LMFDB - Embedded newform 69.8.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$21.5545667584$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 455x^{3} - 474x^{2} + 42284x + 127016 $ x^5 - 455x^3 - 474x^2 + 42284*x + 127016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$12.4672$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} -509.647 q^{10} +500.265 q^{11} -740.672 q^{12} -12895.9 q^{13} +14046.8 q^{14} +1103.73 q^{15} -19142.8 q^{16} -23798.2 q^{17} +9088.62 q^{18} -32879.5 q^{19} -1121.40 q^{20} -30420.8 q^{21} +6236.93 q^{22} +12167.0 q^{23} +33852.7 q^{24} -76453.9 q^{25} -160777. q^{26} -19683.0 q^{27} +30907.8 q^{28} -152213. q^{29} +13760.5 q^{30} +252935. q^{31} -78171.4 q^{32} -13507.2 q^{33} -296698. q^{34} -46058.0 q^{35} +19998.1 q^{36} +89429.5 q^{37} -409916. q^{38} +348190. q^{39} +51254.0 q^{40} +549274. q^{41} -379263. q^{42} -330324. q^{43} +13723.4 q^{44} -29800.7 q^{45} +151689. q^{46} +834394. q^{47} +516856. q^{48} +445899. q^{49} -953170. q^{50} +642551. q^{51} -353765. q^{52} -1.73470e6 q^{53} -245393. q^{54} -20450.3 q^{55} -1.41265e6 q^{56} +887745. q^{57} -1.89768e6 q^{58} -2.58024e6 q^{59} +30277.8 q^{60} -2.40055e6 q^{61} +3.15340e6 q^{62} +821361. q^{63} +1.47570e6 q^{64} +527171. q^{65} -168397. q^{66} +3.57070e6 q^{67} -652839. q^{68} -328509. q^{69} -574216. q^{70} +709607. q^{71} -914022. q^{72} +797398. q^{73} +1.11494e6 q^{74} +2.06426e6 q^{75} -901959. q^{76} +563647. q^{77} +4.34098e6 q^{78} -1.98775e6 q^{79} +782535. q^{80} +531441. q^{81} +6.84793e6 q^{82} +8.20364e6 q^{83} -834512. q^{84} +972842. q^{85} -4.11823e6 q^{86} +4.10976e6 q^{87} -627234. q^{88} +8.23613e6 q^{89} -371532. q^{90} -1.45298e7 q^{91} +333769. q^{92} -6.82924e6 q^{93} +1.04026e7 q^{94} +1.34407e6 q^{95} +2.11063e6 q^{96} +1.54790e6 q^{97} +5.55914e6 q^{98} +364694. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * 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 $

$2$	12.4672	1.10196	0.550980	−	0.834519i	$-0.314254\pi$
$2$	12.4672	1.10196	0.550980	+	0.834519i	$0.314254\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	27.4323	0.214315
$5$	−40.8788	−0.146253	−0.0731263	−	0.997323i	$-0.523298\pi$
$5$	−40.8788	−0.146253	−0.0731263	+	0.997323i	$0.523298\pi$
$6$	−336.616	−0.636217
$7$	1126.70	1.24155	0.620774	−	0.783990i	$-0.286819\pi$
$7$	1126.70	1.24155	0.620774	+	0.783990i	$0.286819\pi$
$8$	−1253.80	−0.865793
$9$	729.000	0.333333
$10$	−509.647	−0.161164
$11$	500.265	0.113325	0.0566626	−	0.998393i	$-0.481954\pi$
$11$	500.265	0.113325	0.0566626	+	0.998393i	$0.481954\pi$
$12$	−740.672	−0.123735
$13$	−12895.9	−1.62799	−0.813994	−	0.580873i	$-0.802711\pi$
$13$	−12895.9	−1.62799	−0.813994	+	0.580873i	$0.802711\pi$
$14$	14046.8	1.36814
$15$	1103.73	0.0844390
$16$	−19142.8	−1.16838
$17$	−23798.2	−1.17482	−0.587412	−	0.809288i	$-0.699853\pi$
$17$	−23798.2	−1.17482	−0.587412	+	0.809288i	$0.699853\pi$
$18$	9088.62	0.367320
$19$	−32879.5	−1.09973	−0.549866	−	0.835253i	$-0.685321\pi$
$19$	−32879.5	−1.09973	−0.549866	+	0.835253i	$0.685321\pi$
$20$	−1121.40	−0.0313441
$21$	−30420.8	−0.716808
$22$	6236.93	0.124880
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	33852.7	0.499866
$25$	−76453.9	−0.978610
$26$	−160777.	−1.79398
$27$	−19683.0	−0.192450
$28$	30907.8	0.266082
$29$	−152213.	−1.15894	−0.579468	−	0.814995i	$-0.696740\pi$
$29$	−152213.	−1.15894	−0.579468	+	0.814995i	$0.696740\pi$
$30$	13760.5	0.0930483
$31$	252935.	1.52490	0.762452	−	0.647045i	$-0.223995\pi$
$31$	252935.	1.52490	0.762452	+	0.647045i	$0.223995\pi$
$32$	−78171.4	−0.421719
$33$	−13507.2	−0.0654283
$34$	−296698.	−1.29461
$35$	−46058.0	−0.181580
$36$	19998.1	0.0714383
$37$	89429.5	0.290252	0.145126	−	0.989413i	$-0.453641\pi$
$37$	89429.5	0.290252	0.145126	+	0.989413i	$0.453641\pi$
$38$	−409916.	−1.21186
$39$	348190.	0.939920
$40$	51254.0	0.126624
$41$	549274.	1.24464	0.622322	−	0.782761i	$-0.286189\pi$
$41$	549274.	1.24464	0.622322	+	0.782761i	$0.286189\pi$
$42$	−379263.	−0.789893
$43$	−330324.	−0.633578	−0.316789	−	0.948496i	$-0.602605\pi$
$43$	−330324.	−0.633578	−0.316789	+	0.948496i	$0.602605\pi$
$44$	13723.4	0.0242873
$45$	−29800.7	−0.0487509
$46$	151689.	0.229774
$47$	834394.	1.17227	0.586136	−	0.810213i	$-0.300648\pi$
$47$	834394.	1.17227	0.586136	+	0.810213i	$0.300648\pi$
$48$	516856.	0.674567
$49$	445899.	0.541440
$50$	−953170.	−1.07839
$51$	642551.	0.678284
$52$	−353765.	−0.348902
$53$	−1.73470e6	−1.60051	−0.800257	−	0.599658i	$-0.795303\pi$
$53$	−1.73470e6	−1.60051	−0.800257	+	0.599658i	$0.795303\pi$
$54$	−245393.	−0.212072
$55$	−20450.3	−0.0165741
$56$	−1.41265e6	−1.07492
$57$	887745.	0.634931
$58$	−1.89768e6	−1.27710
$59$	−2.58024e6	−1.63560	−0.817801	−	0.575502i	$-0.804807\pi$
$59$	−2.58024e6	−1.63560	−0.817801	+	0.575502i	$0.804807\pi$
$60$	30277.8	0.0180965
$61$	−2.40055e6	−1.35412	−0.677060	−	0.735928i	$-0.736746\pi$
$61$	−2.40055e6	−1.35412	−0.677060	+	0.735928i	$0.736746\pi$
$62$	3.15340e6	1.68038
$63$	821361.	0.413849
$64$	1.47570e6	0.703667
$65$	527171.	0.238097
$66$	−168397.	−0.0720993
$67$	3.57070e6	1.45041	0.725206	−	0.688532i	$-0.241744\pi$
$67$	3.57070e6	1.45041	0.725206	+	0.688532i	$0.241744\pi$
$68$	−652839.	−0.251782
$69$	−328509.	−0.120386
$70$	−574216.	−0.200093
$71$	709607.	0.235296	0.117648	−	0.993055i	$-0.462465\pi$
$71$	709607.	0.235296	0.117648	+	0.993055i	$0.462465\pi$
$72$	−914022.	−0.288598
$73$	797398.	0.239908	0.119954	−	0.992779i	$-0.461725\pi$
$73$	797398.	0.239908	0.119954	+	0.992779i	$0.461725\pi$
$74$	1.11494e6	0.319846
$75$	2.06426e6	0.565001
$76$	−901959.	−0.235689
$77$	563647.	0.140699
$78$	4.34098e6	1.03575
$79$	−1.98775e6	−0.453594	−0.226797	−	0.973942i	$-0.572825\pi$
$79$	−1.98775e6	−0.453594	−0.226797	+	0.973942i	$0.572825\pi$
$80$	782535.	0.170879
$81$	531441.	0.111111
$82$	6.84793e6	1.37155
$83$	8.20364e6	1.57483	0.787414	−	0.616425i	$-0.211419\pi$
$83$	8.20364e6	1.57483	0.787414	+	0.616425i	$0.211419\pi$
$84$	−834512.	−0.153623
$85$	972842.	0.171821
$86$	−4.11823e6	−0.698177
$87$	4.10976e6	0.669112
$88$	−627234.	−0.0981161
$89$	8.23613e6	1.23839	0.619196	−	0.785236i	$-0.287459\pi$
$89$	8.23613e6	1.23839	0.619196	+	0.785236i	$0.287459\pi$
$90$	−371532.	−0.0537215
$91$	−1.45298e7	−2.02123
$92$	333769.	0.0446877
$93$	−6.82924e6	−0.880404
$94$	1.04026e7	1.29180
$95$	1.34407e6	0.160839
$96$	2.11063e6	0.243479
$97$	1.54790e6	0.172203	0.0861015	−	0.996286i	$-0.472559\pi$
$97$	1.54790e6	0.172203	0.0861015	+	0.996286i	$0.472559\pi$
$98$	5.55914e6	0.596645
$99$	364694.	0.0377750
$100$	−2.09731e6	−0.209731
$101$	1.26141e7	1.21824	0.609119	−	0.793079i	$-0.291523\pi$
$101$	1.26141e7	1.21824	0.609119	+	0.793079i	$0.291523\pi$
$102$	8.01084e6	0.747442
$103$	−1.53226e7	−1.38166	−0.690830	−	0.723017i	$-0.742755\pi$
$103$	−1.53226e7	−1.38166	−0.690830	+	0.723017i	$0.742755\pi$
$104$	1.61690e7	1.40950
$105$	1.24357e6	0.104835
$106$	−2.16270e7	−1.76370
$107$	3.58516e6	0.282921	0.141461	−	0.989944i	$-0.454820\pi$
$107$	3.58516e6	0.282921	0.141461	+	0.989944i	$0.454820\pi$
$108$	−539950.	−0.0412449
$109$	−1.14000e7	−0.843163	−0.421582	−	0.906790i	$-0.638525\pi$
$109$	−1.14000e7	−0.843163	−0.421582	+	0.906790i	$0.638525\pi$
$110$	−254959.	−0.0182640
$111$	−2.41460e6	−0.167577
$112$	−2.15681e7	−1.45060
$113$	−1.16966e7	−0.762579	−0.381290	−	0.924456i	$-0.624520\pi$
$113$	−1.16966e7	−0.762579	−0.381290	+	0.924456i	$0.624520\pi$
$114$	1.10677e7	0.699668
$115$	−497373.	−0.0304958
$116$	−4.17556e6	−0.248377
$117$	−9.40114e6	−0.542663
$118$	−3.21685e7	−1.80237
$119$	−2.68133e7	−1.45860
$120$	−1.38386e6	−0.0731067
$121$	−1.92369e7	−0.987157
$122$	−2.99283e7	−1.49219
$123$	−1.48304e7	−0.718596
$124$	6.93858e6	0.326810
$125$	6.31901e6	0.289377
$126$	1.02401e7	0.456045
$127$	4.09153e7	1.77244	0.886222	−	0.463262i	$-0.153321\pi$
$127$	4.09153e7	1.77244	0.886222	+	0.463262i	$0.153321\pi$
$128$	2.84038e7	1.19713
$129$	8.91874e6	0.365796
$130$	6.57237e6	0.262374
$131$	9.54150e6	0.370823	0.185412	−	0.982661i	$-0.440638\pi$
$131$	9.54150e6	0.370823	0.185412	+	0.982661i	$0.440638\pi$
$132$	−370533.	−0.0140223
$133$	−3.70451e7	−1.36537
$134$	4.45168e7	1.59830
$135$	804618.	0.0281463
$136$	2.98382e7	1.01715
$137$	−3.72077e7	−1.23626	−0.618132	−	0.786075i	$-0.712110\pi$
$137$	−3.72077e7	−1.23626	−0.618132	+	0.786075i	$0.712110\pi$
$138$	−4.09560e6	−0.132660
$139$	−4.43290e7	−1.40003	−0.700013	−	0.714130i	$-0.746823\pi$
$139$	−4.43290e7	−1.40003	−0.700013	+	0.714130i	$0.746823\pi$
$140$	−1.26348e6	−0.0389152
$141$	−2.25286e7	−0.676812
$142$	8.84685e6	0.259286
$143$	−6.45139e6	−0.184492
$144$	−1.39551e7	−0.389461
$145$	6.22230e6	0.169497
$146$	9.94137e6	0.264369
$147$	−1.20393e7	−0.312601
$148$	2.45326e6	0.0622053
$149$	2.13993e7	0.529965	0.264983	−	0.964253i	$-0.414634\pi$
$149$	2.13993e7	0.529965	0.264983	+	0.964253i	$0.414634\pi$
$150$	2.57356e7	0.622608
$151$	−5.39107e7	−1.27425	−0.637126	−	0.770759i	$-0.719877\pi$
$151$	−5.39107e7	−1.27425	−0.637126	+	0.770759i	$0.719877\pi$
$152$	4.12243e7	0.952141
$153$	−1.73489e7	−0.391608
$154$	7.02712e6	0.155044
$155$	−1.03397e7	−0.223021
$156$	9.55167e6	0.201439
$157$	−4.92467e6	−0.101561	−0.0507807	−	0.998710i	$-0.516171\pi$
$157$	−4.92467e6	−0.101561	−0.0507807	+	0.998710i	$0.516171\pi$
$158$	−2.47818e7	−0.499842
$159$	4.68369e7	0.924057
$160$	3.19555e6	0.0616774
$161$	1.37085e7	0.258881
$162$	6.62561e6	0.122440
$163$	−5.24341e7	−0.948326	−0.474163	−	0.880437i	$-0.657249\pi$
$163$	−5.24341e7	−0.948326	−0.474163	+	0.880437i	$0.657249\pi$
$164$	1.50678e7	0.266746
$165$	552157.	0.00956905
$166$	1.02277e8	1.73540
$167$	5.66790e7	0.941704	0.470852	−	0.882212i	$-0.343947\pi$
$167$	5.66790e7	0.941704	0.470852	+	0.882212i	$0.343947\pi$
$168$	3.81416e7	0.620607
$169$	1.03557e8	1.65035
$170$	1.21287e7	0.189340
$171$	−2.39691e7	−0.366577
$172$	−9.06154e6	−0.135785
$173$	−4.71933e7	−0.692977	−0.346489	−	0.938054i	$-0.612626\pi$
$173$	−4.71933e7	−0.692977	−0.346489	+	0.938054i	$0.612626\pi$
$174$	5.12374e7	0.737335
$175$	−8.61403e7	−1.21499
$176$	−9.57648e6	−0.132407
$177$	6.96664e7	0.944315
$178$	1.02682e8	1.36466
$179$	9.63068e6	0.125508	0.0627540	−	0.998029i	$-0.480012\pi$
$179$	9.63068e6	0.125508	0.0627540	+	0.998029i	$0.480012\pi$
$180$	−817501.	−0.0104480
$181$	1.36975e8	1.71699	0.858494	−	0.512823i	$-0.171400\pi$
$181$	1.36975e8	1.71699	0.858494	+	0.512823i	$0.171400\pi$
$182$	−1.81147e8	−2.22731
$183$	6.48150e7	0.781802
$184$	−1.52550e7	−0.180530
$185$	−3.65577e6	−0.0424501
$186$	−8.51418e7	−0.970169
$187$	−1.19054e7	−0.133137
$188$	2.28893e7	0.251235
$189$	−2.21767e7	−0.238936
$190$	1.67569e7	0.177238
$191$	5.87953e7	0.610556	0.305278	−	0.952263i	$-0.401251\pi$
$191$	5.87953e7	0.610556	0.305278	+	0.952263i	$0.401251\pi$
$192$	−3.98438e7	−0.406262
$193$	1.44972e6	0.0145155	0.00725776	−	0.999974i	$-0.497690\pi$
$193$	1.44972e6	0.0145155	0.00725776	+	0.999974i	$0.497690\pi$
$194$	1.92980e7	0.189761
$195$	−1.42336e7	−0.137466
$196$	1.22320e7	0.116039
$197$	−6.62738e7	−0.617604	−0.308802	−	0.951126i	$-0.599928\pi$
$197$	−6.62738e7	−0.617604	−0.308802	+	0.951126i	$0.599928\pi$
$198$	4.54672e6	0.0416266
$199$	−1.78732e8	−1.60774	−0.803871	−	0.594803i	$-0.797230\pi$
$199$	−1.78732e8	−1.60774	−0.803871	+	0.594803i	$0.797230\pi$
$200$	9.58581e7	0.847274
$201$	−9.64089e7	−0.837396
$202$	1.57263e8	1.34245
$203$	−1.71498e8	−1.43888
$204$	1.76266e7	0.145366
$205$	−2.24537e7	−0.182032
$206$	−1.91030e8	−1.52253
$207$	8.86974e6	0.0695048
$208$	2.46864e8	1.90212
$209$	−1.64485e7	−0.124627
$210$	1.55038e7	0.115524
$211$	−1.07572e7	−0.0788335	−0.0394167	−	0.999223i	$-0.512550\pi$
$211$	−1.07572e7	−0.0788335	−0.0394167	+	0.999223i	$0.512550\pi$
$212$	−4.75869e7	−0.343014
$213$	−1.91594e7	−0.135848
$214$	4.46971e7	0.311768
$215$	1.35032e7	0.0926624
$216$	2.46786e7	0.166622
$217$	2.84980e8	1.89324
$218$	−1.42126e8	−0.929132
$219$	−2.15298e7	−0.138511
$220$	−560998.	−0.00355207
$221$	3.06900e8	1.91260
$222$	−3.01034e7	−0.184663
$223$	−2.14285e8	−1.29397	−0.646986	−	0.762502i	$-0.723971\pi$
$223$	−2.14285e8	−1.29397	−0.646986	+	0.762502i	$0.723971\pi$
$224$	−8.80753e7	−0.523584
$225$	−5.57349e7	−0.326203
$226$	−1.45824e8	−0.840331
$227$	−7.33851e7	−0.416406	−0.208203	−	0.978086i	$-0.566762\pi$
$227$	−7.33851e7	−0.416406	−0.208203	+	0.978086i	$0.566762\pi$
$228$	2.43529e7	0.136075
$229$	−2.83323e7	−0.155904	−0.0779519	−	0.996957i	$-0.524838\pi$
$229$	−2.83323e7	−0.155904	−0.0779519	+	0.996957i	$0.524838\pi$
$230$	−6.20087e6	−0.0336051
$231$	−1.52185e7	−0.0812323
$232$	1.90845e8	1.00340
$233$	1.15251e8	0.596896	0.298448	−	0.954426i	$-0.403531\pi$
$233$	1.15251e8	0.596896	0.298448	+	0.954426i	$0.403531\pi$
$234$	−1.17206e8	−0.597993
$235$	−3.41090e7	−0.171448
$236$	−7.07818e7	−0.350534
$237$	5.36693e7	0.261883
$238$	−3.34288e8	−1.60732
$239$	5.89008e7	0.279080	0.139540	−	0.990216i	$-0.455438\pi$
$239$	5.89008e7	0.279080	0.139540	+	0.990216i	$0.455438\pi$
$240$	−2.11285e7	−0.0986571
$241$	−5.12886e7	−0.236027	−0.118013	−	0.993012i	$-0.537653\pi$
$241$	−5.12886e7	−0.236027	−0.118013	+	0.993012i	$0.537653\pi$
$242$	−2.39831e8	−1.08781
$243$	−1.43489e7	−0.0641500
$244$	−6.58527e7	−0.290208
$245$	−1.82278e7	−0.0791870
$246$	−1.84894e8	−0.791863
$247$	4.24012e8	1.79035
$248$	−3.17130e8	−1.32025
$249$	−2.21498e8	−0.909227
$250$	7.87806e7	0.318882
$251$	−3.11308e7	−0.124260	−0.0621300	−	0.998068i	$-0.519789\pi$
$251$	−3.11308e7	−0.124260	−0.0621300	+	0.998068i	$0.519789\pi$
$252$	2.25318e7	0.0886940
$253$	6.08673e6	0.0236299
$254$	5.10101e8	1.95316
$255$	−2.62667e7	−0.0992008
$256$	1.65228e8	0.615523
$257$	2.38296e8	0.875690	0.437845	−	0.899051i	$-0.355742\pi$
$257$	2.38296e8	0.875690	0.437845	+	0.899051i	$0.355742\pi$
$258$	1.11192e8	0.403093
$259$	1.00760e8	0.360361
$260$	1.44615e7	0.0510278
$261$	−1.10964e8	−0.386312
$262$	1.18956e8	0.408632
$263$	2.84255e8	0.963525	0.481762	−	0.876302i	$-0.339997\pi$
$263$	2.84255e8	0.963525	0.481762	+	0.876302i	$0.339997\pi$
$264$	1.69353e7	0.0566474
$265$	7.09126e7	0.234079
$266$	−4.61851e8	−1.50458
$267$	−2.22376e8	−0.714986
$268$	9.79525e7	0.310845
$269$	−1.94289e8	−0.608576	−0.304288	−	0.952580i	$-0.598419\pi$
$269$	−1.94289e8	−0.608576	−0.304288	+	0.952580i	$0.598419\pi$
$270$	1.00314e7	0.0310161
$271$	−3.69086e8	−1.12651	−0.563254	−	0.826284i	$-0.690451\pi$
$271$	−3.69086e8	−1.12651	−0.563254	+	0.826284i	$0.690451\pi$
$272$	4.55564e8	1.37264
$273$	3.92305e8	1.16695
$274$	−4.63878e8	−1.36231
$275$	−3.82473e7	−0.110901
$276$	−9.01176e6	−0.0258005
$277$	−5.28946e8	−1.49531	−0.747657	−	0.664086i	$-0.768821\pi$
$277$	−5.28946e8	−1.49531	−0.747657	+	0.664086i	$0.768821\pi$
$278$	−5.52661e8	−1.54277
$279$	1.84389e8	0.508301
$280$	5.77476e7	0.157210
$281$	4.54274e7	0.122137	0.0610683	−	0.998134i	$-0.480549\pi$
$281$	4.54274e7	0.122137	0.0610683	+	0.998134i	$0.480549\pi$
$282$	−2.80870e8	−0.745819
$283$	2.28233e8	0.598585	0.299293	−	0.954161i	$-0.403249\pi$
$283$	2.28233e8	0.598585	0.299293	+	0.954161i	$0.403249\pi$
$284$	1.94662e7	0.0504274
$285$	−3.62900e7	−0.0928603
$286$	−8.04311e7	−0.203303
$287$	6.18864e8	1.54528
$288$	−5.69869e7	−0.140573
$289$	1.56015e8	0.380210
$290$	7.75750e7	0.186779
$291$	−4.17932e7	−0.0994214
$292$	2.18745e7	0.0514159
$293$	4.49349e8	1.04363	0.521815	−	0.853058i	$-0.325255\pi$
$293$	4.49349e8	1.04363	0.521815	+	0.853058i	$0.325255\pi$
$294$	−1.50097e8	−0.344473
$295$	1.05477e8	0.239211
$296$	−1.12127e8	−0.251298
$297$	−9.84673e6	−0.0218094
$298$	2.66790e8	0.584000
$299$	−1.56905e8	−0.339459
$300$	5.66273e7	0.121088
$301$	−3.72174e8	−0.786617
$302$	−6.72118e8	−1.40418
$303$	−3.40581e8	−0.703350
$304$	6.29405e8	1.28491
$305$	9.81319e7	0.198044
$306$	−2.16293e8	−0.431536
$307$	−3.56204e8	−0.702609	−0.351305	−	0.936261i	$-0.614262\pi$
$307$	−3.56204e8	−0.702609	−0.351305	+	0.936261i	$0.614262\pi$
$308$	1.54621e7	0.0301538
$309$	4.13709e8	0.797702
$310$	−1.28907e8	−0.245760
$311$	7.14279e8	1.34650	0.673250	−	0.739415i	$-0.264898\pi$
$311$	7.14279e8	1.34650	0.673250	+	0.739415i	$0.264898\pi$
$312$	−4.36562e8	−0.813776
$313$	9.57174e7	0.176435	0.0882177	−	0.996101i	$-0.471883\pi$
$313$	9.57174e7	0.176435	0.0882177	+	0.996101i	$0.471883\pi$
$314$	−6.13971e7	−0.111916
$315$	−3.35763e7	−0.0605265
$316$	−5.45286e7	−0.0972120
$317$	−6.88984e8	−1.21479	−0.607396	−	0.794399i	$-0.707786\pi$
$317$	−6.88984e8	−1.21479	−0.607396	+	0.794399i	$0.707786\pi$
$318$	5.83928e8	1.01827
$319$	−7.61471e7	−0.131337
$320$	−6.03248e7	−0.102913
$321$	−9.67993e7	−0.163345
$322$	1.70907e8	0.285276
$323$	7.82471e8	1.29199
$324$	1.45787e7	0.0238128
$325$	9.85945e8	1.59317
$326$	−6.53709e8	−1.04502
$327$	3.07800e8	0.486801
$328$	−6.88681e8	−1.07760
$329$	9.40108e8	1.45543
$330$	6.88388e6	0.0105447
$331$	8.37330e8	1.26911	0.634554	−	0.772878i	$-0.281184\pi$
$331$	8.37330e8	1.26911	0.634554	+	0.772878i	$0.281184\pi$
$332$	2.25045e8	0.337509
$333$	6.51941e7	0.0967506
$334$	7.06631e8	1.03772
$335$	−1.45966e8	−0.212127
$336$	5.82339e8	0.837507
$337$	−1.24811e9	−1.77643	−0.888213	−	0.459431i	$-0.848053\pi$
$337$	−1.24811e9	−1.77643	−0.888213	+	0.459431i	$0.848053\pi$
$338$	1.29107e9	1.81862
$339$	3.15808e8	0.440275
$340$	2.66873e7	0.0368238
$341$	1.26534e8	0.172810
$342$	−2.98829e8	−0.403954
$343$	−4.25489e8	−0.569324
$344$	4.14161e8	0.548547
$345$	1.34291e7	0.0176067
$346$	−5.88371e8	−0.763633
$347$	−2.11977e8	−0.272356	−0.136178	−	0.990684i	$-0.543482\pi$
$347$	−2.11977e8	−0.272356	−0.136178	+	0.990684i	$0.543482\pi$
$348$	1.12740e8	0.143401
$349$	−1.03849e8	−0.130772	−0.0653858	−	0.997860i	$-0.520828\pi$
$349$	−1.03849e8	−0.130772	−0.0653858	+	0.997860i	$0.520828\pi$
$350$	−1.07393e9	−1.33887
$351$	2.53831e8	0.313307
$352$	−3.91064e7	−0.0477913
$353$	2.33920e8	0.283045	0.141523	−	0.989935i	$-0.454800\pi$
$353$	2.33920e8	0.283045	0.141523	+	0.989935i	$0.454800\pi$
$354$	8.68548e8	1.04060
$355$	−2.90079e7	−0.0344126
$356$	2.25936e8	0.265406
$357$	7.23959e8	0.842122
$358$	1.20068e8	0.138305
$359$	−1.66286e9	−1.89682	−0.948408	−	0.317054i	$-0.897306\pi$
$359$	−1.66286e9	−1.89682	−0.948408	+	0.317054i	$0.897306\pi$
$360$	3.73642e7	0.0422082
$361$	1.87187e8	0.209412
$362$	1.70771e9	1.89205
$363$	5.19396e8	0.569936
$364$	−3.98586e8	−0.433179
$365$	−3.25967e7	−0.0350872
$366$	8.08065e8	0.861514
$367$	−5.37695e8	−0.567812	−0.283906	−	0.958852i	$-0.591630\pi$
$367$	−5.37695e8	−0.567812	−0.283906	+	0.958852i	$0.591630\pi$
$368$	−2.32910e8	−0.243625
$369$	4.00420e8	0.414881
$370$	−4.55774e7	−0.0467782
$371$	−1.95448e9	−1.98711
$372$	−1.87342e8	−0.188684
$373$	−2.19239e8	−0.218744	−0.109372	−	0.994001i	$-0.534884\pi$
$373$	−2.19239e8	−0.218744	−0.109372	+	0.994001i	$0.534884\pi$
$374$	−1.48428e8	−0.146712
$375$	−1.70613e8	−0.167072
$376$	−1.04617e9	−1.01495
$377$	1.96293e9	1.88674
$378$	−2.76483e8	−0.263298
$379$	1.09529e9	1.03346	0.516730	−	0.856148i	$-0.327149\pi$
$379$	1.09529e9	1.03346	0.516730	+	0.856148i	$0.327149\pi$
$380$	3.68710e7	0.0344701
$381$	−1.10471e9	−1.02332
$382$	7.33015e8	0.672808
$383$	−1.49495e9	−1.35966	−0.679831	−	0.733368i	$-0.737947\pi$
$383$	−1.49495e9	−1.35966	−0.679831	+	0.733368i	$0.737947\pi$
$384$	−7.66903e8	−0.691164
$385$	−2.30412e7	−0.0205775
$386$	1.80740e7	0.0159955
$387$	−2.40806e8	−0.211193
$388$	4.24623e7	0.0369057
$389$	−1.06612e9	−0.918296	−0.459148	−	0.888360i	$-0.651845\pi$
$389$	−1.06612e9	−0.918296	−0.459148	+	0.888360i	$0.651845\pi$
$390$	−1.77454e8	−0.151482
$391$	−2.89552e8	−0.244968
$392$	−5.59070e8	−0.468775
$393$	−2.57620e8	−0.214095
$394$	−8.26252e8	−0.680575
$395$	8.12570e7	0.0663393
$396$	1.00044e7	0.00809575
$397$	1.02859e9	0.825044	0.412522	−	0.910948i	$-0.364648\pi$
$397$	1.02859e9	0.825044	0.412522	+	0.910948i	$0.364648\pi$
$398$	−2.22830e9	−1.77167
$399$	1.00022e9	0.788297
$400$	1.46354e9	1.14339
$401$	2.90455e8	0.224943	0.112472	−	0.993655i	$-0.464123\pi$
$401$	2.90455e8	0.224943	0.112472	+	0.993655i	$0.464123\pi$
$402$	−1.20195e9	−0.922776
$403$	−3.26183e9	−2.48253
$404$	3.46034e8	0.261087
$405$	−2.17247e7	−0.0162503
$406$	−2.13811e9	−1.58558
$407$	4.47385e7	0.0328928
$408$	−8.05632e8	−0.587254
$409$	−6.42289e8	−0.464193	−0.232097	−	0.972693i	$-0.574559\pi$
$409$	−6.42289e8	−0.464193	−0.232097	+	0.972693i	$0.574559\pi$
$410$	−2.79935e8	−0.200592
$411$	1.00461e9	0.713757
$412$	−4.20333e8	−0.296110
$413$	−2.90714e9	−2.03068
$414$	1.10581e8	0.0765915
$415$	−3.35355e8	−0.230323
$416$	1.00809e9	0.686553
$417$	1.19688e9	0.808306
$418$	−2.05067e8	−0.137334
$419$	1.95033e9	1.29526	0.647632	−	0.761953i	$-0.275759\pi$
$419$	1.95033e9	1.29526	0.647632	+	0.761953i	$0.275759\pi$
$420$	3.41139e7	0.0224677
$421$	−2.66647e9	−1.74160	−0.870801	−	0.491635i	$-0.836399\pi$
$421$	−2.66647e9	−1.74160	−0.870801	+	0.491635i	$0.836399\pi$
$422$	−1.34113e8	−0.0868713
$423$	6.08273e8	0.390757
$424$	2.17497e9	1.38571
$425$	1.81946e9	1.14969
$426$	−2.38865e8	−0.149699
$427$	−2.70469e9	−1.68120
$428$	9.83492e7	0.0606342
$429$	1.74188e8	0.106516
$430$	1.68348e8	0.102110
$431$	−1.45000e9	−0.872366	−0.436183	−	0.899858i	$-0.643670\pi$
$431$	−1.45000e9	−0.872366	−0.436183	+	0.899858i	$0.643670\pi$
$432$	3.76788e8	0.224856
$433$	8.59337e8	0.508693	0.254347	−	0.967113i	$-0.418140\pi$
$433$	8.59337e8	0.508693	0.254347	+	0.967113i	$0.418140\pi$
$434$	3.55292e9	2.08627
$435$	−1.68002e8	−0.0978594
$436$	−3.12728e8	−0.180702
$437$	−4.00044e8	−0.229310
$438$	−2.68417e8	−0.152634
$439$	−2.27752e9	−1.28480	−0.642402	−	0.766368i	$-0.722062\pi$
$439$	−2.27752e9	−1.28480	−0.642402	+	0.766368i	$0.722062\pi$
$440$	2.56406e7	0.0143497
$441$	3.25061e8	0.180480
$442$	3.82620e9	2.10761
$443$	1.83034e8	0.100027	0.0500136	−	0.998749i	$-0.484074\pi$
$443$	1.83034e8	0.100027	0.0500136	+	0.998749i	$0.484074\pi$
$444$	−6.62379e7	−0.0359142
$445$	−3.36683e8	−0.181118
$446$	−2.67154e9	−1.42590
$447$	−5.77781e8	−0.305976
$448$	1.66266e9	0.873636
$449$	1.10637e9	0.576819	0.288409	−	0.957507i	$-0.406874\pi$
$449$	1.10637e9	0.576819	0.288409	+	0.957507i	$0.406874\pi$
$450$	−6.94861e8	−0.359463
$451$	2.74783e8	0.141049
$452$	−3.20865e8	−0.163432
$453$	1.45559e9	0.735690
$454$	−9.14910e8	−0.458863
$455$	5.93961e8	0.295609
$456$	−1.11306e9	−0.549719
$457$	−1.67986e8	−0.0823314	−0.0411657	−	0.999152i	$-0.513107\pi$
$457$	−1.67986e8	−0.0823314	−0.0411657	+	0.999152i	$0.513107\pi$
$458$	−3.53225e8	−0.171800
$459$	4.68420e8	0.226095
$460$	−1.36441e7	−0.00653570
$461$	2.56850e9	1.22103	0.610516	−	0.792004i	$-0.290962\pi$
$461$	2.56850e9	1.22103	0.610516	+	0.792004i	$0.290962\pi$
$462$	−1.89732e8	−0.0895147
$463$	−1.10660e9	−0.518150	−0.259075	−	0.965857i	$-0.583418\pi$
$463$	−1.10660e9	−0.518150	−0.259075	+	0.965857i	$0.583418\pi$
$464$	2.91379e9	1.35408
$465$	2.79171e8	0.128761
$466$	1.43686e9	0.657756
$467$	2.79649e9	1.27059	0.635294	−	0.772271i	$-0.280879\pi$
$467$	2.79649e9	1.27059	0.635294	+	0.772271i	$0.280879\pi$
$468$	−2.57895e8	−0.116301
$469$	4.02309e9	1.80076
$470$	−4.25246e8	−0.188929
$471$	1.32966e8	0.0586365
$472$	3.23511e9	1.41609
$473$	−1.65249e8	−0.0718003
$474$	6.69109e8	0.288584
$475$	2.51376e9	1.07621
$476$	−7.35551e8	−0.312599
$477$	−1.26460e9	−0.533504
$478$	7.34331e8	0.307535
$479$	2.19334e9	0.911866	0.455933	−	0.890014i	$-0.349306\pi$
$479$	2.19334e9	0.911866	0.455933	+	0.890014i	$0.349306\pi$
$480$	−8.62800e7	−0.0356095
$481$	−1.15328e9	−0.472526
$482$	−6.39428e8	−0.260092
$483$	−3.70130e8	−0.149465
$484$	−5.27713e8	−0.211563
$485$	−6.32762e7	−0.0251851
$486$	−1.78891e8	−0.0706907
$487$	−1.48441e9	−0.582373	−0.291187	−	0.956666i	$-0.594050\pi$
$487$	−1.48441e9	−0.582373	−0.291187	+	0.956666i	$0.594050\pi$
$488$	3.00982e9	1.17239
$489$	1.41572e9	0.547516
$490$	−2.27251e8	−0.0872609
$491$	−3.58617e9	−1.36724	−0.683621	−	0.729837i	$-0.739596\pi$
$491$	−3.58617e9	−1.36724	−0.683621	+	0.729837i	$0.739596\pi$
$492$	−4.06832e8	−0.154006
$493$	3.62240e9	1.36155
$494$	5.28626e9	1.97290
$495$	−1.49082e7	−0.00552470
$496$	−4.84188e9	−1.78167
$497$	7.99511e8	0.292131
$498$	−2.76147e9	−1.00193
$499$	−2.64545e9	−0.953122	−0.476561	−	0.879141i	$-0.658117\pi$
$499$	−2.64545e9	−0.953122	−0.476561	+	0.879141i	$0.658117\pi$
$500$	1.73345e8	0.0620177
$501$	−1.53033e9	−0.543693
$502$	−3.88115e8	−0.136930
$503$	4.47733e8	0.156867	0.0784334	−	0.996919i	$-0.475008\pi$
$503$	4.47733e8	0.156867	0.0784334	+	0.996919i	$0.475008\pi$
$504$	−1.02982e9	−0.358308
$505$	−5.15651e8	−0.178170
$506$	7.58848e7	0.0260392
$507$	−2.79603e9	−0.952828
$508$	1.12240e9	0.379861
$509$	−1.50184e9	−0.504792	−0.252396	−	0.967624i	$-0.581218\pi$
$509$	−1.50184e9	−0.504792	−0.252396	+	0.967624i	$0.581218\pi$
$510$	−3.27474e8	−0.109315
$511$	8.98425e8	0.297858
$512$	−1.57575e9	−0.518850
$513$	6.47166e8	0.211644
$514$	2.97089e9	0.964975
$515$	6.26368e8	0.202071
$516$	2.44661e8	0.0783956
$517$	4.17418e8	0.132848
$518$	1.25620e9	0.397104
$519$	1.27422e9	0.400091
$520$	−6.60968e8	−0.206143
$521$	1.31386e9	0.407020	0.203510	−	0.979073i	$-0.434765\pi$
$521$	1.31386e9	0.407020	0.203510	+	0.979073i	$0.434765\pi$
$522$	−1.38341e9	−0.425700
$523$	4.83261e9	1.47715	0.738577	−	0.674169i	$-0.235498\pi$
$523$	4.83261e9	1.47715	0.738577	+	0.674169i	$0.235498\pi$
$524$	2.61745e8	0.0794729
$525$	2.32579e9	0.701475
$526$	3.54388e9	1.06177
$527$	−6.01938e9	−1.79149
$528$	2.58565e8	0.0764454
$529$	1.48036e8	0.0434783
$530$	8.84085e8	0.257946
$531$	−1.88099e9	−0.545200
$532$	−1.01623e9	−0.292619
$533$	−7.08340e9	−2.02627
$534$	−2.77241e9	−0.787886
$535$	−1.46557e8	−0.0413779
$536$	−4.47695e9	−1.25576
$537$	−2.60028e8	−0.0724621
$538$	−2.42225e9	−0.670626
$539$	2.23068e8	0.0613588
$540$	2.20725e7	0.00603217
$541$	8.09486e8	0.219796	0.109898	−	0.993943i	$-0.464948\pi$
$541$	8.09486e8	0.219796	0.109898	+	0.993943i	$0.464948\pi$
$542$	−4.60148e9	−1.24137
$543$	−3.69833e9	−0.991304
$544$	1.86034e9	0.495445
$545$	4.66018e8	0.123315
$546$	4.89096e9	1.28594
$547$	2.70806e9	0.707460	0.353730	−	0.935348i	$-0.384913\pi$
$547$	2.70806e9	0.707460	0.353730	+	0.935348i	$0.384913\pi$
$548$	−1.02069e9	−0.264950
$549$	−1.75000e9	−0.451373
$550$	−4.76838e8	−0.122209
$551$	5.00469e9	1.27452
$552$	4.11885e8	0.104229
$553$	−2.23959e9	−0.563159
$554$	−6.59450e9	−1.64777
$555$	9.87059e7	0.0245086
$556$	−1.21605e9	−0.300047
$557$	−2.74885e9	−0.673997	−0.336999	−	0.941505i	$-0.609412\pi$
$557$	−2.74885e9	−0.673997	−0.336999	+	0.941505i	$0.609412\pi$
$558$	2.29883e9	0.560127
$559$	4.25983e9	1.03146
$560$	8.81679e8	0.212155
$561$	3.21446e8	0.0768667
$562$	5.66355e8	0.134590
$563$	−4.75796e9	−1.12368	−0.561838	−	0.827247i	$-0.689906\pi$
$563$	−4.75796e9	−1.12368	−0.561838	+	0.827247i	$0.689906\pi$
$564$	−6.18012e8	−0.145051
$565$	4.78143e8	0.111529
$566$	2.84544e9	0.659617
$567$	5.98772e8	0.137950
$568$	−8.89708e8	−0.203717
$569$	3.85644e9	0.877595	0.438797	−	0.898586i	$-0.355405\pi$
$569$	3.85644e9	0.877595	0.438797	+	0.898586i	$0.355405\pi$
$570$	−4.52436e8	−0.102328
$571$	−1.05862e9	−0.237966	−0.118983	−	0.992896i	$-0.537963\pi$
$571$	−1.05862e9	−0.237966	−0.118983	+	0.992896i	$0.537963\pi$
$572$	−1.76977e8	−0.0395394
$573$	−1.58747e9	−0.352505
$574$	7.71553e9	1.70284
$575$	−9.30215e8	−0.204054
$576$	1.07578e9	0.234556
$577$	−5.02825e9	−1.08969	−0.544843	−	0.838538i	$-0.683411\pi$
$577$	−5.02825e9	−1.08969	−0.544843	+	0.838538i	$0.683411\pi$
$578$	1.94507e9	0.418976
$579$	−3.91424e7	−0.00838054
$580$	1.70692e8	0.0363258
$581$	9.24300e9	1.95522
$582$	−5.21046e8	−0.109558
$583$	−8.67811e8	−0.181378
$584$	−9.99780e8	−0.207711
$585$	3.84308e8	0.0793658
$586$	5.60214e9	1.15004
$587$	−8.27222e9	−1.68806	−0.844032	−	0.536293i	$-0.819824\pi$
$587$	−8.27222e9	−1.68806	−0.844032	+	0.536293i	$0.819824\pi$
$588$	−3.30265e8	−0.0669950
$589$	−8.31636e9	−1.67699
$590$	1.31501e9	0.263601
$591$	1.78939e9	0.356574
$592$	−1.71193e9	−0.339125
$593$	−2.13907e9	−0.421245	−0.210622	−	0.977568i	$-0.567549\pi$
$593$	−2.13907e9	−0.421245	−0.210622	+	0.977568i	$0.567549\pi$
$594$	−1.22762e8	−0.0240331
$595$	1.09610e9	0.213324
$596$	5.87032e8	0.113579
$597$	4.82576e9	0.928230
$598$	−1.95617e9	−0.374070
$599$	−6.33073e9	−1.20354	−0.601769	−	0.798670i	$-0.705537\pi$
$599$	−6.33073e9	−1.20354	−0.601769	+	0.798670i	$0.705537\pi$
$600$	−2.58817e9	−0.489174
$601$	−4.50671e8	−0.0846835	−0.0423418	−	0.999103i	$-0.513482\pi$
$601$	−4.50671e8	−0.0846835	−0.0423418	+	0.999103i	$0.513482\pi$
$602$	−4.63999e9	−0.866820
$603$	2.60304e9	0.483471
$604$	−1.47889e9	−0.273091
$605$	7.86382e8	0.144374
$606$	−4.24611e9	−0.775064
$607$	−6.88005e9	−1.24862	−0.624311	−	0.781176i	$-0.714620\pi$
$607$	−6.88005e9	−1.24862	−0.624311	+	0.781176i	$0.714620\pi$
$608$	2.57023e9	0.463778
$609$	4.63045e9	0.830735
$610$	1.22343e9	0.218236
$611$	−1.07603e10	−1.90845
$612$	−4.75920e8	−0.0839274
$613$	8.99737e9	1.57763	0.788813	−	0.614633i	$-0.210696\pi$
$613$	8.99737e9	1.57763	0.788813	+	0.614633i	$0.210696\pi$
$614$	−4.44088e9	−0.774247
$615$	6.06249e8	0.105096
$616$	−7.06702e8	−0.121816
$617$	−2.14587e9	−0.367794	−0.183897	−	0.982946i	$-0.558871\pi$
$617$	−2.14587e9	−0.367794	−0.183897	+	0.982946i	$0.558871\pi$
$618$	5.15782e9	0.879035
$619$	3.60649e9	0.611177	0.305589	−	0.952164i	$-0.401147\pi$
$619$	3.60649e9	0.611177	0.305589	+	0.952164i	$0.401147\pi$
$620$	−2.83641e8	−0.0477967
$621$	−2.39483e8	−0.0401286
$622$	8.90509e9	1.48379
$623$	9.27961e9	1.53752
$624$	−6.66534e9	−1.09819
$625$	5.71465e9	0.936288
$626$	1.19333e9	0.194425
$627$	4.44108e8	0.0719536
$628$	−1.35095e8	−0.0217661
$629$	−2.12826e9	−0.340994
$630$	−4.18604e8	−0.0666978
$631$	−1.96983e9	−0.312123	−0.156062	−	0.987747i	$-0.549880\pi$
$631$	−1.96983e9	−0.312123	−0.156062	+	0.987747i	$0.549880\pi$
$632$	2.49225e9	0.392719
$633$	2.90444e8	0.0455145
$634$	−8.58973e9	−1.33865
$635$	−1.67257e9	−0.259224
$636$	1.28485e9	0.198039
$637$	−5.75029e9	−0.881459
$638$	−9.49345e8	−0.144728
$639$	5.17304e8	0.0784319
$640$	−1.16111e9	−0.175084
$641$	1.07839e9	0.161724	0.0808620	−	0.996725i	$-0.474233\pi$
$641$	1.07839e9	0.161724	0.0808620	+	0.996725i	$0.474233\pi$
$642$	−1.20682e9	−0.179999
$643$	−9.43215e9	−1.39918	−0.699588	−	0.714547i	$-0.746633\pi$
$643$	−9.43215e9	−1.39918	−0.699588	+	0.714547i	$0.746633\pi$
$644$	3.76056e8	0.0554820
$645$	−3.64588e8	−0.0534986
$646$	9.75526e9	1.42372
$647$	−1.91605e9	−0.278126	−0.139063	−	0.990284i	$-0.544409\pi$
$647$	−1.91605e9	−0.278126	−0.139063	+	0.990284i	$0.544409\pi$
$648$	−6.66322e8	−0.0961992
$649$	−1.29080e9	−0.185355
$650$	1.22920e10	1.75560
$651$	−7.69447e9	−1.09306
$652$	−1.43839e9	−0.203240
$653$	1.28782e10	1.80991	0.904956	−	0.425505i	$-0.139904\pi$
$653$	1.28782e10	1.80991	0.904956	+	0.425505i	$0.139904\pi$
$654$	3.83742e9	0.536434
$655$	−3.90045e8	−0.0542339
$656$	−1.05146e10	−1.45422
$657$	5.81303e8	0.0799695
$658$	1.17206e10	1.60383
$659$	−1.04515e10	−1.42259	−0.711294	−	0.702895i	$-0.751890\pi$
$659$	−1.04515e10	−1.42259	−0.711294	+	0.702895i	$0.751890\pi$
$660$	1.51469e7	0.00205079
$661$	9.36029e9	1.26062	0.630310	−	0.776344i	$-0.282928\pi$
$661$	9.36029e9	1.26062	0.630310	+	0.776344i	$0.282928\pi$
$662$	1.04392e10	1.39851
$663$	−8.28630e9	−1.10424
$664$	−1.02857e10	−1.36348
$665$	1.51436e9	0.199689
$666$	8.12791e8	0.106615
$667$	−1.85198e9	−0.241655
$668$	1.55484e9	0.201821
$669$	5.78569e9	0.747075
$670$	−1.81979e9	−0.233755
$671$	−1.20091e9	−0.153456
$672$	2.37803e9	0.302291
$673$	−2.34925e9	−0.297083	−0.148541	−	0.988906i	$-0.547458\pi$
$673$	−2.34925e9	−0.297083	−0.148541	+	0.988906i	$0.547458\pi$
$674$	−1.55605e10	−1.95755
$675$	1.50484e9	0.188334
$676$	2.84080e9	0.353694
$677$	1.18065e10	1.46238	0.731188	−	0.682176i	$-0.238966\pi$
$677$	1.18065e10	1.46238	0.731188	+	0.682176i	$0.238966\pi$
$678$	3.93726e9	0.485166
$679$	1.74401e9	0.213798
$680$	−1.21975e9	−0.148761
$681$	1.98140e9	0.240412
$682$	1.57754e9	0.190429
$683$	5.88145e9	0.706337	0.353169	−	0.935560i	$-0.385104\pi$
$683$	5.88145e9	0.706337	0.353169	+	0.935560i	$0.385104\pi$
$684$	−6.57528e8	−0.0785630
$685$	1.52101e9	0.180807
$686$	−5.30468e9	−0.627372
$687$	7.64971e8	0.0900112
$688$	6.32332e9	0.740262
$689$	2.23706e10	2.60562
$690$	1.67423e8	0.0194019
$691$	−4.16582e9	−0.480316	−0.240158	−	0.970734i	$-0.577199\pi$
$691$	−4.16582e9	−0.480316	−0.240158	+	0.970734i	$0.577199\pi$
$692$	−1.29462e9	−0.148515
$693$	4.10898e8	0.0468995
$694$	−2.64277e9	−0.300125
$695$	1.81212e9	0.204757
$696$	−5.15283e9	−0.579313
$697$	−1.30717e10	−1.46224
$698$	−1.29471e9	−0.144105
$699$	−3.11178e9	−0.344618
$700$	−2.36303e9	−0.260391
$701$	−1.38729e10	−1.52109	−0.760546	−	0.649284i	$-0.775069\pi$
$701$	−1.38729e10	−1.52109	−0.760546	+	0.649284i	$0.775069\pi$
$702$	3.16457e9	0.345251
$703$	−2.94039e9	−0.319199
$704$	7.38240e8	0.0797431
$705$	9.20944e8	0.0989855
$706$	2.91634e9	0.311905
$707$	1.42123e10	1.51250
$708$	1.91111e9	0.202381
$709$	3.27840e9	0.345462	0.172731	−	0.984969i	$-0.444741\pi$
$709$	3.27840e9	0.345462	0.172731	+	0.984969i	$0.444741\pi$
$710$	−3.61649e8	−0.0379213
$711$	−1.44907e9	−0.151198
$712$	−1.03265e10	−1.07219
$713$	3.07746e9	0.317964
$714$	9.02578e9	0.927985
$715$	2.63725e8	0.0269824
$716$	2.64192e8	0.0268982
$717$	−1.59032e9	−0.161127
$718$	−2.07313e10	−2.09021
$719$	−6.37084e9	−0.639213	−0.319607	−	0.947550i	$-0.603551\pi$
$719$	−6.37084e9	−0.639213	−0.319607	+	0.947550i	$0.603551\pi$
$720$	5.70468e8	0.0569597
$721$	−1.72639e10	−1.71540
$722$	2.33371e9	0.230763
$723$	1.38479e9	0.136270
$724$	3.75755e9	0.367976
$725$	1.16373e10	1.13415
$726$	6.47544e9	0.628046
$727$	1.86871e10	1.80373	0.901867	−	0.432013i	$-0.142197\pi$
$727$	1.86871e10	1.80373	0.901867	+	0.432013i	$0.142197\pi$
$728$	1.82175e10	1.74996
$729$	3.87420e8	0.0370370
$730$	−4.06391e8	−0.0386647
$731$	7.86110e9	0.744342
$732$	1.77802e9	0.167552
$733$	1.28678e10	1.20682	0.603409	−	0.797432i	$-0.293809\pi$
$733$	1.28678e10	1.20682	0.603409	+	0.797432i	$0.293809\pi$
$734$	−6.70358e9	−0.625706
$735$	4.92152e8	0.0457187
$736$	−9.51111e8	−0.0879344
$737$	1.78630e9	0.164368
$738$	4.99214e9	0.457182
$739$	−5.37369e9	−0.489798	−0.244899	−	0.969549i	$-0.578755\pi$
$739$	−5.37369e9	−0.489798	−0.244899	+	0.969549i	$0.578755\pi$
$740$	−1.00286e8	−0.00909768
$741$	−1.14483e10	−1.03366
$742$	−2.43670e10	−2.18972
$743$	−5.20876e9	−0.465879	−0.232940	−	0.972491i	$-0.574834\pi$
$743$	−5.20876e9	−0.465879	−0.232940	+	0.972491i	$0.574834\pi$
$744$	8.56251e9	0.762248
$745$	−8.74778e8	−0.0775088
$746$	−2.73331e9	−0.241047
$747$	5.98045e9	0.524943
$748$	−3.26593e8	−0.0285332
$749$	4.03938e9	0.351260
$750$	−2.12708e9	−0.184106
$751$	7.73437e9	0.666324	0.333162	−	0.942870i	$-0.391884\pi$
$751$	7.73437e9	0.666324	0.333162	+	0.942870i	$0.391884\pi$
$752$	−1.59726e10	−1.36966
$753$	8.40530e8	0.0717416
$754$	2.44724e10	2.07911
$755$	2.20381e9	0.186363
$756$	−6.08359e8	−0.0512075
$757$	−1.32525e9	−0.111036	−0.0555178	−	0.998458i	$-0.517681\pi$
$757$	−1.32525e9	−0.111036	−0.0555178	+	0.998458i	$0.517681\pi$
$758$	1.36553e10	1.13883
$759$	−1.64342e8	−0.0136427
$760$	−1.68520e9	−0.139253
$761$	1.43521e10	1.18050	0.590252	−	0.807219i	$-0.299028\pi$
$761$	1.43521e10	1.18050	0.590252	+	0.807219i	$0.299028\pi$
$762$	−1.37727e10	−1.12766
$763$	−1.28443e10	−1.04683
$764$	1.61289e9	0.130851
$765$	7.09202e8	0.0572736
$766$	−1.86379e10	−1.49829
$767$	3.32746e10	2.66274
$768$	−4.46116e9	−0.355373
$769$	1.52167e10	1.20664	0.603320	−	0.797499i	$-0.293844\pi$
$769$	1.52167e10	1.20664	0.603320	+	0.797499i	$0.293844\pi$
$770$	−2.87261e8	−0.0226756
$771$	−6.43398e9	−0.505580
$772$	3.97691e7	0.00311089
$773$	1.74767e10	1.36092	0.680459	−	0.732786i	$-0.261781\pi$
$773$	1.74767e10	1.36092	0.680459	+	0.732786i	$0.261781\pi$
$774$	−3.00219e9	−0.232726
$775$	−1.93378e10	−1.49229
$776$	−1.94076e9	−0.149092
$777$	−2.72051e9	−0.208055
$778$	−1.32916e10	−1.01193
$779$	−1.80598e10	−1.36878
$780$	−3.90461e8	−0.0294609
$781$	3.54992e8	0.0266649
$782$	−3.60992e9	−0.269944
$783$	2.99602e9	0.223037
$784$	−8.53576e9	−0.632610
$785$	2.01315e8	0.0148536
$786$	−3.21182e9	−0.235924
$787$	−7.75898e8	−0.0567405	−0.0283702	−	0.999597i	$-0.509032\pi$
$787$	−7.75898e8	−0.0567405	−0.0283702	+	0.999597i	$0.509032\pi$
$788$	−1.81804e9	−0.132362
$789$	−7.67488e9	−0.556291
$790$	1.01305e9	0.0731032
$791$	−1.31785e10	−0.946778
$792$	−4.57254e8	−0.0327054
$793$	3.09574e10	2.20449
$794$	1.28237e10	0.909165
$795$	−1.91464e9	−0.135146
$796$	−4.90303e9	−0.344563
$797$	−3.17179e9	−0.221922	−0.110961	−	0.993825i	$-0.535393\pi$
$797$	−3.17179e9	−0.221922	−0.110961	+	0.993825i	$0.535393\pi$
$798$	1.24700e10	0.868671
$799$	−1.98571e10	−1.37721
$800$	5.97651e9	0.412698
$801$	6.00414e9	0.412797
$802$	3.62117e9	0.247879
$803$	3.98911e8	0.0271876
$804$	−2.64472e9	−0.179466
$805$	−5.60388e8	−0.0378619
$806$	−4.06661e10	−2.73564
$807$	5.24580e9	0.351362
$808$	−1.58156e10	−1.05474
$809$	−2.96650e10	−1.96981	−0.984905	−	0.173096i	$-0.944623\pi$
$809$	−2.96650e10	−1.96981	−0.984905	+	0.173096i	$0.944623\pi$
$810$	−2.70847e8	−0.0179072
$811$	2.28837e10	1.50645	0.753223	−	0.657765i	$-0.228498\pi$
$811$	2.28837e10	1.50645	0.753223	+	0.657765i	$0.228498\pi$
$812$	−4.70459e9	−0.308372
$813$	9.96531e9	0.650390
$814$	5.57766e8	0.0362465
$815$	2.14345e9	0.138695
$816$	−1.23002e10	−0.792497
$817$	1.08609e10	0.696766
$818$	−8.00757e9	−0.511522
$819$	−1.05922e10	−0.673742
$820$	−6.15956e8	−0.0390122
$821$	−9.31507e9	−0.587469	−0.293734	−	0.955887i	$-0.594898\pi$
$821$	−9.31507e9	−0.587469	−0.293734	+	0.955887i	$0.594898\pi$
$822$	1.25247e10	0.786531
$823$	2.28072e10	1.42618	0.713088	−	0.701074i	$-0.247296\pi$
$823$	2.28072e10	1.42618	0.713088	+	0.701074i	$0.247296\pi$
$824$	1.92115e10	1.19623
$825$	1.03268e9	0.0640288
$826$	−3.62440e10	−2.23772
$827$	1.04877e10	0.644781	0.322391	−	0.946607i	$-0.395514\pi$
$827$	1.04877e10	0.644781	0.322391	+	0.946607i	$0.395514\pi$
$828$	2.43317e8	0.0148959
$829$	−1.95304e10	−1.19061	−0.595307	−	0.803498i	$-0.702970\pi$
$829$	−1.95304e10	−1.19061	−0.595307	+	0.803498i	$0.702970\pi$
$830$	−4.18096e9	−0.253806
$831$	1.42815e10	0.863319
$832$	−1.90305e10	−1.14556
$833$	−1.06116e10	−0.636097
$834$	1.49218e10	0.890720
$835$	−2.31697e9	−0.137727
$836$	−4.51219e8	−0.0267095
$837$	−4.97851e9	−0.293468
$838$	2.43152e10	1.42733
$839$	−1.46981e10	−0.859200	−0.429600	−	0.903019i	$-0.641345\pi$
$839$	−1.46981e10	−0.859200	−0.429600	+	0.903019i	$0.641345\pi$
$840$	−1.55919e9	−0.0907654
$841$	5.91903e9	0.343134
$842$	−3.32435e10	−1.91918
$843$	−1.22654e9	−0.0705156
$844$	−2.95095e8	−0.0168952
$845$	−4.23328e9	−0.241367
$846$	7.58349e9	0.430599
$847$	−2.16741e10	−1.22560
$848$	3.32071e10	1.87001
$849$	−6.16229e9	−0.345593
$850$	2.26837e10	1.26692
$851$	1.08809e9	0.0605217
$852$	−5.25587e8	−0.0291143
$853$	1.80045e10	0.993253	0.496626	−	0.867964i	$-0.334572\pi$
$853$	1.80045e10	0.993253	0.496626	+	0.867964i	$0.334572\pi$
$854$	−3.37201e10	−1.85262
$855$	9.79830e8	0.0536129
$856$	−4.49508e9	−0.244951
$857$	−1.06997e10	−0.580681	−0.290340	−	0.956923i	$-0.593768\pi$
$857$	−1.06997e10	−0.580681	−0.290340	+	0.956923i	$0.593768\pi$
$858$	2.17164e9	0.117377
$859$	2.30270e10	1.23954	0.619771	−	0.784783i	$-0.287225\pi$
$859$	2.30270e10	1.23954	0.619771	+	0.784783i	$0.287225\pi$
$860$	3.70425e8	0.0198589
$861$	−1.67093e10	−0.892171
$862$	−1.80776e10	−0.961312
$863$	1.75568e10	0.929837	0.464919	−	0.885353i	$-0.346084\pi$
$863$	1.75568e10	0.929837	0.464919	+	0.885353i	$0.346084\pi$
$864$	1.53865e9	0.0811598
$865$	1.92921e9	0.101350
$866$	1.07136e10	0.560560
$867$	−4.21240e9	−0.219514
$868$	7.81767e9	0.405750
$869$	−9.94404e8	−0.0514036
$870$	−2.09453e9	−0.107837
$871$	−4.60475e10	−2.36125
$872$	1.42933e10	0.730005
$873$	1.12842e9	0.0574010
$874$	−4.98745e9	−0.252690
$875$	7.11959e9	0.359275
$876$	−5.90611e8	−0.0296850
$877$	−3.55406e10	−1.77920	−0.889602	−	0.456737i	$-0.849018\pi$
$877$	−3.55406e10	−1.77920	−0.889602	+	0.456737i	$0.849018\pi$
$878$	−2.83944e10	−1.41580
$879$	−1.21324e10	−0.602541
$880$	3.91475e8	0.0193649
$881$	−1.43619e10	−0.707616	−0.353808	−	0.935318i	$-0.615113\pi$
$881$	−1.43619e10	−0.707616	−0.353808	+	0.935318i	$0.615113\pi$
$882$	4.05261e9	0.198882
$883$	7.01212e9	0.342757	0.171379	−	0.985205i	$-0.445178\pi$
$883$	7.01212e9	0.342757	0.171379	+	0.985205i	$0.445178\pi$
$884$	8.41897e9	0.409898
$885$	−2.84788e9	−0.138108
$886$	2.28193e9	0.110226
$887$	1.65555e10	0.796546	0.398273	−	0.917267i	$-0.369610\pi$
$887$	1.65555e10	0.796546	0.398273	+	0.917267i	$0.369610\pi$
$888$	3.02743e9	0.145087
$889$	4.60990e10	2.20057
$890$	−4.19752e9	−0.199585
$891$	2.65862e8	0.0125917
$892$	−5.87833e9	−0.277317
$893$	−2.74344e10	−1.28919
$894$	−7.20334e9	−0.337173
$895$	−3.93691e8	−0.0183559
$896$	3.20024e10	1.48630
$897$	4.23643e9	0.195987
$898$	1.37934e10	0.635631
$899$	−3.85000e10	−1.76727
$900$	−1.52894e9	−0.0699102
$901$	4.12827e10	1.88032
$902$	3.42578e9	0.155431
$903$	1.00487e10	0.454154
$904$	1.46652e10	0.660236
$905$	−5.59939e9	−0.251114
$906$	1.81472e10	0.810701
$907$	3.79998e10	1.69105	0.845525	−	0.533936i	$-0.179288\pi$
$907$	3.79998e10	1.69105	0.845525	+	0.533936i	$0.179288\pi$
$908$	−2.01312e9	−0.0892421
$909$	9.19570e9	0.406079
$910$	7.40506e9	0.325750
$911$	9.57936e9	0.419781	0.209890	−	0.977725i	$-0.432689\pi$
$911$	9.57936e9	0.419781	0.209890	+	0.977725i	$0.432689\pi$
$912$	−1.69939e10	−0.741843
$913$	4.10400e9	0.178468
$914$	−2.09432e9	−0.0907258
$915$	−2.64956e9	−0.114340
$916$	−7.77219e8	−0.0334125
$917$	1.07504e10	0.460395
$918$	5.83990e9	0.249147
$919$	2.53060e10	1.07552	0.537760	−	0.843098i	$-0.319271\pi$
$919$	2.53060e10	1.07552	0.537760	+	0.843098i	$0.319271\pi$
$920$	6.23607e8	0.0264030
$921$	9.61750e9	0.405652
$922$	3.20222e10	1.34553
$923$	−9.15106e9	−0.383059
$924$	−4.17477e8	−0.0174093
$925$	−6.83724e9	−0.284043
$926$	−1.37962e10	−0.570980
$927$	−1.11701e10	−0.460553
$928$	1.18987e10	0.488745
$929$	−8.03531e9	−0.328812	−0.164406	−	0.986393i	$-0.552571\pi$
$929$	−8.03531e9	−0.328812	−0.164406	+	0.986393i	$0.552571\pi$
$930$	3.48050e9	0.141890
$931$	−1.46609e10	−0.595440
$932$	3.16160e9	0.127924
$933$	−1.92855e10	−0.777402
$934$	3.48646e10	1.40014
$935$	4.86679e8	0.0194716
$936$	1.17872e10	0.469834
$937$	−1.25462e10	−0.498223	−0.249112	−	0.968475i	$-0.580139\pi$
$937$	−1.25462e10	−0.498223	−0.249112	+	0.968475i	$0.580139\pi$
$938$	5.01569e10	1.98436
$939$	−2.58437e9	−0.101865
$940$	−9.35690e8	−0.0367438
$941$	−1.99670e10	−0.781176	−0.390588	−	0.920566i	$-0.627728\pi$
$941$	−1.99670e10	−0.781176	−0.390588	+	0.920566i	$0.627728\pi$
$942$	1.65772e9	0.0646150
$943$	6.68301e9	0.259526
$944$	4.93930e10	1.91101
$945$	9.06559e8	0.0349450
$946$	−2.06021e9	−0.0791210
$947$	3.32659e10	1.27284	0.636422	−	0.771341i	$-0.280414\pi$
$947$	3.32659e10	1.27284	0.636422	+	0.771341i	$0.280414\pi$
$948$	1.47227e9	0.0561254
$949$	−1.02832e10	−0.390568
$950$	3.13397e10	1.18594
$951$	1.86026e10	0.701360
$952$	3.36186e10	1.26285
$953$	4.41016e8	0.0165055	0.00825277	−	0.999966i	$-0.497373\pi$
$953$	4.41016e8	0.0165055	0.00825277	+	0.999966i	$0.497373\pi$
$954$	−1.57661e10	−0.587900
$955$	−2.40348e9	−0.0892954
$956$	1.61578e9	0.0598109
$957$	2.05597e9	0.0758272
$958$	2.73449e10	1.00484
$959$	−4.19218e10	−1.53488
$960$	1.62877e9	0.0594169
$961$	3.64633e10	1.32533
$962$	−1.43782e10	−0.520705
$963$	2.61358e9	0.0943070
$964$	−1.40696e9	−0.0505840
$965$	−5.92628e7	−0.00212293
$966$	−4.61450e9	−0.164704
$967$	−4.20440e9	−0.149524	−0.0747620	−	0.997201i	$-0.523820\pi$
$967$	−4.20440e9	−0.149524	−0.0747620	+	0.997201i	$0.523820\pi$
$968$	2.41193e10	0.854674
$969$	−2.11267e10	−0.745931
$970$	−7.88880e8	−0.0277530
$971$	−3.02108e10	−1.05900	−0.529499	−	0.848311i	$-0.677620\pi$
$971$	−3.02108e10	−1.05900	−0.529499	+	0.848311i	$0.677620\pi$
$972$	−3.93624e8	−0.0137483
$973$	−4.99453e10	−1.73820
$974$	−1.85065e10	−0.641752
$975$	−2.66205e10	−0.919815
$976$	4.59533e10	1.58213
$977$	−3.60793e10	−1.23773	−0.618866	−	0.785496i	$-0.712408\pi$
$977$	−3.60793e10	−1.23773	−0.618866	+	0.785496i	$0.712408\pi$
$978$	1.76502e10	0.603340
$979$	4.12025e9	0.140341
$980$	−5.00032e8	−0.0169710
$981$	−8.31059e9	−0.281054
$982$	−4.47097e10	−1.50665
$983$	−3.38986e9	−0.113827	−0.0569133	−	0.998379i	$-0.518126\pi$
$983$	−3.38986e9	−0.113827	−0.0569133	+	0.998379i	$0.518126\pi$
$984$	1.85944e10	0.622155
$985$	2.70920e9	0.0903262
$986$	4.51614e10	1.50037
$987$	−2.53829e10	−0.840294
$988$	1.16316e10	0.383699
$989$	−4.01905e9	−0.132110
$990$	−1.85865e8	−0.00608799
$991$	8.93953e9	0.291781	0.145890	−	0.989301i	$-0.453395\pi$
$991$	8.93953e9	0.291781	0.145890	+	0.989301i	$0.453395\pi$
$992$	−1.97723e10	−0.643080
$993$	−2.26079e10	−0.732720
$994$	9.96771e9	0.321916
$995$	7.30635e9	0.235136
$996$	−6.07621e9	−0.194861
$997$	−4.01968e10	−1.28457	−0.642286	−	0.766465i	$-0.722014\pi$
$997$	−4.01968e10	−1.28457	−0.642286	+	0.766465i	$0.722014\pi$
$998$	−3.29815e10	−1.05030
$999$	−1.76024e9	−0.0558590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.a.1.4	✓	5
3.2	odd	2		207.8.a.a.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.4	✓	5	1.1	even	1	trivial
207.8.a.a.1.2		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+12.4672 q^{2} -27.0000 q^{3} +27.4323 q^{4} -40.8788 q^{5} -336.616 q^{6} +1126.70 q^{7} -1253.80 q^{8} +729.000 q^{9} -509.647 q^{10} +500.265 q^{11} -740.672 q^{12} -12895.9 q^{13} +14046.8 q^{14} +1103.73 q^{15} -19142.8 q^{16} -23798.2 q^{17} +9088.62 q^{18} -32879.5 q^{19} -1121.40 q^{20} -30420.8 q^{21} +6236.93 q^{22} +12167.0 q^{23} +33852.7 q^{24} -76453.9 q^{25} -160777. q^{26} -19683.0 q^{27} +30907.8 q^{28} -152213. q^{29} +13760.5 q^{30} +252935. q^{31} -78171.4 q^{32} -13507.2 q^{33} -296698. q^{34} -46058.0 q^{35} +19998.1 q^{36} +89429.5 q^{37} -409916. q^{38} +348190. q^{39} +51254.0 q^{40} +549274. q^{41} -379263. q^{42} -330324. q^{43} +13723.4 q^{44} -29800.7 q^{45} +151689. q^{46} +834394. q^{47} +516856. q^{48} +445899. q^{49} -953170. q^{50} +642551. q^{51} -353765. q^{52} -1.73470e6 q^{53} -245393. q^{54} -20450.3 q^{55} -1.41265e6 q^{56} +887745. q^{57} -1.89768e6 q^{58} -2.58024e6 q^{59} +30277.8 q^{60} -2.40055e6 q^{61} +3.15340e6 q^{62} +821361. q^{63} +1.47570e6 q^{64} +527171. q^{65} -168397. q^{66} +3.57070e6 q^{67} -652839. q^{68} -328509. q^{69} -574216. q^{70} +709607. q^{71} -914022. q^{72} +797398. q^{73} +1.11494e6 q^{74} +2.06426e6 q^{75} -901959. q^{76} +563647. q^{77} +4.34098e6 q^{78} -1.98775e6 q^{79} +782535. q^{80} +531441. q^{81} +6.84793e6 q^{82} +8.20364e6 q^{83} -834512. q^{84} +972842. q^{85} -4.11823e6 q^{86} +4.10976e6 q^{87} -627234. q^{88} +8.23613e6 q^{89} -371532. q^{90} -1.45298e7 q^{91} +333769. q^{92} -6.82924e6 q^{93} +1.04026e7 q^{94} +1.34407e6 q^{95} +2.11063e6 q^{96} +1.54790e6 q^{97} +5.55914e6 q^{98} +364694. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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