LMFDB - Embedded newform 546.8.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

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:	yes
Analytic conductor:	$170.562223914$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 148556x^{3} - 20997404x^{2} - 256427072x + 44264019648 $ x^5 - x^4 - 148556x^3 - 20997404x^2 - 256427072*x + 44264019648
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-88.4385$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +498.821 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +498.821 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -3990.57 q^{10} +6172.03 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} +13468.2 q^{15} +4096.00 q^{16} +6825.78 q^{17} -5832.00 q^{18} -19400.5 q^{19} +31924.5 q^{20} -9261.00 q^{21} -49376.2 q^{22} +4131.06 q^{23} -13824.0 q^{24} +170697. q^{25} -17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +42953.0 q^{29} -107745. q^{30} +32382.2 q^{31} -32768.0 q^{32} +166645. q^{33} -54606.3 q^{34} -171096. q^{35} +46656.0 q^{36} +386429. q^{37} +155204. q^{38} +59319.0 q^{39} -255396. q^{40} -788731. q^{41} +74088.0 q^{42} -143033. q^{43} +395010. q^{44} +363641. q^{45} -33048.5 q^{46} -85215.1 q^{47} +110592. q^{48} +117649. q^{49} -1.36558e6 q^{50} +184296. q^{51} +140608. q^{52} -214552. q^{53} -157464. q^{54} +3.07874e6 q^{55} +175616. q^{56} -523814. q^{57} -343624. q^{58} +1.31215e6 q^{59} +861963. q^{60} +747732. q^{61} -259057. q^{62} -250047. q^{63} +262144. q^{64} +1.09591e6 q^{65} -1.33316e6 q^{66} +2.99635e6 q^{67} +436850. q^{68} +111539. q^{69} +1.36876e6 q^{70} +2.27675e6 q^{71} -373248. q^{72} +6.29248e6 q^{73} -3.09143e6 q^{74} +4.60883e6 q^{75} -1.24163e6 q^{76} -2.11701e6 q^{77} -474552. q^{78} +586881. q^{79} +2.04317e6 q^{80} +531441. q^{81} +6.30985e6 q^{82} +5.44653e6 q^{83} -592704. q^{84} +3.40484e6 q^{85} +1.14426e6 q^{86} +1.15973e6 q^{87} -3.16008e6 q^{88} -2.64024e6 q^{89} -2.90912e6 q^{90} -753571. q^{91} +264388. q^{92} +874319. q^{93} +681721. q^{94} -9.67740e6 q^{95} -884736. q^{96} -1.38182e7 q^{97} -941192. q^{98} +4.49941e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 4072 q^{10} + 958 q^{11} + 8640 q^{12} + 10985 q^{13} + 13720 q^{14} + 13743 q^{15} + 20480 q^{16} + 7864 q^{17} - 29160 q^{18} + 60173 q^{19} + 32576 q^{20} - 46305 q^{21} - 7664 q^{22} + 122869 q^{23} - 69120 q^{24} + 73722 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} - 17317 q^{29} - 109944 q^{30} - 177665 q^{31} - 163840 q^{32} + 25866 q^{33} - 62912 q^{34} - 174587 q^{35} + 233280 q^{36} - 55136 q^{37} - 481384 q^{38} + 296595 q^{39} - 260608 q^{40} - 237570 q^{41} + 370440 q^{42} - 970601 q^{43} + 61312 q^{44} + 371061 q^{45} - 982952 q^{46} - 384035 q^{47} + 552960 q^{48} + 588245 q^{49} - 589776 q^{50} + 212328 q^{51} + 703040 q^{52} - 1977 q^{53} - 787320 q^{54} + 1520014 q^{55} + 878080 q^{56} + 1624671 q^{57} + 138536 q^{58} + 2057936 q^{59} + 879552 q^{60} - 723756 q^{61} + 1421320 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 1118273 q^{65} - 206928 q^{66} + 2695018 q^{67} + 503296 q^{68} + 3317463 q^{69} + 1396696 q^{70} + 7392916 q^{71} - 1866240 q^{72} + 8720441 q^{73} + 441088 q^{74} + 1990494 q^{75} + 3851072 q^{76} - 328594 q^{77} - 2372760 q^{78} + 4646419 q^{79} + 2084864 q^{80} + 2657205 q^{81} + 1900560 q^{82} + 17766733 q^{83} - 2963520 q^{84} + 16495320 q^{85} + 7764808 q^{86} - 467559 q^{87} - 490496 q^{88} + 4692321 q^{89} - 2968488 q^{90} - 3767855 q^{91} + 7863616 q^{92} - 4796955 q^{93} + 3072280 q^{94} - 2355945 q^{95} - 4423680 q^{96} + 15680305 q^{97} - 4705960 q^{98} + 698382 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 4072 * q^10 + 958 * q^11 + 8640 * q^12 + 10985 * q^13 + 13720 * q^14 + 13743 * q^15 + 20480 * q^16 + 7864 * q^17 - 29160 * q^18 + 60173 * q^19 + 32576 * q^20 - 46305 * q^21 - 7664 * q^22 + 122869 * q^23 - 69120 * q^24 + 73722 * q^25 - 87880 * q^26 + 98415 * q^27 - 109760 * q^28 - 17317 * q^29 - 109944 * q^30 - 177665 * q^31 - 163840 * q^32 + 25866 * q^33 - 62912 * q^34 - 174587 * q^35 + 233280 * q^36 - 55136 * q^37 - 481384 * q^38 + 296595 * q^39 - 260608 * q^40 - 237570 * q^41 + 370440 * q^42 - 970601 * q^43 + 61312 * q^44 + 371061 * q^45 - 982952 * q^46 - 384035 * q^47 + 552960 * q^48 + 588245 * q^49 - 589776 * q^50 + 212328 * q^51 + 703040 * q^52 - 1977 * q^53 - 787320 * q^54 + 1520014 * q^55 + 878080 * q^56 + 1624671 * q^57 + 138536 * q^58 + 2057936 * q^59 + 879552 * q^60 - 723756 * q^61 + 1421320 * q^62 - 1250235 * q^63 + 1310720 * q^64 + 1118273 * q^65 - 206928 * q^66 + 2695018 * q^67 + 503296 * q^68 + 3317463 * q^69 + 1396696 * q^70 + 7392916 * q^71 - 1866240 * q^72 + 8720441 * q^73 + 441088 * q^74 + 1990494 * q^75 + 3851072 * q^76 - 328594 * q^77 - 2372760 * q^78 + 4646419 * q^79 + 2084864 * q^80 + 2657205 * q^81 + 1900560 * q^82 + 17766733 * q^83 - 2963520 * q^84 + 16495320 * q^85 + 7764808 * q^86 - 467559 * q^87 - 490496 * q^88 + 4692321 * q^89 - 2968488 * q^90 - 3767855 * q^91 + 7863616 * q^92 - 4796955 * q^93 + 3072280 * q^94 - 2355945 * q^95 - 4423680 * q^96 + 15680305 * q^97 - 4705960 * q^98 + 698382 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	498.821	1.78464	0.892318	−	0.451407i	$-0.149078\pi$
$5$	498.821	1.78464	0.892318	+	0.451407i	$0.149078\pi$
$6$	−216.000	−0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−3990.57	−1.26193
$11$	6172.03	1.39815	0.699075	−	0.715049i	$-0.253595\pi$
$11$	6172.03	1.39815	0.699075	+	0.715049i	$0.253595\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	2744.00	0.267261
$15$	13468.2	1.03036
$16$	4096.00	0.250000
$17$	6825.78	0.336962	0.168481	−	0.985705i	$-0.446114\pi$
$17$	6825.78	0.336962	0.168481	+	0.985705i	$0.446114\pi$
$18$	−5832.00	−0.235702
$19$	−19400.5	−0.648898	−0.324449	−	0.945903i	$-0.605179\pi$
$19$	−19400.5	−0.648898	−0.324449	+	0.945903i	$0.605179\pi$
$20$	31924.5	0.892318
$21$	−9261.00	−0.218218
$22$	−49376.2	−0.988641
$23$	4131.06	0.0707970	0.0353985	−	0.999373i	$-0.488730\pi$
$23$	4131.06	0.0707970	0.0353985	+	0.999373i	$0.488730\pi$
$24$	−13824.0	−0.204124
$25$	170697.	2.18493
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	42953.0	0.327040	0.163520	−	0.986540i	$-0.447715\pi$
$29$	42953.0	0.327040	0.163520	+	0.986540i	$0.447715\pi$
$30$	−107745.	−0.728575
$31$	32382.2	0.195227	0.0976135	−	0.995224i	$-0.468879\pi$
$31$	32382.2	0.195227	0.0976135	+	0.995224i	$0.468879\pi$
$32$	−32768.0	−0.176777
$33$	166645.	0.807222
$34$	−54606.3	−0.238268
$35$	−171096.	−0.674529
$36$	46656.0	0.166667
$37$	386429.	1.25419	0.627095	−	0.778943i	$-0.284244\pi$
$37$	386429.	1.25419	0.627095	+	0.778943i	$0.284244\pi$
$38$	155204.	0.458840
$39$	59319.0	0.160128
$40$	−255396.	−0.630964
$41$	−788731.	−1.78725	−0.893625	−	0.448814i	$-0.851847\pi$
$41$	−788731.	−1.78725	−0.893625	+	0.448814i	$0.851847\pi$
$42$	74088.0	0.154303
$43$	−143033.	−0.274345	−0.137172	−	0.990547i	$-0.543801\pi$
$43$	−143033.	−0.274345	−0.137172	+	0.990547i	$0.543801\pi$
$44$	395010.	0.699075
$45$	363641.	0.594879
$46$	−33048.5	−0.0500610
$47$	−85215.1	−0.119722	−0.0598610	−	0.998207i	$-0.519066\pi$
$47$	−85215.1	−0.119722	−0.0598610	+	0.998207i	$0.519066\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−1.36558e6	−1.54498
$51$	184296.	0.194545
$52$	140608.	0.138675
$53$	−214552.	−0.197955	−0.0989776	−	0.995090i	$-0.531557\pi$
$53$	−214552.	−0.197955	−0.0989776	+	0.995090i	$0.531557\pi$
$54$	−157464.	−0.136083
$55$	3.07874e6	2.49519
$56$	175616.	0.133631
$57$	−523814.	−0.374641
$58$	−343624.	−0.231252
$59$	1.31215e6	0.831766	0.415883	−	0.909418i	$-0.363473\pi$
$59$	1.31215e6	0.831766	0.415883	+	0.909418i	$0.363473\pi$
$60$	861963.	0.515180
$61$	747732.	0.421786	0.210893	−	0.977509i	$-0.432363\pi$
$61$	747732.	0.421786	0.210893	+	0.977509i	$0.432363\pi$
$62$	−259057.	−0.138046
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	1.09591e6	0.494969
$66$	−1.33316e6	−0.570792
$67$	2.99635e6	1.21711	0.608555	−	0.793511i	$-0.291749\pi$
$67$	2.99635e6	1.21711	0.608555	+	0.793511i	$0.291749\pi$
$68$	436850.	0.168481
$69$	111539.	0.0408746
$70$	1.36876e6	0.476964
$71$	2.27675e6	0.754938	0.377469	−	0.926022i	$-0.376794\pi$
$71$	2.27675e6	0.754938	0.377469	+	0.926022i	$0.376794\pi$
$72$	−373248.	−0.117851
$73$	6.29248e6	1.89318	0.946590	−	0.322439i	$-0.104503\pi$
$73$	6.29248e6	1.89318	0.946590	+	0.322439i	$0.104503\pi$
$74$	−3.09143e6	−0.886846
$75$	4.60883e6	1.26147
$76$	−1.24163e6	−0.324449
$77$	−2.11701e6	−0.528451
$78$	−474552.	−0.113228
$79$	586881.	0.133923	0.0669615	−	0.997756i	$-0.478670\pi$
$79$	586881.	0.133923	0.0669615	+	0.997756i	$0.478670\pi$
$80$	2.04317e6	0.446159
$81$	531441.	0.111111
$82$	6.30985e6	1.26378
$83$	5.44653e6	1.04555	0.522777	−	0.852469i	$-0.324896\pi$
$83$	5.44653e6	1.04555	0.522777	+	0.852469i	$0.324896\pi$
$84$	−592704.	−0.109109
$85$	3.40484e6	0.601355
$86$	1.14426e6	0.193991
$87$	1.15973e6	0.188817
$88$	−3.16008e6	−0.494321
$89$	−2.64024e6	−0.396990	−0.198495	−	0.980102i	$-0.563605\pi$
$89$	−2.64024e6	−0.396990	−0.198495	+	0.980102i	$0.563605\pi$
$90$	−2.90912e6	−0.420643
$91$	−753571.	−0.104828
$92$	264388.	0.0353985
$93$	874319.	0.112714
$94$	681721.	0.0846563
$95$	−9.67740e6	−1.15805
$96$	−884736.	−0.102062
$97$	−1.38182e7	−1.53727	−0.768636	−	0.639686i	$-0.779064\pi$
$97$	−1.38182e7	−1.53727	−0.768636	+	0.639686i	$0.779064\pi$
$98$	−941192.	−0.101015
$99$	4.49941e6	0.466050
$100$	1.09246e7	1.09246
$101$	−3.71204e6	−0.358499	−0.179250	−	0.983804i	$-0.557367\pi$
$101$	−3.71204e6	−0.358499	−0.179250	+	0.983804i	$0.557367\pi$
$102$	−1.47437e6	−0.137564
$103$	−1.15964e6	−0.104567	−0.0522834	−	0.998632i	$-0.516650\pi$
$103$	−1.15964e6	−0.104567	−0.0522834	+	0.998632i	$0.516650\pi$
$104$	−1.12486e6	−0.0980581
$105$	−4.61958e6	−0.389440
$106$	1.71642e6	0.139975
$107$	8.56498e6	0.675901	0.337951	−	0.941164i	$-0.390266\pi$
$107$	8.56498e6	0.675901	0.337951	+	0.941164i	$0.390266\pi$
$108$	1.25971e6	0.0962250
$109$	−1.41835e7	−1.04904	−0.524518	−	0.851400i	$-0.675754\pi$
$109$	−1.41835e7	−1.04904	−0.524518	+	0.851400i	$0.675754\pi$
$110$	−2.46299e7	−1.76436
$111$	1.04336e7	0.724107
$112$	−1.40493e6	−0.0944911
$113$	−1.71347e7	−1.11713	−0.558563	−	0.829462i	$-0.688647\pi$
$113$	−1.71347e7	−1.11713	−0.558563	+	0.829462i	$0.688647\pi$
$114$	4.19052e6	0.264911
$115$	2.06066e6	0.126347
$116$	2.74899e6	0.163520
$117$	1.60161e6	0.0924500
$118$	−1.04972e7	−0.588147
$119$	−2.34124e6	−0.127360
$120$	−6.89570e6	−0.364287
$121$	1.86068e7	0.954822
$122$	−5.98186e6	−0.298247
$123$	−2.12957e7	−1.03187
$124$	2.07246e6	0.0976135
$125$	4.61770e7	2.11466
$126$	2.00038e6	0.0890871
$127$	−1.20518e7	−0.522084	−0.261042	−	0.965327i	$-0.584066\pi$
$127$	−1.20518e7	−0.522084	−0.261042	+	0.965327i	$0.584066\pi$
$128$	−2.09715e6	−0.0883883
$129$	−3.86189e6	−0.158393
$130$	−8.76728e6	−0.349996
$131$	−2.88999e7	−1.12318	−0.561588	−	0.827417i	$-0.689809\pi$
$131$	−2.88999e7	−1.12318	−0.561588	+	0.827417i	$0.689809\pi$
$132$	1.06653e7	0.403611
$133$	6.65438e6	0.245260
$134$	−2.39708e7	−0.860627
$135$	9.81829e6	0.343453
$136$	−3.49480e6	−0.119134
$137$	3.10226e6	0.103076	0.0515378	−	0.998671i	$-0.483588\pi$
$137$	3.10226e6	0.103076	0.0515378	+	0.998671i	$0.483588\pi$
$138$	−892310.	−0.0289027
$139$	−1.07134e6	−0.0338358	−0.0169179	−	0.999857i	$-0.505385\pi$
$139$	−1.07134e6	−0.0338358	−0.0169179	+	0.999857i	$0.505385\pi$
$140$	−1.09501e7	−0.337265
$141$	−2.30081e6	−0.0691216
$142$	−1.82140e7	−0.533822
$143$	1.35599e7	0.387777
$144$	2.98598e6	0.0833333
$145$	2.14259e7	0.583647
$146$	−5.03399e7	−1.33868
$147$	3.17652e6	0.0824786
$148$	2.47314e7	0.627095
$149$	−7.45867e7	−1.84718	−0.923590	−	0.383381i	$-0.874760\pi$
$149$	−7.45867e7	−1.84718	−0.923590	+	0.383381i	$0.874760\pi$
$150$	−3.68706e7	−0.891992
$151$	−4.07355e7	−0.962839	−0.481420	−	0.876490i	$-0.659879\pi$
$151$	−4.07355e7	−0.962839	−0.481420	+	0.876490i	$0.659879\pi$
$152$	9.93307e6	0.229420
$153$	4.97600e6	0.112321
$154$	1.69361e7	0.373671
$155$	1.61529e7	0.348409
$156$	3.79642e6	0.0800641
$157$	8.09285e7	1.66899	0.834493	−	0.551018i	$-0.185760\pi$
$157$	8.09285e7	1.66899	0.834493	+	0.551018i	$0.185760\pi$
$158$	−4.69505e6	−0.0946979
$159$	−5.79290e6	−0.114289
$160$	−1.63454e7	−0.315482
$161$	−1.41696e6	−0.0267587
$162$	−4.25153e6	−0.0785674
$163$	−6.71627e7	−1.21471	−0.607354	−	0.794432i	$-0.707769\pi$
$163$	−6.71627e7	−1.21471	−0.607354	+	0.794432i	$0.707769\pi$
$164$	−5.04788e7	−0.893625
$165$	8.31259e7	1.44060
$166$	−4.35723e7	−0.739319
$167$	4.59311e7	0.763131	0.381566	−	0.924342i	$-0.375385\pi$
$167$	4.59311e7	0.763131	0.381566	+	0.924342i	$0.375385\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	−2.72388e7	−0.425222
$171$	−1.41430e7	−0.216299
$172$	−9.15412e6	−0.137172
$173$	−3.53619e7	−0.519247	−0.259624	−	0.965710i	$-0.583599\pi$
$173$	−3.53619e7	−0.519247	−0.259624	+	0.965710i	$0.583599\pi$
$174$	−9.27785e6	−0.133513
$175$	−5.85492e7	−0.825825
$176$	2.52806e7	0.349537
$177$	3.54280e7	0.480220
$178$	2.11220e7	0.280714
$179$	4.14101e7	0.539661	0.269830	−	0.962908i	$-0.413032\pi$
$179$	4.14101e7	0.539661	0.269830	+	0.962908i	$0.413032\pi$
$180$	2.32730e7	0.297439
$181$	5.27375e7	0.661065	0.330533	−	0.943795i	$-0.392772\pi$
$181$	5.27375e7	0.661065	0.330533	+	0.943795i	$0.392772\pi$
$182$	6.02857e6	0.0741249
$183$	2.01888e7	0.243518
$184$	−2.11511e6	−0.0250305
$185$	1.92759e8	2.23827
$186$	−6.99455e6	−0.0797011
$187$	4.21289e7	0.471124
$188$	−5.45377e6	−0.0598610
$189$	−6.75127e6	−0.0727393
$190$	7.74192e7	0.818862
$191$	3.38007e7	0.351001	0.175501	−	0.984479i	$-0.443846\pi$
$191$	3.38007e7	0.351001	0.175501	+	0.984479i	$0.443846\pi$
$192$	7.07789e6	0.0721688
$193$	−6.72708e6	−0.0673560	−0.0336780	−	0.999433i	$-0.510722\pi$
$193$	−6.72708e6	−0.0673560	−0.0336780	+	0.999433i	$0.510722\pi$
$194$	1.10546e8	1.08702
$195$	2.95896e7	0.285771
$196$	7.52954e6	0.0714286
$197$	−2.38479e7	−0.222238	−0.111119	−	0.993807i	$-0.535443\pi$
$197$	−2.38479e7	−0.222238	−0.111119	+	0.993807i	$0.535443\pi$
$198$	−3.59953e7	−0.329547
$199$	1.11506e8	1.00302	0.501512	−	0.865151i	$-0.332777\pi$
$199$	1.11506e8	1.00302	0.501512	+	0.865151i	$0.332777\pi$
$200$	−8.73971e7	−0.772488
$201$	8.09013e7	0.702699
$202$	2.96963e7	0.253497
$203$	−1.47329e7	−0.123609
$204$	1.17950e7	0.0972726
$205$	−3.93436e8	−3.18959
$206$	9.27714e6	0.0739399
$207$	3.01155e6	0.0235990
$208$	8.99891e6	0.0693375
$209$	−1.19741e8	−0.907256
$210$	3.69566e7	0.275375
$211$	4.02304e7	0.294826	0.147413	−	0.989075i	$-0.452905\pi$
$211$	4.02304e7	0.294826	0.147413	+	0.989075i	$0.452905\pi$
$212$	−1.37313e7	−0.0989776
$213$	6.14723e7	0.435864
$214$	−6.85199e7	−0.477934
$215$	−7.13479e7	−0.489606
$216$	−1.00777e7	−0.0680414
$217$	−1.11071e7	−0.0737889
$218$	1.13468e8	0.741780
$219$	1.69897e8	1.09303
$220$	1.97039e8	1.24759
$221$	1.49962e7	0.0934565
$222$	−8.34686e7	−0.512021
$223$	−7.24980e7	−0.437783	−0.218892	−	0.975749i	$-0.570244\pi$
$223$	−7.24980e7	−0.437783	−0.218892	+	0.975749i	$0.570244\pi$
$224$	1.12394e7	0.0668153
$225$	1.24438e8	0.728309
$226$	1.37078e8	0.789927
$227$	9.97657e7	0.566097	0.283049	−	0.959106i	$-0.408654\pi$
$227$	9.97657e7	0.566097	0.283049	+	0.959106i	$0.408654\pi$
$228$	−3.35241e7	−0.187321
$229$	5.48543e7	0.301847	0.150923	−	0.988545i	$-0.451775\pi$
$229$	5.48543e7	0.301847	0.150923	+	0.988545i	$0.451775\pi$
$230$	−1.64853e7	−0.0893407
$231$	−5.71592e7	−0.305101
$232$	−2.19919e7	−0.115626
$233$	2.20920e8	1.14417	0.572084	−	0.820195i	$-0.306135\pi$
$233$	2.20920e8	1.14417	0.572084	+	0.820195i	$0.306135\pi$
$234$	−1.28129e7	−0.0653720
$235$	−4.25071e7	−0.213660
$236$	8.39775e7	0.415883
$237$	1.58458e7	0.0773205
$238$	1.87299e7	0.0900570
$239$	2.13781e8	1.01292	0.506462	−	0.862262i	$-0.330953\pi$
$239$	2.13781e8	1.01292	0.506462	+	0.862262i	$0.330953\pi$
$240$	5.51656e7	0.257590
$241$	3.53348e8	1.62608	0.813041	−	0.582206i	$-0.197810\pi$
$241$	3.53348e8	1.62608	0.813041	+	0.582206i	$0.197810\pi$
$242$	−1.48854e8	−0.675161
$243$	1.43489e7	0.0641500
$244$	4.78549e7	0.210893
$245$	5.86858e7	0.254948
$246$	1.70366e8	0.729642
$247$	−4.26230e7	−0.179972
$248$	−1.65797e7	−0.0690232
$249$	1.47056e8	0.603651
$250$	−3.69416e8	−1.49529
$251$	5.21077e7	0.207991	0.103995	−	0.994578i	$-0.466837\pi$
$251$	5.21077e7	0.207991	0.103995	+	0.994578i	$0.466837\pi$
$252$	−1.60030e7	−0.0629941
$253$	2.54971e7	0.0989847
$254$	9.64147e7	0.369169
$255$	9.19308e7	0.347193
$256$	1.67772e7	0.0625000
$257$	−8.90506e7	−0.327243	−0.163622	−	0.986523i	$-0.552318\pi$
$257$	−8.90506e7	−0.327243	−0.163622	+	0.986523i	$0.552318\pi$
$258$	3.08951e7	0.112001
$259$	−1.32545e8	−0.474039
$260$	7.01382e7	0.247485
$261$	3.13128e7	0.109013
$262$	2.31200e8	0.794205
$263$	−1.33846e8	−0.453691	−0.226846	−	0.973931i	$-0.572841\pi$
$263$	−1.33846e8	−0.453691	−0.226846	+	0.973931i	$0.572841\pi$
$264$	−8.53221e7	−0.285396
$265$	−1.07023e8	−0.353278
$266$	−5.32351e7	−0.173425
$267$	−7.12866e7	−0.229202
$268$	1.91766e8	0.608555
$269$	−2.89716e7	−0.0907484	−0.0453742	−	0.998970i	$-0.514448\pi$
$269$	−2.89716e7	−0.0907484	−0.0453742	+	0.998970i	$0.514448\pi$
$270$	−7.85463e7	−0.242858
$271$	−5.31960e8	−1.62363	−0.811814	−	0.583916i	$-0.801520\pi$
$271$	−5.31960e8	−1.62363	−0.811814	+	0.583916i	$0.801520\pi$
$272$	2.79584e7	0.0842406
$273$	−2.03464e7	−0.0605228
$274$	−2.48180e7	−0.0728854
$275$	1.05355e9	3.05485
$276$	7.13848e6	0.0204373
$277$	3.89094e8	1.09996	0.549978	−	0.835179i	$-0.314636\pi$
$277$	3.89094e8	1.09996	0.549978	+	0.835179i	$0.314636\pi$
$278$	8.57074e6	0.0239255
$279$	2.36066e7	0.0650757
$280$	8.76009e7	0.238482
$281$	−4.28135e8	−1.15109	−0.575544	−	0.817771i	$-0.695210\pi$
$281$	−4.28135e8	−1.15109	−0.575544	+	0.817771i	$0.695210\pi$
$282$	1.84065e7	0.0488763
$283$	6.88971e8	1.80696	0.903480	−	0.428630i	$-0.141004\pi$
$283$	6.88971e8	1.80696	0.903480	+	0.428630i	$0.141004\pi$
$284$	1.45712e8	0.377469
$285$	−2.61290e8	−0.668598
$286$	−1.08480e8	−0.274200
$287$	2.70535e8	0.675517
$288$	−2.38879e7	−0.0589256
$289$	−3.63747e8	−0.886456
$290$	−1.71407e8	−0.412701
$291$	−3.73092e8	−0.887544
$292$	4.02719e8	0.946590
$293$	−6.54225e8	−1.51946	−0.759732	−	0.650236i	$-0.774670\pi$
$293$	−6.54225e8	−1.51946	−0.759732	+	0.650236i	$0.774670\pi$
$294$	−2.54122e7	−0.0583212
$295$	6.54527e8	1.48440
$296$	−1.97851e8	−0.443423
$297$	1.21484e8	0.269074
$298$	5.96693e8	1.30615
$299$	9.07595e6	0.0196355
$300$	2.94965e8	0.630734
$301$	4.90603e7	0.103693
$302$	3.25884e8	0.680830
$303$	−1.00225e8	−0.206980
$304$	−7.94646e7	−0.162224
$305$	3.72985e8	0.752734
$306$	−3.98080e7	−0.0794228
$307$	−5.28996e8	−1.04344	−0.521721	−	0.853116i	$-0.674710\pi$
$307$	−5.28996e8	−1.04344	−0.521721	+	0.853116i	$0.674710\pi$
$308$	−1.35488e8	−0.264225
$309$	−3.13104e7	−0.0603717
$310$	−1.29223e8	−0.246363
$311$	9.39902e8	1.77183	0.885913	−	0.463852i	$-0.153533\pi$
$311$	9.39902e8	1.77183	0.885913	+	0.463852i	$0.153533\pi$
$312$	−3.03713e7	−0.0566139
$313$	5.83621e8	1.07579	0.537893	−	0.843013i	$-0.319220\pi$
$313$	5.83621e8	1.07579	0.537893	+	0.843013i	$0.319220\pi$
$314$	−6.47428e8	−1.18015
$315$	−1.24729e8	−0.224843
$316$	3.75604e7	0.0669615
$317$	−8.69994e8	−1.53394	−0.766971	−	0.641682i	$-0.778237\pi$
$317$	−8.69994e8	−1.53394	−0.766971	+	0.641682i	$0.778237\pi$
$318$	4.63432e7	0.0808149
$319$	2.65107e8	0.457251
$320$	1.30763e8	0.223080
$321$	2.31255e8	0.390232
$322$	1.13356e7	0.0189213
$323$	−1.32424e8	−0.218654
$324$	3.40122e7	0.0555556
$325$	3.75022e8	0.605990
$326$	5.37302e8	0.858928
$327$	−3.82954e8	−0.605661
$328$	4.03830e8	0.631888
$329$	2.92288e7	0.0452507
$330$	−6.65007e8	−1.01866
$331$	−4.04630e8	−0.613282	−0.306641	−	0.951825i	$-0.599205\pi$
$331$	−4.04630e8	−0.613282	−0.306641	+	0.951825i	$0.599205\pi$
$332$	3.48578e8	0.522777
$333$	2.81706e8	0.418063
$334$	−3.67449e8	−0.539615
$335$	1.49464e9	2.17210
$336$	−3.79331e7	−0.0545545
$337$	−7.20141e8	−1.02497	−0.512487	−	0.858695i	$-0.671276\pi$
$337$	−7.20141e8	−1.02497	−0.512487	+	0.858695i	$0.671276\pi$
$338$	−3.86145e7	−0.0543928
$339$	−4.62637e8	−0.644973
$340$	2.17910e8	0.300678
$341$	1.99864e8	0.272957
$342$	1.13144e8	0.152947
$343$	−4.03536e7	−0.0539949
$344$	7.32329e7	0.0969956
$345$	5.56379e7	0.0729464
$346$	2.82895e8	0.367163
$347$	1.03027e9	1.32372	0.661860	−	0.749627i	$-0.269767\pi$
$347$	1.03027e9	1.32372	0.661860	+	0.749627i	$0.269767\pi$
$348$	7.42228e7	0.0944083
$349$	−4.02176e8	−0.506439	−0.253219	−	0.967409i	$-0.581489\pi$
$349$	−4.02176e8	−0.506439	−0.253219	+	0.967409i	$0.581489\pi$
$350$	4.68394e8	0.583946
$351$	4.32436e7	0.0533761
$352$	−2.02245e8	−0.247160
$353$	1.40084e9	1.69502	0.847512	−	0.530776i	$-0.178100\pi$
$353$	1.40084e9	1.69502	0.847512	+	0.530776i	$0.178100\pi$
$354$	−2.83424e8	−0.339567
$355$	1.13569e9	1.34729
$356$	−1.68976e8	−0.198495
$357$	−6.32136e7	−0.0735312
$358$	−3.31281e8	−0.381598
$359$	1.59025e9	1.81399	0.906997	−	0.421137i	$-0.138369\pi$
$359$	1.59025e9	1.81399	0.906997	+	0.421137i	$0.138369\pi$
$360$	−1.86184e8	−0.210321
$361$	−5.17491e8	−0.578932
$362$	−4.21900e8	−0.467444
$363$	5.02383e8	0.551267
$364$	−4.82285e7	−0.0524142
$365$	3.13882e9	3.37864
$366$	−1.61510e8	−0.172193
$367$	−1.54345e9	−1.62990	−0.814951	−	0.579531i	$-0.803236\pi$
$367$	−1.54345e9	−1.62990	−0.814951	+	0.579531i	$0.803236\pi$
$368$	1.69208e7	0.0176992
$369$	−5.74985e8	−0.595750
$370$	−1.54207e9	−1.58270
$371$	7.35913e7	0.0748200
$372$	5.59564e7	0.0563572
$373$	−3.85736e8	−0.384866	−0.192433	−	0.981310i	$-0.561638\pi$
$373$	−3.85736e8	−0.384866	−0.192433	+	0.981310i	$0.561638\pi$
$374$	−3.37032e8	−0.333135
$375$	1.24678e9	1.22090
$376$	4.36301e7	0.0423281
$377$	9.43678e7	0.0907046
$378$	5.40102e7	0.0514344
$379$	1.97743e9	1.86580	0.932899	−	0.360139i	$-0.117271\pi$
$379$	1.97743e9	1.86580	0.932899	+	0.360139i	$0.117271\pi$
$380$	−6.19353e8	−0.579023
$381$	−3.25400e8	−0.301425
$382$	−2.70405e8	−0.248195
$383$	1.26257e9	1.14831	0.574155	−	0.818747i	$-0.305331\pi$
$383$	1.26257e9	1.14831	0.574155	+	0.818747i	$0.305331\pi$
$384$	−5.66231e7	−0.0510310
$385$	−1.05601e9	−0.943093
$386$	5.38167e7	0.0476279
$387$	−1.04271e8	−0.0914483
$388$	−8.84365e8	−0.768636
$389$	1.82412e9	1.57119	0.785597	−	0.618738i	$-0.212356\pi$
$389$	1.82412e9	1.57119	0.785597	+	0.618738i	$0.212356\pi$
$390$	−2.36716e8	−0.202070
$391$	2.81978e7	0.0238559
$392$	−6.02363e7	−0.0505076
$393$	−7.80299e8	−0.648466
$394$	1.90783e8	0.157146
$395$	2.92749e8	0.239004
$396$	2.87962e8	0.233025
$397$	−2.19853e9	−1.76346	−0.881732	−	0.471751i	$-0.843622\pi$
$397$	−2.19853e9	−1.76346	−0.881732	+	0.471751i	$0.843622\pi$
$398$	−8.92046e8	−0.709245
$399$	1.79668e8	0.141601
$400$	6.99176e8	0.546232
$401$	1.66098e8	0.128635	0.0643176	−	0.997929i	$-0.479513\pi$
$401$	1.66098e8	0.128635	0.0643176	+	0.997929i	$0.479513\pi$
$402$	−6.47211e8	−0.496883
$403$	7.11436e7	0.0541462
$404$	−2.37571e8	−0.179250
$405$	2.65094e8	0.198293
$406$	1.17863e8	0.0874051
$407$	2.38505e9	1.75354
$408$	−9.43596e7	−0.0687821
$409$	−1.12841e9	−0.815523	−0.407761	−	0.913089i	$-0.633690\pi$
$409$	−1.12841e9	−0.815523	−0.407761	+	0.913089i	$0.633690\pi$
$410$	3.14748e9	2.25538
$411$	8.37609e7	0.0595107
$412$	−7.42171e7	−0.0522834
$413$	−4.50067e8	−0.314378
$414$	−2.40924e7	−0.0166870
$415$	2.71684e9	1.86593
$416$	−7.19913e7	−0.0490290
$417$	−2.89262e7	−0.0195351
$418$	9.57926e8	0.641527
$419$	1.41326e9	0.938584	0.469292	−	0.883043i	$-0.344509\pi$
$419$	1.41326e9	0.938584	0.469292	+	0.883043i	$0.344509\pi$
$420$	−2.95653e8	−0.194720
$421$	2.36717e9	1.54612	0.773058	−	0.634336i	$-0.218726\pi$
$421$	2.36717e9	1.54612	0.773058	+	0.634336i	$0.218726\pi$
$422$	−3.21843e8	−0.208474
$423$	−6.21218e7	−0.0399073
$424$	1.09851e8	0.0699877
$425$	1.16514e9	0.736238
$426$	−4.91778e8	−0.308202
$427$	−2.56472e8	−0.159420
$428$	5.48159e8	0.337951
$429$	3.66119e8	0.223883
$430$	5.70783e8	0.346204
$431$	2.66247e9	1.60182	0.800912	−	0.598781i	$-0.204348\pi$
$431$	2.66247e9	1.60182	0.800912	+	0.598781i	$0.204348\pi$
$432$	8.06216e7	0.0481125
$433$	−6.83520e7	−0.0404616	−0.0202308	−	0.999795i	$-0.506440\pi$
$433$	−6.83520e7	−0.0404616	−0.0202308	+	0.999795i	$0.506440\pi$
$434$	8.88567e7	0.0521766
$435$	5.78499e8	0.336969
$436$	−9.07743e8	−0.524518
$437$	−8.01449e7	−0.0459400
$438$	−1.35918e9	−0.772888
$439$	1.92349e9	1.08509	0.542543	−	0.840028i	$-0.317462\pi$
$439$	1.92349e9	1.08509	0.542543	+	0.840028i	$0.317462\pi$
$440$	−1.57631e9	−0.882182
$441$	8.57661e7	0.0476190
$442$	−1.19970e8	−0.0660837
$443$	−3.32461e9	−1.81689	−0.908444	−	0.418007i	$-0.862729\pi$
$443$	−3.32461e9	−1.81689	−0.908444	+	0.418007i	$0.862729\pi$
$444$	6.67749e8	0.362053
$445$	−1.31701e9	−0.708482
$446$	5.79984e8	0.309560
$447$	−2.01384e9	−1.06647
$448$	−8.99154e7	−0.0472456
$449$	3.94888e8	0.205879	0.102940	−	0.994688i	$-0.467175\pi$
$449$	3.94888e8	0.205879	0.102940	+	0.994688i	$0.467175\pi$
$450$	−9.95507e8	−0.514992
$451$	−4.86807e9	−2.49884
$452$	−1.09662e9	−0.558563
$453$	−1.09986e9	−0.555895
$454$	−7.98126e8	−0.400291
$455$	−3.75897e8	−0.187081
$456$	2.68193e8	0.132456
$457$	7.57820e8	0.371415	0.185708	−	0.982605i	$-0.440542\pi$
$457$	7.57820e8	0.371415	0.185708	+	0.982605i	$0.440542\pi$
$458$	−4.38834e8	−0.213438
$459$	1.34352e8	0.0648484
$460$	1.31882e8	0.0631734
$461$	−2.98858e9	−1.42073	−0.710365	−	0.703834i	$-0.751470\pi$
$461$	−2.98858e9	−1.42073	−0.710365	+	0.703834i	$0.751470\pi$
$462$	4.57273e8	0.215739
$463$	−8.22388e8	−0.385073	−0.192537	−	0.981290i	$-0.561671\pi$
$463$	−8.22388e8	−0.385073	−0.192537	+	0.981290i	$0.561671\pi$
$464$	1.75936e8	0.0817600
$465$	4.36128e8	0.201154
$466$	−1.76736e9	−0.809050
$467$	−3.46314e9	−1.57348	−0.786741	−	0.617284i	$-0.788233\pi$
$467$	−3.46314e9	−1.57348	−0.786741	+	0.617284i	$0.788233\pi$
$468$	1.02503e8	0.0462250
$469$	−1.02775e9	−0.460025
$470$	3.40057e8	0.151081
$471$	2.18507e9	0.963590
$472$	−6.71820e8	−0.294074
$473$	−8.82804e8	−0.383575
$474$	−1.26766e8	−0.0546739
$475$	−3.31162e9	−1.41779
$476$	−1.49840e8	−0.0636799
$477$	−1.56408e8	−0.0659851
$478$	−1.71025e9	−0.716246
$479$	−3.24230e7	−0.0134797	−0.00673984	−	0.999977i	$-0.502145\pi$
$479$	−3.24230e7	−0.0134797	−0.00673984	+	0.999977i	$0.502145\pi$
$480$	−4.41325e8	−0.182144
$481$	8.48984e8	0.347850
$482$	−2.82678e9	−1.14981
$483$	−3.82578e7	−0.0154492
$484$	1.19083e9	0.477411
$485$	−6.89281e9	−2.74347
$486$	−1.14791e8	−0.0453609
$487$	−1.05645e9	−0.414475	−0.207237	−	0.978291i	$-0.566447\pi$
$487$	−1.05645e9	−0.414475	−0.207237	+	0.978291i	$0.566447\pi$
$488$	−3.82839e8	−0.149124
$489$	−1.81339e9	−0.701311
$490$	−4.69486e8	−0.180275
$491$	−1.18040e9	−0.450034	−0.225017	−	0.974355i	$-0.572244\pi$
$491$	−1.18040e9	−0.450034	−0.225017	+	0.974355i	$0.572244\pi$
$492$	−1.36293e9	−0.515935
$493$	2.93188e8	0.110200
$494$	3.40984e8	0.127259
$495$	2.24440e9	0.831729
$496$	1.32637e8	0.0488068
$497$	−7.80926e8	−0.285340
$498$	−1.17645e9	−0.426846
$499$	−1.23620e9	−0.445385	−0.222693	−	0.974889i	$-0.571485\pi$
$499$	−1.23620e9	−0.445385	−0.222693	+	0.974889i	$0.571485\pi$
$500$	2.95533e9	1.05733
$501$	1.24014e9	0.440594
$502$	−4.16861e8	−0.147072
$503$	−7.91589e8	−0.277340	−0.138670	−	0.990339i	$-0.544283\pi$
$503$	−7.91589e8	−0.277340	−0.138670	+	0.990339i	$0.544283\pi$
$504$	1.28024e8	0.0445435
$505$	−1.85164e9	−0.639790
$506$	−2.03976e8	−0.0699928
$507$	1.30324e8	0.0444116
$508$	−7.71317e8	−0.261042
$509$	2.40309e9	0.807714	0.403857	−	0.914822i	$-0.367669\pi$
$509$	2.40309e9	0.807714	0.403857	+	0.914822i	$0.367669\pi$
$510$	−7.35446e8	−0.245502
$511$	−2.15832e9	−0.715555
$512$	−1.34218e8	−0.0441942
$513$	−3.81861e8	−0.124880
$514$	7.12405e8	0.231396
$515$	−5.78454e8	−0.186614
$516$	−2.47161e8	−0.0791965
$517$	−5.25950e8	−0.167389
$518$	1.06036e9	0.335196
$519$	−9.54772e8	−0.299788
$520$	−5.61106e8	−0.174998
$521$	2.85430e8	0.0884234	0.0442117	−	0.999022i	$-0.485922\pi$
$521$	2.85430e8	0.0884234	0.0442117	+	0.999022i	$0.485922\pi$
$522$	−2.50502e8	−0.0770841
$523$	3.75304e9	1.14717	0.573584	−	0.819147i	$-0.305553\pi$
$523$	3.75304e9	1.14717	0.573584	+	0.819147i	$0.305553\pi$
$524$	−1.84960e9	−0.561588
$525$	−1.58083e9	−0.476790
$526$	1.07077e9	0.320808
$527$	2.21034e8	0.0657842
$528$	6.82577e8	0.201806
$529$	−3.38776e9	−0.994988
$530$	8.56184e8	0.249805
$531$	9.56557e8	0.277255
$532$	4.25881e8	0.122630
$533$	−1.73284e9	−0.495694
$534$	5.70293e8	0.162070
$535$	4.27239e9	1.20624
$536$	−1.53413e9	−0.430314
$537$	1.11807e9	0.311573
$538$	2.31773e8	0.0641688
$539$	7.26133e8	0.199736
$540$	6.28371e8	0.171727
$541$	3.39678e9	0.922311	0.461156	−	0.887319i	$-0.347435\pi$
$541$	3.39678e9	0.922311	0.461156	+	0.887319i	$0.347435\pi$
$542$	4.25568e9	1.14808
$543$	1.42391e9	0.381666
$544$	−2.23667e8	−0.0595671
$545$	−7.07502e9	−1.87215
$546$	1.62771e8	0.0427960
$547$	−1.58435e9	−0.413899	−0.206950	−	0.978352i	$-0.566354\pi$
$547$	−1.58435e9	−0.413899	−0.206950	+	0.978352i	$0.566354\pi$
$548$	1.98544e8	0.0515378
$549$	5.45097e8	0.140595
$550$	−8.42839e9	−2.16011
$551$	−8.33312e8	−0.212215
$552$	−5.71078e7	−0.0144514
$553$	−2.01300e8	−0.0506182
$554$	−3.11275e9	−0.777787
$555$	5.20448e9	1.29227
$556$	−6.85659e7	−0.0169179
$557$	8.03514e8	0.197015	0.0985077	−	0.995136i	$-0.468593\pi$
$557$	8.03514e8	0.197015	0.0985077	+	0.995136i	$0.468593\pi$
$558$	−1.88853e8	−0.0460155
$559$	−3.14244e8	−0.0760896
$560$	−7.00808e8	−0.168632
$561$	1.13748e9	0.272003
$562$	3.42508e9	0.813942
$563$	−3.02961e9	−0.715497	−0.357748	−	0.933818i	$-0.616455\pi$
$563$	−3.02961e9	−0.715497	−0.357748	+	0.933818i	$0.616455\pi$
$564$	−1.47252e8	−0.0345608
$565$	−8.54715e9	−1.99366
$566$	−5.51177e9	−1.27771
$567$	−1.82284e8	−0.0419961
$568$	−1.16570e9	−0.266911
$569$	2.13613e9	0.486110	0.243055	−	0.970013i	$-0.421850\pi$
$569$	2.13613e9	0.486110	0.243055	+	0.970013i	$0.421850\pi$
$570$	2.09032e9	0.472770
$571$	−2.39562e8	−0.0538507	−0.0269254	−	0.999637i	$-0.508572\pi$
$571$	−2.39562e8	−0.0538507	−0.0269254	+	0.999637i	$0.508572\pi$
$572$	8.67837e8	0.193888
$573$	9.12618e8	0.202651
$574$	−2.16428e9	−0.477663
$575$	7.05162e8	0.154686
$576$	1.91103e8	0.0416667
$577$	2.32360e9	0.503555	0.251777	−	0.967785i	$-0.418985\pi$
$577$	2.32360e9	0.503555	0.251777	+	0.967785i	$0.418985\pi$
$578$	2.90998e9	0.626819
$579$	−1.81631e8	−0.0388880
$580$	1.37126e9	0.291824
$581$	−1.86816e9	−0.395183
$582$	2.98473e9	0.627589
$583$	−1.32422e9	−0.276771
$584$	−3.22175e9	−0.669340
$585$	7.98918e8	0.164990
$586$	5.23380e9	1.07442
$587$	8.74855e9	1.78527	0.892633	−	0.450785i	$-0.148856\pi$
$587$	8.74855e9	1.78527	0.892633	+	0.450785i	$0.148856\pi$
$588$	2.03297e8	0.0412393
$589$	−6.28231e8	−0.126682
$590$	−5.23622e9	−1.04963
$591$	−6.43893e8	−0.128309
$592$	1.58281e9	0.313547
$593$	−2.07624e9	−0.408871	−0.204435	−	0.978880i	$-0.565536\pi$
$593$	−2.07624e9	−0.408871	−0.204435	+	0.978880i	$0.565536\pi$
$594$	−9.71873e8	−0.190264
$595$	−1.16786e9	−0.227291
$596$	−4.77355e9	−0.923590
$597$	3.01065e9	0.579096
$598$	−7.26076e7	−0.0138844
$599$	9.79152e8	0.186147	0.0930735	−	0.995659i	$-0.470331\pi$
$599$	9.79152e8	0.186147	0.0930735	+	0.995659i	$0.470331\pi$
$600$	−2.35972e9	−0.445996
$601$	2.82278e9	0.530415	0.265208	−	0.964191i	$-0.414559\pi$
$601$	2.82278e9	0.530415	0.265208	+	0.964191i	$0.414559\pi$
$602$	−3.92483e8	−0.0733217
$603$	2.18434e9	0.405704
$604$	−2.60707e9	−0.481420
$605$	9.28145e9	1.70401
$606$	8.01801e8	0.146357
$607$	−1.00898e10	−1.83113	−0.915567	−	0.402166i	$-0.868258\pi$
$607$	−1.00898e10	−1.83113	−0.915567	+	0.402166i	$0.868258\pi$
$608$	6.35717e8	0.114710
$609$	−3.97788e8	−0.0713660
$610$	−2.98388e9	−0.532263
$611$	−1.87218e8	−0.0332049
$612$	3.18464e8	0.0561604
$613$	−1.69444e9	−0.297108	−0.148554	−	0.988904i	$-0.547462\pi$
$613$	−1.69444e9	−0.297108	−0.148554	+	0.988904i	$0.547462\pi$
$614$	4.23197e9	0.737824
$615$	−1.06228e10	−1.84151
$616$	1.08391e9	0.186836
$617$	1.51863e9	0.260287	0.130144	−	0.991495i	$-0.458456\pi$
$617$	1.51863e9	0.260287	0.130144	+	0.991495i	$0.458456\pi$
$618$	2.50483e8	0.0426892
$619$	−5.38514e9	−0.912598	−0.456299	−	0.889826i	$-0.650825\pi$
$619$	−5.38514e9	−0.912598	−0.456299	+	0.889826i	$0.650825\pi$
$620$	1.03379e9	0.174205
$621$	8.13117e7	0.0136249
$622$	−7.51921e9	−1.25287
$623$	9.05604e8	0.150048
$624$	2.42971e8	0.0400320
$625$	9.69835e9	1.58898
$626$	−4.66897e9	−0.760695
$627$	−3.23300e9	−0.523804
$628$	5.17943e9	0.834493
$629$	2.63768e9	0.422615
$630$	9.97830e8	0.158988
$631$	7.39102e9	1.17112	0.585561	−	0.810629i	$-0.300874\pi$
$631$	7.39102e9	1.17112	0.585561	+	0.810629i	$0.300874\pi$
$632$	−3.00483e8	−0.0473489
$633$	1.08622e9	0.170218
$634$	6.95995e9	1.08466
$635$	−6.01171e9	−0.931729
$636$	−3.70746e8	−0.0571447
$637$	2.58475e8	0.0396214
$638$	−2.12086e9	−0.323325
$639$	1.65975e9	0.251646
$640$	−1.04610e9	−0.157741
$641$	−6.81355e9	−1.02181	−0.510905	−	0.859637i	$-0.670690\pi$
$641$	−6.81355e9	−1.02181	−0.510905	+	0.859637i	$0.670690\pi$
$642$	−1.85004e9	−0.275936
$643$	1.24650e9	0.184907	0.0924537	−	0.995717i	$-0.470529\pi$
$643$	1.24650e9	0.184907	0.0924537	+	0.995717i	$0.470529\pi$
$644$	−9.06851e7	−0.0133794
$645$	−1.92639e9	−0.282674
$646$	1.05939e9	0.154612
$647$	−8.94377e8	−0.129824	−0.0649121	−	0.997891i	$-0.520677\pi$
$647$	−8.94377e8	−0.129824	−0.0649121	+	0.997891i	$0.520677\pi$
$648$	−2.72098e8	−0.0392837
$649$	8.09862e9	1.16293
$650$	−3.00018e9	−0.428499
$651$	−2.99891e8	−0.0426020
$652$	−4.29841e9	−0.607354
$653$	−8.74854e7	−0.0122953	−0.00614766	−	0.999981i	$-0.501957\pi$
$653$	−8.74854e7	−0.0122953	−0.00614766	+	0.999981i	$0.501957\pi$
$654$	3.06363e9	0.428267
$655$	−1.44159e10	−2.00446
$656$	−3.23064e9	−0.446813
$657$	4.58722e9	0.631060
$658$	−2.33830e8	−0.0319971
$659$	5.73683e9	0.780859	0.390430	−	0.920633i	$-0.372326\pi$
$659$	5.73683e9	0.780859	0.390430	+	0.920633i	$0.372326\pi$
$660$	5.32006e9	0.720299
$661$	2.88009e9	0.387884	0.193942	−	0.981013i	$-0.437873\pi$
$661$	2.88009e9	0.387884	0.193942	+	0.981013i	$0.437873\pi$
$662$	3.23704e9	0.433656
$663$	4.04899e8	0.0539571
$664$	−2.78862e9	−0.369659
$665$	3.31935e9	0.437700
$666$	−2.25365e9	−0.295615
$667$	1.77442e8	0.0231534
$668$	2.93959e9	0.381566
$669$	−1.95745e9	−0.252754
$670$	−1.19571e10	−1.53591
$671$	4.61503e9	0.589719
$672$	3.03464e8	0.0385758
$673$	−4.45751e9	−0.563689	−0.281844	−	0.959460i	$-0.590946\pi$
$673$	−4.45751e9	−0.563689	−0.281844	+	0.959460i	$0.590946\pi$
$674$	5.76113e9	0.724766
$675$	3.35984e9	0.420489
$676$	3.08916e8	0.0384615
$677$	−4.92508e9	−0.610033	−0.305017	−	0.952347i	$-0.598662\pi$
$677$	−4.92508e9	−0.610033	−0.305017	+	0.952347i	$0.598662\pi$
$678$	3.70109e9	0.456064
$679$	4.73965e9	0.581034
$680$	−1.74328e9	−0.212611
$681$	2.69367e9	0.326836
$682$	−1.59891e9	−0.193009
$683$	8.09008e9	0.971585	0.485792	−	0.874074i	$-0.338531\pi$
$683$	8.09008e9	0.971585	0.485792	+	0.874074i	$0.338531\pi$
$684$	−9.05151e8	−0.108150
$685$	1.54747e9	0.183952
$686$	3.22829e8	0.0381802
$687$	1.48106e9	0.174271
$688$	−5.85863e8	−0.0685862
$689$	−4.71371e8	−0.0549029
$690$	−4.45103e8	−0.0515809
$691$	1.38080e10	1.59205	0.796027	−	0.605261i	$-0.206931\pi$
$691$	1.38080e10	1.59205	0.796027	+	0.605261i	$0.206931\pi$
$692$	−2.26316e9	−0.259624
$693$	−1.54330e9	−0.176150
$694$	−8.24213e9	−0.936012
$695$	−5.34408e8	−0.0603846
$696$	−5.93783e8	−0.0667567
$697$	−5.38371e9	−0.602236
$698$	3.21741e9	0.358106
$699$	5.96485e9	0.660586
$700$	−3.74715e9	−0.412912
$701$	−9.81601e8	−0.107627	−0.0538136	−	0.998551i	$-0.517138\pi$
$701$	−9.81601e8	−0.107627	−0.0538136	+	0.998551i	$0.517138\pi$
$702$	−3.45948e8	−0.0377426
$703$	−7.49692e9	−0.813841
$704$	1.61796e9	0.174769
$705$	−1.14769e9	−0.123357
$706$	−1.12067e10	−1.19856
$707$	1.27323e9	0.135500
$708$	2.26739e9	0.240110
$709$	1.60805e10	1.69448	0.847242	−	0.531207i	$-0.178261\pi$
$709$	1.60805e10	1.69448	0.847242	+	0.531207i	$0.178261\pi$
$710$	−9.08553e9	−0.952678
$711$	4.27836e8	0.0446410
$712$	1.35181e9	0.140357
$713$	1.33773e8	0.0138215
$714$	5.05709e8	0.0519944
$715$	6.76399e9	0.692041
$716$	2.65025e9	0.269830
$717$	5.77210e9	0.584812
$718$	−1.27220e10	−1.28269
$719$	−1.14584e10	−1.14967	−0.574833	−	0.818271i	$-0.694933\pi$
$719$	−1.14584e10	−1.14967	−0.574833	+	0.818271i	$0.694933\pi$
$720$	1.48947e9	0.148720
$721$	3.97757e8	0.0395225
$722$	4.13993e9	0.409367
$723$	9.54039e9	0.938819
$724$	3.37520e9	0.330533
$725$	7.33197e9	0.714558
$726$	−4.01907e9	−0.389805
$727$	5.87751e9	0.567313	0.283657	−	0.958926i	$-0.408452\pi$
$727$	5.87751e9	0.567313	0.283657	+	0.958926i	$0.408452\pi$
$728$	3.85828e8	0.0370625
$729$	3.87420e8	0.0370370
$730$	−2.51106e10	−2.38906
$731$	−9.76313e8	−0.0924439
$732$	1.29208e9	0.121759
$733$	1.52013e10	1.42566	0.712832	−	0.701335i	$-0.247412\pi$
$733$	1.52013e10	1.42566	0.712832	+	0.701335i	$0.247412\pi$
$734$	1.23476e10	1.15251
$735$	1.58452e9	0.147194
$736$	−1.35367e8	−0.0125153
$737$	1.84935e10	1.70170
$738$	4.59988e9	0.421259
$739$	−1.49863e10	−1.36596	−0.682982	−	0.730435i	$-0.739317\pi$
$739$	−1.49863e10	−1.36596	−0.682982	+	0.730435i	$0.739317\pi$
$740$	1.23366e10	1.11914
$741$	−1.15082e9	−0.103907
$742$	−5.88731e8	−0.0529058
$743$	−5.27323e9	−0.471645	−0.235823	−	0.971796i	$-0.575778\pi$
$743$	−5.27323e9	−0.471645	−0.235823	+	0.971796i	$0.575778\pi$
$744$	−4.47651e8	−0.0398506
$745$	−3.72054e10	−3.29655
$746$	3.08589e9	0.272141
$747$	3.97052e9	0.348518
$748$	2.69625e9	0.235562
$749$	−2.93779e9	−0.255467
$750$	−9.97424e9	−0.863307
$751$	−1.77572e10	−1.52980	−0.764901	−	0.644148i	$-0.777212\pi$
$751$	−1.77572e10	−1.52980	−0.764901	+	0.644148i	$0.777212\pi$
$752$	−3.49041e8	−0.0299305
$753$	1.40691e9	0.120083
$754$	−7.54942e8	−0.0641378
$755$	−2.03197e10	−1.71832
$756$	−4.32081e8	−0.0363696
$757$	3.27789e9	0.274636	0.137318	−	0.990527i	$-0.456152\pi$
$757$	3.27789e9	0.274636	0.137318	+	0.990527i	$0.456152\pi$
$758$	−1.58195e10	−1.31932
$759$	6.88420e8	0.0571489
$760$	4.95483e9	0.409431
$761$	8.74906e9	0.719640	0.359820	−	0.933022i	$-0.382838\pi$
$761$	8.74906e9	0.719640	0.359820	+	0.933022i	$0.382838\pi$
$762$	2.60320e9	0.213140
$763$	4.86493e9	0.396498
$764$	2.16324e9	0.175501
$765$	2.48213e9	0.200452
$766$	−1.01005e10	−0.811978
$767$	2.88279e9	0.230690
$768$	4.52985e8	0.0360844
$769$	−1.92851e10	−1.52925	−0.764627	−	0.644473i	$-0.777077\pi$
$769$	−1.92851e10	−1.52925	−0.764627	+	0.644473i	$0.777077\pi$
$770$	8.44806e9	0.666867
$771$	−2.40437e9	−0.188934
$772$	−4.30533e8	−0.0336780
$773$	−2.89970e9	−0.225800	−0.112900	−	0.993606i	$-0.536014\pi$
$773$	−2.89970e9	−0.225800	−0.112900	+	0.993606i	$0.536014\pi$
$774$	8.34169e8	0.0646637
$775$	5.52755e9	0.426557
$776$	7.07492e9	0.543508
$777$	−3.57872e9	−0.273687
$778$	−1.45930e10	−1.11100
$779$	1.53018e10	1.15974
$780$	1.89373e9	0.142885
$781$	1.40522e10	1.05552
$782$	−2.25582e8	−0.0168687
$783$	8.45444e8	0.0629389
$784$	4.81890e8	0.0357143
$785$	4.03689e10	2.97853
$786$	6.24239e9	0.458534
$787$	−2.80396e7	−0.00205050	−0.00102525	−	0.999999i	$-0.500326\pi$
$787$	−2.80396e7	−0.00205050	−0.00102525	+	0.999999i	$0.500326\pi$
$788$	−1.52627e9	−0.111119
$789$	−3.61384e9	−0.261939
$790$	−2.34199e9	−0.169001
$791$	5.87720e9	0.422234
$792$	−2.30370e9	−0.164774
$793$	1.64277e9	0.116982
$794$	1.75883e10	1.24696
$795$	−2.88962e9	−0.203965
$796$	7.13637e9	0.501512
$797$	−7.65982e9	−0.535938	−0.267969	−	0.963428i	$-0.586352\pi$
$797$	−7.65982e9	−0.535938	−0.267969	+	0.963428i	$0.586352\pi$
$798$	−1.43735e9	−0.100127
$799$	−5.81660e8	−0.0403418
$800$	−5.59341e9	−0.386244
$801$	−1.92474e9	−0.132330
$802$	−1.32879e9	−0.0909588
$803$	3.88374e10	2.64695
$804$	5.17769e9	0.351350
$805$	−7.06807e8	−0.0477546
$806$	−5.69149e8	−0.0382872
$807$	−7.82232e8	−0.0523936
$808$	1.90057e9	0.126749
$809$	−9.25707e9	−0.614687	−0.307343	−	0.951599i	$-0.599440\pi$
$809$	−9.25707e9	−0.614687	−0.307343	+	0.951599i	$0.599440\pi$
$810$	−2.12075e9	−0.140214
$811$	−1.71319e9	−0.112780	−0.0563902	−	0.998409i	$-0.517959\pi$
$811$	−1.71319e9	−0.112780	−0.0563902	+	0.998409i	$0.517959\pi$
$812$	−9.42905e8	−0.0618047
$813$	−1.43629e10	−0.937402
$814$	−1.90804e10	−1.23994
$815$	−3.35022e10	−2.16781
$816$	7.54877e8	0.0486363
$817$	2.77492e9	0.178022
$818$	9.02729e9	0.576662
$819$	−5.49353e8	−0.0349428
$820$	−2.51799e10	−1.59480
$821$	1.69723e10	1.07039	0.535193	−	0.844730i	$-0.320239\pi$
$821$	1.69723e10	1.07039	0.535193	+	0.844730i	$0.320239\pi$
$822$	−6.70087e8	−0.0420804
$823$	2.29672e9	0.143618	0.0718089	−	0.997418i	$-0.477123\pi$
$823$	2.29672e9	0.143618	0.0718089	+	0.997418i	$0.477123\pi$
$824$	5.93737e8	0.0369700
$825$	2.84458e10	1.76372
$826$	3.60054e9	0.222299
$827$	−2.85084e10	−1.75269	−0.876343	−	0.481687i	$-0.840024\pi$
$827$	−2.85084e10	−1.75269	−0.876343	+	0.481687i	$0.840024\pi$
$828$	1.92739e8	0.0117995
$829$	−5.37033e9	−0.327386	−0.163693	−	0.986511i	$-0.552341\pi$
$829$	−5.37033e9	−0.327386	−0.163693	+	0.986511i	$0.552341\pi$
$830$	−2.17348e10	−1.31942
$831$	1.05055e10	0.635060
$832$	5.75930e8	0.0346688
$833$	8.03047e8	0.0481375
$834$	2.31410e8	0.0138134
$835$	2.29114e10	1.36191
$836$	−7.66340e9	−0.453628
$837$	6.37378e8	0.0375715
$838$	−1.13061e10	−0.663679
$839$	−4.82569e9	−0.282093	−0.141047	−	0.990003i	$-0.545047\pi$
$839$	−4.82569e9	−0.282093	−0.141047	+	0.990003i	$0.545047\pi$
$840$	2.36523e9	0.137688
$841$	−1.54049e10	−0.893045
$842$	−1.89374e10	−1.09327
$843$	−1.15596e10	−0.664581
$844$	2.57475e9	0.147413
$845$	2.40771e9	0.137280
$846$	4.96975e8	0.0282188
$847$	−6.38213e9	−0.360889
$848$	−8.78805e8	−0.0494888
$849$	1.86022e10	1.04325
$850$	−9.32115e9	−0.520599
$851$	1.59636e9	0.0887928
$852$	3.93423e9	0.217932
$853$	−1.79604e10	−0.990819	−0.495410	−	0.868660i	$-0.664982\pi$
$853$	−1.79604e10	−0.990819	−0.495410	+	0.868660i	$0.664982\pi$
$854$	2.05178e9	0.112727
$855$	−7.05482e9	−0.386015
$856$	−4.38527e9	−0.238967
$857$	−2.91368e10	−1.58128	−0.790641	−	0.612280i	$-0.790252\pi$
$857$	−2.91368e10	−1.58128	−0.790641	+	0.612280i	$0.790252\pi$
$858$	−2.92895e9	−0.158309
$859$	2.28244e10	1.22864	0.614318	−	0.789059i	$-0.289431\pi$
$859$	2.28244e10	1.22864	0.614318	+	0.789059i	$0.289431\pi$
$860$	−4.56627e9	−0.244803
$861$	7.30444e9	0.390010
$862$	−2.12998e10	−1.13266
$863$	−1.57786e10	−0.835663	−0.417832	−	0.908525i	$-0.637210\pi$
$863$	−1.57786e10	−0.835663	−0.417832	+	0.908525i	$0.637210\pi$
$864$	−6.44973e8	−0.0340207
$865$	−1.76393e10	−0.926668
$866$	5.46816e8	0.0286107
$867$	−9.82118e9	−0.511796
$868$	−7.10853e8	−0.0368944
$869$	3.62225e9	0.187244
$870$	−4.62799e9	−0.238273
$871$	6.58297e9	0.337566
$872$	7.26194e9	0.370890
$873$	−1.00735e10	−0.512424
$874$	6.41159e8	0.0324845
$875$	−1.58387e10	−0.799267
$876$	1.08734e10	0.546514
$877$	2.19891e10	1.10080	0.550402	−	0.834900i	$-0.314475\pi$
$877$	2.19891e10	1.10080	0.550402	+	0.834900i	$0.314475\pi$
$878$	−1.53879e10	−0.767272
$879$	−1.76641e10	−0.877263
$880$	1.26105e10	0.623797
$881$	2.68935e10	1.32505	0.662525	−	0.749040i	$-0.269485\pi$
$881$	2.68935e10	1.32505	0.662525	+	0.749040i	$0.269485\pi$
$882$	−6.86129e8	−0.0336718
$883$	−2.89596e10	−1.41556	−0.707782	−	0.706431i	$-0.750304\pi$
$883$	−2.89596e10	−1.41556	−0.707782	+	0.706431i	$0.750304\pi$
$884$	9.59760e8	0.0467283
$885$	1.76722e10	0.857018
$886$	2.65969e10	1.28473
$887$	−1.08578e10	−0.522405	−0.261203	−	0.965284i	$-0.584119\pi$
$887$	−1.08578e10	−0.522405	−0.261203	+	0.965284i	$0.584119\pi$
$888$	−5.34199e9	−0.256010
$889$	4.13378e9	0.197329
$890$	1.05361e10	0.500972
$891$	3.28007e9	0.155350
$892$	−4.63987e9	−0.218892
$893$	1.65322e9	0.0776873
$894$	1.61107e10	0.754108
$895$	2.06562e10	0.963098
$896$	7.19323e8	0.0334077
$897$	2.45051e8	0.0113366
$898$	−3.15911e9	−0.145578
$899$	1.39091e9	0.0638470
$900$	7.96406e9	0.364154
$901$	−1.46449e9	−0.0667034
$902$	3.89446e10	1.76695
$903$	1.32463e9	0.0598670
$904$	8.77296e9	0.394963
$905$	2.63066e10	1.17976
$906$	8.79887e9	0.393077
$907$	−1.19084e10	−0.529942	−0.264971	−	0.964256i	$-0.585362\pi$
$907$	−1.19084e10	−0.529942	−0.264971	+	0.964256i	$0.585362\pi$
$908$	6.38501e9	0.283049
$909$	−2.70608e9	−0.119500
$910$	3.00718e9	0.132286
$911$	1.72984e10	0.758038	0.379019	−	0.925389i	$-0.376261\pi$
$911$	1.72984e10	0.758038	0.379019	+	0.925389i	$0.376261\pi$
$912$	−2.14554e9	−0.0936603
$913$	3.36162e10	1.46184
$914$	−6.06256e9	−0.262630
$915$	1.00706e10	0.434591
$916$	3.51067e9	0.150923
$917$	9.91268e9	0.424520
$918$	−1.07482e9	−0.0458548
$919$	−2.53872e10	−1.07897	−0.539486	−	0.841995i	$-0.681381\pi$
$919$	−2.53872e10	−1.07897	−0.539486	+	0.841995i	$0.681381\pi$
$920$	−1.05506e9	−0.0446703
$921$	−1.42829e10	−0.602431
$922$	2.39086e10	1.00461
$923$	5.00202e9	0.209382
$924$	−3.65819e9	−0.152551
$925$	6.59624e10	2.74031
$926$	6.57911e9	0.272288
$927$	−8.45380e8	−0.0348556
$928$	−1.40748e9	−0.0578130
$929$	−2.22121e10	−0.908940	−0.454470	−	0.890762i	$-0.650171\pi$
$929$	−2.22121e10	−0.908940	−0.454470	+	0.890762i	$0.650171\pi$
$930$	−3.48903e9	−0.142237
$931$	−2.28245e9	−0.0926997
$932$	1.41389e10	0.572084
$933$	2.53773e10	1.02296
$934$	2.77052e10	1.11262
$935$	2.10148e10	0.840784
$936$	−8.20026e8	−0.0326860
$937$	−1.89528e10	−0.752636	−0.376318	−	0.926491i	$-0.622810\pi$
$937$	−1.89528e10	−0.752636	−0.376318	+	0.926491i	$0.622810\pi$
$938$	8.22197e9	0.325287
$939$	1.57578e10	0.621105
$940$	−2.72045e9	−0.106830
$941$	−1.26156e10	−0.493563	−0.246782	−	0.969071i	$-0.579373\pi$
$941$	−1.26156e10	−0.493563	−0.246782	+	0.969071i	$0.579373\pi$
$942$	−1.74806e10	−0.681361
$943$	−3.25830e9	−0.126532
$944$	5.37456e9	0.207941
$945$	−3.36767e9	−0.129813
$946$	7.06243e9	0.271229
$947$	−1.10895e10	−0.424315	−0.212157	−	0.977236i	$-0.568049\pi$
$947$	−1.10895e10	−0.424315	−0.212157	+	0.977236i	$0.568049\pi$
$948$	1.01413e9	0.0386603
$949$	1.38246e10	0.525074
$950$	2.64930e10	1.00253
$951$	−2.34898e10	−0.885622
$952$	1.19872e9	0.0450285
$953$	−2.54399e10	−0.952118	−0.476059	−	0.879413i	$-0.657935\pi$
$953$	−2.54399e10	−0.952118	−0.476059	+	0.879413i	$0.657935\pi$
$954$	1.25127e9	0.0466585
$955$	1.68605e10	0.626409
$956$	1.36820e10	0.506462
$957$	7.15790e9	0.263994
$958$	2.59384e8	0.00953157
$959$	−1.06407e9	−0.0389589
$960$	3.53060e9	0.128795
$961$	−2.64640e10	−0.961886
$962$	−6.79187e9	−0.245967
$963$	6.24387e9	0.225300
$964$	2.26143e10	0.813041
$965$	−3.35561e9	−0.120206
$966$	3.06062e8	0.0109242
$967$	−4.62477e10	−1.64474	−0.822371	−	0.568951i	$-0.807349\pi$
$967$	−4.62477e10	−1.64474	−0.822371	+	0.568951i	$0.807349\pi$
$968$	−9.52667e9	−0.337581
$969$	−3.57544e9	−0.126240
$970$	5.51425e10	1.93993
$971$	2.06750e9	0.0724734	0.0362367	−	0.999343i	$-0.488463\pi$
$971$	2.06750e9	0.0724734	0.0362367	+	0.999343i	$0.488463\pi$
$972$	9.18330e8	0.0320750
$973$	3.67470e8	0.0127887
$974$	8.45161e9	0.293078
$975$	1.01256e10	0.349868
$976$	3.06271e9	0.105446
$977$	4.86530e10	1.66908	0.834542	−	0.550944i	$-0.185732\pi$
$977$	4.86530e10	1.66908	0.834542	+	0.550944i	$0.185732\pi$
$978$	1.45071e10	0.495902
$979$	−1.62957e10	−0.555051
$980$	3.75589e9	0.127474
$981$	−1.03398e10	−0.349678
$982$	9.44323e9	0.318222
$983$	−1.27022e10	−0.426522	−0.213261	−	0.976995i	$-0.568409\pi$
$983$	−1.27022e10	−0.426522	−0.213261	+	0.976995i	$0.568409\pi$
$984$	1.09034e10	0.364821
$985$	−1.18958e10	−0.396614
$986$	−2.34550e9	−0.0779233
$987$	7.89177e8	0.0261255
$988$	−2.72787e9	−0.0899859
$989$	−5.90879e8	−0.0194228
$990$	−1.79552e10	−0.588122
$991$	−1.86238e10	−0.607871	−0.303935	−	0.952693i	$-0.598301\pi$
$991$	−1.86238e10	−0.607871	−0.303935	+	0.952693i	$0.598301\pi$
$992$	−1.06110e9	−0.0345116
$993$	−1.09250e10	−0.354078
$994$	6.24741e9	0.201766
$995$	5.56214e10	1.79003
$996$	9.41161e9	0.301826
$997$	1.30771e10	0.417905	0.208953	−	0.977926i	$-0.432995\pi$
$997$	1.30771e10	0.417905	0.208953	+	0.977926i	$0.432995\pi$
$998$	9.88958e9	0.314935
$999$	7.60607e9	0.241369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.k.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.k.1.5	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +498.821 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +498.821 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -3990.57 q^{10} +6172.03 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} +13468.2 q^{15} +4096.00 q^{16} +6825.78 q^{17} -5832.00 q^{18} -19400.5 q^{19} +31924.5 q^{20} -9261.00 q^{21} -49376.2 q^{22} +4131.06 q^{23} -13824.0 q^{24} +170697. q^{25} -17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +42953.0 q^{29} -107745. q^{30} +32382.2 q^{31} -32768.0 q^{32} +166645. q^{33} -54606.3 q^{34} -171096. q^{35} +46656.0 q^{36} +386429. q^{37} +155204. q^{38} +59319.0 q^{39} -255396. q^{40} -788731. q^{41} +74088.0 q^{42} -143033. q^{43} +395010. q^{44} +363641. q^{45} -33048.5 q^{46} -85215.1 q^{47} +110592. q^{48} +117649. q^{49} -1.36558e6 q^{50} +184296. q^{51} +140608. q^{52} -214552. q^{53} -157464. q^{54} +3.07874e6 q^{55} +175616. q^{56} -523814. q^{57} -343624. q^{58} +1.31215e6 q^{59} +861963. q^{60} +747732. q^{61} -259057. q^{62} -250047. q^{63} +262144. q^{64} +1.09591e6 q^{65} -1.33316e6 q^{66} +2.99635e6 q^{67} +436850. q^{68} +111539. q^{69} +1.36876e6 q^{70} +2.27675e6 q^{71} -373248. q^{72} +6.29248e6 q^{73} -3.09143e6 q^{74} +4.60883e6 q^{75} -1.24163e6 q^{76} -2.11701e6 q^{77} -474552. q^{78} +586881. q^{79} +2.04317e6 q^{80} +531441. q^{81} +6.30985e6 q^{82} +5.44653e6 q^{83} -592704. q^{84} +3.40484e6 q^{85} +1.14426e6 q^{86} +1.15973e6 q^{87} -3.16008e6 q^{88} -2.64024e6 q^{89} -2.90912e6 q^{90} -753571. q^{91} +264388. q^{92} +874319. q^{93} +681721. q^{94} -9.67740e6 q^{95} -884736. q^{96} -1.38182e7 q^{97} -941192. q^{98} +4.49941e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 509 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 4072 q^{10} + 958 q^{11} + 8640 q^{12} + 10985 q^{13} + 13720 q^{14} + 13743 q^{15} + 20480 q^{16} + 7864 q^{17} - 29160 q^{18} + 60173 q^{19} + 32576 q^{20} - 46305 q^{21} - 7664 q^{22} + 122869 q^{23} - 69120 q^{24} + 73722 q^{25} - 87880 q^{26} + 98415 q^{27} - 109760 q^{28} - 17317 q^{29} - 109944 q^{30} - 177665 q^{31} - 163840 q^{32} + 25866 q^{33} - 62912 q^{34} - 174587 q^{35} + 233280 q^{36} - 55136 q^{37} - 481384 q^{38} + 296595 q^{39} - 260608 q^{40} - 237570 q^{41} + 370440 q^{42} - 970601 q^{43} + 61312 q^{44} + 371061 q^{45} - 982952 q^{46} - 384035 q^{47} + 552960 q^{48} + 588245 q^{49} - 589776 q^{50} + 212328 q^{51} + 703040 q^{52} - 1977 q^{53} - 787320 q^{54} + 1520014 q^{55} + 878080 q^{56} + 1624671 q^{57} + 138536 q^{58} + 2057936 q^{59} + 879552 q^{60} - 723756 q^{61} + 1421320 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 1118273 q^{65} - 206928 q^{66} + 2695018 q^{67} + 503296 q^{68} + 3317463 q^{69} + 1396696 q^{70} + 7392916 q^{71} - 1866240 q^{72} + 8720441 q^{73} + 441088 q^{74} + 1990494 q^{75} + 3851072 q^{76} - 328594 q^{77} - 2372760 q^{78} + 4646419 q^{79} + 2084864 q^{80} + 2657205 q^{81} + 1900560 q^{82} + 17766733 q^{83} - 2963520 q^{84} + 16495320 q^{85} + 7764808 q^{86} - 467559 q^{87} - 490496 q^{88} + 4692321 q^{89} - 2968488 q^{90} - 3767855 q^{91} + 7863616 q^{92} - 4796955 q^{93} + 3072280 q^{94} - 2355945 q^{95} - 4423680 q^{96} + 15680305 q^{97} - 4705960 q^{98} + 698382 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 509 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 4072 * q^10 + 958 * q^11 + 8640 * q^12 + 10985 * q^13 + 13720 * q^14 + 13743 * q^15 + 20480 * q^16 + 7864 * q^17 - 29160 * q^18 + 60173 * q^19 + 32576 * q^20 - 46305 * q^21 - 7664 * q^22 + 122869 * q^23 - 69120 * q^24 + 73722 * q^25 - 87880 * q^26 + 98415 * q^27 - 109760 * q^28 - 17317 * q^29 - 109944 * q^30 - 177665 * q^31 - 163840 * q^32 + 25866 * q^33 - 62912 * q^34 - 174587 * q^35 + 233280 * q^36 - 55136 * q^37 - 481384 * q^38 + 296595 * q^39 - 260608 * q^40 - 237570 * q^41 + 370440 * q^42 - 970601 * q^43 + 61312 * q^44 + 371061 * q^45 - 982952 * q^46 - 384035 * q^47 + 552960 * q^48 + 588245 * q^49 - 589776 * q^50 + 212328 * q^51 + 703040 * q^52 - 1977 * q^53 - 787320 * q^54 + 1520014 * q^55 + 878080 * q^56 + 1624671 * q^57 + 138536 * q^58 + 2057936 * q^59 + 879552 * q^60 - 723756 * q^61 + 1421320 * q^62 - 1250235 * q^63 + 1310720 * q^64 + 1118273 * q^65 - 206928 * q^66 + 2695018 * q^67 + 503296 * q^68 + 3317463 * q^69 + 1396696 * q^70 + 7392916 * q^71 - 1866240 * q^72 + 8720441 * q^73 + 441088 * q^74 + 1990494 * q^75 + 3851072 * q^76 - 328594 * q^77 - 2372760 * q^78 + 4646419 * q^79 + 2084864 * q^80 + 2657205 * q^81 + 1900560 * q^82 + 17766733 * q^83 - 2963520 * q^84 + 16495320 * q^85 + 7764808 * q^86 - 467559 * q^87 - 490496 * q^88 + 4692321 * q^89 - 2968488 * q^90 - 3767855 * q^91 + 7863616 * q^92 - 4796955 * q^93 + 3072280 * q^94 - 2355945 * q^95 - 4423680 * q^96 + 15680305 * q^97 - 4705960 * q^98 + 698382 * q^99

Introduction

L-functions

Modular forms

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