LMFDB - Embedded newform 88.4.i.b.81.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,4,Mod(9,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("88.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	88.i (of order $5$, degree $4$, 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:	$5.19216808051$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{5})$
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} - 3 x^{19} + 120 x^{18} + 34 x^{17} + 8167 x^{16} - 2908 x^{15} + 606151 x^{14} + \cdots + 611123681536 $ x^20 - 3x^19 + 120x^18 + 34x^17 + 8167x^16 - 2908x^15 + 606151x^14 + 2469468x^13 + 58851235x^12 + 38886732x^11 + 519088186x^10 - 25796485x^9 + 13683392843x^8 - 49724730862x^7 + 234118517636x^6 - 459643934160x^5 + 1629579968592x^4 - 2324243526976x^3 + 3035801390848x^2 - 1960238714880*x + 611123681536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			81.5
Root			$-6.46669 - 4.69833i$ of defining polynomial
Character	$\chi$	$=$	88.81
Dual form			88.4.i.b.25.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.97006 - 9.14090i) q^{3} +(13.5223 - 9.82454i) q^{5} +(6.15515 + 18.9436i) q^{7} +(-52.8913 - 38.4278i) q^{9} +(17.1491 + 32.2011i) q^{11} +(-32.0586 - 23.2919i) q^{13} +(-49.6431 - 152.786i) q^{15} +(1.66684 - 1.21103i) q^{17} +(-31.3511 + 96.4889i) q^{19} +191.443 q^{21} +110.549 q^{23} +(47.7045 - 146.819i) q^{25} +(-298.410 + 216.808i) q^{27} +(-10.3095 - 31.7293i) q^{29} +(-140.530 - 102.101i) q^{31} +(345.280 - 61.1193i) q^{33} +(269.344 + 195.690i) q^{35} +(77.6987 + 239.132i) q^{37} +(-308.125 + 223.866i) q^{39} +(-59.5497 + 183.275i) q^{41} +304.185 q^{43} -1092.75 q^{45} +(-10.5409 + 32.4416i) q^{47} +(-43.4814 + 31.5911i) q^{49} +(-6.11930 - 18.8333i) q^{51} +(-175.256 - 127.331i) q^{53} +(548.257 + 266.951i) q^{55} +(788.880 + 573.155i) q^{57} +(-4.96281 - 15.2739i) q^{59} +(429.705 - 312.199i) q^{61} +(402.407 - 1238.48i) q^{63} -662.339 q^{65} +636.628 q^{67} +(328.336 - 1010.51i) q^{69} +(-227.127 + 165.017i) q^{71} +(-202.947 - 624.608i) q^{73} +(-1200.37 - 872.123i) q^{75} +(-504.449 + 523.068i) q^{77} +(-652.080 - 473.764i) q^{79} +(550.049 + 1692.88i) q^{81} +(-516.450 + 375.223i) q^{83} +(10.6417 - 32.7519i) q^{85} -320.654 q^{87} +1144.68 q^{89} +(243.907 - 750.670i) q^{91} +(-1350.68 + 981.324i) q^{93} +(524.019 + 1612.76i) q^{95} +(-679.744 - 493.863i) q^{97} +(330.376 - 2362.16i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3 q^{3} + q^{5} + q^{7} - 58 q^{9} - 72 q^{11} - 95 q^{13} - 225 q^{15} + 41 q^{17} + 191 q^{19} + 262 q^{21} + 328 q^{23} - 172 q^{25} - 315 q^{27} + 169 q^{29} - 825 q^{31} + 1093 q^{33} + 1195 q^{35}+ \cdots + 1840 q^{99}+O(q^{100}) $ 20 * q + 3 * q^3 + q^5 + q^7 - 58 * q^9 - 72 * q^11 - 95 * q^13 - 225 * q^15 + 41 * q^17 + 191 * q^19 + 262 * q^21 + 328 * q^23 - 172 * q^25 - 315 * q^27 + 169 * q^29 - 825 * q^31 + 1093 * q^33 + 1195 * q^35 + 5 * q^37 - 223 * q^39 - 411 * q^41 - 760 * q^43 - 88 * q^45 - 2157 * q^47 - 1694 * q^49 + 2163 * q^51 - 1803 * q^53 + 2067 * q^55 + 1781 * q^57 - 383 * q^59 + 2097 * q^61 + 728 * q^63 + 346 * q^65 + 4520 * q^67 - 2524 * q^69 - 2415 * q^71 + 4517 * q^73 - 7457 * q^75 + 2019 * q^77 + 159 * q^79 - 7492 * q^81 + 701 * q^83 + 243 * q^85 + 2606 * q^87 + 892 * q^89 - 7003 * q^91 - 7817 * q^93 + 5691 * q^95 - 5213 * q^97 + 1840 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{5}\right)$

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			0
$2$		0			0
$3$	2.97006	−	9.14090i	0.571588	−	1.75917i	−0.0759270	−	0.997113i	$-0.524192\pi$
$3$	2.97006	−	9.14090i	0.571588	−	1.75917i	0.647515	−	0.762053i	$-0.275808\pi$
$4$		0			0
$5$	13.5223	−	9.82454i	1.20947	−	0.878734i	0.214291	−	0.976770i	$-0.431256\pi$
$5$	13.5223	−	9.82454i	1.20947	−	0.878734i	0.995183	+	0.0980361i	$0.0312561\pi$
$6$		0			0
$7$	6.15515	+	18.9436i	0.332347	+	1.02286i	0.968014	+	0.250895i	$0.0807250\pi$
$7$	6.15515	+	18.9436i	0.332347	+	1.02286i	−0.635667	+	0.771963i	$0.719275\pi$
$8$		0			0
$9$	−52.8913	−	38.4278i	−1.95894	−	1.42325i
$10$		0			0
$11$	17.1491	+	32.2011i	0.470059	+	0.882635i
$11$	17.1491	+	32.2011i	0.470059	+	0.882635i
$12$		0			0
$13$	−32.0586	−	23.2919i	−0.683957	−	0.496924i	0.190711	−	0.981646i	$-0.438921\pi$
$13$	−32.0586	−	23.2919i	−0.683957	−	0.496924i	−0.874668	+	0.484722i	$0.838921\pi$
$14$		0			0
$15$	−49.6431	−	152.786i	−0.854519	−	2.62994i
$16$		0			0
$17$	1.66684	−	1.21103i	0.0237805	−	0.0172776i	−0.575832	−	0.817568i	$-0.695322\pi$
$17$	1.66684	−	1.21103i	0.0237805	−	0.0172776i	0.599612	+	0.800291i	$0.295322\pi$
$18$		0			0
$19$	−31.3511	+	96.4889i	−0.378550	+	1.16506i	0.562503	+	0.826795i	$0.309839\pi$
$19$	−31.3511	+	96.4889i	−0.378550	+	1.16506i	−0.941052	+	0.338261i	$0.890161\pi$
$20$		0			0
$21$	191.443			1.98934
$22$		0			0
$23$	110.549			1.00222			0.501109	−	0.865384i	$-0.332926\pi$
$23$	110.549			1.00222			0.501109	+	0.865384i	$0.332926\pi$
$24$		0			0
$25$	47.7045	−	146.819i	0.381636	−	1.17455i
$26$		0			0
$27$	−298.410	+	216.808i	−2.12700	+	1.54536i
$28$		0			0
$29$	−10.3095	−	31.7293i	−0.0660145	−	0.203172i	0.912608	−	0.408835i	$-0.134065\pi$
$29$	−10.3095	−	31.7293i	−0.0660145	−	0.203172i	−0.978623	+	0.205663i	$0.934065\pi$
$30$		0			0
$31$	−140.530	−	102.101i	−0.814191	−	0.591545i	0.100852	−	0.994901i	$-0.467843\pi$
$31$	−140.530	−	102.101i	−0.814191	−	0.591545i	−0.915043	+	0.403357i	$0.867843\pi$
$32$		0			0
$33$	345.280	−	61.1193i	1.82138	−	0.322409i
$34$		0			0
$35$	269.344	+	195.690i	1.30078	+	0.945076i
$36$		0			0
$37$	77.6987	+	239.132i	0.345232	+	1.06251i	0.961459	+	0.274947i	$0.0886602\pi$
$37$	77.6987	+	239.132i	0.345232	+	1.06251i	−0.616228	+	0.787568i	$0.711340\pi$
$38$		0			0
$39$	−308.125	+	223.866i	−1.26511	+	0.919159i
$40$		0			0
$41$	−59.5497	+	183.275i	−0.226832	+	0.698116i	0.771269	+	0.636509i	$0.219622\pi$
$41$	−59.5497	+	183.275i	−0.226832	+	0.698116i	−0.998101	+	0.0616065i	$0.980378\pi$
$42$		0			0
$43$	304.185			1.07879			0.539393	−	0.842054i	$-0.318654\pi$
$43$	304.185			1.07879			0.539393	+	0.842054i	$0.318654\pi$
$44$		0			0
$45$	−1092.75			−3.61994
$46$		0			0
$47$	−10.5409	+	32.4416i	−0.0327138	+	0.100683i	−0.966080	−	0.258242i	$-0.916857\pi$
$47$	−10.5409	+	32.4416i	−0.0327138	+	0.100683i	0.933366	+	0.358925i	$0.116857\pi$
$48$		0			0
$49$	−43.4814	+	31.5911i	−0.126768	+	0.0921022i
$50$		0			0
$51$	−6.11930	−	18.8333i	−0.0168014	−	0.0517095i
$52$		0			0
$53$	−175.256	−	127.331i	−0.454211	−	0.330004i	0.337045	−	0.941489i	$-0.390573\pi$
$53$	−175.256	−	127.331i	−0.454211	−	0.330004i	−0.791256	+	0.611485i	$0.790573\pi$
$54$		0			0
$55$	548.257	+	266.951i	1.34413	+	0.654466i
$56$		0			0
$57$	788.880	+	573.155i	1.83315	+	1.33186i
$58$		0			0
$59$	−4.96281	−	15.2739i	−0.0109509	−	0.0337034i	0.945432	−	0.325820i	$-0.105640\pi$
$59$	−4.96281	−	15.2739i	−0.0109509	−	0.0337034i	−0.956383	+	0.292117i	$0.905640\pi$
$60$		0			0
$61$	429.705	−	312.199i	0.901935	−	0.655294i	−0.0370271	−	0.999314i	$-0.511789\pi$
$61$	429.705	−	312.199i	0.901935	−	0.655294i	0.938962	+	0.344020i	$0.111789\pi$
$62$		0			0
$63$	402.407	−	1238.48i	0.804737	−	2.47673i
$64$		0			0
$65$	−662.339			−1.26389
$66$		0			0
$67$	636.628			1.16084			0.580422	−	0.814316i	$-0.302888\pi$
$67$	636.628			1.16084			0.580422	+	0.814316i	$0.302888\pi$
$68$		0			0
$69$	328.336	−	1010.51i	0.572855	−	1.76307i
$70$		0			0
$71$	−227.127	+	165.017i	−0.379648	+	0.275830i	−0.761200	−	0.648517i	$-0.775390\pi$
$71$	−227.127	+	165.017i	−0.379648	+	0.275830i	0.381552	+	0.924347i	$0.375390\pi$
$72$		0			0
$73$	−202.947	−	624.608i	−0.325386	−	1.00144i	−0.971266	−	0.237997i	$-0.923509\pi$
$73$	−202.947	−	624.608i	−0.325386	−	1.00144i	0.645880	−	0.763439i	$-0.276491\pi$
$74$		0			0
$75$	−1200.37	−	872.123i	−1.84810	−	1.34272i
$76$		0			0
$77$	−504.449	+	523.068i	−0.746588	+	0.774145i
$78$		0			0
$79$	−652.080	−	473.764i	−0.928668	−	0.674717i	0.0169983	−	0.999856i	$-0.494589\pi$
$79$	−652.080	−	473.764i	−0.928668	−	0.674717i	−0.945666	+	0.325139i	$0.894589\pi$
$80$		0			0
$81$	550.049	+	1692.88i	0.754525	+	2.32219i
$82$		0			0
$83$	−516.450	+	375.223i	−0.682984	+	0.496217i	−0.874346	−	0.485302i	$-0.838710\pi$
$83$	−516.450	+	375.223i	−0.682984	+	0.496217i	0.191362	+	0.981520i	$0.438710\pi$
$84$		0			0
$85$	10.6417	−	32.7519i	0.0135795	−	0.0417935i
$86$		0			0
$87$	−320.654			−0.395146
$88$		0			0
$89$	1144.68			1.36333			0.681663	−	0.731666i	$-0.261257\pi$
$89$	1144.68			1.36333			0.681663	+	0.731666i	$0.261257\pi$
$90$		0			0
$91$	243.907	−	750.670i	0.280972	−	0.864743i
$92$		0			0
$93$	−1350.68	+	981.324i	−1.50601	+	1.09418i
$94$		0			0
$95$	524.019	+	1612.76i	0.565928	+	1.74175i
$96$		0			0
$97$	−679.744	−	493.863i	−0.711521	−	0.516951i	0.172143	−	0.985072i	$-0.444931\pi$
$97$	−679.744	−	493.863i	−0.711521	−	0.516951i	−0.883664	+	0.468121i	$0.844931\pi$
$98$		0			0
$99$	330.376	−	2362.16i	0.335394	−	2.39804i
$100$		0			0
$101$	−1078.34	−	783.462i	−1.06237	−	0.771855i	−0.0878426	−	0.996134i	$-0.527997\pi$
$101$	−1078.34	−	783.462i	−1.06237	−	0.771855i	−0.974525	+	0.224279i	$0.927997\pi$
$102$		0			0
$103$	29.3623	+	90.3678i	0.0280889	+	0.0864486i	0.964118	−	0.265473i	$-0.0855282\pi$
$103$	29.3623	+	90.3678i	0.0280889	+	0.0864486i	−0.936029	+	0.351922i	$0.885528\pi$
$104$		0			0
$105$	2588.75	−	1880.84i	2.40606	−	1.74810i
$106$		0			0
$107$	−445.652	+	1371.57i	−0.402643	+	1.23921i	0.520205	+	0.854042i	$0.325856\pi$
$107$	−445.652	+	1371.57i	−0.402643	+	1.23921i	−0.922848	+	0.385165i	$0.874144\pi$
$108$		0			0
$109$	−708.872			−0.622914			−0.311457	−	0.950260i	$-0.600817\pi$
$109$	−708.872			−0.622914			−0.311457	+	0.950260i	$0.600817\pi$
$110$		0			0
$111$	2416.65			2.06647
$112$		0			0
$113$	−516.406	+	1589.33i	−0.429906	+	1.32312i	0.468311	+	0.883564i	$0.344863\pi$
$113$	−516.406	+	1589.33i	−0.429906	+	1.32312i	−0.898217	+	0.439552i	$0.855137\pi$
$114$		0			0
$115$	1494.87	−	1086.09i	1.21215	−	0.880682i
$116$		0			0
$117$	800.562	+	2463.88i	0.632581	+	1.94689i
$118$		0			0
$119$	33.2010	+	24.1219i	0.0255759	+	0.0185820i
$120$		0			0
$121$	−742.816	+	1104.44i	−0.558088	+	0.829782i
$122$		0			0
$123$	1498.43	+	1088.67i	1.09845	+	0.798069i
$124$		0			0
$125$	−151.723	−	466.956i	−0.108564	−	0.334127i
$126$		0			0
$127$	−1174.24	+	853.137i	−0.820450	+	0.596092i	−0.916841	−	0.399252i	$-0.869270\pi$
$127$	−1174.24	+	853.137i	−0.820450	+	0.596092i	0.0963912	+	0.995344i	$0.469270\pi$
$128$		0			0
$129$	903.448	−	2780.53i	0.616621	−	1.89777i
$130$		0			0
$131$	−2344.03			−1.56335			−0.781674	−	0.623687i	$-0.785634\pi$
$131$	−2344.03			−1.56335			−0.781674	+	0.623687i	$0.785634\pi$
$132$		0			0
$133$	−2020.82			−1.31750
$134$		0			0
$135$	−1905.16	+	5863.49i	−1.21459	+	3.73814i
$136$		0			0
$137$	−46.6459	+	33.8903i	−0.0290893	+	0.0211346i	−0.602235	−	0.798319i	$-0.705723\pi$
$137$	−46.6459	+	33.8903i	−0.0290893	+	0.0211346i	0.573146	+	0.819454i	$0.305723\pi$
$138$		0			0
$139$	419.220	+	1290.23i	0.255812	+	0.787307i	0.993669	+	0.112350i	$0.0358379\pi$
$139$	419.220	+	1290.23i	0.255812	+	0.787307i	−0.737857	+	0.674957i	$0.764162\pi$
$140$		0			0
$141$	265.238	+	192.707i	0.158419	+	0.115098i
$142$		0			0
$143$	200.248	−	1431.76i	0.117102	−	0.837268i
$144$		0			0
$145$	−451.134	−	327.768i	−0.258377	−	0.187722i
$146$		0			0
$147$	159.629	+	491.286i	0.0895642	+	0.275650i
$148$		0			0
$149$	2147.07	−	1559.94i	1.18050	−	0.857686i	0.188275	−	0.982116i	$-0.439710\pi$
$149$	2147.07	−	1559.94i	1.18050	−	0.857686i	0.992228	+	0.124431i	$0.0397105\pi$
$150$		0			0
$151$	752.073	−	2314.64i	0.405317	−	1.24744i	−0.515314	−	0.857002i	$-0.672325\pi$
$151$	752.073	−	2314.64i	0.405317	−	1.24744i	0.920631	−	0.390435i	$-0.127675\pi$
$152$		0			0
$153$	−134.699			−0.0711748
$154$		0			0
$155$	−2903.39			−1.50455
$156$		0			0
$157$	−427.322	+	1315.16i	−0.217223	+	0.668543i	0.781766	+	0.623572i	$0.214319\pi$
$157$	−427.322	+	1315.16i	−0.217223	+	0.668543i	−0.998988	+	0.0449705i	$0.985681\pi$
$158$		0			0
$159$	−1684.43	+	1223.81i	−0.840153	+	0.610407i
$160$		0			0
$161$	680.444	+	2094.19i	0.333084	+	1.02513i
$162$		0			0
$163$	−1538.86	−	1118.05i	−0.739467	−	0.537254i	0.153077	−	0.988214i	$-0.451082\pi$
$163$	−1538.86	−	1118.05i	−0.739467	−	0.537254i	−0.892544	+	0.450960i	$0.851082\pi$
$164$		0			0
$165$	4068.52	−	4218.70i	1.91960	−	1.99046i
$166$		0			0
$167$	−1739.42	−	1263.76i	−0.805988	−	0.585585i	0.106677	−	0.994294i	$-0.465979\pi$
$167$	−1739.42	−	1263.76i	−0.805988	−	0.585585i	−0.912665	+	0.408709i	$0.865979\pi$
$168$		0			0
$169$	−193.672	−	596.061i	−0.0881529	−	0.271307i
$170$		0			0
$171$	5366.05	−	3898.67i	2.39972	−	1.74350i
$172$		0			0
$173$	163.319	−	502.643i	0.0717739	−	0.220897i	−0.908734	−	0.417375i	$-0.862950\pi$
$173$	163.319	−	502.643i	0.0717739	−	0.220897i	0.980508	+	0.196477i	$0.0629502\pi$
$174$		0			0
$175$	3074.91			1.32824
$176$		0			0
$177$	−154.357			−0.0655492
$178$		0			0
$179$	−1214.27	+	3737.14i	−0.507032	+	1.56048i	0.290295	+	0.956937i	$0.406246\pi$
$179$	−1214.27	+	3737.14i	−0.507032	+	1.56048i	−0.797327	+	0.603547i	$0.793754\pi$
$180$		0			0
$181$	437.542	−	317.893i	0.179681	−	0.130546i	−0.494309	−	0.869286i	$-0.664579\pi$
$181$	437.542	−	317.893i	0.179681	−	0.130546i	0.673990	+	0.738740i	$0.264579\pi$
$182$		0			0
$183$	−1577.53	−	4855.13i	−0.637237	−	1.96121i
$184$		0			0
$185$	3400.03	+	2470.27i	1.35122	+	0.981716i
$186$		0			0
$187$	67.5814	+	32.9060i	0.0264280	+	0.0128680i
$188$		0			0
$189$	−5943.88	−	4318.48i	−2.28758	−	1.66203i
$190$		0			0
$191$	−36.9162	−	113.617i	−0.0139852	−	0.0430419i	0.943820	−	0.330459i	$-0.107203\pi$
$191$	−36.9162	−	113.617i	−0.0139852	−	0.0430419i	−0.957806	+	0.287417i	$0.907203\pi$
$192$		0			0
$193$	528.819	−	384.209i	0.197229	−	0.143295i	−0.484788	−	0.874632i	$-0.661103\pi$
$193$	528.819	−	384.209i	0.197229	−	0.143295i	0.682017	+	0.731337i	$0.261103\pi$
$194$		0			0
$195$	−1967.18	+	6054.37i	−0.722425	+	2.22340i
$196$		0			0
$197$	−749.399			−0.271028			−0.135514	−	0.990775i	$-0.543269\pi$
$197$	−749.399			−0.271028			−0.135514	+	0.990775i	$0.543269\pi$
$198$		0			0
$199$	1836.21			0.654098			0.327049	−	0.945007i	$-0.393946\pi$
$199$	1836.21			0.654098			0.327049	+	0.945007i	$0.393946\pi$
$200$		0			0
$201$	1890.82	−	5819.35i	0.663524	−	2.04212i
$202$		0			0
$203$	537.611	−	390.597i	0.185876	−	0.135047i
$204$		0			0
$205$	995.344	+	3063.35i	0.339111	+	1.04368i
$206$		0			0
$207$	−5847.06	−	4248.14i	−1.96328	−	1.42641i
$208$		0			0
$209$	−3644.69	+	645.159i	−1.20626	+	0.213524i
$210$		0			0
$211$	3238.65	+	2353.02i	1.05667	+	0.767717i	0.973470	−	0.228815i	$-0.0734852\pi$
$211$	3238.65	+	2353.02i	1.05667	+	0.767717i	0.0832024	+	0.996533i	$0.473485\pi$
$212$		0			0
$213$	833.827	+	2566.26i	0.268229	+	0.825525i
$214$		0			0
$215$	4113.29	−	2988.48i	1.30476	−	0.947967i
$216$		0			0
$217$	1069.18	−	3290.59i	0.334472	−	1.02940i
$218$		0			0
$219$	−6312.24			−1.94768
$220$		0			0
$221$	−81.6438			−0.0248505
$222$		0			0
$223$	1190.14	−	3662.86i	0.357387	−	1.09992i	−0.597225	−	0.802074i	$-0.703730\pi$
$223$	1190.14	−	3662.86i	0.357387	−	1.09992i	0.954612	−	0.297851i	$-0.0962699\pi$
$224$		0			0
$225$	−8165.09	+	5932.28i	−2.41928	+	1.75771i
$226$		0			0
$227$	198.497	+	610.911i	0.0580383	+	0.178624i	0.975873	−	0.218340i	$-0.0700640\pi$
$227$	198.497	+	610.911i	0.0580383	+	0.178624i	−0.917835	+	0.396963i	$0.870064\pi$
$228$		0			0
$229$	−3034.69	−	2204.83i	−0.875712	−	0.636242i	0.0564018	−	0.998408i	$-0.482037\pi$
$229$	−3034.69	−	2204.83i	−0.875712	−	0.636242i	−0.932113	+	0.362166i	$0.882037\pi$
$230$		0			0
$231$	3283.07	+	6164.65i	0.935110	+	1.75586i
$232$		0			0
$233$	5612.04	+	4077.38i	1.57793	+	1.14643i	0.919024	+	0.394201i	$0.128979\pi$
$233$	5612.04	+	4077.38i	1.57793	+	1.14643i	0.658901	+	0.752229i	$0.271021\pi$
$234$		0			0
$235$	176.186	+	542.245i	0.0489069	+	0.150520i
$236$		0			0
$237$	−6267.35	+	4553.49i	−1.71775	+	1.24802i
$238$		0			0
$239$	171.161	−	526.780i	0.0463243	−	0.142571i	−0.925219	−	0.379433i	$-0.876119\pi$
$239$	171.161	−	526.780i	0.0463243	−	0.142571i	0.971543	+	0.236862i	$0.0761189\pi$
$240$		0			0
$241$	2532.58			0.676921			0.338460	−	0.940981i	$-0.390094\pi$
$241$	2532.58			0.676921			0.338460	+	0.940981i	$0.390094\pi$
$242$		0			0
$243$	7148.99			1.88728
$244$		0			0
$245$	−277.601	+	854.369i	−0.0723890	+	0.222790i
$246$		0			0
$247$	3252.48	−	2363.07i	0.837856	−	0.608738i
$248$		0			0
$249$	1895.99	+	5835.24i	0.482543	+	1.48511i
$250$		0			0
$251$	2576.82	+	1872.17i	0.647998	+	0.470798i	0.862589	−	0.505906i	$-0.168842\pi$
$251$	2576.82	+	1872.17i	0.647998	+	0.470798i	−0.214591	+	0.976704i	$0.568842\pi$
$252$		0			0
$253$	1895.81	+	3559.78i	0.471102	+	0.884592i
$254$		0			0
$255$	−267.775	−	194.550i	−0.0657598	−	0.0477773i
$256$		0			0
$257$	−1635.39	−	5033.21i	−0.396937	−	1.22165i	−0.927443	−	0.373965i	$-0.877998\pi$
$257$	−1635.39	−	5033.21i	−0.396937	−	1.22165i	0.530506	−	0.847681i	$-0.322002\pi$
$258$		0			0
$259$	−4051.77	+	2943.79i	−0.972065	+	0.706247i
$260$		0			0
$261$	−674.005	+	2074.37i	−0.159846	+	0.491956i
$262$		0			0
$263$	−4256.45			−0.997962			−0.498981	−	0.866613i	$-0.666292\pi$
$263$	−4256.45			−0.997962			−0.498981	+	0.866613i	$0.666292\pi$
$264$		0			0
$265$	−3620.83			−0.839342
$266$		0			0
$267$	3399.77	−	10463.4i	0.779261	−	2.39832i
$268$		0			0
$269$	7015.54	−	5097.09i	1.59013	−	1.15530i	0.686377	−	0.727246i	$-0.259200\pi$
$269$	7015.54	−	5097.09i	1.59013	−	1.15530i	0.903754	−	0.428051i	$-0.140800\pi$
$270$		0			0
$271$	−923.231	−	2841.41i	−0.206946	−	0.636914i	−0.999628	−	0.0272784i	$-0.991316\pi$
$271$	−923.231	−	2841.41i	−0.206946	−	0.636914i	0.792682	−	0.609635i	$-0.208684\pi$
$272$		0			0
$273$	−6137.38	−	4459.07i	−1.36063	−	0.988553i
$274$		0			0
$275$	5545.82	−	981.686i	1.21609	−	0.215265i
$276$		0			0
$277$	6389.90	+	4642.53i	1.38603	+	1.00701i	0.996287	+	0.0860916i	$0.0274378\pi$
$277$	6389.90	+	4642.53i	1.38603	+	1.00701i	0.389748	+	0.920922i	$0.372562\pi$
$278$		0			0
$279$	3509.30	+	10800.5i	0.753033	+	2.31760i
$280$		0			0
$281$	−4459.82	+	3240.25i	−0.946799	+	0.687889i	−0.950048	−	0.312105i	$-0.898966\pi$
$281$	−4459.82	+	3240.25i	−0.946799	+	0.687889i	0.00324900	+	0.999995i	$0.498966\pi$
$282$		0			0
$283$	1011.57	−	3113.28i	0.212478	−	0.653942i	−0.786845	−	0.617151i	$-0.788287\pi$
$283$	1011.57	−	3113.28i	0.212478	−	0.653942i	0.999323	−	0.0367903i	$-0.0117134\pi$
$284$		0			0
$285$	16298.5			3.38750
$286$		0			0
$287$	−3838.43			−0.789461
$288$		0			0
$289$	−1516.89	+	4668.50i	−0.308750	+	0.950235i
$290$		0			0
$291$	−6533.23	+	4746.67i	−1.31610	+	0.956202i
$292$		0			0
$293$	462.333	+	1422.92i	0.0921836	+	0.283712i	0.986509	−	0.163704i	$-0.0523443\pi$
$293$	462.333	+	1422.92i	0.0921836	+	0.283712i	−0.894326	+	0.447416i	$0.852344\pi$
$294$		0			0
$295$	−217.168	−	157.782i	−0.0428611	−	0.0311404i
$296$		0			0
$297$	−12098.9	−	5891.06i	−2.36380	−	1.15096i
$298$		0			0
$299$	−3544.03	−	2574.89i	−0.685474	−	0.498026i
$300$		0			0
$301$	1872.31	+	5762.37i	0.358531	+	1.10345i
$302$		0			0
$303$	−10364.3	+	7530.09i	−1.96506	+	1.42770i
$304$		0			0
$305$	2743.40	−	8443.31i	0.515037	−	1.58512i
$306$		0			0
$307$	−4758.05			−0.884548			−0.442274	−	0.896880i	$-0.645828\pi$
$307$	−4758.05			−0.884548			−0.442274	+	0.896880i	$0.645828\pi$
$308$		0			0
$309$	913.251			0.168133
$310$		0			0
$311$	82.4860	−	253.866i	0.0150397	−	0.0462875i	−0.943255	−	0.332069i	$-0.892253\pi$
$311$	82.4860	−	253.866i	0.0150397	−	0.0462875i	0.958295	+	0.285782i	$0.0922532\pi$
$312$		0			0
$313$	−7237.95	+	5258.68i	−1.30707	+	0.949642i	−0.999998	−	0.00204380i	$-0.999349\pi$
$313$	−7237.95	+	5258.68i	−1.30707	+	0.949642i	−0.307073	+	0.951686i	$0.599349\pi$
$314$		0			0
$315$	−6726.03	−	20700.6i	−1.20308	−	3.70269i
$316$		0			0
$317$	209.750	+	152.392i	0.0371632	+	0.0270007i	0.606212	−	0.795303i	$-0.292688\pi$
$317$	209.750	+	152.392i	0.0371632	+	0.0270007i	−0.569049	+	0.822304i	$0.692688\pi$
$318$		0			0
$319$	844.919	−	876.105i	0.148296	−	0.153770i
$320$		0			0
$321$	11213.8	+	8147.31i	1.94983	+	1.41663i
$322$		0			0
$323$	64.5937	+	198.799i	0.0111272	+	0.0342460i
$324$		0			0
$325$	−4949.04	+	3595.69i	−0.844687	+	0.613701i
$326$		0			0
$327$	−2105.39	+	6479.72i	−0.356050	+	1.09581i
$328$		0			0
$329$	−679.441			−0.113857
$330$		0			0
$331$	−4557.71			−0.756841			−0.378421	−	0.925634i	$-0.623533\pi$
$331$	−4557.71			−0.756841			−0.378421	+	0.925634i	$0.623533\pi$
$332$		0			0
$333$	5079.72	−	15633.8i	0.835937	−	2.57275i
$334$		0			0
$335$	8608.69	−	6254.58i	1.40401	−	1.02007i
$336$		0			0
$337$	−3003.82	−	9244.81i	−0.485545	−	1.49435i	−0.831190	−	0.555988i	$-0.812340\pi$
$337$	−3003.82	−	9244.81i	−0.485545	−	1.49435i	0.345646	−	0.938365i	$-0.387660\pi$
$338$		0			0
$339$	12994.2	+	9440.83i	2.08185	+	1.51255i
$340$		0			0
$341$	877.795	−	6276.15i	0.139400	−	0.996695i
$342$		0			0
$343$	4661.15	+	3386.53i	0.733757	+	0.533106i
$344$		0			0
$345$	−5487.97	−	16890.2i	−0.856413	−	2.63577i
$346$		0			0
$347$	4215.04	−	3062.40i	0.652089	−	0.473771i	−0.211893	−	0.977293i	$-0.567963\pi$
$347$	4215.04	−	3062.40i	0.652089	−	0.473771i	0.863982	+	0.503522i	$0.167963\pi$
$348$		0			0
$349$	−176.166	+	542.185i	−0.0270200	+	0.0831590i	−0.963657	−	0.267142i	$-0.913921\pi$
$349$	−176.166	+	542.185i	−0.0270200	+	0.0831590i	0.936637	+	0.350301i	$0.113921\pi$
$350$		0			0
$351$	14616.5			2.22270
$352$		0			0
$353$	2751.02			0.414793			0.207396	−	0.978257i	$-0.433501\pi$
$353$	2751.02			0.414793			0.207396	+	0.978257i	$0.433501\pi$
$354$		0			0
$355$	−1450.06	+	4462.84i	−0.216793	+	0.667219i
$356$		0			0
$357$	319.105	−	231.843i	0.0473076	−	0.0343710i
$358$		0			0
$359$	1371.91	+	4222.31i	0.201690	+	0.620738i	0.999833	+	0.0182700i	$0.00581586\pi$
$359$	1371.91	+	4222.31i	0.201690	+	0.620738i	−0.798143	+	0.602468i	$0.794184\pi$
$360$		0			0
$361$	−2778.16	−	2018.45i	−0.405039	−	0.294278i
$362$		0			0
$363$	7889.36	+	10070.2i	1.14073	+	1.45606i
$364$		0			0
$365$	−8880.81	−	6452.29i	−1.27354	−	0.925282i
$366$		0			0
$367$	−2881.78	−	8869.21i	−0.409885	−	1.26150i	−0.916747	−	0.399468i	$-0.869195\pi$
$367$	−2881.78	−	8869.21i	−0.409885	−	1.26150i	0.506862	−	0.862027i	$-0.330805\pi$
$368$		0			0
$369$	10192.5	−	7405.29i	1.43794	−	1.04473i
$370$		0			0
$371$	1333.38	−	4103.71i	0.186591	−	0.574269i
$372$		0			0
$373$	−8713.65			−1.20959			−0.604793	−	0.796382i	$-0.706744\pi$
$373$	−8713.65			−1.20959			−0.604793	+	0.796382i	$0.706744\pi$
$374$		0			0
$375$	−4719.02			−0.649838
$376$		0			0
$377$	−408.529	+	1257.32i	−0.0558099	+	0.171765i
$378$		0			0
$379$	1716.17	−	1246.87i	0.232595	−	0.168990i	−0.465383	−	0.885110i	$-0.654083\pi$
$379$	1716.17	−	1246.87i	0.232595	−	0.168990i	0.697978	+	0.716119i	$0.254083\pi$
$380$		0			0
$381$	4310.87	+	13267.5i	0.579666	+	1.78403i
$382$		0			0
$383$	−6020.09	−	4373.85i	−0.803165	−	0.583533i	0.108676	−	0.994077i	$-0.465339\pi$
$383$	−6020.09	−	4373.85i	−0.803165	−	0.583533i	−0.911841	+	0.410544i	$0.865339\pi$
$384$		0			0
$385$	−1682.41	+	12029.1i	−0.222710	+	1.59236i
$386$		0			0
$387$	−16088.8	−	11689.2i	−2.11327	−	1.53538i
$388$		0			0
$389$	−1765.53	−	5433.73i	−0.230118	−	0.708229i	−0.997732	−	0.0673169i	$-0.978556\pi$
$389$	−1765.53	−	5433.73i	−0.230118	−	0.708229i	0.767614	−	0.640912i	$-0.221444\pi$
$390$		0			0
$391$	184.267	−	133.878i	0.0238332	−	0.0173159i
$392$		0			0
$393$	−6961.90	+	21426.5i	−0.893591	+	2.75019i
$394$		0			0
$395$	−13472.2			−1.71610
$396$		0			0
$397$	−7568.62			−0.956821			−0.478411	−	0.878136i	$-0.658787\pi$
$397$	−7568.62			−0.956821			−0.478411	+	0.878136i	$0.658787\pi$
$398$		0			0
$399$	−6001.94	+	18472.1i	−0.753065	+	2.31770i
$400$		0			0
$401$	10188.7	−	7402.50i	1.26882	−	0.921853i	0.269667	−	0.962954i	$-0.413086\pi$
$401$	10188.7	−	7402.50i	1.26882	−	0.921853i	0.999155	+	0.0411003i	$0.0130863\pi$
$402$		0			0
$403$	2127.06	+	6546.42i	0.262919	+	0.809183i
$404$		0			0
$405$	24069.7	+	17487.6i	2.95317	+	2.14560i
$406$		0			0
$407$	−6367.84	+	6602.88i	−0.775533	+	0.804159i
$408$		0			0
$409$	−7971.56	−	5791.68i	−0.963736	−	0.700196i	−0.00972095	−	0.999953i	$-0.503094\pi$
$409$	−7971.56	−	5791.68i	−0.963736	−	0.700196i	−0.954016	+	0.299757i	$0.903094\pi$
$410$		0			0
$411$	171.246	+	527.042i	0.0205522	+	0.0632532i
$412$		0			0
$413$	258.797	−	188.027i	0.0308343	−	0.0224024i
$414$		0			0
$415$	−3297.21	+	10147.8i	−0.390009	+	1.20032i
$416$		0			0
$417$	13038.9			1.53122
$418$		0			0
$419$	7338.49			0.855629			0.427814	−	0.903867i	$-0.359284\pi$
$419$	7338.49			0.855629			0.427814	+	0.903867i	$0.359284\pi$
$420$		0			0
$421$	4787.67	−	14734.9i	0.554244	−	1.70579i	−0.143687	−	0.989623i	$-0.545896\pi$
$421$	4787.67	−	14734.9i	0.554244	−	1.70579i	0.697931	−	0.716165i	$-0.254104\pi$
$422$		0			0
$423$	1804.18	−	1310.81i	0.207381	−	0.150671i
$424$		0			0
$425$	−98.2870	−	302.496i	−0.0112179	−	0.0345252i
$426$		0			0
$427$	8559.07	+	6218.53i	0.970029	+	0.704767i
$428$		0			0
$429$	−12492.8	−	6082.84i	−1.40596	−	0.684574i
$430$		0			0
$431$	5795.68	+	4210.81i	0.647722	+	0.470598i	0.862495	−	0.506066i	$-0.168901\pi$
$431$	5795.68	+	4210.81i	0.647722	+	0.470598i	−0.214772	+	0.976664i	$0.568901\pi$
$432$		0			0
$433$	1069.05	+	3290.21i	0.118650	+	0.365167i	0.992691	−	0.120686i	$-0.0385093\pi$
$433$	1069.05	+	3290.21i	0.118650	+	0.365167i	−0.874041	+	0.485852i	$0.838509\pi$
$434$		0			0
$435$	−4335.99	+	3150.28i	−0.477919	+	0.347228i
$436$		0			0
$437$	−3465.83	+	10666.7i	−0.379389	+	1.16764i
$438$		0			0
$439$	−10534.8			−1.14533			−0.572665	−	0.819790i	$-0.694090\pi$
$439$	−10534.8			−1.14533			−0.572665	+	0.819790i	$0.694090\pi$
$440$		0			0
$441$	3513.76			0.379415
$442$		0			0
$443$	−2103.92	+	6475.21i	−0.225644	+	0.694461i	0.772581	+	0.634916i	$0.218965\pi$
$443$	−2103.92	+	6475.21i	−0.225644	+	0.694461i	−0.998226	+	0.0595455i	$0.981035\pi$
$444$		0			0
$445$	15478.8	−	11246.0i	1.64891	−	1.19800i
$446$		0			0
$447$	−7882.31	−	24259.3i	−0.834051	−	2.56694i
$448$		0			0
$449$	1287.65	+	935.536i	0.135341	+	0.0983311i	0.653396	−	0.757016i	$-0.273344\pi$
$449$	1287.65	+	935.536i	0.135341	+	0.0983311i	−0.518055	+	0.855347i	$0.673344\pi$
$450$		0			0
$451$	−6922.87	+	1225.44i	−0.722806	+	0.127946i
$452$		0			0
$453$	−18924.2	−	13749.2i	−1.96277	−	1.42604i
$454$		0			0
$455$	−4076.79	−	12547.1i	−0.420051	−	1.29278i
$456$		0			0
$457$	7878.87	−	5724.34i	0.806473	−	0.585937i	−0.106333	−	0.994331i	$-0.533911\pi$
$457$	7878.87	−	5724.34i	0.806473	−	0.585937i	0.912806	+	0.408394i	$0.133911\pi$
$458$		0			0
$459$	−234.842	+	722.768i	−0.0238812	+	0.0734988i
$460$		0			0
$461$	16872.1			1.70458			0.852289	−	0.523071i	$-0.175214\pi$
$461$	16872.1			1.70458			0.852289	+	0.523071i	$0.175214\pi$
$462$		0			0
$463$	19730.3			1.98044			0.990221	−	0.139508i	$-0.0445522\pi$
$463$	19730.3			1.98044			0.990221	+	0.139508i	$0.0445522\pi$
$464$		0			0
$465$	−8623.23	+	26539.6i	−0.859984	+	2.64676i
$466$		0			0
$467$	−4728.81	+	3435.68i	−0.468572	+	0.340437i	−0.796884	−	0.604132i	$-0.793520\pi$
$467$	−4728.81	+	3435.68i	−0.468572	+	0.340437i	0.328313	+	0.944569i	$0.393520\pi$
$468$		0			0
$469$	3918.54	+	12060.0i	0.385803	+	1.18738i
$470$		0			0
$471$	10752.6	+	7812.20i	1.05192	+	0.764262i
$472$		0			0
$473$	5216.51	+	9795.09i	0.507094	+	0.952175i
$474$		0			0
$475$	12670.8	+	9205.90i	1.22395	+	0.889254i
$476$		0			0
$477$	4376.46	+	13469.4i	0.420093	+	1.29291i
$478$		0			0
$479$	14668.2	−	10657.1i	1.39918	−	1.01656i	0.404394	−	0.914585i	$-0.367483\pi$
$479$	14668.2	−	10657.1i	1.39918	−	1.01656i	0.994787	−	0.101979i	$-0.0325174\pi$
$480$		0			0
$481$	3078.93	−	9475.98i	0.291865	−	0.898269i
$482$		0			0
$483$	21163.7			1.99375
$484$		0			0
$485$	−14043.7			−1.31483
$486$		0			0
$487$	1069.10	−	3290.36i	0.0994778	−	0.306161i	−0.888917	−	0.458068i	$-0.848541\pi$
$487$	1069.10	−	3290.36i	0.0994778	−	0.306161i	0.988395	+	0.151907i	$0.0485415\pi$
$488$		0			0
$489$	−14790.5	+	10745.9i	−1.36779	+	0.993757i
$490$		0			0
$491$	596.968	+	1837.28i	0.0548692	+	0.168870i	0.974736	−	0.223362i	$-0.0717031\pi$
$491$	596.968	+	1837.28i	0.0548692	+	0.168870i	−0.919866	+	0.392232i	$0.871703\pi$
$492$		0			0
$493$	−55.6095	−	40.4026i	−0.00508017	−	0.00369096i
$494$		0			0
$495$	−18739.7	−	35187.6i	−1.70159	−	3.19508i
$496$		0			0
$497$	−4524.03	−	3286.90i	−0.408310	−	0.296655i
$498$		0			0
$499$	−3620.14	−	11141.6i	−0.324769	−	0.999535i	−0.971545	−	0.236855i	$-0.923883\pi$
$499$	−3620.14	−	11141.6i	−0.324769	−	0.999535i	0.646776	−	0.762680i	$-0.276117\pi$
$500$		0			0
$501$	−16718.1	+	12146.4i	−1.49083	+	1.08315i
$502$		0			0
$503$	−1028.32	+	3164.84i	−0.0911539	+	0.280543i	−0.986232	−	0.165366i	$-0.947119\pi$
$503$	−1028.32	+	3164.84i	−0.0911539	+	0.280543i	0.895078	+	0.445909i	$0.147119\pi$
$504$		0			0
$505$	−22278.9			−1.96316
$506$		0			0
$507$	−6023.75			−0.527661
$508$		0			0
$509$	−3694.18	+	11369.5i	−0.321693	+	0.990070i	0.651218	+	0.758891i	$0.274258\pi$
$509$	−3694.18	+	11369.5i	−0.321693	+	0.990070i	−0.972911	+	0.231179i	$0.925742\pi$
$510$		0			0
$511$	10583.2	−	7689.11i	0.916186	−	0.665648i
$512$		0			0
$513$	−11564.0	−	35590.4i	−0.995252	−	3.06307i
$514$		0			0
$515$	1284.87	+	933.512i	0.109938	+	0.0798747i
$516$		0			0
$517$	−1225.42	+	216.916i	−0.104244	+	0.0184525i
$518$		0			0
$519$	−4109.54	−	2985.76i	−0.347570	−	0.252525i
$520$		0			0
$521$	−3554.62	−	10940.0i	−0.298907	−	0.919941i	−0.981881	−	0.189498i	$-0.939314\pi$
$521$	−3554.62	−	10940.0i	−0.298907	−	0.919941i	0.682974	−	0.730443i	$-0.260686\pi$
$522$		0			0
$523$	−11863.1	+	8619.05i	−0.991850	+	0.720621i	−0.960325	−	0.278882i	$-0.910036\pi$
$523$	−11863.1	+	8619.05i	−0.991850	+	0.720621i	−0.0315245	+	0.999503i	$0.510036\pi$
$524$		0			0
$525$	9132.67	−	28107.5i	0.759204	−	2.33659i
$526$		0			0
$527$	−357.889			−0.0295823
$528$		0			0
$529$	54.0095			0.00443902
$530$		0			0
$531$	−324.455	+	998.568i	−0.0265162	+	0.0816086i
$532$		0			0
$533$	6177.90	−	4488.51i	0.502054	−	0.364763i
$534$		0			0
$535$	7448.85	+	22925.2i	0.601947	+	1.85260i
$536$		0			0
$537$	30554.3	+	22199.0i	2.45534	+	1.78391i
$538$		0			0
$539$	−1762.93	−	858.387i	−0.140881	−	0.0685962i
$540$		0			0
$541$	−2453.96	−	1782.90i	−0.195016	−	0.141688i	0.485992	−	0.873963i	$-0.338458\pi$
$541$	−2453.96	−	1782.90i	−0.195016	−	0.141688i	−0.681008	+	0.732276i	$0.738458\pi$
$542$		0			0
$543$	−1606.30	−	4943.69i	−0.126948	−	0.390707i
$544$		0			0
$545$	−9585.60	+	6964.34i	−0.753398	+	0.547376i
$546$		0			0
$547$	1822.06	−	5607.72i	0.142423	−	0.438334i	−0.854247	−	0.519867i	$-0.825982\pi$
$547$	1822.06	−	5607.72i	0.142423	−	0.438334i	0.996671	+	0.0815329i	$0.0259816\pi$
$548$		0			0
$549$	−34724.7			−2.69948
$550$		0			0
$551$	3384.74			0.261696
$552$		0			0
$553$	4961.15	−	15268.8i	0.381500	−	1.17414i
$554$		0			0
$555$	32678.7	−	23742.5i	2.49934	−	1.81588i
$556$		0			0
$557$	405.592	+	1248.28i	0.0308536	+	0.0949577i	0.965297	−	0.261153i	$-0.0841027\pi$
$557$	405.592	+	1248.28i	0.0308536	+	0.0949577i	−0.934444	+	0.356111i	$0.884103\pi$
$558$		0			0
$559$	−9751.75	−	7085.06i	−0.737844	−	0.536075i
$560$		0			0
$561$	501.510	−	520.022i	0.0377429	−	0.0391361i
$562$		0			0
$563$	9548.34	+	6937.27i	0.714768	+	0.519309i	0.884708	−	0.466145i	$-0.154358\pi$
$563$	9548.34	+	6937.27i	0.714768	+	0.519309i	−0.169940	+	0.985454i	$0.554358\pi$
$564$		0			0
$565$	8631.48	+	26565.0i	0.642706	+	1.97805i
$566$		0			0
$567$	−28683.5	+	20839.8i	−2.12451	+	1.54355i
$568$		0			0
$569$	4759.87	−	14649.4i	0.350693	−	1.07932i	−0.607772	−	0.794111i	$-0.707937\pi$
$569$	4759.87	−	14649.4i	0.350693	−	1.07932i	0.958465	−	0.285210i	$-0.0920633\pi$
$570$		0			0
$571$	−6979.75			−0.511547			−0.255774	−	0.966737i	$-0.582330\pi$
$571$	−6979.75			−0.511547			−0.255774	+	0.966737i	$0.582330\pi$
$572$		0			0
$573$	−1148.20			−0.0837116
$574$		0			0
$575$	5273.67	−	16230.7i	0.382482	−	1.17716i
$576$		0			0
$577$	10481.6	−	7615.30i	0.756245	−	0.549444i	−0.141511	−	0.989937i	$-0.545196\pi$
$577$	10481.6	−	7615.30i	0.756245	−	0.549444i	0.897756	+	0.440493i	$0.145196\pi$
$578$		0			0
$579$	−1941.39	−	5975.00i	−0.139346	−	0.428864i
$580$		0			0
$581$	−10286.9	−	7473.86i	−0.734548	−	0.533680i
$582$		0			0
$583$	1094.70	−	7827.02i	0.0777666	−	0.556024i
$584$		0			0
$585$	35031.9	+	25452.2i	2.47588	+	1.79884i
$586$		0			0
$587$	1862.76	+	5732.97i	0.130978	+	0.403109i	0.994943	−	0.100443i	$-0.0320261\pi$
$587$	1862.76	+	5732.97i	0.130978	+	0.403109i	−0.863965	+	0.503552i	$0.832026\pi$
$588$		0			0
$589$	14257.4	−	10358.6i	0.997394	−	0.724649i
$590$		0			0
$591$	−2225.76	+	6850.17i	−0.154916	+	0.476783i
$592$		0			0
$593$	9230.18			0.639187			0.319594	−	0.947555i	$-0.396454\pi$
$593$	9230.18			0.639187			0.319594	+	0.947555i	$0.396454\pi$
$594$		0			0
$595$	685.941			0.0472619
$596$		0			0
$597$	5453.65	−	16784.6i	0.373874	−	1.15067i
$598$		0			0
$599$	−7234.67	+	5256.29i	−0.493490	+	0.358541i	−0.806525	−	0.591200i	$-0.798654\pi$
$599$	−7234.67	+	5256.29i	−0.493490	+	0.358541i	0.313035	+	0.949742i	$0.398654\pi$
$600$		0			0
$601$	2482.71	+	7640.98i	0.168505	+	0.518606i	0.999277	−	0.0380066i	$-0.0121008\pi$
$601$	2482.71	+	7640.98i	0.168505	+	0.518606i	−0.830772	+	0.556613i	$0.812101\pi$
$602$		0			0
$603$	−33672.1	−	24464.2i	−2.27402	−	1.65217i
$604$		0			0
$605$	806.018	+	22232.4i	0.0541641	+	1.49401i
$606$		0			0
$607$	−2043.54	−	1484.72i	−0.136647	−	0.0992798i	0.517362	−	0.855767i	$-0.326914\pi$
$607$	−2043.54	−	1484.72i	−0.136647	−	0.0992798i	−0.654009	+	0.756487i	$0.726914\pi$
$608$		0			0
$609$	−1973.67	−	6074.34i	−0.131326	−	0.404179i
$610$		0			0
$611$	1093.55	−	794.513i	0.0724066	−	0.0526065i
$612$		0			0
$613$	−2703.55	+	8320.66i	−0.178132	+	0.548235i	−0.999763	−	0.0217855i	$-0.993065\pi$
$613$	−2703.55	+	8320.66i	−0.178132	+	0.548235i	0.821630	+	0.570021i	$0.193065\pi$
$614$		0			0
$615$	30958.0			2.02983
$616$		0			0
$617$	620.479			0.0404855			0.0202427	−	0.999795i	$-0.493556\pi$
$617$	620.479			0.0404855			0.0202427	+	0.999795i	$0.493556\pi$
$618$		0			0
$619$	2569.23	−	7907.26i	0.166827	−	0.513441i	−0.832339	−	0.554266i	$-0.812999\pi$
$619$	2569.23	−	7907.26i	0.166827	−	0.513441i	0.999166	+	0.0408258i	$0.0129989\pi$
$620$		0			0
$621$	−32988.8	+	23967.8i	−2.13172	+	1.54878i
$622$		0			0
$623$	7045.69	+	21684.4i	0.453097	+	1.39449i
$624$		0			0
$625$	8972.21	+	6518.69i	0.574221	+	0.417196i
$626$		0			0
$627$	−4927.60	+	35231.9i	−0.313858	+	2.24406i
$628$		0			0
$629$	419.108	+	304.500i	0.0265674	+	0.0193024i
$630$		0			0
$631$	−2571.38	−	7913.88i	−0.162226	−	0.499282i	0.836595	−	0.547822i	$-0.184543\pi$
$631$	−2571.38	−	7913.88i	−0.162226	−	0.499282i	−0.998821	+	0.0485406i	$0.984543\pi$
$632$		0			0
$633$	31127.7	−	22615.6i	1.95452	−	1.42004i
$634$		0			0
$635$	−7496.80	+	23072.8i	−0.468506	+	1.44191i
$636$		0			0
$637$	2129.77			0.132472
$638$		0			0
$639$	18354.3			1.13628
$640$		0			0
$641$	1491.81	−	4591.30i	0.0919232	−	0.282910i	−0.894516	−	0.447035i	$-0.852480\pi$
$641$	1491.81	−	4591.30i	0.0919232	−	0.282910i	0.986440	+	0.164125i	$0.0524799\pi$
$642$		0			0
$643$	−17241.3	+	12526.5i	−1.05743	+	0.768269i	−0.973612	−	0.228212i	$-0.926712\pi$
$643$	−17241.3	+	12526.5i	−1.05743	+	0.768269i	−0.0838203	+	0.996481i	$0.526712\pi$
$644$		0			0
$645$	−15100.7	−	46475.2i	−0.921844	−	2.83714i
$646$		0			0
$647$	4393.15	+	3191.81i	0.266944	+	0.193946i	0.713203	−	0.700958i	$-0.247244\pi$
$647$	4393.15	+	3191.81i	0.266944	+	0.193946i	−0.446259	+	0.894904i	$0.647244\pi$
$648$		0			0
$649$	406.730	−	421.742i	0.0246002	−	0.0255082i
$650$		0			0
$651$	−26903.4	−	19546.5i	−1.61971	−	1.17679i
$652$		0			0
$653$	435.714	+	1340.99i	0.0261115	+	0.0803630i	0.963263	−	0.268560i	$-0.0865476\pi$
$653$	435.714	+	1340.99i	0.0261115	+	0.0803630i	−0.937152	+	0.348923i	$0.886548\pi$
$654$		0			0
$655$	−31696.7	+	23029.0i	−1.89083	+	1.37377i
$656$		0			0
$657$	−13268.1	+	40835.1i	−0.787883	+	2.42486i
$658$		0			0
$659$	18088.9			1.06926			0.534630	−	0.845086i	$-0.320451\pi$
$659$	18088.9			1.06926			0.534630	+	0.845086i	$0.320451\pi$
$660$		0			0
$661$	16624.5			0.978240			0.489120	−	0.872217i	$-0.337318\pi$
$661$	16624.5			0.978240			0.489120	+	0.872217i	$0.337318\pi$
$662$		0			0
$663$	−242.487	+	746.298i	−0.0142042	+	0.0437161i
$664$		0			0
$665$	−27326.2	+	19853.6i	−1.59348	+	1.15773i
$666$		0			0
$667$	−1139.70	−	3507.63i	−0.0661609	−	0.203622i
$668$		0			0
$669$	−29947.1	−	21757.8i	−1.73067	−	1.25741i
$670$		0			0
$671$	17422.2	+	8483.01i	1.00235	+	0.488052i
$672$		0			0
$673$	−373.402	−	271.293i	−0.0213872	−	0.0155387i	0.577040	−	0.816716i	$-0.304208\pi$
$673$	−373.402	−	271.293i	−0.0213872	−	0.0155387i	−0.598428	+	0.801177i	$0.704208\pi$
$674$		0			0
$675$	17596.0	+	54155.0i	1.00337	+	3.08804i
$676$		0			0
$677$	−12870.2	+	9350.77i	−0.730640	+	0.530841i	−0.889766	−	0.456417i	$-0.849132\pi$
$677$	−12870.2	+	9350.77i	−0.730640	+	0.530841i	0.159126	+	0.987258i	$0.449132\pi$
$678$		0			0
$679$	5171.62	−	15916.6i	0.292295	−	0.899593i
$680$		0			0
$681$	6173.82			0.347403
$682$		0			0
$683$	−32846.5			−1.84017			−0.920084	−	0.391721i	$-0.871880\pi$
$683$	−32846.5			−1.84017			−0.920084	+	0.391721i	$0.871880\pi$
$684$		0			0
$685$	−297.805	+	916.550i	−0.0166110	+	0.0511235i
$686$		0			0
$687$	−29167.3	+	21191.3i	−1.61980	+	1.17685i
$688$		0			0
$689$	2652.67	+	8164.07i	0.146674	+	0.451417i
$690$		0			0
$691$	−1565.49	−	1137.40i	−0.0861855	−	0.0626174i	0.543858	−	0.839177i	$-0.316963\pi$
$691$	−1565.49	−	1137.40i	−0.0861855	−	0.0626174i	−0.630044	+	0.776560i	$0.716963\pi$
$692$		0			0
$693$	46781.3	−	8280.92i	2.56432	−	0.453920i
$694$		0			0
$695$	18344.7	+	13328.2i	1.00123	+	0.727437i
$696$		0			0
$697$	122.692	+	377.607i	0.00666756	+	0.0205206i
$698$		0			0
$699$	53939.0	−	39189.0i	2.91868	−	2.12055i
$700$		0			0
$701$	−1769.44	+	5445.78i	−0.0953365	+	0.293415i	−0.987341	−	0.158611i	$-0.949299\pi$
$701$	−1769.44	+	5445.78i	−0.0953365	+	0.293415i	0.892005	+	0.452026i	$0.149299\pi$
$702$		0			0
$703$	−25509.5			−1.36858
$704$		0			0
$705$	5479.89			0.292744
$706$		0			0
$707$	8204.23	−	25250.0i	0.436424	−	1.34318i
$708$		0			0
$709$	21977.0	−	15967.2i	1.16412	−	0.845784i	0.173829	−	0.984776i	$-0.444386\pi$
$709$	21977.0	−	15967.2i	1.16412	−	0.845784i	0.990294	+	0.138992i	$0.0443861\pi$
$710$		0			0
$711$	16283.7	+	50116.0i	0.858911	+	2.64345i
$712$		0			0
$713$	−15535.4	−	11287.1i	−0.815996	−	0.592856i
$714$		0			0
$715$	−11358.5	−	21328.0i	−0.594104	−	1.11556i
$716$		0			0
$717$	−4306.88	−	3129.13i	−0.224328	−	0.162984i
$718$		0			0
$719$	−3856.55	−	11869.2i	−0.200035	−	0.615643i	−0.999881	−	0.0154382i	$-0.995086\pi$
$719$	−3856.55	−	11869.2i	−0.200035	−	0.615643i	0.799846	−	0.600205i	$-0.204914\pi$
$720$		0			0
$721$	−1531.16	+	1112.46i	−0.0790894	+	0.0574618i
$722$		0			0
$723$	7521.91	−	23150.1i	0.386920	−	1.19082i
$724$		0			0
$725$	−5150.28			−0.263830
$726$		0			0
$727$	4894.88			0.249712			0.124856	−	0.992175i	$-0.460153\pi$
$727$	4894.88			0.249712			0.124856	+	0.992175i	$0.460153\pi$
$728$		0			0
$729$	6381.60	−	19640.5i	0.324219	−	0.997843i
$730$		0			0
$731$	507.029	−	368.378i	0.0256541	−	0.0186388i
$732$		0			0
$733$	−8466.96	−	26058.6i	−0.426650	−	1.31309i	−0.901406	−	0.432976i	$-0.857464\pi$
$733$	−8466.96	−	26058.6i	−0.426650	−	1.31309i	0.474756	−	0.880118i	$-0.342536\pi$
$734$		0			0
$735$	6985.21	+	5075.05i	0.350549	+	0.254689i
$736$		0			0
$737$	10917.6	+	20500.1i	0.545665	+	1.02460i
$738$		0			0
$739$	12028.5	+	8739.19i	0.598747	+	0.435015i	0.845434	−	0.534080i	$-0.179342\pi$
$739$	12028.5	+	8739.19i	0.598747	+	0.435015i	−0.246687	+	0.969095i	$0.579342\pi$
$740$		0			0
$741$	−11940.5	−	36749.0i	−0.591963	−	1.82188i
$742$		0			0
$743$	−26119.5	+	18976.9i	−1.28968	+	0.937007i	−0.999799	−	0.0200565i	$-0.993615\pi$
$743$	−26119.5	+	18976.9i	−1.28968	+	0.937007i	−0.289880	+	0.957063i	$0.593615\pi$
$744$		0			0
$745$	13707.7	−	42188.0i	0.674110	−	2.07470i
$746$		0			0
$747$	41734.6			2.04416
$748$		0			0
$749$	−28725.6			−1.40135
$750$		0			0
$751$	4518.03	−	13905.1i	0.219528	−	0.675636i	−0.779274	−	0.626684i	$-0.784412\pi$
$751$	4518.03	−	13905.1i	0.219528	−	0.675636i	0.998801	−	0.0489525i	$-0.0155883\pi$
$752$		0			0
$753$	24766.6	−	17994.0i	1.19860	−	0.870834i
$754$		0			0
$755$	−12570.5	−	38688.1i	−0.605945	−	1.86491i
$756$		0			0
$757$	5130.04	+	3727.19i	0.246307	+	0.178953i	0.704089	−	0.710112i	$-0.251356\pi$
$757$	5130.04	+	3727.19i	0.246307	+	0.178953i	−0.457781	+	0.889065i	$0.651356\pi$
$758$		0			0
$759$	38170.3	−	6756.66i	1.82542	−	0.323124i
$760$		0			0
$761$	−6442.01	−	4680.39i	−0.306863	−	0.222949i	0.423686	−	0.905809i	$-0.360736\pi$
$761$	−6442.01	−	4680.39i	−0.306863	−	0.222949i	−0.730549	+	0.682860i	$0.760736\pi$
$762$		0			0
$763$	−4363.21	−	13428.6i	−0.207023	−	0.637153i
$764$		0			0
$765$	−1821.44	+	1323.35i	−0.0860840	+	0.0625437i
$766$		0			0
$767$	−196.659	+	605.254i	−0.00925808	+	0.0284934i
$768$		0			0
$769$	19302.7			0.905166			0.452583	−	0.891722i	$-0.350503\pi$
$769$	19302.7			0.905166			0.452583	+	0.891722i	$0.350503\pi$
$770$		0			0
$771$	−50865.3			−2.37596
$772$		0			0
$773$	−11989.6	+	36900.0i	−0.557871	+	1.71695i	0.130368	+	0.991466i	$0.458384\pi$
$773$	−11989.6	+	36900.0i	−0.557871	+	1.71695i	−0.688239	+	0.725484i	$0.741616\pi$
$774$		0			0
$775$	−21694.3	+	15761.8i	−1.00553	+	0.730557i
$776$		0			0
$777$	14874.8	+	45780.0i	0.686785	+	2.11371i
$778$		0			0
$779$	−15817.1	−	11491.8i	−0.727477	−	0.528543i
$780$		0			0
$781$	−9208.76	−	4483.83i	−0.421915	−	0.205434i
$782$		0			0
$783$	9955.61	+	7233.17i	0.454386	+	0.330131i
$784$		0			0
$785$	7142.47	+	21982.3i	0.324746	+	0.999466i
$786$		0			0
$787$	17185.9	−	12486.3i	0.778413	−	0.565550i	−0.126089	−	0.992019i	$-0.540243\pi$
$787$	17185.9	−	12486.3i	0.778413	−	0.565550i	0.904502	+	0.426469i	$0.140243\pi$
$788$		0			0
$789$	−12641.9	+	38907.8i	−0.570423	+	1.75558i
$790$		0			0
$791$	−33286.3			−1.49624
$792$		0			0
$793$	−21047.4			−0.942517
$794$		0			0
$795$	−10754.1	+	33097.6i	−0.479758	+	1.47654i
$796$		0			0
$797$	9806.58	−	7124.90i	0.435843	−	0.316658i	−0.348138	−	0.937443i	$-0.613186\pi$
$797$	9806.58	−	7124.90i	0.435843	−	0.316658i	0.783981	+	0.620785i	$0.213186\pi$
$798$		0			0
$799$	21.7178	+	66.8404i	0.000961601	+	0.00295950i
$800$		0			0
$801$	−60543.7	−	43987.6i	−2.67067	−	1.94035i
$802$		0			0
$803$	16632.7	−	17246.6i	0.730952	−	0.757932i
$804$		0			0
$805$	29775.6	+	21633.3i	1.30367	+	0.947171i
$806$		0			0
$807$	−25755.4	−	79267.0i	−1.12346	−	3.45766i
$808$		0			0
$809$	−16986.1	+	12341.1i	−0.738196	+	0.536331i	−0.892146	−	0.451748i	$-0.850801\pi$
$809$	−16986.1	+	12341.1i	−0.738196	+	0.536331i	0.153950	+	0.988079i	$0.450801\pi$
$810$		0			0
$811$	3610.00	−	11110.4i	0.156306	−	0.481061i	−0.841985	−	0.539501i	$-0.818613\pi$
$811$	3610.00	−	11110.4i	0.156306	−	0.481061i	0.998291	+	0.0584404i	$0.0186128\pi$
$812$		0			0
$813$	−28715.1			−1.23872
$814$		0			0
$815$	−31793.3			−1.36647
$816$		0			0
$817$	−9536.56	+	29350.5i	−0.408374	+	1.25685i
$818$		0			0
$819$	−41747.1	+	30331.1i	−1.78115	+	1.29408i
$820$		0			0
$821$	612.500	+	1885.08i	0.0260370	+	0.0801338i	0.963231	−	0.268676i	$-0.0865860\pi$
$821$	612.500	+	1885.08i	0.0260370	+	0.0801338i	−0.937194	+	0.348810i	$0.886586\pi$
$822$		0			0
$823$	19101.1	+	13877.7i	0.809017	+	0.587785i	0.913546	−	0.406736i	$-0.133333\pi$
$823$	19101.1	+	13877.7i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$824$		0			0
$825$	7497.92	−	53609.5i	0.316417	−	2.26235i
$826$		0			0
$827$	22690.0	+	16485.2i	0.954059	+	0.693165i	0.951763	−	0.306833i	$-0.0992692\pi$
$827$	22690.0	+	16485.2i	0.954059	+	0.693165i	0.00229592	+	0.999997i	$0.499269\pi$
$828$		0			0
$829$	11310.7	+	34810.8i	0.473868	+	1.45842i	0.847478	+	0.530831i	$0.178120\pi$
$829$	11310.7	+	34810.8i	0.473868	+	1.45842i	−0.373609	+	0.927586i	$0.621880\pi$
$830$		0			0
$831$	61415.3	−	44620.8i	2.56374	−	1.86267i
$832$		0			0
$833$	−34.2188	+	105.315i	−0.00142330	+	0.00438048i
$834$		0			0
$835$	−35936.8			−1.48939
$836$		0			0
$837$	64071.8			2.64593
$838$		0			0
$839$	−2092.52	+	6440.13i	−0.0861049	+	0.265004i	−0.984834	−	0.173501i	$-0.944492\pi$
$839$	−2092.52	+	6440.13i	−0.0861049	+	0.265004i	0.898729	+	0.438505i	$0.144492\pi$
$840$		0			0
$841$	18830.7	−	13681.3i	0.772096	−	0.560961i
$842$		0			0
$843$	16372.8	+	50390.4i	0.668933	+	2.05877i
$844$		0			0
$845$	−8474.92	−	6157.39i	−0.345025	−	0.250675i
$846$		0			0
$847$	−25494.2	−	7273.61i	−1.03423	−	0.295070i
$848$		0			0
$849$	−25453.8	−	18493.3i	−1.02894	−	0.747570i
$850$		0			0
$851$	8589.49	+	26435.7i	0.345997	+	1.06487i
$852$		0			0
$853$	14418.5	−	10475.6i	0.578756	−	0.420491i	−0.259519	−	0.965738i	$-0.583564\pi$
$853$	14418.5	−	10475.6i	0.578756	−	0.420491i	0.838276	+	0.545247i	$0.183564\pi$
$854$		0			0
$855$	34258.9	−	105438.i	1.37033	−	4.21743i
$856$		0			0
$857$	28270.3			1.12683			0.563416	−	0.826173i	$-0.309487\pi$
$857$	28270.3			1.12683			0.563416	+	0.826173i	$0.309487\pi$
$858$		0			0
$859$	26699.7			1.06051			0.530257	−	0.847837i	$-0.322095\pi$
$859$	26699.7			1.06051			0.530257	+	0.847837i	$0.322095\pi$
$860$		0			0
$861$	−11400.3	+	35086.7i	−0.451246	+	1.38879i
$862$		0			0
$863$	−10744.9	+	7806.64i	−0.423826	+	0.307927i	−0.779175	−	0.626806i	$-0.784362\pi$
$863$	−10744.9	+	7806.64i	−0.423826	+	0.307927i	0.355350	+	0.934733i	$0.384362\pi$
$864$		0			0
$865$	−2729.79	−	8401.44i	−0.107301	−	0.330240i
$866$		0			0
$867$	38169.1	+	27731.4i	1.49514	+	1.08629i
$868$		0			0
$869$	4073.10	−	29122.3i	0.158999	−	1.13683i
$870$		0			0
$871$	−20409.4	−	14828.3i	−0.793967	−	0.576851i
$872$		0			0
$873$	16974.5	+	52242.1i	0.658075	+	2.02535i
$874$		0			0
$875$	7911.95	−	5748.37i	0.305683	−	0.222092i
$876$		0			0
$877$	−4452.07	+	13702.1i	−0.171421	+	0.527578i	−0.999452	−	0.0331036i	$-0.989461\pi$
$877$	−4452.07	+	13702.1i	−0.171421	+	0.527578i	0.828031	+	0.560682i	$0.189461\pi$
$878$		0			0
$879$	14379.9			0.551788
$880$		0			0
$881$	40967.4			1.56666			0.783330	−	0.621606i	$-0.213519\pi$
$881$	40967.4			1.56666			0.783330	+	0.621606i	$0.213519\pi$
$882$		0			0
$883$	−3507.59	+	10795.2i	−0.133680	+	0.411426i	−0.995382	−	0.0959888i	$-0.969399\pi$
$883$	−3507.59	+	10795.2i	−0.133680	+	0.411426i	0.861702	+	0.507414i	$0.169399\pi$
$884$		0			0
$885$	−2087.27	+	1516.49i	−0.0792800	+	0.0576003i
$886$		0			0
$887$	−7482.35	−	23028.3i	−0.283239	−	0.871720i	−0.986921	−	0.161205i	$-0.948462\pi$
$887$	−7482.35	−	23028.3i	−0.283239	−	0.871720i	0.703682	−	0.710515i	$-0.251538\pi$
$888$		0			0
$889$	−23389.1	−	16993.2i	−0.882392	−	0.641095i
$890$		0			0
$891$	−45079.6	+	46743.5i	−1.69497	+	1.75754i
$892$		0			0
$893$	−2799.78	−	2034.16i	−0.104917	−	0.0762269i
$894$		0			0
$895$	20295.9	+	62464.4i	0.758008	+	2.33291i
$896$		0			0
$897$	−34062.8	+	24748.1i	−1.26792	+	0.921197i
$898$		0			0
$899$	−1790.80	+	5511.53i	−0.0664367	+	0.204471i
$900$		0			0
$901$	−446.325			−0.0165030
$902$		0			0
$903$	58234.1			2.14608
$904$		0			0
$905$	2793.43	−	8597.31i	0.102604	−	0.315784i
$906$		0			0
$907$	36838.6	−	26764.8i	1.34863	−	0.979835i	0.349549	−	0.936918i	$-0.386335\pi$
$907$	36838.6	−	26764.8i	1.34863	−	0.979835i	0.999079	−	0.0429167i	$-0.0136650\pi$
$908$		0			0
$909$	26928.2	+	82876.6i	0.982567	+	3.02403i
$910$		0			0
$911$	6301.96	+	4578.65i	0.229191	+	0.166517i	0.696454	−	0.717601i	$-0.254760\pi$
$911$	6301.96	+	4578.65i	0.229191	+	0.166517i	−0.467263	+	0.884118i	$0.654760\pi$
$912$		0			0
$913$	−20939.2	−	10195.5i	−0.759022	−	0.369574i
$914$		0			0
$915$	−69031.3	−	50154.2i	−2.49410	−	1.81207i
$916$		0			0
$917$	−14427.8	−	44404.3i	−0.519574	−	1.59908i
$918$		0			0
$919$	−16161.5	+	11742.0i	−0.580108	+	0.421473i	−0.838763	−	0.544497i	$-0.816721\pi$
$919$	−16161.5	+	11742.0i	−0.580108	+	0.421473i	0.258655	+	0.965970i	$0.416721\pi$
$920$		0			0
$921$	−14131.7	+	43492.8i	−0.505597	+	1.55607i
$922$		0			0
$923$	11124.9			0.396730
$924$		0			0
$925$	38815.7			1.37973
$926$		0			0
$927$	1919.62	−	5908.00i	0.0680138	−	0.209325i
$928$		0			0
$929$	−38131.4	+	27704.1i	−1.34666	+	0.978409i	−0.347494	+	0.937682i	$0.612967\pi$
$929$	−38131.4	+	27704.1i	−1.34666	+	0.978409i	−0.999170	+	0.0407268i	$0.987033\pi$
$930$		0			0
$931$	−1685.00	−	5185.89i	−0.0593163	−	0.182557i
$932$		0			0
$933$	−2075.57	−	1507.99i	−0.0728309	−	0.0529148i
$934$		0			0
$935$	1237.14	−	218.991i	0.0432716	−	0.00765966i
$936$		0			0
$937$	−38514.3	−	27982.2i	−1.34280	−	0.975603i	−0.999336	−	0.0364383i	$-0.988399\pi$
$937$	−38514.3	−	27982.2i	−1.34280	−	0.975603i	−0.343467	−	0.939165i	$-0.611601\pi$
$938$		0			0
$939$	26571.9	+	81779.9i	0.923473	+	2.84216i
$940$		0			0
$941$	22291.6	−	16195.8i	0.772247	−	0.561071i	−0.130395	−	0.991462i	$-0.541625\pi$
$941$	22291.6	−	16195.8i	0.772247	−	0.561071i	0.902642	+	0.430392i	$0.141625\pi$
$942$		0			0
$943$	−6583.14	+	20260.8i	−0.227335	+	0.699664i
$944$		0			0
$945$	−122802.			−4.22725
$946$		0			0
$947$	−8056.38			−0.276449			−0.138224	−	0.990401i	$-0.544140\pi$
$947$	−8056.38			−0.276449			−0.138224	+	0.990401i	$0.544140\pi$
$948$		0			0
$949$	−8042.11	+	24751.1i	−0.275087	+	0.846632i
$950$		0			0
$951$	2015.97	−	1464.69i	0.0687407	−	0.0499431i
$952$		0			0
$953$	3149.91	+	9694.43i	0.107068	+	0.329521i	0.990210	−	0.139584i	$-0.0445766\pi$
$953$	3149.91	+	9694.43i	0.107068	+	0.329521i	−0.883142	+	0.469105i	$0.844577\pi$
$954$		0			0
$955$	−1615.42	−	1173.67i	−0.0547370	−	0.0397688i
$956$		0			0
$957$	−5498.93	−	10325.4i	−0.185742	−	0.348770i
$958$		0			0
$959$	−929.116	−	675.042i	−0.0312854	−	0.0227302i
$960$		0			0
$961$	118.133	+	363.575i	0.00396538	+	0.0122042i
$962$		0			0
$963$	76277.6	−	55419.0i	2.55245	−	1.85447i
$964$		0			0
$965$	3376.18	−	10390.8i	0.112625	−	0.346624i
$966$		0			0
$967$	−13913.2			−0.462685			−0.231343	−	0.972872i	$-0.574312\pi$
$967$	−13913.2			−0.462685			−0.231343	+	0.972872i	$0.574312\pi$
$968$		0			0
$969$	2009.05			0.0666047
$970$		0			0
$971$	9346.72	−	28766.3i	0.308909	−	0.950724i	−0.669281	−	0.743010i	$-0.733398\pi$
$971$	9346.72	−	28766.3i	0.308909	−	0.950724i	0.978190	−	0.207714i	$-0.0666024\pi$
$972$		0			0
$973$	−21861.2	+	15883.1i	−0.720286	+	0.523318i
$974$		0			0
$975$	18168.9	+	55918.0i	0.596789	+	1.83673i
$976$		0			0
$977$	19677.4	+	14296.5i	0.644356	+	0.468152i	0.861344	−	0.508023i	$-0.169623\pi$
$977$	19677.4	+	14296.5i	0.644356	+	0.468152i	−0.216988	+	0.976174i	$0.569623\pi$
$978$		0			0
$979$	19630.3	+	36860.0i	0.640844	+	1.20332i
$980$		0			0
$981$	37493.1	+	27240.4i	1.22025	+	0.886563i
$982$		0			0
$983$	−5078.63	−	15630.4i	−0.164784	−	0.507154i	0.834236	−	0.551408i	$-0.185909\pi$
$983$	−5078.63	−	15630.4i	−0.164784	−	0.507154i	−0.999020	+	0.0442533i	$0.985909\pi$
$984$		0			0
$985$	−10133.6	+	7362.50i	−0.327801	+	0.238161i
$986$		0			0
$987$	−2017.98	+	6210.70i	−0.0650790	+	0.200293i
$988$		0			0
$989$	33627.3			1.08118
$990$		0			0
$991$	−25604.7			−0.820745			−0.410373	−	0.911918i	$-0.634601\pi$
$991$	−25604.7			−0.820745			−0.410373	+	0.911918i	$0.634601\pi$
$992$		0			0
$993$	−13536.7	+	41661.6i	−0.432601	+	1.33141i
$994$		0			0
$995$	24829.8	−	18039.9i	0.791114	−	0.574778i
$996$		0			0
$997$	−11764.3	−	36206.9i	−0.373701	−	1.15013i	−0.944351	−	0.328940i	$-0.893309\pi$
$997$	−11764.3	−	36206.9i	−0.373701	−	1.15013i	0.570650	−	0.821194i	$-0.306691\pi$
$998$		0			0
$999$	−75031.7	−	54513.7i	−2.37627	−	1.72646i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.4.i.b.81.5	yes	20
4.3	odd	2		176.4.m.f.81.1		20
11.3	even	5	inner	88.4.i.b.25.5	✓	20
11.5	even	5		968.4.a.s.1.9		10
11.6	odd	10		968.4.a.r.1.9		10
44.3	odd	10		176.4.m.f.113.1		20
44.27	odd	10		1936.4.a.bx.1.2		10
44.39	even	10		1936.4.a.by.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.5	✓	20	11.3	even	5	inner
88.4.i.b.81.5	yes	20	1.1	even	1	trivial
176.4.m.f.81.1		20	4.3	odd	2
176.4.m.f.113.1		20	44.3	odd	10
968.4.a.r.1.9		10	11.6	odd	10
968.4.a.s.1.9		10	11.5	even	5
1936.4.a.bx.1.2		10	44.27	odd	10
1936.4.a.by.1.2		10	44.39	even	10

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:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	88.i (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(2.97006 - 9.14090i) q^{3} +(13.5223 - 9.82454i) q^{5} +(6.15515 + 18.9436i) q^{7} +(-52.8913 - 38.4278i) q^{9} +(17.1491 + 32.2011i) q^{11} +(-32.0586 - 23.2919i) q^{13} +(-49.6431 - 152.786i) q^{15} +(1.66684 - 1.21103i) q^{17} +(-31.3511 + 96.4889i) q^{19} +191.443 q^{21} +110.549 q^{23} +(47.7045 - 146.819i) q^{25} +(-298.410 + 216.808i) q^{27} +(-10.3095 - 31.7293i) q^{29} +(-140.530 - 102.101i) q^{31} +(345.280 - 61.1193i) q^{33} +(269.344 + 195.690i) q^{35} +(77.6987 + 239.132i) q^{37} +(-308.125 + 223.866i) q^{39} +(-59.5497 + 183.275i) q^{41} +304.185 q^{43} -1092.75 q^{45} +(-10.5409 + 32.4416i) q^{47} +(-43.4814 + 31.5911i) q^{49} +(-6.11930 - 18.8333i) q^{51} +(-175.256 - 127.331i) q^{53} +(548.257 + 266.951i) q^{55} +(788.880 + 573.155i) q^{57} +(-4.96281 - 15.2739i) q^{59} +(429.705 - 312.199i) q^{61} +(402.407 - 1238.48i) q^{63} -662.339 q^{65} +636.628 q^{67} +(328.336 - 1010.51i) q^{69} +(-227.127 + 165.017i) q^{71} +(-202.947 - 624.608i) q^{73} +(-1200.37 - 872.123i) q^{75} +(-504.449 + 523.068i) q^{77} +(-652.080 - 473.764i) q^{79} +(550.049 + 1692.88i) q^{81} +(-516.450 + 375.223i) q^{83} +(10.6417 - 32.7519i) q^{85} -320.654 q^{87} +1144.68 q^{89} +(243.907 - 750.670i) q^{91} +(-1350.68 + 981.324i) q^{93} +(524.019 + 1612.76i) q^{95} +(-679.744 - 493.863i) q^{97} +(330.376 - 2362.16i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3 q^{3} + q^{5} + q^{7} - 58 q^{9} - 72 q^{11} - 95 q^{13} - 225 q^{15} + 41 q^{17} + 191 q^{19} + 262 q^{21} + 328 q^{23} - 172 q^{25} - 315 q^{27} + 169 q^{29} - 825 q^{31} + 1093 q^{33} + 1195 q^{35}+ \cdots + 1840 q^{99}+O(q^{100}) \) 20 * q + 3 * q^3 + q^5 + q^7 - 58 * q^9 - 72 * q^11 - 95 * q^13 - 225 * q^15 + 41 * q^17 + 191 * q^19 + 262 * q^21 + 328 * q^23 - 172 * q^25 - 315 * q^27 + 169 * q^29 - 825 * q^31 + 1093 * q^33 + 1195 * q^35 + 5 * q^37 - 223 * q^39 - 411 * q^41 - 760 * q^43 - 88 * q^45 - 2157 * q^47 - 1694 * q^49 + 2163 * q^51 - 1803 * q^53 + 2067 * q^55 + 1781 * q^57 - 383 * q^59 + 2097 * q^61 + 728 * q^63 + 346 * q^65 + 4520 * q^67 - 2524 * q^69 - 2415 * q^71 + 4517 * q^73 - 7457 * q^75 + 2019 * q^77 + 159 * q^79 - 7492 * q^81 + 701 * q^83 + 243 * q^85 + 2606 * q^87 + 892 * q^89 - 7003 * q^91 - 7817 * q^93 + 5691 * q^95 - 5213 * q^97 + 1840 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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