LMFDB - Embedded newform 560.4.q.b.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(81,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("560.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.q (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:	$33.0410696032$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	560.81
Dual form			560.4.q.b.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{3} +(-2.50000 - 4.33013i) q^{5} +(14.0000 + 12.1244i) q^{7} +(11.5000 + 19.9186i) q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{3} +(-2.50000 - 4.33013i) q^{5} +(14.0000 + 12.1244i) q^{7} +(11.5000 + 19.9186i) q^{9} +(-22.5000 + 38.9711i) q^{11} +59.0000 q^{13} +10.0000 q^{15} +(27.0000 - 46.7654i) q^{17} +(-60.5000 - 104.789i) q^{19} +(-35.0000 + 12.1244i) q^{21} +(34.5000 + 59.7558i) q^{23} +(-12.5000 + 21.6506i) q^{25} -100.000 q^{27} -162.000 q^{29} +(-44.0000 + 76.2102i) q^{31} +(-45.0000 - 77.9423i) q^{33} +(17.5000 - 90.9327i) q^{35} +(129.500 + 224.301i) q^{37} +(-59.0000 + 102.191i) q^{39} +195.000 q^{41} +286.000 q^{43} +(57.5000 - 99.5929i) q^{45} +(22.5000 + 38.9711i) q^{47} +(49.0000 + 339.482i) q^{49} +(54.0000 + 93.5307i) q^{51} +(-298.500 + 517.017i) q^{53} +225.000 q^{55} +242.000 q^{57} +(-180.000 + 311.769i) q^{59} +(-196.000 - 339.482i) q^{61} +(-80.5000 + 418.290i) q^{63} +(-147.500 - 255.477i) q^{65} +(-140.000 + 242.487i) q^{67} -138.000 q^{69} -48.0000 q^{71} +(-334.000 + 578.505i) q^{73} +(-25.0000 - 43.3013i) q^{75} +(-787.500 + 272.798i) q^{77} +(391.000 + 677.232i) q^{79} +(-210.500 + 364.597i) q^{81} -768.000 q^{83} -270.000 q^{85} +(162.000 - 280.592i) q^{87} +(597.000 + 1034.03i) q^{89} +(826.000 + 715.337i) q^{91} +(-88.0000 - 152.420i) q^{93} +(-302.500 + 523.945i) q^{95} +902.000 q^{97} -1035.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9} - 45 q^{11} + 118 q^{13} + 20 q^{15} + 54 q^{17} - 121 q^{19} - 70 q^{21} + 69 q^{23} - 25 q^{25} - 200 q^{27} - 324 q^{29} - 88 q^{31} - 90 q^{33} + 35 q^{35} + 259 q^{37} - 118 q^{39} + 390 q^{41} + 572 q^{43} + 115 q^{45} + 45 q^{47} + 98 q^{49} + 108 q^{51} - 597 q^{53} + 450 q^{55} + 484 q^{57} - 360 q^{59} - 392 q^{61} - 161 q^{63} - 295 q^{65} - 280 q^{67} - 276 q^{69} - 96 q^{71} - 668 q^{73} - 50 q^{75} - 1575 q^{77} + 782 q^{79} - 421 q^{81} - 1536 q^{83} - 540 q^{85} + 324 q^{87} + 1194 q^{89} + 1652 q^{91} - 176 q^{93} - 605 q^{95} + 1804 q^{97} - 2070 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9 - 45 * q^11 + 118 * q^13 + 20 * q^15 + 54 * q^17 - 121 * q^19 - 70 * q^21 + 69 * q^23 - 25 * q^25 - 200 * q^27 - 324 * q^29 - 88 * q^31 - 90 * q^33 + 35 * q^35 + 259 * q^37 - 118 * q^39 + 390 * q^41 + 572 * q^43 + 115 * q^45 + 45 * q^47 + 98 * q^49 + 108 * q^51 - 597 * q^53 + 450 * q^55 + 484 * q^57 - 360 * q^59 - 392 * q^61 - 161 * q^63 - 295 * q^65 - 280 * q^67 - 276 * q^69 - 96 * q^71 - 668 * q^73 - 50 * q^75 - 1575 * q^77 + 782 * q^79 - 421 * q^81 - 1536 * q^83 - 540 * q^85 + 324 * q^87 + 1194 * q^89 + 1652 * q^91 - 176 * q^93 - 605 * q^95 + 1804 * q^97 - 2070 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000	+	1.73205i	−0.192450	+	0.333333i	−0.946062	−	0.323987i	$-0.894977\pi$
$3$	−1.00000	+	1.73205i	−0.192450	+	0.333333i	0.753612	+	0.657320i	$0.228310\pi$
$4$		0			0
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$6$		0			0
$7$	14.0000	+	12.1244i	0.755929	+	0.654654i
$7$	14.0000	+	12.1244i	0.755929	+	0.654654i
$8$		0			0
$9$	11.5000	+	19.9186i	0.425926	+	0.737725i
$10$		0			0
$11$	−22.5000	+	38.9711i	−0.616728	+	1.06820i	0.373351	+	0.927690i	$0.378209\pi$
$11$	−22.5000	+	38.9711i	−0.616728	+	1.06820i	−0.990079	+	0.140514i	$0.955125\pi$
$12$		0			0
$13$	59.0000			1.25874			0.629371	−	0.777105i	$-0.283312\pi$
$13$	59.0000			1.25874			0.629371	+	0.777105i	$0.283312\pi$
$14$		0			0
$15$	10.0000			0.172133
$16$		0			0
$17$	27.0000	−	46.7654i	0.385204	−	0.667192i	−0.606594	−	0.795012i	$-0.707465\pi$
$17$	27.0000	−	46.7654i	0.385204	−	0.667192i	0.991797	+	0.127820i	$0.0407979\pi$
$18$		0			0
$19$	−60.5000	−	104.789i	−0.730508	−	1.26528i	−0.956666	−	0.291186i	$-0.905950\pi$
$19$	−60.5000	−	104.789i	−0.730508	−	1.26528i	0.226158	−	0.974091i	$-0.427383\pi$
$20$		0			0
$21$	−35.0000	+	12.1244i	−0.363696	+	0.125988i
$22$		0			0
$23$	34.5000	+	59.7558i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	34.5000	+	59.7558i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$	−100.000			−0.712778
$28$		0			0
$29$	−162.000			−1.03733			−0.518666	−	0.854977i	$-0.673571\pi$
$29$	−162.000			−1.03733			−0.518666	+	0.854977i	$0.673571\pi$
$30$		0			0
$31$	−44.0000	+	76.2102i	−0.254924	+	0.441541i	−0.964875	−	0.262710i	$-0.915384\pi$
$31$	−44.0000	+	76.2102i	−0.254924	+	0.441541i	0.709951	+	0.704251i	$0.248717\pi$
$32$		0			0
$33$	−45.0000	−	77.9423i	−0.237379	−	0.411152i
$34$		0			0
$35$	17.5000	−	90.9327i	0.0845154	−	0.439155i
$36$		0			0
$37$	129.500	+	224.301i	0.575396	+	0.996616i	0.995998	+	0.0893706i	$0.0284856\pi$
$37$	129.500	+	224.301i	0.575396	+	0.996616i	−0.420602	+	0.907245i	$0.638181\pi$
$38$		0			0
$39$	−59.0000	+	102.191i	−0.242245	+	0.419581i
$40$		0			0
$41$	195.000			0.742778			0.371389	−	0.928477i	$-0.378882\pi$
$41$	195.000			0.742778			0.371389	+	0.928477i	$0.378882\pi$
$42$		0			0
$43$	286.000			1.01429			0.507146	−	0.861860i	$-0.330700\pi$
$43$	286.000			1.01429			0.507146	+	0.861860i	$0.330700\pi$
$44$		0			0
$45$	57.5000	−	99.5929i	0.190480	−	0.329921i
$46$		0			0
$47$	22.5000	+	38.9711i	0.0698290	+	0.120947i	0.898826	−	0.438306i	$-0.144421\pi$
$47$	22.5000	+	38.9711i	0.0698290	+	0.120947i	−0.828997	+	0.559253i	$0.811088\pi$
$48$		0			0
$49$	49.0000	+	339.482i	0.142857	+	0.989743i
$50$		0			0
$51$	54.0000	+	93.5307i	0.148265	+	0.256802i
$52$		0			0
$53$	−298.500	+	517.017i	−0.773625	+	1.33996i	0.161939	+	0.986801i	$0.448225\pi$
$53$	−298.500	+	517.017i	−0.773625	+	1.33996i	−0.935564	+	0.353157i	$0.885108\pi$
$54$		0			0
$55$	225.000			0.551618
$56$		0			0
$57$	242.000			0.562345
$58$		0			0
$59$	−180.000	+	311.769i	−0.397187	+	0.687947i	−0.993378	−	0.114895i	$-0.963347\pi$
$59$	−180.000	+	311.769i	−0.397187	+	0.687947i	0.596191	+	0.802843i	$0.296680\pi$
$60$		0			0
$61$	−196.000	−	339.482i	−0.411397	−	0.712561i	0.583646	−	0.812009i	$-0.301626\pi$
$61$	−196.000	−	339.482i	−0.411397	−	0.712561i	−0.995043	+	0.0994477i	$0.968292\pi$
$62$		0			0
$63$	−80.5000	+	418.290i	−0.160985	+	0.836502i
$64$		0			0
$65$	−147.500	−	255.477i	−0.281463	−	0.487509i
$66$		0			0
$67$	−140.000	+	242.487i	−0.255279	+	0.442157i	−0.964971	−	0.262355i	$-0.915501\pi$
$67$	−140.000	+	242.487i	−0.255279	+	0.442157i	0.709692	+	0.704512i	$0.248834\pi$
$68$		0			0
$69$	−138.000			−0.240772
$70$		0			0
$71$	−48.0000			−0.0802331			−0.0401166	−	0.999195i	$-0.512773\pi$
$71$	−48.0000			−0.0802331			−0.0401166	+	0.999195i	$0.512773\pi$
$72$		0			0
$73$	−334.000	+	578.505i	−0.535503	+	0.927519i	0.463635	+	0.886026i	$0.346545\pi$
$73$	−334.000	+	578.505i	−0.535503	+	0.927519i	−0.999139	+	0.0414929i	$0.986789\pi$
$74$		0			0
$75$	−25.0000	−	43.3013i	−0.0384900	−	0.0666667i
$76$		0			0
$77$	−787.500	+	272.798i	−1.16551	+	0.403743i
$78$		0			0
$79$	391.000	+	677.232i	0.556847	+	0.964488i	0.997757	+	0.0669365i	$0.0213225\pi$
$79$	391.000	+	677.232i	0.556847	+	0.964488i	−0.440910	+	0.897551i	$0.645344\pi$
$80$		0			0
$81$	−210.500	+	364.597i	−0.288752	+	0.500133i
$82$		0			0
$83$	−768.000			−1.01565			−0.507825	−	0.861460i	$-0.669550\pi$
$83$	−768.000			−1.01565			−0.507825	+	0.861460i	$0.669550\pi$
$84$		0			0
$85$	−270.000			−0.344537
$86$		0			0
$87$	162.000	−	280.592i	0.199635	−	0.345778i
$88$		0			0
$89$	597.000	+	1034.03i	0.711032	+	1.23154i	0.964470	+	0.264192i	$0.0851054\pi$
$89$	597.000	+	1034.03i	0.711032	+	1.23154i	−0.253438	+	0.967352i	$0.581561\pi$
$90$		0			0
$91$	826.000	+	715.337i	0.951520	+	0.824041i
$92$		0			0
$93$	−88.0000	−	152.420i	−0.0981202	−	0.169949i
$94$		0			0
$95$	−302.500	+	523.945i	−0.326693	+	0.565849i
$96$		0			0
$97$	902.000			0.944167			0.472084	−	0.881554i	$-0.343502\pi$
$97$	902.000			0.944167			0.472084	+	0.881554i	$0.343502\pi$
$98$		0			0
$99$	−1035.00			−1.05072
$100$		0			0
$101$	−342.000	+	592.361i	−0.336933	+	0.583586i	−0.983854	−	0.178971i	$-0.942723\pi$
$101$	−342.000	+	592.361i	−0.336933	+	0.583586i	0.646921	+	0.762557i	$0.276056\pi$
$102$		0			0
$103$	−758.000	−	1312.89i	−0.725126	−	1.25595i	−0.958922	−	0.283669i	$-0.908448\pi$
$103$	−758.000	−	1312.89i	−0.725126	−	1.25595i	0.233796	−	0.972286i	$-0.424885\pi$
$104$		0			0
$105$	140.000	+	121.244i	0.130120	+	0.112687i
$106$		0			0
$107$	−366.000	−	633.931i	−0.330678	−	0.572751i	0.651967	−	0.758247i	$-0.273944\pi$
$107$	−366.000	−	633.931i	−0.330678	−	0.572751i	−0.982645	+	0.185496i	$0.940611\pi$
$108$		0			0
$109$	800.000	−	1385.64i	0.702992	−	1.21762i	−0.264420	−	0.964408i	$-0.585180\pi$
$109$	800.000	−	1385.64i	0.702992	−	1.21762i	0.967411	−	0.253210i	$-0.0814863\pi$
$110$		0			0
$111$	−518.000			−0.442940
$112$		0			0
$113$	−1392.00			−1.15883			−0.579417	−	0.815031i	$-0.696720\pi$
$113$	−1392.00			−1.15883			−0.579417	+	0.815031i	$0.696720\pi$
$114$		0			0
$115$	172.500	−	298.779i	0.139876	−	0.242272i
$116$		0			0
$117$	678.500	+	1175.20i	0.536131	+	0.928606i
$118$		0			0
$119$	945.000	−	327.358i	0.727966	−	0.252175i
$120$		0			0
$121$	−347.000	−	601.022i	−0.260706	−	0.451556i
$122$		0			0
$123$	−195.000	+	337.750i	−0.142948	+	0.247593i
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	−803.000			−0.561061			−0.280530	−	0.959845i	$-0.590510\pi$
$127$	−803.000			−0.561061			−0.280530	+	0.959845i	$0.590510\pi$
$128$		0			0
$129$	−286.000	+	495.367i	−0.195201	+	0.338098i
$130$		0			0
$131$	1009.50	+	1748.51i	0.673286	+	1.16617i	0.976967	+	0.213391i	$0.0684509\pi$
$131$	1009.50	+	1748.51i	0.673286	+	1.16617i	−0.303681	+	0.952774i	$0.598216\pi$
$132$		0			0
$133$	423.500	−	2200.57i	0.276106	−	1.43469i
$134$		0			0
$135$	250.000	+	433.013i	0.159382	+	0.276058i
$136$		0			0
$137$	−30.0000	+	51.9615i	−0.0187086	+	0.0324042i	−0.875228	−	0.483710i	$-0.839289\pi$
$137$	−30.0000	+	51.9615i	−0.0187086	+	0.0324042i	0.856520	+	0.516115i	$0.172622\pi$
$138$		0			0
$139$	1708.00			1.04224			0.521118	−	0.853485i	$-0.325515\pi$
$139$	1708.00			1.04224			0.521118	+	0.853485i	$0.325515\pi$
$140$		0			0
$141$	−90.0000			−0.0537544
$142$		0			0
$143$	−1327.50	+	2299.30i	−0.776302	+	1.34459i
$144$		0			0
$145$	405.000	+	701.481i	0.231955	+	0.401757i
$146$		0			0
$147$	−637.000	−	254.611i	−0.357407	−	0.142857i
$148$		0			0
$149$	543.000	+	940.504i	0.298552	+	0.517108i	0.975805	−	0.218643i	$-0.0701629\pi$
$149$	543.000	+	940.504i	0.298552	+	0.517108i	−0.677253	+	0.735751i	$0.736830\pi$
$150$		0			0
$151$	−1433.00	+	2482.03i	−0.772291	+	1.33765i	0.164014	+	0.986458i	$0.447556\pi$
$151$	−1433.00	+	2482.03i	−0.772291	+	1.33765i	−0.936305	+	0.351189i	$0.885778\pi$
$152$		0			0
$153$	1242.00			0.656273
$154$		0			0
$155$	440.000			0.228011
$156$		0			0
$157$	114.500	−	198.320i	0.0582044	−	0.100813i	−0.835455	−	0.549559i	$-0.814796\pi$
$157$	114.500	−	198.320i	0.0582044	−	0.100813i	0.893659	+	0.448746i	$0.148129\pi$
$158$		0			0
$159$	−597.000	−	1034.03i	−0.297768	−	0.515750i
$160$		0			0
$161$	−241.500	+	1254.87i	−0.118217	+	0.614271i
$162$		0			0
$163$	−614.000	−	1063.48i	−0.295044	−	0.511031i	0.679951	−	0.733258i	$-0.262001\pi$
$163$	−614.000	−	1063.48i	−0.295044	−	0.511031i	−0.974995	+	0.222226i	$0.928668\pi$
$164$		0			0
$165$	−225.000	+	389.711i	−0.106159	+	0.183873i
$166$		0			0
$167$	1929.00			0.893835			0.446918	−	0.894575i	$-0.352522\pi$
$167$	1929.00			0.893835			0.446918	+	0.894575i	$0.352522\pi$
$168$		0			0
$169$	1284.00			0.584433
$170$		0			0
$171$	1391.50	−	2410.15i	0.622285	−	1.07783i
$172$		0			0
$173$	349.500	+	605.352i	0.153595	+	0.266035i	0.932547	−	0.361049i	$-0.117581\pi$
$173$	349.500	+	605.352i	0.153595	+	0.266035i	−0.778951	+	0.627084i	$0.784248\pi$
$174$		0			0
$175$	−437.500	+	151.554i	−0.188982	+	0.0654654i
$176$		0			0
$177$	−360.000	−	623.538i	−0.152877	−	0.264791i
$178$		0			0
$179$	1558.50	−	2699.40i	0.650770	−	1.12717i	−0.332167	−	0.943221i	$-0.607780\pi$
$179$	1558.50	−	2699.40i	0.650770	−	1.12717i	0.982936	−	0.183945i	$-0.0588870\pi$
$180$		0			0
$181$	−1798.00			−0.738366			−0.369183	−	0.929357i	$-0.620362\pi$
$181$	−1798.00			−0.738366			−0.369183	+	0.929357i	$0.620362\pi$
$182$		0			0
$183$	784.000			0.316694
$184$		0			0
$185$	647.500	−	1121.50i	0.257325	−	0.445700i
$186$		0			0
$187$	1215.00	+	2104.44i	0.475132	+	0.822952i
$188$		0			0
$189$	−1400.00	−	1212.44i	−0.538810	−	0.466623i
$190$		0			0
$191$	−1194.00	−	2068.07i	−0.452329	−	0.783457i	0.546201	−	0.837654i	$-0.316073\pi$
$191$	−1194.00	−	2068.07i	−0.452329	−	0.783457i	−0.998530	+	0.0541974i	$0.982740\pi$
$192$		0			0
$193$	−136.000	+	235.559i	−0.0507228	+	0.0878544i	−0.890272	−	0.455429i	$-0.849486\pi$
$193$	−136.000	+	235.559i	−0.0507228	+	0.0878544i	0.839549	+	0.543284i	$0.182819\pi$
$194$		0			0
$195$	590.000			0.216671
$196$		0			0
$197$	−2109.00			−0.762741			−0.381371	−	0.924422i	$-0.624548\pi$
$197$	−2109.00			−0.762741			−0.381371	+	0.924422i	$0.624548\pi$
$198$		0			0
$199$	712.000	−	1233.22i	0.253630	−	0.439300i	−0.710893	−	0.703301i	$-0.751709\pi$
$199$	712.000	−	1233.22i	0.253630	−	0.439300i	0.964522	+	0.264001i	$0.0850422\pi$
$200$		0			0
$201$	−280.000	−	484.974i	−0.0982571	−	0.170186i
$202$		0			0
$203$	−2268.00	−	1964.15i	−0.784150	−	0.679094i
$204$		0			0
$205$	−487.500	−	844.375i	−0.166090	−	0.287677i
$206$		0			0
$207$	−793.500	+	1374.38i	−0.266435	+	0.461479i
$208$		0			0
$209$	5445.00			1.80210
$210$		0			0
$211$	3625.00			1.18273			0.591363	−	0.806405i	$-0.298590\pi$
$211$	3625.00			1.18273			0.591363	+	0.806405i	$0.298590\pi$
$212$		0			0
$213$	48.0000	−	83.1384i	0.0154409	−	0.0267444i
$214$		0			0
$215$	−715.000	−	1238.42i	−0.226803	−	0.392834i
$216$		0			0
$217$	−1540.00	+	533.472i	−0.481760	+	0.166887i
$218$		0			0
$219$	−668.000	−	1157.01i	−0.206115	−	0.357002i
$220$		0			0
$221$	1593.00	−	2759.16i	0.484872	−	0.839823i
$222$		0			0
$223$	4960.00			1.48944			0.744722	−	0.667374i	$-0.232582\pi$
$223$	4960.00			1.48944			0.744722	+	0.667374i	$0.232582\pi$
$224$		0			0
$225$	−575.000			−0.170370
$226$		0			0
$227$	−750.000	+	1299.04i	−0.219292	+	0.379825i	−0.954592	−	0.297917i	$-0.903708\pi$
$227$	−750.000	+	1299.04i	−0.219292	+	0.379825i	0.735300	+	0.677742i	$0.237041\pi$
$228$		0			0
$229$	−3046.00	−	5275.83i	−0.878975	−	1.52243i	−0.852467	−	0.522781i	$-0.824894\pi$
$229$	−3046.00	−	5275.83i	−0.878975	−	1.52243i	−0.0265085	−	0.999649i	$-0.508439\pi$
$230$		0			0
$231$	315.000	−	1636.79i	0.0897207	−	0.466202i
$232$		0			0
$233$	−69.0000	−	119.512i	−0.0194006	−	0.0336028i	0.856162	−	0.516707i	$-0.172842\pi$
$233$	−69.0000	−	119.512i	−0.0194006	−	0.0336028i	−0.875563	+	0.483104i	$0.839509\pi$
$234$		0			0
$235$	112.500	−	194.856i	0.0312285	−	0.0540893i
$236$		0			0
$237$	−1564.00			−0.428661
$238$		0			0
$239$	5502.00			1.48910			0.744550	−	0.667567i	$-0.232664\pi$
$239$	5502.00			1.48910			0.744550	+	0.667567i	$0.232664\pi$
$240$		0			0
$241$	−1775.50	+	3075.26i	−0.474564	+	0.821970i	−0.999576	−	0.0291256i	$-0.990728\pi$
$241$	−1775.50	+	3075.26i	−0.474564	+	0.821970i	0.525011	+	0.851095i	$0.324061\pi$
$242$		0			0
$243$	−1771.00	−	3067.46i	−0.467530	−	0.809785i
$244$		0			0
$245$	1347.50	−	1060.88i	0.351382	−	0.276642i
$246$		0			0
$247$	−3569.50	−	6182.56i	−0.919522	−	1.59266i
$248$		0			0
$249$	768.000	−	1330.22i	0.195462	−	0.338550i
$250$		0			0
$251$	−7065.00			−1.77665			−0.888324	−	0.459216i	$-0.848130\pi$
$251$	−7065.00			−1.77665			−0.888324	+	0.459216i	$0.848130\pi$
$252$		0			0
$253$	−3105.00			−0.771580
$254$		0			0
$255$	270.000	−	467.654i	0.0663061	−	0.114846i
$256$		0			0
$257$	2040.00	+	3533.38i	0.495143	+	0.857613i	0.999984	−	0.00559954i	$-0.00178240\pi$
$257$	2040.00	+	3533.38i	0.495143	+	0.857613i	−0.504842	+	0.863212i	$0.668449\pi$
$258$		0			0
$259$	−906.500	+	4710.31i	−0.217479	+	1.13006i
$260$		0			0
$261$	−1863.00	−	3226.81i	−0.441827	−	0.765267i
$262$		0			0
$263$	−1644.00	+	2847.49i	−0.385450	+	0.667619i	−0.991832	−	0.127555i	$-0.959287\pi$
$263$	−1644.00	+	2847.49i	−0.385450	+	0.667619i	0.606381	+	0.795174i	$0.292620\pi$
$264$		0			0
$265$	2985.00			0.691951
$266$		0			0
$267$	−2388.00			−0.547353
$268$		0			0
$269$	1632.00	−	2826.71i	0.369906	−	0.640697i	−0.619644	−	0.784883i	$-0.712723\pi$
$269$	1632.00	−	2826.71i	0.369906	−	0.640697i	0.989551	+	0.144186i	$0.0460564\pi$
$270$		0			0
$271$	−1376.00	−	2383.30i	−0.308436	−	0.534226i	0.669585	−	0.742736i	$-0.266472\pi$
$271$	−1376.00	−	2383.30i	−0.308436	−	0.534226i	−0.978020	+	0.208510i	$0.933139\pi$
$272$		0			0
$273$	−2065.00	+	715.337i	−0.457800	+	0.158587i
$274$		0			0
$275$	−562.500	−	974.279i	−0.123346	−	0.213641i
$276$		0			0
$277$	2345.00	−	4061.66i	0.508655	−	0.881016i	−0.491295	−	0.870993i	$-0.663476\pi$
$277$	2345.00	−	4061.66i	0.508655	−	0.881016i	0.999950	−	0.0100228i	$-0.00319040\pi$
$278$		0			0
$279$	−2024.00			−0.434314
$280$		0			0
$281$	7821.00			1.66036			0.830181	−	0.557494i	$-0.188237\pi$
$281$	7821.00			1.66036			0.830181	+	0.557494i	$0.188237\pi$
$282$		0			0
$283$	−329.000	+	569.845i	−0.0691061	+	0.119695i	−0.898508	−	0.438957i	$-0.855348\pi$
$283$	−329.000	+	569.845i	−0.0691061	+	0.119695i	0.829402	+	0.558652i	$0.188681\pi$
$284$		0			0
$285$	−605.000	−	1047.89i	−0.125744	−	0.217795i
$286$		0			0
$287$	2730.00	+	2364.25i	0.561487	+	0.486262i
$288$		0			0
$289$	998.500	+	1729.45i	0.203236	+	0.352016i
$290$		0			0
$291$	−902.000	+	1562.31i	−0.181705	+	0.314722i
$292$		0			0
$293$	−5997.00			−1.19573			−0.597864	−	0.801597i	$-0.703984\pi$
$293$	−5997.00			−1.19573			−0.597864	+	0.801597i	$0.703984\pi$
$294$		0			0
$295$	1800.00			0.355254
$296$		0			0
$297$	2250.00	−	3897.11i	0.439590	−	0.761392i
$298$		0			0
$299$	2035.50	+	3525.59i	0.393699	+	0.681907i
$300$		0			0
$301$	4004.00	+	3467.57i	0.766733	+	0.664011i
$302$		0			0
$303$	−684.000	−	1184.72i	−0.129686	−	0.224622i
$304$		0			0
$305$	−980.000	+	1697.41i	−0.183982	+	0.318667i
$306$		0			0
$307$	6226.00			1.15745			0.578724	−	0.815523i	$-0.303551\pi$
$307$	6226.00			1.15745			0.578724	+	0.815523i	$0.303551\pi$
$308$		0			0
$309$	3032.00			0.558202
$310$		0			0
$311$	2340.00	−	4053.00i	0.426653	−	0.738985i	−0.569920	−	0.821700i	$-0.693026\pi$
$311$	2340.00	−	4053.00i	0.426653	−	0.738985i	0.996573	+	0.0827149i	$0.0263591\pi$
$312$		0			0
$313$	−514.000	−	890.274i	−0.0928211	−	0.160771i	0.815876	−	0.578227i	$-0.196255\pi$
$313$	−514.000	−	890.274i	−0.0928211	−	0.160771i	−0.908697	+	0.417456i	$0.862922\pi$
$314$		0			0
$315$	2012.50	−	697.150i	0.359973	−	0.124698i
$316$		0			0
$317$	−4311.00	−	7466.87i	−0.763817	−	1.32297i	−0.940870	−	0.338768i	$-0.889990\pi$
$317$	−4311.00	−	7466.87i	−0.763817	−	1.32297i	0.177053	−	0.984201i	$-0.443344\pi$
$318$		0			0
$319$	3645.00	−	6313.33i	0.639752	−	1.10808i
$320$		0			0
$321$	1464.00			0.254556
$322$		0			0
$323$	−6534.00			−1.12558
$324$		0			0
$325$	−737.500	+	1277.39i	−0.125874	+	0.218021i
$326$		0			0
$327$	1600.00	+	2771.28i	0.270582	+	0.468661i
$328$		0			0
$329$	−157.500	+	818.394i	−0.0263929	+	0.137141i
$330$		0			0
$331$	−999.500	−	1731.18i	−0.165974	−	0.287476i	0.771027	−	0.636803i	$-0.219744\pi$
$331$	−999.500	−	1731.18i	−0.165974	−	0.287476i	−0.937001	+	0.349327i	$0.886410\pi$
$332$		0			0
$333$	−2978.50	+	5158.91i	−0.490153	+	0.848969i
$334$		0			0
$335$	1400.00			0.228329
$336$		0			0
$337$	5114.00			0.826639			0.413319	−	0.910586i	$-0.364369\pi$
$337$	5114.00			0.826639			0.413319	+	0.910586i	$0.364369\pi$
$338$		0			0
$339$	1392.00	−	2411.01i	0.223018	−	0.386278i
$340$		0			0
$341$	−1980.00	−	3429.46i	−0.314437	−	0.544621i
$342$		0			0
$343$	−3430.00	+	5346.84i	−0.539949	+	0.841698i
$344$		0			0
$345$	345.000	+	597.558i	0.0538382	+	0.0932505i
$346$		0			0
$347$	2160.00	−	3741.23i	0.334164	−	0.578789i	−0.649160	−	0.760652i	$-0.724879\pi$
$347$	2160.00	−	3741.23i	0.334164	−	0.578789i	0.983324	+	0.181863i	$0.0582128\pi$
$348$		0			0
$349$	7922.00			1.21506			0.607529	−	0.794298i	$-0.292161\pi$
$349$	7922.00			1.21506			0.607529	+	0.794298i	$0.292161\pi$
$350$		0			0
$351$	−5900.00			−0.897204
$352$		0			0
$353$	−414.000	+	717.069i	−0.0624221	+	0.108118i	−0.895548	−	0.444966i	$-0.853216\pi$
$353$	−414.000	+	717.069i	−0.0624221	+	0.108118i	0.833125	+	0.553084i	$0.186549\pi$
$354$		0			0
$355$	120.000	+	207.846i	0.0179407	+	0.0310742i
$356$		0			0
$357$	−378.000	+	1964.15i	−0.0560389	+	0.291187i
$358$		0			0
$359$	−675.000	−	1169.13i	−0.0992344	−	0.171879i	0.812134	−	0.583472i	$-0.198306\pi$
$359$	−675.000	−	1169.13i	−0.0992344	−	0.171879i	−0.911368	+	0.411593i	$0.864973\pi$
$360$		0			0
$361$	−3891.00	+	6739.41i	−0.567284	+	0.982564i
$362$		0			0
$363$	1388.00			0.200692
$364$		0			0
$365$	3340.00			0.478969
$366$		0			0
$367$	1400.50	−	2425.74i	0.199198	−	0.345020i	−0.749071	−	0.662490i	$-0.769500\pi$
$367$	1400.50	−	2425.74i	0.199198	−	0.345020i	0.948268	+	0.317470i	$0.102833\pi$
$368$		0			0
$369$	2242.50	+	3884.12i	0.316368	+	0.547966i
$370$		0			0
$371$	−10447.5	+	3619.12i	−1.46201	+	0.506456i
$372$		0			0
$373$	−3301.00	−	5717.50i	−0.458229	−	0.793675i	0.540639	−	0.841255i	$-0.318183\pi$
$373$	−3301.00	−	5717.50i	−0.458229	−	0.793675i	−0.998867	+	0.0475795i	$0.984849\pi$
$374$		0			0
$375$	−125.000	+	216.506i	−0.0172133	+	0.0298142i
$376$		0			0
$377$	−9558.00			−1.30573
$378$		0			0
$379$	8305.00			1.12559			0.562796	−	0.826596i	$-0.309726\pi$
$379$	8305.00			1.12559			0.562796	+	0.826596i	$0.309726\pi$
$380$		0			0
$381$	803.000	−	1390.84i	0.107976	−	0.187020i
$382$		0			0
$383$	472.500	+	818.394i	0.0630382	+	0.109185i	0.895822	−	0.444413i	$-0.146588\pi$
$383$	472.500	+	818.394i	0.0630382	+	0.109185i	−0.832784	+	0.553598i	$0.813254\pi$
$384$		0			0
$385$	3150.00	+	2727.98i	0.416984	+	0.361119i
$386$		0			0
$387$	3289.00	+	5696.72i	0.432014	+	0.748270i
$388$		0			0
$389$	−6018.00	+	10423.5i	−0.784382	+	1.35859i	0.144985	+	0.989434i	$0.453687\pi$
$389$	−6018.00	+	10423.5i	−0.784382	+	1.35859i	−0.929367	+	0.369156i	$0.879647\pi$
$390$		0			0
$391$	3726.00			0.481923
$392$		0			0
$393$	−4038.00			−0.518296
$394$		0			0
$395$	1955.00	−	3386.16i	0.249030	−	0.431332i
$396$		0			0
$397$	1349.00	+	2336.54i	0.170540	+	0.295384i	0.938609	−	0.344983i	$-0.112115\pi$
$397$	1349.00	+	2336.54i	0.170540	+	0.295384i	−0.768069	+	0.640367i	$0.778782\pi$
$398$		0			0
$399$	3388.00	+	2934.09i	0.425093	+	0.368141i
$400$		0			0
$401$	−3526.50	−	6108.08i	−0.439165	−	0.760655i	0.558461	−	0.829531i	$-0.311392\pi$
$401$	−3526.50	−	6108.08i	−0.439165	−	0.760655i	−0.997625	+	0.0688756i	$0.978059\pi$
$402$		0			0
$403$	−2596.00	+	4496.40i	−0.320883	+	0.555786i
$404$		0			0
$405$	2105.00			0.258267
$406$		0			0
$407$	−11655.0			−1.41945
$408$		0			0
$409$	5435.00	−	9413.70i	0.657074	−	1.13809i	−0.324295	−	0.945956i	$-0.605127\pi$
$409$	5435.00	−	9413.70i	0.657074	−	1.13809i	0.981369	−	0.192130i	$-0.0615396\pi$
$410$		0			0
$411$	−60.0000	−	103.923i	−0.00720093	−	0.0124724i
$412$		0			0
$413$	−6300.00	+	2182.38i	−0.750612	+	0.260020i
$414$		0			0
$415$	1920.00	+	3325.54i	0.227106	+	0.393360i
$416$		0			0
$417$	−1708.00	+	2958.34i	−0.200578	+	0.347412i
$418$		0			0
$419$	9729.00			1.13435			0.567175	−	0.823597i	$-0.308036\pi$
$419$	9729.00			1.13435			0.567175	+	0.823597i	$0.308036\pi$
$420$		0			0
$421$	−12550.0			−1.45285			−0.726425	−	0.687246i	$-0.758819\pi$
$421$	−12550.0			−1.45285			−0.726425	+	0.687246i	$0.758819\pi$
$422$		0			0
$423$	−517.500	+	896.336i	−0.0594840	+	0.103029i
$424$		0			0
$425$	675.000	+	1169.13i	0.0770407	+	0.133438i
$426$		0			0
$427$	1372.00	−	7129.12i	0.155494	−	0.807968i
$428$		0			0
$429$	−2655.00	−	4598.59i	−0.298799	−	0.517534i
$430$		0			0
$431$	1494.00	−	2587.68i	0.166969	−	0.289198i	−0.770384	−	0.637580i	$-0.779935\pi$
$431$	1494.00	−	2587.68i	0.166969	−	0.289198i	0.937353	+	0.348382i	$0.113269\pi$
$432$		0			0
$433$	16616.0			1.84414			0.922072	−	0.387019i	$-0.126495\pi$
$433$	16616.0			1.84414			0.922072	+	0.387019i	$0.126495\pi$
$434$		0			0
$435$	−1620.00			−0.178559
$436$		0			0
$437$	4174.50	−	7230.45i	0.456964	−	0.791485i
$438$		0			0
$439$	3673.00	+	6361.82i	0.399323	+	0.691647i	0.993642	−	0.112581i	$-0.0359119\pi$
$439$	3673.00	+	6361.82i	0.399323	+	0.691647i	−0.594320	+	0.804229i	$0.702579\pi$
$440$		0			0
$441$	−6198.50	+	4880.05i	−0.669312	+	0.526947i
$442$		0			0
$443$	6.00000	+	10.3923i	0.000643496	+	0.00111457i	0.866347	−	0.499443i	$-0.166462\pi$
$443$	6.00000	+	10.3923i	0.000643496	+	0.00111457i	−0.865703	+	0.500557i	$0.833129\pi$
$444$		0			0
$445$	2985.00	−	5170.17i	0.317983	−	0.550763i
$446$		0			0
$447$	−2172.00			−0.229826
$448$		0			0
$449$	9669.00			1.01628			0.508138	−	0.861275i	$-0.330334\pi$
$449$	9669.00			1.01628			0.508138	+	0.861275i	$0.330334\pi$
$450$		0			0
$451$	−4387.50	+	7599.37i	−0.458092	+	0.793438i
$452$		0			0
$453$	−2866.00	−	4964.06i	−0.297255	−	0.514860i
$454$		0			0
$455$	1032.50	−	5365.03i	0.106383	−	0.552783i
$456$		0			0
$457$	4817.00	+	8343.29i	0.493063	+	0.854010i	0.999968	−	0.00799181i	$-0.00254390\pi$
$457$	4817.00	+	8343.29i	0.493063	+	0.854010i	−0.506905	+	0.862002i	$0.669211\pi$
$458$		0			0
$459$	−2700.00	+	4676.54i	−0.274565	+	0.475560i
$460$		0			0
$461$	−342.000			−0.0345521			−0.0172761	−	0.999851i	$-0.505499\pi$
$461$	−342.000			−0.0345521			−0.0172761	+	0.999851i	$0.505499\pi$
$462$		0			0
$463$	−2411.00			−0.242006			−0.121003	−	0.992652i	$-0.538611\pi$
$463$	−2411.00			−0.242006			−0.121003	+	0.992652i	$0.538611\pi$
$464$		0			0
$465$	−440.000	+	762.102i	−0.0438807	+	0.0760035i
$466$		0			0
$467$	−603.000	−	1044.43i	−0.0597506	−	0.103491i	0.834603	−	0.550852i	$-0.185697\pi$
$467$	−603.000	−	1044.43i	−0.0597506	−	0.103491i	−0.894353	+	0.447361i	$0.852364\pi$
$468$		0			0
$469$	−4900.00	+	1697.41i	−0.482433	+	0.167120i
$470$		0			0
$471$	229.000	+	396.640i	0.0224029	+	0.0388030i
$472$		0			0
$473$	−6435.00	+	11145.7i	−0.625543	+	1.08347i
$474$		0			0
$475$	3025.00			0.292203
$476$		0			0
$477$	−13731.0			−1.31803
$478$		0			0
$479$	−216.000	+	374.123i	−0.0206039	+	0.0356871i	−0.876144	−	0.482050i	$-0.839892\pi$
$479$	−216.000	+	374.123i	−0.0206039	+	0.0356871i	0.855540	+	0.517737i	$0.173226\pi$
$480$		0			0
$481$	7640.50	+	13233.7i	0.724276	+	1.25448i
$482$		0			0
$483$	−1932.00	−	1673.16i	−0.182006	−	0.157622i
$484$		0			0
$485$	−2255.00	−	3905.77i	−0.211122	−	0.365674i
$486$		0			0
$487$	−5948.00	+	10302.2i	−0.553449	+	0.958602i	0.444574	+	0.895742i	$0.353355\pi$
$487$	−5948.00	+	10302.2i	−0.553449	+	0.958602i	−0.998022	+	0.0628592i	$0.979978\pi$
$488$		0			0
$489$	2456.00			0.227125
$490$		0			0
$491$	12276.0			1.12833			0.564163	−	0.825663i	$-0.309199\pi$
$491$	12276.0			1.12833			0.564163	+	0.825663i	$0.309199\pi$
$492$		0			0
$493$	−4374.00	+	7575.99i	−0.399584	+	0.692100i
$494$		0			0
$495$	2587.50	+	4481.68i	0.234948	+	0.406943i
$496$		0			0
$497$	−672.000	−	581.969i	−0.0606505	−	0.0525249i
$498$		0			0
$499$	−5438.00	−	9418.89i	−0.487852	−	0.844985i	0.512050	−	0.858956i	$-0.328886\pi$
$499$	−5438.00	−	9418.89i	−0.487852	−	0.844985i	−0.999902	+	0.0139706i	$0.995553\pi$
$500$		0			0
$501$	−1929.00	+	3341.13i	−0.172019	+	0.297945i
$502$		0			0
$503$	−12000.0			−1.06372			−0.531862	−	0.846831i	$-0.678508\pi$
$503$	−12000.0			−1.06372			−0.531862	+	0.846831i	$0.678508\pi$
$504$		0			0
$505$	3420.00			0.301362
$506$		0			0
$507$	−1284.00	+	2223.95i	−0.112474	+	0.194811i
$508$		0			0
$509$	5841.00	+	10116.9i	0.508640	+	0.880990i	0.999950	+	0.0100055i	$0.00318492\pi$
$509$	5841.00	+	10116.9i	0.508640	+	0.880990i	−0.491310	+	0.870985i	$0.663482\pi$
$510$		0			0
$511$	−11690.0	+	4049.53i	−1.01201	+	0.350569i
$512$		0			0
$513$	6050.00	+	10478.9i	0.520690	+	0.901862i
$514$		0			0
$515$	−3790.00	+	6564.47i	−0.324286	+	0.561680i
$516$		0			0
$517$	−2025.00			−0.172262
$518$		0			0
$519$	−1398.00			−0.118238
$520$		0			0
$521$	−4804.50	+	8321.64i	−0.404010	+	0.699765i	−0.994206	−	0.107495i	$-0.965717\pi$
$521$	−4804.50	+	8321.64i	−0.404010	+	0.699765i	0.590196	+	0.807260i	$0.299050\pi$
$522$		0			0
$523$	10594.0	+	18349.3i	0.885742	+	1.53415i	0.844860	+	0.534987i	$0.179683\pi$
$523$	10594.0	+	18349.3i	0.885742	+	1.53415i	0.0408820	+	0.999164i	$0.486983\pi$
$524$		0			0
$525$	175.000	−	909.327i	0.0145479	−	0.0755929i
$526$		0			0
$527$	2376.00	+	4115.35i	0.196395	+	0.340166i
$528$		0			0
$529$	3703.00	−	6413.78i	0.304348	−	0.527146i
$530$		0			0
$531$	−8280.00			−0.676688
$532$		0			0
$533$	11505.0			0.934966
$534$		0			0
$535$	−1830.00	+	3169.65i	−0.147884	+	0.256142i
$536$		0			0
$537$	3117.00	+	5398.80i	0.250481	+	0.433846i
$538$		0			0
$539$	−14332.5	−	5728.76i	−1.14535	−	0.457802i
$540$		0			0
$541$	−4036.00	−	6990.56i	−0.320742	−	0.555541i	0.659900	−	0.751354i	$-0.270599\pi$
$541$	−4036.00	−	6990.56i	−0.320742	−	0.555541i	−0.980641	+	0.195813i	$0.937265\pi$
$542$		0			0
$543$	1798.00	−	3114.23i	0.142099	−	0.246122i
$544$		0			0
$545$	−8000.00			−0.628775
$546$		0			0
$547$	−344.000			−0.0268892			−0.0134446	−	0.999910i	$-0.504280\pi$
$547$	−344.000			−0.0268892			−0.0134446	+	0.999910i	$0.504280\pi$
$548$		0			0
$549$	4508.00	−	7808.09i	0.350449	−	0.606996i
$550$		0			0
$551$	9801.00	+	16975.8i	0.757780	+	1.31251i
$552$		0			0
$553$	−2737.00	+	14221.9i	−0.210468	+	1.09363i
$554$		0			0
$555$	1295.00	+	2243.01i	0.0990445	+	0.171550i
$556$		0			0
$557$	−9181.50	+	15902.8i	−0.698443	+	1.20974i	0.270563	+	0.962702i	$0.412790\pi$
$557$	−9181.50	+	15902.8i	−0.698443	+	1.20974i	−0.969006	+	0.247036i	$0.920543\pi$
$558$		0			0
$559$	16874.0			1.27673
$560$		0			0
$561$	−4860.00			−0.365756
$562$		0			0
$563$	−3147.00	+	5450.76i	−0.235578	+	0.408033i	−0.959440	−	0.281912i	$-0.909032\pi$
$563$	−3147.00	+	5450.76i	−0.235578	+	0.408033i	0.723863	+	0.689944i	$0.242365\pi$
$564$		0			0
$565$	3480.00	+	6027.54i	0.259123	+	0.448815i
$566$		0			0
$567$	−7367.50	+	2552.18i	−0.545689	+	0.189032i
$568$		0			0
$569$	−5866.50	−	10161.1i	−0.432226	−	0.748637i	0.564839	−	0.825201i	$-0.308938\pi$
$569$	−5866.50	−	10161.1i	−0.432226	−	0.748637i	−0.997065	+	0.0765642i	$0.975605\pi$
$570$		0			0
$571$	526.000	−	911.059i	0.0385506	−	0.0667717i	−0.846106	−	0.533014i	$-0.821059\pi$
$571$	526.000	−	911.059i	0.0385506	−	0.0667717i	0.884657	+	0.466242i	$0.154393\pi$
$572$		0			0
$573$	4776.00			0.348203
$574$		0			0
$575$	−1725.00			−0.125109
$576$		0			0
$577$	6578.00	−	11393.4i	0.474603	−	0.822036i	−0.524974	−	0.851118i	$-0.675925\pi$
$577$	6578.00	−	11393.4i	0.474603	−	0.822036i	0.999577	+	0.0290821i	$0.00925844\pi$
$578$		0			0
$579$	−272.000	−	471.118i	−0.0195232	−	0.0338152i
$580$		0			0
$581$	−10752.0	−	9311.51i	−0.767759	−	0.664899i
$582$		0			0
$583$	−13432.5	−	23265.8i	−0.954232	−	1.65278i
$584$		0			0
$585$	3392.50	−	5875.98i	0.239765	−	0.415285i
$586$		0			0
$587$	13368.0			0.939960			0.469980	−	0.882677i	$-0.344261\pi$
$587$	13368.0			0.939960			0.469980	+	0.882677i	$0.344261\pi$
$588$		0			0
$589$	10648.0			0.744895
$590$		0			0
$591$	2109.00	−	3652.90i	0.146790	−	0.254247i
$592$		0			0
$593$	−13332.0	−	23091.7i	−0.923237	−	1.59909i	−0.794372	−	0.607431i	$-0.792200\pi$
$593$	−13332.0	−	23091.7i	−0.923237	−	1.59909i	−0.128865	−	0.991662i	$-0.541133\pi$
$594$		0			0
$595$	−3780.00	−	3273.58i	−0.260445	−	0.225552i
$596$		0			0
$597$	1424.00	+	2466.44i	0.0976222	+	0.169087i
$598$		0			0
$599$	3807.00	−	6593.92i	0.259682	−	0.449783i	−0.706474	−	0.707739i	$-0.749715\pi$
$599$	3807.00	−	6593.92i	0.259682	−	0.449783i	0.966157	+	0.257955i	$0.0830488\pi$
$600$		0			0
$601$	6410.00			0.435057			0.217529	−	0.976054i	$-0.430200\pi$
$601$	6410.00			0.435057			0.217529	+	0.976054i	$0.430200\pi$
$602$		0			0
$603$	−6440.00			−0.434921
$604$		0			0
$605$	−1735.00	+	3005.11i	−0.116591	+	0.201942i
$606$		0			0
$607$	−10734.5	−	18592.7i	−0.717792	−	1.24325i	−0.961873	−	0.273498i	$-0.911819\pi$
$607$	−10734.5	−	18592.7i	−0.717792	−	1.24325i	0.244080	−	0.969755i	$-0.421514\pi$
$608$		0			0
$609$	5670.00	−	1964.15i	0.377274	−	0.130692i
$610$		0			0
$611$	1327.50	+	2299.30i	0.0878967	+	0.152242i
$612$		0			0
$613$	−1868.50	+	3236.34i	−0.123113	+	0.213237i	−0.920994	−	0.389578i	$-0.872621\pi$
$613$	−1868.50	+	3236.34i	−0.123113	+	0.213237i	0.797881	+	0.602815i	$0.205954\pi$
$614$		0			0
$615$	1950.00			0.127856
$616$		0			0
$617$	18078.0			1.17957			0.589784	−	0.807561i	$-0.299213\pi$
$617$	18078.0			1.17957			0.589784	+	0.807561i	$0.299213\pi$
$618$		0			0
$619$	6143.50	−	10640.9i	0.398915	−	0.690940i	−0.594678	−	0.803964i	$-0.702720\pi$
$619$	6143.50	−	10640.9i	0.398915	−	0.690940i	0.993592	+	0.113024i	$0.0360537\pi$
$620$		0			0
$621$	−3450.00	−	5975.58i	−0.222937	−	0.386138i
$622$		0			0
$623$	−4179.00	+	21714.7i	−0.268745	+	1.39644i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	−5445.00	+	9431.02i	−0.346814	+	0.600699i
$628$		0			0
$629$	13986.0			0.886579
$630$		0			0
$631$	9580.00			0.604396			0.302198	−	0.953245i	$-0.402280\pi$
$631$	9580.00			0.604396			0.302198	+	0.953245i	$0.402280\pi$
$632$		0			0
$633$	−3625.00	+	6278.68i	−0.227616	+	0.394242i
$634$		0			0
$635$	2007.50	+	3477.09i	0.125457	+	0.217298i
$636$		0			0
$637$	2891.00	+	20029.4i	0.179820	+	1.24583i
$638$		0			0
$639$	−552.000	−	956.092i	−0.0341734	−	0.0591900i
$640$		0			0
$641$	−5389.50	+	9334.89i	−0.332094	+	0.575204i	−0.982922	−	0.184021i	$-0.941089\pi$
$641$	−5389.50	+	9334.89i	−0.332094	+	0.575204i	0.650828	+	0.759225i	$0.274422\pi$
$642$		0			0
$643$	−8882.00			−0.544746			−0.272373	−	0.962192i	$-0.587809\pi$
$643$	−8882.00			−0.544746			−0.272373	+	0.962192i	$0.587809\pi$
$644$		0			0
$645$	2860.00			0.174593
$646$		0			0
$647$	−5509.50	+	9542.73i	−0.334777	+	0.579851i	−0.983442	−	0.181223i	$-0.941994\pi$
$647$	−5509.50	+	9542.73i	−0.334777	+	0.579851i	0.648665	+	0.761074i	$0.275328\pi$
$648$		0			0
$649$	−8100.00	−	14029.6i	−0.489912	−	0.848552i
$650$		0			0
$651$	616.000	−	3200.83i	0.0370859	−	0.192704i
$652$		0			0
$653$	−11161.5	−	19332.3i	−0.668887	−	1.15855i	−0.978216	−	0.207591i	$-0.933438\pi$
$653$	−11161.5	−	19332.3i	−0.668887	−	1.15855i	0.309329	−	0.950955i	$-0.399896\pi$
$654$		0			0
$655$	5047.50	−	8742.53i	0.301103	−	0.521525i
$656$		0			0
$657$	−15364.0			−0.912339
$658$		0			0
$659$	11856.0			0.700826			0.350413	−	0.936595i	$-0.386041\pi$
$659$	11856.0			0.700826			0.350413	+	0.936595i	$0.386041\pi$
$660$		0			0
$661$	16622.0	−	28790.1i	0.978095	−	1.69411i	0.308777	−	0.951134i	$-0.400080\pi$
$661$	16622.0	−	28790.1i	0.978095	−	1.69411i	0.669318	−	0.742976i	$-0.266586\pi$
$662$		0			0
$663$	3186.00	+	5518.31i	0.186627	+	0.323248i
$664$		0			0
$665$	−10587.5	+	3667.62i	−0.617392	+	0.213871i
$666$		0			0
$667$	−5589.00	−	9680.43i	−0.324448	−	0.561961i
$668$		0			0
$669$	−4960.00	+	8590.97i	−0.286644	+	0.496482i
$670$		0			0
$671$	17640.0			1.01488
$672$		0			0
$673$	−12322.0			−0.705763			−0.352881	−	0.935668i	$-0.614798\pi$
$673$	−12322.0			−0.705763			−0.352881	+	0.935668i	$0.614798\pi$
$674$		0			0
$675$	1250.00	−	2165.06i	0.0712778	−	0.123457i
$676$		0			0
$677$	6298.50	+	10909.3i	0.357564	+	0.619320i	0.987553	−	0.157285i	$-0.0502740\pi$
$677$	6298.50	+	10909.3i	0.357564	+	0.619320i	−0.629989	+	0.776604i	$0.716941\pi$
$678$		0			0
$679$	12628.0	+	10936.2i	0.713723	+	0.618103i
$680$		0			0
$681$	−1500.00	−	2598.08i	−0.0844055	−	0.146195i
$682$		0			0
$683$	−4170.00	+	7222.65i	−0.233617	+	0.404637i	−0.958870	−	0.283846i	$-0.908390\pi$
$683$	−4170.00	+	7222.65i	−0.233617	+	0.404637i	0.725253	+	0.688483i	$0.241723\pi$
$684$		0			0
$685$	300.000			0.0167334
$686$		0			0
$687$	12184.0			0.676636
$688$		0			0
$689$	−17611.5	+	30504.0i	−0.973795	+	1.68666i
$690$		0			0
$691$	−10100.0	−	17493.7i	−0.556038	−	0.963086i	−0.997822	−	0.0659643i	$-0.978988\pi$
$691$	−10100.0	−	17493.7i	−0.556038	−	0.963086i	0.441784	−	0.897121i	$-0.354346\pi$
$692$		0			0
$693$	−14490.0	−	12548.7i	−0.794271	−	0.687859i
$694$		0			0
$695$	−4270.00	−	7395.86i	−0.233051	−	0.403656i
$696$		0			0
$697$	5265.00	−	9119.25i	0.286121	−	0.495576i
$698$		0			0
$699$	276.000			0.0149346
$700$		0			0
$701$	474.000			0.0255388			0.0127694	−	0.999918i	$-0.495935\pi$
$701$	474.000			0.0255388			0.0127694	+	0.999918i	$0.495935\pi$
$702$		0			0
$703$	15669.5	−	27140.4i	0.840663	−	1.45607i
$704$		0			0
$705$	225.000	+	389.711i	0.0120198	+	0.0208190i
$706$		0			0
$707$	−11970.0	+	4146.53i	−0.636744	+	0.220575i
$708$		0			0
$709$	12563.0	+	21759.8i	0.665463	+	1.15262i	0.979160	+	0.203093i	$0.0650993\pi$
$709$	12563.0	+	21759.8i	0.665463	+	1.15262i	−0.313696	+	0.949523i	$0.601567\pi$
$710$		0			0
$711$	−8993.00	+	15576.3i	−0.474351	+	0.821601i
$712$		0			0
$713$	−6072.00			−0.318932
$714$		0			0
$715$	13275.0			0.694345
$716$		0			0
$717$	−5502.00	+	9529.74i	−0.286577	+	0.496367i
$718$		0			0
$719$	−3648.00	−	6318.52i	−0.189218	−	0.327734i	0.755772	−	0.654835i	$-0.227262\pi$
$719$	−3648.00	−	6318.52i	−0.189218	−	0.327734i	−0.944990	+	0.327100i	$0.893928\pi$
$720$		0			0
$721$	5306.00	−	27570.8i	0.274072	−	1.42412i
$722$		0			0
$723$	−3551.00	−	6150.51i	−0.182660	−	0.316376i
$724$		0			0
$725$	2025.00	−	3507.40i	0.103733	−	0.179671i
$726$		0			0
$727$	15421.0			0.786703			0.393352	−	0.919388i	$-0.371316\pi$
$727$	15421.0			0.786703			0.393352	+	0.919388i	$0.371316\pi$
$728$		0			0
$729$	−4283.00			−0.217599
$730$		0			0
$731$	7722.00	−	13374.9i	0.390709	−	0.676728i
$732$		0			0
$733$	14583.5	+	25259.4i	0.734862	+	1.27282i	0.954784	+	0.297301i	$0.0960864\pi$
$733$	14583.5	+	25259.4i	0.734862	+	1.27282i	−0.219922	+	0.975517i	$0.570580\pi$
$734$		0			0
$735$	490.000	+	3394.82i	0.0245904	+	0.170367i
$736$		0			0
$737$	−6300.00	−	10911.9i	−0.314876	−	0.545381i
$738$		0			0
$739$	−6690.50	+	11588.3i	−0.333037	+	0.576836i	−0.983106	−	0.183039i	$-0.941407\pi$
$739$	−6690.50	+	11588.3i	−0.333037	+	0.576836i	0.650069	+	0.759875i	$0.274740\pi$
$740$		0			0
$741$	14278.0			0.707848
$742$		0			0
$743$	−5487.00			−0.270927			−0.135463	−	0.990782i	$-0.543252\pi$
$743$	−5487.00			−0.270927			−0.135463	+	0.990782i	$0.543252\pi$
$744$		0			0
$745$	2715.00	−	4702.52i	0.133517	−	0.231258i
$746$		0			0
$747$	−8832.00	−	15297.5i	−0.432592	−	0.749271i
$748$		0			0
$749$	2562.00	−	13312.5i	0.124985	−	0.649439i
$750$		0			0
$751$	3319.00	+	5748.68i	0.161268	+	0.279324i	0.935324	−	0.353793i	$-0.115108\pi$
$751$	3319.00	+	5748.68i	0.161268	+	0.279324i	−0.774056	+	0.633117i	$0.781775\pi$
$752$		0			0
$753$	7065.00	−	12236.9i	0.341916	−	0.592216i
$754$		0			0
$755$	14330.0			0.690758
$756$		0			0
$757$	14846.0			0.712797			0.356398	−	0.934334i	$-0.384005\pi$
$757$	14846.0			0.712797			0.356398	+	0.934334i	$0.384005\pi$
$758$		0			0
$759$	3105.00	−	5378.02i	0.148491	−	0.257193i
$760$		0			0
$761$	1825.50	+	3161.86i	0.0869571	+	0.150614i	0.906223	−	0.422799i	$-0.138952\pi$
$761$	1825.50	+	3161.86i	0.0869571	+	0.150614i	−0.819266	+	0.573413i	$0.805619\pi$
$762$		0			0
$763$	28000.0	−	9699.48i	1.32853	−	0.460216i
$764$		0			0
$765$	−3105.00	−	5378.02i	−0.146747	−	0.254173i
$766$		0			0
$767$	−10620.0	+	18394.4i	−0.499956	+	0.865949i
$768$		0			0
$769$	29855.0			1.40000			0.699999	−	0.714144i	$-0.253184\pi$
$769$	29855.0			1.40000			0.699999	+	0.714144i	$0.253184\pi$
$770$		0			0
$771$	−8160.00			−0.381161
$772$		0			0
$773$	3259.50	−	5645.62i	0.151664	−	0.262689i	−0.780175	−	0.625561i	$-0.784870\pi$
$773$	3259.50	−	5645.62i	0.151664	−	0.262689i	0.931839	+	0.362871i	$0.118204\pi$
$774$		0			0
$775$	−1100.00	−	1905.26i	−0.0509847	−	0.0883081i
$776$		0			0
$777$	−7252.00	−	6280.42i	−0.334831	−	0.289973i
$778$		0			0
$779$	−11797.5	−	20433.9i	−0.542605	−	0.939819i
$780$		0			0
$781$	1080.00	−	1870.61i	0.0494820	−	0.0857053i
$782$		0			0
$783$	16200.0			0.739388
$784$		0			0
$785$	−1145.00			−0.0520596

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-2.50000 - 4.33013i) q^{5} +(14.0000 + 12.1244i) q^{7} +(11.5000 + 19.9186i) q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{3} +(-2.50000 - 4.33013i) q^{5} +(14.0000 + 12.1244i) q^{7} +(11.5000 + 19.9186i) q^{9} +(-22.5000 + 38.9711i) q^{11} +59.0000 q^{13} +10.0000 q^{15} +(27.0000 - 46.7654i) q^{17} +(-60.5000 - 104.789i) q^{19} +(-35.0000 + 12.1244i) q^{21} +(34.5000 + 59.7558i) q^{23} +(-12.5000 + 21.6506i) q^{25} -100.000 q^{27} -162.000 q^{29} +(-44.0000 + 76.2102i) q^{31} +(-45.0000 - 77.9423i) q^{33} +(17.5000 - 90.9327i) q^{35} +(129.500 + 224.301i) q^{37} +(-59.0000 + 102.191i) q^{39} +195.000 q^{41} +286.000 q^{43} +(57.5000 - 99.5929i) q^{45} +(22.5000 + 38.9711i) q^{47} +(49.0000 + 339.482i) q^{49} +(54.0000 + 93.5307i) q^{51} +(-298.500 + 517.017i) q^{53} +225.000 q^{55} +242.000 q^{57} +(-180.000 + 311.769i) q^{59} +(-196.000 - 339.482i) q^{61} +(-80.5000 + 418.290i) q^{63} +(-147.500 - 255.477i) q^{65} +(-140.000 + 242.487i) q^{67} -138.000 q^{69} -48.0000 q^{71} +(-334.000 + 578.505i) q^{73} +(-25.0000 - 43.3013i) q^{75} +(-787.500 + 272.798i) q^{77} +(391.000 + 677.232i) q^{79} +(-210.500 + 364.597i) q^{81} -768.000 q^{83} -270.000 q^{85} +(162.000 - 280.592i) q^{87} +(597.000 + 1034.03i) q^{89} +(826.000 + 715.337i) q^{91} +(-88.0000 - 152.420i) q^{93} +(-302.500 + 523.945i) q^{95} +902.000 q^{97} -1035.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9} - 45 q^{11} + 118 q^{13} + 20 q^{15} + 54 q^{17} - 121 q^{19} - 70 q^{21} + 69 q^{23} - 25 q^{25} - 200 q^{27} - 324 q^{29} - 88 q^{31} - 90 q^{33} + 35 q^{35} + 259 q^{37} - 118 q^{39} + 390 q^{41} + 572 q^{43} + 115 q^{45} + 45 q^{47} + 98 q^{49} + 108 q^{51} - 597 q^{53} + 450 q^{55} + 484 q^{57} - 360 q^{59} - 392 q^{61} - 161 q^{63} - 295 q^{65} - 280 q^{67} - 276 q^{69} - 96 q^{71} - 668 q^{73} - 50 q^{75} - 1575 q^{77} + 782 q^{79} - 421 q^{81} - 1536 q^{83} - 540 q^{85} + 324 q^{87} + 1194 q^{89} + 1652 q^{91} - 176 q^{93} - 605 q^{95} + 1804 q^{97} - 2070 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9 - 45 * q^11 + 118 * q^13 + 20 * q^15 + 54 * q^17 - 121 * q^19 - 70 * q^21 + 69 * q^23 - 25 * q^25 - 200 * q^27 - 324 * q^29 - 88 * q^31 - 90 * q^33 + 35 * q^35 + 259 * q^37 - 118 * q^39 + 390 * q^41 + 572 * q^43 + 115 * q^45 + 45 * q^47 + 98 * q^49 + 108 * q^51 - 597 * q^53 + 450 * q^55 + 484 * q^57 - 360 * q^59 - 392 * q^61 - 161 * q^63 - 295 * q^65 - 280 * q^67 - 276 * q^69 - 96 * q^71 - 668 * q^73 - 50 * q^75 - 1575 * q^77 + 782 * q^79 - 421 * q^81 - 1536 * q^83 - 540 * q^85 + 324 * q^87 + 1194 * q^89 + 1652 * q^91 - 176 * q^93 - 605 * q^95 + 1804 * q^97 - 2070 * q^99

Introduction

L-functions

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