LMFDB - Embedded newform 280.6.g.a.169.9

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			169.9
Root			$-5.80939i$ of defining polynomial
Character	$\chi$	$=$	280.169
Dual form			280.6.g.a.169.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.80939i q^{3} +(49.5511 + 25.8783i) q^{5} -49.0000i q^{7} +209.251 q^{9} +415.954 q^{11} -419.172i q^{13} +(150.337 - 287.862i) q^{15} +47.1504i q^{17} -1300.43 q^{19} -284.660 q^{21} +887.588i q^{23} +(1785.63 + 2564.60i) q^{25} -2627.30i q^{27} +7316.18 q^{29} -3107.24 q^{31} -2416.44i q^{33} +(1268.04 - 2428.00i) q^{35} +3814.15i q^{37} -2435.13 q^{39} +2049.14 q^{41} +8664.30i q^{43} +(10368.6 + 5415.06i) q^{45} -12342.9i q^{47} -2401.00 q^{49} +273.915 q^{51} -12300.2i q^{53} +(20611.0 + 10764.2i) q^{55} +7554.69i q^{57} -11957.5 q^{59} +16409.6 q^{61} -10253.3i q^{63} +(10847.5 - 20770.4i) q^{65} -28933.1i q^{67} +5156.34 q^{69} +65207.9 q^{71} -54723.1i q^{73} +(14898.7 - 10373.4i) q^{75} -20381.7i q^{77} +27902.0 q^{79} +35585.0 q^{81} -93273.4i q^{83} +(-1220.17 + 2336.36i) q^{85} -42502.5i q^{87} +127515. q^{89} -20539.4 q^{91} +18051.1i q^{93} +(-64437.7 - 33652.9i) q^{95} +102289. i q^{97} +87038.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	5.80939i		−	0.372672i	−0.982486	−	0.186336i	$-0.940339\pi$
$3$		−	5.80939i		−	0.372672i	0.982486	−	0.186336i	$-0.0596614\pi$
$4$		0			0
$5$	49.5511	+	25.8783i	0.886397	+	0.462925i
$5$	49.5511	+	25.8783i	0.886397	+	0.462925i
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	209.251			0.861115
$10$		0			0
$11$	415.954			1.03649			0.518243	−	0.855233i	$-0.326586\pi$
$11$	415.954			1.03649			0.518243	+	0.855233i	$0.326586\pi$
$12$		0			0
$13$		−	419.172i		−	0.687913i	−0.938986	−	0.343957i	$-0.888233\pi$
$13$		−	419.172i		−	0.687913i	0.938986	−	0.343957i	$-0.111767\pi$
$14$		0			0
$15$	150.337	−	287.862i	0.172519	−	0.330336i
$16$		0			0
$17$			47.1504i			0.0395698i	0.999804	+	0.0197849i	$0.00629813\pi$
$17$			47.1504i			0.0395698i	−0.999804	+	0.0197849i	$0.993702\pi$
$18$		0			0
$19$	−1300.43			−0.826423			−0.413212	−	0.910635i	$-0.635593\pi$
$19$	−1300.43			−0.826423			−0.413212	+	0.910635i	$0.635593\pi$
$20$		0			0
$21$	−284.660			−0.140857
$22$		0			0
$23$			887.588i			0.349858i	0.984581	+	0.174929i	$0.0559696\pi$
$23$			887.588i			0.349858i	−0.984581	+	0.174929i	$0.944030\pi$
$24$		0			0
$25$	1785.63	+	2564.60i	0.571400	+	0.820671i
$26$		0			0
$27$		−	2627.30i		−	0.693586i
$28$		0			0
$29$	7316.18			1.61543			0.807717	−	0.589571i	$-0.200703\pi$
$29$	7316.18			1.61543			0.807717	+	0.589571i	$0.200703\pi$
$30$		0			0
$31$	−3107.24			−0.580724			−0.290362	−	0.956917i	$-0.593776\pi$
$31$	−3107.24			−0.580724			−0.290362	+	0.956917i	$0.593776\pi$
$32$		0			0
$33$		−	2416.44i		−	0.386270i
$34$		0			0
$35$	1268.04	−	2428.00i	0.174969	−	0.335027i
$36$		0			0
$37$			3814.15i			0.458029i	0.973423	+	0.229015i	$0.0735503\pi$
$37$			3814.15i			0.458029i	−0.973423	+	0.229015i	$0.926450\pi$
$38$		0			0
$39$	−2435.13			−0.256366
$40$		0			0
$41$	2049.14			0.190376			0.0951878	−	0.995459i	$-0.469655\pi$
$41$	2049.14			0.190376			0.0951878	+	0.995459i	$0.469655\pi$
$42$		0			0
$43$			8664.30i			0.714599i	0.933990	+	0.357300i	$0.116302\pi$
$43$			8664.30i			0.714599i	−0.933990	+	0.357300i	$0.883698\pi$
$44$		0			0
$45$	10368.6	+	5415.06i	0.763290	+	0.398632i
$46$		0			0
$47$		−	12342.9i		−	0.815030i	−0.913199	−	0.407515i	$-0.866395\pi$
$47$		−	12342.9i		−	0.815030i	0.913199	−	0.407515i	$-0.133605\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	273.915			0.0147466
$52$		0			0
$53$		−	12300.2i		−	0.601481i	−0.953706	−	0.300741i	$-0.902766\pi$
$53$		−	12300.2i		−	0.601481i	0.953706	−	0.300741i	$-0.0972339\pi$
$54$		0			0
$55$	20611.0	+	10764.2i	0.918739	+	0.479816i
$56$		0			0
$57$			7554.69i			0.307985i
$58$		0			0
$59$	−11957.5			−0.447208			−0.223604	−	0.974680i	$-0.571782\pi$
$59$	−11957.5			−0.447208			−0.223604	+	0.974680i	$0.571782\pi$
$60$		0			0
$61$	16409.6			0.564641			0.282320	−	0.959320i	$-0.408896\pi$
$61$	16409.6			0.564641			0.282320	+	0.959320i	$0.408896\pi$
$62$		0			0
$63$		−	10253.3i		−	0.325471i
$64$		0			0
$65$	10847.5	−	20770.4i	0.318452	−	0.609765i
$66$		0			0
$67$		−	28933.1i		−	0.787423i	−0.919234	−	0.393711i	$-0.871191\pi$
$67$		−	28933.1i		−	0.787423i	0.919234	−	0.393711i	$-0.128809\pi$
$68$		0			0
$69$	5156.34			0.130382
$70$		0			0
$71$	65207.9			1.53516			0.767581	−	0.640952i	$-0.221460\pi$
$71$	65207.9			1.53516			0.767581	+	0.640952i	$0.221460\pi$
$72$		0			0
$73$		−	54723.1i		−	1.20189i	−0.799292	−	0.600943i	$-0.794792\pi$
$73$		−	54723.1i		−	1.20189i	0.799292	−	0.600943i	$-0.205208\pi$
$74$		0			0
$75$	14898.7	−	10373.4i	0.305842	−	0.212945i
$76$		0			0
$77$		−	20381.7i		−	0.391755i
$78$		0			0
$79$	27902.0			0.503000			0.251500	−	0.967857i	$-0.419076\pi$
$79$	27902.0			0.503000			0.251500	+	0.967857i	$0.419076\pi$
$80$		0			0
$81$	35585.0			0.602635
$82$		0			0
$83$		−	93273.4i		−	1.48615i	−0.669208	−	0.743075i	$-0.733367\pi$
$83$		−	93273.4i		−	1.48615i	0.669208	−	0.743075i	$-0.266633\pi$
$84$		0			0
$85$	−1220.17	+	2336.36i	−0.0183178	+	0.0350745i
$86$		0			0
$87$		−	42502.5i		−	0.602028i
$88$		0			0
$89$	127515.			1.70642			0.853208	−	0.521571i	$-0.174654\pi$
$89$	127515.			1.70642			0.853208	+	0.521571i	$0.174654\pi$
$90$		0			0
$91$	−20539.4			−0.260007
$92$		0			0
$93$			18051.1i			0.216420i
$94$		0			0
$95$	−64437.7	−	33652.9i	−0.732539	−	0.382572i
$96$		0			0
$97$			102289.i			1.10382i	0.833903	+	0.551911i	$0.186101\pi$
$97$			102289.i			1.10382i	−0.833903	+	0.551911i	$0.813899\pi$
$98$		0			0
$99$	87038.8			0.892534
$100$		0			0
$101$	30278.7			0.295348			0.147674	−	0.989036i	$-0.452821\pi$
$101$	30278.7			0.295348			0.147674	+	0.989036i	$0.452821\pi$
$102$		0			0
$103$			140859.i			1.30825i	0.756387	+	0.654124i	$0.226963\pi$
$103$			140859.i			1.30825i	−0.756387	+	0.654124i	$0.773037\pi$
$104$		0			0
$105$	−14105.2	−	7366.52i	−0.124855	−	0.0652062i
$106$		0			0
$107$		−	56163.8i		−	0.474238i	−0.971481	−	0.237119i	$-0.923797\pi$
$107$		−	56163.8i		−	0.474238i	0.971481	−	0.237119i	$-0.0762032\pi$
$108$		0			0
$109$	20718.4			0.167028			0.0835141	−	0.996507i	$-0.473386\pi$
$109$	20718.4			0.167028			0.0835141	+	0.996507i	$0.473386\pi$
$110$		0			0
$111$	22157.9			0.170695
$112$		0			0
$113$			75128.2i			0.553487i	0.960944	+	0.276743i	$0.0892552\pi$
$113$			75128.2i			0.553487i	−0.960944	+	0.276743i	$0.910745\pi$
$114$		0			0
$115$	−22969.3	+	43981.0i	−0.161958	+	0.310113i
$116$		0			0
$117$		−	87712.1i		−	0.592373i
$118$		0			0
$119$	2310.37			0.0149560
$120$		0			0
$121$	11966.7			0.0743036
$122$		0			0
$123$		−	11904.2i		−	0.0709477i
$124$		0			0
$125$	22112.3	+	173288.i	0.126578	+	0.991957i
$126$		0			0
$127$		−	229939.i		−	1.26504i	−0.774545	−	0.632518i	$-0.782021\pi$
$127$		−	229939.i		−	1.26504i	0.774545	−	0.632518i	$-0.217979\pi$
$128$		0			0
$129$	50334.3			0.266311
$130$		0			0
$131$	139960.			0.712569			0.356285	−	0.934378i	$-0.384043\pi$
$131$	139960.			0.712569			0.356285	+	0.934378i	$0.384043\pi$
$132$		0			0
$133$			63721.0i			0.312359i
$134$		0			0
$135$	67990.1	−	130186.i	0.321079	−	0.614793i
$136$		0			0
$137$			38939.8i			0.177253i	0.996065	+	0.0886263i	$0.0282477\pi$
$137$			38939.8i			0.177253i	−0.996065	+	0.0886263i	$0.971752\pi$
$138$		0			0
$139$	−1146.59			−0.00503351			−0.00251676	−	0.999997i	$-0.500801\pi$
$139$	−1146.59			−0.00503351			−0.00251676	+	0.999997i	$0.500801\pi$
$140$		0			0
$141$	−71704.9			−0.303739
$142$		0			0
$143$		−	174356.i		−	0.713013i
$144$		0			0
$145$	362525.	+	189330.i	1.43192	+	0.747825i
$146$		0			0
$147$			13948.3i			0.0532389i
$148$		0			0
$149$	−95068.0			−0.350807			−0.175404	−	0.984497i	$-0.556123\pi$
$149$	−95068.0			−0.350807			−0.175404	+	0.984497i	$0.556123\pi$
$150$		0			0
$151$	−426249.			−1.52132			−0.760661	−	0.649150i	$-0.775125\pi$
$151$	−426249.			−1.52132			−0.760661	+	0.649150i	$0.775125\pi$
$152$		0			0
$153$			9866.28i			0.0340741i
$154$		0			0
$155$	−153967.	−	80410.0i	−0.514752	−	0.268832i
$156$		0			0
$157$			48400.4i			0.156711i	0.996925	+	0.0783555i	$0.0249669\pi$
$157$			48400.4i			0.156711i	−0.996925	+	0.0783555i	$0.975033\pi$
$158$		0			0
$159$	−71456.6			−0.224156
$160$		0			0
$161$	43491.8			0.132234
$162$		0			0
$163$		−	150072.i		−	0.442414i	−0.975227	−	0.221207i	$-0.929000\pi$
$163$		−	150072.i		−	0.442414i	0.975227	−	0.221207i	$-0.0709997\pi$
$164$		0			0
$165$	62533.3	−	119737.i	0.178814	−	0.342389i
$166$		0			0
$167$		−	335508.i		−	0.930918i	−0.885069	−	0.465459i	$-0.845889\pi$
$167$		−	335508.i		−	0.930918i	0.885069	−	0.465459i	$-0.154111\pi$
$168$		0			0
$169$	195588.			0.526775
$170$		0			0
$171$	−272116.			−0.711646
$172$		0			0
$173$			437203.i			1.11063i	0.831641	+	0.555313i	$0.187402\pi$
$173$			437203.i			1.11063i	−0.831641	+	0.555313i	$0.812598\pi$
$174$		0			0
$175$	125665.	−	87495.7i	0.310185	−	0.215969i
$176$		0			0
$177$			69465.6i			0.166662i
$178$		0			0
$179$	123917.			0.289068			0.144534	−	0.989500i	$-0.453832\pi$
$179$	123917.			0.289068			0.144534	+	0.989500i	$0.453832\pi$
$180$		0			0
$181$	−503171.			−1.14161			−0.570806	−	0.821085i	$-0.693369\pi$
$181$	−503171.			−1.14161			−0.570806	+	0.821085i	$0.693369\pi$
$182$		0			0
$183$		−	95329.5i		−	0.210426i
$184$		0			0
$185$	−98703.7	+	188995.i	−0.212033	+	0.405996i
$186$		0			0
$187$			19612.4i			0.0410135i
$188$		0			0
$189$	−128738.			−0.262151
$190$		0			0
$191$	−630079.			−1.24972			−0.624859	−	0.780738i	$-0.714843\pi$
$191$	−630079.			−1.24972			−0.624859	+	0.780738i	$0.714843\pi$
$192$		0			0
$193$			1.03204e6i			1.99436i	0.0750449	+	0.997180i	$0.476090\pi$
$193$			1.03204e6i			1.99436i	−0.0750449	+	0.997180i	$0.523910\pi$
$194$		0			0
$195$	−120664.	−	63017.1i	−0.227242	−	0.118678i
$196$		0			0
$197$			143159.i			0.262817i	0.991328	+	0.131408i	$0.0419499\pi$
$197$			143159.i			0.262817i	−0.991328	+	0.131408i	$0.958050\pi$
$198$		0			0
$199$	−547921.			−0.980811			−0.490405	−	0.871495i	$-0.663151\pi$
$199$	−547921.			−0.980811			−0.490405	+	0.871495i	$0.663151\pi$
$200$		0			0
$201$	−168084.			−0.293451
$202$		0			0
$203$		−	358493.i		−	0.610577i
$204$		0			0
$205$	101537.	+	53028.2i	0.168748	+	0.0881297i
$206$		0			0
$207$			185729.i			0.301268i
$208$		0			0
$209$	−540918.			−0.856576
$210$		0			0
$211$	−225800.			−0.349155			−0.174578	−	0.984643i	$-0.555856\pi$
$211$	−225800.			−0.349155			−0.174578	+	0.984643i	$0.555856\pi$
$212$		0			0
$213$		−	378818.i		−	0.572113i
$214$		0			0
$215$	−224218.	+	429326.i	−0.330806	+	0.633419i
$216$		0			0
$217$			152255.i			0.219493i
$218$		0			0
$219$	−317908.			−0.447910
$220$		0			0
$221$	19764.1			0.0272206
$222$		0			0
$223$			536423.i			0.722346i	0.932499	+	0.361173i	$0.117624\pi$
$223$			536423.i			0.722346i	−0.932499	+	0.361173i	$0.882376\pi$
$224$		0			0
$225$	373644.	+	536645.i	0.492042	+	0.706693i
$226$		0			0
$227$			1.14413e6i			1.47371i	0.676052	+	0.736854i	$0.263690\pi$
$227$			1.14413e6i			1.47371i	−0.676052	+	0.736854i	$0.736310\pi$
$228$		0			0
$229$	−671253.			−0.845858			−0.422929	−	0.906163i	$-0.638998\pi$
$229$	−671253.			−0.845858			−0.422929	+	0.906163i	$0.638998\pi$
$230$		0			0
$231$	−118405.			−0.145996
$232$		0			0
$233$		−	292427.i		−	0.352881i	−0.984311	−	0.176440i	$-0.943542\pi$
$233$		−	292427.i		−	0.352881i	0.984311	−	0.176440i	$-0.0564583\pi$
$234$		0			0
$235$	319414.	−	611606.i	0.377298	−	0.722440i
$236$		0			0
$237$		−	162094.i		−	0.187454i
$238$		0			0
$239$	472673.			0.535262			0.267631	−	0.963522i	$-0.413759\pi$
$239$	472673.			0.535262			0.267631	+	0.963522i	$0.413759\pi$
$240$		0			0
$241$	−575591.			−0.638368			−0.319184	−	0.947693i	$-0.603409\pi$
$241$	−575591.			−0.638368			−0.319184	+	0.947693i	$0.603409\pi$
$242$		0			0
$243$		−	845161.i		−	0.918172i
$244$		0			0
$245$	−118972.	−	62133.8i	−0.126628	−	0.0661322i
$246$		0			0
$247$			545103.i			0.568507i
$248$		0			0
$249$	−541862.			−0.553847
$250$		0			0
$251$	−883595.			−0.885257			−0.442628	−	0.896705i	$-0.645954\pi$
$251$	−883595.			−0.885257			−0.442628	+	0.896705i	$0.645954\pi$
$252$		0			0
$253$			369196.i			0.362623i
$254$		0			0
$255$	13572.8	+	7088.46i	0.0130713	+	0.00682656i
$256$		0			0
$257$		−	1.36145e6i		−	1.28579i	−0.765956	−	0.642893i	$-0.777734\pi$
$257$		−	1.36145e6i		−	1.28579i	0.765956	−	0.642893i	$-0.222266\pi$
$258$		0			0
$259$	186893.			0.173119
$260$		0			0
$261$	1.53092e6			1.39107
$262$		0			0
$263$			510894.i			0.455451i	0.973725	+	0.227725i	$0.0731288\pi$
$263$			510894.i			0.455451i	−0.973725	+	0.227725i	$0.926871\pi$
$264$		0			0
$265$	318308.	−	609488.i	0.278441	−	0.533151i
$266$		0			0
$267$		−	740782.i		−	0.635934i
$268$		0			0
$269$	1.63320e6			1.37613			0.688063	−	0.725651i	$-0.258461\pi$
$269$	1.63320e6			1.37613			0.688063	+	0.725651i	$0.258461\pi$
$270$		0			0
$271$	−641076.			−0.530257			−0.265129	−	0.964213i	$-0.585414\pi$
$271$	−641076.			−0.530257			−0.265129	+	0.964213i	$0.585414\pi$
$272$		0			0
$273$			119321.i			0.0968974i
$274$		0			0
$275$	742738.	+	1.06675e6i	0.592249	+	0.850615i
$276$		0			0
$277$			2.12633e6i			1.66506i	0.553978	+	0.832531i	$0.313109\pi$
$277$			2.12633e6i			1.66506i	−0.553978	+	0.832531i	$0.686891\pi$
$278$		0			0
$279$	−650192.			−0.500070
$280$		0			0
$281$	−2.54572e6			−1.92329			−0.961645	−	0.274296i	$-0.911555\pi$
$281$	−2.54572e6			−1.92329			−0.961645	+	0.274296i	$0.911555\pi$
$282$		0			0
$283$		−	1.19985e6i		−	0.890553i	−0.895393	−	0.445276i	$-0.853105\pi$
$283$		−	1.19985e6i		−	0.890553i	0.895393	−	0.445276i	$-0.146895\pi$
$284$		0			0
$285$	−195503.	+	374344.i	−0.142574	+	0.272997i
$286$		0			0
$287$		−	100408.i		−	0.0719552i
$288$		0			0
$289$	1.41763e6			0.998434
$290$		0			0
$291$	594236.			0.411364
$292$		0			0
$293$			735490.i			0.500504i	0.968181	+	0.250252i	$0.0805135\pi$
$293$			735490.i			0.500504i	−0.968181	+	0.250252i	$0.919487\pi$
$294$		0			0
$295$	−592506.	−	309439.i	−0.396404	−	0.207024i
$296$		0			0
$297$		−	1.09284e6i		−	0.718893i
$298$		0			0
$299$	372052.			0.240672
$300$		0			0
$301$	424551.			0.270093
$302$		0			0
$303$		−	175901.i		−	0.110068i
$304$		0			0
$305$	813112.	+	424652.i	0.500496	+	0.261386i
$306$		0			0
$307$		−	1.14803e6i		−	0.695193i	−0.937644	−	0.347597i	$-0.886998\pi$
$307$		−	1.14803e6i		−	0.695193i	0.937644	−	0.347597i	$-0.113002\pi$
$308$		0			0
$309$	818302.			0.487548
$310$		0			0
$311$	−1.58159e6			−0.927242			−0.463621	−	0.886034i	$-0.653450\pi$
$311$	−1.58159e6			−0.927242			−0.463621	+	0.886034i	$0.653450\pi$
$312$		0			0
$313$		−	523976.i		−	0.302309i	−0.988510	−	0.151154i	$-0.951701\pi$
$313$		−	523976.i		−	0.302309i	0.988510	−	0.151154i	$-0.0482991\pi$
$314$		0			0
$315$	265338.	−	508062.i	0.150669	−	0.288497i
$316$		0			0
$317$			2.67738e6i			1.49645i	0.663446	+	0.748224i	$0.269093\pi$
$317$			2.67738e6i			1.49645i	−0.663446	+	0.748224i	$0.730907\pi$
$318$		0			0
$319$	3.04319e6			1.67437
$320$		0			0
$321$	−326277.			−0.176736
$322$		0			0
$323$		−	61315.8i		−	0.0327014i
$324$		0			0
$325$	1.07501e6	−	748484.i	0.564551	−	0.393074i
$326$		0			0
$327$		−	120361.i		−	0.0622468i
$328$		0			0
$329$	−604803.			−0.308052
$330$		0			0
$331$	−1.23887e6			−0.621520			−0.310760	−	0.950488i	$-0.600583\pi$
$331$	−1.23887e6			−0.621520			−0.310760	+	0.950488i	$0.600583\pi$
$332$		0			0
$333$			798114.i			0.394416i
$334$		0			0
$335$	748739.	−	1.43367e6i	0.364518	−	0.697969i
$336$		0			0
$337$		−	3.52169e6i		−	1.68918i	−0.535414	−	0.844590i	$-0.679844\pi$
$337$		−	3.52169e6i		−	1.68918i	0.535414	−	0.844590i	$-0.320156\pi$
$338$		0			0
$339$	436449.			0.206269
$340$		0			0
$341$	−1.29247e6			−0.601913
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	255503.	+	133437.i	0.115571	+	0.0603573i
$346$		0			0
$347$			2.07966e6i			0.927191i	0.886047	+	0.463595i	$0.153441\pi$
$347$			2.07966e6i			0.927191i	−0.886047	+	0.463595i	$0.846559\pi$
$348$		0			0
$349$	−256053.			−0.112529			−0.0562647	−	0.998416i	$-0.517919\pi$
$349$	−256053.			−0.112529			−0.0562647	+	0.998416i	$0.517919\pi$
$350$		0			0
$351$	−1.10129e6			−0.477127
$352$		0			0
$353$		−	2.84978e6i		−	1.21724i	−0.793463	−	0.608618i	$-0.791724\pi$
$353$		−	2.84978e6i		−	1.21724i	0.793463	−	0.608618i	$-0.208276\pi$
$354$		0			0
$355$	3.23112e6	+	1.68747e6i	1.36076	+	0.710665i
$356$		0			0
$357$		−	13421.8i		−	0.00557368i
$358$		0			0
$359$	3.56014e6			1.45791			0.728956	−	0.684561i	$-0.240006\pi$
$359$	3.56014e6			1.45791			0.728956	+	0.684561i	$0.240006\pi$
$360$		0			0
$361$	−784985.			−0.317025
$362$		0			0
$363$		−	69519.0i		−	0.0276909i
$364$		0			0
$365$	1.41614e6	−	2.71159e6i	0.556384	−	1.06535i
$366$		0			0
$367$			230540.i			0.0893472i	0.999002	+	0.0446736i	$0.0142248\pi$
$367$			230540.i			0.0893472i	−0.999002	+	0.0446736i	$0.985775\pi$
$368$		0			0
$369$	428784.			0.163935
$370$		0			0
$371$	−602709.			−0.227339
$372$		0			0
$373$		−	2.77989e6i		−	1.03456i	−0.855817	−	0.517279i	$-0.826945\pi$
$373$		−	2.77989e6i		−	1.03456i	0.855817	−	0.517279i	$-0.173055\pi$
$374$		0			0
$375$	1.00670e6	−	128459.i	0.369675	−	0.0471722i
$376$		0			0
$377$		−	3.06673e6i		−	1.11128i
$378$		0			0
$379$	620903.			0.222037			0.111019	−	0.993818i	$-0.464589\pi$
$379$	620903.			0.222037			0.111019	+	0.993818i	$0.464589\pi$
$380$		0			0
$381$	−1.33580e6			−0.471444
$382$		0			0
$383$			2.85178e6i			0.993390i	0.867925	+	0.496695i	$0.165453\pi$
$383$			2.85178e6i			0.993390i	−0.867925	+	0.496695i	$0.834547\pi$
$384$		0			0
$385$	527445.	−	1.00994e6i	0.181353	−	0.347251i
$386$		0			0
$387$			1.81301e6i			0.615352i
$388$		0			0
$389$	2.23838e6			0.749996			0.374998	−	0.927026i	$-0.377643\pi$
$389$	2.23838e6			0.749996			0.374998	+	0.927026i	$0.377643\pi$
$390$		0			0
$391$	−41850.2			−0.0138438
$392$		0			0
$393$		−	813084.i		−	0.265555i
$394$		0			0
$395$	1.38258e6	+	722057.i	0.445858	+	0.232851i
$396$		0			0
$397$			693032.i			0.220687i	0.993894	+	0.110344i	$0.0351951\pi$
$397$			693032.i			0.220687i	−0.993894	+	0.110344i	$0.964805\pi$
$398$		0			0
$399$	370180.			0.116407
$400$		0			0
$401$	−4.37580e6			−1.35893			−0.679464	−	0.733709i	$-0.737788\pi$
$401$	−4.37580e6			−1.35893			−0.679464	+	0.733709i	$0.737788\pi$
$402$		0			0
$403$			1.30247e6i			0.399488i
$404$		0			0
$405$	1.76328e6	+	920879.i	0.534174	+	0.278975i
$406$		0			0
$407$			1.58651e6i			0.474741i
$408$		0			0
$409$	−104194.			−0.0307987			−0.0153994	−	0.999881i	$-0.504902\pi$
$409$	−104194.			−0.0307987			−0.0153994	+	0.999881i	$0.504902\pi$
$410$		0			0
$411$	226217.			0.0660572
$412$		0			0
$413$			585916.i			0.169029i
$414$		0			0
$415$	2.41376e6	−	4.62180e6i	0.687976	−	1.31732i
$416$		0			0
$417$			6660.99i			0.00187585i
$418$		0			0
$419$	4.05780e6			1.12916			0.564580	−	0.825379i	$-0.309038\pi$
$419$	4.05780e6			1.12916			0.564580	+	0.825379i	$0.309038\pi$
$420$		0			0
$421$	3.26799e6			0.898617			0.449309	−	0.893377i	$-0.351670\pi$
$421$	3.26799e6			0.898617			0.449309	+	0.893377i	$0.351670\pi$
$422$		0			0
$423$		−	2.58277e6i		−	0.701835i
$424$		0			0
$425$	−120922.	+	84193.1i	−0.0324738	+	0.0226102i
$426$		0			0
$427$		−	804068.i		−	0.213414i
$428$		0			0
$429$	−1.01290e6			−0.265720
$430$		0			0
$431$	1.00601e6			0.260861			0.130431	−	0.991457i	$-0.458364\pi$
$431$	1.00601e6			0.260861			0.130431	+	0.991457i	$0.458364\pi$
$432$		0			0
$433$		−	6.73152e6i		−	1.72541i	−0.505704	−	0.862707i	$-0.668767\pi$
$433$		−	6.73152e6i		−	1.72541i	0.505704	−	0.862707i	$-0.331233\pi$
$434$		0			0
$435$	1.09989e6	−	2.10605e6i	0.278694	−	0.533636i
$436$		0			0
$437$		−	1.15424e6i		−	0.289131i
$438$		0			0
$439$	3.96293e6			0.981420			0.490710	−	0.871323i	$-0.336737\pi$
$439$	3.96293e6			0.981420			0.490710	+	0.871323i	$0.336737\pi$
$440$		0			0
$441$	−502412.			−0.123016
$442$		0			0
$443$			3.58330e6i			0.867508i	0.901031	+	0.433754i	$0.142811\pi$
$443$			3.58330e6i			0.867508i	−0.901031	+	0.433754i	$0.857189\pi$
$444$		0			0
$445$	6.31849e6	+	3.29986e6i	1.51256	+	0.789943i
$446$		0			0
$447$			552287.i			0.130736i
$448$		0			0
$449$	−5.97996e6			−1.39985			−0.699927	−	0.714214i	$-0.746784\pi$
$449$	−5.97996e6			−1.39985			−0.699927	+	0.714214i	$0.746784\pi$
$450$		0			0
$451$	852346.			0.197322
$452$		0			0
$453$			2.47625e6i			0.566955i
$454$		0			0
$455$	−1.01775e6	−	531525.i	−0.230469	−	0.120364i
$456$		0			0
$457$		−	3.26554e6i		−	0.731417i	−0.930730	−	0.365708i	$-0.880827\pi$
$457$		−	3.26554e6i		−	0.731417i	0.930730	−	0.365708i	$-0.119173\pi$
$458$		0			0
$459$	123878.			0.0274450
$460$		0			0
$461$	−6.57854e6			−1.44171			−0.720854	−	0.693087i	$-0.756250\pi$
$461$	−6.57854e6			−1.44171			−0.720854	+	0.693087i	$0.756250\pi$
$462$		0			0
$463$			5.32610e6i			1.15467i	0.816508	+	0.577334i	$0.195907\pi$
$463$			5.32610e6i			1.15467i	−0.816508	+	0.577334i	$0.804093\pi$
$464$		0			0
$465$	−467133.	+	894454.i	−0.100186	+	0.191834i
$466$		0			0
$467$			6.19078e6i			1.31357i	0.754078	+	0.656785i	$0.228084\pi$
$467$			6.19078e6i			1.31357i	−0.754078	+	0.656785i	$0.771916\pi$
$468$		0			0
$469$	−1.41772e6			−0.297618
$470$		0			0
$471$	281177.			0.0584019
$472$		0			0
$473$			3.60395e6i			0.740672i
$474$		0			0
$475$	−2.32208e6	−	3.33508e6i	−0.472219	−	0.678222i
$476$		0			0
$477$		−	2.57383e6i		−	0.517945i
$478$		0			0
$479$	−7.26722e6			−1.44720			−0.723601	−	0.690218i	$-0.757515\pi$
$479$	−7.26722e6			−1.44720			−0.723601	+	0.690218i	$0.757515\pi$
$480$		0			0
$481$	1.59878e6			0.315084
$482$		0			0
$483$		−	252661.i		−	0.0492799i
$484$		0			0
$485$	−2.64706e6	+	5.06853e6i	−0.510987	+	0.978425i
$486$		0			0
$487$			8.70984e6i			1.66413i	0.554676	+	0.832066i	$0.312842\pi$
$487$			8.70984e6i			1.66413i	−0.554676	+	0.832066i	$0.687158\pi$
$488$		0			0
$489$	−871824.			−0.164876
$490$		0			0
$491$	−7.03903e6			−1.31768			−0.658839	−	0.752284i	$-0.728952\pi$
$491$	−7.03903e6			−1.31768			−0.658839	+	0.752284i	$0.728952\pi$
$492$		0			0
$493$			344961.i			0.0639223i
$494$		0			0
$495$	4.31287e6	+	2.25242e6i	0.791140	+	0.413177i
$496$		0			0
$497$		−	3.19519e6i		−	0.580237i
$498$		0			0
$499$	−5.43089e6			−0.976381			−0.488191	−	0.872737i	$-0.662343\pi$
$499$	−5.43089e6			−0.976381			−0.488191	+	0.872737i	$0.662343\pi$
$500$		0			0
$501$	−1.94909e6			−0.346927
$502$		0			0
$503$		−	4.36641e6i		−	0.769492i	−0.923022	−	0.384746i	$-0.874289\pi$
$503$		−	4.36641e6i		−	0.769492i	0.923022	−	0.384746i	$-0.125711\pi$
$504$		0			0
$505$	1.50034e6	+	783562.i	0.261796	+	0.136724i
$506$		0			0
$507$		−	1.13625e6i		−	0.196315i
$508$		0			0
$509$	2.02571e6			0.346564			0.173282	−	0.984872i	$-0.444563\pi$
$509$	2.02571e6			0.346564			0.173282	+	0.984872i	$0.444563\pi$
$510$		0			0
$511$	−2.68143e6			−0.454271
$512$		0			0
$513$			3.41662e6i			0.573196i
$514$		0			0
$515$	−3.64518e6	+	6.97970e6i	−0.605621	+	1.15963i
$516$		0			0
$517$		−	5.13409e6i		−	0.844767i
$518$		0			0
$519$	2.53988e6			0.413900
$520$		0			0
$521$	−8.38319e6			−1.35305			−0.676527	−	0.736418i	$-0.736516\pi$
$521$	−8.38319e6			−1.35305			−0.676527	+	0.736418i	$0.736516\pi$
$522$		0			0
$523$		−	255511.i		−	0.0408466i	−0.999791	−	0.0204233i	$-0.993499\pi$
$523$		−	255511.i		−	0.0408466i	0.999791	−	0.0204233i	$-0.00650139\pi$
$524$		0			0
$525$	−508296.	−	730039.i	−0.0804857	−	0.115597i
$526$		0			0
$527$		−	146508.i		−	0.0229791i
$528$		0			0
$529$	5.64853e6			0.877599
$530$		0			0
$531$	−2.50211e6			−0.385097
$532$		0			0
$533$		−	858940.i		−	0.130962i
$534$		0			0
$535$	1.45342e6	−	2.78298e6i	0.219537	−	0.420364i
$536$		0			0
$537$		−	719884.i		−	0.107728i
$538$		0			0
$539$	−998705.			−0.148069
$540$		0			0
$541$	−6.67280e6			−0.980201			−0.490100	−	0.871666i	$-0.663040\pi$
$541$	−6.67280e6			−0.980201			−0.490100	+	0.871666i	$0.663040\pi$
$542$		0			0
$543$			2.92311e6i			0.425448i
$544$		0			0
$545$	1.02662e6	+	536157.i	0.148053	+	0.0773216i
$546$		0			0
$547$			4.14191e6i			0.591879i	0.955207	+	0.295939i	$0.0956327\pi$
$547$			4.14191e6i			0.591879i	−0.955207	+	0.295939i	$0.904367\pi$
$548$		0			0
$549$	3.43372e6			0.486221
$550$		0			0
$551$	−9.51416e6			−1.33503
$552$		0			0
$553$		−	1.36720e6i		−	0.190116i
$554$		0			0
$555$	1.09795e6	+	573408.i	0.151303	+	0.0790189i
$556$		0			0
$557$		−	1.88106e6i		−	0.256900i	−0.991716	−	0.128450i	$-0.959000\pi$
$557$		−	1.88106e6i		−	0.256900i	0.991716	−	0.128450i	$-0.0410002\pi$
$558$		0			0
$559$	3.63183e6			0.491582
$560$		0			0
$561$	113936.			0.0152846
$562$		0			0
$563$		−	8.65163e6i		−	1.15034i	−0.818033	−	0.575171i	$-0.804936\pi$
$563$		−	8.65163e6i		−	1.15034i	0.818033	−	0.575171i	$-0.195064\pi$
$564$		0			0
$565$	−1.94419e6	+	3.72269e6i	−0.256223	+	0.490609i
$566$		0			0
$567$		−	1.74366e6i		−	0.227775i
$568$		0			0
$569$	5.79055e6			0.749789			0.374894	−	0.927068i	$-0.377679\pi$
$569$	5.79055e6			0.749789			0.374894	+	0.927068i	$0.377679\pi$
$570$		0			0
$571$	−160058.			−0.0205441			−0.0102720	−	0.999947i	$-0.503270\pi$
$571$	−160058.			−0.0205441			−0.0102720	+	0.999947i	$0.503270\pi$
$572$		0			0
$573$			3.66038e6i			0.465735i
$574$		0			0
$575$	−2.27631e6	+	1.58490e6i	−0.287118	+	0.199909i
$576$		0			0
$577$		−	9.18883e6i		−	1.14900i	−0.818504	−	0.574501i	$-0.805196\pi$
$577$		−	9.18883e6i		−	1.14900i	0.818504	−	0.574501i	$-0.194804\pi$
$578$		0			0
$579$	5.99553e6			0.743243
$580$		0			0
$581$	−4.57040e6			−0.561712
$582$		0			0
$583$		−	5.11631e6i		−	0.623427i
$584$		0			0
$585$	2.26984e6	−	4.34623e6i	0.274224	−	0.525078i
$586$		0			0
$587$		−	815904.i		−	0.0977335i	−0.998805	−	0.0488668i	$-0.984439\pi$
$587$		−	815904.i		−	0.0977335i	0.998805	−	0.0488668i	$-0.0155610\pi$
$588$		0			0
$589$	4.04074e6			0.479924
$590$		0			0
$591$	831667.			0.0979446
$592$		0			0
$593$			6.86745e6i			0.801972i	0.916084	+	0.400986i	$0.131332\pi$
$593$			6.86745e6i			0.801972i	−0.916084	+	0.400986i	$0.868668\pi$
$594$		0			0
$595$	114481.	+	59788.5i	0.0132569	+	0.00692349i
$596$		0			0
$597$			3.18308e6i			0.365521i
$598$		0			0
$599$	−9.32027e6			−1.06136			−0.530679	−	0.847573i	$-0.678063\pi$
$599$	−9.32027e6			−1.06136			−0.530679	+	0.847573i	$0.678063\pi$
$600$		0			0
$601$	−1.25201e7			−1.41391			−0.706955	−	0.707258i	$-0.749932\pi$
$601$	−1.25201e7			−1.41391			−0.706955	+	0.707258i	$0.749932\pi$
$602$		0			0
$603$		−	6.05428e6i		−	0.678062i
$604$		0			0
$605$	592962.	+	309677.i	0.0658625	+	0.0343970i
$606$		0			0
$607$			1.54322e7i			1.70002i	0.526764	+	0.850012i	$0.323405\pi$
$607$			1.54322e7i			1.70002i	−0.526764	+	0.850012i	$0.676595\pi$
$608$		0			0
$609$	−2.08262e6			−0.227545
$610$		0			0
$611$	−5.17381e6			−0.560670
$612$		0			0
$613$			1.68625e7i			1.81247i	0.422770	+	0.906237i	$0.361058\pi$
$613$			1.68625e7i			1.81247i	−0.422770	+	0.906237i	$0.638942\pi$
$614$		0			0
$615$	308061.	−	589868.i	0.0328435	−	0.0628879i
$616$		0			0
$617$		−	1.29684e6i		−	0.137143i	−0.997646	−	0.0685713i	$-0.978156\pi$
$617$		−	1.29684e6i		−	0.137143i	0.997646	−	0.0685713i	$-0.0218441\pi$
$618$		0			0
$619$	−1.26790e7			−1.33002			−0.665011	−	0.746834i	$-0.731573\pi$
$619$	−1.26790e7			−1.33002			−0.665011	+	0.746834i	$0.731573\pi$
$620$		0			0
$621$	2.33196e6			0.242657
$622$		0			0
$623$		−	6.24821e6i		−	0.644964i
$624$		0			0
$625$	−3.38870e6	+	9.15883e6i	−0.347003	+	0.937864i
$626$		0			0
$627$			3.14240e6i			0.319222i
$628$		0			0
$629$	−179839.			−0.0181241
$630$		0			0
$631$	−1.73861e7			−1.73832			−0.869160	−	0.494531i	$-0.835340\pi$
$631$	−1.73861e7			−1.73832			−0.869160	+	0.494531i	$0.835340\pi$
$632$		0			0
$633$			1.31176e6i			0.130121i
$634$		0			0
$635$	5.95043e6	−	1.13937e7i	0.585617	−	1.12133i
$636$		0			0
$637$			1.00643e6i			0.0982733i
$638$		0			0
$639$	1.36448e7			1.32195
$640$		0			0
$641$	4.83240e6			0.464535			0.232267	−	0.972652i	$-0.425386\pi$
$641$	4.83240e6			0.464535			0.232267	+	0.972652i	$0.425386\pi$
$642$		0			0
$643$		−	2.67017e6i		−	0.254690i	−0.991859	−	0.127345i	$-0.959355\pi$
$643$		−	2.67017e6i		−	0.254690i	0.991859	−	0.127345i	$-0.0406455\pi$
$644$		0			0
$645$	2.49412e6	+	1.30257e6i	0.236058	+	0.123282i
$646$		0			0
$647$		−	1.27082e7i		−	1.19351i	−0.802425	−	0.596753i	$-0.796457\pi$
$647$		−	1.27082e7i		−	1.19351i	0.802425	−	0.596753i	$-0.203543\pi$
$648$		0			0
$649$	−4.97376e6			−0.463525
$650$		0			0
$651$	884506.			0.0817990
$652$		0			0
$653$			1.59003e7i			1.45922i	0.683863	+	0.729611i	$0.260299\pi$
$653$			1.59003e7i			1.45922i	−0.683863	+	0.729611i	$0.739701\pi$
$654$		0			0
$655$	6.93519e6	+	3.62194e6i	0.631619	+	0.329866i
$656$		0			0
$657$		−	1.14509e7i		−	1.03496i
$658$		0			0
$659$	−1.50434e7			−1.34937			−0.674687	−	0.738104i	$-0.735721\pi$
$659$	−1.50434e7			−1.34937			−0.674687	+	0.738104i	$0.735721\pi$
$660$		0			0
$661$	−9.78584e6			−0.871153			−0.435577	−	0.900152i	$-0.643456\pi$
$661$	−9.78584e6			−0.871153			−0.435577	+	0.900152i	$0.643456\pi$
$662$		0			0
$663$		−	114818.i		−	0.0101444i
$664$		0			0
$665$	−1.64899e6	+	3.15745e6i	−0.144599	+	0.276874i
$666$		0			0
$667$			6.49375e6i			0.565172i
$668$		0			0
$669$	3.11629e6			0.269198
$670$		0			0
$671$	6.82562e6			0.585242
$672$		0			0
$673$		−	2.26085e7i		−	1.92413i	−0.272820	−	0.962065i	$-0.587956\pi$
$673$		−	2.26085e7i		−	1.92413i	0.272820	−	0.962065i	$-0.412044\pi$
$674$		0			0
$675$	6.73797e6	−	4.69138e6i	0.569207	−	0.396316i
$676$		0			0
$677$			2.18844e7i			1.83512i	0.397600	+	0.917559i	$0.369843\pi$
$677$			2.18844e7i			1.83512i	−0.397600	+	0.917559i	$0.630157\pi$
$678$		0			0
$679$	5.01215e6			0.417205
$680$		0			0
$681$	6.64671e6			0.549211
$682$		0			0
$683$			2.06203e7i			1.69139i	0.533666	+	0.845696i	$0.320814\pi$
$683$			2.06203e7i			1.69139i	−0.533666	+	0.845696i	$0.679186\pi$
$684$		0			0
$685$	−1.00770e6	+	1.92951e6i	−0.0820547	+	0.157116i
$686$		0			0
$687$			3.89957e6i			0.315228i
$688$		0			0
$689$	−5.15589e6			−0.413767
$690$		0			0
$691$	2.10211e7			1.67479			0.837396	−	0.546596i	$-0.184077\pi$
$691$	2.10211e7			1.67479			0.837396	+	0.546596i	$0.184077\pi$
$692$		0			0
$693$		−	4.26490e6i		−	0.337346i
$694$		0			0
$695$	−56814.8	−	29671.8i	−0.00446169	−	0.00233014i
$696$		0			0
$697$			96617.7i			0.00753312i
$698$		0			0
$699$	−1.69882e6			−0.131509
$700$		0			0
$701$	1.38535e7			1.06479			0.532397	−	0.846495i	$-0.321291\pi$
$701$	1.38535e7			1.06479			0.532397	+	0.846495i	$0.321291\pi$
$702$		0			0
$703$		−	4.96002e6i		−	0.378526i
$704$		0			0
$705$	−3.55306e6	−	1.85560e6i	−0.269234	−	0.140609i
$706$		0			0
$707$		−	1.48366e6i		−	0.111631i
$708$		0			0
$709$	1.10869e7			0.828315			0.414158	−	0.910205i	$-0.364076\pi$
$709$	1.10869e7			0.828315			0.414158	+	0.910205i	$0.364076\pi$
$710$		0			0
$711$	5.83852e6			0.433141
$712$		0			0
$713$		−	2.75794e6i		−	0.203171i
$714$		0			0
$715$	4.51204e6	−	8.63954e6i	0.330072	−	0.632012i
$716$		0			0
$717$		−	2.74594e6i		−	0.199477i
$718$		0			0
$719$	−6.30329e6			−0.454721			−0.227360	−	0.973811i	$-0.573010\pi$
$719$	−6.30329e6			−0.454721			−0.227360	+	0.973811i	$0.573010\pi$
$720$		0			0
$721$	6.90207e6			0.494472
$722$		0			0
$723$			3.34383e6i			0.237902i
$724$		0			0
$725$	1.30640e7	+	1.87631e7i	0.923060	+	1.32574i
$726$		0			0
$727$		−	9.10582e6i		−	0.638974i	−0.947591	−	0.319487i	$-0.896489\pi$
$727$		−	9.10582e6i		−	0.638974i	0.947591	−	0.319487i	$-0.103511\pi$
$728$		0			0
$729$	3.73728e6			0.260457
$730$		0			0
$731$	−408526.			−0.0282765
$732$		0			0
$733$			6.09212e6i			0.418802i	0.977830	+	0.209401i	$0.0671514\pi$
$733$			6.09212e6i			0.418802i	−0.977830	+	0.209401i	$0.932849\pi$
$734$		0			0
$735$	−360959.	+	691156.i	−0.0246456	+	0.0471908i
$736$		0			0
$737$		−	1.20348e7i		−	0.816153i
$738$		0			0
$739$	2.44066e7			1.64398			0.821990	−	0.569502i	$-0.192864\pi$
$739$	2.44066e7			1.64398			0.821990	+	0.569502i	$0.192864\pi$
$740$		0			0
$741$	3.16671e6			0.211867
$742$		0			0
$743$		−	1.53795e7i		−	1.02205i	−0.859566	−	0.511024i	$-0.829266\pi$
$743$		−	1.53795e7i		−	1.02205i	0.859566	−	0.511024i	$-0.170734\pi$
$744$		0			0
$745$	−4.71073e6	−	2.46020e6i	−0.310955	−	0.162398i
$746$		0			0
$747$		−	1.95176e7i		−	1.27975i
$748$		0			0
$749$	−2.75202e6			−0.179245
$750$		0			0
$751$	4.56800e6			0.295547			0.147773	−	0.989021i	$-0.452789\pi$
$751$	4.56800e6			0.295547			0.147773	+	0.989021i	$0.452789\pi$
$752$		0			0
$753$			5.13315e6i			0.329911i
$754$		0			0
$755$	−2.11211e7	−	1.10306e7i	−1.34850	−	0.704258i
$756$		0			0
$757$		−	1.80153e7i		−	1.14262i	−0.820735	−	0.571309i	$-0.806436\pi$
$757$		−	1.80153e7i		−	1.14262i	0.820735	−	0.571309i	$-0.193564\pi$
$758$		0			0
$759$	2.14480e6			0.135140
$760$		0			0
$761$	−1.55682e7			−0.974489			−0.487245	−	0.873266i	$-0.661998\pi$
$761$	−1.55682e7			−0.974489			−0.487245	+	0.873266i	$0.661998\pi$
$762$		0			0
$763$		−	1.01520e6i		−	0.0631307i
$764$		0			0
$765$	−255323.	+	488885.i	−0.0157738	+	0.0302032i
$766$		0			0
$767$			5.01223e6i			0.307640i
$768$		0			0
$769$	−1.36689e6			−0.0833525			−0.0416762	−	0.999131i	$-0.513270\pi$
$769$	−1.36689e6			−0.0833525			−0.0416762	+	0.999131i	$0.513270\pi$
$770$		0			0
$771$	−7.90919e6			−0.479177
$772$		0			0
$773$			2.19934e7i			1.32386i	0.749564	+	0.661932i	$0.230263\pi$
$773$			2.19934e7i			1.32386i	−0.749564	+	0.661932i	$0.769737\pi$
$774$		0			0
$775$	−5.54836e6	−	7.96881e6i	−0.331826	−	0.476584i
$776$		0			0
$777$		−	1.08574e6i		−	0.0645166i
$778$		0			0
$779$	−2.66475e6			−0.157331
$780$		0			0
$781$	2.71235e7			1.59117
$782$		0			0
$783$		−	1.92218e7i		−	1.12044i
$784$		0			0
$785$	−1.25252e6	+	2.39829e6i	−0.0725455	+	0.138908i
$786$		0			0
$787$			1.21961e6i			0.0701915i	0.999384	+	0.0350957i	$0.0111736\pi$
$787$			1.21961e6i			0.0701915i	−0.999384	+	0.0350957i	$0.988826\pi$
$788$		0			0
$789$	2.96798e6			0.169734
$790$		0			0
$791$	3.68128e6			0.209198
$792$		0			0
$793$		−	6.87842e6i		−	0.388424i
$794$		0			0
$795$	−3.54075e6	−	1.84918e6i	−0.198691	−	0.103767i
$796$		0			0
$797$			4.81890e6i			0.268722i	0.990932	+	0.134361i	$0.0428981\pi$
$797$			4.81890e6i			0.268722i	−0.990932	+	0.134361i	$0.957102\pi$
$798$		0			0
$799$	581974.			0.0322505
$800$		0			0
$801$	2.66825e7			1.46942
$802$		0			0
$803$		−	2.27623e7i		−	1.24574i
$804$		0			0
$805$	2.15507e6	+	1.12549e6i	0.117212	+	0.0612144i
$806$		0			0
$807$		−	9.48788e6i		−	0.512844i
$808$		0			0
$809$	2.17239e6			0.116699			0.0583495	−	0.998296i	$-0.481416\pi$
$809$	2.17239e6			0.116699			0.0583495	+	0.998296i	$0.481416\pi$
$810$		0			0
$811$	1.19038e7			0.635529			0.317764	−	0.948170i	$-0.397068\pi$
$811$	1.19038e7			0.635529			0.317764	+	0.948170i	$0.397068\pi$
$812$		0			0
$813$			3.72426e6i			0.197612i
$814$		0			0
$815$	3.88360e6	−	7.43621e6i	0.204805	−	0.392155i
$816$		0			0
$817$		−	1.12673e7i		−	0.590561i
$818$		0			0
$819$	−4.29789e6			−0.223896
$820$		0			0
$821$	1.71728e7			0.889168			0.444584	−	0.895737i	$-0.353351\pi$
$821$	1.71728e7			0.889168			0.444584	+	0.895737i	$0.353351\pi$
$822$		0			0
$823$		−	1.32227e7i		−	0.680489i	−0.940337	−	0.340244i	$-0.889490\pi$
$823$		−	1.32227e7i		−	0.680489i	0.940337	−	0.340244i	$-0.110510\pi$
$824$		0			0
$825$	6.19719e6	−	4.31486e6i	0.317001	−	0.220715i
$826$		0			0
$827$			2.13177e7i			1.08387i	0.840421	+	0.541934i	$0.182308\pi$
$827$			2.13177e7i			1.08387i	−0.840421	+	0.541934i	$0.817692\pi$
$828$		0			0
$829$	2.14305e7			1.08304			0.541522	−	0.840686i	$-0.317848\pi$
$829$	2.14305e7			1.08304			0.541522	+	0.840686i	$0.317848\pi$
$830$		0			0
$831$	1.23527e7			0.620523
$832$		0			0
$833$		−	113208.i		−	0.00565282i
$834$		0			0
$835$	8.68237e6	−	1.66248e7i	0.430945	−	0.825163i
$836$		0			0
$837$			8.16364e6i			0.402782i
$838$		0			0
$839$	−3.42279e7			−1.67871			−0.839355	−	0.543584i	$-0.817067\pi$
$839$	−3.42279e7			−1.67871			−0.839355	+	0.543584i	$0.817067\pi$
$840$		0			0
$841$	3.30153e7			1.60963
$842$		0			0
$843$			1.47891e7i			0.716757i
$844$		0			0
$845$	9.69160e6	+	5.06149e6i	0.466932	+	0.243858i
$846$		0			0
$847$		−	586367.i		−	0.0280841i
$848$		0			0
$849$	−6.97038e6			−0.331884
$850$		0			0
$851$	−3.38539e6			−0.160245
$852$		0			0
$853$		−	3.84312e6i		−	0.180847i	−0.995903	−	0.0904234i	$-0.971178\pi$
$853$		−	3.84312e6i		−	0.180847i	0.995903	−	0.0904234i	$-0.0288220\pi$
$854$		0			0
$855$	−1.34836e7	−	7.04190e6i	−0.630801	−	0.329439i
$856$		0			0
$857$		−	2.34451e7i		−	1.09043i	−0.838295	−	0.545217i	$-0.816447\pi$
$857$		−	2.34451e7i		−	1.09043i	0.838295	−	0.545217i	$-0.183553\pi$
$858$		0			0
$859$	−2.85004e6			−0.131786			−0.0658928	−	0.997827i	$-0.520990\pi$
$859$	−2.85004e6			−0.131786			−0.0658928	+	0.997827i	$0.520990\pi$
$860$		0			0
$861$	−583307.			−0.0268157
$862$		0			0
$863$			2.37124e6i			0.108380i	0.998531	+	0.0541898i	$0.0172576\pi$
$863$			2.37124e6i			0.108380i	−0.998531	+	0.0541898i	$0.982742\pi$
$864$		0			0
$865$	−1.13141e7	+	2.16639e7i	−0.514137	+	0.984456i
$866$		0			0
$867$		−	8.23559e6i		−	0.372089i
$868$		0			0
$869$	1.16060e7			0.521352
$870$		0			0
$871$	−1.21279e7			−0.541678
$872$		0			0
$873$			2.14040e7i			0.950518i
$874$		0			0
$875$	8.49110e6	−	1.08350e6i	0.374924	−	0.0478421i
$876$		0			0
$877$			7.63907e6i			0.335383i	0.985839	+	0.167692i	$0.0536313\pi$
$877$			7.63907e6i			0.335383i	−0.985839	+	0.167692i	$0.946369\pi$
$878$		0			0
$879$	4.27275e6			0.186524
$880$		0			0
$881$	−3.83404e7			−1.66424			−0.832121	−	0.554594i	$-0.812874\pi$
$881$	−3.83404e7			−1.66424			−0.832121	+	0.554594i	$0.812874\pi$
$882$		0			0
$883$			7.69593e6i			0.332169i	0.986111	+	0.166085i	$0.0531125\pi$
$883$			7.69593e6i			0.332169i	−0.986111	+	0.166085i	$0.946887\pi$
$884$		0			0
$885$	−1.79765e6	+	3.44210e6i	−0.0771520	+	0.147729i
$886$		0			0
$887$			3.71033e7i			1.58345i	0.610880	+	0.791723i	$0.290816\pi$
$887$			3.71033e7i			1.58345i	−0.610880	+	0.791723i	$0.709184\pi$
$888$		0			0
$889$	−1.12670e7			−0.478139
$890$		0			0
$891$	1.48017e7			0.624623
$892$		0			0
$893$			1.60511e7i			0.673559i
$894$		0			0
$895$	6.14024e6	+	3.20677e6i	0.256229	+	0.133817i
$896$		0			0
$897$		−	2.16139e6i		−	0.0896918i
$898$		0			0
$899$	−2.27331e7			−0.938121
$900$		0			0
$901$	579959.			0.0238005
$902$		0			0
$903$		−	2.46638e6i		−	0.100656i
$904$		0			0
$905$	−2.49327e7	−	1.30212e7i	−1.01192	−	0.528481i
$906$		0			0
$907$			4.15393e7i			1.67664i	0.545175	+	0.838322i	$0.316463\pi$
$907$			4.15393e7i			1.67664i	−0.545175	+	0.838322i	$0.683537\pi$
$908$		0			0
$909$	6.33586e6			0.254329
$910$		0			0
$911$	−2.96060e7			−1.18191			−0.590954	−	0.806705i	$-0.701249\pi$
$911$	−2.96060e7			−1.18191			−0.590954	+	0.806705i	$0.701249\pi$
$912$		0			0
$913$		−	3.87974e7i		−	1.54037i
$914$		0			0
$915$	2.46697e6	−	4.72368e6i	0.0974115	−	0.186521i
$916$		0			0
$917$		−	6.85806e6i		−	0.269326i
$918$		0			0
$919$	3.67817e7			1.43662			0.718312	−	0.695721i	$-0.244915\pi$
$919$	3.67817e7			1.43662			0.718312	+	0.695721i	$0.244915\pi$
$920$		0			0
$921$	−6.66933e6			−0.259079
$922$		0			0
$923$		−	2.73333e7i		−	1.05606i
$924$		0			0
$925$	−9.78175e6	+	6.81064e6i	−0.375891	+	0.261718i
$926$		0			0
$927$			2.94748e7i			1.12655i
$928$		0			0
$929$	−1.53870e7			−0.584943			−0.292471	−	0.956274i	$-0.594478\pi$
$929$	−1.53870e7			−0.584943			−0.292471	+	0.956274i	$0.594478\pi$
$930$		0			0
$931$	3.12233e6			0.118060
$932$		0			0
$933$			9.18808e6i			0.345558i
$934$		0			0
$935$	−507536.	+	971817.i	−0.0189862	+	0.0363543i
$936$		0			0
$937$		−	1.48496e6i		−	0.0552543i	−0.999618	−	0.0276272i	$-0.991205\pi$
$937$		−	1.48496e6i		−	0.0552543i	0.999618	−	0.0276272i	$-0.00879512\pi$
$938$		0			0
$939$	−3.04398e6			−0.112662
$940$		0			0
$941$	2.99275e7			1.10178			0.550891	−	0.834577i	$-0.314288\pi$
$941$	2.99275e7			1.10178			0.550891	+	0.834577i	$0.314288\pi$
$942$		0			0
$943$			1.81879e6i			0.0666044i
$944$		0			0
$945$	−6.37910e6	−	3.33152e6i	−0.232370	−	0.121356i
$946$		0			0
$947$			2.52221e7i			0.913917i	0.889488	+	0.456958i	$0.151061\pi$
$947$			2.52221e7i			0.913917i	−0.889488	+	0.456958i	$0.848939\pi$
$948$		0			0
$949$	−2.29384e7			−0.826794
$950$		0			0
$951$	1.55539e7			0.557685
$952$		0			0
$953$		−	1.88094e7i		−	0.670878i	−0.942062	−	0.335439i	$-0.891115\pi$
$953$		−	1.88094e7i		−	0.670878i	0.942062	−	0.335439i	$-0.108885\pi$
$954$		0			0
$955$	−3.12211e7	−	1.63054e7i	−1.10775	−	0.578526i
$956$		0			0
$957$		−	1.76791e7i		−	0.623993i
$958$		0			0
$959$	1.90805e6			0.0669952
$960$		0			0
$961$	−1.89742e7			−0.662759
$962$		0			0
$963$		−	1.17523e7i		−	0.408374i
$964$		0			0
$965$	−2.67075e7	+	5.11388e7i	−0.923240	+	1.76780i
$966$		0			0
$967$			1.20385e7i			0.414004i	0.978340	+	0.207002i	$0.0663707\pi$
$967$			1.20385e7i			0.414004i	−0.978340	+	0.207002i	$0.933629\pi$
$968$		0			0
$969$	−356207.			−0.0121869
$970$		0			0
$971$	3.10000e7			1.05515			0.527575	−	0.849508i	$-0.323101\pi$
$971$	3.10000e7			1.05515			0.527575	+	0.849508i	$0.323101\pi$
$972$		0			0
$973$			56182.9i			0.00190249i
$974$		0			0
$975$	−4.34824e6	−	6.24514e6i	−0.146488	−	0.210393i
$976$		0			0
$977$		−	2.02990e7i		−	0.680358i	−0.940361	−	0.340179i	$-0.889512\pi$
$977$		−	2.02990e7i		−	0.680358i	0.940361	−	0.340179i	$-0.110488\pi$
$978$		0			0
$979$	5.30402e7			1.76868
$980$		0			0
$981$	4.33534e6			0.143831
$982$		0			0
$983$			4.49537e7i			1.48382i	0.670498	+	0.741911i	$0.266080\pi$
$983$			4.49537e7i			1.48382i	−0.670498	+	0.741911i	$0.733920\pi$
$984$		0			0
$985$	−3.70471e6	+	7.09369e6i	−0.121665	+	0.232960i
$986$		0			0
$987$			3.51354e6i			0.114803i
$988$		0			0
$989$	−7.69033e6			−0.250008
$990$		0			0
$991$	3.23706e7			1.04705			0.523525	−	0.852011i	$-0.324617\pi$
$991$	3.23706e7			1.04705			0.523525	+	0.852011i	$0.324617\pi$
$992$		0			0
$993$			7.19706e6i			0.231623i
$994$		0			0
$995$	−2.71501e7	−	1.41793e7i	−0.869388	−	0.454042i
$996$		0			0
$997$		−	4.92250e7i		−	1.56837i	−0.620529	−	0.784183i	$-0.713082\pi$
$997$		−	4.92250e7i		−	1.56837i	0.620529	−	0.784183i	$-0.286918\pi$
$998$		0			0
$999$	1.00209e7			0.317683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.g.a.169.9	✓	20
4.3	odd	2		560.6.g.g.449.12		20
5.4	even	2	inner	280.6.g.a.169.12	yes	20
20.19	odd	2		560.6.g.g.449.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.g.a.169.9	✓	20	1.1	even	1	trivial
280.6.g.a.169.12	yes	20	5.4	even	2	inner
560.6.g.g.449.9		20	20.19	odd	2
560.6.g.g.449.12		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)

\(f(q)\)	\(=\)	\(q-5.80939i q^{3} +(49.5511 + 25.8783i) q^{5} -49.0000i q^{7} +209.251 q^{9} +415.954 q^{11} -419.172i q^{13} +(150.337 - 287.862i) q^{15} +47.1504i q^{17} -1300.43 q^{19} -284.660 q^{21} +887.588i q^{23} +(1785.63 + 2564.60i) q^{25} -2627.30i q^{27} +7316.18 q^{29} -3107.24 q^{31} -2416.44i q^{33} +(1268.04 - 2428.00i) q^{35} +3814.15i q^{37} -2435.13 q^{39} +2049.14 q^{41} +8664.30i q^{43} +(10368.6 + 5415.06i) q^{45} -12342.9i q^{47} -2401.00 q^{49} +273.915 q^{51} -12300.2i q^{53} +(20611.0 + 10764.2i) q^{55} +7554.69i q^{57} -11957.5 q^{59} +16409.6 q^{61} -10253.3i q^{63} +(10847.5 - 20770.4i) q^{65} -28933.1i q^{67} +5156.34 q^{69} +65207.9 q^{71} -54723.1i q^{73} +(14898.7 - 10373.4i) q^{75} -20381.7i q^{77} +27902.0 q^{79} +35585.0 q^{81} -93273.4i q^{83} +(-1220.17 + 2336.36i) q^{85} -42502.5i q^{87} +127515. q^{89} -20539.4 q^{91} +18051.1i q^{93} +(-64437.7 - 33652.9i) q^{95} +102289. i q^{97} +87038.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51}+ \cdots - 630244 q^{99}+O(q^{100}) \) 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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