LMFDB - Embedded newform 567.4.a.g.1.3

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			1.3
Root			$2.61577$ 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)

$f(q)$	$=$	$q-2.61577 q^{2} -1.15774 q^{4} +13.5431 q^{5} -7.00000 q^{7} +23.9546 q^{8} +O(q^{10})$	\(q-2.61577 q^{2} -1.15774 q^{4} +13.5431 q^{5} -7.00000 q^{7} +23.9546 q^{8} -35.4255 q^{10} -24.0692 q^{11} +23.9944 q^{13} +18.3104 q^{14} -53.3978 q^{16} -79.6971 q^{17} -50.0463 q^{19} -15.6793 q^{20} +62.9595 q^{22} +151.608 q^{23} +58.4143 q^{25} -62.7639 q^{26} +8.10415 q^{28} -256.124 q^{29} -2.72508 q^{31} -51.9600 q^{32} +208.469 q^{34} -94.8014 q^{35} +319.617 q^{37} +130.910 q^{38} +324.418 q^{40} -164.778 q^{41} +422.769 q^{43} +27.8657 q^{44} -396.571 q^{46} -100.077 q^{47} +49.0000 q^{49} -152.798 q^{50} -27.7792 q^{52} -194.981 q^{53} -325.970 q^{55} -167.682 q^{56} +669.962 q^{58} -576.515 q^{59} -43.0383 q^{61} +7.12819 q^{62} +563.098 q^{64} +324.958 q^{65} -1016.51 q^{67} +92.2681 q^{68} +247.979 q^{70} +509.305 q^{71} +1039.67 q^{73} -836.046 q^{74} +57.9403 q^{76} +168.484 q^{77} -894.141 q^{79} -723.169 q^{80} +431.023 q^{82} -14.4611 q^{83} -1079.34 q^{85} -1105.87 q^{86} -576.567 q^{88} -1532.12 q^{89} -167.961 q^{91} -175.522 q^{92} +261.779 q^{94} -677.779 q^{95} -1775.15 q^{97} -128.173 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.61577	−0.924815	−0.462408	−	0.886667i	$-0.653014\pi$
$2$	−2.61577	−0.924815	−0.462408	+	0.886667i	$0.653014\pi$
$3$	0	0
$3$	0	0
$4$	−1.15774	−0.144717
$5$	13.5431	1.21133	0.605664	−	0.795721i	$-0.292908\pi$
$5$	13.5431	1.21133	0.605664	+	0.795721i	$0.292908\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	23.9546	1.05865
$9$	0	0
$10$	−35.4255	−1.12025
$11$	−24.0692	−0.659739	−0.329870	−	0.944026i	$-0.607005\pi$
$11$	−24.0692	−0.659739	−0.329870	+	0.944026i	$0.607005\pi$
$12$	0	0
$13$	23.9944	0.511912	0.255956	−	0.966688i	$-0.417610\pi$
$13$	23.9944	0.511912	0.255956	+	0.966688i	$0.417610\pi$
$14$	18.3104	0.349547
$15$	0	0
$16$	−53.3978	−0.834340
$17$	−79.6971	−1.13702	−0.568511	−	0.822675i	$-0.692480\pi$
$17$	−79.6971	−1.13702	−0.568511	+	0.822675i	$0.692480\pi$
$18$	0	0
$19$	−50.0463	−0.604284	−0.302142	−	0.953263i	$-0.597702\pi$
$19$	−50.0463	−0.604284	−0.302142	+	0.953263i	$0.597702\pi$
$20$	−15.6793	−0.175300
$21$	0	0
$22$	62.9595	0.610137
$23$	151.608	1.37445	0.687226	−	0.726444i	$-0.258828\pi$
$23$	151.608	1.37445	0.687226	+	0.726444i	$0.258828\pi$
$24$	0	0
$25$	58.4143	0.467314
$26$	−62.7639	−0.473424
$27$	0	0
$28$	8.10415	0.0546978
$29$	−256.124	−1.64004	−0.820018	−	0.572338i	$-0.806036\pi$
$29$	−256.124	−1.64004	−0.820018	+	0.572338i	$0.806036\pi$
$30$	0	0
$31$	−2.72508	−0.0157883	−0.00789417	−	0.999969i	$-0.502513\pi$
$31$	−2.72508	−0.0157883	−0.00789417	+	0.999969i	$0.502513\pi$
$32$	−51.9600	−0.287041
$33$	0	0
$34$	208.469	1.05154
$35$	−94.8014	−0.457839
$36$	0	0
$37$	319.617	1.42013	0.710065	−	0.704137i	$-0.248666\pi$
$37$	319.617	1.42013	0.710065	+	0.704137i	$0.248666\pi$
$38$	130.910	0.558851
$39$	0	0
$40$	324.418	1.28237
$41$	−164.778	−0.627660	−0.313830	−	0.949479i	$-0.601612\pi$
$41$	−164.778	−0.627660	−0.313830	+	0.949479i	$0.601612\pi$
$42$	0	0
$43$	422.769	1.49934	0.749670	−	0.661811i	$-0.230212\pi$
$43$	422.769	1.49934	0.749670	+	0.661811i	$0.230212\pi$
$44$	27.8657	0.0954754
$45$	0	0
$46$	−396.571	−1.27111
$47$	−100.077	−0.310590	−0.155295	−	0.987868i	$-0.549633\pi$
$47$	−100.077	−0.310590	−0.155295	+	0.987868i	$0.549633\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−152.798	−0.432179
$51$	0	0
$52$	−27.7792	−0.0740823
$53$	−194.981	−0.505333	−0.252667	−	0.967553i	$-0.581308\pi$
$53$	−194.981	−0.505333	−0.252667	+	0.967553i	$0.581308\pi$
$54$	0	0
$55$	−325.970	−0.799160
$56$	−167.682	−0.400133
$57$	0	0
$58$	669.962	1.51673
$59$	−576.515	−1.27213	−0.636067	−	0.771634i	$-0.719440\pi$
$59$	−576.515	−1.27213	−0.636067	+	0.771634i	$0.719440\pi$
$60$	0	0
$61$	−43.0383	−0.0903360	−0.0451680	−	0.998979i	$-0.514382\pi$
$61$	−43.0383	−0.0903360	−0.0451680	+	0.998979i	$0.514382\pi$
$62$	7.12819	0.0146013
$63$	0	0
$64$	563.098	1.09980
$65$	324.958	0.620093
$66$	0	0
$67$	−1016.51	−1.85354	−0.926768	−	0.375635i	$-0.877425\pi$
$67$	−1016.51	−1.85354	−0.926768	+	0.375635i	$0.877425\pi$
$68$	92.2681	0.164546
$69$	0	0
$70$	247.979	0.423416
$71$	509.305	0.851315	0.425657	−	0.904884i	$-0.360043\pi$
$71$	509.305	0.851315	0.425657	+	0.904884i	$0.360043\pi$
$72$	0	0
$73$	1039.67	1.66690	0.833452	−	0.552592i	$-0.186361\pi$
$73$	1039.67	1.66690	0.833452	+	0.552592i	$0.186361\pi$
$74$	−836.046	−1.31336
$75$	0	0
$76$	57.9403	0.0874502
$77$	168.484	0.249358
$78$	0	0
$79$	−894.141	−1.27340	−0.636701	−	0.771111i	$-0.719701\pi$
$79$	−894.141	−1.27340	−0.636701	+	0.771111i	$0.719701\pi$
$80$	−723.169	−1.01066
$81$	0	0
$82$	431.023	0.580470
$83$	−14.4611	−0.0191243	−0.00956214	−	0.999954i	$-0.503044\pi$
$83$	−14.4611	−0.0191243	−0.00956214	+	0.999954i	$0.503044\pi$
$84$	0	0
$85$	−1079.34	−1.37731
$86$	−1105.87	−1.38661
$87$	0	0
$88$	−576.567	−0.698434
$89$	−1532.12	−1.82477	−0.912384	−	0.409336i	$-0.865760\pi$
$89$	−1532.12	−1.82477	−0.912384	+	0.409336i	$0.865760\pi$
$90$	0	0
$91$	−167.961	−0.193484
$92$	−175.522	−0.198906
$93$	0	0
$94$	261.779	0.287239
$95$	−677.779	−0.731986
$96$	0	0
$97$	−1775.15	−1.85813	−0.929067	−	0.369910i	$-0.879388\pi$
$97$	−1775.15	−1.85813	−0.929067	+	0.369910i	$0.879388\pi$
$98$	−128.173	−0.132116
$99$	0	0
$100$	−67.6283	−0.0676283
$101$	−338.697	−0.333680	−0.166840	−	0.985984i	$-0.553356\pi$
$101$	−338.697	−0.333680	−0.166840	+	0.985984i	$0.553356\pi$
$102$	0	0
$103$	−450.471	−0.430934	−0.215467	−	0.976511i	$-0.569127\pi$
$103$	−450.471	−0.430934	−0.215467	+	0.976511i	$0.569127\pi$
$104$	574.775	0.541936
$105$	0	0
$106$	510.025	0.467340
$107$	581.208	0.525117	0.262558	−	0.964916i	$-0.415434\pi$
$107$	581.208	0.525117	0.262558	+	0.964916i	$0.415434\pi$
$108$	0	0
$109$	−1381.38	−1.21388	−0.606938	−	0.794749i	$-0.707602\pi$
$109$	−1381.38	−1.21388	−0.606938	+	0.794749i	$0.707602\pi$
$110$	852.664	0.739076
$111$	0	0
$112$	373.784	0.315351
$113$	−1499.76	−1.24854	−0.624271	−	0.781208i	$-0.714604\pi$
$113$	−1499.76	−1.24854	−0.624271	+	0.781208i	$0.714604\pi$
$114$	0	0
$115$	2053.23	1.66491
$116$	296.524	0.237341
$117$	0	0
$118$	1508.03	1.17649
$119$	557.880	0.429754
$120$	0	0
$121$	−751.674	−0.564744
$122$	112.578	0.0835441
$123$	0	0
$124$	3.15492	0.00228484
$125$	−901.774	−0.645257
$126$	0	0
$127$	−908.229	−0.634585	−0.317292	−	0.948328i	$-0.602774\pi$
$127$	−908.229	−0.634585	−0.317292	+	0.948328i	$0.602774\pi$
$128$	−1057.26	−0.730071
$129$	0	0
$130$	−850.015	−0.573471
$131$	619.864	0.413418	0.206709	−	0.978402i	$-0.433725\pi$
$131$	619.864	0.413418	0.206709	+	0.978402i	$0.433725\pi$
$132$	0	0
$133$	350.324	0.228398
$134$	2658.97	1.71418
$135$	0	0
$136$	−1909.11	−1.20371
$137$	762.575	0.475556	0.237778	−	0.971319i	$-0.423581\pi$
$137$	762.575	0.475556	0.237778	+	0.971319i	$0.423581\pi$
$138$	0	0
$139$	1392.14	0.849492	0.424746	−	0.905313i	$-0.360363\pi$
$139$	1392.14	0.849492	0.424746	+	0.905313i	$0.360363\pi$
$140$	109.755	0.0662570
$141$	0	0
$142$	−1332.23	−0.787309
$143$	−577.526	−0.337728
$144$	0	0
$145$	−3468.70	−1.98662
$146$	−2719.54	−1.54158
$147$	0	0
$148$	−370.032	−0.205517
$149$	−1042.25	−0.573049	−0.286525	−	0.958073i	$-0.592500\pi$
$149$	−1042.25	−0.573049	−0.286525	+	0.958073i	$0.592500\pi$
$150$	0	0
$151$	99.8277	0.0538004	0.0269002	−	0.999638i	$-0.491436\pi$
$151$	99.8277	0.0538004	0.0269002	+	0.999638i	$0.491436\pi$
$152$	−1198.84	−0.639727
$153$	0	0
$154$	−440.717	−0.230610
$155$	−36.9059	−0.0191249
$156$	0	0
$157$	493.442	0.250834	0.125417	−	0.992104i	$-0.459973\pi$
$157$	493.442	0.250834	0.125417	+	0.992104i	$0.459973\pi$
$158$	2338.87	1.17766
$159$	0	0
$160$	−703.697	−0.347701
$161$	−1061.25	−0.519494
$162$	0	0
$163$	2174.81	1.04506	0.522529	−	0.852622i	$-0.324989\pi$
$163$	2174.81	1.04506	0.522529	+	0.852622i	$0.324989\pi$
$164$	190.770	0.0908330
$165$	0	0
$166$	37.8270	0.0176864
$167$	−2300.21	−1.06584	−0.532921	−	0.846165i	$-0.678906\pi$
$167$	−2300.21	−1.06584	−0.532921	+	0.846165i	$0.678906\pi$
$168$	0	0
$169$	−1621.27	−0.737946
$170$	2823.31	1.27375
$171$	0	0
$172$	−489.454	−0.216980
$173$	312.545	0.137355	0.0686773	−	0.997639i	$-0.478122\pi$
$173$	312.545	0.137355	0.0686773	+	0.997639i	$0.478122\pi$
$174$	0	0
$175$	−408.900	−0.176628
$176$	1285.24	0.550447
$177$	0	0
$178$	4007.67	1.68757
$179$	−4345.85	−1.81466	−0.907329	−	0.420421i	$-0.861882\pi$
$179$	−4345.85	−1.81466	−0.907329	+	0.420421i	$0.861882\pi$
$180$	0	0
$181$	−110.995	−0.0455812	−0.0227906	−	0.999740i	$-0.507255\pi$
$181$	−110.995	−0.0455812	−0.0227906	+	0.999740i	$0.507255\pi$
$182$	439.347	0.178937
$183$	0	0
$184$	3631.69	1.45507
$185$	4328.60	1.72024
$186$	0	0
$187$	1918.24	0.750139
$188$	115.863	0.0449477
$189$	0	0
$190$	1772.92	0.676952
$191$	−2205.88	−0.835665	−0.417833	−	0.908524i	$-0.637210\pi$
$191$	−2205.88	−0.835665	−0.417833	+	0.908524i	$0.637210\pi$
$192$	0	0
$193$	−438.517	−0.163550	−0.0817750	−	0.996651i	$-0.526059\pi$
$193$	−438.517	−0.163550	−0.0817750	+	0.996651i	$0.526059\pi$
$194$	4643.39	1.71843
$195$	0	0
$196$	−56.7290	−0.0206738
$197$	−2536.01	−0.917173	−0.458587	−	0.888650i	$-0.651644\pi$
$197$	−2536.01	−0.917173	−0.458587	+	0.888650i	$0.651644\pi$
$198$	0	0
$199$	2797.01	0.996356	0.498178	−	0.867075i	$-0.334003\pi$
$199$	2797.01	0.996356	0.498178	+	0.867075i	$0.334003\pi$
$200$	1399.29	0.494723
$201$	0	0
$202$	885.955	0.308592
$203$	1792.87	0.619875
$204$	0	0
$205$	−2231.60	−0.760302
$206$	1178.33	0.398534
$207$	0	0
$208$	−1281.25	−0.427109
$209$	1204.57	0.398670
$210$	0	0
$211$	1888.16	0.616048	0.308024	−	0.951379i	$-0.400332\pi$
$211$	1888.16	0.616048	0.308024	+	0.951379i	$0.400332\pi$
$212$	225.736	0.0731303
$213$	0	0
$214$	−1520.31	−0.485636
$215$	5725.58	1.81619
$216$	0	0
$217$	19.0756	0.00596743
$218$	3613.38	1.12261
$219$	0	0
$220$	377.387	0.115652
$221$	−1912.28	−0.582055
$222$	0	0
$223$	1349.75	0.405318	0.202659	−	0.979249i	$-0.435042\pi$
$223$	1349.75	0.405318	0.202659	+	0.979249i	$0.435042\pi$
$224$	363.720	0.108491
$225$	0	0
$226$	3923.02	1.15467
$227$	−2485.07	−0.726607	−0.363303	−	0.931671i	$-0.618351\pi$
$227$	−2485.07	−0.726607	−0.363303	+	0.931671i	$0.618351\pi$
$228$	0	0
$229$	−1862.02	−0.537317	−0.268659	−	0.963235i	$-0.586580\pi$
$229$	−1862.02	−0.537317	−0.268659	+	0.963235i	$0.586580\pi$
$230$	−5370.79	−1.53974
$231$	0	0
$232$	−6135.33	−1.73623
$233$	4094.94	1.15137	0.575684	−	0.817672i	$-0.304736\pi$
$233$	4094.94	1.15137	0.575684	+	0.817672i	$0.304736\pi$
$234$	0	0
$235$	−1355.35	−0.376227
$236$	667.452	0.184099
$237$	0	0
$238$	−1459.29	−0.397443
$239$	2122.72	0.574507	0.287254	−	0.957855i	$-0.407258\pi$
$239$	2122.72	0.574507	0.287254	+	0.957855i	$0.407258\pi$
$240$	0	0
$241$	5289.97	1.41393	0.706964	−	0.707249i	$-0.250064\pi$
$241$	5289.97	1.41393	0.706964	+	0.707249i	$0.250064\pi$
$242$	1966.21	0.522284
$243$	0	0
$244$	49.8270	0.0130731
$245$	663.610	0.173047
$246$	0	0
$247$	−1200.83	−0.309340
$248$	−65.2781	−0.0167144
$249$	0	0
$250$	2358.84	0.596743
$251$	4632.56	1.16496	0.582480	−	0.812845i	$-0.302082\pi$
$251$	4632.56	1.16496	0.582480	+	0.812845i	$0.302082\pi$
$252$	0	0
$253$	−3649.07	−0.906780
$254$	2375.72	0.586874
$255$	0	0
$256$	−1739.24	−0.424620
$257$	−3156.47	−0.766129	−0.383065	−	0.923722i	$-0.625131\pi$
$257$	−3156.47	−0.766129	−0.383065	+	0.923722i	$0.625131\pi$
$258$	0	0
$259$	−2237.32	−0.536758
$260$	−376.215	−0.0897379
$261$	0	0
$262$	−1621.42	−0.382335
$263$	1788.57	0.419346	0.209673	−	0.977772i	$-0.432760\pi$
$263$	1788.57	0.419346	0.209673	+	0.977772i	$0.432760\pi$
$264$	0	0
$265$	−2640.64	−0.612124
$266$	−916.368	−0.211226
$267$	0	0
$268$	1176.85	0.268238
$269$	−5780.38	−1.31017	−0.655085	−	0.755555i	$-0.727367\pi$
$269$	−5780.38	−1.31017	−0.655085	+	0.755555i	$0.727367\pi$
$270$	0	0
$271$	5745.97	1.28798	0.643991	−	0.765033i	$-0.277277\pi$
$271$	5745.97	1.28798	0.643991	+	0.765033i	$0.277277\pi$
$272$	4255.65	0.948664
$273$	0	0
$274$	−1994.72	−0.439802
$275$	−1405.98	−0.308306
$276$	0	0
$277$	3493.63	0.757804	0.378902	−	0.925437i	$-0.376302\pi$
$277$	3493.63	0.757804	0.378902	+	0.925437i	$0.376302\pi$
$278$	−3641.51	−0.785623
$279$	0	0
$280$	−2270.92	−0.484692
$281$	−1854.53	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$281$	−1854.53	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$282$	0	0
$283$	5420.32	1.13853	0.569266	−	0.822154i	$-0.307228\pi$
$283$	5420.32	1.13853	0.569266	+	0.822154i	$0.307228\pi$
$284$	−589.640	−0.123200
$285$	0	0
$286$	1510.68	0.312336
$287$	1153.45	0.237233
$288$	0	0
$289$	1438.63	0.292820
$290$	9073.33	1.83726
$291$	0	0
$292$	−1203.66	−0.241229
$293$	−1415.25	−0.282184	−0.141092	−	0.989996i	$-0.545061\pi$
$293$	−1415.25	−0.282184	−0.141092	+	0.989996i	$0.545061\pi$
$294$	0	0
$295$	−7807.78	−1.54097
$296$	7656.29	1.50342
$297$	0	0
$298$	2726.28	0.529964
$299$	3637.74	0.703598
$300$	0	0
$301$	−2959.38	−0.566698
$302$	−261.127	−0.0497555
$303$	0	0
$304$	2672.36	0.504179
$305$	−582.870	−0.109426
$306$	0	0
$307$	6596.15	1.22626	0.613131	−	0.789982i	$-0.289910\pi$
$307$	6596.15	1.22626	0.613131	+	0.789982i	$0.289910\pi$
$308$	−195.060	−0.0360863
$309$	0	0
$310$	96.5374	0.0176870
$311$	758.005	0.138207	0.0691037	−	0.997609i	$-0.477986\pi$
$311$	758.005	0.138207	0.0691037	+	0.997609i	$0.477986\pi$
$312$	0	0
$313$	7105.52	1.28316	0.641578	−	0.767058i	$-0.278280\pi$
$313$	7105.52	1.28316	0.641578	+	0.767058i	$0.278280\pi$
$314$	−1290.73	−0.231975
$315$	0	0
$316$	1035.18	0.184283
$317$	5259.38	0.931849	0.465924	−	0.884825i	$-0.345722\pi$
$317$	5259.38	0.931849	0.465924	+	0.884825i	$0.345722\pi$
$318$	0	0
$319$	6164.69	1.08200
$320$	7626.06	1.33222
$321$	0	0
$322$	2776.00	0.480436
$323$	3988.54	0.687085
$324$	0	0
$325$	1401.62	0.239224
$326$	−5688.81	−0.966485
$327$	0	0
$328$	−3947.19	−0.664473
$329$	700.540	0.117392
$330$	0	0
$331$	−4252.19	−0.706107	−0.353053	−	0.935603i	$-0.614857\pi$
$331$	−4252.19	−0.706107	−0.353053	+	0.935603i	$0.614857\pi$
$332$	16.7422	0.00276761
$333$	0	0
$334$	6016.82	0.985706
$335$	−13766.7	−2.24524
$336$	0	0
$337$	2592.16	0.419002	0.209501	−	0.977808i	$-0.432816\pi$
$337$	2592.16	0.419002	0.209501	+	0.977808i	$0.432816\pi$
$338$	4240.87	0.682464
$339$	0	0
$340$	1249.59	0.199320
$341$	65.5905	0.0104162
$342$	0	0
$343$	−343.000	−0.0539949
$344$	10127.2	1.58728
$345$	0	0
$346$	−817.546	−0.127028
$347$	491.979	0.0761118	0.0380559	−	0.999276i	$-0.487884\pi$
$347$	491.979	0.0761118	0.0380559	+	0.999276i	$0.487884\pi$
$348$	0	0
$349$	−11917.5	−1.82788	−0.913939	−	0.405852i	$-0.866975\pi$
$349$	−11917.5	−1.82788	−0.913939	+	0.405852i	$0.866975\pi$
$350$	1069.59	0.163348
$351$	0	0
$352$	1250.63	0.189372
$353$	−8079.96	−1.21828	−0.609140	−	0.793063i	$-0.708485\pi$
$353$	−8079.96	−1.21828	−0.609140	+	0.793063i	$0.708485\pi$
$354$	0	0
$355$	6897.54	1.03122
$356$	1773.79	0.264075
$357$	0	0
$358$	11367.7	1.67822
$359$	6675.40	0.981376	0.490688	−	0.871335i	$-0.336745\pi$
$359$	6675.40	0.981376	0.490688	+	0.871335i	$0.336745\pi$
$360$	0	0
$361$	−4354.37	−0.634840
$362$	290.338	0.0421542
$363$	0	0
$364$	194.454	0.0280005
$365$	14080.3	2.01917
$366$	0	0
$367$	−11776.9	−1.67506	−0.837532	−	0.546388i	$-0.816002\pi$
$367$	−11776.9	−1.67506	−0.837532	+	0.546388i	$0.816002\pi$
$368$	−8095.51	−1.14676
$369$	0	0
$370$	−11322.6	−1.59091
$371$	1364.87	0.190998
$372$	0	0
$373$	1483.28	0.205901	0.102950	−	0.994686i	$-0.467172\pi$
$373$	1483.28	0.205901	0.102950	+	0.994686i	$0.467172\pi$
$374$	−5017.69	−0.693740
$375$	0	0
$376$	−2397.30	−0.328807
$377$	−6145.54	−0.839553
$378$	0	0
$379$	−13400.9	−1.81625	−0.908124	−	0.418701i	$-0.862486\pi$
$379$	−13400.9	−1.81625	−0.908124	+	0.418701i	$0.862486\pi$
$380$	784.689	0.105931
$381$	0	0
$382$	5770.09	0.772836
$383$	−11767.1	−1.56990	−0.784952	−	0.619557i	$-0.787312\pi$
$383$	−11767.1	−1.56990	−0.784952	+	0.619557i	$0.787312\pi$
$384$	0	0
$385$	2281.79	0.302054
$386$	1147.06	0.151254
$387$	0	0
$388$	2055.15	0.268903
$389$	−7801.94	−1.01690	−0.508450	−	0.861091i	$-0.669781\pi$
$389$	−7801.94	−1.01690	−0.508450	+	0.861091i	$0.669781\pi$
$390$	0	0
$391$	−12082.7	−1.56278
$392$	1173.77	0.151236
$393$	0	0
$394$	6633.62	0.848216
$395$	−12109.4	−1.54251
$396$	0	0
$397$	−6197.81	−0.783525	−0.391762	−	0.920066i	$-0.628134\pi$
$397$	−6197.81	−0.783525	−0.391762	+	0.920066i	$0.628134\pi$
$398$	−7316.35	−0.921446
$399$	0	0
$400$	−3119.19	−0.389899
$401$	−1001.44	−0.124713	−0.0623563	−	0.998054i	$-0.519862\pi$
$401$	−1001.44	−0.124713	−0.0623563	+	0.998054i	$0.519862\pi$
$402$	0	0
$403$	−65.3867	−0.00808224
$404$	392.122	0.0482891
$405$	0	0
$406$	−4689.73	−0.573270
$407$	−7692.93	−0.936915
$408$	0	0
$409$	10109.2	1.22217	0.611085	−	0.791565i	$-0.290733\pi$
$409$	10109.2	1.22217	0.611085	+	0.791565i	$0.290733\pi$
$410$	5837.36	0.703139
$411$	0	0
$412$	521.526	0.0623634
$413$	4035.61	0.480821
$414$	0	0
$415$	−195.848	−0.0231658
$416$	−1246.75	−0.146940
$417$	0	0
$418$	−3150.89	−0.368696
$419$	−356.504	−0.0415665	−0.0207832	−	0.999784i	$-0.506616\pi$
$419$	−356.504	−0.0415665	−0.0207832	+	0.999784i	$0.506616\pi$
$420$	0	0
$421$	−2871.74	−0.332446	−0.166223	−	0.986088i	$-0.553157\pi$
$421$	−2871.74	−0.332446	−0.166223	+	0.986088i	$0.553157\pi$
$422$	−4938.99	−0.569731
$423$	0	0
$424$	−4670.68	−0.534972
$425$	−4655.45	−0.531347
$426$	0	0
$427$	301.268	0.0341438
$428$	−672.885	−0.0759932
$429$	0	0
$430$	−14976.8	−1.67964
$431$	3224.02	0.360314	0.180157	−	0.983638i	$-0.442339\pi$
$431$	3224.02	0.360314	0.180157	+	0.983638i	$0.442339\pi$
$432$	0	0
$433$	−14285.5	−1.58549	−0.792744	−	0.609555i	$-0.791348\pi$
$433$	−14285.5	−1.58549	−0.792744	+	0.609555i	$0.791348\pi$
$434$	−49.8973	−0.00551877
$435$	0	0
$436$	1599.28	0.175668
$437$	−7587.40	−0.830560
$438$	0	0
$439$	3515.13	0.382160	0.191080	−	0.981574i	$-0.438801\pi$
$439$	3515.13	0.382160	0.191080	+	0.981574i	$0.438801\pi$
$440$	−7808.47	−0.846032
$441$	0	0
$442$	5002.10	0.538294
$443$	7982.82	0.856152	0.428076	−	0.903743i	$-0.359192\pi$
$443$	7982.82	0.856152	0.428076	+	0.903743i	$0.359192\pi$
$444$	0	0
$445$	−20749.6	−2.21039
$446$	−3530.63	−0.374844
$447$	0	0
$448$	−3941.68	−0.415685
$449$	−3502.50	−0.368137	−0.184068	−	0.982913i	$-0.558927\pi$
$449$	−3502.50	−0.368137	−0.184068	+	0.982913i	$0.558927\pi$
$450$	0	0
$451$	3966.08	0.414092
$452$	1736.32	0.180685
$453$	0	0
$454$	6500.37	0.671977
$455$	−2274.70	−0.234373
$456$	0	0
$457$	13062.5	1.33706	0.668531	−	0.743684i	$-0.266923\pi$
$457$	13062.5	1.33706	0.668531	+	0.743684i	$0.266923\pi$
$458$	4870.62	0.496919
$459$	0	0
$460$	−2377.10	−0.240941
$461$	−6714.42	−0.678355	−0.339178	−	0.940722i	$-0.610149\pi$
$461$	−6714.42	−0.678355	−0.339178	+	0.940722i	$0.610149\pi$
$462$	0	0
$463$	−6097.68	−0.612059	−0.306029	−	0.952022i	$-0.599001\pi$
$463$	−6097.68	−0.612059	−0.306029	+	0.952022i	$0.599001\pi$
$464$	13676.4	1.36835
$465$	0	0
$466$	−10711.4	−1.06480
$467$	11929.4	1.18207	0.591033	−	0.806647i	$-0.298720\pi$
$467$	11929.4	1.18207	0.591033	+	0.806647i	$0.298720\pi$
$468$	0	0
$469$	7115.59	0.700571
$470$	3545.29	0.347940
$471$	0	0
$472$	−13810.2	−1.34675
$473$	−10175.7	−0.989174
$474$	0	0
$475$	−2923.42	−0.282391
$476$	−645.877	−0.0621927
$477$	0	0
$478$	−5552.55	−0.531313
$479$	7490.01	0.714462	0.357231	−	0.934016i	$-0.383721\pi$
$479$	7490.01	0.714462	0.357231	+	0.934016i	$0.383721\pi$
$480$	0	0
$481$	7669.03	0.726981
$482$	−13837.4	−1.30762
$483$	0	0
$484$	870.240	0.0817280
$485$	−24040.9	−2.25081
$486$	0	0
$487$	14200.8	1.32135	0.660675	−	0.750672i	$-0.270270\pi$
$487$	14200.8	1.32135	0.660675	+	0.750672i	$0.270270\pi$
$488$	−1030.96	−0.0956343
$489$	0	0
$490$	−1735.85	−0.160036
$491$	−10791.9	−0.991915	−0.495957	−	0.868347i	$-0.665183\pi$
$491$	−10791.9	−0.991915	−0.495957	+	0.868347i	$0.665183\pi$
$492$	0	0
$493$	20412.3	1.86476
$494$	3141.10	0.286083
$495$	0	0
$496$	145.513	0.0131729
$497$	−3565.13	−0.321767
$498$	0	0
$499$	6340.17	0.568788	0.284394	−	0.958708i	$-0.408208\pi$
$499$	6340.17	0.568788	0.284394	+	0.958708i	$0.408208\pi$
$500$	1044.02	0.0933796
$501$	0	0
$502$	−12117.7	−1.07737
$503$	−18901.4	−1.67549	−0.837747	−	0.546059i	$-0.816127\pi$
$503$	−18901.4	−1.67549	−0.837747	+	0.546059i	$0.816127\pi$
$504$	0	0
$505$	−4587.00	−0.404195
$506$	9545.15	0.838604
$507$	0	0
$508$	1051.49	0.0918351
$509$	13676.7	1.19098	0.595491	−	0.803362i	$-0.296958\pi$
$509$	13676.7	1.19098	0.595491	+	0.803362i	$0.296958\pi$
$510$	0	0
$511$	−7277.68	−0.630030
$512$	13007.5	1.12277
$513$	0	0
$514$	8256.61	0.708528
$515$	−6100.75	−0.522002
$516$	0	0
$517$	2408.77	0.204909
$518$	5852.32	0.496402
$519$	0	0
$520$	7784.21	0.656462
$521$	10323.9	0.868139	0.434069	−	0.900879i	$-0.357077\pi$
$521$	10323.9	0.868139	0.434069	+	0.900879i	$0.357077\pi$
$522$	0	0
$523$	9294.18	0.777067	0.388534	−	0.921435i	$-0.372982\pi$
$523$	9294.18	0.777067	0.388534	+	0.921435i	$0.372982\pi$
$524$	−717.638	−0.0598285
$525$	0	0
$526$	−4678.49	−0.387817
$527$	217.181	0.0179517
$528$	0	0
$529$	10817.9	0.889118
$530$	6907.30	0.566102
$531$	0	0
$532$	−405.582	−0.0330531
$533$	−3953.76	−0.321307
$534$	0	0
$535$	7871.33	0.636088
$536$	−24350.1	−1.96225
$537$	0	0
$538$	15120.2	1.21167
$539$	−1179.39	−0.0942485
$540$	0	0
$541$	4432.55	0.352256	0.176128	−	0.984367i	$-0.443643\pi$
$541$	4432.55	0.352256	0.176128	+	0.984367i	$0.443643\pi$
$542$	−15030.2	−1.19114
$543$	0	0
$544$	4141.06	0.326372
$545$	−18708.1	−1.47040
$546$	0	0
$547$	4328.37	0.338332	0.169166	−	0.985588i	$-0.445893\pi$
$547$	4328.37	0.338332	0.169166	+	0.985588i	$0.445893\pi$
$548$	−882.860	−0.0688210
$549$	0	0
$550$	3677.74	0.285126
$551$	12818.0	0.991048
$552$	0	0
$553$	6258.99	0.481301
$554$	−9138.54	−0.700829
$555$	0	0
$556$	−1611.72	−0.122936
$557$	−6102.88	−0.464250	−0.232125	−	0.972686i	$-0.574568\pi$
$557$	−6102.88	−0.464250	−0.232125	+	0.972686i	$0.574568\pi$
$558$	0	0
$559$	10144.1	0.767530
$560$	5062.18	0.381993
$561$	0	0
$562$	4851.02	0.364107
$563$	−4857.70	−0.363637	−0.181819	−	0.983332i	$-0.558198\pi$
$563$	−4857.70	−0.363637	−0.181819	+	0.983332i	$0.558198\pi$
$564$	0	0
$565$	−20311.3	−1.51239
$566$	−14178.3	−1.05293
$567$	0	0
$568$	12200.2	0.901246
$569$	−3002.61	−0.221223	−0.110611	−	0.993864i	$-0.535281\pi$
$569$	−3002.61	−0.221223	−0.110611	+	0.993864i	$0.535281\pi$
$570$	0	0
$571$	19365.7	1.41932	0.709658	−	0.704546i	$-0.248850\pi$
$571$	19365.7	1.41932	0.709658	+	0.704546i	$0.248850\pi$
$572$	668.622	0.0488750
$573$	0	0
$574$	−3017.16	−0.219397
$575$	8856.06	0.642301
$576$	0	0
$577$	9688.50	0.699025	0.349512	−	0.936932i	$-0.386347\pi$
$577$	9688.50	0.699025	0.349512	+	0.936932i	$0.386347\pi$
$578$	−3763.12	−0.270805
$579$	0	0
$580$	4015.84	0.287497
$581$	101.228	0.00722830
$582$	0	0
$583$	4693.03	0.333388
$584$	24904.8	1.76467
$585$	0	0
$586$	3701.98	0.260968
$587$	1744.63	0.122672	0.0613360	−	0.998117i	$-0.480464\pi$
$587$	1744.63	0.122672	0.0613360	+	0.998117i	$0.480464\pi$
$588$	0	0
$589$	136.380	0.00954065
$590$	20423.4	1.42511
$591$	0	0
$592$	−17066.9	−1.18487
$593$	−20857.6	−1.44438	−0.722192	−	0.691692i	$-0.756865\pi$
$593$	−20857.6	−1.44438	−0.722192	+	0.691692i	$0.756865\pi$
$594$	0	0
$595$	7555.39	0.520573
$596$	1206.65	0.0829299
$597$	0	0
$598$	−9515.49	−0.650698
$599$	−7196.70	−0.490900	−0.245450	−	0.969409i	$-0.578936\pi$
$599$	−7196.70	−0.490900	−0.245450	+	0.969409i	$0.578936\pi$
$600$	0	0
$601$	11298.5	0.766848	0.383424	−	0.923572i	$-0.374745\pi$
$601$	11298.5	0.766848	0.383424	+	0.923572i	$0.374745\pi$
$602$	7741.07	0.524091
$603$	0	0
$604$	−115.574	−0.00778583
$605$	−10180.0	−0.684090
$606$	0	0
$607$	−18408.6	−1.23094	−0.615472	−	0.788159i	$-0.711035\pi$
$607$	−18408.6	−1.23094	−0.615472	+	0.788159i	$0.711035\pi$
$608$	2600.40	0.173454
$609$	0	0
$610$	1524.66	0.101199
$611$	−2401.29	−0.158995
$612$	0	0
$613$	−8745.01	−0.576195	−0.288098	−	0.957601i	$-0.593023\pi$
$613$	−8745.01	−0.576195	−0.288098	+	0.957601i	$0.593023\pi$
$614$	−17254.0	−1.13406
$615$	0	0
$616$	4035.97	0.263983
$617$	29239.6	1.90785	0.953923	−	0.300051i	$-0.0970036\pi$
$617$	29239.6	1.90785	0.953923	+	0.300051i	$0.0970036\pi$
$618$	0	0
$619$	2579.65	0.167504	0.0837518	−	0.996487i	$-0.473310\pi$
$619$	2579.65	0.167504	0.0837518	+	0.996487i	$0.473310\pi$
$620$	42.7273	0.00276769
$621$	0	0
$622$	−1982.77	−0.127816
$623$	10724.8	0.689697
$624$	0	0
$625$	−19514.6	−1.24893
$626$	−18586.4	−1.18668
$627$	0	0
$628$	−571.275	−0.0362999
$629$	−25472.6	−1.61472
$630$	0	0
$631$	15166.5	0.956847	0.478424	−	0.878129i	$-0.341208\pi$
$631$	15166.5	0.956847	0.478424	+	0.878129i	$0.341208\pi$
$632$	−21418.8	−1.34809
$633$	0	0
$634$	−13757.3	−0.861788
$635$	−12300.2	−0.768690
$636$	0	0
$637$	1175.73	0.0731303
$638$	−16125.4	−1.00065
$639$	0	0
$640$	−14318.5	−0.884355
$641$	11568.1	0.712811	0.356406	−	0.934331i	$-0.384002\pi$
$641$	11568.1	0.712811	0.356406	+	0.934331i	$0.384002\pi$
$642$	0	0
$643$	−52.7280	−0.00323389	−0.00161694	−	0.999999i	$-0.500515\pi$
$643$	−52.7280	−0.00323389	−0.00161694	+	0.999999i	$0.500515\pi$
$644$	1228.65	0.0751795
$645$	0	0
$646$	−10433.1	−0.635427
$647$	13568.7	0.824484	0.412242	−	0.911074i	$-0.364746\pi$
$647$	13568.7	0.824484	0.412242	+	0.911074i	$0.364746\pi$
$648$	0	0
$649$	13876.3	0.839277
$650$	−3666.31	−0.221238
$651$	0	0
$652$	−2517.86	−0.151238
$653$	15414.3	0.923749	0.461875	−	0.886945i	$-0.347177\pi$
$653$	15414.3	0.923749	0.461875	+	0.886945i	$0.347177\pi$
$654$	0	0
$655$	8394.85	0.500784
$656$	8798.80	0.523682
$657$	0	0
$658$	−1832.45	−0.108566
$659$	18835.1	1.11337	0.556684	−	0.830724i	$-0.312073\pi$
$659$	18835.1	1.11337	0.556684	+	0.830724i	$0.312073\pi$
$660$	0	0
$661$	9018.79	0.530696	0.265348	−	0.964153i	$-0.414513\pi$
$661$	9018.79	0.530696	0.265348	+	0.964153i	$0.414513\pi$
$662$	11122.8	0.653018
$663$	0	0
$664$	−346.410	−0.0202460
$665$	4744.46	0.276665
$666$	0	0
$667$	−38830.4	−2.25415
$668$	2663.03	0.154245
$669$	0	0
$670$	36010.5	2.07643
$671$	1035.90	0.0595982
$672$	0	0
$673$	23039.5	1.31962	0.659812	−	0.751431i	$-0.270636\pi$
$673$	23039.5	1.31962	0.659812	+	0.751431i	$0.270636\pi$
$674$	−6780.49	−0.387500
$675$	0	0
$676$	1877.00	0.106793
$677$	29154.2	1.65508	0.827539	−	0.561408i	$-0.189740\pi$
$677$	29154.2	1.65508	0.827539	+	0.561408i	$0.189740\pi$
$678$	0	0
$679$	12426.0	0.702309
$680$	−25855.2	−1.45809
$681$	0	0
$682$	−171.570	−0.00963305
$683$	8818.55	0.494045	0.247022	−	0.969010i	$-0.420548\pi$
$683$	8818.55	0.494045	0.247022	+	0.969010i	$0.420548\pi$
$684$	0	0
$685$	10327.6	0.576054
$686$	897.210	0.0499353
$687$	0	0
$688$	−22574.9	−1.25096
$689$	−4678.45	−0.258686
$690$	0	0
$691$	−9443.00	−0.519868	−0.259934	−	0.965626i	$-0.583701\pi$
$691$	−9443.00	−0.519868	−0.259934	+	0.965626i	$0.583701\pi$
$692$	−361.844	−0.0198775
$693$	0	0
$694$	−1286.90	−0.0703894
$695$	18853.8	1.02901
$696$	0	0
$697$	13132.4	0.713664
$698$	31173.5	1.69045
$699$	0	0
$700$	473.398	0.0255611
$701$	11255.0	0.606411	0.303205	−	0.952925i	$-0.401943\pi$
$701$	11255.0	0.606411	0.303205	+	0.952925i	$0.401943\pi$
$702$	0	0
$703$	−15995.7	−0.858162
$704$	−13553.3	−0.725581
$705$	0	0
$706$	21135.3	1.12668
$707$	2370.88	0.126119
$708$	0	0
$709$	22485.0	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$709$	22485.0	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$710$	−18042.4	−0.953689
$711$	0	0
$712$	−36701.2	−1.93179
$713$	−413.143	−0.0217003
$714$	0	0
$715$	−7821.47	−0.409100
$716$	5031.34	0.262612
$717$	0	0
$718$	−17461.3	−0.907591
$719$	27322.6	1.41719	0.708595	−	0.705615i	$-0.249329\pi$
$719$	27322.6	1.41719	0.708595	+	0.705615i	$0.249329\pi$
$720$	0	0
$721$	3153.30	0.162878
$722$	11390.0	0.587110
$723$	0	0
$724$	128.503	0.00659637
$725$	−14961.3	−0.766412
$726$	0	0
$727$	−777.076	−0.0396426	−0.0198213	−	0.999804i	$-0.506310\pi$
$727$	−777.076	−0.0396426	−0.0198213	+	0.999804i	$0.506310\pi$
$728$	−4023.43	−0.204833
$729$	0	0
$730$	−36830.8	−1.86736
$731$	−33693.5	−1.70478
$732$	0	0
$733$	−25365.2	−1.27815	−0.639075	−	0.769144i	$-0.720683\pi$
$733$	−25365.2	−1.27815	−0.639075	+	0.769144i	$0.720683\pi$
$734$	30805.7	1.54913
$735$	0	0
$736$	−7877.54	−0.394524
$737$	24466.7	1.22285
$738$	0	0
$739$	851.804	0.0424007	0.0212004	−	0.999775i	$-0.493251\pi$
$739$	851.804	0.0424007	0.0212004	+	0.999775i	$0.493251\pi$
$740$	−5011.37	−0.248948
$741$	0	0
$742$	−3570.18	−0.176638
$743$	33820.0	1.66990	0.834950	−	0.550325i	$-0.185496\pi$
$743$	33820.0	1.66990	0.834950	+	0.550325i	$0.185496\pi$
$744$	0	0
$745$	−14115.2	−0.694150
$746$	−3879.91	−0.190420
$747$	0	0
$748$	−2220.82	−0.108558
$749$	−4068.46	−0.198475
$750$	0	0
$751$	−753.259	−0.0366003	−0.0183001	−	0.999833i	$-0.505825\pi$
$751$	−753.259	−0.0366003	−0.0183001	+	0.999833i	$0.505825\pi$
$752$	5343.89	0.259138
$753$	0	0
$754$	16075.3	0.776432
$755$	1351.97	0.0651699
$756$	0	0
$757$	−8507.90	−0.408487	−0.204244	−	0.978920i	$-0.565473\pi$
$757$	−8507.90	−0.408487	−0.204244	+	0.978920i	$0.565473\pi$
$758$	35053.7	1.67969
$759$	0	0
$760$	−16235.9	−0.774918
$761$	−3679.95	−0.175293	−0.0876466	−	0.996152i	$-0.527935\pi$
$761$	−3679.95	−0.175293	−0.0876466	+	0.996152i	$0.527935\pi$
$762$	0	0
$763$	9669.68	0.458802
$764$	2553.83	0.120935
$765$	0	0
$766$	30780.2	1.45187
$767$	−13833.1	−0.651220
$768$	0	0
$769$	21674.1	1.01637	0.508184	−	0.861248i	$-0.330317\pi$
$769$	21674.1	1.01637	0.508184	+	0.861248i	$0.330317\pi$
$770$	−5968.65	−0.279344
$771$	0	0
$772$	507.687	0.0236685
$773$	−16219.4	−0.754685	−0.377342	−	0.926074i	$-0.623162\pi$
$773$	−16219.4	−0.754685	−0.377342	+	0.926074i	$0.623162\pi$
$774$	0	0
$775$	−159.184	−0.00737812
$776$	−42522.9	−1.96712
$777$	0	0
$778$	20408.1	0.940445
$779$	8246.54	0.379285
$780$	0	0
$781$	−12258.6	−0.561646
$782$	31605.6	1.44529
$783$	0	0
$784$	−2616.49	−0.119191
$785$	6682.71	0.303842
$786$	0	0
$787$	−25962.3	−1.17593	−0.587965	−	0.808887i	$-0.700071\pi$
$787$	−25962.3	−1.17593	−0.587965	+	0.808887i	$0.700071\pi$
$788$	2936.03	0.132730
$789$	0	0
$790$	31675.4	1.42653
$791$	10498.3	0.471905
$792$	0	0
$793$	−1032.68	−0.0462440
$794$	16212.1	0.724615
$795$	0	0
$796$	−3238.20	−0.144190
$797$	−27754.5	−1.23352	−0.616759	−	0.787152i	$-0.711555\pi$
$797$	−27754.5	−1.23352	−0.616759	+	0.787152i	$0.711555\pi$
$798$	0	0
$799$	7975.85	0.353148
$800$	−3035.21	−0.134138
$801$	0	0
$802$	2619.55	0.115336
$803$	−25024.0	−1.09972
$804$	0	0
$805$	−14372.6	−0.629277
$806$	171.037	0.00747458
$807$	0	0
$808$	−8113.34	−0.353250
$809$	3816.68	0.165868	0.0829340	−	0.996555i	$-0.473571\pi$
$809$	3816.68	0.165868	0.0829340	+	0.996555i	$0.473571\pi$
$810$	0	0
$811$	20419.0	0.884103	0.442051	−	0.896990i	$-0.354251\pi$
$811$	20419.0	0.884103	0.442051	+	0.896990i	$0.354251\pi$
$812$	−2075.67	−0.0897064
$813$	0	0
$814$	20123.0	0.866473
$815$	29453.6	1.26591
$816$	0	0
$817$	−21158.0	−0.906028
$818$	−26443.4	−1.13028
$819$	0	0
$820$	2583.60	0.110029
$821$	21952.6	0.933194	0.466597	−	0.884470i	$-0.345480\pi$
$821$	21952.6	0.933194	0.466597	+	0.884470i	$0.345480\pi$
$822$	0	0
$823$	11750.4	0.497684	0.248842	−	0.968544i	$-0.419950\pi$
$823$	11750.4	0.497684	0.248842	+	0.968544i	$0.419950\pi$
$824$	−10790.8	−0.456209
$825$	0	0
$826$	−10556.2	−0.444671
$827$	7326.79	0.308074	0.154037	−	0.988065i	$-0.450772\pi$
$827$	7326.79	0.308074	0.154037	+	0.988065i	$0.450772\pi$
$828$	0	0
$829$	−13574.8	−0.568725	−0.284362	−	0.958717i	$-0.591782\pi$
$829$	−13574.8	−0.568725	−0.284362	+	0.958717i	$0.591782\pi$
$830$	512.293	0.0214241
$831$	0	0
$832$	13511.2	0.563001
$833$	−3905.16	−0.162432
$834$	0	0
$835$	−31151.8	−1.29108
$836$	−1394.58	−0.0576943
$837$	0	0
$838$	932.533	0.0384413
$839$	27841.3	1.14564	0.572818	−	0.819683i	$-0.305850\pi$
$839$	27841.3	1.14564	0.572818	+	0.819683i	$0.305850\pi$
$840$	0	0
$841$	41210.5	1.68972
$842$	7511.81	0.307451
$843$	0	0
$844$	−2185.99	−0.0891526
$845$	−21956.9	−0.893895
$846$	0	0
$847$	5261.72	0.213453
$848$	10411.5	0.421620
$849$	0	0
$850$	12177.6	0.491398
$851$	48456.5	1.95190
$852$	0	0
$853$	−21306.1	−0.855227	−0.427613	−	0.903962i	$-0.640646\pi$
$853$	−21306.1	−0.855227	−0.427613	+	0.903962i	$0.640646\pi$
$854$	−788.049	−0.0315767
$855$	0	0
$856$	13922.6	0.555916
$857$	2920.66	0.116415	0.0582077	−	0.998304i	$-0.481461\pi$
$857$	2920.66	0.116415	0.0582077	+	0.998304i	$0.481461\pi$
$858$	0	0
$859$	−27573.1	−1.09520	−0.547602	−	0.836739i	$-0.684459\pi$
$859$	−27573.1	−1.09520	−0.547602	+	0.836739i	$0.684459\pi$
$860$	−6628.71	−0.262834
$861$	0	0
$862$	−8433.29	−0.333224
$863$	−2361.13	−0.0931329	−0.0465664	−	0.998915i	$-0.514828\pi$
$863$	−2361.13	−0.0931329	−0.0465664	+	0.998915i	$0.514828\pi$
$864$	0	0
$865$	4232.81	0.166381
$866$	37367.5	1.46628
$867$	0	0
$868$	−22.0844	−0.000863588 0
$869$	21521.3	0.840113
$870$	0	0
$871$	−24390.6	−0.948847
$872$	−33090.4	−1.28507
$873$	0	0
$874$	19846.9	0.768114
$875$	6312.42	0.243884
$876$	0	0
$877$	−44987.0	−1.73216	−0.866078	−	0.499908i	$-0.833367\pi$
$877$	−44987.0	−1.73216	−0.866078	+	0.499908i	$0.833367\pi$
$878$	−9194.79	−0.353427
$879$	0	0
$880$	17406.1	0.666772
$881$	39465.4	1.50922	0.754610	−	0.656173i	$-0.227826\pi$
$881$	39465.4	1.50922	0.754610	+	0.656173i	$0.227826\pi$
$882$	0	0
$883$	−33583.8	−1.27994	−0.639968	−	0.768402i	$-0.721052\pi$
$883$	−33583.8	−1.27994	−0.639968	+	0.768402i	$0.721052\pi$
$884$	2213.92	0.0842332
$885$	0	0
$886$	−20881.2	−0.791782
$887$	−23421.1	−0.886587	−0.443293	−	0.896377i	$-0.646190\pi$
$887$	−23421.1	−0.886587	−0.443293	+	0.896377i	$0.646190\pi$
$888$	0	0
$889$	6357.60	0.239851
$890$	54276.1	2.04420
$891$	0	0
$892$	−1562.65	−0.0586563
$893$	5008.49	0.187685
$894$	0	0
$895$	−58856.0	−2.19815
$896$	7400.79	0.275941
$897$	0	0
$898$	9161.75	0.340458
$899$	697.958	0.0258934
$900$	0	0
$901$	15539.4	0.574576
$902$	−10374.4	−0.382959
$903$	0	0
$904$	−35926.0	−1.32177
$905$	−1503.21	−0.0552137
$906$	0	0
$907$	4760.17	0.174265	0.0871327	−	0.996197i	$-0.472230\pi$
$907$	4760.17	0.174265	0.0871327	+	0.996197i	$0.472230\pi$
$908$	2877.05	0.105152
$909$	0	0
$910$	5950.11	0.216752
$911$	8807.93	0.320329	0.160164	−	0.987090i	$-0.448798\pi$
$911$	8807.93	0.320329	0.160164	+	0.987090i	$0.448798\pi$
$912$	0	0
$913$	348.068	0.0126170
$914$	−34168.5	−1.23654
$915$	0	0
$916$	2155.72	0.0777589
$917$	−4339.05	−0.156257
$918$	0	0
$919$	50540.1	1.81411	0.907054	−	0.421015i	$-0.138326\pi$
$919$	50540.1	1.81411	0.907054	+	0.421015i	$0.138326\pi$
$920$	49184.2	1.76256
$921$	0	0
$922$	17563.4	0.627353
$923$	12220.5	0.435798
$924$	0	0
$925$	18670.2	0.663647
$926$	15950.1	0.566041
$927$	0	0
$928$	13308.2	0.470757
$929$	−5066.09	−0.178916	−0.0894580	−	0.995991i	$-0.528513\pi$
$929$	−5066.09	−0.178916	−0.0894580	+	0.995991i	$0.528513\pi$
$930$	0	0
$931$	−2452.27	−0.0863263
$932$	−4740.86	−0.166622
$933$	0	0
$934$	−31204.5	−1.09319
$935$	25978.9	0.908663
$936$	0	0
$937$	−17496.0	−0.609999	−0.305000	−	0.952352i	$-0.598656\pi$
$937$	−17496.0	−0.609999	−0.305000	+	0.952352i	$0.598656\pi$
$938$	−18612.8	−0.647898
$939$	0	0
$940$	1569.14	0.0544463
$941$	−31861.6	−1.10378	−0.551890	−	0.833917i	$-0.686093\pi$
$941$	−31861.6	−1.10378	−0.551890	+	0.833917i	$0.686093\pi$
$942$	0	0
$943$	−24981.7	−0.862688
$944$	30784.6	1.06139
$945$	0	0
$946$	26617.3	0.914803
$947$	−12040.7	−0.413168	−0.206584	−	0.978429i	$-0.566235\pi$
$947$	−12040.7	−0.413168	−0.206584	+	0.978429i	$0.566235\pi$
$948$	0	0
$949$	24946.2	0.853308
$950$	7646.99	0.261159
$951$	0	0
$952$	13363.8	0.454960
$953$	−56064.5	−1.90567	−0.952837	−	0.303482i	$-0.901851\pi$
$953$	−56064.5	−1.90567	−0.952837	+	0.303482i	$0.901851\pi$
$954$	0	0
$955$	−29874.4	−1.01226
$956$	−2457.55	−0.0831409
$957$	0	0
$958$	−19592.2	−0.660745
$959$	−5338.03	−0.179743
$960$	0	0
$961$	−29783.6	−0.999751
$962$	−20060.4	−0.672323
$963$	0	0
$964$	−6124.38	−0.204619
$965$	−5938.86	−0.198113
$966$	0	0
$967$	35150.9	1.16895	0.584476	−	0.811411i	$-0.301300\pi$
$967$	35150.9	1.16895	0.584476	+	0.811411i	$0.301300\pi$
$968$	−18006.0	−0.597867
$969$	0	0
$970$	62885.6	2.08158
$971$	−5008.69	−0.165537	−0.0827685	−	0.996569i	$-0.526376\pi$
$971$	−5008.69	−0.165537	−0.0827685	+	0.996569i	$0.526376\pi$
$972$	0	0
$973$	−9744.95	−0.321078
$974$	−37146.0	−1.22201
$975$	0	0
$976$	2298.15	0.0753709
$977$	−15008.2	−0.491457	−0.245729	−	0.969339i	$-0.579027\pi$
$977$	−15008.2	−0.491457	−0.245729	+	0.969339i	$0.579027\pi$
$978$	0	0
$979$	36876.9	1.20387
$980$	−768.284	−0.0250428
$981$	0	0
$982$	28229.1	0.917338
$983$	34987.3	1.13522	0.567611	−	0.823297i	$-0.307868\pi$
$983$	34987.3	1.13522	0.567611	+	0.823297i	$0.307868\pi$
$984$	0	0
$985$	−34345.3	−1.11100
$986$	−53394.0	−1.72456
$987$	0	0
$988$	1390.24	0.0447668
$989$	64095.0	2.06077
$990$	0	0
$991$	−34764.1	−1.11435	−0.557173	−	0.830396i	$-0.688114\pi$
$991$	−34764.1	−1.11435	−0.557173	+	0.830396i	$0.688114\pi$
$992$	141.595	0.00453190
$993$	0	0
$994$	9325.58	0.297575
$995$	37880.1	1.20691
$996$	0	0
$997$	−30328.1	−0.963389	−0.481695	−	0.876339i	$-0.659979\pi$
$997$	−30328.1	−0.963389	−0.481695	+	0.876339i	$0.659979\pi$
$998$	−16584.4	−0.526024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.g.1.3		8
3.2	odd	2		567.4.a.i.1.6		8
9.2	odd	6		63.4.f.b.22.3	✓	16
9.4	even	3		189.4.f.b.127.6		16
9.5	odd	6		63.4.f.b.43.3	yes	16
9.7	even	3		189.4.f.b.64.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.3	✓	16	9.2	odd	6
63.4.f.b.43.3	yes	16	9.5	odd	6
189.4.f.b.64.6		16	9.7	even	3
189.4.f.b.127.6		16	9.4	even	3
567.4.a.g.1.3		8	1.1	even	1	trivial
567.4.a.i.1.6		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)

\(f(q)\)	\(=\)	\(q-2.61577 q^{2} -1.15774 q^{4} +13.5431 q^{5} -7.00000 q^{7} +23.9546 q^{8} +O(q^{10})\)	\(q-2.61577 q^{2} -1.15774 q^{4} +13.5431 q^{5} -7.00000 q^{7} +23.9546 q^{8} -35.4255 q^{10} -24.0692 q^{11} +23.9944 q^{13} +18.3104 q^{14} -53.3978 q^{16} -79.6971 q^{17} -50.0463 q^{19} -15.6793 q^{20} +62.9595 q^{22} +151.608 q^{23} +58.4143 q^{25} -62.7639 q^{26} +8.10415 q^{28} -256.124 q^{29} -2.72508 q^{31} -51.9600 q^{32} +208.469 q^{34} -94.8014 q^{35} +319.617 q^{37} +130.910 q^{38} +324.418 q^{40} -164.778 q^{41} +422.769 q^{43} +27.8657 q^{44} -396.571 q^{46} -100.077 q^{47} +49.0000 q^{49} -152.798 q^{50} -27.7792 q^{52} -194.981 q^{53} -325.970 q^{55} -167.682 q^{56} +669.962 q^{58} -576.515 q^{59} -43.0383 q^{61} +7.12819 q^{62} +563.098 q^{64} +324.958 q^{65} -1016.51 q^{67} +92.2681 q^{68} +247.979 q^{70} +509.305 q^{71} +1039.67 q^{73} -836.046 q^{74} +57.9403 q^{76} +168.484 q^{77} -894.141 q^{79} -723.169 q^{80} +431.023 q^{82} -14.4611 q^{83} -1079.34 q^{85} -1105.87 q^{86} -576.567 q^{88} -1532.12 q^{89} -167.961 q^{91} -175.522 q^{92} +261.779 q^{94} -677.779 q^{95} -1775.15 q^{97} -128.173 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 43 q^{4} - 30 q^{5} - 56 q^{7} - 6 q^{8}+O(q^{10}) \) 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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