LMFDB - Embedded newform 112.6.i.b.81.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{130})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 130x^{2} + 16900 $ x^4 + 130*x^2 + 16900
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.2
Root			$-5.70088 + 9.87421i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.6.i.b.65.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(7.90175 - 13.6862i) q^{3} +(10.5000 + 18.1865i) q^{5} +(-126.411 + 28.7642i) q^{7} +(-3.37544 - 5.84643i) q^{9} +(-312.937 + 542.023i) q^{11} -206.821 q^{13} +331.874 q^{15} +(-530.732 + 919.254i) q^{17} +(-941.989 - 1631.57i) q^{19} +(-605.191 + 1957.37i) q^{21} +(1858.68 + 3219.34i) q^{23} +(1342.00 - 2324.41i) q^{25} +3733.56 q^{27} -123.747 q^{29} +(-4554.63 + 7888.85i) q^{31} +(4945.50 + 8565.86i) q^{33} +(-1850.43 - 1996.95i) q^{35} +(3014.36 + 5221.03i) q^{37} +(-1634.25 + 2830.60i) q^{39} -17201.9 q^{41} -5401.98 q^{43} +(70.8843 - 122.775i) q^{45} +(937.621 + 1624.01i) q^{47} +(15152.2 - 7272.19i) q^{49} +(8387.42 + 14527.4i) q^{51} +(-9353.62 + 16200.9i) q^{53} -13143.3 q^{55} -29773.5 q^{57} +(-1267.39 + 2195.18i) q^{59} +(1047.35 + 1814.07i) q^{61} +(594.859 + 641.959i) q^{63} +(-2171.62 - 3761.36i) q^{65} +(29310.4 - 50767.1i) q^{67} +58747.5 q^{69} +31279.5 q^{71} +(3575.24 - 6192.49i) q^{73} +(-21208.3 - 36733.9i) q^{75} +(23967.7 - 77518.7i) q^{77} +(1489.91 + 2580.59i) q^{79} +(30321.9 - 52519.1i) q^{81} -45954.6 q^{83} -22290.7 q^{85} +(-977.821 + 1693.64i) q^{87} +(-49520.0 - 85771.1i) q^{89} +(26144.4 - 5949.04i) q^{91} +(71979.2 + 124672. i) q^{93} +(19781.8 - 34263.0i) q^{95} -115548. q^{97} +4225.20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{3} + 42 q^{5} - 232 q^{7} - 652 q^{9} - 294 q^{11} - 280 q^{13} - 588 q^{15} - 1302 q^{17} - 1442 q^{19} + 5150 q^{21} + 2646 q^{23} + 5368 q^{25} + 31444 q^{27} + 3336 q^{29} - 14798 q^{31}+ \cdots - 419832 q^{99}+O(q^{100}) $ 4 * q - 14 * q^3 + 42 * q^5 - 232 * q^7 - 652 * q^9 - 294 * q^11 - 280 * q^13 - 588 * q^15 - 1302 * q^17 - 1442 * q^19 + 5150 * q^21 + 2646 * q^23 + 5368 * q^25 + 31444 * q^27 + 3336 * q^29 - 14798 * q^31 + 19782 * q^33 + 1218 * q^35 - 5182 * q^37 - 5260 * q^39 - 10248 * q^41 + 9040 * q^43 + 13692 * q^45 - 14994 * q^47 - 2876 * q^49 + 9606 * q^51 - 24006 * q^53 - 12348 * q^55 - 85892 * q^57 + 38850 * q^59 + 23618 * q^61 - 149948 * q^63 - 2940 * q^65 + 32002 * q^67 + 181356 * q^69 + 178752 * q^71 + 47138 * q^73 + 37576 * q^75 - 22890 * q^77 - 40970 * q^79 - 139858 * q^81 - 136752 * q^83 - 54684 * q^85 - 55356 * q^87 - 123102 * q^89 + 53680 * q^91 - 25586 * q^93 + 30282 * q^95 - 87304 * q^97 - 419832 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	7.90175	−	13.6862i	0.506898	−	0.877973i	−0.493070	−	0.869989i	$-0.664125\pi$
$3$	7.90175	−	13.6862i	0.506898	−	0.877973i	0.999968	−	0.00798326i	$-0.00254118\pi$
$4$		0			0
$5$	10.5000	+	18.1865i	0.187830	+	0.325331i	0.944526	−	0.328436i	$-0.106521\pi$
$5$	10.5000	+	18.1865i	0.187830	+	0.325331i	−0.756697	+	0.653766i	$0.773188\pi$
$6$		0			0
$7$	−126.411	+	28.7642i	−0.975075	+	0.221874i
$7$	−126.411	+	28.7642i	−0.975075	+	0.221874i
$8$		0			0
$9$	−3.37544	−	5.84643i	−0.0138907	−	0.0240594i
$10$		0			0
$11$	−312.937	+	542.023i	−0.779785	+	1.35063i	0.152280	+	0.988337i	$0.451338\pi$
$11$	−312.937	+	542.023i	−0.779785	+	1.35063i	−0.932065	+	0.362290i	$0.881995\pi$
$12$		0			0
$13$	−206.821			−0.339419			−0.169710	−	0.985494i	$-0.554283\pi$
$13$	−206.821			−0.339419			−0.169710	+	0.985494i	$0.554283\pi$
$14$		0			0
$15$	331.874			0.380842
$16$		0			0
$17$	−530.732	+	919.254i	−0.445402	+	0.771460i	−0.998080	−	0.0619354i	$-0.980273\pi$
$17$	−530.732	+	919.254i	−0.445402	+	0.771460i	0.552678	+	0.833395i	$0.313606\pi$
$18$		0			0
$19$	−941.989	−	1631.57i	−0.598635	−	1.03687i	−0.993023	−	0.117922i	$-0.962377\pi$
$19$	−941.989	−	1631.57i	−0.598635	−	1.03687i	0.394388	−	0.918944i	$-0.370957\pi$
$20$		0			0
$21$	−605.191	+	1957.37i	−0.299464	+	0.968557i
$22$		0			0
$23$	1858.68	+	3219.34i	0.732632	+	1.26896i	0.955754	+	0.294166i	$0.0950418\pi$
$23$	1858.68	+	3219.34i	0.732632	+	1.26896i	−0.223122	+	0.974790i	$0.571625\pi$
$24$		0			0
$25$	1342.00	−	2324.41i	0.429440	−	0.743812i
$26$		0			0
$27$	3733.56			0.985631
$28$		0			0
$29$	−123.747			−0.0273238			−0.0136619	−	0.999907i	$-0.504349\pi$
$29$	−123.747			−0.0273238			−0.0136619	+	0.999907i	$0.504349\pi$
$30$		0			0
$31$	−4554.63	+	7888.85i	−0.851234	+	1.47438i	0.0288611	+	0.999583i	$0.490812\pi$
$31$	−4554.63	+	7888.85i	−0.851234	+	1.47438i	−0.880095	+	0.474797i	$0.842521\pi$
$32$		0			0
$33$	4945.50	+	8565.86i	0.790543	+	1.36926i
$34$		0			0
$35$	−1850.43	−	1996.95i	−0.255331	−	0.275547i
$36$		0			0
$37$	3014.36	+	5221.03i	0.361986	+	0.626977i	0.988288	−	0.152603i	$-0.0487656\pi$
$37$	3014.36	+	5221.03i	0.361986	+	0.626977i	−0.626302	+	0.779581i	$0.715432\pi$
$38$		0			0
$39$	−1634.25	+	2830.60i	−0.172051	+	0.298001i
$40$		0			0
$41$	−17201.9			−1.59814			−0.799071	−	0.601236i	$-0.794675\pi$
$41$	−17201.9			−1.59814			−0.799071	+	0.601236i	$0.794675\pi$
$42$		0			0
$43$	−5401.98			−0.445535			−0.222767	−	0.974872i	$-0.571509\pi$
$43$	−5401.98			−0.445535			−0.222767	+	0.974872i	$0.571509\pi$
$44$		0			0
$45$	70.8843	−	122.775i	0.00521817	−	0.00903814i
$46$		0			0
$47$	937.621	+	1624.01i	0.0619131	+	0.107237i	0.895321	−	0.445422i	$-0.146946\pi$
$47$	937.621	+	1624.01i	0.0619131	+	0.107237i	−0.833407	+	0.552659i	$0.813613\pi$
$48$		0			0
$49$	15152.2	−	7272.19i	0.901544	−	0.432688i
$50$		0			0
$51$	8387.42	+	14527.4i	0.451547	+	0.782102i
$52$		0			0
$53$	−9353.62	+	16200.9i	−0.457393	+	0.792229i	−0.998822	−	0.0485180i	$-0.984550\pi$
$53$	−9353.62	+	16200.9i	−0.457393	+	0.792229i	0.541429	+	0.840747i	$0.317884\pi$
$54$		0			0
$55$	−13143.3			−0.585867
$56$		0			0
$57$	−29773.5			−1.21379
$58$		0			0
$59$	−1267.39	+	2195.18i	−0.0474002	+	0.0820995i	−0.888752	−	0.458388i	$-0.848427\pi$
$59$	−1267.39	+	2195.18i	−0.0474002	+	0.0820995i	0.841352	+	0.540488i	$0.181760\pi$
$60$		0			0
$61$	1047.35	+	1814.07i	0.0360386	+	0.0624208i	0.883482	−	0.468465i	$-0.155193\pi$
$61$	1047.35	+	1814.07i	0.0360386	+	0.0624208i	−0.847444	+	0.530886i	$0.821859\pi$
$62$		0			0
$63$	594.859	+	641.959i	0.0188826	+	0.0203777i
$64$		0			0
$65$	−2171.62	−	3761.36i	−0.0637530	−	0.110423i
$66$		0			0
$67$	29310.4	−	50767.1i	0.797691	−	1.38164i	−0.123426	−	0.992354i	$-0.539388\pi$
$67$	29310.4	−	50767.1i	0.797691	−	1.38164i	0.921117	−	0.389287i	$-0.127278\pi$
$68$		0			0
$69$	58747.5			1.48548
$70$		0			0
$71$	31279.5			0.736401			0.368201	−	0.929746i	$-0.379974\pi$
$71$	31279.5			0.736401			0.368201	+	0.929746i	$0.379974\pi$
$72$		0			0
$73$	3575.24	−	6192.49i	0.0785231	−	0.136006i	−0.824090	−	0.566459i	$-0.808313\pi$
$73$	3575.24	−	6192.49i	0.0785231	−	0.136006i	0.902613	+	0.430453i	$0.141646\pi$
$74$		0			0
$75$	−21208.3	−	36733.9i	−0.435364	−	0.754073i
$76$		0			0
$77$	23967.7	−	77518.7i	0.460680	−	1.48998i
$78$		0			0
$79$	1489.91	+	2580.59i	0.0268591	+	0.0465213i	0.879143	−	0.476559i	$-0.158116\pi$
$79$	1489.91	+	2580.59i	0.0268591	+	0.0465213i	−0.852283	+	0.523080i	$0.824783\pi$
$80$		0			0
$81$	30321.9	−	52519.1i	0.513505	−	0.889416i
$82$		0			0
$83$	−45954.6			−0.732207			−0.366103	−	0.930574i	$-0.619308\pi$
$83$	−45954.6			−0.732207			−0.366103	+	0.930574i	$0.619308\pi$
$84$		0			0
$85$	−22290.7			−0.334639
$86$		0			0
$87$	−977.821	+	1693.64i	−0.0138504	+	0.0239895i
$88$		0			0
$89$	−49520.0	−	85771.1i	−0.662682	−	1.14780i	−0.979908	−	0.199450i	$-0.936085\pi$
$89$	−49520.0	−	85771.1i	−0.662682	−	1.14780i	0.317226	−	0.948350i	$-0.397249\pi$
$90$		0			0
$91$	26144.4	−	5949.04i	0.330959	−	0.0753084i
$92$		0			0
$93$	71979.2	+	124672.i	0.862977	+	1.49472i
$94$		0			0
$95$	19781.8	−	34263.0i	0.224883	−	0.389509i
$96$		0			0
$97$	−115548.			−1.24691			−0.623454	−	0.781860i	$-0.714271\pi$
$97$	−115548.			−1.24691			−0.623454	+	0.781860i	$0.714271\pi$
$98$		0			0
$99$	4225.20			0.0433271
$100$		0			0
$101$	−5475.73	+	9484.25i	−0.0534120	+	0.0925123i	−0.891495	−	0.453030i	$-0.850343\pi$
$101$	−5475.73	+	9484.25i	−0.0534120	+	0.0925123i	0.838083	+	0.545542i	$0.183676\pi$
$102$		0			0
$103$	−68862.2	−	119273.i	−0.639570	−	1.10777i	−0.985527	−	0.169517i	$-0.945779\pi$
$103$	−68862.2	−	119273.i	−0.639570	−	1.10777i	0.345957	−	0.938250i	$-0.387554\pi$
$104$		0			0
$105$	−41952.3	+	9546.08i	−0.371349	+	0.0844990i
$106$		0			0
$107$	37786.5	+	65448.1i	0.319064	+	0.552634i	0.980293	−	0.197550i	$-0.0632983\pi$
$107$	37786.5	+	65448.1i	0.319064	+	0.552634i	−0.661229	+	0.750184i	$0.729965\pi$
$108$		0			0
$109$	−22263.1	+	38560.9i	−0.179482	+	0.310871i	−0.941703	−	0.336445i	$-0.890775\pi$
$109$	−22263.1	+	38560.9i	−0.179482	+	0.310871i	0.762221	+	0.647316i	$0.224109\pi$
$110$		0			0
$111$	95275.0			0.733959
$112$		0			0
$113$	90456.5			0.666413			0.333207	−	0.942854i	$-0.391869\pi$
$113$	90456.5			0.666413			0.333207	+	0.942854i	$0.391869\pi$
$114$		0			0
$115$	−39032.4	+	67606.0i	−0.275220	+	0.476695i
$116$		0			0
$117$	698.112	+	1209.17i	0.00471477	+	0.00816622i
$118$		0			0
$119$	40648.5	−	131469.i	0.263134	−	0.851055i
$120$		0			0
$121$	−115333.	−	199763.i	−0.716130	−	1.24037i
$122$		0			0
$123$	−135925.	+	235429.i	−0.810095	+	1.40313i
$124$		0			0
$125$	121989.			0.698306
$126$		0			0
$127$	−187707.			−1.03269			−0.516346	−	0.856380i	$-0.672708\pi$
$127$	−187707.			−1.03269			−0.516346	+	0.856380i	$0.672708\pi$
$128$		0			0
$129$	−42685.1	+	73932.8i	−0.225841	+	0.391167i
$130$		0			0
$131$	77206.1	+	133725.i	0.393073	+	0.680823i	0.992853	−	0.119342i	$-0.0380785\pi$
$131$	77206.1	+	133725.i	0.393073	+	0.680823i	−0.599780	+	0.800165i	$0.704745\pi$
$132$		0			0
$133$	166008.	+	179153.i	0.813768	+	0.878201i
$134$		0			0
$135$	39202.4	+	67900.6i	0.185131	+	0.320656i
$136$		0			0
$137$	−54618.9	+	94602.7i	−0.248623	+	0.430628i	−0.963144	−	0.268986i	$-0.913311\pi$
$137$	−54618.9	+	94602.7i	−0.248623	+	0.430628i	0.714521	+	0.699614i	$0.246645\pi$
$138$		0			0
$139$	204695.			0.898609			0.449305	−	0.893379i	$-0.351672\pi$
$139$	204695.			0.898609			0.449305	+	0.893379i	$0.351672\pi$
$140$		0			0
$141$	29635.4			0.125534
$142$		0			0
$143$	64721.9	−	112102.i	0.264674	−	0.458429i
$144$		0			0
$145$	−1299.35	−	2250.54i	−0.00513222	−	0.00888926i
$146$		0			0
$147$	20200.3	−	264840.i	0.0771018	−	1.01086i
$148$		0			0
$149$	203154.	+	351873.i	0.749651	+	1.29843i	0.947990	+	0.318300i	$0.103112\pi$
$149$	203154.	+	351873.i	0.749651	+	1.29843i	−0.198339	+	0.980134i	$0.563555\pi$
$150$		0			0
$151$	−208419.	+	360992.i	−0.743866	+	1.28841i	0.206856	+	0.978371i	$0.433677\pi$
$151$	−208419.	+	360992.i	−0.743866	+	1.28841i	−0.950723	+	0.310043i	$0.899657\pi$
$152$		0			0
$153$	7165.81			0.0247478
$154$		0			0
$155$	−191295.			−0.639548
$156$		0			0
$157$	−59645.3	+	103309.i	−0.193120	+	0.334493i	−0.946283	−	0.323341i	$-0.895194\pi$
$157$	−59645.3	+	103309.i	−0.193120	+	0.334493i	0.753163	+	0.657834i	$0.228527\pi$
$158$		0			0
$159$	147820.	+	256032.i	0.463703	+	0.803158i
$160$		0			0
$161$	−327559.	−	353494.i	−0.995920	−	1.07478i
$162$		0			0
$163$	23686.2	+	41025.7i	0.0698275	+	0.120945i	0.898825	−	0.438307i	$-0.144422\pi$
$163$	23686.2	+	41025.7i	0.0698275	+	0.120945i	−0.828998	+	0.559252i	$0.811088\pi$
$164$		0			0
$165$	−103856.	+	179883.i	−0.296975	+	0.514375i
$166$		0			0
$167$	231669.			0.642802			0.321401	−	0.946943i	$-0.395846\pi$
$167$	231669.			0.642802			0.321401	+	0.946943i	$0.395846\pi$
$168$		0			0
$169$	−328518.			−0.884795
$170$		0			0
$171$	−6359.26	+	11014.6i	−0.0166309	+	0.0288056i
$172$		0			0
$173$	−67169.6	−	116341.i	−0.170631	−	0.295541i	0.768010	−	0.640438i	$-0.221247\pi$
$173$	−67169.6	−	116341.i	−0.170631	−	0.295541i	−0.938641	+	0.344897i	$0.887914\pi$
$174$		0			0
$175$	−102783.	+	332432.i	−0.253704	+	0.820554i
$176$		0			0
$177$	20029.2	+	34691.6i	0.0480541	+	0.0832321i
$178$		0			0
$179$	23292.2	−	40343.2i	0.0543347	−	0.0941105i	−0.837579	−	0.546317i	$-0.816030\pi$
$179$	23292.2	−	40343.2i	0.0543347	−	0.0941105i	0.891913	+	0.452206i	$0.149363\pi$
$180$		0			0
$181$	829210.			1.88134			0.940672	−	0.339317i	$-0.110196\pi$
$181$	829210.			1.88134			0.940672	+	0.339317i	$0.110196\pi$
$182$		0			0
$183$	33103.7			0.0730716
$184$		0			0
$185$	−63301.6	+	109642.i	−0.135983	+	0.235530i
$186$		0			0
$187$	−332171.	−	575337.i	−0.694636	−	1.20315i
$188$		0			0
$189$	−471962.	+	107393.i	−0.961064	+	0.218686i
$190$		0			0
$191$	−235958.	−	408692.i	−0.468006	−	0.810611i	0.531325	−	0.847168i	$-0.321694\pi$
$191$	−235958.	−	408692.i	−0.468006	−	0.810611i	−0.999332	+	0.0365572i	$0.988361\pi$
$192$		0			0
$193$	−344177.	+	596132.i	−0.665103	+	1.15199i	0.314155	+	0.949372i	$0.398279\pi$
$193$	−344177.	+	596132.i	−0.665103	+	1.15199i	−0.979257	+	0.202620i	$0.935054\pi$
$194$		0			0
$195$	−68638.5			−0.129265
$196$		0			0
$197$	311915.			0.572625			0.286313	−	0.958136i	$-0.407570\pi$
$197$	311915.			0.572625			0.286313	+	0.958136i	$0.407570\pi$
$198$		0			0
$199$	−143606.	+	248733.i	−0.257063	+	0.445246i	−0.965454	−	0.260574i	$-0.916088\pi$
$199$	−143606.	+	248733.i	−0.257063	+	0.445246i	0.708391	+	0.705820i	$0.249421\pi$
$200$		0			0
$201$	−463207.	−	802298.i	−0.808695	−	1.40070i
$202$		0			0
$203$	15643.0	−	3559.49i	0.0266428	−	0.00606245i
$204$		0			0
$205$	−180619.	−	312842.i	−0.300179	−	0.519925i
$206$		0			0
$207$	12547.8	−	21733.4i	0.0203536	−	0.0352534i
$208$		0			0
$209$	1.17913e6			1.86723
$210$		0			0
$211$	460493.			0.712061			0.356031	−	0.934474i	$-0.384130\pi$
$211$	460493.			0.712061			0.356031	+	0.934474i	$0.384130\pi$
$212$		0			0
$213$	247163.	−	428099.i	0.373280	−	0.646540i
$214$		0			0
$215$	−56720.8	−	98243.3i	−0.0836847	−	0.144946i
$216$		0			0
$217$	348837.	−	1.12824e6i	0.502890	−	1.62650i
$218$		0			0
$219$	−56501.3	−	97863.1i	−0.0796064	−	0.137882i
$220$		0			0
$221$	109766.	−	190121.i	0.151178	−	0.261848i
$222$		0			0
$223$	1.19776e6			1.61290			0.806449	−	0.591304i	$-0.201387\pi$
$223$	1.19776e6			1.61290			0.806449	+	0.591304i	$0.201387\pi$
$224$		0			0
$225$	−18119.4			−0.0238609
$226$		0			0
$227$	447281.	−	774713.i	0.576123	−	0.997875i	−0.419795	−	0.907619i	$-0.637898\pi$
$227$	447281.	−	774713.i	0.576123	−	0.997875i	0.995919	−	0.0902559i	$-0.0287685\pi$
$228$		0			0
$229$	129628.	+	224522.i	0.163347	+	0.282925i	0.936067	−	0.351822i	$-0.114438\pi$
$229$	129628.	+	224522.i	0.163347	+	0.282925i	−0.772720	+	0.634747i	$0.781104\pi$
$230$		0			0
$231$	−871553.	−	940561.i	−1.07464	−	1.15973i
$232$		0			0
$233$	−105657.	−	183004.i	−0.127500	−	0.220836i	0.795207	−	0.606337i	$-0.207362\pi$
$233$	−105657.	−	183004.i	−0.127500	−	0.220836i	−0.922707	+	0.385501i	$0.874029\pi$
$234$		0			0
$235$	−19690.0	+	34104.2i	−0.0232582	+	0.0402845i
$236$		0			0
$237$	47091.5			0.0544592
$238$		0			0
$239$	−463.018			−0.000524328			−0.000262164	−	1.00000i	$-0.500083\pi$
$239$	−463.018			−0.000524328			−0.000262164		1.00000i	$0.500083\pi$
$240$		0			0
$241$	−143197.	+	248025.i	−0.158815	+	0.275076i	−0.934442	−	0.356116i	$-0.884101\pi$
$241$	−143197.	+	248025.i	−0.158815	+	0.275076i	0.775626	+	0.631192i	$0.217434\pi$
$242$		0			0
$243$	−25565.0	−	44279.8i	−0.0277734	−	0.0481050i
$244$		0			0
$245$	291355.	+	199209.i	0.310103	+	0.212028i
$246$		0			0
$247$	194823.	+	337444.i	0.203188	+	0.351932i
$248$		0			0
$249$	−363122.	+	628946.i	−0.371154	+	0.642858i
$250$		0			0
$251$	1.37168e6			1.37426			0.687129	−	0.726535i	$-0.258870\pi$
$251$	1.37168e6			1.37426			0.687129	+	0.726535i	$0.258870\pi$
$252$		0			0
$253$	−2.32660e6			−2.28518
$254$		0			0
$255$	−176136.	+	305076.i	−0.169628	+	0.293804i
$256$		0			0
$257$	379612.	+	657507.i	0.358515	+	0.620965i	0.987713	−	0.156279i	$-0.0499500\pi$
$257$	379612.	+	657507.i	0.358515	+	0.620965i	−0.629198	+	0.777245i	$0.716617\pi$
$258$		0			0
$259$	−531226.	−	573287.i	−0.492073	−	0.531035i
$260$		0			0
$261$	417.702	+	723.481i	0.000379547	+	0.000657394i
$262$		0			0
$263$	−536222.	+	928764.i	−0.478030	+	0.827973i	−0.999683	−	0.0251852i	$-0.991982\pi$
$263$	−536222.	+	928764.i	−0.478030	+	0.827973i	0.521652	+	0.853158i	$0.325316\pi$
$264$		0			0
$265$	−392852.			−0.343648
$266$		0			0
$267$	−1.56518e6			−1.34365
$268$		0			0
$269$	883056.	−	1.52950e6i	0.744059	−	1.28875i	−0.206574	−	0.978431i	$-0.566231\pi$
$269$	883056.	−	1.52950e6i	0.744059	−	1.28875i	0.950633	−	0.310317i	$-0.100435\pi$
$270$		0			0
$271$	−1.02586e6	−	1.77684e6i	−0.848522	−	1.46968i	−0.882527	−	0.470262i	$-0.844159\pi$
$271$	−1.02586e6	−	1.77684e6i	−0.848522	−	1.46968i	0.0340046	−	0.999422i	$-0.489174\pi$
$272$		0			0
$273$	125166.	−	404826.i	0.101644	−	0.328747i
$274$		0			0
$275$	839922.	+	1.45479e6i	0.669742	+	1.16003i
$276$		0			0
$277$	−435341.	+	754033.i	−0.340903	+	0.590461i	−0.984601	−	0.174819i	$-0.944066\pi$
$277$	−435341.	+	754033.i	−0.340903	+	0.590461i	0.643698	+	0.765280i	$0.277399\pi$
$278$		0			0
$279$	61495.6			0.0472970
$280$		0			0
$281$	2.35412e6			1.77854			0.889270	−	0.457383i	$-0.151213\pi$
$281$	2.35412e6			1.77854			0.889270	+	0.457383i	$0.151213\pi$
$282$		0			0
$283$	−1.09518e6	+	1.89690e6i	−0.812863	+	1.40792i	0.0979887	+	0.995188i	$0.468759\pi$
$283$	−1.09518e6	+	1.89690e6i	−0.812863	+	1.40792i	−0.910852	+	0.412733i	$0.864574\pi$
$284$		0			0
$285$	−312622.	−	541476.i	−0.227985	−	0.394882i
$286$		0			0
$287$	2.17450e6	−	494797.i	1.55831	−	0.354587i
$288$		0			0
$289$	146576.	+	253878.i	0.103233	+	0.178805i
$290$		0			0
$291$	−913035.	+	1.58142e6i	−0.632055	+	1.09475i
$292$		0			0
$293$	−807700.			−0.549644			−0.274822	−	0.961495i	$-0.588619\pi$
$293$	−807700.			−0.549644			−0.274822	+	0.961495i	$0.588619\pi$
$294$		0			0
$295$	−53230.4			−0.0356127
$296$		0			0
$297$	−1.16837e6	+	2.02368e6i	−0.768580	+	1.33122i
$298$		0			0
$299$	−384415.	−	665826.i	−0.248669	−	0.430708i
$300$		0			0
$301$	682867.	−	155384.i	0.434430	−	0.0988528i
$302$		0			0
$303$	86535.8	+	149884.i	0.0541488	+	0.0937885i
$304$		0			0
$305$	−21994.4	+	38095.4i	−0.0135383	+	0.0234489i
$306$		0			0
$307$	−211516.			−0.128085			−0.0640424	−	0.997947i	$-0.520399\pi$
$307$	−211516.			−0.128085			−0.0640424	+	0.997947i	$0.520399\pi$
$308$		0			0
$309$	−2.17653e6			−1.29679
$310$		0			0
$311$	−145855.	+	252627.i	−0.0855104	+	0.148108i	−0.905609	−	0.424114i	$-0.860585\pi$
$311$	−145855.	+	252627.i	−0.0855104	+	0.148108i	0.820098	+	0.572223i	$0.193919\pi$
$312$		0			0
$313$	997988.	+	1.72857e6i	0.575791	+	0.997299i	0.995955	+	0.0898507i	$0.0286390\pi$
$313$	997988.	+	1.72857e6i	0.575791	+	0.997299i	−0.420165	+	0.907448i	$0.638028\pi$
$314$		0			0
$315$	−5428.99	+	17559.0i	−0.00308278	+	0.00997065i
$316$		0			0
$317$	1.40479e6	+	2.43316e6i	0.785167	+	1.35995i	0.928899	+	0.370332i	$0.120756\pi$
$317$	1.40479e6	+	2.43316e6i	0.785167	+	1.35995i	−0.143733	+	0.989617i	$0.545911\pi$
$318$		0			0
$319$	38725.1	−	67073.9i	0.0213067	−	0.0369043i
$320$		0			0
$321$	1.19432e6			0.646930
$322$		0			0
$323$	1.99977e6			1.06653
$324$		0			0
$325$	−277554.	+	480737.i	−0.145760	+	0.252464i
$326$		0			0
$327$	351836.	+	609397.i	0.181958	+	0.315160i
$328$		0			0
$329$	−165238.	−	178322.i	−0.0841630	−	0.0908269i
$330$		0			0
$331$	690248.	+	1.19554e6i	0.346286	+	0.599785i	0.985587	−	0.169172i	$-0.0541093\pi$
$331$	690248.	+	1.19554e6i	0.346286	+	0.599785i	−0.639301	+	0.768957i	$0.720776\pi$
$332$		0			0
$333$	20349.6	−	35246.6i	0.0100565	−	0.0174183i
$334$		0			0
$335$	1.23104e6			0.599320
$336$		0			0
$337$	566429.			0.271688			0.135844	−	0.990730i	$-0.456625\pi$
$337$	566429.			0.271688			0.135844	+	0.990730i	$0.456625\pi$
$338$		0			0
$339$	714765.	−	1.23801e6i	0.337803	−	0.585092i
$340$		0			0
$341$	−2.85062e6	−	4.93743e6i	−1.32756	−	2.29940i
$342$		0			0
$343$	−1.70622e6	+	1.35512e6i	−0.783070	+	0.621933i
$344$		0			0
$345$	616848.	+	1.06841e6i	0.279017	+	0.483272i
$346$		0			0
$347$	270104.	−	467833.i	0.120422	−	0.208577i	−0.799512	−	0.600650i	$-0.794908\pi$
$347$	270104.	−	467833.i	0.120422	−	0.208577i	0.919934	+	0.392073i	$0.128242\pi$
$348$		0			0
$349$	1.73807e6			0.763841			0.381921	−	0.924195i	$-0.375263\pi$
$349$	1.73807e6			0.763841			0.381921	+	0.924195i	$0.375263\pi$
$350$		0			0
$351$	−772180.			−0.334542
$352$		0			0
$353$	−537109.	+	930300.i	−0.229417	+	0.397362i	−0.957635	−	0.287983i	$-0.907015\pi$
$353$	−537109.	+	930300.i	−0.229417	+	0.397362i	0.728219	+	0.685345i	$0.240349\pi$
$354$		0			0
$355$	328435.	+	568866.i	0.138318	+	0.239574i
$356$		0			0
$357$	−1.47813e6	−	1.59516e6i	−0.613821	−	0.662422i
$358$		0			0
$359$	60601.9	+	104966.i	0.0248170	+	0.0429844i	0.878167	−	0.478354i	$-0.158766\pi$
$359$	60601.9	+	104966.i	0.0248170	+	0.0429844i	−0.853350	+	0.521338i	$0.825433\pi$
$360$		0			0
$361$	−536639.	+	929486.i	−0.216728	+	0.375383i
$362$		0			0
$363$	−3.64535e6			−1.45202
$364$		0			0
$365$	150160.			0.0589959
$366$		0			0
$367$	−276914.	+	479630.i	−0.107320	+	0.185884i	−0.914684	−	0.404171i	$-0.867560\pi$
$367$	−276914.	+	479630.i	−0.107320	+	0.185884i	0.807364	+	0.590054i	$0.200894\pi$
$368$		0			0
$369$	58063.8	+	100570.i	0.0221993	+	0.0384504i
$370$		0			0
$371$	716389.	−	2.31702e6i	0.270218	−	0.873966i
$372$		0			0
$373$	250739.	+	434293.i	0.0933146	+	0.161626i	0.908904	−	0.417006i	$-0.136920\pi$
$373$	250739.	+	434293.i	0.0933146	+	0.161626i	−0.815589	+	0.578631i	$0.803587\pi$
$374$		0			0
$375$	963927.	−	1.66957e6i	0.353970	−	0.613093i
$376$		0			0
$377$	25593.6			0.00927422
$378$		0			0
$379$	−999004.			−0.357248			−0.178624	−	0.983917i	$-0.557165\pi$
$379$	−999004.			−0.357248			−0.178624	+	0.983917i	$0.557165\pi$
$380$		0			0
$381$	−1.48321e6	+	2.56900e6i	−0.523469	+	0.906676i
$382$		0			0
$383$	326232.	+	565050.i	0.113640	+	0.196829i	0.917235	−	0.398346i	$-0.130416\pi$
$383$	326232.	+	565050.i	0.113640	+	0.196829i	−0.803596	+	0.595176i	$0.797082\pi$
$384$		0			0
$385$	1.66146e6	−	378058.i	0.571265	−	0.129989i
$386$		0			0
$387$	18234.1	+	31582.3i	0.00618879	+	0.0107193i
$388$		0			0
$389$	39626.5	−	68635.1i	0.0132774	−	0.0229971i	−0.859310	−	0.511454i	$-0.829107\pi$
$389$	39626.5	−	68635.1i	0.0132774	−	0.0229971i	0.872588	+	0.488457i	$0.162440\pi$
$390$		0			0
$391$	−3.94585e6			−1.30526
$392$		0			0
$393$	2.44026e6			0.796992
$394$		0			0
$395$	−31288.0	+	54192.4i	−0.0100899	+	0.0174762i
$396$		0			0
$397$	−2.00443e6	−	3.47177e6i	−0.638284	−	1.10554i	−0.985809	−	0.167870i	$-0.946311\pi$
$397$	−2.00443e6	−	3.47177e6i	−0.638284	−	1.10554i	0.347525	−	0.937671i	$-0.387022\pi$
$398$		0			0
$399$	3.76368e6	−	856410.i	1.18353	−	0.269308i
$400$		0			0
$401$	−337209.	−	584063.i	−0.104722	−	0.181384i	0.808903	−	0.587943i	$-0.200062\pi$
$401$	−337209.	−	584063.i	−0.104722	−	0.181384i	−0.913625	+	0.406559i	$0.866729\pi$
$402$		0			0
$403$	941994.	−	1.63158e6i	0.288925	−	0.500433i
$404$		0			0
$405$	1.27352e6			0.385806
$406$		0			0
$407$	−3.77322e6			−1.12908
$408$		0			0
$409$	1.42706e6	−	2.47175e6i	0.421828	−	0.730627i	−0.574291	−	0.818652i	$-0.694722\pi$
$409$	1.42706e6	−	2.47175e6i	0.421828	−	0.730627i	0.996118	+	0.0880244i	$0.0280553\pi$
$410$		0			0
$411$	863171.	+	1.49506e6i	0.252053	+	0.436569i
$412$		0			0
$413$	97068.7	−	313950.i	0.0280030	−	0.0905701i
$414$		0			0
$415$	−482523.	−	835755.i	−0.137530	−	0.238209i
$416$		0			0
$417$	1.61745e6	−	2.80151e6i	0.455503	−	0.788954i
$418$		0			0
$419$	4.66553e6			1.29827			0.649136	−	0.760672i	$-0.275130\pi$
$419$	4.66553e6			1.29827			0.649136	+	0.760672i	$0.275130\pi$
$420$		0			0
$421$	−3.73317e6			−1.02653			−0.513266	−	0.858229i	$-0.671565\pi$
$421$	−3.73317e6			−1.02653			−0.513266	+	0.858229i	$0.671565\pi$
$422$		0			0
$423$	6329.77	−	10963.5i	0.00172003	−	0.00297919i
$424$		0			0
$425$	1.42448e6	+	2.46728e6i	0.382547	+	0.662591i
$426$		0			0
$427$	−184577.	−	199191.i	−0.0489899	−	0.0528689i
$428$		0			0
$429$	−1.02283e6	−	1.77160e6i	−0.268325	−	0.464753i
$430$		0			0
$431$	482335.	−	835429.i	0.125071	−	0.216629i	−0.796690	−	0.604388i	$-0.793418\pi$
$431$	482335.	−	835429.i	0.125071	−	0.216629i	0.921761	+	0.387760i	$0.126751\pi$
$432$		0			0
$433$	−6.18096e6			−1.58429			−0.792147	−	0.610330i	$-0.791037\pi$
$433$	−6.18096e6			−1.58429			−0.792147	+	0.610330i	$0.791037\pi$
$434$		0			0
$435$	−41068.5			−0.0104060
$436$		0			0
$437$	3.50172e6	−	6.06516e6i	0.877158	−	1.51928i
$438$		0			0
$439$	−107567.	−	186312.i	−0.0266391	−	0.0461402i	0.852399	−	0.522893i	$-0.175147\pi$
$439$	−107567.	−	186312.i	−0.0266391	−	0.0461402i	−0.879038	+	0.476752i	$0.841814\pi$
$440$		0			0
$441$	−93661.9	−	64039.7i	−0.0229333	−	0.0156803i
$442$		0			0
$443$	−3.59934e6	−	6.23423e6i	−0.871391	−	1.50929i	−0.860558	−	0.509353i	$-0.829885\pi$
$443$	−3.59934e6	−	6.23423e6i	−0.871391	−	1.50929i	−0.0108335	−	0.999941i	$-0.503448\pi$
$444$		0			0
$445$	1.03992e6	−	1.80119e6i	0.248943	−	0.431182i
$446$		0			0
$447$	6.42109e6			1.51999
$448$		0			0
$449$	3.92153e6			0.917994			0.458997	−	0.888438i	$-0.348209\pi$
$449$	3.92153e6			0.917994			0.458997	+	0.888438i	$0.348209\pi$
$450$		0			0
$451$	5.38309e6	−	9.32379e6i	1.24621	−	2.15850i
$452$		0			0
$453$	3.29375e6	+	5.70494e6i	0.754128	+	1.30619i
$454$		0			0
$455$	382708.	+	413010.i	0.0866641	+	0.0935260i
$456$		0			0
$457$	−660839.	−	1.14461e6i	−0.148015	−	0.256369i	0.782479	−	0.622677i	$-0.213955\pi$
$457$	−660839.	−	1.14461e6i	−0.148015	−	0.256369i	−0.930494	+	0.366308i	$0.880622\pi$
$458$		0			0
$459$	−1.98152e6	+	3.43209e6i	−0.439002	+	0.760375i
$460$		0			0
$461$	−75459.1			−0.0165371			−0.00826855	−	0.999966i	$-0.502632\pi$
$461$	−75459.1			−0.0165371			−0.00826855	+	0.999966i	$0.502632\pi$
$462$		0			0
$463$	3.28757e6			0.712727			0.356363	−	0.934347i	$-0.384016\pi$
$463$	3.28757e6			0.712727			0.356363	+	0.934347i	$0.384016\pi$
$464$		0			0
$465$	−1.51156e6	+	2.61810e6i	−0.324186	+	0.561506i
$466$		0			0
$467$	−321691.	−	557185.i	−0.0682569	−	0.118224i	0.829877	−	0.557946i	$-0.188410\pi$
$467$	−321691.	−	557185.i	−0.0682569	−	0.118224i	−0.898134	+	0.439722i	$0.855077\pi$
$468$		0			0
$469$	−2.24487e6	+	7.26058e6i	−0.471258	+	1.52419i
$470$		0			0
$471$	942605.	+	1.63264e6i	0.195784	+	0.339108i
$472$		0			0
$473$	1.69048e6	−	2.92799e6i	0.347422	−	0.601752i
$474$		0			0
$475$	−5.05660e6			−1.02831
$476$		0			0
$477$	126290.			0.0254141
$478$		0			0
$479$	−3.71324e6	+	6.43152e6i	−0.739459	+	1.28078i	0.213280	+	0.976991i	$0.431585\pi$
$479$	−3.71324e6	+	6.43152e6i	−0.739459	+	1.28078i	−0.952739	+	0.303790i	$0.901748\pi$
$480$		0			0
$481$	−623434.	−	1.07982e6i	−0.122865	−	0.212808i
$482$		0			0
$483$	−7.42630e6	+	1.68982e6i	−1.44845	+	0.329590i
$484$		0			0
$485$	−1.21326e6	−	2.10143e6i	−0.234207	−	0.405658i
$486$		0			0
$487$	−2.08542e6	+	3.61206e6i	−0.398448	+	0.690132i	−0.993535	−	0.113529i	$-0.963784\pi$
$487$	−2.08542e6	+	3.61206e6i	−0.398448	+	0.690132i	0.595087	+	0.803662i	$0.297118\pi$
$488$		0			0
$489$	748651.			0.141582
$490$		0			0
$491$	3.73674e6			0.699502			0.349751	−	0.936843i	$-0.386266\pi$
$491$	3.73674e6			0.699502			0.349751	+	0.936843i	$0.386266\pi$
$492$		0			0
$493$	65676.6	−	113755.i	0.0121701	−	0.0210792i
$494$		0			0
$495$	44364.6	+	76841.7i	0.00813811	+	0.0140956i
$496$		0			0
$497$	−3.95406e6	+	899731.i	−0.718047	+	0.163389i
$498$		0			0
$499$	4.35431e6	+	7.54188e6i	0.782831	+	1.35590i	0.930286	+	0.366834i	$0.119558\pi$
$499$	4.35431e6	+	7.54188e6i	0.782831	+	1.35590i	−0.147456	+	0.989069i	$0.547108\pi$
$500$		0			0
$501$	1.83059e6	−	3.17068e6i	0.325835	−	0.564363i
$502$		0			0
$503$	−3.28384e6			−0.578711			−0.289355	−	0.957222i	$-0.593441\pi$
$503$	−3.28384e6			−0.578711			−0.289355	+	0.957222i	$0.593441\pi$
$504$		0			0
$505$	−229981.			−0.0401294
$506$		0			0
$507$	−2.59587e6	+	4.49618e6i	−0.448500	+	0.776826i
$508$		0			0
$509$	4.71851e6	+	8.17271e6i	0.807255	+	1.39821i	0.914758	+	0.404002i	$0.132381\pi$
$509$	4.71851e6	+	8.17271e6i	0.807255	+	1.39821i	−0.107503	+	0.994205i	$0.534286\pi$
$510$		0			0
$511$	−273826.	+	885635.i	−0.0463897	+	0.150038i
$512$		0			0
$513$	−3.51698e6	−	6.09159e6i	−0.590033	−	1.02197i
$514$		0			0
$515$	1.44611e6	−	2.50473e6i	0.240260	−	0.416143i
$516$		0			0
$517$	−1.17366e6			−0.193116
$518$		0			0
$519$	−2.12303e6			−0.345970
$520$		0			0
$521$	−2.95363e6	+	5.11584e6i	−0.476719	+	0.825701i	−0.999644	−	0.0266776i	$-0.991507\pi$
$521$	−2.95363e6	+	5.11584e6i	−0.476719	+	0.825701i	0.522926	+	0.852378i	$0.324841\pi$
$522$		0			0
$523$	327168.	+	566672.i	0.0523019	+	0.0905895i	0.890991	−	0.454021i	$-0.150011\pi$
$523$	327168.	+	566672.i	0.0523019	+	0.0905895i	−0.838689	+	0.544610i	$0.816678\pi$
$524$		0			0
$525$	3.73757e6	+	4.03351e6i	0.591823	+	0.638682i
$526$		0			0
$527$	−4.83457e6	−	8.37373e6i	−0.758284	−	1.31339i
$528$		0			0
$529$	−3.69124e6	+	6.39342e6i	−0.573500	+	0.993331i
$530$		0			0
$531$	17112.0			0.00263369
$532$		0			0
$533$	3.55771e6			0.542440
$534$		0			0
$535$	−793516.	+	1.37441e6i	−0.119859	+	0.207602i
$536$		0			0
$537$	−368098.	−	637565.i	−0.0550843	−	0.0954088i
$538$		0			0
$539$	−800002.	+	1.04886e7i	−0.118609	+	1.55505i
$540$		0			0
$541$	431558.	+	747480.i	0.0633936	+	0.109801i	0.895980	−	0.444094i	$-0.146474\pi$
$541$	431558.	+	747480.i	0.0633936	+	0.109801i	−0.832587	+	0.553895i	$0.813141\pi$
$542$		0			0
$543$	6.55222e6	−	1.13488e7i	0.953649	−	1.65177i
$544$		0			0
$545$	−935052.			−0.134848
$546$		0			0
$547$	5.45692e6			0.779794			0.389897	−	0.920859i	$-0.372511\pi$
$547$	5.45692e6			0.779794			0.389897	+	0.920859i	$0.372511\pi$
$548$		0			0
$549$	7070.55	−	12246.6i	0.00100120	−	0.00173414i
$550$		0			0
$551$	116569.	+	201903.i	0.0163570	+	0.0283311i
$552$		0			0
$553$	−262568.	−	283358.i	−0.0365115	−	0.0394024i
$554$		0			0
$555$	1.00039e6	+	1.73272e6i	0.137859	+	0.238779i
$556$		0			0
$557$	−5.58881e6	+	9.68010e6i	−0.763275	+	1.32203i	0.177879	+	0.984052i	$0.443077\pi$
$557$	−5.58881e6	+	9.68010e6i	−0.763275	+	1.32203i	−0.941154	+	0.337979i	$0.890257\pi$
$558$		0			0
$559$	1.11724e6			0.151223
$560$		0			0
$561$	−1.04989e7			−1.40844
$562$		0			0
$563$	440132.	−	762332.i	0.0585211	−	0.101361i	−0.835281	−	0.549824i	$-0.814695\pi$
$563$	440132.	−	762332.i	0.0585211	−	0.101361i	0.893802	+	0.448462i	$0.148028\pi$
$564$		0			0
$565$	949793.	+	1.64509e6i	0.125172	+	0.216805i
$566$		0			0
$567$	−2.32234e6	+	7.51116e6i	−0.303367	+	0.981181i
$568$		0			0
$569$	774858.	+	1.34209e6i	0.100332	+	0.173781i	0.911822	−	0.410587i	$-0.134676\pi$
$569$	774858.	+	1.34209e6i	0.100332	+	0.173781i	−0.811489	+	0.584367i	$0.801343\pi$
$570$		0			0
$571$	−5.48507e6	+	9.50043e6i	−0.704032	+	1.21942i	0.263008	+	0.964794i	$0.415285\pi$
$571$	−5.48507e6	+	9.50043e6i	−0.704032	+	1.21942i	−0.967040	+	0.254625i	$0.918048\pi$
$572$		0			0
$573$	−7.45794e6			−0.948926
$574$		0			0
$575$	9.97742e6			1.25849
$576$		0			0
$577$	6.29341e6	−	1.09005e7i	0.786948	−	1.36303i	−0.140880	−	0.990027i	$-0.544993\pi$
$577$	6.29341e6	−	1.09005e7i	0.786948	−	1.36303i	0.927828	−	0.373008i	$-0.121674\pi$
$578$		0			0
$579$	5.43921e6	+	9.42098e6i	0.674278	+	1.16788i
$580$		0			0
$581$	5.80915e6	−	1.32185e6i	0.713957	−	0.162458i
$582$		0			0
$583$	−5.85418e6	−	1.01397e7i	−0.713337	−	1.23554i
$584$		0			0
$585$	−14660.4	+	25392.5i	−0.00177115	+	0.00306772i
$586$		0			0
$587$	−1.19962e7			−1.43697			−0.718484	−	0.695544i	$-0.755164\pi$
$587$	−1.19962e7			−1.43697			−0.718484	+	0.695544i	$0.755164\pi$
$588$		0			0
$589$	1.71617e7			2.03831
$590$		0			0
$591$	2.46467e6	−	4.26894e6i	0.290262	−	0.502749i
$592$		0			0
$593$	−2.66154e6	−	4.60993e6i	−0.310811	−	0.538341i	0.667727	−	0.744406i	$-0.267267\pi$
$593$	−2.66154e6	−	4.60993e6i	−0.310811	−	0.538341i	−0.978538	+	0.206065i	$0.933934\pi$
$594$		0			0
$595$	2.81778e6	−	641175.i	0.326298	−	0.0742479i
$596$		0			0
$597$	2.26948e6	+	3.93085e6i	0.260609	+	0.451388i
$598$		0			0
$599$	3.77757e6	−	6.54293e6i	0.430175	−	0.745085i	−0.566713	−	0.823915i	$-0.691785\pi$
$599$	3.77757e6	−	6.54293e6i	0.430175	−	0.745085i	0.996888	+	0.0788305i	$0.0251186\pi$
$600$		0			0
$601$	4.44758e6			0.502270			0.251135	−	0.967952i	$-0.419196\pi$
$601$	4.44758e6			0.502270			0.251135	+	0.967952i	$0.419196\pi$
$602$		0			0
$603$	−395742.			−0.0443219
$604$		0			0
$605$	2.42200e6	−	4.19503e6i	0.269021	−	0.465958i
$606$		0			0
$607$	2.78660e6	+	4.82653e6i	0.306975	+	0.531696i	0.977699	−	0.210011i	$-0.0673499\pi$
$607$	2.78660e6	+	4.82653e6i	0.306975	+	0.531696i	−0.670724	+	0.741707i	$0.734017\pi$
$608$		0			0
$609$	74890.8	−	242220.i	0.00818249	−	0.0264647i
$610$		0			0
$611$	−193920.	−	335879.i	−0.0210145	−	0.0363982i
$612$		0			0
$613$	5.56323e6	−	9.63580e6i	0.597965	−	1.03571i	−0.395156	−	0.918614i	$-0.629309\pi$
$613$	5.56323e6	−	9.63580e6i	0.597965	−	1.03571i	0.993121	−	0.117092i	$-0.0373573\pi$
$614$		0			0
$615$	−5.70884e6			−0.608640
$616$		0			0
$617$	−1.07454e7			−1.13634			−0.568170	−	0.822911i	$-0.692349\pi$
$617$	−1.07454e7			−1.13634			−0.568170	+	0.822911i	$0.692349\pi$
$618$		0			0
$619$	6.78781e6	−	1.17568e7i	0.712038	−	1.23329i	−0.252053	−	0.967713i	$-0.581106\pi$
$619$	6.78781e6	−	1.17568e7i	0.712038	−	1.23329i	0.964091	−	0.265572i	$-0.0855610\pi$
$620$		0			0
$621$	6.93952e6	+	1.20196e7i	0.722105	+	1.25072i
$622$		0			0
$623$	8.72698e6	+	9.41797e6i	0.900833	+	0.972159i
$624$		0			0
$625$	−2.91287e6	−	5.04523e6i	−0.298277	−	0.516632i
$626$		0			0
$627$	9.31722e6	−	1.61379e7i	0.946493	−	1.63937i
$628$		0			0
$629$	−6.39927e6			−0.644917
$630$		0			0
$631$	−1.27986e7			−1.27964			−0.639820	−	0.768525i	$-0.720991\pi$
$631$	−1.27986e7			−1.27964			−0.639820	+	0.768525i	$0.720991\pi$
$632$		0			0
$633$	3.63871e6	−	6.30242e6i	0.360942	−	0.625170i
$634$		0			0
$635$	−1.97092e6	−	3.41374e6i	−0.193970	−	0.335966i
$636$		0			0
$637$	−3.13380e6	+	1.50404e6i	−0.306001	+	0.146863i
$638$		0			0
$639$	−105582.	−	182874.i	−0.0102291	−	0.0177174i
$640$		0			0
$641$	442417.	−	766288.i	0.0425291	−	0.0736626i	−0.843977	−	0.536379i	$-0.819792\pi$
$641$	442417.	−	766288.i	0.0425291	−	0.0736626i	0.886506	+	0.462716i	$0.153125\pi$
$642$		0			0
$643$	−6.66271e6			−0.635511			−0.317756	−	0.948173i	$-0.602929\pi$
$643$	−6.66271e6			−0.635511			−0.317756	+	0.948173i	$0.602929\pi$
$644$		0			0
$645$	−1.79277e6			−0.169678
$646$		0			0
$647$	−8.45887e6	+	1.46512e7i	−0.794423	+	1.37598i	0.128782	+	0.991673i	$0.458893\pi$
$647$	−8.45887e6	+	1.46512e7i	−0.794423	+	1.37598i	−0.923205	+	0.384308i	$0.874440\pi$
$648$		0			0
$649$	−793226.	−	1.37391e6i	−0.0739239	−	0.128040i
$650$		0			0
$651$	−1.26850e7	−	1.36894e7i	−1.17311	−	1.26599i
$652$		0			0
$653$	−4.74654e6	−	8.22124e6i	−0.435606	−	0.754492i	0.561739	−	0.827315i	$-0.310133\pi$
$653$	−4.74654e6	−	8.22124e6i	−0.435606	−	0.754492i	−0.997345	+	0.0728230i	$0.976799\pi$
$654$		0			0
$655$	−1.62133e6	+	2.80822e6i	−0.147662	+	0.255758i
$656$		0			0
$657$	−48272.0			−0.00436297
$658$		0			0
$659$	4.97563e6			0.446308			0.223154	−	0.974783i	$-0.428365\pi$
$659$	4.97563e6			0.446308			0.223154	+	0.974783i	$0.428365\pi$
$660$		0			0
$661$	−1.05765e7	+	1.83191e7i	−0.941543	+	1.63080i	−0.179013	+	0.983847i	$0.557290\pi$
$661$	−1.05765e7	+	1.83191e7i	−0.941543	+	1.63080i	−0.762530	+	0.646953i	$0.776043\pi$
$662$		0			0
$663$	−1.73470e6	−	3.00458e6i	−0.153264	−	0.265461i
$664$		0			0
$665$	−1.51508e6	+	4.90022e6i	−0.132856	+	0.429696i
$666$		0			0
$667$	−230007.	−	398384.i	−0.0200183	−	0.0346727i
$668$		0			0
$669$	9.46438e6	−	1.63928e7i	0.817574	−	1.41608i
$670$		0			0
$671$	−1.31102e6			−0.112410
$672$		0			0
$673$	417573.			0.0355382			0.0177691	−	0.999842i	$-0.494344\pi$
$673$	417573.			0.0355382			0.0177691	+	0.999842i	$0.494344\pi$
$674$		0			0
$675$	5.01044e6	−	8.67834e6i	0.423269	−	0.733124i
$676$		0			0
$677$	−1.31234e6	−	2.27304e6i	−0.110046	−	0.190605i	0.805743	−	0.592266i	$-0.201767\pi$
$677$	−1.31234e6	−	2.27304e6i	−0.110046	−	0.190605i	−0.915789	+	0.401661i	$0.868433\pi$
$678$		0			0
$679$	1.46065e7	−	3.32366e6i	1.21583	−	0.276657i
$680$		0			0
$681$	−7.06860e6	−	1.22432e7i	−0.584071	−	1.01164i
$682$		0			0
$683$	−4.37227e6	+	7.57300e6i	−0.358637	+	0.621178i	−0.987733	−	0.156149i	$-0.950092\pi$
$683$	−4.37227e6	+	7.57300e6i	−0.358637	+	0.621178i	0.629096	+	0.777328i	$0.283425\pi$
$684$		0			0
$685$	−2.29399e6			−0.186795
$686$		0			0
$687$	4.09716e6			0.331200
$688$		0			0
$689$	1.93452e6	−	3.35069e6i	0.155248	−	0.268898i
$690$		0			0
$691$	2.19834e6	+	3.80763e6i	0.175146	+	0.303361i	0.940212	−	0.340591i	$-0.110627\pi$
$691$	2.19834e6	+	3.80763e6i	0.175146	+	0.303361i	−0.765066	+	0.643952i	$0.777294\pi$
$692$		0			0
$693$	−534110.	+	121534.i	−0.0422471	+	0.00961316i
$694$		0			0
$695$	2.14930e6	+	3.72270e6i	0.168786	+	0.292345i
$696$		0			0
$697$	9.12957e6	−	1.58129e7i	0.711817	−	1.23290i
$698$		0			0
$699$	−3.33951e6			−0.258518
$700$		0			0
$701$	6.51339e6			0.500624			0.250312	−	0.968165i	$-0.419467\pi$
$701$	6.51339e6			0.500624			0.250312	+	0.968165i	$0.419467\pi$
$702$		0			0
$703$	5.67900e6	−	9.83631e6i	0.433394	−	0.750661i
$704$		0			0
$705$	311172.	+	538965.i	0.0235791	+	0.0408402i
$706$		0			0
$707$	419383.	−	1.35641e6i	0.0315546	−	0.102057i
$708$		0			0
$709$	5.23256e6	+	9.06305e6i	0.390929	+	0.677110i	0.992572	−	0.121655i	$-0.0388203\pi$
$709$	5.23256e6	+	9.06305e6i	0.390929	+	0.677110i	−0.601643	+	0.798765i	$0.705487\pi$
$710$		0			0
$711$	10058.2	−	17421.3i	0.000746183	−	0.00129243i
$712$		0			0
$713$	−3.38625e7			−2.49457
$714$		0			0
$715$	2.71832e6			0.198855
$716$		0			0
$717$	−3658.66	+	6336.98i	−0.000265781	+	0.000460346i
$718$		0			0
$719$	−8.40746e6	−	1.45621e7i	−0.606516	−	1.05052i	−0.991810	−	0.127723i	$-0.959233\pi$
$719$	−8.40746e6	−	1.45621e7i	−0.606516	−	1.05052i	0.385294	−	0.922794i	$-0.374100\pi$
$720$		0			0
$721$	1.21357e7	+	1.30966e7i	0.869414	+	0.938252i
$722$		0			0
$723$	2.26302e6	+	3.91967e6i	0.161006	+	0.278871i
$724$		0			0
$725$	−166069.	+	287640.i	−0.0117339	+	0.0203238i
$726$		0			0
$727$	1.71928e7			1.20646			0.603228	−	0.797569i	$-0.293881\pi$
$727$	1.71928e7			1.20646			0.603228	+	0.797569i	$0.293881\pi$
$728$		0			0
$729$	1.39284e7			0.970696
$730$		0			0
$731$	2.86700e6	−	4.96579e6i	0.198442	−	0.343712i
$732$		0			0
$733$	−9.86827e6	−	1.70923e7i	−0.678393	−	1.17501i	−0.975465	−	0.220155i	$-0.929344\pi$
$733$	−9.86827e6	−	1.70923e7i	−0.678393	−	1.17501i	0.297072	−	0.954855i	$-0.403990\pi$
$734$		0			0
$735$	5.02863e6	−	2.41345e6i	0.343345	−	0.164786i
$736$		0			0
$737$	1.83446e7	+	3.17738e7i	1.24405	+	2.15477i
$738$		0			0
$739$	6.43593e6	−	1.11474e7i	0.433511	−	0.750863i	−0.563662	−	0.826006i	$-0.690608\pi$
$739$	6.43593e6	−	1.11474e7i	0.433511	−	0.750863i	0.997173	+	0.0751426i	$0.0239412\pi$
$740$		0			0
$741$	6.15778e6			0.411983
$742$		0			0
$743$	−2.45606e7			−1.63218			−0.816089	−	0.577927i	$-0.803862\pi$
$743$	−2.45606e7			−1.63218			−0.816089	+	0.577927i	$0.803862\pi$
$744$		0			0
$745$	−4.26623e6	+	7.38933e6i	−0.281614	+	0.487769i
$746$		0			0
$747$	155117.	+	268671.i	0.0101709	+	0.0176165i
$748$		0			0
$749$	−6.65917e6	−	7.18643e6i	−0.433726	−	0.468068i
$750$		0			0
$751$	−162983.	−	282294.i	−0.0105449	−	0.0182643i	0.860705	−	0.509104i	$-0.170023\pi$
$751$	−162983.	−	282294.i	−0.0105449	−	0.0182643i	−0.871250	+	0.490840i	$0.836690\pi$
$752$		0			0
$753$	1.08387e7	−	1.87731e7i	0.696609	−	1.20656i
$754$		0			0
$755$	−8.75360e6			−0.558881
$756$		0			0
$757$	1.84659e7			1.17120			0.585599	−	0.810601i	$-0.300859\pi$
$757$	1.84659e7			1.17120			0.585599	+	0.810601i	$0.300859\pi$
$758$		0			0
$759$	−1.83842e7	+	3.18424e7i	−1.15835	+	2.00633i
$760$		0			0
$761$	−2.56313e6	−	4.43947e6i	−0.160439	−	0.277888i	0.774587	−	0.632467i	$-0.217958\pi$
$761$	−2.56313e6	−	4.43947e6i	−0.160439	−	0.277888i	−0.935026	+	0.354579i	$0.884624\pi$
$762$		0			0
$763$	1.70512e6	−	5.51488e6i	0.106034	−	0.342945i
$764$		0			0
$765$	75241.0	+	130321.i	0.00464837	+	0.00805122i
$766$		0			0
$767$	262123.	−	454010.i	0.0160885	−	0.0278662i
$768$		0			0
$769$	1.63432e6			0.0996602			0.0498301	−	0.998758i	$-0.484132\pi$
$769$	1.63432e6			0.0996602			0.0498301	+	0.998758i	$0.484132\pi$
$770$		0			0
$771$	1.19984e7			0.726921
$772$		0			0
$773$	2.14128e6	−	3.70881e6i	0.128892	−	0.223247i	−0.794356	−	0.607453i	$-0.792191\pi$
$773$	2.14128e6	−	3.70881e6i	0.128892	−	0.223247i	0.923248	+	0.384206i	$0.125525\pi$
$774$		0			0
$775$	1.22246e7	+	2.11737e7i	0.731108	+	1.26632i
$776$		0			0
$777$	−1.20438e7	+	2.74051e6i	−0.715665	+	0.162847i
$778$		0			0
$779$	1.62040e7	+	2.80661e7i	0.956704	+	1.65706i
$780$		0			0
$781$	−9.78852e6	+	1.69542e7i	−0.574235	+	0.994604i
$782$		0			0
$783$	−462019.			−0.0269312
$784$		0			0
$785$	−2.50510e6			−0.145095
$786$		0			0
$787$	−3.91482e6	+	6.78066e6i	−0.225307	+	0.390243i	−0.956411	−	0.292022i	$-0.905672\pi$
$787$	−3.91482e6	+	6.78066e6i	−0.225307	+	0.390243i	0.731105	+	0.682265i	$0.239005\pi$
$788$		0			0
$789$	8.47419e6	+	1.46777e7i	0.484625	+	0.839395i
$790$		0			0
$791$	−1.14346e7	+	2.60191e6i	−0.649803	+	0.147860i
$792$		0			0
$793$	−216615.	−	375187.i	−0.0122322	−	0.0211868i
$794$		0			0
$795$	−3.10422e6	+	5.37666e6i	−0.174195	+	0.301714i
$796$		0			0
$797$	−3.68238e6			−0.205344			−0.102672	−	0.994715i	$-0.532739\pi$
$797$	−3.68238e6			−0.205344			−0.102672	+	0.994715i	$0.532739\pi$
$798$		0			0
$799$	−1.99050e6			−0.110305
$800$		0			0
$801$	−334304.	+	579031.i	−0.0184102	+	0.0318875i
$802$		0			0
$803$	2.23765e6	+	3.87572e6i	0.122462	+	0.212111i
$804$		0			0
$805$	2.98947e6	−	9.66885e6i	0.162594	−	0.525878i
$806$		0			0
$807$	−1.39554e7	−	2.41714e7i	−0.754324	−	1.30653i
$808$		0			0
$809$	−1.04091e7	+	1.80290e7i	−0.559165	+	0.968502i	0.438402	+	0.898779i	$0.355545\pi$
$809$	−1.04091e7	+	1.80290e7i	−0.559165	+	0.968502i	−0.997566	+	0.0697227i	$0.977789\pi$
$810$		0			0
$811$	−3.03542e7			−1.62057			−0.810283	−	0.586039i	$-0.800687\pi$
$811$	−3.03542e7			−1.62057			−0.810283	+	0.586039i	$0.800687\pi$
$812$		0			0
$813$	−3.24243e7			−1.72046
$814$		0			0
$815$	−497410.	+	861540.i	−0.0262314	+	0.0454341i
$816$		0			0
$817$	5.08861e6	+	8.81373e6i	0.266713	+	0.461960i
$818$		0			0
$819$	−123029.	−	132771.i	−0.00640913	−	0.00691659i
$820$		0			0
$821$	−1.37193e7	−	2.37626e7i	−0.710355	−	1.23037i	−0.964724	−	0.263264i	$-0.915201\pi$
$821$	−1.37193e7	−	2.37626e7i	−0.710355	−	1.23037i	0.254369	−	0.967107i	$-0.418132\pi$
$822$		0			0
$823$	7.95629e6	−	1.37807e7i	0.409459	−	0.709204i	−0.585370	−	0.810766i	$-0.699051\pi$
$823$	7.95629e6	−	1.37807e7i	0.409459	−	0.709204i	0.994829	+	0.101562i	$0.0323840\pi$
$824$		0			0
$825$	2.65474e7			1.35796
$826$		0			0
$827$	−1.40824e7			−0.716001			−0.358001	−	0.933721i	$-0.616541\pi$
$827$	−1.40824e7			−0.716001			−0.358001	+	0.933721i	$0.616541\pi$
$828$		0			0
$829$	−1.09441e7	+	1.89558e7i	−0.553089	+	0.957978i	0.444960	+	0.895550i	$0.353218\pi$
$829$	−1.09441e7	+	1.89558e7i	−0.553089	+	0.957978i	−0.998049	+	0.0624281i	$0.980116\pi$
$830$		0			0
$831$	6.87992e6	+	1.19164e7i	0.345605	+	0.598606i
$832$		0			0
$833$	−1.35678e6	+	1.77883e7i	−0.0677480	+	0.888225i
$834$		0			0
$835$	2.43253e6	+	4.21326e6i	0.120737	+	0.209123i
$836$		0			0
$837$	−1.70050e7	+	2.94535e7i	−0.839003	+	1.45320i
$838$		0			0
$839$	1.98669e7			0.974372			0.487186	−	0.873298i	$-0.338023\pi$
$839$	1.98669e7			0.974372			0.487186	+	0.873298i	$0.338023\pi$
$840$		0			0
$841$	−2.04958e7			−0.999253
$842$		0			0
$843$	1.86017e7	−	3.22191e7i	0.901538	−	1.56151i
$844$		0			0
$845$	−3.44944e6	−	5.97460e6i	−0.166191	−	0.287851i
$846$		0			0
$847$	2.03254e7	+	2.19347e7i	0.973488	+	1.05057i
$848$		0			0
$849$	1.73076e7	+	2.99777e7i	0.824077	+	1.42734i
$850$		0			0
$851$	−1.12055e7	+	1.94085e7i	−0.530405	+	0.918688i
$852$		0			0
$853$	8.75258e6			0.411873			0.205937	−	0.978565i	$-0.433976\pi$
$853$	8.75258e6			0.411873			0.205937	+	0.978565i	$0.433976\pi$
$854$		0			0
$855$	−267089.			−0.0124951
$856$		0			0
$857$	977384.	−	1.69288e6i	0.0454583	−	0.0787361i	−0.842401	−	0.538851i	$-0.818859\pi$
$857$	977384.	−	1.69288e6i	0.0454583	−	0.0787361i	0.887859	+	0.460115i	$0.152192\pi$
$858$		0			0
$859$	2.49225e6	+	4.31670e6i	0.115241	+	0.199604i	0.917876	−	0.396867i	$-0.129902\pi$
$859$	2.49225e6	+	4.31670e6i	0.115241	+	0.199604i	−0.802635	+	0.596471i	$0.796569\pi$
$860$		0			0
$861$	1.04104e7	−	3.36704e7i	0.478586	−	1.54789i
$862$		0			0
$863$	−8.49654e6	−	1.47164e7i	−0.388343	−	0.672629i	0.603884	−	0.797072i	$-0.293619\pi$
$863$	−8.49654e6	−	1.47164e7i	−0.388343	−	0.672629i	−0.992227	+	0.124443i	$0.960286\pi$
$864$		0			0
$865$	1.41056e6	−	2.44316e6i	0.0640991	−	0.111023i
$866$		0			0
$867$	4.63285e6			0.209315
$868$		0			0
$869$	−1.86498e6			−0.0837772
$870$		0			0
$871$	−6.06200e6	+	1.04997e7i	−0.270751	+	0.468955i
$872$		0			0
$873$	390027.	+	675546.i	0.0173204	+	0.0299999i
$874$		0			0
$875$	−1.54207e7	+	3.50892e6i	−0.680901	+	0.154936i
$876$		0			0
$877$	−9.09797e6	−	1.57581e7i	−0.399434	−	0.691840i	0.594222	−	0.804301i	$-0.297460\pi$
$877$	−9.09797e6	−	1.57581e7i	−0.399434	−	0.691840i	−0.993656	+	0.112461i	$0.964127\pi$
$878$		0			0
$879$	−6.38225e6	+	1.10544e7i	−0.278613	+	0.482572i
$880$		0			0
$881$	1.77637e7			0.771068			0.385534	−	0.922694i	$-0.374017\pi$
$881$	1.77637e7			0.771068			0.385534	+	0.922694i	$0.374017\pi$
$882$		0			0
$883$	−1.71479e6			−0.0740131			−0.0370065	−	0.999315i	$-0.511782\pi$
$883$	−1.71479e6			−0.0740131			−0.0370065	+	0.999315i	$0.511782\pi$
$884$		0			0
$885$	−420613.	+	728523.i	−0.0180520	+	0.0312669i
$886$		0			0
$887$	−1.21218e7	−	2.09955e7i	−0.517318	−	0.896021i	−0.999798	−	0.0201139i	$-0.993597\pi$
$887$	−1.21218e7	−	2.09955e7i	−0.517318	−	0.896021i	0.482480	−	0.875907i	$-0.339736\pi$
$888$		0			0
$889$	2.37281e7	−	5.39924e6i	1.00695	−	0.229128i
$890$		0			0
$891$	1.89777e7	+	3.28704e7i	0.800847	+	1.38711i
$892$		0			0
$893$	1.76646e6	−	3.05960e6i	0.0741267	−	0.128391i
$894$		0			0
$895$	978272.			0.0408227
$896$		0			0
$897$	−1.21502e7			−0.504200
$898$		0			0
$899$	563624.	−	976225.i	0.0232589	−	0.0402857i
$900$		0			0
$901$	−9.92852e6	−	1.71967e7i	−0.407448	−	0.705721i
$902$		0			0
$903$	3.26923e6	−	1.05737e7i	0.133422	−	0.431526i
$904$		0			0
$905$	8.70671e6	+	1.50805e7i	0.353372	+	0.612059i
$906$		0			0
$907$	−9.52161e6	+	1.64919e7i	−0.384319	+	0.665661i	−0.991675	−	0.128770i	$-0.958897\pi$
$907$	−9.52161e6	+	1.64919e7i	−0.384319	+	0.665661i	0.607355	+	0.794430i	$0.292231\pi$
$908$		0			0
$909$	73932.0			0.00296772
$910$		0			0
$911$	3.18731e7			1.27242			0.636208	−	0.771518i	$-0.280502\pi$
$911$	3.18731e7			1.27242			0.636208	+	0.771518i	$0.280502\pi$
$912$		0			0
$913$	1.43809e7	−	2.49084e7i	0.570964	−	0.988939i
$914$		0			0
$915$	347589.	+	602041.i	0.0137250	+	0.0237724i
$916$		0			0
$917$	−1.36062e7	−	1.46835e7i	−0.534333	−	0.576641i
$918$		0			0
$919$	−1.47429e7	−	2.55355e7i	−0.575830	−	0.997367i	−0.995951	−	0.0898988i	$-0.971346\pi$
$919$	−1.47429e7	−	2.55355e7i	−0.575830	−	0.997367i	0.420121	−	0.907468i	$-0.361988\pi$
$920$		0			0
$921$	−1.67135e6	+	2.89486e6i	−0.0649259	+	0.112455i
$922$		0			0
$923$	−6.46927e6			−0.249949
$924$		0			0
$925$	1.61811e7			0.621804
$926$		0			0
$927$	−464881.	+	805197.i	−0.0177681	+	0.0307753i
$928$		0			0
$929$	4.86935e6	+	8.43397e6i	0.185111	+	0.320621i	0.943614	−	0.331048i	$-0.107402\pi$
$929$	4.86935e6	+	8.43397e6i	0.185111	+	0.320621i	−0.758503	+	0.651670i	$0.774069\pi$
$930$		0			0
$931$	−2.61384e7	−	1.78717e7i	−0.988335	−	0.675757i
$932$		0			0
$933$	2.30501e6	+	3.99240e6i	0.0866901	+	0.150152i
$934$		0			0
$935$	6.97559e6	−	1.20821e7i	0.260947	−	0.451973i
$936$		0			0
$937$	2.66734e7			0.992498			0.496249	−	0.868180i	$-0.334710\pi$
$937$	2.66734e7			0.992498			0.496249	+	0.868180i	$0.334710\pi$
$938$		0			0
$939$	3.15434e7			1.16747
$940$		0			0
$941$	−1.61432e6	+	2.79609e6i	−0.0594315	+	0.102938i	−0.894210	−	0.447647i	$-0.852262\pi$
$941$	−1.61432e6	+	2.79609e6i	−0.0594315	+	0.102938i	0.834779	+	0.550585i	$0.185595\pi$
$942$		0			0
$943$	−3.19728e7	−	5.53785e7i	−1.17085	−	2.02797i
$944$		0			0
$945$	−6.90871e6	−	7.45572e6i	−0.251662	−	0.271588i
$946$		0			0
$947$	−2.29622e6	−	3.97717e6i	−0.0832029	−	0.144112i	0.821421	−	0.570322i	$-0.193182\pi$
$947$	−2.29622e6	−	3.97717e6i	−0.0832029	−	0.144112i	−0.904624	+	0.426210i	$0.859848\pi$
$948$		0			0
$949$	−739434.	+	1.28074e6i	−0.0266523	+	0.0461631i
$950$		0			0
$951$	4.44011e7			1.59200
$952$		0			0
$953$	2.97125e7			1.05976			0.529880	−	0.848073i	$-0.322237\pi$
$953$	2.97125e7			1.05976			0.529880	+	0.848073i	$0.322237\pi$
$954$		0			0
$955$	4.95512e6	−	8.58253e6i	0.175811	−	0.304514i
$956$		0			0
$957$	−611993.	−	1.06000e6i	−0.0216006	−	0.0374134i
$958$		0			0
$959$	4.18323e6	−	1.35299e7i	0.146881	−	0.475058i
$960$		0			0
$961$	−2.71748e7	−	4.70681e7i	−0.949199	−	1.64406i
$962$		0			0
$963$	255092.	−	441833.i	0.00886403	−	0.0153530i
$964$		0			0
$965$	−1.44554e7			−0.499704
$966$		0			0
$967$	−7.64435e6			−0.262890			−0.131445	−	0.991323i	$-0.541962\pi$
$967$	−7.64435e6			−0.262890			−0.131445	+	0.991323i	$0.541962\pi$
$968$		0			0
$969$	1.58017e7	−	2.73694e7i	0.540624	−	0.936388i
$970$		0			0
$971$	−2.49810e7	−	4.32684e7i	−0.850281	−	1.47273i	−0.880955	−	0.473200i	$-0.843099\pi$
$971$	−2.49810e7	−	4.32684e7i	−0.850281	−	1.47273i	0.0306743	−	0.999529i	$-0.490235\pi$
$972$		0			0
$973$	−2.58756e7	+	5.88790e6i	−0.876212	+	0.199378i
$974$		0			0
$975$	4.38632e6	+	7.59734e6i	0.147771	+	0.255947i
$976$		0			0
$977$	−2.02697e7	+	3.51082e7i	−0.679378	+	1.17672i	0.295791	+	0.955253i	$0.404417\pi$
$977$	−2.02697e7	+	3.51082e7i	−0.679378	+	1.17672i	−0.975169	+	0.221464i	$0.928916\pi$
$978$		0			0
$979$	6.19865e7			2.06700
$980$		0			0
$981$	300592.			0.00997251
$982$		0			0
$983$	−2.06528e7	+	3.57717e7i	−0.681703	+	1.18074i	0.292757	+	0.956187i	$0.405427\pi$
$983$	−2.06528e7	+	3.57717e7i	−0.681703	+	1.18074i	−0.974461	+	0.224558i	$0.927906\pi$
$984$		0			0
$985$	3.27511e6	+	5.67265e6i	0.107556	+	0.186292i
$986$		0			0
$987$	−3.74623e6	+	852439.i	−0.122406	+	0.0278529i
$988$		0			0
$989$	−1.00406e7	−	1.73908e7i	−0.326413	−	0.565364i
$990$		0			0
$991$	−9.20419e6	+	1.59421e7i	−0.297715	+	0.515658i	−0.975613	−	0.219498i	$-0.929558\pi$
$991$	−9.20419e6	+	1.59421e7i	−0.297715	+	0.515658i	0.677897	+	0.735156i	$0.262891\pi$
$992$		0			0
$993$	2.18167e7			0.702126
$994$		0			0
$995$	−6.03144e6			−0.193136
$996$		0			0
$997$	1.69854e7	−	2.94196e7i	0.541175	−	0.937343i	−0.457662	−	0.889126i	$-0.651313\pi$
$997$	1.69854e7	−	2.94196e7i	0.541175	−	0.937343i	0.998837	−	0.0482164i	$-0.0153537\pi$
$998$		0			0
$999$	1.12543e7	+	1.94931e7i	0.356784	+	0.617968i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.i.b.81.2		4
4.3	odd	2		14.6.c.b.11.1	yes	4
7.2	even	3	inner	112.6.i.b.65.2		4
7.3	odd	6		784.6.a.r.1.2		2
7.4	even	3		784.6.a.bc.1.1		2
12.11	even	2		126.6.g.e.109.2		4
28.3	even	6		98.6.a.f.1.1		2
28.11	odd	6		98.6.a.c.1.2		2
28.19	even	6		98.6.c.f.79.2		4
28.23	odd	6		14.6.c.b.9.1	✓	4
28.27	even	2		98.6.c.f.67.2		4
84.11	even	6		882.6.a.bt.1.2		2
84.23	even	6		126.6.g.e.37.2		4
84.59	odd	6		882.6.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.1	✓	4	28.23	odd	6
14.6.c.b.11.1	yes	4	4.3	odd	2
98.6.a.c.1.2		2	28.11	odd	6
98.6.a.f.1.1		2	28.3	even	6
98.6.c.f.67.2		4	28.27	even	2
98.6.c.f.79.2		4	28.19	even	6
112.6.i.b.65.2		4	7.2	even	3	inner
112.6.i.b.81.2		4	1.1	even	1	trivial
126.6.g.e.37.2		4	84.23	even	6
126.6.g.e.109.2		4	12.11	even	2
784.6.a.r.1.2		2	7.3	odd	6
784.6.a.bc.1.1		2	7.4	even	3
882.6.a.bl.1.2		2	84.59	odd	6
882.6.a.bt.1.2		2	84.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.90175 - 13.6862i) q^{3} +(10.5000 + 18.1865i) q^{5} +(-126.411 + 28.7642i) q^{7} +(-3.37544 - 5.84643i) q^{9} +(-312.937 + 542.023i) q^{11} -206.821 q^{13} +331.874 q^{15} +(-530.732 + 919.254i) q^{17} +(-941.989 - 1631.57i) q^{19} +(-605.191 + 1957.37i) q^{21} +(1858.68 + 3219.34i) q^{23} +(1342.00 - 2324.41i) q^{25} +3733.56 q^{27} -123.747 q^{29} +(-4554.63 + 7888.85i) q^{31} +(4945.50 + 8565.86i) q^{33} +(-1850.43 - 1996.95i) q^{35} +(3014.36 + 5221.03i) q^{37} +(-1634.25 + 2830.60i) q^{39} -17201.9 q^{41} -5401.98 q^{43} +(70.8843 - 122.775i) q^{45} +(937.621 + 1624.01i) q^{47} +(15152.2 - 7272.19i) q^{49} +(8387.42 + 14527.4i) q^{51} +(-9353.62 + 16200.9i) q^{53} -13143.3 q^{55} -29773.5 q^{57} +(-1267.39 + 2195.18i) q^{59} +(1047.35 + 1814.07i) q^{61} +(594.859 + 641.959i) q^{63} +(-2171.62 - 3761.36i) q^{65} +(29310.4 - 50767.1i) q^{67} +58747.5 q^{69} +31279.5 q^{71} +(3575.24 - 6192.49i) q^{73} +(-21208.3 - 36733.9i) q^{75} +(23967.7 - 77518.7i) q^{77} +(1489.91 + 2580.59i) q^{79} +(30321.9 - 52519.1i) q^{81} -45954.6 q^{83} -22290.7 q^{85} +(-977.821 + 1693.64i) q^{87} +(-49520.0 - 85771.1i) q^{89} +(26144.4 - 5949.04i) q^{91} +(71979.2 + 124672. i) q^{93} +(19781.8 - 34263.0i) q^{95} -115548. q^{97} +4225.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{3} + 42 q^{5} - 232 q^{7} - 652 q^{9} - 294 q^{11} - 280 q^{13} - 588 q^{15} - 1302 q^{17} - 1442 q^{19} + 5150 q^{21} + 2646 q^{23} + 5368 q^{25} + 31444 q^{27} + 3336 q^{29} - 14798 q^{31}+ \cdots - 419832 q^{99}+O(q^{100}) \) 4 * q - 14 * q^3 + 42 * q^5 - 232 * q^7 - 652 * q^9 - 294 * q^11 - 280 * q^13 - 588 * q^15 - 1302 * q^17 - 1442 * q^19 + 5150 * q^21 + 2646 * q^23 + 5368 * q^25 + 31444 * q^27 + 3336 * q^29 - 14798 * q^31 + 19782 * q^33 + 1218 * q^35 - 5182 * q^37 - 5260 * q^39 - 10248 * q^41 + 9040 * q^43 + 13692 * q^45 - 14994 * q^47 - 2876 * q^49 + 9606 * q^51 - 24006 * q^53 - 12348 * q^55 - 85892 * q^57 + 38850 * q^59 + 23618 * q^61 - 149948 * q^63 - 2940 * q^65 + 32002 * q^67 + 181356 * q^69 + 178752 * q^71 + 47138 * q^73 + 37576 * q^75 - 22890 * q^77 - 40970 * q^79 - 139858 * q^81 - 136752 * q^83 - 54684 * q^85 - 55356 * q^87 - 123102 * q^89 + 53680 * q^91 - 25586 * q^93 + 30282 * q^95 - 87304 * q^97 - 419832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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