LMFDB - Embedded newform 495.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32803 q^{2} +3.07579 q^{4} -5.00000 q^{5} +12.1102 q^{7} -16.3879 q^{8} +O(q^{10})$	\(q+3.32803 q^{2} +3.07579 q^{4} -5.00000 q^{5} +12.1102 q^{7} -16.3879 q^{8} -16.6402 q^{10} +11.0000 q^{11} -57.2210 q^{13} +40.3032 q^{14} -79.1458 q^{16} +22.7802 q^{17} -46.2204 q^{19} -15.3790 q^{20} +36.6083 q^{22} -128.447 q^{23} +25.0000 q^{25} -190.433 q^{26} +37.2485 q^{28} -71.6674 q^{29} -88.7110 q^{31} -132.296 q^{32} +75.8131 q^{34} -60.5511 q^{35} -30.5848 q^{37} -153.823 q^{38} +81.9396 q^{40} -223.209 q^{41} -170.565 q^{43} +33.8337 q^{44} -427.477 q^{46} +247.523 q^{47} -196.343 q^{49} +83.2008 q^{50} -176.000 q^{52} -76.4692 q^{53} -55.0000 q^{55} -198.461 q^{56} -238.511 q^{58} -258.248 q^{59} -97.3761 q^{61} -295.233 q^{62} +192.880 q^{64} +286.105 q^{65} -278.018 q^{67} +70.0671 q^{68} -201.516 q^{70} +292.087 q^{71} -482.377 q^{73} -101.787 q^{74} -142.164 q^{76} +133.212 q^{77} -93.3473 q^{79} +395.729 q^{80} -742.847 q^{82} +1011.13 q^{83} -113.901 q^{85} -567.646 q^{86} -180.267 q^{88} -60.3336 q^{89} -692.959 q^{91} -395.077 q^{92} +823.766 q^{94} +231.102 q^{95} +662.927 q^{97} -653.435 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8	$ 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 66 q^{13} + 88 q^{14} - 103 q^{16} + 100 q^{17} - 70 q^{19} - 5 q^{20} - 11 q^{22} + 10 q^{23} + 75 q^{25} - 88 q^{26} - 64 q^{28} - 34 q^{29} - 104 q^{31} - 37 q^{32} - 278 q^{34} - 10 q^{35} - 270 q^{37} - 154 q^{38} - 15 q^{40} - 82 q^{41} - 428 q^{43} + 11 q^{44} - 706 q^{46} + 246 q^{47} - 685 q^{49} - 25 q^{50} - 528 q^{52} + 298 q^{53} - 165 q^{55} - 288 q^{56} + 114 q^{58} - 204 q^{59} - 492 q^{61} - 584 q^{62} - 319 q^{64} + 330 q^{65} - 812 q^{67} + 798 q^{68} - 440 q^{70} - 442 q^{71} - 1044 q^{73} - 42 q^{74} + 106 q^{76} + 22 q^{77} - 1436 q^{79} + 515 q^{80} - 1246 q^{82} + 734 q^{83} - 500 q^{85} + 82 q^{86} + 33 q^{88} - 556 q^{89} - 176 q^{91} - 630 q^{92} + 38 q^{94} + 350 q^{95} - 1038 q^{97} + 143 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 66 * q^13 + 88 * q^14 - 103 * q^16 + 100 * q^17 - 70 * q^19 - 5 * q^20 - 11 * q^22 + 10 * q^23 + 75 * q^25 - 88 * q^26 - 64 * q^28 - 34 * q^29 - 104 * q^31 - 37 * q^32 - 278 * q^34 - 10 * q^35 - 270 * q^37 - 154 * q^38 - 15 * q^40 - 82 * q^41 - 428 * q^43 + 11 * q^44 - 706 * q^46 + 246 * q^47 - 685 * q^49 - 25 * q^50 - 528 * q^52 + 298 * q^53 - 165 * q^55 - 288 * q^56 + 114 * q^58 - 204 * q^59 - 492 * q^61 - 584 * q^62 - 319 * q^64 + 330 * q^65 - 812 * q^67 + 798 * q^68 - 440 * q^70 - 442 * q^71 - 1044 * q^73 - 42 * q^74 + 106 * q^76 + 22 * q^77 - 1436 * q^79 + 515 * q^80 - 1246 * q^82 + 734 * q^83 - 500 * q^85 + 82 * q^86 + 33 * q^88 - 556 * q^89 - 176 * q^91 - 630 * q^92 + 38 * q^94 + 350 * q^95 - 1038 * q^97 + 143 * 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 $

$2$	3.32803	1.17664	0.588318	−	0.808629i	$-0.299790\pi$
$2$	3.32803	1.17664	0.588318	+	0.808629i	$0.299790\pi$
$3$	0	0
$3$	0	0
$4$	3.07579	0.384474
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	12.1102	0.653890	0.326945	−	0.945043i	$-0.393981\pi$
$7$	12.1102	0.653890	0.326945	+	0.945043i	$0.393981\pi$
$8$	−16.3879	−0.724250
$9$	0	0
$10$	−16.6402	−0.526208
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−57.2210	−1.22079	−0.610395	−	0.792098i	$-0.708989\pi$
$13$	−57.2210	−1.22079	−0.610395	+	0.792098i	$0.708989\pi$
$14$	40.3032	0.769391
$15$	0	0
$16$	−79.1458	−1.23665
$17$	22.7802	0.325000	0.162500	−	0.986709i	$-0.448044\pi$
$17$	22.7802	0.325000	0.162500	+	0.986709i	$0.448044\pi$
$18$	0	0
$19$	−46.2204	−0.558089	−0.279045	−	0.960278i	$-0.590018\pi$
$19$	−46.2204	−0.558089	−0.279045	+	0.960278i	$0.590018\pi$
$20$	−15.3790	−0.171942
$21$	0	0
$22$	36.6083	0.354769
$23$	−128.447	−1.16448	−0.582241	−	0.813016i	$-0.697824\pi$
$23$	−128.447	−1.16448	−0.582241	+	0.813016i	$0.697824\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−190.433	−1.43643
$27$	0	0
$28$	37.2485	0.251404
$29$	−71.6674	−0.458907	−0.229454	−	0.973320i	$-0.573694\pi$
$29$	−71.6674	−0.458907	−0.229454	+	0.973320i	$0.573694\pi$
$30$	0	0
$31$	−88.7110	−0.513967	−0.256983	−	0.966416i	$-0.582729\pi$
$31$	−88.7110	−0.513967	−0.256983	+	0.966416i	$0.582729\pi$
$32$	−132.296	−0.730842
$33$	0	0
$34$	75.8131	0.382407
$35$	−60.5511	−0.292429
$36$	0	0
$37$	−30.5848	−0.135895	−0.0679475	−	0.997689i	$-0.521645\pi$
$37$	−30.5848	−0.135895	−0.0679475	+	0.997689i	$0.521645\pi$
$38$	−153.823	−0.656668
$39$	0	0
$40$	81.9396	0.323895
$41$	−223.209	−0.850230	−0.425115	−	0.905139i	$-0.639766\pi$
$41$	−223.209	−0.850230	−0.425115	+	0.905139i	$0.639766\pi$
$42$	0	0
$43$	−170.565	−0.604905	−0.302452	−	0.953164i	$-0.597805\pi$
$43$	−170.565	−0.604905	−0.302452	+	0.953164i	$0.597805\pi$
$44$	33.8337	0.115923
$45$	0	0
$46$	−427.477	−1.37017
$47$	247.523	0.768192	0.384096	−	0.923293i	$-0.374513\pi$
$47$	247.523	0.768192	0.384096	+	0.923293i	$0.374513\pi$
$48$	0	0
$49$	−196.343	−0.572428
$50$	83.2008	0.235327
$51$	0	0
$52$	−176.000	−0.469362
$53$	−76.4692	−0.198186	−0.0990929	−	0.995078i	$-0.531594\pi$
$53$	−76.4692	−0.198186	−0.0990929	+	0.995078i	$0.531594\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	−198.461	−0.473580
$57$	0	0
$58$	−238.511	−0.539967
$59$	−258.248	−0.569849	−0.284924	−	0.958550i	$-0.591968\pi$
$59$	−258.248	−0.569849	−0.284924	+	0.958550i	$0.591968\pi$
$60$	0	0
$61$	−97.3761	−0.204389	−0.102195	−	0.994764i	$-0.532586\pi$
$61$	−97.3761	−0.204389	−0.102195	+	0.994764i	$0.532586\pi$
$62$	−295.233	−0.604752
$63$	0	0
$64$	192.880	0.376718
$65$	286.105	0.545953
$66$	0	0
$67$	−278.018	−0.506945	−0.253472	−	0.967343i	$-0.581573\pi$
$67$	−278.018	−0.506945	−0.253472	+	0.967343i	$0.581573\pi$
$68$	70.0671	0.124954
$69$	0	0
$70$	−201.516	−0.344082
$71$	292.087	0.488231	0.244116	−	0.969746i	$-0.421502\pi$
$71$	292.087	0.488231	0.244116	+	0.969746i	$0.421502\pi$
$72$	0	0
$73$	−482.377	−0.773397	−0.386699	−	0.922206i	$-0.626385\pi$
$73$	−482.377	−0.773397	−0.386699	+	0.922206i	$0.626385\pi$
$74$	−101.787	−0.159899
$75$	0	0
$76$	−142.164	−0.214571
$77$	133.212	0.197155
$78$	0	0
$79$	−93.3473	−0.132942	−0.0664708	−	0.997788i	$-0.521174\pi$
$79$	−93.3473	−0.132942	−0.0664708	+	0.997788i	$0.521174\pi$
$80$	395.729	0.553048
$81$	0	0
$82$	−742.847	−1.00041
$83$	1011.13	1.33718	0.668588	−	0.743633i	$-0.266899\pi$
$83$	1011.13	1.33718	0.668588	+	0.743633i	$0.266899\pi$
$84$	0	0
$85$	−113.901	−0.145345
$86$	−567.646	−0.711753
$87$	0	0
$88$	−180.267	−0.218370
$89$	−60.3336	−0.0718579	−0.0359290	−	0.999354i	$-0.511439\pi$
$89$	−60.3336	−0.0718579	−0.0359290	+	0.999354i	$0.511439\pi$
$90$	0	0
$91$	−692.959	−0.798262
$92$	−395.077	−0.447713
$93$	0	0
$94$	823.766	0.903883
$95$	231.102	0.249585
$96$	0	0
$97$	662.927	0.693918	0.346959	−	0.937880i	$-0.387214\pi$
$97$	662.927	0.693918	0.346959	+	0.937880i	$0.387214\pi$
$98$	−653.435	−0.673540
$99$	0	0
$100$	76.8948	0.0768948
$101$	1585.81	1.56231	0.781156	−	0.624336i	$-0.214630\pi$
$101$	1585.81	1.56231	0.781156	+	0.624336i	$0.214630\pi$
$102$	0	0
$103$	−2024.01	−1.93623	−0.968115	−	0.250508i	$-0.919402\pi$
$103$	−2024.01	−1.93623	−0.968115	+	0.250508i	$0.919402\pi$
$104$	937.734	0.884157
$105$	0	0
$106$	−254.492	−0.233193
$107$	611.358	0.552357	0.276179	−	0.961106i	$-0.410932\pi$
$107$	611.358	0.552357	0.276179	+	0.961106i	$0.410932\pi$
$108$	0	0
$109$	1180.33	1.03721	0.518604	−	0.855015i	$-0.326452\pi$
$109$	1180.33	1.03721	0.518604	+	0.855015i	$0.326452\pi$
$110$	−183.042	−0.158658
$111$	0	0
$112$	−958.473	−0.808636
$113$	1098.57	0.914557	0.457278	−	0.889324i	$-0.348824\pi$
$113$	1098.57	0.914557	0.457278	+	0.889324i	$0.348824\pi$
$114$	0	0
$115$	642.236	0.520773
$116$	−220.434	−0.176438
$117$	0	0
$118$	−859.459	−0.670505
$119$	275.873	0.212514
$120$	0	0
$121$	121.000	0.0909091
$122$	−324.071	−0.240492
$123$	0	0
$124$	−272.857	−0.197607
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−794.434	−0.555076	−0.277538	−	0.960715i	$-0.589518\pi$
$127$	−794.434	−0.555076	−0.277538	+	0.960715i	$0.589518\pi$
$128$	1700.28	1.17410
$129$	0	0
$130$	952.167	0.642389
$131$	364.955	0.243406	0.121703	−	0.992567i	$-0.461164\pi$
$131$	364.955	0.243406	0.121703	+	0.992567i	$0.461164\pi$
$132$	0	0
$133$	−559.739	−0.364929
$134$	−925.252	−0.596490
$135$	0	0
$136$	−373.320	−0.235382
$137$	2057.56	1.28313	0.641567	−	0.767067i	$-0.278285\pi$
$137$	2057.56	1.28313	0.641567	+	0.767067i	$0.278285\pi$
$138$	0	0
$139$	471.536	0.287735	0.143867	−	0.989597i	$-0.454046\pi$
$139$	471.536	0.287735	0.143867	+	0.989597i	$0.454046\pi$
$140$	−186.243	−0.112431
$141$	0	0
$142$	972.076	0.574471
$143$	−629.431	−0.368082
$144$	0	0
$145$	358.337	0.205230
$146$	−1605.37	−0.910008
$147$	0	0
$148$	−94.0725	−0.0522481
$149$	−135.727	−0.0746255	−0.0373128	−	0.999304i	$-0.511880\pi$
$149$	−135.727	−0.0746255	−0.0373128	+	0.999304i	$0.511880\pi$
$150$	0	0
$151$	−1908.95	−1.02879	−0.514397	−	0.857552i	$-0.671984\pi$
$151$	−1908.95	−1.02879	−0.514397	+	0.857552i	$0.671984\pi$
$152$	757.457	0.404196
$153$	0	0
$154$	443.335	0.231980
$155$	443.555	0.229853
$156$	0	0
$157$	−1028.53	−0.522840	−0.261420	−	0.965225i	$-0.584191\pi$
$157$	−1028.53	−0.522840	−0.261420	+	0.965225i	$0.584191\pi$
$158$	−310.663	−0.156424
$159$	0	0
$160$	661.482	0.326842
$161$	−1555.52	−0.761444
$162$	0	0
$163$	684.628	0.328983	0.164492	−	0.986378i	$-0.447402\pi$
$163$	684.628	0.328983	0.164492	+	0.986378i	$0.447402\pi$
$164$	−686.545	−0.326891
$165$	0	0
$166$	3365.06	1.57337
$167$	−799.231	−0.370337	−0.185169	−	0.982707i	$-0.559283\pi$
$167$	−799.231	−0.370337	−0.185169	+	0.982707i	$0.559283\pi$
$168$	0	0
$169$	1077.25	0.490326
$170$	−379.066	−0.171018
$171$	0	0
$172$	−524.622	−0.232570
$173$	3670.62	1.61313	0.806567	−	0.591143i	$-0.201323\pi$
$173$	3670.62	1.61313	0.806567	+	0.591143i	$0.201323\pi$
$174$	0	0
$175$	302.755	0.130778
$176$	−870.604	−0.372865
$177$	0	0
$178$	−200.792	−0.0845506
$179$	−3971.27	−1.65825	−0.829125	−	0.559064i	$-0.811161\pi$
$179$	−3971.27	−1.65825	−0.829125	+	0.559064i	$0.811161\pi$
$180$	0	0
$181$	2720.15	1.11706	0.558529	−	0.829485i	$-0.311366\pi$
$181$	2720.15	1.11706	0.558529	+	0.829485i	$0.311366\pi$
$182$	−2306.19	−0.939264
$183$	0	0
$184$	2104.98	0.843377
$185$	152.924	0.0607741
$186$	0	0
$187$	250.582	0.0979913
$188$	761.331	0.295350
$189$	0	0
$190$	769.115	0.293671
$191$	23.4786	0.00889450	0.00444725	−	0.999990i	$-0.498584\pi$
$191$	23.4786	0.00889450	0.00444725	+	0.999990i	$0.498584\pi$
$192$	0	0
$193$	−4974.19	−1.85518	−0.927590	−	0.373599i	$-0.878124\pi$
$193$	−4974.19	−1.85518	−0.927590	+	0.373599i	$0.878124\pi$
$194$	2206.24	0.816489
$195$	0	0
$196$	−603.909	−0.220084
$197$	−175.739	−0.0635576	−0.0317788	−	0.999495i	$-0.510117\pi$
$197$	−175.739	−0.0635576	−0.0317788	+	0.999495i	$0.510117\pi$
$198$	0	0
$199$	5328.65	1.89818	0.949091	−	0.315002i	$-0.102005\pi$
$199$	5328.65	1.89818	0.949091	+	0.315002i	$0.102005\pi$
$200$	−409.698	−0.144850
$201$	0	0
$202$	5277.61	1.83827
$203$	−867.908	−0.300075
$204$	0	0
$205$	1116.05	0.380234
$206$	−6735.97	−2.27824
$207$	0	0
$208$	4528.81	1.50969
$209$	−508.425	−0.168270
$210$	0	0
$211$	−4931.62	−1.60904	−0.804518	−	0.593928i	$-0.797576\pi$
$211$	−4931.62	−1.60904	−0.804518	+	0.593928i	$0.797576\pi$
$212$	−235.203	−0.0761973
$213$	0	0
$214$	2034.62	0.649924
$215$	852.825	0.270522
$216$	0	0
$217$	−1074.31	−0.336078
$218$	3928.19	1.22042
$219$	0	0
$220$	−169.169	−0.0518425
$221$	−1303.51	−0.396757
$222$	0	0
$223$	−2070.50	−0.621754	−0.310877	−	0.950450i	$-0.600623\pi$
$223$	−2070.50	−0.621754	−0.310877	+	0.950450i	$0.600623\pi$
$224$	−1602.14	−0.477890
$225$	0	0
$226$	3656.08	1.07610
$227$	−5474.76	−1.60076	−0.800380	−	0.599494i	$-0.795369\pi$
$227$	−5474.76	−1.60076	−0.800380	+	0.599494i	$0.795369\pi$
$228$	0	0
$229$	5061.11	1.46047	0.730234	−	0.683197i	$-0.239411\pi$
$229$	5061.11	1.46047	0.730234	+	0.683197i	$0.239411\pi$
$230$	2137.38	0.612760
$231$	0	0
$232$	1174.48	0.332364
$233$	1740.43	0.489354	0.244677	−	0.969605i	$-0.421318\pi$
$233$	1740.43	0.489354	0.244677	+	0.969605i	$0.421318\pi$
$234$	0	0
$235$	−1237.62	−0.343546
$236$	−794.318	−0.219092
$237$	0	0
$238$	918.113	0.250052
$239$	3557.50	0.962827	0.481413	−	0.876494i	$-0.340124\pi$
$239$	3557.50	0.962827	0.481413	+	0.876494i	$0.340124\pi$
$240$	0	0
$241$	−5027.91	−1.34388	−0.671942	−	0.740603i	$-0.734540\pi$
$241$	−5027.91	−1.34388	−0.671942	+	0.740603i	$0.734540\pi$
$242$	402.692	0.106967
$243$	0	0
$244$	−299.509	−0.0785823
$245$	981.714	0.255997
$246$	0	0
$247$	2644.78	0.681309
$248$	1453.79	0.372241
$249$	0	0
$250$	−416.004	−0.105242
$251$	−1319.91	−0.331919	−0.165960	−	0.986133i	$-0.553072\pi$
$251$	−1319.91	−0.331919	−0.165960	+	0.986133i	$0.553072\pi$
$252$	0	0
$253$	−1412.92	−0.351105
$254$	−2643.90	−0.653123
$255$	0	0
$256$	4115.55	1.00477
$257$	−5827.46	−1.41443	−0.707213	−	0.707001i	$-0.750048\pi$
$257$	−5827.46	−1.41443	−0.707213	+	0.707001i	$0.750048\pi$
$258$	0	0
$259$	−370.389	−0.0888603
$260$	880.000	0.209905
$261$	0	0
$262$	1214.58	0.286401
$263$	4452.54	1.04394	0.521968	−	0.852965i	$-0.325198\pi$
$263$	4452.54	1.04394	0.521968	+	0.852965i	$0.325198\pi$
$264$	0	0
$265$	382.346	0.0886314
$266$	−1862.83	−0.429389
$267$	0	0
$268$	−855.125	−0.194907
$269$	−1558.71	−0.353295	−0.176647	−	0.984274i	$-0.556525\pi$
$269$	−1558.71	−0.353295	−0.176647	+	0.984274i	$0.556525\pi$
$270$	0	0
$271$	−5194.68	−1.16441	−0.582203	−	0.813043i	$-0.697809\pi$
$271$	−5194.68	−1.16441	−0.582203	+	0.813043i	$0.697809\pi$
$272$	−1802.96	−0.401913
$273$	0	0
$274$	6847.63	1.50978
$275$	275.000	0.0603023
$276$	0	0
$277$	−7425.82	−1.61074	−0.805369	−	0.592774i	$-0.798033\pi$
$277$	−7425.82	−1.61074	−0.805369	+	0.592774i	$0.798033\pi$
$278$	1569.29	0.338559
$279$	0	0
$280$	992.306	0.211791
$281$	−2209.95	−0.469163	−0.234581	−	0.972096i	$-0.575372\pi$
$281$	−2209.95	−0.469163	−0.234581	+	0.972096i	$0.575372\pi$
$282$	0	0
$283$	3668.07	0.770475	0.385237	−	0.922817i	$-0.374120\pi$
$283$	3668.07	0.770475	0.385237	+	0.922817i	$0.374120\pi$
$284$	898.400	0.187712
$285$	0	0
$286$	−2094.77	−0.433099
$287$	−2703.11	−0.555957
$288$	0	0
$289$	−4394.06	−0.894375
$290$	1192.56	0.241481
$291$	0	0
$292$	−1483.69	−0.297351
$293$	−1455.91	−0.290291	−0.145145	−	0.989410i	$-0.546365\pi$
$293$	−1455.91	−0.290291	−0.145145	+	0.989410i	$0.546365\pi$
$294$	0	0
$295$	1291.24	0.254844
$296$	501.221	0.0984220
$297$	0	0
$298$	−451.704	−0.0878071
$299$	7349.89	1.42159
$300$	0	0
$301$	−2065.58	−0.395541
$302$	−6353.04	−1.21052
$303$	0	0
$304$	3658.15	0.690163
$305$	486.881	0.0914056
$306$	0	0
$307$	8129.61	1.51134	0.755670	−	0.654953i	$-0.227311\pi$
$307$	8129.61	1.51134	0.755670	+	0.654953i	$0.227311\pi$
$308$	409.734	0.0758011
$309$	0	0
$310$	1476.16	0.270453
$311$	2030.06	0.370142	0.185071	−	0.982725i	$-0.440749\pi$
$311$	2030.06	0.370142	0.185071	+	0.982725i	$0.440749\pi$
$312$	0	0
$313$	−513.357	−0.0927050	−0.0463525	−	0.998925i	$-0.514760\pi$
$313$	−513.357	−0.0927050	−0.0463525	+	0.998925i	$0.514760\pi$
$314$	−3422.99	−0.615192
$315$	0	0
$316$	−287.117	−0.0511126
$317$	5230.37	0.926710	0.463355	−	0.886173i	$-0.346646\pi$
$317$	5230.37	0.926710	0.463355	+	0.886173i	$0.346646\pi$
$318$	0	0
$319$	−788.342	−0.138366
$320$	−964.399	−0.168474
$321$	0	0
$322$	−5176.83	−0.895943
$323$	−1052.91	−0.181379
$324$	0	0
$325$	−1430.53	−0.244158
$326$	2278.46	0.387094
$327$	0	0
$328$	3657.93	0.615779
$329$	2997.56	0.502313
$330$	0	0
$331$	7261.31	1.20579	0.602897	−	0.797819i	$-0.294013\pi$
$331$	7261.31	1.20579	0.602897	+	0.797819i	$0.294013\pi$
$332$	3110.02	0.514110
$333$	0	0
$334$	−2659.87	−0.435753
$335$	1390.09	0.226713
$336$	0	0
$337$	1707.70	0.276037	0.138019	−	0.990430i	$-0.455927\pi$
$337$	1707.70	0.276037	0.138019	+	0.990430i	$0.455927\pi$
$338$	3585.11	0.576936
$339$	0	0
$340$	−350.335	−0.0558812
$341$	−975.821	−0.154967
$342$	0	0
$343$	−6531.56	−1.02819
$344$	2795.20	0.438103
$345$	0	0
$346$	12215.9	1.89807
$347$	−3948.37	−0.610834	−0.305417	−	0.952219i	$-0.598796\pi$
$347$	−3948.37	−0.610834	−0.305417	+	0.952219i	$0.598796\pi$
$348$	0	0
$349$	1623.51	0.249010	0.124505	−	0.992219i	$-0.460266\pi$
$349$	1623.51	0.249010	0.124505	+	0.992219i	$0.460266\pi$
$350$	1007.58	0.153878
$351$	0	0
$352$	−1455.26	−0.220357
$353$	5926.17	0.893535	0.446768	−	0.894650i	$-0.352575\pi$
$353$	5926.17	0.893535	0.446768	+	0.894650i	$0.352575\pi$
$354$	0	0
$355$	−1460.44	−0.218344
$356$	−185.574	−0.0276275
$357$	0	0
$358$	−13216.5	−1.95116
$359$	3545.16	0.521187	0.260594	−	0.965449i	$-0.416082\pi$
$359$	3545.16	0.521187	0.260594	+	0.965449i	$0.416082\pi$
$360$	0	0
$361$	−4722.67	−0.688537
$362$	9052.76	1.31437
$363$	0	0
$364$	−2131.40	−0.306911
$365$	2411.89	0.345874
$366$	0	0
$367$	3429.59	0.487802	0.243901	−	0.969800i	$-0.421573\pi$
$367$	3429.59	0.487802	0.243901	+	0.969800i	$0.421573\pi$
$368$	10166.1	1.44006
$369$	0	0
$370$	508.936	0.0715090
$371$	−926.058	−0.129592
$372$	0	0
$373$	2462.08	0.341773	0.170887	−	0.985291i	$-0.445337\pi$
$373$	2462.08	0.341773	0.170887	+	0.985291i	$0.445337\pi$
$374$	833.945	0.115300
$375$	0	0
$376$	−4056.39	−0.556363
$377$	4100.88	0.560229
$378$	0	0
$379$	−6713.31	−0.909867	−0.454933	−	0.890525i	$-0.650337\pi$
$379$	−6713.31	−0.909867	−0.454933	+	0.890525i	$0.650337\pi$
$380$	710.822	0.0959590
$381$	0	0
$382$	78.1374	0.0104656
$383$	3230.45	0.430988	0.215494	−	0.976505i	$-0.430864\pi$
$383$	3230.45	0.430988	0.215494	+	0.976505i	$0.430864\pi$
$384$	0	0
$385$	−666.062	−0.0881705
$386$	−16554.3	−2.18287
$387$	0	0
$388$	2039.03	0.266793
$389$	3075.88	0.400908	0.200454	−	0.979703i	$-0.435758\pi$
$389$	3075.88	0.400908	0.200454	+	0.979703i	$0.435758\pi$
$390$	0	0
$391$	−2926.05	−0.378457
$392$	3217.65	0.414581
$393$	0	0
$394$	−584.863	−0.0747843
$395$	466.736	0.0594533
$396$	0	0
$397$	4864.47	0.614964	0.307482	−	0.951554i	$-0.400514\pi$
$397$	4864.47	0.614964	0.307482	+	0.951554i	$0.400514\pi$
$398$	17733.9	2.23347
$399$	0	0
$400$	−1978.65	−0.247331
$401$	−11676.4	−1.45409	−0.727044	−	0.686591i	$-0.759107\pi$
$401$	−11676.4	−1.45409	−0.727044	+	0.686591i	$0.759107\pi$
$402$	0	0
$403$	5076.14	0.627445
$404$	4877.61	0.600669
$405$	0	0
$406$	−2888.42	−0.353079
$407$	−336.433	−0.0409739
$408$	0	0
$409$	−16394.0	−1.98199	−0.990993	−	0.133917i	$-0.957244\pi$
$409$	−16394.0	−1.98199	−0.990993	+	0.133917i	$0.957244\pi$
$410$	3714.24	0.447398
$411$	0	0
$412$	−6225.43	−0.744430
$413$	−3127.44	−0.372618
$414$	0	0
$415$	−5055.64	−0.598004
$416$	7570.14	0.892204
$417$	0	0
$418$	−1692.05	−0.197993
$419$	−8451.89	−0.985446	−0.492723	−	0.870186i	$-0.663998\pi$
$419$	−8451.89	−0.985446	−0.492723	+	0.870186i	$0.663998\pi$
$420$	0	0
$421$	−10486.2	−1.21393	−0.606967	−	0.794727i	$-0.707614\pi$
$421$	−10486.2	−1.21393	−0.606967	+	0.794727i	$0.707614\pi$
$422$	−16412.6	−1.89325
$423$	0	0
$424$	1253.17	0.143536
$425$	569.504	0.0650000
$426$	0	0
$427$	−1179.25	−0.133648
$428$	1880.41	0.212367
$429$	0	0
$430$	2838.23	0.318306
$431$	−7443.30	−0.831859	−0.415929	−	0.909397i	$-0.636544\pi$
$431$	−7443.30	−0.831859	−0.415929	+	0.909397i	$0.636544\pi$
$432$	0	0
$433$	−3778.56	−0.419367	−0.209683	−	0.977769i	$-0.567243\pi$
$433$	−3778.56	−0.419367	−0.209683	+	0.977769i	$0.567243\pi$
$434$	−3575.33	−0.395441
$435$	0	0
$436$	3630.46	0.398779
$437$	5936.89	0.649885
$438$	0	0
$439$	−5405.03	−0.587627	−0.293813	−	0.955863i	$-0.594924\pi$
$439$	−5405.03	−0.587627	−0.293813	+	0.955863i	$0.594924\pi$
$440$	901.335	0.0976579
$441$	0	0
$442$	−4338.11	−0.466839
$443$	−2183.94	−0.234226	−0.117113	−	0.993119i	$-0.537364\pi$
$443$	−2183.94	−0.234226	−0.117113	+	0.993119i	$0.537364\pi$
$444$	0	0
$445$	301.668	0.0321358
$446$	−6890.70	−0.731579
$447$	0	0
$448$	2335.82	0.246332
$449$	−13367.2	−1.40498	−0.702492	−	0.711692i	$-0.747929\pi$
$449$	−13367.2	−1.40498	−0.702492	+	0.711692i	$0.747929\pi$
$450$	0	0
$451$	−2455.30	−0.256354
$452$	3378.98	0.351623
$453$	0	0
$454$	−18220.2	−1.88351
$455$	3464.79	0.356994
$456$	0	0
$457$	−13897.3	−1.42251	−0.711256	−	0.702933i	$-0.751873\pi$
$457$	−13897.3	−1.42251	−0.711256	+	0.702933i	$0.751873\pi$
$458$	16843.5	1.71844
$459$	0	0
$460$	1975.39	0.200224
$461$	−12543.9	−1.26730	−0.633652	−	0.773618i	$-0.718445\pi$
$461$	−12543.9	−1.26730	−0.633652	+	0.773618i	$0.718445\pi$
$462$	0	0
$463$	16288.3	1.63495	0.817475	−	0.575965i	$-0.195373\pi$
$463$	16288.3	1.63495	0.817475	+	0.575965i	$0.195373\pi$
$464$	5672.18	0.567509
$465$	0	0
$466$	5792.21	0.575792
$467$	−4689.45	−0.464672	−0.232336	−	0.972636i	$-0.574637\pi$
$467$	−4689.45	−0.464672	−0.232336	+	0.972636i	$0.574637\pi$
$468$	0	0
$469$	−3366.86	−0.331486
$470$	−4118.83	−0.404229
$471$	0	0
$472$	4232.15	0.412713
$473$	−1876.21	−0.182386
$474$	0	0
$475$	−1155.51	−0.111618
$476$	848.527	0.0817063
$477$	0	0
$478$	11839.5	1.13290
$479$	−15228.0	−1.45258	−0.726291	−	0.687387i	$-0.758758\pi$
$479$	−15228.0	−1.45258	−0.726291	+	0.687387i	$0.758758\pi$
$480$	0	0
$481$	1750.09	0.165899
$482$	−16733.0	−1.58126
$483$	0	0
$484$	372.171	0.0349522
$485$	−3314.63	−0.310330
$486$	0	0
$487$	−15805.7	−1.47068	−0.735342	−	0.677697i	$-0.762978\pi$
$487$	−15805.7	−1.47068	−0.735342	+	0.677697i	$0.762978\pi$
$488$	1595.79	0.148029
$489$	0	0
$490$	3267.17	0.301216
$491$	9239.13	0.849198	0.424599	−	0.905382i	$-0.360415\pi$
$491$	9239.13	0.849198	0.424599	+	0.905382i	$0.360415\pi$
$492$	0	0
$493$	−1632.60	−0.149145
$494$	8801.91	0.801653
$495$	0	0
$496$	7021.11	0.635599
$497$	3537.24	0.319249
$498$	0	0
$499$	346.738	0.0311064	0.0155532	−	0.999879i	$-0.495049\pi$
$499$	346.738	0.0311064	0.0155532	+	0.999879i	$0.495049\pi$
$500$	−384.474	−0.0343884
$501$	0	0
$502$	−4392.69	−0.390548
$503$	−12519.4	−1.10977	−0.554884	−	0.831928i	$-0.687237\pi$
$503$	−12519.4	−1.10977	−0.554884	+	0.831928i	$0.687237\pi$
$504$	0	0
$505$	−7929.03	−0.698687
$506$	−4702.24	−0.413123
$507$	0	0
$508$	−2443.51	−0.213412
$509$	−9405.03	−0.818999	−0.409500	−	0.912310i	$-0.634297\pi$
$509$	−9405.03	−0.818999	−0.409500	+	0.912310i	$0.634297\pi$
$510$	0	0
$511$	−5841.69	−0.505717
$512$	94.4331	0.00815116
$513$	0	0
$514$	−19394.0	−1.66426
$515$	10120.0	0.865908
$516$	0	0
$517$	2722.76	0.231619
$518$	−1232.66	−0.104556
$519$	0	0
$520$	−4688.67	−0.395407
$521$	−17884.4	−1.50390	−0.751949	−	0.659222i	$-0.770886\pi$
$521$	−17884.4	−1.50390	−0.751949	+	0.659222i	$0.770886\pi$
$522$	0	0
$523$	7617.35	0.636871	0.318435	−	0.947945i	$-0.396843\pi$
$523$	7617.35	0.636871	0.318435	+	0.947945i	$0.396843\pi$
$524$	1122.52	0.0935834
$525$	0	0
$526$	14818.2	1.22833
$527$	−2020.85	−0.167039
$528$	0	0
$529$	4331.70	0.356021
$530$	1272.46	0.104287
$531$	0	0
$532$	−1721.64	−0.140306
$533$	12772.3	1.03795
$534$	0	0
$535$	−3056.79	−0.247022
$536$	4556.13	0.367155
$537$	0	0
$538$	−5187.44	−0.415700
$539$	−2159.77	−0.172593
$540$	0	0
$541$	−6309.12	−0.501387	−0.250694	−	0.968066i	$-0.580659\pi$
$541$	−6309.12	−0.501387	−0.250694	+	0.968066i	$0.580659\pi$
$542$	−17288.0	−1.37008
$543$	0	0
$544$	−3013.74	−0.237524
$545$	−5901.67	−0.463853
$546$	0	0
$547$	−957.117	−0.0748142	−0.0374071	−	0.999300i	$-0.511910\pi$
$547$	−957.117	−0.0748142	−0.0374071	+	0.999300i	$0.511910\pi$
$548$	6328.63	0.493332
$549$	0	0
$550$	915.209	0.0709539
$551$	3312.50	0.256111
$552$	0	0
$553$	−1130.46	−0.0869292
$554$	−24713.4	−1.89525
$555$	0	0
$556$	1450.35	0.110627
$557$	9284.17	0.706253	0.353127	−	0.935576i	$-0.385119\pi$
$557$	9284.17	0.706253	0.353127	+	0.935576i	$0.385119\pi$
$558$	0	0
$559$	9759.90	0.738461
$560$	4792.37	0.361633
$561$	0	0
$562$	−7354.79	−0.552034
$563$	5842.70	0.437372	0.218686	−	0.975795i	$-0.429823\pi$
$563$	5842.70	0.437372	0.218686	+	0.975795i	$0.429823\pi$
$564$	0	0
$565$	−5492.86	−0.409002
$566$	12207.5	0.906569
$567$	0	0
$568$	−4786.71	−0.353602
$569$	−6805.46	−0.501405	−0.250703	−	0.968064i	$-0.580662\pi$
$569$	−6805.46	−0.501405	−0.250703	+	0.968064i	$0.580662\pi$
$570$	0	0
$571$	21649.6	1.58671	0.793353	−	0.608762i	$-0.208334\pi$
$571$	21649.6	1.58671	0.793353	+	0.608762i	$0.208334\pi$
$572$	−1936.00	−0.141518
$573$	0	0
$574$	−8996.04	−0.654159
$575$	−3211.18	−0.232897
$576$	0	0
$577$	23967.4	1.72925	0.864624	−	0.502420i	$-0.167557\pi$
$577$	23967.4	1.72925	0.864624	+	0.502420i	$0.167557\pi$
$578$	−14623.6	−1.05235
$579$	0	0
$580$	1102.17	0.0789054
$581$	12245.0	0.874367
$582$	0	0
$583$	−841.161	−0.0597553
$584$	7905.16	0.560133
$585$	0	0
$586$	−4845.31	−0.341566
$587$	−7580.37	−0.533007	−0.266504	−	0.963834i	$-0.585868\pi$
$587$	−7580.37	−0.533007	−0.266504	+	0.963834i	$0.585868\pi$
$588$	0	0
$589$	4100.26	0.286839
$590$	4297.29	0.299859
$591$	0	0
$592$	2420.66	0.168055
$593$	2996.14	0.207481	0.103741	−	0.994604i	$-0.466919\pi$
$593$	2996.14	0.207481	0.103741	+	0.994604i	$0.466919\pi$
$594$	0	0
$595$	−1379.36	−0.0950393
$596$	−417.469	−0.0286916
$597$	0	0
$598$	24460.6	1.67269
$599$	−21317.9	−1.45413	−0.727066	−	0.686568i	$-0.759117\pi$
$599$	−21317.9	−1.45413	−0.727066	+	0.686568i	$0.759117\pi$
$600$	0	0
$601$	5703.47	0.387104	0.193552	−	0.981090i	$-0.437999\pi$
$601$	5703.47	0.387104	0.193552	+	0.981090i	$0.437999\pi$
$602$	−6874.31	−0.465408
$603$	0	0
$604$	−5871.53	−0.395545
$605$	−605.000	−0.0406558
$606$	0	0
$607$	23658.6	1.58200	0.790998	−	0.611819i	$-0.209562\pi$
$607$	23658.6	1.58200	0.790998	+	0.611819i	$0.209562\pi$
$608$	6114.80	0.407875
$609$	0	0
$610$	1620.35	0.107551
$611$	−14163.5	−0.937800
$612$	0	0
$613$	8792.89	0.579350	0.289675	−	0.957125i	$-0.406453\pi$
$613$	8792.89	0.579350	0.289675	+	0.957125i	$0.406453\pi$
$614$	27055.6	1.77830
$615$	0	0
$616$	−2183.07	−0.142790
$617$	18080.8	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$617$	18080.8	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$618$	0	0
$619$	−9077.15	−0.589405	−0.294702	−	0.955589i	$-0.595221\pi$
$619$	−9077.15	−0.589405	−0.294702	+	0.955589i	$0.595221\pi$
$620$	1364.28	0.0883725
$621$	0	0
$622$	6756.10	0.435522
$623$	−730.653	−0.0469872
$624$	0	0
$625$	625.000	0.0400000
$626$	−1708.47	−0.109080
$627$	0	0
$628$	−3163.55	−0.201018
$629$	−696.727	−0.0441659
$630$	0	0
$631$	−10646.4	−0.671672	−0.335836	−	0.941920i	$-0.609019\pi$
$631$	−10646.4	−0.671672	−0.335836	+	0.941920i	$0.609019\pi$
$632$	1529.77	0.0962830
$633$	0	0
$634$	17406.8	1.09040
$635$	3972.17	0.248237
$636$	0	0
$637$	11234.9	0.698814
$638$	−2623.63	−0.162806
$639$	0	0
$640$	−8501.41	−0.525075
$641$	14313.4	0.881972	0.440986	−	0.897514i	$-0.354629\pi$
$641$	14313.4	0.881972	0.440986	+	0.897514i	$0.354629\pi$
$642$	0	0
$643$	20481.6	1.25617	0.628084	−	0.778146i	$-0.283839\pi$
$643$	20481.6	1.25617	0.628084	+	0.778146i	$0.283839\pi$
$644$	−4784.47	−0.292755
$645$	0	0
$646$	−3504.12	−0.213417
$647$	13408.9	0.814773	0.407387	−	0.913256i	$-0.366440\pi$
$647$	13408.9	0.814773	0.407387	+	0.913256i	$0.366440\pi$
$648$	0	0
$649$	−2840.73	−0.171816
$650$	−4760.83	−0.287285
$651$	0	0
$652$	2105.77	0.126485
$653$	−27525.1	−1.64953	−0.824764	−	0.565477i	$-0.808692\pi$
$653$	−27525.1	−1.64953	−0.824764	+	0.565477i	$0.808692\pi$
$654$	0	0
$655$	−1824.77	−0.108855
$656$	17666.1	1.05144
$657$	0	0
$658$	9975.98	0.591040
$659$	19977.6	1.18091	0.590453	−	0.807072i	$-0.298949\pi$
$659$	19977.6	1.18091	0.590453	+	0.807072i	$0.298949\pi$
$660$	0	0
$661$	1661.53	0.0977698	0.0488849	−	0.998804i	$-0.484433\pi$
$661$	1661.53	0.0977698	0.0488849	+	0.998804i	$0.484433\pi$
$662$	24165.9	1.41878
$663$	0	0
$664$	−16570.3	−0.968451
$665$	2798.70	0.163201
$666$	0	0
$667$	9205.49	0.534390
$668$	−2458.27	−0.142385
$669$	0	0
$670$	4626.26	0.266758
$671$	−1071.14	−0.0616256
$672$	0	0
$673$	−13546.5	−0.775898	−0.387949	−	0.921681i	$-0.626816\pi$
$673$	−13546.5	−0.775898	−0.387949	+	0.921681i	$0.626816\pi$
$674$	5683.29	0.324796
$675$	0	0
$676$	3313.39	0.188518
$677$	−9859.39	−0.559715	−0.279858	−	0.960042i	$-0.590287\pi$
$677$	−9859.39	−0.559715	−0.279858	+	0.960042i	$0.590287\pi$
$678$	0	0
$679$	8028.19	0.453746
$680$	1866.60	0.105266
$681$	0	0
$682$	−3247.56	−0.182340
$683$	−1182.82	−0.0662654	−0.0331327	−	0.999451i	$-0.510548\pi$
$683$	−1182.82	−0.0662654	−0.0331327	+	0.999451i	$0.510548\pi$
$684$	0	0
$685$	−10287.8	−0.573835
$686$	−21737.2	−1.20981
$687$	0	0
$688$	13499.5	0.748058
$689$	4375.65	0.241943
$690$	0	0
$691$	−4312.15	−0.237398	−0.118699	−	0.992930i	$-0.537872\pi$
$691$	−4312.15	−0.237398	−0.118699	+	0.992930i	$0.537872\pi$
$692$	11290.1	0.620208
$693$	0	0
$694$	−13140.3	−0.718730
$695$	−2357.68	−0.128679
$696$	0	0
$697$	−5084.75	−0.276325
$698$	5403.08	0.292994
$699$	0	0
$700$	931.213	0.0502807
$701$	−23850.3	−1.28504	−0.642519	−	0.766270i	$-0.722111\pi$
$701$	−23850.3	−1.28504	−0.642519	+	0.766270i	$0.722111\pi$
$702$	0	0
$703$	1413.64	0.0758415
$704$	2121.68	0.113585
$705$	0	0
$706$	19722.5	1.05137
$707$	19204.4	1.02158
$708$	0	0
$709$	−28492.0	−1.50922	−0.754612	−	0.656172i	$-0.772175\pi$
$709$	−28492.0	−1.50922	−0.754612	+	0.656172i	$0.772175\pi$
$710$	−4860.38	−0.256911
$711$	0	0
$712$	988.743	0.0520431
$713$	11394.7	0.598505
$714$	0	0
$715$	3147.16	0.164611
$716$	−12214.8	−0.637554
$717$	0	0
$718$	11798.4	0.613248
$719$	14182.8	0.735647	0.367824	−	0.929896i	$-0.380103\pi$
$719$	14182.8	0.735647	0.367824	+	0.929896i	$0.380103\pi$
$720$	0	0
$721$	−24511.2	−1.26608
$722$	−15717.2	−0.810157
$723$	0	0
$724$	8366.63	0.429480
$725$	−1791.69	−0.0917814
$726$	0	0
$727$	31836.8	1.62416	0.812078	−	0.583549i	$-0.198336\pi$
$727$	31836.8	1.62416	0.812078	+	0.583549i	$0.198336\pi$
$728$	11356.2	0.578142
$729$	0	0
$730$	8026.83	0.406968
$731$	−3885.50	−0.196594
$732$	0	0
$733$	10002.6	0.504030	0.252015	−	0.967723i	$-0.418907\pi$
$733$	10002.6	0.504030	0.252015	+	0.967723i	$0.418907\pi$
$734$	11413.8	0.573965
$735$	0	0
$736$	16993.1	0.851053
$737$	−3058.20	−0.152850
$738$	0	0
$739$	18784.8	0.935060	0.467530	−	0.883977i	$-0.345144\pi$
$739$	18784.8	0.935060	0.467530	+	0.883977i	$0.345144\pi$
$740$	470.363	0.0233660
$741$	0	0
$742$	−3081.95	−0.152482
$743$	19455.4	0.960633	0.480317	−	0.877095i	$-0.340522\pi$
$743$	19455.4	0.960633	0.480317	+	0.877095i	$0.340522\pi$
$744$	0	0
$745$	678.636	0.0333735
$746$	8193.87	0.402143
$747$	0	0
$748$	770.738	0.0376751
$749$	7403.68	0.361181
$750$	0	0
$751$	19365.5	0.940953	0.470477	−	0.882413i	$-0.344082\pi$
$751$	19365.5	0.940953	0.470477	+	0.882413i	$0.344082\pi$
$752$	−19590.5	−0.949987
$753$	0	0
$754$	13647.9	0.659186
$755$	9544.74	0.460091
$756$	0	0
$757$	19298.9	0.926595	0.463297	−	0.886203i	$-0.346666\pi$
$757$	19298.9	0.926595	0.463297	+	0.886203i	$0.346666\pi$
$758$	−22342.1	−1.07058
$759$	0	0
$760$	−3787.28	−0.180762
$761$	30510.0	1.45333	0.726666	−	0.686991i	$-0.241069\pi$
$761$	30510.0	1.45333	0.726666	+	0.686991i	$0.241069\pi$
$762$	0	0
$763$	14294.1	0.678219
$764$	72.2152	0.00341970
$765$	0	0
$766$	10751.1	0.507117
$767$	14777.2	0.695665
$768$	0	0
$769$	11569.6	0.542536	0.271268	−	0.962504i	$-0.412557\pi$
$769$	11569.6	0.542536	0.271268	+	0.962504i	$0.412557\pi$
$770$	−2216.67	−0.103745
$771$	0	0
$772$	−15299.6	−0.713269
$773$	42711.1	1.98734	0.993668	−	0.112353i	$-0.0358386\pi$
$773$	42711.1	1.98734	0.993668	+	0.112353i	$0.0358386\pi$
$774$	0	0
$775$	−2217.78	−0.102793
$776$	−10864.0	−0.502570
$777$	0	0
$778$	10236.6	0.471723
$779$	10316.8	0.474504
$780$	0	0
$781$	3212.96	0.147207
$782$	−9737.99	−0.445307
$783$	0	0
$784$	15539.7	0.707895
$785$	5142.66	0.233821
$786$	0	0
$787$	−403.118	−0.0182587	−0.00912936	−	0.999958i	$-0.502906\pi$
$787$	−403.118	−0.0182587	−0.00912936	+	0.999958i	$0.502906\pi$
$788$	−540.535	−0.0244363
$789$	0	0
$790$	1553.31	0.0699549
$791$	13303.9	0.598019
$792$	0	0
$793$	5571.96	0.249516
$794$	16189.1	0.723589
$795$	0	0
$796$	16389.8	0.729802
$797$	11413.0	0.507237	0.253619	−	0.967304i	$-0.418379\pi$
$797$	11413.0	0.507237	0.253619	+	0.967304i	$0.418379\pi$
$798$	0	0
$799$	5638.63	0.249662
$800$	−3307.41	−0.146168
$801$	0	0
$802$	−38859.3	−1.71093
$803$	−5306.15	−0.233188
$804$	0	0
$805$	7777.62	0.340528
$806$	16893.5	0.738275
$807$	0	0
$808$	−25988.1	−1.13151
$809$	−21799.0	−0.947357	−0.473679	−	0.880698i	$-0.657074\pi$
$809$	−21799.0	−0.947357	−0.473679	+	0.880698i	$0.657074\pi$
$810$	0	0
$811$	404.042	0.0174942	0.00874711	−	0.999962i	$-0.497216\pi$
$811$	404.042	0.0174942	0.00874711	+	0.999962i	$0.497216\pi$
$812$	−2669.50	−0.115371
$813$	0	0
$814$	−1119.66	−0.0482113
$815$	−3423.14	−0.147126
$816$	0	0
$817$	7883.59	0.337591
$818$	−54559.8	−2.33208
$819$	0	0
$820$	3432.73	0.146190
$821$	−428.289	−0.0182063	−0.00910316	−	0.999959i	$-0.502898\pi$
$821$	−428.289	−0.0182063	−0.00910316	+	0.999959i	$0.502898\pi$
$822$	0	0
$823$	−21469.0	−0.909309	−0.454655	−	0.890668i	$-0.650237\pi$
$823$	−21469.0	−0.909309	−0.454655	+	0.890668i	$0.650237\pi$
$824$	33169.3	1.40231
$825$	0	0
$826$	−10408.2	−0.438437
$827$	24062.7	1.01178	0.505889	−	0.862599i	$-0.331165\pi$
$827$	24062.7	1.01178	0.505889	+	0.862599i	$0.331165\pi$
$828$	0	0
$829$	−15567.7	−0.652217	−0.326108	−	0.945332i	$-0.605737\pi$
$829$	−15567.7	−0.652217	−0.326108	+	0.945332i	$0.605737\pi$
$830$	−16825.3	−0.703633
$831$	0	0
$832$	−11036.8	−0.459894
$833$	−4472.72	−0.186039
$834$	0	0
$835$	3996.15	0.165620
$836$	−1563.81	−0.0646955
$837$	0	0
$838$	−28128.1	−1.15951
$839$	−11335.4	−0.466437	−0.233218	−	0.972424i	$-0.574926\pi$
$839$	−11335.4	−0.466437	−0.233218	+	0.972424i	$0.574926\pi$
$840$	0	0
$841$	−19252.8	−0.789404
$842$	−34898.4	−1.42836
$843$	0	0
$844$	−15168.6	−0.618633
$845$	−5386.23	−0.219280
$846$	0	0
$847$	1465.34	0.0594446
$848$	6052.22	0.245087
$849$	0	0
$850$	1895.33	0.0764814
$851$	3928.54	0.158247
$852$	0	0
$853$	−5152.34	−0.206814	−0.103407	−	0.994639i	$-0.532974\pi$
$853$	−5152.34	−0.206814	−0.103407	+	0.994639i	$0.532974\pi$
$854$	−3924.57	−0.157255
$855$	0	0
$856$	−10018.9	−0.400045
$857$	−21195.2	−0.844825	−0.422413	−	0.906404i	$-0.638817\pi$
$857$	−21195.2	−0.844825	−0.422413	+	0.906404i	$0.638817\pi$
$858$	0	0
$859$	33887.6	1.34602	0.673010	−	0.739634i	$-0.265001\pi$
$859$	33887.6	1.34602	0.673010	+	0.739634i	$0.265001\pi$
$860$	2623.11	0.104009
$861$	0	0
$862$	−24771.5	−0.978795
$863$	−26947.3	−1.06292	−0.531458	−	0.847085i	$-0.678356\pi$
$863$	−26947.3	−1.06292	−0.531458	+	0.847085i	$0.678356\pi$
$864$	0	0
$865$	−18353.1	−0.721415
$866$	−12575.2	−0.493442
$867$	0	0
$868$	−3304.35	−0.129213
$869$	−1026.82	−0.0400834
$870$	0	0
$871$	15908.5	0.618873
$872$	−19343.2	−0.751198
$873$	0	0
$874$	19758.1	0.764679
$875$	−1513.78	−0.0584857
$876$	0	0
$877$	−15854.7	−0.610462	−0.305231	−	0.952278i	$-0.598734\pi$
$877$	−15854.7	−0.610462	−0.305231	+	0.952278i	$0.598734\pi$
$878$	−17988.1	−0.691423
$879$	0	0
$880$	4353.02	0.166750
$881$	−2198.99	−0.0840930	−0.0420465	−	0.999116i	$-0.513388\pi$
$881$	−2198.99	−0.0840930	−0.0420465	+	0.999116i	$0.513388\pi$
$882$	0	0
$883$	17607.6	0.671058	0.335529	−	0.942030i	$-0.391085\pi$
$883$	17607.6	0.671058	0.335529	+	0.942030i	$0.391085\pi$
$884$	−4009.31	−0.152543
$885$	0	0
$886$	−7268.21	−0.275599
$887$	50391.4	1.90753	0.953764	−	0.300558i	$-0.0971728\pi$
$887$	50391.4	1.90753	0.953764	+	0.300558i	$0.0971728\pi$
$888$	0	0
$889$	−9620.77	−0.362959
$890$	1003.96	0.0378122
$891$	0	0
$892$	−6368.44	−0.239048
$893$	−11440.6	−0.428719
$894$	0	0
$895$	19856.3	0.741592
$896$	20590.8	0.767734
$897$	0	0
$898$	−44486.5	−1.65316
$899$	6357.69	0.235863
$900$	0	0
$901$	−1741.98	−0.0644105
$902$	−8171.32	−0.301635
$903$	0	0
$904$	−18003.3	−0.662368
$905$	−13600.8	−0.499563
$906$	0	0
$907$	20022.6	0.733009	0.366505	−	0.930416i	$-0.380554\pi$
$907$	20022.6	0.733009	0.366505	+	0.930416i	$0.380554\pi$
$908$	−16839.2	−0.615450
$909$	0	0
$910$	11530.9	0.420052
$911$	14270.0	0.518974	0.259487	−	0.965747i	$-0.416447\pi$
$911$	14270.0	0.518974	0.259487	+	0.965747i	$0.416447\pi$
$912$	0	0
$913$	11122.4	0.403174
$914$	−46250.6	−1.67378
$915$	0	0
$916$	15566.9	0.561512
$917$	4419.68	0.159161
$918$	0	0
$919$	36781.2	1.32024	0.660120	−	0.751160i	$-0.270505\pi$
$919$	36781.2	1.32024	0.660120	+	0.751160i	$0.270505\pi$
$920$	−10524.9	−0.377170
$921$	0	0
$922$	−41746.4	−1.49116
$923$	−16713.5	−0.596027
$924$	0	0
$925$	−764.620	−0.0271790
$926$	54208.0	1.92374
$927$	0	0
$928$	9481.35	0.335389
$929$	−35798.6	−1.26428	−0.632139	−	0.774855i	$-0.717823\pi$
$929$	−35798.6	−1.26428	−0.632139	+	0.774855i	$0.717823\pi$
$930$	0	0
$931$	9075.04	0.319466
$932$	5353.21	0.188144
$933$	0	0
$934$	−15606.6	−0.546750
$935$	−1252.91	−0.0438230
$936$	0	0
$937$	49934.6	1.74097	0.870486	−	0.492193i	$-0.163805\pi$
$937$	49934.6	1.74097	0.870486	+	0.492193i	$0.163805\pi$
$938$	−11205.0	−0.390039
$939$	0	0
$940$	−3806.65	−0.132084
$941$	28437.4	0.985156	0.492578	−	0.870268i	$-0.336055\pi$
$941$	28437.4	0.985156	0.492578	+	0.870268i	$0.336055\pi$
$942$	0	0
$943$	28670.6	0.990078
$944$	20439.3	0.704706
$945$	0	0
$946$	−6244.10	−0.214602
$947$	−45305.0	−1.55461	−0.777304	−	0.629125i	$-0.783413\pi$
$947$	−45305.0	−1.55461	−0.777304	+	0.629125i	$0.783413\pi$
$948$	0	0
$949$	27602.1	0.944155
$950$	−3845.58	−0.131334
$951$	0	0
$952$	−4520.98	−0.153914
$953$	−6991.64	−0.237651	−0.118825	−	0.992915i	$-0.537913\pi$
$953$	−6991.64	−0.237651	−0.118825	+	0.992915i	$0.537913\pi$
$954$	0	0
$955$	−117.393	−0.00397774
$956$	10942.1	0.370182
$957$	0	0
$958$	−50679.4	−1.70916
$959$	24917.5	0.839029
$960$	0	0
$961$	−21921.4	−0.735838
$962$	5824.37	0.195203
$963$	0	0
$964$	−15464.8	−0.516689
$965$	24870.9	0.829662
$966$	0	0
$967$	−1119.96	−0.0372444	−0.0186222	−	0.999827i	$-0.505928\pi$
$967$	−1119.96	−0.0372444	−0.0186222	+	0.999827i	$0.505928\pi$
$968$	−1982.94	−0.0658410
$969$	0	0
$970$	−11031.2	−0.365145
$971$	1998.46	0.0660490	0.0330245	−	0.999455i	$-0.489486\pi$
$971$	1998.46	0.0660490	0.0330245	+	0.999455i	$0.489486\pi$
$972$	0	0
$973$	5710.40	0.188147
$974$	−52601.7	−1.73046
$975$	0	0
$976$	7706.92	0.252759
$977$	9456.06	0.309648	0.154824	−	0.987942i	$-0.450519\pi$
$977$	9456.06	0.309648	0.154824	+	0.987942i	$0.450519\pi$
$978$	0	0
$979$	−663.670	−0.0216660
$980$	3019.55	0.0984244
$981$	0	0
$982$	30748.1	0.999197
$983$	−56956.1	−1.84803	−0.924016	−	0.382353i	$-0.875114\pi$
$983$	−56956.1	−1.84803	−0.924016	+	0.382353i	$0.875114\pi$
$984$	0	0
$985$	878.693	0.0284238
$986$	−5433.33	−0.175489
$987$	0	0
$988$	8134.80	0.261946
$989$	21908.6	0.704402
$990$	0	0
$991$	−45020.8	−1.44312	−0.721561	−	0.692351i	$-0.756575\pi$
$991$	−45020.8	−1.44312	−0.721561	+	0.692351i	$0.756575\pi$
$992$	11736.2	0.375628
$993$	0	0
$994$	11772.1	0.375641
$995$	−26643.3	−0.848893
$996$	0	0
$997$	−51620.5	−1.63976	−0.819878	−	0.572538i	$-0.805959\pi$
$997$	−51620.5	−1.63976	−0.819878	+	0.572538i	$0.805959\pi$
$998$	1153.95	0.0366010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.h.1.3	✓	3
3.2	odd	2		495.4.a.j.1.1	yes	3
5.4	even	2		2475.4.a.x.1.1		3
15.14	odd	2		2475.4.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.h.1.3	✓	3	1.1	even	1	trivial
495.4.a.j.1.1	yes	3	3.2	odd	2
2475.4.a.u.1.3		3	15.14	odd	2
2475.4.a.x.1.1		3	5.4	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32803 q^{2} +3.07579 q^{4} -5.00000 q^{5} +12.1102 q^{7} -16.3879 q^{8} +O(q^{10})\)	\(q+3.32803 q^{2} +3.07579 q^{4} -5.00000 q^{5} +12.1102 q^{7} -16.3879 q^{8} -16.6402 q^{10} +11.0000 q^{11} -57.2210 q^{13} +40.3032 q^{14} -79.1458 q^{16} +22.7802 q^{17} -46.2204 q^{19} -15.3790 q^{20} +36.6083 q^{22} -128.447 q^{23} +25.0000 q^{25} -190.433 q^{26} +37.2485 q^{28} -71.6674 q^{29} -88.7110 q^{31} -132.296 q^{32} +75.8131 q^{34} -60.5511 q^{35} -30.5848 q^{37} -153.823 q^{38} +81.9396 q^{40} -223.209 q^{41} -170.565 q^{43} +33.8337 q^{44} -427.477 q^{46} +247.523 q^{47} -196.343 q^{49} +83.2008 q^{50} -176.000 q^{52} -76.4692 q^{53} -55.0000 q^{55} -198.461 q^{56} -238.511 q^{58} -258.248 q^{59} -97.3761 q^{61} -295.233 q^{62} +192.880 q^{64} +286.105 q^{65} -278.018 q^{67} +70.0671 q^{68} -201.516 q^{70} +292.087 q^{71} -482.377 q^{73} -101.787 q^{74} -142.164 q^{76} +133.212 q^{77} -93.3473 q^{79} +395.729 q^{80} -742.847 q^{82} +1011.13 q^{83} -113.901 q^{85} -567.646 q^{86} -180.267 q^{88} -60.3336 q^{89} -692.959 q^{91} -395.077 q^{92} +823.766 q^{94} +231.102 q^{95} +662.927 q^{97} -653.435 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8	\( 3 q - q^{2} + q^{4} - 15 q^{5} + 2 q^{7} + 3 q^{8} + 5 q^{10} + 33 q^{11} - 66 q^{13} + 88 q^{14} - 103 q^{16} + 100 q^{17} - 70 q^{19} - 5 q^{20} - 11 q^{22} + 10 q^{23} + 75 q^{25} - 88 q^{26} - 64 q^{28} - 34 q^{29} - 104 q^{31} - 37 q^{32} - 278 q^{34} - 10 q^{35} - 270 q^{37} - 154 q^{38} - 15 q^{40} - 82 q^{41} - 428 q^{43} + 11 q^{44} - 706 q^{46} + 246 q^{47} - 685 q^{49} - 25 q^{50} - 528 q^{52} + 298 q^{53} - 165 q^{55} - 288 q^{56} + 114 q^{58} - 204 q^{59} - 492 q^{61} - 584 q^{62} - 319 q^{64} + 330 q^{65} - 812 q^{67} + 798 q^{68} - 440 q^{70} - 442 q^{71} - 1044 q^{73} - 42 q^{74} + 106 q^{76} + 22 q^{77} - 1436 q^{79} + 515 q^{80} - 1246 q^{82} + 734 q^{83} - 500 q^{85} + 82 q^{86} + 33 q^{88} - 556 q^{89} - 176 q^{91} - 630 q^{92} + 38 q^{94} + 350 q^{95} - 1038 q^{97} + 143 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 - 15 * q^5 + 2 * q^7 + 3 * q^8 + 5 * q^10 + 33 * q^11 - 66 * q^13 + 88 * q^14 - 103 * q^16 + 100 * q^17 - 70 * q^19 - 5 * q^20 - 11 * q^22 + 10 * q^23 + 75 * q^25 - 88 * q^26 - 64 * q^28 - 34 * q^29 - 104 * q^31 - 37 * q^32 - 278 * q^34 - 10 * q^35 - 270 * q^37 - 154 * q^38 - 15 * q^40 - 82 * q^41 - 428 * q^43 + 11 * q^44 - 706 * q^46 + 246 * q^47 - 685 * q^49 - 25 * q^50 - 528 * q^52 + 298 * q^53 - 165 * q^55 - 288 * q^56 + 114 * q^58 - 204 * q^59 - 492 * q^61 - 584 * q^62 - 319 * q^64 + 330 * q^65 - 812 * q^67 + 798 * q^68 - 440 * q^70 - 442 * q^71 - 1044 * q^73 - 42 * q^74 + 106 * q^76 + 22 * q^77 - 1436 * q^79 + 515 * q^80 - 1246 * q^82 + 734 * q^83 - 500 * q^85 + 82 * q^86 + 33 * q^88 - 556 * q^89 - 176 * q^91 - 630 * q^92 + 38 * q^94 + 350 * q^95 - 1038 * q^97 + 143 * q^98

Introduction

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