LMFDB - Embedded newform 112.7.s.c.17.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,7,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$25.7660573654$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 $ x^8 - 2x^7 + 285x^6 + 282x^5 + 62091x^4 + 29260x^3 + 4838750x^2 + 2401000*x + 294122500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.2
Root			$7.51287 + 13.0127i$ of defining polynomial
Character	$\chi$	$=$	112.17
Dual form			112.7.s.c.33.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-21.8891 - 12.6377i) q^{3} +(-10.7486 + 6.20573i) q^{5} +(-195.705 - 281.689i) q^{7} +(-45.0775 - 78.0766i) q^{9} +(-774.831 + 1342.05i) q^{11} -2770.09i q^{13} +313.705 q^{15} +(-109.615 - 63.2864i) q^{17} +(1147.81 - 662.687i) q^{19} +(723.904 + 8639.18i) q^{21} +(-7119.76 - 12331.8i) q^{23} +(-7735.48 + 13398.2i) q^{25} +20704.5i q^{27} -7479.03 q^{29} +(49241.9 + 28429.8i) q^{31} +(33920.8 - 19584.2i) q^{33} +(3851.65 + 1813.28i) q^{35} +(45005.7 + 77952.2i) q^{37} +(-35007.6 + 60634.9i) q^{39} +35783.9i q^{41} +79422.8 q^{43} +(969.044 + 559.478i) q^{45} +(-126334. + 72939.0i) q^{47} +(-41048.4 + 110256. i) q^{49} +(1599.59 + 2770.57i) q^{51} +(85090.3 - 147381. i) q^{53} -19233.6i q^{55} -33499.3 q^{57} +(187919. + 108495. i) q^{59} +(-172492. + 99588.5i) q^{61} +(-13171.4 + 27977.8i) q^{63} +(17190.5 + 29774.8i) q^{65} +(72219.0 - 125087. i) q^{67} +359909. i q^{69} -407591. q^{71} +(-161733. - 93376.3i) q^{73} +(338646. - 195517. i) q^{75} +(529678. - 44383.4i) q^{77} +(41929.3 + 72623.6i) q^{79} +(228795. - 396285. i) q^{81} -162495. i q^{83} +1570.95 q^{85} +(163709. + 94517.7i) q^{87} +(-439806. + 253922. i) q^{89} +(-780305. + 542120. i) q^{91} +(-718575. - 1.24461e6i) q^{93} +(-8224.91 + 14246.0i) q^{95} -509740. i q^{97} +139710. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) $ 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Character values

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

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

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
$3$	−21.8891	−	12.6377i	−0.810708	−	0.468063i	0.0364935	−	0.999334i	$-0.488381\pi$
$3$	−21.8891	−	12.6377i	−0.810708	−	0.468063i	−0.847202	+	0.531271i	$0.821715\pi$
$4$		0			0
$5$	−10.7486	+	6.20573i	−0.0859892	+	0.0496459i	−0.542378	−	0.840135i	$-0.682476\pi$
$5$	−10.7486	+	6.20573i	−0.0859892	+	0.0496459i	0.456389	+	0.889780i	$0.349143\pi$
$6$		0			0
$7$	−195.705	−	281.689i	−0.570567	−	0.821251i
$7$	−195.705	−	281.689i	−0.570567	−	0.821251i
$8$		0			0
$9$	−45.0775	−	78.0766i	−0.0618347	−	0.107101i
$10$		0			0
$11$	−774.831	+	1342.05i	−0.582142	+	1.00830i	0.413083	+	0.910693i	$0.364452\pi$
$11$	−774.831	+	1342.05i	−0.582142	+	1.00830i	−0.995225	+	0.0976064i	$0.968881\pi$
$12$		0			0
$13$		−	2770.09i		−	1.26085i	−0.776249	−	0.630427i	$-0.782880\pi$
$13$		−	2770.09i		−	1.26085i	0.776249	−	0.630427i	$-0.217120\pi$
$14$		0			0
$15$	313.705			0.0929495
$16$		0			0
$17$	−109.615	−	63.2864i	−0.0223113	−	0.0128814i	0.488803	−	0.872394i	$-0.337434\pi$
$17$	−109.615	−	63.2864i	−0.0223113	−	0.0128814i	−0.511114	+	0.859513i	$0.670767\pi$
$18$		0			0
$19$	1147.81	−	662.687i	0.167343	−	0.0966156i	−0.413989	−	0.910282i	$-0.635865\pi$
$19$	1147.81	−	662.687i	0.167343	−	0.0966156i	0.581332	+	0.813666i	$0.302532\pi$
$20$		0			0
$21$	723.904	+	8639.18i	0.0781669	+	0.932856i
$22$		0			0
$23$	−7119.76	−	12331.8i	−0.585170	−	1.01354i	−0.994854	−	0.101316i	$-0.967695\pi$
$23$	−7119.76	−	12331.8i	−0.585170	−	1.01354i	0.409685	−	0.912227i	$-0.365639\pi$
$24$		0			0
$25$	−7735.48	+	13398.2i	−0.495071	+	0.857487i
$26$		0			0
$27$			20704.5i			1.05190i
$28$		0			0
$29$	−7479.03			−0.306656			−0.153328	−	0.988175i	$-0.548999\pi$
$29$	−7479.03			−0.306656			−0.153328	+	0.988175i	$0.548999\pi$
$30$		0			0
$31$	49241.9	+	28429.8i	1.65291	+	0.954309i	0.975866	+	0.218369i	$0.0700735\pi$
$31$	49241.9	+	28429.8i	1.65291	+	0.954309i	0.677046	+	0.735941i	$0.263260\pi$
$32$		0			0
$33$	33920.8	−	19584.2i	0.943895	−	0.544958i
$34$		0			0
$35$	3851.65	+	1813.28i	0.0898343	+	0.0422924i
$36$		0			0
$37$	45005.7	+	77952.2i	0.888511	+	1.53895i	0.841636	+	0.540046i	$0.181593\pi$
$37$	45005.7	+	77952.2i	0.888511	+	1.53895i	0.0468752	+	0.998901i	$0.485074\pi$
$38$		0			0
$39$	−35007.6	+	60634.9i	−0.590158	+	1.02218i
$40$		0			0
$41$			35783.9i			0.519202i	0.965716	+	0.259601i	$0.0835910\pi$
$41$			35783.9i			0.519202i	−0.965716	+	0.259601i	$0.916409\pi$
$42$		0			0
$43$	79422.8			0.998942			0.499471	−	0.866331i	$-0.333528\pi$
$43$	79422.8			0.998942			0.499471	+	0.866331i	$0.333528\pi$
$44$		0			0
$45$	969.044	+	559.478i	0.0106342	+	0.00613968i
$46$		0			0
$47$	−126334.	+	72939.0i	−1.21682	+	0.702532i	−0.964237	−	0.265043i	$-0.914614\pi$
$47$	−126334.	+	72939.0i	−1.21682	+	0.702532i	−0.252585	+	0.967575i	$0.581281\pi$
$48$		0			0
$49$	−41048.4	+	110256.i	−0.348906	+	0.937158i
$50$		0			0
$51$	1599.59	+	2770.57i	0.0120586	+	0.0208861i
$52$		0			0
$53$	85090.3	−	147381.i	0.571547	−	0.989949i	−0.424860	−	0.905259i	$-0.639677\pi$
$53$	85090.3	−	147381.i	0.571547	−	0.989949i	0.996407	−	0.0846900i	$-0.0269900\pi$
$54$		0			0
$55$		−	19233.6i		−	0.115604i
$56$		0			0
$57$	−33499.3			−0.180889
$58$		0			0
$59$	187919.	+	108495.i	0.914989	+	0.528269i	0.882033	−	0.471188i	$-0.156175\pi$
$59$	187919.	+	108495.i	0.914989	+	0.528269i	0.0329557	+	0.999457i	$0.489508\pi$
$60$		0			0
$61$	−172492.	+	99588.5i	−0.759942	+	0.438753i	−0.829275	−	0.558841i	$-0.811246\pi$
$61$	−172492.	+	99588.5i	−0.759942	+	0.438753i	0.0693329	+	0.997594i	$0.477913\pi$
$62$		0			0
$63$	−13171.4	+	27977.8i	−0.0526758	+	0.111890i
$64$		0			0
$65$	17190.5	+	29774.8i	0.0625962	+	0.108420i
$66$		0			0
$67$	72219.0	−	125087.i	0.240119	−	0.415899i	−0.720629	−	0.693321i	$-0.756147\pi$
$67$	72219.0	−	125087.i	0.240119	−	0.415899i	0.960748	+	0.277422i	$0.0894801\pi$
$68$		0			0
$69$			359909.i			1.09558i
$70$		0			0
$71$	−407591.			−1.13880			−0.569402	−	0.822059i	$-0.692825\pi$
$71$	−407591.			−1.13880			−0.569402	+	0.822059i	$0.692825\pi$
$72$		0			0
$73$	−161733.	−	93376.3i	−0.415747	−	0.240032i	0.277509	−	0.960723i	$-0.410491\pi$
$73$	−161733.	−	93376.3i	−0.415747	−	0.240032i	−0.693256	+	0.720691i	$0.743824\pi$
$74$		0			0
$75$	338646.	−	195517.i	0.802716	−	0.463448i
$76$		0			0
$77$	529678.	−	44383.4i	1.16022	−	0.0972183i
$78$		0			0
$79$	41929.3	+	72623.6i	0.0850425	+	0.147298i	0.905409	−	0.424540i	$-0.139564\pi$
$79$	41929.3	+	72623.6i	0.0850425	+	0.147298i	−0.820367	+	0.571838i	$0.806231\pi$
$80$		0			0
$81$	228795.	−	396285.i	0.430518	−	0.745679i
$82$		0			0
$83$		−	162495.i		−	0.284188i	−0.989853	−	0.142094i	$-0.954616\pi$
$83$		−	162495.i		−	0.284188i	0.989853	−	0.142094i	$-0.0453836\pi$
$84$		0			0
$85$	1570.95			0.00255804
$86$		0			0
$87$	163709.	+	94517.7i	0.248609	+	0.143534i
$88$		0			0
$89$	−439806.	+	253922.i	−0.623866	+	0.360189i	−0.778373	−	0.627803i	$-0.783954\pi$
$89$	−439806.	+	253922.i	−0.623866	+	0.360189i	0.154507	+	0.987992i	$0.450621\pi$
$90$		0			0
$91$	−780305.	+	542120.i	−1.03548	+	0.719402i
$92$		0			0
$93$	−718575.	−	1.24461e6i	−0.893353	−	1.54733i
$94$		0			0
$95$	−8224.91	+	14246.0i	−0.00959313	+	0.0166158i
$96$		0			0
$97$		−	509740.i		−	0.558514i	−0.960216	−	0.279257i	$-0.909912\pi$
$97$		−	509740.i		−	0.558514i	0.960216	−	0.279257i	$-0.0900881\pi$
$98$		0			0
$99$	139710.			0.143986
$100$		0			0
$101$	1.35057e6	+	779754.i	1.31085	+	0.756821i	0.982237	−	0.187642i	$-0.0600846\pi$
$101$	1.35057e6	+	779754.i	1.31085	+	0.756821i	0.328616	+	0.944464i	$0.393418\pi$
$102$		0			0
$103$	629819.	−	363626.i	0.576374	−	0.332769i	−0.183317	−	0.983054i	$-0.558684\pi$
$103$	629819.	−	363626.i	0.576374	−	0.332769i	0.759691	+	0.650284i	$0.225350\pi$
$104$		0			0
$105$	−61393.4	−	88367.1i	−0.0530339	−	0.0763348i
$106$		0			0
$107$	−874822.	−	1.51524e6i	−0.714116	−	1.23688i	−0.963300	−	0.268428i	$-0.913496\pi$
$107$	−874822.	−	1.51524e6i	−0.714116	−	1.23688i	0.249184	−	0.968456i	$-0.419838\pi$
$108$		0			0
$109$	386426.	−	669309.i	0.298392	−	0.516830i	−0.677376	−	0.735637i	$-0.736883\pi$
$109$	386426.	−	669309.i	0.298392	−	0.516830i	0.975768	+	0.218807i	$0.0702165\pi$
$110$		0			0
$111$		−	2.27508e6i		−	1.66352i
$112$		0			0
$113$	−2.26586e6			−1.57035			−0.785176	−	0.619273i	$-0.787428\pi$
$113$	−2.26586e6			−1.57035			−0.785176	+	0.619273i	$0.787428\pi$
$114$		0			0
$115$	153056.	+	88366.7i	0.100637	+	0.0581025i
$116$		0			0
$117$	−216279.	+	124869.i	−0.135039	+	0.0779645i
$118$		0			0
$119$	3625.13	+	43262.8i	0.00215121	+	0.0256729i
$120$		0			0
$121$	−314946.	−	545503.i	−0.177779	−	0.307922i
$122$		0			0
$123$	452226.	−	783278.i	0.243019	−	0.420921i
$124$		0			0
$125$		−	385946.i		−	0.197605i
$126$		0			0
$127$	−813391.			−0.397089			−0.198545	−	0.980092i	$-0.563621\pi$
$127$	−813391.			−0.397089			−0.198545	+	0.980092i	$0.563621\pi$
$128$		0			0
$129$	−1.73850e6	−	1.00372e6i	−0.809850	−	0.467567i
$130$		0			0
$131$	−2.65677e6	+	1.53389e6i	−1.18179	+	0.682307i	−0.956428	−	0.291968i	$-0.905690\pi$
$131$	−2.65677e6	+	1.53389e6i	−1.18179	+	0.682307i	−0.225362	+	0.974275i	$0.572357\pi$
$132$		0			0
$133$	−411303.	−	193634.i	−0.174826	−	0.0823050i
$134$		0			0
$135$	−128486.	−	222545.i	−0.0522223	−	0.0904516i
$136$		0			0
$137$	314000.	−	543865.i	0.122115	−	0.211509i	−0.798487	−	0.602013i	$-0.794366\pi$
$137$	314000.	−	543865.i	0.122115	−	0.211509i	0.920601	+	0.390503i	$0.127699\pi$
$138$		0			0
$139$			2.79572e6i			1.04100i	0.853863	+	0.520498i	$0.174254\pi$
$139$			2.79572e6i			1.04100i	−0.853863	+	0.520498i	$0.825746\pi$
$140$		0			0
$141$	3.68712e6			1.31532
$142$		0			0
$143$	3.71760e6	+	2.14636e6i	1.27132	+	0.733996i
$144$		0			0
$145$	80389.5	−	46412.9i	0.0263691	−	0.0152242i
$146$		0			0
$147$	2.29189e6	−	1.89464e6i	0.721509	−	0.596452i
$148$		0			0
$149$	2.04538e6	+	3.54271e6i	0.618324	+	1.07097i	0.989792	+	0.142522i	$0.0455212\pi$
$149$	2.04538e6	+	3.54271e6i	0.618324	+	1.07097i	−0.371468	+	0.928446i	$0.621145\pi$
$150$		0			0
$151$	819369.	−	1.41919e6i	0.237985	−	0.412201i	−0.722151	−	0.691735i	$-0.756847\pi$
$151$	819369.	−	1.41919e6i	0.237985	−	0.412201i	0.960136	+	0.279534i	$0.0901799\pi$
$152$		0			0
$153$			11411.2i			0.00318607i
$154$		0			0
$155$	−705712.			−0.189510
$156$		0			0
$157$	4.98347e6	+	2.87721e6i	1.28775	+	0.743485i	0.978253	−	0.207414i	$-0.0665046\pi$
$157$	4.98347e6	+	2.87721e6i	1.28775	+	0.743485i	0.309501	+	0.950899i	$0.399838\pi$
$158$		0			0
$159$	−3.72510e6	+	2.15069e6i	−0.926716	+	0.535040i
$160$		0			0
$161$	−2.08036e6	+	4.41895e6i	−0.498495	+	1.05887i
$162$		0			0
$163$	478970.	+	829600.i	0.110597	+	0.191560i	0.916011	−	0.401152i	$-0.131390\pi$
$163$	478970.	+	829600.i	0.110597	+	0.191560i	−0.805414	+	0.592713i	$0.798057\pi$
$164$		0			0
$165$	−243068.	+	421006.i	−0.0541098	+	0.0937210i
$166$		0			0
$167$			6.70923e6i			1.44053i	0.693698	+	0.720266i	$0.255980\pi$
$167$			6.70923e6i			1.44053i	−0.693698	+	0.720266i	$0.744020\pi$
$168$		0			0
$169$	−2.84662e6			−0.589751
$170$		0			0
$171$	−103481.	−	59744.5i	−0.0206952	−	0.0119484i
$172$		0			0
$173$	−300592.	+	173547.i	−0.0580550	+	0.0335181i	−0.528747	−	0.848780i	$-0.677338\pi$
$173$	−300592.	+	173547.i	−0.0580550	+	0.0335181i	0.470692	+	0.882298i	$0.344004\pi$
$174$		0			0
$175$	5.28801e6	−	443098.i	0.986683	−	0.0826772i
$176$		0			0
$177$	−2.74226e6	−	4.74974e6i	−0.494526	−	0.856544i
$178$		0			0
$179$	−1.47270e6	+	2.55079e6i	−0.256777	+	0.444750i	−0.965377	−	0.260860i	$-0.915994\pi$
$179$	−1.47270e6	+	2.55079e6i	−0.256777	+	0.444750i	0.708600	+	0.705611i	$0.249327\pi$
$180$		0			0
$181$			2.88805e6i			0.487046i	0.969895	+	0.243523i	$0.0783031\pi$
$181$			2.88805e6i			0.487046i	−0.969895	+	0.243523i	$0.921697\pi$
$182$		0			0
$183$	5.03428e6			0.821455
$184$		0			0
$185$	−967502.	−	558587.i	−0.152805	−	0.0882218i
$186$		0			0
$187$	169867.	−	98072.5i	0.0259767	−	0.0149976i
$188$		0			0
$189$	5.83222e6	−	4.05196e6i	0.863870	−	0.600177i
$190$		0			0
$191$	−2.83365e6	−	4.90802e6i	−0.406674	−	0.704379i	0.587841	−	0.808976i	$-0.299978\pi$
$191$	−2.83365e6	−	4.90802e6i	−0.406674	−	0.704379i	−0.994515	+	0.104597i	$0.966645\pi$
$192$		0			0
$193$	−6.54313e6	+	1.13330e7i	−0.910152	+	1.57643i	−0.0963038	+	0.995352i	$0.530702\pi$
$193$	−6.54313e6	+	1.13330e7i	−0.910152	+	1.57643i	−0.813848	+	0.581078i	$0.802631\pi$
$194$		0			0
$195$		−	868991.i		−	0.117196i
$196$		0			0
$197$	4.47088e6			0.584783			0.292391	−	0.956299i	$-0.405549\pi$
$197$	4.47088e6			0.584783			0.292391	+	0.956299i	$0.405549\pi$
$198$		0			0
$199$	−6.97809e6	−	4.02880e6i	−0.885477	−	0.511230i	−0.0130165	−	0.999915i	$-0.504143\pi$
$199$	−6.97809e6	−	4.02880e6i	−0.885477	−	0.511230i	−0.872460	+	0.488685i	$0.837477\pi$
$200$		0			0
$201$	−3.16162e6	+	1.82536e6i	−0.389333	+	0.224782i
$202$		0			0
$203$	1.46368e6	+	2.10676e6i	0.174968	+	0.251841i
$204$		0			0
$205$	−222065.	−	384628.i	−0.0257762	−	0.0446457i
$206$		0			0
$207$	−641882.	+	1.11177e6i	−0.0723676	+	0.125344i
$208$		0			0
$209$			2.05388e6i			0.224976i
$210$		0			0
$211$	−421809.			−0.0449023			−0.0224511	−	0.999748i	$-0.507147\pi$
$211$	−421809.			−0.0449023			−0.0224511	+	0.999748i	$0.507147\pi$
$212$		0			0
$213$	8.92180e6	+	5.15101e6i	0.923238	+	0.533032i
$214$		0			0
$215$	−853688.	+	492877.i	−0.0858981	+	0.0495933i
$216$		0			0
$217$	−1.62850e6	−	1.94348e7i	−0.159371	−	1.90195i
$218$		0			0
$219$	2.36012e6	+	4.08785e6i	0.224700	+	0.389191i
$220$		0			0
$221$	−175309.	+	303645.i	−0.0162416	+	0.0281312i
$222$		0			0
$223$			4.49399e6i			0.405245i	0.979257	+	0.202622i	$0.0649464\pi$
$223$			4.49399e6i			0.405245i	−0.979257	+	0.202622i	$0.935054\pi$
$224$		0			0
$225$	1.39478e6			0.122450
$226$		0			0
$227$	3.37074e6	+	1.94610e6i	0.288169	+	0.166375i	0.637116	−	0.770768i	$-0.280127\pi$
$227$	3.37074e6	+	1.94610e6i	0.288169	+	0.166375i	−0.348947	+	0.937143i	$0.613461\pi$
$228$		0			0
$229$	−2.13533e6	+	1.23283e6i	−0.177811	+	0.102659i	−0.586264	−	0.810120i	$-0.699402\pi$
$229$	−2.13533e6	+	1.23283e6i	−0.177811	+	0.102659i	0.408453	+	0.912779i	$0.366068\pi$
$230$		0			0
$231$	−1.21551e7	−	5.72239e6i	−0.986103	−	0.464239i
$232$		0			0
$233$	−1.13486e7	−	1.96564e7i	−0.897170	−	1.55394i	−0.831095	−	0.556130i	$-0.812286\pi$
$233$	−1.13486e7	−	1.96564e7i	−0.897170	−	1.55394i	−0.0660751	−	0.997815i	$-0.521048\pi$
$234$		0			0
$235$	905280.	−	1.56799e6i	0.0697556	−	0.120820i
$236$		0			0
$237$		−	2.11956e6i		−	0.159221i
$238$		0			0
$239$	3.91448e6			0.286734			0.143367	−	0.989670i	$-0.454207\pi$
$239$	3.91448e6			0.286734			0.143367	+	0.989670i	$0.454207\pi$
$240$		0			0
$241$	8.50987e6	+	4.91317e6i	0.607955	+	0.351003i	0.772165	−	0.635422i	$-0.219174\pi$
$241$	8.50987e6	+	4.91317e6i	0.607955	+	0.351003i	−0.164209	+	0.986425i	$0.552507\pi$
$242$		0			0
$243$	3.05515e6	−	1.76389e6i	0.212919	−	0.122929i
$244$		0			0
$245$	−243003.	−	1.43983e6i	−0.0165239	−	0.0979071i
$246$		0			0
$247$	−1.83570e6	−	3.17953e6i	−0.121818	−	0.210995i
$248$		0			0
$249$	−2.05356e6	+	3.55688e6i	−0.133018	+	0.230394i
$250$		0			0
$251$		−	1.12666e7i		−	0.712477i	−0.934395	−	0.356238i	$-0.884059\pi$
$251$		−	1.12666e7i		−	0.712477i	0.934395	−	0.356238i	$-0.115941\pi$
$252$		0			0
$253$	2.20664e7			1.36261
$254$		0			0
$255$	−34386.8	−	19853.2i	−0.00207382	−	0.00119732i
$256$		0			0
$257$	−2.70095e7	+	1.55940e7i	−1.59117	+	0.918665i	−0.598067	+	0.801446i	$0.704065\pi$
$257$	−2.70095e7	+	1.55940e7i	−1.59117	+	0.918665i	−0.993106	+	0.117219i	$0.962602\pi$
$258$		0			0
$259$	1.31505e7	−	2.79332e7i	0.756905	−	1.60776i
$260$		0			0
$261$	337136.	+	583937.i	0.0189620	+	0.0328431i
$262$		0			0
$263$	−6.75928e6	+	1.17074e7i	−0.371564	+	0.643567i	−0.989806	−	0.142420i	$-0.954512\pi$
$263$	−6.75928e6	+	1.17074e7i	−0.371564	+	0.643567i	0.618243	+	0.785987i	$0.287845\pi$
$264$		0			0
$265$			2.11219e6i			0.113500i
$266$		0			0
$267$	1.28360e7			0.674364
$268$		0			0
$269$	−2.25818e7	−	1.30376e7i	−1.16011	−	0.669792i	−0.208783	−	0.977962i	$-0.566950\pi$
$269$	−2.25818e7	−	1.30376e7i	−1.16011	−	0.669792i	−0.951331	+	0.308170i	$0.900284\pi$
$270$		0			0
$271$	−9.56566e6	+	5.52274e6i	−0.480626	+	0.277490i	−0.720677	−	0.693271i	$-0.756169\pi$
$271$	−9.56566e6	+	5.52274e6i	−0.480626	+	0.277490i	0.240051	+	0.970760i	$0.422836\pi$
$272$		0			0
$273$	2.39313e7	−	2.00528e6i	1.17619	−	0.0985570i
$274$		0			0
$275$	−1.19874e7	−	2.07627e7i	−0.576403	−	0.998359i
$276$		0			0
$277$	7.97353e6	−	1.38106e7i	0.375155	−	0.649788i	−0.615195	−	0.788375i	$-0.710923\pi$
$277$	7.97353e6	−	1.38106e7i	0.375155	−	0.649788i	0.990350	+	0.138587i	$0.0442560\pi$
$278$		0			0
$279$		−	5.12618e6i		−	0.236038i
$280$		0			0
$281$	−1.25628e6			−0.0566198			−0.0283099	−	0.999599i	$-0.509013\pi$
$281$	−1.25628e6			−0.0566198			−0.0283099	+	0.999599i	$0.509013\pi$
$282$		0			0
$283$	−292144.	−	168669.i	−0.0128896	−	0.00744179i	0.493541	−	0.869722i	$-0.335702\pi$
$283$	−292144.	−	168669.i	−0.0128896	−	0.00744179i	−0.506431	+	0.862281i	$0.669035\pi$
$284$		0			0
$285$	360072.	−	207888.i	0.0155545	−	0.00898037i
$286$		0			0
$287$	1.00799e7	−	7.00307e6i	0.426395	−	0.296239i
$288$		0			0
$289$	−1.20608e7	−	2.08899e7i	−0.499668	−	0.865451i
$290$		0			0
$291$	−6.44194e6	+	1.11578e7i	−0.261419	+	0.452792i
$292$		0			0
$293$			1.30355e7i			0.518233i	0.965846	+	0.259117i	$0.0834314\pi$
$293$			1.30355e7i			0.518233i	−0.965846	+	0.259117i	$0.916569\pi$
$294$		0			0
$295$	−2.69317e6			−0.104905
$296$		0			0
$297$	−2.77864e7	−	1.60425e7i	−1.06063	−	0.612353i
$298$		0			0
$299$	−3.41602e7	+	1.97224e7i	−1.27793	+	0.737813i
$300$		0			0
$301$	−1.55434e7	−	2.23725e7i	−0.569963	−	0.820382i
$302$		0			0
$303$	−1.97086e7	−	3.41363e7i	−0.708480	−	1.22712i
$304$		0			0
$305$	1.23604e6	−	2.14088e6i	0.0435645	−	0.0754560i
$306$		0			0
$307$			3.48621e7i			1.20487i	0.798170	+	0.602433i	$0.205802\pi$
$307$			3.48621e7i			1.20487i	−0.798170	+	0.602433i	$0.794198\pi$
$308$		0			0
$309$	−1.83816e7			−0.623028
$310$		0			0
$311$	3.81969e7	+	2.20530e7i	1.26983	+	0.733139i	0.974956	−	0.222397i	$-0.0713880\pi$
$311$	3.81969e7	+	2.20530e7i	1.26983	+	0.733139i	0.294877	+	0.955535i	$0.404721\pi$
$312$		0			0
$313$	3.77077e7	−	2.17706e7i	1.22970	−	0.709965i	0.262729	−	0.964870i	$-0.415377\pi$
$313$	3.77077e7	−	2.17706e7i	1.22970	−	0.709965i	0.966966	+	0.254905i	$0.0820441\pi$
$314$		0			0
$315$	−32047.6	−	382462.i	−0.00102533	−	0.0122365i
$316$		0			0
$317$	1.63697e7	+	2.83531e7i	0.513880	+	0.890067i	0.999870	+	0.0161023i	$0.00512574\pi$
$317$	1.63697e7	+	2.83531e7i	0.513880	+	0.890067i	−0.485990	+	0.873964i	$0.661541\pi$
$318$		0			0
$319$	5.79499e6	−	1.00372e7i	0.178517	−	0.309201i
$320$		0			0
$321$			4.42229e7i			1.33700i
$322$		0			0
$323$	−167756.			−0.00497818
$324$		0			0
$325$	3.71144e7	+	2.14280e7i	1.08117	+	0.624211i
$326$		0			0
$327$	−1.69170e7	+	9.76706e6i	−0.483817	+	0.279332i
$328$		0			0
$329$	4.52703e7	+	2.13124e7i	1.27123	+	0.598474i
$330$		0			0
$331$	−1.58127e7	−	2.73884e7i	−0.436036	−	0.755236i	0.561344	−	0.827583i	$-0.310285\pi$
$331$	−1.58127e7	−	2.73884e7i	−0.436036	−	0.755236i	−0.997380	+	0.0723465i	$0.976951\pi$
$332$		0			0
$333$	4.05750e6	−	7.02779e6i	0.109882	−	0.190321i
$334$		0			0
$335$			1.79269e6i			0.0476837i
$336$		0			0
$337$	−8.23591e6			−0.215190			−0.107595	−	0.994195i	$-0.534315\pi$
$337$	−8.23591e6			−0.215190			−0.107595	+	0.994195i	$0.534315\pi$
$338$		0			0
$339$	4.95976e7	+	2.86352e7i	1.27310	+	0.735023i
$340$		0			0
$341$	−7.63083e7	+	4.40566e7i	−1.92446	+	1.11109i
$342$		0			0
$343$	3.90912e7	−	1.00147e7i	0.968716	−	0.248173i
$344$		0			0
$345$	−2.23350e6	−	3.86854e6i	−0.0543912	−	0.0942084i
$346$		0			0
$347$	−3.12505e7	+	5.41274e7i	−0.747943	+	1.29547i	0.200864	+	0.979619i	$0.435625\pi$
$347$	−3.12505e7	+	5.41274e7i	−0.747943	+	1.29547i	−0.948807	+	0.315856i	$0.897708\pi$
$348$		0			0
$349$		−	2.49010e7i		−	0.585788i	−0.956145	−	0.292894i	$-0.905382\pi$
$349$		−	2.49010e7i		−	0.585788i	0.956145	−	0.292894i	$-0.0946183\pi$
$350$		0			0
$351$	5.73533e7			1.32629
$352$		0			0
$353$	5.99767e7	+	3.46276e7i	1.36351	+	0.787224i	0.990089	−	0.140438i	$-0.0448511\pi$
$353$	5.99767e7	+	3.46276e7i	1.36351	+	0.787224i	0.373422	+	0.927662i	$0.378184\pi$
$354$		0			0
$355$	4.38105e6	−	2.52940e6i	0.0979249	−	0.0565369i
$356$		0			0
$357$	467392.	−	992799.i	0.0102725	−	0.0218201i
$358$		0			0
$359$	1.50466e7	+	2.60615e7i	0.325204	+	0.563270i	0.981554	−	0.191187i	$-0.0612337\pi$
$359$	1.50466e7	+	2.60615e7i	0.325204	+	0.563270i	−0.656350	+	0.754457i	$0.727900\pi$
$360$		0			0
$361$	−2.26446e7	+	3.92217e7i	−0.481331	+	0.833689i
$362$		0			0
$363$			1.59208e7i			0.332847i
$364$		0			0
$365$	2.31787e6			0.0476663
$366$		0			0
$367$	1.83705e7	+	1.06062e7i	0.371640	+	0.214566i	0.674175	−	0.738572i	$-0.264500\pi$
$367$	1.83705e7	+	1.06062e7i	0.371640	+	0.214566i	−0.302535	+	0.953138i	$0.597833\pi$
$368$		0			0
$369$	2.79388e6	−	1.61305e6i	0.0556070	−	0.0321047i
$370$		0			0
$371$	−5.81681e7	+	4.87408e6i	−1.13910	+	0.0954489i
$372$		0			0
$373$	−2.44206e7	−	4.22977e7i	−0.470575	−	0.815060i	0.528858	−	0.848710i	$-0.322620\pi$
$373$	−2.44206e7	−	4.22977e7i	−0.470575	−	0.815060i	−0.999434	+	0.0336496i	$0.989287\pi$
$374$		0			0
$375$	−4.87747e6	+	8.44803e6i	−0.0924913	+	0.160200i
$376$		0			0
$377$			2.07176e7i			0.386648i
$378$		0			0
$379$	−8.40563e6			−0.154402			−0.0772009	−	0.997016i	$-0.524598\pi$
$379$	−8.40563e6			−0.154402			−0.0772009	+	0.997016i	$0.524598\pi$
$380$		0			0
$381$	1.78044e7	+	1.02794e7i	0.321924	+	0.185863i
$382$		0			0
$383$	−1.47384e7	+	8.50922e6i	−0.262334	+	0.151458i	−0.625399	−	0.780305i	$-0.715064\pi$
$383$	−1.47384e7	+	8.50922e6i	−0.262334	+	0.151458i	0.363065	+	0.931764i	$0.381730\pi$
$384$		0			0
$385$	−5.41789e6	+	3.76410e6i	−0.0949397	+	0.0659598i
$386$		0			0
$387$	−3.58019e6	−	6.20106e6i	−0.0617693	−	0.106988i
$388$		0			0
$389$	−2.99145e6	+	5.18135e6i	−0.0508199	+	0.0880226i	−0.890316	−	0.455343i	$-0.849517\pi$
$389$	−2.99145e6	+	5.18135e6i	−0.0508199	+	0.0880226i	0.839496	+	0.543365i	$0.182850\pi$
$390$		0			0
$391$			1.80234e6i			0.0301513i
$392$		0			0
$393$	7.75392e7			1.27745
$394$		0			0
$395$	−901366.	−	520404.i	−0.0146255	−	0.00844402i
$396$		0			0
$397$	−9.15067e7	+	5.28314e7i	−1.46245	+	0.844346i	−0.999124	−	0.0418414i	$-0.986678\pi$
$397$	−9.15067e7	+	5.28314e7i	−1.46245	+	0.844346i	−0.463326	+	0.886188i	$0.653344\pi$
$398$		0			0
$399$	6.55597e6	+	9.43639e6i	0.103209	+	0.148555i
$400$		0			0
$401$	−3.06343e7	−	5.30601e7i	−0.475089	−	0.822877i	0.524504	−	0.851408i	$-0.324251\pi$
$401$	−3.06343e7	−	5.30601e7i	−0.475089	−	0.822877i	−0.999593	+	0.0285303i	$0.990917\pi$
$402$		0			0
$403$	7.87533e7	−	1.36405e8i	1.20324	−	2.08408i
$404$		0			0
$405$			5.67936e6i			0.0854938i
$406$		0			0
$407$	−1.39487e8			−2.06896
$408$		0			0
$409$	9.47417e7	+	5.46991e7i	1.38475	+	0.799486i	0.992717	−	0.120466i	$-0.0384388\pi$
$409$	9.47417e7	+	5.46991e7i	1.38475	+	0.799486i	0.392032	+	0.919951i	$0.371772\pi$
$410$		0			0
$411$	−1.37464e7	+	7.93648e6i	−0.197999	+	0.114315i
$412$		0			0
$413$	−6.21476e6	−	7.41679e7i	−0.0882214	−	1.05285i
$414$		0			0
$415$	1.00840e6	+	1.74660e6i	0.0141088	+	0.0244371i
$416$		0			0
$417$	3.53314e7	−	6.11958e7i	0.487251	−	0.843943i
$418$		0			0
$419$			2.04457e7i			0.277946i	0.990296	+	0.138973i	$0.0443802\pi$
$419$			2.04457e7i			0.277946i	−0.990296	+	0.138973i	$0.955620\pi$
$420$		0			0
$421$	−3.19255e7			−0.427850			−0.213925	−	0.976850i	$-0.568625\pi$
$421$	−3.19255e7			−0.427850			−0.213925	+	0.976850i	$0.568625\pi$
$422$		0			0
$423$	1.13897e7	+	6.57582e6i	0.150484	+	0.0868818i
$424$		0			0
$425$	1.69585e6	−	979101.i	0.0220913	−	0.0127544i
$426$		0			0
$427$	6.18106e7	+	2.90993e7i	0.793924	+	0.373765i
$428$		0			0
$429$	−5.42500e7	−	9.39637e7i	−0.687112	−	1.19011i
$430$		0			0
$431$	155642.	−	269579.i	0.00194399	−	0.00336709i	−0.865052	−	0.501683i	$-0.832715\pi$
$431$	155642.	−	269579.i	0.00194399	−	0.00336709i	0.866996	+	0.498316i	$0.166048\pi$
$432$		0			0
$433$		−	5.06252e7i		−	0.623596i	−0.950148	−	0.311798i	$-0.899069\pi$
$433$		−	5.06252e7i		−	0.623596i	0.950148	−	0.311798i	$-0.100931\pi$
$434$		0			0
$435$	−2.34621e6			−0.0285035
$436$		0			0
$437$	−1.63442e7	−	9.43634e6i	−0.195848	−	0.113073i
$438$		0			0
$439$	−7.01109e7	+	4.04786e7i	−0.828690	+	0.478445i	−0.853404	−	0.521250i	$-0.825466\pi$
$439$	−7.01109e7	+	4.04786e7i	−0.828690	+	0.478445i	0.0247137	+	0.999695i	$0.492133\pi$
$440$		0			0
$441$	1.04587e7	−	1.76513e6i	0.121945	−	0.0205808i
$442$		0			0
$443$	−1.28372e6	−	2.22346e6i	−0.0147658	−	0.0255752i	0.858548	−	0.512733i	$-0.171367\pi$
$443$	−1.28372e6	−	2.22346e6i	−0.0147658	−	0.0255752i	−0.873314	+	0.487158i	$0.838034\pi$
$444$		0			0
$445$	3.15155e6	−	5.45864e6i	0.0357638	−	0.0619447i
$446$		0			0
$447$		−	1.03396e8i		−	1.15766i
$448$		0			0
$449$	−6.11330e7			−0.675362			−0.337681	−	0.941261i	$-0.609643\pi$
$449$	−6.11330e7			−0.675362			−0.337681	+	0.941261i	$0.609643\pi$
$450$		0			0
$451$	−4.80237e7	−	2.77265e7i	−0.523511	−	0.302249i
$452$		0			0
$453$	−3.58705e7	+	2.07099e7i	−0.385872	+	0.222783i
$454$		0			0
$455$	5.02297e6	−	1.06694e7i	0.0533245	−	0.113268i
$456$		0			0
$457$	−4.79859e6	−	8.31141e6i	−0.0502765	−	0.0870815i	0.839792	−	0.542908i	$-0.182677\pi$
$457$	−4.79859e6	−	8.31141e6i	−0.0502765	−	0.0870815i	−0.890068	+	0.455827i	$0.849344\pi$
$458$		0			0
$459$	1.31031e6	−	2.26952e6i	0.0135499	−	0.0234691i
$460$		0			0
$461$			6.04509e7i			0.617021i	0.951221	+	0.308510i	$0.0998304\pi$
$461$			6.04509e7i			0.617021i	−0.951221	+	0.308510i	$0.900170\pi$
$462$		0			0
$463$	1.14757e8			1.15621			0.578104	−	0.815963i	$-0.303793\pi$
$463$	1.14757e8			1.15621			0.578104	+	0.815963i	$0.303793\pi$
$464$		0			0
$465$	1.54474e7	+	8.91857e6i	0.153637	+	0.0887026i
$466$		0			0
$467$	5.93604e7	−	3.42717e7i	0.582836	−	0.336500i	−0.179424	−	0.983772i	$-0.557423\pi$
$467$	5.93604e7	−	3.42717e7i	0.582836	−	0.336500i	0.762259	+	0.647272i	$0.224090\pi$
$468$		0			0
$469$	−4.93692e7	+	4.13680e6i	−0.478561	+	0.0401001i
$470$		0			0
$471$	−7.27226e7	−	1.25959e8i	−0.695995	−	1.20550i
$472$		0			0
$473$	−6.15393e7	+	1.06589e8i	−0.581526	+	1.00723i
$474$		0			0
$475$			2.05048e7i			0.191326i
$476$		0			0
$477$	−1.53426e7			−0.141366
$478$		0			0
$479$	−8.46880e7	−	4.88946e7i	−0.770576	−	0.444892i	0.0625041	−	0.998045i	$-0.480091\pi$
$479$	−8.46880e7	−	4.88946e7i	−0.770576	−	0.444892i	−0.833080	+	0.553153i	$0.813425\pi$
$480$		0			0
$481$	2.15935e8	−	1.24670e8i	1.94039	−	1.12028i
$482$		0			0
$483$	1.01383e8	−	7.04359e7i	0.899750	−	0.625105i
$484$		0			0
$485$	3.16331e6	+	5.47902e6i	0.0277279	+	0.0480261i
$486$		0			0
$487$	1.14091e7	−	1.97612e7i	0.0987794	−	0.171091i	−0.812400	−	0.583100i	$-0.801840\pi$
$487$	1.14091e7	−	1.97612e7i	0.0987794	−	0.171091i	0.911180	+	0.412009i	$0.135173\pi$
$488$		0			0
$489$		−	2.42123e7i		−	0.207066i
$490$		0			0
$491$	−1.11615e7			−0.0942930			−0.0471465	−	0.998888i	$-0.515013\pi$
$491$	−1.11615e7			−0.0942930			−0.0471465	+	0.998888i	$0.515013\pi$
$492$		0			0
$493$	819816.	+	473321.i	0.00684188	+	0.00395016i
$494$		0			0
$495$	−1.50169e6	+	867002.i	−0.0123813	+	0.00714833i
$496$		0			0
$497$	7.97674e7	+	1.14814e8i	0.649765	+	0.935244i
$498$		0			0
$499$	−2.16628e7	−	3.75211e7i	−0.174347	−	0.301977i	0.765588	−	0.643331i	$-0.222448\pi$
$499$	−2.16628e7	−	3.75211e7i	−0.174347	−	0.301977i	−0.939935	+	0.341354i	$0.889115\pi$
$500$		0			0
$501$	8.47892e7	−	1.46859e8i	0.674260	−	1.16785i
$502$		0			0
$503$			2.32014e8i			1.82310i	0.411192	+	0.911549i	$0.365113\pi$
$503$			2.32014e8i			1.82310i	−0.411192	+	0.911549i	$0.634887\pi$
$504$		0			0
$505$	−1.93558e7			−0.150292
$506$		0			0
$507$	6.23100e7	+	3.59747e7i	0.478116	+	0.276041i
$508$		0			0
$509$	−1.02119e8	+	5.89587e7i	−0.774382	+	0.447089i	−0.834435	−	0.551106i	$-0.814206\pi$
$509$	−1.02119e8	+	5.89587e7i	−0.774382	+	0.447089i	0.0600538	+	0.998195i	$0.480873\pi$
$510$		0			0
$511$	5.34872e6	+	6.38325e7i	0.0400855	+	0.478387i
$512$		0			0
$513$	1.37206e7	+	2.37647e7i	0.101630	+	0.176028i
$514$		0			0
$515$	−4.51313e6	+	7.81698e6i	−0.0330412	+	0.0572291i
$516$		0			0
$517$		−	2.26062e8i		−	1.63589i
$518$		0			0
$519$	8.77294e6			0.0627542
$520$		0			0
$521$	−1.94107e8	−	1.12068e8i	−1.37255	−	0.792443i	−0.381303	−	0.924450i	$-0.624525\pi$
$521$	−1.94107e8	−	1.12068e8i	−1.37255	−	0.792443i	−0.991249	+	0.132007i	$0.957858\pi$
$522$		0			0
$523$	−1.57711e8	+	9.10546e7i	−1.10245	+	0.636497i	−0.936862	−	0.349698i	$-0.886284\pi$
$523$	−1.57711e8	+	9.10546e7i	−1.10245	+	0.636497i	−0.165584	+	0.986196i	$0.552951\pi$
$524$		0			0
$525$	−1.21350e8	−	5.71292e7i	−0.838610	−	0.394802i
$526$		0			0
$527$	−3.59844e6	−	6.23268e6i	−0.0245857	−	0.0425837i
$528$		0			0
$529$	−2.73640e7	+	4.73959e7i	−0.184847	+	0.320165i
$530$		0			0
$531$		−	1.95628e7i		−	0.130661i
$532$		0			0
$533$	9.91248e7			0.654637
$534$		0			0
$535$	1.88063e7	+	1.08578e7i	0.122812	+	0.0709058i
$536$		0			0
$537$	6.44723e7	−	3.72231e7i	0.416342	−	0.240375i
$538$		0			0
$539$	−1.16163e8	−	1.40518e8i	−0.741823	−	0.897361i
$540$		0			0
$541$	1.22675e8	+	2.12480e8i	0.774756	+	1.34192i	0.934932	+	0.354828i	$0.115461\pi$
$541$	1.22675e8	+	2.12480e8i	0.774756	+	1.34192i	−0.160176	+	0.987089i	$0.551206\pi$
$542$		0			0
$543$	3.64983e7	−	6.32170e7i	0.227968	−	0.394852i
$544$		0			0
$545$			9.59222e6i			0.0592556i
$546$		0			0
$547$	−1.02069e8			−0.623637			−0.311819	−	0.950142i	$-0.600938\pi$
$547$	−1.02069e8			−0.623637			−0.311819	+	0.950142i	$0.600938\pi$
$548$		0			0
$549$	1.55511e7	+	8.97841e6i	0.0939816	+	0.0542603i
$550$		0			0
$551$	−8.58449e6	+	4.95625e6i	−0.0513168	+	0.0296278i
$552$		0			0
$553$	1.22515e7	−	2.60238e7i	0.0724461	−	0.153885i
$554$		0			0
$555$	1.41185e7	+	2.44540e7i	0.0825866	+	0.143044i
$556$		0			0
$557$	−9.57059e7	+	1.65768e8i	−0.553826	+	0.959255i	0.444168	+	0.895944i	$0.353499\pi$
$557$	−9.57059e7	+	1.65768e8i	−0.553826	+	0.959255i	−0.997994	+	0.0633110i	$0.979834\pi$
$558$		0			0
$559$		−	2.20009e8i		−	1.25952i
$560$		0			0
$561$	−4.95764e6			−0.0280793
$562$		0			0
$563$	1.66748e8	+	9.62718e7i	0.934403	+	0.539478i	0.888202	−	0.459454i	$-0.151955\pi$
$563$	1.66748e8	+	9.62718e7i	0.934403	+	0.539478i	0.0462018	+	0.998932i	$0.485288\pi$
$564$		0			0
$565$	2.43549e7	−	1.40613e7i	0.135033	−	0.0779615i
$566$		0			0
$567$	−1.56405e8	+	1.31057e7i	−0.858029	+	0.0718969i
$568$		0			0
$569$	2.99208e7	+	5.18243e7i	0.162419	+	0.281317i	0.935736	−	0.352702i	$-0.114737\pi$
$569$	2.99208e7	+	5.18243e7i	0.162419	+	0.281317i	−0.773317	+	0.634020i	$0.781404\pi$
$570$		0			0
$571$	−1.27846e8	+	2.21435e8i	−0.686716	+	1.18943i	0.286178	+	0.958177i	$0.407615\pi$
$571$	−1.27846e8	+	2.21435e8i	−0.686716	+	1.18943i	−0.972894	+	0.231251i	$0.925718\pi$
$572$		0			0
$573$			1.43243e8i			0.761395i
$574$		0			0
$575$	2.20299e8			1.15880
$576$		0			0
$577$	−2.79824e7	−	1.61556e7i	−0.145666	−	0.0841001i	0.425396	−	0.905007i	$-0.360135\pi$
$577$	−2.79824e7	−	1.61556e7i	−0.145666	−	0.0841001i	−0.571061	+	0.820907i	$0.693468\pi$
$578$		0			0
$579$	2.86447e8	−	1.65380e8i	1.47574	−	0.852016i
$580$		0			0
$581$	−4.57731e7	+	3.18011e7i	−0.233390	+	0.162149i
$582$		0			0
$583$	1.31861e8	+	2.28390e8i	0.665444	+	1.15258i
$584$		0			0
$585$	1.54981e6	−	2.68435e6i	0.00774123	−	0.0134082i
$586$		0			0
$587$			9.56070e7i			0.472689i	0.971669	+	0.236344i	$0.0759493\pi$
$587$			9.56070e7i			0.472689i	−0.971669	+	0.236344i	$0.924051\pi$
$588$		0			0
$589$	7.53603e7			0.368805
$590$		0			0
$591$	−9.78637e7	−	5.65016e7i	−0.474088	−	0.273715i
$592$		0			0
$593$	−1.00699e8	+	5.81386e7i	−0.482905	+	0.278805i	−0.721626	−	0.692283i	$-0.756605\pi$
$593$	−1.00699e8	+	5.81386e7i	−0.482905	+	0.278805i	0.238722	+	0.971088i	$0.423272\pi$
$594$		0			0
$595$	−307443.	−	442520.i	−0.00145953	−	0.00210079i
$596$		0			0
$597$	1.01829e8	+	1.76374e8i	0.478576	+	0.828917i
$598$		0			0
$599$	−1.79992e8	+	3.11756e8i	−0.837477	+	1.45055i	0.0545197	+	0.998513i	$0.482637\pi$
$599$	−1.79992e8	+	3.11756e8i	−0.837477	+	1.45055i	−0.891997	+	0.452041i	$0.850696\pi$
$600$		0			0
$601$			3.47560e8i			1.60105i	0.599297	+	0.800527i	$0.295447\pi$
$601$			3.47560e8i			1.60105i	−0.599297	+	0.800527i	$0.704553\pi$
$602$		0			0
$603$	−1.30218e7			−0.0593908
$604$		0			0
$605$	6.77049e6	+	3.90895e6i	0.0305741	+	0.0176520i
$606$		0			0
$607$	1.84155e8	−	1.06322e8i	0.823414	−	0.475398i	−0.0281785	−	0.999603i	$-0.508971\pi$
$607$	1.84155e8	−	1.06322e8i	0.823414	−	0.475398i	0.851592	+	0.524205i	$0.175637\pi$
$608$		0			0
$609$	−5.41410e6	−	6.46127e7i	−0.0239703	−	0.286066i
$610$		0			0
$611$	2.02048e8	+	3.49957e8i	0.885790	+	1.53423i
$612$		0			0
$613$	3.94274e7	−	6.82903e7i	0.171166	−	0.296468i	−0.767662	−	0.640855i	$-0.778580\pi$
$613$	3.94274e7	−	6.82903e7i	0.171166	−	0.296468i	0.938828	+	0.344387i	$0.111913\pi$
$614$		0			0
$615$			1.12256e7i			0.0482595i
$616$		0			0
$617$	4.19908e6			0.0178772			0.00893859	−	0.999960i	$-0.497155\pi$
$617$	4.19908e6			0.0178772			0.00893859	+	0.999960i	$0.497155\pi$
$618$		0			0
$619$	−2.30232e8	−	1.32925e8i	−0.970721	−	0.560446i	−0.0712651	−	0.997457i	$-0.522704\pi$
$619$	−2.30232e8	−	1.32925e8i	−0.970721	−	0.560446i	−0.899456	+	0.437011i	$0.856037\pi$
$620$		0			0
$621$	2.55323e8	−	1.47411e8i	1.06614	−	0.615537i
$622$		0			0
$623$	1.57599e8	+	7.41948e7i	0.651763	+	0.306838i
$624$		0			0
$625$	−1.18472e8	−	2.05199e8i	−0.485260	−	0.840496i
$626$		0			0
$627$	2.59563e7	−	4.49577e7i	0.105303	−	0.182390i
$628$		0			0
$629$		−	1.13930e7i		−	0.0457811i
$630$		0			0
$631$	2.02533e8			0.806135			0.403068	−	0.915170i	$-0.367944\pi$
$631$	2.02533e8			0.806135			0.403068	+	0.915170i	$0.367944\pi$
$632$		0			0
$633$	9.23303e6	+	5.33069e6i	0.0364027	+	0.0210171i
$634$		0			0
$635$	8.74285e6	−	5.04769e6i	0.0341454	−	0.0197138i
$636$		0			0
$637$	3.05419e8	+	1.13708e8i	1.18162	+	0.439919i
$638$		0			0
$639$	1.83732e7	+	3.18233e7i	0.0704177	+	0.121967i
$640$		0			0
$641$	1.15943e7	−	2.00819e7i	0.0440221	−	0.0762484i	−0.843175	−	0.537640i	$-0.819316\pi$
$641$	1.15943e7	−	2.00819e7i	0.0440221	−	0.0762484i	0.887197	+	0.461391i	$0.152649\pi$
$642$		0			0
$643$		−	3.10031e8i		−	1.16620i	−0.812401	−	0.583100i	$-0.801840\pi$
$643$		−	3.10031e8i		−	1.16620i	0.812401	−	0.583100i	$-0.198160\pi$
$644$		0			0
$645$	2.49153e7			0.0928511
$646$		0			0
$647$	−1.29013e8	−	7.44859e7i	−0.476345	−	0.275018i	0.242547	−	0.970140i	$-0.422017\pi$
$647$	−1.29013e8	−	7.44859e7i	−0.476345	−	0.275018i	−0.718892	+	0.695122i	$0.755350\pi$
$648$		0			0
$649$	−2.91212e8	+	1.68131e8i	−1.06531	+	0.615055i
$650$		0			0
$651$	−2.09964e8	+	4.45990e8i	−0.761030	+	1.61652i
$652$		0			0
$653$	1.15112e8	+	1.99379e8i	0.413408	+	0.716044i	0.995260	−	0.0972506i	$-0.0310048\pi$
$653$	1.15112e8	+	1.99379e8i	0.413408	+	0.716044i	−0.581851	+	0.813295i	$0.697672\pi$
$654$		0			0
$655$	1.90378e7	−	3.29744e7i	0.0677474	−	0.117342i
$656$		0			0
$657$			1.68367e7i			0.0593691i
$658$		0			0
$659$	−4.63051e8			−1.61798			−0.808989	−	0.587824i	$-0.799985\pi$
$659$	−4.63051e8			−1.61798			−0.808989	+	0.587824i	$0.799985\pi$
$660$		0			0
$661$	−2.85316e8	−	1.64727e8i	−0.987920	−	0.570376i	−0.0832680	−	0.996527i	$-0.526536\pi$
$661$	−2.85316e8	−	1.64727e8i	−0.987920	−	0.570376i	−0.904652	+	0.426151i	$0.859869\pi$
$662$		0			0
$663$	7.67473e6	−	4.43101e6i	0.0263344	−	0.0152041i
$664$		0			0
$665$	5.62259e6	−	471134.i	0.0191193	−	0.00160206i
$666$		0			0
$667$	5.32489e7	+	9.22298e7i	0.179446	+	0.310809i
$668$		0			0
$669$	5.67936e7	−	9.83694e7i	0.189680	−	0.328535i
$670$		0			0
$671$		−	3.08657e8i		−	1.02167i
$672$		0			0
$673$	−8.50491e7			−0.279013			−0.139506	−	0.990221i	$-0.544552\pi$
$673$	−8.50491e7			−0.279013			−0.139506	+	0.990221i	$0.544552\pi$
$674$		0			0
$675$	−2.77403e8	−	1.60159e8i	−0.901987	−	0.520762i
$676$		0			0
$677$	3.43079e8	−	1.98077e8i	1.10568	−	0.638362i	0.167970	−	0.985792i	$-0.446279\pi$
$677$	3.43079e8	−	1.98077e8i	1.10568	−	0.638362i	0.937706	+	0.347430i	$0.112945\pi$
$678$		0			0
$679$	−1.43588e8	+	9.97585e7i	−0.458680	+	0.318670i
$680$		0			0
$681$	−4.91883e7	−	8.51967e7i	−0.155747	−	0.269762i
$682$		0			0
$683$	1.61680e8	−	2.80039e8i	0.507452	−	0.878933i	−0.492510	−	0.870307i	$-0.663921\pi$
$683$	1.61680e8	−	2.80039e8i	0.507452	−	0.878933i	0.999963	−	0.00862666i	$-0.00274599\pi$
$684$		0			0
$685$			7.79441e6i			0.0242500i
$686$		0			0
$687$	6.23206e7			0.192204
$688$		0			0
$689$	−4.08258e8	−	2.35708e8i	−1.24818	−	0.720637i
$690$		0			0
$691$	4.13643e8	−	2.38817e8i	1.25369	−	0.723821i	0.281853	−	0.959458i	$-0.409051\pi$
$691$	4.13643e8	−	2.38817e8i	1.25369	−	0.723821i	0.971841	+	0.235637i	$0.0757175\pi$
$692$		0			0
$693$	−2.73419e7	−	3.93547e7i	−0.0821540	−	0.118249i
$694$		0			0
$695$	−1.73495e7	−	3.00502e7i	−0.0516811	−	0.0895143i
$696$		0			0
$697$	2.26463e6	−	3.92246e6i	0.00668805	−	0.0115840i
$698$		0			0
$699$			5.73681e8i			1.67973i
$700$		0			0
$701$	1.25339e7			0.0363859			0.0181929	−	0.999834i	$-0.494209\pi$
$701$	1.25339e7			0.0363859			0.0181929	+	0.999834i	$0.494209\pi$
$702$		0			0
$703$	1.03316e8	+	5.96494e7i	0.297373	+	0.171688i
$704$		0			0
$705$	−3.96316e7	+	2.28813e7i	−0.113103	+	0.0653000i
$706$		0			0
$707$	−4.46653e7	−	5.33043e8i	−0.126390	−	1.50836i
$708$		0			0
$709$	9.71988e7	+	1.68353e8i	0.272723	+	0.472371i	0.969558	−	0.244861i	$-0.0787423\pi$
$709$	9.71988e7	+	1.68353e8i	0.272723	+	0.472371i	−0.696835	+	0.717232i	$0.745409\pi$
$710$		0			0
$711$	3.78013e6	−	6.54739e6i	0.0105172	−	0.0182163i
$712$		0			0
$713$		−	8.09654e8i		−	2.23373i
$714$		0			0
$715$	−5.32788e7			−0.145759
$716$		0			0
$717$	−8.56844e7	−	4.94699e7i	−0.232458	−	0.134210i
$718$		0			0
$719$	2.09066e8	−	1.20704e8i	0.562467	−	0.324741i	−0.191668	−	0.981460i	$-0.561390\pi$
$719$	2.09066e8	−	1.20704e8i	0.562467	−	0.324741i	0.754135	+	0.656719i	$0.228056\pi$
$720$		0			0
$721$	−2.25688e8	−	1.06250e8i	−0.602147	−	0.283480i
$722$		0			0
$723$	−1.24182e8	−	2.15090e8i	−0.328583	−	0.569122i
$724$		0			0
$725$	5.78539e7	−	1.00206e8i	0.151816	−	0.262954i
$726$		0			0
$727$		−	3.89703e8i		−	1.01422i	−0.861882	−	0.507109i	$-0.830714\pi$
$727$		−	3.89703e8i		−	1.01422i	0.861882	−	0.507109i	$-0.169286\pi$
$728$		0			0
$729$	−4.22749e8			−1.09119
$730$		0			0
$731$	−8.70595e6	−	5.02638e6i	−0.0222876	−	0.0128678i
$732$		0			0
$733$	1.51751e8	−	8.76135e7i	0.385319	−	0.222464i	−0.294811	−	0.955556i	$-0.595257\pi$
$733$	1.51751e8	−	8.76135e7i	0.385319	−	0.222464i	0.680130	+	0.733092i	$0.261923\pi$
$734$		0			0
$735$	−1.28771e7	+	3.45877e7i	−0.0324306	+	0.0871083i
$736$		0			0
$737$	1.11915e8	+	1.93843e8i	0.279567	+	0.484224i
$738$		0			0
$739$	3.83128e6	−	6.63597e6i	0.00949315	−	0.0164426i	−0.861240	−	0.508199i	$-0.830312\pi$
$739$	3.83128e6	−	6.63597e6i	0.00949315	−	0.0164426i	0.870733	+	0.491756i	$0.163645\pi$
$740$		0			0
$741$			9.27963e7i			0.228074i
$742$		0			0
$743$	−4.78399e8			−1.16634			−0.583169	−	0.812351i	$-0.698187\pi$
$743$	−4.78399e8			−1.16634			−0.583169	+	0.812351i	$0.698187\pi$
$744$		0			0
$745$	−4.39702e7	−	2.53862e7i	−0.106338	−	0.0613944i
$746$		0			0
$747$	−1.26871e7	+	7.32488e6i	−0.0304368	+	0.0175727i
$748$		0			0
$749$	−2.55619e8	+	5.42967e8i	−0.608341	+	1.29219i
$750$		0			0
$751$	−3.50940e7	−	6.07847e7i	−0.0828540	−	0.143507i	0.821621	−	0.570035i	$-0.193070\pi$
$751$	−3.50940e7	−	6.07847e7i	−0.0828540	−	0.143507i	−0.904475	+	0.426527i	$0.859737\pi$
$752$		0			0
$753$	−1.42384e8	+	2.46615e8i	−0.333484	+	0.577611i
$754$		0			0
$755$			2.03391e7i			0.0472598i
$756$		0			0
$757$	−5.44271e7			−0.125466			−0.0627332	−	0.998030i	$-0.519982\pi$
$757$	−5.44271e7			−0.125466			−0.0627332	+	0.998030i	$0.519982\pi$
$758$		0			0
$759$	−4.83015e8	−	2.78869e8i	−1.10468	−	0.637786i
$760$		0			0
$761$	4.83017e8	−	2.78870e8i	1.09599	−	0.632773i	0.160829	−	0.986982i	$-0.448583\pi$
$761$	4.83017e8	−	2.78870e8i	1.09599	−	0.632773i	0.935166	+	0.354210i	$0.115250\pi$
$762$		0			0
$763$	−2.64162e8	+	2.21350e7i	−0.594699	+	0.0498317i
$764$		0			0
$765$	−70814.7	−	122655.i	−0.000158175	−	0.000273968i
$766$		0			0
$767$	3.00542e8	−	5.20555e8i	0.666070	−	1.15367i
$768$		0			0
$769$		−	3.50325e8i		−	0.770356i	−0.922842	−	0.385178i	$-0.874140\pi$
$769$		−	3.50325e8i		−	0.770356i	0.922842	−	0.385178i	$-0.125860\pi$
$770$		0			0
$771$	7.88286e8			1.71997
$772$		0			0
$773$	4.94039e8	+	2.85234e8i	1.06960	+	0.617536i	0.928073	−	0.372398i	$-0.121464\pi$
$773$	4.94039e8	+	2.85234e8i	1.06960	+	0.617536i	0.141530	+	0.989934i	$0.454798\pi$
$774$		0			0
$775$	−7.61819e8	+	4.39837e8i	−1.63662	+	0.944901i
$776$		0			0
$777$	−6.40864e8	+	4.45243e8i	−1.36616	+	0.949148i
$778$		0			0
$779$	2.37135e7	+	4.10730e7i	0.0501630	+	0.0868848i
$780$		0			0
$781$	3.15814e8	−	5.47006e8i	0.662946	−	1.14826i
$782$		0			0
$783$		−	1.54849e8i		−	0.322570i
$784$		0			0
$785$	−7.14208e7			−0.147644
$786$		0			0
$787$	6.46413e8	+	3.73207e8i	1.32613	+	0.765641i	0.984699	−	0.174266i	$-0.0557553\pi$
$787$	6.46413e8	+	3.73207e8i	1.32613	+	0.765641i	0.341431	+	0.939907i	$0.389089\pi$
$788$		0			0
$789$	2.95909e8	−	1.70843e8i	0.602459	−	0.347830i
$790$		0			0
$791$	4.43438e8	+	6.38267e8i	0.895992	+	1.28965i
$792$		0			0
$793$	2.75870e8	+	4.77820e8i	0.553203	+	0.958175i
$794$		0			0
$795$	2.66932e7	−	4.62340e7i	0.0531250	−	0.0920153i
$796$		0			0
$797$		−	8.52897e8i		−	1.68470i	−0.538934	−	0.842348i	$-0.681173\pi$
$797$		−	8.52897e8i		−	1.68470i	0.538934	−	0.842348i	$-0.318827\pi$
$798$		0			0
$799$	1.84642e7			0.0361984
$800$		0			0
$801$	3.96507e7	+	2.28924e7i	0.0771532	+	0.0445444i
$802$		0			0
$803$	2.50631e8	−	1.44702e8i	0.484047	−	0.279465i
$804$		0			0
$805$	−5.06176e6	−	6.04078e7i	−0.00970317	−	0.115799i
$806$		0			0
$807$	3.29530e8	+	5.70762e8i	0.627010	+	1.08601i
$808$		0			0
$809$	2.68931e8	−	4.65802e8i	0.507920	−	0.879744i	−0.492038	−	0.870574i	$-0.663748\pi$
$809$	2.68931e8	−	4.65802e8i	0.507920	−	0.879744i	0.999958	−	0.00916987i	$-0.00291890\pi$
$810$		0			0
$811$			546840.i			0.00102517i	1.00000		0.000512587i	$0.000163162\pi$
$811$			546840.i			0.00102517i	−1.00000		0.000512587i	$0.999837\pi$
$812$		0			0
$813$	2.79179e8			0.519530
$814$		0			0
$815$	−1.02965e7	−	5.94472e6i	−0.0190204	−	0.0109814i
$816$		0			0
$817$	9.11621e7	−	5.26325e7i	0.167166	−	0.0965134i
$818$		0			0
$819$	7.75011e7	+	3.64861e7i	0.141077	+	0.0664165i
$820$		0			0
$821$	2.66538e7	+	4.61657e7i	0.0481648	+	0.0834238i	0.889103	−	0.457708i	$-0.151329\pi$
$821$	2.66538e7	+	4.61657e7i	0.0481648	+	0.0834238i	−0.840938	+	0.541132i	$0.817996\pi$
$822$		0			0
$823$	−3.83179e8	+	6.63686e8i	−0.687389	+	1.19059i	0.285291	+	0.958441i	$0.407910\pi$
$823$	−3.83179e8	+	6.63686e8i	−0.687389	+	1.19059i	−0.972680	+	0.232151i	$0.925424\pi$
$824$		0			0
$825$			6.05971e8i			1.07917i
$826$		0			0
$827$	−1.00353e9			−1.77425			−0.887127	−	0.461526i	$-0.847302\pi$
$827$	−1.00353e9			−1.77425			−0.887127	+	0.461526i	$0.847302\pi$
$828$		0			0
$829$	−1.28437e8	−	7.41532e7i	−0.225438	−	0.130157i	0.383028	−	0.923737i	$-0.374881\pi$
$829$	−1.28437e8	−	7.41532e7i	−0.225438	−	0.130157i	−0.608466	+	0.793580i	$0.708215\pi$
$830$		0			0
$831$	−3.49067e8	+	2.01534e8i	−0.608283	+	0.351192i
$832$		0			0
$833$	1.14772e7	−	9.48790e6i	0.0198564	−	0.0164148i
$834$		0			0
$835$	−4.16357e7	−	7.21151e7i	−0.0715165	−	0.123870i
$836$		0			0
$837$	−5.88624e8	+	1.01953e9i	−1.00383	+	1.73869i
$838$		0			0
$839$		−	2.21066e8i		−	0.374315i	−0.982330	−	0.187157i	$-0.940073\pi$
$839$		−	2.21066e8i		−	0.374315i	0.982330	−	0.187157i	$-0.0599274\pi$
$840$		0			0
$841$	−5.38887e8			−0.905962
$842$		0			0
$843$	2.74989e7	+	1.58765e7i	0.0459021	+	0.0265016i
$844$		0			0
$845$	3.05973e7	−	1.76653e7i	0.0507122	−	0.0292787i
$846$		0			0
$847$	−9.20258e7	+	1.95474e8i	−0.151446	+	0.321692i
$848$		0			0
$849$	4.26319e6	+	7.38405e6i	0.00696645	+	0.0120662i
$850$		0			0
$851$	6.40860e8	−	1.11000e9i	1.03986	−	1.80109i
$852$		0			0
$853$			9.10570e8i			1.46712i	0.679624	+	0.733561i	$0.262143\pi$
$853$			9.10570e8i			1.46712i	−0.679624	+	0.733561i	$0.737857\pi$
$854$		0			0
$855$	1.48303e6			0.00237276
$856$		0			0
$857$	1.68169e8	+	9.70924e7i	0.267180	+	0.154256i	0.627605	−	0.778532i	$-0.284035\pi$
$857$	1.68169e8	+	9.70924e7i	0.267180	+	0.154256i	−0.360426	+	0.932788i	$0.617369\pi$
$858$		0			0
$859$	6.25498e8	−	3.61132e8i	0.986840	−	0.569752i	0.0825118	−	0.996590i	$-0.473706\pi$
$859$	6.25498e8	−	3.61132e8i	0.986840	−	0.569752i	0.904328	+	0.426838i	$0.140372\pi$
$860$		0			0
$861$	−3.09143e8	+	2.59041e7i	−0.484340	+	0.0405844i
$862$		0			0
$863$	−9.71379e7	−	1.68248e8i	−0.151132	−	0.261768i	0.780512	−	0.625141i	$-0.214958\pi$
$863$	−9.71379e7	−	1.68248e8i	−0.151132	−	0.261768i	−0.931644	+	0.363373i	$0.881625\pi$
$864$		0			0
$865$	2.15397e6	−	3.73079e6i	0.00332807	−	0.00576438i
$866$		0			0
$867$			6.09681e8i			0.935504i
$868$		0			0
$869$	−1.29952e8			−0.198027
$870$		0			0
$871$	−3.46503e8	−	2.00053e8i	−0.524387	−	0.302755i
$872$		0			0
$873$	−3.97988e7	+	2.29778e7i	−0.0598173	+	0.0345355i
$874$		0			0
$875$	−1.08717e8	+	7.55315e7i	−0.162283	+	0.112747i
$876$		0			0
$877$	−2.48367e8	−	4.30184e8i	−0.368210	−	0.637758i	0.621076	−	0.783750i	$-0.286696\pi$
$877$	−2.48367e8	−	4.30184e8i	−0.368210	−	0.637758i	−0.989286	+	0.145992i	$0.953362\pi$
$878$		0			0
$879$	1.64739e8	−	2.85336e8i	0.242566	−	0.420136i
$880$		0			0
$881$			8.63809e7i			0.126325i	0.998003	+	0.0631626i	$0.0201187\pi$
$881$			8.63809e7i			0.126325i	−0.998003	+	0.0631626i	$0.979881\pi$
$882$		0			0
$883$	−6.49219e8			−0.942994			−0.471497	−	0.881868i	$-0.656286\pi$
$883$	−6.49219e8			−0.942994			−0.471497	+	0.881868i	$0.656286\pi$
$884$		0			0
$885$	5.89512e7	+	3.40355e7i	0.0850477	+	0.0491023i
$886$		0			0
$887$	6.17373e8	−	3.56441e8i	0.884661	−	0.510759i	0.0124685	−	0.999922i	$-0.496031\pi$
$887$	6.17373e8	−	3.56441e8i	0.884661	−	0.510759i	0.872192	+	0.489163i	$0.162698\pi$
$888$		0			0
$889$	1.59184e8	+	2.29123e8i	0.226566	+	0.326110i
$890$		0			0
$891$	3.54555e8	+	6.14107e8i	0.501246	+	0.868183i
$892$		0			0
$893$	−9.66714e7	+	1.67440e8i	−0.135751	+	0.235128i
$894$		0			0
$895$		−	3.65568e7i		−	0.0509916i
$896$		0			0
$897$	9.96983e8			1.38137
$898$		0			0
$899$	−3.68282e8	−	2.12628e8i	−0.506875	−	0.292645i
$900$		0			0
$901$	−1.86544e7	+	1.07701e7i	−0.0255039	+	0.0147247i
$902$		0			0
$903$	5.74945e7	+	6.86148e8i	0.0780842	+	0.931869i
$904$		0			0
$905$	−1.79225e7	−	3.10427e7i	−0.0241798	−	0.0418806i
$906$		0			0
$907$	−4.14206e8	+	7.17425e8i	−0.555129	+	0.961512i	0.442764	+	0.896638i	$0.353998\pi$
$907$	−4.14206e8	+	7.17425e8i	−0.555129	+	0.961512i	−0.997893	+	0.0648738i	$0.979336\pi$
$908$		0			0
$909$		−	1.40597e8i		−	0.187191i
$910$		0			0
$911$	1.17587e9			1.55526			0.777631	−	0.628721i	$-0.216421\pi$
$911$	1.17587e9			1.55526			0.777631	+	0.628721i	$0.216421\pi$
$912$		0			0
$913$	2.18076e8	+	1.25906e8i	0.286547	+	0.165438i
$914$		0			0
$915$	−5.41116e7	+	3.12414e7i	−0.0706362	+	0.0407818i
$916$		0			0
$917$	9.52022e8	+	4.48195e8i	1.23464	+	0.581244i
$918$		0			0
$919$	2.25460e8	+	3.90508e8i	0.290485	+	0.503134i	0.973924	−	0.226873i	$-0.0728502\pi$
$919$	2.25460e8	+	3.90508e8i	0.290485	+	0.503134i	−0.683440	+	0.730007i	$0.739517\pi$
$920$		0			0
$921$	4.40576e8	−	7.63101e8i	0.563952	−	0.976794i
$922$		0			0
$923$			1.12907e9i			1.43587i
$924$		0			0
$925$	−1.39256e9			−1.75950
$926$		0			0
$927$	−5.67813e7	−	3.27827e7i	−0.0712798	−	0.0411534i
$928$		0			0
$929$	−1.03420e9	+	5.97094e8i	−1.28990	+	0.744725i	−0.978636	−	0.205599i	$-0.934086\pi$
$929$	−1.03420e9	+	5.97094e8i	−1.28990	+	0.744725i	−0.311264	+	0.950323i	$0.600752\pi$
$930$		0			0
$931$	2.59493e7	+	1.53754e8i	0.0321571	+	0.190537i
$932$		0			0
$933$	−5.57398e8	−	9.65441e8i	−0.686310	−	1.18872i
$934$		0			0
$935$	−1.21722e6	+	2.10829e6i	−0.00148914	+	0.00257927i
$936$		0			0
$937$		−	5.57908e8i		−	0.678178i	−0.940754	−	0.339089i	$-0.889881\pi$
$937$		−	5.57908e8i		−	0.678178i	0.940754	−	0.339089i	$-0.110119\pi$
$938$		0			0
$939$	−1.10052e9			−1.32923
$940$		0			0
$941$	9.18066e8	+	5.30046e8i	1.10181	+	0.636128i	0.936695	−	0.350147i	$-0.113868\pi$
$941$	9.18066e8	+	5.30046e8i	1.10181	+	0.636128i	0.165111	+	0.986275i	$0.447202\pi$
$942$		0			0
$943$	4.41279e8	−	2.54773e8i	0.526234	−	0.303821i
$944$		0			0
$945$	−3.75431e7	+	7.97462e7i	−0.0444871	+	0.0944963i
$946$		0			0
$947$	4.08434e8	+	7.07429e8i	0.480919	+	0.832977i	0.999760	−	0.0218939i	$-0.00696959\pi$
$947$	4.08434e8	+	7.07429e8i	0.480919	+	0.832977i	−0.518841	+	0.854871i	$0.673636\pi$
$948$		0			0
$949$	−2.58661e8	+	4.48015e8i	−0.302645	+	0.524196i
$950$		0			0
$951$		−	8.27499e8i		−	0.962112i
$952$		0			0
$953$	1.23336e9			1.42499			0.712494	−	0.701678i	$-0.247565\pi$
$953$	1.23336e9			1.42499			0.712494	+	0.701678i	$0.247565\pi$
$954$		0			0
$955$	6.09158e7	+	3.51697e7i	0.0699390	+	0.0403793i
$956$		0			0
$957$	−2.53694e8	+	1.46471e8i	−0.289451	+	0.167115i
$958$		0			0
$959$	−2.14652e8	+	1.79864e7i	−0.243377	+	0.0203933i
$960$		0			0
$961$	1.17276e9	+	2.03128e9i	1.32141	+	2.28875i
$962$		0			0
$963$	−7.88697e7	+	1.36606e8i	−0.0883143	+	0.152965i
$964$		0			0
$965$		−	1.62420e8i		−	0.180741i
$966$		0			0
$967$	5.82431e8			0.644118			0.322059	−	0.946720i	$-0.395625\pi$
$967$	5.82431e8			0.644118			0.322059	+	0.946720i	$0.395625\pi$
$968$		0			0
$969$	3.67203e6	+	2.12005e6i	0.00403585	+	0.00233010i
$970$		0			0
$971$	−2.97824e7	+	1.71949e7i	−0.0325314	+	0.0187820i	−0.516177	−	0.856482i	$-0.672646\pi$
$971$	−2.97824e7	+	1.71949e7i	−0.0325314	+	0.0187820i	0.483646	+	0.875264i	$0.339312\pi$
$972$		0			0
$973$	7.87523e8	−	5.47135e8i	0.854918	−	0.593958i
$974$		0			0
$975$	−5.41601e8	−	9.38081e8i	−0.584340	−	1.01211i
$976$		0			0
$977$	2.60603e8	−	4.51377e8i	0.279444	−	0.484012i	−0.691802	−	0.722087i	$-0.743183\pi$
$977$	2.60603e8	−	4.51377e8i	0.279444	−	0.484012i	0.971247	+	0.238075i	$0.0765164\pi$
$978$		0			0
$979$		−	7.86987e8i		−	0.838725i
$980$		0			0
$981$	−6.96765e7			−0.0738039
$982$		0			0
$983$	5.40048e8	+	3.11797e8i	0.568554	+	0.328255i	0.756572	−	0.653911i	$-0.226873\pi$
$983$	5.40048e8	+	3.11797e8i	0.568554	+	0.328255i	−0.188018	+	0.982166i	$0.560206\pi$
$984$		0			0
$985$	−4.80559e7	+	2.77451e7i	−0.0502850	+	0.0290320i
$986$		0			0
$987$	−7.21587e8	−	1.03862e9i	−0.750476	−	1.08020i
$988$		0			0
$989$	−5.65472e8	−	9.79426e8i	−0.584550	−	1.01247i
$990$		0			0
$991$	6.99332e7	−	1.21128e8i	0.0718559	−	0.124458i	−0.827859	−	0.560937i	$-0.810441\pi$
$991$	6.99332e7	−	1.21128e8i	0.0718559	−	0.124458i	0.899715	+	0.436478i	$0.143774\pi$
$992$		0			0
$993$			7.99344e8i			0.816368i
$994$		0			0
$995$	1.00007e8			0.101522
$996$		0			0
$997$	−1.68279e9	−	9.71557e8i	−1.69802	−	0.980353i	−0.947636	−	0.319354i	$-0.896534\pi$
$997$	−1.68279e9	−	9.71557e8i	−1.69802	−	0.980353i	−0.750386	−	0.661000i	$-0.770132\pi$
$998$		0			0
$999$	−1.61396e9	+	9.31820e8i	−1.61881	+	0.934621i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.7.s.c.17.2		8
4.3	odd	2		14.7.d.a.3.4	✓	8
7.5	odd	6	inner	112.7.s.c.33.2		8
12.11	even	2		126.7.n.c.73.1		8
28.3	even	6		98.7.b.c.97.3		8
28.11	odd	6		98.7.b.c.97.2		8
28.19	even	6		14.7.d.a.5.4	yes	8
28.23	odd	6		98.7.d.c.19.3		8
28.27	even	2		98.7.d.c.31.3		8
84.47	odd	6		126.7.n.c.19.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.7.d.a.3.4	✓	8	4.3	odd	2
14.7.d.a.5.4	yes	8	28.19	even	6
98.7.b.c.97.2		8	28.11	odd	6
98.7.b.c.97.3		8	28.3	even	6
98.7.d.c.19.3		8	28.23	odd	6
98.7.d.c.31.3		8	28.27	even	2
112.7.s.c.17.2		8	1.1	even	1	trivial
112.7.s.c.33.2		8	7.5	odd	6	inner
126.7.n.c.19.1		8	84.47	odd	6
126.7.n.c.73.1		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-21.8891 - 12.6377i) q^{3} +(-10.7486 + 6.20573i) q^{5} +(-195.705 - 281.689i) q^{7} +(-45.0775 - 78.0766i) q^{9} +(-774.831 + 1342.05i) q^{11} -2770.09i q^{13} +313.705 q^{15} +(-109.615 - 63.2864i) q^{17} +(1147.81 - 662.687i) q^{19} +(723.904 + 8639.18i) q^{21} +(-7119.76 - 12331.8i) q^{23} +(-7735.48 + 13398.2i) q^{25} +20704.5i q^{27} -7479.03 q^{29} +(49241.9 + 28429.8i) q^{31} +(33920.8 - 19584.2i) q^{33} +(3851.65 + 1813.28i) q^{35} +(45005.7 + 77952.2i) q^{37} +(-35007.6 + 60634.9i) q^{39} +35783.9i q^{41} +79422.8 q^{43} +(969.044 + 559.478i) q^{45} +(-126334. + 72939.0i) q^{47} +(-41048.4 + 110256. i) q^{49} +(1599.59 + 2770.57i) q^{51} +(85090.3 - 147381. i) q^{53} -19233.6i q^{55} -33499.3 q^{57} +(187919. + 108495. i) q^{59} +(-172492. + 99588.5i) q^{61} +(-13171.4 + 27977.8i) q^{63} +(17190.5 + 29774.8i) q^{65} +(72219.0 - 125087. i) q^{67} +359909. i q^{69} -407591. q^{71} +(-161733. - 93376.3i) q^{73} +(338646. - 195517. i) q^{75} +(529678. - 44383.4i) q^{77} +(41929.3 + 72623.6i) q^{79} +(228795. - 396285. i) q^{81} -162495. i q^{83} +1570.95 q^{85} +(163709. + 94517.7i) q^{87} +(-439806. + 253922. i) q^{89} +(-780305. + 542120. i) q^{91} +(-718575. - 1.24461e6i) q^{93} +(-8224.91 + 14246.0i) q^{95} -509740. i q^{97} +139710. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) \) 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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