LMFDB - Embedded newform 495.4.a.i.1.1

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:	3.3.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 10 $ x^3 - x^2 - 9*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.12946$ 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)

$f(q)$	$=$	$q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +O(q^{10})$	\(q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +22.9743 q^{10} -11.0000 q^{11} -15.6584 q^{13} -94.8302 q^{14} +3.04132 q^{16} -72.9507 q^{17} +61.0513 q^{19} -65.5635 q^{20} +50.5434 q^{22} +13.6605 q^{23} +25.0000 q^{25} +71.9483 q^{26} +270.624 q^{28} +31.4663 q^{29} -243.008 q^{31} +173.963 q^{32} +335.198 q^{34} -103.192 q^{35} -65.4018 q^{37} -280.522 q^{38} +117.461 q^{40} +109.087 q^{41} -121.750 q^{43} -144.240 q^{44} -62.7678 q^{46} +519.530 q^{47} +82.9413 q^{49} -114.871 q^{50} -205.324 q^{52} +542.673 q^{53} +55.0000 q^{55} -484.839 q^{56} -144.583 q^{58} -109.478 q^{59} -89.6156 q^{61} +1116.59 q^{62} -823.664 q^{64} +78.2922 q^{65} +488.446 q^{67} -956.581 q^{68} +474.151 q^{70} -837.423 q^{71} +351.216 q^{73} +300.512 q^{74} +800.547 q^{76} -227.022 q^{77} -831.205 q^{79} -15.2066 q^{80} -501.238 q^{82} -1389.13 q^{83} +364.754 q^{85} +559.423 q^{86} +258.413 q^{88} -1523.70 q^{89} -323.164 q^{91} +179.125 q^{92} -2387.17 q^{94} -305.256 q^{95} -426.612 q^{97} -381.103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8	$ 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8} + 5 q^{10} - 33 q^{11} - 20 q^{13} - 144 q^{14} + 25 q^{16} - 32 q^{17} + 116 q^{19} - 85 q^{20} + 11 q^{22} - 240 q^{23} + 75 q^{25} - 302 q^{26} + 160 q^{28} - 238 q^{29} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} - 90 q^{37} - 324 q^{38} - 15 q^{40} + 46 q^{41} - 134 q^{43} - 187 q^{44} - 240 q^{46} + 220 q^{47} - 457 q^{49} - 25 q^{50} - 1530 q^{52} + 798 q^{53} + 165 q^{55} - 688 q^{56} - 978 q^{58} - 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} + 720 q^{70} - 1816 q^{71} + 100 q^{73} + 1874 q^{74} + 396 q^{76} - 66 q^{77} - 96 q^{79} - 125 q^{80} - 910 q^{82} - 858 q^{83} + 160 q^{85} - 188 q^{86} - 33 q^{88} - 838 q^{89} + 332 q^{91} + 688 q^{92} - 3112 q^{94} - 580 q^{95} - 1322 q^{97} - 1017 q^{98}+O(q^{100}) $ 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 + 5 * q^10 - 33 * q^11 - 20 * q^13 - 144 * q^14 + 25 * q^16 - 32 * q^17 + 116 * q^19 - 85 * q^20 + 11 * q^22 - 240 * q^23 + 75 * q^25 - 302 * q^26 + 160 * q^28 - 238 * q^29 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 - 90 * q^37 - 324 * q^38 - 15 * q^40 + 46 * q^41 - 134 * q^43 - 187 * q^44 - 240 * q^46 + 220 * q^47 - 457 * q^49 - 25 * q^50 - 1530 * q^52 + 798 * q^53 + 165 * q^55 - 688 * q^56 - 978 * q^58 - 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 + 720 * q^70 - 1816 * q^71 + 100 * q^73 + 1874 * q^74 + 396 * q^76 - 66 * q^77 - 96 * q^79 - 125 * q^80 - 910 * q^82 - 858 * q^83 + 160 * q^85 - 188 * q^86 - 33 * q^88 - 838 * q^89 + 332 * q^91 + 688 * q^92 - 3112 * q^94 - 580 * q^95 - 1322 * q^97 - 1017 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−4.59486	−1.62453	−0.812263	−	0.583291i	$-0.801765\pi$
$2$	−4.59486	−1.62453	−0.812263	+	0.583291i	$0.801765\pi$
$3$	0	0
$3$	0	0
$4$	13.1127	1.63909
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	20.6383	1.11437	0.557183	−	0.830390i	$-0.311882\pi$
$7$	20.6383	1.11437	0.557183	+	0.830390i	$0.311882\pi$
$8$	−23.4921	−1.03822
$9$	0	0
$10$	22.9743	0.726511
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−15.6584	−0.334067	−0.167033	−	0.985951i	$-0.553419\pi$
$13$	−15.6584	−0.334067	−0.167033	+	0.985951i	$0.553419\pi$
$14$	−94.8302	−1.81032
$15$	0	0
$16$	3.04132	0.0475206
$17$	−72.9507	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$17$	−72.9507	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$18$	0	0
$19$	61.0513	0.737165	0.368582	−	0.929595i	$-0.379843\pi$
$19$	61.0513	0.737165	0.368582	+	0.929595i	$0.379843\pi$
$20$	−65.5635	−0.733022
$21$	0	0
$22$	50.5434	0.489813
$23$	13.6605	0.123844	0.0619218	−	0.998081i	$-0.480277\pi$
$23$	13.6605	0.123844	0.0619218	+	0.998081i	$0.480277\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	71.9483	0.542701
$27$	0	0
$28$	270.624	1.82654
$29$	31.4663	0.201487	0.100744	−	0.994912i	$-0.467878\pi$
$29$	31.4663	0.201487	0.100744	+	0.994912i	$0.467878\pi$
$30$	0	0
$31$	−243.008	−1.40792	−0.703961	−	0.710239i	$-0.748587\pi$
$31$	−243.008	−1.40792	−0.703961	+	0.710239i	$0.748587\pi$
$32$	173.963	0.961016
$33$	0	0
$34$	335.198	1.69076
$35$	−103.192	−0.498360
$36$	0	0
$37$	−65.4018	−0.290594	−0.145297	−	0.989388i	$-0.546414\pi$
$37$	−65.4018	−0.290594	−0.145297	+	0.989388i	$0.546414\pi$
$38$	−280.522	−1.19754
$39$	0	0
$40$	117.461	0.464304
$41$	109.087	0.415524	0.207762	−	0.978179i	$-0.433382\pi$
$41$	109.087	0.415524	0.207762	+	0.978179i	$0.433382\pi$
$42$	0	0
$43$	−121.750	−0.431783	−0.215891	−	0.976417i	$-0.569266\pi$
$43$	−121.750	−0.431783	−0.215891	+	0.976417i	$0.569266\pi$
$44$	−144.240	−0.494204
$45$	0	0
$46$	−62.7678	−0.201187
$47$	519.530	1.61237	0.806184	−	0.591665i	$-0.201529\pi$
$47$	519.530	1.61237	0.806184	+	0.591665i	$0.201529\pi$
$48$	0	0
$49$	82.9413	0.241811
$50$	−114.871	−0.324905
$51$	0	0
$52$	−205.324	−0.547565
$53$	542.673	1.40645	0.703226	−	0.710967i	$-0.251742\pi$
$53$	542.673	1.40645	0.703226	+	0.710967i	$0.251742\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−484.839	−1.15695
$57$	0	0
$58$	−144.583	−0.327322
$59$	−109.478	−0.241574	−0.120787	−	0.992678i	$-0.538542\pi$
$59$	−109.478	−0.241574	−0.120787	+	0.992678i	$0.538542\pi$
$60$	0	0
$61$	−89.6156	−0.188100	−0.0940501	−	0.995567i	$-0.529981\pi$
$61$	−89.6156	−0.188100	−0.0940501	+	0.995567i	$0.529981\pi$
$62$	1116.59	2.28721
$63$	0	0
$64$	−823.664	−1.60872
$65$	78.2922	0.149399
$66$	0	0
$67$	488.446	0.890644	0.445322	−	0.895371i	$-0.353089\pi$
$67$	488.446	0.890644	0.445322	+	0.895371i	$0.353089\pi$
$68$	−956.581	−1.70592
$69$	0	0
$70$	474.151	0.809599
$71$	−837.423	−1.39977	−0.699887	−	0.714254i	$-0.746766\pi$
$71$	−837.423	−1.39977	−0.699887	+	0.714254i	$0.746766\pi$
$72$	0	0
$73$	351.216	0.563105	0.281553	−	0.959546i	$-0.409151\pi$
$73$	351.216	0.563105	0.281553	+	0.959546i	$0.409151\pi$
$74$	300.512	0.472078
$75$	0	0
$76$	800.547	1.20828
$77$	−227.022	−0.335994
$78$	0	0
$79$	−831.205	−1.18377	−0.591885	−	0.806022i	$-0.701616\pi$
$79$	−831.205	−1.18377	−0.591885	+	0.806022i	$0.701616\pi$
$80$	−15.2066	−0.0212519
$81$	0	0
$82$	−501.238	−0.675031
$83$	−1389.13	−1.83707	−0.918537	−	0.395335i	$-0.870629\pi$
$83$	−1389.13	−1.83707	−0.918537	+	0.395335i	$0.870629\pi$
$84$	0	0
$85$	364.754	0.465448
$86$	559.423	0.701443
$87$	0	0
$88$	258.413	0.313034
$89$	−1523.70	−1.81474	−0.907369	−	0.420335i	$-0.861912\pi$
$89$	−1523.70	−1.81474	−0.907369	+	0.420335i	$0.861912\pi$
$90$	0	0
$91$	−323.164	−0.372273
$92$	179.125	0.202990
$93$	0	0
$94$	−2387.17	−2.61933
$95$	−305.256	−0.329670
$96$	0	0
$97$	−426.612	−0.446555	−0.223278	−	0.974755i	$-0.571676\pi$
$97$	−426.612	−0.446555	−0.223278	+	0.974755i	$0.571676\pi$
$98$	−381.103	−0.392829
$99$	0	0
$100$	327.818	0.327818
$101$	−74.1387	−0.0730403	−0.0365202	−	0.999333i	$-0.511627\pi$
$101$	−74.1387	−0.0730403	−0.0365202	+	0.999333i	$0.511627\pi$
$102$	0	0
$103$	−69.3916	−0.0663821	−0.0331911	−	0.999449i	$-0.510567\pi$
$103$	−69.3916	−0.0663821	−0.0331911	+	0.999449i	$0.510567\pi$
$104$	367.850	0.346833
$105$	0	0
$106$	−2493.51	−2.28482
$107$	−1141.71	−1.03152	−0.515761	−	0.856733i	$-0.672491\pi$
$107$	−1141.71	−1.03152	−0.515761	+	0.856733i	$0.672491\pi$
$108$	0	0
$109$	−2226.85	−1.95682	−0.978409	−	0.206680i	$-0.933734\pi$
$109$	−2226.85	−1.95682	−0.978409	+	0.206680i	$0.933734\pi$
$110$	−252.717	−0.219051
$111$	0	0
$112$	62.7678	0.0529554
$113$	1719.76	1.43169	0.715847	−	0.698257i	$-0.246041\pi$
$113$	1719.76	1.43169	0.715847	+	0.698257i	$0.246041\pi$
$114$	0	0
$115$	−68.3023	−0.0553845
$116$	412.608	0.330256
$117$	0	0
$118$	503.038	0.392444
$119$	−1505.58	−1.15980
$120$	0	0
$121$	121.000	0.0909091
$122$	411.771	0.305574
$123$	0	0
$124$	−3186.49	−2.30771
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1601.63	−1.11907	−0.559534	−	0.828807i	$-0.689020\pi$
$127$	−1601.63	−1.11907	−0.559534	+	0.828807i	$0.689020\pi$
$128$	2392.91	1.65239
$129$	0	0
$130$	−359.741	−0.242703
$131$	−2004.13	−1.33665	−0.668327	−	0.743868i	$-0.732989\pi$
$131$	−2004.13	−1.33665	−0.668327	+	0.743868i	$0.732989\pi$
$132$	0	0
$133$	1260.00	0.821471
$134$	−2244.34	−1.44687
$135$	0	0
$136$	1713.77	1.08055
$137$	1672.85	1.04322	0.521610	−	0.853184i	$-0.325331\pi$
$137$	1672.85	1.04322	0.521610	+	0.853184i	$0.325331\pi$
$138$	0	0
$139$	2540.38	1.55016	0.775080	−	0.631863i	$-0.217709\pi$
$139$	2540.38	1.55016	0.775080	+	0.631863i	$0.217709\pi$
$140$	−1353.12	−0.816855
$141$	0	0
$142$	3847.84	2.27397
$143$	172.243	0.100725
$144$	0	0
$145$	−157.331	−0.0901079
$146$	−1613.79	−0.914780
$147$	0	0
$148$	−857.594	−0.476310
$149$	−3090.68	−1.69932	−0.849658	−	0.527334i	$-0.823192\pi$
$149$	−3090.68	−1.69932	−0.849658	+	0.527334i	$0.823192\pi$
$150$	0	0
$151$	1358.74	0.732267	0.366134	−	0.930562i	$-0.380681\pi$
$151$	1358.74	0.732267	0.366134	+	0.930562i	$0.380681\pi$
$152$	−1434.22	−0.765335
$153$	0	0
$154$	1043.13	0.545831
$155$	1215.04	0.629642
$156$	0	0
$157$	−1011.95	−0.514411	−0.257205	−	0.966357i	$-0.582802\pi$
$157$	−1011.95	−0.514411	−0.257205	+	0.966357i	$0.582802\pi$
$158$	3819.27	1.92307
$159$	0	0
$160$	−869.813	−0.429780
$161$	281.929	0.138007
$162$	0	0
$163$	−2816.37	−1.35334	−0.676672	−	0.736285i	$-0.736578\pi$
$163$	−2816.37	−1.35334	−0.676672	+	0.736285i	$0.736578\pi$
$164$	1430.42	0.681081
$165$	0	0
$166$	6382.87	2.98438
$167$	−3448.89	−1.59810	−0.799052	−	0.601262i	$-0.794665\pi$
$167$	−3448.89	−1.59810	−0.799052	+	0.601262i	$0.794665\pi$
$168$	0	0
$169$	−1951.81	−0.888399
$170$	−1675.99	−0.756133
$171$	0	0
$172$	−1596.47	−0.707730
$173$	2287.85	1.00545	0.502723	−	0.864448i	$-0.332332\pi$
$173$	2287.85	1.00545	0.502723	+	0.864448i	$0.332332\pi$
$174$	0	0
$175$	515.959	0.222873
$176$	−33.4545	−0.0143280
$177$	0	0
$178$	7001.17	2.94809
$179$	−3249.06	−1.35668	−0.678340	−	0.734748i	$-0.737300\pi$
$179$	−3249.06	−1.35668	−0.678340	+	0.734748i	$0.737300\pi$
$180$	0	0
$181$	1170.45	0.480655	0.240328	−	0.970692i	$-0.422745\pi$
$181$	1170.45	0.480655	0.240328	+	0.970692i	$0.422745\pi$
$182$	1484.89	0.604767
$183$	0	0
$184$	−320.913	−0.128576
$185$	327.009	0.129958
$186$	0	0
$187$	802.458	0.313805
$188$	6812.44	2.64281
$189$	0	0
$190$	1402.61	0.535558
$191$	2760.35	1.04572	0.522859	−	0.852419i	$-0.324866\pi$
$191$	2760.35	1.04572	0.522859	+	0.852419i	$0.324866\pi$
$192$	0	0
$193$	−1250.61	−0.466430	−0.233215	−	0.972425i	$-0.574925\pi$
$193$	−1250.61	−0.466430	−0.233215	+	0.972425i	$0.574925\pi$
$194$	1960.22	0.725441
$195$	0	0
$196$	1087.58	0.396350
$197$	143.991	0.0520756	0.0260378	−	0.999661i	$-0.491711\pi$
$197$	143.991	0.0520756	0.0260378	+	0.999661i	$0.491711\pi$
$198$	0	0
$199$	761.249	0.271174	0.135587	−	0.990765i	$-0.456708\pi$
$199$	761.249	0.271174	0.135587	+	0.990765i	$0.456708\pi$
$200$	−587.303	−0.207643
$201$	0	0
$202$	340.657	0.118656
$203$	649.411	0.224531
$204$	0	0
$205$	−545.434	−0.185828
$206$	318.844	0.107840
$207$	0	0
$208$	−47.6223	−0.0158751
$209$	−671.564	−0.222263
$210$	0	0
$211$	3976.58	1.29743	0.648717	−	0.761029i	$-0.275306\pi$
$211$	3976.58	1.29743	0.648717	+	0.761029i	$0.275306\pi$
$212$	7115.91	2.30530
$213$	0	0
$214$	5245.97	1.67573
$215$	608.749	0.193099
$216$	0	0
$217$	−5015.29	−1.56894
$218$	10232.0	3.17890
$219$	0	0
$220$	721.199	0.221015
$221$	1142.29	0.347688
$222$	0	0
$223$	908.084	0.272690	0.136345	−	0.990661i	$-0.456464\pi$
$223$	908.084	0.272690	0.136345	+	0.990661i	$0.456464\pi$
$224$	3590.30	1.07092
$225$	0	0
$226$	−7902.05	−2.32583
$227$	2062.15	0.602951	0.301475	−	0.953474i	$-0.402521\pi$
$227$	2062.15	0.602951	0.301475	+	0.953474i	$0.402521\pi$
$228$	0	0
$229$	4077.47	1.17662	0.588312	−	0.808634i	$-0.299793\pi$
$229$	4077.47	1.17662	0.588312	+	0.808634i	$0.299793\pi$
$230$	313.839	0.0899737
$231$	0	0
$232$	−739.209	−0.209187
$233$	−1682.76	−0.473138	−0.236569	−	0.971615i	$-0.576023\pi$
$233$	−1682.76	−0.473138	−0.236569	+	0.971615i	$0.576023\pi$
$234$	0	0
$235$	−2597.65	−0.721073
$236$	−1435.56	−0.395961
$237$	0	0
$238$	6917.93	1.88413
$239$	−4024.96	−1.08934	−0.544672	−	0.838649i	$-0.683346\pi$
$239$	−4024.96	−1.08934	−0.544672	+	0.838649i	$0.683346\pi$
$240$	0	0
$241$	−2784.27	−0.744194	−0.372097	−	0.928194i	$-0.621361\pi$
$241$	−2784.27	−0.744194	−0.372097	+	0.928194i	$0.621361\pi$
$242$	−555.978	−0.147684
$243$	0	0
$244$	−1175.10	−0.308313
$245$	−414.707	−0.108141
$246$	0	0
$247$	−955.968	−0.246262
$248$	5708.78	1.46173
$249$	0	0
$250$	574.357	0.145302
$251$	1827.60	0.459591	0.229796	−	0.973239i	$-0.426194\pi$
$251$	1827.60	0.459591	0.229796	+	0.973239i	$0.426194\pi$
$252$	0	0
$253$	−150.265	−0.0373402
$254$	7359.26	1.81796
$255$	0	0
$256$	−4405.79	−1.07563
$257$	−585.171	−0.142031	−0.0710155	−	0.997475i	$-0.522624\pi$
$257$	−585.171	−0.142031	−0.0710155	+	0.997475i	$0.522624\pi$
$258$	0	0
$259$	−1349.78	−0.323828
$260$	1026.62	0.244878
$261$	0	0
$262$	9208.69	2.17143
$263$	238.098	0.0558241	0.0279120	−	0.999610i	$-0.491114\pi$
$263$	238.098	0.0558241	0.0279120	+	0.999610i	$0.491114\pi$
$264$	0	0
$265$	−2713.37	−0.628984
$266$	−5789.51	−1.33450
$267$	0	0
$268$	6404.84	1.45984
$269$	−4618.46	−1.04681	−0.523406	−	0.852083i	$-0.675339\pi$
$269$	−4618.46	−1.04681	−0.523406	+	0.852083i	$0.675339\pi$
$270$	0	0
$271$	−143.439	−0.0321525	−0.0160762	−	0.999871i	$-0.505117\pi$
$271$	−143.439	−0.0321525	−0.0160762	+	0.999871i	$0.505117\pi$
$272$	−221.867	−0.0494582
$273$	0	0
$274$	−7686.51	−1.69474
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−8602.51	−1.86597	−0.932987	−	0.359911i	$-0.882807\pi$
$277$	−8602.51	−1.86597	−0.932987	+	0.359911i	$0.882807\pi$
$278$	−11672.7	−2.51828
$279$	0	0
$280$	2424.19	0.517404
$281$	2992.81	0.635360	0.317680	−	0.948198i	$-0.397096\pi$
$281$	2992.81	0.635360	0.317680	+	0.948198i	$0.397096\pi$
$282$	0	0
$283$	−6858.89	−1.44070	−0.720351	−	0.693610i	$-0.756019\pi$
$283$	−6858.89	−1.44070	−0.720351	+	0.693610i	$0.756019\pi$
$284$	−10980.9	−2.29435
$285$	0	0
$286$	−791.431	−0.163630
$287$	2251.37	0.463046
$288$	0	0
$289$	408.809	0.0832096
$290$	722.915	0.146383
$291$	0	0
$292$	4605.39	0.922979
$293$	−4049.70	−0.807461	−0.403731	−	0.914878i	$-0.632287\pi$
$293$	−4049.70	−0.807461	−0.403731	+	0.914878i	$0.632287\pi$
$294$	0	0
$295$	547.392	0.108035
$296$	1536.43	0.301699
$297$	0	0
$298$	14201.2	2.76059
$299$	−213.901	−0.0413720
$300$	0	0
$301$	−2512.71	−0.481164
$302$	−6243.20	−1.18959
$303$	0	0
$304$	185.677	0.0350305
$305$	448.078	0.0841209
$306$	0	0
$307$	−9572.69	−1.77962	−0.889808	−	0.456335i	$-0.849162\pi$
$307$	−9572.69	−1.77962	−0.889808	+	0.456335i	$0.849162\pi$
$308$	−2976.87	−0.550724
$309$	0	0
$310$	−5582.94	−1.02287
$311$	−5396.42	−0.983932	−0.491966	−	0.870614i	$-0.663722\pi$
$311$	−5396.42	−0.983932	−0.491966	+	0.870614i	$0.663722\pi$
$312$	0	0
$313$	9755.04	1.76162	0.880811	−	0.473469i	$-0.156998\pi$
$313$	9755.04	1.76162	0.880811	+	0.473469i	$0.156998\pi$
$314$	4649.77	0.835674
$315$	0	0
$316$	−10899.3	−1.94030
$317$	4353.75	0.771391	0.385695	−	0.922626i	$-0.373962\pi$
$317$	4353.75	0.771391	0.385695	+	0.922626i	$0.373962\pi$
$318$	0	0
$319$	−346.129	−0.0607508
$320$	4118.32	0.719440
$321$	0	0
$322$	−1295.42	−0.224196
$323$	−4453.74	−0.767221
$324$	0	0
$325$	−391.461	−0.0668134
$326$	12940.8	2.19854
$327$	0	0
$328$	−2562.68	−0.431404
$329$	10722.2	1.79677
$330$	0	0
$331$	5387.64	0.894656	0.447328	−	0.894370i	$-0.352376\pi$
$331$	5387.64	0.894656	0.447328	+	0.894370i	$0.352376\pi$
$332$	−18215.3	−3.01113
$333$	0	0
$334$	15847.2	2.59616
$335$	−2442.23	−0.398308
$336$	0	0
$337$	−4500.27	−0.727434	−0.363717	−	0.931509i	$-0.618492\pi$
$337$	−4500.27	−0.727434	−0.363717	+	0.931509i	$0.618492\pi$
$338$	8968.30	1.44323
$339$	0	0
$340$	4782.91	0.762910
$341$	2673.09	0.424504
$342$	0	0
$343$	−5367.18	−0.844899
$344$	2860.16	0.448283
$345$	0	0
$346$	−10512.4	−1.63337
$347$	−5906.32	−0.913740	−0.456870	−	0.889533i	$-0.651030\pi$
$347$	−5906.32	−0.913740	−0.456870	+	0.889533i	$0.651030\pi$
$348$	0	0
$349$	3636.26	0.557721	0.278860	−	0.960332i	$-0.410043\pi$
$349$	3636.26	0.557721	0.278860	+	0.960332i	$0.410043\pi$
$350$	−2370.76	−0.362063
$351$	0	0
$352$	−1913.59	−0.289757
$353$	−210.408	−0.0317248	−0.0158624	−	0.999874i	$-0.505049\pi$
$353$	−210.408	−0.0317248	−0.0158624	+	0.999874i	$0.505049\pi$
$354$	0	0
$355$	4187.12	0.625998
$356$	−19979.8	−2.97451
$357$	0	0
$358$	14928.9	2.20396
$359$	−2499.68	−0.367488	−0.183744	−	0.982974i	$-0.558822\pi$
$359$	−2499.68	−0.367488	−0.183744	+	0.982974i	$0.558822\pi$
$360$	0	0
$361$	−3131.74	−0.456588
$362$	−5378.03	−0.780837
$363$	0	0
$364$	−4237.56	−0.610188
$365$	−1756.08	−0.251828
$366$	0	0
$367$	5748.70	0.817656	0.408828	−	0.912612i	$-0.365938\pi$
$367$	5748.70	0.817656	0.408828	+	0.912612i	$0.365938\pi$
$368$	41.5458	0.00588512
$369$	0	0
$370$	−1502.56	−0.211120
$371$	11199.9	1.56730
$372$	0	0
$373$	4467.78	0.620196	0.310098	−	0.950705i	$-0.399638\pi$
$373$	4467.78	0.620196	0.310098	+	0.950705i	$0.399638\pi$
$374$	−3687.18	−0.509785
$375$	0	0
$376$	−12204.9	−1.67398
$377$	−492.712	−0.0673103
$378$	0	0
$379$	7804.08	1.05770	0.528851	−	0.848715i	$-0.322623\pi$
$379$	7804.08	1.05770	0.528851	+	0.848715i	$0.322623\pi$
$380$	−4002.74	−0.540358
$381$	0	0
$382$	−12683.4	−1.69880
$383$	11161.1	1.48904	0.744522	−	0.667597i	$-0.232677\pi$
$383$	11161.1	1.48904	0.744522	+	0.667597i	$0.232677\pi$
$384$	0	0
$385$	1135.11	0.150261
$386$	5746.38	0.757728
$387$	0	0
$388$	−5594.04	−0.731944
$389$	−8490.24	−1.10661	−0.553306	−	0.832978i	$-0.686634\pi$
$389$	−8490.24	−1.10661	−0.553306	+	0.832978i	$0.686634\pi$
$390$	0	0
$391$	−996.540	−0.128893
$392$	−1948.47	−0.251052
$393$	0	0
$394$	−661.616	−0.0845983
$395$	4156.03	0.529398
$396$	0	0
$397$	6019.74	0.761013	0.380507	−	0.924778i	$-0.375750\pi$
$397$	6019.74	0.761013	0.380507	+	0.924778i	$0.375750\pi$
$398$	−3497.83	−0.440529
$399$	0	0
$400$	76.0330	0.00950413
$401$	10398.8	1.29499	0.647495	−	0.762069i	$-0.275816\pi$
$401$	10398.8	1.29499	0.647495	+	0.762069i	$0.275816\pi$
$402$	0	0
$403$	3805.13	0.470340
$404$	−972.158	−0.119720
$405$	0	0
$406$	−2983.95	−0.364756
$407$	719.420	0.0876175
$408$	0	0
$409$	−4733.68	−0.572287	−0.286144	−	0.958187i	$-0.592373\pi$
$409$	−4733.68	−0.572287	−0.286144	+	0.958187i	$0.592373\pi$
$410$	2506.19	0.301883
$411$	0	0
$412$	−909.911	−0.108806
$413$	−2259.45	−0.269202
$414$	0	0
$415$	6945.67	0.821565
$416$	−2723.98	−0.321044
$417$	0	0
$418$	3085.74	0.361073
$419$	8117.57	0.946466	0.473233	−	0.880937i	$-0.343087\pi$
$419$	8117.57	0.946466	0.473233	+	0.880937i	$0.343087\pi$
$420$	0	0
$421$	−9484.27	−1.09795	−0.548973	−	0.835840i	$-0.684981\pi$
$421$	−9484.27	−1.09795	−0.548973	+	0.835840i	$0.684981\pi$
$422$	−18271.8	−2.10772
$423$	0	0
$424$	−12748.5	−1.46020
$425$	−1823.77	−0.208155
$426$	0	0
$427$	−1849.52	−0.209612
$428$	−14970.8	−1.69075
$429$	0	0
$430$	−2797.11	−0.313695
$431$	−9335.16	−1.04329	−0.521646	−	0.853162i	$-0.674682\pi$
$431$	−9335.16	−1.04329	−0.521646	+	0.853162i	$0.674682\pi$
$432$	0	0
$433$	−2983.02	−0.331074	−0.165537	−	0.986204i	$-0.552936\pi$
$433$	−2983.02	−0.331074	−0.165537	+	0.986204i	$0.552936\pi$
$434$	23044.5	2.54878
$435$	0	0
$436$	−29200.0	−3.20739
$437$	833.988	0.0912931
$438$	0	0
$439$	−5232.32	−0.568850	−0.284425	−	0.958698i	$-0.591803\pi$
$439$	−5232.32	−0.568850	−0.284425	+	0.958698i	$0.591803\pi$
$440$	−1292.07	−0.139993
$441$	0	0
$442$	−5248.68	−0.564828
$443$	−7517.71	−0.806269	−0.403135	−	0.915141i	$-0.632079\pi$
$443$	−7517.71	−0.806269	−0.403135	+	0.915141i	$0.632079\pi$
$444$	0	0
$445$	7618.49	0.811575
$446$	−4172.52	−0.442992
$447$	0	0
$448$	−16999.1	−1.79270
$449$	−16070.9	−1.68916	−0.844581	−	0.535428i	$-0.820150\pi$
$449$	−16070.9	−1.68916	−0.844581	+	0.535428i	$0.820150\pi$
$450$	0	0
$451$	−1199.96	−0.125285
$452$	22550.7	2.34667
$453$	0	0
$454$	−9475.29	−0.979510
$455$	1615.82	0.166485
$456$	0	0
$457$	9718.51	0.994776	0.497388	−	0.867528i	$-0.334293\pi$
$457$	9718.51	0.994776	0.497388	+	0.867528i	$0.334293\pi$
$458$	−18735.4	−1.91146
$459$	0	0
$460$	−895.627	−0.0907801
$461$	14538.0	1.46877	0.734385	−	0.678733i	$-0.237471\pi$
$461$	14538.0	1.46877	0.734385	+	0.678733i	$0.237471\pi$
$462$	0	0
$463$	−9978.17	−1.00157	−0.500783	−	0.865573i	$-0.666955\pi$
$463$	−9978.17	−1.00157	−0.500783	+	0.865573i	$0.666955\pi$
$464$	95.6990	0.00957481
$465$	0	0
$466$	7732.03	0.768625
$467$	−15188.2	−1.50498	−0.752489	−	0.658605i	$-0.771147\pi$
$467$	−15188.2	−1.50498	−0.752489	+	0.658605i	$0.771147\pi$
$468$	0	0
$469$	10080.7	0.992503
$470$	11935.8	1.17140
$471$	0	0
$472$	2571.88	0.250806
$473$	1339.25	0.130187
$474$	0	0
$475$	1526.28	0.147433
$476$	−19742.3	−1.90102
$477$	0	0
$478$	18494.1	1.76967
$479$	−11330.8	−1.08083	−0.540415	−	0.841399i	$-0.681733\pi$
$479$	−11330.8	−1.08083	−0.540415	+	0.841399i	$0.681733\pi$
$480$	0	0
$481$	1024.09	0.0970779
$482$	12793.3	1.20896
$483$	0	0
$484$	1586.64	0.149008
$485$	2133.06	0.199706
$486$	0	0
$487$	19086.9	1.77599	0.887997	−	0.459850i	$-0.152097\pi$
$487$	19086.9	1.77599	0.887997	+	0.459850i	$0.152097\pi$
$488$	2105.26	0.195288
$489$	0	0
$490$	1905.52	0.175679
$491$	8112.85	0.745677	0.372839	−	0.927896i	$-0.378384\pi$
$491$	8112.85	0.745677	0.372839	+	0.927896i	$0.378384\pi$
$492$	0	0
$493$	−2295.49	−0.209703
$494$	4392.53	0.400060
$495$	0	0
$496$	−739.066	−0.0669053
$497$	−17283.0	−1.55986
$498$	0	0
$499$	18329.1	1.64433	0.822167	−	0.569246i	$-0.192765\pi$
$499$	18329.1	1.64433	0.822167	+	0.569246i	$0.192765\pi$
$500$	−1639.09	−0.146604
$501$	0	0
$502$	−8397.58	−0.746618
$503$	−7739.57	−0.686064	−0.343032	−	0.939324i	$-0.611454\pi$
$503$	−7739.57	−0.686064	−0.343032	+	0.939324i	$0.611454\pi$
$504$	0	0
$505$	370.693	0.0326646
$506$	690.446	0.0606602
$507$	0	0
$508$	−21001.7	−1.83425
$509$	−15914.9	−1.38589	−0.692943	−	0.720993i	$-0.743686\pi$
$509$	−15914.9	−1.38589	−0.692943	+	0.720993i	$0.743686\pi$
$510$	0	0
$511$	7248.51	0.627505
$512$	1100.65	0.0950048
$513$	0	0
$514$	2688.78	0.230733
$515$	346.958	0.0296870
$516$	0	0
$517$	−5714.83	−0.486147
$518$	6202.07	0.526068
$519$	0	0
$520$	−1839.25	−0.155109
$521$	−2274.50	−0.191262	−0.0956312	−	0.995417i	$-0.530487\pi$
$521$	−2274.50	−0.191262	−0.0956312	+	0.995417i	$0.530487\pi$
$522$	0	0
$523$	10971.1	0.917274	0.458637	−	0.888624i	$-0.348338\pi$
$523$	10971.1	0.917274	0.458637	+	0.888624i	$0.348338\pi$
$524$	−26279.6	−2.19089
$525$	0	0
$526$	−1094.02	−0.0906877
$527$	17727.6	1.46533
$528$	0	0
$529$	−11980.4	−0.984663
$530$	12467.5	1.02180
$531$	0	0
$532$	16522.0	1.34646
$533$	−1708.13	−0.138813
$534$	0	0
$535$	5708.53	0.461310
$536$	−11474.6	−0.924680
$537$	0	0
$538$	21221.2	1.70058
$539$	−912.355	−0.0729089
$540$	0	0
$541$	5313.05	0.422229	0.211115	−	0.977461i	$-0.432291\pi$
$541$	5313.05	0.422229	0.211115	+	0.977461i	$0.432291\pi$
$542$	659.084	0.0522326
$543$	0	0
$544$	−12690.7	−1.00020
$545$	11134.2	0.875115
$546$	0	0
$547$	20685.1	1.61688	0.808439	−	0.588581i	$-0.200313\pi$
$547$	20685.1	1.61688	0.808439	+	0.588581i	$0.200313\pi$
$548$	21935.6	1.70993
$549$	0	0
$550$	1263.59	0.0979627
$551$	1921.06	0.148529
$552$	0	0
$553$	−17154.7	−1.31915
$554$	39527.3	3.03132
$555$	0	0
$556$	33311.2	2.54085
$557$	10853.8	0.825659	0.412830	−	0.910808i	$-0.364540\pi$
$557$	10853.8	0.825659	0.412830	+	0.910808i	$0.364540\pi$
$558$	0	0
$559$	1906.41	0.144244
$560$	−313.839	−0.0236824
$561$	0	0
$562$	−13751.5	−1.03216
$563$	15381.2	1.15141	0.575704	−	0.817658i	$-0.304728\pi$
$563$	15381.2	1.15141	0.575704	+	0.817658i	$0.304728\pi$
$564$	0	0
$565$	−8598.80	−0.640273
$566$	31515.6	2.34046
$567$	0	0
$568$	19672.9	1.45327
$569$	1348.88	0.0993814	0.0496907	−	0.998765i	$-0.484176\pi$
$569$	1348.88	0.0993814	0.0496907	+	0.998765i	$0.484176\pi$
$570$	0	0
$571$	3463.51	0.253841	0.126920	−	0.991913i	$-0.459491\pi$
$571$	3463.51	0.253841	0.126920	+	0.991913i	$0.459491\pi$
$572$	2258.57	0.165097
$573$	0	0
$574$	−10344.7	−0.752231
$575$	341.511	0.0247687
$576$	0	0
$577$	12052.6	0.869598	0.434799	−	0.900528i	$-0.356819\pi$
$577$	12052.6	0.869598	0.434799	+	0.900528i	$0.356819\pi$
$578$	−1878.42	−0.135176
$579$	0	0
$580$	−2063.04	−0.147695
$581$	−28669.4	−2.04717
$582$	0	0
$583$	−5969.41	−0.424061
$584$	−8250.80	−0.584624
$585$	0	0
$586$	18607.8	1.31174
$587$	11133.1	0.782813	0.391407	−	0.920218i	$-0.371989\pi$
$587$	11133.1	0.782813	0.391407	+	0.920218i	$0.371989\pi$
$588$	0	0
$589$	−14836.0	−1.03787
$590$	−2515.19	−0.175506
$591$	0	0
$592$	−198.908	−0.0138092
$593$	7939.69	0.549821	0.274911	−	0.961470i	$-0.411352\pi$
$593$	7939.69	0.549821	0.274911	+	0.961470i	$0.411352\pi$
$594$	0	0
$595$	7527.91	0.518680
$596$	−40527.1	−2.78533
$597$	0	0
$598$	982.846	0.0672100
$599$	19474.7	1.32840	0.664202	−	0.747553i	$-0.268771\pi$
$599$	19474.7	1.32840	0.664202	+	0.747553i	$0.268771\pi$
$600$	0	0
$601$	−19946.1	−1.35377	−0.676887	−	0.736087i	$-0.736671\pi$
$601$	−19946.1	−1.35377	−0.676887	+	0.736087i	$0.736671\pi$
$602$	11545.6	0.781664
$603$	0	0
$604$	17816.7	1.20025
$605$	−605.000	−0.0406558
$606$	0	0
$607$	1427.44	0.0954496	0.0477248	−	0.998861i	$-0.484803\pi$
$607$	1427.44	0.0954496	0.0477248	+	0.998861i	$0.484803\pi$
$608$	10620.6	0.708427
$609$	0	0
$610$	−2058.85	−0.136657
$611$	−8135.03	−0.538638
$612$	0	0
$613$	−8029.40	−0.529045	−0.264522	−	0.964380i	$-0.585214\pi$
$613$	−8029.40	−0.529045	−0.264522	+	0.964380i	$0.585214\pi$
$614$	43985.1	2.89103
$615$	0	0
$616$	5333.22	0.348834
$617$	−20795.5	−1.35688	−0.678440	−	0.734655i	$-0.737344\pi$
$617$	−20795.5	−1.35688	−0.678440	+	0.734655i	$0.737344\pi$
$618$	0	0
$619$	1677.43	0.108920	0.0544602	−	0.998516i	$-0.482656\pi$
$619$	1677.43	0.108920	0.0544602	+	0.998516i	$0.482656\pi$
$620$	15932.5	1.03204
$621$	0	0
$622$	24795.8	1.59842
$623$	−31446.6	−2.02228
$624$	0	0
$625$	625.000	0.0400000
$626$	−44823.0	−2.86180
$627$	0	0
$628$	−13269.4	−0.843165
$629$	4771.11	0.302443
$630$	0	0
$631$	−25225.2	−1.59144	−0.795719	−	0.605666i	$-0.792907\pi$
$631$	−25225.2	−1.59144	−0.795719	+	0.605666i	$0.792907\pi$
$632$	19526.8	1.22901
$633$	0	0
$634$	−20004.8	−1.25315
$635$	8008.15	0.500462
$636$	0	0
$637$	−1298.73	−0.0807812
$638$	1590.41	0.0986912
$639$	0	0
$640$	−11964.6	−0.738971
$641$	−15165.3	−0.934468	−0.467234	−	0.884134i	$-0.654749\pi$
$641$	−15165.3	−0.934468	−0.467234	+	0.884134i	$0.654749\pi$
$642$	0	0
$643$	27156.1	1.66553	0.832763	−	0.553630i	$-0.186758\pi$
$643$	27156.1	1.66553	0.832763	+	0.553630i	$0.186758\pi$
$644$	3696.85	0.226206
$645$	0	0
$646$	20464.3	1.24637
$647$	−29154.9	−1.77156	−0.885778	−	0.464110i	$-0.846374\pi$
$647$	−29154.9	−1.77156	−0.885778	+	0.464110i	$0.846374\pi$
$648$	0	0
$649$	1204.26	0.0728374
$650$	1798.71	0.108540
$651$	0	0
$652$	−36930.2	−2.21825
$653$	19141.7	1.14713	0.573564	−	0.819161i	$-0.305560\pi$
$653$	19141.7	1.14713	0.573564	+	0.819161i	$0.305560\pi$
$654$	0	0
$655$	10020.7	0.597770
$656$	331.768	0.0197460
$657$	0	0
$658$	−49267.2	−2.91890
$659$	24939.6	1.47422	0.737110	−	0.675773i	$-0.236190\pi$
$659$	24939.6	1.47422	0.737110	+	0.675773i	$0.236190\pi$
$660$	0	0
$661$	22617.7	1.33090	0.665452	−	0.746440i	$-0.268239\pi$
$661$	22617.7	1.33090	0.665452	+	0.746440i	$0.268239\pi$
$662$	−24755.4	−1.45339
$663$	0	0
$664$	32633.7	1.90728
$665$	−6299.99	−0.367373
$666$	0	0
$667$	429.843	0.0249529
$668$	−45224.3	−2.61943
$669$	0	0
$670$	11221.7	0.647062
$671$	985.772	0.0567143
$672$	0	0
$673$	−13855.8	−0.793615	−0.396807	−	0.917902i	$-0.629882\pi$
$673$	−13855.8	−0.793615	−0.396807	+	0.917902i	$0.629882\pi$
$674$	20678.1	1.18174
$675$	0	0
$676$	−25593.5	−1.45616
$677$	24992.8	1.41884	0.709419	−	0.704787i	$-0.248957\pi$
$677$	24992.8	1.41884	0.709419	+	0.704787i	$0.248957\pi$
$678$	0	0
$679$	−8804.57	−0.497626
$680$	−8568.84	−0.483235
$681$	0	0
$682$	−12282.5	−0.689619
$683$	14420.5	0.807887	0.403943	−	0.914784i	$-0.367639\pi$
$683$	14420.5	0.807887	0.403943	+	0.914784i	$0.367639\pi$
$684$	0	0
$685$	−8364.25	−0.466543
$686$	24661.4	1.37256
$687$	0	0
$688$	−370.280	−0.0205186
$689$	−8497.42	−0.469849
$690$	0	0
$691$	30552.4	1.68201	0.841005	−	0.541027i	$-0.181964\pi$
$691$	30552.4	1.68201	0.841005	+	0.541027i	$0.181964\pi$
$692$	29999.9	1.64801
$693$	0	0
$694$	27138.7	1.48440
$695$	−12701.9	−0.693253
$696$	0	0
$697$	−7957.96	−0.432467
$698$	−16708.1	−0.906032
$699$	0	0
$700$	6765.61	0.365309
$701$	9151.47	0.493076	0.246538	−	0.969133i	$-0.420707\pi$
$701$	9151.47	0.493076	0.246538	+	0.969133i	$0.420707\pi$
$702$	0	0
$703$	−3992.86	−0.214216
$704$	9060.30	0.485047
$705$	0	0
$706$	966.793	0.0515378
$707$	−1530.10	−0.0813937
$708$	0	0
$709$	−6261.96	−0.331697	−0.165848	−	0.986151i	$-0.553036\pi$
$709$	−6261.96	−0.331697	−0.165848	+	0.986151i	$0.553036\pi$
$710$	−19239.2	−1.01695
$711$	0	0
$712$	35794.9	1.88409
$713$	−3319.60	−0.174362
$714$	0	0
$715$	−861.214	−0.0450456
$716$	−42603.9	−2.22372
$717$	0	0
$718$	11485.7	0.596995
$719$	18228.7	0.945500	0.472750	−	0.881197i	$-0.343261\pi$
$719$	18228.7	0.945500	0.472750	+	0.881197i	$0.343261\pi$
$720$	0	0
$721$	−1432.13	−0.0739740
$722$	14389.9	0.741740
$723$	0	0
$724$	15347.7	0.787836
$725$	786.656	0.0402975
$726$	0	0
$727$	−7233.66	−0.369026	−0.184513	−	0.982830i	$-0.559071\pi$
$727$	−7233.66	−0.369026	−0.184513	+	0.982830i	$0.559071\pi$
$728$	7591.81	0.386499
$729$	0	0
$730$	8068.93	0.409102
$731$	8881.73	0.449388
$732$	0	0
$733$	−13444.8	−0.677485	−0.338743	−	0.940879i	$-0.610002\pi$
$733$	−13444.8	−0.677485	−0.338743	+	0.940879i	$0.610002\pi$
$734$	−26414.4	−1.32830
$735$	0	0
$736$	2376.41	0.119016
$737$	−5372.90	−0.268539
$738$	0	0
$739$	18490.9	0.920432	0.460216	−	0.887807i	$-0.347772\pi$
$739$	18490.9	0.920432	0.460216	+	0.887807i	$0.347772\pi$
$740$	4287.97	0.213012
$741$	0	0
$742$	−51461.8	−2.54612
$743$	25160.9	1.24235	0.621173	−	0.783674i	$-0.286657\pi$
$743$	25160.9	1.24235	0.621173	+	0.783674i	$0.286657\pi$
$744$	0	0
$745$	15453.4	0.759957
$746$	−20528.8	−1.00752
$747$	0	0
$748$	10522.4	0.514354
$749$	−23562.9	−1.14949
$750$	0	0
$751$	−13419.5	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$751$	−13419.5	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$752$	1580.06	0.0766207
$753$	0	0
$754$	2263.94	0.109347
$755$	−6793.68	−0.327480
$756$	0	0
$757$	−7014.90	−0.336804	−0.168402	−	0.985718i	$-0.553861\pi$
$757$	−7014.90	−0.336804	−0.168402	+	0.985718i	$0.553861\pi$
$758$	−35858.6	−1.71826
$759$	0	0
$760$	7171.12	0.342268
$761$	30156.9	1.43651	0.718256	−	0.695779i	$-0.244941\pi$
$761$	30156.9	1.43651	0.718256	+	0.695779i	$0.244941\pi$
$762$	0	0
$763$	−45958.4	−2.18061
$764$	36195.7	1.71402
$765$	0	0
$766$	−51283.5	−2.41899
$767$	1714.26	0.0807019
$768$	0	0
$769$	11292.2	0.529530	0.264765	−	0.964313i	$-0.414706\pi$
$769$	11292.2	0.529530	0.264765	+	0.964313i	$0.414706\pi$
$770$	−5215.66	−0.244103
$771$	0	0
$772$	−16398.9	−0.764519
$773$	−8524.10	−0.396624	−0.198312	−	0.980139i	$-0.563546\pi$
$773$	−8524.10	−0.396624	−0.198312	+	0.980139i	$0.563546\pi$
$774$	0	0
$775$	−6075.21	−0.281584
$776$	10022.0	0.463621
$777$	0	0
$778$	39011.4	1.79772
$779$	6659.89	0.306310
$780$	0	0
$781$	9211.66	0.422047
$782$	4578.96	0.209390
$783$	0	0
$784$	252.251	0.0114910
$785$	5059.76	0.230052
$786$	0	0
$787$	−14983.9	−0.678676	−0.339338	−	0.940665i	$-0.610203\pi$
$787$	−14983.9	−0.678676	−0.339338	+	0.940665i	$0.610203\pi$
$788$	1888.10	0.0853565
$789$	0	0
$790$	−19096.3	−0.860022
$791$	35493.0	1.59543
$792$	0	0
$793$	1403.24	0.0628380
$794$	−27659.9	−1.23629
$795$	0	0
$796$	9982.04	0.444477
$797$	−37172.3	−1.65208	−0.826041	−	0.563610i	$-0.809412\pi$
$797$	−37172.3	−1.65208	−0.826041	+	0.563610i	$0.809412\pi$
$798$	0	0
$799$	−37900.1	−1.67811
$800$	4349.06	0.192203
$801$	0	0
$802$	−47781.0	−2.10375
$803$	−3863.37	−0.169783
$804$	0	0
$805$	−1409.65	−0.0617186
$806$	−17484.0	−0.764080
$807$	0	0
$808$	1741.68	0.0758316
$809$	−23797.1	−1.03419	−0.517096	−	0.855928i	$-0.672987\pi$
$809$	−23797.1	−1.03419	−0.517096	+	0.855928i	$0.672987\pi$
$810$	0	0
$811$	8988.35	0.389178	0.194589	−	0.980885i	$-0.437663\pi$
$811$	8988.35	0.389178	0.194589	+	0.980885i	$0.437663\pi$
$812$	8515.54	0.368026
$813$	0	0
$814$	−3305.63	−0.142337
$815$	14081.8	0.605234
$816$	0	0
$817$	−7432.98	−0.318295
$818$	21750.6	0.929696
$819$	0	0
$820$	−7152.11	−0.304589
$821$	25156.8	1.06940	0.534702	−	0.845041i	$-0.320424\pi$
$821$	25156.8	1.06940	0.534702	+	0.845041i	$0.320424\pi$
$822$	0	0
$823$	1318.51	0.0558447	0.0279224	−	0.999610i	$-0.491111\pi$
$823$	1318.51	0.0558447	0.0279224	+	0.999610i	$0.491111\pi$
$824$	1630.16	0.0689189
$825$	0	0
$826$	10381.9	0.437326
$827$	−124.982	−0.00525519	−0.00262760	−	0.999997i	$-0.500836\pi$
$827$	−124.982	−0.00525519	−0.00262760	+	0.999997i	$0.500836\pi$
$828$	0	0
$829$	−8886.80	−0.372318	−0.186159	−	0.982520i	$-0.559604\pi$
$829$	−8886.80	−0.372318	−0.186159	+	0.982520i	$0.559604\pi$
$830$	−31914.3	−1.33465
$831$	0	0
$832$	12897.3	0.537419
$833$	−6050.63	−0.251671
$834$	0	0
$835$	17244.5	0.714694
$836$	−8806.02	−0.364309
$837$	0	0
$838$	−37299.1	−1.53756
$839$	−2995.21	−0.123249	−0.0616247	−	0.998099i	$-0.519628\pi$
$839$	−2995.21	−0.123249	−0.0616247	+	0.998099i	$0.519628\pi$
$840$	0	0
$841$	−23398.9	−0.959403
$842$	43578.8	1.78364
$843$	0	0
$844$	52143.6	2.12661
$845$	9759.07	0.397304
$846$	0	0
$847$	2497.24	0.101306
$848$	1650.44	0.0668355
$849$	0	0
$850$	8379.95	0.338153
$851$	−893.418	−0.0359882
$852$	0	0
$853$	18130.5	0.727757	0.363878	−	0.931446i	$-0.381452\pi$
$853$	18130.5	0.727757	0.363878	+	0.931446i	$0.381452\pi$
$854$	8498.27	0.340521
$855$	0	0
$856$	26821.1	1.07094
$857$	−26394.1	−1.05205	−0.526024	−	0.850470i	$-0.676318\pi$
$857$	−26394.1	−1.05205	−0.526024	+	0.850470i	$0.676318\pi$
$858$	0	0
$859$	−29456.2	−1.17000	−0.585002	−	0.811032i	$-0.698906\pi$
$859$	−29456.2	−1.17000	−0.585002	+	0.811032i	$0.698906\pi$
$860$	7982.34	0.316506
$861$	0	0
$862$	42893.7	1.69486
$863$	762.616	0.0300808	0.0150404	−	0.999887i	$-0.495212\pi$
$863$	762.616	0.0300808	0.0150404	+	0.999887i	$0.495212\pi$
$864$	0	0
$865$	−11439.3	−0.449649
$866$	13706.6	0.537838
$867$	0	0
$868$	−65764.0	−2.57163
$869$	9143.26	0.356920
$870$	0	0
$871$	−7648.29	−0.297535
$872$	52313.3	2.03160
$873$	0	0
$874$	−3832.06	−0.148308
$875$	−2579.79	−0.0996719
$876$	0	0
$877$	44767.2	1.72369	0.861847	−	0.507168i	$-0.169308\pi$
$877$	44767.2	1.72369	0.861847	+	0.507168i	$0.169308\pi$
$878$	24041.8	0.924112
$879$	0	0
$880$	167.273	0.00640768
$881$	32057.9	1.22595	0.612973	−	0.790104i	$-0.289973\pi$
$881$	32057.9	1.22595	0.612973	+	0.790104i	$0.289973\pi$
$882$	0	0
$883$	7078.95	0.269791	0.134896	−	0.990860i	$-0.456930\pi$
$883$	7078.95	0.269791	0.134896	+	0.990860i	$0.456930\pi$
$884$	14978.6	0.569891
$885$	0	0
$886$	34542.8	1.30981
$887$	25148.1	0.951964	0.475982	−	0.879455i	$-0.342093\pi$
$887$	25148.1	0.951964	0.475982	+	0.879455i	$0.342093\pi$
$888$	0	0
$889$	−33055.0	−1.24705
$890$	−35005.9	−1.31843
$891$	0	0
$892$	11907.4	0.446963
$893$	31718.0	1.18858
$894$	0	0
$895$	16245.3	0.606726
$896$	49385.8	1.84137
$897$	0	0
$898$	73843.6	2.74409
$899$	−7646.56	−0.283679
$900$	0	0
$901$	−39588.4	−1.46380
$902$	5513.62	0.203529
$903$	0	0
$904$	−40400.8	−1.48641
$905$	−5852.23	−0.214955
$906$	0	0
$907$	1269.76	0.0464848	0.0232424	−	0.999730i	$-0.492601\pi$
$907$	1269.76	0.0464848	0.0232424	+	0.999730i	$0.492601\pi$
$908$	27040.4	0.988289
$909$	0	0
$910$	−7424.47	−0.270460
$911$	33783.1	1.22863	0.614316	−	0.789060i	$-0.289432\pi$
$911$	33783.1	1.22863	0.614316	+	0.789060i	$0.289432\pi$
$912$	0	0
$913$	15280.5	0.553899
$914$	−44655.1	−1.61604
$915$	0	0
$916$	53466.6	1.92859
$917$	−41361.9	−1.48952
$918$	0	0
$919$	39262.5	1.40930	0.704652	−	0.709553i	$-0.251103\pi$
$919$	39262.5	1.40930	0.704652	+	0.709553i	$0.251103\pi$
$920$	1604.57	0.0575010
$921$	0	0
$922$	−66800.1	−2.38606
$923$	13112.7	0.467618
$924$	0	0
$925$	−1635.04	−0.0581189
$926$	45848.3	1.62707
$927$	0	0
$928$	5473.95	0.193633
$929$	21175.0	0.747825	0.373913	−	0.927464i	$-0.378016\pi$
$929$	21175.0	0.747825	0.373913	+	0.927464i	$0.378016\pi$
$930$	0	0
$931$	5063.68	0.178255
$932$	−22065.5	−0.775515
$933$	0	0
$934$	69787.5	2.44488
$935$	−4012.29	−0.140338
$936$	0	0
$937$	−5135.11	−0.179036	−0.0895180	−	0.995985i	$-0.528533\pi$
$937$	−5135.11	−0.179036	−0.0895180	+	0.995985i	$0.528533\pi$
$938$	−46319.4	−1.61235
$939$	0	0
$940$	−34062.2	−1.18190
$941$	9702.77	0.336133	0.168067	−	0.985776i	$-0.446248\pi$
$941$	9702.77	0.336133	0.168067	+	0.985776i	$0.446248\pi$
$942$	0	0
$943$	1490.18	0.0514600
$944$	−332.959	−0.0114798
$945$	0	0
$946$	−6153.65	−0.211493
$947$	699.579	0.0240055	0.0120028	−	0.999928i	$-0.496179\pi$
$947$	699.579	0.0240055	0.0120028	+	0.999928i	$0.496179\pi$
$948$	0	0
$949$	−5499.49	−0.188115
$950$	−7013.05	−0.239509
$951$	0	0
$952$	35369.3	1.20412
$953$	−42039.3	−1.42895	−0.714473	−	0.699663i	$-0.753333\pi$
$953$	−42039.3	−1.42895	−0.714473	+	0.699663i	$0.753333\pi$
$954$	0	0
$955$	−13801.8	−0.467659
$956$	−52778.1	−1.78553
$957$	0	0
$958$	52063.4	1.75584
$959$	34524.9	1.16253
$960$	0	0
$961$	29262.0	0.982243
$962$	−4705.55	−0.157706
$963$	0	0
$964$	−36509.3	−1.21980
$965$	6253.05	0.208594
$966$	0	0
$967$	−32794.8	−1.09060	−0.545299	−	0.838242i	$-0.683584\pi$
$967$	−32794.8	−1.09060	−0.545299	+	0.838242i	$0.683584\pi$
$968$	−2842.55	−0.0943832
$969$	0	0
$970$	−9801.10	−0.324427
$971$	3322.53	0.109810	0.0549048	−	0.998492i	$-0.482514\pi$
$971$	3322.53	0.109810	0.0549048	+	0.998492i	$0.482514\pi$
$972$	0	0
$973$	52429.3	1.72745
$974$	−87701.4	−2.88515
$975$	0	0
$976$	−272.550	−0.00893864
$977$	−22192.5	−0.726716	−0.363358	−	0.931650i	$-0.618370\pi$
$977$	−22192.5	−0.726716	−0.363358	+	0.931650i	$0.618370\pi$
$978$	0	0
$979$	16760.7	0.547164
$980$	−5437.92	−0.177253
$981$	0	0
$982$	−37277.4	−1.21137
$983$	−7383.09	−0.239556	−0.119778	−	0.992801i	$-0.538218\pi$
$983$	−7383.09	−0.239556	−0.119778	+	0.992801i	$0.538218\pi$
$984$	0	0
$985$	−719.953	−0.0232889
$986$	10547.4	0.340668
$987$	0	0
$988$	−12535.3	−0.403645
$989$	−1663.16	−0.0534735
$990$	0	0
$991$	46260.8	1.48287	0.741434	−	0.671026i	$-0.234146\pi$
$991$	46260.8	1.48287	0.741434	+	0.671026i	$0.234146\pi$
$992$	−42274.3	−1.35304
$993$	0	0
$994$	79413.1	2.53403
$995$	−3806.25	−0.121273
$996$	0	0
$997$	41196.8	1.30864	0.654320	−	0.756217i	$-0.272955\pi$
$997$	41196.8	1.30864	0.654320	+	0.756217i	$0.272955\pi$
$998$	−84219.5	−2.67127
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.i.1.1		3
3.2	odd	2		165.4.a.g.1.3	✓	3
5.4	even	2		2475.4.a.z.1.3		3
15.2	even	4		825.4.c.m.199.6		6
15.8	even	4		825.4.c.m.199.1		6
15.14	odd	2		825.4.a.p.1.1		3
33.32	even	2		1815.4.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.g.1.3	✓	3	3.2	odd	2
495.4.a.i.1.1		3	1.1	even	1	trivial
825.4.a.p.1.1		3	15.14	odd	2
825.4.c.m.199.1		6	15.8	even	4
825.4.c.m.199.6		6	15.2	even	4
1815.4.a.q.1.1		3	33.32	even	2
2475.4.a.z.1.3		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)

\(f(q)\)	\(=\)	\(q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +O(q^{10})\)	\(q-4.59486 q^{2} +13.1127 q^{4} -5.00000 q^{5} +20.6383 q^{7} -23.4921 q^{8} +22.9743 q^{10} -11.0000 q^{11} -15.6584 q^{13} -94.8302 q^{14} +3.04132 q^{16} -72.9507 q^{17} +61.0513 q^{19} -65.5635 q^{20} +50.5434 q^{22} +13.6605 q^{23} +25.0000 q^{25} +71.9483 q^{26} +270.624 q^{28} +31.4663 q^{29} -243.008 q^{31} +173.963 q^{32} +335.198 q^{34} -103.192 q^{35} -65.4018 q^{37} -280.522 q^{38} +117.461 q^{40} +109.087 q^{41} -121.750 q^{43} -144.240 q^{44} -62.7678 q^{46} +519.530 q^{47} +82.9413 q^{49} -114.871 q^{50} -205.324 q^{52} +542.673 q^{53} +55.0000 q^{55} -484.839 q^{56} -144.583 q^{58} -109.478 q^{59} -89.6156 q^{61} +1116.59 q^{62} -823.664 q^{64} +78.2922 q^{65} +488.446 q^{67} -956.581 q^{68} +474.151 q^{70} -837.423 q^{71} +351.216 q^{73} +300.512 q^{74} +800.547 q^{76} -227.022 q^{77} -831.205 q^{79} -15.2066 q^{80} -501.238 q^{82} -1389.13 q^{83} +364.754 q^{85} +559.423 q^{86} +258.413 q^{88} -1523.70 q^{89} -323.164 q^{91} +179.125 q^{92} -2387.17 q^{94} -305.256 q^{95} -426.612 q^{97} -381.103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8	\( 3 q - q^{2} + 17 q^{4} - 15 q^{5} + 6 q^{7} + 3 q^{8} + 5 q^{10} - 33 q^{11} - 20 q^{13} - 144 q^{14} + 25 q^{16} - 32 q^{17} + 116 q^{19} - 85 q^{20} + 11 q^{22} - 240 q^{23} + 75 q^{25} - 302 q^{26} + 160 q^{28} - 238 q^{29} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} - 90 q^{37} - 324 q^{38} - 15 q^{40} + 46 q^{41} - 134 q^{43} - 187 q^{44} - 240 q^{46} + 220 q^{47} - 457 q^{49} - 25 q^{50} - 1530 q^{52} + 798 q^{53} + 165 q^{55} - 688 q^{56} - 978 q^{58} - 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} + 720 q^{70} - 1816 q^{71} + 100 q^{73} + 1874 q^{74} + 396 q^{76} - 66 q^{77} - 96 q^{79} - 125 q^{80} - 910 q^{82} - 858 q^{83} + 160 q^{85} - 188 q^{86} - 33 q^{88} - 838 q^{89} + 332 q^{91} + 688 q^{92} - 3112 q^{94} - 580 q^{95} - 1322 q^{97} - 1017 q^{98}+O(q^{100}) \) 3 * q - q^2 + 17 * q^4 - 15 * q^5 + 6 * q^7 + 3 * q^8 + 5 * q^10 - 33 * q^11 - 20 * q^13 - 144 * q^14 + 25 * q^16 - 32 * q^17 + 116 * q^19 - 85 * q^20 + 11 * q^22 - 240 * q^23 + 75 * q^25 - 302 * q^26 + 160 * q^28 - 238 * q^29 + 92 * q^31 - 197 * q^32 + 354 * q^34 - 30 * q^35 - 90 * q^37 - 324 * q^38 - 15 * q^40 + 46 * q^41 - 134 * q^43 - 187 * q^44 - 240 * q^46 + 220 * q^47 - 457 * q^49 - 25 * q^50 - 1530 * q^52 + 798 * q^53 + 165 * q^55 - 688 * q^56 - 978 * q^58 - 1236 * q^59 + 342 * q^61 + 1792 * q^62 - 1919 * q^64 + 100 * q^65 + 764 * q^67 - 1074 * q^68 + 720 * q^70 - 1816 * q^71 + 100 * q^73 + 1874 * q^74 + 396 * q^76 - 66 * q^77 - 96 * q^79 - 125 * q^80 - 910 * q^82 - 858 * q^83 + 160 * q^85 - 188 * q^86 - 33 * q^88 - 838 * q^89 + 332 * q^91 + 688 * q^92 - 3112 * q^94 - 580 * q^95 - 1322 * q^97 - 1017 * q^98

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