LMFDB - Embedded newform 560.6.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,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, 6, names="a")

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

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
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:	$89.8149390953$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 463x - 1890 $ x^3 - 463*x - 1890
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-19.0769$ 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)

$f(q)$	$=$	$q-21.0769 q^{3} -25.0000 q^{5} -49.0000 q^{7} +201.235 q^{9} +286.081 q^{11} -751.545 q^{13} +526.922 q^{15} -845.554 q^{17} +1518.68 q^{19} +1032.77 q^{21} -2468.45 q^{23} +625.000 q^{25} +880.283 q^{27} +895.029 q^{29} +9826.32 q^{31} -6029.69 q^{33} +1225.00 q^{35} -2697.90 q^{37} +15840.2 q^{39} +5770.13 q^{41} -3203.78 q^{43} -5030.87 q^{45} +21898.9 q^{47} +2401.00 q^{49} +17821.6 q^{51} +18207.6 q^{53} -7152.02 q^{55} -32009.1 q^{57} +5021.37 q^{59} +22688.3 q^{61} -9860.50 q^{63} +18788.6 q^{65} -15653.8 q^{67} +52027.2 q^{69} +21027.2 q^{71} -52797.0 q^{73} -13173.0 q^{75} -14018.0 q^{77} -80685.6 q^{79} -67453.6 q^{81} +64197.4 q^{83} +21138.9 q^{85} -18864.4 q^{87} -72078.1 q^{89} +36825.7 q^{91} -207108. q^{93} -37967.1 q^{95} +42008.8 q^{97} +57569.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33}+ \cdots + 156568 q^{99}+O(q^{100}) $ 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9 + 578 * q^11 + 80 * q^13 + 150 * q^15 - 1244 * q^17 - 944 * q^19 + 294 * q^21 - 1096 * q^23 + 1875 * q^25 + 3006 * q^27 + 1868 * q^29 + 6620 * q^31 + 2662 * q^33 + 3675 * q^35 - 4058 * q^37 + 20910 * q^39 - 9602 * q^41 + 12340 * q^43 - 5225 * q^45 + 41026 * q^47 + 7203 * q^49 - 10006 * q^51 - 37610 * q^53 - 14450 * q^55 - 95456 * q^57 + 37664 * q^59 - 8386 * q^61 - 10241 * q^63 - 2000 * q^65 - 69340 * q^67 - 37712 * q^69 + 34016 * q^71 - 45314 * q^73 - 3750 * q^75 - 28322 * q^77 - 1382 * q^79 - 101005 * q^81 + 10128 * q^83 + 31100 * q^85 + 41510 * q^87 - 222810 * q^89 - 3920 * q^91 - 174292 * q^93 + 23600 * q^95 - 159476 * q^97 + 156568 * q^99

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
$3$	−21.0769	−1.35208	−0.676041	−	0.736864i	$-0.736306\pi$
$3$	−21.0769	−1.35208	−0.676041	+	0.736864i	$0.736306\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	201.235	0.828126
$10$	0	0
$11$	286.081	0.712865	0.356432	−	0.934321i	$-0.383993\pi$
$11$	286.081	0.712865	0.356432	+	0.934321i	$0.383993\pi$
$12$	0	0
$13$	−751.545	−1.23338	−0.616690	−	0.787206i	$-0.711527\pi$
$13$	−751.545	−1.23338	−0.616690	+	0.787206i	$0.711527\pi$
$14$	0	0
$15$	526.922	0.604670
$16$	0	0
$17$	−845.554	−0.709609	−0.354805	−	0.934940i	$-0.615453\pi$
$17$	−845.554	−0.709609	−0.354805	+	0.934940i	$0.615453\pi$
$18$	0	0
$19$	1518.68	0.965124	0.482562	−	0.875862i	$-0.339706\pi$
$19$	1518.68	0.965124	0.482562	+	0.875862i	$0.339706\pi$
$20$	0	0
$21$	1032.77	0.511039
$22$	0	0
$23$	−2468.45	−0.972982	−0.486491	−	0.873685i	$-0.661723\pi$
$23$	−2468.45	−0.972982	−0.486491	+	0.873685i	$0.661723\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	880.283	0.232387
$28$	0	0
$29$	895.029	0.197625	0.0988125	−	0.995106i	$-0.468496\pi$
$29$	895.029	0.197625	0.0988125	+	0.995106i	$0.468496\pi$
$30$	0	0
$31$	9826.32	1.83648	0.918241	−	0.396023i	$-0.129610\pi$
$31$	9826.32	1.83648	0.918241	+	0.396023i	$0.129610\pi$
$32$	0	0
$33$	−6029.69	−0.963852
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−2697.90	−0.323982	−0.161991	−	0.986792i	$-0.551792\pi$
$37$	−2697.90	−0.323982	−0.161991	+	0.986792i	$0.551792\pi$
$38$	0	0
$39$	15840.2	1.66763
$40$	0	0
$41$	5770.13	0.536075	0.268038	−	0.963408i	$-0.413625\pi$
$41$	5770.13	0.536075	0.268038	+	0.963408i	$0.413625\pi$
$42$	0	0
$43$	−3203.78	−0.264236	−0.132118	−	0.991234i	$-0.542178\pi$
$43$	−3203.78	−0.264236	−0.132118	+	0.991234i	$0.542178\pi$
$44$	0	0
$45$	−5030.87	−0.370349
$46$	0	0
$47$	21898.9	1.44603	0.723016	−	0.690831i	$-0.242755\pi$
$47$	21898.9	1.44603	0.723016	+	0.690831i	$0.242755\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	17821.6	0.959450
$52$	0	0
$53$	18207.6	0.890353	0.445177	−	0.895443i	$-0.353141\pi$
$53$	18207.6	0.890353	0.445177	+	0.895443i	$0.353141\pi$
$54$	0	0
$55$	−7152.02	−0.318803
$56$	0	0
$57$	−32009.1	−1.30493
$58$	0	0
$59$	5021.37	0.187798	0.0938992	−	0.995582i	$-0.470067\pi$
$59$	5021.37	0.187798	0.0938992	+	0.995582i	$0.470067\pi$
$60$	0	0
$61$	22688.3	0.780689	0.390345	−	0.920669i	$-0.372356\pi$
$61$	22688.3	0.780689	0.390345	+	0.920669i	$0.372356\pi$
$62$	0	0
$63$	−9860.50	−0.313002
$64$	0	0
$65$	18788.6	0.551584
$66$	0	0
$67$	−15653.8	−0.426024	−0.213012	−	0.977050i	$-0.568327\pi$
$67$	−15653.8	−0.426024	−0.213012	+	0.977050i	$0.568327\pi$
$68$	0	0
$69$	52027.2	1.31555
$70$	0	0
$71$	21027.2	0.495035	0.247518	−	0.968883i	$-0.420385\pi$
$71$	21027.2	0.495035	0.247518	+	0.968883i	$0.420385\pi$
$72$	0	0
$73$	−52797.0	−1.15958	−0.579792	−	0.814764i	$-0.696866\pi$
$73$	−52797.0	−1.15958	−0.579792	+	0.814764i	$0.696866\pi$
$74$	0	0
$75$	−13173.0	−0.270416
$76$	0	0
$77$	−14018.0	−0.269438
$78$	0	0
$79$	−80685.6	−1.45455	−0.727275	−	0.686347i	$-0.759213\pi$
$79$	−80685.6	−1.45455	−0.727275	+	0.686347i	$0.759213\pi$
$80$	0	0
$81$	−67453.6	−1.14233
$82$	0	0
$83$	64197.4	1.02287	0.511437	−	0.859321i	$-0.329113\pi$
$83$	64197.4	1.02287	0.511437	+	0.859321i	$0.329113\pi$
$84$	0	0
$85$	21138.9	0.317347
$86$	0	0
$87$	−18864.4	−0.267205
$88$	0	0
$89$	−72078.1	−0.964558	−0.482279	−	0.876018i	$-0.660191\pi$
$89$	−72078.1	−0.964558	−0.482279	+	0.876018i	$0.660191\pi$
$90$	0	0
$91$	36825.7	0.466173
$92$	0	0
$93$	−207108.	−2.48307
$94$	0	0
$95$	−37967.1	−0.431617
$96$	0	0
$97$	42008.8	0.453326	0.226663	−	0.973973i	$-0.427218\pi$
$97$	42008.8	0.453326	0.226663	+	0.973973i	$0.427218\pi$
$98$	0	0
$99$	57569.4	0.590342
$100$	0	0
$101$	46466.6	0.453249	0.226625	−	0.973982i	$-0.427231\pi$
$101$	46466.6	0.453249	0.226625	+	0.973982i	$0.427231\pi$
$102$	0	0
$103$	127580.	1.18493	0.592463	−	0.805598i	$-0.298156\pi$
$103$	127580.	1.18493	0.592463	+	0.805598i	$0.298156\pi$
$104$	0	0
$105$	−25819.2	−0.228544
$106$	0	0
$107$	−81394.7	−0.687285	−0.343642	−	0.939101i	$-0.611661\pi$
$107$	−81394.7	−0.687285	−0.343642	+	0.939101i	$0.611661\pi$
$108$	0	0
$109$	43842.4	0.353450	0.176725	−	0.984260i	$-0.443450\pi$
$109$	43842.4	0.353450	0.176725	+	0.984260i	$0.443450\pi$
$110$	0	0
$111$	56863.3	0.438051
$112$	0	0
$113$	−182786.	−1.34663	−0.673313	−	0.739358i	$-0.735129\pi$
$113$	−182786.	−1.34663	−0.673313	+	0.739358i	$0.735129\pi$
$114$	0	0
$115$	61711.3	0.435131
$116$	0	0
$117$	−151237.	−1.02139
$118$	0	0
$119$	41432.2	0.268207
$120$	0	0
$121$	−79208.7	−0.491824
$122$	0	0
$123$	−121616.	−0.724818
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	79395.4	0.436804	0.218402	−	0.975859i	$-0.429916\pi$
$127$	79395.4	0.436804	0.218402	+	0.975859i	$0.429916\pi$
$128$	0	0
$129$	67525.7	0.357268
$130$	0	0
$131$	146610.	0.746422	0.373211	−	0.927747i	$-0.378257\pi$
$131$	146610.	0.746422	0.373211	+	0.927747i	$0.378257\pi$
$132$	0	0
$133$	−74415.5	−0.364783
$134$	0	0
$135$	−22007.1	−0.103927
$136$	0	0
$137$	−292940.	−1.33345	−0.666726	−	0.745303i	$-0.732305\pi$
$137$	−292940.	−1.33345	−0.666726	+	0.745303i	$0.732305\pi$
$138$	0	0
$139$	−227589.	−0.999110	−0.499555	−	0.866282i	$-0.666503\pi$
$139$	−227589.	−0.999110	−0.499555	+	0.866282i	$0.666503\pi$
$140$	0	0
$141$	−461561.	−1.95515
$142$	0	0
$143$	−215003.	−0.879233
$144$	0	0
$145$	−22375.7	−0.0883806
$146$	0	0
$147$	−50605.6	−0.193155
$148$	0	0
$149$	−210681.	−0.777426	−0.388713	−	0.921359i	$-0.627080\pi$
$149$	−210681.	−0.777426	−0.388713	+	0.921359i	$0.627080\pi$
$150$	0	0
$151$	−530330.	−1.89279	−0.946397	−	0.323004i	$-0.895307\pi$
$151$	−530330.	−1.89279	−0.946397	+	0.323004i	$0.895307\pi$
$152$	0	0
$153$	−170155.	−0.587646
$154$	0	0
$155$	−245658.	−0.821299
$156$	0	0
$157$	467511.	1.51371	0.756855	−	0.653582i	$-0.226735\pi$
$157$	467511.	1.51371	0.756855	+	0.653582i	$0.226735\pi$
$158$	0	0
$159$	−383759.	−1.20383
$160$	0	0
$161$	120954.	0.367753
$162$	0	0
$163$	346889.	1.02264	0.511319	−	0.859391i	$-0.329157\pi$
$163$	346889.	1.02264	0.511319	+	0.859391i	$0.329157\pi$
$164$	0	0
$165$	150742.	0.431048
$166$	0	0
$167$	−483668.	−1.34201	−0.671006	−	0.741452i	$-0.734137\pi$
$167$	−483668.	−1.34201	−0.671006	+	0.741452i	$0.734137\pi$
$168$	0	0
$169$	193527.	0.521224
$170$	0	0
$171$	305612.	0.799245
$172$	0	0
$173$	−417227.	−1.05988	−0.529940	−	0.848035i	$-0.677786\pi$
$173$	−417227.	−1.05988	−0.529940	+	0.848035i	$0.677786\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−105835.	−0.253919
$178$	0	0
$179$	−553314.	−1.29074	−0.645371	−	0.763869i	$-0.723297\pi$
$179$	−553314.	−1.29074	−0.645371	+	0.763869i	$0.723297\pi$
$180$	0	0
$181$	188698.	0.428125	0.214062	−	0.976820i	$-0.431330\pi$
$181$	188698.	0.428125	0.214062	+	0.976820i	$0.431330\pi$
$182$	0	0
$183$	−478199.	−1.05556
$184$	0	0
$185$	67447.5	0.144889
$186$	0	0
$187$	−241897.	−0.505855
$188$	0	0
$189$	−43133.8	−0.0878342
$190$	0	0
$191$	−273571.	−0.542608	−0.271304	−	0.962494i	$-0.587455\pi$
$191$	−273571.	−0.542608	−0.271304	+	0.962494i	$0.587455\pi$
$192$	0	0
$193$	312739.	0.604350	0.302175	−	0.953252i	$-0.402287\pi$
$193$	312739.	0.604350	0.302175	+	0.953252i	$0.402287\pi$
$194$	0	0
$195$	−396005.	−0.745787
$196$	0	0
$197$	125978.	0.231274	0.115637	−	0.993292i	$-0.463109\pi$
$197$	125978.	0.231274	0.115637	+	0.993292i	$0.463109\pi$
$198$	0	0
$199$	−701921.	−1.25648	−0.628240	−	0.778019i	$-0.716225\pi$
$199$	−701921.	−1.25648	−0.628240	+	0.778019i	$0.716225\pi$
$200$	0	0
$201$	329934.	0.576019
$202$	0	0
$203$	−43856.4	−0.0746953
$204$	0	0
$205$	−144253.	−0.239740
$206$	0	0
$207$	−496738.	−0.805752
$208$	0	0
$209$	434466.	0.688003
$210$	0	0
$211$	1.01585e6	1.57081	0.785403	−	0.618985i	$-0.212456\pi$
$211$	1.01585e6	1.57081	0.785403	+	0.618985i	$0.212456\pi$
$212$	0	0
$213$	−443188.	−0.669328
$214$	0	0
$215$	80094.5	0.118170
$216$	0	0
$217$	−481490.	−0.694125
$218$	0	0
$219$	1.11280e6	1.56785
$220$	0	0
$221$	635472.	0.875217
$222$	0	0
$223$	−1.02507e6	−1.38035	−0.690177	−	0.723641i	$-0.742467\pi$
$223$	−1.02507e6	−1.38035	−0.690177	+	0.723641i	$0.742467\pi$
$224$	0	0
$225$	125772.	0.165625
$226$	0	0
$227$	501667.	0.646177	0.323088	−	0.946369i	$-0.395279\pi$
$227$	501667.	0.646177	0.323088	+	0.946369i	$0.395279\pi$
$228$	0	0
$229$	1.09069e6	1.37440	0.687200	−	0.726468i	$-0.258840\pi$
$229$	1.09069e6	1.37440	0.687200	+	0.726468i	$0.258840\pi$
$230$	0	0
$231$	295455.	0.364302
$232$	0	0
$233$	−115383.	−0.139237	−0.0696183	−	0.997574i	$-0.522178\pi$
$233$	−115383.	−0.139237	−0.0696183	+	0.997574i	$0.522178\pi$
$234$	0	0
$235$	−547473.	−0.646685
$236$	0	0
$237$	1.70060e6	1.96667
$238$	0	0
$239$	755233.	0.855236	0.427618	−	0.903960i	$-0.359353\pi$
$239$	755233.	0.855236	0.427618	+	0.903960i	$0.359353\pi$
$240$	0	0
$241$	−1.19575e6	−1.32616	−0.663082	−	0.748547i	$-0.730752\pi$
$241$	−1.19575e6	−1.32616	−0.663082	+	0.748547i	$0.730752\pi$
$242$	0	0
$243$	1.20780e6	1.31214
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−1.14136e6	−1.19036
$248$	0	0
$249$	−1.35308e6	−1.38301
$250$	0	0
$251$	−1.43265e6	−1.43535	−0.717673	−	0.696381i	$-0.754793\pi$
$251$	−1.43265e6	−1.43535	−0.717673	+	0.696381i	$0.754793\pi$
$252$	0	0
$253$	−706177.	−0.693605
$254$	0	0
$255$	−445541.	−0.429079
$256$	0	0
$257$	978660.	0.924270	0.462135	−	0.886810i	$-0.347084\pi$
$257$	978660.	0.924270	0.462135	+	0.886810i	$0.347084\pi$
$258$	0	0
$259$	132197.	0.122454
$260$	0	0
$261$	180111.	0.163659
$262$	0	0
$263$	−662781.	−0.590854	−0.295427	−	0.955365i	$-0.595462\pi$
$263$	−662781.	−0.590854	−0.295427	+	0.955365i	$0.595462\pi$
$264$	0	0
$265$	−455189.	−0.398178
$266$	0	0
$267$	1.51918e6	1.30416
$268$	0	0
$269$	−522166.	−0.439975	−0.219988	−	0.975503i	$-0.570602\pi$
$269$	−522166.	−0.439975	−0.219988	+	0.975503i	$0.570602\pi$
$270$	0	0
$271$	999234.	0.826502	0.413251	−	0.910617i	$-0.364393\pi$
$271$	999234.	0.826502	0.413251	+	0.910617i	$0.364393\pi$
$272$	0	0
$273$	−776171.	−0.630305
$274$	0	0
$275$	178801.	0.142573
$276$	0	0
$277$	1.41873e6	1.11097	0.555484	−	0.831527i	$-0.312533\pi$
$277$	1.41873e6	1.11097	0.555484	+	0.831527i	$0.312533\pi$
$278$	0	0
$279$	1.97740e6	1.52084
$280$	0	0
$281$	1.42688e6	1.07800	0.539002	−	0.842305i	$-0.318802\pi$
$281$	1.42688e6	1.07800	0.539002	+	0.842305i	$0.318802\pi$
$282$	0	0
$283$	−2.54018e6	−1.88538	−0.942689	−	0.333674i	$-0.891712\pi$
$283$	−2.54018e6	−1.88538	−0.942689	+	0.333674i	$0.891712\pi$
$284$	0	0
$285$	800228.	0.583581
$286$	0	0
$287$	−282736.	−0.202617
$288$	0	0
$289$	−704895.	−0.496455
$290$	0	0
$291$	−885413.	−0.612934
$292$	0	0
$293$	−2.55139e6	−1.73623	−0.868117	−	0.496359i	$-0.834670\pi$
$293$	−2.55139e6	−1.73623	−0.868117	+	0.496359i	$0.834670\pi$
$294$	0	0
$295$	−125534.	−0.0839860
$296$	0	0
$297$	251832.	0.165661
$298$	0	0
$299$	1.85515e6	1.20006
$300$	0	0
$301$	156985.	0.0998717
$302$	0	0
$303$	−979370.	−0.612830
$304$	0	0
$305$	−567209.	−0.349135
$306$	0	0
$307$	1.82818e6	1.10707	0.553534	−	0.832827i	$-0.313279\pi$
$307$	1.82818e6	1.10707	0.553534	+	0.832827i	$0.313279\pi$
$308$	0	0
$309$	−2.68900e6	−1.60212
$310$	0	0
$311$	2.15857e6	1.26551	0.632754	−	0.774353i	$-0.281925\pi$
$311$	2.15857e6	1.26551	0.632754	+	0.774353i	$0.281925\pi$
$312$	0	0
$313$	151212.	0.0872421	0.0436210	−	0.999048i	$-0.486111\pi$
$313$	151212.	0.0872421	0.0436210	+	0.999048i	$0.486111\pi$
$314$	0	0
$315$	246512.	0.139979
$316$	0	0
$317$	2.59481e6	1.45030	0.725148	−	0.688593i	$-0.241771\pi$
$317$	2.59481e6	1.45030	0.725148	+	0.688593i	$0.241771\pi$
$318$	0	0
$319$	256051.	0.140880
$320$	0	0
$321$	1.71555e6	0.929265
$322$	0	0
$323$	−1.28413e6	−0.684861
$324$	0	0
$325$	−469716.	−0.246676
$326$	0	0
$327$	−924060.	−0.477893
$328$	0	0
$329$	−1.07305e6	−0.546549
$330$	0	0
$331$	351723.	0.176453	0.0882267	−	0.996100i	$-0.471880\pi$
$331$	351723.	0.176453	0.0882267	+	0.996100i	$0.471880\pi$
$332$	0	0
$333$	−542911.	−0.268298
$334$	0	0
$335$	391346.	0.190524
$336$	0	0
$337$	666107.	0.319499	0.159749	−	0.987158i	$-0.448931\pi$
$337$	666107.	0.319499	0.159749	+	0.987158i	$0.448931\pi$
$338$	0	0
$339$	3.85256e6	1.82075
$340$	0	0
$341$	2.81112e6	1.30916
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.30068e6	−0.588333
$346$	0	0
$347$	−428588.	−0.191080	−0.0955402	−	0.995426i	$-0.530458\pi$
$347$	−428588.	−0.191080	−0.0955402	+	0.995426i	$0.530458\pi$
$348$	0	0
$349$	233734.	0.102721	0.0513603	−	0.998680i	$-0.483644\pi$
$349$	233734.	0.102721	0.0513603	+	0.998680i	$0.483644\pi$
$350$	0	0
$351$	−661572.	−0.286622
$352$	0	0
$353$	730202.	0.311893	0.155947	−	0.987765i	$-0.450157\pi$
$353$	730202.	0.311893	0.155947	+	0.987765i	$0.450157\pi$
$354$	0	0
$355$	−525681.	−0.221386
$356$	0	0
$357$	−873260.	−0.362638
$358$	0	0
$359$	1.27637e6	0.522684	0.261342	−	0.965246i	$-0.415835\pi$
$359$	1.27637e6	0.522684	0.261342	+	0.965246i	$0.415835\pi$
$360$	0	0
$361$	−169699.	−0.0685350
$362$	0	0
$363$	1.66947e6	0.664986
$364$	0	0
$365$	1.31993e6	0.518582
$366$	0	0
$367$	−4.92829e6	−1.90999	−0.954995	−	0.296623i	$-0.904139\pi$
$367$	−4.92829e6	−1.90999	−0.954995	+	0.296623i	$0.904139\pi$
$368$	0	0
$369$	1.16115e6	0.443938
$370$	0	0
$371$	−892171.	−0.336522
$372$	0	0
$373$	−2.33084e6	−0.867444	−0.433722	−	0.901047i	$-0.642800\pi$
$373$	−2.33084e6	−0.867444	−0.433722	+	0.901047i	$0.642800\pi$
$374$	0	0
$375$	329326.	0.120934
$376$	0	0
$377$	−672655.	−0.243747
$378$	0	0
$379$	−96821.4	−0.0346237	−0.0173119	−	0.999850i	$-0.505511\pi$
$379$	−96821.4	−0.0346237	−0.0173119	+	0.999850i	$0.505511\pi$
$380$	0	0
$381$	−1.67341e6	−0.590594
$382$	0	0
$383$	−1.02936e6	−0.358566	−0.179283	−	0.983798i	$-0.557378\pi$
$383$	−1.02936e6	−0.358566	−0.179283	+	0.983798i	$0.557378\pi$
$384$	0	0
$385$	350449.	0.120496
$386$	0	0
$387$	−644711.	−0.218820
$388$	0	0
$389$	−1.81126e6	−0.606885	−0.303443	−	0.952850i	$-0.598136\pi$
$389$	−1.81126e6	−0.606885	−0.303443	+	0.952850i	$0.598136\pi$
$390$	0	0
$391$	2.08721e6	0.690437
$392$	0	0
$393$	−3.09007e6	−1.00922
$394$	0	0
$395$	2.01714e6	0.650494
$396$	0	0
$397$	1.80750e6	0.575576	0.287788	−	0.957694i	$-0.407080\pi$
$397$	1.80750e6	0.575576	0.287788	+	0.957694i	$0.407080\pi$
$398$	0	0
$399$	1.56845e6	0.493216
$400$	0	0
$401$	−3.13485e6	−0.973545	−0.486773	−	0.873529i	$-0.661826\pi$
$401$	−3.13485e6	−0.973545	−0.486773	+	0.873529i	$0.661826\pi$
$402$	0	0
$403$	−7.38492e6	−2.26508
$404$	0	0
$405$	1.68634e6	0.510867
$406$	0	0
$407$	−771817.	−0.230956
$408$	0	0
$409$	−6.02204e6	−1.78006	−0.890031	−	0.455900i	$-0.849317\pi$
$409$	−6.02204e6	−1.78006	−0.890031	+	0.455900i	$0.849317\pi$
$410$	0	0
$411$	6.17426e6	1.80294
$412$	0	0
$413$	−246047.	−0.0709811
$414$	0	0
$415$	−1.60494e6	−0.457443
$416$	0	0
$417$	4.79686e6	1.35088
$418$	0	0
$419$	6.77548e6	1.88541	0.942704	−	0.333631i	$-0.108274\pi$
$419$	6.77548e6	1.88541	0.942704	+	0.333631i	$0.108274\pi$
$420$	0	0
$421$	765395.	0.210465	0.105233	−	0.994448i	$-0.466441\pi$
$421$	765395.	0.210465	0.105233	+	0.994448i	$0.466441\pi$
$422$	0	0
$423$	4.40682e6	1.19750
$424$	0	0
$425$	−528471.	−0.141922
$426$	0	0
$427$	−1.11173e6	−0.295073
$428$	0	0
$429$	4.53158e6	1.18879
$430$	0	0
$431$	3.94945e6	1.02410	0.512052	−	0.858955i	$-0.328886\pi$
$431$	3.94945e6	1.02410	0.512052	+	0.858955i	$0.328886\pi$
$432$	0	0
$433$	−4.14225e6	−1.06174	−0.530868	−	0.847454i	$-0.678134\pi$
$433$	−4.14225e6	−1.06174	−0.530868	+	0.847454i	$0.678134\pi$
$434$	0	0
$435$	471610.	0.119498
$436$	0	0
$437$	−3.74880e6	−0.939049
$438$	0	0
$439$	6.06844e6	1.50285	0.751425	−	0.659818i	$-0.229367\pi$
$439$	6.06844e6	1.50285	0.751425	+	0.659818i	$0.229367\pi$
$440$	0	0
$441$	483164.	0.118304
$442$	0	0
$443$	1.61019e6	0.389823	0.194912	−	0.980821i	$-0.437558\pi$
$443$	1.61019e6	0.389823	0.194912	+	0.980821i	$0.437558\pi$
$444$	0	0
$445$	1.80195e6	0.431363
$446$	0	0
$447$	4.44049e6	1.05114
$448$	0	0
$449$	740602.	0.173368	0.0866841	−	0.996236i	$-0.472373\pi$
$449$	740602.	0.173368	0.0866841	+	0.996236i	$0.472373\pi$
$450$	0	0
$451$	1.65072e6	0.382149
$452$	0	0
$453$	1.11777e7	2.55921
$454$	0	0
$455$	−920643.	−0.208479
$456$	0	0
$457$	−44871.6	−0.0100504	−0.00502518	−	0.999987i	$-0.501600\pi$
$457$	−44871.6	−0.0100504	−0.00502518	+	0.999987i	$0.501600\pi$
$458$	0	0
$459$	−744327.	−0.164904
$460$	0	0
$461$	−6.14817e6	−1.34739	−0.673695	−	0.739009i	$-0.735294\pi$
$461$	−6.14817e6	−1.34739	−0.673695	+	0.739009i	$0.735294\pi$
$462$	0	0
$463$	5.91399e6	1.28212	0.641059	−	0.767492i	$-0.278496\pi$
$463$	5.91399e6	1.28212	0.641059	+	0.767492i	$0.278496\pi$
$464$	0	0
$465$	5.17770e6	1.11046
$466$	0	0
$467$	−4.46680e6	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$467$	−4.46680e6	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$468$	0	0
$469$	767038.	0.161022
$470$	0	0
$471$	−9.85368e6	−2.04666
$472$	0	0
$473$	−916540.	−0.188364
$474$	0	0
$475$	949177.	0.193025
$476$	0	0
$477$	3.66399e6	0.737325
$478$	0	0
$479$	3.58911e6	0.714740	0.357370	−	0.933963i	$-0.383674\pi$
$479$	3.58911e6	0.714740	0.357370	+	0.933963i	$0.383674\pi$
$480$	0	0
$481$	2.02759e6	0.399593
$482$	0	0
$483$	−2.54933e6	−0.497232
$484$	0	0
$485$	−1.05022e6	−0.202734
$486$	0	0
$487$	−3.77868e6	−0.721968	−0.360984	−	0.932572i	$-0.617559\pi$
$487$	−3.77868e6	−0.721968	−0.360984	+	0.932572i	$0.617559\pi$
$488$	0	0
$489$	−7.31134e6	−1.38269
$490$	0	0
$491$	−1.51375e6	−0.283369	−0.141684	−	0.989912i	$-0.545252\pi$
$491$	−1.51375e6	−0.283369	−0.141684	+	0.989912i	$0.545252\pi$
$492$	0	0
$493$	−756796.	−0.140237
$494$	0	0
$495$	−1.43924e6	−0.264009
$496$	0	0
$497$	−1.03033e6	−0.187106
$498$	0	0
$499$	−2.20454e6	−0.396338	−0.198169	−	0.980168i	$-0.563500\pi$
$499$	−2.20454e6	−0.396338	−0.198169	+	0.980168i	$0.563500\pi$
$500$	0	0
$501$	1.01942e7	1.81451
$502$	0	0
$503$	9.13351e6	1.60960	0.804799	−	0.593547i	$-0.202273\pi$
$503$	9.13351e6	1.60960	0.804799	+	0.593547i	$0.202273\pi$
$504$	0	0
$505$	−1.16166e6	−0.202699
$506$	0	0
$507$	−4.07894e6	−0.704738
$508$	0	0
$509$	5.86974e6	1.00421	0.502105	−	0.864807i	$-0.332559\pi$
$509$	5.86974e6	1.00421	0.502105	+	0.864807i	$0.332559\pi$
$510$	0	0
$511$	2.58705e6	0.438282
$512$	0	0
$513$	1.33687e6	0.224283
$514$	0	0
$515$	−3.18951e6	−0.529915
$516$	0	0
$517$	6.26486e6	1.03083
$518$	0	0
$519$	8.79383e6	1.43305
$520$	0	0
$521$	−4.28621e6	−0.691797	−0.345899	−	0.938272i	$-0.612426\pi$
$521$	−4.28621e6	−0.691797	−0.345899	+	0.938272i	$0.612426\pi$
$522$	0	0
$523$	412959.	0.0660165	0.0330082	−	0.999455i	$-0.489491\pi$
$523$	412959.	0.0660165	0.0330082	+	0.999455i	$0.489491\pi$
$524$	0	0
$525$	645479.	0.102208
$526$	0	0
$527$	−8.30869e6	−1.30318
$528$	0	0
$529$	−343091.	−0.0533053
$530$	0	0
$531$	1.01047e6	0.155521
$532$	0	0
$533$	−4.33651e6	−0.661184
$534$	0	0
$535$	2.03487e6	0.307363
$536$	0	0
$537$	1.16621e7	1.74519
$538$	0	0
$539$	686880.	0.101838
$540$	0	0
$541$	5.93333e6	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$541$	5.93333e6	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$542$	0	0
$543$	−3.97716e6	−0.578860
$544$	0	0
$545$	−1.09606e6	−0.158068
$546$	0	0
$547$	−3.77825e6	−0.539912	−0.269956	−	0.962873i	$-0.587009\pi$
$547$	−3.77825e6	−0.539912	−0.269956	+	0.962873i	$0.587009\pi$
$548$	0	0
$549$	4.56568e6	0.646509
$550$	0	0
$551$	1.35927e6	0.190733
$552$	0	0
$553$	3.95360e6	0.549768
$554$	0	0
$555$	−1.42158e6	−0.195902
$556$	0	0
$557$	2.28752e6	0.312412	0.156206	−	0.987724i	$-0.450074\pi$
$557$	2.28752e6	0.312412	0.156206	+	0.987724i	$0.450074\pi$
$558$	0	0
$559$	2.40778e6	0.325903
$560$	0	0
$561$	5.09843e6	0.683958
$562$	0	0
$563$	393869.	0.0523698	0.0261849	−	0.999657i	$-0.491664\pi$
$563$	393869.	0.0523698	0.0261849	+	0.999657i	$0.491664\pi$
$564$	0	0
$565$	4.56965e6	0.602229
$566$	0	0
$567$	3.30523e6	0.431761
$568$	0	0
$569$	1.15221e7	1.49194	0.745972	−	0.665977i	$-0.231985\pi$
$569$	1.15221e7	1.49194	0.745972	+	0.665977i	$0.231985\pi$
$570$	0	0
$571$	−8.55465e6	−1.09802	−0.549012	−	0.835814i	$-0.684996\pi$
$571$	−8.55465e6	−1.09802	−0.549012	+	0.835814i	$0.684996\pi$
$572$	0	0
$573$	5.76602e6	0.733651
$574$	0	0
$575$	−1.54278e6	−0.194596
$576$	0	0
$577$	−2.89933e6	−0.362542	−0.181271	−	0.983433i	$-0.558021\pi$
$577$	−2.89933e6	−0.362542	−0.181271	+	0.983433i	$0.558021\pi$
$578$	0	0
$579$	−6.59156e6	−0.817131
$580$	0	0
$581$	−3.14567e6	−0.386610
$582$	0	0
$583$	5.20884e6	0.634701
$584$	0	0
$585$	3.78092e6	0.456781
$586$	0	0
$587$	1.03041e7	1.23428	0.617140	−	0.786854i	$-0.288291\pi$
$587$	1.03041e7	1.23428	0.617140	+	0.786854i	$0.288291\pi$
$588$	0	0
$589$	1.49231e7	1.77243
$590$	0	0
$591$	−2.65521e6	−0.312702
$592$	0	0
$593$	−6.25856e6	−0.730866	−0.365433	−	0.930838i	$-0.619079\pi$
$593$	−6.25856e6	−0.730866	−0.365433	+	0.930838i	$0.619079\pi$
$594$	0	0
$595$	−1.03580e6	−0.119946
$596$	0	0
$597$	1.47943e7	1.69887
$598$	0	0
$599$	−5.46416e6	−0.622238	−0.311119	−	0.950371i	$-0.600704\pi$
$599$	−5.46416e6	−0.622238	−0.311119	+	0.950371i	$0.600704\pi$
$600$	0	0
$601$	−8.28445e6	−0.935572	−0.467786	−	0.883842i	$-0.654948\pi$
$601$	−8.28445e6	−0.935572	−0.467786	+	0.883842i	$0.654948\pi$
$602$	0	0
$603$	−3.15010e6	−0.352802
$604$	0	0
$605$	1.98022e6	0.219950
$606$	0	0
$607$	3.06212e6	0.337326	0.168663	−	0.985674i	$-0.446055\pi$
$607$	3.06212e6	0.337326	0.168663	+	0.985674i	$0.446055\pi$
$608$	0	0
$609$	924356.	0.100994
$610$	0	0
$611$	−1.64580e7	−1.78351
$612$	0	0
$613$	−1.19291e7	−1.28220	−0.641100	−	0.767457i	$-0.721522\pi$
$613$	−1.19291e7	−1.28220	−0.641100	+	0.767457i	$0.721522\pi$
$614$	0	0
$615$	3.04041e6	0.324148
$616$	0	0
$617$	−4.51435e6	−0.477400	−0.238700	−	0.971093i	$-0.576721\pi$
$617$	−4.51435e6	−0.477400	−0.238700	+	0.971093i	$0.576721\pi$
$618$	0	0
$619$	−1.41481e7	−1.48412	−0.742062	−	0.670331i	$-0.766152\pi$
$619$	−1.41481e7	−1.48412	−0.742062	+	0.670331i	$0.766152\pi$
$620$	0	0
$621$	−2.17293e6	−0.226109
$622$	0	0
$623$	3.53183e6	0.364569
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−9.15719e6	−0.930237
$628$	0	0
$629$	2.28122e6	0.229901
$630$	0	0
$631$	−3.95815e6	−0.395748	−0.197874	−	0.980227i	$-0.563404\pi$
$631$	−3.95815e6	−0.395748	−0.197874	+	0.980227i	$0.563404\pi$
$632$	0	0
$633$	−2.14109e7	−2.12386
$634$	0	0
$635$	−1.98489e6	−0.195345
$636$	0	0
$637$	−1.80446e6	−0.176197
$638$	0	0
$639$	4.23141e6	0.409952
$640$	0	0
$641$	−1.87473e7	−1.80217	−0.901083	−	0.433647i	$-0.857226\pi$
$641$	−1.87473e7	−1.80217	−0.901083	+	0.433647i	$0.857226\pi$
$642$	0	0
$643$	−3.25738e6	−0.310700	−0.155350	−	0.987860i	$-0.549650\pi$
$643$	−3.25738e6	−0.310700	−0.155350	+	0.987860i	$0.549650\pi$
$644$	0	0
$645$	−1.68814e6	−0.159775
$646$	0	0
$647$	5.76628e6	0.541546	0.270773	−	0.962643i	$-0.412721\pi$
$647$	5.76628e6	0.541546	0.270773	+	0.962643i	$0.412721\pi$
$648$	0	0
$649$	1.43652e6	0.133875
$650$	0	0
$651$	1.01483e7	0.938514
$652$	0	0
$653$	−9.30278e6	−0.853748	−0.426874	−	0.904311i	$-0.640385\pi$
$653$	−9.30278e6	−0.853748	−0.426874	+	0.904311i	$0.640385\pi$
$654$	0	0
$655$	−3.66524e6	−0.333810
$656$	0	0
$657$	−1.06246e7	−0.960282
$658$	0	0
$659$	1.05250e7	0.944081	0.472040	−	0.881577i	$-0.343518\pi$
$659$	1.05250e7	0.944081	0.472040	+	0.881577i	$0.343518\pi$
$660$	0	0
$661$	−1.73879e7	−1.54790	−0.773950	−	0.633246i	$-0.781722\pi$
$661$	−1.73879e7	−1.54790	−0.773950	+	0.633246i	$0.781722\pi$
$662$	0	0
$663$	−1.33938e7	−1.18337
$664$	0	0
$665$	1.86039e6	0.163136
$666$	0	0
$667$	−2.20934e6	−0.192286
$668$	0	0
$669$	2.16052e7	1.86635
$670$	0	0
$671$	6.49070e6	0.556526
$672$	0	0
$673$	−1.08486e7	−0.923282	−0.461641	−	0.887067i	$-0.652739\pi$
$673$	−1.08486e7	−0.923282	−0.461641	+	0.887067i	$0.652739\pi$
$674$	0	0
$675$	550177.	0.0464775
$676$	0	0
$677$	−1.83721e7	−1.54059	−0.770294	−	0.637689i	$-0.779890\pi$
$677$	−1.83721e7	−1.54059	−0.770294	+	0.637689i	$0.779890\pi$
$678$	0	0
$679$	−2.05843e6	−0.171341
$680$	0	0
$681$	−1.05736e7	−0.873684
$682$	0	0
$683$	−6.45618e6	−0.529571	−0.264785	−	0.964307i	$-0.585301\pi$
$683$	−6.45618e6	−0.529571	−0.264785	+	0.964307i	$0.585301\pi$
$684$	0	0
$685$	7.32350e6	0.596338
$686$	0	0
$687$	−2.29884e7	−1.85830
$688$	0	0
$689$	−1.36838e7	−1.09814
$690$	0	0
$691$	−1.09563e7	−0.872913	−0.436456	−	0.899725i	$-0.643767\pi$
$691$	−1.09563e7	−0.872913	−0.436456	+	0.899725i	$0.643767\pi$
$692$	0	0
$693$	−2.82090e6	−0.223128
$694$	0	0
$695$	5.68971e6	0.446816
$696$	0	0
$697$	−4.87896e6	−0.380404
$698$	0	0
$699$	2.43192e6	0.188259
$700$	0	0
$701$	−2.50941e7	−1.92875	−0.964375	−	0.264538i	$-0.914781\pi$
$701$	−2.50941e7	−1.92875	−0.964375	+	0.264538i	$0.914781\pi$
$702$	0	0
$703$	−4.09725e6	−0.312683
$704$	0	0
$705$	1.15390e7	0.874371
$706$	0	0
$707$	−2.27686e6	−0.171312
$708$	0	0
$709$	−1.52164e7	−1.13683	−0.568416	−	0.822741i	$-0.692444\pi$
$709$	−1.52164e7	−1.13683	−0.568416	+	0.822741i	$0.692444\pi$
$710$	0	0
$711$	−1.62367e7	−1.20455
$712$	0	0
$713$	−2.42558e7	−1.78686
$714$	0	0
$715$	5.37507e6	0.393205
$716$	0	0
$717$	−1.59179e7	−1.15635
$718$	0	0
$719$	2.44444e7	1.76342	0.881712	−	0.471788i	$-0.156391\pi$
$719$	2.44444e7	1.76342	0.881712	+	0.471788i	$0.156391\pi$
$720$	0	0
$721$	−6.25144e6	−0.447860
$722$	0	0
$723$	2.52026e7	1.79308
$724$	0	0
$725$	559393.	0.0395250
$726$	0	0
$727$	−2.58767e7	−1.81582	−0.907912	−	0.419161i	$-0.862324\pi$
$727$	−2.58767e7	−1.81582	−0.907912	+	0.419161i	$0.862324\pi$
$728$	0	0
$729$	−9.06548e6	−0.631789
$730$	0	0
$731$	2.70897e6	0.187504
$732$	0	0
$733$	2.63428e7	1.81093	0.905467	−	0.424416i	$-0.139521\pi$
$733$	2.63428e7	1.81093	0.905467	+	0.424416i	$0.139521\pi$
$734$	0	0
$735$	1.26514e6	0.0863814
$736$	0	0
$737$	−4.47827e6	−0.303697
$738$	0	0
$739$	−1.67769e7	−1.13006	−0.565029	−	0.825071i	$-0.691135\pi$
$739$	−1.67769e7	−1.13006	−0.565029	+	0.825071i	$0.691135\pi$
$740$	0	0
$741$	2.40563e7	1.60947
$742$	0	0
$743$	9.59030e6	0.637324	0.318662	−	0.947868i	$-0.396767\pi$
$743$	9.59030e6	0.637324	0.318662	+	0.947868i	$0.396767\pi$
$744$	0	0
$745$	5.26701e6	0.347675
$746$	0	0
$747$	1.29187e7	0.847069
$748$	0	0
$749$	3.98834e6	0.259769
$750$	0	0
$751$	−1.08659e7	−0.703020	−0.351510	−	0.936184i	$-0.614332\pi$
$751$	−1.08659e7	−0.703020	−0.351510	+	0.936184i	$0.614332\pi$
$752$	0	0
$753$	3.01958e7	1.94070
$754$	0	0
$755$	1.32582e7	0.846484
$756$	0	0
$757$	−1.88916e7	−1.19820	−0.599098	−	0.800676i	$-0.704474\pi$
$757$	−1.88916e7	−1.19820	−0.599098	+	0.800676i	$0.704474\pi$
$758$	0	0
$759$	1.48840e7	0.937811
$760$	0	0
$761$	−2.38282e7	−1.49152	−0.745761	−	0.666214i	$-0.767914\pi$
$761$	−2.38282e7	−1.49152	−0.745761	+	0.666214i	$0.767914\pi$
$762$	0	0
$763$	−2.14828e6	−0.133592
$764$	0	0
$765$	4.25387e6	0.262803
$766$	0	0
$767$	−3.77378e6	−0.231627
$768$	0	0
$769$	−1.44907e7	−0.883638	−0.441819	−	0.897104i	$-0.645667\pi$
$769$	−1.44907e7	−0.883638	−0.441819	+	0.897104i	$0.645667\pi$
$770$	0	0
$771$	−2.06271e7	−1.24969
$772$	0	0
$773$	3.26526e7	1.96548	0.982739	−	0.184995i	$-0.0592269\pi$
$773$	3.26526e7	1.96548	0.982739	+	0.184995i	$0.0592269\pi$
$774$	0	0
$775$	6.14145e6	0.367296
$776$	0	0
$777$	−2.78630e6	−0.165568
$778$	0	0
$779$	8.76300e6	0.517379
$780$	0	0
$781$	6.01549e6	0.352893
$782$	0	0
$783$	787878.	0.0459256
$784$	0	0
$785$	−1.16878e7	−0.676952
$786$	0	0
$787$	3.95371e6	0.227545	0.113773	−	0.993507i	$-0.463706\pi$
$787$	3.95371e6	0.227545	0.113773	+	0.993507i	$0.463706\pi$
$788$	0	0
$789$	1.39693e7	0.798884
$790$	0	0
$791$	8.95651e6	0.508976
$792$	0	0
$793$	−1.70513e7	−0.962886
$794$	0	0
$795$	9.59397e6	0.538369
$796$	0	0
$797$	2.69614e7	1.50348	0.751738	−	0.659462i	$-0.229216\pi$
$797$	2.69614e7	1.50348	0.751738	+	0.659462i	$0.229216\pi$
$798$	0	0
$799$	−1.85167e7	−1.02612
$800$	0	0
$801$	−1.45046e7	−0.798776
$802$	0	0
$803$	−1.51042e7	−0.826627
$804$	0	0
$805$	−3.02385e6	−0.164464
$806$	0	0
$807$	1.10056e7	0.594883
$808$	0	0
$809$	4.71873e6	0.253486	0.126743	−	0.991936i	$-0.459548\pi$
$809$	4.71873e6	0.253486	0.126743	+	0.991936i	$0.459548\pi$
$810$	0	0
$811$	1.31852e7	0.703938	0.351969	−	0.936012i	$-0.385512\pi$
$811$	1.31852e7	0.703938	0.351969	+	0.936012i	$0.385512\pi$
$812$	0	0
$813$	−2.10607e7	−1.11750
$814$	0	0
$815$	−8.67223e6	−0.457338
$816$	0	0
$817$	−4.86553e6	−0.255020
$818$	0	0
$819$	7.41061e6	0.386050
$820$	0	0
$821$	5.15697e6	0.267015	0.133508	−	0.991048i	$-0.457376\pi$
$821$	5.15697e6	0.267015	0.133508	+	0.991048i	$0.457376\pi$
$822$	0	0
$823$	−2.37615e7	−1.22286	−0.611428	−	0.791300i	$-0.709404\pi$
$823$	−2.37615e7	−1.22286	−0.611428	+	0.791300i	$0.709404\pi$
$824$	0	0
$825$	−3.76856e6	−0.192770
$826$	0	0
$827$	5.40811e6	0.274968	0.137484	−	0.990504i	$-0.456098\pi$
$827$	5.40811e6	0.274968	0.137484	+	0.990504i	$0.456098\pi$
$828$	0	0
$829$	−8.68373e6	−0.438854	−0.219427	−	0.975629i	$-0.570419\pi$
$829$	−8.68373e6	−0.438854	−0.219427	+	0.975629i	$0.570419\pi$
$830$	0	0
$831$	−2.99025e7	−1.50212
$832$	0	0
$833$	−2.03018e6	−0.101373
$834$	0	0
$835$	1.20917e7	0.600166
$836$	0	0
$837$	8.64994e6	0.426775
$838$	0	0
$839$	−8.19162e6	−0.401759	−0.200879	−	0.979616i	$-0.564380\pi$
$839$	−8.19162e6	−0.401759	−0.200879	+	0.979616i	$0.564380\pi$
$840$	0	0
$841$	−1.97101e7	−0.960944
$842$	0	0
$843$	−3.00741e7	−1.45755
$844$	0	0
$845$	−4.83817e6	−0.233098
$846$	0	0
$847$	3.88123e6	0.185892
$848$	0	0
$849$	5.35390e7	2.54919
$850$	0	0
$851$	6.65963e6	0.315229
$852$	0	0
$853$	−1.03765e7	−0.488292	−0.244146	−	0.969738i	$-0.578508\pi$
$853$	−1.03765e7	−0.488292	−0.244146	+	0.969738i	$0.578508\pi$
$854$	0	0
$855$	−7.64029e6	−0.357433
$856$	0	0
$857$	−2.83309e7	−1.31768	−0.658838	−	0.752285i	$-0.728952\pi$
$857$	−2.83309e7	−1.31768	−0.658838	+	0.752285i	$0.728952\pi$
$858$	0	0
$859$	−7.91250e6	−0.365874	−0.182937	−	0.983125i	$-0.558560\pi$
$859$	−7.91250e6	−0.365874	−0.182937	+	0.983125i	$0.558560\pi$
$860$	0	0
$861$	5.95920e6	0.273955
$862$	0	0
$863$	3.56407e7	1.62899	0.814495	−	0.580170i	$-0.197014\pi$
$863$	3.56407e7	1.62899	0.814495	+	0.580170i	$0.197014\pi$
$864$	0	0
$865$	1.04307e7	0.473993
$866$	0	0
$867$	1.48570e7	0.671248
$868$	0	0
$869$	−2.30826e7	−1.03690
$870$	0	0
$871$	1.17646e7	0.525449
$872$	0	0
$873$	8.45362e6	0.375411
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	2.22712e6	0.0977789	0.0488895	−	0.998804i	$-0.484432\pi$
$877$	2.22712e6	0.0977789	0.0488895	+	0.998804i	$0.484432\pi$
$878$	0	0
$879$	5.37754e7	2.34753
$880$	0	0
$881$	2.71435e7	1.17822	0.589110	−	0.808053i	$-0.299479\pi$
$881$	2.71435e7	1.17822	0.589110	+	0.808053i	$0.299479\pi$
$882$	0	0
$883$	−8.65272e6	−0.373466	−0.186733	−	0.982411i	$-0.559790\pi$
$883$	−8.65272e6	−0.373466	−0.186733	+	0.982411i	$0.559790\pi$
$884$	0	0
$885$	2.64587e6	0.113556
$886$	0	0
$887$	−2.48135e7	−1.05896	−0.529480	−	0.848323i	$-0.677613\pi$
$887$	−2.48135e7	−1.05896	−0.529480	+	0.848323i	$0.677613\pi$
$888$	0	0
$889$	−3.89038e6	−0.165096
$890$	0	0
$891$	−1.92972e7	−0.814329
$892$	0	0
$893$	3.32575e7	1.39560
$894$	0	0
$895$	1.38329e7	0.577237
$896$	0	0
$897$	−3.91008e7	−1.62257
$898$	0	0
$899$	8.79484e6	0.362935
$900$	0	0
$901$	−1.53955e7	−0.631803
$902$	0	0
$903$	−3.30876e6	−0.135035
$904$	0	0
$905$	−4.71744e6	−0.191463
$906$	0	0
$907$	3.85211e7	1.55482	0.777411	−	0.628993i	$-0.216533\pi$
$907$	3.85211e7	1.55482	0.777411	+	0.628993i	$0.216533\pi$
$908$	0	0
$909$	9.35068e6	0.375348
$910$	0	0
$911$	1.96525e7	0.784554	0.392277	−	0.919847i	$-0.371687\pi$
$911$	1.96525e7	0.784554	0.392277	+	0.919847i	$0.371687\pi$
$912$	0	0
$913$	1.83657e7	0.729171
$914$	0	0
$915$	1.19550e7	0.472059
$916$	0	0
$917$	−7.18387e6	−0.282121
$918$	0	0
$919$	2.65362e7	1.03646	0.518228	−	0.855243i	$-0.326592\pi$
$919$	2.65362e7	1.03646	0.518228	+	0.855243i	$0.326592\pi$
$920$	0	0
$921$	−3.85324e7	−1.49685
$922$	0	0
$923$	−1.58029e7	−0.610566
$924$	0	0
$925$	−1.68619e6	−0.0647965
$926$	0	0
$927$	2.56736e7	0.981268
$928$	0	0
$929$	−6.33956e6	−0.241002	−0.120501	−	0.992713i	$-0.538450\pi$
$929$	−6.33956e6	−0.241002	−0.120501	+	0.992713i	$0.538450\pi$
$930$	0	0
$931$	3.64636e6	0.137875
$932$	0	0
$933$	−4.54959e7	−1.71107
$934$	0	0
$935$	6.04742e6	0.226225
$936$	0	0
$937$	2.07109e7	0.770639	0.385320	−	0.922783i	$-0.374091\pi$
$937$	2.07109e7	0.770639	0.385320	+	0.922783i	$0.374091\pi$
$938$	0	0
$939$	−3.18708e6	−0.117958
$940$	0	0
$941$	4.32264e7	1.59138	0.795692	−	0.605702i	$-0.207108\pi$
$941$	4.32264e7	1.59138	0.795692	+	0.605702i	$0.207108\pi$
$942$	0	0
$943$	−1.42433e7	−0.521592
$944$	0	0
$945$	1.07835e6	0.0392807
$946$	0	0
$947$	3.25763e6	0.118039	0.0590197	−	0.998257i	$-0.481203\pi$
$947$	3.25763e6	0.118039	0.0590197	+	0.998257i	$0.481203\pi$
$948$	0	0
$949$	3.96793e7	1.43021
$950$	0	0
$951$	−5.46904e7	−1.96092
$952$	0	0
$953$	−1.58737e7	−0.566168	−0.283084	−	0.959095i	$-0.591358\pi$
$953$	−1.58737e7	−0.566168	−0.283084	+	0.959095i	$0.591358\pi$
$954$	0	0
$955$	6.83927e6	0.242662
$956$	0	0
$957$	−5.39675e6	−0.190481
$958$	0	0
$959$	1.43541e7	0.503998
$960$	0	0
$961$	6.79274e7	2.37266
$962$	0	0
$963$	−1.63794e7	−0.569158
$964$	0	0
$965$	−7.81847e6	−0.270274
$966$	0	0
$967$	−3.78998e7	−1.30338	−0.651690	−	0.758486i	$-0.725940\pi$
$967$	−3.78998e7	−1.30338	−0.651690	+	0.758486i	$0.725940\pi$
$968$	0	0
$969$	2.70654e7	0.925988
$970$	0	0
$971$	4.44989e7	1.51461	0.757306	−	0.653061i	$-0.226515\pi$
$971$	4.44989e7	1.51461	0.757306	+	0.653061i	$0.226515\pi$
$972$	0	0
$973$	1.11518e7	0.377628
$974$	0	0
$975$	9.90014e6	0.333526
$976$	0	0
$977$	3.58249e7	1.20074	0.600369	−	0.799723i	$-0.295020\pi$
$977$	3.58249e7	1.20074	0.600369	+	0.799723i	$0.295020\pi$
$978$	0	0
$979$	−2.06202e7	−0.687599
$980$	0	0
$981$	8.82261e6	0.292701
$982$	0	0
$983$	−2.97430e7	−0.981749	−0.490874	−	0.871230i	$-0.663323\pi$
$983$	−2.97430e7	−0.981749	−0.490874	+	0.871230i	$0.663323\pi$
$984$	0	0
$985$	−3.14944e6	−0.103429
$986$	0	0
$987$	2.26165e7	0.738979
$988$	0	0
$989$	7.90837e6	0.257097
$990$	0	0
$991$	−4.84017e7	−1.56558	−0.782792	−	0.622284i	$-0.786205\pi$
$991$	−4.84017e7	−1.56558	−0.782792	+	0.622284i	$0.786205\pi$
$992$	0	0
$993$	−7.41321e6	−0.238580
$994$	0	0
$995$	1.75480e7	0.561915
$996$	0	0
$997$	−2.75881e7	−0.878991	−0.439495	−	0.898245i	$-0.644843\pi$
$997$	−2.75881e7	−0.878991	−0.439495	+	0.898245i	$0.644843\pi$
$998$	0	0
$999$	−2.37491e6	−0.0752894

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.s.1.1		3
4.3	odd	2		280.6.a.e.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.e.1.3	✓	3	4.3	odd	2
560.6.a.s.1.1		3	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-21.0769 q^{3} -25.0000 q^{5} -49.0000 q^{7} +201.235 q^{9} +286.081 q^{11} -751.545 q^{13} +526.922 q^{15} -845.554 q^{17} +1518.68 q^{19} +1032.77 q^{21} -2468.45 q^{23} +625.000 q^{25} +880.283 q^{27} +895.029 q^{29} +9826.32 q^{31} -6029.69 q^{33} +1225.00 q^{35} -2697.90 q^{37} +15840.2 q^{39} +5770.13 q^{41} -3203.78 q^{43} -5030.87 q^{45} +21898.9 q^{47} +2401.00 q^{49} +17821.6 q^{51} +18207.6 q^{53} -7152.02 q^{55} -32009.1 q^{57} +5021.37 q^{59} +22688.3 q^{61} -9860.50 q^{63} +18788.6 q^{65} -15653.8 q^{67} +52027.2 q^{69} +21027.2 q^{71} -52797.0 q^{73} -13173.0 q^{75} -14018.0 q^{77} -80685.6 q^{79} -67453.6 q^{81} +64197.4 q^{83} +21138.9 q^{85} -18864.4 q^{87} -72078.1 q^{89} +36825.7 q^{91} -207108. q^{93} -37967.1 q^{95} +42008.8 q^{97} +57569.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33}+ \cdots + 156568 q^{99}+O(q^{100}) \) 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9 + 578 * q^11 + 80 * q^13 + 150 * q^15 - 1244 * q^17 - 944 * q^19 + 294 * q^21 - 1096 * q^23 + 1875 * q^25 + 3006 * q^27 + 1868 * q^29 + 6620 * q^31 + 2662 * q^33 + 3675 * q^35 - 4058 * q^37 + 20910 * q^39 - 9602 * q^41 + 12340 * q^43 - 5225 * q^45 + 41026 * q^47 + 7203 * q^49 - 10006 * q^51 - 37610 * q^53 - 14450 * q^55 - 95456 * q^57 + 37664 * q^59 - 8386 * q^61 - 10241 * q^63 - 2000 * q^65 - 69340 * q^67 - 37712 * q^69 + 34016 * q^71 - 45314 * q^73 - 3750 * q^75 - 28322 * q^77 - 1382 * q^79 - 101005 * q^81 + 10128 * q^83 + 31100 * q^85 + 41510 * q^87 - 222810 * q^89 - 3920 * q^91 - 174292 * q^93 + 23600 * q^95 - 159476 * q^97 + 156568 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more