LMFDB - Embedded newform 23.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,8,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, 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("23.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	23.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:	$7.18485558613$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 $ x^8 - 832x^6 - 1059x^5 + 203052x^4 + 678328x^3 - 13424272x^2 - 73308944x - 37372224
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-21.3077$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.3077 q^{2} +86.9475 q^{3} +326.017 q^{4} -188.239 q^{5} -1852.65 q^{6} +639.155 q^{7} -4219.28 q^{8} +5372.88 q^{9} +O(q^{10})$	\(q-21.3077 q^{2} +86.9475 q^{3} +326.017 q^{4} -188.239 q^{5} -1852.65 q^{6} +639.155 q^{7} -4219.28 q^{8} +5372.88 q^{9} +4010.94 q^{10} -1629.29 q^{11} +28346.4 q^{12} +1138.20 q^{13} -13618.9 q^{14} -16366.9 q^{15} +48172.8 q^{16} +28713.3 q^{17} -114483. q^{18} +43387.0 q^{19} -61369.1 q^{20} +55573.0 q^{21} +34716.4 q^{22} -12167.0 q^{23} -366856. q^{24} -42691.1 q^{25} -24252.3 q^{26} +277004. q^{27} +208375. q^{28} +62583.1 q^{29} +348741. q^{30} +22533.6 q^{31} -486383. q^{32} -141663. q^{33} -611814. q^{34} -120314. q^{35} +1.75165e6 q^{36} -277384. q^{37} -924475. q^{38} +98963.5 q^{39} +794233. q^{40} -130110. q^{41} -1.18413e6 q^{42} +585763. q^{43} -531177. q^{44} -1.01138e6 q^{45} +259250. q^{46} -99374.3 q^{47} +4.18851e6 q^{48} -415024. q^{49} +909647. q^{50} +2.49655e6 q^{51} +371072. q^{52} -636910. q^{53} -5.90231e6 q^{54} +306696. q^{55} -2.69677e6 q^{56} +3.77239e6 q^{57} -1.33350e6 q^{58} -808062. q^{59} -5.33589e6 q^{60} -2.40412e6 q^{61} -480138. q^{62} +3.43410e6 q^{63} +4.19757e6 q^{64} -214253. q^{65} +3.01851e6 q^{66} -2.05060e6 q^{67} +9.36103e6 q^{68} -1.05789e6 q^{69} +2.56361e6 q^{70} -3.83810e6 q^{71} -2.26697e7 q^{72} -2.78541e6 q^{73} +5.91041e6 q^{74} -3.71188e6 q^{75} +1.41449e7 q^{76} -1.04137e6 q^{77} -2.10868e6 q^{78} +7.20890e6 q^{79} -9.06801e6 q^{80} +1.23343e7 q^{81} +2.77235e6 q^{82} -498585. q^{83} +1.81177e7 q^{84} -5.40497e6 q^{85} -1.24813e7 q^{86} +5.44145e6 q^{87} +6.87443e6 q^{88} -1.37216e6 q^{89} +2.15503e7 q^{90} +727485. q^{91} -3.96665e6 q^{92} +1.95924e6 q^{93} +2.11743e6 q^{94} -8.16712e6 q^{95} -4.22898e7 q^{96} +6.57645e6 q^{97} +8.84319e6 q^{98} -8.75398e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) $ 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9} + 19502 q^{10} + 7588 q^{11} + 22733 q^{12} + 19862 q^{13} + 17544 q^{14} - 12770 q^{15} + 64336 q^{16} + 42070 q^{17} - 59129 q^{18} + 1050 q^{19} + 3364 q^{20} - 7698 q^{21} - 128220 q^{22} - 97336 q^{23} - 621188 q^{24} + 49496 q^{25} - 371761 q^{26} - 69500 q^{27} + 143050 q^{28} - 102578 q^{29} - 671470 q^{30} + 304172 q^{31} - 612824 q^{32} + 747242 q^{33} - 524530 q^{34} + 531048 q^{35} + 1868983 q^{36} + 286472 q^{37} - 762932 q^{38} + 1032828 q^{39} + 2105286 q^{40} + 1324414 q^{41} - 1886168 q^{42} + 2052578 q^{43} - 867298 q^{44} + 2087442 q^{45} + 675556 q^{47} + 1411151 q^{48} - 55404 q^{49} + 1458528 q^{50} + 2775482 q^{51} - 1695409 q^{52} + 203654 q^{53} - 9897559 q^{54} - 1024444 q^{55} - 5766846 q^{56} + 3908648 q^{57} - 5039991 q^{58} - 748892 q^{59} - 18153300 q^{60} + 61822 q^{61} - 4939277 q^{62} + 1411632 q^{63} + 2702267 q^{64} - 1571618 q^{65} + 3791866 q^{66} + 3235604 q^{67} + 4914980 q^{68} - 486680 q^{69} + 10871764 q^{70} - 4951664 q^{71} - 7940241 q^{72} + 11019370 q^{73} + 356954 q^{74} - 13607220 q^{75} + 21973240 q^{76} - 5284888 q^{77} - 1506779 q^{78} + 4202464 q^{79} + 8785886 q^{80} + 10294096 q^{81} + 32636759 q^{82} + 518568 q^{83} + 7629190 q^{84} + 9854220 q^{85} - 14681386 q^{86} + 4862532 q^{87} + 20589740 q^{88} + 4203864 q^{89} + 49021076 q^{90} + 2488406 q^{91} - 7786880 q^{92} - 23367842 q^{93} + 12314327 q^{94} - 44485300 q^{95} - 45317009 q^{96} + 18621134 q^{97} + 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9 + 19502 * q^10 + 7588 * q^11 + 22733 * q^12 + 19862 * q^13 + 17544 * q^14 - 12770 * q^15 + 64336 * q^16 + 42070 * q^17 - 59129 * q^18 + 1050 * q^19 + 3364 * q^20 - 7698 * q^21 - 128220 * q^22 - 97336 * q^23 - 621188 * q^24 + 49496 * q^25 - 371761 * q^26 - 69500 * q^27 + 143050 * q^28 - 102578 * q^29 - 671470 * q^30 + 304172 * q^31 - 612824 * q^32 + 747242 * q^33 - 524530 * q^34 + 531048 * q^35 + 1868983 * q^36 + 286472 * q^37 - 762932 * q^38 + 1032828 * q^39 + 2105286 * q^40 + 1324414 * q^41 - 1886168 * q^42 + 2052578 * q^43 - 867298 * q^44 + 2087442 * q^45 + 675556 * q^47 + 1411151 * q^48 - 55404 * q^49 + 1458528 * q^50 + 2775482 * q^51 - 1695409 * q^52 + 203654 * q^53 - 9897559 * q^54 - 1024444 * q^55 - 5766846 * q^56 + 3908648 * q^57 - 5039991 * q^58 - 748892 * q^59 - 18153300 * q^60 + 61822 * q^61 - 4939277 * q^62 + 1411632 * q^63 + 2702267 * q^64 - 1571618 * q^65 + 3791866 * q^66 + 3235604 * q^67 + 4914980 * q^68 - 486680 * q^69 + 10871764 * q^70 - 4951664 * q^71 - 7940241 * q^72 + 11019370 * q^73 + 356954 * q^74 - 13607220 * q^75 + 21973240 * q^76 - 5284888 * q^77 - 1506779 * q^78 + 4202464 * q^79 + 8785886 * q^80 + 10294096 * q^81 + 32636759 * q^82 + 518568 * q^83 + 7629190 * q^84 + 9854220 * q^85 - 14681386 * q^86 + 4862532 * q^87 + 20589740 * q^88 + 4203864 * q^89 + 49021076 * q^90 + 2488406 * q^91 - 7786880 * q^92 - 23367842 * q^93 + 12314327 * q^94 - 44485300 * q^95 - 45317009 * q^96 + 18621134 * q^97 + 35756 * q^98 - 64729930 * 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 $

$2$	−21.3077	−1.88335	−0.941675	−	0.336524i	$-0.890749\pi$
$2$	−21.3077	−1.88335	−0.941675	+	0.336524i	$0.890749\pi$
$3$	86.9475	1.85923	0.929615	−	0.368533i	$-0.120140\pi$
$3$	86.9475	1.85923	0.929615	+	0.368533i	$0.120140\pi$
$4$	326.017	2.54701
$5$	−188.239	−0.673464	−0.336732	−	0.941600i	$-0.609322\pi$
$5$	−188.239	−0.673464	−0.336732	+	0.941600i	$0.609322\pi$
$6$	−1852.65	−3.50158
$7$	639.155	0.704309	0.352154	−	0.935942i	$-0.385449\pi$
$7$	639.155	0.704309	0.352154	+	0.935942i	$0.385449\pi$
$8$	−4219.28	−2.91355
$9$	5372.88	2.45673
$10$	4010.94	1.26837
$11$	−1629.29	−0.369083	−0.184542	−	0.982825i	$-0.559080\pi$
$11$	−1629.29	−0.369083	−0.184542	+	0.982825i	$0.559080\pi$
$12$	28346.4	4.73547
$13$	1138.20	0.143687	0.0718433	−	0.997416i	$-0.477112\pi$
$13$	1138.20	0.143687	0.0718433	+	0.997416i	$0.477112\pi$
$14$	−13618.9	−1.32646
$15$	−16366.9	−1.25212
$16$	48172.8	2.94024
$17$	28713.3	1.41746	0.708732	−	0.705477i	$-0.249267\pi$
$17$	28713.3	1.41746	0.708732	+	0.705477i	$0.249267\pi$
$18$	−114483.	−4.62689
$19$	43387.0	1.45118	0.725591	−	0.688127i	$-0.241567\pi$
$19$	43387.0	1.45118	0.725591	+	0.688127i	$0.241567\pi$
$20$	−61369.1	−1.71532
$21$	55573.0	1.30947
$22$	34716.4	0.695113
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−366856.	−5.41697
$25$	−42691.1	−0.546446
$26$	−24252.3	−0.270612
$27$	277004.	2.70840
$28$	208375.	1.79388
$29$	62583.1	0.476502	0.238251	−	0.971204i	$-0.423426\pi$
$29$	62583.1	0.476502	0.238251	+	0.971204i	$0.423426\pi$
$30$	348741.	2.35819
$31$	22533.6	0.135851	0.0679257	−	0.997690i	$-0.478362\pi$
$31$	22533.6	0.135851	0.0679257	+	0.997690i	$0.478362\pi$
$32$	−486383.	−2.62394
$33$	−141663.	−0.686211
$34$	−611814.	−2.66958
$35$	−120314.	−0.474327
$36$	1.75165e6	6.25732
$37$	−277384.	−0.900276	−0.450138	−	0.892959i	$-0.648625\pi$
$37$	−277384.	−0.900276	−0.450138	+	0.892959i	$0.648625\pi$
$38$	−924475.	−2.73308
$39$	98963.5	0.267146
$40$	794233.	1.96218
$41$	−130110.	−0.294828	−0.147414	−	0.989075i	$-0.547095\pi$
$41$	−130110.	−0.294828	−0.147414	+	0.989075i	$0.547095\pi$
$42$	−1.18413e6	−2.46619
$43$	585763.	1.12352	0.561762	−	0.827299i	$-0.310124\pi$
$43$	585763.	1.12352	0.561762	+	0.827299i	$0.310124\pi$
$44$	−531177.	−0.940058
$45$	−1.01138e6	−1.65452
$46$	259250.	0.392706
$47$	−99374.3	−0.139615	−0.0698074	−	0.997560i	$-0.522238\pi$
$47$	−99374.3	−0.139615	−0.0698074	+	0.997560i	$0.522238\pi$
$48$	4.18851e6	5.46657
$49$	−415024.	−0.503949
$50$	909647.	1.02915
$51$	2.49655e6	2.63539
$52$	371072.	0.365971
$53$	−636910.	−0.587641	−0.293821	−	0.955861i	$-0.594927\pi$
$53$	−636910.	−0.587641	−0.293821	+	0.955861i	$0.594927\pi$
$54$	−5.90231e6	−5.10087
$55$	306696.	0.248564
$56$	−2.69677e6	−2.05204
$57$	3.77239e6	2.69808
$58$	−1.33350e6	−0.897419
$59$	−808062.	−0.512227	−0.256114	−	0.966647i	$-0.582442\pi$
$59$	−808062.	−0.512227	−0.256114	+	0.966647i	$0.582442\pi$
$60$	−5.33589e6	−3.18917
$61$	−2.40412e6	−1.35613	−0.678066	−	0.735001i	$-0.737182\pi$
$61$	−2.40412e6	−1.35613	−0.678066	+	0.735001i	$0.737182\pi$
$62$	−480138.	−0.255856
$63$	3.43410e6	1.73030
$64$	4.19757e6	2.00156
$65$	−214253.	−0.0967678
$66$	3.01851e6	1.29237
$67$	−2.05060e6	−0.832949	−0.416474	−	0.909147i	$-0.636734\pi$
$67$	−2.05060e6	−0.832949	−0.416474	+	0.909147i	$0.636734\pi$
$68$	9.36103e6	3.61029
$69$	−1.05789e6	−0.387676
$70$	2.56361e6	0.893324
$71$	−3.83810e6	−1.27266	−0.636329	−	0.771417i	$-0.719548\pi$
$71$	−3.83810e6	−1.27266	−0.636329	+	0.771417i	$0.719548\pi$
$72$	−2.26697e7	−7.15783
$73$	−2.78541e6	−0.838029	−0.419015	−	0.907979i	$-0.637624\pi$
$73$	−2.78541e6	−0.838029	−0.419015	+	0.907979i	$0.637624\pi$
$74$	5.91041e6	1.69553
$75$	−3.71188e6	−1.01597
$76$	1.41449e7	3.69617
$77$	−1.04137e6	−0.259949
$78$	−2.10868e6	−0.503130
$79$	7.20890e6	1.64503	0.822516	−	0.568742i	$-0.192570\pi$
$79$	7.20890e6	1.64503	0.822516	+	0.568742i	$0.192570\pi$
$80$	−9.06801e6	−1.98014
$81$	1.23343e7	2.57881
$82$	2.77235e6	0.555264
$83$	−498585.	−0.0957120	−0.0478560	−	0.998854i	$-0.515239\pi$
$83$	−498585.	−0.0957120	−0.0478560	+	0.998854i	$0.515239\pi$
$84$	1.81177e7	3.33523
$85$	−5.40497e6	−0.954612
$86$	−1.24813e7	−2.11599
$87$	5.44145e6	0.885926
$88$	6.87443e6	1.07534
$89$	−1.37216e6	−0.206320	−0.103160	−	0.994665i	$-0.532895\pi$
$89$	−1.37216e6	−0.206320	−0.103160	+	0.994665i	$0.532895\pi$
$90$	2.15503e7	3.11604
$91$	727485.	0.101200
$92$	−3.96665e6	−0.531088
$93$	1.95924e6	0.252579
$94$	2.11743e6	0.262944
$95$	−8.16712e6	−0.977319
$96$	−4.22898e7	−4.87850
$97$	6.57645e6	0.731628	0.365814	−	0.930688i	$-0.380791\pi$
$97$	6.57645e6	0.731628	0.365814	+	0.930688i	$0.380791\pi$
$98$	8.84319e6	0.949112
$99$	−8.75398e6	−0.906739
$100$	−1.39180e7	−1.39180
$101$	−2.24529e6	−0.216844	−0.108422	−	0.994105i	$-0.534580\pi$
$101$	−2.24529e6	−0.216844	−0.108422	+	0.994105i	$0.534580\pi$
$102$	−5.31957e7	−4.96336
$103$	2.14203e6	0.193150	0.0965751	−	0.995326i	$-0.469211\pi$
$103$	2.14203e6	0.193150	0.0965751	+	0.995326i	$0.469211\pi$
$104$	−4.80237e6	−0.418639
$105$	−1.04610e7	−0.881883
$106$	1.35711e7	1.10673
$107$	1.72640e7	1.36238	0.681189	−	0.732107i	$-0.261463\pi$
$107$	1.72640e7	1.36238	0.681189	+	0.732107i	$0.261463\pi$
$108$	9.03080e7	6.89832
$109$	−2.52636e7	−1.86854	−0.934269	−	0.356569i	$-0.883947\pi$
$109$	−2.52636e7	−1.86854	−0.934269	+	0.356569i	$0.883947\pi$
$110$	−6.53498e6	−0.468134
$111$	−2.41179e7	−1.67382
$112$	3.07899e7	2.07083
$113$	3.18384e6	0.207576	0.103788	−	0.994599i	$-0.466904\pi$
$113$	3.18384e6	0.207576	0.103788	+	0.994599i	$0.466904\pi$
$114$	−8.03808e7	−5.08143
$115$	2.29030e6	0.140427
$116$	2.04032e7	1.21365
$117$	6.11540e6	0.353000
$118$	1.72179e7	0.964703
$119$	1.83523e7	0.998333
$120$	6.90566e7	3.64813
$121$	−1.68326e7	−0.863778
$122$	5.12263e7	2.55407
$123$	−1.13128e7	−0.548153
$124$	7.34632e6	0.346014
$125$	2.27423e7	1.04148
$126$	−7.31727e7	−3.25876
$127$	−2.02393e7	−0.876763	−0.438381	−	0.898789i	$-0.644448\pi$
$127$	−2.02393e7	−0.876763	−0.438381	+	0.898789i	$0.644448\pi$
$128$	−2.71834e7	−1.14570
$129$	5.09307e7	2.08889
$130$	4.56524e6	0.182248
$131$	−4.46018e7	−1.73342	−0.866708	−	0.498816i	$-0.833768\pi$
$131$	−4.46018e7	−1.73342	−0.866708	+	0.498816i	$0.833768\pi$
$132$	−4.61845e7	−1.74778
$133$	2.77310e7	1.02208
$134$	4.36934e7	1.56873
$135$	−5.21430e7	−1.82401
$136$	−1.21149e8	−4.12986
$137$	2.84803e7	0.946287	0.473144	−	0.880985i	$-0.343119\pi$
$137$	2.84803e7	0.946287	0.473144	+	0.880985i	$0.343119\pi$
$138$	2.25412e7	0.730130
$139$	−4.10033e7	−1.29499	−0.647495	−	0.762070i	$-0.724183\pi$
$139$	−4.10033e7	−1.29499	−0.647495	+	0.762070i	$0.724183\pi$
$140$	−3.92244e7	−1.20811
$141$	−8.64035e6	−0.259576
$142$	8.17810e7	2.39686
$143$	−1.85446e6	−0.0530323
$144$	2.58827e8	7.22338
$145$	−1.17806e7	−0.320907
$146$	5.93506e7	1.57830
$147$	−3.60853e7	−0.936957
$148$	−9.04319e7	−2.29301
$149$	−1.46291e6	−0.0362297	−0.0181148	−	0.999836i	$-0.505766\pi$
$149$	−1.46291e6	−0.0362297	−0.0181148	+	0.999836i	$0.505766\pi$
$150$	7.90916e7	1.91342
$151$	−1.38217e7	−0.326694	−0.163347	−	0.986569i	$-0.552229\pi$
$151$	−1.38217e7	−0.326694	−0.163347	+	0.986569i	$0.552229\pi$
$152$	−1.83062e8	−4.22810
$153$	1.54273e8	3.48233
$154$	2.21892e7	0.489574
$155$	−4.24170e6	−0.0914911
$156$	3.22638e7	0.680423
$157$	2.49008e7	0.513528	0.256764	−	0.966474i	$-0.417344\pi$
$157$	2.49008e7	0.513528	0.256764	+	0.966474i	$0.417344\pi$
$158$	−1.53605e8	−3.09817
$159$	−5.53777e7	−1.09256
$160$	9.15563e7	1.76713
$161$	−7.77660e6	−0.146859
$162$	−2.62816e8	−4.85679
$163$	8.38251e7	1.51606	0.758032	−	0.652218i	$-0.226161\pi$
$163$	8.38251e7	1.51606	0.758032	+	0.652218i	$0.226161\pi$
$164$	−4.24182e7	−0.750929
$165$	2.66665e7	0.462138
$166$	1.06237e7	0.180259
$167$	3.89864e7	0.647747	0.323873	−	0.946100i	$-0.395015\pi$
$167$	3.89864e7	0.647747	0.323873	+	0.946100i	$0.395015\pi$
$168$	−2.34478e8	−3.81522
$169$	−6.14530e7	−0.979354
$170$	1.15167e8	1.79787
$171$	2.33113e8	3.56517
$172$	1.90969e8	2.86162
$173$	9.35688e7	1.37395	0.686973	−	0.726683i	$-0.258939\pi$
$173$	9.35688e7	1.37395	0.686973	+	0.726683i	$0.258939\pi$
$174$	−1.15945e8	−1.66851
$175$	−2.72862e7	−0.384867
$176$	−7.84876e7	−1.08519
$177$	−7.02590e7	−0.952348
$178$	2.92376e7	0.388572
$179$	3.65707e7	0.476592	0.238296	−	0.971193i	$-0.423411\pi$
$179$	3.65707e7	0.476592	0.238296	+	0.971193i	$0.423411\pi$
$180$	−3.29729e8	−4.21408
$181$	8.44628e6	0.105874	0.0529372	−	0.998598i	$-0.483142\pi$
$181$	8.44628e6	0.105874	0.0529372	+	0.998598i	$0.483142\pi$
$182$	−1.55010e7	−0.190594
$183$	−2.09033e8	−2.52136
$184$	5.13360e7	0.607518
$185$	5.22145e7	0.606304
$186$	−4.17468e7	−0.475694
$187$	−4.67824e7	−0.523163
$188$	−3.23977e7	−0.355600
$189$	1.77049e8	1.90755
$190$	1.74022e8	1.84063
$191$	1.13851e8	1.18228	0.591138	−	0.806570i	$-0.298679\pi$
$191$	1.13851e8	1.18228	0.591138	+	0.806570i	$0.298679\pi$
$192$	3.64969e8	3.72136
$193$	1.22314e8	1.22469	0.612345	−	0.790591i	$-0.290226\pi$
$193$	1.22314e8	1.22469	0.612345	+	0.790591i	$0.290226\pi$
$194$	−1.40129e8	−1.37791
$195$	−1.86288e7	−0.179913
$196$	−1.35305e8	−1.28356
$197$	−1.16456e8	−1.08525	−0.542623	−	0.839976i	$-0.682569\pi$
$197$	−1.16456e8	−1.08525	−0.542623	+	0.839976i	$0.682569\pi$
$198$	1.86527e8	1.70771
$199$	−1.76140e8	−1.58443	−0.792216	−	0.610241i	$-0.791072\pi$
$199$	−1.76140e8	−1.58443	−0.792216	+	0.610241i	$0.791072\pi$
$200$	1.80125e8	1.59210
$201$	−1.78294e8	−1.54864
$202$	4.78418e7	0.408393
$203$	4.00003e7	0.335604
$204$	8.13918e8	6.71236
$205$	2.44919e7	0.198556
$206$	−4.56416e7	−0.363769
$207$	−6.53718e7	−0.512264
$208$	5.48302e7	0.422472
$209$	−7.06900e7	−0.535607
$210$	2.22900e8	1.66089
$211$	1.86839e8	1.36924	0.684620	−	0.728901i	$-0.259968\pi$
$211$	1.86839e8	1.36924	0.684620	+	0.728901i	$0.259968\pi$
$212$	−2.07643e8	−1.49673
$213$	−3.33713e8	−2.36616
$214$	−3.67855e8	−2.56584
$215$	−1.10263e8	−0.756654
$216$	−1.16876e9	−7.89108
$217$	1.44024e7	0.0956813
$218$	5.38308e8	3.51911
$219$	−2.42185e8	−1.55809
$220$	9.99881e7	0.633095
$221$	3.26814e7	0.203671
$222$	5.13896e8	3.15239
$223$	−1.16234e8	−0.701884	−0.350942	−	0.936397i	$-0.614139\pi$
$223$	−1.16234e8	−0.701884	−0.350942	+	0.936397i	$0.614139\pi$
$224$	−3.10874e8	−1.84806
$225$	−2.29374e8	−1.34247
$226$	−6.78402e7	−0.390938
$227$	−1.60833e8	−0.912609	−0.456304	−	0.889824i	$-0.650827\pi$
$227$	−1.60833e8	−0.912609	−0.456304	+	0.889824i	$0.650827\pi$
$228$	1.22986e9	6.87202
$229$	−1.58215e8	−0.870610	−0.435305	−	0.900283i	$-0.643359\pi$
$229$	−1.58215e8	−0.870610	−0.435305	+	0.900283i	$0.643359\pi$
$230$	−4.88010e7	−0.264473
$231$	−9.05446e7	−0.483304
$232$	−2.64056e8	−1.38831
$233$	6.74565e7	0.349364	0.174682	−	0.984625i	$-0.444110\pi$
$233$	6.74565e7	0.349364	0.174682	+	0.984625i	$0.444110\pi$
$234$	−1.30305e8	−0.664822
$235$	1.87061e7	0.0940256
$236$	−2.63442e8	−1.30465
$237$	6.26796e8	3.05849
$238$	−3.91044e8	−1.88021
$239$	1.29369e8	0.612968	0.306484	−	0.951876i	$-0.400847\pi$
$239$	1.29369e8	0.612968	0.306484	+	0.951876i	$0.400847\pi$
$240$	−7.88441e8	−3.68154
$241$	4.84417e7	0.222925	0.111463	−	0.993769i	$-0.464446\pi$
$241$	4.84417e7	0.222925	0.111463	+	0.993769i	$0.464446\pi$
$242$	3.58663e8	1.62680
$243$	4.66633e8	2.08619
$244$	−7.83784e8	−3.45408
$245$	7.81236e7	0.339392
$246$	2.41049e8	1.03236
$247$	4.93830e7	0.208515
$248$	−9.50754e7	−0.395810
$249$	−4.33508e7	−0.177951
$250$	−4.84585e8	−1.96146
$251$	3.23668e8	1.29194	0.645968	−	0.763364i	$-0.276454\pi$
$251$	3.23668e8	1.29194	0.645968	+	0.763364i	$0.276454\pi$
$252$	1.11958e9	4.40708
$253$	1.98236e7	0.0769592
$254$	4.31252e8	1.65125
$255$	−4.69949e8	−1.77484
$256$	4.19262e7	0.156187
$257$	3.44328e7	0.126534	0.0632670	−	0.997997i	$-0.479848\pi$
$257$	3.44328e7	0.126534	0.0632670	+	0.997997i	$0.479848\pi$
$258$	−1.08521e9	−3.93411
$259$	−1.77292e8	−0.634072
$260$	−6.98502e7	−0.246468
$261$	3.36251e8	1.17064
$262$	9.50360e8	3.26463
$263$	−7.34996e7	−0.249138	−0.124569	−	0.992211i	$-0.539755\pi$
$263$	−7.34996e7	−0.249138	−0.124569	+	0.992211i	$0.539755\pi$
$264$	5.97715e8	1.99931
$265$	1.19891e8	0.395756
$266$	−5.90883e8	−1.92493
$267$	−1.19306e8	−0.383595
$268$	−6.68529e8	−2.12153
$269$	−3.31972e8	−1.03984	−0.519922	−	0.854213i	$-0.674039\pi$
$269$	−3.31972e8	−1.03984	−0.519922	+	0.854213i	$0.674039\pi$
$270$	1.11105e9	3.43525
$271$	−6.60070e7	−0.201464	−0.100732	−	0.994914i	$-0.532118\pi$
$271$	−6.60070e7	−0.201464	−0.100732	+	0.994914i	$0.532118\pi$
$272$	1.38320e9	4.16768
$273$	6.32531e7	0.188153
$274$	−6.06849e8	−1.78219
$275$	6.95562e7	0.201684
$276$	−3.44890e8	−0.987414
$277$	−5.05242e8	−1.42830	−0.714151	−	0.699991i	$-0.753187\pi$
$277$	−5.05242e8	−1.42830	−0.714151	+	0.699991i	$0.753187\pi$
$278$	8.73684e8	2.43892
$279$	1.21070e8	0.333751
$280$	5.07638e8	1.38198
$281$	6.11069e8	1.64293	0.821463	−	0.570262i	$-0.193158\pi$
$281$	6.11069e8	1.64293	0.821463	+	0.570262i	$0.193158\pi$
$282$	1.84106e8	0.488872
$283$	1.33286e8	0.349567	0.174784	−	0.984607i	$-0.444077\pi$
$283$	1.33286e8	0.349567	0.174784	+	0.984607i	$0.444077\pi$
$284$	−1.25128e9	−3.24147
$285$	−7.10111e8	−1.81706
$286$	3.95141e7	0.0998784
$287$	−8.31608e7	−0.207650
$288$	−2.61328e9	−6.44632
$289$	4.14116e8	1.00921
$290$	2.51017e8	0.604380
$291$	5.71806e8	1.36026
$292$	−9.08091e8	−2.13447
$293$	−4.33046e8	−1.00577	−0.502884	−	0.864354i	$-0.667728\pi$
$293$	−4.33046e8	−1.00577	−0.502884	+	0.864354i	$0.667728\pi$
$294$	7.68893e8	1.76462
$295$	1.52109e8	0.344967
$296$	1.17036e9	2.62300
$297$	−4.51321e8	−0.999626
$298$	3.11711e7	0.0682332
$299$	−1.38485e7	−0.0299607
$300$	−1.21014e9	−2.58768
$301$	3.74394e8	0.791308
$302$	2.94508e8	0.615280
$303$	−1.95222e8	−0.403162
$304$	2.09007e9	4.26681
$305$	4.52550e8	0.913307
$306$	−3.28720e9	−6.55845
$307$	9.27048e8	1.82860	0.914298	−	0.405042i	$-0.132743\pi$
$307$	9.27048e8	1.82860	0.914298	+	0.405042i	$0.132743\pi$
$308$	−3.39504e8	−0.662091
$309$	1.86244e8	0.359110
$310$	9.03807e7	0.172310
$311$	−9.34538e8	−1.76172	−0.880858	−	0.473381i	$-0.843033\pi$
$311$	−9.34538e8	−1.76172	−0.880858	+	0.473381i	$0.843033\pi$
$312$	−4.17555e8	−0.778345
$313$	−7.84627e7	−0.144630	−0.0723149	−	0.997382i	$-0.523039\pi$
$313$	−7.84627e7	−0.144630	−0.0723149	+	0.997382i	$0.523039\pi$
$314$	−5.30578e8	−0.967154
$315$	−6.46432e8	−1.16530
$316$	2.35022e9	4.18991
$317$	1.46811e8	0.258851	0.129426	−	0.991589i	$-0.458687\pi$
$317$	1.46811e8	0.258851	0.129426	+	0.991589i	$0.458687\pi$
$318$	1.17997e9	2.05767
$319$	−1.01966e8	−0.175869
$320$	−7.90147e8	−1.34798
$321$	1.50106e9	2.53297
$322$	1.65701e8	0.276586
$323$	1.24578e9	2.05700
$324$	4.02121e9	6.56824
$325$	−4.85909e7	−0.0785169
$326$	−1.78612e9	−2.85528
$327$	−2.19661e9	−3.47404
$328$	5.48972e8	0.858998
$329$	−6.35156e7	−0.0983320
$330$	−5.68201e8	−0.870368
$331$	−5.09422e8	−0.772111	−0.386055	−	0.922476i	$-0.626163\pi$
$331$	−5.09422e8	−0.772111	−0.386055	+	0.922476i	$0.626163\pi$
$332$	−1.62547e8	−0.243779
$333$	−1.49035e9	−2.21174
$334$	−8.30709e8	−1.21993
$335$	3.86002e8	0.560961
$336$	2.67711e9	3.85016
$337$	−1.03024e9	−1.46633	−0.733166	−	0.680050i	$-0.761958\pi$
$337$	−1.03024e9	−1.46633	−0.733166	+	0.680050i	$0.761958\pi$
$338$	1.30942e9	1.84447
$339$	2.76827e8	0.385931
$340$	−1.76211e9	−2.43140
$341$	−3.67138e7	−0.0501405
$342$	−4.96709e9	−6.71445
$343$	−7.91636e8	−1.05924
$344$	−2.47150e9	−3.27345
$345$	1.99136e8	0.261086
$346$	−1.99373e9	−2.58762
$347$	−1.22094e9	−1.56871	−0.784354	−	0.620313i	$-0.787006\pi$
$347$	−1.22094e9	−1.56871	−0.784354	+	0.620313i	$0.787006\pi$
$348$	1.77400e9	2.25646
$349$	1.05987e9	1.33464	0.667320	−	0.744771i	$-0.267441\pi$
$349$	1.05987e9	1.33464	0.667320	+	0.744771i	$0.267441\pi$
$350$	5.81406e8	0.724838
$351$	3.15286e8	0.389161
$352$	7.92460e8	0.968452
$353$	8.39230e8	1.01548	0.507738	−	0.861512i	$-0.330482\pi$
$353$	8.39230e8	1.01548	0.507738	+	0.861512i	$0.330482\pi$
$354$	1.49706e9	1.79360
$355$	7.22480e8	0.857090
$356$	−4.47348e8	−0.525497
$357$	1.59569e9	1.85613
$358$	−7.79236e8	−0.897590
$359$	7.38651e8	0.842576	0.421288	−	0.906927i	$-0.361578\pi$
$359$	7.38651e8	0.842576	0.421288	+	0.906927i	$0.361578\pi$
$360$	4.26731e9	4.82054
$361$	9.88556e8	1.10593
$362$	−1.79971e8	−0.199398
$363$	−1.46355e9	−1.60596
$364$	2.37172e8	0.257756
$365$	5.24323e8	0.564383
$366$	4.45400e9	4.74861
$367$	1.62094e9	1.71174	0.855868	−	0.517194i	$-0.173024\pi$
$367$	1.62094e9	1.71174	0.855868	+	0.517194i	$0.173024\pi$
$368$	−5.86119e8	−0.613082
$369$	−6.99067e8	−0.724314
$370$	−1.11257e9	−1.14188
$371$	−4.07084e8	−0.413881
$372$	6.38745e8	0.643320
$373$	1.09809e9	1.09562	0.547808	−	0.836604i	$-0.315463\pi$
$373$	1.09809e9	1.09562	0.547808	+	0.836604i	$0.315463\pi$
$374$	9.96824e8	0.985298
$375$	1.97739e9	1.93634
$376$	4.19288e8	0.406775
$377$	7.12320e7	0.0684669
$378$	−3.77249e9	−3.59259
$379$	9.27372e7	0.0875018	0.0437509	−	0.999042i	$-0.486069\pi$
$379$	9.27372e7	0.0875018	0.0437509	+	0.999042i	$0.486069\pi$
$380$	−2.66262e9	−2.48924
$381$	−1.75976e9	−1.63010
$382$	−2.42589e9	−2.22664
$383$	−8.14572e8	−0.740856	−0.370428	−	0.928861i	$-0.620789\pi$
$383$	−8.14572e8	−0.740856	−0.370428	+	0.928861i	$0.620789\pi$
$384$	−2.36353e9	−2.13011
$385$	1.96027e8	0.175066
$386$	−2.60623e9	−2.30652
$387$	3.14723e9	2.76020
$388$	2.14403e9	1.86346
$389$	−1.35871e9	−1.17032	−0.585159	−	0.810919i	$-0.698968\pi$
$389$	−1.35871e9	−1.17032	−0.585159	+	0.810919i	$0.698968\pi$
$390$	3.96936e8	0.338840
$391$	−3.49355e8	−0.295562
$392$	1.75110e9	1.46828
$393$	−3.87802e9	−3.22282
$394$	2.48140e9	2.04390
$395$	−1.35700e9	−1.10787
$396$	−2.85395e9	−2.30947
$397$	2.07777e9	1.66660	0.833300	−	0.552821i	$-0.186449\pi$
$397$	2.07777e9	1.66660	0.833300	+	0.552821i	$0.186449\pi$
$398$	3.75314e9	2.98404
$399$	2.41114e9	1.90028
$400$	−2.05655e9	−1.60668
$401$	−9.70183e8	−0.751361	−0.375680	−	0.926749i	$-0.622591\pi$
$401$	−9.70183e8	−0.751361	−0.375680	+	0.926749i	$0.622591\pi$
$402$	3.79904e9	2.91664
$403$	2.56477e7	0.0195200
$404$	−7.32001e8	−0.552303
$405$	−2.32181e9	−1.73673
$406$	−8.52314e8	−0.632060
$407$	4.51940e8	0.332277
$408$	−1.05337e10	−7.67836
$409$	−4.43084e7	−0.0320224	−0.0160112	−	0.999872i	$-0.505097\pi$
$409$	−4.43084e7	−0.0320224	−0.0160112	+	0.999872i	$0.505097\pi$
$410$	−5.21865e8	−0.373951
$411$	2.47629e9	1.75936
$412$	6.98338e8	0.491955
$413$	−5.16477e8	−0.360766
$414$	1.39292e9	0.964773
$415$	9.38532e7	0.0644586
$416$	−5.53600e8	−0.377025
$417$	−3.56513e9	−2.40768
$418$	1.50624e9	1.00873
$419$	7.38219e8	0.490271	0.245135	−	0.969489i	$-0.421168\pi$
$419$	7.38219e8	0.490271	0.245135	+	0.969489i	$0.421168\pi$
$420$	−3.41046e9	−2.24616
$421$	1.11608e8	0.0728967	0.0364483	−	0.999336i	$-0.488396\pi$
$421$	1.11608e8	0.0728967	0.0364483	+	0.999336i	$0.488396\pi$
$422$	−3.98111e9	−2.57876
$423$	−5.33926e8	−0.342996
$424$	2.68730e9	1.71213
$425$	−1.22580e9	−0.774567
$426$	7.11065e9	4.45632
$427$	−1.53661e9	−0.955136
$428$	5.62835e9	3.46999
$429$	−1.61240e8	−0.0985992
$430$	2.34946e9	1.42504
$431$	1.89063e9	1.13746	0.568730	−	0.822524i	$-0.307435\pi$
$431$	1.89063e9	1.13746	0.568730	+	0.822524i	$0.307435\pi$
$432$	1.33441e10	7.96334
$433$	−7.97013e8	−0.471800	−0.235900	−	0.971777i	$-0.575804\pi$
$433$	−7.97013e8	−0.471800	−0.235900	+	0.971777i	$0.575804\pi$
$434$	−3.06883e8	−0.180201
$435$	−1.02429e9	−0.596639
$436$	−8.23635e9	−4.75918
$437$	−5.27889e8	−0.302592
$438$	5.16039e9	2.93443
$439$	3.82327e8	0.215679	0.107840	−	0.994168i	$-0.465607\pi$
$439$	3.82327e8	0.215679	0.107840	+	0.994168i	$0.465607\pi$
$440$	−1.29404e9	−0.724206
$441$	−2.22987e9	−1.23807
$442$	−6.96366e8	−0.383583
$443$	−2.31678e9	−1.26611	−0.633056	−	0.774106i	$-0.718200\pi$
$443$	−2.31678e9	−1.26611	−0.633056	+	0.774106i	$0.718200\pi$
$444$	−7.86283e9	−4.26323
$445$	2.58294e8	0.138949
$446$	2.47667e9	1.32189
$447$	−1.27196e8	−0.0673593
$448$	2.68290e9	1.40972
$449$	4.47596e8	0.233358	0.116679	−	0.993170i	$-0.462775\pi$
$449$	4.47596e8	0.233358	0.116679	+	0.993170i	$0.462775\pi$
$450$	4.88742e9	2.52834
$451$	2.11988e8	0.108816
$452$	1.03799e9	0.528697
$453$	−1.20176e9	−0.607399
$454$	3.42698e9	1.71876
$455$	−1.36941e8	−0.0681544
$456$	−1.59168e10	−7.86100
$457$	2.31021e9	1.13226	0.566128	−	0.824317i	$-0.308441\pi$
$457$	2.31021e9	1.13226	0.566128	+	0.824317i	$0.308441\pi$
$458$	3.37119e9	1.63966
$459$	7.95371e9	3.83906
$460$	7.46678e8	0.357669
$461$	−1.90114e9	−0.903777	−0.451888	−	0.892075i	$-0.649249\pi$
$461$	−1.90114e9	−0.903777	−0.451888	+	0.892075i	$0.649249\pi$
$462$	1.92929e9	0.910231
$463$	8.12980e8	0.380668	0.190334	−	0.981719i	$-0.439043\pi$
$463$	8.12980e8	0.380668	0.190334	+	0.981719i	$0.439043\pi$
$464$	3.01481e9	1.40103
$465$	−3.68805e8	−0.170103
$466$	−1.43734e9	−0.657975
$467$	−2.31278e9	−1.05081	−0.525406	−	0.850852i	$-0.676086\pi$
$467$	−2.31278e9	−1.05081	−0.525406	+	0.850852i	$0.676086\pi$
$468$	1.99372e9	0.899092
$469$	−1.31065e9	−0.586653
$470$	−3.98584e8	−0.177083
$471$	2.16506e9	0.954767
$472$	3.40944e9	1.49240
$473$	−9.54379e8	−0.414674
$474$	−1.33556e10	−5.76021
$475$	−1.85224e9	−0.792992
$476$	5.98315e9	2.54276
$477$	−3.42204e9	−1.44368
$478$	−2.75655e9	−1.15443
$479$	2.05808e9	0.855634	0.427817	−	0.903865i	$-0.359283\pi$
$479$	2.05808e9	0.855634	0.427817	+	0.903865i	$0.359283\pi$
$480$	7.96060e9	3.28550
$481$	−3.15718e8	−0.129358
$482$	−1.03218e9	−0.419847
$483$	−6.76156e8	−0.273044
$484$	−5.48770e9	−2.20005
$485$	−1.23794e9	−0.492725
$486$	−9.94287e9	−3.92903
$487$	−2.99505e8	−0.117504	−0.0587520	−	0.998273i	$-0.518712\pi$
$487$	−2.99505e8	−0.117504	−0.0587520	+	0.998273i	$0.518712\pi$
$488$	1.01437e10	3.95117
$489$	7.28838e9	2.81871
$490$	−1.66463e9	−0.639193
$491$	−1.40636e9	−0.536182	−0.268091	−	0.963394i	$-0.586393\pi$
$491$	−1.40636e9	−0.536182	−0.268091	+	0.963394i	$0.586393\pi$
$492$	−3.68816e9	−1.39615
$493$	1.79697e9	0.675424
$494$	−1.05224e9	−0.392707
$495$	1.64784e9	0.610657
$496$	1.08551e9	0.399435
$497$	−2.45314e9	−0.896345
$498$	9.23704e8	0.335143
$499$	−6.52753e8	−0.235178	−0.117589	−	0.993062i	$-0.537517\pi$
$499$	−6.52753e8	−0.235178	−0.117589	+	0.993062i	$0.537517\pi$
$500$	7.41437e9	2.65265
$501$	3.38977e9	1.20431
$502$	−6.89660e9	−2.43317
$503$	−4.89782e9	−1.71599	−0.857995	−	0.513658i	$-0.828290\pi$
$503$	−4.89782e9	−1.71599	−0.857995	+	0.513658i	$0.828290\pi$
$504$	−1.44894e10	−5.04132
$505$	4.22651e8	0.146037
$506$	−4.22395e8	−0.144941
$507$	−5.34319e9	−1.82084
$508$	−6.59835e9	−2.23312
$509$	−4.40431e9	−1.48035	−0.740177	−	0.672412i	$-0.765259\pi$
$509$	−4.40431e9	−1.48035	−0.740177	+	0.672412i	$0.765259\pi$
$510$	1.00135e10	3.34265
$511$	−1.78031e9	−0.590231
$512$	2.58613e9	0.851540
$513$	1.20184e10	3.93038
$514$	−7.33684e8	−0.238308
$515$	−4.03213e8	−0.130080
$516$	1.66043e10	5.32042
$517$	1.61910e8	0.0515295
$518$	3.77767e9	1.19418
$519$	8.13557e9	2.55448
$520$	9.03994e8	0.281938
$521$	4.10159e9	1.27063	0.635317	−	0.772251i	$-0.280869\pi$
$521$	4.10159e9	1.27063	0.635317	+	0.772251i	$0.280869\pi$
$522$	−7.16473e9	−2.20472
$523$	−2.81959e8	−0.0861846	−0.0430923	−	0.999071i	$-0.513721\pi$
$523$	−2.81959e8	−0.0861846	−0.0430923	+	0.999071i	$0.513721\pi$
$524$	−1.45409e10	−4.41502
$525$	−2.37247e9	−0.715555
$526$	1.56610e9	0.469214
$527$	6.47014e8	0.192565
$528$	−6.82430e9	−2.01762
$529$	1.48036e8	0.0434783
$530$	−2.55460e9	−0.745346
$531$	−4.34162e9	−1.25841
$532$	9.04077e9	2.60324
$533$	−1.48091e8	−0.0423628
$534$	2.54214e9	0.722444
$535$	−3.24976e9	−0.917513
$536$	8.65204e9	2.42684
$537$	3.17973e9	0.886095
$538$	7.07355e9	1.95839
$539$	6.76194e8	0.185999
$540$	−1.69995e10	−4.64577
$541$	5.49037e9	1.49077	0.745386	−	0.666634i	$-0.232265\pi$
$541$	5.49037e9	1.49077	0.745386	+	0.666634i	$0.232265\pi$
$542$	1.40646e9	0.379427
$543$	7.34384e8	0.196845
$544$	−1.39657e10	−3.71934
$545$	4.75559e9	1.25839
$546$	−1.34778e9	−0.354359
$547$	−1.46859e9	−0.383659	−0.191829	−	0.981428i	$-0.561442\pi$
$547$	−1.46859e9	−0.383659	−0.191829	+	0.981428i	$0.561442\pi$
$548$	9.28507e9	2.41020
$549$	−1.29171e10	−3.33166
$550$	−1.48208e9	−0.379842
$551$	2.71529e9	0.691490
$552$	4.46354e9	1.12952
$553$	4.60761e9	1.15861
$554$	1.07655e10	2.68999
$555$	4.53992e9	1.12726
$556$	−1.33678e10	−3.29835
$557$	9.24360e8	0.226646	0.113323	−	0.993558i	$-0.463851\pi$
$557$	9.24360e8	0.226646	0.113323	+	0.993558i	$0.463851\pi$
$558$	−2.57972e9	−0.628569
$559$	6.66715e8	0.161435
$560$	−5.79586e9	−1.39463
$561$	−4.06761e9	−0.972679
$562$	−1.30205e10	−3.09420
$563$	−6.05124e9	−1.42911	−0.714554	−	0.699581i	$-0.753370\pi$
$563$	−6.05124e9	−1.42911	−0.714554	+	0.699581i	$0.753370\pi$
$564$	−2.81690e9	−0.661142
$565$	−5.99323e8	−0.139795
$566$	−2.84001e9	−0.658357
$567$	7.88356e9	1.81628
$568$	1.61940e10	3.70796
$569$	−2.41253e9	−0.549009	−0.274504	−	0.961586i	$-0.588514\pi$
$569$	−2.41253e9	−0.549009	−0.274504	+	0.961586i	$0.588514\pi$
$570$	1.51308e10	3.42216
$571$	−8.05596e8	−0.181089	−0.0905443	−	0.995892i	$-0.528861\pi$
$571$	−8.05596e8	−0.181089	−0.0905443	+	0.995892i	$0.528861\pi$
$572$	−6.04584e8	−0.135074
$573$	9.89905e9	2.19812
$574$	1.77196e9	0.391078
$575$	5.19422e8	0.113942
$576$	2.25530e10	4.91729
$577$	−5.18931e8	−0.112459	−0.0562295	−	0.998418i	$-0.517908\pi$
$577$	−5.18931e8	−0.112459	−0.0562295	+	0.998418i	$0.517908\pi$
$578$	−8.82385e9	−1.90069
$579$	1.06349e10	2.27698
$580$	−3.84067e9	−0.817352
$581$	−3.18673e8	−0.0674108
$582$	−1.21839e10	−2.56185
$583$	1.03771e9	0.216889
$584$	1.17524e10	2.44164
$585$	−1.15116e9	−0.237733
$586$	9.22720e9	1.89421
$587$	−8.20772e9	−1.67490	−0.837450	−	0.546513i	$-0.815955\pi$
$587$	−8.20772e9	−1.67490	−0.837450	+	0.546513i	$0.815955\pi$
$588$	−1.17644e10	−2.38643
$589$	9.77663e8	0.197145
$590$	−3.24108e9	−0.649693
$591$	−1.01255e10	−2.01772
$592$	−1.33624e10	−2.64702
$593$	−2.50638e9	−0.493577	−0.246789	−	0.969069i	$-0.579375\pi$
$593$	−2.50638e9	−0.493577	−0.246789	+	0.969069i	$0.579375\pi$
$594$	9.61659e9	1.88265
$595$	−3.45461e9	−0.672342
$596$	−4.76932e8	−0.0922772
$597$	−1.53150e10	−2.94582
$598$	2.95078e8	0.0564265
$599$	−6.57651e9	−1.25026	−0.625131	−	0.780519i	$-0.714955\pi$
$599$	−6.57651e9	−1.25026	−0.625131	+	0.780519i	$0.714955\pi$
$600$	1.56615e10	2.96008
$601$	7.94544e9	1.49299	0.746496	−	0.665390i	$-0.231735\pi$
$601$	7.94544e9	1.49299	0.746496	+	0.665390i	$0.231735\pi$
$602$	−7.97746e9	−1.49031
$603$	−1.10176e10	−2.04633
$604$	−4.50610e9	−0.832092
$605$	3.16855e9	0.581723
$606$	4.15973e9	0.759296
$607$	2.68877e9	0.487970	0.243985	−	0.969779i	$-0.421545\pi$
$607$	2.68877e9	0.487970	0.243985	+	0.969779i	$0.421545\pi$
$608$	−2.11027e10	−3.80781
$609$	3.47793e9	0.623965
$610$	−9.64278e9	−1.72008
$611$	−1.13108e8	−0.0200608
$612$	5.02956e10	8.86953
$613$	8.66814e9	1.51990	0.759949	−	0.649983i	$-0.225224\pi$
$613$	8.66814e9	1.51990	0.759949	+	0.649983i	$0.225224\pi$
$614$	−1.97532e10	−3.44389
$615$	2.12951e9	0.369161
$616$	4.39383e9	0.757375
$617$	1.12347e10	1.92559	0.962795	−	0.270232i	$-0.0871003\pi$
$617$	1.12347e10	1.92559	0.962795	+	0.270232i	$0.0871003\pi$
$618$	−3.96843e9	−0.676331
$619$	2.76878e9	0.469215	0.234607	−	0.972090i	$-0.424620\pi$
$619$	2.76878e9	0.469215	0.234607	+	0.972090i	$0.424620\pi$
$620$	−1.38286e9	−0.233028
$621$	−3.37031e9	−0.564741
$622$	1.99128e10	3.31793
$623$	−8.77024e8	−0.145313
$624$	4.76735e9	0.785473
$625$	−9.45748e8	−0.154951
$626$	1.67186e9	0.272389
$627$	−6.14632e9	−0.995816
$628$	8.11808e9	1.30796
$629$	−7.96462e9	−1.27611
$630$	1.37740e10	2.19466
$631$	−2.81973e7	−0.00446791	−0.00223396	−	0.999998i	$-0.500711\pi$
$631$	−2.81973e7	−0.00446791	−0.00223396	+	0.999998i	$0.500711\pi$
$632$	−3.04164e10	−4.79289
$633$	1.62452e10	2.54573
$634$	−3.12820e9	−0.487508
$635$	3.80982e9	0.590469
$636$	−1.80541e10	−2.78276
$637$	−4.72379e8	−0.0724107
$638$	2.17266e9	0.331222
$639$	−2.06216e10	−3.12658
$640$	5.11698e9	0.771585
$641$	6.22935e9	0.934199	0.467100	−	0.884205i	$-0.345299\pi$
$641$	6.22935e9	0.934199	0.467100	+	0.884205i	$0.345299\pi$
$642$	−3.19841e10	−4.77048
$643$	9.36940e8	0.138987	0.0694934	−	0.997582i	$-0.477862\pi$
$643$	9.36940e8	0.138987	0.0694934	+	0.997582i	$0.477862\pi$
$644$	−2.53530e9	−0.374050
$645$	−9.58714e9	−1.40679
$646$	−2.65448e10	−3.87405
$647$	−1.71887e9	−0.249504	−0.124752	−	0.992188i	$-0.539814\pi$
$647$	−1.71887e9	−0.249504	−0.124752	+	0.992188i	$0.539814\pi$
$648$	−5.20420e10	−7.51349
$649$	1.31657e9	0.189054
$650$	1.03536e9	0.147875
$651$	1.25226e9	0.177894
$652$	2.73284e10	3.86142
$653$	1.11139e10	1.56196	0.780981	−	0.624554i	$-0.214719\pi$
$653$	1.11139e10	1.56196	0.780981	+	0.624554i	$0.214719\pi$
$654$	4.68046e10	6.54283
$655$	8.39579e9	1.16739
$656$	−6.26779e9	−0.866864
$657$	−1.49657e10	−2.05881
$658$	1.35337e9	0.185194
$659$	2.55105e9	0.347232	0.173616	−	0.984813i	$-0.444455\pi$
$659$	2.55105e9	0.347232	0.173616	+	0.984813i	$0.444455\pi$
$660$	8.69372e9	1.17707
$661$	−1.34691e10	−1.81399	−0.906993	−	0.421146i	$-0.861628\pi$
$661$	−1.34691e10	−1.81399	−0.906993	+	0.421146i	$0.861628\pi$
$662$	1.08546e10	1.45415
$663$	2.84157e9	0.378670
$664$	2.10367e9	0.278862
$665$	−5.22006e9	−0.688334
$666$	3.17559e10	4.16548
$667$	−7.61449e8	−0.0993574
$668$	1.27102e10	1.64981
$669$	−1.01062e10	−1.30496
$670$	−8.22481e9	−1.05649
$671$	3.91702e9	0.500526
$672$	−2.70298e10	−3.43597
$673$	1.20855e9	0.152831	0.0764154	−	0.997076i	$-0.475652\pi$
$673$	1.20855e9	0.152831	0.0764154	+	0.997076i	$0.475652\pi$
$674$	2.19519e10	2.76162
$675$	−1.18256e10	−1.47999
$676$	−2.00347e10	−2.49442
$677$	1.03356e10	1.28019	0.640097	−	0.768294i	$-0.278894\pi$
$677$	1.03356e10	1.28019	0.640097	+	0.768294i	$0.278894\pi$
$678$	−5.89854e9	−0.726843
$679$	4.20337e9	0.515292
$680$	2.28051e10	2.78131
$681$	−1.39840e10	−1.69675
$682$	7.82285e8	0.0944321
$683$	4.58153e8	0.0550223	0.0275111	−	0.999621i	$-0.491242\pi$
$683$	4.58153e8	0.0550223	0.0275111	+	0.999621i	$0.491242\pi$
$684$	7.59987e10	9.08050
$685$	−5.36111e9	−0.637291
$686$	1.68679e10	1.99493
$687$	−1.37564e10	−1.61866
$688$	2.82179e10	3.30343
$689$	−7.24929e8	−0.0844362
$690$	−4.24313e9	−0.491716
$691$	−6.02963e9	−0.695212	−0.347606	−	0.937641i	$-0.613005\pi$
$691$	−6.02963e9	−0.695212	−0.347606	+	0.937641i	$0.613005\pi$
$692$	3.05050e10	3.49945
$693$	−5.59515e9	−0.638625
$694$	2.60155e10	2.95443
$695$	7.71841e9	0.872130
$696$	−2.29590e10	−2.58119
$697$	−3.73590e9	−0.417908
$698$	−2.25834e10	−2.51360
$699$	5.86518e9	0.649548
$700$	−8.89577e9	−0.980258
$701$	9.96825e9	1.09296	0.546482	−	0.837471i	$-0.315967\pi$
$701$	9.96825e9	1.09296	0.546482	+	0.837471i	$0.315967\pi$
$702$	−6.71800e9	−0.732926
$703$	−1.20349e10	−1.30646
$704$	−6.83907e9	−0.738742
$705$	1.62645e9	0.174815
$706$	−1.78820e10	−1.91250
$707$	−1.43509e9	−0.152725
$708$	−2.29056e10	−2.42564
$709$	4.66397e9	0.491466	0.245733	−	0.969338i	$-0.420971\pi$
$709$	4.66397e9	0.491466	0.245733	+	0.969338i	$0.420971\pi$
$710$	−1.53944e10	−1.61420
$711$	3.87325e10	4.04140
$712$	5.78953e9	0.601123
$713$	−2.74166e8	−0.0283270
$714$	−3.40003e10	−3.49574
$715$	3.49081e8	0.0357154
$716$	1.19227e10	1.21388
$717$	1.12483e10	1.13965
$718$	−1.57389e10	−1.58686
$719$	−3.16368e9	−0.317426	−0.158713	−	0.987325i	$-0.550734\pi$
$719$	−3.16368e9	−0.317426	−0.158713	+	0.987325i	$0.550734\pi$
$720$	−4.87213e10	−4.86469
$721$	1.36909e9	0.136037
$722$	−2.10638e10	−2.08285
$723$	4.21189e9	0.414470
$724$	2.75363e9	0.269663
$725$	−2.67174e9	−0.260382
$726$	3.11849e10	3.02459
$727$	5.00250e9	0.482855	0.241428	−	0.970419i	$-0.422384\pi$
$727$	5.00250e9	0.482855	0.241428	+	0.970419i	$0.422384\pi$
$728$	−3.06946e9	−0.294851
$729$	1.35974e10	1.29990
$730$	−1.11721e10	−1.06293
$731$	1.68192e10	1.59256
$732$	−6.81481e10	−6.42192
$733$	−3.68820e9	−0.345900	−0.172950	−	0.984931i	$-0.555330\pi$
$733$	−3.68820e9	−0.345900	−0.172950	+	0.984931i	$0.555330\pi$
$734$	−3.45385e10	−3.22380
$735$	6.79266e9	0.631007
$736$	5.91782e9	0.547129
$737$	3.34102e9	0.307428
$738$	1.48955e10	1.36414
$739$	−2.90694e9	−0.264960	−0.132480	−	0.991186i	$-0.542294\pi$
$739$	−2.90694e9	−0.264960	−0.132480	+	0.991186i	$0.542294\pi$
$740$	1.70228e10	1.54426
$741$	4.29373e9	0.387678
$742$	8.67402e9	0.779483
$743$	−1.70273e9	−0.152295	−0.0761473	−	0.997097i	$-0.524262\pi$
$743$	−1.70273e9	−0.152295	−0.0761473	+	0.997097i	$0.524262\pi$
$744$	−8.26657e9	−0.735902
$745$	2.75376e8	0.0243994
$746$	−2.33978e10	−2.06343
$747$	−2.67884e9	−0.235139
$748$	−1.52518e10	−1.33250
$749$	1.10344e10	0.959535
$750$	−4.21335e10	−3.64681
$751$	−6.10891e9	−0.526289	−0.263144	−	0.964756i	$-0.584760\pi$
$751$	−6.10891e9	−0.526289	−0.263144	+	0.964756i	$0.584760\pi$
$752$	−4.78714e9	−0.410501
$753$	2.81421e10	2.40201
$754$	−1.51779e9	−0.128947
$755$	2.60178e9	0.220017
$756$	5.77208e10	4.85855
$757$	−4.90563e9	−0.411016	−0.205508	−	0.978655i	$-0.565885\pi$
$757$	−4.90563e9	−0.411016	−0.205508	+	0.978655i	$0.565885\pi$
$758$	−1.97601e9	−0.164796
$759$	1.72361e9	0.143085
$760$	3.44593e10	2.84747
$761$	1.56857e10	1.29020	0.645100	−	0.764098i	$-0.276816\pi$
$761$	1.56857e10	1.29020	0.645100	+	0.764098i	$0.276816\pi$
$762$	3.74963e10	3.07005
$763$	−1.61473e10	−1.31603
$764$	3.71173e10	3.01127
$765$	−2.90402e10	−2.34523
$766$	1.73566e10	1.39529
$767$	−9.19734e8	−0.0736001
$768$	3.64538e9	0.290388
$769$	−8.48511e9	−0.672845	−0.336423	−	0.941711i	$-0.609217\pi$
$769$	−8.48511e9	−0.672845	−0.336423	+	0.941711i	$0.609217\pi$
$770$	−4.17687e9	−0.329711
$771$	2.99385e9	0.235256
$772$	3.98765e10	3.11929
$773$	1.25408e10	0.976558	0.488279	−	0.872688i	$-0.337625\pi$
$773$	1.25408e10	0.976558	0.488279	+	0.872688i	$0.337625\pi$
$774$	−6.70602e10	−5.19842
$775$	−9.61982e8	−0.0742354
$776$	−2.77479e10	−2.13164
$777$	−1.54151e10	−1.17889
$778$	2.89510e10	2.20412
$779$	−5.64510e9	−0.427849
$780$	−6.07330e9	−0.458241
$781$	6.25338e9	0.469717
$782$	7.44394e9	0.556646
$783$	1.73358e10	1.29056
$784$	−1.99929e10	−1.48173
$785$	−4.68730e9	−0.345843
$786$	8.26315e10	6.06969
$787$	−2.44401e10	−1.78728	−0.893639	−	0.448786i	$-0.851857\pi$
$787$	−2.44401e10	−1.78728	−0.893639	+	0.448786i	$0.851857\pi$
$788$	−3.79665e10	−2.76413
$789$	−6.39061e9	−0.463204
$790$	2.89144e10	2.08651
$791$	2.03497e9	0.146198
$792$	3.69355e10	2.64183
$793$	−2.73637e9	−0.194858
$794$	−4.42725e10	−3.13879
$795$	1.04243e10	0.735800
$796$	−5.74248e10	−4.03556
$797$	9.51793e7	0.00665945	0.00332972	−	0.999994i	$-0.498940\pi$
$797$	9.51793e7	0.00665945	0.00332972	+	0.999994i	$0.498940\pi$
$798$	−5.13758e10	−3.57889
$799$	−2.85337e9	−0.197899
$800$	2.07642e10	1.43384
$801$	−7.37246e9	−0.506872
$802$	2.06723e10	1.41508
$803$	4.53825e9	0.309303
$804$	−5.81270e10	−3.94440
$805$	1.46386e9	0.0989040
$806$	−5.46492e8	−0.0367630
$807$	−2.88642e10	−1.93331
$808$	9.47349e9	0.631786
$809$	−1.67314e10	−1.11100	−0.555498	−	0.831518i	$-0.687472\pi$
$809$	−1.67314e10	−1.11100	−0.555498	+	0.831518i	$0.687472\pi$
$810$	4.94723e10	3.27088
$811$	−7.07377e9	−0.465670	−0.232835	−	0.972516i	$-0.574800\pi$
$811$	−7.07377e9	−0.465670	−0.232835	+	0.972516i	$0.574800\pi$
$812$	1.30408e10	0.854786
$813$	−5.73915e9	−0.374568
$814$	−9.62978e9	−0.625793
$815$	−1.57792e10	−1.02101
$816$	1.20266e11	7.74867
$817$	2.54145e10	1.63044
$818$	9.44108e8	0.0603094
$819$	3.90869e9	0.248621
$820$	7.98476e9	0.505724
$821$	−1.14579e9	−0.0722607	−0.0361304	−	0.999347i	$-0.511503\pi$
$821$	−1.14579e9	−0.0722607	−0.0361304	+	0.999347i	$0.511503\pi$
$822$	−5.27641e10	−3.31350
$823$	1.74865e10	1.09346	0.546730	−	0.837309i	$-0.315872\pi$
$823$	1.74865e10	1.09346	0.546730	+	0.837309i	$0.315872\pi$
$824$	−9.03782e9	−0.562754
$825$	6.04774e9	0.374977
$826$	1.10049e10	0.679449
$827$	4.70708e9	0.289389	0.144694	−	0.989476i	$-0.453780\pi$
$827$	4.70708e9	0.289389	0.144694	+	0.989476i	$0.453780\pi$
$828$	−2.13123e10	−1.30474
$829$	−3.27355e9	−0.199562	−0.0997811	−	0.995009i	$-0.531814\pi$
$829$	−3.27355e9	−0.199562	−0.0997811	+	0.995009i	$0.531814\pi$
$830$	−1.99979e9	−0.121398
$831$	−4.39296e10	−2.65554
$832$	4.77767e9	0.287597
$833$	−1.19167e10	−0.714330
$834$	7.59647e10	4.53451
$835$	−7.33876e9	−0.436234
$836$	−2.30461e10	−1.36419
$837$	6.24189e9	0.367940
$838$	−1.57297e10	−0.923352
$839$	−7.60099e9	−0.444328	−0.222164	−	0.975009i	$-0.571312\pi$
$839$	−7.60099e9	−0.444328	−0.222164	+	0.975009i	$0.571312\pi$
$840$	4.41379e10	2.56941
$841$	−1.33332e10	−0.772946
$842$	−2.37810e9	−0.137290
$843$	5.31309e10	3.05458
$844$	6.09127e10	3.48746
$845$	1.15679e10	0.659560
$846$	1.13767e10	0.645982
$847$	−1.07586e10	−0.608366
$848$	−3.06817e10	−1.72780
$849$	1.15889e10	0.649926
$850$	2.61190e10	1.45878
$851$	3.37493e9	0.187720
$852$	−1.08796e11	−6.02664
$853$	1.06027e10	0.584916	0.292458	−	0.956278i	$-0.405527\pi$
$853$	1.06027e10	0.584916	0.292458	+	0.956278i	$0.405527\pi$
$854$	3.27415e10	1.79886
$855$	−4.38809e10	−2.40101
$856$	−7.28415e10	−3.96936
$857$	3.25969e10	1.76906	0.884532	−	0.466479i	$-0.154478\pi$
$857$	3.25969e10	1.76906	0.884532	+	0.466479i	$0.154478\pi$
$858$	3.43566e9	0.185697
$859$	3.28664e10	1.76920	0.884599	−	0.466353i	$-0.154432\pi$
$859$	3.28664e10	1.76920	0.884599	+	0.466353i	$0.154432\pi$
$860$	−3.59478e10	−1.92720
$861$	−7.23063e9	−0.386069
$862$	−4.02849e10	−2.14223
$863$	9.63622e9	0.510351	0.255176	−	0.966895i	$-0.417867\pi$
$863$	9.63622e9	0.510351	0.255176	+	0.966895i	$0.417867\pi$
$864$	−1.34730e11	−7.10668
$865$	−1.76133e10	−0.925303
$866$	1.69825e10	0.888564
$867$	3.60064e10	1.87635
$868$	4.69544e9	0.243701
$869$	−1.17454e10	−0.607154
$870$	2.18253e10	1.12368
$871$	−2.33398e9	−0.119684
$872$	1.06594e11	5.44409
$873$	3.53344e10	1.79741
$874$	1.12481e10	0.569887
$875$	1.45359e10	0.733521
$876$	−7.89563e10	−3.96846
$877$	−9.13957e8	−0.0457538	−0.0228769	−	0.999738i	$-0.507283\pi$
$877$	−9.13957e8	−0.0457538	−0.0228769	+	0.999738i	$0.507283\pi$
$878$	−8.14649e9	−0.406200
$879$	−3.76523e10	−1.86995
$880$	1.47744e10	0.730838
$881$	−2.85437e10	−1.40635	−0.703176	−	0.711016i	$-0.748235\pi$
$881$	−2.85437e10	−1.40635	−0.703176	+	0.711016i	$0.748235\pi$
$882$	4.75133e10	2.33172
$883$	6.16630e9	0.301413	0.150707	−	0.988579i	$-0.451845\pi$
$883$	6.16630e9	0.301413	0.150707	+	0.988579i	$0.451845\pi$
$884$	1.06547e10	0.518750
$885$	1.32255e10	0.641372
$886$	4.93652e10	2.38453
$887$	−4.16328e8	−0.0200310	−0.0100155	−	0.999950i	$-0.503188\pi$
$887$	−4.16328e8	−0.0200310	−0.0100155	+	0.999950i	$0.503188\pi$
$888$	1.01760e11	4.87676
$889$	−1.29360e10	−0.617512
$890$	−5.50365e9	−0.261689
$891$	−2.00963e10	−0.951794
$892$	−3.78942e10	−1.78770
$893$	−4.31155e9	−0.202606
$894$	2.71025e9	0.126861
$895$	−6.88402e9	−0.320968
$896$	−1.73744e10	−0.806924
$897$	−1.20409e9	−0.0557038
$898$	−9.53722e9	−0.439496
$899$	1.41022e9	0.0647334
$900$	−7.47797e10	−3.41928
$901$	−1.82878e10	−0.832961
$902$	−4.51697e9	−0.204939
$903$	3.25526e10	1.47122
$904$	−1.34335e10	−0.604784
$905$	−1.58992e9	−0.0713026
$906$	2.56067e10	1.14395
$907$	−8.01334e9	−0.356605	−0.178303	−	0.983976i	$-0.557061\pi$
$907$	−8.01334e9	−0.356605	−0.178303	+	0.983976i	$0.557061\pi$
$908$	−5.24343e10	−2.32442
$909$	−1.20636e10	−0.532728
$910$	2.91790e9	0.128359
$911$	1.58468e10	0.694426	0.347213	−	0.937786i	$-0.387128\pi$
$911$	1.58468e10	0.694426	0.347213	+	0.937786i	$0.387128\pi$
$912$	1.81727e11	7.93299
$913$	8.12341e8	0.0353257
$914$	−4.92252e10	−2.13244
$915$	3.93481e10	1.69805
$916$	−5.15808e10	−2.21745
$917$	−2.85075e10	−1.22086
$918$	−1.69475e11	−7.23030
$919$	−7.88167e9	−0.334976	−0.167488	−	0.985874i	$-0.553566\pi$
$919$	−7.88167e9	−0.334976	−0.167488	+	0.985874i	$0.553566\pi$
$920$	−9.66343e9	−0.409142
$921$	8.06046e10	3.39978
$922$	4.05089e10	1.70213
$923$	−4.36852e9	−0.182864
$924$	−2.95191e10	−1.23098
$925$	1.18418e10	0.491952
$926$	−1.73227e10	−0.716931
$927$	1.15089e10	0.474518
$928$	−3.04394e10	−1.25031
$929$	1.82739e9	0.0747782	0.0373891	−	0.999301i	$-0.488096\pi$
$929$	1.82739e9	0.0747782	0.0373891	+	0.999301i	$0.488096\pi$
$930$	7.85838e9	0.320363
$931$	−1.80066e10	−0.731321
$932$	2.19920e10	0.889833
$933$	−8.12558e10	−3.27543
$934$	4.92799e10	1.97905
$935$	8.80627e9	0.352331
$936$	−2.58026e10	−1.02848
$937$	1.59441e10	0.633158	0.316579	−	0.948566i	$-0.397466\pi$
$937$	1.59441e10	0.633158	0.316579	+	0.948566i	$0.397466\pi$
$938$	2.79269e10	1.10487
$939$	−6.82214e9	−0.268900
$940$	6.09851e9	0.239484
$941$	1.62795e10	0.636909	0.318455	−	0.947938i	$-0.396836\pi$
$941$	1.62795e10	0.636909	0.318455	+	0.947938i	$0.396836\pi$
$942$	−4.61325e10	−1.79816
$943$	1.58305e9	0.0614759
$944$	−3.89266e10	−1.50607
$945$	−3.33275e10	−1.28467
$946$	2.03356e10	0.780976
$947$	−1.58030e10	−0.604665	−0.302333	−	0.953203i	$-0.597765\pi$
$947$	−1.58030e10	−0.604665	−0.302333	+	0.953203i	$0.597765\pi$
$948$	2.04346e11	7.79000
$949$	−3.17035e9	−0.120414
$950$	3.94668e10	1.49348
$951$	1.27648e10	0.481264
$952$	−7.74333e10	−2.90870
$953$	2.46781e9	0.0923607	0.0461804	−	0.998933i	$-0.485295\pi$
$953$	2.46781e9	0.0923607	0.0461804	+	0.998933i	$0.485295\pi$
$954$	7.29156e10	2.71895
$955$	−2.14312e10	−0.796221
$956$	4.21765e10	1.56123
$957$	−8.86571e9	−0.326980
$958$	−4.38529e10	−1.61146
$959$	1.82033e10	0.666478
$960$	−6.87013e10	−2.50620
$961$	−2.70049e10	−0.981544
$962$	6.72722e9	0.243625
$963$	9.27572e10	3.34700
$964$	1.57928e10	0.567793
$965$	−2.30243e10	−0.824785
$966$	1.44073e10	0.514237
$967$	−2.14651e10	−0.763379	−0.381689	−	0.924291i	$-0.624658\pi$
$967$	−2.14651e10	−0.763379	−0.381689	+	0.924291i	$0.624658\pi$
$968$	7.10213e10	2.51666
$969$	1.08318e11	3.82443
$970$	2.63777e10	0.927974
$971$	−7.97270e9	−0.279472	−0.139736	−	0.990189i	$-0.544625\pi$
$971$	−7.97270e9	−0.279472	−0.139736	+	0.990189i	$0.544625\pi$
$972$	1.52130e11	5.31354
$973$	−2.62074e10	−0.912073
$974$	6.38175e9	0.221301
$975$	−4.22486e9	−0.145981
$976$	−1.15813e11	−3.98735
$977$	9.53879e9	0.327237	0.163618	−	0.986524i	$-0.447683\pi$
$977$	9.53879e9	0.327237	0.163618	+	0.986524i	$0.447683\pi$
$978$	−1.55299e11	−5.30862
$979$	2.23565e9	0.0761491
$980$	2.54696e10	0.864433
$981$	−1.35738e11	−4.59050
$982$	2.99663e10	1.00982
$983$	−9.66381e9	−0.324497	−0.162249	−	0.986750i	$-0.551875\pi$
$983$	−9.66381e9	−0.324497	−0.162249	+	0.986750i	$0.551875\pi$
$984$	4.77318e10	1.59707
$985$	2.19215e10	0.730875
$986$	−3.82892e10	−1.27206
$987$	−5.52253e9	−0.182822
$988$	1.60997e10	0.531090
$989$	−7.12698e9	−0.234271
$990$	−3.51117e10	−1.15008
$991$	2.82627e10	0.922477	0.461238	−	0.887276i	$-0.347405\pi$
$991$	2.82627e10	0.922477	0.461238	+	0.887276i	$0.347405\pi$
$992$	−1.09599e10	−0.356466
$993$	−4.42930e10	−1.43553
$994$	5.22707e10	1.68813
$995$	3.31565e10	1.06706
$996$	−1.41331e10	−0.453241
$997$	2.32367e10	0.742576	0.371288	−	0.928518i	$-0.378916\pi$
$997$	2.32367e10	0.742576	0.371288	+	0.928518i	$0.378916\pi$
$998$	1.39086e10	0.442923
$999$	−7.68365e10	−2.43831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.b.1.1	✓	8
3.2	odd	2		207.8.a.f.1.8		8
4.3	odd	2		368.8.a.h.1.1		8
5.4	even	2		575.8.a.b.1.8		8
23.22	odd	2		529.8.a.c.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.b.1.1	✓	8	1.1	even	1	trivial
207.8.a.f.1.8		8	3.2	odd	2
368.8.a.h.1.1		8	4.3	odd	2
529.8.a.c.1.1		8	23.22	odd	2
575.8.a.b.1.8		8	5.4	even	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q-21.3077 q^{2} +86.9475 q^{3} +326.017 q^{4} -188.239 q^{5} -1852.65 q^{6} +639.155 q^{7} -4219.28 q^{8} +5372.88 q^{9} +O(q^{10})\)	\(q-21.3077 q^{2} +86.9475 q^{3} +326.017 q^{4} -188.239 q^{5} -1852.65 q^{6} +639.155 q^{7} -4219.28 q^{8} +5372.88 q^{9} +4010.94 q^{10} -1629.29 q^{11} +28346.4 q^{12} +1138.20 q^{13} -13618.9 q^{14} -16366.9 q^{15} +48172.8 q^{16} +28713.3 q^{17} -114483. q^{18} +43387.0 q^{19} -61369.1 q^{20} +55573.0 q^{21} +34716.4 q^{22} -12167.0 q^{23} -366856. q^{24} -42691.1 q^{25} -24252.3 q^{26} +277004. q^{27} +208375. q^{28} +62583.1 q^{29} +348741. q^{30} +22533.6 q^{31} -486383. q^{32} -141663. q^{33} -611814. q^{34} -120314. q^{35} +1.75165e6 q^{36} -277384. q^{37} -924475. q^{38} +98963.5 q^{39} +794233. q^{40} -130110. q^{41} -1.18413e6 q^{42} +585763. q^{43} -531177. q^{44} -1.01138e6 q^{45} +259250. q^{46} -99374.3 q^{47} +4.18851e6 q^{48} -415024. q^{49} +909647. q^{50} +2.49655e6 q^{51} +371072. q^{52} -636910. q^{53} -5.90231e6 q^{54} +306696. q^{55} -2.69677e6 q^{56} +3.77239e6 q^{57} -1.33350e6 q^{58} -808062. q^{59} -5.33589e6 q^{60} -2.40412e6 q^{61} -480138. q^{62} +3.43410e6 q^{63} +4.19757e6 q^{64} -214253. q^{65} +3.01851e6 q^{66} -2.05060e6 q^{67} +9.36103e6 q^{68} -1.05789e6 q^{69} +2.56361e6 q^{70} -3.83810e6 q^{71} -2.26697e7 q^{72} -2.78541e6 q^{73} +5.91041e6 q^{74} -3.71188e6 q^{75} +1.41449e7 q^{76} -1.04137e6 q^{77} -2.10868e6 q^{78} +7.20890e6 q^{79} -9.06801e6 q^{80} +1.23343e7 q^{81} +2.77235e6 q^{82} -498585. q^{83} +1.81177e7 q^{84} -5.40497e6 q^{85} -1.24813e7 q^{86} +5.44145e6 q^{87} +6.87443e6 q^{88} -1.37216e6 q^{89} +2.15503e7 q^{90} +727485. q^{91} -3.96665e6 q^{92} +1.95924e6 q^{93} +2.11743e6 q^{94} -8.16712e6 q^{95} -4.22898e7 q^{96} +6.57645e6 q^{97} +8.84319e6 q^{98} -8.75398e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9}+O(q^{10}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9	\( 8 q + 40 q^{3} + 640 q^{4} + 444 q^{5} - 1745 q^{6} + 1446 q^{7} + 3177 q^{8} + 13878 q^{9} + 19502 q^{10} + 7588 q^{11} + 22733 q^{12} + 19862 q^{13} + 17544 q^{14} - 12770 q^{15} + 64336 q^{16} + 42070 q^{17} - 59129 q^{18} + 1050 q^{19} + 3364 q^{20} - 7698 q^{21} - 128220 q^{22} - 97336 q^{23} - 621188 q^{24} + 49496 q^{25} - 371761 q^{26} - 69500 q^{27} + 143050 q^{28} - 102578 q^{29} - 671470 q^{30} + 304172 q^{31} - 612824 q^{32} + 747242 q^{33} - 524530 q^{34} + 531048 q^{35} + 1868983 q^{36} + 286472 q^{37} - 762932 q^{38} + 1032828 q^{39} + 2105286 q^{40} + 1324414 q^{41} - 1886168 q^{42} + 2052578 q^{43} - 867298 q^{44} + 2087442 q^{45} + 675556 q^{47} + 1411151 q^{48} - 55404 q^{49} + 1458528 q^{50} + 2775482 q^{51} - 1695409 q^{52} + 203654 q^{53} - 9897559 q^{54} - 1024444 q^{55} - 5766846 q^{56} + 3908648 q^{57} - 5039991 q^{58} - 748892 q^{59} - 18153300 q^{60} + 61822 q^{61} - 4939277 q^{62} + 1411632 q^{63} + 2702267 q^{64} - 1571618 q^{65} + 3791866 q^{66} + 3235604 q^{67} + 4914980 q^{68} - 486680 q^{69} + 10871764 q^{70} - 4951664 q^{71} - 7940241 q^{72} + 11019370 q^{73} + 356954 q^{74} - 13607220 q^{75} + 21973240 q^{76} - 5284888 q^{77} - 1506779 q^{78} + 4202464 q^{79} + 8785886 q^{80} + 10294096 q^{81} + 32636759 q^{82} + 518568 q^{83} + 7629190 q^{84} + 9854220 q^{85} - 14681386 q^{86} + 4862532 q^{87} + 20589740 q^{88} + 4203864 q^{89} + 49021076 q^{90} + 2488406 q^{91} - 7786880 q^{92} - 23367842 q^{93} + 12314327 q^{94} - 44485300 q^{95} - 45317009 q^{96} + 18621134 q^{97} + 35756 q^{98} - 64729930 q^{99}+O(q^{100}) \) 8 * q + 40 * q^3 + 640 * q^4 + 444 * q^5 - 1745 * q^6 + 1446 * q^7 + 3177 * q^8 + 13878 * q^9 + 19502 * q^10 + 7588 * q^11 + 22733 * q^12 + 19862 * q^13 + 17544 * q^14 - 12770 * q^15 + 64336 * q^16 + 42070 * q^17 - 59129 * q^18 + 1050 * q^19 + 3364 * q^20 - 7698 * q^21 - 128220 * q^22 - 97336 * q^23 - 621188 * q^24 + 49496 * q^25 - 371761 * q^26 - 69500 * q^27 + 143050 * q^28 - 102578 * q^29 - 671470 * q^30 + 304172 * q^31 - 612824 * q^32 + 747242 * q^33 - 524530 * q^34 + 531048 * q^35 + 1868983 * q^36 + 286472 * q^37 - 762932 * q^38 + 1032828 * q^39 + 2105286 * q^40 + 1324414 * q^41 - 1886168 * q^42 + 2052578 * q^43 - 867298 * q^44 + 2087442 * q^45 + 675556 * q^47 + 1411151 * q^48 - 55404 * q^49 + 1458528 * q^50 + 2775482 * q^51 - 1695409 * q^52 + 203654 * q^53 - 9897559 * q^54 - 1024444 * q^55 - 5766846 * q^56 + 3908648 * q^57 - 5039991 * q^58 - 748892 * q^59 - 18153300 * q^60 + 61822 * q^61 - 4939277 * q^62 + 1411632 * q^63 + 2702267 * q^64 - 1571618 * q^65 + 3791866 * q^66 + 3235604 * q^67 + 4914980 * q^68 - 486680 * q^69 + 10871764 * q^70 - 4951664 * q^71 - 7940241 * q^72 + 11019370 * q^73 + 356954 * q^74 - 13607220 * q^75 + 21973240 * q^76 - 5284888 * q^77 - 1506779 * q^78 + 4202464 * q^79 + 8785886 * q^80 + 10294096 * q^81 + 32636759 * q^82 + 518568 * q^83 + 7629190 * q^84 + 9854220 * q^85 - 14681386 * q^86 + 4862532 * q^87 + 20589740 * q^88 + 4203864 * q^89 + 49021076 * q^90 + 2488406 * q^91 - 7786880 * q^92 - 23367842 * q^93 + 12314327 * q^94 - 44485300 * q^95 - 45317009 * q^96 + 18621134 * q^97 + 35756 * q^98 - 64729930 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more