LMFDB - Embedded newform 560.4.g.f.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(449,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.g (of order $2$, degree $1$, 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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$2.67516i$ of defining polynomial
Character	$\chi$	$=$	560.449
Dual form			560.4.g.f.449.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49396i q^{3} +(6.35505 - 9.19855i) q^{5} -7.00000i q^{7} +20.7802 q^{9} +57.5880 q^{11} +45.5159i q^{13} +(-22.9408 - 15.8492i) q^{15} +92.0051i q^{17} +125.177 q^{19} -17.4577 q^{21} +158.496i q^{23} +(-44.2268 - 116.914i) q^{25} -119.162i q^{27} +40.1708 q^{29} -49.5590 q^{31} -143.622i q^{33} +(-64.3899 - 44.4853i) q^{35} +231.307i q^{37} +113.515 q^{39} +169.556 q^{41} -147.428i q^{43} +(132.059 - 191.147i) q^{45} +67.0327i q^{47} -49.0000 q^{49} +229.457 q^{51} -268.647i q^{53} +(365.974 - 529.726i) q^{55} -312.187i q^{57} -240.843 q^{59} +90.4579 q^{61} -145.461i q^{63} +(418.681 + 289.256i) q^{65} +406.498i q^{67} +395.283 q^{69} -330.782 q^{71} -546.255i q^{73} +(-291.580 + 110.300i) q^{75} -403.116i q^{77} -25.3087 q^{79} +263.879 q^{81} +376.255i q^{83} +(846.314 + 584.697i) q^{85} -100.184i q^{87} -1026.44 q^{89} +318.612 q^{91} +123.598i q^{93} +(795.506 - 1151.45i) q^{95} -942.660i q^{97} +1196.69 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} - 46 q^{9} - 84 q^{11} - 8 q^{15} - 72 q^{19} + 140 q^{21} - 362 q^{25} + 88 q^{29} - 120 q^{31} + 28 q^{35} - 212 q^{39} - 852 q^{41} - 510 q^{45} - 490 q^{49} - 1276 q^{51} + 1136 q^{55}+ \cdots + 5304 q^{99}+O(q^{100}) $ 10 * q + 6 * q^5 - 46 * q^9 - 84 * q^11 - 8 * q^15 - 72 * q^19 + 140 * q^21 - 362 * q^25 + 88 * q^29 - 120 * q^31 + 28 * q^35 - 212 * q^39 - 852 * q^41 - 510 * q^45 - 490 * q^49 - 1276 * q^51 + 1136 * q^55 - 864 * q^59 - 884 * q^61 + 520 * q^65 + 4808 * q^69 - 880 * q^71 - 720 * q^75 + 3580 * q^79 - 302 * q^81 + 1572 * q^85 - 1492 * q^89 - 476 * q^91 + 1628 * q^95 + 5304 * q^99

Character values

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

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	2.49396i		−	0.479963i	−0.970777	−	0.239982i	$-0.922859\pi$
$3$		−	2.49396i		−	0.479963i	0.970777	−	0.239982i	$-0.0771414\pi$
$4$		0			0
$5$	6.35505	−	9.19855i	0.568413	−	0.822744i
$5$	6.35505	−	9.19855i	0.568413	−	0.822744i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$		0			0
$9$	20.7802			0.769635
$10$		0			0
$11$	57.5880			1.57849			0.789247	−	0.614076i	$-0.210471\pi$
$11$	57.5880			1.57849			0.789247	+	0.614076i	$0.210471\pi$
$12$		0			0
$13$			45.5159i			0.971066i	0.874218	+	0.485533i	$0.161374\pi$
$13$			45.5159i			0.971066i	−0.874218	+	0.485533i	$0.838626\pi$
$14$		0			0
$15$	−22.9408	−	15.8492i	−0.394887	−	0.272817i
$16$		0			0
$17$			92.0051i			1.31262i	0.754492	+	0.656309i	$0.227883\pi$
$17$			92.0051i			1.31262i	−0.754492	+	0.656309i	$0.772117\pi$
$18$		0			0
$19$	125.177			1.51145			0.755726	−	0.654888i	$-0.227284\pi$
$19$	125.177			1.51145			0.755726	+	0.654888i	$0.227284\pi$
$20$		0			0
$21$	−17.4577			−0.181409
$22$		0			0
$23$			158.496i			1.43690i	0.695578	+	0.718451i	$0.255149\pi$
$23$			158.496i			1.43690i	−0.695578	+	0.718451i	$0.744851\pi$
$24$		0			0
$25$	−44.2268	−	116.914i	−0.353814	−	0.935316i
$26$		0			0
$27$		−	119.162i		−	0.849360i
$28$		0			0
$29$	40.1708			0.257225			0.128613	−	0.991695i	$-0.458948\pi$
$29$	40.1708			0.257225			0.128613	+	0.991695i	$0.458948\pi$
$30$		0			0
$31$	−49.5590			−0.287131			−0.143566	−	0.989641i	$-0.545857\pi$
$31$	−49.5590			−0.287131			−0.143566	+	0.989641i	$0.545857\pi$
$32$		0			0
$33$		−	143.622i		−	0.757619i
$34$		0			0
$35$	−64.3899	−	44.4853i	−0.310968	−	0.214840i
$36$		0			0
$37$			231.307i			1.02775i	0.857866	+	0.513874i	$0.171790\pi$
$37$			231.307i			1.02775i	−0.857866	+	0.513874i	$0.828210\pi$
$38$		0			0
$39$	113.515			0.466076
$40$		0			0
$41$	169.556			0.645859			0.322929	−	0.946423i	$-0.395332\pi$
$41$	169.556			0.645859			0.322929	+	0.946423i	$0.395332\pi$
$42$		0			0
$43$		−	147.428i		−	0.522849i	−0.965224	−	0.261425i	$-0.915808\pi$
$43$		−	147.428i		−	0.522849i	0.965224	−	0.261425i	$-0.0841923\pi$
$44$		0			0
$45$	132.059	−	191.147i	0.437470	−	0.633213i
$46$		0			0
$47$			67.0327i			0.208037i	0.994575	+	0.104018i	$0.0331701\pi$
$47$			67.0327i			0.208037i	−0.994575	+	0.104018i	$0.966830\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	229.457			0.630008
$52$		0			0
$53$		−	268.647i		−	0.696254i	−0.937447	−	0.348127i	$-0.886818\pi$
$53$		−	268.647i		−	0.696254i	0.937447	−	0.348127i	$-0.113182\pi$
$54$		0			0
$55$	365.974	−	529.726i	0.897236	−	1.29870i
$56$		0			0
$57$		−	312.187i		−	0.725441i
$58$		0			0
$59$	−240.843			−0.531442			−0.265721	−	0.964050i	$-0.585610\pi$
$59$	−240.843			−0.531442			−0.265721	+	0.964050i	$0.585610\pi$
$60$		0			0
$61$	90.4579			0.189868			0.0949340	−	0.995484i	$-0.469736\pi$
$61$	90.4579			0.189868			0.0949340	+	0.995484i	$0.469736\pi$
$62$		0			0
$63$		−	145.461i		−	0.290895i
$64$		0			0
$65$	418.681	+	289.256i	0.798938	+	0.551966i
$66$		0			0
$67$			406.498i			0.741218i	0.928789	+	0.370609i	$0.120851\pi$
$67$			406.498i			0.741218i	−0.928789	+	0.370609i	$0.879149\pi$
$68$		0			0
$69$	395.283			0.689660
$70$		0			0
$71$	−330.782			−0.552910			−0.276455	−	0.961027i	$-0.589160\pi$
$71$	−330.782			−0.552910			−0.276455	+	0.961027i	$0.589160\pi$
$72$		0			0
$73$		−	546.255i		−	0.875812i	−0.899021	−	0.437906i	$-0.855720\pi$
$73$		−	546.255i		−	0.875812i	0.899021	−	0.437906i	$-0.144280\pi$
$74$		0			0
$75$	−291.580	+	110.300i	−0.448917	+	0.169818i
$76$		0			0
$77$		−	403.116i		−	0.596615i
$78$		0			0
$79$	−25.3087			−0.0360436			−0.0180218	−	0.999838i	$-0.505737\pi$
$79$	−25.3087			−0.0360436			−0.0180218	+	0.999838i	$0.505737\pi$
$80$		0			0
$81$	263.879			0.361974
$82$		0			0
$83$			376.255i			0.497582i	0.968557	+	0.248791i	$0.0800333\pi$
$83$			376.255i			0.497582i	−0.968557	+	0.248791i	$0.919967\pi$
$84$		0			0
$85$	846.314	+	584.697i	1.07995	+	0.746109i
$86$		0			0
$87$		−	100.184i		−	0.123459i
$88$		0			0
$89$	−1026.44			−1.22250			−0.611248	−	0.791439i	$-0.709332\pi$
$89$	−1026.44			−1.22250			−0.611248	+	0.791439i	$0.709332\pi$
$90$		0			0
$91$	318.612			0.367028
$92$		0			0
$93$			123.598i			0.137812i
$94$		0			0
$95$	795.506	−	1151.45i	0.859128	−	1.24354i
$96$		0			0
$97$		−	942.660i		−	0.986728i	−0.869823	−	0.493364i	$-0.835767\pi$
$97$		−	942.660i		−	0.986728i	0.869823	−	0.493364i	$-0.164233\pi$
$98$		0			0
$99$	1196.69			1.21487
$100$		0			0
$101$	−604.617			−0.595659			−0.297830	−	0.954619i	$-0.596263\pi$
$101$	−604.617			−0.595659			−0.297830	+	0.954619i	$0.596263\pi$
$102$		0			0
$103$		−	300.967i		−	0.287914i	−0.989584	−	0.143957i	$-0.954017\pi$
$103$		−	300.967i		−	0.287914i	0.989584	−	0.143957i	$-0.0459828\pi$
$104$		0			0
$105$	−110.945	+	160.586i	−0.103115	+	0.149253i
$106$		0			0
$107$		−	1511.66i		−	1.36577i	−0.730525	−	0.682886i	$-0.760724\pi$
$107$		−	1511.66i		−	1.36577i	0.730525	−	0.682886i	$-0.239276\pi$
$108$		0			0
$109$	1767.09			1.55281			0.776406	−	0.630233i	$-0.217041\pi$
$109$	1767.09			1.55281			0.776406	+	0.630233i	$0.217041\pi$
$110$		0			0
$111$	576.871			0.493281
$112$		0			0
$113$		−	1045.27i		−	0.870182i	−0.900387	−	0.435091i	$-0.856716\pi$
$113$		−	1045.27i		−	0.870182i	0.900387	−	0.435091i	$-0.143284\pi$
$114$		0			0
$115$	1457.94	+	1007.25i	1.18220	+	0.816753i
$116$		0			0
$117$			945.828i			0.747366i
$118$		0			0
$119$	644.036			0.496123
$120$		0			0
$121$	1985.38			1.49164
$122$		0			0
$123$		−	422.866i		−	0.309988i
$124$		0			0
$125$	−1356.51	−	336.174i	−0.970638	−	0.240547i
$126$		0			0
$127$		−	260.727i		−	0.182171i	−0.995843	−	0.0910857i	$-0.970966\pi$
$127$		−	260.727i		−	0.182171i	0.995843	−	0.0910857i	$-0.0290337\pi$
$128$		0			0
$129$	−367.679			−0.250948
$130$		0			0
$131$	723.522			0.482553			0.241276	−	0.970456i	$-0.422434\pi$
$131$	723.522			0.482553			0.241276	+	0.970456i	$0.422434\pi$
$132$		0			0
$133$		−	876.240i		−	0.571275i
$134$		0			0
$135$	−1096.12	−	757.279i	−0.698805	−	0.482787i
$136$		0			0
$137$			773.693i			0.482490i	0.970464	+	0.241245i	$0.0775557\pi$
$137$			773.693i			0.482490i	−0.970464	+	0.241245i	$0.922444\pi$
$138$		0			0
$139$	−2952.97			−1.80192			−0.900961	−	0.433899i	$-0.857137\pi$
$139$	−2952.97			−1.80192			−0.900961	+	0.433899i	$0.857137\pi$
$140$		0			0
$141$	167.177			0.0998499
$142$		0			0
$143$			2621.17i			1.53282i
$144$		0			0
$145$	255.287	−	369.513i	0.146210	−	0.211630i
$146$		0			0
$147$			122.204i			0.0685662i
$148$		0			0
$149$	−2514.00			−1.38225			−0.691124	−	0.722736i	$-0.742884\pi$
$149$	−2514.00			−1.38225			−0.691124	+	0.722736i	$0.742884\pi$
$150$		0			0
$151$	−101.052			−0.0544605			−0.0272302	−	0.999629i	$-0.508669\pi$
$151$	−101.052			−0.0544605			−0.0272302	+	0.999629i	$0.508669\pi$
$152$		0			0
$153$			1911.88i			1.01024i
$154$		0			0
$155$	−314.950	+	455.872i	−0.163209	+	0.236235i
$156$		0			0
$157$			2338.35i			1.18867i	0.804219	+	0.594333i	$0.202584\pi$
$157$			2338.35i			1.18867i	−0.804219	+	0.594333i	$0.797416\pi$
$158$		0			0
$159$	−669.995			−0.334176
$160$		0			0
$161$	1109.47			0.543098
$162$		0			0
$163$		−	1325.20i		−	0.636798i	−0.947957	−	0.318399i	$-0.896855\pi$
$163$		−	1325.20i		−	0.636798i	0.947957	−	0.318399i	$-0.103145\pi$
$164$		0			0
$165$	−1321.12	−	912.726i	−0.623326	−	0.430640i
$166$		0			0
$167$		−	2086.20i		−	0.966675i	−0.875434	−	0.483338i	$-0.839424\pi$
$167$		−	2086.20i		−	0.966675i	0.875434	−	0.483338i	$-0.160576\pi$
$168$		0			0
$169$	125.299			0.0570317
$170$		0			0
$171$	2601.20			1.16327
$172$		0			0
$173$		−	1918.19i		−	0.842990i	−0.906831	−	0.421495i	$-0.861505\pi$
$173$		−	1918.19i		−	0.842990i	0.906831	−	0.421495i	$-0.138495\pi$
$174$		0			0
$175$	−818.401	+	309.587i	−0.353516	+	0.133729i
$176$		0			0
$177$			600.652i			0.255072i
$178$		0			0
$179$	629.046			0.262665			0.131333	−	0.991338i	$-0.458074\pi$
$179$	629.046			0.262665			0.131333	+	0.991338i	$0.458074\pi$
$180$		0			0
$181$	−2800.85			−1.15020			−0.575099	−	0.818084i	$-0.695036\pi$
$181$	−2800.85			−1.15020			−0.575099	+	0.818084i	$0.695036\pi$
$182$		0			0
$183$		−	225.599i		−	0.0911296i
$184$		0			0
$185$	2127.69	+	1469.97i	0.845573	+	0.584185i
$186$		0			0
$187$			5298.39i			2.07196i
$188$		0			0
$189$	−834.133			−0.321028
$190$		0			0
$191$	−740.255			−0.280434			−0.140217	−	0.990121i	$-0.544780\pi$
$191$	−740.255			−0.280434			−0.140217	+	0.990121i	$0.544780\pi$
$192$		0			0
$193$		−	4082.57i		−	1.52264i	−0.648375	−	0.761321i	$-0.724551\pi$
$193$		−	4082.57i		−	1.52264i	0.648375	−	0.761321i	$-0.275449\pi$
$194$		0			0
$195$	721.393	−	1044.17i	0.264923	−	0.383461i
$196$		0			0
$197$			3414.89i			1.23503i	0.786559	+	0.617515i	$0.211860\pi$
$197$			3414.89i			1.23503i	−0.786559	+	0.617515i	$0.788140\pi$
$198$		0			0
$199$	−3392.44			−1.20846			−0.604231	−	0.796809i	$-0.706520\pi$
$199$	−3392.44			−1.20846			−0.604231	+	0.796809i	$0.706520\pi$
$200$		0			0
$201$	1013.79			0.355757
$202$		0			0
$203$		−	281.196i		−	0.0972220i
$204$		0			0
$205$	1077.54	−	1559.67i	0.367114	−	0.531376i
$206$		0			0
$207$			3293.58i			1.10589i
$208$		0			0
$209$	7208.70			2.38582
$210$		0			0
$211$	−3398.04			−1.10867			−0.554337	−	0.832292i	$-0.687028\pi$
$211$	−3398.04			−1.10867			−0.554337	+	0.832292i	$0.687028\pi$
$212$		0			0
$213$			824.958i			0.265376i
$214$		0			0
$215$	−1356.12	−	936.910i	−0.430171	−	0.297194i
$216$		0			0
$217$			346.913i			0.108525i
$218$		0			0
$219$	−1362.34			−0.420358
$220$		0			0
$221$	−4187.70			−1.27464
$222$		0			0
$223$			182.611i			0.0548365i	0.999624	+	0.0274183i	$0.00872860\pi$
$223$			182.611i			0.0548365i	−0.999624	+	0.0274183i	$0.991271\pi$
$224$		0			0
$225$	−919.040	−	2429.50i	−0.272308	−	0.719852i
$226$		0			0
$227$			3152.33i			0.921707i	0.887476	+	0.460854i	$0.152457\pi$
$227$			3152.33i			0.921707i	−0.887476	+	0.460854i	$0.847543\pi$
$228$		0			0
$229$	−6012.35			−1.73497			−0.867483	−	0.497466i	$-0.834264\pi$
$229$	−6012.35			−1.73497			−0.867483	+	0.497466i	$0.834264\pi$
$230$		0			0
$231$	−1005.36			−0.286353
$232$		0			0
$233$			940.660i			0.264484i	0.991217	+	0.132242i	$0.0422175\pi$
$233$			940.660i			0.264484i	−0.991217	+	0.132242i	$0.957782\pi$
$234$		0			0
$235$	616.604	+	425.996i	0.171161	+	0.118251i
$236$		0			0
$237$			63.1188i			0.0172996i
$238$		0			0
$239$	5158.82			1.39622			0.698109	−	0.715991i	$-0.254025\pi$
$239$	5158.82			1.39622			0.698109	+	0.715991i	$0.254025\pi$
$240$		0			0
$241$	−463.836			−0.123976			−0.0619882	−	0.998077i	$-0.519744\pi$
$241$	−463.836			−0.123976			−0.0619882	+	0.998077i	$0.519744\pi$
$242$		0			0
$243$		−	3875.47i		−	1.02309i
$244$		0			0
$245$	−311.397	+	450.729i	−0.0812018	+	0.117535i
$246$		0			0
$247$			5697.55i			1.46772i
$248$		0			0
$249$	938.365			0.238821
$250$		0			0
$251$	−2290.25			−0.575934			−0.287967	−	0.957640i	$-0.592979\pi$
$251$	−2290.25			−0.575934			−0.287967	+	0.957640i	$0.592979\pi$
$252$		0			0
$253$			9127.48i			2.26814i
$254$		0			0
$255$	1458.21	−	2110.67i	0.358105	−	0.518335i
$256$		0			0
$257$		−	802.202i		−	0.194708i	−0.995250	−	0.0973541i	$-0.968962\pi$
$257$		−	802.202i		−	0.194708i	0.995250	−	0.0973541i	$-0.0310379\pi$
$258$		0			0
$259$	1619.15			0.388452
$260$		0			0
$261$	834.756			0.197970
$262$		0			0
$263$		−	286.978i		−	0.0672845i	−0.999434	−	0.0336423i	$-0.989289\pi$
$263$		−	286.978i		−	0.0672845i	0.999434	−	0.0336423i	$-0.0107107\pi$
$264$		0			0
$265$	−2471.16	−	1707.26i	−0.572839	−	0.395760i
$266$		0			0
$267$			2559.90i			0.586753i
$268$		0			0
$269$	−3561.22			−0.807180			−0.403590	−	0.914940i	$-0.632238\pi$
$269$	−3561.22			−0.807180			−0.403590	+	0.914940i	$0.632238\pi$
$270$		0			0
$271$	−1928.81			−0.432349			−0.216175	−	0.976355i	$-0.569358\pi$
$271$	−1928.81			−0.432349			−0.216175	+	0.976355i	$0.569358\pi$
$272$		0			0
$273$		−	794.605i		−	0.176160i
$274$		0			0
$275$	−2546.93	−	6732.87i	−0.558494	−	1.47639i
$276$		0			0
$277$			6588.69i			1.42916i	0.699556	+	0.714578i	$0.253381\pi$
$277$			6588.69i			1.42916i	−0.699556	+	0.714578i	$0.746619\pi$
$278$		0			0
$279$	−1029.84			−0.220986
$280$		0			0
$281$	815.552			0.173138			0.0865689	−	0.996246i	$-0.472410\pi$
$281$	815.552			0.173138			0.0865689	+	0.996246i	$0.472410\pi$
$282$		0			0
$283$		−	6513.49i		−	1.36815i	−0.729411	−	0.684076i	$-0.760206\pi$
$283$		−	6513.49i		−	1.36815i	0.729411	−	0.684076i	$-0.239794\pi$
$284$		0			0
$285$	−2871.67	−	1983.96i	−0.596852	−	0.412350i
$286$		0			0
$287$		−	1186.89i		−	0.244112i
$288$		0			0
$289$	−3551.94			−0.722967
$290$		0			0
$291$	−2350.96			−0.473593
$292$		0			0
$293$		−	435.520i		−	0.0868373i	−0.999057	−	0.0434186i	$-0.986175\pi$
$293$		−	435.520i		−	0.0868373i	0.999057	−	0.0434186i	$-0.0138249\pi$
$294$		0			0
$295$	−1530.57	+	2215.40i	−0.302078	+	0.437240i
$296$		0			0
$297$		−	6862.29i		−	1.34071i
$298$		0			0
$299$	−7214.10			−1.39533
$300$		0			0
$301$	−1031.99			−0.197618
$302$		0			0
$303$			1507.89i			0.285895i
$304$		0			0
$305$	574.864	−	832.082i	0.107923	−	0.156213i
$306$		0			0
$307$			4915.99i			0.913910i	0.889490	+	0.456955i	$0.151060\pi$
$307$			4915.99i			0.913910i	−0.889490	+	0.456955i	$0.848940\pi$
$308$		0			0
$309$	−750.601			−0.138188
$310$		0			0
$311$	1831.11			0.333868			0.166934	−	0.985968i	$-0.446613\pi$
$311$	1831.11			0.333868			0.166934	+	0.985968i	$0.446613\pi$
$312$		0			0
$313$			2442.96i			0.441163i	0.975369	+	0.220582i	$0.0707955\pi$
$313$			2442.96i			0.441163i	−0.975369	+	0.220582i	$0.929204\pi$
$314$		0			0
$315$	−1338.03	−	924.412i	−0.239332	−	0.165348i
$316$		0			0
$317$			1666.19i			0.295214i	0.989046	+	0.147607i	$0.0471570\pi$
$317$			1666.19i			0.295214i	−0.989046	+	0.147607i	$0.952843\pi$
$318$		0			0
$319$	2313.36			0.406029
$320$		0			0
$321$	−3770.02			−0.655521
$322$		0			0
$323$			11516.9i			1.98396i
$324$		0			0
$325$	5321.47	−	2013.02i	0.908253	−	0.343577i
$326$		0			0
$327$		−	4407.05i		−	0.745292i
$328$		0			0
$329$	469.229			0.0786305
$330$		0			0
$331$	5466.38			0.907732			0.453866	−	0.891070i	$-0.350044\pi$
$331$	5466.38			0.907732			0.453866	+	0.891070i	$0.350044\pi$
$332$		0			0
$333$			4806.60i			0.790991i
$334$		0			0
$335$	3739.19	+	2583.31i	0.609833	+	0.421318i
$336$		0			0
$337$			10650.5i			1.72157i	0.508970	+	0.860784i	$0.330027\pi$
$337$			10650.5i			1.72157i	−0.508970	+	0.860784i	$0.669973\pi$
$338$		0			0
$339$	−2606.86			−0.417655
$340$		0			0
$341$	−2854.01			−0.453235
$342$		0			0
$343$			343.000i			0.0539949i
$344$		0			0
$345$	2512.04	−	3636.04i	0.392011	−	0.567413i
$346$		0			0
$347$		−	4019.13i		−	0.621782i	−0.950446	−	0.310891i	$-0.899373\pi$
$347$		−	4019.13i		−	0.621782i	0.950446	−	0.310891i	$-0.100627\pi$
$348$		0			0
$349$	−10544.9			−1.61735			−0.808674	−	0.588256i	$-0.799815\pi$
$349$	−10544.9			−1.61735			−0.808674	+	0.588256i	$0.799815\pi$
$350$		0			0
$351$	5423.76			0.824784
$352$		0			0
$353$		−	2959.98i		−	0.446300i	−0.974784	−	0.223150i	$-0.928366\pi$
$353$		−	2959.98i		−	0.446300i	0.974784	−	0.223150i	$-0.0716340\pi$
$354$		0			0
$355$	−2102.14	+	3042.72i	−0.314281	+	0.454903i
$356$		0			0
$357$		−	1606.20i		−	0.238121i
$358$		0			0
$359$	2170.17			0.319045			0.159523	−	0.987194i	$-0.449005\pi$
$359$	2170.17			0.319045			0.159523	+	0.987194i	$0.449005\pi$
$360$		0			0
$361$	8810.30			1.28449
$362$		0			0
$363$		−	4951.46i		−	0.715934i
$364$		0			0
$365$	−5024.76	−	3471.48i	−0.720569	−	0.497823i
$366$		0			0
$367$			1252.20i			0.178105i	0.996027	+	0.0890523i	$0.0283838\pi$
$367$			1252.20i			0.178105i	−0.996027	+	0.0890523i	$0.971616\pi$
$368$		0			0
$369$	3523.40			0.497076
$370$		0			0
$371$	−1880.53			−0.263159
$372$		0			0
$373$		−	4646.02i		−	0.644938i	−0.946580	−	0.322469i	$-0.895487\pi$
$373$		−	4646.02i		−	0.644938i	0.946580	−	0.322469i	$-0.104513\pi$
$374$		0			0
$375$	−838.406	+	3383.08i	−0.115454	+	0.465870i
$376$		0			0
$377$			1828.41i			0.249783i
$378$		0			0
$379$	1434.84			0.194466			0.0972331	−	0.995262i	$-0.469001\pi$
$379$	1434.84			0.194466			0.0972331	+	0.995262i	$0.469001\pi$
$380$		0			0
$381$	−650.242			−0.0874355
$382$		0			0
$383$		−	13216.7i		−	1.76330i	−0.471905	−	0.881649i	$-0.656434\pi$
$383$		−	13216.7i		−	1.76330i	0.471905	−	0.881649i	$-0.343566\pi$
$384$		0			0
$385$	−3708.08	−	2561.82i	−0.490861	−	0.339123i
$386$		0			0
$387$		−	3063.57i		−	0.402403i
$388$		0			0
$389$	7755.01			1.01078			0.505391	−	0.862890i	$-0.331348\pi$
$389$	7755.01			1.01078			0.505391	+	0.862890i	$0.331348\pi$
$390$		0			0
$391$	−14582.5			−1.88610
$392$		0			0
$393$		−	1804.44i		−	0.231607i
$394$		0			0
$395$	−160.838	+	232.803i	−0.0204876	+	0.0296547i
$396$		0			0
$397$		−	3560.83i		−	0.450158i	−0.974341	−	0.225079i	$-0.927736\pi$
$397$		−	3560.83i		−	0.450158i	0.974341	−	0.225079i	$-0.0722640\pi$
$398$		0			0
$399$	−2185.31			−0.274191
$400$		0			0
$401$	−5430.61			−0.676288			−0.338144	−	0.941094i	$-0.609799\pi$
$401$	−5430.61			−0.676288			−0.338144	+	0.941094i	$0.609799\pi$
$402$		0			0
$403$		−	2255.73i		−	0.278823i
$404$		0			0
$405$	1676.96	−	2427.31i	0.205751	−	0.297812i
$406$		0			0
$407$			13320.5i			1.62229i
$408$		0			0
$409$	9698.79			1.17255			0.586277	−	0.810111i	$-0.300593\pi$
$409$	9698.79			1.17255			0.586277	+	0.810111i	$0.300593\pi$
$410$		0			0
$411$	1929.56			0.231577
$412$		0			0
$413$			1685.90i			0.200866i
$414$		0			0
$415$	3461.00	+	2391.12i	0.409383	+	0.282832i
$416$		0			0
$417$			7364.58i			0.864856i
$418$		0			0
$419$	13830.9			1.61261			0.806307	−	0.591498i	$-0.201463\pi$
$419$	13830.9			1.61261			0.806307	+	0.591498i	$0.201463\pi$
$420$		0			0
$421$	16703.0			1.93362			0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0			1.93362			0.966810	+	0.255498i	$0.0822393\pi$
$422$		0			0
$423$			1392.95i			0.160112i
$424$		0			0
$425$	10756.7	−	4069.09i	1.22771	−	0.464423i
$426$		0			0
$427$		−	633.205i		−	0.0717634i
$428$		0			0
$429$	6537.10			0.735698
$430$		0			0
$431$	−8174.07			−0.913530			−0.456765	−	0.889588i	$-0.650992\pi$
$431$	−8174.07			−0.913530			−0.456765	+	0.889588i	$0.650992\pi$
$432$		0			0
$433$		−	14222.8i		−	1.57853i	−0.614051	−	0.789267i	$-0.710461\pi$
$433$		−	14222.8i		−	1.57853i	0.614051	−	0.789267i	$-0.289539\pi$
$434$		0			0
$435$	−921.552	−	636.677i	−0.101575	−	0.0701755i
$436$		0			0
$437$			19840.1i			2.17181i
$438$		0			0
$439$	5537.38			0.602016			0.301008	−	0.953622i	$-0.402677\pi$
$439$	5537.38			0.602016			0.301008	+	0.953622i	$0.402677\pi$
$440$		0			0
$441$	−1018.23			−0.109948
$442$		0			0
$443$			3974.09i			0.426218i	0.977028	+	0.213109i	$0.0683589\pi$
$443$			3974.09i			0.426218i	−0.977028	+	0.213109i	$0.931641\pi$
$444$		0			0
$445$	−6523.06	+	9441.74i	−0.694882	+	1.00580i
$446$		0			0
$447$			6269.82i			0.663428i
$448$		0			0
$449$	15243.1			1.60216			0.801078	−	0.598559i	$-0.204260\pi$
$449$	15243.1			1.60216			0.801078	+	0.598559i	$0.204260\pi$
$450$		0			0
$451$	9764.40			1.01948
$452$		0			0
$453$			252.021i			0.0261390i
$454$		0			0
$455$	2024.79	−	2930.77i	0.208623	−	0.301970i
$456$		0			0
$457$		−	10768.9i		−	1.10229i	−0.834410	−	0.551145i	$-0.814191\pi$
$457$		−	10768.9i		−	1.10229i	0.834410	−	0.551145i	$-0.185809\pi$
$458$		0			0
$459$	10963.5			1.11489
$460$		0			0
$461$	−332.605			−0.0336029			−0.0168015	−	0.999859i	$-0.505348\pi$
$461$	−332.605			−0.0336029			−0.0168015	+	0.999859i	$0.505348\pi$
$462$		0			0
$463$		−	8205.35i		−	0.823618i	−0.911270	−	0.411809i	$-0.864897\pi$
$463$		−	8205.35i		−	0.823618i	0.911270	−	0.411809i	$-0.135103\pi$
$464$		0			0
$465$	1136.93	+	785.473i	0.113384	+	0.0783343i
$466$		0			0
$467$			167.628i			0.0166100i	0.999966	+	0.00830501i	$0.00264360\pi$
$467$			167.628i			0.0166100i	−0.999966	+	0.00830501i	$0.997356\pi$
$468$		0			0
$469$	2845.49			0.280154
$470$		0			0
$471$	5831.75			0.570515
$472$		0			0
$473$		−	8490.07i		−	0.825315i
$474$		0			0
$475$	−5536.18	−	14635.0i	−0.534773	−	1.41368i
$476$		0			0
$477$		−	5582.52i		−	0.535862i
$478$		0			0
$479$	6628.58			0.632292			0.316146	−	0.948711i	$-0.397611\pi$
$479$	6628.58			0.632292			0.316146	+	0.948711i	$0.397611\pi$
$480$		0			0
$481$	−10528.2			−0.998010
$482$		0			0
$483$		−	2766.98i		−	0.260667i
$484$		0			0
$485$	−8671.11	−	5990.65i	−0.811824	−	0.560868i
$486$		0			0
$487$			20641.6i			1.92065i	0.278875	+	0.960327i	$0.410039\pi$
$487$			20641.6i			1.92065i	−0.278875	+	0.960327i	$0.589961\pi$
$488$		0			0
$489$	−3305.01			−0.305640
$490$		0			0
$491$	16710.8			1.53594			0.767972	−	0.640484i	$-0.221266\pi$
$491$	16710.8			1.53594			0.767972	+	0.640484i	$0.221266\pi$
$492$		0			0
$493$			3695.92i			0.337639i
$494$		0			0
$495$	7605.01	−	11007.8i	0.690545	−	0.999523i
$496$		0			0
$497$			2315.47i			0.208980i
$498$		0			0
$499$	−13728.7			−1.23162			−0.615812	−	0.787893i	$-0.711172\pi$
$499$	−13728.7			−1.23162			−0.615812	+	0.787893i	$0.711172\pi$
$500$		0			0
$501$	−5202.90			−0.463969
$502$		0			0
$503$		−	19523.7i		−	1.73065i	−0.501209	−	0.865326i	$-0.667111\pi$
$503$		−	19523.7i		−	1.73065i	0.501209	−	0.865326i	$-0.332889\pi$
$504$		0			0
$505$	−3842.37	+	5561.60i	−0.338580	+	0.490075i
$506$		0			0
$507$		−	312.490i		−	0.0273731i
$508$		0			0
$509$	8688.17			0.756574			0.378287	−	0.925688i	$-0.376513\pi$
$509$	8688.17			0.756574			0.378287	+	0.925688i	$0.376513\pi$
$510$		0			0
$511$	−3823.78			−0.331026
$512$		0			0
$513$		−	14916.3i		−	1.28377i
$514$		0			0
$515$	−2768.46	−	1912.66i	−0.236880	−	0.163654i
$516$		0			0
$517$			3860.28i			0.328385i
$518$		0			0
$519$	−4783.89			−0.404604
$520$		0			0
$521$	6771.36			0.569402			0.284701	−	0.958616i	$-0.408106\pi$
$521$	6771.36			0.569402			0.284701	+	0.958616i	$0.408106\pi$
$522$		0			0
$523$		−	1365.89i		−	0.114200i	−0.998368	−	0.0570998i	$-0.981815\pi$
$523$		−	1365.89i		−	0.114200i	0.998368	−	0.0570998i	$-0.0181853\pi$
$524$		0			0
$525$	772.099	+	2041.06i	0.0641851	+	0.169675i
$526$		0			0
$527$		−	4559.68i		−	0.376894i
$528$		0			0
$529$	−12954.0			−1.06469
$530$		0			0
$531$	−5004.75			−0.409016
$532$		0			0
$533$			7717.51i			0.627171i
$534$		0			0
$535$	−13905.1	−	9606.67i	−1.12368	−	0.776322i
$536$		0			0
$537$		−	1568.82i		−	0.126070i
$538$		0			0
$539$	−2821.81			−0.225499
$540$		0			0
$541$	−23250.1			−1.84769			−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1			−1.84769			−0.923844	+	0.382770i	$0.874970\pi$
$542$		0			0
$543$			6985.22i			0.552052i
$544$		0			0
$545$	11229.9	−	16254.7i	0.882638	−	1.27757i
$546$		0			0
$547$			11552.7i			0.903033i	0.892263	+	0.451516i	$0.149117\pi$
$547$			11552.7i			0.903033i	−0.892263	+	0.451516i	$0.850883\pi$
$548$		0			0
$549$	1879.73			0.146129
$550$		0			0
$551$	5028.47			0.388784
$552$		0			0
$553$			177.161i			0.0136232i
$554$		0			0
$555$	3666.04	−	5306.38i	0.280387	−	0.405844i
$556$		0			0
$557$		−	16406.2i		−	1.24803i	−0.781413	−	0.624014i	$-0.785501\pi$
$557$		−	16406.2i		−	1.24803i	0.781413	−	0.624014i	$-0.214499\pi$
$558$		0			0
$559$	6710.31			0.507721
$560$		0			0
$561$	13214.0			0.994465
$562$		0			0
$563$			13631.9i			1.02045i	0.860040	+	0.510227i	$0.170439\pi$
$563$			13631.9i			1.02045i	−0.860040	+	0.510227i	$0.829561\pi$
$564$		0			0
$565$	−9614.95	−	6642.72i	−0.715936	−	0.494622i
$566$		0			0
$567$		−	1847.15i		−	0.136813i
$568$		0			0
$569$	3086.83			0.227428			0.113714	−	0.993514i	$-0.463725\pi$
$569$	3086.83			0.227428			0.113714	+	0.993514i	$0.463725\pi$
$570$		0			0
$571$	3258.06			0.238784			0.119392	−	0.992847i	$-0.461905\pi$
$571$	3258.06			0.238784			0.119392	+	0.992847i	$0.461905\pi$
$572$		0			0
$573$			1846.17i			0.134598i
$574$		0			0
$575$	18530.5	−	7009.78i	1.34396	−	0.508396i
$576$		0			0
$577$			23758.4i			1.71417i	0.515178	+	0.857083i	$0.327726\pi$
$577$			23758.4i			1.71417i	−0.515178	+	0.857083i	$0.672274\pi$
$578$		0			0
$579$	−10181.8			−0.730812
$580$		0			0
$581$	2633.78			0.188068
$582$		0			0
$583$		−	15470.8i		−	1.09903i
$584$		0			0
$585$	8700.25	+	6010.78i	0.614891	+	0.424812i
$586$		0			0
$587$		−	596.893i		−	0.0419701i	−0.999780	−	0.0209850i	$-0.993320\pi$
$587$		−	596.893i		−	0.0419701i	0.999780	−	0.0209850i	$-0.00668023\pi$
$588$		0			0
$589$	−6203.66			−0.433985
$590$		0			0
$591$	8516.60			0.592768
$592$		0			0
$593$		−	19496.3i		−	1.35012i	−0.737765	−	0.675058i	$-0.764119\pi$
$593$		−	19496.3i		−	1.35012i	0.737765	−	0.675058i	$-0.235881\pi$
$594$		0			0
$595$	4092.88	−	5924.20i	0.282003	−	0.408182i
$596$		0			0
$597$			8460.62i			0.580017i
$598$		0			0
$599$	3797.02			0.259001			0.129501	−	0.991579i	$-0.458663\pi$
$599$	3797.02			0.259001			0.129501	+	0.991579i	$0.458663\pi$
$600$		0			0
$601$	5789.33			0.392931			0.196466	−	0.980511i	$-0.437054\pi$
$601$	5789.33			0.392931			0.196466	+	0.980511i	$0.437054\pi$
$602$		0			0
$603$			8447.09i			0.570468i
$604$		0			0
$605$	12617.2	−	18262.6i	0.847869	−	1.22724i
$606$		0			0
$607$		−	18536.4i		−	1.23949i	−0.784803	−	0.619745i	$-0.787236\pi$
$607$		−	18536.4i		−	1.23949i	0.784803	−	0.619745i	$-0.212764\pi$
$608$		0			0
$609$	−701.291			−0.0466630
$610$		0			0
$611$	−3051.06			−0.202017
$612$		0			0
$613$		−	2163.47i		−	0.142548i	−0.997457	−	0.0712738i	$-0.977294\pi$
$613$		−	2163.47i		−	0.142548i	0.997457	−	0.0712738i	$-0.0227064\pi$
$614$		0			0
$615$	−3889.76	−	2687.34i	−0.255041	−	0.176201i
$616$		0			0
$617$			22964.9i			1.49843i	0.662327	+	0.749215i	$0.269569\pi$
$617$			22964.9i			1.49843i	−0.662327	+	0.749215i	$0.730431\pi$
$618$		0			0
$619$	−1386.67			−0.0900401			−0.0450200	−	0.998986i	$-0.514335\pi$
$619$	−1386.67			−0.0900401			−0.0450200	+	0.998986i	$0.514335\pi$
$620$		0			0
$621$	18886.7			1.22045
$622$		0			0
$623$			7185.06i			0.462060i
$624$		0			0
$625$	−11713.0	+	10341.5i	−0.749631	+	0.661856i
$626$		0			0
$627$		−	17978.2i		−	1.14510i
$628$		0			0
$629$	−21281.4			−1.34904
$630$		0			0
$631$	−5969.39			−0.376605			−0.188303	−	0.982111i	$-0.560299\pi$
$631$	−5969.39			−0.376605			−0.188303	+	0.982111i	$0.560299\pi$
$632$		0			0
$633$			8474.57i			0.532123i
$634$		0			0
$635$	−2398.31	−	1656.93i	−0.149880	−	0.103548i
$636$		0			0
$637$		−	2230.28i		−	0.138724i
$638$		0			0
$639$	−6873.70			−0.425539
$640$		0			0
$641$	30367.1			1.87118			0.935592	−	0.353084i	$-0.114867\pi$
$641$	30367.1			1.87118			0.935592	+	0.353084i	$0.114867\pi$
$642$		0			0
$643$			28592.2i			1.75360i	0.480851	+	0.876802i	$0.340328\pi$
$643$			28592.2i			1.75360i	−0.480851	+	0.876802i	$0.659672\pi$
$644$		0			0
$645$	−2336.62	+	3382.12i	−0.142642	+	0.206466i
$646$		0			0
$647$			14507.9i			0.881555i	0.897616	+	0.440778i	$0.145297\pi$
$647$			14507.9i			0.881555i	−0.897616	+	0.440778i	$0.854703\pi$
$648$		0			0
$649$	−13869.7			−0.838877
$650$		0			0
$651$	865.188			0.0520882
$652$		0			0
$653$		−	6999.85i		−	0.419488i	−0.977756	−	0.209744i	$-0.932737\pi$
$653$		−	6999.85i		−	0.419488i	0.977756	−	0.209744i	$-0.0672630\pi$
$654$		0			0
$655$	4598.01	−	6655.35i	0.274289	−	0.397017i
$656$		0			0
$657$		−	11351.3i		−	0.674056i
$658$		0			0
$659$	7308.92			0.432041			0.216020	−	0.976389i	$-0.430692\pi$
$659$	7308.92			0.432041			0.216020	+	0.976389i	$0.430692\pi$
$660$		0			0
$661$	−30097.2			−1.77102			−0.885512	−	0.464617i	$-0.846192\pi$
$661$	−30097.2			−1.77102			−0.885512	+	0.464617i	$0.846192\pi$
$662$		0			0
$663$			10444.0i			0.611779i
$664$		0			0
$665$	−8060.14	−	5568.54i	−0.470013	−	0.324720i
$666$		0			0
$667$			6366.92i			0.369608i
$668$		0			0
$669$	455.425			0.0263195
$670$		0			0
$671$	5209.29			0.299706
$672$		0			0
$673$			5400.26i			0.309309i	0.987969	+	0.154654i	$0.0494264\pi$
$673$			5400.26i			0.309309i	−0.987969	+	0.154654i	$0.950574\pi$
$674$		0			0
$675$	−13931.7	+	5270.15i	−0.794419	+	0.300516i
$676$		0			0
$677$			6431.09i			0.365091i	0.983197	+	0.182546i	$0.0584338\pi$
$677$			6431.09i			0.365091i	−0.983197	+	0.182546i	$0.941566\pi$
$678$		0			0
$679$	−6598.62			−0.372948
$680$		0			0
$681$	7861.79			0.442385
$682$		0			0
$683$		−	20865.8i		−	1.16897i	−0.811404	−	0.584486i	$-0.801296\pi$
$683$		−	20865.8i		−	1.16897i	0.811404	−	0.584486i	$-0.198704\pi$
$684$		0			0
$685$	7116.86	+	4916.86i	0.396965	+	0.274253i
$686$		0			0
$687$			14994.6i			0.832720i
$688$		0			0
$689$	12227.7			0.676109
$690$		0			0
$691$	−18450.3			−1.01575			−0.507873	−	0.861432i	$-0.669568\pi$
$691$	−18450.3			−1.01575			−0.507873	+	0.861432i	$0.669568\pi$
$692$		0			0
$693$		−	8376.81i		−	0.459176i
$694$		0			0
$695$	−18766.2	+	27163.0i	−1.02424	+	1.48252i
$696$		0			0
$697$			15600.0i			0.847766i
$698$		0			0
$699$	2345.97			0.126942
$700$		0			0
$701$	12639.3			0.680996			0.340498	−	0.940245i	$-0.389404\pi$
$701$	12639.3			0.680996			0.340498	+	0.940245i	$0.389404\pi$
$702$		0			0
$703$			28954.4i			1.55339i
$704$		0			0
$705$	1062.42	−	1537.79i	0.0567560	−	0.0821509i
$706$		0			0
$707$			4232.32i			0.225138i
$708$		0			0
$709$	23126.8			1.22503			0.612514	−	0.790460i	$-0.290158\pi$
$709$	23126.8			1.22503			0.612514	+	0.790460i	$0.290158\pi$
$710$		0			0
$711$	−525.918			−0.0277404
$712$		0			0
$713$		−	7854.92i		−	0.412579i
$714$		0			0
$715$	24111.0	+	16657.7i	1.26112	+	0.871275i
$716$		0			0
$717$		−	12865.9i		−	0.670134i
$718$		0			0
$719$	−24093.1			−1.24968			−0.624841	−	0.780752i	$-0.714836\pi$
$719$	−24093.1			−1.24968			−0.624841	+	0.780752i	$0.714836\pi$
$720$		0			0
$721$	−2106.77			−0.108821
$722$		0			0
$723$			1156.79i			0.0595041i
$724$		0			0
$725$	−1776.63	−	4696.55i	−0.0910100	−	0.240587i
$726$		0			0
$727$		−	35983.4i		−	1.83570i	−0.396931	−	0.917849i	$-0.629925\pi$
$727$		−	35983.4i		−	1.83570i	0.396931	−	0.917849i	$-0.370075\pi$
$728$		0			0
$729$	−2540.55			−0.129073
$730$		0			0
$731$	13564.1			0.686302
$732$		0			0
$733$			1451.50i			0.0731413i	0.999331	+	0.0365706i	$0.0116434\pi$
$733$			1451.50i			0.0731413i	−0.999331	+	0.0365706i	$0.988357\pi$
$734$		0			0
$735$	1124.10	+	776.613i	0.0564124	+	0.0389739i
$736$		0			0
$737$			23409.4i			1.17001i
$738$		0			0
$739$	−5891.67			−0.293273			−0.146636	−	0.989190i	$-0.546845\pi$
$739$	−5891.67			−0.293273			−0.146636	+	0.989190i	$0.546845\pi$
$740$		0			0
$741$	14209.5			0.704451
$742$		0			0
$743$			7438.65i			0.367292i	0.982992	+	0.183646i	$0.0587900\pi$
$743$			7438.65i			0.367292i	−0.982992	+	0.183646i	$0.941210\pi$
$744$		0			0
$745$	−15976.6	+	23125.2i	−0.785688	+	1.13724i
$746$		0			0
$747$			7818.64i			0.382957i
$748$		0			0
$749$	−10581.6			−0.516214
$750$		0			0
$751$	−20272.4			−0.985018			−0.492509	−	0.870307i	$-0.663920\pi$
$751$	−20272.4			−0.985018			−0.492509	+	0.870307i	$0.663920\pi$
$752$		0			0
$753$			5711.80i			0.276427i
$754$		0			0
$755$	−642.193	+	929.537i	−0.0309560	+	0.0448070i
$756$		0			0
$757$		−	10193.8i		−	0.489432i	−0.969595	−	0.244716i	$-0.921305\pi$
$757$		−	10193.8i		−	0.489432i	0.969595	−	0.244716i	$-0.0786947\pi$
$758$		0			0
$759$	22763.6			1.08862
$760$		0			0
$761$	−41117.6			−1.95862			−0.979311	−	0.202362i	$-0.935138\pi$
$761$	−41117.6			−1.95862			−0.979311	+	0.202362i	$0.935138\pi$
$762$		0			0
$763$		−	12369.6i		−	0.586908i
$764$		0			0
$765$	17586.5	+	12150.1i	0.831167	+	0.574232i
$766$		0			0
$767$		−	10962.2i		−	0.516065i
$768$		0			0
$769$	−11486.6			−0.538642			−0.269321	−	0.963050i	$-0.586799\pi$
$769$	−11486.6			−0.538642			−0.269321	+	0.963050i	$0.586799\pi$
$770$		0			0
$771$	−2000.66			−0.0934527
$772$		0			0
$773$			21799.2i			1.01431i	0.861854	+	0.507156i	$0.169303\pi$
$773$			21799.2i			1.01431i	−0.861854	+	0.507156i	$0.830697\pi$
$774$		0			0
$775$	2191.84	+	5794.17i	0.101591	+	0.268558i
$776$		0			0
$777$		−	4038.10i		−	0.186443i
$778$		0			0
$779$	21224.5			0.976185
$780$		0			0
$781$	−19049.1			−0.872765
$782$		0			0
$783$		−	4786.83i		−	0.218477i
$784$		0			0
$785$	21509.4	+	14860.3i	0.977967	+	0.675652i
$786$		0			0
$787$			24080.3i			1.09068i	0.838214	+	0.545342i	$0.183600\pi$
$787$			24080.3i			1.09068i	−0.838214	+	0.545342i	$0.816400\pi$
$788$		0			0
$789$	−715.712			−0.0322941
$790$		0			0
$791$	−7316.87			−0.328898
$792$		0			0
$793$			4117.28i			0.184374i
$794$		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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.49396i q^{3} +(6.35505 - 9.19855i) q^{5} -7.00000i q^{7} +20.7802 q^{9} +57.5880 q^{11} +45.5159i q^{13} +(-22.9408 - 15.8492i) q^{15} +92.0051i q^{17} +125.177 q^{19} -17.4577 q^{21} +158.496i q^{23} +(-44.2268 - 116.914i) q^{25} -119.162i q^{27} +40.1708 q^{29} -49.5590 q^{31} -143.622i q^{33} +(-64.3899 - 44.4853i) q^{35} +231.307i q^{37} +113.515 q^{39} +169.556 q^{41} -147.428i q^{43} +(132.059 - 191.147i) q^{45} +67.0327i q^{47} -49.0000 q^{49} +229.457 q^{51} -268.647i q^{53} +(365.974 - 529.726i) q^{55} -312.187i q^{57} -240.843 q^{59} +90.4579 q^{61} -145.461i q^{63} +(418.681 + 289.256i) q^{65} +406.498i q^{67} +395.283 q^{69} -330.782 q^{71} -546.255i q^{73} +(-291.580 + 110.300i) q^{75} -403.116i q^{77} -25.3087 q^{79} +263.879 q^{81} +376.255i q^{83} +(846.314 + 584.697i) q^{85} -100.184i q^{87} -1026.44 q^{89} +318.612 q^{91} +123.598i q^{93} +(795.506 - 1151.45i) q^{95} -942.660i q^{97} +1196.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} - 46 q^{9} - 84 q^{11} - 8 q^{15} - 72 q^{19} + 140 q^{21} - 362 q^{25} + 88 q^{29} - 120 q^{31} + 28 q^{35} - 212 q^{39} - 852 q^{41} - 510 q^{45} - 490 q^{49} - 1276 q^{51} + 1136 q^{55}+ \cdots + 5304 q^{99}+O(q^{100}) \) 10 * q + 6 * q^5 - 46 * q^9 - 84 * q^11 - 8 * q^15 - 72 * q^19 + 140 * q^21 - 362 * q^25 + 88 * q^29 - 120 * q^31 + 28 * q^35 - 212 * q^39 - 852 * q^41 - 510 * q^45 - 490 * q^49 - 1276 * q^51 + 1136 * q^55 - 864 * q^59 - 884 * q^61 + 520 * q^65 + 4808 * q^69 - 880 * q^71 - 720 * q^75 + 3580 * q^79 - 302 * q^81 + 1572 * q^85 - 1492 * q^89 - 476 * q^91 + 1628 * q^95 + 5304 * q^99

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