LMFDB - Embedded newform 78.6.b.a.25.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [78,6,Mod(25,78)] 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("78.25"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	78.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5099379454$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.6
Root			$6.10758 - 6.10758i$ of defining polynomial
Character	$\chi$	$=$	78.25
Dual form			78.6.b.a.25.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +37.1158i q^{5} -36.0000i q^{6} +176.407i q^{7} -64.0000i q^{8} +81.0000 q^{9} -148.463 q^{10} -179.404i q^{11} +144.000 q^{12} +(-554.740 - 252.104i) q^{13} -705.627 q^{14} -334.042i q^{15} +256.000 q^{16} -933.141 q^{17} +324.000i q^{18} -2335.97i q^{19} -593.853i q^{20} -1587.66i q^{21} +717.615 q^{22} -2792.36 q^{23} +576.000i q^{24} +1747.42 q^{25} +(1008.42 - 2218.96i) q^{26} -729.000 q^{27} -2822.51i q^{28} +1503.58 q^{29} +1336.17 q^{30} -737.236i q^{31} +1024.00i q^{32} +1614.63i q^{33} -3732.56i q^{34} -6547.48 q^{35} -1296.00 q^{36} -3775.41i q^{37} +9343.88 q^{38} +(4992.66 + 2268.94i) q^{39} +2375.41 q^{40} +368.848i q^{41} +6350.64 q^{42} -20180.6 q^{43} +2870.46i q^{44} +3006.38i q^{45} -11169.4i q^{46} +20526.5i q^{47} -2304.00 q^{48} -14312.3 q^{49} +6989.67i q^{50} +8398.27 q^{51} +(8875.84 + 4033.67i) q^{52} -25081.9 q^{53} -2916.00i q^{54} +6658.72 q^{55} +11290.0 q^{56} +21023.7i q^{57} +6014.33i q^{58} +35326.3i q^{59} +5344.68i q^{60} +31741.4 q^{61} +2948.94 q^{62} +14288.9i q^{63} -4096.00 q^{64} +(9357.06 - 20589.6i) q^{65} -6458.54 q^{66} -46661.3i q^{67} +14930.3 q^{68} +25131.2 q^{69} -26189.9i q^{70} +58973.4i q^{71} -5184.00i q^{72} +3411.20i q^{73} +15101.6 q^{74} -15726.8 q^{75} +37375.5i q^{76} +31648.1 q^{77} +(-9075.76 + 19970.6i) q^{78} -64977.3 q^{79} +9501.64i q^{80} +6561.00 q^{81} -1475.39 q^{82} +12233.4i q^{83} +25402.6i q^{84} -34634.3i q^{85} -80722.5i q^{86} -13532.2 q^{87} -11481.8 q^{88} -61754.7i q^{89} -12025.5 q^{90} +(44472.9 - 97859.9i) q^{91} +44677.8 q^{92} +6635.12i q^{93} -82106.0 q^{94} +86701.4 q^{95} -9216.00i q^{96} -28030.7i q^{97} -57249.4i q^{98} -14531.7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30}+ \cdots + 567168 q^{95}+O(q^{100}) $ 6 * q - 54 * q^3 - 96 * q^4 + 486 * q^9 + 320 * q^10 + 864 * q^12 + 530 * q^13 - 1360 * q^14 + 1536 * q^16 - 836 * q^17 - 1296 * q^22 - 416 * q^23 + 718 * q^25 - 1360 * q^26 - 4374 * q^27 + 18788 * q^29 - 2880 * q^30 + 6112 * q^35 - 7776 * q^36 + 528 * q^38 - 4770 * q^39 - 5120 * q^40 + 12240 * q^42 - 24200 * q^43 - 13824 * q^48 - 3038 * q^49 + 7524 * q^51 - 8480 * q^52 - 42396 * q^53 + 124656 * q^55 + 21760 * q^56 - 3196 * q^61 - 59344 * q^62 - 24576 * q^64 - 17168 * q^65 + 11664 * q^66 + 13376 * q^68 + 3744 * q^69 - 62240 * q^74 - 6462 * q^75 - 114024 * q^77 + 12240 * q^78 - 169328 * q^79 + 39366 * q^81 + 145120 * q^82 - 169092 * q^87 + 20736 * q^88 + 25920 * q^90 + 236152 * q^91 + 6656 * q^92 - 32688 * q^94 + 567168 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$1$	$-1$

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 $

$2$			4.00000i			0.707107i
$2$			4.00000i			0.707107i
$3$	−9.00000			−0.577350
$3$	−9.00000			−0.577350
$4$	−16.0000			−0.500000
$5$			37.1158i			0.663948i	0.943289	+	0.331974i	$0.107715\pi$
$5$			37.1158i			0.663948i	−0.943289	+	0.331974i	$0.892285\pi$
$6$		−	36.0000i		−	0.408248i
$7$			176.407i			1.36072i	0.732876	+	0.680362i	$0.238177\pi$
$7$			176.407i			1.36072i	−0.732876	+	0.680362i	$0.761823\pi$
$8$		−	64.0000i		−	0.353553i
$9$	81.0000			0.333333
$10$	−148.463			−0.469482
$11$		−	179.404i		−	0.447044i	−0.974699	−	0.223522i	$-0.928245\pi$
$11$		−	179.404i		−	0.447044i	0.974699	−	0.223522i	$-0.0717554\pi$
$12$	144.000			0.288675
$13$	−554.740	−	252.104i	−0.910397	−	0.413735i
$13$	−554.740	−	252.104i	−0.910397	−	0.413735i
$14$	−705.627			−0.962177
$15$		−	334.042i		−	0.383330i
$16$	256.000			0.250000
$17$	−933.141			−0.783114			−0.391557	−	0.920154i	$-0.628063\pi$
$17$	−933.141			−0.783114			−0.391557	+	0.920154i	$0.628063\pi$
$18$			324.000i			0.235702i
$19$		−	2335.97i		−	1.48451i	−0.670117	−	0.742256i	$-0.733756\pi$
$19$		−	2335.97i		−	1.48451i	0.670117	−	0.742256i	$-0.266244\pi$
$20$		−	593.853i		−	0.331974i
$21$		−	1587.66i		−	0.785614i
$22$	717.615			0.316108
$23$	−2792.36			−1.10066			−0.550328	−	0.834948i	$-0.685497\pi$
$23$	−2792.36			−1.10066			−0.550328	+	0.834948i	$0.685497\pi$
$24$			576.000i			0.204124i
$25$	1747.42			0.559174
$26$	1008.42	−	2218.96i	0.292555	−	0.643748i
$27$	−729.000			−0.192450
$28$		−	2822.51i		−	0.680362i
$29$	1503.58			0.331996			0.165998	−	0.986126i	$-0.446916\pi$
$29$	1503.58			0.331996			0.165998	+	0.986126i	$0.446916\pi$
$30$	1336.17			0.271055
$31$		−	737.236i		−	0.137785i	−0.997624	−	0.0688926i	$-0.978053\pi$
$31$		−	737.236i		−	0.137785i	0.997624	−	0.0688926i	$-0.0219466\pi$
$32$			1024.00i			0.176777i
$33$			1614.63i			0.258101i
$34$		−	3732.56i		−	0.553745i
$35$	−6547.48			−0.903450
$36$	−1296.00			−0.166667
$37$		−	3775.41i		−	0.453377i	−0.973967	−	0.226689i	$-0.927210\pi$
$37$		−	3775.41i		−	0.453377i	0.973967	−	0.226689i	$-0.0727900\pi$
$38$	9343.88			1.04971
$39$	4992.66	+	2268.94i	0.525618	+	0.238870i
$40$	2375.41			0.234741
$41$			368.848i			0.0342680i	0.999853	+	0.0171340i	$0.00545418\pi$
$41$			368.848i			0.0342680i	−0.999853	+	0.0171340i	$0.994546\pi$
$42$	6350.64			0.555513
$43$	−20180.6			−1.66442			−0.832211	−	0.554459i	$-0.812925\pi$
$43$	−20180.6			−1.66442			−0.832211	+	0.554459i	$0.812925\pi$
$44$			2870.46i			0.223522i
$45$			3006.38i			0.221316i
$46$		−	11169.4i		−	0.778282i
$47$			20526.5i			1.35541i	0.735335	+	0.677704i	$0.237025\pi$
$47$			20526.5i			1.35541i	−0.735335	+	0.677704i	$0.762975\pi$
$48$	−2304.00			−0.144338
$49$	−14312.3			−0.851570
$50$			6989.67i			0.395395i
$51$	8398.27			0.452131
$52$	8875.84	+	4033.67i	0.455199	+	0.206867i
$53$	−25081.9			−1.22651			−0.613253	−	0.789886i	$-0.710140\pi$
$53$	−25081.9			−1.22651			−0.613253	+	0.789886i	$0.710140\pi$
$54$		−	2916.00i		−	0.136083i
$55$	6658.72			0.296814
$56$	11290.0			0.481089
$57$			21023.7i			0.857083i
$58$			6014.33i			0.234756i
$59$			35326.3i			1.32120i	0.750738	+	0.660600i	$0.229698\pi$
$59$			35326.3i			1.32120i	−0.750738	+	0.660600i	$0.770302\pi$
$60$			5344.68i			0.191665i
$61$	31741.4			1.09220			0.546099	−	0.837721i	$-0.316112\pi$
$61$	31741.4			1.09220			0.546099	+	0.837721i	$0.316112\pi$
$62$	2948.94			0.0974288
$63$			14288.9i			0.453575i
$64$	−4096.00			−0.125000
$65$	9357.06	−	20589.6i	0.274698	−	0.604456i
$66$	−6458.54			−0.182505
$67$		−	46661.3i		−	1.26990i	−0.772552	−	0.634951i	$-0.781020\pi$
$67$		−	46661.3i		−	1.26990i	0.772552	−	0.634951i	$-0.218980\pi$
$68$	14930.3			0.391557
$69$	25131.2			0.635465
$70$		−	26189.9i		−	0.638835i
$71$			58973.4i			1.38839i	0.719789	+	0.694193i	$0.244238\pi$
$71$			58973.4i			1.38839i	−0.719789	+	0.694193i	$0.755762\pi$
$72$		−	5184.00i		−	0.117851i
$73$			3411.20i			0.0749204i	0.999298	+	0.0374602i	$0.0119267\pi$
$73$			3411.20i			0.0749204i	−0.999298	+	0.0374602i	$0.988073\pi$
$74$	15101.6			0.320586
$75$	−15726.8			−0.322839
$76$			37375.5i			0.742256i
$77$	31648.1			0.608303
$78$	−9075.76	+	19970.6i	−0.168907	+	0.371668i
$79$	−64977.3			−1.17137			−0.585685	−	0.810539i	$-0.699174\pi$
$79$	−64977.3			−1.17137			−0.585685	+	0.810539i	$0.699174\pi$
$80$			9501.64i			0.165987i
$81$	6561.00			0.111111
$82$	−1475.39			−0.0242311
$83$			12233.4i			0.194918i	0.995240	+	0.0974591i	$0.0310715\pi$
$83$			12233.4i			0.194918i	−0.995240	+	0.0974591i	$0.968928\pi$
$84$			25402.6i			0.392807i
$85$		−	34634.3i		−	0.519947i
$86$		−	80722.5i		−	1.17692i
$87$	−13532.2			−0.191678
$88$	−11481.8			−0.158054
$89$		−	61754.7i		−	0.826409i	−0.910638	−	0.413204i	$-0.864410\pi$
$89$		−	61754.7i		−	0.826409i	0.910638	−	0.413204i	$-0.135590\pi$
$90$	−12025.5			−0.156494
$91$	44472.9	−	97859.9i	0.562979	−	1.23880i
$92$	44677.8			0.550328
$93$			6635.12i			0.0795503i
$94$	−82106.0			−0.958418
$95$	86701.4			0.985638
$96$		−	9216.00i		−	0.102062i
$97$		−	28030.7i		−	0.302485i	−0.988497	−	0.151243i	$-0.951673\pi$
$97$		−	28030.7i		−	0.302485i	0.988497	−	0.151243i	$-0.0483274\pi$
$98$		−	57249.4i		−	0.602151i
$99$		−	14531.7i		−	0.149015i
$100$	−27958.7			−0.279587
$101$	102600.			1.00080			0.500398	−	0.865796i	$-0.333187\pi$
$101$	102600.			1.00080			0.500398	+	0.865796i	$0.333187\pi$
$102$			33593.1i			0.319705i
$103$	−203803.			−1.89286			−0.946430	−	0.322909i	$-0.895339\pi$
$103$	−203803.			−1.89286			−0.946430	+	0.322909i	$0.895339\pi$
$104$	−16134.7	+	35503.4i	−0.146277	+	0.321874i
$105$	58927.3			0.521607
$106$		−	100327.i		−	0.867271i
$107$	−147356.			−1.24425			−0.622125	−	0.782918i	$-0.713731\pi$
$107$	−147356.			−1.24425			−0.622125	+	0.782918i	$0.713731\pi$
$108$	11664.0			0.0962250
$109$			7846.65i			0.0632584i	0.999500	+	0.0316292i	$0.0100696\pi$
$109$			7846.65i			0.0632584i	−0.999500	+	0.0316292i	$0.989930\pi$
$110$			26634.9i			0.209879i
$111$			33978.7i			0.261758i
$112$			45160.1i			0.340181i
$113$	−192673.			−1.41947			−0.709733	−	0.704471i	$-0.751184\pi$
$113$	−192673.			−1.41947			−0.709733	+	0.704471i	$0.751184\pi$
$114$	−84095.0			−0.606049
$115$		−	103641.i		−	0.730778i
$116$	−24057.3			−0.165998
$117$	−44933.9	−	20420.5i	−0.303466	−	0.137912i
$118$	−141305.			−0.934229
$119$		−	164612.i		−	1.06560i
$120$	−21378.7			−0.135528
$121$	128865.			0.800152
$122$			126966.i			0.772301i
$123$		−	3319.64i		−	0.0197846i
$124$			11795.8i			0.0688926i
$125$			180844.i			1.03521i
$126$	−57155.8			−0.320726
$127$	255644.			1.40646			0.703229	−	0.710964i	$-0.251741\pi$
$127$	255644.			1.40646			0.703229	+	0.710964i	$0.251741\pi$
$128$		−	16384.0i		−	0.0883883i
$129$	181626.			0.960955
$130$	82358.5	+	37428.2i	0.427415	+	0.194241i
$131$	−134257.			−0.683530			−0.341765	−	0.939786i	$-0.611025\pi$
$131$	−134257.			−0.683530			−0.341765	+	0.939786i	$0.611025\pi$
$132$		−	25834.2i		−	0.129050i
$133$	412081.			2.02001
$134$	186645.			0.897956
$135$		−	27057.4i		−	0.127777i
$136$			59721.0i			0.276873i
$137$		−	74373.4i		−	0.338545i	−0.985569	−	0.169272i	$-0.945858\pi$
$137$		−	74373.4i		−	0.338545i	0.985569	−	0.169272i	$-0.0541418\pi$
$138$			100525.i			0.449341i
$139$	36348.4			0.159569			0.0797844	−	0.996812i	$-0.474577\pi$
$139$	36348.4			0.159569			0.0797844	+	0.996812i	$0.474577\pi$
$140$	104760.			0.451725
$141$		−	184738.i		−	0.782545i
$142$	−235893.			−0.981737
$143$	−45228.5	+	99522.5i	−0.184958	+	0.406988i
$144$	20736.0			0.0833333
$145$			55806.7i			0.220428i
$146$	−13644.8			−0.0529767
$147$	128811.			0.491654
$148$			60406.6i			0.226689i
$149$			221253.i			0.816440i	0.912884	+	0.408220i	$0.133850\pi$
$149$			221253.i			0.816440i	−0.912884	+	0.408220i	$0.866150\pi$
$150$		−	62907.0i		−	0.228282i
$151$			482448.i			1.72190i	0.508690	+	0.860950i	$0.330130\pi$
$151$			482448.i			1.72190i	−0.508690	+	0.860950i	$0.669870\pi$
$152$	−149502.			−0.524854
$153$	−75584.4			−0.261038
$154$			126592.i			0.430135i
$155$	27363.1			0.0914821
$156$	−79882.5	−	36303.0i	−0.262809	−	0.119435i
$157$	−110726.			−0.358508			−0.179254	−	0.983803i	$-0.557368\pi$
$157$	−110726.			−0.358508			−0.179254	+	0.983803i	$0.557368\pi$
$158$		−	259909.i		−	0.828283i
$159$	225737.			0.708124
$160$	−38006.6			−0.117370
$161$		−	492591.i		−	1.49769i
$162$			26244.0i			0.0785674i
$163$			161064.i			0.474819i	0.971410	+	0.237410i	$0.0762984\pi$
$163$			161064.i			0.474819i	−0.971410	+	0.237410i	$0.923702\pi$
$164$		−	5901.58i		−	0.0171340i
$165$	−59928.5			−0.171365
$166$	−48933.7			−0.137828
$167$			466578.i			1.29459i	0.762238	+	0.647296i	$0.224100\pi$
$167$			466578.i			1.29459i	−0.762238	+	0.647296i	$0.775900\pi$
$168$	−101610.			−0.277757
$169$	244180.	+	279705.i	0.657647	+	0.753326i
$170$	138537.			0.367658
$171$		−	189214.i		−	0.494837i
$172$	322890.			0.832211
$173$	−240767.			−0.611620			−0.305810	−	0.952093i	$-0.598927\pi$
$173$	−240767.			−0.611620			−0.305810	+	0.952093i	$0.598927\pi$
$174$		−	54129.0i		−	0.135537i
$175$			308256.i			0.760881i
$176$		−	45927.4i		−	0.111761i
$177$		−	317937.i		−	0.762795i
$178$	247019.			0.584359
$179$	−266435.			−0.621526			−0.310763	−	0.950487i	$-0.600585\pi$
$179$	−266435.			−0.621526			−0.310763	+	0.950487i	$0.600585\pi$
$180$		−	48102.1i		−	0.110658i
$181$	64307.2			0.145903			0.0729513	−	0.997336i	$-0.476758\pi$
$181$	64307.2			0.145903			0.0729513	+	0.997336i	$0.476758\pi$
$182$	391439.	+	177892.i	0.875964	+	0.398086i
$183$	−285673.			−0.630581
$184$			178711.i			0.389141i
$185$	140127.			0.301019
$186$	−26540.5			−0.0562505
$187$			167409.i			0.350086i
$188$		−	328424.i		−	0.677704i
$189$		−	128601.i		−	0.261871i
$190$			346806.i			0.696951i
$191$	−925387.			−1.83544			−0.917720	−	0.397228i	$-0.869972\pi$
$191$	−925387.			−1.83544			−0.917720	+	0.397228i	$0.869972\pi$
$192$	36864.0			0.0721688
$193$		−	632306.i		−	1.22190i	−0.791671	−	0.610948i	$-0.790789\pi$
$193$		−	632306.i		−	1.22190i	0.791671	−	0.610948i	$-0.209211\pi$
$194$	112123.			0.213889
$195$	−84213.5	+	185307.i	−0.158597	+	0.348983i
$196$	228997.			0.425785
$197$			100048.i			0.183671i	0.995774	+	0.0918357i	$0.0292735\pi$
$197$			100048.i			0.183671i	−0.995774	+	0.0918357i	$0.970727\pi$
$198$	58126.8			0.105369
$199$	−723076.			−1.29435			−0.647174	−	0.762342i	$-0.724049\pi$
$199$	−723076.			−1.29435			−0.647174	+	0.762342i	$0.724049\pi$
$200$		−	111835.i		−	0.197698i
$201$			419952.i			0.733178i
$202$			410401.i			0.707669i
$203$			265242.i			0.451754i
$204$	−134372.			−0.226066
$205$	−13690.1			−0.0227521
$206$		−	815214.i		−	1.33845i
$207$	−226181.			−0.366886
$208$	−142013.	−	64538.7i	−0.227599	−	0.103434i
$209$	−419082.			−0.663641
$210$			235709.i			0.368832i
$211$	259548.			0.401340			0.200670	−	0.979659i	$-0.435688\pi$
$211$	259548.			0.401340			0.200670	+	0.979659i	$0.435688\pi$
$212$	401310.			0.613253
$213$		−	530760.i		−	0.801585i
$214$		−	589423.i		−	0.879818i
$215$		−	749020.i		−	1.10509i
$216$			46656.0i			0.0680414i
$217$	130053.			0.187488
$218$	−31386.6			−0.0447304
$219$		−	30700.8i		−	0.0432553i
$220$	−106539.			−0.148407
$221$	517651.	+	235249.i	0.712945	+	0.324002i
$222$	−135915.			−0.185091
$223$		−	1.07196e6i		−	1.44349i	−0.692156	−	0.721747i	$-0.743339\pi$
$223$		−	1.07196e6i		−	1.44349i	0.692156	−	0.721747i	$-0.256661\pi$
$224$	−180641.			−0.240544
$225$	141541.			0.186391
$226$		−	770692.i		−	1.00371i
$227$			65210.0i			0.0839942i	0.999118	+	0.0419971i	$0.0133720\pi$
$227$			65210.0i			0.0839942i	−0.999118	+	0.0419971i	$0.986628\pi$
$228$		−	336380.i		−	0.428541i
$229$			191854.i			0.241759i	0.992667	+	0.120879i	$0.0385715\pi$
$229$			191854.i			0.241759i	−0.992667	+	0.120879i	$0.961429\pi$
$230$	414563.			0.516738
$231$	−284832.			−0.351204
$232$		−	96229.3i		−	0.117378i
$233$	353920.			0.427087			0.213543	−	0.976934i	$-0.431500\pi$
$233$	353920.			0.427087			0.213543	+	0.976934i	$0.431500\pi$
$234$	81681.8	−	179736.i	0.0975182	−	0.214583i
$235$	−761857.			−0.899920
$236$		−	565221.i		−	0.660600i
$237$	584796.			0.676291
$238$	658450.			0.753495
$239$		−	1.15374e6i		−	1.30651i	−0.757138	−	0.653255i	$-0.773403\pi$
$239$		−	1.15374e6i		−	1.30651i	0.757138	−	0.653255i	$-0.226597\pi$
$240$		−	85514.8i		−	0.0958326i
$241$			209753.i			0.232630i	0.993212	+	0.116315i	$0.0371083\pi$
$241$			209753.i			0.232630i	−0.993212	+	0.116315i	$0.962892\pi$
$242$			515461.i			0.565793i
$243$	−59049.0			−0.0641500
$244$	−507862.			−0.546099
$245$		−	531214.i		−	0.565398i
$246$	13278.5			0.0139898
$247$	−588909.	+	1.29586e6i	−0.614194	+	1.35150i
$248$	−47183.1			−0.0487144
$249$		−	110101.i		−	0.112536i
$250$	−723375.			−0.732004
$251$	427156.			0.427959			0.213979	−	0.976838i	$-0.431357\pi$
$251$	427156.			0.427959			0.213979	+	0.976838i	$0.431357\pi$
$252$		−	228623.i		−	0.226787i
$253$			500960.i			0.492042i
$254$			1.02258e6i			0.994515i
$255$			311708.i			0.300191i
$256$	65536.0			0.0625000
$257$	207722.			0.196177			0.0980887	−	0.995178i	$-0.468727\pi$
$257$	207722.			0.196177			0.0980887	+	0.995178i	$0.468727\pi$
$258$			726503.i			0.679498i
$259$	666008.			0.616922
$260$	−149713.	+	329434.i	−0.137349	+	0.302228i
$261$	121790.			0.110665
$262$		−	537026.i		−	0.483328i
$263$	−831712.			−0.741453			−0.370726	−	0.928742i	$-0.620891\pi$
$263$	−831712.			−0.741453			−0.370726	+	0.928742i	$0.620891\pi$
$264$	103337.			0.0912524
$265$		−	930933.i		−	0.814336i
$266$			1.64832e6i			1.42836i
$267$			555792.i			0.477127i
$268$			746581.i			0.634951i
$269$	1.67365e6			1.41021			0.705104	−	0.709104i	$-0.250901\pi$
$269$	1.67365e6			1.41021			0.705104	+	0.709104i	$0.250901\pi$
$270$	108230.			0.0903518
$271$			69792.6i			0.0577279i	0.999583	+	0.0288640i	$0.00918896\pi$
$271$			69792.6i			0.0577279i	−0.999583	+	0.0288640i	$0.990811\pi$
$272$	−238884.			−0.195779
$273$	−400256.	+	880739.i	−0.325036	+	0.715221i
$274$	297494.			0.239387
$275$		−	313493.i		−	0.249975i
$276$	−402100.			−0.317732
$277$	1.61380e6			1.26372			0.631859	−	0.775083i	$-0.282292\pi$
$277$	1.61380e6			1.26372			0.631859	+	0.775083i	$0.282292\pi$
$278$			145394.i			0.112832i
$279$		−	59716.1i		−	0.0459284i
$280$			419039.i			0.319418i
$281$			1.05655e6i			0.798224i	0.916902	+	0.399112i	$0.130682\pi$
$281$			1.05655e6i			0.798224i	−0.916902	+	0.399112i	$0.869318\pi$
$282$	738954.			0.553343
$283$	−512741.			−0.380567			−0.190284	−	0.981729i	$-0.560941\pi$
$283$	−512741.			−0.380567			−0.190284	+	0.981729i	$0.560941\pi$
$284$		−	943574.i		−	0.694193i
$285$	−780313.			−0.569058
$286$	−398090.	−	180914.i	−0.287784	−	0.130785i
$287$	−65067.4			−0.0466293
$288$			82944.0i			0.0589256i
$289$	−549105.			−0.386732
$290$	−223227.			−0.155866
$291$			252276.i			0.174640i
$292$		−	54579.2i		−	0.0374602i
$293$			2.71268e6i			1.84599i	0.384809	+	0.922996i	$0.374267\pi$
$293$			2.71268e6i			1.84599i	−0.384809	+	0.922996i	$0.625733\pi$
$294$			515244.i			0.347652i
$295$	−1.31117e6			−0.877207
$296$	−241626.			−0.160293
$297$			130785.i			0.0860336i
$298$	−885014.			−0.577310
$299$	1.54903e6	+	703967.i	1.00204	+	0.455380i
$300$	251628.			0.161420
$301$		−	3.56000e6i		−	2.26482i
$302$	−1.92979e6			−1.21757
$303$	−923403.			−0.577810
$304$		−	598009.i		−	0.371128i
$305$			1.17811e6i			0.725162i
$306$		−	302338.i		−	0.184582i
$307$			2.44475e6i			1.48043i	0.672371	+	0.740215i	$0.265276\pi$
$307$			2.44475e6i			1.48043i	−0.672371	+	0.740215i	$0.734724\pi$
$308$	−506369.			−0.304152
$309$	1.83423e6			1.09284
$310$			109452.i			0.0646876i
$311$	1.99183e6			1.16775			0.583877	−	0.811842i	$-0.301535\pi$
$311$	1.99183e6			1.16775			0.583877	+	0.811842i	$0.301535\pi$
$312$	145212.	−	319530.i	0.0844533	−	0.185834i
$313$	1.66396e6			0.960025			0.480013	−	0.877262i	$-0.340632\pi$
$313$	1.66396e6			0.960025			0.480013	+	0.877262i	$0.340632\pi$
$314$		−	442903.i		−	0.253504i
$315$	−530346.			−0.301150
$316$	1.03964e6			0.585685
$317$		−	3.20482e6i		−	1.79125i	−0.444813	−	0.895624i	$-0.646730\pi$
$317$		−	3.20482e6i		−	1.79125i	0.444813	−	0.895624i	$-0.353270\pi$
$318$			902947.i			0.500719i
$319$		−	269749.i		−	0.148417i
$320$		−	152026.i		−	0.0829935i
$321$	1.32620e6			0.718369
$322$	1.97037e6			1.05903
$323$			2.17979e6i			1.16254i
$324$	−104976.			−0.0555556
$325$	−969362.	−	440532.i	−0.509070	−	0.231350i
$326$	−644254.			−0.335748
$327$		−	70619.9i		−	0.0365222i
$328$	23606.3			0.0121156
$329$	−3.62101e6			−1.84434
$330$		−	239714.i		−	0.121174i
$331$		−	3.53386e6i		−	1.77288i	−0.462844	−	0.886440i	$-0.653171\pi$
$331$		−	3.53386e6i		−	1.77288i	0.462844	−	0.886440i	$-0.346829\pi$
$332$		−	195735.i		−	0.0974591i
$333$		−	305808.i		−	0.151126i
$334$	−1.86631e6			−0.915415
$335$	1.73187e6			0.843148
$336$		−	406441.i		−	0.196404i
$337$	1.10242e6			0.528777			0.264389	−	0.964416i	$-0.414830\pi$
$337$	1.10242e6			0.528777			0.264389	+	0.964416i	$0.414830\pi$
$338$	−1.11882e6	+	976719.i	−0.532682	+	0.465027i
$339$	1.73406e6			0.819529
$340$			554148.i			0.259973i
$341$	−132263.			−0.0615960
$342$	756855.			0.349903
$343$			440075.i			0.201972i
$344$			1.29156e6i			0.588462i
$345$			932766.i			0.421915i
$346$		−	963068.i		−	0.432481i
$347$	3.16432e6			1.41077			0.705386	−	0.708823i	$-0.250774\pi$
$347$	3.16432e6			1.41077			0.705386	+	0.708823i	$0.250774\pi$
$348$	216516.			0.0958389
$349$		−	1.97526e6i		−	0.868080i	−0.900894	−	0.434040i	$-0.857088\pi$
$349$		−	1.97526e6i		−	0.868080i	0.900894	−	0.434040i	$-0.142912\pi$
$350$	−1.23302e6			−0.538024
$351$	404405.	+	183784.i	0.175206	+	0.0796233i
$352$	183710.			0.0790269
$353$		−	2.24015e6i		−	0.956843i	−0.878130	−	0.478422i	$-0.841209\pi$
$353$		−	2.24015e6i		−	0.956843i	0.878130	−	0.478422i	$-0.158791\pi$
$354$	1.27175e6			0.539378
$355$	−2.18884e6			−0.921815
$356$			988075.i			0.413204i
$357$			1.48151e6i			0.615226i
$358$		−	1.06574e6i		−	0.439485i
$359$			2.13770e6i			0.875407i	0.899119	+	0.437704i	$0.144208\pi$
$359$			2.13770e6i			0.875407i	−0.899119	+	0.437704i	$0.855792\pi$
$360$	192408.			0.0782470
$361$	−2.98066e6			−1.20377
$362$			257229.i			0.103169i
$363$	−1.15979e6			−0.461968
$364$	−711567.	+	1.56576e6i	−0.281490	+	0.619400i
$365$	−126609.			−0.0497432
$366$		−	1.14269e6i		−	0.445888i
$367$	1.70502e6			0.660791			0.330395	−	0.943843i	$-0.392818\pi$
$367$	1.70502e6			0.660791			0.330395	+	0.943843i	$0.392818\pi$
$368$	−714844.			−0.275164
$369$			29876.7i			0.0114227i
$370$			560509.i			0.212852i
$371$		−	4.42461e6i		−	1.66894i
$372$		−	106162.i		−	0.0397751i
$373$	−2.31503e6			−0.861558			−0.430779	−	0.902458i	$-0.641761\pi$
$373$	−2.31503e6			−0.861558			−0.430779	+	0.902458i	$0.641761\pi$
$374$	−669636.			−0.247548
$375$		−	1.62759e6i		−	0.597679i
$376$	1.31370e6			0.479209
$377$	−834097.	−	379060.i	−0.302248	−	0.137358i
$378$	514402.			0.185171
$379$		−	4.32354e6i		−	1.54612i	−0.634336	−	0.773058i	$-0.718726\pi$
$379$		−	4.32354e6i		−	1.54612i	0.634336	−	0.773058i	$-0.281274\pi$
$380$	−1.38722e6			−0.492819
$381$	−2.30080e6			−0.812018
$382$		−	3.70155e6i		−	1.29785i
$383$			2.00351e6i			0.697901i	0.937141	+	0.348951i	$0.113462\pi$
$383$			2.00351e6i			0.697901i	−0.937141	+	0.348951i	$0.886538\pi$
$384$			147456.i			0.0510310i
$385$			1.17464e6i			0.403882i
$386$	2.52922e6			0.864011
$387$	−1.63463e6			−0.554807
$388$			448490.i			0.151243i
$389$	1.70696e6			0.571940			0.285970	−	0.958239i	$-0.407684\pi$
$389$	1.70696e6			0.571940			0.285970	+	0.958239i	$0.407684\pi$
$390$	−741226.	−	336854.i	−0.246768	−	0.112145i
$391$	2.60567e6			0.861940
$392$			915990.i			0.301076i
$393$	1.20831e6			0.394636
$394$	−400191.			−0.129875
$395$		−	2.41168e6i		−	0.777728i
$396$			232507.i			0.0745073i
$397$			1.22427e6i			0.389853i	0.980818	+	0.194927i	$0.0624469\pi$
$397$			1.22427e6i			0.389853i	−0.980818	+	0.194927i	$0.937553\pi$
$398$		−	2.89230e6i		−	0.915243i
$399$	−3.70873e6			−1.16625
$400$	447339.			0.139793
$401$		−	4.15598e6i		−	1.29066i	−0.763903	−	0.645332i	$-0.776719\pi$
$401$		−	4.15598e6i		−	1.29066i	0.763903	−	0.645332i	$-0.223281\pi$
$402$	−1.67981e6			−0.518435
$403$	−185860.	+	408974.i	−0.0570065	+	0.125439i
$404$	−1.64161e6			−0.500398
$405$			243517.i			0.0737720i
$406$	−1.06097e6			−0.319439
$407$	−677323.			−0.202680
$408$		−	537489.i		−	0.159852i
$409$			6.52428e6i			1.92852i	0.264958	+	0.964260i	$0.414642\pi$
$409$			6.52428e6i			1.92852i	−0.264958	+	0.964260i	$0.585358\pi$
$410$		−	54760.4i		−	0.0160882i
$411$			669361.i			0.195459i
$412$	3.26086e6			0.946430
$413$	−6.23180e6			−1.79779
$414$		−	904725.i		−	0.259427i
$415$	−454053.			−0.129416
$416$	258155.	−	568054.i	0.0731387	−	0.160937i
$417$	−327136.			−0.0921271
$418$		−	1.67633e6i		−	0.469265i
$419$	4.77950e6			1.32999			0.664994	−	0.746849i	$-0.268434\pi$
$419$	4.77950e6			1.32999			0.664994	+	0.746849i	$0.268434\pi$
$420$	−942837.			−0.260803
$421$			3.41558e6i			0.939203i	0.882878	+	0.469602i	$0.155602\pi$
$421$			3.41558e6i			0.939203i	−0.882878	+	0.469602i	$0.844398\pi$
$422$			1.03819e6i			0.283790i
$423$			1.66265e6i			0.451803i
$424$			1.60524e6i			0.433636i
$425$	−1.63059e6			−0.437897
$426$	2.12304e6			0.566806
$427$			5.59940e6i			1.48618i
$428$	2.35769e6			0.622125
$429$	407057.	−	895702.i	0.106785	−	0.234974i
$430$	2.99608e6			0.781416
$431$		−	6.21413e6i		−	1.61134i	−0.592365	−	0.805670i	$-0.701806\pi$
$431$		−	6.21413e6i		−	1.61134i	0.592365	−	0.805670i	$-0.298194\pi$
$432$	−186624.			−0.0481125
$433$	−5.83362e6			−1.49526			−0.747632	−	0.664113i	$-0.768809\pi$
$433$	−5.83362e6			−1.49526			−0.747632	+	0.664113i	$0.768809\pi$
$434$			520214.i			0.132574i
$435$		−	502260.i		−	0.127264i
$436$		−	125546.i		−	0.0316292i
$437$			6.52287e6i			1.63394i
$438$	122803.			0.0305861
$439$	−766922.			−0.189928			−0.0949642	−	0.995481i	$-0.530274\pi$
$439$	−766922.			−0.189928			−0.0949642	+	0.995481i	$0.530274\pi$
$440$		−	426158.i		−	0.104939i
$441$	−1.15930e6			−0.283857
$442$	−940996.	+	2.07060e6i	−0.229104	+	0.504128i
$443$	1.02042e6			0.247042			0.123521	−	0.992342i	$-0.460581\pi$
$443$	1.02042e6			0.247042			0.123521	+	0.992342i	$0.460581\pi$
$444$		−	543659.i		−	0.130879i
$445$	2.29207e6			0.548692
$446$	4.28783e6			1.02071
$447$		−	1.99128e6i		−	0.471372i
$448$		−	722562.i		−	0.170091i
$449$			4.18123e6i			0.978787i	0.872063	+	0.489394i	$0.162782\pi$
$449$			4.18123e6i			0.978787i	−0.872063	+	0.489394i	$0.837218\pi$
$450$			566163.i			0.131798i
$451$	66172.8			0.0153193
$452$	3.08277e6			0.709733
$453$		−	4.34203e6i		−	0.994139i
$454$	−260840.			−0.0593929
$455$	3.63215e6	+	1.65065e6i	0.822498	+	0.373789i
$456$	1.34552e6			0.303025
$457$			3.58333e6i			0.802594i	0.915948	+	0.401297i	$0.131441\pi$
$457$			3.58333e6i			0.802594i	−0.915948	+	0.401297i	$0.868559\pi$
$458$	−767417.			−0.170949
$459$	680260.			0.150710
$460$			1.65825e6i			0.365389i
$461$		−	1.67459e6i		−	0.366992i	−0.983020	−	0.183496i	$-0.941259\pi$
$461$		−	1.67459e6i		−	0.366992i	0.983020	−	0.183496i	$-0.0587414\pi$
$462$		−	1.13933e6i		−	0.248339i
$463$			6.62362e6i			1.43596i	0.696063	+	0.717981i	$0.254934\pi$
$463$			6.62362e6i			1.43596i	−0.696063	+	0.717981i	$0.745066\pi$
$464$	384917.			0.0829989
$465$	−246268.			−0.0528172
$466$			1.41568e6i			0.301996i
$467$	3.80744e6			0.807868			0.403934	−	0.914788i	$-0.367642\pi$
$467$	3.80744e6			0.807868			0.403934	+	0.914788i	$0.367642\pi$
$468$	718943.	+	326727.i	0.151733	+	0.0689558i
$469$	8.23138e6			1.72799
$470$		−	3.04743e6i		−	0.636339i
$471$	996531.			0.206985
$472$	2.26089e6			0.467115
$473$			3.62048e6i			0.744070i
$474$			2.33918e6i			0.478210i
$475$		−	4.08192e6i		−	0.830099i
$476$			2.63380e6i			0.532801i
$477$	−2.03163e6			−0.408836
$478$	4.61495e6			0.923841
$479$		−	4.85878e6i		−	0.967583i	−0.875183	−	0.483792i	$-0.839259\pi$
$479$		−	4.85878e6i		−	0.967583i	0.875183	−	0.483792i	$-0.160741\pi$
$480$	342059.			0.0677639
$481$	−951798.	+	2.09437e6i	−0.187578	+	0.412754i
$482$	−839013.			−0.164494
$483$			4.43332e6i			0.864692i
$484$	−2.06184e6			−0.400076
$485$	1.04038e6			0.200834
$486$		−	236196.i		−	0.0453609i
$487$		−	71255.1i		−	0.0136142i	−0.999977	−	0.00680712i	$-0.997833\pi$
$487$		−	71255.1i		−	0.0136142i	0.999977	−	0.00680712i	$-0.00216679\pi$
$488$		−	2.03145e6i		−	0.386150i
$489$		−	1.44957e6i		−	0.274137i
$490$	2.12486e6			0.399797
$491$	−6.78617e6			−1.27034			−0.635172	−	0.772371i	$-0.719071\pi$
$491$	−6.78617e6			−1.27034			−0.635172	+	0.772371i	$0.719071\pi$
$492$			53114.2i			0.00989231i
$493$	−1.40305e6			−0.259990
$494$	−5.18343e6	−	2.35563e6i	−0.955651	−	0.434301i
$495$	539356.			0.0989379
$496$		−	188732.i		−	0.0344463i
$497$	−1.04033e7			−1.88921
$498$	440403.			0.0795750
$499$			167487.i			0.0301114i	0.999887	+	0.0150557i	$0.00479256\pi$
$499$			167487.i			0.0301114i	−0.999887	+	0.0150557i	$0.995207\pi$
$500$		−	2.89350e6i		−	0.517605i
$501$		−	4.19920e6i		−	0.747433i
$502$			1.70862e6i			0.302613i
$503$	1.00972e7			1.77942			0.889712	−	0.456522i	$-0.150905\pi$
$503$	1.00972e7			1.77942			0.889712	+	0.456522i	$0.150905\pi$
$504$	914493.			0.160363
$505$			3.80809e6i			0.664476i
$506$	−2.00384e6			−0.347926
$507$	−2.19762e6	−	2.51734e6i	−0.379693	−	0.434933i
$508$	−4.09031e6			−0.703229
$509$			9.79146e6i			1.67515i	0.546324	+	0.837574i	$0.316027\pi$
$509$			9.79146e6i			1.67515i	−0.546324	+	0.837574i	$0.683973\pi$
$510$	−1.24683e6			−0.212267
$511$	−601758.			−0.101946
$512$			262144.i			0.0441942i
$513$			1.70292e6i			0.285694i
$514$			830887.i			0.138718i
$515$		−	7.56433e6i		−	1.25676i
$516$	−2.90601e6			−0.480477
$517$	3.68253e6			0.605927
$518$			2.66403e6i			0.436229i
$519$	2.16690e6			0.353119
$520$	−1.31774e6	−	598852.i	−0.213708	−	0.0971205i
$521$	−9.42352e6			−1.52096			−0.760481	−	0.649360i	$-0.775037\pi$
$521$	−9.42352e6			−1.52096			−0.760481	+	0.649360i	$0.775037\pi$
$522$			487161.i			0.0782521i
$523$	5.16320e6			0.825400			0.412700	−	0.910867i	$-0.364586\pi$
$523$	5.16320e6			0.825400			0.412700	+	0.910867i	$0.364586\pi$
$524$	2.14811e6			0.341765
$525$		−	2.77431e6i		−	0.439295i
$526$		−	3.32685e6i		−	0.524286i
$527$			687945.i			0.107901i
$528$			413346.i			0.0645252i
$529$	1.36094e6			0.211445
$530$	3.72373e6			0.575823
$531$			2.86143e6i			0.440400i
$532$	−6.59330e6			−1.01001
$533$	92988.3	−	204615.i	0.0141779	−	0.0311975i
$534$	−2.22317e6			−0.337380
$535$		−	5.46923e6i		−	0.826117i
$536$	−2.98633e6			−0.448978
$537$	2.39792e6			0.358838
$538$			6.69458e6i			0.997167i
$539$			2.56769e6i			0.380689i
$540$			432919.i			0.0638884i
$541$		−	1.03918e7i		−	1.52650i	−0.646104	−	0.763250i	$-0.723603\pi$
$541$		−	1.03918e7i		−	1.52650i	0.646104	−	0.763250i	$-0.276397\pi$
$542$	−279170.			−0.0408198
$543$	−578765.			−0.0842369
$544$		−	955536.i		−	0.138436i
$545$	−291235.			−0.0420002
$546$	−3.52295e6	−	1.60103e6i	−0.505738	−	0.229835i
$547$	−7.33920e6			−1.04877			−0.524385	−	0.851481i	$-0.675705\pi$
$547$	−7.33920e6			−1.04877			−0.524385	+	0.851481i	$0.675705\pi$
$548$			1.18997e6i			0.169272i
$549$	2.57105e6			0.364066
$550$	1.25397e6			0.176759
$551$		−	3.51233e6i		−	0.492851i
$552$		−	1.60840e6i		−	0.224671i
$553$		−	1.14624e7i		−	1.59391i
$554$			6.45520e6i			0.893584i
$555$	−1.26115e6			−0.173793
$556$	−581574.			−0.0797844
$557$		−	7.97266e6i		−	1.08884i	−0.838812	−	0.544421i	$-0.816749\pi$
$557$		−	7.97266e6i		−	1.08884i	0.838812	−	0.544421i	$-0.183251\pi$
$558$	238864.			0.0324763
$559$	1.11950e7	+	5.08763e6i	1.51529	+	0.688630i
$560$	−1.67615e6			−0.225862
$561$		−	1.50668e6i		−	0.202122i
$562$	−4.22621e6			−0.564430
$563$	5.23721e6			0.696352			0.348176	−	0.937429i	$-0.386801\pi$
$563$	5.23721e6			0.696352			0.348176	+	0.937429i	$0.386801\pi$
$564$			2.95581e6i			0.391273i
$565$		−	7.15122e6i		−	0.942451i
$566$		−	2.05096e6i		−	0.269102i
$567$			1.15740e6i			0.151192i
$568$	3.77430e6			0.490868
$569$	−8.15975e6			−1.05657			−0.528283	−	0.849069i	$-0.677164\pi$
$569$	−8.15975e6			−1.05657			−0.528283	+	0.849069i	$0.677164\pi$
$570$		−	3.12125e6i		−	0.402385i
$571$	−6.61371e6			−0.848897			−0.424448	−	0.905452i	$-0.639532\pi$
$571$	−6.61371e6			−0.848897			−0.424448	+	0.905452i	$0.639532\pi$
$572$	723656.	−	1.59236e6i	0.0924788	−	0.203494i
$573$	8.32849e6			1.05969
$574$		−	260269.i		−	0.0329719i
$575$	−4.87942e6			−0.615458
$576$	−331776.			−0.0416667
$577$			4.82867e6i			0.603793i	0.953341	+	0.301896i	$0.0976196\pi$
$577$			4.82867e6i			0.603793i	−0.953341	+	0.301896i	$0.902380\pi$
$578$		−	2.19642e6i		−	0.273461i
$579$			5.69075e6i			0.705462i
$580$		−	892907.i		−	0.110214i
$581$	−2.15806e6			−0.265230
$582$	−1.00910e6			−0.123489
$583$			4.49978e6i			0.548302i
$584$	218317.			0.0264883
$585$	757922.	−	1.66776e6i	0.0915661	−	0.201485i
$586$	−1.08507e7			−1.30531
$587$			7.52814e6i			0.901763i	0.892584	+	0.450881i	$0.148890\pi$
$587$			7.52814e6i			0.901763i	−0.892584	+	0.450881i	$0.851110\pi$
$588$	−2.06098e6			−0.245827
$589$	−1.72216e6			−0.204544
$590$		−	5.24466e6i		−	0.620279i
$591$		−	900430.i		−	0.106043i
$592$		−	966505.i		−	0.113344i
$593$		−	8.22445e6i		−	0.960439i	−0.877148	−	0.480220i	$-0.840557\pi$
$593$		−	8.22445e6i		−	0.960439i	0.877148	−	0.480220i	$-0.159443\pi$
$594$	−523142.			−0.0608350
$595$	6.10972e6			0.707504
$596$		−	3.54006e6i		−	0.408220i
$597$	6.50769e6			0.747293
$598$	−2.81587e6	+	6.19614e6i	−0.322002	+	0.708546i
$599$	−1.25959e6			−0.143437			−0.0717185	−	0.997425i	$-0.522848\pi$
$599$	−1.25959e6			−0.143437			−0.0717185	+	0.997425i	$0.522848\pi$
$600$			1.00651e6i			0.114141i
$601$	6.56541e6			0.741439			0.370720	−	0.928745i	$-0.379111\pi$
$601$	6.56541e6			0.741439			0.370720	+	0.928745i	$0.379111\pi$
$602$	1.42400e7			1.60147
$603$		−	3.77957e6i		−	0.423301i
$604$		−	7.71916e6i		−	0.860950i
$605$			4.78294e6i			0.531259i
$606$		−	3.69361e6i		−	0.408573i
$607$	−7.92228e6			−0.872727			−0.436364	−	0.899770i	$-0.643734\pi$
$607$	−7.92228e6			−0.872727			−0.436364	+	0.899770i	$0.643734\pi$
$608$	2.39203e6			0.262427
$609$		−	2.38718e6i		−	0.260821i
$610$	−4.71243e6			−0.512767
$611$	5.17482e6	−	1.13869e7i	0.560780	−	1.23396i
$612$	1.20935e6			0.130519
$613$		−	9.74465e6i		−	1.04741i	−0.851901	−	0.523703i	$-0.824550\pi$
$613$		−	9.74465e6i		−	1.04741i	0.851901	−	0.523703i	$-0.175450\pi$
$614$	−9.77898e6			−1.04682
$615$	123211.			0.0131360
$616$		−	2.02548e6i		−	0.215068i
$617$			1.14334e7i			1.20910i	0.796569	+	0.604548i	$0.206646\pi$
$617$			1.14334e7i			1.20910i	−0.796569	+	0.604548i	$0.793354\pi$
$618$			7.33693e6i			0.772757i
$619$		−	1.07067e7i		−	1.12313i	−0.827432	−	0.561566i	$-0.810199\pi$
$619$		−	1.07067e7i		−	1.12313i	0.827432	−	0.561566i	$-0.189801\pi$
$620$	−437810.			−0.0457411
$621$	2.03563e6			0.211822
$622$			7.96732e6i			0.825726i
$623$	1.08939e7			1.12451
$624$	1.27812e6	+	580849.i	0.131405	+	0.0597175i
$625$	−1.25148e6			−0.128151
$626$			6.65585e6i			0.678840i
$627$	3.77174e6			0.383154
$628$	1.77161e6			0.179254
$629$			3.52299e6i			0.355046i
$630$		−	2.12138e6i		−	0.212945i
$631$		−	475976.i		−	0.0475895i	−0.999717	−	0.0237948i	$-0.992425\pi$
$631$		−	475976.i		−	0.0475895i	0.999717	−	0.0237948i	$-0.00757483\pi$
$632$			4.15855e6i			0.414142i
$633$	−2.33593e6			−0.231714
$634$	1.28193e7			1.26660
$635$			9.48844e6i			0.933814i
$636$	−3.61179e6			−0.354062
$637$	7.93963e6	+	3.60820e6i	0.775267	+	0.352324i
$638$	1.07899e6			0.104946
$639$			4.77684e6i			0.462795i
$640$	608105.			0.0586852
$641$	−1.90742e7			−1.83358			−0.916791	−	0.399368i	$-0.869230\pi$
$641$	−1.90742e7			−1.83358			−0.916791	+	0.399368i	$0.869230\pi$
$642$			5.30481e6i			0.507963i
$643$		−	1.16474e7i		−	1.11097i	−0.831528	−	0.555483i	$-0.812534\pi$
$643$		−	1.16474e7i		−	1.11097i	0.831528	−	0.555483i	$-0.187466\pi$
$644$			7.88146e6i			0.748845i
$645$			6.74118e6i			0.638024i
$646$	−8.71916e6			−0.822041
$647$	1.83045e6			0.171908			0.0859540	−	0.996299i	$-0.472606\pi$
$647$	1.83045e6			0.171908			0.0859540	+	0.996299i	$0.472606\pi$
$648$		−	419904.i		−	0.0392837i
$649$	6.33768e6			0.590634
$650$	1.76213e6	−	3.87745e6i	0.163589	−	0.359967i
$651$	−1.17048e6			−0.108246
$652$		−	2.57702e6i		−	0.237410i
$653$	1.75829e7			1.61365			0.806824	−	0.590792i	$-0.201185\pi$
$653$	1.75829e7			1.61365			0.806824	+	0.590792i	$0.201185\pi$
$654$	282479.			0.0258251
$655$		−	4.98304e6i		−	0.453828i
$656$			94425.2i			0.00856699i
$657$			276307.i			0.0249735i
$658$		−	1.44840e7i		−	1.30414i
$659$	−1.84370e7			−1.65378			−0.826888	−	0.562367i	$-0.809891\pi$
$659$	−1.84370e7			−1.65378			−0.826888	+	0.562367i	$0.809891\pi$
$660$	958855.			0.0856827
$661$			1.08046e7i			0.961845i	0.876763	+	0.480922i	$0.159698\pi$
$661$			1.08046e7i			0.961845i	−0.876763	+	0.480922i	$0.840302\pi$
$662$	1.41354e7			1.25362
$663$	−4.65886e6	−	2.11724e6i	−0.411619	−	0.187062i
$664$	782939.			0.0689140
$665$			1.52947e7i			1.34118i
$666$	1.22323e6			0.106862
$667$	−4.19855e6			−0.365413
$668$		−	7.46525e6i		−	0.647296i
$669$			9.64761e6i			0.833402i
$670$			6.92749e6i			0.596196i
$671$		−	5.69453e6i		−	0.488260i
$672$	1.62576e6			0.138878
$673$	−1.57676e7			−1.34193			−0.670964	−	0.741490i	$-0.734119\pi$
$673$	−1.57676e7			−1.34193			−0.670964	+	0.741490i	$0.734119\pi$
$674$			4.40968e6i			0.373902i
$675$	−1.27387e6			−0.107613
$676$	−3.90688e6	−	4.47528e6i	−0.328823	−	0.376663i
$677$	2.86244e6			0.240029			0.120015	−	0.992772i	$-0.461706\pi$
$677$	2.86244e6			0.240029			0.120015	+	0.992772i	$0.461706\pi$
$678$			6.93623e6i			0.579495i
$679$	4.94480e6			0.411599
$680$	−2.21659e6			−0.183829
$681$		−	586890.i		−	0.0484941i
$682$		−	529052.i		−	0.0435549i
$683$			4.91047e6i			0.402783i	0.979511	+	0.201391i	$0.0645463\pi$
$683$			4.91047e6i			0.402783i	−0.979511	+	0.201391i	$0.935454\pi$
$684$			3.02742e6i			0.247419i
$685$	2.76043e6			0.224776
$686$	−1.76030e6			−0.142816
$687$		−	1.72669e6i		−	0.139580i
$688$	−5.16624e6			−0.416106
$689$	1.39139e7	+	6.32325e6i	1.11661	+	0.507449i
$690$	−3.73107e6			−0.298339
$691$			1.79749e7i			1.43209i	0.698052	+	0.716047i	$0.254051\pi$
$691$			1.79749e7i			1.43209i	−0.698052	+	0.716047i	$0.745949\pi$
$692$	3.85227e6			0.305810
$693$	2.56349e6			0.202768
$694$			1.26573e7i			0.997566i
$695$			1.34910e6i			0.105945i
$696$			866064.i			0.0677683i
$697$		−	344188.i		−	0.0268357i
$698$	7.90102e6			0.613825
$699$	−3.18528e6			−0.246579
$700$		−	4.93210e6i		−	0.380440i
$701$	−8.74640e6			−0.672255			−0.336128	−	0.941816i	$-0.609117\pi$
$701$	−8.74640e6			−0.672255			−0.336128	+	0.941816i	$0.609117\pi$
$702$	−735137.	+	1.61762e6i	−0.0563022	+	0.123889i
$703$	−8.81925e6			−0.673044
$704$			734838.i			0.0558805i
$705$	6.85671e6			0.519569
$706$	8.96061e6			0.676590
$707$			1.80994e7i			1.36181i
$708$			5.08699e6i			0.381398i
$709$		−	2.43397e6i		−	0.181844i	−0.995858	−	0.0909221i	$-0.971019\pi$
$709$		−	2.43397e6i		−	0.181844i	0.995858	−	0.0909221i	$-0.0289814\pi$
$710$		−	8.75538e6i		−	0.651822i
$711$	−5.26316e6			−0.390457
$712$	−3.95230e6			−0.292180
$713$			2.05863e6i			0.151654i
$714$	−5.92605e6			−0.435030
$715$	−3.69386e6	−	1.67869e6i	−0.270218	−	0.122802i
$716$	4.26297e6			0.310763
$717$			1.03836e7i			0.754313i
$718$	−8.55079e6			−0.619006
$719$	6.81620e6			0.491723			0.245861	−	0.969305i	$-0.420929\pi$
$719$	6.81620e6			0.491723			0.245861	+	0.969305i	$0.420929\pi$
$720$			769633.i			0.0553290i
$721$		−	3.59523e7i		−	2.57566i
$722$		−	1.19226e7i		−	0.851196i
$723$		−	1.88778e6i		−	0.134309i
$724$	−1.02892e6			−0.0729513
$725$	2.62739e6			0.185643
$726$		−	4.63915e6i		−	0.326661i
$727$	−4.33605e6			−0.304269			−0.152135	−	0.988360i	$-0.548615\pi$
$727$	−4.33605e6			−0.304269			−0.152135	+	0.988360i	$0.548615\pi$
$728$	−6.26303e6	−	2.84627e6i	−0.437982	−	0.199043i
$729$	531441.			0.0370370
$730$		−	506437.i		−	0.0351737i
$731$	1.88314e7			1.30343
$732$	4.57076e6			0.315290
$733$			2.01689e7i			1.38651i	0.720693	+	0.693254i	$0.243824\pi$
$733$			2.01689e7i			1.38651i	−0.720693	+	0.693254i	$0.756176\pi$
$734$			6.82008e6i			0.467250i
$735$			4.78093e6i			0.326433i
$736$		−	2.85938e6i		−	0.194570i
$737$	−8.37122e6			−0.567702
$738$	−119507.			−0.00807704
$739$			2.05423e7i			1.38369i	0.722048	+	0.691843i	$0.243201\pi$
$739$			2.05423e7i			1.38369i	−0.722048	+	0.691843i	$0.756799\pi$
$740$	−2.24204e6			−0.150509
$741$	5.30018e6	−	1.16627e7i	0.354605	−	0.780286i
$742$	1.76984e7			1.18012
$743$			1.48776e7i			0.988689i	0.869266	+	0.494345i	$0.164592\pi$
$743$			1.48776e7i			0.988689i	−0.869266	+	0.494345i	$0.835408\pi$
$744$	424648.			0.0281253
$745$	−8.21200e6			−0.542074
$746$		−	9.26012e6i		−	0.609213i
$747$			990907.i			0.0649728i
$748$		−	2.67855e6i		−	0.175043i
$749$		−	2.59946e7i		−	1.69308i
$750$	6.51037e6			0.422623
$751$	−2.94950e7			−1.90831			−0.954154	−	0.299315i	$-0.903242\pi$
$751$	−2.94950e7			−1.90831			−0.954154	+	0.299315i	$0.903242\pi$
$752$			5.25478e6i			0.338852i
$753$	−3.84440e6			−0.247082
$754$	1.51624e6	−	3.33639e6i	0.0971269	−	0.213722i
$755$	−1.79064e7			−1.14325
$756$			2.05761e6i			0.130936i
$757$	2.01715e7			1.27938			0.639690	−	0.768633i	$-0.279063\pi$
$757$	2.01715e7			1.27938			0.639690	+	0.768633i	$0.279063\pi$
$758$	1.72942e7			1.09327
$759$		−	4.50864e6i		−	0.284080i
$760$		−	5.54889e6i		−	0.348476i
$761$			2.06956e7i			1.29544i	0.761880	+	0.647719i	$0.224277\pi$
$761$			2.06956e7i			1.29544i	−0.761880	+	0.647719i	$0.775723\pi$
$762$		−	9.20319e6i		−	0.574184i
$763$	−1.38420e6			−0.0860772
$764$	1.48062e7			0.917720
$765$		−	2.80538e6i		−	0.173316i
$766$	−8.01403e6			−0.493491
$767$	8.90592e6	−	1.95969e7i	0.546626	−	1.20282i
$768$	−589824.			−0.0360844
$769$		−	1.69130e7i		−	1.03135i	−0.856785	−	0.515674i	$-0.827542\pi$
$769$		−	1.69130e7i		−	1.03135i	0.856785	−	0.515674i	$-0.172458\pi$
$770$	−4.69857e6			−0.285587
$771$	−1.86950e6			−0.113263
$772$			1.01169e7i			0.610948i
$773$		−	3.10969e7i		−	1.87184i	−0.352212	−	0.935920i	$-0.614570\pi$
$773$		−	3.10969e7i		−	1.87184i	0.352212	−	0.935920i	$-0.385430\pi$
$774$		−	6.53852e6i		−	0.392308i
$775$		−	1.28826e6i		−	0.0770458i
$776$	−1.79396e6			−0.106945
$777$	−5.99407e6			−0.356180
$778$			6.82786e6i			0.404423i
$779$	861619.			0.0508712
$780$	1.34742e6	−	2.96490e6i	0.0792986	−	0.174491i
$781$	1.05800e7			0.620669
$782$			1.04227e7i			0.609483i
$783$	−1.09611e6			−0.0638926
$784$	−3.66396e6			−0.212893
$785$		−	4.10967e6i		−	0.238031i
$786$			4.83324e6i			0.279050i
$787$		−	628913.i		−	0.0361954i	−0.999836	−	0.0180977i	$-0.994239\pi$
$787$		−	628913.i		−	0.0361954i	0.999836	−	0.0180977i	$-0.00576099\pi$
$788$		−	1.60076e6i		−	0.0918357i
$789$	7.48541e6			0.428078
$790$	9.64674e6			0.549937
$791$		−	3.39888e7i		−	1.93150i
$792$	−930030.			−0.0526846
$793$	−1.76082e7	−	8.00215e6i	−0.994335	−	0.451881i
$794$	−4.89708e6			−0.275668
$795$			8.37840e6i			0.470157i
$796$	1.15692e7			0.647174
$797$	−2.43866e7			−1.35990			−0.679948	−	0.733260i	$-0.737998\pi$
$797$	−2.43866e7			−1.35990			−0.679948	+	0.733260i	$0.737998\pi$
$798$		−	1.48349e7i		−	0.824666i
$799$		−	1.91541e7i		−	1.06144i
$800$			1.78936e6i			0.0988489i
$801$		−	5.00213e6i		−	0.275470i
$802$	1.66239e7			0.912637
$803$	611982.			0.0334927
$804$		−	6.71923e6i		−	0.366589i
$805$	1.82829e7			0.994388
$806$	−1.63590e6	−	743442.i	−0.0886989	−	0.0403097i
$807$	−1.50628e7			−0.814183
$808$		−	6.56642e6i		−	0.353835i
$809$	−2.86033e7			−1.53654			−0.768272	−	0.640123i	$-0.778883\pi$
$809$	−2.86033e7			−1.53654			−0.768272	+	0.640123i	$0.778883\pi$
$810$	−974067.			−0.0521647
$811$			5.97923e6i			0.319222i	0.987180	+	0.159611i	$0.0510240\pi$
$811$			5.97923e6i			0.319222i	−0.987180	+	0.159611i	$0.948976\pi$
$812$		−	4.24387e6i		−	0.225877i
$813$		−	628133.i		−	0.0333292i
$814$		−	2.70929e6i		−	0.143316i
$815$	−5.97800e6			−0.315255
$816$	2.14996e6			0.113033
$817$			4.71414e7i			2.47085i
$818$	−2.60971e7			−1.36367
$819$	3.60231e6	−	7.92665e6i	0.187660	−	0.412933i
$820$	219042.			0.0113761
$821$		−	1.46642e7i		−	0.759279i	−0.925134	−	0.379640i	$-0.876048\pi$
$821$		−	1.46642e7i		−	0.759279i	0.925134	−	0.379640i	$-0.123952\pi$
$822$	−2.67744e6			−0.138210
$823$	1.80149e7			0.927113			0.463557	−	0.886067i	$-0.346573\pi$
$823$	1.80149e7			0.927113			0.463557	+	0.886067i	$0.346573\pi$
$824$			1.30434e7i			0.669227i
$825$			2.82144e6i			0.144323i
$826$		−	2.49272e7i		−	1.27123i
$827$			9.93027e6i			0.504891i	0.967611	+	0.252445i	$0.0812348\pi$
$827$			9.93027e6i			0.504891i	−0.967611	+	0.252445i	$0.918765\pi$
$828$	3.61890e6			0.183443
$829$	2.97991e7			1.50597			0.752986	−	0.658037i	$-0.228613\pi$
$829$	2.97991e7			1.50597			0.752986	+	0.658037i	$0.228613\pi$
$830$		−	1.81621e6i		−	0.0915106i
$831$	−1.45242e7			−0.729608
$832$	2.27221e6	+	1.03262e6i	0.113800	+	0.0517169i
$833$	1.33554e7			0.666877
$834$		−	1.30854e6i		−	0.0651437i
$835$	−1.73174e7			−0.859542
$836$	6.70532e6			0.331821
$837$			537445.i			0.0265168i
$838$			1.91180e7i			0.940443i
$839$		−	1.35283e7i		−	0.663496i	−0.943368	−	0.331748i	$-0.892362\pi$
$839$		−	1.35283e7i		−	0.663496i	0.943368	−	0.331748i	$-0.107638\pi$
$840$		−	3.77135e6i		−	0.184416i
$841$	−1.82504e7			−0.889779
$842$	−1.36623e7			−0.664117
$843$		−	9.50897e6i		−	0.460855i
$844$	−4.15277e6			−0.200670
$845$	−1.03815e7	+	9.06293e6i	−0.500169	+	0.436643i
$846$	−6.65058e6			−0.319473
$847$			2.27327e7i			1.08879i
$848$	−6.42096e6			−0.306627
$849$	4.61467e6			0.219721
$850$		−	6.52235e6i		−	0.309640i
$851$			1.05423e7i			0.499013i
$852$			8.49217e6i			0.400792i
$853$		−	9.03698e6i		−	0.425257i	−0.977133	−	0.212628i	$-0.931798\pi$
$853$		−	9.03698e6i		−	0.425257i	0.977133	−	0.212628i	$-0.0682023\pi$
$854$	−2.23976e7			−1.05089
$855$	7.02282e6			0.328546
$856$			9.43078e6i			0.439909i
$857$	−9.55364e6			−0.444341			−0.222171	−	0.975008i	$-0.571314\pi$
$857$	−9.55364e6			−0.444341			−0.222171	+	0.975008i	$0.571314\pi$
$858$	3.58281e6	+	1.62823e6i	0.166152	+	0.0755086i
$859$	3.73649e6			0.172775			0.0863874	−	0.996262i	$-0.472468\pi$
$859$	3.73649e6			0.172775			0.0863874	+	0.996262i	$0.472468\pi$
$860$			1.19843e7i			0.552545i
$861$	585606.			0.0269214
$862$	2.48565e7			1.13939
$863$			3.44318e7i			1.57374i	0.617120	+	0.786869i	$0.288299\pi$
$863$			3.44318e7i			1.57374i	−0.617120	+	0.786869i	$0.711701\pi$
$864$		−	746496.i		−	0.0340207i
$865$		−	8.93626e6i		−	0.406084i
$866$		−	2.33345e7i		−	1.05731i
$867$	4.94194e6			0.223280
$868$	−2.08085e6			−0.0937438
$869$			1.16572e7i			0.523653i
$870$	2.00904e6			0.0899892
$871$	−1.17635e7	+	2.58849e7i	−0.525403	+	1.15612i
$872$	502186.			0.0223652
$873$		−	2.27048e6i		−	0.100828i
$874$	−2.60915e7			−1.15537
$875$	−3.19020e7			−1.40863
$876$			491213.i			0.0216276i
$877$		−	2.12195e7i		−	0.931613i	−0.884887	−	0.465807i	$-0.845764\pi$
$877$		−	2.12195e7i		−	0.931613i	0.884887	−	0.465807i	$-0.154236\pi$
$878$		−	3.06769e6i		−	0.134300i
$879$		−	2.44141e7i		−	1.06578i
$880$	1.70463e6			0.0742034
$881$	3.83954e7			1.66663			0.833315	−	0.552798i	$-0.186440\pi$
$881$	3.83954e7			1.66663			0.833315	+	0.552798i	$0.186440\pi$
$882$		−	4.63720e6i		−	0.200717i
$883$	3.58049e7			1.54540			0.772699	−	0.634773i	$-0.218906\pi$
$883$	3.58049e7			1.54540			0.772699	+	0.634773i	$0.218906\pi$
$884$	−8.28241e6	−	3.76398e6i	−0.356472	−	0.162001i
$885$	1.18005e7			0.506456
$886$			4.08168e6i			0.174685i
$887$	2.52692e7			1.07841			0.539203	−	0.842176i	$-0.318726\pi$
$887$	2.52692e7			1.07841			0.539203	+	0.842176i	$0.318726\pi$
$888$	2.17464e6			0.0925453
$889$			4.50973e7i			1.91380i
$890$			9.16830e6i			0.387984i
$891$		−	1.17707e6i		−	0.0496715i
$892$			1.71513e7i			0.721747i
$893$	4.79493e7			2.01212
$894$	7.96512e6			0.333310
$895$		−	9.88896e6i		−	0.412661i
$896$	2.89025e6			0.120272
$897$	−1.39413e7	−	6.33570e6i	−0.578525	−	0.262914i
$898$	−1.67249e7			−0.692107
$899$		−	1.10850e6i		−	0.0457441i
$900$	−2.26465e6			−0.0931956
$901$	2.34049e7			0.960495
$902$			264691.i			0.0108324i
$903$			3.20400e7i			1.30759i
$904$			1.23311e7i			0.501857i
$905$			2.38681e6i			0.0968717i
$906$	1.73681e7			0.702962
$907$	−2.49639e7			−1.00761			−0.503806	−	0.863817i	$-0.668067\pi$
$907$	−2.49639e7			−1.00761			−0.503806	+	0.863817i	$0.668067\pi$
$908$		−	1.04336e6i		−	0.0419971i
$909$	8.31063e6			0.333599
$910$	−6.60259e6	+	1.45286e7i	−0.264308	+	0.581594i
$911$	−807347.			−0.0322303			−0.0161151	−	0.999870i	$-0.505130\pi$
$911$	−807347.			−0.0322303			−0.0161151	+	0.999870i	$0.505130\pi$
$912$			5.38208e6i			0.214271i
$913$	2.19472e6			0.0871370
$914$	−1.43333e7			−0.567520
$915$		−	1.06030e7i		−	0.418673i
$916$		−	3.06967e6i		−	0.120879i
$917$		−	2.36838e7i		−	0.930095i
$918$			2.72104e6i			0.106568i
$919$	−2.92665e6			−0.114309			−0.0571547	−	0.998365i	$-0.518203\pi$
$919$	−2.92665e6			−0.114309			−0.0571547	+	0.998365i	$0.518203\pi$
$920$	−6.63301e6			−0.258369
$921$		−	2.20027e7i		−	0.854726i
$922$	6.69836e6			0.259502
$923$	1.48674e7	−	3.27149e7i	0.574423	−	1.26398i
$924$	4.55732e6			0.175602
$925$		−	6.59722e6i		−	0.253517i
$926$	−2.64945e7			−1.01538
$927$	−1.65081e7			−0.630953
$928$			1.53967e6i			0.0586891i
$929$			2.03515e7i			0.773672i	0.922148	+	0.386836i	$0.126432\pi$
$929$			2.03515e7i			0.773672i	−0.922148	+	0.386836i	$0.873568\pi$
$930$		−	985072.i		−	0.0373474i
$931$			3.34332e7i			1.26417i
$932$	−5.66273e6			−0.213543
$933$	−1.79265e7			−0.674203
$934$			1.52297e7i			0.571249i
$935$	−6.21352e6			−0.232439
$936$	−1.30691e6	+	2.87577e6i	−0.0487591	+	0.107291i
$937$	−659937.			−0.0245558			−0.0122779	−	0.999925i	$-0.503908\pi$
$937$	−659937.			−0.0245558			−0.0122779	+	0.999925i	$0.503908\pi$
$938$			3.29255e7i			1.22187i
$939$	−1.49757e7			−0.554271
$940$	1.21897e7			0.449960
$941$		−	3.40074e7i		−	1.25199i	−0.779828	−	0.625994i	$-0.784694\pi$
$941$		−	3.40074e7i		−	1.25199i	0.779828	−	0.625994i	$-0.215306\pi$
$942$			3.98612e6i			0.146360i
$943$		−	1.02996e6i		−	0.0377173i
$944$			9.04354e6i			0.330300i
$945$	4.77311e6			0.173869
$946$	−1.44819e7			−0.526137
$947$		−	1.44758e7i		−	0.524527i	−0.964996	−	0.262263i	$-0.915531\pi$
$947$		−	1.44758e7i		−	0.524527i	0.964996	−	0.262263i	$-0.0844689\pi$
$948$	−9.35673e6			−0.338145
$949$	859978.	−	1.89233e6i	0.0309972	−	0.0682073i
$950$	1.63277e7			0.586969
$951$			2.88434e7i			1.03418i
$952$	−1.05352e7			−0.376747
$953$	1.88172e7			0.671156			0.335578	−	0.942012i	$-0.391068\pi$
$953$	1.88172e7			0.671156			0.335578	+	0.942012i	$0.391068\pi$
$954$		−	8.12652e6i		−	0.289090i
$955$		−	3.43465e7i		−	1.21864i
$956$			1.84598e7i			0.653255i
$957$			2.42774e6i			0.0856883i
$958$	1.94351e7			0.684185
$959$	1.31200e7			0.460666
$960$			1.36824e6i			0.0479163i
$961$	2.80856e7			0.981015
$962$	−8.37748e6	−	3.80719e6i	−0.291861	−	0.132638i
$963$	−1.19358e7			−0.414750
$964$		−	3.35605e6i		−	0.116315i
$965$	2.34685e7			0.811275
$966$	−1.77333e7			−0.611430
$967$			4.30702e7i			1.48119i	0.671952	+	0.740595i	$0.265456\pi$
$967$			4.30702e7i			1.48119i	−0.671952	+	0.740595i	$0.734544\pi$
$968$		−	8.24738e6i		−	0.282896i
$969$		−	1.96181e7i		−	0.671194i
$970$			4.16152e6i			0.142011i
$971$	2.85474e7			0.971668			0.485834	−	0.874051i	$-0.338516\pi$
$971$	2.85474e7			0.971668			0.485834	+	0.874051i	$0.338516\pi$
$972$	944784.			0.0320750
$973$			6.41210e6i			0.217129i
$974$	285020.			0.00962673
$975$	8.72426e6	+	3.96479e6i	0.293912	+	0.133570i
$976$	8.12580e6			0.273050
$977$			4.23484e7i			1.41939i	0.704510	+	0.709694i	$0.251167\pi$
$977$			4.23484e7i			1.41939i	−0.704510	+	0.709694i	$0.748833\pi$
$978$	5.79829e6			0.193844
$979$	−1.10790e7			−0.369441
$980$			8.49942e6i			0.282699i
$981$			635579.i			0.0210861i
$982$		−	2.71447e7i		−	0.898269i
$983$		−	1.39064e7i		−	0.459020i	−0.973306	−	0.229510i	$-0.926288\pi$
$983$		−	1.39064e7i		−	0.459020i	0.973306	−	0.229510i	$-0.0737124\pi$
$984$	−212457.			−0.00699492
$985$	−3.71335e6			−0.121948
$986$		−	5.61222e6i		−	0.183841i
$987$	3.25891e7			1.06483
$988$	9.42254e6	−	2.07337e7i	0.307097	−	0.675748i
$989$	5.63516e7			1.83196
$990$			2.15742e6i			0.0699597i
$991$	−3.49486e6			−0.113043			−0.0565217	−	0.998401i	$-0.518001\pi$
$991$	−3.49486e6			−0.113043			−0.0565217	+	0.998401i	$0.518001\pi$
$992$	754930.			0.0243572
$993$			3.18047e7i			1.02357i
$994$		−	4.16132e7i		−	1.33587i
$995$		−	2.68376e7i		−	0.859380i
$996$			1.76161e6i			0.0562681i
$997$	77983.9			0.00248466			0.00124233	−	0.999999i	$-0.499605\pi$
$997$	77983.9			0.00248466			0.00124233	+	0.999999i	$0.499605\pi$
$998$	−669949.			−0.0212920
$999$			2.75227e6i			0.0872525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.6.b.a.25.6	yes	6
3.2	odd	2		234.6.b.c.181.1		6
4.3	odd	2		624.6.c.d.337.5		6
13.5	odd	4		1014.6.a.q.1.3		3
13.8	odd	4		1014.6.a.o.1.1		3
13.12	even	2	inner	78.6.b.a.25.1	✓	6
39.38	odd	2		234.6.b.c.181.6		6
52.51	odd	2		624.6.c.d.337.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.a.25.1	✓	6	13.12	even	2	inner
78.6.b.a.25.6	yes	6	1.1	even	1	trivial
234.6.b.c.181.1		6	3.2	odd	2
234.6.b.c.181.6		6	39.38	odd	2
624.6.c.d.337.2		6	52.51	odd	2
624.6.c.d.337.5		6	4.3	odd	2
1014.6.a.o.1.1		3	13.8	odd	4
1014.6.a.q.1.3		3	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	78.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +37.1158i q^{5} -36.0000i q^{6} +176.407i q^{7} -64.0000i q^{8} +81.0000 q^{9} -148.463 q^{10} -179.404i q^{11} +144.000 q^{12} +(-554.740 - 252.104i) q^{13} -705.627 q^{14} -334.042i q^{15} +256.000 q^{16} -933.141 q^{17} +324.000i q^{18} -2335.97i q^{19} -593.853i q^{20} -1587.66i q^{21} +717.615 q^{22} -2792.36 q^{23} +576.000i q^{24} +1747.42 q^{25} +(1008.42 - 2218.96i) q^{26} -729.000 q^{27} -2822.51i q^{28} +1503.58 q^{29} +1336.17 q^{30} -737.236i q^{31} +1024.00i q^{32} +1614.63i q^{33} -3732.56i q^{34} -6547.48 q^{35} -1296.00 q^{36} -3775.41i q^{37} +9343.88 q^{38} +(4992.66 + 2268.94i) q^{39} +2375.41 q^{40} +368.848i q^{41} +6350.64 q^{42} -20180.6 q^{43} +2870.46i q^{44} +3006.38i q^{45} -11169.4i q^{46} +20526.5i q^{47} -2304.00 q^{48} -14312.3 q^{49} +6989.67i q^{50} +8398.27 q^{51} +(8875.84 + 4033.67i) q^{52} -25081.9 q^{53} -2916.00i q^{54} +6658.72 q^{55} +11290.0 q^{56} +21023.7i q^{57} +6014.33i q^{58} +35326.3i q^{59} +5344.68i q^{60} +31741.4 q^{61} +2948.94 q^{62} +14288.9i q^{63} -4096.00 q^{64} +(9357.06 - 20589.6i) q^{65} -6458.54 q^{66} -46661.3i q^{67} +14930.3 q^{68} +25131.2 q^{69} -26189.9i q^{70} +58973.4i q^{71} -5184.00i q^{72} +3411.20i q^{73} +15101.6 q^{74} -15726.8 q^{75} +37375.5i q^{76} +31648.1 q^{77} +(-9075.76 + 19970.6i) q^{78} -64977.3 q^{79} +9501.64i q^{80} +6561.00 q^{81} -1475.39 q^{82} +12233.4i q^{83} +25402.6i q^{84} -34634.3i q^{85} -80722.5i q^{86} -13532.2 q^{87} -11481.8 q^{88} -61754.7i q^{89} -12025.5 q^{90} +(44472.9 - 97859.9i) q^{91} +44677.8 q^{92} +6635.12i q^{93} -82106.0 q^{94} +86701.4 q^{95} -9216.00i q^{96} -28030.7i q^{97} -57249.4i q^{98} -14531.7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30}+ \cdots + 567168 q^{95}+O(q^{100}) \) 6 * q - 54 * q^3 - 96 * q^4 + 486 * q^9 + 320 * q^10 + 864 * q^12 + 530 * q^13 - 1360 * q^14 + 1536 * q^16 - 836 * q^17 - 1296 * q^22 - 416 * q^23 + 718 * q^25 - 1360 * q^26 - 4374 * q^27 + 18788 * q^29 - 2880 * q^30 + 6112 * q^35 - 7776 * q^36 + 528 * q^38 - 4770 * q^39 - 5120 * q^40 + 12240 * q^42 - 24200 * q^43 - 13824 * q^48 - 3038 * q^49 + 7524 * q^51 - 8480 * q^52 - 42396 * q^53 + 124656 * q^55 + 21760 * q^56 - 3196 * q^61 - 59344 * q^62 - 24576 * q^64 - 17168 * q^65 + 11664 * q^66 + 13376 * q^68 + 3744 * q^69 - 62240 * q^74 - 6462 * q^75 - 114024 * q^77 + 12240 * q^78 - 169328 * q^79 + 39366 * q^81 + 145120 * q^82 - 169092 * q^87 + 20736 * q^88 + 25920 * q^90 + 236152 * q^91 + 6656 * q^92 - 32688 * q^94 + 567168 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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