LMFDB - Embedded newform 23.8.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,8,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	23.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:	$7.18485558613$
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} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 $ x^5 - 2x^4 - 104x^3 + 200x^2 + 2037x - 3704
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$8.86257$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.7251 q^{2} -41.9171 q^{3} +60.3796 q^{4} -364.747 q^{5} -575.318 q^{6} +165.110 q^{7} -928.099 q^{8} -429.959 q^{9} +O(q^{10})$	\(q+13.7251 q^{2} -41.9171 q^{3} +60.3796 q^{4} -364.747 q^{5} -575.318 q^{6} +165.110 q^{7} -928.099 q^{8} -429.959 q^{9} -5006.20 q^{10} +31.8310 q^{11} -2530.94 q^{12} +5670.88 q^{13} +2266.15 q^{14} +15289.1 q^{15} -20466.9 q^{16} +10766.2 q^{17} -5901.25 q^{18} -42639.9 q^{19} -22023.3 q^{20} -6920.91 q^{21} +436.885 q^{22} +12167.0 q^{23} +38903.2 q^{24} +54915.2 q^{25} +77833.7 q^{26} +109695. q^{27} +9969.26 q^{28} +102406. q^{29} +209845. q^{30} -202918. q^{31} -162114. q^{32} -1334.26 q^{33} +147768. q^{34} -60223.2 q^{35} -25960.8 q^{36} -165092. q^{37} -585239. q^{38} -237707. q^{39} +338521. q^{40} -499932. q^{41} -94990.6 q^{42} +221332. q^{43} +1921.94 q^{44} +156826. q^{45} +166994. q^{46} -1.16280e6 q^{47} +857912. q^{48} -796282. q^{49} +753719. q^{50} -451288. q^{51} +342406. q^{52} -257314. q^{53} +1.50558e6 q^{54} -11610.2 q^{55} -153238. q^{56} +1.78734e6 q^{57} +1.40553e6 q^{58} +2.24767e6 q^{59} +923151. q^{60} -2.82905e6 q^{61} -2.78508e6 q^{62} -70990.4 q^{63} +394718. q^{64} -2.06844e6 q^{65} -18312.9 q^{66} +3.03807e6 q^{67} +650060. q^{68} -510005. q^{69} -826573. q^{70} +1.49374e6 q^{71} +399045. q^{72} -512282. q^{73} -2.26592e6 q^{74} -2.30188e6 q^{75} -2.57458e6 q^{76} +5255.60 q^{77} -3.26256e6 q^{78} +5.82621e6 q^{79} +7.46523e6 q^{80} -3.65778e6 q^{81} -6.86164e6 q^{82} -1.42467e6 q^{83} -417882. q^{84} -3.92694e6 q^{85} +3.03781e6 q^{86} -4.29255e6 q^{87} -29542.3 q^{88} +7.92006e6 q^{89} +2.15246e6 q^{90} +936318. q^{91} +734639. q^{92} +8.50573e6 q^{93} -1.59596e7 q^{94} +1.55528e7 q^{95} +6.79536e6 q^{96} -1.76056e7 q^{97} -1.09291e7 q^{98} -13686.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9}+O(q^{10}) $ 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9} - 11260 q^{10} - 1318 q^{11} - 28012 q^{12} - 19662 q^{13} - 6848 q^{14} - 24600 q^{15} + 32448 q^{16} - 5002 q^{17} - 24650 q^{18} - 38314 q^{19} + 104440 q^{20} + 33280 q^{21} - 74872 q^{22} + 60835 q^{23} + 361992 q^{24} + 54959 q^{25} + 345430 q^{26} + 100900 q^{27} + 90800 q^{28} - 150634 q^{29} + 515724 q^{30} - 179940 q^{31} - 404032 q^{32} - 619688 q^{33} + 32116 q^{34} - 374032 q^{35} + 510524 q^{36} - 752672 q^{37} + 456808 q^{38} - 207996 q^{39} - 1082576 q^{40} - 1192910 q^{41} - 250520 q^{42} - 932646 q^{43} + 2467104 q^{44} - 389952 q^{45} - 194672 q^{46} - 1008460 q^{47} - 1916464 q^{48} - 2005219 q^{49} + 1571224 q^{50} - 211520 q^{51} - 1740516 q^{52} + 897104 q^{53} + 1844686 q^{54} + 1203168 q^{55} + 3050144 q^{56} + 3137192 q^{57} + 5685090 q^{58} + 1020972 q^{59} - 1479384 q^{60} - 2758364 q^{61} + 2661794 q^{62} + 1135132 q^{63} + 5173248 q^{64} - 1350472 q^{65} + 11693212 q^{66} - 1523138 q^{67} + 2501304 q^{68} - 827356 q^{69} - 2794240 q^{70} + 3044884 q^{71} - 6740904 q^{72} - 8872022 q^{73} + 1408492 q^{74} + 1960276 q^{75} - 17963952 q^{76} - 3501672 q^{77} - 15280362 q^{78} - 4437540 q^{79} + 12197536 q^{80} - 7995203 q^{81} - 7738154 q^{82} - 4637362 q^{83} + 2663744 q^{84} - 8625728 q^{85} - 3025868 q^{86} + 17151068 q^{87} - 41815040 q^{88} + 6381402 q^{89} - 4970376 q^{90} + 3240808 q^{91} + 3114752 q^{92} + 7185076 q^{93} - 13893974 q^{94} + 15762704 q^{95} + 40696544 q^{96} - 6432034 q^{97} + 22652640 q^{98} + 30201754 q^{99}+O(q^{100}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9 - 11260 * q^10 - 1318 * q^11 - 28012 * q^12 - 19662 * q^13 - 6848 * q^14 - 24600 * q^15 + 32448 * q^16 - 5002 * q^17 - 24650 * q^18 - 38314 * q^19 + 104440 * q^20 + 33280 * q^21 - 74872 * q^22 + 60835 * q^23 + 361992 * q^24 + 54959 * q^25 + 345430 * q^26 + 100900 * q^27 + 90800 * q^28 - 150634 * q^29 + 515724 * q^30 - 179940 * q^31 - 404032 * q^32 - 619688 * q^33 + 32116 * q^34 - 374032 * q^35 + 510524 * q^36 - 752672 * q^37 + 456808 * q^38 - 207996 * q^39 - 1082576 * q^40 - 1192910 * q^41 - 250520 * q^42 - 932646 * q^43 + 2467104 * q^44 - 389952 * q^45 - 194672 * q^46 - 1008460 * q^47 - 1916464 * q^48 - 2005219 * q^49 + 1571224 * q^50 - 211520 * q^51 - 1740516 * q^52 + 897104 * q^53 + 1844686 * q^54 + 1203168 * q^55 + 3050144 * q^56 + 3137192 * q^57 + 5685090 * q^58 + 1020972 * q^59 - 1479384 * q^60 - 2758364 * q^61 + 2661794 * q^62 + 1135132 * q^63 + 5173248 * q^64 - 1350472 * q^65 + 11693212 * q^66 - 1523138 * q^67 + 2501304 * q^68 - 827356 * q^69 - 2794240 * q^70 + 3044884 * q^71 - 6740904 * q^72 - 8872022 * q^73 + 1408492 * q^74 + 1960276 * q^75 - 17963952 * q^76 - 3501672 * q^77 - 15280362 * q^78 - 4437540 * q^79 + 12197536 * q^80 - 7995203 * q^81 - 7738154 * q^82 - 4637362 * q^83 + 2663744 * q^84 - 8625728 * q^85 - 3025868 * q^86 + 17151068 * q^87 - 41815040 * q^88 + 6381402 * q^89 - 4970376 * q^90 + 3240808 * q^91 + 3114752 * q^92 + 7185076 * q^93 - 13893974 * q^94 + 15762704 * q^95 + 40696544 * q^96 - 6432034 * q^97 + 22652640 * q^98 + 30201754 * 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 $

$2$	13.7251	1.21314	0.606572	−	0.795029i	$-0.292544\pi$
$2$	13.7251	1.21314	0.606572	+	0.795029i	$0.292544\pi$
$3$	−41.9171	−0.896327	−0.448164	−	0.893952i	$-0.647922\pi$
$3$	−41.9171	−0.896327	−0.448164	+	0.893952i	$0.647922\pi$
$4$	60.3796	0.471716
$5$	−364.747	−1.30496	−0.652479	−	0.757807i	$-0.726271\pi$
$5$	−364.747	−1.30496	−0.652479	+	0.757807i	$0.726271\pi$
$6$	−575.318	−1.08737
$7$	165.110	0.181941	0.0909703	−	0.995854i	$-0.471003\pi$
$7$	165.110	0.181941	0.0909703	+	0.995854i	$0.471003\pi$
$8$	−928.099	−0.640884
$9$	−429.959	−0.196598
$10$	−5006.20	−1.58310
$11$	31.8310	0.00721066	0.00360533	−	0.999994i	$-0.498852\pi$
$11$	31.8310	0.00721066	0.00360533	+	0.999994i	$0.498852\pi$
$12$	−2530.94	−0.422812
$13$	5670.88	0.715894	0.357947	−	0.933742i	$-0.383477\pi$
$13$	5670.88	0.715894	0.357947	+	0.933742i	$0.383477\pi$
$14$	2266.15	0.220720
$15$	15289.1	1.16967
$16$	−20466.9	−1.24920
$17$	10766.2	0.531486	0.265743	−	0.964044i	$-0.414383\pi$
$17$	10766.2	0.531486	0.265743	+	0.964044i	$0.414383\pi$
$18$	−5901.25	−0.238501
$19$	−42639.9	−1.42619	−0.713097	−	0.701066i	$-0.752708\pi$
$19$	−42639.9	−1.42619	−0.713097	+	0.701066i	$0.752708\pi$
$20$	−22023.3	−0.615569
$21$	−6920.91	−0.163078
$22$	436.885	0.00874757
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	38903.2	0.574442
$25$	54915.2	0.702915
$26$	77833.7	0.868482
$27$	109695.	1.07254
$28$	9969.26	0.0858242
$29$	102406.	0.779707	0.389853	−	0.920877i	$-0.372526\pi$
$29$	102406.	0.779707	0.389853	+	0.920877i	$0.372526\pi$
$30$	209845.	1.41898
$31$	−202918.	−1.22336	−0.611681	−	0.791105i	$-0.709506\pi$
$31$	−202918.	−1.22336	−0.611681	+	0.791105i	$0.709506\pi$
$32$	−162114.	−0.874574
$33$	−1334.26	−0.00646311
$34$	147768.	0.644769
$35$	−60223.2	−0.237425
$36$	−25960.8	−0.0927383
$37$	−165092.	−0.535822	−0.267911	−	0.963444i	$-0.586333\pi$
$37$	−165092.	−0.535822	−0.267911	+	0.963444i	$0.586333\pi$
$38$	−585239.	−1.73018
$39$	−237707.	−0.641676
$40$	338521.	0.836327
$41$	−499932.	−1.13284	−0.566419	−	0.824118i	$-0.691671\pi$
$41$	−499932.	−1.13284	−0.566419	+	0.824118i	$0.691671\pi$
$42$	−94990.6	−0.197837
$43$	221332.	0.424526	0.212263	−	0.977213i	$-0.431917\pi$
$43$	221332.	0.424526	0.212263	+	0.977213i	$0.431917\pi$
$44$	1921.94	0.00340139
$45$	156826.	0.256552
$46$	166994.	0.252958
$47$	−1.16280e6	−1.63367	−0.816833	−	0.576874i	$-0.804272\pi$
$47$	−1.16280e6	−1.63367	−0.816833	+	0.576874i	$0.804272\pi$
$48$	857912.	1.11969
$49$	−796282.	−0.966898
$50$	753719.	0.852736
$51$	−451288.	−0.476385
$52$	342406.	0.337699
$53$	−257314.	−0.237409	−0.118705	−	0.992930i	$-0.537874\pi$
$53$	−257314.	−0.237409	−0.118705	+	0.992930i	$0.537874\pi$
$54$	1.50558e6	1.30115
$55$	−11610.2	−0.00940961
$56$	−153238.	−0.116603
$57$	1.78734e6	1.27834
$58$	1.40553e6	0.945896
$59$	2.24767e6	1.42479	0.712395	−	0.701779i	$-0.247610\pi$
$59$	2.24767e6	1.42479	0.712395	+	0.701779i	$0.247610\pi$
$60$	923151.	0.551752
$61$	−2.82905e6	−1.59583	−0.797914	−	0.602771i	$-0.794063\pi$
$61$	−2.82905e6	−1.59583	−0.797914	+	0.602771i	$0.794063\pi$
$62$	−2.78508e6	−1.48411
$63$	−70990.4	−0.0357691
$64$	394718.	0.188216
$65$	−2.06844e6	−0.934212
$66$	−18312.9	−0.00784068
$67$	3.03807e6	1.23406	0.617030	−	0.786940i	$-0.288336\pi$
$67$	3.03807e6	1.23406	0.617030	+	0.786940i	$0.288336\pi$
$68$	650060.	0.250711
$69$	−510005.	−0.186897
$70$	−826573.	−0.288030
$71$	1.49374e6	0.495303	0.247652	−	0.968849i	$-0.420341\pi$
$71$	1.49374e6	0.495303	0.247652	+	0.968849i	$0.420341\pi$
$72$	399045.	0.125996
$73$	−512282.	−0.154127	−0.0770636	−	0.997026i	$-0.524554\pi$
$73$	−512282.	−0.154127	−0.0770636	+	0.997026i	$0.524554\pi$
$74$	−2.26592e6	−0.650029
$75$	−2.30188e6	−0.630041
$76$	−2.57458e6	−0.672758
$77$	5255.60	0.00131191
$78$	−3.26256e6	−0.778444
$79$	5.82621e6	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$79$	5.82621e6	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$80$	7.46523e6	1.63015
$81$	−3.65778e6	−0.764752
$82$	−6.86164e6	−1.37429
$83$	−1.42467e6	−0.273490	−0.136745	−	0.990606i	$-0.543664\pi$
$83$	−1.42467e6	−0.273490	−0.136745	+	0.990606i	$0.543664\pi$
$84$	−417882.	−0.0769266
$85$	−3.92694e6	−0.693567
$86$	3.03781e6	0.515010
$87$	−4.29255e6	−0.698872
$88$	−29542.3	−0.00462120
$89$	7.92006e6	1.19087	0.595434	−	0.803404i	$-0.296980\pi$
$89$	7.92006e6	1.19087	0.595434	+	0.803404i	$0.296980\pi$
$90$	2.15246e6	0.311234
$91$	936318.	0.130250
$92$	734639.	0.0983596
$93$	8.50573e6	1.09653
$94$	−1.59596e7	−1.98187
$95$	1.55528e7	1.86112
$96$	6.79536e6	0.783905
$97$	−1.76056e7	−1.95862	−0.979308	−	0.202374i	$-0.935134\pi$
$97$	−1.76056e7	−1.95862	−0.979308	+	0.202374i	$0.935134\pi$
$98$	−1.09291e7	−1.17299
$99$	−13686.0	−0.00141760
$100$	3.31576e6	0.331576
$101$	828399.	0.0800046	0.0400023	−	0.999200i	$-0.487263\pi$
$101$	828399.	0.0800046	0.0400023	+	0.999200i	$0.487263\pi$
$102$	−6.19400e6	−0.577924
$103$	3.32839e6	0.300126	0.150063	−	0.988676i	$-0.452052\pi$
$103$	3.32839e6	0.300126	0.150063	+	0.988676i	$0.452052\pi$
$104$	−5.26314e6	−0.458805
$105$	2.52438e6	0.212810
$106$	−3.53167e6	−0.288012
$107$	1.21513e7	0.958913	0.479456	−	0.877566i	$-0.340834\pi$
$107$	1.21513e7	0.958913	0.479456	+	0.877566i	$0.340834\pi$
$108$	6.62336e6	0.505936
$109$	6.44543e6	0.476716	0.238358	−	0.971177i	$-0.423391\pi$
$109$	6.44543e6	0.476716	0.238358	+	0.971177i	$0.423391\pi$
$110$	−159352.	−0.0114152
$111$	6.92018e6	0.480272
$112$	−3.37928e6	−0.227280
$113$	1.73550e7	1.13149	0.565743	−	0.824582i	$-0.308589\pi$
$113$	1.73550e7	1.13149	0.565743	+	0.824582i	$0.308589\pi$
$114$	2.45315e7	1.55080
$115$	−4.43787e6	−0.272102
$116$	6.18322e6	0.367800
$117$	−2.43825e6	−0.140743
$118$	3.08496e7	1.72847
$119$	1.77761e6	0.0966988
$120$	−1.41898e7	−0.749622
$121$	−1.94862e7	−0.999948
$122$	−3.88291e7	−1.93597
$123$	2.09557e7	1.01539
$124$	−1.22521e7	−0.577079
$125$	8.46570e6	0.387684
$126$	−974354.	−0.0433930
$127$	−3.56482e7	−1.54427	−0.772137	−	0.635456i	$-0.780812\pi$
$127$	−3.56482e7	−1.54427	−0.772137	+	0.635456i	$0.780812\pi$
$128$	2.61682e7	1.10291
$129$	−9.27758e6	−0.380514
$130$	−2.83896e7	−1.13333
$131$	−2.55304e7	−0.992219	−0.496110	−	0.868260i	$-0.665239\pi$
$131$	−2.55304e7	−0.992219	−0.496110	+	0.868260i	$0.665239\pi$
$132$	−80562.2	−0.00304875
$133$	−7.04026e6	−0.259482
$134$	4.16980e7	1.49709
$135$	−4.00110e7	−1.39962
$136$	−9.99212e6	−0.340621
$137$	−3.24969e7	−1.07974	−0.539871	−	0.841748i	$-0.681527\pi$
$137$	−3.24969e7	−1.07974	−0.539871	+	0.841748i	$0.681527\pi$
$138$	−6.99989e6	−0.226733
$139$	−3.81088e7	−1.20358	−0.601788	−	0.798656i	$-0.705545\pi$
$139$	−3.81088e7	−1.20358	−0.601788	+	0.798656i	$0.705545\pi$
$140$	−3.63626e6	−0.111997
$141$	4.87413e7	1.46430
$142$	2.05018e7	0.600874
$143$	180510.	0.00516207
$144$	8.79993e6	0.245590
$145$	−3.73522e7	−1.01748
$146$	−7.03115e6	−0.186978
$147$	3.33778e7	0.866657
$148$	−9.96821e6	−0.252756
$149$	2.97747e7	0.737388	0.368694	−	0.929551i	$-0.379805\pi$
$149$	2.97747e7	0.737388	0.368694	+	0.929551i	$0.379805\pi$
$150$	−3.15937e7	−0.764330
$151$	−9.80561e6	−0.231769	−0.115884	−	0.993263i	$-0.536970\pi$
$151$	−9.80561e6	−0.231769	−0.115884	+	0.993263i	$0.536970\pi$
$152$	3.95741e7	0.914025
$153$	−4.62903e6	−0.104489
$154$	72133.9	0.00159154
$155$	7.40137e7	1.59644
$156$	−1.43527e7	−0.302689
$157$	5.70394e7	1.17632	0.588161	−	0.808744i	$-0.299852\pi$
$157$	5.70394e7	1.17632	0.588161	+	0.808744i	$0.299852\pi$
$158$	7.99656e7	1.61288
$159$	1.07859e7	0.212797
$160$	5.91307e7	1.14128
$161$	2.00889e6	0.0379372
$162$	−5.02036e7	−0.927753
$163$	1.04716e8	1.89389	0.946947	−	0.321389i	$-0.104150\pi$
$163$	1.04716e8	1.89389	0.946947	+	0.321389i	$0.104150\pi$
$164$	−3.01857e7	−0.534377
$165$	486667.	0.00843409
$166$	−1.95538e7	−0.331782
$167$	−2.05048e7	−0.340681	−0.170340	−	0.985385i	$-0.554487\pi$
$167$	−2.05048e7	−0.340681	−0.170340	+	0.985385i	$0.554487\pi$
$168$	6.42330e6	0.104514
$169$	−3.05896e7	−0.487495
$170$	−5.38979e7	−0.841396
$171$	1.83334e7	0.280386
$172$	1.33639e7	0.200256
$173$	−1.33900e8	−1.96617	−0.983083	−	0.183158i	$-0.941368\pi$
$173$	−1.33900e8	−1.96617	−0.983083	+	0.183158i	$0.941368\pi$
$174$	−5.89159e7	−0.847832
$175$	9.06703e6	0.127889
$176$	−651481.	−0.00900756
$177$	−9.42158e7	−1.27708
$178$	1.08704e8	1.44469
$179$	−8.02166e7	−1.04539	−0.522695	−	0.852519i	$-0.675073\pi$
$179$	−8.02166e7	−1.04539	−0.522695	+	0.852519i	$0.675073\pi$
$180$	9.46911e6	0.121019
$181$	−1.25221e8	−1.56965	−0.784823	−	0.619720i	$-0.787246\pi$
$181$	−1.25221e8	−1.56965	−0.784823	+	0.619720i	$0.787246\pi$
$182$	1.28511e7	0.158012
$183$	1.18585e8	1.43038
$184$	−1.12922e7	−0.133634
$185$	6.02169e7	0.699225
$186$	1.16742e8	1.33025
$187$	342699.	0.00383237
$188$	−7.02096e7	−0.770626
$189$	1.81117e7	0.195139
$190$	2.13464e8	2.25781
$191$	3.24582e6	0.0337060	0.0168530	−	0.999858i	$-0.494635\pi$
$191$	3.24582e6	0.0337060	0.0168530	+	0.999858i	$0.494635\pi$
$192$	−1.65454e7	−0.168703
$193$	−1.07685e8	−1.07822	−0.539109	−	0.842236i	$-0.681239\pi$
$193$	−1.07685e8	−1.07822	−0.539109	+	0.842236i	$0.681239\pi$
$194$	−2.41639e8	−2.37608
$195$	8.67028e7	0.837360
$196$	−4.80792e7	−0.456101
$197$	−6.15574e7	−0.573652	−0.286826	−	0.957983i	$-0.592600\pi$
$197$	−6.15574e7	−0.573652	−0.286826	+	0.957983i	$0.592600\pi$
$198$	−187842.	−0.00171975
$199$	5.44216e7	0.489537	0.244769	−	0.969581i	$-0.421288\pi$
$199$	5.44216e7	0.489537	0.244769	+	0.969581i	$0.421288\pi$
$200$	−5.09668e7	−0.450487
$201$	−1.27347e8	−1.10612
$202$	1.13699e7	0.0970570
$203$	1.69082e7	0.141860
$204$	−2.72486e7	−0.224719
$205$	1.82349e8	1.47830
$206$	4.56826e7	0.364095
$207$	−5.23131e6	−0.0409934
$208$	−1.16065e8	−0.894295
$209$	−1.35727e6	−0.0102838
$210$	3.46475e7	0.258169
$211$	−5.76976e7	−0.422833	−0.211417	−	0.977396i	$-0.567808\pi$
$211$	−5.76976e7	−0.422833	−0.211417	+	0.977396i	$0.567808\pi$
$212$	−1.55365e7	−0.111990
$213$	−6.26132e7	−0.443954
$214$	1.66778e8	1.16330
$215$	−8.07300e7	−0.553988
$216$	−1.01808e8	−0.687376
$217$	−3.35037e7	−0.222579
$218$	8.84645e7	0.578324
$219$	2.14734e7	0.138148
$220$	−701022.	−0.00443866
$221$	6.10540e7	0.380488
$222$	9.49806e7	0.582639
$223$	7.40469e7	0.447136	0.223568	−	0.974688i	$-0.428229\pi$
$223$	7.40469e7	0.447136	0.223568	+	0.974688i	$0.428229\pi$
$224$	−2.67667e7	−0.159120
$225$	−2.36113e7	−0.138191
$226$	2.38199e8	1.37265
$227$	2.72026e8	1.54355	0.771774	−	0.635897i	$-0.219370\pi$
$227$	2.72026e8	1.54355	0.771774	+	0.635897i	$0.219370\pi$
$228$	1.07919e8	0.603012
$229$	−2.28178e8	−1.25559	−0.627797	−	0.778377i	$-0.716043\pi$
$229$	−2.28178e8	−1.25559	−0.627797	+	0.778377i	$0.716043\pi$
$230$	−6.09105e7	−0.330099
$231$	−220299.	−0.00117590
$232$	−9.50427e7	−0.499702
$233$	−3.52957e8	−1.82800	−0.914000	−	0.405714i	$-0.867023\pi$
$233$	−3.52957e8	−1.82800	−0.914000	+	0.405714i	$0.867023\pi$
$234$	−3.34653e7	−0.170742
$235$	4.24128e8	2.13187
$236$	1.35714e8	0.672096
$237$	−2.44218e8	−1.19167
$238$	2.43979e7	0.117310
$239$	1.58503e8	0.751007	0.375504	−	0.926821i	$-0.377470\pi$
$239$	1.58503e8	0.751007	0.375504	+	0.926821i	$0.377470\pi$
$240$	−3.12921e8	−1.46115
$241$	2.05090e8	0.943811	0.471905	−	0.881649i	$-0.343566\pi$
$241$	2.05090e8	0.943811	0.471905	+	0.881649i	$0.343566\pi$
$242$	−2.67450e8	−1.21308
$243$	−8.65799e7	−0.387075
$244$	−1.70817e8	−0.752778
$245$	2.90441e8	1.26176
$246$	2.87620e8	1.23182
$247$	−2.41806e8	−1.02100
$248$	1.88328e8	0.784033
$249$	5.97180e7	0.245136
$250$	1.16193e8	0.470316
$251$	2.93965e8	1.17338	0.586688	−	0.809813i	$-0.300431\pi$
$251$	2.93965e8	1.17338	0.586688	+	0.809813i	$0.300431\pi$
$252$	−4.28638e6	−0.0168728
$253$	387287.	0.00150353
$254$	−4.89276e8	−1.87342
$255$	1.64606e8	0.621663
$256$	3.08639e8	1.14977
$257$	3.01645e8	1.10848	0.554242	−	0.832355i	$-0.313008\pi$
$257$	3.01645e8	1.10848	0.554242	+	0.832355i	$0.313008\pi$
$258$	−1.27336e8	−0.461618
$259$	−2.72583e7	−0.0974878
$260$	−1.24891e8	−0.440683
$261$	−4.40303e7	−0.153289
$262$	−3.50408e8	−1.20370
$263$	−2.44871e8	−0.830027	−0.415013	−	0.909815i	$-0.636223\pi$
$263$	−2.44871e8	−0.830027	−0.415013	+	0.909815i	$0.636223\pi$
$264$	1.23833e6	0.00414211
$265$	9.38545e7	0.309809
$266$	−9.66286e7	−0.314789
$267$	−3.31986e8	−1.06741
$268$	1.83438e8	0.582125
$269$	5.08963e8	1.59424	0.797119	−	0.603823i	$-0.206356\pi$
$269$	5.08963e8	1.59424	0.797119	+	0.603823i	$0.206356\pi$
$270$	−5.49157e8	−1.69794
$271$	2.81631e8	0.859582	0.429791	−	0.902928i	$-0.358587\pi$
$271$	2.81631e8	0.859582	0.429791	+	0.902928i	$0.358587\pi$
$272$	−2.20351e8	−0.663932
$273$	−3.92477e7	−0.116747
$274$	−4.46025e8	−1.30988
$275$	1.74800e6	0.00506848
$276$	−3.07939e7	−0.0881624
$277$	−9.11506e7	−0.257680	−0.128840	−	0.991665i	$-0.541125\pi$
$277$	−9.11506e7	−0.257680	−0.128840	+	0.991665i	$0.541125\pi$
$278$	−5.23049e8	−1.46011
$279$	8.72465e7	0.240510
$280$	5.58931e7	0.152162
$281$	2.84061e8	0.763730	0.381865	−	0.924218i	$-0.375282\pi$
$281$	2.84061e8	0.763730	0.381865	+	0.924218i	$0.375282\pi$
$282$	6.68981e8	1.77640
$283$	−5.73111e8	−1.50309	−0.751547	−	0.659680i	$-0.770692\pi$
$283$	−5.73111e8	−1.50309	−0.751547	+	0.659680i	$0.770692\pi$
$284$	9.01915e7	0.233642
$285$	−6.51926e8	−1.66817
$286$	2.47752e6	0.00626233
$287$	−8.25436e7	−0.206109
$288$	6.97026e7	0.171939
$289$	−2.94427e8	−0.717523
$290$	−5.12664e8	−1.23435
$291$	7.37975e8	1.75556
$292$	−3.09314e7	−0.0727043
$293$	2.76652e8	0.642536	0.321268	−	0.946988i	$-0.395891\pi$
$293$	2.76652e8	0.642536	0.321268	+	0.946988i	$0.395891\pi$
$294$	4.58115e8	1.05138
$295$	−8.19831e8	−1.85929
$296$	1.53222e8	0.343400
$297$	3.49170e6	0.00773375
$298$	4.08663e8	0.894557
$299$	6.89976e7	0.149274
$300$	−1.38987e8	−0.297201
$301$	3.65440e7	0.0772384
$302$	−1.34583e8	−0.281169
$303$	−3.47241e7	−0.0717103
$304$	8.72706e8	1.78160
$305$	1.03189e9	2.08249
$306$	−6.35341e7	−0.126760
$307$	−1.05781e7	−0.0208653	−0.0104327	−	0.999946i	$-0.503321\pi$
$307$	−1.05781e7	−0.0208653	−0.0104327	+	0.999946i	$0.503321\pi$
$308$	317331.	0.000618850 0
$309$	−1.39516e8	−0.269011
$310$	1.01585e9	1.93670
$311$	−5.64572e7	−0.106429	−0.0532143	−	0.998583i	$-0.516947\pi$
$311$	−5.64572e7	−0.106429	−0.0532143	+	0.998583i	$0.516947\pi$
$312$	2.20616e8	0.411240
$313$	−7.00757e8	−1.29170	−0.645851	−	0.763463i	$-0.723497\pi$
$313$	−7.00757e8	−1.29170	−0.645851	+	0.763463i	$0.723497\pi$
$314$	7.82874e8	1.42705
$315$	2.58935e7	0.0466771
$316$	3.51784e8	0.627151
$317$	−4.05544e8	−0.715041	−0.357520	−	0.933905i	$-0.616378\pi$
$317$	−4.05544e8	−0.715041	−0.357520	+	0.933905i	$0.616378\pi$
$318$	1.48037e8	0.258153
$319$	3.25967e6	0.00562220
$320$	−1.43972e8	−0.245614
$321$	−5.09346e8	−0.859499
$322$	2.75723e7	0.0460233
$323$	−4.59070e8	−0.758002
$324$	−2.20856e8	−0.360746
$325$	3.11418e8	0.503213
$326$	1.43724e9	2.29757
$327$	−2.70174e8	−0.427293
$328$	4.63987e8	0.726017
$329$	−1.91990e8	−0.297230
$330$	6.67958e6	0.0102318
$331$	4.73572e8	0.717775	0.358887	−	0.933381i	$-0.383156\pi$
$331$	4.73572e8	0.717775	0.358887	+	0.933381i	$0.383156\pi$
$332$	−8.60210e7	−0.129009
$333$	7.09829e7	0.105341
$334$	−2.81431e8	−0.413294
$335$	−1.10813e9	−1.61039
$336$	1.41650e8	0.203717
$337$	−3.83572e8	−0.545937	−0.272969	−	0.962023i	$-0.588005\pi$
$337$	−3.83572e8	−0.545937	−0.272969	+	0.962023i	$0.588005\pi$
$338$	−4.19847e8	−0.591401
$339$	−7.27469e8	−1.01418
$340$	−2.37107e8	−0.327167
$341$	−6.45908e6	−0.00882125
$342$	2.51629e8	0.340149
$343$	−2.67449e8	−0.357858
$344$	−2.05418e8	−0.272072
$345$	1.86023e8	0.243893
$346$	−1.83780e9	−2.38524
$347$	2.62821e8	0.337681	0.168841	−	0.985643i	$-0.445998\pi$
$347$	2.62821e8	0.337681	0.168841	+	0.985643i	$0.445998\pi$
$348$	−2.59183e8	−0.329669
$349$	−5.33512e8	−0.671823	−0.335912	−	0.941894i	$-0.609044\pi$
$349$	−5.33512e8	−0.671823	−0.335912	+	0.941894i	$0.609044\pi$
$350$	1.24446e8	0.155147
$351$	6.22069e8	0.767828
$352$	−5.16026e6	−0.00630626
$353$	3.57541e8	0.432628	0.216314	−	0.976324i	$-0.430596\pi$
$353$	3.57541e8	0.432628	0.216314	+	0.976324i	$0.430596\pi$
$354$	−1.29313e9	−1.54928
$355$	−5.44837e8	−0.646350
$356$	4.78211e8	0.561751
$357$	−7.45121e7	−0.0866738
$358$	−1.10098e9	−1.26821
$359$	7.42830e8	0.847342	0.423671	−	0.905816i	$-0.360741\pi$
$359$	7.42830e8	0.847342	0.423671	+	0.905816i	$0.360741\pi$
$360$	−1.45550e8	−0.164420
$361$	9.24289e8	1.03403
$362$	−1.71867e9	−1.90420
$363$	8.16803e8	0.896281
$364$	5.65345e7	0.0614411
$365$	1.86853e8	0.201129
$366$	1.62760e9	1.73526
$367$	−9.86306e8	−1.04155	−0.520775	−	0.853694i	$-0.674357\pi$
$367$	−9.86306e8	−1.04155	−0.520775	+	0.853694i	$0.674357\pi$
$368$	−2.49021e8	−0.260476
$369$	2.14950e8	0.222713
$370$	8.26485e8	0.848260
$371$	−4.24850e7	−0.0431944
$372$	5.13573e8	0.517252
$373$	−1.02745e8	−0.102513	−0.0512566	−	0.998686i	$-0.516323\pi$
$373$	−1.02745e8	−0.102513	−0.0512566	+	0.998686i	$0.516323\pi$
$374$	4.70359e6	0.00464921
$375$	−3.54857e8	−0.347492
$376$	1.07920e9	1.04699
$377$	5.80731e8	0.558188
$378$	2.48586e8	0.236732
$379$	1.60090e9	1.51052	0.755262	−	0.655423i	$-0.227510\pi$
$379$	1.60090e9	1.51052	0.755262	+	0.655423i	$0.227510\pi$
$380$	9.39070e8	0.877921
$381$	1.49427e9	1.38417
$382$	4.45494e7	0.0408902
$383$	−6.54385e8	−0.595165	−0.297583	−	0.954696i	$-0.596180\pi$
$383$	−6.54385e8	−0.595165	−0.297583	+	0.954696i	$0.596180\pi$
$384$	−1.09689e9	−0.988566
$385$	−1.91696e6	−0.00171199
$386$	−1.47800e9	−1.30803
$387$	−9.51635e7	−0.0834607
$388$	−1.06302e9	−0.923911
$389$	5.47686e8	0.471746	0.235873	−	0.971784i	$-0.424205\pi$
$389$	5.47686e8	0.471746	0.235873	+	0.971784i	$0.424205\pi$
$390$	1.19001e9	1.01584
$391$	1.30993e8	0.110823
$392$	7.39029e8	0.619669
$393$	1.07016e9	0.889353
$394$	−8.44884e8	−0.695921
$395$	−2.12509e9	−1.73495
$396$	−826356.	−0.000668704 0
$397$	−6.70333e8	−0.537680	−0.268840	−	0.963185i	$-0.586640\pi$
$397$	−6.70333e8	−0.537680	−0.268840	+	0.963185i	$0.586640\pi$
$398$	7.46945e8	0.593879
$399$	2.95107e8	0.232581
$400$	−1.12394e9	−0.878081
$401$	−1.96156e9	−1.51914	−0.759569	−	0.650427i	$-0.774590\pi$
$401$	−1.96156e9	−1.51914	−0.759569	+	0.650427i	$0.774590\pi$
$402$	−1.74786e9	−1.34188
$403$	−1.15072e9	−0.875798
$404$	5.00185e7	0.0377395
$405$	1.33416e9	0.997969
$406$	2.32067e8	0.172097
$407$	−5.25504e6	−0.00386363
$408$	4.18840e8	0.305308
$409$	1.11366e9	0.804858	0.402429	−	0.915451i	$-0.368166\pi$
$409$	1.11366e9	0.804858	0.402429	+	0.915451i	$0.368166\pi$
$410$	2.50276e9	1.79340
$411$	1.36217e9	0.967802
$412$	2.00967e8	0.141574
$413$	3.71112e8	0.259227
$414$	−7.18005e7	−0.0497309
$415$	5.19644e8	0.356892
$416$	−9.19332e8	−0.626103
$417$	1.59741e9	1.07880
$418$	−1.86287e7	−0.0124757
$419$	−1.56650e9	−1.04035	−0.520176	−	0.854059i	$-0.674134\pi$
$419$	−1.56650e9	−1.04035	−0.520176	+	0.854059i	$0.674134\pi$
$420$	1.52421e8	0.100386
$421$	6.66031e8	0.435018	0.217509	−	0.976058i	$-0.430207\pi$
$421$	6.66031e8	0.435018	0.217509	+	0.976058i	$0.430207\pi$
$422$	−7.91908e8	−0.512957
$423$	4.99957e8	0.321175
$424$	2.38813e8	0.152152
$425$	5.91229e8	0.373589
$426$	−8.59376e8	−0.538579
$427$	−4.67104e8	−0.290346
$428$	7.33690e8	0.452334
$429$	−7.56644e6	−0.00462691
$430$	−1.10803e9	−0.672067
$431$	2.09866e9	1.26262	0.631310	−	0.775531i	$-0.282518\pi$
$431$	2.09866e9	1.26262	0.631310	+	0.775531i	$0.282518\pi$
$432$	−2.24512e9	−1.33982
$433$	6.94969e6	0.00411394	0.00205697	−	0.999998i	$-0.499345\pi$
$433$	6.94969e6	0.00411394	0.00205697	+	0.999998i	$0.499345\pi$
$434$	−4.59844e8	−0.270020
$435$	1.56569e9	0.911999
$436$	3.89173e8	0.224874
$437$	−5.18800e8	−0.297382
$438$	2.94725e8	0.167594
$439$	−2.01271e9	−1.13541	−0.567707	−	0.823231i	$-0.692169\pi$
$439$	−2.01271e9	−1.13541	−0.567707	+	0.823231i	$0.692169\pi$
$440$	1.07755e7	0.00603047
$441$	3.42369e8	0.190090
$442$	8.37975e8	0.461586
$443$	−4.14786e7	−0.0226679	−0.0113340	−	0.999936i	$-0.503608\pi$
$443$	−4.14786e7	−0.0226679	−0.0113340	+	0.999936i	$0.503608\pi$
$444$	4.17838e8	0.226552
$445$	−2.88882e9	−1.55403
$446$	1.01631e9	0.542441
$447$	−1.24807e9	−0.660941
$448$	6.51718e7	0.0342442
$449$	−8.30315e8	−0.432893	−0.216447	−	0.976294i	$-0.569447\pi$
$449$	−8.30315e8	−0.432893	−0.216447	+	0.976294i	$0.569447\pi$
$450$	−3.24068e8	−0.167646
$451$	−1.59133e7	−0.00816851
$452$	1.04789e9	0.533740
$453$	4.11022e8	0.207741
$454$	3.73360e9	1.87255
$455$	−3.41519e8	−0.169971
$456$	−1.65883e9	−0.819265
$457$	6.85053e7	0.0335751	0.0167876	−	0.999859i	$-0.494656\pi$
$457$	6.85053e7	0.0335751	0.0167876	+	0.999859i	$0.494656\pi$
$458$	−3.13177e9	−1.52321
$459$	1.18100e9	0.570042
$460$	−2.67957e8	−0.128355
$461$	3.77005e9	1.79223	0.896114	−	0.443824i	$-0.146378\pi$
$461$	3.77005e9	1.79223	0.896114	+	0.443824i	$0.146378\pi$
$462$	−3.02364e6	−0.00142654
$463$	5.54935e8	0.259841	0.129921	−	0.991524i	$-0.458528\pi$
$463$	5.54935e8	0.259841	0.129921	+	0.991524i	$0.458528\pi$
$464$	−2.09593e9	−0.974010
$465$	−3.10244e9	−1.43093
$466$	−4.84439e9	−2.21763
$467$	−2.27560e9	−1.03392	−0.516959	−	0.856010i	$-0.672936\pi$
$467$	−2.27560e9	−1.03392	−0.516959	+	0.856010i	$0.672936\pi$
$468$	−1.47221e8	−0.0663908
$469$	5.01615e8	0.224525
$470$	5.82122e9	2.58626
$471$	−2.39092e9	−1.05437
$472$	−2.08606e9	−0.913125
$473$	7.04520e6	0.00306111
$474$	−3.35192e9	−1.44567
$475$	−2.34158e9	−1.00249
$476$	1.07331e8	0.0456144
$477$	1.10635e8	0.0466741
$478$	2.17547e9	0.911079
$479$	2.27144e9	0.944337	0.472168	−	0.881508i	$-0.343471\pi$
$479$	2.27144e9	0.944337	0.472168	+	0.881508i	$0.343471\pi$
$480$	−2.47859e9	−1.02296
$481$	−9.36219e8	−0.383592
$482$	2.81489e9	1.14498
$483$	−8.42068e7	−0.0340042
$484$	−1.17657e9	−0.471691
$485$	6.42158e9	2.55591
$486$	−1.18832e9	−0.469578
$487$	1.35084e9	0.529972	0.264986	−	0.964252i	$-0.414633\pi$
$487$	1.35084e9	0.529972	0.264986	+	0.964252i	$0.414633\pi$
$488$	2.62564e9	1.02274
$489$	−4.38938e9	−1.69755
$490$	3.98635e9	1.53070
$491$	−4.17684e9	−1.59244	−0.796220	−	0.605007i	$-0.793170\pi$
$491$	−4.17684e9	−1.59244	−0.796220	+	0.605007i	$0.793170\pi$
$492$	1.26530e9	0.478977
$493$	1.10252e9	0.414403
$494$	−3.31882e9	−1.23862
$495$	4.99193e6	0.00184991
$496$	4.15310e9	1.52822
$497$	2.46631e8	0.0901157
$498$	8.19638e8	0.297385
$499$	−4.43292e8	−0.159712	−0.0798562	−	0.996806i	$-0.525446\pi$
$499$	−4.43292e8	−0.159712	−0.0798562	+	0.996806i	$0.525446\pi$
$500$	5.11156e8	0.182877
$501$	8.59500e8	0.305361
$502$	4.03471e9	1.42347
$503$	1.62588e9	0.569640	0.284820	−	0.958581i	$-0.408066\pi$
$503$	1.62588e9	0.569640	0.284820	+	0.958581i	$0.408066\pi$
$504$	6.58861e7	0.0229238
$505$	−3.02156e8	−0.104403
$506$	5.31557e6	0.00182399
$507$	1.28223e9	0.436955
$508$	−2.15242e9	−0.728459
$509$	3.90930e9	1.31398	0.656988	−	0.753901i	$-0.271830\pi$
$509$	3.90930e9	1.31398	0.656988	+	0.753901i	$0.271830\pi$
$510$	2.25924e9	0.754166
$511$	−8.45827e7	−0.0280420
$512$	8.86579e8	0.291926
$513$	−4.67739e9	−1.52965
$514$	4.14012e9	1.34475
$515$	−1.21402e9	−0.391651
$516$	−5.60177e8	−0.179494
$517$	−3.70131e7	−0.0117798
$518$	−3.74125e8	−0.118267
$519$	5.61271e9	1.76233
$520$	1.91971e9	0.598722
$521$	−6.02035e9	−1.86505	−0.932523	−	0.361111i	$-0.882397\pi$
$521$	−6.02035e9	−1.86505	−0.932523	+	0.361111i	$0.882397\pi$
$522$	−6.04322e8	−0.185961
$523$	4.68995e9	1.43355	0.716774	−	0.697306i	$-0.245618\pi$
$523$	4.68995e9	1.43355	0.716774	+	0.697306i	$0.245618\pi$
$524$	−1.54152e9	−0.468046
$525$	−3.80063e8	−0.114630
$526$	−3.36089e9	−1.00694
$527$	−2.18466e9	−0.650200
$528$	2.73082e7	0.00807372
$529$	1.48036e8	0.0434783
$530$	1.28817e9	0.375843
$531$	−9.66407e8	−0.280110
$532$	−4.25088e8	−0.122402
$533$	−2.83506e9	−0.810992
$534$	−4.55655e9	−1.29492
$535$	−4.43214e9	−1.25134
$536$	−2.81963e9	−0.790889
$537$	3.36245e9	0.937012
$538$	6.98559e9	1.93404
$539$	−2.53464e7	−0.00697197
$540$	−2.41585e9	−0.660225
$541$	2.20440e9	0.598550	0.299275	−	0.954167i	$-0.403255\pi$
$541$	2.20440e9	0.598550	0.299275	+	0.954167i	$0.403255\pi$
$542$	3.86542e9	1.04280
$543$	5.24889e9	1.40692
$544$	−1.74536e9	−0.464824
$545$	−2.35095e9	−0.622094
$546$	−5.38680e8	−0.141631
$547$	1.29680e9	0.338779	0.169390	−	0.985549i	$-0.445820\pi$
$547$	1.29680e9	0.338779	0.169390	+	0.985549i	$0.445820\pi$
$548$	−1.96215e9	−0.509331
$549$	1.21638e9	0.313736
$550$	2.39916e7	0.00614879
$551$	−4.36657e9	−1.11201
$552$	4.73335e8	0.119779
$553$	9.61963e8	0.241891
$554$	−1.25105e9	−0.312602
$555$	−2.52412e9	−0.626735
$556$	−2.30100e9	−0.567746
$557$	1.66581e9	0.408444	0.204222	−	0.978925i	$-0.434534\pi$
$557$	1.66581e9	0.408444	0.204222	+	0.978925i	$0.434534\pi$
$558$	1.19747e9	0.291773
$559$	1.25515e9	0.303916
$560$	1.23258e9	0.296591
$561$	−1.43649e7	−0.00343506
$562$	3.89878e9	0.926514
$563$	−2.66413e9	−0.629181	−0.314590	−	0.949228i	$-0.601867\pi$
$563$	−2.66413e9	−0.629181	−0.314590	+	0.949228i	$0.601867\pi$
$564$	2.94298e9	0.690733
$565$	−6.33017e9	−1.47654
$566$	−7.86603e9	−1.82347
$567$	−6.03936e8	−0.139139
$568$	−1.38634e9	−0.317432
$569$	−4.86641e9	−1.10743	−0.553714	−	0.832707i	$-0.686790\pi$
$569$	−4.86641e9	−1.10743	−0.553714	+	0.832707i	$0.686790\pi$
$570$	−8.94778e9	−2.02373
$571$	−4.12268e9	−0.926731	−0.463365	−	0.886167i	$-0.653358\pi$
$571$	−4.12268e9	−0.926731	−0.463365	+	0.886167i	$0.653358\pi$
$572$	1.08991e7	0.00243503
$573$	−1.36055e8	−0.0302116
$574$	−1.13292e9	−0.250040
$575$	6.68153e8	0.146568
$576$	−1.69713e8	−0.0370029
$577$	−4.73370e9	−1.02585	−0.512927	−	0.858432i	$-0.671439\pi$
$577$	−4.73370e9	−1.02585	−0.512927	+	0.858432i	$0.671439\pi$
$578$	−4.04106e9	−0.870457
$579$	4.51386e9	0.966436
$580$	−2.25531e9	−0.479964
$581$	−2.35227e8	−0.0497588
$582$	1.01288e10	2.12975
$583$	−8.19055e6	−0.00171188
$584$	4.75449e8	0.0987776
$585$	8.89343e8	0.183664
$586$	3.79709e9	0.779488
$587$	5.07303e8	0.103522	0.0517612	−	0.998659i	$-0.483517\pi$
$587$	5.07303e8	0.103522	0.0517612	+	0.998659i	$0.483517\pi$
$588$	2.01534e9	0.408816
$589$	8.65240e9	1.74475
$590$	−1.12523e10	−2.25559
$591$	2.58030e9	0.514180
$592$	3.37893e9	0.669349
$593$	2.80919e9	0.553210	0.276605	−	0.960984i	$-0.410791\pi$
$593$	2.80919e9	0.553210	0.276605	+	0.960984i	$0.410791\pi$
$594$	4.79242e7	0.00938214
$595$	−6.48376e8	−0.126188
$596$	1.79779e9	0.347838
$597$	−2.28120e9	−0.438786
$598$	9.47003e8	0.181091
$599$	7.18428e9	1.36581	0.682904	−	0.730508i	$-0.260717\pi$
$599$	7.18428e9	1.36581	0.682904	+	0.730508i	$0.260717\pi$
$600$	2.13638e9	0.403784
$601$	5.67050e9	1.06552	0.532759	−	0.846267i	$-0.321155\pi$
$601$	5.67050e9	1.06552	0.532759	+	0.846267i	$0.321155\pi$
$602$	5.01572e8	0.0937012
$603$	−1.30625e9	−0.242613
$604$	−5.92059e8	−0.109329
$605$	7.10751e9	1.30489
$606$	−4.76593e8	−0.0869948
$607$	9.85285e8	0.178814	0.0894070	−	0.995995i	$-0.471503\pi$
$607$	9.85285e8	0.178814	0.0894070	+	0.995995i	$0.471503\pi$
$608$	6.91254e9	1.24731
$609$	−7.08741e8	−0.127153
$610$	1.41628e10	2.52636
$611$	−6.59411e9	−1.16953
$612$	−2.79499e8	−0.0492891
$613$	8.22759e9	1.44265	0.721325	−	0.692597i	$-0.243533\pi$
$613$	8.22759e9	1.44265	0.721325	+	0.692597i	$0.243533\pi$
$614$	−1.45187e8	−0.0253126
$615$	−7.64352e9	−1.32504
$616$	−4.87772e6	−0.000840783 0
$617$	−6.50643e9	−1.11518	−0.557590	−	0.830117i	$-0.688274\pi$
$617$	−6.50643e9	−1.11518	−0.557590	+	0.830117i	$0.688274\pi$
$618$	−1.91488e9	−0.326349
$619$	−2.33559e9	−0.395803	−0.197901	−	0.980222i	$-0.563413\pi$
$619$	−2.33559e9	−0.395803	−0.197901	+	0.980222i	$0.563413\pi$
$620$	4.46892e9	0.753064
$621$	1.33466e9	0.223641
$622$	−7.74884e8	−0.129113
$623$	1.30768e9	0.216667
$624$	4.86512e9	0.801581
$625$	−7.37809e9	−1.20883
$626$	−9.61800e9	−1.56702
$627$	5.68927e7	0.00921765
$628$	3.44402e9	0.554889
$629$	−1.77742e9	−0.284782
$630$	3.55392e8	0.0566260
$631$	5.24733e9	0.831449	0.415725	−	0.909490i	$-0.363528\pi$
$631$	5.24733e9	0.831449	0.415725	+	0.909490i	$0.363528\pi$
$632$	−5.40730e9	−0.852061
$633$	2.41851e9	0.378997
$634$	−5.56615e9	−0.867447
$635$	1.30026e10	2.01521
$636$	6.51246e8	0.100380
$637$	−4.51562e9	−0.692197
$638$	4.47395e7	0.00682054
$639$	−6.42247e8	−0.0973754
$640$	−9.54477e9	−1.43925
$641$	9.28633e8	0.139265	0.0696324	−	0.997573i	$-0.477817\pi$
$641$	9.28633e8	0.139265	0.0696324	+	0.997573i	$0.477817\pi$
$642$	−6.99085e9	−1.04270
$643$	−1.05720e10	−1.56827	−0.784133	−	0.620593i	$-0.786892\pi$
$643$	−1.05720e10	−1.56827	−0.784133	+	0.620593i	$0.786892\pi$
$644$	1.21296e8	0.0178956
$645$	3.38397e9	0.496555
$646$	−6.30081e9	−0.919565
$647$	−7.21397e9	−1.04715	−0.523576	−	0.851979i	$-0.675402\pi$
$647$	−7.21397e9	−1.04715	−0.523576	+	0.851979i	$0.675402\pi$
$648$	3.39479e9	0.490117
$649$	7.15456e7	0.0102737
$650$	4.27425e9	0.610469
$651$	1.40438e9	0.199504
$652$	6.32271e9	0.893380
$653$	−2.53937e9	−0.356886	−0.178443	−	0.983950i	$-0.557106\pi$
$653$	−2.53937e9	−0.356886	−0.178443	+	0.983950i	$0.557106\pi$
$654$	−3.70817e9	−0.518368
$655$	9.31212e9	1.29480
$656$	1.02321e10	1.41514
$657$	2.20260e8	0.0303010
$658$	−2.63509e9	−0.360583
$659$	8.08492e9	1.10047	0.550233	−	0.835011i	$-0.314539\pi$
$659$	8.08492e9	1.10047	0.550233	+	0.835011i	$0.314539\pi$
$660$	2.93848e7	0.00397850
$661$	−1.08284e10	−1.45834	−0.729169	−	0.684333i	$-0.760093\pi$
$661$	−1.08284e10	−1.45834	−0.729169	+	0.684333i	$0.760093\pi$
$662$	6.49985e9	0.870764
$663$	−2.55920e9	−0.341042
$664$	1.32223e9	0.175275
$665$	2.56791e9	0.338614
$666$	9.74251e8	0.127794
$667$	1.24597e9	0.162580
$668$	−1.23807e9	−0.160705
$669$	−3.10383e9	−0.400781
$670$	−1.52092e10	−1.95364
$671$	−9.00514e7	−0.0115070
$672$	1.12198e9	0.142624
$673$	−2.97284e9	−0.375941	−0.187970	−	0.982175i	$-0.560191\pi$
$673$	−2.97284e9	−0.375941	−0.187970	+	0.982175i	$0.560191\pi$
$674$	−5.26459e9	−0.662300
$675$	6.02394e9	0.753906
$676$	−1.84699e9	−0.229959
$677$	4.49801e9	0.557135	0.278567	−	0.960417i	$-0.410140\pi$
$677$	4.49801e9	0.557135	0.278567	+	0.960417i	$0.410140\pi$
$678$	−9.98462e9	−1.23035
$679$	−2.90685e9	−0.356352
$680$	3.64459e9	0.444496
$681$	−1.14025e10	−1.38352
$682$	−8.86518e7	−0.0107014
$683$	2.98727e9	0.358759	0.179379	−	0.983780i	$-0.442591\pi$
$683$	2.98727e9	0.358759	0.179379	+	0.983780i	$0.442591\pi$
$684$	1.10696e9	0.132263
$685$	1.18531e10	1.40902
$686$	−3.67077e9	−0.434133
$687$	9.56454e9	1.12542
$688$	−4.52997e9	−0.530317
$689$	−1.45920e9	−0.169960
$690$	2.55319e9	0.295877
$691$	1.94984e9	0.224815	0.112408	−	0.993662i	$-0.464144\pi$
$691$	1.94984e9	0.224815	0.112408	+	0.993662i	$0.464144\pi$
$692$	−8.08486e9	−0.927472
$693$	−2.25969e6	−0.000257919 0
$694$	3.60726e9	0.409656
$695$	1.39001e10	1.57062
$696$	3.98391e9	0.447896
$697$	−5.38238e9	−0.602087
$698$	−7.32253e9	−0.815018
$699$	1.47949e10	1.63849
$700$	5.47464e8	0.0603271
$701$	−5.22279e9	−0.572650	−0.286325	−	0.958133i	$-0.592434\pi$
$701$	−5.22279e9	−0.572650	−0.286325	+	0.958133i	$0.592434\pi$
$702$	8.53799e9	0.931485
$703$	7.03952e9	0.764186
$704$	1.25643e7	0.00135716
$705$	−1.77782e10	−1.91085
$706$	4.90731e9	0.524840
$707$	1.36777e8	0.0145561
$708$	−5.68872e9	−0.602418
$709$	−5.14916e9	−0.542594	−0.271297	−	0.962496i	$-0.587452\pi$
$709$	−5.14916e9	−0.542594	−0.271297	+	0.962496i	$0.587452\pi$
$710$	−7.47797e9	−0.784115
$711$	−2.50503e9	−0.261378
$712$	−7.35060e9	−0.763208
$713$	−2.46890e9	−0.255089
$714$	−1.02269e9	−0.105148
$715$	−6.58403e7	−0.00673629
$716$	−4.84345e9	−0.493128
$717$	−6.64397e9	−0.673148
$718$	1.01955e10	1.02795
$719$	1.08945e9	0.109309	0.0546543	−	0.998505i	$-0.482594\pi$
$719$	1.08945e9	0.109309	0.0546543	+	0.998505i	$0.482594\pi$
$720$	−3.20974e9	−0.320484
$721$	5.49549e8	0.0546050
$722$	1.26860e10	1.25442
$723$	−8.59677e9	−0.845963
$724$	−7.56079e9	−0.740427
$725$	5.62363e9	0.548067
$726$	1.12107e10	1.08732
$727$	−7.46975e9	−0.721000	−0.360500	−	0.932759i	$-0.617394\pi$
$727$	−7.46975e9	−0.721000	−0.360500	+	0.932759i	$0.617394\pi$
$728$	−8.68996e8	−0.0834753
$729$	1.16288e10	1.11170
$730$	2.56459e9	0.243999
$731$	2.38290e9	0.225629
$732$	7.16015e9	0.674735
$733$	−5.34556e9	−0.501337	−0.250668	−	0.968073i	$-0.580650\pi$
$733$	−5.34556e9	−0.501337	−0.250668	+	0.968073i	$0.580650\pi$
$734$	−1.35372e10	−1.26355
$735$	−1.21744e10	−1.13095
$736$	−1.97245e9	−0.182361
$737$	9.67047e7	0.00889839
$738$	2.95022e9	0.270183
$739$	−2.81976e9	−0.257014	−0.128507	−	0.991709i	$-0.541019\pi$
$739$	−2.81976e9	−0.257014	−0.128507	+	0.991709i	$0.541019\pi$
$740$	3.63587e9	0.329836
$741$	1.01358e10	0.915154
$742$	−5.83113e8	−0.0524010
$743$	1.06791e10	0.955155	0.477578	−	0.878590i	$-0.341515\pi$
$743$	1.06791e10	0.955155	0.477578	+	0.878590i	$0.341515\pi$
$744$	−7.89416e9	−0.702750
$745$	−1.08602e10	−0.962260
$746$	−1.41019e9	−0.124363
$747$	6.12549e8	0.0537674
$748$	2.06920e7	0.00180779
$749$	2.00629e9	0.174465
$750$	−4.87047e9	−0.421557
$751$	1.65038e10	1.42181	0.710907	−	0.703286i	$-0.248284\pi$
$751$	1.65038e10	1.42181	0.710907	+	0.703286i	$0.248284\pi$
$752$	2.37989e10	2.04078
$753$	−1.23221e10	−1.05173
$754$	7.97062e9	0.677162
$755$	3.57656e9	0.302449
$756$	1.09358e9	0.0920502
$757$	−6.98732e8	−0.0585430	−0.0292715	−	0.999571i	$-0.509319\pi$
$757$	−6.98732e8	−0.0585430	−0.0292715	+	0.999571i	$0.509319\pi$
$758$	2.19726e10	1.83248
$759$	−1.62339e7	−0.00134765
$760$	−1.44345e10	−1.19276
$761$	6.64496e9	0.546570	0.273285	−	0.961933i	$-0.411890\pi$
$761$	6.64496e9	0.546570	0.273285	+	0.961933i	$0.411890\pi$
$762$	2.05090e10	1.67920
$763$	1.06420e9	0.0867339
$764$	1.95981e8	0.0158997
$765$	1.68842e9	0.136354
$766$	−8.98153e9	−0.722021
$767$	1.27463e10	1.02000
$768$	−1.29372e10	−1.03057
$769$	2.08487e10	1.65324	0.826620	−	0.562761i	$-0.190261\pi$
$769$	2.08487e10	1.65324	0.826620	+	0.562761i	$0.190261\pi$
$770$	−2.63106e7	−0.00207689
$771$	−1.26441e10	−0.993565
$772$	−6.50201e9	−0.508613
$773$	−1.95781e10	−1.52455	−0.762274	−	0.647254i	$-0.775917\pi$
$773$	−1.95781e10	−1.52455	−0.762274	+	0.647254i	$0.775917\pi$
$774$	−1.30613e9	−0.101250
$775$	−1.11433e10	−0.859919
$776$	1.63397e10	1.25525
$777$	1.14259e9	0.0873809
$778$	7.51708e9	0.572295
$779$	2.13171e10	1.61565
$780$	5.23508e9	0.394996
$781$	4.75472e7	0.00357146
$782$	1.79789e9	0.134444
$783$	1.12334e10	0.836269
$784$	1.62974e10	1.20785
$785$	−2.08049e10	−1.53505
$786$	1.46881e10	1.07891
$787$	−4.00363e9	−0.292781	−0.146390	−	0.989227i	$-0.546766\pi$
$787$	−4.00363e9	−0.292781	−0.146390	+	0.989227i	$0.546766\pi$
$788$	−3.71681e9	−0.270601
$789$	1.02643e10	0.743975
$790$	−2.91672e10	−2.10475
$791$	2.86547e9	0.205863
$792$	1.27020e7	0.000908517 0
$793$	−1.60432e10	−1.14244
$794$	−9.20041e9	−0.652282
$795$	−3.93410e9	−0.277690
$796$	3.28596e9	0.230923
$797$	2.37163e10	1.65937	0.829685	−	0.558232i	$-0.188520\pi$
$797$	2.37163e10	1.65937	0.829685	+	0.558232i	$0.188520\pi$
$798$	4.05039e9	0.282154
$799$	−1.25190e10	−0.868271
$800$	−8.90255e9	−0.614751
$801$	−3.40530e9	−0.234122
$802$	−2.69227e10	−1.84293
$803$	−1.63064e7	−0.00111136
$804$	−7.68917e9	−0.521775
$805$	−7.32736e8	−0.0495065
$806$	−1.57939e10	−1.06247
$807$	−2.13342e10	−1.42896
$808$	−7.68837e8	−0.0512737
$809$	−1.29604e10	−0.860592	−0.430296	−	0.902688i	$-0.641591\pi$
$809$	−1.29604e10	−0.860592	−0.430296	+	0.902688i	$0.641591\pi$
$810$	1.83116e10	1.21068
$811$	−1.64248e10	−1.08125	−0.540625	−	0.841264i	$-0.681812\pi$
$811$	−1.64248e10	−1.08125	−0.540625	+	0.841264i	$0.681812\pi$
$812$	1.02091e9	0.0669177
$813$	−1.18051e10	−0.770466
$814$	−7.21263e7	−0.00468714
$815$	−3.81948e10	−2.47145
$816$	9.23647e9	0.595101
$817$	−9.43756e9	−0.605456
$818$	1.52851e10	0.976408
$819$	−4.02578e8	−0.0256069
$820$	1.10101e10	0.697340
$821$	−2.05578e10	−1.29651	−0.648256	−	0.761423i	$-0.724501\pi$
$821$	−2.05578e10	−1.29651	−0.648256	+	0.761423i	$0.724501\pi$
$822$	1.86960e10	1.17408
$823$	−1.10026e10	−0.688011	−0.344006	−	0.938968i	$-0.611784\pi$
$823$	−1.10026e10	−0.688011	−0.344006	+	0.938968i	$0.611784\pi$
$824$	−3.08907e9	−0.192346
$825$	−7.32712e7	−0.00454302
$826$	5.09357e9	0.314480
$827$	9.32463e9	0.573274	0.286637	−	0.958039i	$-0.407463\pi$
$827$	9.32463e9	0.573274	0.286637	+	0.958039i	$0.407463\pi$
$828$	−3.15865e8	−0.0193373
$829$	2.18659e10	1.33299	0.666493	−	0.745511i	$-0.267795\pi$
$829$	2.18659e10	1.33299	0.666493	+	0.745511i	$0.267795\pi$
$830$	7.13218e9	0.432961
$831$	3.82076e9	0.230965
$832$	2.23840e9	0.134743
$833$	−8.57294e9	−0.513893
$834$	2.19247e10	1.30874
$835$	7.47905e9	0.444574
$836$	−8.19514e7	−0.00485103
$837$	−2.22592e10	−1.31211
$838$	−2.15004e10	−1.26210
$839$	−1.18840e10	−0.694700	−0.347350	−	0.937736i	$-0.612918\pi$
$839$	−1.18840e10	−0.694700	−0.347350	+	0.937736i	$0.612918\pi$
$840$	−2.34288e9	−0.136387
$841$	−6.76294e9	−0.392057
$842$	9.14138e9	0.527739
$843$	−1.19070e10	−0.684552
$844$	−3.48376e9	−0.199457
$845$	1.11575e10	0.636161
$846$	6.86199e9	0.389631
$847$	−3.21735e9	−0.181931
$848$	5.26642e9	0.296572
$849$	2.40231e10	1.34726
$850$	8.11471e9	0.453217
$851$	−2.00868e9	−0.111727
$852$	−3.78057e9	−0.209420
$853$	2.26976e10	1.25216	0.626078	−	0.779761i	$-0.284659\pi$
$853$	2.26976e10	1.25216	0.626078	+	0.779761i	$0.284659\pi$
$854$	−6.41106e9	−0.352231
$855$	−6.68705e9	−0.365892
$856$	−1.12776e10	−0.614552
$857$	1.71504e10	0.930768	0.465384	−	0.885109i	$-0.345916\pi$
$857$	1.71504e10	0.930768	0.465384	+	0.885109i	$0.345916\pi$
$858$	−1.03850e8	−0.00561310
$859$	2.14474e9	0.115451	0.0577255	−	0.998332i	$-0.481615\pi$
$859$	2.14474e9	0.115451	0.0577255	+	0.998332i	$0.481615\pi$
$860$	−4.87445e9	−0.261325
$861$	3.45999e9	0.184741
$862$	2.88045e10	1.53174
$863$	−2.64750e10	−1.40216	−0.701081	−	0.713082i	$-0.747299\pi$
$863$	−2.64750e10	−1.40216	−0.701081	+	0.713082i	$0.747299\pi$
$864$	−1.77832e10	−0.938018
$865$	4.88397e10	2.56576
$866$	9.53855e7	0.00499080
$867$	1.23415e10	0.643135
$868$	−2.02294e9	−0.104994
$869$	1.85454e8	0.00958664
$870$	2.14894e10	1.10639
$871$	1.72285e10	0.883456
$872$	−5.98200e9	−0.305519
$873$	7.56968e9	0.385059
$874$	−7.12060e9	−0.360767
$875$	1.39777e9	0.0705354
$876$	1.29655e9	0.0651668
$877$	2.88546e10	1.44450	0.722249	−	0.691633i	$-0.243108\pi$
$877$	2.88546e10	1.44450	0.722249	+	0.691633i	$0.243108\pi$
$878$	−2.76247e10	−1.37742
$879$	−1.15964e10	−0.575922
$880$	2.37625e8	0.0117545
$881$	−1.83638e10	−0.904790	−0.452395	−	0.891818i	$-0.649430\pi$
$881$	−1.83638e10	−0.904790	−0.452395	+	0.891818i	$0.649430\pi$
$882$	4.69906e9	0.230606
$883$	1.95197e10	0.954135	0.477067	−	0.878867i	$-0.341700\pi$
$883$	1.95197e10	0.954135	0.477067	+	0.878867i	$0.341700\pi$
$884$	3.68642e9	0.179482
$885$	3.43649e10	1.66653
$886$	−5.69300e8	−0.0274994
$887$	−3.57604e10	−1.72056	−0.860280	−	0.509822i	$-0.829711\pi$
$887$	−3.57604e10	−1.72056	−0.860280	+	0.509822i	$0.829711\pi$
$888$	−6.42262e9	−0.307799
$889$	−5.88586e9	−0.280966
$890$	−3.96494e10	−1.88526
$891$	−1.16431e8	−0.00551437
$892$	4.47093e9	0.210921
$893$	4.95818e10	2.32992
$894$	−1.71299e10	−0.801816
$895$	2.92587e10	1.36419
$896$	4.32062e9	0.200664
$897$	−2.89218e9	−0.133799
$898$	−1.13962e10	−0.525161
$899$	−2.07800e10	−0.953863
$900$	−1.42564e9	−0.0651871
$901$	−2.77030e9	−0.126180
$902$	−2.18413e8	−0.00990957
$903$	−1.53182e9	−0.0692309
$904$	−1.61071e10	−0.725151
$905$	4.56739e10	2.04832
$906$	5.64134e9	0.252019
$907$	1.33486e10	0.594033	0.297016	−	0.954872i	$-0.404008\pi$
$907$	1.33486e10	0.594033	0.297016	+	0.954872i	$0.404008\pi$
$908$	1.64248e10	0.728117
$909$	−3.56178e8	−0.0157287
$910$	−4.68740e9	−0.206199
$911$	1.44202e10	0.631913	0.315957	−	0.948774i	$-0.397675\pi$
$911$	1.44202e10	0.631913	0.315957	+	0.948774i	$0.397675\pi$
$912$	−3.65813e10	−1.59690
$913$	−4.53486e7	−0.00197204
$914$	9.40246e8	0.0407314
$915$	−4.32537e10	−1.86659
$916$	−1.37773e10	−0.592283
$917$	−4.21531e9	−0.180525
$918$	1.62094e10	0.691542
$919$	2.38461e10	1.01347	0.506737	−	0.862101i	$-0.330852\pi$
$919$	2.38461e10	1.01347	0.506737	+	0.862101i	$0.330852\pi$
$920$	4.11879e9	0.174386
$921$	4.43405e8	0.0187022
$922$	5.17444e10	2.17423
$923$	8.47083e9	0.354585
$924$	−1.33016e7	−0.000554692 0
$925$	−9.06608e9	−0.376637
$926$	7.61656e9	0.315225
$927$	−1.43107e9	−0.0590040
$928$	−1.66014e10	−0.681911
$929$	−3.57952e10	−1.46477	−0.732386	−	0.680890i	$-0.761593\pi$
$929$	−3.57952e10	−1.46477	−0.732386	+	0.680890i	$0.761593\pi$
$930$	−4.25814e10	−1.73592
$931$	3.39534e10	1.37898
$932$	−2.13114e10	−0.862297
$933$	2.36652e9	0.0953948
$934$	−3.12329e10	−1.25429
$935$	−1.24998e8	−0.00500108
$936$	2.26294e9	0.0902000
$937$	−3.70969e10	−1.47316	−0.736578	−	0.676352i	$-0.763560\pi$
$937$	−3.70969e10	−1.47316	−0.736578	+	0.676352i	$0.763560\pi$
$938$	6.88474e9	0.272381
$939$	2.93737e10	1.15779
$940$	2.56087e10	1.00563
$941$	1.06625e9	0.0417152	0.0208576	−	0.999782i	$-0.493360\pi$
$941$	1.06625e9	0.0417152	0.0208576	+	0.999782i	$0.493360\pi$
$942$	−3.28158e10	−1.27910
$943$	−6.08267e9	−0.236213
$944$	−4.60029e10	−1.77985
$945$	−6.60620e9	−0.254648
$946$	9.66964e7	0.00371357
$947$	−1.45973e10	−0.558530	−0.279265	−	0.960214i	$-0.590091\pi$
$947$	−1.45973e10	−0.558530	−0.279265	+	0.960214i	$0.590091\pi$
$948$	−1.47458e10	−0.562132
$949$	−2.90509e9	−0.110339
$950$	−3.21385e10	−1.21617
$951$	1.69992e10	0.640910
$952$	−1.64980e9	−0.0619727
$953$	−2.80924e10	−1.05139	−0.525694	−	0.850673i	$-0.676194\pi$
$953$	−2.80924e10	−1.05139	−0.525694	+	0.850673i	$0.676194\pi$
$954$	1.51847e9	0.0566224
$955$	−1.18390e9	−0.0439850
$956$	9.57034e9	0.354262
$957$	−1.36636e8	−0.00503933
$958$	3.11758e10	1.14562
$959$	−5.36555e9	−0.196449
$960$	6.03489e9	0.220151
$961$	1.36631e10	0.496613
$962$	−1.28497e10	−0.465352
$963$	−5.22456e9	−0.188520
$964$	1.23833e10	0.445211
$965$	3.92779e10	1.40703
$966$	−1.15575e9	−0.0412519
$967$	−5.05633e10	−1.79822	−0.899110	−	0.437723i	$-0.855785\pi$
$967$	−5.05633e10	−1.79822	−0.899110	+	0.437723i	$0.855785\pi$
$968$	1.80851e10	0.640851
$969$	1.92429e10	0.679418
$970$	8.81371e10	3.10069
$971$	−3.90879e10	−1.37017	−0.685085	−	0.728463i	$-0.740235\pi$
$971$	−3.90879e10	−1.37017	−0.685085	+	0.728463i	$0.740235\pi$
$972$	−5.22767e9	−0.182590
$973$	−6.29214e9	−0.218979
$974$	1.85405e10	0.642932
$975$	−1.30537e10	−0.451043
$976$	5.79019e10	1.99351
$977$	2.06771e9	0.0709348	0.0354674	−	0.999371i	$-0.488708\pi$
$977$	2.06771e9	0.0709348	0.0354674	+	0.999371i	$0.488708\pi$
$978$	−6.02449e10	−2.05937
$979$	2.52103e8	0.00858695
$980$	1.75367e10	0.595193
$981$	−2.77127e9	−0.0937211
$982$	−5.73278e10	−1.93186
$983$	4.38211e10	1.47145	0.735725	−	0.677280i	$-0.236841\pi$
$983$	4.38211e10	1.47145	0.735725	+	0.677280i	$0.236841\pi$
$984$	−1.94490e10	−0.650749
$985$	2.24529e10	0.748591
$986$	1.51323e10	0.502730
$987$	8.04765e9	0.266415
$988$	−1.46002e10	−0.481624
$989$	2.69294e9	0.0885197
$990$	6.85149e7	0.00224420
$991$	−4.54621e10	−1.48386	−0.741928	−	0.670479i	$-0.766088\pi$
$991$	−4.54621e10	−1.48386	−0.741928	+	0.670479i	$0.766088\pi$
$992$	3.28959e10	1.06992
$993$	−1.98508e10	−0.643361
$994$	3.38505e9	0.109323
$995$	−1.98501e10	−0.638826
$996$	3.60575e9	0.115635
$997$	−2.21452e10	−0.707696	−0.353848	−	0.935303i	$-0.615127\pi$
$997$	−2.21452e10	−0.707696	−0.353848	+	0.935303i	$0.615127\pi$
$998$	−6.08425e9	−0.193754
$999$	−1.81098e10	−0.574692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.8.a.a.1.5	✓	5
3.2	odd	2		207.8.a.b.1.1		5
4.3	odd	2		368.8.a.e.1.4		5
5.4	even	2		575.8.a.a.1.1		5
23.22	odd	2		529.8.a.b.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.8.a.a.1.5	✓	5	1.1	even	1	trivial
207.8.a.b.1.1		5	3.2	odd	2
368.8.a.e.1.4		5	4.3	odd	2
529.8.a.b.1.5		5	23.22	odd	2
575.8.a.a.1.1		5	5.4	even	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q+13.7251 q^{2} -41.9171 q^{3} +60.3796 q^{4} -364.747 q^{5} -575.318 q^{6} +165.110 q^{7} -928.099 q^{8} -429.959 q^{9} +O(q^{10})\)	\(q+13.7251 q^{2} -41.9171 q^{3} +60.3796 q^{4} -364.747 q^{5} -575.318 q^{6} +165.110 q^{7} -928.099 q^{8} -429.959 q^{9} -5006.20 q^{10} +31.8310 q^{11} -2530.94 q^{12} +5670.88 q^{13} +2266.15 q^{14} +15289.1 q^{15} -20466.9 q^{16} +10766.2 q^{17} -5901.25 q^{18} -42639.9 q^{19} -22023.3 q^{20} -6920.91 q^{21} +436.885 q^{22} +12167.0 q^{23} +38903.2 q^{24} +54915.2 q^{25} +77833.7 q^{26} +109695. q^{27} +9969.26 q^{28} +102406. q^{29} +209845. q^{30} -202918. q^{31} -162114. q^{32} -1334.26 q^{33} +147768. q^{34} -60223.2 q^{35} -25960.8 q^{36} -165092. q^{37} -585239. q^{38} -237707. q^{39} +338521. q^{40} -499932. q^{41} -94990.6 q^{42} +221332. q^{43} +1921.94 q^{44} +156826. q^{45} +166994. q^{46} -1.16280e6 q^{47} +857912. q^{48} -796282. q^{49} +753719. q^{50} -451288. q^{51} +342406. q^{52} -257314. q^{53} +1.50558e6 q^{54} -11610.2 q^{55} -153238. q^{56} +1.78734e6 q^{57} +1.40553e6 q^{58} +2.24767e6 q^{59} +923151. q^{60} -2.82905e6 q^{61} -2.78508e6 q^{62} -70990.4 q^{63} +394718. q^{64} -2.06844e6 q^{65} -18312.9 q^{66} +3.03807e6 q^{67} +650060. q^{68} -510005. q^{69} -826573. q^{70} +1.49374e6 q^{71} +399045. q^{72} -512282. q^{73} -2.26592e6 q^{74} -2.30188e6 q^{75} -2.57458e6 q^{76} +5255.60 q^{77} -3.26256e6 q^{78} +5.82621e6 q^{79} +7.46523e6 q^{80} -3.65778e6 q^{81} -6.86164e6 q^{82} -1.42467e6 q^{83} -417882. q^{84} -3.92694e6 q^{85} +3.03781e6 q^{86} -4.29255e6 q^{87} -29542.3 q^{88} +7.92006e6 q^{89} +2.15246e6 q^{90} +936318. q^{91} +734639. q^{92} +8.50573e6 q^{93} -1.59596e7 q^{94} +1.55528e7 q^{95} +6.79536e6 q^{96} -1.76056e7 q^{97} -1.09291e7 q^{98} -13686.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9}+O(q^{10}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9	\( 5 q - 16 q^{2} - 68 q^{3} + 256 q^{4} - 56 q^{5} + 538 q^{6} - 1156 q^{7} - 5952 q^{8} - 2195 q^{9} - 11260 q^{10} - 1318 q^{11} - 28012 q^{12} - 19662 q^{13} - 6848 q^{14} - 24600 q^{15} + 32448 q^{16} - 5002 q^{17} - 24650 q^{18} - 38314 q^{19} + 104440 q^{20} + 33280 q^{21} - 74872 q^{22} + 60835 q^{23} + 361992 q^{24} + 54959 q^{25} + 345430 q^{26} + 100900 q^{27} + 90800 q^{28} - 150634 q^{29} + 515724 q^{30} - 179940 q^{31} - 404032 q^{32} - 619688 q^{33} + 32116 q^{34} - 374032 q^{35} + 510524 q^{36} - 752672 q^{37} + 456808 q^{38} - 207996 q^{39} - 1082576 q^{40} - 1192910 q^{41} - 250520 q^{42} - 932646 q^{43} + 2467104 q^{44} - 389952 q^{45} - 194672 q^{46} - 1008460 q^{47} - 1916464 q^{48} - 2005219 q^{49} + 1571224 q^{50} - 211520 q^{51} - 1740516 q^{52} + 897104 q^{53} + 1844686 q^{54} + 1203168 q^{55} + 3050144 q^{56} + 3137192 q^{57} + 5685090 q^{58} + 1020972 q^{59} - 1479384 q^{60} - 2758364 q^{61} + 2661794 q^{62} + 1135132 q^{63} + 5173248 q^{64} - 1350472 q^{65} + 11693212 q^{66} - 1523138 q^{67} + 2501304 q^{68} - 827356 q^{69} - 2794240 q^{70} + 3044884 q^{71} - 6740904 q^{72} - 8872022 q^{73} + 1408492 q^{74} + 1960276 q^{75} - 17963952 q^{76} - 3501672 q^{77} - 15280362 q^{78} - 4437540 q^{79} + 12197536 q^{80} - 7995203 q^{81} - 7738154 q^{82} - 4637362 q^{83} + 2663744 q^{84} - 8625728 q^{85} - 3025868 q^{86} + 17151068 q^{87} - 41815040 q^{88} + 6381402 q^{89} - 4970376 q^{90} + 3240808 q^{91} + 3114752 q^{92} + 7185076 q^{93} - 13893974 q^{94} + 15762704 q^{95} + 40696544 q^{96} - 6432034 q^{97} + 22652640 q^{98} + 30201754 q^{99}+O(q^{100}) \) 5 * q - 16 * q^2 - 68 * q^3 + 256 * q^4 - 56 * q^5 + 538 * q^6 - 1156 * q^7 - 5952 * q^8 - 2195 * q^9 - 11260 * q^10 - 1318 * q^11 - 28012 * q^12 - 19662 * q^13 - 6848 * q^14 - 24600 * q^15 + 32448 * q^16 - 5002 * q^17 - 24650 * q^18 - 38314 * q^19 + 104440 * q^20 + 33280 * q^21 - 74872 * q^22 + 60835 * q^23 + 361992 * q^24 + 54959 * q^25 + 345430 * q^26 + 100900 * q^27 + 90800 * q^28 - 150634 * q^29 + 515724 * q^30 - 179940 * q^31 - 404032 * q^32 - 619688 * q^33 + 32116 * q^34 - 374032 * q^35 + 510524 * q^36 - 752672 * q^37 + 456808 * q^38 - 207996 * q^39 - 1082576 * q^40 - 1192910 * q^41 - 250520 * q^42 - 932646 * q^43 + 2467104 * q^44 - 389952 * q^45 - 194672 * q^46 - 1008460 * q^47 - 1916464 * q^48 - 2005219 * q^49 + 1571224 * q^50 - 211520 * q^51 - 1740516 * q^52 + 897104 * q^53 + 1844686 * q^54 + 1203168 * q^55 + 3050144 * q^56 + 3137192 * q^57 + 5685090 * q^58 + 1020972 * q^59 - 1479384 * q^60 - 2758364 * q^61 + 2661794 * q^62 + 1135132 * q^63 + 5173248 * q^64 - 1350472 * q^65 + 11693212 * q^66 - 1523138 * q^67 + 2501304 * q^68 - 827356 * q^69 - 2794240 * q^70 + 3044884 * q^71 - 6740904 * q^72 - 8872022 * q^73 + 1408492 * q^74 + 1960276 * q^75 - 17963952 * q^76 - 3501672 * q^77 - 15280362 * q^78 - 4437540 * q^79 + 12197536 * q^80 - 7995203 * q^81 - 7738154 * q^82 - 4637362 * q^83 + 2663744 * q^84 - 8625728 * q^85 - 3025868 * q^86 + 17151068 * q^87 - 41815040 * q^88 + 6381402 * q^89 - 4970376 * q^90 + 3240808 * q^91 + 3114752 * q^92 + 7185076 * q^93 - 13893974 * q^94 + 15762704 * q^95 + 40696544 * q^96 - 6432034 * q^97 + 22652640 * q^98 + 30201754 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more