LMFDB - Embedded newform 546.8.a.n.1.2

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:	$6$
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} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 $ x^6 - x^5 - 309949x^4 - 14548431x^3 + 25221499020x^2 + 1862570808000x - 308009568384000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$348.888$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3031.11 q^{10} +5295.06 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +10230.0 q^{15} +4096.00 q^{16} -1617.93 q^{17} -5832.00 q^{18} +42012.9 q^{19} -24248.8 q^{20} -9261.00 q^{21} -42360.5 q^{22} +101886. q^{23} +13824.0 q^{24} +65431.2 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +65637.5 q^{29} -81839.8 q^{30} +117455. q^{31} -32768.0 q^{32} -142967. q^{33} +12943.4 q^{34} -129959. q^{35} +46656.0 q^{36} +170252. q^{37} -336103. q^{38} -59319.0 q^{39} +193991. q^{40} +305044. q^{41} +74088.0 q^{42} +388832. q^{43} +338884. q^{44} -276209. q^{45} -815087. q^{46} -165989. q^{47} -110592. q^{48} +117649. q^{49} -523450. q^{50} +43684.0 q^{51} +140608. q^{52} +124874. q^{53} +157464. q^{54} -2.00624e6 q^{55} -175616. q^{56} -1.13435e6 q^{57} -525100. q^{58} -1.79173e6 q^{59} +654719. q^{60} +121728. q^{61} -939639. q^{62} +250047. q^{63} +262144. q^{64} -832417. q^{65} +1.14373e6 q^{66} -154439. q^{67} -103547. q^{68} -2.75092e6 q^{69} +1.03967e6 q^{70} -349494. q^{71} -373248. q^{72} +742316. q^{73} -1.36202e6 q^{74} -1.76664e6 q^{75} +2.68883e6 q^{76} +1.81621e6 q^{77} +474552. q^{78} +4.09788e6 q^{79} -1.55193e6 q^{80} +531441. q^{81} -2.44035e6 q^{82} -2.21258e6 q^{83} -592704. q^{84} +613013. q^{85} -3.11065e6 q^{86} -1.77221e6 q^{87} -2.71107e6 q^{88} +585039. q^{89} +2.20968e6 q^{90} +753571. q^{91} +6.52070e6 q^{92} -3.17128e6 q^{93} +1.32792e6 q^{94} -1.59182e7 q^{95} +884736. q^{96} +8.57271e6 q^{97} -941192. q^{98} +3.86010e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 181 * q^5 + 1296 * q^6 + 2058 * q^7 - 3072 * q^8 + 4374 * q^9	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99}+O(q^{100}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 181 * q^5 + 1296 * q^6 + 2058 * q^7 - 3072 * q^8 + 4374 * q^9 + 1448 * q^10 - 6130 * q^11 - 10368 * q^12 + 13182 * q^13 - 16464 * q^14 + 4887 * q^15 + 24576 * q^16 - 34610 * q^17 - 34992 * q^18 - 4085 * q^19 - 11584 * q^20 - 55566 * q^21 + 49040 * q^22 + 1515 * q^23 + 82944 * q^24 + 156609 * q^25 - 105456 * q^26 - 118098 * q^27 + 131712 * q^28 - 59395 * q^29 - 39096 * q^30 + 478241 * q^31 - 196608 * q^32 + 165510 * q^33 + 276880 * q^34 - 62083 * q^35 + 279936 * q^36 + 574310 * q^37 + 32680 * q^38 - 355914 * q^39 + 92672 * q^40 + 201552 * q^41 + 444528 * q^42 + 728605 * q^43 - 392320 * q^44 - 131949 * q^45 - 12120 * q^46 + 227615 * q^47 - 663552 * q^48 + 705894 * q^49 - 1252872 * q^50 + 934470 * q^51 + 843648 * q^52 + 26321 * q^53 + 944784 * q^54 + 2115010 * q^55 - 1053696 * q^56 + 110295 * q^57 + 475160 * q^58 + 478280 * q^59 + 312768 * q^60 - 501406 * q^61 - 3825928 * q^62 + 1500282 * q^63 + 1572864 * q^64 - 397657 * q^65 - 1324080 * q^66 - 3156366 * q^67 - 2215040 * q^68 - 40905 * q^69 + 496664 * q^70 - 2003644 * q^71 - 2239488 * q^72 + 3659111 * q^73 - 4594480 * q^74 - 4228443 * q^75 - 261440 * q^76 - 2102590 * q^77 + 2847312 * q^78 + 1131065 * q^79 - 741376 * q^80 + 3188646 * q^81 - 1612416 * q^82 - 9629297 * q^83 - 3556224 * q^84 + 895068 * q^85 - 5828840 * q^86 + 1603665 * q^87 + 3138560 * q^88 - 21977377 * q^89 + 1055592 * q^90 + 4521426 * q^91 + 96960 * q^92 - 12912507 * q^93 - 1820920 * q^94 - 19325507 * q^95 + 5308416 * q^96 - 26386649 * q^97 - 5647152 * q^98 - 4468770 * 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
$5$	−378.888	−1.35555	−0.677776	−	0.735269i	$-0.737056\pi$
$5$	−378.888	−1.35555	−0.677776	+	0.735269i	$0.737056\pi$
$6$	216.000	0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	3031.11	0.958520
$11$	5295.06	1.19949	0.599745	−	0.800191i	$-0.295269\pi$
$11$	5295.06	1.19949	0.599745	+	0.800191i	$0.295269\pi$
$12$	−1728.00	−0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−2744.00	−0.267261
$15$	10230.0	0.782628
$16$	4096.00	0.250000
$17$	−1617.93	−0.0798707	−0.0399354	−	0.999202i	$-0.512715\pi$
$17$	−1617.93	−0.0798707	−0.0399354	+	0.999202i	$0.512715\pi$
$18$	−5832.00	−0.235702
$19$	42012.9	1.40522	0.702611	−	0.711574i	$-0.252017\pi$
$19$	42012.9	1.40522	0.702611	+	0.711574i	$0.252017\pi$
$20$	−24248.8	−0.677776
$21$	−9261.00	−0.218218
$22$	−42360.5	−0.848167
$23$	101886.	1.74609	0.873045	−	0.487639i	$-0.162142\pi$
$23$	101886.	1.74609	0.873045	+	0.487639i	$0.162142\pi$
$24$	13824.0	0.204124
$25$	65431.2	0.837520
$26$	−17576.0	−0.196116
$27$	−19683.0	−0.192450
$28$	21952.0	0.188982
$29$	65637.5	0.499757	0.249879	−	0.968277i	$-0.419609\pi$
$29$	65637.5	0.499757	0.249879	+	0.968277i	$0.419609\pi$
$30$	−81839.8	−0.553402
$31$	117455.	0.708117	0.354058	−	0.935223i	$-0.384801\pi$
$31$	117455.	0.708117	0.354058	+	0.935223i	$0.384801\pi$
$32$	−32768.0	−0.176777
$33$	−142967.	−0.692526
$34$	12943.4	0.0564771
$35$	−129959.	−0.512350
$36$	46656.0	0.166667
$37$	170252.	0.552568	0.276284	−	0.961076i	$-0.410897\pi$
$37$	170252.	0.552568	0.276284	+	0.961076i	$0.410897\pi$
$38$	−336103.	−0.993643
$39$	−59319.0	−0.160128
$40$	193991.	0.479260
$41$	305044.	0.691225	0.345612	−	0.938377i	$-0.387671\pi$
$41$	305044.	0.691225	0.345612	+	0.938377i	$0.387671\pi$
$42$	74088.0	0.154303
$43$	388832.	0.745800	0.372900	−	0.927872i	$-0.378364\pi$
$43$	388832.	0.745800	0.372900	+	0.927872i	$0.378364\pi$
$44$	338884.	0.599745
$45$	−276209.	−0.451851
$46$	−815087.	−1.23467
$47$	−165989.	−0.233205	−0.116603	−	0.993179i	$-0.537200\pi$
$47$	−165989.	−0.233205	−0.116603	+	0.993179i	$0.537200\pi$
$48$	−110592.	−0.144338
$49$	117649.	0.142857
$50$	−523450.	−0.592216
$51$	43684.0	0.0461134
$52$	140608.	0.138675
$53$	124874.	0.115214	0.0576070	−	0.998339i	$-0.481653\pi$
$53$	124874.	0.115214	0.0576070	+	0.998339i	$0.481653\pi$
$54$	157464.	0.136083
$55$	−2.00624e6	−1.62597
$56$	−175616.	−0.133631
$57$	−1.13435e6	−0.811306
$58$	−525100.	−0.353382
$59$	−1.79173e6	−1.13577	−0.567886	−	0.823107i	$-0.692239\pi$
$59$	−1.79173e6	−1.13577	−0.567886	+	0.823107i	$0.692239\pi$
$60$	654719.	0.391314
$61$	121728.	0.0686651	0.0343326	−	0.999410i	$-0.489069\pi$
$61$	121728.	0.0686651	0.0343326	+	0.999410i	$0.489069\pi$
$62$	−939639.	−0.500714
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	−832417.	−0.375962
$66$	1.14373e6	0.489690
$67$	−154439.	−0.0627328	−0.0313664	−	0.999508i	$-0.509986\pi$
$67$	−154439.	−0.0627328	−0.0313664	+	0.999508i	$0.509986\pi$
$68$	−103547.	−0.0399354
$69$	−2.75092e6	−1.00811
$70$	1.03967e6	0.362286
$71$	−349494.	−0.115887	−0.0579437	−	0.998320i	$-0.518454\pi$
$71$	−349494.	−0.115887	−0.0579437	+	0.998320i	$0.518454\pi$
$72$	−373248.	−0.117851
$73$	742316.	0.223336	0.111668	−	0.993746i	$-0.464381\pi$
$73$	742316.	0.223336	0.111668	+	0.993746i	$0.464381\pi$
$74$	−1.36202e6	−0.390725
$75$	−1.76664e6	−0.483542
$76$	2.68883e6	0.702611
$77$	1.81621e6	0.453364
$78$	474552.	0.113228
$79$	4.09788e6	0.935114	0.467557	−	0.883963i	$-0.345134\pi$
$79$	4.09788e6	0.935114	0.467557	+	0.883963i	$0.345134\pi$
$80$	−1.55193e6	−0.338888
$81$	531441.	0.111111
$82$	−2.44035e6	−0.488770
$83$	−2.21258e6	−0.424742	−0.212371	−	0.977189i	$-0.568119\pi$
$83$	−2.21258e6	−0.424742	−0.212371	+	0.977189i	$0.568119\pi$
$84$	−592704.	−0.109109
$85$	613013.	0.108269
$86$	−3.11065e6	−0.527360
$87$	−1.77221e6	−0.288535
$88$	−2.71107e6	−0.424084
$89$	585039.	0.0879669	0.0439835	−	0.999032i	$-0.485995\pi$
$89$	585039.	0.0879669	0.0439835	+	0.999032i	$0.485995\pi$
$90$	2.20968e6	0.319507
$91$	753571.	0.104828
$92$	6.52070e6	0.873045
$93$	−3.17128e6	−0.408832
$94$	1.32792e6	0.164901
$95$	−1.59182e7	−1.90485
$96$	884736.	0.102062
$97$	8.57271e6	0.953711	0.476856	−	0.878982i	$-0.341776\pi$
$97$	8.57271e6	0.953711	0.476856	+	0.878982i	$0.341776\pi$
$98$	−941192.	−0.101015
$99$	3.86010e6	0.399830
$100$	4.18760e6	0.418760
$101$	−6.25098e6	−0.603703	−0.301852	−	0.953355i	$-0.597605\pi$
$101$	−6.25098e6	−0.603703	−0.301852	+	0.953355i	$0.597605\pi$
$102$	−349472.	−0.0326071
$103$	−1.07333e7	−0.967840	−0.483920	−	0.875112i	$-0.660787\pi$
$103$	−1.07333e7	−0.967840	−0.483920	+	0.875112i	$0.660787\pi$
$104$	−1.12486e6	−0.0980581
$105$	3.50888e6	0.295806
$106$	−998990.	−0.0814686
$107$	−1.82392e6	−0.143934	−0.0719669	−	0.997407i	$-0.522928\pi$
$107$	−1.82392e6	−0.143934	−0.0719669	+	0.997407i	$0.522928\pi$
$108$	−1.25971e6	−0.0962250
$109$	1.27114e7	0.940157	0.470079	−	0.882624i	$-0.344226\pi$
$109$	1.27114e7	0.940157	0.470079	+	0.882624i	$0.344226\pi$
$110$	1.60499e7	1.14973
$111$	−4.59680e6	−0.319025
$112$	1.40493e6	0.0944911
$113$	7.44600e6	0.485454	0.242727	−	0.970095i	$-0.421958\pi$
$113$	7.44600e6	0.485454	0.242727	+	0.970095i	$0.421958\pi$
$114$	9.07479e6	0.573680
$115$	−3.86034e7	−2.36692
$116$	4.20080e6	0.249879
$117$	1.60161e6	0.0924500
$118$	1.43339e7	0.803112
$119$	−554949.	−0.0301883
$120$	−5.23775e6	−0.276701
$121$	8.55049e6	0.438776
$122$	−973824.	−0.0485536
$123$	−8.23619e6	−0.399079
$124$	7.51711e6	0.354058
$125$	4.80951e6	0.220250
$126$	−2.00038e6	−0.0890871
$127$	−5.83896e6	−0.252943	−0.126471	−	0.991970i	$-0.540365\pi$
$127$	−5.83896e6	−0.252943	−0.126471	+	0.991970i	$0.540365\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.04985e7	−0.430588
$130$	6.65934e6	0.265846
$131$	−1.48836e7	−0.578441	−0.289220	−	0.957263i	$-0.593396\pi$
$131$	−1.48836e7	−0.578441	−0.289220	+	0.957263i	$0.593396\pi$
$132$	−9.14986e6	−0.346263
$133$	1.44104e7	0.531124
$134$	1.23551e6	0.0443588
$135$	7.45766e6	0.260876
$136$	828378.	0.0282386
$137$	−4.91113e7	−1.63177	−0.815885	−	0.578214i	$-0.803750\pi$
$137$	−4.91113e7	−1.63177	−0.815885	+	0.578214i	$0.803750\pi$
$138$	2.20074e7	0.712839
$139$	−2.44767e6	−0.0773038	−0.0386519	−	0.999253i	$-0.512306\pi$
$139$	−2.44767e6	−0.0773038	−0.0386519	+	0.999253i	$0.512306\pi$
$140$	−8.31735e6	−0.256175
$141$	4.48171e6	0.134641
$142$	2.79596e6	0.0819447
$143$	1.16332e7	0.332679
$144$	2.98598e6	0.0833333
$145$	−2.48693e7	−0.677447
$146$	−5.93853e6	−0.157922
$147$	−3.17652e6	−0.0824786
$148$	1.08961e7	0.276284
$149$	−5.05865e7	−1.25280	−0.626401	−	0.779501i	$-0.715473\pi$
$149$	−5.05865e7	−1.25280	−0.626401	+	0.779501i	$0.715473\pi$
$150$	1.41331e7	0.341916
$151$	3.76324e7	0.889493	0.444747	−	0.895656i	$-0.353294\pi$
$151$	3.76324e7	0.889493	0.444747	+	0.895656i	$0.353294\pi$
$152$	−2.15106e7	−0.496821
$153$	−1.17947e6	−0.0266236
$154$	−1.45296e7	−0.320577
$155$	−4.45023e7	−0.959889
$156$	−3.79642e6	−0.0800641
$157$	5.73898e7	1.18355	0.591774	−	0.806104i	$-0.298428\pi$
$157$	5.73898e7	1.18355	0.591774	+	0.806104i	$0.298428\pi$
$158$	−3.27831e7	−0.661226
$159$	−3.37159e6	−0.0665189
$160$	1.24154e7	0.239630
$161$	3.49469e7	0.659960
$162$	−4.25153e6	−0.0785674
$163$	−9.76916e6	−0.176685	−0.0883427	−	0.996090i	$-0.528157\pi$
$163$	−9.76916e6	−0.176685	−0.0883427	+	0.996090i	$0.528157\pi$
$164$	1.95228e7	0.345612
$165$	5.41684e7	0.938754
$166$	1.77006e7	0.300338
$167$	−1.14275e8	−1.89865	−0.949324	−	0.314300i	$-0.898230\pi$
$167$	−1.14275e8	−1.89865	−0.949324	+	0.314300i	$0.898230\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	−4.90411e6	−0.0765577
$171$	3.06274e7	0.468408
$172$	2.48852e7	0.372900
$173$	1.54944e7	0.227517	0.113759	−	0.993508i	$-0.463711\pi$
$173$	1.54944e7	0.227517	0.113759	+	0.993508i	$0.463711\pi$
$174$	1.41777e7	0.204025
$175$	2.24429e7	0.316553
$176$	2.16886e7	0.299872
$177$	4.83768e7	0.655738
$178$	−4.68031e6	−0.0622020
$179$	−1.59073e7	−0.207306	−0.103653	−	0.994614i	$-0.533053\pi$
$179$	−1.59073e7	−0.207306	−0.103653	+	0.994614i	$0.533053\pi$
$180$	−1.76774e7	−0.225925
$181$	3.88537e6	0.0487032	0.0243516	−	0.999703i	$-0.492248\pi$
$181$	3.88537e6	0.0487032	0.0243516	+	0.999703i	$0.492248\pi$
$182$	−6.02857e6	−0.0741249
$183$	−3.28666e6	−0.0396438
$184$	−5.21656e7	−0.617336
$185$	−6.45064e7	−0.749035
$186$	2.53702e7	0.289088
$187$	−8.56702e6	−0.0958041
$188$	−1.06233e7	−0.116603
$189$	−6.75127e6	−0.0727393
$190$	1.27346e8	1.34693
$191$	−5.63071e7	−0.584718	−0.292359	−	0.956309i	$-0.594440\pi$
$191$	−5.63071e7	−0.584718	−0.292359	+	0.956309i	$0.594440\pi$
$192$	−7.07789e6	−0.0721688
$193$	1.35067e8	1.35238	0.676188	−	0.736729i	$-0.263631\pi$
$193$	1.35067e8	1.35238	0.676188	+	0.736729i	$0.263631\pi$
$194$	−6.85817e7	−0.674376
$195$	2.24753e7	0.217062
$196$	7.52954e6	0.0714286
$197$	7.06656e7	0.658531	0.329265	−	0.944237i	$-0.393199\pi$
$197$	7.06656e7	0.658531	0.329265	+	0.944237i	$0.393199\pi$
$198$	−3.08808e7	−0.282722
$199$	7.79675e7	0.701339	0.350670	−	0.936499i	$-0.385954\pi$
$199$	7.79675e7	0.701339	0.350670	+	0.936499i	$0.385954\pi$
$200$	−3.35008e7	−0.296108
$201$	4.16985e6	0.0362188
$202$	5.00079e7	0.426883
$203$	2.25137e7	0.188891
$204$	2.79578e6	0.0230567
$205$	−1.15578e8	−0.936991
$206$	8.58665e7	0.684366
$207$	7.42748e7	0.582030
$208$	8.99891e6	0.0693375
$209$	2.22461e8	1.68555
$210$	−2.80711e7	−0.209166
$211$	2.39403e8	1.75445	0.877226	−	0.480077i	$-0.159391\pi$
$211$	2.39403e8	1.75445	0.877226	+	0.480077i	$0.159391\pi$
$212$	7.99192e6	0.0576070
$213$	9.43635e6	0.0669076
$214$	1.45914e7	0.101777
$215$	−1.47324e8	−1.01097
$216$	1.00777e7	0.0680414
$217$	4.02870e7	0.267643
$218$	−1.01691e8	−0.664792
$219$	−2.00425e7	−0.128943
$220$	−1.28399e8	−0.812985
$221$	−3.55458e6	−0.0221521
$222$	3.67744e7	0.225585
$223$	−2.41348e8	−1.45740	−0.728698	−	0.684835i	$-0.759874\pi$
$223$	−2.41348e8	−1.45740	−0.728698	+	0.684835i	$0.759874\pi$
$224$	−1.12394e7	−0.0668153
$225$	4.76994e7	0.279173
$226$	−5.95680e7	−0.343268
$227$	9.31249e7	0.528415	0.264208	−	0.964466i	$-0.414890\pi$
$227$	9.31249e7	0.528415	0.264208	+	0.964466i	$0.414890\pi$
$228$	−7.25983e7	−0.405653
$229$	−2.07153e8	−1.13990	−0.569950	−	0.821680i	$-0.693037\pi$
$229$	−2.07153e8	−1.13990	−0.569950	+	0.821680i	$0.693037\pi$
$230$	3.08827e8	1.67366
$231$	−4.90376e7	−0.261750
$232$	−3.36064e7	−0.176691
$233$	−3.44213e8	−1.78271	−0.891356	−	0.453304i	$-0.850245\pi$
$233$	−3.44213e8	−1.78271	−0.891356	+	0.453304i	$0.850245\pi$
$234$	−1.28129e7	−0.0653720
$235$	6.28914e7	0.316121
$236$	−1.14671e8	−0.567886
$237$	−1.10643e8	−0.539888
$238$	4.43959e6	0.0213463
$239$	2.99906e8	1.42100	0.710498	−	0.703699i	$-0.248470\pi$
$239$	2.99906e8	1.42100	0.710498	+	0.703699i	$0.248470\pi$
$240$	4.19020e7	0.195657
$241$	2.57421e8	1.18463	0.592317	−	0.805705i	$-0.298213\pi$
$241$	2.57421e8	1.18463	0.592317	+	0.805705i	$0.298213\pi$
$242$	−6.84040e7	−0.310261
$243$	−1.43489e7	−0.0641500
$244$	7.79060e6	0.0343326
$245$	−4.45758e7	−0.193650
$246$	6.58895e7	0.282191
$247$	9.23024e7	0.389739
$248$	−6.01369e7	−0.250357
$249$	5.97396e7	0.245225
$250$	−3.84761e7	−0.155740
$251$	2.04128e8	0.814789	0.407394	−	0.913252i	$-0.366437\pi$
$251$	2.04128e8	0.814789	0.407394	+	0.913252i	$0.366437\pi$
$252$	1.60030e7	0.0629941
$253$	5.39492e8	2.09442
$254$	4.67117e7	0.178858
$255$	−1.65514e7	−0.0625091
$256$	1.67772e7	0.0625000
$257$	4.09448e8	1.50464	0.752320	−	0.658798i	$-0.228935\pi$
$257$	4.09448e8	1.50464	0.752320	+	0.658798i	$0.228935\pi$
$258$	8.39877e7	0.304471
$259$	5.83964e7	0.208851
$260$	−5.32747e7	−0.187981
$261$	4.78498e7	0.166586
$262$	1.19069e8	0.409019
$263$	1.00893e8	0.341992	0.170996	−	0.985272i	$-0.445301\pi$
$263$	1.00893e8	0.341992	0.170996	+	0.985272i	$0.445301\pi$
$264$	7.31989e7	0.244845
$265$	−4.73132e7	−0.156179
$266$	−1.15283e8	−0.375562
$267$	−1.57960e7	−0.0507877
$268$	−9.88409e6	−0.0313664
$269$	4.56374e7	0.142951	0.0714756	−	0.997442i	$-0.477229\pi$
$269$	4.56374e7	0.142951	0.0714756	+	0.997442i	$0.477229\pi$
$270$	−5.96612e7	−0.184467
$271$	1.81284e8	0.553309	0.276654	−	0.960969i	$-0.410774\pi$
$271$	1.81284e8	0.553309	0.276654	+	0.960969i	$0.410774\pi$
$272$	−6.62703e6	−0.0199677
$273$	−2.03464e7	−0.0605228
$274$	3.92890e8	1.15384
$275$	3.46462e8	1.00460
$276$	−1.76059e8	−0.504053
$277$	1.89089e8	0.534548	0.267274	−	0.963621i	$-0.413877\pi$
$277$	1.89089e8	0.534548	0.267274	+	0.963621i	$0.413877\pi$
$278$	1.95813e7	0.0546620
$279$	8.56246e7	0.236039
$280$	6.65388e7	0.181143
$281$	6.97181e8	1.87445	0.937224	−	0.348727i	$-0.113386\pi$
$281$	6.97181e8	1.87445	0.937224	+	0.348727i	$0.113386\pi$
$282$	−3.58537e7	−0.0952056
$283$	−1.76668e8	−0.463345	−0.231673	−	0.972794i	$-0.574420\pi$
$283$	−1.76668e8	−0.463345	−0.231673	+	0.972794i	$0.574420\pi$
$284$	−2.23676e7	−0.0579437
$285$	4.29791e8	1.09977
$286$	−9.30660e7	−0.235239
$287$	1.04630e8	0.261258
$288$	−2.38879e7	−0.0589256
$289$	−4.07721e8	−0.993621
$290$	1.98954e8	0.479027
$291$	−2.31463e8	−0.550626
$292$	4.75082e7	0.111668
$293$	1.56661e7	0.0363853	0.0181926	−	0.999835i	$-0.494209\pi$
$293$	1.56661e7	0.0363853	0.0181926	+	0.999835i	$0.494209\pi$
$294$	2.54122e7	0.0583212
$295$	6.78866e8	1.53960
$296$	−8.71690e7	−0.195362
$297$	−1.04223e8	−0.230842
$298$	4.04692e8	0.885865
$299$	2.23843e8	0.484278
$300$	−1.13065e8	−0.241771
$301$	1.33369e8	0.281886
$302$	−3.01059e8	−0.628967
$303$	1.68777e8	0.348548
$304$	1.72085e8	0.351306
$305$	−4.61213e7	−0.0930791
$306$	9.43575e6	0.0188257
$307$	3.01774e8	0.595248	0.297624	−	0.954683i	$-0.403806\pi$
$307$	3.01774e8	0.595248	0.297624	+	0.954683i	$0.403806\pi$
$308$	1.16237e8	0.226682
$309$	2.89799e8	0.558783
$310$	3.56018e8	0.678744
$311$	8.63103e8	1.62705	0.813526	−	0.581528i	$-0.197545\pi$
$311$	8.63103e8	1.62705	0.813526	+	0.581528i	$0.197545\pi$
$312$	3.03713e7	0.0566139
$313$	3.80020e8	0.700490	0.350245	−	0.936658i	$-0.386098\pi$
$313$	3.80020e8	0.700490	0.350245	+	0.936658i	$0.386098\pi$
$314$	−4.59119e8	−0.836895
$315$	−9.47399e7	−0.170783
$316$	2.62264e8	0.467557
$317$	1.43755e8	0.253464	0.126732	−	0.991937i	$-0.459551\pi$
$317$	1.43755e8	0.253464	0.126732	+	0.991937i	$0.459551\pi$
$318$	2.69727e7	0.0470359
$319$	3.47555e8	0.599454
$320$	−9.93233e7	−0.169444
$321$	4.92459e7	0.0831002
$322$	−2.79575e8	−0.466662
$323$	−6.79738e7	−0.112236
$324$	3.40122e7	0.0555556
$325$	1.43752e8	0.232286
$326$	7.81533e7	0.124935
$327$	−3.43208e8	−0.542800
$328$	−1.56183e8	−0.244385
$329$	−5.69344e7	−0.0881432
$330$	−4.33347e8	−0.663800
$331$	3.40991e8	0.516827	0.258413	−	0.966034i	$-0.416800\pi$
$331$	3.40991e8	0.516827	0.258413	+	0.966034i	$0.416800\pi$
$332$	−1.41605e8	−0.212371
$333$	1.24114e8	0.184189
$334$	9.14202e8	1.34255
$335$	5.85151e7	0.0850376
$336$	−3.79331e7	−0.0545545
$337$	−5.50524e7	−0.0783558	−0.0391779	−	0.999232i	$-0.512474\pi$
$337$	−5.50524e7	−0.0783558	−0.0391779	+	0.999232i	$0.512474\pi$
$338$	−3.86145e7	−0.0543928
$339$	−2.01042e8	−0.280277
$340$	3.92328e7	0.0541344
$341$	6.21931e8	0.849379
$342$	−2.45019e8	−0.331214
$343$	4.03536e7	0.0539949
$344$	−1.99082e8	−0.263680
$345$	1.04229e9	1.36654
$346$	−1.23955e8	−0.160879
$347$	−6.65870e8	−0.855532	−0.427766	−	0.903889i	$-0.640699\pi$
$347$	−6.65870e8	−0.855532	−0.427766	+	0.903889i	$0.640699\pi$
$348$	−1.13422e8	−0.144268
$349$	−8.81952e8	−1.11060	−0.555298	−	0.831652i	$-0.687396\pi$
$349$	−8.81952e8	−1.11060	−0.555298	+	0.831652i	$0.687396\pi$
$350$	−1.79543e8	−0.223837
$351$	−4.32436e7	−0.0533761
$352$	−1.73509e8	−0.212042
$353$	6.08276e8	0.736019	0.368010	−	0.929822i	$-0.380039\pi$
$353$	6.08276e8	0.736019	0.368010	+	0.929822i	$0.380039\pi$
$354$	−3.87014e8	−0.463677
$355$	1.32419e8	0.157091
$356$	3.74425e7	0.0439835
$357$	1.49836e7	0.0174292
$358$	1.27259e8	0.146587
$359$	−9.63926e8	−1.09955	−0.549773	−	0.835314i	$-0.685286\pi$
$359$	−9.63926e8	−1.09955	−0.549773	+	0.835314i	$0.685286\pi$
$360$	1.41419e8	0.159753
$361$	8.71213e8	0.974651
$362$	−3.10829e7	−0.0344383
$363$	−2.30863e8	−0.253327
$364$	4.82285e7	0.0524142
$365$	−2.81255e8	−0.302743
$366$	2.62933e7	0.0280324
$367$	−1.50372e9	−1.58795	−0.793975	−	0.607950i	$-0.791992\pi$
$367$	−1.50372e9	−1.58795	−0.793975	+	0.607950i	$0.791992\pi$
$368$	4.17325e8	0.436523
$369$	2.22377e8	0.230408
$370$	5.16051e8	0.529648
$371$	4.28317e7	0.0435468
$372$	−2.02962e8	−0.204416
$373$	1.13197e8	0.112941	0.0564707	−	0.998404i	$-0.482015\pi$
$373$	1.13197e8	0.112941	0.0564707	+	0.998404i	$0.482015\pi$
$374$	6.85361e7	0.0677437
$375$	−1.29857e8	−0.127161
$376$	8.49866e7	0.0824504
$377$	1.44206e8	0.138608
$378$	5.40102e7	0.0514344
$379$	−6.91870e8	−0.652811	−0.326405	−	0.945230i	$-0.605837\pi$
$379$	−6.91870e8	−0.652811	−0.326405	+	0.945230i	$0.605837\pi$
$380$	−1.01876e9	−0.952426
$381$	1.57652e8	0.146037
$382$	4.50457e8	0.413458
$383$	−5.86351e8	−0.533289	−0.266644	−	0.963795i	$-0.585915\pi$
$383$	−5.86351e8	−0.533289	−0.266644	+	0.963795i	$0.585915\pi$
$384$	5.66231e7	0.0510310
$385$	−6.88139e8	−0.614559
$386$	−1.08053e9	−0.956274
$387$	2.83458e8	0.248600
$388$	5.48653e8	0.476856
$389$	6.66861e8	0.574396	0.287198	−	0.957871i	$-0.407276\pi$
$389$	6.66861e8	0.574396	0.287198	+	0.957871i	$0.407276\pi$
$390$	−1.79802e8	−0.153486
$391$	−1.64844e8	−0.139462
$392$	−6.02363e7	−0.0505076
$393$	4.01858e8	0.333963
$394$	−5.65325e8	−0.465652
$395$	−1.55264e9	−1.26760
$396$	2.47046e8	0.199915
$397$	−1.64583e9	−1.32014	−0.660069	−	0.751205i	$-0.729473\pi$
$397$	−1.64583e9	−1.32014	−0.660069	+	0.751205i	$0.729473\pi$
$398$	−6.23740e8	−0.495922
$399$	−3.89082e8	−0.306645
$400$	2.68006e8	0.209380
$401$	2.12678e7	0.0164709	0.00823544	−	0.999966i	$-0.497379\pi$
$401$	2.12678e7	0.0164709	0.00823544	+	0.999966i	$0.497379\pi$
$402$	−3.33588e7	−0.0256106
$403$	2.58048e8	0.196396
$404$	−4.00063e8	−0.301852
$405$	−2.01357e8	−0.150617
$406$	−1.80109e8	−0.133566
$407$	9.01494e8	0.662800
$408$	−2.23662e7	−0.0163035
$409$	−1.10363e9	−0.797611	−0.398805	−	0.917036i	$-0.630575\pi$
$409$	−1.10363e9	−0.797611	−0.398805	+	0.917036i	$0.630575\pi$
$410$	9.24621e8	0.662553
$411$	1.32600e9	0.942103
$412$	−6.86932e8	−0.483920
$413$	−6.14564e8	−0.429281
$414$	−5.94199e8	−0.411558
$415$	8.38320e8	0.575760
$416$	−7.19913e7	−0.0490290
$417$	6.60870e7	0.0446314
$418$	−1.77969e9	−1.19186
$419$	−1.44045e9	−0.956638	−0.478319	−	0.878186i	$-0.658754\pi$
$419$	−1.44045e9	−0.956638	−0.478319	+	0.878186i	$0.658754\pi$
$420$	2.24569e8	0.147903
$421$	−1.45953e8	−0.0953291	−0.0476646	−	0.998863i	$-0.515178\pi$
$421$	−1.45953e8	−0.0953291	−0.0476646	+	0.998863i	$0.515178\pi$
$422$	−1.91523e9	−1.24059
$423$	−1.21006e8	−0.0777350
$424$	−6.39353e7	−0.0407343
$425$	−1.05863e8	−0.0668933
$426$	−7.54908e7	−0.0473108
$427$	4.17527e7	0.0259530
$428$	−1.16731e8	−0.0719669
$429$	−3.14098e8	−0.192072
$430$	1.17859e9	0.714864
$431$	7.55565e8	0.454571	0.227285	−	0.973828i	$-0.427015\pi$
$431$	7.55565e8	0.454571	0.227285	+	0.973828i	$0.427015\pi$
$432$	−8.06216e7	−0.0481125
$433$	1.90514e9	1.12777	0.563883	−	0.825855i	$-0.309307\pi$
$433$	1.90514e9	1.12777	0.563883	+	0.825855i	$0.309307\pi$
$434$	−3.22296e8	−0.189252
$435$	6.71471e8	0.391124
$436$	8.13530e8	0.470079
$437$	4.28053e9	2.45365
$438$	1.60340e8	0.0911765
$439$	8.25574e6	0.00465726	0.00232863	−	0.999997i	$-0.499259\pi$
$439$	8.25574e6	0.00465726	0.00232863	+	0.999997i	$0.499259\pi$
$440$	1.02719e9	0.574867
$441$	8.57661e7	0.0476190
$442$	2.84367e7	0.0156639
$443$	1.87904e9	1.02689	0.513443	−	0.858123i	$-0.328370\pi$
$443$	1.87904e9	1.02689	0.513443	+	0.858123i	$0.328370\pi$
$444$	−2.94195e8	−0.159513
$445$	−2.21664e8	−0.119244
$446$	1.93079e9	1.03053
$447$	1.36584e9	0.723306
$448$	8.99154e7	0.0472456
$449$	2.21500e9	1.15481	0.577406	−	0.816457i	$-0.304065\pi$
$449$	2.21500e9	1.15481	0.577406	+	0.816457i	$0.304065\pi$
$450$	−3.81595e8	−0.197405
$451$	1.61523e9	0.829117
$452$	4.76544e8	0.242727
$453$	−1.01607e9	−0.513549
$454$	−7.44999e8	−0.373646
$455$	−2.85519e8	−0.142100
$456$	5.80787e8	0.286840
$457$	1.02114e9	0.500469	0.250234	−	0.968185i	$-0.419492\pi$
$457$	1.02114e9	0.500469	0.250234	+	0.968185i	$0.419492\pi$
$458$	1.65722e9	0.806031
$459$	3.18456e7	0.0153711
$460$	−2.47062e9	−1.18346
$461$	−2.27846e9	−1.08315	−0.541574	−	0.840653i	$-0.682171\pi$
$461$	−2.27846e9	−1.08315	−0.541574	+	0.840653i	$0.682171\pi$
$462$	3.92300e8	0.185085
$463$	−1.31776e9	−0.617026	−0.308513	−	0.951220i	$-0.599831\pi$
$463$	−1.31776e9	−0.617026	−0.308513	+	0.951220i	$0.599831\pi$
$464$	2.68851e8	0.124939
$465$	1.20156e9	0.554192
$466$	2.75370e9	1.26057
$467$	−4.21868e9	−1.91676	−0.958379	−	0.285498i	$-0.907841\pi$
$467$	−4.21868e9	−1.91676	−0.958379	+	0.285498i	$0.907841\pi$
$468$	1.02503e8	0.0462250
$469$	−5.29726e7	−0.0237108
$470$	−5.03131e8	−0.223532
$471$	−1.54953e9	−0.683322
$472$	9.17367e8	0.401556
$473$	2.05889e9	0.894579
$474$	8.85143e8	0.381759
$475$	2.74896e9	1.17690
$476$	−3.55167e7	−0.0150941
$477$	9.10329e7	0.0384047
$478$	−2.39925e9	−1.00480
$479$	2.47664e9	1.02965	0.514823	−	0.857296i	$-0.327858\pi$
$479$	2.47664e9	1.02965	0.514823	+	0.857296i	$0.327858\pi$
$480$	−3.35216e8	−0.138350
$481$	3.74043e8	0.153255
$482$	−2.05937e9	−0.837663
$483$	−9.43566e8	−0.381028
$484$	5.47232e8	0.219388
$485$	−3.24810e9	−1.29280
$486$	1.14791e8	0.0453609
$487$	1.36666e9	0.536178	0.268089	−	0.963394i	$-0.413608\pi$
$487$	1.36666e9	0.536178	0.268089	+	0.963394i	$0.413608\pi$
$488$	−6.23248e7	−0.0242768
$489$	2.63767e8	0.102009
$490$	3.56607e8	0.136931
$491$	−2.79999e9	−1.06751	−0.533754	−	0.845640i	$-0.679219\pi$
$491$	−2.79999e9	−1.06751	−0.533754	+	0.845640i	$0.679219\pi$
$492$	−5.27116e8	−0.199539
$493$	−1.06197e8	−0.0399160
$494$	−7.38419e8	−0.275587
$495$	−1.46255e9	−0.541990
$496$	4.81095e8	0.177029
$497$	−1.19877e8	−0.0438013
$498$	−4.77917e8	−0.173400
$499$	3.58384e9	1.29121	0.645604	−	0.763672i	$-0.276606\pi$
$499$	3.58384e9	1.29121	0.645604	+	0.763672i	$0.276606\pi$
$500$	3.07809e8	0.110125
$501$	3.08543e9	1.09618
$502$	−1.63303e9	−0.576143
$503$	−3.08646e9	−1.08136	−0.540682	−	0.841227i	$-0.681834\pi$
$503$	−3.08646e9	−1.08136	−0.540682	+	0.841227i	$0.681834\pi$
$504$	−1.28024e8	−0.0445435
$505$	2.36842e9	0.818351
$506$	−4.31594e9	−1.48098
$507$	−1.30324e8	−0.0444116
$508$	−3.73694e8	−0.126471
$509$	−8.26000e8	−0.277631	−0.138815	−	0.990318i	$-0.544329\pi$
$509$	−8.26000e8	−0.277631	−0.138815	+	0.990318i	$0.544329\pi$
$510$	1.32411e8	0.0442006
$511$	2.54614e8	0.0844131
$512$	−1.34218e8	−0.0441942
$513$	−8.26940e8	−0.270435
$514$	−3.27558e9	−1.06394
$515$	4.06673e9	1.31196
$516$	−6.71901e8	−0.215294
$517$	−8.78924e8	−0.279727
$518$	−4.67171e8	−0.147680
$519$	−4.18350e8	−0.131357
$520$	4.26198e8	0.132923
$521$	−6.38043e9	−1.97660	−0.988298	−	0.152539i	$-0.951255\pi$
$521$	−6.38043e9	−1.97660	−0.988298	+	0.152539i	$0.951255\pi$
$522$	−3.82798e8	−0.117794
$523$	−1.94369e9	−0.594117	−0.297058	−	0.954859i	$-0.596006\pi$
$523$	−1.94369e9	−0.594117	−0.297058	+	0.954859i	$0.596006\pi$
$524$	−9.52551e8	−0.289220
$525$	−6.05959e8	−0.182762
$526$	−8.07145e8	−0.241825
$527$	−1.90033e8	−0.0565578
$528$	−5.85591e8	−0.173131
$529$	6.97592e9	2.04883
$530$	3.78505e8	0.110435
$531$	−1.30617e9	−0.378591
$532$	9.22268e8	0.265562
$533$	6.70182e8	0.191711
$534$	1.26368e8	0.0359124
$535$	6.91062e8	0.195110
$536$	7.90727e7	0.0221794
$537$	4.29498e8	0.119688
$538$	−3.65099e8	−0.101082
$539$	6.22959e8	0.171356
$540$	4.77290e8	0.130438
$541$	6.03178e9	1.63778	0.818889	−	0.573952i	$-0.194591\pi$
$541$	6.03178e9	1.63778	0.818889	+	0.573952i	$0.194591\pi$
$542$	−1.45027e9	−0.391248
$543$	−1.04905e8	−0.0281188
$544$	5.30162e7	0.0141193
$545$	−4.81620e9	−1.27443
$546$	1.62771e8	0.0427960
$547$	3.94623e9	1.03092	0.515462	−	0.856913i	$-0.327620\pi$
$547$	3.94623e9	1.03092	0.515462	+	0.856913i	$0.327620\pi$
$548$	−3.14312e9	−0.815885
$549$	8.87398e7	0.0228884
$550$	−2.77170e9	−0.710357
$551$	2.75762e9	0.702270
$552$	1.40847e9	0.356419
$553$	1.40557e9	0.353440
$554$	−1.51271e9	−0.377983
$555$	1.74167e9	0.432455
$556$	−1.56651e8	−0.0386519
$557$	5.88604e9	1.44321	0.721606	−	0.692304i	$-0.243404\pi$
$557$	5.88604e9	1.44321	0.721606	+	0.692304i	$0.243404\pi$
$558$	−6.84997e8	−0.166905
$559$	8.54263e8	0.206848
$560$	−5.32311e8	−0.128088
$561$	2.31309e8	0.0553125
$562$	−5.57745e9	−1.32544
$563$	−3.36222e9	−0.794049	−0.397024	−	0.917808i	$-0.629957\pi$
$563$	−3.36222e9	−0.794049	−0.397024	+	0.917808i	$0.629957\pi$
$564$	2.86830e8	0.0673205
$565$	−2.82120e9	−0.658059
$566$	1.41334e9	0.327634
$567$	1.82284e8	0.0419961
$568$	1.78941e8	0.0409724
$569$	−3.24680e9	−0.738861	−0.369430	−	0.929258i	$-0.620447\pi$
$569$	−3.24680e9	−0.738861	−0.369430	+	0.929258i	$0.620447\pi$
$570$	−3.43833e9	−0.777653
$571$	−5.21927e9	−1.17323	−0.586616	−	0.809865i	$-0.699540\pi$
$571$	−5.21927e9	−1.17323	−0.586616	+	0.809865i	$0.699540\pi$
$572$	7.44528e8	0.166339
$573$	1.52029e9	0.337587
$574$	−8.37041e8	−0.184738
$575$	6.66652e9	1.46239
$576$	1.91103e8	0.0416667
$577$	−6.94575e9	−1.50523	−0.752616	−	0.658460i	$-0.771208\pi$
$577$	−6.94575e9	−1.50523	−0.752616	+	0.658460i	$0.771208\pi$
$578$	3.26177e9	0.702596
$579$	−3.64680e9	−0.780794
$580$	−1.59163e9	−0.338723
$581$	−7.58915e8	−0.160537
$582$	1.85170e9	0.389351
$583$	6.61214e8	0.138198
$584$	−3.80066e8	−0.0789612
$585$	−6.06832e8	−0.125321
$586$	−1.25329e8	−0.0257283
$587$	−6.75547e9	−1.37855	−0.689275	−	0.724500i	$-0.742071\pi$
$587$	−6.75547e9	−1.37855	−0.689275	+	0.724500i	$0.742071\pi$
$588$	−2.03297e8	−0.0412393
$589$	4.93462e9	0.995062
$590$	−5.43093e9	−1.08866
$591$	−1.90797e9	−0.380203
$592$	6.97352e8	0.138142
$593$	−3.66897e8	−0.0722524	−0.0361262	−	0.999347i	$-0.511502\pi$
$593$	−3.66897e8	−0.0722524	−0.0361262	+	0.999347i	$0.511502\pi$
$594$	8.33781e8	0.163230
$595$	2.10264e8	0.0409218
$596$	−3.23754e9	−0.626401
$597$	−2.10512e9	−0.404918
$598$	−1.79075e9	−0.342437
$599$	−3.63967e9	−0.691940	−0.345970	−	0.938246i	$-0.612450\pi$
$599$	−3.63967e9	−0.691940	−0.345970	+	0.938246i	$0.612450\pi$
$600$	9.04522e8	0.170958
$601$	−1.32849e9	−0.249631	−0.124815	−	0.992180i	$-0.539834\pi$
$601$	−1.32849e9	−0.249631	−0.124815	+	0.992180i	$0.539834\pi$
$602$	−1.06695e9	−0.199323
$603$	−1.12586e8	−0.0209109
$604$	2.40847e9	0.444747
$605$	−3.23968e9	−0.594783
$606$	−1.35021e9	−0.246461
$607$	2.37205e9	0.430490	0.215245	−	0.976560i	$-0.430945\pi$
$607$	2.37205e9	0.430490	0.215245	+	0.976560i	$0.430945\pi$
$608$	−1.37668e9	−0.248411
$609$	−6.07869e8	−0.109056
$610$	3.68971e8	0.0658169
$611$	−3.64679e8	−0.0646794
$612$	−7.54860e7	−0.0133118
$613$	−9.60638e8	−0.168441	−0.0842206	−	0.996447i	$-0.526840\pi$
$613$	−9.60638e8	−0.168441	−0.0842206	+	0.996447i	$0.526840\pi$
$614$	−2.41419e9	−0.420904
$615$	3.12060e9	0.540972
$616$	−9.29897e8	−0.160289
$617$	3.07342e8	0.0526773	0.0263386	−	0.999653i	$-0.491615\pi$
$617$	3.07342e8	0.0526773	0.0263386	+	0.999653i	$0.491615\pi$
$618$	−2.31840e9	−0.395119
$619$	3.18881e9	0.540395	0.270197	−	0.962805i	$-0.412911\pi$
$619$	3.18881e9	0.540395	0.270197	+	0.962805i	$0.412911\pi$
$620$	−2.84814e9	−0.479945
$621$	−2.00542e9	−0.336035
$622$	−6.90483e9	−1.15050
$623$	2.00668e8	0.0332484
$624$	−2.42971e8	−0.0400320
$625$	−6.93408e9	−1.13608
$626$	−3.04016e9	−0.495321
$627$	−6.00644e9	−0.973153
$628$	3.67295e9	0.591774
$629$	−2.75455e8	−0.0441340
$630$	7.57919e8	0.120762
$631$	−1.04712e10	−1.65918	−0.829589	−	0.558375i	$-0.811425\pi$
$631$	−1.04712e10	−1.65918	−0.829589	+	0.558375i	$0.811425\pi$
$632$	−2.09812e9	−0.330613
$633$	−6.46389e9	−1.01293
$634$	−1.15004e9	−0.179226
$635$	2.21231e9	0.342877
$636$	−2.15782e8	−0.0332594
$637$	2.58475e8	0.0396214
$638$	−2.78044e9	−0.423878
$639$	−2.54781e8	−0.0386291
$640$	7.94586e8	0.119815
$641$	−3.75555e9	−0.563211	−0.281605	−	0.959530i	$-0.590867\pi$
$641$	−3.75555e9	−0.563211	−0.281605	+	0.959530i	$0.590867\pi$
$642$	−3.93967e8	−0.0587607
$643$	8.85818e9	1.31403	0.657016	−	0.753876i	$-0.271818\pi$
$643$	8.85818e9	1.31403	0.657016	+	0.753876i	$0.271818\pi$
$644$	2.23660e9	0.329980
$645$	3.97774e9	0.583684
$646$	5.43790e8	0.0793629
$647$	−8.35966e9	−1.21345	−0.606727	−	0.794910i	$-0.707518\pi$
$647$	−8.35966e9	−1.21345	−0.606727	+	0.794910i	$0.707518\pi$
$648$	−2.72098e8	−0.0392837
$649$	−9.48733e9	−1.36235
$650$	−1.15002e9	−0.164251
$651$	−1.08775e9	−0.154524
$652$	−6.25226e8	−0.0883427
$653$	−2.32417e9	−0.326642	−0.163321	−	0.986573i	$-0.552221\pi$
$653$	−2.32417e9	−0.326642	−0.163321	+	0.986573i	$0.552221\pi$
$654$	2.74566e9	0.383818
$655$	5.63923e9	0.784106
$656$	1.24946e9	0.172806
$657$	5.41148e8	0.0744453
$658$	4.55475e8	0.0623267
$659$	−1.82416e9	−0.248293	−0.124147	−	0.992264i	$-0.539619\pi$
$659$	−1.82416e9	−0.248293	−0.124147	+	0.992264i	$0.539619\pi$
$660$	3.46678e9	0.469377
$661$	8.34870e9	1.12438	0.562191	−	0.827007i	$-0.309959\pi$
$661$	8.34870e9	1.12438	0.562191	+	0.827007i	$0.309959\pi$
$662$	−2.72793e9	−0.365452
$663$	9.59738e7	0.0127895
$664$	1.13284e9	0.150169
$665$	−5.45994e9	−0.719966
$666$	−9.92909e8	−0.130242
$667$	6.68754e9	0.872622
$668$	−7.31361e9	−0.949324
$669$	6.51641e9	0.841428
$670$	−4.68121e8	−0.0601307
$671$	6.44557e8	0.0823631
$672$	3.03464e8	0.0385758
$673$	−1.11689e10	−1.41240	−0.706200	−	0.708012i	$-0.749592\pi$
$673$	−1.11689e10	−1.41240	−0.706200	+	0.708012i	$0.749592\pi$
$674$	4.40419e8	0.0554059
$675$	−1.28788e9	−0.161181
$676$	3.08916e8	0.0384615
$677$	−1.38490e10	−1.71537	−0.857687	−	0.514173i	$-0.828099\pi$
$677$	−1.38490e10	−1.71537	−0.857687	+	0.514173i	$0.828099\pi$
$678$	1.60834e9	0.198186
$679$	2.94044e9	0.360469
$680$	−3.13863e8	−0.0382788
$681$	−2.51437e9	−0.305081
$682$	−4.97544e9	−0.600602
$683$	−1.46785e10	−1.76283	−0.881414	−	0.472345i	$-0.843408\pi$
$683$	−1.46785e10	−1.76283	−0.881414	+	0.472345i	$0.843408\pi$
$684$	1.96015e9	0.234204
$685$	1.86077e10	2.21195
$686$	−3.22829e8	−0.0381802
$687$	5.59312e9	0.658121
$688$	1.59266e9	0.186450
$689$	2.74348e8	0.0319546
$690$	−8.33833e9	−0.966289
$691$	−4.89134e8	−0.0563969	−0.0281984	−	0.999602i	$-0.508977\pi$
$691$	−4.89134e8	−0.0563969	−0.0281984	+	0.999602i	$0.508977\pi$
$692$	9.91644e8	0.113759
$693$	1.32401e9	0.151121
$694$	5.32696e9	0.604953
$695$	9.27392e8	0.104789
$696$	9.07373e8	0.102013
$697$	−4.93539e8	−0.0552086
$698$	7.05562e9	0.785310
$699$	9.29375e9	1.02925
$700$	1.43635e9	0.158276
$701$	−5.28855e9	−0.579860	−0.289930	−	0.957048i	$-0.593632\pi$
$701$	−5.28855e9	−0.579860	−0.289930	+	0.957048i	$0.593632\pi$
$702$	3.45948e8	0.0377426
$703$	7.15278e9	0.776482
$704$	1.38807e9	0.149936
$705$	−1.69807e9	−0.182513
$706$	−4.86621e9	−0.520444
$707$	−2.14409e9	−0.228178
$708$	3.09611e9	0.327869
$709$	1.15419e10	1.21623	0.608116	−	0.793849i	$-0.291926\pi$
$709$	1.15419e10	1.21623	0.608116	+	0.793849i	$0.291926\pi$
$710$	−1.05935e9	−0.111080
$711$	2.98736e9	0.311705
$712$	−2.99540e8	−0.0311010
$713$	1.19670e10	1.23644
$714$	−1.19869e8	−0.0123243
$715$	−4.40770e9	−0.450963
$716$	−1.01807e9	−0.103653
$717$	−8.09747e9	−0.820412
$718$	7.71141e9	0.777496
$719$	1.19175e10	1.19574	0.597868	−	0.801594i	$-0.296015\pi$
$719$	1.19175e10	1.19574	0.597868	+	0.801594i	$0.296015\pi$
$720$	−1.13135e9	−0.112963
$721$	−3.68153e9	−0.365809
$722$	−6.96971e9	−0.689183
$723$	−6.95037e9	−0.683949
$724$	2.48664e8	0.0243516
$725$	4.29475e9	0.418557
$726$	1.84691e9	0.179129
$727$	3.61469e9	0.348900	0.174450	−	0.984666i	$-0.444185\pi$
$727$	3.61469e9	0.348900	0.174450	+	0.984666i	$0.444185\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	2.25004e9	0.214072
$731$	−6.29101e8	−0.0595676
$732$	−2.10346e8	−0.0198219
$733$	−2.25776e9	−0.211745	−0.105873	−	0.994380i	$-0.533764\pi$
$733$	−2.25776e9	−0.211745	−0.105873	+	0.994380i	$0.533764\pi$
$734$	1.20298e10	1.12285
$735$	1.20355e9	0.111804
$736$	−3.33860e9	−0.308668
$737$	−8.17764e8	−0.0752474
$738$	−1.77902e9	−0.162923
$739$	2.04054e10	1.85990	0.929949	−	0.367688i	$-0.119851\pi$
$739$	2.04054e10	1.85990	0.929949	+	0.367688i	$0.119851\pi$
$740$	−4.12841e9	−0.374517
$741$	−2.49216e9	−0.225016
$742$	−3.42653e8	−0.0307923
$743$	−5.54813e9	−0.496233	−0.248116	−	0.968730i	$-0.579812\pi$
$743$	−5.54813e9	−0.496233	−0.248116	+	0.968730i	$0.579812\pi$
$744$	1.62370e9	0.144544
$745$	1.91666e10	1.69824
$746$	−9.05574e8	−0.0798616
$747$	−1.61297e9	−0.141581
$748$	−5.48289e8	−0.0479020
$749$	−6.25605e8	−0.0544019
$750$	1.03885e9	0.0899167
$751$	−9.72081e9	−0.837457	−0.418729	−	0.908111i	$-0.637524\pi$
$751$	−9.72081e9	−0.837457	−0.418729	+	0.908111i	$0.637524\pi$
$752$	−6.79893e8	−0.0583013
$753$	−5.51146e9	−0.470419
$754$	−1.15365e9	−0.0980105
$755$	−1.42585e10	−1.20575
$756$	−4.32081e8	−0.0363696
$757$	−1.93064e9	−0.161758	−0.0808791	−	0.996724i	$-0.525773\pi$
$757$	−1.93064e9	−0.161758	−0.0808791	+	0.996724i	$0.525773\pi$
$758$	5.53496e9	0.461607
$759$	−1.45663e10	−1.20921
$760$	8.15012e9	0.673467
$761$	−2.22943e9	−0.183378	−0.0916892	−	0.995788i	$-0.529227\pi$
$761$	−2.22943e9	−0.183378	−0.0916892	+	0.995788i	$0.529227\pi$
$762$	−1.26122e9	−0.103264
$763$	4.36001e9	0.355346
$764$	−3.60366e9	−0.292359
$765$	4.46887e8	0.0360896
$766$	4.69081e9	0.377092
$767$	−3.93644e9	−0.315006
$768$	−4.52985e8	−0.0360844
$769$	1.03220e10	0.818506	0.409253	−	0.912421i	$-0.365789\pi$
$769$	1.03220e10	0.818506	0.409253	+	0.912421i	$0.365789\pi$
$770$	5.50511e9	0.434559
$771$	−1.10551e10	−0.868704
$772$	8.64426e9	0.676188
$773$	1.74584e10	1.35949	0.679745	−	0.733449i	$-0.262091\pi$
$773$	1.74584e10	1.35949	0.679745	+	0.733449i	$0.262091\pi$
$774$	−2.26767e9	−0.175787
$775$	7.68522e9	0.593062
$776$	−4.38923e9	−0.337188
$777$	−1.57670e9	−0.120580
$778$	−5.33489e9	−0.406159
$779$	1.28158e10	0.971325
$780$	1.43842e9	0.108531
$781$	−1.85059e9	−0.139006
$782$	1.31875e9	0.0986142
$783$	−1.29194e9	−0.0961783
$784$	4.81890e8	0.0357143
$785$	−2.17443e10	−1.60436
$786$	−3.21486e9	−0.236147
$787$	−9.17018e9	−0.670604	−0.335302	−	0.942111i	$-0.608838\pi$
$787$	−9.17018e9	−0.670604	−0.335302	+	0.942111i	$0.608838\pi$
$788$	4.52260e9	0.329265
$789$	−2.72411e9	−0.197449
$790$	1.24211e10	0.896325
$791$	2.55398e9	0.183485
$792$	−1.97637e9	−0.141361
$793$	2.67437e8	0.0190443
$794$	1.31667e10	0.933478
$795$	1.27746e9	0.0901698
$796$	4.98992e9	0.350670
$797$	1.55129e10	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$797$	1.55129e10	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$798$	3.11265e9	0.216831
$799$	2.68559e8	0.0186263
$800$	−2.14405e9	−0.148054
$801$	4.26493e8	0.0293223
$802$	−1.70142e8	−0.0116467
$803$	3.93061e9	0.267889
$804$	2.66870e8	0.0181094
$805$	−1.32410e10	−0.894610
$806$	−2.06439e9	−0.138873
$807$	−1.23221e9	−0.0825329
$808$	3.20050e9	0.213441
$809$	1.88511e10	1.25175	0.625875	−	0.779924i	$-0.284742\pi$
$809$	1.88511e10	1.25175	0.625875	+	0.779924i	$0.284742\pi$
$810$	1.61085e9	0.106502
$811$	1.37909e10	0.907859	0.453930	−	0.891038i	$-0.350022\pi$
$811$	1.37909e10	0.907859	0.453930	+	0.891038i	$0.350022\pi$
$812$	1.44087e9	0.0944453
$813$	−4.89467e9	−0.319453
$814$	−7.21195e9	−0.468670
$815$	3.70142e9	0.239506
$816$	1.78930e8	0.0115283
$817$	1.63360e10	1.04801
$818$	8.82902e9	0.563996
$819$	5.49353e8	0.0349428
$820$	−7.39697e9	−0.468495
$821$	2.18934e10	1.38074	0.690369	−	0.723457i	$-0.257448\pi$
$821$	2.18934e10	1.38074	0.690369	+	0.723457i	$0.257448\pi$
$822$	−1.06080e10	−0.666168
$823$	1.28762e10	0.805171	0.402586	−	0.915382i	$-0.368112\pi$
$823$	1.28762e10	0.805171	0.402586	+	0.915382i	$0.368112\pi$
$824$	5.49546e9	0.342183
$825$	−9.35449e9	−0.580004
$826$	4.91651e9	0.303548
$827$	−4.75080e9	−0.292077	−0.146039	−	0.989279i	$-0.546652\pi$
$827$	−4.75080e9	−0.292077	−0.146039	+	0.989279i	$0.546652\pi$
$828$	4.75359e9	0.291015
$829$	−2.70326e10	−1.64796	−0.823980	−	0.566619i	$-0.808251\pi$
$829$	−2.70326e10	−1.64796	−0.823980	+	0.566619i	$0.808251\pi$
$830$	−6.70656e9	−0.407124
$831$	−5.10540e9	−0.308622
$832$	5.75930e8	0.0346688
$833$	−1.90347e8	−0.0114101
$834$	−5.28696e8	−0.0315591
$835$	4.32975e10	2.57371
$836$	1.42375e10	0.842775
$837$	−2.31186e9	−0.136277
$838$	1.15236e10	0.676445
$839$	2.00634e10	1.17284	0.586418	−	0.810008i	$-0.300537\pi$
$839$	2.00634e10	1.17284	0.586418	+	0.810008i	$0.300537\pi$
$840$	−1.79655e9	−0.104583
$841$	−1.29416e10	−0.750243
$842$	1.16762e9	0.0674079
$843$	−1.88239e10	−1.08221
$844$	1.53218e10	0.877226
$845$	−1.82882e9	−0.104273
$846$	9.68050e8	0.0549670
$847$	2.93282e9	0.165842
$848$	5.11483e8	0.0288035
$849$	4.77003e9	0.267512
$850$	8.46904e8	0.0473007
$851$	1.73463e10	0.964834
$852$	6.03926e8	0.0334538
$853$	1.10456e10	0.609353	0.304677	−	0.952456i	$-0.401452\pi$
$853$	1.10456e10	0.609353	0.304677	+	0.952456i	$0.401452\pi$
$854$	−3.34022e8	−0.0183515
$855$	−1.16044e10	−0.634951
$856$	9.33848e8	0.0508883
$857$	3.08171e10	1.67247	0.836237	−	0.548368i	$-0.184751\pi$
$857$	3.08171e10	1.67247	0.836237	+	0.548368i	$0.184751\pi$
$858$	2.51278e9	0.135815
$859$	−1.22323e10	−0.658465	−0.329232	−	0.944249i	$-0.606790\pi$
$859$	−1.22323e10	−0.658465	−0.329232	+	0.944249i	$0.606790\pi$
$860$	−9.42872e9	−0.505485
$861$	−2.82501e9	−0.150838
$862$	−6.04452e9	−0.321430
$863$	4.41026e9	0.233575	0.116788	−	0.993157i	$-0.462740\pi$
$863$	4.41026e9	0.233575	0.116788	+	0.993157i	$0.462740\pi$
$864$	6.44973e8	0.0340207
$865$	−5.87066e9	−0.308411
$866$	−1.52411e10	−0.797451
$867$	1.10085e10	0.573667
$868$	2.57837e9	0.133822
$869$	2.16985e10	1.12166
$870$	−5.37176e9	−0.276567
$871$	−3.39302e8	−0.0173990
$872$	−6.50824e9	−0.332396
$873$	6.24950e9	0.317904
$874$	−3.42442e10	−1.73499
$875$	1.64966e9	0.0832467
$876$	−1.28272e9	−0.0644716
$877$	−2.81168e10	−1.40756	−0.703782	−	0.710416i	$-0.748507\pi$
$877$	−2.81168e10	−1.40756	−0.703782	+	0.710416i	$0.748507\pi$
$878$	−6.60459e7	−0.00329318
$879$	−4.22986e8	−0.0210070
$880$	−8.21754e9	−0.406493
$881$	1.66508e10	0.820388	0.410194	−	0.911998i	$-0.365461\pi$
$881$	1.66508e10	0.820388	0.410194	+	0.911998i	$0.365461\pi$
$882$	−6.86129e8	−0.0336718
$883$	−9.07838e8	−0.0443758	−0.0221879	−	0.999754i	$-0.507063\pi$
$883$	−9.07838e8	−0.0443758	−0.0221879	+	0.999754i	$0.507063\pi$
$884$	−2.27493e8	−0.0110761
$885$	−1.83294e10	−0.888887
$886$	−1.50323e10	−0.726119
$887$	1.76337e10	0.848421	0.424211	−	0.905564i	$-0.360552\pi$
$887$	1.76337e10	0.848421	0.424211	+	0.905564i	$0.360552\pi$
$888$	2.35356e9	0.112793
$889$	−2.00276e9	−0.0956034
$890$	1.77331e9	0.0843180
$891$	2.81401e9	0.133277
$892$	−1.54463e10	−0.728698
$893$	−6.97370e9	−0.327705
$894$	−1.09267e10	−0.511455
$895$	6.02710e9	0.281014
$896$	−7.19323e8	−0.0334077
$897$	−6.04377e9	−0.279598
$898$	−1.77200e10	−0.816576
$899$	7.70945e9	0.353887
$900$	3.05276e9	0.139587
$901$	−2.02036e8	−0.00920223
$902$	−1.29218e10	−0.586274
$903$	−3.60097e9	−0.162747
$904$	−3.81235e9	−0.171634
$905$	−1.47212e9	−0.0660196
$906$	8.12860e9	0.363134
$907$	1.18339e10	0.526628	0.263314	−	0.964710i	$-0.415184\pi$
$907$	1.18339e10	0.526628	0.263314	+	0.964710i	$0.415184\pi$
$908$	5.95999e9	0.264208
$909$	−4.55697e9	−0.201234
$910$	2.28415e9	0.100480
$911$	2.18365e10	0.956906	0.478453	−	0.878113i	$-0.341198\pi$
$911$	2.18365e10	0.956906	0.478453	+	0.878113i	$0.341198\pi$
$912$	−4.64629e9	−0.202826
$913$	−1.17157e10	−0.509474
$914$	−8.16909e9	−0.353885
$915$	1.24528e9	0.0537393
$916$	−1.32578e10	−0.569950
$917$	−5.10508e9	−0.218630
$918$	−2.54765e8	−0.0108690
$919$	6.10212e9	0.259344	0.129672	−	0.991557i	$-0.458608\pi$
$919$	6.10212e9	0.259344	0.129672	+	0.991557i	$0.458608\pi$
$920$	1.97649e10	0.836831
$921$	−8.14790e9	−0.343666
$922$	1.82277e10	0.765901
$923$	−7.67839e8	−0.0321414
$924$	−3.13840e9	−0.130875
$925$	1.11398e10	0.462787
$926$	1.05421e10	0.436303
$927$	−7.82459e9	−0.322613
$928$	−2.15081e9	−0.0883455
$929$	−2.88458e10	−1.18039	−0.590197	−	0.807259i	$-0.700950\pi$
$929$	−2.88458e10	−1.18039	−0.590197	+	0.807259i	$0.700950\pi$
$930$	−9.61249e9	−0.391873
$931$	4.94278e9	0.200746
$932$	−2.20296e10	−0.891356
$933$	−2.33038e10	−0.939379
$934$	3.37494e10	1.35535
$935$	3.24594e9	0.129867
$936$	−8.20026e8	−0.0326860
$937$	−5.53047e9	−0.219621	−0.109810	−	0.993953i	$-0.535024\pi$
$937$	−5.53047e9	−0.219621	−0.109810	+	0.993953i	$0.535024\pi$
$938$	4.23780e8	0.0167661
$939$	−1.02606e10	−0.404428
$940$	4.02505e9	0.158061
$941$	−2.60273e10	−1.01828	−0.509139	−	0.860685i	$-0.670036\pi$
$941$	−2.60273e10	−1.01828	−0.509139	+	0.860685i	$0.670036\pi$
$942$	1.23962e10	0.483182
$943$	3.10797e10	1.20694
$944$	−7.33894e9	−0.283943
$945$	2.55798e9	0.0986019
$946$	−1.64711e10	−0.632563
$947$	2.09596e10	0.801972	0.400986	−	0.916084i	$-0.368668\pi$
$947$	2.09596e10	0.801972	0.400986	+	0.916084i	$0.368668\pi$
$948$	−7.08114e9	−0.269944
$949$	1.63087e9	0.0619423
$950$	−2.19917e10	−0.832196
$951$	−3.88139e9	−0.146338
$952$	2.84134e8	0.0106732
$953$	−4.15816e10	−1.55624	−0.778119	−	0.628117i	$-0.783826\pi$
$953$	−4.15816e10	−1.55624	−0.778119	+	0.628117i	$0.783826\pi$
$954$	−7.28264e8	−0.0271562
$955$	2.13341e10	0.792616
$956$	1.91940e10	0.710498
$957$	−9.38398e9	−0.346095
$958$	−1.98131e10	−0.728070
$959$	−1.68452e10	−0.616751
$960$	2.68173e9	0.0978285
$961$	−1.37170e10	−0.498570
$962$	−2.99235e9	−0.108368
$963$	−1.32964e9	−0.0479779
$964$	1.64749e10	0.592317
$965$	−5.11751e10	−1.83321
$966$	7.54853e9	0.269428
$967$	2.40353e10	0.854784	0.427392	−	0.904066i	$-0.359432\pi$
$967$	2.40353e10	0.854784	0.427392	+	0.904066i	$0.359432\pi$
$968$	−4.37785e9	−0.155131
$969$	1.83529e9	0.0647996
$970$	2.59848e10	0.914151
$971$	2.42256e10	0.849194	0.424597	−	0.905382i	$-0.360416\pi$
$971$	2.42256e10	0.849194	0.424597	+	0.905382i	$0.360416\pi$
$972$	−9.18330e8	−0.0320750
$973$	−8.39550e8	−0.0292181
$974$	−1.09333e10	−0.379135
$975$	−3.88132e9	−0.134111
$976$	4.98598e8	0.0171663
$977$	−5.17070e10	−1.77386	−0.886928	−	0.461908i	$-0.847165\pi$
$977$	−5.17070e10	−1.77386	−0.886928	+	0.461908i	$0.847165\pi$
$978$	−2.11014e9	−0.0721315
$979$	3.09781e9	0.105515
$980$	−2.85285e9	−0.0968251
$981$	9.26661e9	0.313386
$982$	2.23999e10	0.754842
$983$	3.76673e10	1.26482	0.632408	−	0.774636i	$-0.282067\pi$
$983$	3.76673e10	1.26482	0.632408	+	0.774636i	$0.282067\pi$
$984$	4.21693e9	0.141096
$985$	−2.67744e10	−0.892672
$986$	8.49574e8	0.0282249
$987$	1.53723e9	0.0508895
$988$	5.90735e9	0.194869
$989$	3.96165e10	1.30223
$990$	1.17004e10	0.383245
$991$	4.35196e10	1.42045	0.710227	−	0.703973i	$-0.248592\pi$
$991$	4.35196e10	1.42045	0.710227	+	0.703973i	$0.248592\pi$
$992$	−3.84876e9	−0.125179
$993$	−9.20675e9	−0.298390
$994$	9.59013e8	0.0309722
$995$	−2.95410e10	−0.950701
$996$	3.82334e9	0.122613
$997$	1.93909e10	0.619676	0.309838	−	0.950789i	$-0.399725\pi$
$997$	1.93909e10	0.619676	0.309838	+	0.950789i	$0.399725\pi$
$998$	−2.86707e10	−0.913022
$999$	−3.35107e9	−0.106342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.n.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.n.1.2	✓	6	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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3031.11 q^{10} +5295.06 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +10230.0 q^{15} +4096.00 q^{16} -1617.93 q^{17} -5832.00 q^{18} +42012.9 q^{19} -24248.8 q^{20} -9261.00 q^{21} -42360.5 q^{22} +101886. q^{23} +13824.0 q^{24} +65431.2 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +65637.5 q^{29} -81839.8 q^{30} +117455. q^{31} -32768.0 q^{32} -142967. q^{33} +12943.4 q^{34} -129959. q^{35} +46656.0 q^{36} +170252. q^{37} -336103. q^{38} -59319.0 q^{39} +193991. q^{40} +305044. q^{41} +74088.0 q^{42} +388832. q^{43} +338884. q^{44} -276209. q^{45} -815087. q^{46} -165989. q^{47} -110592. q^{48} +117649. q^{49} -523450. q^{50} +43684.0 q^{51} +140608. q^{52} +124874. q^{53} +157464. q^{54} -2.00624e6 q^{55} -175616. q^{56} -1.13435e6 q^{57} -525100. q^{58} -1.79173e6 q^{59} +654719. q^{60} +121728. q^{61} -939639. q^{62} +250047. q^{63} +262144. q^{64} -832417. q^{65} +1.14373e6 q^{66} -154439. q^{67} -103547. q^{68} -2.75092e6 q^{69} +1.03967e6 q^{70} -349494. q^{71} -373248. q^{72} +742316. q^{73} -1.36202e6 q^{74} -1.76664e6 q^{75} +2.68883e6 q^{76} +1.81621e6 q^{77} +474552. q^{78} +4.09788e6 q^{79} -1.55193e6 q^{80} +531441. q^{81} -2.44035e6 q^{82} -2.21258e6 q^{83} -592704. q^{84} +613013. q^{85} -3.11065e6 q^{86} -1.77221e6 q^{87} -2.71107e6 q^{88} +585039. q^{89} +2.20968e6 q^{90} +753571. q^{91} +6.52070e6 q^{92} -3.17128e6 q^{93} +1.32792e6 q^{94} -1.59182e7 q^{95} +884736. q^{96} +8.57271e6 q^{97} -941192. q^{98} +3.86010e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 181 * q^5 + 1296 * q^6 + 2058 * q^7 - 3072 * q^8 + 4374 * q^9	\( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99}+O(q^{100}) \) 6 * q - 48 * q^2 - 162 * q^3 + 384 * q^4 - 181 * q^5 + 1296 * q^6 + 2058 * q^7 - 3072 * q^8 + 4374 * q^9 + 1448 * q^10 - 6130 * q^11 - 10368 * q^12 + 13182 * q^13 - 16464 * q^14 + 4887 * q^15 + 24576 * q^16 - 34610 * q^17 - 34992 * q^18 - 4085 * q^19 - 11584 * q^20 - 55566 * q^21 + 49040 * q^22 + 1515 * q^23 + 82944 * q^24 + 156609 * q^25 - 105456 * q^26 - 118098 * q^27 + 131712 * q^28 - 59395 * q^29 - 39096 * q^30 + 478241 * q^31 - 196608 * q^32 + 165510 * q^33 + 276880 * q^34 - 62083 * q^35 + 279936 * q^36 + 574310 * q^37 + 32680 * q^38 - 355914 * q^39 + 92672 * q^40 + 201552 * q^41 + 444528 * q^42 + 728605 * q^43 - 392320 * q^44 - 131949 * q^45 - 12120 * q^46 + 227615 * q^47 - 663552 * q^48 + 705894 * q^49 - 1252872 * q^50 + 934470 * q^51 + 843648 * q^52 + 26321 * q^53 + 944784 * q^54 + 2115010 * q^55 - 1053696 * q^56 + 110295 * q^57 + 475160 * q^58 + 478280 * q^59 + 312768 * q^60 - 501406 * q^61 - 3825928 * q^62 + 1500282 * q^63 + 1572864 * q^64 - 397657 * q^65 - 1324080 * q^66 - 3156366 * q^67 - 2215040 * q^68 - 40905 * q^69 + 496664 * q^70 - 2003644 * q^71 - 2239488 * q^72 + 3659111 * q^73 - 4594480 * q^74 - 4228443 * q^75 - 261440 * q^76 - 2102590 * q^77 + 2847312 * q^78 + 1131065 * q^79 - 741376 * q^80 + 3188646 * q^81 - 1612416 * q^82 - 9629297 * q^83 - 3556224 * q^84 + 895068 * q^85 - 5828840 * q^86 + 1603665 * q^87 + 3138560 * q^88 - 21977377 * q^89 + 1055592 * q^90 + 4521426 * q^91 + 96960 * q^92 - 12912507 * q^93 - 1820920 * q^94 - 19325507 * q^95 + 5308416 * q^96 - 26386649 * q^97 - 5647152 * q^98 - 4468770 * q^99

Introduction

L-functions

Modular forms

Varieties

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