LMFDB - Embedded newform 128.5.f.b.31.3

Newspace parameters

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.f (of order $4$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
Defining polynomial:	$x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + 11456 x^{6} - 19968 x^{5} + 31744 x^{4} - 139264 x^{3} + 491520 x^{2} - 1048576 x + 2097152$
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.3
Root			$-2.15805 + 1.82834i$ of defining polynomial
Character	$\chi$	$=$	128.31
Dual form			128.5.f.b.95.3

$q$-expansion

$f(q)$	$=$	$q+(-3.91498 - 3.91498i) q^{3} +(-4.72348 - 4.72348i) q^{5} +45.3712 q^{7} -50.3458i q^{9} +O(q^{10})$	\(q+(-3.91498 - 3.91498i) q^{3} +(-4.72348 - 4.72348i) q^{5} +45.3712 q^{7} -50.3458i q^{9} +(-110.228 + 110.228i) q^{11} +(157.128 - 157.128i) q^{13} +36.9847i q^{15} -378.592 q^{17} +(-203.127 - 203.127i) q^{19} +(-177.627 - 177.627i) q^{21} -740.444 q^{23} -580.377i q^{25} +(-514.217 + 514.217i) q^{27} +(-82.6244 + 82.6244i) q^{29} +286.217i q^{31} +863.080 q^{33} +(-214.310 - 214.310i) q^{35} +(-1470.39 - 1470.39i) q^{37} -1230.31 q^{39} -1301.21i q^{41} +(366.979 - 366.979i) q^{43} +(-237.808 + 237.808i) q^{45} +751.307i q^{47} -342.455 q^{49} +(1482.18 + 1482.18i) q^{51} +(1929.33 + 1929.33i) q^{53} +1041.32 q^{55} +1590.48i q^{57} +(1357.57 - 1357.57i) q^{59} +(1835.78 - 1835.78i) q^{61} -2284.25i q^{63} -1484.39 q^{65} +(-2205.02 - 2205.02i) q^{67} +(2898.83 + 2898.83i) q^{69} +8970.40 q^{71} +9350.46i q^{73} +(-2272.17 + 2272.17i) q^{75} +(-5001.17 + 5001.17i) q^{77} -2860.08i q^{79} -51.7173 q^{81} +(1036.63 + 1036.63i) q^{83} +(1788.27 + 1788.27i) q^{85} +646.946 q^{87} -5171.06i q^{89} +(7129.10 - 7129.10i) q^{91} +(1120.54 - 1120.54i) q^{93} +1918.94i q^{95} +8539.92 q^{97} +(5549.51 + 5549.51i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14q + 2q^{3} + 2q^{5} - 4q^{7} + O(q^{10}) $	$ 14q + 2q^{3} + 2q^{5} - 4q^{7} - 94q^{11} + 2q^{13} - 4q^{17} + 706q^{19} + 164q^{21} + 1148q^{23} + 1664q^{27} - 862q^{29} - 4q^{33} - 1340q^{35} + 1826q^{37} + 2684q^{39} - 1694q^{43} - 1410q^{45} + 682q^{49} + 3012q^{51} + 482q^{53} - 11780q^{55} + 2786q^{59} + 3778q^{61} - 2020q^{65} - 7998q^{67} - 9628q^{69} + 19964q^{71} - 17570q^{75} + 9508q^{77} + 1454q^{81} + 17282q^{83} - 9948q^{85} - 49284q^{87} + 28036q^{91} - 8896q^{93} - 4q^{97} - 49214q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$	−3.91498	−	3.91498i	−0.434998	−	0.434998i	0.455327	−	0.890324i	$-0.349523\pi$
$3$	−3.91498	−	3.91498i	−0.434998	−	0.434998i	−0.890324	+	0.455327i	$0.849523\pi$
$4$		0			0
$5$	−4.72348	−	4.72348i	−0.188939	−	0.188939i	0.606298	−	0.795237i	$-0.292654\pi$
$5$	−4.72348	−	4.72348i	−0.188939	−	0.188939i	−0.795237	+	0.606298i	$0.792654\pi$
$6$		0			0
$7$	45.3712			0.925943			0.462971	−	0.886373i	$-0.346783\pi$
$7$	45.3712			0.925943			0.462971	+	0.886373i	$0.346783\pi$
$8$		0			0
$9$		−	50.3458i		−	0.621554i
$10$		0			0
$11$	−110.228	+	110.228i	−0.910974	+	0.910974i	−0.996349	−	0.0853752i	$-0.972791\pi$
$11$	−110.228	+	110.228i	−0.910974	+	0.910974i	0.0853752	+	0.996349i	$0.472791\pi$
$12$		0			0
$13$	157.128	−	157.128i	0.929754	−	0.929754i	−0.0679357	−	0.997690i	$-0.521641\pi$
$13$	157.128	−	157.128i	0.929754	−	0.929754i	0.997690	+	0.0679357i	$0.0216413\pi$
$14$		0			0
$15$			36.9847i			0.164376i
$16$		0			0
$17$	−378.592			−1.31001			−0.655003	−	0.755626i	$-0.727333\pi$
$17$	−378.592			−1.31001			−0.655003	+	0.755626i	$0.727333\pi$
$18$		0			0
$19$	−203.127	−	203.127i	−0.562680	−	0.562680i	0.367388	−	0.930068i	$-0.380252\pi$
$19$	−203.127	−	203.127i	−0.562680	−	0.562680i	−0.930068	+	0.367388i	$0.880252\pi$
$20$		0			0
$21$	−177.627	−	177.627i	−0.402783	−	0.402783i
$22$		0			0
$23$	−740.444			−1.39971			−0.699853	−	0.714287i	$-0.746751\pi$
$23$	−740.444			−1.39971			−0.699853	+	0.714287i	$0.746751\pi$
$24$		0			0
$25$		−	580.377i		−	0.928604i
$26$		0			0
$27$	−514.217	+	514.217i	−0.705372	+	0.705372i
$28$		0			0
$29$	−82.6244	+	82.6244i	−0.0982455	+	0.0982455i	−0.754521	−	0.656276i	$-0.772131\pi$
$29$	−82.6244	+	82.6244i	−0.0982455	+	0.0982455i	0.656276	+	0.754521i	$0.272131\pi$
$30$		0			0
$31$			286.217i			0.297833i	0.988850	+	0.148916i	$0.0475785\pi$
$31$			286.217i			0.297833i	−0.988850	+	0.148916i	$0.952421\pi$
$32$		0			0
$33$	863.080			0.792543
$34$		0			0
$35$	−214.310	−	214.310i	−0.174947	−	0.174947i
$36$		0			0
$37$	−1470.39	−	1470.39i	−1.07406	−	1.07406i	−0.997028	−	0.0770352i	$-0.975455\pi$
$37$	−1470.39	−	1470.39i	−1.07406	−	1.07406i	−0.0770352	−	0.997028i	$-0.524545\pi$
$38$		0			0
$39$	−1230.31			−0.808882
$40$		0			0
$41$		−	1301.21i		−	0.774070i	−0.922065	−	0.387035i	$-0.873499\pi$
$41$		−	1301.21i		−	0.774070i	0.922065	−	0.387035i	$-0.126501\pi$
$42$		0			0
$43$	366.979	−	366.979i	0.198474	−	0.198474i	−0.600871	−	0.799346i	$-0.705180\pi$
$43$	366.979	−	366.979i	0.198474	−	0.198474i	0.799346	+	0.600871i	$0.205180\pi$
$44$		0			0
$45$	−237.808	+	237.808i	−0.117436	+	0.117436i
$46$		0			0
$47$			751.307i			0.340112i	0.985434	+	0.170056i	$0.0543948\pi$
$47$			751.307i			0.340112i	−0.985434	+	0.170056i	$0.945605\pi$
$48$		0			0
$49$	−342.455			−0.142630
$50$		0			0
$51$	1482.18	+	1482.18i	0.569850	+	0.569850i
$52$		0			0
$53$	1929.33	+	1929.33i	0.686838	+	0.686838i	0.961532	−	0.274693i	$-0.0885764\pi$
$53$	1929.33	+	1929.33i	0.686838	+	0.686838i	−0.274693	+	0.961532i	$0.588576\pi$
$54$		0			0
$55$	1041.32			0.344238
$56$		0			0
$57$			1590.48i			0.489529i
$58$		0			0
$59$	1357.57	−	1357.57i	0.389994	−	0.389994i	−0.484691	−	0.874685i	$-0.661068\pi$
$59$	1357.57	−	1357.57i	0.389994	−	0.389994i	0.874685	+	0.484691i	$0.161068\pi$
$60$		0			0
$61$	1835.78	−	1835.78i	0.493356	−	0.493356i	−0.416006	−	0.909362i	$-0.636570\pi$
$61$	1835.78	−	1835.78i	0.493356	−	0.493356i	0.909362	+	0.416006i	$0.136570\pi$
$62$		0			0
$63$		−	2284.25i		−	0.575523i
$64$		0			0
$65$	−1484.39			−0.351334
$66$		0			0
$67$	−2205.02	−	2205.02i	−0.491204	−	0.491204i	0.417481	−	0.908686i	$-0.362913\pi$
$67$	−2205.02	−	2205.02i	−0.491204	−	0.491204i	−0.908686	+	0.417481i	$0.862913\pi$
$68$		0			0
$69$	2898.83	+	2898.83i	0.608869	+	0.608869i
$70$		0			0
$71$	8970.40			1.77949			0.889744	−	0.456460i	$-0.150883\pi$
$71$	8970.40			1.77949			0.889744	+	0.456460i	$0.150883\pi$
$72$		0			0
$73$			9350.46i			1.75464i	0.479909	+	0.877318i	$0.340670\pi$
$73$			9350.46i			1.75464i	−0.479909	+	0.877318i	$0.659330\pi$
$74$		0			0
$75$	−2272.17	+	2272.17i	−0.403941	+	0.403941i
$76$		0			0
$77$	−5001.17	+	5001.17i	−0.843509	+	0.843509i
$78$		0			0
$79$		−	2860.08i		−	0.458273i	−0.973394	−	0.229137i	$-0.926410\pi$
$79$		−	2860.08i		−	0.458273i	0.973394	−	0.229137i	$-0.0735903\pi$
$80$		0			0
$81$	−51.7173			−0.00788253
$82$		0			0
$83$	1036.63	+	1036.63i	0.150477	+	0.150477i	0.778331	−	0.627854i	$-0.216067\pi$
$83$	1036.63	+	1036.63i	0.150477	+	0.150477i	−0.627854	+	0.778331i	$0.716067\pi$
$84$		0			0
$85$	1788.27	+	1788.27i	0.247512	+	0.247512i
$86$		0			0
$87$	646.946			0.0854731
$88$		0			0
$89$		−	5171.06i		−	0.652830i	−0.945227	−	0.326415i	$-0.894159\pi$
$89$		−	5171.06i		−	0.652830i	0.945227	−	0.326415i	$-0.105841\pi$
$90$		0			0
$91$	7129.10	−	7129.10i	0.860899	−	0.860899i
$92$		0			0
$93$	1120.54	−	1120.54i	0.129557	−	0.129557i
$94$		0			0
$95$			1918.94i			0.212625i
$96$		0			0
$97$	8539.92			0.907633			0.453817	−	0.891095i	$-0.350062\pi$
$97$	8539.92			0.907633			0.453817	+	0.891095i	$0.350062\pi$
$98$		0			0
$99$	5549.51	+	5549.51i	0.566219	+	0.566219i
$100$		0			0
$101$	−537.642	−	537.642i	−0.0527049	−	0.0527049i	0.680263	−	0.732968i	$-0.261866\pi$
$101$	−537.642	−	537.642i	−0.0527049	−	0.0527049i	−0.732968	+	0.680263i	$0.761866\pi$
$102$		0			0
$103$	−5543.86			−0.522562			−0.261281	−	0.965263i	$-0.584145\pi$
$103$	−5543.86			−0.522562			−0.261281	+	0.965263i	$0.584145\pi$
$104$		0			0
$105$			1678.04i			0.152203i
$106$		0			0
$107$	2989.60	−	2989.60i	0.261123	−	0.261123i	−0.564387	−	0.825510i	$-0.690887\pi$
$107$	2989.60	−	2989.60i	0.261123	−	0.261123i	0.825510	+	0.564387i	$0.190887\pi$
$108$		0			0
$109$	3353.83	−	3353.83i	0.282285	−	0.282285i	−0.551734	−	0.834020i	$-0.686034\pi$
$109$	3353.83	−	3353.83i	0.282285	−	0.282285i	0.834020	+	0.551734i	$0.186034\pi$
$110$		0			0
$111$			11513.1i			0.934431i
$112$		0			0
$113$	−3193.81			−0.250122			−0.125061	−	0.992149i	$-0.539913\pi$
$113$	−3193.81			−0.250122			−0.125061	+	0.992149i	$0.539913\pi$
$114$		0			0
$115$	3497.48	+	3497.48i	0.264459	+	0.264459i
$116$		0			0
$117$	−7910.76	−	7910.76i	−0.577892	−	0.577892i
$118$		0			0
$119$	−17177.2			−1.21299
$120$		0			0
$121$		−	9659.34i		−	0.659746i
$122$		0			0
$123$	−5094.22	+	5094.22i	−0.336719	+	0.336719i
$124$		0			0
$125$	−5693.58	+	5693.58i	−0.364389	+	0.364389i
$126$		0			0
$127$		−	22900.5i		−	1.41983i	−0.704287	−	0.709915i	$-0.748733\pi$
$127$		−	22900.5i		−	1.41983i	0.704287	−	0.709915i	$-0.251267\pi$
$128$		0			0
$129$	−2873.43			−0.172672
$130$		0			0
$131$	−19255.9	−	19255.9i	−1.12207	−	1.12207i	−0.991429	−	0.130644i	$-0.958296\pi$
$131$	−19255.9	−	19255.9i	−1.12207	−	1.12207i	−0.130644	−	0.991429i	$-0.541704\pi$
$132$		0			0
$133$	−9216.13	−	9216.13i	−0.521009	−	0.521009i
$134$		0			0
$135$	4857.79			0.266545
$136$		0			0
$137$		−	19938.2i		−	1.06229i	−0.847280	−	0.531147i	$-0.821761\pi$
$137$		−	19938.2i		−	1.06229i	0.847280	−	0.531147i	$-0.178239\pi$
$138$		0			0
$139$	−25175.4	+	25175.4i	−1.30301	+	1.30301i	−0.376649	+	0.926356i	$0.622924\pi$
$139$	−25175.4	+	25175.4i	−1.30301	+	1.30301i	−0.926356	+	0.376649i	$0.877076\pi$
$140$		0			0
$141$	2941.35	−	2941.35i	0.147948	−	0.147948i
$142$		0			0
$143$			34639.8i			1.69396i
$144$		0			0
$145$	780.550			0.0371249
$146$		0			0
$147$	1340.70	+	1340.70i	0.0620438	+	0.0620438i
$148$		0			0
$149$	3216.99	+	3216.99i	0.144903	+	0.144903i	0.775837	−	0.630934i	$-0.217328\pi$
$149$	3216.99	+	3216.99i	0.144903	+	0.144903i	−0.630934	+	0.775837i	$0.717328\pi$
$150$		0			0
$151$	−14770.5			−0.647802			−0.323901	−	0.946091i	$-0.604994\pi$
$151$	−14770.5			−0.647802			−0.323901	+	0.946091i	$0.604994\pi$
$152$		0			0
$153$			19060.5i			0.814239i
$154$		0			0
$155$	1351.94	−	1351.94i	0.0562723	−	0.0562723i
$156$		0			0
$157$	18014.6	−	18014.6i	0.730845	−	0.730845i	−0.239942	−	0.970787i	$-0.577128\pi$
$157$	18014.6	−	18014.6i	0.730845	−	0.730845i	0.970787	+	0.239942i	$0.0771284\pi$
$158$		0			0
$159$		−	15106.6i		−	0.597547i
$160$		0			0
$161$	−33594.8			−1.29605
$162$		0			0
$163$	−15083.2	−	15083.2i	−0.567700	−	0.567700i	0.363784	−	0.931483i	$-0.381485\pi$
$163$	−15083.2	−	15083.2i	−0.567700	−	0.567700i	−0.931483	+	0.363784i	$0.881485\pi$
$164$		0			0
$165$	−4076.74	−	4076.74i	−0.149743	−	0.149743i
$166$		0			0
$167$	23733.6			0.851001			0.425501	−	0.904958i	$-0.360098\pi$
$167$	23733.6			0.851001			0.425501	+	0.904958i	$0.360098\pi$
$168$		0			0
$169$		−	20817.7i		−	0.728885i
$170$		0			0
$171$	−10226.6	+	10226.6i	−0.349736	+	0.349736i
$172$		0			0
$173$	12072.3	−	12072.3i	0.403363	−	0.403363i	−0.476053	−	0.879416i	$-0.657933\pi$
$173$	12072.3	−	12072.3i	0.403363	−	0.403363i	0.879416	+	0.476053i	$0.157933\pi$
$174$		0			0
$175$		−	26332.4i		−	0.859834i
$176$		0			0
$177$	−10629.7			−0.339293
$178$		0			0
$179$	28111.5	+	28111.5i	0.877361	+	0.877361i	0.993261	−	0.115900i	$-0.0369752\pi$
$179$	28111.5	+	28111.5i	0.877361	+	0.877361i	−0.115900	+	0.993261i	$0.536975\pi$
$180$		0			0
$181$	15906.7	+	15906.7i	0.485539	+	0.485539i	0.906895	−	0.421356i	$-0.138446\pi$
$181$	15906.7	+	15906.7i	0.485539	+	0.485539i	−0.421356	+	0.906895i	$0.638446\pi$
$182$		0			0
$183$	−14374.1			−0.429218
$184$		0			0
$185$			13890.8i			0.405866i
$186$		0			0
$187$	41731.4	−	41731.4i	1.19338	−	1.19338i
$188$		0			0
$189$	−23330.6	+	23330.6i	−0.653134	+	0.653134i
$190$		0			0
$191$			28038.6i			0.768581i	0.923212	+	0.384290i	$0.125554\pi$
$191$			28038.6i			0.768581i	−0.923212	+	0.384290i	$0.874446\pi$
$192$		0			0
$193$	56713.1			1.52254			0.761270	−	0.648436i	$-0.224576\pi$
$193$	56713.1			1.52254			0.761270	+	0.648436i	$0.224576\pi$
$194$		0			0
$195$	5811.35	+	5811.35i	0.152830	+	0.152830i
$196$		0			0
$197$	10179.3	+	10179.3i	0.262292	+	0.262292i	0.825985	−	0.563692i	$-0.190620\pi$
$197$	10179.3	+	10179.3i	0.262292	+	0.262292i	−0.563692	+	0.825985i	$0.690620\pi$
$198$		0			0
$199$	31787.3			0.802689			0.401344	−	0.915927i	$-0.368543\pi$
$199$	31787.3			0.802689			0.401344	+	0.915927i	$0.368543\pi$
$200$		0			0
$201$			17265.2i			0.427346i
$202$		0			0
$203$	−3748.77	+	3748.77i	−0.0909697	+	0.0909697i
$204$		0			0
$205$	−6146.25	+	6146.25i	−0.146252	+	0.146252i
$206$		0			0
$207$			37278.3i			0.869992i
$208$		0			0
$209$	44780.6			1.02517
$210$		0			0
$211$	10699.8	+	10699.8i	0.240332	+	0.240332i	0.816988	−	0.576655i	$-0.195642\pi$
$211$	10699.8	+	10699.8i	0.240332	+	0.240332i	−0.576655	+	0.816988i	$0.695642\pi$
$212$		0			0
$213$	−35118.9	−	35118.9i	−0.774073	−	0.774073i
$214$		0			0
$215$	−3466.84			−0.0749992
$216$		0			0
$217$			12986.0i			0.275776i
$218$		0			0
$219$	36606.9	−	36606.9i	0.763263	−	0.763263i
$220$		0			0
$221$	−59487.5	+	59487.5i	−1.21798	+	1.21798i
$222$		0			0
$223$		−	91700.4i		−	1.84400i	−0.387187	−	0.922001i	$-0.626553\pi$
$223$		−	91700.4i		−	1.84400i	0.387187	−	0.922001i	$-0.373447\pi$
$224$		0			0
$225$	−29219.6			−0.577177
$226$		0			0
$227$	−17499.5	−	17499.5i	−0.339605	−	0.339605i	0.516614	−	0.856218i	$-0.327192\pi$
$227$	−17499.5	−	17499.5i	−0.339605	−	0.339605i	−0.856218	+	0.516614i	$0.827192\pi$
$228$		0			0
$229$	63859.8	+	63859.8i	1.21775	+	1.21775i	0.968420	+	0.249326i	$0.0802091\pi$
$229$	63859.8	+	63859.8i	1.21775	+	1.21775i	0.249326	+	0.968420i	$0.419791\pi$
$230$		0			0
$231$	39159.0			0.733850
$232$		0			0
$233$			63891.7i			1.17688i	0.808540	+	0.588441i	$0.200258\pi$
$233$			63891.7i			1.17688i	−0.808540	+	0.588441i	$0.799742\pi$
$234$		0			0
$235$	3548.79	−	3548.79i	0.0642605	−	0.0642605i
$236$		0			0
$237$	−11197.2	+	11197.2i	−0.199348	+	0.199348i
$238$		0			0
$239$			12096.4i			0.211768i	0.994378	+	0.105884i	$0.0337672\pi$
$239$			12096.4i			0.211768i	−0.994378	+	0.105884i	$0.966233\pi$
$240$		0			0
$241$	27978.6			0.481717			0.240859	−	0.970560i	$-0.422571\pi$
$241$	27978.6			0.481717			0.240859	+	0.970560i	$0.422571\pi$
$242$		0			0
$243$	41854.0	+	41854.0i	0.708801	+	0.708801i
$244$		0			0
$245$	1617.58	+	1617.58i	0.0269484	+	0.0269484i
$246$		0			0
$247$	−63834.2			−1.04631
$248$		0			0
$249$		−	8116.80i		−	0.130914i
$250$		0			0
$251$	−1996.06	+	1996.06i	−0.0316830	+	0.0316830i	−0.722771	−	0.691088i	$-0.757132\pi$
$251$	−1996.06	+	1996.06i	−0.0316830	+	0.0316830i	0.691088	+	0.722771i	$0.257132\pi$
$252$		0			0
$253$	81617.5	−	81617.5i	1.27509	−	1.27509i
$254$		0			0
$255$		−	14002.1i		−	0.215334i
$256$		0			0
$257$	−73510.4			−1.11297			−0.556484	−	0.830859i	$-0.687850\pi$
$257$	−73510.4			−1.11297			−0.556484	+	0.830859i	$0.687850\pi$
$258$		0			0
$259$	−66713.5	−	66713.5i	−0.994521	−	0.994521i
$260$		0			0
$261$	4159.80	+	4159.80i	0.0610648	+	0.0610648i
$262$		0			0
$263$	68109.3			0.984680			0.492340	−	0.870403i	$-0.336142\pi$
$263$	68109.3			0.984680			0.492340	+	0.870403i	$0.336142\pi$
$264$		0			0
$265$		−	18226.3i		−	0.259542i
$266$		0			0
$267$	−20244.6	+	20244.6i	−0.283980	+	0.283980i
$268$		0			0
$269$	15056.1	−	15056.1i	0.208069	−	0.208069i	−0.595377	−	0.803446i	$-0.702997\pi$
$269$	15056.1	−	15056.1i	0.208069	−	0.208069i	0.803446	+	0.595377i	$0.202997\pi$
$270$		0			0
$271$			104320.i			1.42046i	0.703968	+	0.710232i	$0.251410\pi$
$271$			104320.i			1.42046i	−0.703968	+	0.710232i	$0.748590\pi$
$272$		0			0
$273$	−55820.6			−0.748979
$274$		0			0
$275$	63973.7	+	63973.7i	0.845934	+	0.845934i
$276$		0			0
$277$	−2136.36	−	2136.36i	−0.0278430	−	0.0278430i	0.693048	−	0.720891i	$-0.256267\pi$
$277$	−2136.36	−	2136.36i	−0.0278430	−	0.0278430i	−0.720891	+	0.693048i	$0.756267\pi$
$278$		0			0
$279$	14409.8			0.185119
$280$		0			0
$281$		−	36783.7i		−	0.465846i	−0.972495	−	0.232923i	$-0.925171\pi$
$281$		−	36783.7i		−	0.465846i	0.972495	−	0.232923i	$-0.0748290\pi$
$282$		0			0
$283$	−14172.0	+	14172.0i	−0.176953	+	0.176953i	−0.790026	−	0.613073i	$-0.789933\pi$
$283$	−14172.0	+	14172.0i	−0.176953	+	0.176953i	0.613073	+	0.790026i	$0.289933\pi$
$284$		0			0
$285$	7512.60	−	7512.60i	0.0924913	−	0.0924913i
$286$		0			0
$287$		−	59037.5i		−	0.716744i
$288$		0			0
$289$	59810.8			0.716117
$290$		0			0
$291$	−33433.6	−	33433.6i	−0.394819	−	0.394819i
$292$		0			0
$293$	−80721.8	−	80721.8i	−0.940276	−	0.940276i	0.0580384	−	0.998314i	$-0.481515\pi$
$293$	−80721.8	−	80721.8i	−0.940276	−	0.940276i	−0.998314	+	0.0580384i	$0.981515\pi$
$294$		0			0
$295$	−12824.9			−0.147370
$296$		0			0
$297$		−	113362.i		−	1.28515i
$298$		0			0
$299$	−116345.	+	116345.i	−1.30138	+	1.30138i
$300$		0			0
$301$	16650.3	−	16650.3i	0.183776	−	0.183776i
$302$		0			0
$303$			4209.72i			0.0458530i
$304$		0			0
$305$	−17342.5			−0.186429
$306$		0			0
$307$	−113714.	−	113714.i	−1.20653	−	1.20653i	−0.972143	−	0.234389i	$-0.924691\pi$
$307$	−113714.	−	113714.i	−1.20653	−	1.20653i	−0.234389	−	0.972143i	$-0.575309\pi$
$308$		0			0
$309$	21704.1	+	21704.1i	0.227313	+	0.227313i
$310$		0			0
$311$	−93801.8			−0.969819			−0.484909	−	0.874564i	$-0.661147\pi$
$311$	−93801.8			−0.969819			−0.484909	+	0.874564i	$0.661147\pi$
$312$		0			0
$313$		−	9229.52i		−	0.0942086i	−0.998890	−	0.0471043i	$-0.985001\pi$
$313$		−	9229.52i		−	0.0942086i	0.998890	−	0.0471043i	$-0.0149993\pi$
$314$		0			0
$315$	−10789.6	+	10789.6i	−0.108739	+	0.108739i
$316$		0			0
$317$	8447.44	−	8447.44i	0.0840634	−	0.0840634i	−0.663825	−	0.747888i	$-0.731068\pi$
$317$	8447.44	−	8447.44i	0.0840634	−	0.0840634i	0.747888	+	0.663825i	$0.231068\pi$
$318$		0			0
$319$		−	18215.0i		−	0.178998i
$320$		0			0
$321$	−23408.5			−0.227176
$322$		0			0
$323$	76902.4	+	76902.4i	0.737114	+	0.737114i
$324$		0			0
$325$	−91193.8	−	91193.8i	−0.863373	−	0.863373i
$326$		0			0
$327$	−26260.4			−0.245587
$328$		0			0
$329$			34087.7i			0.314924i
$330$		0			0
$331$	46092.1	−	46092.1i	0.420698	−	0.420698i	−0.464746	−	0.885444i	$-0.653854\pi$
$331$	46092.1	−	46092.1i	0.420698	−	0.420698i	0.885444	+	0.464746i	$0.153854\pi$
$332$		0			0
$333$	−74028.2	+	74028.2i	−0.667588	+	0.667588i
$334$		0			0
$335$			20830.7i			0.185616i
$336$		0			0
$337$	39317.8			0.346201			0.173101	−	0.984904i	$-0.444621\pi$
$337$	39317.8			0.346201			0.173101	+	0.984904i	$0.444621\pi$
$338$		0			0
$339$	12503.7	+	12503.7i	0.108803	+	0.108803i
$340$		0			0
$341$	−31549.1	−	31549.1i	−0.271318	−	0.271318i
$342$		0			0
$343$	−124474.			−1.05801
$344$		0			0
$345$		−	27385.1i		−	0.230079i
$346$		0			0
$347$	67694.0	−	67694.0i	0.562201	−	0.562201i	−0.367731	−	0.929932i	$-0.619865\pi$
$347$	67694.0	−	67694.0i	0.562201	−	0.562201i	0.929932	+	0.367731i	$0.119865\pi$
$348$		0			0
$349$	7662.83	−	7662.83i	0.0629127	−	0.0629127i	−0.674950	−	0.737863i	$-0.735835\pi$
$349$	7662.83	−	7662.83i	0.0629127	−	0.0629127i	0.737863	+	0.674950i	$0.235835\pi$
$350$		0			0
$351$			161596.i			1.31165i
$352$		0			0
$353$	70618.0			0.566716			0.283358	−	0.959014i	$-0.408551\pi$
$353$	70618.0			0.566716			0.283358	+	0.959014i	$0.408551\pi$
$354$		0			0
$355$	−42371.5	−	42371.5i	−0.336215	−	0.336215i
$356$		0			0
$357$	67248.3	+	67248.3i	0.527649	+	0.527649i
$358$		0			0
$359$	−168144.			−1.30465			−0.652323	−	0.757941i	$-0.726206\pi$
$359$	−168144.			−1.30465			−0.652323	+	0.757941i	$0.726206\pi$
$360$		0			0
$361$		−	47799.6i		−	0.366783i
$362$		0			0
$363$	−37816.1	+	37816.1i	−0.286988	+	0.286988i
$364$		0			0
$365$	44166.7	−	44166.7i	0.331520	−	0.331520i
$366$		0			0
$367$		−	82596.4i		−	0.613238i	−0.951832	−	0.306619i	$-0.900802\pi$
$367$		−	82596.4i		−	0.613238i	0.951832	−	0.306619i	$-0.0991977\pi$
$368$		0			0
$369$	−65510.6			−0.481126
$370$		0			0
$371$	87536.0	+	87536.0i	0.635973	+	0.635973i
$372$		0			0
$373$	76631.0	+	76631.0i	0.550791	+	0.550791i	0.926669	−	0.375878i	$-0.122659\pi$
$373$	76631.0	+	76631.0i	0.550791	+	0.550791i	−0.375878	+	0.926669i	$0.622659\pi$
$374$		0			0
$375$	44580.5			0.317017
$376$		0			0
$377$			25965.3i			0.182688i
$378$		0			0
$379$	95935.8	−	95935.8i	0.667886	−	0.667886i	−0.289340	−	0.957226i	$-0.593436\pi$
$379$	95935.8	−	95935.8i	0.667886	−	0.667886i	0.957226	+	0.289340i	$0.0934360\pi$
$380$		0			0
$381$	−89654.8	+	89654.8i	−0.617623	+	0.617623i
$382$		0			0
$383$		−	168708.i		−	1.15011i	−0.818116	−	0.575054i	$-0.804981\pi$
$383$		−	168708.i		−	1.15011i	0.818116	−	0.575054i	$-0.195019\pi$
$384$		0			0
$385$	47245.9			0.318744
$386$		0			0
$387$	−18475.9	−	18475.9i	−0.123362	−	0.123362i
$388$		0			0
$389$	−66490.0	−	66490.0i	−0.439397	−	0.439397i	0.452412	−	0.891809i	$-0.350564\pi$
$389$	−66490.0	−	66490.0i	−0.439397	−	0.439397i	−0.891809	+	0.452412i	$0.850564\pi$
$390$		0			0
$391$	280326.			1.83362
$392$		0			0
$393$			150773.i			0.976199i
$394$		0			0
$395$	−13509.6	+	13509.6i	−0.0865859	+	0.0865859i
$396$		0			0
$397$	−18578.8	+	18578.8i	−0.117879	+	0.117879i	−0.763586	−	0.645706i	$-0.776563\pi$
$397$	−18578.8	+	18578.8i	−0.117879	+	0.117879i	0.645706	+	0.763586i	$0.276563\pi$
$398$		0			0
$399$			72162.0i			0.453276i
$400$		0			0
$401$	−129108.			−0.802908			−0.401454	−	0.915879i	$-0.631495\pi$
$401$	−129108.			−0.802908			−0.401454	+	0.915879i	$0.631495\pi$
$402$		0			0
$403$	44972.9	+	44972.9i	0.276911	+	0.276911i
$404$		0			0
$405$	244.286	+	244.286i	0.00148932	+	0.00148932i
$406$		0			0
$407$	324156.			1.95689
$408$		0			0
$409$		−	17952.4i		−	0.107319i	−0.998559	−	0.0536594i	$-0.982911\pi$
$409$		−	17952.4i		−	0.107319i	0.998559	−	0.0536594i	$-0.0170885\pi$
$410$		0			0
$411$	−78057.7	+	78057.7i	−0.462096	+	0.462096i
$412$		0			0
$413$	61594.5	−	61594.5i	0.361112	−	0.361112i
$414$		0			0
$415$		−	9793.04i		−	0.0568619i
$416$		0			0
$417$	197122.			1.13361
$418$		0			0
$419$	−56887.3	−	56887.3i	−0.324031	−	0.324031i	0.526280	−	0.850311i	$-0.323586\pi$
$419$	−56887.3	−	56887.3i	−0.324031	−	0.324031i	−0.850311	+	0.526280i	$0.823586\pi$
$420$		0			0
$421$	144162.	+	144162.i	0.813365	+	0.813365i	0.985137	−	0.171772i	$-0.0549492\pi$
$421$	144162.	+	144162.i	0.813365	+	0.813365i	−0.171772	+	0.985137i	$0.554949\pi$
$422$		0			0
$423$	37825.2			0.211398
$424$		0			0
$425$			219726.i			1.21648i
$426$		0			0
$427$	83291.4	−	83291.4i	0.456819	−	0.456819i
$428$		0			0
$429$	135614.	−	135614.i	0.736870	−	0.736870i
$430$		0			0
$431$			18641.5i			0.100352i	0.998740	+	0.0501760i	$0.0159782\pi$
$431$			18641.5i			0.100352i	−0.998740	+	0.0501760i	$0.984022\pi$
$432$		0			0
$433$	2193.97			0.0117019			0.00585094	−	0.999983i	$-0.498138\pi$
$433$	2193.97			0.0117019			0.00585094	+	0.999983i	$0.498138\pi$
$434$		0			0
$435$	−3055.84	−	3055.84i	−0.0161492	−	0.0161492i
$436$		0			0
$437$	150404.	+	150404.i	0.787586	+	0.787586i
$438$		0			0
$439$	118115.			0.612880			0.306440	−	0.951890i	$-0.400862\pi$
$439$	118115.			0.612880			0.306440	+	0.951890i	$0.400862\pi$
$440$		0			0
$441$			17241.2i			0.0886523i
$442$		0			0
$443$	−37885.8	+	37885.8i	−0.193050	+	0.193050i	−0.797013	−	0.603963i	$-0.793588\pi$
$443$	−37885.8	+	37885.8i	−0.193050	+	0.193050i	0.603963	+	0.797013i	$0.293588\pi$
$444$		0			0
$445$	−24425.4	+	24425.4i	−0.123345	+	0.123345i
$446$		0			0
$447$		−	25188.9i		−	0.126065i
$448$		0			0
$449$	−85186.6			−0.422550			−0.211275	−	0.977427i	$-0.567762\pi$
$449$	−85186.6			−0.422550			−0.211275	+	0.977427i	$0.567762\pi$
$450$		0			0
$451$	143430.	+	143430.i	0.705157	+	0.705157i
$452$		0			0
$453$	57826.4	+	57826.4i	0.281793	+	0.281793i
$454$		0			0
$455$	−67348.4			−0.325315
$456$		0			0
$457$		−	96935.7i		−	0.464142i	−0.972699	−	0.232071i	$-0.925450\pi$
$457$		−	96935.7i		−	0.464142i	0.972699	−	0.232071i	$-0.0745502\pi$
$458$		0			0
$459$	194678.	−	194678.i	0.924042	−	0.924042i
$460$		0			0
$461$	−295014.	+	295014.i	−1.38816	+	1.38816i	−0.558988	+	0.829176i	$0.688810\pi$
$461$	−295014.	+	295014.i	−1.38816	+	1.38816i	−0.829176	+	0.558988i	$0.811190\pi$
$462$		0			0
$463$		−	224348.i		−	1.04655i	−0.852164	−	0.523275i	$-0.824710\pi$
$463$		−	224348.i		−	1.04655i	0.852164	−	0.523275i	$-0.175290\pi$
$464$		0			0
$465$	−10585.7			−0.0489567
$466$		0			0
$467$	−189556.	−	189556.i	−0.869168	−	0.869168i	0.123212	−	0.992380i	$-0.460680\pi$
$467$	−189556.	−	189556.i	−0.869168	−	0.869168i	−0.992380	+	0.123212i	$0.960680\pi$
$468$		0			0
$469$	−100044.	−	100044.i	−0.454827	−	0.454827i
$470$		0			0
$471$	−141054.			−0.635832
$472$		0			0
$473$			80902.6i			0.361610i
$474$		0			0
$475$	−117891.	+	117891.i	−0.522506	+	0.522506i
$476$		0			0
$477$	97133.7	−	97133.7i	0.426907	−	0.426907i
$478$		0			0
$479$			132796.i			0.578781i	0.957211	+	0.289390i	$0.0934526\pi$
$479$			132796.i			0.578781i	−0.957211	+	0.289390i	$0.906547\pi$
$480$		0			0
$481$	−462081.			−1.99723
$482$		0			0
$483$	131523.	+	131523.i	0.563778	+	0.563778i
$484$		0			0
$485$	−40338.2	−	40338.2i	−0.171488	−	0.171488i
$486$		0			0
$487$	−283936.			−1.19719			−0.598595	−	0.801052i	$-0.704274\pi$
$487$	−283936.			−1.19719			−0.598595	+	0.801052i	$0.704274\pi$
$488$		0			0
$489$			118101.i			0.493896i
$490$		0			0
$491$	111725.	−	111725.i	0.463435	−	0.463435i	−0.436345	−	0.899780i	$-0.643727\pi$
$491$	111725.	−	111725.i	0.463435	−	0.463435i	0.899780	+	0.436345i	$0.143727\pi$
$492$		0			0
$493$	31280.9	−	31280.9i	0.128702	−	0.128702i
$494$		0			0
$495$		−	52426.1i		−	0.213962i
$496$		0			0
$497$	406998.			1.64770
$498$		0			0
$499$	−61595.1	−	61595.1i	−0.247369	−	0.247369i	0.572521	−	0.819890i	$-0.305965\pi$
$499$	−61595.1	−	61595.1i	−0.247369	−	0.247369i	−0.819890	+	0.572521i	$0.805965\pi$
$500$		0			0
$501$	−92916.5	−	92916.5i	−0.370184	−	0.370184i
$502$		0			0
$503$	−70874.1			−0.280125			−0.140062	−	0.990143i	$-0.544730\pi$
$503$	−70874.1			−0.280125			−0.140062	+	0.990143i	$0.544730\pi$
$504$		0			0
$505$			5079.09i			0.0199160i
$506$		0			0
$507$	−81500.9	+	81500.9i	−0.317064	+	0.317064i
$508$		0			0
$509$	128181.	−	128181.i	0.494754	−	0.494754i	−0.415046	−	0.909800i	$-0.636235\pi$
$509$	128181.	−	128181.i	0.494754	−	0.494754i	0.909800	+	0.415046i	$0.136235\pi$
$510$		0			0
$511$			424241.i			1.62469i
$512$		0			0
$513$	208903.			0.793797
$514$		0			0
$515$	26186.3	+	26186.3i	0.0987324	+	0.0987324i
$516$		0			0
$517$	−82814.9	−	82814.9i	−0.309833	−	0.309833i
$518$		0			0
$519$	−94525.3			−0.350924
$520$		0			0
$521$		−	494250.i		−	1.82084i	−0.413688	−	0.910419i	$-0.635759\pi$
$521$		−	494250.i		−	1.82084i	0.413688	−	0.910419i	$-0.364241\pi$
$522$		0			0
$523$	−179670.	+	179670.i	−0.656859	+	0.656859i	−0.954636	−	0.297776i	$-0.903755\pi$
$523$	−179670.	+	179670.i	−0.656859	+	0.656859i	0.297776	+	0.954636i	$0.403755\pi$
$524$		0			0
$525$	−103091.	+	103091.i	−0.374026	+	0.374026i
$526$		0			0
$527$		−	108360.i		−	0.390163i
$528$		0			0
$529$	268417.			0.959175
$530$		0			0
$531$	−68347.9	−	68347.9i	−0.242402	−	0.242402i
$532$		0			0
$533$	−204457.	−	204457.i	−0.719694	−	0.719694i
$534$		0			0
$535$	−28242.7			−0.0986729
$536$		0			0
$537$		−	220112.i		−	0.763300i
$538$		0			0
$539$	37748.1	−	37748.1i	0.129932	−	0.129932i
$540$		0			0
$541$	−300961.	+	300961.i	−1.02829	+	1.02829i	−0.0287034	+	0.999588i	$0.509138\pi$
$541$	−300961.	+	300961.i	−1.02829	+	1.02829i	−0.999588	+	0.0287034i	$0.990862\pi$
$542$		0			0
$543$		−	124549.i		−	0.422417i
$544$		0			0
$545$	−31683.6			−0.106670
$546$		0			0
$547$	159057.	+	159057.i	0.531591	+	0.531591i	0.921046	−	0.389455i	$-0.127337\pi$
$547$	159057.	+	159057.i	0.531591	+	0.531591i	−0.389455	+	0.921046i	$0.627337\pi$
$548$		0			0
$549$	−92423.8	−	92423.8i	−0.306647	−	0.306647i
$550$		0			0
$551$	33566.6			0.110561
$552$		0			0
$553$		−	129765.i		−	0.424335i
$554$		0			0
$555$	54382.0	−	54382.0i	0.176551	−	0.176551i
$556$		0			0
$557$	394524.	−	394524.i	1.27164	−	1.27164i	0.326409	−	0.945229i	$-0.394161\pi$
$557$	394524.	−	394524.i	1.27164	−	1.27164i	0.945229	−	0.326409i	$-0.105839\pi$
$558$		0			0
$559$		−	115326.i		−	0.369065i
$560$		0			0
$561$	−326755.			−1.03824
$562$		0			0
$563$	−169317.	−	169317.i	−0.534177	−	0.534177i	0.387636	−	0.921813i	$-0.373292\pi$
$563$	−169317.	−	169317.i	−0.534177	−	0.534177i	−0.921813	+	0.387636i	$0.873292\pi$
$564$		0			0
$565$	15085.9	+	15085.9i	0.0472579	+	0.0472579i
$566$		0			0
$567$	−2346.48			−0.00729877
$568$		0			0
$569$		−	165645.i		−	0.511626i	−0.966726	−	0.255813i	$-0.917657\pi$
$569$		−	165645.i		−	0.511626i	0.966726	−	0.255813i	$-0.0823432\pi$
$570$		0			0
$571$	−335641.	+	335641.i	−1.02945	+	1.02945i	−0.0298925	+	0.999553i	$0.509517\pi$
$571$	−335641.	+	335641.i	−1.02945	+	1.02945i	−0.999553	+	0.0298925i	$0.990483\pi$
$572$		0			0
$573$	109771.	−	109771.i	0.334331	−	0.334331i
$574$		0			0
$575$			429737.i			1.29977i
$576$		0			0
$577$	−99615.9			−0.299211			−0.149605	−	0.988746i	$-0.547800\pi$
$577$	−99615.9			−0.299211			−0.149605	+	0.988746i	$0.547800\pi$
$578$		0			0
$579$	−222031.	−	222031.i	−0.662301	−	0.662301i
$580$		0			0
$581$	47033.3	+	47033.3i	0.139333	+	0.139333i
$582$		0			0
$583$	−425331.			−1.25138
$584$		0			0
$585$			74732.7i			0.218373i
$586$		0			0
$587$	45023.9	−	45023.9i	0.130667	−	0.130667i	−0.638748	−	0.769416i	$-0.720548\pi$
$587$	45023.9	−	45023.9i	0.130667	−	0.130667i	0.769416	+	0.638748i	$0.220548\pi$
$588$		0			0
$589$	58138.5	−	58138.5i	0.167584	−	0.167584i
$590$		0			0
$591$		−	79703.6i		−	0.228193i
$592$		0			0
$593$	−385101.			−1.09513			−0.547564	−	0.836764i	$-0.684445\pi$
$593$	−385101.			−1.09513			−0.547564	+	0.836764i	$0.684445\pi$
$594$		0			0
$595$	81136.1	+	81136.1i	0.229182	+	0.229182i
$596$		0			0
$597$	−124447.	−	124447.i	−0.349168	−	0.349168i
$598$		0			0
$599$	209331.			0.583417			0.291709	−	0.956507i	$-0.405776\pi$
$599$	209331.			0.583417			0.291709	+	0.956507i	$0.405776\pi$
$600$		0			0
$601$			515555.i			1.42734i	0.700484	+	0.713668i	$0.252967\pi$
$601$			515555.i			1.42734i	−0.700484	+	0.713668i	$0.747033\pi$
$602$		0			0
$603$	−111013.	+	111013.i	−0.305310	+	0.305310i
$604$		0			0
$605$	−45625.7	+	45625.7i	−0.124652	+	0.124652i
$606$		0			0
$607$			30292.7i			0.0822167i	0.999155	+	0.0411084i	$0.0130889\pi$
$607$			30292.7i			0.0822167i	−0.999155	+	0.0411084i	$0.986911\pi$
$608$		0			0
$609$	29352.7			0.0791432
$610$		0			0
$611$	118052.	+	118052.i	0.316220	+	0.316220i
$612$		0			0
$613$	99803.4	+	99803.4i	0.265598	+	0.265598i	0.827324	−	0.561726i	$-0.189862\pi$
$613$	99803.4	+	99803.4i	0.265598	+	0.265598i	−0.561726	+	0.827324i	$0.689862\pi$
$614$		0			0
$615$	48124.9			0.127239
$616$		0			0
$617$		−	102196.i		−	0.268451i	−0.990951	−	0.134226i	$-0.957145\pi$
$617$		−	102196.i		−	0.268451i	0.990951	−	0.134226i	$-0.0428547\pi$
$618$		0			0
$619$	105904.	−	105904.i	0.276395	−	0.276395i	−0.555273	−	0.831668i	$-0.687386\pi$
$619$	105904.	−	105904.i	0.276395	−	0.276395i	0.831668	+	0.555273i	$0.187386\pi$
$620$		0			0
$621$	380749.	−	380749.i	0.987314	−	0.987314i
$622$		0			0
$623$		−	234617.i		−	0.604483i
$624$		0			0
$625$	−308949.			−0.790909
$626$		0			0
$627$	−175315.	−	175315.i	−0.445948	−	0.445948i
$628$		0			0
$629$	556679.	+	556679.i	1.40703	+	1.40703i
$630$		0			0
$631$	692480.			1.73920			0.869598	−	0.493760i	$-0.164378\pi$
$631$	692480.			1.73920			0.869598	+	0.493760i	$0.164378\pi$
$632$		0			0
$633$		−	83779.3i		−	0.209088i
$634$		0			0
$635$	−108170.	+	108170.i	−0.268262	+	0.268262i
$636$		0			0
$637$	−53809.4	+	53809.4i	−0.132611	+	0.132611i
$638$		0			0
$639$		−	451622.i		−	1.10605i
$640$		0			0
$641$	49519.0			0.120519			0.0602595	−	0.998183i	$-0.480807\pi$
$641$	49519.0			0.120519			0.0602595	+	0.998183i	$0.480807\pi$
$642$		0			0
$643$	165494.	+	165494.i	0.400276	+	0.400276i	0.878330	−	0.478054i	$-0.158658\pi$
$643$	165494.	+	165494.i	0.400276	+	0.400276i	−0.478054	+	0.878330i	$0.658658\pi$
$644$		0			0
$645$	13572.6	+	13572.6i	0.0326245	+	0.0326245i
$646$		0			0
$647$	−666565.			−1.59233			−0.796166	−	0.605078i	$-0.793142\pi$
$647$	−666565.			−1.59233			−0.796166	+	0.605078i	$0.793142\pi$
$648$		0			0
$649$			299284.i			0.710548i
$650$		0			0
$651$	50840.0	−	50840.0i	0.119962	−	0.119962i
$652$		0			0
$653$	−461118.	+	461118.i	−1.08140	+	1.08140i	−0.0850198	+	0.996379i	$0.527095\pi$
$653$	−461118.	+	461118.i	−1.08140	+	1.08140i	−0.996379	+	0.0850198i	$0.972905\pi$
$654$		0			0
$655$			181910.i			0.424008i
$656$		0			0
$657$	470757.			1.09060
$658$		0			0
$659$	−326883.	−	326883.i	−0.752698	−	0.752698i	0.222284	−	0.974982i	$-0.428649\pi$
$659$	−326883.	−	326883.i	−0.752698	−	0.752698i	−0.974982	+	0.222284i	$0.928649\pi$
$660$		0			0
$661$	−549615.	−	549615.i	−1.25793	−	1.25793i	−0.952080	−	0.305848i	$-0.901060\pi$
$661$	−549615.	−	549615.i	−1.25793	−	1.25793i	−0.305848	−	0.952080i	$-0.598940\pi$
$662$		0			0
$663$	465785.			1.05964
$664$		0			0
$665$			87064.5i			0.196878i
$666$		0			0
$667$	61178.8	−	61178.8i	0.137515	−	0.137515i
$668$		0			0
$669$	−359005.	+	359005.i	−0.802137	+	0.802137i
$670$		0			0
$671$			404708.i			0.898869i
$672$		0			0
$673$	191664.			0.423166			0.211583	−	0.977360i	$-0.432138\pi$
$673$	191664.			0.423166			0.211583	+	0.977360i	$0.432138\pi$
$674$		0			0
$675$	298440.	+	298440.i	0.655012	+	0.655012i
$676$		0			0
$677$	544201.	+	544201.i	1.18736	+	1.18736i	0.977795	+	0.209564i	$0.0672043\pi$
$677$	544201.	+	544201.i	1.18736	+	1.18736i	0.209564	+	0.977795i	$0.432796\pi$
$678$		0			0
$679$	387466.			0.840416
$680$		0			0
$681$			137020.i			0.295455i
$682$		0			0
$683$	570769.	−	570769.i	1.22354	−	1.22354i	0.257179	−	0.966364i	$-0.417207\pi$
$683$	570769.	−	570769.i	1.22354	−	1.22354i	0.966364	−	0.257179i	$-0.0827929\pi$
$684$		0			0
$685$	−94177.8	+	94177.8i	−0.200709	+	0.200709i
$686$		0			0
$687$		−	500020.i		−	1.05943i
$688$		0			0
$689$	606305.			1.27718
$690$		0			0
$691$	333477.	+	333477.i	0.698409	+	0.698409i	0.964067	−	0.265658i	$-0.0855893\pi$
$691$	333477.	+	333477.i	0.698409	+	0.698409i	−0.265658	+	0.964067i	$0.585589\pi$
$692$		0			0
$693$	251788.	+	251788.i	0.524286	+	0.524286i
$694$		0			0
$695$	237831.			0.492378
$696$		0			0
$697$			492628.i			1.01404i
$698$		0			0
$699$	250135.	−	250135.i	0.511941	−	0.511941i
$700$		0			0
$701$	−31503.2	+	31503.2i	−0.0641089	+	0.0641089i	−0.738434	−	0.674325i	$-0.764435\pi$
$701$	−31503.2	+	31503.2i	−0.0641089	+	0.0641089i	0.674325	+	0.738434i	$0.264435\pi$
$702$		0			0
$703$			597354.i			1.20871i
$704$		0			0
$705$	−27786.9			−0.0559064
$706$		0			0
$707$	−24393.5	−	24393.5i	−0.0488017	−	0.0488017i
$708$		0			0
$709$	−609708.	−	609708.i	−1.21291	−	1.21291i	−0.970064	−	0.242847i	$-0.921919\pi$
$709$	−609708.	−	609708.i	−1.21291	−	1.21291i	−0.242847	−	0.970064i	$-0.578081\pi$
$710$		0			0
$711$	−143993.			−0.284841
$712$		0			0
$713$		−	211928.i		−	0.416878i
$714$		0			0
$715$	163621.	−	163621.i	0.320056	−	0.320056i
$716$		0			0
$717$	47357.2	−	47357.2i	0.0921187	−	0.0921187i
$718$		0			0
$719$		−	720908.i		−	1.39451i	−0.716823	−	0.697256i	$-0.754404\pi$
$719$		−	720908.i		−	1.39451i	0.716823	−	0.697256i	$-0.245596\pi$
$720$		0			0
$721$	−251531.			−0.483862
$722$		0			0
$723$	−109536.	−	109536.i	−0.209546	−	0.209546i
$724$		0			0
$725$	47953.4	+	47953.4i	0.0912311	+	0.0912311i
$726$		0			0
$727$	−279904.			−0.529591			−0.264796	−	0.964305i	$-0.585304\pi$
$727$	−279904.			−0.529591			−0.264796	+	0.964305i	$0.585304\pi$
$728$		0			0
$729$		−	323526.i		−	0.608772i
$730$		0			0
$731$	−138935.	+	138935.i	−0.260003	+	0.260003i
$732$		0			0
$733$	266306.	−	266306.i	0.495648	−	0.495648i	−0.414432	−	0.910080i	$-0.636020\pi$
$733$	266306.	−	266306.i	0.495648	−	0.495648i	0.910080	+	0.414432i	$0.136020\pi$
$734$		0			0
$735$		−	12665.6i		−	0.0234450i
$736$		0			0
$737$	486108.			0.894948
$738$		0			0
$739$	594958.	+	594958.i	1.08943	+	1.08943i	0.995587	+	0.0938378i	$0.0299135\pi$
$739$	594958.	+	594958.i	1.08943	+	1.08943i	0.0938378	+	0.995587i	$0.470086\pi$
$740$		0			0
$741$	249910.	+	249910.i	0.455142	+	0.455142i
$742$		0			0
$743$	289367.			0.524170			0.262085	−	0.965045i	$-0.415590\pi$
$743$	289367.			0.524170			0.262085	+	0.965045i	$0.415590\pi$
$744$		0			0
$745$		−	30390.8i		−	0.0547557i
$746$		0			0
$747$	52190.2	−	52190.2i	0.0935293	−	0.0935293i
$748$		0			0
$749$	135642.	−	135642.i	0.241785	−	0.241785i
$750$		0			0
$751$			935795.i			1.65921i	0.558352	+	0.829604i	$0.311434\pi$
$751$			935795.i			1.65921i	−0.558352	+	0.829604i	$0.688566\pi$
$752$		0			0
$753$	15629.1			0.0275641
$754$		0			0
$755$	69768.4	+	69768.4i	0.122395	+	0.122395i
$756$		0			0
$757$	626956.	+	626956.i	1.09407	+	1.09407i	0.995089	+	0.0989818i	$0.0315586\pi$
$757$	626956.	+	626956.i	1.09407	+	1.09407i	0.0989818	+	0.995089i	$0.468441\pi$
$758$		0			0
$759$	−639062.			−1.10933
$760$		0			0
$761$			253505.i			0.437741i	0.975754	+	0.218871i	$0.0702372\pi$
$761$			253505.i			0.437741i	−0.975754	+	0.218871i	$0.929763\pi$
$762$		0			0
$763$	152167.	−	152167.i	0.261380	−	0.261380i
$764$		0			0
$765$	90032.1	−	90032.1i	0.153842	−	0.153842i
$766$		0			0
$767$		−	426625.i		−	0.725197i
$768$		0			0
$769$	−740462.			−1.25213			−0.626066	−	0.779770i	$-0.715336\pi$
$769$	−740462.			−1.25213			−0.626066	+	0.779770i	$0.715336\pi$
$770$		0			0
$771$	287792.	+	287792.i	0.484138	+	0.484138i
$772$		0			0
$773$	−422057.	−	422057.i	−0.706337	−	0.706337i	0.259426	−	0.965763i	$-0.416467\pi$
$773$	−422057.	−	422057.i	−0.706337	−	0.706337i	−0.965763	+	0.259426i	$0.916467\pi$
$774$		0			0
$775$	166114.			0.276569
$776$		0			0
$777$			522364.i			0.865229i
$778$		0			0
$779$	−264312.	+	264312.i	−0.435553	+	0.435553i
$780$		0			0
$781$	−988787.	+	988787.i	−1.62107	+	1.62107i
$782$		0			0
$783$		−	84973.7i		−	0.138599i
$784$		0			0
$785$	−170183.			−0.276171
$786$		0			0
$787$	−658152.	−	658152.i	−1.06262	−	1.06262i	−0.997904	−	0.0647134i	$-0.979387\pi$
$787$	−658152.	−	658152.i	−1.06262	−	1.06262i	−0.0647134	−	0.997904i	$-0.520613\pi$
$788$		0			0
$789$	−266647.	−	266647.i	−0.428334	−	0.428334i
$790$		0			0
$791$	−144907.			−0.231599
$792$		0			0
$793$		−	576906.i		−	0.917400i
$794$		0			0
$795$	−71355.7	+	71355.7i	−0.112900	+	0.112900i
$796$		0			0

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.f (of order \(4\), degree \(2\), not minimal)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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