LMFDB - Embedded newform 31.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [31,8,Mod(1,31)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(31, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("31.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	31.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:	$9.68393579001$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 538x^{5} + 2328x^{4} + 78000x^{3} - 344224x^{2} - 3123712x + 13256192 $ x^7 - 3x^6 - 538x^5 + 2328x^4 + 78000x^3 - 344224x^2 - 3123712x + 13256192
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.7787$ of defining polynomial
Character	$\chi$	$=$	31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.7787 q^{2} -70.5395 q^{3} +188.081 q^{4} +66.6445 q^{5} +1254.10 q^{6} +686.409 q^{7} -1068.16 q^{8} +2788.82 q^{9} +O(q^{10})$	\(q-17.7787 q^{2} -70.5395 q^{3} +188.081 q^{4} +66.6445 q^{5} +1254.10 q^{6} +686.409 q^{7} -1068.16 q^{8} +2788.82 q^{9} -1184.85 q^{10} -2883.59 q^{11} -13267.1 q^{12} +6467.92 q^{13} -12203.4 q^{14} -4701.07 q^{15} -5083.85 q^{16} -15053.0 q^{17} -49581.4 q^{18} +16361.2 q^{19} +12534.6 q^{20} -48418.9 q^{21} +51266.4 q^{22} +82813.3 q^{23} +75347.8 q^{24} -73683.5 q^{25} -114991. q^{26} -42451.8 q^{27} +129101. q^{28} -204845. q^{29} +83578.7 q^{30} +29791.0 q^{31} +227109. q^{32} +203407. q^{33} +267623. q^{34} +45745.4 q^{35} +524524. q^{36} +252609. q^{37} -290880. q^{38} -456244. q^{39} -71187.3 q^{40} -779327. q^{41} +860824. q^{42} -228505. q^{43} -542349. q^{44} +185859. q^{45} -1.47231e6 q^{46} -1.26908e6 q^{47} +358612. q^{48} -352386. q^{49} +1.31000e6 q^{50} +1.06183e6 q^{51} +1.21650e6 q^{52} -987964. q^{53} +754736. q^{54} -192175. q^{55} -733197. q^{56} -1.15411e6 q^{57} +3.64187e6 q^{58} -952323. q^{59} -884182. q^{60} +1.71628e6 q^{61} -529644. q^{62} +1.91427e6 q^{63} -3.38697e6 q^{64} +431051. q^{65} -3.61630e6 q^{66} +1.33563e6 q^{67} -2.83119e6 q^{68} -5.84161e6 q^{69} -813292. q^{70} +4.03243e6 q^{71} -2.97891e6 q^{72} +3.85064e6 q^{73} -4.49106e6 q^{74} +5.19760e6 q^{75} +3.07723e6 q^{76} -1.97932e6 q^{77} +8.11141e6 q^{78} -4.47981e6 q^{79} -338810. q^{80} -3.10462e6 q^{81} +1.38554e7 q^{82} -2.23749e6 q^{83} -9.10669e6 q^{84} -1.00320e6 q^{85} +4.06252e6 q^{86} +1.44497e7 q^{87} +3.08015e6 q^{88} -142190. q^{89} -3.30433e6 q^{90} +4.43964e6 q^{91} +1.55756e7 q^{92} -2.10144e6 q^{93} +2.25625e7 q^{94} +1.09038e6 q^{95} -1.60202e7 q^{96} -1.41852e7 q^{97} +6.26496e6 q^{98} -8.04180e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 17 q^{2} - 14 q^{3} + 229 q^{4} - 430 q^{5} - 528 q^{6} - 832 q^{7} - 135 q^{8} + 2115 q^{9}+O(q^{10}) $ 7 * q - 17 * q^2 - 14 * q^3 + 229 * q^4 - 430 * q^5 - 528 * q^6 - 832 * q^7 - 135 * q^8 + 2115 * q^9	\( 7 q - 17 q^{2} - 14 q^{3} + 229 q^{4} - 430 q^{5} - 528 q^{6} - 832 q^{7} - 135 q^{8} + 2115 q^{9} - 5232 q^{10} - 7886 q^{11} - 24952 q^{12} - 21844 q^{13} - 37902 q^{14} - 37796 q^{15} - 49583 q^{16} - 54822 q^{17} - 57869 q^{18} - 45352 q^{19} - 50110 q^{20} - 90756 q^{21} + 30696 q^{22} - 18464 q^{23} + 84472 q^{24} - 37691 q^{25} + 3906 q^{26} + 31360 q^{27} + 323084 q^{28} - 81488 q^{29} + 549372 q^{30} + 208537 q^{31} + 276513 q^{32} + 266552 q^{33} + 1152342 q^{34} + 154340 q^{35} + 712153 q^{36} + 431648 q^{37} + 798430 q^{38} - 239436 q^{39} + 600366 q^{40} - 1465990 q^{41} + 1324796 q^{42} - 598714 q^{43} - 660872 q^{44} - 728478 q^{45} - 356652 q^{46} - 2003572 q^{47} + 383736 q^{48} - 331317 q^{49} + 91595 q^{50} - 3313124 q^{51} - 28582 q^{52} - 1496844 q^{53} + 1032136 q^{54} - 1414452 q^{55} - 2199880 q^{56} - 6032240 q^{57} + 1517430 q^{58} - 2853828 q^{59} + 546920 q^{60} - 1486900 q^{61} - 506447 q^{62} - 4186384 q^{63} - 1940543 q^{64} - 2252240 q^{65} + 3180552 q^{66} - 5647492 q^{67} + 829234 q^{68} - 1992112 q^{69} + 8307114 q^{70} - 5168828 q^{71} + 5542765 q^{72} + 4710926 q^{73} + 4058410 q^{74} + 5700806 q^{75} + 15097160 q^{76} - 2020724 q^{77} + 20668432 q^{78} + 4582796 q^{79} + 6495030 q^{80} - 5939465 q^{81} + 8295096 q^{82} - 626514 q^{83} + 15579128 q^{84} + 8323116 q^{85} - 3575108 q^{86} + 19337220 q^{87} + 2100840 q^{88} - 13906634 q^{89} + 5754808 q^{90} + 10966300 q^{91} - 2803952 q^{92} - 417074 q^{93} + 21167040 q^{94} - 11547396 q^{95} - 1593224 q^{96} + 2962898 q^{97} - 25061151 q^{98} - 16496442 q^{99}+O(q^{100}) \) 7 * q - 17 * q^2 - 14 * q^3 + 229 * q^4 - 430 * q^5 - 528 * q^6 - 832 * q^7 - 135 * q^8 + 2115 * q^9 - 5232 * q^10 - 7886 * q^11 - 24952 * q^12 - 21844 * q^13 - 37902 * q^14 - 37796 * q^15 - 49583 * q^16 - 54822 * q^17 - 57869 * q^18 - 45352 * q^19 - 50110 * q^20 - 90756 * q^21 + 30696 * q^22 - 18464 * q^23 + 84472 * q^24 - 37691 * q^25 + 3906 * q^26 + 31360 * q^27 + 323084 * q^28 - 81488 * q^29 + 549372 * q^30 + 208537 * q^31 + 276513 * q^32 + 266552 * q^33 + 1152342 * q^34 + 154340 * q^35 + 712153 * q^36 + 431648 * q^37 + 798430 * q^38 - 239436 * q^39 + 600366 * q^40 - 1465990 * q^41 + 1324796 * q^42 - 598714 * q^43 - 660872 * q^44 - 728478 * q^45 - 356652 * q^46 - 2003572 * q^47 + 383736 * q^48 - 331317 * q^49 + 91595 * q^50 - 3313124 * q^51 - 28582 * q^52 - 1496844 * q^53 + 1032136 * q^54 - 1414452 * q^55 - 2199880 * q^56 - 6032240 * q^57 + 1517430 * q^58 - 2853828 * q^59 + 546920 * q^60 - 1486900 * q^61 - 506447 * q^62 - 4186384 * q^63 - 1940543 * q^64 - 2252240 * q^65 + 3180552 * q^66 - 5647492 * q^67 + 829234 * q^68 - 1992112 * q^69 + 8307114 * q^70 - 5168828 * q^71 + 5542765 * q^72 + 4710926 * q^73 + 4058410 * q^74 + 5700806 * q^75 + 15097160 * q^76 - 2020724 * q^77 + 20668432 * q^78 + 4582796 * q^79 + 6495030 * q^80 - 5939465 * q^81 + 8295096 * q^82 - 626514 * q^83 + 15579128 * q^84 + 8323116 * q^85 - 3575108 * q^86 + 19337220 * q^87 + 2100840 * q^88 - 13906634 * q^89 + 5754808 * q^90 + 10966300 * q^91 - 2803952 * q^92 - 417074 * q^93 + 21167040 * q^94 - 11547396 * q^95 - 1593224 * q^96 + 2962898 * q^97 - 25061151 * q^98 - 16496442 * 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 $

$2$	−17.7787	−1.57143	−0.785714	−	0.618590i	$-0.787704\pi$
$2$	−17.7787	−1.57143	−0.785714	+	0.618590i	$0.787704\pi$
$3$	−70.5395	−1.50837	−0.754185	−	0.656662i	$-0.771968\pi$
$3$	−70.5395	−1.50837	−0.754185	+	0.656662i	$0.771968\pi$
$4$	188.081	1.46938
$5$	66.6445	0.238435	0.119217	−	0.992868i	$-0.461961\pi$
$5$	66.6445	0.238435	0.119217	+	0.992868i	$0.461961\pi$
$6$	1254.10	2.37029
$7$	686.409	0.756379	0.378190	−	0.925728i	$-0.376547\pi$
$7$	686.409	0.756379	0.378190	+	0.925728i	$0.376547\pi$
$8$	−1068.16	−0.737604
$9$	2788.82	1.27518
$10$	−1184.85	−0.374683
$11$	−2883.59	−0.653219	−0.326610	−	0.945159i	$-0.605906\pi$
$11$	−2883.59	−0.653219	−0.326610	+	0.945159i	$0.605906\pi$
$12$	−13267.1	−2.21637
$13$	6467.92	0.816513	0.408257	−	0.912867i	$-0.366137\pi$
$13$	6467.92	0.816513	0.408257	+	0.912867i	$0.366137\pi$
$14$	−12203.4	−1.18860
$15$	−4701.07	−0.359647
$16$	−5083.85	−0.310294
$17$	−15053.0	−0.743109	−0.371555	−	0.928411i	$-0.621175\pi$
$17$	−15053.0	−0.743109	−0.371555	+	0.928411i	$0.621175\pi$
$18$	−49581.4	−2.00385
$19$	16361.2	0.547238	0.273619	−	0.961838i	$-0.411779\pi$
$19$	16361.2	0.547238	0.273619	+	0.961838i	$0.411779\pi$
$20$	12534.6	0.350352
$21$	−48418.9	−1.14090
$22$	51266.4	1.02649
$23$	82813.3	1.41923	0.709615	−	0.704590i	$-0.248869\pi$
$23$	82813.3	1.41923	0.709615	+	0.704590i	$0.248869\pi$
$24$	75347.8	1.11258
$25$	−73683.5	−0.943149
$26$	−114991.	−1.28309
$27$	−42451.8	−0.415071
$28$	129101.	1.11141
$29$	−204845.	−1.55967	−0.779834	−	0.625986i	$-0.784697\pi$
$29$	−204845.	−1.55967	−0.779834	+	0.625986i	$0.784697\pi$
$30$	83578.7	0.565160
$31$	29791.0	0.179605
$31$	29791.0	0.179605
$32$	227109.	1.22521
$33$	203407.	0.985296
$34$	267623.	1.16774
$35$	45745.4	0.180347
$36$	524524.	1.87373
$37$	252609.	0.819866	0.409933	−	0.912116i	$-0.365552\pi$
$37$	252609.	0.819866	0.409933	+	0.912116i	$0.365552\pi$
$38$	−290880.	−0.859945
$39$	−456244.	−1.23160
$40$	−71187.3	−0.175870
$41$	−779327.	−1.76594	−0.882971	−	0.469428i	$-0.844460\pi$
$41$	−779327.	−1.76594	−0.882971	+	0.469428i	$0.844460\pi$
$42$	860824.	1.79284
$43$	−228505.	−0.438285	−0.219143	−	0.975693i	$-0.570326\pi$
$43$	−228505.	−0.438285	−0.219143	+	0.975693i	$0.570326\pi$
$44$	−542349.	−0.959830
$45$	185859.	0.304047
$46$	−1.47231e6	−2.23022
$47$	−1.26908e6	−1.78298	−0.891488	−	0.453043i	$-0.850338\pi$
$47$	−1.26908e6	−1.78298	−0.891488	+	0.453043i	$0.850338\pi$
$48$	358612.	0.468037
$49$	−352386.	−0.427890
$50$	1.31000e6	1.48209
$51$	1.06183e6	1.12088
$52$	1.21650e6	1.19977
$53$	−987964.	−0.911540	−0.455770	−	0.890098i	$-0.650636\pi$
$53$	−987964.	−0.911540	−0.455770	+	0.890098i	$0.650636\pi$
$54$	754736.	0.652254
$55$	−192175.	−0.155750
$56$	−733197.	−0.557908
$57$	−1.15411e6	−0.825437
$58$	3.64187e6	2.45091
$59$	−952323.	−0.603674	−0.301837	−	0.953360i	$-0.597600\pi$
$59$	−952323.	−0.603674	−0.301837	+	0.953360i	$0.597600\pi$
$60$	−884182.	−0.528460
$61$	1.71628e6	0.968131	0.484066	−	0.875032i	$-0.339160\pi$
$61$	1.71628e6	0.968131	0.484066	+	0.875032i	$0.339160\pi$
$62$	−529644.	−0.282237
$63$	1.91427e6	0.964519
$64$	−3.38697e6	−1.61503
$65$	431051.	0.194685
$66$	−3.61630e6	−1.54832
$67$	1.33563e6	0.542530	0.271265	−	0.962505i	$-0.412558\pi$
$67$	1.33563e6	0.542530	0.271265	+	0.962505i	$0.412558\pi$
$68$	−2.83119e6	−1.09191
$69$	−5.84161e6	−2.14072
$70$	−813292.	−0.283402
$71$	4.03243e6	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$71$	4.03243e6	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$72$	−2.97891e6	−0.940577
$73$	3.85064e6	1.15852	0.579260	−	0.815143i	$-0.303342\pi$
$73$	3.85064e6	1.15852	0.579260	+	0.815143i	$0.303342\pi$
$74$	−4.49106e6	−1.28836
$75$	5.19760e6	1.42262
$76$	3.07723e6	0.804103
$77$	−1.97932e6	−0.494082
$78$	8.11141e6	1.93538
$79$	−4.47981e6	−1.02227	−0.511134	−	0.859501i	$-0.670774\pi$
$79$	−4.47981e6	−1.02227	−0.511134	+	0.859501i	$0.670774\pi$
$80$	−338810.	−0.0739847
$81$	−3.10462e6	−0.649098
$82$	1.38554e7	2.77505
$83$	−2.23749e6	−0.429524	−0.214762	−	0.976666i	$-0.568898\pi$
$83$	−2.23749e6	−0.429524	−0.214762	+	0.976666i	$0.568898\pi$
$84$	−9.10669e6	−1.67642
$85$	−1.00320e6	−0.177183
$86$	4.06252e6	0.688733
$87$	1.44497e7	2.35256
$88$	3.08015e6	0.481817
$89$	−142190.	−0.0213799	−0.0106899	−	0.999943i	$-0.503403\pi$
$89$	−142190.	−0.0213799	−0.0106899	+	0.999943i	$0.503403\pi$
$90$	−3.30433e6	−0.477787
$91$	4.43964e6	0.617594
$92$	1.55756e7	2.08539
$93$	−2.10144e6	−0.270911
$94$	2.25625e7	2.80182
$95$	1.09038e6	0.130480
$96$	−1.60202e7	−1.84807
$97$	−1.41852e7	−1.57810	−0.789048	−	0.614331i	$-0.789426\pi$
$97$	−1.41852e7	−1.57810	−0.789048	+	0.614331i	$0.789426\pi$
$98$	6.26496e6	0.672399
$99$	−8.04180e6	−0.832971
$100$	−1.38585e7	−1.38585
$101$	−1.27518e7	−1.23154	−0.615768	−	0.787927i	$-0.711154\pi$
$101$	−1.27518e7	−1.23154	−0.615768	+	0.787927i	$0.711154\pi$
$102$	−1.88780e7	−1.76139
$103$	−1.18137e7	−1.06526	−0.532632	−	0.846347i	$-0.678797\pi$
$103$	−1.18137e7	−1.06526	−0.532632	+	0.846347i	$0.678797\pi$
$104$	−6.90881e6	−0.602263
$105$	−3.22685e6	−0.272030
$106$	1.75647e7	1.43242
$107$	7.93394e6	0.626103	0.313052	−	0.949736i	$-0.398649\pi$
$107$	7.93394e6	0.626103	0.313052	+	0.949736i	$0.398649\pi$
$108$	−7.98438e6	−0.609899
$109$	5.32299e6	0.393698	0.196849	−	0.980434i	$-0.436929\pi$
$109$	5.32299e6	0.393698	0.196849	+	0.980434i	$0.436929\pi$
$110$	3.41662e6	0.244750
$111$	−1.78189e7	−1.23666
$112$	−3.48960e6	−0.234700
$113$	−1.96185e7	−1.27906	−0.639531	−	0.768765i	$-0.720871\pi$
$113$	−1.96185e7	−1.27906	−0.639531	+	0.768765i	$0.720871\pi$
$114$	2.05185e7	1.29712
$115$	5.51905e6	0.338393
$116$	−3.85275e7	−2.29175
$117$	1.80379e7	1.04120
$118$	1.69310e7	0.948629
$119$	−1.03325e7	−0.562072
$120$	5.02151e6	0.265277
$121$	−1.11721e7	−0.573304
$122$	−3.05132e7	−1.52135
$123$	5.49733e7	2.66369
$124$	5.60313e6	0.263909
$125$	−1.01172e7	−0.463314
$126$	−3.40331e7	−1.51567
$127$	1.18043e7	0.511360	0.255680	−	0.966761i	$-0.417701\pi$
$127$	1.18043e7	0.511360	0.255680	+	0.966761i	$0.417701\pi$
$128$	3.11458e7	1.31270
$129$	1.61186e7	0.661096
$130$	−7.66352e6	−0.305933
$131$	−1.04384e7	−0.405679	−0.202840	−	0.979212i	$-0.565017\pi$
$131$	−1.04384e7	−0.405679	−0.202840	+	0.979212i	$0.565017\pi$
$132$	3.82570e7	1.44778
$133$	1.12304e7	0.413920
$134$	−2.37457e7	−0.852546
$135$	−2.82918e6	−0.0989673
$136$	1.60791e7	0.548120
$137$	1.59701e7	0.530622	0.265311	−	0.964163i	$-0.414525\pi$
$137$	1.59701e7	0.530622	0.265311	+	0.964163i	$0.414525\pi$
$138$	1.03856e8	3.36399
$139$	2.45850e6	0.0776458	0.0388229	−	0.999246i	$-0.487639\pi$
$139$	2.45850e6	0.0776458	0.0388229	+	0.999246i	$0.487639\pi$
$140$	8.60384e6	0.264999
$141$	8.95201e7	2.68939
$142$	−7.16912e7	−2.10115
$143$	−1.86508e7	−0.533362
$144$	−1.41779e7	−0.395680
$145$	−1.36518e7	−0.371879
$146$	−6.84593e7	−1.82053
$147$	2.48571e7	0.645417
$148$	4.75110e7	1.20470
$149$	−967011.	−0.0239486	−0.0119743	−	0.999928i	$-0.503812\pi$
$149$	−967011.	−0.0239486	−0.0119743	+	0.999928i	$0.503812\pi$
$150$	−9.24064e7	−2.23554
$151$	−4.72383e7	−1.11654	−0.558271	−	0.829658i	$-0.688535\pi$
$151$	−4.72383e7	−1.11654	−0.558271	+	0.829658i	$0.688535\pi$
$152$	−1.74764e7	−0.403645
$153$	−4.19801e7	−0.947597
$154$	3.51897e7	0.776413
$155$	1.98541e6	0.0428241
$156$	−8.58109e7	−1.80970
$157$	−6.10249e7	−1.25851	−0.629257	−	0.777197i	$-0.716641\pi$
$157$	−6.10249e7	−1.25851	−0.629257	+	0.777197i	$0.716641\pi$
$158$	7.96450e7	1.60642
$159$	6.96904e7	1.37494
$160$	1.51356e7	0.292132
$161$	5.68438e7	1.07348
$162$	5.51959e7	1.02001
$163$	397126.	0.00718243	0.00359122	−	0.999994i	$-0.498857\pi$
$163$	397126.	0.00718243	0.00359122	+	0.999994i	$0.498857\pi$
$164$	−1.46577e8	−2.59485
$165$	1.35559e7	0.234929
$166$	3.97796e7	0.674967
$167$	7.47782e7	1.24242	0.621209	−	0.783645i	$-0.286642\pi$
$167$	7.47782e7	1.24242	0.621209	+	0.783645i	$0.286642\pi$
$168$	5.17194e7	0.841532
$169$	−2.09145e7	−0.333306
$170$	1.78356e7	0.278430
$171$	4.56283e7	0.697826
$172$	−4.29776e7	−0.644009
$173$	8.75293e7	1.28526	0.642632	−	0.766175i	$-0.277842\pi$
$173$	8.75293e7	1.28526	0.642632	+	0.766175i	$0.277842\pi$
$174$	−2.56896e8	−3.69687
$175$	−5.05770e7	−0.713378
$176$	1.46597e7	0.202690
$177$	6.71763e7	0.910563
$178$	2.52796e6	0.0335970
$179$	1.14337e8	1.49005	0.745027	−	0.667034i	$-0.232436\pi$
$179$	1.14337e8	1.49005	0.745027	+	0.667034i	$0.232436\pi$
$180$	3.49566e7	0.446761
$181$	−6.25246e7	−0.783747	−0.391873	−	0.920019i	$-0.628173\pi$
$181$	−6.25246e7	−0.783747	−0.391873	+	0.920019i	$0.628173\pi$
$182$	−7.89309e7	−0.970504
$183$	−1.21066e8	−1.46030
$184$	−8.84582e7	−1.04683
$185$	1.68350e7	0.195484
$186$	3.73608e7	0.425717
$187$	4.34068e7	0.485413
$188$	−2.38690e8	−2.61988
$189$	−2.91393e7	−0.313951
$190$	−1.93855e7	−0.205041
$191$	−1.46859e8	−1.52505	−0.762526	−	0.646957i	$-0.776041\pi$
$191$	−1.46859e8	−1.52505	−0.762526	+	0.646957i	$0.776041\pi$
$192$	2.38915e8	2.43606
$193$	−3.29314e7	−0.329731	−0.164865	−	0.986316i	$-0.552719\pi$
$193$	−3.29314e7	−0.329731	−0.164865	+	0.986316i	$0.552719\pi$
$194$	2.52194e8	2.47986
$195$	−3.04061e7	−0.293657
$196$	−6.62772e7	−0.628735
$197$	−1.29534e8	−1.20713	−0.603564	−	0.797315i	$-0.706253\pi$
$197$	−1.29534e8	−1.20713	−0.603564	+	0.797315i	$0.706253\pi$
$198$	1.42973e8	1.30895
$199$	4.53412e7	0.407856	0.203928	−	0.978986i	$-0.434629\pi$
$199$	4.53412e7	0.407856	0.203928	+	0.978986i	$0.434629\pi$
$200$	7.87061e7	0.695670
$201$	−9.42145e7	−0.818335
$202$	2.26710e8	1.93527
$203$	−1.40607e8	−1.17970
$204$	1.99711e8	1.64701
$205$	−5.19379e7	−0.421062
$206$	2.10033e8	1.67398
$207$	2.30951e8	1.80977
$208$	−3.28820e7	−0.253359
$209$	−4.71789e7	−0.357467
$210$	5.73692e7	0.427475
$211$	−2.40362e7	−0.176148	−0.0880739	−	0.996114i	$-0.528071\pi$
$211$	−2.40362e7	−0.176148	−0.0880739	+	0.996114i	$0.528071\pi$
$212$	−1.85817e8	−1.33940
$213$	−2.84445e8	−2.01683
$214$	−1.41055e8	−0.983876
$215$	−1.52286e7	−0.104502
$216$	4.53455e7	0.306158
$217$	2.04488e7	0.135850
$218$	−9.46358e7	−0.618668
$219$	−2.71622e8	−1.74748
$220$	−3.61446e7	−0.228857
$221$	−9.73619e7	−0.606759
$222$	3.16797e8	1.94332
$223$	−1.10811e8	−0.669137	−0.334569	−	0.942371i	$-0.608591\pi$
$223$	−1.10811e8	−0.669137	−0.334569	+	0.942371i	$0.608591\pi$
$224$	1.55890e8	0.926722
$225$	−2.05490e8	−1.20268
$226$	3.48791e8	2.00995
$227$	2.99831e8	1.70132	0.850661	−	0.525715i	$-0.176202\pi$
$227$	2.99831e8	1.70132	0.850661	+	0.525715i	$0.176202\pi$
$228$	−2.17066e8	−1.21288
$229$	3.37036e8	1.85461	0.927305	−	0.374306i	$-0.122119\pi$
$229$	3.37036e8	1.85461	0.927305	+	0.374306i	$0.122119\pi$
$230$	−9.81214e7	−0.531761
$231$	1.39620e8	0.745258
$232$	2.18808e8	1.15042
$233$	−1.18388e8	−0.613145	−0.306572	−	0.951847i	$-0.599182\pi$
$233$	−1.18388e8	−0.613145	−0.306572	+	0.951847i	$0.599182\pi$
$234$	−3.20689e8	−1.63617
$235$	−8.45770e7	−0.425123
$236$	−1.79114e8	−0.887029
$237$	3.16003e8	1.54196
$238$	1.83699e8	0.883256
$239$	−1.66861e8	−0.790610	−0.395305	−	0.918550i	$-0.629361\pi$
$239$	−1.66861e8	−0.790610	−0.395305	+	0.918550i	$0.629361\pi$
$240$	2.38995e7	0.111596
$241$	6.70826e7	0.308710	0.154355	−	0.988015i	$-0.450670\pi$
$241$	6.70826e7	0.308710	0.154355	+	0.988015i	$0.450670\pi$
$242$	1.98625e8	0.900906
$243$	3.11840e8	1.39415
$244$	3.22800e8	1.42256
$245$	−2.34846e7	−0.102024
$246$	−9.77353e8	−4.18580
$247$	1.05823e8	0.446827
$248$	−3.18217e7	−0.132478
$249$	1.57831e8	0.647882
$250$	1.79870e8	0.728064
$251$	1.32817e7	0.0530145	0.0265072	−	0.999649i	$-0.491561\pi$
$251$	1.32817e7	0.0530145	0.0265072	+	0.999649i	$0.491561\pi$
$252$	3.60038e8	1.41725
$253$	−2.38800e8	−0.927068
$254$	−2.09865e8	−0.803566
$255$	7.07653e7	0.267257
$256$	−1.20199e8	−0.447777
$257$	4.76683e8	1.75172	0.875858	−	0.482569i	$-0.160296\pi$
$257$	4.76683e8	1.75172	0.875858	+	0.482569i	$0.160296\pi$
$258$	−2.86568e8	−1.03886
$259$	1.73393e8	0.620130
$260$	8.10727e7	0.286067
$261$	−5.71275e8	−1.98886
$262$	1.85580e8	0.637495
$263$	−2.82902e8	−0.958938	−0.479469	−	0.877559i	$-0.659171\pi$
$263$	−2.82902e8	−0.958938	−0.479469	+	0.877559i	$0.659171\pi$
$264$	−2.17272e8	−0.726758
$265$	−6.58423e7	−0.217343
$266$	−1.99662e8	−0.650445
$267$	1.00300e7	0.0322488
$268$	2.51206e8	0.797185
$269$	−2.01149e8	−0.630064	−0.315032	−	0.949081i	$-0.602015\pi$
$269$	−2.01149e8	−0.630064	−0.315032	+	0.949081i	$0.602015\pi$
$270$	5.02990e7	0.155520
$271$	−1.44594e8	−0.441324	−0.220662	−	0.975350i	$-0.570822\pi$
$271$	−1.44594e8	−0.441324	−0.220662	+	0.975350i	$0.570822\pi$
$272$	7.65273e7	0.230582
$273$	−3.13170e8	−0.931560
$274$	−2.83927e8	−0.833833
$275$	2.12473e8	0.616083
$276$	−1.09870e9	−3.14554
$277$	−6.63088e8	−1.87453	−0.937264	−	0.348620i	$-0.886650\pi$
$277$	−6.63088e8	−1.87453	−0.937264	+	0.348620i	$0.886650\pi$
$278$	−4.37088e7	−0.122015
$279$	8.30816e7	0.229029
$280$	−4.88636e7	−0.133025
$281$	6.17451e8	1.66009	0.830043	−	0.557700i	$-0.188316\pi$
$281$	6.17451e8	1.66009	0.830043	+	0.557700i	$0.188316\pi$
$282$	−1.59155e9	−4.22618
$283$	6.05253e7	0.158739	0.0793697	−	0.996845i	$-0.474709\pi$
$283$	6.05253e7	0.158739	0.0793697	+	0.996845i	$0.474709\pi$
$284$	7.58424e8	1.96471
$285$	−7.69149e7	−0.196813
$286$	3.31587e8	0.838140
$287$	−5.34937e8	−1.33572
$288$	6.33366e8	1.56236
$289$	−1.83745e8	−0.447789
$290$	2.42711e8	0.584381
$291$	1.00061e9	2.38035
$292$	7.24234e8	1.70231
$293$	−7.06523e8	−1.64093	−0.820464	−	0.571698i	$-0.806285\pi$
$293$	−7.06523e8	−1.64093	−0.820464	+	0.571698i	$0.806285\pi$
$294$	−4.41927e8	−1.01423
$295$	−6.34671e7	−0.143937
$296$	−2.69828e8	−0.604737
$297$	1.22413e8	0.271133
$298$	1.71922e7	0.0376334
$299$	5.35630e8	1.15882
$300$	9.77570e8	2.09037
$301$	−1.56848e8	−0.331510
$302$	8.39835e8	1.75457
$303$	8.99506e8	1.85761
$304$	−8.31776e7	−0.169804
$305$	1.14381e8	0.230836
$306$	7.46351e8	1.48908
$307$	8.65076e8	1.70636	0.853179	−	0.521619i	$-0.174672\pi$
$307$	8.65076e8	1.70636	0.853179	+	0.521619i	$0.174672\pi$
$308$	−3.72273e8	−0.725996
$309$	8.33334e8	1.60681
$310$	−3.52979e7	−0.0672950
$311$	6.64291e8	1.25227	0.626134	−	0.779715i	$-0.284636\pi$
$311$	6.64291e8	1.25227	0.626134	+	0.779715i	$0.284636\pi$
$312$	4.87344e8	0.908436
$313$	−7.87347e8	−1.45131	−0.725657	−	0.688057i	$-0.758464\pi$
$313$	−7.87347e8	−1.45131	−0.725657	+	0.688057i	$0.758464\pi$
$314$	1.08494e9	1.97766
$315$	1.27575e8	0.229975
$316$	−8.42567e8	−1.50210
$317$	3.50228e8	0.617510	0.308755	−	0.951142i	$-0.400088\pi$
$317$	3.50228e8	0.617510	0.308755	+	0.951142i	$0.400088\pi$
$318$	−1.23900e9	−2.16062
$319$	5.90689e8	1.01881
$320$	−2.25723e8	−0.385079
$321$	−5.59656e8	−0.944395
$322$	−1.01061e9	−1.68689
$323$	−2.46285e8	−0.406658
$324$	−5.83920e8	−0.953775
$325$	−4.76579e8	−0.770094
$326$	−7.06037e6	−0.0112867
$327$	−3.75481e8	−0.593842
$328$	8.32450e8	1.30257
$329$	−8.71106e8	−1.34861
$330$	−2.41007e8	−0.369173
$331$	−3.82449e8	−0.579663	−0.289831	−	0.957078i	$-0.593599\pi$
$331$	−3.82449e8	−0.579663	−0.289831	+	0.957078i	$0.593599\pi$
$332$	−4.20830e8	−0.631137
$333$	7.04480e8	1.04548
$334$	−1.32946e9	−1.95237
$335$	8.90122e7	0.129358
$336$	2.46154e8	0.354014
$337$	−2.10576e8	−0.299713	−0.149856	−	0.988708i	$-0.547881\pi$
$337$	−2.10576e8	−0.299713	−0.149856	+	0.988708i	$0.547881\pi$
$338$	3.71832e8	0.523767
$339$	1.38388e9	1.92930
$340$	−1.88683e8	−0.260350
$341$	−8.59050e7	−0.117322
$342$	−8.11210e8	−1.09658
$343$	−8.07168e8	−1.08003
$344$	2.44081e8	0.323281
$345$	−3.89311e8	−0.510422
$346$	−1.55616e9	−2.01970
$347$	−1.33619e8	−0.171678	−0.0858388	−	0.996309i	$-0.527357\pi$
$347$	−1.33619e8	−0.171678	−0.0858388	+	0.996309i	$0.527357\pi$
$348$	2.71771e9	3.45681
$349$	−8.50804e7	−0.107137	−0.0535686	−	0.998564i	$-0.517060\pi$
$349$	−8.50804e7	−0.107137	−0.0535686	+	0.998564i	$0.517060\pi$
$350$	8.99192e8	1.12102
$351$	−2.74575e8	−0.338911
$352$	−6.54890e8	−0.800329
$353$	1.00864e9	1.22046	0.610231	−	0.792224i	$-0.291077\pi$
$353$	1.00864e9	1.22046	0.610231	+	0.792224i	$0.291077\pi$
$354$	−1.19431e9	−1.43088
$355$	2.68739e8	0.318810
$356$	−2.67434e7	−0.0314153
$357$	7.28851e8	0.847813
$358$	−2.03276e9	−2.34151
$359$	−6.28832e8	−0.717306	−0.358653	−	0.933471i	$-0.616764\pi$
$359$	−6.28832e8	−0.717306	−0.358653	+	0.933471i	$0.616764\pi$
$360$	−1.98528e8	−0.224266
$361$	−6.26184e8	−0.700530
$362$	1.11160e9	1.23160
$363$	7.88073e8	0.864755
$364$	8.35013e8	0.907483
$365$	2.56624e8	0.276231
$366$	2.15239e9	2.29475
$367$	−1.22300e9	−1.29150	−0.645752	−	0.763547i	$-0.723456\pi$
$367$	−1.22300e9	−1.29150	−0.645752	+	0.763547i	$0.723456\pi$
$368$	−4.21010e8	−0.440378
$369$	−2.17340e9	−2.25189
$370$	−2.99304e8	−0.307190
$371$	−6.78147e8	−0.689470
$372$	−3.95242e8	−0.398073
$373$	9.45435e8	0.943302	0.471651	−	0.881785i	$-0.343658\pi$
$373$	9.45435e8	0.943302	0.471651	+	0.881785i	$0.343658\pi$
$374$	−7.71715e8	−0.762792
$375$	7.13662e8	0.698848
$376$	1.35558e9	1.31513
$377$	−1.32492e9	−1.27349
$378$	5.18057e8	0.493352
$379$	7.06115e8	0.666251	0.333126	−	0.942882i	$-0.391897\pi$
$379$	7.06115e8	0.666251	0.333126	+	0.942882i	$0.391897\pi$
$380$	2.05080e8	0.191726
$381$	−8.32669e8	−0.771320
$382$	2.61096e9	2.39651
$383$	1.45767e9	1.32576	0.662878	−	0.748727i	$-0.269335\pi$
$383$	1.45767e9	1.32576	0.662878	+	0.748727i	$0.269335\pi$
$384$	−2.19701e9	−1.98003
$385$	−1.31911e8	−0.117806
$386$	5.85476e8	0.518148
$387$	−6.37259e8	−0.558892
$388$	−2.66796e9	−2.31883
$389$	1.20483e9	1.03777	0.518885	−	0.854844i	$-0.326347\pi$
$389$	1.20483e9	1.03777	0.518885	+	0.854844i	$0.326347\pi$
$390$	5.40581e8	0.461460
$391$	−1.24659e9	−1.05464
$392$	3.76406e8	0.315614
$393$	7.36316e8	0.611914
$394$	2.30295e9	1.89691
$395$	−2.98554e8	−0.243744
$396$	−1.51251e9	−1.22396
$397$	1.18292e9	0.948827	0.474413	−	0.880302i	$-0.342660\pi$
$397$	1.18292e9	0.948827	0.474413	+	0.880302i	$0.342660\pi$
$398$	−8.06106e8	−0.640916
$399$	−7.92189e8	−0.624344
$400$	3.74596e8	0.292653
$401$	2.40152e9	1.85986	0.929931	−	0.367734i	$-0.119866\pi$
$401$	2.40152e9	1.85986	0.929931	+	0.367734i	$0.119866\pi$
$402$	1.67501e9	1.28595
$403$	1.92686e8	0.146650
$404$	−2.39838e9	−1.80960
$405$	−2.06905e8	−0.154767
$406$	2.49981e9	1.85382
$407$	−7.28421e8	−0.535553
$408$	−1.13421e9	−0.826768
$409$	1.83827e9	1.32855	0.664275	−	0.747488i	$-0.268740\pi$
$409$	1.83827e9	1.32855	0.664275	+	0.747488i	$0.268740\pi$
$410$	9.23386e8	0.661668
$411$	−1.12652e9	−0.800374
$412$	−2.22194e9	−1.56528
$413$	−6.53683e8	−0.456606
$414$	−4.10600e9	−2.84392
$415$	−1.49116e8	−0.102413
$416$	1.46892e9	1.00040
$417$	−1.73421e8	−0.117119
$418$	8.38778e8	0.561733
$419$	−2.30007e9	−1.52754	−0.763768	−	0.645491i	$-0.776653\pi$
$419$	−2.30007e9	−1.52754	−0.763768	+	0.645491i	$0.776653\pi$
$420$	−6.06910e8	−0.399716
$421$	−6.25449e8	−0.408512	−0.204256	−	0.978918i	$-0.565477\pi$
$421$	−6.25449e8	−0.408512	−0.204256	+	0.978918i	$0.565477\pi$
$422$	4.27332e8	0.276804
$423$	−3.53922e9	−2.27361
$424$	1.05531e9	0.672355
$425$	1.10916e9	0.700863
$426$	5.05706e9	3.16931
$427$	1.17807e9	0.732274
$428$	1.49223e9	0.919986
$429$	1.31562e9	0.804507
$430$	2.70745e8	0.164218
$431$	−7.79146e6	−0.00468758	−0.00234379	−	0.999997i	$-0.500746\pi$
$431$	−7.79146e6	−0.00468758	−0.00234379	+	0.999997i	$0.500746\pi$
$432$	2.15818e8	0.128794
$433$	−2.49979e9	−1.47978	−0.739889	−	0.672729i	$-0.765122\pi$
$433$	−2.49979e9	−1.47978	−0.739889	+	0.672729i	$0.765122\pi$
$434$	−3.63553e8	−0.213478
$435$	9.62990e8	0.560931
$436$	1.00116e9	0.578494
$437$	1.35492e9	0.776657
$438$	4.82908e9	2.74603
$439$	−5.26141e8	−0.296809	−0.148404	−	0.988927i	$-0.547414\pi$
$439$	−5.26141e8	−0.296809	−0.148404	+	0.988927i	$0.547414\pi$
$440$	2.05275e8	0.114882
$441$	−9.82740e8	−0.545637
$442$	1.73096e9	0.953477
$443$	−1.79026e9	−0.978370	−0.489185	−	0.872180i	$-0.662706\pi$
$443$	−1.79026e9	−0.978370	−0.489185	+	0.872180i	$0.662706\pi$
$444$	−3.35140e9	−1.81713
$445$	−9.47621e6	−0.00509771
$446$	1.97007e9	1.05150
$447$	6.82124e7	0.0361233
$448$	−2.32484e9	−1.22158
$449$	−2.22533e9	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$449$	−2.22533e9	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$450$	3.65334e9	1.88993
$451$	2.24726e9	1.15355
$452$	−3.68988e9	−1.87943
$453$	3.33217e9	1.68416
$454$	−5.33060e9	−2.67350
$455$	2.95877e8	0.147256
$456$	1.23278e9	0.608846
$457$	3.27707e9	1.60612	0.803061	−	0.595897i	$-0.203203\pi$
$457$	3.27707e9	1.60612	0.803061	+	0.595897i	$0.203203\pi$
$458$	−5.99206e9	−2.91439
$459$	6.39028e8	0.308443
$460$	1.03803e9	0.497230
$461$	−1.26834e9	−0.602952	−0.301476	−	0.953474i	$-0.597479\pi$
$461$	−1.26834e9	−0.602952	−0.301476	+	0.953474i	$0.597479\pi$
$462$	−2.48226e9	−1.17112
$463$	−2.11328e9	−0.989520	−0.494760	−	0.869030i	$-0.664744\pi$
$463$	−2.11328e9	−0.989520	−0.494760	+	0.869030i	$0.664744\pi$
$464$	1.04140e9	0.483955
$465$	−1.40049e8	−0.0645946
$466$	2.10479e9	0.963513
$467$	1.74008e9	0.790608	0.395304	−	0.918550i	$-0.370639\pi$
$467$	1.74008e9	0.790608	0.395304	+	0.918550i	$0.370639\pi$
$468$	3.39258e9	1.52992
$469$	9.16786e8	0.410358
$470$	1.50367e9	0.668050
$471$	4.30466e9	1.89830
$472$	1.01724e9	0.445272
$473$	6.58916e8	0.286296
$474$	−5.61812e9	−2.42307
$475$	−1.20555e9	−0.516127
$476$	−1.94335e9	−0.825901
$477$	−2.75525e9	−1.16238
$478$	2.96657e9	1.24239
$479$	−9.96854e8	−0.414436	−0.207218	−	0.978295i	$-0.566441\pi$
$479$	−9.96854e8	−0.414436	−0.207218	+	0.978295i	$0.566441\pi$
$480$	−1.06766e9	−0.440643
$481$	1.63386e9	0.669432
$482$	−1.19264e9	−0.485115
$483$	−4.00973e9	−1.61920
$484$	−2.10126e9	−0.842405
$485$	−9.45363e8	−0.376273
$486$	−5.54410e9	−2.19081
$487$	−3.93984e9	−1.54571	−0.772853	−	0.634585i	$-0.781171\pi$
$487$	−3.93984e9	−1.54571	−0.772853	+	0.634585i	$0.781171\pi$
$488$	−1.83327e9	−0.714097
$489$	−2.80131e7	−0.0108338
$490$	4.17525e8	0.160323
$491$	−3.93792e8	−0.150135	−0.0750674	−	0.997178i	$-0.523917\pi$
$491$	−3.93792e8	−0.150135	−0.0750674	+	0.997178i	$0.523917\pi$
$492$	1.03395e10	3.91399
$493$	3.08354e9	1.15900
$494$	−1.88139e9	−0.702157
$495$	−5.35942e8	−0.198609
$496$	−1.51453e8	−0.0557304
$497$	2.76789e9	1.01135
$498$	−2.80603e9	−1.01810
$499$	1.38870e9	0.500329	0.250164	−	0.968203i	$-0.419515\pi$
$499$	1.38870e9	0.500329	0.250164	+	0.968203i	$0.419515\pi$
$500$	−1.90286e9	−0.680786
$501$	−5.27482e9	−1.87402
$502$	−2.36130e8	−0.0833084
$503$	−6.25515e8	−0.219154	−0.109577	−	0.993978i	$-0.534950\pi$
$503$	−6.25515e8	−0.219154	−0.109577	+	0.993978i	$0.534950\pi$
$504$	−2.04475e9	−0.711433
$505$	−8.49838e8	−0.293641
$506$	4.24554e9	1.45682
$507$	1.47530e9	0.502749
$508$	2.22017e9	0.751385
$509$	2.21089e9	0.743114	0.371557	−	0.928410i	$-0.378824\pi$
$509$	2.21089e9	0.743114	0.371557	+	0.928410i	$0.378824\pi$
$510$	−1.25811e9	−0.419976
$511$	2.64312e9	0.876280
$512$	−1.84968e9	−0.609048
$513$	−6.94560e8	−0.227143
$514$	−8.47479e9	−2.75269
$515$	−7.87320e8	−0.253996
$516$	3.03161e9	0.971404
$517$	3.65950e9	1.16468
$518$	−3.08270e9	−0.974489
$519$	−6.17427e9	−1.93865
$520$	−4.60434e8	−0.143600
$521$	−1.15858e9	−0.358916	−0.179458	−	0.983766i	$-0.557434\pi$
$521$	−1.15858e9	−0.358916	−0.179458	+	0.983766i	$0.557434\pi$
$522$	1.01565e10	3.12534
$523$	4.79427e9	1.46544	0.732718	−	0.680533i	$-0.238252\pi$
$523$	4.79427e9	1.46544	0.732718	+	0.680533i	$0.238252\pi$
$524$	−1.96326e9	−0.596099
$525$	3.56767e9	1.07604
$526$	5.02962e9	1.50690
$527$	−4.48445e8	−0.133466
$528$	−1.03409e9	−0.305731
$529$	3.45322e9	1.01421
$530$	1.17059e9	0.341538
$531$	−2.65585e9	−0.769792
$532$	2.11223e9	0.608207
$533$	−5.04063e9	−1.44191
$534$	−1.78321e8	−0.0506766
$535$	5.28754e8	0.149285
$536$	−1.42667e9	−0.400172
$537$	−8.06529e9	−2.24755
$538$	3.57616e9	0.990100
$539$	1.01614e9	0.279506
$540$	−5.32115e8	−0.145421
$541$	−3.40517e8	−0.0924587	−0.0462294	−	0.998931i	$-0.514721\pi$
$541$	−3.40517e8	−0.0924587	−0.0462294	+	0.998931i	$0.514721\pi$
$542$	2.57069e9	0.693509
$543$	4.41045e9	1.18218
$544$	−3.41868e9	−0.910463
$545$	3.54748e8	0.0938712
$546$	5.56774e9	1.46388
$547$	5.56092e9	1.45275	0.726376	−	0.687298i	$-0.241203\pi$
$547$	5.56092e9	1.45275	0.726376	+	0.687298i	$0.241203\pi$
$548$	3.00367e9	0.779687
$549$	4.78639e9	1.23454
$550$	−3.77749e9	−0.968130
$551$	−3.35150e9	−0.853510
$552$	6.23980e9	1.57901
$553$	−3.07498e9	−0.773222
$554$	1.17888e10	2.94569
$555$	−1.18753e9	−0.294863
$556$	4.62397e8	0.114091
$557$	−2.55758e8	−0.0627098	−0.0313549	−	0.999508i	$-0.509982\pi$
$557$	−2.55758e8	−0.0627098	−0.0313549	+	0.999508i	$0.509982\pi$
$558$	−1.47708e9	−0.359902
$559$	−1.47796e9	−0.357866
$560$	−2.32562e8	−0.0559605
$561$	−3.06189e9	−0.732183
$562$	−1.09775e10	−2.60870
$563$	−5.94963e9	−1.40511	−0.702555	−	0.711629i	$-0.747958\pi$
$563$	−5.94963e9	−1.40511	−0.702555	+	0.711629i	$0.747958\pi$
$564$	1.68370e10	3.95175
$565$	−1.30747e9	−0.304973
$566$	−1.07606e9	−0.249447
$567$	−2.13104e9	−0.490964
$568$	−4.30730e9	−0.986247
$569$	−7.95486e8	−0.181025	−0.0905127	−	0.995895i	$-0.528851\pi$
$569$	−7.95486e8	−0.181025	−0.0905127	+	0.995895i	$0.528851\pi$
$570$	1.36744e9	0.309277
$571$	−4.03947e9	−0.908026	−0.454013	−	0.890995i	$-0.650008\pi$
$571$	−4.03947e9	−0.908026	−0.454013	+	0.890995i	$0.650008\pi$
$572$	−3.50787e9	−0.783714
$573$	1.03594e10	2.30034
$574$	9.51047e9	2.09899
$575$	−6.10197e9	−1.33854
$576$	−9.44563e9	−2.05945
$577$	6.93338e9	1.50255	0.751276	−	0.659988i	$-0.229439\pi$
$577$	6.93338e9	1.50255	0.751276	+	0.659988i	$0.229439\pi$
$578$	3.26674e9	0.703667
$579$	2.32296e9	0.497356
$580$	−2.56765e9	−0.546433
$581$	−1.53583e9	−0.324883
$582$	−1.77896e10	−3.74055
$583$	2.84888e9	0.595435
$584$	−4.11312e9	−0.854528
$585$	1.20212e9	0.248258
$586$	1.25610e10	2.57860
$587$	7.58879e8	0.154860	0.0774300	−	0.996998i	$-0.475329\pi$
$587$	7.58879e8	0.154860	0.0774300	+	0.996998i	$0.475329\pi$
$588$	4.67516e9	0.948365
$589$	4.87415e8	0.0982869
$590$	1.12836e9	0.226186
$591$	9.13728e9	1.82079
$592$	−1.28423e9	−0.254399
$593$	−1.34569e9	−0.265005	−0.132502	−	0.991183i	$-0.542301\pi$
$593$	−1.34569e9	−0.265005	−0.132502	+	0.991183i	$0.542301\pi$
$594$	−2.17635e9	−0.426065
$595$	−6.88606e8	−0.134017
$596$	−1.81877e8	−0.0351896
$597$	−3.19834e9	−0.615198
$598$	−9.52279e9	−1.82100
$599$	1.17622e9	0.223612	0.111806	−	0.993730i	$-0.464336\pi$
$599$	1.17622e9	0.223612	0.111806	+	0.993730i	$0.464336\pi$
$600$	−5.55189e9	−1.04933
$601$	−8.53093e8	−0.160301	−0.0801504	−	0.996783i	$-0.525540\pi$
$601$	−8.53093e8	−0.160301	−0.0801504	+	0.996783i	$0.525540\pi$
$602$	2.78855e9	0.520944
$603$	3.72482e9	0.691822
$604$	−8.88465e9	−1.64063
$605$	−7.44558e8	−0.136696
$606$	−1.59920e10	−2.91910
$607$	3.51705e9	0.638289	0.319145	−	0.947706i	$-0.396604\pi$
$607$	3.51705e9	0.638289	0.319145	+	0.947706i	$0.396604\pi$
$608$	3.71577e9	0.670480
$609$	9.91837e9	1.77943
$610$	−2.03354e9	−0.362742
$611$	−8.20830e9	−1.45582
$612$	−7.89567e9	−1.39238
$613$	6.20528e9	1.08805	0.544026	−	0.839069i	$-0.316899\pi$
$613$	6.20528e9	1.08805	0.544026	+	0.839069i	$0.316899\pi$
$614$	−1.53799e10	−2.68142
$615$	3.66367e9	0.635116
$616$	2.11424e9	0.364437
$617$	−3.32482e9	−0.569862	−0.284931	−	0.958548i	$-0.591971\pi$
$617$	−3.32482e9	−0.569862	−0.284931	+	0.958548i	$0.591971\pi$
$618$	−1.48156e10	−2.52499
$619$	−6.34229e9	−1.07480	−0.537401	−	0.843327i	$-0.680594\pi$
$619$	−6.34229e9	−1.07480	−0.537401	+	0.843327i	$0.680594\pi$
$620$	3.73418e8	0.0629251
$621$	−3.51557e9	−0.589081
$622$	−1.18102e10	−1.96785
$623$	−9.76008e7	−0.0161713
$624$	2.31948e9	0.382159
$625$	5.08227e9	0.832679
$626$	1.39980e10	2.28063
$627$	3.32797e9	0.539192
$628$	−1.14776e10	−1.84924
$629$	−3.80253e9	−0.609250
$630$	−2.26812e9	−0.361388
$631$	−4.57212e8	−0.0724461	−0.0362230	−	0.999344i	$-0.511533\pi$
$631$	−4.57212e8	−0.0724461	−0.0362230	+	0.999344i	$0.511533\pi$
$632$	4.78517e9	0.754028
$633$	1.69550e9	0.265696
$634$	−6.22660e9	−0.970372
$635$	7.86691e8	0.121926
$636$	1.31075e10	2.02031
$637$	−2.27921e9	−0.349378
$638$	−1.05017e10	−1.60098
$639$	1.12457e10	1.70504
$640$	2.07570e9	0.312992
$641$	−3.40660e9	−0.510880	−0.255440	−	0.966825i	$-0.582220\pi$
$641$	−3.40660e9	−0.510880	−0.255440	+	0.966825i	$0.582220\pi$
$642$	9.94994e9	1.48405
$643$	3.28085e9	0.486685	0.243343	−	0.969940i	$-0.421756\pi$
$643$	3.28085e9	0.486685	0.243343	+	0.969940i	$0.421756\pi$
$644$	1.06912e10	1.57735
$645$	1.07422e9	0.157628
$646$	4.37862e9	0.639033
$647$	1.24280e10	1.80400	0.902001	−	0.431733i	$-0.142098\pi$
$647$	1.24280e10	1.80400	0.902001	+	0.431733i	$0.142098\pi$
$648$	3.31624e9	0.478777
$649$	2.74611e9	0.394331
$650$	8.47295e9	1.21015
$651$	−1.44245e9	−0.204912
$652$	7.46919e7	0.0105538
$653$	3.24629e9	0.456238	0.228119	−	0.973633i	$-0.426742\pi$
$653$	3.24629e9	0.456238	0.228119	+	0.973633i	$0.426742\pi$
$654$	6.67556e9	0.933180
$655$	−6.95659e8	−0.0967279
$656$	3.96198e9	0.547960
$657$	1.07387e10	1.47732
$658$	1.54871e10	2.11924
$659$	1.01236e10	1.37795	0.688976	−	0.724784i	$-0.258060\pi$
$659$	1.01236e10	1.37795	0.688976	+	0.724784i	$0.258060\pi$
$660$	2.54962e9	0.345201
$661$	−5.80319e9	−0.781559	−0.390780	−	0.920484i	$-0.627795\pi$
$661$	−5.80319e9	−0.781559	−0.390780	+	0.920484i	$0.627795\pi$
$662$	6.79943e9	0.910898
$663$	6.86785e9	0.915216
$664$	2.39001e9	0.316819
$665$	7.48447e8	0.0986927
$666$	−1.25247e10	−1.64289
$667$	−1.69639e10	−2.21353
$668$	1.40644e10	1.82559
$669$	7.81654e9	1.00931
$670$	−1.58252e9	−0.203276
$671$	−4.94905e9	−0.632402
$672$	−1.09964e10	−1.39784
$673$	−9.58718e9	−1.21238	−0.606189	−	0.795321i	$-0.707302\pi$
$673$	−9.58718e9	−1.21238	−0.606189	+	0.795321i	$0.707302\pi$
$674$	3.74377e9	0.470977
$675$	3.12800e9	0.391474
$676$	−3.93362e9	−0.489755
$677$	−1.59330e9	−0.197350	−0.0986751	−	0.995120i	$-0.531460\pi$
$677$	−1.59330e9	−0.197350	−0.0986751	+	0.995120i	$0.531460\pi$
$678$	−2.46035e10	−3.03175
$679$	−9.73683e9	−1.19364
$680$	1.07158e9	0.130691
$681$	−2.11499e10	−2.56622
$682$	1.52728e9	0.184362
$683$	−1.11674e10	−1.34116	−0.670579	−	0.741838i	$-0.733954\pi$
$683$	−1.11674e10	−1.34116	−0.670579	+	0.741838i	$0.733954\pi$
$684$	8.58182e9	1.02538
$685$	1.06432e9	0.126519
$686$	1.43504e10	1.69718
$687$	−2.37744e10	−2.79744
$688$	1.16169e9	0.135997
$689$	−6.39008e9	−0.744284
$690$	6.92143e9	0.802092
$691$	−2.43715e9	−0.281001	−0.140501	−	0.990081i	$-0.544871\pi$
$691$	−2.43715e9	−0.281001	−0.140501	+	0.990081i	$0.544871\pi$
$692$	1.64626e10	1.88855
$693$	−5.51996e9	−0.630042
$694$	2.37556e9	0.269779
$695$	1.63845e8	0.0185134
$696$	−1.54346e10	−1.73526
$697$	1.17312e10	1.31229
$698$	1.51262e9	0.168358
$699$	8.35105e9	0.924849
$700$	−9.51258e9	−1.04823
$701$	−1.29449e10	−1.41934	−0.709668	−	0.704536i	$-0.751155\pi$
$701$	−1.29449e10	−1.41934	−0.709668	+	0.704536i	$0.751155\pi$
$702$	4.88158e9	0.532574
$703$	4.13298e9	0.448662
$704$	9.76662e9	1.05497
$705$	5.96602e9	0.641243
$706$	−1.79323e10	−1.91787
$707$	−8.75296e9	−0.931509
$708$	1.26346e10	1.33797
$709$	1.86974e10	1.97024	0.985118	−	0.171877i	$-0.0549831\pi$
$709$	1.86974e10	1.97024	0.985118	+	0.171877i	$0.0549831\pi$
$710$	−4.77782e9	−0.500986
$711$	−1.24934e10	−1.30357
$712$	1.51883e8	0.0157699
$713$	2.46709e9	0.254901
$714$	−1.29580e10	−1.33228
$715$	−1.24298e9	−0.127172
$716$	2.15047e10	2.18946
$717$	1.17703e10	1.19253
$718$	1.11798e10	1.12719
$719$	1.45164e9	0.145649	0.0728246	−	0.997345i	$-0.476799\pi$
$719$	1.45164e9	0.145649	0.0728246	+	0.997345i	$0.476799\pi$
$720$	−9.44880e8	−0.0943437
$721$	−8.10905e9	−0.805743
$722$	1.11327e10	1.10083
$723$	−4.73197e9	−0.465648
$724$	−1.17597e10	−1.15163
$725$	1.50937e10	1.47100
$726$	−1.40109e10	−1.35890
$727$	1.47412e10	1.42286	0.711432	−	0.702755i	$-0.248047\pi$
$727$	1.47412e10	1.42286	0.711432	+	0.702755i	$0.248047\pi$
$728$	−4.74227e9	−0.455539
$729$	−1.52072e10	−1.45380
$730$	−4.56244e9	−0.434077
$731$	3.43970e9	0.325694
$732$	−2.27702e10	−2.14574
$733$	−1.96720e10	−1.84495	−0.922475	−	0.386058i	$-0.873837\pi$
$733$	−1.96720e10	−1.84495	−0.922475	+	0.386058i	$0.873837\pi$
$734$	2.17434e10	2.02951
$735$	1.65659e9	0.153890
$736$	1.88077e10	1.73885
$737$	−3.85140e9	−0.354391
$738$	3.86402e10	3.53868
$739$	−1.48159e10	−1.35043	−0.675215	−	0.737621i	$-0.735949\pi$
$739$	−1.48159e10	−1.35043	−0.675215	+	0.737621i	$0.735949\pi$
$740$	3.16635e9	0.287242
$741$	−7.46468e9	−0.673981
$742$	1.20566e10	1.08345
$743$	−2.66103e9	−0.238007	−0.119003	−	0.992894i	$-0.537970\pi$
$743$	−2.66103e9	−0.238007	−0.119003	+	0.992894i	$0.537970\pi$
$744$	2.24468e9	0.199825
$745$	−6.44459e7	−0.00571016
$746$	−1.68086e10	−1.48233
$747$	−6.23995e9	−0.547720
$748$	8.16400e9	0.713259
$749$	5.44593e9	0.473572
$750$	−1.26880e10	−1.09819
$751$	4.01550e9	0.345940	0.172970	−	0.984927i	$-0.444664\pi$
$751$	4.01550e9	0.345940	0.172970	+	0.984927i	$0.444664\pi$
$752$	6.45180e9	0.553246
$753$	−9.36881e8	−0.0799654
$754$	2.35554e10	2.00120
$755$	−3.14818e9	−0.266222
$756$	−5.48055e9	−0.461315
$757$	2.04832e10	1.71618	0.858090	−	0.513500i	$-0.171651\pi$
$757$	2.04832e10	1.71618	0.858090	+	0.513500i	$0.171651\pi$
$758$	−1.25538e10	−1.04697
$759$	1.68448e10	1.39836
$760$	−1.16471e9	−0.0962429
$761$	−7.81181e9	−0.642548	−0.321274	−	0.946986i	$-0.604111\pi$
$761$	−7.81181e9	−0.642548	−0.321274	+	0.946986i	$0.604111\pi$
$762$	1.48037e10	1.21207
$763$	3.65375e9	0.297785
$764$	−2.76215e10	−2.24089
$765$	−2.79774e9	−0.225940
$766$	−2.59155e10	−2.08333
$767$	−6.15955e9	−0.492907
$768$	8.47880e9	0.675414
$769$	−8.36138e9	−0.663034	−0.331517	−	0.943449i	$-0.607560\pi$
$769$	−8.36138e9	−0.663034	−0.331517	+	0.943449i	$0.607560\pi$
$770$	2.34520e9	0.185124
$771$	−3.36250e10	−2.64224
$772$	−6.19377e9	−0.484501
$773$	1.15155e10	0.896714	0.448357	−	0.893854i	$-0.352009\pi$
$773$	1.15155e10	0.896714	0.448357	+	0.893854i	$0.352009\pi$
$774$	1.13296e10	0.878258
$775$	−2.19511e9	−0.169395
$776$	1.51521e10	1.16401
$777$	−1.22311e10	−0.935385
$778$	−2.14202e10	−1.63078
$779$	−1.27507e10	−0.966391
$780$	−5.71882e9	−0.431495
$781$	−1.16279e10	−0.873417
$782$	2.21627e10	1.65729
$783$	8.69603e9	0.647374
$784$	1.79148e9	0.132772
$785$	−4.06697e9	−0.300073
$786$	−1.30907e10	−0.961578
$787$	−6.25272e9	−0.457254	−0.228627	−	0.973514i	$-0.573424\pi$
$787$	−6.25272e9	−0.457254	−0.228627	+	0.973514i	$0.573424\pi$
$788$	−2.43630e10	−1.77373
$789$	1.99557e10	1.44643
$790$	5.30790e9	0.383026
$791$	−1.34663e10	−0.967456
$792$	8.58997e9	0.614403
$793$	1.11008e10	0.790492
$794$	−2.10307e10	−1.49101
$795$	4.64448e9	0.327833
$796$	8.52782e9	0.599297
$797$	2.43108e10	1.70096	0.850482	−	0.526004i	$-0.176310\pi$
$797$	2.43108e10	1.70096	0.850482	+	0.526004i	$0.176310\pi$
$798$	1.40841e10	0.981111
$799$	1.91035e10	1.32495
$800$	−1.67342e10	−1.15555
$801$	−3.96543e8	−0.0272632
$802$	−4.26958e10	−2.92264
$803$	−1.11037e10	−0.756767
$804$	−1.77200e10	−1.20245
$805$	3.78832e9	0.255954
$806$	−3.42570e9	−0.230450
$807$	1.41889e10	0.950370
$808$	1.36210e10	0.908386
$809$	−2.27903e10	−1.51332	−0.756658	−	0.653811i	$-0.773169\pi$
$809$	−2.27903e10	−1.51332	−0.756658	+	0.653811i	$0.773169\pi$
$810$	3.67851e9	0.243206
$811$	1.74087e10	1.14602	0.573011	−	0.819548i	$-0.305775\pi$
$811$	1.74087e10	1.14602	0.573011	+	0.819548i	$0.305775\pi$
$812$	−2.64456e10	−1.73343
$813$	1.01996e10	0.665680
$814$	1.29504e10	0.841582
$815$	2.64663e7	0.00171254
$816$	−5.39820e9	−0.347803
$817$	−3.73861e9	−0.239846
$818$	−3.26820e10	−2.08772
$819$	1.23813e10	0.787542
$820$	−9.76854e9	−0.618701
$821$	−8.16671e8	−0.0515046	−0.0257523	−	0.999668i	$-0.508198\pi$
$821$	−8.16671e8	−0.0515046	−0.0257523	+	0.999668i	$0.508198\pi$
$822$	2.00280e10	1.25773
$823$	4.40653e9	0.275548	0.137774	−	0.990464i	$-0.456005\pi$
$823$	4.40653e9	0.275548	0.137774	+	0.990464i	$0.456005\pi$
$824$	1.26190e10	0.785742
$825$	−1.49877e10	−0.929281
$826$	1.16216e10	0.717524
$827$	2.64306e10	1.62494	0.812471	−	0.583002i	$-0.198122\pi$
$827$	2.64306e10	1.62494	0.812471	+	0.583002i	$0.198122\pi$
$828$	4.34376e10	2.65925
$829$	1.29309e10	0.788290	0.394145	−	0.919048i	$-0.371041\pi$
$829$	1.29309e10	0.788290	0.394145	+	0.919048i	$0.371041\pi$
$830$	2.65109e9	0.160935
$831$	4.67739e10	2.82748
$832$	−2.19066e10	−1.31869
$833$	5.30448e9	0.317969
$834$	3.08320e9	0.184043
$835$	4.98356e9	0.296235
$836$	−8.87346e9	−0.525256
$837$	−1.26468e9	−0.0745490
$838$	4.08922e10	2.40041
$839$	4.19322e9	0.245121	0.122561	−	0.992461i	$-0.460889\pi$
$839$	4.19322e9	0.245121	0.122561	+	0.992461i	$0.460889\pi$
$840$	3.44681e9	0.200650
$841$	2.47116e10	1.43257
$842$	1.11197e10	0.641946
$843$	−4.35547e10	−2.50402
$844$	−4.52076e9	−0.258829
$845$	−1.39383e9	−0.0794717
$846$	6.29227e10	3.57282
$847$	−7.66861e9	−0.433636
$848$	5.02266e9	0.282845
$849$	−4.26942e9	−0.239438
$850$	−1.97194e10	−1.10136
$851$	2.09194e10	1.16358
$852$	−5.34988e10	−2.96350
$853$	−4.30506e9	−0.237497	−0.118748	−	0.992924i	$-0.537888\pi$
$853$	−4.30506e9	−0.237497	−0.118748	+	0.992924i	$0.537888\pi$
$854$	−2.09445e10	−1.15072
$855$	3.04087e9	0.166386
$856$	−8.47476e9	−0.461816
$857$	−5.75854e9	−0.312521	−0.156261	−	0.987716i	$-0.549944\pi$
$857$	−5.75854e9	−0.312521	−0.156261	+	0.987716i	$0.549944\pi$
$858$	−2.33900e10	−1.26423
$859$	1.98600e9	0.106906	0.0534532	−	0.998570i	$-0.482977\pi$
$859$	1.98600e9	0.106906	0.0534532	+	0.998570i	$0.482977\pi$
$860$	−2.86422e9	−0.153554
$861$	3.77342e10	2.01476
$862$	1.38522e8	0.00736619
$863$	−2.32324e10	−1.23043	−0.615214	−	0.788360i	$-0.710931\pi$
$863$	−2.32324e10	−1.23043	−0.615214	+	0.788360i	$0.710931\pi$
$864$	−9.64118e9	−0.508548
$865$	5.83335e9	0.306451
$866$	4.44430e10	2.32536
$867$	1.29613e10	0.675431
$868$	3.84604e9	0.199616
$869$	1.29179e10	0.667765
$870$	−1.71207e10	−0.881462
$871$	8.63874e9	0.442983
$872$	−5.68583e9	−0.290393
$873$	−3.95598e10	−2.01235
$874$	−2.40887e10	−1.22046
$875$	−6.94453e9	−0.350441
$876$	−5.10871e10	−2.56771
$877$	−3.43141e10	−1.71781	−0.858904	−	0.512137i	$-0.828854\pi$
$877$	−3.43141e10	−1.71781	−0.858904	+	0.512137i	$0.828854\pi$
$878$	9.35410e9	0.466413
$879$	4.98377e10	2.47513
$880$	9.76990e8	0.0483282
$881$	1.07877e10	0.531512	0.265756	−	0.964040i	$-0.414378\pi$
$881$	1.07877e10	0.531512	0.265756	+	0.964040i	$0.414378\pi$
$882$	1.74718e10	0.857429
$883$	7.23350e9	0.353578	0.176789	−	0.984249i	$-0.443429\pi$
$883$	7.23350e9	0.353578	0.176789	+	0.984249i	$0.443429\pi$
$884$	−1.83119e10	−0.891562
$885$	4.47693e9	0.217110
$886$	3.18285e10	1.53744
$887$	8.47997e9	0.408002	0.204001	−	0.978971i	$-0.434605\pi$
$887$	8.47997e9	0.408002	0.204001	+	0.978971i	$0.434605\pi$
$888$	1.90335e10	0.912166
$889$	8.10257e9	0.386782
$890$	1.68474e8	0.00801067
$891$	8.95244e9	0.424003
$892$	−2.08414e10	−0.983220
$893$	−2.07636e10	−0.975713
$894$	−1.21273e9	−0.0567651
$895$	7.61995e9	0.355280
$896$	2.13788e10	0.992897
$897$	−3.77831e10	−1.74793
$898$	3.95634e10	1.82317
$899$	−6.10254e9	−0.280125
$900$	−3.86488e10	−1.76720
$901$	1.48719e10	0.677374
$902$	−3.99533e10	−1.81272
$903$	1.10640e10	0.500039
$904$	2.09558e10	0.943441
$905$	−4.16692e9	−0.186872
$906$	−5.92415e10	−2.64653
$907$	−2.75004e10	−1.22381	−0.611904	−	0.790932i	$-0.709596\pi$
$907$	−2.75004e10	−1.22381	−0.611904	+	0.790932i	$0.709596\pi$
$908$	5.63926e10	2.49990
$909$	−3.55625e10	−1.57043
$910$	−5.26031e9	−0.231402
$911$	3.95002e10	1.73095	0.865475	−	0.500952i	$-0.167017\pi$
$911$	3.95002e10	1.73095	0.865475	+	0.500952i	$0.167017\pi$
$912$	5.86731e9	0.256128
$913$	6.45201e9	0.280574
$914$	−5.82619e10	−2.52390
$915$	−8.06835e9	−0.348186
$916$	6.33902e10	2.72514
$917$	−7.16498e9	−0.306847
$918$	−1.13611e10	−0.484696
$919$	−3.20963e8	−0.0136411	−0.00682057	−	0.999977i	$-0.502171\pi$
$919$	−3.20963e8	−0.0136411	−0.00682057	+	0.999977i	$0.502171\pi$
$920$	−5.89525e9	−0.249600
$921$	−6.10220e10	−2.57382
$922$	2.25494e10	0.947495
$923$	2.60814e10	1.09176
$924$	2.62599e10	1.09507
$925$	−1.86131e10	−0.773256
$926$	3.75714e10	1.55496
$927$	−3.29463e10	−1.35840
$928$	−4.65222e10	−1.91092
$929$	1.64291e10	0.672294	0.336147	−	0.941810i	$-0.390876\pi$
$929$	1.64291e10	0.672294	0.336147	+	0.941810i	$0.390876\pi$
$930$	2.48989e9	0.101506
$931$	−5.76544e9	−0.234158
$932$	−2.22666e10	−0.900946
$933$	−4.68588e10	−1.88888
$934$	−3.09364e10	−1.24238
$935$	2.89282e9	0.115739
$936$	−1.92674e10	−0.767993
$937$	−6.45937e9	−0.256508	−0.128254	−	0.991741i	$-0.540937\pi$
$937$	−6.45937e9	−0.256508	−0.128254	+	0.991741i	$0.540937\pi$
$938$	−1.62992e10	−0.644848
$939$	5.55391e10	2.18912
$940$	−1.59074e10	−0.624670
$941$	1.53005e10	0.598609	0.299304	−	0.954158i	$-0.403245\pi$
$941$	1.53005e10	0.598609	0.299304	+	0.954158i	$0.403245\pi$
$942$	−7.65312e10	−2.98305
$943$	−6.45387e10	−2.50628
$944$	4.84147e9	0.187316
$945$	−1.94197e9	−0.0748568
$946$	−1.17146e10	−0.449894
$947$	1.36187e10	0.521087	0.260543	−	0.965462i	$-0.416098\pi$
$947$	1.36187e10	0.521087	0.260543	+	0.965462i	$0.416098\pi$
$948$	5.94343e10	2.26573
$949$	2.49057e10	0.945946
$950$	2.14330e10	0.811056
$951$	−2.47049e10	−0.931433
$952$	1.10368e10	0.414587
$953$	1.43126e9	0.0535666	0.0267833	−	0.999641i	$-0.491474\pi$
$953$	1.43126e9	0.0535666	0.0267833	+	0.999641i	$0.491474\pi$
$954$	4.89847e10	1.82659
$955$	−9.78737e9	−0.363625
$956$	−3.13834e10	−1.16171
$957$	−4.16669e10	−1.53674
$958$	1.77227e10	0.651256
$959$	1.09620e10	0.401351
$960$	1.59224e10	0.580842
$961$	8.87504e8	0.0322581
$962$	−2.90478e10	−1.05196
$963$	2.21263e10	0.798394
$964$	1.26170e10	0.453613
$965$	−2.19469e9	−0.0786192
$966$	7.12877e10	2.54445
$967$	−9.10409e9	−0.323775	−0.161888	−	0.986809i	$-0.551758\pi$
$967$	−9.10409e9	−0.323775	−0.161888	+	0.986809i	$0.551758\pi$
$968$	1.19336e10	0.422872
$969$	1.73728e10	0.613390
$970$	1.68073e10	0.591285
$971$	−2.51666e9	−0.0882180	−0.0441090	−	0.999027i	$-0.514045\pi$
$971$	−2.51666e9	−0.0882180	−0.0441090	+	0.999027i	$0.514045\pi$
$972$	5.86512e10	2.04854
$973$	1.68753e9	0.0587296
$974$	7.00451e10	2.42897
$975$	3.36177e10	1.16159
$976$	−8.72532e9	−0.300405
$977$	1.45618e10	0.499554	0.249777	−	0.968303i	$-0.419643\pi$
$977$	1.45618e10	0.499554	0.249777	+	0.968303i	$0.419643\pi$
$978$	4.98035e8	0.0170245
$979$	4.10019e8	0.0139658
$980$	−4.41701e9	−0.149912
$981$	1.48448e10	0.502035
$982$	7.00110e9	0.235926
$983$	−1.09489e10	−0.367648	−0.183824	−	0.982959i	$-0.558848\pi$
$983$	−1.09489e10	−0.367648	−0.183824	+	0.982959i	$0.558848\pi$
$984$	−5.87206e10	−1.96475
$985$	−8.63275e9	−0.287821
$986$	−5.48212e10	−1.82129
$987$	6.14473e10	2.03420
$988$	1.99033e10	0.656561
$989$	−1.89233e10	−0.622027
$990$	9.52833e9	0.312100
$991$	−4.82115e9	−0.157360	−0.0786798	−	0.996900i	$-0.525070\pi$
$991$	−4.82115e9	−0.157360	−0.0786798	+	0.996900i	$0.525070\pi$
$992$	6.76581e9	0.220054
$993$	2.69777e10	0.874346
$994$	−4.92095e10	−1.58927
$995$	3.02174e9	0.0972469
$996$	2.96851e10	0.951987
$997$	−1.91710e10	−0.612649	−0.306325	−	0.951927i	$-0.599099\pi$
$997$	−1.91710e10	−0.612649	−0.306325	+	0.951927i	$0.599099\pi$
$998$	−2.46892e10	−0.786231
$999$	−1.07237e10	−0.340303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.8.a.a.1.1	✓	7
3.2	odd	2		279.8.a.b.1.7		7
4.3	odd	2		496.8.a.e.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.8.a.a.1.1	✓	7	1.1	even	1	trivial
279.8.a.b.1.7		7	3.2	odd	2
496.8.a.e.1.7		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	31.a (trivial)

\(f(q)\)	\(=\)	\(q-17.7787 q^{2} -70.5395 q^{3} +188.081 q^{4} +66.6445 q^{5} +1254.10 q^{6} +686.409 q^{7} -1068.16 q^{8} +2788.82 q^{9} +O(q^{10})\)	\(q-17.7787 q^{2} -70.5395 q^{3} +188.081 q^{4} +66.6445 q^{5} +1254.10 q^{6} +686.409 q^{7} -1068.16 q^{8} +2788.82 q^{9} -1184.85 q^{10} -2883.59 q^{11} -13267.1 q^{12} +6467.92 q^{13} -12203.4 q^{14} -4701.07 q^{15} -5083.85 q^{16} -15053.0 q^{17} -49581.4 q^{18} +16361.2 q^{19} +12534.6 q^{20} -48418.9 q^{21} +51266.4 q^{22} +82813.3 q^{23} +75347.8 q^{24} -73683.5 q^{25} -114991. q^{26} -42451.8 q^{27} +129101. q^{28} -204845. q^{29} +83578.7 q^{30} +29791.0 q^{31} +227109. q^{32} +203407. q^{33} +267623. q^{34} +45745.4 q^{35} +524524. q^{36} +252609. q^{37} -290880. q^{38} -456244. q^{39} -71187.3 q^{40} -779327. q^{41} +860824. q^{42} -228505. q^{43} -542349. q^{44} +185859. q^{45} -1.47231e6 q^{46} -1.26908e6 q^{47} +358612. q^{48} -352386. q^{49} +1.31000e6 q^{50} +1.06183e6 q^{51} +1.21650e6 q^{52} -987964. q^{53} +754736. q^{54} -192175. q^{55} -733197. q^{56} -1.15411e6 q^{57} +3.64187e6 q^{58} -952323. q^{59} -884182. q^{60} +1.71628e6 q^{61} -529644. q^{62} +1.91427e6 q^{63} -3.38697e6 q^{64} +431051. q^{65} -3.61630e6 q^{66} +1.33563e6 q^{67} -2.83119e6 q^{68} -5.84161e6 q^{69} -813292. q^{70} +4.03243e6 q^{71} -2.97891e6 q^{72} +3.85064e6 q^{73} -4.49106e6 q^{74} +5.19760e6 q^{75} +3.07723e6 q^{76} -1.97932e6 q^{77} +8.11141e6 q^{78} -4.47981e6 q^{79} -338810. q^{80} -3.10462e6 q^{81} +1.38554e7 q^{82} -2.23749e6 q^{83} -9.10669e6 q^{84} -1.00320e6 q^{85} +4.06252e6 q^{86} +1.44497e7 q^{87} +3.08015e6 q^{88} -142190. q^{89} -3.30433e6 q^{90} +4.43964e6 q^{91} +1.55756e7 q^{92} -2.10144e6 q^{93} +2.25625e7 q^{94} +1.09038e6 q^{95} -1.60202e7 q^{96} -1.41852e7 q^{97} +6.26496e6 q^{98} -8.04180e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 17 q^{2} - 14 q^{3} + 229 q^{4} - 430 q^{5} - 528 q^{6} - 832 q^{7} - 135 q^{8} + 2115 q^{9}+O(q^{10}) \) 7 * q - 17 * q^2 - 14 * q^3 + 229 * q^4 - 430 * q^5 - 528 * q^6 - 832 * q^7 - 135 * q^8 + 2115 * q^9	\( 7 q - 17 q^{2} - 14 q^{3} + 229 q^{4} - 430 q^{5} - 528 q^{6} - 832 q^{7} - 135 q^{8} + 2115 q^{9} - 5232 q^{10} - 7886 q^{11} - 24952 q^{12} - 21844 q^{13} - 37902 q^{14} - 37796 q^{15} - 49583 q^{16} - 54822 q^{17} - 57869 q^{18} - 45352 q^{19} - 50110 q^{20} - 90756 q^{21} + 30696 q^{22} - 18464 q^{23} + 84472 q^{24} - 37691 q^{25} + 3906 q^{26} + 31360 q^{27} + 323084 q^{28} - 81488 q^{29} + 549372 q^{30} + 208537 q^{31} + 276513 q^{32} + 266552 q^{33} + 1152342 q^{34} + 154340 q^{35} + 712153 q^{36} + 431648 q^{37} + 798430 q^{38} - 239436 q^{39} + 600366 q^{40} - 1465990 q^{41} + 1324796 q^{42} - 598714 q^{43} - 660872 q^{44} - 728478 q^{45} - 356652 q^{46} - 2003572 q^{47} + 383736 q^{48} - 331317 q^{49} + 91595 q^{50} - 3313124 q^{51} - 28582 q^{52} - 1496844 q^{53} + 1032136 q^{54} - 1414452 q^{55} - 2199880 q^{56} - 6032240 q^{57} + 1517430 q^{58} - 2853828 q^{59} + 546920 q^{60} - 1486900 q^{61} - 506447 q^{62} - 4186384 q^{63} - 1940543 q^{64} - 2252240 q^{65} + 3180552 q^{66} - 5647492 q^{67} + 829234 q^{68} - 1992112 q^{69} + 8307114 q^{70} - 5168828 q^{71} + 5542765 q^{72} + 4710926 q^{73} + 4058410 q^{74} + 5700806 q^{75} + 15097160 q^{76} - 2020724 q^{77} + 20668432 q^{78} + 4582796 q^{79} + 6495030 q^{80} - 5939465 q^{81} + 8295096 q^{82} - 626514 q^{83} + 15579128 q^{84} + 8323116 q^{85} - 3575108 q^{86} + 19337220 q^{87} + 2100840 q^{88} - 13906634 q^{89} + 5754808 q^{90} + 10966300 q^{91} - 2803952 q^{92} - 417074 q^{93} + 21167040 q^{94} - 11547396 q^{95} - 1593224 q^{96} + 2962898 q^{97} - 25061151 q^{98} - 16496442 q^{99}+O(q^{100}) \) 7 * q - 17 * q^2 - 14 * q^3 + 229 * q^4 - 430 * q^5 - 528 * q^6 - 832 * q^7 - 135 * q^8 + 2115 * q^9 - 5232 * q^10 - 7886 * q^11 - 24952 * q^12 - 21844 * q^13 - 37902 * q^14 - 37796 * q^15 - 49583 * q^16 - 54822 * q^17 - 57869 * q^18 - 45352 * q^19 - 50110 * q^20 - 90756 * q^21 + 30696 * q^22 - 18464 * q^23 + 84472 * q^24 - 37691 * q^25 + 3906 * q^26 + 31360 * q^27 + 323084 * q^28 - 81488 * q^29 + 549372 * q^30 + 208537 * q^31 + 276513 * q^32 + 266552 * q^33 + 1152342 * q^34 + 154340 * q^35 + 712153 * q^36 + 431648 * q^37 + 798430 * q^38 - 239436 * q^39 + 600366 * q^40 - 1465990 * q^41 + 1324796 * q^42 - 598714 * q^43 - 660872 * q^44 - 728478 * q^45 - 356652 * q^46 - 2003572 * q^47 + 383736 * q^48 - 331317 * q^49 + 91595 * q^50 - 3313124 * q^51 - 28582 * q^52 - 1496844 * q^53 + 1032136 * q^54 - 1414452 * q^55 - 2199880 * q^56 - 6032240 * q^57 + 1517430 * q^58 - 2853828 * q^59 + 546920 * q^60 - 1486900 * q^61 - 506447 * q^62 - 4186384 * q^63 - 1940543 * q^64 - 2252240 * q^65 + 3180552 * q^66 - 5647492 * q^67 + 829234 * q^68 - 1992112 * q^69 + 8307114 * q^70 - 5168828 * q^71 + 5542765 * q^72 + 4710926 * q^73 + 4058410 * q^74 + 5700806 * q^75 + 15097160 * q^76 - 2020724 * q^77 + 20668432 * q^78 + 4582796 * q^79 + 6495030 * q^80 - 5939465 * q^81 + 8295096 * q^82 - 626514 * q^83 + 15579128 * q^84 + 8323116 * q^85 - 3575108 * q^86 + 19337220 * q^87 + 2100840 * q^88 - 13906634 * q^89 + 5754808 * q^90 + 10966300 * q^91 - 2803952 * q^92 - 417074 * q^93 + 21167040 * q^94 - 11547396 * q^95 - 1593224 * q^96 + 2962898 * q^97 - 25061151 * q^98 - 16496442 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more