LMFDB - Embedded newform 69.6.a.e.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.40352$ 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)

$f(q)$	$=$	$q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +659.300 q^{10} -436.763 q^{11} +498.647 q^{12} -258.865 q^{13} -838.736 q^{14} +634.683 q^{15} +272.775 q^{16} +591.819 q^{17} +757.275 q^{18} +1663.69 q^{19} +3907.20 q^{20} -807.419 q^{21} -4083.33 q^{22} -529.000 q^{23} +1969.36 q^{24} +1848.12 q^{25} -2420.15 q^{26} +729.000 q^{27} -4970.58 q^{28} -5121.48 q^{29} +5933.70 q^{30} +1871.29 q^{31} -4451.97 q^{32} -3930.87 q^{33} +5532.96 q^{34} -6326.60 q^{35} +4487.83 q^{36} +884.534 q^{37} +15554.0 q^{38} -2329.78 q^{39} +15431.1 q^{40} -18410.5 q^{41} -7548.62 q^{42} +12334.9 q^{43} -24199.0 q^{44} +5712.15 q^{45} -4945.66 q^{46} +23871.6 q^{47} +2454.98 q^{48} -8758.55 q^{49} +17278.2 q^{50} +5326.37 q^{51} -14342.5 q^{52} -14594.8 q^{53} +6815.48 q^{54} -30800.7 q^{55} -19630.8 q^{56} +14973.2 q^{57} -47881.1 q^{58} +47739.1 q^{59} +35164.8 q^{60} -15351.5 q^{61} +17494.8 q^{62} -7266.77 q^{63} -50350.6 q^{64} -18255.2 q^{65} -36750.0 q^{66} -41174.4 q^{67} +32789.9 q^{68} -4761.00 q^{69} -59147.9 q^{70} -45608.3 q^{71} +17724.2 q^{72} -7898.24 q^{73} +8269.57 q^{74} +16633.1 q^{75} +92177.1 q^{76} +39183.4 q^{77} -21781.3 q^{78} +71435.2 q^{79} +19236.2 q^{80} +6561.00 q^{81} -172121. q^{82} +112580. q^{83} -44735.2 q^{84} +41735.3 q^{85} +115320. q^{86} -46093.3 q^{87} -95571.5 q^{88} +137845. q^{89} +53403.3 q^{90} +23223.6 q^{91} -29309.4 q^{92} +16841.6 q^{93} +223177. q^{94} +117324. q^{95} -40067.7 q^{96} +41679.8 q^{97} -81884.3 q^{98} -35377.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * 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 $

$2$	9.34908	1.65270	0.826350	−	0.563157i	$-0.190414\pi$
$2$	9.34908	1.65270	0.826350	+	0.563157i	$0.190414\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	55.4053	1.73141
$5$	70.5203	1.26151	0.630753	−	0.775984i	$-0.282746\pi$
$5$	70.5203	1.26151	0.630753	+	0.775984i	$0.282746\pi$
$6$	84.1417	0.954186
$7$	−89.7132	−0.692008	−0.346004	−	0.938233i	$-0.612462\pi$
$7$	−89.7132	−0.692008	−0.346004	+	0.938233i	$0.612462\pi$
$8$	218.818	1.20881
$9$	81.0000	0.333333
$10$	659.300	2.08489
$11$	−436.763	−1.08834	−0.544170	−	0.838975i	$-0.683155\pi$
$11$	−436.763	−1.08834	−0.544170	+	0.838975i	$0.683155\pi$
$12$	498.647	0.999633
$13$	−258.865	−0.424830	−0.212415	−	0.977180i	$-0.568133\pi$
$13$	−258.865	−0.424830	−0.212415	+	0.977180i	$0.568133\pi$
$14$	−838.736	−1.14368
$15$	634.683	0.728331
$16$	272.775	0.266382
$17$	591.819	0.496669	0.248334	−	0.968674i	$-0.420117\pi$
$17$	591.819	0.496669	0.248334	+	0.968674i	$0.420117\pi$
$18$	757.275	0.550900
$19$	1663.69	1.05727	0.528637	−	0.848848i	$-0.322703\pi$
$19$	1663.69	1.05727	0.528637	+	0.848848i	$0.322703\pi$
$20$	3907.20	2.18419
$21$	−807.419	−0.399531
$22$	−4083.33	−1.79870
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	1969.36	0.697906
$25$	1848.12	0.591398
$26$	−2420.15	−0.702116
$27$	729.000	0.192450
$28$	−4970.58	−1.19815
$29$	−5121.48	−1.13084	−0.565419	−	0.824804i	$-0.691286\pi$
$29$	−5121.48	−1.13084	−0.565419	+	0.824804i	$0.691286\pi$
$30$	5933.70	1.20371
$31$	1871.29	0.349733	0.174867	−	0.984592i	$-0.444051\pi$
$31$	1871.29	0.349733	0.174867	+	0.984592i	$0.444051\pi$
$32$	−4451.97	−0.768559
$33$	−3930.87	−0.628353
$34$	5532.96	0.820844
$35$	−6326.60	−0.872972
$36$	4487.83	0.577138
$37$	884.534	0.106221	0.0531105	−	0.998589i	$-0.483086\pi$
$37$	884.534	0.106221	0.0531105	+	0.998589i	$0.483086\pi$
$38$	15554.0	1.74736
$39$	−2329.78	−0.245275
$40$	15431.1	1.52492
$41$	−18410.5	−1.71043	−0.855216	−	0.518271i	$-0.826576\pi$
$41$	−18410.5	−1.71043	−0.855216	+	0.518271i	$0.826576\pi$
$42$	−7548.62	−0.660305
$43$	12334.9	1.01734	0.508669	−	0.860962i	$-0.330138\pi$
$43$	12334.9	1.01734	0.508669	+	0.860962i	$0.330138\pi$
$44$	−24199.0	−1.88437
$45$	5712.15	0.420502
$46$	−4945.66	−0.344612
$47$	23871.6	1.57629	0.788146	−	0.615489i	$-0.211041\pi$
$47$	23871.6	1.57629	0.788146	+	0.615489i	$0.211041\pi$
$48$	2454.98	0.153796
$49$	−8758.55	−0.521125
$50$	17278.2	0.977403
$51$	5326.37	0.286752
$52$	−14342.5	−0.735556
$53$	−14594.8	−0.713690	−0.356845	−	0.934164i	$-0.616147\pi$
$53$	−14594.8	−0.713690	−0.356845	+	0.934164i	$0.616147\pi$
$54$	6815.48	0.318062
$55$	−30800.7	−1.37295
$56$	−19630.8	−0.836505
$57$	14973.2	0.610418
$58$	−47881.1	−1.86894
$59$	47739.1	1.78544	0.892718	−	0.450616i	$-0.148795\pi$
$59$	47739.1	1.78544	0.892718	+	0.450616i	$0.148795\pi$
$60$	35164.8	1.26104
$61$	−15351.5	−0.528233	−0.264116	−	0.964491i	$-0.585080\pi$
$61$	−15351.5	−0.528233	−0.264116	+	0.964491i	$0.585080\pi$
$62$	17494.8	0.578004
$63$	−7266.77	−0.230669
$64$	−50350.6	−1.53658
$65$	−18255.2	−0.535925
$66$	−36750.0	−1.03848
$67$	−41174.4	−1.12057	−0.560287	−	0.828299i	$-0.689309\pi$
$67$	−41174.4	−1.12057	−0.560287	+	0.828299i	$0.689309\pi$
$68$	32789.9	0.859939
$69$	−4761.00	−0.120386
$70$	−59147.9	−1.44276
$71$	−45608.3	−1.07374	−0.536869	−	0.843666i	$-0.680393\pi$
$71$	−45608.3	−1.07374	−0.536869	+	0.843666i	$0.680393\pi$
$72$	17724.2	0.402936
$73$	−7898.24	−0.173470	−0.0867348	−	0.996231i	$-0.527643\pi$
$73$	−7898.24	−0.173470	−0.0867348	+	0.996231i	$0.527643\pi$
$74$	8269.57	0.175551
$75$	16633.1	0.341444
$76$	92177.1	1.83058
$77$	39183.4	0.753140
$78$	−21781.3	−0.405367
$79$	71435.2	1.28779	0.643894	−	0.765115i	$-0.277318\pi$
$79$	71435.2	1.28779	0.643894	+	0.765115i	$0.277318\pi$
$80$	19236.2	0.336043
$81$	6561.00	0.111111
$82$	−172121.	−2.82683
$83$	112580.	1.79377	0.896885	−	0.442264i	$-0.145825\pi$
$83$	112580.	1.79377	0.896885	+	0.442264i	$0.145825\pi$
$84$	−44735.2	−0.691754
$85$	41735.3	0.626551
$86$	115320.	1.68136
$87$	−46093.3	−0.652890
$88$	−95571.5	−1.31559
$89$	137845.	1.84466	0.922330	−	0.386404i	$-0.126283\pi$
$89$	137845.	1.84466	0.922330	+	0.386404i	$0.126283\pi$
$90$	53403.3	0.694963
$91$	23223.6	0.293985
$92$	−29309.4	−0.361025
$93$	16841.6	0.201918
$94$	223177.	2.60514
$95$	117324.	1.33376
$96$	−40067.7	−0.443728
$97$	41679.8	0.449776	0.224888	−	0.974385i	$-0.427798\pi$
$97$	41679.8	0.449776	0.224888	+	0.974385i	$0.427798\pi$
$98$	−81884.3	−0.861263
$99$	−35377.8	−0.362780
$100$	102395.	1.02395
$101$	37100.3	0.361888	0.180944	−	0.983493i	$-0.442085\pi$
$101$	37100.3	0.361888	0.180944	+	0.983493i	$0.442085\pi$
$102$	49796.7	0.473914
$103$	−123323.	−1.14538	−0.572692	−	0.819771i	$-0.694101\pi$
$103$	−123323.	−1.14538	−0.572692	+	0.819771i	$0.694101\pi$
$104$	−56644.2	−0.513538
$105$	−56939.4	−0.504011
$106$	−136448.	−1.17951
$107$	107455.	0.907338	0.453669	−	0.891170i	$-0.350115\pi$
$107$	107455.	0.907338	0.453669	+	0.891170i	$0.350115\pi$
$108$	40390.4	0.333211
$109$	96652.7	0.779198	0.389599	−	0.920985i	$-0.372614\pi$
$109$	96652.7	0.779198	0.389599	+	0.920985i	$0.372614\pi$
$110$	−287958.	−2.26907
$111$	7960.80	0.0613267
$112$	−24471.5	−0.184338
$113$	55788.7	0.411008	0.205504	−	0.978656i	$-0.434117\pi$
$113$	55788.7	0.411008	0.205504	+	0.978656i	$0.434117\pi$
$114$	139986.	1.00884
$115$	−37305.3	−0.263042
$116$	−283757.	−1.95795
$117$	−20968.1	−0.141610
$118$	446317.	2.95079
$119$	−53094.0	−0.343699
$120$	138880.	0.880413
$121$	29711.1	0.184483
$122$	−143522.	−0.873010
$123$	−165695.	−0.987519
$124$	103679.	0.605533
$125$	−90046.2	−0.515454
$126$	−67937.6	−0.381227
$127$	167839.	0.923388	0.461694	−	0.887039i	$-0.347242\pi$
$127$	167839.	0.923388	0.461694	+	0.887039i	$0.347242\pi$
$128$	−328269.	−1.77094
$129$	111014.	0.587361
$130$	−170670.	−0.885723
$131$	129729.	0.660476	0.330238	−	0.943898i	$-0.392871\pi$
$131$	129729.	0.660476	0.330238	+	0.943898i	$0.392871\pi$
$132$	−217791.	−1.08794
$133$	−149255.	−0.731643
$134$	−384943.	−1.85197
$135$	51409.3	0.242777
$136$	129501.	0.600377
$137$	−226702.	−1.03194	−0.515970	−	0.856607i	$-0.672568\pi$
$137$	−226702.	−1.03194	−0.515970	+	0.856607i	$0.672568\pi$
$138$	−44511.0	−0.198962
$139$	−419175.	−1.84017	−0.920086	−	0.391716i	$-0.871882\pi$
$139$	−419175.	−1.84017	−0.920086	+	0.391716i	$0.871882\pi$
$140$	−350527.	−1.51148
$141$	214844.	0.910072
$142$	−426396.	−1.77457
$143$	113063.	0.462359
$144$	22094.8	0.0887940
$145$	−361169.	−1.42656
$146$	−73841.3	−0.286693
$147$	−78826.9	−0.300872
$148$	49007.8	0.183912
$149$	503270.	1.85710	0.928551	−	0.371206i	$-0.121055\pi$
$149$	503270.	1.85710	0.928551	+	0.371206i	$0.121055\pi$
$150$	155504.	0.564304
$151$	517943.	1.84858	0.924292	−	0.381685i	$-0.124656\pi$
$151$	517943.	1.84858	0.924292	+	0.381685i	$0.124656\pi$
$152$	364044.	1.27804
$153$	47937.4	0.165556
$154$	366329.	1.24471
$155$	131964.	0.441190
$156$	−129082.	−0.424674
$157$	−39747.9	−0.128696	−0.0643480	−	0.997928i	$-0.520497\pi$
$157$	−39747.9	−0.128696	−0.0643480	+	0.997928i	$0.520497\pi$
$158$	667853.	2.12833
$159$	−131353.	−0.412049
$160$	−313954.	−0.969542
$161$	47458.3	0.144294
$162$	61339.3	0.183633
$163$	−157828.	−0.465279	−0.232640	−	0.972563i	$-0.574736\pi$
$163$	−157828.	−0.465279	−0.232640	+	0.972563i	$0.574736\pi$
$164$	−1.02004e6	−2.96147
$165$	−277206.	−0.792671
$166$	1.05252e6	2.96456
$167$	−101000.	−0.280241	−0.140120	−	0.990134i	$-0.544749\pi$
$167$	−101000.	−0.280241	−0.140120	+	0.990134i	$0.544749\pi$
$168$	−176677.	−0.482956
$169$	−304282.	−0.819520
$170$	390187.	1.03550
$171$	134759.	0.352425
$172$	683420.	1.76144
$173$	61568.5	0.156402	0.0782012	−	0.996938i	$-0.475082\pi$
$173$	61568.5	0.156402	0.0782012	+	0.996938i	$0.475082\pi$
$174$	−430930.	−1.07903
$175$	−165801.	−0.409252
$176$	−119138.	−0.289914
$177$	429652.	1.03082
$178$	1.28872e6	3.04867
$179$	−610536.	−1.42423	−0.712113	−	0.702065i	$-0.752262\pi$
$179$	−610536.	−1.42423	−0.712113	+	0.702065i	$0.752262\pi$
$180$	316483.	0.728063
$181$	121378.	0.275387	0.137694	−	0.990475i	$-0.456031\pi$
$181$	121378.	0.275387	0.137694	+	0.990475i	$0.456031\pi$
$182$	217119.	0.485870
$183$	−138163.	−0.304975
$184$	−115755.	−0.252054
$185$	62377.6	0.133998
$186$	157453.	0.333710
$187$	−258485.	−0.540544
$188$	1.32261e6	2.72921
$189$	−65400.9	−0.133177
$190$	1.09687e6	2.20430
$191$	−294581.	−0.584280	−0.292140	−	0.956376i	$-0.594367\pi$
$191$	−294581.	−0.584280	−0.292140	+	0.956376i	$0.594367\pi$
$192$	−453156.	−0.887144
$193$	−8895.53	−0.0171901	−0.00859505	−	0.999963i	$-0.502736\pi$
$193$	−8895.53	−0.0171901	−0.00859505	+	0.999963i	$0.502736\pi$
$194$	389668.	0.743345
$195$	−164297.	−0.309417
$196$	−485270.	−0.902283
$197$	−698113.	−1.28162	−0.640811	−	0.767699i	$-0.721402\pi$
$197$	−698113.	−1.28162	−0.640811	+	0.767699i	$0.721402\pi$
$198$	−330750.	−0.599566
$199$	−533271.	−0.954586	−0.477293	−	0.878744i	$-0.658382\pi$
$199$	−533271.	−0.954586	−0.477293	+	0.878744i	$0.658382\pi$
$200$	404401.	0.714887
$201$	−370570.	−0.646963
$202$	346854.	0.598092
$203$	459464.	0.782550
$204$	295109.	0.496486
$205$	−1.29831e6	−2.15772
$206$	−1.15296e6	−1.89298
$207$	−42849.0	−0.0695048
$208$	−70611.9	−0.113167
$209$	−726638.	−1.15067
$210$	−532331.	−0.832978
$211$	−658407.	−1.01809	−0.509047	−	0.860739i	$-0.670002\pi$
$211$	−658407.	−1.01809	−0.509047	+	0.860739i	$0.670002\pi$
$212$	−808630.	−1.23569
$213$	−410475.	−0.619923
$214$	1.00461e6	1.49956
$215$	869864.	1.28338
$216$	159518.	0.232635
$217$	−167879.	−0.242018
$218$	903614.	1.28778
$219$	−71084.2	−0.100153
$220$	−1.70652e6	−2.37714
$221$	−153201.	−0.211000
$222$	74426.2	0.101355
$223$	79895.6	0.107587	0.0537936	−	0.998552i	$-0.482869\pi$
$223$	79895.6	0.107587	0.0537936	+	0.998552i	$0.482869\pi$
$224$	399400.	0.531849
$225$	149698.	0.197133
$226$	521573.	0.679273
$227$	716111.	0.922393	0.461196	−	0.887298i	$-0.347420\pi$
$227$	716111.	0.922393	0.461196	+	0.887298i	$0.347420\pi$
$228$	829594.	1.05689
$229$	−581299.	−0.732506	−0.366253	−	0.930515i	$-0.619360\pi$
$229$	−581299.	−0.732506	−0.366253	+	0.930515i	$0.619360\pi$
$230$	−348770.	−0.434730
$231$	352651.	0.434825
$232$	−1.12067e6	−1.36697
$233$	122734.	0.148107	0.0740534	−	0.997254i	$-0.476406\pi$
$233$	122734.	0.148107	0.0740534	+	0.997254i	$0.476406\pi$
$234$	−196032.	−0.234039
$235$	1.68343e6	1.98850
$236$	2.64500e6	3.09133
$237$	642917.	0.743505
$238$	−496380.	−0.568031
$239$	210076.	0.237893	0.118946	−	0.992901i	$-0.462048\pi$
$239$	210076.	0.237893	0.118946	+	0.992901i	$0.462048\pi$
$240$	173126.	0.194014
$241$	1.23224e6	1.36664	0.683319	−	0.730120i	$-0.260536\pi$
$241$	1.23224e6	1.36664	0.683319	+	0.730120i	$0.260536\pi$
$242$	277772.	0.304894
$243$	59049.0	0.0641500
$244$	−850553.	−0.914590
$245$	−617656.	−0.657402
$246$	−1.54909e6	−1.63207
$247$	−430670.	−0.449162
$248$	409471.	0.422760
$249$	1.01322e6	1.03563
$250$	−841849.	−0.851891
$251$	−545986.	−0.547012	−0.273506	−	0.961870i	$-0.588183\pi$
$251$	−545986.	−0.547012	−0.273506	+	0.961870i	$0.588183\pi$
$252$	−402617.	−0.399384
$253$	231048.	0.226934
$254$	1.56914e6	1.52608
$255$	375618.	0.361739
$256$	−1.45779e6	−1.39026
$257$	556705.	0.525765	0.262883	−	0.964828i	$-0.415327\pi$
$257$	556705.	0.525765	0.262883	+	0.964828i	$0.415327\pi$
$258$	1.03788e6	0.970731
$259$	−79354.3	−0.0735057
$260$	−1.01144e6	−0.927909
$261$	−414840.	−0.376946
$262$	1.21284e6	1.09157
$263$	1.65392e6	1.47443	0.737217	−	0.675656i	$-0.236140\pi$
$263$	1.65392e6	1.47443	0.737217	+	0.675656i	$0.236140\pi$
$264$	−860144.	−0.759558
$265$	−1.02923e6	−0.900324
$266$	−1.39539e6	−1.20919
$267$	1.24061e6	1.06501
$268$	−2.28128e6	−1.94018
$269$	−2.06545e6	−1.74034	−0.870168	−	0.492755i	$-0.835990\pi$
$269$	−2.06545e6	−1.74034	−0.870168	+	0.492755i	$0.835990\pi$
$270$	480630.	0.401237
$271$	708326.	0.585881	0.292941	−	0.956131i	$-0.405366\pi$
$271$	708326.	0.585881	0.292941	+	0.956131i	$0.405366\pi$
$272$	161434.	0.132304
$273$	209012.	0.169733
$274$	−2.11946e6	−1.70549
$275$	−807190.	−0.643641
$276$	−263784.	−0.208438
$277$	−17729.4	−0.0138833	−0.00694167	−	0.999976i	$-0.502210\pi$
$277$	−17729.4	−0.0138833	−0.00694167	+	0.999976i	$0.502210\pi$
$278$	−3.91890e6	−3.04125
$279$	151574.	0.116578
$280$	−1.38437e6	−1.05526
$281$	754660.	0.570145	0.285073	−	0.958506i	$-0.407982\pi$
$281$	754660.	0.570145	0.285073	+	0.958506i	$0.407982\pi$
$282$	2.00860e6	1.50408
$283$	−417633.	−0.309976	−0.154988	−	0.987916i	$-0.549534\pi$
$283$	−417633.	−0.309976	−0.154988	+	0.987916i	$0.549534\pi$
$284$	−2.52694e6	−1.85909
$285$	1.05591e6	0.770046
$286$	1.05703e6	0.764140
$287$	1.65166e6	1.18363
$288$	−360610.	−0.256186
$289$	−1.06961e6	−0.753320
$290$	−3.37659e6	−2.35767
$291$	375118.	0.259678
$292$	−437604.	−0.300348
$293$	−2.68010e6	−1.82382	−0.911908	−	0.410394i	$-0.865391\pi$
$293$	−2.68010e6	−1.82382	−0.911908	+	0.410394i	$0.865391\pi$
$294$	−736959.	−0.497250
$295$	3.36658e6	2.25234
$296$	193552.	0.128401
$297$	−318400.	−0.209451
$298$	4.70511e6	3.06923
$299$	136940.	0.0885831
$300$	921559.	0.591181
$301$	−1.10661e6	−0.704007
$302$	4.84229e6	3.05515
$303$	333903.	0.208936
$304$	453813.	0.281639
$305$	−1.08259e6	−0.666369
$306$	448170.	0.273615
$307$	−2.42927e6	−1.47106	−0.735528	−	0.677494i	$-0.763066\pi$
$307$	−2.42927e6	−1.47106	−0.735528	+	0.677494i	$0.763066\pi$
$308$	2.17097e6	1.30400
$309$	−1.10991e6	−0.661288
$310$	1.23374e6	0.729155
$311$	1.71783e6	1.00712	0.503559	−	0.863961i	$-0.332024\pi$
$311$	1.71783e6	1.00712	0.503559	+	0.863961i	$0.332024\pi$
$312$	−509798.	−0.296491
$313$	1.56583e6	0.903406	0.451703	−	0.892168i	$-0.350817\pi$
$313$	1.56583e6	0.903406	0.451703	+	0.892168i	$0.350817\pi$
$314$	−371606.	−0.212696
$315$	−512455.	−0.290991
$316$	3.95789e6	2.22970
$317$	673750.	0.376574	0.188287	−	0.982114i	$-0.439706\pi$
$317$	673750.	0.376574	0.188287	+	0.982114i	$0.439706\pi$
$318$	−1.22803e6	−0.680993
$319$	2.23688e6	1.23074
$320$	−3.55074e6	−1.93840
$321$	967099.	0.523852
$322$	443691.	0.238474
$323$	984603.	0.525115
$324$	363514.	0.192379
$325$	−478413.	−0.251243
$326$	−1.47554e6	−0.768967
$327$	869874.	0.449870
$328$	−4.02854e6	−2.06759
$329$	−2.14159e6	−1.09081
$330$	−2.59162e6	−1.31005
$331$	1.69768e6	0.851699	0.425850	−	0.904794i	$-0.359975\pi$
$331$	1.69768e6	0.851699	0.425850	+	0.904794i	$0.359975\pi$
$332$	6.23754e6	3.10576
$333$	71647.2	0.0354070
$334$	−944259.	−0.463153
$335$	−2.90363e6	−1.41361
$336$	−220244.	−0.106428
$337$	−2.37281e6	−1.13812	−0.569061	−	0.822295i	$-0.692693\pi$
$337$	−2.37281e6	−1.13812	−0.569061	+	0.822295i	$0.692693\pi$
$338$	−2.84476e6	−1.35442
$339$	502099.	0.237296
$340$	2.31236e6	1.08482
$341$	−817310.	−0.380628
$342$	1.25987e6	0.582452
$343$	2.29357e6	1.05263
$344$	2.69910e6	1.22977
$345$	−335747.	−0.151867
$346$	575609.	0.258486
$347$	3.98316e6	1.77584	0.887921	−	0.459996i	$-0.152149\pi$
$347$	3.98316e6	1.77584	0.887921	+	0.459996i	$0.152149\pi$
$348$	−2.55381e6	−1.13042
$349$	−2.10463e6	−0.924935	−0.462468	−	0.886636i	$-0.653036\pi$
$349$	−2.10463e6	−0.924935	−0.462468	+	0.886636i	$0.653036\pi$
$350$	−1.55008e6	−0.676370
$351$	−188712.	−0.0817585
$352$	1.94446e6	0.836453
$353$	3.41107e6	1.45698	0.728490	−	0.685056i	$-0.240222\pi$
$353$	3.41107e6	1.45698	0.728490	+	0.685056i	$0.240222\pi$
$354$	4.01685e6	1.70364
$355$	−3.21631e6	−1.35453
$356$	7.63734e6	3.19387
$357$	−477846.	−0.198435
$358$	−5.70795e6	−2.35382
$359$	−1.86760e6	−0.764800	−0.382400	−	0.923997i	$-0.624902\pi$
$359$	−1.86760e6	−0.764800	−0.382400	+	0.923997i	$0.624902\pi$
$360$	1.24992e6	0.508306
$361$	291759.	0.117830
$362$	1.13477e6	0.455132
$363$	267400.	0.106511
$364$	1.28671e6	0.509011
$365$	−556987.	−0.218833
$366$	−1.29170e6	−0.504033
$367$	−437210.	−0.169444	−0.0847218	−	0.996405i	$-0.527000\pi$
$367$	−437210.	−0.169444	−0.0847218	+	0.996405i	$0.527000\pi$
$368$	−144298.	−0.0555445
$369$	−1.49125e6	−0.570144
$370$	583173.	0.221459
$371$	1.30935e6	0.493879
$372$	933113.	0.349605
$373$	−921734.	−0.343031	−0.171516	−	0.985181i	$-0.554866\pi$
$373$	−921734.	−0.343031	−0.171516	+	0.985181i	$0.554866\pi$
$374$	−2.41660e6	−0.893357
$375$	−810415.	−0.297598
$376$	5.22352e6	1.90543
$377$	1.32577e6	0.480414
$378$	−611438.	−0.220102
$379$	−3.80141e6	−1.35940	−0.679699	−	0.733491i	$-0.737889\pi$
$379$	−3.80141e6	−1.35940	−0.679699	+	0.733491i	$0.737889\pi$
$380$	6.50036e6	2.30929
$381$	1.51055e6	0.533118
$382$	−2.75406e6	−0.965639
$383$	−4.52311e6	−1.57558	−0.787790	−	0.615944i	$-0.788775\pi$
$383$	−4.52311e6	−1.57558	−0.787790	+	0.615944i	$0.788775\pi$
$384$	−2.95442e6	−1.02245
$385$	2.76323e6	0.950090
$386$	−83165.0	−0.0284101
$387$	999130.	0.339113
$388$	2.30928e6	0.778749
$389$	4.48168e6	1.50164	0.750822	−	0.660505i	$-0.229658\pi$
$389$	4.48168e6	1.50164	0.750822	+	0.660505i	$0.229658\pi$
$390$	−1.53603e6	−0.511372
$391$	−313072.	−0.103563
$392$	−1.91652e6	−0.629940
$393$	1.16756e6	0.381326
$394$	−6.52671e6	−2.11814
$395$	5.03763e6	1.62455
$396$	−1.96012e6	−0.628122
$397$	1.76656e6	0.562539	0.281269	−	0.959629i	$-0.409245\pi$
$397$	1.76656e6	0.562539	0.281269	+	0.959629i	$0.409245\pi$
$398$	−4.98559e6	−1.57764
$399$	−1.34329e6	−0.422414
$400$	504121.	0.157538
$401$	−522373.	−0.162226	−0.0811128	−	0.996705i	$-0.525847\pi$
$401$	−522373.	−0.162226	−0.0811128	+	0.996705i	$0.525847\pi$
$402$	−3.46448e6	−1.06924
$403$	−484411.	−0.148577
$404$	2.05555e6	0.626579
$405$	462684.	0.140167
$406$	4.29557e6	1.29332
$407$	−386332.	−0.115604
$408$	1.16550e6	0.346628
$409$	−1.60642e6	−0.474845	−0.237422	−	0.971407i	$-0.576303\pi$
$409$	−1.60642e6	−0.474845	−0.237422	+	0.971407i	$0.576303\pi$
$410$	−1.21380e7	−3.56606
$411$	−2.04032e6	−0.595790
$412$	−6.83275e6	−1.98314
$413$	−4.28283e6	−1.23554
$414$	−400599.	−0.114871
$415$	7.93919e6	2.26285
$416$	1.15246e6	0.326507
$417$	−3.77258e6	−1.06242
$418$	−6.79339e6	−1.90172
$419$	−5.30658e6	−1.47666	−0.738329	−	0.674441i	$-0.764385\pi$
$419$	−5.30658e6	−1.47666	−0.738329	+	0.674441i	$0.764385\pi$
$420$	−3.15474e6	−0.872652
$421$	−2.30575e6	−0.634025	−0.317012	−	0.948421i	$-0.602680\pi$
$421$	−2.30575e6	−0.634025	−0.317012	+	0.948421i	$0.602680\pi$
$422$	−6.15549e6	−1.68260
$423$	1.93360e6	0.525430
$424$	−3.19361e6	−0.862714
$425$	1.09375e6	0.293729
$426$	−3.83756e6	−1.02455
$427$	1.37723e6	0.365541
$428$	5.95360e6	1.57098
$429$	1.01756e6	0.266943
$430$	8.13242e6	2.12104
$431$	−1.27091e6	−0.329549	−0.164775	−	0.986331i	$-0.552690\pi$
$431$	−1.27091e6	−0.329549	−0.164775	+	0.986331i	$0.552690\pi$
$432$	198853.	0.0512652
$433$	−3.71880e6	−0.953199	−0.476599	−	0.879121i	$-0.658131\pi$
$433$	−3.71880e6	−0.953199	−0.476599	+	0.879121i	$0.658131\pi$
$434$	−1.56952e6	−0.399983
$435$	−3.25052e6	−0.823625
$436$	5.35507e6	1.34911
$437$	−880091.	−0.220457
$438$	−664571.	−0.165522
$439$	−3.51069e6	−0.869423	−0.434711	−	0.900570i	$-0.643150\pi$
$439$	−3.51069e6	−0.869423	−0.434711	+	0.900570i	$0.643150\pi$
$440$	−6.73974e6	−1.65963
$441$	−709442.	−0.173708
$442$	−1.43229e6	−0.348719
$443$	2.67040e6	0.646497	0.323248	−	0.946314i	$-0.395225\pi$
$443$	2.67040e6	0.646497	0.323248	+	0.946314i	$0.395225\pi$
$444$	441070.	0.106182
$445$	9.72088e6	2.32705
$446$	746950.	0.177809
$447$	4.52943e6	1.07220
$448$	4.51711e6	1.06332
$449$	808771.	0.189326	0.0946628	−	0.995509i	$-0.469823\pi$
$449$	808771.	0.189326	0.0946628	+	0.995509i	$0.469823\pi$
$450$	1.39953e6	0.325801
$451$	8.04103e6	1.86153
$452$	3.09099e6	0.711625
$453$	4.66148e6	1.06728
$454$	6.69498e6	1.52444
$455$	1.63774e6	0.370864
$456$	3.27640e6	0.737878
$457$	−6.03865e6	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$457$	−6.03865e6	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$458$	−5.43461e6	−1.21061
$459$	431436.	0.0955839
$460$	−2.06691e6	−0.455435
$461$	−6.06334e6	−1.32880	−0.664399	−	0.747378i	$-0.731313\pi$
$461$	−6.06334e6	−1.32880	−0.664399	+	0.747378i	$0.731313\pi$
$462$	3.29696e6	0.718636
$463$	−987185.	−0.214016	−0.107008	−	0.994258i	$-0.534127\pi$
$463$	−987185.	−0.214016	−0.107008	+	0.994258i	$0.534127\pi$
$464$	−1.39701e6	−0.301235
$465$	1.18768e6	0.254721
$466$	1.14745e6	0.244776
$467$	−2.75869e6	−0.585344	−0.292672	−	0.956213i	$-0.594544\pi$
$467$	−2.75869e6	−0.585344	−0.292672	+	0.956213i	$0.594544\pi$
$468$	−1.16174e6	−0.245185
$469$	3.69389e6	0.775446
$470$	1.57385e7	3.28639
$471$	−357731.	−0.0743027
$472$	1.04462e7	2.15825
$473$	−5.38745e6	−1.10721
$474$	6.01068e6	1.22879
$475$	3.07469e6	0.625270
$476$	−2.94169e6	−0.595085
$477$	−1.18218e6	−0.237897
$478$	1.96402e6	0.393165
$479$	−1.43871e6	−0.286507	−0.143254	−	0.989686i	$-0.545756\pi$
$479$	−1.43871e6	−0.286507	−0.143254	+	0.989686i	$0.545756\pi$
$480$	−2.82559e6	−0.559765
$481$	−228975.	−0.0451258
$482$	1.15203e7	2.25864
$483$	427124.	0.0833080
$484$	1.64615e6	0.319416
$485$	2.93927e6	0.567395
$486$	552054.	0.106021
$487$	−9.54793e6	−1.82426	−0.912130	−	0.409901i	$-0.865563\pi$
$487$	−9.54793e6	−1.82426	−0.912130	+	0.409901i	$0.865563\pi$
$488$	−3.35917e6	−0.638532
$489$	−1.42045e6	−0.268629
$490$	−5.77451e6	−1.08649
$491$	8.94131e6	1.67378	0.836889	−	0.547373i	$-0.184372\pi$
$491$	8.94131e6	1.67378	0.836889	+	0.547373i	$0.184372\pi$
$492$	−9.18035e6	−1.70980
$493$	−3.03099e6	−0.561652
$494$	−4.02637e6	−0.742329
$495$	−2.49486e6	−0.457649
$496$	510441.	0.0931626
$497$	4.09167e6	0.743035
$498$	9.47269e6	1.71159
$499$	2.34135e6	0.420934	0.210467	−	0.977601i	$-0.432501\pi$
$499$	2.34135e6	0.420934	0.210467	+	0.977601i	$0.432501\pi$
$500$	−4.98903e6	−0.892465
$501$	−909002.	−0.161797
$502$	−5.10446e6	−0.904046
$503$	41307.7	0.00727966	0.00363983	−	0.999993i	$-0.498841\pi$
$503$	41307.7	0.00727966	0.00363983	+	0.999993i	$0.498841\pi$
$504$	−1.59010e6	−0.278835
$505$	2.61633e6	0.456524
$506$	2.16008e6	0.375054
$507$	−2.73854e6	−0.473150
$508$	9.29918e6	1.59877
$509$	3.09166e6	0.528929	0.264464	−	0.964395i	$-0.414805\pi$
$509$	3.09166e6	0.528929	0.264464	+	0.964395i	$0.414805\pi$
$510$	3.51168e6	0.597846
$511$	708576.	0.120042
$512$	−3.12440e6	−0.526735
$513$	1.21283e6	0.203473
$514$	5.20467e6	0.868932
$515$	−8.69678e6	−1.44491
$516$	6.15078e6	1.01697
$517$	−1.04262e7	−1.71554
$518$	−741890.	−0.121483
$519$	554117.	0.0902990
$520$	−3.99457e6	−0.647831
$521$	6.02002e6	0.971635	0.485818	−	0.874060i	$-0.338522\pi$
$521$	6.02002e6	0.971635	0.485818	+	0.874060i	$0.338522\pi$
$522$	−3.87837e6	−0.622979
$523$	−6.39458e6	−1.02225	−0.511125	−	0.859506i	$-0.670771\pi$
$523$	−6.39458e6	−1.02225	−0.511125	+	0.859506i	$0.670771\pi$
$524$	7.18764e6	1.14356
$525$	−1.49220e6	−0.236282
$526$	1.54626e7	2.43679
$527$	1.10746e6	0.173701
$528$	−1.07224e6	−0.167382
$529$	279841.	0.0434783
$530$	−9.62237e6	−1.48796
$531$	3.86687e6	0.595145
$532$	−8.26950e6	−1.26678
$533$	4.76583e6	0.726642
$534$	1.15985e7	1.76015
$535$	7.57779e6	1.14461
$536$	−9.00969e6	−1.35456
$537$	−5.49483e6	−0.822278
$538$	−1.93100e7	−2.87625
$539$	3.82541e6	0.567161
$540$	2.84835e6	0.420348
$541$	−3.24248e6	−0.476304	−0.238152	−	0.971228i	$-0.576542\pi$
$541$	−3.24248e6	−0.476304	−0.238152	+	0.971228i	$0.576542\pi$
$542$	6.62219e6	0.968286
$543$	1.09240e6	0.158995
$544$	−2.63476e6	−0.381719
$545$	6.81598e6	0.982963
$546$	1.95407e6	0.280517
$547$	9.39508e6	1.34255	0.671277	−	0.741206i	$-0.265746\pi$
$547$	9.39508e6	1.34255	0.671277	+	0.741206i	$0.265746\pi$
$548$	−1.25605e7	−1.78671
$549$	−1.24347e6	−0.176078
$550$	−7.54648e6	−1.06375
$551$	−8.52055e6	−1.19561
$552$	−1.04179e6	−0.145523
$553$	−6.40868e6	−0.891160
$554$	−165753.	−0.0229450
$555$	561398.	0.0773640
$556$	−2.32245e7	−3.18610
$557$	2.95696e6	0.403839	0.201919	−	0.979402i	$-0.435282\pi$
$557$	2.95696e6	0.403839	0.201919	+	0.979402i	$0.435282\pi$
$558$	1.41708e6	0.192668
$559$	−3.19308e6	−0.432196
$560$	−1.72574e6	−0.232544
$561$	−2.32636e6	−0.312083
$562$	7.05538e6	0.942279
$563$	−1.35314e7	−1.79917	−0.899584	−	0.436747i	$-0.856130\pi$
$563$	−1.35314e7	−1.79917	−0.899584	+	0.436747i	$0.856130\pi$
$564$	1.19035e7	1.57571
$565$	3.93424e6	0.518489
$566$	−3.90448e6	−0.512297
$567$	−588608.	−0.0768898
$568$	−9.97991e6	−1.29794
$569$	1.32776e7	1.71924	0.859622	−	0.510931i	$-0.170699\pi$
$569$	1.32776e7	1.71924	0.859622	+	0.510931i	$0.170699\pi$
$570$	9.87183e6	1.27265
$571$	7.74960e6	0.994693	0.497346	−	0.867552i	$-0.334308\pi$
$571$	7.74960e6	0.994693	0.497346	+	0.867552i	$0.334308\pi$
$572$	6.26427e6	0.800535
$573$	−2.65123e6	−0.337334
$574$	1.54415e7	1.95619
$575$	−977654.	−0.123315
$576$	−4.07840e6	−0.512193
$577$	−4.83479e6	−0.604558	−0.302279	−	0.953220i	$-0.597747\pi$
$577$	−4.83479e6	−0.604558	−0.302279	+	0.953220i	$0.597747\pi$
$578$	−9.99984e6	−1.24501
$579$	−80059.7	−0.00992471
$580$	−2.00106e7	−2.46997
$581$	−1.00999e7	−1.24130
$582$	3.50701e6	0.429170
$583$	6.37448e6	0.776736
$584$	−1.72827e6	−0.209691
$585$	−1.47867e6	−0.178642
$586$	−2.50564e7	−3.01422
$587$	5.13380e6	0.614955	0.307478	−	0.951555i	$-0.400515\pi$
$587$	5.13380e6	0.614955	0.307478	+	0.951555i	$0.400515\pi$
$588$	−4.36743e6	−0.520934
$589$	3.11324e6	0.369764
$590$	3.14744e7	3.72244
$591$	−6.28301e6	−0.739945
$592$	241279.	0.0282953
$593$	1.29134e7	1.50801	0.754003	−	0.656871i	$-0.228121\pi$
$593$	1.29134e7	1.50801	0.754003	+	0.656871i	$0.228121\pi$
$594$	−2.97675e6	−0.346160
$595$	−3.74421e6	−0.433578
$596$	2.78838e7	3.21541
$597$	−4.79944e6	−0.551130
$598$	1.28026e6	0.146401
$599$	−5.79980e6	−0.660460	−0.330230	−	0.943901i	$-0.607126\pi$
$599$	−5.79980e6	−0.660460	−0.330230	+	0.943901i	$0.607126\pi$
$600$	3.63961e6	0.412740
$601$	1.06993e7	1.20829	0.604144	−	0.796875i	$-0.293515\pi$
$601$	1.06993e7	1.20829	0.604144	+	0.796875i	$0.293515\pi$
$602$	−1.03457e7	−1.16351
$603$	−3.33513e6	−0.373525
$604$	2.86968e7	3.20067
$605$	2.09524e6	0.232726
$606$	3.12169e6	0.345309
$607$	−1.48509e7	−1.63600	−0.817998	−	0.575221i	$-0.804916\pi$
$607$	−1.48509e7	−1.63600	−0.817998	+	0.575221i	$0.804916\pi$
$608$	−7.40669e6	−0.812578
$609$	4.13518e6	0.451805
$610$	−1.01212e7	−1.10131
$611$	−6.17951e6	−0.669655
$612$	2.65598e6	0.286646
$613$	−7.64415e6	−0.821633	−0.410816	−	0.911718i	$-0.634756\pi$
$613$	−7.64415e6	−0.821633	−0.410816	+	0.911718i	$0.634756\pi$
$614$	−2.27114e7	−2.43121
$615$	−1.16848e7	−1.24576
$616$	8.57402e6	0.910401
$617$	−8.02025e6	−0.848155	−0.424077	−	0.905626i	$-0.639402\pi$
$617$	−8.02025e6	−0.848155	−0.424077	+	0.905626i	$0.639402\pi$
$618$	−1.03766e7	−1.09291
$619$	9.46459e6	0.992831	0.496416	−	0.868085i	$-0.334649\pi$
$619$	9.46459e6	0.992831	0.496416	+	0.868085i	$0.334649\pi$
$620$	7.31150e6	0.763883
$621$	−385641.	−0.0401286
$622$	1.60602e7	1.66446
$623$	−1.23665e7	−1.27652
$624$	−635507.	−0.0653370
$625$	−1.21255e7	−1.24165
$626$	1.46390e7	1.49306
$627$	−6.53974e6	−0.664342
$628$	−2.20224e6	−0.222826
$629$	523484.	0.0527566
$630$	−4.79098e6	−0.480920
$631$	8.76525e6	0.876377	0.438189	−	0.898883i	$-0.355620\pi$
$631$	8.76525e6	0.876377	0.438189	+	0.898883i	$0.355620\pi$
$632$	1.56313e7	1.55669
$633$	−5.92566e6	−0.587797
$634$	6.29894e6	0.622364
$635$	1.18361e7	1.16486
$636$	−7.27767e6	−0.713427
$637$	2.26728e6	0.221389
$638$	2.09127e7	2.03404
$639$	−3.69427e6	−0.357913
$640$	−2.31496e7	−2.23406
$641$	−8.04677e6	−0.773529	−0.386764	−	0.922179i	$-0.626407\pi$
$641$	−8.04677e6	−0.773529	−0.386764	+	0.922179i	$0.626407\pi$
$642$	9.04148e6	0.865769
$643$	1.81157e7	1.72794	0.863970	−	0.503543i	$-0.167970\pi$
$643$	1.81157e7	1.72794	0.863970	+	0.503543i	$0.167970\pi$
$644$	2.62944e6	0.249832
$645$	7.82877e6	0.740959
$646$	9.20513e6	0.867858
$647$	6.09047e6	0.571992	0.285996	−	0.958231i	$-0.407676\pi$
$647$	6.09047e6	0.571992	0.285996	+	0.958231i	$0.407676\pi$
$648$	1.43566e6	0.134312
$649$	−2.08507e7	−1.94316
$650$	−4.47272e6	−0.415230
$651$	−1.51091e6	−0.139729
$652$	−8.74448e6	−0.805591
$653$	1.73408e7	1.59142	0.795712	−	0.605675i	$-0.207097\pi$
$653$	1.73408e7	1.59142	0.795712	+	0.605675i	$0.207097\pi$
$654$	8.13252e6	0.743500
$655$	9.14850e6	0.833195
$656$	−5.02193e6	−0.455629
$657$	−639757.	−0.0578232
$658$	−2.00219e7	−1.80277
$659$	1.27618e7	1.14471	0.572357	−	0.820004i	$-0.306029\pi$
$659$	1.27618e7	1.14471	0.572357	+	0.820004i	$0.306029\pi$
$660$	−1.53587e7	−1.37244
$661$	−6.44195e6	−0.573474	−0.286737	−	0.958009i	$-0.592571\pi$
$661$	−6.44195e6	−0.573474	−0.286737	+	0.958009i	$0.592571\pi$
$662$	1.58718e7	1.40760
$663$	−1.37881e6	−0.121821
$664$	2.46345e7	2.16832
$665$	−1.05255e7	−0.922972
$666$	669836.	0.0585171
$667$	2.70926e6	0.235796
$668$	−5.59594e6	−0.485213
$669$	719060.	0.0621155
$670$	−2.71463e7	−2.33627
$671$	6.70496e6	0.574897
$672$	3.59460e6	0.307063
$673$	8.64864e6	0.736055	0.368028	−	0.929815i	$-0.380033\pi$
$673$	8.64864e6	0.736055	0.368028	+	0.929815i	$0.380033\pi$
$674$	−2.21836e7	−1.88097
$675$	1.34728e6	0.113815
$676$	−1.68588e7	−1.41893
$677$	−1.22020e7	−1.02320	−0.511600	−	0.859223i	$-0.670947\pi$
$677$	−1.22020e7	−1.02320	−0.511600	+	0.859223i	$0.670947\pi$
$678$	4.69416e6	0.392178
$679$	−3.73923e6	−0.311249
$680$	9.13242e6	0.757380
$681$	6.44500e6	0.532544
$682$	−7.64110e6	−0.629064
$683$	−1.10784e7	−0.908708	−0.454354	−	0.890821i	$-0.650130\pi$
$683$	−1.10784e7	−0.908708	−0.454354	+	0.890821i	$0.650130\pi$
$684$	7.46634e6	0.610194
$685$	−1.59871e7	−1.30180
$686$	2.14427e7	1.73968
$687$	−5.23170e6	−0.422913
$688$	3.36466e6	0.271001
$689$	3.77809e6	0.303196
$690$	−3.13893e6	−0.250991
$691$	−1.24814e7	−0.994415	−0.497207	−	0.867632i	$-0.665641\pi$
$691$	−1.24814e7	−0.994415	−0.497207	+	0.867632i	$0.665641\pi$
$692$	3.41122e6	0.270797
$693$	3.17386e6	0.251047
$694$	3.72389e7	2.93493
$695$	−2.95604e7	−2.32139
$696$	−1.00860e7	−0.789219
$697$	−1.08957e7	−0.849518
$698$	−1.96763e7	−1.52864
$699$	1.10461e6	0.0855094
$700$	−9.18622e6	−0.708585
$701$	−4.71570e6	−0.362452	−0.181226	−	0.983441i	$-0.558007\pi$
$701$	−4.71570e6	−0.362452	−0.181226	+	0.983441i	$0.558007\pi$
$702$	−1.76429e6	−0.135122
$703$	1.47159e6	0.112305
$704$	2.19913e7	1.67232
$705$	1.51509e7	1.14806
$706$	3.18903e7	2.40795
$707$	−3.32839e6	−0.250430
$708$	2.38050e7	1.78478
$709$	1.05958e7	0.791620	0.395810	−	0.918332i	$-0.370464\pi$
$709$	1.05958e7	0.791620	0.395810	+	0.918332i	$0.370464\pi$
$710$	−3.00696e7	−2.23863
$711$	5.78625e6	0.429263
$712$	3.01629e7	2.22984
$713$	−989912.	−0.0729244
$714$	−4.46742e6	−0.327953
$715$	7.97322e6	0.583268
$716$	−3.38269e7	−2.46593
$717$	1.89068e6	0.137347
$718$	−1.74603e7	−1.26398
$719$	2.95783e6	0.213379	0.106689	−	0.994292i	$-0.465975\pi$
$719$	2.95783e6	0.213379	0.106689	+	0.994292i	$0.465975\pi$
$720$	1.55813e6	0.112014
$721$	1.10637e7	0.792615
$722$	2.72768e6	0.194738
$723$	1.10902e7	0.789029
$724$	6.72499e6	0.476810
$725$	−9.46510e6	−0.668776
$726$	2.49995e6	0.176031
$727$	−1.30768e7	−0.917625	−0.458813	−	0.888533i	$-0.651725\pi$
$727$	−1.30768e7	−0.917625	−0.458813	+	0.888533i	$0.651725\pi$
$728$	5.08173e6	0.355372
$729$	531441.	0.0370370
$730$	−5.20731e6	−0.361665
$731$	7.30005e6	0.505280
$732$	−7.65497e6	−0.528039
$733$	5.06819e6	0.348412	0.174206	−	0.984709i	$-0.444264\pi$
$733$	5.06819e6	0.348412	0.174206	+	0.984709i	$0.444264\pi$
$734$	−4.08751e6	−0.280039
$735$	−5.55890e6	−0.379551
$736$	2.35509e6	0.160256
$737$	1.79835e7	1.21956
$738$	−1.39418e7	−0.942277
$739$	5.39832e6	0.363620	0.181810	−	0.983334i	$-0.441804\pi$
$739$	5.39832e6	0.363620	0.181810	+	0.983334i	$0.441804\pi$
$740$	3.45605e6	0.232007
$741$	−3.87603e6	−0.259324
$742$	1.22412e7	0.816233
$743$	3.71690e6	0.247007	0.123503	−	0.992344i	$-0.460587\pi$
$743$	3.71690e6	0.247007	0.123503	+	0.992344i	$0.460587\pi$
$744$	3.68524e6	0.244081
$745$	3.54908e7	2.34274
$746$	−8.61736e6	−0.566927
$747$	9.11900e6	0.597923
$748$	−1.43214e7	−0.935906
$749$	−9.64017e6	−0.627885
$750$	−7.57664e6	−0.491839
$751$	1.00711e7	0.651594	0.325797	−	0.945440i	$-0.394367\pi$
$751$	1.00711e7	0.651594	0.325797	+	0.945440i	$0.394367\pi$
$752$	6.51157e6	0.419896
$753$	−4.91387e6	−0.315818
$754$	1.23947e7	0.793979
$755$	3.65255e7	2.33200
$756$	−3.62355e6	−0.230585
$757$	−8.49996e6	−0.539110	−0.269555	−	0.962985i	$-0.586877\pi$
$757$	−8.49996e6	−0.539110	−0.269555	+	0.962985i	$0.586877\pi$
$758$	−3.55397e7	−2.24667
$759$	2.07943e6	0.131021
$760$	2.56725e7	1.61226
$761$	2.62780e7	1.64487	0.822434	−	0.568861i	$-0.192616\pi$
$761$	2.62780e7	1.64487	0.822434	+	0.568861i	$0.192616\pi$
$762$	1.41223e7	0.881084
$763$	−8.67102e6	−0.539211
$764$	−1.63213e7	−1.01163
$765$	3.38056e6	0.208850
$766$	−4.22869e7	−2.60396
$767$	−1.23580e7	−0.758506
$768$	−1.31201e7	−0.802666
$769$	899889.	0.0548748	0.0274374	−	0.999624i	$-0.491265\pi$
$769$	899889.	0.0548748	0.0274374	+	0.999624i	$0.491265\pi$
$770$	2.58336e7	1.57021
$771$	5.01034e6	0.303551
$772$	−492859.	−0.0297632
$773$	−1.33758e7	−0.805140	−0.402570	−	0.915389i	$-0.631883\pi$
$773$	−1.33758e7	−0.805140	−0.402570	+	0.915389i	$0.631883\pi$
$774$	9.34094e6	0.560452
$775$	3.45836e6	0.206831
$776$	9.12028e6	0.543693
$777$	−714189.	−0.0424385
$778$	4.18996e7	2.48177
$779$	−3.06293e7	−1.80840
$780$	−9.10293e6	−0.535728
$781$	1.99200e7	1.16859
$782$	−2.92694e6	−0.171158
$783$	−3.73356e6	−0.217630
$784$	−2.38911e6	−0.138818
$785$	−2.80304e6	−0.162351
$786$	1.09156e7	0.630217
$787$	1.67690e7	0.965098	0.482549	−	0.875869i	$-0.339711\pi$
$787$	1.67690e7	0.965098	0.482549	+	0.875869i	$0.339711\pi$
$788$	−3.86791e7	−2.21902
$789$	1.48853e7	0.851264
$790$	4.70972e7	2.68490
$791$	−5.00498e6	−0.284421
$792$	−7.74129e6	−0.438531
$793$	3.97396e6	0.224409
$794$	1.65157e7	0.929707
$795$	−9.26309e6	−0.519802
$796$	−2.95460e7	−1.65278
$797$	2.78280e7	1.55180	0.775902	−	0.630854i	$-0.217295\pi$
$797$	2.78280e7	1.55180	0.775902	+	0.630854i	$0.217295\pi$
$798$	−1.25585e7	−0.698123
$799$	1.41277e7	0.782894
$800$	−8.22776e6	−0.454524
$801$	1.11654e7	0.614886
$802$	−4.88370e6	−0.268110
$803$	3.44966e6	0.188794
$804$	−2.05315e7	−1.12016
$805$	3.34677e6	0.182027
$806$	−4.52880e6	−0.245553
$807$	−1.85890e7	−1.00478
$808$	8.11821e6	0.437454
$809$	−1.98606e7	−1.06689	−0.533447	−	0.845834i	$-0.679103\pi$
$809$	−1.98606e7	−1.06689	−0.533447	+	0.845834i	$0.679103\pi$
$810$	4.32567e6	0.231654
$811$	1.95245e7	1.04238	0.521192	−	0.853439i	$-0.325488\pi$
$811$	1.95245e7	1.04238	0.521192	+	0.853439i	$0.325488\pi$
$812$	2.54568e7	1.35492
$813$	6.37493e6	0.338259
$814$	−3.61185e6	−0.191059
$815$	−1.11301e7	−0.586953
$816$	1.45290e6	0.0763855
$817$	2.05215e7	1.07561
$818$	−1.50186e7	−0.784775
$819$	1.88111e6	0.0979952
$820$	−7.19335e7	−3.73591
$821$	−2.39444e7	−1.23978	−0.619891	−	0.784688i	$-0.712823\pi$
$821$	−2.39444e7	−1.23978	−0.619891	+	0.784688i	$0.712823\pi$
$822$	−1.90751e7	−0.984662
$823$	−2.78310e7	−1.43229	−0.716143	−	0.697953i	$-0.754094\pi$
$823$	−2.78310e7	−1.43229	−0.716143	+	0.697953i	$0.754094\pi$
$824$	−2.69853e7	−1.38455
$825$	−7.26471e6	−0.371607
$826$	−4.00405e7	−2.04197
$827$	−1.94204e7	−0.987405	−0.493703	−	0.869631i	$-0.664357\pi$
$827$	−1.94204e7	−0.987405	−0.493703	+	0.869631i	$0.664357\pi$
$828$	−2.37406e6	−0.120342
$829$	2.66039e7	1.34449	0.672247	−	0.740327i	$-0.265329\pi$
$829$	2.66039e7	1.34449	0.672247	+	0.740327i	$0.265329\pi$
$830$	7.42241e7	3.73981
$831$	−159564.	−0.00801555
$832$	1.30340e7	0.652784
$833$	−5.18348e6	−0.258826
$834$	−3.52701e7	−1.75587
$835$	−7.12257e6	−0.353525
$836$	−4.02596e7	−1.99229
$837$	1.36417e6	0.0673062
$838$	−4.96117e7	−2.44047
$839$	−1.41085e7	−0.691954	−0.345977	−	0.938243i	$-0.612453\pi$
$839$	−1.41085e7	−0.691954	−0.345977	+	0.938243i	$0.612453\pi$
$840$	−1.24594e7	−0.609253
$841$	5.71843e6	0.278796
$842$	−2.15566e7	−1.04785
$843$	6.79194e6	0.329174
$844$	−3.64792e7	−1.76274
$845$	−2.14581e7	−1.03383
$846$	1.80774e7	0.868378
$847$	−2.66548e6	−0.127664
$848$	−3.98111e6	−0.190114
$849$	−3.75869e6	−0.178965
$850$	1.02256e7	0.485445
$851$	−467918.	−0.0221486
$852$	−2.27425e7	−1.07334
$853$	−1.87882e7	−0.884122	−0.442061	−	0.896985i	$-0.645752\pi$
$853$	−1.87882e7	−0.884122	−0.442061	+	0.896985i	$0.645752\pi$
$854$	1.28758e7	0.604130
$855$	9.50323e6	0.444586
$856$	2.35131e7	1.09680
$857$	3.67369e7	1.70864	0.854320	−	0.519748i	$-0.173974\pi$
$857$	3.67369e7	1.70864	0.854320	+	0.519748i	$0.173974\pi$
$858$	9.51329e6	0.441176
$859$	2.03228e6	0.0939723	0.0469861	−	0.998896i	$-0.485038\pi$
$859$	2.03228e6	0.0939723	0.0469861	+	0.998896i	$0.485038\pi$
$860$	4.81950e7	2.22206
$861$	1.48650e7	0.683371
$862$	−1.18818e7	−0.544646
$863$	−1.21735e7	−0.556400	−0.278200	−	0.960523i	$-0.589738\pi$
$863$	−1.21735e7	−0.556400	−0.278200	+	0.960523i	$0.589738\pi$
$864$	−3.24549e6	−0.147909
$865$	4.34183e6	0.197303
$866$	−3.47674e7	−1.57535
$867$	−9.62646e6	−0.434930
$868$	−9.30140e6	−0.419034
$869$	−3.12003e7	−1.40155
$870$	−3.03893e7	−1.36120
$871$	1.06586e7	0.476053
$872$	2.11493e7	0.941901
$873$	3.37606e6	0.149925
$874$	−8.22804e6	−0.364349
$875$	8.07833e6	0.356698
$876$	−3.93844e6	−0.173406
$877$	4.13453e7	1.81521	0.907605	−	0.419825i	$-0.137909\pi$
$877$	4.13453e7	1.81521	0.907605	+	0.419825i	$0.137909\pi$
$878$	−3.28217e7	−1.43689
$879$	−2.41209e7	−1.05298
$880$	−8.40166e6	−0.365728
$881$	1.92417e7	0.835227	0.417613	−	0.908625i	$-0.362867\pi$
$881$	1.92417e7	0.835227	0.417613	+	0.908625i	$0.362867\pi$
$882$	−6.63263e6	−0.287088
$883$	4.59010e6	0.198117	0.0990583	−	0.995082i	$-0.468417\pi$
$883$	4.59010e6	0.198117	0.0990583	+	0.995082i	$0.468417\pi$
$884$	−8.48815e6	−0.365328
$885$	3.02992e7	1.30039
$886$	2.49657e7	1.06846
$887$	−1.34822e7	−0.575374	−0.287687	−	0.957724i	$-0.592886\pi$
$887$	−1.34822e7	−0.575374	−0.287687	+	0.957724i	$0.592886\pi$
$888$	1.74196e6	0.0741322
$889$	−1.50574e7	−0.638992
$890$	9.08813e7	3.84591
$891$	−2.86560e6	−0.120927
$892$	4.42663e6	0.186278
$893$	3.97149e7	1.66657
$894$	4.23460e7	1.77202
$895$	−4.30552e7	−1.79667
$896$	2.94500e7	1.22551
$897$	1.23246e6	0.0511435
$898$	7.56126e6	0.312898
$899$	−9.58377e6	−0.395492
$900$	8.29403e6	0.341318
$901$	−8.63750e6	−0.354467
$902$	7.51762e7	3.07655
$903$	−9.95945e6	−0.406458
$904$	1.22076e7	0.496830
$905$	8.55962e6	0.347403
$906$	4.35806e7	1.76389
$907$	5.13039e6	0.207077	0.103539	−	0.994625i	$-0.466983\pi$
$907$	5.13039e6	0.207077	0.103539	+	0.994625i	$0.466983\pi$
$908$	3.96763e7	1.59704
$909$	3.00513e6	0.120629
$910$	1.53113e7	0.612927
$911$	3.88471e7	1.55082	0.775411	−	0.631456i	$-0.217543\pi$
$911$	3.88471e7	1.55082	0.775411	+	0.631456i	$0.217543\pi$
$912$	4.08432e6	0.162604
$913$	−4.91709e7	−1.95223
$914$	−5.64558e7	−2.23534
$915$	−9.74332e6	−0.384728
$916$	−3.22071e7	−1.26827
$917$	−1.16384e7	−0.457055
$918$	4.03353e6	0.157971
$919$	2.43482e7	0.950993	0.475496	−	0.879718i	$-0.342268\pi$
$919$	2.43482e7	0.950993	0.475496	+	0.879718i	$0.342268\pi$
$920$	−8.16305e6	−0.317968
$921$	−2.18634e7	−0.849315
$922$	−5.66866e7	−2.19610
$923$	1.18064e7	0.456156
$924$	1.95387e7	0.752863
$925$	1.63472e6	0.0628188
$926$	−9.22927e6	−0.353704
$927$	−9.98917e6	−0.381795
$928$	2.28007e7	0.869116
$929$	2.17027e7	0.825040	0.412520	−	0.910948i	$-0.364649\pi$
$929$	2.17027e7	0.825040	0.412520	+	0.910948i	$0.364649\pi$
$930$	1.11037e7	0.420978
$931$	−1.45715e7	−0.550972
$932$	6.80010e6	0.256434
$933$	1.54605e7	0.581460
$934$	−2.57912e7	−0.967397
$935$	−1.82284e7	−0.681900
$936$	−4.58818e6	−0.171179
$937$	−2.01218e6	−0.0748718	−0.0374359	−	0.999299i	$-0.511919\pi$
$937$	−2.01218e6	−0.0748718	−0.0374359	+	0.999299i	$0.511919\pi$
$938$	3.45344e7	1.28158
$939$	1.40925e7	0.521582
$940$	9.32710e7	3.44292
$941$	−1.17136e7	−0.431237	−0.215618	−	0.976478i	$-0.569177\pi$
$941$	−1.17136e7	−0.431237	−0.215618	+	0.976478i	$0.569177\pi$
$942$	−3.34446e6	−0.122800
$943$	9.73916e6	0.356650
$944$	1.30220e7	0.475608
$945$	−4.61209e6	−0.168004
$946$	−5.03676e7	−1.82989
$947$	571455.	0.0207065	0.0103533	−	0.999946i	$-0.496704\pi$
$947$	571455.	0.0207065	0.0103533	+	0.999946i	$0.496704\pi$
$948$	3.56210e7	1.28732
$949$	2.04458e6	0.0736950
$950$	2.87455e7	1.03338
$951$	6.06375e6	0.217415
$952$	−1.16179e7	−0.415466
$953$	2.03678e7	0.726461	0.363231	−	0.931699i	$-0.381674\pi$
$953$	2.03678e7	0.726461	0.363231	+	0.931699i	$0.381674\pi$
$954$	−1.10523e7	−0.393171
$955$	−2.07739e7	−0.737073
$956$	1.16393e7	0.411891
$957$	2.01319e7	0.710566
$958$	−1.34506e7	−0.473510
$959$	2.03382e7	0.714110
$960$	−3.19567e7	−1.11914
$961$	−2.51274e7	−0.877687
$962$	−2.14070e6	−0.0745794
$963$	8.70389e6	0.302446
$964$	6.82727e7	2.36622
$965$	−627316.	−0.0216854
$966$	3.99322e6	0.137683
$967$	−3.93183e7	−1.35216	−0.676081	−	0.736827i	$-0.736323\pi$
$967$	−3.93183e7	−1.35216	−0.676081	+	0.736827i	$0.736323\pi$
$968$	6.50132e6	0.223004
$969$	8.86142e6	0.303175
$970$	2.74795e7	0.937734
$971$	4.86930e7	1.65736	0.828682	−	0.559719i	$-0.189091\pi$
$971$	4.86930e7	1.65736	0.828682	+	0.559719i	$0.189091\pi$
$972$	3.27163e6	0.111070
$973$	3.76055e7	1.27341
$974$	−8.92643e7	−3.01495
$975$	−4.30572e6	−0.145055
$976$	−4.18750e6	−0.140712
$977$	−1.11758e7	−0.374577	−0.187288	−	0.982305i	$-0.559970\pi$
$977$	−1.11758e7	−0.374577	−0.187288	+	0.982305i	$0.559970\pi$
$978$	−1.32799e7	−0.443963
$979$	−6.02056e7	−2.00762
$980$	−3.42214e7	−1.13824
$981$	7.82887e6	0.259733
$982$	8.35930e7	2.76625
$983$	2.09439e7	0.691310	0.345655	−	0.938362i	$-0.387657\pi$
$983$	2.09439e7	0.691310	0.345655	+	0.938362i	$0.387657\pi$
$984$	−3.62569e7	−1.19372
$985$	−4.92311e7	−1.61677
$986$	−2.83370e7	−0.928242
$987$	−1.92744e7	−0.629777
$988$	−2.38614e7	−0.777685
$989$	−6.52518e6	−0.212130
$990$	−2.33246e7	−0.756356
$991$	−1.95125e7	−0.631144	−0.315572	−	0.948902i	$-0.602196\pi$
$991$	−1.95125e7	−0.631144	−0.315572	+	0.948902i	$0.602196\pi$
$992$	−8.33092e6	−0.268790
$993$	1.52791e7	0.491729
$994$	3.82533e7	1.22801
$995$	−3.76064e7	−1.20422
$996$	5.61378e7	1.79311
$997$	1.32696e7	0.422786	0.211393	−	0.977401i	$-0.432200\pi$
$997$	1.32696e7	0.422786	0.211393	+	0.977401i	$0.432200\pi$
$998$	2.18894e7	0.695678
$999$	644825.	0.0204422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.e.1.5	✓	5
3.2	odd	2		207.6.a.f.1.1		5
4.3	odd	2		1104.6.a.r.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.5	✓	5	1.1	even	1	trivial
207.6.a.f.1.1		5	3.2	odd	2
1104.6.a.r.1.4		5	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)

\(f(q)\)	\(=\)	\(q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +659.300 q^{10} -436.763 q^{11} +498.647 q^{12} -258.865 q^{13} -838.736 q^{14} +634.683 q^{15} +272.775 q^{16} +591.819 q^{17} +757.275 q^{18} +1663.69 q^{19} +3907.20 q^{20} -807.419 q^{21} -4083.33 q^{22} -529.000 q^{23} +1969.36 q^{24} +1848.12 q^{25} -2420.15 q^{26} +729.000 q^{27} -4970.58 q^{28} -5121.48 q^{29} +5933.70 q^{30} +1871.29 q^{31} -4451.97 q^{32} -3930.87 q^{33} +5532.96 q^{34} -6326.60 q^{35} +4487.83 q^{36} +884.534 q^{37} +15554.0 q^{38} -2329.78 q^{39} +15431.1 q^{40} -18410.5 q^{41} -7548.62 q^{42} +12334.9 q^{43} -24199.0 q^{44} +5712.15 q^{45} -4945.66 q^{46} +23871.6 q^{47} +2454.98 q^{48} -8758.55 q^{49} +17278.2 q^{50} +5326.37 q^{51} -14342.5 q^{52} -14594.8 q^{53} +6815.48 q^{54} -30800.7 q^{55} -19630.8 q^{56} +14973.2 q^{57} -47881.1 q^{58} +47739.1 q^{59} +35164.8 q^{60} -15351.5 q^{61} +17494.8 q^{62} -7266.77 q^{63} -50350.6 q^{64} -18255.2 q^{65} -36750.0 q^{66} -41174.4 q^{67} +32789.9 q^{68} -4761.00 q^{69} -59147.9 q^{70} -45608.3 q^{71} +17724.2 q^{72} -7898.24 q^{73} +8269.57 q^{74} +16633.1 q^{75} +92177.1 q^{76} +39183.4 q^{77} -21781.3 q^{78} +71435.2 q^{79} +19236.2 q^{80} +6561.00 q^{81} -172121. q^{82} +112580. q^{83} -44735.2 q^{84} +41735.3 q^{85} +115320. q^{86} -46093.3 q^{87} -95571.5 q^{88} +137845. q^{89} +53403.3 q^{90} +23223.6 q^{91} -29309.4 q^{92} +16841.6 q^{93} +223177. q^{94} +117324. q^{95} -40067.7 q^{96} +41679.8 q^{97} -81884.3 q^{98} -35377.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * q^99

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