LMFDB - Embedded newform 560.5.p.h.209.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,5,Mod(209,560)]

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, 1]))

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

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

chi := DirichletCharacter("560.209");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	560.p (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:	$57.8871793270$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 1342 x^{10} + 1715866 x^{8} + 1068594118 x^{6} + 652472238169 x^{4} + 172600636071304 x^{2} + 37\!\cdots\!00 $ x^12 + 1342x^10 + 1715866x^8 + 1068594118x^6 + 652472238169x^4 + 172600636071304*x^2 + 37564835672250000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.10
Root			$9.11517 + 15.9519i$ of defining polynomial
Character	$\chi$	$=$	560.209
Dual form			560.5.p.h.209.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1342 q^{3} +(19.2493 + 15.9519i) q^{5} +(2.89857 + 48.9142i) q^{7} +21.7012 q^{9} +O(q^{10})$	$q+10.1342 q^{3} +(19.2493 + 15.9519i) q^{5} +(2.89857 + 48.9142i) q^{7} +21.7012 q^{9} +97.3493 q^{11} +143.184 q^{13} +(195.076 + 161.659i) q^{15} +146.788 q^{17} -366.102i q^{19} +(29.3746 + 495.704i) q^{21} -256.811i q^{23} +(116.073 + 614.127i) q^{25} -600.944 q^{27} +1522.21 q^{29} -1517.03i q^{31} +986.553 q^{33} +(-724.479 + 987.803i) q^{35} +1259.58i q^{37} +1451.04 q^{39} +2508.44i q^{41} +3047.34i q^{43} +(417.733 + 346.175i) q^{45} -4.96345 q^{47} +(-2384.20 + 283.562i) q^{49} +1487.57 q^{51} +564.960i q^{53} +(1873.91 + 1552.91i) q^{55} -3710.13i q^{57} +2803.59i q^{59} +918.806i q^{61} +(62.9023 + 1061.50i) q^{63} +(2756.19 + 2284.05i) q^{65} -3869.02i q^{67} -2602.56i q^{69} -8704.79 q^{71} -1186.45 q^{73} +(1176.30 + 6223.66i) q^{75} +(282.174 + 4761.76i) q^{77} -3136.71 q^{79} -7847.85 q^{81} +7654.79 q^{83} +(2825.57 + 2341.55i) q^{85} +15426.3 q^{87} -5544.86i q^{89} +(415.027 + 7003.71i) q^{91} -15373.8i q^{93} +(5840.02 - 7047.21i) q^{95} +8296.39 q^{97} +2112.59 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 296 q^{9}+O(q^{10}) $ 12 * q + 296 * q^9	$ 12 q + 296 q^{9} + 20 q^{11} + 672 q^{15} - 1352 q^{21} - 2608 q^{25} - 1796 q^{29} + 2788 q^{35} + 332 q^{39} - 11792 q^{49} - 16844 q^{51} + 6928 q^{65} - 27448 q^{71} + 7948 q^{79} - 14636 q^{81} + 1748 q^{85} - 948 q^{91} + 29940 q^{95} - 67688 q^{99}+O(q^{100}) $ 12 * q + 296 * q^9 + 20 * q^11 + 672 * q^15 - 1352 * q^21 - 2608 * q^25 - 1796 * q^29 + 2788 * q^35 + 332 * q^39 - 11792 * q^49 - 16844 * q^51 + 6928 * q^65 - 27448 * q^71 + 7948 * q^79 - 14636 * q^81 + 1748 * q^85 - 948 * q^91 + 29940 * q^95 - 67688 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	10.1342			1.12602			0.563009	−	0.826451i	$-0.309644\pi$
$3$	10.1342			1.12602			0.563009	+	0.826451i	$0.309644\pi$
$4$		0			0
$5$	19.2493	+	15.9519i	0.769973	+	0.638076i
$5$	19.2493	+	15.9519i	0.769973	+	0.638076i
$6$		0			0
$7$	2.89857	+	48.9142i	0.0591545	+	0.998249i
$7$	2.89857	+	48.9142i	0.0591545	+	0.998249i
$8$		0			0
$9$	21.7012			0.267916
$10$		0			0
$11$	97.3493			0.804540			0.402270	−	0.915521i	$-0.368221\pi$
$11$	97.3493			0.804540			0.402270	+	0.915521i	$0.368221\pi$
$12$		0			0
$13$	143.184			0.847240			0.423620	−	0.905840i	$-0.360759\pi$
$13$	143.184			0.847240			0.423620	+	0.905840i	$0.360759\pi$
$14$		0			0
$15$	195.076	+	161.659i	0.867003	+	0.718485i
$16$		0			0
$17$	146.788			0.507916			0.253958	−	0.967215i	$-0.418267\pi$
$17$	146.788			0.507916			0.253958	+	0.967215i	$0.418267\pi$
$18$		0			0
$19$		−	366.102i		−	1.01413i	−0.861907	−	0.507066i	$-0.830730\pi$
$19$		−	366.102i		−	1.01413i	0.861907	−	0.507066i	$-0.169270\pi$
$20$		0			0
$21$	29.3746	+	495.704i	0.0666090	+	1.12405i
$22$		0			0
$23$		−	256.811i		−	0.485465i	−0.970093	−	0.242733i	$-0.921956\pi$
$23$		−	256.811i		−	0.485465i	0.970093	−	0.242733i	$-0.0780438\pi$
$24$		0			0
$25$	116.073	+	614.127i	0.185717	+	0.982603i
$26$		0			0
$27$	−600.944			−0.824340
$28$		0			0
$29$	1522.21			1.81000			0.904999	−	0.425415i	$-0.139872\pi$
$29$	1522.21			1.81000			0.904999	+	0.425415i	$0.139872\pi$
$30$		0			0
$31$		−	1517.03i		−	1.57859i	−0.614011	−	0.789297i	$-0.710445\pi$
$31$		−	1517.03i		−	1.57859i	0.614011	−	0.789297i	$-0.289555\pi$
$32$		0			0
$33$	986.553			0.905926
$34$		0			0
$35$	−724.479	+	987.803i	−0.591412	+	0.806370i
$36$		0			0
$37$			1259.58i			0.920070i	0.887901	+	0.460035i	$0.152163\pi$
$37$			1259.58i			0.920070i	−0.887901	+	0.460035i	$0.847837\pi$
$38$		0			0
$39$	1451.04			0.954007
$40$		0			0
$41$			2508.44i			1.49223i	0.665818	+	0.746115i	$0.268083\pi$
$41$			2508.44i			1.49223i	−0.665818	+	0.746115i	$0.731917\pi$
$42$		0			0
$43$			3047.34i			1.64810i	0.566515	+	0.824052i	$0.308292\pi$
$43$			3047.34i			1.64810i	−0.566515	+	0.824052i	$0.691708\pi$
$44$		0			0
$45$	417.733	+	346.175i	0.206288	+	0.170951i
$46$		0			0
$47$	−4.96345			−0.00224692			−0.00112346	−	0.999999i	$-0.500358\pi$
$47$	−4.96345			−0.00224692			−0.00112346	+	0.999999i	$0.500358\pi$
$48$		0			0
$49$	−2384.20	+	283.562i	−0.993001	+	0.118102i
$50$		0			0
$51$	1487.57			0.571923
$52$		0			0
$53$			564.960i			0.201125i	0.994931	+	0.100562i	$0.0320642\pi$
$53$			564.960i			0.201125i	−0.994931	+	0.100562i	$0.967936\pi$
$54$		0			0
$55$	1873.91	+	1552.91i	0.619474	+	0.513358i
$56$		0			0
$57$		−	3710.13i		−	1.14193i
$58$		0			0
$59$			2803.59i			0.805397i	0.915333	+	0.402699i	$0.131928\pi$
$59$			2803.59i			0.805397i	−0.915333	+	0.402699i	$0.868072\pi$
$60$		0			0
$61$			918.806i			0.246924i	0.992349	+	0.123462i	$0.0393998\pi$
$61$			918.806i			0.246924i	−0.992349	+	0.123462i	$0.960600\pi$
$62$		0			0
$63$	62.9023	+	1061.50i	0.0158484	+	0.267446i
$64$		0			0
$65$	2756.19	+	2284.05i	0.652352	+	0.540604i
$66$		0			0
$67$		−	3869.02i		−	0.861889i	−0.902378	−	0.430945i	$-0.858180\pi$
$67$		−	3869.02i		−	0.861889i	0.902378	−	0.430945i	$-0.141820\pi$
$68$		0			0
$69$		−	2602.56i		−	0.546642i
$70$		0			0
$71$	−8704.79			−1.72680			−0.863399	−	0.504522i	$-0.831669\pi$
$71$	−8704.79			−1.72680			−0.863399	+	0.504522i	$0.831669\pi$
$72$		0			0
$73$	−1186.45			−0.222640			−0.111320	−	0.993785i	$-0.535508\pi$
$73$	−1186.45			−0.222640			−0.111320	+	0.993785i	$0.535508\pi$
$74$		0			0
$75$	1176.30	+	6223.66i	0.209121	+	1.10643i
$76$		0			0
$77$	282.174	+	4761.76i	0.0475921	+	0.803131i
$78$		0			0
$79$	−3136.71			−0.502597			−0.251299	−	0.967910i	$-0.580858\pi$
$79$	−3136.71			−0.502597			−0.251299	+	0.967910i	$0.580858\pi$
$80$		0			0
$81$	−7847.85			−1.19614
$82$		0			0
$83$	7654.79			1.11116			0.555581	−	0.831463i	$-0.312496\pi$
$83$	7654.79			1.11116			0.555581	+	0.831463i	$0.312496\pi$
$84$		0			0
$85$	2825.57	+	2341.55i	0.391082	+	0.324090i
$86$		0			0
$87$	15426.3			2.03809
$88$		0			0
$89$		−	5544.86i		−	0.700020i	−0.936746	−	0.350010i	$-0.886178\pi$
$89$		−	5544.86i		−	0.700020i	0.936746	−	0.350010i	$-0.113822\pi$
$90$		0			0
$91$	415.027	+	7003.71i	0.0501180	+	0.845756i
$92$		0			0
$93$		−	15373.8i		−	1.77752i
$94$		0			0
$95$	5840.02	−	7047.21i	0.647094	−	0.780854i
$96$		0			0
$97$	8296.39			0.881750			0.440875	−	0.897568i	$-0.354668\pi$
$97$	8296.39			0.881750			0.440875	+	0.897568i	$0.354668\pi$
$98$		0			0
$99$	2112.59			0.215549
$100$		0			0
$101$			4669.92i			0.457791i	0.973451	+	0.228895i	$0.0735113\pi$
$101$			4669.92i			0.457791i	−0.973451	+	0.228895i	$0.926489\pi$
$102$		0			0
$103$	−6010.89			−0.566584			−0.283292	−	0.959034i	$-0.591427\pi$
$103$	−6010.89			−0.566584			−0.283292	+	0.959034i	$0.591427\pi$
$104$		0			0
$105$	−7341.99	+	10010.6i	−0.665940	+	0.907987i
$106$		0			0
$107$			3771.47i			0.329415i	0.986342	+	0.164707i	$0.0526680\pi$
$107$			3771.47i			0.329415i	−0.986342	+	0.164707i	$0.947332\pi$
$108$		0			0
$109$	10053.3			0.846164			0.423082	−	0.906091i	$-0.360948\pi$
$109$	10053.3			0.846164			0.423082	+	0.906091i	$0.360948\pi$
$110$		0			0
$111$			12764.7i			1.03601i
$112$		0			0
$113$		−	17381.5i		−	1.36123i	−0.732642	−	0.680614i	$-0.761713\pi$
$113$		−	17381.5i		−	1.36123i	0.732642	−	0.680614i	$-0.238287\pi$
$114$		0			0
$115$	4096.63	−	4943.44i	0.309764	−	0.373795i
$116$		0			0
$117$	3107.25			0.226989
$118$		0			0
$119$	425.475	+	7180.01i	0.0300455	+	0.507027i
$120$		0			0
$121$	−5164.11			−0.352716
$122$		0			0
$123$			25420.9i			1.68028i
$124$		0			0
$125$	−7562.17	+	13673.1i	−0.483979	+	0.875080i
$126$		0			0
$127$		−	25198.4i		−	1.56230i	−0.624343	−	0.781151i	$-0.714633\pi$
$127$		−	25198.4i		−	1.56230i	0.624343	−	0.781151i	$-0.285367\pi$
$128$		0			0
$129$			30882.3i			1.85579i
$130$		0			0
$131$			18477.7i			1.07673i	0.842712	+	0.538365i	$0.180958\pi$
$131$			18477.7i			1.07673i	−0.842712	+	0.538365i	$0.819042\pi$
$132$		0			0
$133$	17907.6	−	1061.17i	1.01236	−	0.0599904i
$134$		0			0
$135$	−11567.8	−	9586.20i	−0.634720	−	0.525992i
$136$		0			0
$137$			6897.35i			0.367486i	0.982974	+	0.183743i	$0.0588215\pi$
$137$			6897.35i			0.367486i	−0.982974	+	0.183743i	$0.941179\pi$
$138$		0			0
$139$		−	25035.8i		−	1.29578i	−0.761734	−	0.647890i	$-0.775652\pi$
$139$		−	25035.8i		−	1.29578i	0.761734	−	0.647890i	$-0.224348\pi$
$140$		0			0
$141$	−50.3004			−0.00253007
$142$		0			0
$143$	13938.8			0.681638
$144$		0			0
$145$	29301.5	+	24282.1i	1.39365	+	1.15492i
$146$		0			0
$147$	−24161.8	+	2873.67i	−1.11814	+	0.132985i
$148$		0			0
$149$	−11560.2			−0.520708			−0.260354	−	0.965513i	$-0.583839\pi$
$149$	−11560.2			−0.520708			−0.260354	+	0.965513i	$0.583839\pi$
$150$		0			0
$151$	−9146.30			−0.401136			−0.200568	−	0.979680i	$-0.564279\pi$
$151$	−9146.30			−0.401136			−0.200568	+	0.979680i	$0.564279\pi$
$152$		0			0
$153$	3185.47			0.136079
$154$		0			0
$155$	24199.5	−	29201.8i	1.00726	−	1.21548i
$156$		0			0
$157$	−7302.14			−0.296245			−0.148122	−	0.988969i	$-0.547323\pi$
$157$	−7302.14			−0.296245			−0.148122	+	0.988969i	$0.547323\pi$
$158$		0			0
$159$			5725.39i			0.226470i
$160$		0			0
$161$	12561.7	−	744.385i	0.484615	−	0.0287174i
$162$		0			0
$163$			24285.5i			0.914056i	0.889452	+	0.457028i	$0.151086\pi$
$163$			24285.5i			0.914056i	−0.889452	+	0.457028i	$0.848914\pi$
$164$		0			0
$165$	18990.5	+	15737.4i	0.697539	+	0.578050i
$166$		0			0
$167$	−1865.81			−0.0669013			−0.0334506	−	0.999440i	$-0.510650\pi$
$167$	−1865.81			−0.0669013			−0.0334506	+	0.999440i	$0.510650\pi$
$168$		0			0
$169$	−8059.48			−0.282185
$170$		0			0
$171$		−	7944.83i		−	0.271702i
$172$		0			0
$173$	45529.4			1.52125			0.760624	−	0.649193i	$-0.224893\pi$
$173$	45529.4			1.52125			0.760624	+	0.649193i	$0.224893\pi$
$174$		0			0
$175$	−29703.1	+	7457.72i	−0.969897	+	0.243517i
$176$		0			0
$177$			28412.0i			0.906891i
$178$		0			0
$179$	12981.9			0.405166			0.202583	−	0.979265i	$-0.435066\pi$
$179$	12981.9			0.405166			0.202583	+	0.979265i	$0.435066\pi$
$180$		0			0
$181$			40134.4i			1.22507i	0.790445	+	0.612533i	$0.209849\pi$
$181$			40134.4i			1.22507i	−0.790445	+	0.612533i	$0.790151\pi$
$182$		0			0
$183$			9311.32i			0.278041i
$184$		0			0
$185$	−20092.6	+	24246.0i	−0.587075	+	0.708429i
$186$		0			0
$187$	14289.7			0.408639
$188$		0			0
$189$	−1741.88	−	29394.7i	−0.0487634	−	0.822896i
$190$		0			0
$191$	−2653.27			−0.0727302			−0.0363651	−	0.999339i	$-0.511578\pi$
$191$	−2653.27			−0.0727302			−0.0363651	+	0.999339i	$0.511578\pi$
$192$		0			0
$193$		−	47331.0i		−	1.27067i	−0.772238	−	0.635333i	$-0.780863\pi$
$193$		−	47331.0i		−	1.27067i	0.772238	−	0.635333i	$-0.219137\pi$
$194$		0			0
$195$	27931.6	+	23146.9i	0.734560	+	0.608729i
$196$		0			0
$197$		−	12026.4i		−	0.309888i	−0.987923	−	0.154944i	$-0.950480\pi$
$197$		−	12026.4i		−	0.309888i	0.987923	−	0.154944i	$-0.0495197\pi$
$198$		0			0
$199$		−	50808.4i		−	1.28301i	−0.767119	−	0.641504i	$-0.778311\pi$
$199$		−	50808.4i		−	1.28301i	0.767119	−	0.641504i	$-0.221689\pi$
$200$		0			0
$201$		−	39209.3i		−	0.970503i
$202$		0			0
$203$	4412.23	+	74457.6i	0.107069	+	1.80683i
$204$		0			0
$205$	−40014.4	+	48285.7i	−0.952156	+	1.14898i
$206$		0			0
$207$		−	5573.10i		−	0.130064i
$208$		0			0
$209$		−	35639.7i		−	0.815909i
$210$		0			0
$211$	54675.9			1.22809			0.614046	−	0.789270i	$-0.289541\pi$
$211$	54675.9			1.22809			0.614046	+	0.789270i	$0.289541\pi$
$212$		0			0
$213$	−88215.7			−1.94440
$214$		0			0
$215$	−48610.9	+	58659.3i	−1.05162	+	1.26900i
$216$		0			0
$217$	74204.3	−	4397.21i	1.57583	−	0.0933809i
$218$		0			0
$219$	−12023.6			−0.250696
$220$		0			0
$221$	21017.6			0.430327
$222$		0			0
$223$	−285.105			−0.00573318			−0.00286659	−	0.999996i	$-0.500912\pi$
$223$	−285.105			−0.00573318			−0.00286659	+	0.999996i	$0.500912\pi$
$224$		0			0
$225$	2518.92	+	13327.3i	0.0497565	+	0.263255i
$226$		0			0
$227$	92456.8			1.79427			0.897133	−	0.441760i	$-0.145646\pi$
$227$	92456.8			1.79427			0.897133	+	0.441760i	$0.145646\pi$
$228$		0			0
$229$		−	74226.2i		−	1.41542i	−0.706501	−	0.707712i	$-0.749728\pi$
$229$		−	74226.2i		−	1.41542i	0.706501	−	0.707712i	$-0.250272\pi$
$230$		0			0
$231$	2859.59	+	48256.5i	0.0535896	+	0.904340i
$232$		0			0
$233$			16654.3i			0.306772i	0.988166	+	0.153386i	$0.0490178\pi$
$233$			16654.3i			0.306772i	−0.988166	+	0.153386i	$0.950982\pi$
$234$		0			0
$235$	−95.5431	−	79.1765i	−0.00173007	−	0.00143371i
$236$		0			0
$237$	−31787.9			−0.565933
$238$		0			0
$239$	9157.77			0.160322			0.0801611	−	0.996782i	$-0.474457\pi$
$239$	9157.77			0.160322			0.0801611	+	0.996782i	$0.474457\pi$
$240$		0			0
$241$			91995.3i			1.58391i	0.610577	+	0.791957i	$0.290938\pi$
$241$			91995.3i			1.58391i	−0.610577	+	0.791957i	$0.709062\pi$
$242$		0			0
$243$	−30854.9			−0.522531
$244$		0			0
$245$	−50417.5	−	32574.1i	−0.839942	−	0.542676i
$246$		0			0
$247$		−	52419.7i		−	0.859213i
$248$		0			0
$249$	77574.9			1.25119
$250$		0			0
$251$			20767.6i			0.329639i	0.986324	+	0.164820i	$0.0527042\pi$
$251$			20767.6i			0.329639i	−0.986324	+	0.164820i	$0.947296\pi$
$252$		0			0
$253$		−	25000.4i		−	0.390576i
$254$		0			0
$255$	28634.8	+	23729.6i	0.440365	+	0.364930i
$256$		0			0
$257$	−70282.4			−1.06409			−0.532047	−	0.846715i	$-0.678577\pi$
$257$	−70282.4			−1.06409			−0.532047	+	0.846715i	$0.678577\pi$
$258$		0			0
$259$	−61611.1	+	3650.97i	−0.918458	+	0.0544262i
$260$		0			0
$261$	33033.7			0.484927
$262$		0			0
$263$		−	50347.0i		−	0.727884i	−0.931422	−	0.363942i	$-0.881431\pi$
$263$		−	50347.0i		−	0.727884i	0.931422	−	0.363942i	$-0.118569\pi$
$264$		0			0
$265$	−9012.19	+	10875.1i	−0.128333	+	0.154861i
$266$		0			0
$267$		−	56192.5i		−	0.788235i
$268$		0			0
$269$		−	97237.1i		−	1.34378i	−0.740652	−	0.671889i	$-0.765483\pi$
$269$		−	97237.1i		−	1.34378i	0.740652	−	0.671889i	$-0.234517\pi$
$270$		0			0
$271$		−	81838.3i		−	1.11434i	−0.830398	−	0.557171i	$-0.811887\pi$
$271$		−	81838.3i		−	1.11434i	0.830398	−	0.557171i	$-0.188113\pi$
$272$		0			0
$273$	4205.95	+	70976.7i	0.0564338	+	0.952336i
$274$		0			0
$275$	11299.6	+	59784.8i	0.149417	+	0.790543i
$276$		0			0
$277$		−	73995.0i		−	0.964368i	−0.876070	−	0.482184i	$-0.839844\pi$
$277$		−	73995.0i		−	0.964368i	0.876070	−	0.482184i	$-0.160156\pi$
$278$		0			0
$279$		−	32921.3i		−	0.422930i
$280$		0			0
$281$	93051.4			1.17845			0.589224	−	0.807970i	$-0.299434\pi$
$281$	93051.4			1.17845			0.589224	+	0.807970i	$0.299434\pi$
$282$		0			0
$283$	−22936.7			−0.286390			−0.143195	−	0.989694i	$-0.545738\pi$
$283$	−22936.7			−0.286390			−0.143195	+	0.989694i	$0.545738\pi$
$284$		0			0
$285$	59183.7	−	71417.5i	0.728639	−	0.879256i
$286$		0			0
$287$	−122698.	+	7270.88i	−1.48962	+	0.0882721i
$288$		0			0
$289$	−61974.3			−0.742021
$290$		0			0
$291$	84076.9			0.992866
$292$		0			0
$293$	−147843.			−1.72212			−0.861062	−	0.508500i	$-0.830200\pi$
$293$	−147843.			−1.72212			−0.861062	+	0.508500i	$0.830200\pi$
$294$		0			0
$295$	−44722.6	+	53967.2i	−0.513905	+	0.620134i
$296$		0			0
$297$	−58501.5			−0.663214
$298$		0			0
$299$		−	36771.1i		−	0.411305i
$300$		0			0
$301$	−149058.	+	8832.94i	−1.64522	+	0.0974927i
$302$		0			0
$303$			47325.7i			0.515480i
$304$		0			0
$305$	−14656.7	+	17686.4i	−0.157557	+	0.190125i
$306$		0			0
$307$	−169033.			−1.79347			−0.896736	−	0.442565i	$-0.854068\pi$
$307$	−169033.			−1.79347			−0.896736	+	0.442565i	$0.854068\pi$
$308$		0			0
$309$	−60915.3			−0.637984
$310$		0			0
$311$		−	118479.i		−	1.22495i	−0.790489	−	0.612477i	$-0.790173\pi$
$311$		−	118479.i		−	1.22495i	0.790489	−	0.612477i	$-0.209827\pi$
$312$		0			0
$313$	174161.			1.77771			0.888857	−	0.458184i	$-0.151500\pi$
$313$	174161.			1.77771			0.888857	+	0.458184i	$0.151500\pi$
$314$		0			0
$315$	−15722.0	+	21436.5i	−0.158448	+	0.216039i
$316$		0			0
$317$		−	168309.i		−	1.67490i	−0.546517	−	0.837448i	$-0.684046\pi$
$317$		−	168309.i		−	1.67490i	0.546517	−	0.837448i	$-0.315954\pi$
$318$		0			0
$319$	148186.			1.45621
$320$		0			0
$321$			38220.7i			0.370927i
$322$		0			0
$323$		−	53739.3i		−	0.515094i
$324$		0			0
$325$	16619.8	+	87932.9i	0.157347	+	0.832501i
$326$		0			0
$327$	101881.			0.952795
$328$		0			0
$329$	−14.3869	−	242.783i	−0.000132916	−	0.00224299i
$330$		0			0
$331$	−137063.			−1.25102			−0.625510	−	0.780216i	$-0.715109\pi$
$331$	−137063.			−1.25102			−0.625510	+	0.780216i	$0.715109\pi$
$332$		0			0
$333$			27334.3i			0.246501i
$334$		0			0
$335$	61718.3	−	74476.1i	0.549951	−	0.663632i
$336$		0			0
$337$		−	125190.i		−	1.10233i	−0.834398	−	0.551163i	$-0.814184\pi$
$337$		−	125190.i		−	1.10233i	0.834398	−	0.551163i	$-0.185816\pi$
$338$		0			0
$339$		−	176147.i		−	1.53277i
$340$		0			0
$341$		−	147682.i		−	1.27004i
$342$		0			0
$343$	−20781.0	−	115799.i	−0.176635	−	0.984276i
$344$		0			0
$345$	41515.9	−	50097.6i	0.348800	−	0.420900i
$346$		0			0
$347$		−	201097.i		−	1.67012i	−0.550159	−	0.835060i	$-0.685433\pi$
$347$		−	201097.i		−	1.67012i	0.550159	−	0.835060i	$-0.314567\pi$
$348$		0			0
$349$			57214.0i			0.469734i	0.972028	+	0.234867i	$0.0754654\pi$
$349$			57214.0i			0.469734i	−0.972028	+	0.234867i	$0.924535\pi$
$350$		0			0
$351$	−86045.2			−0.698414
$352$		0			0
$353$	24132.9			0.193669			0.0968346	−	0.995300i	$-0.469128\pi$
$353$	24132.9			0.193669			0.0968346	+	0.995300i	$0.469128\pi$
$354$		0			0
$355$	−167561.	−	138858.i	−1.32959	−	1.10183i
$356$		0			0
$357$	4311.83	+	72763.4i	0.0338318	+	0.570921i
$358$		0			0
$359$	−75287.6			−0.584164			−0.292082	−	0.956393i	$-0.594348\pi$
$359$	−75287.6			−0.584164			−0.292082	+	0.956393i	$0.594348\pi$
$360$		0			0
$361$	−3709.38			−0.0284634
$362$		0			0
$363$	−52333.9			−0.397164
$364$		0			0
$365$	−22838.3	−	18926.1i	−0.171427	−	0.142061i
$366$		0			0
$367$	1762.56			0.0130862			0.00654308	−	0.999979i	$-0.497917\pi$
$367$	1762.56			0.0130862			0.00654308	+	0.999979i	$0.497917\pi$
$368$		0			0
$369$			54436.0i			0.399792i
$370$		0			0
$371$	−27634.5	+	1637.58i	−0.200773	+	0.0118974i
$372$		0			0
$373$			188131.i			1.35221i	0.736806	+	0.676104i	$0.236333\pi$
$373$			188131.i			1.35221i	−0.736806	+	0.676104i	$0.763667\pi$
$374$		0			0
$375$	−76636.2	+	138566.i	−0.544969	+	0.985355i
$376$		0			0
$377$	217955.			1.53350
$378$		0			0
$379$	−198575.			−1.38244			−0.691219	−	0.722645i	$-0.742926\pi$
$379$	−198575.			−1.38244			−0.691219	+	0.722645i	$0.742926\pi$
$380$		0			0
$381$		−	255364.i		−	1.75918i
$382$		0			0
$383$	−637.725			−0.00434746			−0.00217373	−	0.999998i	$-0.500692\pi$
$383$	−637.725			−0.00434746			−0.00217373	+	0.999998i	$0.500692\pi$
$384$		0			0
$385$	−70527.6	+	96162.0i	−0.475814	+	0.648757i
$386$		0			0
$387$			66130.9i			0.441553i
$388$		0			0
$389$	7893.41			0.0521633			0.0260817	−	0.999660i	$-0.491697\pi$
$389$	7893.41			0.0521633			0.0260817	+	0.999660i	$0.491697\pi$
$390$		0			0
$391$		−	37696.8i		−	0.246576i
$392$		0			0
$393$			187256.i			1.21242i
$394$		0			0
$395$	−60379.6	−	50036.5i	−0.386986	−	0.320695i
$396$		0			0
$397$	−172079.			−1.09181			−0.545904	−	0.837848i	$-0.683814\pi$
$397$	−172079.			−1.09181			−0.545904	+	0.837848i	$0.683814\pi$
$398$		0			0
$399$	181478.	−	10754.1i	1.13993	−	0.0675503i
$400$		0			0
$401$	−209908.			−1.30539			−0.652695	−	0.757621i	$-0.726362\pi$
$401$	−209908.			−1.30539			−0.652695	+	0.757621i	$0.726362\pi$
$402$		0			0
$403$		−	217214.i		−	1.33745i
$404$		0			0
$405$	−151066.	−	125188.i	−0.920993	−	0.763227i
$406$		0			0
$407$			122619.i			0.740233i
$408$		0			0
$409$			46452.8i			0.277693i	0.990314	+	0.138847i	$0.0443395\pi$
$409$			46452.8i			0.277693i	−0.990314	+	0.138847i	$0.955660\pi$
$410$		0			0
$411$			69898.8i			0.413796i
$412$		0			0
$413$	−137135.	+	8126.39i	−0.803987	+	0.0476429i
$414$		0			0
$415$	147350.	+	122109.i	0.855564	+	0.709006i
$416$		0			0
$417$		−	253716.i		−	1.45907i
$418$		0			0
$419$			151475.i			0.862803i	0.902160	+	0.431402i	$0.141981\pi$
$419$			151475.i			0.862803i	−0.902160	+	0.431402i	$0.858019\pi$
$420$		0			0
$421$	230298.			1.29935			0.649675	−	0.760212i	$-0.274905\pi$
$421$	230298.			1.29935			0.649675	+	0.760212i	$0.274905\pi$
$422$		0			0
$423$	−107.713			−0.000601985
$424$		0			0
$425$	17038.1	+	90146.4i	0.0943288	+	0.499080i
$426$		0			0
$427$	−44942.6	+	2663.22i	−0.246492	+	0.0146067i
$428$		0			0
$429$	141258.			0.767537
$430$		0			0
$431$	234915.			1.26461			0.632304	−	0.774721i	$-0.282110\pi$
$431$	234915.			1.26461			0.632304	+	0.774721i	$0.282110\pi$
$432$		0			0
$433$	−152739.			−0.814658			−0.407329	−	0.913282i	$-0.633540\pi$
$433$	−152739.			−0.814658			−0.407329	+	0.913282i	$0.633540\pi$
$434$		0			0
$435$	296946.	+	246079.i	1.56927	+	1.30046i
$436$		0			0
$437$	−94018.9			−0.492326
$438$		0			0
$439$			46904.2i			0.243379i	0.992568	+	0.121689i	$0.0388312\pi$
$439$			46904.2i			0.243379i	−0.992568	+	0.121689i	$0.961169\pi$
$440$		0			0
$441$	−51739.8	+	6153.64i	−0.266041	+	0.0316413i
$442$		0			0
$443$		−	115014.i		−	0.586064i	−0.956103	−	0.293032i	$-0.905336\pi$
$443$		−	115014.i		−	0.586064i	0.956103	−	0.293032i	$-0.0946642\pi$
$444$		0			0
$445$	88451.1	−	106735.i	0.446666	−	0.538997i
$446$		0			0
$447$	−117153.			−0.586326
$448$		0			0
$449$	−86311.7			−0.428131			−0.214066	−	0.976819i	$-0.568671\pi$
$449$	−86311.7			−0.428131			−0.214066	+	0.976819i	$0.568671\pi$
$450$		0			0
$451$			244195.i			1.20056i
$452$		0			0
$453$	−92690.0			−0.451686
$454$		0			0
$455$	−103733.	+	141437.i	−0.501067	+	0.683189i
$456$		0			0
$457$			130140.i			0.623130i	0.950225	+	0.311565i	$0.100853\pi$
$457$			130140.i			0.623130i	−0.950225	+	0.311565i	$0.899147\pi$
$458$		0			0
$459$	−88211.3			−0.418696
$460$		0			0
$461$		−	293520.i		−	1.38114i	−0.723268	−	0.690568i	$-0.757361\pi$
$461$		−	293520.i		−	1.38114i	0.723268	−	0.690568i	$-0.242639\pi$
$462$		0			0
$463$			113407.i			0.529027i	0.964382	+	0.264513i	$0.0852113\pi$
$463$			113407.i			0.529027i	−0.964382	+	0.264513i	$0.914789\pi$
$464$		0			0
$465$	245242.	−	295936.i	1.13420	−	1.36865i
$466$		0			0
$467$	430178.			1.97249			0.986244	−	0.165297i	$-0.0528583\pi$
$467$	430178.			1.97249			0.986244	+	0.165297i	$0.0528583\pi$
$468$		0			0
$469$	189250.	−	11214.6i	0.860380	−	0.0509846i
$470$		0			0
$471$	−74001.0			−0.333577
$472$		0			0
$473$			296657.i			1.32596i
$474$		0			0
$475$	224833.	−	42494.6i	0.996489	−	0.188342i
$476$		0			0
$477$			12260.3i			0.0538845i
$478$		0			0
$479$			351373.i			1.53143i	0.643179	+	0.765716i	$0.277615\pi$
$479$			351373.i			1.53143i	−0.643179	+	0.765716i	$0.722385\pi$
$480$		0			0
$481$			180350.i			0.779520i
$482$		0			0
$483$	127302.	−	7543.71i	0.545685	−	0.0323363i
$484$		0			0
$485$	159700.	+	132343.i	0.678924	+	0.562624i
$486$		0			0
$487$		−	32302.7i		−	0.136201i	−0.997678	−	0.0681006i	$-0.978306\pi$
$487$		−	32302.7i		−	0.136201i	0.997678	−	0.0681006i	$-0.0216939\pi$
$488$		0			0
$489$			246114.i			1.02924i
$490$		0			0
$491$	−186031.			−0.771655			−0.385827	−	0.922571i	$-0.626084\pi$
$491$	−186031.			−0.771655			−0.385827	+	0.922571i	$0.626084\pi$
$492$		0			0
$493$	223442.			0.919327
$494$		0			0
$495$	40666.0	+	33699.9i	0.165967	+	0.137537i
$496$		0			0
$497$	−25231.4	−	425788.i	−0.102148	−	1.72377i
$498$		0			0
$499$	191131.			0.767593			0.383796	−	0.923418i	$-0.374616\pi$
$499$	191131.			0.767593			0.383796	+	0.923418i	$0.374616\pi$
$500$		0			0
$501$	−18908.4			−0.0753320
$502$		0			0
$503$	−166385.			−0.657623			−0.328812	−	0.944396i	$-0.606648\pi$
$503$	−166385.			−0.657623			−0.328812	+	0.944396i	$0.606648\pi$
$504$		0			0
$505$	−74494.2	+	89892.8i	−0.292105	+	0.352486i
$506$		0			0
$507$	−81676.0			−0.317745
$508$		0			0
$509$		−	100356.i		−	0.387355i	−0.981065	−	0.193677i	$-0.937958\pi$
$509$		−	100356.i		−	0.387355i	0.981065	−	0.193677i	$-0.0620415\pi$
$510$		0			0
$511$	−3439.00	−	58034.1i	−0.0131701	−	0.222250i
$512$		0			0
$513$			220006.i			0.835989i
$514$		0			0
$515$	−115706.	−	95885.2i	−0.436255	−	0.361524i
$516$		0			0
$517$	−483.188			−0.00180774
$518$		0			0
$519$	461402.			1.71295
$520$		0			0
$521$			214836.i			0.791463i	0.918366	+	0.395732i	$0.129509\pi$
$521$			214836.i			0.791463i	−0.918366	+	0.395732i	$0.870491\pi$
$522$		0			0
$523$	373415.			1.36517			0.682587	−	0.730804i	$-0.260855\pi$
$523$	373415.			1.36517			0.682587	+	0.730804i	$0.260855\pi$
$524$		0			0
$525$	−301016.	+	75577.7i	−1.09212	+	0.274205i
$526$		0			0
$527$		−	222681.i		−	0.801794i
$528$		0			0
$529$	213889.			0.764324
$530$		0			0
$531$			60841.1i			0.215778i
$532$		0			0
$533$			359167.i			1.26428i
$534$		0			0
$535$	−60162.2	+	72598.3i	−0.210192	+	0.253641i
$536$		0			0
$537$	131561.			0.456224
$538$		0			0
$539$	−232100.	+	27604.6i	−0.798909	+	0.0950176i
$540$		0			0
$541$	−103486.			−0.353580			−0.176790	−	0.984249i	$-0.556571\pi$
$541$	−103486.			−0.353580			−0.176790	+	0.984249i	$0.556571\pi$
$542$		0			0
$543$			406728.i			1.37945i
$544$		0			0
$545$	193519.	+	160369.i	0.651523	+	0.539917i
$546$		0			0
$547$		−	298032.i		−	0.996066i	−0.867158	−	0.498033i	$-0.834056\pi$
$547$		−	298032.i		−	0.996066i	0.867158	−	0.498033i	$-0.165944\pi$
$548$		0			0
$549$			19939.2i			0.0661549i
$550$		0			0
$551$		−	557283.i		−	1.83558i
$552$		0			0
$553$	−9091.97	−	153430.i	−0.0297309	−	0.501717i
$554$		0			0
$555$	−203622.	+	245713.i	−0.661056	+	0.797703i
$556$		0			0
$557$		−	331497.i		−	1.06849i	−0.845331	−	0.534243i	$-0.820597\pi$
$557$		−	331497.i		−	1.06849i	0.845331	−	0.534243i	$-0.179403\pi$
$558$		0			0
$559$			436329.i			1.39634i
$560$		0			0
$561$	144814.			0.460135
$562$		0			0
$563$	−246442.			−0.777496			−0.388748	−	0.921344i	$-0.627092\pi$
$563$	−246442.			−0.777496			−0.388748	+	0.921344i	$0.627092\pi$
$564$		0			0
$565$	277269.	−	334583.i	0.868568	−	1.04811i
$566$		0			0
$567$	−22747.6	−	383871.i	−0.0707569	−	1.19404i
$568$		0			0
$569$	−344768.			−1.06489			−0.532443	−	0.846466i	$-0.678726\pi$
$569$	−344768.			−1.06489			−0.532443	+	0.846466i	$0.678726\pi$
$570$		0			0
$571$	56617.2			0.173651			0.0868253	−	0.996224i	$-0.472328\pi$
$571$	56617.2			0.173651			0.0868253	+	0.996224i	$0.472328\pi$
$572$		0			0
$573$	−26888.6			−0.0818954
$574$		0			0
$575$	157715.	−	29808.9i	0.477020	−	0.0901592i
$576$		0			0
$577$	586731.			1.76233			0.881165	−	0.472808i	$-0.156760\pi$
$577$	586731.			1.76233			0.881165	+	0.472808i	$0.156760\pi$
$578$		0			0
$579$		−	479660.i		−	1.43079i
$580$		0			0
$581$	22187.9	+	374428.i	0.0657302	+	1.10922i
$582$		0			0
$583$			54998.4i			0.161813i
$584$		0			0
$585$	59812.5	+	49566.6i	0.174775	+	0.144836i
$586$		0			0
$587$	−204909.			−0.594683			−0.297342	−	0.954771i	$-0.596100\pi$
$587$	−204909.			−0.594683			−0.297342	+	0.954771i	$0.596100\pi$
$588$		0			0
$589$	−555387.			−1.60090
$590$		0			0
$591$		−	121878.i		−	0.348939i
$592$		0			0
$593$	355363.			1.01056			0.505280	−	0.862955i	$-0.331389\pi$
$593$	355363.			1.01056			0.505280	+	0.862955i	$0.331389\pi$
$594$		0			0
$595$	−106345.	+	144998.i	−0.300388	+	0.409569i
$596$		0			0
$597$		−	514901.i		−	1.44469i
$598$		0			0
$599$	−556051.			−1.54975			−0.774874	−	0.632115i	$-0.782187\pi$
$599$	−556051.			−1.54975			−0.774874	+	0.632115i	$0.782187\pi$
$600$		0			0
$601$			666007.i			1.84387i	0.387346	+	0.921934i	$0.373392\pi$
$601$			666007.i			1.84387i	−0.387346	+	0.921934i	$0.626608\pi$
$602$		0			0
$603$		−	83962.3i		−	0.230914i
$604$		0			0
$605$	−99405.6	−	82377.4i	−0.271582	−	0.225060i
$606$		0			0
$607$	597917.			1.62279			0.811397	−	0.584495i	$-0.198707\pi$
$607$	597917.			1.62279			0.811397	+	0.584495i	$0.198707\pi$
$608$		0			0
$609$	44714.2	+	754565.i	0.120562	+	2.03452i
$610$		0			0
$611$	−710.684			−0.00190368
$612$		0			0
$613$		−	263964.i		−	0.702463i	−0.936289	−	0.351231i	$-0.885763\pi$
$613$		−	263964.i		−	0.702463i	0.936289	−	0.351231i	$-0.114237\pi$
$614$		0			0
$615$	−405512.	+	489335.i	−1.07214	+	1.29377i
$616$		0			0
$617$		−	266747.i		−	0.700696i	−0.936620	−	0.350348i	$-0.886063\pi$
$617$		−	266747.i		−	0.700696i	0.936620	−	0.350348i	$-0.113937\pi$
$618$		0			0
$619$			704148.i			1.83774i	0.394566	+	0.918868i	$0.370895\pi$
$619$			704148.i			1.83774i	−0.394566	+	0.918868i	$0.629105\pi$
$620$		0			0
$621$			154329.i			0.400188i
$622$		0			0
$623$	271222.	−	16072.2i	0.698794	−	0.0414093i
$624$		0			0
$625$	−363679.	+	142567.i	−0.931018	+	0.364973i
$626$		0			0
$627$		−	361179.i		−	0.918728i
$628$		0			0
$629$			184890.i			0.467318i
$630$		0			0
$631$	−380200.			−0.954890			−0.477445	−	0.878662i	$-0.658437\pi$
$631$	−380200.			−0.954890			−0.477445	+	0.878662i	$0.658437\pi$
$632$		0			0
$633$	554094.			1.38285
$634$		0			0
$635$	401962.	−	485051.i	0.996867	−	1.20293i
$636$		0			0
$637$	−341378.	+	40601.5i	−0.841310	+	0.100061i
$638$		0			0
$639$	−188904.			−0.462636
$640$		0			0
$641$	−730973.			−1.77904			−0.889519	−	0.456898i	$-0.848960\pi$
$641$	−730973.			−1.77904			−0.889519	+	0.456898i	$0.848960\pi$
$642$		0			0
$643$	−279742.			−0.676606			−0.338303	−	0.941037i	$-0.609853\pi$
$643$	−279742.			−0.676606			−0.338303	+	0.941037i	$0.609853\pi$
$644$		0			0
$645$	−492631.	+	594463.i	−1.18414	+	1.42891i
$646$		0			0
$647$	−456117.			−1.08960			−0.544801	−	0.838565i	$-0.683395\pi$
$647$	−456117.			−1.08960			−0.544801	+	0.838565i	$0.683395\pi$
$648$		0			0
$649$			272927.i			0.647974i
$650$		0			0
$651$	751998.	−	44562.1i	1.77441	−	0.105149i
$652$		0			0
$653$		−	528883.i		−	1.24032i	−0.784476	−	0.620159i	$-0.787068\pi$
$653$		−	528883.i		−	1.24032i	0.784476	−	0.620159i	$-0.212932\pi$
$654$		0			0
$655$	−294755.	+	355684.i	−0.687035	+	0.829052i
$656$		0			0
$657$	−25747.3			−0.0596487
$658$		0			0
$659$	−363465.			−0.836936			−0.418468	−	0.908232i	$-0.637433\pi$
$659$	−363465.			−0.836936			−0.418468	+	0.908232i	$0.637433\pi$
$660$		0			0
$661$		−	561323.i		−	1.28472i	−0.766402	−	0.642362i	$-0.777955\pi$
$661$		−	561323.i		−	1.28472i	0.766402	−	0.642362i	$-0.222045\pi$
$662$		0			0
$663$	212996.			0.484556
$664$		0			0
$665$	361636.	+	265233.i	0.817765	+	0.599769i
$666$		0			0
$667$		−	390920.i		−	0.878691i
$668$		0			0
$669$	−2889.30			−0.00645567
$670$		0			0
$671$			89445.1i			0.198661i
$672$		0			0
$673$		−	405247.i		−	0.894725i	−0.894353	−	0.447363i	$-0.852363\pi$
$673$		−	405247.i		−	0.894725i	0.894353	−	0.447363i	$-0.147637\pi$
$674$		0			0
$675$	−69753.5	−	369056.i	−0.153094	−	0.809999i
$676$		0			0
$677$	278140.			0.606858			0.303429	−	0.952854i	$-0.401869\pi$
$677$	278140.			0.606858			0.303429	+	0.952854i	$0.401869\pi$
$678$		0			0
$679$	24047.7	+	405811.i	0.0521595	+	0.880206i
$680$		0			0
$681$	936972.			2.02038
$682$		0			0
$683$			222270.i			0.476475i	0.971207	+	0.238237i	$0.0765696\pi$
$683$			222270.i			0.476475i	−0.971207	+	0.238237i	$0.923430\pi$
$684$		0			0
$685$	−110026.	+	132769.i	−0.234484	+	0.282954i
$686$		0			0
$687$		−	752220.i		−	1.59379i
$688$		0			0
$689$			80892.9i			0.170401i
$690$		0			0
$691$			552562.i			1.15724i	0.815596	+	0.578622i	$0.196409\pi$
$691$			552562.i			1.15724i	−0.815596	+	0.578622i	$0.803591\pi$
$692$		0			0
$693$	6123.50	+	103336.i	0.0127507	+	0.215171i
$694$		0			0
$695$	399368.	−	481922.i	0.826807	−	0.997716i
$696$		0			0
$697$			368208.i			0.757928i
$698$		0			0
$699$			168778.i			0.345431i
$700$		0			0
$701$	409660.			0.833657			0.416828	−	0.908985i	$-0.363142\pi$
$701$	409660.			0.833657			0.416828	+	0.908985i	$0.363142\pi$
$702$		0			0
$703$	461132.			0.933072
$704$		0			0
$705$	−968.249	−	802.387i	−0.00194809	−	0.00161438i
$706$		0			0
$707$	−228425.	+	13536.1i	−0.456989	+	0.0270804i
$708$		0			0
$709$	500981.			0.996618			0.498309	−	0.866999i	$-0.333954\pi$
$709$	500981.			0.996618			0.498309	+	0.866999i	$0.333954\pi$
$710$		0			0
$711$	−68070.3			−0.134654
$712$		0			0
$713$	−389590.			−0.766353
$714$		0			0
$715$	268313.	+	222351.i	0.524843	+	0.434937i
$716$		0			0
$717$	92806.2			0.180526
$718$		0			0
$719$		−	81013.4i		−	0.156711i	−0.996925	−	0.0783555i	$-0.975033\pi$
$719$		−	81013.4i		−	0.156711i	0.996925	−	0.0783555i	$-0.0249669\pi$
$720$		0			0
$721$	−17423.0	−	294018.i	−0.0335160	−	0.565592i
$722$		0			0
$723$			932295.i			1.78351i
$724$		0			0
$725$	176688.	+	934829.i	0.336147	+	1.77851i
$726$		0			0
$727$	957372.			1.81139			0.905695	−	0.423930i	$-0.139350\pi$
$727$	957372.			1.81139			0.905695	+	0.423930i	$0.139350\pi$
$728$		0			0
$729$	322987.			0.607757
$730$		0			0
$731$			447313.i			0.837099i
$732$		0			0
$733$	−486907.			−0.906229			−0.453115	−	0.891452i	$-0.649687\pi$
$733$	−486907.			−0.906229			−0.453115	+	0.891452i	$0.649687\pi$
$734$		0			0
$735$	−510939.	−	330111.i	−0.945790	−	0.611062i
$736$		0			0
$737$		−	376647.i		−	0.693424i
$738$		0			0
$739$	559453.			1.02441			0.512206	−	0.858862i	$-0.328828\pi$
$739$	559453.			1.02441			0.512206	+	0.858862i	$0.328828\pi$
$740$		0			0
$741$		−	531230.i		−	0.967489i
$742$		0			0
$743$			811652.i			1.47025i	0.677930	+	0.735126i	$0.262877\pi$
$743$			811652.i			1.47025i	−0.677930	+	0.735126i	$0.737123\pi$
$744$		0			0
$745$	−222527.	−	184408.i	−0.400931	−	0.332252i
$746$		0			0
$747$	166118.			0.297697
$748$		0			0
$749$	−184478.	+	10931.9i	−0.328838	+	0.0194864i
$750$		0			0
$751$	374124.			0.663339			0.331669	−	0.943396i	$-0.392388\pi$
$751$	374124.			0.663339			0.331669	+	0.943396i	$0.392388\pi$
$752$		0			0
$753$			210462.i			0.371180i
$754$		0			0
$755$	−176060.	−	145901.i	−0.308864	−	0.255955i
$756$		0			0
$757$		−	298016.i		−	0.520054i	−0.965601	−	0.260027i	$-0.916269\pi$
$757$		−	298016.i		−	0.520054i	0.965601	−	0.260027i	$-0.0837314\pi$
$758$		0			0
$759$		−	253358.i		−	0.439796i
$760$		0			0
$761$		−	622134.i		−	1.07427i	−0.843496	−	0.537136i	$-0.819506\pi$
$761$		−	622134.i		−	1.07427i	0.843496	−	0.537136i	$-0.180494\pi$
$762$		0			0
$763$	29140.1	+	491748.i	0.0500544	+	0.844682i
$764$		0			0
$765$	61318.1	+	50814.3i	0.104777	+	0.0868286i
$766$		0			0
$767$			401428.i			0.682364i
$768$		0			0
$769$		−	130966.i		−	0.221465i	−0.993850	−	0.110732i	$-0.964680\pi$
$769$		−	130966.i		−	0.221465i	0.993850	−	0.110732i	$-0.0353196\pi$
$770$		0			0
$771$	−712253.			−1.19819
$772$		0			0
$773$	303462.			0.507862			0.253931	−	0.967222i	$-0.418276\pi$
$773$	303462.			0.507862			0.253931	+	0.967222i	$0.418276\pi$
$774$		0			0
$775$	931649.	−	176086.i	1.55113	−	0.293172i
$776$		0			0
$777$	−624377.	+	36999.5i	−1.03420	+	0.0612849i
$778$		0			0
$779$	918343.			1.51332
$780$		0			0
$781$	−847405.			−1.38928
$782$		0			0
$783$	−914761.			−1.49205
$784$		0			0
$785$	−140561.	−	116483.i	−0.228101	−	0.189027i
$786$		0			0
$787$	325475.			0.525494			0.262747	−	0.964865i	$-0.415372\pi$
$787$	325475.			0.525494			0.262747	+	0.964865i	$0.415372\pi$
$788$		0			0
$789$		−	510224.i		−	0.819610i
$790$		0			0
$791$	850204.	−	50381.6i	1.35885	−	0.0805228i
$792$		0			0
$793$			131558.i			0.209204i
$794$		0			0
$795$	−91330.9	+	110210.i	−0.144505	+	0.174376i
$796$		0			0
$797$	−277890.			−0.437478			−0.218739	−	0.975783i	$-0.570194\pi$
$797$	−277890.			−0.437478			−0.218739	+	0.975783i	$0.570194\pi$
$798$		0			0
$799$	−728.574			−0.00114125
$800$		0			0
$801$		−	120330.i		−	0.187546i
$802$		0			0
$803$	−115500.			−0.179122
$804$		0			0
$805$	253679.	+	186054.i	0.391464	+	0.287110i
$806$		0			0
$807$		−	985416.i		−	1.51312i
$808$		0			0
$809$	−515192.			−0.787177			−0.393589	−	0.919287i	$-0.628767\pi$
$809$	−515192.			−0.787177			−0.393589	+	0.919287i	$0.628767\pi$
$810$		0			0
$811$			1.14259e6i			1.73719i	0.495524	+	0.868594i	$0.334976\pi$
$811$			1.14259e6i			1.73719i	−0.495524	+	0.868594i	$0.665024\pi$
$812$		0			0
$813$		−	829363.i		−	1.25477i
$814$		0			0
$815$	−387401.	+	467480.i	−0.583237	+	0.703798i
$816$		0			0
$817$	1.11564e6			1.67139
$818$		0			0
$819$	9006.58	+	151989.i	0.0134274	+	0.226591i
$820$		0			0
$821$	941956.			1.39748			0.698738	−	0.715378i	$-0.253745\pi$
$821$	941956.			1.39748			0.698738	+	0.715378i	$0.253745\pi$
$822$		0			0
$823$			715694.i			1.05664i	0.849045	+	0.528321i	$0.177178\pi$
$823$			715694.i			1.05664i	−0.849045	+	0.528321i	$0.822822\pi$
$824$		0			0
$825$	114512.	+	605869.i	0.168246	+	0.890166i
$826$		0			0
$827$			600637.i			0.878216i	0.898434	+	0.439108i	$0.144705\pi$
$827$			600637.i			0.878216i	−0.898434	+	0.439108i	$0.855295\pi$
$828$		0			0
$829$		−	440720.i		−	0.641289i	−0.947200	−	0.320645i	$-0.896100\pi$
$829$		−	440720.i		−	0.641289i	0.947200	−	0.320645i	$-0.103900\pi$
$830$		0			0
$831$		−	749877.i		−	1.08590i
$832$		0			0
$833$	−349971.	+	41623.5i	−0.504362	+	0.0599859i
$834$		0			0
$835$	−35915.6	−	29763.2i	−0.0515122	−	0.0426881i
$836$		0			0
$837$			911649.i			1.30130i
$838$		0			0
$839$		−	602252.i		−	0.855568i	−0.903881	−	0.427784i	$-0.859294\pi$
$839$		−	602252.i		−	0.855568i	0.903881	−	0.427784i	$-0.140706\pi$
$840$		0			0
$841$	1.60984e6			2.27609
$842$		0			0
$843$	942998.			1.32695
$844$		0			0
$845$	−155140.	−	128564.i	−0.217275	−	0.180055i
$846$		0			0
$847$	−14968.5	−	252598.i	−0.0208647	−	0.352098i
$848$		0			0
$849$	−232444.			−0.322480
$850$		0			0
$851$	323473.			0.446662
$852$		0			0
$853$	−726758.			−0.998831			−0.499415	−	0.866363i	$-0.666452\pi$
$853$	−726758.			−0.998831			−0.499415	+	0.866363i	$0.666452\pi$
$854$		0			0
$855$	126735.	−	152933.i	0.173366	−	0.209203i
$856$		0			0
$857$	−589208.			−0.802245			−0.401123	−	0.916024i	$-0.631380\pi$
$857$	−589208.			−0.802245			−0.401123	+	0.916024i	$0.631380\pi$
$858$		0			0
$859$		−	103928.i		−	0.140847i	−0.997517	−	0.0704235i	$-0.977565\pi$
$859$		−	103928.i		−	0.140847i	0.997517	−	0.0704235i	$-0.0224351\pi$
$860$		0			0
$861$	−1.24344e6	+	73684.3i	−1.67733	+	0.0993959i
$862$		0			0
$863$		−	1.03405e6i		−	1.38842i	−0.719771	−	0.694211i	$-0.755753\pi$
$863$		−	1.03405e6i		−	1.38842i	0.719771	−	0.694211i	$-0.244247\pi$
$864$		0			0
$865$	876411.	+	726281.i	1.17132	+	0.970672i
$866$		0			0
$867$	−628058.			−0.835529
$868$		0			0
$869$	−305357.			−0.404360
$870$		0			0
$871$		−	553980.i		−	0.730227i
$872$		0			0
$873$	180041.			0.236235
$874$		0			0
$875$	−690729.	−	330265.i	−0.902177	−	0.431366i
$876$		0			0
$877$			90069.0i			0.117105i	0.998284	+	0.0585526i	$0.0186485\pi$
$877$			90069.0i			0.117105i	−0.998284	+	0.0585526i	$0.981351\pi$
$878$		0			0
$879$	−1.49826e6			−1.93914
$880$		0			0
$881$			463279.i			0.596885i	0.954428	+	0.298443i	$0.0964671\pi$
$881$			463279.i			0.596885i	−0.954428	+	0.298443i	$0.903533\pi$
$882$		0			0
$883$			268326.i			0.344145i	0.985084	+	0.172072i	$0.0550463\pi$
$883$			268326.i			0.344145i	−0.985084	+	0.172072i	$0.944954\pi$
$884$		0			0
$885$	−453226.	+	546912.i	−0.578666	+	0.698282i
$886$		0			0
$887$	−1.51729e6			−1.92850			−0.964251	−	0.264991i	$-0.914631\pi$
$887$	−1.51729e6			−1.92850			−0.964251	+	0.264991i	$0.914631\pi$
$888$		0			0
$889$	1.23256e6	−	73039.2i	1.55957	−	0.0924171i
$890$		0			0
$891$	−763983.			−0.962340
$892$		0			0
$893$			1817.13i			0.00227867i
$894$		0			0
$895$	249893.	+	207086.i	0.311967	+	0.258527i
$896$		0			0
$897$		−	372644.i		−	0.463137i
$898$		0			0
$899$		−	2.30923e6i		−	2.85725i
$900$		0			0
$901$			82929.2i			0.102155i
$902$		0			0
$903$	−1.51058e6	+	89514.4i	−1.85254	+	0.109779i
$904$		0			0
$905$	−640220.	+	772560.i	−0.781686	+	0.943268i
$906$		0			0
$907$		−	732396.i		−	0.890290i	−0.895458	−	0.445145i	$-0.853152\pi$
$907$		−	732396.i		−	0.890290i	0.895458	−	0.445145i	$-0.146848\pi$
$908$		0			0
$909$			101343.i			0.122649i
$910$		0			0
$911$	−1.28266e6			−1.54552			−0.772759	−	0.634700i	$-0.781124\pi$
$911$	−1.28266e6			−1.54552			−0.772759	+	0.634700i	$0.781124\pi$
$912$		0			0
$913$	745189.			0.893973
$914$		0			0
$915$	−148533.	+	179237.i	−0.177412	+	0.214084i
$916$		0			0
$917$	−903824.	+	53559.0i	−1.07484	+	0.0636934i
$918$		0			0
$919$	−271044.			−0.320929			−0.160464	−	0.987042i	$-0.551299\pi$
$919$	−271044.			−0.320929			−0.160464	+	0.987042i	$0.551299\pi$
$920$		0			0
$921$	−1.71301e6			−2.01948
$922$		0			0
$923$	−1.24638e6			−1.46301
$924$		0			0
$925$	−773539.	+	146203.i	−0.904063	+	0.170873i
$926$		0			0
$927$	−130443.			−0.151797
$928$		0			0
$929$		−	1.25627e6i		−	1.45563i	−0.685773	−	0.727816i	$-0.740536\pi$
$929$		−	1.25627e6i		−	1.45563i	0.685773	−	0.727816i	$-0.259464\pi$
$930$		0			0
$931$	103813.	+	872858.i	0.119771	+	1.00703i
$932$		0			0
$933$		−	1.20068e6i		−	1.37932i
$934$		0			0
$935$	275067.	+	227948.i	0.314641	+	0.260743i
$936$		0			0
$937$	−1.02005e6			−1.16183			−0.580917	−	0.813963i	$-0.697306\pi$
$937$	−1.02005e6			−1.16183			−0.580917	+	0.813963i	$0.697306\pi$
$938$		0			0
$939$	1.76497e6			2.00174
$940$		0			0
$941$			883973.i			0.998297i	0.866517	+	0.499148i	$0.166354\pi$
$941$			883973.i			0.998297i	−0.866517	+	0.499148i	$0.833646\pi$
$942$		0			0
$943$	644195.			0.724425
$944$		0			0
$945$	435371.	−	593614.i	0.487524	−	0.664723i
$946$		0			0
$947$			316967.i			0.353439i	0.984261	+	0.176719i	$0.0565485\pi$
$947$			316967.i			0.353439i	−0.984261	+	0.176719i	$0.943451\pi$
$948$		0			0
$949$	−169880.			−0.188629
$950$		0			0
$951$		−	1.70567e6i		−	1.88596i
$952$		0			0
$953$			426727.i			0.469856i	0.972013	+	0.234928i	$0.0754854\pi$
$953$			426727.i			0.469856i	−0.972013	+	0.234928i	$0.924515\pi$
$954$		0			0
$955$	−51073.6	−	42324.7i	−0.0560003	−	0.0464074i
$956$		0			0
$957$	1.50174e6			1.63972
$958$		0			0
$959$	−337378.	+	19992.4i	−0.366843	+	0.0217385i
$960$		0			0
$961$	−1.37786e6			−1.49196
$962$		0			0
$963$			81845.3i			0.0882554i
$964$		0			0
$965$	755020.	−	911091.i	0.810782	−	0.978379i
$966$		0			0
$967$		−	774328.i		−	0.828080i	−0.910259	−	0.414040i	$-0.864117\pi$
$967$		−	774328.i		−	0.828080i	0.910259	−	0.414040i	$-0.135883\pi$
$968$		0			0
$969$		−	544602.i		−	0.580005i
$970$		0			0
$971$		−	832664.i		−	0.883144i	−0.897226	−	0.441572i	$-0.854421\pi$
$971$		−	832664.i		−	0.883144i	0.897226	−	0.441572i	$-0.145579\pi$
$972$		0			0
$973$	1.22460e6	−	72567.9i	1.29351	−	0.0766512i
$974$		0			0
$975$	168427.	+	891126.i	0.177175	+	0.937410i
$976$		0			0
$977$		−	1.81838e6i		−	1.90500i	−0.304542	−	0.952499i	$-0.598503\pi$
$977$		−	1.81838e6i		−	1.90500i	0.304542	−	0.952499i	$-0.401497\pi$
$978$		0			0
$979$		−	539788.i		−	0.563194i
$980$		0			0
$981$	218168.			0.226701
$982$		0			0
$983$	1.03560e6			1.07173			0.535864	−	0.844304i	$-0.319986\pi$
$983$	1.03560e6			1.07173			0.535864	+	0.844304i	$0.319986\pi$
$984$		0			0
$985$	191845.	−	231501.i	0.197732	−	0.238605i
$986$		0			0
$987$	−145.799	−	2460.40i	−0.000149665	−	0.00252564i
$988$		0			0
$989$	782592.			0.800097
$990$		0			0
$991$	769409.			0.783447			0.391724	−	0.920083i	$-0.371879\pi$
$991$	769409.			0.783447			0.391724	+	0.920083i	$0.371879\pi$
$992$		0			0
$993$	−1.38902e6			−1.40867
$994$		0			0
$995$	810492.	−	978028.i	0.818658	−	0.987882i
$996$		0			0
$997$	−10925.7			−0.0109916			−0.00549578	−	0.999985i	$-0.501749\pi$
$997$	−10925.7			−0.0109916			−0.00549578	+	0.999985i	$0.501749\pi$
$998$		0			0
$999$		−	756934.i		−	0.758450i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.5.p.h.209.10		12
4.3	odd	2		140.5.h.c.69.4	yes	12
5.4	even	2	inner	560.5.p.h.209.4		12
7.6	odd	2	inner	560.5.p.h.209.3		12
12.11	even	2		1260.5.p.c.1189.1		12
20.3	even	4		700.5.d.e.601.3		12
20.7	even	4		700.5.d.e.601.10		12
20.19	odd	2		140.5.h.c.69.10	yes	12
28.27	even	2		140.5.h.c.69.9	yes	12
35.34	odd	2	inner	560.5.p.h.209.9		12
60.59	even	2		1260.5.p.c.1189.11		12
84.83	odd	2		1260.5.p.c.1189.12		12
140.27	odd	4		700.5.d.e.601.4		12
140.83	odd	4		700.5.d.e.601.9		12
140.139	even	2		140.5.h.c.69.3	✓	12
420.419	odd	2		1260.5.p.c.1189.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.5.h.c.69.3	✓	12	140.139	even	2
140.5.h.c.69.4	yes	12	4.3	odd	2
140.5.h.c.69.9	yes	12	28.27	even	2
140.5.h.c.69.10	yes	12	20.19	odd	2
560.5.p.h.209.3		12	7.6	odd	2	inner
560.5.p.h.209.4		12	5.4	even	2	inner
560.5.p.h.209.9		12	35.34	odd	2	inner
560.5.p.h.209.10		12	1.1	even	1	trivial
700.5.d.e.601.3		12	20.3	even	4
700.5.d.e.601.4		12	140.27	odd	4
700.5.d.e.601.9		12	140.83	odd	4
700.5.d.e.601.10		12	20.7	even	4
1260.5.p.c.1189.1		12	12.11	even	2
1260.5.p.c.1189.2		12	420.419	odd	2
1260.5.p.c.1189.11		12	60.59	even	2
1260.5.p.c.1189.12		12	84.83	odd	2

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	560.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.1342 q^{3} +(19.2493 + 15.9519i) q^{5} +(2.89857 + 48.9142i) q^{7} +21.7012 q^{9} +O(q^{10})\)	\(q+10.1342 q^{3} +(19.2493 + 15.9519i) q^{5} +(2.89857 + 48.9142i) q^{7} +21.7012 q^{9} +97.3493 q^{11} +143.184 q^{13} +(195.076 + 161.659i) q^{15} +146.788 q^{17} -366.102i q^{19} +(29.3746 + 495.704i) q^{21} -256.811i q^{23} +(116.073 + 614.127i) q^{25} -600.944 q^{27} +1522.21 q^{29} -1517.03i q^{31} +986.553 q^{33} +(-724.479 + 987.803i) q^{35} +1259.58i q^{37} +1451.04 q^{39} +2508.44i q^{41} +3047.34i q^{43} +(417.733 + 346.175i) q^{45} -4.96345 q^{47} +(-2384.20 + 283.562i) q^{49} +1487.57 q^{51} +564.960i q^{53} +(1873.91 + 1552.91i) q^{55} -3710.13i q^{57} +2803.59i q^{59} +918.806i q^{61} +(62.9023 + 1061.50i) q^{63} +(2756.19 + 2284.05i) q^{65} -3869.02i q^{67} -2602.56i q^{69} -8704.79 q^{71} -1186.45 q^{73} +(1176.30 + 6223.66i) q^{75} +(282.174 + 4761.76i) q^{77} -3136.71 q^{79} -7847.85 q^{81} +7654.79 q^{83} +(2825.57 + 2341.55i) q^{85} +15426.3 q^{87} -5544.86i q^{89} +(415.027 + 7003.71i) q^{91} -15373.8i q^{93} +(5840.02 - 7047.21i) q^{95} +8296.39 q^{97} +2112.59 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 296 q^{9}+O(q^{10}) \) 12 * q + 296 * q^9	\( 12 q + 296 q^{9} + 20 q^{11} + 672 q^{15} - 1352 q^{21} - 2608 q^{25} - 1796 q^{29} + 2788 q^{35} + 332 q^{39} - 11792 q^{49} - 16844 q^{51} + 6928 q^{65} - 27448 q^{71} + 7948 q^{79} - 14636 q^{81} + 1748 q^{85} - 948 q^{91} + 29940 q^{95} - 67688 q^{99}+O(q^{100}) \) 12 * q + 296 * q^9 + 20 * q^11 + 672 * q^15 - 1352 * q^21 - 2608 * q^25 - 1796 * q^29 + 2788 * q^35 + 332 * q^39 - 11792 * q^49 - 16844 * q^51 + 6928 * q^65 - 27448 * q^71 + 7948 * q^79 - 14636 * q^81 + 1748 * q^85 - 948 * q^91 + 29940 * q^95 - 67688 * q^99

Introduction

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