LMFDB - Embedded newform 135.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(1,135)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("135.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.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:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 23x + 6 $ x^3 - x^2 - 23*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.20067$ of defining polynomial
Character	$\chi$	$=$	135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} +O(q^{10})$	\(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} -26.0034 q^{10} +28.9839 q^{11} -65.3919 q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +95.2349 q^{20} -150.736 q^{22} +154.807 q^{23} +25.0000 q^{25} +340.082 q^{26} +464.772 q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +122.007 q^{35} -337.613 q^{37} -543.076 q^{38} -287.258 q^{40} +195.969 q^{41} +334.882 q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} -130.017 q^{50} -1245.52 q^{52} +319.965 q^{53} +144.919 q^{55} -1401.90 q^{56} -1069.56 q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -326.960 q^{65} +195.876 q^{67} -1298.47 q^{68} -634.517 q^{70} -425.955 q^{71} +929.193 q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +732.057 q^{80} -1019.17 q^{82} +545.859 q^{83} -340.859 q^{85} -1741.61 q^{86} -1665.17 q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} +522.121 q^{95} +827.613 q^{97} -1312.78 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) $ 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8	$ 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) $ 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8 - 5 * q^10 + 38 * q^11 + 28 * q^13 - 108 * q^14 + 191 * q^16 - 19 * q^17 + 187 * q^19 + 115 * q^20 + 122 * q^22 - 81 * q^23 + 75 * q^25 + 416 * q^26 + 410 * q^28 + 160 * q^29 + 227 * q^31 - 569 * q^32 + 17 * q^34 + 220 * q^35 + 78 * q^37 - 757 * q^38 - 180 * q^40 - 338 * q^41 + 22 * q^43 + 1636 * q^44 - 1425 * q^46 - 472 * q^47 - 197 * q^49 - 25 * q^50 - 1566 * q^52 + 521 * q^53 + 190 * q^55 - 1254 * q^56 - 2096 * q^58 + 140 * q^59 + 595 * q^61 + 1407 * q^62 - 918 * q^64 + 140 * q^65 + 878 * q^67 - 3053 * q^68 - 540 * q^70 - 602 * q^71 + 1294 * q^73 + 2878 * q^74 + 525 * q^76 + 288 * q^77 + 629 * q^79 + 955 * q^80 - 1682 * q^82 - 1287 * q^83 - 95 * q^85 - 3730 * q^86 - 858 * q^88 + 2154 * q^89 - 440 * q^91 + 1959 * q^92 - 1108 * q^94 + 935 * q^95 + 1392 * q^97 - 2693 * 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 $

$2$	−5.20067	−1.83871	−0.919357	−	0.393424i	$-0.871291\pi$
$2$	−5.20067	−1.83871	−0.919357	+	0.393424i	$0.871291\pi$
$3$	0	0
$3$	0	0
$4$	19.0470	2.38087
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	24.4013	1.31755	0.658774	−	0.752341i	$-0.271075\pi$
$7$	24.4013	1.31755	0.658774	+	0.752341i	$0.271075\pi$
$8$	−57.4517	−2.53903
$9$	0	0
$10$	−26.0034	−0.822298
$11$	28.9839	0.794451	0.397226	−	0.917721i	$-0.369973\pi$
$11$	28.9839	0.794451	0.397226	+	0.917721i	$0.369973\pi$
$12$	0	0
$13$	−65.3919	−1.39511	−0.697556	−	0.716530i	$-0.745729\pi$
$13$	−65.3919	−1.39511	−0.697556	+	0.716530i	$0.745729\pi$
$14$	−126.903	−2.42260
$15$	0	0
$16$	146.411	2.28768
$17$	−68.1718	−0.972593	−0.486296	−	0.873794i	$-0.661653\pi$
$17$	−68.1718	−0.972593	−0.486296	+	0.873794i	$0.661653\pi$
$18$	0	0
$19$	104.424	1.26087	0.630435	−	0.776242i	$-0.282876\pi$
$19$	104.424	1.26087	0.630435	+	0.776242i	$0.282876\pi$
$20$	95.2349	1.06476
$21$	0	0
$22$	−150.736	−1.46077
$23$	154.807	1.40345	0.701727	−	0.712446i	$-0.252413\pi$
$23$	154.807	1.40345	0.701727	+	0.712446i	$0.252413\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	340.082	2.56521
$27$	0	0
$28$	464.772	3.13691
$29$	205.658	1.31689	0.658443	−	0.752631i	$-0.271215\pi$
$29$	205.658	1.31689	0.658443	+	0.752631i	$0.271215\pi$
$30$	0	0
$31$	−18.2497	−0.105734	−0.0528668	−	0.998602i	$-0.516836\pi$
$31$	−18.2497	−0.105734	−0.0528668	+	0.998602i	$0.516836\pi$
$32$	−301.824	−1.66736
$33$	0	0
$34$	354.539	1.78832
$35$	122.007	0.589226
$36$	0	0
$37$	−337.613	−1.50009	−0.750044	−	0.661387i	$-0.769968\pi$
$37$	−337.613	−1.50009	−0.750044	+	0.661387i	$0.769968\pi$
$38$	−543.076	−2.31838
$39$	0	0
$40$	−287.258	−1.13549
$41$	195.969	0.746469	0.373234	−	0.927737i	$-0.378249\pi$
$41$	195.969	0.746469	0.373234	+	0.927737i	$0.378249\pi$
$42$	0	0
$43$	334.882	1.18765	0.593826	−	0.804594i	$-0.297617\pi$
$43$	334.882	1.18765	0.593826	+	0.804594i	$0.297617\pi$
$44$	552.055	1.89149
$45$	0	0
$46$	−805.098	−2.58055
$47$	5.00398	0.0155299	0.00776496	−	0.999970i	$-0.497528\pi$
$47$	5.00398	0.0155299	0.00776496	+	0.999970i	$0.497528\pi$
$48$	0	0
$49$	252.425	0.735934
$50$	−130.017	−0.367743
$51$	0	0
$52$	−1245.52	−3.32158
$53$	319.965	0.829256	0.414628	−	0.909991i	$-0.363912\pi$
$53$	319.965	0.829256	0.414628	+	0.909991i	$0.363912\pi$
$54$	0	0
$55$	144.919	0.355289
$56$	−1401.90	−3.34529
$57$	0	0
$58$	−1069.56	−2.42138
$59$	−430.611	−0.950182	−0.475091	−	0.879937i	$-0.657585\pi$
$59$	−430.611	−0.950182	−0.475091	+	0.879937i	$0.657585\pi$
$60$	0	0
$61$	594.581	1.24800	0.624002	−	0.781422i	$-0.285505\pi$
$61$	594.581	1.24800	0.624002	+	0.781422i	$0.285505\pi$
$62$	94.9106	0.194414
$63$	0	0
$64$	398.396	0.778118
$65$	−326.960	−0.623913
$66$	0	0
$67$	195.876	0.357166	0.178583	−	0.983925i	$-0.442849\pi$
$67$	195.876	0.357166	0.178583	+	0.983925i	$0.442849\pi$
$68$	−1298.47	−2.31562
$69$	0	0
$70$	−634.517	−1.08342
$71$	−425.955	−0.711994	−0.355997	−	0.934487i	$-0.615859\pi$
$71$	−425.955	−0.711994	−0.355997	+	0.934487i	$0.615859\pi$
$72$	0	0
$73$	929.193	1.48978	0.744889	−	0.667188i	$-0.232502\pi$
$73$	929.193	1.48978	0.744889	+	0.667188i	$0.232502\pi$
$74$	1755.82	2.75824
$75$	0	0
$76$	1988.96	3.00197
$77$	707.245	1.04673
$78$	0	0
$79$	24.4296	0.0347917	0.0173959	−	0.999849i	$-0.494462\pi$
$79$	24.4296	0.0347917	0.0173959	+	0.999849i	$0.494462\pi$
$80$	732.057	1.02308
$81$	0	0
$82$	−1019.17	−1.37254
$83$	545.859	0.721877	0.360938	−	0.932590i	$-0.382456\pi$
$83$	545.859	0.721877	0.360938	+	0.932590i	$0.382456\pi$
$84$	0	0
$85$	−340.859	−0.434957
$86$	−1741.61	−2.18375
$87$	0	0
$88$	−1665.17	−2.01714
$89$	84.1332	0.100203	0.0501017	−	0.998744i	$-0.484045\pi$
$89$	84.1332	0.100203	0.0501017	+	0.998744i	$0.484045\pi$
$90$	0	0
$91$	−1595.65	−1.83813
$92$	2948.60	3.34144
$93$	0	0
$94$	−26.0241	−0.0285551
$95$	522.121	0.563879
$96$	0	0
$97$	827.613	0.866303	0.433152	−	0.901321i	$-0.357401\pi$
$97$	827.613	0.866303	0.433152	+	0.901321i	$0.357401\pi$
$98$	−1312.78	−1.35317
$99$	0	0
$100$	476.174	0.476174
$101$	−823.576	−0.811375	−0.405688	−	0.914012i	$-0.632968\pi$
$101$	−823.576	−0.811375	−0.405688	+	0.914012i	$0.632968\pi$
$102$	0	0
$103$	1171.19	1.12040	0.560198	−	0.828359i	$-0.310725\pi$
$103$	1171.19	1.12040	0.560198	+	0.828359i	$0.310725\pi$
$104$	3756.87	3.54223
$105$	0	0
$106$	−1664.03	−1.52477
$107$	−1023.21	−0.924460	−0.462230	−	0.886760i	$-0.652951\pi$
$107$	−1023.21	−0.924460	−0.462230	+	0.886760i	$0.652951\pi$
$108$	0	0
$109$	−403.647	−0.354700	−0.177350	−	0.984148i	$-0.556753\pi$
$109$	−403.647	−0.354700	−0.177350	+	0.984148i	$0.556753\pi$
$110$	−753.678	−0.653276
$111$	0	0
$112$	3572.63	3.01413
$113$	−1082.20	−0.900931	−0.450465	−	0.892794i	$-0.648742\pi$
$113$	−1082.20	−0.900931	−0.450465	+	0.892794i	$0.648742\pi$
$114$	0	0
$115$	774.033	0.627643
$116$	3917.16	3.13534
$117$	0	0
$118$	2239.46	1.74711
$119$	−1663.48	−1.28144
$120$	0	0
$121$	−490.935	−0.368847
$122$	−3092.22	−2.29473
$123$	0	0
$124$	−347.601	−0.251738
$125$	125.000	0.0894427
$126$	0	0
$127$	−774.132	−0.540890	−0.270445	−	0.962735i	$-0.587171\pi$
$127$	−774.132	−0.540890	−0.270445	+	0.962735i	$0.587171\pi$
$128$	342.664	0.236621
$129$	0	0
$130$	1700.41	1.14720
$131$	1214.04	0.809702	0.404851	−	0.914383i	$-0.367323\pi$
$131$	1214.04	0.809702	0.404851	+	0.914383i	$0.367323\pi$
$132$	0	0
$133$	2548.09	1.66126
$134$	−1018.69	−0.656726
$135$	0	0
$136$	3916.58	2.46944
$137$	−2300.15	−1.43441	−0.717207	−	0.696860i	$-0.754580\pi$
$137$	−2300.15	−1.43441	−0.717207	+	0.696860i	$0.754580\pi$
$138$	0	0
$139$	−1355.93	−0.827396	−0.413698	−	0.910414i	$-0.635763\pi$
$139$	−1355.93	−0.827396	−0.413698	+	0.910414i	$0.635763\pi$
$140$	2323.86	1.40287
$141$	0	0
$142$	2215.25	1.30915
$143$	−1895.31	−1.10835
$144$	0	0
$145$	1028.29	0.588929
$146$	−4832.43	−2.73928
$147$	0	0
$148$	−6430.51	−3.57152
$149$	−259.845	−0.142868	−0.0714340	−	0.997445i	$-0.522758\pi$
$149$	−259.845	−0.142868	−0.0714340	+	0.997445i	$0.522758\pi$
$150$	0	0
$151$	−508.304	−0.273941	−0.136971	−	0.990575i	$-0.543737\pi$
$151$	−508.304	−0.273941	−0.136971	+	0.990575i	$0.543737\pi$
$152$	−5999.34	−3.20139
$153$	0	0
$154$	−3678.15	−1.92463
$155$	−91.2485	−0.0472855
$156$	0	0
$157$	−23.3052	−0.0118468	−0.00592342	−	0.999982i	$-0.501885\pi$
$157$	−23.3052	−0.0118468	−0.00592342	+	0.999982i	$0.501885\pi$
$158$	−127.050	−0.0639721
$159$	0	0
$160$	−1509.12	−0.745665
$161$	3777.49	1.84912
$162$	0	0
$163$	−4032.10	−1.93754	−0.968769	−	0.247964i	$-0.920238\pi$
$163$	−4032.10	−1.93754	−0.968769	+	0.247964i	$0.920238\pi$
$164$	3732.62	1.77725
$165$	0	0
$166$	−2838.83	−1.32733
$167$	671.911	0.311341	0.155671	−	0.987809i	$-0.450246\pi$
$167$	671.911	0.311341	0.155671	+	0.987809i	$0.450246\pi$
$168$	0	0
$169$	2079.10	0.946337
$170$	1772.69	0.799761
$171$	0	0
$172$	6378.49	2.82765
$173$	−1633.53	−0.717889	−0.358944	−	0.933359i	$-0.616863\pi$
$173$	−1633.53	−0.717889	−0.358944	+	0.933359i	$0.616863\pi$
$174$	0	0
$175$	610.034	0.263510
$176$	4243.57	1.81745
$177$	0	0
$178$	−437.549	−0.184245
$179$	−341.260	−0.142497	−0.0712485	−	0.997459i	$-0.522698\pi$
$179$	−341.260	−0.142497	−0.0712485	+	0.997459i	$0.522698\pi$
$180$	0	0
$181$	−1695.92	−0.696447	−0.348223	−	0.937412i	$-0.613215\pi$
$181$	−1695.92	−0.696447	−0.348223	+	0.937412i	$0.613215\pi$
$182$	8298.45	3.37979
$183$	0	0
$184$	−8893.90	−3.56341
$185$	−1688.07	−0.670860
$186$	0	0
$187$	−1975.88	−0.772678
$188$	95.3107	0.0369747
$189$	0	0
$190$	−2715.38	−1.03681
$191$	726.451	0.275205	0.137603	−	0.990488i	$-0.456060\pi$
$191$	726.451	0.275205	0.137603	+	0.990488i	$0.456060\pi$
$192$	0	0
$193$	4247.26	1.58406	0.792032	−	0.610479i	$-0.209023\pi$
$193$	4247.26	1.58406	0.792032	+	0.610479i	$0.209023\pi$
$194$	−4304.14	−1.59288
$195$	0	0
$196$	4807.94	1.75216
$197$	−2678.52	−0.968713	−0.484357	−	0.874871i	$-0.660946\pi$
$197$	−2678.52	−0.968713	−0.484357	+	0.874871i	$0.660946\pi$
$198$	0	0
$199$	1486.48	0.529517	0.264759	−	0.964315i	$-0.414708\pi$
$199$	1486.48	0.529517	0.264759	+	0.964315i	$0.414708\pi$
$200$	−1436.29	−0.507806
$201$	0	0
$202$	4283.15	1.49189
$203$	5018.32	1.73506
$204$	0	0
$205$	979.845	0.333831
$206$	−6090.97	−2.06009
$207$	0	0
$208$	−9574.12	−3.19157
$209$	3026.62	1.00170
$210$	0	0
$211$	−4827.41	−1.57504	−0.787519	−	0.616291i	$-0.788635\pi$
$211$	−4827.41	−1.57504	−0.787519	+	0.616291i	$0.788635\pi$
$212$	6094.37	1.97435
$213$	0	0
$214$	5321.37	1.69982
$215$	1674.41	0.531134
$216$	0	0
$217$	−445.317	−0.139309
$218$	2099.23	0.652193
$219$	0	0
$220$	2760.27	0.845899
$221$	4457.88	1.35688
$222$	0	0
$223$	2774.48	0.833153	0.416576	−	0.909101i	$-0.363230\pi$
$223$	2774.48	0.833153	0.416576	+	0.909101i	$0.363230\pi$
$224$	−7364.91	−2.19683
$225$	0	0
$226$	5628.19	1.65655
$227$	5101.34	1.49158	0.745788	−	0.666184i	$-0.232073\pi$
$227$	5101.34	1.49158	0.745788	+	0.666184i	$0.232073\pi$
$228$	0	0
$229$	−4097.83	−1.18250	−0.591249	−	0.806489i	$-0.701365\pi$
$229$	−4097.83	−1.18250	−0.591249	+	0.806489i	$0.701365\pi$
$230$	−4025.49	−1.15406
$231$	0	0
$232$	−11815.4	−3.34361
$233$	−357.613	−0.100549	−0.0502747	−	0.998735i	$-0.516010\pi$
$233$	−357.613	−0.100549	−0.0502747	+	0.998735i	$0.516010\pi$
$234$	0	0
$235$	25.0199	0.00694519
$236$	−8201.83	−2.26226
$237$	0	0
$238$	8651.22	2.35620
$239$	351.682	0.0951818	0.0475909	−	0.998867i	$-0.484846\pi$
$239$	351.682	0.0951818	0.0475909	+	0.998867i	$0.484846\pi$
$240$	0	0
$241$	−6165.53	−1.64795	−0.823976	−	0.566624i	$-0.808249\pi$
$241$	−6165.53	−1.64795	−0.823976	+	0.566624i	$0.808249\pi$
$242$	2553.19	0.678204
$243$	0	0
$244$	11325.0	2.97134
$245$	1262.13	0.329120
$246$	0	0
$247$	−6828.50	−1.75906
$248$	1048.48	0.268461
$249$	0	0
$250$	−650.084	−0.164460
$251$	−3245.53	−0.816160	−0.408080	−	0.912946i	$-0.633802\pi$
$251$	−3245.53	−0.816160	−0.408080	+	0.912946i	$0.633802\pi$
$252$	0	0
$253$	4486.90	1.11498
$254$	4026.00	0.994543
$255$	0	0
$256$	−4969.25	−1.21320
$257$	−3552.19	−0.862178	−0.431089	−	0.902309i	$-0.641871\pi$
$257$	−3552.19	−0.862178	−0.431089	+	0.902309i	$0.641871\pi$
$258$	0	0
$259$	−8238.22	−1.97644
$260$	−6227.59	−1.48546
$261$	0	0
$262$	−6313.81	−1.48881
$263$	−4416.59	−1.03551	−0.517754	−	0.855530i	$-0.673232\pi$
$263$	−4416.59	−1.03551	−0.517754	+	0.855530i	$0.673232\pi$
$264$	0	0
$265$	1599.83	0.370855
$266$	−13251.8	−3.05458
$267$	0	0
$268$	3730.85	0.850366
$269$	3419.93	0.775155	0.387578	−	0.921837i	$-0.373312\pi$
$269$	3419.93	0.775155	0.387578	+	0.921837i	$0.373312\pi$
$270$	0	0
$271$	716.407	0.160585	0.0802927	−	0.996771i	$-0.474415\pi$
$271$	716.407	0.160585	0.0802927	+	0.996771i	$0.474415\pi$
$272$	−9981.12	−2.22498
$273$	0	0
$274$	11962.3	2.63748
$275$	724.597	0.158890
$276$	0	0
$277$	657.529	0.142625	0.0713124	−	0.997454i	$-0.477281\pi$
$277$	657.529	0.142625	0.0713124	+	0.997454i	$0.477281\pi$
$278$	7051.72	1.52135
$279$	0	0
$280$	−7009.49	−1.49606
$281$	−1513.91	−0.321397	−0.160698	−	0.987004i	$-0.551375\pi$
$281$	−1513.91	−0.321397	−0.160698	+	0.987004i	$0.551375\pi$
$282$	0	0
$283$	3906.38	0.820532	0.410266	−	0.911966i	$-0.365436\pi$
$283$	3906.38	0.820532	0.410266	+	0.911966i	$0.365436\pi$
$284$	−8113.16	−1.69517
$285$	0	0
$286$	9856.89	2.03794
$287$	4781.91	0.983509
$288$	0	0
$289$	−265.611	−0.0540629
$290$	−5347.79	−1.08287
$291$	0	0
$292$	17698.3	3.54697
$293$	8048.76	1.60483	0.802413	−	0.596770i	$-0.203549\pi$
$293$	8048.76	1.60483	0.802413	+	0.596770i	$0.203549\pi$
$294$	0	0
$295$	−2153.05	−0.424934
$296$	19396.4	3.80877
$297$	0	0
$298$	1351.37	0.262693
$299$	−10123.1	−1.95797
$300$	0	0
$301$	8171.57	1.56479
$302$	2643.52	0.503700
$303$	0	0
$304$	15288.9	2.88447
$305$	2972.91	0.558125
$306$	0	0
$307$	101.564	0.0188814	0.00944068	−	0.999955i	$-0.496995\pi$
$307$	101.564	0.0188814	0.00944068	+	0.999955i	$0.496995\pi$
$308$	13470.9	2.49213
$309$	0	0
$310$	474.553	0.0869445
$311$	7684.59	1.40113	0.700567	−	0.713586i	$-0.252930\pi$
$311$	7684.59	1.40113	0.700567	+	0.713586i	$0.252930\pi$
$312$	0	0
$313$	−1345.15	−0.242915	−0.121457	−	0.992597i	$-0.538757\pi$
$313$	−1345.15	−0.242915	−0.121457	+	0.992597i	$0.538757\pi$
$314$	121.202	0.0217830
$315$	0	0
$316$	465.310	0.0828346
$317$	−7622.33	−1.35051	−0.675257	−	0.737583i	$-0.735967\pi$
$317$	−7622.33	−1.35051	−0.675257	+	0.737583i	$0.735967\pi$
$318$	0	0
$319$	5960.76	1.04620
$320$	1991.98	0.347985
$321$	0	0
$322$	−19645.5	−3.40000
$323$	−7118.78	−1.22631
$324$	0	0
$325$	−1634.80	−0.279022
$326$	20969.6	3.56258
$327$	0	0
$328$	−11258.7	−1.89531
$329$	122.104	0.0204614
$330$	0	0
$331$	−6585.09	−1.09350	−0.546751	−	0.837295i	$-0.684136\pi$
$331$	−6585.09	−1.09350	−0.546751	+	0.837295i	$0.684136\pi$
$332$	10397.0	1.71870
$333$	0	0
$334$	−3494.39	−0.572468
$335$	979.382	0.159729
$336$	0	0
$337$	−2946.94	−0.476351	−0.238175	−	0.971222i	$-0.576549\pi$
$337$	−2946.94	−0.476351	−0.238175	+	0.971222i	$0.576549\pi$
$338$	−10812.7	−1.74004
$339$	0	0
$340$	−6492.33	−1.03558
$341$	−528.947	−0.0840002
$342$	0	0
$343$	−2210.14	−0.347919
$344$	−19239.5	−3.01548
$345$	0	0
$346$	8495.44	1.31999
$347$	−8493.48	−1.31399	−0.656994	−	0.753896i	$-0.728172\pi$
$347$	−8493.48	−1.31399	−0.656994	+	0.753896i	$0.728172\pi$
$348$	0	0
$349$	5646.54	0.866053	0.433027	−	0.901381i	$-0.357446\pi$
$349$	5646.54	0.866053	0.433027	+	0.901381i	$0.357446\pi$
$350$	−3172.58	−0.484519
$351$	0	0
$352$	−8748.03	−1.32464
$353$	1221.93	0.184240	0.0921202	−	0.995748i	$-0.470636\pi$
$353$	1221.93	0.184240	0.0921202	+	0.995748i	$0.470636\pi$
$354$	0	0
$355$	−2129.78	−0.318414
$356$	1602.48	0.238571
$357$	0	0
$358$	1774.78	0.262011
$359$	4151.44	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$359$	4151.44	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$360$	0	0
$361$	4045.41	0.589796
$362$	8819.93	1.28057
$363$	0	0
$364$	−30392.3	−4.37635
$365$	4645.96	0.666249
$366$	0	0
$367$	−7038.71	−1.00114	−0.500569	−	0.865696i	$-0.666876\pi$
$367$	−7038.71	−1.00114	−0.500569	+	0.865696i	$0.666876\pi$
$368$	22665.5	3.21065
$369$	0	0
$370$	8779.08	1.23352
$371$	7807.58	1.09259
$372$	0	0
$373$	−7119.57	−0.988303	−0.494152	−	0.869376i	$-0.664521\pi$
$373$	−7119.57	−0.988303	−0.494152	+	0.869376i	$0.664521\pi$
$374$	10275.9	1.42073
$375$	0	0
$376$	−287.487	−0.0394309
$377$	−13448.4	−1.83720
$378$	0	0
$379$	3372.29	0.457053	0.228526	−	0.973538i	$-0.426609\pi$
$379$	3372.29	0.457053	0.228526	+	0.973538i	$0.426609\pi$
$380$	9944.82	1.34252
$381$	0	0
$382$	−3778.03	−0.506024
$383$	−3958.63	−0.528138	−0.264069	−	0.964504i	$-0.585065\pi$
$383$	−3958.63	−0.528138	−0.264069	+	0.964504i	$0.585065\pi$
$384$	0	0
$385$	3536.23	0.468111
$386$	−22088.6	−2.91264
$387$	0	0
$388$	15763.5	2.06256
$389$	−9654.01	−1.25830	−0.629148	−	0.777285i	$-0.716596\pi$
$389$	−9654.01	−1.25830	−0.629148	+	0.777285i	$0.716596\pi$
$390$	0	0
$391$	−10553.4	−1.36499
$392$	−14502.3	−1.86856
$393$	0	0
$394$	13930.1	1.78119
$395$	122.148	0.0155593
$396$	0	0
$397$	10928.3	1.38155	0.690776	−	0.723068i	$-0.257269\pi$
$397$	10928.3	1.38155	0.690776	+	0.723068i	$0.257269\pi$
$398$	−7730.70	−0.973631
$399$	0	0
$400$	3660.28	0.457536
$401$	−4085.57	−0.508787	−0.254393	−	0.967101i	$-0.581876\pi$
$401$	−4085.57	−0.508787	−0.254393	+	0.967101i	$0.581876\pi$
$402$	0	0
$403$	1193.38	0.147510
$404$	−15686.6	−1.93178
$405$	0	0
$406$	−26098.7	−3.19028
$407$	−9785.34	−1.19175
$408$	0	0
$409$	10156.3	1.22786	0.613930	−	0.789361i	$-0.289588\pi$
$409$	10156.3	1.22786	0.613930	+	0.789361i	$0.289588\pi$
$410$	−5095.85	−0.613820
$411$	0	0
$412$	22307.6	2.66752
$413$	−10507.5	−1.25191
$414$	0	0
$415$	2729.29	0.322833
$416$	19736.9	2.32615
$417$	0	0
$418$	−15740.4	−1.84184
$419$	−15878.8	−1.85139	−0.925693	−	0.378275i	$-0.876517\pi$
$419$	−15878.8	−1.85139	−0.925693	+	0.378275i	$0.876517\pi$
$420$	0	0
$421$	−2279.85	−0.263926	−0.131963	−	0.991255i	$-0.542128\pi$
$421$	−2279.85	−0.263926	−0.131963	+	0.991255i	$0.542128\pi$
$422$	25105.8	2.89604
$423$	0	0
$424$	−18382.5	−2.10551
$425$	−1704.29	−0.194519
$426$	0	0
$427$	14508.6	1.64431
$428$	−19489.0	−2.20102
$429$	0	0
$430$	−8708.05	−0.976604
$431$	1947.38	0.217638	0.108819	−	0.994062i	$-0.465293\pi$
$431$	1947.38	0.217638	0.108819	+	0.994062i	$0.465293\pi$
$432$	0	0
$433$	12636.2	1.40244	0.701219	−	0.712946i	$-0.252639\pi$
$433$	12636.2	1.40244	0.701219	+	0.712946i	$0.252639\pi$
$434$	2315.95	0.256150
$435$	0	0
$436$	−7688.25	−0.844496
$437$	16165.6	1.76957
$438$	0	0
$439$	−15849.8	−1.72317	−0.861585	−	0.507614i	$-0.830528\pi$
$439$	−15849.8	−1.72317	−0.861585	+	0.507614i	$0.830528\pi$
$440$	−8325.86	−0.902090
$441$	0	0
$442$	−23184.0	−2.49491
$443$	17455.6	1.87210	0.936048	−	0.351872i	$-0.114455\pi$
$443$	17455.6	1.87210	0.936048	+	0.351872i	$0.114455\pi$
$444$	0	0
$445$	420.666	0.0448123
$446$	−14429.2	−1.53193
$447$	0	0
$448$	9721.41	1.02521
$449$	16068.1	1.68887	0.844435	−	0.535658i	$-0.179936\pi$
$449$	16068.1	1.68887	0.844435	+	0.535658i	$0.179936\pi$
$450$	0	0
$451$	5679.94	0.593033
$452$	−20612.7	−2.14500
$453$	0	0
$454$	−26530.4	−2.74258
$455$	−7978.25	−0.822036
$456$	0	0
$457$	11891.7	1.21722	0.608612	−	0.793468i	$-0.291727\pi$
$457$	11891.7	1.21722	0.608612	+	0.793468i	$0.291727\pi$
$458$	21311.4	2.17428
$459$	0	0
$460$	14743.0	1.49434
$461$	2802.23	0.283108	0.141554	−	0.989931i	$-0.454790\pi$
$461$	2802.23	0.283108	0.141554	+	0.989931i	$0.454790\pi$
$462$	0	0
$463$	−12933.3	−1.29819	−0.649096	−	0.760707i	$-0.724853\pi$
$463$	−12933.3	−1.29819	−0.649096	+	0.760707i	$0.724853\pi$
$464$	30110.6	3.01261
$465$	0	0
$466$	1859.83	0.184882
$467$	−5748.11	−0.569573	−0.284787	−	0.958591i	$-0.591923\pi$
$467$	−5748.11	−0.569573	−0.284787	+	0.958591i	$0.591923\pi$
$468$	0	0
$469$	4779.65	0.470583
$470$	−130.120	−0.0127702
$471$	0	0
$472$	24739.3	2.41254
$473$	9706.17	0.943531
$474$	0	0
$475$	2610.60	0.252174
$476$	−31684.3	−3.05094
$477$	0	0
$478$	−1828.98	−0.175012
$479$	11217.3	1.07000	0.535002	−	0.844851i	$-0.320311\pi$
$479$	11217.3	1.07000	0.535002	+	0.844851i	$0.320311\pi$
$480$	0	0
$481$	22077.2	2.09279
$482$	32064.9	3.03012
$483$	0	0
$484$	−9350.83	−0.878177
$485$	4138.07	0.387423
$486$	0	0
$487$	−8905.12	−0.828603	−0.414301	−	0.910140i	$-0.635974\pi$
$487$	−8905.12	−0.828603	−0.414301	+	0.910140i	$0.635974\pi$
$488$	−34159.7	−3.16872
$489$	0	0
$490$	−6563.91	−0.605157
$491$	6553.10	0.602316	0.301158	−	0.953574i	$-0.402627\pi$
$491$	6553.10	0.602316	0.301158	+	0.953574i	$0.402627\pi$
$492$	0	0
$493$	−14020.1	−1.28079
$494$	35512.8	3.23440
$495$	0	0
$496$	−2671.96	−0.241884
$497$	−10393.9	−0.938087
$498$	0	0
$499$	−4610.09	−0.413579	−0.206789	−	0.978385i	$-0.566302\pi$
$499$	−4610.09	−0.413579	−0.206789	+	0.978385i	$0.566302\pi$
$500$	2380.87	0.212952
$501$	0	0
$502$	16878.9	1.50069
$503$	−13069.1	−1.15850	−0.579249	−	0.815151i	$-0.696654\pi$
$503$	−13069.1	−1.15850	−0.579249	+	0.815151i	$0.696654\pi$
$504$	0	0
$505$	−4117.88	−0.362858
$506$	−23334.9	−2.05012
$507$	0	0
$508$	−14744.9	−1.28779
$509$	−15930.8	−1.38727	−0.693635	−	0.720327i	$-0.743992\pi$
$509$	−15930.8	−1.38727	−0.693635	+	0.720327i	$0.743992\pi$
$510$	0	0
$511$	22673.6	1.96286
$512$	23102.1	1.99410
$513$	0	0
$514$	18473.8	1.58530
$515$	5855.95	0.501056
$516$	0	0
$517$	145.035	0.0123378
$518$	42844.3	3.63411
$519$	0	0
$520$	18784.4	1.58413
$521$	−3654.38	−0.307296	−0.153648	−	0.988126i	$-0.549102\pi$
$521$	−3654.38	−0.307296	−0.153648	+	0.988126i	$0.549102\pi$
$522$	0	0
$523$	−5138.66	−0.429633	−0.214816	−	0.976654i	$-0.568915\pi$
$523$	−5138.66	−0.429633	−0.214816	+	0.976654i	$0.568915\pi$
$524$	23123.7	1.92780
$525$	0	0
$526$	22969.2	1.90400
$527$	1244.11	0.102836
$528$	0	0
$529$	11798.1	0.969681
$530$	−8320.16	−0.681896
$531$	0	0
$532$	48533.4	3.95524
$533$	−12814.8	−1.04141
$534$	0	0
$535$	−5116.04	−0.413431
$536$	−11253.4	−0.906854
$537$	0	0
$538$	−17785.9	−1.42529
$539$	7316.27	0.584664
$540$	0	0
$541$	6932.06	0.550892	0.275446	−	0.961317i	$-0.411175\pi$
$541$	6932.06	0.550892	0.275446	+	0.961317i	$0.411175\pi$
$542$	−3725.80	−0.295271
$543$	0	0
$544$	20575.9	1.62166
$545$	−2018.23	−0.158627
$546$	0	0
$547$	−3423.11	−0.267572	−0.133786	−	0.991010i	$-0.542713\pi$
$547$	−3423.11	−0.267572	−0.133786	+	0.991010i	$0.542713\pi$
$548$	−43810.8	−3.41516
$549$	0	0
$550$	−3768.39	−0.292154
$551$	21475.6	1.66042
$552$	0	0
$553$	596.115	0.0458398
$554$	−3419.59	−0.262246
$555$	0	0
$556$	−25826.3	−1.96992
$557$	24489.2	1.86291	0.931455	−	0.363856i	$-0.118540\pi$
$557$	24489.2	1.86291	0.931455	+	0.363856i	$0.118540\pi$
$558$	0	0
$559$	−21898.6	−1.65691
$560$	17863.2	1.34796
$561$	0	0
$562$	7873.37	0.590957
$563$	10053.1	0.752552	0.376276	−	0.926508i	$-0.377204\pi$
$563$	10053.1	0.752552	0.376276	+	0.926508i	$0.377204\pi$
$564$	0	0
$565$	−5411.02	−0.402908
$566$	−20315.8	−1.50872
$567$	0	0
$568$	24471.8	1.80777
$569$	6670.45	0.491459	0.245729	−	0.969338i	$-0.420973\pi$
$569$	6670.45	0.491459	0.245729	+	0.969338i	$0.420973\pi$
$570$	0	0
$571$	4633.55	0.339594	0.169797	−	0.985479i	$-0.445689\pi$
$571$	4633.55	0.339594	0.169797	+	0.985479i	$0.445689\pi$
$572$	−36099.9	−2.63884
$573$	0	0
$574$	−24869.1	−1.80839
$575$	3870.17	0.280691
$576$	0	0
$577$	7045.15	0.508307	0.254154	−	0.967164i	$-0.418203\pi$
$577$	7045.15	0.508307	0.254154	+	0.967164i	$0.418203\pi$
$578$	1381.35	0.0994062
$579$	0	0
$580$	19585.8	1.40216
$581$	13319.7	0.951108
$582$	0	0
$583$	9273.82	0.658804
$584$	−53383.7	−3.78259
$585$	0	0
$586$	−41859.0	−2.95082
$587$	8001.06	0.562588	0.281294	−	0.959622i	$-0.409236\pi$
$587$	8001.06	0.562588	0.281294	+	0.959622i	$0.409236\pi$
$588$	0	0
$589$	−1905.71	−0.133316
$590$	11197.3	0.781333
$591$	0	0
$592$	−49430.4	−3.43172
$593$	−6747.53	−0.467265	−0.233632	−	0.972325i	$-0.575061\pi$
$593$	−6747.53	−0.467265	−0.233632	+	0.972325i	$0.575061\pi$
$594$	0	0
$595$	−8317.41	−0.573077
$596$	−4949.26	−0.340150
$597$	0	0
$598$	52646.9	3.60016
$599$	−21547.3	−1.46978	−0.734890	−	0.678186i	$-0.762766\pi$
$599$	−21547.3	−1.46978	−0.734890	+	0.678186i	$0.762766\pi$
$600$	0	0
$601$	−12155.1	−0.824983	−0.412492	−	0.910961i	$-0.635341\pi$
$601$	−12155.1	−0.824983	−0.412492	+	0.910961i	$0.635341\pi$
$602$	−42497.6	−2.87720
$603$	0	0
$604$	−9681.64	−0.652219
$605$	−2454.68	−0.164953
$606$	0	0
$607$	16348.9	1.09322	0.546608	−	0.837388i	$-0.315919\pi$
$607$	16348.9	1.09322	0.546608	+	0.837388i	$0.315919\pi$
$608$	−31517.7	−2.10232
$609$	0	0
$610$	−15461.1	−1.02623
$611$	−327.220	−0.0216660
$612$	0	0
$613$	−29955.5	−1.97372	−0.986859	−	0.161581i	$-0.948341\pi$
$613$	−29955.5	−1.97372	−0.986859	+	0.161581i	$0.948341\pi$
$614$	−528.202	−0.0347174
$615$	0	0
$616$	−40632.4	−2.65767
$617$	−2159.74	−0.140921	−0.0704603	−	0.997515i	$-0.522447\pi$
$617$	−2159.74	−0.140921	−0.0704603	+	0.997515i	$0.522447\pi$
$618$	0	0
$619$	22100.8	1.43507	0.717535	−	0.696523i	$-0.245270\pi$
$619$	22100.8	1.43507	0.717535	+	0.696523i	$0.245270\pi$
$620$	−1738.01	−0.112581
$621$	0	0
$622$	−39965.0	−2.57629
$623$	2052.96	0.132023
$624$	0	0
$625$	625.000	0.0400000
$626$	6995.68	0.446651
$627$	0	0
$628$	−443.893	−0.0282058
$629$	23015.7	1.45898
$630$	0	0
$631$	18360.1	1.15833	0.579164	−	0.815211i	$-0.303379\pi$
$631$	18360.1	1.15833	0.579164	+	0.815211i	$0.303379\pi$
$632$	−1403.52	−0.0883372
$633$	0	0
$634$	39641.2	2.48321
$635$	−3870.66	−0.241893
$636$	0	0
$637$	−16506.6	−1.02671
$638$	−30999.9	−1.92367
$639$	0	0
$640$	1713.32	0.105820
$641$	21064.5	1.29797	0.648985	−	0.760802i	$-0.275194\pi$
$641$	21064.5	1.29797	0.648985	+	0.760802i	$0.275194\pi$
$642$	0	0
$643$	−10539.1	−0.646381	−0.323190	−	0.946334i	$-0.604755\pi$
$643$	−10539.1	−0.646381	−0.323190	+	0.946334i	$0.604755\pi$
$644$	71949.8	4.40251
$645$	0	0
$646$	37022.4	2.25484
$647$	−22553.4	−1.37043	−0.685213	−	0.728343i	$-0.740291\pi$
$647$	−22553.4	−1.37043	−0.685213	+	0.728343i	$0.740291\pi$
$648$	0	0
$649$	−12480.8	−0.754873
$650$	8502.04	0.513043
$651$	0	0
$652$	−76799.4	−4.61303
$653$	−22624.0	−1.35582	−0.677908	−	0.735147i	$-0.737113\pi$
$653$	−22624.0	−1.35582	−0.677908	+	0.735147i	$0.737113\pi$
$654$	0	0
$655$	6070.19	0.362110
$656$	28692.1	1.70768
$657$	0	0
$658$	−635.022	−0.0376227
$659$	−6376.60	−0.376930	−0.188465	−	0.982080i	$-0.560351\pi$
$659$	−6376.60	−0.376930	−0.188465	+	0.982080i	$0.560351\pi$
$660$	0	0
$661$	−22097.5	−1.30029	−0.650146	−	0.759809i	$-0.725292\pi$
$661$	−22097.5	−1.30029	−0.650146	+	0.759809i	$0.725292\pi$
$662$	34246.9	2.01064
$663$	0	0
$664$	−31360.5	−1.83287
$665$	12740.4	0.742938
$666$	0	0
$667$	31837.2	1.84819
$668$	12797.9	0.741264
$669$	0	0
$670$	−5093.44	−0.293697
$671$	17233.3	0.991479
$672$	0	0
$673$	24033.0	1.37653	0.688264	−	0.725461i	$-0.258373\pi$
$673$	24033.0	1.37653	0.688264	+	0.725461i	$0.258373\pi$
$674$	15326.1	0.875873
$675$	0	0
$676$	39600.6	2.25311
$677$	−179.638	−0.0101980	−0.00509901	−	0.999987i	$-0.501623\pi$
$677$	−179.638	−0.0101980	−0.00509901	+	0.999987i	$0.501623\pi$
$678$	0	0
$679$	20194.9	1.14140
$680$	19582.9	1.10437
$681$	0	0
$682$	2750.88	0.154452
$683$	30434.2	1.70502	0.852511	−	0.522709i	$-0.175079\pi$
$683$	30434.2	1.70502	0.852511	+	0.522709i	$0.175079\pi$
$684$	0	0
$685$	−11500.7	−0.641490
$686$	11494.2	0.639725
$687$	0	0
$688$	49030.5	2.71696
$689$	−20923.1	−1.15691
$690$	0	0
$691$	9792.73	0.539122	0.269561	−	0.962983i	$-0.413121\pi$
$691$	9792.73	0.539122	0.269561	+	0.962983i	$0.413121\pi$
$692$	−31113.7	−1.70920
$693$	0	0
$694$	44171.8	2.41605
$695$	−6779.63	−0.370023
$696$	0	0
$697$	−13359.6	−0.726010
$698$	−29365.8	−1.59242
$699$	0	0
$700$	11619.3	0.627383
$701$	−8130.47	−0.438065	−0.219032	−	0.975718i	$-0.570290\pi$
$701$	−8130.47	−0.438065	−0.219032	+	0.975718i	$0.570290\pi$
$702$	0	0
$703$	−35255.0	−1.89142
$704$	11547.1	0.618177
$705$	0	0
$706$	−6354.86	−0.338765
$707$	−20096.4	−1.06903
$708$	0	0
$709$	−4859.95	−0.257432	−0.128716	−	0.991681i	$-0.541086\pi$
$709$	−4859.95	−0.257432	−0.128716	+	0.991681i	$0.541086\pi$
$710$	11076.3	0.585472
$711$	0	0
$712$	−4833.59	−0.254419
$713$	−2825.17	−0.148392
$714$	0	0
$715$	−9476.55	−0.495669
$716$	−6499.96	−0.339267
$717$	0	0
$718$	−21590.3	−1.12220
$719$	−19463.0	−1.00952	−0.504762	−	0.863259i	$-0.668420\pi$
$719$	−19463.0	−1.00952	−0.504762	+	0.863259i	$0.668420\pi$
$720$	0	0
$721$	28578.6	1.47618
$722$	−21038.8	−1.08447
$723$	0	0
$724$	−32302.2	−1.65815
$725$	5141.44	0.263377
$726$	0	0
$727$	2432.66	0.124102	0.0620512	−	0.998073i	$-0.480236\pi$
$727$	2432.66	0.124102	0.0620512	+	0.998073i	$0.480236\pi$
$728$	91672.8	4.66706
$729$	0	0
$730$	−24162.1	−1.22504
$731$	−22829.5	−1.15510
$732$	0	0
$733$	−17967.6	−0.905386	−0.452693	−	0.891666i	$-0.649537\pi$
$733$	−17967.6	−0.905386	−0.452693	+	0.891666i	$0.649537\pi$
$734$	36606.0	1.84081
$735$	0	0
$736$	−46724.4	−2.34006
$737$	5677.26	0.283751
$738$	0	0
$739$	23473.0	1.16843	0.584214	−	0.811599i	$-0.301403\pi$
$739$	23473.0	1.16843	0.584214	+	0.811599i	$0.301403\pi$
$740$	−32152.6	−1.59723
$741$	0	0
$742$	−40604.6	−2.00895
$743$	33559.2	1.65702	0.828512	−	0.559971i	$-0.189188\pi$
$743$	33559.2	1.65702	0.828512	+	0.559971i	$0.189188\pi$
$744$	0	0
$745$	−1299.23	−0.0638925
$746$	37026.5	1.81721
$747$	0	0
$748$	−37634.6	−1.83965
$749$	−24967.6	−1.21802
$750$	0	0
$751$	−7782.75	−0.378158	−0.189079	−	0.981962i	$-0.560550\pi$
$751$	−7782.75	−0.378158	−0.189079	+	0.981962i	$0.560550\pi$
$752$	732.640	0.0355274
$753$	0	0
$754$	69940.5	3.37809
$755$	−2541.52	−0.122510
$756$	0	0
$757$	−38154.3	−1.83189	−0.915946	−	0.401300i	$-0.868558\pi$
$757$	−38154.3	−1.83189	−0.915946	+	0.401300i	$0.868558\pi$
$758$	−17538.2	−0.840390
$759$	0	0
$760$	−29996.7	−1.43170
$761$	−19867.1	−0.946363	−0.473182	−	0.880965i	$-0.656895\pi$
$761$	−19867.1	−0.946363	−0.473182	+	0.880965i	$0.656895\pi$
$762$	0	0
$763$	−9849.52	−0.467335
$764$	13836.7	0.655228
$765$	0	0
$766$	20587.5	0.971094
$767$	28158.5	1.32561
$768$	0	0
$769$	−15710.8	−0.736730	−0.368365	−	0.929681i	$-0.620082\pi$
$769$	−15710.8	−0.736730	−0.368365	+	0.929681i	$0.620082\pi$
$770$	−18390.7	−0.860723
$771$	0	0
$772$	80897.5	3.77146
$773$	−25811.9	−1.20102	−0.600510	−	0.799617i	$-0.705036\pi$
$773$	−25811.9	−1.20102	−0.600510	+	0.799617i	$0.705036\pi$
$774$	0	0
$775$	−456.242	−0.0211467
$776$	−47547.8	−2.19957
$777$	0	0
$778$	50207.3	2.31365
$779$	20463.9	0.941201
$780$	0	0
$781$	−12345.8	−0.565645
$782$	54885.0	2.50982
$783$	0	0
$784$	36958.0	1.68358
$785$	−116.526	−0.00529807
$786$	0	0
$787$	−29242.6	−1.32450	−0.662252	−	0.749281i	$-0.730399\pi$
$787$	−29242.6	−1.32450	−0.662252	+	0.749281i	$0.730399\pi$
$788$	−51017.7	−2.30638
$789$	0	0
$790$	−635.252	−0.0286092
$791$	−26407.2	−1.18702
$792$	0	0
$793$	−38880.8	−1.74111
$794$	−56834.6	−2.54028
$795$	0	0
$796$	28313.0	1.26071
$797$	−32573.7	−1.44771	−0.723853	−	0.689955i	$-0.757630\pi$
$797$	−32573.7	−1.44771	−0.723853	+	0.689955i	$0.757630\pi$
$798$	0	0
$799$	−341.130	−0.0151043
$800$	−7545.60	−0.333472
$801$	0	0
$802$	21247.7	0.935514
$803$	26931.6	1.18356
$804$	0	0
$805$	18887.5	0.826951
$806$	−6206.39	−0.271229
$807$	0	0
$808$	47315.8	2.06010
$809$	34644.9	1.50562	0.752812	−	0.658236i	$-0.228697\pi$
$809$	34644.9	1.50562	0.752812	+	0.658236i	$0.228697\pi$
$810$	0	0
$811$	−29057.9	−1.25815	−0.629077	−	0.777343i	$-0.716567\pi$
$811$	−29057.9	−1.25815	−0.629077	+	0.777343i	$0.716567\pi$
$812$	95583.9	4.13096
$813$	0	0
$814$	50890.3	2.19128
$815$	−20160.5	−0.866493
$816$	0	0
$817$	34969.8	1.49748
$818$	−52819.3	−2.25768
$819$	0	0
$820$	18663.1	0.794809
$821$	−46709.5	−1.98560	−0.992798	−	0.119802i	$-0.961774\pi$
$821$	−46709.5	−1.98560	−0.992798	+	0.119802i	$0.961774\pi$
$822$	0	0
$823$	−3468.10	−0.146890	−0.0734450	−	0.997299i	$-0.523399\pi$
$823$	−3468.10	−0.146890	−0.0734450	+	0.997299i	$0.523399\pi$
$824$	−67286.8	−2.84472
$825$	0	0
$826$	54645.9	2.30191
$827$	−42454.9	−1.78513	−0.892564	−	0.450920i	$-0.851096\pi$
$827$	−42454.9	−1.78513	−0.892564	+	0.450920i	$0.851096\pi$
$828$	0	0
$829$	−3933.47	−0.164795	−0.0823975	−	0.996600i	$-0.526258\pi$
$829$	−3933.47	−0.164795	−0.0823975	+	0.996600i	$0.526258\pi$
$830$	−14194.2	−0.593598
$831$	0	0
$832$	−26051.9	−1.08556
$833$	−17208.3	−0.715764
$834$	0	0
$835$	3359.55	0.139236
$836$	57647.9	2.38492
$837$	0	0
$838$	82580.5	3.40417
$839$	32959.6	1.35625	0.678123	−	0.734948i	$-0.262794\pi$
$839$	32959.6	1.35625	0.678123	+	0.734948i	$0.262794\pi$
$840$	0	0
$841$	17906.1	0.734188
$842$	11856.7	0.485285
$843$	0	0
$844$	−91947.6	−3.74996
$845$	10395.5	0.423215
$846$	0	0
$847$	−11979.5	−0.485974
$848$	46846.5	1.89707
$849$	0	0
$850$	8863.47	0.357664
$851$	−52264.8	−2.10530
$852$	0	0
$853$	38845.8	1.55927	0.779634	−	0.626235i	$-0.215405\pi$
$853$	38845.8	1.55927	0.779634	+	0.626235i	$0.215405\pi$
$854$	−75454.3	−3.02341
$855$	0	0
$856$	58785.0	2.34723
$857$	29305.3	1.16809	0.584043	−	0.811723i	$-0.301470\pi$
$857$	29305.3	1.16809	0.584043	+	0.811723i	$0.301470\pi$
$858$	0	0
$859$	−909.659	−0.0361318	−0.0180659	−	0.999837i	$-0.505751\pi$
$859$	−909.659	−0.0361318	−0.0180659	+	0.999837i	$0.505751\pi$
$860$	31892.4	1.26456
$861$	0	0
$862$	−10127.7	−0.400173
$863$	−47998.0	−1.89324	−0.946622	−	0.322346i	$-0.895529\pi$
$863$	−47998.0	−1.89324	−0.946622	+	0.322346i	$0.895529\pi$
$864$	0	0
$865$	−8167.64	−0.321050
$866$	−65716.6	−2.57868
$867$	0	0
$868$	−8481.94	−0.331677
$869$	708.065	0.0276403
$870$	0	0
$871$	−12808.7	−0.498286
$872$	23190.2	0.900594
$873$	0	0
$874$	−84071.7	−3.25374
$875$	3050.17	0.117845
$876$	0	0
$877$	3258.01	0.125445	0.0627225	−	0.998031i	$-0.480022\pi$
$877$	3258.01	0.125445	0.0627225	+	0.998031i	$0.480022\pi$
$878$	82429.8	3.16842
$879$	0	0
$880$	21217.8	0.812788
$881$	33380.2	1.27651	0.638256	−	0.769824i	$-0.279656\pi$
$881$	33380.2	1.27651	0.638256	+	0.769824i	$0.279656\pi$
$882$	0	0
$883$	33714.6	1.28492	0.642460	−	0.766319i	$-0.277914\pi$
$883$	33714.6	1.28492	0.642460	+	0.766319i	$0.277914\pi$
$884$	84909.2	3.23055
$885$	0	0
$886$	−90780.6	−3.44225
$887$	6218.80	0.235408	0.117704	−	0.993049i	$-0.462447\pi$
$887$	6218.80	0.235408	0.117704	+	0.993049i	$0.462447\pi$
$888$	0	0
$889$	−18889.8	−0.712649
$890$	−2187.75	−0.0823971
$891$	0	0
$892$	52845.5	1.98363
$893$	522.537	0.0195812
$894$	0	0
$895$	−1706.30	−0.0637266
$896$	8361.46	0.311760
$897$	0	0
$898$	−83565.1	−3.10535
$899$	−3753.19	−0.139239
$900$	0	0
$901$	−21812.6	−0.806529
$902$	−29539.5	−1.09042
$903$	0	0
$904$	62174.4	2.28749
$905$	−8479.61	−0.311460
$906$	0	0
$907$	−22878.6	−0.837566	−0.418783	−	0.908086i	$-0.637543\pi$
$907$	−22878.6	−0.837566	−0.418783	+	0.908086i	$0.637543\pi$
$908$	97165.0	3.55125
$909$	0	0
$910$	41492.3	1.51149
$911$	30144.8	1.09631	0.548157	−	0.836376i	$-0.315330\pi$
$911$	30144.8	1.09631	0.548157	+	0.836376i	$0.315330\pi$
$912$	0	0
$913$	15821.1	0.573496
$914$	−61844.9	−2.23813
$915$	0	0
$916$	−78051.2	−2.81538
$917$	29624.1	1.06682
$918$	0	0
$919$	3803.52	0.136525	0.0682625	−	0.997667i	$-0.478254\pi$
$919$	3803.52	0.136525	0.0682625	+	0.997667i	$0.478254\pi$
$920$	−44469.5	−1.59360
$921$	0	0
$922$	−14573.5	−0.520555
$923$	27854.0	0.993312
$924$	0	0
$925$	−8440.33	−0.300018
$926$	67262.0	2.38700
$927$	0	0
$928$	−62072.5	−2.19572
$929$	−125.985	−0.00444934	−0.00222467	−	0.999998i	$-0.500708\pi$
$929$	−125.985	−0.00444934	−0.00222467	+	0.999998i	$0.500708\pi$
$930$	0	0
$931$	26359.3	0.927918
$932$	−6811.45	−0.239395
$933$	0	0
$934$	29894.0	1.04728
$935$	−9879.41	−0.345552
$936$	0	0
$937$	28107.9	0.979984	0.489992	−	0.871727i	$-0.337000\pi$
$937$	28107.9	0.979984	0.489992	+	0.871727i	$0.337000\pi$
$938$	−24857.4	−0.865268
$939$	0	0
$940$	476.554	0.0165356
$941$	49194.9	1.70426	0.852130	−	0.523330i	$-0.175311\pi$
$941$	49194.9	1.70426	0.852130	+	0.523330i	$0.175311\pi$
$942$	0	0
$943$	30337.3	1.04763
$944$	−63046.3	−2.17371
$945$	0	0
$946$	−50478.6	−1.73488
$947$	14498.0	0.497490	0.248745	−	0.968569i	$-0.419982\pi$
$947$	14498.0	0.497490	0.248745	+	0.968569i	$0.419982\pi$
$948$	0	0
$949$	−60761.7	−2.07841
$950$	−13576.9	−0.463676
$951$	0	0
$952$	95569.8	3.25361
$953$	−3201.79	−0.108831	−0.0544155	−	0.998518i	$-0.517330\pi$
$953$	−3201.79	−0.108831	−0.0544155	+	0.998518i	$0.517330\pi$
$954$	0	0
$955$	3632.26	0.123075
$956$	6698.48	0.226616
$957$	0	0
$958$	−58337.6	−1.96743
$959$	−56126.7	−1.88991
$960$	0	0
$961$	−29457.9	−0.988820
$962$	−114816.	−3.84805
$963$	0	0
$964$	−117435.	−3.92356
$965$	21236.3	0.708415
$966$	0	0
$967$	57781.9	1.92155	0.960776	−	0.277326i	$-0.0894481\pi$
$967$	57781.9	1.92155	0.960776	+	0.277326i	$0.0894481\pi$
$968$	28205.1	0.936513
$969$	0	0
$970$	−21520.7	−0.712359
$971$	−2611.60	−0.0863133	−0.0431566	−	0.999068i	$-0.513741\pi$
$971$	−2611.60	−0.0863133	−0.0431566	+	0.999068i	$0.513741\pi$
$972$	0	0
$973$	−33086.4	−1.09013
$974$	46312.6	1.52356
$975$	0	0
$976$	87053.4	2.85503
$977$	33598.3	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$977$	33598.3	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$978$	0	0
$979$	2438.51	0.0796067
$980$	24039.7	0.783592
$981$	0	0
$982$	−34080.5	−1.10749
$983$	39484.8	1.28115	0.640575	−	0.767895i	$-0.278696\pi$
$983$	39484.8	1.28115	0.640575	+	0.767895i	$0.278696\pi$
$984$	0	0
$985$	−13392.6	−0.433222
$986$	72913.7	2.35501
$987$	0	0
$988$	−130062.	−4.18809
$989$	51841.9	1.66681
$990$	0	0
$991$	39918.6	1.27957	0.639786	−	0.768553i	$-0.279023\pi$
$991$	39918.6	1.27957	0.639786	+	0.768553i	$0.279023\pi$
$992$	5508.20	0.176296
$993$	0	0
$994$	54055.2	1.72487
$995$	7432.41	0.236807
$996$	0	0
$997$	−25670.3	−0.815432	−0.407716	−	0.913109i	$-0.633675\pi$
$997$	−25670.3	−0.815432	−0.407716	+	0.913109i	$0.633675\pi$
$998$	23975.6	0.760454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.a.f.1.1	✓	3
3.2	odd	2		135.4.a.g.1.3	yes	3
4.3	odd	2		2160.4.a.bm.1.1		3
5.2	odd	4		675.4.b.l.649.1		6
5.3	odd	4		675.4.b.l.649.6		6
5.4	even	2		675.4.a.r.1.3		3
9.2	odd	6		405.4.e.r.271.1		6
9.4	even	3		405.4.e.t.136.3		6
9.5	odd	6		405.4.e.r.136.1		6
9.7	even	3		405.4.e.t.271.3		6
12.11	even	2		2160.4.a.be.1.1		3
15.2	even	4		675.4.b.k.649.6		6
15.8	even	4		675.4.b.k.649.1		6
15.14	odd	2		675.4.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.1	✓	3	1.1	even	1	trivial
135.4.a.g.1.3	yes	3	3.2	odd	2
405.4.e.r.136.1		6	9.5	odd	6
405.4.e.r.271.1		6	9.2	odd	6
405.4.e.t.136.3		6	9.4	even	3
405.4.e.t.271.3		6	9.7	even	3
675.4.a.q.1.1		3	15.14	odd	2
675.4.a.r.1.3		3	5.4	even	2
675.4.b.k.649.1		6	15.8	even	4
675.4.b.k.649.6		6	15.2	even	4
675.4.b.l.649.1		6	5.2	odd	4
675.4.b.l.649.6		6	5.3	odd	4
2160.4.a.be.1.1		3	12.11	even	2
2160.4.a.bm.1.1		3	4.3	odd	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} +O(q^{10})\)	\(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} -26.0034 q^{10} +28.9839 q^{11} -65.3919 q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +95.2349 q^{20} -150.736 q^{22} +154.807 q^{23} +25.0000 q^{25} +340.082 q^{26} +464.772 q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +122.007 q^{35} -337.613 q^{37} -543.076 q^{38} -287.258 q^{40} +195.969 q^{41} +334.882 q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} -130.017 q^{50} -1245.52 q^{52} +319.965 q^{53} +144.919 q^{55} -1401.90 q^{56} -1069.56 q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -326.960 q^{65} +195.876 q^{67} -1298.47 q^{68} -634.517 q^{70} -425.955 q^{71} +929.193 q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +732.057 q^{80} -1019.17 q^{82} +545.859 q^{83} -340.859 q^{85} -1741.61 q^{86} -1665.17 q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} +522.121 q^{95} +827.613 q^{97} -1312.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) \) 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8	\( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8 - 5 * q^10 + 38 * q^11 + 28 * q^13 - 108 * q^14 + 191 * q^16 - 19 * q^17 + 187 * q^19 + 115 * q^20 + 122 * q^22 - 81 * q^23 + 75 * q^25 + 416 * q^26 + 410 * q^28 + 160 * q^29 + 227 * q^31 - 569 * q^32 + 17 * q^34 + 220 * q^35 + 78 * q^37 - 757 * q^38 - 180 * q^40 - 338 * q^41 + 22 * q^43 + 1636 * q^44 - 1425 * q^46 - 472 * q^47 - 197 * q^49 - 25 * q^50 - 1566 * q^52 + 521 * q^53 + 190 * q^55 - 1254 * q^56 - 2096 * q^58 + 140 * q^59 + 595 * q^61 + 1407 * q^62 - 918 * q^64 + 140 * q^65 + 878 * q^67 - 3053 * q^68 - 540 * q^70 - 602 * q^71 + 1294 * q^73 + 2878 * q^74 + 525 * q^76 + 288 * q^77 + 629 * q^79 + 955 * q^80 - 1682 * q^82 - 1287 * q^83 - 95 * q^85 - 3730 * q^86 - 858 * q^88 + 2154 * q^89 - 440 * q^91 + 1959 * q^92 - 1108 * q^94 + 935 * q^95 + 1392 * q^97 - 2693 * q^98

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