LMFDB - Embedded newform 546.8.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$170.562223914$
Analytic rank:	$0$
Dimension:	$7$
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} - 2 x^{6} - 313738 x^{5} + 10691268 x^{4} + 29687523333 x^{3} - 2032748091218 x^{2} + \cdots + 81\!\cdots\!96 $ x^7 - 2x^6 - 313738x^5 + 10691268x^4 + 29687523333x^3 - 2032748091218x^2 - 846237116901940x + 81946324586652296
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-331.377$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -242.377 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -242.377 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1939.02 q^{10} -7077.45 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} -6544.19 q^{15} +4096.00 q^{16} +33069.7 q^{17} +5832.00 q^{18} +48171.3 q^{19} -15512.2 q^{20} +9261.00 q^{21} -56619.6 q^{22} -42567.4 q^{23} +13824.0 q^{24} -19378.2 q^{25} +17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -74402.1 q^{29} -52353.5 q^{30} -49724.0 q^{31} +32768.0 q^{32} -191091. q^{33} +264557. q^{34} -83135.5 q^{35} +46656.0 q^{36} -484106. q^{37} +385371. q^{38} +59319.0 q^{39} -124097. q^{40} +153234. q^{41} +74088.0 q^{42} +127224. q^{43} -452957. q^{44} -176693. q^{45} -340540. q^{46} +184118. q^{47} +110592. q^{48} +117649. q^{49} -155025. q^{50} +892881. q^{51} +140608. q^{52} +1.38323e6 q^{53} +157464. q^{54} +1.71541e6 q^{55} +175616. q^{56} +1.30063e6 q^{57} -595217. q^{58} +2.29540e6 q^{59} -418828. q^{60} +636447. q^{61} -397792. q^{62} +250047. q^{63} +262144. q^{64} -532503. q^{65} -1.52873e6 q^{66} +112178. q^{67} +2.11646e6 q^{68} -1.14932e6 q^{69} -665084. q^{70} -316866. q^{71} +373248. q^{72} +2.53884e6 q^{73} -3.87285e6 q^{74} -523210. q^{75} +3.08297e6 q^{76} -2.42756e6 q^{77} +474552. q^{78} -5.74457e6 q^{79} -992778. q^{80} +531441. q^{81} +1.22587e6 q^{82} +7.10286e6 q^{83} +592704. q^{84} -8.01534e6 q^{85} +1.01779e6 q^{86} -2.00886e6 q^{87} -3.62365e6 q^{88} +1.10627e7 q^{89} -1.41355e6 q^{90} +753571. q^{91} -2.72432e6 q^{92} -1.34255e6 q^{93} +1.47295e6 q^{94} -1.16756e7 q^{95} +884736. q^{96} +9.72667e6 q^{97} +941192. q^{98} -5.15946e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 56 q^{2} + 189 q^{3} + 448 q^{4} + 625 q^{5} + 1512 q^{6} + 2401 q^{7} + 3584 q^{8} + 5103 q^{9}+O(q^{10}) $ 7 * q + 56 * q^2 + 189 * q^3 + 448 * q^4 + 625 * q^5 + 1512 * q^6 + 2401 * q^7 + 3584 * q^8 + 5103 * q^9	\( 7 q + 56 q^{2} + 189 q^{3} + 448 q^{4} + 625 q^{5} + 1512 q^{6} + 2401 q^{7} + 3584 q^{8} + 5103 q^{9} + 5000 q^{10} + 4678 q^{11} + 12096 q^{12} + 15379 q^{13} + 19208 q^{14} + 16875 q^{15} + 28672 q^{16} + 38552 q^{17} + 40824 q^{18} + 60231 q^{19} + 40000 q^{20} + 64827 q^{21} + 37424 q^{22} + 82047 q^{23} + 96768 q^{24} + 136408 q^{25} + 123032 q^{26} + 137781 q^{27} + 153664 q^{28} + 87523 q^{29} + 135000 q^{30} - 3191 q^{31} + 229376 q^{32} + 126306 q^{33} + 308416 q^{34} + 214375 q^{35} + 326592 q^{36} + 360916 q^{37} + 481848 q^{38} + 415233 q^{39} + 320000 q^{40} + 814350 q^{41} + 518616 q^{42} + 840057 q^{43} + 299392 q^{44} + 455625 q^{45} + 656376 q^{46} + 472723 q^{47} + 774144 q^{48} + 823543 q^{49} + 1091264 q^{50} + 1040904 q^{51} + 984256 q^{52} + 2185687 q^{53} + 1102248 q^{54} + 298354 q^{55} + 1229312 q^{56} + 1626237 q^{57} + 700184 q^{58} + 2046232 q^{59} + 1080000 q^{60} + 2744560 q^{61} - 25528 q^{62} + 1750329 q^{63} + 1835008 q^{64} + 1373125 q^{65} + 1010448 q^{66} + 1960358 q^{67} + 2467328 q^{68} + 2215269 q^{69} + 1715000 q^{70} + 2774656 q^{71} + 2612736 q^{72} + 3696313 q^{73} + 2887328 q^{74} + 3683016 q^{75} + 3854784 q^{76} + 1604554 q^{77} + 3321864 q^{78} + 1532089 q^{79} + 2560000 q^{80} + 3720087 q^{81} + 6514800 q^{82} + 7473863 q^{83} + 4148928 q^{84} - 1656624 q^{85} + 6720456 q^{86} + 2363121 q^{87} + 2395136 q^{88} + 14077309 q^{89} + 3645000 q^{90} + 5274997 q^{91} + 5251008 q^{92} - 86157 q^{93} + 3781784 q^{94} - 1704679 q^{95} + 6193152 q^{96} + 8273673 q^{97} + 6588344 q^{98} + 3410262 q^{99}+O(q^{100}) \) 7 * q + 56 * q^2 + 189 * q^3 + 448 * q^4 + 625 * q^5 + 1512 * q^6 + 2401 * q^7 + 3584 * q^8 + 5103 * q^9 + 5000 * q^10 + 4678 * q^11 + 12096 * q^12 + 15379 * q^13 + 19208 * q^14 + 16875 * q^15 + 28672 * q^16 + 38552 * q^17 + 40824 * q^18 + 60231 * q^19 + 40000 * q^20 + 64827 * q^21 + 37424 * q^22 + 82047 * q^23 + 96768 * q^24 + 136408 * q^25 + 123032 * q^26 + 137781 * q^27 + 153664 * q^28 + 87523 * q^29 + 135000 * q^30 - 3191 * q^31 + 229376 * q^32 + 126306 * q^33 + 308416 * q^34 + 214375 * q^35 + 326592 * q^36 + 360916 * q^37 + 481848 * q^38 + 415233 * q^39 + 320000 * q^40 + 814350 * q^41 + 518616 * q^42 + 840057 * q^43 + 299392 * q^44 + 455625 * q^45 + 656376 * q^46 + 472723 * q^47 + 774144 * q^48 + 823543 * q^49 + 1091264 * q^50 + 1040904 * q^51 + 984256 * q^52 + 2185687 * q^53 + 1102248 * q^54 + 298354 * q^55 + 1229312 * q^56 + 1626237 * q^57 + 700184 * q^58 + 2046232 * q^59 + 1080000 * q^60 + 2744560 * q^61 - 25528 * q^62 + 1750329 * q^63 + 1835008 * q^64 + 1373125 * q^65 + 1010448 * q^66 + 1960358 * q^67 + 2467328 * q^68 + 2215269 * q^69 + 1715000 * q^70 + 2774656 * q^71 + 2612736 * q^72 + 3696313 * q^73 + 2887328 * q^74 + 3683016 * q^75 + 3854784 * q^76 + 1604554 * q^77 + 3321864 * q^78 + 1532089 * q^79 + 2560000 * q^80 + 3720087 * q^81 + 6514800 * q^82 + 7473863 * q^83 + 4148928 * q^84 - 1656624 * q^85 + 6720456 * q^86 + 2363121 * q^87 + 2395136 * q^88 + 14077309 * q^89 + 3645000 * q^90 + 5274997 * q^91 + 5251008 * q^92 - 86157 * q^93 + 3781784 * q^94 - 1704679 * q^95 + 6193152 * q^96 + 8273673 * q^97 + 6588344 * q^98 + 3410262 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	−242.377	−0.867156	−0.433578	−	0.901116i	$-0.642749\pi$
$5$	−242.377	−0.867156	−0.433578	+	0.901116i	$0.642749\pi$
$6$	216.000	0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−1939.02	−0.613172
$11$	−7077.45	−1.60325	−0.801627	−	0.597825i	$-0.796032\pi$
$11$	−7077.45	−1.60325	−0.801627	+	0.597825i	$0.796032\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	2744.00	0.267261
$15$	−6544.19	−0.500653
$16$	4096.00	0.250000
$17$	33069.7	1.63252	0.816260	−	0.577684i	$-0.196043\pi$
$17$	33069.7	1.63252	0.816260	+	0.577684i	$0.196043\pi$
$18$	5832.00	0.235702
$19$	48171.3	1.61121	0.805603	−	0.592456i	$-0.201841\pi$
$19$	48171.3	1.61121	0.805603	+	0.592456i	$0.201841\pi$
$20$	−15512.2	−0.433578
$21$	9261.00	0.218218
$22$	−56619.6	−1.13367
$23$	−42567.4	−0.729508	−0.364754	−	0.931104i	$-0.618847\pi$
$23$	−42567.4	−0.729508	−0.364754	+	0.931104i	$0.618847\pi$
$24$	13824.0	0.204124
$25$	−19378.2	−0.248040
$26$	17576.0	0.196116
$27$	19683.0	0.192450
$28$	21952.0	0.188982
$29$	−74402.1	−0.566490	−0.283245	−	0.959048i	$-0.591411\pi$
$29$	−74402.1	−0.566490	−0.283245	+	0.959048i	$0.591411\pi$
$30$	−52353.5	−0.354015
$31$	−49724.0	−0.299778	−0.149889	−	0.988703i	$-0.547892\pi$
$31$	−49724.0	−0.299778	−0.149889	+	0.988703i	$0.547892\pi$
$32$	32768.0	0.176777
$33$	−191091.	−0.925639
$34$	264557.	1.15437
$35$	−83135.5	−0.327754
$36$	46656.0	0.166667
$37$	−484106.	−1.57121	−0.785606	−	0.618727i	$-0.787649\pi$
$37$	−484106.	−1.57121	−0.785606	+	0.618727i	$0.787649\pi$
$38$	385371.	1.13929
$39$	59319.0	0.160128
$40$	−124097.	−0.306586
$41$	153234.	0.347225	0.173612	−	0.984814i	$-0.444456\pi$
$41$	153234.	0.347225	0.173612	+	0.984814i	$0.444456\pi$
$42$	74088.0	0.154303
$43$	127224.	0.244022	0.122011	−	0.992529i	$-0.461066\pi$
$43$	127224.	0.244022	0.122011	+	0.992529i	$0.461066\pi$
$44$	−452957.	−0.801627
$45$	−176693.	−0.289052
$46$	−340540.	−0.515840
$47$	184118.	0.258675	0.129337	−	0.991601i	$-0.458715\pi$
$47$	184118.	0.258675	0.129337	+	0.991601i	$0.458715\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−155025.	−0.175391
$51$	892881.	0.942536
$52$	140608.	0.138675
$53$	1.38323e6	1.27623	0.638114	−	0.769942i	$-0.279715\pi$
$53$	1.38323e6	1.27623	0.638114	+	0.769942i	$0.279715\pi$
$54$	157464.	0.136083
$55$	1.71541e6	1.39027
$56$	175616.	0.133631
$57$	1.30063e6	0.930230
$58$	−595217.	−0.400569
$59$	2.29540e6	1.45505	0.727523	−	0.686083i	$-0.240671\pi$
$59$	2.29540e6	1.45505	0.727523	+	0.686083i	$0.240671\pi$
$60$	−418828.	−0.250326
$61$	636447.	0.359011	0.179505	−	0.983757i	$-0.442550\pi$
$61$	636447.	0.359011	0.179505	+	0.983757i	$0.442550\pi$
$62$	−397792.	−0.211975
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	−532503.	−0.240506
$66$	−1.52873e6	−0.654526
$67$	112178.	0.0455663	0.0227832	−	0.999740i	$-0.492747\pi$
$67$	112178.	0.0455663	0.0227832	+	0.999740i	$0.492747\pi$
$68$	2.11646e6	0.816260
$69$	−1.14932e6	−0.421182
$70$	−665084.	−0.231757
$71$	−316866.	−0.105068	−0.0525341	−	0.998619i	$-0.516730\pi$
$71$	−316866.	−0.105068	−0.0525341	+	0.998619i	$0.516730\pi$
$72$	373248.	0.117851
$73$	2.53884e6	0.763844	0.381922	−	0.924195i	$-0.375262\pi$
$73$	2.53884e6	0.763844	0.381922	+	0.924195i	$0.375262\pi$
$74$	−3.87285e6	−1.11101
$75$	−523210.	−0.143206
$76$	3.08297e6	0.805603
$77$	−2.42756e6	−0.605973
$78$	474552.	0.113228
$79$	−5.74457e6	−1.31088	−0.655440	−	0.755247i	$-0.727517\pi$
$79$	−5.74457e6	−1.31088	−0.655440	+	0.755247i	$0.727517\pi$
$80$	−992778.	−0.216789
$81$	531441.	0.111111
$82$	1.22587e6	0.245525
$83$	7.10286e6	1.36352	0.681758	−	0.731578i	$-0.261216\pi$
$83$	7.10286e6	1.36352	0.681758	+	0.731578i	$0.261216\pi$
$84$	592704.	0.109109
$85$	−8.01534e6	−1.41565
$86$	1.01779e6	0.172549
$87$	−2.00886e6	−0.327063
$88$	−3.62365e6	−0.566836
$89$	1.10627e7	1.66340	0.831701	−	0.555223i	$-0.187367\pi$
$89$	1.10627e7	1.66340	0.831701	+	0.555223i	$0.187367\pi$
$90$	−1.41355e6	−0.204391
$91$	753571.	0.104828
$92$	−2.72432e6	−0.364754
$93$	−1.34255e6	−0.173077
$94$	1.47295e6	0.182911
$95$	−1.16756e7	−1.39717
$96$	884736.	0.102062
$97$	9.72667e6	1.08209	0.541045	−	0.840994i	$-0.318029\pi$
$97$	9.72667e6	1.08209	0.541045	+	0.840994i	$0.318029\pi$
$98$	941192.	0.101015
$99$	−5.15946e6	−0.534418
$100$	−1.24020e6	−0.124020
$101$	−2.09258e6	−0.202095	−0.101048	−	0.994882i	$-0.532219\pi$
$101$	−2.09258e6	−0.202095	−0.101048	+	0.994882i	$0.532219\pi$
$102$	7.14305e6	0.666474
$103$	−803087.	−0.0724156	−0.0362078	−	0.999344i	$-0.511528\pi$
$103$	−803087.	−0.0724156	−0.0362078	+	0.999344i	$0.511528\pi$
$104$	1.12486e6	0.0980581
$105$	−2.24466e6	−0.189229
$106$	1.10658e7	0.902429
$107$	2.31791e7	1.82917	0.914584	−	0.404395i	$-0.132518\pi$
$107$	2.31791e7	1.82917	0.914584	+	0.404395i	$0.132518\pi$
$108$	1.25971e6	0.0962250
$109$	−9.42731e6	−0.697260	−0.348630	−	0.937260i	$-0.613353\pi$
$109$	−9.42731e6	−0.697260	−0.348630	+	0.937260i	$0.613353\pi$
$110$	1.37233e7	0.983070
$111$	−1.30709e7	−0.907139
$112$	1.40493e6	0.0944911
$113$	2.46090e7	1.60442	0.802212	−	0.597039i	$-0.203656\pi$
$113$	2.46090e7	1.60442	0.802212	+	0.597039i	$0.203656\pi$
$114$	1.04050e7	0.657772
$115$	1.03174e7	0.632597
$116$	−4.76173e6	−0.283245
$117$	1.60161e6	0.0924500
$118$	1.83632e7	1.02887
$119$	1.13429e7	0.617035
$120$	−3.35063e6	−0.177007
$121$	3.06031e7	1.57042
$122$	5.09157e6	0.253859
$123$	4.13731e6	0.200470
$124$	−3.18234e6	−0.149889
$125$	2.36326e7	1.08225
$126$	2.00038e6	0.0890871
$127$	−1.63949e7	−0.710225	−0.355112	−	0.934824i	$-0.615557\pi$
$127$	−1.63949e7	−0.710225	−0.355112	+	0.934824i	$0.615557\pi$
$128$	2.09715e6	0.0883883
$129$	3.43504e6	0.140886
$130$	−4.26003e6	−0.170063
$131$	2.57608e7	1.00117	0.500587	−	0.865686i	$-0.333118\pi$
$131$	2.57608e7	1.00117	0.500587	+	0.865686i	$0.333118\pi$
$132$	−1.22298e7	−0.462820
$133$	1.65228e7	0.608979
$134$	897420.	0.0322203
$135$	−4.77072e6	−0.166884
$136$	1.69317e7	0.577183
$137$	−1.99596e7	−0.663177	−0.331589	−	0.943424i	$-0.607585\pi$
$137$	−1.99596e7	−0.663177	−0.331589	+	0.943424i	$0.607585\pi$
$138$	−9.19457e6	−0.297820
$139$	−3.04900e7	−0.962955	−0.481477	−	0.876459i	$-0.659900\pi$
$139$	−3.04900e7	−0.962955	−0.481477	+	0.876459i	$0.659900\pi$
$140$	−5.32067e6	−0.163877
$141$	4.97119e6	0.149346
$142$	−2.53493e6	−0.0742945
$143$	−1.55492e7	−0.444663
$144$	2.98598e6	0.0833333
$145$	1.80334e7	0.491235
$146$	2.03107e7	0.540119
$147$	3.17652e6	0.0824786
$148$	−3.09828e7	−0.785606
$149$	4.21784e7	1.04457	0.522285	−	0.852771i	$-0.325080\pi$
$149$	4.21784e7	1.04457	0.522285	+	0.852771i	$0.325080\pi$
$150$	−4.18568e6	−0.101262
$151$	1.76544e7	0.417287	0.208643	−	0.977992i	$-0.433095\pi$
$151$	1.76544e7	0.417287	0.208643	+	0.977992i	$0.433095\pi$
$152$	2.46637e7	0.569647
$153$	2.41078e7	0.544174
$154$	−1.94205e7	−0.428488
$155$	1.20520e7	0.259955
$156$	3.79642e6	0.0800641
$157$	4.61553e7	0.951860	0.475930	−	0.879483i	$-0.342112\pi$
$157$	4.61553e7	0.951860	0.475930	+	0.879483i	$0.342112\pi$
$158$	−4.59566e7	−0.926932
$159$	3.73471e7	0.736830
$160$	−7.94222e6	−0.153293
$161$	−1.46006e7	−0.275728
$162$	4.25153e6	0.0785674
$163$	−2.29155e7	−0.414450	−0.207225	−	0.978293i	$-0.566443\pi$
$163$	−2.29155e7	−0.414450	−0.207225	+	0.978293i	$0.566443\pi$
$164$	9.80695e6	0.173612
$165$	4.63162e7	0.802674
$166$	5.68229e7	0.964151
$167$	−3.03353e7	−0.504012	−0.252006	−	0.967726i	$-0.581090\pi$
$167$	−3.03353e7	−0.504012	−0.252006	+	0.967726i	$0.581090\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	−6.41228e7	−1.00102
$171$	3.51169e7	0.537069
$172$	8.14231e6	0.122011
$173$	−4.88191e7	−0.716851	−0.358425	−	0.933558i	$-0.616686\pi$
$173$	−4.88191e7	−0.716851	−0.358425	+	0.933558i	$0.616686\pi$
$174$	−1.60709e7	−0.231269
$175$	−6.64671e6	−0.0937505
$176$	−2.89892e7	−0.400814
$177$	6.19759e7	0.840071
$178$	8.85019e7	1.17620
$179$	4.39273e7	0.572466	0.286233	−	0.958160i	$-0.407597\pi$
$179$	4.39273e7	0.572466	0.286233	+	0.958160i	$0.407597\pi$
$180$	−1.13084e7	−0.144526
$181$	−4.96543e7	−0.622417	−0.311208	−	0.950342i	$-0.600734\pi$
$181$	−4.96543e7	−0.622417	−0.311208	+	0.950342i	$0.600734\pi$
$182$	6.02857e6	0.0741249
$183$	1.71841e7	0.207275
$184$	−2.17945e7	−0.257920
$185$	1.17336e8	1.36249
$186$	−1.07404e7	−0.122384
$187$	−2.34049e8	−2.61735
$188$	1.17836e7	0.129337
$189$	6.75127e6	0.0727393
$190$	−9.34052e7	−0.987946
$191$	3.32030e7	0.344795	0.172398	−	0.985027i	$-0.444849\pi$
$191$	3.32030e7	0.344795	0.172398	+	0.985027i	$0.444849\pi$
$192$	7.07789e6	0.0721688
$193$	−2.83349e6	−0.0283707	−0.0141854	−	0.999899i	$-0.504515\pi$
$193$	−2.83349e6	−0.0283707	−0.0141854	+	0.999899i	$0.504515\pi$
$194$	7.78134e7	0.765153
$195$	−1.43776e7	−0.138856
$196$	7.52954e6	0.0714286
$197$	1.15425e8	1.07564	0.537821	−	0.843059i	$-0.319248\pi$
$197$	1.15425e8	1.07564	0.537821	+	0.843059i	$0.319248\pi$
$198$	−4.12757e7	−0.377891
$199$	−7.18597e7	−0.646398	−0.323199	−	0.946331i	$-0.604758\pi$
$199$	−7.18597e7	−0.646398	−0.323199	+	0.946331i	$0.604758\pi$
$200$	−9.92162e6	−0.0876956
$201$	3.02879e6	0.0263077
$202$	−1.67406e7	−0.142903
$203$	−2.55199e7	−0.214113
$204$	5.71444e7	0.471268
$205$	−3.71404e7	−0.301098
$206$	−6.42470e6	−0.0512056
$207$	−3.10317e7	−0.243169
$208$	8.99891e6	0.0693375
$209$	−3.40930e8	−2.58317
$210$	−1.79573e7	−0.133805
$211$	−9.03652e7	−0.662236	−0.331118	−	0.943589i	$-0.607426\pi$
$211$	−9.03652e7	−0.662236	−0.331118	+	0.943589i	$0.607426\pi$
$212$	8.85266e7	0.638114
$213$	−8.55539e6	−0.0606612
$214$	1.85433e8	1.29342
$215$	−3.08361e7	−0.211605
$216$	1.00777e7	0.0680414
$217$	−1.70553e7	−0.113306
$218$	−7.54185e7	−0.493038
$219$	6.85486e7	0.441006
$220$	1.09787e8	0.695136
$221$	7.26541e7	0.452780
$222$	−1.04567e8	−0.641444
$223$	−5.17498e7	−0.312494	−0.156247	−	0.987718i	$-0.549940\pi$
$223$	−5.17498e7	−0.312494	−0.156247	+	0.987718i	$0.549940\pi$
$224$	1.12394e7	0.0668153
$225$	−1.41267e7	−0.0826802
$226$	1.96872e8	1.13450
$227$	1.15272e8	0.654083	0.327041	−	0.945010i	$-0.393948\pi$
$227$	1.15272e8	0.654083	0.327041	+	0.945010i	$0.393948\pi$
$228$	8.32401e7	0.465115
$229$	2.35100e7	0.129369	0.0646843	−	0.997906i	$-0.479396\pi$
$229$	2.35100e7	0.129369	0.0646843	+	0.997906i	$0.479396\pi$
$230$	8.25391e7	0.447314
$231$	−6.55443e7	−0.349859
$232$	−3.80939e7	−0.200284
$233$	−2.23587e8	−1.15798	−0.578991	−	0.815334i	$-0.696553\pi$
$233$	−2.23587e8	−1.15798	−0.578991	+	0.815334i	$0.696553\pi$
$234$	1.28129e7	0.0653720
$235$	−4.46261e7	−0.224311
$236$	1.46906e8	0.727523
$237$	−1.55104e8	−0.756837
$238$	9.07432e7	0.436310
$239$	3.10028e8	1.46895	0.734476	−	0.678634i	$-0.237428\pi$
$239$	3.10028e8	1.46895	0.734476	+	0.678634i	$0.237428\pi$
$240$	−2.68050e7	−0.125163
$241$	2.61146e8	1.20178	0.600888	−	0.799333i	$-0.294814\pi$
$241$	2.61146e8	1.20178	0.600888	+	0.799333i	$0.294814\pi$
$242$	2.44825e8	1.11046
$243$	1.43489e7	0.0641500
$244$	4.07326e7	0.179505
$245$	−2.85155e7	−0.123879
$246$	3.30985e7	0.141754
$247$	1.05832e8	0.446868
$248$	−2.54587e7	−0.105988
$249$	1.91777e8	0.787226
$250$	1.89061e8	0.765263
$251$	9.52874e7	0.380345	0.190172	−	0.981751i	$-0.439095\pi$
$251$	9.52874e7	0.380345	0.190172	+	0.981751i	$0.439095\pi$
$252$	1.60030e7	0.0629941
$253$	3.01269e8	1.16959
$254$	−1.31159e8	−0.502205
$255$	−2.16414e8	−0.817326
$256$	1.67772e7	0.0625000
$257$	5.08997e8	1.87046	0.935232	−	0.354036i	$-0.115191\pi$
$257$	5.08997e8	1.87046	0.935232	+	0.354036i	$0.115191\pi$
$258$	2.74803e7	0.0996214
$259$	−1.66048e8	−0.593862
$260$	−3.40802e7	−0.120253
$261$	−5.42391e7	−0.188830
$262$	2.06086e8	0.707937
$263$	3.29796e8	1.11789	0.558947	−	0.829204i	$-0.311206\pi$
$263$	3.29796e8	1.11789	0.558947	+	0.829204i	$0.311206\pi$
$264$	−9.78387e7	−0.327263
$265$	−3.35263e8	−1.10669
$266$	1.32182e8	0.430613
$267$	2.98694e8	0.960366
$268$	7.17936e6	0.0227832
$269$	2.92871e8	0.917368	0.458684	−	0.888599i	$-0.348321\pi$
$269$	2.92871e8	0.917368	0.458684	+	0.888599i	$0.348321\pi$
$270$	−3.81657e7	−0.118005
$271$	2.12877e8	0.649736	0.324868	−	0.945759i	$-0.394680\pi$
$271$	2.12877e8	0.649736	0.324868	+	0.945759i	$0.394680\pi$
$272$	1.35453e8	0.408130
$273$	2.03464e7	0.0605228
$274$	−1.59677e8	−0.468937
$275$	1.37148e8	0.397672
$276$	−7.35565e7	−0.210591
$277$	−2.36596e8	−0.668850	−0.334425	−	0.942422i	$-0.608542\pi$
$277$	−2.36596e8	−0.668850	−0.334425	+	0.942422i	$0.608542\pi$
$278$	−2.43920e8	−0.680912
$279$	−3.62488e7	−0.0999261
$280$	−4.25654e7	−0.115879
$281$	−8.67972e6	−0.0233364	−0.0116682	−	0.999932i	$-0.503714\pi$
$281$	−8.67972e6	−0.0233364	−0.0116682	+	0.999932i	$0.503714\pi$
$282$	3.97695e7	0.105604
$283$	−5.10463e8	−1.33879	−0.669393	−	0.742908i	$-0.733446\pi$
$283$	−5.10463e8	−1.33879	−0.669393	+	0.742908i	$0.733446\pi$
$284$	−2.02794e7	−0.0525341
$285$	−3.15242e8	−0.806655
$286$	−1.24393e8	−0.314424
$287$	5.25591e7	0.131239
$288$	2.38879e7	0.0589256
$289$	6.83265e8	1.66512
$290$	1.44267e8	0.347356
$291$	2.62620e8	0.624745
$292$	1.62486e8	0.381922
$293$	−6.39595e8	−1.48548	−0.742742	−	0.669577i	$-0.766475\pi$
$293$	−6.39595e8	−1.48548	−0.742742	+	0.669577i	$0.766475\pi$
$294$	2.54122e7	0.0583212
$295$	−5.56354e8	−1.26175
$296$	−2.47862e8	−0.555507
$297$	−1.39305e8	−0.308546
$298$	3.37427e8	0.738623
$299$	−9.35207e7	−0.202329
$300$	−3.34855e7	−0.0716031
$301$	4.36377e7	0.0922315
$302$	1.41235e8	0.295066
$303$	−5.64996e7	−0.116680
$304$	1.97310e8	0.402802
$305$	−1.54260e8	−0.311318
$306$	1.92862e8	0.384789
$307$	3.47945e8	0.686320	0.343160	−	0.939277i	$-0.388503\pi$
$307$	3.47945e8	0.686320	0.343160	+	0.939277i	$0.388503\pi$
$308$	−1.55364e8	−0.302987
$309$	−2.16834e7	−0.0418092
$310$	9.64159e7	0.183816
$311$	−5.92847e8	−1.11759	−0.558794	−	0.829307i	$-0.688736\pi$
$311$	−5.92847e8	−1.11759	−0.558794	+	0.829307i	$0.688736\pi$
$312$	3.03713e7	0.0566139
$313$	−5.10433e8	−0.940879	−0.470439	−	0.882432i	$-0.655905\pi$
$313$	−5.10433e8	−0.940879	−0.470439	+	0.882432i	$0.655905\pi$
$314$	3.69242e8	0.673066
$315$	−6.06058e7	−0.109251
$316$	−3.67653e8	−0.655440
$317$	−4.42779e8	−0.780691	−0.390346	−	0.920668i	$-0.627645\pi$
$317$	−4.42779e8	−0.780691	−0.390346	+	0.920668i	$0.627645\pi$
$318$	2.98777e8	0.521018
$319$	5.26577e8	0.908227
$320$	−6.35378e7	−0.108394
$321$	6.25836e8	1.05607
$322$	−1.16805e8	−0.194969
$323$	1.59301e9	2.63033
$324$	3.40122e7	0.0555556
$325$	−4.25738e7	−0.0687941
$326$	−1.83324e8	−0.293060
$327$	−2.54537e8	−0.402563
$328$	7.84556e7	0.122762
$329$	6.31525e7	0.0977699
$330$	3.70529e8	0.567576
$331$	7.99622e8	1.21196	0.605978	−	0.795482i	$-0.292782\pi$
$331$	7.99622e8	1.21196	0.605978	+	0.795482i	$0.292782\pi$
$332$	4.54583e8	0.681758
$333$	−3.52913e8	−0.523737
$334$	−2.42682e8	−0.356390
$335$	−2.71893e7	−0.0395131
$336$	3.79331e7	0.0545545
$337$	3.58971e7	0.0510922	0.0255461	−	0.999674i	$-0.491868\pi$
$337$	3.58971e7	0.0510922	0.0255461	+	0.999674i	$0.491868\pi$
$338$	3.86145e7	0.0543928
$339$	6.64443e8	0.926315
$340$	−5.12982e8	−0.707825
$341$	3.51919e8	0.480621
$342$	2.80935e8	0.379765
$343$	4.03536e7	0.0539949
$344$	6.51385e7	0.0862747
$345$	2.78570e8	0.365230
$346$	−3.90553e8	−0.506890
$347$	−1.04714e9	−1.34540	−0.672698	−	0.739917i	$-0.734865\pi$
$347$	−1.04714e9	−1.34540	−0.672698	+	0.739917i	$0.734865\pi$
$348$	−1.28567e8	−0.163532
$349$	7.74480e8	0.975262	0.487631	−	0.873050i	$-0.337861\pi$
$349$	7.74480e8	0.975262	0.487631	+	0.873050i	$0.337861\pi$
$350$	−5.31737e7	−0.0662916
$351$	4.32436e7	0.0533761
$352$	−2.31914e8	−0.283418
$353$	−1.03454e9	−1.25180	−0.625900	−	0.779903i	$-0.715268\pi$
$353$	−1.03454e9	−1.25180	−0.625900	+	0.779903i	$0.715268\pi$
$354$	4.95807e8	0.594020
$355$	7.68012e7	0.0911106
$356$	7.08015e8	0.831701
$357$	3.06258e8	0.356245
$358$	3.51419e8	0.404794
$359$	−1.51758e9	−1.73109	−0.865545	−	0.500831i	$-0.833028\pi$
$359$	−1.51758e9	−1.73109	−0.865545	+	0.500831i	$0.833028\pi$
$360$	−9.04669e7	−0.102195
$361$	1.42661e9	1.59599
$362$	−3.97234e8	−0.440115
$363$	8.26284e8	0.906684
$364$	4.82285e7	0.0524142
$365$	−6.15357e8	−0.662372
$366$	1.37472e8	0.146566
$367$	1.82957e9	1.93205	0.966025	−	0.258449i	$-0.0832115\pi$
$367$	1.82957e9	1.93205	0.966025	+	0.258449i	$0.0832115\pi$
$368$	−1.74356e8	−0.182377
$369$	1.11707e8	0.115742
$370$	9.38692e8	0.963423
$371$	4.74447e8	0.482369
$372$	−8.59231e7	−0.0865386
$373$	−1.16099e9	−1.15837	−0.579184	−	0.815197i	$-0.696629\pi$
$373$	−1.16099e9	−1.15837	−0.579184	+	0.815197i	$0.696629\pi$
$374$	−1.87239e9	−1.85074
$375$	6.38079e8	0.624835
$376$	9.42685e7	0.0914554
$377$	−1.63461e8	−0.157116
$378$	5.40102e7	0.0514344
$379$	1.10269e9	1.04043	0.520217	−	0.854034i	$-0.325851\pi$
$379$	1.10269e9	1.04043	0.520217	+	0.854034i	$0.325851\pi$
$380$	−7.47241e8	−0.698584
$381$	−4.42662e8	−0.410048
$382$	2.65624e8	0.243807
$383$	−1.04132e9	−0.947085	−0.473543	−	0.880771i	$-0.657025\pi$
$383$	−1.04132e9	−0.947085	−0.473543	+	0.880771i	$0.657025\pi$
$384$	5.66231e7	0.0510310
$385$	5.88387e8	0.525473
$386$	−2.26679e7	−0.0200611
$387$	9.27460e7	0.0813405
$388$	6.22507e8	0.541045
$389$	−1.45845e9	−1.25623	−0.628114	−	0.778121i	$-0.716173\pi$
$389$	−1.45845e9	−1.25623	−0.628114	+	0.778121i	$0.716173\pi$
$390$	−1.15021e8	−0.0981861
$391$	−1.40769e9	−1.19094
$392$	6.02363e7	0.0505076
$393$	6.95541e8	0.578028
$394$	9.23399e8	0.760594
$395$	1.39236e9	1.13674
$396$	−3.30205e8	−0.267209
$397$	1.74421e8	0.139904	0.0699521	−	0.997550i	$-0.477715\pi$
$397$	1.74421e8	0.139904	0.0699521	+	0.997550i	$0.477715\pi$
$398$	−5.74878e8	−0.457072
$399$	4.46115e8	0.351594
$400$	−7.93730e7	−0.0620101
$401$	−1.58341e9	−1.22628	−0.613140	−	0.789975i	$-0.710094\pi$
$401$	−1.58341e9	−1.22628	−0.613140	+	0.789975i	$0.710094\pi$
$402$	2.42303e7	0.0186024
$403$	−1.09244e8	−0.0831436
$404$	−1.33925e8	−0.101048
$405$	−1.28809e8	−0.0963507
$406$	−2.04159e8	−0.151401
$407$	3.42624e9	2.51905
$408$	4.57155e8	0.333237
$409$	−8.03071e8	−0.580393	−0.290197	−	0.956967i	$-0.593721\pi$
$409$	−8.03071e8	−0.580393	−0.290197	+	0.956967i	$0.593721\pi$
$410$	−2.97123e8	−0.212908
$411$	−5.38909e8	−0.382886
$412$	−5.13976e7	−0.0362078
$413$	7.87323e8	0.549956
$414$	−2.48253e8	−0.171947
$415$	−1.72157e9	−1.18238
$416$	7.19913e7	0.0490290
$417$	−8.23230e8	−0.555962
$418$	−2.72744e9	−1.82658
$419$	−7.25595e8	−0.481887	−0.240944	−	0.970539i	$-0.577457\pi$
$419$	−7.25595e8	−0.481887	−0.240944	+	0.970539i	$0.577457\pi$
$420$	−1.43658e8	−0.0946145
$421$	8.93482e8	0.583577	0.291789	−	0.956483i	$-0.405750\pi$
$421$	8.93482e8	0.583577	0.291789	+	0.956483i	$0.405750\pi$
$422$	−7.22922e8	−0.468272
$423$	1.34222e8	0.0862250
$424$	7.08213e8	0.451215
$425$	−6.40830e8	−0.404931
$426$	−6.84431e7	−0.0428939
$427$	2.18301e8	0.135693
$428$	1.48346e9	0.914584
$429$	−4.19827e8	−0.256726
$430$	−2.46689e8	−0.149627
$431$	1.11544e9	0.671081	0.335540	−	0.942026i	$-0.391081\pi$
$431$	1.11544e9	0.671081	0.335540	+	0.942026i	$0.391081\pi$
$432$	8.06216e7	0.0481125
$433$	−6.58571e8	−0.389848	−0.194924	−	0.980818i	$-0.562446\pi$
$433$	−6.58571e8	−0.389848	−0.194924	+	0.980818i	$0.562446\pi$
$434$	−1.36443e8	−0.0801191
$435$	4.86902e8	0.283615
$436$	−6.03348e8	−0.348630
$437$	−2.05053e9	−1.17539
$438$	5.48389e8	0.311838
$439$	−3.61817e8	−0.204109	−0.102055	−	0.994779i	$-0.532542\pi$
$439$	−3.61817e8	−0.204109	−0.102055	+	0.994779i	$0.532542\pi$
$440$	8.78292e8	0.491535
$441$	8.57661e7	0.0476190
$442$	5.81233e8	0.320164
$443$	−2.09511e9	−1.14497	−0.572486	−	0.819915i	$-0.694021\pi$
$443$	−2.09511e9	−1.14497	−0.572486	+	0.819915i	$0.694021\pi$
$444$	−8.36536e8	−0.453570
$445$	−2.68136e9	−1.44243
$446$	−4.13998e8	−0.220966
$447$	1.13882e9	0.603083
$448$	8.99154e7	0.0472456
$449$	−5.22629e8	−0.272478	−0.136239	−	0.990676i	$-0.543501\pi$
$449$	−5.22629e8	−0.272478	−0.136239	+	0.990676i	$0.543501\pi$
$450$	−1.13013e8	−0.0584637
$451$	−1.08450e9	−0.556689
$452$	1.57498e9	0.802212
$453$	4.76670e8	0.240921
$454$	9.22175e8	0.462506
$455$	−1.82649e8	−0.0909026
$456$	6.65921e8	0.328886
$457$	3.33642e9	1.63521	0.817605	−	0.575780i	$-0.195301\pi$
$457$	3.33642e9	1.63521	0.817605	+	0.575780i	$0.195301\pi$
$458$	1.88080e8	0.0914775
$459$	6.50910e8	0.314179
$460$	6.60313e8	0.316299
$461$	−3.53167e9	−1.67891	−0.839453	−	0.543432i	$-0.817125\pi$
$461$	−3.53167e9	−1.67891	−0.839453	+	0.543432i	$0.817125\pi$
$462$	−5.24354e8	−0.247387
$463$	−9.80570e8	−0.459140	−0.229570	−	0.973292i	$-0.573732\pi$
$463$	−9.80570e8	−0.459140	−0.229570	+	0.973292i	$0.573732\pi$
$464$	−3.04751e8	−0.141622
$465$	3.25404e8	0.150085
$466$	−1.78870e9	−0.818817
$467$	4.18376e9	1.90090	0.950448	−	0.310885i	$-0.100625\pi$
$467$	4.18376e9	1.90090	0.950448	+	0.310885i	$0.100625\pi$
$468$	1.02503e8	0.0462250
$469$	3.84769e7	0.0172225
$470$	−3.57009e8	−0.158612
$471$	1.24619e9	0.549556
$472$	1.17525e9	0.514436
$473$	−9.00419e8	−0.391229
$474$	−1.24083e9	−0.535165
$475$	−9.33472e8	−0.399644
$476$	7.25946e8	0.308517
$477$	1.00837e9	0.425409
$478$	2.48022e9	1.03871
$479$	3.36790e9	1.40018	0.700092	−	0.714053i	$-0.253142\pi$
$479$	3.36790e9	1.40018	0.700092	+	0.714053i	$0.253142\pi$
$480$	−2.14440e8	−0.0885037
$481$	−1.06358e9	−0.435776
$482$	2.08917e9	0.849784
$483$	−3.94217e8	−0.159192
$484$	1.95860e9	0.785212
$485$	−2.35753e9	−0.938340
$486$	1.14791e8	0.0453609
$487$	2.05183e9	0.804989	0.402494	−	0.915423i	$-0.368143\pi$
$487$	2.05183e9	0.804989	0.402494	+	0.915423i	$0.368143\pi$
$488$	3.25861e8	0.126929
$489$	−6.18718e8	−0.239283
$490$	−2.28124e8	−0.0875960
$491$	1.69829e9	0.647479	0.323740	−	0.946146i	$-0.395060\pi$
$491$	1.69829e9	0.647479	0.323740	+	0.946146i	$0.395060\pi$
$492$	2.64788e8	0.100235
$493$	−2.46045e9	−0.924807
$494$	8.46660e8	0.315984
$495$	1.25054e9	0.463424
$496$	−2.03670e8	−0.0749446
$497$	−1.08685e8	−0.0397121
$498$	1.53422e9	0.556653
$499$	−3.75901e9	−1.35432	−0.677160	−	0.735835i	$-0.736790\pi$
$499$	−3.75901e9	−1.35432	−0.677160	+	0.735835i	$0.736790\pi$
$500$	1.51248e9	0.541123
$501$	−8.19053e8	−0.290991
$502$	7.62299e8	0.268944
$503$	4.06158e8	0.142301	0.0711504	−	0.997466i	$-0.477333\pi$
$503$	4.06158e8	0.142301	0.0711504	+	0.997466i	$0.477333\pi$
$504$	1.28024e8	0.0445435
$505$	5.07193e8	0.175248
$506$	2.41015e9	0.827023
$507$	1.30324e8	0.0444116
$508$	−1.04927e9	−0.355112
$509$	3.59143e9	1.20713	0.603566	−	0.797313i	$-0.293746\pi$
$509$	3.59143e9	1.20713	0.603566	+	0.797313i	$0.293746\pi$
$510$	−1.73131e9	−0.577937
$511$	8.70821e8	0.288706
$512$	1.34218e8	0.0441942
$513$	9.48157e8	0.310077
$514$	4.07198e9	1.32262
$515$	1.94650e8	0.0627957
$516$	2.19842e8	0.0704430
$517$	−1.30309e9	−0.414721
$518$	−1.32839e9	−0.419924
$519$	−1.31812e9	−0.413874
$520$	−2.72642e8	−0.0850316
$521$	−6.70225e8	−0.207629	−0.103815	−	0.994597i	$-0.533105\pi$
$521$	−6.70225e8	−0.207629	−0.103815	+	0.994597i	$0.533105\pi$
$522$	−4.33913e8	−0.133523
$523$	−4.62786e9	−1.41457	−0.707284	−	0.706929i	$-0.750080\pi$
$523$	−4.62786e9	−1.41457	−0.707284	+	0.706929i	$0.750080\pi$
$524$	1.64869e9	0.500587
$525$	−1.79461e8	−0.0541269
$526$	2.63837e9	0.790470
$527$	−1.64436e9	−0.489394
$528$	−7.82709e8	−0.231410
$529$	−1.59284e9	−0.467818
$530$	−2.68211e9	−0.782547
$531$	1.67335e9	0.485015
$532$	1.05746e9	0.304489
$533$	3.36654e8	0.0963028
$534$	2.38955e9	0.679081
$535$	−5.61810e9	−1.58617
$536$	5.74349e7	0.0161101
$537$	1.18604e9	0.330513
$538$	2.34297e9	0.648677
$539$	−8.32655e8	−0.229036
$540$	−3.05326e8	−0.0834421
$541$	3.89961e9	1.05884	0.529421	−	0.848359i	$-0.322409\pi$
$541$	3.89961e9	1.05884	0.529421	+	0.848359i	$0.322409\pi$
$542$	1.70302e9	0.459433
$543$	−1.34066e9	−0.359353
$544$	1.08363e9	0.288592
$545$	2.28497e9	0.604633
$546$	1.62771e8	0.0427960
$547$	1.02156e8	0.0266876	0.0133438	−	0.999911i	$-0.495752\pi$
$547$	1.02156e8	0.0266876	0.0133438	+	0.999911i	$0.495752\pi$
$548$	−1.27741e9	−0.331589
$549$	4.63970e8	0.119670
$550$	1.09718e9	0.281197
$551$	−3.58405e9	−0.912732
$552$	−5.88452e8	−0.148910
$553$	−1.97039e9	−0.495466
$554$	−1.89277e9	−0.472948
$555$	3.16808e9	0.786631
$556$	−1.95136e9	−0.481477
$557$	1.52820e9	0.374704	0.187352	−	0.982293i	$-0.440010\pi$
$557$	1.52820e9	0.374704	0.187352	+	0.982293i	$0.440010\pi$
$558$	−2.89990e8	−0.0706584
$559$	2.79510e8	0.0676794
$560$	−3.40523e8	−0.0819385
$561$	−6.31932e9	−1.51113
$562$	−6.94378e7	−0.0165013
$563$	7.08915e9	1.67423	0.837114	−	0.547028i	$-0.184241\pi$
$563$	7.08915e9	1.67423	0.837114	+	0.547028i	$0.184241\pi$
$564$	3.18156e8	0.0746730
$565$	−5.96467e9	−1.39129
$566$	−4.08370e9	−0.946665
$567$	1.82284e8	0.0419961
$568$	−1.62235e8	−0.0371472
$569$	−8.65374e9	−1.96930	−0.984648	−	0.174550i	$-0.944153\pi$
$569$	−8.65374e9	−1.96930	−0.984648	+	0.174550i	$0.944153\pi$
$570$	−2.52194e9	−0.570391
$571$	−2.05815e9	−0.462648	−0.231324	−	0.972877i	$-0.574306\pi$
$571$	−2.05815e9	−0.462648	−0.231324	+	0.972877i	$0.574306\pi$
$572$	−9.95146e8	−0.222331
$573$	8.96482e8	0.199068
$574$	4.20473e8	0.0927997
$575$	8.24879e8	0.180948
$576$	1.91103e8	0.0416667
$577$	5.24193e9	1.13599	0.567997	−	0.823031i	$-0.307718\pi$
$577$	5.24193e9	1.13599	0.567997	+	0.823031i	$0.307718\pi$
$578$	5.46612e9	1.17742
$579$	−7.65041e7	−0.0163798
$580$	1.15414e9	0.245618
$581$	2.43628e9	0.515361
$582$	2.10096e9	0.441761
$583$	−9.78972e9	−2.04612
$584$	1.29988e9	0.270060
$585$	−3.88195e8	−0.0801686
$586$	−5.11676e9	−1.05040
$587$	−4.02819e9	−0.822010	−0.411005	−	0.911633i	$-0.634822\pi$
$587$	−4.02819e9	−0.822010	−0.411005	+	0.911633i	$0.634822\pi$
$588$	2.03297e8	0.0412393
$589$	−2.39527e9	−0.483005
$590$	−4.45083e9	−0.892193
$591$	3.11647e9	0.621022
$592$	−1.98290e9	−0.392803
$593$	5.73356e9	1.12910	0.564550	−	0.825399i	$-0.309050\pi$
$593$	5.73356e9	1.12910	0.564550	+	0.825399i	$0.309050\pi$
$594$	−1.11444e9	−0.218175
$595$	−2.74926e9	−0.535065
$596$	2.69941e9	0.522285
$597$	−1.94021e9	−0.373198
$598$	−7.48165e8	−0.143068
$599$	−9.89135e9	−1.88045	−0.940225	−	0.340554i	$-0.889385\pi$
$599$	−9.89135e9	−1.88045	−0.940225	+	0.340554i	$0.889385\pi$
$600$	−2.67884e8	−0.0506311
$601$	2.56463e8	0.0481908	0.0240954	−	0.999710i	$-0.492329\pi$
$601$	2.56463e8	0.0481908	0.0240954	+	0.999710i	$0.492329\pi$
$602$	3.49102e8	0.0652175
$603$	8.17774e7	0.0151888
$604$	1.12988e9	0.208643
$605$	−7.41750e9	−1.36180
$606$	−4.51996e8	−0.0825051
$607$	−5.60610e8	−0.101742	−0.0508710	−	0.998705i	$-0.516200\pi$
$607$	−5.60610e8	−0.101742	−0.0508710	+	0.998705i	$0.516200\pi$
$608$	1.57848e9	0.284824
$609$	−6.89038e8	−0.123618
$610$	−1.23408e9	−0.220135
$611$	4.04508e8	0.0717435
$612$	1.54290e9	0.272087
$613$	−7.34618e8	−0.128810	−0.0644050	−	0.997924i	$-0.520515\pi$
$613$	−7.34618e8	−0.128810	−0.0644050	+	0.997924i	$0.520515\pi$
$614$	2.78356e9	0.485302
$615$	−1.00279e9	−0.173839
$616$	−1.24291e9	−0.214244
$617$	−6.79850e9	−1.16524	−0.582619	−	0.812745i	$-0.697972\pi$
$617$	−6.79850e9	−1.16524	−0.582619	+	0.812745i	$0.697972\pi$
$618$	−1.73467e8	−0.0295636
$619$	3.08626e9	0.523016	0.261508	−	0.965201i	$-0.415780\pi$
$619$	3.08626e9	0.523016	0.261508	+	0.965201i	$0.415780\pi$
$620$	7.71327e8	0.129977
$621$	−8.37855e8	−0.140394
$622$	−4.74278e9	−0.790254
$623$	3.79452e9	0.628707
$624$	2.42971e8	0.0400320
$625$	−4.21408e9	−0.690435
$626$	−4.08347e9	−0.665302
$627$	−9.20512e9	−1.49140
$628$	2.95394e9	0.475930
$629$	−1.60092e10	−2.56504
$630$	−4.84846e8	−0.0772524
$631$	8.08977e9	1.28184	0.640919	−	0.767608i	$-0.278553\pi$
$631$	8.08977e9	1.28184	0.640919	+	0.767608i	$0.278553\pi$
$632$	−2.94122e9	−0.463466
$633$	−2.43986e9	−0.382342
$634$	−3.54223e9	−0.552032
$635$	3.97375e9	0.615876
$636$	2.39022e9	0.368415
$637$	2.58475e8	0.0396214
$638$	4.21262e9	0.642214
$639$	−2.30995e8	−0.0350228
$640$	−5.08302e8	−0.0766465
$641$	8.34576e9	1.25159	0.625796	−	0.779986i	$-0.284774\pi$
$641$	8.34576e9	1.25159	0.625796	+	0.779986i	$0.284774\pi$
$642$	5.00669e9	0.746755
$643$	−2.43503e8	−0.0361215	−0.0180608	−	0.999837i	$-0.505749\pi$
$643$	−2.43503e8	−0.0361215	−0.0180608	+	0.999837i	$0.505749\pi$
$644$	−9.34441e8	−0.137864
$645$	−8.32576e8	−0.122170
$646$	1.27441e10	1.85992
$647$	−2.96644e9	−0.430597	−0.215298	−	0.976548i	$-0.569072\pi$
$647$	−2.96644e9	−0.430597	−0.215298	+	0.976548i	$0.569072\pi$
$648$	2.72098e8	0.0392837
$649$	−1.62456e10	−2.33281
$650$	−3.40591e8	−0.0486447
$651$	−4.60494e8	−0.0654170
$652$	−1.46659e9	−0.207225
$653$	1.37421e10	1.93133	0.965667	−	0.259783i	$-0.0836509\pi$
$653$	1.37421e10	1.93133	0.965667	+	0.259783i	$0.0836509\pi$
$654$	−2.03630e9	−0.284655
$655$	−6.24383e9	−0.868174
$656$	6.27645e8	0.0868062
$657$	1.85081e9	0.254615
$658$	5.05220e8	0.0691338
$659$	−8.73865e9	−1.18945	−0.594724	−	0.803930i	$-0.702739\pi$
$659$	−8.73865e9	−1.18945	−0.594724	+	0.803930i	$0.702739\pi$
$660$	2.96424e9	0.401337
$661$	−7.98037e9	−1.07478	−0.537388	−	0.843335i	$-0.680589\pi$
$661$	−7.98037e9	−1.07478	−0.537388	+	0.843335i	$0.680589\pi$
$662$	6.39697e9	0.856982
$663$	1.96166e9	0.261413
$664$	3.63667e9	0.482076
$665$	−4.00475e9	−0.528080
$666$	−2.82331e9	−0.370338
$667$	3.16711e9	0.413259
$668$	−1.94146e9	−0.252006
$669$	−1.39724e9	−0.180418
$670$	−2.17514e8	−0.0279400
$671$	−4.50442e9	−0.575585
$672$	3.03464e8	0.0385758
$673$	−6.43539e9	−0.813809	−0.406904	−	0.913471i	$-0.633392\pi$
$673$	−6.43539e9	−0.813809	−0.406904	+	0.913471i	$0.633392\pi$
$674$	2.87176e8	0.0361276
$675$	−3.81420e8	−0.0477354
$676$	3.08916e8	0.0384615
$677$	8.18601e9	1.01394	0.506969	−	0.861964i	$-0.330766\pi$
$677$	8.18601e9	1.01394	0.506969	+	0.861964i	$0.330766\pi$
$678$	5.31554e9	0.655004
$679$	3.33625e9	0.408991
$680$	−4.10386e9	−0.500508
$681$	3.11234e9	0.377635
$682$	2.81535e9	0.339850
$683$	−9.76511e9	−1.17275	−0.586374	−	0.810040i	$-0.699445\pi$
$683$	−9.76511e9	−1.17275	−0.586374	+	0.810040i	$0.699445\pi$
$684$	2.24748e9	0.268534
$685$	4.83775e9	0.575078
$686$	3.22829e8	0.0381802
$687$	6.34771e8	0.0746910
$688$	5.21108e8	0.0610054
$689$	3.03895e9	0.353962
$690$	2.22856e9	0.258257
$691$	−5.27212e9	−0.607872	−0.303936	−	0.952693i	$-0.598301\pi$
$691$	−5.27212e9	−0.607872	−0.303936	+	0.952693i	$0.598301\pi$
$692$	−3.12442e9	−0.358425
$693$	−1.76969e9	−0.201991
$694$	−8.37709e9	−0.951339
$695$	7.39009e9	0.835032
$696$	−1.02853e9	−0.115634
$697$	5.06739e9	0.566851
$698$	6.19584e9	0.689614
$699$	−6.03686e9	−0.668561
$700$	−4.25389e8	−0.0468752
$701$	−1.32430e10	−1.45203	−0.726014	−	0.687680i	$-0.758629\pi$
$701$	−1.32430e10	−1.45203	−0.726014	+	0.687680i	$0.758629\pi$
$702$	3.45948e8	0.0377426
$703$	−2.33200e10	−2.53155
$704$	−1.85531e9	−0.200407
$705$	−1.20490e9	−0.129506
$706$	−8.27630e9	−0.885156
$707$	−7.17754e8	−0.0763849
$708$	3.96645e9	0.420036
$709$	6.88653e9	0.725669	0.362835	−	0.931854i	$-0.381809\pi$
$709$	6.88653e9	0.725669	0.362835	+	0.931854i	$0.381809\pi$
$710$	6.14410e8	0.0644249
$711$	−4.18779e9	−0.436960
$712$	5.66412e9	0.588102
$713$	2.11662e9	0.218691
$714$	2.45007e9	0.251903
$715$	3.76876e9	0.385592
$716$	2.81135e9	0.286233
$717$	8.37075e9	0.848100
$718$	−1.21406e10	−1.22407
$719$	−1.68627e10	−1.69190	−0.845950	−	0.533262i	$-0.820966\pi$
$719$	−1.68627e10	−1.69190	−0.845950	+	0.533262i	$0.820966\pi$
$720$	−7.23735e8	−0.0722630
$721$	−2.75459e8	−0.0273705
$722$	1.14129e10	1.12853
$723$	7.05094e9	0.693846
$724$	−3.17787e9	−0.311208
$725$	1.44178e9	0.140512
$726$	6.61027e9	0.641123
$727$	−2.65933e9	−0.256685	−0.128343	−	0.991730i	$-0.540966\pi$
$727$	−2.65933e9	−0.256685	−0.128343	+	0.991730i	$0.540966\pi$
$728$	3.85828e8	0.0370625
$729$	3.87420e8	0.0370370
$730$	−4.92285e9	−0.468368
$731$	4.20724e9	0.398370
$732$	1.09978e9	0.103637
$733$	−5.67699e9	−0.532420	−0.266210	−	0.963915i	$-0.585771\pi$
$733$	−5.67699e9	−0.532420	−0.266210	+	0.963915i	$0.585771\pi$
$734$	1.46366e10	1.36617
$735$	−7.69918e8	−0.0715218
$736$	−1.39485e9	−0.128960
$737$	−7.93931e8	−0.0730544
$738$	8.93658e8	0.0818416
$739$	7.61406e8	0.0694002	0.0347001	−	0.999398i	$-0.488952\pi$
$739$	7.61406e8	0.0694002	0.0347001	+	0.999398i	$0.488952\pi$
$740$	7.50953e9	0.681243
$741$	2.85748e9	0.257999
$742$	3.79558e9	0.341086
$743$	9.27534e9	0.829601	0.414800	−	0.909912i	$-0.363851\pi$
$743$	9.27534e9	0.829601	0.414800	+	0.909912i	$0.363851\pi$
$744$	−6.87385e8	−0.0611920
$745$	−1.02231e10	−0.905805
$746$	−9.28789e9	−0.819089
$747$	5.17799e9	0.454505
$748$	−1.49791e10	−1.30867
$749$	7.95044e9	0.691361
$750$	5.10464e9	0.441825
$751$	−1.90927e10	−1.64486	−0.822428	−	0.568870i	$-0.807381\pi$
$751$	−1.90927e10	−1.64486	−0.822428	+	0.568870i	$0.807381\pi$
$752$	7.54148e8	0.0646687
$753$	2.57276e9	0.219592
$754$	−1.30769e9	−0.111098
$755$	−4.27904e9	−0.361853
$756$	4.32081e8	0.0363696
$757$	−1.38252e10	−1.15834	−0.579168	−	0.815208i	$-0.696622\pi$
$757$	−1.38252e10	−1.15834	−0.579168	+	0.815208i	$0.696622\pi$
$758$	8.82148e9	0.735698
$759$	8.13426e9	0.675261
$760$	−5.97793e9	−0.493973
$761$	−1.50138e10	−1.23493	−0.617467	−	0.786597i	$-0.711841\pi$
$761$	−1.50138e10	−1.23493	−0.617467	+	0.786597i	$0.711841\pi$
$762$	−3.54130e9	−0.289948
$763$	−3.23357e9	−0.263540
$764$	2.12499e9	0.172398
$765$	−5.84319e9	−0.471883
$766$	−8.33057e9	−0.669690
$767$	5.04300e9	0.403557
$768$	4.52985e8	0.0360844
$769$	−1.21333e10	−0.962137	−0.481068	−	0.876683i	$-0.659751\pi$
$769$	−1.21333e10	−0.962137	−0.481068	+	0.876683i	$0.659751\pi$
$770$	4.70710e9	0.371566
$771$	1.37429e10	1.07991
$772$	−1.81343e8	−0.0141854
$773$	9.58449e9	0.746347	0.373173	−	0.927762i	$-0.378270\pi$
$773$	9.58449e9	0.746347	0.373173	+	0.927762i	$0.378270\pi$
$774$	7.41968e8	0.0575164
$775$	9.63560e8	0.0743572
$776$	4.98006e9	0.382576
$777$	−4.48331e9	−0.342866
$778$	−1.16676e10	−0.888287
$779$	7.38147e9	0.559451
$780$	−9.20166e8	−0.0694280
$781$	2.24260e9	0.168451
$782$	−1.12615e10	−0.842120
$783$	−1.46446e9	−0.109021
$784$	4.81890e8	0.0357143
$785$	−1.11870e10	−0.825411
$786$	5.56433e9	0.408727
$787$	1.81519e10	1.32743	0.663714	−	0.747987i	$-0.268979\pi$
$787$	1.81519e10	1.32743	0.663714	+	0.747987i	$0.268979\pi$
$788$	7.38719e9	0.537821
$789$	8.90449e9	0.645416
$790$	1.11388e10	0.803795
$791$	8.44089e9	0.606416
$792$	−2.64164e9	−0.188945
$793$	1.39827e9	0.0995717
$794$	1.39536e9	0.0989273
$795$	−9.05211e9	−0.638947
$796$	−4.59902e9	−0.323199
$797$	2.63726e10	1.84523	0.922613	−	0.385727i	$-0.126049\pi$
$797$	2.63726e10	1.84523	0.922613	+	0.385727i	$0.126049\pi$
$798$	3.56892e9	0.248615
$799$	6.08873e9	0.422292
$800$	−6.34984e8	−0.0438478
$801$	8.06473e9	0.554468
$802$	−1.26673e10	−0.867110
$803$	−1.79685e10	−1.22464
$804$	1.93843e8	0.0131539
$805$	3.53886e9	0.239099
$806$	−8.73949e8	−0.0587914
$807$	7.90752e9	0.529643
$808$	−1.07140e9	−0.0714515
$809$	−1.13453e10	−0.753350	−0.376675	−	0.926346i	$-0.622933\pi$
$809$	−1.13453e10	−0.753350	−0.376675	+	0.926346i	$0.622933\pi$
$810$	−1.03047e9	−0.0681302
$811$	6.26611e9	0.412501	0.206251	−	0.978499i	$-0.433874\pi$
$811$	6.26611e9	0.412501	0.206251	+	0.978499i	$0.433874\pi$
$812$	−1.63328e9	−0.107057
$813$	5.74769e9	0.375125
$814$	2.74099e10	1.78124
$815$	5.55419e9	0.359393
$816$	3.65724e9	0.235634
$817$	6.12853e9	0.393169
$818$	−6.42457e9	−0.410400
$819$	5.49353e8	0.0349428
$820$	−2.37698e9	−0.150549
$821$	3.39432e8	0.0214068	0.0107034	−	0.999943i	$-0.496593\pi$
$821$	3.39432e8	0.0214068	0.0107034	+	0.999943i	$0.496593\pi$
$822$	−4.31127e9	−0.270741
$823$	−1.47755e10	−0.923939	−0.461969	−	0.886896i	$-0.652857\pi$
$823$	−1.47755e10	−0.923939	−0.461969	+	0.886896i	$0.652857\pi$
$824$	−4.11181e8	−0.0256028
$825$	3.70299e9	0.229596
$826$	6.29858e9	0.388877
$827$	1.71938e10	1.05707	0.528533	−	0.848913i	$-0.322742\pi$
$827$	1.71938e10	1.05707	0.528533	+	0.848913i	$0.322742\pi$
$828$	−1.98603e9	−0.121585
$829$	−8.16491e9	−0.497749	−0.248874	−	0.968536i	$-0.580061\pi$
$829$	−8.16491e9	−0.497749	−0.248874	+	0.968536i	$0.580061\pi$
$830$	−1.37726e10	−0.836070
$831$	−6.38810e9	−0.386161
$832$	5.75930e8	0.0346688
$833$	3.89061e9	0.233217
$834$	−6.58584e9	−0.393125
$835$	7.35260e9	0.437057
$836$	−2.18195e10	−1.29159
$837$	−9.78718e8	−0.0576924
$838$	−5.80476e9	−0.340746
$839$	−2.01798e10	−1.17964	−0.589822	−	0.807533i	$-0.700802\pi$
$839$	−2.01798e10	−1.17964	−0.589822	+	0.807533i	$0.700802\pi$
$840$	−1.14926e9	−0.0669025
$841$	−1.17142e10	−0.679089
$842$	7.14786e9	0.412652
$843$	−2.34352e8	−0.0134733
$844$	−5.78338e9	−0.331118
$845$	−1.16991e9	−0.0667043
$846$	1.07378e9	0.0609702
$847$	1.04969e10	0.593564
$848$	5.66570e9	0.319057
$849$	−1.37825e10	−0.772949
$850$	−5.12664e9	−0.286330
$851$	2.06072e10	1.14621
$852$	−5.47545e8	−0.0303306
$853$	2.70923e9	0.149460	0.0747298	−	0.997204i	$-0.476191\pi$
$853$	2.70923e9	0.149460	0.0747298	+	0.997204i	$0.476191\pi$
$854$	1.74641e9	0.0959497
$855$	−8.51155e9	−0.465722
$856$	1.18677e10	0.646709
$857$	1.15079e10	0.624542	0.312271	−	0.949993i	$-0.398910\pi$
$857$	1.15079e10	0.624542	0.312271	+	0.949993i	$0.398910\pi$
$858$	−3.35862e9	−0.181533
$859$	1.40756e9	0.0757687	0.0378844	−	0.999282i	$-0.487938\pi$
$859$	1.40756e9	0.0757687	0.0378844	+	0.999282i	$0.487938\pi$
$860$	−1.97351e9	−0.105802
$861$	1.41910e9	0.0757706
$862$	8.92350e9	0.474526
$863$	8.94273e9	0.473622	0.236811	−	0.971556i	$-0.423898\pi$
$863$	8.94273e9	0.473622	0.236811	+	0.971556i	$0.423898\pi$
$864$	6.44973e8	0.0340207
$865$	1.18327e10	0.621621
$866$	−5.26857e9	−0.275664
$867$	1.84482e10	0.961360
$868$	−1.09154e9	−0.0566528
$869$	4.06569e10	2.10167
$870$	3.89521e9	0.200546
$871$	2.46454e8	0.0126378
$872$	−4.82678e9	−0.246519
$873$	7.09074e9	0.360697
$874$	−1.64042e10	−0.831125
$875$	8.10597e9	0.409050
$876$	4.38711e9	0.220503
$877$	−1.47232e10	−0.737061	−0.368530	−	0.929616i	$-0.620139\pi$
$877$	−1.47232e10	−0.737061	−0.368530	+	0.929616i	$0.620139\pi$
$878$	−2.89453e9	−0.144327
$879$	−1.72691e10	−0.857645
$880$	7.02634e9	0.347568
$881$	1.64152e10	0.808779	0.404389	−	0.914587i	$-0.367484\pi$
$881$	1.64152e10	0.808779	0.404389	+	0.914587i	$0.367484\pi$
$882$	6.86129e8	0.0336718
$883$	1.35335e10	0.661526	0.330763	−	0.943714i	$-0.392694\pi$
$883$	1.35335e10	0.661526	0.330763	+	0.943714i	$0.392694\pi$
$884$	4.64986e9	0.226390
$885$	−1.50216e10	−0.728473
$886$	−1.67609e10	−0.809617
$887$	−2.49921e10	−1.20246	−0.601230	−	0.799076i	$-0.705322\pi$
$887$	−2.49921e10	−1.20246	−0.601230	+	0.799076i	$0.705322\pi$
$888$	−6.69228e9	−0.320722
$889$	−5.62345e9	−0.268440
$890$	−2.14509e10	−1.01995
$891$	−3.76125e9	−0.178139
$892$	−3.31199e9	−0.156247
$893$	8.86922e9	0.416779
$894$	9.11053e9	0.426444
$895$	−1.06470e10	−0.496417
$896$	7.19323e8	0.0334077
$897$	−2.52506e9	−0.116815
$898$	−4.18103e9	−0.192671
$899$	3.69957e9	0.169821
$900$	−9.04108e8	−0.0413401
$901$	4.57429e10	2.08347
$902$	−8.67602e9	−0.393639
$903$	1.17822e9	0.0532499
$904$	1.25998e10	0.567250
$905$	1.20351e10	0.539733
$906$	3.81336e9	0.170357
$907$	−2.48804e10	−1.10721	−0.553607	−	0.832778i	$-0.686749\pi$
$907$	−2.48804e10	−1.10721	−0.553607	+	0.832778i	$0.686749\pi$
$908$	7.37740e9	0.327041
$909$	−1.52549e9	−0.0673651
$910$	−1.46119e9	−0.0642779
$911$	1.79079e10	0.784749	0.392374	−	0.919806i	$-0.371654\pi$
$911$	1.79079e10	0.784749	0.392374	+	0.919806i	$0.371654\pi$
$912$	5.32737e9	0.232558
$913$	−5.02702e10	−2.18606
$914$	2.66913e10	1.15627
$915$	−4.16503e9	−0.179740
$916$	1.50464e9	0.0646843
$917$	8.83595e9	0.378408
$918$	5.20728e9	0.222158
$919$	−1.51764e10	−0.645005	−0.322503	−	0.946569i	$-0.604524\pi$
$919$	−1.51764e10	−0.645005	−0.322503	+	0.946569i	$0.604524\pi$
$920$	5.28250e9	0.223657
$921$	9.39453e9	0.396247
$922$	−2.82533e10	−1.18717
$923$	−6.96155e8	−0.0291407
$924$	−4.19483e9	−0.174929
$925$	9.38109e9	0.389724
$926$	−7.84456e9	−0.324661
$927$	−5.85450e8	−0.0241385
$928$	−2.43801e9	−0.100142
$929$	1.85426e10	0.758780	0.379390	−	0.925237i	$-0.376134\pi$
$929$	1.85426e10	0.758780	0.379390	+	0.925237i	$0.376134\pi$
$930$	2.60323e9	0.106126
$931$	5.66731e9	0.230172
$932$	−1.43096e10	−0.578991
$933$	−1.60069e10	−0.645239
$934$	3.34701e10	1.34414
$935$	5.67282e10	2.26965
$936$	8.20026e8	0.0326860
$937$	−9.08863e9	−0.360919	−0.180460	−	0.983582i	$-0.557758\pi$
$937$	−9.08863e9	−0.360919	−0.180460	+	0.983582i	$0.557758\pi$
$938$	3.07815e8	0.0121781
$939$	−1.37817e10	−0.543217
$940$	−2.85607e9	−0.112156
$941$	3.95765e10	1.54837	0.774183	−	0.632962i	$-0.218161\pi$
$941$	3.95765e10	1.54837	0.774183	+	0.632962i	$0.218161\pi$
$942$	9.96955e9	0.388595
$943$	−6.52276e9	−0.253303
$944$	9.40197e9	0.363762
$945$	−1.63636e9	−0.0630763
$946$	−7.20335e9	−0.276640
$947$	−1.99816e10	−0.764549	−0.382275	−	0.924049i	$-0.624859\pi$
$947$	−1.99816e10	−0.764549	−0.382275	+	0.924049i	$0.624859\pi$
$948$	−9.92662e9	−0.378419
$949$	5.57782e9	0.211852
$950$	−7.46778e9	−0.282591
$951$	−1.19550e10	−0.450732
$952$	5.80756e9	0.218155
$953$	2.69869e10	1.01002	0.505008	−	0.863114i	$-0.331489\pi$
$953$	2.69869e10	1.01002	0.505008	+	0.863114i	$0.331489\pi$
$954$	8.06698e9	0.300810
$955$	−8.04767e9	−0.298991
$956$	1.98418e10	0.734476
$957$	1.42176e10	0.524365
$958$	2.69432e10	0.990080
$959$	−6.84614e9	−0.250657
$960$	−1.71552e9	−0.0625816
$961$	−2.50401e10	−0.910133
$962$	−8.50865e9	−0.308140
$963$	1.68976e10	0.609723
$964$	1.67133e10	0.600888
$965$	6.86773e8	0.0246018
$966$	−3.15374e9	−0.112566
$967$	−1.66328e7	−0.000591526 0	−0.000295763	−	1.00000i	$-0.500094\pi$
$967$	−1.66328e7	−0.000591526 0	−0.000295763		1.00000i	$0.500094\pi$
$968$	1.56688e10	0.555229
$969$	4.30113e10	1.51862
$970$	−1.88602e10	−0.663507
$971$	−5.94270e9	−0.208313	−0.104157	−	0.994561i	$-0.533214\pi$
$971$	−5.94270e9	−0.208313	−0.104157	+	0.994561i	$0.533214\pi$
$972$	9.18330e8	0.0320750
$973$	−1.04581e10	−0.363963
$974$	1.64146e10	0.569213
$975$	−1.14949e9	−0.0397183
$976$	2.60688e9	0.0897527
$977$	−4.67554e10	−1.60399	−0.801994	−	0.597333i	$-0.796227\pi$
$977$	−4.67554e10	−1.60399	−0.801994	+	0.597333i	$0.796227\pi$
$978$	−4.94974e9	−0.169198
$979$	−7.82959e10	−2.66686
$980$	−1.82499e9	−0.0619397
$981$	−6.87251e9	−0.232420
$982$	1.35863e10	0.457837
$983$	1.42959e10	0.480035	0.240018	−	0.970769i	$-0.422847\pi$
$983$	1.42959e10	0.480035	0.240018	+	0.970769i	$0.422847\pi$
$984$	2.11830e9	0.0708769
$985$	−2.79764e10	−0.932749
$986$	−1.96836e10	−0.653937
$987$	1.70512e9	0.0564475
$988$	6.77328e9	0.223434
$989$	−5.41559e9	−0.178016
$990$	1.00043e10	0.327690
$991$	3.05145e10	0.995974	0.497987	−	0.867185i	$-0.334073\pi$
$991$	3.05145e10	0.995974	0.497987	+	0.867185i	$0.334073\pi$
$992$	−1.62936e9	−0.0529938
$993$	2.15898e10	0.699723
$994$	−8.69481e8	−0.0280807
$995$	1.74172e10	0.560528
$996$	1.22737e10	0.393613
$997$	3.80642e10	1.21642	0.608211	−	0.793776i	$-0.291888\pi$
$997$	3.80642e10	1.21642	0.608211	+	0.793776i	$0.291888\pi$
$998$	−3.00721e10	−0.957649
$999$	−9.52866e9	−0.302380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.s.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.s.1.2	✓	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -242.377 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -242.377 q^{5} +216.000 q^{6} +343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1939.02 q^{10} -7077.45 q^{11} +1728.00 q^{12} +2197.00 q^{13} +2744.00 q^{14} -6544.19 q^{15} +4096.00 q^{16} +33069.7 q^{17} +5832.00 q^{18} +48171.3 q^{19} -15512.2 q^{20} +9261.00 q^{21} -56619.6 q^{22} -42567.4 q^{23} +13824.0 q^{24} -19378.2 q^{25} +17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -74402.1 q^{29} -52353.5 q^{30} -49724.0 q^{31} +32768.0 q^{32} -191091. q^{33} +264557. q^{34} -83135.5 q^{35} +46656.0 q^{36} -484106. q^{37} +385371. q^{38} +59319.0 q^{39} -124097. q^{40} +153234. q^{41} +74088.0 q^{42} +127224. q^{43} -452957. q^{44} -176693. q^{45} -340540. q^{46} +184118. q^{47} +110592. q^{48} +117649. q^{49} -155025. q^{50} +892881. q^{51} +140608. q^{52} +1.38323e6 q^{53} +157464. q^{54} +1.71541e6 q^{55} +175616. q^{56} +1.30063e6 q^{57} -595217. q^{58} +2.29540e6 q^{59} -418828. q^{60} +636447. q^{61} -397792. q^{62} +250047. q^{63} +262144. q^{64} -532503. q^{65} -1.52873e6 q^{66} +112178. q^{67} +2.11646e6 q^{68} -1.14932e6 q^{69} -665084. q^{70} -316866. q^{71} +373248. q^{72} +2.53884e6 q^{73} -3.87285e6 q^{74} -523210. q^{75} +3.08297e6 q^{76} -2.42756e6 q^{77} +474552. q^{78} -5.74457e6 q^{79} -992778. q^{80} +531441. q^{81} +1.22587e6 q^{82} +7.10286e6 q^{83} +592704. q^{84} -8.01534e6 q^{85} +1.01779e6 q^{86} -2.00886e6 q^{87} -3.62365e6 q^{88} +1.10627e7 q^{89} -1.41355e6 q^{90} +753571. q^{91} -2.72432e6 q^{92} -1.34255e6 q^{93} +1.47295e6 q^{94} -1.16756e7 q^{95} +884736. q^{96} +9.72667e6 q^{97} +941192. q^{98} -5.15946e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 56 q^{2} + 189 q^{3} + 448 q^{4} + 625 q^{5} + 1512 q^{6} + 2401 q^{7} + 3584 q^{8} + 5103 q^{9}+O(q^{10}) \) 7 * q + 56 * q^2 + 189 * q^3 + 448 * q^4 + 625 * q^5 + 1512 * q^6 + 2401 * q^7 + 3584 * q^8 + 5103 * q^9	\( 7 q + 56 q^{2} + 189 q^{3} + 448 q^{4} + 625 q^{5} + 1512 q^{6} + 2401 q^{7} + 3584 q^{8} + 5103 q^{9} + 5000 q^{10} + 4678 q^{11} + 12096 q^{12} + 15379 q^{13} + 19208 q^{14} + 16875 q^{15} + 28672 q^{16} + 38552 q^{17} + 40824 q^{18} + 60231 q^{19} + 40000 q^{20} + 64827 q^{21} + 37424 q^{22} + 82047 q^{23} + 96768 q^{24} + 136408 q^{25} + 123032 q^{26} + 137781 q^{27} + 153664 q^{28} + 87523 q^{29} + 135000 q^{30} - 3191 q^{31} + 229376 q^{32} + 126306 q^{33} + 308416 q^{34} + 214375 q^{35} + 326592 q^{36} + 360916 q^{37} + 481848 q^{38} + 415233 q^{39} + 320000 q^{40} + 814350 q^{41} + 518616 q^{42} + 840057 q^{43} + 299392 q^{44} + 455625 q^{45} + 656376 q^{46} + 472723 q^{47} + 774144 q^{48} + 823543 q^{49} + 1091264 q^{50} + 1040904 q^{51} + 984256 q^{52} + 2185687 q^{53} + 1102248 q^{54} + 298354 q^{55} + 1229312 q^{56} + 1626237 q^{57} + 700184 q^{58} + 2046232 q^{59} + 1080000 q^{60} + 2744560 q^{61} - 25528 q^{62} + 1750329 q^{63} + 1835008 q^{64} + 1373125 q^{65} + 1010448 q^{66} + 1960358 q^{67} + 2467328 q^{68} + 2215269 q^{69} + 1715000 q^{70} + 2774656 q^{71} + 2612736 q^{72} + 3696313 q^{73} + 2887328 q^{74} + 3683016 q^{75} + 3854784 q^{76} + 1604554 q^{77} + 3321864 q^{78} + 1532089 q^{79} + 2560000 q^{80} + 3720087 q^{81} + 6514800 q^{82} + 7473863 q^{83} + 4148928 q^{84} - 1656624 q^{85} + 6720456 q^{86} + 2363121 q^{87} + 2395136 q^{88} + 14077309 q^{89} + 3645000 q^{90} + 5274997 q^{91} + 5251008 q^{92} - 86157 q^{93} + 3781784 q^{94} - 1704679 q^{95} + 6193152 q^{96} + 8273673 q^{97} + 6588344 q^{98} + 3410262 q^{99}+O(q^{100}) \) 7 * q + 56 * q^2 + 189 * q^3 + 448 * q^4 + 625 * q^5 + 1512 * q^6 + 2401 * q^7 + 3584 * q^8 + 5103 * q^9 + 5000 * q^10 + 4678 * q^11 + 12096 * q^12 + 15379 * q^13 + 19208 * q^14 + 16875 * q^15 + 28672 * q^16 + 38552 * q^17 + 40824 * q^18 + 60231 * q^19 + 40000 * q^20 + 64827 * q^21 + 37424 * q^22 + 82047 * q^23 + 96768 * q^24 + 136408 * q^25 + 123032 * q^26 + 137781 * q^27 + 153664 * q^28 + 87523 * q^29 + 135000 * q^30 - 3191 * q^31 + 229376 * q^32 + 126306 * q^33 + 308416 * q^34 + 214375 * q^35 + 326592 * q^36 + 360916 * q^37 + 481848 * q^38 + 415233 * q^39 + 320000 * q^40 + 814350 * q^41 + 518616 * q^42 + 840057 * q^43 + 299392 * q^44 + 455625 * q^45 + 656376 * q^46 + 472723 * q^47 + 774144 * q^48 + 823543 * q^49 + 1091264 * q^50 + 1040904 * q^51 + 984256 * q^52 + 2185687 * q^53 + 1102248 * q^54 + 298354 * q^55 + 1229312 * q^56 + 1626237 * q^57 + 700184 * q^58 + 2046232 * q^59 + 1080000 * q^60 + 2744560 * q^61 - 25528 * q^62 + 1750329 * q^63 + 1835008 * q^64 + 1373125 * q^65 + 1010448 * q^66 + 1960358 * q^67 + 2467328 * q^68 + 2215269 * q^69 + 1715000 * q^70 + 2774656 * q^71 + 2612736 * q^72 + 3696313 * q^73 + 2887328 * q^74 + 3683016 * q^75 + 3854784 * q^76 + 1604554 * q^77 + 3321864 * q^78 + 1532089 * q^79 + 2560000 * q^80 + 3720087 * q^81 + 6514800 * q^82 + 7473863 * q^83 + 4148928 * q^84 - 1656624 * q^85 + 6720456 * q^86 + 2363121 * q^87 + 2395136 * q^88 + 14077309 * q^89 + 3645000 * q^90 + 5274997 * q^91 + 5251008 * q^92 - 86157 * q^93 + 3781784 * q^94 - 1704679 * q^95 + 6193152 * q^96 + 8273673 * q^97 + 6588344 * q^98 + 3410262 * q^99

Introduction

L-functions

Modular forms

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Database

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