80 + 1275.80i) q^{46} +(2089.67 + 2089.67i) q^{47} -1813.75i q^{49} +(3057.20 + 3057.20i) q^{50} +(-6034.07 - 1018.03i) q^{52} +663.144 q^{53} +836.222 q^{55} +3538.62i q^{56} +(2354.19 - 2354.19i) q^{58} +(-3213.29 - 3213.29i) q^{59} -1971.21 q^{61} -13792.4i q^{62} +345.097i q^{64} +(505.546 + 710.752i) q^{65} +(2419.73 - 2419.73i) q^{67} +13710.1 q^{68} +(639.005 - 639.005i) q^{70} +(3751.05 - 3751.05i) q^{71} +(3077.50 + 3077.50i) q^{73} -6588.37 q^{74} +(-13571.7 - 13571.7i) q^{76} +3926.46i q^{77} +5600.97 q^{79} +(1736.21 - 1736.21i) q^{80} -1055.72i q^{82} +(-7657.10 + 7657.10i) q^{83} +(-1381.78 - 1381.78i) q^{85} +(9183.09 + 9183.09i) q^{86} +23659.7i q^{88} +(-187.625 - 187.625i) q^{89} +(-3337.32 + 2373.78i) q^{91} +9041.54 q^{92} +21353.3 q^{94} +2735.66i q^{95} +(-1371.54 + 1371.54i) q^{97} +(-9266.91 - 9266.91i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.10926	−	5.10926i	1.27732	−	1.27732i	0.335152	−	0.942164i	$-0.391212\pi$
$2$	5.10926	−	5.10926i	1.27732	−	1.27732i	0.942164	−	0.335152i	$-0.108788\pi$
$3$		0			0
$3$		0			0
$4$		−	36.2092i		−	2.26307i
$5$	−3.64937	+	3.64937i	−0.145975	+	0.145975i	−0.776317	−	0.630342i	$-0.782914\pi$
$5$	−3.64937	+	3.64937i	−0.145975	+	0.145975i	0.630342	+	0.776317i	$0.282914\pi$
$6$		0			0
$7$	−17.1355	−	17.1355i	−0.349705	−	0.349705i	0.510295	−	0.860000i	$-0.329536\pi$
$7$	−17.1355	−	17.1355i	−0.349705	−	0.349705i	−0.860000	+	0.510295i	$0.829536\pi$
$8$	−103.254	−	103.254i	−1.61334	−	1.61334i
$9$		0			0
$10$			37.2912i			0.372912i
$11$	−114.571	−	114.571i	−0.946865	−	0.946865i	0.0517926	−	0.998658i	$-0.483507\pi$
$11$	−114.571	−	114.571i	−0.946865	−	0.946865i	−0.998658	+	0.0517926i	$0.983507\pi$
$12$		0			0
$13$	28.1152	−	166.645i	0.166362	−	0.986065i
$13$	28.1152	−	166.645i	0.166362	−	0.986065i
$14$	−175.100			−0.893367
$15$		0			0
$16$	−475.757			−1.85842
$17$			378.636i			1.31016i	0.755560	+	0.655080i	$0.227365\pi$
$17$			378.636i			1.31016i	−0.755560	+	0.655080i	$0.772635\pi$
$18$		0			0
$19$	374.813	−	374.813i	1.03826	−	1.03826i	0.0390244	−	0.999238i	$-0.487575\pi$
$19$	374.813	−	374.813i	1.03826	−	1.03826i	0.999238	−	0.0390244i	$-0.0124250\pi$
$20$	132.141	+	132.141i	0.330351	+	0.330351i
$21$		0			0
$22$	−1170.74			−2.41889
$23$			249.703i			0.472029i	0.971750	+	0.236014i	$0.0758412\pi$
$23$			249.703i			0.472029i	−0.971750	+	0.236014i	$0.924159\pi$
$24$		0			0
$25$			598.364i			0.957383i
$26$	−707.785	−	995.081i	−1.04702	−	1.47201i
$27$		0			0
$28$	−620.464	+	620.464i	−0.791408	+	0.791408i
$29$	460.768			0.547881			0.273941	−	0.961747i	$-0.411673\pi$
$29$	460.768			0.547881			0.273941	+	0.961747i	$0.411673\pi$
$30$		0			0
$31$	1349.74	−	1349.74i	1.40452	−	1.40452i	0.619611	−	0.784909i	$-0.287290\pi$
$31$	1349.74	−	1349.74i	1.40452	−	1.40452i	0.784909	−	0.619611i	$-0.212710\pi$
$32$	−778.703	+	778.703i	−0.760453	+	0.760453i
$33$		0			0
$34$	1934.55	+	1934.55i	1.67349	+	1.67349i
$35$	125.068			0.102096
$36$		0			0
$37$	−644.748	−	644.748i	−0.470963	−	0.470963i	0.431263	−	0.902226i	$-0.358068\pi$
$37$	−644.748	−	644.748i	−0.470963	−	0.470963i	−0.902226	+	0.431263i	$0.858068\pi$
$38$		−	3830.04i		−	2.65238i
$39$		0			0
$40$	753.623			0.471015
$41$	103.315	−	103.315i	0.0614601	−	0.0614601i	−0.675709	−	0.737169i	$-0.736162\pi$
$41$	103.315	−	103.315i	0.0614601	−	0.0614601i	0.737169	+	0.675709i	$0.236162\pi$
$42$		0			0
$43$			1797.34i			0.972061i	0.873942	+	0.486030i	$0.161556\pi$
$43$			1797.34i			0.972061i	−0.873942	+	0.486030i	$0.838444\pi$
$44$	−4148.51	+	4148.51i	−2.14282	+	2.14282i
$45$		0			0
$46$	1275.80	+	1275.80i	0.602930	+	0.602930i
$47$	2089.67	+	2089.67i	0.945980	+	0.945980i	0.998614	−	0.0526342i	$-0.0167617\pi$
$47$	2089.67	+	2089.67i	0.945980	+	0.945980i	−0.0526342	+	0.998614i	$0.516762\pi$
$48$		0			0
$49$		−	1813.75i		−	0.755413i
$50$	3057.20	+	3057.20i	1.22288	+	1.22288i
$51$		0			0
$52$	−6034.07	−	1018.03i	−2.23154	−	0.376490i
$53$	663.144			0.236078			0.118039	−	0.993009i	$-0.462339\pi$
$53$	663.144			0.236078			0.118039	+	0.993009i	$0.462339\pi$
$54$		0			0
$55$	836.222			0.276437
$56$			3538.62i			1.12839i
$57$		0			0
$58$	2354.19	−	2354.19i	0.699818	−	0.699818i
$59$	−3213.29	−	3213.29i	−0.923093	−	0.923093i	0.0741541	−	0.997247i	$-0.476374\pi$
$59$	−3213.29	−	3213.29i	−0.923093	−	0.923093i	−0.997247	+	0.0741541i	$0.976374\pi$
$60$		0			0
$61$	−1971.21			−0.529751			−0.264876	−	0.964283i	$-0.585331\pi$
$61$	−1971.21			−0.529751			−0.264876	+	0.964283i	$0.585331\pi$
$62$		−	13792.4i		−	3.58803i
$63$		0			0
$64$			345.097i			0.0842521i
$65$	505.546	+	710.752i	0.119656	+	0.168225i
$66$		0			0
$67$	2419.73	−	2419.73i	0.539035	−	0.539035i	−0.384211	−	0.923245i	$-0.625526\pi$
$67$	2419.73	−	2419.73i	0.539035	−	0.539035i	0.923245	+	0.384211i	$0.125526\pi$
$68$	13710.1			2.96499
$69$		0			0
$70$	639.005	−	639.005i	0.130409	−	0.130409i
$71$	3751.05	−	3751.05i	0.744108	−	0.744108i	−0.229257	−	0.973366i	$-0.573630\pi$
$71$	3751.05	−	3751.05i	0.744108	−	0.744108i	0.973366	+	0.229257i	$0.0736298\pi$
$72$		0			0
$73$	3077.50	+	3077.50i	0.577500	+	0.577500i	0.934214	−	0.356714i	$-0.116103\pi$
$73$	3077.50	+	3077.50i	0.577500	+	0.577500i	−0.356714	+	0.934214i	$0.616103\pi$
$74$	−6588.37			−1.20314
$75$		0			0
$76$	−13571.7	−	13571.7i	−2.34966	−	2.34966i
$77$			3926.46i			0.662247i
$78$		0			0
$79$	5600.97			0.897447			0.448724	−	0.893671i	$-0.351879\pi$
$79$	5600.97			0.897447			0.448724	+	0.893671i	$0.351879\pi$
$80$	1736.21	−	1736.21i	0.271283	−	0.271283i
$81$		0			0
$82$		−	1055.72i		−	0.157008i
$83$	−7657.10	+	7657.10i	−1.11150	+	1.11150i	−0.118549	+	0.992948i	$0.537824\pi$
$83$	−7657.10	+	7657.10i	−1.11150	+	1.11150i	−0.992948	+	0.118549i	$0.962176\pi$
$84$		0			0
$85$	−1381.78	−	1381.78i	−0.191250	−	0.191250i
$86$	9183.09	+	9183.09i	1.24163	+	1.24163i
$87$		0			0
$88$			23659.7i			3.05524i
$89$	−187.625	−	187.625i	−0.0236870	−	0.0236870i	0.695164	−	0.718851i	$-0.255332\pi$
$89$	−187.625	−	187.625i	−0.0236870	−	0.0236870i	−0.718851	+	0.695164i	$0.755332\pi$
$90$		0			0
$91$	−3337.32	+	2373.78i	−0.403009	+	0.286654i
$92$	9041.54			1.06824
$93$		0			0
$94$	21353.3			2.41663
$95$			2735.66i			0.303120i
$96$		0			0
$97$	−1371.54	+	1371.54i	−0.145769	+	0.145769i	−0.776225	−	0.630456i	$-0.782868\pi$
$97$	−1371.54	+	1371.54i	−0.145769	+	0.145769i	0.630456	+	0.776225i	$0.282868\pi$
$98$	−9266.91	−	9266.91i	−0.964901	−	0.964901i
$99$		0			0
$100$	21666.3			2.16663
$101$			5770.48i			0.565678i	0.959167	+	0.282839i	$0.0912763\pi$
$101$			5770.48i			0.565678i	−0.959167	+	0.282839i	$0.908724\pi$
$102$		0			0
$103$			10514.1i			0.991053i	0.868593	+	0.495526i	$0.165025\pi$
$103$			10514.1i			0.991053i	−0.868593	+	0.495526i	$0.834975\pi$
$104$	−20109.7	+	14303.7i	−1.85926	+	1.32246i
$105$		0			0
$106$	3388.18	−	3388.18i	0.301547	−	0.301547i
$107$	−14517.5			−1.26801			−0.634006	−	0.773328i	$-0.718591\pi$
$107$	−14517.5			−1.26801			−0.634006	+	0.773328i	$0.718591\pi$
$108$		0			0
$109$	−5657.08	+	5657.08i	−0.476145	+	0.476145i	−0.903897	−	0.427751i	$-0.859306\pi$
$109$	−5657.08	+	5657.08i	−0.476145	+	0.476145i	0.427751	+	0.903897i	$0.359306\pi$
$110$	4272.48	−	4272.48i	0.353097	−	0.353097i
$111$		0			0
$112$	8152.35	+	8152.35i	0.649900	+	0.649900i
$113$	16972.1			1.32917			0.664584	−	0.747213i	$-0.268609\pi$
$113$	16972.1			1.32917			0.664584	+	0.747213i	$0.268609\pi$
$114$		0			0
$115$	−911.259	−	911.259i	−0.0689043	−	0.0689043i
$116$		−	16684.0i		−	1.23990i
$117$		0			0
$118$	−32835.1			−2.35816
$119$	6488.14	−	6488.14i	0.458169	−	0.458169i
$120$		0			0
$121$			11611.9i			0.793108i
$122$	−10071.4	+	10071.4i	−0.676660	+	0.676660i
$123$		0			0
$124$	−48873.1	−	48873.1i	−3.17853	−	3.17853i
$125$	−4464.51	−	4464.51i	−0.285729	−	0.285729i
$126$		0			0
$127$			1663.77i			0.103154i	0.998669	+	0.0515770i	$0.0164247\pi$
$127$			1663.77i			0.103154i	−0.998669	+	0.0515770i	$0.983575\pi$
$128$	−10696.1	−	10696.1i	−0.652836	−	0.652836i
$129$		0			0
$130$	6214.39	+	1048.45i	0.367715	+	0.0620384i
$131$	15728.6			0.916530			0.458265	−	0.888816i	$-0.348471\pi$
$131$	15728.6			0.916530			0.458265	+	0.888816i	$0.348471\pi$
$132$		0			0
$133$	−12845.2			−0.726171
$134$		−	24726.0i		−	1.37703i
$135$		0			0
$136$	39095.7	−	39095.7i	2.11374	−	2.11374i
$137$	4416.77	+	4416.77i	0.235323	+	0.235323i	0.814910	−	0.579587i	$-0.196786\pi$
$137$	4416.77	+	4416.77i	0.235323	+	0.235323i	−0.579587	+	0.814910i	$0.696786\pi$
$138$		0			0
$139$	−25079.5			−1.29804			−0.649021	−	0.760771i	$-0.724821\pi$
$139$	−25079.5			−1.29804			−0.649021	+	0.760771i	$0.724821\pi$
$140$		−	4528.60i		−	0.231051i
$141$		0			0
$142$		−	38330.2i		−	1.90092i
$143$	−22313.8	+	15871.4i	−1.09119	+	0.776148i
$144$		0			0
$145$	−1681.51	+	1681.51i	−0.0799769	+	0.0799769i
$146$	31447.5			1.47530
$147$		0			0
$148$	−23345.8	+	23345.8i	−1.06582	+	1.06582i
$149$	−23320.2	+	23320.2i	−1.05041	+	1.05041i	−0.0517510	+	0.998660i	$0.516480\pi$
$149$	−23320.2	+	23320.2i	−1.05041	+	1.05041i	−0.998660	+	0.0517510i	$0.983520\pi$
$150$		0			0
$151$	20735.4	+	20735.4i	0.909408	+	0.909408i	0.996224	−	0.0868168i	$-0.0276695\pi$
$151$	20735.4	+	20735.4i	0.909408	+	0.909408i	−0.0868168	+	0.996224i	$0.527669\pi$
$152$	−77401.8			−3.35015
$153$		0			0
$154$	20061.3	+	20061.3i	0.845899	+	0.845899i
$155$			9851.43i			0.410049i
$156$		0			0
$157$	−24799.3			−1.00610			−0.503048	−	0.864258i	$-0.667788\pi$
$157$	−24799.3			−1.00610			−0.503048	+	0.864258i	$0.667788\pi$
$158$	28616.8	−	28616.8i	1.14632	−	1.14632i
$159$		0			0
$160$		−	5683.55i		−	0.222014i
$161$	4278.80	−	4278.80i	0.165071	−	0.165071i
$162$		0			0
$163$	6464.65	+	6464.65i	0.243316	+	0.243316i	0.818220	−	0.574905i	$-0.194961\pi$
$163$	6464.65	+	6464.65i	0.243316	+	0.243316i	−0.574905	+	0.818220i	$0.694961\pi$
$164$	−3740.93	−	3740.93i	−0.139089	−	0.139089i
$165$		0			0
$166$			78244.3i			2.83947i
$167$	−7697.87	−	7697.87i	−0.276018	−	0.276018i	0.555499	−	0.831517i	$-0.312527\pi$
$167$	−7697.87	−	7697.87i	−0.276018	−	0.276018i	−0.831517	+	0.555499i	$0.812527\pi$
$168$		0			0
$169$	−26980.1	−	9370.51i	−0.944647	−	0.328088i
$170$	−14119.8			−0.488574
$171$		0			0
$172$	65080.2			2.19984
$173$		−	10901.9i		−	0.364257i	−0.983275	−	0.182129i	$-0.941701\pi$
$173$		−	10901.9i		−	0.364257i	0.983275	−	0.182129i	$-0.0582987\pi$
$174$		0			0
$175$	10253.3	−	10253.3i	0.334801	−	0.334801i
$176$	54507.8	+	54507.8i	1.75968	+	1.75968i
$177$		0			0
$178$	−1917.25			−0.0605116
$179$			47477.0i			1.48176i	0.671638	+	0.740880i	$0.265591\pi$
$179$			47477.0i			1.48176i	−0.671638	+	0.740880i	$0.734409\pi$
$180$		0			0
$181$		−	48805.7i		−	1.48975i	−0.667203	−	0.744876i	$-0.732509\pi$
$181$		−	48805.7i		−	1.48975i	0.667203	−	0.744876i	$-0.267491\pi$
$182$	−4922.97	+	29179.5i	−0.148623	+	0.880918i
$183$		0			0
$184$	25782.8	−	25782.8i	0.761544	−	0.761544i
$185$	4705.85			0.137497
$186$		0			0
$187$	43380.6	−	43380.6i	1.24054	−	1.24054i
$188$	75665.2	−	75665.2i	2.14082	−	2.14082i
$189$		0			0
$190$	13977.2	+	13977.2i	0.387180	+	0.387180i
$191$	5254.59			0.144036			0.0720182	−	0.997403i	$-0.477056\pi$
$191$	5254.59			0.144036			0.0720182	+	0.997403i	$0.477056\pi$
$192$		0			0
$193$	13302.8	+	13302.8i	0.357131	+	0.357131i	0.862754	−	0.505623i	$-0.168737\pi$
$193$	13302.8	+	13302.8i	0.357131	+	0.357131i	−0.505623	+	0.862754i	$0.668737\pi$
$194$			14015.1i			0.372386i
$195$		0			0
$196$	−65674.2			−1.70955
$197$	19229.0	−	19229.0i	0.495477	−	0.495477i	−0.414550	−	0.910027i	$-0.636061\pi$
$197$	19229.0	−	19229.0i	0.495477	−	0.495477i	0.910027	+	0.414550i	$0.136061\pi$
$198$		0			0
$199$			27677.4i			0.698906i	0.936954	+	0.349453i	$0.113632\pi$
$199$			27677.4i			0.698906i	−0.936954	+	0.349453i	$0.886368\pi$
$200$	61783.5	−	61783.5i	1.54459	−	1.54459i
$201$		0			0
$202$	29482.9	+	29482.9i	0.722550	+	0.722550i
$203$	−7895.52	−	7895.52i	−0.191597	−	0.191597i
$204$		0			0
$205$			754.066i			0.0179433i
$206$	53719.2	+	53719.2i	1.26589	+	1.26589i
$207$		0			0
$208$	−13376.0	+	79282.4i	−0.309171	+	1.83253i
$209$	−85885.1			−1.96619
$210$		0			0
$211$	75928.5			1.70545			0.852727	−	0.522357i	$-0.174947\pi$
$211$	75928.5			1.70545			0.852727	+	0.522357i	$0.174947\pi$
$212$		−	24011.9i		−	0.534262i
$213$		0			0
$214$	−74173.6	+	74173.6i	−1.61965	+	1.61965i
$215$	−6559.16	−	6559.16i	−0.141896	−	0.141896i
$216$		0			0
$217$	−46257.2			−0.982335
$218$			57807.1i			1.21638i
$219$		0			0
$220$		−	30278.9i		−	0.625597i
$221$	63097.8	+	10645.4i	1.29190	+	0.217961i
$222$		0			0
$223$	−7655.61	+	7655.61i	−0.153947	+	0.153947i	−0.779878	−	0.625931i	$-0.784719\pi$
$223$	−7655.61	+	7655.61i	−0.153947	+	0.153947i	0.625931	+	0.779878i	$0.284719\pi$
$224$	26687.0			0.531868
$225$		0			0
$226$	86715.2	−	86715.2i	1.69777	−	1.69777i
$227$	13863.1	−	13863.1i	0.269035	−	0.269035i	−0.559676	−	0.828711i	$-0.689074\pi$
$227$	13863.1	−	13863.1i	0.269035	−	0.269035i	0.828711	+	0.559676i	$0.189074\pi$
$228$		0			0
$229$	3756.78	+	3756.78i	0.0716382	+	0.0716382i	0.742018	−	0.670380i	$-0.233869\pi$
$229$	3756.78	+	3756.78i	0.0716382	+	0.0716382i	−0.670380	+	0.742018i	$0.733869\pi$
$230$	−9311.73			−0.176025
$231$		0			0
$232$	−47576.1	−	47576.1i	−0.883921	−	0.883921i
$233$		−	12679.4i		−	0.233554i	−0.993158	−	0.116777i	$-0.962744\pi$
$233$		−	12679.4i		−	0.233554i	0.993158	−	0.116777i	$-0.0372563\pi$
$234$		0			0
$235$	−15251.9			−0.276178
$236$	−116350.	+	116350.i	−2.08903	+	2.08903i
$237$		0			0
$238$		−	66299.2i		−	1.17045i
$239$	61402.0	−	61402.0i	1.07495	−	1.07495i	0.0779924	−	0.996954i	$-0.475149\pi$
$239$	61402.0	−	61402.0i	1.07495	−	1.07495i	0.996954	−	0.0779924i	$-0.0248510\pi$
$240$		0			0
$241$	37991.4	+	37991.4i	0.654111	+	0.654111i	0.953980	−	0.299869i	$-0.0969431\pi$
$241$	37991.4	+	37991.4i	0.654111	+	0.654111i	−0.299869	+	0.953980i	$0.596943\pi$
$242$	59328.2	+	59328.2i	1.01305	+	1.01305i
$243$		0			0
$244$			71375.7i			1.19887i
$245$	6619.03	+	6619.03i	0.110271	+	0.110271i
$246$		0			0
$247$	−51922.7	−	72998.6i	−0.851067	−	1.19652i
$248$	−278733.			−4.53194
$249$		0			0
$250$	−45620.7			−0.729931
$251$		−	86105.8i		−	1.36674i	−0.730074	−	0.683368i	$-0.760514\pi$
$251$		−	86105.8i		−	1.36674i	0.730074	−	0.683368i	$-0.239486\pi$
$252$		0			0
$253$	28608.7	−	28608.7i	0.446948	−	0.446948i
$254$	8500.64	+	8500.64i	0.131760	+	0.131760i
$255$		0			0
$256$	−114820.			−1.75201
$257$		−	23260.9i		−	0.352176i	−0.984374	−	0.176088i	$-0.943656\pi$
$257$		−	23260.9i		−	0.352176i	0.984374	−	0.176088i	$-0.0563443\pi$
$258$		0			0
$259$			22096.2i			0.329396i
$260$	25735.7	−	18305.4i	0.380706	−	0.270790i
$261$		0			0
$262$	80361.5	−	80361.5i	1.17070	−	1.17070i
$263$	32185.6			0.465319			0.232659	−	0.972558i	$-0.425257\pi$
$263$	32185.6			0.465319			0.232659	+	0.972558i	$0.425257\pi$
$264$		0			0
$265$	−2420.06	+	2420.06i	−0.0344615	+	0.0344615i
$266$	−65629.7	+	65629.7i	−0.927550	+	0.927550i
$267$		0			0
$268$	−87616.2	−	87616.2i	−1.21987	−	1.21987i
$269$	60022.3			0.829484			0.414742	−	0.909939i	$-0.363872\pi$
$269$	60022.3			0.829484			0.414742	+	0.909939i	$0.363872\pi$
$270$		0			0
$271$	37263.2	+	37263.2i	0.507390	+	0.507390i	0.913724	−	0.406335i	$-0.133193\pi$
$271$	37263.2	+	37263.2i	0.507390	+	0.507390i	−0.406335	+	0.913724i	$0.633193\pi$
$272$		−	180139.i		−	2.43483i
$273$		0			0
$274$	45132.9			0.601163
$275$	68555.0	−	68555.0i	0.906512	−	0.906512i
$276$		0			0
$277$		−	31412.8i		−	0.409399i	−0.978825	−	0.204699i	$-0.934378\pi$
$277$		−	31412.8i		−	0.409399i	0.978825	−	0.204699i	$-0.0656217\pi$
$278$	−128138.	+	128138.i	−1.65801	+	1.65801i
$279$		0			0
$280$	−12913.7	−	12913.7i	−0.164716	−	0.164716i
$281$	−84576.2	−	84576.2i	−1.07111	−	1.07111i	−0.997270	−	0.0738438i	$-0.976473\pi$
$281$	−84576.2	−	84576.2i	−1.07111	−	1.07111i	−0.0738438	−	0.997270i	$-0.523527\pi$
$282$		0			0
$283$		−	99352.6i		−	1.24053i	−0.784393	−	0.620264i	$-0.787026\pi$
$283$		−	99352.6i		−	1.24053i	0.784393	−	0.620264i	$-0.212974\pi$
$284$	−135822.	−	135822.i	−1.68397	−	1.68397i
$285$		0			0
$286$	−32915.7	+	195099.i	−0.402412	+	2.38518i
$287$	−3540.70			−0.0429858
$288$		0			0
$289$	−59844.3			−0.716518
$290$			17182.6i			0.204311i
$291$		0			0
$292$	111434.	−	111434.i	1.30692	−	1.30692i
$293$	−65772.3	−	65772.3i	−0.766139	−	0.766139i	0.211286	−	0.977424i	$-0.432235\pi$
$293$	−65772.3	−	65772.3i	−0.766139	−	0.766139i	−0.977424	+	0.211286i	$0.932235\pi$
$294$		0			0
$295$	23452.9			0.269497
$296$			133145.i			1.51965i
$297$		0			0
$298$			238298.i			2.68341i
$299$	41611.8	+	7020.45i	0.465451	+	0.0785277i
$300$		0			0
$301$	30798.4	−	30798.4i	0.339934	−	0.339934i
$302$	211885.			2.32320
$303$		0			0
$304$	−178320.	+	178320.i	−1.92953	+	1.92953i
$305$	7193.66	−	7193.66i	0.0773303	−	0.0773303i
$306$		0			0
$307$	78031.0	+	78031.0i	0.827924	+	0.827924i	0.987229	−	0.159305i	$-0.0509254\pi$
$307$	78031.0	+	78031.0i	0.827924	+	0.827924i	−0.159305	+	0.987229i	$0.550925\pi$
$308$	142174.			1.49871
$309$		0			0
$310$	50333.5	+	50333.5i	0.523762	+	0.523762i
$311$			28292.0i			0.292511i	0.989247	+	0.146256i	$0.0467222\pi$
$311$			28292.0i			0.292511i	−0.989247	+	0.146256i	$0.953278\pi$
$312$		0			0
$313$	29449.3			0.300598			0.150299	−	0.988641i	$-0.451976\pi$
$313$	29449.3			0.300598			0.150299	+	0.988641i	$0.451976\pi$
$314$	−126706.	+	126706.i	−1.28510	+	1.28510i
$315$		0			0
$316$		−	202806.i		−	2.03099i
$317$	−109874.	+	109874.i	−1.09339	+	1.09339i	−0.0982270	+	0.995164i	$0.531317\pi$
$317$	−109874.	+	109874.i	−1.09339	+	1.09339i	−0.995164	+	0.0982270i	$0.968683\pi$
$318$		0			0
$319$	−52790.5	−	52790.5i	−0.518770	−	0.518770i
$320$	−1259.39	−	1259.39i	−0.0122987	−	0.0122987i
$321$		0			0
$322$		−	43723.0i		−	0.421695i
$323$	141918.	+	141918.i	1.36029	+	1.36029i
$324$		0			0
$325$	99714.4	+	16823.1i	0.944041	+	0.159272i
$326$	66059.2			0.621582
$327$		0			0
$328$	−21335.3			−0.198313
$329$		−	71615.2i		−	0.661628i
$330$		0			0
$331$	−36941.8	+	36941.8i	−0.337180	+	0.337180i	−0.855305	−	0.518125i	$-0.826630\pi$
$331$	−36941.8	+	36941.8i	−0.337180	+	0.337180i	0.518125	+	0.855305i	$0.326630\pi$
$332$	277257.	+	277257.i	2.51540	+	2.51540i
$333$		0			0
$334$	−78660.9			−0.705125
$335$			17660.9i			0.157371i
$336$		0			0
$337$			68140.0i			0.599988i	0.953941	+	0.299994i	$0.0969847\pi$
$337$			68140.0i			0.599988i	−0.953941	+	0.299994i	$0.903015\pi$
$338$	−185725.	+	89971.9i	−1.62568	+	0.787542i
$339$		0			0
$340$	−50033.2	+	50033.2i	−0.432813	+	0.432813i
$341$	−309282.			−2.65978
$342$		0			0
$343$	−72222.0	+	72222.0i	−0.613877	+	0.613877i
$344$	185582.	−	185582.i	1.56827	−	1.56827i
$345$		0			0
$346$	−55700.4	−	55700.4i	−0.465271	−	0.465271i
$347$	74667.2			0.620113			0.310056	−	0.950718i	$-0.399652\pi$
$347$	74667.2			0.620113			0.310056	+	0.950718i	$0.399652\pi$
$348$		0			0
$349$	−89108.0	−	89108.0i	−0.731587	−	0.731587i	0.239347	−	0.970934i	$-0.423067\pi$
$349$	−89108.0	−	89108.0i	−0.731587	−	0.731587i	−0.970934	+	0.239347i	$0.923067\pi$
$350$		−	104774.i		−	0.855295i
$351$		0			0
$352$	178433.			1.44009
$353$	16373.7	−	16373.7i	0.131400	−	0.131400i	−0.638348	−	0.769748i	$-0.720382\pi$
$353$	16373.7	−	16373.7i	0.131400	−	0.131400i	0.769748	+	0.638348i	$0.220382\pi$
$354$		0			0
$355$			27377.9i			0.217242i
$356$	−6793.73	+	6793.73i	−0.0536054	+	0.0536054i
$357$		0			0
$358$	242573.	+	242573.i	1.89267	+	1.89267i
$359$	81223.2	+	81223.2i	0.630218	+	0.630218i	0.948123	−	0.317905i	$-0.102979\pi$
$359$	81223.2	+	81223.2i	0.630218	+	0.630218i	−0.317905	+	0.948123i	$0.602979\pi$
$360$		0			0
$361$		−	150648.i		−	1.15598i
$362$	−249361.	−	249361.i	−1.90288	−	1.90288i
$363$		0			0
$364$	85952.7	+	120842.i	0.648719	+	0.912039i
$365$	−22461.8			−0.168601
$366$		0			0
$367$	30462.2			0.226167			0.113084	−	0.993585i	$-0.463927\pi$
$367$	30462.2			0.226167			0.113084	+	0.993585i	$0.463927\pi$
$368$		−	118798.i		−	0.877230i
$369$		0			0
$370$	24043.4	−	24043.4i	0.175628	−	0.175628i
$371$	−11363.3	−	11363.3i	−0.0825578	−	0.0825578i
$372$		0			0
$373$	188516.			1.35497			0.677486	−	0.735536i	$-0.263069\pi$
$373$	188516.			1.35497			0.677486	+	0.735536i	$0.263069\pi$
$374$		−	443286.i		−	3.16914i
$375$		0			0
$376$		−	431533.i		−	3.05238i
$377$	12954.6	−	76784.7i	0.0911467	−	0.540247i
$378$		0			0
$379$	−36802.3	+	36802.3i	−0.256210	+	0.256210i	−0.823511	−	0.567300i	$-0.807988\pi$
$379$	−36802.3	+	36802.3i	−0.256210	+	0.256210i	0.567300	+	0.823511i	$0.307988\pi$
$380$	99056.0			0.685983
$381$		0			0
$382$	26847.1	−	26847.1i	0.183980	−	0.183980i
$383$	49201.0	−	49201.0i	0.335410	−	0.335410i	−0.519226	−	0.854637i	$-0.673780\pi$
$383$	49201.0	−	49201.0i	0.335410	−	0.335410i	0.854637	+	0.519226i	$0.173780\pi$
$384$		0			0
$385$	−14329.1	−	14329.1i	−0.0966714	−	0.0966714i
$386$	135935.			0.912338
$387$		0			0
$388$	49662.3	+	49662.3i	0.329886	+	0.329886i
$389$			189271.i			1.25079i	0.780307	+	0.625396i	$0.215063\pi$
$389$			189271.i			1.25079i	−0.780307	+	0.625396i	$0.784937\pi$
$390$		0			0
$391$	−94546.6			−0.618433
$392$	−187276.	+	187276.i	−1.21874	+	1.21874i
$393$		0			0
$394$		−	196492.i		−	1.26576i
$395$	−20440.0	+	20440.0i	−0.131005	+	0.131005i
$396$		0			0
$397$	−53096.3	−	53096.3i	−0.336886	−	0.336886i	0.518308	−	0.855194i	$-0.326562\pi$
$397$	−53096.3	−	53096.3i	−0.336886	−	0.336886i	−0.855194	+	0.518308i	$0.826562\pi$
$398$	141411.	+	141411.i	0.892723	+	0.892723i
$399$		0			0
$400$		−	284676.i		−	1.77922i
$401$	−194820.	−	194820.i	−1.21156	−	1.21156i	−0.970513	−	0.241048i	$-0.922509\pi$
$401$	−194820.	−	194820.i	−1.21156	−	1.21156i	−0.241048	−	0.970513i	$-0.577491\pi$
$402$		0			0
$403$	−186980.	−	262876.i	−1.15129	−	1.61861i
$404$	208944.			1.28017
$405$		0			0
$406$	−80680.5			−0.489459
$407$			147738.i			0.891876i
$408$		0			0
$409$	−183911.	+	183911.i	−1.09941	+	1.09941i	−0.104932	+	0.994479i	$0.533463\pi$
$409$	−183911.	+	183911.i	−1.09941	+	1.09941i	−0.994479	+	0.104932i	$0.966537\pi$
$410$	3852.72	+	3852.72i	0.0229192	+	0.0229192i
$411$		0			0
$412$	380706.			2.24282
$413$			110123.i			0.645620i
$414$		0			0
$415$		−	55887.2i		−	0.324501i
$416$	107874.	+	151660.i	0.623345	+	0.876366i
$417$		0			0
$418$	−438810.	+	438810.i	−2.51145	+	2.51145i
$419$	251451.			1.43227			0.716137	−	0.697960i	$-0.245909\pi$
$419$	251451.			1.43227			0.716137	+	0.697960i	$0.245909\pi$
$420$		0			0
$421$	−244452.	+	244452.i	−1.37921	+	1.37921i	−0.533243	+	0.845962i	$0.679027\pi$
$421$	−244452.	+	244452.i	−1.37921	+	1.37921i	−0.845962	+	0.533243i	$0.820973\pi$
$422$	387939.	−	387939.i	2.17840	−	2.17840i
$423$		0			0
$424$	−68472.2	−	68472.2i	−0.380875	−	0.380875i
$425$	−226562.			−1.25432
$426$		0			0
$427$	33777.7	+	33777.7i	0.185257	+	0.185257i
$428$			525666.i			2.86961i
$429$		0			0
$430$	−67025.0			−0.362493
$431$	16725.2	−	16725.2i	0.0900360	−	0.0900360i	−0.660654	−	0.750690i	$-0.729721\pi$
$431$	16725.2	−	16725.2i	0.0900360	−	0.0900360i	0.750690	+	0.660654i	$0.229721\pi$
$432$		0			0
$433$			301479.i			1.60798i	0.594640	+	0.803992i	$0.297295\pi$
$433$			301479.i			1.60798i	−0.594640	+	0.803992i	$0.702705\pi$
$434$	−236340.	+	236340.i	−1.25475	+	1.25475i
$435$		0			0
$436$	204838.	+	204838.i	1.07755	+	1.07755i
$437$	93592.0	+	93592.0i	0.490090	+	0.490090i
$438$		0			0
$439$			19395.7i			0.100641i	0.998733	+	0.0503207i	$0.0160243\pi$
$439$			19395.7i			0.100641i	−0.998733	+	0.0503207i	$0.983976\pi$
$440$	−86343.2	−	86343.2i	−0.445987	−	0.445987i
$441$		0			0
$442$	376774.	−	267993.i	1.92857	−	1.37176i
$443$	−45077.6			−0.229696			−0.114848	−	0.993383i	$-0.536638\pi$
$443$	−45077.6			−0.229696			−0.114848	+	0.993383i	$0.536638\pi$
$444$		0			0
$445$	1369.42			0.00691541
$446$			78229.0i			0.393277i
$447$		0			0
$448$	5913.42	−	5913.42i	0.0294634	−	0.0294634i
$449$	103299.	+	103299.i	0.512395	+	0.512395i	0.915260	−	0.402864i	$-0.131985\pi$
$449$	103299.	+	103299.i	0.512395	+	0.512395i	−0.402864	+	0.915260i	$0.631985\pi$
$450$		0			0
$451$	−23673.6			−0.116389
$452$		−	614547.i		−	3.00800i
$453$		0			0
$454$		−	141661.i		−	0.687286i
$455$	3516.31	−	20841.9i	0.0169849	−	0.100673i
$456$		0			0
$457$	242426.	−	242426.i	1.16077	−	1.16077i	0.176464	−	0.984307i	$-0.443534\pi$
$457$	242426.	−	242426.i	1.16077	−	1.16077i	0.984307	−	0.176464i	$-0.0564660\pi$
$458$	38388.8			0.183009
$459$		0			0
$460$	−32995.9	+	32995.9i	−0.155935	+	0.155935i
$461$	−263580.	+	263580.i	−1.24026	+	1.24026i	−0.280361	+	0.959895i	$0.590454\pi$
$461$	−263580.	+	263580.i	−1.24026	+	1.24026i	−0.959895	+	0.280361i	$0.909546\pi$
$462$		0			0
$463$	188223.	+	188223.i	0.878034	+	0.878034i	0.993331	−	0.115297i	$-0.0367819\pi$
$463$	188223.	+	188223.i	0.878034	+	0.878034i	−0.115297	+	0.993331i	$0.536782\pi$
$464$	−219214.			−1.01820
$465$		0			0
$466$	−64782.5	−	64782.5i	−0.298322	−	0.298322i
$467$		−	185241.i		−	0.849383i	−0.905338	−	0.424692i	$-0.860383\pi$
$467$		−	185241.i		−	0.849383i	0.905338	−	0.424692i	$-0.139617\pi$
$468$		0			0
$469$	−82926.6			−0.377006
$470$	−77926.2	+	77926.2i	−0.352767	+	0.352767i
$471$		0			0
$472$			663569.i			2.97853i
$473$	205923.	−	205923.i	0.920411	−	0.920411i
$474$		0			0
$475$	224275.	+	224275.i	0.994015	+	0.994015i
$476$	−234930.	−	234930.i	−1.03687	−	1.03687i
$477$		0			0
$478$		−	627438.i		−	2.74609i
$479$	160707.	+	160707.i	0.700429	+	0.700429i	0.964503	−	0.264073i	$-0.0850661\pi$
$479$	160707.	+	160707.i	0.700429	+	0.700429i	−0.264073	+	0.964503i	$0.585066\pi$
$480$		0			0
$481$	−125571.	+	89316.8i	−0.542750	+	0.386049i
$482$	388216.			1.67101
$483$		0			0
$484$	420457.			1.79486
$485$		−	10010.5i		−	0.0425572i
$486$		0			0
$487$	−6674.93	+	6674.93i	−0.0281442	+	0.0281442i	−0.721039	−	0.692895i	$-0.756335\pi$
$487$	−6674.93	+	6674.93i	−0.0281442	+	0.0281442i	0.692895	+	0.721039i	$0.256335\pi$
$488$	203535.	+	203535.i	0.854671	+	0.854671i
$489$		0			0
$490$	67636.7			0.281702
$491$		−	219127.i		−	0.908937i	−0.890763	−	0.454468i	$-0.849829\pi$
$491$		−	219127.i		−	0.908937i	0.890763	−	0.454468i	$-0.150171\pi$
$492$		0			0
$493$			174464.i			0.717812i
$494$	−638256.	−	107682.i	−2.61542	−	0.441255i
$495$		0			0
$496$	−642150.	+	642150.i	−2.61019	+	2.61019i
$497$	−128553.			−0.520437
$498$		0			0
$499$	102393.	−	102393.i	0.411215	−	0.411215i	−0.470947	−	0.882162i	$-0.656088\pi$
$499$	102393.	−	102393.i	0.411215	−	0.411215i	0.882162	+	0.470947i	$0.156088\pi$
$500$	−161656.	+	161656.i	−0.646624	+	0.646624i
$501$		0			0
$502$	−439937.	−	439937.i	−1.74575	−	1.74575i
$503$	−230420.			−0.910717			−0.455358	−	0.890308i	$-0.650489\pi$
$503$	−230420.			−0.910717			−0.455358	+	0.890308i	$0.650489\pi$
$504$		0			0
$505$	−21058.6	−	21058.6i	−0.0825748	−	0.0825748i
$506$		−	292338.i		−	1.14179i
$507$		0			0
$508$	60243.7			0.233445
$509$	8315.17	−	8315.17i	0.0320949	−	0.0320949i	−0.690877	−	0.722972i	$-0.742775\pi$
$509$	8315.17	−	8315.17i	0.0320949	−	0.0320949i	0.722972	+	0.690877i	$0.242775\pi$
$510$		0			0
$511$		−	105469.i		−	0.403909i
$512$	−415507.	+	415507.i	−1.58503	+	1.58503i
$513$		0			0
$514$	−118846.	−	118846.i	−0.449840	−	0.449840i
$515$	−38369.8	−	38369.8i	−0.144669	−	0.144669i
$516$		0			0
$517$		−	478830.i		−	1.79143i
$518$	112895.	+	112895.i	0.420743	+	0.420743i
$519$		0			0
$520$	21188.3	−	125588.i	0.0783590	−	0.464451i
$521$	−142578.			−0.525264			−0.262632	−	0.964896i	$-0.584590\pi$
$521$	−142578.			−0.525264			−0.262632	+	0.964896i	$0.584590\pi$
$522$		0			0
$523$	−299990.			−1.09674			−0.548371	−	0.836235i	$-0.684752\pi$
$523$	−299990.			−1.09674			−0.548371	+	0.836235i	$0.684752\pi$
$524$		−	569519.i		−	2.07417i
$525$		0			0
$526$	164445.	−	164445.i	0.594359	−	0.594359i
$527$	511062.	+	511062.i	1.84015	+	1.84015i
$528$		0			0
$529$	217489.			0.777189
$530$			24729.4i			0.0880364i
$531$		0			0
$532$			465115.i			1.64338i
$533$	−14312.1	−	20121.5i	−0.0503790	−	0.0708283i
$534$		0			0
$535$	52979.6	−	52979.6i	0.185098	−	0.185098i
$536$	−499692.			−1.73929
$537$		0			0
$538$	306670.	−	306670.i	1.05951	−	1.05951i
$539$	−207802.	+	207802.i	−0.715274	+	0.715274i
$540$		0			0
$541$	254443.	+	254443.i	0.869354	+	0.869354i	0.992401	−	0.123047i	$-0.0392665\pi$
$541$	254443.	+	254443.i	0.869354	+	0.869354i	−0.123047	+	0.992401i	$0.539267\pi$
$542$	380775.			1.29619
$543$		0			0
$544$	−294845.	−	294845.i	−0.996314	−	0.996314i
$545$		−	41289.6i		−	0.139010i
$546$		0			0
$547$	230552.			0.770538			0.385269	−	0.922804i	$-0.374109\pi$
$547$	230552.			0.770538			0.385269	+	0.922804i	$0.374109\pi$
$548$	159928.	−	159928.i	0.532552	−	0.532552i
$549$		0			0
$550$		−	700531.i		−	2.31581i
$551$	172702.	−	172702.i	0.568845	−	0.568845i
$552$		0			0
$553$	−95975.6	−	95975.6i	−0.313842	−	0.313842i
$554$	−160496.	−	160496.i	−0.522932	−	0.522932i
$555$		0			0
$556$			908106.i			2.93756i
$557$	−38481.0	−	38481.0i	−0.124033	−	0.124033i	0.642366	−	0.766398i	$-0.277953\pi$
$557$	−38481.0	−	38481.0i	−0.124033	−	0.124033i	−0.766398	+	0.642366i	$0.777953\pi$
$558$		0			0
$559$	299518.	+	50532.6i	0.958515	+	0.161714i
$560$	−59501.9			−0.189738
$561$		0			0
$562$	−864244.			−2.73630
$563$			41906.4i			0.132210i	0.997813	+	0.0661048i	$0.0210572\pi$
$563$			41906.4i			0.132210i	−0.997813	+	0.0661048i	$0.978943\pi$
$564$		0			0
$565$	−61937.6	+	61937.6i	−0.194025	+	0.194025i
$566$	−507619.	−	507619.i	−1.58455	−	1.58455i
$567$		0			0
$568$	−774621.			−2.40100
$569$		−	80590.0i		−	0.248918i	−0.992225	−	0.124459i	$-0.960280\pi$
$569$		−	80590.0i		−	0.248918i	0.992225	−	0.124459i	$-0.0397196\pi$
$570$		0			0
$571$			412664.i			1.26568i	0.774281	+	0.632841i	$0.218111\pi$
$571$			412664.i			1.26568i	−0.774281	+	0.632841i	$0.781889\pi$
$572$	574692.	+	807964.i	1.75648	+	2.46945i
$573$		0			0
$574$	−18090.4	+	18090.4i	−0.0549065	+	0.0549065i
$575$	−149413.			−0.451912
$576$		0			0
$577$	−292623.	+	292623.i	−0.878935	+	0.878935i	−0.993424	−	0.114490i	$-0.963477\pi$
$577$	−292623.	+	292623.i	−0.878935	+	0.878935i	0.114490	+	0.993424i	$0.463477\pi$
$578$	−305760.	+	305760.i	−0.915220	+	0.915220i
$579$		0			0
$580$	60886.2	+	60886.2i	0.180993	+	0.180993i
$581$	262417.			0.777392
$582$		0			0
$583$	−75976.9	−	75976.9i	−0.223534	−	0.223534i
$584$		−	635527.i		−	1.86341i
$585$		0			0
$586$	−672096.			−1.95720
$587$	26371.4	−	26371.4i	0.0765344	−	0.0765344i	−0.667803	−	0.744338i	$-0.732765\pi$
$587$	26371.4	−	26371.4i	0.0765344	−	0.0765344i	0.744338	+	0.667803i	$0.232765\pi$
$588$		0			0
$589$		−	1.01180e6i		−	2.91652i
$590$	119827.	−	119827.i	0.344232	−	0.344232i
$591$		0			0
$592$	306743.	+	306743.i	0.875248	+	0.875248i
$593$	45844.5	+	45844.5i	0.130370	+	0.130370i	0.769281	−	0.638911i	$-0.220615\pi$
$593$	45844.5	+	45844.5i	0.130370	+	0.130370i	−0.638911	+	0.769281i	$0.720615\pi$
$594$		0			0
$595$			47355.2i			0.133762i
$596$	844404.	+	844404.i	2.37716	+	2.37716i
$597$		0			0
$598$	248475.	−	176736.i	0.694833	−	0.494223i
$599$	−459477.			−1.28059			−0.640295	−	0.768129i	$-0.721188\pi$
$599$	−459477.			−1.28059			−0.640295	+	0.768129i	$0.721188\pi$
$600$		0			0
$601$	−103816.			−0.287418			−0.143709	−	0.989620i	$-0.545903\pi$
$601$	−103816.			−0.287418			−0.143709	+	0.989620i	$0.545903\pi$
$602$		−	314714.i		−	0.868408i
$603$		0			0
$604$	750811.	−	750811.i	2.05806	−	2.05806i
$605$	−42376.1	−	42376.1i	−0.115774	−	0.115774i
$606$		0			0
$607$	406615.			1.10359			0.551793	−	0.833981i	$-0.313944\pi$
$607$	406615.			1.10359			0.551793	+	0.833981i	$0.313944\pi$
$608$			583736.i			1.57910i
$609$		0			0
$610$		−	73508.6i		−	0.197551i
$611$	406984.	−	289481.i	1.09017	−	0.775422i
$612$		0			0
$613$	321793.	−	321793.i	0.856358	−	0.856358i	−0.134549	−	0.990907i	$-0.542959\pi$
$613$	321793.	−	321793.i	0.856358	−	0.856358i	0.990907	+	0.134549i	$0.0429586\pi$
$614$	797362.			2.11504
$615$		0			0
$616$	405423.	−	405423.i	1.06843	−	1.06843i
$617$	225476.	−	225476.i	0.592284	−	0.592284i	−0.345964	−	0.938248i	$-0.612448\pi$
$617$	225476.	−	225476.i	0.592284	−	0.592284i	0.938248	+	0.345964i	$0.112448\pi$
$618$		0			0
$619$	−71621.9	−	71621.9i	−0.186924	−	0.186924i	0.607441	−	0.794365i	$-0.292196\pi$
$619$	−71621.9	−	71621.9i	−0.186924	−	0.186924i	−0.794365	+	0.607441i	$0.792196\pi$
$620$	356712.			0.927971
$621$		0			0
$622$	144551.	+	144551.i	0.373629	+	0.373629i
$623$			6430.10i			0.0165669i
$624$		0			0
$625$	−341392.			−0.873964
$626$	150464.	−	150464.i	0.383959	−	0.383959i
$627$		0			0
$628$			897961.i			2.27687i
$629$	244125.	−	244125.i	0.617036	−	0.617036i
$630$		0			0
$631$	77694.6	+	77694.6i	0.195134	+	0.195134i	0.797910	−	0.602776i	$-0.205939\pi$
$631$	77694.6	+	77694.6i	0.195134	+	0.195134i	−0.602776	+	0.797910i	$0.705939\pi$
$632$	−578322.	−	578322.i	−1.44789	−	1.44789i
$633$		0			0
$634$			1.12275e6i			2.79321i
$635$	−6071.71	−	6071.71i	−0.0150579	−	0.0150579i
$636$		0			0
$637$	−302252.	−	50993.8i	−0.744886	−	0.125672i
$638$	−539442.			−1.32527
$639$		0			0
$640$	78067.8			0.190595
$641$			409275.i			0.996092i	0.867151	+	0.498046i	$0.165949\pi$
$641$			409275.i			0.996092i	−0.867151	+	0.498046i	$0.834051\pi$
$642$		0			0
$643$	173185.	−	173185.i	0.418880	−	0.418880i	−0.465938	−	0.884817i	$-0.654283\pi$
$643$	173185.	−	173185.i	0.418880	−	0.418880i	0.884817	+	0.465938i	$0.154283\pi$
$644$	−154932.	−	154932.i	−0.373567	−	0.373567i
$645$		0			0
$646$	1.45019e6			3.47504
$647$			239759.i			0.572752i	0.958117	+	0.286376i	$0.0924507\pi$
$647$			239759.i			0.572752i	−0.958117	+	0.286376i	$0.907549\pi$
$648$		0			0
$649$			736297.i			1.74809i
$650$	595421.	−	423513.i	1.40928	−	1.00240i
$651$		0			0
$652$	234080.	−	234080.i	0.550641	−	0.550641i
$653$	112205.			0.263138			0.131569	−	0.991307i	$-0.457998\pi$
$653$	112205.			0.263138			0.131569	+	0.991307i	$0.457998\pi$
$654$		0			0
$655$	−57399.4	+	57399.4i	−0.133790	+	0.133790i
$656$	−49152.6	+	49152.6i	−0.114219	+	0.114219i
$657$		0			0
$658$	−365901.	−	365901.i	−0.845107	−	0.845107i
$659$	−377953.			−0.870295			−0.435147	−	0.900359i	$-0.643304\pi$
$659$	−377953.			−0.870295			−0.435147	+	0.900359i	$0.643304\pi$
$660$		0			0
$661$	−130696.	−	130696.i	−0.299129	−	0.299129i	0.541544	−	0.840673i	$-0.317840\pi$
$661$	−130696.	−	130696.i	−0.299129	−	0.299129i	−0.840673	+	0.541544i	$0.817840\pi$
$662$			377491.i			0.861371i
$663$		0			0
$664$	1.58125e6			3.58645
$665$	46877.0	−	46877.0i	0.106003	−	0.106003i
$666$		0			0
$667$			115055.i			0.258616i
$668$	−278733.	+	278733.i	−0.624649	+	0.624649i
$669$		0			0
$670$	90234.5	+	90234.5i	0.201012	+	0.201012i
$671$	225842.	+	225842.i	0.501603	+	0.501603i
$672$		0			0
$673$			423645.i			0.935345i	0.883902	+	0.467672i	$0.154907\pi$
$673$			423645.i			0.935345i	−0.883902	+	0.467672i	$0.845093\pi$
$674$	348145.	+	348145.i	0.766374	+	0.766374i
$675$		0			0
$676$	−339298.	+	976926.i	−0.742486	+	2.13781i
$677$	−660988.			−1.44217			−0.721085	−	0.692847i	$-0.756356\pi$
$677$	−660988.			−1.44217			−0.721085	+	0.692847i	$0.756356\pi$
$678$		0			0
$679$	47004.2			0.101952
$680$			285349.i			0.617104i
$681$		0			0
$682$	−1.58020e6	+	1.58020e6i	−3.39738	+	3.39738i
$683$	224471.	+	224471.i	0.481193	+	0.481193i	0.905513	−	0.424319i	$-0.139487\pi$
$683$	224471.	+	224471.i	0.481193	+	0.481193i	−0.424319	+	0.905513i	$0.639487\pi$
$684$		0			0
$685$	−32236.9			−0.0687024
$686$			738002.i			1.56823i
$687$		0			0
$688$		−	855097.i		−	1.80650i
$689$	18644.4	−	110510.i	0.0392745	−	0.232788i
$690$		0			0
$691$	497029.	−	497029.i	1.04094	−	1.04094i	0.0418150	−	0.999125i	$-0.486686\pi$
$691$	497029.	−	497029.i	1.04094	−	1.04094i	0.999125	−	0.0418150i	$-0.0133140\pi$
$692$	−394747.			−0.824340
$693$		0			0
$694$	381494.	−	381494.i	0.792080	−	0.792080i
$695$	91524.2	−	91524.2i	0.189481	−	0.189481i
$696$		0			0
$697$	39118.6	+	39118.6i	0.0805226	+	0.0805226i
$698$	−910553.			−1.86894
$699$		0			0
$700$	−371263.	−	371263.i	−0.757680	−	0.757680i
$701$			187813.i			0.382199i	0.981571	+	0.191099i	$0.0612053\pi$
$701$			187813.i			0.382199i	−0.981571	+	0.191099i	$0.938795\pi$
$702$		0			0
$703$	−483320.			−0.977966
$704$	39538.0	−	39538.0i	0.0797754	−	0.0797754i
$705$		0			0
$706$		−	167315.i		−	0.335680i
$707$	98880.4	−	98880.4i	0.197821	−	0.197821i
$708$		0			0
$709$	−163815.	−	163815.i	−0.325883	−	0.325883i	0.525136	−	0.851019i	$-0.324015\pi$
$709$	−163815.	−	163815.i	−0.325883	−	0.325883i	−0.851019	+	0.525136i	$0.824015\pi$
$710$	139881.	+	139881.i	0.277487	+	0.277487i
$711$		0			0
$712$			38746.0i			0.0764305i
$713$	337035.	+	337035.i	0.662974	+	0.662974i
$714$		0			0
$715$	23510.5	−	139352.i	0.0459886	−	0.272585i
$716$	1.71910e6			3.35333
$717$		0			0
$718$	829981.			1.60998
$719$		−	752295.i		−	1.45523i	−0.685987	−	0.727613i	$-0.740630\pi$
$719$		−	752295.i		−	1.45523i	0.685987	−	0.727613i	$-0.259370\pi$
$720$		0			0
$721$	180164.	−	180164.i	0.346576	−	0.346576i
$722$	−769702.	−	769702.i	−1.47655	−	1.47655i
$723$		0			0
$724$	−1.76722e6			−3.37142
$725$			275707.i			0.524532i
$726$		0			0
$727$		−	267294.i		−	0.505732i	−0.967501	−	0.252866i	$-0.918627\pi$
$727$		−	267294.i		−	0.505732i	0.967501	−	0.252866i	$-0.0813731\pi$
$728$	589694.	+	99489.1i	1.11266	+	0.187721i
$729$		0			0
$730$	−114763.	+	114763.i	−0.215357	+	0.215357i
$731$	−680538.			−1.27355
$732$		0			0
$733$	−8050.21	+	8050.21i	−0.0149830	+	0.0149830i	−0.714559	−	0.699576i	$-0.753372\pi$
$733$	−8050.21	+	8050.21i	−0.0149830	+	0.0149830i	0.699576	+	0.714559i	$0.253372\pi$
$734$	155639.	−	155639.i	0.288887	−	0.288887i
$735$		0			0
$736$	−194445.	−	194445.i	−0.358955	−	0.358955i
$737$	−554459.			−1.02079
$738$		0			0
$739$	−158151.	−	158151.i	−0.289590	−	0.289590i	0.547328	−	0.836918i	$-0.315645\pi$
$739$	−158151.	−	158151.i	−0.289590	−	0.289590i	−0.836918	+	0.547328i	$0.815645\pi$
$740$		−	170395.i		−	0.311166i
$741$		0			0
$742$	−116117.			−0.210905
$743$	361688.	−	361688.i	0.655174	−	0.655174i	−0.299061	−	0.954234i	$-0.596673\pi$
$743$	361688.	−	361688.i	0.655174	−	0.655174i	0.954234	+	0.299061i	$0.0966732\pi$
$744$		0			0
$745$		−	170208.i		−	0.306667i
$746$	963177.	−	963177.i	1.73073	−	1.73073i
$747$		0			0
$748$	−1.57078e6	−	1.57078e6i	−2.80744	−	2.80744i
$749$	248765.	+	248765.i	0.443430	+	0.443430i
$750$		0			0
$751$		−	554610.i		−	0.983349i	−0.870779	−	0.491675i	$-0.836385\pi$
$751$		−	554610.i		−	0.983349i	0.870779	−	0.491675i	$-0.163615\pi$
$752$	−994174.	−	994174.i	−1.75803	−	1.75803i
$753$		0			0
$754$	−326125.	−	458502.i	−0.573642	−	0.806489i
$755$	−151342.			−0.265501
$756$		0			0
$757$	−307275.			−0.536211			−0.268105	−	0.963390i	$-0.586398\pi$
$757$	−307275.			−0.536211			−0.268105	+	0.963390i	$0.586398\pi$
$758$			376066.i			0.654523i
$759$		0			0
$760$	282468.	−	282468.i	0.489037	−	0.489037i
$761$	−428797.	−	428797.i	−0.740427	−	0.740427i	0.232233	−	0.972660i	$-0.425397\pi$
$761$	−428797.	−	428797.i	−0.740427	−	0.740427i	−0.972660	+	0.232233i	$0.925397\pi$
$762$		0			0
$763$	193874.			0.333021
$764$		−	190264.i		−	0.325965i
$765$		0			0
$766$		−	502762.i		−	0.856850i
$767$	−625820.	+	445136.i	−1.06380	+	0.756662i
$768$		0			0
$769$	−95508.3	+	95508.3i	−0.161506	+	0.161506i	−0.783234	−	0.621728i	$-0.786431\pi$
$769$	−95508.3	+	95508.3i	−0.161506	+	0.161506i	0.621728	+	0.783234i	$0.286431\pi$
$770$	−146422.			−0.246960
$771$		0			0
$772$	481682.	−	481682.i	0.808213	−	0.808213i
$773$	−499400.	+	499400.i	−0.835775	+	0.835775i	−0.988300	−	0.152525i	$-0.951260\pi$
$773$	−499400.	+	499400.i	−0.835775	+	0.835775i	0.152525	+	0.988300i	$0.451260\pi$
$774$		0			0
$775$	807638.	+	807638.i	1.34466	+	1.34466i
$776$	283234.			0.470351
$777$		0			0
$778$	967036.	+	967036.i	1.59766	+	1.59766i
$779$		−	77447.2i		−	0.127624i
$780$		0			0
$781$	−859521.			−1.40914
$782$	−483064.	+	483064.i	−0.789934	+	0.789934i
$783$		0			0
$784$			862902.i			1.40388i
$785$	90501.7	−	90501.7i	0.146865	−	0.146865i
$786$		0			0
$787$	−404185.	−	404185.i	−0.652576	−	0.652576i	0.301037	−	0.953613i	$-0.402667\pi$
$787$	−404185.	−	404185.i	−0.652576	−	0.652576i	−0.953613	+	0.301037i	$0.902667\pi$
$788$	−696264.	−	696264.i	−1.12130	−	1.12130i
$789$		0			0
$790$			208867.i			0.334669i
$791$	−290827.	−	290827.i	−0.464817	−	0.464817i
$792$		0			0
$793$	−55420.8	+	328491.i	−0.0881306	+	0.522369i
$794$	−542566.			−0.860620
$795$		0			0
$796$	1.00217e6			1.58167
$797$			95935.2i			0.151029i	0.997145	+	0.0755147i	$0.0240600\pi$
$797$			95935.2i			0.151029i	−0.997145	+	0.0755147i	$0.975940\pi$
$798$		0			0
$799$	−791224.	+	791224.i	−1.23938	+	1.23938i
$800$	−465948.	−	465948.i	−0.728044	−	0.728044i
$801$		0			0
$802$	−1.99078e6			−3.09509
$803$		−	705182.i		−	1.09363i
$804$		0			0
$805$			31229.8i			0.0481923i
$806$	−2.29843e6	−	387776.i	−3.53803	−	0.596913i
$807$		0			0
$808$	595825.	−	595825.i	0.912633	−	0.912633i
$809$	1.07612e6			1.64423			0.822115	−	0.569322i	$-0.192794\pi$
$809$	1.07612e6			1.64423			0.822115	+	0.569322i	$0.192794\pi$
$810$		0			0
$811$	144481.	−	144481.i	0.219669	−	0.219669i	−0.588690	−	0.808359i	$-0.700356\pi$
$811$	144481.	−	144481.i	0.219669	−	0.219669i	0.808359	+	0.588690i	$0.200356\pi$
$812$	−285890.	+	285890.i	−0.433598	+	0.433598i
$813$		0			0
$814$	754835.	+	754835.i	1.13921	+	1.13921i
$815$	−47183.8			−0.0710359
$816$		0			0
$817$	673666.	+	673666.i	1.00925	+	1.00925i
$818$			1.87930e6i			2.80859i
$819$		0			0
$820$	27304.1			0.0406069
$821$	658649.	−	658649.i	0.977165	−	0.977165i	−0.0225802	−	0.999745i	$-0.507188\pi$
$821$	658649.	−	658649.i	0.977165	−	0.977165i	0.999745	+	0.0225802i	$0.00718810\pi$
$822$		0			0
$823$			912410.i			1.34707i	0.739155	+	0.673536i	$0.235225\pi$
$823$			912410.i			1.34707i	−0.739155	+	0.673536i	$0.764775\pi$
$824$	1.08562e6	−	1.08562e6i	1.59891	−	1.59891i
$825$		0			0
$826$	562646.	+	562646.i	0.824661	+	0.824661i
$827$	−136058.	−	136058.i	−0.198936	−	0.198936i	0.600608	−	0.799544i	$-0.294925\pi$
$827$	−136058.	−	136058.i	−0.198936	−	0.198936i	−0.799544	+	0.600608i	$0.794925\pi$
$828$		0			0
$829$		−	548426.i		−	0.798011i	−0.916949	−	0.399005i	$-0.869355\pi$
$829$		−	548426.i		−	0.798011i	0.916949	−	0.399005i	$-0.130645\pi$
$830$	−285542.	−	285542.i	−0.414490	−	0.414490i
$831$		0			0
$832$	57508.6	+	9702.46i	0.0830780	+	0.0140164i
$833$	686750.			0.989711
$834$		0			0
$835$	56184.7			0.0805833
$836$			3.10983e6i			4.44963i
$837$		0			0
$838$	1.28473e6	−	1.28473e6i	1.82947	−	1.82947i
$839$	607302.	+	607302.i	0.862742	+	0.862742i	0.991656	−	0.128914i	$-0.0411490\pi$
$839$	607302.	+	607302.i	0.862742	+	0.862742i	−0.128914	+	0.991656i	$0.541149\pi$
$840$		0			0
$841$	−494974.			−0.699826
$842$			2.49794e6i			3.52336i
$843$		0			0
$844$		−	2.74931e6i		−	3.85957i
$845$	132657.	−	64263.8i	0.185787	−	0.0900022i
$846$		0			0
$847$	198976.	−	198976.i	0.277354	−	0.277354i
$848$	−315495.			−0.438734
$849$		0			0
$850$	−1.15757e6	+	1.15757e6i	−1.60217	+	1.60217i
$851$	160996.	−	160996.i	0.222308	−	0.222308i
$852$		0			0
$853$	70501.4	+	70501.4i	0.0968947	+	0.0968947i	0.753892	−	0.656998i	$-0.228174\pi$
$853$	70501.4	+	70501.4i	0.0968947	+	0.0968947i	−0.656998	+	0.753892i	$0.728174\pi$
$854$	345158.			0.473263
$855$		0			0
$856$	1.49899e6	+	1.49899e6i	2.04574	+	2.04574i
$857$		−	1.39326e6i		−	1.89702i	−0.316753	−	0.948508i	$-0.602593\pi$
$857$		−	1.39326e6i		−	1.89702i	0.316753	−	0.948508i	$-0.397407\pi$
$858$		0			0
$859$	−279190.			−0.378367			−0.189184	−	0.981942i	$-0.560584\pi$
$859$	−279190.			−0.378367			−0.189184	+	0.981942i	$0.560584\pi$
$860$	−237502.	+	237502.i	−0.321122	+	0.321122i
$861$		0			0
$862$		−	170907.i		−	0.230009i
$863$	−648628.	+	648628.i	−0.870912	+	0.870912i	−0.992572	−	0.121660i	$-0.961178\pi$
$863$	−648628.	+	648628.i	−0.870912	+	0.870912i	0.121660	+	0.992572i	$0.461178\pi$
$864$		0			0
$865$	39784.9	+	39784.9i	0.0531723	+	0.0531723i
$866$	1.54034e6	+	1.54034e6i	2.05390	+	2.05390i
$867$		0			0
$868$			1.67493e6i			2.22310i
$869$	−641707.	−	641707.i	−0.849762	−	0.849762i
$870$		0			0
$871$	−335204.	−	471266.i	−0.441848	−	0.621198i
$872$	1.16823e6			1.53637
$873$		0			0
$874$	956372.			1.25200
$875$			153004.i			0.199841i
$876$		0			0
$877$	−168969.	+	168969.i	−0.219689	+	0.219689i	−0.808367	−	0.588678i	$-0.799649\pi$
$877$	−168969.	+	168969.i	−0.219689	+	0.219689i	0.588678	+	0.808367i	$0.299649\pi$
$878$	99097.8	+	99097.8i	0.128551	+	0.128551i
$879$		0			0
$880$	−397838.			−0.513737
$881$		−	325818.i		−	0.419782i	−0.977725	−	0.209891i	$-0.932689\pi$
$881$		−	325818.i		−	0.419782i	0.977725	−	0.209891i	$-0.0673109\pi$
$882$		0			0
$883$			1.09361e6i			1.40262i	0.712858	+	0.701309i	$0.247401\pi$
$883$			1.09361e6i			1.40262i	−0.712858	+	0.701309i	$0.752599\pi$
$884$	385462.	−	2.28472e6i	0.493261	−	2.92367i
$885$		0			0
$886$	−230313.	+	230313.i	−0.293394	+	0.293394i
$887$	−1.24503e6			−1.58246			−0.791228	−	0.611522i	$-0.790558\pi$
$887$	−1.24503e6			−1.58246			−0.791228	+	0.611522i	$0.790558\pi$
$888$		0			0
$889$	28509.6	−	28509.6i	0.0360734	−	0.0360734i
$890$	6996.75	−	6996.75i	0.00883316	−	0.00883316i
$891$		0			0
$892$	277203.	+	277203.i	0.348392	+	0.348392i
$893$	1.56647e6			1.96435
$894$		0			0
$895$	−173261.	−	173261.i	−0.216299	−	0.216299i
$896$			366566.i			0.456600i
$897$		0			0
$898$	1.05557e6			1.30898
$899$	621919.	−	621919.i	0.769510	−	0.769510i
$900$		0			0
$901$			251090.i			0.309300i
$902$	−120955.	+	120955.i	−0.148665	+	0.148665i
$903$		0			0
$904$	−1.75244e6	−	1.75244e6i	−2.14440	−	2.14440i
$905$	178110.	+	178110.i	0.217466	+	0.217466i
$906$		0			0
$907$			141672.i			0.172214i	0.996286	+	0.0861070i	$0.0274427\pi$
$907$			141672.i			0.172214i	−0.996286	+	0.0861070i	$0.972557\pi$
$908$	−501972.	−	501972.i	−0.608846	−	0.608846i
$909$		0			0
$910$	−88521.2	−	124453.i	−0.106897	−	0.150287i
$911$	10630.1			0.0128085			0.00640427	−	0.999979i	$-0.497961\pi$
$911$	10630.1			0.0128085			0.00640427	+	0.999979i	$0.497961\pi$
$912$		0			0
$913$	1.75456e6			2.10488
$914$		−	2.47724e6i		−	2.96534i
$915$		0			0
$916$	136030.	−	136030.i	0.162122	−	0.162122i
$917$	−269518.	−	269518.i	−0.320515	−	0.320515i
$918$		0			0
$919$	−429402.			−0.508432			−0.254216	−	0.967147i	$-0.581817\pi$
$919$	−429402.			−0.508432			−0.254216	+	0.967147i	$0.581817\pi$
$920$			188182.i			0.222332i
$921$		0			0
$922$			2.69340e6i			3.16840i
$923$	−519632.	−	730555.i	−0.609948	−	0.857531i
$924$		0			0
$925$	385794.	−	385794.i	0.450891	−	0.450891i
$926$	1.92337e6			2.24305
$927$		0			0
$928$	−358802.	+	358802.i	−0.416638	+	0.416638i
$929$	241423.	−	241423.i	0.279735	−	0.279735i	−0.553268	−	0.833003i	$-0.686620\pi$
$929$	241423.	−	241423.i	0.279735	−	0.279735i	0.833003	+	0.553268i	$0.186620\pi$
$930$		0			0
$931$	−679815.	−	679815.i	−0.784317	−	0.784317i
$932$	−459111.			−0.528550
$933$		0			0
$934$	−946446.	−	946446.i	−1.08493	−	1.08493i
$935$			316624.i			0.362176i
$936$		0			0
$937$	−646943.			−0.736864			−0.368432	−	0.929655i	$-0.620105\pi$
$937$	−646943.			−0.736864			−0.368432	+	0.929655i	$0.620105\pi$
$938$	−423694.	+	423694.i	−0.481556	+	0.481556i
$939$		0			0
$940$			552260.i			0.625012i
$941$	−603375.	+	603375.i	−0.681410	+	0.681410i	−0.960318	−	0.278908i	$-0.910028\pi$
$941$	−603375.	+	603375.i	−0.681410	+	0.681410i	0.278908	+	0.960318i	$0.410028\pi$
$942$		0			0
$943$	25798.0	+	25798.0i	0.0290110	+	0.0290110i
$944$	1.52874e6	+	1.52874e6i	1.71550	+	1.71550i
$945$		0			0
$946$		−	2.10423e6i		−	2.35131i
$947$	−174701.	−	174701.i	−0.194802	−	0.194802i	0.602965	−	0.797768i	$-0.293986\pi$
$947$	−174701.	−	174701.i	−0.194802	−	0.194802i	−0.797768	+	0.602965i	$0.793986\pi$
$948$		0			0
$949$	599374.	−	426325.i	0.665526	−	0.473378i
$950$	2.29176e6			2.53934
$951$		0			0
$952$	−1.33985e6			−1.47837
$953$			1.45954e6i			1.60705i	0.595273	+	0.803524i	$0.297044\pi$
$953$			1.45954e6i			1.60705i	−0.595273	+	0.803524i	$0.702956\pi$
$954$		0			0
$955$	−19176.0	+	19176.0i	−0.0210257	+	0.0210257i
$956$	−2.22332e6	−	2.22332e6i	−2.43268	−	2.43268i
$957$		0			0
$958$	1.64219e6			1.78934
$959$		−	151368.i		−	0.164587i
$960$		0			0
$961$		−	2.72010e6i		−	2.94535i
$962$	−185233.	+	1.09792e6i	−0.200156	+	1.18637i
$963$		0			0
$964$	1.37564e6	−	1.37564e6i	1.48030	−	1.48030i
$965$	−97093.4			−0.104264
$966$		0			0
$967$	228474.	−	228474.i	0.244334	−	0.244334i	−0.574306	−	0.818640i	$-0.694728\pi$
$967$	228474.	−	228474.i	0.244334	−	0.244334i	0.818640	+	0.574306i	$0.194728\pi$
$968$	1.19897e6	−	1.19897e6i	1.27955	−	1.27955i
$969$		0			0
$970$	−51146.4	−	51146.4i	−0.0543590	−	0.0543590i
$971$	470755.			0.499295			0.249647	−	0.968337i	$-0.419685\pi$
$971$	470755.			0.499295			0.249647	+	0.968337i	$0.419685\pi$
$972$		0			0
$973$	429750.	+	429750.i	0.453931	+	0.453931i
$974$			68208.0i			0.0718981i
$975$		0			0
$976$	937814.			0.984503
$977$	14992.1	−	14992.1i	0.0157063	−	0.0157063i	−0.699210	−	0.714916i	$-0.746465\pi$
$977$	14992.1	−	14992.1i	0.0157063	−	0.0157063i	0.714916	+	0.699210i	$0.246465\pi$
$978$		0			0
$979$			42992.6i			0.0448568i
$980$	239670.	−	239670.i	0.249552	−	0.249552i
$981$		0			0
$982$	−1.11958e6	−	1.11958e6i	−1.16100	−	1.16100i
$983$	196293.	+	196293.i	0.203141	+	0.203141i	0.801344	−	0.598203i	$-0.204118\pi$
$983$	196293.	+	196293.i	0.203141	+	0.203141i	−0.598203	+	0.801344i	$0.704118\pi$
$984$		0			0
$985$			140347.i			0.144654i
$986$	891380.	+	891380.i	0.916873	+	0.916873i
$987$		0			0
$988$	−2.64322e6	+	1.88008e6i	−2.70782	+	1.92603i
$989$	−448802.			−0.458841
$990$		0			0
$991$	−1.70831e6			−1.73948			−0.869741	−	0.493508i	$-0.835714\pi$
$991$	−1.70831e6			−1.73948			−0.869741	+	0.493508i	$0.835714\pi$
$992$			2.10210e6i			2.13614i
$993$		0			0
$994$	−656809.	+	656809.i	−0.664762	+	0.664762i
$995$	−101005.	−	101005.i	−0.102023	−	0.102023i
$996$		0			0
$997$	1.51922e6			1.52838			0.764190	−	0.644992i	$-0.223139\pi$
$997$	1.51922e6			1.52838			0.764190	+	0.644992i	$0.223139\pi$
$998$		−	1.04631e6i		−	1.05050i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.5.j.b.109.10		20
3.2	odd	2		39.5.g.a.31.1	✓	20
13.8	odd	4	inner	117.5.j.b.73.10		20
39.8	even	4		39.5.g.a.34.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.5.g.a.31.1	✓	20	3.2	odd	2
39.5.g.a.34.1	yes	20	39.8	even	4
117.5.j.b.73.10		20	13.8	odd	4	inner
117.5.j.b.109.10		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)

\(f(q)\)	\(=\)	\(q+(5.10926 - 5.10926i) q^{2} -36.2092i q^{4} +(-3.64937 + 3.64937i) q^{5} +(-17.1355 - 17.1355i) q^{7} +(-103.254 - 103.254i) q^{8} +37.2912i q^{10} +(-114.571 - 114.571i) q^{11} +(28.1152 - 166.645i) q^{13} -175.100 q^{14} -475.757 q^{16} +378.636i q^{17} +(374.813 - 374.813i) q^{19} +(132.141 + 132.141i) q^{20} -1170.74 q^{22} +249.703i q^{23} +598.364i q^{25} +(-707.785 - 995.081i) q^{26} +(-620.464 + 620.464i) q^{28} +460.768 q^{29} +(1349.74 - 1349.74i) q^{31} +(-778.703 + 778.703i) q^{32} +(1934.55 + 1934.55i) q^{34} +125.068 q^{35} +(-644.748 - 644.748i) q^{37} -3830.04i q^{38} +753.623 q^{40} +(103.315 - 103.315i) q^{41} +1797.34i q^{43} +(-4148.51 + 4148.51i) q^{44} +(1275.80 + 1275.80i) q^{46} +(2089.67 + 2089.67i) q^{47} -1813.75i q^{49} +(3057.20 + 3057.20i) q^{50} +(-6034.07 - 1018.03i) q^{52} +663.144 q^{53} +836.222 q^{55} +3538.62i q^{56} +(2354.19 - 2354.19i) q^{58} +(-3213.29 - 3213.29i) q^{59} -1971.21 q^{61} -13792.4i q^{62} +345.097i q^{64} +(505.546 + 710.752i) q^{65} +(2419.73 - 2419.73i) q^{67} +13710.1 q^{68} +(639.005 - 639.005i) q^{70} +(3751.05 - 3751.05i) q^{71} +(3077.50 + 3077.50i) q^{73} -6588.37 q^{74} +(-13571.7 - 13571.7i) q^{76} +3926.46i q^{77} +5600.97 q^{79} +(1736.21 - 1736.21i) q^{80} -1055.72i q^{82} +(-7657.10 + 7657.10i) q^{83} +(-1381.78 - 1381.78i) q^{85} +(9183.09 + 9183.09i) q^{86} +23659.7i q^{88} +(-187.625 - 187.625i) q^{89} +(-3337.32 + 2373.78i) q^{91} +9041.54 q^{92} +21353.3 q^{94} +2735.66i q^{95} +(-1371.54 + 1371.54i) q^{97} +(-9266.91 - 9266.91i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) 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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