LMFDB - Embedded newform 495.4.a.m.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:	$4$
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			1.1
Root			$-4.17080$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.17080 q^{2} +18.7372 q^{4} -5.00000 q^{5} -11.1745 q^{7} -55.5199 q^{8} +O(q^{10})$	\(q-5.17080 q^{2} +18.7372 q^{4} -5.00000 q^{5} -11.1745 q^{7} -55.5199 q^{8} +25.8540 q^{10} +11.0000 q^{11} -89.5310 q^{13} +57.7813 q^{14} +137.185 q^{16} -58.3242 q^{17} +24.5575 q^{19} -93.6859 q^{20} -56.8788 q^{22} +111.696 q^{23} +25.0000 q^{25} +462.947 q^{26} -209.379 q^{28} -109.954 q^{29} +119.547 q^{31} -265.196 q^{32} +301.583 q^{34} +55.8726 q^{35} -356.544 q^{37} -126.982 q^{38} +277.599 q^{40} -268.798 q^{41} +263.371 q^{43} +206.109 q^{44} -577.557 q^{46} +206.732 q^{47} -218.130 q^{49} -129.270 q^{50} -1677.56 q^{52} -223.749 q^{53} -55.0000 q^{55} +620.408 q^{56} +568.549 q^{58} -475.000 q^{59} -513.204 q^{61} -618.153 q^{62} +273.798 q^{64} +447.655 q^{65} -264.533 q^{67} -1092.83 q^{68} -288.906 q^{70} +1110.33 q^{71} +893.608 q^{73} +1843.62 q^{74} +460.139 q^{76} -122.920 q^{77} +1303.66 q^{79} -685.923 q^{80} +1389.90 q^{82} -1049.37 q^{83} +291.621 q^{85} -1361.84 q^{86} -610.718 q^{88} +1417.91 q^{89} +1000.47 q^{91} +2092.86 q^{92} -1068.97 q^{94} -122.788 q^{95} +85.8091 q^{97} +1127.91 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.17080	−1.82815	−0.914077	−	0.405540i	$-0.867083\pi$
$2$	−5.17080	−1.82815	−0.914077	+	0.405540i	$0.867083\pi$
$3$	0	0
$3$	0	0
$4$	18.7372	2.34215
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−11.1745	−0.603368	−0.301684	−	0.953408i	$-0.597549\pi$
$7$	−11.1745	−0.603368	−0.301684	+	0.953408i	$0.597549\pi$
$8$	−55.5199	−2.45365
$9$	0	0
$10$	25.8540	0.817575
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−89.5310	−1.91011	−0.955056	−	0.296427i	$-0.904205\pi$
$13$	−89.5310	−1.91011	−0.955056	+	0.296427i	$0.904205\pi$
$14$	57.7813	1.10305
$15$	0	0
$16$	137.185	2.14351
$17$	−58.3242	−0.832100	−0.416050	−	0.909342i	$-0.636586\pi$
$17$	−58.3242	−0.832100	−0.416050	+	0.909342i	$0.636586\pi$
$18$	0	0
$19$	24.5575	0.296520	0.148260	−	0.988948i	$-0.452633\pi$
$19$	24.5575	0.296520	0.148260	+	0.988948i	$0.452633\pi$
$20$	−93.6859	−1.04744
$21$	0	0
$22$	−56.8788	−0.551209
$23$	111.696	1.01262	0.506308	−	0.862353i	$-0.331010\pi$
$23$	111.696	1.01262	0.506308	+	0.862353i	$0.331010\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	462.947	3.49198
$27$	0	0
$28$	−209.379	−1.41318
$29$	−109.954	−0.704066	−0.352033	−	0.935988i	$-0.614510\pi$
$29$	−109.954	−0.704066	−0.352033	+	0.935988i	$0.614510\pi$
$30$	0	0
$31$	119.547	0.692621	0.346310	−	0.938120i	$-0.387434\pi$
$31$	119.547	0.692621	0.346310	+	0.938120i	$0.387434\pi$
$32$	−265.196	−1.46501
$33$	0	0
$34$	301.583	1.52121
$35$	55.8726	0.269834
$36$	0	0
$37$	−356.544	−1.58420	−0.792100	−	0.610391i	$-0.791012\pi$
$37$	−356.544	−1.58420	−0.792100	+	0.610391i	$0.791012\pi$
$38$	−126.982	−0.542085
$39$	0	0
$40$	277.599	1.09731
$41$	−268.798	−1.02388	−0.511941	−	0.859021i	$-0.671073\pi$
$41$	−268.798	−1.02388	−0.511941	+	0.859021i	$0.671073\pi$
$42$	0	0
$43$	263.371	0.934038	0.467019	−	0.884247i	$-0.345328\pi$
$43$	263.371	0.934038	0.467019	+	0.884247i	$0.345328\pi$
$44$	206.109	0.706184
$45$	0	0
$46$	−577.557	−1.85122
$47$	206.732	0.641594	0.320797	−	0.947148i	$-0.396049\pi$
$47$	206.732	0.641594	0.320797	+	0.947148i	$0.396049\pi$
$48$	0	0
$49$	−218.130	−0.635947
$50$	−129.270	−0.365631
$51$	0	0
$52$	−1677.56	−4.47376
$53$	−223.749	−0.579891	−0.289946	−	0.957043i	$-0.593637\pi$
$53$	−223.749	−0.579891	−0.289946	+	0.957043i	$0.593637\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	620.408	1.48046
$57$	0	0
$58$	568.549	1.28714
$59$	−475.000	−1.04813	−0.524066	−	0.851678i	$-0.675585\pi$
$59$	−475.000	−1.04813	−0.524066	+	0.851678i	$0.675585\pi$
$60$	0	0
$61$	−513.204	−1.07720	−0.538598	−	0.842563i	$-0.681046\pi$
$61$	−513.204	−1.07720	−0.538598	+	0.842563i	$0.681046\pi$
$62$	−618.153	−1.26622
$63$	0	0
$64$	273.798	0.534761
$65$	447.655	0.854228
$66$	0	0
$67$	−264.533	−0.482355	−0.241178	−	0.970481i	$-0.577534\pi$
$67$	−264.533	−0.482355	−0.241178	+	0.970481i	$0.577534\pi$
$68$	−1092.83	−1.94890
$69$	0	0
$70$	−288.906	−0.493299
$71$	1110.33	1.85595	0.927973	−	0.372648i	$-0.121550\pi$
$71$	1110.33	1.85595	0.927973	+	0.372648i	$0.121550\pi$
$72$	0	0
$73$	893.608	1.43273	0.716363	−	0.697728i	$-0.245806\pi$
$73$	893.608	1.43273	0.716363	+	0.697728i	$0.245806\pi$
$74$	1843.62	2.89616
$75$	0	0
$76$	460.139	0.694494
$77$	−122.920	−0.181922
$78$	0	0
$79$	1303.66	1.85663	0.928314	−	0.371797i	$-0.121258\pi$
$79$	1303.66	1.85663	0.928314	+	0.371797i	$0.121258\pi$
$80$	−685.923	−0.958607
$81$	0	0
$82$	1389.90	1.87181
$83$	−1049.37	−1.38775	−0.693873	−	0.720098i	$-0.744097\pi$
$83$	−1049.37	−1.38775	−0.693873	+	0.720098i	$0.744097\pi$
$84$	0	0
$85$	291.621	0.372126
$86$	−1361.84	−1.70757
$87$	0	0
$88$	−610.718	−0.739805
$89$	1417.91	1.68875	0.844374	−	0.535754i	$-0.179972\pi$
$89$	1417.91	1.68875	0.844374	+	0.535754i	$0.179972\pi$
$90$	0	0
$91$	1000.47	1.15250
$92$	2092.86	2.37170
$93$	0	0
$94$	−1068.97	−1.17293
$95$	−122.788	−0.132608
$96$	0	0
$97$	85.8091	0.0898206	0.0449103	−	0.998991i	$-0.485700\pi$
$97$	85.8091	0.0898206	0.0449103	+	0.998991i	$0.485700\pi$
$98$	1127.91	1.16261
$99$	0	0
$100$	468.430	0.468430
$101$	−137.761	−0.135721	−0.0678603	−	0.997695i	$-0.521617\pi$
$101$	−137.761	−0.135721	−0.0678603	+	0.997695i	$0.521617\pi$
$102$	0	0
$103$	419.397	0.401208	0.200604	−	0.979672i	$-0.435710\pi$
$103$	419.397	0.401208	0.200604	+	0.979672i	$0.435710\pi$
$104$	4970.75	4.68675
$105$	0	0
$106$	1156.96	1.06013
$107$	1190.45	1.07557	0.537783	−	0.843083i	$-0.319262\pi$
$107$	1190.45	1.07557	0.537783	+	0.843083i	$0.319262\pi$
$108$	0	0
$109$	1033.37	0.908066	0.454033	−	0.890985i	$-0.349985\pi$
$109$	1033.37	0.908066	0.454033	+	0.890985i	$0.349985\pi$
$110$	284.394	0.246508
$111$	0	0
$112$	−1532.97	−1.29333
$113$	930.760	0.774854	0.387427	−	0.921900i	$-0.373364\pi$
$113$	930.760	0.774854	0.387427	+	0.921900i	$0.373364\pi$
$114$	0	0
$115$	−558.479	−0.452856
$116$	−2060.23	−1.64903
$117$	0	0
$118$	2456.13	1.91615
$119$	651.746	0.502062
$120$	0	0
$121$	121.000	0.0909091
$122$	2653.68	1.96928
$123$	0	0
$124$	2239.97	1.62222
$125$	−125.000	−0.0894427
$126$	0	0
$127$	94.4315	0.0659799	0.0329899	−	0.999456i	$-0.489497\pi$
$127$	94.4315	0.0659799	0.0329899	+	0.999456i	$0.489497\pi$
$128$	705.814	0.487389
$129$	0	0
$130$	−2314.74	−1.56166
$131$	−1318.12	−0.879122	−0.439561	−	0.898213i	$-0.644866\pi$
$131$	−1318.12	−0.879122	−0.439561	+	0.898213i	$0.644866\pi$
$132$	0	0
$133$	−274.419	−0.178911
$134$	1367.85	0.881820
$135$	0	0
$136$	3238.15	2.04169
$137$	−2008.43	−1.25249	−0.626247	−	0.779624i	$-0.715410\pi$
$137$	−2008.43	−1.25249	−0.626247	+	0.779624i	$0.715410\pi$
$138$	0	0
$139$	2956.76	1.80424	0.902120	−	0.431486i	$-0.142011\pi$
$139$	2956.76	1.80424	0.902120	+	0.431486i	$0.142011\pi$
$140$	1046.90	0.631992
$141$	0	0
$142$	−5741.30	−3.39295
$143$	−984.841	−0.575920
$144$	0	0
$145$	549.769	0.314868
$146$	−4620.67	−2.61924
$147$	0	0
$148$	−6680.62	−3.71043
$149$	−878.768	−0.483164	−0.241582	−	0.970380i	$-0.577666\pi$
$149$	−878.768	−0.483164	−0.241582	+	0.970380i	$0.577666\pi$
$150$	0	0
$151$	−1679.35	−0.905058	−0.452529	−	0.891750i	$-0.649478\pi$
$151$	−1679.35	−0.905058	−0.452529	+	0.891750i	$0.649478\pi$
$152$	−1363.43	−0.727558
$153$	0	0
$154$	635.594	0.332582
$155$	−597.734	−0.309749
$156$	0	0
$157$	703.051	0.357386	0.178693	−	0.983905i	$-0.442813\pi$
$157$	703.051	0.357386	0.178693	+	0.983905i	$0.442813\pi$
$158$	−6740.99	−3.39420
$159$	0	0
$160$	1325.98	0.655174
$161$	−1248.15	−0.610980
$162$	0	0
$163$	2934.66	1.41019	0.705093	−	0.709114i	$-0.250905\pi$
$163$	2934.66	1.41019	0.705093	+	0.709114i	$0.250905\pi$
$164$	−5036.51	−2.39808
$165$	0	0
$166$	5426.06	2.53701
$167$	2146.05	0.994407	0.497204	−	0.867634i	$-0.334360\pi$
$167$	2146.05	0.994407	0.497204	+	0.867634i	$0.334360\pi$
$168$	0	0
$169$	5818.81	2.64852
$170$	−1507.91	−0.680305
$171$	0	0
$172$	4934.82	2.18766
$173$	−321.356	−0.141227	−0.0706134	−	0.997504i	$-0.522496\pi$
$173$	−321.356	−0.141227	−0.0706134	+	0.997504i	$0.522496\pi$
$174$	0	0
$175$	−279.363	−0.120674
$176$	1509.03	0.646293
$177$	0	0
$178$	−7331.75	−3.08729
$179$	2662.08	1.11158	0.555791	−	0.831322i	$-0.312415\pi$
$179$	2662.08	1.11158	0.555791	+	0.831322i	$0.312415\pi$
$180$	0	0
$181$	3306.02	1.35765	0.678825	−	0.734300i	$-0.262490\pi$
$181$	3306.02	1.35765	0.678825	+	0.734300i	$0.262490\pi$
$182$	−5173.22	−2.10695
$183$	0	0
$184$	−6201.33	−2.48461
$185$	1782.72	0.708476
$186$	0	0
$187$	−641.566	−0.250888
$188$	3873.57	1.50271
$189$	0	0
$190$	634.910	0.242428
$191$	−946.068	−0.358404	−0.179202	−	0.983812i	$-0.557352\pi$
$191$	−946.068	−0.358404	−0.179202	+	0.983812i	$0.557352\pi$
$192$	0	0
$193$	−3692.54	−1.37717	−0.688587	−	0.725154i	$-0.741769\pi$
$193$	−3692.54	−1.37717	−0.688587	+	0.725154i	$0.741769\pi$
$194$	−443.702	−0.164206
$195$	0	0
$196$	−4087.14	−1.48948
$197$	411.503	0.148824	0.0744122	−	0.997228i	$-0.476292\pi$
$197$	411.503	0.148824	0.0744122	+	0.997228i	$0.476292\pi$
$198$	0	0
$199$	1491.28	0.531226	0.265613	−	0.964080i	$-0.414426\pi$
$199$	1491.28	0.531226	0.265613	+	0.964080i	$0.414426\pi$
$200$	−1388.00	−0.490731
$201$	0	0
$202$	712.337	0.248118
$203$	1228.68	0.424811
$204$	0	0
$205$	1343.99	0.457894
$206$	−2168.62	−0.733470
$207$	0	0
$208$	−12282.3	−4.09434
$209$	270.133	0.0894042
$210$	0	0
$211$	−899.947	−0.293625	−0.146813	−	0.989164i	$-0.546901\pi$
$211$	−899.947	−0.293625	−0.146813	+	0.989164i	$0.546901\pi$
$212$	−4192.42	−1.35819
$213$	0	0
$214$	−6155.61	−1.96630
$215$	−1316.85	−0.417715
$216$	0	0
$217$	−1335.88	−0.417905
$218$	−5343.37	−1.66009
$219$	0	0
$220$	−1030.55	−0.315815
$221$	5221.83	1.58940
$222$	0	0
$223$	1935.29	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$223$	1935.29	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$224$	2963.44	0.883942
$225$	0	0
$226$	−4812.77	−1.41655
$227$	2906.26	0.849758	0.424879	−	0.905250i	$-0.360317\pi$
$227$	2906.26	0.849758	0.424879	+	0.905250i	$0.360317\pi$
$228$	0	0
$229$	−5411.89	−1.56169	−0.780846	−	0.624724i	$-0.785212\pi$
$229$	−5411.89	−1.56169	−0.780846	+	0.624724i	$0.785212\pi$
$230$	2887.78	0.827890
$231$	0	0
$232$	6104.62	1.72753
$233$	4778.90	1.34368	0.671838	−	0.740699i	$-0.265505\pi$
$233$	4778.90	1.34368	0.671838	+	0.740699i	$0.265505\pi$
$234$	0	0
$235$	−1033.66	−0.286930
$236$	−8900.17	−2.45488
$237$	0	0
$238$	−3370.05	−0.917848
$239$	−5883.88	−1.59245	−0.796227	−	0.604998i	$-0.793174\pi$
$239$	−5883.88	−1.59245	−0.796227	+	0.604998i	$0.793174\pi$
$240$	0	0
$241$	1806.96	0.482974	0.241487	−	0.970404i	$-0.422365\pi$
$241$	1806.96	0.482974	0.241487	+	0.970404i	$0.422365\pi$
$242$	−625.667	−0.166196
$243$	0	0
$244$	−9616.00	−2.52296
$245$	1090.65	0.284404
$246$	0	0
$247$	−2198.66	−0.566386
$248$	−6637.22	−1.69945
$249$	0	0
$250$	646.350	0.163515
$251$	−1265.10	−0.318137	−0.159069	−	0.987268i	$-0.550849\pi$
$251$	−1265.10	−0.318137	−0.159069	+	0.987268i	$0.550849\pi$
$252$	0	0
$253$	1228.65	0.305315
$254$	−488.287	−0.120621
$255$	0	0
$256$	−5840.00	−1.42578
$257$	4366.53	1.05983	0.529916	−	0.848050i	$-0.322223\pi$
$257$	4366.53	1.05983	0.529916	+	0.848050i	$0.322223\pi$
$258$	0	0
$259$	3984.21	0.955855
$260$	8387.80	2.00073
$261$	0	0
$262$	6815.76	1.60717
$263$	2950.96	0.691878	0.345939	−	0.938257i	$-0.387560\pi$
$263$	2950.96	0.691878	0.345939	+	0.938257i	$0.387560\pi$
$264$	0	0
$265$	1118.74	0.259335
$266$	1418.97	0.327076
$267$	0	0
$268$	−4956.60	−1.12975
$269$	6048.65	1.37098	0.685488	−	0.728084i	$-0.259589\pi$
$269$	6048.65	1.37098	0.685488	+	0.728084i	$0.259589\pi$
$270$	0	0
$271$	−2922.90	−0.655180	−0.327590	−	0.944820i	$-0.606236\pi$
$271$	−2922.90	−0.655180	−0.327590	+	0.944820i	$0.606236\pi$
$272$	−8001.19	−1.78362
$273$	0	0
$274$	10385.2	2.28975
$275$	275.000	0.0603023
$276$	0	0
$277$	−5433.48	−1.17858	−0.589289	−	0.807922i	$-0.700592\pi$
$277$	−5433.48	−1.17858	−0.589289	+	0.807922i	$0.700592\pi$
$278$	−15288.8	−3.29843
$279$	0	0
$280$	−3102.04	−0.662080
$281$	7981.02	1.69433	0.847167	−	0.531327i	$-0.178307\pi$
$281$	7981.02	1.69433	0.847167	+	0.531327i	$0.178307\pi$
$282$	0	0
$283$	−2705.71	−0.568332	−0.284166	−	0.958775i	$-0.591717\pi$
$283$	−2705.71	−0.568332	−0.284166	+	0.958775i	$0.591717\pi$
$284$	20804.5	4.34690
$285$	0	0
$286$	5092.42	1.05287
$287$	3003.69	0.617777
$288$	0	0
$289$	−1511.29	−0.307610
$290$	−2842.75	−0.575627
$291$	0	0
$292$	16743.7	3.35565
$293$	−2207.71	−0.440190	−0.220095	−	0.975478i	$-0.570637\pi$
$293$	−2207.71	−0.440190	−0.220095	+	0.975478i	$0.570637\pi$
$294$	0	0
$295$	2375.00	0.468739
$296$	19795.2	3.88708
$297$	0	0
$298$	4543.94	0.883299
$299$	−10000.2	−1.93421
$300$	0	0
$301$	−2943.04	−0.563569
$302$	8683.60	1.65459
$303$	0	0
$304$	3368.92	0.635594
$305$	2566.02	0.481737
$306$	0	0
$307$	−3918.45	−0.728462	−0.364231	−	0.931309i	$-0.618668\pi$
$307$	−3918.45	−0.728462	−0.364231	+	0.931309i	$0.618668\pi$
$308$	−2303.17	−0.426089
$309$	0	0
$310$	3090.76	0.566270
$311$	−8769.03	−1.59886	−0.799431	−	0.600758i	$-0.794866\pi$
$311$	−8769.03	−1.59886	−0.799431	+	0.600758i	$0.794866\pi$
$312$	0	0
$313$	−5058.92	−0.913569	−0.456785	−	0.889577i	$-0.650999\pi$
$313$	−5058.92	−0.913569	−0.456785	+	0.889577i	$0.650999\pi$
$314$	−3635.34	−0.653357
$315$	0	0
$316$	24427.0	4.34850
$317$	8329.52	1.47581	0.737906	−	0.674904i	$-0.235815\pi$
$317$	8329.52	1.47581	0.737906	+	0.674904i	$0.235815\pi$
$318$	0	0
$319$	−1209.49	−0.212284
$320$	−1368.99	−0.239152
$321$	0	0
$322$	6453.92	1.11697
$323$	−1432.30	−0.246734
$324$	0	0
$325$	−2238.28	−0.382022
$326$	−15174.6	−2.57804
$327$	0	0
$328$	14923.6	2.51225
$329$	−2310.13	−0.387118
$330$	0	0
$331$	2901.85	0.481874	0.240937	−	0.970541i	$-0.422545\pi$
$331$	2901.85	0.481874	0.240937	+	0.970541i	$0.422545\pi$
$332$	−19662.2	−3.25030
$333$	0	0
$334$	−11096.8	−1.81793
$335$	1322.66	0.215716
$336$	0	0
$337$	6000.75	0.969976	0.484988	−	0.874521i	$-0.338824\pi$
$337$	6000.75	0.969976	0.484988	+	0.874521i	$0.338824\pi$
$338$	−30087.9	−4.84191
$339$	0	0
$340$	5464.16	0.871575
$341$	1315.02	0.208833
$342$	0	0
$343$	6270.36	0.987078
$344$	−14622.3	−2.29181
$345$	0	0
$346$	1661.67	0.258185
$347$	9059.04	1.40148	0.700742	−	0.713415i	$-0.252852\pi$
$347$	9059.04	1.40148	0.700742	+	0.713415i	$0.252852\pi$
$348$	0	0
$349$	−9362.35	−1.43597	−0.717987	−	0.696057i	$-0.754936\pi$
$349$	−9362.35	−1.43597	−0.717987	+	0.696057i	$0.754936\pi$
$350$	1444.53	0.220610
$351$	0	0
$352$	−2917.15	−0.441718
$353$	274.260	0.0413524	0.0206762	−	0.999786i	$-0.493418\pi$
$353$	274.260	0.0413524	0.0206762	+	0.999786i	$0.493418\pi$
$354$	0	0
$355$	−5551.66	−0.830004
$356$	26567.7	3.95530
$357$	0	0
$358$	−13765.1	−2.03214
$359$	1394.51	0.205013	0.102506	−	0.994732i	$-0.467314\pi$
$359$	1394.51	0.205013	0.102506	+	0.994732i	$0.467314\pi$
$360$	0	0
$361$	−6255.93	−0.912076
$362$	−17094.8	−2.48199
$363$	0	0
$364$	18745.9	2.69933
$365$	−4468.04	−0.640734
$366$	0	0
$367$	11610.1	1.65134	0.825669	−	0.564155i	$-0.190798\pi$
$367$	11610.1	1.65134	0.825669	+	0.564155i	$0.190798\pi$
$368$	15322.9	2.17055
$369$	0	0
$370$	−9218.08	−1.29520
$371$	2500.29	0.349888
$372$	0	0
$373$	5068.89	0.703639	0.351819	−	0.936068i	$-0.385563\pi$
$373$	5068.89	0.703639	0.351819	+	0.936068i	$0.385563\pi$
$374$	3317.41	0.458661
$375$	0	0
$376$	−11477.7	−1.57425
$377$	9844.28	1.34484
$378$	0	0
$379$	1623.62	0.220052	0.110026	−	0.993929i	$-0.464907\pi$
$379$	1623.62	0.220052	0.110026	+	0.993929i	$0.464907\pi$
$380$	−2300.69	−0.310587
$381$	0	0
$382$	4891.93	0.655217
$383$	−5513.47	−0.735575	−0.367787	−	0.929910i	$-0.619885\pi$
$383$	−5513.47	−0.735575	−0.367787	+	0.929910i	$0.619885\pi$
$384$	0	0
$385$	614.599	0.0813581
$386$	19093.4	2.51769
$387$	0	0
$388$	1607.82	0.210373
$389$	−1591.80	−0.207474	−0.103737	−	0.994605i	$-0.533080\pi$
$389$	−1591.80	−0.207474	−0.103737	+	0.994605i	$0.533080\pi$
$390$	0	0
$391$	−6514.57	−0.842598
$392$	12110.5	1.56039
$393$	0	0
$394$	−2127.80	−0.272074
$395$	−6518.32	−0.830310
$396$	0	0
$397$	−8032.19	−1.01543	−0.507713	−	0.861526i	$-0.669509\pi$
$397$	−8032.19	−1.01543	−0.507713	+	0.861526i	$0.669509\pi$
$398$	−7711.11	−0.971164
$399$	0	0
$400$	3429.62	0.428702
$401$	−588.148	−0.0732436	−0.0366218	−	0.999329i	$-0.511660\pi$
$401$	−588.148	−0.0732436	−0.0366218	+	0.999329i	$0.511660\pi$
$402$	0	0
$403$	−10703.2	−1.32298
$404$	−2581.26	−0.317878
$405$	0	0
$406$	−6353.27	−0.776620
$407$	−3921.98	−0.477654
$408$	0	0
$409$	6945.63	0.839705	0.419853	−	0.907592i	$-0.362082\pi$
$409$	6945.63	0.839705	0.419853	+	0.907592i	$0.362082\pi$
$410$	−6949.49	−0.837100
$411$	0	0
$412$	7858.33	0.939689
$413$	5307.90	0.632409
$414$	0	0
$415$	5246.83	0.620618
$416$	23743.3	2.79834
$417$	0	0
$418$	−1396.80	−0.163445
$419$	3310.93	0.386037	0.193018	−	0.981195i	$-0.438172\pi$
$419$	3310.93	0.386037	0.193018	+	0.981195i	$0.438172\pi$
$420$	0	0
$421$	−13906.6	−1.60989	−0.804946	−	0.593348i	$-0.797806\pi$
$421$	−13906.6	−1.60989	−0.804946	+	0.593348i	$0.797806\pi$
$422$	4653.45	0.536792
$423$	0	0
$424$	12422.5	1.42285
$425$	−1458.11	−0.166420
$426$	0	0
$427$	5734.81	0.649946
$428$	22305.8	2.51914
$429$	0	0
$430$	6809.18	0.763647
$431$	−2713.07	−0.303211	−0.151606	−	0.988441i	$-0.548444\pi$
$431$	−2713.07	−0.303211	−0.151606	+	0.988441i	$0.548444\pi$
$432$	0	0
$433$	668.058	0.0741451	0.0370725	−	0.999313i	$-0.488197\pi$
$433$	668.058	0.0741451	0.0370725	+	0.999313i	$0.488197\pi$
$434$	6907.57	0.763995
$435$	0	0
$436$	19362.5	2.12683
$437$	2742.97	0.300261
$438$	0	0
$439$	16512.6	1.79522	0.897611	−	0.440788i	$-0.145301\pi$
$439$	16512.6	1.79522	0.897611	+	0.440788i	$0.145301\pi$
$440$	3053.59	0.330851
$441$	0	0
$442$	−27001.0	−2.90567
$443$	11305.6	1.21252	0.606258	−	0.795268i	$-0.292670\pi$
$443$	11305.6	1.21252	0.606258	+	0.795268i	$0.292670\pi$
$444$	0	0
$445$	−7089.57	−0.755231
$446$	−10007.0	−1.06244
$447$	0	0
$448$	−3059.56	−0.322657
$449$	−2103.65	−0.221108	−0.110554	−	0.993870i	$-0.535262\pi$
$449$	−2103.65	−0.221108	−0.110554	+	0.993870i	$0.535262\pi$
$450$	0	0
$451$	−2956.77	−0.308712
$452$	17439.8	1.81482
$453$	0	0
$454$	−15027.7	−1.55349
$455$	−5002.34	−0.515414
$456$	0	0
$457$	−16013.6	−1.63914	−0.819570	−	0.572979i	$-0.805788\pi$
$457$	−16013.6	−1.63914	−0.819570	+	0.572979i	$0.805788\pi$
$458$	27983.8	2.85501
$459$	0	0
$460$	−10464.3	−1.06066
$461$	−13332.9	−1.34702	−0.673509	−	0.739179i	$-0.735214\pi$
$461$	−13332.9	−1.34702	−0.673509	+	0.739179i	$0.735214\pi$
$462$	0	0
$463$	2453.48	0.246270	0.123135	−	0.992390i	$-0.460705\pi$
$463$	2453.48	0.246270	0.123135	+	0.992390i	$0.460705\pi$
$464$	−15084.0	−1.50917
$465$	0	0
$466$	−24710.8	−2.45645
$467$	771.767	0.0764735	0.0382367	−	0.999269i	$-0.487826\pi$
$467$	771.767	0.0764735	0.0382367	+	0.999269i	$0.487826\pi$
$468$	0	0
$469$	2956.03	0.291038
$470$	5344.85	0.524552
$471$	0	0
$472$	26371.9	2.57175
$473$	2897.08	0.281623
$474$	0	0
$475$	613.938	0.0593040
$476$	12211.9	1.17590
$477$	0	0
$478$	30424.4	2.91125
$479$	−2782.55	−0.265423	−0.132712	−	0.991155i	$-0.542368\pi$
$479$	−2782.55	−0.265423	−0.132712	+	0.991155i	$0.542368\pi$
$480$	0	0
$481$	31921.7	3.02600
$482$	−9343.44	−0.882950
$483$	0	0
$484$	2267.20	0.212923
$485$	−429.046	−0.0401690
$486$	0	0
$487$	−9716.69	−0.904118	−0.452059	−	0.891988i	$-0.649310\pi$
$487$	−9716.69	−0.904118	−0.452059	+	0.891988i	$0.649310\pi$
$488$	28493.0	2.64307
$489$	0	0
$490$	−5639.53	−0.519935
$491$	−13582.7	−1.24843	−0.624213	−	0.781254i	$-0.714580\pi$
$491$	−13582.7	−1.24843	−0.624213	+	0.781254i	$0.714580\pi$
$492$	0	0
$493$	6412.97	0.585853
$494$	11368.8	1.03544
$495$	0	0
$496$	16400.0	1.48464
$497$	−12407.4	−1.11982
$498$	0	0
$499$	12177.8	1.09249	0.546244	−	0.837626i	$-0.316057\pi$
$499$	12177.8	1.09249	0.546244	+	0.837626i	$0.316057\pi$
$500$	−2342.15	−0.209488
$501$	0	0
$502$	6541.59	0.581604
$503$	−5799.88	−0.514123	−0.257061	−	0.966395i	$-0.582754\pi$
$503$	−5799.88	−0.514123	−0.257061	+	0.966395i	$0.582754\pi$
$504$	0	0
$505$	688.807	0.0606961
$506$	−6353.12	−0.558164
$507$	0	0
$508$	1769.38	0.154535
$509$	−19516.8	−1.69954	−0.849772	−	0.527151i	$-0.823260\pi$
$509$	−19516.8	−1.69954	−0.849772	+	0.527151i	$0.823260\pi$
$510$	0	0
$511$	−9985.65	−0.864460
$512$	24551.0	2.11916
$513$	0	0
$514$	−22578.5	−1.93754
$515$	−2096.99	−0.179426
$516$	0	0
$517$	2274.05	0.193448
$518$	−20601.5	−1.74745
$519$	0	0
$520$	−24853.8	−2.09598
$521$	−9489.76	−0.797993	−0.398996	−	0.916953i	$-0.630641\pi$
$521$	−9489.76	−0.797993	−0.398996	+	0.916953i	$0.630641\pi$
$522$	0	0
$523$	5714.17	0.477750	0.238875	−	0.971050i	$-0.423221\pi$
$523$	5714.17	0.477750	0.238875	+	0.971050i	$0.423221\pi$
$524$	−24697.9	−2.05903
$525$	0	0
$526$	−15258.8	−1.26486
$527$	−6972.47	−0.576330
$528$	0	0
$529$	308.942	0.0253918
$530$	−5784.80	−0.474105
$531$	0	0
$532$	−5141.84	−0.419036
$533$	24065.7	1.95573
$534$	0	0
$535$	−5952.27	−0.481008
$536$	14686.8	1.18353
$537$	0	0
$538$	−31276.4	−2.50636
$539$	−2399.43	−0.191745
$540$	0	0
$541$	−2530.63	−0.201110	−0.100555	−	0.994932i	$-0.532062\pi$
$541$	−2530.63	−0.201110	−0.100555	+	0.994932i	$0.532062\pi$
$542$	15113.8	1.19777
$543$	0	0
$544$	15467.3	1.21904
$545$	−5166.87	−0.406100
$546$	0	0
$547$	18910.4	1.47816	0.739079	−	0.673619i	$-0.235261\pi$
$547$	18910.4	1.47816	0.739079	+	0.673619i	$0.235261\pi$
$548$	−37632.3	−2.93353
$549$	0	0
$550$	−1421.97	−0.110242
$551$	−2700.19	−0.208770
$552$	0	0
$553$	−14567.8	−1.12023
$554$	28095.4	2.15462
$555$	0	0
$556$	55401.4	4.22580
$557$	−4480.34	−0.340823	−0.170411	−	0.985373i	$-0.554510\pi$
$557$	−4480.34	−0.340823	−0.170411	+	0.985373i	$0.554510\pi$
$558$	0	0
$559$	−23579.8	−1.78412
$560$	7664.87	0.578393
$561$	0	0
$562$	−41268.3	−3.09750
$563$	−1726.52	−0.129244	−0.0646218	−	0.997910i	$-0.520584\pi$
$563$	−1726.52	−0.129244	−0.0646218	+	0.997910i	$0.520584\pi$
$564$	0	0
$565$	−4653.80	−0.346525
$566$	13990.7	1.03900
$567$	0	0
$568$	−61645.5	−4.55385
$569$	10862.7	0.800327	0.400164	−	0.916444i	$-0.368953\pi$
$569$	10862.7	0.800327	0.400164	+	0.916444i	$0.368953\pi$
$570$	0	0
$571$	14448.8	1.05895	0.529477	−	0.848324i	$-0.322388\pi$
$571$	14448.8	1.05895	0.529477	+	0.848324i	$0.322388\pi$
$572$	−18453.2	−1.34889
$573$	0	0
$574$	−15531.5	−1.12939
$575$	2792.39	0.202523
$576$	0	0
$577$	−5335.95	−0.384988	−0.192494	−	0.981298i	$-0.561658\pi$
$577$	−5335.95	−0.384988	−0.192494	+	0.981298i	$0.561658\pi$
$578$	7814.56	0.562358
$579$	0	0
$580$	10301.1	0.737467
$581$	11726.2	0.837321
$582$	0	0
$583$	−2461.23	−0.174844
$584$	−49613.0	−3.51541
$585$	0	0
$586$	11415.6	0.804735
$587$	−25633.8	−1.80242	−0.901208	−	0.433386i	$-0.857319\pi$
$587$	−25633.8	−1.80242	−0.901208	+	0.433386i	$0.857319\pi$
$588$	0	0
$589$	2935.77	0.205376
$590$	−12280.7	−0.856926
$591$	0	0
$592$	−48912.3	−3.39575
$593$	16094.8	1.11456	0.557282	−	0.830324i	$-0.311844\pi$
$593$	16094.8	1.11456	0.557282	+	0.830324i	$0.311844\pi$
$594$	0	0
$595$	−3258.73	−0.224529
$596$	−16465.6	−1.13164
$597$	0	0
$598$	51709.2	3.53603
$599$	17160.5	1.17055	0.585275	−	0.810835i	$-0.300986\pi$
$599$	17160.5	1.17055	0.585275	+	0.810835i	$0.300986\pi$
$600$	0	0
$601$	−14563.2	−0.988426	−0.494213	−	0.869341i	$-0.664544\pi$
$601$	−14563.2	−0.988426	−0.494213	+	0.869341i	$0.664544\pi$
$602$	15217.9	1.03029
$603$	0	0
$604$	−31466.3	−2.11978
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−12834.7	−0.858230	−0.429115	−	0.903250i	$-0.641175\pi$
$607$	−12834.7	−0.858230	−0.429115	+	0.903250i	$0.641175\pi$
$608$	−6512.55	−0.434406
$609$	0	0
$610$	−13268.4	−0.880690
$611$	−18508.9	−1.22552
$612$	0	0
$613$	−10814.5	−0.712548	−0.356274	−	0.934381i	$-0.615953\pi$
$613$	−10814.5	−0.712548	−0.356274	+	0.934381i	$0.615953\pi$
$614$	20261.5	1.33174
$615$	0	0
$616$	6824.49	0.446374
$617$	−5118.52	−0.333977	−0.166988	−	0.985959i	$-0.553404\pi$
$617$	−5118.52	−0.333977	−0.166988	+	0.985959i	$0.553404\pi$
$618$	0	0
$619$	26144.1	1.69761	0.848804	−	0.528708i	$-0.177323\pi$
$619$	26144.1	1.69761	0.848804	+	0.528708i	$0.177323\pi$
$620$	−11199.9	−0.725479
$621$	0	0
$622$	45342.9	2.92297
$623$	−15844.5	−1.01894
$624$	0	0
$625$	625.000	0.0400000
$626$	26158.7	1.67015
$627$	0	0
$628$	13173.2	0.837051
$629$	20795.1	1.31821
$630$	0	0
$631$	−15147.2	−0.955626	−0.477813	−	0.878462i	$-0.658570\pi$
$631$	−15147.2	−0.955626	−0.477813	+	0.878462i	$0.658570\pi$
$632$	−72379.2	−4.55552
$633$	0	0
$634$	−43070.3	−2.69801
$635$	−472.158	−0.0295071
$636$	0	0
$637$	19529.4	1.21473
$638$	6254.04	0.388088
$639$	0	0
$640$	−3529.07	−0.217967
$641$	−6959.04	−0.428807	−0.214404	−	0.976745i	$-0.568781\pi$
$641$	−6959.04	−0.428807	−0.214404	+	0.976745i	$0.568781\pi$
$642$	0	0
$643$	50.7662	0.00311357	0.00155678	−	0.999999i	$-0.499504\pi$
$643$	50.7662	0.00311357	0.00155678	+	0.999999i	$0.499504\pi$
$644$	−23386.8	−1.43101
$645$	0	0
$646$	7406.13	0.451069
$647$	14853.2	0.902537	0.451268	−	0.892388i	$-0.350972\pi$
$647$	14853.2	0.902537	0.451268	+	0.892388i	$0.350972\pi$
$648$	0	0
$649$	−5225.00	−0.316023
$650$	11573.7	0.698396
$651$	0	0
$652$	54987.3	3.30287
$653$	−18366.0	−1.10064	−0.550321	−	0.834953i	$-0.685495\pi$
$653$	−18366.0	−1.10064	−0.550321	+	0.834953i	$0.685495\pi$
$654$	0	0
$655$	6590.62	0.393155
$656$	−36874.9	−2.19470
$657$	0	0
$658$	11945.2	0.707711
$659$	8660.13	0.511913	0.255957	−	0.966688i	$-0.417610\pi$
$659$	8660.13	0.511913	0.255957	+	0.966688i	$0.417610\pi$
$660$	0	0
$661$	−11057.5	−0.650661	−0.325331	−	0.945600i	$-0.605476\pi$
$661$	−11057.5	−0.650661	−0.325331	+	0.945600i	$0.605476\pi$
$662$	−15004.9	−0.880940
$663$	0	0
$664$	58260.6	3.40505
$665$	1372.09	0.0800113
$666$	0	0
$667$	−12281.4	−0.712949
$668$	40210.9	2.32905
$669$	0	0
$670$	−6839.23	−0.394362
$671$	−5645.24	−0.324787
$672$	0	0
$673$	1921.35	0.110048	0.0550241	−	0.998485i	$-0.482476\pi$
$673$	1921.35	0.110048	0.0550241	+	0.998485i	$0.482476\pi$
$674$	−31028.7	−1.77327
$675$	0	0
$676$	109028.	6.20324
$677$	−25838.8	−1.46686	−0.733431	−	0.679764i	$-0.762082\pi$
$677$	−25838.8	−1.46686	−0.733431	+	0.679764i	$0.762082\pi$
$678$	0	0
$679$	−958.877	−0.0541948
$680$	−16190.8	−0.913070
$681$	0	0
$682$	−6799.68	−0.381779
$683$	33038.7	1.85094	0.925469	−	0.378823i	$-0.123671\pi$
$683$	33038.7	1.85094	0.925469	+	0.378823i	$0.123671\pi$
$684$	0	0
$685$	10042.2	0.560133
$686$	−32422.8	−1.80453
$687$	0	0
$688$	36130.4	2.00212
$689$	20032.4	1.10766
$690$	0	0
$691$	−2065.11	−0.113691	−0.0568454	−	0.998383i	$-0.518104\pi$
$691$	−2065.11	−0.113691	−0.0568454	+	0.998383i	$0.518104\pi$
$692$	−6021.31	−0.330774
$693$	0	0
$694$	−46842.5	−2.56213
$695$	−14783.8	−0.806880
$696$	0	0
$697$	15677.4	0.851972
$698$	48410.8	2.62518
$699$	0	0
$700$	−5234.48	−0.282635
$701$	23504.1	1.26639	0.633194	−	0.773993i	$-0.281743\pi$
$701$	23504.1	1.26639	0.633194	+	0.773993i	$0.281743\pi$
$702$	0	0
$703$	−8755.83	−0.469747
$704$	3011.77	0.161236
$705$	0	0
$706$	−1418.14	−0.0755985
$707$	1539.42	0.0818894
$708$	0	0
$709$	−135.537	−0.00717943	−0.00358971	−	0.999994i	$-0.501143\pi$
$709$	−135.537	−0.00717943	−0.00358971	+	0.999994i	$0.501143\pi$
$710$	28706.5	1.51738
$711$	0	0
$712$	−78722.4	−4.14361
$713$	13352.9	0.701359
$714$	0	0
$715$	4924.21	0.257559
$716$	49879.9	2.60349
$717$	0	0
$718$	−7210.75	−0.374795
$719$	−10752.3	−0.557711	−0.278856	−	0.960333i	$-0.589955\pi$
$719$	−10752.3	−0.557711	−0.278856	+	0.960333i	$0.589955\pi$
$720$	0	0
$721$	−4686.57	−0.242076
$722$	32348.2	1.66742
$723$	0	0
$724$	61945.5	3.17982
$725$	−2748.85	−0.140813
$726$	0	0
$727$	−7521.06	−0.383687	−0.191844	−	0.981425i	$-0.561447\pi$
$727$	−7521.06	−0.383687	−0.191844	+	0.981425i	$0.561447\pi$
$728$	−55545.8	−2.82784
$729$	0	0
$730$	23103.4	1.17136
$731$	−15360.9	−0.777213
$732$	0	0
$733$	−5330.33	−0.268595	−0.134298	−	0.990941i	$-0.542878\pi$
$733$	−5330.33	−0.268595	−0.134298	+	0.990941i	$0.542878\pi$
$734$	−60033.4	−3.01890
$735$	0	0
$736$	−29621.2	−1.48350
$737$	−2909.86	−0.145436
$738$	0	0
$739$	29328.0	1.45988	0.729938	−	0.683513i	$-0.239549\pi$
$739$	29328.0	1.45988	0.729938	+	0.683513i	$0.239549\pi$
$740$	33403.1	1.65936
$741$	0	0
$742$	−12928.5	−0.639649
$743$	1378.56	0.0680679	0.0340340	−	0.999421i	$-0.489165\pi$
$743$	1378.56	0.0680679	0.0340340	+	0.999421i	$0.489165\pi$
$744$	0	0
$745$	4393.84	0.216078
$746$	−26210.2	−1.28636
$747$	0	0
$748$	−12021.1	−0.587616
$749$	−13302.8	−0.648962
$750$	0	0
$751$	−31066.4	−1.50950	−0.754748	−	0.656015i	$-0.772241\pi$
$751$	−31066.4	−1.50950	−0.754748	+	0.656015i	$0.772241\pi$
$752$	28360.4	1.37526
$753$	0	0
$754$	−50902.8	−2.45858
$755$	8396.76	0.404754
$756$	0	0
$757$	21971.1	1.05489	0.527445	−	0.849589i	$-0.323150\pi$
$757$	21971.1	1.05489	0.527445	+	0.849589i	$0.323150\pi$
$758$	−8395.42	−0.402290
$759$	0	0
$760$	6817.15	0.325374
$761$	35175.8	1.67559	0.837794	−	0.545987i	$-0.183845\pi$
$761$	35175.8	1.67559	0.837794	+	0.545987i	$0.183845\pi$
$762$	0	0
$763$	−11547.5	−0.547898
$764$	−17726.7	−0.839435
$765$	0	0
$766$	28509.1	1.34474
$767$	42527.3	2.00205
$768$	0	0
$769$	32952.5	1.54525	0.772626	−	0.634862i	$-0.218943\pi$
$769$	32952.5	1.54525	0.772626	+	0.634862i	$0.218943\pi$
$770$	−3177.97	−0.148735
$771$	0	0
$772$	−69187.7	−3.22554
$773$	11158.7	0.519213	0.259607	−	0.965714i	$-0.416407\pi$
$773$	11158.7	0.519213	0.259607	+	0.965714i	$0.416407\pi$
$774$	0	0
$775$	2988.67	0.138524
$776$	−4764.11	−0.220389
$777$	0	0
$778$	8230.86	0.379294
$779$	−6601.00	−0.303601
$780$	0	0
$781$	12213.6	0.559589
$782$	33685.5	1.54040
$783$	0	0
$784$	−29924.1	−1.36316
$785$	−3515.26	−0.159828
$786$	0	0
$787$	17308.9	0.783985	0.391993	−	0.919968i	$-0.371786\pi$
$787$	17308.9	0.783985	0.391993	+	0.919968i	$0.371786\pi$
$788$	7710.41	0.348569
$789$	0	0
$790$	33704.9	1.51793
$791$	−10400.8	−0.467522
$792$	0	0
$793$	45947.7	2.05757
$794$	41532.9	1.85636
$795$	0	0
$796$	27942.4	1.24421
$797$	14802.4	0.657876	0.328938	−	0.944351i	$-0.393309\pi$
$797$	14802.4	0.657876	0.328938	+	0.944351i	$0.393309\pi$
$798$	0	0
$799$	−12057.5	−0.533871
$800$	−6629.90	−0.293003
$801$	0	0
$802$	3041.19	0.133901
$803$	9829.69	0.431983
$804$	0	0
$805$	6240.74	0.273239
$806$	55343.9	2.41862
$807$	0	0
$808$	7648.50	0.333011
$809$	8999.53	0.391108	0.195554	−	0.980693i	$-0.437349\pi$
$809$	8999.53	0.391108	0.195554	+	0.980693i	$0.437349\pi$
$810$	0	0
$811$	−41368.5	−1.79117	−0.895587	−	0.444886i	$-0.853244\pi$
$811$	−41368.5	−1.79117	−0.895587	+	0.444886i	$0.853244\pi$
$812$	23022.1	0.994970
$813$	0	0
$814$	20279.8	0.873226
$815$	−14673.3	−0.630655
$816$	0	0
$817$	6467.73	0.276961
$818$	−35914.5	−1.53511
$819$	0	0
$820$	25182.6	1.07245
$821$	−27772.2	−1.18058	−0.590289	−	0.807192i	$-0.700986\pi$
$821$	−27772.2	−1.18058	−0.590289	+	0.807192i	$0.700986\pi$
$822$	0	0
$823$	−9315.89	−0.394571	−0.197285	−	0.980346i	$-0.563213\pi$
$823$	−9315.89	−0.394571	−0.197285	+	0.980346i	$0.563213\pi$
$824$	−23284.9	−0.984426
$825$	0	0
$826$	−27446.1	−1.15614
$827$	20344.6	0.855443	0.427721	−	0.903911i	$-0.359316\pi$
$827$	20344.6	0.855443	0.427721	+	0.903911i	$0.359316\pi$
$828$	0	0
$829$	16916.7	0.708737	0.354368	−	0.935106i	$-0.384696\pi$
$829$	16916.7	0.708737	0.354368	+	0.935106i	$0.384696\pi$
$830$	−27130.3	−1.13459
$831$	0	0
$832$	−24513.4	−1.02145
$833$	12722.3	0.529172
$834$	0	0
$835$	−10730.2	−0.444712
$836$	5061.53	0.209398
$837$	0	0
$838$	−17120.2	−0.705735
$839$	−23690.7	−0.974844	−0.487422	−	0.873167i	$-0.662063\pi$
$839$	−23690.7	−0.974844	−0.487422	+	0.873167i	$0.662063\pi$
$840$	0	0
$841$	−12299.2	−0.504291
$842$	71908.1	2.94313
$843$	0	0
$844$	−16862.5	−0.687714
$845$	−29094.0	−1.18446
$846$	0	0
$847$	−1352.12	−0.0548516
$848$	−30694.9	−1.24300
$849$	0	0
$850$	7539.57	0.304241
$851$	−39824.4	−1.60419
$852$	0	0
$853$	−3804.67	−0.152719	−0.0763596	−	0.997080i	$-0.524330\pi$
$853$	−3804.67	−0.152719	−0.0763596	+	0.997080i	$0.524330\pi$
$854$	−29653.6	−1.18820
$855$	0	0
$856$	−66093.9	−2.63907
$857$	3125.89	0.124596	0.0622978	−	0.998058i	$-0.480157\pi$
$857$	3125.89	0.124596	0.0622978	+	0.998058i	$0.480157\pi$
$858$	0	0
$859$	−38044.2	−1.51112	−0.755559	−	0.655081i	$-0.772635\pi$
$859$	−38044.2	−1.51112	−0.755559	+	0.655081i	$0.772635\pi$
$860$	−24674.1	−0.978349
$861$	0	0
$862$	14028.8	0.554317
$863$	−728.739	−0.0287446	−0.0143723	−	0.999897i	$-0.504575\pi$
$863$	−728.739	−0.0287446	−0.0143723	+	0.999897i	$0.504575\pi$
$864$	0	0
$865$	1606.78	0.0631586
$866$	−3454.39	−0.135549
$867$	0	0
$868$	−25030.6	−0.978796
$869$	14340.3	0.559795
$870$	0	0
$871$	23683.9	0.921352
$872$	−57372.7	−2.22808
$873$	0	0
$874$	−14183.4	−0.548924
$875$	1396.82	0.0539669
$876$	0	0
$877$	18861.7	0.726241	0.363120	−	0.931742i	$-0.381711\pi$
$877$	18861.7	0.726241	0.363120	+	0.931742i	$0.381711\pi$
$878$	−85383.3	−3.28194
$879$	0	0
$880$	−7545.16	−0.289031
$881$	−24435.2	−0.934442	−0.467221	−	0.884141i	$-0.654745\pi$
$881$	−24435.2	−0.934442	−0.467221	+	0.884141i	$0.654745\pi$
$882$	0	0
$883$	3101.93	0.118220	0.0591101	−	0.998251i	$-0.481174\pi$
$883$	3101.93	0.118220	0.0591101	+	0.998251i	$0.481174\pi$
$884$	97842.4	3.72262
$885$	0	0
$886$	−58458.9	−2.21667
$887$	−2129.33	−0.0806043	−0.0403021	−	0.999188i	$-0.512832\pi$
$887$	−2129.33	−0.0806043	−0.0403021	+	0.999188i	$0.512832\pi$
$888$	0	0
$889$	−1055.23	−0.0398101
$890$	36658.8	1.38068
$891$	0	0
$892$	36262.0	1.36114
$893$	5076.82	0.190246
$894$	0	0
$895$	−13310.4	−0.497115
$896$	−7887.14	−0.294075
$897$	0	0
$898$	10877.5	0.404219
$899$	−13144.6	−0.487651
$900$	0	0
$901$	13050.0	0.482527
$902$	15288.9	0.564373
$903$	0	0
$904$	−51675.6	−1.90122
$905$	−16530.1	−0.607159
$906$	0	0
$907$	33678.6	1.23294	0.616471	−	0.787378i	$-0.288562\pi$
$907$	33678.6	1.23294	0.616471	+	0.787378i	$0.288562\pi$
$908$	54455.1	1.99026
$909$	0	0
$910$	25866.1	0.942255
$911$	28917.6	1.05168	0.525841	−	0.850583i	$-0.323751\pi$
$911$	28917.6	1.05168	0.525841	+	0.850583i	$0.323751\pi$
$912$	0	0
$913$	−11543.0	−0.418421
$914$	82803.4	2.99660
$915$	0	0
$916$	−101404.	−3.65771
$917$	14729.4	0.530434
$918$	0	0
$919$	−8611.83	−0.309117	−0.154558	−	0.987984i	$-0.549395\pi$
$919$	−8611.83	−0.309117	−0.154558	+	0.987984i	$0.549395\pi$
$920$	31006.7	1.11115
$921$	0	0
$922$	68941.8	2.46256
$923$	−99409.2	−3.54506
$924$	0	0
$925$	−8913.59	−0.316840
$926$	−12686.5	−0.450219
$927$	0	0
$928$	29159.3	1.03147
$929$	31195.3	1.10171	0.550853	−	0.834602i	$-0.314302\pi$
$929$	31195.3	1.10171	0.550853	+	0.834602i	$0.314302\pi$
$930$	0	0
$931$	−5356.73	−0.188571
$932$	89543.2	3.14709
$933$	0	0
$934$	−3990.65	−0.139805
$935$	3207.83	0.112200
$936$	0	0
$937$	24625.3	0.858561	0.429281	−	0.903171i	$-0.358767\pi$
$937$	24625.3	0.858561	0.429281	+	0.903171i	$0.358767\pi$
$938$	−15285.0	−0.532062
$939$	0	0
$940$	−19367.9	−0.672032
$941$	−3605.28	−0.124898	−0.0624488	−	0.998048i	$-0.519891\pi$
$941$	−3605.28	−0.124898	−0.0624488	+	0.998048i	$0.519891\pi$
$942$	0	0
$943$	−30023.6	−1.03680
$944$	−65162.7	−2.24668
$945$	0	0
$946$	−14980.2	−0.514850
$947$	−11634.6	−0.399231	−0.199616	−	0.979874i	$-0.563969\pi$
$947$	−11634.6	−0.399231	−0.199616	+	0.979874i	$0.563969\pi$
$948$	0	0
$949$	−80005.7	−2.73666
$950$	−3174.55	−0.108417
$951$	0	0
$952$	−36184.8	−1.23189
$953$	−27302.2	−0.928021	−0.464010	−	0.885830i	$-0.653590\pi$
$953$	−27302.2	−0.928021	−0.464010	+	0.885830i	$0.653590\pi$
$954$	0	0
$955$	4730.34	0.160283
$956$	−110247.	−3.72976
$957$	0	0
$958$	14388.0	0.485235
$959$	22443.3	0.755715
$960$	0	0
$961$	−15499.6	−0.520276
$962$	−165061.	−5.53199
$963$	0	0
$964$	33857.4	1.13120
$965$	18462.7	0.615891
$966$	0	0
$967$	−19429.3	−0.646125	−0.323063	−	0.946378i	$-0.604712\pi$
$967$	−19429.3	−0.646125	−0.323063	+	0.946378i	$0.604712\pi$
$968$	−6717.90	−0.223059
$969$	0	0
$970$	2218.51	0.0734351
$971$	−42852.0	−1.41626	−0.708129	−	0.706083i	$-0.750461\pi$
$971$	−42852.0	−1.41626	−0.708129	+	0.706083i	$0.750461\pi$
$972$	0	0
$973$	−33040.4	−1.08862
$974$	50243.1	1.65287
$975$	0	0
$976$	−70403.7	−2.30898
$977$	−6559.02	−0.214782	−0.107391	−	0.994217i	$-0.534250\pi$
$977$	−6559.02	−0.214782	−0.107391	+	0.994217i	$0.534250\pi$
$978$	0	0
$979$	15597.1	0.509177
$980$	20435.7	0.666117
$981$	0	0
$982$	70233.3	2.28232
$983$	15077.3	0.489208	0.244604	−	0.969623i	$-0.421342\pi$
$983$	15077.3	0.489208	0.244604	+	0.969623i	$0.421342\pi$
$984$	0	0
$985$	−2057.52	−0.0665563
$986$	−33160.2	−1.07103
$987$	0	0
$988$	−41196.7	−1.32656
$989$	29417.4	0.945822
$990$	0	0
$991$	45239.3	1.45013	0.725063	−	0.688683i	$-0.241811\pi$
$991$	45239.3	1.45013	0.725063	+	0.688683i	$0.241811\pi$
$992$	−31703.3	−1.01470
$993$	0	0
$994$	64156.4	2.04720
$995$	−7456.40	−0.237572
$996$	0	0
$997$	30975.1	0.983942	0.491971	−	0.870612i	$-0.336277\pi$
$997$	30975.1	0.983942	0.491971	+	0.870612i	$0.336277\pi$
$998$	−62968.8	−1.99724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.m.1.1		4
3.2	odd	2		165.4.a.h.1.4	✓	4
5.4	even	2		2475.4.a.be.1.4		4
15.2	even	4		825.4.c.p.199.8		8
15.8	even	4		825.4.c.p.199.1		8
15.14	odd	2		825.4.a.t.1.1		4
33.32	even	2		1815.4.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.h.1.4	✓	4	3.2	odd	2
495.4.a.m.1.1		4	1.1	even	1	trivial
825.4.a.t.1.1		4	15.14	odd	2
825.4.c.p.199.1		8	15.8	even	4
825.4.c.p.199.8		8	15.2	even	4
1815.4.a.t.1.1		4	33.32	even	2
2475.4.a.be.1.4		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)

\(f(q)\)	\(=\)	\(q-5.17080 q^{2} +18.7372 q^{4} -5.00000 q^{5} -11.1745 q^{7} -55.5199 q^{8} +O(q^{10})\)	\(q-5.17080 q^{2} +18.7372 q^{4} -5.00000 q^{5} -11.1745 q^{7} -55.5199 q^{8} +25.8540 q^{10} +11.0000 q^{11} -89.5310 q^{13} +57.7813 q^{14} +137.185 q^{16} -58.3242 q^{17} +24.5575 q^{19} -93.6859 q^{20} -56.8788 q^{22} +111.696 q^{23} +25.0000 q^{25} +462.947 q^{26} -209.379 q^{28} -109.954 q^{29} +119.547 q^{31} -265.196 q^{32} +301.583 q^{34} +55.8726 q^{35} -356.544 q^{37} -126.982 q^{38} +277.599 q^{40} -268.798 q^{41} +263.371 q^{43} +206.109 q^{44} -577.557 q^{46} +206.732 q^{47} -218.130 q^{49} -129.270 q^{50} -1677.56 q^{52} -223.749 q^{53} -55.0000 q^{55} +620.408 q^{56} +568.549 q^{58} -475.000 q^{59} -513.204 q^{61} -618.153 q^{62} +273.798 q^{64} +447.655 q^{65} -264.533 q^{67} -1092.83 q^{68} -288.906 q^{70} +1110.33 q^{71} +893.608 q^{73} +1843.62 q^{74} +460.139 q^{76} -122.920 q^{77} +1303.66 q^{79} -685.923 q^{80} +1389.90 q^{82} -1049.37 q^{83} +291.621 q^{85} -1361.84 q^{86} -610.718 q^{88} +1417.91 q^{89} +1000.47 q^{91} +2092.86 q^{92} -1068.97 q^{94} -122.788 q^{95} +85.8091 q^{97} +1127.91 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 20 q^{5} + 34 q^{7} - 48 q^{8}+O(q^{10}) \) 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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