LMFDB - Embedded newform 338.8.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.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:	$105.586138614$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6981x^{2} - 35424x + 7188480 $ x^4 - 6981x^2 - 35424x + 7188480
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-71.1584$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -71.1584 q^{3} +64.0000 q^{4} +523.489 q^{5} +569.267 q^{6} -416.801 q^{7} -512.000 q^{8} +2876.52 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -71.1584 q^{3} +64.0000 q^{4} +523.489 q^{5} +569.267 q^{6} -416.801 q^{7} -512.000 q^{8} +2876.52 q^{9} -4187.91 q^{10} +3422.78 q^{11} -4554.14 q^{12} +3334.41 q^{14} -37250.7 q^{15} +4096.00 q^{16} -6538.04 q^{17} -23012.2 q^{18} +27980.5 q^{19} +33503.3 q^{20} +29658.9 q^{21} -27382.2 q^{22} +106178. q^{23} +36433.1 q^{24} +195916. q^{25} -49065.2 q^{27} -26675.3 q^{28} +68546.0 q^{29} +298005. q^{30} +51572.8 q^{31} -32768.0 q^{32} -243560. q^{33} +52304.3 q^{34} -218191. q^{35} +184097. q^{36} +48135.4 q^{37} -223844. q^{38} -268026. q^{40} -602098. q^{41} -237271. q^{42} +916252. q^{43} +219058. q^{44} +1.50583e6 q^{45} -849424. q^{46} -326201. q^{47} -291465. q^{48} -649820. q^{49} -1.56733e6 q^{50} +465237. q^{51} +934068. q^{53} +392522. q^{54} +1.79179e6 q^{55} +213402. q^{56} -1.99105e6 q^{57} -548368. q^{58} +1.17443e6 q^{59} -2.38404e6 q^{60} -2.89090e6 q^{61} -412582. q^{62} -1.19894e6 q^{63} +262144. q^{64} +1.94848e6 q^{66} +318158. q^{67} -418435. q^{68} -7.55546e6 q^{69} +1.74553e6 q^{70} +1.28374e6 q^{71} -1.47278e6 q^{72} -1.67209e6 q^{73} -385083. q^{74} -1.39411e7 q^{75} +1.79075e6 q^{76} -1.42662e6 q^{77} -8.09160e6 q^{79} +2.14421e6 q^{80} -2.79955e6 q^{81} +4.81678e6 q^{82} +5.57141e6 q^{83} +1.89817e6 q^{84} -3.42260e6 q^{85} -7.33001e6 q^{86} -4.87762e6 q^{87} -1.75246e6 q^{88} +7.87572e6 q^{89} -1.20466e7 q^{90} +6.79540e6 q^{92} -3.66984e6 q^{93} +2.60961e6 q^{94} +1.46475e7 q^{95} +2.33172e6 q^{96} -6.80847e6 q^{97} +5.19856e6 q^{98} +9.84569e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9} - 2224 q^{10} + 7392 q^{11} - 4384 q^{14} - 15528 q^{15} + 16384 q^{16} - 28316 q^{17} - 41712 q^{18} + 99888 q^{19} + 17792 q^{20} + 91074 q^{21} - 59136 q^{22} + 33388 q^{23} + 86878 q^{25} + 106272 q^{27} + 35072 q^{28} - 93140 q^{29} + 124224 q^{30} + 311160 q^{31} - 131072 q^{32} - 238638 q^{33} + 226528 q^{34} - 141544 q^{35} + 333696 q^{36} + 9636 q^{37} - 799104 q^{38} - 142336 q^{40} - 82892 q^{41} - 728592 q^{42} + 569264 q^{43} + 473088 q^{44} + 2303394 q^{45} - 267104 q^{46} - 574200 q^{47} + 717798 q^{49} - 695024 q^{50} - 2729928 q^{51} + 1235350 q^{53} - 850176 q^{54} + 1092512 q^{55} - 280576 q^{56} + 3528462 q^{57} + 745120 q^{58} - 231504 q^{59} - 993792 q^{60} - 685684 q^{61} - 2489280 q^{62} + 5951712 q^{63} + 1048576 q^{64} + 1909104 q^{66} - 3271056 q^{67} - 1812224 q^{68} - 5600034 q^{69} + 1132352 q^{70} + 175012 q^{71} - 2669568 q^{72} + 7137890 q^{73} - 77088 q^{74} - 22200960 q^{75} + 6392832 q^{76} - 13915206 q^{77} - 7053952 q^{79} + 1138688 q^{80} - 3758004 q^{81} + 663136 q^{82} + 657288 q^{83} + 5828736 q^{84} - 11814998 q^{85} - 4554112 q^{86} + 7182900 q^{87} - 3784704 q^{88} + 11452234 q^{89} - 18427152 q^{90} + 2136832 q^{92} - 2984688 q^{93} + 4593600 q^{94} + 23334088 q^{95} + 428002 q^{97} - 5742384 q^{98} - 10357656 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9 - 2224 * q^10 + 7392 * q^11 - 4384 * q^14 - 15528 * q^15 + 16384 * q^16 - 28316 * q^17 - 41712 * q^18 + 99888 * q^19 + 17792 * q^20 + 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 + 35072 * q^28 - 93140 * q^29 + 124224 * q^30 + 311160 * q^31 - 131072 * q^32 - 238638 * q^33 + 226528 * q^34 - 141544 * q^35 + 333696 * q^36 + 9636 * q^37 - 799104 * q^38 - 142336 * q^40 - 82892 * q^41 - 728592 * q^42 + 569264 * q^43 + 473088 * q^44 + 2303394 * q^45 - 267104 * q^46 - 574200 * q^47 + 717798 * q^49 - 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 - 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 + 3528462 * q^57 + 745120 * q^58 - 231504 * q^59 - 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 + 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 - 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 + 1132352 * q^70 + 175012 * q^71 - 2669568 * q^72 + 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 + 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 + 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 + 657288 * q^83 + 5828736 * q^84 - 11814998 * q^85 - 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 + 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 - 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 + 428002 * q^97 - 5742384 * q^98 - 10357656 * 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
$3$	−71.1584	−1.52160	−0.760802	−	0.648984i	$-0.775194\pi$
$3$	−71.1584	−1.52160	−0.760802	+	0.648984i	$0.775194\pi$
$4$	64.0000	0.500000
$5$	523.489	1.87289	0.936446	−	0.350812i	$-0.114094\pi$
$5$	523.489	1.87289	0.936446	+	0.350812i	$0.114094\pi$
$6$	569.267	1.07594
$7$	−416.801	−0.459289	−0.229644	−	0.973275i	$-0.573756\pi$
$7$	−416.801	−0.459289	−0.229644	+	0.973275i	$0.573756\pi$
$8$	−512.000	−0.353553
$9$	2876.52	1.31528
$10$	−4187.91	−1.32433
$11$	3422.78	0.775362	0.387681	−	0.921794i	$-0.373276\pi$
$11$	3422.78	0.775362	0.387681	+	0.921794i	$0.373276\pi$
$12$	−4554.14	−0.760802
$13$	0	0
$13$	0	0
$14$	3334.41	0.324766
$15$	−37250.7	−2.84980
$16$	4096.00	0.250000
$17$	−6538.04	−0.322758	−0.161379	−	0.986893i	$-0.551594\pi$
$17$	−6538.04	−0.322758	−0.161379	+	0.986893i	$0.551594\pi$
$18$	−23012.2	−0.930045
$19$	27980.5	0.935875	0.467937	−	0.883762i	$-0.344997\pi$
$19$	27980.5	0.935875	0.467937	+	0.883762i	$0.344997\pi$
$20$	33503.3	0.936446
$21$	29658.9	0.698856
$22$	−27382.2	−0.548264
$23$	106178.	1.81965	0.909824	−	0.414995i	$-0.136217\pi$
$23$	106178.	1.81965	0.909824	+	0.414995i	$0.136217\pi$
$24$	36433.1	0.537969
$25$	195916.	2.50772
$26$	0	0
$27$	−49065.2	−0.479734
$28$	−26675.3	−0.229644
$29$	68546.0	0.521902	0.260951	−	0.965352i	$-0.415964\pi$
$29$	68546.0	0.521902	0.260951	+	0.965352i	$0.415964\pi$
$30$	298005.	2.01511
$31$	51572.8	0.310924	0.155462	−	0.987842i	$-0.450313\pi$
$31$	51572.8	0.310924	0.155462	+	0.987842i	$0.450313\pi$
$32$	−32768.0	−0.176777
$33$	−243560.	−1.17979
$34$	52304.3	0.228224
$35$	−218191.	−0.860198
$36$	184097.	0.657641
$37$	48135.4	0.156228	0.0781140	−	0.996944i	$-0.475110\pi$
$37$	48135.4	0.156228	0.0781140	+	0.996944i	$0.475110\pi$
$38$	−223844.	−0.661763
$39$	0	0
$40$	−268026.	−0.662167
$41$	−602098.	−1.36434	−0.682172	−	0.731192i	$-0.738964\pi$
$41$	−602098.	−1.36434	−0.682172	+	0.731192i	$0.738964\pi$
$42$	−237271.	−0.494166
$43$	916252.	1.75742	0.878709	−	0.477357i	$-0.158405\pi$
$43$	916252.	1.75742	0.878709	+	0.477357i	$0.158405\pi$
$44$	219058.	0.387681
$45$	1.50583e6	2.46338
$46$	−849424.	−1.28669
$47$	−326201.	−0.458293	−0.229146	−	0.973392i	$-0.573593\pi$
$47$	−326201.	−0.458293	−0.229146	+	0.973392i	$0.573593\pi$
$48$	−291465.	−0.380401
$49$	−649820.	−0.789054
$50$	−1.56733e6	−1.77323
$51$	465237.	0.491110
$52$	0	0
$53$	934068.	0.861813	0.430907	−	0.902397i	$-0.358194\pi$
$53$	934068.	0.861813	0.430907	+	0.902397i	$0.358194\pi$
$54$	392522.	0.339223
$55$	1.79179e6	1.45217
$56$	213402.	0.162383
$57$	−1.99105e6	−1.42403
$58$	−548368.	−0.369040
$59$	1.17443e6	0.744469	0.372235	−	0.928139i	$-0.378592\pi$
$59$	1.17443e6	0.744469	0.372235	+	0.928139i	$0.378592\pi$
$60$	−2.38404e6	−1.42490
$61$	−2.89090e6	−1.63072	−0.815358	−	0.578957i	$-0.803460\pi$
$61$	−2.89090e6	−1.63072	−0.815358	+	0.578957i	$0.803460\pi$
$62$	−412582.	−0.219857
$63$	−1.19894e6	−0.604094
$64$	262144.	0.125000
$65$	0	0
$66$	1.94848e6	0.834241
$67$	318158.	0.129235	0.0646176	−	0.997910i	$-0.479417\pi$
$67$	318158.	0.129235	0.0646176	+	0.997910i	$0.479417\pi$
$68$	−418435.	−0.161379
$69$	−7.55546e6	−2.76878
$70$	1.74553e6	0.608252
$71$	1.28374e6	0.425669	0.212834	−	0.977088i	$-0.431730\pi$
$71$	1.28374e6	0.425669	0.212834	+	0.977088i	$0.431730\pi$
$72$	−1.47278e6	−0.465022
$73$	−1.67209e6	−0.503070	−0.251535	−	0.967848i	$-0.580935\pi$
$73$	−1.67209e6	−0.503070	−0.251535	+	0.967848i	$0.580935\pi$
$74$	−385083.	−0.110470
$75$	−1.39411e7	−3.81577
$76$	1.79075e6	0.467937
$77$	−1.42662e6	−0.356115
$78$	0	0
$79$	−8.09160e6	−1.84646	−0.923229	−	0.384250i	$-0.874460\pi$
$79$	−8.09160e6	−1.84646	−0.923229	+	0.384250i	$0.874460\pi$
$80$	2.14421e6	0.468223
$81$	−2.79955e6	−0.585316
$82$	4.81678e6	0.964736
$83$	5.57141e6	1.06953	0.534764	−	0.845002i	$-0.320401\pi$
$83$	5.57141e6	1.06953	0.534764	+	0.845002i	$0.320401\pi$
$84$	1.89817e6	0.349428
$85$	−3.42260e6	−0.604490
$86$	−7.33001e6	−1.24268
$87$	−4.87762e6	−0.794129
$88$	−1.75246e6	−0.274132
$89$	7.87572e6	1.18420	0.592100	−	0.805864i	$-0.298299\pi$
$89$	7.87572e6	1.18420	0.592100	+	0.805864i	$0.298299\pi$
$90$	−1.20466e7	−1.74187
$91$	0	0
$92$	6.79540e6	0.909824
$93$	−3.66984e6	−0.473104
$94$	2.60961e6	0.324062
$95$	1.46475e7	1.75279
$96$	2.33172e6	0.268984
$97$	−6.80847e6	−0.757441	−0.378720	−	0.925511i	$-0.623636\pi$
$97$	−6.80847e6	−0.757441	−0.378720	+	0.925511i	$0.623636\pi$
$98$	5.19856e6	0.557945
$99$	9.84569e6	1.01982
$100$	1.25386e7	1.25386
$101$	1.61012e6	0.155501	0.0777506	−	0.996973i	$-0.475226\pi$
$101$	1.61012e6	0.155501	0.0777506	+	0.996973i	$0.475226\pi$
$102$	−3.72189e6	−0.347267
$103$	4.93664e6	0.445145	0.222572	−	0.974916i	$-0.428555\pi$
$103$	4.93664e6	0.445145	0.222572	+	0.974916i	$0.428555\pi$
$104$	0	0
$105$	1.55261e7	1.30888
$106$	−7.47255e6	−0.609394
$107$	−1.22703e7	−0.968304	−0.484152	−	0.874984i	$-0.660872\pi$
$107$	−1.22703e7	−0.968304	−0.484152	+	0.874984i	$0.660872\pi$
$108$	−3.14017e6	−0.239867
$109$	−1.65278e7	−1.22243	−0.611213	−	0.791467i	$-0.709318\pi$
$109$	−1.65278e7	−1.22243	−0.611213	+	0.791467i	$0.709318\pi$
$110$	−1.43343e7	−1.02684
$111$	−3.42524e6	−0.237717
$112$	−1.70722e6	−0.114822
$113$	1.32876e7	0.866307	0.433154	−	0.901320i	$-0.357401\pi$
$113$	1.32876e7	0.866307	0.433154	+	0.901320i	$0.357401\pi$
$114$	1.59284e7	1.00694
$115$	5.55831e7	3.40800
$116$	4.38694e6	0.260951
$117$	0	0
$118$	−9.39547e6	−0.526419
$119$	2.72506e6	0.148239
$120$	1.90723e7	1.00756
$121$	−7.77175e6	−0.398814
$122$	2.31272e7	1.15309
$123$	4.28443e7	2.07599
$124$	3.30066e6	0.155462
$125$	6.16623e7	2.82381
$126$	9.59150e6	0.427159
$127$	2.72872e6	0.118208	0.0591040	−	0.998252i	$-0.481176\pi$
$127$	2.72872e6	0.118208	0.0591040	+	0.998252i	$0.481176\pi$
$128$	−2.09715e6	−0.0883883
$129$	−6.51990e7	−2.67410
$130$	0	0
$131$	1.96041e7	0.761898	0.380949	−	0.924596i	$-0.375597\pi$
$131$	1.96041e7	0.761898	0.380949	+	0.924596i	$0.375597\pi$
$132$	−1.55878e7	−0.589897
$133$	−1.16623e7	−0.429837
$134$	−2.54526e6	−0.0913831
$135$	−2.56851e7	−0.898490
$136$	3.34748e6	0.114112
$137$	652397.	0.0216765	0.0108383	−	0.999941i	$-0.496550\pi$
$137$	652397.	0.0216765	0.0108383	+	0.999941i	$0.496550\pi$
$138$	6.04437e7	1.95783
$139$	9.21661e6	0.291085	0.145542	−	0.989352i	$-0.453507\pi$
$139$	9.21661e6	0.291085	0.145542	+	0.989352i	$0.453507\pi$
$140$	−1.39642e7	−0.430099
$141$	2.32120e7	0.697341
$142$	−1.02699e7	−0.300993
$143$	0	0
$144$	1.17822e7	0.328820
$145$	3.58831e7	0.977466
$146$	1.33767e7	0.355724
$147$	4.62401e7	1.20063
$148$	3.08067e6	0.0781140
$149$	−2.44462e6	−0.0605424	−0.0302712	−	0.999542i	$-0.509637\pi$
$149$	−2.44462e6	−0.0605424	−0.0302712	+	0.999542i	$0.509637\pi$
$150$	1.11529e8	2.69815
$151$	−2.73196e7	−0.645736	−0.322868	−	0.946444i	$-0.604647\pi$
$151$	−2.73196e7	−0.645736	−0.322868	+	0.946444i	$0.604647\pi$
$152$	−1.43260e7	−0.330882
$153$	−1.88068e7	−0.424517
$154$	1.14129e7	0.251811
$155$	2.69978e7	0.582328
$156$	0	0
$157$	5.54490e7	1.14352	0.571761	−	0.820420i	$-0.306260\pi$
$157$	5.54490e7	1.14352	0.571761	+	0.820420i	$0.306260\pi$
$158$	6.47328e7	1.30564
$159$	−6.64668e7	−1.31134
$160$	−1.71537e7	−0.331084
$161$	−4.42551e7	−0.835744
$162$	2.23964e7	0.413881
$163$	3.20013e7	0.578777	0.289388	−	0.957212i	$-0.406548\pi$
$163$	3.20013e7	0.578777	0.289388	+	0.957212i	$0.406548\pi$
$164$	−3.85343e7	−0.682172
$165$	−1.27501e8	−2.20963
$166$	−4.45713e7	−0.756270
$167$	−1.93461e7	−0.321429	−0.160715	−	0.987001i	$-0.551380\pi$
$167$	−1.93461e7	−0.321429	−0.160715	+	0.987001i	$0.551380\pi$
$168$	−1.51854e7	−0.247083
$169$	0	0
$170$	2.73808e7	0.427439
$171$	8.04865e7	1.23094
$172$	5.86401e7	0.878709
$173$	4.13749e7	0.607540	0.303770	−	0.952745i	$-0.401754\pi$
$173$	4.13749e7	0.607540	0.303770	+	0.952745i	$0.401754\pi$
$174$	3.90210e7	0.561534
$175$	−8.16580e7	−1.15177
$176$	1.40197e7	0.193840
$177$	−8.35709e7	−1.13279
$178$	−6.30058e7	−0.837356
$179$	−3.22877e7	−0.420776	−0.210388	−	0.977618i	$-0.567473\pi$
$179$	−3.22877e7	−0.420776	−0.210388	+	0.977618i	$0.567473\pi$
$180$	9.63730e7	1.23169
$181$	−8.08975e7	−1.01405	−0.507026	−	0.861931i	$-0.669255\pi$
$181$	−8.08975e7	−1.01405	−0.507026	+	0.861931i	$0.669255\pi$
$182$	0	0
$183$	2.05712e8	2.48130
$184$	−5.43632e7	−0.643343
$185$	2.51984e7	0.292598
$186$	2.93587e7	0.334535
$187$	−2.23783e7	−0.250254
$188$	−2.08769e7	−0.229146
$189$	2.04504e7	0.220336
$190$	−1.17180e8	−1.23941
$191$	−1.36342e7	−0.141584	−0.0707919	−	0.997491i	$-0.522553\pi$
$191$	−1.36342e7	−0.141584	−0.0707919	+	0.997491i	$0.522553\pi$
$192$	−1.86538e7	−0.190201
$193$	1.73244e7	0.173463	0.0867316	−	0.996232i	$-0.472358\pi$
$193$	1.73244e7	0.173463	0.0867316	+	0.996232i	$0.472358\pi$
$194$	5.44678e7	0.535591
$195$	0	0
$196$	−4.15885e7	−0.394527
$197$	6.23747e7	0.581269	0.290634	−	0.956834i	$-0.406134\pi$
$197$	6.23747e7	0.581269	0.290634	+	0.956834i	$0.406134\pi$
$198$	−7.87656e7	−0.721121
$199$	−2.41707e6	−0.0217422	−0.0108711	−	0.999941i	$-0.503460\pi$
$199$	−2.41707e6	−0.0217422	−0.0108711	+	0.999941i	$0.503460\pi$
$200$	−1.00309e8	−0.886615
$201$	−2.26396e7	−0.196645
$202$	−1.28810e7	−0.109956
$203$	−2.85700e7	−0.239704
$204$	2.97752e7	0.245555
$205$	−3.15192e8	−2.55527
$206$	−3.94931e7	−0.314765
$207$	3.05423e8	2.39335
$208$	0	0
$209$	9.57711e7	0.725642
$210$	−1.24209e8	−0.925519
$211$	9.72678e7	0.712821	0.356411	−	0.934329i	$-0.384000\pi$
$211$	9.72678e7	0.712821	0.356411	+	0.934329i	$0.384000\pi$
$212$	5.97804e7	0.430907
$213$	−9.13487e7	−0.647700
$214$	9.81624e7	0.684694
$215$	4.79648e8	3.29146
$216$	2.51214e7	0.169612
$217$	−2.14956e7	−0.142804
$218$	1.32222e8	0.864385
$219$	1.18983e8	0.765474
$220$	1.14674e8	0.726085
$221$	0	0
$222$	2.74019e7	0.168091
$223$	1.67120e8	1.00917	0.504583	−	0.863363i	$-0.331646\pi$
$223$	1.67120e8	1.00917	0.504583	+	0.863363i	$0.331646\pi$
$224$	1.36577e7	0.0811915
$225$	5.63556e8	3.29836
$226$	−1.06301e8	−0.612572
$227$	5.70679e7	0.323818	0.161909	−	0.986806i	$-0.448235\pi$
$227$	5.70679e7	0.323818	0.161909	+	0.986806i	$0.448235\pi$
$228$	−1.27427e8	−0.712016
$229$	2.68753e8	1.47886	0.739432	−	0.673231i	$-0.235094\pi$
$229$	2.68753e8	1.47886	0.739432	+	0.673231i	$0.235094\pi$
$230$	−4.44665e8	−2.40982
$231$	1.01516e8	0.541866
$232$	−3.50955e7	−0.184520
$233$	−6.53750e7	−0.338584	−0.169292	−	0.985566i	$-0.554148\pi$
$233$	−6.53750e7	−0.338584	−0.169292	+	0.985566i	$0.554148\pi$
$234$	0	0
$235$	−1.70763e8	−0.858333
$236$	7.51638e7	0.372235
$237$	5.75785e8	2.80958
$238$	−2.18005e7	−0.104821
$239$	−2.06797e8	−0.979832	−0.489916	−	0.871770i	$-0.662973\pi$
$239$	−2.06797e8	−0.979832	−0.489916	+	0.871770i	$0.662973\pi$
$240$	−1.52579e8	−0.712450
$241$	−2.80476e8	−1.29073	−0.645365	−	0.763874i	$-0.723295\pi$
$241$	−2.80476e8	−1.29073	−0.645365	+	0.763874i	$0.723295\pi$
$242$	6.21740e7	0.282004
$243$	3.06517e8	1.37035
$244$	−1.85017e8	−0.815358
$245$	−3.40174e8	−1.47781
$246$	−3.42755e8	−1.46795
$247$	0	0
$248$	−2.64053e7	−0.109928
$249$	−3.96453e8	−1.62740
$250$	−4.93299e8	−1.99673
$251$	−3.11949e8	−1.24516	−0.622580	−	0.782556i	$-0.713916\pi$
$251$	−3.11949e8	−1.24516	−0.622580	+	0.782556i	$0.713916\pi$
$252$	−7.67320e7	−0.302047
$253$	3.63424e8	1.41089
$254$	−2.18298e7	−0.0835856
$255$	2.43546e8	0.919795
$256$	1.67772e7	0.0625000
$257$	2.14289e8	0.787471	0.393736	−	0.919224i	$-0.371183\pi$
$257$	2.14289e8	0.787471	0.393736	+	0.919224i	$0.371183\pi$
$258$	5.21592e8	1.89087
$259$	−2.00629e7	−0.0717537
$260$	0	0
$261$	1.97174e8	0.686448
$262$	−1.56833e8	−0.538743
$263$	1.74837e8	0.592637	0.296318	−	0.955089i	$-0.404241\pi$
$263$	1.74837e8	0.592637	0.296318	+	0.955089i	$0.404241\pi$
$264$	1.24702e8	0.417120
$265$	4.88975e8	1.61408
$266$	9.32984e7	0.303941
$267$	−5.60424e8	−1.80189
$268$	2.03621e7	0.0646176
$269$	2.83096e8	0.886749	0.443374	−	0.896336i	$-0.353781\pi$
$269$	2.83096e8	0.886749	0.443374	+	0.896336i	$0.353781\pi$
$270$	2.05481e8	0.635328
$271$	2.17755e8	0.664624	0.332312	−	0.943170i	$-0.392171\pi$
$271$	2.17755e8	0.664624	0.332312	+	0.943170i	$0.392171\pi$
$272$	−2.67798e7	−0.0806894
$273$	0	0
$274$	−5.21918e6	−0.0153276
$275$	6.70577e8	1.94439
$276$	−4.83550e8	−1.38439
$277$	−5.50259e8	−1.55556	−0.777782	−	0.628534i	$-0.783655\pi$
$277$	−5.50259e8	−1.55556	−0.777782	+	0.628534i	$0.783655\pi$
$278$	−7.37329e7	−0.205828
$279$	1.48350e8	0.408953
$280$	1.11714e8	0.304126
$281$	−1.42673e8	−0.383591	−0.191796	−	0.981435i	$-0.561431\pi$
$281$	−1.42673e8	−0.383591	−0.191796	+	0.981435i	$0.561431\pi$
$282$	−1.85696e8	−0.493094
$283$	−2.59247e7	−0.0679926	−0.0339963	−	0.999422i	$-0.510823\pi$
$283$	−2.59247e7	−0.0679926	−0.0339963	+	0.999422i	$0.510823\pi$
$284$	8.21592e7	0.212834
$285$	−1.04229e9	−2.66706
$286$	0	0
$287$	2.50955e8	0.626627
$288$	−9.42578e7	−0.232511
$289$	−3.67593e8	−0.895827
$290$	−2.87065e8	−0.691173
$291$	4.84480e8	1.15253
$292$	−1.07014e8	−0.251535
$293$	7.43864e8	1.72765	0.863827	−	0.503788i	$-0.168061\pi$
$293$	7.43864e8	1.72765	0.863827	+	0.503788i	$0.168061\pi$
$294$	−3.69921e8	−0.848972
$295$	6.14804e8	1.39431
$296$	−2.46453e7	−0.0552349
$297$	−1.67939e8	−0.371967
$298$	1.95570e7	0.0428099
$299$	0	0
$300$	−8.92229e8	−1.90788
$301$	−3.81895e8	−0.807163
$302$	2.18557e8	0.456605
$303$	−1.14574e8	−0.236611
$304$	1.14608e8	0.233969
$305$	−1.51335e9	−3.05415
$306$	1.50455e8	0.300179
$307$	2.86830e8	0.565771	0.282886	−	0.959154i	$-0.408708\pi$
$307$	2.86830e8	0.565771	0.282886	+	0.959154i	$0.408708\pi$
$308$	−9.13036e7	−0.178058
$309$	−3.51284e8	−0.677335
$310$	−2.15982e8	−0.411768
$311$	7.18890e8	1.35519	0.677596	−	0.735434i	$-0.263022\pi$
$311$	7.18890e8	1.35519	0.677596	+	0.735434i	$0.263022\pi$
$312$	0	0
$313$	6.20382e8	1.14355	0.571774	−	0.820411i	$-0.306255\pi$
$313$	6.20382e8	1.14355	0.571774	+	0.820411i	$0.306255\pi$
$314$	−4.43592e8	−0.808593
$315$	−6.27631e8	−1.13140
$316$	−5.17862e8	−0.923229
$317$	1.67254e8	0.294897	0.147448	−	0.989070i	$-0.452894\pi$
$317$	1.67254e8	0.294897	0.147448	+	0.989070i	$0.452894\pi$
$318$	5.31735e8	0.927257
$319$	2.34618e8	0.404663
$320$	1.37230e8	0.234112
$321$	8.73135e8	1.47338
$322$	3.54041e8	0.590960
$323$	−1.82938e8	−0.302061
$324$	−1.79171e8	−0.292658
$325$	0	0
$326$	−2.56010e8	−0.409257
$327$	1.17609e9	1.86005
$328$	3.08274e8	0.482368
$329$	1.35961e8	0.210489
$330$	1.02001e9	1.56244
$331$	7.89285e8	1.19629	0.598144	−	0.801388i	$-0.295905\pi$
$331$	7.89285e8	1.19629	0.598144	+	0.801388i	$0.295905\pi$
$332$	3.56570e8	0.534764
$333$	1.38463e8	0.205484
$334$	1.54769e8	0.227285
$335$	1.66552e8	0.242044
$336$	1.21483e8	0.174714
$337$	−6.94195e8	−0.988046	−0.494023	−	0.869449i	$-0.664474\pi$
$337$	−6.94195e8	−0.988046	−0.494023	+	0.869449i	$0.664474\pi$
$338$	0	0
$339$	−9.45525e8	−1.31818
$340$	−2.19046e8	−0.302245
$341$	1.76522e8	0.241079
$342$	−6.43892e8	−0.870405
$343$	6.14099e8	0.821692
$344$	−4.69121e8	−0.621341
$345$	−3.95520e9	−5.18564
$346$	−3.30999e8	−0.429596
$347$	−2.64528e7	−0.0339875	−0.0169937	−	0.999856i	$-0.505410\pi$
$347$	−2.64528e7	−0.0339875	−0.0169937	+	0.999856i	$0.505410\pi$
$348$	−3.12168e8	−0.397064
$349$	7.79391e8	0.981446	0.490723	−	0.871316i	$-0.336733\pi$
$349$	7.79391e8	0.981446	0.490723	+	0.871316i	$0.336733\pi$
$350$	6.53264e8	0.814424
$351$	0	0
$352$	−1.12158e8	−0.137066
$353$	−7.29412e8	−0.882595	−0.441297	−	0.897361i	$-0.645482\pi$
$353$	−7.29412e8	−0.882595	−0.441297	+	0.897361i	$0.645482\pi$
$354$	6.68567e8	0.801002
$355$	6.72023e8	0.797232
$356$	5.04046e8	0.592100
$357$	−1.93911e8	−0.225561
$358$	2.58301e8	0.297533
$359$	−5.70058e8	−0.650262	−0.325131	−	0.945669i	$-0.605408\pi$
$359$	−5.70058e8	−0.650262	−0.325131	+	0.945669i	$0.605408\pi$
$360$	−7.70984e8	−0.870936
$361$	−1.10964e8	−0.124138
$362$	6.47180e8	0.717042
$363$	5.53026e8	0.606837
$364$	0	0
$365$	−8.75319e8	−0.942196
$366$	−1.64569e9	−1.75455
$367$	−4.82133e8	−0.509139	−0.254569	−	0.967055i	$-0.581934\pi$
$367$	−4.82133e8	−0.509139	−0.254569	+	0.967055i	$0.581934\pi$
$368$	4.34905e8	0.454912
$369$	−1.73195e9	−1.79450
$370$	−2.01587e8	−0.206898
$371$	−3.89321e8	−0.395821
$372$	−2.34870e8	−0.236552
$373$	−2.99359e8	−0.298684	−0.149342	−	0.988786i	$-0.547716\pi$
$373$	−2.99359e8	−0.298684	−0.149342	+	0.988786i	$0.547716\pi$
$374$	1.79026e8	0.176956
$375$	−4.38779e9	−4.29672
$376$	1.67015e8	0.162031
$377$	0	0
$378$	−1.63604e8	−0.155801
$379$	1.01064e9	0.953587	0.476793	−	0.879015i	$-0.341799\pi$
$379$	1.01064e9	0.953587	0.476793	+	0.879015i	$0.341799\pi$
$380$	9.37439e8	0.876396
$381$	−1.94172e8	−0.179866
$382$	1.09074e8	0.100115
$383$	8.85754e8	0.805596	0.402798	−	0.915289i	$-0.368038\pi$
$383$	8.85754e8	0.805596	0.402798	+	0.915289i	$0.368038\pi$
$384$	1.49230e8	0.134492
$385$	−7.46819e8	−0.666965
$386$	−1.38595e8	−0.122657
$387$	2.63562e9	2.31150
$388$	−4.35742e8	−0.378720
$389$	2.15563e9	1.85674	0.928370	−	0.371656i	$-0.121210\pi$
$389$	2.15563e9	1.85674	0.928370	+	0.371656i	$0.121210\pi$
$390$	0	0
$391$	−6.94197e8	−0.587305
$392$	3.32708e8	0.278973
$393$	−1.39500e9	−1.15931
$394$	−4.98998e8	−0.411019
$395$	−4.23586e9	−3.45822
$396$	6.30124e8	0.509910
$397$	−9.47072e8	−0.759654	−0.379827	−	0.925057i	$-0.624017\pi$
$397$	−9.47072e8	−0.759654	−0.379827	+	0.925057i	$0.624017\pi$
$398$	1.93366e7	0.0153741
$399$	8.29871e8	0.654042
$400$	8.02472e8	0.626931
$401$	−1.53757e9	−1.19077	−0.595386	−	0.803440i	$-0.703001\pi$
$401$	−1.53757e9	−1.19077	−0.595386	+	0.803440i	$0.703001\pi$
$402$	1.81117e8	0.139049
$403$	0	0
$404$	1.03048e8	0.0777506
$405$	−1.46553e9	−1.09623
$406$	2.28560e8	0.169496
$407$	1.64757e8	0.121133
$408$	−2.38201e8	−0.173633
$409$	9.56109e8	0.690997	0.345498	−	0.938419i	$-0.387710\pi$
$409$	9.56109e8	0.690997	0.345498	+	0.938419i	$0.387710\pi$
$410$	2.52153e9	1.80685
$411$	−4.64236e7	−0.0329831
$412$	3.15945e8	0.222572
$413$	−4.89506e8	−0.341926
$414$	−2.44339e9	−1.69235
$415$	2.91657e9	2.00311
$416$	0	0
$417$	−6.55839e8	−0.442916
$418$	−7.66168e8	−0.513106
$419$	2.00130e9	1.32912	0.664559	−	0.747236i	$-0.268620\pi$
$419$	2.00130e9	1.32912	0.664559	+	0.747236i	$0.268620\pi$
$420$	9.93672e8	0.654441
$421$	2.08228e9	1.36004	0.680020	−	0.733194i	$-0.261971\pi$
$421$	2.08228e9	1.36004	0.680020	+	0.733194i	$0.261971\pi$
$422$	−7.78143e8	−0.504041
$423$	−9.38325e8	−0.602784
$424$	−4.78243e8	−0.304697
$425$	−1.28091e9	−0.809387
$426$	7.30790e8	0.457993
$427$	1.20493e9	0.748969
$428$	−7.85299e8	−0.484152
$429$	0	0
$430$	−3.83718e9	−2.32741
$431$	−7.37289e8	−0.443575	−0.221788	−	0.975095i	$-0.571189\pi$
$431$	−7.37289e8	−0.443575	−0.221788	+	0.975095i	$0.571189\pi$
$432$	−2.00971e8	−0.119933
$433$	−4.40104e8	−0.260524	−0.130262	−	0.991480i	$-0.541582\pi$
$433$	−4.40104e8	−0.260524	−0.130262	+	0.991480i	$0.541582\pi$
$434$	1.71965e8	0.100978
$435$	−2.55338e9	−1.48732
$436$	−1.05778e9	−0.611213
$437$	2.97091e9	1.70296
$438$	−9.51864e8	−0.541272
$439$	−2.17758e9	−1.22842	−0.614212	−	0.789141i	$-0.710526\pi$
$439$	−2.17758e9	−1.22842	−0.614212	+	0.789141i	$0.710526\pi$
$440$	−9.17395e8	−0.513419
$441$	−1.86922e9	−1.03783
$442$	0	0
$443$	1.78526e9	0.975639	0.487819	−	0.872945i	$-0.337793\pi$
$443$	1.78526e9	0.975639	0.487819	+	0.872945i	$0.337793\pi$
$444$	−2.19215e8	−0.118859
$445$	4.12285e9	2.21788
$446$	−1.33696e9	−0.713588
$447$	1.73955e8	0.0921216
$448$	−1.09262e8	−0.0574111
$449$	−9.37542e8	−0.488797	−0.244398	−	0.969675i	$-0.578590\pi$
$449$	−9.37542e8	−0.488797	−0.244398	+	0.969675i	$0.578590\pi$
$450$	−4.50845e9	−2.33230
$451$	−2.06085e9	−1.05786
$452$	8.50406e8	0.433154
$453$	1.94402e9	0.982556
$454$	−4.56543e8	−0.228974
$455$	0	0
$456$	1.01942e9	0.503471
$457$	3.07434e9	1.50676	0.753381	−	0.657584i	$-0.228421\pi$
$457$	3.07434e9	1.50676	0.753381	+	0.657584i	$0.228421\pi$
$458$	−2.15002e9	−1.04572
$459$	3.20790e8	0.154838
$460$	3.55732e9	1.70400
$461$	3.28755e9	1.56286	0.781428	−	0.623995i	$-0.214491\pi$
$461$	3.28755e9	1.56286	0.781428	+	0.623995i	$0.214491\pi$
$462$	−8.12127e8	−0.383157
$463$	8.40096e7	0.0393365	0.0196682	−	0.999807i	$-0.493739\pi$
$463$	8.40096e7	0.0393365	0.0196682	+	0.999807i	$0.493739\pi$
$464$	2.80764e8	0.130475
$465$	−1.92112e9	−0.886072
$466$	5.23000e8	0.239415
$467$	3.33082e9	1.51336	0.756681	−	0.653785i	$-0.226820\pi$
$467$	3.33082e9	1.51336	0.756681	+	0.653785i	$0.226820\pi$
$468$	0	0
$469$	−1.32609e8	−0.0593563
$470$	1.36610e9	0.606933
$471$	−3.94566e9	−1.73999
$472$	−6.01310e8	−0.263210
$473$	3.13613e9	1.36264
$474$	−4.60628e9	−1.98667
$475$	5.48183e9	2.34692
$476$	1.74404e8	0.0741195
$477$	2.68687e9	1.13353
$478$	1.65438e9	0.692846
$479$	−8.98971e8	−0.373741	−0.186871	−	0.982385i	$-0.559835\pi$
$479$	−8.98971e8	−0.373741	−0.186871	+	0.982385i	$0.559835\pi$
$480$	1.22063e9	0.503779
$481$	0	0
$482$	2.24380e9	0.912684
$483$	3.14913e9	1.27167
$484$	−4.97392e8	−0.199407
$485$	−3.56416e9	−1.41860
$486$	−2.45214e9	−0.968986
$487$	−2.66963e9	−1.04737	−0.523685	−	0.851912i	$-0.675443\pi$
$487$	−2.66963e9	−1.04737	−0.523685	+	0.851912i	$0.675443\pi$
$488$	1.48014e9	0.576545
$489$	−2.27716e9	−0.880670
$490$	2.72139e9	1.04497
$491$	3.41098e9	1.30045	0.650225	−	0.759741i	$-0.274674\pi$
$491$	3.41098e9	1.30045	0.650225	+	0.759741i	$0.274674\pi$
$492$	2.74204e9	1.03800
$493$	−4.48157e8	−0.168448
$494$	0	0
$495$	5.15412e9	1.91001
$496$	2.11242e8	0.0777311
$497$	−5.35063e8	−0.195505
$498$	3.17162e9	1.15074
$499$	−5.43259e9	−1.95729	−0.978645	−	0.205559i	$-0.934099\pi$
$499$	−5.43259e9	−1.95729	−0.978645	+	0.205559i	$0.934099\pi$
$500$	3.94639e9	1.41190
$501$	1.37664e9	0.489088
$502$	2.49559e9	0.880461
$503$	1.16913e9	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$503$	1.16913e9	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$504$	6.13856e8	0.213579
$505$	8.42881e8	0.291237
$506$	−2.90739e9	−0.997647
$507$	0	0
$508$	1.74638e8	0.0591040
$509$	2.17002e9	0.729375	0.364688	−	0.931130i	$-0.381176\pi$
$509$	2.17002e9	0.729375	0.364688	+	0.931130i	$0.381176\pi$
$510$	−1.94837e9	−0.650394
$511$	6.96928e8	0.231055
$512$	−1.34218e8	−0.0441942
$513$	−1.37287e9	−0.448971
$514$	−1.71432e9	−0.556826
$515$	2.58428e9	0.833708
$516$	−4.17274e9	−1.33705
$517$	−1.11651e9	−0.355343
$518$	1.60503e8	0.0507376
$519$	−2.94417e9	−0.924436
$520$	0	0
$521$	−5.57218e9	−1.72621	−0.863103	−	0.505027i	$-0.831482\pi$
$521$	−5.57218e9	−1.72621	−0.863103	+	0.505027i	$0.831482\pi$
$522$	−1.57739e9	−0.485392
$523$	−6.67518e8	−0.204036	−0.102018	−	0.994783i	$-0.532530\pi$
$523$	−6.67518e8	−0.204036	−0.102018	+	0.994783i	$0.532530\pi$
$524$	1.25466e9	0.380949
$525$	5.81066e9	1.75254
$526$	−1.39870e9	−0.419058
$527$	−3.37185e8	−0.100353
$528$	−9.97620e8	−0.294949
$529$	7.86895e9	2.31112
$530$	−3.91180e9	−1.14133
$531$	3.37828e9	0.979187
$532$	−7.46387e8	−0.214918
$533$	0	0
$534$	4.48339e9	1.27413
$535$	−6.42337e9	−1.81353
$536$	−1.62897e8	−0.0456916
$537$	2.29754e9	0.640255
$538$	−2.26477e9	−0.627026
$539$	−2.22419e9	−0.611802
$540$	−1.64385e9	−0.449245
$541$	4.87524e9	1.32375	0.661875	−	0.749614i	$-0.269761\pi$
$541$	4.87524e9	1.32375	0.661875	+	0.749614i	$0.269761\pi$
$542$	−1.74204e9	−0.469960
$543$	5.75654e9	1.54299
$544$	2.14239e8	0.0570560
$545$	−8.65212e9	−2.28947
$546$	0	0
$547$	6.55899e8	0.171349	0.0856744	−	0.996323i	$-0.472696\pi$
$547$	6.55899e8	0.171349	0.0856744	+	0.996323i	$0.472696\pi$
$548$	4.17534e7	0.0108383
$549$	−8.31573e9	−2.14485
$550$	−5.36462e9	−1.37489
$551$	1.91795e9	0.488435
$552$	3.86840e9	0.978913
$553$	3.37259e9	0.848057
$554$	4.40207e9	1.09995
$555$	−1.79308e9	−0.445219
$556$	5.89863e8	0.145542
$557$	−1.46638e9	−0.359545	−0.179773	−	0.983708i	$-0.557536\pi$
$557$	−1.46638e9	−0.359545	−0.179773	+	0.983708i	$0.557536\pi$
$558$	−1.18680e9	−0.289173
$559$	0	0
$560$	−8.93710e8	−0.215050
$561$	1.59240e9	0.380788
$562$	1.14138e9	0.271240
$563$	−3.01372e9	−0.711743	−0.355872	−	0.934535i	$-0.615816\pi$
$563$	−3.01372e9	−0.711743	−0.355872	+	0.934535i	$0.615816\pi$
$564$	1.48557e9	0.348670
$565$	6.95592e9	1.62250
$566$	2.07398e8	0.0480780
$567$	1.16686e9	0.268829
$568$	−6.57273e8	−0.150497
$569$	−7.50194e9	−1.70718	−0.853592	−	0.520942i	$-0.825581\pi$
$569$	−7.50194e9	−1.70718	−0.853592	+	0.520942i	$0.825581\pi$
$570$	8.33834e9	1.88589
$571$	−1.19452e9	−0.268514	−0.134257	−	0.990947i	$-0.542865\pi$
$571$	−1.19452e9	−0.268514	−0.134257	+	0.990947i	$0.542865\pi$
$572$	0	0
$573$	9.70190e8	0.215435
$574$	−2.00764e9	−0.443093
$575$	2.08020e10	4.56318
$576$	7.54063e8	0.164410
$577$	−4.85637e9	−1.05244	−0.526219	−	0.850349i	$-0.676391\pi$
$577$	−4.85637e9	−1.05244	−0.526219	+	0.850349i	$0.676391\pi$
$578$	2.94074e9	0.633446
$579$	−1.23278e9	−0.263943
$580$	2.29652e9	0.488733
$581$	−2.32217e9	−0.491222
$582$	−3.87584e9	−0.814959
$583$	3.19711e9	0.668217
$584$	8.56108e8	0.177862
$585$	0	0
$586$	−5.95091e9	−1.22164
$587$	−2.02484e9	−0.413196	−0.206598	−	0.978426i	$-0.566239\pi$
$587$	−2.02484e9	−0.413196	−0.206598	+	0.978426i	$0.566239\pi$
$588$	2.95937e9	0.600314
$589$	1.44303e9	0.290986
$590$	−4.91843e9	−0.985926
$591$	−4.43849e9	−0.884461
$592$	1.97163e8	0.0390570
$593$	9.30040e8	0.183151	0.0915757	−	0.995798i	$-0.470810\pi$
$593$	9.30040e8	0.183151	0.0915757	+	0.995798i	$0.470810\pi$
$594$	1.34351e9	0.263021
$595$	1.42654e9	0.277636
$596$	−1.56456e8	−0.0302712
$597$	1.71995e8	0.0330831
$598$	0	0
$599$	2.25978e9	0.429608	0.214804	−	0.976657i	$-0.431089\pi$
$599$	2.25978e9	0.429608	0.214804	+	0.976657i	$0.431089\pi$
$600$	7.13783e9	1.34908
$601$	1.45670e9	0.273722	0.136861	−	0.990590i	$-0.456299\pi$
$601$	1.45670e9	0.273722	0.136861	+	0.990590i	$0.456299\pi$
$602$	3.05516e9	0.570750
$603$	9.15188e8	0.169981
$604$	−1.74846e9	−0.322868
$605$	−4.06843e9	−0.746935
$606$	9.16589e8	0.167309
$607$	−5.28287e9	−0.958758	−0.479379	−	0.877608i	$-0.659138\pi$
$607$	−5.28287e9	−0.958758	−0.479379	+	0.877608i	$0.659138\pi$
$608$	−9.16865e8	−0.165441
$609$	2.03300e9	0.364734
$610$	1.21068e10	2.15961
$611$	0	0
$612$	−1.20364e9	−0.212259
$613$	−2.41287e9	−0.423079	−0.211540	−	0.977369i	$-0.567848\pi$
$613$	−2.41287e9	−0.423079	−0.211540	+	0.977369i	$0.567848\pi$
$614$	−2.29464e9	−0.400061
$615$	2.24285e10	3.88811
$616$	7.30429e8	0.125906
$617$	9.37862e9	1.60746	0.803732	−	0.594992i	$-0.202845\pi$
$617$	9.37862e9	1.60746	0.803732	+	0.594992i	$0.202845\pi$
$618$	2.81027e9	0.478948
$619$	2.73222e9	0.463019	0.231510	−	0.972833i	$-0.425634\pi$
$619$	2.73222e9	0.463019	0.231510	+	0.972833i	$0.425634\pi$
$620$	1.72786e9	0.291164
$621$	−5.20965e9	−0.872947
$622$	−5.75112e9	−0.958266
$623$	−3.28261e9	−0.543890
$624$	0	0
$625$	1.69736e10	2.78096
$626$	−4.96306e9	−0.808610
$627$	−6.81492e9	−1.10414
$628$	3.54873e9	0.571761
$629$	−3.14711e8	−0.0504238
$630$	5.02105e9	0.800023
$631$	9.05266e9	1.43441	0.717205	−	0.696862i	$-0.245421\pi$
$631$	9.05266e9	1.43441	0.717205	+	0.696862i	$0.245421\pi$
$632$	4.14290e9	0.652821
$633$	−6.92142e9	−1.08463
$634$	−1.33803e9	−0.208523
$635$	1.42846e9	0.221391
$636$	−4.25388e9	−0.655670
$637$	0	0
$638$	−1.87694e9	−0.286140
$639$	3.69270e9	0.559874
$640$	−1.09784e9	−0.165542
$641$	−1.03610e10	−1.55382	−0.776910	−	0.629612i	$-0.783214\pi$
$641$	−1.03610e10	−1.55382	−0.776910	+	0.629612i	$0.783214\pi$
$642$	−6.98508e9	−1.04183
$643$	−3.92800e9	−0.582684	−0.291342	−	0.956619i	$-0.594102\pi$
$643$	−3.92800e9	−0.582684	−0.291342	+	0.956619i	$0.594102\pi$
$644$	−2.83233e9	−0.417872
$645$	−3.41310e10	−5.00829
$646$	1.46350e9	0.213589
$647$	−1.12735e10	−1.63641	−0.818206	−	0.574926i	$-0.805031\pi$
$647$	−1.12735e10	−1.63641	−0.818206	+	0.574926i	$0.805031\pi$
$648$	1.43337e9	0.206940
$649$	4.01983e9	0.577233
$650$	0	0
$651$	1.52959e9	0.217291
$652$	2.04808e9	0.289388
$653$	−1.38664e10	−1.94880	−0.974401	−	0.224815i	$-0.927822\pi$
$653$	−1.38664e10	−1.94880	−0.974401	+	0.224815i	$0.927822\pi$
$654$	−9.40874e9	−1.31525
$655$	1.02625e10	1.42695
$656$	−2.46619e9	−0.341086
$657$	−4.80979e9	−0.661679
$658$	−1.08769e9	−0.148838
$659$	−1.58849e8	−0.0216215	−0.0108107	−	0.999942i	$-0.503441\pi$
$659$	−1.58849e8	−0.0216215	−0.0108107	+	0.999942i	$0.503441\pi$
$660$	−8.16005e9	−1.10481
$661$	1.11754e10	1.50508	0.752539	−	0.658548i	$-0.228829\pi$
$661$	1.11754e10	1.50508	0.752539	+	0.658548i	$0.228829\pi$
$662$	−6.31428e9	−0.845904
$663$	0	0
$664$	−2.85256e9	−0.378135
$665$	−6.10509e9	−0.805038
$666$	−1.10770e9	−0.145299
$667$	7.27808e9	0.949678
$668$	−1.23815e9	−0.160715
$669$	−1.18920e10	−1.53555
$670$	−1.33242e9	−0.171151
$671$	−9.89490e9	−1.26439
$672$	−9.71863e8	−0.123541
$673$	−5.39911e9	−0.682762	−0.341381	−	0.939925i	$-0.610895\pi$
$673$	−5.39911e9	−0.682762	−0.341381	+	0.939925i	$0.610895\pi$
$674$	5.55356e9	0.698654
$675$	−9.61266e9	−1.20304
$676$	0	0
$677$	−7.69486e9	−0.953104	−0.476552	−	0.879146i	$-0.658114\pi$
$677$	−7.69486e9	−0.953104	−0.476552	+	0.879146i	$0.658114\pi$
$678$	7.56420e9	0.932092
$679$	2.83778e9	0.347884
$680$	1.75237e9	0.213720
$681$	−4.06086e9	−0.492724
$682$	−1.41218e9	−0.170468
$683$	−5.09851e9	−0.612309	−0.306155	−	0.951982i	$-0.599042\pi$
$683$	−5.09851e9	−0.612309	−0.306155	+	0.951982i	$0.599042\pi$
$684$	5.15113e9	0.615469
$685$	3.41523e8	0.0405978
$686$	−4.91279e9	−0.581024
$687$	−1.91240e10	−2.25025
$688$	3.75297e9	0.439355
$689$	0	0
$690$	3.16416e10	3.66680
$691$	−3.07222e8	−0.0354225	−0.0177112	−	0.999843i	$-0.505638\pi$
$691$	−3.07222e8	−0.0354225	−0.0177112	+	0.999843i	$0.505638\pi$
$692$	2.64799e9	0.303770
$693$	−4.10370e9	−0.468392
$694$	2.11623e8	0.0240328
$695$	4.82480e9	0.545170
$696$	2.49734e9	0.280767
$697$	3.93654e9	0.440352
$698$	−6.23513e9	−0.693987
$699$	4.65198e9	0.515191
$700$	−5.22611e9	−0.575885
$701$	1.35330e9	0.148382	0.0741910	−	0.997244i	$-0.476363\pi$
$701$	1.35330e9	0.148382	0.0741910	+	0.997244i	$0.476363\pi$
$702$	0	0
$703$	1.34685e9	0.146210
$704$	8.97261e8	0.0969202
$705$	1.21512e10	1.30604
$706$	5.83529e9	0.624089
$707$	−6.71100e8	−0.0714199
$708$	−5.34854e9	−0.566394
$709$	1.09112e10	1.14977	0.574887	−	0.818233i	$-0.305046\pi$
$709$	1.09112e10	1.14977	0.574887	+	0.818233i	$0.305046\pi$
$710$	−5.37618e9	−0.563728
$711$	−2.32756e10	−2.42861
$712$	−4.03237e9	−0.418678
$713$	5.47590e9	0.565773
$714$	1.55129e9	0.159496
$715$	0	0
$716$	−2.06641e9	−0.210388
$717$	1.47154e10	1.49092
$718$	4.56047e9	0.459805
$719$	−1.02932e10	−1.03276	−0.516379	−	0.856360i	$-0.672720\pi$
$719$	−1.02932e10	−1.03276	−0.516379	+	0.856360i	$0.672720\pi$
$720$	6.16787e9	0.615845
$721$	−2.05760e9	−0.204450
$722$	8.87709e8	0.0877790
$723$	1.99582e10	1.96398
$724$	−5.17744e9	−0.507026
$725$	1.34293e10	1.30879
$726$	−4.42421e9	−0.429099
$727$	7.61641e9	0.735156	0.367578	−	0.929993i	$-0.380187\pi$
$727$	7.61641e9	0.735156	0.367578	+	0.929993i	$0.380187\pi$
$728$	0	0
$729$	−1.56887e10	−1.49982
$730$	7.00255e9	0.666233
$731$	−5.99049e9	−0.567220
$732$	1.31655e10	1.24065
$733$	5.63920e9	0.528875	0.264438	−	0.964403i	$-0.414814\pi$
$733$	5.63920e9	0.528875	0.264438	+	0.964403i	$0.414814\pi$
$734$	3.85707e9	0.360015
$735$	2.42062e10	2.24865
$736$	−3.47924e9	−0.321671
$737$	1.08898e9	0.100204
$738$	1.38556e10	1.26890
$739$	2.18346e10	1.99017	0.995083	−	0.0990445i	$-0.0315786\pi$
$739$	2.18346e10	1.99017	0.995083	+	0.0990445i	$0.0315786\pi$
$740$	1.61270e9	0.146299
$741$	0	0
$742$	3.11457e9	0.279888
$743$	−1.14865e10	−1.02737	−0.513684	−	0.857979i	$-0.671720\pi$
$743$	−1.14865e10	−1.02737	−0.513684	+	0.857979i	$0.671720\pi$
$744$	1.87896e9	0.167267
$745$	−1.27973e9	−0.113389
$746$	2.39487e9	0.211201
$747$	1.60263e10	1.40673
$748$	−1.43221e9	−0.125127
$749$	5.11427e9	0.444731
$750$	3.51023e10	3.03824
$751$	6.97497e9	0.600900	0.300450	−	0.953798i	$-0.402863\pi$
$751$	6.97497e9	0.600900	0.300450	+	0.953798i	$0.402863\pi$
$752$	−1.33612e9	−0.114573
$753$	2.21978e10	1.89464
$754$	0	0
$755$	−1.43015e10	−1.20939
$756$	1.30883e9	0.110168
$757$	−3.17517e9	−0.266030	−0.133015	−	0.991114i	$-0.542466\pi$
$757$	−3.17517e9	−0.266030	−0.133015	+	0.991114i	$0.542466\pi$
$758$	−8.08514e9	−0.674288
$759$	−2.58607e10	−2.14681
$760$	−7.49951e9	−0.619706
$761$	1.53073e10	1.25908	0.629540	−	0.776968i	$-0.283243\pi$
$761$	1.53073e10	1.25908	0.629540	+	0.776968i	$0.283243\pi$
$762$	1.55337e9	0.127184
$763$	6.88881e9	0.561446
$764$	−8.72590e8	−0.0707919
$765$	−9.84517e9	−0.795075
$766$	−7.08603e9	−0.569642
$767$	0	0
$768$	−1.19384e9	−0.0951003
$769$	2.95358e9	0.234211	0.117105	−	0.993119i	$-0.462638\pi$
$769$	2.95358e9	0.234211	0.117105	+	0.993119i	$0.462638\pi$
$770$	5.97456e9	0.471615
$771$	−1.52485e10	−1.19822
$772$	1.10876e9	0.0867316
$773$	9.84588e9	0.766701	0.383351	−	0.923603i	$-0.374770\pi$
$773$	9.84588e9	0.766701	0.383351	+	0.923603i	$0.374770\pi$
$774$	−2.10849e10	−1.63448
$775$	1.01039e10	0.779712
$776$	3.48594e9	0.267796
$777$	1.42764e9	0.109181
$778$	−1.72451e10	−1.31291
$779$	−1.68470e10	−1.27685
$780$	0	0
$781$	4.39395e9	0.330047
$782$	5.55357e9	0.415288
$783$	−3.36322e9	−0.250374
$784$	−2.66166e9	−0.197263
$785$	2.90269e10	2.14169
$786$	1.11600e10	0.819755
$787$	−5.21130e9	−0.381096	−0.190548	−	0.981678i	$-0.561026\pi$
$787$	−5.21130e9	−0.381096	−0.190548	+	0.981678i	$0.561026\pi$
$788$	3.99198e9	0.290634
$789$	−1.24411e10	−0.901759
$790$	3.38869e10	2.44533
$791$	−5.53829e9	−0.397885
$792$	−5.04100e9	−0.360561
$793$	0	0
$794$	7.57657e9	0.537157
$795$	−3.47947e10	−2.45600
$796$	−1.54693e8	−0.0108711
$797$	7.85960e9	0.549916	0.274958	−	0.961456i	$-0.411336\pi$
$797$	7.85960e9	0.549916	0.274958	+	0.961456i	$0.411336\pi$
$798$	−6.63897e9	−0.462477
$799$	2.13272e9	0.147918
$800$	−6.41978e9	−0.443307
$801$	2.26547e10	1.55756
$802$	1.23005e10	0.842003
$803$	−5.72318e9	−0.390062
$804$	−1.44894e9	−0.0983225
$805$	−2.31671e10	−1.56526
$806$	0	0
$807$	−2.01447e10	−1.34928
$808$	−8.24382e8	−0.0549780
$809$	−6.45868e9	−0.428868	−0.214434	−	0.976738i	$-0.568791\pi$
$809$	−6.45868e9	−0.428868	−0.214434	+	0.976738i	$0.568791\pi$
$810$	1.17243e10	0.775154
$811$	−2.35320e10	−1.54912	−0.774562	−	0.632498i	$-0.782030\pi$
$811$	−2.35320e10	−1.54912	−0.774562	+	0.632498i	$0.782030\pi$
$812$	−1.82848e9	−0.119852
$813$	−1.54951e10	−1.01130
$814$	−1.31806e9	−0.0856541
$815$	1.67523e10	1.08399
$816$	1.90561e9	0.122777
$817$	2.56372e10	1.64472
$818$	−7.64887e9	−0.488608
$819$	0	0
$820$	−2.01723e10	−1.27763
$821$	−2.52398e9	−0.159179	−0.0795894	−	0.996828i	$-0.525361\pi$
$821$	−2.52398e9	−0.159179	−0.0795894	+	0.996828i	$0.525361\pi$
$822$	3.71388e8	0.0233226
$823$	−1.73764e10	−1.08658	−0.543289	−	0.839546i	$-0.682821\pi$
$823$	−1.73764e10	−1.08658	−0.543289	+	0.839546i	$0.682821\pi$
$824$	−2.52756e9	−0.157382
$825$	−4.77172e10	−2.95860
$826$	3.91605e9	0.241778
$827$	1.96660e10	1.20906	0.604530	−	0.796583i	$-0.293361\pi$
$827$	1.96660e10	1.20906	0.604530	+	0.796583i	$0.293361\pi$
$828$	1.95471e10	1.19667
$829$	3.06688e10	1.86963	0.934817	−	0.355130i	$-0.115563\pi$
$829$	3.06688e10	1.86963	0.934817	+	0.355130i	$0.115563\pi$
$830$	−2.33326e10	−1.41641
$831$	3.91555e10	2.36695
$832$	0	0
$833$	4.24855e9	0.254673
$834$	5.24672e9	0.313189
$835$	−1.01275e10	−0.602002
$836$	6.12935e9	0.362821
$837$	−2.53043e9	−0.149161
$838$	−1.60104e10	−0.939828
$839$	−3.00838e10	−1.75859	−0.879297	−	0.476275i	$-0.841987\pi$
$839$	−3.00838e10	−1.75859	−0.879297	+	0.476275i	$0.841987\pi$
$840$	−7.94937e9	−0.462760
$841$	−1.25513e10	−0.727618
$842$	−1.66582e10	−0.961693
$843$	1.01524e10	0.583674
$844$	6.22514e9	0.356411
$845$	0	0
$846$	7.50660e9	0.426233
$847$	3.23928e9	0.183171
$848$	3.82594e9	0.215453
$849$	1.84476e9	0.103458
$850$	1.02473e10	0.572323
$851$	5.11092e9	0.284280
$852$	−5.84632e9	−0.323850
$853$	−8.80922e9	−0.485977	−0.242989	−	0.970029i	$-0.578128\pi$
$853$	−8.80922e9	−0.485977	−0.242989	+	0.970029i	$0.578128\pi$
$854$	−9.63943e9	−0.529601
$855$	4.21338e10	2.30542
$856$	6.28239e9	0.342347
$857$	1.71306e10	0.929696	0.464848	−	0.885391i	$-0.346109\pi$
$857$	1.71306e10	0.929696	0.464848	+	0.885391i	$0.346109\pi$
$858$	0	0
$859$	9.23130e9	0.496921	0.248460	−	0.968642i	$-0.420075\pi$
$859$	9.23130e9	0.496921	0.248460	+	0.968642i	$0.420075\pi$
$860$	3.06975e10	1.64573
$861$	−1.78576e10	−0.953479
$862$	5.89831e9	0.313655
$863$	−2.02167e10	−1.07071	−0.535356	−	0.844627i	$-0.679822\pi$
$863$	−2.02167e10	−1.07071	−0.535356	+	0.844627i	$0.679822\pi$
$864$	1.60777e9	0.0848058
$865$	2.16593e10	1.13786
$866$	3.52083e9	0.184218
$867$	2.61573e10	1.36310
$868$	−1.37572e9	−0.0714020
$869$	−2.76957e10	−1.43167
$870$	2.04271e10	1.05169
$871$	0	0
$872$	8.46223e9	0.432193
$873$	−1.95847e10	−0.996248
$874$	−2.37673e10	−1.20418
$875$	−2.57009e10	−1.29694
$876$	7.61491e9	0.382737
$877$	−1.42090e10	−0.711318	−0.355659	−	0.934616i	$-0.615744\pi$
$877$	−1.42090e10	−0.711318	−0.355659	+	0.934616i	$0.615744\pi$
$878$	1.74207e10	0.868628
$879$	−5.29322e10	−2.62881
$880$	7.33916e9	0.363042
$881$	−1.51920e10	−0.748511	−0.374256	−	0.927326i	$-0.622102\pi$
$881$	−1.51920e10	−0.748511	−0.374256	+	0.927326i	$0.622102\pi$
$882$	1.49538e10	0.733855
$883$	5.15139e8	0.0251803	0.0125902	−	0.999921i	$-0.495992\pi$
$883$	5.15139e8	0.0251803	0.0125902	+	0.999921i	$0.495992\pi$
$884$	0	0
$885$	−4.37485e10	−2.12159
$886$	−1.42821e10	−0.689881
$887$	2.75987e10	1.32787	0.663936	−	0.747789i	$-0.268885\pi$
$887$	2.75987e10	1.32787	0.663936	+	0.747789i	$0.268885\pi$
$888$	1.75372e9	0.0840457
$889$	−1.13734e9	−0.0542916
$890$	−3.29828e10	−1.56828
$891$	−9.58223e9	−0.453832
$892$	1.06957e10	0.504583
$893$	−9.12727e9	−0.428905
$894$	−1.39164e9	−0.0651398
$895$	−1.69022e10	−0.788068
$896$	8.74095e8	0.0405958
$897$	0	0
$898$	7.50033e9	0.345632
$899$	3.53511e9	0.162272
$900$	3.60676e10	1.64918
$901$	−6.10698e9	−0.278157
$902$	1.64868e10	0.748020
$903$	2.71750e10	1.22818
$904$	−6.80325e9	−0.306286
$905$	−4.23489e10	−1.89921
$906$	−1.55522e10	−0.694772
$907$	−2.39342e10	−1.06511	−0.532555	−	0.846396i	$-0.678768\pi$
$907$	−2.39342e10	−1.06511	−0.532555	+	0.846396i	$0.678768\pi$
$908$	3.65235e9	0.161909
$909$	4.63155e9	0.204528
$910$	0	0
$911$	−2.98472e10	−1.30794	−0.653972	−	0.756519i	$-0.726898\pi$
$911$	−2.98472e10	−1.30794	−0.653972	+	0.756519i	$0.726898\pi$
$912$	−8.15533e9	−0.356008
$913$	1.90697e10	0.829271
$914$	−2.45947e10	−1.06544
$915$	1.07688e11	4.64722
$916$	1.72002e10	0.739432
$917$	−8.17100e9	−0.349931
$918$	−2.56632e9	−0.109487
$919$	−8.23752e9	−0.350100	−0.175050	−	0.984560i	$-0.556009\pi$
$919$	−8.23752e9	−0.350100	−0.175050	+	0.984560i	$0.556009\pi$
$920$	−2.84585e10	−1.20491
$921$	−2.04104e10	−0.860880
$922$	−2.63004e10	−1.10511
$923$	0	0
$924$	6.49702e9	0.270933
$925$	9.43050e9	0.391777
$926$	−6.72077e8	−0.0278151
$927$	1.42004e10	0.585491
$928$	−2.24611e9	−0.0922601
$929$	3.30769e10	1.35353	0.676767	−	0.736197i	$-0.263380\pi$
$929$	3.30769e10	1.35353	0.676767	+	0.736197i	$0.263380\pi$
$930$	1.53690e10	0.626548
$931$	−1.81823e10	−0.738456
$932$	−4.18400e9	−0.169292
$933$	−5.11551e10	−2.06207
$934$	−2.66466e10	−1.07011
$935$	−1.17148e10	−0.468699
$936$	0	0
$937$	6.86759e9	0.272719	0.136360	−	0.990659i	$-0.456460\pi$
$937$	6.86759e9	0.272719	0.136360	+	0.990659i	$0.456460\pi$
$938$	1.06087e9	0.0419712
$939$	−4.41454e10	−1.74003
$940$	−1.09288e10	−0.429167
$941$	−3.10041e10	−1.21298	−0.606492	−	0.795090i	$-0.707424\pi$
$941$	−3.10041e10	−1.21298	−0.606492	+	0.795090i	$0.707424\pi$
$942$	3.15653e10	1.23036
$943$	−6.39296e10	−2.48262
$944$	4.81048e9	0.186117
$945$	1.07056e10	0.412666
$946$	−2.50890e10	−0.963529
$947$	−2.16546e10	−0.828561	−0.414280	−	0.910149i	$-0.635967\pi$
$947$	−2.16546e10	−0.828561	−0.414280	+	0.910149i	$0.635967\pi$
$948$	3.68503e10	1.40479
$949$	0	0
$950$	−4.38546e10	−1.65952
$951$	−1.19016e10	−0.448716
$952$	−1.39523e9	−0.0524104
$953$	−1.23673e10	−0.462862	−0.231431	−	0.972851i	$-0.574341\pi$
$953$	−1.23673e10	−0.462862	−0.231431	+	0.972851i	$0.574341\pi$
$954$	−2.14949e10	−0.801525
$955$	−7.13737e9	−0.265171
$956$	−1.32350e10	−0.489916
$957$	−1.66950e10	−0.615737
$958$	7.19177e9	0.264275
$959$	−2.71920e8	−0.00995579
$960$	−9.76504e9	−0.356225
$961$	−2.48529e10	−0.903326
$962$	0	0
$963$	−3.52958e10	−1.27359
$964$	−1.79504e10	−0.645365
$965$	9.06914e9	0.324878
$966$	−2.51930e10	−0.899208
$967$	3.71093e10	1.31974	0.659872	−	0.751378i	$-0.270610\pi$
$967$	3.71093e10	1.31974	0.659872	+	0.751378i	$0.270610\pi$
$968$	3.97914e9	0.141002
$969$	1.30176e10	0.459617
$970$	2.85133e10	1.00310
$971$	−2.70025e10	−0.946534	−0.473267	−	0.880919i	$-0.656925\pi$
$971$	−2.70025e10	−0.946534	−0.473267	+	0.880919i	$0.656925\pi$
$972$	1.96171e10	0.685177
$973$	−3.84149e9	−0.133692
$974$	2.13570e10	0.740602
$975$	0	0
$976$	−1.18411e10	−0.407679
$977$	3.00925e10	1.03235	0.516175	−	0.856483i	$-0.327356\pi$
$977$	3.00925e10	1.03235	0.516175	+	0.856483i	$0.327356\pi$
$978$	1.82173e10	0.622727
$979$	2.69568e10	0.918184
$980$	−2.17711e10	−0.738906
$981$	−4.75426e10	−1.60783
$982$	−2.72878e10	−0.919557
$983$	−5.51343e10	−1.85133	−0.925667	−	0.378340i	$-0.876495\pi$
$983$	−5.51343e10	−1.85133	−0.925667	+	0.378340i	$0.876495\pi$
$984$	−2.19363e10	−0.733974
$985$	3.26525e10	1.08865
$986$	3.58525e9	0.119111
$987$	−9.67477e9	−0.320281
$988$	0	0
$989$	9.72858e10	3.19788
$990$	−4.12329e10	−1.35058
$991$	−1.53856e10	−0.502177	−0.251089	−	0.967964i	$-0.580789\pi$
$991$	−1.53856e10	−0.502177	−0.251089	+	0.967964i	$0.580789\pi$
$992$	−1.68994e9	−0.0549642
$993$	−5.61643e10	−1.82028
$994$	4.28051e9	0.138243
$995$	−1.26531e9	−0.0407209
$996$	−2.53730e10	−0.813699
$997$	−2.79656e10	−0.893700	−0.446850	−	0.894609i	$-0.647454\pi$
$997$	−2.79656e10	−0.893700	−0.446850	+	0.894609i	$0.647454\pi$
$998$	4.34607e10	1.38401
$999$	−2.36177e9	−0.0749478

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.i.1.1		4
13.3	even	3		26.8.c.b.9.4	yes	8
13.5	odd	4		338.8.b.h.337.5		8
13.8	odd	4		338.8.b.h.337.1		8
13.9	even	3		26.8.c.b.3.4	✓	8
13.12	even	2		338.8.a.j.1.1		4
39.29	odd	6		234.8.h.b.217.1		8
39.35	odd	6		234.8.h.b.55.1		8
52.3	odd	6		208.8.i.b.113.1		8
52.35	odd	6		208.8.i.b.81.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.c.b.3.4	✓	8	13.9	even	3
26.8.c.b.9.4	yes	8	13.3	even	3
208.8.i.b.81.1		8	52.35	odd	6
208.8.i.b.113.1		8	52.3	odd	6
234.8.h.b.55.1		8	39.35	odd	6
234.8.h.b.217.1		8	39.29	odd	6
338.8.a.i.1.1		4	1.1	even	1	trivial
338.8.a.j.1.1		4	13.12	even	2
338.8.b.h.337.1		8	13.8	odd	4
338.8.b.h.337.5		8	13.5	odd	4

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -71.1584 q^{3} +64.0000 q^{4} +523.489 q^{5} +569.267 q^{6} -416.801 q^{7} -512.000 q^{8} +2876.52 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -71.1584 q^{3} +64.0000 q^{4} +523.489 q^{5} +569.267 q^{6} -416.801 q^{7} -512.000 q^{8} +2876.52 q^{9} -4187.91 q^{10} +3422.78 q^{11} -4554.14 q^{12} +3334.41 q^{14} -37250.7 q^{15} +4096.00 q^{16} -6538.04 q^{17} -23012.2 q^{18} +27980.5 q^{19} +33503.3 q^{20} +29658.9 q^{21} -27382.2 q^{22} +106178. q^{23} +36433.1 q^{24} +195916. q^{25} -49065.2 q^{27} -26675.3 q^{28} +68546.0 q^{29} +298005. q^{30} +51572.8 q^{31} -32768.0 q^{32} -243560. q^{33} +52304.3 q^{34} -218191. q^{35} +184097. q^{36} +48135.4 q^{37} -223844. q^{38} -268026. q^{40} -602098. q^{41} -237271. q^{42} +916252. q^{43} +219058. q^{44} +1.50583e6 q^{45} -849424. q^{46} -326201. q^{47} -291465. q^{48} -649820. q^{49} -1.56733e6 q^{50} +465237. q^{51} +934068. q^{53} +392522. q^{54} +1.79179e6 q^{55} +213402. q^{56} -1.99105e6 q^{57} -548368. q^{58} +1.17443e6 q^{59} -2.38404e6 q^{60} -2.89090e6 q^{61} -412582. q^{62} -1.19894e6 q^{63} +262144. q^{64} +1.94848e6 q^{66} +318158. q^{67} -418435. q^{68} -7.55546e6 q^{69} +1.74553e6 q^{70} +1.28374e6 q^{71} -1.47278e6 q^{72} -1.67209e6 q^{73} -385083. q^{74} -1.39411e7 q^{75} +1.79075e6 q^{76} -1.42662e6 q^{77} -8.09160e6 q^{79} +2.14421e6 q^{80} -2.79955e6 q^{81} +4.81678e6 q^{82} +5.57141e6 q^{83} +1.89817e6 q^{84} -3.42260e6 q^{85} -7.33001e6 q^{86} -4.87762e6 q^{87} -1.75246e6 q^{88} +7.87572e6 q^{89} -1.20466e7 q^{90} +6.79540e6 q^{92} -3.66984e6 q^{93} +2.60961e6 q^{94} +1.46475e7 q^{95} +2.33172e6 q^{96} -6.80847e6 q^{97} +5.19856e6 q^{98} +9.84569e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9	\( 4 q - 32 q^{2} + 256 q^{4} + 278 q^{5} + 548 q^{7} - 2048 q^{8} + 5214 q^{9} - 2224 q^{10} + 7392 q^{11} - 4384 q^{14} - 15528 q^{15} + 16384 q^{16} - 28316 q^{17} - 41712 q^{18} + 99888 q^{19} + 17792 q^{20} + 91074 q^{21} - 59136 q^{22} + 33388 q^{23} + 86878 q^{25} + 106272 q^{27} + 35072 q^{28} - 93140 q^{29} + 124224 q^{30} + 311160 q^{31} - 131072 q^{32} - 238638 q^{33} + 226528 q^{34} - 141544 q^{35} + 333696 q^{36} + 9636 q^{37} - 799104 q^{38} - 142336 q^{40} - 82892 q^{41} - 728592 q^{42} + 569264 q^{43} + 473088 q^{44} + 2303394 q^{45} - 267104 q^{46} - 574200 q^{47} + 717798 q^{49} - 695024 q^{50} - 2729928 q^{51} + 1235350 q^{53} - 850176 q^{54} + 1092512 q^{55} - 280576 q^{56} + 3528462 q^{57} + 745120 q^{58} - 231504 q^{59} - 993792 q^{60} - 685684 q^{61} - 2489280 q^{62} + 5951712 q^{63} + 1048576 q^{64} + 1909104 q^{66} - 3271056 q^{67} - 1812224 q^{68} - 5600034 q^{69} + 1132352 q^{70} + 175012 q^{71} - 2669568 q^{72} + 7137890 q^{73} - 77088 q^{74} - 22200960 q^{75} + 6392832 q^{76} - 13915206 q^{77} - 7053952 q^{79} + 1138688 q^{80} - 3758004 q^{81} + 663136 q^{82} + 657288 q^{83} + 5828736 q^{84} - 11814998 q^{85} - 4554112 q^{86} + 7182900 q^{87} - 3784704 q^{88} + 11452234 q^{89} - 18427152 q^{90} + 2136832 q^{92} - 2984688 q^{93} + 4593600 q^{94} + 23334088 q^{95} + 428002 q^{97} - 5742384 q^{98} - 10357656 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 + 278 * q^5 + 548 * q^7 - 2048 * q^8 + 5214 * q^9 - 2224 * q^10 + 7392 * q^11 - 4384 * q^14 - 15528 * q^15 + 16384 * q^16 - 28316 * q^17 - 41712 * q^18 + 99888 * q^19 + 17792 * q^20 + 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 + 35072 * q^28 - 93140 * q^29 + 124224 * q^30 + 311160 * q^31 - 131072 * q^32 - 238638 * q^33 + 226528 * q^34 - 141544 * q^35 + 333696 * q^36 + 9636 * q^37 - 799104 * q^38 - 142336 * q^40 - 82892 * q^41 - 728592 * q^42 + 569264 * q^43 + 473088 * q^44 + 2303394 * q^45 - 267104 * q^46 - 574200 * q^47 + 717798 * q^49 - 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 - 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 + 3528462 * q^57 + 745120 * q^58 - 231504 * q^59 - 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 + 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 - 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 + 1132352 * q^70 + 175012 * q^71 - 2669568 * q^72 + 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 + 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 + 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 + 657288 * q^83 + 5828736 * q^84 - 11814998 * q^85 - 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 + 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 - 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 + 428002 * q^97 - 5742384 * q^98 - 10357656 * q^99

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