LMFDB - Embedded newform 76.6.e.a.45.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,6,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, degree $2$, 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:	$12.1891703058$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} + \cdots + 80\!\cdots\!36 $ x^18 - 2x^17 + 1540x^16 - 768x^15 + 1608492x^14 - 1027368x^13 + 897054160x^12 - 1275481376x^11 + 361098181456x^10 - 863969476320x^9 + 79755165392064x^8 - 375077568148992x^7 + 12736924096193536x^6 - 57314532742553600x^5 + 977121800205220864x^4 - 4977732006498379776x^3 + 53672321824823513088x^2 - 185653809995679793152*x + 804303742853852430336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			45.9
Root			$-12.9311 + 22.3974i$ of defining polynomial
Character	$\chi$	$=$	76.45
Dual form			76.6.e.a.49.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.4311 + 21.5314i) q^{3} +(-18.0963 - 31.3437i) q^{5} -208.885 q^{7} +(-187.566 + 324.875i) q^{9} +O(q^{10})$	\(q+(12.4311 + 21.5314i) q^{3} +(-18.0963 - 31.3437i) q^{5} -208.885 q^{7} +(-187.566 + 324.875i) q^{9} -204.444 q^{11} +(-146.736 + 254.155i) q^{13} +(449.916 - 779.277i) q^{15} +(-850.109 - 1472.43i) q^{17} +(1329.36 + 841.960i) q^{19} +(-2596.68 - 4497.58i) q^{21} +(-1768.22 + 3062.65i) q^{23} +(907.547 - 1571.92i) q^{25} -3285.13 q^{27} +(-3204.96 + 5551.15i) q^{29} -2731.24 q^{31} +(-2541.47 - 4401.96i) q^{33} +(3780.05 + 6547.24i) q^{35} -5427.82 q^{37} -7296.40 q^{39} +(-3692.75 - 6396.03i) q^{41} +(7870.13 + 13631.5i) q^{43} +13577.1 q^{45} +(9723.97 - 16842.4i) q^{47} +26826.0 q^{49} +(21135.7 - 36608.0i) q^{51} +(-4290.51 + 7431.39i) q^{53} +(3699.69 + 6408.05i) q^{55} +(-1603.08 + 39089.5i) q^{57} +(13969.1 + 24195.1i) q^{59} +(-15366.2 + 26615.0i) q^{61} +(39179.8 - 67861.5i) q^{63} +10621.5 q^{65} +(-24126.9 + 41788.9i) q^{67} -87924.1 q^{69} +(-3430.92 - 5942.53i) q^{71} +(19300.4 + 33429.3i) q^{73} +45127.4 q^{75} +42705.4 q^{77} +(-32155.8 - 55695.5i) q^{79} +(4740.78 + 8211.28i) q^{81} +71525.2 q^{83} +(-30767.7 + 53291.2i) q^{85} -159365. q^{87} +(-23256.6 + 40281.7i) q^{89} +(30651.0 - 53089.2i) q^{91} +(-33952.4 - 58807.3i) q^{93} +(2333.64 - 56903.5i) q^{95} +(-45206.3 - 78299.6i) q^{97} +(38346.9 - 66418.7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	$ 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) $ 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	12.4311	+	21.5314i	0.797458	+	1.38124i	0.921267	+	0.388932i	$0.127156\pi$
$3$	12.4311	+	21.5314i	0.797458	+	1.38124i	−0.123809	+	0.992306i	$0.539511\pi$
$4$		0			0
$5$	−18.0963	−	31.3437i	−0.323717	−	0.560694i	0.657535	−	0.753424i	$-0.271599\pi$
$5$	−18.0963	−	31.3437i	−0.323717	−	0.560694i	−0.981252	+	0.192730i	$0.938266\pi$
$6$		0			0
$7$	−208.885			−1.61125			−0.805624	−	0.592427i	$-0.798170\pi$
$7$	−208.885			−1.61125			−0.805624	+	0.592427i	$0.798170\pi$
$8$		0			0
$9$	−187.566	+	324.875i	−0.771879	+	1.33693i
$10$		0			0
$11$	−204.444			−0.509440			−0.254720	−	0.967015i	$-0.581983\pi$
$11$	−204.444			−0.509440			−0.254720	+	0.967015i	$0.581983\pi$
$12$		0			0
$13$	−146.736	+	254.155i	−0.240813	+	0.417100i	−0.960946	−	0.276736i	$-0.910747\pi$
$13$	−146.736	+	254.155i	−0.240813	+	0.417100i	0.720133	+	0.693836i	$0.244081\pi$
$14$		0			0
$15$	449.916	−	779.277i	0.516301	−	0.894260i
$16$		0			0
$17$	−850.109	−	1472.43i	−0.713432	−	1.23570i	−0.963561	−	0.267488i	$-0.913807\pi$
$17$	−850.109	−	1472.43i	−0.713432	−	1.23570i	0.250130	−	0.968212i	$-0.419527\pi$
$18$		0			0
$19$	1329.36	+	841.960i	0.844810	+	0.535066i
$19$	1329.36	+	841.960i	0.844810	+	0.535066i
$20$		0			0
$21$	−2596.68	−	4497.58i	−1.28490	−	2.22552i
$22$		0			0
$23$	−1768.22	+	3062.65i	−0.696975	+	1.20720i	0.272535	+	0.962146i	$0.412138\pi$
$23$	−1768.22	+	3062.65i	−0.696975	+	1.20720i	−0.969510	+	0.245051i	$0.921195\pi$
$24$		0			0
$25$	907.547	−	1571.92i	0.290415	−	0.503013i
$26$		0			0
$27$	−3285.13			−0.867247
$28$		0			0
$29$	−3204.96	+	5551.15i	−0.707664	+	1.22571i	0.258058	+	0.966130i	$0.416918\pi$
$29$	−3204.96	+	5551.15i	−0.707664	+	1.22571i	−0.965722	+	0.259580i	$0.916416\pi$
$30$		0			0
$31$	−2731.24			−0.510452			−0.255226	−	0.966881i	$-0.582150\pi$
$31$	−2731.24			−0.510452			−0.255226	+	0.966881i	$0.582150\pi$
$32$		0			0
$33$	−2541.47	−	4401.96i	−0.406257	−	0.703658i
$34$		0			0
$35$	3780.05	+	6547.24i	0.521588	+	0.903417i
$36$		0			0
$37$	−5427.82			−0.651811			−0.325905	−	0.945402i	$-0.605669\pi$
$37$	−5427.82			−0.651811			−0.325905	+	0.945402i	$0.605669\pi$
$38$		0			0
$39$	−7296.40			−0.768152
$40$		0			0
$41$	−3692.75	−	6396.03i	−0.343076	−	0.594225i	0.641926	−	0.766766i	$-0.278135\pi$
$41$	−3692.75	−	6396.03i	−0.343076	−	0.594225i	−0.985002	+	0.172541i	$0.944802\pi$
$42$		0			0
$43$	7870.13	+	13631.5i	0.649099	+	1.12427i	0.983339	+	0.181784i	$0.0581871\pi$
$43$	7870.13	+	13631.5i	0.649099	+	1.12427i	−0.334240	+	0.942488i	$0.608480\pi$
$44$		0			0
$45$	13577.1			0.999480
$46$		0			0
$47$	9723.97	−	16842.4i	0.642094	−	1.11214i	−0.342870	−	0.939383i	$-0.611399\pi$
$47$	9723.97	−	16842.4i	0.642094	−	1.11214i	0.984964	−	0.172757i	$-0.0552676\pi$
$48$		0			0
$49$	26826.0			1.59612
$50$		0			0
$51$	21135.7	−	36608.0i	1.13786	−	1.97084i
$52$		0			0
$53$	−4290.51	+	7431.39i	−0.209807	+	0.363396i	−0.951654	−	0.307173i	$-0.900617\pi$
$53$	−4290.51	+	7431.39i	−0.209807	+	0.363396i	0.741847	+	0.670569i	$0.233950\pi$
$54$		0			0
$55$	3699.69	+	6408.05i	0.164914	+	0.285640i
$56$		0			0
$57$	−1603.08	+	39089.5i	−0.0653532	+	1.59358i
$58$		0			0
$59$	13969.1	+	24195.1i	0.522441	+	0.904894i	0.999659	+	0.0261092i	$0.00831175\pi$
$59$	13969.1	+	24195.1i	0.522441	+	0.904894i	−0.477218	+	0.878785i	$0.658355\pi$
$60$		0			0
$61$	−15366.2	+	26615.0i	−0.528739	+	0.915803i	0.470700	+	0.882294i	$0.344002\pi$
$61$	−15366.2	+	26615.0i	−0.528739	+	0.915803i	−0.999438	+	0.0335090i	$0.989332\pi$
$62$		0			0
$63$	39179.8	−	67861.5i	1.24369	−	2.15413i
$64$		0			0
$65$	10621.5			0.311820
$66$		0			0
$67$	−24126.9	+	41788.9i	−0.656619	+	1.13730i	0.324866	+	0.945760i	$0.394681\pi$
$67$	−24126.9	+	41788.9i	−0.656619	+	1.13730i	−0.981485	+	0.191538i	$0.938652\pi$
$68$		0			0
$69$	−87924.1			−2.22323
$70$		0			0
$71$	−3430.92	−	5942.53i	−0.0807727	−	0.139902i	0.822809	−	0.568317i	$-0.192405\pi$
$71$	−3430.92	−	5942.53i	−0.0807727	−	0.139902i	−0.903582	+	0.428415i	$0.859072\pi$
$72$		0			0
$73$	19300.4	+	33429.3i	0.423897	+	0.734210i	0.996317	−	0.0857501i	$-0.0273287\pi$
$73$	19300.4	+	33429.3i	0.423897	+	0.734210i	−0.572420	+	0.819961i	$0.693995\pi$
$74$		0			0
$75$	45127.4			0.926375
$76$		0			0
$77$	42705.4			0.820834
$78$		0			0
$79$	−32155.8	−	55695.5i	−0.579684	−	1.00404i	−0.995515	−	0.0946008i	$-0.969843\pi$
$79$	−32155.8	−	55695.5i	−0.579684	−	1.00404i	0.415831	−	0.909442i	$-0.363491\pi$
$80$		0			0
$81$	4740.78	+	8211.28i	0.0802856	+	0.139059i
$82$		0			0
$83$	71525.2			1.13963			0.569815	−	0.821773i	$-0.307015\pi$
$83$	71525.2			1.13963			0.569815	+	0.821773i	$0.307015\pi$
$84$		0			0
$85$	−30767.7	+	53291.2i	−0.461900	+	0.800034i
$86$		0			0
$87$	−159365.			−2.25733
$88$		0			0
$89$	−23256.6	+	40281.7i	−0.311223	+	0.539054i	−0.978627	−	0.205641i	$-0.934072\pi$
$89$	−23256.6	+	40281.7i	−0.311223	+	0.539054i	0.667404	+	0.744696i	$0.267405\pi$
$90$		0			0
$91$	30651.0	−	53089.2i	0.388009	−	0.672051i
$92$		0			0
$93$	−33952.4	−	58807.3i	−0.407064	−	0.705056i
$94$		0			0
$95$	2333.64	−	56903.5i	0.0265292	−	0.646890i
$96$		0			0
$97$	−45206.3	−	78299.6i	−0.487831	−	0.844949i	0.512071	−	0.858943i	$-0.328879\pi$
$97$	−45206.3	−	78299.6i	−0.487831	−	0.844949i	−0.999902	+	0.0139945i	$0.995545\pi$
$98$		0			0
$99$	38346.9	−	66418.7i	0.393226	−	0.681087i
$100$		0			0
$101$	73036.4	−	126503.i	0.712420	−	1.23395i	−0.251527	−	0.967850i	$-0.580933\pi$
$101$	73036.4	−	126503.i	0.712420	−	1.23395i	0.963946	−	0.266097i	$-0.0857340\pi$
$102$		0			0
$103$	−85248.6			−0.791761			−0.395880	−	0.918302i	$-0.629561\pi$
$103$	−85248.6			−0.791761			−0.395880	+	0.918302i	$0.629561\pi$
$104$		0			0
$105$	−93980.7	+	162779.i	−0.831889	+	1.44087i
$106$		0			0
$107$	−28226.4			−0.238339			−0.119170	−	0.992874i	$-0.538023\pi$
$107$	−28226.4			−0.238339			−0.119170	+	0.992874i	$0.538023\pi$
$108$		0			0
$109$	109018.	+	188825.i	0.878888	+	1.52228i	0.852563	+	0.522625i	$0.175047\pi$
$109$	109018.	+	188825.i	0.878888	+	1.52228i	0.0263250	+	0.999653i	$0.491620\pi$
$110$		0			0
$111$	−67474.0	−	116868.i	−0.519792	−	0.900305i
$112$		0			0
$113$	−174546.			−1.28592			−0.642958	−	0.765901i	$-0.722293\pi$
$113$	−174546.			−1.28592			−0.642958	+	0.765901i	$0.722293\pi$
$114$		0			0
$115$	127993.			0.902490
$116$		0			0
$117$	−55045.6	−	95341.8i	−0.371756	−	0.643901i
$118$		0			0
$119$	177575.	+	307569.i	1.14952	+	1.99102i
$120$		0			0
$121$	−119254.			−0.740471
$122$		0			0
$123$	91810.2	−	159020.i	0.547177	−	0.947739i
$124$		0			0
$125$	−178795.			−1.02348
$126$		0			0
$127$	−71846.8	+	124442.i	−0.395274	+	0.684634i	−0.993136	−	0.116965i	$-0.962684\pi$
$127$	−71846.8	+	124442.i	−0.395274	+	0.684634i	0.597862	+	0.801599i	$0.296017\pi$
$128$		0			0
$129$	−195669.	+	338909.i	−1.03526	+	1.79312i
$130$		0			0
$131$	2867.13	+	4966.01i	0.0145972	+	0.0252830i	0.873232	−	0.487305i	$-0.162020\pi$
$131$	2867.13	+	4966.01i	0.0145972	+	0.0252830i	−0.858635	+	0.512588i	$0.828687\pi$
$132$		0			0
$133$	−277684.	−	175873.i	−1.36120	−	0.862125i
$134$		0			0
$135$	59448.7	+	102968.i	0.280742	+	0.486260i
$136$		0			0
$137$	150187.	−	260131.i	0.683645	−	1.18411i	−0.290216	−	0.956961i	$-0.593727\pi$
$137$	150187.	−	260131.i	0.683645	−	1.18411i	0.973861	−	0.227146i	$-0.0729394\pi$
$138$		0			0
$139$	218624.	−	378668.i	0.959757	−	1.66235i	0.236670	−	0.971590i	$-0.423944\pi$
$139$	218624.	−	378668.i	0.959757	−	1.66235i	0.723087	−	0.690757i	$-0.242723\pi$
$140$		0			0
$141$	483520.			2.04817
$142$		0			0
$143$	29999.4	−	51960.5i	0.122680	−	0.212487i
$144$		0			0
$145$	231992.			0.916331
$146$		0			0
$147$	333478.	+	577600.i	1.27284	+	2.20462i
$148$		0			0
$149$	4540.46	+	7864.31i	0.0167546	+	0.0290198i	0.874281	−	0.485420i	$-0.161333\pi$
$149$	4540.46	+	7864.31i	0.0167546	+	0.0290198i	−0.857527	+	0.514440i	$0.828000\pi$
$150$		0			0
$151$	43328.1			0.154642			0.0773209	−	0.997006i	$-0.475363\pi$
$151$	43328.1			0.154642			0.0773209	+	0.997006i	$0.475363\pi$
$152$		0			0
$153$	637808.			2.20273
$154$		0			0
$155$	49425.3	+	85607.2i	0.165242	+	0.286208i
$156$		0			0
$157$	−50402.1	−	87299.0i	−0.163192	−	0.282657i	0.772820	−	0.634626i	$-0.218846\pi$
$157$	−50402.1	−	87299.0i	−0.163192	−	0.282657i	−0.936012	+	0.351968i	$0.885512\pi$
$158$		0			0
$159$	−213344.			−0.669248
$160$		0			0
$161$	369355.	−	639742.i	1.12300	−	1.94509i
$162$		0			0
$163$	−315707.			−0.930712			−0.465356	−	0.885124i	$-0.654074\pi$
$163$	−315707.			−0.930712			−0.465356	+	0.885124i	$0.654074\pi$
$164$		0			0
$165$	−91982.7	+	159319.i	−0.263024	+	0.455572i
$166$		0			0
$167$	280911.	−	486551.i	0.779430	−	1.35001i	−0.152841	−	0.988251i	$-0.548842\pi$
$167$	280911.	−	486551.i	0.779430	−	1.35001i	0.932271	−	0.361761i	$-0.117824\pi$
$168$		0			0
$169$	142583.	+	246962.i	0.384019	+	0.665140i
$170$		0			0
$171$	−522875.	+	273952.i	−1.36744	+	0.716448i
$172$		0			0
$173$	−323128.	−	559674.i	−0.820842	−	1.42174i	−0.905056	−	0.425292i	$-0.860171\pi$
$173$	−323128.	−	559674.i	−0.820842	−	1.42174i	0.0842143	−	0.996448i	$-0.473162\pi$
$174$		0			0
$175$	−189573.	+	328350.i	−0.467930	+	0.810479i
$176$		0			0
$177$	−347303.	+	601546.i	−0.833249	+	1.44323i
$178$		0			0
$179$	391686.			0.913704			0.456852	−	0.889543i	$-0.348977\pi$
$179$	391686.			0.913704			0.456852	+	0.889543i	$0.348977\pi$
$180$		0			0
$181$	−356595.	+	617641.i	−0.809057	+	1.40133i	0.104461	+	0.994529i	$0.466688\pi$
$181$	−356595.	+	617641.i	−0.809057	+	1.40133i	−0.913518	+	0.406798i	$0.866645\pi$
$182$		0			0
$183$	−764076.			−1.68659
$184$		0			0
$185$	98223.6	+	170128.i	0.211002	+	0.365466i
$186$		0			0
$187$	173800.	+	301030.i	0.363451	+	0.629515i
$188$		0			0
$189$	686214.			1.39735
$190$		0			0
$191$	73696.8			0.146172			0.0730862	−	0.997326i	$-0.476715\pi$
$191$	73696.8			0.146172			0.0730862	+	0.997326i	$0.476715\pi$
$192$		0			0
$193$	−281706.	−	487929.i	−0.544381	−	0.942896i	−0.998646	−	0.0520289i	$-0.983431\pi$
$193$	−281706.	−	487929.i	−0.544381	−	0.942896i	0.454264	−	0.890867i	$-0.349902\pi$
$194$		0			0
$195$	132038.	+	228696.i	0.248664	+	0.430698i
$196$		0			0
$197$	227957.			0.418491			0.209246	−	0.977863i	$-0.432899\pi$
$197$	227957.			0.418491			0.209246	+	0.977863i	$0.432899\pi$
$198$		0			0
$199$	−305231.	+	528676.i	−0.546382	+	0.946361i	0.452137	+	0.891949i	$0.350662\pi$
$199$	−305231.	+	528676.i	−0.546382	+	0.946361i	−0.998519	+	0.0544126i	$0.982671\pi$
$200$		0			0
$201$	−1.19970e6			−2.09451
$202$		0			0
$203$	669468.	−	1.15955e6i	1.14022	−	1.97492i
$204$		0			0
$205$	−133650.	+	231489.i	−0.222119	+	0.384721i
$206$		0			0
$207$	−663319.	−	1.14890e6i	−1.07596	−	1.86362i
$208$		0			0
$209$	−271780.	−	172134.i	−0.430380	−	0.272584i
$210$		0			0
$211$	−340640.	−	590006.i	−0.526732	−	0.912327i	−0.999515	−	0.0311480i	$-0.990084\pi$
$211$	−340640.	−	590006.i	−0.526732	−	0.912327i	0.472782	−	0.881179i	$-0.343250\pi$
$212$		0			0
$213$	85300.5	−	147745.i	0.128826	−	0.223133i
$214$		0			0
$215$	284841.	−	493359.i	0.420248	−	0.727891i
$216$		0			0
$217$	570515.			0.822466
$218$		0			0
$219$	−479853.	+	831129.i	−0.676079	+	1.17100i
$220$		0			0
$221$	498968.			0.687214
$222$		0			0
$223$	−464769.	−	805003.i	−0.625856	−	1.08402i	−0.988375	−	0.152038i	$-0.951416\pi$
$223$	−464769.	−	805003.i	−0.625856	−	1.08402i	0.362518	−	0.931977i	$-0.381917\pi$
$224$		0			0
$225$	340451.	+	589678.i	0.448330	+	0.776530i
$226$		0			0
$227$	−836852.			−1.07791			−0.538957	−	0.842333i	$-0.681181\pi$
$227$	−836852.			−1.07791			−0.538957	+	0.842333i	$0.681181\pi$
$228$		0			0
$229$	−43953.2			−0.0553862			−0.0276931	−	0.999616i	$-0.508816\pi$
$229$	−43953.2			−0.0553862			−0.0276931	+	0.999616i	$0.508816\pi$
$230$		0			0
$231$	530876.	+	919505.i	0.654581	+	1.13377i
$232$		0			0
$233$	19649.6	+	34034.0i	0.0237117	+	0.0410699i	0.877638	−	0.479324i	$-0.159118\pi$
$233$	19649.6	+	34034.0i	0.0237117	+	0.0410699i	−0.853926	+	0.520394i	$0.825785\pi$
$234$		0			0
$235$	−703872.			−0.831427
$236$		0			0
$237$	799466.	−	1.38472e6i	0.924548	−	1.60136i
$238$		0			0
$239$	1.14239e6			1.29365			0.646827	−	0.762637i	$-0.276096\pi$
$239$	1.14239e6			1.29365			0.646827	+	0.762637i	$0.276096\pi$
$240$		0			0
$241$	−142528.	+	246866.i	−0.158073	+	0.273790i	−0.934174	−	0.356819i	$-0.883861\pi$
$241$	−142528.	+	246866.i	−0.158073	+	0.273790i	0.776101	+	0.630609i	$0.217195\pi$
$242$		0			0
$243$	−517010.	+	895487.i	−0.561672	+	0.972845i
$244$		0			0
$245$	−485452.	−	840827.i	−0.516691	−	0.894935i
$246$		0			0
$247$	−409054.	+	214317.i	−0.426617	+	0.223519i
$248$		0			0
$249$	889139.	+	1.54003e6i	0.908806	+	1.57410i
$250$		0			0
$251$	−787154.	+	1.36339e6i	−0.788634	+	1.36595i	0.138169	+	0.990409i	$0.455878\pi$
$251$	−787154.	+	1.36339e6i	−0.788634	+	1.36595i	−0.926804	+	0.375546i	$0.877455\pi$
$252$		0			0
$253$	361503.	−	626141.i	0.355067	−	0.614994i
$254$		0			0
$255$	−1.52991e6			−1.47338
$256$		0			0
$257$	172908.	−	299486.i	0.163299	−	0.282842i	−0.772751	−	0.634709i	$-0.781120\pi$
$257$	172908.	−	299486.i	0.163299	−	0.282842i	0.936050	+	0.351867i	$0.114453\pi$
$258$		0			0
$259$	1.13379e6			1.05023
$260$		0			0
$261$	−1.20228e6	−	2.08242e6i	−1.09246	−	1.89220i
$262$		0			0
$263$	494023.	+	855673.i	0.440411	+	0.762814i	0.997720	−	0.0674913i	$-0.0214995\pi$
$263$	494023.	+	855673.i	0.440411	+	0.762814i	−0.557309	+	0.830305i	$0.688166\pi$
$264$		0			0
$265$	310570.			0.271672
$266$		0			0
$267$	−1.15643e6			−0.992750
$268$		0			0
$269$	418483.	+	724833.i	0.352612	+	0.610741i	0.986706	−	0.162514i	$-0.0519603\pi$
$269$	418483.	+	724833.i	0.352612	+	0.610741i	−0.634094	+	0.773256i	$0.718627\pi$
$270$		0			0
$271$	1.04254e6	+	1.80574e6i	0.862325	+	1.49359i	0.869679	+	0.493618i	$0.164326\pi$
$271$	1.04254e6	+	1.80574e6i	0.862325	+	1.49359i	−0.00735345	+	0.999973i	$0.502341\pi$
$272$		0			0
$273$	1.52411e6			1.23768
$274$		0			0
$275$	−185543.	+	321369.i	−0.147949	+	0.256255i
$276$		0			0
$277$	−654742.			−0.512709			−0.256355	−	0.966583i	$-0.582521\pi$
$277$	−654742.			−0.512709			−0.256355	+	0.966583i	$0.582521\pi$
$278$		0			0
$279$	512289.	−	887310.i	0.394007	−	0.682441i
$280$		0			0
$281$	432664.	−	749397.i	0.326878	−	0.566169i	−0.655013	−	0.755618i	$-0.727337\pi$
$281$	432664.	−	749397.i	0.326878	−	0.566169i	0.981891	+	0.189449i	$0.0606702\pi$
$282$		0			0
$283$	840551.	+	1.45588e6i	0.623876	+	1.08058i	0.988757	+	0.149530i	$0.0477761\pi$
$283$	840551.	+	1.45588e6i	0.623876	+	1.08058i	−0.364882	+	0.931054i	$0.618891\pi$
$284$		0			0
$285$	1.25422e6	−	657129.i	0.914665	−	0.479224i
$286$		0			0
$287$	771360.	+	1.33604e6i	0.552780	+	0.957444i
$288$		0			0
$289$	−735443.	+	1.27382e6i	−0.517970	+	0.897150i
$290$		0			0
$291$	1.12393e6	−	1.94671e6i	0.778050	−	1.34762i
$292$		0			0
$293$	495850.			0.337428			0.168714	−	0.985665i	$-0.446039\pi$
$293$	495850.			0.337428			0.168714	+	0.985665i	$0.446039\pi$
$294$		0			0
$295$	505577.	−	875685.i	0.338246	−	0.585859i
$296$		0			0
$297$	671625.			0.441810
$298$		0			0
$299$	−518925.	−	898805.i	−0.335681	−	0.581416i
$300$		0			0
$301$	−1.64395e6	−	2.84741e6i	−1.04586	−	1.81148i
$302$		0			0
$303$	3.63170e6			2.27250
$304$		0			0
$305$	1.11228e6			0.684647
$306$		0			0
$307$	1.22956e6	+	2.12967e6i	0.744569	+	1.28963i	0.950396	+	0.311043i	$0.100678\pi$
$307$	1.22956e6	+	2.12967e6i	0.744569	+	1.28963i	−0.205826	+	0.978589i	$0.565988\pi$
$308$		0			0
$309$	−1.05974e6	−	1.83552e6i	−0.631396	−	1.09361i
$310$		0			0
$311$	1.82000e6			1.06702			0.533508	−	0.845795i	$-0.320873\pi$
$311$	1.82000e6			1.06702			0.533508	+	0.845795i	$0.320873\pi$
$312$		0			0
$313$	−847350.	+	1.46765e6i	−0.488880	+	0.846765i	−0.999918	−	0.0127931i	$-0.995928\pi$
$313$	−847350.	+	1.46765e6i	−0.488880	+	0.846765i	0.511038	+	0.859558i	$0.329261\pi$
$314$		0			0
$315$	−2.83604e6			−1.61041
$316$		0			0
$317$	−785090.	+	1.35982e6i	−0.438805	+	0.760033i	−0.997598	−	0.0692749i	$-0.977931\pi$
$317$	−785090.	+	1.35982e6i	−0.438805	+	0.760033i	0.558793	+	0.829307i	$0.311265\pi$
$318$		0			0
$319$	655235.	−	1.13490e6i	0.360512	−	0.624426i
$320$		0			0
$321$	−350886.	−	607753.i	−0.190066	−	0.329203i
$322$		0			0
$323$	109627.	−	2.67315e6i	0.0584671	−	1.42567i
$324$		0			0
$325$	266340.	+	461315.i	0.139871	+	0.242264i
$326$		0			0
$327$	−2.71045e6	+	4.69463e6i	−1.40175	+	2.42791i
$328$		0			0
$329$	−2.03119e6	+	3.51813e6i	−1.03457	+	1.79193i
$330$		0			0
$331$	−1.47267e6			−0.738816			−0.369408	−	0.929267i	$-0.620440\pi$
$331$	−1.47267e6			−0.738816			−0.369408	+	0.929267i	$0.620440\pi$
$332$		0			0
$333$	1.01808e6	−	1.76336e6i	0.503119	−	0.871427i
$334$		0			0
$335$	1.74643e6			0.850235
$336$		0			0
$337$	−1.04852e6	−	1.81609e6i	−0.502922	−	0.871087i	−0.999994	−	0.00337788i	$-0.998925\pi$
$337$	−1.04852e6	−	1.81609e6i	−0.502922	−	0.871087i	0.497072	−	0.867709i	$-0.334409\pi$
$338$		0			0
$339$	−2.16980e6	−	3.75820e6i	−1.02546	−	1.77616i
$340$		0			0
$341$	558386.			0.260045
$342$		0			0
$343$	−2.09282e6			−0.960498
$344$		0			0
$345$	1.59110e6	+	2.75587e6i	0.719698	+	1.24655i
$346$		0			0
$347$	−1.08983e6	−	1.88764e6i	−0.485887	−	0.841582i	0.513981	−	0.857802i	$-0.328170\pi$
$347$	−1.08983e6	−	1.88764e6i	−0.485887	−	0.841582i	−0.999868	+	0.0162199i	$0.994837\pi$
$348$		0			0
$349$	1.80723e6			0.794238			0.397119	−	0.917767i	$-0.370010\pi$
$349$	1.80723e6			0.794238			0.397119	+	0.917767i	$0.370010\pi$
$350$		0			0
$351$	482047.	−	834931.i	0.208844	−	0.361728i
$352$		0			0
$353$	−2.11807e6			−0.904696			−0.452348	−	0.891842i	$-0.649414\pi$
$353$	−2.11807e6			−0.904696			−0.452348	+	0.891842i	$0.649414\pi$
$354$		0			0
$355$	−124174.	+	215076.i	−0.0522950	+	0.0905775i
$356$		0			0
$357$	−4.41492e6	+	7.64687e6i	−1.83338	+	3.17551i
$358$		0			0
$359$	−1.04610e6	−	1.81190e6i	−0.428388	−	0.741990i	0.568342	−	0.822793i	$-0.307585\pi$
$359$	−1.04610e6	−	1.81190e6i	−0.428388	−	0.741990i	−0.996730	+	0.0808022i	$0.974252\pi$
$360$		0			0
$361$	1.05830e6	+	2.23854e6i	0.427408	+	0.904059i
$362$		0			0
$363$	−1.48246e6	−	2.56769e6i	−0.590494	−	1.02277i
$364$		0			0
$365$	698534.	−	1.20990e6i	0.274445	−	0.475352i
$366$		0			0
$367$	−1.45084e6	+	2.51293e6i	−0.562282	+	0.973901i	0.435015	+	0.900423i	$0.356743\pi$
$367$	−1.45084e6	+	2.51293e6i	−0.562282	+	0.973901i	−0.997297	+	0.0734776i	$0.976590\pi$
$368$		0			0
$369$	2.77054e6			1.05925
$370$		0			0
$371$	896224.	−	1.55231e6i	0.338051	−	0.585521i
$372$		0			0
$373$	−3.15657e6			−1.17474			−0.587372	−	0.809317i	$-0.699837\pi$
$373$	−3.15657e6			−1.17474			−0.587372	+	0.809317i	$0.699837\pi$
$374$		0			0
$375$	−2.22263e6	−	3.84970e6i	−0.816184	−	1.41367i
$376$		0			0
$377$	−940567.	−	1.62911e6i	−0.340829	−	0.590333i
$378$		0			0
$379$	3.95334e6			1.41373			0.706864	−	0.707350i	$-0.250109\pi$
$379$	3.95334e6			1.41373			0.706864	+	0.707350i	$0.250109\pi$
$380$		0			0
$381$	−3.57255e6			−1.26086
$382$		0			0
$383$	1.93528e6	+	3.35201e6i	0.674136	+	1.16764i	0.976720	+	0.214516i	$0.0688174\pi$
$383$	1.93528e6	+	3.35201e6i	0.674136	+	1.16764i	−0.302584	+	0.953123i	$0.597849\pi$
$384$		0			0
$385$	−772810.	−	1.33855e6i	−0.265718	−	0.460237i
$386$		0			0
$387$	−5.90469e6			−2.00410
$388$		0			0
$389$	1.11143e6	−	1.92505e6i	0.372399	−	0.645013i	−0.617535	−	0.786543i	$-0.711869\pi$
$389$	1.11143e6	−	1.92505e6i	0.372399	−	0.645013i	0.989934	+	0.141530i	$0.0452021\pi$
$390$		0			0
$391$	6.01273e6			1.98898
$392$		0			0
$393$	−71283.3	+	123466.i	−0.0232813	+	0.0403243i
$394$		0			0
$395$	−1.16380e6	+	2.01577e6i	−0.375307	+	0.650051i
$396$		0			0
$397$	−207721.	−	359784.i	−0.0661462	−	0.114569i	0.831056	−	0.556189i	$-0.187737\pi$
$397$	−207721.	−	359784.i	−0.0661462	−	0.114569i	−0.897202	+	0.441621i	$0.854404\pi$
$398$		0			0
$399$	334859.	−	8.16521e6i	0.105300	−	2.56765i
$400$		0			0
$401$	1.50770e6	+	2.61141e6i	0.468224	+	0.810988i	0.999341	−	0.0363107i	$-0.0115606\pi$
$401$	1.50770e6	+	2.61141e6i	0.468224	+	0.810988i	−0.531116	+	0.847299i	$0.678227\pi$
$402$		0			0
$403$	400772.	−	694157.i	0.122923	−	0.212910i
$404$		0			0
$405$	171581.	−	297188.i	0.0519796	−	0.0900313i
$406$		0			0
$407$	1.10969e6			0.332058
$408$		0			0
$409$	−1.89456e6	+	3.28148e6i	−0.560016	+	0.969976i	0.437478	+	0.899229i	$0.355872\pi$
$409$	−1.89456e6	+	3.28148e6i	−0.560016	+	0.969976i	−0.997494	+	0.0707474i	$0.977462\pi$
$410$		0			0
$411$	7.46797e6			2.18071
$412$		0			0
$413$	−2.91793e6	−	5.05400e6i	−0.841782	−	1.45801i
$414$		0			0
$415$	−1.29434e6	−	2.24187e6i	−0.368917	−	0.638983i
$416$		0			0
$417$	1.08710e7			3.06146
$418$		0			0
$419$	359202.			0.0999549			0.0499774	−	0.998750i	$-0.484085\pi$
$419$	359202.			0.0999549			0.0499774	+	0.998750i	$0.484085\pi$
$420$		0			0
$421$	−272896.	−	472669.i	−0.0750398	−	0.129973i	0.826064	−	0.563577i	$-0.190575\pi$
$421$	−272896.	−	472669.i	−0.0750398	−	0.129973i	−0.901104	+	0.433604i	$0.857242\pi$
$422$		0			0
$423$	3.64778e6	+	6.31814e6i	0.991238	+	1.71687i
$424$		0			0
$425$	−3.08606e6			−0.828765
$426$		0			0
$427$	3.20977e6	−	5.55948e6i	0.851930	−	1.47559i
$428$		0			0
$429$	1.49171e6			0.391327
$430$		0			0
$431$	−1.14797e6	+	1.98834e6i	−0.297671	+	0.515581i	−0.975603	−	0.219544i	$-0.929543\pi$
$431$	−1.14797e6	+	1.98834e6i	−0.297671	+	0.515581i	0.677932	+	0.735125i	$0.262876\pi$
$432$		0			0
$433$	−2.52128e6	+	4.36698e6i	−0.646251	+	1.11934i	0.337760	+	0.941232i	$0.390331\pi$
$433$	−2.52128e6	+	4.36698e6i	−0.646251	+	1.11934i	−0.984011	+	0.178107i	$0.943003\pi$
$434$		0			0
$435$	2.88392e6	+	4.99510e6i	0.730735	+	1.26567i
$436$		0			0
$437$	−4.92924e6	+	2.58260e6i	−1.23474	+	0.646924i
$438$		0			0
$439$	−3.91644e6	−	6.78347e6i	−0.969907	−	1.67993i	−0.695810	−	0.718226i	$-0.744955\pi$
$439$	−3.91644e6	−	6.78347e6i	−0.969907	−	1.67993i	−0.274097	−	0.961702i	$-0.588379\pi$
$440$		0			0
$441$	−5.03166e6	+	8.71509e6i	−1.23201	+	2.13391i
$442$		0			0
$443$	−2.75654e6	+	4.77446e6i	−0.667351	+	1.15589i	0.311291	+	0.950315i	$0.399239\pi$
$443$	−2.75654e6	+	4.77446e6i	−0.667351	+	1.15589i	−0.978642	+	0.205571i	$0.934095\pi$
$444$		0			0
$445$	1.68344e6			0.402993
$446$		0			0
$447$	−112886.	+	195525.i	−0.0267222	+	0.0462842i
$448$		0			0
$449$	5.99055e6			1.40233			0.701166	−	0.712998i	$-0.252663\pi$
$449$	5.99055e6			1.40233			0.701166	+	0.712998i	$0.252663\pi$
$450$		0			0
$451$	754961.	+	1.30763e6i	0.174777	+	0.302722i
$452$		0			0
$453$	538617.	+	932913.i	0.123320	+	0.213597i
$454$		0			0
$455$	−2.21868e6			−0.502420
$456$		0			0
$457$	289587.			0.0648617			0.0324308	−	0.999474i	$-0.489675\pi$
$457$	289587.			0.0648617			0.0324308	+	0.999474i	$0.489675\pi$
$458$		0			0
$459$	2.79272e6	+	4.83713e6i	0.618721	+	1.07166i
$460$		0			0
$461$	−1.79574e6	−	3.11031e6i	−0.393542	−	0.681635i	0.599372	−	0.800471i	$-0.295417\pi$
$461$	−1.79574e6	−	3.11031e6i	−0.393542	−	0.681635i	−0.992914	+	0.118836i	$0.962084\pi$
$462$		0			0
$463$	108518.			0.0235260			0.0117630	−	0.999931i	$-0.496256\pi$
$463$	108518.			0.0235260			0.0117630	+	0.999931i	$0.496256\pi$
$464$		0			0
$465$	−1.22883e6	+	2.12839e6i	−0.263547	+	0.456477i
$466$		0			0
$467$	−4.47855e6			−0.950267			−0.475133	−	0.879914i	$-0.657600\pi$
$467$	−4.47855e6			−0.950267			−0.475133	+	0.879914i	$0.657600\pi$
$468$		0			0
$469$	5.03974e6	−	8.72909e6i	1.05798	−	1.83247i
$470$		0			0
$471$	1.25311e6	−	2.17045e6i	0.260278	−	0.450815i
$472$		0			0
$473$	−1.60900e6	−	2.78687e6i	−0.330677	−	0.572749i
$474$		0			0
$475$	2.52995e6	−	1.32553e6i	0.514491	−	0.269560i
$476$		0			0
$477$	−1.60951e6	−	2.78776e6i	−0.323891	−	0.560995i
$478$		0			0
$479$	−3.21830e6	+	5.57425e6i	−0.640896	+	1.11006i	0.344337	+	0.938846i	$0.388104\pi$
$479$	−3.21830e6	+	5.57425e6i	−0.640896	+	1.11006i	−0.985233	+	0.171218i	$0.945230\pi$
$480$		0			0
$481$	796459.	−	1.37951e6i	0.156964	−	0.271870i
$482$		0			0
$483$	1.83660e7			3.58218
$484$		0			0
$485$	−1.63614e6	+	2.83387e6i	−0.315838	+	0.547048i
$486$		0			0
$487$	−983792.			−0.187967			−0.0939833	−	0.995574i	$-0.529960\pi$
$487$	−983792.			−0.187967			−0.0939833	+	0.995574i	$0.529960\pi$
$488$		0			0
$489$	−3.92460e6	−	6.79761e6i	−0.742204	−	1.28553i
$490$		0			0
$491$	3.28089e6	+	5.68267e6i	0.614170	+	1.06377i	0.990530	+	0.137300i	$0.0438423\pi$
$491$	3.28089e6	+	5.68267e6i	0.614170	+	1.06377i	−0.376360	+	0.926474i	$0.622824\pi$
$492$		0			0
$493$	1.08983e7			2.01948
$494$		0			0
$495$	−2.77575e6			−0.509175
$496$		0			0
$497$	716668.	+	1.24131e6i	0.130145	+	0.225418i
$498$		0			0
$499$	−2.23268e6	−	3.86711e6i	−0.401398	−	0.695241i	0.592497	−	0.805572i	$-0.298142\pi$
$499$	−2.23268e6	−	3.86711e6i	−0.401398	−	0.695241i	−0.993895	+	0.110331i	$0.964809\pi$
$500$		0			0
$501$	1.39682e7			2.48625
$502$		0			0
$503$	−1.44875e6	+	2.50931e6i	−0.255313	+	0.442215i	−0.964981	−	0.262322i	$-0.915512\pi$
$503$	−1.44875e6	+	2.50931e6i	−0.255313	+	0.442215i	0.709667	+	0.704537i	$0.248845\pi$
$504$		0			0
$505$	−5.28676e6			−0.922489
$506$		0			0
$507$	−3.54495e6	+	6.14003e6i	−0.612477	+	1.06084i
$508$		0			0
$509$	3.19919e6	−	5.54116e6i	0.547325	−	0.947995i	−0.451131	−	0.892458i	$-0.648979\pi$
$509$	3.19919e6	−	5.54116e6i	0.547325	−	0.947995i	0.998457	−	0.0555374i	$-0.0176872\pi$
$510$		0			0
$511$	−4.03157e6	−	6.98289e6i	−0.683003	−	1.18300i
$512$		0			0
$513$	−4.36712e6	−	2.76595e6i	−0.732659	−	0.464035i
$514$		0			0
$515$	1.54269e6	+	2.67201e6i	0.256306	+	0.443936i
$516$		0			0
$517$	−1.98801e6	+	3.44333e6i	−0.327109	+	0.566569i
$518$		0			0
$519$	8.03370e6	−	1.39148e7i	1.30917	−	2.26756i
$520$		0			0
$521$	−1.13178e7			−1.82671			−0.913353	−	0.407168i	$-0.866516\pi$
$521$	−1.13178e7			−1.82671			−0.913353	+	0.407168i	$0.866516\pi$
$522$		0			0
$523$	−214962.	+	372326.i	−0.0343643	+	0.0595208i	−0.882696	−	0.469944i	$-0.844274\pi$
$523$	−214962.	+	372326.i	−0.0343643	+	0.0595208i	0.848332	+	0.529465i	$0.177607\pi$
$524$		0			0
$525$	−9.42643e6			−1.49262
$526$		0			0
$527$	2.32185e6	+	4.02156e6i	0.364173	+	0.630766i
$528$		0			0
$529$	−3.03505e6	−	5.25687e6i	−0.471549	−	0.816747i
$530$		0			0
$531$	−1.04805e7			−1.61304
$532$		0			0
$533$	2.16744e6			0.330468
$534$		0			0
$535$	510794.	+	884721.i	0.0771545	+	0.133635i
$536$		0			0
$537$	4.86910e6	+	8.43354e6i	0.728641	+	1.26204i
$538$		0			0
$539$	−5.48442e6			−0.813128
$540$		0			0
$541$	4.05574e6	−	7.02475e6i	0.595768	−	1.03190i	−0.397670	−	0.917529i	$-0.630181\pi$
$541$	4.05574e6	−	7.02475e6i	0.595768	−	1.03190i	0.993438	−	0.114372i	$-0.0364855\pi$
$542$		0			0
$543$	−1.77315e7			−2.58075
$544$		0			0
$545$	3.94566e6	−	6.83409e6i	0.569021	−	0.985574i
$546$		0			0
$547$	−4.15311e6	+	7.19340e6i	−0.593479	+	1.02794i	0.400281	+	0.916393i	$0.368913\pi$
$547$	−4.15311e6	+	7.19340e6i	−0.593479	+	1.02794i	−0.993760	+	0.111543i	$0.964421\pi$
$548$		0			0
$549$	−5.76436e6	−	9.98416e6i	−0.816244	−	1.41378i
$550$		0			0
$551$	−8.93439e6	+	4.68103e6i	−1.25368	+	0.656845i
$552$		0			0
$553$	6.71687e6	+	1.16340e7i	0.934015	+	1.61776i
$554$		0			0
$555$	−2.44206e6	+	4.22978e6i	−0.336531	+	0.582888i
$556$		0			0
$557$	−1.99228e6	+	3.45072e6i	−0.272089	+	0.471272i	−0.969397	−	0.245500i	$-0.921048\pi$
$557$	−1.99228e6	+	3.45072e6i	−0.272089	+	0.471272i	0.697307	+	0.716772i	$0.254381\pi$
$558$		0			0
$559$	−4.61934e6			−0.625245
$560$		0			0
$561$	−4.32106e6	+	7.48430e6i	−0.579673	+	1.00402i
$562$		0			0
$563$	6.11824e6			0.813496			0.406748	−	0.913540i	$-0.366663\pi$
$563$	6.11824e6			0.813496			0.406748	+	0.913540i	$0.366663\pi$
$564$		0			0
$565$	3.15863e6	+	5.47091e6i	0.416273	+	0.721005i
$566$		0			0
$567$	−990279.	−	1.71521e6i	−0.129360	−	0.224058i
$568$		0			0
$569$	908888.			0.117687			0.0588436	−	0.998267i	$-0.481259\pi$
$569$	908888.			0.117687			0.0588436	+	0.998267i	$0.481259\pi$
$570$		0			0
$571$	1.83881e6			0.236019			0.118010	−	0.993012i	$-0.462349\pi$
$571$	1.83881e6			0.236019			0.118010	+	0.993012i	$0.462349\pi$
$572$		0			0
$573$	916135.	+	1.58679e6i	0.116566	+	0.201899i
$574$		0			0
$575$	3.20949e6	+	5.55900e6i	0.404824	+	0.701176i
$576$		0			0
$577$	5.44901e6			0.681362			0.340681	−	0.940179i	$-0.389342\pi$
$577$	5.44901e6			0.681362			0.340681	+	0.940179i	$0.389342\pi$
$578$		0			0
$579$	7.00386e6	−	1.21310e7i	0.868242	−	1.50384i
$580$		0			0
$581$	−1.49405e7			−1.83623
$582$		0			0
$583$	877170.	−	1.51930e6i	0.106884	−	0.185128i
$584$		0			0
$585$	−1.99225e6	+	3.45067e6i	−0.240687	+	0.416883i
$586$		0			0
$587$	−1.89239e6	−	3.27772e6i	−0.226681	−	0.392623i	0.730141	−	0.683296i	$-0.239454\pi$
$587$	−1.89239e6	−	3.27772e6i	−0.226681	−	0.392623i	−0.956823	+	0.290673i	$0.906121\pi$
$588$		0			0
$589$	−3.63080e6	−	2.29959e6i	−0.431235	−	0.273126i
$590$		0			0
$591$	2.83376e6	+	4.90822e6i	0.333729	+	0.578036i
$592$		0			0
$593$	5.72339e6	−	9.91320e6i	0.668369	−	1.15765i	−0.309990	−	0.950740i	$-0.600326\pi$
$593$	5.72339e6	−	9.91320e6i	0.668369	−	1.15765i	0.978360	−	0.206910i	$-0.0663408\pi$
$594$		0			0
$595$	6.42691e6	−	1.11317e7i	0.744235	−	1.28905i
$596$		0			0
$597$	−1.51775e7			−1.74287
$598$		0			0
$599$	−3.14807e6	+	5.45262e6i	−0.358491	+	0.620924i	−0.987709	−	0.156305i	$-0.950042\pi$
$599$	−3.14807e6	+	5.45262e6i	−0.358491	+	0.620924i	0.629218	+	0.777229i	$0.283375\pi$
$600$		0			0
$601$	−3.03647e6			−0.342913			−0.171456	−	0.985192i	$-0.554847\pi$
$601$	−3.03647e6			−0.342913			−0.171456	+	0.985192i	$0.554847\pi$
$602$		0			0
$603$	−9.05078e6	−	1.56764e7i	−1.01366	−	1.75571i
$604$		0			0
$605$	2.15805e6	+	3.73785e6i	0.239703	+	0.415177i
$606$		0			0
$607$	−1.08106e7			−1.19091			−0.595454	−	0.803389i	$-0.703028\pi$
$607$	−1.08106e7			−1.19091			−0.595454	+	0.803389i	$0.703028\pi$
$608$		0			0
$609$	3.32890e7			3.63712
$610$		0			0
$611$	2.85372e6	+	4.94279e6i	0.309249	+	0.535635i
$612$		0			0
$613$	4.94957e6	+	8.57291e6i	0.532006	+	0.921461i	0.999302	+	0.0373600i	$0.0118948\pi$
$613$	4.94957e6	+	8.57291e6i	0.532006	+	0.921461i	−0.467296	+	0.884101i	$0.654772\pi$
$614$		0			0
$615$	−6.64570e6			−0.708522
$616$		0			0
$617$	973579.	−	1.68629e6i	0.102958	−	0.178328i	−0.809944	−	0.586507i	$-0.800503\pi$
$617$	973579.	−	1.68629e6i	0.102958	−	0.178328i	0.912902	+	0.408179i	$0.133836\pi$
$618$		0			0
$619$	3.48976e6			0.366074			0.183037	−	0.983106i	$-0.441407\pi$
$619$	3.48976e6			0.366074			0.183037	+	0.983106i	$0.441407\pi$
$620$		0			0
$621$	5.80884e6	−	1.00612e7i	0.604450	−	1.04694i
$622$		0			0
$623$	4.85797e6	−	8.41424e6i	0.501458	−	0.868550i
$624$		0			0
$625$	399448.	+	691864.i	0.0409034	+	0.0708468i
$626$		0			0
$627$	327740.	−	7.99162e6i	0.0332936	−	0.811832i
$628$		0			0
$629$	4.61424e6	+	7.99210e6i	0.465022	+	0.805442i
$630$		0			0
$631$	−4.65587e6	+	8.06421e6i	−0.465509	+	0.806285i	−0.999224	−	0.0393793i	$-0.987462\pi$
$631$	−4.65587e6	+	8.06421e6i	−0.465509	+	0.806285i	0.533716	+	0.845664i	$0.320795\pi$
$632$		0			0
$633$	8.46910e6	−	1.46689e7i	0.840094	−	1.45509i
$634$		0			0
$635$	5.20065e6			0.511827
$636$		0			0
$637$	−3.93635e6	+	6.81796e6i	−0.384366	+	0.665741i
$638$		0			0
$639$	2.57410e6			0.249387
$640$		0			0
$641$	2.98479e6	+	5.16981e6i	0.286925	+	0.496969i	0.973074	−	0.230492i	$-0.0740336\pi$
$641$	2.98479e6	+	5.16981e6i	0.286925	+	0.496969i	−0.686149	+	0.727461i	$0.740700\pi$
$642$		0			0
$643$	9.07253e6	+	1.57141e7i	0.865368	+	1.49886i	0.866681	+	0.498863i	$0.166249\pi$
$643$	9.07253e6	+	1.57141e7i	0.865368	+	1.49886i	−0.00131239	+	0.999999i	$0.500418\pi$
$644$		0			0
$645$	1.41636e7			1.34052
$646$		0			0
$647$	1.03554e7			0.972537			0.486269	−	0.873809i	$-0.338358\pi$
$647$	1.03554e7			0.972537			0.486269	+	0.873809i	$0.338358\pi$
$648$		0			0
$649$	−2.85589e6	−	4.94655e6i	−0.266152	−	0.460989i
$650$		0			0
$651$	7.09215e6	+	1.22840e7i	0.655882	+	1.13602i
$652$		0			0
$653$	4.38119e6			0.402077			0.201038	−	0.979583i	$-0.435568\pi$
$653$	4.38119e6			0.402077			0.201038	+	0.979583i	$0.435568\pi$
$654$		0			0
$655$	103769.	−	179733.i	0.00945069	−	0.0163691i
$656$		0			0
$657$	−1.44805e7			−1.30879
$658$		0			0
$659$	−7.34358e6	+	1.27195e7i	−0.658710	+	1.14092i	0.322239	+	0.946658i	$0.395564\pi$
$659$	−7.34358e6	+	1.27195e7i	−0.658710	+	1.14092i	−0.980950	+	0.194262i	$0.937769\pi$
$660$		0			0
$661$	−1.90277e6	+	3.29569e6i	−0.169388	+	0.293388i	−0.938205	−	0.346081i	$-0.887512\pi$
$661$	−1.90277e6	+	3.29569e6i	−0.169388	+	0.293388i	0.768817	+	0.639469i	$0.220846\pi$
$662$		0			0
$663$	6.20274e6	+	1.07435e7i	0.548024	+	0.949205i
$664$		0			0
$665$	−487462.	+	1.18863e7i	−0.0427452	+	1.04230i
$666$		0			0
$667$	−1.13342e7	−	1.96313e7i	−0.986449	−	1.70858i
$668$		0			0
$669$	1.15552e7	−	2.00142e7i	0.998188	−	1.72891i
$670$		0			0
$671$	3.14153e6	−	5.44128e6i	0.269361	−	0.466547i
$672$		0			0
$673$	2.15454e7			1.83365			0.916824	−	0.399291i	$-0.130744\pi$
$673$	2.15454e7			1.83365			0.916824	+	0.399291i	$0.130744\pi$
$674$		0			0
$675$	−2.98141e6	+	5.16395e6i	−0.251861	+	0.436237i
$676$		0			0
$677$	−182950.			−0.0153412			−0.00767061	−	0.999971i	$-0.502442\pi$
$677$	−182950.			−0.0153412			−0.00767061	+	0.999971i	$0.502442\pi$
$678$		0			0
$679$	9.44293e6	+	1.63556e7i	0.786017	+	1.36142i
$680$		0			0
$681$	−1.04030e7	−	1.80186e7i	−0.859591	−	1.48885i
$682$		0			0
$683$	−1.90678e7			−1.56404			−0.782021	−	0.623252i	$-0.785811\pi$
$683$	−1.90678e7			−1.56404			−0.782021	+	0.623252i	$0.785811\pi$
$684$		0			0
$685$	−1.08713e7			−0.885229
$686$		0			0
$687$	−546388.	−	946372.i	−0.0441682	−	0.0765015i
$688$		0			0
$689$	−1.25915e6	−	2.18091e6i	−0.101048	−	0.175021i
$690$		0			0
$691$	1.57718e7			1.25657			0.628285	−	0.777983i	$-0.283757\pi$
$691$	1.57718e7			1.25657			0.628285	+	0.777983i	$0.283757\pi$
$692$		0			0
$693$	−8.01009e6	+	1.38739e7i	−0.633584	+	1.09740i
$694$		0			0
$695$	−1.58252e7			−1.24276
$696$		0			0
$697$	−6.27848e6	+	1.08746e7i	−0.489522	+	0.847878i
$698$		0			0
$699$	−488533.	+	846164.i	−0.0378182	+	0.0655030i
$700$		0			0
$701$	−8.89878e6	−	1.54131e7i	−0.683967	−	1.18467i	−0.973760	−	0.227577i	$-0.926920\pi$
$701$	−8.89878e6	−	1.54131e7i	−0.683967	−	1.18467i	0.289793	−	0.957089i	$-0.406414\pi$
$702$		0			0
$703$	−7.21554e6	−	4.57001e6i	−0.550656	−	0.348762i
$704$		0			0
$705$	−8.74993e6	−	1.51553e7i	−0.663028	−	1.14840i
$706$		0			0
$707$	−1.52562e7	+	2.64245e7i	−1.14788	+	1.98819i
$708$		0			0
$709$	−1.02786e6	+	1.78031e6i	−0.0767924	+	0.133008i	−0.901864	−	0.432019i	$-0.857801\pi$
$709$	−1.02786e6	+	1.78031e6i	−0.0767924	+	0.133008i	0.825072	+	0.565028i	$0.191135\pi$
$710$		0			0
$711$	2.41254e7			1.78978
$712$		0			0
$713$	4.82944e6	−	8.36483e6i	0.355773	−	0.616217i
$714$		0			0
$715$	−2.17151e6			−0.158854
$716$		0			0
$717$	1.42012e7	+	2.45971e7i	1.03163	+	1.78684i
$718$		0			0
$719$	1.09362e7	+	1.89420e7i	0.788940	+	1.36648i	0.926617	+	0.376007i	$0.122703\pi$
$719$	1.09362e7	+	1.89420e7i	0.788940	+	1.36648i	−0.137676	+	0.990477i	$0.543963\pi$
$720$		0			0
$721$	1.78072e7			1.27572
$722$		0			0
$723$	−7.08714e6			−0.504226
$724$		0			0
$725$	5.81729e6	+	1.00758e7i	0.411032	+	0.711929i
$726$		0			0
$727$	1.17837e6	+	2.04099e6i	0.0826884	+	0.143221i	0.904404	−	0.426677i	$-0.140316\pi$
$727$	1.17837e6	+	2.04099e6i	0.0826884	+	0.143221i	−0.821715	+	0.569898i	$0.806983\pi$
$728$		0			0
$729$	−2.34041e7			−1.63107
$730$		0			0
$731$	1.33809e7	−	2.31765e7i	0.926175	−	1.60418i
$732$		0			0
$733$	2.16077e7			1.48541			0.742707	−	0.669616i	$-0.233541\pi$
$733$	2.16077e7			1.48541			0.742707	+	0.669616i	$0.233541\pi$
$734$		0			0
$735$	1.20694e7	−	2.09049e7i	0.824079	−	1.42735i
$736$		0			0
$737$	4.93260e6	−	8.54351e6i	0.334508	−	0.579385i
$738$		0			0
$739$	−2.23082e6	−	3.86389e6i	−0.150263	−	0.260264i	0.781061	−	0.624455i	$-0.214679\pi$
$739$	−2.23082e6	−	3.86389e6i	−0.150263	−	0.260264i	−0.931324	+	0.364191i	$0.881345\pi$
$740$		0			0
$741$	−9.69955e6	−	6.14328e6i	−0.648942	−	0.411012i
$742$		0			0
$743$	−7.70913e6	−	1.33526e7i	−0.512311	−	0.887348i	−0.999898	−	0.0142742i	$-0.995456\pi$
$743$	−7.70913e6	−	1.33526e7i	−0.512311	−	0.887348i	0.487587	−	0.873074i	$-0.337877\pi$
$744$		0			0
$745$	164331.	−	284630.i	0.0108475	−	0.0187884i
$746$		0			0
$747$	−1.34157e7	+	2.32367e7i	−0.879655	+	1.52361i
$748$		0			0
$749$	5.89607e6			0.384024
$750$		0			0
$751$	−1.64269e6	+	2.84522e6i	−0.106281	+	0.184084i	−0.914261	−	0.405126i	$-0.867228\pi$
$751$	−1.64269e6	+	2.84522e6i	−0.106281	+	0.184084i	0.807980	+	0.589210i	$0.200561\pi$
$752$		0			0
$753$	−3.91409e7			−2.51561
$754$		0			0
$755$	−784079.	−	1.35806e6i	−0.0500602	−	0.0867068i
$756$		0			0
$757$	−5.40092e6	−	9.35466e6i	−0.342553	−	0.593319i	0.642353	−	0.766409i	$-0.277958\pi$
$757$	−5.40092e6	−	9.35466e6i	−0.342553	−	0.593319i	−0.984906	+	0.173090i	$0.944625\pi$
$758$		0			0
$759$	1.79756e7			1.13260
$760$		0			0
$761$	−9.38280e6			−0.587315			−0.293657	−	0.955911i	$-0.594872\pi$
$761$	−9.38280e6			−0.587315			−0.293657	+	0.955911i	$0.594872\pi$
$762$		0			0
$763$	−2.27723e7	−	3.94428e7i	−1.41611	−	2.45277i
$764$		0			0
$765$	−1.15420e7	−	1.99913e7i	−0.713061	−	1.23506i
$766$		0			0
$767$	−8.19907e6			−0.503241
$768$		0			0
$769$	2.58311e6	−	4.47407e6i	0.157517	−	0.272827i	−0.776456	−	0.630172i	$-0.782985\pi$
$769$	2.58311e6	−	4.47407e6i	0.157517	−	0.272827i	0.933973	+	0.357345i	$0.116318\pi$
$770$		0			0
$771$	8.59780e6			0.520896
$772$		0			0
$773$	2.46904e6	−	4.27650e6i	0.148621	−	0.257419i	−0.782097	−	0.623156i	$-0.785850\pi$
$773$	2.46904e6	−	4.27650e6i	0.148621	−	0.257419i	0.930718	+	0.365738i	$0.119183\pi$
$774$		0			0
$775$	−2.47873e6	+	4.29328e6i	−0.148243	+	0.256764i
$776$		0			0
$777$	1.40943e7	+	2.44121e7i	0.837513	+	1.45062i
$778$		0			0
$779$	476204.	−	1.16118e7i	0.0281157	−	0.685575i
$780$		0			0
$781$	701432.	+	1.21492e6i	0.0411489	+	0.0712719i
$782$		0			0
$783$	1.05287e7	−	1.82362e7i	0.613719	−	1.06299i
$784$		0			0
$785$	−1.82419e6	+	3.15958e6i	−0.105656	+	0.183002i
$786$		0			0
$787$	−7.65272e6			−0.440432			−0.220216	−	0.975451i	$-0.570676\pi$
$787$	−7.65272e6			−0.440432			−0.220216	+	0.975451i	$0.570676\pi$
$788$		0			0
$789$	−1.22825e7	+	2.12740e7i	−0.702418	+	1.21662i
$790$		0			0
$791$	3.64600e7			2.07193
$792$		0			0
$793$	−4.50955e6	−	7.81077e6i	−0.254654	−	0.441074i
$794$		0			0
$795$	3.86074e6	+	6.68699e6i	0.216647	+	0.375243i
$796$		0			0
$797$	−1.09086e7			−0.608306			−0.304153	−	0.952623i	$-0.598373\pi$
$797$	−1.09086e7			−0.608306			−0.304153	+	0.952623i	$0.598373\pi$
$798$		0			0
$799$	−3.30657e7			−1.83236
$800$		0			0
$801$	−8.72433e6	−	1.51110e7i	−0.480453	−	0.832169i
$802$		0			0
$803$	−3.94586e6	−	6.83443e6i	−0.215950	−	0.374036i
$804$		0			0
$805$	−2.67359e7			−1.45414
$806$		0			0
$807$	−1.04044e7	+	1.80210e7i	−0.562386	+	0.974081i
$808$		0			0
$809$	2.87426e7			1.54403			0.772013	−	0.635606i	$-0.219250\pi$
$809$	2.87426e7			1.54403			0.772013	+	0.635606i	$0.219250\pi$
$810$		0			0
$811$	−9.14408e6	+	1.58380e7i	−0.488189	+	0.845568i	−0.999908	−	0.0135852i	$-0.995676\pi$
$811$	−9.14408e6	+	1.58380e7i	−0.488189	+	0.845568i	0.511719	+	0.859153i	$0.329009\pi$
$812$		0			0
$813$	−2.59200e7	+	4.48948e7i	−1.37534	+	2.38215i
$814$		0			0
$815$	5.71314e6	+	9.89544e6i	0.301287	+	0.521845i
$816$		0			0
$817$	−1.01490e6	+	2.47475e7i	−0.0531949	+	1.29711i
$818$		0			0
$819$	1.14982e7	+	1.99155e7i	0.598991	+	1.03748i
$820$		0			0
$821$	7.63449e6	−	1.32233e7i	0.395296	−	0.684672i	−0.597843	−	0.801613i	$-0.703975\pi$
$821$	7.63449e6	−	1.32233e7i	0.395296	−	0.684672i	0.993139	+	0.116941i	$0.0373088\pi$
$822$		0			0
$823$	3.66967e6	−	6.35605e6i	0.188854	−	0.327105i	−0.756014	−	0.654555i	$-0.772856\pi$
$823$	3.66967e6	−	6.35605e6i	0.188854	−	0.327105i	0.944869	+	0.327450i	$0.106189\pi$
$824$		0			0
$825$	−9.22603e6			−0.471932
$826$		0			0
$827$	9.35237e6	−	1.61988e7i	0.475508	−	0.823605i	−0.524098	−	0.851658i	$-0.675597\pi$
$827$	9.35237e6	−	1.61988e7i	0.475508	−	0.823605i	0.999606	+	0.0280533i	$0.00893082\pi$
$828$		0			0
$829$	1.92855e7			0.974641			0.487320	−	0.873223i	$-0.337974\pi$
$829$	1.92855e7			0.974641			0.487320	+	0.873223i	$0.337974\pi$
$830$		0			0
$831$	−8.13919e6	−	1.40975e7i	−0.408864	−	0.708173i
$832$		0			0
$833$	−2.28050e7	−	3.94995e7i	−1.13872	−	1.97233i
$834$		0			0
$835$	−2.03338e7			−1.00926
$836$		0			0
$837$	8.97246e6			0.442688
$838$		0			0
$839$	6.24428e6	+	1.08154e7i	0.306251	+	0.530442i	0.977539	−	0.210754i	$-0.0675920\pi$
$839$	6.24428e6	+	1.08154e7i	0.306251	+	0.530442i	−0.671288	+	0.741196i	$0.734259\pi$
$840$		0			0
$841$	−1.02879e7	−	1.78192e7i	−0.501576	−	0.868756i
$842$		0			0
$843$	2.15140e7			1.04268
$844$		0			0
$845$	5.16047e6	−	8.93819e6i	0.248626	−	0.430634i
$846$		0			0
$847$	2.49103e7			1.19308
$848$		0			0
$849$	−2.08980e7	+	3.61964e7i	−0.995029	+	1.72344i
$850$		0			0
$851$	9.59760e6	−	1.66235e7i	0.454296	−	0.786864i
$852$		0			0
$853$	3.42614e6	+	5.93425e6i	0.161225	+	0.279250i	0.935308	−	0.353834i	$-0.115122\pi$
$853$	3.42614e6	+	5.93425e6i	0.161225	+	0.279250i	−0.774083	+	0.633084i	$0.781789\pi$
$854$		0			0
$855$	1.80488e7	+	1.14313e7i	0.844371	+	0.534788i
$856$		0			0
$857$	−5.61768e6	−	9.73011e6i	−0.261279	−	0.452549i	0.705303	−	0.708906i	$-0.250811\pi$
$857$	−5.61768e6	−	9.73011e6i	−0.261279	−	0.452549i	−0.966582	+	0.256357i	$0.917478\pi$
$858$		0			0
$859$	1.89450e6	−	3.28138e6i	0.0876017	−	0.151731i	−0.818895	−	0.573943i	$-0.805413\pi$
$859$	1.89450e6	−	3.28138e6i	0.0876017	−	0.151731i	0.906497	+	0.422212i	$0.138746\pi$
$860$		0			0
$861$	−1.91778e7	+	3.32169e7i	−0.881638	+	1.52704i
$862$		0			0
$863$	−3.08965e7			−1.41215			−0.706077	−	0.708135i	$-0.749537\pi$
$863$	−3.08965e7			−1.41215			−0.706077	+	0.708135i	$0.749537\pi$
$864$		0			0
$865$	−1.16949e7	+	2.02561e7i	−0.531441	+	0.920482i
$866$		0			0
$867$	−3.65696e7			−1.65224
$868$		0			0
$869$	6.57407e6	+	1.13866e7i	0.295314	+	0.511500i
$870$		0			0
$871$	−7.08057e6	−	1.22639e7i	−0.316244	−	0.547752i
$872$		0			0
$873$	3.39168e7			1.50619
$874$		0			0
$875$	3.73476e7			1.64908
$876$		0			0
$877$	1.06989e7	+	1.85310e7i	0.469719	+	0.813577i	0.999401	−	0.0346194i	$-0.0110219\pi$
$877$	1.06989e7	+	1.85310e7i	0.469719	+	0.813577i	−0.529682	+	0.848197i	$0.677689\pi$
$878$		0			0
$879$	6.16398e6	+	1.06763e7i	0.269085	+	0.466068i
$880$		0			0
$881$	3.47758e6			0.150951			0.0754756	−	0.997148i	$-0.475952\pi$
$881$	3.47758e6			0.150951			0.0754756	+	0.997148i	$0.475952\pi$
$882$		0			0
$883$	−7.52985e6	+	1.30421e7i	−0.325001	+	0.562918i	−0.981513	−	0.191398i	$-0.938698\pi$
$883$	−7.52985e6	+	1.30421e7i	−0.325001	+	0.562918i	0.656512	+	0.754316i	$0.272031\pi$
$884$		0			0
$885$	2.51396e7			1.07895
$886$		0			0
$887$	1.37599e7	−	2.38328e7i	0.587225	−	1.01710i	−0.407369	−	0.913264i	$-0.633554\pi$
$887$	1.37599e7	−	2.38328e7i	0.587225	−	1.01710i	0.994594	−	0.103840i	$-0.0331131\pi$
$888$		0			0
$889$	1.50077e7	−	2.59941e7i	0.636884	−	1.10312i
$890$		0			0
$891$	−969226.	−	1.67875e6i	−0.0409007	−	0.0708421i
$892$		0			0
$893$	2.71073e7	−	1.42025e7i	1.13752	−	0.595984i
$894$		0			0
$895$	−7.08807e6	−	1.22769e7i	−0.295781	−	0.512308i
$896$		0			0
$897$	1.29017e7	−	2.23463e7i	0.535383	−	0.927310i
$898$		0			0
$899$	8.75350e6	−	1.51615e7i	0.361229	−	0.625667i
$900$		0			0
$901$	1.45896e7			0.598731
$902$		0			0
$903$	4.08724e7	−	7.07931e7i	1.66806	−	2.88916i
$904$		0			0
$905$	2.58122e7			1.04762
$906$		0			0
$907$	−1.31824e7	−	2.28326e7i	−0.532079	−	0.921588i	−0.999299	−	0.0374469i	$-0.988077\pi$
$907$	−1.31824e7	−	2.28326e7i	−0.532079	−	0.921588i	0.467219	−	0.884141i	$-0.345256\pi$
$908$		0			0
$909$	2.73984e7	+	4.74553e7i	1.09980	+	1.90491i
$910$		0			0
$911$	−6.00749e6			−0.239827			−0.119913	−	0.992784i	$-0.538262\pi$
$911$	−6.00749e6			−0.239827			−0.119913	+	0.992784i	$0.538262\pi$
$912$		0			0
$913$	−1.46229e7			−0.580573
$914$		0			0
$915$	1.38270e7	+	2.39490e7i	0.545977	+	0.945660i
$916$		0			0
$917$	−598900.	−	1.03733e6i	−0.0235197	−	0.0407372i
$918$		0			0
$919$	1.21396e7			0.474150			0.237075	−	0.971491i	$-0.423811\pi$
$919$	1.21396e7			0.474150			0.237075	+	0.971491i	$0.423811\pi$
$920$		0			0
$921$	−3.05698e7	+	5.29484e7i	−1.18753	+	2.05685i
$922$		0			0
$923$	2.01376e6			0.0778044
$924$		0			0
$925$	−4.92600e6	+	8.53209e6i	−0.189296	+	0.327869i
$926$		0			0
$927$	1.59898e7	−	2.76951e7i	0.611143	−	1.05853i
$928$		0			0
$929$	−4.94616e6	−	8.56699e6i	−0.188031	−	0.325679i	0.756563	−	0.653921i	$-0.226877\pi$
$929$	−4.94616e6	−	8.56699e6i	−0.188031	−	0.325679i	−0.944594	+	0.328242i	$0.893544\pi$
$930$		0			0
$931$	3.56614e7	+	2.25864e7i	1.34842	+	0.854030i
$932$		0			0
$933$	2.26247e7	+	3.91871e7i	0.850901	+	1.47380i
$934$		0			0
$935$	6.29028e6	−	1.08951e7i	0.235310	−	0.407569i
$936$		0			0
$937$	−1.02564e7	+	1.77646e7i	−0.381632	+	0.661007i	−0.991296	−	0.131654i	$-0.957971\pi$
$937$	−1.02564e7	+	1.77646e7i	−0.381632	+	0.661007i	0.609663	+	0.792660i	$0.291305\pi$
$938$		0			0
$939$	−4.21341e7			−1.55944
$940$		0			0
$941$	2.28320e7	−	3.95462e7i	0.840562	−	1.45590i	−0.0488582	−	0.998806i	$-0.515558\pi$
$941$	2.28320e7	−	3.95462e7i	0.840562	−	1.45590i	0.889420	−	0.457090i	$-0.151108\pi$
$942$		0			0
$943$	2.61184e7			0.956462
$944$		0			0
$945$	−1.24179e7	−	2.15085e7i	−0.452346	−	0.783486i
$946$		0			0
$947$	−5.04770e6	−	8.74287e6i	−0.182902	−	0.316796i	0.759966	−	0.649963i	$-0.225216\pi$
$947$	−5.04770e6	−	8.74287e6i	−0.182902	−	0.316796i	−0.942868	+	0.333168i	$0.891882\pi$
$948$		0			0
$949$	−1.13283e7			−0.408319
$950$		0			0
$951$	−3.90383e7			−1.39971
$952$		0			0
$953$	−1.33676e7	−	2.31533e7i	−0.476782	−	0.825811i	0.522864	−	0.852416i	$-0.324864\pi$
$953$	−1.33676e7	−	2.31533e7i	−0.476782	−	0.825811i	−0.999646	+	0.0266050i	$0.991530\pi$
$954$		0			0
$955$	−1.33364e6	−	2.30993e6i	−0.0473184	−	0.0819579i
$956$		0			0
$957$	3.25813e7			1.14997
$958$		0			0
$959$	−3.13718e7	+	5.43375e7i	−1.10152	+	1.90789i
$960$		0			0
$961$	−2.11695e7			−0.739438
$962$		0			0
$963$	5.29433e6	−	9.17004e6i	0.183969	−	0.318644i
$964$		0			0
$965$	−1.01957e7	+	1.76594e7i	−0.352451	+	0.610462i
$966$		0			0
$967$	4.40549e6	+	7.63054e6i	0.151505	+	0.262415i	0.931781	−	0.363021i	$-0.118255\pi$
$967$	4.40549e6	+	7.63054e6i	0.151505	+	0.262415i	−0.780276	+	0.625436i	$0.784921\pi$
$968$		0			0
$969$	5.89194e7	−	3.08699e7i	2.01581	−	1.05615i
$970$		0			0
$971$	1.90262e7	+	3.29543e7i	0.647594	+	1.12167i	0.983696	+	0.179840i	$0.0575581\pi$
$971$	1.90262e7	+	3.29543e7i	0.647594	+	1.12167i	−0.336102	+	0.941826i	$0.609109\pi$
$972$		0			0
$973$	−4.56673e7	+	7.90981e7i	−1.54641	+	2.67845i
$974$		0			0
$975$	−6.62182e6	+	1.14693e7i	−0.223083	+	0.386391i
$976$		0			0
$977$	−4.16722e7			−1.39672			−0.698361	−	0.715746i	$-0.746087\pi$
$977$	−4.16722e7			−1.39672			−0.698361	+	0.715746i	$0.746087\pi$
$978$		0			0
$979$	4.75469e6	−	8.23536e6i	0.158550	−	0.274616i
$980$		0			0
$981$	−8.17928e7			−2.71358
$982$		0			0
$983$	2.16371e7	+	3.74766e7i	0.714194	+	1.23702i	0.963270	+	0.268536i	$0.0865397\pi$
$983$	2.16371e7	+	3.74766e7i	0.714194	+	1.23702i	−0.249076	+	0.968484i	$0.580127\pi$
$984$		0			0
$985$	−4.12517e6	−	7.14501e6i	−0.135473	−	0.234646i
$986$		0			0
$987$	−1.01000e8			−3.30012
$988$		0			0
$989$	−5.56646e7			−1.80962
$990$		0			0
$991$	1.59147e7	+	2.75651e7i	0.514772	+	0.891611i	0.999853	+	0.0171418i	$0.00545668\pi$
$991$	1.59147e7	+	2.75651e7i	0.514772	+	0.891611i	−0.485081	+	0.874469i	$0.661210\pi$
$992$		0			0
$993$	−1.83070e7	−	3.17087e7i	−0.589175	−	1.02048i
$994$		0			0
$995$	2.20942e7			0.707492
$996$		0			0
$997$	2.67257e7	−	4.62903e7i	0.851514	−	1.47487i	−0.0283272	−	0.999599i	$-0.509018\pi$
$997$	2.67257e7	−	4.62903e7i	0.851514	−	1.47487i	0.879841	−	0.475267i	$-0.157649\pi$
$998$		0			0
$999$	1.78311e7			0.565281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.e.a.45.9	✓	18
3.2	odd	2		684.6.k.f.577.6		18
4.3	odd	2		304.6.i.d.273.1		18
19.11	even	3	inner	76.6.e.a.49.9	yes	18
57.11	odd	6		684.6.k.f.505.6		18
76.11	odd	6		304.6.i.d.49.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.9	✓	18	1.1	even	1	trivial
76.6.e.a.49.9	yes	18	19.11	even	3	inner
304.6.i.d.49.1		18	76.11	odd	6
304.6.i.d.273.1		18	4.3	odd	2
684.6.k.f.505.6		18	57.11	odd	6
684.6.k.f.577.6		18	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(12.4311 + 21.5314i) q^{3} +(-18.0963 - 31.3437i) q^{5} -208.885 q^{7} +(-187.566 + 324.875i) q^{9} +O(q^{10})\)	\(q+(12.4311 + 21.5314i) q^{3} +(-18.0963 - 31.3437i) q^{5} -208.885 q^{7} +(-187.566 + 324.875i) q^{9} -204.444 q^{11} +(-146.736 + 254.155i) q^{13} +(449.916 - 779.277i) q^{15} +(-850.109 - 1472.43i) q^{17} +(1329.36 + 841.960i) q^{19} +(-2596.68 - 4497.58i) q^{21} +(-1768.22 + 3062.65i) q^{23} +(907.547 - 1571.92i) q^{25} -3285.13 q^{27} +(-3204.96 + 5551.15i) q^{29} -2731.24 q^{31} +(-2541.47 - 4401.96i) q^{33} +(3780.05 + 6547.24i) q^{35} -5427.82 q^{37} -7296.40 q^{39} +(-3692.75 - 6396.03i) q^{41} +(7870.13 + 13631.5i) q^{43} +13577.1 q^{45} +(9723.97 - 16842.4i) q^{47} +26826.0 q^{49} +(21135.7 - 36608.0i) q^{51} +(-4290.51 + 7431.39i) q^{53} +(3699.69 + 6408.05i) q^{55} +(-1603.08 + 39089.5i) q^{57} +(13969.1 + 24195.1i) q^{59} +(-15366.2 + 26615.0i) q^{61} +(39179.8 - 67861.5i) q^{63} +10621.5 q^{65} +(-24126.9 + 41788.9i) q^{67} -87924.1 q^{69} +(-3430.92 - 5942.53i) q^{71} +(19300.4 + 33429.3i) q^{73} +45127.4 q^{75} +42705.4 q^{77} +(-32155.8 - 55695.5i) q^{79} +(4740.78 + 8211.28i) q^{81} +71525.2 q^{83} +(-30767.7 + 53291.2i) q^{85} -159365. q^{87} +(-23256.6 + 40281.7i) q^{89} +(30651.0 - 53089.2i) q^{91} +(-33952.4 - 58807.3i) q^{93} +(2333.64 - 56903.5i) q^{95} +(-45206.3 - 78299.6i) q^{97} +(38346.9 - 66418.7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9}+O(q^{10}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9	\( 18 q - 11 q^{3} + 11 q^{5} + 336 q^{7} - 902 q^{9} - 320 q^{11} + 227 q^{13} - 101 q^{15} + 179 q^{17} - 868 q^{19} - 5700 q^{21} - 3425 q^{23} - 7054 q^{25} + 14722 q^{27} - 7349 q^{29} - 9960 q^{31} - 2998 q^{33} + 15888 q^{35} + 26444 q^{37} - 30246 q^{39} - 7311 q^{41} - 8283 q^{43} - 62164 q^{45} + 37603 q^{47} + 124738 q^{49} + 47227 q^{51} - 20337 q^{53} + 716 q^{55} - 57555 q^{57} - 74455 q^{59} - 7569 q^{61} - 52544 q^{63} + 188998 q^{65} - 26177 q^{67} + 116282 q^{69} - 53463 q^{71} - 14103 q^{73} + 120912 q^{75} - 31960 q^{77} + 31825 q^{79} - 21137 q^{81} + 82600 q^{83} - 50787 q^{85} - 339766 q^{87} - 155197 q^{89} - 2800 q^{91} - 46460 q^{93} + 49315 q^{95} + 111241 q^{97} - 193544 q^{99}+O(q^{100}) \) 18 * q - 11 * q^3 + 11 * q^5 + 336 * q^7 - 902 * q^9 - 320 * q^11 + 227 * q^13 - 101 * q^15 + 179 * q^17 - 868 * q^19 - 5700 * q^21 - 3425 * q^23 - 7054 * q^25 + 14722 * q^27 - 7349 * q^29 - 9960 * q^31 - 2998 * q^33 + 15888 * q^35 + 26444 * q^37 - 30246 * q^39 - 7311 * q^41 - 8283 * q^43 - 62164 * q^45 + 37603 * q^47 + 124738 * q^49 + 47227 * q^51 - 20337 * q^53 + 716 * q^55 - 57555 * q^57 - 74455 * q^59 - 7569 * q^61 - 52544 * q^63 + 188998 * q^65 - 26177 * q^67 + 116282 * q^69 - 53463 * q^71 - 14103 * q^73 + 120912 * q^75 - 31960 * q^77 + 31825 * q^79 - 21137 * q^81 + 82600 * q^83 - 50787 * q^85 - 339766 * q^87 - 155197 * q^89 - 2800 * q^91 - 46460 * q^93 + 49315 * q^95 + 111241 * q^97 - 193544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more