LMFDB - Embedded newform 495.4.a.l.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
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			1.1
Root			$-4.59056$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} +O(q^{10})$	\(q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} -17.9528 q^{10} -11.0000 q^{11} -54.1214 q^{13} +57.9749 q^{14} -79.2041 q^{16} +107.010 q^{17} +48.7496 q^{19} +24.4606 q^{20} +39.4962 q^{22} -11.9498 q^{23} +25.0000 q^{25} +194.326 q^{26} -78.9905 q^{28} -239.733 q^{29} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} -21.7573 q^{37} -175.038 q^{38} +55.7952 q^{40} +124.835 q^{41} +224.459 q^{43} -53.8133 q^{44} +42.9064 q^{46} +186.832 q^{47} -82.2913 q^{49} -89.7640 q^{50} -264.768 q^{52} -233.997 q^{53} -55.0000 q^{55} -180.179 q^{56} +860.774 q^{58} -232.936 q^{59} +163.849 q^{61} +294.731 q^{62} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} +289.874 q^{70} +733.141 q^{71} +1161.97 q^{73} +78.1208 q^{74} +238.489 q^{76} +177.611 q^{77} -588.831 q^{79} -396.021 q^{80} -448.226 q^{82} +1161.06 q^{83} +535.048 q^{85} -805.933 q^{86} -122.749 q^{88} +1042.16 q^{89} +873.869 q^{91} -58.4597 q^{92} -670.831 q^{94} +243.748 q^{95} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8}+O(q^{10}) $ 3 * q + 4 * q^2 + 22 * q^4 + 15 * q^5 - 4 * q^7 + 48 * q^8	$ 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8} + 20 q^{10} - 33 q^{11} + 56 q^{14} + 50 q^{16} + 218 q^{17} + 146 q^{19} + 110 q^{20} - 44 q^{22} + 200 q^{23} + 75 q^{25} + 508 q^{26} - 340 q^{28} - 68 q^{29} - 68 q^{31} + 688 q^{32} - 176 q^{34} - 20 q^{35} - 390 q^{37} + 316 q^{38} + 240 q^{40} + 196 q^{41} - 524 q^{43} - 242 q^{44} + 1160 q^{46} + 60 q^{47} - 157 q^{49} + 100 q^{50} + 1020 q^{52} + 158 q^{53} - 165 q^{55} - 1368 q^{56} + 1092 q^{58} + 1044 q^{59} + 642 q^{61} - 88 q^{62} + 1166 q^{64} - 236 q^{67} - 144 q^{68} + 280 q^{70} + 544 q^{71} + 900 q^{73} - 1536 q^{74} + 1996 q^{76} + 44 q^{77} - 1586 q^{79} + 250 q^{80} + 380 q^{82} + 1582 q^{83} + 1090 q^{85} - 3568 q^{86} - 528 q^{88} + 2122 q^{89} - 8 q^{91} + 4128 q^{92} - 2152 q^{94} + 730 q^{95} + 618 q^{97} - 572 q^{98}+O(q^{100}) $ 3 * q + 4 * q^2 + 22 * q^4 + 15 * q^5 - 4 * q^7 + 48 * q^8 + 20 * q^10 - 33 * q^11 + 56 * q^14 + 50 * q^16 + 218 * q^17 + 146 * q^19 + 110 * q^20 - 44 * q^22 + 200 * q^23 + 75 * q^25 + 508 * q^26 - 340 * q^28 - 68 * q^29 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 - 390 * q^37 + 316 * q^38 + 240 * q^40 + 196 * q^41 - 524 * q^43 - 242 * q^44 + 1160 * q^46 + 60 * q^47 - 157 * q^49 + 100 * q^50 + 1020 * q^52 + 158 * q^53 - 165 * q^55 - 1368 * q^56 + 1092 * q^58 + 1044 * q^59 + 642 * q^61 - 88 * q^62 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 280 * q^70 + 544 * q^71 + 900 * q^73 - 1536 * q^74 + 1996 * q^76 + 44 * q^77 - 1586 * q^79 + 250 * q^80 + 380 * q^82 + 1582 * q^83 + 1090 * q^85 - 3568 * q^86 - 528 * q^88 + 2122 * q^89 - 8 * q^91 + 4128 * q^92 - 2152 * q^94 + 730 * q^95 + 618 * q^97 - 572 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.59056	−1.26945	−0.634727	−	0.772736i	$-0.718888\pi$
$2$	−3.59056	−1.26945	−0.634727	+	0.772736i	$0.718888\pi$
$3$	0	0
$3$	0	0
$4$	4.89212	0.611515
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−16.1465	−0.871828	−0.435914	−	0.899988i	$-0.643575\pi$
$7$	−16.1465	−0.871828	−0.435914	+	0.899988i	$0.643575\pi$
$8$	11.1590	0.493164
$9$	0	0
$10$	−17.9528	−0.567717
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−54.1214	−1.15466	−0.577329	−	0.816511i	$-0.695905\pi$
$13$	−54.1214	−1.15466	−0.577329	+	0.816511i	$0.695905\pi$
$14$	57.9749	1.10675
$15$	0	0
$16$	−79.2041	−1.23756
$17$	107.010	1.52668	0.763342	−	0.645995i	$-0.223557\pi$
$17$	107.010	1.52668	0.763342	+	0.645995i	$0.223557\pi$
$18$	0	0
$19$	48.7496	0.588628	0.294314	−	0.955709i	$-0.404909\pi$
$19$	48.7496	0.588628	0.294314	+	0.955709i	$0.404909\pi$
$20$	24.4606	0.273478
$21$	0	0
$22$	39.4962	0.382755
$23$	−11.9498	−0.108335	−0.0541674	−	0.998532i	$-0.517250\pi$
$23$	−11.9498	−0.108335	−0.0541674	+	0.998532i	$0.517250\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	194.326	1.46579
$27$	0	0
$28$	−78.9905	−0.533136
$29$	−239.733	−1.53508	−0.767538	−	0.641003i	$-0.778519\pi$
$29$	−239.733	−1.53508	−0.767538	+	0.641003i	$0.778519\pi$
$30$	0	0
$31$	−82.0851	−0.475578	−0.237789	−	0.971317i	$-0.576423\pi$
$31$	−82.0851	−0.475578	−0.237789	+	0.971317i	$0.576423\pi$
$32$	195.115	1.07787
$33$	0	0
$34$	−384.224	−1.93806
$35$	−80.7324	−0.389893
$36$	0	0
$37$	−21.7573	−0.0966723	−0.0483361	−	0.998831i	$-0.515392\pi$
$37$	−21.7573	−0.0966723	−0.0483361	+	0.998831i	$0.515392\pi$
$38$	−175.038	−0.747236
$39$	0	0
$40$	55.7952	0.220550
$41$	124.835	0.475510	0.237755	−	0.971325i	$-0.423588\pi$
$41$	124.835	0.475510	0.237755	+	0.971325i	$0.423588\pi$
$42$	0	0
$43$	224.459	0.796039	0.398019	−	0.917377i	$-0.369698\pi$
$43$	224.459	0.796039	0.398019	+	0.917377i	$0.369698\pi$
$44$	−53.8133	−0.184379
$45$	0	0
$46$	42.9064	0.137526
$47$	186.832	0.579835	0.289917	−	0.957052i	$-0.406372\pi$
$47$	186.832	0.579835	0.289917	+	0.957052i	$0.406372\pi$
$48$	0	0
$49$	−82.2913	−0.239916
$50$	−89.7640	−0.253891
$51$	0	0
$52$	−264.768	−0.706091
$53$	−233.997	−0.606453	−0.303226	−	0.952919i	$-0.598064\pi$
$53$	−233.997	−0.606453	−0.303226	+	0.952919i	$0.598064\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	−180.179	−0.429954
$57$	0	0
$58$	860.774	1.94871
$59$	−232.936	−0.513996	−0.256998	−	0.966412i	$-0.582733\pi$
$59$	−232.936	−0.513996	−0.256998	+	0.966412i	$0.582733\pi$
$60$	0	0
$61$	163.849	0.343913	0.171957	−	0.985105i	$-0.444991\pi$
$61$	163.849	0.343913	0.171957	+	0.985105i	$0.444991\pi$
$62$	294.731	0.603725
$63$	0	0
$64$	−66.9386	−0.130739
$65$	−270.607	−0.516379
$66$	0	0
$67$	−876.918	−1.59899	−0.799497	−	0.600670i	$-0.794900\pi$
$67$	−876.918	−1.59899	−0.799497	+	0.600670i	$0.794900\pi$
$68$	523.503	0.933590
$69$	0	0
$70$	289.874	0.494952
$71$	733.141	1.22546	0.612731	−	0.790291i	$-0.290071\pi$
$71$	733.141	1.22546	0.612731	+	0.790291i	$0.290071\pi$
$72$	0	0
$73$	1161.97	1.86299	0.931496	−	0.363750i	$-0.118504\pi$
$73$	1161.97	1.86299	0.931496	+	0.363750i	$0.118504\pi$
$74$	78.1208	0.122721
$75$	0	0
$76$	238.489	0.359954
$77$	177.611	0.262866
$78$	0	0
$79$	−588.831	−0.838591	−0.419296	−	0.907850i	$-0.637723\pi$
$79$	−588.831	−0.838591	−0.419296	+	0.907850i	$0.637723\pi$
$80$	−396.021	−0.553456
$81$	0	0
$82$	−448.226	−0.603638
$83$	1161.06	1.53546	0.767731	−	0.640772i	$-0.221386\pi$
$83$	1161.06	1.53546	0.767731	+	0.640772i	$0.221386\pi$
$84$	0	0
$85$	535.048	0.682754
$86$	−805.933	−1.01053
$87$	0	0
$88$	−122.749	−0.148695
$89$	1042.16	1.24122	0.620610	−	0.784120i	$-0.286885\pi$
$89$	1042.16	1.24122	0.620610	+	0.784120i	$0.286885\pi$
$90$	0	0
$91$	873.869	1.00666
$92$	−58.4597	−0.0662483
$93$	0	0
$94$	−670.831	−0.736074
$95$	243.748	0.263242
$96$	0	0
$97$	1546.63	1.61893	0.809464	−	0.587169i	$-0.199758\pi$
$97$	1546.63	1.61893	0.809464	+	0.587169i	$0.199758\pi$
$98$	295.472	0.304563
$99$	0	0
$100$	122.303	0.122303
$101$	662.282	0.652470	0.326235	−	0.945289i	$-0.394220\pi$
$101$	662.282	0.652470	0.326235	+	0.945289i	$0.394220\pi$
$102$	0	0
$103$	399.592	0.382262	0.191131	−	0.981565i	$-0.438784\pi$
$103$	399.592	0.382262	0.191131	+	0.981565i	$0.438784\pi$
$104$	−603.942	−0.569436
$105$	0	0
$106$	840.181	0.769864
$107$	1591.22	1.43765	0.718827	−	0.695189i	$-0.244679\pi$
$107$	1591.22	1.43765	0.718827	+	0.695189i	$0.244679\pi$
$108$	0	0
$109$	755.128	0.663561	0.331780	−	0.943357i	$-0.392351\pi$
$109$	755.128	0.663561	0.331780	+	0.943357i	$0.392351\pi$
$110$	197.481	0.171173
$111$	0	0
$112$	1278.87	1.07894
$113$	1145.65	0.953753	0.476876	−	0.878970i	$-0.341769\pi$
$113$	1145.65	0.953753	0.476876	+	0.878970i	$0.341769\pi$
$114$	0	0
$115$	−59.7488	−0.0484488
$116$	−1172.80	−0.938722
$117$	0	0
$118$	836.372	0.652494
$119$	−1727.83	−1.33101
$120$	0	0
$121$	121.000	0.0909091
$122$	−588.309	−0.436582
$123$	0	0
$124$	−401.570	−0.290823
$125$	125.000	0.0894427
$126$	0	0
$127$	1461.29	1.02101	0.510505	−	0.859875i	$-0.329458\pi$
$127$	1461.29	1.02101	0.510505	+	0.859875i	$0.329458\pi$
$128$	−1320.57	−0.911900
$129$	0	0
$130$	971.630	0.655520
$131$	1524.94	1.01706	0.508528	−	0.861045i	$-0.330190\pi$
$131$	1524.94	1.01706	0.508528	+	0.861045i	$0.330190\pi$
$132$	0	0
$133$	−787.134	−0.513182
$134$	3148.63	2.02985
$135$	0	0
$136$	1194.12	0.752906
$137$	2125.68	1.32561	0.662805	−	0.748792i	$-0.269366\pi$
$137$	2125.68	1.32561	0.662805	+	0.748792i	$0.269366\pi$
$138$	0	0
$139$	−1774.28	−1.08268	−0.541339	−	0.840805i	$-0.682082\pi$
$139$	−1774.28	−1.08268	−0.541339	+	0.840805i	$0.682082\pi$
$140$	−394.952	−0.238425
$141$	0	0
$142$	−2632.39	−1.55567
$143$	595.335	0.348143
$144$	0	0
$145$	−1198.66	−0.686507
$146$	−4172.13	−2.36498
$147$	0	0
$148$	−106.439	−0.0591165
$149$	−1575.78	−0.866393	−0.433197	−	0.901299i	$-0.642614\pi$
$149$	−1575.78	−0.866393	−0.433197	+	0.901299i	$0.642614\pi$
$150$	0	0
$151$	420.978	0.226879	0.113439	−	0.993545i	$-0.463813\pi$
$151$	420.978	0.226879	0.113439	+	0.993545i	$0.463813\pi$
$152$	543.998	0.290290
$153$	0	0
$154$	−637.724	−0.333696
$155$	−410.425	−0.212685
$156$	0	0
$157$	−2224.30	−1.13069	−0.565345	−	0.824854i	$-0.691257\pi$
$157$	−2224.30	−1.13069	−0.565345	+	0.824854i	$0.691257\pi$
$158$	2114.23	1.06455
$159$	0	0
$160$	975.574	0.482037
$161$	192.947	0.0944492
$162$	0	0
$163$	−3093.37	−1.48645	−0.743226	−	0.669040i	$-0.766705\pi$
$163$	−3093.37	−1.48645	−0.743226	+	0.669040i	$0.766705\pi$
$164$	610.706	0.290781
$165$	0	0
$166$	−4168.87	−1.94920
$167$	2416.43	1.11970	0.559848	−	0.828595i	$-0.310860\pi$
$167$	2416.43	1.11970	0.559848	+	0.828595i	$0.310860\pi$
$168$	0	0
$169$	732.122	0.333237
$170$	−1921.12	−0.866725
$171$	0	0
$172$	1098.08	0.486789
$173$	3758.02	1.65154	0.825771	−	0.564005i	$-0.190740\pi$
$173$	3758.02	1.65154	0.825771	+	0.564005i	$0.190740\pi$
$174$	0	0
$175$	−403.662	−0.174366
$176$	871.245	0.373140
$177$	0	0
$178$	−3741.93	−1.57567
$179$	−2533.99	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$179$	−2533.99	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$180$	0	0
$181$	−13.8995	−0.00570798	−0.00285399	−	0.999996i	$-0.500908\pi$
$181$	−13.8995	−0.00570798	−0.00285399	+	0.999996i	$0.500908\pi$
$182$	−3137.68	−1.27791
$183$	0	0
$184$	−133.348	−0.0534268
$185$	−108.786	−0.0432332
$186$	0	0
$187$	−1177.10	−0.460312
$188$	914.004	0.354577
$189$	0	0
$190$	−875.192	−0.334174
$191$	−3495.39	−1.32417	−0.662087	−	0.749427i	$-0.730329\pi$
$191$	−3495.39	−1.32417	−0.662087	+	0.749427i	$0.730329\pi$
$192$	0	0
$193$	3469.33	1.29393	0.646963	−	0.762522i	$-0.276039\pi$
$193$	3469.33	1.29393	0.646963	+	0.762522i	$0.276039\pi$
$194$	−5553.25	−2.05516
$195$	0	0
$196$	−402.579	−0.146712
$197$	−3638.39	−1.31586	−0.657930	−	0.753079i	$-0.728568\pi$
$197$	−3638.39	−1.31586	−0.657930	+	0.753079i	$0.728568\pi$
$198$	0	0
$199$	51.6049	0.0183828	0.00919140	−	0.999958i	$-0.497074\pi$
$199$	51.6049	0.0183828	0.00919140	+	0.999958i	$0.497074\pi$
$200$	278.976	0.0986329
$201$	0	0
$202$	−2377.96	−0.828281
$203$	3870.84	1.33832
$204$	0	0
$205$	624.173	0.212654
$206$	−1434.76	−0.485265
$207$	0	0
$208$	4286.63	1.42896
$209$	−536.246	−0.177478
$210$	0	0
$211$	2084.57	0.680131	0.340065	−	0.940402i	$-0.389551\pi$
$211$	2084.57	0.680131	0.340065	+	0.940402i	$0.389551\pi$
$212$	−1144.74	−0.370855
$213$	0	0
$214$	−5713.37	−1.82504
$215$	1122.29	0.355999
$216$	0	0
$217$	1325.38	0.414622
$218$	−2711.33	−0.842361
$219$	0	0
$220$	−269.067	−0.0824566
$221$	−5791.50	−1.76280
$222$	0	0
$223$	1887.36	0.566757	0.283378	−	0.959008i	$-0.408545\pi$
$223$	1887.36	0.566757	0.283378	+	0.959008i	$0.408545\pi$
$224$	−3150.42	−0.939715
$225$	0	0
$226$	−4113.54	−1.21075
$227$	1150.73	0.336462	0.168231	−	0.985748i	$-0.446194\pi$
$227$	1150.73	0.336462	0.168231	+	0.985748i	$0.446194\pi$
$228$	0	0
$229$	4106.79	1.18508	0.592542	−	0.805540i	$-0.298125\pi$
$229$	4106.79	1.18508	0.592542	+	0.805540i	$0.298125\pi$
$230$	214.532	0.0615035
$231$	0	0
$232$	−2675.18	−0.757045
$233$	5733.58	1.61210	0.806050	−	0.591848i	$-0.201601\pi$
$233$	5733.58	1.61210	0.806050	+	0.591848i	$0.201601\pi$
$234$	0	0
$235$	934.159	0.259310
$236$	−1139.55	−0.314316
$237$	0	0
$238$	6203.86	1.68965
$239$	−6036.18	−1.63367	−0.816837	−	0.576868i	$-0.804275\pi$
$239$	−6036.18	−1.63367	−0.816837	+	0.576868i	$0.804275\pi$
$240$	0	0
$241$	3720.90	0.994540	0.497270	−	0.867596i	$-0.334336\pi$
$241$	3720.90	0.994540	0.497270	+	0.867596i	$0.334336\pi$
$242$	−434.458	−0.115405
$243$	0	0
$244$	801.568	0.210308
$245$	−411.457	−0.107294
$246$	0	0
$247$	−2638.39	−0.679664
$248$	−915.990	−0.234538
$249$	0	0
$250$	−448.820	−0.113543
$251$	3809.88	0.958077	0.479038	−	0.877794i	$-0.340985\pi$
$251$	3809.88	0.958077	0.479038	+	0.877794i	$0.340985\pi$
$252$	0	0
$253$	131.447	0.0326641
$254$	−5246.84	−1.29613
$255$	0	0
$256$	5277.10	1.28835
$257$	−1225.95	−0.297559	−0.148780	−	0.988870i	$-0.547535\pi$
$257$	−1225.95	−0.297559	−0.148780	+	0.988870i	$0.547535\pi$
$258$	0	0
$259$	351.303	0.0842816
$260$	−1323.84	−0.315773
$261$	0	0
$262$	−5475.38	−1.29111
$263$	−7397.68	−1.73445	−0.867225	−	0.497916i	$-0.834099\pi$
$263$	−7397.68	−1.73445	−0.867225	+	0.497916i	$0.834099\pi$
$264$	0	0
$265$	−1169.99	−0.271214
$266$	2826.25	0.651461
$267$	0	0
$268$	−4289.99	−0.977808
$269$	3214.48	0.728589	0.364295	−	0.931284i	$-0.381310\pi$
$269$	3214.48	0.728589	0.364295	+	0.931284i	$0.381310\pi$
$270$	0	0
$271$	−7377.37	−1.65367	−0.826833	−	0.562448i	$-0.809860\pi$
$271$	−7377.37	−1.65367	−0.826833	+	0.562448i	$0.809860\pi$
$272$	−8475.60	−1.88937
$273$	0	0
$274$	−7632.36	−1.68280
$275$	−275.000	−0.0603023
$276$	0	0
$277$	−810.606	−0.175829	−0.0879144	−	0.996128i	$-0.528020\pi$
$277$	−810.606	−0.175829	−0.0879144	+	0.996128i	$0.528020\pi$
$278$	6370.65	1.37441
$279$	0	0
$280$	−900.895	−0.192281
$281$	−1114.72	−0.236651	−0.118325	−	0.992975i	$-0.537753\pi$
$281$	−1114.72	−0.236651	−0.118325	+	0.992975i	$0.537753\pi$
$282$	0	0
$283$	2265.40	0.475844	0.237922	−	0.971284i	$-0.423534\pi$
$283$	2265.40	0.475844	0.237922	+	0.971284i	$0.423534\pi$
$284$	3586.61	0.749388
$285$	0	0
$286$	−2137.59	−0.441951
$287$	−2015.64	−0.414563
$288$	0	0
$289$	6538.04	1.33076
$290$	4303.87	0.871490
$291$	0	0
$292$	5684.50	1.13925
$293$	−3802.06	−0.758084	−0.379042	−	0.925380i	$-0.623746\pi$
$293$	−3802.06	−0.758084	−0.379042	+	0.925380i	$0.623746\pi$
$294$	0	0
$295$	−1164.68	−0.229866
$296$	−242.790	−0.0476753
$297$	0	0
$298$	5657.92	1.09985
$299$	646.738	0.125090
$300$	0	0
$301$	−3624.22	−0.694009
$302$	−1511.55	−0.288013
$303$	0	0
$304$	−3861.17	−0.728465
$305$	819.244	0.153803
$306$	0	0
$307$	−1356.35	−0.252153	−0.126077	−	0.992021i	$-0.540239\pi$
$307$	−1356.35	−0.252153	−0.126077	+	0.992021i	$0.540239\pi$
$308$	868.895	0.160746
$309$	0	0
$310$	1473.66	0.269994
$311$	8078.07	1.47288	0.736440	−	0.676503i	$-0.236505\pi$
$311$	8078.07	1.47288	0.736440	+	0.676503i	$0.236505\pi$
$312$	0	0
$313$	5761.54	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$313$	5761.54	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$314$	7986.48	1.43536
$315$	0	0
$316$	−2880.63	−0.512811
$317$	−5107.00	−0.904851	−0.452426	−	0.891802i	$-0.649441\pi$
$317$	−5107.00	−0.904851	−0.452426	+	0.891802i	$0.649441\pi$
$318$	0	0
$319$	2637.06	0.462843
$320$	−334.693	−0.0584684
$321$	0	0
$322$	−692.786	−0.119899
$323$	5216.67	0.898648
$324$	0	0
$325$	−1353.03	−0.230932
$326$	11106.9	1.88698
$327$	0	0
$328$	1393.03	0.234504
$329$	−3016.68	−0.505516
$330$	0	0
$331$	−2780.94	−0.461796	−0.230898	−	0.972978i	$-0.574166\pi$
$331$	−2780.94	−0.461796	−0.230898	+	0.972978i	$0.574166\pi$
$332$	5680.06	0.938958
$333$	0	0
$334$	−8676.35	−1.42140
$335$	−4384.59	−0.715092
$336$	0	0
$337$	4939.28	0.798397	0.399198	−	0.916865i	$-0.369288\pi$
$337$	4939.28	0.798397	0.399198	+	0.916865i	$0.369288\pi$
$338$	−2628.73	−0.423029
$339$	0	0
$340$	2617.52	0.417514
$341$	902.936	0.143392
$342$	0	0
$343$	6866.96	1.08099
$344$	2504.74	0.392578
$345$	0	0
$346$	−13493.4	−2.09656
$347$	2711.58	0.419496	0.209748	−	0.977755i	$-0.432736\pi$
$347$	2711.58	0.419496	0.209748	+	0.977755i	$0.432736\pi$
$348$	0	0
$349$	5496.03	0.842967	0.421484	−	0.906836i	$-0.361510\pi$
$349$	5496.03	0.842967	0.421484	+	0.906836i	$0.361510\pi$
$350$	1449.37	0.221349
$351$	0	0
$352$	−2146.26	−0.324989
$353$	−6372.23	−0.960792	−0.480396	−	0.877052i	$-0.659507\pi$
$353$	−6372.23	−0.960792	−0.480396	+	0.877052i	$0.659507\pi$
$354$	0	0
$355$	3665.71	0.548043
$356$	5098.36	0.759024
$357$	0	0
$358$	9098.46	1.34321
$359$	−630.622	−0.0927101	−0.0463551	−	0.998925i	$-0.514761\pi$
$359$	−630.622	−0.0927101	−0.0463551	+	0.998925i	$0.514761\pi$
$360$	0	0
$361$	−4482.48	−0.653518
$362$	49.9071	0.00724602
$363$	0	0
$364$	4275.07	0.615590
$365$	5809.86	0.833156
$366$	0	0
$367$	−5374.49	−0.764431	−0.382216	−	0.924073i	$-0.624839\pi$
$367$	−5374.49	−0.764431	−0.382216	+	0.924073i	$0.624839\pi$
$368$	946.471	0.134071
$369$	0	0
$370$	390.604	0.0548825
$371$	3778.23	0.528722
$372$	0	0
$373$	7520.12	1.04391	0.521953	−	0.852974i	$-0.325204\pi$
$373$	7520.12	1.04391	0.521953	+	0.852974i	$0.325204\pi$
$374$	4226.46	0.584346
$375$	0	0
$376$	2084.86	0.285954
$377$	12974.7	1.77249
$378$	0	0
$379$	−12509.1	−1.69538	−0.847690	−	0.530492i	$-0.822007\pi$
$379$	−12509.1	−1.69538	−0.847690	+	0.530492i	$0.822007\pi$
$380$	1192.44	0.160977
$381$	0	0
$382$	12550.4	1.68098
$383$	11149.9	1.48755	0.743775	−	0.668430i	$-0.233033\pi$
$383$	11149.9	1.48755	0.743775	+	0.668430i	$0.233033\pi$
$384$	0	0
$385$	888.056	0.117557
$386$	−12456.8	−1.64258
$387$	0	0
$388$	7566.28	0.989999
$389$	−3194.22	−0.416333	−0.208166	−	0.978093i	$-0.566750\pi$
$389$	−3194.22	−0.416333	−0.208166	+	0.978093i	$0.566750\pi$
$390$	0	0
$391$	−1278.74	−0.165393
$392$	−918.292	−0.118318
$393$	0	0
$394$	13063.8	1.67042
$395$	−2944.16	−0.375029
$396$	0	0
$397$	−584.410	−0.0738809	−0.0369404	−	0.999317i	$-0.511761\pi$
$397$	−584.410	−0.0738809	−0.0369404	+	0.999317i	$0.511761\pi$
$398$	−185.291	−0.0233361
$399$	0	0
$400$	−1980.10	−0.247513
$401$	6951.24	0.865657	0.432829	−	0.901476i	$-0.357516\pi$
$401$	6951.24	0.865657	0.432829	+	0.901476i	$0.357516\pi$
$402$	0	0
$403$	4442.56	0.549130
$404$	3239.96	0.398995
$405$	0	0
$406$	−13898.5	−1.69894
$407$	239.330	0.0291478
$408$	0	0
$409$	11754.1	1.42104	0.710519	−	0.703678i	$-0.248460\pi$
$409$	11754.1	1.42104	0.710519	+	0.703678i	$0.248460\pi$
$410$	−2241.13	−0.269955
$411$	0	0
$412$	1954.85	0.233759
$413$	3761.10	0.448116
$414$	0	0
$415$	5805.32	0.686680
$416$	−10559.9	−1.24457
$417$	0	0
$418$	1925.42	0.225300
$419$	−11829.8	−1.37929	−0.689646	−	0.724146i	$-0.742234\pi$
$419$	−11829.8	−1.37929	−0.689646	+	0.724146i	$0.742234\pi$
$420$	0	0
$421$	−2000.07	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$421$	−2000.07	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$422$	−7484.76	−0.863395
$423$	0	0
$424$	−2611.18	−0.299081
$425$	2675.24	0.305337
$426$	0	0
$427$	−2645.58	−0.299833
$428$	7784.43	0.879147
$429$	0	0
$430$	−4029.67	−0.451925
$431$	8064.11	0.901240	0.450620	−	0.892716i	$-0.351203\pi$
$431$	8064.11	0.901240	0.450620	+	0.892716i	$0.351203\pi$
$432$	0	0
$433$	−10710.3	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$433$	−10710.3	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$434$	−4758.87	−0.526344
$435$	0	0
$436$	3694.18	0.405777
$437$	−582.546	−0.0637688
$438$	0	0
$439$	4658.08	0.506419	0.253210	−	0.967411i	$-0.418514\pi$
$439$	4658.08	0.506419	0.253210	+	0.967411i	$0.418514\pi$
$440$	−613.747	−0.0664983
$441$	0	0
$442$	20794.7	2.23779
$443$	−2094.73	−0.224658	−0.112329	−	0.993671i	$-0.535831\pi$
$443$	−2094.73	−0.224658	−0.112329	+	0.993671i	$0.535831\pi$
$444$	0	0
$445$	5210.79	0.555090
$446$	−6776.67	−0.719472
$447$	0	0
$448$	1080.82	0.113982
$449$	1402.21	0.147382	0.0736909	−	0.997281i	$-0.476522\pi$
$449$	1402.21	0.147382	0.0736909	+	0.997281i	$0.476522\pi$
$450$	0	0
$451$	−1373.18	−0.143372
$452$	5604.67	0.583234
$453$	0	0
$454$	−4131.78	−0.427124
$455$	4369.35	0.450194
$456$	0	0
$457$	−4914.19	−0.503011	−0.251506	−	0.967856i	$-0.580926\pi$
$457$	−4914.19	−0.503011	−0.251506	+	0.967856i	$0.580926\pi$
$458$	−14745.7	−1.50441
$459$	0	0
$460$	−292.298	−0.0296271
$461$	2214.08	0.223688	0.111844	−	0.993726i	$-0.464324\pi$
$461$	2214.08	0.223688	0.111844	+	0.993726i	$0.464324\pi$
$462$	0	0
$463$	−5567.02	−0.558793	−0.279396	−	0.960176i	$-0.590134\pi$
$463$	−5567.02	−0.558793	−0.279396	+	0.960176i	$0.590134\pi$
$464$	18987.8	1.89976
$465$	0	0
$466$	−20586.8	−2.04649
$467$	−497.054	−0.0492525	−0.0246263	−	0.999697i	$-0.507840\pi$
$467$	−497.054	−0.0492525	−0.0246263	+	0.999697i	$0.507840\pi$
$468$	0	0
$469$	14159.1	1.39405
$470$	−3354.15	−0.329182
$471$	0	0
$472$	−2599.35	−0.253484
$473$	−2469.05	−0.240015
$474$	0	0
$475$	1218.74	0.117726
$476$	−8452.73	−0.813929
$477$	0	0
$478$	21673.3	2.07388
$479$	9349.28	0.891815	0.445908	−	0.895079i	$-0.352881\pi$
$479$	9349.28	0.891815	0.445908	+	0.895079i	$0.352881\pi$
$480$	0	0
$481$	1177.53	0.111624
$482$	−13360.1	−1.26252
$483$	0	0
$484$	591.946	0.0555923
$485$	7733.13	0.724007
$486$	0	0
$487$	197.750	0.0184003	0.00920013	−	0.999958i	$-0.497071\pi$
$487$	197.750	0.0184003	0.00920013	+	0.999958i	$0.497071\pi$
$488$	1828.40	0.169606
$489$	0	0
$490$	1477.36	0.136205
$491$	9997.05	0.918861	0.459430	−	0.888214i	$-0.348054\pi$
$491$	9997.05	0.918861	0.459430	+	0.888214i	$0.348054\pi$
$492$	0	0
$493$	−25653.7	−2.34358
$494$	9473.31	0.862803
$495$	0	0
$496$	6501.48	0.588558
$497$	−11837.6	−1.06839
$498$	0	0
$499$	−8714.73	−0.781813	−0.390907	−	0.920430i	$-0.627838\pi$
$499$	−8714.73	−0.781813	−0.390907	+	0.920430i	$0.627838\pi$
$500$	611.515	0.0546955
$501$	0	0
$502$	−13679.6	−1.21623
$503$	−5978.53	−0.529959	−0.264979	−	0.964254i	$-0.585365\pi$
$503$	−5978.53	−0.529959	−0.264979	+	0.964254i	$0.585365\pi$
$504$	0	0
$505$	3311.41	0.291794
$506$	−471.970	−0.0414657
$507$	0	0
$508$	7148.79	0.624363
$509$	8205.79	0.714569	0.357284	−	0.933996i	$-0.383703\pi$
$509$	8205.79	0.714569	0.357284	+	0.933996i	$0.383703\pi$
$510$	0	0
$511$	−18761.7	−1.62421
$512$	−8383.17	−0.723608
$513$	0	0
$514$	4401.85	0.377738
$515$	1997.96	0.170953
$516$	0	0
$517$	−2055.15	−0.174827
$518$	−1261.38	−0.106992
$519$	0	0
$520$	−3019.71	−0.254660
$521$	5266.06	0.442822	0.221411	−	0.975181i	$-0.428934\pi$
$521$	5266.06	0.442822	0.221411	+	0.975181i	$0.428934\pi$
$522$	0	0
$523$	−22398.6	−1.87270	−0.936350	−	0.351068i	$-0.885819\pi$
$523$	−22398.6	−1.87270	−0.936350	+	0.351068i	$0.885819\pi$
$524$	7460.18	0.621945
$525$	0	0
$526$	26561.8	2.20181
$527$	−8783.89	−0.726057
$528$	0	0
$529$	−12024.2	−0.988264
$530$	4200.90	0.344294
$531$	0	0
$532$	−3850.75	−0.313818
$533$	−6756.22	−0.549052
$534$	0	0
$535$	7956.10	0.642938
$536$	−9785.56	−0.788566
$537$	0	0
$538$	−11541.8	−0.924911
$539$	905.205	0.0723375
$540$	0	0
$541$	−13030.2	−1.03551	−0.517757	−	0.855528i	$-0.673233\pi$
$541$	−13030.2	−1.03551	−0.517757	+	0.855528i	$0.673233\pi$
$542$	26488.9	2.09925
$543$	0	0
$544$	20879.1	1.64556
$545$	3775.64	0.296753
$546$	0	0
$547$	10448.9	0.816747	0.408374	−	0.912815i	$-0.366096\pi$
$547$	10448.9	0.816747	0.408374	+	0.912815i	$0.366096\pi$
$548$	10399.1	0.810631
$549$	0	0
$550$	987.404	0.0765510
$551$	−11686.9	−0.903588
$552$	0	0
$553$	9507.55	0.731107
$554$	2910.53	0.223207
$555$	0	0
$556$	−8679.97	−0.662073
$557$	11448.7	0.870911	0.435456	−	0.900210i	$-0.356587\pi$
$557$	11448.7	0.870911	0.435456	+	0.900210i	$0.356587\pi$
$558$	0	0
$559$	−12148.0	−0.919153
$560$	6394.34	0.482518
$561$	0	0
$562$	4002.49	0.300418
$563$	26035.8	1.94898	0.974492	−	0.224422i	$-0.0720494\pi$
$563$	26035.8	1.94898	0.974492	+	0.224422i	$0.0720494\pi$
$564$	0	0
$565$	5728.27	0.426531
$566$	−8134.05	−0.604063
$567$	0	0
$568$	8181.15	0.604354
$569$	−25075.0	−1.84745	−0.923725	−	0.383057i	$-0.874871\pi$
$569$	−25075.0	−1.84745	−0.923725	+	0.383057i	$0.874871\pi$
$570$	0	0
$571$	19056.6	1.39666	0.698330	−	0.715776i	$-0.253927\pi$
$571$	19056.6	1.39666	0.698330	+	0.715776i	$0.253927\pi$
$572$	2912.45	0.212894
$573$	0	0
$574$	7237.28	0.526268
$575$	−298.744	−0.0216669
$576$	0	0
$577$	−6482.55	−0.467716	−0.233858	−	0.972271i	$-0.575135\pi$
$577$	−6482.55	−0.467716	−0.233858	+	0.972271i	$0.575135\pi$
$578$	−23475.2	−1.68934
$579$	0	0
$580$	−5864.00	−0.419809
$581$	−18747.1	−1.33866
$582$	0	0
$583$	2573.97	0.182852
$584$	12966.5	0.918762
$585$	0	0
$586$	13651.5	0.962353
$587$	−24650.3	−1.73327	−0.866633	−	0.498946i	$-0.833720\pi$
$587$	−24650.3	−1.73327	−0.866633	+	0.498946i	$0.833720\pi$
$588$	0	0
$589$	−4001.61	−0.279938
$590$	4181.86	0.291804
$591$	0	0
$592$	1723.27	0.119638
$593$	−23163.2	−1.60405	−0.802024	−	0.597292i	$-0.796243\pi$
$593$	−23163.2	−1.60405	−0.802024	+	0.597292i	$0.796243\pi$
$594$	0	0
$595$	−8639.13	−0.595244
$596$	−7708.88	−0.529812
$597$	0	0
$598$	−2322.15	−0.158796
$599$	−5355.44	−0.365304	−0.182652	−	0.983178i	$-0.558468\pi$
$599$	−5355.44	−0.365304	−0.182652	+	0.983178i	$0.558468\pi$
$600$	0	0
$601$	−20417.6	−1.38578	−0.692889	−	0.721045i	$-0.743662\pi$
$601$	−20417.6	−1.38578	−0.692889	+	0.721045i	$0.743662\pi$
$602$	13013.0	0.881012
$603$	0	0
$604$	2059.48	0.138740
$605$	605.000	0.0406558
$606$	0	0
$607$	−6749.81	−0.451345	−0.225673	−	0.974203i	$-0.572458\pi$
$607$	−6749.81	−0.451345	−0.225673	+	0.974203i	$0.572458\pi$
$608$	9511.77	0.634463
$609$	0	0
$610$	−2941.55	−0.195245
$611$	−10111.6	−0.669511
$612$	0	0
$613$	−30321.0	−1.99780	−0.998901	−	0.0468613i	$-0.985078\pi$
$613$	−30321.0	−1.99780	−0.998901	+	0.0468613i	$0.985078\pi$
$614$	4870.06	0.320097
$615$	0	0
$616$	1981.97	0.129636
$617$	−15236.8	−0.994182	−0.497091	−	0.867699i	$-0.665598\pi$
$617$	−15236.8	−0.994182	−0.497091	+	0.867699i	$0.665598\pi$
$618$	0	0
$619$	−20875.4	−1.35550	−0.677749	−	0.735293i	$-0.737044\pi$
$619$	−20875.4	−1.35550	−0.677749	+	0.735293i	$0.737044\pi$
$620$	−2007.85	−0.130060
$621$	0	0
$622$	−29004.8	−1.86975
$623$	−16827.2	−1.08213
$624$	0	0
$625$	625.000	0.0400000
$626$	−20687.2	−1.32081
$627$	0	0
$628$	−10881.5	−0.691434
$629$	−2328.24	−0.147588
$630$	0	0
$631$	27966.1	1.76437	0.882183	−	0.470907i	$-0.156073\pi$
$631$	27966.1	1.76437	0.882183	+	0.470907i	$0.156073\pi$
$632$	−6570.79	−0.413563
$633$	0	0
$634$	18337.0	1.14867
$635$	7306.44	0.456610
$636$	0	0
$637$	4453.72	0.277022
$638$	−9468.52	−0.587558
$639$	0	0
$640$	−6602.86	−0.407814
$641$	17992.0	1.10865	0.554323	−	0.832301i	$-0.312977\pi$
$641$	17992.0	1.10865	0.554323	+	0.832301i	$0.312977\pi$
$642$	0	0
$643$	−9448.64	−0.579499	−0.289750	−	0.957102i	$-0.593572\pi$
$643$	−9448.64	−0.579499	−0.289750	+	0.957102i	$0.593572\pi$
$644$	943.918	0.0577571
$645$	0	0
$646$	−18730.8	−1.14079
$647$	7429.22	0.451426	0.225713	−	0.974194i	$-0.427529\pi$
$647$	7429.22	0.451426	0.225713	+	0.974194i	$0.427529\pi$
$648$	0	0
$649$	2562.30	0.154976
$650$	4858.15	0.293157
$651$	0	0
$652$	−15133.2	−0.908988
$653$	4488.63	0.268995	0.134497	−	0.990914i	$-0.457058\pi$
$653$	4488.63	0.268995	0.134497	+	0.990914i	$0.457058\pi$
$654$	0	0
$655$	7624.69	0.454842
$656$	−9887.42	−0.588474
$657$	0	0
$658$	10831.6	0.641730
$659$	25326.7	1.49710	0.748550	−	0.663078i	$-0.230750\pi$
$659$	25326.7	1.49710	0.748550	+	0.663078i	$0.230750\pi$
$660$	0	0
$661$	15192.3	0.893969	0.446984	−	0.894542i	$-0.352498\pi$
$661$	15192.3	0.893969	0.446984	+	0.894542i	$0.352498\pi$
$662$	9985.15	0.586229
$663$	0	0
$664$	12956.4	0.757235
$665$	−3935.67	−0.229502
$666$	0	0
$667$	2864.75	0.166302
$668$	11821.5	0.684711
$669$	0	0
$670$	15743.1	0.907776
$671$	−1802.34	−0.103694
$672$	0	0
$673$	11718.2	0.671180	0.335590	−	0.942008i	$-0.391064\pi$
$673$	11718.2	0.671180	0.335590	+	0.942008i	$0.391064\pi$
$674$	−17734.8	−1.01353
$675$	0	0
$676$	3581.63	0.203779
$677$	15649.1	0.888397	0.444198	−	0.895928i	$-0.353489\pi$
$677$	15649.1	0.888397	0.444198	+	0.895928i	$0.353489\pi$
$678$	0	0
$679$	−24972.6	−1.41143
$680$	5970.61	0.336710
$681$	0	0
$682$	−3242.05	−0.182030
$683$	18162.8	1.01754	0.508770	−	0.860903i	$-0.330101\pi$
$683$	18162.8	1.01754	0.508770	+	0.860903i	$0.330101\pi$
$684$	0	0
$685$	10628.4	0.592831
$686$	−24656.2	−1.37227
$687$	0	0
$688$	−17778.1	−0.985149
$689$	12664.2	0.700246
$690$	0	0
$691$	29606.7	1.62995	0.814973	−	0.579500i	$-0.196752\pi$
$691$	29606.7	1.62995	0.814973	+	0.579500i	$0.196752\pi$
$692$	18384.7	1.00994
$693$	0	0
$694$	−9736.08	−0.532531
$695$	−8871.38	−0.484188
$696$	0	0
$697$	13358.5	0.725953
$698$	−19733.8	−1.07011
$699$	0	0
$700$	−1974.76	−0.106627
$701$	26164.7	1.40974	0.704870	−	0.709336i	$-0.251005\pi$
$701$	26164.7	1.40974	0.704870	+	0.709336i	$0.251005\pi$
$702$	0	0
$703$	−1060.66	−0.0569040
$704$	736.324	0.0394194
$705$	0	0
$706$	22879.9	1.21968
$707$	−10693.5	−0.568842
$708$	0	0
$709$	−14508.9	−0.768535	−0.384268	−	0.923222i	$-0.625546\pi$
$709$	−14508.9	−0.768535	−0.384268	+	0.923222i	$0.625546\pi$
$710$	−13161.9	−0.695716
$711$	0	0
$712$	11629.5	0.612125
$713$	980.898	0.0515216
$714$	0	0
$715$	2976.67	0.155694
$716$	−12396.6	−0.647043
$717$	0	0
$718$	2264.28	0.117691
$719$	2545.80	0.132048	0.0660239	−	0.997818i	$-0.478969\pi$
$719$	2545.80	0.132048	0.0660239	+	0.997818i	$0.478969\pi$
$720$	0	0
$721$	−6452.01	−0.333267
$722$	16094.6	0.829611
$723$	0	0
$724$	−67.9982	−0.00349051
$725$	−5993.31	−0.307015
$726$	0	0
$727$	−18984.8	−0.968511	−0.484255	−	0.874927i	$-0.660909\pi$
$727$	−18984.8	−0.968511	−0.484255	+	0.874927i	$0.660909\pi$
$728$	9751.54	0.496451
$729$	0	0
$730$	−20860.6	−1.05765
$731$	24019.2	1.21530
$732$	0	0
$733$	36740.4	1.85134	0.925672	−	0.378326i	$-0.123500\pi$
$733$	36740.4	1.85134	0.925672	+	0.378326i	$0.123500\pi$
$734$	19297.4	0.970411
$735$	0	0
$736$	−2331.58	−0.116770
$737$	9646.10	0.482115
$738$	0	0
$739$	1755.17	0.0873682	0.0436841	−	0.999045i	$-0.486090\pi$
$739$	1755.17	0.0873682	0.0436841	+	0.999045i	$0.486090\pi$
$740$	−532.196	−0.0264377
$741$	0	0
$742$	−13566.0	−0.671189
$743$	−17972.6	−0.887419	−0.443709	−	0.896171i	$-0.646338\pi$
$743$	−17972.6	−0.887419	−0.443709	+	0.896171i	$0.646338\pi$
$744$	0	0
$745$	−7878.88	−0.387463
$746$	−27001.4	−1.32519
$747$	0	0
$748$	−5758.54	−0.281488
$749$	−25692.6	−1.25339
$750$	0	0
$751$	22704.9	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$751$	22704.9	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$752$	−14797.9	−0.717583
$753$	0	0
$754$	−46586.3	−2.25010
$755$	2104.89	0.101463
$756$	0	0
$757$	39860.3	1.91380	0.956901	−	0.290414i	$-0.0937929\pi$
$757$	39860.3	1.91380	0.956901	+	0.290414i	$0.0937929\pi$
$758$	44914.6	2.15221
$759$	0	0
$760$	2719.99	0.129822
$761$	−19631.4	−0.935135	−0.467568	−	0.883957i	$-0.654870\pi$
$761$	−19631.4	−0.935135	−0.467568	+	0.883957i	$0.654870\pi$
$762$	0	0
$763$	−12192.7	−0.578511
$764$	−17099.8	−0.809752
$765$	0	0
$766$	−40034.3	−1.88838
$767$	12606.8	0.593490
$768$	0	0
$769$	35029.9	1.64267	0.821333	−	0.570449i	$-0.193231\pi$
$769$	35029.9	1.64267	0.821333	+	0.570449i	$0.193231\pi$
$770$	−3188.62	−0.149234
$771$	0	0
$772$	16972.4	0.791255
$773$	26736.9	1.24406	0.622032	−	0.782992i	$-0.286307\pi$
$773$	26736.9	1.24406	0.622032	+	0.782992i	$0.286307\pi$
$774$	0	0
$775$	−2052.13	−0.0951156
$776$	17258.8	0.798398
$777$	0	0
$778$	11469.0	0.528515
$779$	6085.64	0.279898
$780$	0	0
$781$	−8064.55	−0.369491
$782$	4591.39	0.209959
$783$	0	0
$784$	6517.81	0.296912
$785$	−11121.5	−0.505660
$786$	0	0
$787$	4799.45	0.217385	0.108693	−	0.994075i	$-0.465334\pi$
$787$	4799.45	0.217385	0.108693	+	0.994075i	$0.465334\pi$
$788$	−17799.4	−0.804668
$789$	0	0
$790$	10571.2	0.476083
$791$	−18498.3	−0.831508
$792$	0	0
$793$	−8867.72	−0.397102
$794$	2098.36	0.0937884
$795$	0	0
$796$	252.457	0.0112413
$797$	−38438.9	−1.70838	−0.854189	−	0.519963i	$-0.825946\pi$
$797$	−38438.9	−1.70838	−0.854189	+	0.519963i	$0.825946\pi$
$798$	0	0
$799$	19992.8	0.885224
$800$	4877.87	0.215574
$801$	0	0
$802$	−24958.9	−1.09891
$803$	−12781.7	−0.561713
$804$	0	0
$805$	964.733	0.0422390
$806$	−15951.3	−0.697096
$807$	0	0
$808$	7390.42	0.321775
$809$	−9960.91	−0.432889	−0.216444	−	0.976295i	$-0.569446\pi$
$809$	−9960.91	−0.432889	−0.216444	+	0.976295i	$0.569446\pi$
$810$	0	0
$811$	−34199.9	−1.48079	−0.740396	−	0.672171i	$-0.765362\pi$
$811$	−34199.9	−1.48079	−0.740396	+	0.672171i	$0.765362\pi$
$812$	18936.6	0.818404
$813$	0	0
$814$	−859.329	−0.0370018
$815$	−15466.9	−0.664762
$816$	0	0
$817$	10942.3	0.468570
$818$	−42203.9	−1.80394
$819$	0	0
$820$	3053.53	0.130041
$821$	31796.4	1.35165	0.675823	−	0.737064i	$-0.263788\pi$
$821$	31796.4	1.35165	0.675823	+	0.737064i	$0.263788\pi$
$822$	0	0
$823$	−10783.6	−0.456733	−0.228366	−	0.973575i	$-0.573338\pi$
$823$	−10783.6	−0.456733	−0.228366	+	0.973575i	$0.573338\pi$
$824$	4459.07	0.188518
$825$	0	0
$826$	−13504.5	−0.568862
$827$	−44197.5	−1.85840	−0.929201	−	0.369576i	$-0.879503\pi$
$827$	−44197.5	−1.85840	−0.929201	+	0.369576i	$0.879503\pi$
$828$	0	0
$829$	22487.7	0.942137	0.471069	−	0.882097i	$-0.343868\pi$
$829$	22487.7	0.942137	0.471069	+	0.882097i	$0.343868\pi$
$830$	−20844.4	−0.871709
$831$	0	0
$832$	3622.81	0.150959
$833$	−8805.96	−0.366276
$834$	0	0
$835$	12082.2	0.500743
$836$	−2623.38	−0.108530
$837$	0	0
$838$	42475.6	1.75095
$839$	−22491.0	−0.925477	−0.462738	−	0.886495i	$-0.653133\pi$
$839$	−22491.0	−0.925477	−0.462738	+	0.886495i	$0.653133\pi$
$840$	0	0
$841$	33082.7	1.35646
$842$	7181.37	0.293927
$843$	0	0
$844$	10198.0	0.415910
$845$	3660.61	0.149028
$846$	0	0
$847$	−1953.72	−0.0792571
$848$	18533.5	0.750524
$849$	0	0
$850$	−9605.60	−0.387611
$851$	259.994	0.0104730
$852$	0	0
$853$	22327.9	0.896241	0.448120	−	0.893973i	$-0.352094\pi$
$853$	22327.9	0.896241	0.448120	+	0.893973i	$0.352094\pi$
$854$	9499.12	0.380624
$855$	0	0
$856$	17756.5	0.708999
$857$	14505.9	0.578193	0.289096	−	0.957300i	$-0.406645\pi$
$857$	14505.9	0.578193	0.289096	+	0.957300i	$0.406645\pi$
$858$	0	0
$859$	−8411.45	−0.334104	−0.167052	−	0.985948i	$-0.553425\pi$
$859$	−8411.45	−0.334104	−0.167052	+	0.985948i	$0.553425\pi$
$860$	5490.40	0.217699
$861$	0	0
$862$	−28954.7	−1.14408
$863$	−32499.6	−1.28192	−0.640961	−	0.767573i	$-0.721464\pi$
$863$	−32499.6	−1.28192	−0.640961	+	0.767573i	$0.721464\pi$
$864$	0	0
$865$	18790.1	0.738592
$866$	38455.9	1.50899
$867$	0	0
$868$	6483.94	0.253548
$869$	6477.15	0.252845
$870$	0	0
$871$	47460.0	1.84629
$872$	8426.50	0.327245
$873$	0	0
$874$	2091.67	0.0809516
$875$	−2018.31	−0.0779786
$876$	0	0
$877$	−32183.1	−1.23916	−0.619581	−	0.784933i	$-0.712697\pi$
$877$	−32183.1	−1.23916	−0.619581	+	0.784933i	$0.712697\pi$
$878$	−16725.1	−0.642876
$879$	0	0
$880$	4356.23	0.166873
$881$	6246.34	0.238870	0.119435	−	0.992842i	$-0.461892\pi$
$881$	6246.34	0.238870	0.119435	+	0.992842i	$0.461892\pi$
$882$	0	0
$883$	−11801.1	−0.449762	−0.224881	−	0.974386i	$-0.572199\pi$
$883$	−11801.1	−0.449762	−0.224881	+	0.974386i	$0.572199\pi$
$884$	−28332.7	−1.07798
$885$	0	0
$886$	7521.24	0.285193
$887$	−32375.1	−1.22553	−0.612767	−	0.790264i	$-0.709944\pi$
$887$	−32375.1	−1.22553	−0.612767	+	0.790264i	$0.709944\pi$
$888$	0	0
$889$	−23594.6	−0.890145
$890$	−18709.7	−0.704662
$891$	0	0
$892$	9233.17	0.346580
$893$	9107.98	0.341307
$894$	0	0
$895$	−12670.0	−0.473196
$896$	21322.6	0.795020
$897$	0	0
$898$	−5034.72	−0.187095
$899$	19678.5	0.730049
$900$	0	0
$901$	−25039.9	−0.925861
$902$	4930.49	0.182004
$903$	0	0
$904$	12784.4	0.470357
$905$	−69.4977	−0.00255269
$906$	0	0
$907$	−19592.8	−0.717277	−0.358638	−	0.933477i	$-0.616759\pi$
$907$	−19592.8	−0.717277	−0.358638	+	0.933477i	$0.616759\pi$
$908$	5629.53	0.205752
$909$	0	0
$910$	−15688.4	−0.571500
$911$	−18673.8	−0.679133	−0.339567	−	0.940582i	$-0.610280\pi$
$911$	−18673.8	−0.679133	−0.339567	+	0.940582i	$0.610280\pi$
$912$	0	0
$913$	−12771.7	−0.462959
$914$	17644.7	0.638550
$915$	0	0
$916$	20090.9	0.724696
$917$	−24622.4	−0.886698
$918$	0	0
$919$	−4572.90	−0.164142	−0.0820708	−	0.996627i	$-0.526153\pi$
$919$	−4572.90	−0.164142	−0.0820708	+	0.996627i	$0.526153\pi$
$920$	−666.739	−0.0238932
$921$	0	0
$922$	−7949.78	−0.283961
$923$	−39678.6	−1.41499
$924$	0	0
$925$	−543.932	−0.0193345
$926$	19988.7	0.709362
$927$	0	0
$928$	−46775.4	−1.65461
$929$	44222.1	1.56176	0.780882	−	0.624679i	$-0.214770\pi$
$929$	44222.1	1.56176	0.780882	+	0.624679i	$0.214770\pi$
$930$	0	0
$931$	−4011.67	−0.141221
$932$	28049.3	0.985823
$933$	0	0
$934$	1784.70	0.0625238
$935$	−5885.52	−0.205858
$936$	0	0
$937$	9218.62	0.321408	0.160704	−	0.987003i	$-0.448624\pi$
$937$	9218.62	0.321408	0.160704	+	0.987003i	$0.448624\pi$
$938$	−50839.2	−1.76968
$939$	0	0
$940$	4570.02	0.158572
$941$	26484.9	0.917516	0.458758	−	0.888561i	$-0.348294\pi$
$941$	26484.9	0.917516	0.458758	+	0.888561i	$0.348294\pi$
$942$	0	0
$943$	−1491.75	−0.0515142
$944$	18449.5	0.636103
$945$	0	0
$946$	8865.26	0.304688
$947$	44972.4	1.54320	0.771599	−	0.636110i	$-0.219457\pi$
$947$	44972.4	1.54320	0.771599	+	0.636110i	$0.219457\pi$
$948$	0	0
$949$	−62887.5	−2.15112
$950$	−4375.96	−0.149447
$951$	0	0
$952$	−19280.9	−0.656404
$953$	2052.50	0.0697659	0.0348829	−	0.999391i	$-0.488894\pi$
$953$	2052.50	0.0697659	0.0348829	+	0.999391i	$0.488894\pi$
$954$	0	0
$955$	−17476.9	−0.592189
$956$	−29529.7	−0.999016
$957$	0	0
$958$	−33569.2	−1.13212
$959$	−34322.2	−1.15570
$960$	0	0
$961$	−23053.0	−0.773826
$962$	−4228.00	−0.141701
$963$	0	0
$964$	18203.1	0.608176
$965$	17346.6	0.578661
$966$	0	0
$967$	14950.9	0.497195	0.248598	−	0.968607i	$-0.420030\pi$
$967$	14950.9	0.497195	0.248598	+	0.968607i	$0.420030\pi$
$968$	1350.24	0.0448331
$969$	0	0
$970$	−27766.3	−0.919094
$971$	−26751.9	−0.884151	−0.442076	−	0.896978i	$-0.645758\pi$
$971$	−26751.9	−0.884151	−0.442076	+	0.896978i	$0.645758\pi$
$972$	0	0
$973$	28648.3	0.943908
$974$	−710.034	−0.0233583
$975$	0	0
$976$	−12977.5	−0.425615
$977$	56893.7	1.86304	0.931520	−	0.363690i	$-0.118483\pi$
$977$	56893.7	1.86304	0.931520	+	0.363690i	$0.118483\pi$
$978$	0	0
$979$	−11463.7	−0.374242
$980$	−2012.89	−0.0656118
$981$	0	0
$982$	−35895.0	−1.16645
$983$	34633.1	1.12373	0.561863	−	0.827230i	$-0.310085\pi$
$983$	34633.1	1.12373	0.561863	+	0.827230i	$0.310085\pi$
$984$	0	0
$985$	−18191.9	−0.588470
$986$	92111.0	2.97506
$987$	0	0
$988$	−12907.3	−0.415625
$989$	−2682.23	−0.0862386
$990$	0	0
$991$	1961.79	0.0628843	0.0314422	−	0.999506i	$-0.489990\pi$
$991$	1961.79	0.0628843	0.0314422	+	0.999506i	$0.489990\pi$
$992$	−16016.0	−0.512610
$993$	0	0
$994$	42503.8	1.35628
$995$	258.025	0.00822103
$996$	0	0
$997$	13096.5	0.416017	0.208009	−	0.978127i	$-0.433302\pi$
$997$	13096.5	0.416017	0.208009	+	0.978127i	$0.433302\pi$
$998$	31290.8	0.992476
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.l.1.1		3
3.2	odd	2		165.4.a.d.1.3	✓	3
5.4	even	2		2475.4.a.s.1.3		3
15.2	even	4		825.4.c.l.199.5		6
15.8	even	4		825.4.c.l.199.2		6
15.14	odd	2		825.4.a.s.1.1		3
33.32	even	2		1815.4.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.3	✓	3	3.2	odd	2
495.4.a.l.1.1		3	1.1	even	1	trivial
825.4.a.s.1.1		3	15.14	odd	2
825.4.c.l.199.2		6	15.8	even	4
825.4.c.l.199.5		6	15.2	even	4
1815.4.a.s.1.1		3	33.32	even	2
2475.4.a.s.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)

\(f(q)\)	\(=\)	\(q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} +O(q^{10})\)	\(q-3.59056 q^{2} +4.89212 q^{4} +5.00000 q^{5} -16.1465 q^{7} +11.1590 q^{8} -17.9528 q^{10} -11.0000 q^{11} -54.1214 q^{13} +57.9749 q^{14} -79.2041 q^{16} +107.010 q^{17} +48.7496 q^{19} +24.4606 q^{20} +39.4962 q^{22} -11.9498 q^{23} +25.0000 q^{25} +194.326 q^{26} -78.9905 q^{28} -239.733 q^{29} -82.0851 q^{31} +195.115 q^{32} -384.224 q^{34} -80.7324 q^{35} -21.7573 q^{37} -175.038 q^{38} +55.7952 q^{40} +124.835 q^{41} +224.459 q^{43} -53.8133 q^{44} +42.9064 q^{46} +186.832 q^{47} -82.2913 q^{49} -89.7640 q^{50} -264.768 q^{52} -233.997 q^{53} -55.0000 q^{55} -180.179 q^{56} +860.774 q^{58} -232.936 q^{59} +163.849 q^{61} +294.731 q^{62} -66.9386 q^{64} -270.607 q^{65} -876.918 q^{67} +523.503 q^{68} +289.874 q^{70} +733.141 q^{71} +1161.97 q^{73} +78.1208 q^{74} +238.489 q^{76} +177.611 q^{77} -588.831 q^{79} -396.021 q^{80} -448.226 q^{82} +1161.06 q^{83} +535.048 q^{85} -805.933 q^{86} -122.749 q^{88} +1042.16 q^{89} +873.869 q^{91} -58.4597 q^{92} -670.831 q^{94} +243.748 q^{95} +1546.63 q^{97} +295.472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8}+O(q^{10}) \) 3 * q + 4 * q^2 + 22 * q^4 + 15 * q^5 - 4 * q^7 + 48 * q^8	\( 3 q + 4 q^{2} + 22 q^{4} + 15 q^{5} - 4 q^{7} + 48 q^{8} + 20 q^{10} - 33 q^{11} + 56 q^{14} + 50 q^{16} + 218 q^{17} + 146 q^{19} + 110 q^{20} - 44 q^{22} + 200 q^{23} + 75 q^{25} + 508 q^{26} - 340 q^{28} - 68 q^{29} - 68 q^{31} + 688 q^{32} - 176 q^{34} - 20 q^{35} - 390 q^{37} + 316 q^{38} + 240 q^{40} + 196 q^{41} - 524 q^{43} - 242 q^{44} + 1160 q^{46} + 60 q^{47} - 157 q^{49} + 100 q^{50} + 1020 q^{52} + 158 q^{53} - 165 q^{55} - 1368 q^{56} + 1092 q^{58} + 1044 q^{59} + 642 q^{61} - 88 q^{62} + 1166 q^{64} - 236 q^{67} - 144 q^{68} + 280 q^{70} + 544 q^{71} + 900 q^{73} - 1536 q^{74} + 1996 q^{76} + 44 q^{77} - 1586 q^{79} + 250 q^{80} + 380 q^{82} + 1582 q^{83} + 1090 q^{85} - 3568 q^{86} - 528 q^{88} + 2122 q^{89} - 8 q^{91} + 4128 q^{92} - 2152 q^{94} + 730 q^{95} + 618 q^{97} - 572 q^{98}+O(q^{100}) \) 3 * q + 4 * q^2 + 22 * q^4 + 15 * q^5 - 4 * q^7 + 48 * q^8 + 20 * q^10 - 33 * q^11 + 56 * q^14 + 50 * q^16 + 218 * q^17 + 146 * q^19 + 110 * q^20 - 44 * q^22 + 200 * q^23 + 75 * q^25 + 508 * q^26 - 340 * q^28 - 68 * q^29 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 - 390 * q^37 + 316 * q^38 + 240 * q^40 + 196 * q^41 - 524 * q^43 - 242 * q^44 + 1160 * q^46 + 60 * q^47 - 157 * q^49 + 100 * q^50 + 1020 * q^52 + 158 * q^53 - 165 * q^55 - 1368 * q^56 + 1092 * q^58 + 1044 * q^59 + 642 * q^61 - 88 * q^62 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 280 * q^70 + 544 * q^71 + 900 * q^73 - 1536 * q^74 + 1996 * q^76 + 44 * q^77 - 1586 * q^79 + 250 * q^80 + 380 * q^82 + 1582 * q^83 + 1090 * q^85 - 3568 * q^86 - 528 * q^88 + 2122 * q^89 - 8 * q^91 + 4128 * q^92 - 2152 * q^94 + 730 * q^95 + 618 * q^97 - 572 * q^98

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