LMFDB - Embedded newform 729.4.a.b.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,4,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	729.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:	$43.0123923942$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 48 x^{10} + 269 x^{9} + 900 x^{8} - 4059 x^{7} - 8325 x^{6} + 23940 x^{5} + \cdots - 3392 $ x^12 - 6x^11 - 48x^10 + 269x^9 + 900x^8 - 4059x^7 - 8325x^6 + 23940x^5 + 32976x^4 - 55288x^3 - 40224x^2 + 45408*x - 3392
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-3.32122$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.32122 q^{2} +10.6730 q^{4} -3.36114 q^{5} -22.2331 q^{7} +11.5505 q^{8} -14.5243 q^{10} +32.7665 q^{11} +17.2460 q^{13} -96.0744 q^{14} -35.4713 q^{16} -104.698 q^{17} +86.2360 q^{19} -35.8734 q^{20} +141.591 q^{22} +37.2508 q^{23} -113.703 q^{25} +74.5237 q^{26} -237.294 q^{28} -179.556 q^{29} -251.179 q^{31} -245.684 q^{32} -452.423 q^{34} +74.7288 q^{35} -347.233 q^{37} +372.645 q^{38} -38.8231 q^{40} -466.757 q^{41} +375.073 q^{43} +349.716 q^{44} +160.969 q^{46} +181.852 q^{47} +151.312 q^{49} -491.335 q^{50} +184.066 q^{52} -148.879 q^{53} -110.133 q^{55} -256.805 q^{56} -775.901 q^{58} -377.724 q^{59} -287.938 q^{61} -1085.40 q^{62} -777.885 q^{64} -57.9661 q^{65} +27.6743 q^{67} -1117.44 q^{68} +322.920 q^{70} +759.978 q^{71} +120.862 q^{73} -1500.47 q^{74} +920.395 q^{76} -728.502 q^{77} +1115.08 q^{79} +119.224 q^{80} -2016.96 q^{82} +736.208 q^{83} +351.904 q^{85} +1620.77 q^{86} +378.471 q^{88} -321.649 q^{89} -383.432 q^{91} +397.577 q^{92} +785.822 q^{94} -289.852 q^{95} +1548.41 q^{97} +653.854 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 36 q^{4} + 12 q^{5} - 42 q^{7} + 21 q^{8} - 60 q^{10} + 42 q^{11} - 78 q^{13} - 312 q^{14} + 48 q^{16} - 18 q^{17} - 228 q^{19} - 69 q^{20} - 309 q^{22} - 114 q^{23} - 18 q^{25} + 30 q^{26}+ \cdots + 9567 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 + 36 * q^4 + 12 * q^5 - 42 * q^7 + 21 * q^8 - 60 * q^10 + 42 * q^11 - 78 * q^13 - 312 * q^14 + 48 * q^16 - 18 * q^17 - 228 * q^19 - 69 * q^20 - 309 * q^22 - 114 * q^23 - 18 * q^25 + 30 * q^26 - 813 * q^28 - 660 * q^29 - 708 * q^31 + 729 * q^32 - 972 * q^34 - 624 * q^35 - 354 * q^37 - 213 * q^38 - 1335 * q^40 - 1032 * q^41 - 744 * q^43 + 1653 * q^44 - 1077 * q^46 + 942 * q^47 - 192 * q^49 + 1905 * q^50 - 1371 * q^52 - 828 * q^53 - 1554 * q^55 - 3324 * q^56 - 831 * q^58 - 24 * q^59 - 1698 * q^61 - 2184 * q^62 - 933 * q^64 + 294 * q^65 - 1266 * q^67 - 4734 * q^68 - 1137 * q^70 + 3888 * q^71 - 1164 * q^73 + 5448 * q^74 - 2385 * q^76 + 3018 * q^77 - 2382 * q^79 - 4677 * q^80 - 276 * q^82 - 4008 * q^83 - 1116 * q^85 - 1176 * q^86 - 2769 * q^88 - 3582 * q^89 - 3222 * q^91 - 2958 * q^92 - 3324 * q^94 - 2784 * q^95 - 2958 * q^97 + 9567 * q^98

Coefficient data

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

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

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

$2$	4.32122	1.52778	0.763892	−	0.645344i	$-0.223286\pi$
$2$	4.32122	1.52778	0.763892	+	0.645344i	$0.223286\pi$
$3$	0	0
$3$	0	0
$4$	10.6730	1.33412
$5$	−3.36114	−0.300630	−0.150315	−	0.988638i	$-0.548029\pi$
$5$	−3.36114	−0.300630	−0.150315	+	0.988638i	$0.548029\pi$
$6$	0	0
$7$	−22.2331	−1.20048	−0.600238	−	0.799821i	$-0.704928\pi$
$7$	−22.2331	−1.20048	−0.600238	+	0.799821i	$0.704928\pi$
$8$	11.5505	0.510467
$9$	0	0
$10$	−14.5243	−0.459297
$11$	32.7665	0.898133	0.449067	−	0.893498i	$-0.351757\pi$
$11$	32.7665	0.898133	0.449067	+	0.893498i	$0.351757\pi$
$12$	0	0
$13$	17.2460	0.367936	0.183968	−	0.982932i	$-0.441106\pi$
$13$	17.2460	0.367936	0.183968	+	0.982932i	$0.441106\pi$
$14$	−96.0744	−1.83407
$15$	0	0
$16$	−35.4713	−0.554240
$17$	−104.698	−1.49370	−0.746851	−	0.664991i	$-0.768435\pi$
$17$	−104.698	−1.49370	−0.746851	+	0.664991i	$0.768435\pi$
$18$	0	0
$19$	86.2360	1.04126	0.520629	−	0.853783i	$-0.325698\pi$
$19$	86.2360	1.04126	0.520629	+	0.853783i	$0.325698\pi$
$20$	−35.8734	−0.401077
$21$	0	0
$22$	141.591	1.37215
$23$	37.2508	0.337710	0.168855	−	0.985641i	$-0.445993\pi$
$23$	37.2508	0.337710	0.168855	+	0.985641i	$0.445993\pi$
$24$	0	0
$25$	−113.703	−0.909622
$26$	74.5237	0.562127
$27$	0	0
$28$	−237.294	−1.60158
$29$	−179.556	−1.14975	−0.574874	−	0.818242i	$-0.694949\pi$
$29$	−179.556	−1.14975	−0.574874	+	0.818242i	$0.694949\pi$
$30$	0	0
$31$	−251.179	−1.45526	−0.727630	−	0.685970i	$-0.759378\pi$
$31$	−251.179	−1.45526	−0.727630	+	0.685970i	$0.759378\pi$
$32$	−245.684	−1.35723
$33$	0	0
$34$	−452.423	−2.28205
$35$	74.7288	0.360899
$36$	0	0
$37$	−347.233	−1.54283	−0.771416	−	0.636331i	$-0.780451\pi$
$37$	−347.233	−1.54283	−0.771416	+	0.636331i	$0.780451\pi$
$38$	372.645	1.59082
$39$	0	0
$40$	−38.8231	−0.153462
$41$	−466.757	−1.77793	−0.888966	−	0.457973i	$-0.848576\pi$
$41$	−466.757	−1.77793	−0.888966	+	0.457973i	$0.848576\pi$
$42$	0	0
$43$	375.073	1.33019	0.665094	−	0.746760i	$-0.268391\pi$
$43$	375.073	1.33019	0.665094	+	0.746760i	$0.268391\pi$
$44$	349.716	1.19822
$45$	0	0
$46$	160.969	0.515947
$47$	181.852	0.564379	0.282189	−	0.959359i	$-0.408939\pi$
$47$	181.852	0.564379	0.282189	+	0.959359i	$0.408939\pi$
$48$	0	0
$49$	151.312	0.441144
$50$	−491.335	−1.38971
$51$	0	0
$52$	184.066	0.490872
$53$	−148.879	−0.385850	−0.192925	−	0.981214i	$-0.561797\pi$
$53$	−148.879	−0.385850	−0.192925	+	0.981214i	$0.561797\pi$
$54$	0	0
$55$	−110.133	−0.270006
$56$	−256.805	−0.612804
$57$	0	0
$58$	−775.901	−1.75657
$59$	−377.724	−0.833483	−0.416742	−	0.909025i	$-0.636828\pi$
$59$	−377.724	−0.833483	−0.416742	+	0.909025i	$0.636828\pi$
$60$	0	0
$61$	−287.938	−0.604371	−0.302185	−	0.953249i	$-0.597716\pi$
$61$	−287.938	−0.604371	−0.302185	+	0.953249i	$0.597716\pi$
$62$	−1085.40	−2.22332
$63$	0	0
$64$	−777.885	−1.51931
$65$	−57.9661	−0.110613
$66$	0	0
$67$	27.6743	0.0504619	0.0252310	−	0.999682i	$-0.491968\pi$
$67$	27.6743	0.0504619	0.0252310	+	0.999682i	$0.491968\pi$
$68$	−1117.44	−1.99278
$69$	0	0
$70$	322.920	0.551376
$71$	759.978	1.27032	0.635161	−	0.772380i	$-0.280934\pi$
$71$	759.978	1.27032	0.635161	+	0.772380i	$0.280934\pi$
$72$	0	0
$73$	120.862	0.193778	0.0968892	−	0.995295i	$-0.469111\pi$
$73$	120.862	0.193778	0.0968892	+	0.995295i	$0.469111\pi$
$74$	−1500.47	−2.35711
$75$	0	0
$76$	920.395	1.38917
$77$	−728.502	−1.07819
$78$	0	0
$79$	1115.08	1.58806	0.794028	−	0.607882i	$-0.207981\pi$
$79$	1115.08	1.58806	0.794028	+	0.607882i	$0.207981\pi$
$80$	119.224	0.166621
$81$	0	0
$82$	−2016.96	−2.71629
$83$	736.208	0.973606	0.486803	−	0.873512i	$-0.338163\pi$
$83$	736.208	0.973606	0.486803	+	0.873512i	$0.338163\pi$
$84$	0	0
$85$	351.904	0.449051
$86$	1620.77	2.03224
$87$	0	0
$88$	378.471	0.458467
$89$	−321.649	−0.383087	−0.191544	−	0.981484i	$-0.561349\pi$
$89$	−321.649	−0.383087	−0.191544	+	0.981484i	$0.561349\pi$
$90$	0	0
$91$	−383.432	−0.441698
$92$	397.577	0.450546
$93$	0	0
$94$	785.822	0.862249
$95$	−289.852	−0.313033
$96$	0	0
$97$	1548.41	1.62080	0.810401	−	0.585876i	$-0.199250\pi$
$97$	1548.41	1.62080	0.810401	+	0.585876i	$0.199250\pi$
$98$	653.854	0.673972
$99$	0	0
$100$	−1213.55	−1.21355
$101$	807.686	0.795721	0.397860	−	0.917446i	$-0.369753\pi$
$101$	807.686	0.795721	0.397860	+	0.917446i	$0.369753\pi$
$102$	0	0
$103$	−815.358	−0.779996	−0.389998	−	0.920816i	$-0.627524\pi$
$103$	−815.358	−0.779996	−0.389998	+	0.920816i	$0.627524\pi$
$104$	199.200	0.187819
$105$	0	0
$106$	−643.338	−0.589495
$107$	361.979	0.327045	0.163522	−	0.986540i	$-0.447714\pi$
$107$	361.979	0.327045	0.163522	+	0.986540i	$0.447714\pi$
$108$	0	0
$109$	888.751	0.780981	0.390490	−	0.920607i	$-0.372305\pi$
$109$	888.751	0.780981	0.390490	+	0.920607i	$0.372305\pi$
$110$	−475.909	−0.412510
$111$	0	0
$112$	788.639	0.665351
$113$	1144.99	0.953198	0.476599	−	0.879121i	$-0.341869\pi$
$113$	1144.99	0.953198	0.476599	+	0.879121i	$0.341869\pi$
$114$	0	0
$115$	−125.205	−0.101526
$116$	−1916.40	−1.53391
$117$	0	0
$118$	−1632.23	−1.27338
$119$	2327.76	1.79315
$120$	0	0
$121$	−257.357	−0.193356
$122$	−1244.24	−0.923348
$123$	0	0
$124$	−2680.83	−1.94149
$125$	802.314	0.574089
$126$	0	0
$127$	−2109.78	−1.47412	−0.737058	−	0.675830i	$-0.763785\pi$
$127$	−2109.78	−1.47412	−0.737058	+	0.675830i	$0.763785\pi$
$128$	−1395.94	−0.963946
$129$	0	0
$130$	−250.485	−0.168992
$131$	1958.10	1.30595	0.652977	−	0.757378i	$-0.273520\pi$
$131$	1958.10	1.30595	0.652977	+	0.757378i	$0.273520\pi$
$132$	0	0
$133$	−1917.30	−1.25001
$134$	119.587	0.0770949
$135$	0	0
$136$	−1209.32	−0.762486
$137$	−1076.89	−0.671570	−0.335785	−	0.941939i	$-0.609002\pi$
$137$	−1076.89	−0.671570	−0.335785	+	0.941939i	$0.609002\pi$
$138$	0	0
$139$	−138.169	−0.0843119	−0.0421560	−	0.999111i	$-0.513423\pi$
$139$	−138.169	−0.0843119	−0.0421560	+	0.999111i	$0.513423\pi$
$140$	797.579	0.481483
$141$	0	0
$142$	3284.04	1.94078
$143$	565.089	0.330456
$144$	0	0
$145$	603.513	0.345649
$146$	522.272	0.296052
$147$	0	0
$148$	−3706.01	−2.05833
$149$	2225.74	1.22376	0.611878	−	0.790952i	$-0.290414\pi$
$149$	2225.74	1.22376	0.611878	+	0.790952i	$0.290414\pi$
$150$	0	0
$151$	−2783.61	−1.50018	−0.750088	−	0.661338i	$-0.769989\pi$
$151$	−2783.61	−1.50018	−0.750088	+	0.661338i	$0.769989\pi$
$152$	996.073	0.531528
$153$	0	0
$154$	−3148.02	−1.64724
$155$	844.248	0.437494
$156$	0	0
$157$	−3194.84	−1.62405	−0.812026	−	0.583621i	$-0.801635\pi$
$157$	−3194.84	−1.62405	−0.812026	+	0.583621i	$0.801635\pi$
$158$	4818.51	2.42620
$159$	0	0
$160$	825.779	0.408022
$161$	−828.201	−0.405412
$162$	0	0
$163$	883.995	0.424784	0.212392	−	0.977185i	$-0.431875\pi$
$163$	883.995	0.424784	0.212392	+	0.977185i	$0.431875\pi$
$164$	−4981.69	−2.37198
$165$	0	0
$166$	3181.32	1.48746
$167$	−800.402	−0.370880	−0.185440	−	0.982656i	$-0.559371\pi$
$167$	−800.402	−0.370880	−0.185440	+	0.982656i	$0.559371\pi$
$168$	0	0
$169$	−1899.58	−0.864623
$170$	1520.66	0.686053
$171$	0	0
$172$	4003.15	1.77463
$173$	3252.60	1.42943	0.714714	−	0.699417i	$-0.246557\pi$
$173$	3252.60	1.42943	0.714714	+	0.699417i	$0.246557\pi$
$174$	0	0
$175$	2527.97	1.09198
$176$	−1162.27	−0.497781
$177$	0	0
$178$	−1389.92	−0.585274
$179$	−1274.44	−0.532157	−0.266079	−	0.963951i	$-0.585728\pi$
$179$	−1274.44	−0.532157	−0.266079	+	0.963951i	$0.585728\pi$
$180$	0	0
$181$	−833.975	−0.342480	−0.171240	−	0.985229i	$-0.554777\pi$
$181$	−833.975	−0.342480	−0.171240	+	0.985229i	$0.554777\pi$
$182$	−1656.89	−0.674820
$183$	0	0
$184$	430.267	0.172390
$185$	1167.10	0.463821
$186$	0	0
$187$	−3430.58	−1.34154
$188$	1940.90	0.752951
$189$	0	0
$190$	−1252.51	−0.478247
$191$	3342.52	1.26626	0.633132	−	0.774044i	$-0.281769\pi$
$191$	3342.52	1.26626	0.633132	+	0.774044i	$0.281769\pi$
$192$	0	0
$193$	1718.02	0.640754	0.320377	−	0.947290i	$-0.396190\pi$
$193$	1718.02	0.640754	0.320377	+	0.947290i	$0.396190\pi$
$194$	6691.05	2.47623
$195$	0	0
$196$	1614.95	0.588540
$197$	−4254.29	−1.53861	−0.769304	−	0.638883i	$-0.779397\pi$
$197$	−4254.29	−1.53861	−0.769304	+	0.638883i	$0.779397\pi$
$198$	0	0
$199$	−5356.72	−1.90818	−0.954091	−	0.299518i	$-0.903174\pi$
$199$	−5356.72	−1.90818	−0.954091	+	0.299518i	$0.903174\pi$
$200$	−1313.33	−0.464332
$201$	0	0
$202$	3490.19	1.21569
$203$	3992.09	1.38025
$204$	0	0
$205$	1568.84	0.534499
$206$	−3523.35	−1.19167
$207$	0	0
$208$	−611.737	−0.203925
$209$	2825.65	0.935188
$210$	0	0
$211$	−5774.35	−1.88399	−0.941997	−	0.335620i	$-0.891054\pi$
$211$	−5774.35	−1.88399	−0.941997	+	0.335620i	$0.891054\pi$
$212$	−1588.98	−0.514771
$213$	0	0
$214$	1564.19	0.499654
$215$	−1260.67	−0.399894
$216$	0	0
$217$	5584.49	1.74700
$218$	3840.49	1.19317
$219$	0	0
$220$	−1175.45	−0.360221
$221$	−1805.61	−0.549587
$222$	0	0
$223$	144.030	0.0432511	0.0216255	−	0.999766i	$-0.493116\pi$
$223$	144.030	0.0432511	0.0216255	+	0.999766i	$0.493116\pi$
$224$	5462.32	1.62932
$225$	0	0
$226$	4947.75	1.45628
$227$	−4209.03	−1.23068	−0.615338	−	0.788264i	$-0.710980\pi$
$227$	−4209.03	−1.23068	−0.615338	+	0.788264i	$0.710980\pi$
$228$	0	0
$229$	−2433.68	−0.702280	−0.351140	−	0.936323i	$-0.614206\pi$
$229$	−2433.68	−0.702280	−0.351140	+	0.936323i	$0.614206\pi$
$230$	−541.040	−0.155109
$231$	0	0
$232$	−2073.97	−0.586908
$233$	1809.70	0.508829	0.254414	−	0.967095i	$-0.418117\pi$
$233$	1809.70	0.508829	0.254414	+	0.967095i	$0.418117\pi$
$234$	0	0
$235$	−611.230	−0.169669
$236$	−4031.44	−1.11197
$237$	0	0
$238$	10058.8	2.73955
$239$	−1938.19	−0.524566	−0.262283	−	0.964991i	$-0.584475\pi$
$239$	−1938.19	−0.524566	−0.262283	+	0.964991i	$0.584475\pi$
$240$	0	0
$241$	5357.82	1.43207	0.716033	−	0.698067i	$-0.245956\pi$
$241$	5357.82	1.43207	0.716033	+	0.698067i	$0.245956\pi$
$242$	−1112.10	−0.295407
$243$	0	0
$244$	−3073.15	−0.806305
$245$	−508.582	−0.132621
$246$	0	0
$247$	1487.22	0.383116
$248$	−2901.25	−0.742862
$249$	0	0
$250$	3466.98	0.877084
$251$	−15.6125	−0.00392610	−0.00196305	−	0.999998i	$-0.500625\pi$
$251$	−15.6125	−0.00392610	−0.00196305	+	0.999998i	$0.500625\pi$
$252$	0	0
$253$	1220.58	0.303308
$254$	−9116.83	−2.25213
$255$	0	0
$256$	190.894	0.0466050
$257$	−3500.11	−0.849537	−0.424768	−	0.905302i	$-0.639644\pi$
$257$	−3500.11	−0.849537	−0.424768	+	0.905302i	$0.639644\pi$
$258$	0	0
$259$	7720.08	1.85213
$260$	−618.671	−0.147571
$261$	0	0
$262$	8461.38	1.99521
$263$	2513.37	0.589281	0.294641	−	0.955608i	$-0.404800\pi$
$263$	2513.37	0.589281	0.294641	+	0.955608i	$0.404800\pi$
$264$	0	0
$265$	500.402	0.115998
$266$	−8285.07	−1.90974
$267$	0	0
$268$	295.367	0.0673224
$269$	3522.65	0.798437	0.399218	−	0.916856i	$-0.369282\pi$
$269$	3522.65	0.798437	0.399218	+	0.916856i	$0.369282\pi$
$270$	0	0
$271$	−448.194	−0.100464	−0.0502321	−	0.998738i	$-0.515996\pi$
$271$	−448.194	−0.100464	−0.0502321	+	0.998738i	$0.515996\pi$
$272$	3713.77	0.827869
$273$	0	0
$274$	−4653.49	−1.02601
$275$	−3725.64	−0.816962
$276$	0	0
$277$	−2867.90	−0.622078	−0.311039	−	0.950397i	$-0.600677\pi$
$277$	−2867.90	−0.622078	−0.311039	+	0.950397i	$0.600677\pi$
$278$	−597.060	−0.128810
$279$	0	0
$280$	863.158	0.184227
$281$	6067.10	1.28802	0.644009	−	0.765018i	$-0.277270\pi$
$281$	6067.10	1.28802	0.644009	+	0.765018i	$0.277270\pi$
$282$	0	0
$283$	−4710.89	−0.989517	−0.494759	−	0.869030i	$-0.664744\pi$
$283$	−4710.89	−0.989517	−0.494759	+	0.869030i	$0.664744\pi$
$284$	8111.23	1.69476
$285$	0	0
$286$	2441.88	0.504865
$287$	10377.5	2.13437
$288$	0	0
$289$	6048.62	1.23115
$290$	2607.92	0.528076
$291$	0	0
$292$	1289.96	0.258524
$293$	−7210.41	−1.43767	−0.718834	−	0.695182i	$-0.755324\pi$
$293$	−7210.41	−1.43767	−0.718834	+	0.695182i	$0.755324\pi$
$294$	0	0
$295$	1269.59	0.250570
$296$	−4010.74	−0.787565
$297$	0	0
$298$	9617.91	1.86963
$299$	642.425	0.124256
$300$	0	0
$301$	−8339.05	−1.59686
$302$	−12028.6	−2.29194
$303$	0	0
$304$	−3058.91	−0.577106
$305$	967.800	0.181692
$306$	0	0
$307$	8.62224	0.00160292	0.000801462	−	1.00000i	$-0.499745\pi$
$307$	8.62224	0.00160292	0.000801462		1.00000i	$0.499745\pi$
$308$	−7775.28	−1.43843
$309$	0	0
$310$	3648.19	0.668397
$311$	3260.63	0.594513	0.297256	−	0.954798i	$-0.403928\pi$
$311$	3260.63	0.594513	0.297256	+	0.954798i	$0.403928\pi$
$312$	0	0
$313$	3987.70	0.720122	0.360061	−	0.932929i	$-0.382756\pi$
$313$	3987.70	0.720122	0.360061	+	0.932929i	$0.382756\pi$
$314$	−13805.6	−2.48120
$315$	0	0
$316$	11901.2	2.11866
$317$	7302.14	1.29378	0.646891	−	0.762582i	$-0.276069\pi$
$317$	7302.14	1.29378	0.646891	+	0.762582i	$0.276069\pi$
$318$	0	0
$319$	−5883.42	−1.03263
$320$	2614.58	0.456749
$321$	0	0
$322$	−3578.84	−0.619382
$323$	−9028.72	−1.55533
$324$	0	0
$325$	−1960.91	−0.334683
$326$	3819.94	0.648978
$327$	0	0
$328$	−5391.30	−0.907575
$329$	−4043.13	−0.677523
$330$	0	0
$331$	−10675.5	−1.77275	−0.886376	−	0.462966i	$-0.846785\pi$
$331$	−10675.5	−1.77275	−0.886376	+	0.462966i	$0.846785\pi$
$332$	7857.53	1.29891
$333$	0	0
$334$	−3458.72	−0.566625
$335$	−93.0172	−0.0151704
$336$	0	0
$337$	4177.51	0.675263	0.337631	−	0.941278i	$-0.390374\pi$
$337$	4177.51	0.675263	0.337631	+	0.941278i	$0.390374\pi$
$338$	−8208.50	−1.32096
$339$	0	0
$340$	3755.87	0.599090
$341$	−8230.25	−1.30702
$342$	0	0
$343$	4261.82	0.670894
$344$	4332.30	0.679017
$345$	0	0
$346$	14055.2	2.18386
$347$	3953.54	0.611635	0.305817	−	0.952090i	$-0.401070\pi$
$347$	3953.54	0.611635	0.305817	+	0.952090i	$0.401070\pi$
$348$	0	0
$349$	10.7075	0.00164229	0.000821147	−	1.00000i	$-0.499739\pi$
$349$	10.7075	0.00164229	0.000821147		1.00000i	$0.499739\pi$
$350$	10923.9	1.66831
$351$	0	0
$352$	−8050.20	−1.21897
$353$	−3203.52	−0.483020	−0.241510	−	0.970398i	$-0.577643\pi$
$353$	−3203.52	−0.483020	−0.241510	+	0.970398i	$0.577643\pi$
$354$	0	0
$355$	−2554.40	−0.381896
$356$	−3432.96	−0.511085
$357$	0	0
$358$	−5507.14	−0.813021
$359$	4093.88	0.601857	0.300929	−	0.953647i	$-0.402703\pi$
$359$	4093.88	0.601857	0.300929	+	0.953647i	$0.402703\pi$
$360$	0	0
$361$	577.650	0.0842178
$362$	−3603.79	−0.523235
$363$	0	0
$364$	−4092.36	−0.589280
$365$	−406.235	−0.0582556
$366$	0	0
$367$	−2022.81	−0.287711	−0.143855	−	0.989599i	$-0.545950\pi$
$367$	−2022.81	−0.287711	−0.143855	+	0.989599i	$0.545950\pi$
$368$	−1321.33	−0.187172
$369$	0	0
$370$	5043.30	0.708619
$371$	3310.04	0.463204
$372$	0	0
$373$	−2520.20	−0.349842	−0.174921	−	0.984583i	$-0.555967\pi$
$373$	−2520.20	−0.349842	−0.174921	+	0.984583i	$0.555967\pi$
$374$	−14824.3	−2.04959
$375$	0	0
$376$	2100.49	0.288097
$377$	−3096.61	−0.423034
$378$	0	0
$379$	−456.488	−0.0618686	−0.0309343	−	0.999521i	$-0.509848\pi$
$379$	−456.488	−0.0618686	−0.0309343	+	0.999521i	$0.509848\pi$
$380$	−3093.58	−0.417625
$381$	0	0
$382$	14443.8	1.93458
$383$	−11401.7	−1.52115	−0.760575	−	0.649250i	$-0.775083\pi$
$383$	−11401.7	−1.52115	−0.760575	+	0.649250i	$0.775083\pi$
$384$	0	0
$385$	2448.60	0.324135
$386$	7423.93	0.978933
$387$	0	0
$388$	16526.2	2.16235
$389$	−834.987	−0.108832	−0.0544158	−	0.998518i	$-0.517330\pi$
$389$	−834.987	−0.108832	−0.0544158	+	0.998518i	$0.517330\pi$
$390$	0	0
$391$	−3900.07	−0.504438
$392$	1747.74	0.225189
$393$	0	0
$394$	−18383.8	−2.35066
$395$	−3747.95	−0.477417
$396$	0	0
$397$	1528.12	0.193184	0.0965920	−	0.995324i	$-0.469206\pi$
$397$	1528.12	0.193184	0.0965920	+	0.995324i	$0.469206\pi$
$398$	−23147.6	−2.91529
$399$	0	0
$400$	4033.19	0.504148
$401$	11674.5	1.45386	0.726928	−	0.686713i	$-0.240947\pi$
$401$	11674.5	1.45386	0.726928	+	0.686713i	$0.240947\pi$
$402$	0	0
$403$	−4331.82	−0.535442
$404$	8620.42	1.06159
$405$	0	0
$406$	17250.7	2.10872
$407$	−11377.6	−1.38567
$408$	0	0
$409$	−10228.7	−1.23661	−0.618306	−	0.785937i	$-0.712181\pi$
$409$	−10228.7	−1.23661	−0.618306	+	0.785937i	$0.712181\pi$
$410$	6779.30	0.816599
$411$	0	0
$412$	−8702.30	−1.04061
$413$	8398.00	1.00058
$414$	0	0
$415$	−2474.50	−0.292695
$416$	−4237.06	−0.499372
$417$	0	0
$418$	12210.3	1.42877
$419$	−1135.49	−0.132393	−0.0661963	−	0.997807i	$-0.521086\pi$
$419$	−1135.49	−0.132393	−0.0661963	+	0.997807i	$0.521086\pi$
$420$	0	0
$421$	−252.776	−0.0292626	−0.0146313	−	0.999893i	$-0.504657\pi$
$421$	−252.776	−0.0292626	−0.0146313	+	0.999893i	$0.504657\pi$
$422$	−24952.3	−2.87834
$423$	0	0
$424$	−1719.63	−0.196964
$425$	11904.4	1.35870
$426$	0	0
$427$	6401.75	0.725533
$428$	3863.39	0.436318
$429$	0	0
$430$	−5447.66	−0.610952
$431$	−447.152	−0.0499734	−0.0249867	−	0.999688i	$-0.507954\pi$
$431$	−447.152	−0.0499734	−0.0249867	+	0.999688i	$0.507954\pi$
$432$	0	0
$433$	4400.35	0.488377	0.244189	−	0.969728i	$-0.421478\pi$
$433$	4400.35	0.488377	0.244189	+	0.969728i	$0.421478\pi$
$434$	24131.8	2.66905
$435$	0	0
$436$	9485.62	1.04192
$437$	3212.36	0.351643
$438$	0	0
$439$	9773.16	1.06252	0.531261	−	0.847208i	$-0.321718\pi$
$439$	9773.16	1.06252	0.531261	+	0.847208i	$0.321718\pi$
$440$	−1272.10	−0.137829
$441$	0	0
$442$	−7802.46	−0.839650
$443$	12083.8	1.29598	0.647990	−	0.761649i	$-0.275610\pi$
$443$	12083.8	1.29598	0.647990	+	0.761649i	$0.275610\pi$
$444$	0	0
$445$	1081.11	0.115167
$446$	622.388	0.0660783
$447$	0	0
$448$	17294.8	1.82389
$449$	16507.7	1.73507	0.867535	−	0.497375i	$-0.165703\pi$
$449$	16507.7	1.73507	0.867535	+	0.497375i	$0.165703\pi$
$450$	0	0
$451$	−15294.0	−1.59682
$452$	12220.4	1.27168
$453$	0	0
$454$	−18188.2	−1.88021
$455$	1288.77	0.132788
$456$	0	0
$457$	4918.71	0.503474	0.251737	−	0.967796i	$-0.418998\pi$
$457$	4918.71	0.503474	0.251737	+	0.967796i	$0.418998\pi$
$458$	−10516.5	−1.07293
$459$	0	0
$460$	−1336.31	−0.135448
$461$	−8126.57	−0.821024	−0.410512	−	0.911855i	$-0.634650\pi$
$461$	−8126.57	−0.821024	−0.410512	+	0.911855i	$0.634650\pi$
$462$	0	0
$463$	2460.50	0.246974	0.123487	−	0.992346i	$-0.460592\pi$
$463$	2460.50	0.246974	0.123487	+	0.992346i	$0.460592\pi$
$464$	6369.09	0.637236
$465$	0	0
$466$	7820.10	0.777380
$467$	−15753.5	−1.56100	−0.780500	−	0.625156i	$-0.785035\pi$
$467$	−15753.5	−1.56100	−0.780500	+	0.625156i	$0.785035\pi$
$468$	0	0
$469$	−615.286	−0.0605784
$470$	−2641.26	−0.259218
$471$	0	0
$472$	−4362.92	−0.425466
$473$	12289.8	1.19469
$474$	0	0
$475$	−9805.27	−0.947151
$476$	24844.1	2.39229
$477$	0	0
$478$	−8375.37	−0.801423
$479$	−11202.5	−1.06859	−0.534294	−	0.845299i	$-0.679422\pi$
$479$	−11202.5	−1.06859	−0.534294	+	0.845299i	$0.679422\pi$
$480$	0	0
$481$	−5988.37	−0.567664
$482$	23152.4	2.18789
$483$	0	0
$484$	−2746.77	−0.257961
$485$	−5204.44	−0.487261
$486$	0	0
$487$	7312.98	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$487$	7312.98	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$488$	−3325.84	−0.308511
$489$	0	0
$490$	−2197.70	−0.202616
$491$	3742.40	0.343975	0.171988	−	0.985099i	$-0.444981\pi$
$491$	3742.40	0.343975	0.171988	+	0.985099i	$0.444981\pi$
$492$	0	0
$493$	18799.1	1.71738
$494$	6426.62	0.585319
$495$	0	0
$496$	8909.65	0.806562
$497$	−16896.7	−1.52499
$498$	0	0
$499$	−12595.3	−1.12995	−0.564974	−	0.825109i	$-0.691114\pi$
$499$	−12595.3	−1.12995	−0.564974	+	0.825109i	$0.691114\pi$
$500$	8563.08	0.765905
$501$	0	0
$502$	−67.4651	−0.00599824
$503$	1573.72	0.139500	0.0697501	−	0.997564i	$-0.477780\pi$
$503$	1573.72	0.139500	0.0697501	+	0.997564i	$0.477780\pi$
$504$	0	0
$505$	−2714.75	−0.239217
$506$	5274.39	0.463389
$507$	0	0
$508$	−22517.6	−1.96665
$509$	−16893.8	−1.47113	−0.735564	−	0.677455i	$-0.763083\pi$
$509$	−16893.8	−1.47113	−0.735564	+	0.677455i	$0.763083\pi$
$510$	0	0
$511$	−2687.14	−0.232626
$512$	11992.4	1.03515
$513$	0	0
$514$	−15124.8	−1.29791
$515$	2740.54	0.234490
$516$	0	0
$517$	5958.64	0.506888
$518$	33360.2	2.82966
$519$	0	0
$520$	−669.541	−0.0564640
$521$	−17746.8	−1.49233	−0.746165	−	0.665762i	$-0.768107\pi$
$521$	−17746.8	−1.49233	−0.746165	+	0.665762i	$0.768107\pi$
$522$	0	0
$523$	20042.4	1.67570	0.837850	−	0.545900i	$-0.183812\pi$
$523$	20042.4	1.67570	0.837850	+	0.545900i	$0.183812\pi$
$524$	20898.7	1.74230
$525$	0	0
$526$	10860.8	0.900294
$527$	26297.9	2.17372
$528$	0	0
$529$	−10779.4	−0.885952
$530$	2162.35	0.177220
$531$	0	0
$532$	−20463.3	−1.66766
$533$	−8049.67	−0.654165
$534$	0	0
$535$	−1216.66	−0.0983194
$536$	319.653	0.0257591
$537$	0	0
$538$	15222.1	1.21984
$539$	4957.97	0.396206
$540$	0	0
$541$	−15921.2	−1.26526	−0.632630	−	0.774455i	$-0.718024\pi$
$541$	−15921.2	−1.26526	−0.632630	+	0.774455i	$0.718024\pi$
$542$	−1936.74	−0.153488
$543$	0	0
$544$	25722.6	2.02729
$545$	−2987.22	−0.234786
$546$	0	0
$547$	8082.66	0.631791	0.315896	−	0.948794i	$-0.397695\pi$
$547$	8082.66	0.631791	0.315896	+	0.948794i	$0.397695\pi$
$548$	−11493.7	−0.895957
$549$	0	0
$550$	−16099.3	−1.24814
$551$	−15484.2	−1.19718
$552$	0	0
$553$	−24791.7	−1.90642
$554$	−12392.9	−0.950401
$555$	0	0
$556$	−1474.68	−0.112482
$557$	17943.5	1.36498	0.682488	−	0.730897i	$-0.260898\pi$
$557$	17943.5	1.36498	0.682488	+	0.730897i	$0.260898\pi$
$558$	0	0
$559$	6468.49	0.489424
$560$	−2650.73	−0.200024
$561$	0	0
$562$	26217.3	1.96781
$563$	−609.231	−0.0456057	−0.0228029	−	0.999740i	$-0.507259\pi$
$563$	−609.231	−0.0456057	−0.0228029	+	0.999740i	$0.507259\pi$
$564$	0	0
$565$	−3848.47	−0.286560
$566$	−20356.8	−1.51177
$567$	0	0
$568$	8778.16	0.648457
$569$	−16787.5	−1.23685	−0.618424	−	0.785844i	$-0.712229\pi$
$569$	−16787.5	−1.23685	−0.618424	+	0.785844i	$0.712229\pi$
$570$	0	0
$571$	−8312.94	−0.609257	−0.304628	−	0.952471i	$-0.598532\pi$
$571$	−8312.94	−0.609257	−0.304628	+	0.952471i	$0.598532\pi$
$572$	6031.19	0.440868
$573$	0	0
$574$	44843.4	3.26085
$575$	−4235.51	−0.307188
$576$	0	0
$577$	−10217.8	−0.737212	−0.368606	−	0.929586i	$-0.620165\pi$
$577$	−10217.8	−0.737212	−0.368606	+	0.929586i	$0.620165\pi$
$578$	26137.5	1.88092
$579$	0	0
$580$	6441.28	0.461138
$581$	−16368.2	−1.16879
$582$	0	0
$583$	−4878.23	−0.346545
$584$	1396.02	0.0989175
$585$	0	0
$586$	−31157.8	−2.19644
$587$	−4471.30	−0.314395	−0.157198	−	0.987567i	$-0.550246\pi$
$587$	−4471.30	−0.314395	−0.157198	+	0.987567i	$0.550246\pi$
$588$	0	0
$589$	−21660.7	−1.51530
$590$	5486.16	0.382817
$591$	0	0
$592$	12316.8	0.855099
$593$	−1365.36	−0.0945506	−0.0472753	−	0.998882i	$-0.515054\pi$
$593$	−1365.36	−0.0945506	−0.0472753	+	0.998882i	$0.515054\pi$
$594$	0	0
$595$	−7823.93	−0.539076
$596$	23755.3	1.63264
$597$	0	0
$598$	2776.06	0.189836
$599$	11156.0	0.760971	0.380486	−	0.924787i	$-0.375757\pi$
$599$	11156.0	0.760971	0.380486	+	0.924787i	$0.375757\pi$
$600$	0	0
$601$	−10329.1	−0.701055	−0.350527	−	0.936553i	$-0.613998\pi$
$601$	−10329.1	−0.701055	−0.350527	+	0.936553i	$0.613998\pi$
$602$	−36034.9	−2.43966
$603$	0	0
$604$	−29709.4	−2.00142
$605$	865.015	0.0581287
$606$	0	0
$607$	6616.59	0.442436	0.221218	−	0.975224i	$-0.428997\pi$
$607$	6616.59	0.442436	0.221218	+	0.975224i	$0.428997\pi$
$608$	−21186.8	−1.41322
$609$	0	0
$610$	4182.08	0.277586
$611$	3136.21	0.207655
$612$	0	0
$613$	−5058.25	−0.333280	−0.166640	−	0.986018i	$-0.553292\pi$
$613$	−5058.25	−0.333280	−0.166640	+	0.986018i	$0.553292\pi$
$614$	37.2586	0.00244892
$615$	0	0
$616$	−8414.59	−0.550379
$617$	27350.4	1.78458	0.892290	−	0.451462i	$-0.149098\pi$
$617$	27350.4	1.78458	0.892290	+	0.451462i	$0.149098\pi$
$618$	0	0
$619$	812.301	0.0527449	0.0263725	−	0.999652i	$-0.491604\pi$
$619$	812.301	0.0527449	0.0263725	+	0.999652i	$0.491604\pi$
$620$	9010.64	0.583671
$621$	0	0
$622$	14089.9	0.908286
$623$	7151.27	0.459887
$624$	0	0
$625$	11516.1	0.737033
$626$	17231.7	1.10019
$627$	0	0
$628$	−34098.5	−2.16669
$629$	36354.5	2.30453
$630$	0	0
$631$	15522.5	0.979307	0.489653	−	0.871917i	$-0.337123\pi$
$631$	15522.5	0.979307	0.489653	+	0.871917i	$0.337123\pi$
$632$	12879.8	0.810650
$633$	0	0
$634$	31554.2	1.97662
$635$	7091.27	0.443163
$636$	0	0
$637$	2609.52	0.162313
$638$	−25423.6	−1.57763
$639$	0	0
$640$	4691.97	0.289791
$641$	−17574.3	−1.08291	−0.541454	−	0.840731i	$-0.682126\pi$
$641$	−17574.3	−1.08291	−0.541454	+	0.840731i	$0.682126\pi$
$642$	0	0
$643$	13127.3	0.805119	0.402560	−	0.915394i	$-0.368121\pi$
$643$	13127.3	0.805119	0.402560	+	0.915394i	$0.368121\pi$
$644$	−8839.38	−0.540870
$645$	0	0
$646$	−39015.1	−2.37621
$647$	19927.3	1.21085	0.605426	−	0.795901i	$-0.293003\pi$
$647$	19927.3	1.21085	0.605426	+	0.795901i	$0.293003\pi$
$648$	0	0
$649$	−12376.7	−0.748579
$650$	−8473.54	−0.511323
$651$	0	0
$652$	9434.86	0.566714
$653$	−10185.0	−0.610369	−0.305185	−	0.952293i	$-0.598718\pi$
$653$	−10185.0	−0.610369	−0.305185	+	0.952293i	$0.598718\pi$
$654$	0	0
$655$	−6581.45	−0.392608
$656$	16556.5	0.985400
$657$	0	0
$658$	−17471.3	−1.03511
$659$	19821.1	1.17165	0.585827	−	0.810436i	$-0.300770\pi$
$659$	19821.1	1.17165	0.585827	+	0.810436i	$0.300770\pi$
$660$	0	0
$661$	−13456.9	−0.791850	−0.395925	−	0.918283i	$-0.629576\pi$
$661$	−13456.9	−0.791850	−0.395925	+	0.918283i	$0.629576\pi$
$662$	−46131.4	−2.70838
$663$	0	0
$664$	8503.61	0.496994
$665$	6444.31	0.375789
$666$	0	0
$667$	−6688.60	−0.388281
$668$	−8542.68	−0.494800
$669$	0	0
$670$	−401.948	−0.0231770
$671$	−9434.70	−0.542806
$672$	0	0
$673$	−9632.15	−0.551697	−0.275848	−	0.961201i	$-0.588959\pi$
$673$	−9632.15	−0.551697	−0.275848	+	0.961201i	$0.588959\pi$
$674$	18052.0	1.03166
$675$	0	0
$676$	−20274.1	−1.15351
$677$	−16689.5	−0.947456	−0.473728	−	0.880671i	$-0.657092\pi$
$677$	−16689.5	−0.947456	−0.473728	+	0.880671i	$0.657092\pi$
$678$	0	0
$679$	−34426.1	−1.94573
$680$	4064.69	0.229226
$681$	0	0
$682$	−35564.7	−1.99684
$683$	−29437.9	−1.64921	−0.824604	−	0.565710i	$-0.808602\pi$
$683$	−29437.9	−1.64921	−0.824604	+	0.565710i	$0.808602\pi$
$684$	0	0
$685$	3619.59	0.201894
$686$	18416.3	1.02498
$687$	0	0
$688$	−13304.3	−0.737243
$689$	−2567.55	−0.141968
$690$	0	0
$691$	−16775.2	−0.923530	−0.461765	−	0.887002i	$-0.652784\pi$
$691$	−16775.2	−0.923530	−0.461765	+	0.887002i	$0.652784\pi$
$692$	34715.0	1.90703
$693$	0	0
$694$	17084.1	0.934446
$695$	464.406	0.0253467
$696$	0	0
$697$	48868.4	2.65570
$698$	46.2696	0.00250907
$699$	0	0
$700$	26980.9	1.45683
$701$	−22612.6	−1.21835	−0.609176	−	0.793035i	$-0.708500\pi$
$701$	−22612.6	−1.21835	−0.609176	+	0.793035i	$0.708500\pi$
$702$	0	0
$703$	−29944.0	−1.60649
$704$	−25488.6	−1.36454
$705$	0	0
$706$	−13843.1	−0.737950
$707$	−17957.4	−0.955244
$708$	0	0
$709$	4601.19	0.243726	0.121863	−	0.992547i	$-0.461113\pi$
$709$	4601.19	0.243726	0.121863	+	0.992547i	$0.461113\pi$
$710$	−11038.1	−0.583455
$711$	0	0
$712$	−3715.23	−0.195553
$713$	−9356.60	−0.491455
$714$	0	0
$715$	−1899.35	−0.0993448
$716$	−13602.1	−0.709963
$717$	0	0
$718$	17690.6	0.919508
$719$	−2324.99	−0.120595	−0.0602973	−	0.998180i	$-0.519205\pi$
$719$	−2324.99	−0.120595	−0.0602973	+	0.998180i	$0.519205\pi$
$720$	0	0
$721$	18128.0	0.936367
$722$	2496.15	0.128667
$723$	0	0
$724$	−8901.00	−0.456910
$725$	20416.0	1.04584
$726$	0	0
$727$	−4915.99	−0.250790	−0.125395	−	0.992107i	$-0.540020\pi$
$727$	−4915.99	−0.250790	−0.125395	+	0.992107i	$0.540020\pi$
$728$	−4428.85	−0.225472
$729$	0	0
$730$	−1755.43	−0.0890019
$731$	−39269.3	−1.98690
$732$	0	0
$733$	−33675.5	−1.69691	−0.848454	−	0.529269i	$-0.822466\pi$
$733$	−33675.5	−1.69691	−0.848454	+	0.529269i	$0.822466\pi$
$734$	−8741.01	−0.439560
$735$	0	0
$736$	−9151.92	−0.458348
$737$	906.788	0.0453215
$738$	0	0
$739$	−12008.7	−0.597762	−0.298881	−	0.954290i	$-0.596613\pi$
$739$	−12008.7	−0.597762	−0.298881	+	0.954290i	$0.596613\pi$
$740$	12456.4	0.618795
$741$	0	0
$742$	14303.4	0.707675
$743$	−20885.8	−1.03126	−0.515629	−	0.856812i	$-0.672442\pi$
$743$	−20885.8	−1.03126	−0.515629	+	0.856812i	$0.672442\pi$
$744$	0	0
$745$	−7481.02	−0.367897
$746$	−10890.3	−0.534482
$747$	0	0
$748$	−36614.5	−1.78978
$749$	−8047.92	−0.392609
$750$	0	0
$751$	2210.99	0.107431	0.0537153	−	0.998556i	$-0.482894\pi$
$751$	2210.99	0.107431	0.0537153	+	0.998556i	$0.482894\pi$
$752$	−6450.52	−0.312801
$753$	0	0
$754$	−13381.2	−0.646304
$755$	9356.10	0.450998
$756$	0	0
$757$	−7793.92	−0.374207	−0.187104	−	0.982340i	$-0.559910\pi$
$757$	−7793.92	−0.374207	−0.187104	+	0.982340i	$0.559910\pi$
$758$	−1972.59	−0.0945218
$759$	0	0
$760$	−3347.95	−0.159793
$761$	−19565.7	−0.932005	−0.466003	−	0.884783i	$-0.654306\pi$
$761$	−19565.7	−0.932005	−0.466003	+	0.884783i	$0.654306\pi$
$762$	0	0
$763$	−19759.7	−0.937549
$764$	35674.6	1.68935
$765$	0	0
$766$	−49269.4	−2.32399
$767$	−6514.22	−0.306669
$768$	0	0
$769$	28969.5	1.35847	0.679237	−	0.733919i	$-0.262311\pi$
$769$	28969.5	1.35847	0.679237	+	0.733919i	$0.262311\pi$
$770$	10580.9	0.495209
$771$	0	0
$772$	18336.4	0.854844
$773$	−39776.6	−1.85079	−0.925397	−	0.379000i	$-0.876268\pi$
$773$	−39776.6	−1.85079	−0.925397	+	0.379000i	$0.876268\pi$
$774$	0	0
$775$	28559.7	1.32374
$776$	17885.0	0.827365
$777$	0	0
$778$	−3608.17	−0.166271
$779$	−40251.3	−1.85129
$780$	0	0
$781$	24901.8	1.14092
$782$	−16853.1	−0.770671
$783$	0	0
$784$	−5367.25	−0.244499
$785$	10738.3	0.488239
$786$	0	0
$787$	11873.5	0.537795	0.268898	−	0.963169i	$-0.413341\pi$
$787$	11873.5	0.537795	0.268898	+	0.963169i	$0.413341\pi$
$788$	−45406.0	−2.05269
$789$	0	0
$790$	−16195.7	−0.729389
$791$	−25456.6	−1.14429
$792$	0	0
$793$	−4965.76	−0.222370
$794$	6603.34	0.295143
$795$	0	0
$796$	−57172.2	−2.54575
$797$	−12939.1	−0.575066	−0.287533	−	0.957771i	$-0.592835\pi$
$797$	−12939.1	−0.575066	−0.287533	+	0.957771i	$0.592835\pi$
$798$	0	0
$799$	−19039.5	−0.843014
$800$	27934.9	1.23456
$801$	0	0
$802$	50448.1	2.22118
$803$	3960.22	0.174039
$804$	0	0
$805$	2783.70	0.121879
$806$	−18718.8	−0.818040
$807$	0	0
$808$	9329.22	0.406189
$809$	−18133.3	−0.788049	−0.394024	−	0.919100i	$-0.628917\pi$
$809$	−18133.3	−0.788049	−0.394024	+	0.919100i	$0.628917\pi$
$810$	0	0
$811$	26902.2	1.16481	0.582407	−	0.812897i	$-0.302111\pi$
$811$	26902.2	1.16481	0.582407	+	0.812897i	$0.302111\pi$
$812$	42607.5	1.84142
$813$	0	0
$814$	−49165.2	−2.11700
$815$	−2971.23	−0.127703
$816$	0	0
$817$	32344.8	1.38507
$818$	−44200.3	−1.88928
$819$	0	0
$820$	16744.2	0.713088
$821$	−2255.42	−0.0958768	−0.0479384	−	0.998850i	$-0.515265\pi$
$821$	−2255.42	−0.0958768	−0.0479384	+	0.998850i	$0.515265\pi$
$822$	0	0
$823$	−22738.1	−0.963063	−0.481531	−	0.876429i	$-0.659919\pi$
$823$	−22738.1	−0.963063	−0.481531	+	0.876429i	$0.659919\pi$
$824$	−9417.83	−0.398162
$825$	0	0
$826$	36289.6	1.52867
$827$	28656.1	1.20492	0.602462	−	0.798148i	$-0.294187\pi$
$827$	28656.1	1.20492	0.602462	+	0.798148i	$0.294187\pi$
$828$	0	0
$829$	−19121.7	−0.801114	−0.400557	−	0.916272i	$-0.631183\pi$
$829$	−19121.7	−0.801114	−0.400557	+	0.916272i	$0.631183\pi$
$830$	−10692.9	−0.447175
$831$	0	0
$832$	−13415.4	−0.559008
$833$	−15842.1	−0.658937
$834$	0	0
$835$	2690.27	0.111498
$836$	30158.1	1.24766
$837$	0	0
$838$	−4906.72	−0.202267
$839$	3406.70	0.140182	0.0700909	−	0.997541i	$-0.477671\pi$
$839$	3406.70	0.140182	0.0700909	+	0.997541i	$0.477671\pi$
$840$	0	0
$841$	7851.33	0.321921
$842$	−1092.30	−0.0447069
$843$	0	0
$844$	−61629.6	−2.51348
$845$	6384.75	0.259931
$846$	0	0
$847$	5721.86	0.232120
$848$	5280.92	0.213853
$849$	0	0
$850$	51441.7	2.07581
$851$	−12934.7	−0.521029
$852$	0	0
$853$	43325.4	1.73908	0.869538	−	0.493865i	$-0.164416\pi$
$853$	43325.4	1.73908	0.869538	+	0.493865i	$0.164416\pi$
$854$	27663.4	1.10846
$855$	0	0
$856$	4181.05	0.166946
$857$	36770.9	1.46566	0.732829	−	0.680413i	$-0.238200\pi$
$857$	36770.9	1.46566	0.732829	+	0.680413i	$0.238200\pi$
$858$	0	0
$859$	−9578.35	−0.380453	−0.190226	−	0.981740i	$-0.560922\pi$
$859$	−9578.35	−0.380453	−0.190226	+	0.981740i	$0.560922\pi$
$860$	−13455.1	−0.533508
$861$	0	0
$862$	−1932.24	−0.0763485
$863$	2609.85	0.102944	0.0514719	−	0.998674i	$-0.483609\pi$
$863$	2609.85	0.102944	0.0514719	+	0.998674i	$0.483609\pi$
$864$	0	0
$865$	−10932.5	−0.429728
$866$	19014.9	0.746135
$867$	0	0
$868$	59603.2	2.33072
$869$	36537.3	1.42629
$870$	0	0
$871$	477.269	0.0185668
$872$	10265.6	0.398665
$873$	0	0
$874$	13881.3	0.537234
$875$	−17838.0	−0.689181
$876$	0	0
$877$	−577.965	−0.0222537	−0.0111268	−	0.999938i	$-0.503542\pi$
$877$	−577.965	−0.0222537	−0.0111268	+	0.999938i	$0.503542\pi$
$878$	42232.0	1.62330
$879$	0	0
$880$	3906.56	0.149648
$881$	−17578.3	−0.672223	−0.336112	−	0.941822i	$-0.609112\pi$
$881$	−17578.3	−0.672223	−0.336112	+	0.941822i	$0.609112\pi$
$882$	0	0
$883$	35663.8	1.35921	0.679604	−	0.733579i	$-0.262152\pi$
$883$	35663.8	1.35921	0.679604	+	0.733579i	$0.262152\pi$
$884$	−19271.3	−0.733216
$885$	0	0
$886$	52216.8	1.97998
$887$	−4177.67	−0.158143	−0.0790713	−	0.996869i	$-0.525195\pi$
$887$	−4177.67	−0.158143	−0.0790713	+	0.996869i	$0.525195\pi$
$888$	0	0
$889$	46907.0	1.76964
$890$	4671.72	0.175951
$891$	0	0
$892$	1537.23	0.0577023
$893$	15682.2	0.587664
$894$	0	0
$895$	4283.58	0.159982
$896$	31036.2	1.15719
$897$	0	0
$898$	71333.5	2.65081
$899$	45100.6	1.67318
$900$	0	0
$901$	15587.2	0.576345
$902$	−66088.8	−2.43960
$903$	0	0
$904$	13225.2	0.486576
$905$	2803.11	0.102960
$906$	0	0
$907$	−3530.69	−0.129255	−0.0646277	−	0.997909i	$-0.520586\pi$
$907$	−3530.69	−0.129255	−0.0646277	+	0.997909i	$0.520586\pi$
$908$	−44922.9	−1.64187
$909$	0	0
$910$	5569.06	0.202871
$911$	−19956.9	−0.725797	−0.362898	−	0.931829i	$-0.618213\pi$
$911$	−19956.9	−0.725797	−0.362898	+	0.931829i	$0.618213\pi$
$912$	0	0
$913$	24123.0	0.874428
$914$	21254.9	0.769200
$915$	0	0
$916$	−25974.6	−0.936928
$917$	−43534.7	−1.56777
$918$	0	0
$919$	1053.74	0.0378235	0.0189117	−	0.999821i	$-0.493980\pi$
$919$	1053.74	0.0378235	0.0189117	+	0.999821i	$0.493980\pi$
$920$	−1446.19	−0.0518255
$921$	0	0
$922$	−35116.7	−1.25435
$923$	13106.5	0.467397
$924$	0	0
$925$	39481.4	1.40339
$926$	10632.4	0.377323
$927$	0	0
$928$	44114.0	1.56047
$929$	−14474.4	−0.511183	−0.255592	−	0.966785i	$-0.582270\pi$
$929$	−14474.4	−0.511183	−0.255592	+	0.966785i	$0.582270\pi$
$930$	0	0
$931$	13048.6	0.459344
$932$	19314.8	0.678840
$933$	0	0
$934$	−68074.6	−2.38487
$935$	11530.7	0.403308
$936$	0	0
$937$	−36085.8	−1.25813	−0.629067	−	0.777351i	$-0.716563\pi$
$937$	−36085.8	−1.25813	−0.629067	+	0.777351i	$0.716563\pi$
$938$	−2658.79	−0.0925506
$939$	0	0
$940$	−6523.64	−0.226359
$941$	−6939.63	−0.240410	−0.120205	−	0.992749i	$-0.538355\pi$
$941$	−6939.63	−0.240410	−0.120205	+	0.992749i	$0.538355\pi$
$942$	0	0
$943$	−17387.1	−0.600425
$944$	13398.4	0.461949
$945$	0	0
$946$	53107.1	1.82522
$947$	28579.7	0.980693	0.490346	−	0.871528i	$-0.336870\pi$
$947$	28579.7	0.980693	0.490346	+	0.871528i	$0.336870\pi$
$948$	0	0
$949$	2084.38	0.0712981
$950$	−42370.8	−1.44704
$951$	0	0
$952$	26886.9	0.915346
$953$	11960.8	0.406557	0.203278	−	0.979121i	$-0.434840\pi$
$953$	11960.8	0.406557	0.203278	+	0.979121i	$0.434840\pi$
$954$	0	0
$955$	−11234.7	−0.380676
$956$	−20686.3	−0.699835
$957$	0	0
$958$	−48408.3	−1.63257
$959$	23942.7	0.806204
$960$	0	0
$961$	33299.8	1.11778
$962$	−25877.1	−0.867267
$963$	0	0
$964$	57184.0	1.91055
$965$	−5774.50	−0.192630
$966$	0	0
$967$	7031.84	0.233846	0.116923	−	0.993141i	$-0.462697\pi$
$967$	7031.84	0.233846	0.116923	+	0.993141i	$0.462697\pi$
$968$	−2972.62	−0.0987020
$969$	0	0
$970$	−22489.6	−0.744429
$971$	28959.5	0.957110	0.478555	−	0.878057i	$-0.341161\pi$
$971$	28959.5	0.957110	0.478555	+	0.878057i	$0.341161\pi$
$972$	0	0
$973$	3071.93	0.101214
$974$	31601.0	1.03959
$975$	0	0
$976$	10213.5	0.334966
$977$	7934.65	0.259828	0.129914	−	0.991525i	$-0.458530\pi$
$977$	7934.65	0.259828	0.129914	+	0.991525i	$0.458530\pi$
$978$	0	0
$979$	−10539.3	−0.344063
$980$	−5428.09	−0.176933
$981$	0	0
$982$	16171.7	0.525520
$983$	2948.06	0.0956545	0.0478272	−	0.998856i	$-0.484770\pi$
$983$	2948.06	0.0956545	0.0478272	+	0.998856i	$0.484770\pi$
$984$	0	0
$985$	14299.3	0.462551
$986$	81235.1	2.62379
$987$	0	0
$988$	15873.1	0.511124
$989$	13971.8	0.449217
$990$	0	0
$991$	−28435.4	−0.911482	−0.455741	−	0.890112i	$-0.650626\pi$
$991$	−28435.4	−0.911482	−0.455741	+	0.890112i	$0.650626\pi$
$992$	61710.6	1.97511
$993$	0	0
$994$	−73014.4	−2.32986
$995$	18004.7	0.573656
$996$	0	0
$997$	−6736.12	−0.213977	−0.106988	−	0.994260i	$-0.534121\pi$
$997$	−6736.12	−0.213977	−0.106988	+	0.994260i	$0.534121\pi$
$998$	−54427.2	−1.72632
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.4.a.b.1.11	yes	12
3.2	odd	2		729.4.a.a.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.4.a.a.1.2	✓	12	3.2	odd	2
729.4.a.b.1.11	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+4.32122 q^{2} +10.6730 q^{4} -3.36114 q^{5} -22.2331 q^{7} +11.5505 q^{8} -14.5243 q^{10} +32.7665 q^{11} +17.2460 q^{13} -96.0744 q^{14} -35.4713 q^{16} -104.698 q^{17} +86.2360 q^{19} -35.8734 q^{20} +141.591 q^{22} +37.2508 q^{23} -113.703 q^{25} +74.5237 q^{26} -237.294 q^{28} -179.556 q^{29} -251.179 q^{31} -245.684 q^{32} -452.423 q^{34} +74.7288 q^{35} -347.233 q^{37} +372.645 q^{38} -38.8231 q^{40} -466.757 q^{41} +375.073 q^{43} +349.716 q^{44} +160.969 q^{46} +181.852 q^{47} +151.312 q^{49} -491.335 q^{50} +184.066 q^{52} -148.879 q^{53} -110.133 q^{55} -256.805 q^{56} -775.901 q^{58} -377.724 q^{59} -287.938 q^{61} -1085.40 q^{62} -777.885 q^{64} -57.9661 q^{65} +27.6743 q^{67} -1117.44 q^{68} +322.920 q^{70} +759.978 q^{71} +120.862 q^{73} -1500.47 q^{74} +920.395 q^{76} -728.502 q^{77} +1115.08 q^{79} +119.224 q^{80} -2016.96 q^{82} +736.208 q^{83} +351.904 q^{85} +1620.77 q^{86} +378.471 q^{88} -321.649 q^{89} -383.432 q^{91} +397.577 q^{92} +785.822 q^{94} -289.852 q^{95} +1548.41 q^{97} +653.854 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 36 q^{4} + 12 q^{5} - 42 q^{7} + 21 q^{8} - 60 q^{10} + 42 q^{11} - 78 q^{13} - 312 q^{14} + 48 q^{16} - 18 q^{17} - 228 q^{19} - 69 q^{20} - 309 q^{22} - 114 q^{23} - 18 q^{25} + 30 q^{26}+ \cdots + 9567 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 + 36 * q^4 + 12 * q^5 - 42 * q^7 + 21 * q^8 - 60 * q^10 + 42 * q^11 - 78 * q^13 - 312 * q^14 + 48 * q^16 - 18 * q^17 - 228 * q^19 - 69 * q^20 - 309 * q^22 - 114 * q^23 - 18 * q^25 + 30 * q^26 - 813 * q^28 - 660 * q^29 - 708 * q^31 + 729 * q^32 - 972 * q^34 - 624 * q^35 - 354 * q^37 - 213 * q^38 - 1335 * q^40 - 1032 * q^41 - 744 * q^43 + 1653 * q^44 - 1077 * q^46 + 942 * q^47 - 192 * q^49 + 1905 * q^50 - 1371 * q^52 - 828 * q^53 - 1554 * q^55 - 3324 * q^56 - 831 * q^58 - 24 * q^59 - 1698 * q^61 - 2184 * q^62 - 933 * q^64 + 294 * q^65 - 1266 * q^67 - 4734 * q^68 - 1137 * q^70 + 3888 * q^71 - 1164 * q^73 + 5448 * q^74 - 2385 * q^76 + 3018 * q^77 - 2382 * q^79 - 4677 * q^80 - 276 * q^82 - 4008 * q^83 - 1116 * q^85 - 1176 * q^86 - 2769 * q^88 - 3582 * q^89 - 3222 * q^91 - 2958 * q^92 - 3324 * q^94 - 2784 * q^95 - 2958 * q^97 + 9567 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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