LMFDB - Embedded newform 354.8.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} + \cdots - 22\!\cdots\!40 $ x^8 - x^7 - 103558x^6 + 5805883x^5 + 2559087821x^4 - 196601024266x^3 - 17055507548256x^2 + 1662873307981032x - 22179668074797840
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{7}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$137.528$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -490.703 q^{5} -216.000 q^{6} +975.190 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -490.703 q^{5} -216.000 q^{6} +975.190 q^{7} +512.000 q^{8} +729.000 q^{9} -3925.63 q^{10} -7237.82 q^{11} -1728.00 q^{12} +5317.06 q^{13} +7801.52 q^{14} +13249.0 q^{15} +4096.00 q^{16} +6977.85 q^{17} +5832.00 q^{18} +41206.4 q^{19} -31405.0 q^{20} -26330.1 q^{21} -57902.5 q^{22} -33889.7 q^{23} -13824.0 q^{24} +162665. q^{25} +42536.5 q^{26} -19683.0 q^{27} +62412.1 q^{28} +65498.6 q^{29} +105992. q^{30} +58908.3 q^{31} +32768.0 q^{32} +195421. q^{33} +55822.8 q^{34} -478529. q^{35} +46656.0 q^{36} -271425. q^{37} +329651. q^{38} -143561. q^{39} -251240. q^{40} -531831. q^{41} -210641. q^{42} +269416. q^{43} -463220. q^{44} -357723. q^{45} -271117. q^{46} +840158. q^{47} -110592. q^{48} +127452. q^{49} +1.30132e6 q^{50} -188402. q^{51} +340292. q^{52} +430795. q^{53} -157464. q^{54} +3.55162e6 q^{55} +499297. q^{56} -1.11257e6 q^{57} +523989. q^{58} -205379. q^{59} +847935. q^{60} -269547. q^{61} +471266. q^{62} +710913. q^{63} +262144. q^{64} -2.60910e6 q^{65} +1.56337e6 q^{66} +1.59548e6 q^{67} +446582. q^{68} +915021. q^{69} -3.82823e6 q^{70} -2.78830e6 q^{71} +373248. q^{72} -1.90543e6 q^{73} -2.17140e6 q^{74} -4.39194e6 q^{75} +2.63721e6 q^{76} -7.05824e6 q^{77} -1.14849e6 q^{78} -4.73641e6 q^{79} -2.00992e6 q^{80} +531441. q^{81} -4.25465e6 q^{82} -8.24608e6 q^{83} -1.68513e6 q^{84} -3.42405e6 q^{85} +2.15533e6 q^{86} -1.76846e6 q^{87} -3.70576e6 q^{88} +2.74603e6 q^{89} -2.86178e6 q^{90} +5.18514e6 q^{91} -2.16894e6 q^{92} -1.59052e6 q^{93} +6.72127e6 q^{94} -2.02201e7 q^{95} -884736. q^{96} +7.05661e6 q^{97} +1.01962e6 q^{98} -5.27637e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * 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
$5$	−490.703	−1.75559	−0.877796	−	0.479034i	$-0.840987\pi$
$5$	−490.703	−1.75559	−0.877796	+	0.479034i	$0.840987\pi$
$6$	−216.000	−0.408248
$7$	975.190	1.07460	0.537299	−	0.843392i	$-0.319445\pi$
$7$	975.190	1.07460	0.537299	+	0.843392i	$0.319445\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−3925.63	−1.24139
$11$	−7237.82	−1.63958	−0.819791	−	0.572663i	$-0.805910\pi$
$11$	−7237.82	−1.63958	−0.819791	+	0.572663i	$0.805910\pi$
$12$	−1728.00	−0.288675
$13$	5317.06	0.671228	0.335614	−	0.942000i	$-0.391056\pi$
$13$	5317.06	0.671228	0.335614	+	0.942000i	$0.391056\pi$
$14$	7801.52	0.759855
$15$	13249.0	1.01359
$16$	4096.00	0.250000
$17$	6977.85	0.344469	0.172235	−	0.985056i	$-0.444901\pi$
$17$	6977.85	0.344469	0.172235	+	0.985056i	$0.444901\pi$
$18$	5832.00	0.235702
$19$	41206.4	1.37825	0.689123	−	0.724644i	$-0.257996\pi$
$19$	41206.4	1.37825	0.689123	+	0.724644i	$0.257996\pi$
$20$	−31405.0	−0.877796
$21$	−26330.1	−0.620419
$22$	−57902.5	−1.15936
$23$	−33889.7	−0.580791	−0.290395	−	0.956907i	$-0.593787\pi$
$23$	−33889.7	−0.580791	−0.290395	+	0.956907i	$0.593787\pi$
$24$	−13824.0	−0.204124
$25$	162665.	2.08211
$26$	42536.5	0.474630
$27$	−19683.0	−0.192450
$28$	62412.1	0.537299
$29$	65498.6	0.498699	0.249350	−	0.968413i	$-0.419783\pi$
$29$	65498.6	0.498699	0.249350	+	0.968413i	$0.419783\pi$
$30$	105992.	0.716718
$31$	58908.3	0.355149	0.177574	−	0.984107i	$-0.443175\pi$
$31$	58908.3	0.355149	0.177574	+	0.984107i	$0.443175\pi$
$32$	32768.0	0.176777
$33$	195421.	0.946613
$34$	55822.8	0.243577
$35$	−478529.	−1.88656
$36$	46656.0	0.166667
$37$	−271425.	−0.880936	−0.440468	−	0.897768i	$-0.645188\pi$
$37$	−271425.	−0.880936	−0.440468	+	0.897768i	$0.645188\pi$
$38$	329651.	0.974567
$39$	−143561.	−0.387534
$40$	−251240.	−0.620696
$41$	−531831.	−1.20512	−0.602559	−	0.798074i	$-0.705852\pi$
$41$	−531831.	−1.20512	−0.602559	+	0.798074i	$0.705852\pi$
$42$	−210641.	−0.438703
$43$	269416.	0.516754	0.258377	−	0.966044i	$-0.416812\pi$
$43$	269416.	0.516754	0.258377	+	0.966044i	$0.416812\pi$
$44$	−463220.	−0.819791
$45$	−357723.	−0.585198
$46$	−271117.	−0.410681
$47$	840158.	1.18037	0.590186	−	0.807267i	$-0.299054\pi$
$47$	840158.	1.18037	0.590186	+	0.807267i	$0.299054\pi$
$48$	−110592.	−0.144338
$49$	127452.	0.154761
$50$	1.30132e6	1.47227
$51$	−188402.	−0.198879
$52$	340292.	0.335614
$53$	430795.	0.397471	0.198735	−	0.980053i	$-0.436317\pi$
$53$	430795.	0.397471	0.198735	+	0.980053i	$0.436317\pi$
$54$	−157464.	−0.136083
$55$	3.55162e6	2.87844
$56$	499297.	0.379928
$57$	−1.11257e6	−0.795731
$58$	523989.	0.352634
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	847935.	0.506796
$61$	−269547.	−0.152048	−0.0760238	−	0.997106i	$-0.524222\pi$
$61$	−269547.	−0.152048	−0.0760238	+	0.997106i	$0.524222\pi$
$62$	471266.	0.251128
$63$	710913.	0.358199
$64$	262144.	0.125000
$65$	−2.60910e6	−1.17840
$66$	1.56337e6	0.669357
$67$	1.59548e6	0.648083	0.324041	−	0.946043i	$-0.394958\pi$
$67$	1.59548e6	0.648083	0.324041	+	0.946043i	$0.394958\pi$
$68$	446582.	0.172235
$69$	915021.	0.335320
$70$	−3.82823e6	−1.33400
$71$	−2.78830e6	−0.924560	−0.462280	−	0.886734i	$-0.652968\pi$
$71$	−2.78830e6	−0.924560	−0.462280	+	0.886734i	$0.652968\pi$
$72$	373248.	0.117851
$73$	−1.90543e6	−0.573276	−0.286638	−	0.958039i	$-0.592538\pi$
$73$	−1.90543e6	−0.573276	−0.286638	+	0.958039i	$0.592538\pi$
$74$	−2.17140e6	−0.622916
$75$	−4.39194e6	−1.20210
$76$	2.63721e6	0.689123
$77$	−7.05824e6	−1.76189
$78$	−1.14849e6	−0.274028
$79$	−4.73641e6	−1.08082	−0.540412	−	0.841401i	$-0.681732\pi$
$79$	−4.73641e6	−1.08082	−0.540412	+	0.841401i	$0.681732\pi$
$80$	−2.00992e6	−0.438898
$81$	531441.	0.111111
$82$	−4.25465e6	−0.852148
$83$	−8.24608e6	−1.58298	−0.791488	−	0.611185i	$-0.790693\pi$
$83$	−8.24608e6	−1.58298	−0.791488	+	0.611185i	$0.790693\pi$
$84$	−1.68513e6	−0.310210
$85$	−3.42405e6	−0.604748
$86$	2.15533e6	0.365400
$87$	−1.76846e6	−0.287924
$88$	−3.70576e6	−0.579680
$89$	2.74603e6	0.412895	0.206447	−	0.978458i	$-0.433810\pi$
$89$	2.74603e6	0.412895	0.206447	+	0.978458i	$0.433810\pi$
$90$	−2.86178e6	−0.413797
$91$	5.18514e6	0.721300
$92$	−2.16894e6	−0.290395
$93$	−1.59052e6	−0.205045
$94$	6.72127e6	0.834649
$95$	−2.02201e7	−2.41964
$96$	−884736.	−0.102062
$97$	7.05661e6	0.785046	0.392523	−	0.919742i	$-0.371602\pi$
$97$	7.05661e6	0.785046	0.392523	+	0.919742i	$0.371602\pi$
$98$	1.01962e6	0.109432
$99$	−5.27637e6	−0.546527
$100$	1.04105e7	1.04105
$101$	−3.83873e6	−0.370734	−0.185367	−	0.982669i	$-0.559347\pi$
$101$	−3.83873e6	−0.370734	−0.185367	+	0.982669i	$0.559347\pi$
$102$	−1.50722e6	−0.140629
$103$	−4.34883e6	−0.392141	−0.196070	−	0.980590i	$-0.562818\pi$
$103$	−4.34883e6	−0.392141	−0.196070	+	0.980590i	$0.562818\pi$
$104$	2.72234e6	0.237315
$105$	1.29203e7	1.08920
$106$	3.44636e6	0.281054
$107$	−1.89763e7	−1.49751	−0.748753	−	0.662849i	$-0.769347\pi$
$107$	−1.89763e7	−1.49751	−0.748753	+	0.662849i	$0.769347\pi$
$108$	−1.25971e6	−0.0962250
$109$	−2.13320e7	−1.57775	−0.788877	−	0.614551i	$-0.789337\pi$
$109$	−2.13320e7	−1.57775	−0.788877	+	0.614551i	$0.789337\pi$
$110$	2.84130e7	2.03536
$111$	7.32849e6	0.508609
$112$	3.99438e6	0.268649
$113$	−1.59478e7	−1.03974	−0.519872	−	0.854244i	$-0.674020\pi$
$113$	−1.59478e7	−1.03974	−0.519872	+	0.854244i	$0.674020\pi$
$114$	−8.90058e6	−0.562667
$115$	1.66298e7	1.01963
$116$	4.19191e6	0.249350
$117$	3.87614e6	0.223743
$118$	−1.64303e6	−0.0920575
$119$	6.80473e6	0.370166
$120$	6.78348e6	0.358359
$121$	3.28988e7	1.68823
$122$	−2.15637e6	−0.107514
$123$	1.43594e7	0.695776
$124$	3.77013e6	0.177574
$125$	−4.14838e7	−1.89974
$126$	5.68731e6	0.253285
$127$	−2.83880e7	−1.22977	−0.614883	−	0.788619i	$-0.710797\pi$
$127$	−2.83880e7	−1.22977	−0.614883	+	0.788619i	$0.710797\pi$
$128$	2.09715e6	0.0883883
$129$	−7.27423e6	−0.298348
$130$	−2.08728e7	−0.833257
$131$	2.46988e6	0.0959901	0.0479951	−	0.998848i	$-0.484717\pi$
$131$	2.46988e6	0.0959901	0.0479951	+	0.998848i	$0.484717\pi$
$132$	1.25069e7	0.473307
$133$	4.01840e7	1.48106
$134$	1.27639e7	0.458264
$135$	9.65851e6	0.337864
$136$	3.57266e6	0.121788
$137$	2.68213e7	0.891164	0.445582	−	0.895241i	$-0.352997\pi$
$137$	2.68213e7	0.891164	0.445582	+	0.895241i	$0.352997\pi$
$138$	7.32016e6	0.237107
$139$	−9.94950e6	−0.314231	−0.157116	−	0.987580i	$-0.550220\pi$
$139$	−9.94950e6	−0.314231	−0.157116	+	0.987580i	$0.550220\pi$
$140$	−3.06258e7	−0.943278
$141$	−2.26843e7	−0.681488
$142$	−2.23064e7	−0.653762
$143$	−3.84839e7	−1.10053
$144$	2.98598e6	0.0833333
$145$	−3.21404e7	−0.875513
$146$	−1.52435e7	−0.405367
$147$	−3.44120e6	−0.0893511
$148$	−1.73712e7	−0.440468
$149$	−2.86458e7	−0.709428	−0.354714	−	0.934975i	$-0.615422\pi$
$149$	−2.86458e7	−0.709428	−0.354714	+	0.934975i	$0.615422\pi$
$150$	−3.51355e7	−0.850016
$151$	7.99725e7	1.89026	0.945130	−	0.326695i	$-0.105935\pi$
$151$	7.99725e7	1.89026	0.945130	+	0.326695i	$0.105935\pi$
$152$	2.10977e7	0.487284
$153$	5.08685e6	0.114823
$154$	−5.64660e7	−1.24585
$155$	−2.89065e7	−0.623497
$156$	−9.18788e6	−0.193767
$157$	−1.85923e7	−0.383429	−0.191714	−	0.981451i	$-0.561405\pi$
$157$	−1.85923e7	−0.383429	−0.191714	+	0.981451i	$0.561405\pi$
$158$	−3.78913e7	−0.764258
$159$	−1.16315e7	−0.229480
$160$	−1.60794e7	−0.310348
$161$	−3.30488e7	−0.624116
$162$	4.25153e6	0.0785674
$163$	−4.51506e7	−0.816595	−0.408297	−	0.912849i	$-0.633877\pi$
$163$	−4.51506e7	−0.816595	−0.408297	+	0.912849i	$0.633877\pi$
$164$	−3.40372e7	−0.602559
$165$	−9.58937e7	−1.66187
$166$	−6.59687e7	−1.11933
$167$	−3.44909e7	−0.573056	−0.286528	−	0.958072i	$-0.592501\pi$
$167$	−3.44909e7	−0.573056	−0.286528	+	0.958072i	$0.592501\pi$
$168$	−1.34810e7	−0.219351
$169$	−3.44774e7	−0.549453
$170$	−2.73924e7	−0.427621
$171$	3.00394e7	0.459415
$172$	1.72426e7	0.258377
$173$	1.71119e7	0.251268	0.125634	−	0.992077i	$-0.459903\pi$
$173$	1.71119e7	0.251268	0.125634	+	0.992077i	$0.459903\pi$
$174$	−1.41477e7	−0.203593
$175$	1.58629e8	2.23743
$176$	−2.96461e7	−0.409896
$177$	5.54523e6	0.0751646
$178$	2.19682e7	0.291961
$179$	−1.03658e8	−1.35088	−0.675442	−	0.737413i	$-0.736047\pi$
$179$	−1.03658e8	−1.35088	−0.675442	+	0.737413i	$0.736047\pi$
$180$	−2.28942e7	−0.292599
$181$	−5.57138e7	−0.698373	−0.349187	−	0.937053i	$-0.613542\pi$
$181$	−5.57138e7	−0.698373	−0.349187	+	0.937053i	$0.613542\pi$
$182$	4.14812e7	0.510036
$183$	7.27776e6	0.0877847
$184$	−1.73515e7	−0.205341
$185$	1.33189e8	1.54657
$186$	−1.27242e7	−0.144989
$187$	−5.05044e7	−0.564786
$188$	5.37701e7	0.590186
$189$	−1.91947e7	−0.206806
$190$	−1.61761e8	−1.71094
$191$	9.80880e7	1.01859	0.509295	−	0.860592i	$-0.329906\pi$
$191$	9.80880e7	1.01859	0.509295	+	0.860592i	$0.329906\pi$
$192$	−7.07789e6	−0.0721688
$193$	−4.11622e7	−0.412143	−0.206071	−	0.978537i	$-0.566068\pi$
$193$	−4.11622e7	−0.412143	−0.206071	+	0.978537i	$0.566068\pi$
$194$	5.64529e7	0.555112
$195$	7.04457e7	0.680351
$196$	8.15693e6	0.0773803
$197$	1.70172e8	1.58583	0.792913	−	0.609335i	$-0.208563\pi$
$197$	1.70172e8	1.58583	0.792913	+	0.609335i	$0.208563\pi$
$198$	−4.22109e7	−0.386453
$199$	8.82297e7	0.793650	0.396825	−	0.917894i	$-0.370112\pi$
$199$	8.82297e7	0.793650	0.396825	+	0.917894i	$0.370112\pi$
$200$	8.32843e7	0.736136
$201$	−4.30781e7	−0.374171
$202$	−3.07098e7	−0.262148
$203$	6.38735e7	0.535901
$204$	−1.20577e7	−0.0994397
$205$	2.60971e8	2.11570
$206$	−3.47906e7	−0.277285
$207$	−2.47056e7	−0.193597
$208$	2.17787e7	0.167807
$209$	−2.98244e8	−2.25975
$210$	1.03362e8	0.770183
$211$	−2.63411e8	−1.93039	−0.965195	−	0.261530i	$-0.915773\pi$
$211$	−2.63411e8	−1.93039	−0.965195	+	0.261530i	$0.915773\pi$
$212$	2.75709e7	0.198735
$213$	7.52840e7	0.533795
$214$	−1.51810e8	−1.05890
$215$	−1.32203e8	−0.907210
$216$	−1.00777e7	−0.0680414
$217$	5.74468e7	0.381642
$218$	−1.70656e8	−1.11564
$219$	5.14467e7	0.330981
$220$	2.27304e8	1.43922
$221$	3.71017e7	0.231217
$222$	5.86279e7	0.359641
$223$	−2.91164e8	−1.75821	−0.879104	−	0.476631i	$-0.841858\pi$
$223$	−2.91164e8	−1.75821	−0.879104	+	0.476631i	$0.841858\pi$
$224$	3.19550e7	0.189964
$225$	1.18582e8	0.694036
$226$	−1.27583e8	−0.735210
$227$	−2.45618e8	−1.39370	−0.696849	−	0.717218i	$-0.745415\pi$
$227$	−2.45618e8	−1.39370	−0.696849	+	0.717218i	$0.745415\pi$
$228$	−7.12046e7	−0.397865
$229$	−3.16384e8	−1.74097	−0.870483	−	0.492198i	$-0.836194\pi$
$229$	−3.16384e8	−1.74097	−0.870483	+	0.492198i	$0.836194\pi$
$230$	1.33038e8	0.720989
$231$	1.90573e8	1.01723
$232$	3.35353e7	0.176317
$233$	3.51827e8	1.82215	0.911073	−	0.412245i	$-0.135255\pi$
$233$	3.51827e8	1.82215	0.911073	+	0.412245i	$0.135255\pi$
$234$	3.10091e7	0.158210
$235$	−4.12268e8	−2.07225
$236$	−1.31443e7	−0.0650945
$237$	1.27883e8	0.624014
$238$	5.44378e7	0.261747
$239$	3.02685e8	1.43416	0.717081	−	0.696989i	$-0.245478\pi$
$239$	3.02685e8	1.43416	0.717081	+	0.696989i	$0.245478\pi$
$240$	5.42678e7	0.253398
$241$	7.44801e7	0.342752	0.171376	−	0.985206i	$-0.445179\pi$
$241$	7.44801e7	0.342752	0.171376	+	0.985206i	$0.445179\pi$
$242$	2.63190e8	1.19376
$243$	−1.43489e7	−0.0641500
$244$	−1.72510e7	−0.0760238
$245$	−6.25411e7	−0.271697
$246$	1.14875e8	0.491988
$247$	2.19097e8	0.925117
$248$	3.01610e7	0.125564
$249$	2.22644e8	0.913932
$250$	−3.31871e8	−1.34332
$251$	4.03601e8	1.61100	0.805498	−	0.592598i	$-0.201898\pi$
$251$	4.03601e8	1.61100	0.805498	+	0.592598i	$0.201898\pi$
$252$	4.54985e7	0.179100
$253$	2.45287e8	0.952254
$254$	−2.27104e8	−0.869575
$255$	9.24494e7	0.349151
$256$	1.67772e7	0.0625000
$257$	2.07525e8	0.762614	0.381307	−	0.924448i	$-0.375474\pi$
$257$	2.07525e8	0.762614	0.381307	+	0.924448i	$0.375474\pi$
$258$	−5.81939e7	−0.210964
$259$	−2.64691e8	−0.946653
$260$	−1.66982e8	−0.589201
$261$	4.77485e7	0.166233
$262$	1.97591e7	0.0678753
$263$	−3.34743e8	−1.13466	−0.567331	−	0.823490i	$-0.692024\pi$
$263$	−3.34743e8	−1.13466	−0.567331	+	0.823490i	$0.692024\pi$
$264$	1.00056e8	0.334678
$265$	−2.11393e8	−0.697797
$266$	3.21472e8	1.04727
$267$	−7.41427e7	−0.238385
$268$	1.02111e8	0.324041
$269$	3.88757e8	1.21771	0.608856	−	0.793280i	$-0.291629\pi$
$269$	3.88757e8	1.21771	0.608856	+	0.793280i	$0.291629\pi$
$270$	7.72681e7	0.238906
$271$	−3.70374e8	−1.13044	−0.565221	−	0.824939i	$-0.691209\pi$
$271$	−3.70374e8	−1.13044	−0.565221	+	0.824939i	$0.691209\pi$
$272$	2.85813e7	0.0861173
$273$	−1.39999e8	−0.416443
$274$	2.14570e8	0.630148
$275$	−1.17734e9	−3.41378
$276$	5.85613e7	0.167660
$277$	3.25943e8	0.921429	0.460714	−	0.887548i	$-0.347593\pi$
$277$	3.25943e8	0.921429	0.460714	+	0.887548i	$0.347593\pi$
$278$	−7.95960e7	−0.222195
$279$	4.29441e7	0.118383
$280$	−2.45007e8	−0.666998
$281$	4.10932e8	1.10483	0.552417	−	0.833568i	$-0.313705\pi$
$281$	4.10932e8	1.10483	0.552417	+	0.833568i	$0.313705\pi$
$282$	−1.81474e8	−0.481885
$283$	−1.68483e8	−0.441880	−0.220940	−	0.975287i	$-0.570912\pi$
$283$	−1.68483e8	−0.441880	−0.220940	+	0.975287i	$0.570912\pi$
$284$	−1.78451e8	−0.462280
$285$	5.45943e8	1.39698
$286$	−3.07871e8	−0.778194
$287$	−5.18636e8	−1.29502
$288$	2.38879e7	0.0589256
$289$	−3.61648e8	−0.881341
$290$	−2.57123e8	−0.619081
$291$	−1.90529e8	−0.453247
$292$	−1.21948e8	−0.286638
$293$	−1.01134e8	−0.234888	−0.117444	−	0.993079i	$-0.537470\pi$
$293$	−1.01134e8	−0.234888	−0.117444	+	0.993079i	$0.537470\pi$
$294$	−2.75296e7	−0.0631808
$295$	1.00780e8	0.228559
$296$	−1.38970e8	−0.311458
$297$	1.42462e8	0.315538
$298$	−2.29166e8	−0.501641
$299$	−1.80193e8	−0.389843
$300$	−2.81084e8	−0.601052
$301$	2.62732e8	0.555303
$302$	6.39780e8	1.33662
$303$	1.03646e8	0.214043
$304$	1.68781e8	0.344562
$305$	1.32267e8	0.266934
$306$	4.06948e7	0.0811922
$307$	8.63290e8	1.70283	0.851417	−	0.524489i	$-0.175743\pi$
$307$	8.63290e8	1.70283	0.851417	+	0.524489i	$0.175743\pi$
$308$	−4.51728e8	−0.880946
$309$	1.17418e8	0.226403
$310$	−2.31252e8	−0.440879
$311$	−3.89880e8	−0.734969	−0.367485	−	0.930030i	$-0.619781\pi$
$311$	−3.89880e8	−0.734969	−0.367485	+	0.930030i	$0.619781\pi$
$312$	−7.35031e7	−0.137014
$313$	−7.63476e8	−1.40731	−0.703656	−	0.710541i	$-0.748450\pi$
$313$	−7.63476e8	−1.40731	−0.703656	+	0.710541i	$0.748450\pi$
$314$	−1.48739e8	−0.271125
$315$	−3.48847e8	−0.628852
$316$	−3.03130e8	−0.540412
$317$	−8.48026e8	−1.49521	−0.747604	−	0.664145i	$-0.768796\pi$
$317$	−8.48026e8	−1.49521	−0.747604	+	0.664145i	$0.768796\pi$
$318$	−9.30518e7	−0.162267
$319$	−4.74067e8	−0.817659
$320$	−1.28635e8	−0.219449
$321$	5.12360e8	0.864585
$322$	−2.64391e8	−0.441317
$323$	2.87532e8	0.474763
$324$	3.40122e7	0.0555556
$325$	8.64898e8	1.39757
$326$	−3.61205e8	−0.577420
$327$	5.75965e8	0.910917
$328$	−2.72297e8	−0.426074
$329$	8.19314e8	1.26842
$330$	−7.67150e8	−1.17512
$331$	7.89819e8	1.19710	0.598549	−	0.801086i	$-0.295744\pi$
$331$	7.89819e8	1.19710	0.598549	+	0.801086i	$0.295744\pi$
$332$	−5.27749e8	−0.791488
$333$	−1.97869e8	−0.293645
$334$	−2.75927e8	−0.405211
$335$	−7.82909e8	−1.13777
$336$	−1.07848e8	−0.155105
$337$	9.19957e8	1.30937	0.654686	−	0.755901i	$-0.272801\pi$
$337$	9.19957e8	1.30937	0.654686	+	0.755901i	$0.272801\pi$
$338$	−2.75819e8	−0.388522
$339$	4.30591e8	0.600297
$340$	−2.19139e8	−0.302374
$341$	−4.26367e8	−0.582296
$342$	2.40316e8	0.324856
$343$	−6.78821e8	−0.908292
$344$	1.37941e8	0.182700
$345$	−4.49003e8	−0.588685
$346$	1.36895e8	0.177674
$347$	−6.50840e8	−0.836221	−0.418111	−	0.908396i	$-0.637308\pi$
$347$	−6.50840e8	−0.836221	−0.418111	+	0.908396i	$0.637308\pi$
$348$	−1.13182e8	−0.143962
$349$	−4.03839e8	−0.508533	−0.254266	−	0.967134i	$-0.581834\pi$
$349$	−4.03839e8	−0.508533	−0.254266	+	0.967134i	$0.581834\pi$
$350$	1.26903e9	1.58210
$351$	−1.04656e8	−0.129178
$352$	−2.37169e8	−0.289840
$353$	−3.01757e8	−0.365128	−0.182564	−	0.983194i	$-0.558440\pi$
$353$	−3.01757e8	−0.365128	−0.182564	+	0.983194i	$0.558440\pi$
$354$	4.43619e7	0.0531494
$355$	1.36823e9	1.62315
$356$	1.75746e8	0.206447
$357$	−1.83728e8	−0.213715
$358$	−8.29265e8	−0.955219
$359$	7.10294e8	0.810229	0.405115	−	0.914266i	$-0.367232\pi$
$359$	7.10294e8	0.810229	0.405115	+	0.914266i	$0.367232\pi$
$360$	−1.83154e8	−0.206899
$361$	8.04093e8	0.899562
$362$	−4.45710e8	−0.493824
$363$	−8.88268e8	−0.974700
$364$	3.31849e8	0.360650
$365$	9.35002e8	1.00644
$366$	5.82221e7	0.0620732
$367$	1.32362e9	1.39776	0.698879	−	0.715240i	$-0.253683\pi$
$367$	1.32362e9	1.39776	0.698879	+	0.715240i	$0.253683\pi$
$368$	−1.38812e8	−0.145198
$369$	−3.87705e8	−0.401706
$370$	1.06551e9	1.09359
$371$	4.20107e8	0.427121
$372$	−1.01794e8	−0.102523
$373$	1.28625e8	0.128335	0.0641675	−	0.997939i	$-0.479561\pi$
$373$	1.28625e8	0.128335	0.0641675	+	0.997939i	$0.479561\pi$
$374$	−4.04035e8	−0.399364
$375$	1.12006e9	1.09681
$376$	4.30161e8	0.417324
$377$	3.48260e8	0.334741
$378$	−1.53557e8	−0.146234
$379$	−7.57457e8	−0.714695	−0.357347	−	0.933972i	$-0.616319\pi$
$379$	−7.57457e8	−0.714695	−0.357347	+	0.933972i	$0.616319\pi$
$380$	−1.29409e9	−1.20982
$381$	7.66477e8	0.710005
$382$	7.84704e8	0.720251
$383$	−1.32695e7	−0.0120687	−0.00603434	−	0.999982i	$-0.501921\pi$
$383$	−1.32695e7	−0.0120687	−0.00603434	+	0.999982i	$0.501921\pi$
$384$	−5.66231e7	−0.0510310
$385$	3.46350e9	3.09316
$386$	−3.29297e8	−0.291429
$387$	1.96404e8	0.172251
$388$	4.51623e8	0.392523
$389$	−2.95113e8	−0.254193	−0.127097	−	0.991890i	$-0.540566\pi$
$389$	−2.95113e8	−0.254193	−0.127097	+	0.991890i	$0.540566\pi$
$390$	5.63565e8	0.481081
$391$	−2.36477e8	−0.200065
$392$	6.52554e7	0.0547161
$393$	−6.66868e7	−0.0554199
$394$	1.36137e9	1.12135
$395$	2.32417e9	1.89749
$396$	−3.37688e8	−0.273264
$397$	−1.35162e9	−1.08414	−0.542072	−	0.840332i	$-0.682360\pi$
$397$	−1.35162e9	−1.08414	−0.542072	+	0.840332i	$0.682360\pi$
$398$	7.05837e8	0.561195
$399$	−1.08497e9	−0.855091
$400$	6.66274e8	0.520527
$401$	1.93272e9	1.49680	0.748400	−	0.663248i	$-0.230823\pi$
$401$	1.93272e9	1.49680	0.748400	+	0.663248i	$0.230823\pi$
$402$	−3.44624e8	−0.264579
$403$	3.13219e8	0.238386
$404$	−2.45678e8	−0.185367
$405$	−2.60780e8	−0.195066
$406$	5.10988e8	0.378940
$407$	1.96453e9	1.44437
$408$	−9.64618e7	−0.0703145
$409$	−1.57487e9	−1.13818	−0.569092	−	0.822274i	$-0.692705\pi$
$409$	−1.57487e9	−1.13818	−0.569092	+	0.822274i	$0.692705\pi$
$410$	2.08777e9	1.49602
$411$	−7.24175e8	−0.514514
$412$	−2.78325e8	−0.196070
$413$	−2.00283e8	−0.139901
$414$	−1.97644e8	−0.136894
$415$	4.04638e9	2.77906
$416$	1.74229e8	0.118657
$417$	2.68636e8	0.181422
$418$	−2.38595e9	−1.59788
$419$	1.36240e9	0.904805	0.452403	−	0.891814i	$-0.350567\pi$
$419$	1.36240e9	0.904805	0.452403	+	0.891814i	$0.350567\pi$
$420$	8.26898e8	0.544602
$421$	2.65278e8	0.173266	0.0866331	−	0.996240i	$-0.472389\pi$
$421$	2.65278e8	0.173266	0.0866331	+	0.996240i	$0.472389\pi$
$422$	−2.10729e9	−1.36499
$423$	6.12476e8	0.393457
$424$	2.20567e8	0.140527
$425$	1.13505e9	0.717222
$426$	6.02272e8	0.377450
$427$	−2.62859e8	−0.163390
$428$	−1.21448e9	−0.748753
$429$	1.03907e9	0.635393
$430$	−1.05763e9	−0.641494
$431$	2.87491e9	1.72963	0.864815	−	0.502090i	$-0.167436\pi$
$431$	2.87491e9	1.72963	0.864815	+	0.502090i	$0.167436\pi$
$432$	−8.06216e7	−0.0481125
$433$	−2.04359e8	−0.120972	−0.0604861	−	0.998169i	$-0.519265\pi$
$433$	−2.04359e8	−0.120972	−0.0604861	+	0.998169i	$0.519265\pi$
$434$	4.59574e8	0.269862
$435$	8.67790e8	0.505478
$436$	−1.36525e9	−0.788877
$437$	−1.39647e9	−0.800473
$438$	4.11574e8	0.234039
$439$	2.10483e6	0.00118739	0.000593693	−	1.00000i	$-0.499811\pi$
$439$	2.10483e6	0.00118739	0.000593693		1.00000i	$0.499811\pi$
$440$	1.81843e9	1.01768
$441$	9.29125e7	0.0515869
$442$	2.96813e8	0.163495
$443$	2.32260e9	1.26929	0.634646	−	0.772803i	$-0.281146\pi$
$443$	2.32260e9	1.26929	0.634646	+	0.772803i	$0.281146\pi$
$444$	4.69023e8	0.254304
$445$	−1.34748e9	−0.724876
$446$	−2.32931e9	−1.24324
$447$	7.73435e8	0.409589
$448$	2.55640e8	0.134325
$449$	9.73065e8	0.507317	0.253659	−	0.967294i	$-0.418366\pi$
$449$	9.73065e8	0.507317	0.253659	+	0.967294i	$0.418366\pi$
$450$	9.48660e8	0.490757
$451$	3.84929e9	1.97589
$452$	−1.02066e9	−0.519872
$453$	−2.15926e9	−1.09134
$454$	−1.96494e9	−0.985494
$455$	−2.54437e9	−1.26631
$456$	−5.69637e8	−0.281333
$457$	−2.40313e9	−1.17780	−0.588898	−	0.808207i	$-0.700438\pi$
$457$	−2.40313e9	−1.17780	−0.588898	+	0.808207i	$0.700438\pi$
$458$	−2.53107e9	−1.23105
$459$	−1.37345e8	−0.0662931
$460$	1.06430e9	0.509816
$461$	−7.12825e8	−0.338867	−0.169434	−	0.985542i	$-0.554194\pi$
$461$	−7.12825e8	−0.338867	−0.169434	+	0.985542i	$0.554194\pi$
$462$	1.52458e9	0.719289
$463$	−4.38861e8	−0.205491	−0.102746	−	0.994708i	$-0.532763\pi$
$463$	−4.38861e8	−0.205491	−0.102746	+	0.994708i	$0.532763\pi$
$464$	2.68282e8	0.124675
$465$	7.80475e8	0.359976
$466$	2.81461e9	1.28845
$467$	−1.86916e9	−0.849252	−0.424626	−	0.905369i	$-0.639594\pi$
$467$	−1.86916e9	−0.849252	−0.424626	+	0.905369i	$0.639594\pi$
$468$	2.48073e8	0.111871
$469$	1.55590e9	0.696428
$470$	−3.29815e9	−1.46530
$471$	5.01993e8	0.221373
$472$	−1.05154e8	−0.0460287
$473$	−1.94998e9	−0.847260
$474$	1.02307e9	0.441244
$475$	6.70282e9	2.86965
$476$	4.35503e8	0.185083
$477$	3.14050e8	0.132490
$478$	2.42148e9	1.01411
$479$	−3.86437e9	−1.60659	−0.803295	−	0.595582i	$-0.796922\pi$
$479$	−3.86437e9	−1.60659	−0.803295	+	0.595582i	$0.796922\pi$
$480$	4.34143e8	0.179179
$481$	−1.44319e9	−0.591309
$482$	5.95840e8	0.242362
$483$	8.92319e8	0.360334
$484$	2.10552e9	0.844115
$485$	−3.46270e9	−1.37822
$486$	−1.14791e8	−0.0453609
$487$	1.33479e9	0.523675	0.261837	−	0.965112i	$-0.415672\pi$
$487$	1.33479e9	0.523675	0.261837	+	0.965112i	$0.415672\pi$
$488$	−1.38008e8	−0.0537569
$489$	1.21907e9	0.471461
$490$	−5.00329e8	−0.192119
$491$	−2.18071e9	−0.831406	−0.415703	−	0.909500i	$-0.636464\pi$
$491$	−2.18071e9	−0.831406	−0.415703	+	0.909500i	$0.636464\pi$
$492$	9.19003e8	0.347888
$493$	4.57039e8	0.171787
$494$	1.75277e9	0.654157
$495$	2.58913e9	0.959480
$496$	2.41288e8	0.0887872
$497$	−2.71912e9	−0.993530
$498$	1.78115e9	0.646247
$499$	−1.42092e9	−0.511937	−0.255969	−	0.966685i	$-0.582394\pi$
$499$	−1.42092e9	−0.511937	−0.255969	+	0.966685i	$0.582394\pi$
$500$	−2.65497e9	−0.949869
$501$	9.31254e8	0.330854
$502$	3.22881e9	1.13915
$503$	−4.86664e9	−1.70507	−0.852533	−	0.522673i	$-0.824935\pi$
$503$	−4.86664e9	−1.70507	−0.852533	+	0.522673i	$0.824935\pi$
$504$	3.63988e8	0.126643
$505$	1.88368e9	0.650858
$506$	1.96230e9	0.673345
$507$	9.30889e8	0.317227
$508$	−1.81683e9	−0.614883
$509$	−2.44770e9	−0.822709	−0.411354	−	0.911475i	$-0.634944\pi$
$509$	−2.44770e9	−0.822709	−0.411354	+	0.911475i	$0.634944\pi$
$510$	7.39595e8	0.246887
$511$	−1.85816e9	−0.616041
$512$	1.34218e8	0.0441942
$513$	−8.11065e8	−0.265244
$514$	1.66020e9	0.539249
$515$	2.13398e9	0.688440
$516$	−4.65551e8	−0.149174
$517$	−6.08091e9	−1.93532
$518$	−2.11753e9	−0.669384
$519$	−4.62022e8	−0.145070
$520$	−1.33586e9	−0.416628
$521$	1.88873e8	0.0585112	0.0292556	−	0.999572i	$-0.490686\pi$
$521$	1.88873e8	0.0585112	0.0292556	+	0.999572i	$0.490686\pi$
$522$	3.81988e8	0.117545
$523$	−5.31937e9	−1.62594	−0.812970	−	0.582306i	$-0.802151\pi$
$523$	−5.31937e9	−1.62594	−0.812970	+	0.582306i	$0.802151\pi$
$524$	1.58072e8	0.0479951
$525$	−4.28298e9	−1.29178
$526$	−2.67794e9	−0.802327
$527$	4.11053e8	0.122338
$528$	8.00445e8	0.236653
$529$	−2.25632e9	−0.662682
$530$	−1.69114e9	−0.493417
$531$	−1.49721e8	−0.0433963
$532$	2.57178e9	0.740530
$533$	−2.82778e9	−0.808909
$534$	−5.93142e8	−0.168564
$535$	9.31173e9	2.62901
$536$	8.16888e8	0.229132
$537$	2.79877e9	0.779933
$538$	3.11005e9	0.861053
$539$	−9.22474e8	−0.253743
$540$	6.18145e8	0.168932
$541$	3.08999e9	0.839008	0.419504	−	0.907754i	$-0.362204\pi$
$541$	3.08999e9	0.839008	0.419504	+	0.907754i	$0.362204\pi$
$542$	−2.96300e9	−0.799344
$543$	1.50427e9	0.403206
$544$	2.28650e8	0.0608941
$545$	1.04677e10	2.76989
$546$	−1.11999e9	−0.294470
$547$	−8.36929e8	−0.218642	−0.109321	−	0.994007i	$-0.534868\pi$
$547$	−8.36929e8	−0.218642	−0.109321	+	0.994007i	$0.534868\pi$
$548$	1.71656e9	0.445582
$549$	−1.96499e8	−0.0506825
$550$	−9.41869e9	−2.41391
$551$	2.69896e9	0.687331
$552$	4.68491e8	0.118553
$553$	−4.61890e9	−1.16145
$554$	2.60754e9	0.651549
$555$	−3.59611e9	−0.892910
$556$	−6.36768e8	−0.157116
$557$	−4.78126e9	−1.17233	−0.586164	−	0.810192i	$-0.699363\pi$
$557$	−4.78126e9	−1.17233	−0.586164	+	0.810192i	$0.699363\pi$
$558$	3.43553e8	0.0837094
$559$	1.43250e9	0.346860
$560$	−1.96005e9	−0.471639
$561$	1.36362e9	0.326079
$562$	3.28745e9	0.781236
$563$	−7.31218e8	−0.172690	−0.0863451	−	0.996265i	$-0.527519\pi$
$563$	−7.31218e8	−0.172690	−0.0863451	+	0.996265i	$0.527519\pi$
$564$	−1.45179e9	−0.340744
$565$	7.82564e9	1.82537
$566$	−1.34787e9	−0.312456
$567$	5.18256e8	0.119400
$568$	−1.42761e9	−0.326881
$569$	−3.66047e9	−0.832998	−0.416499	−	0.909136i	$-0.636743\pi$
$569$	−3.66047e9	−0.832998	−0.416499	+	0.909136i	$0.636743\pi$
$570$	4.36754e9	0.987813
$571$	−2.67216e9	−0.600669	−0.300335	−	0.953834i	$-0.597098\pi$
$571$	−2.67216e9	−0.600669	−0.300335	+	0.953834i	$0.597098\pi$
$572$	−2.46297e9	−0.550267
$573$	−2.64838e9	−0.588083
$574$	−4.14909e9	−0.915716
$575$	−5.51265e9	−1.20927
$576$	1.91103e8	0.0416667
$577$	−3.68219e9	−0.797978	−0.398989	−	0.916956i	$-0.630639\pi$
$577$	−3.68219e9	−0.797978	−0.398989	+	0.916956i	$0.630639\pi$
$578$	−2.89319e9	−0.623202
$579$	1.11138e9	0.237951
$580$	−2.05698e9	−0.437757
$581$	−8.04150e9	−1.70106
$582$	−1.52423e9	−0.320494
$583$	−3.11802e9	−0.651686
$584$	−9.75582e8	−0.202684
$585$	−1.90203e9	−0.392801
$586$	−8.09073e8	−0.166091
$587$	−2.52669e7	−0.00515607	−0.00257804	−	0.999997i	$-0.500821\pi$
$587$	−2.52669e7	−0.00515607	−0.00257804	+	0.999997i	$0.500821\pi$
$588$	−2.20237e8	−0.0446755
$589$	2.42740e9	0.489483
$590$	8.06241e8	0.161615
$591$	−4.59464e9	−0.915577
$592$	−1.11176e9	−0.220234
$593$	7.05079e9	1.38850	0.694251	−	0.719733i	$-0.255736\pi$
$593$	7.05079e9	1.38850	0.694251	+	0.719733i	$0.255736\pi$
$594$	1.13970e9	0.223119
$595$	−3.33910e9	−0.649861
$596$	−1.83333e9	−0.354714
$597$	−2.38220e9	−0.458214
$598$	−1.44155e9	−0.275661
$599$	−2.72805e8	−0.0518631	−0.0259315	−	0.999664i	$-0.508255\pi$
$599$	−2.72805e8	−0.0518631	−0.0259315	+	0.999664i	$0.508255\pi$
$600$	−2.24868e9	−0.425008
$601$	9.03849e9	1.69838	0.849191	−	0.528086i	$-0.177090\pi$
$601$	9.03849e9	1.69838	0.849191	+	0.528086i	$0.177090\pi$
$602$	2.10185e9	0.392658
$603$	1.16311e9	0.216028
$604$	5.11824e9	0.945130
$605$	−1.61436e10	−2.96384
$606$	8.29165e8	0.151351
$607$	7.30011e9	1.32486	0.662429	−	0.749125i	$-0.269526\pi$
$607$	7.30011e9	1.32486	0.662429	+	0.749125i	$0.269526\pi$
$608$	1.35025e9	0.243642
$609$	−1.72459e9	−0.309403
$610$	1.05814e9	0.188751
$611$	4.46717e9	0.792298
$612$	3.25559e8	0.0574115
$613$	−1.70082e9	−0.298228	−0.149114	−	0.988820i	$-0.547642\pi$
$613$	−1.70082e9	−0.298228	−0.149114	+	0.988820i	$0.547642\pi$
$614$	6.90632e9	1.20409
$615$	−7.04622e9	−1.22150
$616$	−3.61382e9	−0.622923
$617$	4.71199e9	0.807619	0.403810	−	0.914843i	$-0.367686\pi$
$617$	4.71199e9	0.807619	0.403810	+	0.914843i	$0.367686\pi$
$618$	9.39347e8	0.160091
$619$	5.54630e8	0.0939910	0.0469955	−	0.998895i	$-0.485035\pi$
$619$	5.54630e8	0.0939910	0.0469955	+	0.998895i	$0.485035\pi$
$620$	−1.85001e9	−0.311748
$621$	6.67050e8	0.111773
$622$	−3.11904e9	−0.519702
$623$	2.67790e9	0.443696
$624$	−5.88024e8	−0.0968834
$625$	7.64808e9	1.25306
$626$	−6.10781e9	−0.995119
$627$	8.05259e9	1.30467
$628$	−1.18991e9	−0.191714
$629$	−1.89397e9	−0.303455
$630$	−2.79078e9	−0.444666
$631$	5.90180e9	0.935151	0.467576	−	0.883953i	$-0.345128\pi$
$631$	5.90180e9	0.935151	0.467576	+	0.883953i	$0.345128\pi$
$632$	−2.42504e9	−0.382129
$633$	7.11210e9	1.11451
$634$	−6.78421e9	−1.05727
$635$	1.39301e10	2.15897
$636$	−7.44414e8	−0.114740
$637$	6.77670e8	0.103880
$638$	−3.79253e9	−0.578172
$639$	−2.03267e9	−0.308187
$640$	−1.02908e9	−0.155174
$641$	−1.08485e10	−1.62693	−0.813463	−	0.581616i	$-0.802421\pi$
$641$	−1.08485e10	−1.62693	−0.813463	+	0.581616i	$0.802421\pi$
$642$	4.09888e9	0.611354
$643$	−1.89051e9	−0.280440	−0.140220	−	0.990120i	$-0.544781\pi$
$643$	−1.89051e9	−0.280440	−0.140220	+	0.990120i	$0.544781\pi$
$644$	−2.11513e9	−0.312058
$645$	3.56949e9	0.523778
$646$	2.30025e9	0.335708
$647$	−6.27959e9	−0.911520	−0.455760	−	0.890103i	$-0.650632\pi$
$647$	−6.27959e9	−0.911520	−0.455760	+	0.890103i	$0.650632\pi$
$648$	2.72098e8	0.0392837
$649$	1.48650e9	0.213455
$650$	6.91918e9	0.988230
$651$	−1.55106e9	−0.220341
$652$	−2.88964e9	−0.408297
$653$	−5.08475e9	−0.714618	−0.357309	−	0.933986i	$-0.616306\pi$
$653$	−5.08475e9	−0.714618	−0.357309	+	0.933986i	$0.616306\pi$
$654$	4.60772e9	0.644115
$655$	−1.21198e9	−0.168520
$656$	−2.17838e9	−0.301280
$657$	−1.38906e9	−0.191092
$658$	6.55451e9	0.896912
$659$	1.10831e10	1.50856	0.754279	−	0.656554i	$-0.227986\pi$
$659$	1.10831e10	1.50856	0.754279	+	0.656554i	$0.227986\pi$
$660$	−6.13720e9	−0.830934
$661$	−9.40868e9	−1.26714	−0.633569	−	0.773687i	$-0.718411\pi$
$661$	−9.40868e9	−1.26714	−0.633569	+	0.773687i	$0.718411\pi$
$662$	6.31856e9	0.846476
$663$	−1.00174e9	−0.133493
$664$	−4.22199e9	−0.559667
$665$	−1.97184e10	−2.60014
$666$	−1.58295e9	−0.207639
$667$	−2.21972e9	−0.289640
$668$	−2.20742e9	−0.286528
$669$	7.86142e9	1.01510
$670$	−6.26327e9	−0.804525
$671$	1.95093e9	0.249294
$672$	−8.62785e8	−0.109676
$673$	4.57943e9	0.579107	0.289554	−	0.957162i	$-0.406493\pi$
$673$	4.57943e9	0.579107	0.289554	+	0.957162i	$0.406493\pi$
$674$	7.35966e9	0.925865
$675$	−3.20173e9	−0.400702
$676$	−2.20655e9	−0.274727
$677$	−1.30063e10	−1.61100	−0.805499	−	0.592597i	$-0.798103\pi$
$677$	−1.30063e10	−1.61100	−0.805499	+	0.592597i	$0.798103\pi$
$678$	3.44473e9	0.424474
$679$	6.88154e9	0.843609
$680$	−1.75312e9	−0.213811
$681$	6.63167e9	0.804652
$682$	−3.41094e9	−0.411745
$683$	−1.31732e10	−1.58205	−0.791025	−	0.611783i	$-0.790452\pi$
$683$	−1.31732e10	−1.58205	−0.791025	+	0.611783i	$0.790452\pi$
$684$	1.92252e9	0.229708
$685$	−1.31613e10	−1.56452
$686$	−5.43057e9	−0.642260
$687$	8.54237e9	1.00515
$688$	1.10353e9	0.129188
$689$	2.29056e9	0.266794
$690$	−3.59203e9	−0.416263
$691$	3.66953e9	0.423094	0.211547	−	0.977368i	$-0.432150\pi$
$691$	3.66953e9	0.423094	0.211547	+	0.977368i	$0.432150\pi$
$692$	1.09516e9	0.125634
$693$	−5.14546e9	−0.587297
$694$	−5.20672e9	−0.591298
$695$	4.88225e9	0.551662
$696$	−9.05452e8	−0.101797
$697$	−3.71103e9	−0.415126
$698$	−3.23071e9	−0.359587
$699$	−9.49932e9	−1.05202
$700$	1.01522e10	1.11871
$701$	−4.48301e9	−0.491537	−0.245769	−	0.969329i	$-0.579040\pi$
$701$	−4.48301e9	−0.491537	−0.245769	+	0.969329i	$0.579040\pi$
$702$	−8.37246e8	−0.0913425
$703$	−1.11845e10	−1.21415
$704$	−1.89735e9	−0.204948
$705$	1.11312e10	1.19642
$706$	−2.41405e9	−0.258185
$707$	−3.74349e9	−0.398390
$708$	3.54895e8	0.0375823
$709$	7.89270e9	0.831694	0.415847	−	0.909435i	$-0.363485\pi$
$709$	7.89270e9	0.831694	0.415847	+	0.909435i	$0.363485\pi$
$710$	1.09458e10	1.14774
$711$	−3.45285e9	−0.360275
$712$	1.40597e9	0.145980
$713$	−1.99638e9	−0.206267
$714$	−1.46982e9	−0.151120
$715$	1.88842e10	1.93209
$716$	−6.63412e9	−0.675442
$717$	−8.17250e9	−0.828014
$718$	5.68236e9	0.572918
$719$	3.65046e9	0.366266	0.183133	−	0.983088i	$-0.441376\pi$
$719$	3.65046e9	0.366266	0.183133	+	0.983088i	$0.441376\pi$
$720$	−1.46523e9	−0.146299
$721$	−4.24093e9	−0.421394
$722$	6.43274e9	0.636086
$723$	−2.01096e9	−0.197888
$724$	−3.56568e9	−0.349187
$725$	1.06543e10	1.03835
$726$	−7.10614e9	−0.689217
$727$	1.28461e10	1.23994	0.619972	−	0.784624i	$-0.287144\pi$
$727$	1.28461e10	1.23994	0.619972	+	0.784624i	$0.287144\pi$
$728$	2.65479e9	0.255018
$729$	3.87420e8	0.0370370
$730$	7.48002e9	0.711660
$731$	1.87994e9	0.178006
$732$	4.65777e8	0.0438924
$733$	−1.11207e10	−1.04296	−0.521479	−	0.853264i	$-0.674620\pi$
$733$	−1.11207e10	−1.04296	−0.521479	+	0.853264i	$0.674620\pi$
$734$	1.05890e10	0.988364
$735$	1.68861e9	0.156864
$736$	−1.11050e9	−0.102670
$737$	−1.15478e10	−1.06258
$738$	−3.10164e9	−0.284049
$739$	−1.27938e10	−1.16612	−0.583062	−	0.812428i	$-0.698146\pi$
$739$	−1.27938e10	−1.16612	−0.583062	+	0.812428i	$0.698146\pi$
$740$	8.52412e9	0.773283
$741$	−5.91561e9	−0.534117
$742$	3.36086e9	0.302020
$743$	5.14733e8	0.0460385	0.0230192	−	0.999735i	$-0.492672\pi$
$743$	5.14733e8	0.0460385	0.0230192	+	0.999735i	$0.492672\pi$
$744$	−8.14348e8	−0.0724945
$745$	1.40566e10	1.24547
$746$	1.02900e9	0.0907466
$747$	−6.01139e9	−0.527659
$748$	−3.23228e9	−0.282393
$749$	−1.85055e10	−1.60922
$750$	8.96051e9	0.775565
$751$	5.25875e9	0.453047	0.226523	−	0.974006i	$-0.427264\pi$
$751$	5.25875e9	0.453047	0.226523	+	0.974006i	$0.427264\pi$
$752$	3.44129e9	0.295093
$753$	−1.08972e10	−0.930109
$754$	2.78608e9	0.236698
$755$	−3.92428e10	−3.31853
$756$	−1.22846e9	−0.103403
$757$	−6.26707e9	−0.525084	−0.262542	−	0.964921i	$-0.584561\pi$
$757$	−6.26707e9	−0.525084	−0.262542	+	0.964921i	$0.584561\pi$
$758$	−6.05965e9	−0.505366
$759$	−6.62275e9	−0.549784
$760$	−1.03527e10	−0.855472
$761$	1.18861e10	0.977675	0.488837	−	0.872375i	$-0.337421\pi$
$761$	1.18861e10	0.977675	0.488837	+	0.872375i	$0.337421\pi$
$762$	6.13181e9	0.502050
$763$	−2.08028e10	−1.69545
$764$	6.27763e9	0.509295
$765$	−2.49613e9	−0.201583
$766$	−1.06156e8	−0.00853385
$767$	−1.09201e9	−0.0873864
$768$	−4.52985e8	−0.0360844
$769$	7.68247e9	0.609198	0.304599	−	0.952481i	$-0.401478\pi$
$769$	7.68247e9	0.609198	0.304599	+	0.952481i	$0.401478\pi$
$770$	2.77080e10	2.18720
$771$	−5.60318e9	−0.440295
$772$	−2.63438e9	−0.206071
$773$	−2.05490e10	−1.60016	−0.800079	−	0.599894i	$-0.795209\pi$
$773$	−2.05490e10	−1.60016	−0.800079	+	0.599894i	$0.795209\pi$
$774$	1.57123e9	0.121800
$775$	9.58229e9	0.739458
$776$	3.61299e9	0.277556
$777$	7.14667e9	0.546550
$778$	−2.36090e9	−0.179742
$779$	−2.19148e10	−1.66095
$780$	4.50852e9	0.340176
$781$	2.01812e10	1.51589
$782$	−1.89182e9	−0.141467
$783$	−1.28921e9	−0.0959748
$784$	5.22044e8	0.0386902
$785$	9.12331e9	0.673145
$786$	−5.33494e8	−0.0391878
$787$	−4.74048e9	−0.346665	−0.173333	−	0.984863i	$-0.555454\pi$
$787$	−4.74048e9	−0.346665	−0.173333	+	0.984863i	$0.555454\pi$
$788$	1.08910e10	0.792913
$789$	9.03806e9	0.655097
$790$	1.85934e10	1.34173
$791$	−1.55521e10	−1.11731
$792$	−2.70150e9	−0.193227
$793$	−1.43320e9	−0.102059
$794$	−1.08129e10	−0.766605
$795$	5.70760e9	0.402873
$796$	5.64670e9	0.396825
$797$	−2.66301e10	−1.86324	−0.931620	−	0.363435i	$-0.881604\pi$
$797$	−2.66301e10	−1.86324	−0.931620	+	0.363435i	$0.881604\pi$
$798$	−8.67975e9	−0.604640
$799$	5.86250e9	0.406602
$800$	5.33019e9	0.368068
$801$	2.00185e9	0.137632
$802$	1.54618e10	1.05840
$803$	1.37912e10	0.939933
$804$	−2.75700e9	−0.187085
$805$	1.62172e10	1.09569
$806$	2.50575e9	0.168564
$807$	−1.04964e10	−0.703047
$808$	−1.96543e9	−0.131074
$809$	−2.51764e10	−1.67176	−0.835879	−	0.548913i	$-0.815042\pi$
$809$	−2.51764e10	−1.67176	−0.835879	+	0.548913i	$0.815042\pi$
$810$	−2.08624e9	−0.137932
$811$	−2.66147e10	−1.75206	−0.876030	−	0.482256i	$-0.839817\pi$
$811$	−2.66147e10	−1.75206	−0.876030	+	0.482256i	$0.839817\pi$
$812$	4.08791e9	0.267951
$813$	1.00001e10	0.652661
$814$	1.57162e10	1.02132
$815$	2.21555e10	1.43361
$816$	−7.71694e8	−0.0497198
$817$	1.11017e10	0.712214
$818$	−1.25989e10	−0.804818
$819$	3.77997e9	0.240433
$820$	1.67021e10	1.05785
$821$	1.76717e10	1.11449	0.557247	−	0.830347i	$-0.311858\pi$
$821$	1.76717e10	1.11449	0.557247	+	0.830347i	$0.311858\pi$
$822$	−5.79340e9	−0.363816
$823$	−9.57328e9	−0.598634	−0.299317	−	0.954154i	$-0.596759\pi$
$823$	−9.57328e9	−0.598634	−0.299317	+	0.954154i	$0.596759\pi$
$824$	−2.22660e9	−0.138643
$825$	3.17881e10	1.97095
$826$	−1.60227e9	−0.0989248
$827$	9.48426e9	0.583088	0.291544	−	0.956557i	$-0.405831\pi$
$827$	9.48426e9	0.583088	0.291544	+	0.956557i	$0.405831\pi$
$828$	−1.58116e9	−0.0967985
$829$	−1.48996e10	−0.908310	−0.454155	−	0.890923i	$-0.650059\pi$
$829$	−1.48996e10	−0.908310	−0.454155	+	0.890923i	$0.650059\pi$
$830$	3.23710e10	1.96509
$831$	−8.80045e9	−0.531987
$832$	1.39384e9	0.0839035
$833$	8.89341e8	0.0533103
$834$	2.14909e9	0.128284
$835$	1.69248e10	1.00605
$836$	−1.90876e10	−1.12987
$837$	−1.15949e9	−0.0683484
$838$	1.08992e10	0.639794
$839$	4.89284e9	0.286019	0.143009	−	0.989721i	$-0.454322\pi$
$839$	4.89284e9	0.286019	0.143009	+	0.989721i	$0.454322\pi$
$840$	6.61518e9	0.385092
$841$	−1.29598e10	−0.751299
$842$	2.12222e9	0.122518
$843$	−1.10952e10	−0.637877
$844$	−1.68583e10	−0.965195
$845$	1.69182e10	0.964616
$846$	4.89980e9	0.278216
$847$	3.20826e10	1.81417
$848$	1.76454e9	0.0993677
$849$	4.54905e9	0.255119
$850$	9.08039e9	0.507152
$851$	9.19852e9	0.511640
$852$	4.81818e9	0.266897
$853$	−7.34131e9	−0.404997	−0.202499	−	0.979283i	$-0.564906\pi$
$853$	−7.34131e9	−0.404997	−0.202499	+	0.979283i	$0.564906\pi$
$854$	−2.10287e9	−0.115534
$855$	−1.47404e10	−0.806546
$856$	−9.71587e9	−0.529448
$857$	−1.53723e10	−0.834268	−0.417134	−	0.908845i	$-0.636965\pi$
$857$	−1.53723e10	−0.834268	−0.417134	+	0.908845i	$0.636965\pi$
$858$	8.31253e9	0.449291
$859$	3.14541e10	1.69317	0.846585	−	0.532253i	$-0.178654\pi$
$859$	3.14541e10	1.69317	0.846585	+	0.532253i	$0.178654\pi$
$860$	−8.46101e9	−0.453605
$861$	1.40032e10	0.747679
$862$	2.29993e10	1.22303
$863$	1.64811e10	0.872867	0.436434	−	0.899736i	$-0.356241\pi$
$863$	1.64811e10	0.872867	0.436434	+	0.899736i	$0.356241\pi$
$864$	−6.44973e8	−0.0340207
$865$	−8.39688e9	−0.441125
$866$	−1.63487e9	−0.0855403
$867$	9.76450e9	0.508842
$868$	3.67659e9	0.190821
$869$	3.42813e10	1.77210
$870$	6.94232e9	0.357427
$871$	8.48328e9	0.435011
$872$	−1.09220e10	−0.557820
$873$	5.14427e9	0.261682
$874$	−1.11718e10	−0.566020
$875$	−4.04546e10	−2.04146
$876$	3.29259e9	0.165491
$877$	−2.37167e10	−1.18728	−0.593642	−	0.804729i	$-0.702311\pi$
$877$	−2.37167e10	−1.18728	−0.593642	+	0.804729i	$0.702311\pi$
$878$	1.68387e7	0.000839608 0
$879$	2.73062e9	0.135613
$880$	1.45474e10	0.719610
$881$	−1.65327e10	−0.814569	−0.407284	−	0.913301i	$-0.633524\pi$
$881$	−1.65327e10	−0.814569	−0.407284	+	0.913301i	$0.633524\pi$
$882$	7.43300e8	0.0364774
$883$	1.42317e10	0.695654	0.347827	−	0.937559i	$-0.386920\pi$
$883$	1.42317e10	0.695654	0.347827	+	0.937559i	$0.386920\pi$
$884$	2.37451e9	0.115609
$885$	−2.72106e9	−0.131958
$886$	1.85808e10	0.897524
$887$	3.50765e10	1.68765	0.843827	−	0.536616i	$-0.180298\pi$
$887$	3.50765e10	1.68765	0.843827	+	0.536616i	$0.180298\pi$
$888$	3.75219e9	0.179820
$889$	−2.76837e10	−1.32150
$890$	−1.07799e10	−0.512564
$891$	−3.84647e9	−0.182176
$892$	−1.86345e10	−0.879104
$893$	3.46199e10	1.62684
$894$	6.18748e9	0.289623
$895$	5.08654e10	2.37160
$896$	2.04512e9	0.0949819
$897$	4.86522e9	0.225076
$898$	7.78452e9	0.358727
$899$	3.85841e9	0.177113
$900$	7.58928e9	0.347018
$901$	3.00602e9	0.136916
$902$	3.07943e10	1.39717
$903$	−7.09376e9	−0.320604
$904$	−8.16528e9	−0.367605
$905$	2.73389e10	1.22606
$906$	−1.72741e10	−0.771695
$907$	−3.82236e10	−1.70101	−0.850503	−	0.525971i	$-0.823702\pi$
$907$	−3.82236e10	−1.70101	−0.850503	+	0.525971i	$0.823702\pi$
$908$	−1.57195e10	−0.696849
$909$	−2.79843e9	−0.123578
$910$	−2.03549e10	−0.895416
$911$	−3.09862e10	−1.35786	−0.678929	−	0.734204i	$-0.737556\pi$
$911$	−3.09862e10	−1.35786	−0.678929	+	0.734204i	$0.737556\pi$
$912$	−4.55709e9	−0.198933
$913$	5.96836e10	2.59542
$914$	−1.92250e10	−0.832828
$915$	−3.57122e9	−0.154114
$916$	−2.02486e10	−0.870483
$917$	2.40860e9	0.103151
$918$	−1.09876e9	−0.0468763
$919$	1.19668e10	0.508599	0.254299	−	0.967126i	$-0.418155\pi$
$919$	1.19668e10	0.508599	0.254299	+	0.967126i	$0.418155\pi$
$920$	8.51444e9	0.360494
$921$	−2.33088e10	−0.983132
$922$	−5.70260e9	−0.239615
$923$	−1.48256e10	−0.620590
$924$	1.21966e10	0.508614
$925$	−4.41513e10	−1.83420
$926$	−3.51089e9	−0.145304
$927$	−3.17030e9	−0.130714
$928$	2.14626e9	0.0881584
$929$	−2.02181e10	−0.827344	−0.413672	−	0.910426i	$-0.635754\pi$
$929$	−2.02181e10	−0.827344	−0.413672	+	0.910426i	$0.635754\pi$
$930$	6.24380e9	0.254542
$931$	5.25184e9	0.213298
$932$	2.25169e10	0.911073
$933$	1.05268e10	0.424335
$934$	−1.49532e10	−0.600512
$935$	2.47827e10	0.991533
$936$	1.98458e9	0.0791050
$937$	−4.40105e9	−0.174770	−0.0873851	−	0.996175i	$-0.527851\pi$
$937$	−4.40105e9	−0.174770	−0.0873851	+	0.996175i	$0.527851\pi$
$938$	1.24472e10	0.492449
$939$	2.06138e10	0.812512
$940$	−2.63852e10	−1.03613
$941$	−1.00676e10	−0.393878	−0.196939	−	0.980416i	$-0.563100\pi$
$941$	−1.00676e10	−0.393878	−0.196939	+	0.980416i	$0.563100\pi$
$942$	4.01594e9	0.156534
$943$	1.80236e10	0.699922
$944$	−8.41232e8	−0.0325472
$945$	9.41888e9	0.363068
$946$	−1.55999e10	−0.599104
$947$	4.35749e10	1.66729	0.833646	−	0.552299i	$-0.186249\pi$
$947$	4.35749e10	1.66729	0.833646	+	0.552299i	$0.186249\pi$
$948$	8.18452e9	0.312007
$949$	−1.01313e10	−0.384799
$950$	5.36225e10	2.02915
$951$	2.28967e10	0.863259
$952$	3.48402e9	0.130873
$953$	−1.12868e10	−0.422421	−0.211211	−	0.977441i	$-0.567741\pi$
$953$	−1.12868e10	−0.422421	−0.211211	+	0.977441i	$0.567741\pi$
$954$	2.51240e9	0.0936848
$955$	−4.81321e10	−1.78823
$956$	1.93719e10	0.717081
$957$	1.27998e10	0.472075
$958$	−3.09150e10	−1.13603
$959$	2.61558e10	0.957643
$960$	3.47314e9	0.126699
$961$	−2.40424e10	−0.873869
$962$	−1.15455e10	−0.418119
$963$	−1.38337e10	−0.499169
$964$	4.76672e9	0.171376
$965$	2.01984e10	0.723555
$966$	7.13855e9	0.254794
$967$	−1.93991e10	−0.689904	−0.344952	−	0.938620i	$-0.612105\pi$
$967$	−1.93991e10	−0.689904	−0.344952	+	0.938620i	$0.612105\pi$
$968$	1.68442e10	0.596879
$969$	−7.76336e9	−0.274105
$970$	−2.77016e10	−0.974550
$971$	4.91304e10	1.72220	0.861099	−	0.508437i	$-0.169777\pi$
$971$	4.91304e10	1.72220	0.861099	+	0.508437i	$0.169777\pi$
$972$	−9.18330e8	−0.0320750
$973$	−9.70265e9	−0.337672
$974$	1.06783e10	0.370294
$975$	−2.33522e10	−0.806886
$976$	−1.10406e9	−0.0380119
$977$	−1.30482e10	−0.447630	−0.223815	−	0.974632i	$-0.571851\pi$
$977$	−1.30482e10	−0.447630	−0.223815	+	0.974632i	$0.571851\pi$
$978$	9.75252e9	0.333373
$979$	−1.98752e10	−0.676975
$980$	−4.00263e9	−0.135848
$981$	−1.55510e10	−0.525918
$982$	−1.74457e10	−0.587893
$983$	1.82318e10	0.612197	0.306098	−	0.952000i	$-0.400976\pi$
$983$	1.82318e10	0.612197	0.306098	+	0.952000i	$0.400976\pi$
$984$	7.35203e9	0.245994
$985$	−8.35038e10	−2.78407
$986$	3.65631e9	0.121471
$987$	−2.21215e10	−0.732325
$988$	1.40222e10	0.462559
$989$	−9.13041e9	−0.300126
$990$	2.07130e10	0.678454
$991$	2.07318e10	0.676674	0.338337	−	0.941025i	$-0.390136\pi$
$991$	2.07318e10	0.676674	0.338337	+	0.941025i	$0.390136\pi$
$992$	1.93031e9	0.0627820
$993$	−2.13251e10	−0.691145
$994$	−2.17530e10	−0.702532
$995$	−4.32946e10	−1.39333
$996$	1.42492e10	0.456966
$997$	3.75433e10	1.19977	0.599887	−	0.800085i	$-0.295212\pi$
$997$	3.75433e10	1.19977	0.599887	+	0.800085i	$0.295212\pi$
$998$	−1.13673e10	−0.361994
$999$	5.34247e9	0.169536

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.d.1.2	✓	8	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -490.703 q^{5} -216.000 q^{6} +975.190 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -490.703 q^{5} -216.000 q^{6} +975.190 q^{7} +512.000 q^{8} +729.000 q^{9} -3925.63 q^{10} -7237.82 q^{11} -1728.00 q^{12} +5317.06 q^{13} +7801.52 q^{14} +13249.0 q^{15} +4096.00 q^{16} +6977.85 q^{17} +5832.00 q^{18} +41206.4 q^{19} -31405.0 q^{20} -26330.1 q^{21} -57902.5 q^{22} -33889.7 q^{23} -13824.0 q^{24} +162665. q^{25} +42536.5 q^{26} -19683.0 q^{27} +62412.1 q^{28} +65498.6 q^{29} +105992. q^{30} +58908.3 q^{31} +32768.0 q^{32} +195421. q^{33} +55822.8 q^{34} -478529. q^{35} +46656.0 q^{36} -271425. q^{37} +329651. q^{38} -143561. q^{39} -251240. q^{40} -531831. q^{41} -210641. q^{42} +269416. q^{43} -463220. q^{44} -357723. q^{45} -271117. q^{46} +840158. q^{47} -110592. q^{48} +127452. q^{49} +1.30132e6 q^{50} -188402. q^{51} +340292. q^{52} +430795. q^{53} -157464. q^{54} +3.55162e6 q^{55} +499297. q^{56} -1.11257e6 q^{57} +523989. q^{58} -205379. q^{59} +847935. q^{60} -269547. q^{61} +471266. q^{62} +710913. q^{63} +262144. q^{64} -2.60910e6 q^{65} +1.56337e6 q^{66} +1.59548e6 q^{67} +446582. q^{68} +915021. q^{69} -3.82823e6 q^{70} -2.78830e6 q^{71} +373248. q^{72} -1.90543e6 q^{73} -2.17140e6 q^{74} -4.39194e6 q^{75} +2.63721e6 q^{76} -7.05824e6 q^{77} -1.14849e6 q^{78} -4.73641e6 q^{79} -2.00992e6 q^{80} +531441. q^{81} -4.25465e6 q^{82} -8.24608e6 q^{83} -1.68513e6 q^{84} -3.42405e6 q^{85} +2.15533e6 q^{86} -1.76846e6 q^{87} -3.70576e6 q^{88} +2.74603e6 q^{89} -2.86178e6 q^{90} +5.18514e6 q^{91} -2.16894e6 q^{92} -1.59052e6 q^{93} +6.72127e6 q^{94} -2.02201e7 q^{95} -884736. q^{96} +7.05661e6 q^{97} +1.01962e6 q^{98} -5.27637e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * q^99

Introduction

L-functions

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