LMFDB - Embedded newform 495.4.a.m.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:	$4$
Coefficient field:	4.4.1540841.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 27x^{2} - 18x + 92 $ x^4 - 27x^2 - 18x + 92
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.60719$ 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)

$f(q)$	$=$	$q+0.607192 q^{2} -7.63132 q^{4} -5.00000 q^{5} +8.95080 q^{7} -9.49121 q^{8} +O(q^{10})$	\(q+0.607192 q^{2} -7.63132 q^{4} -5.00000 q^{5} +8.95080 q^{7} -9.49121 q^{8} -3.03596 q^{10} +11.0000 q^{11} -0.460387 q^{13} +5.43485 q^{14} +55.2876 q^{16} -128.395 q^{17} -0.0245858 q^{19} +38.1566 q^{20} +6.67911 q^{22} -171.528 q^{23} +25.0000 q^{25} -0.279543 q^{26} -68.3064 q^{28} +226.938 q^{29} +195.637 q^{31} +109.500 q^{32} -77.9604 q^{34} -44.7540 q^{35} +338.584 q^{37} -0.0149283 q^{38} +47.4560 q^{40} -136.972 q^{41} +336.083 q^{43} -83.9445 q^{44} -104.151 q^{46} +540.292 q^{47} -262.883 q^{49} +15.1798 q^{50} +3.51336 q^{52} +622.387 q^{53} -55.0000 q^{55} -84.9539 q^{56} +137.795 q^{58} +9.86955 q^{59} +902.712 q^{61} +118.789 q^{62} -375.813 q^{64} +2.30193 q^{65} +146.979 q^{67} +979.823 q^{68} -27.1743 q^{70} +893.798 q^{71} -1149.71 q^{73} +205.585 q^{74} +0.187622 q^{76} +98.4588 q^{77} -459.528 q^{79} -276.438 q^{80} -83.1686 q^{82} +125.876 q^{83} +641.975 q^{85} +204.067 q^{86} -104.403 q^{88} -150.461 q^{89} -4.12083 q^{91} +1308.99 q^{92} +328.061 q^{94} +0.122929 q^{95} -1264.58 q^{97} -159.621 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 20 q^{5} + 34 q^{7} - 48 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 20 * q^5 + 34 * q^7 - 48 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 20 q^{5} + 34 q^{7} - 48 q^{8} + 20 q^{10} + 44 q^{11} + 2 q^{13} + 52 q^{14} + 66 q^{16} - 74 q^{17} + 136 q^{19} - 130 q^{20} - 44 q^{22} + 64 q^{23} + 100 q^{25} + 320 q^{26} - 20 q^{28} - 52 q^{29} + 492 q^{31} - 208 q^{32} + 244 q^{34} - 170 q^{35} - 4 q^{37} + 404 q^{38} + 240 q^{40} - 268 q^{41} + 546 q^{43} + 286 q^{44} + 368 q^{46} + 276 q^{47} - 496 q^{49} - 100 q^{50} - 1084 q^{52} + 184 q^{53} - 220 q^{55} + 852 q^{56} - 444 q^{58} + 1032 q^{59} + 116 q^{61} + 1240 q^{62} - 918 q^{64} - 10 q^{65} - 552 q^{67} + 720 q^{68} - 260 q^{70} + 920 q^{71} + 926 q^{73} + 2856 q^{74} + 1572 q^{76} + 374 q^{77} + 1152 q^{79} - 330 q^{80} - 1924 q^{82} + 134 q^{83} + 370 q^{85} - 236 q^{86} - 528 q^{88} + 1064 q^{89} + 2780 q^{91} + 4896 q^{92} - 1432 q^{94} - 680 q^{95} - 1648 q^{97} + 188 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 20 * q^5 + 34 * q^7 - 48 * q^8 + 20 * q^10 + 44 * q^11 + 2 * q^13 + 52 * q^14 + 66 * q^16 - 74 * q^17 + 136 * q^19 - 130 * q^20 - 44 * q^22 + 64 * q^23 + 100 * q^25 + 320 * q^26 - 20 * q^28 - 52 * q^29 + 492 * q^31 - 208 * q^32 + 244 * q^34 - 170 * q^35 - 4 * q^37 + 404 * q^38 + 240 * q^40 - 268 * q^41 + 546 * q^43 + 286 * q^44 + 368 * q^46 + 276 * q^47 - 496 * q^49 - 100 * q^50 - 1084 * q^52 + 184 * q^53 - 220 * q^55 + 852 * q^56 - 444 * q^58 + 1032 * q^59 + 116 * q^61 + 1240 * q^62 - 918 * q^64 - 10 * q^65 - 552 * q^67 + 720 * q^68 - 260 * q^70 + 920 * q^71 + 926 * q^73 + 2856 * q^74 + 1572 * q^76 + 374 * q^77 + 1152 * q^79 - 330 * q^80 - 1924 * q^82 + 134 * q^83 + 370 * q^85 - 236 * q^86 - 528 * q^88 + 1064 * q^89 + 2780 * q^91 + 4896 * q^92 - 1432 * q^94 - 680 * q^95 - 1648 * q^97 + 188 * 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 $

$2$	0.607192	0.214675	0.107337	−	0.994223i	$-0.465768\pi$
$2$	0.607192	0.214675	0.107337	+	0.994223i	$0.465768\pi$
$3$	0	0
$3$	0	0
$4$	−7.63132	−0.953915
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	8.95080	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$7$	8.95080	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$8$	−9.49121	−0.419456
$9$	0	0
$10$	−3.03596	−0.0960055
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−0.460387	−0.00982218	−0.00491109	−	0.999988i	$-0.501563\pi$
$13$	−0.460387	−0.00982218	−0.00491109	+	0.999988i	$0.501563\pi$
$14$	5.43485	0.103752
$15$	0	0
$16$	55.2876	0.863868
$17$	−128.395	−1.83179	−0.915893	−	0.401423i	$-0.868516\pi$
$17$	−128.395	−1.83179	−0.915893	+	0.401423i	$0.868516\pi$
$18$	0	0
$19$	−0.0245858	−0.000296862 0	−0.000148431	−	1.00000i	$-0.500047\pi$
$19$	−0.0245858	−0.000296862 0	−0.000148431		1.00000i	$0.500047\pi$
$20$	38.1566	0.426604
$21$	0	0
$22$	6.67911	0.0647269
$23$	−171.528	−1.55505	−0.777525	−	0.628852i	$-0.783525\pi$
$23$	−171.528	−1.55505	−0.777525	+	0.628852i	$0.783525\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−0.279543	−0.00210857
$27$	0	0
$28$	−68.3064	−0.461025
$29$	226.938	1.45315	0.726574	−	0.687088i	$-0.241111\pi$
$29$	226.938	1.45315	0.726574	+	0.687088i	$0.241111\pi$
$30$	0	0
$31$	195.637	1.13347	0.566734	−	0.823901i	$-0.308207\pi$
$31$	195.637	1.13347	0.566734	+	0.823901i	$0.308207\pi$
$32$	109.500	0.604907
$33$	0	0
$34$	−77.9604	−0.393238
$35$	−44.7540	−0.216137
$36$	0	0
$37$	338.584	1.50440	0.752200	−	0.658935i	$-0.228993\pi$
$37$	338.584	1.50440	0.752200	+	0.658935i	$0.228993\pi$
$38$	−0.0149283	−6.37287e−5 0
$39$	0	0
$40$	47.4560	0.187586
$41$	−136.972	−0.521744	−0.260872	−	0.965373i	$-0.584010\pi$
$41$	−136.972	−0.521744	−0.260872	+	0.965373i	$0.584010\pi$
$42$	0	0
$43$	336.083	1.19191	0.595956	−	0.803017i	$-0.296773\pi$
$43$	336.083	1.19191	0.595956	+	0.803017i	$0.296773\pi$
$44$	−83.9445	−0.287616
$45$	0	0
$46$	−104.151	−0.333830
$47$	540.292	1.67680	0.838400	−	0.545055i	$-0.183491\pi$
$47$	540.292	1.67680	0.838400	+	0.545055i	$0.183491\pi$
$48$	0	0
$49$	−262.883	−0.766423
$50$	15.1798	0.0429349
$51$	0	0
$52$	3.51336	0.00936952
$53$	622.387	1.61305	0.806523	−	0.591203i	$-0.201347\pi$
$53$	622.387	1.61305	0.806523	+	0.591203i	$0.201347\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	−84.9539	−0.202722
$57$	0	0
$58$	137.795	0.311954
$59$	9.86955	0.0217781	0.0108890	−	0.999941i	$-0.496534\pi$
$59$	9.86955	0.0217781	0.0108890	+	0.999941i	$0.496534\pi$
$60$	0	0
$61$	902.712	1.89476	0.947380	−	0.320110i	$-0.103720\pi$
$61$	902.712	1.89476	0.947380	+	0.320110i	$0.103720\pi$
$62$	118.789	0.243327
$63$	0	0
$64$	−375.813	−0.734010
$65$	2.30193	0.00439261
$66$	0	0
$67$	146.979	0.268005	0.134003	−	0.990981i	$-0.457217\pi$
$67$	146.979	0.268005	0.134003	+	0.990981i	$0.457217\pi$
$68$	979.823	1.74737
$69$	0	0
$70$	−27.1743	−0.0463992
$71$	893.798	1.49400	0.747002	−	0.664821i	$-0.231492\pi$
$71$	893.798	1.49400	0.747002	+	0.664821i	$0.231492\pi$
$72$	0	0
$73$	−1149.71	−1.84333	−0.921664	−	0.387988i	$-0.873170\pi$
$73$	−1149.71	−1.84333	−0.921664	+	0.387988i	$0.873170\pi$
$74$	205.585	0.322957
$75$	0	0
$76$	0.187622	0.000283181 0
$77$	98.4588	0.145720
$78$	0	0
$79$	−459.528	−0.654443	−0.327221	−	0.944948i	$-0.606112\pi$
$79$	−459.528	−0.654443	−0.327221	+	0.944948i	$0.606112\pi$
$80$	−276.438	−0.386334
$81$	0	0
$82$	−83.1686	−0.112005
$83$	125.876	0.166466	0.0832331	−	0.996530i	$-0.473475\pi$
$83$	125.876	0.166466	0.0832331	+	0.996530i	$0.473475\pi$
$84$	0	0
$85$	641.975	0.819199
$86$	204.067	0.255873
$87$	0	0
$88$	−104.403	−0.126471
$89$	−150.461	−0.179201	−0.0896003	−	0.995978i	$-0.528559\pi$
$89$	−150.461	−0.179201	−0.0896003	+	0.995978i	$0.528559\pi$
$90$	0	0
$91$	−4.12083	−0.00474704
$92$	1308.99	1.48339
$93$	0	0
$94$	328.061	0.359967
$95$	0.122929	0.000132761 0
$96$	0	0
$97$	−1264.58	−1.32370	−0.661851	−	0.749635i	$-0.730229\pi$
$97$	−1264.58	−1.32370	−0.661851	+	0.749635i	$0.730229\pi$
$98$	−159.621	−0.164532
$99$	0	0
$100$	−190.783	−0.190783
$101$	−690.792	−0.680558	−0.340279	−	0.940324i	$-0.610522\pi$
$101$	−690.792	−0.680558	−0.340279	+	0.940324i	$0.610522\pi$
$102$	0	0
$103$	−11.7823	−0.0112713	−0.00563563	−	0.999984i	$-0.501794\pi$
$103$	−11.7823	−0.0112713	−0.00563563	+	0.999984i	$0.501794\pi$
$104$	4.36963	0.00411997
$105$	0	0
$106$	377.908	0.346280
$107$	462.884	0.418212	0.209106	−	0.977893i	$-0.432945\pi$
$107$	462.884	0.418212	0.209106	+	0.977893i	$0.432945\pi$
$108$	0	0
$109$	330.525	0.290446	0.145223	−	0.989399i	$-0.453610\pi$
$109$	330.525	0.290446	0.145223	+	0.989399i	$0.453610\pi$
$110$	−33.3956	−0.0289467
$111$	0	0
$112$	494.868	0.417506
$113$	−1142.31	−0.950965	−0.475483	−	0.879725i	$-0.657727\pi$
$113$	−1142.31	−0.950965	−0.475483	+	0.879725i	$0.657727\pi$
$114$	0	0
$115$	857.642	0.695440
$116$	−1731.83	−1.38618
$117$	0	0
$118$	5.99271	0.00467520
$119$	−1149.24	−0.885298
$120$	0	0
$121$	121.000	0.0909091
$122$	548.119	0.406757
$123$	0	0
$124$	−1492.97	−1.08123
$125$	−125.000	−0.0894427
$126$	0	0
$127$	629.589	0.439898	0.219949	−	0.975511i	$-0.429411\pi$
$127$	629.589	0.439898	0.219949	+	0.975511i	$0.429411\pi$
$128$	−1104.19	−0.762480
$129$	0	0
$130$	1.39772	0.000942983 0
$131$	−572.566	−0.381873	−0.190937	−	0.981602i	$-0.561152\pi$
$131$	−572.566	−0.381873	−0.190937	+	0.981602i	$0.561152\pi$
$132$	0	0
$133$	−0.220063	−0.000143473 0
$134$	89.2446	0.0575340
$135$	0	0
$136$	1218.62	0.768354
$137$	948.680	0.591615	0.295807	−	0.955248i	$-0.404411\pi$
$137$	948.680	0.591615	0.295807	+	0.955248i	$0.404411\pi$
$138$	0	0
$139$	2488.86	1.51872	0.759362	−	0.650668i	$-0.225511\pi$
$139$	2488.86	1.51872	0.759362	+	0.650668i	$0.225511\pi$
$140$	341.532	0.206177
$141$	0	0
$142$	542.707	0.320725
$143$	−5.06425	−0.00296150
$144$	0	0
$145$	−1134.69	−0.649867
$146$	−698.093	−0.395716
$147$	0	0
$148$	−2583.84	−1.43507
$149$	2186.43	1.20214	0.601072	−	0.799195i	$-0.294741\pi$
$149$	2186.43	1.20214	0.601072	+	0.799195i	$0.294741\pi$
$150$	0	0
$151$	3668.21	1.97692	0.988459	−	0.151488i	$-0.0484064\pi$
$151$	3668.21	1.97692	0.988459	+	0.151488i	$0.0484064\pi$
$152$	0.233349	0.000124520 0
$153$	0	0
$154$	59.7834	0.0312824
$155$	−978.186	−0.506902
$156$	0	0
$157$	1418.28	0.720960	0.360480	−	0.932767i	$-0.382613\pi$
$157$	1418.28	0.720960	0.360480	+	0.932767i	$0.382613\pi$
$158$	−279.022	−0.140492
$159$	0	0
$160$	−547.499	−0.270523
$161$	−1535.32	−0.751552
$162$	0	0
$163$	−601.009	−0.288802	−0.144401	−	0.989519i	$-0.546125\pi$
$163$	−601.009	−0.288802	−0.144401	+	0.989519i	$0.546125\pi$
$164$	1045.28	0.497699
$165$	0	0
$166$	76.4309	0.0357361
$167$	2221.19	1.02923	0.514614	−	0.857422i	$-0.327935\pi$
$167$	2221.19	1.02923	0.514614	+	0.857422i	$0.327935\pi$
$168$	0	0
$169$	−2196.79	−0.999904
$170$	389.802	0.175861
$171$	0	0
$172$	−2564.76	−1.13698
$173$	237.280	0.104278	0.0521390	−	0.998640i	$-0.483396\pi$
$173$	237.280	0.104278	0.0521390	+	0.998640i	$0.483396\pi$
$174$	0	0
$175$	223.770	0.0966595
$176$	608.163	0.260466
$177$	0	0
$178$	−91.3588	−0.0384698
$179$	−1065.77	−0.445023	−0.222512	−	0.974930i	$-0.571426\pi$
$179$	−1065.77	−0.445023	−0.222512	+	0.974930i	$0.571426\pi$
$180$	0	0
$181$	−1241.00	−0.509627	−0.254813	−	0.966990i	$-0.582014\pi$
$181$	−1241.00	−0.509627	−0.254813	+	0.966990i	$0.582014\pi$
$182$	−2.50213	−0.00101907
$183$	0	0
$184$	1628.01	0.652275
$185$	−1692.92	−0.672788
$186$	0	0
$187$	−1412.34	−0.552304
$188$	−4123.14	−1.59952
$189$	0	0
$190$	0.0746415	2.85003e−5 0
$191$	1956.80	0.741305	0.370653	−	0.928772i	$-0.379134\pi$
$191$	1956.80	0.741305	0.370653	+	0.928772i	$0.379134\pi$
$192$	0	0
$193$	2778.02	1.03610	0.518048	−	0.855352i	$-0.326659\pi$
$193$	2778.02	1.03610	0.518048	+	0.855352i	$0.326659\pi$
$194$	−767.846	−0.284165
$195$	0	0
$196$	2006.15	0.731102
$197$	−800.242	−0.289416	−0.144708	−	0.989474i	$-0.546224\pi$
$197$	−800.242	−0.289416	−0.144708	+	0.989474i	$0.546224\pi$
$198$	0	0
$199$	−2536.78	−0.903657	−0.451829	−	0.892105i	$-0.649228\pi$
$199$	−2536.78	−0.903657	−0.451829	+	0.892105i	$0.649228\pi$
$200$	−237.280	−0.0838912
$201$	0	0
$202$	−419.443	−0.146099
$203$	2031.27	0.702303
$204$	0	0
$205$	684.862	0.233331
$206$	−7.15409	−0.00241966
$207$	0	0
$208$	−25.4537	−0.00848507
$209$	−0.270444	−8.95071e−5 0
$210$	0	0
$211$	1598.95	0.521688	0.260844	−	0.965381i	$-0.415999\pi$
$211$	1598.95	0.521688	0.260844	+	0.965381i	$0.415999\pi$
$212$	−4749.63	−1.53871
$213$	0	0
$214$	281.060	0.0897797
$215$	−1680.42	−0.533039
$216$	0	0
$217$	1751.11	0.547802
$218$	200.692	0.0623513
$219$	0	0
$220$	419.722	0.128626
$221$	59.1113	0.0179921
$222$	0	0
$223$	−5074.55	−1.52384	−0.761922	−	0.647669i	$-0.775744\pi$
$223$	−5074.55	−1.52384	−0.761922	+	0.647669i	$0.775744\pi$
$224$	980.111	0.292350
$225$	0	0
$226$	−693.599	−0.204148
$227$	−1048.19	−0.306480	−0.153240	−	0.988189i	$-0.548971\pi$
$227$	−1048.19	−0.306480	−0.153240	+	0.988189i	$0.548971\pi$
$228$	0	0
$229$	−734.662	−0.211999	−0.106000	−	0.994366i	$-0.533804\pi$
$229$	−734.662	−0.211999	−0.106000	+	0.994366i	$0.533804\pi$
$230$	520.754	0.149293
$231$	0	0
$232$	−2153.91	−0.609532
$233$	2012.78	0.565929	0.282965	−	0.959130i	$-0.408682\pi$
$233$	2012.78	0.565929	0.282965	+	0.959130i	$0.408682\pi$
$234$	0	0
$235$	−2701.46	−0.749888
$236$	−75.3176	−0.0207744
$237$	0	0
$238$	−697.808	−0.190051
$239$	−5566.41	−1.50653	−0.753267	−	0.657715i	$-0.771523\pi$
$239$	−5566.41	−1.50653	−0.753267	+	0.657715i	$0.771523\pi$
$240$	0	0
$241$	−3361.25	−0.898412	−0.449206	−	0.893428i	$-0.648293\pi$
$241$	−3361.25	−0.898412	−0.449206	+	0.893428i	$0.648293\pi$
$242$	73.4702	0.0195159
$243$	0	0
$244$	−6888.88	−1.80744
$245$	1314.42	0.342755
$246$	0	0
$247$	0.0113190	2.91583e−6 0
$248$	−1856.83	−0.475440
$249$	0	0
$250$	−75.8990	−0.0192011
$251$	2930.46	0.736929	0.368464	−	0.929642i	$-0.379884\pi$
$251$	2930.46	0.736929	0.368464	+	0.929642i	$0.379884\pi$
$252$	0	0
$253$	−1886.81	−0.468865
$254$	382.281	0.0944349
$255$	0	0
$256$	2336.05	0.570325
$257$	3412.89	0.828367	0.414184	−	0.910193i	$-0.364067\pi$
$257$	3412.89	0.828367	0.414184	+	0.910193i	$0.364067\pi$
$258$	0	0
$259$	3030.59	0.727073
$260$	−17.5668	−0.00419018
$261$	0	0
$262$	−347.658	−0.0819785
$263$	−3709.06	−0.869622	−0.434811	−	0.900522i	$-0.643185\pi$
$263$	−3709.06	−0.869622	−0.434811	+	0.900522i	$0.643185\pi$
$264$	0	0
$265$	−3111.94	−0.721376
$266$	−0.133620	−3.07999e−5 0
$267$	0	0
$268$	−1121.64	−0.255654
$269$	5764.39	1.30655	0.653273	−	0.757122i	$-0.273395\pi$
$269$	5764.39	1.30655	0.653273	+	0.757122i	$0.273395\pi$
$270$	0	0
$271$	−1886.33	−0.422827	−0.211414	−	0.977397i	$-0.567807\pi$
$271$	−1886.33	−0.422827	−0.211414	+	0.977397i	$0.567807\pi$
$272$	−7098.64	−1.58242
$273$	0	0
$274$	576.031	0.127005
$275$	275.000	0.0603023
$276$	0	0
$277$	−4095.99	−0.888462	−0.444231	−	0.895912i	$-0.646523\pi$
$277$	−4095.99	−0.888462	−0.444231	+	0.895912i	$0.646523\pi$
$278$	1511.22	0.326032
$279$	0	0
$280$	424.770	0.0906601
$281$	8788.69	1.86580	0.932899	−	0.360138i	$-0.117271\pi$
$281$	8788.69	1.86580	0.932899	+	0.360138i	$0.117271\pi$
$282$	0	0
$283$	−1767.75	−0.371314	−0.185657	−	0.982615i	$-0.559441\pi$
$283$	−1767.75	−0.371314	−0.185657	+	0.982615i	$0.559441\pi$
$284$	−6820.86	−1.42515
$285$	0	0
$286$	−3.07497	−0.000635759 0
$287$	−1226.01	−0.252158
$288$	0	0
$289$	11572.3	2.35544
$290$	−688.974	−0.139510
$291$	0	0
$292$	8773.78	1.75838
$293$	−3079.85	−0.614085	−0.307043	−	0.951696i	$-0.599339\pi$
$293$	−3079.85	−0.614085	−0.307043	+	0.951696i	$0.599339\pi$
$294$	0	0
$295$	−49.3477	−0.00973944
$296$	−3213.57	−0.631030
$297$	0	0
$298$	1327.58	0.258070
$299$	78.9695	0.0152740
$300$	0	0
$301$	3008.21	0.576048
$302$	2227.31	0.424394
$303$	0	0
$304$	−1.35929	−0.000256449 0
$305$	−4513.56	−0.847363
$306$	0	0
$307$	2872.16	0.533950	0.266975	−	0.963703i	$-0.413976\pi$
$307$	2872.16	0.533950	0.266975	+	0.963703i	$0.413976\pi$
$308$	−751.370	−0.139004
$309$	0	0
$310$	−593.947	−0.108819
$311$	317.978	0.0579771	0.0289885	−	0.999580i	$-0.490771\pi$
$311$	317.978	0.0579771	0.0289885	+	0.999580i	$0.490771\pi$
$312$	0	0
$313$	−8274.81	−1.49431	−0.747156	−	0.664648i	$-0.768581\pi$
$313$	−8274.81	−1.49431	−0.747156	+	0.664648i	$0.768581\pi$
$314$	861.166	0.154772
$315$	0	0
$316$	3506.81	0.624283
$317$	−1024.34	−0.181492	−0.0907459	−	0.995874i	$-0.528925\pi$
$317$	−1024.34	−0.181492	−0.0907459	+	0.995874i	$0.528925\pi$
$318$	0	0
$319$	2496.32	0.438141
$320$	1879.07	0.328259
$321$	0	0
$322$	−932.232	−0.161339
$323$	3.15669	0.000543787 0
$324$	0	0
$325$	−11.5097	−0.00196444
$326$	−364.928	−0.0619984
$327$	0	0
$328$	1300.03	0.218849
$329$	4836.04	0.810394
$330$	0	0
$331$	4575.83	0.759850	0.379925	−	0.925017i	$-0.375950\pi$
$331$	4575.83	0.759850	0.379925	+	0.925017i	$0.375950\pi$
$332$	−960.600	−0.158795
$333$	0	0
$334$	1348.69	0.220949
$335$	−734.896	−0.119856
$336$	0	0
$337$	−917.888	−0.148370	−0.0741848	−	0.997245i	$-0.523635\pi$
$337$	−917.888	−0.148370	−0.0741848	+	0.997245i	$0.523635\pi$
$338$	−1333.87	−0.214654
$339$	0	0
$340$	−4899.11	−0.781446
$341$	2152.01	0.341753
$342$	0	0
$343$	−5423.14	−0.853708
$344$	−3189.84	−0.499955
$345$	0	0
$346$	144.075	0.0223859
$347$	−5402.26	−0.835760	−0.417880	−	0.908502i	$-0.637227\pi$
$347$	−5402.26	−0.835760	−0.417880	+	0.908502i	$0.637227\pi$
$348$	0	0
$349$	5115.85	0.784656	0.392328	−	0.919825i	$-0.371670\pi$
$349$	5115.85	0.784656	0.392328	+	0.919825i	$0.371670\pi$
$350$	135.871	0.0207504
$351$	0	0
$352$	1204.50	0.182386
$353$	3102.03	0.467718	0.233859	−	0.972270i	$-0.424865\pi$
$353$	3102.03	0.467718	0.233859	+	0.972270i	$0.424865\pi$
$354$	0	0
$355$	−4468.99	−0.668139
$356$	1148.22	0.170942
$357$	0	0
$358$	−647.125	−0.0955352
$359$	−3496.32	−0.514008	−0.257004	−	0.966410i	$-0.582735\pi$
$359$	−3496.32	−0.514008	−0.257004	+	0.966410i	$0.582735\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	−753.522	−0.109404
$363$	0	0
$364$	31.4474	0.00452827
$365$	5748.53	0.824362
$366$	0	0
$367$	7644.54	1.08731	0.543654	−	0.839309i	$-0.317040\pi$
$367$	7644.54	1.08731	0.543654	+	0.839309i	$0.317040\pi$
$368$	−9483.39	−1.34336
$369$	0	0
$370$	−1027.93	−0.144431
$371$	5570.86	0.779582
$372$	0	0
$373$	−5189.90	−0.720437	−0.360218	−	0.932868i	$-0.617298\pi$
$373$	−5189.90	−0.720437	−0.360218	+	0.932868i	$0.617298\pi$
$374$	−857.564	−0.118566
$375$	0	0
$376$	−5128.02	−0.703344
$377$	−104.479	−0.0142731
$378$	0	0
$379$	8573.91	1.16204	0.581019	−	0.813890i	$-0.302654\pi$
$379$	8573.91	1.16204	0.581019	+	0.813890i	$0.302654\pi$
$380$	−0.938110	−0.000126642 0
$381$	0	0
$382$	1188.15	0.159140
$383$	11740.2	1.56631	0.783157	−	0.621824i	$-0.213608\pi$
$383$	11740.2	1.56631	0.783157	+	0.621824i	$0.213608\pi$
$384$	0	0
$385$	−492.294	−0.0651679
$386$	1686.79	0.222424
$387$	0	0
$388$	9650.45	1.26270
$389$	3763.68	0.490556	0.245278	−	0.969453i	$-0.421121\pi$
$389$	3763.68	0.490556	0.245278	+	0.969453i	$0.421121\pi$
$390$	0	0
$391$	22023.4	2.84852
$392$	2495.08	0.321481
$393$	0	0
$394$	−485.900	−0.0621302
$395$	2297.64	0.292676
$396$	0	0
$397$	1151.35	0.145554	0.0727768	−	0.997348i	$-0.476814\pi$
$397$	1151.35	0.145554	0.0727768	+	0.997348i	$0.476814\pi$
$398$	−1540.31	−0.193992
$399$	0	0
$400$	1382.19	0.172774
$401$	10051.0	1.25168	0.625842	−	0.779950i	$-0.284755\pi$
$401$	10051.0	1.25168	0.625842	+	0.779950i	$0.284755\pi$
$402$	0	0
$403$	−90.0688	−0.0111331
$404$	5271.66	0.649195
$405$	0	0
$406$	1233.37	0.150767
$407$	3724.42	0.453594
$408$	0	0
$409$	−6263.39	−0.757225	−0.378612	−	0.925555i	$-0.623599\pi$
$409$	−6263.39	−0.757225	−0.378612	+	0.925555i	$0.623599\pi$
$410$	415.843	0.0500903
$411$	0	0
$412$	89.9142	0.0107518
$413$	88.3403	0.0105253
$414$	0	0
$415$	−629.380	−0.0744459
$416$	−50.4123	−0.00594150
$417$	0	0
$418$	−0.164211	−1.92149e−5 0
$419$	−8625.67	−1.00571	−0.502854	−	0.864372i	$-0.667717\pi$
$419$	−8625.67	−1.00571	−0.502854	+	0.864372i	$0.667717\pi$
$420$	0	0
$421$	−9095.24	−1.05291	−0.526455	−	0.850203i	$-0.676479\pi$
$421$	−9095.24	−1.05291	−0.526455	+	0.850203i	$0.676479\pi$
$422$	970.868	0.111993
$423$	0	0
$424$	−5907.21	−0.676602
$425$	−3209.87	−0.366357
$426$	0	0
$427$	8079.99	0.915733
$428$	−3532.42	−0.398939
$429$	0	0
$430$	−1020.34	−0.114430
$431$	4008.72	0.448013	0.224006	−	0.974588i	$-0.428086\pi$
$431$	4008.72	0.448013	0.224006	+	0.974588i	$0.428086\pi$
$432$	0	0
$433$	2715.38	0.301369	0.150684	−	0.988582i	$-0.451852\pi$
$433$	2715.38	0.301369	0.150684	+	0.988582i	$0.451852\pi$
$434$	1063.26	0.117599
$435$	0	0
$436$	−2522.34	−0.277060
$437$	4.21717	0.000461635 0
$438$	0	0
$439$	−11087.5	−1.20541	−0.602707	−	0.797962i	$-0.705911\pi$
$439$	−11087.5	−1.20541	−0.602707	+	0.797962i	$0.705911\pi$
$440$	522.017	0.0565595
$441$	0	0
$442$	35.8919	0.00386245
$443$	−7132.98	−0.765007	−0.382503	−	0.923954i	$-0.624938\pi$
$443$	−7132.98	−0.765007	−0.382503	+	0.923954i	$0.624938\pi$
$444$	0	0
$445$	752.306	0.0801410
$446$	−3081.23	−0.327131
$447$	0	0
$448$	−3363.83	−0.354745
$449$	−6291.28	−0.661255	−0.330628	−	0.943761i	$-0.607260\pi$
$449$	−6291.28	−0.661255	−0.330628	+	0.943761i	$0.607260\pi$
$450$	0	0
$451$	−1506.70	−0.157312
$452$	8717.30	0.907140
$453$	0	0
$454$	−636.455	−0.0657936
$455$	20.6042	0.00212294
$456$	0	0
$457$	7291.98	0.746400	0.373200	−	0.927751i	$-0.378261\pi$
$457$	7291.98	0.746400	0.373200	+	0.927751i	$0.378261\pi$
$458$	−446.081	−0.0455109
$459$	0	0
$460$	−6544.94	−0.663390
$461$	16805.4	1.69784	0.848922	−	0.528519i	$-0.177252\pi$
$461$	16805.4	1.69784	0.848922	+	0.528519i	$0.177252\pi$
$462$	0	0
$463$	7478.48	0.750658	0.375329	−	0.926892i	$-0.377530\pi$
$463$	7478.48	0.750658	0.375329	+	0.926892i	$0.377530\pi$
$464$	12546.8	1.25533
$465$	0	0
$466$	1222.14	0.121491
$467$	7526.83	0.745824	0.372912	−	0.927867i	$-0.378359\pi$
$467$	7526.83	0.745824	0.372912	+	0.927867i	$0.378359\pi$
$468$	0	0
$469$	1315.58	0.129526
$470$	−1640.30	−0.160982
$471$	0	0
$472$	−93.6739	−0.00913494
$473$	3696.92	0.359375
$474$	0	0
$475$	−0.614645	−5.93723e−5 0
$476$	8770.20	0.844499
$477$	0	0
$478$	−3379.88	−0.323415
$479$	−1258.80	−0.120075	−0.0600376	−	0.998196i	$-0.519122\pi$
$479$	−1258.80	−0.120075	−0.0600376	+	0.998196i	$0.519122\pi$
$480$	0	0
$481$	−155.879	−0.0147765
$482$	−2040.93	−0.192866
$483$	0	0
$484$	−923.389	−0.0867195
$485$	6322.92	0.591978
$486$	0	0
$487$	5127.97	0.477147	0.238574	−	0.971124i	$-0.423320\pi$
$487$	5127.97	0.477147	0.238574	+	0.971124i	$0.423320\pi$
$488$	−8567.83	−0.794769
$489$	0	0
$490$	798.103	0.0735808
$491$	−11932.3	−1.09673	−0.548366	−	0.836238i	$-0.684750\pi$
$491$	−11932.3	−1.09673	−0.548366	+	0.836238i	$0.684750\pi$
$492$	0	0
$493$	−29137.7	−2.66185
$494$	0.00687279	6.25955e−7 0
$495$	0	0
$496$	10816.3	0.979166
$497$	8000.21	0.722049
$498$	0	0
$499$	14716.3	1.32022	0.660111	−	0.751168i	$-0.270509\pi$
$499$	14716.3	1.32022	0.660111	+	0.751168i	$0.270509\pi$
$500$	953.915	0.0853207
$501$	0	0
$502$	1779.35	0.158200
$503$	5598.95	0.496312	0.248156	−	0.968720i	$-0.420175\pi$
$503$	5598.95	0.496312	0.248156	+	0.968720i	$0.420175\pi$
$504$	0	0
$505$	3453.96	0.304355
$506$	−1145.66	−0.100654
$507$	0	0
$508$	−4804.60	−0.419625
$509$	16720.0	1.45600	0.727998	−	0.685579i	$-0.240451\pi$
$509$	16720.0	1.45600	0.727998	+	0.685579i	$0.240451\pi$
$510$	0	0
$511$	−10290.8	−0.890877
$512$	10251.9	0.884915
$513$	0	0
$514$	2072.28	0.177830
$515$	58.9113	0.00504066
$516$	0	0
$517$	5943.21	0.505574
$518$	1840.15	0.156084
$519$	0	0
$520$	−21.8481	−0.00184251
$521$	−3498.81	−0.294214	−0.147107	−	0.989121i	$-0.546996\pi$
$521$	−3498.81	−0.294214	−0.147107	+	0.989121i	$0.546996\pi$
$522$	0	0
$523$	−5681.71	−0.475036	−0.237518	−	0.971383i	$-0.576334\pi$
$523$	−5681.71	−0.475036	−0.237518	+	0.971383i	$0.576334\pi$
$524$	4369.44	0.364274
$525$	0	0
$526$	−2252.11	−0.186686
$527$	−25118.8	−2.07627
$528$	0	0
$529$	17255.0	1.41818
$530$	−1889.54	−0.154861
$531$	0	0
$532$	1.67937	0.000136861 0
$533$	63.0603	0.00512466
$534$	0	0
$535$	−2314.42	−0.187030
$536$	−1395.01	−0.112417
$537$	0	0
$538$	3500.09	0.280482
$539$	−2891.72	−0.231085
$540$	0	0
$541$	11641.7	0.925169	0.462585	−	0.886575i	$-0.346922\pi$
$541$	11641.7	0.925169	0.462585	+	0.886575i	$0.346922\pi$
$542$	−1145.36	−0.0907704
$543$	0	0
$544$	−14059.2	−1.10806
$545$	−1652.63	−0.129891
$546$	0	0
$547$	−16460.2	−1.28663	−0.643315	−	0.765602i	$-0.722441\pi$
$547$	−16460.2	−1.28663	−0.643315	+	0.765602i	$0.722441\pi$
$548$	−7239.68	−0.564350
$549$	0	0
$550$	166.978	0.0129454
$551$	−5.57945	−0.000431384 0
$552$	0	0
$553$	−4113.15	−0.316291
$554$	−2487.05	−0.190730
$555$	0	0
$556$	−18993.3	−1.44873
$557$	11769.7	0.895331	0.447666	−	0.894201i	$-0.352255\pi$
$557$	11769.7	0.895331	0.447666	+	0.894201i	$0.352255\pi$
$558$	0	0
$559$	−154.728	−0.0117072
$560$	−2474.34	−0.186714
$561$	0	0
$562$	5336.42	0.400540
$563$	−21123.9	−1.58129	−0.790644	−	0.612276i	$-0.790254\pi$
$563$	−21123.9	−1.58129	−0.790644	+	0.612276i	$0.790254\pi$
$564$	0	0
$565$	5711.53	0.425285
$566$	−1073.36	−0.0797118
$567$	0	0
$568$	−8483.23	−0.626670
$569$	21890.4	1.61282	0.806410	−	0.591357i	$-0.201408\pi$
$569$	21890.4	1.61282	0.806410	+	0.591357i	$0.201408\pi$
$570$	0	0
$571$	6630.99	0.485986	0.242993	−	0.970028i	$-0.421871\pi$
$571$	6630.99	0.485986	0.242993	+	0.970028i	$0.421871\pi$
$572$	38.6469	0.00282502
$573$	0	0
$574$	−744.425	−0.0541319
$575$	−4288.21	−0.311010
$576$	0	0
$577$	13361.3	0.964015	0.482007	−	0.876167i	$-0.339908\pi$
$577$	13361.3	0.964015	0.482007	+	0.876167i	$0.339908\pi$
$578$	7026.58	0.505653
$579$	0	0
$580$	8659.17	0.619918
$581$	1126.69	0.0804527
$582$	0	0
$583$	6846.26	0.486352
$584$	10912.1	0.773196
$585$	0	0
$586$	−1870.06	−0.131829
$587$	9451.34	0.664563	0.332282	−	0.943180i	$-0.392182\pi$
$587$	9451.34	0.664563	0.332282	+	0.943180i	$0.392182\pi$
$588$	0	0
$589$	−4.80990	−0.000336483 0
$590$	−29.9635	−0.00209081
$591$	0	0
$592$	18719.5	1.29960
$593$	−712.344	−0.0493296	−0.0246648	−	0.999696i	$-0.507852\pi$
$593$	−712.344	−0.0493296	−0.0246648	+	0.999696i	$0.507852\pi$
$594$	0	0
$595$	5746.19	0.395917
$596$	−16685.3	−1.14674
$597$	0	0
$598$	47.9496	0.00327894
$599$	−23951.9	−1.63380	−0.816901	−	0.576777i	$-0.804310\pi$
$599$	−23951.9	−1.63380	−0.816901	+	0.576777i	$0.804310\pi$
$600$	0	0
$601$	18577.3	1.26087	0.630434	−	0.776243i	$-0.282877\pi$
$601$	18577.3	1.26087	0.630434	+	0.776243i	$0.282877\pi$
$602$	1826.56	0.123663
$603$	0	0
$604$	−27993.3	−1.88581
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−6230.72	−0.416634	−0.208317	−	0.978061i	$-0.566799\pi$
$607$	−6230.72	−0.416634	−0.208317	+	0.978061i	$0.566799\pi$
$608$	−2.69214	−0.000179574 0
$609$	0	0
$610$	−2740.60	−0.181907
$611$	−248.743	−0.0164698
$612$	0	0
$613$	−17744.8	−1.16918	−0.584588	−	0.811330i	$-0.698744\pi$
$613$	−17744.8	−1.16918	−0.584588	+	0.811330i	$0.698744\pi$
$614$	1743.95	0.114626
$615$	0	0
$616$	−934.493	−0.0611230
$617$	−15305.4	−0.998656	−0.499328	−	0.866413i	$-0.666420\pi$
$617$	−15305.4	−0.998656	−0.499328	+	0.866413i	$0.666420\pi$
$618$	0	0
$619$	17372.3	1.12803	0.564015	−	0.825764i	$-0.309256\pi$
$619$	17372.3	1.12803	0.564015	+	0.825764i	$0.309256\pi$
$620$	7464.85	0.483541
$621$	0	0
$622$	193.074	0.0124462
$623$	−1346.75	−0.0866073
$624$	0	0
$625$	625.000	0.0400000
$626$	−5024.40	−0.320791
$627$	0	0
$628$	−10823.3	−0.687735
$629$	−43472.4	−2.75574
$630$	0	0
$631$	23699.3	1.49517	0.747586	−	0.664165i	$-0.231213\pi$
$631$	23699.3	1.49517	0.747586	+	0.664165i	$0.231213\pi$
$632$	4361.48	0.274510
$633$	0	0
$634$	−621.973	−0.0389617
$635$	−3147.95	−0.196728
$636$	0	0
$637$	121.028	0.00752795
$638$	1515.74	0.0940577
$639$	0	0
$640$	5520.95	0.340992
$641$	15423.9	0.950400	0.475200	−	0.879878i	$-0.342376\pi$
$641$	15423.9	0.950400	0.475200	+	0.879878i	$0.342376\pi$
$642$	0	0
$643$	13152.3	0.806652	0.403326	−	0.915056i	$-0.367854\pi$
$643$	13152.3	0.806652	0.403326	+	0.915056i	$0.367854\pi$
$644$	11716.5	0.716917
$645$	0	0
$646$	1.91672	0.000116737 0
$647$	5601.87	0.340390	0.170195	−	0.985410i	$-0.445560\pi$
$647$	5601.87	0.340390	0.170195	+	0.985410i	$0.445560\pi$
$648$	0	0
$649$	108.565	0.00656633
$650$	−6.98858	−0.000421715 0
$651$	0	0
$652$	4586.49	0.275492
$653$	27505.9	1.64837	0.824186	−	0.566319i	$-0.191633\pi$
$653$	27505.9	1.64837	0.824186	+	0.566319i	$0.191633\pi$
$654$	0	0
$655$	2862.83	0.170779
$656$	−7572.87	−0.450718
$657$	0	0
$658$	2936.41	0.173971
$659$	−16490.6	−0.974785	−0.487392	−	0.873183i	$-0.662052\pi$
$659$	−16490.6	−0.974785	−0.487392	+	0.873183i	$0.662052\pi$
$660$	0	0
$661$	12184.4	0.716973	0.358487	−	0.933535i	$-0.383293\pi$
$661$	12184.4	0.716973	0.358487	+	0.933535i	$0.383293\pi$
$662$	2778.41	0.163121
$663$	0	0
$664$	−1194.72	−0.0698253
$665$	1.10031	6.41629e−5 0
$666$	0	0
$667$	−38926.3	−2.25972
$668$	−16950.6	−0.981795
$669$	0	0
$670$	−446.223	−0.0257300
$671$	9929.83	0.571292
$672$	0	0
$673$	−13598.5	−0.778876	−0.389438	−	0.921053i	$-0.627331\pi$
$673$	−13598.5	−0.778876	−0.389438	+	0.921053i	$0.627331\pi$
$674$	−557.334	−0.0318512
$675$	0	0
$676$	16764.4	0.953823
$677$	−23318.1	−1.32377	−0.661883	−	0.749607i	$-0.730242\pi$
$677$	−23318.1	−1.32377	−0.661883	+	0.749607i	$0.730242\pi$
$678$	0	0
$679$	−11319.0	−0.639742
$680$	−6093.12	−0.343618
$681$	0	0
$682$	1306.68	0.0733658
$683$	−11493.5	−0.643906	−0.321953	−	0.946756i	$-0.604339\pi$
$683$	−11493.5	−0.643906	−0.321953	+	0.946756i	$0.604339\pi$
$684$	0	0
$685$	−4743.40	−0.264578
$686$	−3292.89	−0.183270
$687$	0	0
$688$	18581.2	1.02965
$689$	−286.539	−0.0158436
$690$	0	0
$691$	−20253.1	−1.11500	−0.557501	−	0.830176i	$-0.688240\pi$
$691$	−20253.1	−1.11500	−0.557501	+	0.830176i	$0.688240\pi$
$692$	−1810.76	−0.0994724
$693$	0	0
$694$	−3280.21	−0.179416
$695$	−12444.3	−0.679194
$696$	0	0
$697$	17586.6	0.955723
$698$	3106.30	0.168446
$699$	0	0
$700$	−1707.66	−0.0922050
$701$	−26895.3	−1.44910	−0.724551	−	0.689221i	$-0.757953\pi$
$701$	−26895.3	−1.44910	−0.724551	+	0.689221i	$0.757953\pi$
$702$	0	0
$703$	−8.32435	−0.000446599 0
$704$	−4133.94	−0.221312
$705$	0	0
$706$	1883.53	0.100407
$707$	−6183.14	−0.328912
$708$	0	0
$709$	−18567.3	−0.983512	−0.491756	−	0.870733i	$-0.663645\pi$
$709$	−18567.3	−0.983512	−0.491756	+	0.870733i	$0.663645\pi$
$710$	−2713.54	−0.143433
$711$	0	0
$712$	1428.06	0.0751668
$713$	−33557.4	−1.76260
$714$	0	0
$715$	25.3213	0.00132442
$716$	8133.21	0.424514
$717$	0	0
$718$	−2122.94	−0.110344
$719$	19223.5	0.997104	0.498552	−	0.866860i	$-0.333865\pi$
$719$	19223.5	0.997104	0.498552	+	0.866860i	$0.333865\pi$
$720$	0	0
$721$	−105.461	−0.00544738
$722$	−4164.73	−0.214675
$723$	0	0
$724$	9470.43	0.486140
$725$	5673.44	0.290630
$726$	0	0
$727$	−33855.9	−1.72716	−0.863581	−	0.504210i	$-0.831784\pi$
$727$	−33855.9	−1.72716	−0.863581	+	0.504210i	$0.831784\pi$
$728$	39.1117	0.00199117
$729$	0	0
$730$	3490.46	0.176970
$731$	−43151.4	−2.18333
$732$	0	0
$733$	26532.1	1.33695	0.668477	−	0.743733i	$-0.266947\pi$
$733$	26532.1	1.33695	0.668477	+	0.743733i	$0.266947\pi$
$734$	4641.71	0.233418
$735$	0	0
$736$	−18782.3	−0.940661
$737$	1616.77	0.0808067
$738$	0	0
$739$	−15358.9	−0.764528	−0.382264	−	0.924053i	$-0.624856\pi$
$739$	−15358.9	−0.764528	−0.382264	+	0.924053i	$0.624856\pi$
$740$	12919.2	0.641783
$741$	0	0
$742$	3382.58	0.167356
$743$	30995.4	1.53043	0.765216	−	0.643774i	$-0.222632\pi$
$743$	30995.4	1.53043	0.765216	+	0.643774i	$0.222632\pi$
$744$	0	0
$745$	−10932.2	−0.537615
$746$	−3151.27	−0.154660
$747$	0	0
$748$	10778.0	0.526851
$749$	4143.19	0.202121
$750$	0	0
$751$	−32222.4	−1.56566	−0.782831	−	0.622234i	$-0.786225\pi$
$751$	−32222.4	−1.56566	−0.782831	+	0.622234i	$0.786225\pi$
$752$	29871.4	1.44853
$753$	0	0
$754$	−63.4389	−0.00306407
$755$	−18341.0	−0.884105
$756$	0	0
$757$	−36596.5	−1.75710	−0.878549	−	0.477651i	$-0.841488\pi$
$757$	−36596.5	−1.75710	−0.878549	+	0.477651i	$0.841488\pi$
$758$	5206.01	0.249460
$759$	0	0
$760$	−1.16675	−5.56872e−5 0
$761$	−24718.0	−1.17743	−0.588716	−	0.808340i	$-0.700366\pi$
$761$	−24718.0	−1.17743	−0.588716	+	0.808340i	$0.700366\pi$
$762$	0	0
$763$	2958.46	0.140372
$764$	−14933.0	−0.707142
$765$	0	0
$766$	7128.58	0.336248
$767$	−4.54381	−0.000213908 0
$768$	0	0
$769$	36376.7	1.70582	0.852910	−	0.522058i	$-0.174836\pi$
$769$	36376.7	1.70582	0.852910	+	0.522058i	$0.174836\pi$
$770$	−298.917	−0.0139899
$771$	0	0
$772$	−21200.0	−0.988347
$773$	−7525.40	−0.350155	−0.175078	−	0.984555i	$-0.556018\pi$
$773$	−7525.40	−0.350155	−0.175078	+	0.984555i	$0.556018\pi$
$774$	0	0
$775$	4890.93	0.226693
$776$	12002.4	0.555235
$777$	0	0
$778$	2285.28	0.105310
$779$	3.36758	0.000154886 0
$780$	0	0
$781$	9831.78	0.450459
$782$	13372.4	0.611505
$783$	0	0
$784$	−14534.2	−0.662089
$785$	−7091.38	−0.322423
$786$	0	0
$787$	−41044.5	−1.85906	−0.929530	−	0.368747i	$-0.879787\pi$
$787$	−41044.5	−1.85906	−0.929530	+	0.368747i	$0.879787\pi$
$788$	6106.90	0.276078
$789$	0	0
$790$	1395.11	0.0628301
$791$	−10224.5	−0.459599
$792$	0	0
$793$	−415.597	−0.0186107
$794$	699.093	0.0312467
$795$	0	0
$796$	19359.0	0.862012
$797$	10836.9	0.481635	0.240817	−	0.970570i	$-0.422584\pi$
$797$	10836.9	0.481635	0.240817	+	0.970570i	$0.422584\pi$
$798$	0	0
$799$	−69370.7	−3.07154
$800$	2737.50	0.120981
$801$	0	0
$802$	6102.91	0.268705
$803$	−12646.8	−0.555785
$804$	0	0
$805$	7676.59	0.336104
$806$	−54.6890	−0.00239000
$807$	0	0
$808$	6556.45	0.285464
$809$	16093.5	0.699402	0.349701	−	0.936861i	$-0.386283\pi$
$809$	16093.5	0.699402	0.349701	+	0.936861i	$0.386283\pi$
$810$	0	0
$811$	7289.26	0.315611	0.157805	−	0.987470i	$-0.449558\pi$
$811$	7289.26	0.315611	0.157805	+	0.987470i	$0.449558\pi$
$812$	−15501.3	−0.669937
$813$	0	0
$814$	2261.44	0.0973751
$815$	3005.05	0.129156
$816$	0	0
$817$	−8.26288	−0.000353833 0
$818$	−3803.08	−0.162557
$819$	0	0
$820$	−5226.40	−0.222578
$821$	33153.4	1.40933	0.704666	−	0.709539i	$-0.251097\pi$
$821$	33153.4	1.40933	0.704666	+	0.709539i	$0.251097\pi$
$822$	0	0
$823$	−2781.57	−0.117812	−0.0589060	−	0.998264i	$-0.518761\pi$
$823$	−2781.57	−0.117812	−0.0589060	+	0.998264i	$0.518761\pi$
$824$	111.828	0.00472780
$825$	0	0
$826$	53.6395	0.00225951
$827$	4498.61	0.189156	0.0945780	−	0.995517i	$-0.469850\pi$
$827$	4498.61	0.189156	0.0945780	+	0.995517i	$0.469850\pi$
$828$	0	0
$829$	15630.3	0.654843	0.327421	−	0.944878i	$-0.393820\pi$
$829$	15630.3	0.654843	0.327421	+	0.944878i	$0.393820\pi$
$830$	−382.155	−0.0159817
$831$	0	0
$832$	173.019	0.00720958
$833$	33752.9	1.40392
$834$	0	0
$835$	−11106.0	−0.460284
$836$	2.06384	8.53822e−5 0
$837$	0	0
$838$	−5237.43	−0.215900
$839$	6182.34	0.254396	0.127198	−	0.991877i	$-0.459402\pi$
$839$	6182.34	0.254396	0.127198	+	0.991877i	$0.459402\pi$
$840$	0	0
$841$	27111.8	1.11164
$842$	−5522.56	−0.226033
$843$	0	0
$844$	−12202.1	−0.497646
$845$	10983.9	0.447170
$846$	0	0
$847$	1083.05	0.0439362
$848$	34410.3	1.39346
$849$	0	0
$850$	−1949.01	−0.0786476
$851$	−58076.7	−2.33942
$852$	0	0
$853$	31979.3	1.28365	0.641824	−	0.766852i	$-0.278178\pi$
$853$	31979.3	1.28365	0.641824	+	0.766852i	$0.278178\pi$
$854$	4906.11	0.196585
$855$	0	0
$856$	−4393.33	−0.175422
$857$	8439.63	0.336397	0.168199	−	0.985753i	$-0.446205\pi$
$857$	8439.63	0.336397	0.168199	+	0.985753i	$0.446205\pi$
$858$	0	0
$859$	47640.1	1.89227	0.946134	−	0.323774i	$-0.104952\pi$
$859$	47640.1	1.89227	0.946134	+	0.323774i	$0.104952\pi$
$860$	12823.8	0.508474
$861$	0	0
$862$	2434.06	0.0961770
$863$	−27963.3	−1.10299	−0.551496	−	0.834177i	$-0.685943\pi$
$863$	−27963.3	−1.10299	−0.551496	+	0.834177i	$0.685943\pi$
$864$	0	0
$865$	−1186.40	−0.0466346
$866$	1648.75	0.0646962
$867$	0	0
$868$	−13363.3	−0.522557
$869$	−5054.81	−0.197322
$870$	0	0
$871$	−67.6673	−0.00263240
$872$	−3137.08	−0.121829
$873$	0	0
$874$	2.56063	9.91013e−5 0
$875$	−1118.85	−0.0432275
$876$	0	0
$877$	−30584.0	−1.17759	−0.588796	−	0.808282i	$-0.700398\pi$
$877$	−30584.0	−1.17759	−0.588796	+	0.808282i	$0.700398\pi$
$878$	−6732.23	−0.258772
$879$	0	0
$880$	−3040.82	−0.116484
$881$	−14444.3	−0.552372	−0.276186	−	0.961104i	$-0.589071\pi$
$881$	−14444.3	−0.552372	−0.276186	+	0.961104i	$0.589071\pi$
$882$	0	0
$883$	−19921.4	−0.759241	−0.379620	−	0.925142i	$-0.623945\pi$
$883$	−19921.4	−0.759241	−0.379620	+	0.925142i	$0.623945\pi$
$884$	−451.097	−0.0171630
$885$	0	0
$886$	−4331.09	−0.164228
$887$	13172.5	0.498635	0.249317	−	0.968422i	$-0.419794\pi$
$887$	13172.5	0.498635	0.249317	+	0.968422i	$0.419794\pi$
$888$	0	0
$889$	5635.33	0.212602
$890$	456.794	0.0172042
$891$	0	0
$892$	38725.5	1.45362
$893$	−13.2835	−0.000497778 0
$894$	0	0
$895$	5328.83	0.199020
$896$	−9883.38	−0.368505
$897$	0	0
$898$	−3820.01	−0.141955
$899$	44397.5	1.64710
$900$	0	0
$901$	−79911.4	−2.95475
$902$	−914.854	−0.0337709
$903$	0	0
$904$	10841.9	0.398888
$905$	6204.98	0.227912
$906$	0	0
$907$	16309.7	0.597084	0.298542	−	0.954396i	$-0.403500\pi$
$907$	16309.7	0.597084	0.298542	+	0.954396i	$0.403500\pi$
$908$	7999.10	0.292356
$909$	0	0
$910$	12.5107	0.000455741 0
$911$	14994.4	0.545320	0.272660	−	0.962110i	$-0.412097\pi$
$911$	14994.4	0.545320	0.272660	+	0.962110i	$0.412097\pi$
$912$	0	0
$913$	1384.64	0.0501914
$914$	4427.63	0.160233
$915$	0	0
$916$	5606.44	0.202229
$917$	−5124.93	−0.184558
$918$	0	0
$919$	40263.5	1.44524	0.722618	−	0.691248i	$-0.242939\pi$
$919$	40263.5	1.44524	0.722618	+	0.691248i	$0.242939\pi$
$920$	−8140.06	−0.291706
$921$	0	0
$922$	10204.1	0.364484
$923$	−411.493	−0.0146744
$924$	0	0
$925$	8464.59	0.300880
$926$	4540.87	0.161147
$927$	0	0
$928$	24849.6	0.879019
$929$	9469.52	0.334429	0.167215	−	0.985921i	$-0.446523\pi$
$929$	9469.52	0.334429	0.167215	+	0.985921i	$0.446523\pi$
$930$	0	0
$931$	6.46319	0.000227522 0
$932$	−15360.2	−0.539848
$933$	0	0
$934$	4570.23	0.160110
$935$	7061.72	0.246998
$936$	0	0
$937$	6356.51	0.221620	0.110810	−	0.993842i	$-0.464655\pi$
$937$	6356.51	0.221620	0.110810	+	0.993842i	$0.464655\pi$
$938$	798.810	0.0278061
$939$	0	0
$940$	20615.7	0.715329
$941$	5764.45	0.199698	0.0998489	−	0.995003i	$-0.468164\pi$
$941$	5764.45	0.199698	0.0998489	+	0.995003i	$0.468164\pi$
$942$	0	0
$943$	23494.7	0.811338
$944$	545.663	0.0188134
$945$	0	0
$946$	2244.74	0.0771487
$947$	2284.74	0.0783993	0.0391997	−	0.999231i	$-0.487519\pi$
$947$	2284.74	0.0783993	0.0391997	+	0.999231i	$0.487519\pi$
$948$	0	0
$949$	529.310	0.0181055
$950$	−0.373207	−1.27457e−5 0
$951$	0	0
$952$	10907.7	0.371344
$953$	17209.0	0.584947	0.292474	−	0.956274i	$-0.405522\pi$
$953$	17209.0	0.584947	0.292474	+	0.956274i	$0.405522\pi$
$954$	0	0
$955$	−9784.02	−0.331522
$956$	42479.1	1.43710
$957$	0	0
$958$	−764.333	−0.0257771
$959$	8491.45	0.285926
$960$	0	0
$961$	8482.92	0.284748
$962$	−94.6487	−0.00317214
$963$	0	0
$964$	25650.8	0.857008
$965$	−13890.1	−0.463356
$966$	0	0
$967$	−7966.25	−0.264920	−0.132460	−	0.991188i	$-0.542288\pi$
$967$	−7966.25	−0.264920	−0.132460	+	0.991188i	$0.542288\pi$
$968$	−1148.44	−0.0381324
$969$	0	0
$970$	3839.23	0.127083
$971$	25547.5	0.844343	0.422172	−	0.906516i	$-0.361268\pi$
$971$	25547.5	0.844343	0.422172	+	0.906516i	$0.361268\pi$
$972$	0	0
$973$	22277.3	0.733996
$974$	3113.66	0.102431
$975$	0	0
$976$	49908.7	1.63682
$977$	−12899.8	−0.422417	−0.211209	−	0.977441i	$-0.567740\pi$
$977$	−12899.8	−0.422417	−0.211209	+	0.977441i	$0.567740\pi$
$978$	0	0
$979$	−1655.07	−0.0540310
$980$	−10030.7	−0.326959
$981$	0	0
$982$	−7245.18	−0.235441
$983$	7399.28	0.240082	0.120041	−	0.992769i	$-0.461697\pi$
$983$	7399.28	0.240082	0.120041	+	0.992769i	$0.461697\pi$
$984$	0	0
$985$	4001.21	0.129431
$986$	−17692.2	−0.571433
$987$	0	0
$988$	−0.0863787	−2.78145e−6 0
$989$	−57647.8	−1.85348
$990$	0	0
$991$	−38466.4	−1.23302	−0.616511	−	0.787346i	$-0.711454\pi$
$991$	−38466.4	−1.23302	−0.616511	+	0.787346i	$0.711454\pi$
$992$	21422.2	0.685642
$993$	0	0
$994$	4857.66	0.155006
$995$	12683.9	0.404128
$996$	0	0
$997$	−62092.5	−1.97241	−0.986203	−	0.165538i	$-0.947064\pi$
$997$	−62092.5	−1.97241	−0.986203	+	0.165538i	$0.947064\pi$
$998$	8935.60	0.283418
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.m.1.3		4
3.2	odd	2		165.4.a.h.1.2	✓	4
5.4	even	2		2475.4.a.be.1.2		4
15.2	even	4		825.4.c.p.199.4		8
15.8	even	4		825.4.c.p.199.5		8
15.14	odd	2		825.4.a.t.1.3		4
33.32	even	2		1815.4.a.t.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.h.1.2	✓	4	3.2	odd	2
495.4.a.m.1.3		4	1.1	even	1	trivial
825.4.a.t.1.3		4	15.14	odd	2
825.4.c.p.199.4		8	15.2	even	4
825.4.c.p.199.5		8	15.8	even	4
1815.4.a.t.1.3		4	33.32	even	2
2475.4.a.be.1.2		4	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)

\(f(q)\)	\(=\)	\(q+0.607192 q^{2} -7.63132 q^{4} -5.00000 q^{5} +8.95080 q^{7} -9.49121 q^{8} +O(q^{10})\)	\(q+0.607192 q^{2} -7.63132 q^{4} -5.00000 q^{5} +8.95080 q^{7} -9.49121 q^{8} -3.03596 q^{10} +11.0000 q^{11} -0.460387 q^{13} +5.43485 q^{14} +55.2876 q^{16} -128.395 q^{17} -0.0245858 q^{19} +38.1566 q^{20} +6.67911 q^{22} -171.528 q^{23} +25.0000 q^{25} -0.279543 q^{26} -68.3064 q^{28} +226.938 q^{29} +195.637 q^{31} +109.500 q^{32} -77.9604 q^{34} -44.7540 q^{35} +338.584 q^{37} -0.0149283 q^{38} +47.4560 q^{40} -136.972 q^{41} +336.083 q^{43} -83.9445 q^{44} -104.151 q^{46} +540.292 q^{47} -262.883 q^{49} +15.1798 q^{50} +3.51336 q^{52} +622.387 q^{53} -55.0000 q^{55} -84.9539 q^{56} +137.795 q^{58} +9.86955 q^{59} +902.712 q^{61} +118.789 q^{62} -375.813 q^{64} +2.30193 q^{65} +146.979 q^{67} +979.823 q^{68} -27.1743 q^{70} +893.798 q^{71} -1149.71 q^{73} +205.585 q^{74} +0.187622 q^{76} +98.4588 q^{77} -459.528 q^{79} -276.438 q^{80} -83.1686 q^{82} +125.876 q^{83} +641.975 q^{85} +204.067 q^{86} -104.403 q^{88} -150.461 q^{89} -4.12083 q^{91} +1308.99 q^{92} +328.061 q^{94} +0.122929 q^{95} -1264.58 q^{97} -159.621 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 20 q^{5} + 34 q^{7} - 48 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 20 * q^5 + 34 * q^7 - 48 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 20 q^{5} + 34 q^{7} - 48 q^{8} + 20 q^{10} + 44 q^{11} + 2 q^{13} + 52 q^{14} + 66 q^{16} - 74 q^{17} + 136 q^{19} - 130 q^{20} - 44 q^{22} + 64 q^{23} + 100 q^{25} + 320 q^{26} - 20 q^{28} - 52 q^{29} + 492 q^{31} - 208 q^{32} + 244 q^{34} - 170 q^{35} - 4 q^{37} + 404 q^{38} + 240 q^{40} - 268 q^{41} + 546 q^{43} + 286 q^{44} + 368 q^{46} + 276 q^{47} - 496 q^{49} - 100 q^{50} - 1084 q^{52} + 184 q^{53} - 220 q^{55} + 852 q^{56} - 444 q^{58} + 1032 q^{59} + 116 q^{61} + 1240 q^{62} - 918 q^{64} - 10 q^{65} - 552 q^{67} + 720 q^{68} - 260 q^{70} + 920 q^{71} + 926 q^{73} + 2856 q^{74} + 1572 q^{76} + 374 q^{77} + 1152 q^{79} - 330 q^{80} - 1924 q^{82} + 134 q^{83} + 370 q^{85} - 236 q^{86} - 528 q^{88} + 1064 q^{89} + 2780 q^{91} + 4896 q^{92} - 1432 q^{94} - 680 q^{95} - 1648 q^{97} + 188 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 20 * q^5 + 34 * q^7 - 48 * q^8 + 20 * q^10 + 44 * q^11 + 2 * q^13 + 52 * q^14 + 66 * q^16 - 74 * q^17 + 136 * q^19 - 130 * q^20 - 44 * q^22 + 64 * q^23 + 100 * q^25 + 320 * q^26 - 20 * q^28 - 52 * q^29 + 492 * q^31 - 208 * q^32 + 244 * q^34 - 170 * q^35 - 4 * q^37 + 404 * q^38 + 240 * q^40 - 268 * q^41 + 546 * q^43 + 286 * q^44 + 368 * q^46 + 276 * q^47 - 496 * q^49 - 100 * q^50 - 1084 * q^52 + 184 * q^53 - 220 * q^55 + 852 * q^56 - 444 * q^58 + 1032 * q^59 + 116 * q^61 + 1240 * q^62 - 918 * q^64 - 10 * q^65 - 552 * q^67 + 720 * q^68 - 260 * q^70 + 920 * q^71 + 926 * q^73 + 2856 * q^74 + 1572 * q^76 + 374 * q^77 + 1152 * q^79 - 330 * q^80 - 1924 * q^82 + 134 * q^83 + 370 * q^85 - 236 * q^86 - 528 * q^88 + 1064 * q^89 + 2780 * q^91 + 4896 * q^92 - 1432 * q^94 - 680 * q^95 - 1648 * q^97 + 188 * q^98

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