LMFDB - Embedded newform 69.6.a.d.1.2

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:	$1$
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} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.04157$ 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)

$f(q)$	$=$	$q-2.04157 q^{2} -9.00000 q^{3} -27.8320 q^{4} +42.3660 q^{5} +18.3742 q^{6} +191.647 q^{7} +122.151 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-2.04157 q^{2} -9.00000 q^{3} -27.8320 q^{4} +42.3660 q^{5} +18.3742 q^{6} +191.647 q^{7} +122.151 q^{8} +81.0000 q^{9} -86.4934 q^{10} -655.589 q^{11} +250.488 q^{12} -932.384 q^{13} -391.262 q^{14} -381.294 q^{15} +641.242 q^{16} -344.188 q^{17} -165.368 q^{18} -548.438 q^{19} -1179.13 q^{20} -1724.83 q^{21} +1338.43 q^{22} -529.000 q^{23} -1099.36 q^{24} -1330.12 q^{25} +1903.53 q^{26} -729.000 q^{27} -5333.92 q^{28} +4399.54 q^{29} +778.440 q^{30} -4434.84 q^{31} -5217.99 q^{32} +5900.30 q^{33} +702.685 q^{34} +8119.33 q^{35} -2254.39 q^{36} -11414.8 q^{37} +1119.68 q^{38} +8391.46 q^{39} +5175.07 q^{40} -6661.33 q^{41} +3521.36 q^{42} -22878.8 q^{43} +18246.3 q^{44} +3431.65 q^{45} +1079.99 q^{46} +17936.0 q^{47} -5771.18 q^{48} +19921.7 q^{49} +2715.54 q^{50} +3097.69 q^{51} +25950.1 q^{52} +4562.43 q^{53} +1488.31 q^{54} -27774.7 q^{55} +23410.0 q^{56} +4935.95 q^{57} -8981.98 q^{58} -23244.6 q^{59} +10612.2 q^{60} +25147.4 q^{61} +9054.05 q^{62} +15523.4 q^{63} -9866.83 q^{64} -39501.4 q^{65} -12045.9 q^{66} -6212.65 q^{67} +9579.42 q^{68} +4761.00 q^{69} -16576.2 q^{70} +6800.71 q^{71} +9894.27 q^{72} -32996.3 q^{73} +23304.2 q^{74} +11971.1 q^{75} +15264.1 q^{76} -125642. q^{77} -17131.8 q^{78} -21087.6 q^{79} +27166.9 q^{80} +6561.00 q^{81} +13599.6 q^{82} +28330.8 q^{83} +48005.3 q^{84} -14581.9 q^{85} +46708.8 q^{86} -39595.8 q^{87} -80081.2 q^{88} +137589. q^{89} -7005.96 q^{90} -178689. q^{91} +14723.1 q^{92} +39913.5 q^{93} -36617.7 q^{94} -23235.1 q^{95} +46961.9 q^{96} -32286.5 q^{97} -40671.6 q^{98} -53102.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 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 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9} - 496 q^{10} - 1076 q^{11} - 234 q^{12} - 396 q^{13} - 1806 q^{14} - 198 q^{15} - 1982 q^{16} + 70 q^{17} + 324 q^{18} - 6366 q^{19} - 5240 q^{20} + 558 q^{21} - 6974 q^{22} - 2116 q^{23} - 648 q^{24} + 1264 q^{25} + 2464 q^{26} - 2916 q^{27} - 6474 q^{28} + 3948 q^{29} + 4464 q^{30} + 3092 q^{31} - 3672 q^{32} + 9684 q^{33} + 11682 q^{34} + 1304 q^{35} + 2106 q^{36} - 17464 q^{37} - 12628 q^{38} + 3564 q^{39} - 14108 q^{40} + 18680 q^{41} + 16254 q^{42} - 25846 q^{43} + 20746 q^{44} + 1782 q^{45} - 2116 q^{46} + 18392 q^{47} + 17838 q^{48} + 7952 q^{49} + 69444 q^{50} - 630 q^{51} + 8844 q^{52} - 26518 q^{53} - 2916 q^{54} - 40848 q^{55} + 54890 q^{56} + 57294 q^{57} + 568 q^{58} - 14520 q^{59} + 47160 q^{60} - 13688 q^{61} + 120136 q^{62} - 5022 q^{63} - 30190 q^{64} + 38324 q^{65} + 62766 q^{66} - 11098 q^{67} + 112138 q^{68} + 19044 q^{69} - 29596 q^{70} - 57496 q^{71} + 5832 q^{72} - 112272 q^{73} - 21226 q^{74} - 11376 q^{75} - 76240 q^{76} - 4792 q^{77} - 22176 q^{78} - 240754 q^{79} + 41200 q^{80} + 26244 q^{81} + 49976 q^{82} - 93268 q^{83} + 58266 q^{84} - 323204 q^{85} - 88224 q^{86} - 35532 q^{87} + 42382 q^{88} - 107582 q^{89} - 40176 q^{90} - 301532 q^{91} - 13754 q^{92} - 27828 q^{93} + 79360 q^{94} - 18640 q^{95} + 33048 q^{96} - 53076 q^{97} + 59664 q^{98} - 87156 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9 - 496 * q^10 - 1076 * q^11 - 234 * q^12 - 396 * q^13 - 1806 * q^14 - 198 * q^15 - 1982 * q^16 + 70 * q^17 + 324 * q^18 - 6366 * q^19 - 5240 * q^20 + 558 * q^21 - 6974 * q^22 - 2116 * q^23 - 648 * q^24 + 1264 * q^25 + 2464 * q^26 - 2916 * q^27 - 6474 * q^28 + 3948 * q^29 + 4464 * q^30 + 3092 * q^31 - 3672 * q^32 + 9684 * q^33 + 11682 * q^34 + 1304 * q^35 + 2106 * q^36 - 17464 * q^37 - 12628 * q^38 + 3564 * q^39 - 14108 * q^40 + 18680 * q^41 + 16254 * q^42 - 25846 * q^43 + 20746 * q^44 + 1782 * q^45 - 2116 * q^46 + 18392 * q^47 + 17838 * q^48 + 7952 * q^49 + 69444 * q^50 - 630 * q^51 + 8844 * q^52 - 26518 * q^53 - 2916 * q^54 - 40848 * q^55 + 54890 * q^56 + 57294 * q^57 + 568 * q^58 - 14520 * q^59 + 47160 * q^60 - 13688 * q^61 + 120136 * q^62 - 5022 * q^63 - 30190 * q^64 + 38324 * q^65 + 62766 * q^66 - 11098 * q^67 + 112138 * q^68 + 19044 * q^69 - 29596 * q^70 - 57496 * q^71 + 5832 * q^72 - 112272 * q^73 - 21226 * q^74 - 11376 * q^75 - 76240 * q^76 - 4792 * q^77 - 22176 * q^78 - 240754 * q^79 + 41200 * q^80 + 26244 * q^81 + 49976 * q^82 - 93268 * q^83 + 58266 * q^84 - 323204 * q^85 - 88224 * q^86 - 35532 * q^87 + 42382 * q^88 - 107582 * q^89 - 40176 * q^90 - 301532 * q^91 - 13754 * q^92 - 27828 * q^93 + 79360 * q^94 - 18640 * q^95 + 33048 * q^96 - 53076 * q^97 + 59664 * q^98 - 87156 * 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 $

$2$	−2.04157	−0.360903	−0.180451	−	0.983584i	$-0.557756\pi$
$2$	−2.04157	−0.360903	−0.180451	+	0.983584i	$0.557756\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−27.8320	−0.869749
$5$	42.3660	0.757866	0.378933	−	0.925424i	$-0.376291\pi$
$5$	42.3660	0.757866	0.378933	+	0.925424i	$0.376291\pi$
$6$	18.3742	0.208367
$7$	191.647	1.47828	0.739142	−	0.673550i	$-0.235231\pi$
$7$	191.647	1.47828	0.739142	+	0.673550i	$0.235231\pi$
$8$	122.151	0.674798
$9$	81.0000	0.333333
$10$	−86.4934	−0.273516
$11$	−655.589	−1.63362	−0.816808	−	0.576909i	$-0.804259\pi$
$11$	−655.589	−1.63362	−0.816808	+	0.576909i	$0.804259\pi$
$12$	250.488	0.502150
$13$	−932.384	−1.53016	−0.765079	−	0.643936i	$-0.777300\pi$
$13$	−932.384	−1.53016	−0.765079	+	0.643936i	$0.777300\pi$
$14$	−391.262	−0.533517
$15$	−381.294	−0.437554
$16$	641.242	0.626213
$17$	−344.188	−0.288850	−0.144425	−	0.989516i	$-0.546133\pi$
$17$	−344.188	−0.288850	−0.144425	+	0.989516i	$0.546133\pi$
$18$	−165.368	−0.120301
$19$	−548.438	−0.348533	−0.174266	−	0.984699i	$-0.555755\pi$
$19$	−548.438	−0.348533	−0.174266	+	0.984699i	$0.555755\pi$
$20$	−1179.13	−0.659154
$21$	−1724.83	−0.853487
$22$	1338.43	0.589577
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	−1099.36	−0.389595
$25$	−1330.12	−0.425639
$26$	1903.53	0.552238
$27$	−729.000	−0.192450
$28$	−5333.92	−1.28574
$29$	4399.54	0.971431	0.485715	−	0.874117i	$-0.338559\pi$
$29$	4399.54	0.971431	0.485715	+	0.874117i	$0.338559\pi$
$30$	778.440	0.157915
$31$	−4434.84	−0.828845	−0.414422	−	0.910085i	$-0.636016\pi$
$31$	−4434.84	−0.828845	−0.414422	+	0.910085i	$0.636016\pi$
$32$	−5217.99	−0.900800
$33$	5900.30	0.943169
$34$	702.685	0.104247
$35$	8119.33	1.12034
$36$	−2254.39	−0.289916
$37$	−11414.8	−1.37077	−0.685384	−	0.728182i	$-0.740366\pi$
$37$	−11414.8	−1.37077	−0.685384	+	0.728182i	$0.740366\pi$
$38$	1119.68	0.125786
$39$	8391.46	0.883437
$40$	5175.07	0.511406
$41$	−6661.33	−0.618872	−0.309436	−	0.950920i	$-0.600140\pi$
$41$	−6661.33	−0.618872	−0.309436	+	0.950920i	$0.600140\pi$
$42$	3521.36	0.308026
$43$	−22878.8	−1.88696	−0.943478	−	0.331434i	$-0.892467\pi$
$43$	−22878.8	−1.88696	−0.943478	+	0.331434i	$0.892467\pi$
$44$	18246.3	1.42084
$45$	3431.65	0.252622
$46$	1079.99	0.0752534
$47$	17936.0	1.18435	0.592176	−	0.805808i	$-0.298269\pi$
$47$	17936.0	1.18435	0.592176	+	0.805808i	$0.298269\pi$
$48$	−5771.18	−0.361544
$49$	19921.7	1.18532
$50$	2715.54	0.153614
$51$	3097.69	0.166768
$52$	25950.1	1.33085
$53$	4562.43	0.223103	0.111552	−	0.993759i	$-0.464418\pi$
$53$	4562.43	0.223103	0.111552	+	0.993759i	$0.464418\pi$
$54$	1488.31	0.0694558
$55$	−27774.7	−1.23806
$56$	23410.0	0.997542
$57$	4935.95	0.201226
$58$	−8981.98	−0.350592
$59$	−23244.6	−0.869345	−0.434673	−	0.900589i	$-0.643136\pi$
$59$	−23244.6	−0.869345	−0.434673	+	0.900589i	$0.643136\pi$
$60$	10612.2	0.380562
$61$	25147.4	0.865304	0.432652	−	0.901561i	$-0.357578\pi$
$61$	25147.4	0.865304	0.432652	+	0.901561i	$0.357578\pi$
$62$	9054.05	0.299132
$63$	15523.4	0.492761
$64$	−9866.83	−0.301112
$65$	−39501.4	−1.15966
$66$	−12045.9	−0.340392
$67$	−6212.65	−0.169079	−0.0845396	−	0.996420i	$-0.526942\pi$
$67$	−6212.65	−0.169079	−0.0845396	+	0.996420i	$0.526942\pi$
$68$	9579.42	0.251227
$69$	4761.00	0.120386
$70$	−16576.2	−0.404334
$71$	6800.71	0.160106	0.0800531	−	0.996791i	$-0.474491\pi$
$71$	6800.71	0.160106	0.0800531	+	0.996791i	$0.474491\pi$
$72$	9894.27	0.224933
$73$	−32996.3	−0.724700	−0.362350	−	0.932042i	$-0.618026\pi$
$73$	−32996.3	−0.724700	−0.362350	+	0.932042i	$0.618026\pi$
$74$	23304.2	0.494714
$75$	11971.1	0.245743
$76$	15264.1	0.303136
$77$	−125642.	−2.41495
$78$	−17131.8	−0.318835
$79$	−21087.6	−0.380154	−0.190077	−	0.981769i	$-0.560874\pi$
$79$	−21087.6	−0.380154	−0.190077	+	0.981769i	$0.560874\pi$
$80$	27166.9	0.474586
$81$	6561.00	0.111111
$82$	13599.6	0.223353
$83$	28330.8	0.451402	0.225701	−	0.974197i	$-0.427533\pi$
$83$	28330.8	0.451402	0.225701	+	0.974197i	$0.427533\pi$
$84$	48005.3	0.742320
$85$	−14581.9	−0.218910
$86$	46708.8	0.681008
$87$	−39595.8	−0.560856
$88$	−80081.2	−1.10236
$89$	137589.	1.84123	0.920616	−	0.390469i	$-0.127687\pi$
$89$	137589.	1.84123	0.920616	+	0.390469i	$0.127687\pi$
$90$	−7005.96	−0.0911720
$91$	−178689.	−2.26201
$92$	14723.1	0.181355
$93$	39913.5	0.478534
$94$	−36617.7	−0.427436
$95$	−23235.1	−0.264141
$96$	46961.9	0.520077
$97$	−32286.5	−0.348411	−0.174206	−	0.984709i	$-0.555736\pi$
$97$	−32286.5	−0.348411	−0.174206	+	0.984709i	$0.555736\pi$
$98$	−40671.6	−0.427786
$99$	−53102.7	−0.544539
$100$	37019.9	0.370199
$101$	−163233.	−1.59222	−0.796112	−	0.605149i	$-0.793113\pi$
$101$	−163233.	−1.59222	−0.796112	+	0.605149i	$0.793113\pi$
$102$	−6324.16	−0.0601870
$103$	−68514.0	−0.636336	−0.318168	−	0.948034i	$-0.603068\pi$
$103$	−68514.0	−0.636336	−0.318168	+	0.948034i	$0.603068\pi$
$104$	−113892.	−1.03255
$105$	−73074.0	−0.646829
$106$	−9314.53	−0.0805186
$107$	−20369.5	−0.171997	−0.0859985	−	0.996295i	$-0.527408\pi$
$107$	−20369.5	−0.171997	−0.0859985	+	0.996295i	$0.527408\pi$
$108$	20289.5	0.167383
$109$	245848.	1.98198	0.990992	−	0.133920i	$-0.0427567\pi$
$109$	245848.	1.98198	0.990992	+	0.133920i	$0.0427567\pi$
$110$	56704.1	0.446820
$111$	102733.	0.791413
$112$	122892.	0.925720
$113$	258501.	1.90444	0.952218	−	0.305419i	$-0.0987966\pi$
$113$	258501.	1.90444	0.952218	+	0.305419i	$0.0987966\pi$
$114$	−10077.1	−0.0726229
$115$	−22411.6	−0.158026
$116$	−122448.	−0.844901
$117$	−75523.1	−0.510053
$118$	47455.6	0.313749
$119$	−65962.6	−0.427003
$120$	−46575.6	−0.295261
$121$	268746.	1.66870
$122$	−51340.3	−0.312291
$123$	59951.9	0.357306
$124$	123430.	0.720887
$125$	−188746.	−1.08044
$126$	−31692.2	−0.177839
$127$	−151439.	−0.833160	−0.416580	−	0.909099i	$-0.636771\pi$
$127$	−151439.	−0.833160	−0.416580	+	0.909099i	$0.636771\pi$
$128$	187120.	1.00947
$129$	205909.	1.08943
$130$	80645.0	0.418523
$131$	54236.2	0.276128	0.138064	−	0.990423i	$-0.455912\pi$
$131$	54236.2	0.276128	0.138064	+	0.990423i	$0.455912\pi$
$132$	−164217.	−0.820320
$133$	−105107.	−0.515230
$134$	12683.6	0.0610211
$135$	−30884.8	−0.145851
$136$	−42043.0	−0.194916
$137$	210827.	0.959677	0.479839	−	0.877357i	$-0.340695\pi$
$137$	210827.	0.959677	0.479839	+	0.877357i	$0.340695\pi$
$138$	−9719.94	−0.0434476
$139$	−319153.	−1.40108	−0.700539	−	0.713614i	$-0.747057\pi$
$139$	−319153.	−1.40108	−0.700539	+	0.713614i	$0.747057\pi$
$140$	−225977.	−0.974416
$141$	−161424.	−0.683786
$142$	−13884.2	−0.0577828
$143$	611261.	2.49969
$144$	51940.6	0.208738
$145$	186391.	0.736215
$146$	67364.5	0.261546
$147$	−179295.	−0.684346
$148$	317696.	1.19222
$149$	−319057.	−1.17734	−0.588671	−	0.808373i	$-0.700349\pi$
$149$	−319057.	−1.17734	−0.588671	+	0.808373i	$0.700349\pi$
$150$	−24439.9	−0.0886892
$151$	100255.	0.357820	0.178910	−	0.983865i	$-0.442743\pi$
$151$	100255.	0.357820	0.178910	+	0.983865i	$0.442743\pi$
$152$	−66992.5	−0.235189
$153$	−27879.2	−0.0962835
$154$	256507.	0.871561
$155$	−187886.	−0.628154
$156$	−233551.	−0.768369
$157$	334886.	1.08430	0.542149	−	0.840283i	$-0.317611\pi$
$157$	334886.	1.08430	0.542149	+	0.840283i	$0.317611\pi$
$158$	43051.9	0.137198
$159$	−41061.8	−0.128809
$160$	−221065.	−0.682686
$161$	−101381.	−0.308243
$162$	−13394.8	−0.0401003
$163$	268471.	0.791460	0.395730	−	0.918367i	$-0.370492\pi$
$163$	268471.	0.791460	0.395730	+	0.918367i	$0.370492\pi$
$164$	185398.	0.538264
$165$	249972.	0.714796
$166$	−57839.5	−0.162912
$167$	−250738.	−0.695711	−0.347855	−	0.937548i	$-0.613090\pi$
$167$	−250738.	−0.695711	−0.347855	+	0.937548i	$0.613090\pi$
$168$	−210690.	−0.575931
$169$	498047.	1.34138
$170$	29769.9	0.0790052
$171$	−44423.5	−0.116178
$172$	636762.	1.64118
$173$	478595.	1.21577	0.607887	−	0.794023i	$-0.292017\pi$
$173$	478595.	1.21577	0.607887	+	0.794023i	$0.292017\pi$
$174$	80837.8	0.202414
$175$	−254914.	−0.629215
$176$	−420391.	−1.02299
$177$	209202.	0.501917
$178$	−280898.	−0.664506
$179$	−416312.	−0.971150	−0.485575	−	0.874195i	$-0.661390\pi$
$179$	−416312.	−0.971150	−0.485575	+	0.874195i	$0.661390\pi$
$180$	−95509.5	−0.219718
$181$	−517731.	−1.17465	−0.587324	−	0.809352i	$-0.699818\pi$
$181$	−517731.	−1.17465	−0.587324	+	0.809352i	$0.699818\pi$
$182$	364807.	0.816365
$183$	−226327.	−0.499583
$184$	−64618.1	−0.140705
$185$	−483600.	−1.03886
$186$	−81486.4	−0.172704
$187$	225646.	0.471871
$188$	−499194.	−1.03009
$189$	−139711.	−0.284496
$190$	47436.3	0.0953293
$191$	−162153.	−0.321618	−0.160809	−	0.986986i	$-0.551410\pi$
$191$	−162153.	−0.321618	−0.160809	+	0.986986i	$0.551410\pi$
$192$	88801.5	0.173847
$193$	−385440.	−0.744840	−0.372420	−	0.928064i	$-0.621472\pi$
$193$	−385440.	−0.744840	−0.372420	+	0.928064i	$0.621472\pi$
$194$	65915.3	0.125743
$195$	355512.	0.669527
$196$	−554460.	−1.03093
$197$	−788562.	−1.44767	−0.723836	−	0.689972i	$-0.757623\pi$
$197$	−788562.	−1.44767	−0.723836	+	0.689972i	$0.757623\pi$
$198$	108413.	0.196526
$199$	−556981.	−0.997028	−0.498514	−	0.866882i	$-0.666121\pi$
$199$	−556981.	−0.997028	−0.498514	+	0.866882i	$0.666121\pi$
$200$	−162476.	−0.287220
$201$	55913.9	0.0976179
$202$	333252.	0.574638
$203$	843159.	1.43605
$204$	−86214.8	−0.145046
$205$	−282214.	−0.469022
$206$	139876.	0.229655
$207$	−42849.0	−0.0695048
$208$	−597884.	−0.958205
$209$	359550.	0.569369
$210$	149186.	0.233442
$211$	617009.	0.954081	0.477041	−	0.878881i	$-0.341709\pi$
$211$	617009.	0.954081	0.477041	+	0.878881i	$0.341709\pi$
$212$	−126981.	−0.194044
$213$	−61206.4	−0.0924374
$214$	41585.8	0.0620742
$215$	−969283.	−1.43006
$216$	−89048.4	−0.129865
$217$	−849925.	−1.22527
$218$	−501917.	−0.715304
$219$	296967.	0.418406
$220$	773025.	1.07680
$221$	320915.	0.441987
$222$	−209738.	−0.285623
$223$	373644.	0.503148	0.251574	−	0.967838i	$-0.419052\pi$
$223$	373644.	0.503148	0.251574	+	0.967838i	$0.419052\pi$
$224$	−1.00001e6	−1.33164
$225$	−107740.	−0.141880
$226$	−527749.	−0.687316
$227$	550317.	0.708841	0.354420	−	0.935086i	$-0.384678\pi$
$227$	550317.	0.708841	0.354420	+	0.935086i	$0.384678\pi$
$228$	−137377.	−0.175016
$229$	−830028.	−1.04593	−0.522966	−	0.852353i	$-0.675175\pi$
$229$	−830028.	−1.04593	−0.522966	+	0.852353i	$0.675175\pi$
$230$	45755.0	0.0570320
$231$	1.13078e6	1.39427
$232$	537410.	0.655519
$233$	1.54010e6	1.85848	0.929242	−	0.369472i	$-0.120461\pi$
$233$	1.54010e6	1.85848	0.929242	+	0.369472i	$0.120461\pi$
$234$	154186.	0.184079
$235$	759877.	0.897581
$236$	646943.	0.756112
$237$	189788.	0.219482
$238$	134668.	0.154106
$239$	1.35520e6	1.53465	0.767323	−	0.641261i	$-0.221588\pi$
$239$	1.35520e6	1.53465	0.767323	+	0.641261i	$0.221588\pi$
$240$	−244502.	−0.274002
$241$	1.20119e6	1.33220	0.666099	−	0.745864i	$-0.267963\pi$
$241$	1.20119e6	1.33220	0.666099	+	0.745864i	$0.267963\pi$
$242$	−548665.	−0.602239
$243$	−59049.0	−0.0641500
$244$	−699902.	−0.752597
$245$	844003.	0.898315
$246$	−122396.	−0.128953
$247$	511355.	0.533311
$248$	−541722.	−0.559303
$249$	−254977.	−0.260617
$250$	385338.	0.389935
$251$	523235.	0.524218	0.262109	−	0.965038i	$-0.415582\pi$
$251$	523235.	0.524218	0.262109	+	0.965038i	$0.415582\pi$
$252$	−432048.	−0.428579
$253$	346807.	0.340633
$254$	309174.	0.300690
$255$	131237.	0.126388
$256$	−66279.8	−0.0632094
$257$	667770.	0.630658	0.315329	−	0.948982i	$-0.397885\pi$
$257$	667770.	0.630658	0.315329	+	0.948982i	$0.397885\pi$
$258$	−420379.	−0.393180
$259$	−2.18762e6	−2.02638
$260$	1.09940e6	1.00861
$261$	356362.	0.323810
$262$	−110727.	−0.0996555
$263$	−2.15898e6	−1.92469	−0.962343	−	0.271837i	$-0.912369\pi$
$263$	−2.15898e6	−1.92469	−0.962343	+	0.271837i	$0.912369\pi$
$264$	720730.	0.636448
$265$	193292.	0.169083
$266$	214583.	0.185948
$267$	−1.23830e6	−1.06304
$268$	172910.	0.147056
$269$	−89814.4	−0.0756772	−0.0378386	−	0.999284i	$-0.512047\pi$
$269$	−89814.4	−0.0756772	−0.0378386	+	0.999284i	$0.512047\pi$
$270$	63053.7	0.0526382
$271$	−1.74478e6	−1.44317	−0.721583	−	0.692328i	$-0.756585\pi$
$271$	−1.74478e6	−1.44317	−0.721583	+	0.692328i	$0.756585\pi$
$272$	−220708.	−0.180882
$273$	1.60820e6	1.30597
$274$	−430419.	−0.346350
$275$	872013.	0.695331
$276$	−132508.	−0.104705
$277$	2.19896e6	1.72194	0.860971	−	0.508654i	$-0.169857\pi$
$277$	2.19896e6	1.72194	0.860971	+	0.508654i	$0.169857\pi$
$278$	651575.	0.505653
$279$	−359222.	−0.276282
$280$	991788.	0.756004
$281$	857174.	0.647595	0.323797	−	0.946126i	$-0.395040\pi$
$281$	857174.	0.647595	0.323797	+	0.946126i	$0.395040\pi$
$282$	329559.	0.246780
$283$	500871.	0.371757	0.185879	−	0.982573i	$-0.440487\pi$
$283$	500871.	0.371757	0.185879	+	0.982573i	$0.440487\pi$
$284$	−189277.	−0.139252
$285$	209116.	0.152502
$286$	−1.24793e6	−0.902146
$287$	−1.27663e6	−0.914869
$288$	−422657.	−0.300267
$289$	−1.30139e6	−0.916565
$290$	−380531.	−0.265702
$291$	290579.	0.201155
$292$	918353.	0.630307
$293$	543998.	0.370193	0.185096	−	0.982720i	$-0.440740\pi$
$293$	543998.	0.370193	0.185096	+	0.982720i	$0.440740\pi$
$294$	366045.	0.246982
$295$	−984782.	−0.658847
$296$	−1.39433e6	−0.924991
$297$	477925.	0.314390
$298$	651379.	0.424906
$299$	493231.	0.319060
$300$	−333179.	−0.213735
$301$	−4.38466e6	−2.78946
$302$	−204678.	−0.129138
$303$	1.46910e6	0.919271
$304$	−351682.	−0.218256
$305$	1.06540e6	0.655785
$306$	56917.5	0.0347490
$307$	−2.32077e6	−1.40536	−0.702678	−	0.711508i	$-0.748013\pi$
$307$	−2.32077e6	−1.40536	−0.702678	+	0.711508i	$0.748013\pi$
$308$	3.49686e6	2.10040
$309$	616626.	0.367389
$310$	383584.	0.226702
$311$	−1.61095e6	−0.944455	−0.472227	−	0.881477i	$-0.656550\pi$
$311$	−1.61095e6	−0.944455	−0.472227	+	0.881477i	$0.656550\pi$
$312$	1.02503e6	0.596141
$313$	458852.	0.264735	0.132368	−	0.991201i	$-0.457742\pi$
$313$	458852.	0.264735	0.132368	+	0.991201i	$0.457742\pi$
$314$	−683695.	−0.391326
$315$	657666.	0.373447
$316$	586909.	0.330638
$317$	−1.32737e6	−0.741900	−0.370950	−	0.928653i	$-0.620968\pi$
$317$	−1.32737e6	−0.741900	−0.370950	+	0.928653i	$0.620968\pi$
$318$	83830.8	0.0464875
$319$	−2.88429e6	−1.58695
$320$	−418018.	−0.228202
$321$	183325.	0.0993025
$322$	206978.	0.111246
$323$	188766.	0.100674
$324$	−182606.	−0.0966388
$325$	1.24018e6	0.651295
$326$	−548104.	−0.285640
$327$	−2.21263e6	−1.14430
$328$	−813690.	−0.417614
$329$	3.43739e6	1.75081
$330$	−510337.	−0.257972
$331$	−1.53447e6	−0.769819	−0.384910	−	0.922954i	$-0.625767\pi$
$331$	−1.53447e6	−0.769819	−0.384910	+	0.922954i	$0.625767\pi$
$332$	−788503.	−0.392607
$333$	−924599.	−0.456923
$334$	511900.	0.251084
$335$	−263205.	−0.128139
$336$	−1.10603e6	−0.534465
$337$	−1.62882e6	−0.781265	−0.390632	−	0.920547i	$-0.627744\pi$
$337$	−1.62882e6	−0.781265	−0.390632	+	0.920547i	$0.627744\pi$
$338$	−1.01680e6	−0.484110
$339$	−2.32651e6	−1.09953
$340$	405842.	0.190397
$341$	2.90743e6	1.35401
$342$	90693.9	0.0419288
$343$	596925.	0.273958
$344$	−2.79468e6	−1.27331
$345$	201705.	0.0912364
$346$	−977088.	−0.438776
$347$	−645289.	−0.287694	−0.143847	−	0.989600i	$-0.545947\pi$
$347$	−645289.	−0.287694	−0.143847	+	0.989600i	$0.545947\pi$
$348$	1.10203e6	0.487804
$349$	3.55973e6	1.56442	0.782211	−	0.623013i	$-0.214092\pi$
$349$	3.55973e6	1.56442	0.782211	+	0.623013i	$0.214092\pi$
$350$	520426.	0.227085
$351$	679708.	0.294479
$352$	3.42086e6	1.47156
$353$	−1.45690e6	−0.622292	−0.311146	−	0.950362i	$-0.600713\pi$
$353$	−1.45690e6	−0.622292	−0.311146	+	0.950362i	$0.600713\pi$
$354$	−427100.	−0.181143
$355$	288119.	0.121339
$356$	−3.82937e6	−1.60141
$357$	593664.	0.246530
$358$	849932.	0.350491
$359$	−1.36735e6	−0.559942	−0.279971	−	0.960008i	$-0.590325\pi$
$359$	−1.36735e6	−0.559942	−0.279971	+	0.960008i	$0.590325\pi$
$360$	419181.	0.170469
$361$	−2.17531e6	−0.878525
$362$	1.05699e6	0.423933
$363$	−2.41872e6	−0.963426
$364$	4.97327e6	1.96738
$365$	−1.39792e6	−0.549226
$366$	462063.	0.180301
$367$	−469454.	−0.181940	−0.0909700	−	0.995854i	$-0.528997\pi$
$367$	−469454.	−0.181940	−0.0909700	+	0.995854i	$0.528997\pi$
$368$	−339217.	−0.130574
$369$	−539567.	−0.206291
$370$	987305.	0.374927
$371$	874377.	0.329810
$372$	−1.11087e6	−0.416204
$373$	3.72580e6	1.38659	0.693295	−	0.720654i	$-0.256159\pi$
$373$	3.72580e6	1.38659	0.693295	+	0.720654i	$0.256159\pi$
$374$	−460672.	−0.170299
$375$	1.69871e6	0.623794
$376$	2.19091e6	0.799198
$377$	−4.10206e6	−1.48644
$378$	285230.	0.102675
$379$	298202.	0.106638	0.0533191	−	0.998578i	$-0.483020\pi$
$379$	298202.	0.106638	0.0533191	+	0.998578i	$0.483020\pi$
$380$	646680.	0.229737
$381$	1.36295e6	0.481025
$382$	331047.	0.116073
$383$	−666922.	−0.232315	−0.116158	−	0.993231i	$-0.537058\pi$
$383$	−666922.	−0.232315	−0.116158	+	0.993231i	$0.537058\pi$
$384$	−1.68408e6	−0.582819
$385$	−5.32295e6	−1.83021
$386$	786904.	0.268815
$387$	−1.85318e6	−0.628986
$388$	898598.	0.303030
$389$	784906.	0.262993	0.131496	−	0.991317i	$-0.458022\pi$
$389$	784906.	0.262993	0.131496	+	0.991317i	$0.458022\pi$
$390$	−725805.	−0.241634
$391$	182075.	0.0602295
$392$	2.43346e6	0.799852
$393$	−488126.	−0.159423
$394$	1.60991e6	0.522469
$395$	−893397.	−0.288106
$396$	1.47795e6	0.473612
$397$	−940631.	−0.299532	−0.149766	−	0.988721i	$-0.547852\pi$
$397$	−940631.	−0.299532	−0.149766	+	0.988721i	$0.547852\pi$
$398$	1.13712e6	0.359830
$399$	945961.	0.297468
$400$	−852930.	−0.266540
$401$	−1.47321e6	−0.457514	−0.228757	−	0.973484i	$-0.573466\pi$
$401$	−1.47321e6	−0.457514	−0.228757	+	0.973484i	$0.573466\pi$
$402$	−114152.	−0.0352306
$403$	4.13497e6	1.26826
$404$	4.54309e6	1.38484
$405$	277963.	0.0842074
$406$	−1.72137e6	−0.518274
$407$	7.48342e6	2.23931
$408$	378387.	0.112535
$409$	2.05532e6	0.607533	0.303767	−	0.952746i	$-0.401756\pi$
$409$	2.05532e6	0.607533	0.303767	+	0.952746i	$0.401756\pi$
$410$	576160.	0.169271
$411$	−1.89744e6	−0.554070
$412$	1.90688e6	0.553453
$413$	−4.45477e6	−1.28514
$414$	87479.4	0.0250845
$415$	1.20026e6	0.342103
$416$	4.86517e6	1.37837
$417$	2.87238e6	0.808913
$418$	−734049.	−0.205487
$419$	−2.01044e6	−0.559442	−0.279721	−	0.960081i	$-0.590242\pi$
$419$	−2.01044e6	−0.559442	−0.279721	+	0.960081i	$0.590242\pi$
$420$	2.03379e6	0.562579
$421$	1.78234e6	0.490100	0.245050	−	0.969510i	$-0.421196\pi$
$421$	1.78234e6	0.490100	0.245050	+	0.969510i	$0.421196\pi$
$422$	−1.25967e6	−0.344331
$423$	1.45282e6	0.394784
$424$	557307.	0.150550
$425$	457811.	0.122946
$426$	124957.	0.0333609
$427$	4.81943e6	1.27916
$428$	566923.	0.149594
$429$	−5.50135e6	−1.44320
$430$	1.97886e6	0.516113
$431$	−5.62870e6	−1.45954	−0.729768	−	0.683695i	$-0.760372\pi$
$431$	−5.62870e6	−1.45954	−0.729768	+	0.683695i	$0.760372\pi$
$432$	−467465.	−0.120515
$433$	−432214.	−0.110785	−0.0553923	−	0.998465i	$-0.517641\pi$
$433$	−432214.	−0.110785	−0.0553923	+	0.998465i	$0.517641\pi$
$434$	1.73518e6	0.442203
$435$	−1.67752e6	−0.425054
$436$	−6.84243e6	−1.72383
$437$	290124.	0.0726741
$438$	−606280.	−0.151004
$439$	−2.99306e6	−0.741233	−0.370616	−	0.928786i	$-0.620854\pi$
$439$	−2.99306e6	−0.741233	−0.370616	+	0.928786i	$0.620854\pi$
$440$	−3.39272e6	−0.835442
$441$	1.61366e6	0.395107
$442$	−655172.	−0.159514
$443$	−1.63968e6	−0.396963	−0.198481	−	0.980105i	$-0.563601\pi$
$443$	−1.63968e6	−0.396963	−0.198481	+	0.980105i	$0.563601\pi$
$444$	−2.85927e6	−0.688331
$445$	5.82909e6	1.39541
$446$	−762822.	−0.181587
$447$	2.87151e6	0.679739
$448$	−1.89095e6	−0.445129
$449$	−3.99192e6	−0.934471	−0.467236	−	0.884133i	$-0.654750\pi$
$449$	−3.99192e6	−0.934471	−0.467236	+	0.884133i	$0.654750\pi$
$450$	219959.	0.0512047
$451$	4.36709e6	1.01100
$452$	−7.19460e6	−1.65638
$453$	−902297.	−0.206587
$454$	−1.12351e6	−0.255823
$455$	−7.57034e6	−1.71430
$456$	602933.	0.135787
$457$	−1.36122e6	−0.304887	−0.152444	−	0.988312i	$-0.548714\pi$
$457$	−1.36122e6	−0.304887	−0.152444	+	0.988312i	$0.548714\pi$
$458$	1.69456e6	0.377480
$459$	250913.	0.0555893
$460$	623760.	0.137443
$461$	5.00721e6	1.09734	0.548672	−	0.836037i	$-0.315133\pi$
$461$	5.00721e6	1.09734	0.548672	+	0.836037i	$0.315133\pi$
$462$	−2.30857e6	−0.503196
$463$	7.87132e6	1.70646	0.853228	−	0.521538i	$-0.174641\pi$
$463$	7.87132e6	1.70646	0.853228	+	0.521538i	$0.174641\pi$
$464$	2.82117e6	0.608322
$465$	1.69098e6	0.362665
$466$	−3.14423e6	−0.670732
$467$	−1.70051e6	−0.360817	−0.180409	−	0.983592i	$-0.557742\pi$
$467$	−1.70051e6	−0.360817	−0.180409	+	0.983592i	$0.557742\pi$
$468$	2.10196e6	0.443618
$469$	−1.19064e6	−0.249947
$470$	−1.55134e6	−0.323939
$471$	−3.01398e6	−0.626019
$472$	−2.83936e6	−0.586632
$473$	1.49991e7	3.08256
$474$	−387467.	−0.0792116
$475$	729490.	0.148349
$476$	1.83587e6	0.371385
$477$	369557.	0.0743678
$478$	−2.76674e6	−0.553858
$479$	4.15702e6	0.827834	0.413917	−	0.910315i	$-0.364160\pi$
$479$	4.15702e6	0.827834	0.413917	+	0.910315i	$0.364160\pi$
$480$	1.98959e6	0.394149
$481$	1.06430e7	2.09749
$482$	−2.45232e6	−0.480794
$483$	912433.	0.177964
$484$	−7.47974e6	−1.45135
$485$	−1.36785e6	−0.264049
$486$	120553.	0.0231519
$487$	−2.37037e6	−0.452891	−0.226445	−	0.974024i	$-0.572710\pi$
$487$	−2.37037e6	−0.452891	−0.226445	+	0.974024i	$0.572710\pi$
$488$	3.07179e6	0.583905
$489$	−2.41624e6	−0.456950
$490$	−1.72310e6	−0.324205
$491$	−5.11264e6	−0.957064	−0.478532	−	0.878070i	$-0.658831\pi$
$491$	−5.11264e6	−0.957064	−0.478532	+	0.878070i	$0.658831\pi$
$492$	−1.66858e6	−0.310767
$493$	−1.51427e6	−0.280598
$494$	−1.04397e6	−0.192473
$495$	−2.24975e6	−0.412688
$496$	−2.84380e6	−0.519033
$497$	1.30334e6	0.236682
$498$	520555.	0.0940575
$499$	1.81109e6	0.325603	0.162802	−	0.986659i	$-0.447947\pi$
$499$	1.81109e6	0.325603	0.162802	+	0.986659i	$0.447947\pi$
$500$	5.25317e6	0.939715
$501$	2.25664e6	0.401669
$502$	−1.06822e6	−0.189192
$503$	1.30925e6	0.230730	0.115365	−	0.993323i	$-0.463196\pi$
$503$	1.30925e6	0.230730	0.115365	+	0.993323i	$0.463196\pi$
$504$	1.89621e6	0.332514
$505$	−6.91553e6	−1.20669
$506$	−708032.	−0.122935
$507$	−4.48242e6	−0.774449
$508$	4.21485e6	0.724641
$509$	−6.70636e6	−1.14734	−0.573671	−	0.819086i	$-0.694481\pi$
$509$	−6.70636e6	−1.14734	−0.573671	+	0.819086i	$0.694481\pi$
$510$	−267929.	−0.0456137
$511$	−6.32366e6	−1.07131
$512$	−5.85251e6	−0.986659
$513$	399812.	0.0670752
$514$	−1.36330e6	−0.227606
$515$	−2.90267e6	−0.482257
$516$	−5.73086e6	−0.947535
$517$	−1.17586e7	−1.93478
$518$	4.46618e6	0.731328
$519$	−4.30736e6	−0.701928
$520$	−4.82515e6	−0.782533
$521$	6.03711e6	0.974394	0.487197	−	0.873292i	$-0.338019\pi$
$521$	6.03711e6	0.974394	0.487197	+	0.873292i	$0.338019\pi$
$522$	−727540.	−0.116864
$523$	−200223.	−0.0320081	−0.0160041	−	0.999872i	$-0.505094\pi$
$523$	−200223.	−0.0320081	−0.0160041	+	0.999872i	$0.505094\pi$
$524$	−1.50950e6	−0.240162
$525$	2.29423e6	0.363277
$526$	4.40773e6	0.694625
$527$	1.52642e6	0.239412
$528$	3.78352e6	0.590624
$529$	279841.	0.0434783
$530$	−394620.	−0.0610224
$531$	−1.88281e6	−0.289782
$532$	2.92533e6	0.448121
$533$	6.21091e6	0.946973
$534$	2.52808e6	0.383653
$535$	−862974.	−0.130351
$536$	−758885.	−0.114094
$537$	3.74681e6	0.560693
$538$	183363.	0.0273121
$539$	−1.30605e7	−1.93636
$540$	859586.	0.126854
$541$	−7.38505e6	−1.08483	−0.542413	−	0.840112i	$-0.682489\pi$
$541$	−7.38505e6	−1.08483	−0.542413	+	0.840112i	$0.682489\pi$
$542$	3.56209e6	0.520843
$543$	4.65958e6	0.678183
$544$	1.79597e6	0.260196
$545$	1.04156e7	1.50208
$546$	−3.28326e6	−0.471329
$547$	−5.16128e6	−0.737546	−0.368773	−	0.929519i	$-0.620222\pi$
$547$	−5.16128e6	−0.737546	−0.368773	+	0.929519i	$0.620222\pi$
$548$	−5.86774e6	−0.834678
$549$	2.03694e6	0.288435
$550$	−1.78028e6	−0.250947
$551$	−2.41287e6	−0.338576
$552$	581563.	0.0812361
$553$	−4.04138e6	−0.561975
$554$	−4.48935e6	−0.621454
$555$	4.35240e6	0.599786
$556$	8.88267e6	1.21859
$557$	−4.53224e6	−0.618978	−0.309489	−	0.950903i	$-0.600158\pi$
$557$	−4.53224e6	−0.618978	−0.309489	+	0.950903i	$0.600158\pi$
$558$	733378.	0.0997108
$559$	2.13318e7	2.88734
$560$	5.20646e6	0.701572
$561$	−2.03081e6	−0.272435
$562$	−1.74999e6	−0.233719
$563$	6.52075e6	0.867014	0.433507	−	0.901150i	$-0.357276\pi$
$563$	6.52075e6	0.867014	0.433507	+	0.901150i	$0.357276\pi$
$564$	4.49275e6	0.594723
$565$	1.09517e7	1.44331
$566$	−1.02256e6	−0.134168
$567$	1.25740e6	0.164254
$568$	830716.	0.108039
$569$	9.01923e6	1.16785	0.583927	−	0.811806i	$-0.301515\pi$
$569$	9.01923e6	1.16785	0.583927	+	0.811806i	$0.301515\pi$
$570$	−426926.	−0.0550384
$571$	−1.28625e7	−1.65095	−0.825476	−	0.564437i	$-0.809093\pi$
$571$	−1.28625e7	−1.65095	−0.825476	+	0.564437i	$0.809093\pi$
$572$	−1.70126e7	−2.17410
$573$	1.45937e6	0.185686
$574$	2.60633e6	0.330179
$575$	703634.	0.0887518
$576$	−799213.	−0.100371
$577$	−5.08026e6	−0.635252	−0.317626	−	0.948216i	$-0.602886\pi$
$577$	−5.08026e6	−0.635252	−0.317626	+	0.948216i	$0.602886\pi$
$578$	2.65689e6	0.330791
$579$	3.46896e6	0.430034
$580$	−5.18762e6	−0.640322
$581$	5.42953e6	0.667301
$582$	−593238.	−0.0725975
$583$	−2.99108e6	−0.364465
$584$	−4.03055e6	−0.489026
$585$	−3.19961e6	−0.386552
$586$	−1.11061e6	−0.133604
$587$	6.83820e6	0.819118	0.409559	−	0.912284i	$-0.365683\pi$
$587$	6.83820e6	0.819118	0.409559	+	0.912284i	$0.365683\pi$
$588$	4.99014e6	0.595209
$589$	2.43223e6	0.288880
$590$	2.01050e6	0.237780
$591$	7.09706e6	0.835814
$592$	−7.31965e6	−0.858393
$593$	−1.32165e7	−1.54341	−0.771705	−	0.635981i	$-0.780596\pi$
$593$	−1.32165e7	−1.54341	−0.771705	+	0.635981i	$0.780596\pi$
$594$	−975718.	−0.113464
$595$	−2.79457e6	−0.323611
$596$	8.87999e6	1.02399
$597$	5.01283e6	0.575635
$598$	−1.00697e6	−0.115150
$599$	−7.22555e6	−0.822818	−0.411409	−	0.911451i	$-0.634963\pi$
$599$	−7.22555e6	−0.822818	−0.411409	+	0.911451i	$0.634963\pi$
$600$	1.46229e6	0.165827
$601$	1.03172e7	1.16514	0.582568	−	0.812782i	$-0.302048\pi$
$601$	1.03172e7	1.16514	0.582568	+	0.812782i	$0.302048\pi$
$602$	8.95161e6	1.00672
$603$	−503225.	−0.0563597
$604$	−2.79030e6	−0.311214
$605$	1.13857e7	1.26465
$606$	−2.99927e6	−0.331767
$607$	1.41293e7	1.55650	0.778248	−	0.627957i	$-0.216109\pi$
$607$	1.41293e7	1.55650	0.778248	+	0.627957i	$0.216109\pi$
$608$	2.86175e6	0.313958
$609$	−7.58843e6	−0.829104
$610$	−2.17508e6	−0.236674
$611$	−1.67232e7	−1.81225
$612$	775933.	0.0837425
$613$	1.24624e7	1.33952	0.669760	−	0.742578i	$-0.266397\pi$
$613$	1.24624e7	1.33952	0.669760	+	0.742578i	$0.266397\pi$
$614$	4.73803e6	0.507197
$615$	2.53992e6	0.270790
$616$	−1.53473e7	−1.62960
$617$	5.63363e6	0.595765	0.297883	−	0.954602i	$-0.403720\pi$
$617$	5.63363e6	0.595765	0.297883	+	0.954602i	$0.403720\pi$
$618$	−1.25889e6	−0.132592
$619$	7.52500e6	0.789369	0.394684	−	0.918817i	$-0.370854\pi$
$619$	7.52500e6	0.789369	0.394684	+	0.918817i	$0.370854\pi$
$620$	5.22925e6	0.546336
$621$	385641.	0.0401286
$622$	3.28888e6	0.340856
$623$	2.63686e7	2.72186
$624$	5.38095e6	0.553220
$625$	−3.83977e6	−0.393193
$626$	−936780.	−0.0955437
$627$	−3.23595e6	−0.328725
$628$	−9.32055e6	−0.943066
$629$	3.92883e6	0.395947
$630$	−1.34267e6	−0.134778
$631$	−8.39408e6	−0.839267	−0.419633	−	0.907694i	$-0.637841\pi$
$631$	−8.39408e6	−0.839267	−0.419633	+	0.907694i	$0.637841\pi$
$632$	−2.57588e6	−0.256527
$633$	−5.55308e6	−0.550839
$634$	2.70993e6	0.267754
$635$	−6.41587e6	−0.631424
$636$	1.14283e6	0.112031
$637$	−1.85747e7	−1.81373
$638$	5.88849e6	0.572733
$639$	550857.	0.0533688
$640$	7.92751e6	0.765044
$641$	−8.76329e6	−0.842407	−0.421204	−	0.906966i	$-0.638392\pi$
$641$	−8.76329e6	−0.842407	−0.421204	+	0.906966i	$0.638392\pi$
$642$	−374272.	−0.0358385
$643$	−4.80058e6	−0.457896	−0.228948	−	0.973439i	$-0.573529\pi$
$643$	−4.80058e6	−0.457896	−0.228948	+	0.973439i	$0.573529\pi$
$644$	2.82165e6	0.268094
$645$	8.72355e6	0.825646
$646$	−385379.	−0.0363335
$647$	1.74670e7	1.64043	0.820217	−	0.572053i	$-0.193853\pi$
$647$	1.74670e7	1.64043	0.820217	+	0.572053i	$0.193853\pi$
$648$	801436.	0.0749775
$649$	1.52389e7	1.42018
$650$	−2.53193e6	−0.235054
$651$	7.64932e6	0.707409
$652$	−7.47209e6	−0.688372
$653$	−1.60101e7	−1.46930	−0.734651	−	0.678446i	$-0.762654\pi$
$653$	−1.60101e7	−1.46930	−0.734651	+	0.678446i	$0.762654\pi$
$654$	4.51725e6	0.412981
$655$	2.29777e6	0.209268
$656$	−4.27152e6	−0.387546
$657$	−2.67270e6	−0.241567
$658$	−7.01768e6	−0.631872
$659$	1.08861e7	0.976472	0.488236	−	0.872712i	$-0.337640\pi$
$659$	1.08861e7	0.976472	0.488236	+	0.872712i	$0.337640\pi$
$660$	−6.95722e6	−0.621693
$661$	−1.86000e7	−1.65580	−0.827902	−	0.560873i	$-0.810466\pi$
$661$	−1.86000e7	−1.65580	−0.827902	+	0.560873i	$0.810466\pi$
$662$	3.13274e6	0.277830
$663$	−2.88824e6	−0.255181
$664$	3.46065e6	0.304605
$665$	−4.45295e6	−0.390476
$666$	1.88764e6	0.164905
$667$	−2.32735e6	−0.202557
$668$	6.97853e6	0.605094
$669$	−3.36279e6	−0.290493
$670$	537353.	0.0462459
$671$	−1.64864e7	−1.41357
$672$	9.00012e6	0.768821
$673$	2.01949e7	1.71872	0.859358	−	0.511374i	$-0.170863\pi$
$673$	2.01949e7	1.71872	0.859358	+	0.511374i	$0.170863\pi$
$674$	3.32536e6	0.281961
$675$	969658.	0.0819142
$676$	−1.38616e7	−1.16667
$677$	−1.55789e7	−1.30637	−0.653184	−	0.757200i	$-0.726567\pi$
$677$	−1.55789e7	−1.30637	−0.653184	+	0.757200i	$0.726567\pi$
$678$	4.74974e6	0.396822
$679$	−6.18763e6	−0.515050
$680$	−1.78119e6	−0.147720
$681$	−4.95286e6	−0.409249
$682$	−5.93574e6	−0.488668
$683$	9.66927e6	0.793126	0.396563	−	0.918008i	$-0.370203\pi$
$683$	9.66927e6	0.793126	0.396563	+	0.918008i	$0.370203\pi$
$684$	1.23639e6	0.101045
$685$	8.93191e6	0.727307
$686$	−1.21867e6	−0.0988723
$687$	7.47025e6	0.603870
$688$	−1.46708e7	−1.18164
$689$	−4.25393e6	−0.341384
$690$	−411795.	−0.0329275
$691$	6.67472e6	0.531787	0.265894	−	0.964002i	$-0.414333\pi$
$691$	6.67472e6	0.531787	0.265894	+	0.964002i	$0.414333\pi$
$692$	−1.33202e7	−1.05742
$693$	−1.01770e7	−0.804983
$694$	1.31741e6	0.103830
$695$	−1.35213e7	−1.06183
$696$	−4.83669e6	−0.378464
$697$	2.29275e6	0.178762
$698$	−7.26746e6	−0.564604
$699$	−1.38609e7	−1.07300
$700$	7.09477e6	0.547259
$701$	−1.95292e6	−0.150103	−0.0750516	−	0.997180i	$-0.523912\pi$
$701$	−1.95292e6	−0.150103	−0.0750516	+	0.997180i	$0.523912\pi$
$702$	−1.38767e6	−0.106278
$703$	6.26031e6	0.477758
$704$	6.46859e6	0.491901
$705$	−6.83889e6	−0.518218
$706$	2.97438e6	0.224587
$707$	−3.12832e7	−2.35376
$708$	−5.82249e6	−0.436542
$709$	−1.62124e6	−0.121125	−0.0605623	−	0.998164i	$-0.519289\pi$
$709$	−1.62124e6	−0.121125	−0.0605623	+	0.998164i	$0.519289\pi$
$710$	−588216.	−0.0437916
$711$	−1.70809e6	−0.126718
$712$	1.68067e7	1.24246
$713$	2.34603e6	0.172826
$714$	−1.21201e6	−0.0889734
$715$	2.58967e7	1.89443
$716$	1.15868e7	0.844657
$717$	−1.21968e7	−0.886028
$718$	2.79154e6	0.202085
$719$	−6.94284e6	−0.500858	−0.250429	−	0.968135i	$-0.580572\pi$
$719$	−6.94284e6	−0.500858	−0.250429	+	0.968135i	$0.580572\pi$
$720$	2.20052e6	0.158195
$721$	−1.31305e7	−0.940685
$722$	4.44107e6	0.317062
$723$	−1.08107e7	−0.769145
$724$	1.44095e7	1.02165
$725$	−5.85192e6	−0.413479
$726$	4.93799e6	0.347703
$727$	−7.93350e6	−0.556710	−0.278355	−	0.960478i	$-0.589789\pi$
$727$	−7.93350e6	−0.556710	−0.278355	+	0.960478i	$0.589789\pi$
$728$	−2.18271e7	−1.52640
$729$	531441.	0.0370370
$730$	2.85396e6	0.198217
$731$	7.87460e6	0.545048
$732$	6.29912e6	0.434512
$733$	−364053.	−0.0250267	−0.0125134	−	0.999922i	$-0.503983\pi$
$733$	−364053.	−0.0250267	−0.0125134	+	0.999922i	$0.503983\pi$
$734$	958426.	0.0656627
$735$	−7.59603e6	−0.518643
$736$	2.76032e6	0.187830
$737$	4.07295e6	0.276211
$738$	1.10157e6	0.0744509
$739$	−5.17273e6	−0.348424	−0.174212	−	0.984708i	$-0.555738\pi$
$739$	−5.17273e6	−0.348424	−0.174212	+	0.984708i	$0.555738\pi$
$740$	1.34595e7	0.903547
$741$	−4.60220e6	−0.307907
$742$	−1.78511e6	−0.119029
$743$	−1.68796e7	−1.12173	−0.560866	−	0.827907i	$-0.689532\pi$
$743$	−1.68796e7	−1.12173	−0.560866	+	0.827907i	$0.689532\pi$
$744$	4.87549e6	0.322914
$745$	−1.35172e7	−0.892268
$746$	−7.60650e6	−0.500424
$747$	2.29480e6	0.150467
$748$	−6.28016e6	−0.410409
$749$	−3.90376e6	−0.254260
$750$	−3.46805e6	−0.225129
$751$	4.65656e6	0.301277	0.150638	−	0.988589i	$-0.451867\pi$
$751$	4.65656e6	0.301277	0.150638	+	0.988589i	$0.451867\pi$
$752$	1.15013e7	0.741657
$753$	−4.70911e6	−0.302658
$754$	8.37465e6	0.536461
$755$	4.24741e6	0.271180
$756$	3.88843e6	0.247440
$757$	−1.10890e6	−0.0703322	−0.0351661	−	0.999381i	$-0.511196\pi$
$757$	−1.10890e6	−0.0703322	−0.0351661	+	0.999381i	$0.511196\pi$
$758$	−608802.	−0.0384860
$759$	−3.12126e6	−0.196664
$760$	−2.83821e6	−0.178242
$761$	−1.61024e7	−1.00793	−0.503964	−	0.863725i	$-0.668126\pi$
$761$	−1.61024e7	−1.00793	−0.503964	+	0.863725i	$0.668126\pi$
$762$	−2.78257e6	−0.173603
$763$	4.71161e7	2.92993
$764$	4.51303e6	0.279727
$765$	−1.18113e6	−0.0729700
$766$	1.36157e6	0.0838432
$767$	2.16729e7	1.33024
$768$	596519.	0.0364940
$769$	−1.10052e7	−0.671091	−0.335545	−	0.942024i	$-0.608921\pi$
$769$	−1.10052e7	−0.671091	−0.335545	+	0.942024i	$0.608921\pi$
$770$	1.08672e7	0.660527
$771$	−6.00993e6	−0.364111
$772$	1.07275e7	0.647824
$773$	−1.33435e7	−0.803193	−0.401596	−	0.915817i	$-0.631545\pi$
$773$	−1.33435e7	−0.803193	−0.401596	+	0.915817i	$0.631545\pi$
$774$	3.78341e6	0.227003
$775$	5.89887e6	0.352789
$776$	−3.94384e6	−0.235107
$777$	1.96885e7	1.16993
$778$	−1.60244e6	−0.0949148
$779$	3.65333e6	0.215697
$780$	−9.89461e6	−0.582321
$781$	−4.45847e6	−0.261552
$782$	−371720.	−0.0217370
$783$	−3.20726e6	−0.186952
$784$	1.27746e7	0.742264
$785$	1.41878e7	0.821752
$786$	996545.	0.0575361
$787$	8.81426e6	0.507281	0.253641	−	0.967299i	$-0.418372\pi$
$787$	8.81426e6	0.507281	0.253641	+	0.967299i	$0.418372\pi$
$788$	2.19472e7	1.25911
$789$	1.94309e7	1.11122
$790$	1.82394e6	0.103978
$791$	4.95411e7	2.81530
$792$	−6.48657e6	−0.367453
$793$	−2.34470e7	−1.32405
$794$	1.92037e6	0.108102
$795$	−1.73963e6	−0.0976198
$796$	1.55019e7	0.867165
$797$	3.20515e6	0.178732	0.0893660	−	0.995999i	$-0.471516\pi$
$797$	3.20515e6	0.178732	0.0893660	+	0.995999i	$0.471516\pi$
$798$	−1.93125e6	−0.107357
$799$	−6.17335e6	−0.342101
$800$	6.94056e6	0.383415
$801$	1.11447e7	0.613744
$802$	3.00767e6	0.165118
$803$	2.16320e7	1.18388
$804$	−1.55619e6	−0.0849031
$805$	−4.29513e6	−0.233607
$806$	−8.44185e6	−0.457720
$807$	808330.	0.0436923
$808$	−1.99391e7	−1.07443
$809$	−2.41440e7	−1.29700	−0.648498	−	0.761216i	$-0.724603\pi$
$809$	−2.41440e7	−1.29700	−0.648498	+	0.761216i	$0.724603\pi$
$810$	−567483.	−0.0303907
$811$	−1.49460e7	−0.797946	−0.398973	−	0.916963i	$-0.630633\pi$
$811$	−1.49460e7	−0.797946	−0.398973	+	0.916963i	$0.630633\pi$
$812$	−2.34668e7	−1.24900
$813$	1.57030e7	0.833212
$814$	−1.52780e7	−0.808173
$815$	1.13741e7	0.599821
$816$	1.98637e6	0.104432
$817$	1.25476e7	0.657666
$818$	−4.19608e6	−0.219261
$819$	−1.44738e7	−0.754003
$820$	7.85457e6	0.407932
$821$	−3.27314e7	−1.69476	−0.847378	−	0.530991i	$-0.821820\pi$
$821$	−3.27314e7	−1.69476	−0.847378	+	0.530991i	$0.821820\pi$
$822$	3.87377e6	0.199965
$823$	4.46307e6	0.229686	0.114843	−	0.993384i	$-0.463364\pi$
$823$	4.46307e6	0.229686	0.114843	+	0.993384i	$0.463364\pi$
$824$	−8.36909e6	−0.429398
$825$	−7.84812e6	−0.401449
$826$	9.09474e6	0.463810
$827$	−2.80337e7	−1.42533	−0.712667	−	0.701502i	$-0.752513\pi$
$827$	−2.80337e7	−1.42533	−0.712667	+	0.701502i	$0.752513\pi$
$828$	1.19257e6	0.0604517
$829$	−3.09291e6	−0.156308	−0.0781540	−	0.996941i	$-0.524903\pi$
$829$	−3.09291e6	−0.156308	−0.0781540	+	0.996941i	$0.524903\pi$
$830$	−2.45043e6	−0.123466
$831$	−1.97907e7	−0.994164
$832$	9.19967e6	0.460749
$833$	−6.85680e6	−0.342381
$834$	−5.86418e6	−0.291939
$835$	−1.06228e7	−0.527256
$836$	−1.00070e7	−0.495208
$837$	3.23300e6	0.159511
$838$	4.10446e6	0.201904
$839$	1.38702e7	0.680263	0.340131	−	0.940378i	$-0.389528\pi$
$839$	1.38702e7	0.680263	0.340131	+	0.940378i	$0.389528\pi$
$840$	−8.92609e6	−0.436479
$841$	−1.15523e6	−0.0563222
$842$	−3.63878e6	−0.176879
$843$	−7.71457e6	−0.373889
$844$	−1.71726e7	−0.829812
$845$	2.11003e7	1.01659
$846$	−2.96603e6	−0.142479
$847$	5.15045e7	2.46682
$848$	2.92562e6	0.139710
$849$	−4.50784e6	−0.214634
$850$	−934656.	−0.0443715
$851$	6.03843e6	0.285825
$852$	1.70349e6	0.0803974
$853$	−4.04588e7	−1.90388	−0.951942	−	0.306278i	$-0.900916\pi$
$853$	−4.04588e7	−1.90388	−0.951942	+	0.306278i	$0.900916\pi$
$854$	−9.83923e6	−0.461654
$855$	−1.88205e6	−0.0880471
$856$	−2.48816e6	−0.116063
$857$	1.82715e6	0.0849808	0.0424904	−	0.999097i	$-0.486471\pi$
$857$	1.82715e6	0.0849808	0.0424904	+	0.999097i	$0.486471\pi$
$858$	1.12314e7	0.520854
$859$	2.08415e7	0.963710	0.481855	−	0.876251i	$-0.339963\pi$
$859$	2.08415e7	0.963710	0.481855	+	0.876251i	$0.339963\pi$
$860$	2.69771e7	1.24379
$861$	1.14896e7	0.528200
$862$	1.14914e7	0.526750
$863$	4.06174e7	1.85646	0.928230	−	0.372007i	$-0.121330\pi$
$863$	4.06174e7	1.85646	0.928230	+	0.372007i	$0.121330\pi$
$864$	3.80391e6	0.173359
$865$	2.02762e7	0.921394
$866$	882397.	0.0399824
$867$	1.17125e7	0.529179
$868$	2.36551e7	1.06568
$869$	1.38248e7	0.621025
$870$	3.42478e6	0.153403
$871$	5.79258e6	0.258718
$872$	3.00307e7	1.33744
$873$	−2.61521e6	−0.116137
$874$	−592309.	−0.0262283
$875$	−3.61726e7	−1.59720
$876$	−8.26518e6	−0.363908
$877$	−2.96392e7	−1.30127	−0.650636	−	0.759390i	$-0.725498\pi$
$877$	−2.96392e7	−1.30127	−0.650636	+	0.759390i	$0.725498\pi$
$878$	6.11056e6	0.267513
$879$	−4.89598e6	−0.213731
$880$	−1.78103e7	−0.775291
$881$	−3.53245e7	−1.53333	−0.766665	−	0.642047i	$-0.778085\pi$
$881$	−3.53245e7	−1.53333	−0.766665	+	0.642047i	$0.778085\pi$
$882$	−3.29440e6	−0.142595
$883$	2.32881e7	1.00515	0.502576	−	0.864533i	$-0.332386\pi$
$883$	2.32881e7	1.00515	0.502576	+	0.864533i	$0.332386\pi$
$884$	−8.93170e6	−0.384418
$885$	8.86303e6	0.380386
$886$	3.34753e6	0.143265
$887$	1.05116e7	0.448600	0.224300	−	0.974520i	$-0.427991\pi$
$887$	1.05116e7	0.448600	0.224300	+	0.974520i	$0.427991\pi$
$888$	1.25490e7	0.534044
$889$	−2.90229e7	−1.23165
$890$	−1.19005e7	−0.503606
$891$	−4.30132e6	−0.181513
$892$	−1.03992e7	−0.437612
$893$	−9.83679e6	−0.412786
$894$	−5.86241e6	−0.245320
$895$	−1.76375e7	−0.736001
$896$	3.58610e7	1.49229
$897$	−4.43908e6	−0.184209
$898$	8.14980e6	0.337253
$899$	−1.95112e7	−0.805166
$900$	2.99861e6	0.123400
$901$	−1.57033e6	−0.0644435
$902$	−8.91575e6	−0.364873
$903$	3.94619e7	1.61049
$904$	3.15763e7	1.28511
$905$	−2.19342e7	−0.890225
$906$	1.84211e6	0.0745580
$907$	3.13327e7	1.26468	0.632339	−	0.774692i	$-0.282095\pi$
$907$	3.13327e7	1.26468	0.632339	+	0.774692i	$0.282095\pi$
$908$	−1.53164e7	−0.616514
$909$	−1.32219e7	−0.530741
$910$	1.54554e7	0.618695
$911$	−2.74109e7	−1.09428	−0.547139	−	0.837042i	$-0.684283\pi$
$911$	−2.74109e7	−1.09428	−0.547139	+	0.837042i	$0.684283\pi$
$912$	3.16513e6	0.126010
$913$	−1.85734e7	−0.737418
$914$	2.77904e6	0.110035
$915$	−9.58856e6	−0.378617
$916$	2.31013e7	0.909699
$917$	1.03942e7	0.408196
$918$	−512257.	−0.0200623
$919$	−6.27955e6	−0.245267	−0.122634	−	0.992452i	$-0.539134\pi$
$919$	−6.27955e6	−0.245267	−0.122634	+	0.992452i	$0.539134\pi$
$920$	−2.73761e6	−0.106636
$921$	2.08869e7	0.811383
$922$	−1.02226e7	−0.396035
$923$	−6.34087e6	−0.244988
$924$	−3.14718e7	−1.21267
$925$	1.51831e7	0.583452
$926$	−1.60699e7	−0.615865
$927$	−5.54964e6	−0.212112
$928$	−2.29567e7	−0.875065
$929$	6.33332e6	0.240764	0.120382	−	0.992728i	$-0.461588\pi$
$929$	6.33332e6	0.240764	0.120382	+	0.992728i	$0.461588\pi$
$930$	−3.45225e6	−0.130887
$931$	−1.09258e7	−0.413124
$932$	−4.28640e7	−1.61641
$933$	1.44986e7	0.545281
$934$	3.47172e6	0.130220
$935$	9.55971e6	0.357615
$936$	−9.22525e6	−0.344182
$937$	−2.00147e7	−0.744732	−0.372366	−	0.928086i	$-0.621453\pi$
$937$	−2.00147e7	−0.744732	−0.372366	+	0.928086i	$0.621453\pi$
$938$	2.43078e6	0.0902066
$939$	−4.12967e6	−0.152845
$940$	−2.11489e7	−0.780670
$941$	2.61878e7	0.964106	0.482053	−	0.876142i	$-0.339891\pi$
$941$	2.61878e7	0.964106	0.482053	+	0.876142i	$0.339891\pi$
$942$	6.15326e6	0.225932
$943$	3.52384e6	0.129044
$944$	−1.49054e7	−0.544395
$945$	−5.91899e6	−0.215610
$946$	−3.06218e7	−1.11251
$947$	−2.31722e7	−0.839640	−0.419820	−	0.907607i	$-0.637907\pi$
$947$	−2.31722e7	−0.839640	−0.419820	+	0.907607i	$0.637907\pi$
$948$	−5.28218e6	−0.190894
$949$	3.07652e7	1.10891
$950$	−1.48931e6	−0.0535396
$951$	1.19464e7	0.428336
$952$	−8.05743e6	−0.288140
$953$	2.57957e7	0.920057	0.460028	−	0.887904i	$-0.347839\pi$
$953$	2.57957e7	0.920057	0.460028	+	0.887904i	$0.347839\pi$
$954$	−754477.	−0.0268395
$955$	−6.86976e6	−0.243743
$956$	−3.77179e7	−1.33476
$957$	2.59586e7	0.916223
$958$	−8.48687e6	−0.298768
$959$	4.04045e7	1.41867
$960$	3.76216e6	0.131753
$961$	−8.96138e6	−0.313016
$962$	−2.17284e7	−0.756991
$963$	−1.64993e6	−0.0573323
$964$	−3.34315e7	−1.15868
$965$	−1.63295e7	−0.564489
$966$	−1.86280e6	−0.0642279
$967$	−1.19166e7	−0.409814	−0.204907	−	0.978781i	$-0.565689\pi$
$967$	−1.19166e7	−0.409814	−0.204907	+	0.978781i	$0.565689\pi$
$968$	3.28277e7	1.12604
$969$	−1.69889e6	−0.0581241
$970$	2.79257e6	0.0952960
$971$	−2.11925e7	−0.721331	−0.360666	−	0.932695i	$-0.617450\pi$
$971$	−2.11925e7	−0.721331	−0.360666	+	0.932695i	$0.617450\pi$
$972$	1.64345e6	0.0557944
$973$	−6.11649e7	−2.07119
$974$	4.83928e6	0.163449
$975$	−1.11617e7	−0.376025
$976$	1.61256e7	0.541864
$977$	4.08942e7	1.37064	0.685322	−	0.728240i	$-0.259661\pi$
$977$	4.08942e7	1.37064	0.685322	+	0.728240i	$0.259661\pi$
$978$	4.93294e6	0.164914
$979$	−9.02018e7	−3.00787
$980$	−2.34903e7	−0.781309
$981$	1.99137e7	0.660661
$982$	1.04378e7	0.345407
$983$	3.71260e7	1.22545	0.612724	−	0.790297i	$-0.290074\pi$
$983$	3.71260e7	1.22545	0.612724	+	0.790297i	$0.290074\pi$
$984$	7.32321e6	0.241109
$985$	−3.34082e7	−1.09714
$986$	3.09149e6	0.101269
$987$	−3.09365e7	−1.01083
$988$	−1.42320e7	−0.463846
$989$	1.21029e7	0.393458
$990$	4.59303e6	0.148940
$991$	3.44781e7	1.11522	0.557609	−	0.830104i	$-0.311719\pi$
$991$	3.44781e7	1.11522	0.557609	+	0.830104i	$0.311719\pi$
$992$	2.31409e7	0.746623
$993$	1.38102e7	0.444455
$994$	−2.66086e6	−0.0854194
$995$	−2.35971e7	−0.755614
$996$	7.09652e6	0.226672
$997$	624230.	0.0198887	0.00994435	−	0.999951i	$-0.496835\pi$
$997$	624230.	0.0198887	0.00994435	+	0.999951i	$0.496835\pi$
$998$	−3.69748e6	−0.117511
$999$	8.32139e6	0.263804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.d.1.2	✓	4
3.2	odd	2		207.6.a.e.1.3		4
4.3	odd	2		1104.6.a.o.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.2	✓	4	1.1	even	1	trivial
207.6.a.e.1.3		4	3.2	odd	2
1104.6.a.o.1.3		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)

\(f(q)\)	\(=\)	\(q-2.04157 q^{2} -9.00000 q^{3} -27.8320 q^{4} +42.3660 q^{5} +18.3742 q^{6} +191.647 q^{7} +122.151 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-2.04157 q^{2} -9.00000 q^{3} -27.8320 q^{4} +42.3660 q^{5} +18.3742 q^{6} +191.647 q^{7} +122.151 q^{8} +81.0000 q^{9} -86.4934 q^{10} -655.589 q^{11} +250.488 q^{12} -932.384 q^{13} -391.262 q^{14} -381.294 q^{15} +641.242 q^{16} -344.188 q^{17} -165.368 q^{18} -548.438 q^{19} -1179.13 q^{20} -1724.83 q^{21} +1338.43 q^{22} -529.000 q^{23} -1099.36 q^{24} -1330.12 q^{25} +1903.53 q^{26} -729.000 q^{27} -5333.92 q^{28} +4399.54 q^{29} +778.440 q^{30} -4434.84 q^{31} -5217.99 q^{32} +5900.30 q^{33} +702.685 q^{34} +8119.33 q^{35} -2254.39 q^{36} -11414.8 q^{37} +1119.68 q^{38} +8391.46 q^{39} +5175.07 q^{40} -6661.33 q^{41} +3521.36 q^{42} -22878.8 q^{43} +18246.3 q^{44} +3431.65 q^{45} +1079.99 q^{46} +17936.0 q^{47} -5771.18 q^{48} +19921.7 q^{49} +2715.54 q^{50} +3097.69 q^{51} +25950.1 q^{52} +4562.43 q^{53} +1488.31 q^{54} -27774.7 q^{55} +23410.0 q^{56} +4935.95 q^{57} -8981.98 q^{58} -23244.6 q^{59} +10612.2 q^{60} +25147.4 q^{61} +9054.05 q^{62} +15523.4 q^{63} -9866.83 q^{64} -39501.4 q^{65} -12045.9 q^{66} -6212.65 q^{67} +9579.42 q^{68} +4761.00 q^{69} -16576.2 q^{70} +6800.71 q^{71} +9894.27 q^{72} -32996.3 q^{73} +23304.2 q^{74} +11971.1 q^{75} +15264.1 q^{76} -125642. q^{77} -17131.8 q^{78} -21087.6 q^{79} +27166.9 q^{80} +6561.00 q^{81} +13599.6 q^{82} +28330.8 q^{83} +48005.3 q^{84} -14581.9 q^{85} +46708.8 q^{86} -39595.8 q^{87} -80081.2 q^{88} +137589. q^{89} -7005.96 q^{90} -178689. q^{91} +14723.1 q^{92} +39913.5 q^{93} -36617.7 q^{94} -23235.1 q^{95} +46961.9 q^{96} -32286.5 q^{97} -40671.6 q^{98} -53102.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9	\( 4 q + 4 q^{2} - 36 q^{3} + 26 q^{4} + 22 q^{5} - 36 q^{6} - 62 q^{7} + 72 q^{8} + 324 q^{9} - 496 q^{10} - 1076 q^{11} - 234 q^{12} - 396 q^{13} - 1806 q^{14} - 198 q^{15} - 1982 q^{16} + 70 q^{17} + 324 q^{18} - 6366 q^{19} - 5240 q^{20} + 558 q^{21} - 6974 q^{22} - 2116 q^{23} - 648 q^{24} + 1264 q^{25} + 2464 q^{26} - 2916 q^{27} - 6474 q^{28} + 3948 q^{29} + 4464 q^{30} + 3092 q^{31} - 3672 q^{32} + 9684 q^{33} + 11682 q^{34} + 1304 q^{35} + 2106 q^{36} - 17464 q^{37} - 12628 q^{38} + 3564 q^{39} - 14108 q^{40} + 18680 q^{41} + 16254 q^{42} - 25846 q^{43} + 20746 q^{44} + 1782 q^{45} - 2116 q^{46} + 18392 q^{47} + 17838 q^{48} + 7952 q^{49} + 69444 q^{50} - 630 q^{51} + 8844 q^{52} - 26518 q^{53} - 2916 q^{54} - 40848 q^{55} + 54890 q^{56} + 57294 q^{57} + 568 q^{58} - 14520 q^{59} + 47160 q^{60} - 13688 q^{61} + 120136 q^{62} - 5022 q^{63} - 30190 q^{64} + 38324 q^{65} + 62766 q^{66} - 11098 q^{67} + 112138 q^{68} + 19044 q^{69} - 29596 q^{70} - 57496 q^{71} + 5832 q^{72} - 112272 q^{73} - 21226 q^{74} - 11376 q^{75} - 76240 q^{76} - 4792 q^{77} - 22176 q^{78} - 240754 q^{79} + 41200 q^{80} + 26244 q^{81} + 49976 q^{82} - 93268 q^{83} + 58266 q^{84} - 323204 q^{85} - 88224 q^{86} - 35532 q^{87} + 42382 q^{88} - 107582 q^{89} - 40176 q^{90} - 301532 q^{91} - 13754 q^{92} - 27828 q^{93} + 79360 q^{94} - 18640 q^{95} + 33048 q^{96} - 53076 q^{97} + 59664 q^{98} - 87156 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 36 * q^3 + 26 * q^4 + 22 * q^5 - 36 * q^6 - 62 * q^7 + 72 * q^8 + 324 * q^9 - 496 * q^10 - 1076 * q^11 - 234 * q^12 - 396 * q^13 - 1806 * q^14 - 198 * q^15 - 1982 * q^16 + 70 * q^17 + 324 * q^18 - 6366 * q^19 - 5240 * q^20 + 558 * q^21 - 6974 * q^22 - 2116 * q^23 - 648 * q^24 + 1264 * q^25 + 2464 * q^26 - 2916 * q^27 - 6474 * q^28 + 3948 * q^29 + 4464 * q^30 + 3092 * q^31 - 3672 * q^32 + 9684 * q^33 + 11682 * q^34 + 1304 * q^35 + 2106 * q^36 - 17464 * q^37 - 12628 * q^38 + 3564 * q^39 - 14108 * q^40 + 18680 * q^41 + 16254 * q^42 - 25846 * q^43 + 20746 * q^44 + 1782 * q^45 - 2116 * q^46 + 18392 * q^47 + 17838 * q^48 + 7952 * q^49 + 69444 * q^50 - 630 * q^51 + 8844 * q^52 - 26518 * q^53 - 2916 * q^54 - 40848 * q^55 + 54890 * q^56 + 57294 * q^57 + 568 * q^58 - 14520 * q^59 + 47160 * q^60 - 13688 * q^61 + 120136 * q^62 - 5022 * q^63 - 30190 * q^64 + 38324 * q^65 + 62766 * q^66 - 11098 * q^67 + 112138 * q^68 + 19044 * q^69 - 29596 * q^70 - 57496 * q^71 + 5832 * q^72 - 112272 * q^73 - 21226 * q^74 - 11376 * q^75 - 76240 * q^76 - 4792 * q^77 - 22176 * q^78 - 240754 * q^79 + 41200 * q^80 + 26244 * q^81 + 49976 * q^82 - 93268 * q^83 + 58266 * q^84 - 323204 * q^85 - 88224 * q^86 - 35532 * q^87 + 42382 * q^88 - 107582 * q^89 - 40176 * q^90 - 301532 * q^91 - 13754 * q^92 - 27828 * q^93 + 79360 * q^94 - 18640 * q^95 + 33048 * q^96 - 53076 * q^97 + 59664 * q^98 - 87156 * q^99

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