LMFDB - Embedded newform 560.8.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,8,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("560.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	560.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:	$174.935614271$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2283x^{2} - 2749x + 794610 $ x^4 - x^3 - 2283x^2 - 2749x + 794610
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.6591$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-45.4647 q^{3} +125.000 q^{5} -343.000 q^{7} -119.962 q^{9} +O(q^{10})$	$q-45.4647 q^{3} +125.000 q^{5} -343.000 q^{7} -119.962 q^{9} +3210.39 q^{11} +544.208 q^{13} -5683.09 q^{15} -17000.0 q^{17} -20648.4 q^{19} +15594.4 q^{21} +94486.0 q^{23} +15625.0 q^{25} +104885. q^{27} -83775.3 q^{29} -24882.0 q^{31} -145959. q^{33} -42875.0 q^{35} -215157. q^{37} -24742.2 q^{39} -109255. q^{41} +21472.6 q^{43} -14995.3 q^{45} -164409. q^{47} +117649. q^{49} +772899. q^{51} -1.04717e6 q^{53} +401298. q^{55} +938775. q^{57} +2.46158e6 q^{59} +2.81068e6 q^{61} +41147.0 q^{63} +68026.0 q^{65} -761881. q^{67} -4.29578e6 q^{69} +5.02959e6 q^{71} +1.46692e6 q^{73} -710386. q^{75} -1.10116e6 q^{77} -4.00013e6 q^{79} -4.50622e6 q^{81} -3.02833e6 q^{83} -2.12500e6 q^{85} +3.80882e6 q^{87} -1.26511e6 q^{89} -186663. q^{91} +1.13125e6 q^{93} -2.58105e6 q^{95} +1.10133e7 q^{97} -385125. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 55 q^{3} + 500 q^{5} - 1372 q^{7} + 1063 q^{9}+O(q^{10}) $ 4 * q + 55 * q^3 + 500 * q^5 - 1372 * q^7 + 1063 * q^9	$ 4 q + 55 q^{3} + 500 q^{5} - 1372 q^{7} + 1063 q^{9} + 633 q^{11} - 5397 q^{13} + 6875 q^{15} - 41987 q^{17} + 41958 q^{19} - 18865 q^{21} + 63650 q^{23} + 62500 q^{25} + 101737 q^{27} + 1071 q^{29} - 79700 q^{31} - 92499 q^{33} - 171500 q^{35} - 131804 q^{37} - 168611 q^{39} - 313834 q^{41} - 604338 q^{43} + 132875 q^{45} - 2429 q^{47} + 470596 q^{49} - 1178733 q^{51} - 590946 q^{53} + 79125 q^{55} + 1307410 q^{57} - 437296 q^{59} - 369710 q^{61} - 364609 q^{63} - 674625 q^{65} - 537812 q^{67} - 4360534 q^{69} + 2096808 q^{71} - 6033412 q^{73} + 859375 q^{75} - 217119 q^{77} - 5554681 q^{79} - 13721804 q^{81} + 2234760 q^{83} - 5248375 q^{85} + 5900337 q^{87} - 5347574 q^{89} + 1851171 q^{91} - 20106036 q^{93} + 5244750 q^{95} - 2106023 q^{97} + 3460842 q^{99}+O(q^{100}) $ 4 * q + 55 * q^3 + 500 * q^5 - 1372 * q^7 + 1063 * q^9 + 633 * q^11 - 5397 * q^13 + 6875 * q^15 - 41987 * q^17 + 41958 * q^19 - 18865 * q^21 + 63650 * q^23 + 62500 * q^25 + 101737 * q^27 + 1071 * q^29 - 79700 * q^31 - 92499 * q^33 - 171500 * q^35 - 131804 * q^37 - 168611 * q^39 - 313834 * q^41 - 604338 * q^43 + 132875 * q^45 - 2429 * q^47 + 470596 * q^49 - 1178733 * q^51 - 590946 * q^53 + 79125 * q^55 + 1307410 * q^57 - 437296 * q^59 - 369710 * q^61 - 364609 * q^63 - 674625 * q^65 - 537812 * q^67 - 4360534 * q^69 + 2096808 * q^71 - 6033412 * q^73 + 859375 * q^75 - 217119 * q^77 - 5554681 * q^79 - 13721804 * q^81 + 2234760 * q^83 - 5248375 * q^85 + 5900337 * q^87 - 5347574 * q^89 + 1851171 * q^91 - 20106036 * q^93 + 5244750 * q^95 - 2106023 * q^97 + 3460842 * 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 $

$2$	0	0
$2$	0	0
$3$	−45.4647	−0.972187	−0.486094	−	0.873907i	$-0.661579\pi$
$3$	−45.4647	−0.972187	−0.486094	+	0.873907i	$0.661579\pi$
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	−119.962	−0.0548524
$10$	0	0
$11$	3210.39	0.727249	0.363624	−	0.931546i	$-0.381539\pi$
$11$	3210.39	0.727249	0.363624	+	0.931546i	$0.381539\pi$
$12$	0	0
$13$	544.208	0.0687010	0.0343505	−	0.999410i	$-0.489064\pi$
$13$	544.208	0.0687010	0.0343505	+	0.999410i	$0.489064\pi$
$14$	0	0
$15$	−5683.09	−0.434775
$16$	0	0
$17$	−17000.0	−0.839223	−0.419612	−	0.907704i	$-0.637834\pi$
$17$	−17000.0	−0.839223	−0.419612	+	0.907704i	$0.637834\pi$
$18$	0	0
$19$	−20648.4	−0.690636	−0.345318	−	0.938486i	$-0.612229\pi$
$19$	−20648.4	−0.690636	−0.345318	+	0.938486i	$0.612229\pi$
$20$	0	0
$21$	15594.4	0.367452
$22$	0	0
$23$	94486.0	1.61927	0.809636	−	0.586932i	$-0.199664\pi$
$23$	94486.0	1.61927	0.809636	+	0.586932i	$0.199664\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	104885.	1.02551
$28$	0	0
$29$	−83775.3	−0.637856	−0.318928	−	0.947779i	$-0.603323\pi$
$29$	−83775.3	−0.637856	−0.318928	+	0.947779i	$0.603323\pi$
$30$	0	0
$31$	−24882.0	−0.150010	−0.0750049	−	0.997183i	$-0.523897\pi$
$31$	−24882.0	−0.150010	−0.0750049	+	0.997183i	$0.523897\pi$
$32$	0	0
$33$	−145959.	−0.707022
$34$	0	0
$35$	−42875.0	−0.169031
$36$	0	0
$37$	−215157.	−0.698310	−0.349155	−	0.937065i	$-0.613531\pi$
$37$	−215157.	−0.698310	−0.349155	+	0.937065i	$0.613531\pi$
$38$	0	0
$39$	−24742.2	−0.0667902
$40$	0	0
$41$	−109255.	−0.247569	−0.123785	−	0.992309i	$-0.539503\pi$
$41$	−109255.	−0.247569	−0.123785	+	0.992309i	$0.539503\pi$
$42$	0	0
$43$	21472.6	0.0411855	0.0205927	−	0.999788i	$-0.493445\pi$
$43$	21472.6	0.0411855	0.0205927	+	0.999788i	$0.493445\pi$
$44$	0	0
$45$	−14995.3	−0.0245307
$46$	0	0
$47$	−164409.	−0.230985	−0.115492	−	0.993308i	$-0.536845\pi$
$47$	−164409.	−0.230985	−0.115492	+	0.993308i	$0.536845\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	772899.	0.815882
$52$	0	0
$53$	−1.04717e6	−0.966166	−0.483083	−	0.875574i	$-0.660483\pi$
$53$	−1.04717e6	−0.966166	−0.483083	+	0.875574i	$0.660483\pi$
$54$	0	0
$55$	401298.	0.325235
$56$	0	0
$57$	938775.	0.671428
$58$	0	0
$59$	2.46158e6	1.56038	0.780192	−	0.625541i	$-0.215122\pi$
$59$	2.46158e6	1.56038	0.780192	+	0.625541i	$0.215122\pi$
$60$	0	0
$61$	2.81068e6	1.58546	0.792732	−	0.609570i	$-0.208658\pi$
$61$	2.81068e6	1.58546	0.792732	+	0.609570i	$0.208658\pi$
$62$	0	0
$63$	41147.0	0.0207323
$64$	0	0
$65$	68026.0	0.0307240
$66$	0	0
$67$	−761881.	−0.309475	−0.154737	−	0.987956i	$-0.549453\pi$
$67$	−761881.	−0.309475	−0.154737	+	0.987956i	$0.549453\pi$
$68$	0	0
$69$	−4.29578e6	−1.57424
$70$	0	0
$71$	5.02959e6	1.66774	0.833870	−	0.551961i	$-0.186120\pi$
$71$	5.02959e6	1.66774	0.833870	+	0.551961i	$0.186120\pi$
$72$	0	0
$73$	1.46692e6	0.441342	0.220671	−	0.975348i	$-0.429175\pi$
$73$	1.46692e6	0.441342	0.220671	+	0.975348i	$0.429175\pi$
$74$	0	0
$75$	−710386.	−0.194437
$76$	0	0
$77$	−1.10116e6	−0.274874
$78$	0	0
$79$	−4.00013e6	−0.912808	−0.456404	−	0.889773i	$-0.650863\pi$
$79$	−4.00013e6	−0.912808	−0.456404	+	0.889773i	$0.650863\pi$
$80$	0	0
$81$	−4.50622e6	−0.942139
$82$	0	0
$83$	−3.02833e6	−0.581340	−0.290670	−	0.956823i	$-0.593878\pi$
$83$	−3.02833e6	−0.581340	−0.290670	+	0.956823i	$0.593878\pi$
$84$	0	0
$85$	−2.12500e6	−0.375312
$86$	0	0
$87$	3.80882e6	0.620116
$88$	0	0
$89$	−1.26511e6	−0.190224	−0.0951119	−	0.995467i	$-0.530321\pi$
$89$	−1.26511e6	−0.190224	−0.0951119	+	0.995467i	$0.530321\pi$
$90$	0	0
$91$	−186663.	−0.0259665
$92$	0	0
$93$	1.13125e6	0.145838
$94$	0	0
$95$	−2.58105e6	−0.308862
$96$	0	0
$97$	1.10133e7	1.22523	0.612616	−	0.790381i	$-0.290117\pi$
$97$	1.10133e7	1.22523	0.612616	+	0.790381i	$0.290117\pi$
$98$	0	0
$99$	−385125.	−0.0398913
$100$	0	0
$101$	1.19027e6	0.114953	0.0574767	−	0.998347i	$-0.481695\pi$
$101$	1.19027e6	0.114953	0.0574767	+	0.998347i	$0.481695\pi$
$102$	0	0
$103$	1.10519e7	0.996565	0.498283	−	0.867015i	$-0.333964\pi$
$103$	1.10519e7	0.996565	0.498283	+	0.867015i	$0.333964\pi$
$104$	0	0
$105$	1.94930e6	0.164330
$106$	0	0
$107$	−537042.	−0.0423804	−0.0211902	−	0.999775i	$-0.506746\pi$
$107$	−537042.	−0.0423804	−0.0211902	+	0.999775i	$0.506746\pi$
$108$	0	0
$109$	−1.64733e6	−0.121839	−0.0609196	−	0.998143i	$-0.519403\pi$
$109$	−1.64733e6	−0.121839	−0.0609196	+	0.998143i	$0.519403\pi$
$110$	0	0
$111$	9.78203e6	0.678888
$112$	0	0
$113$	−1.22085e6	−0.0795951	−0.0397975	−	0.999208i	$-0.512671\pi$
$113$	−1.22085e6	−0.0795951	−0.0397975	+	0.999208i	$0.512671\pi$
$114$	0	0
$115$	1.18107e7	0.724161
$116$	0	0
$117$	−65284.4	−0.00376841
$118$	0	0
$119$	5.83100e6	0.317197
$120$	0	0
$121$	−9.18059e6	−0.471109
$122$	0	0
$123$	4.96723e6	0.240683
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	3.51712e7	1.52361	0.761805	−	0.647806i	$-0.224313\pi$
$127$	3.51712e7	1.52361	0.761805	+	0.647806i	$0.224313\pi$
$128$	0	0
$129$	−976243.	−0.0400400
$130$	0	0
$131$	−4.20205e6	−0.163309	−0.0816547	−	0.996661i	$-0.526020\pi$
$131$	−4.20205e6	−0.163309	−0.0816547	+	0.996661i	$0.526020\pi$
$132$	0	0
$133$	7.08241e6	0.261036
$134$	0	0
$135$	1.31107e7	0.458624
$136$	0	0
$137$	−1.55435e7	−0.516447	−0.258223	−	0.966085i	$-0.583137\pi$
$137$	−1.55435e7	−0.516447	−0.258223	+	0.966085i	$0.583137\pi$
$138$	0	0
$139$	−2.21085e7	−0.698244	−0.349122	−	0.937077i	$-0.613520\pi$
$139$	−2.21085e7	−0.698244	−0.349122	+	0.937077i	$0.613520\pi$
$140$	0	0
$141$	7.47481e6	0.224561
$142$	0	0
$143$	1.74712e6	0.0499627
$144$	0	0
$145$	−1.04719e7	−0.285258
$146$	0	0
$147$	−5.34888e6	−0.138884
$148$	0	0
$149$	−6.35020e7	−1.57266	−0.786331	−	0.617805i	$-0.788022\pi$
$149$	−6.35020e7	−1.57266	−0.786331	+	0.617805i	$0.788022\pi$
$150$	0	0
$151$	−1.01009e7	−0.238748	−0.119374	−	0.992849i	$-0.538089\pi$
$151$	−1.01009e7	−0.238748	−0.119374	+	0.992849i	$0.538089\pi$
$152$	0	0
$153$	2.03936e6	0.0460334
$154$	0	0
$155$	−3.11025e6	−0.0670864
$156$	0	0
$157$	−1.60843e7	−0.331706	−0.165853	−	0.986150i	$-0.553038\pi$
$157$	−1.60843e7	−0.331706	−0.165853	+	0.986150i	$0.553038\pi$
$158$	0	0
$159$	4.76093e7	0.939294
$160$	0	0
$161$	−3.24087e7	−0.612027
$162$	0	0
$163$	6.51042e7	1.17748	0.588738	−	0.808324i	$-0.299625\pi$
$163$	6.51042e7	1.17748	0.588738	+	0.808324i	$0.299625\pi$
$164$	0	0
$165$	−1.82449e7	−0.316190
$166$	0	0
$167$	1.74774e7	0.290381	0.145191	−	0.989404i	$-0.453620\pi$
$167$	1.74774e7	0.290381	0.145191	+	0.989404i	$0.453620\pi$
$168$	0	0
$169$	−6.24524e7	−0.995280
$170$	0	0
$171$	2.47703e6	0.0378831
$172$	0	0
$173$	2.00103e7	0.293828	0.146914	−	0.989149i	$-0.453066\pi$
$173$	2.00103e7	0.293828	0.146914	+	0.989149i	$0.453066\pi$
$174$	0	0
$175$	−5.35938e6	−0.0755929
$176$	0	0
$177$	−1.11915e8	−1.51698
$178$	0	0
$179$	1.09241e8	1.42364	0.711818	−	0.702364i	$-0.247872\pi$
$179$	1.09241e8	1.42364	0.711818	+	0.702364i	$0.247872\pi$
$180$	0	0
$181$	−1.08439e8	−1.35928	−0.679641	−	0.733545i	$-0.737865\pi$
$181$	−1.08439e8	−1.35928	−0.679641	+	0.733545i	$0.737865\pi$
$182$	0	0
$183$	−1.27787e8	−1.54137
$184$	0	0
$185$	−2.68946e7	−0.312294
$186$	0	0
$187$	−5.45765e7	−0.610324
$188$	0	0
$189$	−3.59757e7	−0.387608
$190$	0	0
$191$	−1.22600e8	−1.27314	−0.636568	−	0.771220i	$-0.719647\pi$
$191$	−1.22600e8	−1.27314	−0.636568	+	0.771220i	$0.719647\pi$
$192$	0	0
$193$	4.08805e7	0.409323	0.204661	−	0.978833i	$-0.434391\pi$
$193$	4.08805e7	0.409323	0.204661	+	0.978833i	$0.434391\pi$
$194$	0	0
$195$	−3.09278e6	−0.0298695
$196$	0	0
$197$	−8.95402e7	−0.834423	−0.417212	−	0.908809i	$-0.636993\pi$
$197$	−8.95402e7	−0.834423	−0.417212	+	0.908809i	$0.636993\pi$
$198$	0	0
$199$	−4.38218e7	−0.394189	−0.197094	−	0.980385i	$-0.563151\pi$
$199$	−4.38218e7	−0.394189	−0.197094	+	0.980385i	$0.563151\pi$
$200$	0	0
$201$	3.46387e7	0.300867
$202$	0	0
$203$	2.87349e7	0.241087
$204$	0	0
$205$	−1.36568e7	−0.110716
$206$	0	0
$207$	−1.13347e7	−0.0888210
$208$	0	0
$209$	−6.62895e7	−0.502264
$210$	0	0
$211$	7.67543e7	0.562489	0.281245	−	0.959636i	$-0.409253\pi$
$211$	7.67543e7	0.562489	0.281245	+	0.959636i	$0.409253\pi$
$212$	0	0
$213$	−2.28669e8	−1.62136
$214$	0	0
$215$	2.68407e6	0.0184187
$216$	0	0
$217$	8.53453e6	0.0566984
$218$	0	0
$219$	−6.66929e7	−0.429067
$220$	0	0
$221$	−9.25153e6	−0.0576555
$222$	0	0
$223$	−2.59819e8	−1.56893	−0.784467	−	0.620171i	$-0.787063\pi$
$223$	−2.59819e8	−1.56893	−0.784467	+	0.620171i	$0.787063\pi$
$224$	0	0
$225$	−1.87441e6	−0.0109705
$226$	0	0
$227$	−1.96531e8	−1.11517	−0.557583	−	0.830121i	$-0.688271\pi$
$227$	−1.96531e8	−1.11517	−0.557583	+	0.830121i	$0.688271\pi$
$228$	0	0
$229$	−2.66432e8	−1.46610	−0.733049	−	0.680176i	$-0.761903\pi$
$229$	−2.66432e8	−1.46610	−0.733049	+	0.680176i	$0.761903\pi$
$230$	0	0
$231$	5.00640e7	0.267229
$232$	0	0
$233$	−7.99943e7	−0.414298	−0.207149	−	0.978309i	$-0.566419\pi$
$233$	−7.99943e7	−0.414298	−0.207149	+	0.978309i	$0.566419\pi$
$234$	0	0
$235$	−2.05511e7	−0.103300
$236$	0	0
$237$	1.81865e8	0.887420
$238$	0	0
$239$	−1.68154e8	−0.796736	−0.398368	−	0.917226i	$-0.630423\pi$
$239$	−1.68154e8	−0.796736	−0.398368	+	0.917226i	$0.630423\pi$
$240$	0	0
$241$	1.50643e8	0.693251	0.346625	−	0.938004i	$-0.387328\pi$
$241$	1.50643e8	0.693251	0.346625	+	0.938004i	$0.387328\pi$
$242$	0	0
$243$	−2.45103e7	−0.109579
$244$	0	0
$245$	1.47061e7	0.0638877
$246$	0	0
$247$	−1.12370e7	−0.0474474
$248$	0	0
$249$	1.37682e8	0.565171
$250$	0	0
$251$	−3.37833e8	−1.34848	−0.674239	−	0.738513i	$-0.735528\pi$
$251$	−3.37833e8	−1.34848	−0.674239	+	0.738513i	$0.735528\pi$
$252$	0	0
$253$	3.03337e8	1.17761
$254$	0	0
$255$	9.66124e7	0.364873
$256$	0	0
$257$	5.52025e7	0.202858	0.101429	−	0.994843i	$-0.467658\pi$
$257$	5.52025e7	0.202858	0.101429	+	0.994843i	$0.467658\pi$
$258$	0	0
$259$	7.37987e7	0.263937
$260$	0	0
$261$	1.00499e7	0.0349880
$262$	0	0
$263$	−4.00605e8	−1.35791	−0.678956	−	0.734179i	$-0.737567\pi$
$263$	−4.00605e8	−1.35791	−0.678956	+	0.734179i	$0.737567\pi$
$264$	0	0
$265$	−1.30896e8	−0.432083
$266$	0	0
$267$	5.75181e7	0.184933
$268$	0	0
$269$	−1.74470e8	−0.546497	−0.273248	−	0.961944i	$-0.588098\pi$
$269$	−1.74470e8	−0.546497	−0.273248	+	0.961944i	$0.588098\pi$
$270$	0	0
$271$	−4.11225e8	−1.25513	−0.627563	−	0.778566i	$-0.715947\pi$
$271$	−4.11225e8	−1.25513	−0.627563	+	0.778566i	$0.715947\pi$
$272$	0	0
$273$	8.48659e6	0.0252443
$274$	0	0
$275$	5.01623e7	0.145450
$276$	0	0
$277$	−2.15901e8	−0.610345	−0.305173	−	0.952297i	$-0.598714\pi$
$277$	−2.15901e8	−0.610345	−0.305173	+	0.952297i	$0.598714\pi$
$278$	0	0
$279$	2.98490e6	0.00822840
$280$	0	0
$281$	−2.18626e8	−0.587800	−0.293900	−	0.955836i	$-0.594953\pi$
$281$	−2.18626e8	−0.587800	−0.293900	+	0.955836i	$0.594953\pi$
$282$	0	0
$283$	−1.75909e8	−0.461354	−0.230677	−	0.973030i	$-0.574094\pi$
$283$	−1.75909e8	−0.461354	−0.230677	+	0.973030i	$0.574094\pi$
$284$	0	0
$285$	1.17347e8	0.300272
$286$	0	0
$287$	3.74744e7	0.0935723
$288$	0	0
$289$	−1.21339e8	−0.295704
$290$	0	0
$291$	−5.00718e8	−1.19115
$292$	0	0
$293$	7.37770e8	1.71350	0.856750	−	0.515732i	$-0.172480\pi$
$293$	7.37770e8	1.71350	0.856750	+	0.515732i	$0.172480\pi$
$294$	0	0
$295$	3.07697e8	0.697825
$296$	0	0
$297$	3.36722e8	0.745804
$298$	0	0
$299$	5.14200e7	0.111246
$300$	0	0
$301$	−7.36509e6	−0.0155667
$302$	0	0
$303$	−5.41153e7	−0.111756
$304$	0	0
$305$	3.51335e8	0.709041
$306$	0	0
$307$	−2.18670e8	−0.431325	−0.215663	−	0.976468i	$-0.569191\pi$
$307$	−2.18670e8	−0.431325	−0.215663	+	0.976468i	$0.569191\pi$
$308$	0	0
$309$	−5.02470e8	−0.968848
$310$	0	0
$311$	−5.81203e8	−1.09564	−0.547818	−	0.836598i	$-0.684541\pi$
$311$	−5.81203e8	−1.09564	−0.547818	+	0.836598i	$0.684541\pi$
$312$	0	0
$313$	5.61741e7	0.103545	0.0517727	−	0.998659i	$-0.483513\pi$
$313$	5.61741e7	0.103545	0.0517727	+	0.998659i	$0.483513\pi$
$314$	0	0
$315$	5.14338e6	0.00927175
$316$	0	0
$317$	−6.35251e8	−1.12005	−0.560025	−	0.828475i	$-0.689209\pi$
$317$	−6.35251e8	−1.12005	−0.560025	+	0.828475i	$0.689209\pi$
$318$	0	0
$319$	−2.68951e8	−0.463880
$320$	0	0
$321$	2.44164e7	0.0412017
$322$	0	0
$323$	3.51023e8	0.579598
$324$	0	0
$325$	8.50325e6	0.0137402
$326$	0	0
$327$	7.48952e7	0.118451
$328$	0	0
$329$	5.63923e7	0.0873041
$330$	0	0
$331$	5.50411e8	0.834236	0.417118	−	0.908852i	$-0.363040\pi$
$331$	5.50411e8	0.834236	0.417118	+	0.908852i	$0.363040\pi$
$332$	0	0
$333$	2.58107e7	0.0383040
$334$	0	0
$335$	−9.52351e7	−0.138401
$336$	0	0
$337$	−5.33773e8	−0.759717	−0.379859	−	0.925045i	$-0.624027\pi$
$337$	−5.33773e8	−0.759717	−0.379859	+	0.925045i	$0.624027\pi$
$338$	0	0
$339$	5.55054e7	0.0773813
$340$	0	0
$341$	−7.98809e7	−0.109094
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	−5.36972e8	−0.704020
$346$	0	0
$347$	−4.60114e7	−0.0591169	−0.0295585	−	0.999563i	$-0.509410\pi$
$347$	−4.60114e7	−0.0591169	−0.0295585	+	0.999563i	$0.509410\pi$
$348$	0	0
$349$	1.55807e9	1.96199	0.980996	−	0.194026i	$-0.0621545\pi$
$349$	1.55807e9	1.96199	0.980996	+	0.194026i	$0.0621545\pi$
$350$	0	0
$351$	5.70794e7	0.0704538
$352$	0	0
$353$	3.65423e7	0.0442165	0.0221082	−	0.999756i	$-0.492962\pi$
$353$	3.65423e7	0.0442165	0.0221082	+	0.999756i	$0.492962\pi$
$354$	0	0
$355$	6.28699e8	0.745836
$356$	0	0
$357$	−2.65104e8	−0.308374
$358$	0	0
$359$	−1.10300e9	−1.25818	−0.629092	−	0.777331i	$-0.716573\pi$
$359$	−1.10300e9	−1.25818	−0.629092	+	0.777331i	$0.716573\pi$
$360$	0	0
$361$	−4.67514e8	−0.523021
$362$	0	0
$363$	4.17393e8	0.458006
$364$	0	0
$365$	1.83365e8	0.197374
$366$	0	0
$367$	−5.62242e8	−0.593735	−0.296867	−	0.954919i	$-0.595942\pi$
$367$	−5.62242e8	−0.593735	−0.296867	+	0.954919i	$0.595942\pi$
$368$	0	0
$369$	1.31064e7	0.0135798
$370$	0	0
$371$	3.59180e8	0.365177
$372$	0	0
$373$	1.13360e9	1.13105	0.565523	−	0.824733i	$-0.308675\pi$
$373$	1.13360e9	1.13105	0.565523	+	0.824733i	$0.308675\pi$
$374$	0	0
$375$	−8.87982e7	−0.0869551
$376$	0	0
$377$	−4.55912e7	−0.0438214
$378$	0	0
$379$	−7.94090e7	−0.0749260	−0.0374630	−	0.999298i	$-0.511928\pi$
$379$	−7.94090e7	−0.0749260	−0.0374630	+	0.999298i	$0.511928\pi$
$380$	0	0
$381$	−1.59905e9	−1.48123
$382$	0	0
$383$	−1.01351e9	−0.921787	−0.460894	−	0.887455i	$-0.652471\pi$
$383$	−1.01351e9	−0.921787	−0.460894	+	0.887455i	$0.652471\pi$
$384$	0	0
$385$	−1.37645e8	−0.122927
$386$	0	0
$387$	−2.57590e6	−0.00225912
$388$	0	0
$389$	5.08827e8	0.438275	0.219138	−	0.975694i	$-0.429676\pi$
$389$	5.08827e8	0.438275	0.219138	+	0.975694i	$0.429676\pi$
$390$	0	0
$391$	−1.60626e9	−1.35893
$392$	0	0
$393$	1.91045e8	0.158767
$394$	0	0
$395$	−5.00017e8	−0.408220
$396$	0	0
$397$	−9.27941e7	−0.0744310	−0.0372155	−	0.999307i	$-0.511849\pi$
$397$	−9.27941e7	−0.0744310	−0.0372155	+	0.999307i	$0.511849\pi$
$398$	0	0
$399$	−3.22000e8	−0.253776
$400$	0	0
$401$	2.11608e8	0.163880	0.0819400	−	0.996637i	$-0.473888\pi$
$401$	2.11608e8	0.163880	0.0819400	+	0.996637i	$0.473888\pi$
$402$	0	0
$403$	−1.35410e7	−0.0103058
$404$	0	0
$405$	−5.63278e8	−0.421337
$406$	0	0
$407$	−6.90736e8	−0.507845
$408$	0	0
$409$	4.50896e8	0.325871	0.162935	−	0.986637i	$-0.447904\pi$
$409$	4.50896e8	0.325871	0.162935	+	0.986637i	$0.447904\pi$
$410$	0	0
$411$	7.06678e8	0.502083
$412$	0	0
$413$	−8.44321e8	−0.589769
$414$	0	0
$415$	−3.78541e8	−0.259983
$416$	0	0
$417$	1.00516e9	0.678824
$418$	0	0
$419$	−9.10448e8	−0.604653	−0.302326	−	0.953204i	$-0.597763\pi$
$419$	−9.10448e8	−0.604653	−0.302326	+	0.953204i	$0.597763\pi$
$420$	0	0
$421$	1.12247e9	0.733141	0.366570	−	0.930390i	$-0.380532\pi$
$421$	1.12247e9	0.733141	0.366570	+	0.930390i	$0.380532\pi$
$422$	0	0
$423$	1.97229e7	0.0126701
$424$	0	0
$425$	−2.65625e8	−0.167845
$426$	0	0
$427$	−9.64062e8	−0.599249
$428$	0	0
$429$	−7.94321e7	−0.0485731
$430$	0	0
$431$	−2.33902e9	−1.40723	−0.703614	−	0.710582i	$-0.748432\pi$
$431$	−2.33902e9	−1.40723	−0.703614	+	0.710582i	$0.748432\pi$
$432$	0	0
$433$	7.17166e8	0.424534	0.212267	−	0.977212i	$-0.431915\pi$
$433$	7.17166e8	0.424534	0.212267	+	0.977212i	$0.431915\pi$
$434$	0	0
$435$	4.76102e8	0.277324
$436$	0	0
$437$	−1.95099e9	−1.11833
$438$	0	0
$439$	2.06948e9	1.16745	0.583723	−	0.811953i	$-0.301596\pi$
$439$	2.06948e9	1.16745	0.583723	+	0.811953i	$0.301596\pi$
$440$	0	0
$441$	−1.41134e7	−0.00783606
$442$	0	0
$443$	1.37967e9	0.753985	0.376993	−	0.926216i	$-0.376958\pi$
$443$	1.37967e9	0.753985	0.376993	+	0.926216i	$0.376958\pi$
$444$	0	0
$445$	−1.58139e8	−0.0850707
$446$	0	0
$447$	2.88710e9	1.52892
$448$	0	0
$449$	−1.54855e9	−0.807351	−0.403675	−	0.914902i	$-0.632267\pi$
$449$	−1.54855e9	−0.807351	−0.403675	+	0.914902i	$0.632267\pi$
$450$	0	0
$451$	−3.50750e8	−0.180044
$452$	0	0
$453$	4.59234e8	0.232108
$454$	0	0
$455$	−2.33329e7	−0.0116126
$456$	0	0
$457$	−3.40911e9	−1.67084	−0.835420	−	0.549613i	$-0.814775\pi$
$457$	−3.40911e9	−1.67084	−0.835420	+	0.549613i	$0.814775\pi$
$458$	0	0
$459$	−1.78305e9	−0.860635
$460$	0	0
$461$	−8.59095e8	−0.408402	−0.204201	−	0.978929i	$-0.565460\pi$
$461$	−8.59095e8	−0.408402	−0.204201	+	0.978929i	$0.565460\pi$
$462$	0	0
$463$	2.31257e9	1.08283	0.541417	−	0.840754i	$-0.317888\pi$
$463$	2.31257e9	1.08283	0.541417	+	0.840754i	$0.317888\pi$
$464$	0	0
$465$	1.41407e8	0.0652205
$466$	0	0
$467$	1.49834e9	0.680773	0.340386	−	0.940286i	$-0.389442\pi$
$467$	1.49834e9	0.680773	0.340386	+	0.940286i	$0.389442\pi$
$468$	0	0
$469$	2.61325e8	0.116970
$470$	0	0
$471$	7.31269e8	0.322481
$472$	0	0
$473$	6.89352e7	0.0299521
$474$	0	0
$475$	−3.22632e8	−0.138127
$476$	0	0
$477$	1.25621e8	0.0529965
$478$	0	0
$479$	−3.27196e9	−1.36030	−0.680148	−	0.733075i	$-0.738084\pi$
$479$	−3.27196e9	−1.36030	−0.680148	+	0.733075i	$0.738084\pi$
$480$	0	0
$481$	−1.17090e8	−0.0479746
$482$	0	0
$483$	1.47345e9	0.595005
$484$	0	0
$485$	1.37667e9	0.547940
$486$	0	0
$487$	−1.77949e9	−0.698142	−0.349071	−	0.937096i	$-0.613503\pi$
$487$	−1.77949e9	−0.698142	−0.349071	+	0.937096i	$0.613503\pi$
$488$	0	0
$489$	−2.95994e9	−1.14473
$490$	0	0
$491$	3.77985e9	1.44108	0.720542	−	0.693412i	$-0.243893\pi$
$491$	3.77985e9	1.44108	0.720542	+	0.693412i	$0.243893\pi$
$492$	0	0
$493$	1.42418e9	0.535304
$494$	0	0
$495$	−4.81406e7	−0.0178399
$496$	0	0
$497$	−1.72515e9	−0.630346
$498$	0	0
$499$	−2.59630e9	−0.935412	−0.467706	−	0.883884i	$-0.654919\pi$
$499$	−2.59630e9	−0.935412	−0.467706	+	0.883884i	$0.654919\pi$
$500$	0	0
$501$	−7.94603e8	−0.282305
$502$	0	0
$503$	3.32057e9	1.16339	0.581694	−	0.813408i	$-0.302390\pi$
$503$	3.32057e9	1.16339	0.581694	+	0.813408i	$0.302390\pi$
$504$	0	0
$505$	1.48784e8	0.0514087
$506$	0	0
$507$	2.83938e9	0.967598
$508$	0	0
$509$	3.07278e9	1.03281	0.516404	−	0.856345i	$-0.327270\pi$
$509$	3.07278e9	1.03281	0.516404	+	0.856345i	$0.327270\pi$
$510$	0	0
$511$	−5.03153e8	−0.166812
$512$	0	0
$513$	−2.16572e9	−0.708257
$514$	0	0
$515$	1.38148e9	0.445678
$516$	0	0
$517$	−5.27817e8	−0.167983
$518$	0	0
$519$	−9.09764e8	−0.285656
$520$	0	0
$521$	−3.38456e9	−1.04850	−0.524252	−	0.851563i	$-0.675655\pi$
$521$	−3.38456e9	−1.04850	−0.524252	+	0.851563i	$0.675655\pi$
$522$	0	0
$523$	2.43049e9	0.742912	0.371456	−	0.928451i	$-0.378859\pi$
$523$	2.43049e9	0.742912	0.371456	+	0.928451i	$0.378859\pi$
$524$	0	0
$525$	2.43662e8	0.0734904
$526$	0	0
$527$	4.22994e8	0.125892
$528$	0	0
$529$	5.52277e9	1.62204
$530$	0	0
$531$	−2.95296e8	−0.0855908
$532$	0	0
$533$	−5.94572e7	−0.0170082
$534$	0	0
$535$	−6.71302e7	−0.0189531
$536$	0	0
$537$	−4.96659e9	−1.38404
$538$	0	0
$539$	3.77699e8	0.103893
$540$	0	0
$541$	−3.94638e9	−1.07154	−0.535770	−	0.844364i	$-0.679979\pi$
$541$	−3.94638e9	−1.07154	−0.535770	+	0.844364i	$0.679979\pi$
$542$	0	0
$543$	4.93014e9	1.32148
$544$	0	0
$545$	−2.05916e8	−0.0544882
$546$	0	0
$547$	6.24321e9	1.63099	0.815496	−	0.578762i	$-0.196464\pi$
$547$	6.24321e9	1.63099	0.815496	+	0.578762i	$0.196464\pi$
$548$	0	0
$549$	−3.37175e8	−0.0869665
$550$	0	0
$551$	1.72983e9	0.440527
$552$	0	0
$553$	1.37205e9	0.345009
$554$	0	0
$555$	1.22275e9	0.303608
$556$	0	0
$557$	−3.46765e9	−0.850242	−0.425121	−	0.905137i	$-0.639768\pi$
$557$	−3.46765e9	−0.850242	−0.425121	+	0.905137i	$0.639768\pi$
$558$	0	0
$559$	1.16855e7	0.00282948
$560$	0	0
$561$	2.48131e9	0.593349
$562$	0	0
$563$	−5.07752e7	−0.0119915	−0.00599573	−	0.999982i	$-0.501909\pi$
$563$	−5.07752e7	−0.0119915	−0.00599573	+	0.999982i	$0.501909\pi$
$564$	0	0
$565$	−1.52606e8	−0.0355960
$566$	0	0
$567$	1.54563e9	0.356095
$568$	0	0
$569$	3.91117e9	0.890048	0.445024	−	0.895519i	$-0.353195\pi$
$569$	3.91117e9	0.890048	0.445024	+	0.895519i	$0.353195\pi$
$570$	0	0
$571$	−2.80703e9	−0.630987	−0.315494	−	0.948928i	$-0.602170\pi$
$571$	−2.80703e9	−0.630987	−0.315494	+	0.948928i	$0.602170\pi$
$572$	0	0
$573$	5.57399e9	1.23773
$574$	0	0
$575$	1.47634e9	0.323854
$576$	0	0
$577$	−4.56328e9	−0.988922	−0.494461	−	0.869200i	$-0.664635\pi$
$577$	−4.56328e9	−0.988922	−0.494461	+	0.869200i	$0.664635\pi$
$578$	0	0
$579$	−1.85862e9	−0.397938
$580$	0	0
$581$	1.03872e9	0.219726
$582$	0	0
$583$	−3.36182e9	−0.702643
$584$	0	0
$585$	−8.16054e6	−0.00168529
$586$	0	0
$587$	2.45292e9	0.500552	0.250276	−	0.968175i	$-0.419479\pi$
$587$	2.45292e9	0.500552	0.250276	+	0.968175i	$0.419479\pi$
$588$	0	0
$589$	5.13775e8	0.103602
$590$	0	0
$591$	4.07092e9	0.811215
$592$	0	0
$593$	7.17725e9	1.41340	0.706702	−	0.707511i	$-0.250182\pi$
$593$	7.17725e9	1.41340	0.706702	+	0.707511i	$0.250182\pi$
$594$	0	0
$595$	7.28875e8	0.141855
$596$	0	0
$597$	1.99234e9	0.383225
$598$	0	0
$599$	−3.02663e9	−0.575393	−0.287697	−	0.957722i	$-0.592890\pi$
$599$	−3.02663e9	−0.575393	−0.287697	+	0.957722i	$0.592890\pi$
$600$	0	0
$601$	−4.03920e9	−0.758988	−0.379494	−	0.925194i	$-0.623902\pi$
$601$	−4.03920e9	−0.758988	−0.379494	+	0.925194i	$0.623902\pi$
$602$	0	0
$603$	9.13969e7	0.0169754
$604$	0	0
$605$	−1.14757e9	−0.210687
$606$	0	0
$607$	−2.17668e9	−0.395034	−0.197517	−	0.980299i	$-0.563288\pi$
$607$	−2.17668e9	−0.395034	−0.197517	+	0.980299i	$0.563288\pi$
$608$	0	0
$609$	−1.30642e9	−0.234382
$610$	0	0
$611$	−8.94728e7	−0.0158689
$612$	0	0
$613$	−8.94803e8	−0.156897	−0.0784487	−	0.996918i	$-0.524997\pi$
$613$	−8.94803e8	−0.156897	−0.0784487	+	0.996918i	$0.524997\pi$
$614$	0	0
$615$	6.20904e8	0.107637
$616$	0	0
$617$	−7.74799e8	−0.132798	−0.0663989	−	0.997793i	$-0.521151\pi$
$617$	−7.74799e8	−0.132798	−0.0663989	+	0.997793i	$0.521151\pi$
$618$	0	0
$619$	−1.08482e9	−0.183841	−0.0919203	−	0.995766i	$-0.529301\pi$
$619$	−1.08482e9	−0.183841	−0.0919203	+	0.995766i	$0.529301\pi$
$620$	0	0
$621$	9.91019e9	1.66059
$622$	0	0
$623$	4.33934e8	0.0718979
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	3.01383e9	0.488295
$628$	0	0
$629$	3.65766e9	0.586038
$630$	0	0
$631$	−9.02289e9	−1.42969	−0.714847	−	0.699281i	$-0.753504\pi$
$631$	−9.02289e9	−1.42969	−0.714847	+	0.699281i	$0.753504\pi$
$632$	0	0
$633$	−3.48961e9	−0.546845
$634$	0	0
$635$	4.39640e9	0.681379
$636$	0	0
$637$	6.40255e7	0.00981443
$638$	0	0
$639$	−6.03361e8	−0.0914795
$640$	0	0
$641$	−6.21202e9	−0.931600	−0.465800	−	0.884890i	$-0.654233\pi$
$641$	−6.21202e9	−0.931600	−0.465800	+	0.884890i	$0.654233\pi$
$642$	0	0
$643$	1.10084e10	1.63300	0.816502	−	0.577342i	$-0.195910\pi$
$643$	1.10084e10	1.63300	0.816502	+	0.577342i	$0.195910\pi$
$644$	0	0
$645$	−1.22030e8	−0.0179064
$646$	0	0
$647$	1.28642e10	1.86732	0.933660	−	0.358162i	$-0.116596\pi$
$647$	1.28642e10	1.86732	0.933660	+	0.358162i	$0.116596\pi$
$648$	0	0
$649$	7.90261e9	1.13479
$650$	0	0
$651$	−3.88020e8	−0.0551214
$652$	0	0
$653$	4.08300e9	0.573830	0.286915	−	0.957956i	$-0.407370\pi$
$653$	4.08300e9	0.573830	0.286915	+	0.957956i	$0.407370\pi$
$654$	0	0
$655$	−5.25256e8	−0.0730342
$656$	0	0
$657$	−1.75975e8	−0.0242087
$658$	0	0
$659$	−7.07190e9	−0.962581	−0.481290	−	0.876561i	$-0.659832\pi$
$659$	−7.07190e9	−0.962581	−0.481290	+	0.876561i	$0.659832\pi$
$660$	0	0
$661$	1.11678e10	1.50405	0.752025	−	0.659135i	$-0.229077\pi$
$661$	1.11678e10	1.50405	0.752025	+	0.659135i	$0.229077\pi$
$662$	0	0
$663$	4.20618e8	0.0560519
$664$	0	0
$665$	8.85302e8	0.116739
$666$	0	0
$667$	−7.91559e9	−1.03286
$668$	0	0
$669$	1.18126e10	1.52530
$670$	0	0
$671$	9.02336e9	1.15303
$672$	0	0
$673$	1.00591e10	1.27205	0.636027	−	0.771667i	$-0.280577\pi$
$673$	1.00591e10	1.27205	0.636027	+	0.771667i	$0.280577\pi$
$674$	0	0
$675$	1.63883e9	0.205103
$676$	0	0
$677$	1.09282e10	1.35359	0.676796	−	0.736170i	$-0.263368\pi$
$677$	1.09282e10	1.35359	0.676796	+	0.736170i	$0.263368\pi$
$678$	0	0
$679$	−3.77758e9	−0.463094
$680$	0	0
$681$	8.93520e9	1.08415
$682$	0	0
$683$	−1.61883e10	−1.94414	−0.972070	−	0.234689i	$-0.924593\pi$
$683$	−1.61883e10	−1.94414	−0.972070	+	0.234689i	$0.924593\pi$
$684$	0	0
$685$	−1.94293e9	−0.230962
$686$	0	0
$687$	1.21133e10	1.42532
$688$	0	0
$689$	−5.69878e8	−0.0663766
$690$	0	0
$691$	−3.84848e9	−0.443728	−0.221864	−	0.975078i	$-0.571214\pi$
$691$	−3.84848e9	−0.443728	−0.221864	+	0.975078i	$0.571214\pi$
$692$	0	0
$693$	1.32098e8	0.0150775
$694$	0	0
$695$	−2.76356e9	−0.312264
$696$	0	0
$697$	1.85733e9	0.207766
$698$	0	0
$699$	3.63691e9	0.402775
$700$	0	0
$701$	−9.13118e9	−1.00118	−0.500592	−	0.865684i	$-0.666884\pi$
$701$	−9.13118e9	−1.00118	−0.500592	+	0.865684i	$0.666884\pi$
$702$	0	0
$703$	4.44265e9	0.482279
$704$	0	0
$705$	9.34352e8	0.100427
$706$	0	0
$707$	−4.08263e8	−0.0434483
$708$	0	0
$709$	2.52188e8	0.0265744	0.0132872	−	0.999912i	$-0.495770\pi$
$709$	2.52188e8	0.0265744	0.0132872	+	0.999912i	$0.495770\pi$
$710$	0	0
$711$	4.79865e8	0.0500697
$712$	0	0
$713$	−2.35100e9	−0.242907
$714$	0	0
$715$	2.18390e8	0.0223440
$716$	0	0
$717$	7.64506e9	0.774576
$718$	0	0
$719$	2.79286e9	0.280219	0.140109	−	0.990136i	$-0.455255\pi$
$719$	2.79286e9	0.280219	0.140109	+	0.990136i	$0.455255\pi$
$720$	0	0
$721$	−3.79079e9	−0.376666
$722$	0	0
$723$	−6.84895e9	−0.673969
$724$	0	0
$725$	−1.30899e9	−0.127571
$726$	0	0
$727$	1.08562e10	1.04787	0.523933	−	0.851760i	$-0.324464\pi$
$727$	1.08562e10	1.04787	0.523933	+	0.851760i	$0.324464\pi$
$728$	0	0
$729$	1.09695e10	1.04867
$730$	0	0
$731$	−3.65033e8	−0.0345638
$732$	0	0
$733$	1.62067e10	1.51996	0.759979	−	0.649948i	$-0.225209\pi$
$733$	1.62067e10	1.51996	0.759979	+	0.649948i	$0.225209\pi$
$734$	0	0
$735$	−6.68609e8	−0.0621108
$736$	0	0
$737$	−2.44593e9	−0.225065
$738$	0	0
$739$	−9.19324e9	−0.837940	−0.418970	−	0.908000i	$-0.637609\pi$
$739$	−9.19324e9	−0.837940	−0.418970	+	0.908000i	$0.637609\pi$
$740$	0	0
$741$	5.10888e8	0.0461278
$742$	0	0
$743$	−8.56205e8	−0.0765802	−0.0382901	−	0.999267i	$-0.512191\pi$
$743$	−8.56205e8	−0.0765802	−0.0382901	+	0.999267i	$0.512191\pi$
$744$	0	0
$745$	−7.93775e9	−0.703316
$746$	0	0
$747$	3.63285e8	0.0318879
$748$	0	0
$749$	1.84205e8	0.0160183
$750$	0	0
$751$	−3.55645e9	−0.306392	−0.153196	−	0.988196i	$-0.548957\pi$
$751$	−3.55645e9	−0.306392	−0.153196	+	0.988196i	$0.548957\pi$
$752$	0	0
$753$	1.53595e10	1.31097
$754$	0	0
$755$	−1.26261e9	−0.106771
$756$	0	0
$757$	−6.70680e9	−0.561927	−0.280963	−	0.959718i	$-0.590654\pi$
$757$	−6.70680e9	−0.561927	−0.280963	+	0.959718i	$0.590654\pi$
$758$	0	0
$759$	−1.37911e10	−1.14486
$760$	0	0
$761$	1.18178e10	0.972051	0.486025	−	0.873945i	$-0.338446\pi$
$761$	1.18178e10	0.972051	0.486025	+	0.873945i	$0.338446\pi$
$762$	0	0
$763$	5.65033e8	0.0460509
$764$	0	0
$765$	2.54920e8	0.0205868
$766$	0	0
$767$	1.33961e9	0.107200
$768$	0	0
$769$	−1.33613e10	−1.05951	−0.529757	−	0.848149i	$-0.677717\pi$
$769$	−1.33613e10	−1.05951	−0.529757	+	0.848149i	$0.677717\pi$
$770$	0	0
$771$	−2.50977e9	−0.197216
$772$	0	0
$773$	−1.05943e10	−0.824979	−0.412489	−	0.910962i	$-0.635341\pi$
$773$	−1.05943e10	−0.824979	−0.412489	+	0.910962i	$0.635341\pi$
$774$	0	0
$775$	−3.88781e8	−0.0300020
$776$	0	0
$777$	−3.35523e9	−0.256596
$778$	0	0
$779$	2.25594e9	0.170980
$780$	0	0
$781$	1.61469e10	1.21286
$782$	0	0
$783$	−8.78680e9	−0.654131
$784$	0	0
$785$	−2.01054e9	−0.148344
$786$	0	0
$787$	−2.02495e9	−0.148082	−0.0740411	−	0.997255i	$-0.523590\pi$
$787$	−2.02495e9	−0.148082	−0.0740411	+	0.997255i	$0.523590\pi$
$788$	0	0
$789$	1.82134e10	1.32014
$790$	0	0
$791$	4.18750e8	0.0300841
$792$	0	0
$793$	1.52959e9	0.108923
$794$	0	0
$795$	5.95116e9	0.420065
$796$	0	0
$797$	−2.69141e10	−1.88311	−0.941556	−	0.336855i	$-0.890637\pi$
$797$	−2.69141e10	−1.88311	−0.941556	+	0.336855i	$0.890637\pi$
$798$	0	0
$799$	2.79495e9	0.193848
$800$	0	0
$801$	1.51766e8	0.0104342
$802$	0	0
$803$	4.70937e9	0.320966
$804$	0	0
$805$	−4.05109e9	−0.273707
$806$	0	0
$807$	7.93222e9	0.531297
$808$	0	0
$809$	−2.62338e10	−1.74197	−0.870986	−	0.491308i	$-0.836519\pi$
$809$	−2.62338e10	−1.74197	−0.870986	+	0.491308i	$0.836519\pi$
$810$	0	0
$811$	1.01056e10	0.665257	0.332629	−	0.943058i	$-0.392064\pi$
$811$	1.01056e10	0.665257	0.332629	+	0.943058i	$0.392064\pi$
$812$	0	0
$813$	1.86962e10	1.22022
$814$	0	0
$815$	8.13802e9	0.526584
$816$	0	0
$817$	−4.43375e8	−0.0284442
$818$	0	0
$819$	2.23925e7	0.00142433
$820$	0	0
$821$	1.62669e9	0.102590	0.0512950	−	0.998684i	$-0.483665\pi$
$821$	1.62669e9	0.102590	0.0512950	+	0.998684i	$0.483665\pi$
$822$	0	0
$823$	−1.57698e10	−0.986111	−0.493055	−	0.869998i	$-0.664120\pi$
$823$	−1.57698e10	−0.986111	−0.493055	+	0.869998i	$0.664120\pi$
$824$	0	0
$825$	−2.28061e9	−0.141404
$826$	0	0
$827$	−2.00193e10	−1.23078	−0.615390	−	0.788223i	$-0.711002\pi$
$827$	−2.00193e10	−1.23078	−0.615390	+	0.788223i	$0.711002\pi$
$828$	0	0
$829$	−1.70550e10	−1.03971	−0.519855	−	0.854255i	$-0.674014\pi$
$829$	−1.70550e10	−1.03971	−0.519855	+	0.854255i	$0.674014\pi$
$830$	0	0
$831$	9.81587e9	0.593370
$832$	0	0
$833$	−2.00003e9	−0.119889
$834$	0	0
$835$	2.18467e9	0.129862
$836$	0	0
$837$	−2.60976e9	−0.153837
$838$	0	0
$839$	3.43532e9	0.200817	0.100409	−	0.994946i	$-0.467985\pi$
$839$	3.43532e9	0.200817	0.100409	+	0.994946i	$0.467985\pi$
$840$	0	0
$841$	−1.02316e10	−0.593139
$842$	0	0
$843$	9.93976e9	0.571452
$844$	0	0
$845$	−7.80654e9	−0.445103
$846$	0	0
$847$	3.14894e9	0.178063
$848$	0	0
$849$	7.99763e9	0.448523
$850$	0	0
$851$	−2.03293e10	−1.13075
$852$	0	0
$853$	8.19010e9	0.451822	0.225911	−	0.974148i	$-0.427464\pi$
$853$	8.19010e9	0.451822	0.225911	+	0.974148i	$0.427464\pi$
$854$	0	0
$855$	3.09629e8	0.0169418
$856$	0	0
$857$	−2.05644e10	−1.11605	−0.558024	−	0.829825i	$-0.688440\pi$
$857$	−2.05644e10	−1.11605	−0.558024	+	0.829825i	$0.688440\pi$
$858$	0	0
$859$	−2.18059e10	−1.17381	−0.586904	−	0.809656i	$-0.699653\pi$
$859$	−2.18059e10	−1.17381	−0.586904	+	0.809656i	$0.699653\pi$
$860$	0	0
$861$	−1.70376e9	−0.0909698
$862$	0	0
$863$	−1.98316e10	−1.05031	−0.525157	−	0.851005i	$-0.675993\pi$
$863$	−1.98316e10	−1.05031	−0.525157	+	0.851005i	$0.675993\pi$
$864$	0	0
$865$	2.50129e9	0.131404
$866$	0	0
$867$	5.51664e9	0.287480
$868$	0	0
$869$	−1.28420e10	−0.663839
$870$	0	0
$871$	−4.14621e8	−0.0212612
$872$	0	0
$873$	−1.32119e9	−0.0672069
$874$	0	0
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	3.35166e9	0.167788	0.0838941	−	0.996475i	$-0.473264\pi$
$877$	3.35166e9	0.167788	0.0838941	+	0.996475i	$0.473264\pi$
$878$	0	0
$879$	−3.35425e10	−1.66584
$880$	0	0
$881$	3.65392e9	0.180029	0.0900147	−	0.995940i	$-0.471309\pi$
$881$	3.65392e9	0.180029	0.0900147	+	0.995940i	$0.471309\pi$
$882$	0	0
$883$	4.38823e9	0.214500	0.107250	−	0.994232i	$-0.465795\pi$
$883$	4.38823e9	0.214500	0.107250	+	0.994232i	$0.465795\pi$
$884$	0	0
$885$	−1.39893e10	−0.678416
$886$	0	0
$887$	3.08901e10	1.48623	0.743115	−	0.669164i	$-0.233347\pi$
$887$	3.08901e10	1.48623	0.743115	+	0.669164i	$0.233347\pi$
$888$	0	0
$889$	−1.20637e10	−0.575871
$890$	0	0
$891$	−1.44667e10	−0.685169
$892$	0	0
$893$	3.39479e9	0.159527
$894$	0	0
$895$	1.36551e10	0.636669
$896$	0	0
$897$	−2.33779e9	−0.108152
$898$	0	0
$899$	2.08450e9	0.0956847
$900$	0	0
$901$	1.78019e10	0.810829
$902$	0	0
$903$	3.34851e8	0.0151337
$904$	0	0
$905$	−1.35549e10	−0.607890
$906$	0	0
$907$	−8.78490e9	−0.390941	−0.195470	−	0.980710i	$-0.562623\pi$
$907$	−8.78490e9	−0.390941	−0.195470	+	0.980710i	$0.562623\pi$
$908$	0	0
$909$	−1.42788e8	−0.00630546
$910$	0	0
$911$	1.01855e10	0.446340	0.223170	−	0.974780i	$-0.428359\pi$
$911$	1.01855e10	0.446340	0.223170	+	0.974780i	$0.428359\pi$
$912$	0	0
$913$	−9.72211e9	−0.422778
$914$	0	0
$915$	−1.59733e10	−0.689321
$916$	0	0
$917$	1.44130e9	0.0617252
$918$	0	0
$919$	−2.14516e10	−0.911707	−0.455854	−	0.890055i	$-0.650666\pi$
$919$	−2.14516e10	−0.911707	−0.455854	+	0.890055i	$0.650666\pi$
$920$	0	0
$921$	9.94176e9	0.419329
$922$	0	0
$923$	2.73714e9	0.114575
$924$	0	0
$925$	−3.36182e9	−0.139662
$926$	0	0
$927$	−1.32581e9	−0.0546640
$928$	0	0
$929$	−5.24496e9	−0.214628	−0.107314	−	0.994225i	$-0.534225\pi$
$929$	−5.24496e9	−0.214628	−0.107314	+	0.994225i	$0.534225\pi$
$930$	0	0
$931$	−2.42927e9	−0.0986624
$932$	0	0
$933$	2.64242e10	1.06516
$934$	0	0
$935$	−6.82207e9	−0.272945
$936$	0	0
$937$	−3.22894e10	−1.28225	−0.641124	−	0.767437i	$-0.721532\pi$
$937$	−3.22894e10	−1.28225	−0.641124	+	0.767437i	$0.721532\pi$
$938$	0	0
$939$	−2.55394e9	−0.100666
$940$	0	0
$941$	1.12478e10	0.440054	0.220027	−	0.975494i	$-0.429385\pi$
$941$	1.12478e10	0.440054	0.220027	+	0.975494i	$0.429385\pi$
$942$	0	0
$943$	−1.03230e10	−0.400882
$944$	0	0
$945$	−4.49696e9	−0.173343
$946$	0	0
$947$	3.54636e10	1.35693	0.678466	−	0.734632i	$-0.262645\pi$
$947$	3.54636e10	1.35693	0.678466	+	0.734632i	$0.262645\pi$
$948$	0	0
$949$	7.98308e8	0.0303206
$950$	0	0
$951$	2.88815e10	1.08890
$952$	0	0
$953$	−1.41063e10	−0.527942	−0.263971	−	0.964531i	$-0.585032\pi$
$953$	−1.41063e10	−0.527942	−0.263971	+	0.964531i	$0.585032\pi$
$954$	0	0
$955$	−1.53250e10	−0.569364
$956$	0	0
$957$	1.22278e10	0.450978
$958$	0	0
$959$	5.33140e9	0.195199
$960$	0	0
$961$	−2.68935e10	−0.977497
$962$	0	0
$963$	6.44247e7	0.00232467
$964$	0	0
$965$	5.11007e9	0.183055
$966$	0	0
$967$	2.67518e10	0.951395	0.475697	−	0.879609i	$-0.342196\pi$
$967$	2.67518e10	0.951395	0.475697	+	0.879609i	$0.342196\pi$
$968$	0	0
$969$	−1.59592e10	−0.563478
$970$	0	0
$971$	−1.35657e10	−0.475527	−0.237763	−	0.971323i	$-0.576414\pi$
$971$	−1.35657e10	−0.475527	−0.237763	+	0.971323i	$0.576414\pi$
$972$	0	0
$973$	7.58321e9	0.263911
$974$	0	0
$975$	−3.86597e8	−0.0133580
$976$	0	0
$977$	−1.75830e10	−0.603200	−0.301600	−	0.953435i	$-0.597521\pi$
$977$	−1.75830e10	−0.603200	−0.301600	+	0.953435i	$0.597521\pi$
$978$	0	0
$979$	−4.06151e9	−0.138340
$980$	0	0
$981$	1.97617e8	0.00668318
$982$	0	0
$983$	−8.63801e9	−0.290052	−0.145026	−	0.989428i	$-0.546327\pi$
$983$	−8.63801e9	−0.290052	−0.145026	+	0.989428i	$0.546327\pi$
$984$	0	0
$985$	−1.11925e10	−0.373165
$986$	0	0
$987$	−2.56386e9	−0.0848759
$988$	0	0
$989$	2.02886e9	0.0666905
$990$	0	0
$991$	−3.61292e10	−1.17923	−0.589617	−	0.807683i	$-0.700721\pi$
$991$	−3.61292e10	−1.17923	−0.589617	+	0.807683i	$0.700721\pi$
$992$	0	0
$993$	−2.50242e10	−0.811033
$994$	0	0
$995$	−5.47772e9	−0.176287
$996$	0	0
$997$	−9.48913e9	−0.303245	−0.151622	−	0.988438i	$-0.548450\pi$
$997$	−9.48913e9	−0.303245	−0.151622	+	0.988438i	$0.548450\pi$
$998$	0	0
$999$	−2.25668e10	−0.716127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.8.a.q.1.1		4
4.3	odd	2		280.8.a.a.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.8.a.a.1.4	✓	4	4.3	odd	2
560.8.a.q.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-45.4647 q^{3} +125.000 q^{5} -343.000 q^{7} -119.962 q^{9} +O(q^{10})\)	\(q-45.4647 q^{3} +125.000 q^{5} -343.000 q^{7} -119.962 q^{9} +3210.39 q^{11} +544.208 q^{13} -5683.09 q^{15} -17000.0 q^{17} -20648.4 q^{19} +15594.4 q^{21} +94486.0 q^{23} +15625.0 q^{25} +104885. q^{27} -83775.3 q^{29} -24882.0 q^{31} -145959. q^{33} -42875.0 q^{35} -215157. q^{37} -24742.2 q^{39} -109255. q^{41} +21472.6 q^{43} -14995.3 q^{45} -164409. q^{47} +117649. q^{49} +772899. q^{51} -1.04717e6 q^{53} +401298. q^{55} +938775. q^{57} +2.46158e6 q^{59} +2.81068e6 q^{61} +41147.0 q^{63} +68026.0 q^{65} -761881. q^{67} -4.29578e6 q^{69} +5.02959e6 q^{71} +1.46692e6 q^{73} -710386. q^{75} -1.10116e6 q^{77} -4.00013e6 q^{79} -4.50622e6 q^{81} -3.02833e6 q^{83} -2.12500e6 q^{85} +3.80882e6 q^{87} -1.26511e6 q^{89} -186663. q^{91} +1.13125e6 q^{93} -2.58105e6 q^{95} +1.10133e7 q^{97} -385125. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 55 q^{3} + 500 q^{5} - 1372 q^{7} + 1063 q^{9}+O(q^{10}) \) 4 * q + 55 * q^3 + 500 * q^5 - 1372 * q^7 + 1063 * q^9	\( 4 q + 55 q^{3} + 500 q^{5} - 1372 q^{7} + 1063 q^{9} + 633 q^{11} - 5397 q^{13} + 6875 q^{15} - 41987 q^{17} + 41958 q^{19} - 18865 q^{21} + 63650 q^{23} + 62500 q^{25} + 101737 q^{27} + 1071 q^{29} - 79700 q^{31} - 92499 q^{33} - 171500 q^{35} - 131804 q^{37} - 168611 q^{39} - 313834 q^{41} - 604338 q^{43} + 132875 q^{45} - 2429 q^{47} + 470596 q^{49} - 1178733 q^{51} - 590946 q^{53} + 79125 q^{55} + 1307410 q^{57} - 437296 q^{59} - 369710 q^{61} - 364609 q^{63} - 674625 q^{65} - 537812 q^{67} - 4360534 q^{69} + 2096808 q^{71} - 6033412 q^{73} + 859375 q^{75} - 217119 q^{77} - 5554681 q^{79} - 13721804 q^{81} + 2234760 q^{83} - 5248375 q^{85} + 5900337 q^{87} - 5347574 q^{89} + 1851171 q^{91} - 20106036 q^{93} + 5244750 q^{95} - 2106023 q^{97} + 3460842 q^{99}+O(q^{100}) \) 4 * q + 55 * q^3 + 500 * q^5 - 1372 * q^7 + 1063 * q^9 + 633 * q^11 - 5397 * q^13 + 6875 * q^15 - 41987 * q^17 + 41958 * q^19 - 18865 * q^21 + 63650 * q^23 + 62500 * q^25 + 101737 * q^27 + 1071 * q^29 - 79700 * q^31 - 92499 * q^33 - 171500 * q^35 - 131804 * q^37 - 168611 * q^39 - 313834 * q^41 - 604338 * q^43 + 132875 * q^45 - 2429 * q^47 + 470596 * q^49 - 1178733 * q^51 - 590946 * q^53 + 79125 * q^55 + 1307410 * q^57 - 437296 * q^59 - 369710 * q^61 - 364609 * q^63 - 674625 * q^65 - 537812 * q^67 - 4360534 * q^69 + 2096808 * q^71 - 6033412 * q^73 + 859375 * q^75 - 217119 * q^77 - 5554681 * q^79 - 13721804 * q^81 + 2234760 * q^83 - 5248375 * q^85 + 5900337 * q^87 - 5347574 * q^89 + 1851171 * q^91 - 20106036 * q^93 + 5244750 * q^95 - 2106023 * q^97 + 3460842 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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