LMFDB - Embedded newform 588.4.i.f.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	588.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.4.i.f.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.50000 - 2.59808i) q^{3} +(-3.00000 - 5.19615i) q^{5} +(-4.50000 - 7.79423i) q^{9} +(-18.0000 + 31.1769i) q^{11} +62.0000 q^{13} -18.0000 q^{15} +(-57.0000 + 98.7269i) q^{17} +(38.0000 + 65.8179i) q^{19} +(12.0000 + 20.7846i) q^{23} +(44.5000 - 77.0763i) q^{25} -27.0000 q^{27} +54.0000 q^{29} +(56.0000 - 96.9948i) q^{31} +(54.0000 + 93.5307i) q^{33} +(89.0000 + 154.153i) q^{37} +(93.0000 - 161.081i) q^{39} +378.000 q^{41} -172.000 q^{43} +(-27.0000 + 46.7654i) q^{45} +(96.0000 + 166.277i) q^{47} +(171.000 + 296.181i) q^{51} +(201.000 - 348.142i) q^{53} +216.000 q^{55} +228.000 q^{57} +(-198.000 + 342.946i) q^{59} +(-127.000 - 219.970i) q^{61} +(-186.000 - 322.161i) q^{65} +(506.000 - 876.418i) q^{67} +72.0000 q^{69} +840.000 q^{71} +(-445.000 + 770.763i) q^{73} +(-133.500 - 231.229i) q^{75} +(-40.0000 - 69.2820i) q^{79} +(-40.5000 + 70.1481i) q^{81} -108.000 q^{83} +684.000 q^{85} +(81.0000 - 140.296i) q^{87} +(819.000 + 1418.55i) q^{89} +(-168.000 - 290.985i) q^{93} +(228.000 - 394.908i) q^{95} +1010.00 q^{97} +324.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 6 q^{5} - 9 q^{9} - 36 q^{11} + 124 q^{13} - 36 q^{15} - 114 q^{17} + 76 q^{19} + 24 q^{23} + 89 q^{25} - 54 q^{27} + 108 q^{29} + 112 q^{31} + 108 q^{33} + 178 q^{37} + 186 q^{39} + 756 q^{41}+ \cdots + 648 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 6 * q^5 - 9 * q^9 - 36 * q^11 + 124 * q^13 - 36 * q^15 - 114 * q^17 + 76 * q^19 + 24 * q^23 + 89 * q^25 - 54 * q^27 + 108 * q^29 + 112 * q^31 + 108 * q^33 + 178 * q^37 + 186 * q^39 + 756 * q^41 - 344 * q^43 - 54 * q^45 + 192 * q^47 + 342 * q^51 + 402 * q^53 + 432 * q^55 + 456 * q^57 - 396 * q^59 - 254 * q^61 - 372 * q^65 + 1012 * q^67 + 144 * q^69 + 1680 * q^71 - 890 * q^73 - 267 * q^75 - 80 * q^79 - 81 * q^81 - 216 * q^83 + 1368 * q^85 + 162 * q^87 + 1638 * q^89 - 336 * q^93 + 456 * q^95 + 2020 * q^97 + 648 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.50000	−	2.59808i	0.288675	−	0.500000i
$3$	1.50000	−	2.59808i	0.288675	−	0.500000i
$4$		0			0
$5$	−3.00000	−	5.19615i	−0.268328	−	0.464758i	0.700102	−	0.714043i	$-0.253138\pi$
$5$	−3.00000	−	5.19615i	−0.268328	−	0.464758i	−0.968430	+	0.249285i	$0.919804\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.50000	−	7.79423i	−0.166667	−	0.288675i
$10$		0			0
$11$	−18.0000	+	31.1769i	−0.493382	+	0.854563i	−0.999971	−	0.00762479i	$-0.997573\pi$
$11$	−18.0000	+	31.1769i	−0.493382	+	0.854563i	0.506589	+	0.862188i	$0.330906\pi$
$12$		0			0
$13$	62.0000			1.32275			0.661373	−	0.750057i	$-0.269974\pi$
$13$	62.0000			1.32275			0.661373	+	0.750057i	$0.269974\pi$
$14$		0			0
$15$	−18.0000			−0.309839
$16$		0			0
$17$	−57.0000	+	98.7269i	−0.813208	+	1.40852i	0.0974001	+	0.995245i	$0.468947\pi$
$17$	−57.0000	+	98.7269i	−0.813208	+	1.40852i	−0.910608	+	0.413272i	$0.864386\pi$
$18$		0			0
$19$	38.0000	+	65.8179i	0.458831	+	0.794719i	0.998899	−	0.0469020i	$-0.0149348\pi$
$19$	38.0000	+	65.8179i	0.458831	+	0.794719i	−0.540068	+	0.841621i	$0.681602\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	12.0000	+	20.7846i	0.108790	+	0.188430i	0.915280	−	0.402817i	$-0.131969\pi$
$23$	12.0000	+	20.7846i	0.108790	+	0.188430i	−0.806490	+	0.591247i	$0.798636\pi$
$24$		0			0
$25$	44.5000	−	77.0763i	0.356000	−	0.616610i
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	54.0000			0.345778			0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000			0.345778			0.172889	+	0.984941i	$0.444690\pi$
$30$		0			0
$31$	56.0000	−	96.9948i	0.324448	−	0.561961i	−0.656952	−	0.753932i	$-0.728155\pi$
$31$	56.0000	−	96.9948i	0.324448	−	0.561961i	0.981401	+	0.191971i	$0.0614880\pi$
$32$		0			0
$33$	54.0000	+	93.5307i	0.284854	+	0.493382i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	89.0000	+	154.153i	0.395446	+	0.684933i	0.993158	−	0.116778i	$-0.0372566\pi$
$37$	89.0000	+	154.153i	0.395446	+	0.684933i	−0.597712	+	0.801711i	$0.703923\pi$
$38$		0			0
$39$	93.0000	−	161.081i	0.381844	−	0.661373i
$40$		0			0
$41$	378.000			1.43985			0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000			1.43985			0.719923	+	0.694054i	$0.244177\pi$
$42$		0			0
$43$	−172.000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−172.000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	−27.0000	+	46.7654i	−0.0894427	+	0.154919i
$46$		0			0
$47$	96.0000	+	166.277i	0.297937	+	0.516042i	0.975664	−	0.219272i	$-0.0703681\pi$
$47$	96.0000	+	166.277i	0.297937	+	0.516042i	−0.677727	+	0.735314i	$0.737035\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	171.000	+	296.181i	0.469506	+	0.813208i
$52$		0			0
$53$	201.000	−	348.142i	0.520933	−	0.902283i	−0.478770	−	0.877940i	$-0.658917\pi$
$53$	201.000	−	348.142i	0.520933	−	0.902283i	0.999704	−	0.0243430i	$-0.00774937\pi$
$54$		0			0
$55$	216.000			0.529553
$56$		0			0
$57$	228.000			0.529813
$58$		0			0
$59$	−198.000	+	342.946i	−0.436905	+	0.756742i	−0.997449	−	0.0713828i	$-0.977259\pi$
$59$	−198.000	+	342.946i	−0.436905	+	0.756742i	0.560544	+	0.828125i	$0.310592\pi$
$60$		0			0
$61$	−127.000	−	219.970i	−0.266569	−	0.461710i	0.701405	−	0.712763i	$-0.252557\pi$
$61$	−127.000	−	219.970i	−0.266569	−	0.461710i	−0.967973	+	0.251053i	$0.919223\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−186.000	−	322.161i	−0.354930	−	0.614757i
$66$		0			0
$67$	506.000	−	876.418i	0.922653	−	1.59808i	0.127360	−	0.991857i	$-0.459350\pi$
$67$	506.000	−	876.418i	0.922653	−	1.59808i	0.795293	−	0.606225i	$-0.207317\pi$
$68$		0			0
$69$	72.0000			0.125620
$70$		0			0
$71$	840.000			1.40408			0.702040	−	0.712138i	$-0.252273\pi$
$71$	840.000			1.40408			0.702040	+	0.712138i	$0.252273\pi$
$72$		0			0
$73$	−445.000	+	770.763i	−0.713470	+	1.23577i	0.250077	+	0.968226i	$0.419544\pi$
$73$	−445.000	+	770.763i	−0.713470	+	1.23577i	−0.963547	+	0.267540i	$0.913789\pi$
$74$		0			0
$75$	−133.500	−	231.229i	−0.205537	−	0.356000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−40.0000	−	69.2820i	−0.0569665	−	0.0986688i	0.836136	−	0.548522i	$-0.184809\pi$
$79$	−40.0000	−	69.2820i	−0.0569665	−	0.0986688i	−0.893102	+	0.449854i	$0.851476\pi$
$80$		0			0
$81$	−40.5000	+	70.1481i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−108.000			−0.142826			−0.0714129	−	0.997447i	$-0.522751\pi$
$83$	−108.000			−0.142826			−0.0714129	+	0.997447i	$0.522751\pi$
$84$		0			0
$85$	684.000			0.872826
$86$		0			0
$87$	81.0000	−	140.296i	0.0998174	−	0.172889i
$88$		0			0
$89$	819.000	+	1418.55i	0.975436	+	1.68951i	0.678488	+	0.734612i	$0.262636\pi$
$89$	819.000	+	1418.55i	0.975436	+	1.68951i	0.296948	+	0.954894i	$0.404031\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−168.000	−	290.985i	−0.187320	−	0.324448i
$94$		0			0
$95$	228.000	−	394.908i	0.246235	−	0.426491i
$96$		0			0
$97$	1010.00			1.05722			0.528608	−	0.848866i	$-0.322714\pi$
$97$	1010.00			1.05722			0.528608	+	0.848866i	$0.322714\pi$
$98$		0			0
$99$	324.000			0.328921
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.00295556	+	0.00511917i	−0.867499	−	0.497438i	$-0.834274\pi$
$101$	−3.00000	+	5.19615i	−0.00295556	+	0.00511917i	0.864544	+	0.502557i	$0.167607\pi$
$102$		0			0
$103$	236.000	+	408.764i	0.225765	+	0.391036i	0.956549	−	0.291573i	$-0.0941786\pi$
$103$	236.000	+	408.764i	0.225765	+	0.391036i	−0.730784	+	0.682609i	$0.760845\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	486.000	+	841.777i	0.439097	+	0.760539i	0.997620	−	0.0689505i	$-0.0219651\pi$
$107$	486.000	+	841.777i	0.439097	+	0.760539i	−0.558523	+	0.829489i	$0.688632\pi$
$108$		0			0
$109$	893.000	−	1546.72i	0.784715	−	1.35917i	−0.144455	−	0.989511i	$-0.546143\pi$
$109$	893.000	−	1546.72i	0.784715	−	1.35917i	0.929169	−	0.369654i	$-0.120524\pi$
$110$		0			0
$111$	534.000			0.456622
$112$		0			0
$113$	−2286.00			−1.90309			−0.951543	−	0.307515i	$-0.900503\pi$
$113$	−2286.00			−1.90309			−0.951543	+	0.307515i	$0.900503\pi$
$114$		0			0
$115$	72.0000	−	124.708i	0.0583829	−	0.101122i
$116$		0			0
$117$	−279.000	−	483.242i	−0.220458	−	0.381844i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	17.5000	+	30.3109i	0.0131480	+	0.0227730i
$122$		0			0
$123$	567.000	−	982.073i	0.415648	−	0.719923i
$124$		0			0
$125$	−1284.00			−0.918756
$126$		0			0
$127$	1328.00			0.927881			0.463941	−	0.885866i	$-0.346435\pi$
$127$	1328.00			0.927881			0.463941	+	0.885866i	$0.346435\pi$
$128$		0			0
$129$	−258.000	+	446.869i	−0.176090	+	0.304997i
$130$		0			0
$131$	606.000	+	1049.62i	0.404171	+	0.700046i	0.994225	−	0.107319i	$-0.0342266\pi$
$131$	606.000	+	1049.62i	0.404171	+	0.700046i	−0.590053	+	0.807364i	$0.700893\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	81.0000	+	140.296i	0.0516398	+	0.0894427i
$136$		0			0
$137$	627.000	−	1086.00i	0.391009	−	0.677247i	−0.601574	−	0.798817i	$-0.705459\pi$
$137$	627.000	−	1086.00i	0.391009	−	0.677247i	0.992583	+	0.121570i	$0.0387928\pi$
$138$		0			0
$139$	−340.000			−0.207471			−0.103735	−	0.994605i	$-0.533079\pi$
$139$	−340.000			−0.207471			−0.103735	+	0.994605i	$0.533079\pi$
$140$		0			0
$141$	576.000			0.344028
$142$		0			0
$143$	−1116.00	+	1932.97i	−0.652620	+	1.13037i
$144$		0			0
$145$	−162.000	−	280.592i	−0.0927818	−	0.160703i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−519.000	−	898.934i	−0.285357	−	0.494252i	0.687339	−	0.726337i	$-0.258779\pi$
$149$	−519.000	−	898.934i	−0.285357	−	0.494252i	−0.972696	+	0.232085i	$0.925445\pi$
$150$		0			0
$151$	−1468.00	+	2542.65i	−0.791153	+	1.37032i	0.134100	+	0.990968i	$0.457186\pi$
$151$	−1468.00	+	2542.65i	−0.791153	+	1.37032i	−0.925253	+	0.379350i	$0.876148\pi$
$152$		0			0
$153$	1026.00			0.542138
$154$		0			0
$155$	−672.000			−0.348234
$156$		0			0
$157$	665.000	−	1151.81i	0.338043	−	0.585508i	−0.646021	−	0.763319i	$-0.723568\pi$
$157$	665.000	−	1151.81i	0.338043	−	0.585508i	0.984065	+	0.177811i	$0.0569017\pi$
$158$		0			0
$159$	−603.000	−	1044.43i	−0.300761	−	0.520933i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1682.00	+	2913.31i	0.808248	+	1.39993i	0.914076	+	0.405542i	$0.132917\pi$
$163$	1682.00	+	2913.31i	0.808248	+	1.39993i	−0.105828	+	0.994384i	$0.533749\pi$
$164$		0			0
$165$	324.000	−	561.184i	0.152869	−	0.264777i
$166$		0			0
$167$	3048.00			1.41234			0.706172	−	0.708041i	$-0.250421\pi$
$167$	3048.00			1.41234			0.706172	+	0.708041i	$0.250421\pi$
$168$		0			0
$169$	1647.00			0.749659
$170$		0			0
$171$	342.000	−	592.361i	0.152944	−	0.264906i
$172$		0			0
$173$	1353.00	+	2343.46i	0.594605	+	1.02989i	0.993602	+	0.112934i	$0.0360249\pi$
$173$	1353.00	+	2343.46i	0.594605	+	1.02989i	−0.398997	+	0.916952i	$0.630642\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	594.000	+	1028.84i	0.252247	+	0.436905i
$178$		0			0
$179$	−2358.00	+	4084.18i	−0.984610	+	1.70539i	−0.340953	+	0.940080i	$0.610750\pi$
$179$	−2358.00	+	4084.18i	−0.984610	+	1.70539i	−0.643657	+	0.765314i	$0.722584\pi$
$180$		0			0
$181$	1910.00			0.784360			0.392180	−	0.919888i	$-0.371721\pi$
$181$	1910.00			0.784360			0.392180	+	0.919888i	$0.371721\pi$
$182$		0			0
$183$	−762.000			−0.307807
$184$		0			0
$185$	534.000	−	924.915i	0.212219	−	0.367574i
$186$		0			0
$187$	−2052.00	−	3554.17i	−0.802444	−	1.38987i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2040.00	−	3533.38i	−0.772823	−	1.33857i	−0.936010	−	0.351974i	$-0.885511\pi$
$191$	−2040.00	−	3533.38i	−0.772823	−	1.33857i	0.163187	−	0.986595i	$-0.447823\pi$
$192$		0			0
$193$	1343.00	−	2326.14i	0.500887	−	0.867562i	−0.499112	−	0.866537i	$-0.666340\pi$
$193$	1343.00	−	2326.14i	0.500887	−	0.867562i	0.999999	−	0.00102491i	$-0.000326238\pi$
$194$		0			0
$195$	−1116.00			−0.409838
$196$		0			0
$197$	510.000			0.184447			0.0922233	−	0.995738i	$-0.470603\pi$
$197$	510.000			0.184447			0.0922233	+	0.995738i	$0.470603\pi$
$198$		0			0
$199$	−676.000	+	1170.87i	−0.240806	+	0.417088i	−0.960944	−	0.276743i	$-0.910745\pi$
$199$	−676.000	+	1170.87i	−0.240806	+	0.417088i	0.720138	+	0.693831i	$0.244078\pi$
$200$		0			0
$201$	−1518.00	−	2629.25i	−0.532694	−	0.922653i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1134.00	−	1964.15i	−0.386351	−	0.669180i
$206$		0			0
$207$	108.000	−	187.061i	0.0362634	−	0.0628100i
$208$		0			0
$209$	−2736.00			−0.905517
$210$		0			0
$211$	−3364.00			−1.09757			−0.548785	−	0.835963i	$-0.684909\pi$
$211$	−3364.00			−1.09757			−0.548785	+	0.835963i	$0.684909\pi$
$212$		0			0
$213$	1260.00	−	2182.38i	0.405323	−	0.702040i
$214$		0			0
$215$	516.000	+	893.738i	0.163679	+	0.283500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1335.00	+	2312.29i	0.411922	+	0.713470i
$220$		0			0
$221$	−3534.00	+	6121.07i	−1.07567	+	1.86311i
$222$		0			0
$223$	−4768.00			−1.43179			−0.715894	−	0.698209i	$-0.753981\pi$
$223$	−4768.00			−1.43179			−0.715894	+	0.698209i	$0.753981\pi$
$224$		0			0
$225$	−801.000			−0.237333
$226$		0			0
$227$	−210.000	+	363.731i	−0.0614017	+	0.106351i	−0.895092	−	0.445881i	$-0.852890\pi$
$227$	−210.000	+	363.731i	−0.0614017	+	0.106351i	0.833690	+	0.552232i	$0.186224\pi$
$228$		0			0
$229$	941.000	+	1629.86i	0.271542	+	0.470324i	0.969257	−	0.246051i	$-0.0791332\pi$
$229$	941.000	+	1629.86i	0.271542	+	0.470324i	−0.697715	+	0.716375i	$0.745800\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2541.00	−	4401.14i	−0.714448	−	1.23746i	−0.963172	−	0.268886i	$-0.913344\pi$
$233$	−2541.00	−	4401.14i	−0.714448	−	1.23746i	0.248724	−	0.968574i	$-0.419989\pi$
$234$		0			0
$235$	576.000	−	997.661i	0.159890	−	0.276937i
$236$		0			0
$237$	−240.000			−0.0657792
$238$		0			0
$239$	−5424.00			−1.46799			−0.733995	−	0.679155i	$-0.762346\pi$
$239$	−5424.00			−1.46799			−0.733995	+	0.679155i	$0.762346\pi$
$240$		0			0
$241$	1295.00	−	2243.01i	0.346134	−	0.599522i	−0.639425	−	0.768853i	$-0.720828\pi$
$241$	1295.00	−	2243.01i	0.346134	−	0.599522i	0.985559	+	0.169332i	$0.0541609\pi$
$242$		0			0
$243$	121.500	+	210.444i	0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2356.00	+	4080.71i	0.606918	+	1.05121i
$248$		0			0
$249$	−162.000	+	280.592i	−0.0412303	+	0.0714129i
$250$		0			0
$251$	−4932.00			−1.24026			−0.620130	−	0.784499i	$-0.712920\pi$
$251$	−4932.00			−1.24026			−0.620130	+	0.784499i	$0.712920\pi$
$252$		0			0
$253$	−864.000			−0.214700
$254$		0			0
$255$	1026.00	−	1777.08i	0.251963	−	0.436413i
$256$		0			0
$257$	1719.00	+	2977.40i	0.417231	+	0.722665i	0.995660	−	0.0930680i	$-0.0296674\pi$
$257$	1719.00	+	2977.40i	0.417231	+	0.722665i	−0.578429	+	0.815733i	$0.696334\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−243.000	−	420.888i	−0.0576296	−	0.0998174i
$262$		0			0
$263$	−3060.00	+	5300.08i	−0.717444	+	1.24265i	0.244566	+	0.969633i	$0.421355\pi$
$263$	−3060.00	+	5300.08i	−0.717444	+	1.24265i	−0.962009	+	0.273016i	$0.911979\pi$
$264$		0			0
$265$	−2412.00			−0.559124
$266$		0			0
$267$	4914.00			1.12634
$268$		0			0
$269$	9.00000	−	15.5885i	0.00203992	−	0.00353325i	−0.865004	−	0.501766i	$-0.832684\pi$
$269$	9.00000	−	15.5885i	0.00203992	−	0.00353325i	0.867044	+	0.498232i	$0.166017\pi$
$270$		0			0
$271$	−3448.00	−	5972.11i	−0.772882	−	1.33867i	−0.935977	−	0.352061i	$-0.885481\pi$
$271$	−3448.00	−	5972.11i	−0.772882	−	1.33867i	0.163095	−	0.986610i	$-0.447852\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1602.00	+	2774.75i	0.351288	+	0.608449i
$276$		0			0
$277$	−3127.00	+	5416.12i	−0.678279	+	1.17481i	0.297220	+	0.954809i	$0.403940\pi$
$277$	−3127.00	+	5416.12i	−0.678279	+	1.17481i	−0.975499	+	0.220004i	$0.929393\pi$
$278$		0			0
$279$	−1008.00			−0.216299
$280$		0			0
$281$	1194.00			0.253481			0.126740	−	0.991936i	$-0.459549\pi$
$281$	1194.00			0.253481			0.126740	+	0.991936i	$0.459549\pi$
$282$		0			0
$283$	3578.00	−	6197.28i	0.751555	−	1.30173i	−0.195514	−	0.980701i	$-0.562638\pi$
$283$	3578.00	−	6197.28i	0.751555	−	1.30173i	0.947069	−	0.321030i	$-0.104029\pi$
$284$		0			0
$285$	−684.000	−	1184.72i	−0.142164	−	0.246235i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4041.50	−	7000.08i	−0.822613	−	1.42481i
$290$		0			0
$291$	1515.00	−	2624.06i	0.305192	−	0.528608i
$292$		0			0
$293$	−3738.00			−0.745312			−0.372656	−	0.927970i	$-0.621553\pi$
$293$	−3738.00			−0.745312			−0.372656	+	0.927970i	$0.621553\pi$
$294$		0			0
$295$	2376.00			0.468936
$296$		0			0
$297$	486.000	−	841.777i	0.0949514	−	0.164461i
$298$		0			0
$299$	744.000	+	1288.65i	0.143902	+	0.249245i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	9.00000	+	15.5885i	0.00170639	+	0.00295556i
$304$		0			0
$305$	−762.000	+	1319.82i	−0.143056	+	0.247780i
$306$		0			0
$307$	−844.000			−0.156904			−0.0784522	−	0.996918i	$-0.524998\pi$
$307$	−844.000			−0.156904			−0.0784522	+	0.996918i	$0.524998\pi$
$308$		0			0
$309$	1416.00			0.260691
$310$		0			0
$311$	−3156.00	+	5466.35i	−0.575435	+	0.996683i	0.420559	+	0.907265i	$0.361834\pi$
$311$	−3156.00	+	5466.35i	−0.575435	+	0.996683i	−0.995994	+	0.0894178i	$0.971499\pi$
$312$		0			0
$313$	−4141.00	−	7172.42i	−0.747806	−	1.29524i	−0.948872	−	0.315660i	$-0.897774\pi$
$313$	−4141.00	−	7172.42i	−0.747806	−	1.29524i	0.201067	−	0.979578i	$-0.435559\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4659.00	−	8069.62i	−0.825475	−	1.42976i	−0.901556	−	0.432663i	$-0.857574\pi$
$317$	−4659.00	−	8069.62i	−0.825475	−	1.42976i	0.0760811	−	0.997102i	$-0.475759\pi$
$318$		0			0
$319$	−972.000	+	1683.55i	−0.170600	+	0.295489i
$320$		0			0
$321$	2916.00			0.507026
$322$		0			0
$323$	−8664.00			−1.49250
$324$		0			0
$325$	2759.00	−	4778.73i	0.470898	−	0.815619i
$326$		0			0
$327$	−2679.00	−	4640.16i	−0.453055	−	0.784715i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−826.000	−	1430.67i	−0.137163	−	0.237574i	0.789258	−	0.614061i	$-0.210465\pi$
$331$	−826.000	−	1430.67i	−0.137163	−	0.237574i	−0.926422	+	0.376487i	$0.877132\pi$
$332$		0			0
$333$	801.000	−	1387.37i	0.131815	−	0.228311i
$334$		0			0
$335$	−6072.00			−0.990295
$336$		0			0
$337$	−1294.00			−0.209165			−0.104583	−	0.994516i	$-0.533351\pi$
$337$	−1294.00			−0.209165			−0.104583	+	0.994516i	$0.533351\pi$
$338$		0			0
$339$	−3429.00	+	5939.20i	−0.549374	+	0.951543i
$340$		0			0
$341$	2016.00	+	3491.81i	0.320154	+	0.554523i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−216.000	−	374.123i	−0.0337074	−	0.0583829i
$346$		0			0
$347$	−1818.00	+	3148.87i	−0.281255	+	0.487147i	−0.971694	−	0.236243i	$-0.924084\pi$
$347$	−1818.00	+	3148.87i	−0.281255	+	0.487147i	0.690439	+	0.723390i	$0.257417\pi$
$348$		0			0
$349$	10478.0			1.60709			0.803545	−	0.595244i	$-0.202945\pi$
$349$	10478.0			1.60709			0.803545	+	0.595244i	$0.202945\pi$
$350$		0			0
$351$	−1674.00			−0.254563
$352$		0			0
$353$	3783.00	−	6552.35i	0.570393	−	0.987950i	−0.426132	−	0.904661i	$-0.640124\pi$
$353$	3783.00	−	6552.35i	0.570393	−	0.987950i	0.996525	−	0.0832890i	$-0.0265425\pi$
$354$		0			0
$355$	−2520.00	−	4364.77i	−0.376754	−	0.652557i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4020.00	+	6962.84i	0.590996	+	1.02363i	0.994099	+	0.108480i	$0.0345984\pi$
$359$	4020.00	+	6962.84i	0.590996	+	1.02363i	−0.403103	+	0.915155i	$0.632068\pi$
$360$		0			0
$361$	541.500	−	937.906i	0.0789474	−	0.136741i
$362$		0			0
$363$	105.000			0.0151820
$364$		0			0
$365$	5340.00			0.765776
$366$		0			0
$367$	−3784.00	+	6554.08i	−0.538210	+	0.932208i	0.460790	+	0.887509i	$0.347566\pi$
$367$	−3784.00	+	6554.08i	−0.538210	+	0.932208i	−0.999001	+	0.0446985i	$0.985767\pi$
$368$		0			0
$369$	−1701.00	−	2946.22i	−0.239974	−	0.415648i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6761.00	+	11710.4i	0.938529	+	1.62558i	0.768217	+	0.640190i	$0.221144\pi$
$373$	6761.00	+	11710.4i	0.938529	+	1.62558i	0.170312	+	0.985390i	$0.445522\pi$
$374$		0			0
$375$	−1926.00	+	3335.93i	−0.265222	+	0.459378i
$376$		0			0
$377$	3348.00			0.457376
$378$		0			0
$379$	2468.00			0.334492			0.167246	−	0.985915i	$-0.446513\pi$
$379$	2468.00			0.334492			0.167246	+	0.985915i	$0.446513\pi$
$380$		0			0
$381$	1992.00	−	3450.25i	0.267856	−	0.463941i
$382$		0			0
$383$	6168.00	+	10683.3i	0.822898	+	1.42530i	0.903515	+	0.428557i	$0.140978\pi$
$383$	6168.00	+	10683.3i	0.822898	+	1.42530i	−0.0806166	+	0.996745i	$0.525689\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	774.000	+	1340.61i	0.101666	+	0.176090i
$388$		0			0
$389$	1881.00	−	3257.99i	0.245168	−	0.424644i	−0.717011	−	0.697062i	$-0.754490\pi$
$389$	1881.00	−	3257.99i	0.245168	−	0.424644i	0.962179	+	0.272418i	$0.0878234\pi$
$390$		0			0
$391$	−2736.00			−0.353876
$392$		0			0
$393$	3636.00			0.466697
$394$		0			0
$395$	−240.000	+	415.692i	−0.0305714	+	0.0529513i
$396$		0			0
$397$	4385.00	+	7595.04i	0.554350	+	0.960162i	0.997954	+	0.0639390i	$0.0203663\pi$
$397$	4385.00	+	7595.04i	0.554350	+	0.960162i	−0.443604	+	0.896223i	$0.646300\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3321.00	−	5752.14i	−0.413573	−	0.716330i	0.581704	−	0.813400i	$-0.302386\pi$
$401$	−3321.00	−	5752.14i	−0.413573	−	0.716330i	−0.995277	+	0.0970706i	$0.969053\pi$
$402$		0			0
$403$	3472.00	−	6013.68i	0.429163	−	0.743332i
$404$		0			0
$405$	486.000			0.0596285
$406$		0			0
$407$	−6408.00			−0.780424
$408$		0			0
$409$	755.000	−	1307.70i	0.0912771	−	0.158097i	−0.816772	−	0.576961i	$-0.804238\pi$
$409$	755.000	−	1307.70i	0.0912771	−	0.158097i	0.908049	+	0.418864i	$0.137572\pi$
$410$		0			0
$411$	−1881.00	−	3257.99i	−0.225749	−	0.391009i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	324.000	+	561.184i	0.0383242	+	0.0663794i
$416$		0			0
$417$	−510.000	+	883.346i	−0.0598916	+	0.103735i
$418$		0			0
$419$	−1260.00			−0.146909			−0.0734547	−	0.997299i	$-0.523402\pi$
$419$	−1260.00			−0.146909			−0.0734547	+	0.997299i	$0.523402\pi$
$420$		0			0
$421$	3998.00			0.462828			0.231414	−	0.972855i	$-0.425665\pi$
$421$	3998.00			0.462828			0.231414	+	0.972855i	$0.425665\pi$
$422$		0			0
$423$	864.000	−	1496.49i	0.0993123	−	0.172014i
$424$		0			0
$425$	5073.00	+	8786.69i	0.579004	+	1.00286i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	3348.00	+	5798.91i	0.376790	+	0.652620i
$430$		0			0
$431$	1368.00	−	2369.45i	0.152887	−	0.264808i	−0.779401	−	0.626526i	$-0.784476\pi$
$431$	1368.00	−	2369.45i	0.152887	−	0.264808i	0.932288	+	0.361718i	$0.117810\pi$
$432$		0			0
$433$	2690.00			0.298552			0.149276	−	0.988796i	$-0.452306\pi$
$433$	2690.00			0.298552			0.149276	+	0.988796i	$0.452306\pi$
$434$		0			0
$435$	−972.000			−0.107135
$436$		0			0
$437$	−912.000	+	1579.63i	−0.0998327	+	0.172915i
$438$		0			0
$439$	620.000	+	1073.87i	0.0674054	+	0.116750i	0.897758	−	0.440488i	$-0.145195\pi$
$439$	620.000	+	1073.87i	0.0674054	+	0.116750i	−0.830353	+	0.557238i	$0.811861\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1950.00	+	3377.50i	0.209136	+	0.362234i	0.951443	−	0.307826i	$-0.0996015\pi$
$443$	1950.00	+	3377.50i	0.209136	+	0.362234i	−0.742307	+	0.670060i	$0.766268\pi$
$444$		0			0
$445$	4914.00	−	8511.30i	0.523474	−	0.906684i
$446$		0			0
$447$	−3114.00			−0.329501
$448$		0			0
$449$	−10878.0			−1.14335			−0.571675	−	0.820480i	$-0.693706\pi$
$449$	−10878.0			−1.14335			−0.571675	+	0.820480i	$0.693706\pi$
$450$		0			0
$451$	−6804.00	+	11784.9i	−0.710394	+	1.23044i
$452$		0			0
$453$	4404.00	+	7627.95i	0.456773	+	0.791153i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1165.00	−	2017.84i	−0.119248	−	0.206544i	0.800222	−	0.599704i	$-0.204715\pi$
$457$	−1165.00	−	2017.84i	−0.119248	−	0.206544i	−0.919470	+	0.393160i	$0.871382\pi$
$458$		0			0
$459$	1539.00	−	2665.63i	0.156502	−	0.271069i
$460$		0			0
$461$	15150.0			1.53060			0.765299	−	0.643675i	$-0.222591\pi$
$461$	15150.0			1.53060			0.765299	+	0.643675i	$0.222591\pi$
$462$		0			0
$463$	−2992.00			−0.300324			−0.150162	−	0.988661i	$-0.547980\pi$
$463$	−2992.00			−0.300324			−0.150162	+	0.988661i	$0.547980\pi$
$464$		0			0
$465$	−1008.00	+	1745.91i	−0.100527	+	0.174117i
$466$		0			0
$467$	−4362.00	−	7555.21i	−0.432225	−	0.748636i	0.564839	−	0.825201i	$-0.308938\pi$
$467$	−4362.00	−	7555.21i	−0.432225	−	0.748636i	−0.997065	+	0.0765646i	$0.975605\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1995.00	−	3455.44i	−0.195169	−	0.338043i
$472$		0			0
$473$	3096.00	−	5362.43i	0.300960	−	0.521279i
$474$		0			0
$475$	6764.00			0.653376
$476$		0			0
$477$	−3618.00			−0.347289
$478$		0			0
$479$	−4872.00	+	8438.55i	−0.464734	+	0.804942i	−0.999189	−	0.0402542i	$-0.987183\pi$
$479$	−4872.00	+	8438.55i	−0.464734	+	0.804942i	0.534456	+	0.845196i	$0.320517\pi$
$480$		0			0
$481$	5518.00	+	9557.46i	0.523075	+	0.905993i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3030.00	−	5248.11i	−0.283681	−	0.491350i
$486$		0			0
$487$	−2068.00	+	3581.88i	−0.192423	+	0.333286i	−0.946053	−	0.324013i	$-0.894968\pi$
$487$	−2068.00	+	3581.88i	−0.192423	+	0.333286i	0.753630	+	0.657299i	$0.228301\pi$
$488$		0			0
$489$	10092.0			0.933284
$490$		0			0
$491$	16212.0			1.49010			0.745048	−	0.667011i	$-0.232426\pi$
$491$	16212.0			1.49010			0.745048	+	0.667011i	$0.232426\pi$
$492$		0			0
$493$	−3078.00	+	5331.25i	−0.281189	+	0.487034i
$494$		0			0
$495$	−972.000	−	1683.55i	−0.0882589	−	0.152869i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	0.807272	−	0.590180i	$-0.200943\pi$
$499$	−1198.00	−	2075.00i	−0.107475	−	0.186152i	−0.914747	+	0.404028i	$0.867610\pi$
$500$		0			0
$501$	4572.00	−	7918.94i	0.407708	−	0.706172i
$502$		0			0
$503$	13128.0			1.16371			0.581857	−	0.813291i	$-0.302326\pi$
$503$	13128.0			1.16371			0.581857	+	0.813291i	$0.302326\pi$
$504$		0			0
$505$	36.0000			0.00317224
$506$		0			0
$507$	2470.50	−	4279.03i	0.216408	−	0.374829i
$508$		0			0
$509$	−6399.00	−	11083.4i	−0.557231	−	0.965153i	−0.997726	−	0.0673977i	$-0.978530\pi$
$509$	−6399.00	−	11083.4i	−0.557231	−	0.965153i	0.440495	−	0.897755i	$-0.354803\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1026.00	−	1777.08i	−0.0883022	−	0.152944i
$514$		0			0
$515$	1416.00	−	2452.58i	0.121158	−	0.209852i
$516$		0			0
$517$	−6912.00			−0.587987
$518$		0			0
$519$	8118.00			0.686591
$520$		0			0
$521$	−3693.00	+	6396.46i	−0.310544	+	0.537877i	−0.978480	−	0.206341i	$-0.933844\pi$
$521$	−3693.00	+	6396.46i	−0.310544	+	0.537877i	0.667936	+	0.744218i	$0.267178\pi$
$522$		0			0
$523$	−2590.00	−	4486.01i	−0.216545	−	0.375066i	0.737205	−	0.675669i	$-0.236145\pi$
$523$	−2590.00	−	4486.01i	−0.216545	−	0.375066i	−0.953749	+	0.300603i	$0.902812\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6384.00	+	11057.4i	0.527688	+	0.913982i
$528$		0			0
$529$	5795.50	−	10038.1i	0.476329	−	0.825027i
$530$		0			0
$531$	3564.00			0.291270
$532$		0			0
$533$	23436.0			1.90455
$534$		0			0
$535$	2916.00	−	5050.66i	0.235644	−	0.408148i
$536$		0			0
$537$	7074.00	+	12252.5i	0.568465	+	0.984610i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2035.00	−	3524.72i	−0.161722	−	0.280110i	0.773764	−	0.633473i	$-0.218371\pi$
$541$	−2035.00	−	3524.72i	−0.161722	−	0.280110i	−0.935486	+	0.353363i	$0.885038\pi$
$542$		0			0
$543$	2865.00	−	4962.33i	0.226425	−	0.392180i
$544$		0			0
$545$	−10716.0			−0.842244
$546$		0			0
$547$	14780.0			1.15530			0.577648	−	0.816286i	$-0.303971\pi$
$547$	14780.0			1.15530			0.577648	+	0.816286i	$0.303971\pi$
$548$		0			0
$549$	−1143.00	+	1979.73i	−0.0888562	+	0.153903i
$550$		0			0
$551$	2052.00	+	3554.17i	0.158654	+	0.274796i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1602.00	−	2774.75i	−0.122525	−	0.212219i
$556$		0			0
$557$	3429.00	−	5939.20i	0.260846	−	0.451799i	−0.705621	−	0.708590i	$-0.749332\pi$
$557$	3429.00	−	5939.20i	0.260846	−	0.451799i	0.966467	+	0.256791i	$0.0826651\pi$
$558$		0			0
$559$	−10664.0			−0.806868
$560$		0			0
$561$	−12312.0			−0.926583
$562$		0			0
$563$	−3330.00	+	5767.73i	−0.249277	+	0.431760i	−0.963325	−	0.268336i	$-0.913526\pi$
$563$	−3330.00	+	5767.73i	−0.249277	+	0.431760i	0.714049	+	0.700096i	$0.246860\pi$
$564$		0			0
$565$	6858.00	+	11878.4i	0.510652	+	0.884475i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	75.0000	+	129.904i	0.00552577	+	0.00957092i	0.868775	−	0.495207i	$-0.164908\pi$
$569$	75.0000	+	129.904i	0.00552577	+	0.00957092i	−0.863249	+	0.504778i	$0.831574\pi$
$570$		0			0
$571$	4094.00	−	7091.02i	0.300050	−	0.519702i	−0.676097	−	0.736813i	$-0.736330\pi$
$571$	4094.00	−	7091.02i	0.300050	−	0.519702i	0.976147	+	0.217111i	$0.0696633\pi$
$572$		0			0
$573$	−12240.0			−0.892379
$574$		0			0
$575$	2136.00			0.154917
$576$		0			0
$577$	2927.00	−	5069.71i	0.211183	−	0.365780i	−0.740902	−	0.671613i	$-0.765602\pi$
$577$	2927.00	−	5069.71i	0.211183	−	0.365780i	0.952085	+	0.305833i	$0.0989351\pi$
$578$		0			0
$579$	−4029.00	−	6978.43i	−0.289187	−	0.500887i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	7236.00	+	12533.1i	0.514039	+	0.890341i
$584$		0			0
$585$	−1674.00	+	2899.45i	−0.118310	+	0.204919i
$586$		0			0
$587$	17580.0			1.23612			0.618062	−	0.786130i	$-0.287918\pi$
$587$	17580.0			1.23612			0.618062	+	0.786130i	$0.287918\pi$
$588$		0			0
$589$	8512.00			0.595468
$590$		0			0
$591$	765.000	−	1325.02i	0.0532452	−	0.0922233i
$592$		0			0
$593$	−8577.00	−	14855.8i	−0.593955	−	1.02876i	−0.993693	−	0.112131i	$-0.964232\pi$
$593$	−8577.00	−	14855.8i	−0.593955	−	1.02876i	0.399739	−	0.916629i	$-0.369101\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2028.00	+	3512.60i	0.139029	+	0.240806i
$598$		0			0
$599$	9060.00	−	15692.4i	0.617999	−	1.07041i	−0.371851	−	0.928292i	$-0.621277\pi$
$599$	9060.00	−	15692.4i	0.617999	−	1.07041i	0.989850	−	0.142114i	$-0.0453899\pi$
$600$		0			0
$601$	17546.0			1.19088			0.595438	−	0.803401i	$-0.296979\pi$
$601$	17546.0			1.19088			0.595438	+	0.803401i	$0.296979\pi$
$602$		0			0
$603$	−9108.00			−0.615102
$604$		0			0
$605$	105.000	−	181.865i	0.00705596	−	0.0122213i
$606$		0			0
$607$	7280.00	+	12609.3i	0.486798	+	0.843158i	0.999885	−	0.0151784i	$-0.00483163\pi$
$607$	7280.00	+	12609.3i	0.486798	+	0.843158i	−0.513087	+	0.858336i	$0.671498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5952.00	+	10309.2i	0.394095	+	0.682593i
$612$		0			0
$613$	2249.00	−	3895.38i	0.148183	−	0.256661i	−0.782373	−	0.622810i	$-0.785991\pi$
$613$	2249.00	−	3895.38i	0.148183	−	0.256661i	0.930556	+	0.366150i	$0.119324\pi$
$614$		0			0
$615$	−6804.00			−0.446120
$616$		0			0
$617$	−5478.00			−0.357433			−0.178716	−	0.983901i	$-0.557194\pi$
$617$	−5478.00			−0.357433			−0.178716	+	0.983901i	$0.557194\pi$
$618$		0			0
$619$	−3022.00	+	5234.26i	−0.196227	+	0.339875i	−0.947302	−	0.320342i	$-0.896202\pi$
$619$	−3022.00	+	5234.26i	−0.196227	+	0.339875i	0.751075	+	0.660217i	$0.229536\pi$
$620$		0			0
$621$	−324.000	−	561.184i	−0.0209367	−	0.0362634i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1710.50	−	2962.67i	−0.109472	−	0.189611i
$626$		0			0
$627$	−4104.00	+	7108.34i	−0.261400	+	0.452759i
$628$		0			0
$629$	−20292.0			−1.28632
$630$		0			0
$631$	−15352.0			−0.968547			−0.484274	−	0.874917i	$-0.660916\pi$
$631$	−15352.0			−0.968547			−0.484274	+	0.874917i	$0.660916\pi$
$632$		0			0
$633$	−5046.00	+	8739.93i	−0.316841	+	0.548785i
$634$		0			0
$635$	−3984.00	−	6900.49i	−0.248977	−	0.431240i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−3780.00	−	6547.15i	−0.234013	−	0.405323i
$640$		0			0
$641$	11199.0	−	19397.2i	0.690068	−	1.19523i	−0.281746	−	0.959489i	$-0.590914\pi$
$641$	11199.0	−	19397.2i	0.690068	−	1.19523i	0.971815	−	0.235745i	$-0.0757530\pi$
$642$		0			0
$643$	3764.00			0.230852			0.115426	−	0.993316i	$-0.463177\pi$
$643$	3764.00			0.230852			0.115426	+	0.993316i	$0.463177\pi$
$644$		0			0
$645$	3096.00			0.189000
$646$		0			0
$647$	8844.00	−	15318.3i	0.537393	−	0.930793i	−0.461650	−	0.887062i	$-0.652742\pi$
$647$	8844.00	−	15318.3i	0.537393	−	0.930793i	0.999043	−	0.0437305i	$-0.0139243\pi$
$648$		0			0
$649$	−7128.00	−	12346.1i	−0.431122	−	0.746726i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9939.00	−	17214.9i	−0.595625	−	1.03165i	−0.993458	−	0.114195i	$-0.963571\pi$
$653$	−9939.00	−	17214.9i	−0.595625	−	1.03165i	0.397833	−	0.917458i	$-0.369762\pi$
$654$		0			0
$655$	3636.00	−	6297.74i	0.216901	−	0.375684i
$656$		0			0
$657$	8010.00			0.475647
$658$		0			0
$659$	−20004.0			−1.18247			−0.591233	−	0.806501i	$-0.701359\pi$
$659$	−20004.0			−1.18247			−0.591233	+	0.806501i	$0.701359\pi$
$660$		0			0
$661$	653.000	−	1131.03i	0.0384247	−	0.0665536i	−0.846173	−	0.532908i	$-0.821099\pi$
$661$	653.000	−	1131.03i	0.0384247	−	0.0665536i	0.884598	+	0.466354i	$0.154433\pi$
$662$		0			0
$663$	10602.0	+	18363.2i	0.621037	+	1.07567i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	648.000	+	1122.37i	0.0376172	+	0.0651549i
$668$		0			0
$669$	−7152.00	+	12387.6i	−0.413322	+	0.715894i
$670$		0			0
$671$	9144.00			0.526081
$672$		0			0
$673$	−13054.0			−0.747689			−0.373845	−	0.927491i	$-0.621961\pi$
$673$	−13054.0			−0.747689			−0.373845	+	0.927491i	$0.621961\pi$
$674$		0			0
$675$	−1201.50	+	2081.06i	−0.0685122	+	0.118667i
$676$		0			0
$677$	−2523.00	−	4369.96i	−0.143230	−	0.248082i	0.785481	−	0.618886i	$-0.212416\pi$
$677$	−2523.00	−	4369.96i	−0.143230	−	0.248082i	−0.928711	+	0.370804i	$0.879082\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	630.000	+	1091.19i	0.0354503	+	0.0614017i
$682$		0			0
$683$	−6234.00	+	10797.6i	−0.349249	+	0.604918i	−0.986116	−	0.166056i	$-0.946897\pi$
$683$	−6234.00	+	10797.6i	−0.349249	+	0.604918i	0.636867	+	0.770974i	$0.280230\pi$
$684$		0			0
$685$	−7524.00			−0.419675
$686$		0			0
$687$	5646.00			0.313549
$688$		0			0
$689$	12462.0	−	21584.8i	0.689063	−	1.19349i
$690$		0			0
$691$	11606.0	+	20102.2i	0.638948	+	1.10669i	0.985664	+	0.168720i	$0.0539634\pi$
$691$	11606.0	+	20102.2i	0.638948	+	1.10669i	−0.346716	+	0.937970i	$0.612703\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1020.00	+	1766.69i	0.0556702	+	0.0964237i
$696$		0			0
$697$	−21546.0	+	37318.8i	−1.17089	+	2.02805i
$698$		0			0
$699$	−15246.0			−0.824974
$700$		0			0
$701$	35958.0			1.93740			0.968698	−	0.248241i	$-0.0798526\pi$
$701$	35958.0			1.93740			0.968698	+	0.248241i	$0.0798526\pi$
$702$		0			0
$703$	−6764.00	+	11715.6i	−0.362886	+	0.628538i
$704$		0			0
$705$	−1728.00	−	2992.98i	−0.0923124	−	0.159890i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3223.00	−	5582.40i	−0.170723	−	0.295700i	0.767950	−	0.640510i	$-0.221277\pi$
$709$	−3223.00	−	5582.40i	−0.170723	−	0.295700i	−0.938673	+	0.344809i	$0.887944\pi$
$710$		0			0
$711$	−360.000	+	623.538i	−0.0189888	+	0.0328896i
$712$		0			0
$713$	2688.00			0.141187
$714$		0			0
$715$	13392.0			0.700465
$716$		0			0
$717$	−8136.00	+	14092.0i	−0.423772	+	0.733995i
$718$		0			0
$719$	2352.00	+	4073.78i	0.121996	+	0.211302i	0.920555	−	0.390614i	$-0.127737\pi$
$719$	2352.00	+	4073.78i	0.121996	+	0.211302i	−0.798559	+	0.601917i	$0.794404\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−3885.00	−	6729.02i	−0.199841	−	0.346134i
$724$		0			0
$725$	2403.00	−	4162.12i	0.123097	−	0.213210i
$726$		0			0
$727$	−10600.0			−0.540760			−0.270380	−	0.962754i	$-0.587149\pi$
$727$	−10600.0			−0.540760			−0.270380	+	0.962754i	$0.587149\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	9804.00	−	16981.0i	0.496052	−	0.859187i
$732$		0			0
$733$	−6271.00	−	10861.7i	−0.315995	−	0.547320i	0.663653	−	0.748040i	$-0.269005\pi$
$733$	−6271.00	−	10861.7i	−0.315995	−	0.547320i	−0.979649	+	0.200720i	$0.935672\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18216.0	+	31551.0i	0.910441	+	1.57693i
$738$		0			0
$739$	−11662.0	+	20199.2i	−0.580506	+	1.00547i	0.414914	+	0.909861i	$0.363812\pi$
$739$	−11662.0	+	20199.2i	−0.580506	+	1.00547i	−0.995419	+	0.0956044i	$0.969522\pi$
$740$		0			0
$741$	14136.0			0.700808
$742$		0			0
$743$	−6312.00			−0.311662			−0.155831	−	0.987784i	$-0.549806\pi$
$743$	−6312.00			−0.311662			−0.155831	+	0.987784i	$0.549806\pi$
$744$		0			0
$745$	−3114.00	+	5393.61i	−0.153138	+	0.265244i
$746$		0			0
$747$	486.000	+	841.777i	0.0238043	+	0.0412303i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−17920.0	−	31038.4i	−0.870719	−	1.50813i	−0.861254	−	0.508175i	$-0.830320\pi$
$751$	−17920.0	−	31038.4i	−0.870719	−	1.50813i	−0.00946509	−	0.999955i	$-0.503013\pi$
$752$		0			0
$753$	−7398.00	+	12813.7i	−0.358032	+	0.620130i
$754$		0			0
$755$	17616.0			0.849155
$756$		0			0
$757$	−34594.0			−1.66095			−0.830476	−	0.557055i	$-0.811931\pi$
$757$	−34594.0			−1.66095			−0.830476	+	0.557055i	$0.811931\pi$
$758$		0			0
$759$	−1296.00	+	2244.74i	−0.0619787	+	0.107350i
$760$		0			0
$761$	−11973.0	−	20737.8i	−0.570330	−	0.987840i	−0.996532	−	0.0832121i	$-0.973482\pi$
$761$	−11973.0	−	20737.8i	−0.570330	−	0.987840i	0.426202	−	0.904628i	$-0.359851\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3078.00	−	5331.25i	−0.145471	−	0.251963i
$766$		0			0
$767$	−12276.0	+	21262.7i	−0.577915	+	1.00098i
$768$		0			0
$769$	18770.0			0.880187			0.440093	−	0.897952i	$-0.354945\pi$
$769$	18770.0			0.880187			0.440093	+	0.897952i	$0.354945\pi$
$770$		0			0
$771$	10314.0			0.481776
$772$		0			0
$773$	−15171.0	+	26276.9i	−0.705903	+	1.22266i	0.260462	+	0.965484i	$0.416125\pi$
$773$	−15171.0	+	26276.9i	−0.705903	+	1.22266i	−0.966365	+	0.257176i	$0.917208\pi$
$774$		0			0
$775$	−4984.00	−	8632.54i	−0.231007	−	0.400116i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14364.0	+	24879.2i	0.660647	+	1.14427i
$780$		0			0
$781$	−15120.0	+	26188.6i	−0.692748	+	1.19987i
$782$		0			0
$783$	−1458.00			−0.0665449
$784$		0			0
$785$	−7980.00			−0.362826
$786$		0			0
$787$	13094.0	−	22679.5i	0.593076	−	1.02724i	−0.400739	−	0.916192i	$-0.631247\pi$
$787$	13094.0	−	22679.5i	0.593076	−	1.02724i	0.993815	−	0.111045i	$-0.0354199\pi$
$788$		0			0
$789$	9180.00	+	15900.2i	0.414216	+	0.717444i
$790$		0			0
$791$		0			0

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{3} +(-3.00000 - 5.19615i) q^{5} +(-4.50000 - 7.79423i) q^{9} +(-18.0000 + 31.1769i) q^{11} +62.0000 q^{13} -18.0000 q^{15} +(-57.0000 + 98.7269i) q^{17} +(38.0000 + 65.8179i) q^{19} +(12.0000 + 20.7846i) q^{23} +(44.5000 - 77.0763i) q^{25} -27.0000 q^{27} +54.0000 q^{29} +(56.0000 - 96.9948i) q^{31} +(54.0000 + 93.5307i) q^{33} +(89.0000 + 154.153i) q^{37} +(93.0000 - 161.081i) q^{39} +378.000 q^{41} -172.000 q^{43} +(-27.0000 + 46.7654i) q^{45} +(96.0000 + 166.277i) q^{47} +(171.000 + 296.181i) q^{51} +(201.000 - 348.142i) q^{53} +216.000 q^{55} +228.000 q^{57} +(-198.000 + 342.946i) q^{59} +(-127.000 - 219.970i) q^{61} +(-186.000 - 322.161i) q^{65} +(506.000 - 876.418i) q^{67} +72.0000 q^{69} +840.000 q^{71} +(-445.000 + 770.763i) q^{73} +(-133.500 - 231.229i) q^{75} +(-40.0000 - 69.2820i) q^{79} +(-40.5000 + 70.1481i) q^{81} -108.000 q^{83} +684.000 q^{85} +(81.0000 - 140.296i) q^{87} +(819.000 + 1418.55i) q^{89} +(-168.000 - 290.985i) q^{93} +(228.000 - 394.908i) q^{95} +1010.00 q^{97} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 6 q^{5} - 9 q^{9} - 36 q^{11} + 124 q^{13} - 36 q^{15} - 114 q^{17} + 76 q^{19} + 24 q^{23} + 89 q^{25} - 54 q^{27} + 108 q^{29} + 112 q^{31} + 108 q^{33} + 178 q^{37} + 186 q^{39} + 756 q^{41}+ \cdots + 648 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 6 * q^5 - 9 * q^9 - 36 * q^11 + 124 * q^13 - 36 * q^15 - 114 * q^17 + 76 * q^19 + 24 * q^23 + 89 * q^25 - 54 * q^27 + 108 * q^29 + 112 * q^31 + 108 * q^33 + 178 * q^37 + 186 * q^39 + 756 * q^41 - 344 * q^43 - 54 * q^45 + 192 * q^47 + 342 * q^51 + 402 * q^53 + 432 * q^55 + 456 * q^57 - 396 * q^59 - 254 * q^61 - 372 * q^65 + 1012 * q^67 + 144 * q^69 + 1680 * q^71 - 890 * q^73 - 267 * q^75 - 80 * q^79 - 81 * q^81 - 216 * q^83 + 1368 * q^85 + 162 * q^87 + 1638 * q^89 - 336 * q^93 + 456 * q^95 + 2020 * q^97 + 648 * q^99

Introduction

L-functions

Modular forms

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