LMFDB - Embedded newform 68.6.e.a.13.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,6,Mod(13,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("68.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	68.e (of order $4$, 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:	$10.9060997473$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1473 x^{14} + 868792 x^{12} + 259909217 x^{10} + 41026119309 x^{8} + 3204542941640 x^{6} + \cdots + 12\!\cdots\!36 $ x^16 + 1473x^14 + 868792x^12 + 259909217x^10 + 41026119309x^8 + 3204542941640x^6 + 103639704706420x^4 + 1172643744770256*x^2 + 1279875238318336
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.6
Root			$5.68063i$ of defining polynomial
Character	$\chi$	$=$	68.13
Dual form			68.6.e.a.21.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.68063 + 6.68063i) q^{3} +(-20.9824 - 20.9824i) q^{5} +(78.1871 - 78.1871i) q^{7} -153.738i q^{9} +O(q^{10})$	\(q+(6.68063 + 6.68063i) q^{3} +(-20.9824 - 20.9824i) q^{5} +(78.1871 - 78.1871i) q^{7} -153.738i q^{9} +(-31.9317 + 31.9317i) q^{11} +1090.59 q^{13} -280.351i q^{15} +(-114.574 - 1186.06i) q^{17} +2445.74i q^{19} +1044.68 q^{21} +(3027.16 - 3027.16i) q^{23} -2244.48i q^{25} +(2650.46 - 2650.46i) q^{27} +(800.821 + 800.821i) q^{29} +(65.8039 + 65.8039i) q^{31} -426.648 q^{33} -3281.11 q^{35} +(-4177.22 - 4177.22i) q^{37} +(7285.84 + 7285.84i) q^{39} +(-8237.48 + 8237.48i) q^{41} +5135.23i q^{43} +(-3225.80 + 3225.80i) q^{45} -23070.1 q^{47} +4580.55i q^{49} +(7158.18 - 8689.04i) q^{51} +13037.6i q^{53} +1340.01 q^{55} +(-16339.1 + 16339.1i) q^{57} -29456.0i q^{59} +(-33553.4 + 33553.4i) q^{61} +(-12020.4 - 12020.4i) q^{63} +(-22883.2 - 22883.2i) q^{65} +42275.1 q^{67} +40446.7 q^{69} +(37038.6 + 37038.6i) q^{71} +(-28791.5 - 28791.5i) q^{73} +(14994.5 - 14994.5i) q^{75} +4993.29i q^{77} +(21535.0 - 21535.0i) q^{79} -1944.84 q^{81} -20019.2i q^{83} +(-22482.3 + 27290.3i) q^{85} +10700.0i q^{87} +36289.2 q^{89} +(85270.2 - 85270.2i) q^{91} +879.223i q^{93} +(51317.5 - 51317.5i) q^{95} +(45948.3 + 45948.3i) q^{97} +(4909.12 + 4909.12i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7}+O(q^{10}) $ 16 * q + 22 * q^3 - 44 * q^5 - 118 * q^7	$ 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7} + 414 q^{11} + 252 q^{13} - 2784 q^{17} + 5068 q^{21} + 1102 q^{23} - 14612 q^{27} + 9672 q^{29} + 2162 q^{31} - 20108 q^{33} + 59228 q^{35} - 26148 q^{37} - 42488 q^{39} + 18032 q^{41} + 40544 q^{45} - 12144 q^{47} - 57046 q^{51} + 17284 q^{55} + 31700 q^{57} - 9724 q^{61} - 20022 q^{63} - 34212 q^{65} + 47080 q^{67} - 17796 q^{69} + 117038 q^{71} + 37912 q^{73} - 82670 q^{75} + 129822 q^{79} - 197196 q^{81} + 113848 q^{85} - 456228 q^{89} - 263080 q^{91} + 393180 q^{95} + 209900 q^{97} + 222358 q^{99}+O(q^{100}) $ 16 * q + 22 * q^3 - 44 * q^5 - 118 * q^7 + 414 * q^11 + 252 * q^13 - 2784 * q^17 + 5068 * q^21 + 1102 * q^23 - 14612 * q^27 + 9672 * q^29 + 2162 * q^31 - 20108 * q^33 + 59228 * q^35 - 26148 * q^37 - 42488 * q^39 + 18032 * q^41 + 40544 * q^45 - 12144 * q^47 - 57046 * q^51 + 17284 * q^55 + 31700 * q^57 - 9724 * q^61 - 20022 * q^63 - 34212 * q^65 + 47080 * q^67 - 17796 * q^69 + 117038 * q^71 + 37912 * q^73 - 82670 * q^75 + 129822 * q^79 - 197196 * q^81 + 113848 * q^85 - 456228 * q^89 - 263080 * q^91 + 393180 * q^95 + 209900 * q^97 + 222358 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	6.68063	+	6.68063i	0.428563	+	0.428563i	0.888139	−	0.459576i	$-0.151999\pi$
$3$	6.68063	+	6.68063i	0.428563	+	0.428563i	−0.459576	+	0.888139i	$0.651999\pi$
$4$		0			0
$5$	−20.9824	−	20.9824i	−0.375344	−	0.375344i	0.494075	−	0.869419i	$-0.335507\pi$
$5$	−20.9824	−	20.9824i	−0.375344	−	0.375344i	−0.869419	+	0.494075i	$0.835507\pi$
$6$		0			0
$7$	78.1871	−	78.1871i	0.603101	−	0.603101i	−0.338033	−	0.941134i	$-0.609762\pi$
$7$	78.1871	−	78.1871i	0.603101	−	0.603101i	0.941134	+	0.338033i	$0.109762\pi$
$8$		0			0
$9$		−	153.738i		−	0.632668i
$10$		0			0
$11$	−31.9317	+	31.9317i	−0.0795683	+	0.0795683i	−0.745771	−	0.666203i	$-0.767919\pi$
$11$	−31.9317	+	31.9317i	−0.0795683	+	0.0795683i	0.666203	+	0.745771i	$0.267919\pi$
$12$		0			0
$13$	1090.59			1.78980			0.894898	−	0.446270i	$-0.147248\pi$
$13$	1090.59			1.78980			0.894898	+	0.446270i	$0.147248\pi$
$14$		0			0
$15$		−	280.351i		−	0.321717i
$16$		0			0
$17$	−114.574	−	1186.06i	−0.0961536	−	0.995367i
$17$	−114.574	−	1186.06i	−0.0961536	−	0.995367i
$18$		0			0
$19$			2445.74i			1.55427i	0.629335	+	0.777134i	$0.283328\pi$
$19$			2445.74i			1.55427i	−0.629335	+	0.777134i	$0.716672\pi$
$20$		0			0
$21$	1044.68			0.516934
$22$		0			0
$23$	3027.16	−	3027.16i	1.19321	−	1.19321i	0.217046	−	0.976161i	$-0.430358\pi$
$23$	3027.16	−	3027.16i	1.19321	−	1.19321i	0.976161	−	0.217046i	$-0.0696423\pi$
$24$		0			0
$25$		−	2244.48i		−	0.718233i
$26$		0			0
$27$	2650.46	−	2650.46i	0.699701	−	0.699701i
$28$		0			0
$29$	800.821	+	800.821i	0.176824	+	0.176824i	0.789970	−	0.613146i	$-0.210096\pi$
$29$	800.821	+	800.821i	0.176824	+	0.176824i	−0.613146	+	0.789970i	$0.710096\pi$
$30$		0			0
$31$	65.8039	+	65.8039i	0.0122984	+	0.0122984i	0.713229	−	0.700931i	$-0.247232\pi$
$31$	65.8039	+	65.8039i	0.0122984	+	0.0122984i	−0.700931	+	0.713229i	$0.747232\pi$
$32$		0			0
$33$	−426.648			−0.0682001
$34$		0			0
$35$	−3281.11			−0.452741
$36$		0			0
$37$	−4177.22	−	4177.22i	−0.501630	−	0.501630i	0.410314	−	0.911944i	$-0.365419\pi$
$37$	−4177.22	−	4177.22i	−0.501630	−	0.501630i	−0.911944	+	0.410314i	$0.865419\pi$
$38$		0			0
$39$	7285.84	+	7285.84i	0.767041	+	0.767041i
$40$		0			0
$41$	−8237.48	+	8237.48i	−0.765305	+	0.765305i	−0.977276	−	0.211971i	$-0.932012\pi$
$41$	−8237.48	+	8237.48i	−0.765305	+	0.765305i	0.211971	+	0.977276i	$0.432012\pi$
$42$		0			0
$43$			5135.23i			0.423535i	0.977320	+	0.211767i	$0.0679219\pi$
$43$			5135.23i			0.423535i	−0.977320	+	0.211767i	$0.932078\pi$
$44$		0			0
$45$	−3225.80	+	3225.80i	−0.237468	+	0.237468i
$46$		0			0
$47$	−23070.1			−1.52337			−0.761683	−	0.647949i	$-0.775627\pi$
$47$	−23070.1			−1.52337			−0.761683	+	0.647949i	$0.775627\pi$
$48$		0			0
$49$			4580.55i			0.272538i
$50$		0			0
$51$	7158.18	−	8689.04i	0.385369	−	0.467785i
$52$		0			0
$53$			13037.6i			0.637539i	0.947832	+	0.318770i	$0.103270\pi$
$53$			13037.6i			0.637539i	−0.947832	+	0.318770i	$0.896730\pi$
$54$		0			0
$55$	1340.01			0.0597310
$56$		0			0
$57$	−16339.1	+	16339.1i	−0.666102	+	0.666102i
$58$		0			0
$59$		−	29456.0i		−	1.10165i	−0.834620	−	0.550826i	$-0.814313\pi$
$59$		−	29456.0i		−	1.10165i	0.834620	−	0.550826i	$-0.185687\pi$
$60$		0			0
$61$	−33553.4	+	33553.4i	−1.15455	+	1.15455i	−0.168919	+	0.985630i	$0.554028\pi$
$61$	−33553.4	+	33553.4i	−1.15455	+	1.15455i	−0.985630	+	0.168919i	$0.945972\pi$
$62$		0			0
$63$	−12020.4	−	12020.4i	−0.381563	−	0.381563i
$64$		0			0
$65$	−22883.2	−	22883.2i	−0.671790	−	0.671790i
$66$		0			0
$67$	42275.1			1.15053			0.575264	−	0.817967i	$-0.304899\pi$
$67$	42275.1			1.15053			0.575264	+	0.817967i	$0.304899\pi$
$68$		0			0
$69$	40446.7			1.02273
$70$		0			0
$71$	37038.6	+	37038.6i	0.871985	+	0.871985i	0.992689	−	0.120704i	$-0.0385150\pi$
$71$	37038.6	+	37038.6i	0.871985	+	0.871985i	−0.120704	+	0.992689i	$0.538515\pi$
$72$		0			0
$73$	−28791.5	−	28791.5i	−0.632350	−	0.632350i	0.316307	−	0.948657i	$-0.397557\pi$
$73$	−28791.5	−	28791.5i	−0.632350	−	0.632350i	−0.948657	+	0.316307i	$0.897557\pi$
$74$		0			0
$75$	14994.5	−	14994.5i	0.307808	−	0.307808i
$76$		0			0
$77$			4993.29i			0.0959755i
$78$		0			0
$79$	21535.0	−	21535.0i	0.388220	−	0.388220i	−0.485832	−	0.874052i	$-0.661483\pi$
$79$	21535.0	−	21535.0i	0.388220	−	0.388220i	0.874052	+	0.485832i	$0.161483\pi$
$80$		0			0
$81$	−1944.84			−0.0329360
$82$		0			0
$83$		−	20019.2i		−	0.318972i	−0.987200	−	0.159486i	$-0.949016\pi$
$83$		−	20019.2i		−	0.318972i	0.987200	−	0.159486i	$-0.0509836\pi$
$84$		0			0
$85$	−22482.3	+	27290.3i	−0.337515	+	0.409696i
$86$		0			0
$87$			10700.0i			0.151560i
$88$		0			0
$89$	36289.2			0.485626			0.242813	−	0.970073i	$-0.421930\pi$
$89$	36289.2			0.485626			0.242813	+	0.970073i	$0.421930\pi$
$90$		0			0
$91$	85270.2	−	85270.2i	1.07943	−	1.07943i
$92$		0			0
$93$			879.223i			0.0105412i
$94$		0			0
$95$	51317.5	−	51317.5i	0.583386	−	0.583386i
$96$		0			0
$97$	45948.3	+	45948.3i	0.495839	+	0.495839i	0.910140	−	0.414301i	$-0.135974\pi$
$97$	45948.3	+	45948.3i	0.495839	+	0.495839i	−0.414301	+	0.910140i	$0.635974\pi$
$98$		0			0
$99$	4909.12	+	4909.12i	0.0503403	+	0.0503403i
$100$		0			0
$101$	−86175.8			−0.840586			−0.420293	−	0.907389i	$-0.638073\pi$
$101$	−86175.8			−0.840586			−0.420293	+	0.907389i	$0.638073\pi$
$102$		0			0
$103$	97808.3			0.908411			0.454206	−	0.890897i	$-0.349923\pi$
$103$	97808.3			0.908411			0.454206	+	0.890897i	$0.349923\pi$
$104$		0			0
$105$	−21919.9	−	21919.9i	−0.194028	−	0.194028i
$106$		0			0
$107$	30065.6	+	30065.6i	0.253870	+	0.253870i	0.822555	−	0.568686i	$-0.192548\pi$
$107$	30065.6	+	30065.6i	0.253870	+	0.253870i	−0.568686	+	0.822555i	$0.692548\pi$
$108$		0			0
$109$	−107107.	+	107107.i	−0.863481	+	0.863481i	−0.991741	−	0.128259i	$-0.959061\pi$
$109$	−107107.	+	107107.i	−0.863481	+	0.863481i	0.128259	+	0.991741i	$0.459061\pi$
$110$		0			0
$111$		−	55813.0i		−	0.429960i
$112$		0			0
$113$	148586.	−	148586.i	1.09467	−	1.09467i	0.0996427	−	0.995023i	$-0.468230\pi$
$113$	148586.	−	148586.i	1.09467	−	1.09467i	0.995023	−	0.0996427i	$-0.0317700\pi$
$114$		0			0
$115$	−127034.			−0.895728
$116$		0			0
$117$		−	167666.i		−	1.13235i
$118$		0			0
$119$	−101693.	−	83776.1i	−0.658297	−	0.542316i
$120$		0			0
$121$			159012.i			0.987338i
$122$		0			0
$123$	−110063.			−0.655963
$124$		0			0
$125$	−112665.	+	112665.i	−0.644929	+	0.644929i
$126$		0			0
$127$			278106.i			1.53003i	0.644010	+	0.765017i	$0.277269\pi$
$127$			278106.i			1.53003i	−0.644010	+	0.765017i	$0.722731\pi$
$128$		0			0
$129$	−34306.6	+	34306.6i	−0.181511	+	0.181511i
$130$		0			0
$131$	34179.1	+	34179.1i	0.174013	+	0.174013i	0.788740	−	0.614727i	$-0.210734\pi$
$131$	34179.1	+	34179.1i	0.174013	+	0.174013i	−0.614727	+	0.788740i	$0.710734\pi$
$132$		0			0
$133$	191225.	+	191225.i	0.937381	+	0.937381i
$134$		0			0
$135$	−111226.			−0.525258
$136$		0			0
$137$	−343492.			−1.56356			−0.781781	−	0.623554i	$-0.785688\pi$
$137$	−343492.			−1.56356			−0.781781	+	0.623554i	$0.785688\pi$
$138$		0			0
$139$	142267.	+	142267.i	0.624551	+	0.624551i	0.946692	−	0.322141i	$-0.104402\pi$
$139$	142267.	+	142267.i	0.624551	+	0.624551i	−0.322141	+	0.946692i	$0.604402\pi$
$140$		0			0
$141$	−154123.	−	154123.i	−0.652858	−	0.652858i
$142$		0			0
$143$	−34824.4	+	34824.4i	−0.142411	+	0.142411i
$144$		0			0
$145$		−	33606.3i		−	0.132740i
$146$		0			0
$147$	−30601.0	+	30601.0i	−0.116800	+	0.116800i
$148$		0			0
$149$	31591.4			0.116575			0.0582873	−	0.998300i	$-0.481436\pi$
$149$	31591.4			0.116575			0.0582873	+	0.998300i	$0.481436\pi$
$150$		0			0
$151$			102119.i			0.364471i	0.983255	+	0.182235i	$0.0583333\pi$
$151$			102119.i			0.364471i	−0.983255	+	0.182235i	$0.941667\pi$
$152$		0			0
$153$	−182342.	+	17614.5i	−0.629736	+	0.0608333i
$154$		0			0
$155$		−	2761.45i		−	0.00923224i
$156$		0			0
$157$	−80708.1			−0.261317			−0.130659	−	0.991427i	$-0.541709\pi$
$157$	−80708.1			−0.261317			−0.130659	+	0.991427i	$0.541709\pi$
$158$		0			0
$159$	−87099.2	+	87099.2i	−0.273226	+	0.273226i
$160$		0			0
$161$		−	473370.i		−	1.43925i
$162$		0			0
$163$	178196.	−	178196.i	0.525325	−	0.525325i	−0.393849	−	0.919175i	$-0.628857\pi$
$163$	178196.	−	178196.i	0.525325	−	0.525325i	0.919175	+	0.393849i	$0.128857\pi$
$164$		0			0
$165$	8952.09	+	8952.09i	0.0255985	+	0.0255985i
$166$		0			0
$167$	−336216.	−	336216.i	−0.932884	−	0.932884i	0.0650014	−	0.997885i	$-0.479295\pi$
$167$	−336216.	−	336216.i	−0.932884	−	0.932884i	−0.997885	+	0.0650014i	$0.979295\pi$
$168$		0			0
$169$	818097.			2.20337
$170$		0			0
$171$	376004.			0.983336
$172$		0			0
$173$	448934.	+	448934.i	1.14043	+	1.14043i	0.988372	+	0.152054i	$0.0485889\pi$
$173$	448934.	+	448934.i	1.14043	+	1.14043i	0.152054	+	0.988372i	$0.451411\pi$
$174$		0			0
$175$	−175489.	−	175489.i	−0.433167	−	0.433167i
$176$		0			0
$177$	196785.	−	196785.i	0.472127	−	0.472127i
$178$		0			0
$179$			82488.1i			0.192424i	0.995361	+	0.0962119i	$0.0306726\pi$
$179$			82488.1i			0.192424i	−0.995361	+	0.0962119i	$0.969327\pi$
$180$		0			0
$181$	18250.1	−	18250.1i	0.0414066	−	0.0414066i	−0.686100	−	0.727507i	$-0.740679\pi$
$181$	18250.1	−	18250.1i	0.0414066	−	0.0414066i	0.727507	+	0.686100i	$0.240679\pi$
$182$		0			0
$183$	−448316.			−0.989594
$184$		0			0
$185$			175296.i			0.376568i
$186$		0			0
$187$	41531.3	+	34214.2i	0.0868504	+	0.0715488i
$188$		0			0
$189$		−	414464.i		−	0.843981i
$190$		0			0
$191$	−225699.			−0.447658			−0.223829	−	0.974628i	$-0.571856\pi$
$191$	−225699.			−0.447658			−0.223829	+	0.974628i	$0.571856\pi$
$192$		0			0
$193$	−166008.	+	166008.i	−0.320802	+	0.320802i	−0.849075	−	0.528273i	$-0.822840\pi$
$193$	−166008.	+	166008.i	−0.320802	+	0.320802i	0.528273	+	0.849075i	$0.322840\pi$
$194$		0			0
$195$		−	305749.i		−	0.575809i
$196$		0			0
$197$	519537.	−	519537.i	0.953786	−	0.953786i	−0.0451925	−	0.998978i	$-0.514390\pi$
$197$	519537.	−	519537.i	0.953786	−	0.953786i	0.998978	+	0.0451925i	$0.0143901\pi$
$198$		0			0
$199$	708286.	+	708286.i	1.26787	+	1.26787i	0.947185	+	0.320689i	$0.103914\pi$
$199$	708286.	+	708286.i	1.26787	+	1.26787i	0.320689	+	0.947185i	$0.396086\pi$
$200$		0			0
$201$	282424.	+	282424.i	0.493074	+	0.493074i
$202$		0			0
$203$	125228.			0.213285
$204$		0			0
$205$	345684.			0.574506
$206$		0			0
$207$	−465391.	−	465391.i	−0.754904	−	0.754904i
$208$		0			0
$209$	−78096.6	−	78096.6i	−0.123671	−	0.123671i
$210$		0			0
$211$	−44792.1	+	44792.1i	−0.0692621	+	0.0692621i	−0.740889	−	0.671627i	$-0.765596\pi$
$211$	−44792.1	+	44792.1i	−0.0692621	+	0.0692621i	0.671627	+	0.740889i	$0.265596\pi$
$212$		0			0
$213$			494883.i			0.747401i
$214$		0			0
$215$	107750.	−	107750.i	0.158971	−	0.158971i
$216$		0			0
$217$	10290.0			0.0148343
$218$		0			0
$219$		−	384691.i		−	0.542003i
$220$		0			0
$221$	−124954.	−	1.29350e6i	−0.172095	−	1.78150i
$222$		0			0
$223$			574132.i			0.773125i	0.922263	+	0.386562i	$0.126338\pi$
$223$			574132.i			0.773125i	−0.922263	+	0.386562i	$0.873662\pi$
$224$		0			0
$225$	−345062.			−0.454403
$226$		0			0
$227$	−271053.	+	271053.i	−0.349132	+	0.349132i	−0.859786	−	0.510654i	$-0.829403\pi$
$227$	−271053.	+	271053.i	−0.349132	+	0.349132i	0.510654	+	0.859786i	$0.329403\pi$
$228$		0			0
$229$			537112.i			0.676825i	0.940998	+	0.338413i	$0.109890\pi$
$229$			537112.i			0.676825i	−0.940998	+	0.338413i	$0.890110\pi$
$230$		0			0
$231$	−33358.4	+	33358.4i	−0.0411315	+	0.0411315i
$232$		0			0
$233$	−148193.	−	148193.i	−0.178829	−	0.178829i	0.612016	−	0.790845i	$-0.290359\pi$
$233$	−148193.	−	148193.i	−0.178829	−	0.178829i	−0.790845	+	0.612016i	$0.790359\pi$
$234$		0			0
$235$	484065.	+	484065.i	0.571787	+	0.571787i
$236$		0			0
$237$	287735.			0.332753
$238$		0			0
$239$	−980883.			−1.11077			−0.555383	−	0.831595i	$-0.687428\pi$
$239$	−980883.			−1.11077			−0.555383	+	0.831595i	$0.687428\pi$
$240$		0			0
$241$	−830101.	−	830101.i	−0.920637	−	0.920637i	0.0764375	−	0.997074i	$-0.475645\pi$
$241$	−830101.	−	830101.i	−0.920637	−	0.920637i	−0.997074	+	0.0764375i	$0.975645\pi$
$242$		0			0
$243$	−657055.	−	657055.i	−0.713816	−	0.713816i
$244$		0			0
$245$	96110.8	−	96110.8i	0.102296	−	0.102296i
$246$		0			0
$247$			2.66730e6i			2.78183i
$248$		0			0
$249$	133741.	−	133741.i	0.136699	−	0.136699i
$250$		0			0
$251$	−610435.			−0.611583			−0.305792	−	0.952098i	$-0.598921\pi$
$251$	−610435.			−0.611583			−0.305792	+	0.952098i	$0.598921\pi$
$252$		0			0
$253$			193325.i			0.189883i
$254$		0			0
$255$	−332513.	+	32121.1i	−0.320227	+	0.0309343i
$256$		0			0
$257$			1.33043e6i			1.25649i	0.778016	+	0.628245i	$0.216226\pi$
$257$			1.33043e6i			1.25649i	−0.778016	+	0.628245i	$0.783774\pi$
$258$		0			0
$259$	−653210.			−0.605067
$260$		0			0
$261$	123117.	−	123117.i	0.111871	−	0.111871i
$262$		0			0
$263$		−	395748.i		−	0.352801i	−0.984318	−	0.176400i	$-0.943555\pi$
$263$		−	395748.i		−	0.352801i	0.984318	−	0.176400i	$-0.0564454\pi$
$264$		0			0
$265$	273559.	−	273559.i	0.239297	−	0.239297i
$266$		0			0
$267$	242435.	+	242435.i	0.208121	+	0.208121i
$268$		0			0
$269$	18608.3	+	18608.3i	0.0156793	+	0.0156793i	0.714903	−	0.699224i	$-0.246471\pi$
$269$	18608.3	+	18608.3i	0.0156793	+	0.0156793i	−0.699224	+	0.714903i	$0.746471\pi$
$270$		0			0
$271$	−1.17418e6			−0.971207			−0.485604	−	0.874179i	$-0.661400\pi$
$271$	−1.17418e6			−0.971207			−0.485604	+	0.874179i	$0.661400\pi$
$272$		0			0
$273$	1.13932e6			0.925206
$274$		0			0
$275$	71670.0	+	71670.0i	0.0571486	+	0.0571486i
$276$		0			0
$277$	−913582.	−	913582.i	−0.715398	−	0.715398i	0.252261	−	0.967659i	$-0.418826\pi$
$277$	−913582.	−	913582.i	−0.715398	−	0.715398i	−0.967659	+	0.252261i	$0.918826\pi$
$278$		0			0
$279$	10116.6	−	10116.6i	0.00778077	−	0.00778077i
$280$		0			0
$281$			1.09392e6i			0.826456i	0.910628	+	0.413228i	$0.135599\pi$
$281$			1.09392e6i			0.826456i	−0.910628	+	0.413228i	$0.864401\pi$
$282$		0			0
$283$	1.18772e6	−	1.18772e6i	0.881555	−	0.881555i	−0.112137	−	0.993693i	$-0.535770\pi$
$283$	1.18772e6	−	1.18772e6i	0.881555	−	0.881555i	0.993693	+	0.112137i	$0.0357696\pi$
$284$		0			0
$285$	685666.			0.500035
$286$		0			0
$287$			1.28813e6i			0.923113i
$288$		0			0
$289$	−1.39360e6	+	271784.i	−0.981509	+	0.191416i
$290$		0			0
$291$			613928.i			0.424996i
$292$		0			0
$293$	−2.91429e6			−1.98319			−0.991595	−	0.129380i	$-0.958701\pi$
$293$	−2.91429e6			−1.98319			−0.991595	+	0.129380i	$0.958701\pi$
$294$		0			0
$295$	−618058.	+	618058.i	−0.413499	+	0.413499i
$296$		0			0
$297$			169268.i			0.111348i
$298$		0			0
$299$	3.30140e6	−	3.30140e6i	2.13560	−	2.13560i
$300$		0			0
$301$	401509.	+	401509.i	0.255434	+	0.255434i
$302$		0			0
$303$	−575709.	−	575709.i	−0.360244	−	0.360244i
$304$		0			0
$305$	1.40806e6			0.866707
$306$		0			0
$307$	2.48916e6			1.50733			0.753664	−	0.657260i	$-0.228285\pi$
$307$	2.48916e6			1.50733			0.753664	+	0.657260i	$0.228285\pi$
$308$		0			0
$309$	653421.	+	653421.i	0.389311	+	0.389311i
$310$		0			0
$311$	−1.24278e6	−	1.24278e6i	−0.728605	−	0.728605i	0.241737	−	0.970342i	$-0.422283\pi$
$311$	−1.24278e6	−	1.24278e6i	−0.728605	−	0.728605i	−0.970342	+	0.241737i	$0.922283\pi$
$312$		0			0
$313$	1.71156e6	−	1.71156e6i	0.987488	−	0.987488i	−0.0124348	−	0.999923i	$-0.503958\pi$
$313$	1.71156e6	−	1.71156e6i	0.987488	−	0.987488i	0.999923	+	0.0124348i	$0.00395823\pi$
$314$		0			0
$315$			504431.i			0.286435i
$316$		0			0
$317$	714036.	−	714036.i	0.399091	−	0.399091i	−0.478821	−	0.877912i	$-0.658936\pi$
$317$	714036.	−	714036.i	0.399091	−	0.399091i	0.877912	+	0.478821i	$0.158936\pi$
$318$		0			0
$319$	−51143.1			−0.0281391
$320$		0			0
$321$			401715.i			0.217598i
$322$		0			0
$323$	2.90078e6	−	280219.i	1.54707	−	0.149449i
$324$		0			0
$325$		−	2.44781e6i		−	1.28549i
$326$		0			0
$327$	−1.43109e6			−0.740112
$328$		0			0
$329$	−1.80378e6	+	1.80378e6i	−0.918744	+	0.918744i
$330$		0			0
$331$		−	2.57359e6i		−	1.29113i	−0.763705	−	0.645565i	$-0.776622\pi$
$331$		−	2.57359e6i		−	1.29113i	0.763705	−	0.645565i	$-0.223378\pi$
$332$		0			0
$333$	−642199.	+	642199.i	−0.317365	+	0.317365i
$334$		0			0
$335$	−887033.	−	887033.i	−0.431845	−	0.431845i
$336$		0			0
$337$	1.69127e6	+	1.69127e6i	0.811217	+	0.811217i	0.984816	−	0.173599i	$-0.0555397\pi$
$337$	1.69127e6	+	1.69127e6i	0.811217	+	0.811217i	−0.173599	+	0.984816i	$0.555540\pi$
$338$		0			0
$339$	1.98530e6			0.938267
$340$		0			0
$341$	−4202.46			−0.00195712
$342$		0			0
$343$	1.67223e6	+	1.67223e6i	0.767469	+	0.767469i
$344$		0			0
$345$	−848669.	−	848669.i	−0.383876	−	0.383876i
$346$		0			0
$347$	301180.	−	301180.i	0.134277	−	0.134277i	−0.636774	−	0.771051i	$-0.719731\pi$
$347$	301180.	−	301180.i	0.134277	−	0.134277i	0.771051	+	0.636774i	$0.219731\pi$
$348$		0			0
$349$			3.22588e6i			1.41770i	0.705359	+	0.708850i	$0.250786\pi$
$349$			3.22588e6i			1.41770i	−0.705359	+	0.708850i	$0.749214\pi$
$350$		0			0
$351$	2.89057e6	−	2.89057e6i	1.25232	−	1.25232i
$352$		0			0
$353$	−3.02924e6			−1.29389			−0.646944	−	0.762537i	$-0.723953\pi$
$353$	−3.02924e6			−1.29389			−0.646944	+	0.762537i	$0.723953\pi$
$354$		0			0
$355$		−	1.55432e6i		−	0.654590i
$356$		0			0
$357$	−119694.	−	1.23905e6i	−0.0497050	−	0.514538i
$358$		0			0
$359$			2.45437e6i			1.00509i	0.864552	+	0.502543i	$0.167602\pi$
$359$			2.45437e6i			1.00509i	−0.864552	+	0.502543i	$0.832398\pi$
$360$		0			0
$361$	−3.50554e6			−1.41575
$362$		0			0
$363$	−1.06230e6	+	1.06230e6i	−0.423136	+	0.423136i
$364$		0			0
$365$			1.20823e6i			0.474698i
$366$		0			0
$367$	1.22992e6	−	1.22992e6i	0.476662	−	0.476662i	−0.427400	−	0.904062i	$-0.640571\pi$
$367$	1.22992e6	−	1.22992e6i	0.476662	−	0.476662i	0.904062	+	0.427400i	$0.140571\pi$
$368$		0			0
$369$	1.26642e6	+	1.26642e6i	0.484184	+	0.484184i
$370$		0			0
$371$	1.01937e6	+	1.01937e6i	0.384501	+	0.384501i
$372$		0			0
$373$	−436958.			−0.162618			−0.0813088	−	0.996689i	$-0.525910\pi$
$373$	−436958.			−0.162618			−0.0813088	+	0.996689i	$0.525910\pi$
$374$		0			0
$375$	−1.50534e6			−0.552786
$376$		0			0
$377$	873368.	+	873368.i	0.316478	+	0.316478i
$378$		0			0
$379$	−1.15692e6	−	1.15692e6i	−0.413718	−	0.413718i	0.469313	−	0.883032i	$-0.344501\pi$
$379$	−1.15692e6	−	1.15692e6i	−0.413718	−	0.413718i	−0.883032	+	0.469313i	$0.844501\pi$
$380$		0			0
$381$	−1.85792e6	+	1.85792e6i	−0.655716	+	0.655716i
$382$		0			0
$383$			3.00410e6i			1.04645i	0.852196	+	0.523223i	$0.175271\pi$
$383$			3.00410e6i			1.04645i	−0.852196	+	0.523223i	$0.824729\pi$
$384$		0			0
$385$	104771.	−	104771.i	0.0360239	−	0.0360239i
$386$		0			0
$387$	789482.			0.267957
$388$		0			0
$389$		−	594718.i		−	0.199268i	−0.995024	−	0.0996340i	$-0.968233\pi$
$389$		−	594718.i		−	0.199268i	0.995024	−	0.0996340i	$-0.0317672\pi$
$390$		0			0
$391$	−3.93722e6	−	3.24355e6i	−1.30241	−	1.07295i
$392$		0			0
$393$			456677.i			0.149151i
$394$		0			0
$395$	−903713.			−0.291432
$396$		0			0
$397$	−376833.	+	376833.i	−0.119998	+	0.119998i	−0.764555	−	0.644558i	$-0.777041\pi$
$397$	−376833.	+	376833.i	−0.119998	+	0.119998i	0.644558	+	0.764555i	$0.277041\pi$
$398$		0			0
$399$			2.55501e6i			0.803454i
$400$		0			0
$401$	3.82760e6	−	3.82760e6i	1.18868	−	1.18868i	0.211251	−	0.977432i	$-0.432246\pi$
$401$	3.82760e6	−	3.82760e6i	1.18868	−	1.18868i	0.977432	−	0.211251i	$-0.0677539\pi$
$402$		0			0
$403$	71765.1	+	71765.1i	0.0220116	+	0.0220116i
$404$		0			0
$405$	40807.4	+	40807.4i	0.0123624	+	0.0123624i
$406$		0			0
$407$	266771.			0.0798276
$408$		0			0
$409$	−3.77449e6			−1.11571			−0.557853	−	0.829940i	$-0.688375\pi$
$409$	−3.77449e6			−1.11571			−0.557853	+	0.829940i	$0.688375\pi$
$410$		0			0
$411$	−2.29474e6	−	2.29474e6i	−0.670084	−	0.670084i
$412$		0			0
$413$	−2.30308e6	−	2.30308e6i	−0.664407	−	0.664407i
$414$		0			0
$415$	−420051.	+	420051.i	−0.119724	+	0.119724i
$416$		0			0
$417$			1.90087e6i			0.535319i
$418$		0			0
$419$	−1.33638e6	+	1.33638e6i	−0.371872	+	0.371872i	−0.868159	−	0.496287i	$-0.834696\pi$
$419$	−1.33638e6	+	1.33638e6i	−0.371872	+	0.371872i	0.496287	+	0.868159i	$0.334696\pi$
$420$		0			0
$421$	−2.31787e6			−0.637358			−0.318679	−	0.947863i	$-0.603239\pi$
$421$	−2.31787e6			−0.637358			−0.318679	+	0.947863i	$0.603239\pi$
$422$		0			0
$423$			3.54675e6i			0.963785i
$424$		0			0
$425$	−2.66208e6	+	257160.i	−0.714905	+	0.0690607i
$426$		0			0
$427$			5.24689e6i			1.39262i
$428$		0			0
$429$	−465298.			−0.122064
$430$		0			0
$431$	553421.	−	553421.i	0.143503	−	0.143503i	−0.631705	−	0.775209i	$-0.717645\pi$
$431$	553421.	−	553421.i	0.143503	−	0.143503i	0.775209	+	0.631705i	$0.217645\pi$
$432$		0			0
$433$		−	6.44505e6i		−	1.65199i	−0.563681	−	0.825993i	$-0.690615\pi$
$433$		−	6.44505e6i		−	1.65199i	0.563681	−	0.825993i	$-0.309385\pi$
$434$		0			0
$435$	224511.	−	224511.i	0.0568872	−	0.0568872i
$436$		0			0
$437$	7.40365e6	+	7.40365e6i	1.85457	+	1.85457i
$438$		0			0
$439$	−1.11437e6	−	1.11437e6i	−0.275975	−	0.275975i	0.555525	−	0.831500i	$-0.312517\pi$
$439$	−1.11437e6	−	1.11437e6i	−0.275975	−	0.275975i	−0.831500	+	0.555525i	$0.812517\pi$
$440$		0			0
$441$	704205.			0.172426
$442$		0			0
$443$	7.79871e6			1.88805			0.944025	−	0.329874i	$-0.107006\pi$
$443$	7.79871e6			1.88805			0.944025	+	0.329874i	$0.107006\pi$
$444$		0			0
$445$	−761434.	−	761434.i	−0.182277	−	0.182277i
$446$		0			0
$447$	211051.	+	211051.i	0.0499595	+	0.0499595i
$448$		0			0
$449$	−990031.	+	990031.i	−0.231757	+	0.231757i	−0.813426	−	0.581669i	$-0.802400\pi$
$449$	−990031.	+	990031.i	−0.231757	+	0.231757i	0.581669	+	0.813426i	$0.302400\pi$
$450$		0			0
$451$		−	526073.i		−	0.121788i
$452$		0			0
$453$	−682217.	+	682217.i	−0.156199	+	0.156199i
$454$		0			0
$455$	−3.57835e6			−0.810315
$456$		0			0
$457$			610705.i			0.136786i	0.997658	+	0.0683929i	$0.0217872\pi$
$457$			610705.i			0.136786i	−0.997658	+	0.0683929i	$0.978213\pi$
$458$		0			0
$459$	−3.44727e6	−	2.83992e6i	−0.763738	−	0.629180i
$460$		0			0
$461$			1.61790e6i			0.354568i	0.984160	+	0.177284i	$0.0567312\pi$
$461$			1.61790e6i			0.354568i	−0.984160	+	0.177284i	$0.943269\pi$
$462$		0			0
$463$	−55759.6			−0.0120883			−0.00604417	−	0.999982i	$-0.501924\pi$
$463$	−55759.6			−0.0120883			−0.00604417	+	0.999982i	$0.501924\pi$
$464$		0			0
$465$	18448.2	−	18448.2i	0.00395660	−	0.00395660i
$466$		0			0
$467$			4.04306e6i			0.857862i	0.903337	+	0.428931i	$0.141110\pi$
$467$			4.04306e6i			0.857862i	−0.903337	+	0.428931i	$0.858890\pi$
$468$		0			0
$469$	3.30537e6	−	3.30537e6i	0.693885	−	0.693885i
$470$		0			0
$471$	−539181.	−	539181.i	−0.111991	−	0.111991i
$472$		0			0
$473$	−163977.	−	163977.i	−0.0337000	−	0.0337000i
$474$		0			0
$475$	5.48941e6			1.11633
$476$		0			0
$477$	2.00437e6			0.403350
$478$		0			0
$479$	−417965.	−	417965.i	−0.0832340	−	0.0832340i	0.664264	−	0.747498i	$-0.268745\pi$
$479$	−417965.	−	417965.i	−0.0832340	−	0.0832340i	−0.747498	+	0.664264i	$0.768745\pi$
$480$		0			0
$481$	−4.55564e6	−	4.55564e6i	−0.897815	−	0.897815i
$482$		0			0
$483$	3.16241e6	−	3.16241e6i	0.616809	−	0.616809i
$484$		0			0
$485$		−	1.92821e6i		−	0.372221i
$486$		0			0
$487$	3.62866e6	−	3.62866e6i	0.693303	−	0.693303i	−0.269654	−	0.962957i	$-0.586909\pi$
$487$	3.62866e6	−	3.62866e6i	0.693303	−	0.693303i	0.962957	+	0.269654i	$0.0869093\pi$
$488$		0			0
$489$	2.38092e6			0.450270
$490$		0			0
$491$		−	9.97056e6i		−	1.86645i	−0.359296	−	0.933224i	$-0.616983\pi$
$491$		−	9.97056e6i		−	1.86645i	0.359296	−	0.933224i	$-0.383017\pi$
$492$		0			0
$493$	858065.	−	1.04157e6i	0.159002	−	0.193007i
$494$		0			0
$495$		−	206010.i		−	0.0377899i
$496$		0			0
$497$	5.79189e6			1.05179
$498$		0			0
$499$	1.42169e6	−	1.42169e6i	0.255596	−	0.255596i	−0.567664	−	0.823260i	$-0.692153\pi$
$499$	1.42169e6	−	1.42169e6i	0.255596	−	0.255596i	0.823260	+	0.567664i	$0.192153\pi$
$500$		0			0
$501$		−	4.49228e6i		−	0.799599i
$502$		0			0
$503$	−4.92523e6	+	4.92523e6i	−0.867974	+	0.867974i	−0.992248	−	0.124274i	$-0.960340\pi$
$503$	−4.92523e6	+	4.92523e6i	−0.867974	+	0.867974i	0.124274	+	0.992248i	$0.460340\pi$
$504$		0			0
$505$	1.80817e6	+	1.80817e6i	0.315509	+	0.315509i
$506$		0			0
$507$	5.46541e6	+	5.46541e6i	0.944284	+	0.944284i
$508$		0			0
$509$	9.64341e6			1.64982			0.824909	−	0.565266i	$-0.191226\pi$
$509$	9.64341e6			1.64982			0.824909	+	0.565266i	$0.191226\pi$
$510$		0			0
$511$	−4.50225e6			−0.762742
$512$		0			0
$513$	6.48234e6	+	6.48234e6i	1.08752	+	1.08752i
$514$		0			0
$515$	−2.05225e6	−	2.05225e6i	−0.340967	−	0.340967i
$516$		0			0
$517$	736666.	−	736666.i	0.121212	−	0.121212i
$518$		0			0
$519$			5.99833e6i			0.977489i
$520$		0			0
$521$	3.44833e6	−	3.44833e6i	0.556563	−	0.556563i	−0.371764	−	0.928327i	$-0.621247\pi$
$521$	3.44833e6	−	3.44833e6i	0.556563	−	0.556563i	0.928327	+	0.371764i	$0.121247\pi$
$522$		0			0
$523$	−1.03636e7			−1.65675			−0.828377	−	0.560171i	$-0.810736\pi$
$523$	−1.03636e7			−1.65675			−0.828377	+	0.560171i	$0.810736\pi$
$524$		0			0
$525$		−	2.34476e6i		−	0.371279i
$526$		0			0
$527$	70507.7	−	85586.5i	0.0110588	−	0.0134239i
$528$		0			0
$529$		−	1.18911e7i		−	1.84749i
$530$		0			0
$531$	−4.52852e6			−0.696979
$532$		0			0
$533$	−8.98373e6	+	8.98373e6i	−1.36974	+	1.36974i
$534$		0			0
$535$		−	1.26170e6i		−	0.190577i
$536$		0			0
$537$	−551073.	+	551073.i	−0.0824657	+	0.0824657i
$538$		0			0
$539$	−146265.	−	146265.i	−0.0216854	−	0.0216854i
$540$		0			0
$541$	−3.55992e6	−	3.55992e6i	−0.522934	−	0.522934i	0.395523	−	0.918456i	$-0.370564\pi$
$541$	−3.55992e6	−	3.55992e6i	−0.522934	−	0.522934i	−0.918456	+	0.395523i	$0.870564\pi$
$542$		0			0
$543$	243845.			0.0354907
$544$		0			0
$545$	4.49474e6			0.648206
$546$		0			0
$547$	3.19710e6	+	3.19710e6i	0.456865	+	0.456865i	0.897625	−	0.440760i	$-0.145291\pi$
$547$	3.19710e6	+	3.19710e6i	0.456865	+	0.456865i	−0.440760	+	0.897625i	$0.645291\pi$
$548$		0			0
$549$	5.15844e6	+	5.15844e6i	0.730446	+	0.730446i
$550$		0			0
$551$	−1.95860e6	+	1.95860e6i	−0.274831	+	0.274831i
$552$		0			0
$553$		−	3.36752e6i		−	0.468272i
$554$		0			0
$555$	−1.17109e6	+	1.17109e6i	−0.161383	+	0.161383i
$556$		0			0
$557$	−2.32531e6			−0.317572			−0.158786	−	0.987313i	$-0.550758\pi$
$557$	−2.32531e6			−0.317572			−0.158786	+	0.987313i	$0.550758\pi$
$558$		0			0
$559$			5.60044e6i			0.758041i
$560$		0			0
$561$	48883.0	+	506028.i	0.00655768	+	0.0678840i
$562$		0			0
$563$			4.34331e6i			0.577497i	0.957405	+	0.288748i	$0.0932391\pi$
$563$			4.34331e6i			0.577497i	−0.957405	+	0.288748i	$0.906761\pi$
$564$		0			0
$565$	−6.23538e6			−0.821754
$566$		0			0
$567$	−152061.	+	152061.i	−0.0198638	+	0.0198638i
$568$		0			0
$569$			8.18803e6i			1.06023i	0.847927	+	0.530113i	$0.177851\pi$
$569$			8.18803e6i			1.06023i	−0.847927	+	0.530113i	$0.822149\pi$
$570$		0			0
$571$	−863893.	+	863893.i	−0.110884	+	0.110884i	−0.760372	−	0.649488i	$-0.774983\pi$
$571$	−863893.	+	863893.i	−0.110884	+	0.110884i	0.649488	+	0.760372i	$0.274983\pi$
$572$		0			0
$573$	−1.50781e6	−	1.50781e6i	−0.191850	−	0.191850i
$574$		0			0
$575$	−6.79440e6	−	6.79440e6i	−0.857001	−	0.857001i
$576$		0			0
$577$	−4.39226e6			−0.549223			−0.274612	−	0.961555i	$-0.588549\pi$
$577$	−4.39226e6			−0.549223			−0.274612	+	0.961555i	$0.588549\pi$
$578$		0			0
$579$	−2.21808e6			−0.274967
$580$		0			0
$581$	−1.56525e6	−	1.56525e6i	−0.192372	−	0.192372i
$582$		0			0
$583$	−416312.	−	416312.i	−0.0507279	−	0.0507279i
$584$		0			0
$585$	−3.51803e6	+	3.51803e6i	−0.425020	+	0.425020i
$586$		0			0
$587$			2.70622e6i			0.324166i	0.986777	+	0.162083i	$0.0518212\pi$
$587$			2.70622e6i			0.324166i	−0.986777	+	0.162083i	$0.948179\pi$
$588$		0			0
$589$	−160939.	+	160939.i	−0.0191150	+	0.0191150i
$590$		0			0
$591$	6.94167e6			0.817514
$592$		0			0
$593$			5.35588e6i			0.625452i	0.949843	+	0.312726i	$0.101242\pi$
$593$			5.35588e6i			0.625452i	−0.949843	+	0.312726i	$0.898758\pi$
$594$		0			0
$595$	375931.	+	3.89158e6i	0.0435327	+	0.450644i
$596$		0			0
$597$			9.46360e6i			1.08673i
$598$		0			0
$599$	−499444.			−0.0568748			−0.0284374	−	0.999596i	$-0.509053\pi$
$599$	−499444.			−0.0568748			−0.0284374	+	0.999596i	$0.509053\pi$
$600$		0			0
$601$	7.77455e6	−	7.77455e6i	0.877989	−	0.877989i	−0.115338	−	0.993326i	$-0.536795\pi$
$601$	7.77455e6	−	7.77455e6i	0.877989	−	0.877989i	0.993326	+	0.115338i	$0.0367950\pi$
$602$		0			0
$603$		−	6.49930e6i		−	0.727902i
$604$		0			0
$605$	3.33645e6	−	3.33645e6i	0.370592	−	0.370592i
$606$		0			0
$607$	−2.78761e6	−	2.78761e6i	−0.307086	−	0.307086i	0.536692	−	0.843778i	$-0.319674\pi$
$607$	−2.78761e6	−	2.78761e6i	−0.307086	−	0.307086i	−0.843778	+	0.536692i	$0.819674\pi$
$608$		0			0
$609$	836601.	+	836601.i	0.0914061	+	0.0914061i
$610$		0			0
$611$	−2.51600e7			−2.72652
$612$		0			0
$613$	−242001.			−0.0260115			−0.0130057	−	0.999915i	$-0.504140\pi$
$613$	−242001.			−0.0260115			−0.0130057	+	0.999915i	$0.504140\pi$
$614$		0			0
$615$	2.30939e6	+	2.30939e6i	0.246212	+	0.246212i
$616$		0			0
$617$	−1.43397e6	−	1.43397e6i	−0.151645	−	0.151645i	0.627207	−	0.778852i	$-0.284198\pi$
$617$	−1.43397e6	−	1.43397e6i	−0.151645	−	0.151645i	−0.778852	+	0.627207i	$0.784198\pi$
$618$		0			0
$619$	−1.07178e7	+	1.07178e7i	−1.12429	+	1.12429i	−0.133204	+	0.991089i	$0.542527\pi$
$619$	−1.07178e7	+	1.07178e7i	−1.12429	+	1.12429i	−0.991089	+	0.133204i	$0.957473\pi$
$620$		0			0
$621$		−	1.60468e7i		−	1.66978i
$622$		0			0
$623$	2.83735e6	−	2.83735e6i	0.292882	−	0.292882i
$624$		0			0
$625$	−2.28605e6			−0.234092
$626$		0			0
$627$		−	1.04347e6i		−	0.106001i
$628$		0			0
$629$	−4.47582e6	+	5.43302e6i	−0.451072	+	0.547539i
$630$		0			0
$631$			1.10691e7i			1.10673i	0.832940	+	0.553363i	$0.186656\pi$
$631$			1.10691e7i			1.10673i	−0.832940	+	0.553363i	$0.813344\pi$
$632$		0			0
$633$	−598479.			−0.0593663
$634$		0			0
$635$	5.83533e6	−	5.83533e6i	0.574290	−	0.574290i
$636$		0			0
$637$			4.99550e6i			0.487788i
$638$		0			0
$639$	5.69425e6	−	5.69425e6i	0.551677	−	0.551677i
$640$		0			0
$641$	−2.77622e6	−	2.77622e6i	−0.266875	−	0.266875i	0.560965	−	0.827840i	$-0.310430\pi$
$641$	−2.77622e6	−	2.77622e6i	−0.266875	−	0.266875i	−0.827840	+	0.560965i	$0.810430\pi$
$642$		0			0
$643$	1.31429e7	+	1.31429e7i	1.25361	+	1.25361i	0.954089	+	0.299525i	$0.0968281\pi$
$643$	1.31429e7	+	1.31429e7i	1.25361	+	1.25361i	0.299525	+	0.954089i	$0.403172\pi$
$644$		0			0
$645$	1.43967e6			0.136259
$646$		0			0
$647$	−2.67901e6			−0.251602			−0.125801	−	0.992055i	$-0.540150\pi$
$647$	−2.67901e6			−0.251602			−0.125801	+	0.992055i	$0.540150\pi$
$648$		0			0
$649$	940581.	+	940581.i	0.0876565	+	0.0876565i
$650$		0			0
$651$	68743.9	+	68743.9i	0.00635744	+	0.00635744i
$652$		0			0
$653$	1.33266e7	−	1.33266e7i	1.22303	−	1.22303i	0.256483	−	0.966549i	$-0.417436\pi$
$653$	1.33266e7	−	1.33266e7i	1.22303	−	1.22303i	0.966549	−	0.256483i	$-0.0825638\pi$
$654$		0			0
$655$		−	1.43432e6i		−	0.130630i
$656$		0			0
$657$	−4.42636e6	+	4.42636e6i	−0.400067	+	0.400067i
$658$		0			0
$659$	1.75722e7			1.57620			0.788102	−	0.615544i	$-0.211064\pi$
$659$	1.75722e7			1.57620			0.788102	+	0.615544i	$0.211064\pi$
$660$		0			0
$661$		−	622812.i		−	0.0554438i	−0.999616	−	0.0277219i	$-0.991175\pi$
$661$		−	622812.i		−	0.0554438i	0.999616	−	0.0277219i	$-0.00882529\pi$
$662$		0			0
$663$	7.80665e6	−	9.47619e6i	0.689733	−	0.837240i
$664$		0			0
$665$		−	8.02473e6i		−	0.703682i
$666$		0			0
$667$	4.84843e6			0.421975
$668$		0			0
$669$	−3.83557e6	+	3.83557e6i	−0.331333	+	0.331333i
$670$		0			0
$671$		−	2.14283e6i		−	0.183731i
$672$		0			0
$673$	−3.87560e6	+	3.87560e6i	−0.329838	+	0.329838i	−0.852525	−	0.522687i	$-0.824930\pi$
$673$	−3.87560e6	+	3.87560e6i	−0.329838	+	0.329838i	0.522687	+	0.852525i	$0.324930\pi$
$674$		0			0
$675$	−5.94891e6	−	5.94891e6i	−0.502548	−	0.502548i
$676$		0			0
$677$	−4.04301e6	−	4.04301e6i	−0.339026	−	0.339026i	0.516975	−	0.856001i	$-0.327058\pi$
$677$	−4.04301e6	−	4.04301e6i	−0.339026	−	0.339026i	−0.856001	+	0.516975i	$0.827058\pi$
$678$		0			0
$679$	7.18514e6			0.598082
$680$		0			0
$681$	−3.62161e6			−0.299250
$682$		0			0
$683$	−9.79430e6	−	9.79430e6i	−0.803381	−	0.803381i	0.180242	−	0.983622i	$-0.442312\pi$
$683$	−9.79430e6	−	9.79430e6i	−0.803381	−	0.803381i	−0.983622	+	0.180242i	$0.942312\pi$
$684$		0			0
$685$	7.20728e6	+	7.20728e6i	0.586874	+	0.586874i
$686$		0			0
$687$	−3.58825e6	+	3.58825e6i	−0.290062	+	0.290062i
$688$		0			0
$689$			1.42187e7i			1.14107i
$690$		0			0
$691$	−131918.	+	131918.i	−0.0105102	+	0.0105102i	−0.712342	−	0.701832i	$-0.752366\pi$
$691$	−131918.	+	131918.i	−0.0105102	+	0.0105102i	0.701832	+	0.712342i	$0.252366\pi$
$692$		0			0
$693$	767660.			0.0607206
$694$		0			0
$695$		−	5.97021e6i		−	0.468843i
$696$		0			0
$697$	1.07139e7	+	8.82631e6i	0.835346	+	0.688173i
$698$		0			0
$699$		−	1.98005e6i		−	0.153279i
$700$		0			0
$701$	1.90602e7			1.46498			0.732490	−	0.680778i	$-0.238358\pi$
$701$	1.90602e7			1.46498			0.732490	+	0.680778i	$0.238358\pi$
$702$		0			0
$703$	1.02164e7	−	1.02164e7i	0.779667	−	0.779667i
$704$		0			0
$705$			6.46773e6i			0.490094i
$706$		0			0
$707$	−6.73784e6	+	6.73784e6i	−0.506958	+	0.506958i
$708$		0			0
$709$	−1.32768e7	−	1.32768e7i	−0.991924	−	0.991924i	0.00804353	−	0.999968i	$-0.497440\pi$
$709$	−1.32768e7	−	1.32768e7i	−0.991924	−	0.991924i	−0.999968	+	0.00804353i	$0.997440\pi$
$710$		0			0
$711$	−3.31076e6	−	3.31076e6i	−0.245614	−	0.245614i
$712$		0			0
$713$	398398.			0.0293490
$714$		0			0
$715$	1.46140e6			0.106906
$716$		0			0
$717$	−6.55292e6	−	6.55292e6i	−0.476033	−	0.476033i
$718$		0			0
$719$	1.46971e7	+	1.46971e7i	1.06026	+	1.06026i	0.998064	+	0.0621912i	$0.0198089\pi$
$719$	1.46971e7	+	1.46971e7i	1.06026	+	1.06026i	0.0621912	+	0.998064i	$0.480191\pi$
$720$		0			0
$721$	7.64735e6	−	7.64735e6i	0.547864	−	0.547864i
$722$		0			0
$723$		−	1.10912e7i		−	0.789102i
$724$		0			0
$725$	1.79742e6	−	1.79742e6i	0.127001	−	0.127001i
$726$		0			0
$727$	1.71776e7			1.20539			0.602693	−	0.797973i	$-0.294094\pi$
$727$	1.71776e7			1.20539			0.602693	+	0.797973i	$0.294094\pi$
$728$		0			0
$729$		−	8.30650e6i		−	0.578894i
$730$		0			0
$731$	6.09068e6	−	588367.i	0.421572	−	0.0407244i
$732$		0			0
$733$		−	1.37018e7i		−	0.941927i	−0.882153	−	0.470963i	$-0.843906\pi$
$733$		−	1.37018e7i		−	0.941927i	0.882153	−	0.470963i	$-0.156094\pi$
$734$		0			0
$735$	1.28416e6			0.0876802
$736$		0			0
$737$	−1.34991e6	+	1.34991e6i	−0.0915456	+	0.0915456i
$738$		0			0
$739$		−	7.36851e6i		−	0.496328i	−0.968718	−	0.248164i	$-0.920173\pi$
$739$		−	7.36851e6i		−	0.496328i	0.968718	−	0.248164i	$-0.0798272\pi$
$740$		0			0
$741$	−1.78193e7	+	1.78193e7i	−1.19219	+	1.19219i
$742$		0			0
$743$	9.73835e6	+	9.73835e6i	0.647163	+	0.647163i	0.952306	−	0.305144i	$-0.0987045\pi$
$743$	9.73835e6	+	9.73835e6i	0.647163	+	0.647163i	−0.305144	+	0.952306i	$0.598705\pi$
$744$		0			0
$745$	−662864.	−	662864.i	−0.0437556	−	0.0437556i
$746$		0			0
$747$	−3.07772e6			−0.201803
$748$		0			0
$749$	4.70149e6			0.306218
$750$		0			0
$751$	−5.25774e6	−	5.25774e6i	−0.340172	−	0.340172i	0.516260	−	0.856432i	$-0.327324\pi$
$751$	−5.25774e6	−	5.25774e6i	−0.340172	−	0.340172i	−0.856432	+	0.516260i	$0.827324\pi$
$752$		0			0
$753$	−4.07810e6	−	4.07810e6i	−0.262102	−	0.262102i
$754$		0			0
$755$	2.14269e6	−	2.14269e6i	0.136802	−	0.136802i
$756$		0			0
$757$			4.01275e6i			0.254509i	0.991870	+	0.127254i	$0.0406165\pi$
$757$			4.01275e6i			0.254509i	−0.991870	+	0.127254i	$0.959384\pi$
$758$		0			0
$759$	−1.29153e6	+	1.29153e6i	−0.0813768	+	0.0813768i
$760$		0			0
$761$	−2.29021e7			−1.43355			−0.716775	−	0.697304i	$-0.754383\pi$
$761$	−2.29021e7			−1.43355			−0.716775	+	0.697304i	$0.754383\pi$
$762$		0			0
$763$			1.67488e7i			1.04153i
$764$		0			0
$765$	4.19557e6	+	3.45638e6i	0.259201	+	0.213535i
$766$		0			0
$767$		−	3.21245e7i		−	1.97173i
$768$		0			0
$769$	3.75957e6			0.229257			0.114628	−	0.993408i	$-0.463432\pi$
$769$	3.75957e6			0.229257			0.114628	+	0.993408i	$0.463432\pi$
$770$		0			0
$771$	−8.88811e6	+	8.88811e6i	−0.538485	+	0.538485i
$772$		0			0
$773$			1.10332e7i			0.664130i	0.943257	+	0.332065i	$0.107745\pi$
$773$			1.10332e7i			0.664130i	−0.943257	+	0.332065i	$0.892255\pi$
$774$		0			0
$775$	147695.	−	147695.i	0.00883309	−	0.00883309i
$776$		0			0
$777$	−4.36386e6	−	4.36386e6i	−0.259309	−	0.259309i
$778$		0			0
$779$	−2.01467e7	−	2.01467e7i	−1.18949	−	1.18949i
$780$		0			0
$781$	−2.36541e6			−0.138765
$782$		0			0
$783$	4.24509e6			0.247447
$784$		0			0
$785$	1.69345e6	+	1.69345e6i	0.0980839	+	0.0980839i
$786$		0			0
$787$	1.48312e7	+	1.48312e7i	0.853569	+	0.853569i	0.990571	−	0.137001i	$-0.0437465\pi$
$787$	1.48312e7	+	1.48312e7i	0.853569	+	0.853569i	−0.137001	+	0.990571i	$0.543746\pi$
$788$		0			0
$789$	2.64385e6	−	2.64385e6i	0.151197	−	0.151197i
$790$		0			0
$791$		−	2.32350e7i		−	1.32039i
$792$		0			0
$793$	−3.65931e7	+	3.65931e7i	−2.06641	+	2.06641i
$794$		0			0
$795$	3.65510e6			0.205107
$796$		0			0
$797$			3.18453e7i			1.77582i	0.460015	+	0.887911i	$0.347844\pi$
$797$			3.18453e7i			1.77582i	−0.460015	+	0.887911i	$0.652156\pi$
$798$		0			0
$799$	2.64324e6	+	2.73624e7i	0.146477	+	1.51631i
$800$		0			0
$801$		−	5.57904e6i		−	0.307240i
$802$		0			0
$803$	1.83872e6			0.100630
$804$		0			0
$805$	−9.93244e6	+	9.93244e6i	−0.540214	+	0.540214i
$806$		0			0
$807$			248631.i			0.0134391i
$808$		0			0
$809$	−1.82717e7	+	1.82717e7i	−0.981537	+	0.981537i	−0.999833	−	0.0182952i	$-0.994176\pi$
$809$	−1.82717e7	+	1.82717e7i	−0.981537	+	0.981537i	0.0182952	+	0.999833i	$0.494176\pi$
$810$		0			0
$811$	1.48508e7	+	1.48508e7i	0.792865	+	0.792865i	0.981959	−	0.189094i	$-0.0605552\pi$
$811$	1.48508e7	+	1.48508e7i	0.792865	+	0.792865i	−0.189094	+	0.981959i	$0.560555\pi$
$812$		0			0
$813$	−7.84428e6	−	7.84428e6i	−0.416223	−	0.416223i
$814$		0			0
$815$	−7.47795e6			−0.394356
$816$		0			0
$817$	−1.25594e7			−0.658287
$818$		0			0
$819$	−1.31093e7	−	1.31093e7i	−0.682919	−	0.682919i
$820$		0			0
$821$	−3.16281e6	−	3.16281e6i	−0.163763	−	0.163763i	0.620468	−	0.784231i	$-0.286942\pi$
$821$	−3.16281e6	−	3.16281e6i	−0.163763	−	0.163763i	−0.784231	+	0.620468i	$0.786942\pi$
$822$		0			0
$823$	4.62500e6	−	4.62500e6i	0.238019	−	0.238019i	−0.578010	−	0.816029i	$-0.696171\pi$
$823$	4.62500e6	−	4.62500e6i	0.238019	−	0.238019i	0.816029	+	0.578010i	$0.196171\pi$
$824$		0			0
$825$			957602.i			0.0489835i
$826$		0			0
$827$	−2.16267e7	+	2.16267e7i	−1.09958	+	1.09958i	−0.105117	+	0.994460i	$0.533522\pi$
$827$	−2.16267e7	+	2.16267e7i	−1.09958	+	1.09958i	−0.994460	+	0.105117i	$0.966478\pi$
$828$		0			0
$829$	1.96609e7			0.993613			0.496806	−	0.867861i	$-0.334506\pi$
$829$	1.96609e7			0.993613			0.496806	+	0.867861i	$0.334506\pi$
$830$		0			0
$831$		−	1.22066e7i		−	0.613186i
$832$		0			0
$833$	5.43279e6	−	524814.i	0.271275	−	0.0262055i
$834$		0			0
$835$			1.41092e7i			0.700306i
$836$		0			0
$837$	348821.			0.0172103
$838$		0			0
$839$	2.75859e7	−	2.75859e7i	1.35295	−	1.35295i	0.470610	−	0.882341i	$-0.344034\pi$
$839$	2.75859e7	−	2.75859e7i	1.35295	−	1.35295i	0.882341	−	0.470610i	$-0.155966\pi$
$840$		0			0
$841$		−	1.92285e7i		−	0.937467i
$842$		0			0
$843$	−7.30808e6	+	7.30808e6i	−0.354188	+	0.354188i
$844$		0			0
$845$	−1.71656e7	−	1.71656e7i	−0.827024	−	0.827024i
$846$		0			0
$847$	1.24327e7	+	1.24327e7i	0.595465	+	0.595465i
$848$		0			0
$849$	1.58695e7			0.755604
$850$		0			0
$851$	−2.52902e7			−1.19710
$852$		0			0
$853$	7.63619e6	+	7.63619e6i	0.359339	+	0.359339i	0.863569	−	0.504230i	$-0.168224\pi$
$853$	7.63619e6	+	7.63619e6i	0.359339	+	0.359339i	−0.504230	+	0.863569i	$0.668224\pi$
$854$		0			0
$855$	−7.88946e6	−	7.88946e6i	−0.369090	−	0.369090i
$856$		0			0
$857$	−3.62557e6	+	3.62557e6i	−0.168626	+	0.168626i	−0.786375	−	0.617749i	$-0.788045\pi$
$857$	−3.62557e6	+	3.62557e6i	−0.168626	+	0.168626i	0.617749	+	0.786375i	$0.288045\pi$
$858$		0			0
$859$			6.35583e6i			0.293893i	0.989144	+	0.146947i	$0.0469446\pi$
$859$			6.35583e6i			0.293893i	−0.989144	+	0.146947i	$0.953055\pi$
$860$		0			0
$861$	−8.60553e6	+	8.60553e6i	−0.395612	+	0.395612i
$862$		0			0
$863$	1.46095e7			0.667743			0.333871	−	0.942619i	$-0.391645\pi$
$863$	1.46095e7			0.667743			0.333871	+	0.942619i	$0.391645\pi$
$864$		0			0
$865$		−	1.88394e7i		−	0.856105i
$866$		0			0
$867$	−1.11258e7	−	7.49446e6i	−0.502672	−	0.338604i
$868$		0			0
$869$			1.37530e6i			0.0617800i
$870$		0			0
$871$	4.61049e7			2.05921
$872$		0			0
$873$	7.06401e6	−	7.06401e6i	0.313701	−	0.313701i
$874$		0			0
$875$			1.76178e7i			0.777915i
$876$		0			0
$877$	1.41877e6	−	1.41877e6i	0.0622892	−	0.0622892i	−0.675276	−	0.737565i	$-0.735975\pi$
$877$	1.41877e6	−	1.41877e6i	0.0622892	−	0.0622892i	0.737565	+	0.675276i	$0.235975\pi$
$878$		0			0
$879$	−1.94693e7	−	1.94693e7i	−0.849922	−	0.849922i
$880$		0			0
$881$	1.26688e7	+	1.26688e7i	0.549915	+	0.549915i	0.926416	−	0.376501i	$-0.122873\pi$
$881$	1.26688e7	+	1.26688e7i	0.549915	+	0.549915i	−0.376501	+	0.926416i	$0.622873\pi$
$882$		0			0
$883$	−4.50342e7			−1.94375			−0.971875	−	0.235497i	$-0.924328\pi$
$883$	−4.50342e7			−1.94375			−0.971875	+	0.235497i	$0.924328\pi$
$884$		0			0
$885$	−8.25804e6			−0.354420
$886$		0			0
$887$	−2.61699e7	−	2.61699e7i	−1.11684	−	1.11684i	−0.992202	−	0.124641i	$-0.960222\pi$
$887$	−2.61699e7	−	2.61699e7i	−1.11684	−	1.11684i	−0.124641	−	0.992202i	$-0.539778\pi$
$888$		0			0
$889$	2.17443e7	+	2.17443e7i	0.922765	+	0.922765i
$890$		0			0
$891$	62102.0	−	62102.0i	0.00262066	−	0.00262066i
$892$		0			0
$893$		−	5.64234e7i		−	2.36772i
$894$		0			0
$895$	1.73080e6	−	1.73080e6i	0.0722252	−	0.0722252i
$896$		0			0
$897$	4.41109e7			1.83048
$898$		0			0
$899$			105394.i			0.00434928i
$900$		0			0
$901$	1.54633e7	−	1.49377e6i	0.634585	−	0.0613017i
$902$		0			0
$903$			5.36467e6i			0.218939i
$904$		0			0
$905$	−765863.			−0.0310835
$906$		0			0
$907$	1.18119e7	−	1.18119e7i	0.476763	−	0.476763i	−0.427332	−	0.904095i	$-0.640546\pi$
$907$	1.18119e7	−	1.18119e7i	0.476763	−	0.476763i	0.904095	+	0.427332i	$0.140546\pi$
$908$		0			0
$909$			1.32485e7i			0.531811i
$910$		0			0
$911$	1.34194e7	−	1.34194e7i	0.535718	−	0.535718i	−0.386551	−	0.922268i	$-0.626334\pi$
$911$	1.34194e7	−	1.34194e7i	0.535718	−	0.535718i	0.922268	+	0.386551i	$0.126334\pi$
$912$		0			0
$913$	639248.	+	639248.i	0.0253800	+	0.0253800i
$914$		0			0
$915$	9.40675e6	+	9.40675e6i	0.371438	+	0.371438i
$916$		0			0
$917$	5.34474e6			0.209895
$918$		0			0
$919$	−2.20135e6			−0.0859807			−0.0429903	−	0.999075i	$-0.513688\pi$
$919$	−2.20135e6			−0.0859807			−0.0429903	+	0.999075i	$0.513688\pi$
$920$		0			0
$921$	1.66292e7	+	1.66292e7i	0.645985	+	0.645985i
$922$		0			0
$923$	4.03940e7	+	4.03940e7i	1.56068	+	1.56068i
$924$		0			0
$925$	−9.37568e6	+	9.37568e6i	−0.360287	+	0.360287i
$926$		0			0
$927$		−	1.50369e7i		−	0.574722i
$928$		0			0
$929$	2.37613e7	−	2.37613e7i	0.903296	−	0.903296i	−0.0924236	−	0.995720i	$-0.529461\pi$
$929$	2.37613e7	−	2.37613e7i	0.903296	−	0.903296i	0.995720	+	0.0924236i	$0.0294614\pi$
$930$		0			0
$931$	−1.12028e7			−0.423597
$932$		0			0
$933$		−	1.66051e7i		−	0.624506i
$934$		0			0
$935$	−153531.	−	1.58932e6i	−0.00574336	−	0.0594543i
$936$		0			0
$937$		−	2.79874e7i		−	1.04139i	−0.853743	−	0.520695i	$-0.825673\pi$
$937$		−	2.79874e7i		−	1.04139i	0.853743	−	0.520695i	$-0.174327\pi$
$938$		0			0
$939$	2.28686e7			0.846401
$940$		0			0
$941$	−1.34398e7	+	1.34398e7i	−0.494786	+	0.494786i	−0.909810	−	0.415024i	$-0.863773\pi$
$941$	−1.34398e7	+	1.34398e7i	−0.494786	+	0.494786i	0.415024	+	0.909810i	$0.363773\pi$
$942$		0			0
$943$			4.98724e7i			1.82634i
$944$		0			0
$945$	−8.69645e6	+	8.69645e6i	−0.316783	+	0.316783i
$946$		0			0
$947$	−1.01889e7	−	1.01889e7i	−0.369191	−	0.369191i	0.497991	−	0.867182i	$-0.334071\pi$
$947$	−1.01889e7	−	1.01889e7i	−0.369191	−	0.369191i	−0.867182	+	0.497991i	$0.834071\pi$
$948$		0			0
$949$	−3.13998e7	−	3.13998e7i	−1.13178	−	1.13178i
$950$		0			0
$951$	9.54043e6			0.342071
$952$		0			0
$953$	−1.47497e7			−0.526077			−0.263039	−	0.964785i	$-0.584725\pi$
$953$	−1.47497e7			−0.526077			−0.263039	+	0.964785i	$0.584725\pi$
$954$		0			0
$955$	4.73571e6	+	4.73571e6i	0.168026	+	0.168026i
$956$		0			0
$957$	−341668.	−	341668.i	−0.0120594	−	0.0120594i
$958$		0			0
$959$	−2.68566e7	+	2.68566e7i	−0.942985	+	0.942985i
$960$		0			0
$961$		−	2.86205e7i		−	0.999698i
$962$		0			0
$963$	4.62223e6	−	4.62223e6i	0.160615	−	0.160615i
$964$		0			0
$965$	6.96650e6			0.240822
$966$		0			0
$967$			9.31587e6i			0.320374i	0.987087	+	0.160187i	$0.0512097\pi$
$967$			9.31587e6i			0.320374i	−0.987087	+	0.160187i	$0.948790\pi$
$968$		0			0
$969$	2.12511e7	+	1.75070e7i	0.727064	+	0.598968i
$970$		0			0
$971$		−	1.53590e7i		−	0.522775i	−0.965234	−	0.261387i	$-0.915820\pi$
$971$		−	1.53590e7i		−	0.522775i	0.965234	−	0.261387i	$-0.0841800\pi$
$972$		0			0
$973$	2.22469e7			0.753335
$974$		0			0
$975$	1.63529e7	−	1.63529e7i	0.550914	−	0.550914i
$976$		0			0
$977$		−	5.82883e7i		−	1.95364i	−0.214060	−	0.976821i	$-0.568669\pi$
$977$		−	5.82883e7i		−	1.95364i	0.214060	−	0.976821i	$-0.431331\pi$
$978$		0			0
$979$	−1.15878e6	+	1.15878e6i	−0.0386405	+	0.0386405i
$980$		0			0
$981$	1.64665e7	+	1.64665e7i	0.546297	+	0.546297i
$982$		0			0
$983$	3.47079e7	+	3.47079e7i	1.14563	+	1.14563i	0.987403	+	0.158227i	$0.0505779\pi$
$983$	3.47079e7	+	3.47079e7i	1.14563	+	1.14563i	0.158227	+	0.987403i	$0.449422\pi$
$984$		0			0
$985$	−2.18023e7			−0.715996
$986$		0			0
$987$	−2.41008e7			−0.787479
$988$		0			0
$989$	1.55452e7	+	1.55452e7i	0.505365	+	0.505365i
$990$		0			0
$991$	−3.38943e6	−	3.38943e6i	−0.109633	−	0.109633i	0.650162	−	0.759795i	$-0.274701\pi$
$991$	−3.38943e6	−	3.38943e6i	−0.109633	−	0.109633i	−0.759795	+	0.650162i	$0.774701\pi$
$992$		0			0
$993$	1.71932e7	−	1.71932e7i	0.553331	−	0.553331i
$994$		0			0
$995$		−	2.97231e7i		−	0.951779i
$996$		0			0
$997$	6.24525e6	−	6.24525e6i	0.198981	−	0.198981i	−0.600582	−	0.799563i	$-0.705064\pi$
$997$	6.24525e6	−	6.24525e6i	0.198981	−	0.198981i	0.799563	+	0.600582i	$0.205064\pi$
$998$		0			0
$999$	−2.21431e7			−0.701981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.6.e.a.13.6	✓	16
3.2	odd	2		612.6.k.a.217.6		16
4.3	odd	2		272.6.o.d.81.3		16
17.4	even	4	inner	68.6.e.a.21.6	yes	16
51.38	odd	4		612.6.k.a.361.6		16
68.55	odd	4		272.6.o.d.225.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.6.e.a.13.6	✓	16	1.1	even	1	trivial
68.6.e.a.21.6	yes	16	17.4	even	4	inner
272.6.o.d.81.3		16	4.3	odd	2
272.6.o.d.225.3		16	68.55	odd	4
612.6.k.a.217.6		16	3.2	odd	2
612.6.k.a.361.6		16	51.38	odd	4

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:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	68.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(6.68063 + 6.68063i) q^{3} +(-20.9824 - 20.9824i) q^{5} +(78.1871 - 78.1871i) q^{7} -153.738i q^{9} +O(q^{10})\)	\(q+(6.68063 + 6.68063i) q^{3} +(-20.9824 - 20.9824i) q^{5} +(78.1871 - 78.1871i) q^{7} -153.738i q^{9} +(-31.9317 + 31.9317i) q^{11} +1090.59 q^{13} -280.351i q^{15} +(-114.574 - 1186.06i) q^{17} +2445.74i q^{19} +1044.68 q^{21} +(3027.16 - 3027.16i) q^{23} -2244.48i q^{25} +(2650.46 - 2650.46i) q^{27} +(800.821 + 800.821i) q^{29} +(65.8039 + 65.8039i) q^{31} -426.648 q^{33} -3281.11 q^{35} +(-4177.22 - 4177.22i) q^{37} +(7285.84 + 7285.84i) q^{39} +(-8237.48 + 8237.48i) q^{41} +5135.23i q^{43} +(-3225.80 + 3225.80i) q^{45} -23070.1 q^{47} +4580.55i q^{49} +(7158.18 - 8689.04i) q^{51} +13037.6i q^{53} +1340.01 q^{55} +(-16339.1 + 16339.1i) q^{57} -29456.0i q^{59} +(-33553.4 + 33553.4i) q^{61} +(-12020.4 - 12020.4i) q^{63} +(-22883.2 - 22883.2i) q^{65} +42275.1 q^{67} +40446.7 q^{69} +(37038.6 + 37038.6i) q^{71} +(-28791.5 - 28791.5i) q^{73} +(14994.5 - 14994.5i) q^{75} +4993.29i q^{77} +(21535.0 - 21535.0i) q^{79} -1944.84 q^{81} -20019.2i q^{83} +(-22482.3 + 27290.3i) q^{85} +10700.0i q^{87} +36289.2 q^{89} +(85270.2 - 85270.2i) q^{91} +879.223i q^{93} +(51317.5 - 51317.5i) q^{95} +(45948.3 + 45948.3i) q^{97} +(4909.12 + 4909.12i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7}+O(q^{10}) \) 16 * q + 22 * q^3 - 44 * q^5 - 118 * q^7	\( 16 q + 22 q^{3} - 44 q^{5} - 118 q^{7} + 414 q^{11} + 252 q^{13} - 2784 q^{17} + 5068 q^{21} + 1102 q^{23} - 14612 q^{27} + 9672 q^{29} + 2162 q^{31} - 20108 q^{33} + 59228 q^{35} - 26148 q^{37} - 42488 q^{39} + 18032 q^{41} + 40544 q^{45} - 12144 q^{47} - 57046 q^{51} + 17284 q^{55} + 31700 q^{57} - 9724 q^{61} - 20022 q^{63} - 34212 q^{65} + 47080 q^{67} - 17796 q^{69} + 117038 q^{71} + 37912 q^{73} - 82670 q^{75} + 129822 q^{79} - 197196 q^{81} + 113848 q^{85} - 456228 q^{89} - 263080 q^{91} + 393180 q^{95} + 209900 q^{97} + 222358 q^{99}+O(q^{100}) \) 16 * q + 22 * q^3 - 44 * q^5 - 118 * q^7 + 414 * q^11 + 252 * q^13 - 2784 * q^17 + 5068 * q^21 + 1102 * q^23 - 14612 * q^27 + 9672 * q^29 + 2162 * q^31 - 20108 * q^33 + 59228 * q^35 - 26148 * q^37 - 42488 * q^39 + 18032 * q^41 + 40544 * q^45 - 12144 * q^47 - 57046 * q^51 + 17284 * q^55 + 31700 * q^57 - 9724 * q^61 - 20022 * q^63 - 34212 * q^65 + 47080 * q^67 - 17796 * q^69 + 117038 * q^71 + 37912 * q^73 - 82670 * q^75 + 129822 * q^79 - 197196 * q^81 + 113848 * q^85 - 456228 * q^89 - 263080 * q^91 + 393180 * q^95 + 209900 * q^97 + 222358 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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