LMFDB - Embedded newform 78.6.b.a.25.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,6,Mod(25,78)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			25.5
Root			$-15.0768 + 15.0768i$ of defining polynomial
Character	$\chi$	$=$	78.25
Dual form			78.6.b.a.25.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +9.73803i q^{5} -36.0000i q^{6} -105.184i q^{7} -64.0000i q^{8} +81.0000 q^{9} -38.9521 q^{10} -269.350i q^{11} +144.000 q^{12} +(227.464 + 565.290i) q^{13} +420.736 q^{14} -87.6423i q^{15} +256.000 q^{16} +1663.75 q^{17} +324.000i q^{18} +2.82742i q^{19} -155.809i q^{20} +946.657i q^{21} +1077.40 q^{22} +2151.14 q^{23} +576.000i q^{24} +3030.17 q^{25} +(-2261.16 + 909.856i) q^{26} -729.000 q^{27} +1682.95i q^{28} +220.829 q^{29} +350.569 q^{30} +788.106i q^{31} +1024.00i q^{32} +2424.15i q^{33} +6654.98i q^{34} +1024.29 q^{35} -1296.00 q^{36} +980.445i q^{37} -11.3097 q^{38} +(-2047.18 - 5087.61i) q^{39} +623.234 q^{40} -14809.7i q^{41} -3786.63 q^{42} +14142.2 q^{43} +4309.61i q^{44} +788.781i q^{45} +8604.55i q^{46} -25181.2i q^{47} -2304.00 q^{48} +5743.31 q^{49} +12120.7i q^{50} -14973.7 q^{51} +(-3639.42 - 9044.65i) q^{52} -30900.9 q^{53} -2916.00i q^{54} +2622.94 q^{55} -6731.78 q^{56} -25.4468i q^{57} +883.317i q^{58} -25094.7i q^{59} +1402.28i q^{60} +12060.8 q^{61} -3152.42 q^{62} -8519.91i q^{63} -4096.00 q^{64} +(-5504.81 + 2215.05i) q^{65} -9696.61 q^{66} -14450.5i q^{67} -26619.9 q^{68} -19360.2 q^{69} +4097.14i q^{70} +33547.5i q^{71} -5184.00i q^{72} +25805.7i q^{73} -3921.78 q^{74} -27271.5 q^{75} -45.2388i q^{76} -28331.4 q^{77} +(20350.5 - 8188.70i) q^{78} +15620.3 q^{79} +2492.94i q^{80} +6561.00 q^{81} +59238.7 q^{82} -2819.78i q^{83} -15146.5i q^{84} +16201.6i q^{85} +56568.8i q^{86} -1987.46 q^{87} -17238.4 q^{88} +16227.2i q^{89} -3155.12 q^{90} +(59459.6 - 23925.6i) q^{91} -34418.2 q^{92} -7092.95i q^{93} +100725. q^{94} -27.5335 q^{95} -9216.00i q^{96} +126289. i q^{97} +22973.2i q^{98} -21817.4i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			9.73803i			0.174199i	0.996200	+	0.0870996i	$0.0277598\pi$
$5$			9.73803i			0.174199i	−0.996200	+	0.0870996i	$0.972240\pi$
$6$		−	36.0000i		−	0.408248i
$7$		−	105.184i		−	0.811344i	−0.914019	−	0.405672i	$-0.867038\pi$
$7$		−	105.184i		−	0.811344i	0.914019	−	0.405672i	$-0.132962\pi$
$8$		−	64.0000i		−	0.353553i
$9$	81.0000			0.333333
$10$	−38.9521			−0.123177
$11$		−	269.350i		−	0.671175i	−0.942009	−	0.335588i	$-0.891065\pi$
$11$		−	269.350i		−	0.671175i	0.942009	−	0.335588i	$-0.108935\pi$
$12$	144.000			0.288675
$13$	227.464	+	565.290i	0.373297	+	0.927712i
$13$	227.464	+	565.290i	0.373297	+	0.927712i
$14$	420.736			0.573707
$15$		−	87.6423i		−	0.100574i
$16$	256.000			0.250000
$17$	1663.75			1.39625			0.698127	−	0.715974i	$-0.254017\pi$
$17$	1663.75			1.39625			0.698127	+	0.715974i	$0.254017\pi$
$18$			324.000i			0.235702i
$19$			2.82742i			0.00179683i	1.00000		0.000898415i	$0.000285974\pi$
$19$			2.82742i			0.00179683i	−1.00000		0.000898415i	$0.999714\pi$
$20$		−	155.809i		−	0.0870996i
$21$			946.657i			0.468430i
$22$	1077.40			0.474592
$23$	2151.14			0.847908			0.423954	−	0.905684i	$-0.360642\pi$
$23$	2151.14			0.847908			0.423954	+	0.905684i	$0.360642\pi$
$24$			576.000i			0.204124i
$25$	3030.17			0.969655
$26$	−2261.16	+	909.856i	−0.655991	+	0.263961i
$27$	−729.000			−0.192450
$28$			1682.95i			0.405672i
$29$	220.829			0.0487598			0.0243799	−	0.999703i	$-0.492239\pi$
$29$	220.829			0.0487598			0.0243799	+	0.999703i	$0.492239\pi$
$30$	350.569			0.0711165
$31$			788.106i			0.147292i	0.997284	+	0.0736462i	$0.0234636\pi$
$31$			788.106i			0.147292i	−0.997284	+	0.0736462i	$0.976536\pi$
$32$			1024.00i			0.176777i
$33$			2424.15i			0.387503i
$34$			6654.98i			0.987301i
$35$	1024.29			0.141335
$36$	−1296.00			−0.166667
$37$			980.445i			0.117739i	0.998266	+	0.0588693i	$0.0187495\pi$
$37$			980.445i			0.117739i	−0.998266	+	0.0588693i	$0.981250\pi$
$38$	−11.3097			−0.00127055
$39$	−2047.18	−	5087.61i	−0.215523	−	0.535615i
$40$	623.234			0.0615887
$41$		−	14809.7i		−	1.37590i	−0.725759	−	0.687949i	$-0.758511\pi$
$41$		−	14809.7i		−	1.37590i	0.725759	−	0.687949i	$-0.241489\pi$
$42$	−3786.63			−0.331230
$43$	14142.2			1.16640			0.583198	−	0.812330i	$-0.301801\pi$
$43$	14142.2			1.16640			0.583198	+	0.812330i	$0.301801\pi$
$44$			4309.61i			0.335588i
$45$			788.781i			0.0580664i
$46$			8604.55i			0.599561i
$47$		−	25181.2i		−	1.66277i	−0.555699	−	0.831383i	$-0.687549\pi$
$47$		−	25181.2i		−	1.66277i	0.555699	−	0.831383i	$-0.312451\pi$
$48$	−2304.00			−0.144338
$49$	5743.31			0.341721
$50$			12120.7i			0.685649i
$51$	−14973.7			−0.806128
$52$	−3639.42	−	9044.65i	−0.186648	−	0.463856i
$53$	−30900.9			−1.51106			−0.755530	−	0.655114i	$-0.772621\pi$
$53$	−30900.9			−1.51106			−0.755530	+	0.655114i	$0.772621\pi$
$54$		−	2916.00i		−	0.136083i
$55$	2622.94			0.116918
$56$	−6731.78			−0.286853
$57$		−	25.4468i		−	0.00103740i
$58$			883.317i			0.0344784i
$59$		−	25094.7i		−	0.938538i	−0.883055	−	0.469269i	$-0.844517\pi$
$59$		−	25094.7i		−	0.938538i	0.883055	−	0.469269i	$-0.155483\pi$
$60$			1402.28i			0.0502870i
$61$	12060.8			0.415003			0.207501	−	0.978235i	$-0.433467\pi$
$61$	12060.8			0.415003			0.207501	+	0.978235i	$0.433467\pi$
$62$	−3152.42			−0.104151
$63$		−	8519.91i		−	0.270448i
$64$	−4096.00			−0.125000
$65$	−5504.81	+	2215.05i	−0.161607	+	0.0650280i
$66$	−9696.61			−0.274006
$67$		−	14450.5i		−	0.393276i	−0.980476	−	0.196638i	$-0.936998\pi$
$67$		−	14450.5i		−	0.393276i	0.980476	−	0.196638i	$-0.0630023\pi$
$68$	−26619.9			−0.698127
$69$	−19360.2			−0.489540
$70$			4097.14i			0.0999393i
$71$			33547.5i			0.789795i	0.918725	+	0.394897i	$0.129220\pi$
$71$			33547.5i			0.789795i	−0.918725	+	0.394897i	$0.870780\pi$
$72$		−	5184.00i		−	0.117851i
$73$			25805.7i			0.566772i	0.959006	+	0.283386i	$0.0914578\pi$
$73$			25805.7i			0.566772i	−0.959006	+	0.283386i	$0.908542\pi$
$74$	−3921.78			−0.0832538
$75$	−27271.5			−0.559830
$76$		−	45.2388i		−	0.000898415i
$77$	−28331.4			−0.544554
$78$	20350.5	−	8188.70i	0.378737	−	0.152398i
$79$	15620.3			0.281593			0.140797	−	0.990039i	$-0.455034\pi$
$79$	15620.3			0.281593			0.140797	+	0.990039i	$0.455034\pi$
$80$			2492.94i			0.0435498i
$81$	6561.00			0.111111
$82$	59238.7			0.972906
$83$		−	2819.78i		−	0.0449284i	−0.999748	−	0.0224642i	$-0.992849\pi$
$83$		−	2819.78i		−	0.0449284i	0.999748	−	0.0224642i	$-0.00715117\pi$
$84$		−	15146.5i		−	0.234215i
$85$			16201.6i			0.243226i
$86$			56568.8i			0.824766i
$87$	−1987.46			−0.0281515
$88$	−17238.4			−0.237296
$89$			16227.2i			0.217154i	0.994088	+	0.108577i	$0.0346294\pi$
$89$			16227.2i			0.217154i	−0.994088	+	0.108577i	$0.965371\pi$
$90$	−3155.12			−0.0410591
$91$	59459.6	−	23925.6i	0.752693	−	0.302872i
$92$	−34418.2			−0.423954
$93$		−	7092.95i		−	0.0850393i
$94$	100725.			1.17575
$95$	−27.5335			−0.000313006
$96$		−	9216.00i		−	0.102062i
$97$			126289.i			1.36281i	0.731906	+	0.681406i	$0.238631\pi$
$97$			126289.i			1.36281i	−0.731906	+	0.681406i	$0.761369\pi$
$98$			22973.2i			0.241633i
$99$		−	21817.4i		−	0.223725i
$100$	−48482.7			−0.484827
$101$	92398.8			0.901287			0.450643	−	0.892704i	$-0.351195\pi$
$101$	92398.8			0.901287			0.450643	+	0.892704i	$0.351195\pi$
$102$		−	59894.8i		−	0.570018i
$103$	148091.			1.37542			0.687711	−	0.725985i	$-0.258616\pi$
$103$	148091.			1.37542			0.687711	+	0.725985i	$0.258616\pi$
$104$	36178.6	−	14557.7i	0.327996	−	0.131980i
$105$	−9218.57			−0.0816001
$106$		−	123604.i		−	1.06848i
$107$	88180.1			0.744580			0.372290	−	0.928117i	$-0.378573\pi$
$107$	88180.1			0.744580			0.372290	+	0.928117i	$0.378573\pi$
$108$	11664.0			0.0962250
$109$			61467.2i			0.495538i	0.968819	+	0.247769i	$0.0796974\pi$
$109$			61467.2i			0.495538i	−0.968819	+	0.247769i	$0.920303\pi$
$110$			10491.8i			0.0826736i
$111$		−	8824.01i		−	0.0679764i
$112$		−	26927.1i		−	0.202836i
$113$	−82256.1			−0.605999			−0.303000	−	0.952991i	$-0.597988\pi$
$113$	−82256.1			−0.605999			−0.303000	+	0.952991i	$0.597988\pi$
$114$	101.787			0.000733553
$115$			20947.8i			0.147705i
$116$	−3533.27			−0.0243799
$117$	18424.6	+	45788.5i	0.124432	+	0.309237i
$118$	100379.			0.663647
$119$		−	175000.i		−	1.13284i
$120$	−5609.11			−0.0355583
$121$	88501.4			0.549524
$122$			48243.1i			0.293451i
$123$			133287.i			0.794375i
$124$		−	12609.7i		−	0.0736462i
$125$			59939.2i			0.343112i
$126$	34079.6			0.191236
$127$	−143150.			−0.787556			−0.393778	−	0.919206i	$-0.628832\pi$
$127$	−143150.			−0.787556			−0.393778	+	0.919206i	$0.628832\pi$
$128$		−	16384.0i		−	0.0883883i
$129$	−127280.			−0.673419
$130$	−8860.20	−	22019.3i	−0.0459817	−	0.114273i
$131$	−141329.			−0.719539			−0.359769	−	0.933041i	$-0.617145\pi$
$131$	−141329.			−0.719539			−0.359769	+	0.933041i	$0.617145\pi$
$132$		−	38786.4i		−	0.193752i
$133$	297.400			0.00145785
$134$	57802.2			0.278088
$135$		−	7099.02i		−	0.0335247i
$136$		−	106480.i		−	0.493650i
$137$		−	3958.33i		−	0.0180182i	−0.999959	−	0.00900909i	$-0.997132\pi$
$137$		−	3958.33i		−	0.0180182i	0.999959	−	0.00900909i	$-0.00286772\pi$
$138$		−	77440.9i		−	0.346157i
$139$	−445579.			−1.95608			−0.978042	−	0.208408i	$-0.933172\pi$
$139$	−445579.			−1.95608			−0.978042	+	0.208408i	$0.933172\pi$
$140$	−16388.6			−0.0706677
$141$			226631.i			0.959999i
$142$	−134190.			−0.558469
$143$	152261.	−	61267.5i	0.622657	−	0.250547i
$144$	20736.0			0.0833333
$145$			2150.44i			0.00849391i
$146$	−103223.			−0.400769
$147$	−51689.7			−0.197293
$148$		−	15687.1i		−	0.0588693i
$149$			49619.3i			0.183098i	0.995801	+	0.0915492i	$0.0291819\pi$
$149$			49619.3i			0.183098i	−0.995801	+	0.0915492i	$0.970818\pi$
$150$		−	109086.i		−	0.395860i
$151$			402323.i			1.43593i	0.696081	+	0.717964i	$0.254926\pi$
$151$			402323.i			1.43593i	−0.696081	+	0.717964i	$0.745074\pi$
$152$	180.955			0.000635275
$153$	134763.			0.465418
$154$		−	113325.i		−	0.385058i
$155$	−7674.60			−0.0256582
$156$	32754.8	+	81401.8i	0.107761	+	0.267807i
$157$	−562429.			−1.82104			−0.910519	−	0.413468i	$-0.864317\pi$
$157$	−562429.			−1.82104			−0.910519	+	0.413468i	$0.864317\pi$
$158$			62481.3i			0.199116i
$159$	278108.			0.872411
$160$	−9971.74			−0.0307944
$161$		−	226265.i		−	0.687945i
$162$			26244.0i			0.0785674i
$163$		−	313878.i		−	0.925319i	−0.886536	−	0.462659i	$-0.846895\pi$
$163$		−	313878.i		−	0.925319i	0.886536	−	0.462659i	$-0.153105\pi$
$164$			236955.i			0.687949i
$165$	−23606.5			−0.0675027
$166$	11279.1			0.0317691
$167$		−	602228.i		−	1.67097i	−0.549510	−	0.835487i	$-0.685186\pi$
$167$		−	602228.i		−	1.67097i	0.549510	−	0.835487i	$-0.314814\pi$
$168$	60586.0			0.165615
$169$	−267813.	+	257166.i	−0.721299	+	0.692624i
$170$	−64806.4			−0.171987
$171$			229.021i			0.000598943i
$172$	−226275.			−0.583198
$173$	304021.			0.772304			0.386152	−	0.922435i	$-0.373804\pi$
$173$	304021.			0.772304			0.386152	+	0.922435i	$0.373804\pi$
$174$		−	7949.85i		−	0.0199061i
$175$		−	318726.i		−	0.786723i
$176$		−	68953.7i		−	0.167794i
$177$			225852.i			0.541865i
$178$	−64908.7			−0.153551
$179$	406190.			0.947539			0.473769	−	0.880649i	$-0.342893\pi$
$179$	406190.			0.947539			0.473769	+	0.880649i	$0.342893\pi$
$180$		−	12620.5i		−	0.0290332i
$181$	−240967.			−0.546714			−0.273357	−	0.961913i	$-0.588134\pi$
$181$	−240967.			−0.546714			−0.273357	+	0.961913i	$0.588134\pi$
$182$	95702.4	+	237838.i	0.214163	+	0.532235i
$183$	−108547.			−0.239602
$184$		−	137673.i		−	0.299781i
$185$	−9547.60			−0.0205100
$186$	28371.8			0.0601319
$187$		−	448130.i		−	0.937131i
$188$			402899.i			0.831383i
$189$			76679.2i			0.156143i
$190$		−	110.134i		−	0.000221329i
$191$	226140.			0.448532			0.224266	−	0.974528i	$-0.428001\pi$
$191$	226140.			0.448532			0.224266	+	0.974528i	$0.428001\pi$
$192$	36864.0			0.0721688
$193$		−	406075.i		−	0.784717i	−0.919812	−	0.392359i	$-0.871659\pi$
$193$		−	406075.i		−	0.784717i	0.919812	−	0.392359i	$-0.128341\pi$
$194$	−505156.			−0.963653
$195$	49543.3	−	19935.5i	0.0933037	−	0.0375439i
$196$	−91892.9			−0.170861
$197$		−	407390.i		−	0.747902i	−0.927449	−	0.373951i	$-0.878003\pi$
$197$		−	407390.i		−	0.747902i	0.927449	−	0.373951i	$-0.121997\pi$
$198$	87269.5			0.158197
$199$	202865.			0.363140			0.181570	−	0.983378i	$-0.441882\pi$
$199$	202865.			0.363140			0.181570	+	0.983378i	$0.441882\pi$
$200$		−	193931.i		−	0.342825i
$201$			130055.i			0.227058i
$202$			369595.i			0.637306i
$203$		−	23227.7i		−	0.0395609i
$204$	239579.			0.403064
$205$	144217.			0.239680
$206$			592364.i			0.972570i
$207$	174242.			0.282636
$208$	58230.8	+	144714.i	0.0933242	+	0.231928i
$209$	761.568			0.00120599
$210$		−	36874.3i		−	0.0577000i
$211$	657970.			1.01742			0.508710	−	0.860938i	$-0.330123\pi$
$211$	657970.			1.01742			0.508710	+	0.860938i	$0.330123\pi$
$212$	494415.			0.755530
$213$		−	301927.i		−	0.455988i
$214$			352720.i			0.526497i
$215$			137717.i			0.203185i
$216$			46656.0i			0.0680414i
$217$	82896.2			0.119505
$218$	−245869.			−0.350398
$219$		−	232251.i		−	0.327226i
$220$	−41967.1			−0.0584591
$221$	378442.	+	940499.i	0.521217	+	1.29532i
$222$	35296.0			0.0480666
$223$		−	978060.i		−	1.31705i	−0.752558	−	0.658527i	$-0.771180\pi$
$223$		−	978060.i		−	1.31705i	0.752558	−	0.658527i	$-0.228820\pi$
$224$	107709.			0.143427
$225$	245444.			0.323218
$226$		−	329024.i		−	0.428506i
$227$			812137.i			1.04608i	0.852308	+	0.523040i	$0.175202\pi$
$227$			812137.i			1.04608i	−0.852308	+	0.523040i	$0.824798\pi$
$228$			407.149i			0.000518700i
$229$			1.36329e6i			1.71790i	0.512058	+	0.858951i	$0.328883\pi$
$229$			1.36329e6i			1.71790i	−0.512058	+	0.858951i	$0.671117\pi$
$230$	−83791.4			−0.104443
$231$	254982.			0.314398
$232$		−	14133.1i		−	0.0172392i
$233$	931926.			1.12458			0.562292	−	0.826939i	$-0.309920\pi$
$233$	931926.			1.12458			0.562292	+	0.826939i	$0.309920\pi$
$234$	−183154.	+	73698.3i	−0.218664	+	0.0879869i
$235$	245215.			0.289653
$236$			401515.i			0.469269i
$237$	−140583.			−0.162578
$238$	699998.			0.801041
$239$		−	885874.i		−	1.00318i	−0.865107	−	0.501588i	$-0.832749\pi$
$239$		−	885874.i		−	1.00318i	0.865107	−	0.501588i	$-0.167251\pi$
$240$		−	22436.4i		−	0.0251435i
$241$			1.32690e6i			1.47161i	0.677191	+	0.735807i	$0.263197\pi$
$241$			1.32690e6i			1.47161i	−0.677191	+	0.735807i	$0.736803\pi$
$242$			354006.i			0.388572i
$243$	−59049.0			−0.0641500
$244$	−192973.			−0.207501
$245$			55928.5i			0.0595275i
$246$	−533148.			−0.561708
$247$	−1598.32	+	643.137i	−0.00166694	+	0.000670751i
$248$	50438.8			0.0520757
$249$			25378.1i			0.0259394i
$250$	−239757.			−0.242617
$251$	−547596.			−0.548625			−0.274313	−	0.961641i	$-0.588450\pi$
$251$	−547596.			−0.548625			−0.274313	+	0.961641i	$0.588450\pi$
$252$			136319.i			0.135224i
$253$		−	579409.i		−	0.569094i
$254$		−	572599.i		−	0.556886i
$255$		−	145814.i		−	0.140427i
$256$	65536.0			0.0625000
$257$	−1.65509e6			−1.56311			−0.781554	−	0.623837i	$-0.785573\pi$
$257$	−1.65509e6			−1.56311			−0.781554	+	0.623837i	$0.785573\pi$
$258$		−	509119.i		−	0.476179i
$259$	103127.			0.0955265
$260$	88077.0	−	35440.8i	0.0808033	−	0.0325140i
$261$	17887.2			0.0162533
$262$		−	565317.i		−	0.508791i
$263$	−668188.			−0.595675			−0.297838	−	0.954617i	$-0.596265\pi$
$263$	−668188.			−0.595675			−0.297838	+	0.954617i	$0.596265\pi$
$264$	155146.			0.137003
$265$		−	300914.i		−	0.263226i
$266$			1189.60i			0.00103085i
$267$		−	146045.i		−	0.125374i
$268$			231209.i			0.196638i
$269$	−384942.			−0.324350			−0.162175	−	0.986762i	$-0.551851\pi$
$269$	−384942.			−0.324350			−0.162175	+	0.986762i	$0.551851\pi$
$270$	28396.1			0.0237055
$271$		−	1.05344e6i		−	0.871334i	−0.900108	−	0.435667i	$-0.856513\pi$
$271$		−	1.05344e6i		−	0.871334i	0.900108	−	0.435667i	$-0.143487\pi$
$272$	425919.			0.349064
$273$	−535136.	+	215330.i	−0.434568	+	0.174863i
$274$	15833.3			0.0127408
$275$		−	816177.i		−	0.650808i
$276$	309764.			0.244770
$277$	−1.62051e6			−1.26897			−0.634485	−	0.772936i	$-0.718788\pi$
$277$	−1.62051e6			−1.26897			−0.634485	+	0.772936i	$0.718788\pi$
$278$		−	1.78231e6i		−	1.38316i
$279$			63836.6i			0.0490975i
$280$		−	65554.3i		−	0.0499696i
$281$			563009.i			0.425353i	0.977123	+	0.212677i	$0.0682181\pi$
$281$			563009.i			0.425353i	−0.977123	+	0.212677i	$0.931782\pi$
$282$	−906522.			−0.678822
$283$	361892.			0.268604			0.134302	−	0.990940i	$-0.457121\pi$
$283$	361892.			0.268604			0.134302	+	0.990940i	$0.457121\pi$
$284$		−	536760.i		−	0.394897i
$285$	247.802			0.000180714
$286$	245070.	+	609045.i	0.177164	+	0.440285i
$287$	−1.55774e6			−1.11633
$288$			82944.0i			0.0589256i
$289$	1.34819e6			0.949526
$290$	−8601.77			−0.00600610
$291$		−	1.13660e6i		−	0.786820i
$292$		−	412891.i		−	0.283386i
$293$			757318.i			0.515358i	0.966231	+	0.257679i	$0.0829577\pi$
$293$			757318.i			0.515358i	−0.966231	+	0.257679i	$0.917042\pi$
$294$		−	206759.i		−	0.139507i
$295$	244373.			0.163493
$296$	62748.5			0.0416269
$297$			196356.i			0.129168i
$298$	−198477.			−0.129470
$299$	489306.	+	1.21602e6i	0.316521	+	0.786614i
$300$	436345.			0.279915
$301$		−	1.48753e6i		−	0.946348i
$302$	−1.60929e6			−1.01535
$303$	−831589.			−0.520358
$304$			723.821i			0.000449207i
$305$			117448.i			0.0722932i
$306$			539053.i			0.329100i
$307$			1.96280e6i			1.18858i	0.804249	+	0.594292i	$0.202568\pi$
$307$			1.96280e6i			1.18858i	−0.804249	+	0.594292i	$0.797432\pi$
$308$	453302.			0.272277
$309$	−1.33282e6			−0.794100
$310$		−	30698.4i		−	0.0181431i
$311$	834419.			0.489197			0.244598	−	0.969624i	$-0.421344\pi$
$311$	834419.			0.489197			0.244598	+	0.969624i	$0.421344\pi$
$312$	−325607.	+	131019.i	−0.189368	+	0.0761989i
$313$	1.61834e6			0.933703			0.466851	−	0.884336i	$-0.345388\pi$
$313$	1.61834e6			0.933703			0.466851	+	0.884336i	$0.345388\pi$
$314$		−	2.24972e6i		−	1.28767i
$315$	82967.2			0.0471118
$316$	−249925.			−0.140797
$317$		−	660304.i		−	0.369059i	−0.982827	−	0.184529i	$-0.940924\pi$
$317$		−	660304.i		−	0.369059i	0.982827	−	0.184529i	$-0.0590761\pi$
$318$			1.11243e6i			0.616888i
$319$		−	59480.4i		−	0.0327263i
$320$		−	39887.0i		−	0.0217749i
$321$	−793621.			−0.429883
$322$	905062.			0.486450
$323$			4704.11i			0.00250883i
$324$	−104976.			−0.0555556
$325$	689255.	+	1.71293e6i	0.361969	+	0.899560i
$326$	1.25551e6			0.654299
$327$		−	553205.i		−	0.286099i
$328$	−947819.			−0.486453
$329$	−2.64866e6			−1.34908
$330$		−	94425.9i		−	0.0477316i
$331$			2.73497e6i			1.37209i	0.727558	+	0.686046i	$0.240655\pi$
$331$			2.73497e6i			1.37209i	−0.727558	+	0.686046i	$0.759345\pi$
$332$			45116.5i			0.0224642i
$333$			79416.1i			0.0392462i
$334$	2.40891e6			1.18156
$335$	140720.			0.0685083
$336$			242344.i			0.117107i
$337$	−880707.			−0.422432			−0.211216	−	0.977439i	$-0.567742\pi$
$337$	−880707.			−0.422432			−0.211216	+	0.977439i	$0.567742\pi$
$338$	−1.02867e6	−	1.07125e6i	−0.489759	−	0.510035i
$339$	740305.			0.349874
$340$		−	259226.i		−	0.121613i
$341$	212277.			0.0988590
$342$	−916.085			−0.000423517
$343$		−	2.37193e6i		−	1.08860i
$344$		−	905101.i		−	0.412383i
$345$		−	188531.i		−	0.0852774i
$346$			1.21608e6i			0.546102i
$347$	−33079.1			−0.0147479			−0.00737395	−	0.999973i	$-0.502347\pi$
$347$	−33079.1			−0.0147479			−0.00737395	+	0.999973i	$0.502347\pi$
$348$	31799.4			0.0140757
$349$			4.06448e6i			1.78624i	0.449814	+	0.893122i	$0.351490\pi$
$349$			4.06448e6i			1.78624i	−0.449814	+	0.893122i	$0.648510\pi$
$350$	1.27490e6			0.556297
$351$	−165821.	−	412097.i	−0.0718410	−	0.178538i
$352$	275815.			0.118648
$353$		−	3.57770e6i		−	1.52815i	−0.645125	−	0.764077i	$-0.723195\pi$
$353$		−	3.57770e6i		−	1.52815i	0.645125	−	0.764077i	$-0.276805\pi$
$354$	−903409.			−0.383157
$355$	−326686.			−0.137582
$356$		−	259635.i		−	0.108577i
$357$			1.57500e6i			0.654047i
$358$			1.62476e6i			0.670011i
$359$			3.31455e6i			1.35734i	0.734444	+	0.678669i	$0.237443\pi$
$359$			3.31455e6i			1.35734i	−0.734444	+	0.678669i	$0.762557\pi$
$360$	50482.0			0.0205296
$361$	2.47609e6			0.999997
$362$		−	963867.i		−	0.386585i
$363$	−796513.			−0.317268
$364$	−951353.	+	382809.i	−0.376347	+	0.151436i
$365$	−251297.			−0.0987313
$366$		−	434188.i		−	0.169424i
$367$	192406.			0.0745681			0.0372840	−	0.999305i	$-0.488129\pi$
$367$	192406.			0.0745681			0.0372840	+	0.999305i	$0.488129\pi$
$368$	550691.			0.211977
$369$		−	1.19958e6i		−	0.458632i
$370$		−	38190.4i		−	0.0145027i
$371$			3.25029e6i			1.22599i
$372$			113487.i			0.0425197i
$373$	1.60161e6			0.596053			0.298027	−	0.954558i	$-0.403672\pi$
$373$	1.60161e6			0.596053			0.298027	+	0.954558i	$0.403672\pi$
$374$	1.79252e6			0.662652
$375$		−	539453.i		−	0.198096i
$376$	−1.61160e6			−0.587877
$377$	50230.7	+	124833.i	0.0182019	+	0.0452350i
$378$	−306717.			−0.110410
$379$		−	3.76006e6i		−	1.34461i	−0.740274	−	0.672305i	$-0.765304\pi$
$379$		−	3.76006e6i		−	1.34461i	0.740274	−	0.672305i	$-0.234696\pi$
$380$	440.537			0.000156503
$381$	1.28835e6			0.454695
$382$			904560.i			0.317160i
$383$			4.69693e6i			1.63613i	0.575128	+	0.818063i	$0.304952\pi$
$383$			4.69693e6i			1.63613i	−0.575128	+	0.818063i	$0.695048\pi$
$384$			147456.i			0.0510310i
$385$		−	275892.i		−	0.0948608i
$386$	1.62430e6			0.554879
$387$	1.14552e6			0.388799
$388$		−	2.02062e6i		−	0.681406i
$389$	4.85881e6			1.62801			0.814003	−	0.580860i	$-0.197284\pi$
$389$	4.85881e6			1.62801			0.814003	+	0.580860i	$0.197284\pi$
$390$	79741.8	+	198173.i	0.0265476	+	0.0659757i
$391$	3.57894e6			1.18389
$392$		−	367572.i		−	0.120817i
$393$	1.27196e6			0.415426
$394$	1.62956e6			0.528846
$395$			152111.i			0.0490533i
$396$			349078.i			0.111863i
$397$		−	3.91926e6i		−	1.24804i	−0.781409	−	0.624019i	$-0.785499\pi$
$397$		−	3.91926e6i		−	1.24804i	0.781409	−	0.624019i	$-0.214501\pi$
$398$			811460.i			0.256779i
$399$	−2676.60			−0.000841688
$400$	775724.			0.242414
$401$		−	5.01898e6i		−	1.55867i	−0.626606	−	0.779336i	$-0.715557\pi$
$401$		−	5.01898e6i		−	1.55867i	0.626606	−	0.779336i	$-0.284443\pi$
$402$	−520219.			−0.160554
$403$	−445509.	+	179266.i	−0.136645	+	0.0549838i
$404$	−1.47838e6			−0.450643
$405$			63891.2i			0.0193555i
$406$	92910.9			0.0279738
$407$	264083.			0.0790232
$408$			958317.i			0.285009i
$409$		−	2.97886e6i		−	0.880526i	−0.897869	−	0.440263i	$-0.854885\pi$
$409$		−	2.97886e6i		−	0.880526i	0.897869	−	0.440263i	$-0.145115\pi$
$410$			576868.i			0.169480i
$411$			35625.0i			0.0104028i
$412$	−2.36946e6			−0.687711
$413$	−2.63956e6			−0.761477
$414$			696968.i			0.199854i
$415$	27459.1			0.00782648
$416$	−578857.	+	232923.i	−0.163998	+	0.0659902i
$417$	4.01021e6			1.12935
$418$			3046.27i			0.000852762i
$419$	−4.76887e6			−1.32703			−0.663515	−	0.748163i	$-0.730936\pi$
$419$	−4.76887e6			−1.32703			−0.663515	+	0.748163i	$0.730936\pi$
$420$	147497.			0.0408000
$421$			651294.i			0.179090i	0.995983	+	0.0895451i	$0.0285413\pi$
$421$			651294.i			0.179090i	−0.995983	+	0.0895451i	$0.971459\pi$
$422$			2.63188e6i			0.719424i
$423$		−	2.03968e6i		−	0.554256i
$424$			1.97766e6i			0.534241i
$425$	5.04143e6			1.35388
$426$	1.20771e6			0.322432
$427$		−	1.26860e6i		−	0.336710i
$428$	−1.41088e6			−0.372290
$429$	−1.37035e6	+	551407.i	−0.359491	+	0.144654i
$430$	−550869.			−0.143674
$431$			5.22621e6i			1.35517i	0.735444	+	0.677585i	$0.236973\pi$
$431$			5.22621e6i			1.35517i	−0.735444	+	0.677585i	$0.763027\pi$
$432$	−186624.			−0.0481125
$433$	−6.33506e6			−1.62379			−0.811897	−	0.583801i	$-0.801565\pi$
$433$	−6.33506e6			−1.62379			−0.811897	+	0.583801i	$0.801565\pi$
$434$			331585.i			0.0845026i
$435$		−	19354.0i		−	0.00490396i
$436$		−	983475.i		−	0.247769i
$437$			6082.18i			0.00152355i
$438$	929005.			0.231384
$439$	−7.44321e6			−1.84331			−0.921657	−	0.388006i	$-0.873164\pi$
$439$	−7.44321e6			−1.84331			−0.921657	+	0.388006i	$0.873164\pi$
$440$		−	167868.i		−	0.0413368i
$441$	465208.			0.113907
$442$	−3.76200e6	+	1.51377e6i	−0.915931	+	0.368556i
$443$	327031.			0.0791734			0.0395867	−	0.999216i	$-0.487396\pi$
$443$	327031.			0.0791734			0.0395867	+	0.999216i	$0.487396\pi$
$444$			141184.i			0.0339882i
$445$	−158021.			−0.0378281
$446$	3.91224e6			0.931297
$447$		−	446574.i		−	0.105712i
$448$			430834.i			0.101418i
$449$		−	5.82118e6i		−	1.36268i	−0.731965	−	0.681342i	$-0.761396\pi$
$449$		−	5.82118e6i		−	1.36268i	0.731965	−	0.681342i	$-0.238604\pi$
$450$			981775.i			0.228550i
$451$	−3.98899e6			−0.923468
$452$	1.31610e6			0.303000
$453$		−	3.62091e6i		−	0.829033i
$454$	−3.24855e6			−0.739690
$455$	232988.	+	579019.i	0.0527601	+	0.131119i
$456$	−1628.60			−0.000366776
$457$		−	8.15788e6i		−	1.82720i	−0.406612	−	0.913601i	$-0.633290\pi$
$457$		−	8.15788e6i		−	1.82720i	0.406612	−	0.913601i	$-0.366710\pi$
$458$	−5.45315e6			−1.21474
$459$	−1.21287e6			−0.268709
$460$		−	335165.i		−	0.0738524i
$461$			3.15011e6i			0.690357i	0.938537	+	0.345179i	$0.112182\pi$
$461$			3.15011e6i			0.690357i	−0.938537	+	0.345179i	$0.887818\pi$
$462$			1.01993e6i			0.222313i
$463$			202622.i			0.0439273i	0.999759	+	0.0219637i	$0.00699182\pi$
$463$			202622.i			0.0439273i	−0.999759	+	0.0219637i	$0.993008\pi$
$464$	56532.3			0.0121899
$465$	69071.4			0.0148138
$466$			3.72771e6i			0.795201i
$467$	91749.8			0.0194676			0.00973382	−	0.999953i	$-0.496902\pi$
$467$	91749.8			0.0194676			0.00973382	+	0.999953i	$0.496902\pi$
$468$	−294793.	−	732616.i	−0.0622161	−	0.154619i
$469$	−1.51997e6			−0.319082
$470$			980860.i			0.204815i
$471$	5.06186e6			1.05138
$472$	−1.60606e6			−0.331823
$473$		−	3.80921e6i		−	0.782856i
$474$		−	562331.i		−	0.114960i
$475$			8567.58i			0.00174230i
$476$			2.79999e6i			0.566421i
$477$	−2.50298e6			−0.503687
$478$	3.54350e6			0.709353
$479$			2.82910e6i			0.563390i	0.959504	+	0.281695i	$0.0908967\pi$
$479$			2.82910e6i			0.563390i	−0.959504	+	0.281695i	$0.909103\pi$
$480$	89745.7			0.0177791
$481$	−554236.	+	223016.i	−0.109228	+	0.0439514i
$482$	−5.30758e6			−1.04059
$483$			2.03639e6i			0.397185i
$484$	−1.41602e6			−0.274762
$485$	−1.22981e6			−0.237401
$486$		−	236196.i		−	0.0453609i
$487$		−	1.30176e6i		−	0.248719i	−0.992237	−	0.124359i	$-0.960312\pi$
$487$		−	1.30176e6i		−	0.248719i	0.992237	−	0.124359i	$-0.0396876\pi$
$488$		−	771890.i		−	0.146726i
$489$			2.82490e6i			0.534233i
$490$	−223714.			−0.0420923
$491$	−6.07696e6			−1.13758			−0.568791	−	0.822482i	$-0.692588\pi$
$491$	−6.07696e6			−1.13758			−0.568791	+	0.822482i	$0.692588\pi$
$492$		−	2.13259e6i		−	0.397187i
$493$	367404.			0.0680810
$494$	−2572.55	−	6393.26i	−0.000474292	−	0.00117870i
$495$	212458.			0.0389727
$496$			201755.i			0.0368231i
$497$	3.52866e6			0.640795
$498$	−101512.			−0.0183419
$499$			3.78708e6i			0.680853i	0.940271	+	0.340426i	$0.110571\pi$
$499$			3.78708e6i			0.680853i	−0.940271	+	0.340426i	$0.889429\pi$
$500$		−	959028.i		−	0.171556i
$501$			5.42005e6i			0.964737i
$502$		−	2.19038e6i		−	0.387937i
$503$	2.28832e6			0.403271			0.201636	−	0.979461i	$-0.435374\pi$
$503$	2.28832e6			0.403271			0.201636	+	0.979461i	$0.435374\pi$
$504$	−545274.			−0.0956178
$505$			899783.i			0.157003i
$506$	2.31764e6			0.402410
$507$	2.41032e6	−	2.31450e6i	0.416442	−	0.399886i
$508$	2.29040e6			0.393778
$509$		−	3.70029e6i		−	0.633055i	−0.948583	−	0.316528i	$-0.897483\pi$
$509$		−	3.70029e6i		−	0.633055i	0.948583	−	0.316528i	$-0.102517\pi$
$510$	583258.			0.0992968
$511$	2.71435e6			0.459847
$512$			262144.i			0.0441942i
$513$		−	2061.19i		−	0.000345800i
$514$		−	6.62037e6i		−	1.10528i
$515$			1.44212e6i			0.239597i
$516$	2.03648e6			0.336709
$517$	−6.78256e6			−1.11601
$518$			412509.i			0.0675474i
$519$	−2.73619e6			−0.445890
$520$	141763.	+	352308.i	0.0229909	+	0.0571366i
$521$	6.13756e6			0.990608			0.495304	−	0.868720i	$-0.335057\pi$
$521$	6.13756e6			0.990608			0.495304	+	0.868720i	$0.335057\pi$
$522$			71548.7i			0.0114928i
$523$	4.10572e6			0.656349			0.328175	−	0.944617i	$-0.393567\pi$
$523$	4.10572e6			0.656349			0.328175	+	0.944617i	$0.393567\pi$
$524$	2.26127e6			0.359769
$525$			2.86853e6i			0.454215i
$526$		−	2.67275e6i		−	0.421206i
$527$			1.31121e6i			0.205658i
$528$			620583.i			0.0968758i
$529$	−1.80895e6			−0.281053
$530$	1.20366e6			0.186129
$531$		−	2.03267e6i		−	0.312846i
$532$	−4758.40			−0.000728923
$533$	8.37177e6	−	3.36867e6i	1.27644	−	0.513618i
$534$	584178.			0.0886528
$535$			858701.i			0.129705i
$536$	−924834.			−0.139044
$537$	−3.65571e6			−0.547062
$538$		−	1.53977e6i		−	0.229350i
$539$		−	1.54696e6i		−	0.229355i
$540$			113584.i			0.0167623i
$541$			2.51266e6i			0.369098i	0.982823	+	0.184549i	$0.0590824\pi$
$541$			2.51266e6i			0.369098i	−0.982823	+	0.184549i	$0.940918\pi$
$542$	4.21374e6			0.616126
$543$	2.16870e6			0.315646
$544$			1.70368e6i			0.246825i
$545$	−598569.			−0.0863223
$546$	−861321.	−	2.14054e6i	−0.123647	−	0.307286i
$547$	−2.29454e6			−0.327889			−0.163945	−	0.986470i	$-0.552422\pi$
$547$	−2.29454e6			−0.327889			−0.163945	+	0.986470i	$0.552422\pi$
$548$			63333.4i			0.00900909i
$549$	976924.			0.138334
$550$	3.26471e6			0.460191
$551$			624.378i			8.76130e-5i
$552$			1.23905e6i			0.173078i
$553$		−	1.64301e6i		−	0.228469i
$554$		−	6.48202e6i		−	0.897297i
$555$	85928.4			0.0118414
$556$	7.12926e6			0.978042
$557$		−	5.96324e6i		−	0.814412i	−0.913336	−	0.407206i	$-0.866503\pi$
$557$		−	5.96324e6i		−	0.814412i	0.913336	−	0.407206i	$-0.133497\pi$
$558$	−255346.			−0.0347171
$559$	3.21684e6	+	7.99445e6i	0.435412	+	1.08208i
$560$	262217.			0.0353339
$561$			4.03317e6i			0.541053i
$562$	−2.25204e6			−0.300770
$563$	−7.37754e6			−0.980935			−0.490468	−	0.871459i	$-0.663174\pi$
$563$	−7.37754e6			−0.980935			−0.490468	+	0.871459i	$0.663174\pi$
$564$		−	3.62609e6i		−	0.479999i
$565$		−	801012.i		−	0.105565i
$566$			1.44757e6i			0.189932i
$567$		−	690113.i		−	0.0901493i
$568$	2.14704e6			0.279235
$569$	2.60845e6			0.337756			0.168878	−	0.985637i	$-0.445986\pi$
$569$	2.60845e6			0.337756			0.168878	+	0.985637i	$0.445986\pi$
$570$			991.208i			0.000127784i
$571$	1.09867e7			1.41019			0.705095	−	0.709113i	$-0.250905\pi$
$571$	1.09867e7			1.41019			0.705095	+	0.709113i	$0.250905\pi$
$572$	−2.43618e6	+	980280.i	−0.311329	+	0.125274i
$573$	−2.03526e6			−0.258960
$574$		−	6.23097e6i		−	0.789362i
$575$	6.51831e6			0.822177
$576$	−331776.			−0.0416667
$577$			1.10048e7i			1.37607i	0.725677	+	0.688036i	$0.241527\pi$
$577$			1.10048e7i			1.37607i	−0.725677	+	0.688036i	$0.758473\pi$
$578$			5.39276e6i			0.671416i
$579$			3.65468e6i			0.453057i
$580$		−	34407.1i		−	0.00424696i
$581$	−296596.			−0.0364523
$582$	4.54640e6			0.556365
$583$			8.32318e6i			1.01419i
$584$	1.65157e6			0.200384
$585$	−445890.	+	179419.i	−0.0538689	+	0.0216760i
$586$	−3.02927e6			−0.364413
$587$			1.09851e7i			1.31586i	0.753080	+	0.657929i	$0.228567\pi$
$587$			1.09851e7i			1.31586i	−0.753080	+	0.657929i	$0.771433\pi$
$588$	827036.			0.0986464
$589$	−2228.31			−0.000264659
$590$			977492.i			0.115607i
$591$			3.66651e6i			0.431801i
$592$			250994.i			0.0294347i
$593$		−	3.90620e6i		−	0.456161i	−0.973642	−	0.228081i	$-0.926755\pi$
$593$		−	3.90620e6i		−	0.456161i	0.973642	−	0.228081i	$-0.0732449\pi$
$594$	−785426.			−0.0913354
$595$	1.70415e6			0.197340
$596$		−	793908.i		−	0.0915492i
$597$	−1.82578e6			−0.209659
$598$	−4.86407e6	+	1.95722e6i	−0.556220	+	0.223814i
$599$	−1.69973e7			−1.93559			−0.967797	−	0.251733i	$-0.918999\pi$
$599$	−1.69973e7			−1.93559			−0.967797	+	0.251733i	$0.918999\pi$
$600$			1.74538e6i			0.197930i
$601$	1.07099e7			1.20948			0.604739	−	0.796424i	$-0.293278\pi$
$601$	1.07099e7			1.20948			0.604739	+	0.796424i	$0.293278\pi$
$602$	5.95014e6			0.669169
$603$		−	1.17049e6i		−	0.131092i
$604$		−	6.43717e6i		−	0.717964i
$605$			861829.i			0.0957267i
$606$		−	3.32636e6i		−	0.367949i
$607$	−1.43102e7			−1.57642			−0.788212	−	0.615404i	$-0.788993\pi$
$607$	−1.43102e7			−1.57642			−0.788212	+	0.615404i	$0.788993\pi$
$608$	−2895.28			−0.000317638
$609$			209050.i			0.0228405i
$610$	−469793.			−0.0511190
$611$	1.42347e7	−	5.72781e6i	1.54257	−	0.620705i
$612$	−2.15621e6			−0.232709
$613$			1.67594e7i			1.80139i	0.434450	+	0.900696i	$0.356943\pi$
$613$			1.67594e7i			1.80139i	−0.434450	+	0.900696i	$0.643057\pi$
$614$	−7.85120e6			−0.840456
$615$	−1.29795e6			−0.138379
$616$			1.81321e6i			0.192529i
$617$		−	1.26793e7i		−	1.34085i	−0.741976	−	0.670427i	$-0.766111\pi$
$617$		−	1.26793e7i		−	1.34085i	0.741976	−	0.670427i	$-0.233889\pi$
$618$		−	5.33128e6i		−	0.561513i
$619$		−	5.43928e6i		−	0.570577i	−0.958442	−	0.285289i	$-0.907911\pi$
$619$		−	5.43928e6i		−	0.570577i	0.958442	−	0.285289i	$-0.0920895\pi$
$620$	122794.			0.0128291
$621$	−1.56818e6			−0.163180
$622$			3.33768e6i			0.345914i
$623$	1.70684e6			0.176187
$624$	−524077.	−	1.30243e6i	−0.0538807	−	0.133904i
$625$	8.88559e6			0.909885
$626$			6.47335e6i			0.660227i
$627$	−6854.11			−0.000696277
$628$	8.99887e6			0.910519
$629$			1.63121e6i			0.164393i
$630$			331869.i			0.0333131i
$631$		−	5.28439e6i		−	0.528350i	−0.964475	−	0.264175i	$-0.914900\pi$
$631$		−	5.28439e6i		−	0.528350i	0.964475	−	0.264175i	$-0.0850996\pi$
$632$		−	999700.i		−	0.0995582i
$633$	−5.92173e6			−0.587407
$634$	2.64121e6			0.260964
$635$		−	1.39400e6i		−	0.137192i
$636$	−4.44973e6			−0.436206
$637$	1.30639e6	+	3.24664e6i	0.127563	+	0.317019i
$638$	237922.			0.0231410
$639$			2.71735e6i			0.263265i
$640$	159548.			0.0153972
$641$	8.01855e6			0.770816			0.385408	−	0.922746i	$-0.374061\pi$
$641$	8.01855e6			0.770816			0.385408	+	0.922746i	$0.374061\pi$
$642$		−	3.17448e6i		−	0.303973i
$643$		−	1.39227e7i		−	1.32799i	−0.747735	−	0.663997i	$-0.768859\pi$
$643$		−	1.39227e7i		−	1.32799i	0.747735	−	0.663997i	$-0.231141\pi$
$644$			3.62025e6i			0.343972i
$645$		−	1.23945e6i		−	0.117309i
$646$	−18816.5			−0.00177401
$647$	−9.55517e6			−0.897383			−0.448692	−	0.893687i	$-0.648110\pi$
$647$	−9.55517e6			−0.897383			−0.448692	+	0.893687i	$0.648110\pi$
$648$		−	419904.i		−	0.0392837i
$649$	−6.75926e6			−0.629923
$650$	−6.85170e6	+	2.75702e6i	−0.636085	+	0.255951i
$651$	−746066.			−0.0689961
$652$			5.02204e6i			0.462659i
$653$	−1.87340e7			−1.71928			−0.859640	−	0.510901i	$-0.829312\pi$
$653$	−1.87340e7			−1.71928			−0.859640	+	0.510901i	$0.829312\pi$
$654$	2.21282e6			0.202303
$655$		−	1.37627e6i		−	0.125343i
$656$		−	3.79128e6i		−	0.343974i
$657$			2.09026e6i			0.188924i
$658$		−	1.05946e7i		−	0.953941i
$659$	−3.39787e6			−0.304785			−0.152392	−	0.988320i	$-0.548698\pi$
$659$	−3.39787e6			−0.304785			−0.152392	+	0.988320i	$0.548698\pi$
$660$	377704.			0.0337514
$661$			1.81823e7i			1.61862i	0.587382	+	0.809310i	$0.300159\pi$
$661$			1.81823e7i			1.61862i	−0.587382	+	0.809310i	$0.699841\pi$
$662$	−1.09399e7			−0.970215
$663$	−3.40598e6	−	8.46449e6i	−0.300925	−	0.747854i
$664$	−180466.			−0.0158846
$665$			2896.09i			0.000253956i
$666$	−317664.			−0.0277513
$667$	475034.			0.0413438
$668$			9.63564e6i			0.835487i
$669$			8.80254e6i			0.760401i
$670$			562879.i			0.0484427i
$671$		−	3.24858e6i		−	0.278540i
$672$	−969377.			−0.0828074
$673$	686844.			0.0584548			0.0292274	−	0.999573i	$-0.490695\pi$
$673$	686844.			0.0584548			0.0292274	+	0.999573i	$0.490695\pi$
$674$		−	3.52283e6i		−	0.298704i
$675$	−2.20899e6			−0.186610
$676$	4.28501e6	−	4.11466e6i	0.360650	−	0.346312i
$677$	−6.64858e6			−0.557516			−0.278758	−	0.960361i	$-0.589923\pi$
$677$	−6.64858e6			−0.557516			−0.278758	+	0.960361i	$0.589923\pi$
$678$			2.96122e6i			0.247398i
$679$	1.32836e7			1.10571
$680$	1.03690e6			0.0859935
$681$		−	7.30923e6i		−	0.603954i
$682$			849106.i			0.0699038i
$683$			1.66853e7i			1.36862i	0.729193	+	0.684308i	$0.239896\pi$
$683$			1.66853e7i			1.36862i	−0.729193	+	0.684308i	$0.760104\pi$
$684$		−	3664.34i		−	0.000299472i
$685$	38546.4			0.00313875
$686$	9.48773e6			0.769754
$687$		−	1.22696e7i		−	0.991831i
$688$	3.62040e6			0.291599
$689$	−7.02885e6	−	1.74680e7i	−0.564074	−	1.40183i
$690$	754122.			0.0603002
$691$		−	1.75290e7i		−	1.39657i	−0.715822	−	0.698283i	$-0.753948\pi$
$691$		−	1.75290e7i		−	1.39657i	0.715822	−	0.698283i	$-0.246052\pi$
$692$	−4.86434e6			−0.386152
$693$	−2.29484e6			−0.181518
$694$		−	132317.i		−	0.0104283i
$695$		−	4.33906e6i		−	0.340748i
$696$			127198.i			0.00995305i
$697$		−	2.46395e7i		−	1.92110i
$698$	−1.62579e7			−1.26307
$699$	−8.38734e6			−0.649279
$700$			5.09961e6i			0.393362i
$701$	7.06379e6			0.542929			0.271464	−	0.962449i	$-0.412492\pi$
$701$	7.06379e6			0.542929			0.271464	+	0.962449i	$0.412492\pi$
$702$	1.64839e6	−	663285.i	0.126246	−	0.0507993i
$703$	−2772.13			−0.000211556
$704$			1.10326e6i			0.0838969i
$705$	−2.20694e6			−0.167231
$706$	1.43108e7			1.08057
$707$		−	9.71889e6i		−	0.731253i
$708$		−	3.61364e6i		−	0.270933i
$709$			9.07731e6i			0.678175i	0.940755	+	0.339087i	$0.110118\pi$
$709$			9.07731e6i			0.678175i	−0.940755	+	0.339087i	$0.889882\pi$
$710$		−	1.30675e6i		−	0.0972849i
$711$	1.26525e6			0.0938644
$712$	1.03854e6			0.0767755
$713$			1.69532e6i			0.124890i
$714$	−6.29998e6			−0.462481
$715$	596625.	+	1.48272e6i	0.0436452	+	0.108466i
$716$	−6.49904e6			−0.473769
$717$			7.97287e6i			0.579184i
$718$	−1.32582e7			−0.959783
$719$	−1.03755e7			−0.748492			−0.374246	−	0.927330i	$-0.622098\pi$
$719$	−1.03755e7			−0.748492			−0.374246	+	0.927330i	$0.622098\pi$
$720$			201928.i			0.0145166i
$721$		−	1.55768e7i		−	1.11594i
$722$			9.90436e6i			0.707104i
$723$		−	1.19421e7i		−	0.849637i
$724$	3.85547e6			0.273357
$725$	669150.			0.0472801
$726$		−	3.18605e6i		−	0.224342i
$727$	−2.94915e6			−0.206948			−0.103474	−	0.994632i	$-0.532996\pi$
$727$	−2.94915e6			−0.206948			−0.103474	+	0.994632i	$0.532996\pi$
$728$	−1.53124e6	−	3.80541e6i	−0.107081	−	0.266117i
$729$	531441.			0.0370370
$730$		−	1.00519e6i		−	0.0698136i
$731$	2.35290e7			1.62858
$732$	1.73675e6			0.119801
$733$			1.71605e7i			1.17970i	0.807513	+	0.589849i	$0.200813\pi$
$733$			1.71605e7i			1.17970i	−0.807513	+	0.589849i	$0.799187\pi$
$734$			769623.i			0.0527276i
$735$		−	503356.i		−	0.0343682i
$736$			2.20276e6i			0.149890i
$737$	−3.89226e6			−0.263957
$738$	4.79834e6			0.324302
$739$			2.20849e7i			1.48760i	0.668405	+	0.743798i	$0.266977\pi$
$739$			2.20849e7i			1.48760i	−0.668405	+	0.743798i	$0.733023\pi$
$740$	152762.			0.0102550
$741$	14384.8	−	5788.23i	0.000962409	−	0.000387258i
$742$	−1.30011e7			−0.866906
$743$			2.21459e7i			1.47171i	0.677140	+	0.735854i	$0.263219\pi$
$743$			2.21459e7i			1.47171i	−0.677140	+	0.735854i	$0.736781\pi$
$744$	−453949.			−0.0300659
$745$	−483194.			−0.0318956
$746$			6.40644e6i			0.421473i
$747$		−	228402.i		−	0.0149761i
$748$			7.17008e6i			0.468565i
$749$		−	9.27514e6i		−	0.604110i
$750$	2.15781e6			0.140075
$751$	6.38710e6			0.413242			0.206621	−	0.978421i	$-0.433753\pi$
$751$	6.38710e6			0.413242			0.206621	+	0.978421i	$0.433753\pi$
$752$		−	6.44638e6i		−	0.415692i
$753$	4.92836e6			0.316749
$754$	−499331.	+	200923.i	−0.0319860	+	0.0128707i
$755$	−3.91783e6			−0.250137
$756$		−	1.22687e6i		−	0.0780716i
$757$	−2.47266e6			−0.156828			−0.0784141	−	0.996921i	$-0.524986\pi$
$757$	−2.47266e6			−0.156828			−0.0784141	+	0.996921i	$0.524986\pi$
$758$	1.50402e7			0.950783
$759$			5.21469e6i			0.328567i
$760$			1762.15i			0.000110664i
$761$		−	1.59009e7i		−	0.995315i	−0.867374	−	0.497657i	$-0.834194\pi$
$761$		−	1.59009e7i		−	0.995315i	0.867374	−	0.497657i	$-0.165806\pi$
$762$			5.15339e6i			0.321518i
$763$	6.46537e6			0.402052
$764$	−3.61824e6			−0.224266
$765$			1.31233e6i			0.0810755i
$766$	−1.87877e7			−1.15692
$767$	1.41858e7	−	5.70814e6i	0.870693	−	0.350353i
$768$	−589824.			−0.0360844
$769$			4.80537e6i			0.293029i	0.989209	+	0.146515i	$0.0468055\pi$
$769$			4.80537e6i			0.293029i	−0.989209	+	0.146515i	$0.953194\pi$
$770$	1.10357e6			0.0670767
$771$	1.48958e7			0.902461
$772$			6.49720e6i			0.392359i
$773$			2.22008e7i			1.33635i	0.744004	+	0.668175i	$0.232924\pi$
$773$			2.22008e7i			1.33635i	−0.744004	+	0.668175i	$0.767076\pi$
$774$			4.58207e6i			0.274922i
$775$			2.38810e6i			0.142823i
$776$	8.08249e6			0.481827
$777$	−928145.			−0.0551523
$778$			1.94353e7i			1.15117i
$779$	41873.2			0.00247225
$780$	−792693.	+	318967.i	−0.0466518	+	0.0187720i
$781$	9.03603e6			0.530090
$782$			1.43158e7i			0.837140i
$783$	−160985.			−0.00938382
$784$	1.47029e6			0.0854303
$785$		−	5.47695e6i		−	0.317223i
$786$			5.08786e6i			0.293750i
$787$		−	2.85397e7i		−	1.64253i	−0.570548	−	0.821264i	$-0.693269\pi$
$787$		−	2.85397e7i		−	1.64253i	0.570548	−	0.821264i	$-0.306731\pi$
$788$			6.51824e6i			0.373951i
$789$	6.01370e6			0.343913
$790$	−608444.			−0.0346859
$791$			8.65203e6i			0.491674i
$792$	−1.39631e6			−0.0790987
$793$	2.74339e6	+	6.81785e6i	0.154919	+	0.385003i
$794$	1.56770e7			0.882496
$795$			2.70823e6i			0.151973i
$796$	−3.24584e6			−0.181570
$797$	−2.10013e7			−1.17112			−0.585558	−	0.810630i	$-0.699125\pi$
$797$	−2.10013e7			−1.17112			−0.585558	+	0.810630i	$0.699125\pi$
$798$		−	10706.4i		−	0.000595163i
$799$		−	4.18951e7i		−	2.32165i
$800$			3.10289e6i			0.171412i
$801$			1.31440e6i			0.0723847i
$802$	2.00759e7			1.10215
$803$	6.95078e6			0.380403
$804$		−	2.08088e6i		−	0.113529i
$805$	2.20338e6			0.119839
$806$	−717063.	−	1.78203e6i	−0.0388794	−	0.0966225i
$807$	3.46448e6			0.187264
$808$		−	5.91352e6i		−	0.318653i
$809$	3.59907e7			1.93339			0.966694	−	0.255935i	$-0.0823834\pi$
$809$	3.59907e7			1.93339			0.966694	+	0.255935i	$0.0823834\pi$
$810$	−255565.			−0.0136864
$811$			3.20415e7i			1.71065i	0.518093	+	0.855324i	$0.326642\pi$
$811$			3.20415e7i			1.71065i	−0.518093	+	0.855324i	$0.673358\pi$
$812$			371644.i			0.0197805i
$813$			9.48092e6i			0.503065i
$814$			1.05633e6i			0.0558779i
$815$	3.05655e6			0.161190
$816$	−3.83327e6			−0.201532
$817$			39986.0i			0.00209581i
$818$	1.19154e7			0.622626
$819$	4.81622e6	−	1.93797e6i	0.250898	−	0.100957i
$820$	−2.30747e6			−0.119840
$821$			2.92849e7i			1.51630i	0.652079	+	0.758151i	$0.273897\pi$
$821$			2.92849e7i			1.51630i	−0.652079	+	0.758151i	$0.726103\pi$
$822$	−142500.			−0.00735589
$823$	−1.27788e6			−0.0657646			−0.0328823	−	0.999459i	$-0.510469\pi$
$823$	−1.27788e6			−0.0657646			−0.0328823	+	0.999459i	$0.510469\pi$
$824$		−	9.47783e6i		−	0.486285i
$825$			7.34560e6i			0.375744i
$826$		−	1.05583e7i		−	0.538446i
$827$		−	3.34508e7i		−	1.70076i	−0.526169	−	0.850380i	$-0.676372\pi$
$827$		−	3.34508e7i		−	1.70076i	0.526169	−	0.850380i	$-0.323628\pi$
$828$	−2.78787e6			−0.141318
$829$	−1.65727e7			−0.837541			−0.418770	−	0.908092i	$-0.637539\pi$
$829$	−1.65727e7			−0.837541			−0.418770	+	0.908092i	$0.637539\pi$
$830$			109837.i			0.00553416i
$831$	1.45845e7			0.732640
$832$	−931692.	−	2.31543e6i	−0.0466621	−	0.115964i
$833$	9.55540e6			0.477129
$834$			1.60408e7i			0.798568i
$835$	5.86451e6			0.291082
$836$	−12185.1			−0.000602994
$837$		−	574529.i		−	0.0283464i
$838$		−	1.90755e7i		−	0.938352i
$839$		−	7.23411e6i		−	0.354797i	−0.984139	−	0.177399i	$-0.943232\pi$
$839$		−	7.23411e6i		−	0.354797i	0.984139	−	0.177399i	$-0.0567682\pi$
$840$			589989.i			0.0288500i
$841$	−2.04624e7			−0.997622
$842$	−2.60518e6			−0.126636
$843$		−	5.06708e6i		−	0.245578i
$844$	−1.05275e7			−0.508710
$845$	−2.50429e6	−	2.60797e6i	−0.120654	−	0.125650i
$846$	8.15870e6			0.391918
$847$		−	9.30894e6i		−	0.445853i
$848$	−7.91064e6			−0.377765
$849$	−3.25703e6			−0.155079
$850$			2.01657e7i			0.957341i
$851$			2.10907e6i			0.0998315i
$852$			4.83084e6i			0.227994i
$853$			3.08382e6i			0.145116i	0.997364	+	0.0725582i	$0.0231163\pi$
$853$			3.08382e6i			0.145116i	−0.997364	+	0.0725582i	$0.976884\pi$
$854$	5.07441e6			0.238090
$855$	−2230.22			−0.000104335
$856$		−	5.64353e6i		−	0.263249i
$857$	9.15615e6			0.425854			0.212927	−	0.977068i	$-0.431700\pi$
$857$	9.15615e6			0.425854			0.212927	+	0.977068i	$0.431700\pi$
$858$	−2.20563e6	−	5.48140e6i	−0.102286	−	0.254199i
$859$	−3.40397e7			−1.57399			−0.786996	−	0.616958i	$-0.788365\pi$
$859$	−3.40397e7			−1.57399			−0.786996	+	0.616958i	$0.788365\pi$
$860$		−	2.20348e6i		−	0.101593i
$861$	1.40197e7			0.644511
$862$	−2.09048e7			−0.958250
$863$			1.92683e7i			0.880676i	0.897832	+	0.440338i	$0.145141\pi$
$863$			1.92683e7i			0.880676i	−0.897832	+	0.440338i	$0.854859\pi$
$864$		−	746496.i		−	0.0340207i
$865$			2.96057e6i			0.134535i
$866$		−	2.53402e7i		−	1.14820i
$867$	−1.21337e7			−0.548209
$868$	−1.32634e6			−0.0597524
$869$		−	4.20734e6i		−	0.188998i
$870$	77415.9			0.00346763
$871$	8.16875e6	−	3.28698e6i	0.364846	−	0.146808i
$872$	3.93390e6			0.175199
$873$			1.02294e7i			0.454270i
$874$	−24328.7			−0.00107731
$875$	6.30466e6			0.278382
$876$			3.71602e6i			0.163613i
$877$		−	2.63479e7i		−	1.15677i	−0.815765	−	0.578384i	$-0.803684\pi$
$877$		−	2.63479e7i		−	1.15677i	0.815765	−	0.578384i	$-0.196316\pi$
$878$		−	2.97729e7i		−	1.30342i
$879$		−	6.81586e6i		−	0.297542i
$880$	671473.			0.0292295
$881$	4.12657e6			0.179122			0.0895610	−	0.995981i	$-0.471454\pi$
$881$	4.12657e6			0.179122			0.0895610	+	0.995981i	$0.471454\pi$
$882$			1.86083e6i			0.0805444i
$883$	1.36506e6			0.0589182			0.0294591	−	0.999566i	$-0.490622\pi$
$883$	1.36506e6			0.0589182			0.0294591	+	0.999566i	$0.490622\pi$
$884$	−6.05507e6	−	1.50480e7i	−0.260609	−	0.647661i
$885$	−2.19936e6			−0.0943925
$886$			1.30812e6i			0.0559841i
$887$	−2.77026e7			−1.18225			−0.591127	−	0.806578i	$-0.701317\pi$
$887$	−2.77026e7			−1.18225			−0.591127	+	0.806578i	$0.701317\pi$
$888$	−564736.			−0.0240333
$889$			1.50571e7i			0.638978i
$890$		−	632083.i		−	0.0267485i
$891$		−	1.76721e6i		−	0.0745750i
$892$			1.56490e7i			0.658527i
$893$	71197.9			0.00298771
$894$	1.78629e6			0.0747496
$895$			3.95549e6i			0.165060i
$896$	−1.72334e6			−0.0717133
$897$	−4.40376e6	−	1.09442e7i	−0.182744	−	0.454152i
$898$	2.32847e7			0.963564
$899$			174037.i			0.00718194i
$900$	−3.92710e6			−0.161609
$901$	−5.14113e7			−2.10983
$902$		−	1.59560e7i		−	0.652990i
$903$			1.33878e7i			0.546374i
$904$			5.26439e6i			0.214253i
$905$		−	2.34654e6i		−	0.0952372i
$906$	1.44836e7			0.586215
$907$	−2.08574e7			−0.841863			−0.420931	−	0.907093i	$-0.638297\pi$
$907$	−2.08574e7			−0.841863			−0.420931	+	0.907093i	$0.638297\pi$
$908$		−	1.29942e7i		−	0.523040i
$909$	7.48430e6			0.300429
$910$	−2.31608e6	+	931953.i	−0.0927149	+	0.0373070i
$911$	3.95317e6			0.157815			0.0789077	−	0.996882i	$-0.474857\pi$
$911$	3.95317e6			0.157815			0.0789077	+	0.996882i	$0.474857\pi$
$912$		−	6514.38i		−	0.000259350i
$913$	−759510.			−0.0301548
$914$	3.26315e7			1.29203
$915$		−	1.05703e6i		−	0.0417385i
$916$		−	2.18126e7i		−	0.858951i
$917$			1.48656e7i			0.583793i
$918$		−	4.85148e6i		−	0.190006i
$919$	1.66130e7			0.648873			0.324436	−	0.945907i	$-0.394825\pi$
$919$	1.66130e7			0.648873			0.324436	+	0.945907i	$0.394825\pi$
$920$	1.34066e6			0.0522215
$921$		−	1.76652e7i		−	0.686230i
$922$	−1.26005e7			−0.488156
$923$	−1.89641e7	+	7.63084e6i	−0.732702	+	0.294828i
$924$	−4.07972e6			−0.157199
$925$			2.97092e6i			0.114166i
$926$	−810490.			−0.0310613
$927$	1.19954e7			0.458474
$928$			226129.i			0.00861959i
$929$			4.33406e7i			1.64762i	0.566869	+	0.823808i	$0.308155\pi$
$929$			4.33406e7i			1.64762i	−0.566869	+	0.823808i	$0.691845\pi$
$930$			276286.i			0.0104749i
$931$			16238.8i			0.000614014i
$932$	−1.49108e7			−0.562292
$933$	−7.50978e6			−0.282438
$934$			366999.i			0.0137657i
$935$	4.36391e6			0.163247
$936$	2.93047e6	−	1.17917e6i	0.109332	−	0.0439934i
$937$	−5.84680e6			−0.217555			−0.108778	−	0.994066i	$-0.534694\pi$
$937$	−5.84680e6			−0.217555			−0.108778	+	0.994066i	$0.534694\pi$
$938$		−	6.07987e6i		−	0.225625i
$939$	−1.45650e7			−0.539073
$940$	−3.92344e6			−0.144826
$941$			3.23340e7i			1.19038i	0.803585	+	0.595190i	$0.202923\pi$
$941$			3.23340e7i			1.19038i	−0.803585	+	0.595190i	$0.797077\pi$
$942$			2.02475e7i			0.743435i
$943$		−	3.18577e7i		−	1.16663i
$944$		−	6.42424e6i		−	0.234634i
$945$	−746705.			−0.0272000
$946$	1.52368e7			0.553563
$947$		−	3.74163e6i		−	0.135577i	−0.997700	−	0.0677884i	$-0.978406\pi$
$947$		−	3.74163e6i		−	0.135577i	0.997700	−	0.0677884i	$-0.0215943\pi$
$948$	2.24933e6			0.0812889
$949$	−1.45877e7	+	5.86987e6i	−0.525801	+	0.211574i
$950$	−34270.3			−0.00123200
$951$			5.94273e6i			0.213076i
$952$	−1.12000e7			−0.400520
$953$	−5.30640e7			−1.89264			−0.946319	−	0.323234i	$-0.895230\pi$
$953$	−5.30640e7			−1.89264			−0.946319	+	0.323234i	$0.895230\pi$
$954$		−	1.00119e7i		−	0.356160i
$955$			2.20216e6i			0.0781340i
$956$			1.41740e7i			0.501588i
$957$			535324.i			0.0188946i
$958$	−1.13164e7			−0.398377
$959$	−416354.			−0.0146189
$960$			358983.i			0.0125717i
$961$	2.80080e7			0.978305
$962$	−892064.	−	2.21694e6i	−0.0310784	−	0.0772355i
$963$	7.14259e6			0.248193
$964$		−	2.12303e7i		−	0.735807i
$965$	3.95437e6			0.136697
$966$	−8.14555e6			−0.280852
$967$			888502.i			0.0305557i	0.999883	+	0.0152779i	$0.00486328\pi$
$967$			888502.i			0.0305557i	−0.999883	+	0.0152779i	$0.995137\pi$
$968$		−	5.66409e6i		−	0.194286i
$969$		−	42337.0i		−	0.00144847i
$970$		−	4.91922e6i		−	0.167868i
$971$	−1.92989e7			−0.656878			−0.328439	−	0.944525i	$-0.606523\pi$
$971$	−1.92989e7			−0.656878			−0.328439	+	0.944525i	$0.606523\pi$
$972$	944784.			0.0320750
$973$			4.68678e7i			1.58706i
$974$	5.20704e6			0.175871
$975$	−6.20329e6	−	1.54163e7i	−0.208983	−	0.519361i
$976$	3.08756e6			0.103751
$977$			2.73358e7i			0.916212i	0.888898	+	0.458106i	$0.151472\pi$
$977$			2.73358e7i			0.916212i	−0.888898	+	0.458106i	$0.848528\pi$
$978$	−1.12996e7			−0.377760
$979$	4.37079e6			0.145748
$980$		−	894856.i		−	0.0297638i
$981$			4.97884e6i			0.165179i
$982$		−	2.43078e7i		−	0.804391i
$983$			2.36204e7i			0.779656i	0.920888	+	0.389828i	$0.127466\pi$
$983$			2.36204e7i			0.779656i	−0.920888	+	0.389828i	$0.872534\pi$
$984$	8.53038e6			0.280854
$985$	3.96717e6			0.130284
$986$			1.46961e6i			0.0481406i
$987$	2.38379e7			0.778889
$988$	25573.0	−	10290.2i	0.000833470	−	0.000335375i
$989$	3.04218e7			0.988996
$990$			849833.i			0.0275579i
$991$	−3.49750e7			−1.13129			−0.565644	−	0.824649i	$-0.691372\pi$
$991$	−3.49750e7			−1.13129			−0.565644	+	0.824649i	$0.691372\pi$
$992$	−807020.			−0.0260379
$993$		−	2.46148e7i		−	0.792177i
$994$			1.41146e7i			0.453110i
$995$			1.97550e6i			0.0632587i
$996$		−	406049.i		−	0.0129697i
$997$	1.96013e7			0.624521			0.312261	−	0.949996i	$-0.398914\pi$
$997$	1.96013e7			0.624521			0.312261	+	0.949996i	$0.398914\pi$
$998$	−1.51483e7			−0.481436
$999$		−	714744.i		−	0.0226588i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.6.b.a.25.5	yes	6
3.2	odd	2		234.6.b.c.181.2		6
4.3	odd	2		624.6.c.d.337.4		6
13.5	odd	4		1014.6.a.q.1.2		3
13.8	odd	4		1014.6.a.o.1.2		3
13.12	even	2	inner	78.6.b.a.25.2	✓	6
39.38	odd	2		234.6.b.c.181.5		6
52.51	odd	2		624.6.c.d.337.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.a.25.2	✓	6	13.12	even	2	inner
78.6.b.a.25.5	yes	6	1.1	even	1	trivial
234.6.b.c.181.2		6	3.2	odd	2
234.6.b.c.181.5		6	39.38	odd	2
624.6.c.d.337.3		6	52.51	odd	2
624.6.c.d.337.4		6	4.3	odd	2
1014.6.a.o.1.2		3	13.8	odd	4
1014.6.a.q.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000 q^{3} -16.0000 q^{4} +9.73803i q^{5} -36.0000i q^{6} -105.184i q^{7} -64.0000i q^{8} +81.0000 q^{9} -38.9521 q^{10} -269.350i q^{11} +144.000 q^{12} +(227.464 + 565.290i) q^{13} +420.736 q^{14} -87.6423i q^{15} +256.000 q^{16} +1663.75 q^{17} +324.000i q^{18} +2.82742i q^{19} -155.809i q^{20} +946.657i q^{21} +1077.40 q^{22} +2151.14 q^{23} +576.000i q^{24} +3030.17 q^{25} +(-2261.16 + 909.856i) q^{26} -729.000 q^{27} +1682.95i q^{28} +220.829 q^{29} +350.569 q^{30} +788.106i q^{31} +1024.00i q^{32} +2424.15i q^{33} +6654.98i q^{34} +1024.29 q^{35} -1296.00 q^{36} +980.445i q^{37} -11.3097 q^{38} +(-2047.18 - 5087.61i) q^{39} +623.234 q^{40} -14809.7i q^{41} -3786.63 q^{42} +14142.2 q^{43} +4309.61i q^{44} +788.781i q^{45} +8604.55i q^{46} -25181.2i q^{47} -2304.00 q^{48} +5743.31 q^{49} +12120.7i q^{50} -14973.7 q^{51} +(-3639.42 - 9044.65i) q^{52} -30900.9 q^{53} -2916.00i q^{54} +2622.94 q^{55} -6731.78 q^{56} -25.4468i q^{57} +883.317i q^{58} -25094.7i q^{59} +1402.28i q^{60} +12060.8 q^{61} -3152.42 q^{62} -8519.91i q^{63} -4096.00 q^{64} +(-5504.81 + 2215.05i) q^{65} -9696.61 q^{66} -14450.5i q^{67} -26619.9 q^{68} -19360.2 q^{69} +4097.14i q^{70} +33547.5i q^{71} -5184.00i q^{72} +25805.7i q^{73} -3921.78 q^{74} -27271.5 q^{75} -45.2388i q^{76} -28331.4 q^{77} +(20350.5 - 8188.70i) q^{78} +15620.3 q^{79} +2492.94i q^{80} +6561.00 q^{81} +59238.7 q^{82} -2819.78i q^{83} -15146.5i q^{84} +16201.6i q^{85} +56568.8i q^{86} -1987.46 q^{87} -17238.4 q^{88} +16227.2i q^{89} -3155.12 q^{90} +(59459.6 - 23925.6i) q^{91} -34418.2 q^{92} -7092.95i q^{93} +100725. q^{94} -27.5335 q^{95} -9216.00i q^{96} +126289. i q^{97} +22973.2i q^{98} -21817.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 96 q^{4} + 486 q^{9} + 320 q^{10} + 864 q^{12} + 530 q^{13} - 1360 q^{14} + 1536 q^{16} - 836 q^{17} - 1296 q^{22} - 416 q^{23} + 718 q^{25} - 1360 q^{26} - 4374 q^{27} + 18788 q^{29} - 2880 q^{30}+ \cdots + 567168 q^{95}+O(q^{100}) \) 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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