LMFDB - Embedded newform 112.4.p.g.47.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,4,Mod(31,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("112.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, 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:	$6.60821392064$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.12258833328.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 29x^{4} - 20x^{3} + 808x^{2} - 672x + 576 $ x^6 - x^5 + 29x^4 - 20x^3 + 808x^2 - 672x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.3
Root			$-2.61524 - 4.52973i$ of defining polynomial
Character	$\chi$	$=$	112.47
Dual form			112.4.p.g.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.64712 + 8.04905i) q^{3} +(-17.1915 - 9.92549i) q^{5} +(-18.3972 + 2.13127i) q^{7} +(-29.6915 + 51.4271i) q^{9} +O(q^{10})$	\(q+(4.64712 + 8.04905i) q^{3} +(-17.1915 - 9.92549i) q^{5} +(-18.3972 + 2.13127i) q^{7} +(-29.6915 + 51.4271i) q^{9} +(-10.2499 + 5.91778i) q^{11} +19.9862i q^{13} -184.500i q^{15} +(1.57359 - 0.908512i) q^{17} +(-3.66142 + 6.34176i) q^{19} +(-102.649 - 138.176i) q^{21} +(92.2074 + 53.2360i) q^{23} +(134.531 + 233.014i) q^{25} -300.975 q^{27} -191.679 q^{29} +(62.5752 + 108.383i) q^{31} +(-95.2651 - 55.0013i) q^{33} +(337.429 + 145.962i) q^{35} +(-158.414 + 274.381i) q^{37} +(-160.870 + 92.8782i) q^{39} -321.806i q^{41} +74.3089i q^{43} +(1020.88 - 589.405i) q^{45} +(77.2810 - 133.855i) q^{47} +(333.915 - 78.4187i) q^{49} +(14.6253 + 8.44393i) q^{51} +(159.711 + 276.627i) q^{53} +234.948 q^{55} -68.0602 q^{57} +(-30.5584 - 52.9288i) q^{59} +(-267.882 - 154.662i) q^{61} +(436.635 - 1009.40i) q^{63} +(198.373 - 343.592i) q^{65} +(-514.507 + 297.051i) q^{67} +989.576i q^{69} +48.6774i q^{71} +(-667.072 + 385.134i) q^{73} +(-1250.36 + 2165.69i) q^{75} +(175.957 - 130.716i) q^{77} +(831.564 + 480.103i) q^{79} +(-596.996 - 1034.03i) q^{81} +1064.55 q^{83} -36.0697 q^{85} +(-890.755 - 1542.83i) q^{87} +(-567.737 - 327.783i) q^{89} +(-42.5959 - 367.690i) q^{91} +(-581.589 + 1007.34i) q^{93} +(125.890 - 72.6828i) q^{95} -704.225i q^{97} -702.830i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9}+O(q^{10}) $ 6 * q + 7 * q^3 - 3 * q^5 - 52 * q^7 - 78 * q^9	$ 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} - 15 q^{21} + 15 q^{23} + 306 q^{25} - 362 q^{27} - 348 q^{29} - 205 q^{31} - 471 q^{33} + 1185 q^{35} - 249 q^{37} - 288 q^{39} + 2118 q^{45} - 75 q^{47} + 702 q^{49} + 2505 q^{51} - 645 q^{53} - 918 q^{55} - 6 q^{57} - 321 q^{59} - 1707 q^{61} + 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} + 555 q^{77} + 3447 q^{79} + 225 q^{81} - 24 q^{83} + 3786 q^{85} - 3642 q^{87} - 2607 q^{89} + 2448 q^{91} - 2991 q^{93} - 2085 q^{95}+O(q^{100}) $ 6 * q + 7 * q^3 - 3 * q^5 - 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 - 15 * q^21 + 15 * q^23 + 306 * q^25 - 362 * q^27 - 348 * q^29 - 205 * q^31 - 471 * q^33 + 1185 * q^35 - 249 * q^37 - 288 * q^39 + 2118 * q^45 - 75 * q^47 + 702 * q^49 + 2505 * q^51 - 645 * q^53 - 918 * q^55 - 6 * q^57 - 321 * q^59 - 1707 * q^61 + 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 + 555 * q^77 + 3447 * q^79 + 225 * q^81 - 24 * q^83 + 3786 * q^85 - 3642 * q^87 - 2607 * q^89 + 2448 * q^91 - 2991 * q^93 - 2085 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	4.64712	+	8.04905i	0.894339	+	1.54904i	0.834621	+	0.550825i	$0.185687\pi$
$3$	4.64712	+	8.04905i	0.894339	+	1.54904i	0.0597180	+	0.998215i	$0.480980\pi$
$4$		0			0
$5$	−17.1915	−	9.92549i	−1.53765	−	0.887763i	−0.998976	−	0.0452498i	$-0.985592\pi$
$5$	−17.1915	−	9.92549i	−1.53765	−	0.887763i	−0.538675	−	0.842513i	$-0.681075\pi$
$6$		0			0
$7$	−18.3972	+	2.13127i	−0.993357	+	0.115078i
$7$	−18.3972	+	2.13127i	−0.993357	+	0.115078i
$8$		0			0
$9$	−29.6915	+	51.4271i	−1.09968	+	1.90471i
$10$		0			0
$11$	−10.2499	+	5.91778i	−0.280951	+	0.162207i	−0.633854	−	0.773453i	$-0.718528\pi$
$11$	−10.2499	+	5.91778i	−0.280951	+	0.162207i	0.352903	+	0.935660i	$0.385195\pi$
$12$		0			0
$13$			19.9862i			0.426398i	0.977009	+	0.213199i	$0.0683882\pi$
$13$			19.9862i			0.426398i	−0.977009	+	0.213199i	$0.931612\pi$
$14$		0			0
$15$		−	184.500i		−	3.17584i
$16$		0			0
$17$	1.57359	−	0.908512i	0.0224501	−	0.0129616i	−0.488733	−	0.872433i	$-0.662541\pi$
$17$	1.57359	−	0.908512i	0.0224501	−	0.0129616i	0.511183	+	0.859472i	$0.329207\pi$
$18$		0			0
$19$	−3.66142	+	6.34176i	−0.0442098	+	0.0765737i	−0.887284	−	0.461224i	$-0.847410\pi$
$19$	−3.66142	+	6.34176i	−0.0442098	+	0.0765737i	0.843074	+	0.537798i	$0.180744\pi$
$20$		0			0
$21$	−102.649	−	138.176i	−1.06666	−	1.43583i
$22$		0			0
$23$	92.2074	+	53.2360i	0.835938	+	0.482629i	0.855881	−	0.517172i	$-0.173015\pi$
$23$	92.2074	+	53.2360i	0.835938	+	0.482629i	−0.0199435	+	0.999801i	$0.506349\pi$
$24$		0			0
$25$	134.531	+	233.014i	1.07625	+	1.86411i
$26$		0			0
$27$	−300.975			−2.14528
$28$		0			0
$29$	−191.679			−1.22737			−0.613687	−	0.789549i	$-0.710315\pi$
$29$	−191.679			−1.22737			−0.613687	+	0.789549i	$0.710315\pi$
$30$		0			0
$31$	62.5752	+	108.383i	0.362543	+	0.627944i	0.988379	−	0.152012i	$-0.0485751\pi$
$31$	62.5752	+	108.383i	0.362543	+	0.627944i	−0.625835	+	0.779955i	$0.715242\pi$
$32$		0			0
$33$	−95.2651	−	55.0013i	−0.502531	−	0.290136i
$34$		0			0
$35$	337.429	+	145.962i	1.62960	+	0.704916i
$36$		0			0
$37$	−158.414	+	274.381i	−0.703867	+	1.21913i	0.263232	+	0.964733i	$0.415211\pi$
$37$	−158.414	+	274.381i	−0.703867	+	1.21913i	−0.967099	+	0.254401i	$0.918122\pi$
$38$		0			0
$39$	−160.870	+	92.8782i	−0.660507	+	0.381344i
$40$		0			0
$41$		−	321.806i		−	1.22580i	−0.790162	−	0.612898i	$-0.790003\pi$
$41$		−	321.806i		−	1.22580i	0.790162	−	0.612898i	$-0.209997\pi$
$42$		0			0
$43$			74.3089i			0.263535i	0.991281	+	0.131767i	$0.0420652\pi$
$43$			74.3089i			0.263535i	−0.991281	+	0.131767i	$0.957935\pi$
$44$		0			0
$45$	1020.88	−	589.405i	3.38186	−	1.95252i
$46$		0			0
$47$	77.2810	−	133.855i	0.239842	−	0.415419i	−0.720827	−	0.693116i	$-0.756238\pi$
$47$	77.2810	−	133.855i	0.239842	−	0.415419i	0.960669	+	0.277696i	$0.0895709\pi$
$48$		0			0
$49$	333.915	−	78.4187i	0.973514	−	0.228626i
$50$		0			0
$51$	14.6253	+	8.44393i	0.0401560	+	0.0231841i
$52$		0			0
$53$	159.711	+	276.627i	0.413923	+	0.716936i	0.995315	−	0.0966878i	$-0.0308248\pi$
$53$	159.711	+	276.627i	0.413923	+	0.716936i	−0.581391	+	0.813624i	$0.697492\pi$
$54$		0			0
$55$	234.948			0.576006
$56$		0			0
$57$	−68.0602			−0.158154
$58$		0			0
$59$	−30.5584	−	52.9288i	−0.0674300	−	0.116792i	0.830339	−	0.557258i	$-0.188147\pi$
$59$	−30.5584	−	52.9288i	−0.0674300	−	0.116792i	−0.897769	+	0.440466i	$0.854813\pi$
$60$		0			0
$61$	−267.882	−	154.662i	−0.562275	−	0.324630i	0.191783	−	0.981437i	$-0.438573\pi$
$61$	−267.882	−	154.662i	−0.562275	−	0.324630i	−0.754058	+	0.656808i	$0.771906\pi$
$62$		0			0
$63$	436.635	−	1009.40i	0.873189	−	2.01860i
$64$		0			0
$65$	198.373	−	343.592i	0.378540	−	0.655651i
$66$		0			0
$67$	−514.507	+	297.051i	−0.938165	+	0.541650i	−0.889385	−	0.457160i	$-0.848867\pi$
$67$	−514.507	+	297.051i	−0.938165	+	0.541650i	−0.0487806	+	0.998810i	$0.515534\pi$
$68$		0			0
$69$			989.576i			1.72654i
$70$		0			0
$71$			48.6774i			0.0813654i	0.999172	+	0.0406827i	$0.0129533\pi$
$71$			48.6774i			0.0813654i	−0.999172	+	0.0406827i	$0.987047\pi$
$72$		0			0
$73$	−667.072	+	385.134i	−1.06952	+	0.617487i	−0.928050	−	0.372455i	$-0.878516\pi$
$73$	−667.072	+	385.134i	−1.06952	+	0.617487i	−0.141469	+	0.989943i	$0.545183\pi$
$74$		0			0
$75$	−1250.36	+	2165.69i	−1.92506	+	3.33430i
$76$		0			0
$77$	175.957	−	130.716i	0.260418	−	0.193461i
$78$		0			0
$79$	831.564	+	480.103i	1.18428	+	0.683745i	0.957001	−	0.290084i	$-0.0936834\pi$
$79$	831.564	+	480.103i	1.18428	+	0.683745i	0.227280	+	0.973829i	$0.427017\pi$
$80$		0			0
$81$	−596.996	−	1034.03i	−0.818925	−	1.41842i
$82$		0			0
$83$	1064.55			1.40782			0.703912	−	0.710288i	$-0.251435\pi$
$83$	1064.55			1.40782			0.703912	+	0.710288i	$0.251435\pi$
$84$		0			0
$85$	−36.0697			−0.0460272
$86$		0			0
$87$	−890.755	−	1542.83i	−1.09769	−	1.90125i
$88$		0			0
$89$	−567.737	−	327.783i	−0.676180	−	0.390393i	0.122234	−	0.992501i	$-0.460994\pi$
$89$	−567.737	−	327.783i	−0.676180	−	0.390393i	−0.798414	+	0.602108i	$0.794327\pi$
$90$		0			0
$91$	−42.5959	−	367.690i	−0.0490688	−	0.423565i
$92$		0			0
$93$	−581.589	+	1007.34i	−0.648473	+	1.12319i
$94$		0			0
$95$	125.890	−	72.6828i	0.135959	−	0.0784957i
$96$		0			0
$97$		−	704.225i		−	0.737147i	−0.929599	−	0.368573i	$-0.879846\pi$
$97$		−	704.225i		−	0.737147i	0.929599	−	0.368573i	$-0.120154\pi$
$98$		0			0
$99$		−	702.830i		−	0.713506i
$100$		0			0
$101$	−597.608	+	345.029i	−0.588754	+	0.339918i	−0.764605	−	0.644499i	$-0.777066\pi$
$101$	−597.608	+	345.029i	−0.588754	+	0.339918i	0.175850	+	0.984417i	$0.443733\pi$
$102$		0			0
$103$	699.136	−	1210.94i	0.668814	−	1.15842i	−0.309421	−	0.950925i	$-0.600135\pi$
$103$	699.136	−	1210.94i	0.668814	−	1.15842i	0.978236	−	0.207496i	$-0.0665313\pi$
$104$		0			0
$105$	393.218	+	3394.29i	0.365468	+	3.15475i
$106$		0			0
$107$	−26.9985	−	15.5876i	−0.0243929	−	0.0140832i	0.487754	−	0.872981i	$-0.337816\pi$
$107$	−26.9985	−	15.5876i	−0.0243929	−	0.0140832i	−0.512147	+	0.858898i	$0.671150\pi$
$108$		0			0
$109$	−239.079	−	414.096i	−0.210088	−	0.363883i	0.741654	−	0.670783i	$-0.234042\pi$
$109$	−239.079	−	414.096i	−0.210088	−	0.363883i	−0.951742	+	0.306900i	$0.900708\pi$
$110$		0			0
$111$	−2944.67			−2.51798
$112$		0			0
$113$	1861.07			1.54933			0.774666	−	0.632371i	$-0.217918\pi$
$113$	1861.07			1.54933			0.774666	+	0.632371i	$0.217918\pi$
$114$		0			0
$115$	−1056.79	−	1830.41i	−0.856921	−	1.48423i
$116$		0			0
$117$	−1027.83	−	593.419i	−0.812163	−	0.468903i
$118$		0			0
$119$	−27.0134	+	20.0678i	−0.0208094	+	0.0154590i
$120$		0			0
$121$	−595.460	+	1031.37i	−0.447378	+	0.774881i
$122$		0			0
$123$	2590.23	−	1495.47i	1.89881	−	1.09628i
$124$		0			0
$125$		−	2859.77i		−	2.04628i
$126$		0			0
$127$			1583.04i			1.10608i	0.833155	+	0.553040i	$0.186532\pi$
$127$			1583.04i			1.10608i	−0.833155	+	0.553040i	$0.813468\pi$
$128$		0			0
$129$	−598.116	+	345.323i	−0.408226	+	0.235690i
$130$		0			0
$131$	−1098.60	+	1902.83i	−0.732712	+	1.26909i	0.223008	+	0.974817i	$0.428412\pi$
$131$	−1098.60	+	1902.83i	−0.732712	+	1.26909i	−0.955720	+	0.294278i	$0.904921\pi$
$132$		0			0
$133$	53.8439	−	124.474i	0.0351042	−	0.0811525i
$134$		0			0
$135$	5174.19	+	2987.32i	3.29869	+	1.90450i
$136$		0			0
$137$	−1249.70	−	2164.54i	−0.779336	−	1.34985i	−0.932325	−	0.361621i	$-0.882223\pi$
$137$	−1249.70	−	2164.54i	−0.779336	−	1.34985i	0.152990	−	0.988228i	$-0.451110\pi$
$138$		0			0
$139$	−2709.21			−1.65318			−0.826591	−	0.562804i	$-0.809723\pi$
$139$	−2709.21			−1.65318			−0.826591	+	0.562804i	$0.809723\pi$
$140$		0			0
$141$	1436.54			0.858001
$142$		0			0
$143$	−118.274	−	204.856i	−0.0691647	−	0.119797i
$144$		0			0
$145$	3295.24	+	1902.51i	1.88727	+	1.08962i
$146$		0			0
$147$	2182.94	+	2323.28i	1.22480	+	1.30354i
$148$		0			0
$149$	843.791	−	1461.49i	0.463933	−	0.803556i	−0.535219	−	0.844713i	$-0.679771\pi$
$149$	843.791	−	1461.49i	0.463933	−	0.803556i	0.999153	+	0.0411569i	$0.0131044\pi$
$150$		0			0
$151$	−2212.49	+	1277.38i	−1.19239	+	0.688424i	−0.958847	−	0.283924i	$-0.908364\pi$
$151$	−2212.49	+	1277.38i	−1.19239	+	0.688424i	−0.233538	+	0.972348i	$0.575030\pi$
$152$		0			0
$153$			107.900i			0.0570145i
$154$		0			0
$155$		−	2484.36i		−	1.28741i
$156$		0			0
$157$	−249.833	+	144.241i	−0.126999	+	0.0733230i	−0.562154	−	0.827033i	$-0.690027\pi$
$157$	−249.833	+	144.241i	−0.126999	+	0.0733230i	0.435155	+	0.900356i	$0.356694\pi$
$158$		0			0
$159$	−1484.39	+	2571.04i	−0.740375	+	1.28237i
$160$		0			0
$161$	−1809.82	−	782.875i	−0.885924	−	0.383225i
$162$		0			0
$163$	633.656	+	365.841i	0.304489	+	0.175797i	0.644458	−	0.764640i	$-0.277083\pi$
$163$	633.656	+	365.841i	0.304489	+	0.175797i	−0.339969	+	0.940437i	$0.610416\pi$
$164$		0			0
$165$	1091.83	+	1891.11i	0.515145	+	0.892257i
$166$		0			0
$167$	2121.63			0.983092			0.491546	−	0.870852i	$-0.336432\pi$
$167$	2121.63			0.983092			0.491546	+	0.870852i	$0.336432\pi$
$168$		0			0
$169$	1797.55			0.818185
$170$		0			0
$171$	−217.426	−	376.592i	−0.0972337	−	0.168414i
$172$		0			0
$173$	−2753.74	−	1589.87i	−1.21019	−	0.698703i	−0.247388	−	0.968916i	$-0.579572\pi$
$173$	−2753.74	−	1589.87i	−1.21019	−	0.698703i	−0.962800	+	0.270214i	$0.912906\pi$
$174$		0			0
$175$	−2971.61	−	4000.10i	−1.28361	−	1.72788i
$176$		0			0
$177$	284.018	−	491.933i	0.120611	−	0.208904i
$178$		0			0
$179$	1903.63	−	1099.06i	0.794881	−	0.458924i	−0.0467974	−	0.998904i	$-0.514902\pi$
$179$	1903.63	−	1099.06i	0.794881	−	0.458924i	0.841678	+	0.539980i	$0.181568\pi$
$180$		0			0
$181$			1266.98i			0.520299i	0.965568	+	0.260150i	$0.0837719\pi$
$181$			1266.98i			0.520299i	−0.965568	+	0.260150i	$0.916228\pi$
$182$		0			0
$183$		−	2874.93i		−	1.16132i
$184$		0			0
$185$	5446.73	−	3144.67i	2.16460	−	1.24973i
$186$		0			0
$187$	−10.7528	+	18.6243i	−0.00420492	+	0.00728313i
$188$		0			0
$189$	5537.10	−	641.457i	2.13103	−	0.246874i
$190$		0			0
$191$	−1508.17	−	870.745i	−0.571349	−	0.329869i	0.186339	−	0.982486i	$-0.440338\pi$
$191$	−1508.17	−	870.745i	−0.571349	−	0.329869i	−0.757688	+	0.652617i	$0.773671\pi$
$192$		0			0
$193$	639.307	+	1107.31i	0.238437	+	0.412985i	0.960266	−	0.279087i	$-0.0900316\pi$
$193$	639.307	+	1107.31i	0.238437	+	0.412985i	−0.721829	+	0.692071i	$0.756698\pi$
$194$		0			0
$195$	3687.45			1.35417
$196$		0			0
$197$	760.622			0.275087			0.137543	−	0.990496i	$-0.456079\pi$
$197$	760.622			0.275087			0.137543	+	0.990496i	$0.456079\pi$
$198$		0			0
$199$	2219.45	+	3844.19i	0.790615	+	1.36939i	0.925587	+	0.378536i	$0.123572\pi$
$199$	2219.45	+	3844.19i	0.790615	+	1.36939i	−0.134971	+	0.990850i	$0.543094\pi$
$200$		0			0
$201$	−4781.96	−	2760.86i	−1.67808	−	0.968837i
$202$		0			0
$203$	3526.36	−	408.519i	1.21922	−	0.141243i
$204$		0			0
$205$	−3194.08	+	5532.32i	−1.08822	+	1.88485i
$206$		0			0
$207$	−5475.55	+	3161.31i	−1.83853	+	1.06148i
$208$		0			0
$209$		−	86.6699i		−	0.0286846i
$210$		0			0
$211$			2084.90i			0.680239i	0.940382	+	0.340119i	$0.110467\pi$
$211$			2084.90i			0.680239i	−0.940382	+	0.340119i	$0.889533\pi$
$212$		0			0
$213$	−391.807	+	226.210i	−0.126038	+	0.0727683i
$214$		0			0
$215$	737.553	−	1277.48i	0.233957	−	0.405225i
$216$		0			0
$217$	−1382.20	−	1860.59i	−0.432397	−	0.582051i
$218$		0			0
$219$	−6199.93	−	3579.53i	−1.91303	−	1.10449i
$220$		0			0
$221$	18.1577	+	31.4500i	0.00552678	+	0.00957266i
$222$		0			0
$223$	−399.456			−0.119953			−0.0599766	−	0.998200i	$-0.519103\pi$
$223$	−399.456			−0.119953			−0.0599766	+	0.998200i	$0.519103\pi$
$224$		0			0
$225$	−15977.7			−4.73413
$226$		0			0
$227$	−1466.31	−	2539.72i	−0.428733	−	0.742587i	0.568028	−	0.823009i	$-0.307706\pi$
$227$	−1466.31	−	2539.72i	−0.428733	−	0.742587i	−0.996761	+	0.0804222i	$0.974373\pi$
$228$		0			0
$229$	1303.58	+	752.621i	0.376170	+	0.217182i	0.676150	−	0.736764i	$-0.263647\pi$
$229$	1303.58	+	752.621i	0.376170	+	0.217182i	−0.299981	+	0.953945i	$0.596980\pi$
$230$		0			0
$231$	1869.83	+	808.836i	0.532580	+	0.230379i
$232$		0			0
$233$	−467.095	+	809.032i	−0.131332	+	0.227474i	−0.924190	−	0.381932i	$-0.875259\pi$
$233$	−467.095	+	809.032i	−0.131332	+	0.227474i	0.792858	+	0.609406i	$0.208592\pi$
$234$		0			0
$235$	−2657.15	+	1534.10i	−0.737588	+	0.425846i
$236$		0			0
$237$			8924.39i			2.44600i
$238$		0			0
$239$			6859.78i			1.85658i	0.371860	+	0.928289i	$0.378720\pi$
$239$			6859.78i			1.85658i	−0.371860	+	0.928289i	$0.621280\pi$
$240$		0			0
$241$	1120.85	−	647.123i	0.299586	−	0.172966i	−0.342671	−	0.939456i	$-0.611331\pi$
$241$	1120.85	−	647.123i	0.299586	−	0.172966i	0.642257	+	0.766489i	$0.277998\pi$
$242$		0			0
$243$	1485.47	−	2572.91i	0.392152	−	0.679227i
$244$		0			0
$245$	−6518.84	−	1966.14i	−1.69989	−	0.512703i
$246$		0			0
$247$	−126.748	−	73.1778i	−0.0326508	−	0.0188510i
$248$		0			0
$249$	4947.08	+	8568.60i	1.25907	+	2.18077i
$250$		0			0
$251$	−178.426			−0.0448691			−0.0224345	−	0.999748i	$-0.507142\pi$
$251$	−178.426			−0.0448691			−0.0224345	+	0.999748i	$0.507142\pi$
$252$		0			0
$253$	−1260.16			−0.313144
$254$		0			0
$255$	−167.620	−	290.327i	−0.0411639	−	0.0712980i
$256$		0			0
$257$	5109.33	+	2949.87i	1.24012	+	0.715984i	0.969119	−	0.246595i	$-0.0793116\pi$
$257$	5109.33	+	2949.87i	1.24012	+	0.715984i	0.271002	+	0.962579i	$0.412645\pi$
$258$		0			0
$259$	2329.60	−	5385.47i	0.558896	−	1.29203i
$260$		0			0
$261$	5691.23	−	9857.49i	1.34972	−	2.33779i
$262$		0			0
$263$	1748.87	−	1009.71i	0.410037	−	0.236735i	−0.280768	−	0.959776i	$-0.590589\pi$
$263$	1748.87	−	1009.71i	0.410037	−	0.236735i	0.690806	+	0.723040i	$0.257256\pi$
$264$		0			0
$265$		−	6340.83i		−	1.46986i
$266$		0			0
$267$		−	6092.99i		−	1.39657i
$268$		0			0
$269$	−1601.94	+	924.883i	−0.363094	+	0.209632i	−0.670437	−	0.741966i	$-0.733893\pi$
$269$	−1601.94	+	924.883i	−0.363094	+	0.209632i	0.307343	+	0.951599i	$0.400560\pi$
$270$		0			0
$271$	3599.43	−	6234.39i	0.806826	−	1.39746i	−0.108226	−	0.994126i	$-0.534517\pi$
$271$	3599.43	−	6234.39i	0.806826	−	1.39746i	0.915052	−	0.403337i	$-0.132150\pi$
$272$		0			0
$273$	2761.61	−	2051.56i	0.612235	−	0.454820i
$274$		0			0
$275$	−2757.86	−	1592.25i	−0.604746	−	0.349150i
$276$		0			0
$277$	−2767.41	−	4793.30i	−0.600281	−	1.03972i	−0.992778	−	0.119964i	$-0.961722\pi$
$277$	−2767.41	−	4793.30i	−0.600281	−	1.03972i	0.392498	−	0.919753i	$-0.371611\pi$
$278$		0			0
$279$	−7431.80			−1.59473
$280$		0			0
$281$	752.919			0.159841			0.0799207	−	0.996801i	$-0.474533\pi$
$281$	752.919			0.159841			0.0799207	+	0.996801i	$0.474533\pi$
$282$		0			0
$283$	2811.79	+	4870.17i	0.590614	+	1.02297i	0.994150	+	0.108009i	$0.0344476\pi$
$283$	2811.79	+	4870.17i	0.590614	+	1.02297i	−0.403536	+	0.914964i	$0.632219\pi$
$284$		0			0
$285$	1170.05	+	675.531i	0.243186	+	0.140404i
$286$		0			0
$287$	685.854	+	5920.34i	0.141062	+	1.21765i
$288$		0			0
$289$	−2454.85	+	4251.92i	−0.499664	+	0.865443i
$290$		0			0
$291$	5668.34	−	3272.62i	1.14187	−	0.659259i
$292$		0			0
$293$			507.111i			0.101112i	0.998721	+	0.0505559i	$0.0160993\pi$
$293$			507.111i			0.101112i	−0.998721	+	0.0505559i	$0.983901\pi$
$294$		0			0
$295$			1213.23i			0.239447i
$296$		0			0
$297$	3084.96	−	1781.10i	0.602719	−	0.347980i
$298$		0			0
$299$	−1063.98	+	1842.87i	−0.205792	+	0.356442i
$300$		0			0
$301$	−158.372	−	1367.08i	−0.0303270	−	0.261784i
$302$		0			0
$303$	−5554.31	−	3206.78i	−1.05309	−	0.608003i
$304$		0			0
$305$	3070.19	+	5317.73i	0.576389	+	0.998335i
$306$		0			0
$307$	7402.96			1.37625			0.688126	−	0.725591i	$-0.258434\pi$
$307$	7402.96			1.37625			0.688126	+	0.725591i	$0.258434\pi$
$308$		0			0
$309$	12995.9			2.39259
$310$		0			0
$311$	2142.39	+	3710.72i	0.390623	+	0.676578i	0.992532	−	0.121986i	$-0.0389264\pi$
$311$	2142.39	+	3710.72i	0.390623	+	0.676578i	−0.601909	+	0.798565i	$0.705593\pi$
$312$		0			0
$313$	−1300.77	−	750.998i	−0.234900	−	0.135620i	0.377930	−	0.925834i	$-0.376636\pi$
$313$	−1300.77	−	750.998i	−0.234900	−	0.135620i	−0.612830	+	0.790214i	$0.709969\pi$
$314$		0			0
$315$	−17525.2	+	13019.2i	−3.13470	+	2.32872i
$316$		0			0
$317$	−3548.77	+	6146.66i	−0.628767	+	1.08906i	0.359033	+	0.933325i	$0.383107\pi$
$317$	−3548.77	+	6146.66i	−0.628767	+	1.08906i	−0.987800	+	0.155731i	$0.950227\pi$
$318$		0			0
$319$	1964.69	−	1134.31i	0.344832	−	0.199089i
$320$		0			0
$321$		−	289.749i		−	0.0503808i
$322$		0			0
$323$			13.3058i			0.00229211i
$324$		0			0
$325$	−4657.07	+	2688.76i	−0.794854	+	0.458909i
$326$		0			0
$327$	2222.05	−	3848.71i	0.375779	−	0.650869i
$328$		0			0
$329$	−1136.48	+	2627.26i	−0.190444	+	0.440260i
$330$		0			0
$331$	9956.85	+	5748.59i	1.65341	+	0.954595i	0.975657	+	0.219303i	$0.0703782\pi$
$331$	9956.85	+	5748.59i	1.65341	+	0.954595i	0.677750	+	0.735292i	$0.262955\pi$
$332$		0			0
$333$	−9407.08	−	16293.5i	−1.54806	−	2.68132i
$334$		0			0
$335$	11793.5			1.92343
$336$		0			0
$337$	6020.26			0.973128			0.486564	−	0.873645i	$-0.338250\pi$
$337$	6020.26			0.973128			0.486564	+	0.873645i	$0.338250\pi$
$338$		0			0
$339$	8648.60	+	14979.8i	1.38563	+	2.39998i
$340$		0			0
$341$	−1282.78	−	740.613i	−0.203714	−	0.117614i
$342$		0			0
$343$	−5975.98	+	2154.35i	−0.940737	+	0.339137i
$344$		0			0
$345$	9822.03	−	17012.3i	1.53275	−	2.65481i
$346$		0			0
$347$	−7699.30	+	4445.19i	−1.19112	+	0.687696i	−0.958562	−	0.284886i	$-0.908044\pi$
$347$	−7699.30	+	4445.19i	−1.19112	+	0.687696i	−0.232563	+	0.972581i	$0.574711\pi$
$348$		0			0
$349$		−	6396.47i		−	0.981075i	−0.871420	−	0.490538i	$-0.836800\pi$
$349$		−	6396.47i		−	0.981075i	0.871420	−	0.490538i	$-0.163200\pi$
$350$		0			0
$351$		−	6015.33i		−	0.914743i
$352$		0			0
$353$	−9935.50	+	5736.27i	−1.49806	+	0.864903i	−0.999997	−	0.00224128i	$-0.999287\pi$
$353$	−9935.50	+	5736.27i	−1.49806	+	0.864903i	−0.498058	+	0.867144i	$0.665953\pi$
$354$		0			0
$355$	483.147	−	836.836i	0.0722332	−	0.125112i
$356$		0			0
$357$	−287.062	−	124.174i	−0.0425572	−	0.0184090i
$358$		0			0
$359$	−169.226	−	97.7025i	−0.0248785	−	0.0143636i	0.487509	−	0.873118i	$-0.337906\pi$
$359$	−169.226	−	97.7025i	−0.0248785	−	0.0143636i	−0.512388	+	0.858754i	$0.671239\pi$
$360$		0			0
$361$	3402.69	+	5893.63i	0.496091	+	0.859255i
$362$		0			0
$363$	−11068.7			−1.60043
$364$		0			0
$365$	15290.6			2.19273
$366$		0			0
$367$	−5113.87	−	8857.49i	−0.727362	−	1.25983i	−0.957994	−	0.286787i	$-0.907413\pi$
$367$	−5113.87	−	8857.49i	−0.727362	−	1.25983i	0.230632	−	0.973041i	$-0.425921\pi$
$368$		0			0
$369$	16549.6	+	9554.89i	2.33479	+	1.34799i
$370$		0			0
$371$	−3527.80	−	4748.78i	−0.493677	−	0.664540i
$372$		0			0
$373$	3867.07	−	6697.97i	0.536808	−	0.929779i	−0.462265	−	0.886742i	$-0.652963\pi$
$373$	3867.07	−	6697.97i	0.536808	−	0.929779i	0.999073	−	0.0430373i	$-0.0137034\pi$
$374$		0			0
$375$	23018.4	−	13289.7i	3.16978	−	1.83007i
$376$		0			0
$377$		−	3830.93i		−	0.523350i
$378$		0			0
$379$			9739.99i			1.32008i	0.751231	+	0.660039i	$0.229460\pi$
$379$			9739.99i			1.32008i	−0.751231	+	0.660039i	$0.770540\pi$
$380$		0			0
$381$	−12742.0	+	7356.58i	−1.71336	+	0.989211i
$382$		0			0
$383$	−1262.59	+	2186.86i	−0.168447	+	0.291759i	−0.937874	−	0.346976i	$-0.887209\pi$
$383$	−1262.59	+	2186.86i	−0.168447	+	0.291759i	0.769427	+	0.638735i	$0.220542\pi$
$384$		0			0
$385$	−4322.38	+	500.736i	−0.572180	+	0.0662854i
$386$		0			0
$387$	−3821.49	−	2206.34i	−0.501957	−	0.289805i
$388$		0			0
$389$	1332.63	+	2308.18i	0.173694	+	0.300847i	0.939709	−	0.341976i	$-0.111096\pi$
$389$	1332.63	+	2308.18i	0.173694	+	0.300847i	−0.766014	+	0.642823i	$0.777763\pi$
$390$		0			0
$391$	193.462			0.0250225
$392$		0			0
$393$	−20421.3			−2.62117
$394$		0			0
$395$	−9530.53	−	16507.4i	−1.21401	−	2.10272i
$396$		0			0
$397$	6899.43	+	3983.39i	0.872223	+	0.503578i	0.868086	−	0.496413i	$-0.165350\pi$
$397$	6899.43	+	3983.39i	0.872223	+	0.503578i	0.00413668	+	0.999991i	$0.498683\pi$
$398$		0			0
$399$	1252.12	−	145.054i	0.157104	−	0.0182000i
$400$		0			0
$401$	−1303.95	+	2258.50i	−0.162384	+	0.281257i	−0.935723	−	0.352735i	$-0.885252\pi$
$401$	−1303.95	+	2258.50i	−0.162384	+	0.281257i	0.773339	+	0.633992i	$0.218585\pi$
$402$		0			0
$403$	−2166.17	+	1250.64i	−0.267754	+	0.154588i
$404$		0			0
$405$			23701.9i			2.90805i
$406$		0			0
$407$		−	3749.83i		−	0.456689i
$408$		0			0
$409$	−7616.13	+	4397.17i	−0.920766	+	0.531605i	−0.883879	−	0.467715i	$-0.845077\pi$
$409$	−7616.13	+	4397.17i	−0.920766	+	0.531605i	−0.0368867	+	0.999319i	$0.511744\pi$
$410$		0			0
$411$	11615.0	−	20117.8i	1.39398	−	2.41444i
$412$		0			0
$413$	674.996	+	908.614i	0.0804222	+	0.108257i
$414$		0			0
$415$	−18301.1	−	10566.2i	−2.16474	−	1.24981i
$416$		0			0
$417$	−12590.0	−	21806.6i	−1.47850	−	2.56084i
$418$		0			0
$419$	333.409			0.0388737			0.0194368	−	0.999811i	$-0.493813\pi$
$419$	333.409			0.0388737			0.0194368	+	0.999811i	$0.493813\pi$
$420$		0			0
$421$	−1186.07			−0.137305			−0.0686526	−	0.997641i	$-0.521870\pi$
$421$	−1186.07			−0.137305			−0.0686526	+	0.997641i	$0.521870\pi$
$422$		0			0
$423$	4589.17	+	7948.68i	0.527502	+	0.913659i
$424$		0			0
$425$	423.393	+	244.446i	0.0483237	+	0.0278997i
$426$		0			0
$427$	5257.91	+	2274.42i	0.595897	+	0.257768i
$428$		0			0
$429$	1099.27	−	1903.98i	0.123713	−	0.214278i
$430$		0			0
$431$	−10533.5	+	6081.53i	−1.17722	+	0.679669i	−0.955370	−	0.295412i	$-0.904543\pi$
$431$	−10533.5	+	6081.53i	−1.17722	+	0.679669i	−0.221851	+	0.975081i	$0.571210\pi$
$432$		0			0
$433$		−	14229.4i		−	1.57926i	−0.613582	−	0.789631i	$-0.710272\pi$
$433$		−	14229.4i		−	1.57926i	0.613582	−	0.789631i	$-0.289728\pi$
$434$		0			0
$435$			35364.7i			3.89795i
$436$		0			0
$437$	−675.220	+	389.838i	−0.0739134	+	0.0426739i
$438$		0			0
$439$	−3977.54	+	6889.31i	−0.432432	+	0.748995i	−0.997082	−	0.0763357i	$-0.975678\pi$
$439$	−3977.54	+	6889.31i	−0.432432	+	0.748995i	0.564650	+	0.825331i	$0.309011\pi$
$440$		0			0
$441$	−5881.59	+	19500.7i	−0.635092	+	2.10568i
$442$		0			0
$443$	−5552.88	−	3205.96i	−0.595542	−	0.343837i	0.171744	−	0.985142i	$-0.445060\pi$
$443$	−5552.88	−	3205.96i	−0.595542	−	0.343837i	−0.767286	+	0.641305i	$0.778393\pi$
$444$		0			0
$445$	6506.82	+	11270.1i	0.693153	+	1.20058i
$446$		0			0
$447$	15684.8			1.65965
$448$		0			0
$449$	−5918.99			−0.622125			−0.311063	−	0.950389i	$-0.600685\pi$
$449$	−5918.99			−0.622125			−0.311063	+	0.950389i	$0.600685\pi$
$450$		0			0
$451$	1904.38	+	3298.48i	0.198833	+	0.344389i
$452$		0			0
$453$	−20563.4	−	11872.3i	−2.13279	−	1.23137i
$454$		0			0
$455$	−2917.22	+	6743.92i	−0.300575	+	0.694856i
$456$		0			0
$457$	−434.306	+	752.241i	−0.0444551	+	0.0769986i	−0.887397	−	0.461006i	$-0.847489\pi$
$457$	−434.306	+	752.241i	−0.0444551	+	0.0769986i	0.842942	+	0.538005i	$0.180822\pi$
$458$		0			0
$459$	−473.611	+	273.439i	−0.0481618	+	0.0278062i
$460$		0			0
$461$			4645.89i			0.469372i	0.972071	+	0.234686i	$0.0754062\pi$
$461$			4645.89i			0.469372i	−0.972071	+	0.234686i	$0.924594\pi$
$462$		0			0
$463$			288.831i			0.0289916i	0.999895	+	0.0144958i	$0.00461433\pi$
$463$			288.831i			0.0289916i	−0.999895	+	0.0144958i	$0.995386\pi$
$464$		0			0
$465$	19996.7	−	11545.1i	1.99425	−	1.15138i
$466$		0			0
$467$	5658.72	−	9801.20i	0.560716	−	0.971189i	−0.436718	−	0.899599i	$-0.643859\pi$
$467$	5658.72	−	9801.20i	0.560716	−	0.971189i	0.997434	−	0.0715906i	$-0.0228075\pi$
$468$		0			0
$469$	8832.41	−	6561.46i	0.869601	−	0.646013i
$470$		0			0
$471$	−2322.01	−	1340.61i	−0.227161	−	0.131151i
$472$		0			0
$473$	−439.744	−	761.659i	−0.0427473	−	0.0740404i
$474$		0			0
$475$	−1970.30			−0.190323
$476$		0			0
$477$	−18968.2			−1.82074
$478$		0			0
$479$	−2162.11	−	3744.89i	−0.206241	−	0.357220i	0.744286	−	0.667860i	$-0.232790\pi$
$479$	−2162.11	−	3744.89i	−0.206241	−	0.357220i	−0.950527	+	0.310641i	$0.899456\pi$
$480$		0			0
$481$	−5483.82	−	3166.09i	−0.519835	−	0.300127i
$482$		0			0
$483$	−2109.05	−	18205.4i	−0.198685	−	1.71507i
$484$		0			0
$485$	−6989.78	+	12106.7i	−0.654412	+	1.13347i
$486$		0			0
$487$	9942.41	−	5740.26i	0.925121	−	0.534119i	0.0398556	−	0.999205i	$-0.487310\pi$
$487$	9942.41	−	5740.26i	0.925121	−	0.534119i	0.885265	+	0.465087i	$0.153977\pi$
$488$		0			0
$489$			6800.44i			0.628888i
$490$		0			0
$491$		−	10052.9i		−	0.923994i	−0.886881	−	0.461997i	$-0.847133\pi$
$491$		−	10052.9i		−	0.923994i	0.886881	−	0.461997i	$-0.152867\pi$
$492$		0			0
$493$	−301.624	+	174.143i	−0.0275547	+	0.0159087i
$494$		0			0
$495$	−6975.94	+	12082.7i	−0.633425	+	1.09712i
$496$		0			0
$497$	−103.745	−	895.529i	−0.00936334	−	0.0808249i
$498$		0			0
$499$	−6238.96	−	3602.06i	−0.559708	−	0.323147i	0.193320	−	0.981136i	$-0.438074\pi$
$499$	−6238.96	−	3602.06i	−0.559708	−	0.323147i	−0.753028	+	0.657988i	$0.771408\pi$
$500$		0			0
$501$	9859.45	+	17077.1i	0.879217	+	1.52285i
$502$		0			0
$503$	6434.36			0.570366			0.285183	−	0.958473i	$-0.407946\pi$
$503$	6434.36			0.570366			0.285183	+	0.958473i	$0.407946\pi$
$504$		0			0
$505$	13698.3			1.20707
$506$		0			0
$507$	8353.44	+	14468.6i	0.731735	+	1.26740i
$508$		0			0
$509$	11271.3	+	6507.49i	0.981517	+	0.566679i	0.902728	−	0.430213i	$-0.141561\pi$
$509$	11271.3	+	6507.49i	0.981517	+	0.566679i	0.0787889	+	0.996891i	$0.474895\pi$
$510$		0			0
$511$	11451.5	−	8507.11i	0.991355	−	0.736463i
$512$		0			0
$513$	1101.99	−	1908.71i	0.0948426	−	0.164272i
$514$		0			0
$515$	−24038.3	+	13878.5i	−2.05681	+	1.18750i
$516$		0			0
$517$			1829.33i			0.155617i
$518$		0			0
$519$		−	29553.3i		−	2.49951i
$520$		0			0
$521$	4455.62	−	2572.46i	0.374673	−	0.216317i	−0.300825	−	0.953679i	$-0.597262\pi$
$521$	4455.62	−	2572.46i	0.374673	−	0.216317i	0.675498	+	0.737362i	$0.263929\pi$
$522$		0			0
$523$	−5598.83	+	9697.46i	−0.468106	+	0.810784i	−0.999336	−	0.0364439i	$-0.988397\pi$
$523$	−5598.83	+	9697.46i	−0.468106	+	0.810784i	0.531229	+	0.847228i	$0.321730\pi$
$524$		0			0
$525$	18387.5	−	42507.6i	1.52857	−	3.53368i
$526$		0			0
$527$	196.935	+	113.701i	0.0162783	+	0.00939826i
$528$		0			0
$529$	−415.363	−	719.430i	−0.0341385	−	0.0591296i
$530$		0			0
$531$	3629.30			0.296607
$532$		0			0
$533$	6431.67			0.522677
$534$		0			0
$535$	309.429	+	535.946i	0.0250052	+	0.0433102i
$536$		0			0
$537$	17692.8	+	10214.9i	1.42179	+	0.820868i
$538$		0			0
$539$	−2958.53	+	2779.82i	−0.236425	+	0.222144i
$540$		0			0
$541$	10913.7	−	18903.1i	0.867313	−	1.50223i	0.00258182	−	0.999997i	$-0.499178\pi$
$541$	10913.7	−	18903.1i	0.867313	−	1.50223i	0.864732	−	0.502234i	$-0.167488\pi$
$542$		0			0
$543$	−10198.0	+	5887.82i	−0.805964	+	0.465324i
$544$		0			0
$545$			9491.89i			0.746033i
$546$		0			0
$547$		−	20005.0i		−	1.56371i	−0.623459	−	0.781856i	$-0.714273\pi$
$547$		−	20005.0i		−	1.56371i	0.623459	−	0.781856i	$-0.285727\pi$
$548$		0			0
$549$	15907.6	−	9184.27i	1.23665	−	0.713980i
$550$		0			0
$551$	701.816	−	1215.58i	0.0542620	−	0.0939846i
$552$		0			0
$553$	−16321.7	−	7060.29i	−1.25510	−	0.542918i
$554$		0			0
$555$	50623.2	+	29227.3i	3.87178	+	2.23537i
$556$		0			0
$557$	3700.46	+	6409.38i	0.281496	+	0.487566i	0.971753	−	0.235998i	$-0.0758360\pi$
$557$	3700.46	+	6409.38i	0.281496	+	0.487566i	−0.690257	+	0.723564i	$0.742503\pi$
$558$		0			0
$559$	−1485.15			−0.112371
$560$		0			0
$561$	−199.877			−0.0150425
$562$		0			0
$563$	−540.397	−	935.995i	−0.0404529	−	0.0700666i	0.845090	−	0.534624i	$-0.179547\pi$
$563$	−540.397	−	935.995i	−0.0404529	−	0.0700666i	−0.885543	+	0.464557i	$0.846213\pi$
$564$		0			0
$565$	−31994.5	−	18472.0i	−2.38233	−	1.37544i
$566$		0			0
$567$	13186.9	+	17750.9i	0.976713	+	1.31476i
$568$		0			0
$569$	−2104.06	+	3644.35i	−0.155021	+	0.268504i	−0.933067	−	0.359703i	$-0.882878\pi$
$569$	−2104.06	+	3644.35i	−0.155021	+	0.268504i	0.778046	+	0.628208i	$0.216211\pi$
$570$		0			0
$571$	11155.9	−	6440.87i	0.817619	−	0.472052i	−0.0319758	−	0.999489i	$-0.510180\pi$
$571$	11155.9	−	6440.87i	0.817619	−	0.472052i	0.849595	+	0.527436i	$0.176847\pi$
$572$		0			0
$573$		−	16185.8i		−	1.18006i
$574$		0			0
$575$			28647.5i			2.07771i
$576$		0			0
$577$	11493.5	−	6635.80i	0.829259	−	0.478773i	−0.0243396	−	0.999704i	$-0.507748\pi$
$577$	11493.5	−	6635.80i	0.829259	−	0.478773i	0.853599	+	0.520931i	$0.174415\pi$
$578$		0			0
$579$	−5941.87	+	10291.6i	−0.426487	+	0.738697i
$580$		0			0
$581$	−19584.7	+	2268.83i	−1.39847	+	0.162009i
$582$		0			0
$583$	−3274.03	−	1890.26i	−0.232584	−	0.134283i
$584$		0			0
$585$	11780.0	+	20403.5i	0.832549	+	1.44202i
$586$		0			0
$587$	−16892.3			−1.18777			−0.593885	−	0.804550i	$-0.702407\pi$
$587$	−16892.3			−1.18777			−0.593885	+	0.804550i	$0.702407\pi$
$588$		0			0
$589$	−916.456			−0.0641119
$590$		0			0
$591$	3534.70	+	6122.28i	0.246021	+	0.426120i
$592$		0			0
$593$	9489.27	+	5478.63i	0.657129	+	0.379394i	0.791182	−	0.611580i	$-0.209466\pi$
$593$	9489.27	+	5478.63i	0.657129	+	0.379394i	−0.134053	+	0.990974i	$0.542799\pi$
$594$		0			0
$595$	663.583	−	76.8742i	0.0457214	−	0.00529670i
$596$		0			0
$597$	−20628.1	+	35728.9i	−1.41416	+	2.44939i
$598$		0			0
$599$	−10024.8	+	5787.82i	−0.683810	+	0.394798i	−0.801289	−	0.598277i	$-0.795852\pi$
$599$	−10024.8	+	5787.82i	−0.683810	+	0.394798i	0.117479	+	0.993075i	$0.462519\pi$
$600$		0			0
$601$		−	17612.6i		−	1.19540i	−0.801721	−	0.597699i	$-0.796082\pi$
$601$		−	17612.6i		−	1.19540i	0.801721	−	0.597699i	$-0.203918\pi$
$602$		0			0
$603$		−	35279.5i		−	2.38257i
$604$		0			0
$605$	20473.6	−	11820.5i	1.37582	−	0.794331i
$606$		0			0
$607$	−8181.52	+	14170.8i	−0.547080	+	0.947570i	0.451393	+	0.892325i	$0.350927\pi$
$607$	−8181.52	+	14170.8i	−0.547080	+	0.947570i	−0.998473	+	0.0552449i	$0.982406\pi$
$608$		0			0
$609$	19675.6	+	26485.4i	1.30919	+	1.76230i
$610$		0			0
$611$	2675.24	+	1544.55i	0.177134	+	0.102268i
$612$		0			0
$613$	−503.862	−	872.715i	−0.0331987	−	0.0575018i	0.848949	−	0.528475i	$-0.177236\pi$
$613$	−503.862	−	872.715i	−0.0331987	−	0.0575018i	−0.882147	+	0.470973i	$0.843903\pi$
$614$		0			0
$615$	−59373.2			−3.89294
$616$		0			0
$617$	−14986.9			−0.977876			−0.488938	−	0.872318i	$-0.662616\pi$
$617$	−14986.9			−0.977876			−0.488938	+	0.872318i	$0.662616\pi$
$618$		0			0
$619$	8834.12	+	15301.2i	0.573624	+	0.993546i	0.996190	+	0.0872139i	$0.0277964\pi$
$619$	8834.12	+	15301.2i	0.573624	+	0.993546i	−0.422565	+	0.906332i	$0.638870\pi$
$620$		0			0
$621$	−27752.1	−	16022.7i	−1.79332	−	1.03538i
$622$		0			0
$623$	11143.4	+	4820.30i	0.716613	+	0.309986i
$624$		0			0
$625$	−11568.3	+	20036.8i	−0.740369	+	1.28236i
$626$		0			0
$627$	697.610	−	402.766i	0.0444336	−	0.0256538i
$628$		0			0
$629$			575.684i			0.0364929i
$630$		0			0
$631$			3794.54i			0.239395i	0.992810	+	0.119698i	$0.0381925\pi$
$631$			3794.54i			0.239395i	−0.992810	+	0.119698i	$0.961807\pi$
$632$		0			0
$633$	−16781.5	+	9688.78i	−1.05372	+	0.608364i
$634$		0			0
$635$	15712.5	−	27214.8i	0.981937	−	1.70077i
$636$		0			0
$637$	1567.29	+	6673.69i	0.0974856	+	0.415104i
$638$		0			0
$639$	−2503.34	−	1445.30i	−0.154977	−	0.0894762i
$640$		0			0
$641$	−7125.73	−	12342.1i	−0.439079	−	0.760507i	0.558540	−	0.829478i	$-0.311362\pi$
$641$	−7125.73	−	12342.1i	−0.439079	−	0.760507i	−0.997619	+	0.0689710i	$0.978028\pi$
$642$		0			0
$643$	10643.5			0.652782			0.326391	−	0.945235i	$-0.394167\pi$
$643$	10643.5			0.652782			0.326391	+	0.945235i	$0.394167\pi$
$644$		0			0
$645$	13710.0			0.836946
$646$		0			0
$647$	−9474.19	−	16409.8i	−0.575686	−	0.997118i	−0.995967	−	0.0897234i	$-0.971402\pi$
$647$	−9474.19	−	16409.8i	−0.575686	−	0.997118i	0.420281	−	0.907394i	$-0.361932\pi$
$648$		0			0
$649$	626.442	+	361.676i	0.0378891	+	0.0218753i
$650$		0			0
$651$	8552.71	−	19771.8i	0.514911	−	1.19035i
$652$		0			0
$653$	−10673.1	+	18486.4i	−0.639620	+	1.10785i	0.345896	+	0.938273i	$0.387575\pi$
$653$	−10673.1	+	18486.4i	−0.639620	+	1.10785i	−0.985516	+	0.169582i	$0.945758\pi$
$654$		0			0
$655$	37773.1	−	21808.3i	2.25331	−	1.30095i
$656$		0			0
$657$		−	45740.8i		−	2.71616i
$658$		0			0
$659$		−	3245.68i		−	0.191857i	−0.995388	−	0.0959284i	$-0.969418\pi$
$659$		−	3245.68i		−	0.191857i	0.995388	−	0.0959284i	$-0.0305820\pi$
$660$		0			0
$661$	13847.6	−	7994.93i	0.814841	−	0.470449i	−0.0337929	−	0.999429i	$-0.510759\pi$
$661$	13847.6	−	7994.93i	0.814841	−	0.470449i	0.848634	+	0.528980i	$0.177425\pi$
$662$		0			0
$663$	−168.762	+	292.304i	−0.00988563	+	0.0171224i
$664$		0			0
$665$	−2161.12	+	1605.47i	−0.126022	+	0.0936200i
$666$		0			0
$667$	−17674.2	−	10204.2i	−1.02601	−	0.592367i
$668$		0			0
$669$	−1856.32	−	3215.24i	−0.107279	−	0.185812i
$670$		0			0
$671$	3661.02			0.210629
$672$		0			0
$673$	−3473.65			−0.198959			−0.0994796	−	0.995040i	$-0.531718\pi$
$673$	−3473.65			−0.198959			−0.0994796	+	0.995040i	$0.531718\pi$
$674$		0			0
$675$	−40490.4	−	70131.4i	−2.30885	−	3.99905i
$676$		0			0
$677$	23449.5	+	13538.6i	1.33122	+	0.768583i	0.985487	−	0.169748i	$-0.0542954\pi$
$677$	23449.5	+	13538.6i	1.33122	+	0.768583i	0.345737	+	0.938331i	$0.387629\pi$
$678$		0			0
$679$	1500.89	+	12955.8i	0.0848290	+	0.732249i
$680$		0			0
$681$	13628.2	−	23604.8i	0.766865	−	1.32825i
$682$		0			0
$683$	5728.91	−	3307.59i	0.320952	−	0.185302i	−0.330865	−	0.943678i	$-0.607340\pi$
$683$	5728.91	−	3307.59i	0.320952	−	0.185302i	0.651817	+	0.758376i	$0.274007\pi$
$684$		0			0
$685$			49615.5i			2.76746i
$686$		0			0
$687$			13990.1i			0.776936i
$688$		0			0
$689$	−5528.71	+	3192.00i	−0.305700	+	0.176496i
$690$		0			0
$691$	−7904.11	+	13690.3i	−0.435147	+	0.753697i	−0.997308	−	0.0733313i	$-0.976637\pi$
$691$	−7904.11	+	13690.3i	−0.435147	+	0.753697i	0.562161	+	0.827028i	$0.309970\pi$
$692$		0			0
$693$	1497.92	+	12930.1i	0.0821086	+	0.708766i
$694$		0			0
$695$	46575.3	+	26890.2i	2.54202	+	1.46763i
$696$		0			0
$697$	−292.365	−	506.391i	−0.0158882	−	0.0275192i
$698$		0			0
$699$	−8682.59			−0.469822
$700$		0			0
$701$	9851.70			0.530804			0.265402	−	0.964138i	$-0.414495\pi$
$701$	9851.70			0.530804			0.265402	+	0.964138i	$0.414495\pi$
$702$		0			0
$703$	−1160.04	−	2009.25i	−0.0622357	−	0.107795i
$704$		0			0
$705$	−24696.2	−	14258.3i	−1.31931	−	0.761702i
$706$		0			0
$707$	10259.0	−	7621.24i	0.545726	−	0.405412i
$708$		0			0
$709$	5383.62	−	9324.70i	0.285171	−	0.493930i	−0.687480	−	0.726204i	$-0.741283\pi$
$709$	5383.62	−	9324.70i	0.285171	−	0.493930i	0.972651	+	0.232273i	$0.0746163\pi$
$710$		0			0
$711$	−49380.7	+	28509.9i	−2.60467	+	1.50381i
$712$		0			0
$713$			13325.0i			0.699896i
$714$		0			0
$715$			4695.71i			0.245608i
$716$		0			0
$717$	−55214.7	+	31878.2i	−2.87591	+	1.66041i
$718$		0			0
$719$	−10369.0	+	17959.6i	−0.537827	+	0.931544i	0.461193	+	0.887300i	$0.347421\pi$
$719$	−10369.0	+	17959.6i	−0.537827	+	0.931544i	−0.999021	+	0.0442447i	$0.985912\pi$
$720$		0			0
$721$	−10281.3	+	23767.9i	−0.531063	+	1.22769i
$722$		0			0
$723$	10417.5	+	6014.52i	0.535864	+	0.309381i
$724$		0			0
$725$	−25786.7	−	44663.9i	−1.32096	−	2.28797i
$726$		0			0
$727$	−2738.00			−0.139679			−0.0698396	−	0.997558i	$-0.522249\pi$
$727$	−2738.00			−0.139679			−0.0698396	+	0.997558i	$0.522249\pi$
$728$		0			0
$729$	−4625.18			−0.234984
$730$		0			0
$731$	67.5106	+	116.932i	0.00341583	+	0.00591638i
$732$		0			0
$733$	14909.6	+	8608.06i	0.751294	+	0.433760i	0.826161	−	0.563434i	$-0.190520\pi$
$733$	14909.6	+	8608.06i	0.751294	+	0.433760i	−0.0748673	+	0.997194i	$0.523853\pi$
$734$		0			0
$735$	−14468.3	−	61607.4i	−0.726081	−	3.09173i
$736$		0			0
$737$	3515.77	−	6089.49i	0.175719	−	0.304354i
$738$		0			0
$739$	−4783.71	+	2761.87i	−0.238121	+	0.137479i	−0.614313	−	0.789063i	$-0.710567\pi$
$739$	−4783.71	+	2761.87i	−0.238121	+	0.137479i	0.376192	+	0.926542i	$0.377233\pi$
$740$		0			0
$741$		−	1360.26i		−	0.0674366i
$742$		0			0
$743$		−	20424.6i		−	1.00849i	−0.863561	−	0.504244i	$-0.831771\pi$
$743$		−	20424.6i		−	1.00849i	0.863561	−	0.504244i	$-0.168229\pi$
$744$		0			0
$745$	−29012.0	+	16750.1i	−1.42674	+	0.823726i
$746$		0			0
$747$	−31608.0	+	54746.6i	−1.54816	+	2.68149i
$748$		0			0
$749$	529.918	+	229.227i	0.0258515	+	0.0111826i
$750$		0			0
$751$	9321.70	+	5381.88i	0.452934	+	0.261502i	0.709069	−	0.705140i	$-0.249116\pi$
$751$	9321.70	+	5381.88i	0.452934	+	0.261502i	−0.256134	+	0.966641i	$0.582449\pi$
$752$		0			0
$753$	−829.166	−	1436.16i	−0.0401281	−	0.0695040i
$754$		0			0
$755$	50714.7			2.44463
$756$		0			0
$757$	8591.14			0.412484			0.206242	−	0.978501i	$-0.433877\pi$
$757$	8591.14			0.412484			0.206242	+	0.978501i	$0.433877\pi$
$758$		0			0
$759$	−5856.10	−	10143.1i	−0.280056	−	0.485072i
$760$		0			0
$761$	−25875.0	−	14938.9i	−1.23254	−	0.711610i	−0.264985	−	0.964252i	$-0.585367\pi$
$761$	−25875.0	−	14938.9i	−1.23254	−	0.711610i	−0.967560	+	0.252642i	$0.918700\pi$
$762$		0			0
$763$	5280.93	+	7108.68i	0.250567	+	0.337289i
$764$		0			0
$765$	1070.96	−	1854.96i	0.0506154	−	0.0876684i
$766$		0			0
$767$	1057.84	−	610.746i	0.0497999	−	0.0287520i
$768$		0			0
$769$			3093.61i			0.145070i	0.997366	+	0.0725348i	$0.0231089\pi$
$769$			3093.61i			0.145070i	−0.997366	+	0.0725348i	$0.976891\pi$
$770$		0			0
$771$			54833.6i			2.56133i
$772$		0			0
$773$	20812.1	−	12015.9i	0.968381	−	0.559095i	0.0696388	−	0.997572i	$-0.477815\pi$
$773$	20812.1	−	12015.9i	0.968381	−	0.559095i	0.898742	+	0.438477i	$0.144482\pi$
$774$		0			0
$775$	−16836.6	+	29161.9i	−0.780373	+	1.35164i
$776$		0			0
$777$	54173.8	−	6275.88i	2.50125	−	0.289763i
$778$		0			0
$779$	2040.82	+	1178.27i	0.0938638	+	0.0541923i
$780$		0			0
$781$	−288.062	−	498.939i	−0.0131981	−	0.0228597i
$782$		0			0
$783$	57690.5			2.63307
$784$		0			0
$785$	5726.67			0.260374
$786$		0			0
$787$	5796.60	+	10040.0i	0.262550	+	0.454749i	0.966919	−	0.255085i	$-0.0821033\pi$
$787$	5796.60	+	10040.0i	0.262550	+	0.454749i	−0.704369	+	0.709834i	$0.748770\pi$
$788$		0			0
$789$	16254.4	+	9384.48i	0.733425	+	0.423443i
$790$		0			0
$791$	−34238.5	+	3966.43i	−1.53904	+	0.178293i
$792$		0			0
$793$	3091.10	−	5353.94i	0.138421	−	0.239753i
$794$		0			0
$795$	51037.6	−	29466.6i	2.27688	−	1.31456i
$796$		0			0
$797$		−	11467.4i		−	0.509658i	−0.966986	−	0.254829i	$-0.917981\pi$
$797$		−	11467.4i		−	0.509658i	0.966986	−	0.254829i	$-0.0820192\pi$
$798$		0			0
$799$		−	280.843i		−	0.0124349i
$800$		0			0
$801$	33713.9	−	19464.7i	1.48717	−	0.858617i
$802$		0			0
$803$	4558.28	−	7895.18i	0.200322	−	0.346967i
$804$		0			0
$805$	23343.0	+	31422.1i	1.02203	+	1.37576i
$806$		0			0
$807$	−14888.9	−	8596.09i	−0.649458	−	0.374965i
$808$		0			0
$809$	−18190.8	−	31507.5i	−0.790551	−	1.36928i	−0.925626	−	0.378440i	$-0.876461\pi$
$809$	−18190.8	−	31507.5i	−0.790551	−	1.36928i	0.135074	−	0.990835i	$-0.456873\pi$
$810$		0			0
$811$	−35717.1			−1.54648			−0.773241	−	0.634113i	$-0.781366\pi$
$811$	−35717.1			−1.54648			−0.773241	+	0.634113i	$0.781366\pi$
$812$		0			0
$813$	66907.9			2.88630
$814$		0			0
$815$	−7262.31	−	12578.7i	−0.312132	−	0.540629i
$816$		0			0
$817$	−471.250	−	272.076i	−0.0201798	−	0.0116508i
$818$		0			0
$819$	20174.0	+	8726.67i	0.860728	+	0.372326i
$820$		0			0
$821$	15659.5	−	27123.1i	0.665678	−	1.15299i	−0.313423	−	0.949614i	$-0.601476\pi$
$821$	15659.5	−	27123.1i	0.665678	−	1.15299i	0.979101	−	0.203375i	$-0.0651909\pi$
$822$		0			0
$823$	−3616.81	+	2088.17i	−0.153189	+	0.0884435i	−0.574635	−	0.818410i	$-0.694856\pi$
$823$	−3616.81	+	2088.17i	−0.153189	+	0.0884435i	0.421446	+	0.906853i	$0.361523\pi$
$824$		0			0
$825$		−	29597.5i		−	1.24903i
$826$		0			0
$827$		−	7930.70i		−	0.333467i	−0.986002	−	0.166734i	$-0.946678\pi$
$827$		−	7930.70i		−	0.333467i	0.986002	−	0.166734i	$-0.0533220\pi$
$828$		0			0
$829$	2465.72	−	1423.58i	0.103303	−	0.0596419i	−0.447459	−	0.894305i	$-0.647671\pi$
$829$	2465.72	−	1423.58i	0.103303	−	0.0596419i	0.550761	+	0.834663i	$0.314337\pi$
$830$		0			0
$831$	25721.0	−	44550.1i	1.07371	−	1.85972i
$832$		0			0
$833$	454.201	−	426.765i	0.0188921	−	0.0177509i
$834$		0			0
$835$	−36473.9	−	21058.2i	−1.51165	−	0.872753i
$836$		0			0
$837$	−18833.6	−	32620.7i	−0.777758	−	1.34712i
$838$		0			0
$839$	25648.9			1.05542			0.527711	−	0.849424i	$-0.323051\pi$
$839$	25648.9			1.05542			0.527711	+	0.849424i	$0.323051\pi$
$840$		0			0
$841$	12351.8			0.506449
$842$		0			0
$843$	3498.91	+	6060.29i	0.142952	+	0.247601i
$844$		0			0
$845$	−30902.6	−	17841.6i	−1.25808	−	0.726355i
$846$		0			0
$847$	8756.69	−	20243.4i	0.355234	−	0.821216i
$848$		0			0
$849$	−26133.5	+	45264.5i	−1.05642	+	1.82977i
$850$		0			0
$851$	−29213.9	+	16866.6i	−1.17678	+	0.679413i
$852$		0			0
$853$			1004.85i			0.0403348i	0.999797	+	0.0201674i	$0.00641991\pi$
$853$			1004.85i			0.0403348i	−0.999797	+	0.0201674i	$0.993580\pi$
$854$		0			0
$855$			8632.23i			0.345282i
$856$		0			0
$857$	30363.3	−	17530.2i	1.21026	−	0.698742i	0.247441	−	0.968903i	$-0.420410\pi$
$857$	30363.3	−	17530.2i	1.21026	−	0.698742i	0.962815	+	0.270161i	$0.0870770\pi$
$858$		0			0
$859$	4503.21	−	7799.79i	0.178868	−	0.309808i	−0.762625	−	0.646841i	$-0.776090\pi$
$859$	4503.21	−	7799.79i	0.178868	−	0.309808i	0.941493	+	0.337032i	$0.109423\pi$
$860$		0			0
$861$	−44465.8	+	33033.0i	−1.76004	+	1.30750i
$862$		0			0
$863$	37701.0	+	21766.7i	1.48709	+	0.858570i	0.999892	−	0.0147228i	$-0.00468658\pi$
$863$	37701.0	+	21766.7i	1.48709	+	0.858570i	0.487195	+	0.873293i	$0.338020\pi$
$864$		0			0
$865$	31560.5	+	54664.4i	1.24057	+	2.14872i
$866$		0			0
$867$	−45631.9			−1.78748
$868$		0			0
$869$	−11364.6			−0.443633
$870$		0			0
$871$	−5936.91	−	10283.0i	−0.230958	−	0.400031i
$872$		0			0
$873$	36216.3	+	20909.5i	1.40405	+	0.810628i
$874$		0			0
$875$	6094.93	+	52611.8i	0.235481	+	2.03269i
$876$		0			0
$877$	11768.8	−	20384.1i	0.453140	−	0.784862i	−0.545439	−	0.838151i	$-0.683637\pi$
$877$	11768.8	−	20384.1i	0.453140	−	0.784862i	0.998579	+	0.0532888i	$0.0169704\pi$
$878$		0			0
$879$	−4081.76	+	2356.61i	−0.156626	+	0.0904281i
$880$		0			0
$881$			41234.7i			1.57688i	0.615111	+	0.788440i	$0.289111\pi$
$881$			41234.7i			1.57688i	−0.615111	+	0.788440i	$0.710889\pi$
$882$		0			0
$883$		−	2981.04i		−	0.113613i	−0.998385	−	0.0568063i	$-0.981908\pi$
$883$		−	2981.04i		−	0.113613i	0.998385	−	0.0568063i	$-0.0180917\pi$
$884$		0			0
$885$	−9765.35	+	5638.03i	−0.370914	+	0.214147i
$886$		0			0
$887$	11920.6	−	20647.1i	0.451245	−	0.781580i	−0.547218	−	0.836990i	$-0.684313\pi$
$887$	11920.6	−	20647.1i	0.451245	−	0.781580i	0.998464	+	0.0554102i	$0.0176466\pi$
$888$		0			0
$889$	−3373.88	−	29123.6i	−0.127285	−	1.09873i
$890$		0			0
$891$	12238.3	+	7065.79i	0.460156	+	0.265671i
$892$		0			0
$893$	565.916	+	980.195i	0.0212068	+	0.0367312i
$894$		0			0
$895$	−43634.8			−1.62967
$896$		0			0
$897$	−19777.8			−0.736191
$898$		0			0
$899$	−11994.4	−	20774.8i	−0.444977	−	0.770722i
$900$		0			0
$901$	502.638	+	290.198i	0.0185852	+	0.0107302i
$902$		0			0
$903$	10267.7	−	7627.72i	0.378392	−	0.281101i
$904$		0			0
$905$	12575.4	−	21781.3i	0.461902	−	0.800038i
$906$		0			0
$907$	29395.0	−	16971.2i	1.07612	−	0.621301i	0.146276	−	0.989244i	$-0.453271\pi$
$907$	29395.0	−	16971.2i	1.07612	−	0.621301i	0.929848	+	0.367943i	$0.119938\pi$
$908$		0			0
$909$		−	40977.7i		−	1.49521i
$910$		0			0
$911$			18217.3i			0.662531i	0.943538	+	0.331265i	$0.107476\pi$
$911$			18217.3i			0.662531i	−0.943538	+	0.331265i	$0.892524\pi$
$912$		0			0
$913$	−10911.5	+	6299.76i	−0.395529	+	0.228359i
$914$		0			0
$915$	−28535.1	+	49424.2i	−1.03097	+	1.78570i
$916$		0			0
$917$	16155.8	−	37348.3i	0.581800	−	1.34498i
$918$		0			0
$919$	35386.4	+	20430.3i	1.27017	+	0.733335i	0.975021	−	0.222115i	$-0.0712960\pi$
$919$	35386.4	+	20430.3i	1.27017	+	0.733335i	0.295153	+	0.955450i	$0.404629\pi$
$920$		0			0
$921$	34402.5	+	59586.8i	1.23084	+	2.13187i
$922$		0			0
$923$	−972.875			−0.0346940
$924$		0			0
$925$	−85246.2			−3.03014
$926$		0			0
$927$	41516.7	+	71909.1i	1.47097	+	2.54779i
$928$		0			0
$929$	26497.2	+	15298.2i	0.935786	+	0.540276i	0.888637	−	0.458612i	$-0.151653\pi$
$929$	26497.2	+	15298.2i	0.935786	+	0.540276i	0.0471490	+	0.998888i	$0.484986\pi$
$930$		0			0
$931$	−725.291	+	2404.74i	−0.0255322	+	0.0846531i
$932$		0			0
$933$	−19911.9	+	34488.4i	−0.698698	+	1.21018i
$934$		0			0
$935$	369.711	−	213.453i	0.0129314	−	0.00746594i
$936$		0			0
$937$		−	28125.2i		−	0.980586i	−0.871558	−	0.490293i	$-0.836890\pi$
$937$		−	28125.2i		−	0.980586i	0.871558	−	0.490293i	$-0.163110\pi$
$938$		0			0
$939$		−	13959.9i		−	0.485159i
$940$		0			0
$941$	−26065.7	+	15049.0i	−0.902994	+	0.521344i	−0.878170	−	0.478348i	$-0.841236\pi$
$941$	−26065.7	+	15049.0i	−0.902994	+	0.521344i	−0.0248237	+	0.999692i	$0.507902\pi$
$942$		0			0
$943$	17131.7	−	29672.9i	0.591605	−	1.02469i
$944$		0			0
$945$	−101558.	−	43930.8i	−3.49595	−	1.51224i
$946$		0			0
$947$	−35865.6	−	20707.0i	−1.23070	−	0.710547i	−0.263526	−	0.964652i	$-0.584886\pi$
$947$	−35865.6	−	20707.0i	−1.23070	−	0.710547i	−0.967176	+	0.254106i	$0.918219\pi$
$948$		0			0
$949$	−7697.36	−	13332.2i	−0.263295	−	0.456040i
$950$		0			0
$951$	−65966.3			−2.24932
$952$		0			0
$953$	−33248.1			−1.13013			−0.565063	−	0.825047i	$-0.691148\pi$
$953$	−33248.1			−1.13013			−0.565063	+	0.825047i	$0.691148\pi$
$954$		0			0
$955$	17285.1	+	29938.8i	0.585690	+	1.01445i
$956$		0			0
$957$	18260.3	+	10542.6i	0.616794	+	0.356106i
$958$		0			0
$959$	27604.2	+	37158.1i	0.929495	+	1.25120i
$960$		0			0
$961$	7064.18	−	12235.5i	0.237125	−	0.410712i
$962$		0			0
$963$	1603.25	−	925.635i	0.0536489	−	0.0309742i
$964$		0			0
$965$		−	25381.8i		−	0.846702i
$966$		0			0
$967$		−	39769.8i		−	1.32255i	−0.750142	−	0.661277i	$-0.770015\pi$
$967$		−	39769.8i		−	1.32255i	0.750142	−	0.661277i	$-0.229985\pi$
$968$		0			0
$969$	−107.099	+	61.8335i	−0.00355058	+	0.00204993i
$970$		0			0
$971$	13406.3	−	23220.3i	0.443077	−	0.767431i	−0.554839	−	0.831957i	$-0.687220\pi$
$971$	13406.3	−	23220.3i	0.443077	−	0.767431i	0.997916	+	0.0645263i	$0.0205536\pi$
$972$		0			0
$973$	49841.9	−	5774.05i	1.64220	−	0.190244i
$974$		0			0
$975$	−43283.9	−	24990.0i	−1.42174	−	0.820841i
$976$		0			0
$977$	−7910.98	−	13702.2i	−0.259053	−	0.448693i	0.706935	−	0.707278i	$-0.250077\pi$
$977$	−7910.98	−	13702.2i	−0.259053	−	0.448693i	−0.965988	+	0.258585i	$0.916744\pi$
$978$		0			0
$979$	7759.00			0.253298
$980$		0			0
$981$	28394.4			0.924120
$982$		0			0
$983$	−17516.6	−	30339.7i	−0.568356	−	0.984421i	−0.996729	−	0.0808192i	$-0.974246\pi$
$983$	−17516.6	−	30339.7i	−0.568356	−	0.984421i	0.428373	−	0.903602i	$-0.359087\pi$
$984$		0			0
$985$	−13076.2	−	7549.55i	−0.422987	−	0.244212i
$986$		0			0
$987$	−26428.3	+	3061.64i	−0.852301	+	0.0987367i
$988$		0			0
$989$	−3955.91	+	6851.83i	−0.127190	+	0.220299i
$990$		0			0
$991$	−22133.0	+	12778.5i	−0.709463	+	0.409608i	−0.810862	−	0.585237i	$-0.801001\pi$
$991$	−22133.0	+	12778.5i	−0.709463	+	0.409608i	0.101399	+	0.994846i	$0.467668\pi$
$992$		0			0
$993$			106858.i			3.41493i
$994$		0			0
$995$		−	88116.4i		−	2.80752i
$996$		0			0
$997$	−25633.4	+	14799.4i	−0.814259	+	0.470113i	−0.848433	−	0.529303i	$-0.822453\pi$
$997$	−25633.4	+	14799.4i	−0.814259	+	0.470113i	0.0341734	+	0.999416i	$0.489120\pi$
$998$		0			0
$999$	47678.6	−	82581.7i	1.50999	−	2.61538i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.4.p.g.47.3	yes	6
4.3	odd	2		112.4.p.f.47.1	yes	6
7.2	even	3		784.4.f.g.783.2		6
7.3	odd	6		112.4.p.f.31.1	✓	6
7.5	odd	6		784.4.f.h.783.5		6
8.3	odd	2		448.4.p.g.383.3		6
8.5	even	2		448.4.p.f.383.1		6
28.3	even	6	inner	112.4.p.g.31.3	yes	6
28.19	even	6		784.4.f.g.783.1		6
28.23	odd	6		784.4.f.h.783.6		6
56.3	even	6		448.4.p.f.255.1		6
56.45	odd	6		448.4.p.g.255.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.p.f.31.1	✓	6	7.3	odd	6
112.4.p.f.47.1	yes	6	4.3	odd	2
112.4.p.g.31.3	yes	6	28.3	even	6	inner
112.4.p.g.47.3	yes	6	1.1	even	1	trivial
448.4.p.f.255.1		6	56.3	even	6
448.4.p.f.383.1		6	8.5	even	2
448.4.p.g.255.3		6	56.45	odd	6
448.4.p.g.383.3		6	8.3	odd	2
784.4.f.g.783.1		6	28.19	even	6
784.4.f.g.783.2		6	7.2	even	3
784.4.f.h.783.5		6	7.5	odd	6
784.4.f.h.783.6		6	28.23	odd	6

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.64712 + 8.04905i) q^{3} +(-17.1915 - 9.92549i) q^{5} +(-18.3972 + 2.13127i) q^{7} +(-29.6915 + 51.4271i) q^{9} +O(q^{10})\)	\(q+(4.64712 + 8.04905i) q^{3} +(-17.1915 - 9.92549i) q^{5} +(-18.3972 + 2.13127i) q^{7} +(-29.6915 + 51.4271i) q^{9} +(-10.2499 + 5.91778i) q^{11} +19.9862i q^{13} -184.500i q^{15} +(1.57359 - 0.908512i) q^{17} +(-3.66142 + 6.34176i) q^{19} +(-102.649 - 138.176i) q^{21} +(92.2074 + 53.2360i) q^{23} +(134.531 + 233.014i) q^{25} -300.975 q^{27} -191.679 q^{29} +(62.5752 + 108.383i) q^{31} +(-95.2651 - 55.0013i) q^{33} +(337.429 + 145.962i) q^{35} +(-158.414 + 274.381i) q^{37} +(-160.870 + 92.8782i) q^{39} -321.806i q^{41} +74.3089i q^{43} +(1020.88 - 589.405i) q^{45} +(77.2810 - 133.855i) q^{47} +(333.915 - 78.4187i) q^{49} +(14.6253 + 8.44393i) q^{51} +(159.711 + 276.627i) q^{53} +234.948 q^{55} -68.0602 q^{57} +(-30.5584 - 52.9288i) q^{59} +(-267.882 - 154.662i) q^{61} +(436.635 - 1009.40i) q^{63} +(198.373 - 343.592i) q^{65} +(-514.507 + 297.051i) q^{67} +989.576i q^{69} +48.6774i q^{71} +(-667.072 + 385.134i) q^{73} +(-1250.36 + 2165.69i) q^{75} +(175.957 - 130.716i) q^{77} +(831.564 + 480.103i) q^{79} +(-596.996 - 1034.03i) q^{81} +1064.55 q^{83} -36.0697 q^{85} +(-890.755 - 1542.83i) q^{87} +(-567.737 - 327.783i) q^{89} +(-42.5959 - 367.690i) q^{91} +(-581.589 + 1007.34i) q^{93} +(125.890 - 72.6828i) q^{95} -704.225i q^{97} -702.830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9}+O(q^{10}) \) 6 * q + 7 * q^3 - 3 * q^5 - 52 * q^7 - 78 * q^9	\( 6 q + 7 q^{3} - 3 q^{5} - 52 q^{7} - 78 q^{9} - 99 q^{11} + 9 q^{17} - 143 q^{19} - 15 q^{21} + 15 q^{23} + 306 q^{25} - 362 q^{27} - 348 q^{29} - 205 q^{31} - 471 q^{33} + 1185 q^{35} - 249 q^{37} - 288 q^{39} + 2118 q^{45} - 75 q^{47} + 702 q^{49} + 2505 q^{51} - 645 q^{53} - 918 q^{55} - 6 q^{57} - 321 q^{59} - 1707 q^{61} + 1502 q^{63} - 612 q^{65} - 447 q^{67} + 705 q^{73} - 4138 q^{75} + 555 q^{77} + 3447 q^{79} + 225 q^{81} - 24 q^{83} + 3786 q^{85} - 3642 q^{87} - 2607 q^{89} + 2448 q^{91} - 2991 q^{93} - 2085 q^{95}+O(q^{100}) \) 6 * q + 7 * q^3 - 3 * q^5 - 52 * q^7 - 78 * q^9 - 99 * q^11 + 9 * q^17 - 143 * q^19 - 15 * q^21 + 15 * q^23 + 306 * q^25 - 362 * q^27 - 348 * q^29 - 205 * q^31 - 471 * q^33 + 1185 * q^35 - 249 * q^37 - 288 * q^39 + 2118 * q^45 - 75 * q^47 + 702 * q^49 + 2505 * q^51 - 645 * q^53 - 918 * q^55 - 6 * q^57 - 321 * q^59 - 1707 * q^61 + 1502 * q^63 - 612 * q^65 - 447 * q^67 + 705 * q^73 - 4138 * q^75 + 555 * q^77 + 3447 * q^79 + 225 * q^81 - 24 * q^83 + 3786 * q^85 - 3642 * q^87 - 2607 * q^89 + 2448 * q^91 - 2991 * q^93 - 2085 * q^95

Introduction

L-functions

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