LMFDB - Embedded newform 69.6.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,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, 6, names="a")

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
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:	$11.0664835671$
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} - 39x^{2} - 30x + 20 $ x^4 - 39x^2 - 30x + 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.56547$ 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)

$f(q)$	$=$	$q+7.56547 q^{2} -9.00000 q^{3} +25.2363 q^{4} +40.1532 q^{5} -68.0892 q^{6} +194.921 q^{7} -51.1704 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+7.56547 q^{2} -9.00000 q^{3} +25.2363 q^{4} +40.1532 q^{5} -68.0892 q^{6} +194.921 q^{7} -51.1704 q^{8} +81.0000 q^{9} +303.778 q^{10} -39.1915 q^{11} -227.127 q^{12} +705.021 q^{13} +1474.67 q^{14} -361.379 q^{15} -1194.69 q^{16} +1222.51 q^{17} +612.803 q^{18} +1891.46 q^{19} +1013.32 q^{20} -1754.29 q^{21} -296.502 q^{22} +529.000 q^{23} +460.534 q^{24} -1512.72 q^{25} +5333.81 q^{26} -729.000 q^{27} +4919.10 q^{28} -8579.39 q^{29} -2734.00 q^{30} -9582.72 q^{31} -7400.94 q^{32} +352.723 q^{33} +9248.86 q^{34} +7826.73 q^{35} +2044.14 q^{36} -8144.49 q^{37} +14309.8 q^{38} -6345.19 q^{39} -2054.66 q^{40} +4834.24 q^{41} -13272.0 q^{42} +10631.5 q^{43} -989.049 q^{44} +3252.41 q^{45} +4002.13 q^{46} +222.835 q^{47} +10752.2 q^{48} +21187.4 q^{49} -11444.4 q^{50} -11002.6 q^{51} +17792.1 q^{52} +6577.44 q^{53} -5515.23 q^{54} -1573.66 q^{55} -9974.21 q^{56} -17023.1 q^{57} -64907.1 q^{58} -30780.8 q^{59} -9119.88 q^{60} -8695.20 q^{61} -72497.8 q^{62} +15788.6 q^{63} -17761.5 q^{64} +28308.9 q^{65} +2668.52 q^{66} +48398.0 q^{67} +30851.6 q^{68} -4761.00 q^{69} +59212.8 q^{70} -60285.7 q^{71} -4144.80 q^{72} +30638.2 q^{73} -61616.8 q^{74} +13614.5 q^{75} +47733.4 q^{76} -7639.26 q^{77} -48004.3 q^{78} +50285.3 q^{79} -47970.7 q^{80} +6561.00 q^{81} +36573.3 q^{82} -35671.7 q^{83} -44271.9 q^{84} +49087.7 q^{85} +80432.0 q^{86} +77214.5 q^{87} +2005.44 q^{88} -77619.2 q^{89} +24606.0 q^{90} +137424. q^{91} +13350.0 q^{92} +86244.5 q^{93} +1685.85 q^{94} +75948.1 q^{95} +66608.5 q^{96} -83069.4 q^{97} +160292. q^{98} -3174.51 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9} + 642 q^{10} + 32 q^{11} + 414 q^{12} + 1364 q^{13} + 2754 q^{14} + 1098 q^{15} + 18 q^{16} + 278 q^{17} + 324 q^{18} + 2862 q^{19} + 3830 q^{20} - 558 q^{21} + 3176 q^{22} + 2116 q^{23} - 648 q^{24} + 5944 q^{25} + 6996 q^{26} - 2916 q^{27} + 4738 q^{28} - 5180 q^{29} - 5778 q^{30} - 1788 q^{31} - 7352 q^{32} - 288 q^{33} + 15818 q^{34} + 11768 q^{35} - 3726 q^{36} + 3348 q^{37} + 1050 q^{38} - 12276 q^{39} - 13462 q^{40} - 17664 q^{41} - 24786 q^{42} + 25398 q^{43} - 16848 q^{44} - 9882 q^{45} + 2116 q^{46} - 26040 q^{47} - 162 q^{48} + 55720 q^{49} - 35256 q^{50} - 2502 q^{51} - 2752 q^{52} - 32006 q^{53} - 2916 q^{54} + 34904 q^{55} - 68542 q^{56} - 25758 q^{57} - 40804 q^{58} - 61136 q^{59} - 34470 q^{60} + 35844 q^{61} - 47524 q^{62} + 5022 q^{63} - 35142 q^{64} - 48036 q^{65} - 28584 q^{66} + 73458 q^{67} + 17910 q^{68} - 19044 q^{69} - 59104 q^{70} + 24432 q^{71} + 5832 q^{72} + 122512 q^{73} - 20828 q^{74} - 53496 q^{75} + 56834 q^{76} + 159496 q^{77} - 62964 q^{78} + 90170 q^{79} - 36546 q^{80} + 26244 q^{81} + 84144 q^{82} + 28592 q^{83} - 42642 q^{84} + 355124 q^{85} + 103778 q^{86} + 46620 q^{87} - 150776 q^{88} - 27926 q^{89} + 52002 q^{90} + 334180 q^{91} - 24334 q^{92} + 16092 q^{93} + 113632 q^{94} + 113392 q^{95} + 66168 q^{96} + 16580 q^{97} + 49688 q^{98} + 2592 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9 + 642 * q^10 + 32 * q^11 + 414 * q^12 + 1364 * q^13 + 2754 * q^14 + 1098 * q^15 + 18 * q^16 + 278 * q^17 + 324 * q^18 + 2862 * q^19 + 3830 * q^20 - 558 * q^21 + 3176 * q^22 + 2116 * q^23 - 648 * q^24 + 5944 * q^25 + 6996 * q^26 - 2916 * q^27 + 4738 * q^28 - 5180 * q^29 - 5778 * q^30 - 1788 * q^31 - 7352 * q^32 - 288 * q^33 + 15818 * q^34 + 11768 * q^35 - 3726 * q^36 + 3348 * q^37 + 1050 * q^38 - 12276 * q^39 - 13462 * q^40 - 17664 * q^41 - 24786 * q^42 + 25398 * q^43 - 16848 * q^44 - 9882 * q^45 + 2116 * q^46 - 26040 * q^47 - 162 * q^48 + 55720 * q^49 - 35256 * q^50 - 2502 * q^51 - 2752 * q^52 - 32006 * q^53 - 2916 * q^54 + 34904 * q^55 - 68542 * q^56 - 25758 * q^57 - 40804 * q^58 - 61136 * q^59 - 34470 * q^60 + 35844 * q^61 - 47524 * q^62 + 5022 * q^63 - 35142 * q^64 - 48036 * q^65 - 28584 * q^66 + 73458 * q^67 + 17910 * q^68 - 19044 * q^69 - 59104 * q^70 + 24432 * q^71 + 5832 * q^72 + 122512 * q^73 - 20828 * q^74 - 53496 * q^75 + 56834 * q^76 + 159496 * q^77 - 62964 * q^78 + 90170 * q^79 - 36546 * q^80 + 26244 * q^81 + 84144 * q^82 + 28592 * q^83 - 42642 * q^84 + 355124 * q^85 + 103778 * q^86 + 46620 * q^87 - 150776 * q^88 - 27926 * q^89 + 52002 * q^90 + 334180 * q^91 - 24334 * q^92 + 16092 * q^93 + 113632 * q^94 + 113392 * q^95 + 66168 * q^96 + 16580 * q^97 + 49688 * q^98 + 2592 * 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 $

$2$	7.56547	1.33740	0.668699	−	0.743533i	$-0.266851\pi$
$2$	7.56547	1.33740	0.668699	+	0.743533i	$0.266851\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	25.2363	0.788635
$5$	40.1532	0.718283	0.359141	−	0.933283i	$-0.383070\pi$
$5$	40.1532	0.718283	0.359141	+	0.933283i	$0.383070\pi$
$6$	−68.0892	−0.772147
$7$	194.921	1.50354	0.751769	−	0.659426i	$-0.229201\pi$
$7$	194.921	1.50354	0.751769	+	0.659426i	$0.229201\pi$
$8$	−51.1704	−0.282679
$9$	81.0000	0.333333
$10$	303.778	0.960630
$11$	−39.1915	−0.0976585	−0.0488292	−	0.998807i	$-0.515549\pi$
$11$	−39.1915	−0.0976585	−0.0488292	+	0.998807i	$0.515549\pi$
$12$	−227.127	−0.455319
$13$	705.021	1.15703	0.578514	−	0.815673i	$-0.303633\pi$
$13$	705.021	1.15703	0.578514	+	0.815673i	$0.303633\pi$
$14$	1474.67	2.01083
$15$	−361.379	−0.414701
$16$	−1194.69	−1.16669
$17$	1222.51	1.02596	0.512979	−	0.858401i	$-0.328542\pi$
$17$	1222.51	1.02596	0.512979	+	0.858401i	$0.328542\pi$
$18$	612.803	0.445800
$19$	1891.46	1.20202	0.601011	−	0.799241i	$-0.294765\pi$
$19$	1891.46	1.20202	0.601011	+	0.799241i	$0.294765\pi$
$20$	1013.32	0.566463
$21$	−1754.29	−0.868068
$22$	−296.502	−0.130608
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	460.534	0.163205
$25$	−1512.72	−0.484070
$26$	5333.81	1.54741
$27$	−729.000	−0.192450
$28$	4919.10	1.18574
$29$	−8579.39	−1.89436	−0.947178	−	0.320709i	$-0.896079\pi$
$29$	−8579.39	−1.89436	−0.947178	+	0.320709i	$0.896079\pi$
$30$	−2734.00	−0.554620
$31$	−9582.72	−1.79095	−0.895477	−	0.445107i	$-0.853165\pi$
$31$	−9582.72	−1.79095	−0.895477	+	0.445107i	$0.853165\pi$
$32$	−7400.94	−1.27765
$33$	352.723	0.0563832
$34$	9248.86	1.37212
$35$	7826.73	1.07997
$36$	2044.14	0.262878
$37$	−8144.49	−0.978046	−0.489023	−	0.872271i	$-0.662647\pi$
$37$	−8144.49	−0.978046	−0.489023	+	0.872271i	$0.662647\pi$
$38$	14309.8	1.60758
$39$	−6345.19	−0.668010
$40$	−2054.66	−0.203044
$41$	4834.24	0.449127	0.224563	−	0.974459i	$-0.427904\pi$
$41$	4834.24	0.449127	0.224563	+	0.974459i	$0.427904\pi$
$42$	−13272.0	−1.16095
$43$	10631.5	0.876843	0.438422	−	0.898769i	$-0.355538\pi$
$43$	10631.5	0.876843	0.438422	+	0.898769i	$0.355538\pi$
$44$	−989.049	−0.0770169
$45$	3252.41	0.239428
$46$	4002.13	0.278867
$47$	222.835	0.0147142	0.00735712	−	0.999973i	$-0.497658\pi$
$47$	222.835	0.0147142	0.00735712	+	0.999973i	$0.497658\pi$
$48$	10752.2	0.673589
$49$	21187.4	1.26063
$50$	−11444.4	−0.647394
$51$	−11002.6	−0.592338
$52$	17792.1	0.912472
$53$	6577.44	0.321638	0.160819	−	0.986984i	$-0.448586\pi$
$53$	6577.44	0.321638	0.160819	+	0.986984i	$0.448586\pi$
$54$	−5515.23	−0.257382
$55$	−1573.66	−0.0701464
$56$	−9974.21	−0.425019
$57$	−17023.1	−0.693988
$58$	−64907.1	−2.53351
$59$	−30780.8	−1.15120	−0.575600	−	0.817732i	$-0.695231\pi$
$59$	−30780.8	−1.15120	−0.575600	+	0.817732i	$0.695231\pi$
$60$	−9119.88	−0.327048
$61$	−8695.20	−0.299196	−0.149598	−	0.988747i	$-0.547798\pi$
$61$	−8695.20	−0.299196	−0.149598	+	0.988747i	$0.547798\pi$
$62$	−72497.8	−2.39522
$63$	15788.6	0.501179
$64$	−17761.5	−0.542038
$65$	28308.9	0.831073
$66$	2668.52	0.0754067
$67$	48398.0	1.31716	0.658582	−	0.752509i	$-0.271156\pi$
$67$	48398.0	1.31716	0.658582	+	0.752509i	$0.271156\pi$
$68$	30851.6	0.809107
$69$	−4761.00	−0.120386
$70$	59212.8	1.44434
$71$	−60285.7	−1.41928	−0.709640	−	0.704564i	$-0.751142\pi$
$71$	−60285.7	−1.41928	−0.709640	+	0.704564i	$0.751142\pi$
$72$	−4144.80	−0.0942264
$73$	30638.2	0.672908	0.336454	−	0.941700i	$-0.390772\pi$
$73$	30638.2	0.672908	0.336454	+	0.941700i	$0.390772\pi$
$74$	−61616.8	−1.30804
$75$	13614.5	0.279478
$76$	47733.4	0.947957
$77$	−7639.26	−0.146833
$78$	−48004.3	−0.893396
$79$	50285.3	0.906512	0.453256	−	0.891380i	$-0.350262\pi$
$79$	50285.3	0.906512	0.453256	+	0.891380i	$0.350262\pi$
$80$	−47970.7	−0.838013
$81$	6561.00	0.111111
$82$	36573.3	0.600661
$83$	−35671.7	−0.568366	−0.284183	−	0.958770i	$-0.591722\pi$
$83$	−35671.7	−0.568366	−0.284183	+	0.958770i	$0.591722\pi$
$84$	−44271.9	−0.684589
$85$	49087.7	0.736929
$86$	80432.0	1.17269
$87$	77214.5	1.09371
$88$	2005.44	0.0276060
$89$	−77619.2	−1.03871	−0.519355	−	0.854559i	$-0.673828\pi$
$89$	−77619.2	−1.03871	−0.519355	+	0.854559i	$0.673828\pi$
$90$	24606.0	0.320210
$91$	137424.	1.73963
$92$	13350.0	0.164442
$93$	86244.5	1.03401
$94$	1685.85	0.0196788
$95$	75948.1	0.863392
$96$	66608.5	0.737652
$97$	−83069.4	−0.896421	−0.448210	−	0.893928i	$-0.647938\pi$
$97$	−83069.4	−0.896421	−0.448210	+	0.893928i	$0.647938\pi$
$98$	160292.	1.68596
$99$	−3174.51	−0.0325528
$100$	−38175.4	−0.381754
$101$	30834.9	0.300773	0.150387	−	0.988627i	$-0.451948\pi$
$101$	30834.9	0.300773	0.150387	+	0.988627i	$0.451948\pi$
$102$	−83239.7	−0.792191
$103$	141687.	1.31594	0.657970	−	0.753044i	$-0.271415\pi$
$103$	141687.	1.31594	0.657970	+	0.753044i	$0.271415\pi$
$104$	−36076.2	−0.327068
$105$	−70440.5	−0.623518
$106$	49761.4	0.430158
$107$	132757.	1.12098	0.560488	−	0.828162i	$-0.310614\pi$
$107$	132757.	1.12098	0.560488	+	0.828162i	$0.310614\pi$
$108$	−18397.3	−0.151773
$109$	−113030.	−0.911232	−0.455616	−	0.890176i	$-0.650581\pi$
$109$	−113030.	−0.911232	−0.455616	+	0.890176i	$0.650581\pi$
$110$	−11905.5	−0.0938137
$111$	73300.4	0.564675
$112$	−232871.	−1.75416
$113$	46155.5	0.340038	0.170019	−	0.985441i	$-0.445617\pi$
$113$	46155.5	0.340038	0.170019	+	0.985441i	$0.445617\pi$
$114$	−128788.	−0.928138
$115$	21241.1	0.149772
$116$	−216512.	−1.49396
$117$	57106.7	0.385676
$118$	−232872.	−1.53961
$119$	238293.	1.54257
$120$	18491.9	0.117227
$121$	−159515.	−0.990463
$122$	−65783.3	−0.400144
$123$	−43508.2	−0.259303
$124$	−241833.	−1.41241
$125$	−186219.	−1.06598
$126$	119448.	0.670277
$127$	−223792.	−1.23122	−0.615609	−	0.788051i	$-0.711090\pi$
$127$	−223792.	−1.23122	−0.615609	+	0.788051i	$0.711090\pi$
$128$	102456.	0.552730
$129$	−95683.2	−0.506246
$130$	214170.	1.11148
$131$	−249499.	−1.27025	−0.635127	−	0.772407i	$-0.719052\pi$
$131$	−249499.	−1.27025	−0.635127	+	0.772407i	$0.719052\pi$
$132$	8901.44	0.0444657
$133$	368685.	1.80729
$134$	366153.	1.76157
$135$	−29271.7	−0.138234
$136$	−62556.3	−0.290017
$137$	−25829.5	−0.117575	−0.0587875	−	0.998271i	$-0.518723\pi$
$137$	−25829.5	−0.117575	−0.0587875	+	0.998271i	$0.518723\pi$
$138$	−36019.2	−0.161004
$139$	79301.9	0.348134	0.174067	−	0.984734i	$-0.444309\pi$
$139$	79301.9	0.348134	0.174067	+	0.984734i	$0.444309\pi$
$140$	197518.	0.851699
$141$	−2005.51	−0.00849527
$142$	−456089.	−1.89814
$143$	−27630.8	−0.112994
$144$	−96769.9	−0.388897
$145$	−344490.	−1.36068
$146$	231792.	0.899947
$147$	−190686.	−0.727823
$148$	−205537.	−0.771321
$149$	−338110.	−1.24765	−0.623825	−	0.781564i	$-0.714422\pi$
$149$	−338110.	−1.24765	−0.623825	+	0.781564i	$0.714422\pi$
$150$	103000.	0.373773
$151$	494788.	1.76594	0.882971	−	0.469427i	$-0.155539\pi$
$151$	494788.	1.76594	0.882971	+	0.469427i	$0.155539\pi$
$152$	−96786.6	−0.339787
$153$	99023.3	0.341986
$154$	−57794.6	−0.196375
$155$	−384777.	−1.28641
$156$	−160129.	−0.526816
$157$	195963.	0.634490	0.317245	−	0.948344i	$-0.397242\pi$
$157$	195963.	0.634490	0.317245	+	0.948344i	$0.397242\pi$
$158$	380432.	1.21237
$159$	−59197.0	−0.185698
$160$	−297172.	−0.917714
$161$	103113.	0.313509
$162$	49637.0	0.148600
$163$	223223.	0.658067	0.329034	−	0.944318i	$-0.393277\pi$
$163$	223223.	0.658067	0.329034	+	0.944318i	$0.393277\pi$
$164$	121999.	0.354197
$165$	14163.0	0.0404991
$166$	−269873.	−0.760132
$167$	−232525.	−0.645176	−0.322588	−	0.946539i	$-0.604553\pi$
$167$	−232525.	−0.645176	−0.322588	+	0.946539i	$0.604553\pi$
$168$	89767.9	0.245385
$169$	125762.	0.338713
$170$	371372.	0.985567
$171$	153208.	0.400674
$172$	268299.	0.691509
$173$	129688.	0.329445	0.164723	−	0.986340i	$-0.447327\pi$
$173$	129688.	0.329445	0.164723	+	0.986340i	$0.447327\pi$
$174$	584164.	1.46272
$175$	−294861.	−0.727817
$176$	46821.7	0.113937
$177$	277028.	0.664645
$178$	−587225.	−1.38917
$179$	722275.	1.68489	0.842443	−	0.538786i	$-0.181117\pi$
$179$	722275.	1.68489	0.842443	+	0.538786i	$0.181117\pi$
$180$	82078.9	0.188821
$181$	−126793.	−0.287672	−0.143836	−	0.989602i	$-0.545944\pi$
$181$	−126793.	−0.287672	−0.143836	+	0.989602i	$0.545944\pi$
$182$	1.03967e6	2.32659
$183$	78256.8	0.172741
$184$	−27069.1	−0.0589427
$185$	−327027.	−0.702514
$186$	652480.	1.38288
$187$	−47912.0	−0.100194
$188$	5623.53	0.0116042
$189$	−142098.	−0.289356
$190$	574583.	1.15470
$191$	−582441.	−1.15523	−0.577615	−	0.816310i	$-0.696016\pi$
$191$	−582441.	−1.15523	−0.577615	+	0.816310i	$0.696016\pi$
$192$	159853.	0.312946
$193$	−441777.	−0.853709	−0.426855	−	0.904320i	$-0.640378\pi$
$193$	−441777.	−0.853709	−0.426855	+	0.904320i	$0.640378\pi$
$194$	−628459.	−1.19887
$195$	−254780.	−0.479820
$196$	534691.	0.994175
$197$	−149838.	−0.275078	−0.137539	−	0.990496i	$-0.543919\pi$
$197$	−149838.	−0.275078	−0.137539	+	0.990496i	$0.543919\pi$
$198$	−24016.7	−0.0435361
$199$	−48644.4	−0.0870764	−0.0435382	−	0.999052i	$-0.513863\pi$
$199$	−48644.4	−0.0870764	−0.0435382	+	0.999052i	$0.513863\pi$
$200$	77406.4	0.136836
$201$	−435582.	−0.760465
$202$	233280.	0.402254
$203$	−1.67231e6	−2.84824
$204$	−277665.	−0.467138
$205$	194111.	0.322600
$206$	1.07193e6	1.75994
$207$	42849.0	0.0695048
$208$	−842282.	−1.34989
$209$	−74129.0	−0.117388
$210$	−532916.	−0.833893
$211$	921315.	1.42463	0.712315	−	0.701860i	$-0.247647\pi$
$211$	921315.	1.42463	0.712315	+	0.701860i	$0.247647\pi$
$212$	165990.	0.253655
$213$	542571.	0.819422
$214$	1.00437e6	1.49919
$215$	426888.	0.629822
$216$	37303.2	0.0544016
$217$	−1.86788e6	−2.69277
$218$	−855128.	−1.21868
$219$	−275744.	−0.388504
$220$	−39713.5	−0.0553199
$221$	861895.	1.18706
$222$	554552.	0.755196
$223$	998446.	1.34450	0.672252	−	0.740322i	$-0.265327\pi$
$223$	998446.	1.34450	0.672252	+	0.740322i	$0.265327\pi$
$224$	−1.44260e6	−1.92100
$225$	−122530.	−0.161357
$226$	349188.	0.454766
$227$	−376703.	−0.485215	−0.242608	−	0.970125i	$-0.578003\pi$
$227$	−376703.	−0.485215	−0.242608	+	0.970125i	$0.578003\pi$
$228$	−429601.	−0.547303
$229$	814218.	1.02601	0.513005	−	0.858385i	$-0.328532\pi$
$229$	814218.	1.02601	0.513005	+	0.858385i	$0.328532\pi$
$230$	160699.	0.200305
$231$	68753.3	0.0847742
$232$	439011.	0.535495
$233$	255689.	0.308547	0.154274	−	0.988028i	$-0.450696\pi$
$233$	255689.	0.308547	0.154274	+	0.988028i	$0.450696\pi$
$234$	432039.	0.515802
$235$	8947.53	0.0105690
$236$	−776795.	−0.907876
$237$	−452568.	−0.523375
$238$	1.80280e6	2.06303
$239$	−1.27483e6	−1.44364	−0.721818	−	0.692083i	$-0.756693\pi$
$239$	−1.27483e6	−1.44364	−0.721818	+	0.692083i	$0.756693\pi$
$240$	431736.	0.483827
$241$	−1.47036e6	−1.63073	−0.815365	−	0.578948i	$-0.803464\pi$
$241$	−1.47036e6	−1.63073	−0.815365	+	0.578948i	$0.803464\pi$
$242$	−1.20681e6	−1.32464
$243$	−59049.0	−0.0641500
$244$	−219435.	−0.235956
$245$	850741.	0.905487
$246$	−329160.	−0.346792
$247$	1.33352e6	1.39077
$248$	490352.	0.506266
$249$	321045.	0.328146
$250$	−1.40884e6	−1.42564
$251$	1.25961e6	1.26197	0.630987	−	0.775793i	$-0.282650\pi$
$251$	1.25961e6	1.26197	0.630987	+	0.775793i	$0.282650\pi$
$252$	398447.	0.395248
$253$	−20732.3	−0.0203632
$254$	−1.69309e6	−1.64663
$255$	−441789.	−0.425466
$256$	1.34350e6	1.28126
$257$	400710.	0.378440	0.189220	−	0.981935i	$-0.439404\pi$
$257$	400710.	0.378440	0.189220	+	0.981935i	$0.439404\pi$
$258$	−723888.	−0.677052
$259$	−1.58753e6	−1.47053
$260$	714412.	0.655413
$261$	−694931.	−0.631452
$262$	−1.88758e6	−1.69884
$263$	1.10828e6	0.988009	0.494004	−	0.869459i	$-0.335533\pi$
$263$	1.10828e6	0.988009	0.494004	+	0.869459i	$0.335533\pi$
$264$	−18049.0	−0.0159383
$265$	264105.	0.231027
$266$	2.78928e6	2.41706
$267$	698573.	0.599699
$268$	1.22139e6	1.03876
$269$	−1.22073e6	−1.02858	−0.514289	−	0.857617i	$-0.671944\pi$
$269$	−1.22073e6	−1.02858	−0.514289	+	0.857617i	$0.671944\pi$
$270$	−221454.	−0.184873
$271$	−606547.	−0.501697	−0.250848	−	0.968026i	$-0.580710\pi$
$271$	−606547.	−0.501697	−0.250848	+	0.968026i	$0.580710\pi$
$272$	−1.46052e6	−1.19698
$273$	−1.23681e6	−1.00438
$274$	−195413.	−0.157245
$275$	59285.7	0.0472735
$276$	−120150.	−0.0949405
$277$	1.73194e6	1.35623	0.678116	−	0.734955i	$-0.262797\pi$
$277$	1.73194e6	1.35623	0.678116	+	0.734955i	$0.262797\pi$
$278$	599956.	0.465594
$279$	−776200.	−0.596985
$280$	−400497.	−0.305284
$281$	−1.03139e6	−0.779212	−0.389606	−	0.920982i	$-0.627389\pi$
$281$	−1.03139e6	−0.779212	−0.389606	+	0.920982i	$0.627389\pi$
$282$	−15172.6	−0.0113616
$283$	1.83231e6	1.35998	0.679990	−	0.733221i	$-0.261984\pi$
$283$	1.83231e6	1.35998	0.679990	+	0.733221i	$0.261984\pi$
$284$	−1.52139e6	−1.11929
$285$	−683533.	−0.498480
$286$	−209040.	−0.151117
$287$	942298.	0.675279
$288$	−599476.	−0.425883
$289$	74672.5	0.0525915
$290$	−2.60623e6	−1.81978
$291$	747625.	0.517549
$292$	773195.	0.530679
$293$	−277123.	−0.188584	−0.0942919	−	0.995545i	$-0.530059\pi$
$293$	−277123.	−0.188584	−0.0942919	+	0.995545i	$0.530059\pi$
$294$	−1.44263e6	−0.973390
$295$	−1.23595e6	−0.826887
$296$	416757.	0.276473
$297$	28570.6	0.0187944
$298$	−2.55796e6	−1.66860
$299$	372956.	0.241257
$300$	343579.	0.220406
$301$	2.07230e6	1.31837
$302$	3.74330e6	2.36177
$303$	−277514.	−0.173651
$304$	−2.25971e6	−1.40239
$305$	−349141.	−0.214907
$306$	749157.	0.457372
$307$	1.98065e6	1.19939	0.599696	−	0.800228i	$-0.295288\pi$
$307$	1.98065e6	1.19939	0.599696	+	0.800228i	$0.295288\pi$
$308$	−192787.	−0.115798
$309$	−1.27518e6	−0.759758
$310$	−2.91102e6	−1.72045
$311$	2.56169e6	1.50185	0.750925	−	0.660388i	$-0.229608\pi$
$311$	2.56169e6	1.50185	0.750925	+	0.660388i	$0.229608\pi$
$312$	324686.	0.188833
$313$	2.46321e6	1.42115	0.710576	−	0.703621i	$-0.248435\pi$
$313$	2.46321e6	1.42115	0.710576	+	0.703621i	$0.248435\pi$
$314$	1.48255e6	0.848566
$315$	633965.	0.359989
$316$	1.26902e6	0.714907
$317$	−3.04669e6	−1.70287	−0.851433	−	0.524464i	$-0.824266\pi$
$317$	−3.04669e6	−1.70287	−0.851433	+	0.524464i	$0.824266\pi$
$318$	−447853.	−0.248352
$319$	336239.	0.185000
$320$	−713181.	−0.389336
$321$	−1.19481e6	−0.647196
$322$	780101.	0.419287
$323$	2.31232e6	1.23323
$324$	165575.	0.0876261
$325$	−1.06650e6	−0.560082
$326$	1.68879e6	0.880098
$327$	1.01727e6	0.526100
$328$	−247370.	−0.126959
$329$	43435.2	0.0221234
$330$	107150.	0.0541634
$331$	−2.59437e6	−1.30155	−0.650776	−	0.759270i	$-0.725556\pi$
$331$	−2.59437e6	−1.30155	−0.650776	+	0.759270i	$0.725556\pi$
$332$	−900222.	−0.448234
$333$	−659703.	−0.326015
$334$	−1.75916e6	−0.862857
$335$	1.94333e6	0.946097
$336$	2.09584e6	1.01277
$337$	2.23509e6	1.07206	0.536032	−	0.844198i	$-0.319923\pi$
$337$	2.23509e6	1.07206	0.536032	+	0.844198i	$0.319923\pi$
$338$	951445.	0.452994
$339$	−415400.	−0.196321
$340$	1.23879e6	0.581168
$341$	375561.	0.174902
$342$	1.15909e6	0.535861
$343$	853826.	0.391863
$344$	−544017.	−0.247865
$345$	−191170.	−0.0864711
$346$	981148.	0.440600
$347$	1.61546e6	0.720230	0.360115	−	0.932908i	$-0.382737\pi$
$347$	1.61546e6	0.720230	0.360115	+	0.932908i	$0.382737\pi$
$348$	1.94861e6	0.862535
$349$	−2.56381e6	−1.12674	−0.563369	−	0.826205i	$-0.690495\pi$
$349$	−2.56381e6	−1.12674	−0.563369	+	0.826205i	$0.690495\pi$
$350$	−2.23076e6	−0.973382
$351$	−513960.	−0.222670
$352$	290054.	0.124773
$353$	−1.64946e6	−0.704541	−0.352271	−	0.935898i	$-0.614590\pi$
$353$	−1.64946e6	−0.704541	−0.352271	+	0.935898i	$0.614590\pi$
$354$	2.09584e6	0.888896
$355$	−2.42066e6	−1.01944
$356$	−1.95882e6	−0.819162
$357$	−2.14464e6	−0.890602
$358$	5.46435e6	2.25336
$359$	1.30522e6	0.534500	0.267250	−	0.963627i	$-0.413885\pi$
$359$	1.30522e6	0.534500	0.267250	+	0.963627i	$0.413885\pi$
$360$	−166427.	−0.0676812
$361$	1.10151e6	0.444857
$362$	−959247.	−0.384732
$363$	1.43564e6	0.571844
$364$	3.46807e6	1.37194
$365$	1.23022e6	0.483338
$366$	592050.	0.231023
$367$	5.04859e6	1.95661	0.978306	−	0.207164i	$-0.0664233\pi$
$367$	5.04859e6	1.95661	0.978306	+	0.207164i	$0.0664233\pi$
$368$	−631991.	−0.243272
$369$	391574.	0.149709
$370$	−2.47412e6	−0.939541
$371$	1.28208e6	0.483595
$372$	2.17649e6	0.815455
$373$	512860.	0.190865	0.0954326	−	0.995436i	$-0.469577\pi$
$373$	512860.	0.190865	0.0954326	+	0.995436i	$0.469577\pi$
$374$	−362476.	−0.133999
$375$	1.67597e6	0.615445
$376$	−11402.5	−0.00415941
$377$	−6.04865e6	−2.19182
$378$	−1.07504e6	−0.386984
$379$	1.34067e6	0.479429	0.239714	−	0.970843i	$-0.422946\pi$
$379$	1.34067e6	0.479429	0.239714	+	0.970843i	$0.422946\pi$
$380$	1.91665e6	0.680901
$381$	2.01413e6	0.710844
$382$	−4.40644e6	−1.54500
$383$	−5.32498e6	−1.85490	−0.927451	−	0.373944i	$-0.878005\pi$
$383$	−5.32498e6	−1.85490	−0.927451	+	0.373944i	$0.878005\pi$
$384$	−922105.	−0.319119
$385$	−306741.	−0.105468
$386$	−3.34225e6	−1.14175
$387$	861149.	0.292281
$388$	−2.09637e6	−0.706949
$389$	4.22472e6	1.41554	0.707772	−	0.706441i	$-0.249700\pi$
$389$	4.22472e6	1.41554	0.707772	+	0.706441i	$0.249700\pi$
$390$	−1.92753e6	−0.641711
$391$	646708.	0.213927
$392$	−1.08417e6	−0.356353
$393$	2.24549e6	0.733382
$394$	−1.13359e6	−0.367889
$395$	2.01912e6	0.651132
$396$	−80113.0	−0.0256723
$397$	−451747.	−0.143853	−0.0719266	−	0.997410i	$-0.522915\pi$
$397$	−451747.	−0.143853	−0.0719266	+	0.997410i	$0.522915\pi$
$398$	−368018.	−0.116456
$399$	−3.31817e6	−1.04344
$400$	1.80723e6	0.564759
$401$	1.72721e6	0.536395	0.268197	−	0.963364i	$-0.413572\pi$
$401$	1.72721e6	0.536395	0.268197	+	0.963364i	$0.413572\pi$
$402$	−3.29538e6	−1.01705
$403$	−6.75602e6	−2.07218
$404$	778159.	0.237200
$405$	263445.	0.0798092
$406$	−1.26518e7	−3.80923
$407$	319194.	0.0955145
$408$	563007.	0.167442
$409$	−1.57103e6	−0.464384	−0.232192	−	0.972670i	$-0.574590\pi$
$409$	−1.57103e6	−0.464384	−0.232192	+	0.972670i	$0.574590\pi$
$410$	1.46854e6	0.431445
$411$	232466.	0.0678820
$412$	3.57565e6	1.03780
$413$	−5.99985e6	−1.73087
$414$	324173.	0.0929556
$415$	−1.43233e6	−0.408248
$416$	−5.21782e6	−1.47828
$417$	−713717.	−0.200995
$418$	−560821.	−0.156994
$419$	−1.93316e6	−0.537938	−0.268969	−	0.963149i	$-0.586683\pi$
$419$	−1.93316e6	−0.537938	−0.268969	+	0.963149i	$0.586683\pi$
$420$	−1.77766e6	−0.491729
$421$	−188965.	−0.0519610	−0.0259805	−	0.999662i	$-0.508271\pi$
$421$	−188965.	−0.0519610	−0.0259805	+	0.999662i	$0.508271\pi$
$422$	6.97018e6	1.90530
$423$	18049.6	0.00490475
$424$	−336570.	−0.0909204
$425$	−1.84931e6	−0.496636
$426$	4.10480e6	1.09589
$427$	−1.69488e6	−0.449852
$428$	3.35029e6	0.884041
$429$	248677.	0.0652369
$430$	3.22961e6	0.842323
$431$	−3.11533e6	−0.807812	−0.403906	−	0.914800i	$-0.632348\pi$
$431$	−3.11533e6	−0.807812	−0.403906	+	0.914800i	$0.632348\pi$
$432$	870929.	0.224530
$433$	−2.18147e6	−0.559151	−0.279576	−	0.960124i	$-0.590194\pi$
$433$	−2.18147e6	−0.559151	−0.279576	+	0.960124i	$0.590194\pi$
$434$	−1.41314e7	−3.60130
$435$	3.10041e6	0.785591
$436$	−2.85247e6	−0.718629
$437$	1.00058e6	0.250639
$438$	−2.08613e6	−0.519584
$439$	−3.03543e6	−0.751725	−0.375863	−	0.926675i	$-0.622654\pi$
$439$	−3.03543e6	−0.751725	−0.375863	+	0.926675i	$0.622654\pi$
$440$	80525.1	0.0198289
$441$	1.71618e6	0.420209
$442$	6.52064e6	1.58758
$443$	−3.62547e6	−0.877719	−0.438859	−	0.898556i	$-0.644617\pi$
$443$	−3.62547e6	−0.877719	−0.438859	+	0.898556i	$0.644617\pi$
$444$	1.84983e6	0.445323
$445$	−3.11666e6	−0.746087
$446$	7.55371e6	1.79814
$447$	3.04299e6	0.720331
$448$	−3.46209e6	−0.814974
$449$	6.84914e6	1.60332	0.801660	−	0.597781i	$-0.203951\pi$
$449$	6.84914e6	1.60332	0.801660	+	0.597781i	$0.203951\pi$
$450$	−926998.	−0.215798
$451$	−189461.	−0.0438610
$452$	1.16480e6	0.268166
$453$	−4.45309e6	−1.01957
$454$	−2.84993e6	−0.648926
$455$	5.51801e6	1.24955
$456$	871080.	0.196176
$457$	4.07502e6	0.912725	0.456362	−	0.889794i	$-0.349152\pi$
$457$	4.07502e6	0.912725	0.456362	+	0.889794i	$0.349152\pi$
$458$	6.15994e6	1.37219
$459$	−891209.	−0.197446
$460$	536046.	0.118116
$461$	1.10815e6	0.242854	0.121427	−	0.992600i	$-0.461253\pi$
$461$	1.10815e6	0.242854	0.121427	+	0.992600i	$0.461253\pi$
$462$	520151.	0.113377
$463$	3.77893e6	0.819251	0.409625	−	0.912254i	$-0.365659\pi$
$463$	3.77893e6	0.819251	0.409625	+	0.912254i	$0.365659\pi$
$464$	1.02497e7	2.21013
$465$	3.46299e6	0.742710
$466$	1.93440e6	0.412650
$467$	8.40396e6	1.78317	0.891583	−	0.452857i	$-0.149595\pi$
$467$	8.40396e6	1.78317	0.891583	+	0.452857i	$0.149595\pi$
$468$	1.44116e6	0.304157
$469$	9.43380e6	1.98041
$470$	67692.3	0.0141350
$471$	−1.76367e6	−0.366323
$472$	1.57507e6	0.325420
$473$	−416663.	−0.0856312
$474$	−3.42389e6	−0.699961
$475$	−2.86124e6	−0.581863
$476$	6.01365e6	1.21652
$477$	532773.	0.107213
$478$	−9.64470e6	−1.93072
$479$	6.44233e6	1.28293	0.641467	−	0.767151i	$-0.278326\pi$
$479$	6.44233e6	1.28293	0.641467	+	0.767151i	$0.278326\pi$
$480$	2.67454e6	0.529843
$481$	−5.74203e6	−1.13163
$482$	−1.11240e7	−2.18094
$483$	−928021.	−0.181005
$484$	−4.02557e6	−0.781114
$485$	−3.33551e6	−0.643884
$486$	−446733.	−0.0857942
$487$	−2.57622e6	−0.492221	−0.246111	−	0.969242i	$-0.579153\pi$
$487$	−2.57622e6	−0.492221	−0.246111	+	0.969242i	$0.579153\pi$
$488$	444937.	0.0845764
$489$	−2.00901e6	−0.379935
$490$	6.43625e6	1.21100
$491$	−443058.	−0.0829386	−0.0414693	−	0.999140i	$-0.513204\pi$
$491$	−443058.	−0.0829386	−0.0414693	+	0.999140i	$0.513204\pi$
$492$	−1.09799e6	−0.204496
$493$	−1.04884e7	−1.94353
$494$	1.00887e7	1.86002
$495$	−127467.	−0.0233821
$496$	1.14484e7	2.08949
$497$	−1.17510e7	−2.13394
$498$	2.42886e6	0.438863
$499$	6.36169e6	1.14372	0.571862	−	0.820350i	$-0.306221\pi$
$499$	6.36169e6	1.14372	0.571862	+	0.820350i	$0.306221\pi$
$500$	−4.69949e6	−0.840671
$501$	2.09272e6	0.372492
$502$	9.52952e6	1.68776
$503$	4.69139e6	0.826763	0.413382	−	0.910558i	$-0.364348\pi$
$503$	4.69139e6	0.826763	0.413382	+	0.910558i	$0.364348\pi$
$504$	−807911.	−0.141673
$505$	1.23812e6	0.216040
$506$	−156850.	−0.0272337
$507$	−1.13185e6	−0.195556
$508$	−5.64769e6	−0.970982
$509$	3.35385e6	0.573785	0.286893	−	0.957963i	$-0.407378\pi$
$509$	3.35385e6	0.573785	0.286893	+	0.957963i	$0.407378\pi$
$510$	−3.34234e6	−0.569018
$511$	5.97204e6	1.01174
$512$	6.88558e6	1.16082
$513$	−1.37887e6	−0.231329
$514$	3.03156e6	0.506125
$515$	5.68918e6	0.945217
$516$	−2.41469e6	−0.399243
$517$	−8733.22	−0.00143697
$518$	−1.20104e7	−1.96668
$519$	−1.16719e6	−0.190205
$520$	−1.44858e6	−0.234927
$521$	3.64554e6	0.588393	0.294196	−	0.955745i	$-0.404948\pi$
$521$	3.64554e6	0.588393	0.294196	+	0.955745i	$0.404948\pi$
$522$	−5.25748e6	−0.844503
$523$	−9.63934e6	−1.54097	−0.770483	−	0.637461i	$-0.779985\pi$
$523$	−9.63934e6	−1.54097	−0.770483	+	0.637461i	$0.779985\pi$
$524$	−6.29644e6	−1.00177
$525$	2.65375e6	0.420206
$526$	8.38467e6	1.32136
$527$	−1.17150e7	−1.83745
$528$	−421395.	−0.0657816
$529$	279841.	0.0434783
$530$	1.99808e6	0.308975
$531$	−2.49325e6	−0.383733
$532$	9.30426e6	1.42529
$533$	3.40824e6	0.519652
$534$	5.28503e6	0.802037
$535$	5.33060e6	0.805178
$536$	−2.47654e6	−0.372335
$537$	−6.50048e6	−0.972769
$538$	−9.23537e6	−1.37562
$539$	−830364.	−0.123111
$540$	−738710.	−0.109016
$541$	1.04511e7	1.53521	0.767604	−	0.640925i	$-0.221449\pi$
$541$	1.04511e7	1.53521	0.767604	+	0.640925i	$0.221449\pi$
$542$	−4.58881e6	−0.670969
$543$	1.14113e6	0.166088
$544$	−9.04772e6	−1.31082
$545$	−4.53854e6	−0.654522
$546$	−9.35707e6	−1.34325
$547$	−6.35425e6	−0.908021	−0.454011	−	0.890996i	$-0.650007\pi$
$547$	−6.35425e6	−0.908021	−0.454011	+	0.890996i	$0.650007\pi$
$548$	−651843.	−0.0927238
$549$	−704312.	−0.0997319
$550$	448524.	0.0632235
$551$	−1.62276e7	−2.27706
$552$	243622.	0.0340306
$553$	9.80169e6	1.36298
$554$	1.31030e7	1.81382
$555$	2.94325e6	0.405596
$556$	2.00129e6	0.274551
$557$	−283483.	−0.0387159	−0.0193580	−	0.999813i	$-0.506162\pi$
$557$	−283483.	−0.0387159	−0.0193580	+	0.999813i	$0.506162\pi$
$558$	−5.87232e6	−0.798407
$559$	7.49541e6	1.01453
$560$	−9.35051e6	−1.25999
$561$	431208.	0.0578468
$562$	−7.80292e6	−1.04212
$563$	1.05154e7	1.39815	0.699076	−	0.715047i	$-0.253595\pi$
$563$	1.05154e7	1.39815	0.699076	+	0.715047i	$0.253595\pi$
$564$	−50611.7	−0.00669967
$565$	1.85329e6	0.244243
$566$	1.38623e7	1.81884
$567$	1.27888e6	0.167060
$568$	3.08484e6	0.401201
$569$	6.83667e6	0.885246	0.442623	−	0.896708i	$-0.354048\pi$
$569$	6.83667e6	0.885246	0.442623	+	0.896708i	$0.354048\pi$
$570$	−5.17125e6	−0.666666
$571$	3.11227e6	0.399473	0.199736	−	0.979850i	$-0.435991\pi$
$571$	3.11227e6	0.399473	0.199736	+	0.979850i	$0.435991\pi$
$572$	−697300.	−0.0891107
$573$	5.24197e6	0.666972
$574$	7.12892e6	0.903117
$575$	−800228.	−0.100936
$576$	−1.43868e6	−0.180679
$577$	−1.10234e7	−1.37841	−0.689203	−	0.724569i	$-0.742039\pi$
$577$	−1.10234e7	−1.37841	−0.689203	+	0.724569i	$0.742039\pi$
$578$	564932.	0.0703358
$579$	3.97599e6	0.492889
$580$	−8.69367e6	−1.07308
$581$	−6.95317e6	−0.854560
$582$	5.65613e6	0.692169
$583$	−257780.	−0.0314107
$584$	−1.56777e6	−0.190217
$585$	2.29302e6	0.277024
$586$	−2.09657e6	−0.252212
$587$	−2.79249e6	−0.334500	−0.167250	−	0.985915i	$-0.553489\pi$
$587$	−2.79249e6	−0.334500	−0.167250	+	0.985915i	$0.553489\pi$
$588$	−4.81222e6	−0.573987
$589$	−1.81253e7	−2.15277
$590$	−9.35054e6	−1.10588
$591$	1.34854e6	0.158816
$592$	9.73014e6	1.14108
$593$	−1.65811e7	−1.93632	−0.968161	−	0.250328i	$-0.919462\pi$
$593$	−1.65811e7	−1.93632	−0.968161	+	0.250328i	$0.919462\pi$
$594$	216150.	0.0251356
$595$	9.56825e6	1.10800
$596$	−8.53266e6	−0.983940
$597$	437800.	0.0502736
$598$	2.82159e6	0.322657
$599$	6.89907e6	0.785640	0.392820	−	0.919615i	$-0.371500\pi$
$599$	6.89907e6	0.785640	0.392820	+	0.919615i	$0.371500\pi$
$600$	−696658.	−0.0790026
$601$	−807441.	−0.0911853	−0.0455927	−	0.998960i	$-0.514518\pi$
$601$	−807441.	−0.0911853	−0.0455927	+	0.998960i	$0.514518\pi$
$602$	1.56779e7	1.76318
$603$	3.92023e6	0.439055
$604$	1.24866e7	1.39268
$605$	−6.40504e6	−0.711432
$606$	−2.09952e6	−0.232241
$607$	−1.46288e6	−0.161153	−0.0805765	−	0.996748i	$-0.525676\pi$
$607$	−1.46288e6	−0.161153	−0.0805765	+	0.996748i	$0.525676\pi$
$608$	−1.39986e7	−1.53576
$609$	1.50508e7	1.64443
$610$	−2.64141e6	−0.287416
$611$	157103.	0.0170248
$612$	2.49898e6	0.269702
$613$	−1.41124e7	−1.51687	−0.758435	−	0.651749i	$-0.774036\pi$
$613$	−1.41124e7	−1.51687	−0.758435	+	0.651749i	$0.774036\pi$
$614$	1.49845e7	1.60406
$615$	−1.74699e6	−0.186253
$616$	390904.	0.0415067
$617$	−1.32361e7	−1.39974	−0.699871	−	0.714269i	$-0.746759\pi$
$617$	−1.32361e7	−1.39974	−0.699871	+	0.714269i	$0.746759\pi$
$618$	−9.64734e6	−1.01610
$619$	3.89949e6	0.409055	0.204527	−	0.978861i	$-0.434434\pi$
$619$	3.89949e6	0.409055	0.204527	+	0.978861i	$0.434434\pi$
$620$	−9.71036e6	−1.01451
$621$	−385641.	−0.0401286
$622$	1.93804e7	2.00857
$623$	−1.51296e7	−1.56174
$624$	7.58054e6	0.779361
$625$	−2.75007e6	−0.281607
$626$	1.86353e7	1.90065
$627$	667161.	0.0677738
$628$	4.94538e6	0.500381
$629$	−9.95671e6	−1.00344
$630$	4.79624e6	0.481448
$631$	−7.32659e6	−0.732535	−0.366268	−	0.930510i	$-0.619365\pi$
$631$	−7.32659e6	−0.732535	−0.366268	+	0.930510i	$0.619365\pi$
$632$	−2.57312e6	−0.256252
$633$	−8.29184e6	−0.822511
$634$	−2.30497e7	−2.27741
$635$	−8.98597e6	−0.884363
$636$	−1.49391e6	−0.146448
$637$	1.49375e7	1.45858
$638$	2.54381e6	0.247419
$639$	−4.88314e6	−0.473093
$640$	4.11394e6	0.397016
$641$	4.68243e6	0.450118	0.225059	−	0.974345i	$-0.427742\pi$
$641$	4.68243e6	0.450118	0.225059	+	0.974345i	$0.427742\pi$
$642$	−9.03929e6	−0.865559
$643$	5.15232e6	0.491445	0.245723	−	0.969340i	$-0.420975\pi$
$643$	5.15232e6	0.491445	0.245723	+	0.969340i	$0.420975\pi$
$644$	2.60220e6	0.247244
$645$	−3.84199e6	−0.363628
$646$	1.74938e7	1.64931
$647$	−5.60768e6	−0.526650	−0.263325	−	0.964707i	$-0.584819\pi$
$647$	−5.60768e6	−0.526650	−0.263325	+	0.964707i	$0.584819\pi$
$648$	−335729.	−0.0314088
$649$	1.20635e6	0.112424
$650$	−8.06856e6	−0.749053
$651$	1.68109e7	1.55467
$652$	5.63333e6	0.518975
$653$	7.57877e6	0.695530	0.347765	−	0.937582i	$-0.386941\pi$
$653$	7.57877e6	0.695530	0.347765	+	0.937582i	$0.386941\pi$
$654$	7.69615e6	0.703605
$655$	−1.00182e7	−0.912402
$656$	−5.77542e6	−0.523992
$657$	2.48169e6	0.224303
$658$	328608.	0.0295878
$659$	1.64530e7	1.47581	0.737905	−	0.674904i	$-0.235815\pi$
$659$	1.64530e7	1.47581	0.737905	+	0.674904i	$0.235815\pi$
$660$	357422.	0.0319390
$661$	1.72074e6	0.153184	0.0765918	−	0.997063i	$-0.475596\pi$
$661$	1.72074e6	0.153184	0.0765918	+	0.997063i	$0.475596\pi$
$662$	−1.96276e7	−1.74069
$663$	−7.75705e6	−0.685351
$664$	1.82533e6	0.160665
$665$	1.48039e7	1.29814
$666$	−4.99096e6	−0.436012
$667$	−4.53850e6	−0.395000
$668$	−5.86807e6	−0.508808
$669$	−8.98601e6	−0.776250
$670$	1.47022e7	1.26531
$671$	340778.	0.0292190
$672$	1.29834e7	1.10909
$673$	8.54705e6	0.727408	0.363704	−	0.931514i	$-0.381512\pi$
$673$	8.54705e6	0.727408	0.363704	+	0.931514i	$0.381512\pi$
$674$	1.69095e7	1.43378
$675$	1.10277e6	0.0931593
$676$	3.17376e6	0.267121
$677$	−8.31269e6	−0.697060	−0.348530	−	0.937298i	$-0.613319\pi$
$677$	−8.31269e6	−0.697060	−0.348530	+	0.937298i	$0.613319\pi$
$678$	−3.14269e6	−0.262559
$679$	−1.61920e7	−1.34780
$680$	−2.51184e6	−0.208314
$681$	3.39033e6	0.280139
$682$	2.84129e6	0.233914
$683$	−1.80054e6	−0.147690	−0.0738451	−	0.997270i	$-0.523527\pi$
$683$	−1.80054e6	−0.147690	−0.0738451	+	0.997270i	$0.523527\pi$
$684$	3.86641e6	0.315986
$685$	−1.03714e6	−0.0844522
$686$	6.45960e6	0.524077
$687$	−7.32796e6	−0.592368
$688$	−1.27013e7	−1.02300
$689$	4.63723e6	0.372144
$690$	−1.44629e6	−0.115646
$691$	−1.38259e7	−1.10153	−0.550766	−	0.834660i	$-0.685664\pi$
$691$	−1.38259e7	−1.10153	−0.550766	+	0.834660i	$0.685664\pi$
$692$	3.27284e6	0.259812
$693$	−618780.	−0.0489444
$694$	1.22217e7	0.963235
$695$	3.18423e6	0.250059
$696$	−3.95110e6	−0.309168
$697$	5.90991e6	0.460786
$698$	−1.93965e7	−1.50690
$699$	−2.30120e6	−0.178140
$700$	−7.44121e6	−0.573982
$701$	2.42009e7	1.86010	0.930051	−	0.367430i	$-0.119762\pi$
$701$	2.42009e7	1.86010	0.930051	+	0.367430i	$0.119762\pi$
$702$	−3.88835e6	−0.297799
$703$	−1.54049e7	−1.17563
$704$	696099.	0.0529346
$705$	−80527.8	−0.00610201
$706$	−1.24790e7	−0.942252
$707$	6.01038e6	0.452224
$708$	6.99116e6	0.524162
$709$	−1.13875e7	−0.850768	−0.425384	−	0.905013i	$-0.639861\pi$
$709$	−1.13875e7	−0.850768	−0.425384	+	0.905013i	$0.639861\pi$
$710$	−1.83135e7	−1.36340
$711$	4.07311e6	0.302171
$712$	3.97181e6	0.293622
$713$	−5.06926e6	−0.373440
$714$	−1.62252e7	−1.19109
$715$	−1.10947e6	−0.0811613
$716$	1.82276e7	1.32876
$717$	1.14735e7	0.833484
$718$	9.87459e6	0.714839
$719$	−1.24869e7	−0.900807	−0.450403	−	0.892825i	$-0.648720\pi$
$719$	−1.24869e7	−0.900807	−0.450403	+	0.892825i	$0.648720\pi$
$720$	−3.88563e6	−0.279338
$721$	2.76178e7	1.97857
$722$	8.33344e6	0.594951
$723$	1.32333e7	0.941502
$724$	−3.19978e6	−0.226868
$725$	1.29782e7	0.917000
$726$	1.08613e7	0.764783
$727$	−1.69620e7	−1.19026	−0.595128	−	0.803631i	$-0.702899\pi$
$727$	−1.69620e7	−1.19026	−0.595128	+	0.803631i	$0.702899\pi$
$728$	−7.03203e6	−0.491759
$729$	531441.	0.0370370
$730$	9.30721e6	0.646416
$731$	1.29971e7	0.899605
$732$	1.97491e6	0.136229
$733$	1.73348e7	1.19168	0.595839	−	0.803104i	$-0.296820\pi$
$733$	1.73348e7	1.19168	0.595839	+	0.803104i	$0.296820\pi$
$734$	3.81949e7	2.61677
$735$	−7.65667e6	−0.522783
$736$	−3.91510e6	−0.266408
$737$	−1.89679e6	−0.128632
$738$	2.96244e6	0.200220
$739$	2.67348e6	0.180080	0.0900400	−	0.995938i	$-0.471301\pi$
$739$	2.67348e6	0.180080	0.0900400	+	0.995938i	$0.471301\pi$
$740$	−8.25297e6	−0.554027
$741$	−1.20017e7	−0.802963
$742$	9.69957e6	0.646759
$743$	1.67454e7	1.11281	0.556407	−	0.830910i	$-0.312180\pi$
$743$	1.67454e7	1.11281	0.556407	+	0.830910i	$0.312180\pi$
$744$	−4.41316e6	−0.292293
$745$	−1.35762e7	−0.896165
$746$	3.88002e6	0.255263
$747$	−2.88941e6	−0.189455
$748$	−1.20912e6	−0.0790162
$749$	2.58771e7	1.68543
$750$	1.26795e7	0.823095
$751$	−2.86431e7	−1.85319	−0.926595	−	0.376062i	$-0.877278\pi$
$751$	−2.86431e7	−1.85319	−0.926595	+	0.376062i	$0.877278\pi$
$752$	−266218.	−0.0171670
$753$	−1.13365e7	−0.728601
$754$	−4.57609e7	−2.93134
$755$	1.98673e7	1.26845
$756$	−3.58602e6	−0.228196
$757$	2.69646e7	1.71023	0.855113	−	0.518441i	$-0.173487\pi$
$757$	2.69646e7	1.71023	0.855113	+	0.518441i	$0.173487\pi$
$758$	1.01428e7	0.641188
$759$	186591.	0.0117567
$760$	−3.88630e6	−0.244063
$761$	4.94085e6	0.309272	0.154636	−	0.987972i	$-0.450580\pi$
$761$	4.94085e6	0.309272	0.154636	+	0.987972i	$0.450580\pi$
$762$	1.52378e7	0.950682
$763$	−2.20320e7	−1.37007
$764$	−1.46987e7	−0.911054
$765$	3.97610e6	0.245643
$766$	−4.02860e7	−2.48074
$767$	−2.17011e7	−1.33197
$768$	−1.20915e7	−0.739734
$769$	3.74373e6	0.228291	0.114145	−	0.993464i	$-0.463587\pi$
$769$	3.74373e6	0.228291	0.114145	+	0.993464i	$0.463587\pi$
$770$	−2.32064e6	−0.141053
$771$	−3.60639e6	−0.218492
$772$	−1.11488e7	−0.673265
$773$	2.17718e7	1.31053	0.655264	−	0.755400i	$-0.272557\pi$
$773$	2.17718e7	1.31053	0.655264	+	0.755400i	$0.272557\pi$
$774$	6.51500e6	0.390896
$775$	1.44960e7	0.866947
$776$	4.25070e6	0.253400
$777$	1.42878e7	0.849011
$778$	3.19620e7	1.89315
$779$	9.14376e6	0.539860
$780$	−6.42971e6	−0.378403
$781$	2.36268e6	0.138605
$782$	4.89265e6	0.286106
$783$	6.25438e6	0.364569
$784$	−2.53123e7	−1.47076
$785$	7.86855e6	0.455743
$786$	1.69882e7	0.980824
$787$	−2.13696e7	−1.22987	−0.614936	−	0.788577i	$-0.710818\pi$
$787$	−2.13696e7	−1.22987	−0.614936	+	0.788577i	$0.710818\pi$
$788$	−3.78135e6	−0.216936
$789$	−9.97454e6	−0.570427
$790$	1.52756e7	0.870823
$791$	8.99670e6	0.511260
$792$	162441.	0.00920201
$793$	−6.13030e6	−0.346178
$794$	−3.41768e6	−0.192389
$795$	−2.37695e6	−0.133384
$796$	−1.22761e6	−0.0686715
$797$	2.80148e7	1.56222	0.781108	−	0.624396i	$-0.214655\pi$
$797$	2.80148e7	1.56222	0.781108	+	0.624396i	$0.214655\pi$
$798$	−2.51035e7	−1.39549
$799$	272417.	0.0150962
$800$	1.11955e7	0.618472
$801$	−6.28715e6	−0.346236
$802$	1.30672e7	0.717374
$803$	−1.20076e6	−0.0657152
$804$	−1.09925e7	−0.599730
$805$	4.14034e6	0.225188
$806$	−5.11124e7	−2.77134
$807$	1.09865e7	0.593850
$808$	−1.57783e6	−0.0850223
$809$	−3.12241e7	−1.67733	−0.838665	−	0.544648i	$-0.816663\pi$
$809$	−3.12241e7	−1.67733	−0.838665	+	0.544648i	$0.816663\pi$
$810$	1.99309e6	0.106737
$811$	1.30079e7	0.694471	0.347235	−	0.937778i	$-0.387120\pi$
$811$	1.30079e7	0.694471	0.347235	+	0.937778i	$0.387120\pi$
$812$	−4.22029e7	−2.24622
$813$	5.45892e6	0.289655
$814$	2.41486e6	0.127741
$815$	8.96313e6	0.472678
$816$	1.31447e7	0.691074
$817$	2.01090e7	1.05399
$818$	−1.18856e7	−0.621066
$819$	1.11313e7	0.579878
$820$	4.89863e6	0.254414
$821$	9.48385e6	0.491051	0.245526	−	0.969390i	$-0.421039\pi$
$821$	9.48385e6	0.491051	0.245526	+	0.969390i	$0.421039\pi$
$822$	1.75871e6	0.0907853
$823$	−1.57317e7	−0.809610	−0.404805	−	0.914403i	$-0.632661\pi$
$823$	−1.57317e7	−0.809610	−0.404805	+	0.914403i	$0.632661\pi$
$824$	−7.25017e6	−0.371989
$825$	−533571.	−0.0272934
$826$	−4.53916e7	−2.31487
$827$	−1.68809e7	−0.858284	−0.429142	−	0.903237i	$-0.641184\pi$
$827$	−1.68809e7	−0.858284	−0.429142	+	0.903237i	$0.641184\pi$
$828$	1.08135e6	0.0548139
$829$	2.04228e7	1.03212	0.516059	−	0.856553i	$-0.327399\pi$
$829$	2.04228e7	1.03212	0.516059	+	0.856553i	$0.327399\pi$
$830$	−1.08363e7	−0.545990
$831$	−1.55875e7	−0.783021
$832$	−1.25222e7	−0.627152
$833$	2.59018e7	1.29335
$834$	−5.39961e6	−0.268811
$835$	−9.33662e6	−0.463419
$836$	−1.87074e6	−0.0925760
$837$	6.98580e6	0.344669
$838$	−1.46252e7	−0.719437
$839$	−3.17685e7	−1.55809	−0.779044	−	0.626970i	$-0.784295\pi$
$839$	−3.17685e7	−1.55809	−0.779044	+	0.626970i	$0.784295\pi$
$840$	3.60447e6	0.176256
$841$	5.30948e7	2.58858
$842$	−1.42961e6	−0.0694925
$843$	9.28248e6	0.449878
$844$	2.32506e7	1.12351
$845$	5.04973e6	0.243291
$846$	136554.	0.00655960
$847$	−3.10929e7	−1.48920
$848$	−7.85800e6	−0.375252
$849$	−1.64908e7	−0.785185
$850$	−1.39909e7	−0.664200
$851$	−4.30843e6	−0.203937
$852$	1.36925e7	0.646225
$853$	−1.26754e7	−0.596471	−0.298236	−	0.954492i	$-0.596398\pi$
$853$	−1.26754e7	−0.596471	−0.298236	+	0.954492i	$0.596398\pi$
$854$	−1.28226e7	−0.601631
$855$	6.15180e6	0.287797
$856$	−6.79321e6	−0.316877
$857$	−1.77408e7	−0.825126	−0.412563	−	0.910929i	$-0.635366\pi$
$857$	−1.77408e7	−0.825126	−0.412563	+	0.910929i	$0.635366\pi$
$858$	1.88136e6	0.0872477
$859$	3.98531e6	0.184280	0.0921401	−	0.995746i	$-0.470629\pi$
$859$	3.98531e6	0.184280	0.0921401	+	0.995746i	$0.470629\pi$
$860$	1.07731e7	0.496699
$861$	−8.48068e6	−0.389873
$862$	−2.35689e7	−1.08037
$863$	3.42751e7	1.56658	0.783289	−	0.621658i	$-0.213541\pi$
$863$	3.42751e7	1.56658	0.783289	+	0.621658i	$0.213541\pi$
$864$	5.39529e6	0.245884
$865$	5.20738e6	0.236635
$866$	−1.65038e7	−0.747808
$867$	−672052.	−0.0303637
$868$	−4.71383e7	−2.12361
$869$	−1.97076e6	−0.0885286
$870$	2.34561e7	1.05065
$871$	3.41216e7	1.52400
$872$	5.78381e6	0.257586
$873$	−6.72862e6	−0.298807
$874$	7.56986e6	0.335204
$875$	−3.62981e7	−1.60274
$876$	−6.95875e6	−0.306388
$877$	−3.57889e7	−1.57126	−0.785632	−	0.618694i	$-0.787662\pi$
$877$	−3.57889e7	−1.57126	−0.785632	+	0.618694i	$0.787662\pi$
$878$	−2.29645e7	−1.00536
$879$	2.49411e6	0.108879
$880$	1.88004e6	0.0818391
$881$	5.76071e6	0.250055	0.125028	−	0.992153i	$-0.460098\pi$
$881$	5.76071e6	0.250055	0.125028	+	0.992153i	$0.460098\pi$
$882$	1.29837e7	0.561987
$883$	−4.41397e7	−1.90514	−0.952572	−	0.304312i	$-0.901573\pi$
$883$	−4.41397e7	−1.90514	−0.952572	+	0.304312i	$0.901573\pi$
$884$	2.17511e7	0.936159
$885$	1.11236e7	0.477403
$886$	−2.74284e7	−1.17386
$887$	−9.63127e6	−0.411031	−0.205515	−	0.978654i	$-0.565887\pi$
$887$	−9.63127e6	−0.411031	−0.205515	+	0.978654i	$0.565887\pi$
$888$	−3.75081e6	−0.159622
$889$	−4.36218e7	−1.85118
$890$	−2.35790e7	−0.997816
$891$	−257135.	−0.0108509
$892$	2.51971e7	1.06032
$893$	421482.	0.0176869
$894$	2.30217e7	0.963369
$895$	2.90017e7	1.21022
$896$	1.99709e7	0.831050
$897$	−3.35660e6	−0.139290
$898$	5.18169e7	2.14428
$899$	8.22139e7	3.39270
$900$	−3.09221e6	−0.127251
$901$	8.04098e6	0.329987
$902$	−1.43336e6	−0.0586597
$903$	−1.86507e7	−0.761160
$904$	−2.36180e6	−0.0961217
$905$	−5.09114e6	−0.206630
$906$	−3.36897e7	−1.36357
$907$	1.72941e7	0.698037	0.349019	−	0.937116i	$-0.386515\pi$
$907$	1.72941e7	0.698037	0.349019	+	0.937116i	$0.386515\pi$
$908$	−9.50659e6	−0.382658
$909$	2.49763e6	0.100258
$910$	4.17463e7	1.67115
$911$	−3.64206e7	−1.45396	−0.726978	−	0.686660i	$-0.759076\pi$
$911$	−3.64206e7	−1.45396	−0.726978	+	0.686660i	$0.759076\pi$
$912$	2.03373e7	0.809669
$913$	1.39803e6	0.0555058
$914$	3.08295e7	1.22068
$915$	3.14227e6	0.124077
$916$	2.05479e7	0.809148
$917$	−4.86327e7	−1.90988
$918$	−6.74242e6	−0.264064
$919$	3.59707e7	1.40495	0.702474	−	0.711710i	$-0.252079\pi$
$919$	3.59707e7	1.40495	0.702474	+	0.711710i	$0.252079\pi$
$920$	−1.08691e6	−0.0423375
$921$	−1.78258e7	−0.692469
$922$	8.38366e6	0.324793
$923$	−4.25027e7	−1.64215
$924$	1.73508e6	0.0668559
$925$	1.23203e7	0.473443
$926$	2.85894e7	1.09566
$927$	1.14766e7	0.438647
$928$	6.34956e7	2.42032
$929$	−1.66420e7	−0.632654	−0.316327	−	0.948650i	$-0.602450\pi$
$929$	−1.66420e7	−0.632654	−0.316327	+	0.948650i	$0.602450\pi$
$930$	2.61992e7	0.993300
$931$	4.00750e7	1.51530
$932$	6.45264e6	0.243331
$933$	−2.30552e7	−0.867093
$934$	6.35799e7	2.38480
$935$	−1.92382e6	−0.0719673
$936$	−2.92217e6	−0.109023
$937$	1.14846e7	0.427332	0.213666	−	0.976907i	$-0.431460\pi$
$937$	1.14846e7	0.427332	0.213666	+	0.976907i	$0.431460\pi$
$938$	7.13711e7	2.64859
$939$	−2.21689e7	−0.820502
$940$	225803.	0.00833508
$941$	4.50411e7	1.65819	0.829095	−	0.559107i	$-0.188856\pi$
$941$	4.50411e7	1.65819	0.829095	+	0.559107i	$0.188856\pi$
$942$	−1.33430e7	−0.489920
$943$	2.55731e6	0.0936494
$944$	3.67736e7	1.34309
$945$	−5.70568e6	−0.207839
$946$	−3.15225e6	−0.114523
$947$	3.18096e7	1.15261	0.576307	−	0.817234i	$-0.304493\pi$
$947$	3.18096e7	1.15261	0.576307	+	0.817234i	$0.304493\pi$
$948$	−1.14211e7	−0.412752
$949$	2.16006e7	0.778573
$950$	−2.16466e7	−0.778182
$951$	2.74202e7	0.983150
$952$	−1.21936e7	−0.436052
$953$	−3.14562e7	−1.12195	−0.560975	−	0.827833i	$-0.689574\pi$
$953$	−3.14562e7	−1.12195	−0.560975	+	0.827833i	$0.689574\pi$
$954$	4.03068e6	0.143386
$955$	−2.33869e7	−0.829781
$956$	−3.21720e7	−1.13850
$957$	−3.02615e6	−0.106810
$958$	4.87393e7	1.71579
$959$	−5.03473e6	−0.176779
$960$	6.41863e6	0.224783
$961$	6.31993e7	2.20752
$962$	−4.34412e7	−1.51344
$963$	1.07533e7	0.373659
$964$	−3.71065e7	−1.28605
$965$	−1.77388e7	−0.613205
$966$	−7.02091e6	−0.242075
$967$	−1.39347e7	−0.479216	−0.239608	−	0.970870i	$-0.577019\pi$
$967$	−1.39347e7	−0.479216	−0.239608	+	0.970870i	$0.577019\pi$
$968$	8.16245e6	0.279983
$969$	−2.08109e7	−0.712003
$970$	−2.52347e7	−0.861129
$971$	−6.17798e6	−0.210280	−0.105140	−	0.994457i	$-0.533529\pi$
$971$	−6.17798e6	−0.210280	−0.105140	+	0.994457i	$0.533529\pi$
$972$	−1.49018e6	−0.0505910
$973$	1.54576e7	0.523433
$974$	−1.94903e7	−0.658296
$975$	9.59848e6	0.323363
$976$	1.03881e7	0.349068
$977$	−2.28927e7	−0.767293	−0.383646	−	0.923480i	$-0.625332\pi$
$977$	−2.28927e7	−0.767293	−0.383646	+	0.923480i	$0.625332\pi$
$978$	−1.51991e7	−0.508125
$979$	3.04201e6	0.101439
$980$	2.14696e7	0.714099
$981$	−9.15546e6	−0.303744
$982$	−3.35194e6	−0.110922
$983$	−3.76941e7	−1.24420	−0.622100	−	0.782938i	$-0.713720\pi$
$983$	−3.76941e7	−1.24420	−0.622100	+	0.782938i	$0.713720\pi$
$984$	2.22633e6	0.0732997
$985$	−6.01647e6	−0.197584
$986$	−7.93496e7	−2.59928
$987$	−390917.	−0.0127730
$988$	3.36531e7	1.09681
$989$	5.62405e6	0.182835
$990$	−964346.	−0.0312712
$991$	−2.55042e7	−0.824951	−0.412475	−	0.910969i	$-0.635336\pi$
$991$	−2.55042e7	−0.824951	−0.412475	+	0.910969i	$0.635336\pi$
$992$	7.09211e7	2.28821
$993$	2.33493e7	0.751451
$994$	−8.89016e7	−2.85393
$995$	−1.95323e6	−0.0625455
$996$	8.10200e6	0.258788
$997$	−2.40517e7	−0.766315	−0.383157	−	0.923683i	$-0.625163\pi$
$997$	−2.40517e7	−0.766315	−0.383157	+	0.923683i	$0.625163\pi$
$998$	4.81292e7	1.52961
$999$	5.93733e6	0.188225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.c.1.4	✓	4
3.2	odd	2		207.6.a.d.1.1		4
4.3	odd	2		1104.6.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.c.1.4	✓	4	1.1	even	1	trivial
207.6.a.d.1.1		4	3.2	odd	2
1104.6.a.n.1.4		4	4.3	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+7.56547 q^{2} -9.00000 q^{3} +25.2363 q^{4} +40.1532 q^{5} -68.0892 q^{6} +194.921 q^{7} -51.1704 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+7.56547 q^{2} -9.00000 q^{3} +25.2363 q^{4} +40.1532 q^{5} -68.0892 q^{6} +194.921 q^{7} -51.1704 q^{8} +81.0000 q^{9} +303.778 q^{10} -39.1915 q^{11} -227.127 q^{12} +705.021 q^{13} +1474.67 q^{14} -361.379 q^{15} -1194.69 q^{16} +1222.51 q^{17} +612.803 q^{18} +1891.46 q^{19} +1013.32 q^{20} -1754.29 q^{21} -296.502 q^{22} +529.000 q^{23} +460.534 q^{24} -1512.72 q^{25} +5333.81 q^{26} -729.000 q^{27} +4919.10 q^{28} -8579.39 q^{29} -2734.00 q^{30} -9582.72 q^{31} -7400.94 q^{32} +352.723 q^{33} +9248.86 q^{34} +7826.73 q^{35} +2044.14 q^{36} -8144.49 q^{37} +14309.8 q^{38} -6345.19 q^{39} -2054.66 q^{40} +4834.24 q^{41} -13272.0 q^{42} +10631.5 q^{43} -989.049 q^{44} +3252.41 q^{45} +4002.13 q^{46} +222.835 q^{47} +10752.2 q^{48} +21187.4 q^{49} -11444.4 q^{50} -11002.6 q^{51} +17792.1 q^{52} +6577.44 q^{53} -5515.23 q^{54} -1573.66 q^{55} -9974.21 q^{56} -17023.1 q^{57} -64907.1 q^{58} -30780.8 q^{59} -9119.88 q^{60} -8695.20 q^{61} -72497.8 q^{62} +15788.6 q^{63} -17761.5 q^{64} +28308.9 q^{65} +2668.52 q^{66} +48398.0 q^{67} +30851.6 q^{68} -4761.00 q^{69} +59212.8 q^{70} -60285.7 q^{71} -4144.80 q^{72} +30638.2 q^{73} -61616.8 q^{74} +13614.5 q^{75} +47733.4 q^{76} -7639.26 q^{77} -48004.3 q^{78} +50285.3 q^{79} -47970.7 q^{80} +6561.00 q^{81} +36573.3 q^{82} -35671.7 q^{83} -44271.9 q^{84} +49087.7 q^{85} +80432.0 q^{86} +77214.5 q^{87} +2005.44 q^{88} -77619.2 q^{89} +24606.0 q^{90} +137424. q^{91} +13350.0 q^{92} +86244.5 q^{93} +1685.85 q^{94} +75948.1 q^{95} +66608.5 q^{96} -83069.4 q^{97} +160292. q^{98} -3174.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} - 46 q^{4} - 122 q^{5} - 36 q^{6} + 62 q^{7} + 72 q^{8} + 324 q^{9} + 642 q^{10} + 32 q^{11} + 414 q^{12} + 1364 q^{13} + 2754 q^{14} + 1098 q^{15} + 18 q^{16} + 278 q^{17} + 324 q^{18} + 2862 q^{19} + 3830 q^{20} - 558 q^{21} + 3176 q^{22} + 2116 q^{23} - 648 q^{24} + 5944 q^{25} + 6996 q^{26} - 2916 q^{27} + 4738 q^{28} - 5180 q^{29} - 5778 q^{30} - 1788 q^{31} - 7352 q^{32} - 288 q^{33} + 15818 q^{34} + 11768 q^{35} - 3726 q^{36} + 3348 q^{37} + 1050 q^{38} - 12276 q^{39} - 13462 q^{40} - 17664 q^{41} - 24786 q^{42} + 25398 q^{43} - 16848 q^{44} - 9882 q^{45} + 2116 q^{46} - 26040 q^{47} - 162 q^{48} + 55720 q^{49} - 35256 q^{50} - 2502 q^{51} - 2752 q^{52} - 32006 q^{53} - 2916 q^{54} + 34904 q^{55} - 68542 q^{56} - 25758 q^{57} - 40804 q^{58} - 61136 q^{59} - 34470 q^{60} + 35844 q^{61} - 47524 q^{62} + 5022 q^{63} - 35142 q^{64} - 48036 q^{65} - 28584 q^{66} + 73458 q^{67} + 17910 q^{68} - 19044 q^{69} - 59104 q^{70} + 24432 q^{71} + 5832 q^{72} + 122512 q^{73} - 20828 q^{74} - 53496 q^{75} + 56834 q^{76} + 159496 q^{77} - 62964 q^{78} + 90170 q^{79} - 36546 q^{80} + 26244 q^{81} + 84144 q^{82} + 28592 q^{83} - 42642 q^{84} + 355124 q^{85} + 103778 q^{86} + 46620 q^{87} - 150776 q^{88} - 27926 q^{89} + 52002 q^{90} + 334180 q^{91} - 24334 q^{92} + 16092 q^{93} + 113632 q^{94} + 113392 q^{95} + 66168 q^{96} + 16580 q^{97} + 49688 q^{98} + 2592 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 - 46 * q^4 - 122 * q^5 - 36 * q^6 + 62 * q^7 + 72 * q^8 + 324 * q^9 + 642 * q^10 + 32 * q^11 + 414 * q^12 + 1364 * q^13 + 2754 * q^14 + 1098 * q^15 + 18 * q^16 + 278 * q^17 + 324 * q^18 + 2862 * q^19 + 3830 * q^20 - 558 * q^21 + 3176 * q^22 + 2116 * q^23 - 648 * q^24 + 5944 * q^25 + 6996 * q^26 - 2916 * q^27 + 4738 * q^28 - 5180 * q^29 - 5778 * q^30 - 1788 * q^31 - 7352 * q^32 - 288 * q^33 + 15818 * q^34 + 11768 * q^35 - 3726 * q^36 + 3348 * q^37 + 1050 * q^38 - 12276 * q^39 - 13462 * q^40 - 17664 * q^41 - 24786 * q^42 + 25398 * q^43 - 16848 * q^44 - 9882 * q^45 + 2116 * q^46 - 26040 * q^47 - 162 * q^48 + 55720 * q^49 - 35256 * q^50 - 2502 * q^51 - 2752 * q^52 - 32006 * q^53 - 2916 * q^54 + 34904 * q^55 - 68542 * q^56 - 25758 * q^57 - 40804 * q^58 - 61136 * q^59 - 34470 * q^60 + 35844 * q^61 - 47524 * q^62 + 5022 * q^63 - 35142 * q^64 - 48036 * q^65 - 28584 * q^66 + 73458 * q^67 + 17910 * q^68 - 19044 * q^69 - 59104 * q^70 + 24432 * q^71 + 5832 * q^72 + 122512 * q^73 - 20828 * q^74 - 53496 * q^75 + 56834 * q^76 + 159496 * q^77 - 62964 * q^78 + 90170 * q^79 - 36546 * q^80 + 26244 * q^81 + 84144 * q^82 + 28592 * q^83 - 42642 * q^84 + 355124 * q^85 + 103778 * q^86 + 46620 * q^87 - 150776 * q^88 - 27926 * q^89 + 52002 * q^90 + 334180 * q^91 - 24334 * q^92 + 16092 * q^93 + 113632 * q^94 + 113392 * q^95 + 66168 * q^96 + 16580 * q^97 + 49688 * q^98 + 2592 * q^99

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