LMFDB - Embedded newform 567.4.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(1,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("567.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.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:	$33.4540829733$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 $ x^8 - 3x^7 - 49x^6 + 138x^5 + 708x^4 - 1941x^3 - 2506x^2 + 8592*x - 4616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.59498$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59498 q^{2} -5.45602 q^{4} +2.55632 q^{5} -7.00000 q^{7} +21.4622 q^{8} +O(q^{10})$	\(q-1.59498 q^{2} -5.45602 q^{4} +2.55632 q^{5} -7.00000 q^{7} +21.4622 q^{8} -4.07730 q^{10} -8.09077 q^{11} -26.2373 q^{13} +11.1649 q^{14} +9.41638 q^{16} +69.7407 q^{17} +105.751 q^{19} -13.9474 q^{20} +12.9047 q^{22} -154.317 q^{23} -118.465 q^{25} +41.8481 q^{26} +38.1922 q^{28} +72.6547 q^{29} +282.799 q^{31} -186.716 q^{32} -111.235 q^{34} -17.8943 q^{35} +25.7974 q^{37} -168.671 q^{38} +54.8642 q^{40} +87.1089 q^{41} +89.1106 q^{43} +44.1434 q^{44} +246.133 q^{46} -314.617 q^{47} +49.0000 q^{49} +188.950 q^{50} +143.151 q^{52} -356.536 q^{53} -20.6826 q^{55} -150.235 q^{56} -115.883 q^{58} -412.489 q^{59} +146.756 q^{61} -451.060 q^{62} +222.479 q^{64} -67.0710 q^{65} -306.402 q^{67} -380.507 q^{68} +28.5411 q^{70} -1038.77 q^{71} -1157.10 q^{73} -41.1464 q^{74} -576.979 q^{76} +56.6354 q^{77} +746.293 q^{79} +24.0713 q^{80} -138.937 q^{82} -525.424 q^{83} +178.280 q^{85} -142.130 q^{86} -173.645 q^{88} +643.894 q^{89} +183.661 q^{91} +841.957 q^{92} +501.810 q^{94} +270.333 q^{95} +309.162 q^{97} -78.1543 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 43 * q^4 - 30 * q^5 - 56 * q^7 - 6 * q^8	$ 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8} - 14 q^{10} - 24 q^{11} + 68 q^{13} + 21 q^{14} + 103 q^{16} - 168 q^{17} + 176 q^{19} - 330 q^{20} + 151 q^{22} - 228 q^{23} + 244 q^{25} - 795 q^{26} - 301 q^{28} - 618 q^{29} + 72 q^{31} - 786 q^{32} - 261 q^{34} + 210 q^{35} + 210 q^{37} - 1032 q^{38} - 375 q^{40} - 420 q^{41} - 2 q^{43} - 387 q^{44} + 402 q^{46} - 570 q^{47} + 392 q^{49} - 1110 q^{50} - 431 q^{52} - 528 q^{53} - 838 q^{55} + 42 q^{56} + 37 q^{58} + 150 q^{59} + 578 q^{61} - 1170 q^{62} - 112 q^{64} + 366 q^{65} - 898 q^{67} - 2526 q^{68} + 98 q^{70} - 882 q^{71} + 972 q^{73} + 222 q^{74} + 1423 q^{76} + 168 q^{77} - 158 q^{79} - 2475 q^{80} - 211 q^{82} - 2958 q^{83} - 774 q^{85} + 114 q^{86} + 1317 q^{88} - 4380 q^{89} - 476 q^{91} - 4629 q^{92} - 3234 q^{94} - 930 q^{95} - 60 q^{97} - 147 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 43 * q^4 - 30 * q^5 - 56 * q^7 - 6 * q^8 - 14 * q^10 - 24 * q^11 + 68 * q^13 + 21 * q^14 + 103 * q^16 - 168 * q^17 + 176 * q^19 - 330 * q^20 + 151 * q^22 - 228 * q^23 + 244 * q^25 - 795 * q^26 - 301 * q^28 - 618 * q^29 + 72 * q^31 - 786 * q^32 - 261 * q^34 + 210 * q^35 + 210 * q^37 - 1032 * q^38 - 375 * q^40 - 420 * q^41 - 2 * q^43 - 387 * q^44 + 402 * q^46 - 570 * q^47 + 392 * q^49 - 1110 * q^50 - 431 * q^52 - 528 * q^53 - 838 * q^55 + 42 * q^56 + 37 * q^58 + 150 * q^59 + 578 * q^61 - 1170 * q^62 - 112 * q^64 + 366 * q^65 - 898 * q^67 - 2526 * q^68 + 98 * q^70 - 882 * q^71 + 972 * q^73 + 222 * q^74 + 1423 * q^76 + 168 * q^77 - 158 * q^79 - 2475 * q^80 - 211 * q^82 - 2958 * q^83 - 774 * q^85 + 114 * q^86 + 1317 * q^88 - 4380 * q^89 - 476 * q^91 - 4629 * q^92 - 3234 * q^94 - 930 * q^95 - 60 * q^97 - 147 * q^98

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 $

$2$	−1.59498	−0.563912	−0.281956	−	0.959427i	$-0.590983\pi$
$2$	−1.59498	−0.563912	−0.281956	+	0.959427i	$0.590983\pi$
$3$	0	0
$3$	0	0
$4$	−5.45602	−0.682003
$5$	2.55632	0.228644	0.114322	−	0.993444i	$-0.463530\pi$
$5$	2.55632	0.228644	0.114322	+	0.993444i	$0.463530\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.4622	0.948502
$9$	0	0
$10$	−4.07730	−0.128935
$11$	−8.09077	−0.221769	−0.110885	−	0.993833i	$-0.535368\pi$
$11$	−8.09077	−0.221769	−0.110885	+	0.993833i	$0.535368\pi$
$12$	0	0
$13$	−26.2373	−0.559763	−0.279882	−	0.960035i	$-0.590295\pi$
$13$	−26.2373	−0.559763	−0.279882	+	0.960035i	$0.590295\pi$
$14$	11.1649	0.213139
$15$	0	0
$16$	9.41638	0.147131
$17$	69.7407	0.994977	0.497489	−	0.867471i	$-0.334256\pi$
$17$	69.7407	0.994977	0.497489	+	0.867471i	$0.334256\pi$
$18$	0	0
$19$	105.751	1.27689	0.638445	−	0.769667i	$-0.279578\pi$
$19$	105.751	1.27689	0.638445	+	0.769667i	$0.279578\pi$
$20$	−13.9474	−0.155936
$21$	0	0
$22$	12.9047	0.125058
$23$	−154.317	−1.39901	−0.699507	−	0.714626i	$-0.746597\pi$
$23$	−154.317	−1.39901	−0.699507	+	0.714626i	$0.746597\pi$
$24$	0	0
$25$	−118.465	−0.947722
$26$	41.8481	0.315657
$27$	0	0
$28$	38.1922	0.257773
$29$	72.6547	0.465229	0.232615	−	0.972569i	$-0.425272\pi$
$29$	72.6547	0.465229	0.232615	+	0.972569i	$0.425272\pi$
$30$	0	0
$31$	282.799	1.63846	0.819230	−	0.573466i	$-0.194401\pi$
$31$	282.799	1.63846	0.819230	+	0.573466i	$0.194401\pi$
$32$	−186.716	−1.03147
$33$	0	0
$34$	−111.235	−0.561080
$35$	−17.8943	−0.0864195
$36$	0	0
$37$	25.7974	0.114623	0.0573117	−	0.998356i	$-0.481747\pi$
$37$	25.7974	0.114623	0.0573117	+	0.998356i	$0.481747\pi$
$38$	−168.671	−0.720054
$39$	0	0
$40$	54.8642	0.216870
$41$	87.1089	0.331808	0.165904	−	0.986142i	$-0.446946\pi$
$41$	87.1089	0.331808	0.165904	+	0.986142i	$0.446946\pi$
$42$	0	0
$43$	89.1106	0.316029	0.158014	−	0.987437i	$-0.449491\pi$
$43$	89.1106	0.316029	0.158014	+	0.987437i	$0.449491\pi$
$44$	44.1434	0.151247
$45$	0	0
$46$	246.133	0.788921
$47$	−314.617	−0.976418	−0.488209	−	0.872727i	$-0.662350\pi$
$47$	−314.617	−0.976418	−0.488209	+	0.872727i	$0.662350\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	188.950	0.534432
$51$	0	0
$52$	143.151	0.381760
$53$	−356.536	−0.924038	−0.462019	−	0.886870i	$-0.652875\pi$
$53$	−356.536	−0.924038	−0.462019	+	0.886870i	$0.652875\pi$
$54$	0	0
$55$	−20.6826	−0.0507063
$56$	−150.235	−0.358500
$57$	0	0
$58$	−115.883	−0.262348
$59$	−412.489	−0.910195	−0.455098	−	0.890442i	$-0.650396\pi$
$59$	−412.489	−0.910195	−0.455098	+	0.890442i	$0.650396\pi$
$60$	0	0
$61$	146.756	0.308036	0.154018	−	0.988068i	$-0.450779\pi$
$61$	146.756	0.308036	0.154018	+	0.988068i	$0.450779\pi$
$62$	−451.060	−0.923947
$63$	0	0
$64$	222.479	0.434528
$65$	−67.0710	−0.127987
$66$	0	0
$67$	−306.402	−0.558701	−0.279350	−	0.960189i	$-0.590119\pi$
$67$	−306.402	−0.558701	−0.279350	+	0.960189i	$0.590119\pi$
$68$	−380.507	−0.678577
$69$	0	0
$70$	28.5411	0.0487330
$71$	−1038.77	−1.73632	−0.868160	−	0.496284i	$-0.834698\pi$
$71$	−1038.77	−1.73632	−0.868160	+	0.496284i	$0.834698\pi$
$72$	0	0
$73$	−1157.10	−1.85518	−0.927590	−	0.373600i	$-0.878123\pi$
$73$	−1157.10	−1.85518	−0.927590	+	0.373600i	$0.878123\pi$
$74$	−41.1464	−0.0646375
$75$	0	0
$76$	−576.979	−0.870843
$77$	56.6354	0.0838208
$78$	0	0
$79$	746.293	1.06284	0.531421	−	0.847108i	$-0.321658\pi$
$79$	746.293	1.06284	0.531421	+	0.847108i	$0.321658\pi$
$80$	24.0713	0.0336407
$81$	0	0
$82$	−138.937	−0.187111
$83$	−525.424	−0.694853	−0.347427	−	0.937707i	$-0.612944\pi$
$83$	−525.424	−0.694853	−0.347427	+	0.937707i	$0.612944\pi$
$84$	0	0
$85$	178.280	0.227496
$86$	−142.130	−0.178213
$87$	0	0
$88$	−173.645	−0.210348
$89$	643.894	0.766883	0.383441	−	0.923565i	$-0.374739\pi$
$89$	643.894	0.766883	0.383441	+	0.923565i	$0.374739\pi$
$90$	0	0
$91$	183.661	0.211571
$92$	841.957	0.954131
$93$	0	0
$94$	501.810	0.550614
$95$	270.333	0.291954
$96$	0	0
$97$	309.162	0.323615	0.161808	−	0.986822i	$-0.448268\pi$
$97$	309.162	0.323615	0.161808	+	0.986822i	$0.448268\pi$
$98$	−78.1543	−0.0805589
$99$	0	0
$100$	646.349	0.646349
$101$	−1120.57	−1.10397	−0.551983	−	0.833856i	$-0.686129\pi$
$101$	−1120.57	−1.10397	−0.551983	+	0.833856i	$0.686129\pi$
$102$	0	0
$103$	565.226	0.540713	0.270356	−	0.962760i	$-0.412858\pi$
$103$	565.226	0.540713	0.270356	+	0.962760i	$0.412858\pi$
$104$	−563.109	−0.530937
$105$	0	0
$106$	568.670	0.521077
$107$	−1593.24	−1.43948	−0.719738	−	0.694246i	$-0.755738\pi$
$107$	−1593.24	−1.43948	−0.719738	+	0.694246i	$0.755738\pi$
$108$	0	0
$109$	−1498.64	−1.31691	−0.658457	−	0.752619i	$-0.728790\pi$
$109$	−1498.64	−1.31691	−0.658457	+	0.752619i	$0.728790\pi$
$110$	32.9885	0.0285939
$111$	0	0
$112$	−65.9146	−0.0556102
$113$	−695.278	−0.578817	−0.289408	−	0.957206i	$-0.593459\pi$
$113$	−695.278	−0.578817	−0.289408	+	0.957206i	$0.593459\pi$
$114$	0	0
$115$	−394.484	−0.319877
$116$	−396.406	−0.317288
$117$	0	0
$118$	657.914	0.513270
$119$	−488.185	−0.376066
$120$	0	0
$121$	−1265.54	−0.950819
$122$	−234.074	−0.173705
$123$	0	0
$124$	−1542.96	−1.11743
$125$	−622.376	−0.445336
$126$	0	0
$127$	−1377.51	−0.962475	−0.481237	−	0.876590i	$-0.659812\pi$
$127$	−1377.51	−0.962475	−0.481237	+	0.876590i	$0.659812\pi$
$128$	1138.88	0.786435
$129$	0	0
$130$	106.977	0.0721733
$131$	2722.27	1.81562	0.907809	−	0.419384i	$-0.137754\pi$
$131$	2722.27	1.81562	0.907809	+	0.419384i	$0.137754\pi$
$132$	0	0
$133$	−740.256	−0.482619
$134$	488.706	0.315058
$135$	0	0
$136$	1496.79	0.943738
$137$	−2352.62	−1.46714	−0.733570	−	0.679614i	$-0.762147\pi$
$137$	−2352.62	−1.46714	−0.733570	+	0.679614i	$0.762147\pi$
$138$	0	0
$139$	−2048.69	−1.25013	−0.625065	−	0.780573i	$-0.714927\pi$
$139$	−2048.69	−1.25013	−0.625065	+	0.780573i	$0.714927\pi$
$140$	97.6315	0.0589383
$141$	0	0
$142$	1656.82	0.979133
$143$	212.280	0.124138
$144$	0	0
$145$	185.729	0.106372
$146$	1845.55	1.04616
$147$	0	0
$148$	−140.751	−0.0781735
$149$	−1293.99	−0.711460	−0.355730	−	0.934589i	$-0.615768\pi$
$149$	−1293.99	−0.711460	−0.355730	+	0.934589i	$0.615768\pi$
$150$	0	0
$151$	−2141.69	−1.15423	−0.577113	−	0.816664i	$-0.695821\pi$
$151$	−2141.69	−1.15423	−0.577113	+	0.816664i	$0.695821\pi$
$152$	2269.64	1.21113
$153$	0	0
$154$	−90.3326	−0.0472676
$155$	722.926	0.374625
$156$	0	0
$157$	2588.19	1.31567	0.657835	−	0.753162i	$-0.271472\pi$
$157$	2588.19	1.31567	0.657835	+	0.753162i	$0.271472\pi$
$158$	−1190.33	−0.599350
$159$	0	0
$160$	−477.307	−0.235840
$161$	1080.22	0.528777
$162$	0	0
$163$	842.940	0.405056	0.202528	−	0.979276i	$-0.435084\pi$
$163$	842.940	0.405056	0.202528	+	0.979276i	$0.435084\pi$
$164$	−475.268	−0.226294
$165$	0	0
$166$	838.044	0.391836
$167$	−1657.39	−0.767979	−0.383990	−	0.923337i	$-0.625450\pi$
$167$	−1657.39	−0.767979	−0.383990	+	0.923337i	$0.625450\pi$
$168$	0	0
$169$	−1508.60	−0.686665
$170$	−284.354	−0.128288
$171$	0	0
$172$	−486.190	−0.215533
$173$	−1947.10	−0.855694	−0.427847	−	0.903851i	$-0.640728\pi$
$173$	−1947.10	−0.855694	−0.427847	+	0.903851i	$0.640728\pi$
$174$	0	0
$175$	829.256	0.358205
$176$	−76.1857	−0.0326291
$177$	0	0
$178$	−1027.00	−0.432455
$179$	841.904	0.351547	0.175773	−	0.984431i	$-0.443757\pi$
$179$	841.904	0.351547	0.175773	+	0.984431i	$0.443757\pi$
$180$	0	0
$181$	158.260	0.0649912	0.0324956	−	0.999472i	$-0.489655\pi$
$181$	158.260	0.0649912	0.0324956	+	0.999472i	$0.489655\pi$
$182$	−292.937	−0.119307
$183$	0	0
$184$	−3311.97	−1.32697
$185$	65.9464	0.0262080
$186$	0	0
$187$	−564.256	−0.220655
$188$	1716.56	0.665920
$189$	0	0
$190$	−431.178	−0.164636
$191$	4618.96	1.74982	0.874912	−	0.484282i	$-0.160919\pi$
$191$	4618.96	1.74982	0.874912	+	0.484282i	$0.160919\pi$
$192$	0	0
$193$	4590.96	1.71225	0.856126	−	0.516767i	$-0.172865\pi$
$193$	4590.96	1.71225	0.856126	+	0.516767i	$0.172865\pi$
$194$	−493.109	−0.182491
$195$	0	0
$196$	−267.345	−0.0974290
$197$	2646.03	0.956965	0.478482	−	0.878097i	$-0.341187\pi$
$197$	2646.03	0.956965	0.478482	+	0.878097i	$0.341187\pi$
$198$	0	0
$199$	4741.58	1.68905	0.844527	−	0.535513i	$-0.179882\pi$
$199$	4741.58	1.68905	0.844527	+	0.535513i	$0.179882\pi$
$200$	−2542.52	−0.898916
$201$	0	0
$202$	1787.29	0.622540
$203$	−508.583	−0.175840
$204$	0	0
$205$	222.678	0.0758660
$206$	−901.527	−0.304915
$207$	0	0
$208$	−247.060	−0.0823585
$209$	−855.606	−0.283175
$210$	0	0
$211$	−4580.79	−1.49457	−0.747285	−	0.664503i	$-0.768643\pi$
$211$	−4580.79	−1.49457	−0.747285	+	0.664503i	$0.768643\pi$
$212$	1945.27	0.630197
$213$	0	0
$214$	2541.19	0.811738
$215$	227.796	0.0722583
$216$	0	0
$217$	−1979.59	−0.619279
$218$	2390.31	0.742624
$219$	0	0
$220$	112.845	0.0345818
$221$	−1829.81	−0.556952
$222$	0	0
$223$	4476.28	1.34419	0.672093	−	0.740467i	$-0.265395\pi$
$223$	4476.28	1.34419	0.672093	+	0.740467i	$0.265395\pi$
$224$	1307.01	0.389859
$225$	0	0
$226$	1108.96	0.326402
$227$	−1574.88	−0.460477	−0.230239	−	0.973134i	$-0.573951\pi$
$227$	−1574.88	−0.460477	−0.230239	+	0.973134i	$0.573951\pi$
$228$	0	0
$229$	447.154	0.129034	0.0645170	−	0.997917i	$-0.479449\pi$
$229$	447.154	0.129034	0.0645170	+	0.997917i	$0.479449\pi$
$230$	629.196	0.180382
$231$	0	0
$232$	1559.33	0.441271
$233$	−3933.72	−1.10604	−0.553018	−	0.833169i	$-0.686524\pi$
$233$	−3933.72	−1.10604	−0.553018	+	0.833169i	$0.686524\pi$
$234$	0	0
$235$	−804.263	−0.223253
$236$	2250.55	0.620756
$237$	0	0
$238$	778.648	0.212068
$239$	3679.05	0.995724	0.497862	−	0.867256i	$-0.334119\pi$
$239$	3679.05	0.995724	0.497862	+	0.867256i	$0.334119\pi$
$240$	0	0
$241$	833.009	0.222651	0.111325	−	0.993784i	$-0.464490\pi$
$241$	833.009	0.222651	0.111325	+	0.993784i	$0.464490\pi$
$242$	2018.52	0.536178
$243$	0	0
$244$	−800.705	−0.210081
$245$	125.260	0.0326635
$246$	0	0
$247$	−2774.62	−0.714756
$248$	6069.48	1.55408
$249$	0	0
$250$	992.680	0.251130
$251$	−7237.27	−1.81997	−0.909985	−	0.414640i	$-0.863907\pi$
$251$	−7237.27	−1.81997	−0.909985	+	0.414640i	$0.863907\pi$
$252$	0	0
$253$	1248.54	0.310258
$254$	2197.11	0.542751
$255$	0	0
$256$	−3596.32	−0.878009
$257$	3078.77	0.747270	0.373635	−	0.927576i	$-0.378111\pi$
$257$	3078.77	0.747270	0.373635	+	0.927576i	$0.378111\pi$
$258$	0	0
$259$	−180.582	−0.0433236
$260$	365.941	0.0872873
$261$	0	0
$262$	−4341.98	−1.02385
$263$	3957.15	0.927788	0.463894	−	0.885891i	$-0.346452\pi$
$263$	3957.15	0.927788	0.463894	+	0.885891i	$0.346452\pi$
$264$	0	0
$265$	−911.422	−0.211276
$266$	1180.70	0.272155
$267$	0	0
$268$	1671.74	0.381035
$269$	2552.30	0.578499	0.289249	−	0.957254i	$-0.406594\pi$
$269$	2552.30	0.578499	0.289249	+	0.957254i	$0.406594\pi$
$270$	0	0
$271$	−1888.99	−0.423424	−0.211712	−	0.977332i	$-0.567904\pi$
$271$	−1888.99	−0.423424	−0.211712	+	0.977332i	$0.567904\pi$
$272$	656.705	0.146392
$273$	0	0
$274$	3752.40	0.827339
$275$	958.475	0.210175
$276$	0	0
$277$	−2952.96	−0.640528	−0.320264	−	0.947328i	$-0.603772\pi$
$277$	−2952.96	−0.640528	−0.320264	+	0.947328i	$0.603772\pi$
$278$	3267.64	0.704963
$279$	0	0
$280$	−384.049	−0.0819691
$281$	−1206.48	−0.256129	−0.128065	−	0.991766i	$-0.540877\pi$
$281$	−1206.48	−0.256129	−0.128065	+	0.991766i	$0.540877\pi$
$282$	0	0
$283$	3697.51	0.776658	0.388329	−	0.921521i	$-0.373052\pi$
$283$	3697.51	0.776658	0.388329	+	0.921521i	$0.373052\pi$
$284$	5667.53	1.18418
$285$	0	0
$286$	−338.584	−0.0700030
$287$	−609.762	−0.125412
$288$	0	0
$289$	−49.2304	−0.0100204
$290$	−296.235	−0.0599845
$291$	0	0
$292$	6313.16	1.26524
$293$	−2485.98	−0.495674	−0.247837	−	0.968802i	$-0.579720\pi$
$293$	−2485.98	−0.495674	−0.247837	+	0.968802i	$0.579720\pi$
$294$	0	0
$295$	−1054.46	−0.208111
$296$	553.667	0.108720
$297$	0	0
$298$	2063.89	0.401201
$299$	4048.86	0.783116
$300$	0	0
$301$	−623.775	−0.119448
$302$	3415.96	0.650882
$303$	0	0
$304$	995.790	0.187870
$305$	375.156	0.0704307
$306$	0	0
$307$	−2167.98	−0.403040	−0.201520	−	0.979484i	$-0.564588\pi$
$307$	−2167.98	−0.403040	−0.201520	+	0.979484i	$0.564588\pi$
$308$	−309.004	−0.0571660
$309$	0	0
$310$	−1153.06	−0.211255
$311$	−4661.98	−0.850021	−0.425010	−	0.905188i	$-0.639730\pi$
$311$	−4661.98	−0.850021	−0.425010	+	0.905188i	$0.639730\pi$
$312$	0	0
$313$	499.190	0.0901466	0.0450733	−	0.998984i	$-0.485648\pi$
$313$	499.190	0.0901466	0.0450733	+	0.998984i	$0.485648\pi$
$314$	−4128.13	−0.741923
$315$	0	0
$316$	−4071.79	−0.724862
$317$	9813.65	1.73877	0.869384	−	0.494138i	$-0.164516\pi$
$317$	9813.65	1.73877	0.869384	+	0.494138i	$0.164516\pi$
$318$	0	0
$319$	−587.833	−0.103173
$320$	568.727	0.0993525
$321$	0	0
$322$	−1722.93	−0.298184
$323$	7375.14	1.27048
$324$	0	0
$325$	3108.21	0.530500
$326$	−1344.48	−0.228416
$327$	0	0
$328$	1869.54	0.314720
$329$	2202.32	0.369051
$330$	0	0
$331$	10030.4	1.66561	0.832807	−	0.553563i	$-0.186732\pi$
$331$	10030.4	1.66561	0.832807	+	0.553563i	$0.186732\pi$
$332$	2866.73	0.473892
$333$	0	0
$334$	2643.51	0.433073
$335$	−783.262	−0.127744
$336$	0	0
$337$	−8805.71	−1.42338	−0.711688	−	0.702496i	$-0.752069\pi$
$337$	−8805.71	−1.42338	−0.711688	+	0.702496i	$0.752069\pi$
$338$	2406.20	0.387219
$339$	0	0
$340$	−972.699	−0.155153
$341$	−2288.06	−0.363359
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1912.51	0.299754
$345$	0	0
$346$	3105.59	0.482536
$347$	−6372.55	−0.985869	−0.492934	−	0.870066i	$-0.664076\pi$
$347$	−6372.55	−0.985869	−0.492934	+	0.870066i	$0.664076\pi$
$348$	0	0
$349$	−2747.33	−0.421380	−0.210690	−	0.977553i	$-0.567571\pi$
$349$	−2747.33	−0.421380	−0.210690	+	0.977553i	$0.567571\pi$
$350$	−1322.65	−0.201996
$351$	0	0
$352$	1510.68	0.228748
$353$	−10279.3	−1.54989	−0.774947	−	0.632027i	$-0.782223\pi$
$353$	−10279.3	−1.54989	−0.774947	+	0.632027i	$0.782223\pi$
$354$	0	0
$355$	−2655.42	−0.397000
$356$	−3513.10	−0.523016
$357$	0	0
$358$	−1342.82	−0.198242
$359$	10473.7	1.53978	0.769891	−	0.638176i	$-0.220311\pi$
$359$	10473.7	1.53978	0.769891	+	0.638176i	$0.220311\pi$
$360$	0	0
$361$	4324.25	0.630448
$362$	−252.423	−0.0366493
$363$	0	0
$364$	−1002.06	−0.144292
$365$	−2957.92	−0.424177
$366$	0	0
$367$	1982.53	0.281981	0.140990	−	0.990011i	$-0.454971\pi$
$367$	1982.53	0.281981	0.140990	+	0.990011i	$0.454971\pi$
$368$	−1453.11	−0.205838
$369$	0	0
$370$	−105.184	−0.0147790
$371$	2495.75	0.349254
$372$	0	0
$373$	9341.67	1.29676	0.648382	−	0.761315i	$-0.275446\pi$
$373$	9341.67	1.29676	0.648382	+	0.761315i	$0.275446\pi$
$374$	899.980	0.124430
$375$	0	0
$376$	−6752.36	−0.926134
$377$	−1906.27	−0.260418
$378$	0	0
$379$	8964.44	1.21497	0.607483	−	0.794333i	$-0.292179\pi$
$379$	8964.44	1.21497	0.607483	+	0.794333i	$0.292179\pi$
$380$	−1474.95	−0.199113
$381$	0	0
$382$	−7367.17	−0.986747
$383$	−4571.49	−0.609902	−0.304951	−	0.952368i	$-0.598640\pi$
$383$	−4571.49	−0.609902	−0.304951	+	0.952368i	$0.598640\pi$
$384$	0	0
$385$	144.778	0.0191652
$386$	−7322.52	−0.965560
$387$	0	0
$388$	−1686.80	−0.220706
$389$	−8837.19	−1.15183	−0.575917	−	0.817508i	$-0.695355\pi$
$389$	−8837.19	−1.15183	−0.575917	+	0.817508i	$0.695355\pi$
$390$	0	0
$391$	−10762.2	−1.39199
$392$	1051.65	0.135500
$393$	0	0
$394$	−4220.38	−0.539644
$395$	1907.77	0.243013
$396$	0	0
$397$	−3649.31	−0.461344	−0.230672	−	0.973032i	$-0.574092\pi$
$397$	−3649.31	−0.461344	−0.230672	+	0.973032i	$0.574092\pi$
$398$	−7562.75	−0.952478
$399$	0	0
$400$	−1115.51	−0.139439
$401$	1019.46	0.126957	0.0634784	−	0.997983i	$-0.479781\pi$
$401$	1019.46	0.126957	0.0634784	+	0.997983i	$0.479781\pi$
$402$	0	0
$403$	−7419.89	−0.917149
$404$	6113.84	0.752908
$405$	0	0
$406$	811.182	0.0991584
$407$	−208.721	−0.0254199
$408$	0	0
$409$	1380.17	0.166858	0.0834289	−	0.996514i	$-0.473413\pi$
$409$	1380.17	0.166858	0.0834289	+	0.996514i	$0.473413\pi$
$410$	−355.169	−0.0427818
$411$	0	0
$412$	−3083.89	−0.368768
$413$	2887.42	0.344022
$414$	0	0
$415$	−1343.15	−0.158874
$416$	4898.93	0.577380
$417$	0	0
$418$	1364.68	0.159686
$419$	−11289.5	−1.31630	−0.658149	−	0.752888i	$-0.728660\pi$
$419$	−11289.5	−1.31630	−0.658149	+	0.752888i	$0.728660\pi$
$420$	0	0
$421$	−11340.1	−1.31279	−0.656394	−	0.754418i	$-0.727919\pi$
$421$	−11340.1	−1.31279	−0.656394	+	0.754418i	$0.727919\pi$
$422$	7306.29	0.842807
$423$	0	0
$424$	−7652.04	−0.876452
$425$	−8261.85	−0.942961
$426$	0	0
$427$	−1027.29	−0.116427
$428$	8692.73	0.981727
$429$	0	0
$430$	−363.331	−0.0407473
$431$	−15264.9	−1.70600	−0.853000	−	0.521911i	$-0.825219\pi$
$431$	−15264.9	−1.70600	−0.853000	+	0.521911i	$0.825219\pi$
$432$	0	0
$433$	−3367.10	−0.373701	−0.186851	−	0.982388i	$-0.559828\pi$
$433$	−3367.10	−0.373701	−0.186851	+	0.982388i	$0.559828\pi$
$434$	3157.42	0.349219
$435$	0	0
$436$	8176.61	0.898139
$437$	−16319.1	−1.78639
$438$	0	0
$439$	11053.4	1.20170	0.600852	−	0.799360i	$-0.294828\pi$
$439$	11053.4	1.20170	0.600852	+	0.799360i	$0.294828\pi$
$440$	−443.894	−0.0480950
$441$	0	0
$442$	2918.52	0.314072
$443$	−7738.17	−0.829913	−0.414957	−	0.909841i	$-0.636203\pi$
$443$	−7738.17	−0.829913	−0.414957	+	0.909841i	$0.636203\pi$
$444$	0	0
$445$	1646.00	0.175344
$446$	−7139.59	−0.758003
$447$	0	0
$448$	−1557.35	−0.164236
$449$	−1209.56	−0.127133	−0.0635665	−	0.997978i	$-0.520248\pi$
$449$	−1209.56	−0.127133	−0.0635665	+	0.997978i	$0.520248\pi$
$450$	0	0
$451$	−704.778	−0.0735847
$452$	3793.45	0.394755
$453$	0	0
$454$	2511.91	0.259669
$455$	469.497	0.0483744
$456$	0	0
$457$	−14257.6	−1.45939	−0.729697	−	0.683771i	$-0.760339\pi$
$457$	−14257.6	−1.45939	−0.729697	+	0.683771i	$0.760339\pi$
$458$	−713.204	−0.0727638
$459$	0	0
$460$	2152.31	0.218157
$461$	−11918.0	−1.20407	−0.602033	−	0.798471i	$-0.705642\pi$
$461$	−11918.0	−1.20407	−0.602033	+	0.798471i	$0.705642\pi$
$462$	0	0
$463$	7860.87	0.789040	0.394520	−	0.918887i	$-0.370911\pi$
$463$	7860.87	0.789040	0.394520	+	0.918887i	$0.370911\pi$
$464$	684.144	0.0684496
$465$	0	0
$466$	6274.22	0.623707
$467$	−16152.3	−1.60051	−0.800257	−	0.599657i	$-0.795304\pi$
$467$	−16152.3	−1.60051	−0.800257	+	0.599657i	$0.795304\pi$
$468$	0	0
$469$	2144.81	0.211169
$470$	1282.79	0.125895
$471$	0	0
$472$	−8852.91	−0.863322
$473$	−720.974	−0.0700854
$474$	0	0
$475$	−12527.8	−1.21014
$476$	2663.55	0.256478
$477$	0	0
$478$	−5868.03	−0.561501
$479$	11703.9	1.11641	0.558207	−	0.829701i	$-0.311489\pi$
$479$	11703.9	1.11641	0.558207	+	0.829701i	$0.311489\pi$
$480$	0	0
$481$	−676.854	−0.0641619
$482$	−1328.64	−0.125555
$483$	0	0
$484$	6904.81	0.648461
$485$	790.318	0.0739928
$486$	0	0
$487$	41.1258	0.00382667	0.00191333	−	0.999998i	$-0.499391\pi$
$487$	41.1258	0.00382667	0.00191333	+	0.999998i	$0.499391\pi$
$488$	3149.70	0.292173
$489$	0	0
$490$	−199.788	−0.0184193
$491$	−8435.65	−0.775348	−0.387674	−	0.921797i	$-0.626721\pi$
$491$	−8435.65	−0.775348	−0.387674	+	0.921797i	$0.626721\pi$
$492$	0	0
$493$	5066.99	0.462892
$494$	4425.48	0.403060
$495$	0	0
$496$	2662.94	0.241068
$497$	7271.36	0.656268
$498$	0	0
$499$	2186.15	0.196124	0.0980618	−	0.995180i	$-0.468736\pi$
$499$	2186.15	0.196124	0.0980618	+	0.995180i	$0.468736\pi$
$500$	3395.70	0.303720
$501$	0	0
$502$	11543.3	1.02630
$503$	−14676.6	−1.30098	−0.650492	−	0.759513i	$-0.725437\pi$
$503$	−14676.6	−1.30098	−0.650492	+	0.759513i	$0.725437\pi$
$504$	0	0
$505$	−2864.53	−0.252416
$506$	−1991.41	−0.174958
$507$	0	0
$508$	7515.73	0.656411
$509$	12899.6	1.12331	0.561654	−	0.827372i	$-0.310165\pi$
$509$	12899.6	1.12331	0.561654	+	0.827372i	$0.310165\pi$
$510$	0	0
$511$	8099.69	0.701192
$512$	−3374.96	−0.291315
$513$	0	0
$514$	−4910.59	−0.421395
$515$	1444.90	0.123631
$516$	0	0
$517$	2545.50	0.216539
$518$	288.025	0.0244307
$519$	0	0
$520$	−1439.49	−0.121396
$521$	−11771.7	−0.989880	−0.494940	−	0.868927i	$-0.664810\pi$
$521$	−11771.7	−0.989880	−0.494940	+	0.868927i	$0.664810\pi$
$522$	0	0
$523$	5339.23	0.446402	0.223201	−	0.974772i	$-0.428349\pi$
$523$	5339.23	0.446402	0.223201	+	0.974772i	$0.428349\pi$
$524$	−14852.8	−1.23826
$525$	0	0
$526$	−6311.59	−0.523191
$527$	19722.6	1.63023
$528$	0	0
$529$	11646.7	0.957238
$530$	1453.70	0.119141
$531$	0	0
$532$	4038.85	0.329148
$533$	−2285.50	−0.185734
$534$	0	0
$535$	−4072.83	−0.329128
$536$	−6576.04	−0.529929
$537$	0	0
$538$	−4070.87	−0.326223
$539$	−396.448	−0.0316813
$540$	0	0
$541$	828.599	0.0658489	0.0329244	−	0.999458i	$-0.489518\pi$
$541$	828.599	0.0658489	0.0329244	+	0.999458i	$0.489518\pi$
$542$	3012.91	0.238774
$543$	0	0
$544$	−13021.7	−1.02629
$545$	−3831.00	−0.301105
$546$	0	0
$547$	13897.4	1.08631	0.543153	−	0.839634i	$-0.317230\pi$
$547$	13897.4	1.08631	0.543153	+	0.839634i	$0.317230\pi$
$548$	12836.0	1.00059
$549$	0	0
$550$	−1528.75	−0.118520
$551$	7683.30	0.594047
$552$	0	0
$553$	−5224.05	−0.401717
$554$	4709.93	0.361202
$555$	0	0
$556$	11177.7	0.852592
$557$	−11168.5	−0.849595	−0.424798	−	0.905288i	$-0.639655\pi$
$557$	−11168.5	−0.849595	−0.424798	+	0.905288i	$0.639655\pi$
$558$	0	0
$559$	−2338.02	−0.176901
$560$	−168.499	−0.0127150
$561$	0	0
$562$	1924.31	0.144434
$563$	645.729	0.0483379	0.0241689	−	0.999708i	$-0.492306\pi$
$563$	645.729	0.0483379	0.0241689	+	0.999708i	$0.492306\pi$
$564$	0	0
$565$	−1777.36	−0.132343
$566$	−5897.47	−0.437967
$567$	0	0
$568$	−22294.1	−1.64690
$569$	−19085.0	−1.40612	−0.703062	−	0.711128i	$-0.748185\pi$
$569$	−19085.0	−1.40612	−0.703062	+	0.711128i	$0.748185\pi$
$570$	0	0
$571$	−2818.29	−0.206553	−0.103277	−	0.994653i	$-0.532933\pi$
$571$	−2818.29	−0.206553	−0.103277	+	0.994653i	$0.532933\pi$
$572$	−1158.21	−0.0846626
$573$	0	0
$574$	972.561	0.0707211
$575$	18281.2	1.32588
$576$	0	0
$577$	16153.2	1.16546	0.582728	−	0.812667i	$-0.301985\pi$
$577$	16153.2	1.16546	0.582728	+	0.812667i	$0.301985\pi$
$578$	78.5218	0.00565065
$579$	0	0
$580$	−1013.34	−0.0725461
$581$	3677.97	0.262630
$582$	0	0
$583$	2884.65	0.204923
$584$	−24833.8	−1.75964
$585$	0	0
$586$	3965.10	0.279517
$587$	−13941.0	−0.980249	−0.490125	−	0.871652i	$-0.663049\pi$
$587$	−13941.0	−0.980249	−0.490125	+	0.871652i	$0.663049\pi$
$588$	0	0
$589$	29906.3	2.09213
$590$	1681.84	0.117356
$591$	0	0
$592$	242.918	0.0168646
$593$	−203.913	−0.0141209	−0.00706046	−	0.999975i	$-0.502247\pi$
$593$	−203.913	−0.0141209	−0.00706046	+	0.999975i	$0.502247\pi$
$594$	0	0
$595$	−1247.96	−0.0859854
$596$	7060.02	0.485218
$597$	0	0
$598$	−6457.87	−0.441609
$599$	2309.02	0.157503	0.0787514	−	0.996894i	$-0.474907\pi$
$599$	2309.02	0.157503	0.0787514	+	0.996894i	$0.474907\pi$
$600$	0	0
$601$	22350.3	1.51695	0.758477	−	0.651700i	$-0.225944\pi$
$601$	22350.3	1.51695	0.758477	+	0.651700i	$0.225944\pi$
$602$	994.911	0.0673580
$603$	0	0
$604$	11685.1	0.787186
$605$	−3235.13	−0.217399
$606$	0	0
$607$	−17105.3	−1.14379	−0.571895	−	0.820327i	$-0.693792\pi$
$607$	−17105.3	−1.14379	−0.571895	+	0.820327i	$0.693792\pi$
$608$	−19745.4	−1.31708
$609$	0	0
$610$	−598.368	−0.0397167
$611$	8254.71	0.546563
$612$	0	0
$613$	−7474.70	−0.492496	−0.246248	−	0.969207i	$-0.579198\pi$
$613$	−7474.70	−0.492496	−0.246248	+	0.969207i	$0.579198\pi$
$614$	3457.90	0.227279
$615$	0	0
$616$	1215.52	0.0795042
$617$	−3604.25	−0.235173	−0.117586	−	0.993063i	$-0.537516\pi$
$617$	−3604.25	−0.235173	−0.117586	+	0.993063i	$0.537516\pi$
$618$	0	0
$619$	−4542.61	−0.294964	−0.147482	−	0.989065i	$-0.547117\pi$
$619$	−4542.61	−0.294964	−0.147482	+	0.989065i	$0.547117\pi$
$620$	−3944.30	−0.255495
$621$	0	0
$622$	7435.78	0.479337
$623$	−4507.26	−0.289855
$624$	0	0
$625$	13217.2	0.845898
$626$	−796.201	−0.0508348
$627$	0	0
$628$	−14121.2	−0.897291
$629$	1799.13	0.114048
$630$	0	0
$631$	−11564.2	−0.729578	−0.364789	−	0.931090i	$-0.618859\pi$
$631$	−11564.2	−0.729578	−0.364789	+	0.931090i	$0.618859\pi$
$632$	16017.1	1.00811
$633$	0	0
$634$	−15652.6	−0.980512
$635$	−3521.36	−0.220064
$636$	0	0
$637$	−1285.63	−0.0799662
$638$	937.585	0.0581808
$639$	0	0
$640$	2911.34	0.179814
$641$	−1699.17	−0.104701	−0.0523503	−	0.998629i	$-0.516671\pi$
$641$	−1699.17	−0.104701	−0.0523503	+	0.998629i	$0.516671\pi$
$642$	0	0
$643$	29267.4	1.79501	0.897506	−	0.441002i	$-0.145377\pi$
$643$	29267.4	1.79501	0.897506	+	0.441002i	$0.145377\pi$
$644$	−5893.70	−0.360628
$645$	0	0
$646$	−11763.2	−0.716437
$647$	−19018.7	−1.15564	−0.577822	−	0.816163i	$-0.696097\pi$
$647$	−19018.7	−1.15564	−0.577822	+	0.816163i	$0.696097\pi$
$648$	0	0
$649$	3337.36	0.201853
$650$	−4957.55	−0.299155
$651$	0	0
$652$	−4599.10	−0.276249
$653$	9469.77	0.567505	0.283753	−	0.958898i	$-0.408421\pi$
$653$	9469.77	0.567505	0.283753	+	0.958898i	$0.408421\pi$
$654$	0	0
$655$	6959.00	0.415131
$656$	820.250	0.0488192
$657$	0	0
$658$	−3512.67	−0.208113
$659$	17225.1	1.01820	0.509100	−	0.860707i	$-0.329978\pi$
$659$	17225.1	1.01820	0.509100	+	0.860707i	$0.329978\pi$
$660$	0	0
$661$	−9879.81	−0.581362	−0.290681	−	0.956820i	$-0.593882\pi$
$661$	−9879.81	−0.581362	−0.290681	+	0.956820i	$0.593882\pi$
$662$	−15998.3	−0.939260
$663$	0	0
$664$	−11276.7	−0.659070
$665$	−1892.33	−0.110348
$666$	0	0
$667$	−11211.9	−0.650862
$668$	9042.75	0.523764
$669$	0	0
$670$	1249.29	0.0720363
$671$	−1187.37	−0.0683128
$672$	0	0
$673$	−30181.0	−1.72867	−0.864334	−	0.502918i	$-0.832260\pi$
$673$	−30181.0	−1.72867	−0.864334	+	0.502918i	$0.832260\pi$
$674$	14045.0	0.802659
$675$	0	0
$676$	8230.97	0.468308
$677$	22450.9	1.27453	0.637265	−	0.770644i	$-0.280065\pi$
$677$	22450.9	1.27453	0.637265	+	0.770644i	$0.280065\pi$
$678$	0	0
$679$	−2164.14	−0.122315
$680$	3826.27	0.215780
$681$	0	0
$682$	3649.43	0.204903
$683$	12957.7	0.725933	0.362966	−	0.931802i	$-0.381764\pi$
$683$	12957.7	0.725933	0.362966	+	0.931802i	$0.381764\pi$
$684$	0	0
$685$	−6014.07	−0.335454
$686$	547.080	0.0304484
$687$	0	0
$688$	839.099	0.0464976
$689$	9354.56	0.517243
$690$	0	0
$691$	571.056	0.0314385	0.0157192	−	0.999876i	$-0.494996\pi$
$691$	571.056	0.0314385	0.0157192	+	0.999876i	$0.494996\pi$
$692$	10623.4	0.583586
$693$	0	0
$694$	10164.1	0.555944
$695$	−5237.12	−0.285835
$696$	0	0
$697$	6075.04	0.330141
$698$	4381.96	0.237621
$699$	0	0
$700$	−4524.44	−0.244297
$701$	26258.8	1.41481	0.707404	−	0.706810i	$-0.249866\pi$
$701$	26258.8	1.41481	0.707404	+	0.706810i	$0.249866\pi$
$702$	0	0
$703$	2728.10	0.146361
$704$	−1800.02	−0.0963649
$705$	0	0
$706$	16395.3	0.874004
$707$	7843.96	0.417260
$708$	0	0
$709$	−10241.3	−0.542482	−0.271241	−	0.962511i	$-0.587434\pi$
$709$	−10241.3	−0.542482	−0.271241	+	0.962511i	$0.587434\pi$
$710$	4235.35	0.223873
$711$	0	0
$712$	13819.3	0.727390
$713$	−43640.7	−2.29223
$714$	0	0
$715$	542.656	0.0283835
$716$	−4593.45	−0.239756
$717$	0	0
$718$	−16705.4	−0.868301
$719$	9398.00	0.487464	0.243732	−	0.969843i	$-0.421628\pi$
$719$	9398.00	0.487464	0.243732	+	0.969843i	$0.421628\pi$
$720$	0	0
$721$	−3956.58	−0.204370
$722$	−6897.11	−0.355518
$723$	0	0
$724$	−863.473	−0.0443242
$725$	−8607.06	−0.440908
$726$	0	0
$727$	6280.69	0.320410	0.160205	−	0.987084i	$-0.448785\pi$
$727$	6280.69	0.320410	0.160205	+	0.987084i	$0.448785\pi$
$728$	3941.77	0.200675
$729$	0	0
$730$	4717.83	0.239198
$731$	6214.64	0.314442
$732$	0	0
$733$	10537.9	0.531007	0.265503	−	0.964110i	$-0.414462\pi$
$733$	10537.9	0.531007	0.265503	+	0.964110i	$0.414462\pi$
$734$	−3162.10	−0.159013
$735$	0	0
$736$	28813.5	1.44304
$737$	2479.03	0.123902
$738$	0	0
$739$	20509.6	1.02092	0.510459	−	0.859902i	$-0.329475\pi$
$739$	20509.6	1.02092	0.510459	+	0.859902i	$0.329475\pi$
$740$	−359.805	−0.0178739
$741$	0	0
$742$	−3980.69	−0.196948
$743$	26278.9	1.29755	0.648776	−	0.760980i	$-0.275281\pi$
$743$	26278.9	1.29755	0.648776	+	0.760980i	$0.275281\pi$
$744$	0	0
$745$	−3307.85	−0.162671
$746$	−14899.8	−0.731261
$747$	0	0
$748$	3078.60	0.150487
$749$	11152.7	0.544071
$750$	0	0
$751$	−1377.54	−0.0669337	−0.0334668	−	0.999440i	$-0.510655\pi$
$751$	−1377.54	−0.0669337	−0.0334668	+	0.999440i	$0.510655\pi$
$752$	−2962.55	−0.143661
$753$	0	0
$754$	3040.46	0.146853
$755$	−5474.85	−0.263907
$756$	0	0
$757$	13632.8	0.654549	0.327275	−	0.944929i	$-0.393870\pi$
$757$	13632.8	0.654549	0.327275	+	0.944929i	$0.393870\pi$
$758$	−14298.1	−0.685135
$759$	0	0
$760$	5801.94	0.276919
$761$	−1552.11	−0.0739344	−0.0369672	−	0.999316i	$-0.511770\pi$
$761$	−1552.11	−0.0739344	−0.0369672	+	0.999316i	$0.511770\pi$
$762$	0	0
$763$	10490.5	0.497746
$764$	−25201.2	−1.19338
$765$	0	0
$766$	7291.46	0.343931
$767$	10822.6	0.509494
$768$	0	0
$769$	−10628.4	−0.498400	−0.249200	−	0.968452i	$-0.580168\pi$
$769$	−10628.4	−0.498400	−0.249200	+	0.968452i	$0.580168\pi$
$770$	−230.919	−0.0108075
$771$	0	0
$772$	−25048.4	−1.16776
$773$	20748.7	0.965432	0.482716	−	0.875777i	$-0.339650\pi$
$773$	20748.7	0.965432	0.482716	+	0.875777i	$0.339650\pi$
$774$	0	0
$775$	−33501.9	−1.55280
$776$	6635.29	0.306950
$777$	0	0
$778$	14095.2	0.649533
$779$	9211.84	0.423682
$780$	0	0
$781$	8404.41	0.385062
$782$	17165.5	0.784958
$783$	0	0
$784$	461.402	0.0210187
$785$	6616.26	0.300821
$786$	0	0
$787$	−22386.0	−1.01395	−0.506973	−	0.861962i	$-0.669236\pi$
$787$	−22386.0	−1.01395	−0.506973	+	0.861962i	$0.669236\pi$
$788$	−14436.8	−0.652653
$789$	0	0
$790$	−3042.86	−0.137038
$791$	4866.95	0.218772
$792$	0	0
$793$	−3850.49	−0.172427
$794$	5820.59	0.260158
$795$	0	0
$796$	−25870.2	−1.15194
$797$	1037.05	0.0460906	0.0230453	−	0.999734i	$-0.492664\pi$
$797$	1037.05	0.0460906	0.0230453	+	0.999734i	$0.492664\pi$
$798$	0	0
$799$	−21941.6	−0.971513
$800$	22119.4	0.977548
$801$	0	0
$802$	−1626.03	−0.0715925
$803$	9361.82	0.411421
$804$	0	0
$805$	2761.39	0.120902
$806$	11834.6	0.517192
$807$	0	0
$808$	−24049.8	−1.04711
$809$	10345.1	0.449586	0.224793	−	0.974407i	$-0.427829\pi$
$809$	10345.1	0.449586	0.224793	+	0.974407i	$0.427829\pi$
$810$	0	0
$811$	8037.38	0.348003	0.174002	−	0.984745i	$-0.444330\pi$
$811$	8037.38	0.348003	0.174002	+	0.984745i	$0.444330\pi$
$812$	2774.84	0.119923
$813$	0	0
$814$	332.906	0.0143346
$815$	2154.83	0.0926138
$816$	0	0
$817$	9423.53	0.403534
$818$	−2201.34	−0.0940931
$819$	0	0
$820$	−1214.94	−0.0517408
$821$	12189.7	0.518177	0.259089	−	0.965854i	$-0.416578\pi$
$821$	12189.7	0.518177	0.259089	+	0.965854i	$0.416578\pi$
$822$	0	0
$823$	−17826.7	−0.755043	−0.377521	−	0.926001i	$-0.623224\pi$
$823$	−17826.7	−0.755043	−0.377521	+	0.926001i	$0.623224\pi$
$824$	12131.0	0.512867
$825$	0	0
$826$	−4605.40	−0.193998
$827$	41268.6	1.73525	0.867624	−	0.497220i	$-0.165646\pi$
$827$	41268.6	1.73525	0.867624	+	0.497220i	$0.165646\pi$
$828$	0	0
$829$	−42762.2	−1.79155	−0.895774	−	0.444509i	$-0.853378\pi$
$829$	−42762.2	−1.79155	−0.895774	+	0.444509i	$0.853378\pi$
$830$	2142.31	0.0895912
$831$	0	0
$832$	−5837.24	−0.243233
$833$	3417.30	0.142140
$834$	0	0
$835$	−4236.82	−0.175594
$836$	4668.21	0.193126
$837$	0	0
$838$	18006.6	0.742277
$839$	44415.8	1.82766	0.913828	−	0.406101i	$-0.133112\pi$
$839$	44415.8	1.82766	0.913828	+	0.406101i	$0.133112\pi$
$840$	0	0
$841$	−19110.3	−0.783562
$842$	18087.3	0.740298
$843$	0	0
$844$	24992.9	1.01930
$845$	−3856.48	−0.157002
$846$	0	0
$847$	8858.78	0.359376
$848$	−3357.28	−0.135955
$849$	0	0
$850$	13177.5	0.531748
$851$	−3980.97	−0.160360
$852$	0	0
$853$	9171.20	0.368131	0.184066	−	0.982914i	$-0.441074\pi$
$853$	9171.20	0.368131	0.184066	+	0.982914i	$0.441074\pi$
$854$	1638.52	0.0656544
$855$	0	0
$856$	−34194.3	−1.36535
$857$	−11823.2	−0.471265	−0.235633	−	0.971842i	$-0.575716\pi$
$857$	−11823.2	−0.471265	−0.235633	+	0.971842i	$0.575716\pi$
$858$	0	0
$859$	2772.40	0.110120	0.0550600	−	0.998483i	$-0.482465\pi$
$859$	2772.40	0.110120	0.0550600	+	0.998483i	$0.482465\pi$
$860$	−1242.86	−0.0492804
$861$	0	0
$862$	24347.3	0.962034
$863$	−7669.34	−0.302512	−0.151256	−	0.988495i	$-0.548332\pi$
$863$	−7669.34	−0.302512	−0.151256	+	0.988495i	$0.548332\pi$
$864$	0	0
$865$	−4977.41	−0.195650
$866$	5370.48	0.210735
$867$	0	0
$868$	10800.7	0.422350
$869$	−6038.09	−0.235706
$870$	0	0
$871$	8039.16	0.312740
$872$	−32164.0	−1.24910
$873$	0	0
$874$	26028.8	1.00737
$875$	4356.63	0.168321
$876$	0	0
$877$	−13867.7	−0.533956	−0.266978	−	0.963703i	$-0.586025\pi$
$877$	−13867.7	−0.533956	−0.266978	+	0.963703i	$0.586025\pi$
$878$	−17630.0	−0.677656
$879$	0	0
$880$	−194.755	−0.00746046
$881$	−19974.2	−0.763846	−0.381923	−	0.924194i	$-0.624738\pi$
$881$	−19974.2	−0.763846	−0.381923	+	0.924194i	$0.624738\pi$
$882$	0	0
$883$	22197.0	0.845969	0.422984	−	0.906137i	$-0.360983\pi$
$883$	22197.0	0.845969	0.422984	+	0.906137i	$0.360983\pi$
$884$	9983.48	0.379843
$885$	0	0
$886$	12342.3	0.467998
$887$	33701.8	1.27575	0.637877	−	0.770138i	$-0.279813\pi$
$887$	33701.8	1.27575	0.637877	+	0.770138i	$0.279813\pi$
$888$	0	0
$889$	9642.57	0.363781
$890$	−2625.34	−0.0988784
$891$	0	0
$892$	−24422.7	−0.916739
$893$	−33271.0	−1.24678
$894$	0	0
$895$	2152.18	0.0803792
$896$	−7972.16	−0.297245
$897$	0	0
$898$	1929.23	0.0716919
$899$	20546.7	0.762259
$900$	0	0
$901$	−24865.1	−0.919397
$902$	1124.11	0.0414953
$903$	0	0
$904$	−14922.2	−0.549009
$905$	404.565	0.0148599
$906$	0	0
$907$	−23900.3	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$907$	−23900.3	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$908$	8592.58	0.314047
$909$	0	0
$910$	−748.841	−0.0272789
$911$	12753.6	0.463826	0.231913	−	0.972737i	$-0.425502\pi$
$911$	12753.6	0.463826	0.231913	+	0.972737i	$0.425502\pi$
$912$	0	0
$913$	4251.09	0.154097
$914$	22740.7	0.822970
$915$	0	0
$916$	−2439.68	−0.0880015
$917$	−19055.9	−0.686239
$918$	0	0
$919$	−10491.5	−0.376585	−0.188293	−	0.982113i	$-0.560295\pi$
$919$	−10491.5	−0.376585	−0.188293	+	0.982113i	$0.560295\pi$
$920$	−8466.47	−0.303404
$921$	0	0
$922$	19009.0	0.678988
$923$	27254.4	0.971929
$924$	0	0
$925$	−3056.09	−0.108631
$926$	−12538.0	−0.444949
$927$	0	0
$928$	−13565.8	−0.479870
$929$	50045.9	1.76744	0.883720	−	0.468016i	$-0.155031\pi$
$929$	50045.9	1.76744	0.883720	+	0.468016i	$0.155031\pi$
$930$	0	0
$931$	5181.79	0.182413
$932$	21462.4	0.754320
$933$	0	0
$934$	25762.7	0.902550
$935$	−1442.42	−0.0504516
$936$	0	0
$937$	−35500.0	−1.23771	−0.618855	−	0.785505i	$-0.712403\pi$
$937$	−35500.0	−1.23771	−0.618855	+	0.785505i	$0.712403\pi$
$938$	−3420.94	−0.119081
$939$	0	0
$940$	4388.08	0.152259
$941$	−42264.1	−1.46416	−0.732078	−	0.681221i	$-0.761449\pi$
$941$	−42264.1	−1.46416	−0.732078	+	0.681221i	$0.761449\pi$
$942$	0	0
$943$	−13442.4	−0.464204
$944$	−3884.15	−0.133918
$945$	0	0
$946$	1149.94	0.0395220
$947$	48995.6	1.68125	0.840624	−	0.541619i	$-0.182188\pi$
$947$	48995.6	1.68125	0.840624	+	0.541619i	$0.182188\pi$
$948$	0	0
$949$	30359.2	1.03846
$950$	19981.6	0.682411
$951$	0	0
$952$	−10477.5	−0.356699
$953$	−5041.88	−0.171377	−0.0856886	−	0.996322i	$-0.527309\pi$
$953$	−5041.88	−0.171377	−0.0856886	+	0.996322i	$0.527309\pi$
$954$	0	0
$955$	11807.6	0.400087
$956$	−20073.0	−0.679087
$957$	0	0
$958$	−18667.5	−0.629560
$959$	16468.4	0.554527
$960$	0	0
$961$	50184.4	1.68455
$962$	1079.57	0.0361817
$963$	0	0
$964$	−4544.91	−0.151848
$965$	11736.0	0.391497
$966$	0	0
$967$	−6959.37	−0.231436	−0.115718	−	0.993282i	$-0.536917\pi$
$967$	−6959.37	−0.231436	−0.115718	+	0.993282i	$0.536917\pi$
$968$	−27161.2	−0.901853
$969$	0	0
$970$	−1260.55	−0.0417255
$971$	12979.8	0.428983	0.214492	−	0.976726i	$-0.431191\pi$
$971$	12979.8	0.428983	0.214492	+	0.976726i	$0.431191\pi$
$972$	0	0
$973$	14340.9	0.472505
$974$	−65.5950	−0.00215791
$975$	0	0
$976$	1381.91	0.0453216
$977$	42262.1	1.38392	0.691958	−	0.721938i	$-0.256748\pi$
$977$	42262.1	1.38392	0.691958	+	0.721938i	$0.256748\pi$
$978$	0	0
$979$	−5209.60	−0.170071
$980$	−683.420	−0.0222766
$981$	0	0
$982$	13454.7	0.437228
$983$	−54187.7	−1.75821	−0.879105	−	0.476629i	$-0.841859\pi$
$983$	−54187.7	−1.75821	−0.879105	+	0.476629i	$0.841859\pi$
$984$	0	0
$985$	6764.12	0.218805
$986$	−8081.78	−0.261031
$987$	0	0
$988$	15138.4	0.487466
$989$	−13751.3	−0.442129
$990$	0	0
$991$	37682.0	1.20788	0.603940	−	0.797030i	$-0.293597\pi$
$991$	37682.0	1.20788	0.603940	+	0.797030i	$0.293597\pi$
$992$	−52803.2	−1.69002
$993$	0	0
$994$	−11597.7	−0.370077
$995$	12121.0	0.386193
$996$	0	0
$997$	−49864.7	−1.58398	−0.791991	−	0.610533i	$-0.790955\pi$
$997$	−49864.7	−1.58398	−0.791991	+	0.610533i	$0.790955\pi$
$998$	−3486.88	−0.110597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.g.1.4		8
3.2	odd	2		567.4.a.i.1.5		8
9.2	odd	6		63.4.f.b.22.4	✓	16
9.4	even	3		189.4.f.b.127.5		16
9.5	odd	6		63.4.f.b.43.4	yes	16
9.7	even	3		189.4.f.b.64.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.4	✓	16	9.2	odd	6
63.4.f.b.43.4	yes	16	9.5	odd	6
189.4.f.b.64.5		16	9.7	even	3
189.4.f.b.127.5		16	9.4	even	3
567.4.a.g.1.4		8	1.1	even	1	trivial
567.4.a.i.1.5		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q-1.59498 q^{2} -5.45602 q^{4} +2.55632 q^{5} -7.00000 q^{7} +21.4622 q^{8} +O(q^{10})\)	\(q-1.59498 q^{2} -5.45602 q^{4} +2.55632 q^{5} -7.00000 q^{7} +21.4622 q^{8} -4.07730 q^{10} -8.09077 q^{11} -26.2373 q^{13} +11.1649 q^{14} +9.41638 q^{16} +69.7407 q^{17} +105.751 q^{19} -13.9474 q^{20} +12.9047 q^{22} -154.317 q^{23} -118.465 q^{25} +41.8481 q^{26} +38.1922 q^{28} +72.6547 q^{29} +282.799 q^{31} -186.716 q^{32} -111.235 q^{34} -17.8943 q^{35} +25.7974 q^{37} -168.671 q^{38} +54.8642 q^{40} +87.1089 q^{41} +89.1106 q^{43} +44.1434 q^{44} +246.133 q^{46} -314.617 q^{47} +49.0000 q^{49} +188.950 q^{50} +143.151 q^{52} -356.536 q^{53} -20.6826 q^{55} -150.235 q^{56} -115.883 q^{58} -412.489 q^{59} +146.756 q^{61} -451.060 q^{62} +222.479 q^{64} -67.0710 q^{65} -306.402 q^{67} -380.507 q^{68} +28.5411 q^{70} -1038.77 q^{71} -1157.10 q^{73} -41.1464 q^{74} -576.979 q^{76} +56.6354 q^{77} +746.293 q^{79} +24.0713 q^{80} -138.937 q^{82} -525.424 q^{83} +178.280 q^{85} -142.130 q^{86} -173.645 q^{88} +643.894 q^{89} +183.661 q^{91} +841.957 q^{92} +501.810 q^{94} +270.333 q^{95} +309.162 q^{97} -78.1543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 43 * q^4 - 30 * q^5 - 56 * q^7 - 6 * q^8	\( 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8} - 14 q^{10} - 24 q^{11} + 68 q^{13} + 21 q^{14} + 103 q^{16} - 168 q^{17} + 176 q^{19} - 330 q^{20} + 151 q^{22} - 228 q^{23} + 244 q^{25} - 795 q^{26} - 301 q^{28} - 618 q^{29} + 72 q^{31} - 786 q^{32} - 261 q^{34} + 210 q^{35} + 210 q^{37} - 1032 q^{38} - 375 q^{40} - 420 q^{41} - 2 q^{43} - 387 q^{44} + 402 q^{46} - 570 q^{47} + 392 q^{49} - 1110 q^{50} - 431 q^{52} - 528 q^{53} - 838 q^{55} + 42 q^{56} + 37 q^{58} + 150 q^{59} + 578 q^{61} - 1170 q^{62} - 112 q^{64} + 366 q^{65} - 898 q^{67} - 2526 q^{68} + 98 q^{70} - 882 q^{71} + 972 q^{73} + 222 q^{74} + 1423 q^{76} + 168 q^{77} - 158 q^{79} - 2475 q^{80} - 211 q^{82} - 2958 q^{83} - 774 q^{85} + 114 q^{86} + 1317 q^{88} - 4380 q^{89} - 476 q^{91} - 4629 q^{92} - 3234 q^{94} - 930 q^{95} - 60 q^{97} - 147 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 43 * q^4 - 30 * q^5 - 56 * q^7 - 6 * q^8 - 14 * q^10 - 24 * q^11 + 68 * q^13 + 21 * q^14 + 103 * q^16 - 168 * q^17 + 176 * q^19 - 330 * q^20 + 151 * q^22 - 228 * q^23 + 244 * q^25 - 795 * q^26 - 301 * q^28 - 618 * q^29 + 72 * q^31 - 786 * q^32 - 261 * q^34 + 210 * q^35 + 210 * q^37 - 1032 * q^38 - 375 * q^40 - 420 * q^41 - 2 * q^43 - 387 * q^44 + 402 * q^46 - 570 * q^47 + 392 * q^49 - 1110 * q^50 - 431 * q^52 - 528 * q^53 - 838 * q^55 + 42 * q^56 + 37 * q^58 + 150 * q^59 + 578 * q^61 - 1170 * q^62 - 112 * q^64 + 366 * q^65 - 898 * q^67 - 2526 * q^68 + 98 * q^70 - 882 * q^71 + 972 * q^73 + 222 * q^74 + 1423 * q^76 + 168 * q^77 - 158 * q^79 - 2475 * q^80 - 211 * q^82 - 2958 * q^83 - 774 * q^85 + 114 * q^86 + 1317 * q^88 - 4380 * q^89 - 476 * q^91 - 4629 * q^92 - 3234 * q^94 - 930 * q^95 - 60 * q^97 - 147 * q^98

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