LMFDB - Embedded newform 729.4.a.c.1.7

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:	$24$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-3.18772 q^{2} +2.16159 q^{4} +7.17014 q^{5} +4.85729 q^{7} +18.6113 q^{8} +O(q^{10})$	\(q-3.18772 q^{2} +2.16159 q^{4} +7.17014 q^{5} +4.85729 q^{7} +18.6113 q^{8} -22.8564 q^{10} +21.4831 q^{11} -34.9058 q^{13} -15.4837 q^{14} -76.6202 q^{16} -114.386 q^{17} +151.792 q^{19} +15.4989 q^{20} -68.4821 q^{22} -92.8819 q^{23} -73.5892 q^{25} +111.270 q^{26} +10.4994 q^{28} -188.247 q^{29} -6.38716 q^{31} +95.3541 q^{32} +364.630 q^{34} +34.8274 q^{35} +392.782 q^{37} -483.872 q^{38} +133.445 q^{40} -208.753 q^{41} +151.908 q^{43} +46.4375 q^{44} +296.082 q^{46} +168.461 q^{47} -319.407 q^{49} +234.582 q^{50} -75.4519 q^{52} +2.81647 q^{53} +154.037 q^{55} +90.4002 q^{56} +600.081 q^{58} +245.968 q^{59} -712.982 q^{61} +20.3605 q^{62} +308.999 q^{64} -250.279 q^{65} +237.177 q^{67} -247.254 q^{68} -111.020 q^{70} -122.465 q^{71} +275.306 q^{73} -1252.08 q^{74} +328.112 q^{76} +104.350 q^{77} +258.287 q^{79} -549.377 q^{80} +665.446 q^{82} -1330.37 q^{83} -820.161 q^{85} -484.242 q^{86} +399.827 q^{88} -83.1666 q^{89} -169.548 q^{91} -200.772 q^{92} -537.008 q^{94} +1088.37 q^{95} -37.0722 q^{97} +1018.18 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8}+O(q^{10}) $ 24 * q - 6 * q^2 + 84 * q^4 - 30 * q^5 - 75 * q^8	$ 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8} + 3 q^{10} - 132 q^{11} - 192 q^{14} + 240 q^{16} - 207 q^{17} + 3 q^{19} - 552 q^{20} + 24 q^{22} - 588 q^{23} + 300 q^{25} - 957 q^{26} - 6 q^{28} - 834 q^{29} - 1431 q^{32} - 1257 q^{35} + 3 q^{37} - 1587 q^{38} + 24 q^{40} - 1344 q^{41} - 2211 q^{44} + 3 q^{46} - 1716 q^{47} + 294 q^{49} - 1932 q^{50} + 192 q^{52} - 1368 q^{53} - 6 q^{55} - 2958 q^{56} - 48 q^{58} - 2388 q^{59} + 540 q^{61} - 2118 q^{62} + 195 q^{64} - 2175 q^{65} + 378 q^{67} - 2979 q^{68} + 375 q^{70} - 3105 q^{71} + 219 q^{73} - 2514 q^{74} + 24 q^{76} - 2937 q^{77} - 4935 q^{80} - 6 q^{82} - 3669 q^{83} - 3837 q^{86} + 192 q^{88} - 5202 q^{89} - 267 q^{91} - 5250 q^{92} + 24 q^{94} - 6972 q^{95} - 4392 q^{98}+O(q^{100}) $ 24 * q - 6 * q^2 + 84 * q^4 - 30 * q^5 - 75 * q^8 + 3 * q^10 - 132 * q^11 - 192 * q^14 + 240 * q^16 - 207 * q^17 + 3 * q^19 - 552 * q^20 + 24 * q^22 - 588 * q^23 + 300 * q^25 - 957 * q^26 - 6 * q^28 - 834 * q^29 - 1431 * q^32 - 1257 * q^35 + 3 * q^37 - 1587 * q^38 + 24 * q^40 - 1344 * q^41 - 2211 * q^44 + 3 * q^46 - 1716 * q^47 + 294 * q^49 - 1932 * q^50 + 192 * q^52 - 1368 * q^53 - 6 * q^55 - 2958 * q^56 - 48 * q^58 - 2388 * q^59 + 540 * q^61 - 2118 * q^62 + 195 * q^64 - 2175 * q^65 + 378 * q^67 - 2979 * q^68 + 375 * q^70 - 3105 * q^71 + 219 * q^73 - 2514 * q^74 + 24 * q^76 - 2937 * q^77 - 4935 * q^80 - 6 * q^82 - 3669 * q^83 - 3837 * q^86 + 192 * q^88 - 5202 * q^89 - 267 * q^91 - 5250 * q^92 + 24 * q^94 - 6972 * q^95 - 4392 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−3.18772	−1.12703	−0.563515	−	0.826106i	$-0.690551\pi$
$2$	−3.18772	−1.12703	−0.563515	+	0.826106i	$0.690551\pi$
$3$	0	0
$3$	0	0
$4$	2.16159	0.270198
$5$	7.17014	0.641316	0.320658	−	0.947195i	$-0.396096\pi$
$5$	7.17014	0.641316	0.320658	+	0.947195i	$0.396096\pi$
$6$	0	0
$7$	4.85729	0.262269	0.131134	−	0.991365i	$-0.458138\pi$
$7$	4.85729	0.262269	0.131134	+	0.991365i	$0.458138\pi$
$8$	18.6113	0.822509
$9$	0	0
$10$	−22.8564	−0.722783
$11$	21.4831	0.588854	0.294427	−	0.955674i	$-0.404871\pi$
$11$	21.4831	0.588854	0.294427	+	0.955674i	$0.404871\pi$
$12$	0	0
$13$	−34.9058	−0.744702	−0.372351	−	0.928092i	$-0.621448\pi$
$13$	−34.9058	−0.744702	−0.372351	+	0.928092i	$0.621448\pi$
$14$	−15.4837	−0.295585
$15$	0	0
$16$	−76.6202	−1.19719
$17$	−114.386	−1.63192	−0.815959	−	0.578110i	$-0.803791\pi$
$17$	−114.386	−1.63192	−0.815959	+	0.578110i	$0.803791\pi$
$18$	0	0
$19$	151.792	1.83282	0.916409	−	0.400244i	$-0.131075\pi$
$19$	151.792	1.83282	0.916409	+	0.400244i	$0.131075\pi$
$20$	15.4989	0.173283
$21$	0	0
$22$	−68.4821	−0.663656
$23$	−92.8819	−0.842053	−0.421026	−	0.907048i	$-0.638330\pi$
$23$	−92.8819	−0.842053	−0.421026	+	0.907048i	$0.638330\pi$
$24$	0	0
$25$	−73.5892	−0.588713
$26$	111.270	0.839302
$27$	0	0
$28$	10.4994	0.0708646
$29$	−188.247	−1.20540	−0.602701	−	0.797967i	$-0.705909\pi$
$29$	−188.247	−1.20540	−0.602701	+	0.797967i	$0.705909\pi$
$30$	0	0
$31$	−6.38716	−0.0370054	−0.0185027	−	0.999829i	$-0.505890\pi$
$31$	−6.38716	−0.0370054	−0.0185027	+	0.999829i	$0.505890\pi$
$32$	95.3541	0.526762
$33$	0	0
$34$	364.630	1.83922
$35$	34.8274	0.168197
$36$	0	0
$37$	392.782	1.74521	0.872607	−	0.488424i	$-0.162428\pi$
$37$	392.782	1.74521	0.872607	+	0.488424i	$0.162428\pi$
$38$	−483.872	−2.06564
$39$	0	0
$40$	133.445	0.527489
$41$	−208.753	−0.795164	−0.397582	−	0.917567i	$-0.630151\pi$
$41$	−208.753	−0.795164	−0.397582	+	0.917567i	$0.630151\pi$
$42$	0	0
$43$	151.908	0.538740	0.269370	−	0.963037i	$-0.413185\pi$
$43$	151.908	0.538740	0.269370	+	0.963037i	$0.413185\pi$
$44$	46.4375	0.159107
$45$	0	0
$46$	296.082	0.949019
$47$	168.461	0.522821	0.261411	−	0.965228i	$-0.415812\pi$
$47$	168.461	0.522821	0.261411	+	0.965228i	$0.415812\pi$
$48$	0	0
$49$	−319.407	−0.931215
$50$	234.582	0.663498
$51$	0	0
$52$	−75.4519	−0.201217
$53$	2.81647	0.00729946	0.00364973	−	0.999993i	$-0.498838\pi$
$53$	2.81647	0.00729946	0.00364973	+	0.999993i	$0.498838\pi$
$54$	0	0
$55$	154.037	0.377642
$56$	90.4002	0.215719
$57$	0	0
$58$	600.081	1.35853
$59$	245.968	0.542751	0.271376	−	0.962473i	$-0.412521\pi$
$59$	245.968	0.542751	0.271376	+	0.962473i	$0.412521\pi$
$60$	0	0
$61$	−712.982	−1.49652	−0.748262	−	0.663404i	$-0.769111\pi$
$61$	−712.982	−1.49652	−0.748262	+	0.663404i	$0.769111\pi$
$62$	20.3605	0.0417062
$63$	0	0
$64$	308.999	0.603514
$65$	−250.279	−0.477590
$66$	0	0
$67$	237.177	0.432475	0.216237	−	0.976341i	$-0.430622\pi$
$67$	237.177	0.432475	0.216237	+	0.976341i	$0.430622\pi$
$68$	−247.254	−0.440941
$69$	0	0
$70$	−111.020	−0.189564
$71$	−122.465	−0.204703	−0.102351	−	0.994748i	$-0.532637\pi$
$71$	−122.465	−0.204703	−0.102351	+	0.994748i	$0.532637\pi$
$72$	0	0
$73$	275.306	0.441399	0.220700	−	0.975342i	$-0.429166\pi$
$73$	275.306	0.441399	0.220700	+	0.975342i	$0.429166\pi$
$74$	−1252.08	−1.96691
$75$	0	0
$76$	328.112	0.495224
$77$	104.350	0.154438
$78$	0	0
$79$	258.287	0.367842	0.183921	−	0.982941i	$-0.441121\pi$
$79$	258.287	0.367842	0.183921	+	0.982941i	$0.441121\pi$
$80$	−549.377	−0.767778
$81$	0	0
$82$	665.446	0.896174
$83$	−1330.37	−1.75937	−0.879684	−	0.475559i	$-0.842246\pi$
$83$	−1330.37	−1.75937	−0.879684	+	0.475559i	$0.842246\pi$
$84$	0	0
$85$	−820.161	−1.04658
$86$	−484.242	−0.607176
$87$	0	0
$88$	399.827	0.484338
$89$	−83.1666	−0.0990521	−0.0495261	−	0.998773i	$-0.515771\pi$
$89$	−83.1666	−0.0990521	−0.0495261	+	0.998773i	$0.515771\pi$
$90$	0	0
$91$	−169.548	−0.195312
$92$	−200.772	−0.227521
$93$	0	0
$94$	−537.008	−0.589236
$95$	1088.37	1.17542
$96$	0	0
$97$	−37.0722	−0.0388052	−0.0194026	−	0.999812i	$-0.506176\pi$
$97$	−37.0722	−0.0388052	−0.0194026	+	0.999812i	$0.506176\pi$
$98$	1018.18	1.04951
$99$	0	0
$100$	−159.069	−0.159069
$101$	−1116.17	−1.09964	−0.549818	−	0.835285i	$-0.685303\pi$
$101$	−1116.17	−1.09964	−0.549818	+	0.835285i	$0.685303\pi$
$102$	0	0
$103$	1235.47	1.18188	0.590942	−	0.806714i	$-0.298756\pi$
$103$	1235.47	1.18188	0.590942	+	0.806714i	$0.298756\pi$
$104$	−649.641	−0.612524
$105$	0	0
$106$	−8.97812	−0.00822671
$107$	738.624	0.667341	0.333670	−	0.942690i	$-0.391713\pi$
$107$	738.624	0.667341	0.333670	+	0.942690i	$0.391713\pi$
$108$	0	0
$109$	879.581	0.772923	0.386461	−	0.922306i	$-0.373697\pi$
$109$	879.581	0.772923	0.386461	+	0.922306i	$0.373697\pi$
$110$	−491.026	−0.425614
$111$	0	0
$112$	−372.167	−0.313986
$113$	−891.279	−0.741986	−0.370993	−	0.928636i	$-0.620983\pi$
$113$	−891.279	−0.741986	−0.370993	+	0.928636i	$0.620983\pi$
$114$	0	0
$115$	−665.976	−0.540022
$116$	−406.913	−0.325698
$117$	0	0
$118$	−784.079	−0.611697
$119$	−555.604	−0.428001
$120$	0	0
$121$	−869.477	−0.653251
$122$	2272.79	1.68663
$123$	0	0
$124$	−13.8064	−0.00999879
$125$	−1423.91	−1.01887
$126$	0	0
$127$	−2204.59	−1.54036	−0.770179	−	0.637827i	$-0.779833\pi$
$127$	−2204.59	−1.54036	−0.770179	+	0.637827i	$0.779833\pi$
$128$	−1747.84	−1.20694
$129$	0	0
$130$	797.821	0.538258
$131$	367.729	0.245257	0.122629	−	0.992453i	$-0.460868\pi$
$131$	367.729	0.245257	0.122629	+	0.992453i	$0.460868\pi$
$132$	0	0
$133$	737.299	0.480691
$134$	−756.055	−0.487412
$135$	0	0
$136$	−2128.86	−1.34227
$137$	−998.068	−0.622414	−0.311207	−	0.950342i	$-0.600733\pi$
$137$	−998.068	−0.622414	−0.311207	+	0.950342i	$0.600733\pi$
$138$	0	0
$139$	−2530.57	−1.54417	−0.772087	−	0.635517i	$-0.780787\pi$
$139$	−2530.57	−1.54417	−0.772087	+	0.635517i	$0.780787\pi$
$140$	75.2824	0.0454466
$141$	0	0
$142$	390.384	0.230706
$143$	−749.884	−0.438521
$144$	0	0
$145$	−1349.76	−0.773044
$146$	−877.600	−0.497470
$147$	0	0
$148$	849.031	0.471553
$149$	−3373.24	−1.85468	−0.927339	−	0.374223i	$-0.877909\pi$
$149$	−3373.24	−1.85468	−0.927339	+	0.374223i	$0.877909\pi$
$150$	0	0
$151$	1073.28	0.578428	0.289214	−	0.957264i	$-0.406606\pi$
$151$	1073.28	0.578428	0.289214	+	0.957264i	$0.406606\pi$
$152$	2825.04	1.50751
$153$	0	0
$154$	−332.637	−0.174056
$155$	−45.7968	−0.0237322
$156$	0	0
$157$	−2734.52	−1.39005	−0.695026	−	0.718985i	$-0.744607\pi$
$157$	−2734.52	−1.39005	−0.695026	+	0.718985i	$0.744607\pi$
$158$	−823.348	−0.414570
$159$	0	0
$160$	683.702	0.337821
$161$	−451.154	−0.220844
$162$	0	0
$163$	−503.590	−0.241989	−0.120995	−	0.992653i	$-0.538608\pi$
$163$	−503.590	−0.241989	−0.120995	+	0.992653i	$0.538608\pi$
$164$	−451.237	−0.214852
$165$	0	0
$166$	4240.86	1.98286
$167$	529.172	0.245201	0.122600	−	0.992456i	$-0.460877\pi$
$167$	529.172	0.245201	0.122600	+	0.992456i	$0.460877\pi$
$168$	0	0
$169$	−978.585	−0.445419
$170$	2614.45	1.17952
$171$	0	0
$172$	328.363	0.145566
$173$	−921.574	−0.405006	−0.202503	−	0.979282i	$-0.564908\pi$
$173$	−921.574	−0.405006	−0.202503	+	0.979282i	$0.564908\pi$
$174$	0	0
$175$	−357.444	−0.154401
$176$	−1646.04	−0.704971
$177$	0	0
$178$	265.112	0.111635
$179$	−515.461	−0.215237	−0.107618	−	0.994192i	$-0.534322\pi$
$179$	−515.461	−0.215237	−0.107618	+	0.994192i	$0.534322\pi$
$180$	0	0
$181$	−1681.34	−0.690459	−0.345230	−	0.938518i	$-0.612199\pi$
$181$	−1681.34	−0.690459	−0.345230	+	0.938518i	$0.612199\pi$
$182$	540.471	0.220123
$183$	0	0
$184$	−1728.65	−0.692596
$185$	2816.30	1.11923
$186$	0	0
$187$	−2457.36	−0.960961
$188$	364.143	0.141265
$189$	0	0
$190$	−3469.43	−1.32473
$191$	2536.30	0.960837	0.480419	−	0.877039i	$-0.340485\pi$
$191$	2536.30	0.960837	0.480419	+	0.877039i	$0.340485\pi$
$192$	0	0
$193$	1882.92	0.702256	0.351128	−	0.936327i	$-0.385798\pi$
$193$	1882.92	0.702256	0.351128	+	0.936327i	$0.385798\pi$
$194$	118.176	0.0437347
$195$	0	0
$196$	−690.425	−0.251613
$197$	−1561.96	−0.564898	−0.282449	−	0.959282i	$-0.591147\pi$
$197$	−1561.96	−0.564898	−0.282449	+	0.959282i	$0.591147\pi$
$198$	0	0
$199$	−2016.21	−0.718219	−0.359110	−	0.933295i	$-0.616920\pi$
$199$	−2016.21	−0.718219	−0.359110	+	0.933295i	$0.616920\pi$
$200$	−1369.59	−0.484222
$201$	0	0
$202$	3558.04	1.23932
$203$	−914.372	−0.316140
$204$	0	0
$205$	−1496.79	−0.509951
$206$	−3938.33	−1.33202
$207$	0	0
$208$	2674.49	0.891551
$209$	3260.96	1.07926
$210$	0	0
$211$	766.588	0.250114	0.125057	−	0.992150i	$-0.460089\pi$
$211$	766.588	0.250114	0.125057	+	0.992150i	$0.460089\pi$
$212$	6.08803	0.00197230
$213$	0	0
$214$	−2354.53	−0.752113
$215$	1089.20	0.345503
$216$	0	0
$217$	−31.0243	−0.00970537
$218$	−2803.86	−0.871107
$219$	0	0
$220$	332.963	0.102038
$221$	3992.72	1.21529
$222$	0	0
$223$	3858.77	1.15876	0.579378	−	0.815059i	$-0.303296\pi$
$223$	3858.77	1.15876	0.579378	+	0.815059i	$0.303296\pi$
$224$	463.162	0.138153
$225$	0	0
$226$	2841.15	0.836241
$227$	−3108.52	−0.908899	−0.454449	−	0.890773i	$-0.650164\pi$
$227$	−3108.52	−0.908899	−0.454449	+	0.890773i	$0.650164\pi$
$228$	0	0
$229$	−2653.80	−0.765799	−0.382899	−	0.923790i	$-0.625074\pi$
$229$	−2653.80	−0.765799	−0.382899	+	0.923790i	$0.625074\pi$
$230$	2122.95	0.608622
$231$	0	0
$232$	−3503.52	−0.991454
$233$	−1573.43	−0.442399	−0.221200	−	0.975229i	$-0.570997\pi$
$233$	−1573.43	−0.442399	−0.221200	+	0.975229i	$0.570997\pi$
$234$	0	0
$235$	1207.89	0.335294
$236$	531.681	0.146650
$237$	0	0
$238$	1771.11	0.482371
$239$	−5027.79	−1.36076	−0.680378	−	0.732862i	$-0.738184\pi$
$239$	−5027.79	−1.36076	−0.680378	+	0.732862i	$0.738184\pi$
$240$	0	0
$241$	1975.78	0.528096	0.264048	−	0.964510i	$-0.414942\pi$
$241$	1975.78	0.528096	0.264048	+	0.964510i	$0.414942\pi$
$242$	2771.65	0.736234
$243$	0	0
$244$	−1541.17	−0.404358
$245$	−2290.19	−0.597203
$246$	0	0
$247$	−5298.43	−1.36490
$248$	−118.873	−0.0304373
$249$	0	0
$250$	4539.04	1.14830
$251$	−2196.36	−0.552323	−0.276162	−	0.961111i	$-0.589063\pi$
$251$	−2196.36	−0.552323	−0.276162	+	0.961111i	$0.589063\pi$
$252$	0	0
$253$	−1995.39	−0.495846
$254$	7027.62	1.73603
$255$	0	0
$256$	3099.63	0.756745
$257$	2403.23	0.583305	0.291653	−	0.956524i	$-0.405795\pi$
$257$	2403.23	0.583305	0.291653	+	0.956524i	$0.405795\pi$
$258$	0	0
$259$	1907.85	0.457715
$260$	−541.000	−0.129044
$261$	0	0
$262$	−1172.22	−0.276412
$263$	4082.86	0.957263	0.478631	−	0.878016i	$-0.341133\pi$
$263$	4082.86	0.957263	0.478631	+	0.878016i	$0.341133\pi$
$264$	0	0
$265$	20.1944	0.00468126
$266$	−2350.30	−0.541754
$267$	0	0
$268$	512.679	0.116854
$269$	−1962.92	−0.444911	−0.222456	−	0.974943i	$-0.571407\pi$
$269$	−1962.92	−0.444911	−0.222456	+	0.974943i	$0.571407\pi$
$270$	0	0
$271$	1628.04	0.364930	0.182465	−	0.983212i	$-0.441592\pi$
$271$	1628.04	0.364930	0.182465	+	0.983212i	$0.441592\pi$
$272$	8764.26	1.95372
$273$	0	0
$274$	3181.57	0.701480
$275$	−1580.92	−0.346666
$276$	0	0
$277$	3631.88	0.787793	0.393897	−	0.919155i	$-0.371127\pi$
$277$	3631.88	0.787793	0.393897	+	0.919155i	$0.371127\pi$
$278$	8066.76	1.74033
$279$	0	0
$280$	648.182	0.138344
$281$	8359.04	1.77459	0.887293	−	0.461206i	$-0.152583\pi$
$281$	8359.04	1.77459	0.887293	+	0.461206i	$0.152583\pi$
$282$	0	0
$283$	−6823.07	−1.43318	−0.716589	−	0.697496i	$-0.754298\pi$
$283$	−6823.07	−1.43318	−0.716589	+	0.697496i	$0.754298\pi$
$284$	−264.718	−0.0553103
$285$	0	0
$286$	2390.42	0.494226
$287$	−1013.97	−0.208547
$288$	0	0
$289$	8171.09	1.66316
$290$	4302.66	0.871245
$291$	0	0
$292$	595.098	0.119265
$293$	8143.56	1.62373	0.811864	−	0.583847i	$-0.198453\pi$
$293$	8143.56	1.62373	0.811864	+	0.583847i	$0.198453\pi$
$294$	0	0
$295$	1763.63	0.348075
$296$	7310.16	1.43545
$297$	0	0
$298$	10753.0	2.09028
$299$	3242.12	0.627078
$300$	0	0
$301$	737.863	0.141295
$302$	−3421.34	−0.651907
$303$	0	0
$304$	−11630.4	−2.19423
$305$	−5112.17	−0.959745
$306$	0	0
$307$	4650.72	0.864594	0.432297	−	0.901731i	$-0.357703\pi$
$307$	4650.72	0.864594	0.432297	+	0.901731i	$0.357703\pi$
$308$	225.560	0.0417289
$309$	0	0
$310$	145.988	0.0267469
$311$	−2879.46	−0.525014	−0.262507	−	0.964930i	$-0.584549\pi$
$311$	−2879.46	−0.525014	−0.262507	+	0.964930i	$0.584549\pi$
$312$	0	0
$313$	−871.240	−0.157333	−0.0786667	−	0.996901i	$-0.525066\pi$
$313$	−871.240	−0.157333	−0.0786667	+	0.996901i	$0.525066\pi$
$314$	8716.88	1.56663
$315$	0	0
$316$	558.309	0.0993904
$317$	−6655.07	−1.17914	−0.589568	−	0.807719i	$-0.700702\pi$
$317$	−6655.07	−1.17914	−0.589568	+	0.807719i	$0.700702\pi$
$318$	0	0
$319$	−4044.13	−0.709806
$320$	2215.57	0.387044
$321$	0	0
$322$	1438.15	0.248898
$323$	−17362.9	−2.99101
$324$	0	0
$325$	2568.69	0.438416
$326$	1605.31	0.272729
$327$	0	0
$328$	−3885.15	−0.654029
$329$	818.265	0.137120
$330$	0	0
$331$	−8929.44	−1.48280	−0.741399	−	0.671064i	$-0.765837\pi$
$331$	−8929.44	−1.48280	−0.741399	+	0.671064i	$0.765837\pi$
$332$	−2875.72	−0.475378
$333$	0	0
$334$	−1686.85	−0.276349
$335$	1700.59	0.277353
$336$	0	0
$337$	1156.13	0.186879	0.0934394	−	0.995625i	$-0.470214\pi$
$337$	1156.13	0.186879	0.0934394	+	0.995625i	$0.470214\pi$
$338$	3119.46	0.502001
$339$	0	0
$340$	−1772.85	−0.282783
$341$	−137.216	−0.0217908
$342$	0	0
$343$	−3217.50	−0.506498
$344$	2827.20	0.443118
$345$	0	0
$346$	2937.73	0.456454
$347$	−6846.90	−1.05925	−0.529627	−	0.848231i	$-0.677668\pi$
$347$	−6846.90	−1.05925	−0.529627	+	0.848231i	$0.677668\pi$
$348$	0	0
$349$	−2261.23	−0.346822	−0.173411	−	0.984850i	$-0.555479\pi$
$349$	−2261.23	−0.346822	−0.173411	+	0.984850i	$0.555479\pi$
$350$	1139.43	0.174015
$351$	0	0
$352$	2048.50	0.310186
$353$	5343.77	0.805723	0.402861	−	0.915261i	$-0.368016\pi$
$353$	5343.77	0.805723	0.402861	+	0.915261i	$0.368016\pi$
$354$	0	0
$355$	−878.090	−0.131279
$356$	−179.772	−0.0267637
$357$	0	0
$358$	1643.15	0.242579
$359$	366.248	0.0538435	0.0269217	−	0.999638i	$-0.491430\pi$
$359$	366.248	0.0538435	0.0269217	+	0.999638i	$0.491430\pi$
$360$	0	0
$361$	16181.9	2.35922
$362$	5359.65	0.778169
$363$	0	0
$364$	−366.491	−0.0527730
$365$	1973.98	0.283077
$366$	0	0
$367$	8193.66	1.16541	0.582705	−	0.812684i	$-0.301994\pi$
$367$	8193.66	1.16541	0.582705	+	0.812684i	$0.301994\pi$
$368$	7116.63	1.00810
$369$	0	0
$370$	−8977.58	−1.26141
$371$	13.6804	0.00191442
$372$	0	0
$373$	2578.36	0.357916	0.178958	−	0.983857i	$-0.442727\pi$
$373$	2578.36	0.357916	0.178958	+	0.983857i	$0.442727\pi$
$374$	7833.38	1.08303
$375$	0	0
$376$	3135.28	0.430025
$377$	6570.93	0.897666
$378$	0	0
$379$	1973.04	0.267410	0.133705	−	0.991021i	$-0.457313\pi$
$379$	1973.04	0.267410	0.133705	+	0.991021i	$0.457313\pi$
$380$	2352.61	0.317595
$381$	0	0
$382$	−8085.01	−1.08289
$383$	−7944.47	−1.05990	−0.529952	−	0.848027i	$-0.677790\pi$
$383$	−7944.47	−1.05990	−0.529952	+	0.848027i	$0.677790\pi$
$384$	0	0
$385$	748.200	0.0990437
$386$	−6002.22	−0.791464
$387$	0	0
$388$	−80.1346	−0.0104851
$389$	−326.501	−0.0425560	−0.0212780	−	0.999774i	$-0.506774\pi$
$389$	−326.501	−0.0425560	−0.0212780	+	0.999774i	$0.506774\pi$
$390$	0	0
$391$	10624.4	1.37416
$392$	−5944.56	−0.765933
$393$	0	0
$394$	4979.09	0.636658
$395$	1851.95	0.235903
$396$	0	0
$397$	4407.94	0.557250	0.278625	−	0.960400i	$-0.410121\pi$
$397$	4407.94	0.557250	0.278625	+	0.960400i	$0.410121\pi$
$398$	6427.13	0.809455
$399$	0	0
$400$	5638.42	0.704802
$401$	−8193.20	−1.02032	−0.510160	−	0.860079i	$-0.670414\pi$
$401$	−8193.20	−1.02032	−0.510160	+	0.860079i	$0.670414\pi$
$402$	0	0
$403$	222.949	0.0275580
$404$	−2412.70	−0.297119
$405$	0	0
$406$	2914.77	0.356299
$407$	8438.16	1.02768
$408$	0	0
$409$	9627.55	1.16394	0.581970	−	0.813210i	$-0.302282\pi$
$409$	9627.55	1.16394	0.581970	+	0.813210i	$0.302282\pi$
$410$	4771.34	0.574731
$411$	0	0
$412$	2670.57	0.319343
$413$	1194.74	0.142347
$414$	0	0
$415$	−9538.96	−1.12831
$416$	−3328.41	−0.392281
$417$	0	0
$418$	−10395.1	−1.21636
$419$	13734.6	1.60138	0.800692	−	0.599077i	$-0.204466\pi$
$419$	13734.6	1.60138	0.800692	+	0.599077i	$0.204466\pi$
$420$	0	0
$421$	−10728.3	−1.24196	−0.620981	−	0.783826i	$-0.713266\pi$
$421$	−10728.3	−1.24196	−0.620981	+	0.783826i	$0.713266\pi$
$422$	−2443.67	−0.281886
$423$	0	0
$424$	52.4180	0.00600387
$425$	8417.55	0.960732
$426$	0	0
$427$	−3463.16	−0.392492
$428$	1596.60	0.180314
$429$	0	0
$430$	−3472.08	−0.389392
$431$	−1902.52	−0.212624	−0.106312	−	0.994333i	$-0.533904\pi$
$431$	−1902.52	−0.212624	−0.106312	+	0.994333i	$0.533904\pi$
$432$	0	0
$433$	−12218.4	−1.35607	−0.678034	−	0.735031i	$-0.737168\pi$
$433$	−12218.4	−1.35607	−0.678034	+	0.735031i	$0.737168\pi$
$434$	98.8968	0.0109382
$435$	0	0
$436$	1901.29	0.208842
$437$	−14098.7	−1.54333
$438$	0	0
$439$	1231.79	0.133918	0.0669591	−	0.997756i	$-0.478670\pi$
$439$	1231.79	0.133918	0.0669591	+	0.997756i	$0.478670\pi$
$440$	2866.81	0.310614
$441$	0	0
$442$	−12727.7	−1.36967
$443$	11037.5	1.18376	0.591881	−	0.806025i	$-0.298385\pi$
$443$	11037.5	1.18376	0.591881	+	0.806025i	$0.298385\pi$
$444$	0	0
$445$	−596.315	−0.0635237
$446$	−12300.7	−1.30595
$447$	0	0
$448$	1500.90	0.158283
$449$	−13280.8	−1.39590	−0.697950	−	0.716147i	$-0.745904\pi$
$449$	−13280.8	−1.39590	−0.697950	+	0.716147i	$0.745904\pi$
$450$	0	0
$451$	−4484.65	−0.468235
$452$	−1926.58	−0.200483
$453$	0	0
$454$	9909.12	1.02436
$455$	−1215.68	−0.125257
$456$	0	0
$457$	829.795	0.0849369	0.0424684	−	0.999098i	$-0.486478\pi$
$457$	829.795	0.0849369	0.0424684	+	0.999098i	$0.486478\pi$
$458$	8459.58	0.863079
$459$	0	0
$460$	−1439.56	−0.145913
$461$	−14071.2	−1.42161	−0.710805	−	0.703389i	$-0.751669\pi$
$461$	−14071.2	−1.42161	−0.710805	+	0.703389i	$0.751669\pi$
$462$	0	0
$463$	−1260.39	−0.126512	−0.0632560	−	0.997997i	$-0.520148\pi$
$463$	−1260.39	−0.126512	−0.0632560	+	0.997997i	$0.520148\pi$
$464$	14423.6	1.44310
$465$	0	0
$466$	5015.67	0.498597
$467$	1047.03	0.103749	0.0518743	−	0.998654i	$-0.483480\pi$
$467$	1047.03	0.103749	0.0518743	+	0.998654i	$0.483480\pi$
$468$	0	0
$469$	1152.04	0.113425
$470$	−3850.42	−0.377887
$471$	0	0
$472$	4577.78	0.446418
$473$	3263.46	0.317239
$474$	0	0
$475$	−11170.3	−1.07900
$476$	−1200.99	−0.115645
$477$	0	0
$478$	16027.2	1.53361
$479$	−14886.7	−1.42003	−0.710013	−	0.704189i	$-0.751311\pi$
$479$	−14886.7	−1.42003	−0.710013	+	0.704189i	$0.751311\pi$
$480$	0	0
$481$	−13710.4	−1.29966
$482$	−6298.24	−0.595180
$483$	0	0
$484$	−1879.45	−0.176507
$485$	−265.812	−0.0248864
$486$	0	0
$487$	−5554.29	−0.516815	−0.258408	−	0.966036i	$-0.583198\pi$
$487$	−5554.29	−0.516815	−0.258408	+	0.966036i	$0.583198\pi$
$488$	−13269.5	−1.23090
$489$	0	0
$490$	7300.49	0.673067
$491$	20841.1	1.91557	0.957787	−	0.287478i	$-0.0928168\pi$
$491$	20841.1	1.91557	0.957787	+	0.287478i	$0.0928168\pi$
$492$	0	0
$493$	21532.8	1.96712
$494$	16889.9	1.53829
$495$	0	0
$496$	489.386	0.0443025
$497$	−594.847	−0.0536872
$498$	0	0
$499$	6062.21	0.543852	0.271926	−	0.962318i	$-0.412339\pi$
$499$	6062.21	0.543852	0.271926	+	0.962318i	$0.412339\pi$
$500$	−3077.91	−0.275296
$501$	0	0
$502$	7001.40	0.622486
$503$	−14901.8	−1.32095	−0.660477	−	0.750847i	$-0.729646\pi$
$503$	−14901.8	−1.32095	−0.660477	+	0.750847i	$0.729646\pi$
$504$	0	0
$505$	−8003.10	−0.705214
$506$	6360.75	0.558834
$507$	0	0
$508$	−4765.41	−0.416202
$509$	5598.74	0.487544	0.243772	−	0.969833i	$-0.421615\pi$
$509$	5598.74	0.487544	0.243772	+	0.969833i	$0.421615\pi$
$510$	0	0
$511$	1337.24	0.115765
$512$	4101.93	0.354066
$513$	0	0
$514$	−7660.84	−0.657403
$515$	8858.46	0.757962
$516$	0	0
$517$	3619.07	0.307865
$518$	−6081.71	−0.515859
$519$	0	0
$520$	−4658.01	−0.392822
$521$	−6168.96	−0.518746	−0.259373	−	0.965777i	$-0.583516\pi$
$521$	−6168.96	−0.518746	−0.259373	+	0.965777i	$0.583516\pi$
$522$	0	0
$523$	−23425.6	−1.95857	−0.979285	−	0.202487i	$-0.935097\pi$
$523$	−23425.6	−1.95857	−0.979285	+	0.202487i	$0.935097\pi$
$524$	794.879	0.0662680
$525$	0	0
$526$	−13015.0	−1.07886
$527$	730.600	0.0603898
$528$	0	0
$529$	−3539.96	−0.290947
$530$	−64.3743	−0.00527593
$531$	0	0
$532$	1593.73	0.129882
$533$	7286.68	0.592160
$534$	0	0
$535$	5296.03	0.427976
$536$	4414.16	0.355714
$537$	0	0
$538$	6257.24	0.501429
$539$	−6861.84	−0.548350
$540$	0	0
$541$	−17748.3	−1.41046	−0.705230	−	0.708979i	$-0.749156\pi$
$541$	−17748.3	−1.41046	−0.705230	+	0.708979i	$0.749156\pi$
$542$	−5189.73	−0.411288
$543$	0	0
$544$	−10907.1	−0.859632
$545$	6306.71	0.495688
$546$	0	0
$547$	−20089.7	−1.57033	−0.785167	−	0.619284i	$-0.787423\pi$
$547$	−20089.7	−1.57033	−0.785167	+	0.619284i	$0.787423\pi$
$548$	−2157.41	−0.168175
$549$	0	0
$550$	5039.54	0.390703
$551$	−28574.5	−2.20928
$552$	0	0
$553$	1254.57	0.0964737
$554$	−11577.4	−0.887867
$555$	0	0
$556$	−5470.04	−0.417233
$557$	13936.8	1.06019	0.530093	−	0.847940i	$-0.322157\pi$
$557$	13936.8	1.06019	0.530093	+	0.847940i	$0.322157\pi$
$558$	0	0
$559$	−5302.48	−0.401201
$560$	−2668.48	−0.201364
$561$	0	0
$562$	−26646.3	−2.00001
$563$	3074.49	0.230150	0.115075	−	0.993357i	$-0.463289\pi$
$563$	3074.49	0.230150	0.115075	+	0.993357i	$0.463289\pi$
$564$	0	0
$565$	−6390.59	−0.475848
$566$	21750.1	1.61524
$567$	0	0
$568$	−2279.22	−0.168370
$569$	−6326.06	−0.466084	−0.233042	−	0.972467i	$-0.574868\pi$
$569$	−6326.06	−0.466084	−0.233042	+	0.972467i	$0.574868\pi$
$570$	0	0
$571$	12423.4	0.910512	0.455256	−	0.890361i	$-0.349548\pi$
$571$	12423.4	0.910512	0.455256	+	0.890361i	$0.349548\pi$
$572$	−1620.94	−0.118487
$573$	0	0
$574$	3232.26	0.235039
$575$	6835.10	0.495727
$576$	0	0
$577$	9332.17	0.673316	0.336658	−	0.941627i	$-0.390703\pi$
$577$	9332.17	0.673316	0.336658	+	0.941627i	$0.390703\pi$
$578$	−26047.2	−1.87443
$579$	0	0
$580$	−2917.62	−0.208875
$581$	−6462.01	−0.461427
$582$	0	0
$583$	60.5064	0.00429831
$584$	5123.79	0.363055
$585$	0	0
$586$	−25959.4	−1.82999
$587$	−9474.35	−0.666181	−0.333090	−	0.942895i	$-0.608091\pi$
$587$	−9474.35	−0.666181	−0.333090	+	0.942895i	$0.608091\pi$
$588$	0	0
$589$	−969.521	−0.0678241
$590$	−5621.95	−0.392292
$591$	0	0
$592$	−30095.0	−2.08935
$593$	−15625.3	−1.08204	−0.541022	−	0.841008i	$-0.681963\pi$
$593$	−15625.3	−1.08204	−0.541022	+	0.841008i	$0.681963\pi$
$594$	0	0
$595$	−3983.76	−0.274484
$596$	−7291.56	−0.501130
$597$	0	0
$598$	−10335.0	−0.706737
$599$	−19563.7	−1.33447	−0.667237	−	0.744845i	$-0.732523\pi$
$599$	−19563.7	−1.33447	−0.667237	+	0.744845i	$0.732523\pi$
$600$	0	0
$601$	6738.79	0.457373	0.228686	−	0.973500i	$-0.426557\pi$
$601$	6738.79	0.457373	0.228686	+	0.973500i	$0.426557\pi$
$602$	−2352.10	−0.159243
$603$	0	0
$604$	2320.00	0.156290
$605$	−6234.27	−0.418941
$606$	0	0
$607$	8159.39	0.545601	0.272800	−	0.962071i	$-0.412050\pi$
$607$	8159.39	0.545601	0.272800	+	0.962071i	$0.412050\pi$
$608$	14474.0	0.965459
$609$	0	0
$610$	16296.2	1.08166
$611$	−5880.28	−0.389346
$612$	0	0
$613$	4201.72	0.276845	0.138423	−	0.990373i	$-0.455797\pi$
$613$	4201.72	0.276845	0.138423	+	0.990373i	$0.455797\pi$
$614$	−14825.2	−0.974425
$615$	0	0
$616$	1942.08	0.127027
$617$	9759.59	0.636801	0.318401	−	0.947956i	$-0.396854\pi$
$617$	9759.59	0.636801	0.318401	+	0.947956i	$0.396854\pi$
$618$	0	0
$619$	−13203.5	−0.857341	−0.428670	−	0.903461i	$-0.641018\pi$
$619$	−13203.5	−0.857341	−0.428670	+	0.903461i	$0.641018\pi$
$620$	−98.9937	−0.00641239
$621$	0	0
$622$	9178.93	0.591707
$623$	−403.964	−0.0259783
$624$	0	0
$625$	−1010.99	−0.0647035
$626$	2777.27	0.177320
$627$	0	0
$628$	−5910.89	−0.375590
$629$	−44928.6	−2.84804
$630$	0	0
$631$	16267.8	1.02632	0.513162	−	0.858292i	$-0.328474\pi$
$631$	16267.8	1.02632	0.513162	+	0.858292i	$0.328474\pi$
$632$	4807.05	0.302554
$633$	0	0
$634$	21214.5	1.32892
$635$	−15807.2	−0.987857
$636$	0	0
$637$	11149.1	0.693478
$638$	12891.6	0.799973
$639$	0	0
$640$	−12532.2	−0.774031
$641$	−5573.20	−0.343414	−0.171707	−	0.985148i	$-0.554928\pi$
$641$	−5573.20	−0.343414	−0.171707	+	0.985148i	$0.554928\pi$
$642$	0	0
$643$	24426.0	1.49808	0.749042	−	0.662523i	$-0.230514\pi$
$643$	24426.0	1.49808	0.749042	+	0.662523i	$0.230514\pi$
$644$	−975.208	−0.0596717
$645$	0	0
$646$	55348.0	3.37096
$647$	−25936.4	−1.57599	−0.787994	−	0.615683i	$-0.788880\pi$
$647$	−25936.4	−1.57599	−0.787994	+	0.615683i	$0.788880\pi$
$648$	0	0
$649$	5284.15	0.319601
$650$	−8188.27	−0.494108
$651$	0	0
$652$	−1088.55	−0.0653850
$653$	15039.1	0.901266	0.450633	−	0.892709i	$-0.351198\pi$
$653$	15039.1	0.901266	0.450633	+	0.892709i	$0.351198\pi$
$654$	0	0
$655$	2636.67	0.157287
$656$	15994.7	0.951963
$657$	0	0
$658$	−2608.40	−0.154538
$659$	8762.95	0.517991	0.258995	−	0.965879i	$-0.416609\pi$
$659$	8762.95	0.517991	0.258995	+	0.965879i	$0.416609\pi$
$660$	0	0
$661$	−14431.1	−0.849174	−0.424587	−	0.905387i	$-0.639581\pi$
$661$	−14431.1	−0.849174	−0.424587	+	0.905387i	$0.639581\pi$
$662$	28464.6	1.67116
$663$	0	0
$664$	−24759.9	−1.44710
$665$	5286.53	0.308275
$666$	0	0
$667$	17484.8	1.01501
$668$	1143.85	0.0662528
$669$	0	0
$670$	−5421.02	−0.312585
$671$	−15317.0	−0.881234
$672$	0	0
$673$	6445.23	0.369161	0.184581	−	0.982817i	$-0.440907\pi$
$673$	6445.23	0.369161	0.184581	+	0.982817i	$0.440907\pi$
$674$	−3685.41	−0.210618
$675$	0	0
$676$	−2115.30	−0.120351
$677$	−7084.53	−0.402187	−0.201093	−	0.979572i	$-0.564449\pi$
$677$	−7084.53	−0.402187	−0.201093	+	0.979572i	$0.564449\pi$
$678$	0	0
$679$	−180.070	−0.0101774
$680$	−15264.2	−0.860818
$681$	0	0
$682$	437.406	0.0245589
$683$	19404.1	1.08708	0.543542	−	0.839382i	$-0.317083\pi$
$683$	19404.1	1.08708	0.543542	+	0.839382i	$0.317083\pi$
$684$	0	0
$685$	−7156.28	−0.399164
$686$	10256.5	0.570838
$687$	0	0
$688$	−11639.3	−0.644974
$689$	−98.3110	−0.00543592
$690$	0	0
$691$	29661.6	1.63297	0.816484	−	0.577369i	$-0.195920\pi$
$691$	29661.6	1.63297	0.816484	+	0.577369i	$0.195920\pi$
$692$	−1992.06	−0.109432
$693$	0	0
$694$	21826.0	1.19381
$695$	−18144.5	−0.990304
$696$	0	0
$697$	23878.3	1.29764
$698$	7208.19	0.390880
$699$	0	0
$700$	−772.645	−0.0417189
$701$	29123.5	1.56916	0.784579	−	0.620029i	$-0.212879\pi$
$701$	29123.5	1.56916	0.784579	+	0.620029i	$0.212879\pi$
$702$	0	0
$703$	59621.2	3.19866
$704$	6638.25	0.355382
$705$	0	0
$706$	−17034.5	−0.908074
$707$	−5421.56	−0.288400
$708$	0	0
$709$	−20337.7	−1.07729	−0.538644	−	0.842534i	$-0.681063\pi$
$709$	−20337.7	−1.07729	−0.538644	+	0.842534i	$0.681063\pi$
$710$	2799.11	0.147956
$711$	0	0
$712$	−1547.83	−0.0814713
$713$	593.251	0.0311605
$714$	0	0
$715$	−5376.77	−0.281231
$716$	−1114.21	−0.0581566
$717$	0	0
$718$	−1167.50	−0.0606833
$719$	1455.53	0.0754965	0.0377482	−	0.999287i	$-0.487982\pi$
$719$	1455.53	0.0754965	0.0377482	+	0.999287i	$0.487982\pi$
$720$	0	0
$721$	6001.02	0.309972
$722$	−51583.4	−2.65891
$723$	0	0
$724$	−3634.36	−0.186561
$725$	13853.0	0.709636
$726$	0	0
$727$	16201.1	0.826500	0.413250	−	0.910618i	$-0.364394\pi$
$727$	16201.1	0.826500	0.413250	+	0.910618i	$0.364394\pi$
$728$	−3155.49	−0.160646
$729$	0	0
$730$	−6292.51	−0.319036
$731$	−17376.1	−0.879179
$732$	0	0
$733$	13630.4	0.686835	0.343418	−	0.939183i	$-0.388415\pi$
$733$	13630.4	0.686835	0.343418	+	0.939183i	$0.388415\pi$
$734$	−26119.1	−1.31345
$735$	0	0
$736$	−8856.67	−0.443561
$737$	5095.29	0.254664
$738$	0	0
$739$	−19299.8	−0.960698	−0.480349	−	0.877077i	$-0.659490\pi$
$739$	−19299.8	−0.960698	−0.480349	+	0.877077i	$0.659490\pi$
$740$	6087.67	0.302415
$741$	0	0
$742$	−43.6093	−0.00215761
$743$	6147.47	0.303538	0.151769	−	0.988416i	$-0.451503\pi$
$743$	6147.47	0.303538	0.151769	+	0.988416i	$0.451503\pi$
$744$	0	0
$745$	−24186.6	−1.18944
$746$	−8219.11	−0.403382
$747$	0	0
$748$	−5311.79	−0.259650
$749$	3587.71	0.175023
$750$	0	0
$751$	27266.7	1.32487	0.662433	−	0.749121i	$-0.269524\pi$
$751$	27266.7	1.32487	0.662433	+	0.749121i	$0.269524\pi$
$752$	−12907.5	−0.625917
$753$	0	0
$754$	−20946.3	−1.01170
$755$	7695.60	0.370956
$756$	0	0
$757$	4412.15	0.211839	0.105920	−	0.994375i	$-0.466221\pi$
$757$	4412.15	0.211839	0.105920	+	0.994375i	$0.466221\pi$
$758$	−6289.52	−0.301379
$759$	0	0
$760$	20256.0	0.966790
$761$	8668.93	0.412941	0.206471	−	0.978453i	$-0.433802\pi$
$761$	8668.93	0.412941	0.206471	+	0.978453i	$0.433802\pi$
$762$	0	0
$763$	4272.38	0.202714
$764$	5482.42	0.259616
$765$	0	0
$766$	25324.8	1.19455
$767$	−8585.72	−0.404188
$768$	0	0
$769$	−3305.36	−0.154999	−0.0774996	−	0.996992i	$-0.524694\pi$
$769$	−3305.36	−0.154999	−0.0774996	+	0.996992i	$0.524694\pi$
$770$	−2385.06	−0.111625
$771$	0	0
$772$	4070.09	0.189748
$773$	2960.57	0.137755	0.0688774	−	0.997625i	$-0.478058\pi$
$773$	2960.57	0.137755	0.0688774	+	0.997625i	$0.478058\pi$
$774$	0	0
$775$	470.026	0.0217856
$776$	−689.959	−0.0319177
$777$	0	0
$778$	1040.80	0.0479619
$779$	−31687.1	−1.45739
$780$	0	0
$781$	−2630.92	−0.120540
$782$	−33867.5	−1.54872
$783$	0	0
$784$	24473.0	1.11484
$785$	−19606.9	−0.891463
$786$	0	0
$787$	−19734.9	−0.893869	−0.446934	−	0.894567i	$-0.647484\pi$
$787$	−19734.9	−0.893869	−0.446934	+	0.894567i	$0.647484\pi$
$788$	−3376.31	−0.152634
$789$	0	0
$790$	−5903.51	−0.265870
$791$	−4329.20	−0.194600
$792$	0	0
$793$	24887.2	1.11446
$794$	−14051.3	−0.628038
$795$	0	0
$796$	−4358.22	−0.194062
$797$	8587.75	0.381673	0.190837	−	0.981622i	$-0.438880\pi$
$797$	8587.75	0.381673	0.190837	+	0.981622i	$0.438880\pi$
$798$	0	0
$799$	−19269.6	−0.853202
$800$	−7017.03	−0.310112
$801$	0	0
$802$	26117.6	1.14993
$803$	5914.42	0.259920
$804$	0	0
$805$	−3234.84	−0.141631
$806$	−710.700	−0.0310587
$807$	0	0
$808$	−20773.3	−0.904460
$809$	2823.95	0.122725	0.0613626	−	0.998116i	$-0.480455\pi$
$809$	2823.95	0.122725	0.0613626	+	0.998116i	$0.480455\pi$
$810$	0	0
$811$	8173.40	0.353892	0.176946	−	0.984221i	$-0.443378\pi$
$811$	8173.40	0.353892	0.176946	+	0.984221i	$0.443378\pi$
$812$	−1976.49	−0.0854203
$813$	0	0
$814$	−26898.5	−1.15822
$815$	−3610.81	−0.155192
$816$	0	0
$817$	23058.5	0.987411
$818$	−30690.0	−1.31180
$819$	0	0
$820$	−3235.43	−0.137788
$821$	9917.01	0.421566	0.210783	−	0.977533i	$-0.432399\pi$
$821$	9917.01	0.421566	0.210783	+	0.977533i	$0.432399\pi$
$822$	0	0
$823$	28421.7	1.20379	0.601895	−	0.798575i	$-0.294412\pi$
$823$	28421.7	1.20379	0.601895	+	0.798575i	$0.294412\pi$
$824$	22993.6	0.972111
$825$	0	0
$826$	−3808.50	−0.160429
$827$	26236.5	1.10318	0.551591	−	0.834115i	$-0.314021\pi$
$827$	26236.5	1.10318	0.551591	+	0.834115i	$0.314021\pi$
$828$	0	0
$829$	−26182.8	−1.09694	−0.548472	−	0.836169i	$-0.684790\pi$
$829$	−26182.8	−1.09694	−0.548472	+	0.836169i	$0.684790\pi$
$830$	30407.6	1.27164
$831$	0	0
$832$	−10785.9	−0.449438
$833$	36535.6	1.51967
$834$	0	0
$835$	3794.23	0.157251
$836$	7048.85	0.291614
$837$	0	0
$838$	−43782.1	−1.80481
$839$	23247.6	0.956611	0.478305	−	0.878194i	$-0.341251\pi$
$839$	23247.6	0.956611	0.478305	+	0.878194i	$0.341251\pi$
$840$	0	0
$841$	11048.1	0.452995
$842$	34198.9	1.39973
$843$	0	0
$844$	1657.05	0.0675804
$845$	−7016.59	−0.285654
$846$	0	0
$847$	−4223.30	−0.171327
$848$	−215.798	−0.00873885
$849$	0	0
$850$	−26832.8	−1.08277
$851$	−36482.3	−1.46956
$852$	0	0
$853$	29985.1	1.20360	0.601799	−	0.798647i	$-0.294451\pi$
$853$	29985.1	1.20360	0.601799	+	0.798647i	$0.294451\pi$
$854$	11039.6	0.442350
$855$	0	0
$856$	13746.7	0.548894
$857$	31686.7	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$857$	31686.7	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$858$	0	0
$859$	−16789.0	−0.666861	−0.333430	−	0.942775i	$-0.608206\pi$
$859$	−16789.0	−0.666861	−0.333430	+	0.942775i	$0.608206\pi$
$860$	2354.41	0.0933542
$861$	0	0
$862$	6064.70	0.239634
$863$	30224.1	1.19217	0.596083	−	0.802923i	$-0.296723\pi$
$863$	30224.1	1.19217	0.596083	+	0.802923i	$0.296723\pi$
$864$	0	0
$865$	−6607.81	−0.259737
$866$	38948.8	1.52833
$867$	0	0
$868$	−67.0616	−0.00262237
$869$	5548.80	0.216605
$870$	0	0
$871$	−8278.86	−0.322065
$872$	16370.1	0.635736
$873$	0	0
$874$	44942.9	1.73938
$875$	−6916.35	−0.267217
$876$	0	0
$877$	46037.5	1.77261	0.886303	−	0.463106i	$-0.153265\pi$
$877$	46037.5	1.77261	0.886303	+	0.463106i	$0.153265\pi$
$878$	−3926.61	−0.150930
$879$	0	0
$880$	−11802.3	−0.452109
$881$	−41491.6	−1.58671	−0.793354	−	0.608761i	$-0.791667\pi$
$881$	−41491.6	−1.58671	−0.793354	+	0.608761i	$0.791667\pi$
$882$	0	0
$883$	8761.29	0.333908	0.166954	−	0.985965i	$-0.446607\pi$
$883$	8761.29	0.333908	0.166954	+	0.985965i	$0.446607\pi$
$884$	8630.61	0.328370
$885$	0	0
$886$	−35184.5	−1.33414
$887$	−449.267	−0.0170067	−0.00850333	−	0.999964i	$-0.502707\pi$
$887$	−449.267	−0.0170067	−0.00850333	+	0.999964i	$0.502707\pi$
$888$	0	0
$889$	−10708.3	−0.403988
$890$	1900.89	0.0715932
$891$	0	0
$892$	8341.06	0.313094
$893$	25571.1	0.958236
$894$	0	0
$895$	−3695.93	−0.138035
$896$	−8489.75	−0.316543
$897$	0	0
$898$	42335.5	1.57322
$899$	1202.37	0.0446064
$900$	0	0
$901$	−322.163	−0.0119121
$902$	14295.8	0.527715
$903$	0	0
$904$	−16587.8	−0.610290
$905$	−12055.4	−0.442803
$906$	0	0
$907$	38245.6	1.40014	0.700069	−	0.714075i	$-0.253152\pi$
$907$	38245.6	1.40014	0.700069	+	0.714075i	$0.253152\pi$
$908$	−6719.34	−0.245583
$909$	0	0
$910$	3875.25	0.141168
$911$	28323.1	1.03006	0.515030	−	0.857172i	$-0.327781\pi$
$911$	28323.1	1.03006	0.515030	+	0.857172i	$0.327781\pi$
$912$	0	0
$913$	−28580.5	−1.03601
$914$	−2645.16	−0.0957265
$915$	0	0
$916$	−5736.41	−0.206917
$917$	1786.17	0.0643233
$918$	0	0
$919$	−15767.9	−0.565981	−0.282990	−	0.959123i	$-0.591326\pi$
$919$	−15767.9	−0.565981	−0.282990	+	0.959123i	$0.591326\pi$
$920$	−12394.6	−0.444173
$921$	0	0
$922$	44855.2	1.60220
$923$	4274.73	0.152443
$924$	0	0
$925$	−28904.5	−1.02743
$926$	4017.76	0.142583
$927$	0	0
$928$	−17950.2	−0.634960
$929$	−19605.0	−0.692377	−0.346188	−	0.938165i	$-0.612524\pi$
$929$	−19605.0	−0.692377	−0.346188	+	0.938165i	$0.612524\pi$
$930$	0	0
$931$	−48483.5	−1.70675
$932$	−3401.11	−0.119535
$933$	0	0
$934$	−3337.63	−0.116928
$935$	−17619.6	−0.616280
$936$	0	0
$937$	7734.46	0.269663	0.134831	−	0.990869i	$-0.456951\pi$
$937$	7734.46	0.269663	0.134831	+	0.990869i	$0.456951\pi$
$938$	−3672.38	−0.127833
$939$	0	0
$940$	2610.96	0.0905958
$941$	35963.2	1.24587	0.622936	−	0.782273i	$-0.285940\pi$
$941$	35963.2	1.24587	0.622936	+	0.782273i	$0.285940\pi$
$942$	0	0
$943$	19389.4	0.669570
$944$	−18846.1	−0.649777
$945$	0	0
$946$	−10403.0	−0.357538
$947$	−8661.79	−0.297223	−0.148612	−	0.988896i	$-0.547480\pi$
$947$	−8661.79	−0.297223	−0.148612	+	0.988896i	$0.547480\pi$
$948$	0	0
$949$	−9609.78	−0.328711
$950$	35607.7	1.21607
$951$	0	0
$952$	−10340.5	−0.352035
$953$	30365.2	1.03213	0.516067	−	0.856548i	$-0.327395\pi$
$953$	30365.2	1.03213	0.516067	+	0.856548i	$0.327395\pi$
$954$	0	0
$955$	18185.6	0.616201
$956$	−10868.0	−0.367674
$957$	0	0
$958$	47454.8	1.60041
$959$	−4847.90	−0.163240
$960$	0	0
$961$	−29750.2	−0.998631
$962$	43704.8	1.46476
$963$	0	0
$964$	4270.81	0.142691
$965$	13500.8	0.450368
$966$	0	0
$967$	−24260.5	−0.806788	−0.403394	−	0.915026i	$-0.632170\pi$
$967$	−24260.5	−0.806788	−0.403394	+	0.915026i	$0.632170\pi$
$968$	−16182.1	−0.537305
$969$	0	0
$970$	847.337	0.0280478
$971$	−5488.24	−0.181386	−0.0906931	−	0.995879i	$-0.528908\pi$
$971$	−5488.24	−0.181386	−0.0906931	+	0.995879i	$0.528908\pi$
$972$	0	0
$973$	−12291.7	−0.404989
$974$	17705.5	0.582466
$975$	0	0
$976$	54628.8	1.79162
$977$	17438.2	0.571030	0.285515	−	0.958374i	$-0.407835\pi$
$977$	17438.2	0.571030	0.285515	+	0.958374i	$0.407835\pi$
$978$	0	0
$979$	−1786.67	−0.0583272
$980$	−4950.44	−0.161363
$981$	0	0
$982$	−66435.8	−2.15891
$983$	9970.79	0.323519	0.161759	−	0.986830i	$-0.448283\pi$
$983$	9970.79	0.323519	0.161759	+	0.986830i	$0.448283\pi$
$984$	0	0
$985$	−11199.5	−0.362279
$986$	−68640.7	−2.21700
$987$	0	0
$988$	−11453.0	−0.368794
$989$	−14109.5	−0.453647
$990$	0	0
$991$	−34797.3	−1.11541	−0.557706	−	0.830039i	$-0.688318\pi$
$991$	−34797.3	−1.11541	−0.557706	+	0.830039i	$0.688318\pi$
$992$	−609.042	−0.0194930
$993$	0	0
$994$	1896.21	0.0605071
$995$	−14456.5	−0.460606
$996$	0	0
$997$	−8341.71	−0.264980	−0.132490	−	0.991184i	$-0.542297\pi$
$997$	−8341.71	−0.264980	−0.132490	+	0.991184i	$0.542297\pi$
$998$	−19324.7	−0.612938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.4.a.c.1.7		24
3.2	odd	2		729.4.a.d.1.18		24
27.2	odd	18		243.4.e.c.190.2		48
27.4	even	9		81.4.e.a.46.2		48
27.5	odd	18		243.4.e.d.217.7		48
27.7	even	9		81.4.e.a.37.2		48
27.11	odd	18		243.4.e.d.28.7		48
27.13	even	9		243.4.e.b.55.7		48
27.14	odd	18		243.4.e.c.55.2		48
27.16	even	9		243.4.e.a.28.2		48
27.20	odd	18		27.4.e.a.22.7	yes	48
27.22	even	9		243.4.e.a.217.2		48
27.23	odd	18		27.4.e.a.16.7	✓	48
27.25	even	9		243.4.e.b.190.7		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.7	✓	48	27.23	odd	18
27.4.e.a.22.7	yes	48	27.20	odd	18
81.4.e.a.37.2		48	27.7	even	9
81.4.e.a.46.2		48	27.4	even	9
243.4.e.a.28.2		48	27.16	even	9
243.4.e.a.217.2		48	27.22	even	9
243.4.e.b.55.7		48	27.13	even	9
243.4.e.b.190.7		48	27.25	even	9
243.4.e.c.55.2		48	27.14	odd	18
243.4.e.c.190.2		48	27.2	odd	18
243.4.e.d.28.7		48	27.11	odd	18
243.4.e.d.217.7		48	27.5	odd	18
729.4.a.c.1.7		24	1.1	even	1	trivial
729.4.a.d.1.18		24	3.2	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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q-3.18772 q^{2} +2.16159 q^{4} +7.17014 q^{5} +4.85729 q^{7} +18.6113 q^{8} +O(q^{10})\)	\(q-3.18772 q^{2} +2.16159 q^{4} +7.17014 q^{5} +4.85729 q^{7} +18.6113 q^{8} -22.8564 q^{10} +21.4831 q^{11} -34.9058 q^{13} -15.4837 q^{14} -76.6202 q^{16} -114.386 q^{17} +151.792 q^{19} +15.4989 q^{20} -68.4821 q^{22} -92.8819 q^{23} -73.5892 q^{25} +111.270 q^{26} +10.4994 q^{28} -188.247 q^{29} -6.38716 q^{31} +95.3541 q^{32} +364.630 q^{34} +34.8274 q^{35} +392.782 q^{37} -483.872 q^{38} +133.445 q^{40} -208.753 q^{41} +151.908 q^{43} +46.4375 q^{44} +296.082 q^{46} +168.461 q^{47} -319.407 q^{49} +234.582 q^{50} -75.4519 q^{52} +2.81647 q^{53} +154.037 q^{55} +90.4002 q^{56} +600.081 q^{58} +245.968 q^{59} -712.982 q^{61} +20.3605 q^{62} +308.999 q^{64} -250.279 q^{65} +237.177 q^{67} -247.254 q^{68} -111.020 q^{70} -122.465 q^{71} +275.306 q^{73} -1252.08 q^{74} +328.112 q^{76} +104.350 q^{77} +258.287 q^{79} -549.377 q^{80} +665.446 q^{82} -1330.37 q^{83} -820.161 q^{85} -484.242 q^{86} +399.827 q^{88} -83.1666 q^{89} -169.548 q^{91} -200.772 q^{92} -537.008 q^{94} +1088.37 q^{95} -37.0722 q^{97} +1018.18 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8}+O(q^{10}) \) 24 * q - 6 * q^2 + 84 * q^4 - 30 * q^5 - 75 * q^8	\( 24 q - 6 q^{2} + 84 q^{4} - 30 q^{5} - 75 q^{8} + 3 q^{10} - 132 q^{11} - 192 q^{14} + 240 q^{16} - 207 q^{17} + 3 q^{19} - 552 q^{20} + 24 q^{22} - 588 q^{23} + 300 q^{25} - 957 q^{26} - 6 q^{28} - 834 q^{29} - 1431 q^{32} - 1257 q^{35} + 3 q^{37} - 1587 q^{38} + 24 q^{40} - 1344 q^{41} - 2211 q^{44} + 3 q^{46} - 1716 q^{47} + 294 q^{49} - 1932 q^{50} + 192 q^{52} - 1368 q^{53} - 6 q^{55} - 2958 q^{56} - 48 q^{58} - 2388 q^{59} + 540 q^{61} - 2118 q^{62} + 195 q^{64} - 2175 q^{65} + 378 q^{67} - 2979 q^{68} + 375 q^{70} - 3105 q^{71} + 219 q^{73} - 2514 q^{74} + 24 q^{76} - 2937 q^{77} - 4935 q^{80} - 6 q^{82} - 3669 q^{83} - 3837 q^{86} + 192 q^{88} - 5202 q^{89} - 267 q^{91} - 5250 q^{92} + 24 q^{94} - 6972 q^{95} - 4392 q^{98}+O(q^{100}) \) 24 * q - 6 * q^2 + 84 * q^4 - 30 * q^5 - 75 * q^8 + 3 * q^10 - 132 * q^11 - 192 * q^14 + 240 * q^16 - 207 * q^17 + 3 * q^19 - 552 * q^20 + 24 * q^22 - 588 * q^23 + 300 * q^25 - 957 * q^26 - 6 * q^28 - 834 * q^29 - 1431 * q^32 - 1257 * q^35 + 3 * q^37 - 1587 * q^38 + 24 * q^40 - 1344 * q^41 - 2211 * q^44 + 3 * q^46 - 1716 * q^47 + 294 * q^49 - 1932 * q^50 + 192 * q^52 - 1368 * q^53 - 6 * q^55 - 2958 * q^56 - 48 * q^58 - 2388 * q^59 + 540 * q^61 - 2118 * q^62 + 195 * q^64 - 2175 * q^65 + 378 * q^67 - 2979 * q^68 + 375 * q^70 - 3105 * q^71 + 219 * q^73 - 2514 * q^74 + 24 * q^76 - 2937 * q^77 - 4935 * q^80 - 6 * q^82 - 3669 * q^83 - 3837 * q^86 + 192 * q^88 - 5202 * q^89 - 267 * q^91 - 5250 * q^92 + 24 * q^94 - 6972 * q^95 - 4392 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more