LMFDB - Embedded newform 567.4.a.i.1.7

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:	$0$
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			1.7
Root			$4.56358$ 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)

$f(q)$	$=$	$q+4.56358 q^{2} +12.8263 q^{4} +20.7910 q^{5} -7.00000 q^{7} +22.0250 q^{8} +O(q^{10})$	\(q+4.56358 q^{2} +12.8263 q^{4} +20.7910 q^{5} -7.00000 q^{7} +22.0250 q^{8} +94.8813 q^{10} +45.3985 q^{11} +29.3042 q^{13} -31.9451 q^{14} -2.09727 q^{16} -98.0555 q^{17} +31.1554 q^{19} +266.670 q^{20} +207.180 q^{22} +8.39191 q^{23} +307.265 q^{25} +133.732 q^{26} -89.7838 q^{28} -72.2664 q^{29} -30.8598 q^{31} -185.771 q^{32} -447.484 q^{34} -145.537 q^{35} -196.369 q^{37} +142.180 q^{38} +457.921 q^{40} +212.673 q^{41} +236.877 q^{43} +582.293 q^{44} +38.2972 q^{46} -220.036 q^{47} +49.0000 q^{49} +1402.23 q^{50} +375.863 q^{52} +55.9028 q^{53} +943.880 q^{55} -154.175 q^{56} -329.793 q^{58} -654.652 q^{59} +348.783 q^{61} -140.831 q^{62} -831.002 q^{64} +609.263 q^{65} +210.494 q^{67} -1257.68 q^{68} -664.169 q^{70} -548.252 q^{71} -266.888 q^{73} -896.146 q^{74} +399.607 q^{76} -317.790 q^{77} -269.892 q^{79} -43.6042 q^{80} +970.550 q^{82} +625.269 q^{83} -2038.67 q^{85} +1081.01 q^{86} +999.902 q^{88} +1605.63 q^{89} -205.129 q^{91} +107.637 q^{92} -1004.15 q^{94} +647.751 q^{95} -291.099 q^{97} +223.615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.56358	1.61347	0.806734	−	0.590914i	$-0.201233\pi$
$2$	4.56358	1.61347	0.806734	+	0.590914i	$0.201233\pi$
$3$	0	0
$3$	0	0
$4$	12.8263	1.60328
$5$	20.7910	1.85960	0.929801	−	0.368063i	$-0.119979\pi$
$5$	20.7910	1.85960	0.929801	+	0.368063i	$0.119979\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	22.0250	0.973376
$9$	0	0
$10$	94.8813	3.00041
$11$	45.3985	1.24438	0.622190	−	0.782867i	$-0.286243\pi$
$11$	45.3985	1.24438	0.622190	+	0.782867i	$0.286243\pi$
$12$	0	0
$13$	29.3042	0.625194	0.312597	−	0.949886i	$-0.398801\pi$
$13$	29.3042	0.625194	0.312597	+	0.949886i	$0.398801\pi$
$14$	−31.9451	−0.609834
$15$	0	0
$16$	−2.09727	−0.0327698
$17$	−98.0555	−1.39894	−0.699469	−	0.714663i	$-0.746580\pi$
$17$	−98.0555	−1.39894	−0.699469	+	0.714663i	$0.746580\pi$
$18$	0	0
$19$	31.1554	0.376186	0.188093	−	0.982151i	$-0.439769\pi$
$19$	31.1554	0.376186	0.188093	+	0.982151i	$0.439769\pi$
$20$	266.670	2.98147
$21$	0	0
$22$	207.180	2.00777
$23$	8.39191	0.0760798	0.0380399	−	0.999276i	$-0.487889\pi$
$23$	8.39191	0.0760798	0.0380399	+	0.999276i	$0.487889\pi$
$24$	0	0
$25$	307.265	2.45812
$26$	133.732	1.00873
$27$	0	0
$28$	−89.7838	−0.605983
$29$	−72.2664	−0.462742	−0.231371	−	0.972866i	$-0.574321\pi$
$29$	−72.2664	−0.462742	−0.231371	+	0.972866i	$0.574321\pi$
$30$	0	0
$31$	−30.8598	−0.178793	−0.0893965	−	0.995996i	$-0.528494\pi$
$31$	−30.8598	−0.178793	−0.0893965	+	0.995996i	$0.528494\pi$
$32$	−185.771	−1.02625
$33$	0	0
$34$	−447.484	−2.25714
$35$	−145.537	−0.702863
$36$	0	0
$37$	−196.369	−0.872511	−0.436255	−	0.899823i	$-0.643696\pi$
$37$	−196.369	−0.872511	−0.436255	+	0.899823i	$0.643696\pi$
$38$	142.180	0.606964
$39$	0	0
$40$	457.921	1.81009
$41$	212.673	0.810096	0.405048	−	0.914295i	$-0.367255\pi$
$41$	212.673	0.810096	0.405048	+	0.914295i	$0.367255\pi$
$42$	0	0
$43$	236.877	0.840078	0.420039	−	0.907506i	$-0.362016\pi$
$43$	236.877	0.840078	0.420039	+	0.907506i	$0.362016\pi$
$44$	582.293	1.99509
$45$	0	0
$46$	38.2972	0.122752
$47$	−220.036	−0.682884	−0.341442	−	0.939903i	$-0.610915\pi$
$47$	−220.036	−0.682884	−0.341442	+	0.939903i	$0.610915\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	1402.23	3.96610
$51$	0	0
$52$	375.863	1.00236
$53$	55.9028	0.144884	0.0724419	−	0.997373i	$-0.476921\pi$
$53$	55.9028	0.144884	0.0724419	+	0.997373i	$0.476921\pi$
$54$	0	0
$55$	943.880	2.31405
$56$	−154.175	−0.367902
$57$	0	0
$58$	−329.793	−0.746620
$59$	−654.652	−1.44455	−0.722275	−	0.691606i	$-0.756904\pi$
$59$	−654.652	−1.44455	−0.722275	+	0.691606i	$0.756904\pi$
$60$	0	0
$61$	348.783	0.732083	0.366041	−	0.930599i	$-0.380713\pi$
$61$	348.783	0.732083	0.366041	+	0.930599i	$0.380713\pi$
$62$	−140.831	−0.288477
$63$	0	0
$64$	−831.002	−1.62305
$65$	609.263	1.16261
$66$	0	0
$67$	210.494	0.383820	0.191910	−	0.981413i	$-0.438532\pi$
$67$	210.494	0.383820	0.191910	+	0.981413i	$0.438532\pi$
$68$	−1257.68	−2.24289
$69$	0	0
$70$	−664.169	−1.13405
$71$	−548.252	−0.916416	−0.458208	−	0.888845i	$-0.651508\pi$
$71$	−548.252	−0.916416	−0.458208	+	0.888845i	$0.651508\pi$
$72$	0	0
$73$	−266.888	−0.427902	−0.213951	−	0.976844i	$-0.568633\pi$
$73$	−266.888	−0.427902	−0.213951	+	0.976844i	$0.568633\pi$
$74$	−896.146	−1.40777
$75$	0	0
$76$	399.607	0.603132
$77$	−317.790	−0.470331
$78$	0	0
$79$	−269.892	−0.384370	−0.192185	−	0.981359i	$-0.561557\pi$
$79$	−269.892	−0.384370	−0.192185	+	0.981359i	$0.561557\pi$
$80$	−43.6042	−0.0609388
$81$	0	0
$82$	970.550	1.30706
$83$	625.269	0.826893	0.413447	−	0.910528i	$-0.364325\pi$
$83$	625.269	0.826893	0.413447	+	0.910528i	$0.364325\pi$
$84$	0	0
$85$	−2038.67	−2.60147
$86$	1081.01	1.35544
$87$	0	0
$88$	999.902	1.21125
$89$	1605.63	1.91232	0.956162	−	0.292838i	$-0.0945996\pi$
$89$	1605.63	1.91232	0.956162	+	0.292838i	$0.0945996\pi$
$90$	0	0
$91$	−205.129	−0.236301
$92$	107.637	0.121977
$93$	0	0
$94$	−1004.15	−1.10181
$95$	647.751	0.699556
$96$	0	0
$97$	−291.099	−0.304707	−0.152354	−	0.988326i	$-0.548685\pi$
$97$	−291.099	−0.304707	−0.152354	+	0.988326i	$0.548685\pi$
$98$	223.615	0.230496
$99$	0	0
$100$	3941.06	3.94106
$101$	−1452.61	−1.43109	−0.715547	−	0.698565i	$-0.753822\pi$
$101$	−1452.61	−1.43109	−0.715547	+	0.698565i	$0.753822\pi$
$102$	0	0
$103$	−56.2681	−0.0538278	−0.0269139	−	0.999638i	$-0.508568\pi$
$103$	−56.2681	−0.0538278	−0.0269139	+	0.999638i	$0.508568\pi$
$104$	645.425	0.608549
$105$	0	0
$106$	255.117	0.233765
$107$	−300.124	−0.271160	−0.135580	−	0.990766i	$-0.543290\pi$
$107$	−300.124	−0.271160	−0.135580	+	0.990766i	$0.543290\pi$
$108$	0	0
$109$	−936.906	−0.823296	−0.411648	−	0.911343i	$-0.635047\pi$
$109$	−936.906	−0.823296	−0.411648	+	0.911343i	$0.635047\pi$
$110$	4307.47	3.73365
$111$	0	0
$112$	14.6809	0.0123858
$113$	1394.59	1.16100	0.580498	−	0.814262i	$-0.302858\pi$
$113$	1394.59	1.16100	0.580498	+	0.814262i	$0.302858\pi$
$114$	0	0
$115$	174.476	0.141478
$116$	−926.907	−0.741906
$117$	0	0
$118$	−2987.56	−2.33074
$119$	686.389	0.528749
$120$	0	0
$121$	730.027	0.548480
$122$	1591.70	1.18119
$123$	0	0
$124$	−395.815	−0.286655
$125$	3789.47	2.71152
$126$	0	0
$127$	−2387.37	−1.66807	−0.834034	−	0.551714i	$-0.813974\pi$
$127$	−2387.37	−1.66807	−0.834034	+	0.551714i	$0.813974\pi$
$128$	−2306.18	−1.59249
$129$	0	0
$130$	2780.42	1.87584
$131$	2119.99	1.41393	0.706963	−	0.707250i	$-0.250065\pi$
$131$	2119.99	1.41393	0.706963	+	0.707250i	$0.250065\pi$
$132$	0	0
$133$	−218.088	−0.142185
$134$	960.606	0.619282
$135$	0	0
$136$	−2159.67	−1.36169
$137$	−65.9016	−0.0410975	−0.0205487	−	0.999789i	$-0.506541\pi$
$137$	−65.9016	−0.0410975	−0.0205487	+	0.999789i	$0.506541\pi$
$138$	0	0
$139$	−1963.28	−1.19801	−0.599005	−	0.800745i	$-0.704437\pi$
$139$	−1963.28	−1.19801	−0.599005	+	0.800745i	$0.704437\pi$
$140$	−1866.69	−1.12689
$141$	0	0
$142$	−2501.99	−1.47861
$143$	1330.37	0.777979
$144$	0	0
$145$	−1502.49	−0.860517
$146$	−1217.96	−0.690407
$147$	0	0
$148$	−2518.68	−1.39888
$149$	−2633.82	−1.44813	−0.724063	−	0.689734i	$-0.757727\pi$
$149$	−2633.82	−1.44813	−0.724063	+	0.689734i	$0.757727\pi$
$150$	0	0
$151$	−737.473	−0.397448	−0.198724	−	0.980055i	$-0.563680\pi$
$151$	−737.473	−0.397448	−0.198724	+	0.980055i	$0.563680\pi$
$152$	686.197	0.366170
$153$	0	0
$154$	−1450.26	−0.758865
$155$	−641.605	−0.332484
$156$	0	0
$157$	2346.72	1.19292	0.596462	−	0.802641i	$-0.296573\pi$
$157$	2346.72	1.19292	0.596462	+	0.802641i	$0.296573\pi$
$158$	−1231.67	−0.620169
$159$	0	0
$160$	−3862.36	−1.90841
$161$	−58.7434	−0.0287555
$162$	0	0
$163$	−572.002	−0.274863	−0.137431	−	0.990511i	$-0.543885\pi$
$163$	−572.002	−0.274863	−0.137431	+	0.990511i	$0.543885\pi$
$164$	2727.80	1.29881
$165$	0	0
$166$	2853.46	1.33417
$167$	543.772	0.251966	0.125983	−	0.992032i	$-0.459792\pi$
$167$	543.772	0.251966	0.125983	+	0.992032i	$0.459792\pi$
$168$	0	0
$169$	−1338.26	−0.609132
$170$	−9303.63	−4.19739
$171$	0	0
$172$	3038.24	1.34688
$173$	2034.45	0.894083	0.447042	−	0.894513i	$-0.352478\pi$
$173$	2034.45	0.894083	0.447042	+	0.894513i	$0.352478\pi$
$174$	0	0
$175$	−2150.85	−0.929082
$176$	−95.2129	−0.0407781
$177$	0	0
$178$	7327.44	3.08548
$179$	−931.218	−0.388841	−0.194420	−	0.980918i	$-0.562283\pi$
$179$	−931.218	−0.388841	−0.194420	+	0.980918i	$0.562283\pi$
$180$	0	0
$181$	1002.74	0.411785	0.205892	−	0.978575i	$-0.433990\pi$
$181$	1002.74	0.411785	0.205892	+	0.978575i	$0.433990\pi$
$182$	−936.125	−0.381265
$183$	0	0
$184$	184.832	0.0740542
$185$	−4082.71	−1.62252
$186$	0	0
$187$	−4451.58	−1.74081
$188$	−2822.24	−1.09486
$189$	0	0
$190$	2956.06	1.12871
$191$	−3843.08	−1.45589	−0.727947	−	0.685634i	$-0.759525\pi$
$191$	−3843.08	−1.45589	−0.727947	+	0.685634i	$0.759525\pi$
$192$	0	0
$193$	−3466.10	−1.29272	−0.646361	−	0.763031i	$-0.723710\pi$
$193$	−3466.10	−1.29272	−0.646361	+	0.763031i	$0.723710\pi$
$194$	−1328.45	−0.491636
$195$	0	0
$196$	628.486	0.229040
$197$	−3691.99	−1.33525	−0.667623	−	0.744500i	$-0.732688\pi$
$197$	−3691.99	−1.33525	−0.667623	+	0.744500i	$0.732688\pi$
$198$	0	0
$199$	−481.783	−0.171622	−0.0858108	−	0.996311i	$-0.527348\pi$
$199$	−481.783	−0.171622	−0.0858108	+	0.996311i	$0.527348\pi$
$200$	6767.51	2.39267
$201$	0	0
$202$	−6629.12	−2.30902
$203$	505.865	0.174900
$204$	0	0
$205$	4421.68	1.50646
$206$	−256.784	−0.0868494
$207$	0	0
$208$	−61.4588	−0.0204875
$209$	1414.41	0.468118
$210$	0	0
$211$	1149.78	0.375139	0.187570	−	0.982251i	$-0.439939\pi$
$211$	1149.78	0.375139	0.187570	+	0.982251i	$0.439939\pi$
$212$	717.023	0.232289
$213$	0	0
$214$	−1369.64	−0.437508
$215$	4924.90	1.56221
$216$	0	0
$217$	216.018	0.0675774
$218$	−4275.64	−1.32836
$219$	0	0
$220$	12106.4	3.71007
$221$	−2873.44	−0.874608
$222$	0	0
$223$	−2015.96	−0.605374	−0.302687	−	0.953090i	$-0.597884\pi$
$223$	−2015.96	−0.605374	−0.302687	+	0.953090i	$0.597884\pi$
$224$	1300.40	0.387886
$225$	0	0
$226$	6364.34	1.87323
$227$	3042.20	0.889506	0.444753	−	0.895653i	$-0.353291\pi$
$227$	3042.20	0.889506	0.444753	+	0.895653i	$0.353291\pi$
$228$	0	0
$229$	4081.34	1.17774	0.588870	−	0.808228i	$-0.299573\pi$
$229$	4081.34	1.17774	0.588870	+	0.808228i	$0.299573\pi$
$230$	796.236	0.228271
$231$	0	0
$232$	−1591.67	−0.450422
$233$	2667.57	0.750035	0.375017	−	0.927018i	$-0.377637\pi$
$233$	2667.57	0.750035	0.375017	+	0.927018i	$0.377637\pi$
$234$	0	0
$235$	−4574.77	−1.26989
$236$	−8396.74	−2.31602
$237$	0	0
$238$	3132.39	0.853120
$239$	−5972.63	−1.61647	−0.808237	−	0.588858i	$-0.799578\pi$
$239$	−5972.63	−1.61647	−0.808237	+	0.588858i	$0.799578\pi$
$240$	0	0
$241$	−2641.26	−0.705969	−0.352984	−	0.935629i	$-0.614833\pi$
$241$	−2641.26	−0.705969	−0.352984	+	0.935629i	$0.614833\pi$
$242$	3331.54	0.884956
$243$	0	0
$244$	4473.57	1.17373
$245$	1018.76	0.265657
$246$	0	0
$247$	912.984	0.235189
$248$	−679.686	−0.174033
$249$	0	0
$250$	17293.5	4.37496
$251$	−7001.16	−1.76060	−0.880298	−	0.474422i	$-0.842657\pi$
$251$	−7001.16	−1.76060	−0.880298	+	0.474422i	$0.842657\pi$
$252$	0	0
$253$	380.981	0.0946721
$254$	−10894.9	−2.69137
$255$	0	0
$256$	−3876.40	−0.946387
$257$	5787.71	1.40478	0.702388	−	0.711795i	$-0.252117\pi$
$257$	5787.71	1.40478	0.702388	+	0.711795i	$0.252117\pi$
$258$	0	0
$259$	1374.58	0.329778
$260$	7814.57	1.86400
$261$	0	0
$262$	9674.74	2.28133
$263$	−7486.63	−1.75531	−0.877653	−	0.479296i	$-0.840892\pi$
$263$	−7486.63	−1.75531	−0.877653	+	0.479296i	$0.840892\pi$
$264$	0	0
$265$	1162.27	0.269426
$266$	−995.260	−0.229411
$267$	0	0
$268$	2699.85	0.615372
$269$	−3249.42	−0.736508	−0.368254	−	0.929725i	$-0.620044\pi$
$269$	−3249.42	−0.736508	−0.368254	+	0.929725i	$0.620044\pi$
$270$	0	0
$271$	−3946.32	−0.884581	−0.442291	−	0.896872i	$-0.645834\pi$
$271$	−3946.32	−0.884581	−0.442291	+	0.896872i	$0.645834\pi$
$272$	205.649	0.0458429
$273$	0	0
$274$	−300.747	−0.0663095
$275$	13949.4	3.05883
$276$	0	0
$277$	4654.26	1.00956	0.504779	−	0.863249i	$-0.331574\pi$
$277$	4654.26	1.00956	0.504779	+	0.863249i	$0.331574\pi$
$278$	−8959.60	−1.93295
$279$	0	0
$280$	−3205.45	−0.684150
$281$	6446.89	1.36864	0.684322	−	0.729180i	$-0.260098\pi$
$281$	6446.89	1.36864	0.684322	+	0.729180i	$0.260098\pi$
$282$	0	0
$283$	−3638.46	−0.764254	−0.382127	−	0.924110i	$-0.624808\pi$
$283$	−3638.46	−0.764254	−0.382127	+	0.924110i	$0.624808\pi$
$284$	−7032.02	−1.46927
$285$	0	0
$286$	6071.24	1.25524
$287$	−1488.71	−0.306187
$288$	0	0
$289$	4701.88	0.957029
$290$	−6856.73	−1.38842
$291$	0	0
$292$	−3423.17	−0.686048
$293$	6327.46	1.26162	0.630809	−	0.775938i	$-0.282723\pi$
$293$	6327.46	1.26162	0.630809	+	0.775938i	$0.282723\pi$
$294$	0	0
$295$	−13610.9	−2.68629
$296$	−4325.03	−0.849281
$297$	0	0
$298$	−12019.6	−2.33651
$299$	245.918	0.0475646
$300$	0	0
$301$	−1658.14	−0.317520
$302$	−3365.52	−0.641271
$303$	0	0
$304$	−65.3412	−0.0123275
$305$	7251.53	1.36138
$306$	0	0
$307$	2712.80	0.504324	0.252162	−	0.967685i	$-0.418858\pi$
$307$	2712.80	0.504324	0.252162	+	0.967685i	$0.418858\pi$
$308$	−4076.05	−0.754073
$309$	0	0
$310$	−2928.02	−0.536452
$311$	3046.07	0.555392	0.277696	−	0.960669i	$-0.410429\pi$
$311$	3046.07	0.555392	0.277696	+	0.960669i	$0.410429\pi$
$312$	0	0
$313$	4380.53	0.791061	0.395531	−	0.918453i	$-0.370561\pi$
$313$	4380.53	0.791061	0.395531	+	0.918453i	$0.370561\pi$
$314$	10709.5	1.92475
$315$	0	0
$316$	−3461.70	−0.616253
$317$	6906.25	1.22364	0.611820	−	0.790997i	$-0.290438\pi$
$317$	6906.25	1.22364	0.611820	+	0.790997i	$0.290438\pi$
$318$	0	0
$319$	−3280.79	−0.575827
$320$	−17277.3	−3.01823
$321$	0	0
$322$	−268.080	−0.0463960
$323$	−3054.96	−0.526261
$324$	0	0
$325$	9004.16	1.53680
$326$	−2610.38	−0.443483
$327$	0	0
$328$	4684.12	0.788528
$329$	1540.25	0.258106
$330$	0	0
$331$	5702.52	0.946945	0.473472	−	0.880809i	$-0.343000\pi$
$331$	5702.52	0.946945	0.473472	+	0.880809i	$0.343000\pi$
$332$	8019.85	1.32574
$333$	0	0
$334$	2481.54	0.406539
$335$	4376.38	0.713753
$336$	0	0
$337$	1229.61	0.198758	0.0993788	−	0.995050i	$-0.468314\pi$
$337$	1229.61	0.198758	0.0993788	+	0.995050i	$0.468314\pi$
$338$	−6107.27	−0.982816
$339$	0	0
$340$	−26148.5	−4.17089
$341$	−1400.99	−0.222486
$342$	0	0
$343$	−343.000	−0.0539949
$344$	5217.20	0.817712
$345$	0	0
$346$	9284.37	1.44258
$347$	4385.48	0.678458	0.339229	−	0.940704i	$-0.389834\pi$
$347$	4385.48	0.678458	0.339229	+	0.940704i	$0.389834\pi$
$348$	0	0
$349$	2920.03	0.447868	0.223934	−	0.974604i	$-0.428110\pi$
$349$	2920.03	0.447868	0.223934	+	0.974604i	$0.428110\pi$
$350$	−9815.59	−1.49904
$351$	0	0
$352$	−8433.73	−1.27704
$353$	11131.9	1.67845	0.839223	−	0.543787i	$-0.183010\pi$
$353$	11131.9	1.67845	0.839223	+	0.543787i	$0.183010\pi$
$354$	0	0
$355$	−11398.7	−1.70417
$356$	20594.3	3.06599
$357$	0	0
$358$	−4249.69	−0.627383
$359$	7640.81	1.12330	0.561652	−	0.827373i	$-0.310166\pi$
$359$	7640.81	1.12330	0.561652	+	0.827373i	$0.310166\pi$
$360$	0	0
$361$	−5888.34	−0.858484
$362$	4576.08	0.664402
$363$	0	0
$364$	−2631.04	−0.378857
$365$	−5548.86	−0.795728
$366$	0	0
$367$	4408.50	0.627035	0.313518	−	0.949582i	$-0.398493\pi$
$367$	4408.50	0.627035	0.313518	+	0.949582i	$0.398493\pi$
$368$	−17.6001	−0.00249312
$369$	0	0
$370$	−18631.8	−2.61789
$371$	−391.319	−0.0547609
$372$	0	0
$373$	4604.33	0.639151	0.319575	−	0.947561i	$-0.396460\pi$
$373$	4604.33	0.639151	0.319575	+	0.947561i	$0.396460\pi$
$374$	−20315.1	−2.80874
$375$	0	0
$376$	−4846.29	−0.664703
$377$	−2117.71	−0.289304
$378$	0	0
$379$	14679.5	1.98954	0.994770	−	0.102143i	$-0.0325700\pi$
$379$	14679.5	1.98954	0.994770	+	0.102143i	$0.0325700\pi$
$380$	8308.22	1.12159
$381$	0	0
$382$	−17538.2	−2.34904
$383$	1823.75	0.243313	0.121657	−	0.992572i	$-0.461179\pi$
$383$	1823.75	0.243313	0.121657	+	0.992572i	$0.461179\pi$
$384$	0	0
$385$	−6607.16	−0.874629
$386$	−15817.8	−2.08577
$387$	0	0
$388$	−3733.71	−0.488532
$389$	8613.59	1.12269	0.561345	−	0.827582i	$-0.310284\pi$
$389$	8613.59	1.12269	0.561345	+	0.827582i	$0.310284\pi$
$390$	0	0
$391$	−822.873	−0.106431
$392$	1079.22	0.139054
$393$	0	0
$394$	−16848.7	−2.15438
$395$	−5611.32	−0.714775
$396$	0	0
$397$	−5497.04	−0.694933	−0.347467	−	0.937692i	$-0.612958\pi$
$397$	−5497.04	−0.694933	−0.347467	+	0.937692i	$0.612958\pi$
$398$	−2198.66	−0.276906
$399$	0	0
$400$	−644.417	−0.0805521
$401$	−8621.04	−1.07360	−0.536801	−	0.843709i	$-0.680367\pi$
$401$	−8621.04	−1.07360	−0.536801	+	0.843709i	$0.680367\pi$
$402$	0	0
$403$	−904.322	−0.111780
$404$	−18631.6	−2.29445
$405$	0	0
$406$	2308.55	0.282196
$407$	−8914.87	−1.08573
$408$	0	0
$409$	9398.90	1.13630	0.568149	−	0.822926i	$-0.307660\pi$
$409$	9398.90	1.13630	0.568149	+	0.822926i	$0.307660\pi$
$410$	20178.7	2.43062
$411$	0	0
$412$	−721.709	−0.0863011
$413$	4582.57	0.545989
$414$	0	0
$415$	13000.0	1.53769
$416$	−5443.87	−0.641605
$417$	0	0
$418$	6454.76	0.755294
$419$	−594.726	−0.0693420	−0.0346710	−	0.999399i	$-0.511038\pi$
$419$	−594.726	−0.0693420	−0.0346710	+	0.999399i	$0.511038\pi$
$420$	0	0
$421$	3117.77	0.360929	0.180464	−	0.983582i	$-0.442240\pi$
$421$	3117.77	0.360929	0.180464	+	0.983582i	$0.442240\pi$
$422$	5247.13	0.605276
$423$	0	0
$424$	1231.26	0.141026
$425$	−30129.0	−3.43876
$426$	0	0
$427$	−2441.48	−0.276701
$428$	−3849.47	−0.434745
$429$	0	0
$430$	22475.2	2.52058
$431$	−8465.93	−0.946148	−0.473074	−	0.881023i	$-0.656856\pi$
$431$	−8465.93	−0.946148	−0.473074	+	0.881023i	$0.656856\pi$
$432$	0	0
$433$	5368.98	0.595881	0.297941	−	0.954584i	$-0.403700\pi$
$433$	5368.98	0.595881	0.297941	+	0.954584i	$0.403700\pi$
$434$	985.817	0.109034
$435$	0	0
$436$	−12017.0	−1.31998
$437$	261.453	0.0286201
$438$	0	0
$439$	−17792.7	−1.93440	−0.967198	−	0.254023i	$-0.918246\pi$
$439$	−17792.7	−1.93440	−0.967198	+	0.254023i	$0.918246\pi$
$440$	20788.9	2.25244
$441$	0	0
$442$	−13113.2	−1.41115
$443$	8863.59	0.950614	0.475307	−	0.879820i	$-0.342337\pi$
$443$	8863.59	0.950614	0.475307	+	0.879820i	$0.342337\pi$
$444$	0	0
$445$	33382.7	3.55616
$446$	−9199.97	−0.976752
$447$	0	0
$448$	5817.01	0.613456
$449$	114.115	0.0119943	0.00599714	−	0.999982i	$-0.498091\pi$
$449$	114.115	0.0119943	0.00599714	+	0.999982i	$0.498091\pi$
$450$	0	0
$451$	9655.04	1.00807
$452$	17887.4	1.86140
$453$	0	0
$454$	13883.3	1.43519
$455$	−4264.84	−0.439426
$456$	0	0
$457$	8062.56	0.825275	0.412638	−	0.910895i	$-0.364608\pi$
$457$	8062.56	0.825275	0.412638	+	0.910895i	$0.364608\pi$
$458$	18625.5	1.90025
$459$	0	0
$460$	2237.87	0.226829
$461$	4129.87	0.417239	0.208620	−	0.977997i	$-0.433103\pi$
$461$	4129.87	0.417239	0.208620	+	0.977997i	$0.433103\pi$
$462$	0	0
$463$	11818.5	1.18629	0.593144	−	0.805096i	$-0.297886\pi$
$463$	11818.5	1.18629	0.593144	+	0.805096i	$0.297886\pi$
$464$	151.562	0.0151640
$465$	0	0
$466$	12173.7	1.21016
$467$	1590.26	0.157577	0.0787884	−	0.996891i	$-0.474895\pi$
$467$	1590.26	0.157577	0.0787884	+	0.996891i	$0.474895\pi$
$468$	0	0
$469$	−1473.46	−0.145070
$470$	−20877.3	−2.04893
$471$	0	0
$472$	−14418.7	−1.40609
$473$	10753.9	1.04538
$474$	0	0
$475$	9572.95	0.924710
$476$	8803.79	0.847734
$477$	0	0
$478$	−27256.6	−2.60813
$479$	13134.6	1.25289	0.626444	−	0.779467i	$-0.284510\pi$
$479$	13134.6	1.25289	0.626444	+	0.779467i	$0.284510\pi$
$480$	0	0
$481$	−5754.44	−0.545489
$482$	−12053.6	−1.13906
$483$	0	0
$484$	9363.51	0.879368
$485$	−6052.23	−0.566635
$486$	0	0
$487$	−18337.9	−1.70631	−0.853153	−	0.521661i	$-0.825313\pi$
$487$	−18337.9	−1.70631	−0.853153	+	0.521661i	$0.825313\pi$
$488$	7681.93	0.712592
$489$	0	0
$490$	4649.18	0.428630
$491$	−8097.18	−0.744237	−0.372119	−	0.928185i	$-0.621369\pi$
$491$	−8097.18	−0.744237	−0.372119	+	0.928185i	$0.621369\pi$
$492$	0	0
$493$	7086.12	0.647348
$494$	4166.47	0.379471
$495$	0	0
$496$	64.7212	0.00585901
$497$	3837.76	0.346373
$498$	0	0
$499$	−602.355	−0.0540383	−0.0270191	−	0.999635i	$-0.508602\pi$
$499$	−602.355	−0.0540383	−0.0270191	+	0.999635i	$0.508602\pi$
$500$	48604.7	4.34733
$501$	0	0
$502$	−31950.4	−2.84067
$503$	11416.3	1.01198	0.505992	−	0.862538i	$-0.331126\pi$
$503$	11416.3	1.01198	0.505992	+	0.862538i	$0.331126\pi$
$504$	0	0
$505$	−30201.3	−2.66126
$506$	1738.63	0.152750
$507$	0	0
$508$	−30621.0	−2.67438
$509$	12812.1	1.11569	0.557847	−	0.829944i	$-0.311628\pi$
$509$	12812.1	1.11569	0.557847	+	0.829944i	$0.311628\pi$
$510$	0	0
$511$	1868.22	0.161732
$512$	759.150	0.0655273
$513$	0	0
$514$	26412.7	2.26656
$515$	−1169.87	−0.100098
$516$	0	0
$517$	−9989.32	−0.849767
$518$	6273.02	0.532087
$519$	0	0
$520$	13419.0	1.13166
$521$	3033.32	0.255071	0.127536	−	0.991834i	$-0.459293\pi$
$521$	3033.32	0.255071	0.127536	+	0.991834i	$0.459293\pi$
$522$	0	0
$523$	12789.3	1.06928	0.534642	−	0.845078i	$-0.320446\pi$
$523$	12789.3	1.06928	0.534642	+	0.845078i	$0.320446\pi$
$524$	27191.5	2.26692
$525$	0	0
$526$	−34165.8	−2.83213
$527$	3025.97	0.250120
$528$	0	0
$529$	−12096.6	−0.994212
$530$	5304.13	0.434711
$531$	0	0
$532$	−2797.25	−0.227962
$533$	6232.21	0.506467
$534$	0	0
$535$	−6239.87	−0.504249
$536$	4636.13	0.373601
$537$	0	0
$538$	−14829.0	−1.18833
$539$	2224.53	0.177768
$540$	0	0
$541$	3836.05	0.304851	0.152426	−	0.988315i	$-0.451292\pi$
$541$	3836.05	0.304851	0.152426	+	0.988315i	$0.451292\pi$
$542$	−18009.3	−1.42724
$543$	0	0
$544$	18215.9	1.43566
$545$	−19479.2	−1.53100
$546$	0	0
$547$	4876.53	0.381180	0.190590	−	0.981670i	$-0.438960\pi$
$547$	4876.53	0.381180	0.190590	+	0.981670i	$0.438960\pi$
$548$	−845.271	−0.0658908
$549$	0	0
$550$	63659.1	4.93533
$551$	−2251.49	−0.174077
$552$	0	0
$553$	1889.24	0.145278
$554$	21240.1	1.62889
$555$	0	0
$556$	−25181.6	−1.92075
$557$	26004.6	1.97819	0.989095	−	0.147276i	$-0.0470506\pi$
$557$	26004.6	1.97819	0.989095	+	0.147276i	$0.0470506\pi$
$558$	0	0
$559$	6941.48	0.525212
$560$	305.230	0.0230327
$561$	0	0
$562$	29420.9	2.20827
$563$	15350.2	1.14908	0.574542	−	0.818475i	$-0.305180\pi$
$563$	15350.2	1.14908	0.574542	+	0.818475i	$0.305180\pi$
$564$	0	0
$565$	28995.0	2.15899
$566$	−16604.4	−1.23310
$567$	0	0
$568$	−12075.2	−0.892017
$569$	−20873.5	−1.53789	−0.768947	−	0.639312i	$-0.779219\pi$
$569$	−20873.5	−1.53789	−0.768947	+	0.639312i	$0.779219\pi$
$570$	0	0
$571$	−7.06942	−0.000518119 0	−0.000259059	−	1.00000i	$-0.500082\pi$
$571$	−7.06942	−0.000518119 0	−0.000259059		1.00000i	$0.500082\pi$
$572$	17063.6	1.24732
$573$	0	0
$574$	−6793.85	−0.494024
$575$	2578.54	0.187013
$576$	0	0
$577$	−13121.5	−0.946715	−0.473357	−	0.880870i	$-0.656958\pi$
$577$	−13121.5	−0.946715	−0.473357	+	0.880870i	$0.656958\pi$
$578$	21457.4	1.54414
$579$	0	0
$580$	−19271.3	−1.37965
$581$	−4376.88	−0.312536
$582$	0	0
$583$	2537.90	0.180290
$584$	−5878.20	−0.416510
$585$	0	0
$586$	28875.9	2.03558
$587$	−3836.30	−0.269747	−0.134873	−	0.990863i	$-0.543063\pi$
$587$	−3836.30	−0.269747	−0.134873	+	0.990863i	$0.543063\pi$
$588$	0	0
$589$	−961.448	−0.0672594
$590$	−62114.3	−4.33424
$591$	0	0
$592$	411.839	0.0285920
$593$	2999.00	0.207680	0.103840	−	0.994594i	$-0.466887\pi$
$593$	2999.00	0.207680	0.103840	+	0.994594i	$0.466887\pi$
$594$	0	0
$595$	14270.7	0.983263
$596$	−33782.0	−2.32175
$597$	0	0
$598$	1122.27	0.0767441
$599$	8362.03	0.570389	0.285195	−	0.958470i	$-0.407942\pi$
$599$	8362.03	0.570389	0.285195	+	0.958470i	$0.407942\pi$
$600$	0	0
$601$	−14530.1	−0.986185	−0.493092	−	0.869977i	$-0.664134\pi$
$601$	−14530.1	−0.986185	−0.493092	+	0.869977i	$0.664134\pi$
$602$	−7567.04	−0.512308
$603$	0	0
$604$	−9459.02	−0.637222
$605$	15178.0	1.01995
$606$	0	0
$607$	−13009.3	−0.869902	−0.434951	−	0.900454i	$-0.643234\pi$
$607$	−13009.3	−0.869902	−0.434951	+	0.900454i	$0.643234\pi$
$608$	−5787.76	−0.386060
$609$	0	0
$610$	33092.9	2.19655
$611$	−6447.98	−0.426935
$612$	0	0
$613$	3740.67	0.246467	0.123233	−	0.992378i	$-0.460674\pi$
$613$	3740.67	0.246467	0.123233	+	0.992378i	$0.460674\pi$
$614$	12380.1	0.813712
$615$	0	0
$616$	−6999.31	−0.457809
$617$	22685.6	1.48021	0.740104	−	0.672493i	$-0.234776\pi$
$617$	22685.6	1.48021	0.740104	+	0.672493i	$0.234776\pi$
$618$	0	0
$619$	−1189.05	−0.0772080	−0.0386040	−	0.999255i	$-0.512291\pi$
$619$	−1189.05	−0.0772080	−0.0386040	+	0.999255i	$0.512291\pi$
$620$	−8229.39	−0.533065
$621$	0	0
$622$	13901.0	0.896107
$623$	−11239.4	−0.722791
$624$	0	0
$625$	40378.6	2.58423
$626$	19990.9	1.27635
$627$	0	0
$628$	30099.7	1.91259
$629$	19255.1	1.22059
$630$	0	0
$631$	23574.4	1.48729	0.743646	−	0.668574i	$-0.233095\pi$
$631$	23574.4	1.48729	0.743646	+	0.668574i	$0.233095\pi$
$632$	−5944.37	−0.374136
$633$	0	0
$634$	31517.2	1.97430
$635$	−49635.7	−3.10194
$636$	0	0
$637$	1435.91	0.0893135
$638$	−14972.1	−0.929079
$639$	0	0
$640$	−47947.7	−2.96140
$641$	−13992.7	−0.862211	−0.431105	−	0.902302i	$-0.641876\pi$
$641$	−13992.7	−0.862211	−0.431105	+	0.902302i	$0.641876\pi$
$642$	0	0
$643$	11896.7	0.729639	0.364820	−	0.931078i	$-0.381131\pi$
$643$	11896.7	0.729639	0.364820	+	0.931078i	$0.381131\pi$
$644$	−753.458	−0.0461031
$645$	0	0
$646$	−13941.5	−0.849106
$647$	−5011.63	−0.304525	−0.152262	−	0.988340i	$-0.548656\pi$
$647$	−5011.63	−0.304525	−0.152262	+	0.988340i	$0.548656\pi$
$648$	0	0
$649$	−29720.3	−1.79757
$650$	41091.2	2.47958
$651$	0	0
$652$	−7336.64	−0.440683
$653$	−12477.2	−0.747735	−0.373868	−	0.927482i	$-0.621969\pi$
$653$	−12477.2	−0.747735	−0.373868	+	0.927482i	$0.621969\pi$
$654$	0	0
$655$	44076.7	2.62934
$656$	−446.032	−0.0265467
$657$	0	0
$658$	7029.06	0.416446
$659$	28927.1	1.70992	0.854962	−	0.518691i	$-0.173580\pi$
$659$	28927.1	1.70992	0.854962	+	0.518691i	$0.173580\pi$
$660$	0	0
$661$	−24679.8	−1.45224	−0.726121	−	0.687567i	$-0.758679\pi$
$661$	−24679.8	−1.45224	−0.726121	+	0.687567i	$0.758679\pi$
$662$	26023.9	1.52787
$663$	0	0
$664$	13771.5	0.804878
$665$	−4534.26	−0.264407
$666$	0	0
$667$	−606.453	−0.0352053
$668$	6974.55	0.403972
$669$	0	0
$670$	19972.0	1.15162
$671$	15834.2	0.910989
$672$	0	0
$673$	−1075.62	−0.0616079	−0.0308040	−	0.999525i	$-0.509807\pi$
$673$	−1075.62	−0.0616079	−0.0308040	+	0.999525i	$0.509807\pi$
$674$	5611.44	0.320689
$675$	0	0
$676$	−17164.9	−0.976610
$677$	23037.2	1.30782	0.653908	−	0.756574i	$-0.273129\pi$
$677$	23037.2	1.30782	0.653908	+	0.756574i	$0.273129\pi$
$678$	0	0
$679$	2037.69	0.115169
$680$	−44901.7	−2.53221
$681$	0	0
$682$	−6393.52	−0.358975
$683$	−1233.44	−0.0691015	−0.0345508	−	0.999403i	$-0.511000\pi$
$683$	−1233.44	−0.0691015	−0.0345508	+	0.999403i	$0.511000\pi$
$684$	0	0
$685$	−1370.16	−0.0764249
$686$	−1565.31	−0.0871191
$687$	0	0
$688$	−496.794	−0.0275292
$689$	1638.19	0.0905805
$690$	0	0
$691$	−30427.9	−1.67516	−0.837578	−	0.546317i	$-0.816029\pi$
$691$	−30427.9	−1.67516	−0.837578	+	0.546317i	$0.816029\pi$
$692$	26094.4	1.43347
$693$	0	0
$694$	20013.5	1.09467
$695$	−40818.6	−2.22782
$696$	0	0
$697$	−20853.8	−1.13327
$698$	13325.8	0.722620
$699$	0	0
$700$	−27587.4	−1.48958
$701$	359.627	0.0193765	0.00968825	−	0.999953i	$-0.496916\pi$
$701$	359.627	0.0193765	0.00968825	+	0.999953i	$0.496916\pi$
$702$	0	0
$703$	−6117.96	−0.328226
$704$	−37726.3	−2.01969
$705$	0	0
$706$	50801.3	2.70812
$707$	10168.3	0.540903
$708$	0	0
$709$	33267.9	1.76221	0.881103	−	0.472925i	$-0.156802\pi$
$709$	33267.9	1.76221	0.881103	+	0.472925i	$0.156802\pi$
$710$	−52018.8	−2.74962
$711$	0	0
$712$	35364.1	1.86141
$713$	−258.973	−0.0136025
$714$	0	0
$715$	27659.7	1.44673
$716$	−11944.0	−0.623422
$717$	0	0
$718$	34869.4	1.81242
$719$	−8882.66	−0.460733	−0.230367	−	0.973104i	$-0.573993\pi$
$719$	−8882.66	−0.460733	−0.230367	+	0.973104i	$0.573993\pi$
$720$	0	0
$721$	393.877	0.0203450
$722$	−26871.9	−1.38514
$723$	0	0
$724$	12861.4	0.660207
$725$	−22204.9	−1.13748
$726$	0	0
$727$	−16988.8	−0.866687	−0.433343	−	0.901229i	$-0.642666\pi$
$727$	−16988.8	−0.866687	−0.433343	+	0.901229i	$0.642666\pi$
$728$	−4517.97	−0.230010
$729$	0	0
$730$	−25322.7	−1.28388
$731$	−23227.1	−1.17522
$732$	0	0
$733$	−21983.8	−1.10776	−0.553882	−	0.832595i	$-0.686854\pi$
$733$	−21983.8	−1.10776	−0.553882	+	0.832595i	$0.686854\pi$
$734$	20118.5	1.01170
$735$	0	0
$736$	−1558.97	−0.0780768
$737$	9556.12	0.477618
$738$	0	0
$739$	18909.8	0.941284	0.470642	−	0.882324i	$-0.344022\pi$
$739$	18909.8	0.941284	0.470642	+	0.882324i	$0.344022\pi$
$740$	−52365.9	−2.60136
$741$	0	0
$742$	−1785.82	−0.0883550
$743$	−10009.3	−0.494221	−0.247111	−	0.968987i	$-0.579481\pi$
$743$	−10009.3	−0.494221	−0.247111	+	0.968987i	$0.579481\pi$
$744$	0	0
$745$	−54759.6	−2.69294
$746$	21012.2	1.03125
$747$	0	0
$748$	−57097.0	−2.79101
$749$	2100.87	0.102489
$750$	0	0
$751$	−29253.7	−1.42142	−0.710708	−	0.703487i	$-0.751625\pi$
$751$	−29253.7	−1.42142	−0.710708	+	0.703487i	$0.751625\pi$
$752$	461.475	0.0223780
$753$	0	0
$754$	−9664.33	−0.466783
$755$	−15332.8	−0.739096
$756$	0	0
$757$	−20805.2	−0.998912	−0.499456	−	0.866339i	$-0.666467\pi$
$757$	−20805.2	−0.998912	−0.499456	+	0.866339i	$0.666467\pi$
$758$	66991.1	3.21006
$759$	0	0
$760$	14266.7	0.680931
$761$	10125.0	0.482303	0.241151	−	0.970487i	$-0.422475\pi$
$761$	10125.0	0.482303	0.241151	+	0.970487i	$0.422475\pi$
$762$	0	0
$763$	6558.34	0.311177
$764$	−49292.3	−2.33421
$765$	0	0
$766$	8322.81	0.392579
$767$	−19184.1	−0.903125
$768$	0	0
$769$	−37264.8	−1.74747	−0.873735	−	0.486402i	$-0.838309\pi$
$769$	−37264.8	−1.74747	−0.873735	+	0.486402i	$0.838309\pi$
$770$	−30152.3	−1.41119
$771$	0	0
$772$	−44457.1	−2.07260
$773$	16637.3	0.774130	0.387065	−	0.922052i	$-0.373489\pi$
$773$	16637.3	0.774130	0.387065	+	0.922052i	$0.373489\pi$
$774$	0	0
$775$	−9482.13	−0.439494
$776$	−6411.45	−0.296595
$777$	0	0
$778$	39308.8	1.81143
$779$	6625.90	0.304747
$780$	0	0
$781$	−24889.8	−1.14037
$782$	−3755.25	−0.171723
$783$	0	0
$784$	−102.766	−0.00468140
$785$	48790.7	2.21836
$786$	0	0
$787$	13119.8	0.594247	0.297123	−	0.954839i	$-0.403973\pi$
$787$	13119.8	0.594247	0.297123	+	0.954839i	$0.403973\pi$
$788$	−47354.4	−2.14077
$789$	0	0
$790$	−25607.7	−1.15327
$791$	−9762.16	−0.438815
$792$	0	0
$793$	10220.8	0.457694
$794$	−25086.2	−1.12125
$795$	0	0
$796$	−6179.47	−0.275158
$797$	−382.506	−0.0170001	−0.00850004	−	0.999964i	$-0.502706\pi$
$797$	−382.506	−0.0170001	−0.00850004	+	0.999964i	$0.502706\pi$
$798$	0	0
$799$	21575.8	0.955313
$800$	−57080.9	−2.52264
$801$	0	0
$802$	−39342.8	−1.73222
$803$	−12116.3	−0.532473
$804$	0	0
$805$	−1221.33	−0.0534737
$806$	−4126.94	−0.180354
$807$	0	0
$808$	−31993.8	−1.39299
$809$	2590.19	0.112566	0.0562831	−	0.998415i	$-0.482075\pi$
$809$	2590.19	0.112566	0.0562831	+	0.998415i	$0.482075\pi$
$810$	0	0
$811$	−20167.1	−0.873199	−0.436599	−	0.899656i	$-0.643817\pi$
$811$	−20167.1	−0.873199	−0.436599	+	0.899656i	$0.643817\pi$
$812$	6488.35	0.280414
$813$	0	0
$814$	−40683.7	−1.75180
$815$	−11892.5	−0.511136
$816$	0	0
$817$	7379.98	0.316025
$818$	42892.6	1.83338
$819$	0	0
$820$	56713.6	2.41527
$821$	29612.6	1.25881	0.629407	−	0.777076i	$-0.283298\pi$
$821$	29612.6	1.25881	0.629407	+	0.777076i	$0.283298\pi$
$822$	0	0
$823$	10236.6	0.433567	0.216784	−	0.976220i	$-0.430443\pi$
$823$	10236.6	0.433567	0.216784	+	0.976220i	$0.430443\pi$
$824$	−1239.30	−0.0523947
$825$	0	0
$826$	20912.9	0.880936
$827$	−8733.00	−0.367202	−0.183601	−	0.983001i	$-0.558775\pi$
$827$	−8733.00	−0.367202	−0.183601	+	0.983001i	$0.558775\pi$
$828$	0	0
$829$	10394.4	0.435481	0.217741	−	0.976007i	$-0.430131\pi$
$829$	10394.4	0.435481	0.217741	+	0.976007i	$0.430131\pi$
$830$	59326.3	2.48102
$831$	0	0
$832$	−24351.9	−1.01472
$833$	−4804.72	−0.199848
$834$	0	0
$835$	11305.5	0.468556
$836$	18141.6	0.750525
$837$	0	0
$838$	−2714.08	−0.111881
$839$	−28689.6	−1.18054	−0.590272	−	0.807205i	$-0.700979\pi$
$839$	−28689.6	−1.18054	−0.590272	+	0.807205i	$0.700979\pi$
$840$	0	0
$841$	−19166.6	−0.785869
$842$	14228.2	0.582347
$843$	0	0
$844$	14747.4	0.601454
$845$	−27823.8	−1.13274
$846$	0	0
$847$	−5110.19	−0.207306
$848$	−117.243	−0.00474781
$849$	0	0
$850$	−137496.	−5.54833
$851$	−1647.91	−0.0663804
$852$	0	0
$853$	22128.6	0.888240	0.444120	−	0.895967i	$-0.353516\pi$
$853$	22128.6	0.888240	0.444120	+	0.895967i	$0.353516\pi$
$854$	−11141.9	−0.446449
$855$	0	0
$856$	−6610.22	−0.263940
$857$	4787.51	0.190826	0.0954132	−	0.995438i	$-0.469583\pi$
$857$	4787.51	0.190826	0.0954132	+	0.995438i	$0.469583\pi$
$858$	0	0
$859$	11575.9	0.459794	0.229897	−	0.973215i	$-0.426161\pi$
$859$	11575.9	0.459794	0.229897	+	0.973215i	$0.426161\pi$
$860$	63168.0	2.50466
$861$	0	0
$862$	−38634.9	−1.52658
$863$	4683.61	0.184741	0.0923707	−	0.995725i	$-0.470556\pi$
$863$	4683.61	0.184741	0.0923707	+	0.995725i	$0.470556\pi$
$864$	0	0
$865$	42298.2	1.66264
$866$	24501.7	0.961436
$867$	0	0
$868$	2770.71	0.108346
$869$	−12252.7	−0.478302
$870$	0	0
$871$	6168.36	0.239962
$872$	−20635.3	−0.801377
$873$	0	0
$874$	1193.16	0.0461777
$875$	−26526.3	−1.02486
$876$	0	0
$877$	−22871.3	−0.880626	−0.440313	−	0.897844i	$-0.645132\pi$
$877$	−22871.3	−0.880626	−0.440313	+	0.897844i	$0.645132\pi$
$878$	−81198.5	−3.12109
$879$	0	0
$880$	−1979.57	−0.0758310
$881$	−20712.8	−0.792091	−0.396045	−	0.918231i	$-0.629618\pi$
$881$	−20712.8	−0.792091	−0.396045	+	0.918231i	$0.629618\pi$
$882$	0	0
$883$	−21641.0	−0.824775	−0.412388	−	0.911008i	$-0.635305\pi$
$883$	−21641.0	−0.824775	−0.412388	+	0.911008i	$0.635305\pi$
$884$	−36855.5	−1.40224
$885$	0	0
$886$	40449.7	1.53379
$887$	26958.9	1.02051	0.510255	−	0.860023i	$-0.329551\pi$
$887$	26958.9	1.02051	0.510255	+	0.860023i	$0.329551\pi$
$888$	0	0
$889$	16711.6	0.630470
$890$	152345.	5.73776
$891$	0	0
$892$	−25857.1	−0.970585
$893$	−6855.31	−0.256891
$894$	0	0
$895$	−19360.9	−0.723089
$896$	16143.2	0.601906
$897$	0	0
$898$	520.774	0.0193524
$899$	2230.12	0.0827351
$900$	0	0
$901$	−5481.58	−0.202683
$902$	44061.5	1.62648
$903$	0	0
$904$	30715.9	1.13008
$905$	20847.9	0.765755
$906$	0	0
$907$	−20368.3	−0.745665	−0.372833	−	0.927899i	$-0.621613\pi$
$907$	−20368.3	−0.745665	−0.372833	+	0.927899i	$0.621613\pi$
$908$	39020.0	1.42613
$909$	0	0
$910$	−19462.9	−0.709001
$911$	21501.8	0.781983	0.390991	−	0.920394i	$-0.372132\pi$
$911$	21501.8	0.781983	0.390991	+	0.920394i	$0.372132\pi$
$912$	0	0
$913$	28386.3	1.02897
$914$	36794.2	1.33156
$915$	0	0
$916$	52348.3	1.88825
$917$	−14839.9	−0.534414
$918$	0	0
$919$	−2461.02	−0.0883369	−0.0441685	−	0.999024i	$-0.514064\pi$
$919$	−2461.02	−0.0883369	−0.0441685	+	0.999024i	$0.514064\pi$
$920$	3842.83	0.137711
$921$	0	0
$922$	18847.0	0.673202
$923$	−16066.1	−0.572938
$924$	0	0
$925$	−60337.4	−2.14474
$926$	53934.6	1.91404
$927$	0	0
$928$	13425.0	0.474889
$929$	1732.09	0.0611713	0.0305857	−	0.999532i	$-0.490263\pi$
$929$	1732.09	0.0611713	0.0305857	+	0.999532i	$0.490263\pi$
$930$	0	0
$931$	1526.61	0.0537409
$932$	34214.9	1.20252
$933$	0	0
$934$	7257.26	0.254245
$935$	−92552.7	−3.23721
$936$	0	0
$937$	35120.9	1.22449	0.612247	−	0.790667i	$-0.290266\pi$
$937$	35120.9	1.22449	0.612247	+	0.790667i	$0.290266\pi$
$938$	−6724.24	−0.234067
$939$	0	0
$940$	−58677.1	−2.03600
$941$	−12928.0	−0.447864	−0.223932	−	0.974605i	$-0.571889\pi$
$941$	−12928.0	−0.447864	−0.223932	+	0.974605i	$0.571889\pi$
$942$	0	0
$943$	1784.73	0.0616319
$944$	1372.98	0.0473376
$945$	0	0
$946$	49076.1	1.68668
$947$	−3202.94	−0.109907	−0.0549533	−	0.998489i	$-0.517501\pi$
$947$	−3202.94	−0.109907	−0.0549533	+	0.998489i	$0.517501\pi$
$948$	0	0
$949$	−7820.94	−0.267522
$950$	43686.9	1.49199
$951$	0	0
$952$	15117.7	0.514672
$953$	−36827.2	−1.25178	−0.625891	−	0.779910i	$-0.715265\pi$
$953$	−36827.2	−1.25178	−0.625891	+	0.779910i	$0.715265\pi$
$954$	0	0
$955$	−79901.5	−2.70738
$956$	−76606.4	−2.59166
$957$	0	0
$958$	59940.6	2.02149
$959$	461.311	0.0155334
$960$	0	0
$961$	−28838.7	−0.968033
$962$	−26260.9	−0.880129
$963$	0	0
$964$	−33877.5	−1.13187
$965$	−72063.7	−2.40395
$966$	0	0
$967$	981.048	0.0326250	0.0163125	−	0.999867i	$-0.494807\pi$
$967$	981.048	0.0326250	0.0163125	+	0.999867i	$0.494807\pi$
$968$	16078.8	0.533877
$969$	0	0
$970$	−27619.8	−0.914247
$971$	35299.7	1.16666	0.583328	−	0.812237i	$-0.301750\pi$
$971$	35299.7	1.16666	0.583328	+	0.812237i	$0.301750\pi$
$972$	0	0
$973$	13743.0	0.452805
$974$	−83686.6	−2.75307
$975$	0	0
$976$	−731.490	−0.0239902
$977$	−39605.6	−1.29692	−0.648462	−	0.761247i	$-0.724587\pi$
$977$	−39605.6	−1.29692	−0.648462	+	0.761247i	$0.724587\pi$
$978$	0	0
$979$	72893.4	2.37966
$980$	13066.8	0.425924
$981$	0	0
$982$	−36952.1	−1.20080
$983$	−11669.4	−0.378633	−0.189317	−	0.981916i	$-0.560627\pi$
$983$	−11669.4	−0.378633	−0.189317	+	0.981916i	$0.560627\pi$
$984$	0	0
$985$	−76760.1	−2.48303
$986$	32338.1	1.04448
$987$	0	0
$988$	11710.2	0.377075
$989$	1987.85	0.0639129
$990$	0	0
$991$	37428.6	1.19976	0.599879	−	0.800091i	$-0.295215\pi$
$991$	37428.6	1.19976	0.599879	+	0.800091i	$0.295215\pi$
$992$	5732.85	0.183486
$993$	0	0
$994$	17513.9	0.558861
$995$	−10016.7	−0.319148
$996$	0	0
$997$	10612.5	0.337111	0.168556	−	0.985692i	$-0.446090\pi$
$997$	10612.5	0.337111	0.168556	+	0.985692i	$0.446090\pi$
$998$	−2748.89	−0.0871891
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.i.1.7		8
3.2	odd	2		567.4.a.g.1.2		8
9.2	odd	6		189.4.f.b.64.7		16
9.4	even	3		63.4.f.b.43.2	yes	16
9.5	odd	6		189.4.f.b.127.7		16
9.7	even	3		63.4.f.b.22.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.2	✓	16	9.7	even	3
63.4.f.b.43.2	yes	16	9.4	even	3
189.4.f.b.64.7		16	9.2	odd	6
189.4.f.b.127.7		16	9.5	odd	6
567.4.a.g.1.2		8	3.2	odd	2
567.4.a.i.1.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.56358 q^{2} +12.8263 q^{4} +20.7910 q^{5} -7.00000 q^{7} +22.0250 q^{8} +O(q^{10})\)	\(q+4.56358 q^{2} +12.8263 q^{4} +20.7910 q^{5} -7.00000 q^{7} +22.0250 q^{8} +94.8813 q^{10} +45.3985 q^{11} +29.3042 q^{13} -31.9451 q^{14} -2.09727 q^{16} -98.0555 q^{17} +31.1554 q^{19} +266.670 q^{20} +207.180 q^{22} +8.39191 q^{23} +307.265 q^{25} +133.732 q^{26} -89.7838 q^{28} -72.2664 q^{29} -30.8598 q^{31} -185.771 q^{32} -447.484 q^{34} -145.537 q^{35} -196.369 q^{37} +142.180 q^{38} +457.921 q^{40} +212.673 q^{41} +236.877 q^{43} +582.293 q^{44} +38.2972 q^{46} -220.036 q^{47} +49.0000 q^{49} +1402.23 q^{50} +375.863 q^{52} +55.9028 q^{53} +943.880 q^{55} -154.175 q^{56} -329.793 q^{58} -654.652 q^{59} +348.783 q^{61} -140.831 q^{62} -831.002 q^{64} +609.263 q^{65} +210.494 q^{67} -1257.68 q^{68} -664.169 q^{70} -548.252 q^{71} -266.888 q^{73} -896.146 q^{74} +399.607 q^{76} -317.790 q^{77} -269.892 q^{79} -43.6042 q^{80} +970.550 q^{82} +625.269 q^{83} -2038.67 q^{85} +1081.01 q^{86} +999.902 q^{88} +1605.63 q^{89} -205.129 q^{91} +107.637 q^{92} -1004.15 q^{94} +647.751 q^{95} -291.099 q^{97} +223.615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) \) 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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