LMFDB - Embedded newform 495.4.a.o.1.7

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$6.03128$ 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)

$f(q)$	$=$	$q+5.03128 q^{2} +17.3138 q^{4} -5.00000 q^{5} +18.5765 q^{7} +46.8604 q^{8} +O(q^{10})$	\(q+5.03128 q^{2} +17.3138 q^{4} -5.00000 q^{5} +18.5765 q^{7} +46.8604 q^{8} -25.1564 q^{10} -11.0000 q^{11} +47.0274 q^{13} +93.4634 q^{14} +97.2576 q^{16} +2.62785 q^{17} +157.253 q^{19} -86.5691 q^{20} -55.3441 q^{22} -186.809 q^{23} +25.0000 q^{25} +236.608 q^{26} +321.629 q^{28} +270.325 q^{29} -25.7761 q^{31} +114.447 q^{32} +13.2215 q^{34} -92.8823 q^{35} +228.781 q^{37} +791.187 q^{38} -234.302 q^{40} -6.85273 q^{41} -386.113 q^{43} -190.452 q^{44} -939.888 q^{46} -283.719 q^{47} +2.08486 q^{49} +125.782 q^{50} +814.223 q^{52} +452.807 q^{53} +55.0000 q^{55} +870.501 q^{56} +1360.08 q^{58} +88.1209 q^{59} -829.267 q^{61} -129.687 q^{62} -202.245 q^{64} -235.137 q^{65} +489.909 q^{67} +45.4982 q^{68} -467.317 q^{70} +63.5781 q^{71} -78.3019 q^{73} +1151.06 q^{74} +2722.66 q^{76} -204.341 q^{77} -203.812 q^{79} -486.288 q^{80} -34.4780 q^{82} -287.947 q^{83} -13.1393 q^{85} -1942.65 q^{86} -515.465 q^{88} -1329.42 q^{89} +873.602 q^{91} -3234.37 q^{92} -1427.47 q^{94} -786.267 q^{95} -62.6842 q^{97} +10.4895 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8}+O(q^{10}) $ 7 * q - 5 * q^2 + 33 * q^4 - 35 * q^5 + 30 * q^7 - 45 * q^8	$ 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8} + 25 q^{10} - 77 q^{11} + 38 q^{13} - 20 q^{14} + 309 q^{16} - 12 q^{17} + 226 q^{19} - 165 q^{20} + 55 q^{22} - 334 q^{23} + 175 q^{25} + 372 q^{26} + 812 q^{28} + 258 q^{29} + 336 q^{31} - 485 q^{32} + 78 q^{34} - 150 q^{35} + 466 q^{37} + 494 q^{38} + 225 q^{40} + 258 q^{41} + 308 q^{43} - 363 q^{44} + 98 q^{46} - 546 q^{47} + 735 q^{49} - 125 q^{50} + 512 q^{52} - 110 q^{53} + 385 q^{55} - 20 q^{56} + 1362 q^{58} + 68 q^{59} + 1096 q^{61} - 356 q^{62} + 2761 q^{64} - 190 q^{65} + 2268 q^{67} + 1186 q^{68} + 100 q^{70} + 166 q^{71} + 200 q^{73} + 1710 q^{74} + 3310 q^{76} - 330 q^{77} + 2152 q^{79} - 1545 q^{80} - 1006 q^{82} - 370 q^{83} + 60 q^{85} - 106 q^{86} + 495 q^{88} + 252 q^{89} + 2768 q^{91} - 3774 q^{92} + 2218 q^{94} - 1130 q^{95} + 3698 q^{97} - 697 q^{98}+O(q^{100}) $ 7 * q - 5 * q^2 + 33 * q^4 - 35 * q^5 + 30 * q^7 - 45 * q^8 + 25 * q^10 - 77 * q^11 + 38 * q^13 - 20 * q^14 + 309 * q^16 - 12 * q^17 + 226 * q^19 - 165 * q^20 + 55 * q^22 - 334 * q^23 + 175 * q^25 + 372 * q^26 + 812 * q^28 + 258 * q^29 + 336 * q^31 - 485 * q^32 + 78 * q^34 - 150 * q^35 + 466 * q^37 + 494 * q^38 + 225 * q^40 + 258 * q^41 + 308 * q^43 - 363 * q^44 + 98 * q^46 - 546 * q^47 + 735 * q^49 - 125 * q^50 + 512 * q^52 - 110 * q^53 + 385 * q^55 - 20 * q^56 + 1362 * q^58 + 68 * q^59 + 1096 * q^61 - 356 * q^62 + 2761 * q^64 - 190 * q^65 + 2268 * q^67 + 1186 * q^68 + 100 * q^70 + 166 * q^71 + 200 * q^73 + 1710 * q^74 + 3310 * q^76 - 330 * q^77 + 2152 * q^79 - 1545 * q^80 - 1006 * q^82 - 370 * q^83 + 60 * q^85 - 106 * q^86 + 495 * q^88 + 252 * q^89 + 2768 * q^91 - 3774 * q^92 + 2218 * q^94 - 1130 * q^95 + 3698 * q^97 - 697 * 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 $

$2$	5.03128	1.77883	0.889414	−	0.457103i	$-0.151113\pi$
$2$	5.03128	1.77883	0.889414	+	0.457103i	$0.151113\pi$
$3$	0	0
$3$	0	0
$4$	17.3138	2.16423
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	18.5765	1.00303	0.501517	−	0.865148i	$-0.332775\pi$
$7$	18.5765	1.00303	0.501517	+	0.865148i	$0.332775\pi$
$8$	46.8604	2.07096
$9$	0	0
$10$	−25.1564	−0.795516
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	47.0274	1.00331	0.501656	−	0.865067i	$-0.332724\pi$
$13$	47.0274	1.00331	0.501656	+	0.865067i	$0.332724\pi$
$14$	93.4634	1.78423
$15$	0	0
$16$	97.2576	1.51965
$17$	2.62785	0.0374911	0.0187455	−	0.999824i	$-0.494033\pi$
$17$	2.62785	0.0374911	0.0187455	+	0.999824i	$0.494033\pi$
$18$	0	0
$19$	157.253	1.89876	0.949379	−	0.314132i	$-0.101713\pi$
$19$	157.253	1.89876	0.949379	+	0.314132i	$0.101713\pi$
$20$	−86.5691	−0.967871
$21$	0	0
$22$	−55.3441	−0.536337
$23$	−186.809	−1.69358	−0.846790	−	0.531927i	$-0.821468\pi$
$23$	−186.809	−1.69358	−0.846790	+	0.531927i	$0.821468\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	236.608	1.78472
$27$	0	0
$28$	321.629	2.17079
$29$	270.325	1.73097	0.865483	−	0.500938i	$-0.167011\pi$
$29$	270.325	1.73097	0.865483	+	0.500938i	$0.167011\pi$
$30$	0	0
$31$	−25.7761	−0.149340	−0.0746698	−	0.997208i	$-0.523790\pi$
$31$	−25.7761	−0.149340	−0.0746698	+	0.997208i	$0.523790\pi$
$32$	114.447	0.632236
$33$	0	0
$34$	13.2215	0.0666901
$35$	−92.8823	−0.448571
$36$	0	0
$37$	228.781	1.01652	0.508262	−	0.861202i	$-0.330288\pi$
$37$	228.781	1.01652	0.508262	+	0.861202i	$0.330288\pi$
$38$	791.187	3.37756
$39$	0	0
$40$	−234.302	−0.926160
$41$	−6.85273	−0.0261029	−0.0130514	−	0.999915i	$-0.504155\pi$
$41$	−6.85273	−0.0261029	−0.0130514	+	0.999915i	$0.504155\pi$
$42$	0	0
$43$	−386.113	−1.36934	−0.684671	−	0.728852i	$-0.740054\pi$
$43$	−386.113	−1.36934	−0.684671	+	0.728852i	$0.740054\pi$
$44$	−190.452	−0.652539
$45$	0	0
$46$	−939.888	−3.01259
$47$	−283.719	−0.880525	−0.440262	−	0.897869i	$-0.645115\pi$
$47$	−283.719	−0.880525	−0.440262	+	0.897869i	$0.645115\pi$
$48$	0	0
$49$	2.08486	0.00607831
$50$	125.782	0.355765
$51$	0	0
$52$	814.223	2.17139
$53$	452.807	1.17354	0.586772	−	0.809753i	$-0.300399\pi$
$53$	452.807	1.17354	0.586772	+	0.809753i	$0.300399\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	870.501	2.07724
$57$	0	0
$58$	1360.08	3.07909
$59$	88.1209	0.194447	0.0972234	−	0.995263i	$-0.469004\pi$
$59$	88.1209	0.194447	0.0972234	+	0.995263i	$0.469004\pi$
$60$	0	0
$61$	−829.267	−1.74060	−0.870301	−	0.492520i	$-0.836076\pi$
$61$	−829.267	−1.74060	−0.870301	+	0.492520i	$0.836076\pi$
$62$	−129.687	−0.265649
$63$	0	0
$64$	−202.245	−0.395010
$65$	−235.137	−0.448694
$66$	0	0
$67$	489.909	0.893313	0.446656	−	0.894706i	$-0.352615\pi$
$67$	489.909	0.893313	0.446656	+	0.894706i	$0.352615\pi$
$68$	45.4982	0.0811391
$69$	0	0
$70$	−467.317	−0.797930
$71$	63.5781	0.106272	0.0531361	−	0.998587i	$-0.483078\pi$
$71$	63.5781	0.106272	0.0531361	+	0.998587i	$0.483078\pi$
$72$	0	0
$73$	−78.3019	−0.125542	−0.0627709	−	0.998028i	$-0.519994\pi$
$73$	−78.3019	−0.125542	−0.0627709	+	0.998028i	$0.519994\pi$
$74$	1151.06	1.80822
$75$	0	0
$76$	2722.66	4.10934
$77$	−204.341	−0.302426
$78$	0	0
$79$	−203.812	−0.290262	−0.145131	−	0.989412i	$-0.546360\pi$
$79$	−203.812	−0.290262	−0.145131	+	0.989412i	$0.546360\pi$
$80$	−486.288	−0.679608
$81$	0	0
$82$	−34.4780	−0.0464325
$83$	−287.947	−0.380799	−0.190399	−	0.981707i	$-0.560978\pi$
$83$	−287.947	−0.380799	−0.190399	+	0.981707i	$0.560978\pi$
$84$	0	0
$85$	−13.1393	−0.0167665
$86$	−1942.65	−2.43582
$87$	0	0
$88$	−515.465	−0.624417
$89$	−1329.42	−1.58335	−0.791677	−	0.610940i	$-0.790792\pi$
$89$	−1329.42	−1.58335	−0.791677	+	0.610940i	$0.790792\pi$
$90$	0	0
$91$	873.602	1.00636
$92$	−3234.37	−3.66529
$93$	0	0
$94$	−1427.47	−1.56630
$95$	−786.267	−0.849151
$96$	0	0
$97$	−62.6842	−0.0656146	−0.0328073	−	0.999462i	$-0.510445\pi$
$97$	−62.6842	−0.0656146	−0.0328073	+	0.999462i	$0.510445\pi$
$98$	10.4895	0.0108123
$99$	0	0
$100$	432.845	0.432845
$101$	−1206.36	−1.18848	−0.594242	−	0.804286i	$-0.702548\pi$
$101$	−1206.36	−1.18848	−0.594242	+	0.804286i	$0.702548\pi$
$102$	0	0
$103$	1815.37	1.73664	0.868319	−	0.496006i	$-0.165201\pi$
$103$	1815.37	1.73664	0.868319	+	0.496006i	$0.165201\pi$
$104$	2203.72	2.07782
$105$	0	0
$106$	2278.20	2.08753
$107$	−861.230	−0.778114	−0.389057	−	0.921214i	$-0.627199\pi$
$107$	−861.230	−0.778114	−0.389057	+	0.921214i	$0.627199\pi$
$108$	0	0
$109$	−739.420	−0.649758	−0.324879	−	0.945756i	$-0.605324\pi$
$109$	−739.420	−0.649758	−0.324879	+	0.945756i	$0.605324\pi$
$110$	276.721	0.239857
$111$	0	0
$112$	1806.70	1.52426
$113$	−955.065	−0.795088	−0.397544	−	0.917583i	$-0.630137\pi$
$113$	−955.065	−0.795088	−0.397544	+	0.917583i	$0.630137\pi$
$114$	0	0
$115$	934.044	0.757392
$116$	4680.35	3.74620
$117$	0	0
$118$	443.361	0.345887
$119$	48.8162	0.0376048
$120$	0	0
$121$	121.000	0.0909091
$122$	−4172.28	−3.09623
$123$	0	0
$124$	−446.283	−0.323205
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2243.09	−1.56726	−0.783631	−	0.621227i	$-0.786635\pi$
$127$	−2243.09	−1.56726	−0.783631	+	0.621227i	$0.786635\pi$
$128$	−1933.13	−1.33489
$129$	0	0
$130$	−1183.04	−0.798150
$131$	1785.43	1.19079	0.595397	−	0.803432i	$-0.296995\pi$
$131$	1785.43	1.19079	0.595397	+	0.803432i	$0.296995\pi$
$132$	0	0
$133$	2921.21	1.90452
$134$	2464.87	1.58905
$135$	0	0
$136$	123.142	0.0776424
$137$	−251.359	−0.156752	−0.0783761	−	0.996924i	$-0.524973\pi$
$137$	−251.359	−0.156752	−0.0783761	+	0.996924i	$0.524973\pi$
$138$	0	0
$139$	−2363.05	−1.44195	−0.720976	−	0.692960i	$-0.756306\pi$
$139$	−2363.05	−1.44195	−0.720976	+	0.692960i	$0.756306\pi$
$140$	−1608.15	−0.970809
$141$	0	0
$142$	319.879	0.189040
$143$	−517.301	−0.302510
$144$	0	0
$145$	−1351.62	−0.774112
$146$	−393.959	−0.223317
$147$	0	0
$148$	3961.08	2.19999
$149$	600.435	0.330131	0.165065	−	0.986283i	$-0.447216\pi$
$149$	600.435	0.330131	0.165065	+	0.986283i	$0.447216\pi$
$150$	0	0
$151$	1329.67	0.716604	0.358302	−	0.933606i	$-0.383356\pi$
$151$	1329.67	0.716604	0.358302	+	0.933606i	$0.383356\pi$
$152$	7368.96	3.93225
$153$	0	0
$154$	−1028.10	−0.537964
$155$	128.881	0.0667867
$156$	0	0
$157$	−524.797	−0.266773	−0.133386	−	0.991064i	$-0.542585\pi$
$157$	−524.797	−0.266773	−0.133386	+	0.991064i	$0.542585\pi$
$158$	−1025.44	−0.516325
$159$	0	0
$160$	−572.235	−0.282745
$161$	−3470.25	−1.69872
$162$	0	0
$163$	1159.74	0.557287	0.278643	−	0.960395i	$-0.410115\pi$
$163$	1159.74	0.557287	0.278643	+	0.960395i	$0.410115\pi$
$164$	−118.647	−0.0564925
$165$	0	0
$166$	−1448.74	−0.677375
$167$	−2452.36	−1.13634	−0.568171	−	0.822910i	$-0.692349\pi$
$167$	−2452.36	−1.13634	−0.568171	+	0.822910i	$0.692349\pi$
$168$	0	0
$169$	14.5740	0.00663359
$170$	−66.1074	−0.0298247
$171$	0	0
$172$	−6685.09	−2.96357
$173$	−1660.70	−0.729831	−0.364916	−	0.931041i	$-0.618902\pi$
$173$	−1660.70	−0.729831	−0.364916	+	0.931041i	$0.618902\pi$
$174$	0	0
$175$	464.411	0.200607
$176$	−1069.83	−0.458192
$177$	0	0
$178$	−6688.70	−2.81651
$179$	−216.967	−0.0905972	−0.0452986	−	0.998973i	$-0.514424\pi$
$179$	−216.967	−0.0905972	−0.0452986	+	0.998973i	$0.514424\pi$
$180$	0	0
$181$	2803.62	1.15133	0.575667	−	0.817684i	$-0.304742\pi$
$181$	2803.62	1.15133	0.575667	+	0.817684i	$0.304742\pi$
$182$	4395.34	1.79013
$183$	0	0
$184$	−8753.94	−3.50733
$185$	−1143.91	−0.454604
$186$	0	0
$187$	−28.9064	−0.0113040
$188$	−4912.26	−1.90566
$189$	0	0
$190$	−3955.93	−1.51049
$191$	−2595.38	−0.983222	−0.491611	−	0.870815i	$-0.663592\pi$
$191$	−2595.38	−0.983222	−0.491611	+	0.870815i	$0.663592\pi$
$192$	0	0
$193$	1564.28	0.583416	0.291708	−	0.956507i	$-0.405776\pi$
$193$	1564.28	0.583416	0.291708	+	0.956507i	$0.405776\pi$
$194$	−315.382	−0.116717
$195$	0	0
$196$	36.0969	0.0131548
$197$	2521.45	0.911906	0.455953	−	0.890004i	$-0.349298\pi$
$197$	2521.45	0.911906	0.455953	+	0.890004i	$0.349298\pi$
$198$	0	0
$199$	−3080.80	−1.09745	−0.548725	−	0.836003i	$-0.684887\pi$
$199$	−3080.80	−1.09745	−0.548725	+	0.836003i	$0.684887\pi$
$200$	1171.51	0.414192
$201$	0	0
$202$	−6069.52	−2.11411
$203$	5021.68	1.73622
$204$	0	0
$205$	34.2637	0.0116736
$206$	9133.64	3.08918
$207$	0	0
$208$	4573.77	1.52468
$209$	−1729.79	−0.572497
$210$	0	0
$211$	1779.21	0.580503	0.290252	−	0.956950i	$-0.406261\pi$
$211$	1779.21	0.580503	0.290252	+	0.956950i	$0.406261\pi$
$212$	7839.81	2.53981
$213$	0	0
$214$	−4333.09	−1.38413
$215$	1930.57	0.612389
$216$	0	0
$217$	−478.829	−0.149793
$218$	−3720.23	−1.15581
$219$	0	0
$220$	952.260	0.291824
$221$	123.581	0.0376152
$222$	0	0
$223$	−2534.63	−0.761126	−0.380563	−	0.924755i	$-0.624270\pi$
$223$	−2534.63	−0.761126	−0.380563	+	0.924755i	$0.624270\pi$
$224$	2126.02	0.634155
$225$	0	0
$226$	−4805.20	−1.41432
$227$	−2838.70	−0.830006	−0.415003	−	0.909820i	$-0.636219\pi$
$227$	−2838.70	−0.830006	−0.415003	+	0.909820i	$0.636219\pi$
$228$	0	0
$229$	6450.97	1.86154	0.930769	−	0.365607i	$-0.119139\pi$
$229$	6450.97	1.86154	0.930769	+	0.365607i	$0.119139\pi$
$230$	4699.44	1.34727
$231$	0	0
$232$	12667.5	3.58476
$233$	4705.33	1.32299	0.661494	−	0.749950i	$-0.269923\pi$
$233$	4705.33	1.32299	0.661494	+	0.749950i	$0.269923\pi$
$234$	0	0
$235$	1418.60	0.393783
$236$	1525.71	0.420827
$237$	0	0
$238$	245.608	0.0668925
$239$	5684.89	1.53860	0.769299	−	0.638889i	$-0.220606\pi$
$239$	5684.89	1.53860	0.769299	+	0.638889i	$0.220606\pi$
$240$	0	0
$241$	883.354	0.236107	0.118054	−	0.993007i	$-0.462335\pi$
$241$	883.354	0.236107	0.118054	+	0.993007i	$0.462335\pi$
$242$	608.785	0.161712
$243$	0	0
$244$	−14357.8	−3.76706
$245$	−10.4243	−0.00271830
$246$	0	0
$247$	7395.22	1.90505
$248$	−1207.88	−0.309276
$249$	0	0
$250$	−628.910	−0.159103
$251$	893.330	0.224647	0.112324	−	0.993672i	$-0.464171\pi$
$251$	893.330	0.224647	0.112324	+	0.993672i	$0.464171\pi$
$252$	0	0
$253$	2054.90	0.510634
$254$	−11285.6	−2.78789
$255$	0	0
$256$	−8108.16	−1.97953
$257$	7675.88	1.86307	0.931533	−	0.363656i	$-0.118472\pi$
$257$	7675.88	1.86307	0.931533	+	0.363656i	$0.118472\pi$
$258$	0	0
$259$	4249.95	1.01961
$260$	−4071.12	−0.971076
$261$	0	0
$262$	8983.01	2.11822
$263$	−2218.64	−0.520180	−0.260090	−	0.965584i	$-0.583752\pi$
$263$	−2218.64	−0.520180	−0.260090	+	0.965584i	$0.583752\pi$
$264$	0	0
$265$	−2264.03	−0.524824
$266$	14697.4	3.38781
$267$	0	0
$268$	8482.20	1.93333
$269$	−3636.28	−0.824194	−0.412097	−	0.911140i	$-0.635203\pi$
$269$	−3636.28	−0.824194	−0.412097	+	0.911140i	$0.635203\pi$
$270$	0	0
$271$	4676.98	1.04836	0.524181	−	0.851607i	$-0.324372\pi$
$271$	4676.98	1.04836	0.524181	+	0.851607i	$0.324372\pi$
$272$	255.579	0.0569733
$273$	0	0
$274$	−1264.66	−0.278835
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−1796.79	−0.389743	−0.194872	−	0.980829i	$-0.562429\pi$
$277$	−1796.79	−0.389743	−0.194872	+	0.980829i	$0.562429\pi$
$278$	−11889.2	−2.56499
$279$	0	0
$280$	−4352.50	−0.928971
$281$	−4275.90	−0.907754	−0.453877	−	0.891064i	$-0.649959\pi$
$281$	−4275.90	−0.907754	−0.453877	+	0.891064i	$0.649959\pi$
$282$	0	0
$283$	3700.60	0.777306	0.388653	−	0.921384i	$-0.372941\pi$
$283$	3700.60	0.777306	0.388653	+	0.921384i	$0.372941\pi$
$284$	1100.78	0.229997
$285$	0	0
$286$	−2602.69	−0.538113
$287$	−127.300	−0.0261821
$288$	0	0
$289$	−4906.09	−0.998594
$290$	−6800.40	−1.37701
$291$	0	0
$292$	−1355.70	−0.271701
$293$	−5675.57	−1.13164	−0.565820	−	0.824529i	$-0.691440\pi$
$293$	−5675.57	−1.13164	−0.565820	+	0.824529i	$0.691440\pi$
$294$	0	0
$295$	−440.605	−0.0869593
$296$	10720.8	2.10518
$297$	0	0
$298$	3020.96	0.587246
$299$	−8785.13	−1.69919
$300$	0	0
$301$	−7172.62	−1.37350
$302$	6689.96	1.27471
$303$	0	0
$304$	15294.1	2.88545
$305$	4146.33	0.778421
$306$	0	0
$307$	3024.92	0.562350	0.281175	−	0.959656i	$-0.409276\pi$
$307$	3024.92	0.562350	0.281175	+	0.959656i	$0.409276\pi$
$308$	−3537.92	−0.654519
$309$	0	0
$310$	648.435	0.118802
$311$	−7871.93	−1.43529	−0.717647	−	0.696407i	$-0.754781\pi$
$311$	−7871.93	−1.43529	−0.717647	+	0.696407i	$0.754781\pi$
$312$	0	0
$313$	−7516.44	−1.35736	−0.678681	−	0.734433i	$-0.737448\pi$
$313$	−7516.44	−1.35736	−0.678681	+	0.734433i	$0.737448\pi$
$314$	−2640.40	−0.474543
$315$	0	0
$316$	−3528.77	−0.628192
$317$	2754.21	0.487987	0.243993	−	0.969777i	$-0.421542\pi$
$317$	2754.21	0.487987	0.243993	+	0.969777i	$0.421542\pi$
$318$	0	0
$319$	−2973.57	−0.521906
$320$	1011.23	0.176654
$321$	0	0
$322$	−17459.8	−3.02173
$323$	413.239	0.0711865
$324$	0	0
$325$	1175.68	0.200662
$326$	5834.97	0.991316
$327$	0	0
$328$	−321.122	−0.0540579
$329$	−5270.50	−0.883197
$330$	0	0
$331$	9581.98	1.59116	0.795579	−	0.605850i	$-0.207167\pi$
$331$	9581.98	1.59116	0.795579	+	0.605850i	$0.207167\pi$
$332$	−4985.46	−0.824134
$333$	0	0
$334$	−12338.5	−2.02136
$335$	−2449.55	−0.399502
$336$	0	0
$337$	9012.50	1.45680	0.728401	−	0.685151i	$-0.240264\pi$
$337$	9012.50	1.45680	0.728401	+	0.685151i	$0.240264\pi$
$338$	73.3259	0.0118000
$339$	0	0
$340$	−227.491	−0.0362865
$341$	283.537	0.0450276
$342$	0	0
$343$	−6333.00	−0.996938
$344$	−18093.4	−2.83585
$345$	0	0
$346$	−8355.46	−1.29824
$347$	1823.58	0.282117	0.141059	−	0.990001i	$-0.454949\pi$
$347$	1823.58	0.282117	0.141059	+	0.990001i	$0.454949\pi$
$348$	0	0
$349$	4446.17	0.681942	0.340971	−	0.940074i	$-0.389244\pi$
$349$	4446.17	0.681942	0.340971	+	0.940074i	$0.389244\pi$
$350$	2336.59	0.356845
$351$	0	0
$352$	−1258.92	−0.190626
$353$	−11347.3	−1.71092	−0.855462	−	0.517865i	$-0.826727\pi$
$353$	−11347.3	−1.71092	−0.855462	+	0.517865i	$0.826727\pi$
$354$	0	0
$355$	−317.890	−0.0475264
$356$	−23017.4	−3.42674
$357$	0	0
$358$	−1091.62	−0.161157
$359$	10034.0	1.47514	0.737568	−	0.675273i	$-0.235974\pi$
$359$	10034.0	1.47514	0.737568	+	0.675273i	$0.235974\pi$
$360$	0	0
$361$	17869.7	2.60529
$362$	14105.8	2.04803
$363$	0	0
$364$	15125.4	2.17798
$365$	391.510	0.0561440
$366$	0	0
$367$	−9745.44	−1.38613	−0.693063	−	0.720877i	$-0.743739\pi$
$367$	−9745.44	−1.38613	−0.693063	+	0.720877i	$0.743739\pi$
$368$	−18168.6	−2.57365
$369$	0	0
$370$	−5755.32	−0.808661
$371$	8411.55	1.17710
$372$	0	0
$373$	−926.803	−0.128654	−0.0643272	−	0.997929i	$-0.520490\pi$
$373$	−926.803	−0.128654	−0.0643272	+	0.997929i	$0.520490\pi$
$374$	−145.436	−0.0201078
$375$	0	0
$376$	−13295.2	−1.82353
$377$	12712.7	1.73670
$378$	0	0
$379$	−5164.90	−0.700008	−0.350004	−	0.936748i	$-0.613820\pi$
$379$	−5164.90	−0.700008	−0.350004	+	0.936748i	$0.613820\pi$
$380$	−13613.3	−1.83775
$381$	0	0
$382$	−13058.1	−1.74898
$383$	5260.08	0.701770	0.350885	−	0.936419i	$-0.385881\pi$
$383$	5260.08	0.701770	0.350885	+	0.936419i	$0.385881\pi$
$384$	0	0
$385$	1021.71	0.135249
$386$	7870.33	1.03780
$387$	0	0
$388$	−1085.30	−0.142005
$389$	13650.6	1.77921	0.889606	−	0.456729i	$-0.150979\pi$
$389$	13650.6	1.77921	0.889606	+	0.456729i	$0.150979\pi$
$390$	0	0
$391$	−490.906	−0.0634941
$392$	97.6974	0.0125879
$393$	0	0
$394$	12686.1	1.62212
$395$	1019.06	0.129809
$396$	0	0
$397$	−1974.69	−0.249640	−0.124820	−	0.992179i	$-0.539835\pi$
$397$	−1974.69	−0.249640	−0.124820	+	0.992179i	$0.539835\pi$
$398$	−15500.4	−1.95217
$399$	0	0
$400$	2431.44	0.303930
$401$	4277.62	0.532704	0.266352	−	0.963876i	$-0.414182\pi$
$401$	4277.62	0.532704	0.266352	+	0.963876i	$0.414182\pi$
$402$	0	0
$403$	−1212.18	−0.149834
$404$	−20886.6	−2.57215
$405$	0	0
$406$	25265.5	3.08843
$407$	−2516.59	−0.306494
$408$	0	0
$409$	−7578.82	−0.916256	−0.458128	−	0.888886i	$-0.651480\pi$
$409$	−7578.82	−0.916256	−0.458128	+	0.888886i	$0.651480\pi$
$410$	172.390	0.0207652
$411$	0	0
$412$	31431.0	3.75848
$413$	1636.97	0.195037
$414$	0	0
$415$	1439.73	0.170298
$416$	5382.14	0.634330
$417$	0	0
$418$	−8703.05	−1.01837
$419$	3895.92	0.454244	0.227122	−	0.973866i	$-0.427068\pi$
$419$	3895.92	0.454244	0.227122	+	0.973866i	$0.427068\pi$
$420$	0	0
$421$	6153.56	0.712366	0.356183	−	0.934416i	$-0.384078\pi$
$421$	6153.56	0.712366	0.356183	+	0.934416i	$0.384078\pi$
$422$	8951.73	1.03262
$423$	0	0
$424$	21218.7	2.43036
$425$	65.6963	0.00749821
$426$	0	0
$427$	−15404.8	−1.74588
$428$	−14911.2	−1.68402
$429$	0	0
$430$	9713.23	1.08933
$431$	14957.7	1.67166	0.835831	−	0.548987i	$-0.184987\pi$
$431$	14957.7	1.67166	0.835831	+	0.548987i	$0.184987\pi$
$432$	0	0
$433$	1622.09	0.180029	0.0900145	−	0.995940i	$-0.471309\pi$
$433$	1622.09	0.180029	0.0900145	+	0.995940i	$0.471309\pi$
$434$	−2409.12	−0.266455
$435$	0	0
$436$	−12802.2	−1.40622
$437$	−29376.3	−3.21570
$438$	0	0
$439$	7316.66	0.795456	0.397728	−	0.917503i	$-0.369799\pi$
$439$	7316.66	0.795456	0.397728	+	0.917503i	$0.369799\pi$
$440$	2577.32	0.279248
$441$	0	0
$442$	621.771	0.0669110
$443$	17181.1	1.84267	0.921333	−	0.388775i	$-0.127102\pi$
$443$	17181.1	1.84267	0.921333	+	0.388775i	$0.127102\pi$
$444$	0	0
$445$	6647.11	0.708097
$446$	−12752.4	−1.35391
$447$	0	0
$448$	−3757.00	−0.396209
$449$	7924.26	0.832893	0.416447	−	0.909160i	$-0.363275\pi$
$449$	7924.26	0.832893	0.416447	+	0.909160i	$0.363275\pi$
$450$	0	0
$451$	75.3801	0.00787031
$452$	−16535.8	−1.72075
$453$	0	0
$454$	−14282.3	−1.47644
$455$	−4368.01	−0.450056
$456$	0	0
$457$	−10739.7	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$457$	−10739.7	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$458$	32456.7	3.31136
$459$	0	0
$460$	16171.9	1.63917
$461$	−1002.93	−0.101325	−0.0506626	−	0.998716i	$-0.516133\pi$
$461$	−1002.93	−0.101325	−0.0506626	+	0.998716i	$0.516133\pi$
$462$	0	0
$463$	16766.1	1.68291	0.841454	−	0.540328i	$-0.181700\pi$
$463$	16766.1	1.68291	0.841454	+	0.540328i	$0.181700\pi$
$464$	26291.1	2.63046
$465$	0	0
$466$	23673.8	2.35337
$467$	−6558.63	−0.649887	−0.324944	−	0.945733i	$-0.605345\pi$
$467$	−6558.63	−0.649887	−0.324944	+	0.945733i	$0.605345\pi$
$468$	0	0
$469$	9100.78	0.896024
$470$	7137.35	0.700471
$471$	0	0
$472$	4129.38	0.402691
$473$	4247.25	0.412872
$474$	0	0
$475$	3931.34	0.379752
$476$	845.195	0.0813854
$477$	0	0
$478$	28602.3	2.73690
$479$	5073.87	0.483990	0.241995	−	0.970277i	$-0.422198\pi$
$479$	5073.87	0.483990	0.241995	+	0.970277i	$0.422198\pi$
$480$	0	0
$481$	10759.0	1.01989
$482$	4444.41	0.419994
$483$	0	0
$484$	2094.97	0.196748
$485$	313.421	0.0293437
$486$	0	0
$487$	−9958.28	−0.926597	−0.463299	−	0.886202i	$-0.653334\pi$
$487$	−9958.28	−0.926597	−0.463299	+	0.886202i	$0.653334\pi$
$488$	−38859.8	−3.60471
$489$	0	0
$490$	−52.4476	−0.00483539
$491$	−8235.93	−0.756991	−0.378495	−	0.925603i	$-0.623558\pi$
$491$	−8235.93	−0.756991	−0.378495	+	0.925603i	$0.623558\pi$
$492$	0	0
$493$	710.374	0.0648958
$494$	37207.4	3.38875
$495$	0	0
$496$	−2506.92	−0.226944
$497$	1181.06	0.106595
$498$	0	0
$499$	−3551.70	−0.318629	−0.159314	−	0.987228i	$-0.550928\pi$
$499$	−3551.70	−0.318629	−0.159314	+	0.987228i	$0.550928\pi$
$500$	−2164.23	−0.193574
$501$	0	0
$502$	4494.60	0.399609
$503$	−1569.35	−0.139113	−0.0695566	−	0.997578i	$-0.522158\pi$
$503$	−1569.35	−0.139113	−0.0695566	+	0.997578i	$0.522158\pi$
$504$	0	0
$505$	6031.78	0.531507
$506$	10338.8	0.908329
$507$	0	0
$508$	−38836.5	−3.39191
$509$	9448.27	0.822765	0.411382	−	0.911463i	$-0.365046\pi$
$509$	9448.27	0.822765	0.411382	+	0.911463i	$0.365046\pi$
$510$	0	0
$511$	−1454.57	−0.125923
$512$	−25329.4	−2.18635
$513$	0	0
$514$	38619.5	3.31407
$515$	−9076.85	−0.776648
$516$	0	0
$517$	3120.91	0.265488
$518$	21382.7	1.81371
$519$	0	0
$520$	−11018.6	−0.929227
$521$	−1920.03	−0.161455	−0.0807274	−	0.996736i	$-0.525724\pi$
$521$	−1920.03	−0.161455	−0.0807274	+	0.996736i	$0.525724\pi$
$522$	0	0
$523$	−21937.1	−1.83412	−0.917058	−	0.398755i	$-0.869442\pi$
$523$	−21937.1	−1.83412	−0.917058	+	0.398755i	$0.869442\pi$
$524$	30912.6	2.57715
$525$	0	0
$526$	−11162.6	−0.925310
$527$	−67.7358	−0.00559890
$528$	0	0
$529$	22730.5	1.86821
$530$	−11391.0	−0.933572
$531$	0	0
$532$	50577.3	4.12181
$533$	−322.266	−0.0261893
$534$	0	0
$535$	4306.15	0.347983
$536$	22957.4	1.85001
$537$	0	0
$538$	−18295.2	−1.46610
$539$	−22.9335	−0.00183268
$540$	0	0
$541$	3574.21	0.284043	0.142022	−	0.989864i	$-0.454640\pi$
$541$	3574.21	0.284043	0.142022	+	0.989864i	$0.454640\pi$
$542$	23531.2	1.86486
$543$	0	0
$544$	300.750	0.0237032
$545$	3697.10	0.290581
$546$	0	0
$547$	−23100.2	−1.80566	−0.902828	−	0.430003i	$-0.858513\pi$
$547$	−23100.2	−1.80566	−0.902828	+	0.430003i	$0.858513\pi$
$548$	−4351.98	−0.339247
$549$	0	0
$550$	−1383.60	−0.107267
$551$	42509.5	3.28669
$552$	0	0
$553$	−3786.11	−0.291142
$554$	−9040.18	−0.693286
$555$	0	0
$556$	−40913.4	−3.12071
$557$	19446.0	1.47927	0.739636	−	0.673007i	$-0.234998\pi$
$557$	19446.0	1.47927	0.739636	+	0.673007i	$0.234998\pi$
$558$	0	0
$559$	−18157.9	−1.37388
$560$	−9033.51	−0.681670
$561$	0	0
$562$	−21513.3	−1.61474
$563$	16150.8	1.20901	0.604506	−	0.796600i	$-0.293370\pi$
$563$	16150.8	1.20901	0.604506	+	0.796600i	$0.293370\pi$
$564$	0	0
$565$	4775.33	0.355574
$566$	18618.7	1.38269
$567$	0	0
$568$	2979.29	0.220085
$569$	22829.6	1.68202	0.841008	−	0.541023i	$-0.181963\pi$
$569$	22829.6	1.68202	0.841008	+	0.541023i	$0.181963\pi$
$570$	0	0
$571$	−15659.3	−1.14767	−0.573836	−	0.818970i	$-0.694545\pi$
$571$	−15659.3	−1.14767	−0.573836	+	0.818970i	$0.694545\pi$
$572$	−8956.45	−0.654700
$573$	0	0
$574$	−640.480	−0.0465734
$575$	−4670.22	−0.338716
$576$	0	0
$577$	13787.7	0.994783	0.497391	−	0.867526i	$-0.334291\pi$
$577$	13787.7	0.994783	0.497391	+	0.867526i	$0.334291\pi$
$578$	−24684.0	−1.77633
$579$	0	0
$580$	−23401.7	−1.67535
$581$	−5349.03	−0.381954
$582$	0	0
$583$	−4980.87	−0.353837
$584$	−3669.26	−0.259992
$585$	0	0
$586$	−28555.4	−2.01299
$587$	−10191.4	−0.716602	−0.358301	−	0.933606i	$-0.616644\pi$
$587$	−10191.4	−0.716602	−0.358301	+	0.933606i	$0.616644\pi$
$588$	0	0
$589$	−4053.38	−0.283560
$590$	−2216.81	−0.154686
$591$	0	0
$592$	22250.7	1.54476
$593$	4805.09	0.332751	0.166376	−	0.986062i	$-0.446794\pi$
$593$	4805.09	0.332751	0.166376	+	0.986062i	$0.446794\pi$
$594$	0	0
$595$	−244.081	−0.0168174
$596$	10395.8	0.714478
$597$	0	0
$598$	−44200.5	−3.02256
$599$	−22011.1	−1.50142	−0.750709	−	0.660632i	$-0.770288\pi$
$599$	−22011.1	−1.50142	−0.750709	+	0.660632i	$0.770288\pi$
$600$	0	0
$601$	−10795.0	−0.732671	−0.366336	−	0.930483i	$-0.619388\pi$
$601$	−10795.0	−0.732671	−0.366336	+	0.930483i	$0.619388\pi$
$602$	−36087.5	−2.44322
$603$	0	0
$604$	23021.7	1.55089
$605$	−605.000	−0.0406558
$606$	0	0
$607$	7739.23	0.517505	0.258753	−	0.965944i	$-0.416689\pi$
$607$	7739.23	0.517505	0.258753	+	0.965944i	$0.416689\pi$
$608$	17997.2	1.20046
$609$	0	0
$610$	20861.4	1.38468
$611$	−13342.6	−0.883441
$612$	0	0
$613$	23617.2	1.55610	0.778051	−	0.628201i	$-0.216208\pi$
$613$	23617.2	1.55610	0.778051	+	0.628201i	$0.216208\pi$
$614$	15219.2	1.00032
$615$	0	0
$616$	−9575.51	−0.626312
$617$	2695.50	0.175878	0.0879389	−	0.996126i	$-0.471972\pi$
$617$	2695.50	0.175878	0.0879389	+	0.996126i	$0.471972\pi$
$618$	0	0
$619$	18595.3	1.20745	0.603724	−	0.797194i	$-0.293683\pi$
$619$	18595.3	1.20745	0.603724	+	0.797194i	$0.293683\pi$
$620$	2231.41	0.144541
$621$	0	0
$622$	−39605.9	−2.55314
$623$	−24696.0	−1.58816
$624$	0	0
$625$	625.000	0.0400000
$626$	−37817.3	−2.41451
$627$	0	0
$628$	−9086.23	−0.577357
$629$	601.204	0.0381106
$630$	0	0
$631$	13987.2	0.882440	0.441220	−	0.897399i	$-0.354546\pi$
$631$	13987.2	0.882440	0.441220	+	0.897399i	$0.354546\pi$
$632$	−9550.73	−0.601120
$633$	0	0
$634$	13857.2	0.868045
$635$	11215.5	0.700901
$636$	0	0
$637$	98.0455	0.00609843
$638$	−14960.9	−0.928381
$639$	0	0
$640$	9665.65	0.596982
$641$	3944.64	0.243064	0.121532	−	0.992588i	$-0.461219\pi$
$641$	3944.64	0.243064	0.121532	+	0.992588i	$0.461219\pi$
$642$	0	0
$643$	−20619.4	−1.26462	−0.632309	−	0.774717i	$-0.717893\pi$
$643$	−20619.4	−1.26462	−0.632309	+	0.774717i	$0.717893\pi$
$644$	−60083.2	−3.67641
$645$	0	0
$646$	2079.12	0.126628
$647$	−14495.3	−0.880784	−0.440392	−	0.897806i	$-0.645161\pi$
$647$	−14495.3	−0.880784	−0.440392	+	0.897806i	$0.645161\pi$
$648$	0	0
$649$	−969.330	−0.0586279
$650$	5915.20	0.356944
$651$	0	0
$652$	20079.5	1.20609
$653$	8173.25	0.489807	0.244903	−	0.969547i	$-0.421244\pi$
$653$	8173.25	0.489807	0.244903	+	0.969547i	$0.421244\pi$
$654$	0	0
$655$	−8927.16	−0.532539
$656$	−666.480	−0.0396672
$657$	0	0
$658$	−26517.4	−1.57105
$659$	−29275.6	−1.73053	−0.865263	−	0.501317i	$-0.832849\pi$
$659$	−29275.6	−1.73053	−0.865263	+	0.501317i	$0.832849\pi$
$660$	0	0
$661$	25648.3	1.50924	0.754618	−	0.656164i	$-0.227822\pi$
$661$	25648.3	1.50924	0.754618	+	0.656164i	$0.227822\pi$
$662$	48209.7	2.83040
$663$	0	0
$664$	−13493.3	−0.788618
$665$	−14606.1	−0.851728
$666$	0	0
$667$	−50499.0	−2.93153
$668$	−42459.7	−2.45930
$669$	0	0
$670$	−12324.4	−0.710644
$671$	9121.94	0.524811
$672$	0	0
$673$	11239.4	0.643754	0.321877	−	0.946781i	$-0.395686\pi$
$673$	11239.4	0.643754	0.321877	+	0.946781i	$0.395686\pi$
$674$	45344.4	2.59140
$675$	0	0
$676$	252.331	0.0143566
$677$	−29268.1	−1.66154	−0.830770	−	0.556615i	$-0.812100\pi$
$677$	−29268.1	−1.66154	−0.830770	+	0.556615i	$0.812100\pi$
$678$	0	0
$679$	−1164.45	−0.0658137
$680$	−615.712	−0.0347227
$681$	0	0
$682$	1426.56	0.0800963
$683$	3045.63	0.170626	0.0853131	−	0.996354i	$-0.472811\pi$
$683$	3045.63	0.170626	0.0853131	+	0.996354i	$0.472811\pi$
$684$	0	0
$685$	1256.80	0.0701017
$686$	−31863.1	−1.77338
$687$	0	0
$688$	−37552.4	−2.08092
$689$	21294.3	1.17743
$690$	0	0
$691$	−32593.4	−1.79437	−0.897185	−	0.441654i	$-0.854392\pi$
$691$	−32593.4	−1.79437	−0.897185	+	0.441654i	$0.854392\pi$
$692$	−28753.1	−1.57952
$693$	0	0
$694$	9174.92	0.501838
$695$	11815.3	0.644861
$696$	0	0
$697$	−18.0080	−0.000978624 0
$698$	22369.9	1.21306
$699$	0	0
$700$	8040.73	0.434159
$701$	6375.19	0.343491	0.171746	−	0.985141i	$-0.445059\pi$
$701$	6375.19	0.343491	0.171746	+	0.985141i	$0.445059\pi$
$702$	0	0
$703$	35976.6	1.93013
$704$	2224.70	0.119100
$705$	0	0
$706$	−57091.5	−3.04344
$707$	−22409.8	−1.19209
$708$	0	0
$709$	−20886.1	−1.10634	−0.553169	−	0.833069i	$-0.686582\pi$
$709$	−20886.1	−1.10634	−0.553169	+	0.833069i	$0.686582\pi$
$710$	−1599.40	−0.0845412
$711$	0	0
$712$	−62297.3	−3.27906
$713$	4815.21	0.252918
$714$	0	0
$715$	2586.51	0.135286
$716$	−3756.53	−0.196073
$717$	0	0
$718$	50483.9	2.62401
$719$	4812.72	0.249630	0.124815	−	0.992180i	$-0.460166\pi$
$719$	4812.72	0.249630	0.124815	+	0.992180i	$0.460166\pi$
$720$	0	0
$721$	33723.1	1.74191
$722$	89907.3	4.63435
$723$	0	0
$724$	48541.4	2.49175
$725$	6758.12	0.346193
$726$	0	0
$727$	23018.4	1.17429	0.587143	−	0.809483i	$-0.300253\pi$
$727$	23018.4	1.17429	0.587143	+	0.809483i	$0.300253\pi$
$728$	40937.4	2.08412
$729$	0	0
$730$	1969.80	0.0998704
$731$	−1014.65	−0.0513381
$732$	0	0
$733$	−12757.0	−0.642826	−0.321413	−	0.946939i	$-0.604158\pi$
$733$	−12757.0	−0.642826	−0.321413	+	0.946939i	$0.604158\pi$
$734$	−49032.1	−2.46568
$735$	0	0
$736$	−21379.7	−1.07074
$737$	−5389.00	−0.269344
$738$	0	0
$739$	−28292.8	−1.40835	−0.704173	−	0.710029i	$-0.748682\pi$
$739$	−28292.8	−1.40835	−0.704173	+	0.710029i	$0.748682\pi$
$740$	−19805.4	−0.983865
$741$	0	0
$742$	42320.9	2.09387
$743$	17460.5	0.862131	0.431065	−	0.902321i	$-0.358138\pi$
$743$	17460.5	0.862131	0.431065	+	0.902321i	$0.358138\pi$
$744$	0	0
$745$	−3002.17	−0.147639
$746$	−4663.01	−0.228854
$747$	0	0
$748$	−500.480	−0.0244644
$749$	−15998.6	−0.780476
$750$	0	0
$751$	−3040.84	−0.147752	−0.0738762	−	0.997267i	$-0.523537\pi$
$751$	−3040.84	−0.147752	−0.0738762	+	0.997267i	$0.523537\pi$
$752$	−27593.8	−1.33809
$753$	0	0
$754$	63961.0	3.08929
$755$	−6648.36	−0.320475
$756$	0	0
$757$	10802.4	0.518653	0.259327	−	0.965790i	$-0.416499\pi$
$757$	10802.4	0.518653	0.259327	+	0.965790i	$0.416499\pi$
$758$	−25986.1	−1.24519
$759$	0	0
$760$	−36844.8	−1.75856
$761$	−32751.1	−1.56009	−0.780043	−	0.625725i	$-0.784803\pi$
$761$	−32751.1	−1.56009	−0.780043	+	0.625725i	$0.784803\pi$
$762$	0	0
$763$	−13735.8	−0.651730
$764$	−44936.0	−2.12792
$765$	0	0
$766$	26465.0	1.24833
$767$	4144.09	0.195091
$768$	0	0
$769$	−16826.0	−0.789025	−0.394512	−	0.918891i	$-0.629086\pi$
$769$	−16826.0	−0.789025	−0.394512	+	0.918891i	$0.629086\pi$
$770$	5140.49	0.240585
$771$	0	0
$772$	27083.6	1.26264
$773$	3163.75	0.147208	0.0736042	−	0.997288i	$-0.476550\pi$
$773$	3163.75	0.147208	0.0736042	+	0.997288i	$0.476550\pi$
$774$	0	0
$775$	−644.403	−0.0298679
$776$	−2937.41	−0.135885
$777$	0	0
$778$	68680.1	3.16491
$779$	−1077.62	−0.0495630
$780$	0	0
$781$	−699.359	−0.0320423
$782$	−2469.89	−0.112945
$783$	0	0
$784$	202.768	0.00923690
$785$	2623.98	0.119304
$786$	0	0
$787$	2470.36	0.111892	0.0559459	−	0.998434i	$-0.482183\pi$
$787$	2470.36	0.111892	0.0559459	+	0.998434i	$0.482183\pi$
$788$	43655.8	1.97357
$789$	0	0
$790$	5127.19	0.230908
$791$	−17741.7	−0.797501
$792$	0	0
$793$	−38998.2	−1.74637
$794$	−9935.24	−0.444066
$795$	0	0
$796$	−53340.5	−2.37513
$797$	−23636.7	−1.05051	−0.525255	−	0.850945i	$-0.676030\pi$
$797$	−23636.7	−1.05051	−0.525255	+	0.850945i	$0.676030\pi$
$798$	0	0
$799$	−745.572	−0.0330118
$800$	2861.17	0.126447
$801$	0	0
$802$	21521.9	0.947588
$803$	861.321	0.0378523
$804$	0	0
$805$	17351.2	0.759690
$806$	−6098.83	−0.266529
$807$	0	0
$808$	−56530.4	−2.46130
$809$	−18945.4	−0.823343	−0.411671	−	0.911332i	$-0.635055\pi$
$809$	−18945.4	−0.823343	−0.411671	+	0.911332i	$0.635055\pi$
$810$	0	0
$811$	14514.9	0.628466	0.314233	−	0.949346i	$-0.398253\pi$
$811$	14514.9	0.628466	0.314233	+	0.949346i	$0.398253\pi$
$812$	86944.3	3.75757
$813$	0	0
$814$	−12661.7	−0.545199
$815$	−5798.69	−0.249226
$816$	0	0
$817$	−60717.7	−2.60005
$818$	−38131.2	−1.62986
$819$	0	0
$820$	593.235	0.0252642
$821$	1084.32	0.0460940	0.0230470	−	0.999734i	$-0.492663\pi$
$821$	1084.32	0.0460940	0.0230470	+	0.999734i	$0.492663\pi$
$822$	0	0
$823$	−12462.6	−0.527846	−0.263923	−	0.964544i	$-0.585017\pi$
$823$	−12462.6	−0.527846	−0.263923	+	0.964544i	$0.585017\pi$
$824$	85069.0	3.59650
$825$	0	0
$826$	8236.08	0.346937
$827$	−4247.48	−0.178597	−0.0892984	−	0.996005i	$-0.528462\pi$
$827$	−4247.48	−0.178597	−0.0892984	+	0.996005i	$0.528462\pi$
$828$	0	0
$829$	−28657.3	−1.20061	−0.600306	−	0.799770i	$-0.704955\pi$
$829$	−28657.3	−1.20061	−0.600306	+	0.799770i	$0.704955\pi$
$830$	7243.71	0.302931
$831$	0	0
$832$	−9511.07	−0.396318
$833$	5.47870	0.000227882 0
$834$	0	0
$835$	12261.8	0.508188
$836$	−29949.2	−1.23901
$837$	0	0
$838$	19601.5	0.808021
$839$	−22588.4	−0.929486	−0.464743	−	0.885446i	$-0.653853\pi$
$839$	−22588.4	−0.929486	−0.464743	+	0.885446i	$0.653853\pi$
$840$	0	0
$841$	48686.4	1.99624
$842$	30960.3	1.26718
$843$	0	0
$844$	30805.0	1.25634
$845$	−72.8700	−0.00296663
$846$	0	0
$847$	2247.75	0.0911850
$848$	44038.9	1.78337
$849$	0	0
$850$	330.537	0.0133380
$851$	−42738.4	−1.72157
$852$	0	0
$853$	15866.1	0.636863	0.318432	−	0.947946i	$-0.396844\pi$
$853$	15866.1	0.636863	0.318432	+	0.947946i	$0.396844\pi$
$854$	−77506.1	−3.10563
$855$	0	0
$856$	−40357.6	−1.61144
$857$	−21458.5	−0.855321	−0.427660	−	0.903940i	$-0.640662\pi$
$857$	−21458.5	−0.855321	−0.427660	+	0.903940i	$0.640662\pi$
$858$	0	0
$859$	2634.31	0.104635	0.0523174	−	0.998631i	$-0.483339\pi$
$859$	2634.31	0.104635	0.0523174	+	0.998631i	$0.483339\pi$
$860$	33425.5	1.32535
$861$	0	0
$862$	75256.3	2.97360
$863$	11174.9	0.440787	0.220394	−	0.975411i	$-0.429266\pi$
$863$	11174.9	0.440787	0.220394	+	0.975411i	$0.429266\pi$
$864$	0	0
$865$	8303.51	0.326391
$866$	8161.18	0.320240
$867$	0	0
$868$	−8290.35	−0.324185
$869$	2241.93	0.0875172
$870$	0	0
$871$	23039.2	0.896271
$872$	−34649.5	−1.34562
$873$	0	0
$874$	−147801.	−5.72017
$875$	−2322.06	−0.0897141
$876$	0	0
$877$	−31858.8	−1.22668	−0.613338	−	0.789821i	$-0.710174\pi$
$877$	−31858.8	−1.22668	−0.613338	+	0.789821i	$0.710174\pi$
$878$	36812.2	1.41498
$879$	0	0
$880$	5349.17	0.204909
$881$	37097.8	1.41868	0.709340	−	0.704866i	$-0.248993\pi$
$881$	37097.8	1.41868	0.709340	+	0.704866i	$0.248993\pi$
$882$	0	0
$883$	−42861.6	−1.63353	−0.816766	−	0.576969i	$-0.804235\pi$
$883$	−42861.6	−1.63353	−0.816766	+	0.576969i	$0.804235\pi$
$884$	2139.66	0.0814078
$885$	0	0
$886$	86443.2	3.27778
$887$	−13189.2	−0.499266	−0.249633	−	0.968341i	$-0.580310\pi$
$887$	−13189.2	−0.499266	−0.249633	+	0.968341i	$0.580310\pi$
$888$	0	0
$889$	−41668.7	−1.57202
$890$	33443.5	1.25958
$891$	0	0
$892$	−43884.0	−1.64725
$893$	−44615.8	−1.67190
$894$	0	0
$895$	1084.84	0.0405163
$896$	−35910.7	−1.33894
$897$	0	0
$898$	39869.2	1.48157
$899$	−6967.92	−0.258502
$900$	0	0
$901$	1189.91	0.0439974
$902$	379.258	0.0139999
$903$	0	0
$904$	−44754.8	−1.64659
$905$	−14018.1	−0.514893
$906$	0	0
$907$	−174.880	−0.00640219	−0.00320109	−	0.999995i	$-0.501019\pi$
$907$	−174.880	−0.00640219	−0.00320109	+	0.999995i	$0.501019\pi$
$908$	−49148.8	−1.79632
$909$	0	0
$910$	−21976.7	−0.800572
$911$	22777.7	0.828387	0.414193	−	0.910189i	$-0.364064\pi$
$911$	22777.7	0.828387	0.414193	+	0.910189i	$0.364064\pi$
$912$	0	0
$913$	3167.42	0.114815
$914$	−54034.3	−1.95546
$915$	0	0
$916$	111691.	4.02879
$917$	33167.0	1.19441
$918$	0	0
$919$	−7744.17	−0.277972	−0.138986	−	0.990294i	$-0.544384\pi$
$919$	−7744.17	−0.277972	−0.138986	+	0.990294i	$0.544384\pi$
$920$	43769.7	1.56853
$921$	0	0
$922$	−5046.01	−0.180240
$923$	2989.91	0.106624
$924$	0	0
$925$	5719.53	0.203305
$926$	84355.0	2.99360
$927$	0	0
$928$	30937.8	1.09438
$929$	18043.2	0.637221	0.318610	−	0.947886i	$-0.396784\pi$
$929$	18043.2	0.637221	0.318610	+	0.947886i	$0.396784\pi$
$930$	0	0
$931$	327.851	0.0115412
$932$	81467.1	2.86325
$933$	0	0
$934$	−32998.3	−1.15604
$935$	144.532	0.00505529
$936$	0	0
$937$	30305.0	1.05658	0.528292	−	0.849063i	$-0.322833\pi$
$937$	30305.0	1.05658	0.528292	+	0.849063i	$0.322833\pi$
$938$	45788.6	1.59387
$939$	0	0
$940$	24561.3	0.852235
$941$	4363.67	0.151170	0.0755852	−	0.997139i	$-0.475918\pi$
$941$	4363.67	0.151170	0.0755852	+	0.997139i	$0.475918\pi$
$942$	0	0
$943$	1280.15	0.0442073
$944$	8570.42	0.295491
$945$	0	0
$946$	21369.1	0.734429
$947$	17184.1	0.589659	0.294830	−	0.955550i	$-0.404737\pi$
$947$	17184.1	0.589659	0.294830	+	0.955550i	$0.404737\pi$
$948$	0	0
$949$	−3682.33	−0.125957
$950$	19779.7	0.675513
$951$	0	0
$952$	2287.55	0.0778780
$953$	−24180.9	−0.821926	−0.410963	−	0.911652i	$-0.634807\pi$
$953$	−24180.9	−0.821926	−0.410963	+	0.911652i	$0.634807\pi$
$954$	0	0
$955$	12976.9	0.439710
$956$	98427.1	3.32988
$957$	0	0
$958$	25528.1	0.860935
$959$	−4669.36	−0.157228
$960$	0	0
$961$	−29126.6	−0.977698
$962$	54131.5	1.81421
$963$	0	0
$964$	15294.2	0.510990
$965$	−7821.40	−0.260912
$966$	0	0
$967$	−57283.3	−1.90497	−0.952485	−	0.304585i	$-0.901482\pi$
$967$	−57283.3	−1.90497	−0.952485	+	0.304585i	$0.901482\pi$
$968$	5670.11	0.188269
$969$	0	0
$970$	1576.91	0.0521974
$971$	19631.8	0.648830	0.324415	−	0.945915i	$-0.394833\pi$
$971$	19631.8	0.648830	0.324415	+	0.945915i	$0.394833\pi$
$972$	0	0
$973$	−43897.1	−1.44633
$974$	−50102.9	−1.64826
$975$	0	0
$976$	−80652.5	−2.64511
$977$	−28373.4	−0.929114	−0.464557	−	0.885543i	$-0.653786\pi$
$977$	−28373.4	−0.929114	−0.464557	+	0.885543i	$0.653786\pi$
$978$	0	0
$979$	14623.6	0.477399
$980$	−180.484	−0.00588302
$981$	0	0
$982$	−41437.3	−1.34656
$983$	−50486.2	−1.63811	−0.819054	−	0.573716i	$-0.805501\pi$
$983$	−50486.2	−1.63811	−0.819054	+	0.573716i	$0.805501\pi$
$984$	0	0
$985$	−12607.2	−0.407817
$986$	3574.09	0.115438
$987$	0	0
$988$	128039.	4.12295
$989$	72129.4	2.31909
$990$	0	0
$991$	−10061.4	−0.322513	−0.161257	−	0.986913i	$-0.551555\pi$
$991$	−10061.4	−0.322513	−0.161257	+	0.986913i	$0.551555\pi$
$992$	−2950.00	−0.0944179
$993$	0	0
$994$	5942.22	0.189614
$995$	15404.0	0.490794
$996$	0	0
$997$	−37628.5	−1.19529	−0.597646	−	0.801760i	$-0.703897\pi$
$997$	−37628.5	−1.19529	−0.597646	+	0.801760i	$0.703897\pi$
$998$	−17869.6	−0.566786
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.o.1.7	✓	7
3.2	odd	2		495.4.a.p.1.1	yes	7
5.4	even	2		2475.4.a.bt.1.1		7
15.14	odd	2		2475.4.a.bp.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.7	✓	7	1.1	even	1	trivial
495.4.a.p.1.1	yes	7	3.2	odd	2
2475.4.a.bp.1.7		7	15.14	odd	2
2475.4.a.bt.1.1		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+5.03128 q^{2} +17.3138 q^{4} -5.00000 q^{5} +18.5765 q^{7} +46.8604 q^{8} +O(q^{10})\)	\(q+5.03128 q^{2} +17.3138 q^{4} -5.00000 q^{5} +18.5765 q^{7} +46.8604 q^{8} -25.1564 q^{10} -11.0000 q^{11} +47.0274 q^{13} +93.4634 q^{14} +97.2576 q^{16} +2.62785 q^{17} +157.253 q^{19} -86.5691 q^{20} -55.3441 q^{22} -186.809 q^{23} +25.0000 q^{25} +236.608 q^{26} +321.629 q^{28} +270.325 q^{29} -25.7761 q^{31} +114.447 q^{32} +13.2215 q^{34} -92.8823 q^{35} +228.781 q^{37} +791.187 q^{38} -234.302 q^{40} -6.85273 q^{41} -386.113 q^{43} -190.452 q^{44} -939.888 q^{46} -283.719 q^{47} +2.08486 q^{49} +125.782 q^{50} +814.223 q^{52} +452.807 q^{53} +55.0000 q^{55} +870.501 q^{56} +1360.08 q^{58} +88.1209 q^{59} -829.267 q^{61} -129.687 q^{62} -202.245 q^{64} -235.137 q^{65} +489.909 q^{67} +45.4982 q^{68} -467.317 q^{70} +63.5781 q^{71} -78.3019 q^{73} +1151.06 q^{74} +2722.66 q^{76} -204.341 q^{77} -203.812 q^{79} -486.288 q^{80} -34.4780 q^{82} -287.947 q^{83} -13.1393 q^{85} -1942.65 q^{86} -515.465 q^{88} -1329.42 q^{89} +873.602 q^{91} -3234.37 q^{92} -1427.47 q^{94} -786.267 q^{95} -62.6842 q^{97} +10.4895 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8}+O(q^{10}) \) 7 * q - 5 * q^2 + 33 * q^4 - 35 * q^5 + 30 * q^7 - 45 * q^8	\( 7 q - 5 q^{2} + 33 q^{4} - 35 q^{5} + 30 q^{7} - 45 q^{8} + 25 q^{10} - 77 q^{11} + 38 q^{13} - 20 q^{14} + 309 q^{16} - 12 q^{17} + 226 q^{19} - 165 q^{20} + 55 q^{22} - 334 q^{23} + 175 q^{25} + 372 q^{26} + 812 q^{28} + 258 q^{29} + 336 q^{31} - 485 q^{32} + 78 q^{34} - 150 q^{35} + 466 q^{37} + 494 q^{38} + 225 q^{40} + 258 q^{41} + 308 q^{43} - 363 q^{44} + 98 q^{46} - 546 q^{47} + 735 q^{49} - 125 q^{50} + 512 q^{52} - 110 q^{53} + 385 q^{55} - 20 q^{56} + 1362 q^{58} + 68 q^{59} + 1096 q^{61} - 356 q^{62} + 2761 q^{64} - 190 q^{65} + 2268 q^{67} + 1186 q^{68} + 100 q^{70} + 166 q^{71} + 200 q^{73} + 1710 q^{74} + 3310 q^{76} - 330 q^{77} + 2152 q^{79} - 1545 q^{80} - 1006 q^{82} - 370 q^{83} + 60 q^{85} - 106 q^{86} + 495 q^{88} + 252 q^{89} + 2768 q^{91} - 3774 q^{92} + 2218 q^{94} - 1130 q^{95} + 3698 q^{97} - 697 q^{98}+O(q^{100}) \) 7 * q - 5 * q^2 + 33 * q^4 - 35 * q^5 + 30 * q^7 - 45 * q^8 + 25 * q^10 - 77 * q^11 + 38 * q^13 - 20 * q^14 + 309 * q^16 - 12 * q^17 + 226 * q^19 - 165 * q^20 + 55 * q^22 - 334 * q^23 + 175 * q^25 + 372 * q^26 + 812 * q^28 + 258 * q^29 + 336 * q^31 - 485 * q^32 + 78 * q^34 - 150 * q^35 + 466 * q^37 + 494 * q^38 + 225 * q^40 + 258 * q^41 + 308 * q^43 - 363 * q^44 + 98 * q^46 - 546 * q^47 + 735 * q^49 - 125 * q^50 + 512 * q^52 - 110 * q^53 + 385 * q^55 - 20 * q^56 + 1362 * q^58 + 68 * q^59 + 1096 * q^61 - 356 * q^62 + 2761 * q^64 - 190 * q^65 + 2268 * q^67 + 1186 * q^68 + 100 * q^70 + 166 * q^71 + 200 * q^73 + 1710 * q^74 + 3310 * q^76 - 330 * q^77 + 2152 * q^79 - 1545 * q^80 - 1006 * q^82 - 370 * q^83 + 60 * q^85 - 106 * q^86 + 495 * q^88 + 252 * q^89 + 2768 * q^91 - 3774 * q^92 + 2218 * q^94 - 1130 * q^95 + 3698 * q^97 - 697 * q^98

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