LMFDB - Embedded newform 280.6.g.a.169.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [280,6,Mod(169,280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("280.169"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$44.9074695476$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 2997 x^{18} + 3735306 x^{16} + 2520827714 x^{14} + 1008202629141 x^{12} + 246520004342481 x^{10} + \cdots + 81\!\cdots\!00 $ x^20 + 2997x^18 + 3735306x^16 + 2520827714x^14 + 1008202629141x^12 + 246520004342481x^10 + 36629357739714496x^8 + 3153081134047078008x^6 + 139141465378172063856x^4 + 2257384477735837731600*x^2 + 810880935783229440000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.7
Root			$-12.7619i$ of defining polynomial
Character	$\chi$	$=$	280.169
Dual form			280.6.g.a.169.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.7619i q^{3} +(3.33191 - 55.8023i) q^{5} +49.0000i q^{7} +80.1346 q^{9} -501.355 q^{11} -19.2939i q^{13} +(-712.142 - 42.5214i) q^{15} +1308.13i q^{17} -989.445 q^{19} +625.332 q^{21} +1316.61i q^{23} +(-3102.80 - 371.857i) q^{25} -4123.80i q^{27} -4496.56 q^{29} -2726.50 q^{31} +6398.23i q^{33} +(2734.31 + 163.264i) q^{35} +962.659i q^{37} -246.226 q^{39} +12225.2 q^{41} +12086.6i q^{43} +(267.001 - 4471.70i) q^{45} +7532.17i q^{47} -2401.00 q^{49} +16694.2 q^{51} -56.2085i q^{53} +(-1670.47 + 27976.8i) q^{55} +12627.2i q^{57} -5748.75 q^{59} +21159.2 q^{61} +3926.60i q^{63} +(-1076.64 - 64.2854i) q^{65} +17484.8i q^{67} +16802.3 q^{69} -40905.9 q^{71} +18739.2i q^{73} +(-4745.59 + 39597.5i) q^{75} -24566.4i q^{77} +4249.94 q^{79} -33154.7 q^{81} +78472.0i q^{83} +(72996.9 + 4358.58i) q^{85} +57384.5i q^{87} +49177.9 q^{89} +945.399 q^{91} +34795.3i q^{93} +(-3296.74 + 55213.3i) q^{95} +84998.7i q^{97} -40175.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51}+ \cdots - 630244 q^{99}+O(q^{100}) $ 20 * q + 14 * q^5 - 1134 * q^9 + 822 * q^11 + 1322 * q^15 + 3000 * q^19 - 882 * q^21 - 1944 * q^25 + 10406 * q^29 - 8532 * q^31 - 2058 * q^35 + 71066 * q^39 - 28880 * q^41 - 3922 * q^45 - 48020 * q^49 + 34770 * q^51 - 8844 * q^55 + 61196 * q^59 - 145052 * q^61 - 12722 * q^65 + 64980 * q^69 + 41464 * q^71 - 159032 * q^75 + 97174 * q^79 - 165852 * q^81 + 319566 * q^85 - 84604 * q^89 + 61642 * q^91 + 33416 * q^95 - 630244 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	12.7619i		−	0.818675i	−0.912383	−	0.409337i	$-0.865760\pi$
$3$		−	12.7619i		−	0.818675i	0.912383	−	0.409337i	$-0.134240\pi$
$4$		0			0
$5$	3.33191	−	55.8023i	0.0596030	−	0.998222i
$5$	3.33191	−	55.8023i	0.0596030	−	0.998222i
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	80.1346			0.329772
$10$		0			0
$11$	−501.355			−1.24929			−0.624646	−	0.780908i	$-0.714757\pi$
$11$	−501.355			−1.24929			−0.624646	+	0.780908i	$0.714757\pi$
$12$		0			0
$13$		−	19.2939i		−	0.0316636i	−0.999875	−	0.0158318i	$-0.994960\pi$
$13$		−	19.2939i		−	0.0316636i	0.999875	−	0.0158318i	$-0.00503963\pi$
$14$		0			0
$15$	−712.142	−	42.5214i	−0.817219	−	0.0487955i
$16$		0			0
$17$			1308.13i			1.09782i	0.835883	+	0.548908i	$0.184956\pi$
$17$			1308.13i			1.09782i	−0.835883	+	0.548908i	$0.815044\pi$
$18$		0			0
$19$	−989.445			−0.628793			−0.314396	−	0.949292i	$-0.601802\pi$
$19$	−989.445			−0.628793			−0.314396	+	0.949292i	$0.601802\pi$
$20$		0			0
$21$	625.332			0.309430
$22$		0			0
$23$			1316.61i			0.518962i	0.965748	+	0.259481i	$0.0835516\pi$
$23$			1316.61i			0.518962i	−0.965748	+	0.259481i	$0.916448\pi$
$24$		0			0
$25$	−3102.80	−	371.857i	−0.992895	−	0.118994i
$26$		0			0
$27$		−	4123.80i		−	1.08865i
$28$		0			0
$29$	−4496.56			−0.992854			−0.496427	−	0.868078i	$-0.665355\pi$
$29$	−4496.56			−0.992854			−0.496427	+	0.868078i	$0.665355\pi$
$30$		0			0
$31$	−2726.50			−0.509567			−0.254784	−	0.966998i	$-0.582004\pi$
$31$	−2726.50			−0.509567			−0.254784	+	0.966998i	$0.582004\pi$
$32$		0			0
$33$			6398.23i			1.02276i
$34$		0			0
$35$	2734.31	+	163.264i	0.377293	+	0.0225278i
$36$		0			0
$37$			962.659i			0.115603i	0.998328	+	0.0578014i	$0.0184090\pi$
$37$			962.659i			0.115603i	−0.998328	+	0.0578014i	$0.981591\pi$
$38$		0			0
$39$	−246.226			−0.0259222
$40$		0			0
$41$	12225.2			1.13578			0.567892	−	0.823103i	$-0.307759\pi$
$41$	12225.2			1.13578			0.567892	+	0.823103i	$0.307759\pi$
$42$		0			0
$43$			12086.6i			0.996855i	0.866931	+	0.498427i	$0.166089\pi$
$43$			12086.6i			0.996855i	−0.866931	+	0.498427i	$0.833911\pi$
$44$		0			0
$45$	267.001	−	4471.70i	0.0196554	−	0.329186i
$46$		0			0
$47$			7532.17i			0.497365i	0.968585	+	0.248683i	$0.0799976\pi$
$47$			7532.17i			0.497365i	−0.968585	+	0.248683i	$0.920002\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	16694.2			0.898754
$52$		0			0
$53$		−	56.2085i		−	0.00274861i	−0.999999	−	0.00137430i	$-0.999563\pi$
$53$		−	56.2085i		−	0.00274861i	0.999999	−	0.00137430i	$-0.000437454\pi$
$54$		0			0
$55$	−1670.47	+	27976.8i	−0.0744616	+	1.24707i
$56$		0			0
$57$			12627.2i			0.514777i
$58$		0			0
$59$	−5748.75			−0.215003			−0.107501	−	0.994205i	$-0.534285\pi$
$59$	−5748.75			−0.215003			−0.107501	+	0.994205i	$0.534285\pi$
$60$		0			0
$61$	21159.2			0.728071			0.364035	−	0.931385i	$-0.381399\pi$
$61$	21159.2			0.728071			0.364035	+	0.931385i	$0.381399\pi$
$62$		0			0
$63$			3926.60i			0.124642i
$64$		0			0
$65$	−1076.64	−	64.2854i	−0.0316073	−	0.00188725i
$66$		0			0
$67$			17484.8i			0.475854i	0.971283	+	0.237927i	$0.0764679\pi$
$67$			17484.8i			0.475854i	−0.971283	+	0.237927i	$0.923532\pi$
$68$		0			0
$69$	16802.3			0.424861
$70$		0			0
$71$	−40905.9			−0.963031			−0.481515	−	0.876438i	$-0.659913\pi$
$71$	−40905.9			−0.963031			−0.481515	+	0.876438i	$0.659913\pi$
$72$		0			0
$73$			18739.2i			0.411570i	0.978597	+	0.205785i	$0.0659747\pi$
$73$			18739.2i			0.411570i	−0.978597	+	0.205785i	$0.934025\pi$
$74$		0			0
$75$	−4745.59	+	39597.5i	−0.0974175	+	0.812858i
$76$		0			0
$77$		−	24566.4i		−	0.472188i
$78$		0			0
$79$	4249.94			0.0766151			0.0383076	−	0.999266i	$-0.487803\pi$
$79$	4249.94			0.0766151			0.0383076	+	0.999266i	$0.487803\pi$
$80$		0			0
$81$	−33154.7			−0.561478
$82$		0			0
$83$			78472.0i			1.25032i	0.780498	+	0.625158i	$0.214965\pi$
$83$			78472.0i			1.25032i	−0.780498	+	0.625158i	$0.785035\pi$
$84$		0			0
$85$	72996.9	+	4358.58i	1.09586	+	0.0654332i
$86$		0			0
$87$			57384.5i			0.812825i
$88$		0			0
$89$	49177.9			0.658104			0.329052	−	0.944312i	$-0.393271\pi$
$89$	49177.9			0.658104			0.329052	+	0.944312i	$0.393271\pi$
$90$		0			0
$91$	945.399			0.0119677
$92$		0			0
$93$			34795.3i			0.417170i
$94$		0			0
$95$	−3296.74	+	55213.3i	−0.0374779	+	0.627675i
$96$		0			0
$97$			84998.7i			0.917240i	0.888632	+	0.458620i	$0.151656\pi$
$97$			84998.7i			0.917240i	−0.888632	+	0.458620i	$0.848344\pi$
$98$		0			0
$99$	−40175.9			−0.411982
$100$		0			0
$101$	−124429.			−1.21372			−0.606861	−	0.794808i	$-0.707571\pi$
$101$	−124429.			−1.21372			−0.606861	+	0.794808i	$0.707571\pi$
$102$		0			0
$103$			179896.i			1.67081i	0.549634	+	0.835406i	$0.314767\pi$
$103$			179896.i			1.67081i	−0.549634	+	0.835406i	$0.685233\pi$
$104$		0			0
$105$	2083.55	−	34895.0i	0.0184430	−	0.308880i
$106$		0			0
$107$		−	188900.i		−	1.59505i	−0.603288	−	0.797524i	$-0.706143\pi$
$107$		−	188900.i		−	1.59505i	0.603288	−	0.797524i	$-0.293857\pi$
$108$		0			0
$109$	153193.			1.23502			0.617509	−	0.786564i	$-0.288142\pi$
$109$	153193.			1.23502			0.617509	+	0.786564i	$0.288142\pi$
$110$		0			0
$111$	12285.3			0.0946410
$112$		0			0
$113$			142026.i			1.04634i	0.852229	+	0.523170i	$0.175251\pi$
$113$			142026.i			1.04634i	−0.852229	+	0.523170i	$0.824749\pi$
$114$		0			0
$115$	73469.6	+	4386.81i	0.518040	+	0.0309317i
$116$		0			0
$117$		−	1546.11i		−	0.0104418i
$118$		0			0
$119$	−64098.5			−0.414935
$120$		0			0
$121$	90306.2			0.560731
$122$		0			0
$123$		−	156016.i		−	0.929838i
$124$		0			0
$125$	−31088.7	+	171904.i	−0.177962	+	0.984037i
$126$		0			0
$127$		−	85958.2i		−	0.472909i	−0.971643	−	0.236455i	$-0.924015\pi$
$127$		−	85958.2i		−	0.472909i	0.971643	−	0.236455i	$-0.0759855\pi$
$128$		0			0
$129$	154247.			0.816100
$130$		0			0
$131$	−186106.			−0.947506			−0.473753	−	0.880658i	$-0.657101\pi$
$131$	−186106.			−0.947506			−0.473753	+	0.880658i	$0.657101\pi$
$132$		0			0
$133$		−	48482.8i		−	0.237661i
$134$		0			0
$135$	−230118.	−	13740.1i	−1.08672	−	0.0648869i
$136$		0			0
$137$			62849.6i			0.286089i	0.989716	+	0.143045i	$0.0456892\pi$
$137$			62849.6i			0.286089i	−0.989716	+	0.143045i	$0.954311\pi$
$138$		0			0
$139$	−337229.			−1.48043			−0.740215	−	0.672371i	$-0.765276\pi$
$139$	−337229.			−1.48043			−0.740215	+	0.672371i	$0.765276\pi$
$140$		0			0
$141$	96124.6			0.407180
$142$		0			0
$143$			9673.08i			0.0395571i
$144$		0			0
$145$	−14982.1	+	250919.i	−0.0591771	+	0.991089i
$146$		0			0
$147$			30641.3i			0.116954i
$148$		0			0
$149$	−135566.			−0.500248			−0.250124	−	0.968214i	$-0.580471\pi$
$149$	−135566.			−0.500248			−0.250124	+	0.968214i	$0.580471\pi$
$150$		0			0
$151$	−98649.4			−0.352089			−0.176044	−	0.984382i	$-0.556330\pi$
$151$	−98649.4			−0.352089			−0.176044	+	0.984382i	$0.556330\pi$
$152$		0			0
$153$			104827.i			0.362029i
$154$		0			0
$155$	−9084.46	+	152145.i	−0.0303718	+	0.508661i
$156$		0			0
$157$		−	360552.i		−	1.16740i	−0.811970	−	0.583699i	$-0.801605\pi$
$157$		−	360552.i		−	1.16740i	0.811970	−	0.583699i	$-0.198395\pi$
$158$		0			0
$159$	−717.326			−0.00225021
$160$		0			0
$161$	−64513.7			−0.196149
$162$		0			0
$163$		−	338233.i		−	0.997120i	−0.866855	−	0.498560i	$-0.833862\pi$
$163$		−	338233.i		−	0.997120i	0.866855	−	0.498560i	$-0.166138\pi$
$164$		0			0
$165$	357036.	+	21318.3i	1.02095	+	0.0609598i
$166$		0			0
$167$			25703.4i			0.0713181i	0.999364	+	0.0356590i	$0.0113530\pi$
$167$			25703.4i			0.0713181i	−0.999364	+	0.0356590i	$0.988647\pi$
$168$		0			0
$169$	370921.			0.998997
$170$		0			0
$171$	−79288.7			−0.207358
$172$		0			0
$173$		−	166733.i		−	0.423553i	−0.977318	−	0.211776i	$-0.932075\pi$
$173$		−	166733.i		−	0.423553i	0.977318	−	0.211776i	$-0.0679248\pi$
$174$		0			0
$175$	18221.0	−	152037.i	0.0449756	−	0.375279i
$176$		0			0
$177$			73364.9i			0.176017i
$178$		0			0
$179$	−332531.			−0.775711			−0.387855	−	0.921720i	$-0.626784\pi$
$179$	−332531.			−0.775711			−0.387855	+	0.921720i	$0.626784\pi$
$180$		0			0
$181$	315882.			0.716686			0.358343	−	0.933590i	$-0.383342\pi$
$181$	315882.			0.716686			0.358343	+	0.933590i	$0.383342\pi$
$182$		0			0
$183$		−	270030.i		−	0.596053i
$184$		0			0
$185$	53718.6	+	3207.49i	0.115397	+	0.00689027i
$186$		0			0
$187$		−	655840.i		−	1.37149i
$188$		0			0
$189$	202066.			0.411471
$190$		0			0
$191$	193935.			0.384656			0.192328	−	0.981331i	$-0.438396\pi$
$191$	193935.			0.384656			0.192328	+	0.981331i	$0.438396\pi$
$192$		0			0
$193$		−	258933.i		−	0.500374i	−0.968198	−	0.250187i	$-0.919508\pi$
$193$		−	258933.i		−	0.500374i	0.968198	−	0.250187i	$-0.0804921\pi$
$194$		0			0
$195$	−820.402	+	13740.0i	−0.00154504	+	0.0258761i
$196$		0			0
$197$		−	1.03919e6i		−	1.90778i	−0.300153	−	0.953891i	$-0.597038\pi$
$197$		−	1.03919e6i		−	1.90778i	0.300153	−	0.953891i	$-0.402962\pi$
$198$		0			0
$199$	171300.			0.306638			0.153319	−	0.988177i	$-0.451004\pi$
$199$	171300.			0.306638			0.153319	+	0.988177i	$0.451004\pi$
$200$		0			0
$201$	223139.			0.389569
$202$		0			0
$203$		−	220331.i		−	0.375264i
$204$		0			0
$205$	40733.2	−	682194.i	0.0676962	−	1.13377i
$206$		0			0
$207$			105506.i			0.171139i
$208$		0			0
$209$	496063.			0.785546
$210$		0			0
$211$	−574831.			−0.888861			−0.444430	−	0.895813i	$-0.646594\pi$
$211$	−574831.			−0.888861			−0.444430	+	0.895813i	$0.646594\pi$
$212$		0			0
$213$			522036.i			0.788409i
$214$		0			0
$215$	674459.	+	40271.4i	0.995083	+	0.0594156i
$216$		0			0
$217$		−	133599.i		−	0.192598i
$218$		0			0
$219$	239147.			0.336941
$220$		0			0
$221$	25238.9			0.0347608
$222$		0			0
$223$			1.08708e6i			1.46386i	0.681378	+	0.731932i	$0.261381\pi$
$223$			1.08708e6i			1.46386i	−0.681378	+	0.731932i	$0.738619\pi$
$224$		0			0
$225$	−248641.	−	29798.6i	−0.327429	−	0.0392409i
$226$		0			0
$227$			250824.i			0.323076i	0.986867	+	0.161538i	$0.0516454\pi$
$227$			250824.i			0.323076i	−0.986867	+	0.161538i	$0.948355\pi$
$228$		0			0
$229$	799016.			1.00685			0.503427	−	0.864038i	$-0.332072\pi$
$229$	799016.			1.00685			0.503427	+	0.864038i	$0.332072\pi$
$230$		0			0
$231$	−313513.			−0.386568
$232$		0			0
$233$		−	1.15788e6i		−	1.39725i	−0.715490	−	0.698623i	$-0.753796\pi$
$233$		−	1.15788e6i		−	1.39725i	0.715490	−	0.698623i	$-0.246204\pi$
$234$		0			0
$235$	420312.	+	25096.5i	0.496481	+	0.0296445i
$236$		0			0
$237$		−	54237.1i		−	0.0627229i
$238$		0			0
$239$	−1.73551e6			−1.96531			−0.982656	−	0.185436i	$-0.940630\pi$
$239$	−1.73551e6			−1.96531			−0.982656	+	0.185436i	$0.940630\pi$
$240$		0			0
$241$	−1.71193e6			−1.89864			−0.949321	−	0.314308i	$-0.898227\pi$
$241$	−1.71193e6			−1.89864			−0.949321	+	0.314308i	$0.898227\pi$
$242$		0			0
$243$		−	578967.i		−	0.628982i
$244$		0			0
$245$	−7999.92	+	133981.i	−0.00851472	+	0.142603i
$246$		0			0
$247$			19090.2i			0.0199099i
$248$		0			0
$249$	1.00145e6			1.02360
$250$		0			0
$251$	−1.49387e6			−1.49668			−0.748342	−	0.663313i	$-0.769150\pi$
$251$	−1.49387e6			−1.49668			−0.748342	+	0.663313i	$0.769150\pi$
$252$		0			0
$253$		−	660087.i		−	0.648336i
$254$		0			0
$255$	55623.7	−	931577.i	0.0535685	−	0.897156i
$256$		0			0
$257$		−	2.04206e6i		−	1.92858i	−0.264858	−	0.964288i	$-0.585325\pi$
$257$		−	2.04206e6i		−	1.92858i	0.264858	−	0.964288i	$-0.414675\pi$
$258$		0			0
$259$	−47170.3			−0.0436937
$260$		0			0
$261$	−360330.			−0.327416
$262$		0			0
$263$			671803.i			0.598898i	0.954112	+	0.299449i	$0.0968028\pi$
$263$			671803.i			0.598898i	−0.954112	+	0.299449i	$0.903197\pi$
$264$		0			0
$265$	−3136.57	−	187.282i	−0.00274372	−	0.000163825i
$266$		0			0
$267$		−	627602.i		−	0.538773i
$268$		0			0
$269$	−993623.			−0.837223			−0.418611	−	0.908165i	$-0.637483\pi$
$269$	−993623.			−0.837223			−0.418611	+	0.908165i	$0.637483\pi$
$270$		0			0
$271$	−2.09638e6			−1.73399			−0.866996	−	0.498314i	$-0.833953\pi$
$271$	−2.09638e6			−1.73399			−0.866996	+	0.498314i	$0.833953\pi$
$272$		0			0
$273$		−	12065.1i		−	0.00979767i
$274$		0			0
$275$	1.55560e6	+	186432.i	1.24042	+	0.148658i
$276$		0			0
$277$			2.13109e6i			1.66880i	0.551162	+	0.834398i	$0.314184\pi$
$277$			2.13109e6i			1.66880i	−0.551162	+	0.834398i	$0.685816\pi$
$278$		0			0
$279$	−218487.			−0.168041
$280$		0			0
$281$	−263911.			−0.199385			−0.0996924	−	0.995018i	$-0.531786\pi$
$281$	−263911.			−0.199385			−0.0996924	+	0.995018i	$0.531786\pi$
$282$		0			0
$283$		−	158115.i		−	0.117356i	−0.998277	−	0.0586782i	$-0.981311\pi$
$283$		−	158115.i		−	0.117356i	0.998277	−	0.0586782i	$-0.0186886\pi$
$284$		0			0
$285$	704625.	+	42072.6i	0.513861	+	0.0306822i
$286$		0			0
$287$			599034.i			0.429286i
$288$		0			0
$289$	−291355.			−0.205200
$290$		0			0
$291$	1.08474e6			0.750921
$292$		0			0
$293$		−	315168.i		−	0.214473i	−0.994234	−	0.107237i	$-0.965800\pi$
$293$		−	315168.i		−	0.214473i	0.994234	−	0.107237i	$-0.0342003\pi$
$294$		0			0
$295$	−19154.3	+	320794.i	−0.0128148	+	0.214620i
$296$		0			0
$297$			2.06749e6i			1.36004i
$298$		0			0
$299$	25402.4			0.0164322
$300$		0			0
$301$	−592242.			−0.376776
$302$		0			0
$303$			1.58795e6i			0.993643i
$304$		0			0
$305$	70500.4	−	1.18073e6i	0.0433952	−	0.726777i
$306$		0			0
$307$			406602.i			0.246220i	0.992393	+	0.123110i	$0.0392868\pi$
$307$			406602.i			0.246220i	−0.992393	+	0.123110i	$0.960713\pi$
$308$		0			0
$309$	2.29580e6			1.36785
$310$		0			0
$311$	555573.			0.325717			0.162859	−	0.986649i	$-0.447929\pi$
$311$	555573.			0.325717			0.162859	+	0.986649i	$0.447929\pi$
$312$		0			0
$313$			491547.i			0.283599i	0.989895	+	0.141799i	$0.0452888\pi$
$313$			491547.i			0.283599i	−0.989895	+	0.141799i	$0.954711\pi$
$314$		0			0
$315$	219113.	+	13083.1i	0.124421	+	0.00742905i
$316$		0			0
$317$			2.56154e6i			1.43170i	0.698253	+	0.715851i	$0.253961\pi$
$317$			2.56154e6i			1.43170i	−0.698253	+	0.715851i	$0.746039\pi$
$318$		0			0
$319$	2.25438e6			1.24036
$320$		0			0
$321$	−2.41072e6			−1.30582
$322$		0			0
$323$		−	1.29433e6i		−	0.690299i
$324$		0			0
$325$	−7174.55	+	59864.9i	−0.00376779	+	0.0314387i
$326$		0			0
$327$		−	1.95503e6i		−	1.01108i
$328$		0			0
$329$	−369076.			−0.187986
$330$		0			0
$331$	591755.			0.296874			0.148437	−	0.988922i	$-0.452576\pi$
$331$	591755.			0.296874			0.148437	+	0.988922i	$0.452576\pi$
$332$		0			0
$333$			77142.3i			0.0381225i
$334$		0			0
$335$	975692.	+	58257.8i	0.475008	+	0.0283623i
$336$		0			0
$337$			2.78632e6i			1.33646i	0.743954	+	0.668231i	$0.232948\pi$
$337$			2.78632e6i			1.33646i	−0.743954	+	0.668231i	$0.767052\pi$
$338$		0			0
$339$	1.81252e6			0.856611
$340$		0			0
$341$	1.36695e6			0.636598
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	55983.9	−	937610.i	0.0253230	−	0.424106i
$346$		0			0
$347$			387553.i			0.172786i	0.996261	+	0.0863929i	$0.0275340\pi$
$347$			387553.i			0.172786i	−0.996261	+	0.0863929i	$0.972466\pi$
$348$		0			0
$349$	325051.			0.142853			0.0714263	−	0.997446i	$-0.477245\pi$
$349$	325051.			0.142853			0.0714263	+	0.997446i	$0.477245\pi$
$350$		0			0
$351$	−79564.1			−0.0344706
$352$		0			0
$353$			828937.i			0.354067i	0.984205	+	0.177033i	$0.0566500\pi$
$353$			828937.i			0.354067i	−0.984205	+	0.177033i	$0.943350\pi$
$354$		0			0
$355$	−136295.	+	2.28264e6i	−0.0573996	+	0.961319i
$356$		0			0
$357$			818017.i			0.339697i
$358$		0			0
$359$	847241.			0.346953			0.173477	−	0.984838i	$-0.444500\pi$
$359$	847241.			0.346953			0.173477	+	0.984838i	$0.444500\pi$
$360$		0			0
$361$	−1.49710e6			−0.604620
$362$		0			0
$363$		−	1.15248e6i		−	0.459056i
$364$		0			0
$365$	1.04569e6	+	62437.2i	0.410838	+	0.0245308i
$366$		0			0
$367$			1.25278e6i			0.485522i	0.970086	+	0.242761i	$0.0780531\pi$
$367$			1.25278e6i			0.485522i	−0.970086	+	0.242761i	$0.921947\pi$
$368$		0			0
$369$	979661.			0.374550
$370$		0			0
$371$	2754.22			0.00103888
$372$		0			0
$373$			1.69236e6i			0.629828i	0.949120	+	0.314914i	$0.101976\pi$
$373$			1.69236e6i			0.629828i	−0.949120	+	0.314914i	$0.898024\pi$
$374$		0			0
$375$	2.19382e6	+	396750.i	0.805606	+	0.145693i
$376$		0			0
$377$			86756.0i			0.0314374i
$378$		0			0
$379$	−826577.			−0.295587			−0.147793	−	0.989018i	$-0.547217\pi$
$379$	−826577.			−0.295587			−0.147793	+	0.989018i	$0.547217\pi$
$380$		0			0
$381$	−1.09699e6			−0.387159
$382$		0			0
$383$			1.89387e6i			0.659709i	0.944032	+	0.329855i	$0.107000\pi$
$383$			1.89387e6i			0.659709i	−0.944032	+	0.329855i	$0.893000\pi$
$384$		0			0
$385$	−1.37086e6	−	81853.1i	−0.471349	−	0.0281438i
$386$		0			0
$387$			968552.i			0.328735i
$388$		0			0
$389$	4.37435e6			1.46568			0.732840	−	0.680401i	$-0.238194\pi$
$389$	4.37435e6			1.46568			0.732840	+	0.680401i	$0.238194\pi$
$390$		0			0
$391$	−1.72229e6			−0.569725
$392$		0			0
$393$			2.37506e6i			0.775699i
$394$		0			0
$395$	14160.4	−	237156.i	0.00456649	−	0.0764789i
$396$		0			0
$397$			2.59203e6i			0.825399i	0.910867	+	0.412699i	$0.135414\pi$
$397$			2.59203e6i			0.825399i	−0.910867	+	0.412699i	$0.864586\pi$
$398$		0			0
$399$	−618731.			−0.194567
$400$		0			0
$401$	−5.17451e6			−1.60697			−0.803487	−	0.595323i	$-0.797024\pi$
$401$	−5.17451e6			−1.60697			−0.803487	+	0.595323i	$0.797024\pi$
$402$		0			0
$403$			52604.7i			0.0161348i
$404$		0			0
$405$	−110469.	+	1.85011e6i	−0.0334658	+	0.560480i
$406$		0			0
$407$		−	482634.i		−	0.144422i
$408$		0			0
$409$	−5.89146e6			−1.74146			−0.870732	−	0.491758i	$-0.836354\pi$
$409$	−5.89146e6			−1.74146			−0.870732	+	0.491758i	$0.836354\pi$
$410$		0			0
$411$	802079.			0.234214
$412$		0			0
$413$		−	281689.i		−	0.0812633i
$414$		0			0
$415$	4.37892e6	+	261462.i	1.24809	+	0.0745226i
$416$		0			0
$417$			4.30367e6i			1.21199i
$418$		0			0
$419$	−5.58215e6			−1.55334			−0.776670	−	0.629908i	$-0.783093\pi$
$419$	−5.58215e6			−1.55334			−0.776670	+	0.629908i	$0.783093\pi$
$420$		0			0
$421$	−6.01010e6			−1.65263			−0.826317	−	0.563205i	$-0.809568\pi$
$421$	−6.01010e6			−1.65263			−0.826317	+	0.563205i	$0.809568\pi$
$422$		0			0
$423$			603587.i			0.164017i
$424$		0			0
$425$	486438.	−	4.05887e6i	0.130634	−	1.09002i
$426$		0			0
$427$			1.03680e6i			0.275185i
$428$		0			0
$429$	123447.			0.0323844
$430$		0			0
$431$	4.04322e6			1.04842			0.524209	−	0.851590i	$-0.324361\pi$
$431$	4.04322e6			1.04842			0.524209	+	0.851590i	$0.324361\pi$
$432$		0			0
$433$		−	2.28104e6i		−	0.584673i	−0.956316	−	0.292336i	$-0.905567\pi$
$433$		−	2.28104e6i		−	0.584673i	0.956316	−	0.292336i	$-0.0944327\pi$
$434$		0			0
$435$	3.20219e6	+	191200.i	0.811379	+	0.0484468i
$436$		0			0
$437$		−	1.30271e6i		−	0.326320i
$438$		0			0
$439$	4.80560e6			1.19011			0.595054	−	0.803686i	$-0.297131\pi$
$439$	4.80560e6			1.19011			0.595054	+	0.803686i	$0.297131\pi$
$440$		0			0
$441$	−192403.			−0.0471103
$442$		0			0
$443$		−	1.15791e6i		−	0.280327i	−0.990128	−	0.140163i	$-0.955237\pi$
$443$		−	1.15791e6i		−	0.280327i	0.990128	−	0.140163i	$-0.0447628\pi$
$444$		0			0
$445$	163856.	−	2.74424e6i	0.0392250	−	0.656934i
$446$		0			0
$447$			1.73008e6i			0.409541i
$448$		0			0
$449$	−2.34434e6			−0.548788			−0.274394	−	0.961617i	$-0.588477\pi$
$449$	−2.34434e6			−0.548788			−0.274394	+	0.961617i	$0.588477\pi$
$450$		0			0
$451$	−6.12916e6			−1.41893
$452$		0			0
$453$			1.25895e6i			0.288246i
$454$		0			0
$455$	3149.98	−	52755.5i	0.000713313	−	0.0119464i
$456$		0			0
$457$		−	2.17240e6i		−	0.486575i	−0.969954	−	0.243288i	$-0.921774\pi$
$457$		−	2.17240e6i		−	0.486575i	0.969954	−	0.243288i	$-0.0782259\pi$
$458$		0			0
$459$	5.39448e6			1.19514
$460$		0			0
$461$	−6.07435e6			−1.33121			−0.665606	−	0.746303i	$-0.731827\pi$
$461$	−6.07435e6			−1.33121			−0.665606	+	0.746303i	$0.731827\pi$
$462$		0			0
$463$			3.25155e6i			0.704917i	0.935827	+	0.352458i	$0.114654\pi$
$463$			3.25155e6i			0.704917i	−0.935827	+	0.352458i	$0.885346\pi$
$464$		0			0
$465$	1.94166e6	+	115935.i	0.416428	+	0.0248646i
$466$		0			0
$467$			3.96756e6i			0.841844i	0.907097	+	0.420922i	$0.138293\pi$
$467$			3.96756e6i			0.841844i	−0.907097	+	0.420922i	$0.861707\pi$
$468$		0			0
$469$	−856755.			−0.179856
$470$		0			0
$471$	−4.60132e6			−0.955720
$472$		0			0
$473$		−	6.05967e6i		−	1.24536i
$474$		0			0
$475$	3.07005e6	+	367932.i	0.624325	+	0.0748226i
$476$		0			0
$477$		−	4504.25i		−	0.000906413i
$478$		0			0
$479$	3.23390e6			0.644003			0.322002	−	0.946739i	$-0.395644\pi$
$479$	3.23390e6			0.644003			0.322002	+	0.946739i	$0.395644\pi$
$480$		0			0
$481$	18573.4			0.00366040
$482$		0			0
$483$			823315.i			0.160583i
$484$		0			0
$485$	4.74313e6	+	283208.i	0.915610	+	0.0546703i
$486$		0			0
$487$		−	9.62313e6i		−	1.83863i	−0.393523	−	0.919315i	$-0.628744\pi$
$487$		−	9.62313e6i		−	1.83863i	0.393523	−	0.919315i	$-0.371256\pi$
$488$		0			0
$489$	−4.31649e6			−0.816316
$490$		0			0
$491$	−8.52958e6			−1.59670			−0.798351	−	0.602193i	$-0.794294\pi$
$491$	−8.52958e6			−1.59670			−0.798351	+	0.602193i	$0.794294\pi$
$492$		0			0
$493$		−	5.88210e6i		−	1.08997i
$494$		0			0
$495$	−133863.	+	2.24191e6i	−0.0245553	+	0.411249i
$496$		0			0
$497$		−	2.00439e6i		−	0.363991i
$498$		0			0
$499$	7.08560e6			1.27387			0.636935	−	0.770917i	$-0.280202\pi$
$499$	7.08560e6			1.27387			0.636935	+	0.770917i	$0.280202\pi$
$500$		0			0
$501$	328024.			0.0583863
$502$		0			0
$503$		−	7.61574e6i		−	1.34212i	−0.741402	−	0.671061i	$-0.765839\pi$
$503$		−	7.61574e6i		−	1.34212i	0.741402	−	0.671061i	$-0.234161\pi$
$504$		0			0
$505$	−414587.	+	6.94344e6i	−0.0723415	+	1.21156i
$506$		0			0
$507$		−	4.73364e6i		−	0.817854i
$508$		0			0
$509$	9.17677e6			1.56998			0.784992	−	0.619506i	$-0.212667\pi$
$509$	9.17677e6			1.56998			0.784992	+	0.619506i	$0.212667\pi$
$510$		0			0
$511$	−918219.			−0.155559
$512$		0			0
$513$			4.08027e6i			0.684535i
$514$		0			0
$515$	1.00386e7	+	599396.i	1.66784	+	0.0995854i
$516$		0			0
$517$		−	3.77629e6i		−	0.621354i
$518$		0			0
$519$	−2.12783e6			−0.346752
$520$		0			0
$521$	−6.73255e6			−1.08664			−0.543320	−	0.839526i	$-0.682833\pi$
$521$	−6.73255e6			−1.08664			−0.543320	+	0.839526i	$0.682833\pi$
$522$		0			0
$523$			5.53809e6i			0.885331i	0.896687	+	0.442665i	$0.145967\pi$
$523$			5.53809e6i			0.885331i	−0.896687	+	0.442665i	$0.854033\pi$
$524$		0			0
$525$	−1.94028e6	−	232534.i	−0.307231	−	0.0368203i
$526$		0			0
$527$		−	3.56663e6i		−	0.559411i
$528$		0			0
$529$	4.70289e6			0.730678
$530$		0			0
$531$	−460674.			−0.0709018
$532$		0			0
$533$		−	235871.i		−	0.0359631i
$534$		0			0
$535$	−1.05411e7	−	629399.i	−1.59221	−	0.0950696i
$536$		0			0
$537$			4.24372e6i			0.635055i
$538$		0			0
$539$	1.20375e6			0.178470
$540$		0			0
$541$	1.08520e7			1.59410			0.797052	−	0.603911i	$-0.206392\pi$
$541$	1.08520e7			1.59410			0.797052	+	0.603911i	$0.206392\pi$
$542$		0			0
$543$		−	4.03125e6i		−	0.586733i
$544$		0			0
$545$	510426.	−	8.54854e6i	0.0736109	−	1.23282i
$546$		0			0
$547$		−	2.14366e6i		−	0.306328i	−0.988201	−	0.153164i	$-0.951054\pi$
$547$		−	2.14366e6i		−	0.306328i	0.988201	−	0.153164i	$-0.0489463\pi$
$548$		0			0
$549$	1.69558e6			0.240097
$550$		0			0
$551$	4.44910e6			0.624299
$552$		0			0
$553$			208247.i			0.0289578i
$554$		0			0
$555$	40933.6	−	685550.i	0.00564089	−	0.0944728i
$556$		0			0
$557$		−	1.19370e7i		−	1.63026i	−0.579277	−	0.815131i	$-0.696665\pi$
$557$		−	1.19370e7i		−	1.63026i	0.579277	−	0.815131i	$-0.303335\pi$
$558$		0			0
$559$	233197.			0.0315640
$560$		0			0
$561$	−8.36974e6			−1.12281
$562$		0			0
$563$			316392.i			0.0420683i	0.999779	+	0.0210341i	$0.00669587\pi$
$563$			316392.i			0.0420683i	−0.999779	+	0.0210341i	$0.993304\pi$
$564$		0			0
$565$	7.92539e6	+	473219.i	1.04448	+	0.0623650i
$566$		0			0
$567$		−	1.62458e6i		−	0.212219i
$568$		0			0
$569$	−1.05638e7			−1.36785			−0.683927	−	0.729551i	$-0.739729\pi$
$569$	−1.05638e7			−1.36785			−0.683927	+	0.729551i	$0.739729\pi$
$570$		0			0
$571$	1.04764e7			1.34469			0.672343	−	0.740240i	$-0.265288\pi$
$571$	1.04764e7			1.34469			0.672343	+	0.740240i	$0.265288\pi$
$572$		0			0
$573$		−	2.47497e6i		−	0.314908i
$574$		0			0
$575$	489588.	−	4.08516e6i	0.0617535	−	0.515275i
$576$		0			0
$577$		−	1.04055e6i		−	0.130114i	−0.997882	−	0.0650568i	$-0.979277\pi$
$577$		−	1.04055e6i		−	0.130114i	0.997882	−	0.0650568i	$-0.0207228\pi$
$578$		0			0
$579$	−3.30447e6			−0.409643
$580$		0			0
$581$	−3.84513e6			−0.472575
$582$		0			0
$583$			28180.4i			0.00343381i
$584$		0			0
$585$	−86276.3	−	5151.48i	−0.0104232	−	0.000622362i
$586$		0			0
$587$		−	7.32194e6i		−	0.877063i	−0.898716	−	0.438532i	$-0.855499\pi$
$587$		−	7.32194e6i		−	0.877063i	0.898716	−	0.438532i	$-0.144501\pi$
$588$		0			0
$589$	2.69772e6			0.320412
$590$		0			0
$591$	−1.32620e7			−1.56185
$592$		0			0
$593$		−	7.78789e6i		−	0.909459i	−0.890630	−	0.454729i	$-0.849736\pi$
$593$		−	7.78789e6i		−	0.909459i	0.890630	−	0.454729i	$-0.150264\pi$
$594$		0			0
$595$	−213571.	+	3.57685e6i	−0.0247314	+	0.414198i
$596$		0			0
$597$		−	2.18611e6i		−	0.251037i
$598$		0			0
$599$	3.61688e6			0.411876			0.205938	−	0.978565i	$-0.433975\pi$
$599$	3.61688e6			0.411876			0.205938	+	0.978565i	$0.433975\pi$
$600$		0			0
$601$	−1.00550e7			−1.13553			−0.567763	−	0.823192i	$-0.692191\pi$
$601$	−1.00550e7			−1.13553			−0.567763	+	0.823192i	$0.692191\pi$
$602$		0			0
$603$			1.40114e6i			0.156923i
$604$		0			0
$605$	300892.	−	5.03930e6i	0.0334212	−	0.559734i
$606$		0			0
$607$		−	4.67650e6i		−	0.515168i	−0.966256	−	0.257584i	$-0.917074\pi$
$607$		−	4.67650e6i		−	0.515168i	0.966256	−	0.257584i	$-0.0829265\pi$
$608$		0			0
$609$	−2.81184e6			−0.307219
$610$		0			0
$611$	145325.			0.0157484
$612$		0			0
$613$			8.08638e6i			0.869166i	0.900632	+	0.434583i	$0.143104\pi$
$613$			8.08638e6i			0.869166i	−0.900632	+	0.434583i	$0.856896\pi$
$614$		0			0
$615$	−8.70607e6	−	519832.i	−0.928185	−	0.0554211i
$616$		0			0
$617$			1.10272e7i			1.16615i	0.812419	+	0.583074i	$0.198150\pi$
$617$			1.10272e7i			1.16615i	−0.812419	+	0.583074i	$0.801850\pi$
$618$		0			0
$619$	3.65723e6			0.383641			0.191821	−	0.981430i	$-0.438561\pi$
$619$	3.65723e6			0.383641			0.191821	+	0.981430i	$0.438561\pi$
$620$		0			0
$621$	5.42942e6			0.564969
$622$		0			0
$623$			2.40972e6i			0.248740i
$624$		0			0
$625$	9.48907e6	+	2.30759e6i	0.971681	+	0.236297i
$626$		0			0
$627$		−	6.33070e6i		−	0.643106i
$628$		0			0
$629$	−1.25929e6			−0.126911
$630$		0			0
$631$	1.09862e7			1.09843			0.549215	−	0.835681i	$-0.314927\pi$
$631$	1.09862e7			1.09843			0.549215	+	0.835681i	$0.314927\pi$
$632$		0			0
$633$			7.33592e6i			0.727688i
$634$		0			0
$635$	−4.79667e6	−	286405.i	−0.472069	−	0.0281868i
$636$		0			0
$637$			46324.5i			0.00452338i
$638$		0			0
$639$	−3.27798e6			−0.317581
$640$		0			0
$641$	4.58921e6			0.441157			0.220578	−	0.975369i	$-0.429206\pi$
$641$	4.58921e6			0.441157			0.220578	+	0.975369i	$0.429206\pi$
$642$		0			0
$643$			1.99793e7i			1.90569i	0.303456	+	0.952845i	$0.401859\pi$
$643$			1.99793e7i			1.90569i	−0.303456	+	0.952845i	$0.598141\pi$
$644$		0			0
$645$	513938.	−	8.60735e6i	0.0486420	−	0.814649i
$646$		0			0
$647$		−	1.73498e7i		−	1.62942i	−0.579866	−	0.814712i	$-0.696895\pi$
$647$		−	1.73498e7i		−	1.62942i	0.579866	−	0.814712i	$-0.303105\pi$
$648$		0			0
$649$	2.88217e6			0.268601
$650$		0			0
$651$	−1.70497e6			−0.157675
$652$		0			0
$653$		−	1.62393e7i		−	1.49034i	−0.666875	−	0.745170i	$-0.732368\pi$
$653$		−	1.62393e7i		−	1.49034i	0.666875	−	0.745170i	$-0.267632\pi$
$654$		0			0
$655$	−620088.	+	1.03851e7i	−0.0564742	+	0.945821i
$656$		0			0
$657$			1.50166e6i			0.135724i
$658$		0			0
$659$	−7.37588e6			−0.661607			−0.330804	−	0.943700i	$-0.607320\pi$
$659$	−7.37588e6			−0.661607			−0.330804	+	0.943700i	$0.607320\pi$
$660$		0			0
$661$	4.46654e6			0.397620			0.198810	−	0.980038i	$-0.436292\pi$
$661$	4.46654e6			0.397620			0.198810	+	0.980038i	$0.436292\pi$
$662$		0			0
$663$		−	322096.i		−	0.0284578i
$664$		0			0
$665$	−2.70545e6	−	161540.i	−0.237239	−	0.0141653i
$666$		0			0
$667$		−	5.92020e6i		−	0.515254i
$668$		0			0
$669$	1.38732e7			1.19843
$670$		0			0
$671$	−1.06083e7			−0.909573
$672$		0			0
$673$		−	507076.i		−	0.0431554i	−0.999767	−	0.0215777i	$-0.993131\pi$
$673$		−	507076.i		−	0.0431554i	0.999767	−	0.0215777i	$-0.00686894\pi$
$674$		0			0
$675$	−1.53346e6	+	1.27953e7i	−0.129543	+	1.08092i
$676$		0			0
$677$			6.96739e6i			0.584250i	0.956380	+	0.292125i	$0.0943623\pi$
$677$			6.96739e6i			0.584250i	−0.956380	+	0.292125i	$0.905638\pi$
$678$		0			0
$679$	−4.16494e6			−0.346684
$680$		0			0
$681$	3.20098e6			0.264494
$682$		0			0
$683$		−	8.59041e6i		−	0.704632i	−0.935881	−	0.352316i	$-0.885394\pi$
$683$		−	8.59041e6i		−	0.704632i	0.935881	−	0.352316i	$-0.114606\pi$
$684$		0			0
$685$	3.50715e6	+	209409.i	0.285580	+	0.0170518i
$686$		0			0
$687$		−	1.01969e7i		−	0.824286i
$688$		0			0
$689$	−1084.48			−8.70308e−5
$690$		0			0
$691$	−2.24356e7			−1.78749			−0.893743	−	0.448578i	$-0.851930\pi$
$691$	−2.24356e7			−1.78749			−0.893743	+	0.448578i	$0.851930\pi$
$692$		0			0
$693$		−	1.96862e6i		−	0.155714i
$694$		0			0
$695$	−1.12362e6	+	1.88181e7i	−0.0882380	+	1.47780i
$696$		0			0
$697$			1.59922e7i			1.24688i
$698$		0			0
$699$	−1.47767e7			−1.14389
$700$		0			0
$701$	1.99956e7			1.53688			0.768440	−	0.639921i	$-0.221033\pi$
$701$	1.99956e7			1.53688			0.768440	+	0.639921i	$0.221033\pi$
$702$		0			0
$703$		−	952498.i		−	0.0726901i
$704$		0			0
$705$	320278.	−	5.36397e6i	0.0242692	−	0.406456i
$706$		0			0
$707$		−	6.09703e6i		−	0.458744i
$708$		0			0
$709$	−5.91142e6			−0.441648			−0.220824	−	0.975314i	$-0.570875\pi$
$709$	−5.91142e6			−0.441648			−0.220824	+	0.975314i	$0.570875\pi$
$710$		0			0
$711$	340567.			0.0252655
$712$		0			0
$713$		−	3.58973e6i		−	0.264446i
$714$		0			0
$715$	539780.	+	32229.8i	0.0394868	+	0.00235772i
$716$		0			0
$717$			2.21483e7i			1.60895i
$718$		0			0
$719$	2.07736e7			1.49861			0.749306	−	0.662224i	$-0.230387\pi$
$719$	2.07736e7			1.49861			0.749306	+	0.662224i	$0.230387\pi$
$720$		0			0
$721$	−8.81488e6			−0.631507
$722$		0			0
$723$			2.18474e7i			1.55437i
$724$		0			0
$725$	1.39519e7	+	1.67208e6i	0.985800	+	0.118144i
$726$		0			0
$727$			1.11990e7i			0.785860i	0.919568	+	0.392930i	$0.128538\pi$
$727$			1.11990e7i			0.785860i	−0.919568	+	0.392930i	$0.871462\pi$
$728$		0			0
$729$	−1.54453e7			−1.07641
$730$		0			0
$731$	−1.58108e7			−1.09436
$732$		0			0
$733$		−	2.06164e7i		−	1.41727i	−0.705573	−	0.708637i	$-0.749310\pi$
$733$		−	2.06164e7i		−	1.41727i	0.705573	−	0.708637i	$-0.250690\pi$
$734$		0			0
$735$	1.70985e6	+	102094.i	0.116746	+	0.00697078i
$736$		0			0
$737$		−	8.76610e6i		−	0.594480i
$738$		0			0
$739$	2.32124e7			1.56354			0.781771	−	0.623566i	$-0.214317\pi$
$739$	2.32124e7			1.56354			0.781771	+	0.623566i	$0.214317\pi$
$740$		0			0
$741$	243627.			0.0162997
$742$		0			0
$743$		−	2.38953e7i		−	1.58796i	−0.607943	−	0.793981i	$-0.708005\pi$
$743$		−	2.38953e7i		−	1.58796i	0.607943	−	0.793981i	$-0.291995\pi$
$744$		0			0
$745$	−451694.	+	7.56491e6i	−0.0298163	+	0.499359i
$746$		0			0
$747$			6.28832e6i			0.412319i
$748$		0			0
$749$	9.25612e6			0.602871
$750$		0			0
$751$	8.66074e6			0.560345			0.280172	−	0.959950i	$-0.409608\pi$
$751$	8.66074e6			0.560345			0.280172	+	0.959950i	$0.409608\pi$
$752$		0			0
$753$			1.90646e7i			1.22530i
$754$		0			0
$755$	−328691.	+	5.50487e6i	−0.0209856	+	0.351463i
$756$		0			0
$757$			2.49562e7i			1.58285i	0.611269	+	0.791423i	$0.290659\pi$
$757$			2.49562e7i			1.58285i	−0.611269	+	0.791423i	$0.709341\pi$
$758$		0			0
$759$	−8.42395e6			−0.530776
$760$		0			0
$761$	−1.36077e6			−0.0851770			−0.0425885	−	0.999093i	$-0.513560\pi$
$761$	−1.36077e6			−0.0851770			−0.0425885	+	0.999093i	$0.513560\pi$
$762$		0			0
$763$			7.50647e6i			0.466793i
$764$		0			0
$765$	5.84957e6	+	349273.i	0.361385	+	0.0215780i
$766$		0			0
$767$			110916.i			0.00680776i
$768$		0			0
$769$	−2.11438e7			−1.28934			−0.644670	−	0.764461i	$-0.723005\pi$
$769$	−2.11438e7			−1.28934			−0.644670	+	0.764461i	$0.723005\pi$
$770$		0			0
$771$	−2.60606e7			−1.57888
$772$		0			0
$773$			1.59665e7i			0.961085i	0.876972	+	0.480542i	$0.159560\pi$
$773$			1.59665e7i			0.961085i	−0.876972	+	0.480542i	$0.840440\pi$
$774$		0			0
$775$	8.45978e6	+	1.01387e6i	0.505947	+	0.0606355i
$776$		0			0
$777$			601981.i			0.0357709i
$778$		0			0
$779$	−1.20961e7			−0.714173
$780$		0			0
$781$	2.05084e7			1.20311
$782$		0			0
$783$			1.85429e7i			1.08087i
$784$		0			0
$785$	−2.01197e7	−	1.20133e6i	−1.16532	−	0.0695805i
$786$		0			0
$787$		−	5.28802e6i		−	0.304338i	−0.988354	−	0.152169i	$-0.951374\pi$
$787$		−	5.28802e6i		−	0.304338i	0.988354	−	0.152169i	$-0.0486258\pi$
$788$		0			0
$789$	8.57347e6			0.490302
$790$		0			0
$791$	−6.95929e6			−0.395479
$792$		0			0
$793$		−	408242.i		−	0.0230534i
$794$		0			0
$795$	−2390.07	+	40028.5i	−0.000134120	+	0.00224621i
$796$		0			0
$797$			2.37918e7i			1.32672i	0.748299	+	0.663362i	$0.230871\pi$
$797$			2.37918e7i			1.32672i	−0.748299	+	0.663362i	$0.769129\pi$
$798$		0			0
$799$	−9.85308e6			−0.546016
$800$		0			0
$801$	3.94085e6			0.217024
$802$		0			0
$803$		−	9.39498e6i		−	0.514171i
$804$		0			0
$805$	−214954.	+	3.60001e6i	−0.0116911	+	0.195801i
$806$		0			0
$807$			1.26805e7i			0.685413i
$808$		0			0
$809$	2.52592e7			1.35690			0.678452	−	0.734645i	$-0.262651\pi$
$809$	2.52592e7			1.35690			0.678452	+	0.734645i	$0.262651\pi$
$810$		0			0
$811$	1.31418e7			0.701624			0.350812	−	0.936446i	$-0.385906\pi$
$811$	1.31418e7			0.701624			0.350812	+	0.936446i	$0.385906\pi$
$812$		0			0
$813$			2.67538e7i			1.41958i
$814$		0			0
$815$	−1.88742e7	−	1.12696e6i	−0.995347	−	0.0594313i
$816$		0			0
$817$		−	1.19590e7i		−	0.626815i
$818$		0			0
$819$	75759.2			0.00394662
$820$		0			0
$821$	3.67102e7			1.90077			0.950385	−	0.311077i	$-0.100690\pi$
$821$	3.67102e7			1.90077			0.950385	+	0.311077i	$0.100690\pi$
$822$		0			0
$823$			1.71569e7i			0.882955i	0.897272	+	0.441478i	$0.145545\pi$
$823$			1.71569e7i			0.882955i	−0.897272	+	0.441478i	$0.854455\pi$
$824$		0			0
$825$	2.37923e6	−	1.98524e7i	0.121703	−	1.01550i
$826$		0			0
$827$		−	3.78073e7i		−	1.92226i	−0.276097	−	0.961130i	$-0.589041\pi$
$827$		−	3.78073e7i		−	1.92226i	0.276097	−	0.961130i	$-0.410959\pi$
$828$		0			0
$829$	−5.24151e6			−0.264893			−0.132446	−	0.991190i	$-0.542283\pi$
$829$	−5.24151e6			−0.264893			−0.132446	+	0.991190i	$0.542283\pi$
$830$		0			0
$831$	2.71968e7			1.36620
$832$		0			0
$833$		−	3.14083e6i		−	0.156831i
$834$		0			0
$835$	1.43431e6	+	85641.4i	0.0711913	+	0.00425077i
$836$		0			0
$837$			1.12436e7i			0.554741i
$838$		0			0
$839$	−1.11536e7			−0.547028			−0.273514	−	0.961868i	$-0.588186\pi$
$839$	−1.11536e7			−0.547028			−0.273514	+	0.961868i	$0.588186\pi$
$840$		0			0
$841$	−292087.			−0.0142404
$842$		0			0
$843$			3.36800e6i			0.163231i
$844$		0			0
$845$	1.23587e6	−	2.06982e7i	0.0595433	−	0.997221i
$846$		0			0
$847$			4.42500e6i			0.211936i
$848$		0			0
$849$	−2.01784e6			−0.0960767
$850$		0			0
$851$	−1.26744e6			−0.0599935
$852$		0			0
$853$		−	1.58246e7i		−	0.744664i	−0.928100	−	0.372332i	$-0.878558\pi$
$853$		−	1.58246e7i		−	0.744664i	0.928100	−	0.372332i	$-0.121442\pi$
$854$		0			0
$855$	−264183.	+	4.42450e6i	−0.0123592	+	0.206990i
$856$		0			0
$857$			2.01607e7i			0.937679i	0.883283	+	0.468839i	$0.155328\pi$
$857$			2.01607e7i			0.937679i	−0.883283	+	0.468839i	$0.844672\pi$
$858$		0			0
$859$	3.27621e7			1.51492			0.757458	−	0.652883i	$-0.226441\pi$
$859$	3.27621e7			1.51492			0.757458	+	0.652883i	$0.226441\pi$
$860$		0			0
$861$	7.64480e6			0.351446
$862$		0			0
$863$		−	7.18471e6i		−	0.328384i	−0.986428	−	0.164192i	$-0.947498\pi$
$863$		−	7.18471e6i		−	0.328384i	0.986428	−	0.164192i	$-0.0525017\pi$
$864$		0			0
$865$	−9.30411e6	−	555541.i	−0.422800	−	0.0252450i
$866$		0			0
$867$			3.71824e6i			0.167992i
$868$		0			0
$869$	−2.13073e6			−0.0957147
$870$		0			0
$871$	337349.			0.0150673
$872$		0			0
$873$			6.81134e6i			0.302480i
$874$		0			0
$875$	−8.42331e6	−	1.52335e6i	−0.371931	−	0.0672634i
$876$		0			0
$877$			7.95215e6i			0.349129i	0.984646	+	0.174564i	$0.0558517\pi$
$877$			7.95215e6i			0.349129i	−0.984646	+	0.174564i	$0.944148\pi$
$878$		0			0
$879$	−4.02214e6			−0.175584
$880$		0			0
$881$	1.79413e7			0.778779			0.389389	−	0.921073i	$-0.372686\pi$
$881$	1.79413e7			0.778779			0.389389	+	0.921073i	$0.372686\pi$
$882$		0			0
$883$			6.00912e6i			0.259364i	0.991556	+	0.129682i	$0.0413956\pi$
$883$			6.00912e6i			0.259364i	−0.991556	+	0.129682i	$0.958604\pi$
$884$		0			0
$885$	4.09393e6	+	244445.i	0.175704	+	0.0104912i
$886$		0			0
$887$			1.90391e7i			0.812526i	0.913756	+	0.406263i	$0.133168\pi$
$887$			1.90391e7i			0.812526i	−0.913756	+	0.406263i	$0.866832\pi$
$888$		0			0
$889$	4.21195e6			0.178743
$890$		0			0
$891$	1.66223e7			0.701451
$892$		0			0
$893$		−	7.45266e6i		−	0.312740i
$894$		0			0
$895$	−1.10796e6	+	1.85560e7i	−0.0462347	+	0.774332i
$896$		0			0
$897$		−	324182.i		−	0.0134527i
$898$		0			0
$899$	1.22599e7			0.505926
$900$		0			0
$901$	73528.2			0.00301746
$902$		0			0
$903$			7.55812e6i			0.308457i
$904$		0			0
$905$	1.05249e6	−	1.76270e7i	0.0427167	−	0.715412i
$906$		0			0
$907$			3.04513e7i			1.22910i	0.788877	+	0.614551i	$0.210663\pi$
$907$			3.04513e7i			1.22910i	−0.788877	+	0.614551i	$0.789337\pi$
$908$		0			0
$909$	−9.97109e6			−0.400251
$910$		0			0
$911$	1.10797e7			0.442314			0.221157	−	0.975238i	$-0.429017\pi$
$911$	1.10797e7			0.442314			0.221157	+	0.975238i	$0.429017\pi$
$912$		0			0
$913$		−	3.93424e7i		−	1.56201i
$914$		0			0
$915$	−1.50683e7	−	899717.i	−0.594993	−	0.0355266i
$916$		0			0
$917$		−	9.11919e6i		−	0.358124i
$918$		0			0
$919$	−4.47897e7			−1.74940			−0.874700	−	0.484664i	$-0.838942\pi$
$919$	−4.47897e7			−1.74940			−0.874700	+	0.484664i	$0.838942\pi$
$920$		0			0
$921$	5.18900e6			0.201574
$922$		0			0
$923$			789233.i			0.0304930i
$924$		0			0
$925$	357971.	−	2.98693e6i	0.0137560	−	0.114781i
$926$		0			0
$927$			1.44159e7i			0.550987i
$928$		0			0
$929$	−2.61361e7			−0.993576			−0.496788	−	0.867872i	$-0.665487\pi$
$929$	−2.61361e7			−0.993576			−0.496788	+	0.867872i	$0.665487\pi$
$930$		0			0
$931$	2.37566e6			0.0898275
$932$		0			0
$933$		−	7.09016e6i		−	0.266656i
$934$		0			0
$935$	−3.65974e7	−	2.18520e6i	−1.36905	−	0.0817451i
$936$		0			0
$937$			3.85137e7i			1.43306i	0.697554	+	0.716532i	$0.254272\pi$
$937$			3.85137e7i			1.43306i	−0.697554	+	0.716532i	$0.745728\pi$
$938$		0			0
$939$	6.27306e6			0.232175
$940$		0			0
$941$	−1.19244e7			−0.439000			−0.219500	−	0.975613i	$-0.570443\pi$
$941$	−1.19244e7			−0.439000			−0.219500	+	0.975613i	$0.570443\pi$
$942$		0			0
$943$			1.60957e7i			0.589429i
$944$		0			0
$945$	673267.	−	1.12758e7i	0.0245249	−	0.410740i
$946$		0			0
$947$		−	4.18411e7i		−	1.51610i	−0.652195	−	0.758051i	$-0.726152\pi$
$947$		−	4.18411e7i		−	1.51610i	0.652195	−	0.758051i	$-0.273848\pi$
$948$		0			0
$949$	361551.			0.0130318
$950$		0			0
$951$	3.26900e7			1.17210
$952$		0			0
$953$		−	1.62515e7i		−	0.579645i	−0.957080	−	0.289822i	$-0.906404\pi$
$953$		−	1.62515e7i		−	0.579645i	0.957080	−	0.289822i	$-0.0935962\pi$
$954$		0			0
$955$	646173.	−	1.08220e7i	0.0229267	−	0.383972i
$956$		0			0
$957$		−	2.87700e7i		−	1.01546i
$958$		0			0
$959$	−3.07963e6			−0.108131
$960$		0			0
$961$	−2.11953e7			−0.740341
$962$		0			0
$963$		−	1.51375e7i		−	0.526002i
$964$		0			0
$965$	−1.44491e7	−	862742.i	−0.499484	−	0.0298238i
$966$		0			0
$967$		−	3.10364e7i		−	1.06735i	−0.845690	−	0.533674i	$-0.820811\pi$
$967$		−	3.10364e7i		−	1.06735i	0.845690	−	0.533674i	$-0.179189\pi$
$968$		0			0
$969$	−1.65180e7			−0.565130
$970$		0			0
$971$	6.62731e6			0.225574			0.112787	−	0.993619i	$-0.464022\pi$
$971$	6.62731e6			0.225574			0.112787	+	0.993619i	$0.464022\pi$
$972$		0			0
$973$		−	1.65242e7i		−	0.559550i
$974$		0			0
$975$	763988.	+	91560.7i	0.0257380	+	0.00308459i
$976$		0			0
$977$		−	3.40946e7i		−	1.14275i	−0.820691	−	0.571373i	$-0.806411\pi$
$977$		−	3.40946e7i		−	1.14275i	0.820691	−	0.571373i	$-0.193589\pi$
$978$		0			0
$979$	−2.46556e7			−0.822164
$980$		0			0
$981$	1.22761e7			0.407275
$982$		0			0
$983$		−	3.70245e6i		−	0.122210i	−0.998131	−	0.0611048i	$-0.980538\pi$
$983$		−	3.70245e6i		−	0.122210i	0.998131	−	0.0611048i	$-0.0194624\pi$
$984$		0			0
$985$	−5.79891e7	−	3.46248e6i	−1.90439	−	0.113710i
$986$		0			0
$987$			4.71010e6i			0.153900i
$988$		0			0
$989$	−1.59132e7			−0.517330
$990$		0			0
$991$	1.07105e7			0.346438			0.173219	−	0.984883i	$-0.444583\pi$
$991$	1.07105e7			0.346438			0.173219	+	0.984883i	$0.444583\pi$
$992$		0			0
$993$		−	7.55191e6i		−	0.243043i
$994$		0			0
$995$	570758.	−	9.55896e6i	0.0182765	−	0.306093i
$996$		0			0
$997$			2.31495e7i			0.737572i	0.929514	+	0.368786i	$0.120226\pi$
$997$			2.31495e7i			0.737572i	−0.929514	+	0.368786i	$0.879774\pi$
$998$		0			0
$999$	3.96982e6			0.125851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.g.a.169.7	✓	20
4.3	odd	2		560.6.g.g.449.14		20
5.4	even	2	inner	280.6.g.a.169.14	yes	20
20.19	odd	2		560.6.g.g.449.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.g.a.169.7	✓	20	1.1	even	1	trivial
280.6.g.a.169.14	yes	20	5.4	even	2	inner
560.6.g.g.449.7		20	20.19	odd	2
560.6.g.g.449.14		20	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.7619i q^{3} +(3.33191 - 55.8023i) q^{5} +49.0000i q^{7} +80.1346 q^{9} -501.355 q^{11} -19.2939i q^{13} +(-712.142 - 42.5214i) q^{15} +1308.13i q^{17} -989.445 q^{19} +625.332 q^{21} +1316.61i q^{23} +(-3102.80 - 371.857i) q^{25} -4123.80i q^{27} -4496.56 q^{29} -2726.50 q^{31} +6398.23i q^{33} +(2734.31 + 163.264i) q^{35} +962.659i q^{37} -246.226 q^{39} +12225.2 q^{41} +12086.6i q^{43} +(267.001 - 4471.70i) q^{45} +7532.17i q^{47} -2401.00 q^{49} +16694.2 q^{51} -56.2085i q^{53} +(-1670.47 + 27976.8i) q^{55} +12627.2i q^{57} -5748.75 q^{59} +21159.2 q^{61} +3926.60i q^{63} +(-1076.64 - 64.2854i) q^{65} +17484.8i q^{67} +16802.3 q^{69} -40905.9 q^{71} +18739.2i q^{73} +(-4745.59 + 39597.5i) q^{75} -24566.4i q^{77} +4249.94 q^{79} -33154.7 q^{81} +78472.0i q^{83} +(72996.9 + 4358.58i) q^{85} +57384.5i q^{87} +49177.9 q^{89} +945.399 q^{91} +34795.3i q^{93} +(-3296.74 + 55213.3i) q^{95} +84998.7i q^{97} -40175.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51}+ \cdots - 630244 q^{99}+O(q^{100}) \) 20 * q + 14 * q^5 - 1134 * q^9 + 822 * q^11 + 1322 * q^15 + 3000 * q^19 - 882 * q^21 - 1944 * q^25 + 10406 * q^29 - 8532 * q^31 - 2058 * q^35 + 71066 * q^39 - 28880 * q^41 - 3922 * q^45 - 48020 * q^49 + 34770 * q^51 - 8844 * q^55 + 61196 * q^59 - 145052 * q^61 - 12722 * q^65 + 64980 * q^69 + 41464 * q^71 - 159032 * q^75 + 97174 * q^79 - 165852 * q^81 + 319566 * q^85 - 84604 * q^89 + 61642 * q^91 + 33416 * q^95 - 630244 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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