LMFDB - Embedded newform 280.6.g.a.169.5

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)

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);

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")

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			169.5
Root			$-14.6692i$ of defining polynomial
Character	$\chi$	$=$	280.169
Dual form			280.6.g.a.169.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-14.6692i q^{3} +(-55.0879 + 9.50366i) q^{5} +49.0000i q^{7} +27.8149 q^{9} +341.248 q^{11} -439.638i q^{13} +(139.411 + 808.095i) q^{15} -602.434i q^{17} +649.259 q^{19} +718.790 q^{21} -191.245i q^{23} +(2944.36 - 1047.07i) q^{25} -3972.63i q^{27} -614.500 q^{29} -8957.89 q^{31} -5005.83i q^{33} +(-465.679 - 2699.31i) q^{35} -5406.81i q^{37} -6449.13 q^{39} -7440.24 q^{41} +5330.00i q^{43} +(-1532.27 + 264.343i) q^{45} -5230.38i q^{47} -2401.00 q^{49} -8837.22 q^{51} +18342.2i q^{53} +(-18798.6 + 3243.10i) q^{55} -9524.10i q^{57} -19693.0 q^{59} +834.097 q^{61} +1362.93i q^{63} +(4178.17 + 24218.7i) q^{65} +8377.91i q^{67} -2805.41 q^{69} -25484.4 q^{71} +25333.9i q^{73} +(-15359.7 - 43191.4i) q^{75} +16721.2i q^{77} -52783.7 q^{79} -51516.3 q^{81} -34973.5i q^{83} +(5725.33 + 33186.9i) q^{85} +9014.22i q^{87} -26289.1 q^{89} +21542.3 q^{91} +131405. i q^{93} +(-35766.3 + 6170.33i) q^{95} -67525.5i q^{97} +9491.78 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	14.6692i		−	0.941029i	−0.882392	−	0.470514i	$-0.844068\pi$
$3$		−	14.6692i		−	0.941029i	0.882392	−	0.470514i	$-0.155932\pi$
$4$		0			0
$5$	−55.0879	+	9.50366i	−0.985443	+	0.170007i
$5$	−55.0879	+	9.50366i	−0.985443	+	0.170007i
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	27.8149			0.114465
$10$		0			0
$11$	341.248			0.850332			0.425166	−	0.905115i	$-0.360216\pi$
$11$	341.248			0.850332			0.425166	+	0.905115i	$0.360216\pi$
$12$		0			0
$13$		−	439.638i		−	0.721501i	−0.932662	−	0.360750i	$-0.882521\pi$
$13$		−	439.638i		−	0.721501i	0.932662	−	0.360750i	$-0.117479\pi$
$14$		0			0
$15$	139.411	+	808.095i	0.159981	+	0.927330i
$16$		0			0
$17$		−	602.434i		−	0.505577i	−0.967522	−	0.252789i	$-0.918652\pi$
$17$		−	602.434i		−	0.505577i	0.967522	−	0.252789i	$-0.0813477\pi$
$18$		0			0
$19$	649.259			0.412604			0.206302	−	0.978488i	$-0.433857\pi$
$19$	649.259			0.412604			0.206302	+	0.978488i	$0.433857\pi$
$20$		0			0
$21$	718.790			0.355675
$22$		0			0
$23$		−	191.245i		−	0.0753826i	−0.999289	−	0.0376913i	$-0.988000\pi$
$23$		−	191.245i		−	0.0753826i	0.999289	−	0.0376913i	$-0.0120004\pi$
$24$		0			0
$25$	2944.36	−	1047.07i	0.942196	−	0.335064i
$26$		0			0
$27$		−	3972.63i		−	1.04874i
$28$		0			0
$29$	−614.500			−0.135683			−0.0678417	−	0.997696i	$-0.521611\pi$
$29$	−614.500			−0.135683			−0.0678417	+	0.997696i	$0.521611\pi$
$30$		0			0
$31$	−8957.89			−1.67418			−0.837089	−	0.547067i	$-0.815744\pi$
$31$	−8957.89			−1.67418			−0.837089	+	0.547067i	$0.815744\pi$
$32$		0			0
$33$		−	5005.83i		−	0.800187i
$34$		0			0
$35$	−465.679	−	2699.31i	−0.0642564	−	0.372462i
$36$		0			0
$37$		−	5406.81i		−	0.649287i	−0.945836	−	0.324644i	$-0.894756\pi$
$37$		−	5406.81i		−	0.649287i	0.945836	−	0.324644i	$-0.105244\pi$
$38$		0			0
$39$	−6449.13			−0.678953
$40$		0			0
$41$	−7440.24			−0.691238			−0.345619	−	0.938375i	$-0.612331\pi$
$41$	−7440.24			−0.691238			−0.345619	+	0.938375i	$0.612331\pi$
$42$		0			0
$43$			5330.00i			0.439598i	0.975545	+	0.219799i	$0.0705402\pi$
$43$			5330.00i			0.439598i	−0.975545	+	0.219799i	$0.929460\pi$
$44$		0			0
$45$	−1532.27	+	264.343i	−0.112798	+	0.0194597i
$46$		0			0
$47$		−	5230.38i		−	0.345373i	−0.984977	−	0.172687i	$-0.944755\pi$
$47$		−	5230.38i		−	0.345373i	0.984977	−	0.172687i	$-0.0552448\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−8837.22			−0.475763
$52$		0			0
$53$			18342.2i			0.896938i	0.893798	+	0.448469i	$0.148031\pi$
$53$			18342.2i			0.896938i	−0.893798	+	0.448469i	$0.851969\pi$
$54$		0			0
$55$	−18798.6	+	3243.10i	−0.837953	+	0.144562i
$56$		0			0
$57$		−	9524.10i		−	0.388272i
$58$		0			0
$59$	−19693.0			−0.736515			−0.368258	−	0.929724i	$-0.620046\pi$
$59$	−19693.0			−0.736515			−0.368258	+	0.929724i	$0.620046\pi$
$60$		0			0
$61$	834.097			0.0287007			0.0143503	−	0.999897i	$-0.495432\pi$
$61$	834.097			0.0287007			0.0143503	+	0.999897i	$0.495432\pi$
$62$		0			0
$63$			1362.93i			0.0432636i
$64$		0			0
$65$	4178.17	+	24218.7i	0.122660	+	0.710998i
$66$		0			0
$67$			8377.91i			0.228007i	0.993480	+	0.114004i	$0.0363675\pi$
$67$			8377.91i			0.228007i	−0.993480	+	0.114004i	$0.963632\pi$
$68$		0			0
$69$	−2805.41			−0.0709372
$70$		0			0
$71$	−25484.4			−0.599968			−0.299984	−	0.953944i	$-0.596981\pi$
$71$	−25484.4			−0.599968			−0.299984	+	0.953944i	$0.596981\pi$
$72$		0			0
$73$			25333.9i			0.556410i	0.960522	+	0.278205i	$0.0897395\pi$
$73$			25333.9i			0.556410i	−0.960522	+	0.278205i	$0.910261\pi$
$74$		0			0
$75$	−15359.7	−	43191.4i	−0.315304	−	0.886633i
$76$		0			0
$77$			16721.2i			0.321395i
$78$		0			0
$79$	−52783.7			−0.951551			−0.475776	−	0.879567i	$-0.657833\pi$
$79$	−52783.7			−0.951551			−0.475776	+	0.879567i	$0.657833\pi$
$80$		0			0
$81$	−51516.3			−0.872433
$82$		0			0
$83$		−	34973.5i		−	0.557242i	−0.960401	−	0.278621i	$-0.910123\pi$
$83$		−	34973.5i		−	0.557242i	0.960401	−	0.278621i	$-0.0898773\pi$
$84$		0			0
$85$	5725.33	+	33186.9i	0.0859514	+	0.498217i
$86$		0			0
$87$			9014.22i			0.127682i
$88$		0			0
$89$	−26289.1			−0.351804			−0.175902	−	0.984408i	$-0.556284\pi$
$89$	−26289.1			−0.351804			−0.175902	+	0.984408i	$0.556284\pi$
$90$		0			0
$91$	21542.3			0.272702
$92$		0			0
$93$			131405.i			1.57545i
$94$		0			0
$95$	−35766.3	+	6170.33i	−0.406598	+	0.0701454i
$96$		0			0
$97$		−	67525.5i		−	0.728682i	−0.931266	−	0.364341i	$-0.881294\pi$
$97$		−	67525.5i		−	0.728682i	0.931266	−	0.364341i	$-0.118706\pi$
$98$		0			0
$99$	9491.78			0.0973329
$100$		0			0
$101$	−93789.1			−0.914848			−0.457424	−	0.889249i	$-0.651228\pi$
$101$	−93789.1			−0.914848			−0.457424	+	0.889249i	$0.651228\pi$
$102$		0			0
$103$		−	150808.i		−	1.40066i	−0.713820	−	0.700329i	$-0.753037\pi$
$103$		−	150808.i		−	1.40066i	0.713820	−	0.700329i	$-0.246963\pi$
$104$		0			0
$105$	−39596.7	+	6831.14i	−0.350498	+	0.0604672i
$106$		0			0
$107$		−	179343.i		−	1.51435i	−0.653213	−	0.757174i	$-0.726580\pi$
$107$		−	179343.i		−	1.51435i	0.653213	−	0.757174i	$-0.273420\pi$
$108$		0			0
$109$	−211337.			−1.70376			−0.851881	−	0.523736i	$-0.824538\pi$
$109$	−211337.			−1.70376			−0.851881	+	0.523736i	$0.824538\pi$
$110$		0			0
$111$	−79313.5			−0.610998
$112$		0			0
$113$		−	90264.5i		−	0.664999i	−0.943103	−	0.332499i	$-0.892108\pi$
$113$		−	90264.5i		−	0.664999i	0.943103	−	0.332499i	$-0.107892\pi$
$114$		0			0
$115$	1817.53	+	10535.3i	0.0128155	+	0.0742853i
$116$		0			0
$117$		−	12228.5i		−	0.0825863i
$118$		0			0
$119$	29519.3			0.191090
$120$		0			0
$121$	−44600.8			−0.276936
$122$		0			0
$123$			109142.i			0.650474i
$124$		0			0
$125$	−152248.	+	85663.3i	−0.871517	+	0.490365i
$126$		0			0
$127$		−	286965.i		−	1.57877i	−0.613896	−	0.789387i	$-0.710398\pi$
$127$		−	286965.i		−	1.57877i	0.613896	−	0.789387i	$-0.289602\pi$
$128$		0			0
$129$	78186.7			0.413674
$130$		0			0
$131$	166317.			0.846758			0.423379	−	0.905953i	$-0.360844\pi$
$131$	166317.			0.846758			0.423379	+	0.905953i	$0.360844\pi$
$132$		0			0
$133$			31813.7i			0.155950i
$134$		0			0
$135$	37754.6	+	218844.i	0.178293	+	1.03348i
$136$		0			0
$137$		−	79463.7i		−	0.361716i	−0.983509	−	0.180858i	$-0.942113\pi$
$137$		−	79463.7i		−	0.361716i	0.983509	−	0.180858i	$-0.0578874\pi$
$138$		0			0
$139$	−215530.			−0.946172			−0.473086	−	0.881016i	$-0.656860\pi$
$139$	−215530.			−0.946172			−0.473086	+	0.881016i	$0.656860\pi$
$140$		0			0
$141$	−76725.4			−0.325006
$142$		0			0
$143$		−	150026.i		−	0.613515i
$144$		0			0
$145$	33851.5	−	5840.00i	0.133708	−	0.0230671i
$146$		0			0
$147$			35220.7i			0.134433i
$148$		0			0
$149$	412265.			1.52128			0.760642	−	0.649171i	$-0.224884\pi$
$149$	412265.			1.52128			0.760642	+	0.649171i	$0.224884\pi$
$150$		0			0
$151$	−119643.			−0.427018			−0.213509	−	0.976941i	$-0.568489\pi$
$151$	−119643.			−0.427018			−0.213509	+	0.976941i	$0.568489\pi$
$152$		0			0
$153$		−	16756.7i		−	0.0578707i
$154$		0			0
$155$	493472.	−	85132.7i	1.64981	−	0.284621i
$156$		0			0
$157$		−	27450.5i		−	0.0888795i	−0.999012	−	0.0444397i	$-0.985850\pi$
$157$		−	27450.5i		−	0.0888795i	0.999012	−	0.0444397i	$-0.0141503\pi$
$158$		0			0
$159$	269066.			0.844045
$160$		0			0
$161$	9371.02			0.0284920
$162$		0			0
$163$			175366.i			0.516982i	0.966014	+	0.258491i	$0.0832253\pi$
$163$			175366.i			0.516982i	−0.966014	+	0.258491i	$0.916775\pi$
$164$		0			0
$165$	47573.7	+	275761.i	0.136037	+	0.788538i
$166$		0			0
$167$		−	358292.i		−	0.994137i	−0.867711	−	0.497068i	$-0.834410\pi$
$167$		−	358292.i		−	0.994137i	0.867711	−	0.497068i	$-0.165590\pi$
$168$		0			0
$169$	178011.			0.479437
$170$		0			0
$171$	18059.1			0.0472286
$172$		0			0
$173$			426234.i			1.08276i	0.840777	+	0.541381i	$0.182098\pi$
$173$			426234.i			1.08276i	−0.840777	+	0.541381i	$0.817902\pi$
$174$		0			0
$175$	51306.6	+	144274.i	0.126642	+	0.356116i
$176$		0			0
$177$			288880.i			0.693082i
$178$		0			0
$179$	385606.			0.899522			0.449761	−	0.893149i	$-0.351509\pi$
$179$	385606.			0.899522			0.449761	+	0.893149i	$0.351509\pi$
$180$		0			0
$181$	−672846.			−1.52658			−0.763290	−	0.646056i	$-0.776417\pi$
$181$	−672846.			−1.52658			−0.763290	+	0.646056i	$0.776417\pi$
$182$		0			0
$183$		−	12235.5i		−	0.0270081i
$184$		0			0
$185$	51384.5	+	297850.i	0.110383	+	0.639835i
$186$		0			0
$187$		−	205580.i		−	0.429908i
$188$		0			0
$189$	194659.			0.396388
$190$		0			0
$191$	91941.4			0.182359			0.0911796	−	0.995834i	$-0.470936\pi$
$191$	91941.4			0.182359			0.0911796	+	0.995834i	$0.470936\pi$
$192$		0			0
$193$			471730.i			0.911591i	0.890084	+	0.455796i	$0.150645\pi$
$193$			471730.i			0.911591i	−0.890084	+	0.455796i	$0.849355\pi$
$194$		0			0
$195$	355269.	−	61290.3i	0.669069	−	0.115426i
$196$		0			0
$197$			768387.i			1.41063i	0.708892	+	0.705317i	$0.249195\pi$
$197$			768387.i			1.41063i	−0.708892	+	0.705317i	$0.750805\pi$
$198$		0			0
$199$	−929517.			−1.66389			−0.831945	−	0.554858i	$-0.812773\pi$
$199$	−929517.			−1.66389			−0.831945	+	0.554858i	$0.812773\pi$
$200$		0			0
$201$	122897.			0.214561
$202$		0			0
$203$		−	30110.5i		−	0.0512835i
$204$		0			0
$205$	409867.	−	70709.5i	0.681175	−	0.117515i
$206$		0			0
$207$		−	5319.47i		−	0.00862865i
$208$		0			0
$209$	221558.			0.350850
$210$		0			0
$211$	748945.			1.15809			0.579047	−	0.815294i	$-0.303425\pi$
$211$	748945.			1.15809			0.579047	+	0.815294i	$0.303425\pi$
$212$		0			0
$213$			373835.i			0.564587i
$214$		0			0
$215$	−50654.5	−	293618.i	−0.0747346	−	0.433199i
$216$		0			0
$217$		−	438937.i		−	0.632780i
$218$		0			0
$219$	371628.			0.523598
$220$		0			0
$221$	−264853.			−0.364774
$222$		0			0
$223$		−	634384.i		−	0.854261i	−0.904190	−	0.427130i	$-0.859525\pi$
$223$		−	634384.i		−	0.854261i	0.904190	−	0.427130i	$-0.140475\pi$
$224$		0			0
$225$	81897.1	−	29124.3i	0.107848	−	0.0383529i
$226$		0			0
$227$		−	1.00381e6i		−	1.29297i	−0.762928	−	0.646484i	$-0.776239\pi$
$227$		−	1.00381e6i		−	1.29297i	0.762928	−	0.646484i	$-0.223761\pi$
$228$		0			0
$229$	1.17402e6			1.47941			0.739704	−	0.672932i	$-0.234966\pi$
$229$	1.17402e6			1.47941			0.739704	+	0.672932i	$0.234966\pi$
$230$		0			0
$231$	245286.			0.302442
$232$		0			0
$233$		−	572322.i		−	0.690638i	−0.938485	−	0.345319i	$-0.887771\pi$
$233$		−	572322.i		−	0.690638i	0.938485	−	0.345319i	$-0.112229\pi$
$234$		0			0
$235$	49707.7	+	288131.i	0.0587157	+	0.340346i
$236$		0			0
$237$			774294.i			0.895437i
$238$		0			0
$239$	197064.			0.223158			0.111579	−	0.993756i	$-0.464409\pi$
$239$	197064.			0.223158			0.111579	+	0.993756i	$0.464409\pi$
$240$		0			0
$241$	648792.			0.719553			0.359777	−	0.933038i	$-0.382853\pi$
$241$	648792.			0.719553			0.359777	+	0.933038i	$0.382853\pi$
$242$		0			0
$243$		−	209648.i		−	0.227759i
$244$		0			0
$245$	132266.	−	22818.3i	0.140778	−	0.0242867i
$246$		0			0
$247$		−	285439.i		−	0.297694i
$248$		0			0
$249$	−513033.			−0.524381
$250$		0			0
$251$	−424915.			−0.425714			−0.212857	−	0.977083i	$-0.568277\pi$
$251$	−424915.			−0.425714			−0.212857	+	0.977083i	$0.568277\pi$
$252$		0			0
$253$		−	65262.1i		−	0.0641003i
$254$		0			0
$255$	486824.	−	83985.9i	0.468837	−	0.0808828i
$256$		0			0
$257$			1.67677e6i			1.58358i	0.610792	+	0.791791i	$0.290851\pi$
$257$			1.67677e6i			1.58358i	−0.610792	+	0.791791i	$0.709149\pi$
$258$		0			0
$259$	264934.			0.245407
$260$		0			0
$261$	−17092.3			−0.0155310
$262$		0			0
$263$			293550.i			0.261694i	0.991403	+	0.130847i	$0.0417696\pi$
$263$			293550.i			0.261694i	−0.991403	+	0.130847i	$0.958230\pi$
$264$		0			0
$265$	−174318.	−	1.01044e6i	−0.152485	−	0.883881i
$266$		0			0
$267$			385640.i			0.331058i
$268$		0			0
$269$	1.93470e6			1.63017			0.815086	−	0.579340i	$-0.196690\pi$
$269$	1.93470e6			1.63017			0.815086	+	0.579340i	$0.196690\pi$
$270$		0			0
$271$	−1.40684e6			−1.16365			−0.581825	−	0.813314i	$-0.697661\pi$
$271$	−1.40684e6			−1.16365			−0.581825	+	0.813314i	$0.697661\pi$
$272$		0			0
$273$		−	316007.i		−	0.256620i
$274$		0			0
$275$	1.00476e6	−	357312.i	0.801179	−	0.284915i
$276$		0			0
$277$		−	1.16389e6i		−	0.911406i	−0.890132	−	0.455703i	$-0.849388\pi$
$277$		−	1.16389e6i		−	0.911406i	0.890132	−	0.455703i	$-0.150612\pi$
$278$		0			0
$279$	−249163.			−0.191634
$280$		0			0
$281$	199201.			0.150496			0.0752482	−	0.997165i	$-0.476025\pi$
$281$	199201.			0.150496			0.0752482	+	0.997165i	$0.476025\pi$
$282$		0			0
$283$		−	107163.i		−	0.0795389i	−0.999209	−	0.0397694i	$-0.987338\pi$
$283$		−	107163.i		−	0.0795389i	0.999209	−	0.0397694i	$-0.0126623\pi$
$284$		0			0
$285$	90513.7	+	524663.i	0.0660089	+	0.382620i
$286$		0			0
$287$		−	364572.i		−	0.261263i
$288$		0			0
$289$	1.05693e6			0.744392
$290$		0			0
$291$	−990544.			−0.685711
$292$		0			0
$293$		−	553524.i		−	0.376675i	−0.982104	−	0.188338i	$-0.939690\pi$
$293$		−	553524.i		−	0.376675i	0.982104	−	0.188338i	$-0.0603099\pi$
$294$		0			0
$295$	1.08485e6	−	187156.i	0.725794	−	0.125212i
$296$		0			0
$297$		−	1.35565e6i		−	0.891780i
$298$		0			0
$299$	−84078.7			−0.0543886
$300$		0			0
$301$	−261170.			−0.166152
$302$		0			0
$303$			1.37581e6i			0.860899i
$304$		0			0
$305$	−45948.7	+	7926.97i	−0.0282829	+	0.00487930i
$306$		0			0
$307$		−	315776.i		−	0.191220i	−0.995419	−	0.0956100i	$-0.969520\pi$
$307$		−	315776.i		−	0.191220i	0.995419	−	0.0956100i	$-0.0304802\pi$
$308$		0			0
$309$	−2.21223e6			−1.31806
$310$		0			0
$311$	−1.48740e6			−0.872023			−0.436011	−	0.899941i	$-0.643609\pi$
$311$	−1.48740e6			−0.872023			−0.436011	+	0.899941i	$0.643609\pi$
$312$		0			0
$313$			1.99829e6i			1.15291i	0.817128	+	0.576457i	$0.195565\pi$
$313$			1.99829e6i			1.15291i	−0.817128	+	0.576457i	$0.804435\pi$
$314$		0			0
$315$	−12952.8	−	75081.0i	−0.00735509	−	0.0426338i
$316$		0			0
$317$			319663.i			0.178667i	0.996002	+	0.0893334i	$0.0284737\pi$
$317$			319663.i			0.178667i	−0.996002	+	0.0893334i	$0.971526\pi$
$318$		0			0
$319$	−209697.			−0.115376
$320$		0			0
$321$	−2.63082e6			−1.42504
$322$		0			0
$323$		−	391136.i		−	0.208603i
$324$		0			0
$325$	−460333.	−	1.29445e6i	−0.241749	−	0.679795i
$326$		0			0
$327$			3.10014e6i			1.60329i
$328$		0			0
$329$	256289.			0.130539
$330$		0			0
$331$	−1.86050e6			−0.933383			−0.466691	−	0.884420i	$-0.654554\pi$
$331$	−1.86050e6			−0.933383			−0.466691	+	0.884420i	$0.654554\pi$
$332$		0			0
$333$		−	150390.i		−	0.0743204i
$334$		0			0
$335$	−79620.7	−	461522.i	−0.0387627	−	0.224688i
$336$		0			0
$337$		−	272158.i		−	0.130541i	−0.997868	−	0.0652705i	$-0.979209\pi$
$337$		−	272158.i		−	0.130541i	0.997868	−	0.0652705i	$-0.0207910\pi$
$338$		0			0
$339$	−1.32411e6			−0.625783
$340$		0			0
$341$	−3.05686e6			−1.42361
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	154544.	−	26661.7i	0.0699046	−	0.0120598i
$346$		0			0
$347$			3.84561e6i			1.71451i	0.514889	+	0.857257i	$0.327833\pi$
$347$			3.84561e6i			1.71451i	−0.514889	+	0.857257i	$0.672167\pi$
$348$		0			0
$349$	159406.			0.0700555			0.0350277	−	0.999386i	$-0.488848\pi$
$349$	159406.			0.0700555			0.0350277	+	0.999386i	$0.488848\pi$
$350$		0			0
$351$	−1.74652e6			−0.756669
$352$		0			0
$353$		−	3.05511e6i		−	1.30494i	−0.757815	−	0.652470i	$-0.773733\pi$
$353$		−	3.05511e6i		−	1.30494i	0.757815	−	0.652470i	$-0.226267\pi$
$354$		0			0
$355$	1.40388e6	−	242195.i	0.591234	−	0.101998i
$356$		0			0
$357$		−	433024.i		−	0.179821i
$358$		0			0
$359$	−2.23826e6			−0.916589			−0.458295	−	0.888800i	$-0.651540\pi$
$359$	−2.23826e6			−0.916589			−0.458295	+	0.888800i	$0.651540\pi$
$360$		0			0
$361$	−2.05456e6			−0.829758
$362$		0			0
$363$			654257.i			0.260605i
$364$		0			0
$365$	−240765.	−	1.39559e6i	−0.0945934	−	0.548310i
$366$		0			0
$367$			995191.i			0.385693i	0.981229	+	0.192846i	$0.0617719\pi$
$367$			995191.i			0.385693i	−0.981229	+	0.192846i	$0.938228\pi$
$368$		0			0
$369$	−206950.			−0.0791223
$370$		0			0
$371$	−898769.			−0.339011
$372$		0			0
$373$		−	1.46662e6i		−	0.545816i	−0.962040	−	0.272908i	$-0.912014\pi$
$373$		−	1.46662e6i		−	0.545816i	0.962040	−	0.272908i	$-0.0879855\pi$
$374$		0			0
$375$	1.25661e6	+	2.23335e6i	0.461448	+	0.820123i
$376$		0			0
$377$			270158.i			0.0978957i
$378$		0			0
$379$	1.13954e6			0.407505			0.203752	−	0.979022i	$-0.434686\pi$
$379$	1.13954e6			0.407505			0.203752	+	0.979022i	$0.434686\pi$
$380$		0			0
$381$	−4.20955e6			−1.48567
$382$		0			0
$383$		−	3.21168e6i		−	1.11876i	−0.828913	−	0.559378i	$-0.811040\pi$
$383$		−	3.21168e6i		−	1.11876i	0.828913	−	0.559378i	$-0.188960\pi$
$384$		0			0
$385$	−158912.	−	921134.i	−0.0546393	−	0.316717i
$386$		0			0
$387$			148253.i			0.0503184i
$388$		0			0
$389$	1.91138e6			0.640432			0.320216	−	0.947345i	$-0.396245\pi$
$389$	1.91138e6			0.640432			0.320216	+	0.947345i	$0.396245\pi$
$390$		0			0
$391$	−115213.			−0.0381117
$392$		0			0
$393$		−	2.43974e6i		−	0.796824i
$394$		0			0
$395$	2.90775e6	−	501638.i	0.937699	−	0.161770i
$396$		0			0
$397$			5.18750e6i			1.65189i	0.563749	+	0.825946i	$0.309358\pi$
$397$			5.18750e6i			1.65189i	−0.563749	+	0.825946i	$0.690642\pi$
$398$		0			0
$399$	466681.			0.146753
$400$		0			0
$401$	−2.36540e6			−0.734586			−0.367293	−	0.930105i	$-0.619715\pi$
$401$	−2.36540e6			−0.734586			−0.367293	+	0.930105i	$0.619715\pi$
$402$		0			0
$403$			3.93823e6i			1.20792i
$404$		0			0
$405$	2.83793e6	−	489593.i	0.859733	−	0.148319i
$406$		0			0
$407$		−	1.84506e6i		−	0.552110i
$408$		0			0
$409$	2.41270e6			0.713172			0.356586	−	0.934263i	$-0.383941\pi$
$409$	2.41270e6			0.713172			0.356586	+	0.934263i	$0.383941\pi$
$410$		0			0
$411$	−1.16567e6			−0.340385
$412$		0			0
$413$		−	964957.i		−	0.278377i
$414$		0			0
$415$	332376.	+	1.92662e6i	0.0947348	+	0.549130i
$416$		0			0
$417$			3.16165e6i			0.890375i
$418$		0			0
$419$	2.17863e6			0.606244			0.303122	−	0.952952i	$-0.401971\pi$
$419$	2.17863e6			0.606244			0.303122	+	0.952952i	$0.401971\pi$
$420$		0			0
$421$	−4.59686e6			−1.26402			−0.632012	−	0.774958i	$-0.717771\pi$
$421$	−4.59686e6			−1.26402			−0.632012	+	0.774958i	$0.717771\pi$
$422$		0			0
$423$		−	145483.i		−	0.0395330i
$424$		0			0
$425$	−630793.	−	1.77378e6i	−0.169400	−	0.476353i
$426$		0			0
$427$			40870.7i			0.0108478i
$428$		0			0
$429$	−2.20075e6			−0.577335
$430$		0			0
$431$	−236140.			−0.0612317			−0.0306158	−	0.999531i	$-0.509747\pi$
$431$	−236140.			−0.0612317			−0.0306158	+	0.999531i	$0.509747\pi$
$432$		0			0
$433$		−	6.41329e6i		−	1.64385i	−0.569599	−	0.821923i	$-0.692902\pi$
$433$		−	6.41329e6i		−	1.64385i	0.569599	−	0.821923i	$-0.307098\pi$
$434$		0			0
$435$	−85668.0	−	496575.i	−0.0217068	−	0.125823i
$436$		0			0
$437$		−	124168.i		−	0.0311032i
$438$		0			0
$439$	−3.52684e6			−0.873422			−0.436711	−	0.899602i	$-0.643857\pi$
$439$	−3.52684e6			−0.873422			−0.436711	+	0.899602i	$0.643857\pi$
$440$		0			0
$441$	−66783.6			−0.0163521
$442$		0			0
$443$			4.85321e6i			1.17495i	0.809242	+	0.587475i	$0.199878\pi$
$443$			4.85321e6i			1.17495i	−0.809242	+	0.587475i	$0.800122\pi$
$444$		0			0
$445$	1.44821e6	−	249843.i	0.346683	−	0.0598091i
$446$		0			0
$447$		−	6.04759e6i		−	1.43157i
$448$		0			0
$449$	1.45115e6			0.339700			0.169850	−	0.985470i	$-0.445672\pi$
$449$	1.45115e6			0.339700			0.169850	+	0.985470i	$0.445672\pi$
$450$		0			0
$451$	−2.53897e6			−0.587781
$452$		0			0
$453$			1.75507e6i			0.401837i
$454$		0			0
$455$	−1.18672e6	+	204730.i	−0.268732	+	0.0463611i
$456$		0			0
$457$		−	4.50725e6i		−	1.00953i	−0.863255	−	0.504767i	$-0.831578\pi$
$457$		−	4.50725e6i		−	1.00953i	0.863255	−	0.504767i	$-0.168422\pi$
$458$		0			0
$459$	−2.39325e6			−0.530221
$460$		0			0
$461$	8.83196e6			1.93555			0.967776	−	0.251814i	$-0.0810272\pi$
$461$	8.83196e6			1.93555			0.967776	+	0.251814i	$0.0810272\pi$
$462$		0			0
$463$		−	3.07452e6i		−	0.666537i	−0.942832	−	0.333268i	$-0.891848\pi$
$463$		−	3.07452e6i		−	0.666537i	0.942832	−	0.333268i	$-0.108152\pi$
$464$		0			0
$465$	−1.24883e6	−	7.23883e6i	−0.267837	−	1.55252i
$466$		0			0
$467$		−	5.14269e6i		−	1.09119i	−0.838051	−	0.545593i	$-0.816305\pi$
$467$		−	5.14269e6i		−	1.09119i	0.838051	−	0.545593i	$-0.183695\pi$
$468$		0			0
$469$	−410517.			−0.0861786
$470$		0			0
$471$	−402677.			−0.0836382
$472$		0			0
$473$			1.81885e6i			0.373804i
$474$		0			0
$475$	1.91165e6	−	679821.i	0.388754	−	0.138249i
$476$		0			0
$477$			510188.i			0.102668i
$478$		0			0
$479$	9.10692e6			1.81356			0.906782	−	0.421600i	$-0.138532\pi$
$479$	9.10692e6			1.81356			0.906782	+	0.421600i	$0.138532\pi$
$480$		0			0
$481$	−2.37704e6			−0.468461
$482$		0			0
$483$		−	137465.i		−	0.0268118i
$484$		0			0
$485$	641739.	+	3.71984e6i	0.123881	+	0.718075i
$486$		0			0
$487$			8.05908e6i			1.53980i	0.638167	+	0.769898i	$0.279693\pi$
$487$			8.05908e6i			1.53980i	−0.638167	+	0.769898i	$0.720307\pi$
$488$		0			0
$489$	2.57247e6			0.486495
$490$		0			0
$491$	−3.90044e6			−0.730147			−0.365073	−	0.930979i	$-0.618956\pi$
$491$	−3.90044e6			−0.730147			−0.365073	+	0.930979i	$0.618956\pi$
$492$		0			0
$493$			370196.i			0.0685985i
$494$		0			0
$495$	−522883.	+	90206.7i	−0.0959161	+	0.0165472i
$496$		0			0
$497$		−	1.24873e6i		−	0.226767i
$498$		0			0
$499$	−9.64079e6			−1.73325			−0.866625	−	0.498960i	$-0.833715\pi$
$499$	−9.64079e6			−1.73325			−0.866625	+	0.498960i	$0.833715\pi$
$500$		0			0
$501$	−5.25585e6			−0.935511
$502$		0			0
$503$			1.16340e6i			0.205026i	0.994732	+	0.102513i	$0.0326883\pi$
$503$			1.16340e6i			0.205026i	−0.994732	+	0.102513i	$0.967312\pi$
$504$		0			0
$505$	5.16665e6	−	891340.i	0.901531	−	0.155530i
$506$		0			0
$507$		−	2.61128e6i		−	0.451164i
$508$		0			0
$509$	7.73793e6			1.32382			0.661912	−	0.749582i	$-0.269745\pi$
$509$	7.73793e6			1.32382			0.661912	+	0.749582i	$0.269745\pi$
$510$		0			0
$511$	−1.24136e6			−0.210303
$512$		0			0
$513$		−	2.57927e6i		−	0.432716i
$514$		0			0
$515$	1.43323e6	+	8.30771e6i	0.238121	+	1.38027i
$516$		0			0
$517$		−	1.78486e6i		−	0.293682i
$518$		0			0
$519$	6.25251e6			1.01891
$520$		0			0
$521$	−248838.			−0.0401627			−0.0200813	−	0.999798i	$-0.506393\pi$
$521$	−248838.			−0.0401627			−0.0200813	+	0.999798i	$0.506393\pi$
$522$		0			0
$523$			5.45353e6i			0.871813i	0.899992	+	0.435906i	$0.143572\pi$
$523$			5.45353e6i			0.871813i	−0.899992	+	0.435906i	$0.856428\pi$
$524$		0			0
$525$	2.11638e6	−	752626.i	0.335116	−	0.119174i
$526$		0			0
$527$			5.39654e6i			0.846426i
$528$		0			0
$529$	6.39977e6			0.994317
$530$		0			0
$531$	−547759.			−0.0843050
$532$		0			0
$533$			3.27101e6i			0.498728i
$534$		0			0
$535$	1.70442e6	+	9.87965e6i	0.257449	+	1.49230i
$536$		0			0
$537$		−	5.65653e6i		−	0.846476i
$538$		0			0
$539$	−819336.			−0.121476
$540$		0			0
$541$	4.32656e6			0.635550			0.317775	−	0.948166i	$-0.397064\pi$
$541$	4.32656e6			0.635550			0.317775	+	0.948166i	$0.397064\pi$
$542$		0			0
$543$			9.87011e6i			1.43656i
$544$		0			0
$545$	1.16421e7	−	2.00847e6i	1.67896	−	0.289651i
$546$		0			0
$547$		−	9.03707e6i		−	1.29140i	−0.763593	−	0.645698i	$-0.776566\pi$
$547$		−	9.03707e6i		−	1.29140i	0.763593	−	0.645698i	$-0.223434\pi$
$548$		0			0
$549$	23200.3			0.00328521
$550$		0			0
$551$	−398969.			−0.0559836
$552$		0			0
$553$		−	2.58640e6i		−	0.359653i
$554$		0			0
$555$	4.36922e6	−	753768.i	0.602104	−	0.103874i
$556$		0			0
$557$			1.33231e7i			1.81956i	0.415092	+	0.909779i	$0.363749\pi$
$557$			1.33231e7i			1.81956i	−0.415092	+	0.909779i	$0.636251\pi$
$558$		0			0
$559$	2.34327e6			0.317170
$560$		0			0
$561$	−3.01568e6			−0.404556
$562$		0			0
$563$		−	1.87300e6i		−	0.249039i	−0.992217	−	0.124519i	$-0.960261\pi$
$563$		−	1.87300e6i		−	0.249039i	0.992217	−	0.124519i	$-0.0397389\pi$
$564$		0			0
$565$	857843.	+	4.97249e6i	0.113054	+	0.655318i
$566$		0			0
$567$		−	2.52430e6i		−	0.329749i
$568$		0			0
$569$	1.04724e6			0.135602			0.0678008	−	0.997699i	$-0.478402\pi$
$569$	1.04724e6			0.135602			0.0678008	+	0.997699i	$0.478402\pi$
$570$		0			0
$571$	−8.44858e6			−1.08441			−0.542205	−	0.840246i	$-0.682410\pi$
$571$	−8.44858e6			−1.08441			−0.542205	+	0.840246i	$0.682410\pi$
$572$		0			0
$573$		−	1.34871e6i		−	0.171605i
$574$		0			0
$575$	−200248.	−	563095.i	−0.0252580	−	0.0710252i
$576$		0			0
$577$		−	2.10637e6i		−	0.263387i	−0.991290	−	0.131693i	$-0.957959\pi$
$577$		−	2.10637e6i		−	0.263387i	0.991290	−	0.131693i	$-0.0420415\pi$
$578$		0			0
$579$	6.91990e6			0.857834
$580$		0			0
$581$	1.71370e6			0.210618
$582$		0			0
$583$			6.25925e6i			0.762695i
$584$		0			0
$585$	116215.	+	673642.i	0.0140402	+	0.0813841i
$586$		0			0
$587$		−	5.90675e6i		−	0.707543i	−0.935332	−	0.353772i	$-0.884899\pi$
$587$		−	5.90675e6i		−	0.707543i	0.935332	−	0.353772i	$-0.115101\pi$
$588$		0			0
$589$	−5.81599e6			−0.690773
$590$		0			0
$591$	1.12716e7			1.32745
$592$		0			0
$593$			3.75821e6i			0.438878i	0.975626	+	0.219439i	$0.0704228\pi$
$593$			3.75821e6i			0.438878i	−0.975626	+	0.219439i	$0.929577\pi$
$594$		0			0
$595$	−1.62616e6	+	280541.i	−0.188308	+	0.0324866i
$596$		0			0
$597$			1.36353e7i			1.56577i
$598$		0			0
$599$	1.58162e7			1.80109			0.900545	−	0.434762i	$-0.143168\pi$
$599$	1.58162e7			1.80109			0.900545	+	0.434762i	$0.143168\pi$
$600$		0			0
$601$	1.79730e6			0.202971			0.101485	−	0.994837i	$-0.467640\pi$
$601$	1.79730e6			0.202971			0.101485	+	0.994837i	$0.467640\pi$
$602$		0			0
$603$			233031.i			0.0260988i
$604$		0			0
$605$	2.45697e6	−	423871.i	0.272904	−	0.0470809i
$606$		0			0
$607$		−	1.01472e7i		−	1.11783i	−0.829226	−	0.558913i	$-0.811218\pi$
$607$		−	1.01472e7i		−	1.11783i	0.829226	−	0.558913i	$-0.188782\pi$
$608$		0			0
$609$	−441697.			−0.0482593
$610$		0			0
$611$	−2.29947e6			−0.249187
$612$		0			0
$613$			1.86067e7i			1.99994i	0.00764220	+	0.999971i	$0.497567\pi$
$613$			1.86067e7i			1.99994i	−0.00764220	+	0.999971i	$0.502433\pi$
$614$		0			0
$615$	−1.03725e6	−	6.01242e6i	−0.110585	−	0.641005i
$616$		0			0
$617$		−	5.37524e6i		−	0.568441i	−0.958759	−	0.284220i	$-0.908265\pi$
$617$		−	5.37524e6i		−	0.568441i	0.958759	−	0.284220i	$-0.0917347\pi$
$618$		0			0
$619$	6.42372e6			0.673845			0.336922	−	0.941532i	$-0.390614\pi$
$619$	6.42372e6			0.673845			0.336922	+	0.941532i	$0.390614\pi$
$620$		0			0
$621$	−759748.			−0.0790571
$622$		0			0
$623$		−	1.28817e6i		−	0.132970i
$624$		0			0
$625$	7.57290e6	−	6.16593e6i	0.775465	−	0.631391i
$626$		0			0
$627$		−	3.25008e6i		−	0.330160i
$628$		0			0
$629$	−3.25725e6			−0.328265
$630$		0			0
$631$	614335.			0.0614232			0.0307116	−	0.999528i	$-0.490223\pi$
$631$	614335.			0.0614232			0.0307116	+	0.999528i	$0.490223\pi$
$632$		0			0
$633$		−	1.09864e7i		−	1.08980i
$634$		0			0
$635$	2.72722e6	+	1.58083e7i	0.268402	+	1.55579i
$636$		0			0
$637$			1.05557e6i			0.103072i
$638$		0			0
$639$	−708846.			−0.0686751
$640$		0			0
$641$	7.86089e6			0.755660			0.377830	−	0.925875i	$-0.376670\pi$
$641$	7.86089e6			0.755660			0.377830	+	0.925875i	$0.376670\pi$
$642$		0			0
$643$		−	1.52696e7i		−	1.45646i	−0.685330	−	0.728232i	$-0.740342\pi$
$643$		−	1.52696e7i		−	1.45646i	0.685330	−	0.728232i	$-0.259658\pi$
$644$		0			0
$645$	−4.30715e6	+	743060.i	−0.407653	+	0.0703274i
$646$		0			0
$647$		−	1.82230e7i		−	1.71143i	−0.517451	−	0.855713i	$-0.673119\pi$
$647$		−	1.82230e7i		−	1.71143i	0.517451	−	0.855713i	$-0.326881\pi$
$648$		0			0
$649$	−6.72020e6			−0.626282
$650$		0			0
$651$	−6.43884e6			−0.595464
$652$		0			0
$653$			4.47978e6i			0.411125i	0.978644	+	0.205563i	$0.0659024\pi$
$653$			4.47978e6i			0.411125i	−0.978644	+	0.205563i	$0.934098\pi$
$654$		0			0
$655$	−9.16208e6	+	1.58062e6i	−0.834432	+	0.143954i
$656$		0			0
$657$			704660.i			0.0636893i
$658$		0			0
$659$	1.10772e7			0.993611			0.496806	−	0.867862i	$-0.334506\pi$
$659$	1.10772e7			0.993611			0.496806	+	0.867862i	$0.334506\pi$
$660$		0			0
$661$	−1.87751e7			−1.67139			−0.835695	−	0.549194i	$-0.814935\pi$
$661$	−1.87751e7			−1.67139			−0.835695	+	0.549194i	$0.814935\pi$
$662$		0			0
$663$			3.88518e6i			0.343263i
$664$		0			0
$665$	−302346.	−	1.75255e6i	−0.0265125	−	0.153680i
$666$		0			0
$667$			117520.i			0.0102282i
$668$		0			0
$669$	−9.30590e6			−0.803884
$670$		0			0
$671$	284634.			0.0244051
$672$		0			0
$673$		−	2.16795e7i		−	1.84507i	−0.385918	−	0.922533i	$-0.626115\pi$
$673$		−	2.16795e7i		−	1.84507i	0.385918	−	0.922533i	$-0.373885\pi$
$674$		0			0
$675$	−4.15964e6	−	1.16969e7i	−0.351396	−	0.988121i
$676$		0			0
$677$			1.78783e7i			1.49918i	0.661900	+	0.749592i	$0.269750\pi$
$677$			1.78783e7i			1.49918i	−0.661900	+	0.749592i	$0.730250\pi$
$678$		0			0
$679$	3.30875e6			0.275416
$680$		0			0
$681$	−1.47251e7			−1.21672
$682$		0			0
$683$		−	300512.i		−	0.0246496i	−0.999924	−	0.0123248i	$-0.996077\pi$
$683$		−	300512.i		−	0.0246496i	0.999924	−	0.0123248i	$-0.00392320\pi$
$684$		0			0
$685$	755196.	+	4.37749e6i	0.0614941	+	0.356450i
$686$		0			0
$687$		−	1.72220e7i		−	1.39217i
$688$		0			0
$689$	8.06394e6			0.647142
$690$		0			0
$691$	−1.56756e7			−1.24890			−0.624452	−	0.781063i	$-0.714678\pi$
$691$	−1.56756e7			−1.24890			−0.624452	+	0.781063i	$0.714678\pi$
$692$		0			0
$693$			465097.i			0.0367884i
$694$		0			0
$695$	1.18731e7	−	2.04832e6i	0.932398	−	0.160855i
$696$		0			0
$697$			4.48226e6i			0.349474i
$698$		0			0
$699$	−8.39549e6			−0.649910
$700$		0			0
$701$	7.51766e6			0.577813			0.288907	−	0.957357i	$-0.406708\pi$
$701$	7.51766e6			0.577813			0.288907	+	0.957357i	$0.406708\pi$
$702$		0			0
$703$		−	3.51042e6i		−	0.267899i
$704$		0			0
$705$	4.22665e6	−	729172.i	0.320275	−	0.0552532i
$706$		0			0
$707$		−	4.59567e6i		−	0.345780i
$708$		0			0
$709$	−2.07266e7			−1.54850			−0.774252	−	0.632878i	$-0.781874\pi$
$709$	−2.07266e7			−1.54850			−0.774252	+	0.632878i	$0.781874\pi$
$710$		0			0
$711$	−1.46817e6			−0.108919
$712$		0			0
$713$			1.71316e6i			0.126204i
$714$		0			0
$715$	1.42579e6	+	8.26460e6i	0.104302	+	0.604584i
$716$		0			0
$717$		−	2.89076e6i		−	0.209998i
$718$		0			0
$719$	−6.05962e6			−0.437142			−0.218571	−	0.975821i	$-0.570140\pi$
$719$	−6.05962e6			−0.437142			−0.218571	+	0.975821i	$0.570140\pi$
$720$		0			0
$721$	7.38960e6			0.529399
$722$		0			0
$723$		−	9.51725e6i		−	0.677120i
$724$		0			0
$725$	−1.80931e6	+	643427.i	−0.127840	+	0.0454626i
$726$		0			0
$727$		−	8.76278e6i		−	0.614902i	−0.951564	−	0.307451i	$-0.900524\pi$
$727$		−	8.76278e6i		−	0.614902i	0.951564	−	0.307451i	$-0.0994761\pi$
$728$		0			0
$729$	−1.55938e7			−1.08676
$730$		0			0
$731$	3.21097e6			0.222251
$732$		0			0
$733$			1.93206e7i			1.32819i	0.747649	+	0.664094i	$0.231183\pi$
$733$			1.93206e7i			1.32819i	−0.747649	+	0.664094i	$0.768817\pi$
$734$		0			0
$735$	−334726.	−	1.94024e6i	−0.0228544	−	0.132476i
$736$		0			0
$737$			2.85894e6i			0.193882i
$738$		0			0
$739$	669807.			0.0451169			0.0225584	−	0.999746i	$-0.492819\pi$
$739$	669807.			0.0451169			0.0225584	+	0.999746i	$0.492819\pi$
$740$		0			0
$741$	−4.18715e6			−0.280139
$742$		0			0
$743$			1.06451e7i			0.707418i	0.935356	+	0.353709i	$0.115080\pi$
$743$			1.06451e7i			0.707418i	−0.935356	+	0.353709i	$0.884920\pi$
$744$		0			0
$745$	−2.27108e7	+	3.91802e6i	−1.49914	+	0.258628i
$746$		0			0
$747$		−	972785.i		−	0.0637845i
$748$		0			0
$749$	8.78782e6			0.572370
$750$		0			0
$751$	1.02189e7			0.661154			0.330577	−	0.943779i	$-0.392757\pi$
$751$	1.02189e7			0.661154			0.330577	+	0.943779i	$0.392757\pi$
$752$		0			0
$753$			6.23315e6i			0.400609i
$754$		0			0
$755$	6.59091e6	−	1.13705e6i	0.420802	−	0.0725959i
$756$		0			0
$757$			5.86387e6i			0.371916i	0.982558	+	0.185958i	$0.0595388\pi$
$757$			5.86387e6i			0.371916i	−0.982558	+	0.185958i	$0.940461\pi$
$758$		0			0
$759$	−957342.			−0.0603202
$760$		0			0
$761$	1.39721e7			0.874579			0.437290	−	0.899321i	$-0.355938\pi$
$761$	1.39721e7			0.874579			0.437290	+	0.899321i	$0.355938\pi$
$762$		0			0
$763$		−	1.03555e7i		−	0.643961i
$764$		0			0
$765$	159250.	+	923090.i	0.00983840	+	0.0570283i
$766$		0			0
$767$			8.65779e6i			0.531396i
$768$		0			0
$769$	−7.82547e6			−0.477193			−0.238597	−	0.971119i	$-0.576687\pi$
$769$	−7.82547e6			−0.477193			−0.238597	+	0.971119i	$0.576687\pi$
$770$		0			0
$771$	2.45969e7			1.49020
$772$		0			0
$773$		−	1.27309e7i		−	0.766322i	−0.923682	−	0.383161i	$-0.874835\pi$
$773$		−	1.27309e7i		−	0.766322i	0.923682	−	0.383161i	$-0.125165\pi$
$774$		0			0
$775$	−2.63753e7	+	9.37957e6i	−1.57740	+	0.560956i
$776$		0			0
$777$		−	3.88636e6i		−	0.230936i
$778$		0			0
$779$	−4.83064e6			−0.285208
$780$		0			0
$781$	−8.69649e6			−0.510172
$782$		0			0
$783$			2.44118e6i			0.142297i
$784$		0			0
$785$	260880.	+	1.51219e6i	0.0151101	+	0.0875857i
$786$		0			0
$787$			4.45153e6i			0.256196i	0.991762	+	0.128098i	$0.0408872\pi$
$787$			4.45153e6i			0.256196i	−0.991762	+	0.128098i	$0.959113\pi$
$788$		0			0
$789$	4.30615e6			0.246261
$790$		0			0
$791$	4.42296e6			0.251346
$792$		0			0
$793$		−	366700.i		−	0.0207075i
$794$		0			0
$795$	−1.48223e7	+	2.55711e6i	−0.831758	+	0.143493i
$796$		0			0
$797$			3.47382e6i			0.193714i	0.995298	+	0.0968570i	$0.0308790\pi$
$797$			3.47382e6i			0.193714i	−0.995298	+	0.0968570i	$0.969121\pi$
$798$		0			0
$799$	−3.15096e6			−0.174613
$800$		0			0
$801$	−731230.			−0.0402692
$802$		0			0
$803$			8.64514e6i			0.473133i
$804$		0			0
$805$	−516230.	+	89059.0i	−0.0280772	+	0.00484382i
$806$		0			0
$807$		−	2.83805e7i		−	1.53404i
$808$		0			0
$809$	−2.80278e7			−1.50563			−0.752814	−	0.658233i	$-0.771304\pi$
$809$	−2.80278e7			−1.50563			−0.752814	+	0.658233i	$0.771304\pi$
$810$		0			0
$811$	−2.46216e7			−1.31451			−0.657256	−	0.753668i	$-0.728283\pi$
$811$	−2.46216e7			−1.31451			−0.657256	+	0.753668i	$0.728283\pi$
$812$		0			0
$813$			2.06372e7i			1.09503i
$814$		0			0
$815$	−1.66662e6	−	9.66054e6i	−0.0878904	−	0.509457i
$816$		0			0
$817$			3.46055e6i			0.181380i
$818$		0			0
$819$	599196.			0.0312147
$820$		0			0
$821$	3.44740e7			1.78498			0.892490	−	0.451066i	$-0.148956\pi$
$821$	3.44740e7			1.78498			0.892490	+	0.451066i	$0.148956\pi$
$822$		0			0
$823$		−	1.00349e6i		−	0.0516430i	−0.999667	−	0.0258215i	$-0.991780\pi$
$823$		−	1.00349e6i		−	0.0516430i	0.999667	−	0.0258215i	$-0.00822015\pi$
$824$		0			0
$825$	−5.24147e6	−	1.47390e7i	−0.268113	−	0.753932i
$826$		0			0
$827$			1.17957e7i			0.599738i	0.953980	+	0.299869i	$0.0969429\pi$
$827$			1.17957e7i			0.599738i	−0.953980	+	0.299869i	$0.903057\pi$
$828$		0			0
$829$	3.56920e6			0.180379			0.0901893	−	0.995925i	$-0.471253\pi$
$829$	3.56920e6			0.180379			0.0901893	+	0.995925i	$0.471253\pi$
$830$		0			0
$831$	−1.70733e7			−0.857660
$832$		0			0
$833$			1.44644e6i			0.0722253i
$834$		0			0
$835$	3.40509e6	+	1.97376e7i	0.169010	+	0.979665i
$836$		0			0
$837$			3.55864e7i			1.75578i
$838$		0			0
$839$	−3.39622e7			−1.66568			−0.832840	−	0.553514i	$-0.813286\pi$
$839$	−3.39622e7			−1.66568			−0.832840	+	0.553514i	$0.813286\pi$
$840$		0			0
$841$	−2.01335e7			−0.981590
$842$		0			0
$843$		−	2.92212e6i		−	0.141621i
$844$		0			0
$845$	−9.80628e6	+	1.69176e6i	−0.472457	+	0.0815074i
$846$		0			0
$847$		−	2.18544e6i		−	0.104672i
$848$		0			0
$849$	−1.57200e6			−0.0748484
$850$		0			0
$851$	−1.03403e6			−0.0489450
$852$		0			0
$853$		−	3.91329e7i		−	1.84149i	−0.390165	−	0.920745i	$-0.627582\pi$
$853$		−	3.91329e7i		−	1.84149i	0.390165	−	0.920745i	$-0.372418\pi$
$854$		0			0
$855$	−994837.	+	171627.i	−0.0465411	+	0.00802917i
$856$		0			0
$857$			2.09215e7i			0.973061i	0.873664	+	0.486530i	$0.161738\pi$
$857$			2.09215e7i			0.973061i	−0.873664	+	0.486530i	$0.838262\pi$
$858$		0			0
$859$	2.06467e7			0.954702			0.477351	−	0.878713i	$-0.341597\pi$
$859$	2.06467e7			0.954702			0.477351	+	0.878713i	$0.341597\pi$
$860$		0			0
$861$	−5.34797e6			−0.245856
$862$		0			0
$863$		−	5.12173e6i		−	0.234094i	−0.993126	−	0.117047i	$-0.962657\pi$
$863$		−	5.12173e6i		−	0.234094i	0.993126	−	0.117047i	$-0.0373428\pi$
$864$		0			0
$865$	−4.05078e6	−	2.34804e7i	−0.184077	−	1.06700i
$866$		0			0
$867$		−	1.55043e7i		−	0.700494i
$868$		0			0
$869$	−1.80123e7			−0.809134
$870$		0			0
$871$	3.68325e6			0.164507
$872$		0			0
$873$		−	1.87822e6i		−	0.0834084i
$874$		0			0
$875$	−4.19750e6	−	7.46014e6i	−0.185341	−	0.329402i
$876$		0			0
$877$			2.87856e7i			1.26379i	0.775052	+	0.631897i	$0.217723\pi$
$877$			2.87856e7i			1.26379i	−0.775052	+	0.631897i	$0.782277\pi$
$878$		0			0
$879$	−8.11975e6			−0.354462
$880$		0			0
$881$	4.86756e6			0.211287			0.105643	−	0.994404i	$-0.466310\pi$
$881$	4.86756e6			0.211287			0.105643	+	0.994404i	$0.466310\pi$
$882$		0			0
$883$		−	3.46194e6i		−	0.149423i	−0.997205	−	0.0747116i	$-0.976196\pi$
$883$		−	3.46194e6i		−	0.149423i	0.997205	−	0.0747116i	$-0.0238036\pi$
$884$		0			0
$885$	−2.74542e6	−	1.59138e7i	−0.117829	−	0.682993i
$886$		0			0
$887$			1.36901e7i			0.584246i	0.956381	+	0.292123i	$0.0943617\pi$
$887$			1.36901e7i			0.584246i	−0.956381	+	0.292123i	$0.905638\pi$
$888$		0			0
$889$	1.40613e7			0.596721
$890$		0			0
$891$	−1.75798e7			−0.741858
$892$		0			0
$893$		−	3.39587e6i		−	0.142502i
$894$		0			0
$895$	−2.12423e7	+	3.66467e6i	−0.886427	+	0.152925i
$896$		0			0
$897$			1.23337e6i			0.0511813i
$898$		0			0
$899$	5.50463e6			0.227158
$900$		0			0
$901$	1.10500e7			0.453471
$902$		0			0
$903$			3.83115e6i			0.156354i
$904$		0			0
$905$	3.70657e7	−	6.39450e6i	1.50436	−	0.259529i
$906$		0			0
$907$		−	9.04499e6i		−	0.365081i	−0.983198	−	0.182541i	$-0.941568\pi$
$907$		−	9.04499e6i		−	0.365081i	0.983198	−	0.182541i	$-0.0584321\pi$
$908$		0			0
$909$	−2.60874e6			−0.104718
$910$		0			0
$911$	2.85718e7			1.14062			0.570311	−	0.821429i	$-0.306823\pi$
$911$	2.85718e7			1.14062			0.570311	+	0.821429i	$0.306823\pi$
$912$		0			0
$913$		−	1.19346e7i		−	0.473840i
$914$		0			0
$915$	116282.	+	674029.i	0.00459156	+	0.0266150i
$916$		0			0
$917$			8.14955e6i			0.320044i
$918$		0			0
$919$	−2.62607e7			−1.02569			−0.512846	−	0.858481i	$-0.671409\pi$
$919$	−2.62607e7			−1.02569			−0.512846	+	0.858481i	$0.671409\pi$
$920$		0			0
$921$	−4.63218e6			−0.179944
$922$		0			0
$923$			1.12039e7i			0.432877i
$924$		0			0
$925$	−5.66133e6	−	1.59196e7i	−0.217552	−	0.611755i
$926$		0			0
$927$		−	4.19472e6i		−	0.160326i
$928$		0			0
$929$	4.01791e7			1.52743			0.763714	−	0.645555i	$-0.223374\pi$
$929$	4.01791e7			1.52743			0.763714	+	0.645555i	$0.223374\pi$
$930$		0			0
$931$	−1.55887e6			−0.0589435
$932$		0			0
$933$			2.18190e7i			0.820599i
$934$		0			0
$935$	1.95376e6	+	1.13250e7i	0.0730872	+	0.423650i
$936$		0			0
$937$			1.16005e7i			0.431644i	0.976433	+	0.215822i	$0.0692431\pi$
$937$			1.16005e7i			0.431644i	−0.976433	+	0.215822i	$0.930757\pi$
$938$		0			0
$939$	2.93132e7			1.08493
$940$		0			0
$941$	1.70775e7			0.628709			0.314354	−	0.949306i	$-0.398212\pi$
$941$	1.70775e7			0.628709			0.314354	+	0.949306i	$0.398212\pi$
$942$		0			0
$943$			1.42291e6i			0.0521073i
$944$		0			0
$945$	−1.07234e7	+	1.84997e6i	−0.390618	+	0.0673885i
$946$		0			0
$947$		−	3.73013e7i		−	1.35160i	−0.737083	−	0.675802i	$-0.763797\pi$
$947$		−	3.73013e7i		−	1.35160i	0.737083	−	0.675802i	$-0.236203\pi$
$948$		0			0
$949$	1.11377e7			0.401450
$950$		0			0
$951$	4.68919e6			0.168131
$952$		0			0
$953$		−	3.25762e7i		−	1.16190i	−0.813940	−	0.580949i	$-0.802682\pi$
$953$		−	3.25762e7i		−	1.16190i	0.813940	−	0.580949i	$-0.197318\pi$
$954$		0			0
$955$	−5.06486e6	+	873779.i	−0.179705	+	0.0310023i
$956$		0			0
$957$			3.07608e6i			0.108572i
$958$		0			0
$959$	3.89372e6			0.136716
$960$		0			0
$961$	5.16146e7			1.80287
$962$		0			0
$963$		−	4.98842e6i		−	0.173339i
$964$		0			0
$965$	−4.48316e6	−	2.59866e7i	−0.154977	−	0.898321i
$966$		0			0
$967$		−	1.67956e7i		−	0.577603i	−0.957389	−	0.288801i	$-0.906743\pi$
$967$		−	1.67956e7i		−	0.577603i	0.957389	−	0.288801i	$-0.0932567\pi$
$968$		0			0
$969$	−5.73764e6			−0.196302
$970$		0			0
$971$	2.23930e7			0.762191			0.381095	−	0.924536i	$-0.375547\pi$
$971$	2.23930e7			0.762191			0.381095	+	0.924536i	$0.375547\pi$
$972$		0			0
$973$		−	1.05610e7i		−	0.357619i
$974$		0			0
$975$	−1.89886e7	+	6.75272e6i	−0.639707	+	0.227492i
$976$		0			0
$977$			3.37475e7i			1.13111i	0.824710	+	0.565556i	$0.191338\pi$
$977$			3.37475e7i			1.13111i	−0.824710	+	0.565556i	$0.808662\pi$
$978$		0			0
$979$	−8.97111e6			−0.299150
$980$		0			0
$981$	−5.87831e6			−0.195021
$982$		0			0
$983$		−	1.53504e7i		−	0.506683i	−0.967377	−	0.253342i	$-0.918470\pi$
$983$		−	1.53504e7i		−	0.506683i	0.967377	−	0.253342i	$-0.0815296\pi$
$984$		0			0
$985$	−7.30249e6	−	4.23289e7i	−0.239817	−	1.39010i
$986$		0			0
$987$		−	3.75955e6i		−	0.122841i
$988$		0			0
$989$	1.01934e6			0.0331381
$990$		0			0
$991$	4.37551e7			1.41529			0.707644	−	0.706570i	$-0.249758\pi$
$991$	4.37551e7			1.41529			0.707644	+	0.706570i	$0.249758\pi$
$992$		0			0
$993$			2.72920e7i			0.878340i
$994$		0			0
$995$	5.12052e7	−	8.83381e6i	1.63967	−	0.282872i
$996$		0			0
$997$			2.40759e7i			0.767087i	0.923523	+	0.383544i	$0.125296\pi$
$997$			2.40759e7i			0.767087i	−0.923523	+	0.383544i	$0.874704\pi$
$998$		0			0
$999$	−2.14793e7			−0.680936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.g.a.169.5	✓	20
4.3	odd	2		560.6.g.g.449.16		20
5.4	even	2	inner	280.6.g.a.169.16	yes	20
20.19	odd	2		560.6.g.g.449.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.g.a.169.5	✓	20	1.1	even	1	trivial
280.6.g.a.169.16	yes	20	5.4	even	2	inner
560.6.g.g.449.5		20	20.19	odd	2
560.6.g.g.449.16		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)

\(f(q)\)	\(=\)	\(q-14.6692i q^{3} +(-55.0879 + 9.50366i) q^{5} +49.0000i q^{7} +27.8149 q^{9} +341.248 q^{11} -439.638i q^{13} +(139.411 + 808.095i) q^{15} -602.434i q^{17} +649.259 q^{19} +718.790 q^{21} -191.245i q^{23} +(2944.36 - 1047.07i) q^{25} -3972.63i q^{27} -614.500 q^{29} -8957.89 q^{31} -5005.83i q^{33} +(-465.679 - 2699.31i) q^{35} -5406.81i q^{37} -6449.13 q^{39} -7440.24 q^{41} +5330.00i q^{43} +(-1532.27 + 264.343i) q^{45} -5230.38i q^{47} -2401.00 q^{49} -8837.22 q^{51} +18342.2i q^{53} +(-18798.6 + 3243.10i) q^{55} -9524.10i q^{57} -19693.0 q^{59} +834.097 q^{61} +1362.93i q^{63} +(4178.17 + 24218.7i) q^{65} +8377.91i q^{67} -2805.41 q^{69} -25484.4 q^{71} +25333.9i q^{73} +(-15359.7 - 43191.4i) q^{75} +16721.2i q^{77} -52783.7 q^{79} -51516.3 q^{81} -34973.5i q^{83} +(5725.33 + 33186.9i) q^{85} +9014.22i q^{87} -26289.1 q^{89} +21542.3 q^{91} +131405. i q^{93} +(-35766.3 + 6170.33i) q^{95} -67525.5i q^{97} +9491.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51}+ \cdots - 630244 q^{99}+O(q^{100}) \) 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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