LMFDB - Embedded newform 531.4.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
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} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.26905$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} +O(q^{10})$	\(q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} -18.8859 q^{10} +70.2303 q^{11} -0.125351 q^{13} +28.5413 q^{14} +27.9422 q^{16} -57.2801 q^{17} +40.8191 q^{19} +95.0887 q^{20} +89.1255 q^{22} -190.065 q^{23} +96.4733 q^{25} -0.159076 q^{26} -143.702 q^{28} -133.399 q^{29} +129.061 q^{31} +181.548 q^{32} -72.6911 q^{34} -334.701 q^{35} -364.329 q^{37} +51.8013 q^{38} +271.759 q^{40} -195.632 q^{41} +31.6882 q^{43} -448.738 q^{44} -241.201 q^{46} -479.330 q^{47} +162.815 q^{49} +122.429 q^{50} +0.800930 q^{52} +402.892 q^{53} -1045.17 q^{55} -410.695 q^{56} -169.290 q^{58} -59.0000 q^{59} -209.849 q^{61} +163.785 q^{62} +6.85497 q^{64} +1.86547 q^{65} -455.024 q^{67} +365.992 q^{68} -424.750 q^{70} -203.497 q^{71} -177.712 q^{73} -462.350 q^{74} -260.815 q^{76} +1579.50 q^{77} -491.575 q^{79} -415.835 q^{80} -248.266 q^{82} +717.862 q^{83} +852.441 q^{85} +40.2137 q^{86} -1282.47 q^{88} -1306.23 q^{89} -2.81918 q^{91} +1214.42 q^{92} -608.291 q^{94} -607.469 q^{95} +538.879 q^{97} +206.620 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * 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 $

$2$	1.26905	0.448676	0.224338	−	0.974511i	$-0.427978\pi$
$2$	1.26905	0.448676	0.224338	+	0.974511i	$0.427978\pi$
$3$	0	0
$3$	0	0
$4$	−6.38952	−0.798690
$5$	−14.8820	−1.33108	−0.665542	−	0.746360i	$-0.731800\pi$
$5$	−14.8820	−1.33108	−0.665542	+	0.746360i	$0.731800\pi$
$6$	0	0
$7$	22.4903	1.21436	0.607182	−	0.794563i	$-0.292300\pi$
$7$	22.4903	1.21436	0.607182	+	0.794563i	$0.292300\pi$
$8$	−18.2610	−0.807028
$9$	0	0
$10$	−18.8859	−0.597225
$11$	70.2303	1.92502	0.962510	−	0.271244i	$-0.0874352\pi$
$11$	70.2303	1.92502	0.962510	+	0.271244i	$0.0874352\pi$
$12$	0	0
$13$	−0.125351	−0.00267431	−0.00133715	−	0.999999i	$-0.500426\pi$
$13$	−0.125351	−0.00267431	−0.00133715	+	0.999999i	$0.500426\pi$
$14$	28.5413	0.544855
$15$	0	0
$16$	27.9422	0.436596
$17$	−57.2801	−0.817203	−0.408602	−	0.912713i	$-0.633983\pi$
$17$	−57.2801	−0.817203	−0.408602	+	0.912713i	$0.633983\pi$
$18$	0	0
$19$	40.8191	0.492871	0.246435	−	0.969159i	$-0.420741\pi$
$19$	40.8191	0.492871	0.246435	+	0.969159i	$0.420741\pi$
$20$	95.0887	1.06312
$21$	0	0
$22$	89.1255	0.863710
$23$	−190.065	−1.72310	−0.861550	−	0.507673i	$-0.830506\pi$
$23$	−190.065	−1.72310	−0.861550	+	0.507673i	$0.830506\pi$
$24$	0	0
$25$	96.4733	0.771787
$26$	−0.159076	−0.00119990
$27$	0	0
$28$	−143.702	−0.969900
$29$	−133.399	−0.854193	−0.427096	−	0.904206i	$-0.640463\pi$
$29$	−133.399	−0.854193	−0.427096	+	0.904206i	$0.640463\pi$
$30$	0	0
$31$	129.061	0.747745	0.373872	−	0.927480i	$-0.378030\pi$
$31$	129.061	0.747745	0.373872	+	0.927480i	$0.378030\pi$
$32$	181.548	1.00292
$33$	0	0
$34$	−72.6911	−0.366659
$35$	−334.701	−1.61642
$36$	0	0
$37$	−364.329	−1.61879	−0.809396	−	0.587263i	$-0.800205\pi$
$37$	−364.329	−1.61879	−0.809396	+	0.587263i	$0.800205\pi$
$38$	51.8013	0.221139
$39$	0	0
$40$	271.759	1.07422
$41$	−195.632	−0.745184	−0.372592	−	0.927995i	$-0.621531\pi$
$41$	−195.632	−0.745184	−0.372592	+	0.927995i	$0.621531\pi$
$42$	0	0
$43$	31.6882	0.112381	0.0561907	−	0.998420i	$-0.482105\pi$
$43$	31.6882	0.112381	0.0561907	+	0.998420i	$0.482105\pi$
$44$	−448.738	−1.53750
$45$	0	0
$46$	−241.201	−0.773113
$47$	−479.330	−1.48760	−0.743802	−	0.668400i	$-0.766980\pi$
$47$	−479.330	−1.48760	−0.743802	+	0.668400i	$0.766980\pi$
$48$	0	0
$49$	162.815	0.474679
$50$	122.429	0.346282
$51$	0	0
$52$	0.800930	0.00213594
$53$	402.892	1.04418	0.522090	−	0.852891i	$-0.325153\pi$
$53$	402.892	1.04418	0.522090	+	0.852891i	$0.325153\pi$
$54$	0	0
$55$	−1045.17	−2.56237
$56$	−410.695	−0.980026
$57$	0	0
$58$	−169.290	−0.383255
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−209.849	−0.440465	−0.220232	−	0.975447i	$-0.570682\pi$
$61$	−209.849	−0.440465	−0.220232	+	0.975447i	$0.570682\pi$
$62$	163.785	0.335495
$63$	0	0
$64$	6.85497	0.0133886
$65$	1.86547	0.00355973
$66$	0	0
$67$	−455.024	−0.829702	−0.414851	−	0.909889i	$-0.636166\pi$
$67$	−455.024	−0.829702	−0.414851	+	0.909889i	$0.636166\pi$
$68$	365.992	0.652692
$69$	0	0
$70$	−424.750	−0.725248
$71$	−203.497	−0.340151	−0.170075	−	0.985431i	$-0.554401\pi$
$71$	−203.497	−0.340151	−0.170075	+	0.985431i	$0.554401\pi$
$72$	0	0
$73$	−177.712	−0.284927	−0.142463	−	0.989800i	$-0.545502\pi$
$73$	−177.712	−0.284927	−0.142463	+	0.989800i	$0.545502\pi$
$74$	−462.350	−0.726313
$75$	0	0
$76$	−260.815	−0.393651
$77$	1579.50	2.33768
$78$	0	0
$79$	−491.575	−0.700082	−0.350041	−	0.936734i	$-0.613832\pi$
$79$	−491.575	−0.700082	−0.350041	+	0.936734i	$0.613832\pi$
$80$	−415.835	−0.581147
$81$	0	0
$82$	−248.266	−0.334346
$83$	717.862	0.949345	0.474673	−	0.880162i	$-0.342567\pi$
$83$	717.862	0.949345	0.474673	+	0.880162i	$0.342567\pi$
$84$	0	0
$85$	852.441	1.08777
$86$	40.2137	0.0504228
$87$	0	0
$88$	−1282.47	−1.55355
$89$	−1306.23	−1.55573	−0.777865	−	0.628431i	$-0.783697\pi$
$89$	−1306.23	−1.55573	−0.777865	+	0.628431i	$0.783697\pi$
$90$	0	0
$91$	−2.81918	−0.00324758
$92$	1214.42	1.37622
$93$	0	0
$94$	−608.291	−0.667452
$95$	−607.469	−0.656053
$96$	0	0
$97$	538.879	0.564071	0.282036	−	0.959404i	$-0.408990\pi$
$97$	538.879	0.564071	0.282036	+	0.959404i	$0.408990\pi$
$98$	206.620	0.212977
$99$	0	0
$100$	−616.419	−0.616419
$101$	1585.19	1.56171	0.780854	−	0.624714i	$-0.214784\pi$
$101$	1585.19	1.56171	0.780854	+	0.624714i	$0.214784\pi$
$102$	0	0
$103$	−512.174	−0.489961	−0.244980	−	0.969528i	$-0.578782\pi$
$103$	−512.174	−0.489961	−0.244980	+	0.969528i	$0.578782\pi$
$104$	2.28902	0.00215824
$105$	0	0
$106$	511.289	0.468498
$107$	−90.3234	−0.0816065	−0.0408033	−	0.999167i	$-0.512992\pi$
$107$	−90.3234	−0.0816065	−0.0408033	+	0.999167i	$0.512992\pi$
$108$	0	0
$109$	−2241.55	−1.96974	−0.984870	−	0.173297i	$-0.944558\pi$
$109$	−2241.55	−1.96974	−0.984870	+	0.173297i	$0.944558\pi$
$110$	−1326.36	−1.14967
$111$	0	0
$112$	628.428	0.530187
$113$	1538.31	1.28064	0.640319	−	0.768109i	$-0.278802\pi$
$113$	1538.31	1.28064	0.640319	+	0.768109i	$0.278802\pi$
$114$	0	0
$115$	2828.54	2.29359
$116$	852.356	0.682235
$117$	0	0
$118$	−74.8737	−0.0584126
$119$	−1288.25	−0.992382
$120$	0	0
$121$	3601.29	2.70571
$122$	−266.308	−0.197626
$123$	0	0
$124$	−824.640	−0.597216
$125$	424.533	0.303771
$126$	0	0
$127$	1230.53	0.859780	0.429890	−	0.902881i	$-0.358552\pi$
$127$	1230.53	0.859780	0.429890	+	0.902881i	$0.358552\pi$
$128$	−1443.68	−0.996911
$129$	0	0
$130$	2.36736	0.00159716
$131$	−972.455	−0.648578	−0.324289	−	0.945958i	$-0.605125\pi$
$131$	−972.455	−0.648578	−0.324289	+	0.945958i	$0.605125\pi$
$132$	0	0
$133$	918.035	0.598525
$134$	−577.447	−0.372267
$135$	0	0
$136$	1045.99	0.659506
$137$	335.709	0.209355	0.104677	−	0.994506i	$-0.466619\pi$
$137$	335.709	0.209355	0.104677	+	0.994506i	$0.466619\pi$
$138$	0	0
$139$	−2222.42	−1.35614	−0.678068	−	0.734999i	$-0.737183\pi$
$139$	−2222.42	−1.35614	−0.678068	+	0.734999i	$0.737183\pi$
$140$	2138.58	1.29102
$141$	0	0
$142$	−258.248	−0.152617
$143$	−8.80341	−0.00514810
$144$	0	0
$145$	1985.24	1.13700
$146$	−225.525	−0.127840
$147$	0	0
$148$	2327.89	1.29291
$149$	−1467.58	−0.806906	−0.403453	−	0.915000i	$-0.632190\pi$
$149$	−1467.58	−0.806906	−0.403453	+	0.915000i	$0.632190\pi$
$150$	0	0
$151$	−2400.31	−1.29361	−0.646804	−	0.762656i	$-0.723895\pi$
$151$	−2400.31	−1.29361	−0.646804	+	0.762656i	$0.723895\pi$
$152$	−745.397	−0.397761
$153$	0	0
$154$	2004.46	1.04886
$155$	−1920.69	−0.995312
$156$	0	0
$157$	−3571.25	−1.81539	−0.907697	−	0.419627i	$-0.862161\pi$
$157$	−3571.25	−1.81539	−0.907697	+	0.419627i	$0.862161\pi$
$158$	−623.831	−0.314110
$159$	0	0
$160$	−2701.79	−1.33497
$161$	−4274.62	−2.09247
$162$	0	0
$163$	−1305.57	−0.627365	−0.313683	−	0.949528i	$-0.601563\pi$
$163$	−1305.57	−0.627365	−0.313683	+	0.949528i	$0.601563\pi$
$164$	1249.99	0.595172
$165$	0	0
$166$	911.000	0.425948
$167$	−156.555	−0.0725423	−0.0362711	−	0.999342i	$-0.511548\pi$
$167$	−156.555	−0.0725423	−0.0362711	+	0.999342i	$0.511548\pi$
$168$	0	0
$169$	−2196.98	−0.999993
$170$	1081.79	0.488054
$171$	0	0
$172$	−202.472	−0.0897579
$173$	−357.266	−0.157008	−0.0785042	−	0.996914i	$-0.525014\pi$
$173$	−357.266	−0.157008	−0.0785042	+	0.996914i	$0.525014\pi$
$174$	0	0
$175$	2169.72	0.937230
$176$	1962.39	0.840457
$177$	0	0
$178$	−1657.66	−0.698018
$179$	−1581.04	−0.660183	−0.330092	−	0.943949i	$-0.607080\pi$
$179$	−1581.04	−0.660183	−0.330092	+	0.943949i	$0.607080\pi$
$180$	0	0
$181$	1674.78	0.687766	0.343883	−	0.939012i	$-0.388258\pi$
$181$	1674.78	0.687766	0.343883	+	0.939012i	$0.388258\pi$
$182$	−3.57766	−0.00145711
$183$	0	0
$184$	3470.77	1.39059
$185$	5421.94	2.15475
$186$	0	0
$187$	−4022.80	−1.57313
$188$	3062.69	1.18814
$189$	0	0
$190$	−770.907	−0.294355
$191$	−5015.44	−1.90002	−0.950011	−	0.312217i	$-0.898929\pi$
$191$	−5015.44	−1.90002	−0.950011	+	0.312217i	$0.898929\pi$
$192$	0	0
$193$	1296.37	0.483496	0.241748	−	0.970339i	$-0.422279\pi$
$193$	1296.37	0.483496	0.241748	+	0.970339i	$0.422279\pi$
$194$	683.863	0.253085
$195$	0	0
$196$	−1040.31	−0.379121
$197$	2894.70	1.04690	0.523449	−	0.852057i	$-0.324645\pi$
$197$	2894.70	1.04690	0.523449	+	0.852057i	$0.324645\pi$
$198$	0	0
$199$	−2835.15	−1.00994	−0.504971	−	0.863136i	$-0.668497\pi$
$199$	−2835.15	−1.00994	−0.504971	+	0.863136i	$0.668497\pi$
$200$	−1761.70	−0.622854
$201$	0	0
$202$	2011.68	0.700700
$203$	−3000.19	−1.03730
$204$	0	0
$205$	2911.39	0.991904
$206$	−649.972	−0.219833
$207$	0	0
$208$	−3.50257	−0.00116759
$209$	2866.74	0.948787
$210$	0	0
$211$	894.249	0.291766	0.145883	−	0.989302i	$-0.453398\pi$
$211$	894.249	0.291766	0.145883	+	0.989302i	$0.453398\pi$
$212$	−2574.29	−0.833976
$213$	0	0
$214$	−114.625	−0.0366148
$215$	−471.583	−0.149589
$216$	0	0
$217$	2902.63	0.908034
$218$	−2844.63	−0.883774
$219$	0	0
$220$	6678.11	2.04654
$221$	7.18009	0.00218545
$222$	0	0
$223$	2418.78	0.726338	0.363169	−	0.931723i	$-0.381695\pi$
$223$	2418.78	0.726338	0.363169	+	0.931723i	$0.381695\pi$
$224$	4083.07	1.21791
$225$	0	0
$226$	1952.19	0.574591
$227$	1116.89	0.326565	0.163283	−	0.986579i	$-0.447792\pi$
$227$	1116.89	0.326565	0.163283	+	0.986579i	$0.447792\pi$
$228$	0	0
$229$	−4003.70	−1.15534	−0.577668	−	0.816272i	$-0.696037\pi$
$229$	−4003.70	−1.15534	−0.577668	+	0.816272i	$0.696037\pi$
$230$	3589.55	1.02908
$231$	0	0
$232$	2436.00	0.689358
$233$	3831.54	1.07731	0.538654	−	0.842527i	$-0.318933\pi$
$233$	3831.54	1.07731	0.538654	+	0.842527i	$0.318933\pi$
$234$	0	0
$235$	7133.37	1.98013
$236$	376.982	0.103981
$237$	0	0
$238$	−1634.85	−0.445258
$239$	665.336	0.180071	0.0900356	−	0.995939i	$-0.471302\pi$
$239$	665.336	0.180071	0.0900356	+	0.995939i	$0.471302\pi$
$240$	0	0
$241$	6021.52	1.60946	0.804730	−	0.593640i	$-0.202310\pi$
$241$	6021.52	1.60946	0.804730	+	0.593640i	$0.202310\pi$
$242$	4570.21	1.21398
$243$	0	0
$244$	1340.83	0.351795
$245$	−2423.01	−0.631838
$246$	0	0
$247$	−5.11670	−0.00131809
$248$	−2356.78	−0.603451
$249$	0	0
$250$	538.752	0.136295
$251$	1854.47	0.466347	0.233173	−	0.972435i	$-0.425089\pi$
$251$	1854.47	0.466347	0.233173	+	0.972435i	$0.425089\pi$
$252$	0	0
$253$	−13348.3	−3.31700
$254$	1561.60	0.385762
$255$	0	0
$256$	−1886.94	−0.460678
$257$	3407.16	0.826976	0.413488	−	0.910510i	$-0.364310\pi$
$257$	3407.16	0.826976	0.413488	+	0.910510i	$0.364310\pi$
$258$	0	0
$259$	−8193.88	−1.96580
$260$	−11.9194	−0.00284312
$261$	0	0
$262$	−1234.09	−0.291001
$263$	2085.60	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$263$	2085.60	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$264$	0	0
$265$	−5995.83	−1.38989
$266$	1165.03	0.268543
$267$	0	0
$268$	2907.39	0.662675
$269$	6495.06	1.47216	0.736080	−	0.676895i	$-0.236675\pi$
$269$	6495.06	1.47216	0.736080	+	0.676895i	$0.236675\pi$
$270$	0	0
$271$	−3252.61	−0.729085	−0.364543	−	0.931187i	$-0.618775\pi$
$271$	−3252.61	−0.729085	−0.364543	+	0.931187i	$0.618775\pi$
$272$	−1600.53	−0.356788
$273$	0	0
$274$	426.031	0.0939323
$275$	6775.35	1.48571
$276$	0	0
$277$	5905.21	1.28090	0.640451	−	0.767999i	$-0.278747\pi$
$277$	5905.21	1.28090	0.640451	+	0.767999i	$0.278747\pi$
$278$	−2820.35	−0.608465
$279$	0	0
$280$	6111.96	1.30450
$281$	−6504.05	−1.38078	−0.690390	−	0.723437i	$-0.742561\pi$
$281$	−6504.05	−1.38078	−0.690390	+	0.723437i	$0.742561\pi$
$282$	0	0
$283$	6587.83	1.38377	0.691883	−	0.722010i	$-0.256782\pi$
$283$	6587.83	1.38377	0.691883	+	0.722010i	$0.256782\pi$
$284$	1300.25	0.271675
$285$	0	0
$286$	−11.1719	−0.00230983
$287$	−4399.82	−0.904925
$288$	0	0
$289$	−1631.99	−0.332179
$290$	2519.36	0.510145
$291$	0	0
$292$	1135.50	0.227568
$293$	−4655.68	−0.928285	−0.464142	−	0.885761i	$-0.653637\pi$
$293$	−4655.68	−0.928285	−0.464142	+	0.885761i	$0.653637\pi$
$294$	0	0
$295$	878.037	0.173292
$296$	6653.00	1.30641
$297$	0	0
$298$	−1862.43	−0.362039
$299$	23.8248	0.00460810
$300$	0	0
$301$	712.677	0.136472
$302$	−3046.11	−0.580410
$303$	0	0
$304$	1140.57	0.215186
$305$	3122.96	0.586296
$306$	0	0
$307$	−1381.12	−0.256758	−0.128379	−	0.991725i	$-0.540977\pi$
$307$	−1381.12	−0.256758	−0.128379	+	0.991725i	$0.540977\pi$
$308$	−10092.3	−1.86708
$309$	0	0
$310$	−2437.44	−0.446572
$311$	−2269.90	−0.413873	−0.206936	−	0.978354i	$-0.566349\pi$
$311$	−2269.90	−0.413873	−0.206936	+	0.978354i	$0.566349\pi$
$312$	0	0
$313$	−487.625	−0.0880581	−0.0440290	−	0.999030i	$-0.514019\pi$
$313$	−487.625	−0.0880581	−0.0440290	+	0.999030i	$0.514019\pi$
$314$	−4532.08	−0.814523
$315$	0	0
$316$	3140.93	0.559149
$317$	−8517.00	−1.50903	−0.754515	−	0.656283i	$-0.772128\pi$
$317$	−8517.00	−1.50903	−0.754515	+	0.656283i	$0.772128\pi$
$318$	0	0
$319$	−9368.66	−1.64434
$320$	−102.015	−0.0178214
$321$	0	0
$322$	−5424.69	−0.938840
$323$	−2338.12	−0.402776
$324$	0	0
$325$	−12.0930	−0.00206400
$326$	−1656.83	−0.281483
$327$	0	0
$328$	3572.43	0.601385
$329$	−10780.3	−1.80649
$330$	0	0
$331$	10338.9	1.71685	0.858423	−	0.512942i	$-0.171444\pi$
$331$	10338.9	1.71685	0.858423	+	0.512942i	$0.171444\pi$
$332$	−4586.80	−0.758233
$333$	0	0
$334$	−198.675	−0.0325480
$335$	6771.66	1.10440
$336$	0	0
$337$	−344.378	−0.0556660	−0.0278330	−	0.999613i	$-0.508861\pi$
$337$	−344.378	−0.0556660	−0.0278330	+	0.999613i	$0.508861\pi$
$338$	−2788.07	−0.448672
$339$	0	0
$340$	−5446.69	−0.868789
$341$	9064.01	1.43942
$342$	0	0
$343$	−4052.42	−0.637931
$344$	−578.656	−0.0906949
$345$	0	0
$346$	−453.387	−0.0704458
$347$	−4032.71	−0.623883	−0.311942	−	0.950101i	$-0.600979\pi$
$347$	−4032.71	−0.623883	−0.311942	+	0.950101i	$0.600979\pi$
$348$	0	0
$349$	6299.32	0.966175	0.483087	−	0.875572i	$-0.339515\pi$
$349$	6299.32	0.966175	0.483087	+	0.875572i	$0.339515\pi$
$350$	2753.47	0.420512
$351$	0	0
$352$	12750.1	1.93064
$353$	8240.96	1.24256	0.621278	−	0.783590i	$-0.286614\pi$
$353$	8240.96	1.24256	0.621278	+	0.783590i	$0.286614\pi$
$354$	0	0
$355$	3028.44	0.452769
$356$	8346.18	1.24255
$357$	0	0
$358$	−2006.42	−0.296208
$359$	6605.66	0.971124	0.485562	−	0.874202i	$-0.338615\pi$
$359$	6605.66	0.971124	0.485562	+	0.874202i	$0.338615\pi$
$360$	0	0
$361$	−5192.80	−0.757078
$362$	2125.38	0.308584
$363$	0	0
$364$	18.0132	0.00259381
$365$	2644.71	0.379262
$366$	0	0
$367$	10563.8	1.50253	0.751263	−	0.660003i	$-0.229445\pi$
$367$	10563.8	1.50253	0.751263	+	0.660003i	$0.229445\pi$
$368$	−5310.83	−0.752299
$369$	0	0
$370$	6880.69	0.966784
$371$	9061.18	1.26801
$372$	0	0
$373$	−9588.55	−1.33104	−0.665518	−	0.746382i	$-0.731789\pi$
$373$	−9588.55	−1.33104	−0.665518	+	0.746382i	$0.731789\pi$
$374$	−5105.11	−0.705827
$375$	0	0
$376$	8753.02	1.20054
$377$	16.7217	0.00228437
$378$	0	0
$379$	8045.04	1.09036	0.545180	−	0.838319i	$-0.316461\pi$
$379$	8045.04	1.09036	0.545180	+	0.838319i	$0.316461\pi$
$380$	3881.44	0.523983
$381$	0	0
$382$	−6364.82	−0.852493
$383$	13181.7	1.75863	0.879314	−	0.476242i	$-0.158001\pi$
$383$	13181.7	1.75863	0.879314	+	0.476242i	$0.158001\pi$
$384$	0	0
$385$	−23506.1	−3.11164
$386$	1645.15	0.216933
$387$	0	0
$388$	−3443.18	−0.450518
$389$	−1629.56	−0.212396	−0.106198	−	0.994345i	$-0.533868\pi$
$389$	−1629.56	−0.212396	−0.106198	+	0.994345i	$0.533868\pi$
$390$	0	0
$391$	10886.9	1.40812
$392$	−2973.16	−0.383079
$393$	0	0
$394$	3673.51	0.469718
$395$	7315.61	0.931869
$396$	0	0
$397$	2405.86	0.304148	0.152074	−	0.988369i	$-0.451405\pi$
$397$	2405.86	0.304148	0.152074	+	0.988369i	$0.451405\pi$
$398$	−3597.94	−0.453136
$399$	0	0
$400$	2695.67	0.336959
$401$	−4383.33	−0.545868	−0.272934	−	0.962033i	$-0.587994\pi$
$401$	−4383.33	−0.545868	−0.272934	+	0.962033i	$0.587994\pi$
$402$	0	0
$403$	−16.1779	−0.00199970
$404$	−10128.6	−1.24732
$405$	0	0
$406$	−3807.38	−0.465411
$407$	−25586.9	−3.11621
$408$	0	0
$409$	−8567.44	−1.03578	−0.517888	−	0.855448i	$-0.673282\pi$
$409$	−8567.44	−1.03578	−0.517888	+	0.855448i	$0.673282\pi$
$410$	3694.69	0.445043
$411$	0	0
$412$	3272.54	0.391327
$413$	−1326.93	−0.158097
$414$	0	0
$415$	−10683.2	−1.26366
$416$	−22.7571	−0.00268211
$417$	0	0
$418$	3638.02	0.425698
$419$	−7908.61	−0.922102	−0.461051	−	0.887374i	$-0.652528\pi$
$419$	−7908.61	−0.922102	−0.461051	+	0.887374i	$0.652528\pi$
$420$	0	0
$421$	4262.15	0.493407	0.246703	−	0.969091i	$-0.420653\pi$
$421$	4262.15	0.493407	0.246703	+	0.969091i	$0.420653\pi$
$422$	1134.84	0.130908
$423$	0	0
$424$	−7357.20	−0.842682
$425$	−5526.00	−0.630707
$426$	0	0
$427$	−4719.56	−0.534885
$428$	577.124	0.0651783
$429$	0	0
$430$	−598.460	−0.0671170
$431$	−12851.4	−1.43627	−0.718134	−	0.695905i	$-0.755003\pi$
$431$	−12851.4	−1.43627	−0.718134	+	0.695905i	$0.755003\pi$
$432$	0	0
$433$	−10952.4	−1.21556	−0.607782	−	0.794104i	$-0.707940\pi$
$433$	−10952.4	−1.21556	−0.607782	+	0.794104i	$0.707940\pi$
$434$	3683.57	0.407413
$435$	0	0
$436$	14322.4	1.57321
$437$	−7758.28	−0.849266
$438$	0	0
$439$	−12153.2	−1.32128	−0.660638	−	0.750705i	$-0.729714\pi$
$439$	−12153.2	−1.32128	−0.660638	+	0.750705i	$0.729714\pi$
$440$	19085.7	2.06790
$441$	0	0
$442$	9.11187	0.000980560 0
$443$	−11624.3	−1.24670	−0.623348	−	0.781944i	$-0.714228\pi$
$443$	−11624.3	−1.24670	−0.623348	+	0.781944i	$0.714228\pi$
$444$	0	0
$445$	19439.3	2.07081
$446$	3069.54	0.325890
$447$	0	0
$448$	154.170	0.0162586
$449$	9152.69	0.962009	0.481005	−	0.876718i	$-0.340272\pi$
$449$	9152.69	0.962009	0.481005	+	0.876718i	$0.340272\pi$
$450$	0	0
$451$	−13739.3	−1.43450
$452$	−9829.08	−1.02283
$453$	0	0
$454$	1417.38	0.146522
$455$	41.9549	0.00432281
$456$	0	0
$457$	−1864.25	−0.190822	−0.0954112	−	0.995438i	$-0.530417\pi$
$457$	−1864.25	−0.190822	−0.0954112	+	0.995438i	$0.530417\pi$
$458$	−5080.88	−0.518371
$459$	0	0
$460$	−18073.0	−1.83187
$461$	19215.7	1.94136	0.970679	−	0.240377i	$-0.0772712\pi$
$461$	19215.7	1.94136	0.970679	+	0.240377i	$0.0772712\pi$
$462$	0	0
$463$	7787.00	0.781626	0.390813	−	0.920470i	$-0.372194\pi$
$463$	7787.00	0.781626	0.390813	+	0.920470i	$0.372194\pi$
$464$	−3727.46	−0.372937
$465$	0	0
$466$	4862.41	0.483362
$467$	9537.66	0.945076	0.472538	−	0.881310i	$-0.343338\pi$
$467$	9537.66	0.945076	0.472538	+	0.881310i	$0.343338\pi$
$468$	0	0
$469$	−10233.6	−1.00756
$470$	9052.58	0.888435
$471$	0	0
$472$	1077.40	0.105066
$473$	2225.47	0.216336
$474$	0	0
$475$	3937.96	0.380391
$476$	8231.29	0.792606
$477$	0	0
$478$	844.343	0.0807936
$479$	2560.30	0.244223	0.122112	−	0.992516i	$-0.461033\pi$
$479$	2560.30	0.244223	0.122112	+	0.992516i	$0.461033\pi$
$480$	0	0
$481$	45.6689	0.00432915
$482$	7641.58	0.722126
$483$	0	0
$484$	−23010.5	−2.16102
$485$	−8019.59	−0.750827
$486$	0	0
$487$	4855.22	0.451768	0.225884	−	0.974154i	$-0.427473\pi$
$487$	4855.22	0.451768	0.225884	+	0.974154i	$0.427473\pi$
$488$	3832.04	0.355468
$489$	0	0
$490$	−3074.91	−0.283490
$491$	−18773.3	−1.72551	−0.862755	−	0.505622i	$-0.831263\pi$
$491$	−18773.3	−1.72551	−0.862755	+	0.505622i	$0.831263\pi$
$492$	0	0
$493$	7641.11	0.698049
$494$	−6.49333	−0.000591394 0
$495$	0	0
$496$	3606.25	0.326463
$497$	−4576.72	−0.413067
$498$	0	0
$499$	−8138.08	−0.730081	−0.365040	−	0.930992i	$-0.618945\pi$
$499$	−8138.08	−0.730081	−0.365040	+	0.930992i	$0.618945\pi$
$500$	−2712.56	−0.242619
$501$	0	0
$502$	2353.41	0.209238
$503$	−7732.32	−0.685422	−0.342711	−	0.939441i	$-0.611345\pi$
$503$	−7732.32	−0.685422	−0.342711	+	0.939441i	$0.611345\pi$
$504$	0	0
$505$	−23590.8	−2.07877
$506$	−16939.6	−1.48826
$507$	0	0
$508$	−7862.51	−0.686698
$509$	−1365.77	−0.118933	−0.0594665	−	0.998230i	$-0.518940\pi$
$509$	−1365.77	−0.118933	−0.0594665	+	0.998230i	$0.518940\pi$
$510$	0	0
$511$	−3996.81	−0.346005
$512$	9154.84	0.790216
$513$	0	0
$514$	4323.84	0.371044
$515$	7622.16	0.652179
$516$	0	0
$517$	−33663.5	−2.86367
$518$	−10398.4	−0.882008
$519$	0	0
$520$	−34.0652	−0.00287280
$521$	5123.71	0.430852	0.215426	−	0.976520i	$-0.430886\pi$
$521$	5123.71	0.430852	0.215426	+	0.976520i	$0.430886\pi$
$522$	0	0
$523$	20522.1	1.71581	0.857907	−	0.513805i	$-0.171765\pi$
$523$	20522.1	1.71581	0.857907	+	0.513805i	$0.171765\pi$
$524$	6213.52	0.518013
$525$	0	0
$526$	2646.72	0.219396
$527$	−7392.64	−0.611060
$528$	0	0
$529$	23957.7	1.96907
$530$	−7608.99	−0.623610
$531$	0	0
$532$	−5865.81	−0.478036
$533$	24.5226	0.00199285
$534$	0	0
$535$	1344.19	0.108625
$536$	8309.18	0.669593
$537$	0	0
$538$	8242.53	0.660522
$539$	11434.5	0.913767
$540$	0	0
$541$	−17106.3	−1.35944	−0.679720	−	0.733471i	$-0.737899\pi$
$541$	−17106.3	−1.35944	−0.679720	+	0.733471i	$0.737899\pi$
$542$	−4127.71	−0.327123
$543$	0	0
$544$	−10399.1	−0.819588
$545$	33358.7	2.62189
$546$	0	0
$547$	4023.62	0.314511	0.157255	−	0.987558i	$-0.449735\pi$
$547$	4023.62	0.314511	0.157255	+	0.987558i	$0.449735\pi$
$548$	−2145.02	−0.167209
$549$	0	0
$550$	8598.23	0.666600
$551$	−5445.23	−0.421007
$552$	0	0
$553$	−11055.7	−0.850154
$554$	7493.99	0.574709
$555$	0	0
$556$	14200.2	1.08313
$557$	−18019.0	−1.37072	−0.685360	−	0.728204i	$-0.740355\pi$
$557$	−18019.0	−1.37072	−0.685360	+	0.728204i	$0.740355\pi$
$558$	0	0
$559$	−3.97213	−0.000300542 0
$560$	−9352.26	−0.705723
$561$	0	0
$562$	−8253.94	−0.619522
$563$	−25706.0	−1.92430	−0.962148	−	0.272526i	$-0.912141\pi$
$563$	−25706.0	−1.92430	−0.962148	+	0.272526i	$0.912141\pi$
$564$	0	0
$565$	−22893.1	−1.70464
$566$	8360.25	0.620862
$567$	0	0
$568$	3716.06	0.274511
$569$	3090.70	0.227713	0.113857	−	0.993497i	$-0.463680\pi$
$569$	3090.70	0.227713	0.113857	+	0.993497i	$0.463680\pi$
$570$	0	0
$571$	7296.29	0.534746	0.267373	−	0.963593i	$-0.413844\pi$
$571$	7296.29	0.534746	0.267373	+	0.963593i	$0.413844\pi$
$572$	56.2496	0.00411174
$573$	0	0
$574$	−5583.58	−0.406018
$575$	−18336.2	−1.32987
$576$	0	0
$577$	−1681.44	−0.121316	−0.0606579	−	0.998159i	$-0.519320\pi$
$577$	−1681.44	−0.121316	−0.0606579	+	0.998159i	$0.519320\pi$
$578$	−2071.07	−0.149040
$579$	0	0
$580$	−12684.7	−0.908113
$581$	16145.0	1.15285
$582$	0	0
$583$	28295.2	2.01007
$584$	3245.20	0.229944
$585$	0	0
$586$	−5908.27	−0.416499
$587$	6885.79	0.484169	0.242084	−	0.970255i	$-0.422169\pi$
$587$	6885.79	0.484169	0.242084	+	0.970255i	$0.422169\pi$
$588$	0	0
$589$	5268.17	0.368542
$590$	1114.27	0.0777521
$591$	0	0
$592$	−10180.1	−0.706759
$593$	24880.0	1.72293	0.861467	−	0.507813i	$-0.169546\pi$
$593$	24880.0	1.72293	0.861467	+	0.507813i	$0.169546\pi$
$594$	0	0
$595$	19171.7	1.32094
$596$	9377.15	0.644468
$597$	0	0
$598$	30.2347	0.00206754
$599$	−5068.39	−0.345724	−0.172862	−	0.984946i	$-0.555302\pi$
$599$	−5068.39	−0.345724	−0.172862	+	0.984946i	$0.555302\pi$
$600$	0	0
$601$	−5800.47	−0.393687	−0.196844	−	0.980435i	$-0.563069\pi$
$601$	−5800.47	−0.393687	−0.196844	+	0.980435i	$0.563069\pi$
$602$	904.420	0.0612316
$603$	0	0
$604$	15336.9	1.03319
$605$	−53594.4	−3.60152
$606$	0	0
$607$	8659.84	0.579064	0.289532	−	0.957168i	$-0.406500\pi$
$607$	8659.84	0.579064	0.289532	+	0.957168i	$0.406500\pi$
$608$	7410.61	0.494309
$609$	0	0
$610$	3963.18	0.263057
$611$	60.0843	0.00397831
$612$	0	0
$613$	−3303.27	−0.217647	−0.108824	−	0.994061i	$-0.534708\pi$
$613$	−3303.27	−0.217647	−0.108824	+	0.994061i	$0.534708\pi$
$614$	−1752.70	−0.115201
$615$	0	0
$616$	−28843.2	−1.88657
$617$	9356.43	0.610495	0.305248	−	0.952273i	$-0.401261\pi$
$617$	9356.43	0.610495	0.305248	+	0.952273i	$0.401261\pi$
$618$	0	0
$619$	1106.92	0.0718756	0.0359378	−	0.999354i	$-0.488558\pi$
$619$	1106.92	0.0718756	0.0359378	+	0.999354i	$0.488558\pi$
$620$	12272.3	0.794946
$621$	0	0
$622$	−2880.61	−0.185695
$623$	−29377.5	−1.88922
$624$	0	0
$625$	−18377.1	−1.17613
$626$	−618.818	−0.0395095
$627$	0	0
$628$	22818.6	1.44994
$629$	20868.8	1.32288
$630$	0	0
$631$	23600.6	1.48895	0.744473	−	0.667653i	$-0.232701\pi$
$631$	23600.6	1.48895	0.744473	+	0.667653i	$0.232701\pi$
$632$	8976.63	0.564986
$633$	0	0
$634$	−10808.5	−0.677065
$635$	−18312.7	−1.14444
$636$	0	0
$637$	−20.4089	−0.00126944
$638$	−11889.3	−0.737774
$639$	0	0
$640$	21484.8	1.32697
$641$	3922.67	0.241710	0.120855	−	0.992670i	$-0.461436\pi$
$641$	3922.67	0.241710	0.120855	+	0.992670i	$0.461436\pi$
$642$	0	0
$643$	−11078.8	−0.679482	−0.339741	−	0.940519i	$-0.610339\pi$
$643$	−11078.8	−0.679482	−0.339741	+	0.940519i	$0.610339\pi$
$644$	27312.8	1.67123
$645$	0	0
$646$	−2967.18	−0.180716
$647$	18371.0	1.11629	0.558143	−	0.829745i	$-0.311514\pi$
$647$	18371.0	1.11629	0.558143	+	0.829745i	$0.311514\pi$
$648$	0	0
$649$	−4143.59	−0.250616
$650$	−15.3466	−0.000926064 0
$651$	0	0
$652$	8342.00	0.501070
$653$	−12179.0	−0.729862	−0.364931	−	0.931035i	$-0.618907\pi$
$653$	−12179.0	−0.729862	−0.364931	+	0.931035i	$0.618907\pi$
$654$	0	0
$655$	14472.1	0.863313
$656$	−5466.38	−0.325345
$657$	0	0
$658$	−13680.7	−0.810529
$659$	18245.7	1.07853	0.539266	−	0.842135i	$-0.318702\pi$
$659$	18245.7	1.07853	0.539266	+	0.842135i	$0.318702\pi$
$660$	0	0
$661$	20360.6	1.19809	0.599043	−	0.800717i	$-0.295548\pi$
$661$	20360.6	1.19809	0.599043	+	0.800717i	$0.295548\pi$
$662$	13120.5	0.770307
$663$	0	0
$664$	−13108.9	−0.766148
$665$	−13662.2	−0.796687
$666$	0	0
$667$	25354.5	1.47186
$668$	1000.31	0.0579388
$669$	0	0
$670$	8593.55	0.495519
$671$	−14737.7	−0.847904
$672$	0	0
$673$	−3894.49	−0.223063	−0.111531	−	0.993761i	$-0.535576\pi$
$673$	−3894.49	−0.223063	−0.111531	+	0.993761i	$0.535576\pi$
$674$	−437.031	−0.0249760
$675$	0	0
$676$	14037.7	0.798685
$677$	−23571.6	−1.33815	−0.669077	−	0.743194i	$-0.733310\pi$
$677$	−23571.6	−1.33815	−0.669077	+	0.743194i	$0.733310\pi$
$678$	0	0
$679$	12119.6	0.684987
$680$	−15566.4	−0.877859
$681$	0	0
$682$	11502.6	0.645834
$683$	7789.31	0.436383	0.218192	−	0.975906i	$-0.429984\pi$
$683$	7789.31	0.436383	0.218192	+	0.975906i	$0.429984\pi$
$684$	0	0
$685$	−4996.02	−0.278669
$686$	−5142.71	−0.286224
$687$	0	0
$688$	885.436	0.0490653
$689$	−50.5028	−0.00279246
$690$	0	0
$691$	−8889.84	−0.489414	−0.244707	−	0.969597i	$-0.578692\pi$
$691$	−8889.84	−0.489414	−0.244707	+	0.969597i	$0.578692\pi$
$692$	2282.76	0.125401
$693$	0	0
$694$	−5117.70	−0.279921
$695$	33074.0	1.80513
$696$	0	0
$697$	11205.8	0.608967
$698$	7994.13	0.433499
$699$	0	0
$700$	−13863.5	−0.748556
$701$	−32472.7	−1.74961	−0.874804	−	0.484477i	$-0.839010\pi$
$701$	−32472.7	−1.74961	−0.874804	+	0.484477i	$0.839010\pi$
$702$	0	0
$703$	−14871.6	−0.797856
$704$	481.426	0.0257734
$705$	0	0
$706$	10458.2	0.557504
$707$	35651.5	1.89648
$708$	0	0
$709$	4596.06	0.243454	0.121727	−	0.992564i	$-0.461157\pi$
$709$	4596.06	0.243454	0.121727	+	0.992564i	$0.461157\pi$
$710$	3843.24	0.203147
$711$	0	0
$712$	23853.0	1.25552
$713$	−24530.0	−1.28844
$714$	0	0
$715$	131.012	0.00685256
$716$	10102.1	0.527282
$717$	0	0
$718$	8382.89	0.435720
$719$	81.6045	0.00423273	0.00211637	−	0.999998i	$-0.499326\pi$
$719$	81.6045	0.00423273	0.00211637	+	0.999998i	$0.499326\pi$
$720$	0	0
$721$	−11519.0	−0.594991
$722$	−6589.90	−0.339682
$723$	0	0
$724$	−10701.1	−0.549312
$725$	−12869.5	−0.659254
$726$	0	0
$727$	−5240.27	−0.267333	−0.133666	−	0.991026i	$-0.542675\pi$
$727$	−5240.27	−0.267333	−0.133666	+	0.991026i	$0.542675\pi$
$728$	51.4809	0.00262089
$729$	0	0
$730$	3356.26	0.170166
$731$	−1815.10	−0.0918384
$732$	0	0
$733$	27362.1	1.37877	0.689386	−	0.724394i	$-0.257880\pi$
$733$	27362.1	1.37877	0.689386	+	0.724394i	$0.257880\pi$
$734$	13406.0	0.674147
$735$	0	0
$736$	−34505.8	−1.72813
$737$	−31956.5	−1.59719
$738$	0	0
$739$	15458.7	0.769497	0.384748	−	0.923021i	$-0.374288\pi$
$739$	15458.7	0.769497	0.384748	+	0.923021i	$0.374288\pi$
$740$	−34643.6	−1.72098
$741$	0	0
$742$	11499.1	0.568927
$743$	33376.7	1.64801	0.824006	−	0.566581i	$-0.191734\pi$
$743$	33376.7	1.64801	0.824006	+	0.566581i	$0.191734\pi$
$744$	0	0
$745$	21840.5	1.07406
$746$	−12168.3	−0.597203
$747$	0	0
$748$	25703.7	1.25645
$749$	−2031.40	−0.0991000
$750$	0	0
$751$	−8832.20	−0.429150	−0.214575	−	0.976708i	$-0.568837\pi$
$751$	−8832.20	−0.429150	−0.214575	+	0.976708i	$0.568837\pi$
$752$	−13393.5	−0.649483
$753$	0	0
$754$	21.2205	0.00102494
$755$	35721.4	1.72190
$756$	0	0
$757$	−4971.26	−0.238684	−0.119342	−	0.992853i	$-0.538078\pi$
$757$	−4971.26	−0.238684	−0.119342	+	0.992853i	$0.538078\pi$
$758$	10209.5	0.489218
$759$	0	0
$760$	11093.0	0.529453
$761$	−25414.3	−1.21060	−0.605300	−	0.795997i	$-0.706947\pi$
$761$	−25414.3	−1.21060	−0.605300	+	0.795997i	$0.706947\pi$
$762$	0	0
$763$	−50413.2	−2.39198
$764$	32046.2	1.51753
$765$	0	0
$766$	16728.2	0.789054
$767$	7.39569	0.000348165 0
$768$	0	0
$769$	30483.0	1.42945	0.714725	−	0.699406i	$-0.246552\pi$
$769$	30483.0	1.42945	0.714725	+	0.699406i	$0.246552\pi$
$770$	−29830.4	−1.39612
$771$	0	0
$772$	−8283.19	−0.386164
$773$	−18113.2	−0.842802	−0.421401	−	0.906874i	$-0.638462\pi$
$773$	−18113.2	−0.842802	−0.421401	+	0.906874i	$0.638462\pi$
$774$	0	0
$775$	12451.0	0.577099
$776$	−9840.46	−0.455221
$777$	0	0
$778$	−2067.99	−0.0952969
$779$	−7985.52	−0.367280
$780$	0	0
$781$	−14291.7	−0.654797
$782$	13816.0	0.631790
$783$	0	0
$784$	4549.40	0.207243
$785$	53147.3	2.41644
$786$	0	0
$787$	3681.38	0.166743	0.0833717	−	0.996519i	$-0.473431\pi$
$787$	3681.38	0.166743	0.0833717	+	0.996519i	$0.473431\pi$
$788$	−18495.8	−0.836148
$789$	0	0
$790$	9283.84	0.418107
$791$	34597.1	1.55516
$792$	0	0
$793$	26.3046	0.00117794
$794$	3053.15	0.136464
$795$	0	0
$796$	18115.3	0.806631
$797$	40435.4	1.79711	0.898554	−	0.438863i	$-0.144619\pi$
$797$	40435.4	1.79711	0.898554	+	0.438863i	$0.144619\pi$
$798$	0	0
$799$	27456.0	1.21568
$800$	17514.5	0.774039
$801$	0	0
$802$	−5562.65	−0.244918
$803$	−12480.8	−0.548490
$804$	0	0
$805$	63614.9	2.78525
$806$	−20.5305	−0.000897216 0
$807$	0	0
$808$	−28947.1	−1.26034
$809$	36431.5	1.58327	0.791633	−	0.610997i	$-0.209231\pi$
$809$	36431.5	1.58327	0.791633	+	0.610997i	$0.209231\pi$
$810$	0	0
$811$	−30923.2	−1.33891	−0.669457	−	0.742851i	$-0.733473\pi$
$811$	−30923.2	−1.33891	−0.669457	+	0.742851i	$0.733473\pi$
$812$	19169.8	0.828482
$813$	0	0
$814$	−32471.0	−1.39817
$815$	19429.5	0.835076
$816$	0	0
$817$	1293.48	0.0553895
$818$	−10872.5	−0.464728
$819$	0	0
$820$	−18602.4	−0.792224
$821$	−3516.20	−0.149472	−0.0747359	−	0.997203i	$-0.523811\pi$
$821$	−3516.20	−0.149472	−0.0747359	+	0.997203i	$0.523811\pi$
$822$	0	0
$823$	39234.9	1.66178	0.830889	−	0.556439i	$-0.187833\pi$
$823$	39234.9	1.66178	0.830889	+	0.556439i	$0.187833\pi$
$824$	9352.78	0.395412
$825$	0	0
$826$	−1683.93	−0.0709341
$827$	−20321.9	−0.854486	−0.427243	−	0.904137i	$-0.640515\pi$
$827$	−20321.9	−0.854486	−0.427243	+	0.904137i	$0.640515\pi$
$828$	0	0
$829$	−36098.3	−1.51236	−0.756180	−	0.654364i	$-0.772936\pi$
$829$	−36098.3	−1.51236	−0.756180	+	0.654364i	$0.772936\pi$
$830$	−13557.5	−0.566973
$831$	0	0
$832$	−0.859274	−3.58053e−5 0
$833$	−9326.04	−0.387909
$834$	0	0
$835$	2329.84	0.0965600
$836$	−18317.1	−0.757787
$837$	0	0
$838$	−10036.4	−0.413725
$839$	−19751.4	−0.812746	−0.406373	−	0.913707i	$-0.633207\pi$
$839$	−19751.4	−0.812746	−0.406373	+	0.913707i	$0.633207\pi$
$840$	0	0
$841$	−6593.69	−0.270355
$842$	5408.86	0.221380
$843$	0	0
$844$	−5713.82	−0.233031
$845$	32695.5	1.33108
$846$	0	0
$847$	80994.3	3.28571
$848$	11257.7	0.455885
$849$	0	0
$850$	−7012.75	−0.282983
$851$	69246.2	2.78934
$852$	0	0
$853$	13812.9	0.554448	0.277224	−	0.960805i	$-0.410586\pi$
$853$	13812.9	0.554448	0.277224	+	0.960805i	$0.410586\pi$
$854$	−5989.34	−0.239990
$855$	0	0
$856$	1649.39	0.0658588
$857$	−10194.0	−0.406323	−0.203162	−	0.979145i	$-0.565122\pi$
$857$	−10194.0	−0.406323	−0.203162	+	0.979145i	$0.565122\pi$
$858$	0	0
$859$	−11608.7	−0.461099	−0.230549	−	0.973061i	$-0.574052\pi$
$859$	−11608.7	−0.461099	−0.230549	+	0.973061i	$0.574052\pi$
$860$	3013.19	0.119475
$861$	0	0
$862$	−16309.0	−0.644418
$863$	45095.9	1.77877	0.889387	−	0.457155i	$-0.151132\pi$
$863$	45095.9	1.77877	0.889387	+	0.457155i	$0.151132\pi$
$864$	0	0
$865$	5316.83	0.208991
$866$	−13899.1	−0.545393
$867$	0	0
$868$	−18546.4	−0.725238
$869$	−34523.5	−1.34767
$870$	0	0
$871$	57.0376	0.00221888
$872$	40932.9	1.58964
$873$	0	0
$874$	−9845.62	−0.381045
$875$	9547.89	0.368889
$876$	0	0
$877$	38676.9	1.48920	0.744599	−	0.667512i	$-0.232641\pi$
$877$	38676.9	1.48920	0.744599	+	0.667512i	$0.232641\pi$
$878$	−15423.0	−0.592824
$879$	0	0
$880$	−29204.2	−1.11872
$881$	34130.6	1.30521	0.652604	−	0.757699i	$-0.273677\pi$
$881$	34130.6	1.30521	0.652604	+	0.757699i	$0.273677\pi$
$882$	0	0
$883$	−39285.3	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$883$	−39285.3	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$884$	−45.8774	−0.00174550
$885$	0	0
$886$	−14751.8	−0.559362
$887$	27255.1	1.03172	0.515860	−	0.856673i	$-0.327472\pi$
$887$	27255.1	1.03172	0.515860	+	0.856673i	$0.327472\pi$
$888$	0	0
$889$	27675.1	1.04409
$890$	24669.3	0.929121
$891$	0	0
$892$	−15454.8	−0.580119
$893$	−19565.8	−0.733197
$894$	0	0
$895$	23529.1	0.878760
$896$	−32468.9	−1.21061
$897$	0	0
$898$	11615.2	0.431630
$899$	−17216.6	−0.638718
$900$	0	0
$901$	−23077.7	−0.853307
$902$	−17435.8	−0.643623
$903$	0	0
$904$	−28091.1	−1.03351
$905$	−24924.1	−0.915475
$906$	0	0
$907$	−40763.1	−1.49230	−0.746151	−	0.665777i	$-0.768100\pi$
$907$	−40763.1	−1.49230	−0.746151	+	0.665777i	$0.768100\pi$
$908$	−7136.36	−0.260824
$909$	0	0
$910$	53.2427	0.00193954
$911$	−9175.77	−0.333707	−0.166853	−	0.985982i	$-0.553361\pi$
$911$	−9175.77	−0.333707	−0.166853	+	0.985982i	$0.553361\pi$
$912$	0	0
$913$	50415.7	1.82751
$914$	−2365.82	−0.0856174
$915$	0	0
$916$	25581.7	0.922756
$917$	−21870.8	−0.787610
$918$	0	0
$919$	−33814.3	−1.21374	−0.606871	−	0.794800i	$-0.707576\pi$
$919$	−33814.3	−1.21374	−0.606871	+	0.794800i	$0.707576\pi$
$920$	−51651.9	−1.85099
$921$	0	0
$922$	24385.7	0.871040
$923$	25.5085	0.000909668 0
$924$	0	0
$925$	−35148.0	−1.24936
$926$	9882.07	0.350696
$927$	0	0
$928$	−24218.3	−0.856685
$929$	−428.503	−0.0151332	−0.00756659	−	0.999971i	$-0.502409\pi$
$929$	−428.503	−0.0151332	−0.00756659	+	0.999971i	$0.502409\pi$
$930$	0	0
$931$	6645.96	0.233955
$932$	−24481.7	−0.860435
$933$	0	0
$934$	12103.7	0.424032
$935$	59867.2	2.09397
$936$	0	0
$937$	32339.2	1.12751	0.563755	−	0.825942i	$-0.309356\pi$
$937$	32339.2	1.12751	0.563755	+	0.825942i	$0.309356\pi$
$938$	−12987.0	−0.452068
$939$	0	0
$940$	−45578.8	−1.58151
$941$	−3308.12	−0.114603	−0.0573016	−	0.998357i	$-0.518250\pi$
$941$	−3308.12	−0.114603	−0.0573016	+	0.998357i	$0.518250\pi$
$942$	0	0
$943$	37182.8	1.28403
$944$	−1648.59	−0.0568400
$945$	0	0
$946$	2824.22	0.0970649
$947$	−2319.50	−0.0795921	−0.0397961	−	0.999208i	$-0.512671\pi$
$947$	−2319.50	−0.0795921	−0.0397961	+	0.999208i	$0.512671\pi$
$948$	0	0
$949$	22.2764	0.000761982 0
$950$	4997.45	0.170672
$951$	0	0
$952$	23524.6	0.800880
$953$	−8790.44	−0.298793	−0.149397	−	0.988777i	$-0.547733\pi$
$953$	−8790.44	−0.298793	−0.149397	+	0.988777i	$0.547733\pi$
$954$	0	0
$955$	74639.6	2.52909
$956$	−4251.18	−0.143821
$957$	0	0
$958$	3249.13	0.109577
$959$	7550.21	0.254233
$960$	0	0
$961$	−13134.2	−0.440878
$962$	57.9559	0.00194238
$963$	0	0
$964$	−38474.6	−1.28546
$965$	−19292.6	−0.643575
$966$	0	0
$967$	−20867.5	−0.693953	−0.346976	−	0.937874i	$-0.612792\pi$
$967$	−20867.5	−0.693953	−0.346976	+	0.937874i	$0.612792\pi$
$968$	−65763.1	−2.18358
$969$	0	0
$970$	−10177.2	−0.336877
$971$	−2174.66	−0.0718726	−0.0359363	−	0.999354i	$-0.511441\pi$
$971$	−2174.66	−0.0718726	−0.0359363	+	0.999354i	$0.511441\pi$
$972$	0	0
$973$	−49982.9	−1.64684
$974$	6161.49	0.202697
$975$	0	0
$976$	−5863.62	−0.192305
$977$	6656.04	0.217959	0.108979	−	0.994044i	$-0.465242\pi$
$977$	6656.04	0.217959	0.108979	+	0.994044i	$0.465242\pi$
$978$	0	0
$979$	−91736.8	−2.99481
$980$	15481.9	0.504643
$981$	0	0
$982$	−23824.1	−0.774194
$983$	−52028.5	−1.68815	−0.844075	−	0.536224i	$-0.819850\pi$
$983$	−52028.5	−1.68815	−0.844075	+	0.536224i	$0.819850\pi$
$984$	0	0
$985$	−43078.9	−1.39351
$986$	9696.92	0.313198
$987$	0	0
$988$	32.6933	0.00105274
$989$	−6022.81	−0.193644
$990$	0	0
$991$	9152.99	0.293395	0.146697	−	0.989181i	$-0.453136\pi$
$991$	9152.99	0.293395	0.146697	+	0.989181i	$0.453136\pi$
$992$	23430.8	0.749927
$993$	0	0
$994$	−5808.07	−0.185333
$995$	42192.7	1.34432
$996$	0	0
$997$	−43711.9	−1.38854	−0.694268	−	0.719717i	$-0.744272\pi$
$997$	−43711.9	−1.38854	−0.694268	+	0.719717i	$0.744272\pi$
$998$	−10327.6	−0.327569
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.e.1.6		8
3.2	odd	2		177.4.a.d.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.3	✓	8	3.2	odd	2
531.4.a.e.1.6		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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} +O(q^{10})\)	\(q+1.26905 q^{2} -6.38952 q^{4} -14.8820 q^{5} +22.4903 q^{7} -18.2610 q^{8} -18.8859 q^{10} +70.2303 q^{11} -0.125351 q^{13} +28.5413 q^{14} +27.9422 q^{16} -57.2801 q^{17} +40.8191 q^{19} +95.0887 q^{20} +89.1255 q^{22} -190.065 q^{23} +96.4733 q^{25} -0.159076 q^{26} -143.702 q^{28} -133.399 q^{29} +129.061 q^{31} +181.548 q^{32} -72.6911 q^{34} -334.701 q^{35} -364.329 q^{37} +51.8013 q^{38} +271.759 q^{40} -195.632 q^{41} +31.6882 q^{43} -448.738 q^{44} -241.201 q^{46} -479.330 q^{47} +162.815 q^{49} +122.429 q^{50} +0.800930 q^{52} +402.892 q^{53} -1045.17 q^{55} -410.695 q^{56} -169.290 q^{58} -59.0000 q^{59} -209.849 q^{61} +163.785 q^{62} +6.85497 q^{64} +1.86547 q^{65} -455.024 q^{67} +365.992 q^{68} -424.750 q^{70} -203.497 q^{71} -177.712 q^{73} -462.350 q^{74} -260.815 q^{76} +1579.50 q^{77} -491.575 q^{79} -415.835 q^{80} -248.266 q^{82} +717.862 q^{83} +852.441 q^{85} +40.2137 q^{86} -1282.47 q^{88} -1306.23 q^{89} -2.81918 q^{91} +1214.42 q^{92} -608.291 q^{94} -607.469 q^{95} +538.879 q^{97} +206.620 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

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Database

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