LMFDB - Embedded newform 69.8.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.5
Root			$18.2528$ 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)

$f(q)$	$=$	$q+18.2528 q^{2} -27.0000 q^{3} +205.166 q^{4} -55.2224 q^{5} -492.827 q^{6} -1258.24 q^{7} +1408.51 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+18.2528 q^{2} -27.0000 q^{3} +205.166 q^{4} -55.2224 q^{5} -492.827 q^{6} -1258.24 q^{7} +1408.51 q^{8} +729.000 q^{9} -1007.97 q^{10} -4573.07 q^{11} -5539.50 q^{12} -2124.92 q^{13} -22966.4 q^{14} +1491.01 q^{15} -552.024 q^{16} -4071.48 q^{17} +13306.3 q^{18} +13281.5 q^{19} -11329.8 q^{20} +33972.4 q^{21} -83471.6 q^{22} +12167.0 q^{23} -38029.7 q^{24} -75075.5 q^{25} -38785.9 q^{26} -19683.0 q^{27} -258148. q^{28} -15902.7 q^{29} +27215.1 q^{30} -147371. q^{31} -190365. q^{32} +123473. q^{33} -74316.2 q^{34} +69482.8 q^{35} +149566. q^{36} +279922. q^{37} +242425. q^{38} +57372.9 q^{39} -77781.2 q^{40} +34648.2 q^{41} +620092. q^{42} +753667. q^{43} -938241. q^{44} -40257.1 q^{45} +222082. q^{46} -646376. q^{47} +14904.7 q^{48} +759614. q^{49} -1.37034e6 q^{50} +109930. q^{51} -435963. q^{52} +528832. q^{53} -359271. q^{54} +252536. q^{55} -1.77224e6 q^{56} -358600. q^{57} -290269. q^{58} -943560. q^{59} +305904. q^{60} +3.51338e6 q^{61} -2.68993e6 q^{62} -917254. q^{63} -3.40405e6 q^{64} +117343. q^{65} +2.25373e6 q^{66} -3.74373e6 q^{67} -835332. q^{68} -328509. q^{69} +1.26826e6 q^{70} -501066. q^{71} +1.02680e6 q^{72} -524118. q^{73} +5.10937e6 q^{74} +2.02704e6 q^{75} +2.72491e6 q^{76} +5.75400e6 q^{77} +1.04722e6 q^{78} +2.45466e6 q^{79} +30484.1 q^{80} +531441. q^{81} +632429. q^{82} +3.22642e6 q^{83} +6.96999e6 q^{84} +224837. q^{85} +1.37566e7 q^{86} +429373. q^{87} -6.44121e6 q^{88} +1.95540e6 q^{89} -734808. q^{90} +2.67365e6 q^{91} +2.49626e6 q^{92} +3.97900e6 q^{93} -1.17982e7 q^{94} -733435. q^{95} +5.13986e6 q^{96} +5.62884e6 q^{97} +1.38651e7 q^{98} -3.33377e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	18.2528	1.61334	0.806670	−	0.591003i	$-0.201268\pi$
$2$	18.2528	1.61334	0.806670	+	0.591003i	$0.201268\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	205.166	1.60286
$5$	−55.2224	−0.197570	−0.0987849	−	0.995109i	$-0.531496\pi$
$5$	−55.2224	−0.197570	−0.0987849	+	0.995109i	$0.531496\pi$
$6$	−492.827	−0.931462
$7$	−1258.24	−1.38650	−0.693248	−	0.720699i	$-0.743821\pi$
$7$	−1258.24	−1.38650	−0.693248	+	0.720699i	$0.743821\pi$
$8$	1408.51	0.972623
$9$	729.000	0.333333
$10$	−1007.97	−0.318747
$11$	−4573.07	−1.03594	−0.517969	−	0.855399i	$-0.673312\pi$
$11$	−4573.07	−1.03594	−0.517969	+	0.855399i	$0.673312\pi$
$12$	−5539.50	−0.925413
$13$	−2124.92	−0.268251	−0.134125	−	0.990964i	$-0.542823\pi$
$13$	−2124.92	−0.268251	−0.134125	+	0.990964i	$0.542823\pi$
$14$	−22966.4	−2.23689
$15$	1491.01	0.114067
$16$	−552.024	−0.0336929
$17$	−4071.48	−0.200993	−0.100497	−	0.994937i	$-0.532043\pi$
$17$	−4071.48	−0.200993	−0.100497	+	0.994937i	$0.532043\pi$
$18$	13306.3	0.537780
$19$	13281.5	0.444231	0.222115	−	0.975020i	$-0.428704\pi$
$19$	13281.5	0.444231	0.222115	+	0.975020i	$0.428704\pi$
$20$	−11329.8	−0.316677
$21$	33972.4	0.800494
$22$	−83471.6	−1.67132
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−38029.7	−0.561544
$25$	−75075.5	−0.960966
$26$	−38785.9	−0.432780
$27$	−19683.0	−0.192450
$28$	−258148.	−2.22236
$29$	−15902.7	−0.121081	−0.0605407	−	0.998166i	$-0.519282\pi$
$29$	−15902.7	−0.121081	−0.0605407	+	0.998166i	$0.519282\pi$
$30$	27215.1	0.184029
$31$	−147371.	−0.888474	−0.444237	−	0.895909i	$-0.646525\pi$
$31$	−147371.	−0.888474	−0.444237	+	0.895909i	$0.646525\pi$
$32$	−190365.	−1.02698
$33$	123473.	0.598099
$34$	−74316.2	−0.324270
$35$	69482.8	0.273930
$36$	149566.	0.534288
$37$	279922.	0.908512	0.454256	−	0.890871i	$-0.349905\pi$
$37$	279922.	0.908512	0.454256	+	0.890871i	$0.349905\pi$
$38$	242425.	0.716694
$39$	57372.9	0.154875
$40$	−77781.2	−0.192161
$41$	34648.2	0.0785122	0.0392561	−	0.999229i	$-0.487501\pi$
$41$	34648.2	0.0785122	0.0392561	+	0.999229i	$0.487501\pi$
$42$	620092.	1.29147
$43$	753667.	1.44557	0.722786	−	0.691071i	$-0.242861\pi$
$43$	753667.	1.44557	0.722786	+	0.691071i	$0.242861\pi$
$44$	−938241.	−1.66047
$45$	−40257.1	−0.0658566
$46$	222082.	0.336404
$47$	−646376.	−0.908119	−0.454059	−	0.890971i	$-0.650025\pi$
$47$	−646376.	−0.908119	−0.454059	+	0.890971i	$0.650025\pi$
$48$	14904.7	0.0194526
$49$	759614.	0.922373
$50$	−1.37034e6	−1.55036
$51$	109930.	0.116044
$52$	−435963.	−0.429969
$53$	528832.	0.487924	0.243962	−	0.969785i	$-0.421553\pi$
$53$	528832.	0.487924	0.243962	+	0.969785i	$0.421553\pi$
$54$	−359271.	−0.310487
$55$	252536.	0.204670
$56$	−1.77224e6	−1.34854
$57$	−358600.	−0.256477
$58$	−290269.	−0.195345
$59$	−943560.	−0.598119	−0.299059	−	0.954235i	$-0.596673\pi$
$59$	−943560.	−0.598119	−0.299059	+	0.954235i	$0.596673\pi$
$60$	305904.	0.182834
$61$	3.51338e6	1.98185	0.990925	−	0.134416i	$-0.0429157\pi$
$61$	3.51338e6	1.98185	0.990925	+	0.134416i	$0.0429157\pi$
$62$	−2.68993e6	−1.43341
$63$	−917254.	−0.462166
$64$	−3.40405e6	−1.62318
$65$	117343.	0.0529983
$66$	2.25373e6	0.964936
$67$	−3.74373e6	−1.52070	−0.760350	−	0.649514i	$-0.774972\pi$
$67$	−3.74373e6	−1.52070	−0.760350	+	0.649514i	$0.774972\pi$
$68$	−835332.	−0.322165
$69$	−328509.	−0.120386
$70$	1.26826e6	0.441942
$71$	−501066.	−0.166146	−0.0830732	−	0.996543i	$-0.526474\pi$
$71$	−501066.	−0.166146	−0.0830732	+	0.996543i	$0.526474\pi$
$72$	1.02680e6	0.324208
$73$	−524118.	−0.157688	−0.0788441	−	0.996887i	$-0.525123\pi$
$73$	−524118.	−0.157688	−0.0788441	+	0.996887i	$0.525123\pi$
$74$	5.10937e6	1.46574
$75$	2.02704e6	0.554814
$76$	2.72491e6	0.712041
$77$	5.75400e6	1.43632
$78$	1.04722e6	0.249865
$79$	2.45466e6	0.560140	0.280070	−	0.959980i	$-0.409642\pi$
$79$	2.45466e6	0.560140	0.280070	+	0.959980i	$0.409642\pi$
$80$	30484.1	0.00665669
$81$	531441.	0.111111
$82$	632429.	0.126667
$83$	3.22642e6	0.619366	0.309683	−	0.950840i	$-0.399777\pi$
$83$	3.22642e6	0.619366	0.309683	+	0.950840i	$0.399777\pi$
$84$	6.96999e6	1.28308
$85$	224837.	0.0397102
$86$	1.37566e7	2.33220
$87$	429373.	0.0699064
$88$	−6.44121e6	−1.00758
$89$	1.95540e6	0.294016	0.147008	−	0.989135i	$-0.453036\pi$
$89$	1.95540e6	0.294016	0.147008	+	0.989135i	$0.453036\pi$
$90$	−734808.	−0.106249
$91$	2.67365e6	0.371929
$92$	2.49626e6	0.334220
$93$	3.97900e6	0.512961
$94$	−1.17982e7	−1.46510
$95$	−733435.	−0.0877665
$96$	5.13986e6	0.592928
$97$	5.62884e6	0.626207	0.313104	−	0.949719i	$-0.398631\pi$
$97$	5.62884e6	0.626207	0.313104	+	0.949719i	$0.398631\pi$
$98$	1.38651e7	1.48810
$99$	−3.33377e6	−0.345313
$100$	−1.54030e7	−1.54030
$101$	−1.17210e7	−1.13198	−0.565990	−	0.824412i	$-0.691506\pi$
$101$	−1.17210e7	−1.13198	−0.565990	+	0.824412i	$0.691506\pi$
$102$	2.00654e6	0.187218
$103$	−1.95882e7	−1.76630	−0.883148	−	0.469095i	$-0.844580\pi$
$103$	−1.95882e7	−1.76630	−0.883148	+	0.469095i	$0.844580\pi$
$104$	−2.99297e6	−0.260907
$105$	−1.87604e6	−0.158153
$106$	9.65269e6	0.787187
$107$	−4.65881e6	−0.367648	−0.183824	−	0.982959i	$-0.558848\pi$
$107$	−4.65881e6	−0.367648	−0.183824	+	0.982959i	$0.558848\pi$
$108$	−4.03829e6	−0.308471
$109$	−2.00461e7	−1.48265	−0.741324	−	0.671148i	$-0.765802\pi$
$109$	−2.00461e7	−1.48265	−0.741324	+	0.671148i	$0.765802\pi$
$110$	4.60950e6	0.330202
$111$	−7.55789e6	−0.524530
$112$	694576.	0.0467151
$113$	2.90290e6	0.189260	0.0946298	−	0.995513i	$-0.469833\pi$
$113$	2.90290e6	0.189260	0.0946298	+	0.995513i	$0.469833\pi$
$114$	−6.54546e6	−0.413784
$115$	−671891.	−0.0411961
$116$	−3.26270e6	−0.194077
$117$	−1.54907e6	−0.0894169
$118$	−1.72227e7	−0.964968
$119$	5.12289e6	0.278677
$120$	2.10009e6	0.110944
$121$	1.42582e6	0.0731669
$122$	6.41292e7	3.19740
$123$	−935502.	−0.0453291
$124$	−3.02355e7	−1.42410
$125$	8.46010e6	0.387428
$126$	−1.67425e7	−0.745630
$127$	−3.34220e7	−1.44784	−0.723918	−	0.689886i	$-0.757661\pi$
$127$	−3.34220e7	−1.44784	−0.723918	+	0.689886i	$0.757661\pi$
$128$	−3.77668e7	−1.59175
$129$	−2.03490e7	−0.834602
$130$	2.14185e6	0.0855042
$131$	2.49474e7	0.969561	0.484780	−	0.874636i	$-0.338900\pi$
$131$	2.49474e7	0.969561	0.484780	+	0.874636i	$0.338900\pi$
$132$	2.53325e7	0.958671
$133$	−1.67112e7	−0.615924
$134$	−6.83338e7	−2.45340
$135$	1.08694e6	0.0380223
$136$	−5.73472e6	−0.195491
$137$	6.84572e6	0.227456	0.113728	−	0.993512i	$-0.463721\pi$
$137$	6.84572e6	0.227456	0.113728	+	0.993512i	$0.463721\pi$
$138$	−5.99623e6	−0.194223
$139$	−1.70159e6	−0.0537408	−0.0268704	−	0.999639i	$-0.508554\pi$
$139$	−1.70159e6	−0.0537408	−0.0268704	+	0.999639i	$0.508554\pi$
$140$	1.42555e7	0.439072
$141$	1.74522e7	0.524303
$142$	−9.14589e6	−0.268050
$143$	9.71742e6	0.277891
$144$	−402426.	−0.0112310
$145$	878185.	0.0239220
$146$	−9.56665e6	−0.254405
$147$	−2.05096e7	−0.532532
$148$	5.74306e7	1.45622
$149$	−5.37112e7	−1.33019	−0.665094	−	0.746760i	$-0.731609\pi$
$149$	−5.37112e7	−1.33019	−0.665094	+	0.746760i	$0.731609\pi$
$150$	3.69992e7	0.895103
$151$	8.21927e6	0.194274	0.0971368	−	0.995271i	$-0.469032\pi$
$151$	8.21927e6	0.194274	0.0971368	+	0.995271i	$0.469032\pi$
$152$	1.87071e7	0.432069
$153$	−2.96811e6	−0.0669978
$154$	1.05027e8	2.31728
$155$	8.13816e6	0.175536
$156$	1.17710e7	0.248243
$157$	−6.78723e7	−1.39973	−0.699864	−	0.714276i	$-0.746756\pi$
$157$	−6.78723e7	−1.39973	−0.699864	+	0.714276i	$0.746756\pi$
$158$	4.48046e7	0.903696
$159$	−1.42785e7	−0.281703
$160$	1.05124e7	0.202900
$161$	−1.53090e7	−0.289105
$162$	9.70031e6	0.179260
$163$	−9.92510e7	−1.79506	−0.897528	−	0.440957i	$-0.854639\pi$
$163$	−9.92510e7	−1.79506	−0.897528	+	0.440957i	$0.854639\pi$
$164$	7.10865e6	0.125844
$165$	−6.81847e6	−0.118166
$166$	5.88914e7	0.999248
$167$	4.93170e7	0.819386	0.409693	−	0.912223i	$-0.365636\pi$
$167$	4.93170e7	0.819386	0.409693	+	0.912223i	$0.365636\pi$
$168$	4.78503e7	0.778579
$169$	−5.82332e7	−0.928041
$170$	4.10392e6	0.0640660
$171$	9.68219e6	0.148077
$172$	1.54627e8	2.31706
$173$	9.85108e7	1.44651	0.723256	−	0.690580i	$-0.242644\pi$
$173$	9.85108e7	1.44651	0.723256	+	0.690580i	$0.242644\pi$
$174$	7.83727e6	0.112783
$175$	9.44627e7	1.33238
$176$	2.52445e6	0.0349037
$177$	2.54761e7	0.345324
$178$	3.56916e7	0.474347
$179$	−9.93526e7	−1.29477	−0.647386	−	0.762162i	$-0.724138\pi$
$179$	−9.93526e7	−1.29477	−0.647386	+	0.762162i	$0.724138\pi$
$180$	−8.25942e6	−0.105559
$181$	−2.51054e6	−0.0314697	−0.0157348	−	0.999876i	$-0.505009\pi$
$181$	−2.51054e6	−0.0314697	−0.0157348	+	0.999876i	$0.505009\pi$
$182$	4.88018e7	0.600047
$183$	−9.48613e7	−1.14422
$184$	1.71373e7	0.202806
$185$	−1.54580e7	−0.179494
$186$	7.26281e7	0.827579
$187$	1.86192e7	0.208217
$188$	−1.32615e8	−1.45559
$189$	2.47659e7	0.266831
$190$	−1.33873e7	−0.141597
$191$	−1.82296e8	−1.89304	−0.946519	−	0.322649i	$-0.895427\pi$
$191$	−1.82296e8	−1.89304	−0.946519	+	0.322649i	$0.895427\pi$
$192$	9.19092e7	0.937141
$193$	−9.63233e7	−0.964452	−0.482226	−	0.876047i	$-0.660172\pi$
$193$	−9.63233e7	−0.964452	−0.482226	+	0.876047i	$0.660172\pi$
$194$	1.02742e8	1.01028
$195$	−3.16827e6	−0.0305986
$196$	1.55847e8	1.47844
$197$	8.55300e7	0.797052	0.398526	−	0.917157i	$-0.369522\pi$
$197$	8.55300e7	0.797052	0.398526	+	0.917157i	$0.369522\pi$
$198$	−6.08508e7	−0.557106
$199$	4.61517e7	0.415147	0.207574	−	0.978219i	$-0.433443\pi$
$199$	4.61517e7	0.415147	0.207574	+	0.978219i	$0.433443\pi$
$200$	−1.05744e8	−0.934658
$201$	1.01081e8	0.877976
$202$	−2.13941e8	−1.82627
$203$	2.00093e7	0.167879
$204$	2.25540e7	0.186002
$205$	−1.91336e6	−0.0155116
$206$	−3.57540e8	−2.84963
$207$	8.86974e6	0.0695048
$208$	1.17301e6	0.00903814
$209$	−6.07371e7	−0.460195
$210$	−3.42430e7	−0.255155
$211$	−1.94632e8	−1.42635	−0.713175	−	0.700986i	$-0.752744\pi$
$211$	−1.94632e8	−1.42635	−0.713175	+	0.700986i	$0.752744\pi$
$212$	1.08499e8	0.782076
$213$	1.35288e7	0.0959247
$214$	−8.50366e7	−0.593141
$215$	−4.16193e7	−0.285601
$216$	−2.77237e7	−0.187181
$217$	1.85427e8	1.23187
$218$	−3.65899e8	−2.39201
$219$	1.41512e7	0.0910414
$220$	5.18119e7	0.328058
$221$	8.65158e6	0.0539166
$222$	−1.37953e8	−0.846244
$223$	3.19359e8	1.92847	0.964235	−	0.265049i	$-0.0853883\pi$
$223$	3.19359e8	1.92847	0.964235	+	0.265049i	$0.0853883\pi$
$224$	2.39524e8	1.42391
$225$	−5.47300e7	−0.320322
$226$	5.29862e7	0.305340
$227$	1.49456e8	0.848051	0.424025	−	0.905650i	$-0.360617\pi$
$227$	1.49456e8	0.848051	0.424025	+	0.905650i	$0.360617\pi$
$228$	−7.35726e7	−0.411097
$229$	6.02340e7	0.331450	0.165725	−	0.986172i	$-0.447004\pi$
$229$	6.02340e7	0.331450	0.165725	+	0.986172i	$0.447004\pi$
$230$	−1.22639e7	−0.0664633
$231$	−1.55358e8	−0.829262
$232$	−2.23991e7	−0.117767
$233$	2.26979e8	1.17555	0.587773	−	0.809026i	$-0.300005\pi$
$233$	2.26979e8	1.17555	0.587773	+	0.809026i	$0.300005\pi$
$234$	−2.82749e7	−0.144260
$235$	3.56945e7	0.179417
$236$	−1.93587e8	−0.958702
$237$	−6.62759e7	−0.323397
$238$	9.35073e7	0.449600
$239$	3.42126e8	1.62104	0.810519	−	0.585712i	$-0.199185\pi$
$239$	3.42126e8	1.62104	0.810519	+	0.585712i	$0.199185\pi$
$240$	−823071.	−0.00384324
$241$	4.68382e7	0.215546	0.107773	−	0.994176i	$-0.465628\pi$
$241$	4.68382e7	0.215546	0.107773	+	0.994176i	$0.465628\pi$
$242$	2.60252e7	0.118043
$243$	−1.43489e7	−0.0641500
$244$	7.20828e8	3.17663
$245$	−4.19477e7	−0.182233
$246$	−1.70756e7	−0.0731311
$247$	−2.82221e7	−0.119165
$248$	−2.07573e8	−0.864150
$249$	−8.71133e7	−0.357591
$250$	1.54421e8	0.625052
$251$	−1.23613e8	−0.493409	−0.246705	−	0.969091i	$-0.579348\pi$
$251$	−1.23613e8	−0.493409	−0.246705	+	0.969091i	$0.579348\pi$
$252$	−1.88190e8	−0.740788
$253$	−5.56406e7	−0.216008
$254$	−6.10047e8	−2.33585
$255$	−6.07061e6	−0.0229267
$256$	−2.53634e8	−0.944860
$257$	1.64775e8	0.605514	0.302757	−	0.953068i	$-0.402093\pi$
$257$	1.64775e8	0.605514	0.302757	+	0.953068i	$0.402093\pi$
$258$	−3.71427e8	−1.34650
$259$	−3.52208e8	−1.25965
$260$	2.40749e7	0.0849489
$261$	−1.15931e7	−0.0403605
$262$	4.55360e8	1.56423
$263$	7.07371e7	0.239774	0.119887	−	0.992788i	$-0.461747\pi$
$263$	7.07371e7	0.239774	0.119887	+	0.992788i	$0.461747\pi$
$264$	1.73913e8	0.581725
$265$	−2.92034e7	−0.0963990
$266$	−3.05027e8	−0.993695
$267$	−5.27958e7	−0.169750
$268$	−7.68089e8	−2.43747
$269$	5.02046e8	1.57257	0.786286	−	0.617863i	$-0.212001\pi$
$269$	5.02046e8	1.57257	0.786286	+	0.617863i	$0.212001\pi$
$270$	1.98398e7	0.0613429
$271$	−3.00790e8	−0.918059	−0.459030	−	0.888421i	$-0.651803\pi$
$271$	−3.00790e8	−0.918059	−0.459030	+	0.888421i	$0.651803\pi$
$272$	2.24756e6	0.00677204
$273$	−7.21886e7	−0.214733
$274$	1.24954e8	0.366964
$275$	3.43326e8	0.995501
$276$	−6.73990e7	−0.192962
$277$	1.45667e8	0.411797	0.205899	−	0.978573i	$-0.433988\pi$
$277$	1.45667e8	0.411797	0.205899	+	0.978573i	$0.433988\pi$
$278$	−3.10589e7	−0.0867021
$279$	−1.07433e8	−0.296158
$280$	9.78671e7	0.266430
$281$	5.23161e8	1.40658	0.703288	−	0.710905i	$-0.251715\pi$
$281$	5.23161e8	1.40658	0.703288	+	0.710905i	$0.251715\pi$
$282$	3.18552e8	0.845878
$283$	−1.24633e8	−0.326875	−0.163438	−	0.986554i	$-0.552258\pi$
$283$	−1.24633e8	−0.326875	−0.163438	+	0.986554i	$0.552258\pi$
$284$	−1.02802e8	−0.266310
$285$	1.98027e7	0.0506720
$286$	1.77371e8	0.448333
$287$	−4.35956e7	−0.108857
$288$	−1.38776e8	−0.342327
$289$	−3.93762e8	−0.959602
$290$	1.60294e7	0.0385943
$291$	−1.51979e8	−0.361541
$292$	−1.07532e8	−0.252753
$293$	7.01514e8	1.62930	0.814648	−	0.579956i	$-0.196930\pi$
$293$	7.01514e8	1.62930	0.814648	+	0.579956i	$0.196930\pi$
$294$	−3.74358e8	−0.859155
$295$	5.21056e7	0.118170
$296$	3.94272e8	0.883639
$297$	9.00118e7	0.199366
$298$	−9.80383e8	−2.14604
$299$	−2.58539e7	−0.0559342
$300$	4.15880e8	0.889291
$301$	−9.48291e8	−2.00428
$302$	1.50025e8	0.313429
$303$	3.16466e8	0.653549
$304$	−7.33169e6	−0.0149674
$305$	−1.94017e8	−0.391554
$306$	−5.41765e7	−0.108090
$307$	3.91398e8	0.772029	0.386015	−	0.922493i	$-0.373851\pi$
$307$	3.91398e8	0.772029	0.386015	+	0.922493i	$0.373851\pi$
$308$	1.18053e9	2.30223
$309$	5.28880e8	1.01977
$310$	1.48545e8	0.283198
$311$	−6.93505e8	−1.30734	−0.653669	−	0.756780i	$-0.726771\pi$
$311$	−6.93505e8	−1.30734	−0.653669	+	0.756780i	$0.726771\pi$
$312$	8.08102e7	0.150635
$313$	1.14627e8	0.211292	0.105646	−	0.994404i	$-0.466309\pi$
$313$	1.14627e8	0.211292	0.105646	+	0.994404i	$0.466309\pi$
$314$	−1.23886e9	−2.25824
$315$	5.06530e7	0.0913099
$316$	5.03614e8	0.897828
$317$	2.96539e8	0.522847	0.261424	−	0.965224i	$-0.415808\pi$
$317$	2.96539e8	0.522847	0.261424	+	0.965224i	$0.415808\pi$
$318$	−2.60623e8	−0.454483
$319$	7.27241e7	0.125433
$320$	1.87980e8	0.320690
$321$	1.25788e8	0.212262
$322$	−2.79432e8	−0.466424
$323$	−5.40753e7	−0.0892874
$324$	1.09034e8	0.178096
$325$	1.59529e8	0.257780
$326$	−1.81161e9	−2.89603
$327$	5.41246e8	0.856007
$328$	4.88023e7	0.0763628
$329$	8.13293e8	1.25910
$330$	−1.24457e8	−0.190642
$331$	−1.84341e8	−0.279399	−0.139700	−	0.990194i	$-0.544614\pi$
$331$	−1.84341e8	−0.279399	−0.139700	+	0.990194i	$0.544614\pi$
$332$	6.61953e8	0.992759
$333$	2.04063e8	0.302837
$334$	9.00175e8	1.32195
$335$	2.06738e8	0.300444
$336$	−1.87536e7	−0.0269710
$337$	−5.02145e7	−0.0714702	−0.0357351	−	0.999361i	$-0.511377\pi$
$337$	−5.02145e7	−0.0714702	−0.0357351	+	0.999361i	$0.511377\pi$
$338$	−1.06292e9	−1.49725
$339$	−7.83784e7	−0.109269
$340$	4.61291e7	0.0636500
$341$	6.73936e8	0.920404
$342$	1.76728e8	0.238898
$343$	8.04378e7	0.107629
$344$	1.06155e9	1.40600
$345$	1.81411e7	0.0237846
$346$	1.79810e9	2.33372
$347$	−6.94065e8	−0.891758	−0.445879	−	0.895093i	$-0.647109\pi$
$347$	−6.94065e8	−0.891758	−0.445879	+	0.895093i	$0.647109\pi$
$348$	8.80928e7	0.112050
$349$	9.76747e8	1.22997	0.614983	−	0.788540i	$-0.289163\pi$
$349$	9.76747e8	1.22997	0.614983	+	0.788540i	$0.289163\pi$
$350$	1.72421e9	2.14958
$351$	4.18248e7	0.0516249
$352$	8.70553e8	1.06389
$353$	−1.26629e9	−1.53222	−0.766112	−	0.642708i	$-0.777811\pi$
$353$	−1.26629e9	−1.53222	−0.766112	+	0.642708i	$0.777811\pi$
$354$	4.65012e8	0.557125
$355$	2.76701e7	0.0328255
$356$	4.01182e8	0.471267
$357$	−1.38318e8	−0.160894
$358$	−1.81347e9	−2.08891
$359$	2.38669e8	0.272248	0.136124	−	0.990692i	$-0.456535\pi$
$359$	2.38669e8	0.272248	0.136124	+	0.990692i	$0.456535\pi$
$360$	−5.67025e7	−0.0640536
$361$	−7.17474e8	−0.802659
$362$	−4.58245e7	−0.0507712
$363$	−3.84970e7	−0.0422429
$364$	5.48544e8	0.596151
$365$	2.89431e7	0.0311544
$366$	−1.73149e9	−1.84602
$367$	−2.75310e8	−0.290730	−0.145365	−	0.989378i	$-0.546436\pi$
$367$	−2.75310e8	−0.290730	−0.145365	+	0.989378i	$0.546436\pi$
$368$	−6.71648e6	−0.00702545
$369$	2.52585e7	0.0261707
$370$	−2.82152e8	−0.289585
$371$	−6.65395e8	−0.676505
$372$	8.16358e8	0.822206
$373$	5.15794e7	0.0514631	0.0257315	−	0.999669i	$-0.491808\pi$
$373$	5.15794e7	0.0514631	0.0257315	+	0.999669i	$0.491808\pi$
$374$	3.39853e8	0.335924
$375$	−2.28423e8	−0.223681
$376$	−9.10426e8	−0.883257
$377$	3.37920e7	0.0324802
$378$	4.52047e8	0.430490
$379$	1.37438e9	1.29679	0.648394	−	0.761305i	$-0.275441\pi$
$379$	1.37438e9	1.29679	0.648394	+	0.761305i	$0.275441\pi$
$380$	−1.50476e8	−0.140678
$381$	9.02394e8	0.835909
$382$	−3.32741e9	−3.05411
$383$	−1.39122e9	−1.26532	−0.632660	−	0.774430i	$-0.718037\pi$
$383$	−1.39122e9	−1.26532	−0.632660	+	0.774430i	$0.718037\pi$
$384$	1.01970e9	0.918998
$385$	−3.17750e8	−0.283774
$386$	−1.75817e9	−1.55599
$387$	5.49423e8	0.481858
$388$	1.15485e9	1.00372
$389$	−2.13078e9	−1.83533	−0.917666	−	0.397352i	$-0.869929\pi$
$389$	−2.13078e9	−1.83533	−0.917666	+	0.397352i	$0.869929\pi$
$390$	−5.78299e7	−0.0493658
$391$	−4.95378e7	−0.0419100
$392$	1.06992e9	0.897121
$393$	−6.73579e8	−0.559776
$394$	1.56117e9	1.28591
$395$	−1.35552e8	−0.110667
$396$	−6.83978e8	−0.553489
$397$	−1.37134e9	−1.09997	−0.549984	−	0.835176i	$-0.685366\pi$
$397$	−1.37134e9	−1.09997	−0.549984	+	0.835176i	$0.685366\pi$
$398$	8.42400e8	0.669773
$399$	4.51203e8	0.355604
$400$	4.14435e7	0.0323777
$401$	−1.18545e9	−0.918073	−0.459037	−	0.888417i	$-0.651805\pi$
$401$	−1.18545e9	−0.918073	−0.459037	+	0.888417i	$0.651805\pi$
$402$	1.84501e9	1.41647
$403$	3.13151e8	0.238334
$404$	−2.40475e9	−1.81441
$405$	−2.93475e7	−0.0219522
$406$	3.65227e8	0.270846
$407$	−1.28010e9	−0.941162
$408$	1.54837e8	0.112867
$409$	−1.29049e9	−0.932658	−0.466329	−	0.884611i	$-0.654424\pi$
$409$	−1.29049e9	−0.932658	−0.466329	+	0.884611i	$0.654424\pi$
$410$	−3.49242e7	−0.0250255
$411$	−1.84834e8	−0.131322
$412$	−4.01883e9	−2.83113
$413$	1.18722e9	0.829289
$414$	1.61898e8	0.112135
$415$	−1.78171e8	−0.122368
$416$	4.04511e8	0.275488
$417$	4.59430e7	0.0310272
$418$	−1.10863e9	−0.742451
$419$	1.41402e9	0.939085	0.469543	−	0.882910i	$-0.344419\pi$
$419$	1.41402e9	0.939085	0.469543	+	0.882910i	$0.344419\pi$
$420$	−3.84900e8	−0.253498
$421$	5.22633e8	0.341357	0.170679	−	0.985327i	$-0.445404\pi$
$421$	5.22633e8	0.341357	0.170679	+	0.985327i	$0.445404\pi$
$422$	−3.55259e9	−2.30119
$423$	−4.71208e8	−0.302706
$424$	7.44864e8	0.474566
$425$	3.05669e8	0.193148
$426$	2.46939e8	0.154759
$427$	−4.42066e9	−2.74783
$428$	−9.55832e8	−0.589289
$429$	−2.62370e8	−0.160441
$430$	−7.59671e8	−0.460772
$431$	−1.51985e9	−0.914386	−0.457193	−	0.889367i	$-0.651145\pi$
$431$	−1.51985e9	−0.914386	−0.457193	+	0.889367i	$0.651145\pi$
$432$	1.08655e7	0.00648420
$433$	8.00200e8	0.473686	0.236843	−	0.971548i	$-0.423887\pi$
$433$	8.00200e8	0.473686	0.236843	+	0.971548i	$0.423887\pi$
$434$	3.38457e9	1.98742
$435$	−2.37110e7	−0.0138114
$436$	−4.11279e9	−2.37648
$437$	1.61596e8	0.0926285
$438$	2.58300e8	0.146881
$439$	−4.16973e8	−0.235224	−0.117612	−	0.993060i	$-0.537524\pi$
$439$	−4.16973e8	−0.235224	−0.117612	+	0.993060i	$0.537524\pi$
$440$	3.55699e8	0.199067
$441$	5.53759e8	0.307458
$442$	1.57916e8	0.0869858
$443$	3.45365e9	1.88740	0.943702	−	0.330797i	$-0.107317\pi$
$443$	3.45365e9	1.88740	0.943702	+	0.330797i	$0.107317\pi$
$444$	−1.55063e9	−0.840749
$445$	−1.07982e8	−0.0580886
$446$	5.82922e9	3.11128
$447$	1.45020e9	0.767984
$448$	4.28309e9	2.25053
$449$	2.45695e9	1.28096	0.640478	−	0.767977i	$-0.278736\pi$
$449$	2.45695e9	1.28096	0.640478	+	0.767977i	$0.278736\pi$
$450$	−9.98979e8	−0.516788
$451$	−1.58449e8	−0.0813338
$452$	5.95578e8	0.303357
$453$	−2.21920e8	−0.112164
$454$	2.72799e9	1.36819
$455$	−1.47646e8	−0.0734819
$456$	−5.05090e8	−0.249455
$457$	−1.91347e8	−0.0937810	−0.0468905	−	0.998900i	$-0.514931\pi$
$457$	−1.91347e8	−0.0937810	−0.0468905	+	0.998900i	$0.514931\pi$
$458$	1.09944e9	0.534741
$459$	8.01390e7	0.0386812
$460$	−1.37850e8	−0.0660318
$461$	1.96253e9	0.932961	0.466481	−	0.884531i	$-0.345522\pi$
$461$	1.96253e9	0.932961	0.466481	+	0.884531i	$0.345522\pi$
$462$	−2.83573e9	−1.33788
$463$	−1.29413e9	−0.605962	−0.302981	−	0.952997i	$-0.597982\pi$
$463$	−1.29413e9	−0.605962	−0.302981	+	0.952997i	$0.597982\pi$
$464$	8.77867e6	0.00407958
$465$	−2.19730e8	−0.101345
$466$	4.14301e9	1.89655
$467$	9.86202e8	0.448081	0.224041	−	0.974580i	$-0.428075\pi$
$467$	9.86202e8	0.448081	0.224041	+	0.974580i	$0.428075\pi$
$468$	−3.17817e8	−0.143323
$469$	4.71050e9	2.10844
$470$	6.51525e8	0.289460
$471$	1.83255e9	0.808134
$472$	−1.32901e9	−0.581744
$473$	−3.44657e9	−1.49752
$474$	−1.20972e9	−0.521749
$475$	−9.97112e8	−0.426891
$476$	1.05104e9	0.446680
$477$	3.85519e8	0.162641
$478$	6.24477e9	2.61528
$479$	4.76901e9	1.98268	0.991342	−	0.131302i	$-0.0419159\pi$
$479$	4.76901e9	1.98268	0.991342	+	0.131302i	$0.0419159\pi$
$480$	−2.83835e8	−0.117145
$481$	−5.94812e8	−0.243709
$482$	8.54930e8	0.347749
$483$	4.13342e8	0.166915
$484$	2.92529e8	0.117276
$485$	−3.10838e8	−0.123720
$486$	−2.61908e8	−0.103496
$487$	1.99662e9	0.783329	0.391664	−	0.920108i	$-0.371899\pi$
$487$	1.99662e9	0.783329	0.391664	+	0.920108i	$0.371899\pi$
$488$	4.94863e9	1.92759
$489$	2.67978e9	1.03638
$490$	−7.65665e8	−0.294004
$491$	−4.21304e9	−1.60624	−0.803119	−	0.595818i	$-0.796828\pi$
$491$	−4.21304e9	−1.60624	−0.803119	+	0.595818i	$0.796828\pi$
$492$	−1.91934e8	−0.0726563
$493$	6.47475e7	0.0243366
$494$	−5.15133e8	−0.192254
$495$	1.84099e8	0.0682233
$496$	8.13521e7	0.0299352
$497$	6.30459e8	0.230361
$498$	−1.59007e9	−0.576916
$499$	−3.85300e8	−0.138818	−0.0694092	−	0.997588i	$-0.522111\pi$
$499$	−3.85300e8	−0.138818	−0.0694092	+	0.997588i	$0.522111\pi$
$500$	1.73573e9	0.620993
$501$	−1.33156e9	−0.473073
$502$	−2.25630e9	−0.796037
$503$	−4.50683e9	−1.57900	−0.789502	−	0.613749i	$-0.789661\pi$
$503$	−4.50683e9	−1.57900	−0.789502	+	0.613749i	$0.789661\pi$
$504$	−1.29196e9	−0.449513
$505$	6.47260e8	0.223645
$506$	−1.01560e9	−0.348494
$507$	1.57230e9	0.535805
$508$	−6.85708e9	−2.32068
$509$	−3.82145e9	−1.28445	−0.642223	−	0.766518i	$-0.721988\pi$
$509$	−3.82145e9	−1.28445	−0.642223	+	0.766518i	$0.721988\pi$
$510$	−1.10806e8	−0.0369885
$511$	6.59464e8	0.218634
$512$	2.04610e8	0.0673724
$513$	−2.61419e8	−0.0854922
$514$	3.00761e9	0.976900
$515$	1.08171e9	0.348967
$516$	−4.17494e9	−1.33775
$517$	2.95592e9	0.940755
$518$	−6.42879e9	−2.03224
$519$	−2.65979e9	−0.835145
$520$	1.65279e8	0.0515473
$521$	−1.94335e9	−0.602032	−0.301016	−	0.953619i	$-0.597326\pi$
$521$	−1.94335e9	−0.602032	−0.301016	+	0.953619i	$0.597326\pi$
$522$	−2.11606e8	−0.0651151
$523$	−2.76162e9	−0.844129	−0.422064	−	0.906566i	$-0.638694\pi$
$523$	−2.76162e9	−0.844129	−0.422064	+	0.906566i	$0.638694\pi$
$524$	5.11836e9	1.55407
$525$	−2.55049e9	−0.769248
$526$	1.29115e9	0.386837
$527$	6.00017e8	0.178577
$528$	−6.81600e7	−0.0201517
$529$	1.48036e8	0.0434783
$530$	−5.33045e8	−0.155524
$531$	−6.87855e8	−0.199373
$532$	−3.42858e9	−0.987242
$533$	−7.36247e7	−0.0210610
$534$	−9.63673e8	−0.273864
$535$	2.57271e8	0.0726361
$536$	−5.27308e9	−1.47907
$537$	2.68252e9	0.747537
$538$	9.16377e9	2.53709
$539$	−3.47377e9	−0.955521
$540$	2.23004e8	0.0609446
$541$	3.42796e9	0.930776	0.465388	−	0.885107i	$-0.345915\pi$
$541$	3.42796e9	0.930776	0.465388	+	0.885107i	$0.345915\pi$
$542$	−5.49027e9	−1.48114
$543$	6.77846e7	0.0181690
$544$	7.75068e8	0.206416
$545$	1.10700e9	0.292926
$546$	−1.31765e9	−0.346438
$547$	−7.51988e9	−1.96451	−0.982257	−	0.187537i	$-0.939949\pi$
$547$	−7.51988e9	−1.96451	−0.982257	+	0.187537i	$0.939949\pi$
$548$	1.40451e9	0.364581
$549$	2.56125e9	0.660617
$550$	6.26667e9	1.60608
$551$	−2.11211e8	−0.0537881
$552$	−4.62708e8	−0.117090
$553$	−3.08854e9	−0.776633
$554$	2.65885e9	0.664368
$555$	4.17365e8	0.103631
$556$	−3.49110e8	−0.0861391
$557$	3.43622e9	0.842534	0.421267	−	0.906937i	$-0.361586\pi$
$557$	3.43622e9	0.842534	0.421267	+	0.906937i	$0.361586\pi$
$558$	−1.96096e9	−0.477803
$559$	−1.60148e9	−0.387776
$560$	−3.83562e7	−0.00922948
$561$	−5.02718e8	−0.120214
$562$	9.54917e9	2.26928
$563$	−4.79808e9	−1.13315	−0.566576	−	0.824009i	$-0.691732\pi$
$563$	−4.79808e9	−1.13315	−0.566576	+	0.824009i	$0.691732\pi$
$564$	3.58060e9	0.840386
$565$	−1.60305e8	−0.0373920
$566$	−2.27491e9	−0.527361
$567$	−6.68678e8	−0.154055
$568$	−7.05756e8	−0.161598
$569$	6.93537e9	1.57825	0.789127	−	0.614231i	$-0.210534\pi$
$569$	6.93537e9	1.57825	0.789127	+	0.614231i	$0.210534\pi$
$570$	3.61456e8	0.0817512
$571$	−4.55930e9	−1.02488	−0.512439	−	0.858724i	$-0.671258\pi$
$571$	−4.55930e9	−1.02488	−0.512439	+	0.858724i	$0.671258\pi$
$572$	1.99369e9	0.445421
$573$	4.92198e9	1.09295
$574$	−7.95744e8	−0.175623
$575$	−9.13443e8	−0.200375
$576$	−2.48155e9	−0.541058
$577$	6.25462e9	1.35546	0.677729	−	0.735312i	$-0.262964\pi$
$577$	6.25462e9	1.35546	0.677729	+	0.735312i	$0.262964\pi$
$578$	−7.18727e9	−1.54816
$579$	2.60073e9	0.556827
$580$	1.80174e8	0.0383437
$581$	−4.05960e9	−0.858749
$582$	−2.77404e9	−0.583288
$583$	−2.41839e9	−0.505459
$584$	−7.38225e8	−0.153371
$585$	8.55433e7	0.0176661
$586$	1.28046e10	2.62861
$587$	1.68140e9	0.343114	0.171557	−	0.985174i	$-0.445120\pi$
$587$	1.68140e9	0.343114	0.171557	+	0.985174i	$0.445120\pi$
$588$	−4.20788e9	−0.853577
$589$	−1.95730e9	−0.394687
$590$	9.51077e8	0.190649
$591$	−2.30931e9	−0.460178
$592$	−1.54524e8	−0.0306104
$593$	1.03834e9	0.204478	0.102239	−	0.994760i	$-0.467399\pi$
$593$	1.03834e9	0.204478	0.102239	+	0.994760i	$0.467399\pi$
$594$	1.64297e9	0.321645
$595$	−2.82898e8	−0.0550581
$596$	−1.10197e10	−2.13211
$597$	−1.24610e9	−0.239685
$598$	−4.71908e8	−0.0902408
$599$	−4.78677e9	−0.910015	−0.455007	−	0.890488i	$-0.650363\pi$
$599$	−4.78677e9	−0.910015	−0.455007	+	0.890488i	$0.650363\pi$
$600$	2.85510e9	0.539625
$601$	2.09790e9	0.394208	0.197104	−	0.980383i	$-0.436846\pi$
$601$	2.09790e9	0.394208	0.197104	+	0.980383i	$0.436846\pi$
$602$	−1.73090e10	−3.23359
$603$	−2.72918e9	−0.506900
$604$	1.68632e9	0.311394
$605$	−7.87370e7	−0.0144556
$606$	5.77641e9	1.05440
$607$	−6.24951e9	−1.13419	−0.567095	−	0.823652i	$-0.691933\pi$
$607$	−6.24951e9	−1.13419	−0.567095	+	0.823652i	$0.691933\pi$
$608$	−2.52833e9	−0.456216
$609$	−5.40252e8	−0.0969250
$610$	−3.54137e9	−0.631709
$611$	1.37350e9	0.243604
$612$	−6.08957e8	−0.107388
$613$	−5.00469e9	−0.877538	−0.438769	−	0.898600i	$-0.644585\pi$
$613$	−5.00469e9	−0.877538	−0.438769	+	0.898600i	$0.644585\pi$
$614$	7.14412e9	1.24555
$615$	5.16607e7	0.00895565
$616$	8.10456e9	1.39700
$617$	1.14177e10	1.95695	0.978476	−	0.206360i	$-0.0661618\pi$
$617$	1.14177e10	1.95695	0.978476	+	0.206360i	$0.0661618\pi$
$618$	9.65357e9	1.64524
$619$	−6.53729e9	−1.10785	−0.553924	−	0.832567i	$-0.686870\pi$
$619$	−6.53729e9	−1.10785	−0.553924	+	0.832567i	$0.686870\pi$
$620$	1.66968e9	0.281359
$621$	−2.39483e8	−0.0401286
$622$	−1.26584e10	−2.10918
$623$	−2.46035e9	−0.407652
$624$	−3.16712e7	−0.00521817
$625$	5.39808e9	0.884422
$626$	2.09227e9	0.340886
$627$	1.63990e9	0.265694
$628$	−1.39251e10	−2.24357
$629$	−1.13970e9	−0.182605
$630$	9.24561e8	0.147314
$631$	5.62904e9	0.891932	0.445966	−	0.895050i	$-0.352860\pi$
$631$	5.62904e9	0.891932	0.445966	+	0.895050i	$0.352860\pi$
$632$	3.45741e9	0.544805
$633$	5.25507e9	0.823504
$634$	5.41269e9	0.843530
$635$	1.84564e9	0.286049
$636$	−2.92946e9	−0.451532
$637$	−1.61412e9	−0.247427
$638$	1.32742e9	0.202366
$639$	−3.65277e8	−0.0553821
$640$	2.08557e9	0.314482
$641$	3.22138e9	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$641$	3.22138e9	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$642$	2.29599e9	0.342450
$643$	8.84948e9	1.31274	0.656371	−	0.754438i	$-0.272091\pi$
$643$	8.84948e9	1.31274	0.656371	+	0.754438i	$0.272091\pi$
$644$	−3.14088e9	−0.463395
$645$	1.12372e9	0.164892
$646$	−9.87028e8	−0.144051
$647$	−3.18486e8	−0.0462302	−0.0231151	−	0.999733i	$-0.507358\pi$
$647$	−3.18486e8	−0.0462302	−0.0231151	+	0.999733i	$0.507358\pi$
$648$	7.48539e8	0.108069
$649$	4.31497e9	0.619614
$650$	2.91187e9	0.415887
$651$	−5.00652e9	−0.711218
$652$	−2.03630e10	−2.87723
$653$	−7.67323e9	−1.07841	−0.539203	−	0.842176i	$-0.681274\pi$
$653$	−7.67323e9	−1.07841	−0.539203	+	0.842176i	$0.681274\pi$
$654$	9.87927e9	1.38103
$655$	−1.37765e9	−0.191556
$656$	−1.91266e7	−0.00264530
$657$	−3.82082e8	−0.0525628
$658$	1.48449e10	2.03136
$659$	5.93429e9	0.807736	0.403868	−	0.914817i	$-0.367665\pi$
$659$	5.93429e9	0.807736	0.403868	+	0.914817i	$0.367665\pi$
$660$	−1.39892e9	−0.189404
$661$	4.68411e9	0.630844	0.315422	−	0.948951i	$-0.397854\pi$
$661$	4.68411e9	0.630844	0.315422	+	0.948951i	$0.397854\pi$
$662$	−3.36476e9	−0.450765
$663$	−2.33593e8	−0.0311288
$664$	4.54444e9	0.602410
$665$	9.22834e8	0.121688
$666$	3.72473e9	0.488579
$667$	−1.93488e8	−0.0252472
$668$	1.01182e10	1.31336
$669$	−8.62271e9	−1.11340
$670$	3.77356e9	0.484718
$671$	−1.60669e10	−2.05307
$672$	−6.46715e9	−0.822092
$673$	−2.25019e9	−0.284555	−0.142278	−	0.989827i	$-0.545443\pi$
$673$	−2.25019e9	−0.284555	−0.142278	+	0.989827i	$0.545443\pi$
$674$	−9.16558e8	−0.115306
$675$	1.47771e9	0.184938
$676$	−1.19475e10	−1.48752
$677$	4.51199e9	0.558867	0.279433	−	0.960165i	$-0.409853\pi$
$677$	4.51199e9	0.558867	0.279433	+	0.960165i	$0.409853\pi$
$678$	−1.43063e9	−0.176288
$679$	−7.08241e9	−0.868234
$680$	3.16685e8	0.0386230
$681$	−4.03530e9	−0.489622
$682$	1.23012e10	1.48492
$683$	−2.46140e9	−0.295604	−0.147802	−	0.989017i	$-0.547220\pi$
$683$	−2.46140e9	−0.295604	−0.147802	+	0.989017i	$0.547220\pi$
$684$	1.98646e9	0.237347
$685$	−3.78037e8	−0.0449384
$686$	1.46822e9	0.173643
$687$	−1.62632e9	−0.191363
$688$	−4.16042e8	−0.0487055
$689$	−1.12373e9	−0.130886
$690$	3.31126e8	0.0383726
$691$	−1.19297e10	−1.37549	−0.687744	−	0.725954i	$-0.741399\pi$
$691$	−1.19297e10	−1.37549	−0.687744	+	0.725954i	$0.741399\pi$
$692$	2.02111e10	2.31856
$693$	4.19467e9	0.478775
$694$	−1.26687e10	−1.43871
$695$	9.39661e7	0.0106175
$696$	6.04775e8	0.0679925
$697$	−1.41070e8	−0.0157804
$698$	1.78284e10	1.98435
$699$	−6.12843e9	−0.678702
$700$	1.93806e10	2.13562
$701$	−8.67060e9	−0.950683	−0.475342	−	0.879801i	$-0.657676\pi$
$701$	−8.67060e9	−0.950683	−0.475342	+	0.879801i	$0.657676\pi$
$702$	7.63422e8	0.0832885
$703$	3.71777e9	0.403589
$704$	1.55669e10	1.68151
$705$	−9.63750e8	−0.103586
$706$	−2.31134e10	−2.47200
$707$	1.47477e10	1.56949
$708$	5.22684e9	0.553507
$709$	−1.23643e10	−1.30289	−0.651443	−	0.758698i	$-0.725836\pi$
$709$	−1.23643e10	−1.30289	−0.651443	+	0.758698i	$0.725836\pi$
$710$	5.05058e8	0.0529587
$711$	1.78945e9	0.186713
$712$	2.75420e9	0.285966
$713$	−1.79306e9	−0.185260
$714$	−2.52470e9	−0.259577
$715$	−5.36619e8	−0.0549029
$716$	−2.03838e10	−2.07534
$717$	−9.23740e9	−0.935907
$718$	4.35639e9	0.439229
$719$	1.16695e10	1.17085	0.585424	−	0.810727i	$-0.300928\pi$
$719$	1.16695e10	1.17085	0.585424	+	0.810727i	$0.300928\pi$
$720$	2.22229e7	0.00221890
$721$	2.46465e10	2.44896
$722$	−1.30960e10	−1.29496
$723$	−1.26463e9	−0.124446
$724$	−5.15079e8	−0.0504416
$725$	1.19390e9	0.116355
$726$	−7.02680e8	−0.0681521
$727$	4.17740e9	0.403214	0.201607	−	0.979466i	$-0.435384\pi$
$727$	4.17740e9	0.403214	0.201607	+	0.979466i	$0.435384\pi$
$728$	3.76586e9	0.361747
$729$	3.87420e8	0.0370370
$730$	5.28294e8	0.0502627
$731$	−3.06854e9	−0.290550
$732$	−1.94624e10	−1.83403
$733$	−1.01595e10	−0.952819	−0.476410	−	0.879223i	$-0.658062\pi$
$733$	−1.01595e10	−0.952819	−0.476410	+	0.879223i	$0.658062\pi$
$734$	−5.02519e9	−0.469047
$735$	1.13259e9	0.105212
$736$	−2.31617e9	−0.214140
$737$	1.71204e10	1.57535
$738$	4.61040e8	0.0422223
$739$	2.00545e9	0.182791	0.0913956	−	0.995815i	$-0.470867\pi$
$739$	2.00545e9	0.182791	0.0913956	+	0.995815i	$0.470867\pi$
$740$	−3.17145e9	−0.287705
$741$	7.61996e8	0.0688001
$742$	−1.21454e10	−1.09143
$743$	3.87120e9	0.346246	0.173123	−	0.984900i	$-0.444614\pi$
$743$	3.87120e9	0.346246	0.173123	+	0.984900i	$0.444614\pi$
$744$	5.60446e9	0.498917
$745$	2.96606e9	0.262805
$746$	9.41471e8	0.0830274
$747$	2.35206e9	0.206455
$748$	3.82003e9	0.333743
$749$	5.86189e9	0.509743
$750$	−4.16937e9	−0.360874
$751$	1.85856e10	1.60117	0.800583	−	0.599222i	$-0.204523\pi$
$751$	1.85856e10	1.60117	0.800583	+	0.599222i	$0.204523\pi$
$752$	3.56815e8	0.0305971
$753$	3.33756e9	0.284870
$754$	6.16799e8	0.0524016
$755$	−4.53888e8	−0.0383826
$756$	5.08112e9	0.427694
$757$	−1.16626e10	−0.977150	−0.488575	−	0.872522i	$-0.662483\pi$
$757$	−1.16626e10	−0.977150	−0.488575	+	0.872522i	$0.662483\pi$
$758$	2.50863e10	2.09216
$759$	1.50230e9	0.124712
$760$	−1.03305e9	−0.0853637
$761$	1.73542e10	1.42744	0.713719	−	0.700432i	$-0.247009\pi$
$761$	1.73542e10	1.42744	0.713719	+	0.700432i	$0.247009\pi$
$762$	1.64713e10	1.34860
$763$	2.52228e10	2.05569
$764$	−3.74009e10	−3.03428
$765$	1.63906e8	0.0132367
$766$	−2.53937e10	−2.04139
$767$	2.00499e9	0.160446
$768$	6.84811e9	0.545515
$769$	4.61561e8	0.0366005	0.0183002	−	0.999833i	$-0.494175\pi$
$769$	4.61561e8	0.0366005	0.0183002	+	0.999833i	$0.494175\pi$
$770$	−5.79984e9	−0.457824
$771$	−4.44892e9	−0.349594
$772$	−1.97623e10	−1.54589
$773$	1.82090e10	1.41794	0.708971	−	0.705237i	$-0.249160\pi$
$773$	1.82090e10	1.41794	0.708971	+	0.705237i	$0.249160\pi$
$774$	1.00285e10	0.777400
$775$	1.10639e10	0.853793
$776$	7.92827e9	0.609063
$777$	9.50960e9	0.727259
$778$	−3.88928e10	−2.96101
$779$	4.60179e8	0.0348775
$780$	−6.50023e8	−0.0490453
$781$	2.29141e9	0.172117
$782$	−9.04205e8	−0.0676150
$783$	3.13013e8	0.0233021
$784$	−4.19325e8	−0.0310774
$785$	3.74807e9	0.276544
$786$	−1.22947e10	−0.903109
$787$	−1.17211e10	−0.857149	−0.428574	−	0.903507i	$-0.640984\pi$
$787$	−1.17211e10	−0.857149	−0.428574	+	0.903507i	$0.640984\pi$
$788$	1.75479e10	1.27756
$789$	−1.90990e9	−0.138434
$790$	−2.47422e9	−0.178543
$791$	−3.65254e9	−0.262408
$792$	−4.69564e9	−0.335859
$793$	−7.46566e9	−0.531633
$794$	−2.50309e10	−1.77462
$795$	7.88492e8	0.0556560
$796$	9.46879e9	0.665424
$797$	2.23652e10	1.56483	0.782417	−	0.622755i	$-0.213987\pi$
$797$	2.23652e10	1.56483	0.782417	+	0.622755i	$0.213987\pi$
$798$	8.23574e9	0.573710
$799$	2.63171e9	0.182526
$800$	1.42917e10	0.986894
$801$	1.42549e9	0.0980052
$802$	−2.16378e10	−1.48116
$803$	2.39683e9	0.163355
$804$	2.07384e10	1.40728
$805$	8.45398e8	0.0571183
$806$	5.71589e9	0.384513
$807$	−1.35552e10	−0.907925
$808$	−1.65091e10	−1.10099
$809$	−5.03940e9	−0.334625	−0.167313	−	0.985904i	$-0.553509\pi$
$809$	−5.03940e9	−0.334625	−0.167313	+	0.985904i	$0.553509\pi$
$810$	−5.35675e8	−0.0354163
$811$	−2.50963e10	−1.65210	−0.826050	−	0.563597i	$-0.809417\pi$
$811$	−2.50963e10	−1.65210	−0.826050	+	0.563597i	$0.809417\pi$
$812$	4.10524e9	0.269087
$813$	8.12133e9	0.530042
$814$	−2.33655e10	−1.51841
$815$	5.48088e9	0.354649
$816$	−6.06841e7	−0.00390984
$817$	1.00098e10	0.642168
$818$	−2.35551e10	−1.50469
$819$	1.94909e9	0.123976
$820$	−3.92557e8	−0.0248630
$821$	8.10280e9	0.511016	0.255508	−	0.966807i	$-0.417757\pi$
$821$	8.10280e9	0.511016	0.255508	+	0.966807i	$0.417757\pi$
$822$	−3.37376e9	−0.211867
$823$	2.86095e10	1.78900	0.894501	−	0.447066i	$-0.147531\pi$
$823$	2.86095e10	1.78900	0.894501	+	0.447066i	$0.147531\pi$
$824$	−2.75901e10	−1.71794
$825$	−9.26979e9	−0.574753
$826$	2.16702e10	1.33793
$827$	−8.62907e8	−0.0530512	−0.0265256	−	0.999648i	$-0.508444\pi$
$827$	−8.62907e8	−0.0530512	−0.0265256	+	0.999648i	$0.508444\pi$
$828$	1.81977e9	0.111407
$829$	−3.20682e10	−1.95494	−0.977470	−	0.211076i	$-0.932303\pi$
$829$	−3.20682e10	−1.95494	−0.977470	+	0.211076i	$0.932303\pi$
$830$	−3.25212e9	−0.197421
$831$	−3.93302e9	−0.237751
$832$	7.23333e9	0.435418
$833$	−3.09276e9	−0.185391
$834$	8.38591e8	0.0500575
$835$	−2.72340e9	−0.161886
$836$	−1.24612e10	−0.737630
$837$	2.90069e9	0.170987
$838$	2.58098e10	1.51506
$839$	−3.04096e10	−1.77764	−0.888820	−	0.458257i	$-0.848474\pi$
$839$	−3.04096e10	−1.77764	−0.888820	+	0.458257i	$0.848474\pi$
$840$	−2.64241e9	−0.153824
$841$	−1.69970e10	−0.985339
$842$	9.53954e9	0.550725
$843$	−1.41253e10	−0.812087
$844$	−3.99320e10	−2.28625
$845$	3.21578e9	0.183353
$846$	−8.60089e9	−0.488368
$847$	−1.79401e9	−0.101446
$848$	−2.91928e8	−0.0164396
$849$	3.36510e9	0.188721
$850$	5.57932e9	0.311613
$851$	3.40581e9	0.189438
$852$	2.77565e9	0.153754
$853$	1.60575e10	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$853$	1.60575e10	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$854$	−8.06897e10	−4.43318
$855$	−5.34674e8	−0.0292555
$856$	−6.56198e9	−0.357583
$857$	5.64916e9	0.306585	0.153293	−	0.988181i	$-0.451012\pi$
$857$	5.64916e9	0.306585	0.153293	+	0.988181i	$0.451012\pi$
$858$	−4.78900e9	−0.258845
$859$	−1.90033e10	−1.02295	−0.511474	−	0.859299i	$-0.670900\pi$
$859$	−1.90033e10	−1.02295	−0.511474	+	0.859299i	$0.670900\pi$
$860$	−8.53889e9	−0.457780
$861$	1.17708e9	0.0628486
$862$	−2.77416e10	−1.47522
$863$	−1.38204e10	−0.731951	−0.365975	−	0.930625i	$-0.619265\pi$
$863$	−1.38204e10	−0.731951	−0.365975	+	0.930625i	$0.619265\pi$
$864$	3.74696e9	0.197643
$865$	−5.44000e9	−0.285787
$866$	1.46059e10	0.764217
$867$	1.06316e10	0.554026
$868$	3.80434e10	1.97451
$869$	−1.12253e10	−0.580271
$870$	−4.32793e8	−0.0222825
$871$	7.95514e9	0.407929
$872$	−2.82351e10	−1.44206
$873$	4.10343e9	0.208736
$874$	2.94958e9	0.149441
$875$	−1.06448e10	−0.537167
$876$	2.90335e9	0.145927
$877$	2.63591e10	1.31957	0.659784	−	0.751455i	$-0.270648\pi$
$877$	2.63591e10	1.31957	0.659784	+	0.751455i	$0.270648\pi$
$878$	−7.61095e9	−0.379497
$879$	−1.89409e10	−0.940674
$880$	−1.39406e8	−0.00689592
$881$	−1.92456e10	−0.948233	−0.474117	−	0.880462i	$-0.657232\pi$
$881$	−1.92456e10	−0.948233	−0.474117	+	0.880462i	$0.657232\pi$
$882$	1.01077e10	0.496034
$883$	1.47880e10	0.722849	0.361424	−	0.932401i	$-0.382291\pi$
$883$	1.47880e10	0.722849	0.361424	+	0.932401i	$0.382291\pi$
$884$	1.77501e9	0.0864210
$885$	−1.40685e9	−0.0682256
$886$	6.30389e10	3.04502
$887$	2.45474e10	1.18106	0.590531	−	0.807015i	$-0.298919\pi$
$887$	2.45474e10	1.18106	0.590531	+	0.807015i	$0.298919\pi$
$888$	−1.06453e10	−0.510169
$889$	4.20528e10	2.00742
$890$	−1.97098e9	−0.0937166
$891$	−2.43032e9	−0.115104
$892$	6.55219e10	3.09107
$893$	−8.58482e9	−0.403414
$894$	2.64703e10	1.23902
$895$	5.48649e9	0.255808
$896$	4.75195e10	2.20696
$897$	6.98056e8	0.0322936
$898$	4.48463e10	2.06662
$899$	2.34359e9	0.107578
$900$	−1.12288e10	−0.513432
$901$	−2.15313e9	−0.0980695
$902$	−2.89214e9	−0.131219
$903$	2.56039e10	1.15717
$904$	4.08876e9	0.184078
$905$	1.38638e8	0.00621745
$906$	−4.05068e9	−0.180958
$907$	2.15113e10	0.957285	0.478643	−	0.878010i	$-0.341129\pi$
$907$	2.15113e10	0.957285	0.478643	+	0.878010i	$0.341129\pi$
$908$	3.06633e10	1.35931
$909$	−8.54458e9	−0.377327
$910$	−2.69495e9	−0.118551
$911$	−3.30530e10	−1.44843	−0.724214	−	0.689575i	$-0.757797\pi$
$911$	−3.30530e10	−1.44843	−0.724214	+	0.689575i	$0.757797\pi$
$912$	1.97956e8	0.00864144
$913$	−1.47546e10	−0.641625
$914$	−3.49263e9	−0.151300
$915$	5.23847e9	0.226064
$916$	1.23580e10	0.531269
$917$	−3.13897e10	−1.34429
$918$	1.46277e9	0.0624059
$919$	−1.79392e10	−0.762426	−0.381213	−	0.924487i	$-0.624494\pi$
$919$	−1.79392e10	−0.762426	−0.381213	+	0.924487i	$0.624494\pi$
$920$	−9.46364e8	−0.0400683
$921$	−1.05677e10	−0.445731
$922$	3.58218e10	1.50518
$923$	1.06473e9	0.0445689
$924$	−3.18743e10	−1.32919
$925$	−2.10153e10	−0.873049
$926$	−2.36216e10	−0.977623
$927$	−1.42798e10	−0.588765
$928$	3.02732e9	0.124348
$929$	2.09235e10	0.856207	0.428104	−	0.903730i	$-0.359182\pi$
$929$	2.09235e10	0.856207	0.428104	+	0.903730i	$0.359182\pi$
$930$	−4.01070e9	−0.163505
$931$	1.00888e10	0.409746
$932$	4.65685e10	1.88424
$933$	1.87246e10	0.754792
$934$	1.80010e10	0.722907
$935$	−1.02820e9	−0.0411373
$936$	−2.18187e9	−0.0869690
$937$	−2.70877e10	−1.07568	−0.537842	−	0.843046i	$-0.680760\pi$
$937$	−2.70877e10	−1.07568	−0.537842	+	0.843046i	$0.680760\pi$
$938$	8.59801e10	3.40164
$939$	−3.09494e9	−0.121989
$940$	7.32331e9	0.287581
$941$	−4.47844e10	−1.75212	−0.876058	−	0.482205i	$-0.839836\pi$
$941$	−4.47844e10	−1.75212	−0.876058	+	0.482205i	$0.839836\pi$
$942$	3.34493e10	1.30379
$943$	4.21565e8	0.0163709
$944$	5.20868e8	0.0201523
$945$	−1.36763e9	−0.0527178
$946$	−6.29098e10	−2.41601
$947$	−2.58624e10	−0.989563	−0.494782	−	0.869017i	$-0.664752\pi$
$947$	−2.58624e10	−0.989563	−0.494782	+	0.869017i	$0.664752\pi$
$948$	−1.35976e10	−0.518361
$949$	1.11371e9	0.0423000
$950$	−1.82001e10	−0.688719
$951$	−8.00656e9	−0.301866
$952$	7.21563e9	0.271047
$953$	−2.50276e10	−0.936685	−0.468342	−	0.883547i	$-0.655149\pi$
$953$	−2.50276e10	−0.936685	−0.468342	+	0.883547i	$0.655149\pi$
$954$	7.03681e9	0.262396
$955$	1.00668e10	0.374007
$956$	7.01928e10	2.59830
$957$	−1.96355e9	−0.0724187
$958$	8.70480e10	3.19874
$959$	−8.61353e9	−0.315367
$960$	−5.07545e9	−0.185151
$961$	−5.79455e9	−0.210614
$962$	−1.08570e10	−0.393185
$963$	−3.39628e9	−0.122549
$964$	9.60963e9	0.345491
$965$	5.31921e9	0.190547
$966$	7.54466e9	0.269290
$967$	3.09247e10	1.09980	0.549898	−	0.835232i	$-0.314667\pi$
$967$	3.09247e10	1.09980	0.549898	+	0.835232i	$0.314667\pi$
$968$	2.00827e9	0.0711638
$969$	1.46003e9	0.0515501
$970$	−5.67368e9	−0.199602
$971$	−1.36720e10	−0.479253	−0.239627	−	0.970865i	$-0.577025\pi$
$971$	−1.36720e10	−0.479253	−0.239627	+	0.970865i	$0.577025\pi$
$972$	−2.94391e9	−0.102824
$973$	2.14100e9	0.0745114
$974$	3.64440e10	1.26377
$975$	−4.30730e9	−0.148829
$976$	−1.93947e9	−0.0667742
$977$	2.07571e10	0.712090	0.356045	−	0.934469i	$-0.384125\pi$
$977$	2.07571e10	0.712090	0.356045	+	0.934469i	$0.384125\pi$
$978$	4.89135e10	1.67203
$979$	−8.94218e9	−0.304582
$980$	−8.60627e9	−0.292095
$981$	−1.46136e10	−0.494216
$982$	−7.68999e10	−2.59141
$983$	−2.15864e10	−0.724842	−0.362421	−	0.932015i	$-0.618050\pi$
$983$	−2.15864e10	−0.724842	−0.362421	+	0.932015i	$0.618050\pi$
$984$	−1.31766e9	−0.0440881
$985$	−4.72317e9	−0.157473
$986$	1.18183e9	0.0392631
$987$	−2.19589e10	−0.726944
$988$	−5.79022e9	−0.191006
$989$	9.16987e9	0.301423
$990$	3.36033e9	0.110067
$991$	4.79952e10	1.56654	0.783268	−	0.621685i	$-0.213551\pi$
$991$	4.79952e10	1.56654	0.783268	+	0.621685i	$0.213551\pi$
$992$	2.80542e10	0.912446
$993$	4.97722e9	0.161311
$994$	1.15077e10	0.371651
$995$	−2.54861e9	−0.0820205
$996$	−1.78727e10	−0.573170
$997$	−3.34275e10	−1.06824	−0.534122	−	0.845408i	$-0.679358\pi$
$997$	−3.34275e10	−1.06824	−0.534122	+	0.845408i	$0.679358\pi$
$998$	−7.03282e9	−0.223961
$999$	−5.50970e9	−0.174843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.a.1.5	✓	5
3.2	odd	2		207.8.a.a.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.5	✓	5	1.1	even	1	trivial
207.8.a.a.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+18.2528 q^{2} -27.0000 q^{3} +205.166 q^{4} -55.2224 q^{5} -492.827 q^{6} -1258.24 q^{7} +1408.51 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+18.2528 q^{2} -27.0000 q^{3} +205.166 q^{4} -55.2224 q^{5} -492.827 q^{6} -1258.24 q^{7} +1408.51 q^{8} +729.000 q^{9} -1007.97 q^{10} -4573.07 q^{11} -5539.50 q^{12} -2124.92 q^{13} -22966.4 q^{14} +1491.01 q^{15} -552.024 q^{16} -4071.48 q^{17} +13306.3 q^{18} +13281.5 q^{19} -11329.8 q^{20} +33972.4 q^{21} -83471.6 q^{22} +12167.0 q^{23} -38029.7 q^{24} -75075.5 q^{25} -38785.9 q^{26} -19683.0 q^{27} -258148. q^{28} -15902.7 q^{29} +27215.1 q^{30} -147371. q^{31} -190365. q^{32} +123473. q^{33} -74316.2 q^{34} +69482.8 q^{35} +149566. q^{36} +279922. q^{37} +242425. q^{38} +57372.9 q^{39} -77781.2 q^{40} +34648.2 q^{41} +620092. q^{42} +753667. q^{43} -938241. q^{44} -40257.1 q^{45} +222082. q^{46} -646376. q^{47} +14904.7 q^{48} +759614. q^{49} -1.37034e6 q^{50} +109930. q^{51} -435963. q^{52} +528832. q^{53} -359271. q^{54} +252536. q^{55} -1.77224e6 q^{56} -358600. q^{57} -290269. q^{58} -943560. q^{59} +305904. q^{60} +3.51338e6 q^{61} -2.68993e6 q^{62} -917254. q^{63} -3.40405e6 q^{64} +117343. q^{65} +2.25373e6 q^{66} -3.74373e6 q^{67} -835332. q^{68} -328509. q^{69} +1.26826e6 q^{70} -501066. q^{71} +1.02680e6 q^{72} -524118. q^{73} +5.10937e6 q^{74} +2.02704e6 q^{75} +2.72491e6 q^{76} +5.75400e6 q^{77} +1.04722e6 q^{78} +2.45466e6 q^{79} +30484.1 q^{80} +531441. q^{81} +632429. q^{82} +3.22642e6 q^{83} +6.96999e6 q^{84} +224837. q^{85} +1.37566e7 q^{86} +429373. q^{87} -6.44121e6 q^{88} +1.95540e6 q^{89} -734808. q^{90} +2.67365e6 q^{91} +2.49626e6 q^{92} +3.97900e6 q^{93} -1.17982e7 q^{94} -733435. q^{95} +5.13986e6 q^{96} +5.62884e6 q^{97} +1.38651e7 q^{98} -3.33377e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more