LMFDB - Embedded newform 280.3.c.f.69.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,3,Mod(69,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("280.69");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$7.62944740209$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49 $ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			69.4
Root			$-1.87083 + 1.87083i$ of defining polynomial
Character	$\chi$	$=$	280.69
Dual form			280.3.c.f.69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} +1.74166i q^{3} -4.00000 q^{4} +(4.87083 + 1.12917i) q^{5} -3.48331 q^{6} +7.00000i q^{7} -8.00000i q^{8} +5.96663 q^{9} +O(q^{10})$	\(q+2.00000i q^{2} +1.74166i q^{3} -4.00000 q^{4} +(4.87083 + 1.12917i) q^{5} -3.48331 q^{6} +7.00000i q^{7} -8.00000i q^{8} +5.96663 q^{9} +(-2.25834 + 9.74166i) q^{10} -6.96663i q^{12} +21.7417i q^{13} -14.0000 q^{14} +(-1.96663 + 8.48331i) q^{15} +16.0000 q^{16} +11.9333i q^{18} -32.1916 q^{19} +(-19.4833 - 4.51669i) q^{20} -12.1916 q^{21} -44.8999i q^{23} +13.9333 q^{24} +(22.4499 + 11.0000i) q^{25} -43.4833 q^{26} +26.0667i q^{27} -28.0000i q^{28} +(-16.9666 - 3.93326i) q^{30} +32.0000i q^{32} +(-7.90420 + 34.0958i) q^{35} -23.8665 q^{36} -64.3832i q^{38} -37.8665 q^{39} +(9.03337 - 38.9666i) q^{40} -24.3832i q^{42} +(29.0624 + 6.73735i) q^{45} +89.7998 q^{46} +27.8665i q^{48} -49.0000 q^{49} +(-22.0000 + 44.8999i) q^{50} -86.9666i q^{52} -52.1335 q^{54} +56.0000 q^{56} -56.0667i q^{57} -9.60818 q^{59} +(7.86652 - 33.9333i) q^{60} +80.0581 q^{61} +41.7664i q^{63} -64.0000 q^{64} +(-24.5501 + 105.900i) q^{65} +78.2002 q^{69} +(-68.1916 - 15.8084i) q^{70} +89.7998 q^{71} -47.7330i q^{72} +(-19.1582 + 39.1001i) q^{75} +128.766 q^{76} -75.7330i q^{78} +89.7998 q^{79} +(77.9333 + 18.0667i) q^{80} +8.30033 q^{81} +133.991i q^{83} +48.7664 q^{84} +(-13.4747 + 58.1249i) q^{90} -152.192 q^{91} +179.600i q^{92} +(-156.800 - 36.3498i) q^{95} -55.7330 q^{96} -98.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9	$ 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9} - 24 q^{10} - 56 q^{14} + 52 q^{15} + 64 q^{16} - 24 q^{19} - 48 q^{20} + 56 q^{21} - 64 q^{24} - 144 q^{26} - 8 q^{30} - 84 q^{35} + 144 q^{36} + 88 q^{39} + 96 q^{40} + 4 q^{45} - 196 q^{49} - 88 q^{50} - 448 q^{54} + 224 q^{56} + 216 q^{59} - 208 q^{60} - 24 q^{61} - 256 q^{64} - 188 q^{65} + 672 q^{69} - 168 q^{70} + 88 q^{75} + 96 q^{76} + 192 q^{80} + 572 q^{81} - 224 q^{84} + 440 q^{90} - 504 q^{91} - 268 q^{95} + 256 q^{96}+O(q^{100}) $ 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9 - 24 * q^10 - 56 * q^14 + 52 * q^15 + 64 * q^16 - 24 * q^19 - 48 * q^20 + 56 * q^21 - 64 * q^24 - 144 * q^26 - 8 * q^30 - 84 * q^35 + 144 * q^36 + 88 * q^39 + 96 * q^40 + 4 * q^45 - 196 * q^49 - 88 * q^50 - 448 * q^54 + 224 * q^56 + 216 * q^59 - 208 * q^60 - 24 * q^61 - 256 * q^64 - 188 * q^65 + 672 * q^69 - 168 * q^70 + 88 * q^75 + 96 * q^76 + 192 * q^80 + 572 * q^81 - 224 * q^84 + 440 * q^90 - 504 * q^91 - 268 * q^95 + 256 * q^96

Display 100 coefficients 10 coefficients

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 $

$2$			2.00000i			1.00000i
$2$			2.00000i			1.00000i
$3$			1.74166i			0.580552i	0.956943	+	0.290276i	$0.0937472\pi$
$3$			1.74166i			0.580552i	−0.956943	+	0.290276i	$0.906253\pi$
$4$	−4.00000			−1.00000
$5$	4.87083	+	1.12917i	0.974166	+	0.225834i
$5$	4.87083	+	1.12917i	0.974166	+	0.225834i
$6$	−3.48331			−0.580552
$7$			7.00000i			1.00000i
$7$			7.00000i			1.00000i
$8$		−	8.00000i		−	1.00000i
$9$	5.96663			0.662959
$10$	−2.25834	+	9.74166i	−0.225834	+	0.974166i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	6.96663i		−	0.580552i
$13$			21.7417i			1.67244i	0.548398	+	0.836218i	$0.315238\pi$
$13$			21.7417i			1.67244i	−0.548398	+	0.836218i	$0.684762\pi$
$14$	−14.0000			−1.00000
$15$	−1.96663	+	8.48331i	−0.131109	+	0.565554i
$16$	16.0000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$			11.9333i			0.662959i
$19$	−32.1916			−1.69429			−0.847147	−	0.531358i	$-0.821682\pi$
$19$	−32.1916			−1.69429			−0.847147	+	0.531358i	$0.821682\pi$
$20$	−19.4833	−	4.51669i	−0.974166	−	0.225834i
$21$	−12.1916			−0.580552
$22$		0			0
$23$		−	44.8999i		−	1.95217i	−0.217391	−	0.976085i	$-0.569755\pi$
$23$		−	44.8999i		−	1.95217i	0.217391	−	0.976085i	$-0.430245\pi$
$24$	13.9333			0.580552
$25$	22.4499	+	11.0000i	0.897998	+	0.440000i
$26$	−43.4833			−1.67244
$27$			26.0667i			0.965435i
$28$		−	28.0000i		−	1.00000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−16.9666	−	3.93326i	−0.565554	−	0.131109i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			32.0000i			1.00000i
$33$		0			0
$34$		0			0
$35$	−7.90420	+	34.0958i	−0.225834	+	0.974166i
$36$	−23.8665			−0.662959
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		−	64.3832i		−	1.69429i
$39$	−37.8665			−0.970936
$40$	9.03337	−	38.9666i	0.225834	−	0.974166i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		−	24.3832i		−	0.580552i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	29.0624	+	6.73735i	0.645832	+	0.149719i
$46$	89.7998			1.95217
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			27.8665i			0.580552i
$49$	−49.0000			−1.00000
$50$	−22.0000	+	44.8999i	−0.440000	+	0.897998i
$51$		0			0
$52$		−	86.9666i		−	1.67244i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−52.1335			−0.965435
$55$		0			0
$56$	56.0000			1.00000
$57$		−	56.0667i		−	0.983627i
$58$		0			0
$59$	−9.60818			−0.162850			−0.0814252	−	0.996679i	$-0.525947\pi$
$59$	−9.60818			−0.162850			−0.0814252	+	0.996679i	$0.525947\pi$
$60$	7.86652	−	33.9333i	0.131109	−	0.565554i
$61$	80.0581			1.31243			0.656214	−	0.754575i	$-0.272157\pi$
$61$	80.0581			1.31243			0.656214	+	0.754575i	$0.272157\pi$
$62$		0			0
$63$			41.7664i			0.662959i
$64$	−64.0000			−1.00000
$65$	−24.5501	+	105.900i	−0.377693	+	1.62923i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	78.2002			1.13334
$70$	−68.1916	−	15.8084i	−0.974166	−	0.225834i
$71$	89.7998			1.26479			0.632393	−	0.774648i	$-0.282073\pi$
$71$	89.7998			1.26479			0.632393	+	0.774648i	$0.282073\pi$
$72$		−	47.7330i		−	0.662959i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−19.1582	+	39.1001i	−0.255443	+	0.521335i
$76$	128.766			1.69429
$77$		0			0
$78$		−	75.7330i		−	0.970936i
$79$	89.7998			1.13671			0.568353	−	0.822785i	$-0.307581\pi$
$79$	89.7998			1.13671			0.568353	+	0.822785i	$0.307581\pi$
$80$	77.9333	+	18.0667i	0.974166	+	0.225834i
$81$	8.30033			0.102473
$82$		0			0
$83$			133.991i			1.61435i	0.590310	+	0.807177i	$0.299006\pi$
$83$			133.991i			1.61435i	−0.590310	+	0.807177i	$0.700994\pi$
$84$	48.7664			0.580552
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−13.4747	+	58.1249i	−0.149719	+	0.645832i
$91$	−152.192			−1.67244
$92$			179.600i			1.95217i
$93$		0			0
$94$		0			0
$95$	−156.800	−	36.3498i	−1.65052	−	0.382630i
$96$	−55.7330			−0.580552
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		−	98.0000i		−	1.00000i
$99$		0			0
$100$	−89.7998	−	44.0000i	−0.897998	−	0.440000i
$101$	200.058			1.98077			0.990387	−	0.138326i	$-0.0441722\pi$
$101$	200.058			1.98077			0.990387	+	0.138326i	$0.0441722\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	173.933			1.67244
$105$	−59.3832	−	13.7664i	−0.565554	−	0.131109i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		−	104.267i		−	0.965435i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$			112.000i			1.00000i
$113$		−	26.0000i		−	0.230088i	−0.993360	−	0.115044i	$-0.963299\pi$
$113$		−	26.0000i		−	0.230088i	0.993360	−	0.115044i	$-0.0367010\pi$
$114$	112.133			0.983627
$115$	50.6997	−	218.700i	0.440867	−	1.90174i
$116$		0			0
$117$			129.724i			1.10876i
$118$		−	19.2164i		−	0.162850i
$119$		0			0
$120$	67.8665	+	15.7330i	0.565554	+	0.131109i
$121$	121.000			1.00000
$122$			160.116i			1.31243i
$123$		0			0
$124$		0			0
$125$	96.9289	+	78.9289i	0.775432	+	0.631432i
$126$	−83.5328			−0.662959
$127$		−	44.8999i		−	0.353542i	−0.984252	−	0.176771i	$-0.943435\pi$
$127$		−	44.8999i		−	0.353542i	0.984252	−	0.176771i	$-0.0565653\pi$
$128$		−	128.000i		−	1.00000i
$129$		0			0
$130$	−211.800	−	49.1001i	−1.62923	−	0.377693i
$131$	110.392			0.842686			0.421343	−	0.906901i	$-0.361559\pi$
$131$	110.392			0.842686			0.421343	+	0.906901i	$0.361559\pi$
$132$		0			0
$133$		−	225.341i		−	1.69429i
$134$		0			0
$135$	−29.4338	+	126.967i	−0.218028	+	0.940494i
$136$		0			0
$137$		−	269.399i		−	1.96642i	−0.182482	−	0.983209i	$-0.558413\pi$
$137$		−	269.399i		−	1.96642i	0.182482	−	0.983209i	$-0.441587\pi$
$138$			156.400i			1.13334i
$139$	−249.608			−1.79574			−0.897871	−	0.440258i	$-0.854887\pi$
$139$	−249.608			−1.79574			−0.897871	+	0.440258i	$0.854887\pi$
$140$	31.6168	−	136.383i	0.225834	−	0.974166i
$141$		0			0
$142$			179.600i			1.26479i
$143$		0			0
$144$	95.4661			0.662959
$145$		0			0
$146$		0			0
$147$		−	85.3412i		−	0.580552i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−78.2002	−	38.3165i	−0.521335	−	0.255443i
$151$	202.000			1.33775			0.668874	−	0.743376i	$-0.266776\pi$
$151$	202.000			1.33775			0.668874	+	0.743376i	$0.266776\pi$
$152$			257.533i			1.69429i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	151.466			0.970936
$157$		−	218.258i		−	1.39018i	−0.718922	−	0.695090i	$-0.755364\pi$
$157$		−	218.258i		−	1.39018i	0.718922	−	0.695090i	$-0.244636\pi$
$158$			179.600i			1.13671i
$159$		0			0
$160$	−36.1335	+	155.867i	−0.225834	+	0.974166i
$161$	314.299			1.95217
$162$			16.6007i			0.102473i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−267.983			−1.61435
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$			97.5328i			0.580552i
$169$	−303.700			−1.79704
$170$		0			0
$171$	−192.075			−1.12325
$172$		0			0
$173$		−	345.341i		−	1.99619i	−0.0616798	−	0.998096i	$-0.519646\pi$
$173$		−	345.341i		−	1.99619i	0.0616798	−	0.998096i	$-0.480354\pi$
$174$		0			0
$175$	−77.0000	+	157.150i	−0.440000	+	0.897998i
$176$		0			0
$177$		−	16.7341i		−	0.0945432i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−116.250	−	26.9494i	−0.645832	−	0.149719i
$181$	−361.858			−1.99921			−0.999607	−	0.0280169i	$-0.991081\pi$
$181$	−361.858			−1.99921			−0.999607	+	0.0280169i	$0.991081\pi$
$182$		−	304.383i		−	1.67244i
$183$			139.434i			0.761933i
$184$	−359.199			−1.95217
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−182.467			−0.965435
$190$	72.6997	−	313.600i	0.382630	−	1.65052i
$191$	−359.199			−1.88062			−0.940312	−	0.340314i	$-0.889467\pi$
$191$	−359.199			−1.88062			−0.940312	+	0.340314i	$0.889467\pi$
$192$		−	111.466i		−	0.580552i
$193$			314.000i			1.62694i	0.581605	+	0.813472i	$0.302425\pi$
$193$			314.000i			1.62694i	−0.581605	+	0.813472i	$0.697575\pi$
$194$		0			0
$195$	−184.441	−	42.7578i	−0.945853	−	0.219271i
$196$	196.000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	88.0000	−	179.600i	0.440000	−	0.897998i
$201$		0			0
$202$			400.116i			1.98077i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	267.901i		−	1.29421i
$208$			347.867i			1.67244i
$209$		0			0
$210$	27.5328	−	118.766i	0.131109	−	0.565554i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$			156.400i			0.734274i
$214$		0			0
$215$		0			0
$216$	208.534			0.965435
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−224.000			−1.00000
$225$	133.951	+	65.6329i	0.595336	+	0.291702i
$226$	52.0000			0.230088
$227$		−	128.592i		−	0.566485i	−0.959048	−	0.283242i	$-0.908590\pi$
$227$		−	128.592i		−	0.566485i	0.959048	−	0.283242i	$-0.0914101\pi$
$228$			224.267i			0.983627i
$229$	447.141			1.95258			0.976290	−	0.216465i	$-0.0694527\pi$
$229$	447.141			1.95258			0.976290	+	0.216465i	$0.0694527\pi$
$230$	437.399	+	101.399i	1.90174	+	0.440867i
$231$		0			0
$232$		0			0
$233$			179.600i			0.770814i	0.922747	+	0.385407i	$0.125939\pi$
$233$			179.600i			0.770814i	−0.922747	+	0.385407i	$0.874061\pi$
$234$	−259.449			−1.10876
$235$		0			0
$236$	38.4327			0.162850
$237$			156.400i			0.659917i
$238$		0			0
$239$	422.000			1.76569			0.882845	−	0.469664i	$-0.155625\pi$
$239$	422.000			1.76569			0.882845	+	0.469664i	$0.155625\pi$
$240$	−31.4661	+	135.733i	−0.131109	+	0.565554i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			242.000i			1.00000i
$243$			249.057i			1.02493i
$244$	−320.232			−1.31243
$245$	−238.671	−	55.3294i	−0.974166	−	0.225834i
$246$		0			0
$247$		−	699.899i		−	2.83360i
$248$		0			0
$249$	−233.367			−0.937217
$250$	−157.858	+	193.858i	−0.631432	+	0.775432i
$251$	327.808			1.30601			0.653005	−	0.757354i	$-0.273508\pi$
$251$	327.808			1.30601			0.653005	+	0.757354i	$0.273508\pi$
$252$		−	167.066i		−	0.662959i
$253$		0			0
$254$	89.7998			0.353542
$255$		0			0
$256$	256.000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$	98.2002	−	423.600i	0.377693	−	1.62923i
$261$		0			0
$262$			220.784i			0.842686i
$263$			274.000i			1.04183i	0.853610	+	0.520913i	$0.174408\pi$
$263$			274.000i			1.04183i	−0.853610	+	0.520913i	$0.825592\pi$
$264$		0			0
$265$		0			0
$266$	450.682			1.69429
$267$		0			0
$268$		0			0
$269$	−248.941			−0.925430			−0.462715	−	0.886507i	$-0.653125\pi$
$269$	−248.941			−0.925430			−0.462715	+	0.886507i	$0.653125\pi$
$270$	−253.933	−	58.8676i	−0.940494	−	0.218028i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		−	265.066i		−	0.970936i
$274$	538.799			1.96642
$275$		0			0
$276$	−312.801			−1.13334
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		−	499.216i		−	1.79574i
$279$		0			0
$280$	272.766	+	63.2336i	0.974166	+	0.225834i
$281$	−338.000			−1.20285			−0.601423	−	0.798930i	$-0.705400\pi$
$281$	−338.000			−1.20285			−0.601423	+	0.798930i	$0.705400\pi$
$282$		0			0
$283$		−	555.008i		−	1.96116i	−0.196126	−	0.980579i	$-0.562836\pi$
$283$		−	555.008i		−	1.96116i	0.196126	−	0.980579i	$-0.437164\pi$
$284$	−359.199			−1.26479
$285$	63.3090	−	273.091i	0.222137	−	0.958216i
$286$		0			0
$287$		0			0
$288$			190.932i			0.662959i
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			583.658i			1.99201i	0.0893214	+	0.996003i	$0.471530\pi$
$293$			583.658i			1.99201i	−0.0893214	+	0.996003i	$0.528470\pi$
$294$	170.682			0.580552
$295$	−46.7998	−	10.8493i	−0.158643	−	0.0367772i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	976.198			3.26488
$300$	76.6329	−	156.400i	0.255443	−	0.521335i
$301$		0			0
$302$			404.000i			1.33775i
$303$			348.433i			1.14994i
$304$	−515.066			−1.69429
$305$	389.949	+	90.3993i	1.27852	+	0.296391i
$306$		0			0
$307$			22.4091i			0.0729937i	0.999334	+	0.0364969i	$0.0116199\pi$
$307$			22.4091i			0.0729937i	−0.999334	+	0.0364969i	$0.988380\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$			302.932i			0.970936i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$	436.517			1.39018
$315$	−47.1614	+	203.437i	−0.149719	+	0.645832i
$316$	−359.199			−1.13671
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	−311.733	−	72.2670i	−0.974166	−	0.225834i
$321$		0			0
$322$			628.598i			1.95217i
$323$		0			0
$324$	−33.2013			−0.102473
$325$	−239.158	+	488.099i	−0.735871	+	1.50184i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	535.966i		−	1.61435i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−195.066			−0.580552
$337$		−	26.0000i		−	0.0771513i	−0.999256	−	0.0385757i	$-0.987718\pi$
$337$		−	26.0000i		−	0.0771513i	0.999256	−	0.0385757i	$-0.0122821\pi$
$338$		−	607.399i		−	1.79704i
$339$	45.2831			0.133578
$340$		0			0
$341$		0			0
$342$		−	384.151i		−	1.12325i
$343$		−	343.000i		−	1.00000i
$344$		0			0
$345$	380.900	+	88.3014i	1.10406	+	0.255946i
$346$	690.682			1.99619
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−399.942			−1.14597			−0.572983	−	0.819568i	$-0.694214\pi$
$349$	−399.942			−1.14597			−0.572983	+	0.819568i	$0.694214\pi$
$350$	−314.299	−	154.000i	−0.897998	−	0.440000i
$351$	−566.734			−1.61463
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	33.4683			0.0945432
$355$	437.399	+	101.399i	1.23211	+	0.285632i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	682.000			1.89972			0.949861	−	0.312673i	$-0.101225\pi$
$359$	682.000			1.89972			0.949861	+	0.312673i	$0.101225\pi$
$360$	53.8988	−	232.499i	0.149719	−	0.645832i
$361$	675.299			1.87063
$362$		−	723.716i		−	1.99921i
$363$			210.741i			0.580552i
$364$	608.766			1.67244
$365$		0			0
$366$	−278.868			−0.761933
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		−	718.398i		−	1.95217i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−137.467	+	168.817i	−0.366579	+	0.450179i
$376$		0			0
$377$		0			0
$378$		−	364.934i		−	0.965435i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	627.199	+	145.399i	1.65052	+	0.382630i
$381$	78.2002			0.205250
$382$		−	718.398i		−	1.88062i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	222.932			0.580552
$385$		0			0
$386$	−628.000			−1.62694
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	85.5156	−	368.883i	0.219271	−	0.945853i
$391$		0			0
$392$			392.000i			1.00000i
$393$			192.265i			0.489223i
$394$		0			0
$395$	437.399	+	101.399i	1.10734	+	0.256707i
$396$		0			0
$397$		−	787.257i		−	1.98302i	−0.130047	−	0.991508i	$-0.541513\pi$
$397$		−	787.257i		−	1.98302i	0.130047	−	0.991508i	$-0.458487\pi$
$398$		0			0
$399$	392.467			0.983627
$400$	359.199	+	176.000i	0.897998	+	0.440000i
$401$	−583.699			−1.45561			−0.727804	−	0.685786i	$-0.759459\pi$
$401$	−583.699			−1.45561			−0.727804	+	0.685786i	$0.759459\pi$
$402$		0			0
$403$		0			0
$404$	−800.232			−1.98077
$405$	40.4295	+	9.37250i	0.0998259	+	0.0231420i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	469.201			1.14161
$412$		0			0
$413$		−	67.2572i		−	0.162850i
$414$	535.802			1.29421
$415$	−151.299	+	652.649i	−0.364576	+	1.57265i
$416$	−695.733			−1.67244
$417$		−	434.732i		−	1.04252i
$418$		0			0
$419$	−121.190			−0.289237			−0.144619	−	0.989487i	$-0.546196\pi$
$419$	−121.190			−0.289237			−0.144619	+	0.989487i	$0.546196\pi$
$420$	237.533	+	55.0656i	0.565554	+	0.131109i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−312.801			−0.734274
$427$			560.407i			1.31243i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−538.000			−1.24826			−0.624130	−	0.781321i	$-0.714546\pi$
$431$	−538.000			−1.24826			−0.624130	+	0.781321i	$0.714546\pi$
$432$			417.068i			0.965435i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1445.40i			3.30755i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−292.365			−0.662959
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	448.000i		−	1.00000i
$449$	2.00000			0.00445434			0.00222717	−	0.999998i	$-0.499291\pi$
$449$	2.00000			0.00445434			0.00222717	+	0.999998i	$0.499291\pi$
$450$	−131.266	+	267.901i	−0.291702	+	0.595336i
$451$		0			0
$452$			104.000i			0.230088i
$453$			351.815i			0.776633i
$454$	257.184			0.566485
$455$	−741.299	−	171.850i	−1.62923	−	0.377693i
$456$	−448.534			−0.983627
$457$		−	886.000i		−	1.93873i	−0.245623	−	0.969365i	$-0.578993\pi$
$457$		−	886.000i		−	1.93873i	0.245623	−	0.969365i	$-0.421007\pi$
$458$			894.282i			1.95258i
$459$		0			0
$460$	−202.799	+	874.799i	−0.440867	+	1.90174i
$461$	−810.857			−1.75891			−0.879454	−	0.475983i	$-0.842092\pi$
$461$	−810.857			−1.75891			−0.879454	+	0.475983i	$0.842092\pi$
$462$		0			0
$463$		−	226.000i		−	0.488121i	−0.969760	−	0.244060i	$-0.921520\pi$
$463$		−	226.000i		−	0.488121i	0.969760	−	0.244060i	$-0.0784795\pi$
$464$		0			0
$465$		0			0
$466$	−359.199			−0.770814
$467$			222.990i			0.477495i	0.971082	+	0.238748i	$0.0767369\pi$
$467$			222.990i			0.477495i	−0.971082	+	0.238748i	$0.923263\pi$
$468$		−	518.898i		−	1.10876i
$469$		0			0
$470$		0			0
$471$	380.131			0.807073
$472$			76.8654i			0.162850i
$473$		0			0
$474$	−312.801			−0.659917
$475$	−722.700	−	354.108i	−1.52147	−	0.745490i
$476$		0			0
$477$		0			0
$478$			844.000i			1.76569i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$	−271.466	−	62.9321i	−0.565554	−	0.131109i
$481$		0			0
$482$		0			0
$483$			547.402i			1.13334i
$484$	−484.000			−1.00000
$485$		0			0
$486$	−498.114			−1.02493
$487$			853.098i			1.75174i	0.482546	+	0.875871i	$0.339712\pi$
$487$			853.098i			1.75174i	−0.482546	+	0.875871i	$0.660288\pi$
$488$		−	640.465i		−	1.31243i
$489$		0			0
$490$	110.659	−	477.341i	0.225834	−	0.974166i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	1399.80			2.83360
$495$		0			0
$496$		0			0
$497$			628.598i			1.26479i
$498$		−	466.734i		−	0.937217i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−387.716	−	315.716i	−0.775432	−	0.631432i
$501$		0			0
$502$			655.617i			1.30601i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	334.131			0.662959
$505$	974.449	+	225.900i	1.92960	+	0.447327i
$506$		0			0
$507$		−	528.941i		−	1.04328i
$508$			179.600i			0.353542i
$509$	−392.859			−0.771825			−0.385913	−	0.922535i	$-0.626113\pi$
$509$	−392.859			−0.771825			−0.385913	+	0.922535i	$0.626113\pi$
$510$		0			0
$511$		0			0
$512$			512.000i			1.00000i
$513$		−	839.130i		−	1.63573i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	601.466			1.15889
$520$	847.199	+	196.400i	1.62923	+	0.377693i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			382.409i			0.731184i	0.930775	+	0.365592i	$0.119133\pi$
$523$			382.409i			0.731184i	−0.930775	+	0.365592i	$0.880867\pi$
$524$	−441.567			−0.842686
$525$	−273.701	−	134.108i	−0.521335	−	0.255443i
$526$	−548.000			−1.04183
$527$		0			0
$528$		0			0
$529$	−1487.00			−2.81096
$530$		0			0
$531$	−57.3284			−0.107963
$532$			901.365i			1.69429i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		−	497.882i		−	0.925430i
$539$		0			0
$540$	117.735	−	507.867i	0.218028	−	0.940494i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	630.232i		−	1.16065i
$544$		0			0
$545$		0			0
$546$	530.131			0.970936
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$			1077.60i			1.96642i
$549$	477.677			0.870086
$550$		0			0
$551$		0			0
$552$		−	625.602i		−	1.13334i
$553$			628.598i			1.13671i
$554$		0			0
$555$		0			0
$556$	998.433			1.79574
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−126.467	+	545.533i	−0.225834	+	0.974166i
$561$		0			0
$562$		−	676.000i		−	1.20285i
$563$		−	1026.59i		−	1.82343i	−0.410826	−	0.911714i	$-0.634760\pi$
$563$		−	1026.59i		−	1.82343i	0.410826	−	0.911714i	$-0.365240\pi$
$564$		0			0
$565$	29.3585	−	126.642i	0.0519619	−	0.224144i
$566$	1110.02			1.96116
$567$			58.1023i			0.102473i
$568$		−	718.398i		−	1.26479i
$569$	−134.700			−0.236731			−0.118365	−	0.992970i	$-0.537765\pi$
$569$	−134.700			−0.236731			−0.118365	+	0.992970i	$0.537765\pi$
$570$	546.183	+	126.618i	0.958216	+	0.222137i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		−	625.602i		−	1.09180i
$574$		0			0
$575$	493.899	−	1008.00i	0.858954	−	1.75304i
$576$	−381.864			−0.662959
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		−	578.000i		−	1.00000i
$579$	−546.880			−0.944526
$580$		0			0
$581$	−937.940			−1.61435
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−146.481	+	631.865i	−0.250395	+	1.08011i
$586$	−1167.32			−1.99201
$587$		−	706.009i		−	1.20274i	−0.798971	−	0.601370i	$-0.794622\pi$
$587$		−	706.009i		−	1.20274i	0.798971	−	0.601370i	$-0.205378\pi$
$588$			341.365i			0.580552i
$589$		0			0
$590$	21.6986	−	93.5996i	0.0367772	−	0.158643i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$			1952.40i			3.26488i
$599$	538.799			0.899497			0.449748	−	0.893155i	$-0.351514\pi$
$599$	538.799			0.899497			0.449748	+	0.893155i	$0.351514\pi$
$600$	312.801	+	153.266i	0.521335	+	0.255443i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−808.000			−1.33775
$605$	589.370	+	136.630i	0.974166	+	0.225834i
$606$	−696.865			−1.14994
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		−	1030.13i		−	1.69429i
$609$		0			0
$610$	−180.799	+	779.899i	−0.296391	+	1.27852i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−44.8181			−0.0729937
$615$		0			0
$616$		0			0
$617$			1034.00i			1.67585i	0.545785	+	0.837925i	$0.316232\pi$
$617$			1034.00i			1.67585i	−0.545785	+	0.837925i	$0.683768\pi$
$618$		0			0
$619$	−400.609			−0.647188			−0.323594	−	0.946196i	$-0.604891\pi$
$619$	−400.609			−0.647188			−0.323594	+	0.946196i	$0.604891\pi$
$620$		0			0
$621$	1170.39			1.88469
$622$		0			0
$623$		0			0
$624$	−605.864			−0.970936
$625$	383.000	+	493.899i	0.612800	+	0.790238i
$626$		0			0
$627$		0			0
$628$			873.033i			1.39018i
$629$		0			0
$630$	−406.874	−	94.3229i	−0.645832	−	0.149719i
$631$	−1257.20			−1.99239			−0.996194	−	0.0871632i	$-0.972220\pi$
$631$	−1257.20			−1.99239			−0.996194	+	0.0871632i	$0.972220\pi$
$632$		−	718.398i		−	1.13671i
$633$		0			0
$634$		0			0
$635$	50.6997	−	218.700i	0.0798420	−	0.344409i
$636$		0			0
$637$		−	1065.34i		−	1.67244i
$638$		0			0
$639$	535.802			0.838501
$640$	144.534	−	623.466i	0.225834	−	0.974166i
$641$	763.298			1.19079			0.595396	−	0.803432i	$-0.296995\pi$
$641$	763.298			1.19079			0.595396	+	0.803432i	$0.296995\pi$
$642$		0			0
$643$		−	1266.59i		−	1.96981i	−0.173087	−	0.984907i	$-0.555374\pi$
$643$		−	1266.59i		−	1.96981i	0.173087	−	0.984907i	$-0.444626\pi$
$644$	−1257.20			−1.95217
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	66.4027i		−	0.102473i
$649$		0			0
$650$	−976.198	−	478.316i	−1.50184	−	0.735871i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	537.700	+	124.651i	0.820916	+	0.190307i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1167.14			1.76572			0.882860	−	0.469636i	$-0.155615\pi$
$661$	1167.14			1.76572			0.882860	+	0.469636i	$0.155615\pi$
$662$		0			0
$663$		0			0
$664$	1071.93			1.61435
$665$	254.449	−	1097.60i	0.382630	−	1.65052i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		−	390.131i		−	0.580552i
$673$		−	1167.40i		−	1.73462i	−0.497771	−	0.867308i	$-0.665848\pi$
$673$		−	1167.40i		−	1.73462i	0.497771	−	0.867308i	$-0.334152\pi$
$674$	52.0000			0.0771513
$675$	−286.734	+	585.197i	−0.424791	+	0.866958i
$676$	1214.80			1.79704
$677$		−	1236.26i		−	1.82608i	−0.407870	−	0.913040i	$-0.633728\pi$
$677$		−	1236.26i		−	1.82608i	0.407870	−	0.913040i	$-0.366272\pi$
$678$			90.5662i			0.133578i
$679$		0			0
$680$		0			0
$681$	223.963			0.328874
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	768.301			1.12325
$685$	304.198	−	1312.20i	0.444085	−	1.91562i
$686$	686.000			1.00000
$687$			778.766i			1.13358i
$688$		0			0
$689$		0			0
$690$	−176.603	+	761.800i	−0.255946	+	1.10406i
$691$	−787.606			−1.13981			−0.569903	−	0.821712i	$-0.693019\pi$
$691$	−787.606			−1.13981			−0.569903	+	0.821712i	$0.693019\pi$
$692$			1381.36i			1.99619i
$693$		0			0
$694$		0			0
$695$	−1215.80	−	281.850i	−1.74935	−	0.405540i
$696$		0			0
$697$		0			0
$698$		−	799.884i		−	1.14597i
$699$	−312.801			−0.447498
$700$	308.000	−	628.598i	0.440000	−	0.897998i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		−	1133.47i		−	1.61463i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1400.41i			1.98077i
$708$			66.9366i			0.0945432i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	−202.799	+	874.799i	−0.285632	+	1.23211i
$711$	535.802			0.753589
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$			734.979i			1.02508i
$718$			1364.00i			1.89972i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	464.999	+	107.798i	0.645832	+	0.149719i
$721$		0			0
$722$			1350.60i			1.87063i
$723$		0			0
$724$	1447.43			1.99921
$725$		0			0
$726$	−421.481			−0.580552
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$			1217.53i			1.67244i
$729$	−359.069			−0.492550
$730$		0			0
$731$		0			0
$732$		−	557.735i		−	0.761933i
$733$		−	736.342i		−	1.00456i	−0.864705	−	0.502280i	$-0.832495\pi$
$733$		−	736.342i		−	1.00456i	0.864705	−	0.502280i	$-0.167505\pi$
$734$		0			0
$735$	96.3648	−	415.682i	0.131109	−	0.565554i
$736$	1436.80			1.95217
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	1218.98			1.64505
$742$		0			0
$743$			404.099i			0.543875i	0.962315	+	0.271937i	$0.0876644\pi$
$743$			404.099i			0.543875i	−0.962315	+	0.271937i	$0.912336\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			799.477i			1.07025i
$748$		0			0
$749$		0			0
$750$	−337.634	−	274.934i	−0.450179	−	0.366579i
$751$	−998.000			−1.32889			−0.664447	−	0.747335i	$-0.731333\pi$
$751$	−998.000			−1.32889			−0.664447	+	0.747335i	$0.731333\pi$
$752$		0			0
$753$			570.930i			0.758207i
$754$		0			0
$755$	983.907	+	228.093i	1.30319	+	0.302109i
$756$	729.869			0.965435
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	−290.799	+	1254.40i	−0.382630	+	1.65052i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$			156.400i			0.205250i
$763$		0			0
$764$	1436.80			1.88062
$765$		0			0
$766$		0			0
$767$		−	208.898i		−	0.272357i
$768$			445.864i			0.580552i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		−	1256.00i		−	1.62694i
$773$			1519.74i			1.96603i	0.183531	+	0.983014i	$0.441247\pi$
$773$			1519.74i			1.96603i	−0.183531	+	0.983014i	$0.558753\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	737.765	+	171.031i	0.945853	+	0.219271i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−784.000			−1.00000
$785$	246.451	−	1063.10i	0.313950	−	1.35427i
$786$	−384.529			−0.489223
$787$		−	1395.01i		−	1.77256i	−0.463146	−	0.886282i	$-0.653280\pi$
$787$		−	1395.01i		−	1.77256i	0.463146	−	0.886282i	$-0.346720\pi$
$788$		0			0
$789$	−477.214			−0.604834
$790$	−202.799	+	874.799i	−0.256707	+	1.10734i
$791$	182.000			0.230088
$792$		0			0
$793$			1740.60i			2.19495i
$794$	1574.51			1.98302
$795$		0			0
$796$		0			0
$797$		−	763.339i		−	0.957765i	−0.877879	−	0.478883i	$-0.841042\pi$
$797$		−	763.339i		−	0.957765i	0.877879	−	0.478883i	$-0.158958\pi$
$798$			784.934i			0.983627i
$799$		0			0
$800$	−352.000	+	718.398i	−0.440000	+	0.897998i
$801$		0			0
$802$		−	1167.40i		−	1.45561i
$803$		0			0
$804$		0			0
$805$	1530.90	+	354.898i	1.90174	+	0.440867i
$806$		0			0
$807$		−	433.570i		−	0.537261i
$808$		−	1600.46i		−	1.98077i
$809$	−1481.70			−1.83152			−0.915758	−	0.401731i	$-0.868409\pi$
$809$	−1481.70			−1.83152			−0.915758	+	0.401731i	$0.868409\pi$
$810$	−18.7450	+	80.8590i	−0.0231420	+	0.0998259i
$811$	−1538.61			−1.89717			−0.948586	−	0.316518i	$-0.897486\pi$
$811$	−1538.61			−1.89717			−0.948586	+	0.316518i	$0.897486\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−908.071			−1.10876
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$			938.403i			1.14161i
$823$		−	946.000i		−	1.14945i	−0.818345	−	0.574727i	$-0.805108\pi$
$823$		−	946.000i		−	1.14945i	0.818345	−	0.574727i	$-0.194892\pi$
$824$		0			0
$825$		0			0
$826$	134.514			0.162850
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$			1071.60i			1.29421i
$829$	1225.14			1.47785			0.738926	−	0.673787i	$-0.235333\pi$
$829$	1225.14			1.47785			0.738926	+	0.673787i	$0.235333\pi$
$830$	−1305.30	−	302.598i	−1.57265	−	0.364576i
$831$		0			0
$832$		−	1391.47i		−	1.67244i
$833$		0			0
$834$	869.464			1.04252
$835$		0			0
$836$		0			0
$837$		0			0
$838$		−	242.381i		−	0.289237i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	−110.131	+	475.066i	−0.131109	+	0.565554i
$841$	841.000			1.00000
$842$		0			0
$843$		−	588.680i		−	0.698316i
$844$		0			0
$845$	−1479.27	−	342.929i	−1.75061	−	0.405833i
$846$		0			0
$847$			847.000i			1.00000i
$848$		0			0
$849$	966.633			1.13855
$850$		0			0
$851$		0			0
$852$		−	625.602i		−	0.734274i
$853$			1481.66i			1.73699i	0.495695	+	0.868497i	$0.334913\pi$
$853$			1481.66i			1.73699i	−0.495695	+	0.868497i	$0.665087\pi$
$854$	−1120.81			−1.31243
$855$	−935.566	−	216.886i	−1.09423	−	0.253668i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	25.8062			0.0300421			0.0150211	−	0.999887i	$-0.495218\pi$
$859$	25.8062			0.0300421			0.0150211	+	0.999887i	$0.495218\pi$
$860$		0			0
$861$		0			0
$862$		−	1076.00i		−	1.24826i
$863$			1474.00i			1.70800i	0.520277	+	0.853998i	$0.325829\pi$
$863$			1474.00i			1.70800i	−0.520277	+	0.853998i	$0.674171\pi$
$864$	−834.136			−0.965435
$865$	389.949	−	1682.10i	0.450809	−	1.94462i
$866$		0			0
$867$		−	503.339i		−	0.580552i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−2890.80			−3.30755
$875$	−552.503	+	678.503i	−0.631432	+	0.775432i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$	−1016.53			−1.15646
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		−	584.730i		−	0.662959i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$	18.8957	−	81.5092i	0.0213511	−	0.0921008i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	314.299			0.353542
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	896.000			1.00000
$897$			1700.20i			1.89543i
$898$			4.00000i			0.00445434i
$899$		0			0
$900$	−535.802	−	262.532i	−0.595336	−	0.291702i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−208.000			−0.230088
$905$	−1762.55	−	408.600i	−1.94757	−	0.451491i
$906$	−703.630			−0.776633
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$			514.368i			0.566485i
$909$	1193.67			1.31317
$910$	343.701	−	1482.60i	0.377693	−	1.62923i
$911$	922.000			1.01207			0.506037	−	0.862512i	$-0.331110\pi$
$911$	922.000			1.01207			0.506037	+	0.862512i	$0.331110\pi$
$912$		−	897.068i		−	0.983627i
$913$		0			0
$914$	1772.00			1.93873
$915$	−157.445	+	679.158i	−0.172071	+	0.742249i
$916$	−1788.56			−1.95258
$917$			772.743i			0.842686i
$918$		0			0
$919$	987.798			1.07486			0.537431	−	0.843308i	$-0.319395\pi$
$919$	987.798			1.07486			0.537431	+	0.843308i	$0.319395\pi$
$920$	−1749.60	−	405.597i	−1.90174	−	0.440867i
$921$	−39.0289			−0.0423767
$922$		−	1621.71i		−	1.75891i
$923$			1952.40i			2.11527i
$924$		0			0
$925$		0			0
$926$	452.000			0.488121
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	1577.39			1.69429
$932$		−	718.398i		−	0.770814i
$933$		0			0
$934$	−445.981			−0.477495
$935$		0			0
$936$	1037.80			1.10876
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1859.86			−1.97647			−0.988234	−	0.152952i	$-0.951122\pi$
$941$	−1859.86			−1.97647			−0.988234	+	0.152952i	$0.951122\pi$
$942$			760.263i			0.807073i
$943$		0			0
$944$	−153.731			−0.162850
$945$	−888.766	−	206.037i	−0.940494	−	0.218028i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		−	625.602i		−	0.659917i
$949$		0			0
$950$	708.215	−	1445.40i	0.745490	−	1.52147i
$951$		0			0
$952$		0			0
$953$		−	1616.40i		−	1.69611i	−0.529906	−	0.848057i	$-0.677773\pi$
$953$		−	1616.40i		−	1.69611i	0.529906	−	0.848057i	$-0.322227\pi$
$954$		0			0
$955$	−1749.60	−	405.597i	−1.83204	−	0.424709i
$956$	−1688.00			−1.76569
$957$		0			0
$958$		0			0
$959$	1885.80			1.96642
$960$	125.864	−	542.932i	0.131109	−	0.565554i
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−354.560	+	1529.44i	−0.367419	+	1.58491i
$966$	−1094.80			−1.13334
$967$			1302.10i			1.34653i	0.739400	+	0.673266i	$0.235109\pi$
$967$			1302.10i			1.34653i	−0.739400	+	0.673266i	$0.764891\pi$
$968$		−	968.000i		−	1.00000i
$969$		0			0
$970$		0			0
$971$	−1379.19			−1.42038			−0.710190	−	0.704010i	$-0.751391\pi$
$971$	−1379.19			−1.42038			−0.710190	+	0.704010i	$0.751391\pi$
$972$		−	996.228i		−	1.02493i
$973$		−	1747.26i		−	1.79574i
$974$	−1706.20			−1.75174
$975$	−850.101	−	416.532i	−0.871899	−	0.427212i
$976$	1280.93			1.31243
$977$			1077.60i			1.10297i	0.834186	+	0.551483i	$0.185938\pi$
$977$			1077.60i			1.10297i	−0.834186	+	0.551483i	$0.814062\pi$
$978$		0			0
$979$		0			0
$980$	954.682	+	221.318i	0.974166	+	0.225834i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$			2799.60i			2.83360i
$989$		0			0
$990$		0			0
$991$	1885.80			1.90292			0.951461	−	0.307770i	$-0.0995827\pi$
$991$	1885.80			1.90292			0.951461	+	0.307770i	$0.0995827\pi$
$992$		0			0
$993$		0			0
$994$	−1257.20			−1.26479
$995$		0			0
$996$	933.468			0.937217
$997$			1214.66i			1.21831i	0.793050	+	0.609157i	$0.208492\pi$
$997$			1214.66i			1.21831i	−0.793050	+	0.609157i	$0.791508\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.3.c.f.69.4	yes	4
4.3	odd	2		1120.3.c.f.209.2		4
5.4	even	2	inner	280.3.c.f.69.1	yes	4
7.6	odd	2		280.3.c.e.69.3	yes	4
8.3	odd	2		1120.3.c.e.209.3		4
8.5	even	2		280.3.c.e.69.3	yes	4
20.19	odd	2		1120.3.c.f.209.3		4
28.27	even	2		1120.3.c.e.209.3		4
35.34	odd	2		280.3.c.e.69.2	✓	4
40.19	odd	2		1120.3.c.e.209.2		4
40.29	even	2		280.3.c.e.69.2	✓	4
56.13	odd	2	CM	280.3.c.f.69.4	yes	4
56.27	even	2		1120.3.c.f.209.2		4
140.139	even	2		1120.3.c.e.209.2		4
280.69	odd	2	inner	280.3.c.f.69.1	yes	4
280.139	even	2		1120.3.c.f.209.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.e.69.2	✓	4	35.34	odd	2
280.3.c.e.69.2	✓	4	40.29	even	2
280.3.c.e.69.3	yes	4	7.6	odd	2
280.3.c.e.69.3	yes	4	8.5	even	2
280.3.c.f.69.1	yes	4	5.4	even	2	inner
280.3.c.f.69.1	yes	4	280.69	odd	2	inner
280.3.c.f.69.4	yes	4	1.1	even	1	trivial
280.3.c.f.69.4	yes	4	56.13	odd	2	CM
1120.3.c.e.209.2		4	40.19	odd	2
1120.3.c.e.209.2		4	140.139	even	2
1120.3.c.e.209.3		4	8.3	odd	2
1120.3.c.e.209.3		4	28.27	even	2
1120.3.c.f.209.2		4	4.3	odd	2
1120.3.c.f.209.2		4	56.27	even	2
1120.3.c.f.209.3		4	20.19	odd	2
1120.3.c.f.209.3		4	280.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +1.74166i q^{3} -4.00000 q^{4} +(4.87083 + 1.12917i) q^{5} -3.48331 q^{6} +7.00000i q^{7} -8.00000i q^{8} +5.96663 q^{9} +O(q^{10})\)	\(q+2.00000i q^{2} +1.74166i q^{3} -4.00000 q^{4} +(4.87083 + 1.12917i) q^{5} -3.48331 q^{6} +7.00000i q^{7} -8.00000i q^{8} +5.96663 q^{9} +(-2.25834 + 9.74166i) q^{10} -6.96663i q^{12} +21.7417i q^{13} -14.0000 q^{14} +(-1.96663 + 8.48331i) q^{15} +16.0000 q^{16} +11.9333i q^{18} -32.1916 q^{19} +(-19.4833 - 4.51669i) q^{20} -12.1916 q^{21} -44.8999i q^{23} +13.9333 q^{24} +(22.4499 + 11.0000i) q^{25} -43.4833 q^{26} +26.0667i q^{27} -28.0000i q^{28} +(-16.9666 - 3.93326i) q^{30} +32.0000i q^{32} +(-7.90420 + 34.0958i) q^{35} -23.8665 q^{36} -64.3832i q^{38} -37.8665 q^{39} +(9.03337 - 38.9666i) q^{40} -24.3832i q^{42} +(29.0624 + 6.73735i) q^{45} +89.7998 q^{46} +27.8665i q^{48} -49.0000 q^{49} +(-22.0000 + 44.8999i) q^{50} -86.9666i q^{52} -52.1335 q^{54} +56.0000 q^{56} -56.0667i q^{57} -9.60818 q^{59} +(7.86652 - 33.9333i) q^{60} +80.0581 q^{61} +41.7664i q^{63} -64.0000 q^{64} +(-24.5501 + 105.900i) q^{65} +78.2002 q^{69} +(-68.1916 - 15.8084i) q^{70} +89.7998 q^{71} -47.7330i q^{72} +(-19.1582 + 39.1001i) q^{75} +128.766 q^{76} -75.7330i q^{78} +89.7998 q^{79} +(77.9333 + 18.0667i) q^{80} +8.30033 q^{81} +133.991i q^{83} +48.7664 q^{84} +(-13.4747 + 58.1249i) q^{90} -152.192 q^{91} +179.600i q^{92} +(-156.800 - 36.3498i) q^{95} -55.7330 q^{96} -98.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9	\( 4 q - 16 q^{4} + 12 q^{5} + 16 q^{6} - 36 q^{9} - 24 q^{10} - 56 q^{14} + 52 q^{15} + 64 q^{16} - 24 q^{19} - 48 q^{20} + 56 q^{21} - 64 q^{24} - 144 q^{26} - 8 q^{30} - 84 q^{35} + 144 q^{36} + 88 q^{39} + 96 q^{40} + 4 q^{45} - 196 q^{49} - 88 q^{50} - 448 q^{54} + 224 q^{56} + 216 q^{59} - 208 q^{60} - 24 q^{61} - 256 q^{64} - 188 q^{65} + 672 q^{69} - 168 q^{70} + 88 q^{75} + 96 q^{76} + 192 q^{80} + 572 q^{81} - 224 q^{84} + 440 q^{90} - 504 q^{91} - 268 q^{95} + 256 q^{96}+O(q^{100}) \) 4 * q - 16 * q^4 + 12 * q^5 + 16 * q^6 - 36 * q^9 - 24 * q^10 - 56 * q^14 + 52 * q^15 + 64 * q^16 - 24 * q^19 - 48 * q^20 + 56 * q^21 - 64 * q^24 - 144 * q^26 - 8 * q^30 - 84 * q^35 + 144 * q^36 + 88 * q^39 + 96 * q^40 + 4 * q^45 - 196 * q^49 - 88 * q^50 - 448 * q^54 + 224 * q^56 + 216 * q^59 - 208 * q^60 - 24 * q^61 - 256 * q^64 - 188 * q^65 + 672 * q^69 - 168 * q^70 + 88 * q^75 + 96 * q^76 + 192 * q^80 + 572 * q^81 - 224 * q^84 + 440 * q^90 - 504 * q^91 - 268 * q^95 + 256 * q^96

Introduction

L-functions

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