LMFDB - Embedded newform 69.8.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$0$
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} - 775x^{5} - 474x^{4} + 167184x^{3} - 33920x^{2} - 9348928x + 28965760 $ x^7 - 775x^5 - 474x^4 + 167184x^3 - 33920x^2 - 9348928*x + 28965760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.2020$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.2020 q^{2} -27.0000 q^{3} +134.506 q^{4} +395.951 q^{5} +437.455 q^{6} +1527.96 q^{7} -105.404 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-16.2020 q^{2} -27.0000 q^{3} +134.506 q^{4} +395.951 q^{5} +437.455 q^{6} +1527.96 q^{7} -105.404 q^{8} +729.000 q^{9} -6415.21 q^{10} -977.157 q^{11} -3631.65 q^{12} +7625.78 q^{13} -24756.0 q^{14} -10690.7 q^{15} -15509.0 q^{16} +13547.0 q^{17} -11811.3 q^{18} +25.2152 q^{19} +53257.7 q^{20} -41254.8 q^{21} +15831.9 q^{22} -12167.0 q^{23} +2845.91 q^{24} +78652.3 q^{25} -123553. q^{26} -19683.0 q^{27} +205519. q^{28} -96873.1 q^{29} +173211. q^{30} +266572. q^{31} +264768. q^{32} +26383.2 q^{33} -219489. q^{34} +604996. q^{35} +98054.6 q^{36} +39869.6 q^{37} -408.538 q^{38} -205896. q^{39} -41734.9 q^{40} -746001. q^{41} +668412. q^{42} +579650. q^{43} -131433. q^{44} +288648. q^{45} +197130. q^{46} -84917.9 q^{47} +418742. q^{48} +1.51111e6 q^{49} -1.27433e6 q^{50} -365769. q^{51} +1.02571e6 q^{52} -1.55729e6 q^{53} +318904. q^{54} -386906. q^{55} -161053. q^{56} -680.811 q^{57} +1.56954e6 q^{58} -1.35938e6 q^{59} -1.43796e6 q^{60} +1.13718e6 q^{61} -4.31901e6 q^{62} +1.11388e6 q^{63} -2.30464e6 q^{64} +3.01944e6 q^{65} -427462. q^{66} +2.47935e6 q^{67} +1.82215e6 q^{68} +328509. q^{69} -9.80216e6 q^{70} -263847. q^{71} -76839.7 q^{72} -4.65126e6 q^{73} -645968. q^{74} -2.12361e6 q^{75} +3391.59 q^{76} -1.49305e6 q^{77} +3.33593e6 q^{78} -4.49927e6 q^{79} -6.14079e6 q^{80} +531441. q^{81} +1.20867e7 q^{82} +5.06957e6 q^{83} -5.54901e6 q^{84} +5.36395e6 q^{85} -9.39150e6 q^{86} +2.61557e6 q^{87} +102996. q^{88} +1.01382e7 q^{89} -4.67669e6 q^{90} +1.16519e7 q^{91} -1.63653e6 q^{92} -7.19745e6 q^{93} +1.37584e6 q^{94} +9984.00 q^{95} -7.14874e6 q^{96} +1.22771e7 q^{97} -2.44830e7 q^{98} -712347. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) $ 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * 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 $

$2$	−16.2020	−1.43207	−0.716035	−	0.698064i	$-0.754045\pi$
$2$	−16.2020	−1.43207	−0.716035	+	0.698064i	$0.754045\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	134.506	1.05083
$5$	395.951	1.41660	0.708299	−	0.705913i	$-0.249463\pi$
$5$	395.951	1.41660	0.708299	+	0.705913i	$0.249463\pi$
$6$	437.455	0.826806
$7$	1527.96	1.68371	0.841856	−	0.539702i	$-0.181463\pi$
$7$	1527.96	1.68371	0.841856	+	0.539702i	$0.181463\pi$
$8$	−105.404	−0.0727852
$9$	729.000	0.333333
$10$	−6415.21	−2.02867
$11$	−977.157	−0.221355	−0.110678	−	0.993856i	$-0.535302\pi$
$11$	−977.157	−0.221355	−0.110678	+	0.993856i	$0.535302\pi$
$12$	−3631.65	−0.606694
$13$	7625.78	0.962682	0.481341	−	0.876533i	$-0.340150\pi$
$13$	7625.78	0.962682	0.481341	+	0.876533i	$0.340150\pi$
$14$	−24756.0	−2.41119
$15$	−10690.7	−0.817873
$16$	−15509.0	−0.946592
$17$	13547.0	0.668763	0.334381	−	0.942438i	$-0.391473\pi$
$17$	13547.0	0.668763	0.334381	+	0.942438i	$0.391473\pi$
$18$	−11811.3	−0.477357
$19$	25.2152	0.000843384 0	0.000421692	−	1.00000i	$-0.499866\pi$
$19$	25.2152	0.000843384 0	0.000421692		1.00000i	$0.499866\pi$
$20$	53257.7	1.48860
$21$	−41254.8	−0.972092
$22$	15831.9	0.316996
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	2845.91	0.0420226
$25$	78652.3	1.00675
$26$	−123553.	−1.37863
$27$	−19683.0	−0.192450
$28$	205519.	1.76929
$29$	−96873.1	−0.737582	−0.368791	−	0.929512i	$-0.620228\pi$
$29$	−96873.1	−0.737582	−0.368791	+	0.929512i	$0.620228\pi$
$30$	173211.	1.17125
$31$	266572.	1.60712	0.803561	−	0.595222i	$-0.202936\pi$
$31$	266572.	1.60712	0.803561	+	0.595222i	$0.202936\pi$
$32$	264768.	1.42837
$33$	26383.2	0.127800
$34$	−219489.	−0.957715
$35$	604996.	2.38514
$36$	98054.6	0.350275
$37$	39869.6	0.129400	0.0647002	−	0.997905i	$-0.479391\pi$
$37$	39869.6	0.129400	0.0647002	+	0.997905i	$0.479391\pi$
$38$	−408.538	−0.00120779
$39$	−205896.	−0.555805
$40$	−41734.9	−0.103107
$41$	−746001.	−1.69042	−0.845212	−	0.534431i	$-0.820526\pi$
$41$	−746001.	−1.69042	−0.845212	+	0.534431i	$0.820526\pi$
$42$	668412.	1.39210
$43$	579650.	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$43$	579650.	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$44$	−131433.	−0.232606
$45$	288648.	0.472199
$46$	197130.	0.298607
$47$	−84917.9	−0.119304	−0.0596522	−	0.998219i	$-0.518999\pi$
$47$	−84917.9	−0.119304	−0.0596522	+	0.998219i	$0.518999\pi$
$48$	418742.	0.546515
$49$	1.51111e6	1.83489
$50$	−1.27433e6	−1.44174
$51$	−365769.	−0.386110
$52$	1.02571e6	1.01161
$53$	−1.55729e6	−1.43683	−0.718413	−	0.695616i	$-0.755131\pi$
$53$	−1.55729e6	−1.43683	−0.718413	+	0.695616i	$0.755131\pi$
$54$	318904.	0.275602
$55$	−386906.	−0.313571
$56$	−161053.	−0.122549
$57$	−680.811	−0.000486928 0
$58$	1.56954e6	1.05627
$59$	−1.35938e6	−0.861702	−0.430851	−	0.902423i	$-0.641787\pi$
$59$	−1.35938e6	−0.861702	−0.430851	+	0.902423i	$0.641787\pi$
$60$	−1.43796e6	−0.859442
$61$	1.13718e6	0.641470	0.320735	−	0.947169i	$-0.396070\pi$
$61$	1.13718e6	0.641470	0.320735	+	0.947169i	$0.396070\pi$
$62$	−4.31901e6	−2.30151
$63$	1.11388e6	0.561237
$64$	−2.30464e6	−1.09894
$65$	3.01944e6	1.36373
$66$	−427462.	−0.183018
$67$	2.47935e6	1.00711	0.503555	−	0.863963i	$-0.332025\pi$
$67$	2.47935e6	1.00711	0.503555	+	0.863963i	$0.332025\pi$
$68$	1.82215e6	0.702753
$69$	328509.	0.120386
$70$	−9.80216e6	−3.41569
$71$	−263847.	−0.0874879	−0.0437439	−	0.999043i	$-0.513929\pi$
$71$	−263847.	−0.0874879	−0.0437439	+	0.999043i	$0.513929\pi$
$72$	−76839.7	−0.0242617
$73$	−4.65126e6	−1.39939	−0.699697	−	0.714439i	$-0.746682\pi$
$73$	−4.65126e6	−1.39939	−0.699697	+	0.714439i	$0.746682\pi$
$74$	−645968.	−0.185311
$75$	−2.12361e6	−0.581247
$76$	3391.59	0.000886249 0
$77$	−1.49305e6	−0.372699
$78$	3.33593e6	0.795951
$79$	−4.49927e6	−1.02671	−0.513354	−	0.858177i	$-0.671597\pi$
$79$	−4.49927e6	−1.02671	−0.513354	+	0.858177i	$0.671597\pi$
$80$	−6.14079e6	−1.34094
$81$	531441.	0.111111
$82$	1.20867e7	2.42081
$83$	5.06957e6	0.973191	0.486595	−	0.873627i	$-0.338239\pi$
$83$	5.06957e6	0.973191	0.486595	+	0.873627i	$0.338239\pi$
$84$	−5.54901e6	−1.02150
$85$	5.36395e6	0.947368
$86$	−9.39150e6	−1.59217
$87$	2.61557e6	0.425843
$88$	102996.	0.0161114
$89$	1.01382e7	1.52438	0.762191	−	0.647352i	$-0.224124\pi$
$89$	1.01382e7	1.52438	0.762191	+	0.647352i	$0.224124\pi$
$90$	−4.67669e6	−0.676223
$91$	1.16519e7	1.62088
$92$	−1.63653e6	−0.219112
$93$	−7.19745e6	−0.927872
$94$	1.37584e6	0.170852
$95$	9984.00	0.00119474
$96$	−7.14874e6	−0.824670
$97$	1.22771e7	1.36583	0.682914	−	0.730499i	$-0.260713\pi$
$97$	1.22771e7	1.36583	0.682914	+	0.730499i	$0.260713\pi$
$98$	−2.44830e7	−2.62769
$99$	−712347.	−0.0737851
$100$	1.05792e7	1.05792
$101$	1.93887e7	1.87250	0.936252	−	0.351329i	$-0.114270\pi$
$101$	1.93887e7	1.87250	0.936252	+	0.351329i	$0.114270\pi$
$102$	5.92620e6	0.552937
$103$	1.89953e6	0.171284	0.0856419	−	0.996326i	$-0.472706\pi$
$103$	1.89953e6	0.171284	0.0856419	+	0.996326i	$0.472706\pi$
$104$	−803790.	−0.0700690
$105$	−1.63349e7	−1.37706
$106$	2.52313e7	2.05764
$107$	−2.31415e6	−0.182620	−0.0913101	−	0.995823i	$-0.529105\pi$
$107$	−2.31415e6	−0.182620	−0.0913101	+	0.995823i	$0.529105\pi$
$108$	−2.64747e6	−0.202231
$109$	−7.11020e6	−0.525883	−0.262942	−	0.964812i	$-0.584693\pi$
$109$	−7.11020e6	−0.525883	−0.262942	+	0.964812i	$0.584693\pi$
$110$	6.26867e6	0.449056
$111$	−1.07648e6	−0.0747094
$112$	−2.36970e7	−1.59379
$113$	1.94480e7	1.26794	0.633971	−	0.773357i	$-0.281424\pi$
$113$	1.94480e7	1.26794	0.633971	+	0.773357i	$0.281424\pi$
$114$	11030.5	0.000697315 0
$115$	−4.81754e6	−0.295381
$116$	−1.30300e7	−0.775069
$117$	5.55920e6	0.320894
$118$	2.20246e7	1.23402
$119$	2.06992e7	1.12600
$120$	1.12684e6	0.0595291
$121$	−1.85323e7	−0.951002
$122$	−1.84247e7	−0.918631
$123$	2.01420e7	0.975967
$124$	3.58555e7	1.68880
$125$	208794.	0.00956168
$126$	−1.80471e7	−0.803731
$127$	−2.18116e7	−0.944877	−0.472439	−	0.881364i	$-0.656626\pi$
$127$	−2.18116e7	−0.944877	−0.472439	+	0.881364i	$0.656626\pi$
$128$	3.44943e6	0.145382
$129$	−1.56505e7	−0.641897
$130$	−4.89210e7	−1.95296
$131$	3.40934e7	1.32502	0.662508	−	0.749054i	$-0.269492\pi$
$131$	3.40934e7	1.32502	0.662508	+	0.749054i	$0.269492\pi$
$132$	3.54869e6	0.134295
$133$	38527.8	0.00142002
$134$	−4.01705e7	−1.44225
$135$	−7.79351e6	−0.272624
$136$	−1.42791e6	−0.0486760
$137$	−4.69929e7	−1.56139	−0.780693	−	0.624915i	$-0.785133\pi$
$137$	−4.69929e7	−1.56139	−0.780693	+	0.624915i	$0.785133\pi$
$138$	−5.32251e6	−0.172401
$139$	2.34403e7	0.740305	0.370153	−	0.928971i	$-0.379305\pi$
$139$	2.34403e7	0.740305	0.370153	+	0.928971i	$0.379305\pi$
$140$	8.13754e7	2.50637
$141$	2.29278e6	0.0688805
$142$	4.27485e6	0.125289
$143$	−7.45159e6	−0.213095
$144$	−1.13060e7	−0.315531
$145$	−3.83570e7	−1.04486
$146$	7.53598e7	2.00403
$147$	−4.07999e7	−1.05937
$148$	5.36269e6	0.135977
$149$	−1.58376e7	−0.392227	−0.196113	−	0.980581i	$-0.562832\pi$
$149$	−1.58376e7	−0.392227	−0.196113	+	0.980581i	$0.562832\pi$
$150$	3.44068e7	0.832387
$151$	8.30927e7	1.96401	0.982005	−	0.188856i	$-0.0604779\pi$
$151$	8.30927e7	1.96401	0.982005	+	0.188856i	$0.0604779\pi$
$152$	−2657.79	−6.13859e−5 0
$153$	9.87577e6	0.222921
$154$	2.41905e7	0.533730
$155$	1.05550e8	2.27665
$156$	−2.76942e7	−0.584053
$157$	−3.85096e7	−0.794182	−0.397091	−	0.917779i	$-0.629980\pi$
$157$	−3.85096e7	−0.794182	−0.397091	+	0.917779i	$0.629980\pi$
$158$	7.28973e7	1.47032
$159$	4.20469e7	0.829552
$160$	1.04835e8	2.02343
$161$	−1.85906e7	−0.351078
$162$	−8.61042e6	−0.159119
$163$	8.47299e6	0.153243	0.0766214	−	0.997060i	$-0.475587\pi$
$163$	8.47299e6	0.153243	0.0766214	+	0.997060i	$0.475587\pi$
$164$	−1.00341e8	−1.77634
$165$	1.04465e7	0.181041
$166$	−8.21373e7	−1.39368
$167$	−1.12087e8	−1.86228	−0.931142	−	0.364656i	$-0.881187\pi$
$167$	−1.12087e8	−1.86228	−0.931142	+	0.364656i	$0.881187\pi$
$168$	4.34843e6	0.0707539
$169$	−4.59595e6	−0.0732439
$170$	−8.69069e7	−1.35670
$171$	18381.9	0.000281128 0
$172$	7.79662e7	1.16831
$173$	−7.91886e7	−1.16279	−0.581395	−	0.813622i	$-0.697493\pi$
$173$	−7.91886e7	−1.16279	−0.581395	+	0.813622i	$0.697493\pi$
$174$	−4.23776e7	−0.609837
$175$	1.20177e8	1.69508
$176$	1.51547e7	0.209533
$177$	3.67031e7	0.497504
$178$	−1.64259e8	−2.18302
$179$	−7.84923e7	−1.02292	−0.511459	−	0.859307i	$-0.670895\pi$
$179$	−7.84923e7	−1.02292	−0.511459	+	0.859307i	$0.670895\pi$
$180$	3.88248e7	0.496199
$181$	−4.42911e7	−0.555190	−0.277595	−	0.960698i	$-0.589537\pi$
$181$	−4.42911e7	−0.555190	−0.277595	+	0.960698i	$0.589537\pi$
$182$	−1.88784e8	−2.32121
$183$	−3.07040e7	−0.370353
$184$	1.28245e6	0.0151768
$185$	1.57864e7	0.183308
$186$	1.16613e8	1.32878
$187$	−1.32375e7	−0.148034
$188$	−1.14219e7	−0.125368
$189$	−3.00748e7	−0.324031
$190$	−161761.	−0.00171095
$191$	4.58471e7	0.476096	0.238048	−	0.971253i	$-0.423492\pi$
$191$	4.58471e7	0.476096	0.238048	+	0.971253i	$0.423492\pi$
$192$	6.22252e7	0.634471
$193$	−2.93112e7	−0.293483	−0.146741	−	0.989175i	$-0.546879\pi$
$193$	−2.93112e7	−0.293483	−0.146741	+	0.989175i	$0.546879\pi$
$194$	−1.98914e8	−1.95596
$195$	−8.15248e7	−0.787352
$196$	2.03253e8	1.92815
$197$	1.71466e8	1.59789	0.798943	−	0.601407i	$-0.205393\pi$
$197$	1.71466e8	1.59789	0.798943	+	0.601407i	$0.205393\pi$
$198$	1.15415e7	0.105665
$199$	1.13235e8	1.01858	0.509288	−	0.860596i	$-0.329909\pi$
$199$	1.13235e8	1.01858	0.509288	+	0.860596i	$0.329909\pi$
$200$	−8.29029e6	−0.0732765
$201$	−6.69425e7	−0.581455
$202$	−3.14135e8	−2.68156
$203$	−1.48018e8	−1.24188
$204$	−4.91980e7	−0.405734
$205$	−2.95380e8	−2.39465
$206$	−3.07763e7	−0.245290
$207$	−8.86974e6	−0.0695048
$208$	−1.18268e8	−0.911266
$209$	−24639.2	−0.000186687 0
$210$	2.64658e8	1.97205
$211$	−1.96433e8	−1.43954	−0.719772	−	0.694211i	$-0.755754\pi$
$211$	−1.96433e8	−1.43954	−0.719772	+	0.694211i	$0.755754\pi$
$212$	−2.09464e8	−1.50985
$213$	7.12387e6	0.0505111
$214$	3.74940e7	0.261525
$215$	2.29513e8	1.57497
$216$	2.07467e6	0.0140075
$217$	4.07311e8	2.70593
$218$	1.15200e8	0.753102
$219$	1.25584e8	0.807941
$220$	−5.20411e7	−0.329509
$221$	1.03307e8	0.643806
$222$	1.74411e7	0.106989
$223$	2.38859e8	1.44236	0.721180	−	0.692747i	$-0.243600\pi$
$223$	2.38859e8	1.44236	0.721180	+	0.692747i	$0.243600\pi$
$224$	4.04554e8	2.40497
$225$	5.73375e7	0.335583
$226$	−3.15096e8	−1.81578
$227$	−8.13545e6	−0.0461627	−0.0230813	−	0.999734i	$-0.507348\pi$
$227$	−8.13545e6	−0.0461627	−0.0230813	+	0.999734i	$0.507348\pi$
$228$	−91572.9	−0.000511676 0
$229$	2.29391e8	1.26227	0.631134	−	0.775674i	$-0.282590\pi$
$229$	2.29391e8	1.26227	0.631134	+	0.775674i	$0.282590\pi$
$230$	7.80539e7	0.423006
$231$	4.03124e7	0.215178
$232$	1.02108e7	0.0536850
$233$	−1.37622e8	−0.712756	−0.356378	−	0.934342i	$-0.615989\pi$
$233$	−1.37622e8	−0.712756	−0.356378	+	0.934342i	$0.615989\pi$
$234$	−9.00702e7	−0.459543
$235$	−3.36233e7	−0.169006
$236$	−1.82844e8	−0.905499
$237$	1.21480e8	0.592771
$238$	−3.35370e8	−1.61252
$239$	4.68408e7	0.221938	0.110969	−	0.993824i	$-0.464605\pi$
$239$	4.68408e7	0.221938	0.110969	+	0.993824i	$0.464605\pi$
$240$	1.65801e8	0.774192
$241$	−4.15160e8	−1.91054	−0.955270	−	0.295735i	$-0.904436\pi$
$241$	−4.15160e8	−1.91054	−0.955270	+	0.295735i	$0.904436\pi$
$242$	3.00261e8	1.36190
$243$	−1.43489e7	−0.0641500
$244$	1.52958e8	0.674073
$245$	5.98325e8	2.59930
$246$	−3.26341e8	−1.39765
$247$	192286.	0.000811910 0
$248$	−2.80978e7	−0.116975
$249$	−1.36878e8	−0.561872
$250$	−3.38289e6	−0.0136930
$251$	3.43110e8	1.36954	0.684771	−	0.728759i	$-0.259902\pi$
$251$	3.43110e8	1.36954	0.684771	+	0.728759i	$0.259902\pi$
$252$	1.49823e8	0.589762
$253$	1.18891e7	0.0461558
$254$	3.53393e8	1.35313
$255$	−1.44827e8	−0.546963
$256$	2.39106e8	0.890738
$257$	3.10904e8	1.14251	0.571255	−	0.820773i	$-0.306457\pi$
$257$	3.10904e8	1.14251	0.571255	+	0.820773i	$0.306457\pi$
$258$	2.53571e8	0.919242
$259$	6.09190e7	0.217873
$260$	4.06131e8	1.43304
$261$	−7.06205e7	−0.245861
$262$	−5.52383e8	−1.89752
$263$	−1.90372e8	−0.645294	−0.322647	−	0.946519i	$-0.604573\pi$
$263$	−1.90372e8	−0.645294	−0.322647	+	0.946519i	$0.604573\pi$
$264$	−2.78090e6	−0.00930191
$265$	−6.16611e8	−2.03541
$266$	−624228.	−0.00203356
$267$	−2.73730e8	−0.880103
$268$	3.33487e8	1.05830
$269$	1.01060e8	0.316553	0.158277	−	0.987395i	$-0.449406\pi$
$269$	1.01060e8	0.316553	0.158277	+	0.987395i	$0.449406\pi$
$270$	1.26271e8	0.390417
$271$	5.89781e8	1.80011	0.900053	−	0.435780i	$-0.143527\pi$
$271$	5.89781e8	1.80011	0.900053	+	0.435780i	$0.143527\pi$
$272$	−2.10100e8	−0.633045
$273$	−3.14600e8	−0.935815
$274$	7.61380e8	2.23601
$275$	−7.68556e7	−0.222849
$276$	4.41863e7	0.126504
$277$	−2.50265e8	−0.707490	−0.353745	−	0.935342i	$-0.615092\pi$
$277$	−2.50265e8	−0.707490	−0.353745	+	0.935342i	$0.615092\pi$
$278$	−3.79780e8	−1.06017
$279$	1.94331e8	0.535707
$280$	−6.37692e7	−0.173603
$281$	−2.97671e7	−0.0800322	−0.0400161	−	0.999199i	$-0.512741\pi$
$281$	−2.97671e7	−0.0800322	−0.0400161	+	0.999199i	$0.512741\pi$
$282$	−3.71477e7	−0.0986417
$283$	−3.16316e8	−0.829600	−0.414800	−	0.909913i	$-0.636148\pi$
$283$	−3.16316e8	−0.829600	−0.414800	+	0.909913i	$0.636148\pi$
$284$	−3.54889e7	−0.0919344
$285$	−269568.	−0.000689781 0
$286$	1.20731e8	0.305166
$287$	−1.13986e9	−2.84619
$288$	1.93016e8	0.476124
$289$	−2.26817e8	−0.552757
$290$	6.21461e8	1.49631
$291$	−3.31483e8	−0.788561
$292$	−6.25620e8	−1.47052
$293$	−2.41195e7	−0.0560185	−0.0280092	−	0.999608i	$-0.508917\pi$
$293$	−2.41195e7	−0.0560185	−0.0280092	+	0.999608i	$0.508917\pi$
$294$	6.61041e8	1.51710
$295$	−5.38246e8	−1.22069
$296$	−4.20242e6	−0.00941844
$297$	1.92334e7	0.0425998
$298$	2.56601e8	0.561696
$299$	−9.27829e7	−0.200733
$300$	−2.85638e8	−0.610789
$301$	8.85680e8	1.87195
$302$	−1.34627e9	−2.81260
$303$	−5.23494e8	−1.08109
$304$	−391062.	−0.000798340 0
$305$	4.50270e8	0.908706
$306$	−1.60007e8	−0.319238
$307$	4.01012e8	0.790993	0.395496	−	0.918468i	$-0.370573\pi$
$307$	4.01012e8	0.790993	0.395496	+	0.918468i	$0.370573\pi$
$308$	−2.00824e8	−0.391641
$309$	−5.12874e7	−0.0988908
$310$	−1.71012e9	−3.26032
$311$	−7.78223e8	−1.46704	−0.733522	−	0.679666i	$-0.762125\pi$
$311$	−7.78223e8	−1.46704	−0.733522	+	0.679666i	$0.762125\pi$
$312$	2.17023e7	0.0404544
$313$	−7.95716e8	−1.46674	−0.733370	−	0.679830i	$-0.762054\pi$
$313$	−7.95716e8	−1.46674	−0.733370	+	0.679830i	$0.762054\pi$
$314$	6.23933e8	1.13732
$315$	4.41042e8	0.795048
$316$	−6.05177e8	−1.07889
$317$	8.52074e8	1.50235	0.751173	−	0.660106i	$-0.229488\pi$
$317$	8.52074e8	1.50235	0.751173	+	0.660106i	$0.229488\pi$
$318$	−6.81245e8	−1.18798
$319$	9.46602e7	0.163268
$320$	−9.12523e8	−1.55675
$321$	6.24821e7	0.105436
$322$	3.01206e8	0.502769
$323$	341591.	0.000564024 0
$324$	7.14818e7	0.116758
$325$	5.99786e8	0.969180
$326$	−1.37280e8	−0.219454
$327$	1.91975e8	0.303619
$328$	7.86316e7	0.123038
$329$	−1.29751e8	−0.200874
$330$	−1.69254e8	−0.259263
$331$	2.06965e8	0.313689	0.156845	−	0.987623i	$-0.449868\pi$
$331$	2.06965e8	0.313689	0.156845	+	0.987623i	$0.449868\pi$
$332$	6.81886e8	1.02265
$333$	2.90649e7	0.0431335
$334$	1.81603e9	2.66692
$335$	9.81703e8	1.42667
$336$	6.39819e8	0.920174
$337$	−1.17012e9	−1.66543	−0.832713	−	0.553705i	$-0.813213\pi$
$337$	−1.17012e9	−1.66543	−0.832713	+	0.553705i	$0.813213\pi$
$338$	7.44636e7	0.104890
$339$	−5.25095e8	−0.732047
$340$	7.21482e8	0.995518
$341$	−2.60483e8	−0.355745
$342$	−297824.	−0.000402595 0
$343$	1.05057e9	1.40571
$344$	−6.10975e7	−0.0809225
$345$	1.30074e8	0.170538
$346$	1.28302e9	1.66520
$347$	1.74281e8	0.223921	0.111961	−	0.993713i	$-0.464287\pi$
$347$	1.74281e8	0.223921	0.111961	+	0.993713i	$0.464287\pi$
$348$	3.51809e8	0.447486
$349$	5.92942e8	0.746661	0.373330	−	0.927698i	$-0.378216\pi$
$349$	5.92942e8	0.746661	0.373330	+	0.927698i	$0.378216\pi$
$350$	−1.94712e9	−2.42747
$351$	−1.50098e8	−0.185268
$352$	−2.58720e8	−0.316177
$353$	−5.68861e8	−0.688327	−0.344164	−	0.938910i	$-0.611838\pi$
$353$	−5.68861e8	−0.688327	−0.344164	+	0.938910i	$0.611838\pi$
$354$	−5.94665e8	−0.712461
$355$	−1.04470e8	−0.123935
$356$	1.36364e9	1.60186
$357$	−5.58879e8	−0.650099
$358$	1.27173e9	1.46489
$359$	6.61337e8	0.754384	0.377192	−	0.926135i	$-0.376890\pi$
$359$	6.61337e8	0.754384	0.377192	+	0.926135i	$0.376890\pi$
$360$	−3.04248e7	−0.0343691
$361$	−8.93871e8	−0.999999
$362$	7.17606e8	0.795071
$363$	5.00373e8	0.549061
$364$	1.56724e9	1.70326
$365$	−1.84167e9	−1.98238
$366$	4.97467e8	0.530372
$367$	3.27583e8	0.345932	0.172966	−	0.984928i	$-0.444665\pi$
$367$	3.27583e8	0.345932	0.172966	+	0.984928i	$0.444665\pi$
$368$	1.88697e8	0.197378
$369$	−5.43834e8	−0.563475
$370$	−2.55772e8	−0.262511
$371$	−2.37947e9	−2.41920
$372$	−9.68097e8	−0.975032
$373$	−1.50919e9	−1.50578	−0.752890	−	0.658146i	$-0.771341\pi$
$373$	−1.50919e9	−1.50578	−0.752890	+	0.658146i	$0.771341\pi$
$374$	2.14475e8	0.211995
$375$	−5.63745e6	−0.00552044
$376$	8.95071e6	0.00868360
$377$	−7.38733e8	−0.710056
$378$	4.87272e8	0.464035
$379$	1.60365e8	0.151312	0.0756560	−	0.997134i	$-0.475895\pi$
$379$	1.60365e8	0.151312	0.0756560	+	0.997134i	$0.475895\pi$
$380$	1.34290e6	0.00125546
$381$	5.88914e8	0.545525
$382$	−7.42815e8	−0.681803
$383$	6.86470e8	0.624347	0.312174	−	0.950025i	$-0.398943\pi$
$383$	6.86470e8	0.624347	0.312174	+	0.950025i	$0.398943\pi$
$384$	−9.31345e7	−0.0839366
$385$	−5.91176e8	−0.527964
$386$	4.74901e8	0.420288
$387$	4.22565e8	0.370599
$388$	1.65134e9	1.43525
$389$	−5.97929e8	−0.515022	−0.257511	−	0.966275i	$-0.582902\pi$
$389$	−5.97929e8	−0.515022	−0.257511	+	0.966275i	$0.582902\pi$
$390$	1.32087e9	1.12754
$391$	−1.64826e8	−0.139447
$392$	−1.59277e8	−0.133553
$393$	−9.20523e8	−0.764999
$394$	−2.77809e9	−2.28828
$395$	−1.78149e9	−1.45443
$396$	−9.58147e7	−0.0775352
$397$	1.06536e9	0.854534	0.427267	−	0.904125i	$-0.359476\pi$
$397$	1.06536e9	0.854534	0.427267	+	0.904125i	$0.359476\pi$
$398$	−1.83463e9	−1.45867
$399$	−1.04025e6	−0.000819847 0
$400$	−1.21982e9	−0.952981
$401$	−2.90355e8	−0.224866	−0.112433	−	0.993659i	$-0.535864\pi$
$401$	−2.90355e8	−0.224866	−0.112433	+	0.993659i	$0.535864\pi$
$402$	1.08460e9	0.832684
$403$	2.03282e9	1.54715
$404$	2.60788e9	1.96767
$405$	2.10425e8	0.157400
$406$	2.39819e9	1.77845
$407$	−3.89588e7	−0.0286435
$408$	3.85536e7	0.0281031
$409$	1.99399e9	1.44109	0.720545	−	0.693408i	$-0.243892\pi$
$409$	1.99399e9	1.44109	0.720545	+	0.693408i	$0.243892\pi$
$410$	4.78575e9	3.42931
$411$	1.26881e9	0.901466
$412$	2.55498e8	0.179989
$413$	−2.07707e9	−1.45086
$414$	1.43708e8	0.0995358
$415$	2.00730e9	1.37862
$416$	2.01907e9	1.37507
$417$	−6.32887e8	−0.427415
$418$	399205.	0.000267350 0
$419$	−8.88709e8	−0.590215	−0.295108	−	0.955464i	$-0.595356\pi$
$419$	−8.88709e8	−0.590215	−0.295108	+	0.955464i	$0.595356\pi$
$420$	−2.19714e9	−1.44705
$421$	−1.77426e9	−1.15886	−0.579430	−	0.815022i	$-0.696725\pi$
$421$	−1.77426e9	−1.15886	−0.579430	+	0.815022i	$0.696725\pi$
$422$	3.18261e9	2.06153
$423$	−6.19051e7	−0.0397682
$424$	1.64145e8	0.104580
$425$	1.06550e9	0.673277
$426$	−1.15421e8	−0.0723355
$427$	1.73757e9	1.08005
$428$	−3.11267e8	−0.191902
$429$	2.01193e8	0.123030
$430$	−3.71858e9	−2.25547
$431$	−1.50258e9	−0.904000	−0.452000	−	0.892018i	$-0.649289\pi$
$431$	−1.50258e9	−0.904000	−0.452000	+	0.892018i	$0.649289\pi$
$432$	3.05263e8	0.182172
$433$	−2.55359e8	−0.151162	−0.0755812	−	0.997140i	$-0.524081\pi$
$433$	−2.55359e8	−0.151162	−0.0755812	+	0.997140i	$0.524081\pi$
$434$	−6.59926e9	−3.87508
$435$	1.03564e9	0.603248
$436$	−9.56362e8	−0.552611
$437$	−306794.	−0.000175858 0
$438$	−2.03471e9	−1.15703
$439$	2.86807e9	1.61795	0.808973	−	0.587846i	$-0.200024\pi$
$439$	2.86807e9	1.61795	0.808973	+	0.587846i	$0.200024\pi$
$440$	4.07816e7	0.0228234
$441$	1.10160e9	0.611629
$442$	−1.67377e9	−0.921975
$443$	−1.44103e9	−0.787520	−0.393760	−	0.919213i	$-0.628826\pi$
$443$	−1.44103e9	−0.787520	−0.393760	+	0.919213i	$0.628826\pi$
$444$	−1.44793e8	−0.0785065
$445$	4.01421e9	2.15944
$446$	−3.86999e9	−2.06556
$447$	4.27615e8	0.226452
$448$	−3.52138e9	−1.85029
$449$	−1.78535e9	−0.930812	−0.465406	−	0.885097i	$-0.654092\pi$
$449$	−1.78535e9	−0.930812	−0.465406	+	0.885097i	$0.654092\pi$
$450$	−9.28984e8	−0.480579
$451$	7.28959e8	0.374184
$452$	2.61586e9	1.33239
$453$	−2.24350e9	−1.13392
$454$	1.31811e8	0.0661082
$455$	4.61357e9	2.29613
$456$	71760.4	3.54412e−5 0
$457$	−2.88464e8	−0.141379	−0.0706895	−	0.997498i	$-0.522520\pi$
$457$	−2.88464e8	−0.141379	−0.0706895	+	0.997498i	$0.522520\pi$
$458$	−3.71659e9	−1.80766
$459$	−2.66646e8	−0.128703
$460$	−6.47986e8	−0.310394
$461$	−3.59417e8	−0.170862	−0.0854310	−	0.996344i	$-0.527227\pi$
$461$	−3.59417e8	−0.170862	−0.0854310	+	0.996344i	$0.527227\pi$
$462$	−6.53143e8	−0.308149
$463$	2.93665e8	0.137505	0.0687526	−	0.997634i	$-0.478098\pi$
$463$	2.93665e8	0.137505	0.0687526	+	0.997634i	$0.478098\pi$
$464$	1.50240e9	0.698189
$465$	−2.84984e9	−1.31442
$466$	2.22975e9	1.02072
$467$	3.19317e9	1.45082	0.725409	−	0.688318i	$-0.241651\pi$
$467$	3.19317e9	1.45082	0.725409	+	0.688318i	$0.241651\pi$
$468$	7.47743e8	0.337203
$469$	3.78834e9	1.69568
$470$	5.44766e8	0.242029
$471$	1.03976e9	0.458521
$472$	1.43284e8	0.0627192
$473$	−5.66409e8	−0.246102
$474$	−1.96823e9	−0.848889
$475$	1.98324e6	0.000849077 0
$476$	2.78416e9	1.18323
$477$	−1.13527e9	−0.478942
$478$	−7.58917e8	−0.317831
$479$	1.00355e9	0.417220	0.208610	−	0.977999i	$-0.433106\pi$
$479$	1.00355e9	0.417220	0.208610	+	0.977999i	$0.433106\pi$
$480$	−2.83055e9	−1.16823
$481$	3.04037e8	0.124571
$482$	6.72644e9	2.73603
$483$	5.01947e8	0.202695
$484$	−2.49270e9	−0.999337
$485$	4.86114e9	1.93483
$486$	2.32481e8	0.0918673
$487$	1.69071e9	0.663312	0.331656	−	0.943400i	$-0.392393\pi$
$487$	1.69071e9	0.663312	0.331656	+	0.943400i	$0.392393\pi$
$488$	−1.19864e8	−0.0466896
$489$	−2.28771e8	−0.0884747
$490$	−9.69408e9	−3.72238
$491$	1.44669e9	0.551559	0.275779	−	0.961221i	$-0.411064\pi$
$491$	1.44669e9	0.551559	0.275779	+	0.961221i	$0.411064\pi$
$492$	2.70921e9	1.02557
$493$	−1.31234e9	−0.493267
$494$	−3.11542e6	−0.00116271
$495$	−2.82055e8	−0.104524
$496$	−4.13426e9	−1.52129
$497$	−4.03147e8	−0.147304
$498$	2.21771e9	0.804640
$499$	−2.74672e9	−0.989605	−0.494803	−	0.869005i	$-0.664760\pi$
$499$	−2.74672e9	−0.989605	−0.494803	+	0.869005i	$0.664760\pi$
$500$	2.80840e7	0.0100476
$501$	3.02634e9	1.07519
$502$	−5.55907e9	−1.96128
$503$	6.14861e8	0.215422	0.107711	−	0.994182i	$-0.465648\pi$
$503$	6.14861e8	0.215422	0.107711	+	0.994182i	$0.465648\pi$
$504$	−1.17408e8	−0.0408498
$505$	7.67696e9	2.65259
$506$	−1.92627e8	−0.0660983
$507$	1.24091e8	0.0422874
$508$	−2.93379e9	−0.992901
$509$	−7.09303e7	−0.0238407	−0.0119204	−	0.999929i	$-0.503794\pi$
$509$	−7.09303e7	−0.0238407	−0.0119204	+	0.999929i	$0.503794\pi$
$510$	2.34649e9	0.783290
$511$	−7.10692e9	−2.35618
$512$	−4.31552e9	−1.42098
$513$	−496311.	−0.000162309 0
$514$	−5.03727e9	−1.63615
$515$	7.52122e8	0.242640
$516$	−2.10509e9	−0.674522
$517$	8.29781e7	0.0264087
$518$	−9.87011e8	−0.312010
$519$	2.13809e9	0.671337
$520$	−3.18262e8	−0.0992596
$521$	−5.06862e9	−1.57021	−0.785104	−	0.619364i	$-0.787391\pi$
$521$	−5.06862e9	−1.57021	−0.785104	+	0.619364i	$0.787391\pi$
$522$	1.14419e9	0.352090
$523$	−2.63435e9	−0.805227	−0.402613	−	0.915370i	$-0.631898\pi$
$523$	−2.63435e9	−0.805227	−0.402613	+	0.915370i	$0.631898\pi$
$524$	4.58576e9	1.39236
$525$	−3.24479e9	−0.978653
$526$	3.08441e9	0.924106
$527$	3.61125e9	1.07478
$528$	−4.09176e8	−0.120974
$529$	1.48036e8	0.0434783
$530$	9.99035e9	2.91484
$531$	−9.90984e8	−0.287234
$532$	5.18220e6	0.00149219
$533$	−5.68884e9	−1.62734
$534$	4.43498e9	1.26037
$535$	−9.16292e8	−0.258699
$536$	−2.61334e8	−0.0733027
$537$	2.11929e9	0.590583
$538$	−1.63738e9	−0.453327
$539$	−1.47659e9	−0.406162
$540$	−1.04827e9	−0.286481
$541$	−5.42527e9	−1.47310	−0.736548	−	0.676385i	$-0.763545\pi$
$541$	−5.42527e9	−1.47310	−0.736548	+	0.676385i	$0.763545\pi$
$542$	−9.55564e9	−2.57788
$543$	1.19586e9	0.320539
$544$	3.58682e9	0.955241
$545$	−2.81529e9	−0.744965
$546$	5.09716e9	1.34015
$547$	−8.73776e8	−0.228268	−0.114134	−	0.993465i	$-0.536409\pi$
$547$	−8.73776e8	−0.228268	−0.114134	+	0.993465i	$0.536409\pi$
$548$	−6.32081e9	−1.64074
$549$	8.29008e8	0.213823
$550$	1.24522e9	0.319136
$551$	−2.44268e6	−0.000622065 0
$552$	−3.46262e7	−0.00876231
$553$	−6.87469e9	−1.72868
$554$	4.05480e9	1.01318
$555$	−4.26233e8	−0.105833
$556$	3.15285e9	0.777931
$557$	2.63557e9	0.646222	0.323111	−	0.946361i	$-0.395271\pi$
$557$	2.63557e9	0.646222	0.323111	+	0.946361i	$0.395271\pi$
$558$	−3.14856e9	−0.767171
$559$	4.42028e9	1.07031
$560$	−9.38286e9	−2.25776
$561$	3.57414e8	0.0854675
$562$	4.82288e8	0.114612
$563$	−2.80648e9	−0.662800	−0.331400	−	0.943490i	$-0.607521\pi$
$563$	−2.80648e9	−0.662800	−0.331400	+	0.943490i	$0.607521\pi$
$564$	3.08392e8	0.0723813
$565$	7.70045e9	1.79617
$566$	5.12496e9	1.18805
$567$	8.12019e8	0.187079
$568$	2.78106e7	0.00636782
$569$	2.66168e8	0.0605706	0.0302853	−	0.999541i	$-0.490358\pi$
$569$	2.66168e8	0.0605706	0.0302853	+	0.999541i	$0.490358\pi$
$570$	4.36755e6	0.000987815 0
$571$	−4.42686e9	−0.995107	−0.497554	−	0.867433i	$-0.665768\pi$
$571$	−4.42686e9	−0.995107	−0.497554	+	0.867433i	$0.665768\pi$
$572$	−1.00228e9	−0.223925
$573$	−1.23787e9	−0.274874
$574$	1.84680e10	4.07594
$575$	−9.56963e8	−0.209922
$576$	−1.68008e9	−0.366312
$577$	2.24293e9	0.486072	0.243036	−	0.970017i	$-0.421857\pi$
$577$	2.24293e9	0.486072	0.243036	+	0.970017i	$0.421857\pi$
$578$	3.67490e9	0.791586
$579$	7.91402e8	0.169442
$580$	−5.15923e9	−1.09796
$581$	7.74609e9	1.63857
$582$	5.37069e9	1.12927
$583$	1.52172e9	0.318049
$584$	4.90262e8	0.101855
$585$	2.20117e9	0.454578
$586$	3.90784e8	0.0802224
$587$	8.36007e8	0.170599	0.0852995	−	0.996355i	$-0.472815\pi$
$587$	8.36007e8	0.170599	0.0852995	+	0.996355i	$0.472815\pi$
$588$	−5.48782e9	−1.11322
$589$	6.72168e6	0.00135542
$590$	8.72068e9	1.74811
$591$	−4.62958e9	−0.922540
$592$	−6.18336e8	−0.122489
$593$	3.15873e9	0.622044	0.311022	−	0.950403i	$-0.399329\pi$
$593$	3.15873e9	0.622044	0.311022	+	0.950403i	$0.399329\pi$
$594$	−3.11620e8	−0.0610060
$595$	8.19589e9	1.59509
$596$	−2.13024e9	−0.412162
$597$	−3.05734e9	−0.588075
$598$	1.50327e9	0.287464
$599$	−6.25625e9	−1.18938	−0.594689	−	0.803956i	$-0.702725\pi$
$599$	−6.25625e9	−1.18938	−0.594689	+	0.803956i	$0.702725\pi$
$600$	2.23838e8	0.0423062
$601$	6.43863e8	0.120985	0.0604927	−	0.998169i	$-0.480733\pi$
$601$	6.43863e8	0.120985	0.0604927	+	0.998169i	$0.480733\pi$
$602$	−1.43498e10	−2.68076
$603$	1.80745e9	0.335703
$604$	1.11764e10	2.06383
$605$	−7.33790e9	−1.34719
$606$	8.48166e9	1.54820
$607$	2.21655e9	0.402270	0.201135	−	0.979564i	$-0.435537\pi$
$607$	2.21655e9	0.402270	0.201135	+	0.979564i	$0.435537\pi$
$608$	6.67619e6	0.00120467
$609$	3.99648e9	0.716997
$610$	−7.29528e9	−1.30133
$611$	−6.47565e8	−0.114852
$612$	1.32835e9	0.234251
$613$	3.90045e9	0.683916	0.341958	−	0.939715i	$-0.388910\pi$
$613$	3.90045e9	0.683916	0.341958	+	0.939715i	$0.388910\pi$
$614$	−6.49720e9	−1.13276
$615$	7.97525e9	1.38255
$616$	1.57374e8	0.0271269
$617$	3.39587e8	0.0582040	0.0291020	−	0.999576i	$-0.490735\pi$
$617$	3.39587e8	0.0582040	0.0291020	+	0.999576i	$0.490735\pi$
$618$	8.30959e8	0.141619
$619$	−8.96930e9	−1.51999	−0.759996	−	0.649928i	$-0.774799\pi$
$619$	−8.96930e9	−1.51999	−0.759996	+	0.649928i	$0.774799\pi$
$620$	1.41970e10	2.39236
$621$	2.39483e8	0.0401286
$622$	1.26088e10	2.10091
$623$	1.54907e10	2.56662
$624$	3.19323e9	0.526120
$625$	−6.06204e9	−0.993205
$626$	1.28922e10	2.10047
$627$	665259.	0.000107784 0
$628$	−5.17975e9	−0.834546
$629$	5.40114e8	0.0865382
$630$	−7.14578e9	−1.13856
$631$	−1.32361e9	−0.209728	−0.104864	−	0.994487i	$-0.533441\pi$
$631$	−1.32361e9	−0.209728	−0.104864	+	0.994487i	$0.533441\pi$
$632$	4.74242e8	0.0747292
$633$	5.30368e9	0.831121
$634$	−1.38053e10	−2.15146
$635$	−8.63635e9	−1.33851
$636$	5.65554e9	0.871714
$637$	1.15234e10	1.76641
$638$	−1.53369e9	−0.233811
$639$	−1.92344e8	−0.0291626
$640$	1.36580e9	0.205948
$641$	3.41095e9	0.511531	0.255765	−	0.966739i	$-0.417673\pi$
$641$	3.41095e9	0.511531	0.255765	+	0.966739i	$0.417673\pi$
$642$	−1.01234e9	−0.150992
$643$	−9.01499e9	−1.33729	−0.668647	−	0.743580i	$-0.733126\pi$
$643$	−9.01499e9	−1.33729	−0.668647	+	0.743580i	$0.733126\pi$
$644$	−2.50055e9	−0.368922
$645$	−6.19685e9	−0.909310
$646$	−5.53446e6	−0.000807722 0
$647$	−3.32698e9	−0.482932	−0.241466	−	0.970409i	$-0.577628\pi$
$647$	−3.32698e9	−0.482932	−0.241466	+	0.970409i	$0.577628\pi$
$648$	−5.60161e7	−0.00808725
$649$	1.32832e9	0.190742
$650$	−9.71774e9	−1.38793
$651$	−1.09974e10	−1.56227
$652$	1.13966e9	0.161031
$653$	−1.27437e10	−1.79102	−0.895510	−	0.445042i	$-0.853189\pi$
$653$	−1.27437e10	−1.79102	−0.895510	+	0.445042i	$0.853189\pi$
$654$	−3.11039e9	−0.434803
$655$	1.34993e10	1.87702
$656$	1.15697e10	1.60014
$657$	−3.39077e9	−0.466465
$658$	2.10223e9	0.287666
$659$	1.10663e10	1.50628	0.753138	−	0.657863i	$-0.228539\pi$
$659$	1.10663e10	1.50628	0.753138	+	0.657863i	$0.228539\pi$
$660$	1.40511e9	0.190242
$661$	−1.44103e10	−1.94074	−0.970370	−	0.241624i	$-0.922320\pi$
$661$	−1.44103e10	−1.94074	−0.970370	+	0.241624i	$0.922320\pi$
$662$	−3.35326e9	−0.449225
$663$	−2.78928e9	−0.371701
$664$	−5.34354e8	−0.0708339
$665$	1.52551e7	0.00201159
$666$	−4.70911e8	−0.0617702
$667$	1.17865e9	0.153796
$668$	−1.50763e10	−1.95694
$669$	−6.44918e9	−0.832747
$670$	−1.59056e10	−2.04309
$671$	−1.11121e9	−0.141993
$672$	−1.09230e10	−1.38851
$673$	−9.63239e9	−1.21809	−0.609047	−	0.793134i	$-0.708448\pi$
$673$	−9.63239e9	−1.21809	−0.609047	+	0.793134i	$0.708448\pi$
$674$	1.89583e10	2.38501
$675$	−1.54811e9	−0.193749
$676$	−6.18181e8	−0.0769665
$677$	−1.92427e9	−0.238344	−0.119172	−	0.992874i	$-0.538024\pi$
$677$	−1.92427e9	−0.238344	−0.119172	+	0.992874i	$0.538024\pi$
$678$	8.50760e9	1.04834
$679$	1.87589e10	2.29966
$680$	−5.65383e8	−0.0689544
$681$	2.19657e8	0.0266520
$682$	4.22035e9	0.509452
$683$	−8.38945e9	−1.00754	−0.503769	−	0.863839i	$-0.668054\pi$
$683$	−8.38945e9	−1.00754	−0.503769	+	0.863839i	$0.668054\pi$
$684$	2.47247e6	0.000295416 0
$685$	−1.86069e10	−2.21185
$686$	−1.70214e10	−2.01307
$687$	−6.19355e9	−0.728771
$688$	−8.98976e9	−1.05242
$689$	−1.18756e10	−1.38321
$690$	−2.10745e9	−0.244223
$691$	1.77731e9	0.204922	0.102461	−	0.994737i	$-0.467328\pi$
$691$	1.77731e9	0.204922	0.102461	+	0.994737i	$0.467328\pi$
$692$	−1.06513e10	−1.22189
$693$	−1.08844e9	−0.124233
$694$	−2.82370e9	−0.320671
$695$	9.28120e9	1.04871
$696$	−2.75692e8	−0.0309951
$697$	−1.01061e10	−1.13049
$698$	−9.60686e9	−1.06927
$699$	3.71579e9	0.411510
$700$	1.61645e10	1.78123
$701$	−7.77870e9	−0.852891	−0.426446	−	0.904513i	$-0.640234\pi$
$701$	−7.77870e9	−0.852891	−0.426446	+	0.904513i	$0.640234\pi$
$702$	2.43190e9	0.265317
$703$	1.00532e6	0.000109134 0
$704$	2.25199e9	0.243255
$705$	9.07830e8	0.0975759
$706$	9.21671e9	0.985733
$707$	2.96250e10	3.15276
$708$	4.93678e9	0.522790
$709$	−3.46303e9	−0.364917	−0.182459	−	0.983214i	$-0.558406\pi$
$709$	−3.46303e9	−0.364917	−0.182459	+	0.983214i	$0.558406\pi$
$710$	1.69263e9	0.177484
$711$	−3.27997e9	−0.342236
$712$	−1.06860e9	−0.110952
$713$	−3.24338e9	−0.335108
$714$	9.05498e9	0.930987
$715$	−2.95046e9	−0.301869
$716$	−1.05577e10	−1.07491
$717$	−1.26470e9	−0.128136
$718$	−1.07150e10	−1.08033
$719$	−3.37019e9	−0.338145	−0.169072	−	0.985604i	$-0.554077\pi$
$719$	−3.37019e9	−0.338145	−0.169072	+	0.985604i	$0.554077\pi$
$720$	−4.47664e9	−0.446980
$721$	2.90240e9	0.288393
$722$	1.44825e10	1.43207
$723$	1.12093e10	1.10305
$724$	−5.95741e9	−0.583408
$725$	−7.61929e9	−0.742560
$726$	−8.10706e9	−0.786294
$727$	−6.61011e8	−0.0638026	−0.0319013	−	0.999491i	$-0.510156\pi$
$727$	−6.61011e8	−0.0638026	−0.0319013	+	0.999491i	$0.510156\pi$
$728$	−1.22816e9	−0.117976
$729$	3.87420e8	0.0370370
$730$	2.98388e10	2.83891
$731$	7.85252e9	0.743529
$732$	−4.12986e9	−0.389176
$733$	1.05019e10	0.984931	0.492465	−	0.870332i	$-0.336096\pi$
$733$	1.05019e10	0.984931	0.492465	+	0.870332i	$0.336096\pi$
$734$	−5.30751e9	−0.495398
$735$	−1.61548e10	−1.50070
$736$	−3.22144e9	−0.297836
$737$	−2.42272e9	−0.222929
$738$	8.81122e9	0.806935
$739$	−4.71267e9	−0.429547	−0.214774	−	0.976664i	$-0.568901\pi$
$739$	−4.71267e9	−0.429547	−0.214774	+	0.976664i	$0.568901\pi$
$740$	2.12336e9	0.192625
$741$	−5.19172e6	−0.000468757 0
$742$	3.85523e10	3.46447
$743$	−1.38869e10	−1.24206	−0.621031	−	0.783786i	$-0.713286\pi$
$743$	−1.38869e10	−1.24206	−0.621031	+	0.783786i	$0.713286\pi$
$744$	7.58642e8	0.0675354
$745$	−6.27091e9	−0.555627
$746$	2.44519e10	2.15638
$747$	3.69572e9	0.324397
$748$	−1.78052e9	−0.155558
$749$	−3.53593e9	−0.307480
$750$	9.13381e7	0.00790565
$751$	−1.37115e10	−1.18126	−0.590632	−	0.806941i	$-0.701121\pi$
$751$	−1.37115e10	−1.18126	−0.590632	+	0.806941i	$0.701121\pi$
$752$	1.31699e9	0.112933
$753$	−9.26396e9	−0.790705
$754$	1.19690e10	1.01685
$755$	3.29007e10	2.78221
$756$	−4.04523e9	−0.340499
$757$	−1.08694e10	−0.910685	−0.455343	−	0.890316i	$-0.650483\pi$
$757$	−1.08694e10	−0.910685	−0.455343	+	0.890316i	$0.650483\pi$
$758$	−2.59824e9	−0.216689
$759$	−3.21005e8	−0.0266480
$760$	−1.05236e6	−8.69591e−5 0
$761$	−8.48694e9	−0.698079	−0.349040	−	0.937108i	$-0.613492\pi$
$761$	−8.48694e9	−0.698079	−0.349040	+	0.937108i	$0.613492\pi$
$762$	−9.54161e9	−0.781230
$763$	−1.08641e10	−0.885436
$764$	6.16669e9	0.500294
$765$	3.91032e9	0.315789
$766$	−1.11222e10	−0.894109
$767$	−1.03663e10	−0.829545
$768$	−6.45585e9	−0.514268
$769$	1.66331e10	1.31896	0.659479	−	0.751723i	$-0.270777\pi$
$769$	1.66331e10	1.31896	0.659479	+	0.751723i	$0.270777\pi$
$770$	9.57825e9	0.756082
$771$	−8.39440e9	−0.659628
$772$	−3.94252e9	−0.308399
$773$	1.70231e10	1.32559	0.662796	−	0.748800i	$-0.269370\pi$
$773$	1.70231e10	1.32559	0.662796	+	0.748800i	$0.269370\pi$
$774$	−6.84640e9	−0.530725
$775$	2.09665e10	1.61797
$776$	−1.29406e9	−0.0994120
$777$	−1.64481e9	−0.125789
$778$	9.68765e9	0.737548
$779$	−1.88106e7	−0.00142568
$780$	−1.09655e10	−0.827369
$781$	2.57820e8	0.0193659
$782$	2.67052e9	0.199697
$783$	1.90675e9	0.141948
$784$	−2.34357e10	−1.73689
$785$	−1.52479e10	−1.12504
$786$	1.49143e10	1.09553
$787$	1.89829e10	1.38820	0.694100	−	0.719879i	$-0.255803\pi$
$787$	1.89829e10	1.38820	0.694100	+	0.719879i	$0.255803\pi$
$788$	2.30631e10	1.67910
$789$	5.14004e9	0.372560
$790$	2.88638e10	2.08285
$791$	2.97156e10	2.13485
$792$	7.50844e7	0.00537046
$793$	8.67192e9	0.617532
$794$	−1.72610e10	−1.22375
$795$	1.66485e10	1.17514
$796$	1.52307e10	1.07035
$797$	−1.55539e10	−1.08826	−0.544131	−	0.839000i	$-0.683141\pi$
$797$	−1.55539e10	−1.08826	−0.544131	+	0.839000i	$0.683141\pi$
$798$	1.68542e7	0.00117408
$799$	−1.15038e9	−0.0797864
$800$	2.08246e10	1.43801
$801$	7.39071e9	0.508127
$802$	4.70435e9	0.322025
$803$	4.54501e9	0.309763
$804$	−9.00415e9	−0.611007
$805$	−7.36099e9	−0.497337
$806$	−3.29358e10	−2.21562
$807$	−2.72862e9	−0.182762
$808$	−2.04365e9	−0.136291
$809$	−3.30592e9	−0.219519	−0.109760	−	0.993958i	$-0.535008\pi$
$809$	−3.30592e9	−0.219519	−0.109760	+	0.993958i	$0.535008\pi$
$810$	−3.40931e9	−0.225408
$811$	7.80890e9	0.514064	0.257032	−	0.966403i	$-0.417256\pi$
$811$	7.80890e9	0.514064	0.257032	+	0.966403i	$0.417256\pi$
$812$	−1.99092e10	−1.30499
$813$	−1.59241e10	−1.03929
$814$	6.31212e8	0.0410195
$815$	3.35489e9	0.217083
$816$	5.67270e9	0.365489
$817$	1.46160e7	0.000937673 0
$818$	−3.23066e10	−2.06374
$819$	8.49421e9	0.540293
$820$	−3.97302e10	−2.51636
$821$	2.03343e10	1.28242	0.641208	−	0.767367i	$-0.278434\pi$
$821$	2.03343e10	1.28242	0.641208	+	0.767367i	$0.278434\pi$
$822$	−2.05573e10	−1.29096
$823$	−5.44877e9	−0.340721	−0.170360	−	0.985382i	$-0.554493\pi$
$823$	−5.44877e9	−0.340721	−0.170360	+	0.985382i	$0.554493\pi$
$824$	−2.00219e8	−0.0124669
$825$	2.07510e9	0.128662
$826$	3.36527e10	2.07773
$827$	1.46646e10	0.901571	0.450785	−	0.892632i	$-0.351144\pi$
$827$	1.46646e10	0.901571	0.450785	+	0.892632i	$0.351144\pi$
$828$	−1.19303e9	−0.0730374
$829$	1.62752e10	0.992171	0.496086	−	0.868274i	$-0.334770\pi$
$829$	1.62752e10	0.992171	0.496086	+	0.868274i	$0.334770\pi$
$830$	−3.25224e10	−1.97428
$831$	6.75715e9	0.408470
$832$	−1.75747e10	−1.05793
$833$	2.04710e10	1.22710
$834$	1.02541e10	0.612089
$835$	−4.43808e10	−2.63811
$836$	−3.31412e6	−0.000196176 0
$837$	−5.24694e9	−0.309291
$838$	1.43989e10	0.845230
$839$	−2.08897e10	−1.22114	−0.610570	−	0.791962i	$-0.709060\pi$
$839$	−2.08897e10	−1.22114	−0.610570	+	0.791962i	$0.709060\pi$
$840$	1.72177e9	0.100230
$841$	−7.86548e9	−0.455973
$842$	2.87467e10	1.65957
$843$	8.03712e8	0.0462066
$844$	−2.64213e10	−1.51271
$845$	−1.81977e9	−0.103757
$846$	1.00299e9	0.0569508
$847$	−2.83166e10	−1.60121
$848$	2.41520e10	1.36009
$849$	8.54053e9	0.478970
$850$	−1.72633e10	−0.964179
$851$	−4.85093e8	−0.0269819
$852$	9.58200e8	0.0530784
$853$	8.63798e9	0.476530	0.238265	−	0.971200i	$-0.423421\pi$
$853$	8.63798e9	0.476530	0.238265	+	0.971200i	$0.423421\pi$
$854$	−2.81521e10	−1.54671
$855$	7.27834e6	0.000398245 0
$856$	2.43922e8	0.0132921
$857$	−2.73211e10	−1.48274	−0.741370	−	0.671097i	$-0.765823\pi$
$857$	−2.73211e10	−1.48274	−0.741370	+	0.671097i	$0.765823\pi$
$858$	−3.25973e9	−0.176188
$859$	2.72853e10	1.46877	0.734383	−	0.678735i	$-0.237471\pi$
$859$	2.72853e10	1.46877	0.734383	+	0.678735i	$0.237471\pi$
$860$	3.08708e10	1.65502
$861$	3.07761e10	1.64325
$862$	2.43449e10	1.29459
$863$	1.11895e10	0.592617	0.296308	−	0.955092i	$-0.404244\pi$
$863$	1.11895e10	0.592617	0.296308	+	0.955092i	$0.404244\pi$
$864$	−5.21143e9	−0.274890
$865$	−3.13548e10	−1.64721
$866$	4.13733e9	0.216475
$867$	6.12407e9	0.319134
$868$	5.47856e10	2.84346
$869$	4.39649e9	0.227267
$870$	−1.67795e10	−0.863894
$871$	1.89070e10	0.969526
$872$	7.49446e8	0.0382765
$873$	8.95003e9	0.455276
$874$	4.97068e6	0.000251841 0
$875$	3.19029e8	0.0160991
$876$	1.68917e10	0.849005
$877$	2.05215e9	0.102733	0.0513667	−	0.998680i	$-0.483642\pi$
$877$	2.05215e9	0.102733	0.0513667	+	0.998680i	$0.483642\pi$
$878$	−4.64685e10	−2.31701
$879$	6.51226e8	0.0323423
$880$	6.00051e9	0.296824
$881$	2.13115e10	1.05002	0.525010	−	0.851096i	$-0.324062\pi$
$881$	2.13115e10	1.05002	0.525010	+	0.851096i	$0.324062\pi$
$882$	−1.78481e10	−0.875896
$883$	−3.57525e9	−0.174761	−0.0873804	−	0.996175i	$-0.527850\pi$
$883$	−3.57525e9	−0.174761	−0.0873804	+	0.996175i	$0.527850\pi$
$884$	1.38953e10	0.676527
$885$	1.45326e10	0.704763
$886$	2.33477e10	1.12778
$887$	−1.76896e9	−0.0851107	−0.0425554	−	0.999094i	$-0.513550\pi$
$887$	−1.76896e9	−0.0851107	−0.0425554	+	0.999094i	$0.513550\pi$
$888$	1.13465e8	0.00543774
$889$	−3.33273e10	−1.59090
$890$	−6.50384e10	−3.09247
$891$	−5.19301e8	−0.0245950
$892$	3.21278e10	1.51567
$893$	−2.14122e6	−0.000100619 0
$894$	−6.92823e9	−0.324295
$895$	−3.10791e10	−1.44906
$896$	5.27057e9	0.244782
$897$	2.50514e9	0.115893
$898$	2.89263e10	1.33299
$899$	−2.58237e10	−1.18538
$900$	7.71222e9	0.352639
$901$	−2.10966e10	−0.960896
$902$	−1.18106e10	−0.535858
$903$	−2.39134e10	−1.08077
$904$	−2.04990e9	−0.0922875
$905$	−1.75371e10	−0.786481
$906$	3.63493e10	1.62386
$907$	−3.18196e10	−1.41602	−0.708009	−	0.706203i	$-0.750407\pi$
$907$	−3.18196e10	−1.41602	−0.708009	+	0.706203i	$0.750407\pi$
$908$	−1.09426e9	−0.0485089
$909$	1.41343e10	0.624168
$910$	−7.47492e10	−3.28823
$911$	−2.03709e10	−0.892680	−0.446340	−	0.894863i	$-0.647273\pi$
$911$	−2.03709e10	−0.892680	−0.446340	+	0.894863i	$0.647273\pi$
$912$	1.05587e7	0.000460922 0
$913$	−4.95377e9	−0.215421
$914$	4.67370e9	0.202465
$915$	−1.21573e10	−0.524641
$916$	3.08543e10	1.32642
$917$	5.20933e10	2.23095
$918$	4.32020e9	0.184312
$919$	2.27511e10	0.966936	0.483468	−	0.875362i	$-0.339377\pi$
$919$	2.27511e10	0.966936	0.483468	+	0.875362i	$0.339377\pi$
$920$	5.07789e8	0.0214994
$921$	−1.08273e10	−0.456680
$922$	5.82328e9	0.244686
$923$	−2.01204e9	−0.0842230
$924$	5.42225e9	0.226114
$925$	3.13584e9	0.130274
$926$	−4.75797e9	−0.196917
$927$	1.38476e9	0.0570946
$928$	−2.56489e10	−1.05354
$929$	2.74318e10	1.12253	0.561267	−	0.827635i	$-0.310314\pi$
$929$	2.74318e10	1.12253	0.561267	+	0.827635i	$0.310314\pi$
$930$	4.61732e10	1.88234
$931$	3.81030e7	0.00154751
$932$	−1.85109e10	−0.748982
$933$	2.10120e10	0.846998
$934$	−5.17358e10	−2.07767
$935$	−5.24142e9	−0.209705
$936$	−5.85963e8	−0.0233563
$937$	4.61149e10	1.83127	0.915635	−	0.402011i	$-0.131689\pi$
$937$	4.61149e10	1.83127	0.915635	+	0.402011i	$0.131689\pi$
$938$	−6.13788e10	−2.42834
$939$	2.14843e10	0.846822
$940$	−4.52253e9	−0.177596
$941$	2.61990e10	1.02499	0.512497	−	0.858689i	$-0.328720\pi$
$941$	2.61990e10	1.02499	0.512497	+	0.858689i	$0.328720\pi$
$942$	−1.68462e10	−0.656634
$943$	9.07659e9	0.352478
$944$	2.10825e10	0.815680
$945$	−1.19081e10	−0.459021
$946$	9.17697e9	0.352436
$947$	3.00640e10	1.15033	0.575165	−	0.818038i	$-0.304938\pi$
$947$	3.00640e10	1.15033	0.575165	+	0.818038i	$0.304938\pi$
$948$	1.63398e10	0.622898
$949$	−3.54695e10	−1.34717
$950$	−3.21325e7	−0.00121594
$951$	−2.30060e10	−0.867380
$952$	−2.18179e9	−0.0819564
$953$	6.39450e9	0.239321	0.119661	−	0.992815i	$-0.461819\pi$
$953$	6.39450e9	0.239321	0.119661	+	0.992815i	$0.461819\pi$
$954$	1.83936e10	0.685879
$955$	1.81532e10	0.674437
$956$	6.30036e9	0.233218
$957$	−2.55582e9	−0.0942626
$958$	−1.62595e10	−0.597488
$959$	−7.18031e10	−2.62892
$960$	2.46381e10	0.898790
$961$	4.35481e10	1.58284
$962$	−4.92601e9	−0.178395
$963$	−1.68702e9	−0.0608734
$964$	−5.58414e10	−2.00764
$965$	−1.16058e10	−0.415747
$966$	−8.13257e9	−0.290274
$967$	−1.61416e10	−0.574056	−0.287028	−	0.957922i	$-0.592667\pi$
$967$	−1.61416e10	−0.574056	−0.287028	+	0.957922i	$0.592667\pi$
$968$	1.95339e9	0.0692189
$969$	−9.22295e6	−0.000325639 0
$970$	−7.87604e10	−2.77081
$971$	3.16346e10	1.10891	0.554454	−	0.832214i	$-0.312927\pi$
$971$	3.16346e10	1.10891	0.554454	+	0.832214i	$0.312927\pi$
$972$	−1.93001e9	−0.0674105
$973$	3.58157e10	1.24646
$974$	−2.73929e10	−0.949910
$975$	−1.61942e10	−0.559556
$976$	−1.76365e10	−0.607210
$977$	1.02749e10	0.352490	0.176245	−	0.984346i	$-0.443605\pi$
$977$	1.02749e10	0.352490	0.176245	+	0.984346i	$0.443605\pi$
$978$	3.70655e9	0.126702
$979$	−9.90657e9	−0.337430
$980$	8.04781e10	2.73141
$981$	−5.18334e9	−0.175294
$982$	−2.34394e10	−0.789871
$983$	1.83595e10	0.616487	0.308244	−	0.951307i	$-0.400259\pi$
$983$	1.83595e10	0.616487	0.308244	+	0.951307i	$0.400259\pi$
$984$	−2.12305e9	−0.0710359
$985$	6.78921e10	2.26356
$986$	2.12626e10	0.706393
$987$	3.50327e9	0.115975
$988$	2.58635e7	0.000853176 0
$989$	−7.05260e9	−0.231826
$990$	4.56986e9	0.149685
$991$	−2.66502e10	−0.869847	−0.434924	−	0.900467i	$-0.643225\pi$
$991$	−2.66502e10	−0.869847	−0.434924	+	0.900467i	$0.643225\pi$
$992$	7.05799e10	2.29557
$993$	−5.58806e9	−0.181109
$994$	6.53179e9	0.210950
$995$	4.48354e10	1.44291
$996$	−1.84109e10	−0.590429
$997$	−3.57015e10	−1.14092	−0.570458	−	0.821327i	$-0.693234\pi$
$997$	−3.57015e10	−1.14092	−0.570458	+	0.821327i	$0.693234\pi$
$998$	4.45024e10	1.41718
$999$	−7.84753e8	−0.0249031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.c.1.2	✓	7
3.2	odd	2		207.8.a.d.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.c.1.2	✓	7	1.1	even	1	trivial
207.8.a.d.1.6		7	3.2	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-16.2020 q^{2} -27.0000 q^{3} +134.506 q^{4} +395.951 q^{5} +437.455 q^{6} +1527.96 q^{7} -105.404 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-16.2020 q^{2} -27.0000 q^{3} +134.506 q^{4} +395.951 q^{5} +437.455 q^{6} +1527.96 q^{7} -105.404 q^{8} +729.000 q^{9} -6415.21 q^{10} -977.157 q^{11} -3631.65 q^{12} +7625.78 q^{13} -24756.0 q^{14} -10690.7 q^{15} -15509.0 q^{16} +13547.0 q^{17} -11811.3 q^{18} +25.2152 q^{19} +53257.7 q^{20} -41254.8 q^{21} +15831.9 q^{22} -12167.0 q^{23} +2845.91 q^{24} +78652.3 q^{25} -123553. q^{26} -19683.0 q^{27} +205519. q^{28} -96873.1 q^{29} +173211. q^{30} +266572. q^{31} +264768. q^{32} +26383.2 q^{33} -219489. q^{34} +604996. q^{35} +98054.6 q^{36} +39869.6 q^{37} -408.538 q^{38} -205896. q^{39} -41734.9 q^{40} -746001. q^{41} +668412. q^{42} +579650. q^{43} -131433. q^{44} +288648. q^{45} +197130. q^{46} -84917.9 q^{47} +418742. q^{48} +1.51111e6 q^{49} -1.27433e6 q^{50} -365769. q^{51} +1.02571e6 q^{52} -1.55729e6 q^{53} +318904. q^{54} -386906. q^{55} -161053. q^{56} -680.811 q^{57} +1.56954e6 q^{58} -1.35938e6 q^{59} -1.43796e6 q^{60} +1.13718e6 q^{61} -4.31901e6 q^{62} +1.11388e6 q^{63} -2.30464e6 q^{64} +3.01944e6 q^{65} -427462. q^{66} +2.47935e6 q^{67} +1.82215e6 q^{68} +328509. q^{69} -9.80216e6 q^{70} -263847. q^{71} -76839.7 q^{72} -4.65126e6 q^{73} -645968. q^{74} -2.12361e6 q^{75} +3391.59 q^{76} -1.49305e6 q^{77} +3.33593e6 q^{78} -4.49927e6 q^{79} -6.14079e6 q^{80} +531441. q^{81} +1.20867e7 q^{82} +5.06957e6 q^{83} -5.54901e6 q^{84} +5.36395e6 q^{85} -9.39150e6 q^{86} +2.61557e6 q^{87} +102996. q^{88} +1.01382e7 q^{89} -4.67669e6 q^{90} +1.16519e7 q^{91} -1.63653e6 q^{92} -7.19745e6 q^{93} +1.37584e6 q^{94} +9984.00 q^{95} -7.14874e6 q^{96} +1.22771e7 q^{97} -2.44830e7 q^{98} -712347. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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