LMFDB - Embedded newform 567.4.c.c.566.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(566,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("567.566");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.c (of order $2$, degree $1$, 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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			566.7
Character	$\chi$	$=$	567.566
Dual form			567.4.c.c.566.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.99283i q^{2} -0.957005 q^{4} -15.6029 q^{5} +(-4.16816 + 18.0451i) q^{7} -21.0785i q^{8} +O(q^{10})$	\(q-2.99283i q^{2} -0.957005 q^{4} -15.6029 q^{5} +(-4.16816 + 18.0451i) q^{7} -21.0785i q^{8} +46.6969i q^{10} -53.6590i q^{11} +33.6519i q^{13} +(54.0059 + 12.4746i) q^{14} -70.7402 q^{16} -43.5488 q^{17} -32.2524i q^{19} +14.9321 q^{20} -160.592 q^{22} +149.313i q^{23} +118.452 q^{25} +100.714 q^{26} +(3.98894 - 17.2693i) q^{28} +147.097i q^{29} +70.6305i q^{31} +43.0854i q^{32} +130.334i q^{34} +(65.0355 - 281.557i) q^{35} +40.3540 q^{37} -96.5258 q^{38} +328.886i q^{40} +358.909 q^{41} +507.771 q^{43} +51.3519i q^{44} +446.867 q^{46} +456.321 q^{47} +(-308.253 - 150.430i) q^{49} -354.506i q^{50} -32.2051i q^{52} -213.420i q^{53} +837.238i q^{55} +(380.363 + 87.8583i) q^{56} +440.237 q^{58} -319.186 q^{59} +350.638i q^{61} +211.385 q^{62} -436.974 q^{64} -525.069i q^{65} +578.516 q^{67} +41.6764 q^{68} +(-842.651 - 194.640i) q^{70} +787.243i q^{71} -146.373i q^{73} -120.773i q^{74} +30.8657i q^{76} +(968.283 + 223.659i) q^{77} +386.191 q^{79} +1103.76 q^{80} -1074.15i q^{82} -197.809 q^{83} +679.490 q^{85} -1519.67i q^{86} -1131.05 q^{88} +596.020 q^{89} +(-607.253 - 140.267i) q^{91} -142.893i q^{92} -1365.69i q^{94} +503.233i q^{95} +729.176i q^{97} +(-450.210 + 922.547i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) $ 44 * q - 156 * q^4 - 10 * q^7	$ 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) $ 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.99283i		−	1.05812i	−0.848583	−	0.529062i	$-0.822544\pi$
$2$		−	2.99283i		−	1.05812i	0.848583	−	0.529062i	$-0.177456\pi$
$3$		0			0
$3$		0			0
$4$	−0.957005			−0.119626
$5$	−15.6029			−1.39557			−0.697785	−	0.716307i	$-0.745831\pi$
$5$	−15.6029			−1.39557			−0.697785	+	0.716307i	$0.745831\pi$
$6$		0			0
$7$	−4.16816	+	18.0451i	−0.225059	+	0.974345i
$7$	−4.16816	+	18.0451i	−0.225059	+	0.974345i
$8$		−	21.0785i		−	0.931545i
$9$		0			0
$10$			46.6969i			1.47669i
$11$		−	53.6590i		−	1.47080i	−0.677634	−	0.735400i	$-0.736994\pi$
$11$		−	53.6590i		−	1.47080i	0.677634	−	0.735400i	$-0.263006\pi$
$12$		0			0
$13$			33.6519i			0.717951i	0.933347	+	0.358976i	$0.116874\pi$
$13$			33.6519i			0.717951i	−0.933347	+	0.358976i	$0.883126\pi$
$14$	54.0059	+	12.4746i	1.03098	+	0.238141i
$15$		0			0
$16$	−70.7402			−1.10532
$17$	−43.5488			−0.621302			−0.310651	−	0.950524i	$-0.600547\pi$
$17$	−43.5488			−0.621302			−0.310651	+	0.950524i	$0.600547\pi$
$18$		0			0
$19$		−	32.2524i		−	0.389432i	−0.980860	−	0.194716i	$-0.937621\pi$
$19$		−	32.2524i		−	0.389432i	0.980860	−	0.194716i	$-0.0623785\pi$
$20$	14.9321			0.166946
$21$		0			0
$22$	−160.592			−1.55629
$23$			149.313i			1.35365i	0.736145	+	0.676824i	$0.236644\pi$
$23$			149.313i			1.35365i	−0.736145	+	0.676824i	$0.763356\pi$
$24$		0			0
$25$	118.452			0.947615
$26$	100.714			0.759681
$27$		0			0
$28$	3.98894	−	17.2693i	0.0269228	−	0.116557i
$29$			147.097i			0.941906i	0.882158	+	0.470953i	$0.156090\pi$
$29$			147.097i			0.941906i	−0.882158	+	0.470953i	$0.843910\pi$
$30$		0			0
$31$			70.6305i			0.409213i	0.978844	+	0.204607i	$0.0655915\pi$
$31$			70.6305i			0.409213i	−0.978844	+	0.204607i	$0.934408\pi$
$32$			43.0854i			0.238015i
$33$		0			0
$34$			130.334i			0.657415i
$35$	65.0355	−	281.557i	0.314086	−	1.35977i
$36$		0			0
$37$	40.3540			0.179302			0.0896509	−	0.995973i	$-0.471425\pi$
$37$	40.3540			0.179302			0.0896509	+	0.995973i	$0.471425\pi$
$38$	−96.5258			−0.412067
$39$		0			0
$40$			328.886i			1.30004i
$41$	358.909			1.36712			0.683562	−	0.729892i	$-0.260430\pi$
$41$	358.909			1.36712			0.683562	+	0.729892i	$0.260430\pi$
$42$		0			0
$43$	507.771			1.80080			0.900400	−	0.435064i	$-0.143274\pi$
$43$	507.771			1.80080			0.900400	+	0.435064i	$0.143274\pi$
$44$			51.3519i			0.175945i
$45$		0			0
$46$	446.867			1.43233
$47$	456.321			1.41620			0.708098	−	0.706114i	$-0.249553\pi$
$47$	456.321			1.41620			0.708098	+	0.706114i	$0.249553\pi$
$48$		0			0
$49$	−308.253	−	150.430i	−0.898697	−	0.438571i
$50$		−	354.506i		−	1.00269i
$51$		0			0
$52$		−	32.2051i		−	0.0858853i
$53$		−	213.420i		−	0.553123i	−0.960996	−	0.276561i	$-0.910805\pi$
$53$		−	213.420i		−	0.553123i	0.960996	−	0.276561i	$-0.0891949\pi$
$54$		0			0
$55$			837.238i			2.05260i
$56$	380.363	+	87.8583i	0.907646	+	0.209653i
$57$		0			0
$58$	440.237			0.996653
$59$	−319.186			−0.704314			−0.352157	−	0.935941i	$-0.614552\pi$
$59$	−319.186			−0.704314			−0.352157	+	0.935941i	$0.614552\pi$
$60$		0			0
$61$			350.638i			0.735978i	0.929830	+	0.367989i	$0.119954\pi$
$61$			350.638i			0.735978i	−0.929830	+	0.367989i	$0.880046\pi$
$62$	211.385			0.432998
$63$		0			0
$64$	−436.974			−0.853466
$65$		−	525.069i		−	1.00195i
$66$		0			0
$67$	578.516			1.05488			0.527440	−	0.849592i	$-0.323152\pi$
$67$	578.516			1.05488			0.527440	+	0.849592i	$0.323152\pi$
$68$	41.6764			0.0743237
$69$		0			0
$70$	−842.651	−	194.640i	−1.43880	−	0.332342i
$71$			787.243i			1.31589i	0.753064	+	0.657947i	$0.228575\pi$
$71$			787.243i			1.31589i	−0.753064	+	0.657947i	$0.771425\pi$
$72$		0			0
$73$		−	146.373i		−	0.234681i	−0.993092	−	0.117340i	$-0.962563\pi$
$73$		−	146.373i		−	0.234681i	0.993092	−	0.117340i	$-0.0374368\pi$
$74$		−	120.773i		−	0.189723i
$75$		0			0
$76$			30.8657i			0.0465860i
$77$	968.283	+	223.659i	1.43307	+	0.331017i
$78$		0			0
$79$	386.191			0.549998			0.274999	−	0.961445i	$-0.411322\pi$
$79$	386.191			0.549998			0.274999	+	0.961445i	$0.411322\pi$
$80$	1103.76			1.54254
$81$		0			0
$82$		−	1074.15i		−	1.44659i
$83$	−197.809			−0.261594			−0.130797	−	0.991409i	$-0.541754\pi$
$83$	−197.809			−0.261594			−0.130797	+	0.991409i	$0.541754\pi$
$84$		0			0
$85$	679.490			0.867071
$86$		−	1519.67i		−	1.90547i
$87$		0			0
$88$	−1131.05			−1.37012
$89$	596.020			0.709865			0.354932	−	0.934892i	$-0.384504\pi$
$89$	596.020			0.709865			0.354932	+	0.934892i	$0.384504\pi$
$90$		0			0
$91$	−607.253	−	140.267i	−0.699532	−	0.161582i
$92$		−	142.893i		−	0.161931i
$93$		0			0
$94$		−	1365.69i		−	1.49851i
$95$			503.233i			0.543480i
$96$		0			0
$97$			729.176i			0.763264i	0.924314	+	0.381632i	$0.124638\pi$
$97$			729.176i			0.763264i	−0.924314	+	0.381632i	$0.875362\pi$
$98$	−450.210	+	922.547i	−0.464062	+	0.950932i
$99$		0			0
$100$	−113.359			−0.113359
$101$	−573.447			−0.564951			−0.282476	−	0.959274i	$-0.591156\pi$
$101$	−573.447			−0.564951			−0.282476	+	0.959274i	$0.591156\pi$
$102$		0			0
$103$			1639.72i			1.56861i	0.620378	+	0.784303i	$0.286979\pi$
$103$			1639.72i			1.56861i	−0.620378	+	0.784303i	$0.713021\pi$
$104$	709.331			0.668804
$105$		0			0
$106$	−638.729			−0.585272
$107$		−	377.596i		−	0.341155i	−0.985344	−	0.170577i	$-0.945437\pi$
$107$		−	377.596i		−	0.341155i	0.985344	−	0.170577i	$-0.0545633\pi$
$108$		0			0
$109$	−1166.06			−1.02467			−0.512334	−	0.858787i	$-0.671219\pi$
$109$	−1166.06			−1.02467			−0.512334	+	0.858787i	$0.671219\pi$
$110$	2505.71			2.17191
$111$		0			0
$112$	294.856	−	1276.52i	0.248761	−	1.07696i
$113$		−	322.241i		−	0.268265i	−0.990963	−	0.134132i	$-0.957175\pi$
$113$		−	322.241i		−	0.268265i	0.990963	−	0.134132i	$-0.0428247\pi$
$114$		0			0
$115$		−	2329.72i		−	1.88911i
$116$		−	140.773i		−	0.112676i
$117$		0			0
$118$			955.269i			0.745252i
$119$	181.518	−	785.844i	0.139830	−	0.605363i
$120$		0			0
$121$	−1548.29			−1.16325
$122$	1049.40			0.778755
$123$		0			0
$124$		−	67.5937i		−	0.0489524i
$125$	102.169			0.0731064
$126$		0			0
$127$	1356.31			0.947663			0.473832	−	0.880615i	$-0.342871\pi$
$127$	1356.31			0.947663			0.473832	+	0.880615i	$0.342871\pi$
$128$			1652.47i			1.14109i
$129$		0			0
$130$	−1571.44			−1.06019
$131$	−1958.00			−1.30589			−0.652943	−	0.757407i	$-0.726466\pi$
$131$	−1958.00			−1.30589			−0.652943	+	0.757407i	$0.726466\pi$
$132$		0			0
$133$	581.999	+	134.433i	0.379441	+	0.0876453i
$134$		−	1731.40i		−	1.11619i
$135$		0			0
$136$			917.942i			0.578771i
$137$			992.344i			0.618844i	0.950925	+	0.309422i	$0.100136\pi$
$137$			992.344i			0.618844i	−0.950925	+	0.309422i	$0.899864\pi$
$138$		0			0
$139$		−	833.880i		−	0.508840i	−0.967094	−	0.254420i	$-0.918115\pi$
$139$		−	833.880i		−	0.508840i	0.967094	−	0.254420i	$-0.0818845\pi$
$140$	−62.2393	+	269.451i	−0.0375727	+	0.162663i
$141$		0			0
$142$	2356.08			1.39238
$143$	1805.73			1.05596
$144$		0			0
$145$		−	2295.15i		−	1.31450i
$146$	−438.070			−0.248321
$147$		0			0
$148$	−38.6190			−0.0214491
$149$			1528.86i			0.840596i	0.907386	+	0.420298i	$0.138075\pi$
$149$			1528.86i			0.840596i	−0.907386	+	0.420298i	$0.861925\pi$
$150$		0			0
$151$	747.142			0.402659			0.201330	−	0.979524i	$-0.435474\pi$
$151$	747.142			0.402659			0.201330	+	0.979524i	$0.435474\pi$
$152$	−679.831			−0.362774
$153$		0			0
$154$	669.373	−	2897.90i	0.350257	−	1.51636i
$155$		−	1102.04i		−	0.571086i
$156$		0			0
$157$		−	1377.80i		−	0.700384i	−0.936678	−	0.350192i	$-0.886116\pi$
$157$		−	1377.80i		−	0.700384i	0.936678	−	0.350192i	$-0.113884\pi$
$158$		−	1155.80i		−	0.581966i
$159$		0			0
$160$		−	672.259i		−	0.332167i
$161$	−2694.37	−	622.360i	−1.31892	−	0.304651i
$162$		0			0
$163$	1900.28			0.913139			0.456569	−	0.889688i	$-0.349078\pi$
$163$	1900.28			0.913139			0.456569	+	0.889688i	$0.349078\pi$
$164$	−343.477			−0.163543
$165$		0			0
$166$			592.007i			0.276799i
$167$	1672.81			0.775124			0.387562	−	0.921844i	$-0.373317\pi$
$167$	1672.81			0.775124			0.387562	+	0.921844i	$0.373317\pi$
$168$		0			0
$169$	1064.55			0.484546
$170$		−	2033.59i		−	0.917468i
$171$		0			0
$172$	−485.939			−0.215422
$173$	−992.796			−0.436306			−0.218153	−	0.975915i	$-0.570003\pi$
$173$	−992.796			−0.436306			−0.218153	+	0.975915i	$0.570003\pi$
$174$		0			0
$175$	−493.726	+	2137.48i	−0.213270	+	0.923304i
$176$			3795.85i			1.62570i
$177$		0			0
$178$		−	1783.78i		−	0.751125i
$179$			145.772i			0.0608687i	0.999537	+	0.0304343i	$0.00968905\pi$
$179$			145.772i			0.0608687i	−0.999537	+	0.0304343i	$0.990311\pi$
$180$		0			0
$181$			2253.56i			0.925448i	0.886502	+	0.462724i	$0.153128\pi$
$181$			2253.56i			0.925448i	−0.886502	+	0.462724i	$0.846872\pi$
$182$	−419.793	+	1817.40i	−0.170973	+	0.740192i
$183$		0			0
$184$	3147.29			1.26098
$185$	−629.642			−0.250228
$186$		0			0
$187$			2336.79i			0.913811i
$188$	−436.701			−0.169413
$189$		0			0
$190$	1506.09			0.575069
$191$		−	3696.72i		−	1.40044i	−0.713925	−	0.700222i	$-0.753084\pi$
$191$		−	3696.72i		−	1.40044i	0.713925	−	0.700222i	$-0.246916\pi$
$192$		0			0
$193$	−4389.51			−1.63712			−0.818559	−	0.574423i	$-0.805226\pi$
$193$	−4389.51			−1.63712			−0.818559	+	0.574423i	$0.805226\pi$
$194$	2182.30			0.807628
$195$		0			0
$196$	294.999	+	143.962i	0.107507	+	0.0524643i
$197$			2110.35i			0.763229i	0.924322	+	0.381615i	$0.124632\pi$
$197$			2110.35i			0.763229i	−0.924322	+	0.381615i	$0.875368\pi$
$198$		0			0
$199$			3443.20i			1.22654i	0.789873	+	0.613271i	$0.210146\pi$
$199$			3443.20i			1.22654i	−0.789873	+	0.613271i	$0.789854\pi$
$200$		−	2496.78i		−	0.882746i
$201$		0			0
$202$			1716.23i			0.597788i
$203$	−2654.39	−	613.125i	−0.917742	−	0.211985i
$204$		0			0
$205$	−5600.03			−1.90792
$206$	4907.40			1.65978
$207$		0			0
$208$		−	2380.54i		−	0.793562i
$209$	−1730.63			−0.572777
$210$		0			0
$211$	3730.16			1.21704			0.608518	−	0.793540i	$-0.291764\pi$
$211$	3730.16			1.21704			0.608518	+	0.793540i	$0.291764\pi$
$212$			204.244i			0.0661676i
$213$		0			0
$214$	−1130.08			−0.360984
$215$	−7922.73			−2.51314
$216$		0			0
$217$	−1274.54	−	294.399i	−0.398715	−	0.0920973i
$218$			3489.83i			1.08422i
$219$		0			0
$220$		−	801.241i		−	0.245544i
$221$		−	1465.50i		−	0.446065i
$222$		0			0
$223$			2724.70i			0.818203i	0.912489	+	0.409101i	$0.134158\pi$
$223$			2724.70i			0.818203i	−0.912489	+	0.409101i	$0.865842\pi$
$224$	−777.481	−	179.587i	−0.231909	−	0.0535675i
$225$		0			0
$226$	−964.411			−0.283857
$227$	−5665.44			−1.65651			−0.828256	−	0.560350i	$-0.810667\pi$
$227$	−5665.44			−1.65651			−0.828256	+	0.560350i	$0.810667\pi$
$228$		0			0
$229$		−	423.575i		−	0.122230i	−0.998131	−	0.0611149i	$-0.980534\pi$
$229$		−	423.575i		−	0.122230i	0.998131	−	0.0611149i	$-0.0194656\pi$
$230$	−6972.45			−1.99891
$231$		0			0
$232$	3100.58			0.877428
$233$			234.803i			0.0660192i	0.999455	+	0.0330096i	$0.0105092\pi$
$233$			234.803i			0.0660192i	−0.999455	+	0.0330096i	$0.989491\pi$
$234$		0			0
$235$	−7119.95			−1.97640
$236$	305.463			0.0842540
$237$		0			0
$238$	−2351.89	−	543.253i	−0.640549	−	0.147957i
$239$			1819.69i			0.492493i	0.969207	+	0.246246i	$0.0791973\pi$
$239$			1819.69i			0.492493i	−0.969207	+	0.246246i	$0.920803\pi$
$240$		0			0
$241$			2723.30i			0.727896i	0.931419	+	0.363948i	$0.118571\pi$
$241$			2723.30i			0.727896i	−0.931419	+	0.363948i	$0.881429\pi$
$242$			4633.75i			1.23086i
$243$		0			0
$244$		−	335.562i		−	0.0880417i
$245$	4809.65	+	2347.15i	1.25419	+	0.612056i
$246$		0			0
$247$	1085.36			0.279593
$248$	1488.78			0.381201
$249$		0			0
$250$		−	305.775i		−	0.0773556i
$251$	4214.66			1.05987			0.529934	−	0.848039i	$-0.322217\pi$
$251$	4214.66			1.05987			0.529934	+	0.848039i	$0.322217\pi$
$252$		0			0
$253$	8011.98			1.99094
$254$		−	4059.21i		−	1.00274i
$255$		0			0
$256$	1449.76			0.353946
$257$	−2777.98			−0.674263			−0.337132	−	0.941457i	$-0.609457\pi$
$257$	−2777.98			−0.674263			−0.337132	+	0.941457i	$0.609457\pi$
$258$		0			0
$259$	−168.202	+	728.194i	−0.0403535	+	0.174702i
$260$			502.494i			0.119859i
$261$		0			0
$262$			5859.95i			1.38179i
$263$			3160.88i			0.741096i	0.928813	+	0.370548i	$0.120830\pi$
$263$			3160.88i			0.741096i	−0.928813	+	0.370548i	$0.879170\pi$
$264$		0			0
$265$			3329.98i			0.771922i
$266$	402.335	−	1741.82i	0.0927396	−	0.401496i
$267$		0			0
$268$	−553.642			−0.126191
$269$	−4337.32			−0.983088			−0.491544	−	0.870853i	$-0.663567\pi$
$269$	−4337.32			−0.983088			−0.491544	+	0.870853i	$0.663567\pi$
$270$		0			0
$271$			8399.64i			1.88281i	0.337276	+	0.941406i	$0.390495\pi$
$271$			8399.64i			1.88281i	−0.337276	+	0.941406i	$0.609505\pi$
$272$	3080.65			0.686735
$273$		0			0
$274$	2969.91			0.654814
$275$		−	6356.01i		−	1.39375i
$276$		0			0
$277$	−3652.90			−0.792351			−0.396176	−	0.918175i	$-0.629663\pi$
$277$	−3652.90			−0.792351			−0.396176	+	0.918175i	$0.629663\pi$
$278$	−2495.66			−0.538415
$279$		0			0
$280$	−5934.79	−	1370.85i	−1.26668	−	0.292585i
$281$			8312.55i			1.76472i	0.470579	+	0.882358i	$0.344045\pi$
$281$			8312.55i			1.76472i	−0.470579	+	0.882358i	$0.655955\pi$
$282$		0			0
$283$		−	1494.92i		−	0.314005i	−0.987598	−	0.157003i	$-0.949817\pi$
$283$		−	1494.92i		−	0.314005i	0.987598	−	0.157003i	$-0.0501831\pi$
$284$		−	753.395i		−	0.157415i
$285$		0			0
$286$		−	5404.23i		−	1.11734i
$287$	−1495.99	+	6476.55i	−0.307684	+	1.33205i
$288$		0			0
$289$	−3016.50			−0.613983
$290$	−6868.99			−1.39090
$291$		0			0
$292$			140.080i			0.0280738i
$293$	−279.676			−0.0557641			−0.0278820	−	0.999611i	$-0.508876\pi$
$293$	−279.676			−0.0557641			−0.0278820	+	0.999611i	$0.508876\pi$
$294$		0			0
$295$	4980.25			0.982920
$296$		−	850.601i		−	0.167028i
$297$		0			0
$298$	4575.60			0.889455
$299$	−5024.67			−0.971853
$300$		0			0
$301$	−2116.47	+	9162.79i	−0.405287	+	1.75460i
$302$		−	2236.07i		−	0.426063i
$303$		0			0
$304$			2281.54i			0.430445i
$305$		−	5470.99i		−	1.02711i
$306$		0			0
$307$		−	10727.5i		−	1.99430i	−0.0754451	−	0.997150i	$-0.524038\pi$
$307$		−	10727.5i		−	1.99430i	0.0754451	−	0.997150i	$-0.475962\pi$
$308$	−926.651	−	214.043i	−0.171431	−	0.0395981i
$309$		0			0
$310$	−3298.23			−0.604280
$311$	−9552.69			−1.74175			−0.870873	−	0.491507i	$-0.836446\pi$
$311$	−9552.69			−1.74175			−0.870873	+	0.491507i	$0.836446\pi$
$312$		0			0
$313$			3689.69i			0.666305i	0.942873	+	0.333153i	$0.108112\pi$
$313$			3689.69i			0.666305i	−0.942873	+	0.333153i	$0.891888\pi$
$314$	−4123.51			−0.741093
$315$		0			0
$316$	−369.586			−0.0657938
$317$			753.424i			0.133491i	0.997770	+	0.0667453i	$0.0212615\pi$
$317$			753.424i			0.133491i	−0.997770	+	0.0667453i	$0.978739\pi$
$318$		0			0
$319$	7893.09			1.38536
$320$	6818.09			1.19107
$321$		0			0
$322$	−1862.61	+	8063.78i	−0.322358	+	1.39558i
$323$			1404.55i			0.241955i
$324$		0			0
$325$			3986.14i			0.680342i
$326$		−	5687.21i		−	0.966214i
$327$		0			0
$328$		−	7565.24i		−	1.27354i
$329$	−1902.02	+	8234.37i	−0.318728	+	1.37986i
$330$		0			0
$331$	6238.68			1.03598			0.517989	−	0.855387i	$-0.326681\pi$
$331$	6238.68			1.03598			0.517989	+	0.855387i	$0.326681\pi$
$332$	189.304			0.0312933
$333$		0			0
$334$		−	5006.42i		−	0.820177i
$335$	−9026.55			−1.47216
$336$		0			0
$337$	7859.52			1.27043			0.635216	−	0.772335i	$-0.280911\pi$
$337$	7859.52			1.27043			0.635216	+	0.772335i	$0.280911\pi$
$338$		−	3186.01i		−	0.512710i
$339$		0			0
$340$	−650.275			−0.103724
$341$	3789.96			0.601871
$342$		0			0
$343$	3999.37	−	4935.45i	0.629579	−	0.776936i
$344$		−	10703.0i		−	1.67753i
$345$		0			0
$346$			2971.27i			0.461665i
$347$			1319.21i			0.204090i	0.994780	+	0.102045i	$0.0325385\pi$
$347$			1319.21i			0.204090i	−0.994780	+	0.102045i	$0.967461\pi$
$348$		0			0
$349$		−	4954.90i		−	0.759970i	−0.924993	−	0.379985i	$-0.875929\pi$
$349$		−	4954.90i		−	0.759970i	0.924993	−	0.379985i	$-0.124071\pi$
$350$	6397.10	+	1477.64i	0.976970	+	0.225666i
$351$		0			0
$352$	2311.92			0.350073
$353$	5381.52			0.811415			0.405707	−	0.914003i	$-0.367025\pi$
$353$	5381.52			0.811415			0.405707	+	0.914003i	$0.367025\pi$
$354$		0			0
$355$		−	12283.3i		−	1.83642i
$356$	−570.394			−0.0849180
$357$		0			0
$358$	436.269			0.0644066
$359$		−	3814.03i		−	0.560715i	−0.959896	−	0.280357i	$-0.909547\pi$
$359$		−	3814.03i		−	0.560715i	0.959896	−	0.280357i	$-0.0904530\pi$
$360$		0			0
$361$	5818.78			0.848343
$362$	6744.53			0.979239
$363$		0			0
$364$	581.144	+	134.236i	0.0836819	+	0.0193293i
$365$			2283.85i			0.327513i
$366$		0			0
$367$		−	12566.7i		−	1.78741i	−0.448657	−	0.893704i	$-0.648097\pi$
$367$		−	12566.7i		−	1.78741i	0.448657	−	0.893704i	$-0.351903\pi$
$368$		−	10562.4i		−	1.49621i
$369$		0			0
$370$			1884.41i			0.264772i
$371$	3851.19	+	889.568i	0.538932	+	0.124485i
$372$		0			0
$373$	−4677.20			−0.649266			−0.324633	−	0.945840i	$-0.605241\pi$
$373$	−4677.20			−0.649266			−0.324633	+	0.945840i	$0.605241\pi$
$374$	6993.59			0.966925
$375$		0			0
$376$		−	9618.54i		−	1.31925i
$377$	−4950.11			−0.676243
$378$		0			0
$379$	−4851.71			−0.657561			−0.328780	−	0.944406i	$-0.606638\pi$
$379$	−4851.71			−0.657561			−0.328780	+	0.944406i	$0.606638\pi$
$380$		−	481.596i		−	0.0650141i
$381$		0			0
$382$	−11063.6			−1.48184
$383$	4912.35			0.655377			0.327689	−	0.944786i	$-0.393730\pi$
$383$	4912.35			0.655377			0.327689	+	0.944786i	$0.393730\pi$
$384$		0			0
$385$	−15108.1	−	3489.74i	−1.99994	−	0.461958i
$386$			13137.0i			1.73227i
$387$		0			0
$388$		−	697.825i		−	0.0913059i
$389$			5336.96i			0.695616i	0.937566	+	0.347808i	$0.113074\pi$
$389$			5336.96i			0.695616i	−0.937566	+	0.347808i	$0.886926\pi$
$390$		0			0
$391$		−	6502.40i		−	0.841024i
$392$	−3170.83	+	6497.50i	−0.408548	+	0.837176i
$393$		0			0
$394$	6315.91			0.807591
$395$	−6025.71			−0.767561
$396$		0			0
$397$		−	1308.17i		−	0.165378i	−0.996575	−	0.0826889i	$-0.973649\pi$
$397$		−	1308.17i		−	0.165378i	0.996575	−	0.0826889i	$-0.0263508\pi$
$398$	10304.9			1.29783
$399$		0			0
$400$	−8379.31			−1.04741
$401$		−	2011.73i		−	0.250526i	−0.992124	−	0.125263i	$-0.960023\pi$
$401$		−	2011.73i		−	0.250526i	0.992124	−	0.125263i	$-0.0399774\pi$
$402$		0			0
$403$	−2376.85			−0.293795
$404$	548.791			0.0675826
$405$		0			0
$406$	−1834.98	+	7944.12i	−0.224306	+	0.971084i
$407$		−	2165.36i		−	0.263717i
$408$		0			0
$409$		−	71.0430i		−	0.00858887i	−0.999991	−	0.00429444i	$-0.998633\pi$
$409$		−	71.0430i		−	0.00858887i	0.999991	−	0.00429444i	$-0.00136697\pi$
$410$			16759.9i			2.01881i
$411$		0			0
$412$		−	1569.22i		−	0.187645i
$413$	1330.42	−	5759.76i	0.158512	−	0.686245i
$414$		0			0
$415$	3086.40			0.365073
$416$	−1449.91			−0.170883
$417$		0			0
$418$			5179.48i			0.606068i
$419$	−3650.29			−0.425605			−0.212803	−	0.977095i	$-0.568259\pi$
$419$	−3650.29			−0.425605			−0.212803	+	0.977095i	$0.568259\pi$
$420$		0			0
$421$	5940.24			0.687671			0.343836	−	0.939030i	$-0.388274\pi$
$421$	5940.24			0.687671			0.343836	+	0.939030i	$0.388274\pi$
$422$		−	11163.7i		−	1.28778i
$423$		0			0
$424$	−4498.57			−0.515259
$425$	−5158.44			−0.588756
$426$		0			0
$427$	−6327.31	−	1461.52i	−0.717096	−	0.165639i
$428$			361.361i			0.0408108i
$429$		0			0
$430$			23711.3i			2.65921i
$431$			7508.08i			0.839098i	0.907733	+	0.419549i	$0.137812\pi$
$431$			7508.08i			0.839098i	−0.907733	+	0.419549i	$0.862188\pi$
$432$		0			0
$433$		−	6880.95i		−	0.763689i	−0.924227	−	0.381845i	$-0.875289\pi$
$433$		−	6880.95i		−	0.763689i	0.924227	−	0.381845i	$-0.124711\pi$
$434$	−881.085	+	3814.47i	−0.0974503	+	0.421890i
$435$		0			0
$436$	1115.93			0.122576
$437$	4815.70			0.527154
$438$		0			0
$439$			5190.54i			0.564307i	0.959369	+	0.282154i	$0.0910488\pi$
$439$			5190.54i			0.564307i	−0.959369	+	0.282154i	$0.908951\pi$
$440$	17647.7			1.91209
$441$		0			0
$442$	−4385.99			−0.471992
$443$			1014.42i			0.108795i	0.998519	+	0.0543977i	$0.0173239\pi$
$443$			1014.42i			0.108795i	−0.998519	+	0.0543977i	$0.982676\pi$
$444$		0			0
$445$	−9299.66			−0.990666
$446$	8154.54			0.865760
$447$		0			0
$448$	1821.38	−	7885.26i	0.192080	−	0.831570i
$449$		−	6361.05i		−	0.668589i	−0.942469	−	0.334295i	$-0.891502\pi$
$449$		−	6361.05i		−	0.668589i	0.942469	−	0.334295i	$-0.108498\pi$
$450$		0			0
$451$		−	19258.7i		−	2.01077i
$452$			308.386i			0.0320913i
$453$		0			0
$454$			16955.7i			1.75279i
$455$	9474.94	+	2188.57i	0.976246	+	0.225498i
$456$		0			0
$457$	5616.06			0.574854			0.287427	−	0.957803i	$-0.407200\pi$
$457$	5616.06			0.574854			0.287427	+	0.957803i	$0.407200\pi$
$458$	−1267.69			−0.129334
$459$		0			0
$460$			2229.55i			0.225986i
$461$	4623.05			0.467065			0.233532	−	0.972349i	$-0.424972\pi$
$461$	4623.05			0.467065			0.233532	+	0.972349i	$0.424972\pi$
$462$		0			0
$463$	−4360.08			−0.437646			−0.218823	−	0.975765i	$-0.570222\pi$
$463$	−4360.08			−0.437646			−0.218823	+	0.975765i	$0.570222\pi$
$464$		−	10405.7i		−	1.04110i
$465$		0			0
$466$	702.725			0.0698565
$467$	10923.9			1.08243			0.541216	−	0.840884i	$-0.317964\pi$
$467$	10923.9			1.08243			0.541216	+	0.840884i	$0.317964\pi$
$468$		0			0
$469$	−2411.34	+	10439.4i	−0.237410	+	1.02782i
$470$			21308.8i			2.09128i
$471$		0			0
$472$			6727.96i			0.656100i
$473$		−	27246.5i		−	2.64861i
$474$		0			0
$475$		−	3820.36i		−	0.369032i
$476$	−173.714	+	752.056i	−0.0167272	+	0.0724169i
$477$		0			0
$478$	5446.01			0.521118
$479$	9595.14			0.915268			0.457634	−	0.889141i	$-0.348697\pi$
$479$	9595.14			0.915268			0.457634	+	0.889141i	$0.348697\pi$
$480$		0			0
$481$			1357.99i			0.128730i
$482$	8150.35			0.770204
$483$		0			0
$484$	1481.72			0.139155
$485$		−	11377.3i		−	1.06519i
$486$		0			0
$487$	−8240.46			−0.766758			−0.383379	−	0.923591i	$-0.625240\pi$
$487$	−8240.46			−0.766758			−0.383379	+	0.923591i	$0.625240\pi$
$488$	7390.91			0.685596
$489$		0			0
$490$	7024.60	−	14394.5i	0.647631	−	1.32709i
$491$			2012.00i			0.184929i	0.995716	+	0.0924645i	$0.0294745\pi$
$491$			2012.00i			0.184929i	−0.995716	+	0.0924645i	$0.970526\pi$
$492$		0			0
$493$		−	6405.91i		−	0.585209i
$494$		−	3248.28i		−	0.295844i
$495$		0			0
$496$		−	4996.42i		−	0.452310i
$497$	−14205.9	−	3281.35i	−1.28214	−	0.296154i
$498$		0			0
$499$	14373.0			1.28943			0.644714	−	0.764424i	$-0.276976\pi$
$499$	14373.0			1.28943			0.644714	+	0.764424i	$0.276976\pi$
$500$	−97.7765			−0.00874540
$501$		0			0
$502$		−	12613.7i		−	1.12147i
$503$	−12261.8			−1.08693			−0.543467	−	0.839430i	$-0.682889\pi$
$503$	−12261.8			−1.08693			−0.543467	+	0.839430i	$0.682889\pi$
$504$		0			0
$505$	8947.46			0.788429
$506$		−	23978.5i		−	2.10667i
$507$		0			0
$508$	−1298.00			−0.113365
$509$	−21970.4			−1.91320			−0.956602	−	0.291396i	$-0.905880\pi$
$509$	−21970.4			−1.91320			−0.956602	+	0.291396i	$0.905880\pi$
$510$		0			0
$511$	2641.32	+	610.107i	0.228660	+	0.0528171i
$512$			8880.88i			0.766569i
$513$		0			0
$514$			8314.01i			0.713454i
$515$		−	25584.5i		−	2.18910i
$516$		0			0
$517$		−	24485.7i		−	2.08294i
$518$	2179.36	+	503.399i	0.184856	+	0.0426990i
$519$		0			0
$520$	−11067.6			−0.933363
$521$	9869.79			0.829949			0.414974	−	0.909833i	$-0.363791\pi$
$521$	9869.79			0.829949			0.414974	+	0.909833i	$0.363791\pi$
$522$		0			0
$523$			4243.21i			0.354766i	0.984142	+	0.177383i	$0.0567632\pi$
$523$			4243.21i			0.354766i	−0.984142	+	0.177383i	$0.943237\pi$
$524$	1873.81			0.156217
$525$		0			0
$526$	9459.97			0.784171
$527$		−	3075.88i		−	0.254245i
$528$		0			0
$529$	−10127.3			−0.832362
$530$	9966.06			0.816788
$531$		0			0
$532$	−556.976	−	128.653i	−0.0453909	−	0.0104846i
$533$			12078.0i			0.981529i
$534$		0			0
$535$			5891.60i			0.476105i
$536$		−	12194.2i		−	0.982668i
$537$		0			0
$538$			12980.8i			1.04023i
$539$	−8071.91	+	16540.5i	−0.645050	+	1.32180i
$540$		0			0
$541$	3649.63			0.290037			0.145018	−	0.989429i	$-0.453676\pi$
$541$	3649.63			0.290037			0.145018	+	0.989429i	$0.453676\pi$
$542$	25138.7			1.99225
$543$		0			0
$544$		−	1876.32i		−	0.147879i
$545$	18194.0			1.42999
$546$		0			0
$547$	−1517.82			−0.118642			−0.0593212	−	0.998239i	$-0.518894\pi$
$547$	−1517.82			−0.118642			−0.0593212	+	0.998239i	$0.518894\pi$
$548$		−	949.678i		−	0.0740296i
$549$		0			0
$550$	−19022.4			−1.47476
$551$	4744.24			0.366809
$552$		0			0
$553$	−1609.70	+	6968.86i	−0.123782	+	0.535888i
$554$			10932.5i			0.838406i
$555$		0			0
$556$			798.027i			0.0608703i
$557$			3101.35i			0.235922i	0.993018	+	0.117961i	$0.0376357\pi$
$557$			3101.35i			0.235922i	−0.993018	+	0.117961i	$0.962364\pi$
$558$		0			0
$559$			17087.5i			1.29289i
$560$	−4600.62	+	19917.4i	−0.347164	+	1.50297i
$561$		0			0
$562$	24878.0			1.86729
$563$	−976.931			−0.0731309			−0.0365655	−	0.999331i	$-0.511642\pi$
$563$	−976.931			−0.0731309			−0.0365655	+	0.999331i	$0.511642\pi$
$564$		0			0
$565$			5027.91i			0.374382i
$566$	−4474.02			−0.332256
$567$		0			0
$568$	16593.9			1.22581
$569$		−	17317.1i		−	1.27587i	−0.770091	−	0.637934i	$-0.779789\pi$
$569$		−	17317.1i		−	1.27587i	0.770091	−	0.637934i	$-0.220211\pi$
$570$		0			0
$571$	19205.4			1.40757			0.703785	−	0.710413i	$-0.251492\pi$
$571$	19205.4			1.40757			0.703785	+	0.710413i	$0.251492\pi$
$572$	−1728.09			−0.126320
$573$		0			0
$574$	19383.2	+	4477.23i	1.40948	+	0.325568i
$575$			17686.4i			1.28274i
$576$		0			0
$577$		−	19299.1i		−	1.39243i	−0.717832	−	0.696216i	$-0.754866\pi$
$577$		−	19299.1i		−	1.39243i	0.717832	−	0.696216i	$-0.245134\pi$
$578$			9027.86i			0.649670i
$579$		0			0
$580$			2196.47i			0.157247i
$581$	824.497	−	3569.48i	0.0588742	−	0.254883i
$582$		0			0
$583$	−11451.9			−0.813533
$584$	−3085.32			−0.218616
$585$		0			0
$586$			837.023i			0.0590053i
$587$	−9381.24			−0.659634			−0.329817	−	0.944045i	$-0.606987\pi$
$587$	−9381.24			−0.659634			−0.329817	+	0.944045i	$0.606987\pi$
$588$		0			0
$589$	2278.00			0.159361
$590$		−	14905.0i		−	1.04005i
$591$		0			0
$592$	−2854.65			−0.198185
$593$	21351.5			1.47859			0.739294	−	0.673383i	$-0.235159\pi$
$593$	21351.5			1.47859			0.739294	+	0.673383i	$0.235159\pi$
$594$		0			0
$595$	−2832.22	+	12261.5i	−0.195142	+	0.844826i
$596$		−	1463.12i		−	0.100557i
$597$		0			0
$598$			15038.0i			1.02834i
$599$			17985.6i			1.22683i	0.789759	+	0.613417i	$0.210206\pi$
$599$			17985.6i			1.22683i	−0.789759	+	0.613417i	$0.789794\pi$
$600$		0			0
$601$			16224.8i			1.10120i	0.834769	+	0.550601i	$0.185601\pi$
$601$			16224.8i			1.10120i	−0.834769	+	0.550601i	$0.814399\pi$
$602$	27422.6	+	6334.22i	1.85658	+	0.428843i
$603$		0			0
$604$	−715.019			−0.0481684
$605$	24157.8			1.62340
$606$		0			0
$607$			11187.9i			0.748111i	0.927406	+	0.374055i	$0.122033\pi$
$607$			11187.9i			0.748111i	−0.927406	+	0.374055i	$0.877967\pi$
$608$	1389.61			0.0926908
$609$		0			0
$610$	−16373.7			−1.08681
$611$			15356.1i			1.01676i
$612$		0			0
$613$	26379.5			1.73810			0.869052	−	0.494720i	$-0.164729\pi$
$613$	26379.5			1.73810			0.869052	+	0.494720i	$0.164729\pi$
$614$	−32105.5			−2.11022
$615$		0			0
$616$	4714.39	−	20409.9i	0.308357	−	1.33497i
$617$			21927.5i			1.43074i	0.698745	+	0.715371i	$0.253742\pi$
$617$			21927.5i			1.43074i	−0.698745	+	0.715371i	$0.746258\pi$
$618$		0			0
$619$			17343.2i			1.12614i	0.826409	+	0.563071i	$0.190380\pi$
$619$			17343.2i			1.12614i	−0.826409	+	0.563071i	$0.809620\pi$
$620$			1054.66i			0.0683165i
$621$		0			0
$622$			28589.5i			1.84298i
$623$	−2484.30	+	10755.2i	−0.159762	+	0.691653i
$624$		0			0
$625$	−16400.6			−1.04964
$626$	11042.6			0.705033
$627$		0			0
$628$			1318.56i			0.0837839i
$629$	−1757.37			−0.111401
$630$		0			0
$631$	−30543.8			−1.92699			−0.963495	−	0.267727i	$-0.913727\pi$
$631$	−30543.8			−1.92699			−0.963495	+	0.267727i	$0.913727\pi$
$632$		−	8140.30i		−	0.512348i
$633$		0			0
$634$	2254.87			0.141250
$635$	−21162.5			−1.32253
$636$		0			0
$637$	5062.25	−	10373.3i	0.314872	−	0.645220i
$638$		−	23622.7i		−	1.46588i
$639$		0			0
$640$		−	25783.4i		−	1.59247i
$641$			592.867i			0.0365317i	0.999833	+	0.0182659i	$0.00581452\pi$
$641$			592.867i			0.0365317i	−0.999833	+	0.0182659i	$0.994185\pi$
$642$		0			0
$643$			31816.0i			1.95132i	0.219282	+	0.975661i	$0.429628\pi$
$643$			31816.0i			1.95132i	−0.219282	+	0.975661i	$0.570372\pi$
$644$	2578.52	+	595.601i	0.157777	+	0.0364440i
$645$		0			0
$646$	4203.59			0.256018
$647$	−14337.8			−0.871218			−0.435609	−	0.900136i	$-0.643467\pi$
$647$	−14337.8			−0.871218			−0.435609	+	0.900136i	$0.643467\pi$
$648$		0			0
$649$			17127.2i			1.03591i
$650$	11929.8			0.719886
$651$		0			0
$652$	−1818.58			−0.109235
$653$			1513.23i			0.0906852i	0.998971	+	0.0453426i	$0.0144379\pi$
$653$			1513.23i			0.0906852i	−0.998971	+	0.0453426i	$0.985562\pi$
$654$		0			0
$655$	30550.5			1.82246
$656$	−25389.3			−1.51110
$657$		0			0
$658$	24644.0	+	5692.40i	1.46007	+	0.337254i
$659$		−	3167.89i		−	0.187259i	−0.995607	−	0.0936293i	$-0.970153\pi$
$659$		−	3167.89i		−	0.187259i	0.995607	−	0.0936293i	$-0.0298468\pi$
$660$		0			0
$661$		−	5720.72i		−	0.336626i	−0.985734	−	0.168313i	$-0.946168\pi$
$661$		−	5720.72i		−	0.336626i	0.985734	−	0.168313i	$-0.0538320\pi$
$662$		−	18671.3i		−	1.09619i
$663$		0			0
$664$			4169.50i			0.243687i
$665$	−9080.90	−	2097.55i	−0.529537	−	0.122315i
$666$		0			0
$667$	−21963.5			−1.27501
$668$	−1600.88			−0.0927247
$669$		0			0
$670$			27014.9i			1.55773i
$671$	18814.9			1.08248
$672$		0			0
$673$	−6740.56			−0.386077			−0.193038	−	0.981191i	$-0.561834\pi$
$673$	−6740.56			−0.386077			−0.193038	+	0.981191i	$0.561834\pi$
$674$		−	23522.2i		−	1.34427i
$675$		0			0
$676$	−1018.78			−0.0579641
$677$	4419.27			0.250881			0.125440	−	0.992101i	$-0.459966\pi$
$677$	4419.27			0.250881			0.125440	+	0.992101i	$0.459966\pi$
$678$		0			0
$679$	−13158.1	−	3039.32i	−0.743682	−	0.171780i
$680$		−	14322.6i		−	0.807716i
$681$		0			0
$682$		−	11342.7i		−	0.636854i
$683$		−	25188.7i		−	1.41115i	−0.708633	−	0.705577i	$-0.750688\pi$
$683$		−	25188.7i		−	1.41115i	0.708633	−	0.705577i	$-0.249312\pi$
$684$		0			0
$685$		−	15483.5i		−	0.863640i
$686$	−14770.9	−	11969.4i	−0.822095	−	0.666173i
$687$		0			0
$688$	−35919.8			−1.99045
$689$	7182.00			0.397115
$690$		0			0
$691$		−	24.4326i		−	0.00134510i	−1.00000		0.000672548i	$-0.999786\pi$
$691$		−	24.4326i		−	0.00134510i	1.00000		0.000672548i	$-0.000214079\pi$
$692$	950.111			0.0521933
$693$		0			0
$694$	3948.18			0.215952
$695$			13011.0i			0.710122i
$696$		0			0
$697$	−15630.0			−0.849398
$698$	−14829.1			−0.804142
$699$		0			0
$700$	472.498	−	2045.58i	0.0255125	−	0.110451i
$701$		−	5989.84i		−	0.322729i	−0.986895	−	0.161365i	$-0.948411\pi$
$701$		−	5989.84i		−	0.322729i	0.986895	−	0.161365i	$-0.0515895\pi$
$702$		0			0
$703$		−	1301.52i		−	0.0698259i
$704$			23447.6i			1.25528i
$705$		0			0
$706$		−	16106.0i		−	0.858577i
$707$	2390.22	−	10347.9i	0.127148	−	0.550457i
$708$		0			0
$709$	4428.28			0.234566			0.117283	−	0.993099i	$-0.462582\pi$
$709$	4428.28			0.234566			0.117283	+	0.993099i	$0.462582\pi$
$710$	−36761.8			−1.94316
$711$		0			0
$712$		−	12563.2i		−	0.661271i
$713$	−10546.0			−0.553931
$714$		0			0
$715$	−28174.7			−1.47367
$716$		−	139.504i		−	0.00728145i
$717$		0			0
$718$	−11414.7			−0.593306
$719$	28053.6			1.45511			0.727554	−	0.686050i	$-0.240657\pi$
$719$	28053.6			1.45511			0.727554	+	0.686050i	$0.240657\pi$
$720$		0			0
$721$	−29589.0	−	6834.61i	−1.52836	−	0.353030i
$722$		−	17414.6i		−	0.897651i
$723$		0			0
$724$		−	2156.67i		−	0.110707i
$725$			17424.0i			0.892565i
$726$		0			0
$727$		−	15727.1i		−	0.802318i	−0.916008	−	0.401159i	$-0.868607\pi$
$727$		−	15727.1i		−	0.802318i	0.916008	−	0.401159i	$-0.131393\pi$
$728$	−2956.60	+	12800.0i	−0.150521	+	0.651646i
$729$		0			0
$730$	6835.18			0.346550
$731$	−22112.8			−1.11884
$732$		0			0
$733$		−	32959.7i		−	1.66084i	−0.557141	−	0.830418i	$-0.688102\pi$
$733$		−	32959.7i		−	1.66084i	0.557141	−	0.830418i	$-0.311898\pi$
$734$	−37610.1			−1.89130
$735$		0			0
$736$	−6433.20			−0.322189
$737$		−	31042.6i		−	1.55152i
$738$		0			0
$739$	18483.4			0.920056			0.460028	−	0.887905i	$-0.347839\pi$
$739$	18483.4			0.920056			0.460028	+	0.887905i	$0.347839\pi$
$740$	602.570			0.0299337
$741$		0			0
$742$	2662.32	−	11525.9i	0.131721	−	0.570257i
$743$			32277.2i			1.59372i	0.604163	+	0.796861i	$0.293508\pi$
$743$			32277.2i			1.59372i	−0.604163	+	0.796861i	$0.706492\pi$
$744$		0			0
$745$		−	23854.7i		−	1.17311i
$746$			13998.0i			0.687004i
$747$		0			0
$748$		−	2236.31i		−	0.109315i
$749$	6813.76	+	1573.88i	0.332402	+	0.0767800i
$750$		0			0
$751$	−25750.6			−1.25120			−0.625601	−	0.780143i	$-0.715146\pi$
$751$	−25750.6			−1.25120			−0.625601	+	0.780143i	$0.715146\pi$
$752$	−32280.2			−1.56534
$753$		0			0
$754$			14814.8i			0.715549i
$755$	−11657.6			−0.561939
$756$		0			0
$757$	14678.2			0.704741			0.352371	−	0.935861i	$-0.385376\pi$
$757$	14678.2			0.704741			0.352371	+	0.935861i	$0.385376\pi$
$758$			14520.3i			0.695781i
$759$		0			0
$760$	10607.4			0.506276
$761$	30497.5			1.45274			0.726369	−	0.687305i	$-0.241207\pi$
$761$	30497.5			1.45274			0.726369	+	0.687305i	$0.241207\pi$
$762$		0			0
$763$	4860.34	−	21041.8i	0.230611	−	0.998379i
$764$			3537.77i			0.167529i
$765$		0			0
$766$		−	14701.8i		−	0.693470i
$767$		−	10741.2i		−	0.505663i
$768$		0			0
$769$			35356.4i			1.65798i	0.559264	+	0.828989i	$0.311084\pi$
$769$			35356.4i			1.65798i	−0.559264	+	0.828989i	$0.688916\pi$
$770$	−10444.2	+	45215.8i	−0.488808	+	2.11619i
$771$		0			0
$772$	4200.78			0.195841
$773$	4097.47			0.190654			0.0953272	−	0.995446i	$-0.469610\pi$
$773$	4097.47			0.190654			0.0953272	+	0.995446i	$0.469610\pi$
$774$		0			0
$775$			8366.32i			0.387777i
$776$	15369.9			0.711015
$777$		0			0
$778$	15972.6			0.736048
$779$		−	11575.7i		−	0.532402i
$780$		0			0
$781$	42242.6			1.93542
$782$	−19460.6			−0.889908
$783$		0			0
$784$	21805.9	+	10641.4i	0.993343	+	0.484759i
$785$			21497.7i			0.977435i
$786$		0			0
$787$		−	33723.5i		−	1.52746i	−0.645535	−	0.763730i	$-0.723366\pi$
$787$		−	33723.5i		−	1.52746i	0.645535	−	0.763730i	$-0.276634\pi$
$788$		−	2019.61i		−	0.0913017i
$789$		0			0
$790$			18033.9i			0.812174i
$791$	5814.88	+	1343.15i	0.261382	+	0.0603754i
$792$		0			0
$793$	−11799.7			−0.528396
$794$	−3915.11			−0.174990
$795$		0			0
$796$		−	3295.15i		−	0.146726i
$797$	−14265.6			−0.634020			−0.317010	−	0.948422i	$-0.602679\pi$
$797$	−14265.6			−0.634020			−0.317010	+	0.948422i	$0.602679\pi$
$798$		0			0
$799$	−19872.2			−0.879886
$800$			5103.54i			0.225547i
$801$		0			0
$802$	−6020.75			−0.265087
$803$	−7854.24			−0.345168
$804$		0			0
$805$	42040.1	+	9710.64i	1.84064	+	0.425162i
$806$			7113.51i			0.310872i
$807$		0			0
$808$			12087.4i			0.526277i
$809$			13916.9i			0.604812i	0.953179	+	0.302406i	$0.0977898\pi$
$809$			13916.9i			0.604812i	−0.953179	+	0.302406i	$0.902210\pi$
$810$		0			0
$811$			7296.14i			0.315909i	0.987446	+	0.157954i	$0.0504899\pi$
$811$			7296.14i			0.315909i	−0.987446	+	0.157954i	$0.949510\pi$
$812$	2540.26	+	586.763i	0.109785	+	0.0253588i
$813$		0			0
$814$	−6480.54			−0.279045
$815$	−29650.0			−1.27435
$816$		0			0
$817$		−	16376.8i		−	0.701289i
$818$	−212.619			−0.00908809
$819$		0			0
$820$	5359.26			0.228236
$821$		−	34952.1i		−	1.48579i	−0.669406	−	0.742897i	$-0.733451\pi$
$821$		−	34952.1i		−	1.48579i	0.669406	−	0.742897i	$-0.266549\pi$
$822$		0			0
$823$	23530.6			0.996628			0.498314	−	0.866997i	$-0.333953\pi$
$823$	23530.6			0.996628			0.498314	+	0.866997i	$0.333953\pi$
$824$	34562.8			1.46123
$825$		0			0
$826$	−17238.0	−	3981.71i	−0.726132	−	0.167726i
$827$			40158.9i			1.68859i	0.535880	+	0.844294i	$0.319980\pi$
$827$			40158.9i			1.68859i	−0.535880	+	0.844294i	$0.680020\pi$
$828$		0			0
$829$		−	10739.4i		−	0.449934i	−0.974366	−	0.224967i	$-0.927773\pi$
$829$		−	10739.4i		−	0.449934i	0.974366	−	0.224967i	$-0.0722275\pi$
$830$		−	9237.05i		−	0.386292i
$831$		0			0
$832$		−	14705.0i		−	0.612747i
$833$	13424.1	+	6551.04i	0.558362	+	0.272485i
$834$		0			0
$835$	−26100.7			−1.08174
$836$	1656.22			0.0685187
$837$		0			0
$838$			10924.7i			0.450343i
$839$	−18570.1			−0.764137			−0.382068	−	0.924134i	$-0.624788\pi$
$839$	−18570.1			−0.764137			−0.382068	+	0.924134i	$0.624788\pi$
$840$		0			0
$841$	2751.38			0.112812
$842$		−	17778.1i		−	0.727641i
$843$		0			0
$844$	−3569.78			−0.145589
$845$	−16610.1			−0.676218
$846$		0			0
$847$	6453.50	−	27939.0i	0.261801	−	1.13341i
$848$			15097.4i			0.611375i
$849$		0			0
$850$			15438.3i			0.622976i
$851$			6025.38i			0.242711i
$852$		0			0
$853$			5384.45i			0.216131i	0.994144	+	0.108066i	$0.0344657\pi$
$853$			5384.45i			0.216131i	−0.994144	+	0.108066i	$0.965534\pi$
$854$	−4374.06	+	18936.5i	−0.175266	+	0.758776i
$855$		0			0
$856$	−7959.13			−0.317801
$857$	−11057.6			−0.440747			−0.220373	−	0.975416i	$-0.570728\pi$
$857$	−11057.6			−0.440747			−0.220373	+	0.975416i	$0.570728\pi$
$858$		0			0
$859$		−	16323.9i		−	0.648386i	−0.945991	−	0.324193i	$-0.894907\pi$
$859$		−	16323.9i		−	0.648386i	0.945991	−	0.324193i	$-0.105093\pi$
$860$	7582.08			0.300636
$861$		0			0
$862$	22470.4			0.887869
$863$		−	37617.5i		−	1.48379i	−0.670515	−	0.741896i	$-0.733927\pi$
$863$		−	37617.5i		−	1.48379i	0.670515	−	0.741896i	$-0.266073\pi$
$864$		0			0
$865$	15490.5			0.608895
$866$	−20593.5			−0.808078
$867$		0			0
$868$	1219.74	+	281.741i	0.0476965	+	0.0110172i
$869$		−	20722.6i		−	0.808937i
$870$		0			0
$871$			19468.2i			0.757352i
$872$			24578.8i			0.954523i
$873$		0			0
$874$		−	14412.6i		−	0.557794i
$875$	−425.858	+	1843.66i	−0.0164533	+	0.0712309i
$876$		0			0
$877$	42140.3			1.62255			0.811275	−	0.584664i	$-0.198774\pi$
$877$	42140.3			1.62255			0.811275	+	0.584664i	$0.198774\pi$
$878$	15534.4			0.597107
$879$		0			0
$880$		−	59226.4i		−	2.26877i
$881$	−17255.0			−0.659861			−0.329930	−	0.944005i	$-0.607025\pi$
$881$	−17255.0			−0.659861			−0.329930	+	0.944005i	$0.607025\pi$
$882$		0			0
$883$	−26461.1			−1.00848			−0.504240	−	0.863564i	$-0.668227\pi$
$883$	−26461.1			−1.00848			−0.504240	+	0.863564i	$0.668227\pi$
$884$			1402.49i			0.0533608i
$885$		0			0
$886$	3035.97			0.115119
$887$	−19366.5			−0.733103			−0.366552	−	0.930398i	$-0.619462\pi$
$887$	−19366.5			−0.733103			−0.366552	+	0.930398i	$0.619462\pi$
$888$		0			0
$889$	−5653.32	+	24474.8i	−0.213280	+	0.923351i
$890$			27832.3i			1.04825i
$891$		0			0
$892$		−	2607.55i		−	0.0978780i
$893$		−	14717.4i		−	0.551512i
$894$		0			0
$895$		−	2274.47i		−	0.0849465i
$896$	−29819.0	−	6887.76i	−1.11181	−	0.256812i
$897$		0			0
$898$	−19037.5			−0.707450
$899$	−10389.6			−0.385441
$900$		0			0
$901$			9294.19i			0.343656i
$902$	−57637.9			−2.12764
$903$		0			0
$904$	−6792.34			−0.249900
$905$		−	35162.2i		−	1.29153i
$906$		0			0
$907$	−1139.38			−0.0417115			−0.0208558	−	0.999782i	$-0.506639\pi$
$907$	−1139.38			−0.0417115			−0.0208558	+	0.999782i	$0.506639\pi$
$908$	5421.85			0.198161
$909$		0			0
$910$	6550.01	−	28356.8i	0.238605	−	1.03299i
$911$			8696.33i			0.316270i	0.987417	+	0.158135i	$0.0505482\pi$
$911$			8696.33i			0.316270i	−0.987417	+	0.158135i	$0.949452\pi$
$912$		0			0
$913$			10614.2i			0.384753i
$914$		−	16807.9i		−	0.608267i
$915$		0			0
$916$			405.363i			0.0146218i
$917$	8161.24	−	35332.3i	0.293902	−	1.27238i
$918$		0			0
$919$	−2297.46			−0.0824659			−0.0412330	−	0.999150i	$-0.513129\pi$
$919$	−2297.46			−0.0824659			−0.0412330	+	0.999150i	$0.513129\pi$
$920$	−49106.9			−1.75979
$921$		0			0
$922$		−	13836.0i		−	0.494212i
$923$	−26492.2			−0.944748
$924$		0			0
$925$	4780.01			0.169909
$926$			13048.9i			0.463083i
$927$		0			0
$928$	−6337.74			−0.224188
$929$	48395.7			1.70916			0.854581	−	0.519318i	$-0.173814\pi$
$929$	48395.7			1.70916			0.854581	+	0.519318i	$0.173814\pi$
$930$		0			0
$931$	−4851.72	+	9941.90i	−0.170794	+	0.349981i
$932$		−	224.708i		−	0.00789759i
$933$		0			0
$934$		−	32693.2i		−	1.14535i
$935$		−	36460.7i		−	1.27529i
$936$		0			0
$937$		−	23667.6i		−	0.825172i	−0.910919	−	0.412586i	$-0.864626\pi$
$937$		−	23667.6i		−	0.825172i	0.910919	−	0.412586i	$-0.135374\pi$
$938$	31243.3	+	7216.73i	1.08756	+	0.251210i
$939$		0			0
$940$	6813.82			0.236428
$941$	10199.0			0.353323			0.176661	−	0.984272i	$-0.443470\pi$
$941$	10199.0			0.353323			0.176661	+	0.984272i	$0.443470\pi$
$942$		0			0
$943$			53589.7i			1.85061i
$944$	22579.3			0.778489
$945$		0			0
$946$	−81544.0			−2.80256
$947$			3398.95i			0.116633i	0.998298	+	0.0583163i	$0.0185732\pi$
$947$			3398.95i			0.116633i	−0.998298	+	0.0583163i	$0.981427\pi$
$948$		0			0
$949$	4925.74			0.168489
$950$	−11433.7			−0.390481
$951$		0			0
$952$	−16564.4	−	3826.13i	−0.563923	−	0.130258i
$953$			27410.8i			0.931714i	0.884860	+	0.465857i	$0.154254\pi$
$953$			27410.8i			0.931714i	−0.884860	+	0.465857i	$0.845746\pi$
$954$		0			0
$955$			57679.7i			1.95442i
$956$		−	1741.45i		−	0.0589147i
$957$		0			0
$958$		−	28716.6i		−	0.968466i
$959$	−17907.0	−	4136.24i	−0.602968	−	0.139277i
$960$		0			0
$961$	24802.3			0.832544
$962$	4064.23			0.136212
$963$		0			0
$964$		−	2606.21i		−	0.0870750i
$965$	68489.2			2.28471
$966$		0			0
$967$	−37023.1			−1.23121			−0.615606	−	0.788054i	$-0.711089\pi$
$967$	−37023.1			−1.23121			−0.615606	+	0.788054i	$0.711089\pi$
$968$			32635.5i			1.08362i
$969$		0			0
$970$	−34050.2			−1.12710
$971$	17050.2			0.563509			0.281754	−	0.959487i	$-0.409084\pi$
$971$	17050.2			0.563509			0.281754	+	0.959487i	$0.409084\pi$
$972$		0			0
$973$	15047.5	+	3475.74i	0.495786	+	0.114519i
$974$			24662.3i			0.811325i
$975$		0			0
$976$		−	24804.2i		−	0.813487i
$977$		−	16373.7i		−	0.536171i	−0.963395	−	0.268086i	$-0.913609\pi$
$977$		−	16373.7i		−	0.536171i	0.963395	−	0.268086i	$-0.0863910\pi$
$978$		0			0
$979$		−	31981.8i		−	1.04407i
$980$	−4602.86	−	2246.23i	−0.150034	−	0.0732176i
$981$		0			0
$982$	6021.56			0.195678
$983$	21538.3			0.698845			0.349423	−	0.936965i	$-0.386378\pi$
$983$	21538.3			0.698845			0.349423	+	0.936965i	$0.386378\pi$
$984$		0			0
$985$		−	32927.7i		−	1.06514i
$986$	−19171.8			−0.619223
$987$		0			0
$988$	−1038.69			−0.0334465
$989$			75816.8i			2.43765i
$990$		0			0
$991$	39248.1			1.25808			0.629040	−	0.777373i	$-0.283448\pi$
$991$	39248.1			1.25808			0.629040	+	0.777373i	$0.283448\pi$
$992$	−3043.14			−0.0973990
$993$		0			0
$994$	−9820.51	+	42515.8i	−0.313368	+	1.35666i
$995$		−	53724.0i		−	1.71172i
$996$		0			0
$997$			3986.86i			0.126645i	0.997993	+	0.0633226i	$0.0201697\pi$
$997$			3986.86i			0.126645i	−0.997993	+	0.0633226i	$0.979830\pi$
$998$		−	43015.9i		−	1.36437i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.c.c.566.7		44
3.2	odd	2	inner	567.4.c.c.566.38		44
7.6	odd	2	inner	567.4.c.c.566.37		44
9.2	odd	6		63.4.o.a.41.17	yes	44
9.4	even	3		63.4.o.a.20.18	yes	44
9.5	odd	6		189.4.o.a.62.5		44
9.7	even	3		189.4.o.a.125.6		44
21.20	even	2	inner	567.4.c.c.566.8		44
63.13	odd	6		63.4.o.a.20.17	✓	44
63.20	even	6		63.4.o.a.41.18	yes	44
63.34	odd	6		189.4.o.a.125.5		44
63.41	even	6		189.4.o.a.62.6		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.o.a.20.17	✓	44	63.13	odd	6
63.4.o.a.20.18	yes	44	9.4	even	3
63.4.o.a.41.17	yes	44	9.2	odd	6
63.4.o.a.41.18	yes	44	63.20	even	6
189.4.o.a.62.5		44	9.5	odd	6
189.4.o.a.62.6		44	63.41	even	6
189.4.o.a.125.5		44	63.34	odd	6
189.4.o.a.125.6		44	9.7	even	3
567.4.c.c.566.7		44	1.1	even	1	trivial
567.4.c.c.566.8		44	21.20	even	2	inner
567.4.c.c.566.37		44	7.6	odd	2	inner
567.4.c.c.566.38		44	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.99283i q^{2} -0.957005 q^{4} -15.6029 q^{5} +(-4.16816 + 18.0451i) q^{7} -21.0785i q^{8} +O(q^{10})\)	\(q-2.99283i q^{2} -0.957005 q^{4} -15.6029 q^{5} +(-4.16816 + 18.0451i) q^{7} -21.0785i q^{8} +46.6969i q^{10} -53.6590i q^{11} +33.6519i q^{13} +(54.0059 + 12.4746i) q^{14} -70.7402 q^{16} -43.5488 q^{17} -32.2524i q^{19} +14.9321 q^{20} -160.592 q^{22} +149.313i q^{23} +118.452 q^{25} +100.714 q^{26} +(3.98894 - 17.2693i) q^{28} +147.097i q^{29} +70.6305i q^{31} +43.0854i q^{32} +130.334i q^{34} +(65.0355 - 281.557i) q^{35} +40.3540 q^{37} -96.5258 q^{38} +328.886i q^{40} +358.909 q^{41} +507.771 q^{43} +51.3519i q^{44} +446.867 q^{46} +456.321 q^{47} +(-308.253 - 150.430i) q^{49} -354.506i q^{50} -32.2051i q^{52} -213.420i q^{53} +837.238i q^{55} +(380.363 + 87.8583i) q^{56} +440.237 q^{58} -319.186 q^{59} +350.638i q^{61} +211.385 q^{62} -436.974 q^{64} -525.069i q^{65} +578.516 q^{67} +41.6764 q^{68} +(-842.651 - 194.640i) q^{70} +787.243i q^{71} -146.373i q^{73} -120.773i q^{74} +30.8657i q^{76} +(968.283 + 223.659i) q^{77} +386.191 q^{79} +1103.76 q^{80} -1074.15i q^{82} -197.809 q^{83} +679.490 q^{85} -1519.67i q^{86} -1131.05 q^{88} +596.020 q^{89} +(-607.253 - 140.267i) q^{91} -142.893i q^{92} -1365.69i q^{94} +503.233i q^{95} +729.176i q^{97} +(-450.210 + 922.547i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 156 q^{4} - 10 q^{7}+O(q^{10}) \) 44 * q - 156 * q^4 - 10 * q^7	\( 44 q - 156 q^{4} - 10 q^{7} + 484 q^{16} + 68 q^{22} + 704 q^{25} + 300 q^{28} + 328 q^{37} + 340 q^{43} + 968 q^{46} + 158 q^{49} + 1076 q^{58} - 808 q^{64} + 1180 q^{67} - 768 q^{70} + 604 q^{79} + 1224 q^{85} - 2588 q^{88} + 210 q^{91}+O(q^{100}) \) 44 * q - 156 * q^4 - 10 * q^7 + 484 * q^16 + 68 * q^22 + 704 * q^25 + 300 * q^28 + 328 * q^37 + 340 * q^43 + 968 * q^46 + 158 * q^49 + 1076 * q^58 - 808 * q^64 + 1180 * q^67 - 768 * q^70 + 604 * q^79 + 1224 * q^85 - 2588 * q^88 + 210 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more