LMFDB - Embedded newform 686.2.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [686,2,Mod(1,686)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("686.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(686, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 686 = 2 \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	686.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.47773757866$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	686.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +0.445042 q^{3} +1.00000 q^{4} -3.04892 q^{5} -0.445042 q^{6} -1.00000 q^{8} -2.80194 q^{9} +3.04892 q^{10} -4.40581 q^{11} +0.445042 q^{12} +5.71379 q^{13} -1.35690 q^{15} +1.00000 q^{16} +7.54288 q^{17} +2.80194 q^{18} +1.39612 q^{19} -3.04892 q^{20} +4.40581 q^{22} +7.18598 q^{23} -0.445042 q^{24} +4.29590 q^{25} -5.71379 q^{26} -2.58211 q^{27} -0.692021 q^{29} +1.35690 q^{30} +7.13706 q^{31} -1.00000 q^{32} -1.96077 q^{33} -7.54288 q^{34} -2.80194 q^{36} +1.66487 q^{37} -1.39612 q^{38} +2.54288 q^{39} +3.04892 q^{40} +0.506041 q^{41} +0.850855 q^{43} -4.40581 q^{44} +8.54288 q^{45} -7.18598 q^{46} +0.158834 q^{47} +0.445042 q^{48} -4.29590 q^{50} +3.35690 q^{51} +5.71379 q^{52} -6.41119 q^{53} +2.58211 q^{54} +13.4330 q^{55} +0.621334 q^{57} +0.692021 q^{58} -10.9269 q^{59} -1.35690 q^{60} +5.70410 q^{61} -7.13706 q^{62} +1.00000 q^{64} -17.4209 q^{65} +1.96077 q^{66} +10.9390 q^{67} +7.54288 q^{68} +3.19806 q^{69} +6.18060 q^{71} +2.80194 q^{72} +1.54527 q^{73} -1.66487 q^{74} +1.91185 q^{75} +1.39612 q^{76} -2.54288 q^{78} -7.84117 q^{79} -3.04892 q^{80} +7.25667 q^{81} -0.506041 q^{82} +3.21313 q^{83} -22.9976 q^{85} -0.850855 q^{86} -0.307979 q^{87} +4.40581 q^{88} +13.9487 q^{89} -8.54288 q^{90} +7.18598 q^{92} +3.17629 q^{93} -0.158834 q^{94} -4.25667 q^{95} -0.445042 q^{96} +1.55496 q^{97} +12.3448 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 3 q^{8} - 4 q^{9} + q^{12} + 9 q^{13} + 3 q^{16} + 4 q^{17} + 4 q^{18} + 13 q^{19} + 7 q^{23} - q^{24} - q^{25} - 9 q^{26} - 2 q^{27} + 3 q^{29} + 16 q^{31}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 3 * q^8 - 4 * q^9 + q^12 + 9 * q^13 + 3 * q^16 + 4 * q^17 + 4 * q^18 + 13 * q^19 + 7 * q^23 - q^24 - q^25 - 9 * q^26 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 3 * q^32 + 7 * q^33 - 4 * q^34 - 4 * q^36 + 6 * q^37 - 13 * q^38 - 11 * q^39 + 11 * q^41 - 11 * q^43 + 7 * q^45 - 7 * q^46 - 8 * q^47 + q^48 + q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 + 2 * q^54 + 21 * q^55 + 9 * q^57 - 3 * q^58 - 4 * q^59 + 31 * q^61 - 16 * q^62 + 3 * q^64 - 14 * q^65 - 7 * q^66 + 23 * q^67 + 4 * q^68 + 14 * q^69 + 7 * q^71 + 4 * q^72 + 27 * q^73 - 6 * q^74 + 2 * q^75 + 13 * q^76 + 11 * q^78 - 32 * q^79 - 5 * q^81 - 11 * q^82 - 11 * q^83 - 28 * q^85 + 11 * q^86 - 6 * q^87 + 10 * q^89 - 7 * q^90 + 7 * q^92 + 17 * q^93 + 8 * q^94 + 14 * q^95 - q^96 + 5 * q^97 + 14 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.445042	0.256945	0.128473	−	0.991713i	$-0.458993\pi$
$3$	0.445042	0.256945	0.128473	+	0.991713i	$0.458993\pi$
$4$	1.00000	0.500000
$5$	−3.04892	−1.36352	−0.681759	−	0.731577i	$-0.738785\pi$
$5$	−3.04892	−1.36352	−0.681759	+	0.731577i	$0.738785\pi$
$6$	−0.445042	−0.181688
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−2.80194	−0.933979
$10$	3.04892	0.964152
$11$	−4.40581	−1.32840	−0.664201	−	0.747554i	$-0.731228\pi$
$11$	−4.40581	−1.32840	−0.664201	+	0.747554i	$0.731228\pi$
$12$	0.445042	0.128473
$13$	5.71379	1.58472	0.792360	−	0.610053i	$-0.208852\pi$
$13$	5.71379	1.58472	0.792360	+	0.610053i	$0.208852\pi$
$14$	0	0
$15$	−1.35690	−0.350349
$16$	1.00000	0.250000
$17$	7.54288	1.82942	0.914708	−	0.404115i	$-0.132420\pi$
$17$	7.54288	1.82942	0.914708	+	0.404115i	$0.132420\pi$
$18$	2.80194	0.660423
$19$	1.39612	0.320293	0.160146	−	0.987093i	$-0.448803\pi$
$19$	1.39612	0.320293	0.160146	+	0.987093i	$0.448803\pi$
$20$	−3.04892	−0.681759
$21$	0	0
$22$	4.40581	0.939323
$23$	7.18598	1.49838	0.749190	−	0.662355i	$-0.230443\pi$
$23$	7.18598	1.49838	0.749190	+	0.662355i	$0.230443\pi$
$24$	−0.445042	−0.0908438
$25$	4.29590	0.859179
$26$	−5.71379	−1.12057
$27$	−2.58211	−0.496926
$28$	0	0
$29$	−0.692021	−0.128505	−0.0642526	−	0.997934i	$-0.520466\pi$
$29$	−0.692021	−0.128505	−0.0642526	+	0.997934i	$0.520466\pi$
$30$	1.35690	0.247734
$31$	7.13706	1.28185	0.640927	−	0.767602i	$-0.278550\pi$
$31$	7.13706	1.28185	0.640927	+	0.767602i	$0.278550\pi$
$32$	−1.00000	−0.176777
$33$	−1.96077	−0.341326
$34$	−7.54288	−1.29359
$35$	0	0
$36$	−2.80194	−0.466990
$37$	1.66487	0.273704	0.136852	−	0.990592i	$-0.456302\pi$
$37$	1.66487	0.273704	0.136852	+	0.990592i	$0.456302\pi$
$38$	−1.39612	−0.226481
$39$	2.54288	0.407186
$40$	3.04892	0.482076
$41$	0.506041	0.0790303	0.0395151	−	0.999219i	$-0.487419\pi$
$41$	0.506041	0.0790303	0.0395151	+	0.999219i	$0.487419\pi$
$42$	0	0
$43$	0.850855	0.129754	0.0648771	−	0.997893i	$-0.479334\pi$
$43$	0.850855	0.129754	0.0648771	+	0.997893i	$0.479334\pi$
$44$	−4.40581	−0.664201
$45$	8.54288	1.27350
$46$	−7.18598	−1.05952
$47$	0.158834	0.0231683	0.0115841	−	0.999933i	$-0.496313\pi$
$47$	0.158834	0.0231683	0.0115841	+	0.999933i	$0.496313\pi$
$48$	0.445042	0.0642363
$49$	0	0
$50$	−4.29590	−0.607532
$51$	3.35690	0.470059
$52$	5.71379	0.792360
$53$	−6.41119	−0.880645	−0.440322	−	0.897840i	$-0.645136\pi$
$53$	−6.41119	−0.880645	−0.440322	+	0.897840i	$0.645136\pi$
$54$	2.58211	0.351380
$55$	13.4330	1.81130
$56$	0	0
$57$	0.621334	0.0822977
$58$	0.692021	0.0908669
$59$	−10.9269	−1.42256	−0.711282	−	0.702907i	$-0.751885\pi$
$59$	−10.9269	−1.42256	−0.711282	+	0.702907i	$0.751885\pi$
$60$	−1.35690	−0.175175
$61$	5.70410	0.730336	0.365168	−	0.930942i	$-0.381012\pi$
$61$	5.70410	0.730336	0.365168	+	0.930942i	$0.381012\pi$
$62$	−7.13706	−0.906408
$63$	0	0
$64$	1.00000	0.125000
$65$	−17.4209	−2.16079
$66$	1.96077	0.241354
$67$	10.9390	1.33641	0.668206	−	0.743976i	$-0.267063\pi$
$67$	10.9390	1.33641	0.668206	+	0.743976i	$0.267063\pi$
$68$	7.54288	0.914708
$69$	3.19806	0.385001
$70$	0	0
$71$	6.18060	0.733503	0.366751	−	0.930319i	$-0.380470\pi$
$71$	6.18060	0.733503	0.366751	+	0.930319i	$0.380470\pi$
$72$	2.80194	0.330212
$73$	1.54527	0.180860	0.0904301	−	0.995903i	$-0.471176\pi$
$73$	1.54527	0.180860	0.0904301	+	0.995903i	$0.471176\pi$
$74$	−1.66487	−0.193538
$75$	1.91185	0.220762
$76$	1.39612	0.160146
$77$	0	0
$78$	−2.54288	−0.287924
$79$	−7.84117	−0.882200	−0.441100	−	0.897458i	$-0.645412\pi$
$79$	−7.84117	−0.882200	−0.441100	+	0.897458i	$0.645412\pi$
$80$	−3.04892	−0.340879
$81$	7.25667	0.806296
$82$	−0.506041	−0.0558829
$83$	3.21313	0.352687	0.176343	−	0.984329i	$-0.443573\pi$
$83$	3.21313	0.352687	0.176343	+	0.984329i	$0.443573\pi$
$84$	0	0
$85$	−22.9976	−2.49444
$86$	−0.850855	−0.0917501
$87$	−0.307979	−0.0330188
$88$	4.40581	0.469661
$89$	13.9487	1.47856	0.739279	−	0.673399i	$-0.235166\pi$
$89$	13.9487	1.47856	0.739279	+	0.673399i	$0.235166\pi$
$90$	−8.54288	−0.900498
$91$	0	0
$92$	7.18598	0.749190
$93$	3.17629	0.329366
$94$	−0.158834	−0.0163824
$95$	−4.25667	−0.436725
$96$	−0.445042	−0.0454219
$97$	1.55496	0.157882	0.0789410	−	0.996879i	$-0.474846\pi$
$97$	1.55496	0.157882	0.0789410	+	0.996879i	$0.474846\pi$
$98$	0	0
$99$	12.3448	1.24070
$100$	4.29590	0.429590
$101$	−5.00969	−0.498483	−0.249241	−	0.968441i	$-0.580181\pi$
$101$	−5.00969	−0.498483	−0.249241	+	0.968441i	$0.580181\pi$
$102$	−3.35690	−0.332382
$103$	−2.89546	−0.285298	−0.142649	−	0.989773i	$-0.545562\pi$
$103$	−2.89546	−0.285298	−0.142649	+	0.989773i	$0.545562\pi$
$104$	−5.71379	−0.560283
$105$	0	0
$106$	6.41119	0.622710
$107$	9.99330	0.966088	0.483044	−	0.875596i	$-0.339531\pi$
$107$	9.99330	0.966088	0.483044	+	0.875596i	$0.339531\pi$
$108$	−2.58211	−0.248463
$109$	−14.6015	−1.39857	−0.699284	−	0.714844i	$-0.746498\pi$
$109$	−14.6015	−1.39857	−0.699284	+	0.714844i	$0.746498\pi$
$110$	−13.4330	−1.28078
$111$	0.740939	0.0703268
$112$	0	0
$113$	16.5405	1.55600	0.777999	−	0.628266i	$-0.216235\pi$
$113$	16.5405	1.55600	0.777999	+	0.628266i	$0.216235\pi$
$114$	−0.621334	−0.0581932
$115$	−21.9095	−2.04307
$116$	−0.692021	−0.0642526
$117$	−16.0097	−1.48010
$118$	10.9269	1.00590
$119$	0	0
$120$	1.35690	0.123867
$121$	8.41119	0.764654
$122$	−5.70410	−0.516425
$123$	0.225209	0.0203064
$124$	7.13706	0.640927
$125$	2.14675	0.192011
$126$	0	0
$127$	−6.32304	−0.561079	−0.280540	−	0.959842i	$-0.590513\pi$
$127$	−6.32304	−0.561079	−0.280540	+	0.959842i	$0.590513\pi$
$128$	−1.00000	−0.0883883
$129$	0.378666	0.0333397
$130$	17.4209	1.52791
$131$	6.41550	0.560525	0.280263	−	0.959923i	$-0.409578\pi$
$131$	6.41550	0.560525	0.280263	+	0.959923i	$0.409578\pi$
$132$	−1.96077	−0.170663
$133$	0	0
$134$	−10.9390	−0.944986
$135$	7.87263	0.677568
$136$	−7.54288	−0.646796
$137$	−2.21014	−0.188825	−0.0944127	−	0.995533i	$-0.530097\pi$
$137$	−2.21014	−0.188825	−0.0944127	+	0.995533i	$0.530097\pi$
$138$	−3.19806	−0.272237
$139$	14.5579	1.23479	0.617394	−	0.786654i	$-0.288188\pi$
$139$	14.5579	1.23479	0.617394	+	0.786654i	$0.288188\pi$
$140$	0	0
$141$	0.0706876	0.00595297
$142$	−6.18060	−0.518665
$143$	−25.1739	−2.10515
$144$	−2.80194	−0.233495
$145$	2.10992	0.175219
$146$	−1.54527	−0.127887
$147$	0	0
$148$	1.66487	0.136852
$149$	10.0532	0.823593	0.411796	−	0.911276i	$-0.364902\pi$
$149$	10.0532	0.823593	0.411796	+	0.911276i	$0.364902\pi$
$150$	−1.91185	−0.156102
$151$	−4.67994	−0.380848	−0.190424	−	0.981702i	$-0.560986\pi$
$151$	−4.67994	−0.380848	−0.190424	+	0.981702i	$0.560986\pi$
$152$	−1.39612	−0.113241
$153$	−21.1347	−1.70864
$154$	0	0
$155$	−21.7603	−1.74783
$156$	2.54288	0.203593
$157$	13.0030	1.03775	0.518876	−	0.854850i	$-0.326351\pi$
$157$	13.0030	1.03775	0.518876	+	0.854850i	$0.326351\pi$
$158$	7.84117	0.623810
$159$	−2.85325	−0.226277
$160$	3.04892	0.241038
$161$	0	0
$162$	−7.25667	−0.570138
$163$	−19.1226	−1.49780	−0.748898	−	0.662685i	$-0.769417\pi$
$163$	−19.1226	−1.49780	−0.748898	+	0.662685i	$0.769417\pi$
$164$	0.506041	0.0395151
$165$	5.97823	0.465405
$166$	−3.21313	−0.249387
$167$	−22.3177	−1.72699	−0.863496	−	0.504355i	$-0.831730\pi$
$167$	−22.3177	−1.72699	−0.863496	+	0.504355i	$0.831730\pi$
$168$	0	0
$169$	19.6474	1.51134
$170$	22.9976	1.76384
$171$	−3.91185	−0.299147
$172$	0.850855	0.0648771
$173$	−1.95108	−0.148338	−0.0741690	−	0.997246i	$-0.523630\pi$
$173$	−1.95108	−0.148338	−0.0741690	+	0.997246i	$0.523630\pi$
$174$	0.307979	0.0233478
$175$	0	0
$176$	−4.40581	−0.332101
$177$	−4.86294	−0.365521
$178$	−13.9487	−1.04550
$179$	1.78448	0.133378	0.0666891	−	0.997774i	$-0.478756\pi$
$179$	1.78448	0.133378	0.0666891	+	0.997774i	$0.478756\pi$
$180$	8.54288	0.636748
$181$	10.3394	0.768524	0.384262	−	0.923224i	$-0.374456\pi$
$181$	10.3394	0.768524	0.384262	+	0.923224i	$0.374456\pi$
$182$	0	0
$183$	2.53856	0.187656
$184$	−7.18598	−0.529758
$185$	−5.07606	−0.373200
$186$	−3.17629	−0.232897
$187$	−33.2325	−2.43020
$188$	0.158834	0.0115841
$189$	0	0
$190$	4.25667	0.308811
$191$	−20.1468	−1.45777	−0.728884	−	0.684637i	$-0.759961\pi$
$191$	−20.1468	−1.45777	−0.728884	+	0.684637i	$0.759961\pi$
$192$	0.445042	0.0321181
$193$	−13.9172	−1.00178	−0.500892	−	0.865510i	$-0.666995\pi$
$193$	−13.9172	−1.00178	−0.500892	+	0.865510i	$0.666995\pi$
$194$	−1.55496	−0.111639
$195$	−7.75302	−0.555205
$196$	0	0
$197$	4.08815	0.291268	0.145634	−	0.989339i	$-0.453478\pi$
$197$	4.08815	0.291268	0.145634	+	0.989339i	$0.453478\pi$
$198$	−12.3448	−0.877308
$199$	−8.45473	−0.599340	−0.299670	−	0.954043i	$-0.596877\pi$
$199$	−8.45473	−0.599340	−0.299670	+	0.954043i	$0.596877\pi$
$200$	−4.29590	−0.303766
$201$	4.86831	0.343384
$202$	5.00969	0.352480
$203$	0	0
$204$	3.35690	0.235030
$205$	−1.54288	−0.107759
$206$	2.89546	0.201736
$207$	−20.1347	−1.39946
$208$	5.71379	0.396180
$209$	−6.15106	−0.425478
$210$	0	0
$211$	4.04221	0.278277	0.139139	−	0.990273i	$-0.455567\pi$
$211$	4.04221	0.278277	0.139139	+	0.990273i	$0.455567\pi$
$212$	−6.41119	−0.440322
$213$	2.75063	0.188470
$214$	−9.99330	−0.683128
$215$	−2.59419	−0.176922
$216$	2.58211	0.175690
$217$	0	0
$218$	14.6015	0.988937
$219$	0.687710	0.0464711
$220$	13.4330	0.905650
$221$	43.0984	2.89911
$222$	−0.740939	−0.0497286
$223$	2.87800	0.192725	0.0963626	−	0.995346i	$-0.469279\pi$
$223$	2.87800	0.192725	0.0963626	+	0.995346i	$0.469279\pi$
$224$	0	0
$225$	−12.0368	−0.802456
$226$	−16.5405	−1.10026
$227$	−25.1250	−1.66760	−0.833802	−	0.552064i	$-0.813840\pi$
$227$	−25.1250	−1.66760	−0.833802	+	0.552064i	$0.813840\pi$
$228$	0.621334	0.0411488
$229$	−5.87800	−0.388429	−0.194215	−	0.980959i	$-0.562216\pi$
$229$	−5.87800	−0.388429	−0.194215	+	0.980959i	$0.562216\pi$
$230$	21.9095	1.44467
$231$	0	0
$232$	0.692021	0.0454334
$233$	17.7506	1.16288	0.581441	−	0.813588i	$-0.302489\pi$
$233$	17.7506	1.16288	0.581441	+	0.813588i	$0.302489\pi$
$234$	16.0097	1.04659
$235$	−0.484271	−0.0315903
$236$	−10.9269	−0.711282
$237$	−3.48965	−0.226677
$238$	0	0
$239$	−10.7681	−0.696530	−0.348265	−	0.937396i	$-0.613229\pi$
$239$	−10.7681	−0.696530	−0.348265	+	0.937396i	$0.613229\pi$
$240$	−1.35690	−0.0875873
$241$	18.7899	1.21036	0.605181	−	0.796088i	$-0.293101\pi$
$241$	18.7899	1.21036	0.605181	+	0.796088i	$0.293101\pi$
$242$	−8.41119	−0.540692
$243$	10.9758	0.704100
$244$	5.70410	0.365168
$245$	0	0
$246$	−0.225209	−0.0143588
$247$	7.97716	0.507575
$248$	−7.13706	−0.453204
$249$	1.42998	0.0906211
$250$	−2.14675	−0.135773
$251$	−2.76271	−0.174381	−0.0871903	−	0.996192i	$-0.527789\pi$
$251$	−2.76271	−0.174381	−0.0871903	+	0.996192i	$0.527789\pi$
$252$	0	0
$253$	−31.6601	−1.99045
$254$	6.32304	0.396743
$255$	−10.2349	−0.640934
$256$	1.00000	0.0625000
$257$	2.89546	0.180614	0.0903069	−	0.995914i	$-0.471215\pi$
$257$	2.89546	0.180614	0.0903069	+	0.995914i	$0.471215\pi$
$258$	−0.378666	−0.0235747
$259$	0	0
$260$	−17.4209	−1.08040
$261$	1.93900	0.120021
$262$	−6.41550	−0.396351
$263$	20.0519	1.23645	0.618227	−	0.786000i	$-0.287851\pi$
$263$	20.0519	1.23645	0.618227	+	0.786000i	$0.287851\pi$
$264$	1.96077	0.120677
$265$	19.5472	1.20077
$266$	0	0
$267$	6.20775	0.379908
$268$	10.9390	0.668206
$269$	17.5961	1.07285	0.536427	−	0.843947i	$-0.319774\pi$
$269$	17.5961	1.07285	0.536427	+	0.843947i	$0.319774\pi$
$270$	−7.87263	−0.479113
$271$	14.3787	0.873442	0.436721	−	0.899597i	$-0.356140\pi$
$271$	14.3787	0.873442	0.436721	+	0.899597i	$0.356140\pi$
$272$	7.54288	0.457354
$273$	0	0
$274$	2.21014	0.133520
$275$	−18.9269	−1.14134
$276$	3.19806	0.192501
$277$	3.95407	0.237577	0.118788	−	0.992920i	$-0.462099\pi$
$277$	3.95407	0.237577	0.118788	+	0.992920i	$0.462099\pi$
$278$	−14.5579	−0.873127
$279$	−19.9976	−1.19723
$280$	0	0
$281$	9.94438	0.593232	0.296616	−	0.954997i	$-0.404142\pi$
$281$	9.94438	0.593232	0.296616	+	0.954997i	$0.404142\pi$
$282$	−0.0706876	−0.00420939
$283$	21.0054	1.24864	0.624320	−	0.781169i	$-0.285376\pi$
$283$	21.0054	1.24864	0.624320	+	0.781169i	$0.285376\pi$
$284$	6.18060	0.366751
$285$	−1.89440	−0.112214
$286$	25.1739	1.48856
$287$	0	0
$288$	2.80194	0.165106
$289$	39.8950	2.34676
$290$	−2.10992	−0.123899
$291$	0.692021	0.0405670
$292$	1.54527	0.0904301
$293$	−18.7972	−1.09814	−0.549071	−	0.835776i	$-0.685018\pi$
$293$	−18.7972	−1.09814	−0.549071	+	0.835776i	$0.685018\pi$
$294$	0	0
$295$	33.3153	1.93969
$296$	−1.66487	−0.0967689
$297$	11.3763	0.660118
$298$	−10.0532	−0.582368
$299$	41.0592	2.37451
$300$	1.91185	0.110381
$301$	0	0
$302$	4.67994	0.269300
$303$	−2.22952	−0.128083
$304$	1.39612	0.0800732
$305$	−17.3913	−0.995825
$306$	21.1347	1.20819
$307$	−13.4601	−0.768209	−0.384104	−	0.923290i	$-0.625490\pi$
$307$	−13.4601	−0.768209	−0.384104	+	0.923290i	$0.625490\pi$
$308$	0	0
$309$	−1.28860	−0.0733060
$310$	21.7603	1.23590
$311$	20.7966	1.17926	0.589632	−	0.807672i	$-0.299273\pi$
$311$	20.7966	1.17926	0.589632	+	0.807672i	$0.299273\pi$
$312$	−2.54288	−0.143962
$313$	25.8931	1.46356	0.731781	−	0.681539i	$-0.238689\pi$
$313$	25.8931	1.46356	0.731781	+	0.681539i	$0.238689\pi$
$314$	−13.0030	−0.733801
$315$	0	0
$316$	−7.84117	−0.441100
$317$	−7.51573	−0.422125	−0.211063	−	0.977473i	$-0.567692\pi$
$317$	−7.51573	−0.422125	−0.211063	+	0.977473i	$0.567692\pi$
$318$	2.85325	0.160002
$319$	3.04892	0.170707
$320$	−3.04892	−0.170440
$321$	4.44743	0.248232
$322$	0	0
$323$	10.5308	0.585949
$324$	7.25667	0.403148
$325$	24.5459	1.36156
$326$	19.1226	1.05910
$327$	−6.49827	−0.359355
$328$	−0.506041	−0.0279414
$329$	0	0
$330$	−5.97823	−0.329091
$331$	16.9148	0.929724	0.464862	−	0.885383i	$-0.346104\pi$
$331$	16.9148	0.929724	0.464862	+	0.885383i	$0.346104\pi$
$332$	3.21313	0.176343
$333$	−4.66487	−0.255634
$334$	22.3177	1.22117
$335$	−33.3521	−1.82222
$336$	0	0
$337$	20.6722	1.12608	0.563042	−	0.826428i	$-0.309631\pi$
$337$	20.6722	1.12608	0.563042	+	0.826428i	$0.309631\pi$
$338$	−19.6474	−1.06868
$339$	7.36121	0.399806
$340$	−22.9976	−1.24722
$341$	−31.4446	−1.70282
$342$	3.91185	0.211529
$343$	0	0
$344$	−0.850855	−0.0458750
$345$	−9.75063	−0.524956
$346$	1.95108	0.104891
$347$	−14.3599	−0.770879	−0.385439	−	0.922733i	$-0.625950\pi$
$347$	−14.3599	−0.770879	−0.385439	+	0.922733i	$0.625950\pi$
$348$	−0.307979	−0.0165094
$349$	−4.80731	−0.257330	−0.128665	−	0.991688i	$-0.541069\pi$
$349$	−4.80731	−0.257330	−0.128665	+	0.991688i	$0.541069\pi$
$350$	0	0
$351$	−14.7536	−0.787490
$352$	4.40581	0.234831
$353$	29.1239	1.55011	0.775055	−	0.631894i	$-0.217722\pi$
$353$	29.1239	1.55011	0.775055	+	0.631894i	$0.217722\pi$
$354$	4.86294	0.258462
$355$	−18.8442	−1.00014
$356$	13.9487	0.739279
$357$	0	0
$358$	−1.78448	−0.0943127
$359$	−9.98361	−0.526915	−0.263457	−	0.964671i	$-0.584863\pi$
$359$	−9.98361	−0.526915	−0.263457	+	0.964671i	$0.584863\pi$
$360$	−8.54288	−0.450249
$361$	−17.0508	−0.897412
$362$	−10.3394	−0.543429
$363$	3.74333	0.196474
$364$	0	0
$365$	−4.71140	−0.246606
$366$	−2.53856	−0.132693
$367$	−23.6160	−1.23274	−0.616371	−	0.787456i	$-0.711398\pi$
$367$	−23.6160	−1.23274	−0.616371	+	0.787456i	$0.711398\pi$
$368$	7.18598	0.374595
$369$	−1.41789	−0.0738127
$370$	5.07606	0.263892
$371$	0	0
$372$	3.17629	0.164683
$373$	−26.3980	−1.36684	−0.683419	−	0.730026i	$-0.739508\pi$
$373$	−26.3980	−1.36684	−0.683419	+	0.730026i	$0.739508\pi$
$374$	33.2325	1.71841
$375$	0.955395	0.0493364
$376$	−0.158834	−0.00819122
$377$	−3.95407	−0.203645
$378$	0	0
$379$	−26.3913	−1.35563	−0.677816	−	0.735232i	$-0.737073\pi$
$379$	−26.3913	−1.35563	−0.677816	+	0.735232i	$0.737073\pi$
$380$	−4.25667	−0.218362
$381$	−2.81402	−0.144167
$382$	20.1468	1.03080
$383$	28.9081	1.47714	0.738568	−	0.674179i	$-0.235502\pi$
$383$	28.9081	1.47714	0.738568	+	0.674179i	$0.235502\pi$
$384$	−0.445042	−0.0227109
$385$	0	0
$386$	13.9172	0.708368
$387$	−2.38404	−0.121188
$388$	1.55496	0.0789410
$389$	3.46250	0.175556	0.0877779	−	0.996140i	$-0.472023\pi$
$389$	3.46250	0.175556	0.0877779	+	0.996140i	$0.472023\pi$
$390$	7.75302	0.392589
$391$	54.2030	2.74116
$392$	0	0
$393$	2.85517	0.144024
$394$	−4.08815	−0.205958
$395$	23.9071	1.20290
$396$	12.3448	0.620350
$397$	−38.2368	−1.91905	−0.959525	−	0.281622i	$-0.909128\pi$
$397$	−38.2368	−1.91905	−0.959525	+	0.281622i	$0.909128\pi$
$398$	8.45473	0.423797
$399$	0	0
$400$	4.29590	0.214795
$401$	9.49934	0.474374	0.237187	−	0.971464i	$-0.423775\pi$
$401$	9.49934	0.474374	0.237187	+	0.971464i	$0.423775\pi$
$402$	−4.86831	−0.242809
$403$	40.7797	2.03138
$404$	−5.00969	−0.249241
$405$	−22.1250	−1.09940
$406$	0	0
$407$	−7.33513	−0.363589
$408$	−3.35690	−0.166191
$409$	9.25906	0.457831	0.228916	−	0.973446i	$-0.426482\pi$
$409$	9.25906	0.457831	0.228916	+	0.973446i	$0.426482\pi$
$410$	1.54288	0.0761972
$411$	−0.983607	−0.0485177
$412$	−2.89546	−0.142649
$413$	0	0
$414$	20.1347	0.989565
$415$	−9.79656	−0.480894
$416$	−5.71379	−0.280142
$417$	6.47889	0.317273
$418$	6.15106	0.300858
$419$	−26.8310	−1.31078	−0.655390	−	0.755291i	$-0.727496\pi$
$419$	−26.8310	−1.31078	−0.655390	+	0.755291i	$0.727496\pi$
$420$	0	0
$421$	14.7560	0.719164	0.359582	−	0.933114i	$-0.382919\pi$
$421$	14.7560	0.719164	0.359582	+	0.933114i	$0.382919\pi$
$422$	−4.04221	−0.196772
$423$	−0.445042	−0.0216387
$424$	6.41119	0.311355
$425$	32.4034	1.57180
$426$	−2.75063	−0.133268
$427$	0	0
$428$	9.99330	0.483044
$429$	−11.2034	−0.540907
$430$	2.59419	0.125103
$431$	−31.8582	−1.53455	−0.767277	−	0.641316i	$-0.778389\pi$
$431$	−31.8582	−1.53455	−0.767277	+	0.641316i	$0.778389\pi$
$432$	−2.58211	−0.124232
$433$	−16.0194	−0.769842	−0.384921	−	0.922949i	$-0.625771\pi$
$433$	−16.0194	−0.769842	−0.384921	+	0.922949i	$0.625771\pi$
$434$	0	0
$435$	0.939001	0.0450217
$436$	−14.6015	−0.699284
$437$	10.0325	0.479921
$438$	−0.687710	−0.0328600
$439$	2.35690	0.112489	0.0562443	−	0.998417i	$-0.482087\pi$
$439$	2.35690	0.112489	0.0562443	+	0.998417i	$0.482087\pi$
$440$	−13.4330	−0.640391
$441$	0	0
$442$	−43.0984	−2.04998
$443$	13.1830	0.626343	0.313172	−	0.949697i	$-0.398609\pi$
$443$	13.1830	0.626343	0.313172	+	0.949697i	$0.398609\pi$
$444$	0.740939	0.0351634
$445$	−42.5284	−2.01604
$446$	−2.87800	−0.136277
$447$	4.47411	0.211618
$448$	0	0
$449$	−15.3448	−0.724167	−0.362083	−	0.932146i	$-0.617934\pi$
$449$	−15.3448	−0.724167	−0.362083	+	0.932146i	$0.617934\pi$
$450$	12.0368	0.567422
$451$	−2.22952	−0.104984
$452$	16.5405	0.777999
$453$	−2.08277	−0.0978570
$454$	25.1250	1.17917
$455$	0	0
$456$	−0.621334	−0.0290966
$457$	−22.8159	−1.06728	−0.533642	−	0.845710i	$-0.679177\pi$
$457$	−22.8159	−1.06728	−0.533642	+	0.845710i	$0.679177\pi$
$458$	5.87800	0.274661
$459$	−19.4765	−0.909085
$460$	−21.9095	−1.02153
$461$	−0.440730	−0.0205268	−0.0102634	−	0.999947i	$-0.503267\pi$
$461$	−0.440730	−0.0205268	−0.0102634	+	0.999947i	$0.503267\pi$
$462$	0	0
$463$	−25.5133	−1.18571	−0.592853	−	0.805311i	$-0.701998\pi$
$463$	−25.5133	−1.18571	−0.592853	+	0.805311i	$0.701998\pi$
$464$	−0.692021	−0.0321263
$465$	−9.68425	−0.449096
$466$	−17.7506	−0.822282
$467$	−21.4101	−0.990742	−0.495371	−	0.868681i	$-0.664968\pi$
$467$	−21.4101	−0.990742	−0.495371	+	0.868681i	$0.664968\pi$
$468$	−16.0097	−0.740048
$469$	0	0
$470$	0.484271	0.0223377
$471$	5.78687	0.266645
$472$	10.9269	0.502952
$473$	−3.74871	−0.172366
$474$	3.48965	0.160285
$475$	5.99761	0.275189
$476$	0	0
$477$	17.9638	0.822504
$478$	10.7681	0.492521
$479$	−32.6437	−1.49153	−0.745764	−	0.666210i	$-0.767915\pi$
$479$	−32.6437	−1.49153	−0.745764	+	0.666210i	$0.767915\pi$
$480$	1.35690	0.0619335
$481$	9.51275	0.433744
$482$	−18.7899	−0.855854
$483$	0	0
$484$	8.41119	0.382327
$485$	−4.74094	−0.215275
$486$	−10.9758	−0.497874
$487$	41.4010	1.87606	0.938030	−	0.346555i	$-0.112648\pi$
$487$	41.4010	1.87606	0.938030	+	0.346555i	$0.112648\pi$
$488$	−5.70410	−0.258213
$489$	−8.51035	−0.384851
$490$	0	0
$491$	1.97046	0.0889256	0.0444628	−	0.999011i	$-0.485842\pi$
$491$	1.97046	0.0889256	0.0444628	+	0.999011i	$0.485842\pi$
$492$	0.225209	0.0101532
$493$	−5.21983	−0.235089
$494$	−7.97716	−0.358910
$495$	−37.6383	−1.69172
$496$	7.13706	0.320464
$497$	0	0
$498$	−1.42998	−0.0640788
$499$	4.32006	0.193392	0.0966962	−	0.995314i	$-0.469172\pi$
$499$	4.32006	0.193392	0.0966962	+	0.995314i	$0.469172\pi$
$500$	2.14675	0.0960057
$501$	−9.93230	−0.443742
$502$	2.76271	0.123306
$503$	29.8649	1.33161	0.665804	−	0.746127i	$-0.268089\pi$
$503$	29.8649	1.33161	0.665804	+	0.746127i	$0.268089\pi$
$504$	0	0
$505$	15.2741	0.679690
$506$	31.6601	1.40746
$507$	8.74392	0.388331
$508$	−6.32304	−0.280540
$509$	37.9017	1.67996	0.839981	−	0.542615i	$-0.182566\pi$
$509$	37.9017	1.67996	0.839981	+	0.542615i	$0.182566\pi$
$510$	10.2349	0.453209
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−3.60494	−0.159162
$514$	−2.89546	−0.127713
$515$	8.82802	0.389009
$516$	0.378666	0.0166698
$517$	−0.699791	−0.0307768
$518$	0	0
$519$	−0.868313	−0.0381147
$520$	17.4209	0.763956
$521$	4.97823	0.218100	0.109050	−	0.994036i	$-0.465219\pi$
$521$	4.97823	0.218100	0.109050	+	0.994036i	$0.465219\pi$
$522$	−1.93900	−0.0848678
$523$	−16.8810	−0.738154	−0.369077	−	0.929399i	$-0.620326\pi$
$523$	−16.8810	−0.738154	−0.369077	+	0.929399i	$0.620326\pi$
$524$	6.41550	0.280263
$525$	0	0
$526$	−20.0519	−0.874304
$527$	53.8340	2.34505
$528$	−1.96077	−0.0853316
$529$	28.6383	1.24514
$530$	−19.5472	−0.849076
$531$	30.6165	1.32865
$532$	0	0
$533$	2.89141	0.125241
$534$	−6.20775	−0.268636
$535$	−30.4687	−1.31728
$536$	−10.9390	−0.472493
$537$	0.794168	0.0342709
$538$	−17.5961	−0.758622
$539$	0	0
$540$	7.87263	0.338784
$541$	−11.3720	−0.488919	−0.244459	−	0.969660i	$-0.578610\pi$
$541$	−11.3720	−0.488919	−0.244459	+	0.969660i	$0.578610\pi$
$542$	−14.3787	−0.617617
$543$	4.60148	0.197469
$544$	−7.54288	−0.323398
$545$	44.5187	1.90697
$546$	0	0
$547$	20.8528	0.891600	0.445800	−	0.895133i	$-0.352919\pi$
$547$	20.8528	0.891600	0.445800	+	0.895133i	$0.352919\pi$
$548$	−2.21014	−0.0944127
$549$	−15.9825	−0.682118
$550$	18.9269	0.807047
$551$	−0.966148	−0.0411593
$552$	−3.19806	−0.136119
$553$	0	0
$554$	−3.95407	−0.167992
$555$	−2.25906	−0.0958918
$556$	14.5579	0.617394
$557$	22.1672	0.939254	0.469627	−	0.882865i	$-0.344388\pi$
$557$	22.1672	0.939254	0.469627	+	0.882865i	$0.344388\pi$
$558$	19.9976	0.846566
$559$	4.86161	0.205624
$560$	0	0
$561$	−14.7899	−0.624428
$562$	−9.94438	−0.419478
$563$	19.0358	0.802262	0.401131	−	0.916021i	$-0.368617\pi$
$563$	19.0358	0.802262	0.401131	+	0.916021i	$0.368617\pi$
$564$	0.0706876	0.00297649
$565$	−50.4306	−2.12163
$566$	−21.0054	−0.882922
$567$	0	0
$568$	−6.18060	−0.259332
$569$	16.4209	0.688399	0.344200	−	0.938897i	$-0.388150\pi$
$569$	16.4209	0.688399	0.344200	+	0.938897i	$0.388150\pi$
$570$	1.89440	0.0793475
$571$	−27.0726	−1.13295	−0.566477	−	0.824078i	$-0.691694\pi$
$571$	−27.0726	−1.13295	−0.566477	+	0.824078i	$0.691694\pi$
$572$	−25.1739	−1.05257
$573$	−8.96615	−0.374566
$574$	0	0
$575$	30.8702	1.28738
$576$	−2.80194	−0.116747
$577$	−10.1056	−0.420702	−0.210351	−	0.977626i	$-0.567461\pi$
$577$	−10.1056	−0.420702	−0.210351	+	0.977626i	$0.567461\pi$
$578$	−39.8950	−1.65941
$579$	−6.19375	−0.257403
$580$	2.10992	0.0876095
$581$	0	0
$582$	−0.692021	−0.0286852
$583$	28.2465	1.16985
$584$	−1.54527	−0.0639437
$585$	48.8122	2.01814
$586$	18.7972	0.776503
$587$	−10.4964	−0.433231	−0.216615	−	0.976257i	$-0.569502\pi$
$587$	−10.4964	−0.433231	−0.216615	+	0.976257i	$0.569502\pi$
$588$	0	0
$589$	9.96423	0.410569
$590$	−33.3153	−1.37157
$591$	1.81940	0.0748400
$592$	1.66487	0.0684259
$593$	16.0218	0.657935	0.328968	−	0.944341i	$-0.393299\pi$
$593$	16.0218	0.657935	0.328968	+	0.944341i	$0.393299\pi$
$594$	−11.3763	−0.466774
$595$	0	0
$596$	10.0532	0.411796
$597$	−3.76271	−0.153997
$598$	−41.0592	−1.67904
$599$	−27.7356	−1.13324	−0.566622	−	0.823978i	$-0.691750\pi$
$599$	−27.7356	−1.13324	−0.566622	+	0.823978i	$0.691750\pi$
$600$	−1.91185	−0.0780511
$601$	11.7885	0.480864	0.240432	−	0.970666i	$-0.422711\pi$
$601$	11.7885	0.480864	0.240432	+	0.970666i	$0.422711\pi$
$602$	0	0
$603$	−30.6504	−1.24818
$604$	−4.67994	−0.190424
$605$	−25.6450	−1.04262
$606$	2.22952	0.0905681
$607$	−26.7875	−1.08727	−0.543635	−	0.839322i	$-0.682952\pi$
$607$	−26.7875	−1.08727	−0.543635	+	0.839322i	$0.682952\pi$
$608$	−1.39612	−0.0566203
$609$	0	0
$610$	17.3913	0.704155
$611$	0.907542	0.0367152
$612$	−21.1347	−0.854318
$613$	−3.90276	−0.157631	−0.0788154	−	0.996889i	$-0.525114\pi$
$613$	−3.90276	−0.157631	−0.0788154	+	0.996889i	$0.525114\pi$
$614$	13.4601	0.543206
$615$	−0.686645	−0.0276882
$616$	0	0
$617$	−45.5883	−1.83532	−0.917659	−	0.397370i	$-0.869923\pi$
$617$	−45.5883	−1.83532	−0.917659	+	0.397370i	$0.869923\pi$
$618$	1.28860	0.0518351
$619$	−45.7434	−1.83858	−0.919292	−	0.393576i	$-0.871238\pi$
$619$	−45.7434	−1.83858	−0.919292	+	0.393576i	$0.871238\pi$
$620$	−21.7603	−0.873915
$621$	−18.5550	−0.744585
$622$	−20.7966	−0.833866
$623$	0	0
$624$	2.54288	0.101797
$625$	−28.0248	−1.12099
$626$	−25.8931	−1.03490
$627$	−2.73748	−0.109324
$628$	13.0030	0.518876
$629$	12.5579	0.500718
$630$	0	0
$631$	−19.4832	−0.775614	−0.387807	−	0.921741i	$-0.626767\pi$
$631$	−19.4832	−0.775614	−0.387807	+	0.921741i	$0.626767\pi$
$632$	7.84117	0.311905
$633$	1.79895	0.0715020
$634$	7.51573	0.298488
$635$	19.2784	0.765041
$636$	−2.85325	−0.113139
$637$	0	0
$638$	−3.04892	−0.120708
$639$	−17.3177	−0.685076
$640$	3.04892	0.120519
$641$	3.11231	0.122929	0.0614644	−	0.998109i	$-0.480423\pi$
$641$	3.11231	0.122929	0.0614644	+	0.998109i	$0.480423\pi$
$642$	−4.44743	−0.175526
$643$	−19.4644	−0.767602	−0.383801	−	0.923416i	$-0.625385\pi$
$643$	−19.4644	−0.767602	−0.383801	+	0.923416i	$0.625385\pi$
$644$	0	0
$645$	−1.15452	−0.0454592
$646$	−10.5308	−0.414329
$647$	13.0084	0.511411	0.255706	−	0.966755i	$-0.417692\pi$
$647$	13.0084	0.511411	0.255706	+	0.966755i	$0.417692\pi$
$648$	−7.25667	−0.285069
$649$	48.1420	1.88974
$650$	−24.5459	−0.962768
$651$	0	0
$652$	−19.1226	−0.748898
$653$	27.9571	1.09404	0.547022	−	0.837118i	$-0.315761\pi$
$653$	27.9571	1.09404	0.547022	+	0.837118i	$0.315761\pi$
$654$	6.49827	0.254103
$655$	−19.5603	−0.764286
$656$	0.506041	0.0197576
$657$	−4.32975	−0.168920
$658$	0	0
$659$	−12.3655	−0.481692	−0.240846	−	0.970563i	$-0.577425\pi$
$659$	−12.3655	−0.481692	−0.240846	+	0.970563i	$0.577425\pi$
$660$	5.97823	0.232702
$661$	−29.6233	−1.15221	−0.576105	−	0.817375i	$-0.695428\pi$
$661$	−29.6233	−1.15221	−0.576105	+	0.817375i	$0.695428\pi$
$662$	−16.9148	−0.657414
$663$	19.1806	0.744913
$664$	−3.21313	−0.124694
$665$	0	0
$666$	4.66487	0.180760
$667$	−4.97285	−0.192550
$668$	−22.3177	−0.863496
$669$	1.28083	0.0495198
$670$	33.3521	1.28850
$671$	−25.1312	−0.970180
$672$	0	0
$673$	29.6528	1.14303	0.571516	−	0.820591i	$-0.306356\pi$
$673$	29.6528	1.14303	0.571516	+	0.820591i	$0.306356\pi$
$674$	−20.6722	−0.796262
$675$	−11.0925	−0.426949
$676$	19.6474	0.755670
$677$	−19.0868	−0.733566	−0.366783	−	0.930307i	$-0.619541\pi$
$677$	−19.0868	−0.733566	−0.366783	+	0.930307i	$0.619541\pi$
$678$	−7.36121	−0.282705
$679$	0	0
$680$	22.9976	0.881918
$681$	−11.1817	−0.428482
$682$	31.4446	1.20407
$683$	8.31336	0.318102	0.159051	−	0.987270i	$-0.449157\pi$
$683$	8.31336	0.318102	0.159051	+	0.987270i	$0.449157\pi$
$684$	−3.91185	−0.149573
$685$	6.73855	0.257467
$686$	0	0
$687$	−2.61596	−0.0998050
$688$	0.850855	0.0324385
$689$	−36.6322	−1.39558
$690$	9.75063	0.371200
$691$	42.4868	1.61627	0.808137	−	0.588995i	$-0.200476\pi$
$691$	42.4868	1.61627	0.808137	+	0.588995i	$0.200476\pi$
$692$	−1.95108	−0.0741690
$693$	0	0
$694$	14.3599	0.545094
$695$	−44.3860	−1.68366
$696$	0.307979	0.0116739
$697$	3.81700	0.144579
$698$	4.80731	0.181960
$699$	7.89977	0.298797
$700$	0	0
$701$	14.7356	0.556554	0.278277	−	0.960501i	$-0.410237\pi$
$701$	14.7356	0.556554	0.278277	+	0.960501i	$0.410237\pi$
$702$	14.7536	0.556839
$703$	2.32437	0.0876653
$704$	−4.40581	−0.166050
$705$	−0.215521	−0.00811698
$706$	−29.1239	−1.09609
$707$	0	0
$708$	−4.86294	−0.182760
$709$	−44.8732	−1.68525	−0.842625	−	0.538502i	$-0.818991\pi$
$709$	−44.8732	−1.68525	−0.842625	+	0.538502i	$0.818991\pi$
$710$	18.8442	0.707208
$711$	21.9705	0.823957
$712$	−13.9487	−0.522749
$713$	51.2868	1.92071
$714$	0	0
$715$	76.7531	2.87040
$716$	1.78448	0.0666891
$717$	−4.79225	−0.178970
$718$	9.98361	0.372585
$719$	0.0639828	0.00238616	0.00119308	−	0.999999i	$-0.499620\pi$
$719$	0.0639828	0.00238616	0.00119308	+	0.999999i	$0.499620\pi$
$720$	8.54288	0.318374
$721$	0	0
$722$	17.0508	0.634566
$723$	8.36227	0.310996
$724$	10.3394	0.384262
$725$	−2.97285	−0.110409
$726$	−3.74333	−0.138928
$727$	9.80625	0.363694	0.181847	−	0.983327i	$-0.441792\pi$
$727$	9.80625	0.363694	0.181847	+	0.983327i	$0.441792\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	4.71140	0.174377
$731$	6.41789	0.237374
$732$	2.53856	0.0938281
$733$	−2.92692	−0.108108	−0.0540541	−	0.998538i	$-0.517214\pi$
$733$	−2.92692	−0.108108	−0.0540541	+	0.998538i	$0.517214\pi$
$734$	23.6160	0.871681
$735$	0	0
$736$	−7.18598	−0.264879
$737$	−48.1952	−1.77529
$738$	1.41789	0.0521934
$739$	46.3116	1.70360	0.851799	−	0.523869i	$-0.175512\pi$
$739$	46.3116	1.70360	0.851799	+	0.523869i	$0.175512\pi$
$740$	−5.07606	−0.186600
$741$	3.55017	0.130419
$742$	0	0
$743$	20.6692	0.758279	0.379139	−	0.925340i	$-0.376220\pi$
$743$	20.6692	0.758279	0.379139	+	0.925340i	$0.376220\pi$
$744$	−3.17629	−0.116449
$745$	−30.6515	−1.12298
$746$	26.3980	0.966501
$747$	−9.00298	−0.329402
$748$	−33.2325	−1.21510
$749$	0	0
$750$	−0.955395	−0.0348861
$751$	26.2185	0.956727	0.478363	−	0.878162i	$-0.341230\pi$
$751$	26.2185	0.956727	0.478363	+	0.878162i	$0.341230\pi$
$752$	0.158834	0.00579207
$753$	−1.22952	−0.0448062
$754$	3.95407	0.143999
$755$	14.2687	0.519293
$756$	0	0
$757$	−10.5603	−0.383822	−0.191911	−	0.981412i	$-0.561468\pi$
$757$	−10.5603	−0.383822	−0.191911	+	0.981412i	$0.561468\pi$
$758$	26.3913	0.958577
$759$	−14.0901	−0.511437
$760$	4.25667	0.154406
$761$	40.4819	1.46747	0.733733	−	0.679437i	$-0.237776\pi$
$761$	40.4819	1.46747	0.733733	+	0.679437i	$0.237776\pi$
$762$	2.81402	0.101941
$763$	0	0
$764$	−20.1468	−0.728884
$765$	64.4379	2.32976
$766$	−28.9081	−1.04449
$767$	−62.4341	−2.25437
$768$	0.445042	0.0160591
$769$	23.8670	0.860666	0.430333	−	0.902670i	$-0.358396\pi$
$769$	23.8670	0.860666	0.430333	+	0.902670i	$0.358396\pi$
$770$	0	0
$771$	1.28860	0.0464078
$772$	−13.9172	−0.500892
$773$	27.8297	1.00096	0.500482	−	0.865747i	$-0.333156\pi$
$773$	27.8297	1.00096	0.500482	+	0.865747i	$0.333156\pi$
$774$	2.38404	0.0856927
$775$	30.6601	1.10134
$776$	−1.55496	−0.0558197
$777$	0	0
$778$	−3.46250	−0.124137
$779$	0.706496	0.0253128
$780$	−7.75302	−0.277603
$781$	−27.2306	−0.974387
$782$	−54.2030	−1.93829
$783$	1.78687	0.0638576
$784$	0	0
$785$	−39.6450	−1.41499
$786$	−2.85517	−0.101840
$787$	−3.91617	−0.139596	−0.0697981	−	0.997561i	$-0.522236\pi$
$787$	−3.91617	−0.139596	−0.0697981	+	0.997561i	$0.522236\pi$
$788$	4.08815	0.145634
$789$	8.92394	0.317701
$790$	−23.9071	−0.850575
$791$	0	0
$792$	−12.3448	−0.438654
$793$	32.5921	1.15738
$794$	38.2368	1.35697
$795$	8.69932	0.308533
$796$	−8.45473	−0.299670
$797$	10.3110	0.365233	0.182617	−	0.983184i	$-0.441543\pi$
$797$	10.3110	0.365233	0.182617	+	0.983184i	$0.441543\pi$
$798$	0	0
$799$	1.19806	0.0423844
$800$	−4.29590	−0.151883
$801$	−39.0834	−1.38094
$802$	−9.49934	−0.335433
$803$	−6.80817	−0.240255
$804$	4.86831	0.171692
$805$	0	0
$806$	−40.7797	−1.43640
$807$	7.83100	0.275664
$808$	5.00969	0.176240
$809$	−39.5200	−1.38945	−0.694725	−	0.719275i	$-0.744474\pi$
$809$	−39.5200	−1.38945	−0.694725	+	0.719275i	$0.744474\pi$
$810$	22.1250	0.777393
$811$	27.8769	0.978892	0.489446	−	0.872034i	$-0.337199\pi$
$811$	27.8769	0.978892	0.489446	+	0.872034i	$0.337199\pi$
$812$	0	0
$813$	6.39911	0.224427
$814$	7.33513	0.257096
$815$	58.3032	2.04227
$816$	3.35690	0.117515
$817$	1.18790	0.0415593
$818$	−9.25906	−0.323735
$819$	0	0
$820$	−1.54288	−0.0538796
$821$	6.76032	0.235937	0.117968	−	0.993017i	$-0.462362\pi$
$821$	6.76032	0.235937	0.117968	+	0.993017i	$0.462362\pi$
$822$	0.983607	0.0343072
$823$	−4.70901	−0.164146	−0.0820728	−	0.996626i	$-0.526154\pi$
$823$	−4.70901	−0.164146	−0.0820728	+	0.996626i	$0.526154\pi$
$824$	2.89546	0.100868
$825$	−8.42327	−0.293261
$826$	0	0
$827$	−23.4010	−0.813733	−0.406867	−	0.913488i	$-0.633379\pi$
$827$	−23.4010	−0.813733	−0.406867	+	0.913488i	$0.633379\pi$
$828$	−20.1347	−0.699728
$829$	−18.6746	−0.648594	−0.324297	−	0.945955i	$-0.605128\pi$
$829$	−18.6746	−0.648594	−0.324297	+	0.945955i	$0.605128\pi$
$830$	9.79656	0.340044
$831$	1.75973	0.0610442
$832$	5.71379	0.198090
$833$	0	0
$834$	−6.47889	−0.224346
$835$	68.0447	2.35478
$836$	−6.15106	−0.212739
$837$	−18.4286	−0.636987
$838$	26.8310	0.926862
$839$	−21.0334	−0.726153	−0.363076	−	0.931759i	$-0.618274\pi$
$839$	−21.0334	−0.726153	−0.363076	+	0.931759i	$0.618274\pi$
$840$	0	0
$841$	−28.5211	−0.983486
$842$	−14.7560	−0.508525
$843$	4.42566	0.152428
$844$	4.04221	0.139139
$845$	−59.9033	−2.06074
$846$	0.445042	0.0153009
$847$	0	0
$848$	−6.41119	−0.220161
$849$	9.34827	0.320832
$850$	−32.4034	−1.11143
$851$	11.9638	0.410112
$852$	2.75063	0.0942349
$853$	−5.53750	−0.189600	−0.0948002	−	0.995496i	$-0.530221\pi$
$853$	−5.53750	−0.189600	−0.0948002	+	0.995496i	$0.530221\pi$
$854$	0	0
$855$	11.9269	0.407892
$856$	−9.99330	−0.341564
$857$	29.8955	1.02121	0.510605	−	0.859816i	$-0.329422\pi$
$857$	29.8955	1.02121	0.510605	+	0.859816i	$0.329422\pi$
$858$	11.2034	0.382479
$859$	33.5603	1.14506	0.572532	−	0.819882i	$-0.305961\pi$
$859$	33.5603	1.14506	0.572532	+	0.819882i	$0.305961\pi$
$860$	−2.59419	−0.0884610
$861$	0	0
$862$	31.8582	1.08509
$863$	−31.6980	−1.07901	−0.539506	−	0.841982i	$-0.681389\pi$
$863$	−31.6980	−1.07901	−0.539506	+	0.841982i	$0.681389\pi$
$864$	2.58211	0.0878450
$865$	5.94869	0.202262
$866$	16.0194	0.544361
$867$	17.7549	0.602989
$868$	0	0
$869$	34.5467	1.17192
$870$	−0.939001	−0.0318351
$871$	62.5032	2.11784
$872$	14.6015	0.494469
$873$	−4.35690	−0.147459
$874$	−10.0325	−0.339355
$875$	0	0
$876$	0.687710	0.0232356
$877$	39.4644	1.33262	0.666309	−	0.745675i	$-0.267873\pi$
$877$	39.4644	1.33262	0.666309	+	0.745675i	$0.267873\pi$
$878$	−2.35690	−0.0795414
$879$	−8.36552	−0.282162
$880$	13.4330	0.452825
$881$	−7.31634	−0.246494	−0.123247	−	0.992376i	$-0.539331\pi$
$881$	−7.31634	−0.246494	−0.123247	+	0.992376i	$0.539331\pi$
$882$	0	0
$883$	−44.8321	−1.50872	−0.754360	−	0.656461i	$-0.772052\pi$
$883$	−44.8321	−1.50872	−0.754360	+	0.656461i	$0.772052\pi$
$884$	43.0984	1.44956
$885$	14.8267	0.498394
$886$	−13.1830	−0.442891
$887$	−23.0790	−0.774919	−0.387459	−	0.921887i	$-0.626647\pi$
$887$	−23.0790	−0.774919	−0.387459	+	0.921887i	$0.626647\pi$
$888$	−0.740939	−0.0248643
$889$	0	0
$890$	42.5284	1.42556
$891$	−31.9715	−1.07109
$892$	2.87800	0.0963626
$893$	0.221751	0.00742063
$894$	−4.47411	−0.149637
$895$	−5.44073	−0.181864
$896$	0	0
$897$	18.2731	0.610120
$898$	15.3448	0.512063
$899$	−4.93900	−0.164725
$900$	−12.0368	−0.401228
$901$	−48.3588	−1.61107
$902$	2.22952	0.0742349
$903$	0	0
$904$	−16.5405	−0.550128
$905$	−31.5241	−1.04790
$906$	2.08277	0.0691954
$907$	−44.6031	−1.48102	−0.740511	−	0.672044i	$-0.765417\pi$
$907$	−44.6031	−1.48102	−0.740511	+	0.672044i	$0.765417\pi$
$908$	−25.1250	−0.833802
$909$	14.0368	0.465572
$910$	0	0
$911$	7.19460	0.238368	0.119184	−	0.992872i	$-0.461972\pi$
$911$	7.19460	0.238368	0.119184	+	0.992872i	$0.461972\pi$
$912$	0.621334	0.0205744
$913$	−14.1564	−0.468510
$914$	22.8159	0.754684
$915$	−7.73987	−0.255872
$916$	−5.87800	−0.194215
$917$	0	0
$918$	19.4765	0.642820
$919$	−45.6469	−1.50575	−0.752877	−	0.658161i	$-0.771334\pi$
$919$	−45.6469	−1.50575	−0.752877	+	0.658161i	$0.771334\pi$
$920$	21.9095	0.722334
$921$	−5.99031	−0.197387
$922$	0.440730	0.0145147
$923$	35.3147	1.16240
$924$	0	0
$925$	7.15213	0.235161
$926$	25.5133	0.838420
$927$	8.11290	0.266463
$928$	0.692021	0.0227167
$929$	54.6631	1.79344	0.896719	−	0.442601i	$-0.145944\pi$
$929$	54.6631	1.79344	0.896719	+	0.442601i	$0.145944\pi$
$930$	9.68425	0.317559
$931$	0	0
$932$	17.7506	0.581441
$933$	9.25534	0.303006
$934$	21.4101	0.700561
$935$	101.323	3.31362
$936$	16.0097	0.523293
$937$	−49.8316	−1.62793	−0.813964	−	0.580916i	$-0.802695\pi$
$937$	−49.8316	−1.62793	−0.813964	+	0.580916i	$0.802695\pi$
$938$	0	0
$939$	11.5235	0.376055
$940$	−0.484271	−0.0157952
$941$	−25.0086	−0.815258	−0.407629	−	0.913148i	$-0.633644\pi$
$941$	−25.0086	−0.815258	−0.407629	+	0.913148i	$0.633644\pi$
$942$	−5.78687	−0.188546
$943$	3.63640	0.118417
$944$	−10.9269	−0.355641
$945$	0	0
$946$	3.74871	0.121881
$947$	−9.35796	−0.304093	−0.152046	−	0.988373i	$-0.548586\pi$
$947$	−9.35796	−0.304093	−0.152046	+	0.988373i	$0.548586\pi$
$948$	−3.48965	−0.113338
$949$	8.82935	0.286613
$950$	−5.99761	−0.194588
$951$	−3.34481	−0.108463
$952$	0	0
$953$	−32.2411	−1.04439	−0.522196	−	0.852825i	$-0.674887\pi$
$953$	−32.2411	−1.04439	−0.522196	+	0.852825i	$0.674887\pi$
$954$	−17.9638	−0.581598
$955$	61.4258	1.98769
$956$	−10.7681	−0.348265
$957$	1.35690	0.0438622
$958$	32.6437	1.05467
$959$	0	0
$960$	−1.35690	−0.0437936
$961$	19.9377	0.643151
$962$	−9.51275	−0.306703
$963$	−28.0006	−0.902306
$964$	18.7899	0.605181
$965$	42.4325	1.36595
$966$	0	0
$967$	−51.8859	−1.66854	−0.834269	−	0.551358i	$-0.814110\pi$
$967$	−51.8859	−1.66854	−0.834269	+	0.551358i	$0.814110\pi$
$968$	−8.41119	−0.270346
$969$	4.68664	0.150557
$970$	4.74094	0.152222
$971$	−31.1976	−1.00118	−0.500589	−	0.865685i	$-0.666883\pi$
$971$	−31.1976	−1.00118	−0.500589	+	0.865685i	$0.666883\pi$
$972$	10.9758	0.352050
$973$	0	0
$974$	−41.4010	−1.32657
$975$	10.9239	0.349846
$976$	5.70410	0.182584
$977$	9.30367	0.297651	0.148825	−	0.988863i	$-0.452451\pi$
$977$	9.30367	0.297651	0.148825	+	0.988863i	$0.452451\pi$
$978$	8.51035	0.272131
$979$	−61.4553	−1.96412
$980$	0	0
$981$	40.9124	1.30623
$982$	−1.97046	−0.0628799
$983$	30.5109	0.973148	0.486574	−	0.873639i	$-0.338246\pi$
$983$	30.5109	0.973148	0.486574	+	0.873639i	$0.338246\pi$
$984$	−0.225209	−0.00717941
$985$	−12.4644	−0.397149
$986$	5.21983	0.166233
$987$	0	0
$988$	7.97716	0.253787
$989$	6.11423	0.194421
$990$	37.6383	1.19622
$991$	50.5609	1.60612	0.803060	−	0.595898i	$-0.203204\pi$
$991$	50.5609	1.60612	0.803060	+	0.595898i	$0.203204\pi$
$992$	−7.13706	−0.226602
$993$	7.52781	0.238888
$994$	0	0
$995$	25.7778	0.817210
$996$	1.42998	0.0453105
$997$	34.0907	1.07966	0.539831	−	0.841773i	$-0.318488\pi$
$997$	34.0907	1.07966	0.539831	+	0.841773i	$0.318488\pi$
$998$	−4.32006	−0.136749
$999$	−4.29888	−0.136011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	686.2.a.b.1.2	yes	3
3.2	odd	2		6174.2.a.l.1.3		3
4.3	odd	2		5488.2.a.b.1.2		3
7.2	even	3		686.2.c.c.361.2		6
7.3	odd	6		686.2.c.d.667.2		6
7.4	even	3		686.2.c.c.667.2		6
7.5	odd	6		686.2.c.d.361.2		6
7.6	odd	2		686.2.a.a.1.2	✓	3
21.20	even	2		6174.2.a.k.1.1		3
28.27	even	2		5488.2.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
686.2.a.a.1.2	✓	3	7.6	odd	2
686.2.a.b.1.2	yes	3	1.1	even	1	trivial
686.2.c.c.361.2		6	7.2	even	3
686.2.c.c.667.2		6	7.4	even	3
686.2.c.d.361.2		6	7.5	odd	6
686.2.c.d.667.2		6	7.3	odd	6
5488.2.a.b.1.2		3	4.3	odd	2
5488.2.a.e.1.2		3	28.27	even	2
6174.2.a.k.1.1		3	21.20	even	2
6174.2.a.l.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 686 = 2 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	686.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.445042 q^{3} +1.00000 q^{4} -3.04892 q^{5} -0.445042 q^{6} -1.00000 q^{8} -2.80194 q^{9} +3.04892 q^{10} -4.40581 q^{11} +0.445042 q^{12} +5.71379 q^{13} -1.35690 q^{15} +1.00000 q^{16} +7.54288 q^{17} +2.80194 q^{18} +1.39612 q^{19} -3.04892 q^{20} +4.40581 q^{22} +7.18598 q^{23} -0.445042 q^{24} +4.29590 q^{25} -5.71379 q^{26} -2.58211 q^{27} -0.692021 q^{29} +1.35690 q^{30} +7.13706 q^{31} -1.00000 q^{32} -1.96077 q^{33} -7.54288 q^{34} -2.80194 q^{36} +1.66487 q^{37} -1.39612 q^{38} +2.54288 q^{39} +3.04892 q^{40} +0.506041 q^{41} +0.850855 q^{43} -4.40581 q^{44} +8.54288 q^{45} -7.18598 q^{46} +0.158834 q^{47} +0.445042 q^{48} -4.29590 q^{50} +3.35690 q^{51} +5.71379 q^{52} -6.41119 q^{53} +2.58211 q^{54} +13.4330 q^{55} +0.621334 q^{57} +0.692021 q^{58} -10.9269 q^{59} -1.35690 q^{60} +5.70410 q^{61} -7.13706 q^{62} +1.00000 q^{64} -17.4209 q^{65} +1.96077 q^{66} +10.9390 q^{67} +7.54288 q^{68} +3.19806 q^{69} +6.18060 q^{71} +2.80194 q^{72} +1.54527 q^{73} -1.66487 q^{74} +1.91185 q^{75} +1.39612 q^{76} -2.54288 q^{78} -7.84117 q^{79} -3.04892 q^{80} +7.25667 q^{81} -0.506041 q^{82} +3.21313 q^{83} -22.9976 q^{85} -0.850855 q^{86} -0.307979 q^{87} +4.40581 q^{88} +13.9487 q^{89} -8.54288 q^{90} +7.18598 q^{92} +3.17629 q^{93} -0.158834 q^{94} -4.25667 q^{95} -0.445042 q^{96} +1.55496 q^{97} +12.3448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 3 q^{8} - 4 q^{9} + q^{12} + 9 q^{13} + 3 q^{16} + 4 q^{17} + 4 q^{18} + 13 q^{19} + 7 q^{23} - q^{24} - q^{25} - 9 q^{26} - 2 q^{27} + 3 q^{29} + 16 q^{31}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 3 * q^8 - 4 * q^9 + q^12 + 9 * q^13 + 3 * q^16 + 4 * q^17 + 4 * q^18 + 13 * q^19 + 7 * q^23 - q^24 - q^25 - 9 * q^26 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 3 * q^32 + 7 * q^33 - 4 * q^34 - 4 * q^36 + 6 * q^37 - 13 * q^38 - 11 * q^39 + 11 * q^41 - 11 * q^43 + 7 * q^45 - 7 * q^46 - 8 * q^47 + q^48 + q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 + 2 * q^54 + 21 * q^55 + 9 * q^57 - 3 * q^58 - 4 * q^59 + 31 * q^61 - 16 * q^62 + 3 * q^64 - 14 * q^65 - 7 * q^66 + 23 * q^67 + 4 * q^68 + 14 * q^69 + 7 * q^71 + 4 * q^72 + 27 * q^73 - 6 * q^74 + 2 * q^75 + 13 * q^76 + 11 * q^78 - 32 * q^79 - 5 * q^81 - 11 * q^82 - 11 * q^83 - 28 * q^85 + 11 * q^86 - 6 * q^87 + 10 * q^89 - 7 * q^90 + 7 * q^92 + 17 * q^93 + 8 * q^94 + 14 * q^95 - q^96 + 5 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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