LMFDB - Embedded newform 495.4.a.k.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.47528.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x - 22 $ x^3 - x^2 - 26*x - 22
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.06484$ 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)

$f(q)$	$=$	$q+5.06484 q^{2} +17.6526 q^{4} +5.00000 q^{5} +27.4348 q^{7} +48.8887 q^{8} +O(q^{10})$	\(q+5.06484 q^{2} +17.6526 q^{4} +5.00000 q^{5} +27.4348 q^{7} +48.8887 q^{8} +25.3242 q^{10} -11.0000 q^{11} +22.6949 q^{13} +138.953 q^{14} +106.392 q^{16} +41.1755 q^{17} -142.128 q^{19} +88.2628 q^{20} -55.7132 q^{22} -176.166 q^{23} +25.0000 q^{25} +114.946 q^{26} +484.295 q^{28} -76.2044 q^{29} +197.373 q^{31} +147.751 q^{32} +208.547 q^{34} +137.174 q^{35} +367.297 q^{37} -719.856 q^{38} +244.443 q^{40} +238.279 q^{41} +30.2905 q^{43} -194.178 q^{44} -892.254 q^{46} -137.390 q^{47} +409.668 q^{49} +126.621 q^{50} +400.623 q^{52} -638.665 q^{53} -55.0000 q^{55} +1341.25 q^{56} -385.963 q^{58} -103.146 q^{59} +605.596 q^{61} +999.662 q^{62} -102.804 q^{64} +113.474 q^{65} -704.925 q^{67} +726.852 q^{68} +694.764 q^{70} +782.162 q^{71} -243.132 q^{73} +1860.30 q^{74} -2508.93 q^{76} -301.783 q^{77} -532.874 q^{79} +531.962 q^{80} +1206.84 q^{82} -1204.91 q^{83} +205.877 q^{85} +153.416 q^{86} -537.775 q^{88} -1058.49 q^{89} +622.629 q^{91} -3109.79 q^{92} -695.857 q^{94} -710.641 q^{95} -85.1964 q^{97} +2074.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 30 q^{4} + 15 q^{5} + 10 q^{7} - 18 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 30 * q^4 + 15 * q^5 + 10 * q^7 - 18 * q^8	$ 3 q + 2 q^{2} + 30 q^{4} + 15 q^{5} + 10 q^{7} - 18 q^{8} + 10 q^{10} - 33 q^{11} + 114 q^{13} + 68 q^{14} + 178 q^{16} + 104 q^{17} - 58 q^{19} + 150 q^{20} - 22 q^{22} - 120 q^{23} + 75 q^{25} + 120 q^{26} + 676 q^{28} + 220 q^{29} + 248 q^{31} + 258 q^{32} - 80 q^{34} + 50 q^{35} + 838 q^{37} - 600 q^{38} - 90 q^{40} - 156 q^{41} + 122 q^{43} - 330 q^{44} - 1256 q^{46} - 504 q^{47} + 279 q^{49} + 50 q^{50} + 520 q^{52} - 282 q^{53} - 165 q^{55} + 1644 q^{56} - 1644 q^{58} - 548 q^{59} + 414 q^{61} + 2448 q^{62} - 58 q^{64} + 570 q^{65} - 428 q^{67} + 1704 q^{68} + 340 q^{70} + 912 q^{71} + 618 q^{73} + 1612 q^{74} - 2752 q^{76} - 110 q^{77} - 542 q^{79} + 890 q^{80} + 3372 q^{82} + 520 q^{85} + 1548 q^{86} + 198 q^{88} - 790 q^{89} - 772 q^{91} - 1912 q^{92} - 424 q^{94} - 290 q^{95} + 2074 q^{97} + 3978 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 30 * q^4 + 15 * q^5 + 10 * q^7 - 18 * q^8 + 10 * q^10 - 33 * q^11 + 114 * q^13 + 68 * q^14 + 178 * q^16 + 104 * q^17 - 58 * q^19 + 150 * q^20 - 22 * q^22 - 120 * q^23 + 75 * q^25 + 120 * q^26 + 676 * q^28 + 220 * q^29 + 248 * q^31 + 258 * q^32 - 80 * q^34 + 50 * q^35 + 838 * q^37 - 600 * q^38 - 90 * q^40 - 156 * q^41 + 122 * q^43 - 330 * q^44 - 1256 * q^46 - 504 * q^47 + 279 * q^49 + 50 * q^50 + 520 * q^52 - 282 * q^53 - 165 * q^55 + 1644 * q^56 - 1644 * q^58 - 548 * q^59 + 414 * q^61 + 2448 * q^62 - 58 * q^64 + 570 * q^65 - 428 * q^67 + 1704 * q^68 + 340 * q^70 + 912 * q^71 + 618 * q^73 + 1612 * q^74 - 2752 * q^76 - 110 * q^77 - 542 * q^79 + 890 * q^80 + 3372 * q^82 + 520 * q^85 + 1548 * q^86 + 198 * q^88 - 790 * q^89 - 772 * q^91 - 1912 * q^92 - 424 * q^94 - 290 * q^95 + 2074 * q^97 + 3978 * 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 $

$2$	5.06484	1.79069	0.895345	−	0.445373i	$-0.146929\pi$
$2$	5.06484	1.79069	0.895345	+	0.445373i	$0.146929\pi$
$3$	0	0
$3$	0	0
$4$	17.6526	2.20657
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	27.4348	1.48134	0.740670	−	0.671869i	$-0.234508\pi$
$7$	27.4348	1.48134	0.740670	+	0.671869i	$0.234508\pi$
$8$	48.8887	2.16059
$9$	0	0
$10$	25.3242	0.800821
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	22.6949	0.484187	0.242093	−	0.970253i	$-0.422166\pi$
$13$	22.6949	0.484187	0.242093	+	0.970253i	$0.422166\pi$
$14$	138.953	2.65262
$15$	0	0
$16$	106.392	1.66238
$17$	41.1755	0.587442	0.293721	−	0.955891i	$-0.405106\pi$
$17$	41.1755	0.587442	0.293721	+	0.955891i	$0.405106\pi$
$18$	0	0
$19$	−142.128	−1.71613	−0.858064	−	0.513542i	$-0.828333\pi$
$19$	−142.128	−1.71613	−0.858064	+	0.513542i	$0.828333\pi$
$20$	88.2628	0.986808
$21$	0	0
$22$	−55.7132	−0.539913
$23$	−176.166	−1.59710	−0.798548	−	0.601931i	$-0.794398\pi$
$23$	−176.166	−1.59710	−0.798548	+	0.601931i	$0.794398\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	114.946	0.867028
$27$	0	0
$28$	484.295	3.26868
$29$	−76.2044	−0.487959	−0.243979	−	0.969780i	$-0.578453\pi$
$29$	−76.2044	−0.487959	−0.243979	+	0.969780i	$0.578453\pi$
$30$	0	0
$31$	197.373	1.14352	0.571762	−	0.820420i	$-0.306260\pi$
$31$	197.373	1.14352	0.571762	+	0.820420i	$0.306260\pi$
$32$	147.751	0.816218
$33$	0	0
$34$	208.547	1.05193
$35$	137.174	0.662475
$36$	0	0
$37$	367.297	1.63198	0.815989	−	0.578067i	$-0.196193\pi$
$37$	367.297	1.63198	0.815989	+	0.578067i	$0.196193\pi$
$38$	−719.856	−3.07305
$39$	0	0
$40$	244.443	0.966247
$41$	238.279	0.907633	0.453817	−	0.891095i	$-0.350062\pi$
$41$	238.279	0.907633	0.453817	+	0.891095i	$0.350062\pi$
$42$	0	0
$43$	30.2905	0.107425	0.0537123	−	0.998556i	$-0.482895\pi$
$43$	30.2905	0.107425	0.0537123	+	0.998556i	$0.482895\pi$
$44$	−194.178	−0.665306
$45$	0	0
$46$	−892.254	−2.85990
$47$	−137.390	−0.426391	−0.213195	−	0.977010i	$-0.568387\pi$
$47$	−137.390	−0.426391	−0.213195	+	0.977010i	$0.568387\pi$
$48$	0	0
$49$	409.668	1.19437
$50$	126.621	0.358138
$51$	0	0
$52$	400.623	1.06839
$53$	−638.665	−1.65523	−0.827617	−	0.561293i	$-0.810304\pi$
$53$	−638.665	−1.65523	−0.827617	+	0.561293i	$0.810304\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	1341.25	3.20057
$57$	0	0
$58$	−385.963	−0.873783
$59$	−103.146	−0.227601	−0.113800	−	0.993504i	$-0.536302\pi$
$59$	−103.146	−0.227601	−0.113800	+	0.993504i	$0.536302\pi$
$60$	0	0
$61$	605.596	1.27113	0.635563	−	0.772049i	$-0.280768\pi$
$61$	605.596	1.27113	0.635563	+	0.772049i	$0.280768\pi$
$62$	999.662	2.04770
$63$	0	0
$64$	−102.804	−0.200789
$65$	113.474	0.216535
$66$	0	0
$67$	−704.925	−1.28538	−0.642688	−	0.766128i	$-0.722181\pi$
$67$	−704.925	−1.28538	−0.642688	+	0.766128i	$0.722181\pi$
$68$	726.852	1.29623
$69$	0	0
$70$	694.764	1.18629
$71$	782.162	1.30740	0.653701	−	0.756753i	$-0.273215\pi$
$71$	782.162	1.30740	0.653701	+	0.756753i	$0.273215\pi$
$72$	0	0
$73$	−243.132	−0.389814	−0.194907	−	0.980822i	$-0.562440\pi$
$73$	−243.132	−0.389814	−0.194907	+	0.980822i	$0.562440\pi$
$74$	1860.30	2.92237
$75$	0	0
$76$	−2508.93	−3.78676
$77$	−301.783	−0.446641
$78$	0	0
$79$	−532.874	−0.758899	−0.379449	−	0.925212i	$-0.623887\pi$
$79$	−532.874	−0.758899	−0.379449	+	0.925212i	$0.623887\pi$
$80$	531.962	0.743440
$81$	0	0
$82$	1206.84	1.62529
$83$	−1204.91	−1.59344	−0.796722	−	0.604345i	$-0.793435\pi$
$83$	−1204.91	−1.59344	−0.796722	+	0.604345i	$0.793435\pi$
$84$	0	0
$85$	205.877	0.262712
$86$	153.416	0.192364
$87$	0	0
$88$	−537.775	−0.651443
$89$	−1058.49	−1.26067	−0.630337	−	0.776322i	$-0.717083\pi$
$89$	−1058.49	−1.26067	−0.630337	+	0.776322i	$0.717083\pi$
$90$	0	0
$91$	622.629	0.717245
$92$	−3109.79	−3.52411
$93$	0	0
$94$	−695.857	−0.763533
$95$	−710.641	−0.767476
$96$	0	0
$97$	−85.1964	−0.0891792	−0.0445896	−	0.999005i	$-0.514198\pi$
$97$	−85.1964	−0.0891792	−0.0445896	+	0.999005i	$0.514198\pi$
$98$	2074.90	2.13874
$99$	0	0
$100$	441.314	0.441314
$101$	7.81823	0.00770241	0.00385120	−	0.999993i	$-0.498774\pi$
$101$	7.81823	0.00770241	0.00385120	+	0.999993i	$0.498774\pi$
$102$	0	0
$103$	12.4770	0.0119359	0.00596794	−	0.999982i	$-0.498100\pi$
$103$	12.4770	0.0119359	0.00596794	+	0.999982i	$0.498100\pi$
$104$	1109.52	1.04613
$105$	0	0
$106$	−3234.73	−2.96401
$107$	1376.81	1.24393	0.621967	−	0.783043i	$-0.286334\pi$
$107$	1376.81	1.24393	0.621967	+	0.783043i	$0.286334\pi$
$108$	0	0
$109$	610.189	0.536197	0.268099	−	0.963391i	$-0.413605\pi$
$109$	610.189	0.536197	0.268099	+	0.963391i	$0.413605\pi$
$110$	−278.566	−0.241457
$111$	0	0
$112$	2918.86	2.46255
$113$	36.7727	0.0306132	0.0153066	−	0.999883i	$-0.495128\pi$
$113$	36.7727	0.0306132	0.0153066	+	0.999883i	$0.495128\pi$
$114$	0	0
$115$	−880.832	−0.714243
$116$	−1345.20	−1.07672
$117$	0	0
$118$	−522.416	−0.407562
$119$	1129.64	0.870201
$120$	0	0
$121$	121.000	0.0909091
$122$	3067.25	2.27619
$123$	0	0
$124$	3484.14	2.52326
$125$	125.000	0.0894427
$126$	0	0
$127$	−54.5310	−0.0381011	−0.0190506	−	0.999819i	$-0.506064\pi$
$127$	−54.5310	−0.0381011	−0.0190506	+	0.999819i	$0.506064\pi$
$128$	−1702.69	−1.17577
$129$	0	0
$130$	574.729	0.387747
$131$	−2377.83	−1.58589	−0.792947	−	0.609290i	$-0.791454\pi$
$131$	−2377.83	−1.58589	−0.792947	+	0.609290i	$0.791454\pi$
$132$	0	0
$133$	−3899.26	−2.54217
$134$	−3570.33	−2.30171
$135$	0	0
$136$	2013.01	1.26922
$137$	3078.41	1.91976	0.959878	−	0.280417i	$-0.0904729\pi$
$137$	3078.41	1.91976	0.959878	+	0.280417i	$0.0904729\pi$
$138$	0	0
$139$	−1197.18	−0.730530	−0.365265	−	0.930904i	$-0.619022\pi$
$139$	−1197.18	−0.730530	−0.365265	+	0.930904i	$0.619022\pi$
$140$	2421.47	1.46180
$141$	0	0
$142$	3961.52	2.34115
$143$	−249.644	−0.145988
$144$	0	0
$145$	−381.022	−0.218222
$146$	−1231.42	−0.698035
$147$	0	0
$148$	6483.73	3.60108
$149$	749.456	0.412066	0.206033	−	0.978545i	$-0.433945\pi$
$149$	749.456	0.412066	0.206033	+	0.978545i	$0.433945\pi$
$150$	0	0
$151$	−2645.72	−1.42586	−0.712931	−	0.701234i	$-0.752633\pi$
$151$	−2645.72	−1.42586	−0.712931	+	0.701234i	$0.752633\pi$
$152$	−6948.46	−3.70786
$153$	0	0
$154$	−1528.48	−0.799795
$155$	986.865	0.511399
$156$	0	0
$157$	−3475.74	−1.76684	−0.883422	−	0.468578i	$-0.844767\pi$
$157$	−3475.74	−1.76684	−0.883422	+	0.468578i	$0.844767\pi$
$158$	−2698.92	−1.35895
$159$	0	0
$160$	738.757	0.365024
$161$	−4833.09	−2.36584
$162$	0	0
$163$	3518.89	1.69092	0.845462	−	0.534035i	$-0.179325\pi$
$163$	3518.89	1.69092	0.845462	+	0.534035i	$0.179325\pi$
$164$	4206.24	2.00276
$165$	0	0
$166$	−6102.67	−2.85337
$167$	−250.304	−0.115983	−0.0579914	−	0.998317i	$-0.518470\pi$
$167$	−250.304	−0.115983	−0.0579914	+	0.998317i	$0.518470\pi$
$168$	0	0
$169$	−1681.94	−0.765563
$170$	1042.73	0.470436
$171$	0	0
$172$	534.705	0.237040
$173$	−941.985	−0.413976	−0.206988	−	0.978344i	$-0.566366\pi$
$173$	−941.985	−0.413976	−0.206988	+	0.978344i	$0.566366\pi$
$174$	0	0
$175$	685.870	0.296268
$176$	−1170.32	−0.501227
$177$	0	0
$178$	−5361.09	−2.25748
$179$	−336.037	−0.140316	−0.0701582	−	0.997536i	$-0.522350\pi$
$179$	−336.037	−0.140316	−0.0701582	+	0.997536i	$0.522350\pi$
$180$	0	0
$181$	1107.45	0.454784	0.227392	−	0.973803i	$-0.426980\pi$
$181$	1107.45	0.454784	0.227392	+	0.973803i	$0.426980\pi$
$182$	3153.52	1.28436
$183$	0	0
$184$	−8612.53	−3.45068
$185$	1836.48	0.729843
$186$	0	0
$187$	−452.930	−0.177120
$188$	−2425.28	−0.940861
$189$	0	0
$190$	−3599.28	−1.37431
$191$	4243.01	1.60740	0.803700	−	0.595035i	$-0.202862\pi$
$191$	4243.01	1.60740	0.803700	+	0.595035i	$0.202862\pi$
$192$	0	0
$193$	−3324.23	−1.23981	−0.619905	−	0.784677i	$-0.712829\pi$
$193$	−3324.23	−1.23981	−0.619905	+	0.784677i	$0.712829\pi$
$194$	−431.506	−0.159692
$195$	0	0
$196$	7231.69	2.63546
$197$	−2677.14	−0.968213	−0.484107	−	0.875009i	$-0.660855\pi$
$197$	−2677.14	−0.968213	−0.484107	+	0.875009i	$0.660855\pi$
$198$	0	0
$199$	2779.90	0.990261	0.495131	−	0.868819i	$-0.335120\pi$
$199$	2779.90	0.990261	0.495131	+	0.868819i	$0.335120\pi$
$200$	1222.22	0.432119
$201$	0	0
$202$	39.5981	0.0137926
$203$	−2090.65	−0.722833
$204$	0	0
$205$	1191.40	0.405906
$206$	63.1940	0.0213735
$207$	0	0
$208$	2414.56	0.804903
$209$	1563.41	0.517432
$210$	0	0
$211$	3056.15	0.997129	0.498564	−	0.866853i	$-0.333861\pi$
$211$	3056.15	0.997129	0.498564	+	0.866853i	$0.333861\pi$
$212$	−11274.1	−3.65239
$213$	0	0
$214$	6973.30	2.22750
$215$	151.453	0.0480417
$216$	0	0
$217$	5414.89	1.69395
$218$	3090.51	0.960163
$219$	0	0
$220$	−970.891	−0.297534
$221$	934.472	0.284432
$222$	0	0
$223$	571.375	0.171579	0.0857895	−	0.996313i	$-0.472659\pi$
$223$	571.375	0.171579	0.0857895	+	0.996313i	$0.472659\pi$
$224$	4053.53	1.20910
$225$	0	0
$226$	186.248	0.0548187
$227$	1094.40	0.319989	0.159995	−	0.987118i	$-0.448852\pi$
$227$	1094.40	0.319989	0.159995	+	0.987118i	$0.448852\pi$
$228$	0	0
$229$	−645.240	−0.186195	−0.0930975	−	0.995657i	$-0.529677\pi$
$229$	−645.240	−0.186195	−0.0930975	+	0.995657i	$0.529677\pi$
$230$	−4461.27	−1.27899
$231$	0	0
$232$	−3725.53	−1.05428
$233$	2337.39	0.657201	0.328600	−	0.944469i	$-0.393423\pi$
$233$	2337.39	0.657201	0.328600	+	0.944469i	$0.393423\pi$
$234$	0	0
$235$	−686.949	−0.190688
$236$	−1820.79	−0.502217
$237$	0	0
$238$	5721.44	1.55826
$239$	3656.91	0.989733	0.494866	−	0.868969i	$-0.335217\pi$
$239$	3656.91	0.989733	0.494866	+	0.868969i	$0.335217\pi$
$240$	0	0
$241$	−389.034	−0.103983	−0.0519914	−	0.998648i	$-0.516557\pi$
$241$	−389.034	−0.103983	−0.0519914	+	0.998648i	$0.516557\pi$
$242$	612.845	0.162790
$243$	0	0
$244$	10690.3	2.80483
$245$	2048.34	0.534138
$246$	0	0
$247$	−3225.58	−0.830927
$248$	9649.30	2.47069
$249$	0	0
$250$	633.104	0.160164
$251$	−2299.55	−0.578271	−0.289135	−	0.957288i	$-0.593368\pi$
$251$	−2299.55	−0.578271	−0.289135	+	0.957288i	$0.593368\pi$
$252$	0	0
$253$	1937.83	0.481543
$254$	−276.191	−0.0682273
$255$	0	0
$256$	−7801.44	−1.90465
$257$	−4921.61	−1.19456	−0.597279	−	0.802033i	$-0.703752\pi$
$257$	−4921.61	−1.19456	−0.597279	+	0.802033i	$0.703752\pi$
$258$	0	0
$259$	10076.7	2.41752
$260$	2003.11	0.477799
$261$	0	0
$262$	−12043.3	−2.83985
$263$	2575.61	0.603875	0.301938	−	0.953328i	$-0.402367\pi$
$263$	2575.61	0.603875	0.301938	+	0.953328i	$0.402367\pi$
$264$	0	0
$265$	−3193.33	−0.740243
$266$	−19749.1	−4.55224
$267$	0	0
$268$	−12443.7	−2.83627
$269$	4794.97	1.08682	0.543410	−	0.839468i	$-0.317133\pi$
$269$	4794.97	1.08682	0.543410	+	0.839468i	$0.317133\pi$
$270$	0	0
$271$	2729.47	0.611821	0.305910	−	0.952060i	$-0.401039\pi$
$271$	2729.47	0.611821	0.305910	+	0.952060i	$0.401039\pi$
$272$	4380.76	0.976553
$273$	0	0
$274$	15591.7	3.43769
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−3761.45	−0.815898	−0.407949	−	0.913005i	$-0.633756\pi$
$277$	−3761.45	−0.815898	−0.407949	+	0.913005i	$0.633756\pi$
$278$	−6063.53	−1.30815
$279$	0	0
$280$	6706.25	1.43134
$281$	6434.87	1.36609	0.683046	−	0.730375i	$-0.260655\pi$
$281$	6434.87	1.36609	0.683046	+	0.730375i	$0.260655\pi$
$282$	0	0
$283$	3335.65	0.700649	0.350325	−	0.936628i	$-0.386071\pi$
$283$	3335.65	0.700649	0.350325	+	0.936628i	$0.386071\pi$
$284$	13807.2	2.88487
$285$	0	0
$286$	−1264.40	−0.261419
$287$	6537.14	1.34451
$288$	0	0
$289$	−3217.58	−0.654912
$290$	−1929.81	−0.390768
$291$	0	0
$292$	−4291.89	−0.860151
$293$	2878.93	0.574023	0.287012	−	0.957927i	$-0.407338\pi$
$293$	2878.93	0.574023	0.287012	+	0.957927i	$0.407338\pi$
$294$	0	0
$295$	−515.729	−0.101786
$296$	17956.6	3.52604
$297$	0	0
$298$	3795.87	0.737883
$299$	−3998.07	−0.773293
$300$	0	0
$301$	831.014	0.159132
$302$	−13400.1	−2.55328
$303$	0	0
$304$	−15121.4	−2.85286
$305$	3027.98	0.568465
$306$	0	0
$307$	−8154.28	−1.51593	−0.757963	−	0.652297i	$-0.773805\pi$
$307$	−8154.28	−1.51593	−0.757963	+	0.652297i	$0.773805\pi$
$308$	−5327.24	−0.985544
$309$	0	0
$310$	4998.31	0.915757
$311$	−3818.24	−0.696182	−0.348091	−	0.937461i	$-0.613170\pi$
$311$	−3818.24	−0.696182	−0.348091	+	0.937461i	$0.613170\pi$
$312$	0	0
$313$	2527.23	0.456381	0.228191	−	0.973616i	$-0.426719\pi$
$313$	2527.23	0.456381	0.228191	+	0.973616i	$0.426719\pi$
$314$	−17604.1	−3.16387
$315$	0	0
$316$	−9406.59	−1.67456
$317$	11084.7	1.96398	0.981989	−	0.188937i	$-0.0605042\pi$
$317$	11084.7	1.96398	0.981989	+	0.188937i	$0.0605042\pi$
$318$	0	0
$319$	838.249	0.147125
$320$	−514.019	−0.0897954
$321$	0	0
$322$	−24478.8	−4.23649
$323$	−5852.19	−1.00813
$324$	0	0
$325$	567.372	0.0968373
$326$	17822.6	3.02792
$327$	0	0
$328$	11649.1	1.96103
$329$	−3769.26	−0.631629
$330$	0	0
$331$	−9417.70	−1.56388	−0.781939	−	0.623355i	$-0.785769\pi$
$331$	−9417.70	−1.56388	−0.781939	+	0.623355i	$0.785769\pi$
$332$	−21269.7	−3.51605
$333$	0	0
$334$	−1267.75	−0.207689
$335$	−3524.62	−0.574838
$336$	0	0
$337$	−11263.2	−1.82061	−0.910305	−	0.413938i	$-0.864153\pi$
$337$	−11263.2	−1.82061	−0.910305	+	0.413938i	$0.864153\pi$
$338$	−8518.76	−1.37089
$339$	0	0
$340$	3634.26	0.579693
$341$	−2171.10	−0.344785
$342$	0	0
$343$	1829.03	0.287925
$344$	1480.86	0.232101
$345$	0	0
$346$	−4771.00	−0.741302
$347$	4213.25	0.651814	0.325907	−	0.945402i	$-0.394330\pi$
$347$	4213.25	0.651814	0.325907	+	0.945402i	$0.394330\pi$
$348$	0	0
$349$	4987.07	0.764905	0.382452	−	0.923975i	$-0.375080\pi$
$349$	4987.07	0.764905	0.382452	+	0.923975i	$0.375080\pi$
$350$	3473.82	0.530524
$351$	0	0
$352$	−1625.26	−0.246099
$353$	7633.47	1.15096	0.575480	−	0.817816i	$-0.304815\pi$
$353$	7633.47	1.15096	0.575480	+	0.817816i	$0.304815\pi$
$354$	0	0
$355$	3910.81	0.584688
$356$	−18685.1	−2.78177
$357$	0	0
$358$	−1701.97	−0.251263
$359$	3114.76	0.457913	0.228957	−	0.973437i	$-0.426469\pi$
$359$	3114.76	0.457913	0.228957	+	0.973437i	$0.426469\pi$
$360$	0	0
$361$	13341.4	1.94510
$362$	5609.04	0.814378
$363$	0	0
$364$	10991.0	1.58265
$365$	−1215.66	−0.174330
$366$	0	0
$367$	5931.48	0.843653	0.421827	−	0.906677i	$-0.361389\pi$
$367$	5931.48	0.843653	0.421827	+	0.906677i	$0.361389\pi$
$368$	−18742.8	−2.65499
$369$	0	0
$370$	9301.49	1.30692
$371$	−17521.7	−2.45196
$372$	0	0
$373$	13918.9	1.93215	0.966073	−	0.258267i	$-0.0831515\pi$
$373$	13918.9	1.93215	0.966073	+	0.258267i	$0.0831515\pi$
$374$	−2294.02	−0.317168
$375$	0	0
$376$	−6716.80	−0.921257
$377$	−1729.45	−0.236263
$378$	0	0
$379$	−12267.3	−1.66261	−0.831303	−	0.555820i	$-0.812404\pi$
$379$	−12267.3	−1.66261	−0.831303	+	0.555820i	$0.812404\pi$
$380$	−12544.6	−1.69349
$381$	0	0
$382$	21490.1	2.87835
$383$	6935.04	0.925233	0.462616	−	0.886559i	$-0.346911\pi$
$383$	6935.04	0.925233	0.462616	+	0.886559i	$0.346911\pi$
$384$	0	0
$385$	−1508.91	−0.199744
$386$	−16836.7	−2.22012
$387$	0	0
$388$	−1503.93	−0.196780
$389$	−2775.18	−0.361715	−0.180858	−	0.983509i	$-0.557887\pi$
$389$	−2775.18	−0.361715	−0.180858	+	0.983509i	$0.557887\pi$
$390$	0	0
$391$	−7253.73	−0.938202
$392$	20028.1	2.58054
$393$	0	0
$394$	−13559.3	−1.73377
$395$	−2664.37	−0.339390
$396$	0	0
$397$	10539.5	1.33240	0.666198	−	0.745775i	$-0.267921\pi$
$397$	10539.5	1.33240	0.666198	+	0.745775i	$0.267921\pi$
$398$	14079.7	1.77325
$399$	0	0
$400$	2659.81	0.332477
$401$	12295.3	1.53117	0.765585	−	0.643334i	$-0.222450\pi$
$401$	12295.3	1.53117	0.765585	+	0.643334i	$0.222450\pi$
$402$	0	0
$403$	4479.35	0.553679
$404$	138.012	0.0169959
$405$	0	0
$406$	−10588.8	−1.29437
$407$	−4040.26	−0.492060
$408$	0	0
$409$	−11545.7	−1.39584	−0.697918	−	0.716178i	$-0.745890\pi$
$409$	−11545.7	−1.39584	−0.697918	+	0.716178i	$0.745890\pi$
$410$	6034.22	0.726852
$411$	0	0
$412$	220.251	0.0263374
$413$	−2829.78	−0.337154
$414$	0	0
$415$	−6024.54	−0.712610
$416$	3353.20	0.395202
$417$	0	0
$418$	7918.42	0.926561
$419$	−1851.51	−0.215876	−0.107938	−	0.994158i	$-0.534425\pi$
$419$	−1851.51	−0.215876	−0.107938	+	0.994158i	$0.534425\pi$
$420$	0	0
$421$	−1303.60	−0.150911	−0.0754557	−	0.997149i	$-0.524041\pi$
$421$	−1303.60	−0.150911	−0.0754557	+	0.997149i	$0.524041\pi$
$422$	15478.9	1.78555
$423$	0	0
$424$	−31223.5	−3.57629
$425$	1029.39	0.117488
$426$	0	0
$427$	16614.4	1.88297
$428$	24304.2	2.74483
$429$	0	0
$430$	767.082	0.0860279
$431$	−8228.85	−0.919652	−0.459826	−	0.888009i	$-0.652088\pi$
$431$	−8228.85	−0.919652	−0.459826	+	0.888009i	$0.652088\pi$
$432$	0	0
$433$	5830.21	0.647072	0.323536	−	0.946216i	$-0.395128\pi$
$433$	5830.21	0.647072	0.323536	+	0.946216i	$0.395128\pi$
$434$	27425.5	3.03333
$435$	0	0
$436$	10771.4	1.18316
$437$	25038.2	2.74082
$438$	0	0
$439$	14261.0	1.55044	0.775218	−	0.631694i	$-0.217640\pi$
$439$	14261.0	1.55044	0.775218	+	0.631694i	$0.217640\pi$
$440$	−2688.88	−0.291334
$441$	0	0
$442$	4732.95	0.509329
$443$	−12025.7	−1.28975	−0.644875	−	0.764288i	$-0.723090\pi$
$443$	−12025.7	−1.28975	−0.644875	+	0.764288i	$0.723090\pi$
$444$	0	0
$445$	−5292.46	−0.563790
$446$	2893.92	0.307245
$447$	0	0
$448$	−2820.40	−0.297436
$449$	7073.28	0.743449	0.371725	−	0.928343i	$-0.378767\pi$
$449$	7073.28	0.743449	0.371725	+	0.928343i	$0.378767\pi$
$450$	0	0
$451$	−2621.07	−0.273662
$452$	649.133	0.0675501
$453$	0	0
$454$	5542.93	0.573001
$455$	3113.15	0.320762
$456$	0	0
$457$	2732.31	0.279677	0.139838	−	0.990174i	$-0.455342\pi$
$457$	2732.31	0.279677	0.139838	+	0.990174i	$0.455342\pi$
$458$	−3268.04	−0.333418
$459$	0	0
$460$	−15548.9	−1.57603
$461$	332.708	0.0336134	0.0168067	−	0.999859i	$-0.494650\pi$
$461$	332.708	0.0336134	0.0168067	+	0.999859i	$0.494650\pi$
$462$	0	0
$463$	8248.39	0.827937	0.413969	−	0.910291i	$-0.364142\pi$
$463$	8248.39	0.827937	0.413969	+	0.910291i	$0.364142\pi$
$464$	−8107.58	−0.811174
$465$	0	0
$466$	11838.5	1.17684
$467$	−7359.35	−0.729229	−0.364615	−	0.931158i	$-0.618799\pi$
$467$	−7359.35	−0.729229	−0.364615	+	0.931158i	$0.618799\pi$
$468$	0	0
$469$	−19339.5	−1.90408
$470$	−3479.28	−0.341462
$471$	0	0
$472$	−5042.66	−0.491752
$473$	−333.196	−0.0323897
$474$	0	0
$475$	−3553.21	−0.343226
$476$	19941.0	1.92016
$477$	0	0
$478$	18521.7	1.77230
$479$	10327.6	0.985140	0.492570	−	0.870273i	$-0.336058\pi$
$479$	10327.6	0.985140	0.492570	+	0.870273i	$0.336058\pi$
$480$	0	0
$481$	8335.75	0.790182
$482$	−1970.39	−0.186201
$483$	0	0
$484$	2135.96	0.200597
$485$	−425.982	−0.0398822
$486$	0	0
$487$	9622.57	0.895360	0.447680	−	0.894194i	$-0.352250\pi$
$487$	9622.57	0.895360	0.447680	+	0.894194i	$0.352250\pi$
$488$	29606.8	2.74639
$489$	0	0
$490$	10374.5	0.956475
$491$	8993.59	0.826629	0.413315	−	0.910588i	$-0.364371\pi$
$491$	8993.59	0.826629	0.413315	+	0.910588i	$0.364371\pi$
$492$	0	0
$493$	−3137.75	−0.286648
$494$	−16337.0	−1.48793
$495$	0	0
$496$	20999.0	1.90097
$497$	21458.5	1.93671
$498$	0	0
$499$	2623.70	0.235377	0.117688	−	0.993051i	$-0.462452\pi$
$499$	2623.70	0.235377	0.117688	+	0.993051i	$0.462452\pi$
$500$	2206.57	0.197362
$501$	0	0
$502$	−11646.8	−1.03550
$503$	−13234.5	−1.17316	−0.586579	−	0.809892i	$-0.699526\pi$
$503$	−13234.5	−1.17316	−0.586579	+	0.809892i	$0.699526\pi$
$504$	0	0
$505$	39.0912	0.00344462
$506$	9814.79	0.862294
$507$	0	0
$508$	−962.612	−0.0840728
$509$	12810.3	1.11553	0.557765	−	0.829999i	$-0.311659\pi$
$509$	12810.3	1.11553	0.557765	+	0.829999i	$0.311659\pi$
$510$	0	0
$511$	−6670.26	−0.577446
$512$	−25891.5	−2.23487
$513$	0	0
$514$	−24927.1	−2.13908
$515$	62.3850	0.00533789
$516$	0	0
$517$	1511.29	0.128562
$518$	51036.9	4.32902
$519$	0	0
$520$	5547.61	0.467844
$521$	−1151.47	−0.0968268	−0.0484134	−	0.998827i	$-0.515416\pi$
$521$	−1151.47	−0.0968268	−0.0484134	+	0.998827i	$0.515416\pi$
$522$	0	0
$523$	−17319.1	−1.44801	−0.724005	−	0.689794i	$-0.757701\pi$
$523$	−17319.1	−1.44801	−0.724005	+	0.689794i	$0.757701\pi$
$524$	−41974.8	−3.49939
$525$	0	0
$526$	13045.1	1.08135
$527$	8126.92	0.671754
$528$	0	0
$529$	18867.6	1.55072
$530$	−16173.7	−1.32555
$531$	0	0
$532$	−68831.9	−5.60948
$533$	5407.72	0.439464
$534$	0	0
$535$	6884.03	0.556304
$536$	−34462.8	−2.77718
$537$	0	0
$538$	24285.7	1.94616
$539$	−4506.35	−0.360115
$540$	0	0
$541$	−7190.73	−0.571449	−0.285724	−	0.958312i	$-0.592234\pi$
$541$	−7190.73	−0.571449	−0.285724	+	0.958312i	$0.592234\pi$
$542$	13824.3	1.09558
$543$	0	0
$544$	6083.73	0.479481
$545$	3050.94	0.239795
$546$	0	0
$547$	18670.5	1.45940	0.729701	−	0.683766i	$-0.239659\pi$
$547$	18670.5	1.45940	0.729701	+	0.683766i	$0.239659\pi$
$548$	54341.9	4.23608
$549$	0	0
$550$	−1392.83	−0.107983
$551$	10830.8	0.837400
$552$	0	0
$553$	−14619.3	−1.12419
$554$	−19051.1	−1.46102
$555$	0	0
$556$	−21133.3	−1.61197
$557$	5510.44	0.419183	0.209591	−	0.977789i	$-0.432787\pi$
$557$	5510.44	0.419183	0.209591	+	0.977789i	$0.432787\pi$
$558$	0	0
$559$	687.439	0.0520136
$560$	14594.3	1.10129
$561$	0	0
$562$	32591.5	2.44625
$563$	−3576.57	−0.267734	−0.133867	−	0.990999i	$-0.542740\pi$
$563$	−3576.57	−0.267734	−0.133867	+	0.990999i	$0.542740\pi$
$564$	0	0
$565$	183.864	0.0136906
$566$	16894.5	1.25465
$567$	0	0
$568$	38238.8	2.82476
$569$	−12285.2	−0.905138	−0.452569	−	0.891729i	$-0.649492\pi$
$569$	−12285.2	−0.905138	−0.452569	+	0.891729i	$0.649492\pi$
$570$	0	0
$571$	13889.5	1.01797	0.508983	−	0.860777i	$-0.330022\pi$
$571$	13889.5	1.01797	0.508983	+	0.860777i	$0.330022\pi$
$572$	−4406.85	−0.322132
$573$	0	0
$574$	33109.5	2.40761
$575$	−4404.16	−0.319419
$576$	0	0
$577$	11579.4	0.835457	0.417728	−	0.908572i	$-0.362826\pi$
$577$	11579.4	0.835457	0.417728	+	0.908572i	$0.362826\pi$
$578$	−16296.5	−1.17274
$579$	0	0
$580$	−6726.02	−0.481522
$581$	−33056.4	−2.36043
$582$	0	0
$583$	7025.32	0.499072
$584$	−11886.4	−0.842229
$585$	0	0
$586$	14581.3	1.02790
$587$	−26468.0	−1.86107	−0.930537	−	0.366199i	$-0.880659\pi$
$587$	−26468.0	−1.86107	−0.930537	+	0.366199i	$0.880659\pi$
$588$	0	0
$589$	−28052.3	−1.96243
$590$	−2612.08	−0.182267
$591$	0	0
$592$	39077.6	2.71297
$593$	−1059.52	−0.0733716	−0.0366858	−	0.999327i	$-0.511680\pi$
$593$	−1059.52	−0.0733716	−0.0366858	+	0.999327i	$0.511680\pi$
$594$	0	0
$595$	5648.20	0.389166
$596$	13229.8	0.909253
$597$	0	0
$598$	−20249.6	−1.38473
$599$	17858.7	1.21817	0.609086	−	0.793104i	$-0.291536\pi$
$599$	17858.7	1.21817	0.609086	+	0.793104i	$0.291536\pi$
$600$	0	0
$601$	−9650.91	−0.655023	−0.327511	−	0.944847i	$-0.606210\pi$
$601$	−9650.91	−0.655023	−0.327511	+	0.944847i	$0.606210\pi$
$602$	4208.95	0.284957
$603$	0	0
$604$	−46703.7	−3.14627
$605$	605.000	0.0406558
$606$	0	0
$607$	22754.9	1.52157	0.760786	−	0.649002i	$-0.224813\pi$
$607$	22754.9	1.52157	0.760786	+	0.649002i	$0.224813\pi$
$608$	−20999.6	−1.40074
$609$	0	0
$610$	15336.2	1.01794
$611$	−3118.04	−0.206453
$612$	0	0
$613$	13074.5	0.861459	0.430729	−	0.902481i	$-0.358256\pi$
$613$	13074.5	0.861459	0.430729	+	0.902481i	$0.358256\pi$
$614$	−41300.1	−2.71455
$615$	0	0
$616$	−14753.8	−0.965009
$617$	13393.0	0.873878	0.436939	−	0.899491i	$-0.356063\pi$
$617$	13393.0	0.873878	0.436939	+	0.899491i	$0.356063\pi$
$618$	0	0
$619$	−15965.3	−1.03667	−0.518336	−	0.855177i	$-0.673448\pi$
$619$	−15965.3	−1.03667	−0.518336	+	0.855177i	$0.673448\pi$
$620$	17420.7	1.12844
$621$	0	0
$622$	−19338.8	−1.24665
$623$	−29039.5	−1.86749
$624$	0	0
$625$	625.000	0.0400000
$626$	12800.0	0.817237
$627$	0	0
$628$	−61355.8	−3.89867
$629$	15123.6	0.958693
$630$	0	0
$631$	17698.3	1.11657	0.558287	−	0.829648i	$-0.311459\pi$
$631$	17698.3	1.11657	0.558287	+	0.829648i	$0.311459\pi$
$632$	−26051.5	−1.63967
$633$	0	0
$634$	56142.4	3.51688
$635$	−272.655	−0.0170393
$636$	0	0
$637$	9297.37	0.578297
$638$	4245.59	0.263455
$639$	0	0
$640$	−8513.47	−0.525820
$641$	−18264.8	−1.12546	−0.562728	−	0.826642i	$-0.690248\pi$
$641$	−18264.8	−1.12546	−0.562728	+	0.826642i	$0.690248\pi$
$642$	0	0
$643$	−15730.5	−0.964778	−0.482389	−	0.875957i	$-0.660231\pi$
$643$	−15730.5	−0.964778	−0.482389	+	0.875957i	$0.660231\pi$
$644$	−85316.4	−5.22040
$645$	0	0
$646$	−29640.4	−1.80524
$647$	−21176.6	−1.28676	−0.643382	−	0.765545i	$-0.722469\pi$
$647$	−21176.6	−1.28676	−0.643382	+	0.765545i	$0.722469\pi$
$648$	0	0
$649$	1134.60	0.0686242
$650$	2873.65	0.173406
$651$	0	0
$652$	62117.4	3.73114
$653$	28293.6	1.69558	0.847791	−	0.530331i	$-0.177932\pi$
$653$	28293.6	1.69558	0.847791	+	0.530331i	$0.177932\pi$
$654$	0	0
$655$	−11889.2	−0.709234
$656$	25351.1	1.50883
$657$	0	0
$658$	−19090.7	−1.13105
$659$	3894.19	0.230191	0.115096	−	0.993354i	$-0.463283\pi$
$659$	3894.19	0.230191	0.115096	+	0.993354i	$0.463283\pi$
$660$	0	0
$661$	−6063.63	−0.356805	−0.178402	−	0.983958i	$-0.557093\pi$
$661$	−6063.63	−0.356805	−0.178402	+	0.983958i	$0.557093\pi$
$662$	−47699.1	−2.80042
$663$	0	0
$664$	−58906.4	−3.44279
$665$	−19496.3	−1.13689
$666$	0	0
$667$	13424.7	0.779317
$668$	−4418.51	−0.255924
$669$	0	0
$670$	−17851.6	−1.02936
$671$	−6661.56	−0.383259
$672$	0	0
$673$	17297.1	0.990719	0.495360	−	0.868688i	$-0.335036\pi$
$673$	17297.1	0.990719	0.495360	+	0.868688i	$0.335036\pi$
$674$	−57046.3	−3.26015
$675$	0	0
$676$	−29690.6	−1.68927
$677$	4640.36	0.263432	0.131716	−	0.991287i	$-0.457951\pi$
$677$	4640.36	0.263432	0.131716	+	0.991287i	$0.457951\pi$
$678$	0	0
$679$	−2337.35	−0.132105
$680$	10065.1	0.567614
$681$	0	0
$682$	−10996.3	−0.617404
$683$	14694.9	0.823256	0.411628	−	0.911352i	$-0.364960\pi$
$683$	14694.9	0.823256	0.411628	+	0.911352i	$0.364960\pi$
$684$	0	0
$685$	15392.1	0.858541
$686$	9263.73	0.515584
$687$	0	0
$688$	3222.68	0.178581
$689$	−14494.4	−0.801442
$690$	0	0
$691$	−9905.09	−0.545307	−0.272654	−	0.962112i	$-0.587901\pi$
$691$	−9905.09	−0.545307	−0.272654	+	0.962112i	$0.587901\pi$
$692$	−16628.4	−0.913466
$693$	0	0
$694$	21339.4	1.16720
$695$	−5985.91	−0.326703
$696$	0	0
$697$	9811.25	0.533182
$698$	25258.7	1.36971
$699$	0	0
$700$	12107.4	0.653736
$701$	947.946	0.0510748	0.0255374	−	0.999674i	$-0.491870\pi$
$701$	947.946	0.0510748	0.0255374	+	0.999674i	$0.491870\pi$
$702$	0	0
$703$	−52203.2	−2.80069
$704$	1130.84	0.0605401
$705$	0	0
$706$	38662.3	2.06101
$707$	214.492	0.0114099
$708$	0	0
$709$	−3310.76	−0.175371	−0.0876855	−	0.996148i	$-0.527947\pi$
$709$	−3310.76	−0.175371	−0.0876855	+	0.996148i	$0.527947\pi$
$710$	19807.6	1.04699
$711$	0	0
$712$	−51748.3	−2.72380
$713$	−34770.5	−1.82632
$714$	0	0
$715$	−1248.22	−0.0652877
$716$	−5931.92	−0.309618
$717$	0	0
$718$	15775.8	0.819981
$719$	−3061.15	−0.158778	−0.0793892	−	0.996844i	$-0.525297\pi$
$719$	−3061.15	−0.158778	−0.0793892	+	0.996844i	$0.525297\pi$
$720$	0	0
$721$	342.304	0.0176811
$722$	67572.1	3.48307
$723$	0	0
$724$	19549.3	1.00351
$725$	−1905.11	−0.0975918
$726$	0	0
$727$	−7405.09	−0.377771	−0.188886	−	0.981999i	$-0.560488\pi$
$727$	−7405.09	−0.377771	−0.188886	+	0.981999i	$0.560488\pi$
$728$	30439.5	1.54967
$729$	0	0
$730$	−6157.11	−0.312171
$731$	1247.23	0.0631057
$732$	0	0
$733$	29791.4	1.50119	0.750593	−	0.660765i	$-0.229768\pi$
$733$	29791.4	1.50119	0.750593	+	0.660765i	$0.229768\pi$
$734$	30042.0	1.51072
$735$	0	0
$736$	−26028.8	−1.30358
$737$	7754.17	0.387556
$738$	0	0
$739$	−7150.93	−0.355955	−0.177978	−	0.984035i	$-0.556955\pi$
$739$	−7150.93	−0.355955	−0.177978	+	0.984035i	$0.556955\pi$
$740$	32418.6	1.61045
$741$	0	0
$742$	−88744.3	−4.39071
$743$	1940.69	0.0958239	0.0479120	−	0.998852i	$-0.484743\pi$
$743$	1940.69	0.0958239	0.0479120	+	0.998852i	$0.484743\pi$
$744$	0	0
$745$	3747.28	0.184282
$746$	70496.7	3.45988
$747$	0	0
$748$	−7995.38	−0.390829
$749$	37772.4	1.84269
$750$	0	0
$751$	−29491.2	−1.43295	−0.716476	−	0.697611i	$-0.754246\pi$
$751$	−29491.2	−1.43295	−0.716476	+	0.697611i	$0.754246\pi$
$752$	−14617.2	−0.708824
$753$	0	0
$754$	−8759.38	−0.423074
$755$	−13228.6	−0.637665
$756$	0	0
$757$	3542.54	0.170087	0.0850436	−	0.996377i	$-0.472897\pi$
$757$	3542.54	0.170087	0.0850436	+	0.996377i	$0.472897\pi$
$758$	−62131.7	−2.97721
$759$	0	0
$760$	−34742.3	−1.65820
$761$	−8552.86	−0.407412	−0.203706	−	0.979032i	$-0.565299\pi$
$761$	−8552.86	−0.407412	−0.203706	+	0.979032i	$0.565299\pi$
$762$	0	0
$763$	16740.4	0.794290
$764$	74900.0	3.54684
$765$	0	0
$766$	35124.9	1.65680
$767$	−2340.88	−0.110201
$768$	0	0
$769$	−3128.73	−0.146716	−0.0733582	−	0.997306i	$-0.523372\pi$
$769$	−3128.73	−0.146716	−0.0733582	+	0.997306i	$0.523372\pi$
$770$	−7642.40	−0.357679
$771$	0	0
$772$	−58681.2	−2.73573
$773$	5364.51	0.249609	0.124805	−	0.992181i	$-0.460170\pi$
$773$	5364.51	0.249609	0.124805	+	0.992181i	$0.460170\pi$
$774$	0	0
$775$	4934.32	0.228705
$776$	−4165.14	−0.192680
$777$	0	0
$778$	−14055.8	−0.647719
$779$	−33866.2	−1.55762
$780$	0	0
$781$	−8603.78	−0.394197
$782$	−36738.9	−1.68003
$783$	0	0
$784$	43585.6	1.98550
$785$	−17378.7	−0.790157
$786$	0	0
$787$	12160.9	0.550813	0.275406	−	0.961328i	$-0.411188\pi$
$787$	12160.9	0.550813	0.275406	+	0.961328i	$0.411188\pi$
$788$	−47258.3	−2.13643
$789$	0	0
$790$	−13494.6	−0.607742
$791$	1008.85	0.0453485
$792$	0	0
$793$	13743.9	0.615462
$794$	53380.7	2.38591
$795$	0	0
$796$	49072.4	2.18508
$797$	581.179	0.0258299	0.0129149	−	0.999917i	$-0.495889\pi$
$797$	581.179	0.0258299	0.0129149	+	0.999917i	$0.495889\pi$
$798$	0	0
$799$	−5657.09	−0.250480
$800$	3693.78	0.163244
$801$	0	0
$802$	62273.8	2.74185
$803$	2674.45	0.117533
$804$	0	0
$805$	−24165.4	−1.05804
$806$	22687.2	0.991467
$807$	0	0
$808$	382.223	0.0166418
$809$	−4763.37	−0.207010	−0.103505	−	0.994629i	$-0.533006\pi$
$809$	−4763.37	−0.207010	−0.103505	+	0.994629i	$0.533006\pi$
$810$	0	0
$811$	−18055.4	−0.781762	−0.390881	−	0.920441i	$-0.627830\pi$
$811$	−18055.4	−0.781762	−0.390881	+	0.920441i	$0.627830\pi$
$812$	−36905.4	−1.59498
$813$	0	0
$814$	−20463.3	−0.881127
$815$	17594.4	0.756205
$816$	0	0
$817$	−4305.14	−0.184354
$818$	−58476.9	−2.49951
$819$	0	0
$820$	21031.2	0.895660
$821$	−1128.04	−0.0479522	−0.0239761	−	0.999713i	$-0.507633\pi$
$821$	−1128.04	−0.0479522	−0.0239761	+	0.999713i	$0.507633\pi$
$822$	0	0
$823$	−32124.2	−1.36061	−0.680304	−	0.732930i	$-0.738152\pi$
$823$	−32124.2	−1.36061	−0.680304	+	0.732930i	$0.738152\pi$
$824$	609.984	0.0257886
$825$	0	0
$826$	−14332.4	−0.603738
$827$	−11914.2	−0.500964	−0.250482	−	0.968121i	$-0.580589\pi$
$827$	−11914.2	−0.500964	−0.250482	+	0.968121i	$0.580589\pi$
$828$	0	0
$829$	−37721.6	−1.58037	−0.790185	−	0.612868i	$-0.790016\pi$
$829$	−37721.6	−1.58037	−0.790185	+	0.612868i	$0.790016\pi$
$830$	−30513.3	−1.27606
$831$	0	0
$832$	−2333.12	−0.0972192
$833$	16868.3	0.701622
$834$	0	0
$835$	−1251.52	−0.0518690
$836$	27598.2	1.14175
$837$	0	0
$838$	−9377.57	−0.386567
$839$	16550.5	0.681034	0.340517	−	0.940238i	$-0.389398\pi$
$839$	16550.5	0.681034	0.340517	+	0.940238i	$0.389398\pi$
$840$	0	0
$841$	−18581.9	−0.761896
$842$	−6602.53	−0.270236
$843$	0	0
$844$	53948.9	2.20023
$845$	−8409.71	−0.342370
$846$	0	0
$847$	3319.61	0.134667
$848$	−67949.2	−2.75163
$849$	0	0
$850$	5213.67	0.210385
$851$	−64705.3	−2.60643
$852$	0	0
$853$	1045.52	0.0419672	0.0209836	−	0.999780i	$-0.493320\pi$
$853$	1045.52	0.0419672	0.0209836	+	0.999780i	$0.493320\pi$
$854$	84149.3	3.37181
$855$	0	0
$856$	67310.2	2.68764
$857$	14016.5	0.558688	0.279344	−	0.960191i	$-0.409883\pi$
$857$	14016.5	0.558688	0.279344	+	0.960191i	$0.409883\pi$
$858$	0	0
$859$	−20476.3	−0.813319	−0.406660	−	0.913580i	$-0.633306\pi$
$859$	−20476.3	−0.813319	−0.406660	+	0.913580i	$0.633306\pi$
$860$	2673.53	0.106008
$861$	0	0
$862$	−41677.8	−1.64681
$863$	−24083.0	−0.949936	−0.474968	−	0.880003i	$-0.657540\pi$
$863$	−24083.0	−0.949936	−0.474968	+	0.880003i	$0.657540\pi$
$864$	0	0
$865$	−4709.92	−0.185135
$866$	29529.1	1.15871
$867$	0	0
$868$	95586.6	3.73781
$869$	5861.61	0.228817
$870$	0	0
$871$	−15998.2	−0.622362
$872$	29831.3	1.15850
$873$	0	0
$874$	126814.	4.90796
$875$	3429.35	0.132495
$876$	0	0
$877$	30432.5	1.17176	0.585879	−	0.810399i	$-0.300750\pi$
$877$	30432.5	1.17176	0.585879	+	0.810399i	$0.300750\pi$
$878$	72229.7	2.77635
$879$	0	0
$880$	−5851.59	−0.224156
$881$	−24559.2	−0.939183	−0.469592	−	0.882884i	$-0.655599\pi$
$881$	−24559.2	−0.939183	−0.469592	+	0.882884i	$0.655599\pi$
$882$	0	0
$883$	6013.88	0.229200	0.114600	−	0.993412i	$-0.463441\pi$
$883$	6013.88	0.229200	0.114600	+	0.993412i	$0.463441\pi$
$884$	16495.8	0.627618
$885$	0	0
$886$	−60908.3	−2.30954
$887$	−13395.5	−0.507075	−0.253538	−	0.967325i	$-0.581594\pi$
$887$	−13395.5	−0.507075	−0.253538	+	0.967325i	$0.581594\pi$
$888$	0	0
$889$	−1496.05	−0.0564407
$890$	−26805.4	−1.00957
$891$	0	0
$892$	10086.2	0.378601
$893$	19527.0	0.731741
$894$	0	0
$895$	−1680.19	−0.0627514
$896$	−46713.1	−1.74171
$897$	0	0
$898$	35825.0	1.33129
$899$	−15040.7	−0.557992
$900$	0	0
$901$	−26297.3	−0.972354
$902$	−13275.3	−0.490043
$903$	0	0
$904$	1797.77	0.0661426
$905$	5537.24	0.203386
$906$	0	0
$907$	−32078.9	−1.17438	−0.587189	−	0.809450i	$-0.699766\pi$
$907$	−32078.9	−1.17438	−0.587189	+	0.809450i	$0.699766\pi$
$908$	19318.9	0.706079
$909$	0	0
$910$	15767.6	0.574385
$911$	15992.0	0.581600	0.290800	−	0.956784i	$-0.406079\pi$
$911$	15992.0	0.581600	0.290800	+	0.956784i	$0.406079\pi$
$912$	0	0
$913$	13254.0	0.480442
$914$	13838.7	0.500814
$915$	0	0
$916$	−11390.1	−0.410852
$917$	−65235.4	−2.34925
$918$	0	0
$919$	33876.1	1.21596	0.607980	−	0.793952i	$-0.291980\pi$
$919$	33876.1	1.21596	0.607980	+	0.793952i	$0.291980\pi$
$920$	−43062.7	−1.54319
$921$	0	0
$922$	1685.11	0.0601912
$923$	17751.1	0.633026
$924$	0	0
$925$	9182.42	0.326396
$926$	41776.7	1.48258
$927$	0	0
$928$	−11259.3	−0.398281
$929$	−21163.4	−0.747416	−0.373708	−	0.927546i	$-0.621914\pi$
$929$	−21163.4	−0.747416	−0.373708	+	0.927546i	$0.621914\pi$
$930$	0	0
$931$	−58225.4	−2.04969
$932$	41261.0	1.45016
$933$	0	0
$934$	−37273.9	−1.30582
$935$	−2264.65	−0.0792107
$936$	0	0
$937$	−49771.9	−1.73530	−0.867650	−	0.497175i	$-0.834371\pi$
$937$	−49771.9	−1.73530	−0.867650	+	0.497175i	$0.834371\pi$
$938$	−97951.2	−3.40962
$939$	0	0
$940$	−12126.4	−0.420766
$941$	−32194.6	−1.11532	−0.557659	−	0.830070i	$-0.688300\pi$
$941$	−32194.6	−1.11532	−0.557659	+	0.830070i	$0.688300\pi$
$942$	0	0
$943$	−41976.8	−1.44958
$944$	−10973.9	−0.378359
$945$	0	0
$946$	−1687.58	−0.0580000
$947$	30091.9	1.03258	0.516291	−	0.856413i	$-0.327312\pi$
$947$	30091.9	1.03258	0.516291	+	0.856413i	$0.327312\pi$
$948$	0	0
$949$	−5517.84	−0.188742
$950$	−17996.4	−0.614611
$951$	0	0
$952$	55226.6	1.88015
$953$	−5710.17	−0.194093	−0.0970465	−	0.995280i	$-0.530940\pi$
$953$	−5710.17	−0.194093	−0.0970465	+	0.995280i	$0.530940\pi$
$954$	0	0
$955$	21215.0	0.718851
$956$	64553.9	2.18392
$957$	0	0
$958$	52307.8	1.76408
$959$	84455.7	2.84381
$960$	0	0
$961$	9165.07	0.307646
$962$	42219.2	1.41497
$963$	0	0
$964$	−6867.44	−0.229445
$965$	−16621.2	−0.554460
$966$	0	0
$967$	29638.4	0.985634	0.492817	−	0.870133i	$-0.335967\pi$
$967$	29638.4	0.985634	0.492817	+	0.870133i	$0.335967\pi$
$968$	5915.53	0.196418
$969$	0	0
$970$	−2157.53	−0.0714166
$971$	−21416.3	−0.707808	−0.353904	−	0.935282i	$-0.615146\pi$
$971$	−21416.3	−0.707808	−0.353904	+	0.935282i	$0.615146\pi$
$972$	0	0
$973$	−32844.5	−1.08216
$974$	48736.7	1.60331
$975$	0	0
$976$	64430.9	2.11310
$977$	−1038.84	−0.0340177	−0.0170088	−	0.999855i	$-0.505414\pi$
$977$	−1038.84	−0.0340177	−0.0170088	+	0.999855i	$0.505414\pi$
$978$	0	0
$979$	11643.4	0.380107
$980$	36158.5	1.17861
$981$	0	0
$982$	45551.1	1.48024
$983$	44173.6	1.43329	0.716643	−	0.697440i	$-0.245678\pi$
$983$	44173.6	1.43329	0.716643	+	0.697440i	$0.245678\pi$
$984$	0	0
$985$	−13385.7	−0.432998
$986$	−15892.2	−0.513297
$987$	0	0
$988$	−56939.8	−1.83350
$989$	−5336.17	−0.171567
$990$	0	0
$991$	−23940.9	−0.767414	−0.383707	−	0.923455i	$-0.625353\pi$
$991$	−23940.9	−0.767414	−0.383707	+	0.923455i	$0.625353\pi$
$992$	29162.1	0.933365
$993$	0	0
$994$	108684.	3.46804
$995$	13899.5	0.442858
$996$	0	0
$997$	13557.9	0.430674	0.215337	−	0.976540i	$-0.430915\pi$
$997$	13557.9	0.430674	0.215337	+	0.976540i	$0.430915\pi$
$998$	13288.6	0.421487
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.k.1.3		3
3.2	odd	2		165.4.a.e.1.1	✓	3
5.4	even	2		2475.4.a.t.1.1		3
15.2	even	4		825.4.c.k.199.1		6
15.8	even	4		825.4.c.k.199.6		6
15.14	odd	2		825.4.a.r.1.3		3
33.32	even	2		1815.4.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.e.1.1	✓	3	3.2	odd	2
495.4.a.k.1.3		3	1.1	even	1	trivial
825.4.a.r.1.3		3	15.14	odd	2
825.4.c.k.199.1		6	15.2	even	4
825.4.c.k.199.6		6	15.8	even	4
1815.4.a.r.1.3		3	33.32	even	2
2475.4.a.t.1.1		3	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)

\(f(q)\)	\(=\)	\(q+5.06484 q^{2} +17.6526 q^{4} +5.00000 q^{5} +27.4348 q^{7} +48.8887 q^{8} +O(q^{10})\)	\(q+5.06484 q^{2} +17.6526 q^{4} +5.00000 q^{5} +27.4348 q^{7} +48.8887 q^{8} +25.3242 q^{10} -11.0000 q^{11} +22.6949 q^{13} +138.953 q^{14} +106.392 q^{16} +41.1755 q^{17} -142.128 q^{19} +88.2628 q^{20} -55.7132 q^{22} -176.166 q^{23} +25.0000 q^{25} +114.946 q^{26} +484.295 q^{28} -76.2044 q^{29} +197.373 q^{31} +147.751 q^{32} +208.547 q^{34} +137.174 q^{35} +367.297 q^{37} -719.856 q^{38} +244.443 q^{40} +238.279 q^{41} +30.2905 q^{43} -194.178 q^{44} -892.254 q^{46} -137.390 q^{47} +409.668 q^{49} +126.621 q^{50} +400.623 q^{52} -638.665 q^{53} -55.0000 q^{55} +1341.25 q^{56} -385.963 q^{58} -103.146 q^{59} +605.596 q^{61} +999.662 q^{62} -102.804 q^{64} +113.474 q^{65} -704.925 q^{67} +726.852 q^{68} +694.764 q^{70} +782.162 q^{71} -243.132 q^{73} +1860.30 q^{74} -2508.93 q^{76} -301.783 q^{77} -532.874 q^{79} +531.962 q^{80} +1206.84 q^{82} -1204.91 q^{83} +205.877 q^{85} +153.416 q^{86} -537.775 q^{88} -1058.49 q^{89} +622.629 q^{91} -3109.79 q^{92} -695.857 q^{94} -710.641 q^{95} -85.1964 q^{97} +2074.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 30 q^{4} + 15 q^{5} + 10 q^{7} - 18 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 30 * q^4 + 15 * q^5 + 10 * q^7 - 18 * q^8	\( 3 q + 2 q^{2} + 30 q^{4} + 15 q^{5} + 10 q^{7} - 18 q^{8} + 10 q^{10} - 33 q^{11} + 114 q^{13} + 68 q^{14} + 178 q^{16} + 104 q^{17} - 58 q^{19} + 150 q^{20} - 22 q^{22} - 120 q^{23} + 75 q^{25} + 120 q^{26} + 676 q^{28} + 220 q^{29} + 248 q^{31} + 258 q^{32} - 80 q^{34} + 50 q^{35} + 838 q^{37} - 600 q^{38} - 90 q^{40} - 156 q^{41} + 122 q^{43} - 330 q^{44} - 1256 q^{46} - 504 q^{47} + 279 q^{49} + 50 q^{50} + 520 q^{52} - 282 q^{53} - 165 q^{55} + 1644 q^{56} - 1644 q^{58} - 548 q^{59} + 414 q^{61} + 2448 q^{62} - 58 q^{64} + 570 q^{65} - 428 q^{67} + 1704 q^{68} + 340 q^{70} + 912 q^{71} + 618 q^{73} + 1612 q^{74} - 2752 q^{76} - 110 q^{77} - 542 q^{79} + 890 q^{80} + 3372 q^{82} + 520 q^{85} + 1548 q^{86} + 198 q^{88} - 790 q^{89} - 772 q^{91} - 1912 q^{92} - 424 q^{94} - 290 q^{95} + 2074 q^{97} + 3978 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 30 * q^4 + 15 * q^5 + 10 * q^7 - 18 * q^8 + 10 * q^10 - 33 * q^11 + 114 * q^13 + 68 * q^14 + 178 * q^16 + 104 * q^17 - 58 * q^19 + 150 * q^20 - 22 * q^22 - 120 * q^23 + 75 * q^25 + 120 * q^26 + 676 * q^28 + 220 * q^29 + 248 * q^31 + 258 * q^32 - 80 * q^34 + 50 * q^35 + 838 * q^37 - 600 * q^38 - 90 * q^40 - 156 * q^41 + 122 * q^43 - 330 * q^44 - 1256 * q^46 - 504 * q^47 + 279 * q^49 + 50 * q^50 + 520 * q^52 - 282 * q^53 - 165 * q^55 + 1644 * q^56 - 1644 * q^58 - 548 * q^59 + 414 * q^61 + 2448 * q^62 - 58 * q^64 + 570 * q^65 - 428 * q^67 + 1704 * q^68 + 340 * q^70 + 912 * q^71 + 618 * q^73 + 1612 * q^74 - 2752 * q^76 - 110 * q^77 - 542 * q^79 + 890 * q^80 + 3372 * q^82 + 520 * q^85 + 1548 * q^86 + 198 * q^88 - 790 * q^89 - 772 * q^91 - 1912 * q^92 - 424 * q^94 - 290 * q^95 + 2074 * q^97 + 3978 * q^98

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