LMFDB - Embedded newform 128.4.g.a.113.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(17,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("128.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	128.g (of order $8$, degree $4$, not 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:	$7.55224448073$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$11$ over $\Q(\zeta_{8})$
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			113.8
Character	$\chi$	$=$	128.113
Dual form			128.4.g.a.17.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.20185 + 2.90153i) q^{3} +(-3.98512 - 1.65069i) q^{5} +(-22.4050 - 22.4050i) q^{7} +(12.1174 - 12.1174i) q^{9} +O(q^{10})$	\(q+(1.20185 + 2.90153i) q^{3} +(-3.98512 - 1.65069i) q^{5} +(-22.4050 - 22.4050i) q^{7} +(12.1174 - 12.1174i) q^{9} +(16.5161 - 39.8735i) q^{11} +(17.9201 - 7.42277i) q^{13} -13.5469i q^{15} +45.9852i q^{17} +(25.0023 - 10.3563i) q^{19} +(38.0813 - 91.9364i) q^{21} +(40.3415 - 40.3415i) q^{23} +(-75.2319 - 75.2319i) q^{25} +(128.064 + 53.0458i) q^{27} +(-88.6955 - 214.130i) q^{29} -260.478 q^{31} +135.544 q^{33} +(52.3029 + 126.270i) q^{35} +(70.4470 + 29.1801i) q^{37} +(43.0748 + 43.0748i) q^{39} +(-251.246 + 251.246i) q^{41} +(95.7143 - 231.075i) q^{43} +(-68.2916 + 28.2873i) q^{45} +15.5684i q^{47} +660.968i q^{49} +(-133.428 + 55.2676i) q^{51} +(171.815 - 414.797i) q^{53} +(-131.638 + 131.638i) q^{55} +(60.0981 + 60.0981i) q^{57} +(-53.3294 - 22.0897i) q^{59} +(297.690 + 718.686i) q^{61} -542.983 q^{63} -83.6667 q^{65} +(377.382 + 911.080i) q^{67} +(165.537 + 68.5675i) q^{69} +(359.297 + 359.297i) q^{71} +(605.446 - 605.446i) q^{73} +(127.870 - 308.706i) q^{75} +(-1263.41 + 523.321i) q^{77} -380.220i q^{79} -27.3544i q^{81} +(-235.183 + 97.4158i) q^{83} +(75.9074 - 183.257i) q^{85} +(514.706 - 514.706i) q^{87} +(-949.793 - 949.793i) q^{89} +(-567.808 - 235.194i) q^{91} +(-313.056 - 755.785i) q^{93} -116.732 q^{95} +663.589 q^{97} +(-283.031 - 683.298i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 44 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9	$ 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) $ 44 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^19 - 4 * q^21 - 324 * q^23 - 4 * q^25 + 268 * q^27 - 4 * q^29 + 752 * q^31 - 8 * q^33 + 460 * q^35 - 4 * q^37 - 596 * q^39 - 4 * q^41 - 804 * q^43 + 104 * q^45 + 1384 * q^51 + 748 * q^53 + 292 * q^55 - 4 * q^57 - 1372 * q^59 - 1828 * q^61 - 2512 * q^63 - 8 * q^65 - 2036 * q^67 - 1060 * q^69 - 220 * q^71 - 4 * q^73 + 1712 * q^75 + 1900 * q^77 - 2436 * q^83 + 496 * q^85 + 1292 * q^87 - 4 * q^89 + 3604 * q^91 - 112 * q^93 + 6088 * q^95 - 8 * q^97 + 5424 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$e\left(\frac{1}{8}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.20185	+	2.90153i	0.231297	+	0.558400i	0.996330	−	0.0855902i	$-0.0272776\pi$
$3$	1.20185	+	2.90153i	0.231297	+	0.558400i	−0.765033	+	0.643991i	$0.777278\pi$
$4$		0			0
$5$	−3.98512	−	1.65069i	−0.356440	−	0.147642i	0.197276	−	0.980348i	$-0.436791\pi$
$5$	−3.98512	−	1.65069i	−0.356440	−	0.147642i	−0.553716	+	0.832706i	$0.686791\pi$
$6$		0			0
$7$	−22.4050	−	22.4050i	−1.20976	−	1.20976i	−0.971105	−	0.238651i	$-0.923295\pi$
$7$	−22.4050	−	22.4050i	−1.20976	−	1.20976i	−0.238651	−	0.971105i	$-0.576705\pi$
$8$		0			0
$9$	12.1174	−	12.1174i	0.448794	−	0.448794i
$10$		0			0
$11$	16.5161	−	39.8735i	0.452709	−	1.09294i	−0.518578	−	0.855030i	$-0.673539\pi$
$11$	16.5161	−	39.8735i	0.452709	−	1.09294i	0.971288	−	0.237907i	$-0.0764614\pi$
$12$		0			0
$13$	17.9201	−	7.42277i	0.382320	−	0.158362i	−0.183242	−	0.983068i	$-0.558659\pi$
$13$	17.9201	−	7.42277i	0.382320	−	0.158362i	0.565562	+	0.824706i	$0.308659\pi$
$14$		0			0
$15$		−	13.5469i		−	0.233185i
$16$		0			0
$17$			45.9852i			0.656062i	0.944667	+	0.328031i	$0.106385\pi$
$17$			45.9852i			0.656062i	−0.944667	+	0.328031i	$0.893615\pi$
$18$		0			0
$19$	25.0023	−	10.3563i	0.301890	−	0.125047i	−0.226596	−	0.973989i	$-0.572760\pi$
$19$	25.0023	−	10.3563i	0.301890	−	0.125047i	0.528486	+	0.848942i	$0.322760\pi$
$20$		0			0
$21$	38.0813	−	91.9364i	0.395715	−	0.955341i
$22$		0			0
$23$	40.3415	−	40.3415i	0.365729	−	0.365729i	−0.500188	−	0.865917i	$-0.666736\pi$
$23$	40.3415	−	40.3415i	0.365729	−	0.365729i	0.865917	+	0.500188i	$0.166736\pi$
$24$		0			0
$25$	−75.2319	−	75.2319i	−0.601856	−	0.601856i
$26$		0			0
$27$	128.064	+	53.0458i	0.912812	+	0.378099i
$28$		0			0
$29$	−88.6955	−	214.130i	−0.567943	−	1.37113i	−0.903286	−	0.429039i	$-0.858852\pi$
$29$	−88.6955	−	214.130i	−0.567943	−	1.37113i	0.335344	−	0.942096i	$-0.391148\pi$
$30$		0			0
$31$	−260.478			−1.50913			−0.754567	−	0.656223i	$-0.772153\pi$
$31$	−260.478			−1.50913			−0.754567	+	0.656223i	$0.772153\pi$
$32$		0			0
$33$	135.544			0.715007
$34$		0			0
$35$	52.3029	+	126.270i	0.252594	+	0.609817i
$36$		0			0
$37$	70.4470	+	29.1801i	0.313011	+	0.129654i	0.533658	−	0.845700i	$-0.320817\pi$
$37$	70.4470	+	29.1801i	0.313011	+	0.129654i	−0.220647	+	0.975354i	$0.570817\pi$
$38$		0			0
$39$	43.0748	+	43.0748i	0.176859	+	0.176859i
$40$		0			0
$41$	−251.246	+	251.246i	−0.957026	+	0.957026i	−0.999114	−	0.0420884i	$-0.986599\pi$
$41$	−251.246	+	251.246i	−0.957026	+	0.957026i	0.0420884	+	0.999114i	$0.486599\pi$
$42$		0			0
$43$	95.7143	−	231.075i	0.339449	−	0.819501i	−0.658320	−	0.752738i	$-0.728733\pi$
$43$	95.7143	−	231.075i	0.339449	−	0.819501i	0.997769	−	0.0667635i	$-0.0212673\pi$
$44$		0			0
$45$	−68.2916	+	28.2873i	−0.226229	+	0.0937072i
$46$		0			0
$47$			15.5684i			0.0483167i	0.999708	+	0.0241584i	$0.00769059\pi$
$47$			15.5684i			0.0483167i	−0.999708	+	0.0241584i	$0.992309\pi$
$48$		0			0
$49$			660.968i			1.92702i
$50$		0			0
$51$	−133.428	+	55.2676i	−0.366345	+	0.151745i
$52$		0			0
$53$	171.815	−	414.797i	0.445294	−	1.07503i	−0.528771	−	0.848765i	$-0.677347\pi$
$53$	171.815	−	414.797i	0.445294	−	1.07503i	0.974065	−	0.226270i	$-0.0726531\pi$
$54$		0			0
$55$	−131.638	+	131.638i	−0.322728	+	0.322728i
$56$		0			0
$57$	60.0981	+	60.0981i	0.139652	+	0.139652i
$58$		0			0
$59$	−53.3294	−	22.0897i	−0.117676	−	0.0487431i	0.323069	−	0.946376i	$-0.395286\pi$
$59$	−53.3294	−	22.0897i	−0.117676	−	0.0487431i	−0.440745	+	0.897633i	$0.645286\pi$
$60$		0			0
$61$	297.690	+	718.686i	0.624840	+	1.50850i	0.845958	+	0.533250i	$0.179029\pi$
$61$	297.690	+	718.686i	0.624840	+	1.50850i	−0.221118	+	0.975247i	$0.570971\pi$
$62$		0			0
$63$	−542.983			−1.08586
$64$		0			0
$65$	−83.6667			−0.159655
$66$		0			0
$67$	377.382	+	911.080i	0.688127	+	1.66128i	0.748522	+	0.663110i	$0.230764\pi$
$67$	377.382	+	911.080i	0.688127	+	1.66128i	−0.0603949	+	0.998175i	$0.519236\pi$
$68$		0			0
$69$	165.537	+	68.5675i	0.288815	+	0.119631i
$70$		0			0
$71$	359.297	+	359.297i	0.600574	+	0.600574i	0.940465	−	0.339891i	$-0.110390\pi$
$71$	359.297	+	359.297i	0.600574	+	0.600574i	−0.339891	+	0.940465i	$0.610390\pi$
$72$		0			0
$73$	605.446	−	605.446i	0.970714	−	0.970714i	−0.0288694	−	0.999583i	$-0.509191\pi$
$73$	605.446	−	605.446i	0.970714	−	0.970714i	0.999583	+	0.0288694i	$0.00919069\pi$
$74$		0			0
$75$	127.870	−	308.706i	0.196869	−	0.475284i
$76$		0			0
$77$	−1263.41	+	523.321i	−1.86986	+	0.774520i
$78$		0			0
$79$		−	380.220i		−	0.541494i	−0.962650	−	0.270747i	$-0.912729\pi$
$79$		−	380.220i		−	0.541494i	0.962650	−	0.270747i	$-0.0872707\pi$
$80$		0			0
$81$		−	27.3544i		−	0.0375232i
$82$		0			0
$83$	−235.183	+	97.4158i	−0.311020	+	0.128829i	−0.532733	−	0.846283i	$-0.678835\pi$
$83$	−235.183	+	97.4158i	−0.311020	+	0.128829i	0.221713	+	0.975112i	$0.428835\pi$
$84$		0			0
$85$	75.9074	−	183.257i	0.0968625	−	0.233847i
$86$		0			0
$87$	514.706	−	514.706i	0.634279	−	0.634279i
$88$		0			0
$89$	−949.793	−	949.793i	−1.13121	−	1.13121i	−0.989976	−	0.141237i	$-0.954892\pi$
$89$	−949.793	−	949.793i	−1.13121	−	1.13121i	−0.141237	−	0.989976i	$-0.545108\pi$
$90$		0			0
$91$	−567.808	−	235.194i	−0.654093	−	0.270934i
$92$		0			0
$93$	−313.056	−	755.785i	−0.349058	−	0.842701i
$94$		0			0
$95$	−116.732			−0.126068
$96$		0			0
$97$	663.589			0.694611			0.347305	−	0.937752i	$-0.387097\pi$
$97$	663.589			0.694611			0.347305	+	0.937752i	$0.387097\pi$
$98$		0			0
$99$	−283.031	−	683.298i	−0.287331	−	0.693677i
$100$		0			0
$101$	937.978	+	388.523i	0.924082	+	0.382767i	0.793430	−	0.608661i	$-0.208293\pi$
$101$	937.978	+	388.523i	0.924082	+	0.382767i	0.130651	+	0.991428i	$0.458293\pi$
$102$		0			0
$103$	954.756	+	954.756i	0.913348	+	0.913348i	0.996534	−	0.0831859i	$-0.0265095\pi$
$103$	954.756	+	954.756i	0.913348	+	0.913348i	−0.0831859	+	0.996534i	$0.526510\pi$
$104$		0			0
$105$	−303.517	+	303.517i	−0.282098	+	0.282098i
$106$		0			0
$107$	172.355	−	416.102i	0.155721	−	0.375945i	−0.826694	−	0.562651i	$-0.809781\pi$
$107$	172.355	−	416.102i	0.155721	−	0.375945i	0.982416	+	0.186706i	$0.0597813\pi$
$108$		0			0
$109$	253.241	−	104.896i	0.222533	−	0.0921761i	−0.268632	−	0.963243i	$-0.586571\pi$
$109$	253.241	−	104.896i	0.222533	−	0.0921761i	0.491164	+	0.871067i	$0.336571\pi$
$110$		0			0
$111$			239.475i			0.204774i
$112$		0			0
$113$		−	1164.80i		−	0.969690i	−0.874600	−	0.484845i	$-0.838876\pi$
$113$		−	1164.80i		−	0.969690i	0.874600	−	0.484845i	$-0.161124\pi$
$114$		0			0
$115$	−227.357	+	94.1743i	−0.184358	+	0.0763635i
$116$		0			0
$117$	127.201	−	307.091i	0.100511	−	0.242655i
$118$		0			0
$119$	1030.30	−	1030.30i	0.793675	−	0.793675i
$120$		0			0
$121$	−375.954	−	375.954i	−0.282459	−	0.282459i
$122$		0			0
$123$	−1030.96	−	427.038i	−0.755760	−	0.313046i
$124$		0			0
$125$	381.960	+	922.133i	0.273308	+	0.659825i
$126$		0			0
$127$	624.520			0.436356			0.218178	−	0.975909i	$-0.429989\pi$
$127$	624.520			0.436356			0.218178	+	0.975909i	$0.429989\pi$
$128$		0			0
$129$	785.506			0.536123
$130$		0			0
$131$	−370.999	−	895.672i	−0.247438	−	0.597368i	0.750547	−	0.660817i	$-0.229790\pi$
$131$	−370.999	−	895.672i	−0.247438	−	0.597368i	−0.997985	+	0.0634490i	$0.979790\pi$
$132$		0			0
$133$	−792.208	−	328.143i	−0.516490	−	0.213937i
$134$		0			0
$135$	−422.788	−	422.788i	−0.269539	−	0.269539i
$136$		0			0
$137$	1408.41	−	1408.41i	0.878308	−	0.878308i	−0.115051	−	0.993360i	$-0.536703\pi$
$137$	1408.41	−	1408.41i	0.878308	−	0.878308i	0.993360	+	0.115051i	$0.0367032\pi$
$138$		0			0
$139$	127.491	−	307.790i	0.0777958	−	0.187816i	−0.880197	−	0.474609i	$-0.842589\pi$
$139$	127.491	−	307.790i	0.0777958	−	0.187816i	0.957992	+	0.286794i	$0.0925894\pi$
$140$		0			0
$141$	−45.1722	+	18.7110i	−0.0269801	+	0.0111755i
$142$		0			0
$143$		−	837.134i		−	0.489543i
$144$		0			0
$145$			999.742i			0.572580i
$146$		0			0
$147$	−1917.82	+	794.387i	−1.07605	+	0.445714i
$148$		0			0
$149$	−765.994	+	1849.27i	−0.421159	+	1.01677i	0.560847	+	0.827919i	$0.310476\pi$
$149$	−765.994	+	1849.27i	−0.421159	+	1.01677i	−0.982006	+	0.188849i	$0.939524\pi$
$150$		0			0
$151$	−773.796	+	773.796i	−0.417024	+	0.417024i	−0.884177	−	0.467153i	$-0.845280\pi$
$151$	−773.796	+	773.796i	−0.417024	+	0.417024i	0.467153	+	0.884177i	$0.345280\pi$
$152$		0			0
$153$	557.223	+	557.223i	0.294437	+	0.294437i
$154$		0			0
$155$	1038.04	+	429.968i	0.537916	+	0.222812i
$156$		0			0
$157$	640.806	+	1547.04i	0.325745	+	0.786417i	0.998899	+	0.0469150i	$0.0149390\pi$
$157$	640.806	+	1547.04i	0.325745	+	0.786417i	−0.673154	+	0.739502i	$0.735061\pi$
$158$		0			0
$159$	1410.04			0.703295
$160$		0			0
$161$	−1807.70			−0.884887
$162$		0			0
$163$	636.715	+	1537.17i	0.305959	+	0.738651i	0.999828	+	0.0185496i	$0.00590487\pi$
$163$	636.715	+	1537.17i	0.305959	+	0.738651i	−0.693869	+	0.720102i	$0.744095\pi$
$164$		0			0
$165$	−540.160	−	223.742i	−0.254857	−	0.105565i
$166$		0			0
$167$	−1241.93	−	1241.93i	−0.575469	−	0.575469i	0.358183	−	0.933651i	$-0.383396\pi$
$167$	−1241.93	−	1241.93i	−0.575469	−	0.575469i	−0.933651	+	0.358183i	$0.883396\pi$
$168$		0			0
$169$	−1287.48	+	1287.48i	−0.586017	+	0.586017i
$170$		0			0
$171$	177.472	−	428.455i	0.0793661	−	0.191607i
$172$		0			0
$173$	1247.56	−	516.757i	0.548268	−	0.227100i	−0.0913149	−	0.995822i	$-0.529107\pi$
$173$	1247.56	−	516.757i	0.548268	−	0.227100i	0.639583	+	0.768722i	$0.279107\pi$
$174$		0			0
$175$			3371.14i			1.45620i
$176$		0			0
$177$		−	181.286i		−	0.0769845i
$178$		0			0
$179$	985.168	−	408.070i	0.411368	−	0.170394i	−0.167395	−	0.985890i	$-0.553536\pi$
$179$	985.168	−	408.070i	0.411368	−	0.170394i	0.578763	+	0.815496i	$0.303536\pi$
$180$		0			0
$181$	−120.962	+	292.028i	−0.0496742	+	0.119924i	−0.946769	−	0.321914i	$-0.895674\pi$
$181$	−120.962	+	292.028i	−0.0496742	+	0.119924i	0.897095	+	0.441839i	$0.145674\pi$
$182$		0			0
$183$	−1727.51	+	1727.51i	−0.697822	+	0.697822i
$184$		0			0
$185$	−232.573	−	232.573i	−0.0924274	−	0.0924274i
$186$		0			0
$187$	1833.59	+	759.499i	0.717035	+	0.297006i
$188$		0			0
$189$	−1680.78	−	4057.76i	−0.646872	−	1.56169i
$190$		0			0
$191$	−584.594			−0.221465			−0.110732	−	0.993850i	$-0.535320\pi$
$191$	−584.594			−0.221465			−0.110732	+	0.993850i	$0.535320\pi$
$192$		0			0
$193$	660.360			0.246289			0.123145	−	0.992389i	$-0.460702\pi$
$193$	660.360			0.246289			0.123145	+	0.992389i	$0.460702\pi$
$194$		0			0
$195$	−100.555	−	242.762i	−0.0369277	−	0.0891514i
$196$		0			0
$197$	−827.877	−	342.918i	−0.299410	−	0.124020i	0.227921	−	0.973680i	$-0.426807\pi$
$197$	−827.877	−	342.918i	−0.299410	−	0.124020i	−0.527331	+	0.849660i	$0.676807\pi$
$198$		0			0
$199$	153.764	+	153.764i	0.0547742	+	0.0547742i	0.733963	−	0.679189i	$-0.237668\pi$
$199$	153.764	+	153.764i	0.0547742	+	0.0547742i	−0.679189	+	0.733963i	$0.737668\pi$
$200$		0			0
$201$	−2189.97	+	2189.97i	−0.768500	+	0.768500i
$202$		0			0
$203$	−2810.36	+	6784.80i	−0.971667	+	2.34581i
$204$		0			0
$205$	1415.98	−	586.516i	0.482420	−	0.199825i
$206$		0			0
$207$		−	977.671i		−	0.328274i
$208$		0			0
$209$		−	1167.97i		−	0.386557i
$210$		0			0
$211$	5199.68	−	2153.78i	1.69650	−	0.702712i	0.696606	−	0.717454i	$-0.254692\pi$
$211$	5199.68	−	2153.78i	1.69650	−	0.702712i	0.999891	+	0.0147412i	$0.00469245\pi$
$212$		0			0
$213$	−610.690	+	1474.34i	−0.196450	+	0.474272i
$214$		0			0
$215$	−762.866	+	762.866i	−0.241986	+	0.241986i
$216$		0			0
$217$	5836.00	+	5836.00i	1.82568	+	1.82568i
$218$		0			0
$219$	2484.38	+	1029.06i	0.766570	+	0.317524i
$220$		0			0
$221$	341.338	+	824.062i	0.103895	+	0.250825i
$222$		0			0
$223$	−998.982			−0.299986			−0.149993	−	0.988687i	$-0.547925\pi$
$223$	−998.982			−0.299986			−0.149993	+	0.988687i	$0.547925\pi$
$224$		0			0
$225$	−1823.24			−0.540218
$226$		0			0
$227$	−1393.39	−	3363.95i	−0.407413	−	0.983583i	−0.985816	−	0.167831i	$-0.946324\pi$
$227$	−1393.39	−	3363.95i	−0.407413	−	0.983583i	0.578402	−	0.815752i	$-0.303676\pi$
$228$		0			0
$229$	941.495	+	389.980i	0.271684	+	0.112535i	0.514366	−	0.857571i	$-0.328027\pi$
$229$	941.495	+	389.980i	0.271684	+	0.112535i	−0.242682	+	0.970106i	$0.578027\pi$
$230$		0			0
$231$	−3036.87	−	3036.87i	−0.864984	−	0.864984i
$232$		0			0
$233$	540.109	−	540.109i	0.151861	−	0.151861i	−0.627088	−	0.778949i	$-0.715753\pi$
$233$	540.109	−	540.109i	0.151861	−	0.151861i	0.778949	+	0.627088i	$0.215753\pi$
$234$		0			0
$235$	25.6986	−	62.0420i	0.00713359	−	0.0172220i
$236$		0			0
$237$	1103.22	−	456.969i	0.302371	−	0.125246i
$238$		0			0
$239$		−	2875.66i		−	0.778287i	−0.921177	−	0.389144i	$-0.872771\pi$
$239$		−	2875.66i		−	0.778287i	0.921177	−	0.389144i	$-0.127229\pi$
$240$		0			0
$241$			179.902i			0.0480851i	0.999711	+	0.0240425i	$0.00765371\pi$
$241$			179.902i			0.0480851i	−0.999711	+	0.0240425i	$0.992346\pi$
$242$		0			0
$243$	3537.10	−	1465.11i	0.933765	−	0.386778i
$244$		0			0
$245$	1091.05	−	2634.04i	0.284510	−	0.686867i
$246$		0			0
$247$	371.172	−	371.172i	0.0956158	−	0.0956158i
$248$		0			0
$249$	−565.311	−	565.311i	−0.143876	−	0.143876i
$250$		0			0
$251$	−5834.77	−	2416.84i	−1.46728	−	0.607767i	−0.501043	−	0.865422i	$-0.667050\pi$
$251$	−5834.77	−	2416.84i	−1.46728	−	0.607767i	−0.966237	+	0.257655i	$0.917050\pi$
$252$		0			0
$253$	−942.270	−	2274.84i	−0.234150	−	0.565288i
$254$		0			0
$255$	622.955			0.152984
$256$		0			0
$257$	−5382.92			−1.30653			−0.653263	−	0.757131i	$-0.726600\pi$
$257$	−5382.92			−1.30653			−0.653263	+	0.757131i	$0.726600\pi$
$258$		0			0
$259$	−924.585	−	2232.15i	−0.221818	−	0.535516i
$260$		0			0
$261$	−3669.47	−	1519.94i	−0.870247	−	0.360468i
$262$		0			0
$263$	−1725.28	−	1725.28i	−0.404508	−	0.404508i	0.475311	−	0.879818i	$-0.342336\pi$
$263$	−1725.28	−	1725.28i	−0.404508	−	0.404508i	−0.879818	+	0.475311i	$0.842336\pi$
$264$		0			0
$265$	−1369.40	+	1369.40i	−0.317441	+	0.317441i
$266$		0			0
$267$	1614.34	−	3897.37i	0.370023	−	0.893315i
$268$		0			0
$269$	7959.26	−	3296.83i	1.80403	−	0.747255i	0.819256	−	0.573428i	$-0.194387\pi$
$269$	7959.26	−	3296.83i	1.80403	−	0.747255i	0.984776	−	0.173826i	$-0.0556130\pi$
$270$		0			0
$271$		−	1049.81i		−	0.235320i	−0.993054	−	0.117660i	$-0.962461\pi$
$271$		−	1049.81i		−	0.235320i	0.993054	−	0.117660i	$-0.0375393\pi$
$272$		0			0
$273$		−	1930.18i		−	0.427912i
$274$		0			0
$275$	−4242.30	+	1757.22i	−0.930256	+	0.385325i
$276$		0			0
$277$	1433.09	−	3459.78i	0.310852	−	0.750463i	−0.688822	−	0.724931i	$-0.741872\pi$
$277$	1433.09	−	3459.78i	0.310852	−	0.750463i	0.999674	−	0.0255326i	$-0.00812818\pi$
$278$		0			0
$279$	−3156.32	+	3156.32i	−0.677291	+	0.677291i
$280$		0			0
$281$	1634.94	+	1634.94i	0.347089	+	0.347089i	0.859024	−	0.511935i	$-0.171071\pi$
$281$	1634.94	+	1634.94i	0.347089	+	0.347089i	−0.511935	+	0.859024i	$0.671071\pi$
$282$		0			0
$283$	4367.76	+	1809.19i	0.917443	+	0.380017i	0.790901	−	0.611944i	$-0.209612\pi$
$283$	4367.76	+	1809.19i	0.917443	+	0.380017i	0.126542	+	0.991961i	$0.459612\pi$
$284$		0			0
$285$	−140.295	−	338.702i	−0.0291591	−	0.0703963i
$286$		0			0
$287$	11258.3			2.31553
$288$		0			0
$289$	2798.36			0.569582
$290$		0			0
$291$	797.537	+	1925.43i	0.160661	+	0.387871i
$292$		0			0
$293$	−440.173	−	182.326i	−0.0877651	−	0.0363535i	0.338369	−	0.941014i	$-0.390125\pi$
$293$	−440.173	−	182.326i	−0.0877651	−	0.0363535i	−0.426134	+	0.904660i	$0.640125\pi$
$294$		0			0
$295$	176.061	+	176.061i	0.0347480	+	0.0347480i
$296$		0			0
$297$	4230.25	−	4230.25i	0.826477	−	0.826477i
$298$		0			0
$299$	423.480	−	1022.37i	0.0819079	−	0.197743i
$300$		0			0
$301$	−7321.71	+	3032.75i	−1.40205	+	0.580747i
$302$		0			0
$303$			3188.52i			0.604540i
$304$		0			0
$305$		−	3355.44i		−	0.629942i
$306$		0			0
$307$	−20.2214	+	8.37598i	−0.00375927	+	0.00155714i	−0.384562	−	0.923099i	$-0.625648\pi$
$307$	−20.2214	+	8.37598i	−0.00375927	+	0.00155714i	0.380803	+	0.924656i	$0.375648\pi$
$308$		0			0
$309$	−1622.78	+	3917.73i	−0.298759	+	0.721269i
$310$		0			0
$311$	390.934	−	390.934i	0.0712791	−	0.0712791i	−0.670568	−	0.741848i	$-0.733950\pi$
$311$	390.934	−	390.934i	0.0712791	−	0.0712791i	0.741848	+	0.670568i	$0.233950\pi$
$312$		0			0
$313$	−3353.24	−	3353.24i	−0.605548	−	0.605548i	0.336231	−	0.941779i	$-0.390848\pi$
$313$	−3353.24	−	3353.24i	−0.605548	−	0.605548i	−0.941779	+	0.336231i	$0.890848\pi$
$314$		0			0
$315$	2163.85	+	896.296i	0.387045	+	0.160319i
$316$		0			0
$317$	1362.73	+	3289.92i	0.241447	+	0.582904i	0.997427	−	0.0716906i	$-0.0228394\pi$
$317$	1362.73	+	3289.92i	0.241447	+	0.582904i	−0.755980	+	0.654594i	$0.772839\pi$
$318$		0			0
$319$	−10003.0			−1.75568
$320$		0			0
$321$	1414.48			0.245946
$322$		0			0
$323$	476.236	+	1149.73i	0.0820386	+	0.198059i
$324$		0			0
$325$	−1906.60	−	789.738i	−0.325412	−	0.134790i
$326$		0			0
$327$	608.717	+	608.717i	0.102942	+	0.102942i
$328$		0			0
$329$	348.810	−	348.810i	0.0584514	−	0.0584514i
$330$		0			0
$331$	−3035.45	+	7328.23i	−0.504059	+	1.21691i	0.443196	+	0.896425i	$0.353845\pi$
$331$	−3035.45	+	7328.23i	−0.504059	+	1.21691i	−0.947255	+	0.320481i	$0.896155\pi$
$332$		0			0
$333$	1207.23	−	500.049i	0.198665	−	0.0822899i
$334$		0			0
$335$		−	4253.70i		−	0.693745i
$336$		0			0
$337$		−	2973.12i		−	0.480582i	−0.970701	−	0.240291i	$-0.922757\pi$
$337$		−	2973.12i		−	0.480582i	0.970701	−	0.240291i	$-0.0772429\pi$
$338$		0			0
$339$	3379.70	−	1399.92i	0.541475	−	0.224286i
$340$		0			0
$341$	−4302.09	+	10386.2i	−0.683200	+	1.64939i
$342$		0			0
$343$	7124.07	−	7124.07i	1.12147	−	1.12147i
$344$		0			0
$345$	−546.500	−	546.500i	−0.0852828	−	0.0852828i
$346$		0			0
$347$	−1239.20	−	513.294i	−0.191711	−	0.0794094i	0.284762	−	0.958598i	$-0.408085\pi$
$347$	−1239.20	−	513.294i	−0.191711	−	0.0794094i	−0.476474	+	0.879189i	$0.658085\pi$
$348$		0			0
$349$	618.263	+	1492.62i	0.0948276	+	0.228934i	0.964175	−	0.265268i	$-0.0854604\pi$
$349$	618.263	+	1492.62i	0.0948276	+	0.228934i	−0.869347	+	0.494202i	$0.835460\pi$
$350$		0			0
$351$	2688.67			0.408862
$352$		0			0
$353$	2669.91			0.402563			0.201282	−	0.979533i	$-0.435489\pi$
$353$	2669.91			0.402563			0.201282	+	0.979533i	$0.435489\pi$
$354$		0			0
$355$	−838.754	−	2024.93i	−0.125398	−	0.302739i
$356$		0			0
$357$	4227.72	+	1751.18i	0.626763	+	0.259614i
$358$		0			0
$359$	6834.40	+	6834.40i	1.00475	+	1.00475i	0.999989	+	0.00476246i	$0.00151595\pi$
$359$	6834.40	+	6834.40i	1.00475	+	1.00475i	0.00476246	+	0.999989i	$0.498484\pi$
$360$		0			0
$361$	−4332.19	+	4332.19i	−0.631606	+	0.631606i
$362$		0			0
$363$	639.000	−	1542.68i	0.0923934	−	0.223057i
$364$		0			0
$365$	−3412.18	+	1413.37i	−0.489320	+	0.202683i
$366$		0			0
$367$			12688.1i			1.80467i	0.431034	+	0.902336i	$0.358149\pi$
$367$			12688.1i			1.80467i	−0.431034	+	0.902336i	$0.641851\pi$
$368$		0			0
$369$			6088.92i			0.859015i
$370$		0			0
$371$	−13143.0	+	5444.03i	−1.83923	+	0.761832i
$372$		0			0
$373$	921.648	−	2225.05i	0.127939	−	0.308871i	−0.846911	−	0.531735i	$-0.821540\pi$
$373$	921.648	−	2225.05i	0.127939	−	0.308871i	0.974850	+	0.222863i	$0.0715404\pi$
$374$		0			0
$375$	−2216.54	+	2216.54i	−0.305231	+	0.305231i
$376$		0			0
$377$	−3178.87	−	3178.87i	−0.434271	−	0.434271i
$378$		0			0
$379$	−13206.6	−	5470.36i	−1.78992	−	0.741407i	−0.989960	−	0.141348i	$-0.954857\pi$
$379$	−13206.6	−	5470.36i	−1.78992	−	0.741407i	−0.799955	−	0.600060i	$-0.795143\pi$
$380$		0			0
$381$	750.582	+	1812.06i	0.100928	+	0.243661i
$382$		0			0
$383$	−1489.39			−0.198706			−0.0993529	−	0.995052i	$-0.531677\pi$
$383$	−1489.39			−0.198706			−0.0993529	+	0.995052i	$0.531677\pi$
$384$		0			0
$385$	5898.68			0.780843
$386$		0			0
$387$	−1640.22	−	3959.85i	−0.215445	−	0.520130i
$388$		0			0
$389$	11998.3	+	4969.86i	1.56385	+	0.647768i	0.985753	−	0.168200i	$-0.0537954\pi$
$389$	11998.3	+	4969.86i	1.56385	+	0.647768i	0.578097	+	0.815968i	$0.303795\pi$
$390$		0			0
$391$	1855.11	+	1855.11i	0.239941	+	0.239941i
$392$		0			0
$393$	2152.93	−	2152.93i	0.276339	−	0.276339i
$394$		0			0
$395$	−627.625	+	1515.22i	−0.0799475	+	0.193010i
$396$		0			0
$397$	−8675.83	+	3593.64i	−1.09679	+	0.454307i	−0.856371	−	0.516361i	$-0.827286\pi$
$397$	−8675.83	+	3593.64i	−1.09679	+	0.454307i	−0.240423	+	0.970668i	$0.577286\pi$
$398$		0			0
$399$		−	2693.00i		−	0.337891i
$400$		0			0
$401$			3772.26i			0.469769i	0.972023	+	0.234885i	$0.0754713\pi$
$401$			3772.26i			0.469769i	−0.972023	+	0.234885i	$0.924529\pi$
$402$		0			0
$403$	−4667.80	+	1933.47i	−0.576972	+	0.238990i
$404$		0			0
$405$	−45.1537	+	109.011i	−0.00554001	+	0.0133748i
$406$		0			0
$407$	2327.03	−	2327.03i	0.283406	−	0.283406i
$408$		0			0
$409$	−9020.80	−	9020.80i	−1.09059	−	1.09059i	−0.995466	−	0.0951210i	$-0.969676\pi$
$409$	−9020.80	−	9020.80i	−1.09059	−	1.09059i	−0.0951210	−	0.995466i	$-0.530324\pi$
$410$		0			0
$411$	5779.24	+	2393.84i	0.693598	+	0.287298i
$412$		0			0
$413$	699.924	+	1689.77i	0.0833922	+	0.201327i
$414$		0			0
$415$	1098.03			0.129880
$416$		0			0
$417$	1046.29			0.122870
$418$		0			0
$419$	3925.38	+	9476.72i	0.457679	+	1.10494i	0.969335	+	0.245745i	$0.0790325\pi$
$419$	3925.38	+	9476.72i	0.457679	+	1.10494i	−0.511655	+	0.859191i	$0.670968\pi$
$420$		0			0
$421$	−4298.89	−	1780.66i	−0.497660	−	0.206138i	0.119712	−	0.992809i	$-0.461803\pi$
$421$	−4298.89	−	1780.66i	−0.497660	−	0.206138i	−0.617372	+	0.786671i	$0.711803\pi$
$422$		0			0
$423$	188.649	+	188.649i	0.0216843	+	0.0216843i
$424$		0			0
$425$	3459.56	−	3459.56i	0.394855	−	0.394855i
$426$		0			0
$427$	9432.43	−	22771.9i	1.06901	−	2.58082i
$428$		0			0
$429$	2428.97	−	1006.11i	0.273361	−	0.113230i
$430$		0			0
$431$		−	2281.76i		−	0.255008i	−0.991838	−	0.127504i	$-0.959303\pi$
$431$		−	2281.76i		−	0.255008i	0.991838	−	0.127504i	$-0.0406966\pi$
$432$		0			0
$433$			8491.53i			0.942441i	0.882016	+	0.471220i	$0.156186\pi$
$433$			8491.53i			0.942441i	−0.882016	+	0.471220i	$0.843814\pi$
$434$		0			0
$435$	−2900.78	+	1201.54i	−0.319729	+	0.132436i
$436$		0			0
$437$	590.840	−	1426.41i	0.0646767	−	0.156143i
$438$		0			0
$439$	3417.05	−	3417.05i	0.371496	−	0.371496i	−0.496526	−	0.868022i	$-0.665391\pi$
$439$	3417.05	−	3417.05i	0.371496	−	0.371496i	0.868022	+	0.496526i	$0.165391\pi$
$440$		0			0
$441$	8009.24	+	8009.24i	0.864835	+	0.864835i
$442$		0			0
$443$	984.558	+	407.817i	0.105593	+	0.0437381i	0.434855	−	0.900501i	$-0.356800\pi$
$443$	984.558	+	407.817i	0.105593	+	0.0437381i	−0.329261	+	0.944239i	$0.606800\pi$
$444$		0			0
$445$	2217.23	+	5352.86i	0.236195	+	0.570224i
$446$		0			0
$447$	−6286.34			−0.665176
$448$		0			0
$449$	−6407.14			−0.673434			−0.336717	−	0.941606i	$-0.609317\pi$
$449$	−6407.14			−0.673434			−0.336717	+	0.941606i	$0.609317\pi$
$450$		0			0
$451$	5868.44	+	14167.7i	0.612714	+	1.47922i
$452$		0			0
$453$	−3175.18	−	1315.20i	−0.329323	−	0.136410i
$454$		0			0
$455$	1874.55	+	1874.55i	0.193144	+	0.193144i
$456$		0			0
$457$	6405.90	−	6405.90i	0.655701	−	0.655701i	−0.298659	−	0.954360i	$-0.596539\pi$
$457$	6405.90	−	6405.90i	0.655701	−	0.655701i	0.954360	+	0.298659i	$0.0965393\pi$
$458$		0			0
$459$	−2439.32	+	5889.05i	−0.248057	+	0.598861i
$460$		0			0
$461$	−11209.2	+	4642.99i	−1.13246	+	0.469080i	−0.868616	−	0.495486i	$-0.834990\pi$
$461$	−11209.2	+	4642.99i	−1.13246	+	0.469080i	−0.263843	+	0.964566i	$0.584990\pi$
$462$		0			0
$463$		−	10213.2i		−	1.02515i	−0.858642	−	0.512576i	$-0.828691\pi$
$463$		−	10213.2i		−	1.02515i	0.858642	−	0.512576i	$-0.171309\pi$
$464$		0			0
$465$			3528.65i			0.351908i
$466$		0			0
$467$	8301.10	−	3438.43i	0.822546	−	0.340710i	0.0685986	−	0.997644i	$-0.478147\pi$
$467$	8301.10	−	3438.43i	0.822546	−	0.340710i	0.753948	+	0.656935i	$0.228147\pi$
$468$		0			0
$469$	11957.5	−	28868.0i	1.17728	−	2.84222i
$470$		0			0
$471$	−3718.64	+	3718.64i	−0.363792	+	0.363792i
$472$		0			0
$473$	−7632.93	−	7632.93i	−0.741992	−	0.741992i
$474$		0			0
$475$	−2660.09	−	1101.85i	−0.256954	−	0.106434i
$476$		0			0
$477$	−2944.33	−	7108.24i	−0.282624	−	0.682314i
$478$		0			0
$479$	1983.62			0.189215			0.0946075	−	0.995515i	$-0.469840\pi$
$479$	1983.62			0.189215			0.0946075	+	0.995515i	$0.469840\pi$
$480$		0			0
$481$	1479.02			0.140203
$482$		0			0
$483$	−2172.59	−	5245.10i	−0.204672	−	0.494121i
$484$		0			0
$485$	−2644.48	−	1095.38i	−0.247587	−	0.102554i
$486$		0			0
$487$	2241.43	+	2241.43i	0.208561	+	0.208561i	0.803655	−	0.595095i	$-0.202886\pi$
$487$	2241.43	+	2241.43i	0.208561	+	0.208561i	−0.595095	+	0.803655i	$0.702886\pi$
$488$		0			0
$489$	−3694.90	+	3694.90i	−0.341696	+	0.341696i
$490$		0			0
$491$	1553.51	−	3750.50i	0.142788	−	0.344720i	−0.836265	−	0.548325i	$-0.815266\pi$
$491$	1553.51	−	3750.50i	0.142788	−	0.344720i	0.979053	+	0.203605i	$0.0652657\pi$
$492$		0			0
$493$	9846.81	−	4078.68i	0.899550	−	0.372606i
$494$		0			0
$495$			3190.22i			0.289677i
$496$		0			0
$497$		−	16100.1i		−	1.45310i
$498$		0			0
$499$	8740.71	−	3620.52i	0.784144	−	0.324803i	0.0455575	−	0.998962i	$-0.485494\pi$
$499$	8740.71	−	3620.52i	0.784144	−	0.324803i	0.738587	+	0.674159i	$0.235494\pi$
$500$		0			0
$501$	2110.88	−	5096.11i	0.188238	−	0.454446i
$502$		0			0
$503$	3856.31	−	3856.31i	0.341838	−	0.341838i	−0.515220	−	0.857058i	$-0.672290\pi$
$503$	3856.31	−	3856.31i	0.341838	−	0.341838i	0.857058	+	0.515220i	$0.172290\pi$
$504$		0			0
$505$	−3096.62	−	3096.62i	−0.272867	−	0.272867i
$506$		0			0
$507$	−5283.03	−	2188.30i	−0.462776	−	0.191688i
$508$		0			0
$509$	−776.546	−	1874.75i	−0.0676224	−	0.163255i	0.886455	−	0.462814i	$-0.153160\pi$
$509$	−776.546	−	1874.75i	−0.0676224	−	0.163255i	−0.954078	+	0.299559i	$0.903160\pi$
$510$		0			0
$511$	−27130.0			−2.34865
$512$		0			0
$513$	3751.24			0.322849
$514$		0			0
$515$	−2228.81	−	5380.82i	−0.190705	−	0.460403i
$516$		0			0
$517$	620.767	+	257.130i	0.0528071	+	0.0218734i
$518$		0			0
$519$	2998.78	+	2998.78i	0.253626	+	0.253626i
$520$		0			0
$521$	−4528.26	+	4528.26i	−0.380780	+	0.380780i	−0.871383	−	0.490603i	$-0.836777\pi$
$521$	−4528.26	+	4528.26i	−0.380780	+	0.380780i	0.490603	+	0.871383i	$0.336777\pi$
$522$		0			0
$523$	1376.98	−	3324.33i	0.115127	−	0.277940i	−0.855804	−	0.517300i	$-0.826937\pi$
$523$	1376.98	−	3324.33i	0.115127	−	0.277940i	0.970931	+	0.239359i	$0.0769373\pi$
$524$		0			0
$525$	−9781.48	+	4051.62i	−0.813141	+	0.336814i
$526$		0			0
$527$		−	11978.1i		−	0.990086i
$528$		0			0
$529$			8912.13i			0.732484i
$530$		0			0
$531$	−913.887	+	378.544i	−0.0746880	+	0.0309368i
$532$		0			0
$533$	−2637.43	+	6367.31i	−0.214333	+	0.517446i
$534$		0			0
$535$	−1373.71	+	1373.71i	−0.111011	+	0.111011i
$536$		0			0
$537$	2368.06	+	2368.06i	0.190296	+	0.190296i
$538$		0			0
$539$	26355.1	+	10916.6i	2.10611	+	0.872380i
$540$		0			0
$541$	−3507.90	−	8468.82i	−0.278773	−	0.673018i	0.721029	−	0.692905i	$-0.243669\pi$
$541$	−3507.90	−	8468.82i	−0.278773	−	0.673018i	−0.999802	+	0.0198866i	$0.993669\pi$
$542$		0			0
$543$	−992.708			−0.0784552
$544$		0			0
$545$	−1182.35			−0.0929287
$546$		0			0
$547$	2578.26	+	6224.46i	0.201532	+	0.486542i	0.992042	−	0.125907i	$-0.0401842\pi$
$547$	2578.26	+	6224.46i	0.201532	+	0.486542i	−0.790510	+	0.612450i	$0.790184\pi$
$548$		0			0
$549$	12315.9	+	5101.40i	0.957429	+	0.396580i
$550$		0			0
$551$	−4435.17	−	4435.17i	−0.342912	−	0.342912i
$552$		0			0
$553$	−8518.82	+	8518.82i	−0.655076	+	0.655076i
$554$		0			0
$555$	395.299	−	954.335i	0.0302333	−	0.0729897i
$556$		0			0
$557$	8187.07	−	3391.19i	0.622796	−	0.257970i	−0.0488930	−	0.998804i	$-0.515569\pi$
$557$	8187.07	−	3391.19i	0.622796	−	0.257970i	0.671689	+	0.740834i	$0.265569\pi$
$558$		0			0
$559$		−	4851.36i		−	0.367067i
$560$		0			0
$561$			6233.04i			0.469089i
$562$		0			0
$563$	5205.06	−	2156.00i	0.389639	−	0.161394i	−0.179259	−	0.983802i	$-0.557370\pi$
$563$	5205.06	−	2156.00i	0.389639	−	0.161394i	0.568898	+	0.822408i	$0.307370\pi$
$564$		0			0
$565$	−1922.72	+	4641.86i	−0.143167	+	0.345636i
$566$		0			0
$567$	−612.876	+	612.876i	−0.0453939	+	0.0453939i
$568$		0			0
$569$	13921.3	+	13921.3i	1.02568	+	1.02568i	0.999662	+	0.0260140i	$0.00828146\pi$
$569$	13921.3	+	13921.3i	1.02568	+	1.02568i	0.0260140	+	0.999662i	$0.491719\pi$
$570$		0			0
$571$	−18276.2	−	7570.27i	−1.33947	−	0.554826i	−0.406128	−	0.913816i	$-0.633121\pi$
$571$	−18276.2	−	7570.27i	−1.33947	−	0.554826i	−0.933341	+	0.358990i	$0.883121\pi$
$572$		0			0
$573$	−702.597	−	1696.22i	−0.0512241	−	0.123666i
$574$		0			0
$575$	−6069.93			−0.440232
$576$		0			0
$577$	24309.7			1.75394			0.876972	−	0.480542i	$-0.159560\pi$
$577$	24309.7			1.75394			0.876972	+	0.480542i	$0.159560\pi$
$578$		0			0
$579$	793.657	+	1916.06i	0.0569659	+	0.137528i
$580$		0			0
$581$	7451.87	+	3086.66i	0.532109	+	0.220407i
$582$		0			0
$583$	−13701.7	−	13701.7i	−0.973356	−	0.973356i
$584$		0			0
$585$	−1013.83	+	1013.83i	−0.0716522	+	0.0716522i
$586$		0			0
$587$	−6035.14	+	14570.1i	−0.424356	+	1.02449i	0.556691	+	0.830719i	$0.312071\pi$
$587$	−6035.14	+	14570.1i	−0.424356	+	1.02449i	−0.981048	+	0.193767i	$0.937929\pi$
$588$		0			0
$589$	−6512.53	+	2697.58i	−0.455593	+	0.188713i
$590$		0			0
$591$		−	2814.25i		−	0.195876i
$592$		0			0
$593$			14290.4i			0.989608i	0.869005	+	0.494804i	$0.164760\pi$
$593$			14290.4i			0.989608i	−0.869005	+	0.494804i	$0.835240\pi$
$594$		0			0
$595$	−5806.57	+	2405.16i	−0.400078	+	0.165718i
$596$		0			0
$597$	−261.350	+	630.954i	−0.0179168	+	0.0432550i
$598$		0			0
$599$	−2752.47	+	2752.47i	−0.187751	+	0.187751i	−0.794723	−	0.606972i	$-0.792384\pi$
$599$	−2752.47	+	2752.47i	−0.187751	+	0.187751i	0.606972	+	0.794723i	$0.292384\pi$
$600$		0			0
$601$	−10680.0	−	10680.0i	−0.724871	−	0.724871i	0.244722	−	0.969593i	$-0.421303\pi$
$601$	−10680.0	−	10680.0i	−0.724871	−	0.724871i	−0.969593	+	0.244722i	$0.921303\pi$
$602$		0			0
$603$	15612.9	+	6467.06i	1.05440	+	0.436748i
$604$		0			0
$605$	877.637	+	2118.80i	0.0589769	+	0.142383i
$606$		0			0
$607$	13933.7			0.931719			0.465859	−	0.884859i	$-0.345745\pi$
$607$	13933.7			0.931719			0.465859	+	0.884859i	$0.345745\pi$
$608$		0			0
$609$	−23064.0			−1.53464
$610$		0			0
$611$	115.561	+	278.988i	0.00765153	+	0.0184724i
$612$		0			0
$613$	−20129.3	−	8337.81i	−1.32629	−	0.549366i	−0.396692	−	0.917952i	$-0.629842\pi$
$613$	−20129.3	−	8337.81i	−1.32629	−	0.549366i	−0.929594	+	0.368586i	$0.879842\pi$
$614$		0			0
$615$	3403.59	+	3403.59i	0.223164	+	0.223164i
$616$		0			0
$617$	4886.46	−	4886.46i	0.318836	−	0.318836i	−0.529484	−	0.848320i	$-0.677615\pi$
$617$	4886.46	−	4886.46i	0.318836	−	0.318836i	0.848320	+	0.529484i	$0.177615\pi$
$618$		0			0
$619$	7135.77	−	17227.3i	0.463345	−	1.11861i	−0.503670	−	0.863896i	$-0.668017\pi$
$619$	7135.77	−	17227.3i	0.463345	−	1.11861i	0.967015	−	0.254718i	$-0.0819827\pi$
$620$		0			0
$621$	7306.23	−	3026.34i	0.472124	−	0.195560i
$622$		0			0
$623$			42560.2i			2.73698i
$624$		0			0
$625$			8993.94i			0.575612i
$626$		0			0
$627$	3388.91	−	1403.73i	0.215853	−	0.0894094i
$628$		0			0
$629$	−1341.85	+	3239.52i	−0.0850608	+	0.205355i
$630$		0			0
$631$	−9484.77	+	9484.77i	−0.598388	+	0.598388i	−0.939883	−	0.341496i	$-0.889067\pi$
$631$	−9484.77	+	9484.77i	−0.598388	+	0.598388i	0.341496	+	0.939883i	$0.389067\pi$
$632$		0			0
$633$	12498.5	+	12498.5i	0.784790	+	0.784790i
$634$		0			0
$635$	−2488.79	−	1030.89i	−0.155535	−	0.0644245i
$636$		0			0
$637$	4906.21	+	11844.6i	0.305167	+	0.736737i
$638$		0			0
$639$	8707.53			0.539068
$640$		0			0
$641$	−26621.4			−1.64038			−0.820189	−	0.572092i	$-0.806132\pi$
$641$	−26621.4			−1.64038			−0.820189	+	0.572092i	$0.806132\pi$
$642$		0			0
$643$	3206.34	+	7740.79i	0.196650	+	0.474754i	0.991188	−	0.132459i	$-0.0422873\pi$
$643$	3206.34	+	7740.79i	0.196650	+	0.474754i	−0.794539	+	0.607213i	$0.792287\pi$
$644$		0			0
$645$	−3130.33	−	1296.63i	−0.191096	−	0.0791545i
$646$		0			0
$647$	11415.8	+	11415.8i	0.693664	+	0.693664i	0.963036	−	0.269373i	$-0.0868163\pi$
$647$	11415.8	+	11415.8i	0.693664	+	0.693664i	−0.269373	+	0.963036i	$0.586816\pi$
$648$		0			0
$649$	−1761.59	+	1761.59i	−0.106546	+	0.106546i
$650$		0			0
$651$	−9919.33	+	23947.4i	−0.597188	+	1.44174i
$652$		0			0
$653$	24638.3	−	10205.5i	1.47652	−	0.611596i	0.508188	−	0.861246i	$-0.330316\pi$
$653$	24638.3	−	10205.5i	1.47652	−	0.611596i	0.968336	+	0.249650i	$0.0803156\pi$
$654$		0			0
$655$			4181.76i			0.249458i
$656$		0			0
$657$		−	14672.9i		−	0.871301i
$658$		0			0
$659$	−13287.3	+	5503.77i	−0.785430	+	0.325336i	−0.739105	−	0.673591i	$-0.764751\pi$
$659$	−13287.3	+	5503.77i	−0.785430	+	0.325336i	−0.0463255	+	0.998926i	$0.514751\pi$
$660$		0			0
$661$	−1075.25	+	2595.89i	−0.0632716	+	0.152751i	−0.952353	−	0.304998i	$-0.901344\pi$
$661$	−1075.25	+	2595.89i	−0.0632716	+	0.152751i	0.889081	+	0.457749i	$0.151344\pi$
$662$		0			0
$663$	−1980.81	+	1980.81i	−0.116030	+	0.116030i
$664$		0			0
$665$	2615.38	+	2615.38i	0.152511	+	0.152511i
$666$		0			0
$667$	−12216.4	−	5060.20i	−0.709177	−	0.293751i
$668$		0			0
$669$	−1200.63	−	2898.58i	−0.0693858	−	0.167512i
$670$		0			0
$671$	33573.2			1.93156
$672$		0			0
$673$	2416.06			0.138384			0.0691920	−	0.997603i	$-0.477958\pi$
$673$	2416.06			0.138384			0.0691920	+	0.997603i	$0.477958\pi$
$674$		0			0
$675$	−5643.76	−	13625.2i	−0.321820	−	0.776942i
$676$		0			0
$677$	−10314.0	−	4272.22i	−0.585526	−	0.242533i	0.0701988	−	0.997533i	$-0.477637\pi$
$677$	−10314.0	−	4272.22i	−0.585526	−	0.242533i	−0.655724	+	0.755000i	$0.727637\pi$
$678$		0			0
$679$	−14867.7	−	14867.7i	−0.840310	−	0.840310i
$680$		0			0
$681$	8085.96	−	8085.96i	0.455000	−	0.455000i
$682$		0			0
$683$	−3410.41	+	8233.45i	−0.191062	+	0.461265i	−0.990161	−	0.139935i	$-0.955311\pi$
$683$	−3410.41	+	8233.45i	−0.191062	+	0.461265i	0.799098	+	0.601200i	$0.205311\pi$
$684$		0			0
$685$	−7937.51	+	3287.83i	−0.442740	+	0.183389i
$686$		0			0
$687$			3200.48i			0.177738i
$688$		0			0
$689$		−	8708.57i		−	0.481524i
$690$		0			0
$691$	3706.50	−	1535.28i	0.204055	−	0.0845224i	−0.278315	−	0.960490i	$-0.589776\pi$
$691$	3706.50	−	1535.28i	0.204055	−	0.0845224i	0.482370	+	0.875967i	$0.339776\pi$
$692$		0			0
$693$	−8967.98	+	21650.6i	−0.491580	+	1.18678i
$694$		0			0
$695$	−1016.13	+	1016.13i	−0.0554591	+	0.0554591i
$696$		0			0
$697$	−11553.6	−	11553.6i	−0.627868	−	0.627868i
$698$		0			0
$699$	2216.27	+	918.011i	0.119924	+	0.0496743i
$700$		0			0
$701$	−6159.04	−	14869.2i	−0.331846	−	0.801146i	−0.998446	−	0.0557306i	$-0.982251\pi$
$701$	−6159.04	−	14869.2i	−0.331846	−	0.801146i	0.666600	−	0.745415i	$-0.267749\pi$
$702$		0			0
$703$	2063.53			0.110708
$704$		0			0
$705$	210.903			0.0112668
$706$		0			0
$707$	−12310.5	−	29720.2i	−0.654859	−	1.58097i
$708$		0			0
$709$	−1554.84	−	644.034i	−0.0823598	−	0.0341145i	0.341123	−	0.940019i	$-0.389193\pi$
$709$	−1554.84	−	644.034i	−0.0823598	−	0.0341145i	−0.423483	+	0.905904i	$0.639193\pi$
$710$		0			0
$711$	−4607.29	−	4607.29i	−0.243020	−	0.243020i
$712$		0			0
$713$	−10508.0	+	10508.0i	−0.551935	+	0.551935i
$714$		0			0
$715$	−1381.85	+	3336.08i	−0.0722773	+	0.174493i
$716$		0			0
$717$	8343.81	−	3456.12i	0.434596	−	0.180016i
$718$		0			0
$719$		−	33535.5i		−	1.73945i	−0.493540	−	0.869723i	$-0.664297\pi$
$719$		−	33535.5i		−	1.73945i	0.493540	−	0.869723i	$-0.335703\pi$
$720$		0			0
$721$		−	42782.6i		−	2.20986i
$722$		0			0
$723$	−521.991	+	216.216i	−0.0268507	+	0.0111219i
$724$		0			0
$725$	−9436.67	+	22782.1i	−0.483406	+	1.16704i
$726$		0			0
$727$	15842.6	−	15842.6i	0.808210	−	0.808210i	−0.176153	−	0.984363i	$-0.556365\pi$
$727$	15842.6	−	15842.6i	0.808210	−	0.808210i	0.984363	+	0.176153i	$0.0563654\pi$
$728$		0			0
$729$	7979.90	+	7979.90i	0.405421	+	0.405421i
$730$		0			0
$731$	10626.0	+	4401.44i	0.537644	+	0.222699i
$732$		0			0
$733$	10693.1	+	25815.3i	0.538823	+	1.30083i	0.925545	+	0.378638i	$0.123607\pi$
$733$	10693.1	+	25815.3i	0.538823	+	1.30083i	−0.386722	+	0.922196i	$0.626393\pi$
$734$		0			0
$735$	8954.03			0.449353
$736$		0			0
$737$	42560.8			2.12720
$738$		0			0
$739$	2894.05	+	6986.86i	0.144059	+	0.347789i	0.979396	−	0.201949i	$-0.0647276\pi$
$739$	2894.05	+	6986.86i	0.144059	+	0.347789i	−0.835337	+	0.549738i	$0.814728\pi$
$740$		0			0
$741$	1523.06	+	630.873i	0.0755075	+	0.0312762i
$742$		0			0
$743$	−15149.2	−	15149.2i	−0.748009	−	0.748009i	0.226096	−	0.974105i	$-0.427404\pi$
$743$	−15149.2	−	15149.2i	−0.748009	−	0.748009i	−0.974105	+	0.226096i	$0.927404\pi$
$744$		0			0
$745$	6105.16	−	6105.16i	0.300236	−	0.300236i
$746$		0			0
$747$	−1669.38	+	4030.24i	−0.0817664	+	0.197401i
$748$		0			0
$749$	−13184.4	+	5461.15i	−0.643187	+	0.266417i
$750$		0			0
$751$			23633.7i			1.14834i	0.818735	+	0.574171i	$0.194676\pi$
$751$			23633.7i			1.14834i	−0.818735	+	0.574171i	$0.805324\pi$
$752$		0			0
$753$		−	19834.5i		−	0.959905i
$754$		0			0
$755$	4360.97	−	1806.37i	0.210214	−	0.0870736i
$756$		0			0
$757$	−8369.97	+	20206.9i	−0.401865	+	0.970188i	0.585348	+	0.810782i	$0.300958\pi$
$757$	−8369.97	+	20206.9i	−0.401865	+	0.970188i	−0.987213	+	0.159406i	$0.949042\pi$
$758$		0			0
$759$	5468.05	−	5468.05i	0.261499	−	0.261499i
$760$		0			0
$761$	4475.11	+	4475.11i	0.213170	+	0.213170i	0.805613	−	0.592442i	$-0.201836\pi$
$761$	4475.11	+	4475.11i	0.213170	+	0.213170i	−0.592442	+	0.805613i	$0.701836\pi$
$762$		0			0
$763$	−8024.05	−	3323.67i	−0.380721	−	0.157700i
$764$		0			0
$765$	−1300.80	−	3140.41i	−0.0614778	−	0.148420i
$766$		0			0
$767$	−1119.64			−0.0527089
$768$		0			0
$769$	−6556.87			−0.307473			−0.153736	−	0.988112i	$-0.549131\pi$
$769$	−6556.87			−0.307473			−0.153736	+	0.988112i	$0.549131\pi$
$770$		0			0
$771$	−6469.48	−	15618.7i	−0.302195	−	0.729564i
$772$		0			0
$773$	16437.5	+	6808.62i	0.764832	+	0.316804i	0.730777	−	0.682616i	$-0.239158\pi$
$773$	16437.5	+	6808.62i	0.764832	+	0.316804i	0.0340547	+	0.999420i	$0.489158\pi$
$774$		0			0
$775$	19596.2	+	19596.2i	0.908281	+	0.908281i
$776$		0			0
$777$	5365.43	−	5365.43i	0.247727	−	0.247727i
$778$		0			0
$779$	−3679.74	+	8883.69i	−0.169243	+	0.408590i
$780$		0			0
$781$	20260.6	−	8392.23i	0.928275	−	0.384504i
$782$		0			0
$783$		−	32127.2i		−	1.46633i
$784$		0			0
$785$		−	7222.93i		−	0.328404i
$786$		0			0
$787$	20223.1	−	8376.67i	0.915978	−	0.379410i	0.125636	−	0.992076i	$-0.459903\pi$
$787$	20223.1	−	8376.67i	0.915978	−	0.379410i	0.790342	+	0.612666i	$0.209903\pi$
$788$		0			0
$789$	2932.43	−	7079.50i	0.132316	−	0.319439i
$790$		0			0
$791$	−26097.3	+	26097.3i	−1.17309	+	1.17309i
$792$		0			0
$793$	10669.3	+	10669.3i	0.477777	+	0.477777i
$794$		0			0
$795$	−5619.20	−	2327.55i	−0.250682	−	0.103836i
$796$		0			0
$797$	−9378.36	−	22641.4i	−0.416811	−	1.00627i	−0.983266	−	0.182178i	$-0.941685\pi$
$797$	−9378.36	−	22641.4i	−0.416811	−	1.00627i	0.566454	−	0.824093i	$-0.308315\pi$
$798$		0			0
$799$	−715.917			−0.0316988
$800$		0			0
$801$	−23018.1			−1.01536
$802$		0			0
$803$	−14141.6	−	34140.9i	−0.621478	−	1.50038i
$804$		0			0
$805$	7203.90	+	2983.96i	0.315409	+	0.130647i
$806$		0			0
$807$	19131.7	+	19131.7i	0.834534	+	0.834534i
$808$		0			0
$809$	19542.5	−	19542.5i	0.849293	−	0.849293i	−0.140752	−	0.990045i	$-0.544952\pi$
$809$	19542.5	−	19542.5i	0.849293	−	0.849293i	0.990045	+	0.140752i	$0.0449520\pi$
$810$		0			0
$811$	−718.754	+	1735.23i	−0.0311207	+	0.0751319i	−0.938676	−	0.344801i	$-0.887946\pi$
$811$	−718.754	+	1735.23i	−0.0311207	+	0.0751319i	0.907555	+	0.419933i	$0.137946\pi$
$812$		0			0
$813$	3046.07	−	1261.72i	0.131403	−	0.0544287i
$814$		0			0
$815$		−	7176.81i		−	0.308457i
$816$		0			0
$817$		−	6768.63i		−	0.289846i
$818$		0			0
$819$	−9730.33	+	4030.43i	−0.415147	+	0.171959i
$820$		0			0
$821$	7505.22	−	18119.2i	0.319043	−	0.770237i	−0.680263	−	0.732969i	$-0.738134\pi$
$821$	7505.22	−	18119.2i	0.319043	−	0.770237i	0.999305	−	0.0372687i	$-0.0118658\pi$
$822$		0			0
$823$	21223.2	−	21223.2i	0.898901	−	0.898901i	−0.0964384	−	0.995339i	$-0.530745\pi$
$823$	21223.2	−	21223.2i	0.898901	−	0.898901i	0.995339	+	0.0964384i	$0.0307451\pi$
$824$		0			0
$825$	−10197.3	−	10197.3i	−0.430331	−	0.430331i
$826$		0			0
$827$	−38266.7	−	15850.6i	−1.60902	−	0.666479i	−0.616368	−	0.787458i	$-0.711397\pi$
$827$	−38266.7	−	15850.6i	−1.60902	−	0.666479i	−0.992655	+	0.120979i	$0.961397\pi$
$828$		0			0
$829$	204.285	+	493.187i	0.00855862	+	0.0206623i	0.928101	−	0.372328i	$-0.121440\pi$
$829$	204.285	+	493.187i	0.00855862	+	0.0206623i	−0.919542	+	0.392991i	$0.871440\pi$
$830$		0			0
$831$	11761.0			0.490958
$832$		0			0
$833$	−30394.8			−1.26424
$834$		0			0
$835$	2899.19	+	6999.27i	0.120157	+	0.290084i
$836$		0			0
$837$	−33357.8	−	13817.3i	−1.37756	−	0.570602i
$838$		0			0
$839$	−29085.3	−	29085.3i	−1.19682	−	1.19682i	−0.975112	−	0.221711i	$-0.928836\pi$
$839$	−29085.3	−	29085.3i	−1.19682	−	1.19682i	−0.221711	−	0.975112i	$-0.571164\pi$
$840$		0			0
$841$	−20739.1	+	20739.1i	−0.850345	+	0.850345i
$842$		0			0
$843$	−2778.87	+	6708.78i	−0.113534	+	0.274096i
$844$		0			0
$845$	7255.99	−	3005.53i	0.295401	−	0.122359i
$846$		0			0
$847$			16846.5i			0.683414i
$848$		0			0
$849$			14847.6i			0.600197i
$850$		0			0
$851$	4019.10	−	1664.77i	0.161896	−	0.0670593i
$852$		0			0
$853$	12547.7	−	30292.7i	0.503662	−	1.21595i	−0.443814	−	0.896119i	$-0.646375\pi$
$853$	12547.7	−	30292.7i	0.503662	−	1.21595i	0.947476	−	0.319828i	$-0.103625\pi$
$854$		0			0
$855$	−1414.49	+	1414.49i	−0.0565785	+	0.0565785i
$856$		0			0
$857$	23194.0	+	23194.0i	0.924495	+	0.924495i	0.997343	−	0.0728478i	$-0.0232087\pi$
$857$	23194.0	+	23194.0i	0.924495	+	0.924495i	−0.0728478	+	0.997343i	$0.523209\pi$
$858$		0			0
$859$	31813.5	+	13177.6i	1.26363	+	0.523415i	0.911023	−	0.412355i	$-0.135294\pi$
$859$	31813.5	+	13177.6i	1.26363	+	0.523415i	0.352612	+	0.935770i	$0.385294\pi$
$860$		0			0
$861$	13530.9	+	32666.4i	0.535576	+	1.29300i
$862$		0			0
$863$	11588.4			0.457096			0.228548	−	0.973533i	$-0.426602\pi$
$863$	11588.4			0.457096			0.228548	+	0.973533i	$0.426602\pi$
$864$		0			0
$865$	−5824.69			−0.228954
$866$		0			0
$867$	3363.22	+	8119.53i	0.131743	+	0.318055i
$868$		0			0
$869$	−15160.7	−	6279.76i	−0.591819	−	0.245140i
$870$		0			0
$871$	13525.5	+	13525.5i	0.526169	+	0.526169i
$872$		0			0
$873$	8041.00	−	8041.00i	0.311737	−	0.311737i
$874$		0			0
$875$	12102.6	−	29218.2i	0.467591	−	1.12886i
$876$		0			0
$877$	−33657.5	+	13941.4i	−1.29593	+	0.536794i	−0.920749	−	0.390156i	$-0.872421\pi$
$877$	−33657.5	+	13941.4i	−1.29593	+	0.536794i	−0.375186	+	0.926950i	$0.622421\pi$
$878$		0			0
$879$		−	1496.31i		−	0.0574165i
$880$		0			0
$881$			12045.9i			0.460656i	0.973113	+	0.230328i	$0.0739799\pi$
$881$			12045.9i			0.460656i	−0.973113	+	0.230328i	$0.926020\pi$
$882$		0			0
$883$	−875.893	+	362.807i	−0.0333818	+	0.0138272i	−0.399312	−	0.916815i	$-0.630751\pi$
$883$	−875.893	+	362.807i	−0.0333818	+	0.0138272i	0.365930	+	0.930642i	$0.380751\pi$
$884$		0			0
$885$	−299.247	+	722.445i	−0.0113662	+	0.0274404i
$886$		0			0
$887$	−34149.9	+	34149.9i	−1.29272	+	1.29272i	−0.359622	+	0.933098i	$0.617094\pi$
$887$	−34149.9	+	34149.9i	−1.29272	+	1.29272i	−0.933098	+	0.359622i	$0.882906\pi$
$888$		0			0
$889$	−13992.4	−	13992.4i	−0.527884	−	0.527884i
$890$		0			0
$891$	−1090.72	−	451.789i	−0.0410105	−	0.0169871i
$892$		0			0
$893$	161.231	+	389.245i	0.00604186	+	0.0145863i
$894$		0			0
$895$	−4599.61			−0.171785
$896$		0			0
$897$	3475.40			0.129365
$898$		0			0
$899$	23103.2	+	55776.0i	0.857102	+	2.06923i
$900$		0			0
$901$	19074.6	+	7900.94i	0.705289	+	0.292140i
$902$		0			0
$903$	−17599.2	−	17599.2i	−0.648578	−	0.648578i
$904$		0			0
$905$	964.096	−	964.096i	0.0354118	−	0.0354118i
$906$		0			0
$907$	−15871.7	+	38317.6i	−0.581048	+	1.40277i	0.310816	+	0.950470i	$0.399398\pi$
$907$	−15871.7	+	38317.6i	−0.581048	+	1.40277i	−0.891864	+	0.452304i	$0.850602\pi$
$908$		0			0
$909$	16073.8	−	6657.98i	0.586506	−	0.242939i
$910$		0			0
$911$			19522.9i			0.710015i	0.934863	+	0.355008i	$0.115522\pi$
$911$			19522.9i			0.710015i	−0.934863	+	0.355008i	$0.884478\pi$
$912$		0			0
$913$			10986.5i			0.398247i
$914$		0			0
$915$	9735.93	−	4032.76i	0.351760	−	0.145704i
$916$		0			0
$917$	−11755.3	+	28379.8i	−0.423330	+	1.02201i
$918$		0			0
$919$	15534.4	−	15534.4i	0.557597	−	0.557597i	−0.371026	−	0.928623i	$-0.620994\pi$
$919$	15534.4	−	15534.4i	0.557597	−	0.557597i	0.928623	+	0.371026i	$0.120994\pi$
$920$		0			0
$921$	−48.6064	−	48.6064i	−0.00173902	−	0.00173902i
$922$		0			0
$923$	9105.64	+	3771.68i	0.324719	+	0.134503i
$924$		0			0
$925$	−3104.59	−	7495.14i	−0.110355	−	0.266420i
$926$		0			0
$927$	23138.4			0.819811
$928$		0			0
$929$	4318.29			0.152506			0.0762532	−	0.997088i	$-0.475704\pi$
$929$	4318.29			0.152506			0.0762532	+	0.997088i	$0.475704\pi$
$930$		0			0
$931$	6845.16	+	16525.7i	0.240968	+	0.581748i
$932$		0			0
$933$	1604.15	+	664.462i	0.0562889	+	0.0233156i
$934$		0			0
$935$	−6053.39	−	6053.39i	−0.211729	−	0.211729i
$936$		0			0
$937$	−18383.8	+	18383.8i	−0.640953	+	0.640953i	−0.950790	−	0.309837i	$-0.899726\pi$
$937$	−18383.8	+	18383.8i	−0.640953	+	0.640953i	0.309837	+	0.950790i	$0.399726\pi$
$938$		0			0
$939$	5699.44	−	13759.7i	0.198077	−	0.478200i
$940$		0			0
$941$	−1124.52	+	465.790i	−0.0389566	+	0.0161364i	−0.402077	−	0.915606i	$-0.631711\pi$
$941$	−1124.52	+	465.790i	−0.0389566	+	0.0161364i	0.363120	+	0.931742i	$0.381711\pi$
$942$		0			0
$943$			20271.3i			0.700025i
$944$		0			0
$945$			18945.1i			0.652154i
$946$		0			0
$947$	9864.83	−	4086.15i	0.338505	−	0.140213i	−0.206955	−	0.978351i	$-0.566355\pi$
$947$	9864.83	−	4086.15i	0.338505	−	0.140213i	0.545459	+	0.838137i	$0.316355\pi$
$948$		0			0
$949$	6355.60	−	15343.8i	0.217399	−	0.524847i
$950$		0			0
$951$	−7908.02	+	7908.02i	−0.269648	+	0.269648i
$952$		0			0
$953$	2417.47	+	2417.47i	0.0821717	+	0.0821717i	0.746998	−	0.664826i	$-0.231494\pi$
$953$	2417.47	+	2417.47i	0.0821717	+	0.0821717i	−0.664826	+	0.746998i	$0.731494\pi$
$954$		0			0
$955$	2329.68	+	964.984i	0.0789388	+	0.0326975i
$956$		0			0
$957$	−12022.2	−	29024.1i	−0.406083	−	0.980371i
$958$		0			0
$959$	−63110.7			−2.12508
$960$		0			0
$961$	38057.6			1.27749
$962$		0			0
$963$	−2953.59	−	7130.60i	−0.0988350	−	0.238609i
$964$		0			0
$965$	−2631.62	−	1090.05i	−0.0877873	−	0.0363627i
$966$		0			0
$967$	25629.3	+	25629.3i	0.852309	+	0.852309i	0.990417	−	0.138108i	$-0.0441021\pi$
$967$	25629.3	+	25629.3i	0.852309	+	0.852309i	−0.138108	+	0.990417i	$0.544102\pi$
$968$		0			0
$969$	−2763.63	+	2763.63i	−0.0916207	+	0.0916207i
$970$		0			0
$971$	11851.2	−	28611.4i	0.391683	−	0.945607i	−0.597890	−	0.801578i	$-0.703994\pi$
$971$	11851.2	−	28611.4i	0.391683	−	0.945607i	0.989573	−	0.144029i	$-0.0460059\pi$
$972$		0			0
$973$	−9752.46	+	4039.60i	−0.321325	+	0.133097i
$974$		0			0
$975$		−	6481.20i		−	0.212887i
$976$		0			0
$977$			39097.2i			1.28028i	0.768259	+	0.640139i	$0.221123\pi$
$977$			39097.2i			1.28028i	−0.768259	+	0.640139i	$0.778877\pi$
$978$		0			0
$979$	−53558.5	+	22184.7i	−1.74845	+	0.724234i
$980$		0			0
$981$	1797.56	−	4339.70i	0.0585033	−	0.141239i
$982$		0			0
$983$	2866.32	−	2866.32i	0.0930026	−	0.0930026i	−0.659075	−	0.752077i	$-0.729052\pi$
$983$	2866.32	−	2866.32i	0.0930026	−	0.0930026i	0.752077	+	0.659075i	$0.229052\pi$
$984$		0			0
$985$	2733.14	+	2733.14i	0.0884112	+	0.0884112i
$986$		0			0
$987$	1431.30	+	592.865i	0.0461589	+	0.0191197i
$988$		0			0
$989$	−5460.64	−	13183.1i	−0.175569	−	0.423862i
$990$		0			0
$991$	−49476.9			−1.58596			−0.792979	−	0.609249i	$-0.791471\pi$
$991$	−49476.9			−1.58596			−0.792979	+	0.609249i	$0.791471\pi$
$992$		0			0
$993$	−24911.3			−0.796108
$994$		0			0
$995$	−358.952	−	866.586i	−0.0114367	−	0.0276107i
$996$		0			0
$997$	45633.0	+	18901.8i	1.44956	+	0.600427i	0.962095	−	0.272716i	$-0.0879218\pi$
$997$	45633.0	+	18901.8i	1.44956	+	0.600427i	0.487465	+	0.873143i	$0.337922\pi$
$998$		0			0
$999$	7473.84	+	7473.84i	0.236699	+	0.236699i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.g.a.113.8		44
4.3	odd	2		32.4.g.a.5.9	✓	44
8.3	odd	2		256.4.g.b.225.8		44
8.5	even	2		256.4.g.a.225.4		44
32.3	odd	8		256.4.g.b.33.8		44
32.13	even	8	inner	128.4.g.a.17.8		44
32.19	odd	8		32.4.g.a.13.9	yes	44
32.29	even	8		256.4.g.a.33.4		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.4.g.a.5.9	✓	44	4.3	odd	2
32.4.g.a.13.9	yes	44	32.19	odd	8
128.4.g.a.17.8		44	32.13	even	8	inner
128.4.g.a.113.8		44	1.1	even	1	trivial
256.4.g.a.33.4		44	32.29	even	8
256.4.g.a.225.4		44	8.5	even	2
256.4.g.b.33.8		44	32.3	odd	8
256.4.g.b.225.8		44	8.3	odd	2

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.g (of order \(8\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.20185 + 2.90153i) q^{3} +(-3.98512 - 1.65069i) q^{5} +(-22.4050 - 22.4050i) q^{7} +(12.1174 - 12.1174i) q^{9} +O(q^{10})\)	\(q+(1.20185 + 2.90153i) q^{3} +(-3.98512 - 1.65069i) q^{5} +(-22.4050 - 22.4050i) q^{7} +(12.1174 - 12.1174i) q^{9} +(16.5161 - 39.8735i) q^{11} +(17.9201 - 7.42277i) q^{13} -13.5469i q^{15} +45.9852i q^{17} +(25.0023 - 10.3563i) q^{19} +(38.0813 - 91.9364i) q^{21} +(40.3415 - 40.3415i) q^{23} +(-75.2319 - 75.2319i) q^{25} +(128.064 + 53.0458i) q^{27} +(-88.6955 - 214.130i) q^{29} -260.478 q^{31} +135.544 q^{33} +(52.3029 + 126.270i) q^{35} +(70.4470 + 29.1801i) q^{37} +(43.0748 + 43.0748i) q^{39} +(-251.246 + 251.246i) q^{41} +(95.7143 - 231.075i) q^{43} +(-68.2916 + 28.2873i) q^{45} +15.5684i q^{47} +660.968i q^{49} +(-133.428 + 55.2676i) q^{51} +(171.815 - 414.797i) q^{53} +(-131.638 + 131.638i) q^{55} +(60.0981 + 60.0981i) q^{57} +(-53.3294 - 22.0897i) q^{59} +(297.690 + 718.686i) q^{61} -542.983 q^{63} -83.6667 q^{65} +(377.382 + 911.080i) q^{67} +(165.537 + 68.5675i) q^{69} +(359.297 + 359.297i) q^{71} +(605.446 - 605.446i) q^{73} +(127.870 - 308.706i) q^{75} +(-1263.41 + 523.321i) q^{77} -380.220i q^{79} -27.3544i q^{81} +(-235.183 + 97.4158i) q^{83} +(75.9074 - 183.257i) q^{85} +(514.706 - 514.706i) q^{87} +(-949.793 - 949.793i) q^{89} +(-567.808 - 235.194i) q^{91} +(-313.056 - 755.785i) q^{93} -116.732 q^{95} +663.589 q^{97} +(-283.031 - 683.298i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 44 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9	\( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) \) 44 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 - 4 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^19 - 4 * q^21 - 324 * q^23 - 4 * q^25 + 268 * q^27 - 4 * q^29 + 752 * q^31 - 8 * q^33 + 460 * q^35 - 4 * q^37 - 596 * q^39 - 4 * q^41 - 804 * q^43 + 104 * q^45 + 1384 * q^51 + 748 * q^53 + 292 * q^55 - 4 * q^57 - 1372 * q^59 - 1828 * q^61 - 2512 * q^63 - 8 * q^65 - 2036 * q^67 - 1060 * q^69 - 220 * q^71 - 4 * q^73 + 1712 * q^75 + 1900 * q^77 - 2436 * q^83 + 496 * q^85 + 1292 * q^87 - 4 * q^89 + 3604 * q^91 - 112 * q^93 + 6088 * q^95 - 8 * q^97 + 5424 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more