LMFDB - Embedded newform 76.5.j.a.53.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,5,Mod(13,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("76.13");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	76.j (of order $18$, degree $6$, 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:	$7.85611719437$
Analytic rank:	$0$
Dimension:	$42$
Relative dimension:	$7$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			53.4
Character	$\chi$	$=$	76.53
Dual form			76.5.j.a.33.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.693233 - 0.122236i) q^{3} +(-24.2132 + 20.3173i) q^{5} +(32.5833 - 56.4359i) q^{7} +(-75.6495 + 27.5342i) q^{9} +O(q^{10})$	\(q+(0.693233 - 0.122236i) q^{3} +(-24.2132 + 20.3173i) q^{5} +(32.5833 - 56.4359i) q^{7} +(-75.6495 + 27.5342i) q^{9} +(-96.0822 - 166.419i) q^{11} +(-85.9617 - 15.1574i) q^{13} +(-14.3019 + 17.0443i) q^{15} +(-103.134 - 37.5377i) q^{17} +(-84.7252 + 350.917i) q^{19} +(15.6893 - 43.1061i) q^{21} +(-676.013 - 567.242i) q^{23} +(64.9565 - 368.387i) q^{25} +(-98.4562 + 56.8437i) q^{27} +(-495.669 - 1361.84i) q^{29} +(1387.38 + 801.005i) q^{31} +(-86.9497 - 103.623i) q^{33} +(357.679 + 2028.50i) q^{35} +2397.08i q^{37} -61.4442 q^{39} +(2312.10 - 407.686i) q^{41} +(175.985 - 147.669i) q^{43} +(1272.30 - 2203.68i) q^{45} +(344.708 - 125.464i) q^{47} +(-922.842 - 1598.41i) q^{49} +(-76.0843 - 13.4157i) q^{51} +(-2078.65 + 2477.24i) q^{53} +(5707.64 + 2077.41i) q^{55} +(-15.8397 + 253.624i) q^{57} +(-624.116 + 1714.75i) q^{59} +(-2111.67 - 1771.90i) q^{61} +(-910.993 + 5166.50i) q^{63} +(2389.36 - 1379.50i) q^{65} +(-2010.34 - 5523.36i) q^{67} +(-537.972 - 310.598i) q^{69} +(876.187 + 1044.20i) q^{71} +(-985.246 - 5587.61i) q^{73} -263.318i q^{75} -12522.7 q^{77} +(-4620.95 + 814.799i) q^{79} +(4933.97 - 4140.09i) q^{81} +(-375.320 + 650.073i) q^{83} +(3259.87 - 1186.49i) q^{85} +(-510.079 - 883.483i) q^{87} +(-7669.75 - 1352.38i) q^{89} +(-3656.33 + 4357.45i) q^{91} +(1059.69 + 385.695i) q^{93} +(-5078.21 - 10218.2i) q^{95} +(2188.55 - 6012.99i) q^{97} +(11850.8 + 9943.99i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	$ 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) $ 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{11}{18}\right)$	$1$

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
$3$	0.693233	−	0.122236i	0.0770259	−	0.0135817i	−0.135002	−	0.990845i	$-0.543104\pi$
$3$	0.693233	−	0.122236i	0.0770259	−	0.0135817i	0.212028	+	0.977264i	$0.431993\pi$
$4$		0			0
$5$	−24.2132	+	20.3173i	−0.968527	+	0.812691i	−0.982319	−	0.187214i	$-0.940054\pi$
$5$	−24.2132	+	20.3173i	−0.968527	+	0.812691i	0.0137919	+	0.999905i	$0.495610\pi$
$6$		0			0
$7$	32.5833	−	56.4359i	0.664965	−	1.15175i	−0.314330	−	0.949314i	$-0.601780\pi$
$7$	32.5833	−	56.4359i	0.664965	−	1.15175i	0.979295	−	0.202439i	$-0.0648869\pi$
$8$		0			0
$9$	−75.6495	+	27.5342i	−0.933944	+	0.339928i
$10$		0			0
$11$	−96.0822	−	166.419i	−0.794068	−	1.37537i	−0.923429	−	0.383768i	$-0.874626\pi$
$11$	−96.0822	−	166.419i	−0.794068	−	1.37537i	0.129362	−	0.991597i	$-0.458707\pi$
$12$		0			0
$13$	−85.9617	−	15.1574i	−0.508649	−	0.0896885i	−0.0865667	−	0.996246i	$-0.527590\pi$
$13$	−85.9617	−	15.1574i	−0.508649	−	0.0896885i	−0.422082	+	0.906558i	$0.638701\pi$
$14$		0			0
$15$	−14.3019	+	17.0443i	−0.0635639	+	0.0757525i
$16$		0			0
$17$	−103.134	−	37.5377i	−0.356865	−	0.129888i	0.157365	−	0.987540i	$-0.449700\pi$
$17$	−103.134	−	37.5377i	−0.356865	−	0.129888i	−0.514230	+	0.857652i	$0.671922\pi$
$18$		0			0
$19$	−84.7252	+	350.917i	−0.234696	+	0.972069i
$19$	−84.7252	+	350.917i	−0.234696	+	0.972069i
$20$		0			0
$21$	15.6893	−	43.1061i	0.0355767	−	0.0977462i
$22$		0			0
$23$	−676.013	−	567.242i	−1.27791	−	1.07229i	−0.993529	−	0.113580i	$-0.963768\pi$
$23$	−676.013	−	567.242i	−1.27791	−	1.07229i	−0.284379	−	0.958712i	$-0.591787\pi$
$24$		0			0
$25$	64.9565	−	368.387i	0.103930	−	0.589419i
$26$		0			0
$27$	−98.4562	+	56.8437i	−0.135056	+	0.0779749i
$28$		0			0
$29$	−495.669	−	1361.84i	−0.589380	−	1.61931i	−0.771640	−	0.636059i	$-0.780564\pi$
$29$	−495.669	−	1361.84i	−0.589380	−	1.61931i	0.182260	−	0.983250i	$-0.441659\pi$
$30$		0			0
$31$	1387.38	+	801.005i	1.44368	+	0.833512i	0.998093	−	0.0617246i	$-0.0196601\pi$
$31$	1387.38	+	801.005i	1.44368	+	0.833512i	0.445592	+	0.895236i	$0.352993\pi$
$32$		0			0
$33$	−86.9497	−	103.623i	−0.0798436	−	0.0951539i
$34$		0			0
$35$	357.679	+	2028.50i	0.291983	+	1.65592i
$36$		0			0
$37$			2397.08i			1.75097i	0.483245	+	0.875485i	$0.339458\pi$
$37$			2397.08i			1.75097i	−0.483245	+	0.875485i	$0.660542\pi$
$38$		0			0
$39$	−61.4442			−0.0403973
$40$		0			0
$41$	2312.10	−	407.686i	1.37543	−	0.242526i	0.563422	−	0.826169i	$-0.309485\pi$
$41$	2312.10	−	407.686i	1.37543	−	0.242526i	0.812010	+	0.583643i	$0.198373\pi$
$42$		0			0
$43$	175.985	−	147.669i	0.0951785	−	0.0798642i	−0.593957	−	0.804497i	$-0.702435\pi$
$43$	175.985	−	147.669i	0.0951785	−	0.0798642i	0.689135	+	0.724633i	$0.257991\pi$
$44$		0			0
$45$	1272.30	−	2203.68i	0.628294	−	1.08824i
$46$		0			0
$47$	344.708	−	125.464i	0.156047	−	0.0567966i	−0.262816	−	0.964846i	$-0.584651\pi$
$47$	344.708	−	125.464i	0.156047	−	0.0567966i	0.418863	+	0.908050i	$0.362429\pi$
$48$		0			0
$49$	−922.842	−	1598.41i	−0.384357	−	0.665726i
$50$		0			0
$51$	−76.0843	−	13.4157i	−0.0292519	−	0.00515791i
$52$		0			0
$53$	−2078.65	+	2477.24i	−0.739996	+	0.881892i	−0.996409	−	0.0846749i	$-0.973015\pi$
$53$	−2078.65	+	2477.24i	−0.739996	+	0.881892i	0.256413	+	0.966567i	$0.417459\pi$
$54$		0			0
$55$	5707.64	+	2077.41i	1.88682	+	0.686748i
$56$		0			0
$57$	−15.8397	+	253.624i	−0.00487526	+	0.0780620i
$58$		0			0
$59$	−624.116	+	1714.75i	−0.179292	+	0.492602i	−0.996486	−	0.0837619i	$-0.973306\pi$
$59$	−624.116	+	1714.75i	−0.179292	+	0.492602i	0.817194	+	0.576363i	$0.195529\pi$
$60$		0			0
$61$	−2111.67	−	1771.90i	−0.567502	−	0.476191i	0.313314	−	0.949650i	$-0.398561\pi$
$61$	−2111.67	−	1771.90i	−0.567502	−	0.476191i	−0.880816	+	0.473459i	$0.843005\pi$
$62$		0			0
$63$	−910.993	+	5166.50i	−0.229527	+	1.30171i
$64$		0			0
$65$	2389.36	−	1379.50i	0.565529	−	0.326509i
$66$		0			0
$67$	−2010.34	−	5523.36i	−0.447837	−	1.23042i	−0.934227	−	0.356680i	$-0.883909\pi$
$67$	−2010.34	−	5523.36i	−0.447837	−	1.23042i	0.486390	−	0.873742i	$-0.338313\pi$
$68$		0			0
$69$	−537.972	−	310.598i	−0.112996	−	0.0652380i
$70$		0			0
$71$	876.187	+	1044.20i	0.173812	+	0.207141i	0.845917	−	0.533315i	$-0.179054\pi$
$71$	876.187	+	1044.20i	0.173812	+	0.207141i	−0.672105	+	0.740456i	$0.734609\pi$
$72$		0			0
$73$	−985.246	−	5587.61i	−0.184884	−	1.04853i	−0.926105	−	0.377267i	$-0.876864\pi$
$73$	−985.246	−	5587.61i	−0.184884	−	1.04853i	0.741221	−	0.671261i	$-0.234247\pi$
$74$		0			0
$75$		−	263.318i		−	0.0468121i
$76$		0			0
$77$	−12522.7			−2.11211
$78$		0			0
$79$	−4620.95	+	814.799i	−0.740419	+	0.130556i	−0.531121	−	0.847296i	$-0.678229\pi$
$79$	−4620.95	+	814.799i	−0.740419	+	0.130556i	−0.209298	+	0.977852i	$0.567118\pi$
$80$		0			0
$81$	4933.97	−	4140.09i	0.752014	−	0.631015i
$82$		0			0
$83$	−375.320	+	650.073i	−0.0544810	+	0.0943639i	−0.891980	−	0.452075i	$-0.850684\pi$
$83$	−375.320	+	650.073i	−0.0544810	+	0.0943639i	0.837499	+	0.546439i	$0.184017\pi$
$84$		0			0
$85$	3259.87	−	1186.49i	0.451192	−	0.164221i
$86$		0			0
$87$	−510.079	−	883.483i	−0.0673906	−	0.116724i
$88$		0			0
$89$	−7669.75	−	1352.38i	−0.968281	−	0.170734i	−0.332925	−	0.942953i	$-0.608036\pi$
$89$	−7669.75	−	1352.38i	−0.968281	−	0.170734i	−0.635356	+	0.772219i	$0.719147\pi$
$90$		0			0
$91$	−3656.33	+	4357.45i	−0.441533	+	0.526198i
$92$		0			0
$93$	1059.69	+	385.695i	0.122522	+	0.0445942i
$94$		0			0
$95$	−5078.21	−	10218.2i	−0.562682	−	1.13221i
$96$		0			0
$97$	2188.55	−	6012.99i	0.232602	−	0.639068i	−0.767396	−	0.641173i	$-0.778448\pi$
$97$	2188.55	−	6012.99i	0.232602	−	0.639068i	0.999998	+	0.00210552i	$0.000670207\pi$
$98$		0			0
$99$	11850.8	+	9943.99i	1.20914	+	1.01459i
$100$		0			0
$101$	−683.910	+	3878.65i	−0.0670435	+	0.380222i	0.932762	+	0.360493i	$0.117392\pi$
$101$	−683.910	+	3878.65i	−0.0670435	+	0.380222i	−0.999805	+	0.0197293i	$0.993720\pi$
$102$		0			0
$103$	−10478.4	+	6049.68i	−0.987685	+	0.570240i	−0.904582	−	0.426301i	$-0.859817\pi$
$103$	−10478.4	+	6049.68i	−0.987685	+	0.570240i	−0.0831035	+	0.996541i	$0.526483\pi$
$104$		0			0
$105$	495.909	+	1362.50i	0.0449804	+	0.123583i
$106$		0			0
$107$	−66.4411	−	38.3598i	−0.00580322	−	0.00335049i	0.497096	−	0.867696i	$-0.334400\pi$
$107$	−66.4411	−	38.3598i	−0.00580322	−	0.00335049i	−0.502899	+	0.864345i	$0.667733\pi$
$108$		0			0
$109$	4428.59	+	5277.79i	0.372746	+	0.444221i	0.919511	−	0.393065i	$-0.128585\pi$
$109$	4428.59	+	5277.79i	0.372746	+	0.444221i	−0.546765	+	0.837286i	$0.684141\pi$
$110$		0			0
$111$	293.008	+	1661.73i	0.0237812	+	0.134870i
$112$		0			0
$113$		−	18412.6i		−	1.44197i	−0.692948	−	0.720987i	$-0.743689\pi$
$113$		−	18412.6i		−	1.44197i	0.692948	−	0.720987i	$-0.256311\pi$
$114$		0			0
$115$	27893.2			2.10913
$116$		0			0
$117$	6920.30	−	1220.24i	0.505537	−	0.0891399i
$118$		0			0
$119$	−5478.92	+	4597.36i	−0.386902	+	0.324649i
$120$		0			0
$121$	−11143.1	+	19300.4i	−0.761087	+	1.31824i
$122$		0			0
$123$	1552.99	−	565.243i	0.102650	−	0.0373615i
$124$		0			0
$125$	−3965.71	−	6868.81i	−0.253805	−	0.439604i
$126$		0			0
$127$	9768.85	+	1722.51i	0.605670	+	0.106796i	0.468069	−	0.883692i	$-0.344950\pi$
$127$	9768.85	+	1722.51i	0.605670	+	0.106796i	0.137601	+	0.990488i	$0.456061\pi$
$128$		0			0
$129$	103.948	−	123.881i	0.00624651	−	0.00744430i
$130$		0			0
$131$	−6000.91	−	2184.15i	−0.349683	−	0.127274i	0.161206	−	0.986921i	$-0.448462\pi$
$131$	−6000.91	−	2184.15i	−0.349683	−	0.127274i	−0.510889	+	0.859647i	$0.670684\pi$
$132$		0			0
$133$	17043.7	+	16215.6i	0.963519	+	0.916703i
$134$		0			0
$135$	1229.03	−	3376.73i	0.0674364	−	0.185280i
$136$		0			0
$137$	3234.82	+	2714.34i	0.172349	+	0.144618i	0.724882	−	0.688873i	$-0.241894\pi$
$137$	3234.82	+	2714.34i	0.172349	+	0.144618i	−0.552533	+	0.833491i	$0.686339\pi$
$138$		0			0
$139$	2765.31	−	15682.9i	0.143125	−	0.811700i	−0.825729	−	0.564067i	$-0.809236\pi$
$139$	2765.31	−	15682.9i	0.143125	−	0.811700i	0.968854	−	0.247634i	$-0.0796529\pi$
$140$		0			0
$141$	223.627	−	129.111i	0.0112483	−	0.00649420i
$142$		0			0
$143$	5736.91	+	15762.0i	0.280547	+	0.770797i
$144$		0			0
$145$	39670.6	+	22903.8i	1.88683	+	1.08936i
$146$		0			0
$147$	−835.127	−	995.265i	−0.0386472	−	0.0460579i
$148$		0			0
$149$	2527.38	+	14333.5i	0.113841	+	0.645622i	0.987318	+	0.158757i	$0.0507486\pi$
$149$	2527.38	+	14333.5i	0.113841	+	0.645622i	−0.873477	+	0.486865i	$0.838140\pi$
$150$		0			0
$151$		−	29961.2i		−	1.31403i	−0.753878	−	0.657014i	$-0.771819\pi$
$151$		−	29961.2i		−	1.31403i	0.753878	−	0.657014i	$-0.228181\pi$
$152$		0			0
$153$	8835.60			0.377445
$154$		0			0
$155$	−49867.1	+	8792.92i	−2.07564	+	0.365991i
$156$		0			0
$157$	9032.84	−	7579.45i	0.366459	−	0.307495i	−0.440900	−	0.897556i	$-0.645341\pi$
$157$	9032.84	−	7579.45i	0.366459	−	0.307495i	0.807359	+	0.590061i	$0.200896\pi$
$158$		0			0
$159$	−1138.18	+	1971.39i	−0.0450212	+	0.0779790i
$160$		0			0
$161$	−54039.6	+	19668.8i	−2.08478	+	0.758798i
$162$		0			0
$163$	−2502.29	−	4334.09i	−0.0941807	−	0.163126i	0.815086	−	0.579340i	$-0.196690\pi$
$163$	−2502.29	−	4334.09i	−0.0941807	−	0.163126i	−0.909266	+	0.416215i	$0.863356\pi$
$164$		0			0
$165$	4210.66	+	742.453i	0.154661	+	0.0272710i
$166$		0			0
$167$	24384.0	−	29059.7i	0.874323	−	1.04198i	−0.124439	−	0.992227i	$-0.539713\pi$
$167$	24384.0	−	29059.7i	0.874323	−	1.04198i	0.998762	−	0.0497499i	$-0.0158424\pi$
$168$		0			0
$169$	−19678.9	−	7162.53i	−0.689013	−	0.250780i
$170$		0			0
$171$	−3252.79	−	28879.5i	−0.111241	−	0.987638i
$172$		0			0
$173$	1148.94	−	3156.69i	0.0383889	−	0.105473i	−0.919017	−	0.394217i	$-0.871016\pi$
$173$	1148.94	−	3156.69i	0.0383889	−	0.105473i	0.957406	+	0.288745i	$0.0932379\pi$
$174$		0			0
$175$	−18673.7	−	15669.1i	−0.609755	−	0.511645i
$176$		0			0
$177$	−223.055	+	1265.01i	−0.00711976	+	0.0403782i
$178$		0			0
$179$	4338.08	−	2504.59i	0.135392	−	0.0781684i	−0.430774	−	0.902460i	$-0.641759\pi$
$179$	4338.08	−	2504.59i	0.135392	−	0.0781684i	0.566166	+	0.824291i	$0.308426\pi$
$180$		0			0
$181$	4371.44	+	12010.4i	0.133434	+	0.366608i	0.988358	−	0.152146i	$-0.0486184\pi$
$181$	4371.44	+	12010.4i	0.133434	+	0.366608i	−0.854924	+	0.518754i	$0.826396\pi$
$182$		0			0
$183$	−1680.47	−	970.221i	−0.0501798	−	0.0289713i
$184$		0			0
$185$	−48702.1	−	58040.9i	−1.42300	−	1.69586i
$186$		0			0
$187$	3662.34	+	20770.2i	0.104731	+	0.593960i
$188$		0			0
$189$			7408.62i			0.207402i
$190$		0			0
$191$	17413.4			0.477327			0.238664	−	0.971102i	$-0.423291\pi$
$191$	17413.4			0.477327			0.238664	+	0.971102i	$0.423291\pi$
$192$		0			0
$193$	30690.7	−	5411.60i	0.823933	−	0.145282i	0.254242	−	0.967141i	$-0.418174\pi$
$193$	30690.7	−	5411.60i	0.823933	−	0.145282i	0.569691	+	0.821859i	$0.307063\pi$
$194$		0			0
$195$	1487.76	−	1248.38i	0.0391258	−	0.0328305i
$196$		0			0
$197$	−23799.8	+	41222.5i	−0.613255	+	1.06219i	0.377433	+	0.926037i	$0.376807\pi$
$197$	−23799.8	+	41222.5i	−0.613255	+	1.06219i	−0.990688	+	0.136152i	$0.956526\pi$
$198$		0			0
$199$	−49214.5	+	17912.6i	−1.24276	+	0.452327i	−0.877948	−	0.478755i	$-0.841088\pi$
$199$	−49214.5	+	17912.6i	−1.24276	+	0.452327i	−0.364810	+	0.931082i	$0.618866\pi$
$200$		0			0
$201$	−2068.78	−	3583.24i	−0.0512063	−	0.0886919i
$202$		0			0
$203$	−93007.2	−	16399.7i	−2.25696	−	0.397963i
$204$		0			0
$205$	−47700.3	+	56847.0i	−1.13505	+	1.35269i
$206$		0			0
$207$	66758.6	+	24298.1i	1.55800	+	0.567064i
$208$		0			0
$209$	66539.9	−	19617.0i	1.52331	−	0.449096i
$210$		0			0
$211$	15504.7	−	42598.8i	0.348255	−	0.956824i	−0.634664	−	0.772788i	$-0.718862\pi$
$211$	15504.7	−	42598.8i	0.348255	−	0.956824i	0.982920	−	0.184036i	$-0.0589163\pi$
$212$		0			0
$213$	735.040	+	616.772i	0.0162014	+	0.0135946i
$214$		0			0
$215$	−1260.93	+	7151.07i	−0.0272780	+	0.154701i
$216$		0			0
$217$	90410.9	−	52198.7i	1.92000	−	1.10851i
$218$		0			0
$219$	−1366.01	−	3753.08i	−0.0284817	−	0.0782528i
$220$		0			0
$221$	8296.60	+	4790.04i	0.169870	+	0.0980742i
$222$		0			0
$223$	−30355.7	−	36176.5i	−0.610422	−	0.727473i	0.368970	−	0.929441i	$-0.379711\pi$
$223$	−30355.7	−	36176.5i	−0.610422	−	0.727473i	−0.979392	+	0.201969i	$0.935266\pi$
$224$		0			0
$225$	5229.29	+	29656.8i	0.103295	+	0.585813i
$226$		0			0
$227$		−	12826.9i		−	0.248926i	−0.992224	−	0.124463i	$-0.960279\pi$
$227$		−	12826.9i		−	0.248926i	0.992224	−	0.124463i	$-0.0397207\pi$
$228$		0			0
$229$	−70554.2			−1.34540			−0.672701	−	0.739915i	$-0.734866\pi$
$229$	−70554.2			−1.34540			−0.672701	+	0.739915i	$0.734866\pi$
$230$		0			0
$231$	−8681.15	+	1530.72i	−0.162687	+	0.0286861i
$232$		0			0
$233$	17559.4	−	14734.1i	0.323443	−	0.271401i	−0.466579	−	0.884480i	$-0.654514\pi$
$233$	17559.4	−	14734.1i	0.323443	−	0.271401i	0.790022	+	0.613079i	$0.210069\pi$
$234$		0			0
$235$	−5797.41	+	10041.4i	−0.104978	+	0.181827i
$236$		0			0
$237$	−3103.80	+	1129.69i	−0.0552582	+	0.0201124i
$238$		0			0
$239$	21648.2	+	37495.7i	0.378988	+	0.656426i	0.990915	−	0.134488i	$-0.0429388\pi$
$239$	21648.2	+	37495.7i	0.378988	+	0.656426i	−0.611927	+	0.790914i	$0.709606\pi$
$240$		0			0
$241$	38715.2	+	6826.54i	0.666573	+	0.117535i	0.496688	−	0.867929i	$-0.334549\pi$
$241$	38715.2	+	6826.54i	0.666573	+	0.117535i	0.169885	+	0.985464i	$0.445660\pi$
$242$		0			0
$243$	8833.55	−	10527.4i	0.149597	−	0.178283i
$244$		0			0
$245$	54820.2	+	19952.9i	0.913290	+	0.332410i
$246$		0			0
$247$	12602.1	−	28881.2i	0.206561	−	0.473392i
$248$		0			0
$249$	−180.722	+	496.529i	−0.00291482	+	0.00800841i
$250$		0			0
$251$	48070.5	+	40336.0i	0.763012	+	0.640243i	0.938909	−	0.344165i	$-0.111838\pi$
$251$	48070.5	+	40336.0i	0.763012	+	0.640243i	−0.175897	+	0.984409i	$0.556282\pi$
$252$		0			0
$253$	−29447.2	+	167003.i	−0.460048	+	2.60906i
$254$		0			0
$255$	2114.81	−	1220.99i	0.0325231	−	0.0187772i
$256$		0			0
$257$	−27288.3	−	74974.0i	−0.413152	−	1.13513i	−0.955505	−	0.294976i	$-0.904688\pi$
$257$	−27288.3	−	74974.0i	−0.413152	−	1.13513i	0.542352	−	0.840151i	$-0.317534\pi$
$258$		0			0
$259$	135281.	+	78104.7i	2.01669	+	1.16433i
$260$		0			0
$261$	74994.2	+	89374.6i	1.10090	+	1.31200i
$262$		0			0
$263$	9564.63	+	54243.7i	0.138279	+	0.784220i	0.972520	+	0.232819i	$0.0747951\pi$
$263$	9564.63	+	54243.7i	0.138279	+	0.784220i	−0.834241	+	0.551400i	$0.814094\pi$
$264$		0			0
$265$		−	102214.i		−	1.45552i
$266$		0			0
$267$	−5482.24			−0.0769016
$268$		0			0
$269$	−31645.8	+	5580.00i	−0.437332	+	0.0771134i	−0.387979	−	0.921668i	$-0.626827\pi$
$269$	−31645.8	+	5580.00i	−0.437332	+	0.0771134i	−0.0493523	+	0.998781i	$0.515716\pi$
$270$		0			0
$271$	40261.4	−	33783.3i	0.548214	−	0.460006i	−0.326121	−	0.945328i	$-0.605742\pi$
$271$	40261.4	−	33783.3i	0.548214	−	0.460006i	0.874336	+	0.485321i	$0.161297\pi$
$272$		0			0
$273$	−2002.06	+	3467.66i	−0.0268628	+	0.0465277i
$274$		0			0
$275$	−67547.8	+	24585.4i	−0.893194	+	0.325096i
$276$		0			0
$277$	−64329.5	−	111422.i	−0.838399	−	1.45215i	−0.891233	−	0.453546i	$-0.850159\pi$
$277$	−64329.5	−	111422.i	−0.838399	−	1.45215i	0.0528337	−	0.998603i	$-0.483175\pi$
$278$		0			0
$279$	−127010.	−	22395.2i	−1.63165	−	0.287705i
$280$		0			0
$281$	18278.5	−	21783.4i	0.231487	−	0.275876i	−0.637780	−	0.770219i	$-0.720147\pi$
$281$	18278.5	−	21783.4i	0.231487	−	0.275876i	0.869267	+	0.494343i	$0.164591\pi$
$282$		0			0
$283$	105516.	+	38404.9i	1.31749	+	0.479527i	0.902653	−	0.430369i	$-0.141616\pi$
$283$	105516.	+	38404.9i	1.31749	+	0.479527i	0.414837	+	0.909896i	$0.363839\pi$
$284$		0			0
$285$	−4769.41	−	6462.85i	−0.0587185	−	0.0795673i
$286$		0			0
$287$	52327.7	−	143769.i	0.635284	−	1.74543i
$288$		0			0
$289$	−54753.3	−	45943.4i	−0.655563	−	0.550082i
$290$		0			0
$291$	782.172	−	4435.92i	0.00923669	−	0.0523839i
$292$		0			0
$293$	−115335.	+	66588.6i	−1.34346	+	0.775647i	−0.987314	−	0.158782i	$-0.949243\pi$
$293$	−115335.	+	66588.6i	−1.34346	+	0.775647i	−0.356147	+	0.934430i	$0.615910\pi$
$294$		0			0
$295$	−19727.1	−	54199.8i	−0.226683	−	0.622807i
$296$		0			0
$297$	18919.8	+	10923.3i	0.214488	+	0.123835i
$298$		0			0
$299$	49513.3	+	59007.7i	0.553834	+	0.660034i
$300$		0			0
$301$	−2599.66	−	14743.4i	−0.0286935	−	0.162729i
$302$		0			0
$303$			2772.41i			0.0301975i
$304$		0			0
$305$	87130.6			0.936637
$306$		0			0
$307$	−5220.79	+	920.566i	−0.0553936	+	0.00976738i	−0.201276	−	0.979535i	$-0.564509\pi$
$307$	−5220.79	+	920.566i	−0.0553936	+	0.00976738i	0.145883	+	0.989302i	$0.453398\pi$
$308$		0			0
$309$	−6524.45	+	5474.66i	−0.0683324	+	0.0573377i
$310$		0			0
$311$	−74776.9	+	129517.i	−0.773120	+	1.33908i	0.162725	+	0.986671i	$0.447972\pi$
$311$	−74776.9	+	129517.i	−0.773120	+	1.33908i	−0.935845	+	0.352411i	$0.885362\pi$
$312$		0			0
$313$	30999.8	−	11283.0i	0.316424	−	0.115169i	−0.178926	−	0.983863i	$-0.557262\pi$
$313$	30999.8	−	11283.0i	0.316424	−	0.115169i	0.495350	+	0.868694i	$0.335040\pi$
$314$		0			0
$315$	−82911.1	−	143606.i	−0.835587	−	1.44728i
$316$		0			0
$317$	−407.501	−	71.8534i	−0.00405518	−	0.000715038i	0.171620	−	0.985163i	$-0.445100\pi$
$317$	−407.501	−	71.8534i	−0.00405518	−	0.000715038i	−0.175675	+	0.984448i	$0.556211\pi$
$318$		0			0
$319$	−179011.	+	213337.i	−1.75913	+	2.09646i
$320$		0			0
$321$	−50.7481	−	18.4708i	−0.000492504	−	0.000179257i
$322$		0			0
$323$	21910.7	−	33011.1i	0.210015	−	0.316413i
$324$		0			0
$325$	−11167.5	+	30682.6i	−0.105728	+	0.290486i
$326$		0			0
$327$	3715.18	+	3117.41i	0.0347444	+	0.0291540i
$328$		0			0
$329$	4151.08	−	23541.9i	0.0383504	−	0.217496i
$330$		0			0
$331$	−114465.	+	66086.3i	−1.04476	+	0.603191i	−0.921177	−	0.389143i	$-0.872771\pi$
$331$	−114465.	+	66086.3i	−1.04476	+	0.603191i	−0.123581	+	0.992335i	$0.539438\pi$
$332$		0			0
$333$	−66001.5	−	181338.i	−0.595204	−	1.63531i
$334$		0			0
$335$	160896.	+	92893.5i	1.43369	+	0.827744i
$336$		0			0
$337$	−111287.	−	132626.i	−0.979905	−	1.16780i	−0.985817	−	0.167822i	$-0.946326\pi$
$337$	−111287.	−	132626.i	−0.979905	−	1.16780i	0.00591259	−	0.999983i	$-0.498118\pi$
$338$		0			0
$339$	−2250.67	−	12764.2i	−0.0195845	−	0.111069i
$340$		0			0
$341$		−	307849.i		−	2.64746i
$342$		0			0
$343$	36188.1			0.307594
$344$		0			0
$345$	19336.5	−	3409.55i	0.162458	−	0.0286457i
$346$		0			0
$347$	30558.2	−	25641.4i	0.253787	−	0.212953i	−0.507014	−	0.861938i	$-0.669251\pi$
$347$	30558.2	−	25641.4i	0.253787	−	0.212953i	0.760801	+	0.648985i	$0.224806\pi$
$348$		0			0
$349$	−50857.7	+	88088.0i	−0.417547	+	0.723213i	−0.995692	−	0.0927212i	$-0.970443\pi$
$349$	−50857.7	+	88088.0i	−0.417547	+	0.723213i	0.578145	+	0.815934i	$0.303777\pi$
$350$		0			0
$351$	9325.06	−	3394.04i	0.0756898	−	0.0275488i
$352$		0			0
$353$	−69157.7	−	119785.i	−0.554998	−	0.961284i	−0.997904	−	0.0647160i	$-0.979386\pi$
$353$	−69157.7	−	119785.i	−0.554998	−	0.961284i	0.442906	−	0.896568i	$-0.353947\pi$
$354$		0			0
$355$	−42430.5	−	7481.65i	−0.336684	−	0.0593664i
$356$		0			0
$357$	−3236.21	+	3856.76i	−0.0253922	+	0.0302612i
$358$		0			0
$359$	44126.7	+	16060.8i	0.342384	+	0.124617i	0.507488	−	0.861659i	$-0.330574\pi$
$359$	44126.7	+	16060.8i	0.342384	+	0.124617i	−0.165105	+	0.986276i	$0.552796\pi$
$360$		0			0
$361$	−115964.	−	59463.0i	−0.889836	−	0.456281i
$362$		0			0
$363$	−5365.55	+	14741.7i	−0.0407194	+	0.111876i
$364$		0			0
$365$	137381.	+	115276.i	1.03119	+	0.865275i
$366$		0			0
$367$	1580.16	−	8961.54i	0.0117319	−	0.0665351i	−0.978380	−	0.206817i	$-0.933690\pi$
$367$	1580.16	−	8961.54i	0.0117319	−	0.0665351i	0.990112	+	0.140282i	$0.0448008\pi$
$368$		0			0
$369$	−163684.	+	94503.0i	−1.20214	+	0.694053i
$370$		0			0
$371$	72075.9	+	198027.i	0.523651	+	1.43872i
$372$		0			0
$373$	190559.	+	110019.i	1.36966	+	0.790771i	0.990884	−	0.134719i	$-0.0430133\pi$
$373$	190559.	+	110019.i	1.36966	+	0.790771i	0.378771	+	0.925490i	$0.376347\pi$
$374$		0			0
$375$	−3588.77	−	4276.93i	−0.0255202	−	0.0304138i
$376$		0			0
$377$	21966.6	+	124579.i	0.154554	+	0.876521i
$378$		0			0
$379$		−	2519.18i		−	0.0175380i	−0.999962	−	0.00876900i	$-0.997209\pi$
$379$		−	2519.18i		−	0.0175380i	0.999962	−	0.00876900i	$-0.00279130\pi$
$380$		0			0
$381$	6982.64			0.0481027
$382$		0			0
$383$	−82852.9	+	14609.2i	−0.564820	+	0.0995930i	−0.448765	−	0.893650i	$-0.648136\pi$
$383$	−82852.9	+	14609.2i	−0.564820	+	0.0995930i	−0.116055	+	0.993243i	$0.537025\pi$
$384$		0			0
$385$	303214.	−	254427.i	2.04564	−	1.71649i
$386$		0			0
$387$	−9247.23	+	16016.7i	−0.0617433	+	0.106943i
$388$		0			0
$389$	−190672.	+	69399.0i	−1.26005	+	0.458621i	−0.883786	−	0.467891i	$-0.845014\pi$
$389$	−190672.	+	69399.0i	−1.26005	+	0.458621i	−0.376265	+	0.926512i	$0.622792\pi$
$390$		0			0
$391$	48427.0	+	83877.9i	0.316762	+	0.548649i
$392$		0			0
$393$	−4427.01	−	780.601i	−0.0286632	−	0.00505410i
$394$		0			0
$395$	95333.5	−	113614.i	0.611014	−	0.728179i
$396$		0			0
$397$	173205.	+	63041.5i	1.09896	+	0.399987i	0.826931	−	0.562303i	$-0.190085\pi$
$397$	173205.	+	63041.5i	1.09896	+	0.399987i	0.272024	+	0.962290i	$0.412307\pi$
$398$		0			0
$399$	13797.4	+	9157.82i	0.0866663	+	0.0575236i
$400$		0			0
$401$	−19800.2	+	54400.7i	−0.123135	+	0.338311i	−0.985910	−	0.167278i	$-0.946502\pi$
$401$	−19800.2	+	54400.7i	−0.123135	+	0.338311i	0.862775	+	0.505588i	$0.168725\pi$
$402$		0			0
$403$	−107120.	−	89884.8i	−0.659572	−	0.553447i
$404$		0			0
$405$	−35351.7	+	200489.i	−0.215526	+	1.22231i
$406$		0			0
$407$	398920.	−	230317.i	2.40822	−	1.39039i
$408$		0			0
$409$	15173.5	+	41688.8i	0.0907065	+	0.249214i	0.976748	−	0.214393i	$-0.0687772\pi$
$409$	15173.5	+	41688.8i	0.0907065	+	0.249214i	−0.886041	+	0.463607i	$0.846555\pi$
$410$		0			0
$411$	2574.28	+	1486.26i	0.0152395	+	0.00879854i
$412$		0			0
$413$	76437.5	+	91094.6i	0.448132	+	0.534063i
$414$		0			0
$415$	−4120.02	−	23365.8i	−0.0239223	−	0.135670i
$416$		0			0
$417$		−	11209.9i		−	0.0644658i
$418$		0			0
$419$	107796.			0.614006			0.307003	−	0.951709i	$-0.400674\pi$
$419$	107796.			0.614006			0.307003	+	0.951709i	$0.400674\pi$
$420$		0			0
$421$	−185201.	+	32655.9i	−1.04491	+	0.184246i	−0.669652	−	0.742675i	$-0.733557\pi$
$421$	−185201.	+	32655.9i	−1.04491	+	0.184246i	−0.375257	+	0.926921i	$0.622446\pi$
$422$		0			0
$423$	−22622.5	+	18982.5i	−0.126433	+	0.106090i
$424$		0			0
$425$	−20527.6	+	35554.9i	−0.113648	+	0.196844i
$426$		0			0
$427$	−168804.	+	61439.8i	−0.925823	+	0.336972i
$428$		0			0
$429$	5903.70	+	10225.5i	0.0320782	+	0.0555610i
$430$		0			0
$431$	83201.5	+	14670.7i	0.447895	+	0.0789760i	0.393046	−	0.919519i	$-0.371421\pi$
$431$	83201.5	+	14670.7i	0.447895	+	0.0789760i	0.0548489	+	0.998495i	$0.482532\pi$
$432$		0			0
$433$	−197252.	+	235076.i	−1.05207	+	1.25381i	−0.0857960	+	0.996313i	$0.527343\pi$
$433$	−197252.	+	235076.i	−1.05207	+	1.25381i	−0.966278	+	0.257500i	$0.917101\pi$
$434$		0			0
$435$	30300.6	+	11028.5i	0.160130	+	0.0582826i
$436$		0			0
$437$	256330.	−	189165.i	1.34226	−	0.990552i
$438$		0			0
$439$	11453.1	−	31467.2i	0.0594286	−	0.163279i	−0.906425	−	0.422366i	$-0.861200\pi$
$439$	11453.1	−	31467.2i	0.0594286	−	0.163279i	0.965854	+	0.259088i	$0.0834218\pi$
$440$		0			0
$441$	113823.	+	95509.1i	0.585267	+	0.491097i
$442$		0			0
$443$	−19837.3	+	112503.i	−0.101082	+	0.573265i	0.891631	+	0.452763i	$0.149562\pi$
$443$	−19837.3	+	112503.i	−0.101082	+	0.573265i	−0.992713	+	0.120502i	$0.961549\pi$
$444$		0			0
$445$	213186.	−	123083.i	1.07656	−	0.621553i
$446$		0			0
$447$	3504.12	+	9627.49i	0.0175373	+	0.0481835i
$448$		0			0
$449$	−28145.4	−	16249.7i	−0.139609	−	0.0806035i	0.428568	−	0.903509i	$-0.359018\pi$
$449$	−28145.4	−	16249.7i	−0.139609	−	0.0806035i	−0.568178	+	0.822906i	$0.692351\pi$
$450$		0			0
$451$	−289999.	−	345607.i	−1.42575	−	1.69914i
$452$		0			0
$453$	−3662.32	−	20770.1i	−0.0178468	−	0.101214i
$454$		0			0
$455$		−	179794.i		−	0.868467i
$456$		0			0
$457$	180019.			0.861956			0.430978	−	0.902362i	$-0.358169\pi$
$457$	180019.			0.861956			0.430978	+	0.902362i	$0.358169\pi$
$458$		0			0
$459$	12288.0	−	2166.70i	0.0583250	−	0.0102843i
$460$		0			0
$461$	−227440.	+	190845.i	−1.07020	+	0.898005i	−0.995071	−	0.0991692i	$-0.968381\pi$
$461$	−227440.	+	190845.i	−1.07020	+	0.898005i	−0.0751296	+	0.997174i	$0.523937\pi$
$462$		0			0
$463$	−30464.3	+	52765.7i	−0.142112	+	0.246144i	−0.928292	−	0.371853i	$-0.878723\pi$
$463$	−30464.3	+	52765.7i	−0.142112	+	0.246144i	0.786180	+	0.617998i	$0.212056\pi$
$464$		0			0
$465$	−33494.7	+	12191.1i	−0.154907	+	0.0563815i
$466$		0			0
$467$	127334.	+	220550.i	0.583864	+	1.01128i	0.995016	+	0.0997156i	$0.0317933\pi$
$467$	127334.	+	220550.i	0.583864	+	1.01128i	−0.411152	+	0.911567i	$0.634873\pi$
$468$		0			0
$469$	−377219.	−	66514.0i	−1.71494	−	0.302390i
$470$		0			0
$471$	5335.38	−	6358.46i	0.0240505	−	0.0286622i
$472$		0			0
$473$	−41484.0	−	15098.9i	−0.185421	−	0.0674876i
$474$		0			0
$475$	123770.	+	54006.0i	0.548564	+	0.239362i
$476$		0			0
$477$	89040.0	−	244635.i	0.391335	−	1.07518i
$478$		0			0
$479$	−44696.4	−	37504.7i	−0.194806	−	0.163461i	0.540168	−	0.841558i	$-0.318361\pi$
$479$	−44696.4	−	37504.7i	−0.194806	−	0.163461i	−0.734973	+	0.678096i	$0.762805\pi$
$480$		0			0
$481$	36333.4	−	206057.i	0.157042	−	0.890629i
$482$		0			0
$483$	−35057.8	+	20240.6i	−0.150276	+	0.0867620i
$484$		0			0
$485$	69175.8	+	190059.i	0.294084	+	0.807988i
$486$		0			0
$487$	−64126.5	−	37023.4i	−0.270383	−	0.156106i	0.358679	−	0.933461i	$-0.383227\pi$
$487$	−64126.5	−	37023.4i	−0.270383	−	0.156106i	−0.629062	+	0.777355i	$0.716561\pi$
$488$		0			0
$489$	−2264.45	−	2698.66i	−0.00946988	−	0.0112858i
$490$		0			0
$491$	34828.9	+	197524.i	0.144470	+	0.819328i	0.967792	+	0.251753i	$0.0810071\pi$
$491$	34828.9	+	197524.i	0.144470	+	0.819328i	−0.823322	+	0.567575i	$0.807882\pi$
$492$		0			0
$493$			159058.i			0.654428i
$494$		0			0
$495$	−488980.			−1.99563
$496$		0			0
$497$	87479.4	−	15425.0i	0.354155	−	0.0624470i
$498$		0			0
$499$	−129561.	+	108714.i	−0.520322	+	0.436602i	−0.864744	−	0.502213i	$-0.832519\pi$
$499$	−129561.	+	108714.i	−0.520322	+	0.436602i	0.344422	+	0.938815i	$0.388075\pi$
$500$		0			0
$501$	13351.6	−	23125.7i	0.0531936	−	0.0921340i
$502$		0			0
$503$	−128080.	+	46617.4i	−0.506228	+	0.184252i	−0.582493	−	0.812836i	$-0.697923\pi$
$503$	−128080.	+	46617.4i	−0.506228	+	0.184252i	0.0762650	+	0.997088i	$0.475700\pi$
$504$		0			0
$505$	−62243.9	−	107810.i	−0.244070	−	0.422741i
$506$		0			0
$507$	−14517.6	−	2559.84i	−0.0564779	−	0.00995857i
$508$		0			0
$509$	301408.	−	359204.i	1.16337	−	1.38645i	0.255708	−	0.966754i	$-0.417691\pi$
$509$	301408.	−	359204.i	1.16337	−	1.38645i	0.907663	−	0.419699i	$-0.137864\pi$
$510$		0			0
$511$	−347444.	−	126459.i	−1.33059	−	0.484294i
$512$		0			0
$513$	−11605.7	−	39366.0i	−0.0440998	−	0.149585i
$514$		0			0
$515$	130801.	−	359373.i	0.493171	−	1.35498i
$516$		0			0
$517$	−53999.9	−	45311.3i	−0.202028	−	0.169522i
$518$		0			0
$519$	410.623	−	2328.76i	0.00152444	−	0.00864550i
$520$		0			0
$521$	−72663.0	+	41952.0i	−0.267694	+	0.154553i	−0.627839	−	0.778343i	$-0.716060\pi$
$521$	−72663.0	+	41952.0i	−0.267694	+	0.154553i	0.360145	+	0.932896i	$0.382727\pi$
$522$		0			0
$523$	−150695.	−	414032.i	−0.550930	−	1.51367i	−0.832443	−	0.554110i	$-0.813059\pi$
$523$	−150695.	−	414032.i	−0.550930	−	1.51367i	0.281514	−	0.959557i	$-0.409164\pi$
$524$		0			0
$525$	−14860.6	−	8579.76i	−0.0539160	−	0.0311284i
$526$		0			0
$527$	−113018.	−	134690.i	−0.406937	−	0.484969i
$528$		0			0
$529$	86636.0	+	491337.i	0.309590	+	1.75577i
$530$		0			0
$531$		−	146904.i		−	0.521009i
$532$		0			0
$533$	−204932.			−0.721364
$534$		0			0
$535$	2388.12	−	421.089i	0.00834349	−	0.00147118i
$536$		0			0
$537$	2701.15	−	2266.54i	0.00936700	−	0.00785985i
$538$		0			0
$539$	−177337.	+	307157.i	−0.610411	+	1.05726i
$540$		0			0
$541$	−234761.	+	85446.1i	−0.802106	+	0.291943i	−0.710359	−	0.703840i	$-0.751467\pi$
$541$	−234761.	+	85446.1i	−0.802106	+	0.291943i	−0.0917472	+	0.995782i	$0.529245\pi$
$542$		0			0
$543$	4498.53	+	7791.68i	0.0152571	+	0.0264260i
$544$		0			0
$545$	−214461.	−	37815.2i	−0.722029	−	0.127313i
$546$		0			0
$547$	284604.	−	339177.i	0.951187	−	1.13358i	−0.0397445	−	0.999210i	$-0.512654\pi$
$547$	284604.	−	339177.i	0.951187	−	1.13358i	0.990931	−	0.134370i	$-0.0429011\pi$
$548$		0			0
$549$	208535.	+	75900.5i	0.691885	+	0.251826i
$550$		0			0
$551$	519888.	−	58556.6i	1.71241	−	0.192873i
$552$		0			0
$553$	−104582.	+	287337.i	−0.341985	+	0.939595i
$554$		0			0
$555$	−40856.6	−	34282.7i	−0.132640	−	0.111299i
$556$		0			0
$557$	18753.5	−	106357.i	0.0604467	−	0.342810i	−0.939553	−	0.342403i	$-0.888759\pi$
$557$	18753.5	−	106357.i	0.0604467	−	0.342810i	1.00000		0.000407388i	$-0.000129676\pi$
$558$		0			0
$559$	−17366.2	+	10026.4i	−0.0555753	+	0.0320864i
$560$		0			0
$561$	5077.72	+	13950.9i	0.0161340	+	0.0443279i
$562$		0			0
$563$	17056.8	+	9847.76i	0.0538123	+	0.0310685i	0.526665	−	0.850073i	$-0.323442\pi$
$563$	17056.8	+	9847.76i	0.0538123	+	0.0310685i	−0.472852	+	0.881142i	$0.656776\pi$
$564$		0			0
$565$	374093.	+	445827.i	1.17188	+	1.39659i
$566$		0			0
$567$	−72884.9	−	413351.i	−0.226710	−	1.28574i
$568$		0			0
$569$			558758.i			1.72584i	0.505344	+	0.862918i	$0.331365\pi$
$569$			558758.i			1.72584i	−0.505344	+	0.862918i	$0.668635\pi$
$570$		0			0
$571$	538854.			1.65272			0.826359	−	0.563144i	$-0.190408\pi$
$571$	538854.			1.65272			0.826359	+	0.563144i	$0.190408\pi$
$572$		0			0
$573$	12071.5	−	2128.53i	0.0367665	−	0.00648293i
$574$		0			0
$575$	−252876.	+	212188.i	−0.764843	+	0.641779i
$576$		0			0
$577$	140861.	−	243978.i	0.423096	−	0.732824i	−0.573145	−	0.819454i	$-0.694277\pi$
$577$	140861.	−	243978.i	0.423096	−	0.732824i	0.996240	+	0.0866306i	$0.0276100\pi$
$578$		0			0
$579$	20614.3	−	7502.99i	0.0614910	−	0.0223809i
$580$		0			0
$581$	24458.3	+	42363.0i	0.0724559	+	0.125497i
$582$		0			0
$583$	611981.	+	107909.i	1.80053	+	0.317482i
$584$		0			0
$585$	−142771.	+	170147.i	−0.417184	+	0.497180i
$586$		0			0
$587$	−303711.	−	110542.i	−0.881423	−	0.320812i	−0.138639	−	0.990343i	$-0.544273\pi$
$587$	−303711.	−	110542.i	−0.881423	−	0.320812i	−0.742784	+	0.669531i	$0.766495\pi$
$588$		0			0
$589$	−398632.	+	418990.i	−1.14906	+	1.20774i
$590$		0			0
$591$	−11460.0	+	31486.0i	−0.0328101	+	0.0901451i
$592$		0			0
$593$	−65213.6	−	54720.7i	−0.185451	−	0.155612i	0.545336	−	0.838218i	$-0.316402\pi$
$593$	−65213.6	−	54720.7i	−0.185451	−	0.155612i	−0.730787	+	0.682606i	$0.760847\pi$
$594$		0			0
$595$	39256.3	−	222633.i	0.110886	−	0.628863i
$596$		0			0
$597$	−31927.5	+	18433.4i	−0.0895811	+	0.0517197i
$598$		0			0
$599$	129481.	+	355746.i	0.360871	+	0.991485i	0.978722	+	0.205190i	$0.0657813\pi$
$599$	129481.	+	355746.i	0.360871	+	0.991485i	−0.617851	+	0.786295i	$0.711997\pi$
$600$		0			0
$601$	334921.	+	193367.i	0.927244	+	0.535345i	0.885939	−	0.463802i	$-0.153515\pi$
$601$	334921.	+	193367.i	0.927244	+	0.535345i	0.0413051	+	0.999147i	$0.486848\pi$
$602$		0			0
$603$	304162.	+	362486.i	0.836509	+	0.996913i
$604$		0			0
$605$	−122322.	−	693721.i	−0.334189	−	1.89528i
$606$		0			0
$607$		−	243144.i		−	0.659912i	−0.943996	−	0.329956i	$-0.892966\pi$
$607$		−	243144.i		−	0.659912i	0.943996	−	0.329956i	$-0.107034\pi$
$608$		0			0
$609$	−66480.3			−0.179250
$610$		0			0
$611$	−31533.4	+	5560.19i	−0.0844673	+	0.0148939i
$612$		0			0
$613$	177368.	−	148829.i	0.472012	−	0.396065i	−0.375516	−	0.926816i	$-0.622534\pi$
$613$	177368.	−	148829.i	0.472012	−	0.396065i	0.847528	+	0.530751i	$0.178090\pi$
$614$		0			0
$615$	−26118.7	+	45238.9i	−0.0690559	+	0.119608i
$616$		0			0
$617$	512867.	−	186668.i	1.34721	−	0.490343i	0.435132	−	0.900366i	$-0.356702\pi$
$617$	512867.	−	186668.i	1.34721	−	0.490343i	0.912075	+	0.410023i	$0.134479\pi$
$618$		0			0
$619$	−265793.	−	460367.i	−0.693685	−	1.20150i	−0.970622	−	0.240610i	$-0.922653\pi$
$619$	−265793.	−	460367.i	−0.693685	−	1.20150i	0.276937	−	0.960888i	$-0.410681\pi$
$620$		0			0
$621$	98801.8	+	17421.4i	0.256202	+	0.0451752i
$622$		0			0
$623$	−326229.	+	388784.i	−0.840517	+	1.00169i
$624$		0			0
$625$	455272.	+	165705.i	1.16550	+	0.424206i
$626$		0			0
$627$	43729.8	−	21732.7i	0.111235	−	0.0552813i
$628$		0			0
$629$	89980.8	−	247220.i	0.227430	−	0.624860i
$630$		0			0
$631$	208648.	+	175076.i	0.524029	+	0.439712i	0.866033	−	0.499986i	$-0.166662\pi$
$631$	208648.	+	175076.i	0.524029	+	0.439712i	−0.342005	+	0.939698i	$0.611106\pi$
$632$		0			0
$633$	5541.27	−	31426.1i	0.0138293	−	0.0784301i
$634$		0			0
$635$	−271532.	+	156769.i	−0.673400	+	0.388788i
$636$		0			0
$637$	55101.4	+	151390.i	0.135795	+	0.373093i
$638$		0			0
$639$	−95034.2	−	54868.0i	−0.232744	−	0.134375i
$640$		0			0
$641$	−277020.	−	330140.i	−0.674210	−	0.803492i	0.315140	−	0.949045i	$-0.397948\pi$
$641$	−277020.	−	330140.i	−0.674210	−	0.803492i	−0.989351	+	0.145553i	$0.953504\pi$
$642$		0			0
$643$	84349.0	+	478367.i	0.204013	+	1.15702i	0.898985	+	0.437979i	$0.144306\pi$
$643$	84349.0	+	478367.i	0.204013	+	1.15702i	−0.694972	+	0.719037i	$0.744583\pi$
$644$		0			0
$645$			5111.49i			0.0122865i
$646$		0			0
$647$	95597.5			0.228369			0.114185	−	0.993460i	$-0.463574\pi$
$647$	95597.5			0.228369			0.114185	+	0.993460i	$0.463574\pi$
$648$		0			0
$649$	345333.	−	60891.6i	0.819877	−	0.144567i
$650$		0			0
$651$	56295.2	−	47237.3i	0.132834	−	0.111461i
$652$		0			0
$653$	251342.	−	435337.i	0.589438	−	1.02094i	−0.404868	−	0.914375i	$-0.632683\pi$
$653$	251342.	−	435337.i	0.589438	−	1.02094i	0.994306	−	0.106561i	$-0.0339841\pi$
$654$		0			0
$655$	189677.	−	69036.8i	0.442112	−	0.160916i
$656$		0			0
$657$	228383.	+	395572.i	0.529095	+	0.916419i
$658$		0			0
$659$	240961.	+	42488.0i	0.554851	+	0.0978352i	0.444041	−	0.896006i	$-0.353544\pi$
$659$	240961.	+	42488.0i	0.554851	+	0.0978352i	0.110810	+	0.993842i	$0.464656\pi$
$660$		0			0
$661$	35563.9	−	42383.4i	0.0813967	−	0.0970048i	−0.723809	−	0.690000i	$-0.757610\pi$
$661$	35563.9	−	42383.4i	0.0813967	−	0.0970048i	0.805206	+	0.592996i	$0.202055\pi$
$662$		0			0
$663$	6336.99	+	2306.48i	0.0144164	+	0.00524713i
$664$		0			0
$665$	−742138.	−	46349.2i	−1.67819	−	0.104809i
$666$		0			0
$667$	−437414.	+	1.20179e6i	−0.983198	+	2.70132i
$668$		0			0
$669$	−25465.6	−	21368.2i	−0.0568987	−	0.0477436i
$670$		0			0
$671$	−91984.8	+	521672.i	−0.204301	+	1.15865i
$672$		0			0
$673$	−371361.	+	214405.i	−0.819909	+	0.473375i	−0.850385	−	0.526161i	$-0.823631\pi$
$673$	−371361.	+	214405.i	−0.819909	+	0.473375i	0.0304761	+	0.999535i	$0.490298\pi$
$674$		0			0
$675$	14545.1	+	39962.3i	0.0319234	+	0.0877088i
$676$		0			0
$677$	−743727.	−	429391.i	−1.62269	−	0.936862i	−0.986196	−	0.165583i	$-0.947049\pi$
$677$	−743727.	−	429391.i	−1.62269	−	0.936862i	−0.636497	−	0.771279i	$-0.719617\pi$
$678$		0			0
$679$	−268038.	−	319436.i	−0.581376	−	0.692857i
$680$		0			0
$681$	−1567.90	−	8892.02i	−0.00338084	−	0.0191737i
$682$		0			0
$683$		−	231818.i		−	0.496942i	−0.968639	−	0.248471i	$-0.920072\pi$
$683$		−	231818.i		−	0.496942i	0.968639	−	0.248471i	$-0.0799281\pi$
$684$		0			0
$685$	−133473.			−0.284455
$686$		0			0
$687$	−48910.5	+	8624.24i	−0.103631	+	0.0182729i
$688$		0			0
$689$	216232.	−	181441.i	0.455494	−	0.382205i
$690$		0			0
$691$	300258.	−	520062.i	0.628837	−	1.08918i	−0.358948	−	0.933358i	$-0.616864\pi$
$691$	300258.	−	520062.i	0.628837	−	1.08918i	0.987785	−	0.155821i	$-0.0498022\pi$
$692$		0			0
$693$	947335.	−	344802.i	1.97259	−	0.717965i
$694$		0			0
$695$	251676.	+	435916.i	0.521041	+	0.902470i
$696$		0			0
$697$	−253760.	−	44744.7i	−0.522345	−	0.0921035i
$698$		0			0
$699$	10371.7	−	12360.5i	0.0212274	−	0.0252978i
$700$		0			0
$701$	−256747.	−	93448.3i	−0.522480	−	0.190167i	0.0672977	−	0.997733i	$-0.478562\pi$
$701$	−256747.	−	93448.3i	−0.522480	−	0.190167i	−0.589777	+	0.807566i	$0.700785\pi$
$702$		0			0
$703$	−841175.	−	203093.i	−1.70206	−	0.410945i
$704$		0			0
$705$	−2791.54	+	7669.68i	−0.00561649	+	0.0154312i
$706$		0			0
$707$	196611.	+	164976.i	0.393341	+	0.330052i
$708$		0			0
$709$	71493.5	−	405460.i	0.142224	−	0.806595i	−0.827330	−	0.561717i	$-0.810141\pi$
$709$	71493.5	−	405460.i	0.142224	−	0.806595i	0.969554	−	0.244878i	$-0.0787478\pi$
$710$		0			0
$711$	327138.	−	188873.i	0.647130	−	0.373621i
$712$		0			0
$713$	−483524.	−	1.32847e6i	−0.951128	−	2.61320i
$714$		0			0
$715$	−459150.	−	265091.i	−0.898137	−	0.518540i
$716$		0			0
$717$	19590.5	+	23347.1i	0.0381073	+	0.0454145i
$718$		0			0
$719$	−123960.	−	703011.i	−0.239786	−	1.35989i	−0.832297	−	0.554330i	$-0.812975\pi$
$719$	−123960.	−	703011.i	−0.239786	−	1.35989i	0.592511	−	0.805562i	$-0.298137\pi$
$720$		0			0
$721$			788474.i			1.51676i
$722$		0			0
$723$	27673.1			0.0529397
$724$		0			0
$725$	−533880.	+	94137.5i	−1.01571	+	0.179096i
$726$		0			0
$727$	767353.	−	643885.i	1.45186	−	1.21826i	0.520660	−	0.853764i	$-0.325686\pi$
$727$	767353.	−	643885.i	1.45186	−	1.21826i	0.931205	−	0.364495i	$-0.118758\pi$
$728$		0			0
$729$	−256017.	+	443434.i	−0.481741	+	0.834400i
$730$		0			0
$731$	−23693.2	+	8623.61i	−0.0443393	+	0.0161382i
$732$		0			0
$733$	−201153.	−	348407.i	−0.374385	−	0.648454i	0.615850	−	0.787864i	$-0.288813\pi$
$733$	−201153.	−	348407.i	−0.374385	−	0.648454i	−0.990235	+	0.139410i	$0.955480\pi$
$734$		0			0
$735$	40442.2	+	7131.04i	0.0748617	+	0.0132001i
$736$		0			0
$737$	−726036.	+	865256.i	−1.33667	+	1.59298i
$738$		0			0
$739$	−840729.	−	306000.i	−1.53946	−	0.560316i	−0.573541	−	0.819177i	$-0.694431\pi$
$739$	−840729.	−	306000.i	−1.53946	−	0.560316i	−0.965915	+	0.258860i	$0.916653\pi$
$740$		0			0
$741$	5205.87	−	21561.8i	0.00948106	−	0.0392689i
$742$		0			0
$743$	−188375.	+	517555.i	−0.341228	+	0.937516i	0.643811	+	0.765185i	$0.277352\pi$
$743$	−188375.	+	517555.i	−0.341228	+	0.937516i	−0.985039	+	0.172332i	$0.944870\pi$
$744$		0			0
$745$	−352413.	−	295709.i	−0.634949	−	0.532785i
$746$		0			0
$747$	10493.5	−	59511.8i	0.0188053	−	0.106650i
$748$		0			0
$749$	−4329.74	+	2499.77i	−0.00771788	+	0.00445592i
$750$		0			0
$751$	348702.	+	958051.i	0.618265	+	1.69867i	0.711193	+	0.702997i	$0.248155\pi$
$751$	348702.	+	958051.i	0.618265	+	1.69867i	−0.0929277	+	0.995673i	$0.529623\pi$
$752$		0			0
$753$	38254.6	+	22086.3i	0.0674673	+	0.0389523i
$754$		0			0
$755$	608729.	+	725455.i	1.06790	+	1.27267i
$756$		0			0
$757$	−81811.7	−	463977.i	−0.142766	−	0.809664i	−0.969134	−	0.246534i	$-0.920708\pi$
$757$	−81811.7	−	463977.i	−0.142766	−	0.809664i	0.826368	−	0.563130i	$-0.190403\pi$
$758$		0			0
$759$			119372.i			0.207214i
$760$		0			0
$761$	−371141.			−0.640869			−0.320434	−	0.947271i	$-0.603829\pi$
$761$	−371141.			−0.640869			−0.320434	+	0.947271i	$0.603829\pi$
$762$		0			0
$763$	442155.	−	77963.9i	0.759496	−	0.133920i
$764$		0			0
$765$	−213938.	+	179515.i	−0.365565	+	0.306746i
$766$		0			0
$767$	79641.1	−	137942.i	0.135378	−	0.234481i
$768$		0			0
$769$	443948.	−	161584.i	0.750723	−	0.273241i	0.0618129	−	0.998088i	$-0.480312\pi$
$769$	443948.	−	161584.i	0.750723	−	0.273241i	0.688910	+	0.724847i	$0.258090\pi$
$770$		0			0
$771$	−28081.6	−	48638.8i	−0.0472404	−	0.0818228i
$772$		0			0
$773$	891114.	+	157127.i	1.49133	+	0.262962i	0.859096	−	0.511814i	$-0.171026\pi$
$773$	891114.	+	157127.i	1.49133	+	0.262962i	0.632236	+	0.774776i	$0.282137\pi$
$774$		0			0
$775$	385199.	−	459062.i	0.641330	−	0.764308i
$776$		0			0
$777$	103329.	+	37608.6i	0.171151	+	0.0622938i
$778$		0			0
$779$	−52829.3	+	845897.i	−0.0870562	+	1.39393i
$780$		0			0
$781$	89588.8	−	246143.i	0.146876	−	0.403539i
$782$		0			0
$783$	126214.	+	105906.i	0.205865	+	0.172741i
$784$		0			0
$785$	−64720.0	+	367045.i	−0.105027	+	0.595635i
$786$		0			0
$787$	27445.8	−	15845.8i	0.0443125	−	0.0255838i	−0.477680	−	0.878534i	$-0.658522\pi$
$787$	27445.8	−	15845.8i	0.0443125	−	0.0255838i	0.521993	+	0.852950i	$0.325189\pi$
$788$		0			0
$789$	13261.0	+	36434.4i	0.0213021	+	0.0585271i
$790$		0			0
$791$	−1.03913e6	−	599942.i	−1.66080	−	0.958863i
$792$		0			0
$793$	154666.	+	184323.i	0.245950	+	0.293112i
$794$		0			0
$795$	−12494.2	−	70858.3i	−0.0197686	−	0.112113i
$796$		0			0
$797$			145767.i			0.229478i	0.993396	+	0.114739i	$0.0366032\pi$
$797$			145767.i			0.229478i	−0.993396	+	0.114739i	$0.963397\pi$
$798$		0			0
$799$	−40260.8			−0.0630650
$800$		0			0
$801$	617450.	−	108873.i	0.962358	−	0.169690i
$802$		0			0
$803$	−835221.	+	700833.i	−1.29530	+	1.08689i
$804$		0			0
$805$	908854.	−	1.57418e6i	1.40250	−	2.42920i
$806$		0			0
$807$	−21255.8	+	7736.48i	−0.0326385	+	0.0118794i
$808$		0			0
$809$	−35669.8	−	61781.9i	−0.0545009	−	0.0943984i	0.837488	−	0.546456i	$-0.184023\pi$
$809$	−35669.8	−	61781.9i	−0.0545009	−	0.0943984i	−0.891989	+	0.452058i	$0.850690\pi$
$810$		0			0
$811$	−67074.6	−	11827.1i	−0.101980	−	0.0179819i	0.122425	−	0.992478i	$-0.460933\pi$
$811$	−67074.6	−	11827.1i	−0.101980	−	0.0179819i	−0.224405	+	0.974496i	$0.572044\pi$
$812$		0			0
$813$	23781.0	−	28341.1i	0.0359790	−	0.0428781i
$814$		0			0
$815$	148645.	+	54102.4i	0.223787	+	0.0814520i
$816$		0			0
$817$	36909.2	+	74267.4i	0.0552955	+	0.111264i
$818$		0			0
$819$	156621.	−	430313.i	0.233498	−	0.641529i
$820$		0			0
$821$	−749298.	−	628736.i	−1.11165	−	0.932785i	−0.113498	−	0.993538i	$-0.536205\pi$
$821$	−749298.	−	628736.i	−1.11165	−	0.932785i	−0.998153	+	0.0607529i	$0.980650\pi$
$822$		0			0
$823$	73885.5	−	419026.i	0.109084	−	0.618644i	−0.880427	−	0.474182i	$-0.842744\pi$
$823$	73885.5	−	419026.i	0.109084	−	0.618644i	0.989510	−	0.144462i	$-0.0461451\pi$
$824$		0			0
$825$	−43821.2	+	25300.2i	−0.0643837	+	0.0371720i
$826$		0			0
$827$	105592.	+	290112.i	0.154391	+	0.424185i	0.992640	−	0.121102i	$-0.0386429\pi$
$827$	105592.	+	290112.i	0.154391	+	0.424185i	−0.838249	+	0.545287i	$0.816421\pi$
$828$		0			0
$829$	327216.	+	188918.i	0.476130	+	0.274894i	0.718802	−	0.695214i	$-0.244691\pi$
$829$	327216.	+	188918.i	0.476130	+	0.274894i	−0.242672	+	0.970108i	$0.578024\pi$
$830$		0			0
$831$	−58215.1	−	69378.0i	−0.0843011	−	0.100466i
$832$		0			0
$833$	35175.8	+	199492.i	0.0506936	+	0.287498i
$834$		0			0
$835$			1.19904e6i			1.71974i
$836$		0			0
$837$	−182128.			−0.259972
$838$		0			0
$839$	665042.	−	117265.i	0.944768	−	0.166588i	0.320016	−	0.947412i	$-0.396312\pi$
$839$	665042.	−	117265.i	0.944768	−	0.166588i	0.624751	+	0.780824i	$0.285200\pi$
$840$		0			0
$841$	−1.06711e6	+	895411.i	−1.50875	+	1.26599i
$842$		0			0
$843$	10008.5	−	17335.3i	0.0140836	−	0.0243936i
$844$		0			0
$845$	622012.	−	226394.i	0.871135	−	0.317067i
$846$		0			0
$847$	726156.	+	1.25774e6i	1.01219	+	1.75317i
$848$		0			0
$849$	77841.9	+	13725.6i	0.107994	+	0.0190422i
$850$		0			0
$851$	1.35972e6	−	1.62046e6i	1.87755	−	2.23758i
$852$		0			0
$853$	728297.	+	265078.i	1.00095	+	0.364314i	0.789949	−	0.613173i	$-0.210107\pi$
$853$	728297.	+	265078.i	1.00095	+	0.364314i	0.210996	+	0.977487i	$0.432329\pi$
$854$		0			0
$855$	665513.	+	633177.i	0.910384	+	0.866150i
$856$		0			0
$857$	−52799.5	+	145066.i	−0.0718900	+	0.197516i	−0.970434	−	0.241368i	$-0.922404\pi$
$857$	−52799.5	+	145066.i	−0.0718900	+	0.197516i	0.898544	+	0.438884i	$0.144626\pi$
$858$		0			0
$859$	498289.	+	418114.i	0.675297	+	0.566641i	0.914628	−	0.404297i	$-0.132484\pi$
$859$	498289.	+	418114.i	0.675297	+	0.566641i	−0.239331	+	0.970938i	$0.576928\pi$
$860$		0			0
$861$	18701.6	−	106062.i	0.0252274	−	0.143072i
$862$		0			0
$863$	−365470.	+	211004.i	−0.490715	+	0.283315i	−0.724871	−	0.688884i	$-0.758101\pi$
$863$	−365470.	+	211004.i	−0.490715	+	0.283315i	0.234156	+	0.972199i	$0.424767\pi$
$864$		0			0
$865$	36315.8	+	99776.8i	0.0485359	+	0.133351i
$866$		0			0
$867$	−43572.7	−	25156.7i	−0.0579664	−	0.0334669i
$868$		0			0
$869$	579590.	+	690728.i	0.767505	+	0.914677i
$870$		0			0
$871$	89092.5	+	505269.i	0.117437	+	0.666018i
$872$		0			0
$873$			515139.i			0.675921i
$874$		0			0
$875$	−516864.			−0.675087
$876$		0			0
$877$	−240096.	+	42335.4i	−0.312166	+	0.0550432i	−0.327536	−	0.944839i	$-0.606218\pi$
$877$	−240096.	+	42335.4i	−0.312166	+	0.0550432i	0.0153705	+	0.999882i	$0.495107\pi$
$878$		0			0
$879$	−71814.4	+	60259.4i	−0.0929466	+	0.0779915i
$880$		0			0
$881$	−203870.	+	353113.i	−0.262664	+	0.454948i	−0.966949	−	0.254970i	$-0.917934\pi$
$881$	−203870.	+	353113.i	−0.262664	+	0.454948i	0.704285	+	0.709917i	$0.251268\pi$
$882$		0			0
$883$	−236698.	+	86151.1i	−0.303580	+	0.110494i	−0.489319	−	0.872105i	$-0.662754\pi$
$883$	−236698.	+	86151.1i	−0.303580	+	0.110494i	0.185739	+	0.982599i	$0.440532\pi$
$884$		0			0
$885$	−20300.6	−	35161.7i	−0.0259193	−	0.0448935i
$886$		0			0
$887$	−187497.	−	33060.7i	−0.238312	−	0.0420209i	0.0532163	−	0.998583i	$-0.483053\pi$
$887$	−187497.	−	33060.7i	−0.238312	−	0.0420209i	−0.291529	+	0.956562i	$0.594164\pi$
$888$		0			0
$889$	415513.	−	495189.i	0.525752	−	0.626567i
$890$		0			0
$891$	−1.16306e6	−	423318.i	−1.46503	−	0.533226i
$892$		0			0
$893$	14821.8	+	131594.i	0.0185865	+	0.165019i
$894$		0			0
$895$	−54152.3	+	148782.i	−0.0676038	+	0.185740i
$896$		0			0
$897$	41537.1	+	34853.8i	0.0516240	+	0.0433176i
$898$		0			0
$899$	403158.	−	2.28642e6i	0.498834	−	2.82903i
$900$		0			0
$901$	307369.	−	177460.i	0.378626	−	0.218600i
$902$		0			0
$903$	−3604.34	−	9902.85i	−0.00442029	−	0.0121446i
$904$		0			0
$905$	−349866.	−	201995.i	−0.427173	−	0.246629i
$906$		0			0
$907$	−351542.	−	418951.i	−0.427329	−	0.509271i	0.508821	−	0.860873i	$-0.330082\pi$
$907$	−351542.	−	418951.i	−0.427329	−	0.509271i	−0.936150	+	0.351601i	$0.885637\pi$
$908$		0			0
$909$	−55057.9	−	312249.i	−0.0666333	−	0.377896i
$910$		0			0
$911$			921192.i			1.10998i	0.831858	+	0.554988i	$0.187277\pi$
$911$			921192.i			1.10998i	−0.831858	+	0.554988i	$0.812723\pi$
$912$		0			0
$913$	144246.			0.173046
$914$		0			0
$915$	60401.8	−	10650.5i	0.0721453	−	0.0127212i
$916$		0			0
$917$	−318794.	+	267500.i	−0.379115	+	0.318115i
$918$		0			0
$919$	−246580.	+	427090.i	−0.291963	+	0.505694i	−0.974274	−	0.225368i	$-0.927642\pi$
$919$	−246580.	+	427090.i	−0.291963	+	0.505694i	0.682311	+	0.731062i	$0.260975\pi$
$920$		0			0
$921$	−3506.70	+	1276.33i	−0.00413408	+	0.00150468i
$922$		0			0
$923$	−59491.2	−	103042.i	−0.0698312	−	0.120951i
$924$		0			0
$925$	883052.	+	155706.i	1.03205	+	0.181979i
$926$		0			0
$927$	626109.	−	746167.i	0.728602	−	0.868314i
$928$		0			0
$929$	−1.22258e6	−	444983.i	−1.41660	−	0.515599i	−0.483538	−	0.875323i	$-0.660648\pi$
$929$	−1.22258e6	−	444983.i	−1.41660	−	0.515599i	−0.933059	+	0.359724i	$0.882871\pi$
$930$		0			0
$931$	639096.	−	188415.i	0.737339	−	0.217379i
$932$		0			0
$933$	−36006.2	+	98926.2i	−0.0413632	+	0.113644i
$934$		0			0
$935$	−510671.	−	428503.i	−0.584141	−	0.490152i
$936$		0			0
$937$	−74205.2	+	420839.i	−0.0845192	+	0.479332i	0.912940	+	0.408094i	$0.133806\pi$
$937$	−74205.2	+	420839.i	−0.0845192	+	0.479332i	−0.997459	+	0.0712385i	$0.977305\pi$
$938$		0			0
$939$	20110.9	−	11611.0i	0.0228087	−	0.0131686i
$940$		0			0
$941$	−359469.	−	987633.i	−0.405959	−	1.11536i	−0.959295	−	0.282405i	$-0.908868\pi$
$941$	−359469.	−	987633.i	−0.405959	−	1.11536i	0.553336	−	0.832958i	$-0.313355\pi$
$942$		0			0
$943$	−1.79427e6	−	1.03592e6i	−2.01773	−	1.16494i
$944$		0			0
$945$	−150523.	−	179386.i	−0.168554	−	0.200875i
$946$		0			0
$947$	78149.7	+	443209.i	0.0871419	+	0.494207i	0.996874	+	0.0790103i	$0.0251760\pi$
$947$	78149.7	+	443209.i	0.0871419	+	0.494207i	−0.909732	+	0.415196i	$0.863713\pi$
$948$		0			0
$949$			495254.i			0.549915i
$950$		0			0
$951$	−291.276			−0.000322065
$952$		0			0
$953$	1.00103e6	−	176509.i	1.10221	−	0.194349i	0.407191	−	0.913343i	$-0.366508\pi$
$953$	1.00103e6	−	176509.i	1.10221	−	0.194349i	0.695016	+	0.718994i	$0.255397\pi$
$954$		0			0
$955$	−421633.	+	353792.i	−0.462304	+	0.387919i
$956$		0			0
$957$	−98019.1	+	169774.i	−0.107025	+	0.185373i
$958$		0			0
$959$	258587.	−	94118.1i	0.281171	−	0.102338i
$960$		0			0
$961$	821457.	+	1.42281e6i	0.889484	+	1.54063i
$962$		0			0
$963$	6082.44	+	1072.50i	0.00655881	+	0.00115649i
$964$		0			0
$965$	−633170.	+	754583.i	−0.679933	+	0.810312i
$966$		0			0
$967$	−1.51176e6	−	550235.i	−1.61670	−	0.588431i	−0.633951	−	0.773373i	$-0.718568\pi$
$967$	−1.51176e6	−	550235.i	−1.61670	−	0.588431i	−0.982749	+	0.184942i	$0.940790\pi$
$968$		0			0
$969$	11154.1	−	25562.6i	0.0118791	−	0.0272244i
$970$		0			0
$971$	16122.4	−	44295.9i	0.0170998	−	0.0469813i	−0.930849	−	0.365404i	$-0.880931\pi$
$971$	16122.4	−	44295.9i	0.0170998	−	0.0469813i	0.947949	+	0.318423i	$0.103153\pi$
$972$		0			0
$973$	−794974.	−	667062.i	−0.839706	−	0.704597i
$974$		0			0
$975$	−3991.20	+	22635.2i	−0.00419851	+	0.0238109i
$976$		0			0
$977$	−515574.	+	297667.i	−0.540135	+	0.311847i	−0.745134	−	0.666915i	$-0.767614\pi$
$977$	−515574.	+	297667.i	−0.540135	+	0.311847i	0.204999	+	0.978762i	$0.434281\pi$
$978$		0			0
$979$	511864.	+	1.40634e6i	0.534059	+	1.46732i
$980$		0			0
$981$	−480340.	−	277325.i	−0.499127	−	0.288171i
$982$		0			0
$983$	−634961.	−	756716.i	−0.657112	−	0.783116i	0.329856	−	0.944031i	$-0.393000\pi$
$983$	−634961.	−	756716.i	−0.657112	−	0.783116i	−0.986968	+	0.160915i	$0.948555\pi$
$984$		0			0
$985$	−261259.	−	1.48168e6i	−0.269277	−	1.52715i
$986$		0			0
$987$		−	16827.5i		−	0.0172737i
$988$		0			0
$989$	−202732.			−0.207267
$990$		0			0
$991$	−1.09494e6	+	193068.i	−1.11492	+	0.196591i	−0.700611	−	0.713544i	$-0.747089\pi$
$991$	−1.09494e6	+	193068.i	−1.11492	+	0.196591i	−0.414312	+	0.910135i	$0.635978\pi$
$992$		0			0
$993$	−71272.6	+	59804.8i	−0.0722810	+	0.0606510i
$994$		0			0
$995$	827703.	−	1.43362e6i	0.836043	−	1.44807i
$996$		0			0
$997$	−185329.	+	67454.3i	−0.186446	+	0.0678608i	−0.433556	−	0.901127i	$-0.642741\pi$
$997$	−185329.	+	67454.3i	−0.186446	+	0.0678608i	0.247110	+	0.968987i	$0.420519\pi$
$998$		0			0
$999$	−136259.	−	236007.i	−0.136532	−	0.236480i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.j.a.53.4	yes	42
19.14	odd	18	inner	76.5.j.a.33.4	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.j.a.33.4	✓	42	19.14	odd	18	inner
76.5.j.a.53.4	yes	42	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.693233 - 0.122236i) q^{3} +(-24.2132 + 20.3173i) q^{5} +(32.5833 - 56.4359i) q^{7} +(-75.6495 + 27.5342i) q^{9} +O(q^{10})\)	\(q+(0.693233 - 0.122236i) q^{3} +(-24.2132 + 20.3173i) q^{5} +(32.5833 - 56.4359i) q^{7} +(-75.6495 + 27.5342i) q^{9} +(-96.0822 - 166.419i) q^{11} +(-85.9617 - 15.1574i) q^{13} +(-14.3019 + 17.0443i) q^{15} +(-103.134 - 37.5377i) q^{17} +(-84.7252 + 350.917i) q^{19} +(15.6893 - 43.1061i) q^{21} +(-676.013 - 567.242i) q^{23} +(64.9565 - 368.387i) q^{25} +(-98.4562 + 56.8437i) q^{27} +(-495.669 - 1361.84i) q^{29} +(1387.38 + 801.005i) q^{31} +(-86.9497 - 103.623i) q^{33} +(357.679 + 2028.50i) q^{35} +2397.08i q^{37} -61.4442 q^{39} +(2312.10 - 407.686i) q^{41} +(175.985 - 147.669i) q^{43} +(1272.30 - 2203.68i) q^{45} +(344.708 - 125.464i) q^{47} +(-922.842 - 1598.41i) q^{49} +(-76.0843 - 13.4157i) q^{51} +(-2078.65 + 2477.24i) q^{53} +(5707.64 + 2077.41i) q^{55} +(-15.8397 + 253.624i) q^{57} +(-624.116 + 1714.75i) q^{59} +(-2111.67 - 1771.90i) q^{61} +(-910.993 + 5166.50i) q^{63} +(2389.36 - 1379.50i) q^{65} +(-2010.34 - 5523.36i) q^{67} +(-537.972 - 310.598i) q^{69} +(876.187 + 1044.20i) q^{71} +(-985.246 - 5587.61i) q^{73} -263.318i q^{75} -12522.7 q^{77} +(-4620.95 + 814.799i) q^{79} +(4933.97 - 4140.09i) q^{81} +(-375.320 + 650.073i) q^{83} +(3259.87 - 1186.49i) q^{85} +(-510.079 - 883.483i) q^{87} +(-7669.75 - 1352.38i) q^{89} +(-3656.33 + 4357.45i) q^{91} +(1059.69 + 385.695i) q^{93} +(-5078.21 - 10218.2i) q^{95} +(2188.55 - 6012.99i) q^{97} +(11850.8 + 9943.99i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9}+O(q^{10}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9	\( 42 q + 12 q^{3} - 45 q^{7} - 84 q^{9} - 45 q^{11} + 33 q^{13} - 393 q^{15} + 909 q^{17} + 1242 q^{19} + 1107 q^{21} - 360 q^{23} - 810 q^{25} - 7056 q^{27} - 2889 q^{29} + 2808 q^{31} + 10875 q^{33} + 6741 q^{35} - 3480 q^{39} - 3060 q^{41} - 8079 q^{43} - 4320 q^{45} - 2655 q^{47} - 474 q^{49} - 12222 q^{51} - 6705 q^{53} + 4623 q^{55} - 8022 q^{57} + 24309 q^{59} + 7104 q^{61} + 12063 q^{63} + 25245 q^{65} + 15573 q^{67} - 10881 q^{69} - 25506 q^{71} + 3036 q^{73} + 12924 q^{77} - 16839 q^{79} - 2208 q^{81} - 6363 q^{83} - 37890 q^{85} - 21924 q^{87} - 22644 q^{89} + 17418 q^{91} + 8184 q^{93} - 82413 q^{95} + 13383 q^{97} + 23565 q^{99}+O(q^{100}) \) 42 * q + 12 * q^3 - 45 * q^7 - 84 * q^9 - 45 * q^11 + 33 * q^13 - 393 * q^15 + 909 * q^17 + 1242 * q^19 + 1107 * q^21 - 360 * q^23 - 810 * q^25 - 7056 * q^27 - 2889 * q^29 + 2808 * q^31 + 10875 * q^33 + 6741 * q^35 - 3480 * q^39 - 3060 * q^41 - 8079 * q^43 - 4320 * q^45 - 2655 * q^47 - 474 * q^49 - 12222 * q^51 - 6705 * q^53 + 4623 * q^55 - 8022 * q^57 + 24309 * q^59 + 7104 * q^61 + 12063 * q^63 + 25245 * q^65 + 15573 * q^67 - 10881 * q^69 - 25506 * q^71 + 3036 * q^73 + 12924 * q^77 - 16839 * q^79 - 2208 * q^81 - 6363 * q^83 - 37890 * q^85 - 21924 * q^87 - 22644 * q^89 + 17418 * q^91 + 8184 * q^93 - 82413 * q^95 + 13383 * q^97 + 23565 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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