LMFDB - Embedded newform 81.4.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,4,Mod(1,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, 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("81.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	81.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:	$4.77915471046$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.37228 q^{2} -6.11684 q^{4} -10.3723 q^{5} -5.11684 q^{7} -19.3723 q^{8} +O(q^{10})$	\(q+1.37228 q^{2} -6.11684 q^{4} -10.3723 q^{5} -5.11684 q^{7} -19.3723 q^{8} -14.2337 q^{10} -55.9783 q^{11} +37.5842 q^{13} -7.02175 q^{14} +22.3505 q^{16} -23.6495 q^{17} +39.0516 q^{19} +63.4456 q^{20} -76.8179 q^{22} -71.0733 q^{23} -17.4158 q^{25} +51.5761 q^{26} +31.2989 q^{28} +28.3723 q^{29} +12.8832 q^{31} +185.649 q^{32} -32.4537 q^{34} +53.0733 q^{35} -180.103 q^{37} +53.5898 q^{38} +200.935 q^{40} +215.484 q^{41} +61.2337 q^{43} +342.410 q^{44} -97.5326 q^{46} +61.8776 q^{47} -316.818 q^{49} -23.8993 q^{50} -229.897 q^{52} -492.310 q^{53} +580.622 q^{55} +99.1249 q^{56} +38.9348 q^{58} -789.630 q^{59} +521.090 q^{61} +17.6793 q^{62} +75.9590 q^{64} -389.834 q^{65} +304.429 q^{67} +144.660 q^{68} +72.8316 q^{70} -270.391 q^{71} -925.464 q^{73} -247.152 q^{74} -238.873 q^{76} +286.432 q^{77} -1289.03 q^{79} -231.826 q^{80} +295.704 q^{82} -713.834 q^{83} +245.299 q^{85} +84.0298 q^{86} +1084.43 q^{88} +404.804 q^{89} -192.313 q^{91} +434.745 q^{92} +84.9135 q^{94} -405.054 q^{95} +75.0273 q^{97} -434.763 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 5 q^{4} - 15 q^{5} + 7 q^{7} - 33 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 5 * q^4 - 15 * q^5 + 7 * q^7 - 33 * q^8	$ 2 q - 3 q^{2} + 5 q^{4} - 15 q^{5} + 7 q^{7} - 33 q^{8} + 6 q^{10} - 66 q^{11} - 11 q^{13} - 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 12 q^{20} - 33 q^{22} - 33 q^{23} - 121 q^{25} + 264 q^{26} + 166 q^{28} + 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} - 3 q^{35} - 50 q^{37} + 561 q^{38} + 264 q^{40} - 132 q^{41} + 88 q^{43} + 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 429 q^{50} - 770 q^{52} - 54 q^{53} + 627 q^{55} - 66 q^{56} - 60 q^{58} - 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 165 q^{65} + 988 q^{67} - 693 q^{68} + 318 q^{70} - 1368 q^{71} - 455 q^{73} - 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 96 q^{80} + 1815 q^{82} - 813 q^{83} + 594 q^{85} - 33 q^{86} + 1221 q^{88} + 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} + 132 q^{95} + 736 q^{97} + 423 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 5 * q^4 - 15 * q^5 + 7 * q^7 - 33 * q^8 + 6 * q^10 - 66 * q^11 - 11 * q^13 - 60 * q^14 - 7 * q^16 - 99 * q^17 - 77 * q^19 + 12 * q^20 - 33 * q^22 - 33 * q^23 - 121 * q^25 + 264 * q^26 + 166 * q^28 + 51 * q^29 + 43 * q^31 + 423 * q^32 + 297 * q^34 - 3 * q^35 - 50 * q^37 + 561 * q^38 + 264 * q^40 - 132 * q^41 + 88 * q^43 + 231 * q^44 - 264 * q^46 - 399 * q^47 - 513 * q^49 + 429 * q^50 - 770 * q^52 - 54 * q^53 + 627 * q^55 - 66 * q^56 - 60 * q^58 - 798 * q^59 + 439 * q^61 - 114 * q^62 - 727 * q^64 - 165 * q^65 + 988 * q^67 - 693 * q^68 + 318 * q^70 - 1368 * q^71 - 455 * q^73 - 816 * q^74 - 1529 * q^76 + 165 * q^77 - 803 * q^79 - 96 * q^80 + 1815 * q^82 - 813 * q^83 + 594 * q^85 - 33 * q^86 + 1221 * q^88 + 396 * q^89 - 781 * q^91 + 858 * q^92 + 2100 * q^94 + 132 * q^95 + 736 * q^97 + 423 * 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 $

$2$	1.37228	0.485175	0.242587	−	0.970130i	$-0.422004\pi$
$2$	1.37228	0.485175	0.242587	+	0.970130i	$0.422004\pi$
$3$	0	0
$3$	0	0
$4$	−6.11684	−0.764605
$5$	−10.3723	−0.927725	−0.463863	−	0.885907i	$-0.653537\pi$
$5$	−10.3723	−0.927725	−0.463863	+	0.885907i	$0.653537\pi$
$6$	0	0
$7$	−5.11684	−0.276284	−0.138142	−	0.990412i	$-0.544113\pi$
$7$	−5.11684	−0.276284	−0.138142	+	0.990412i	$0.544113\pi$
$8$	−19.3723	−0.856142
$9$	0	0
$10$	−14.2337	−0.450109
$11$	−55.9783	−1.53437	−0.767185	−	0.641425i	$-0.778343\pi$
$11$	−55.9783	−1.53437	−0.767185	+	0.641425i	$0.778343\pi$
$12$	0	0
$13$	37.5842	0.801845	0.400923	−	0.916112i	$-0.368690\pi$
$13$	37.5842	0.801845	0.400923	+	0.916112i	$0.368690\pi$
$14$	−7.02175	−0.134046
$15$	0	0
$16$	22.3505	0.349227
$17$	−23.6495	−0.337402	−0.168701	−	0.985667i	$-0.553957\pi$
$17$	−23.6495	−0.337402	−0.168701	+	0.985667i	$0.553957\pi$
$18$	0	0
$19$	39.0516	0.471529	0.235764	−	0.971810i	$-0.424241\pi$
$19$	39.0516	0.471529	0.235764	+	0.971810i	$0.424241\pi$
$20$	63.4456	0.709344
$21$	0	0
$22$	−76.8179	−0.744438
$23$	−71.0733	−0.644340	−0.322170	−	0.946682i	$-0.604412\pi$
$23$	−71.0733	−0.644340	−0.322170	+	0.946682i	$0.604412\pi$
$24$	0	0
$25$	−17.4158	−0.139326
$26$	51.5761	0.389035
$27$	0	0
$28$	31.2989	0.211248
$29$	28.3723	0.181676	0.0908379	−	0.995866i	$-0.471045\pi$
$29$	28.3723	0.181676	0.0908379	+	0.995866i	$0.471045\pi$
$30$	0	0
$31$	12.8832	0.0746414	0.0373207	−	0.999303i	$-0.488118\pi$
$31$	12.8832	0.0746414	0.0373207	+	0.999303i	$0.488118\pi$
$32$	185.649	1.02558
$33$	0	0
$34$	−32.4537	−0.163699
$35$	53.0733	0.256315
$36$	0	0
$37$	−180.103	−0.800237	−0.400119	−	0.916463i	$-0.631031\pi$
$37$	−180.103	−0.800237	−0.400119	+	0.916463i	$0.631031\pi$
$38$	53.5898	0.228774
$39$	0	0
$40$	200.935	0.794264
$41$	215.484	0.820802	0.410401	−	0.911905i	$-0.365389\pi$
$41$	215.484	0.820802	0.410401	+	0.911905i	$0.365389\pi$
$42$	0	0
$43$	61.2337	0.217164	0.108582	−	0.994087i	$-0.465369\pi$
$43$	61.2337	0.217164	0.108582	+	0.994087i	$0.465369\pi$
$44$	342.410	1.17319
$45$	0	0
$46$	−97.5326	−0.312617
$47$	61.8776	0.192038	0.0960189	−	0.995380i	$-0.469389\pi$
$47$	61.8776	0.192038	0.0960189	+	0.995380i	$0.469389\pi$
$48$	0	0
$49$	−316.818	−0.923667
$50$	−23.8993	−0.0675976
$51$	0	0
$52$	−229.897	−0.613095
$53$	−492.310	−1.27592	−0.637962	−	0.770068i	$-0.720222\pi$
$53$	−492.310	−1.27592	−0.637962	+	0.770068i	$0.720222\pi$
$54$	0	0
$55$	580.622	1.42347
$56$	99.1249	0.236538
$57$	0	0
$58$	38.9348	0.0881445
$59$	−789.630	−1.74239	−0.871196	−	0.490936i	$-0.836655\pi$
$59$	−789.630	−1.74239	−0.871196	+	0.490936i	$0.836655\pi$
$60$	0	0
$61$	521.090	1.09375	0.546874	−	0.837215i	$-0.315818\pi$
$61$	521.090	1.09375	0.546874	+	0.837215i	$0.315818\pi$
$62$	17.6793	0.0362141
$63$	0	0
$64$	75.9590	0.148358
$65$	−389.834	−0.743892
$66$	0	0
$67$	304.429	0.555104	0.277552	−	0.960711i	$-0.410477\pi$
$67$	304.429	0.555104	0.277552	+	0.960711i	$0.410477\pi$
$68$	144.660	0.257980
$69$	0	0
$70$	72.8316	0.124358
$71$	−270.391	−0.451966	−0.225983	−	0.974131i	$-0.572559\pi$
$71$	−270.391	−0.451966	−0.225983	+	0.974131i	$0.572559\pi$
$72$	0	0
$73$	−925.464	−1.48380	−0.741900	−	0.670510i	$-0.766075\pi$
$73$	−925.464	−1.48380	−0.741900	+	0.670510i	$0.766075\pi$
$74$	−247.152	−0.388255
$75$	0	0
$76$	−238.873	−0.360534
$77$	286.432	0.423921
$78$	0	0
$79$	−1289.03	−1.83579	−0.917897	−	0.396818i	$-0.870114\pi$
$79$	−1289.03	−1.83579	−0.917897	+	0.396818i	$0.870114\pi$
$80$	−231.826	−0.323987
$81$	0	0
$82$	295.704	0.398232
$83$	−713.834	−0.944018	−0.472009	−	0.881594i	$-0.656471\pi$
$83$	−713.834	−0.944018	−0.472009	+	0.881594i	$0.656471\pi$
$84$	0	0
$85$	245.299	0.313017
$86$	84.0298	0.105362
$87$	0	0
$88$	1084.43	1.31364
$89$	404.804	0.482125	0.241063	−	0.970510i	$-0.422504\pi$
$89$	404.804	0.482125	0.241063	+	0.970510i	$0.422504\pi$
$90$	0	0
$91$	−192.313	−0.221537
$92$	434.745	0.492666
$93$	0	0
$94$	84.9135	0.0931719
$95$	−405.054	−0.437449
$96$	0	0
$97$	75.0273	0.0785347	0.0392674	−	0.999229i	$-0.487498\pi$
$97$	75.0273	0.0785347	0.0392674	+	0.999229i	$0.487498\pi$
$98$	−434.763	−0.448140
$99$	0	0
$100$	106.530	0.106530
$101$	1087.88	1.07176	0.535881	−	0.844294i	$-0.319980\pi$
$101$	1087.88	1.07176	0.535881	+	0.844294i	$0.319980\pi$
$102$	0	0
$103$	−1091.82	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$103$	−1091.82	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$104$	−728.092	−0.686493
$105$	0	0
$106$	−675.587	−0.619046
$107$	1029.15	0.929833	0.464917	−	0.885354i	$-0.346084\pi$
$107$	1029.15	0.929833	0.464917	+	0.885354i	$0.346084\pi$
$108$	0	0
$109$	1776.52	1.56110	0.780548	−	0.625096i	$-0.214940\pi$
$109$	1776.52	1.56110	0.780548	+	0.625096i	$0.214940\pi$
$110$	796.777	0.690634
$111$	0	0
$112$	−114.364	−0.0964857
$113$	1615.94	1.34526	0.672631	−	0.739978i	$-0.265164\pi$
$113$	1615.94	1.34526	0.672631	+	0.739978i	$0.265164\pi$
$114$	0	0
$115$	737.193	0.597770
$116$	−173.549	−0.138910
$117$	0	0
$118$	−1083.59	−0.845364
$119$	121.011	0.0932187
$120$	0	0
$121$	1802.56	1.35429
$122$	715.081	0.530659
$123$	0	0
$124$	−78.8043	−0.0570712
$125$	1477.18	1.05698
$126$	0	0
$127$	−1206.10	−0.842711	−0.421356	−	0.906895i	$-0.638446\pi$
$127$	−1206.10	−0.842711	−0.421356	+	0.906895i	$0.638446\pi$
$128$	−1380.96	−0.953599
$129$	0	0
$130$	−534.962	−0.360918
$131$	1027.86	0.685528	0.342764	−	0.939422i	$-0.388637\pi$
$131$	1027.86	0.685528	0.342764	+	0.939422i	$0.388637\pi$
$132$	0	0
$133$	−199.821	−0.130276
$134$	417.763	0.269322
$135$	0	0
$136$	458.144	0.288864
$137$	1260.91	0.786326	0.393163	−	0.919469i	$-0.371381\pi$
$137$	1260.91	0.786326	0.393163	+	0.919469i	$0.371381\pi$
$138$	0	0
$139$	461.832	0.281813	0.140907	−	0.990023i	$-0.454998\pi$
$139$	461.832	0.281813	0.140907	+	0.990023i	$0.454998\pi$
$140$	−324.641	−0.195980
$141$	0	0
$142$	−371.053	−0.219282
$143$	−2103.90	−1.23033
$144$	0	0
$145$	−294.285	−0.168545
$146$	−1270.00	−0.719902
$147$	0	0
$148$	1101.66	0.611866
$149$	−1459.32	−0.802365	−0.401182	−	0.915998i	$-0.631401\pi$
$149$	−1459.32	−0.802365	−0.401182	+	0.915998i	$0.631401\pi$
$150$	0	0
$151$	1541.32	0.830666	0.415333	−	0.909669i	$-0.363665\pi$
$151$	1541.32	0.830666	0.415333	+	0.909669i	$0.363665\pi$
$152$	−756.518	−0.403696
$153$	0	0
$154$	393.065	0.205676
$155$	−133.628	−0.0692467
$156$	0	0
$157$	−3215.57	−1.63459	−0.817295	−	0.576220i	$-0.804527\pi$
$157$	−3215.57	−1.63459	−0.817295	+	0.576220i	$0.804527\pi$
$158$	−1768.92	−0.890681
$159$	0	0
$160$	−1925.61	−0.951454
$161$	363.671	0.178021
$162$	0	0
$163$	947.587	0.455342	0.227671	−	0.973738i	$-0.426889\pi$
$163$	947.587	0.455342	0.227671	+	0.973738i	$0.426889\pi$
$164$	−1318.08	−0.627590
$165$	0	0
$166$	−979.581	−0.458014
$167$	−685.960	−0.317851	−0.158926	−	0.987291i	$-0.550803\pi$
$167$	−685.960	−0.317851	−0.158926	+	0.987291i	$0.550803\pi$
$168$	0	0
$169$	−784.426	−0.357044
$170$	336.619	0.151868
$171$	0	0
$172$	−374.557	−0.166045
$173$	2212.83	0.972475	0.486237	−	0.873827i	$-0.338369\pi$
$173$	2212.83	0.972475	0.486237	+	0.873827i	$0.338369\pi$
$174$	0	0
$175$	89.1138	0.0384936
$176$	−1251.14	−0.535844
$177$	0	0
$178$	555.505	0.233915
$179$	−3023.22	−1.26238	−0.631190	−	0.775629i	$-0.717433\pi$
$179$	−3023.22	−1.26238	−0.631190	+	0.775629i	$0.717433\pi$
$180$	0	0
$181$	391.445	0.160751	0.0803753	−	0.996765i	$-0.474388\pi$
$181$	391.445	0.160751	0.0803753	+	0.996765i	$0.474388\pi$
$182$	−263.907	−0.107484
$183$	0	0
$184$	1376.85	0.551646
$185$	1868.08	0.742400
$186$	0	0
$187$	1323.86	0.517700
$188$	−378.496	−0.146833
$189$	0	0
$190$	−555.848	−0.212239
$191$	−3485.59	−1.32046	−0.660231	−	0.751062i	$-0.729542\pi$
$191$	−3485.59	−1.32046	−0.660231	+	0.751062i	$0.729542\pi$
$192$	0	0
$193$	2215.07	0.826136	0.413068	−	0.910700i	$-0.364457\pi$
$193$	2215.07	0.826136	0.413068	+	0.910700i	$0.364457\pi$
$194$	102.959	0.0381031
$195$	0	0
$196$	1937.93	0.706241
$197$	3975.11	1.43764	0.718820	−	0.695196i	$-0.244682\pi$
$197$	3975.11	1.43764	0.718820	+	0.695196i	$0.244682\pi$
$198$	0	0
$199$	−1555.34	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$199$	−1555.34	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$200$	337.383	0.119283
$201$	0	0
$202$	1492.87	0.519991
$203$	−145.177	−0.0501941
$204$	0	0
$205$	−2235.06	−0.761479
$206$	−1498.28	−0.506749
$207$	0	0
$208$	840.027	0.280026
$209$	−2186.04	−0.723500
$210$	0	0
$211$	1747.73	0.570231	0.285115	−	0.958493i	$-0.407968\pi$
$211$	1747.73	0.570231	0.285115	+	0.958493i	$0.407968\pi$
$212$	3011.38	0.975578
$213$	0	0
$214$	1412.29	0.451132
$215$	−635.133	−0.201468
$216$	0	0
$217$	−65.9211	−0.0206222
$218$	2437.88	0.757404
$219$	0	0
$220$	−3551.58	−1.08840
$221$	−888.847	−0.270544
$222$	0	0
$223$	−2541.94	−0.763323	−0.381662	−	0.924302i	$-0.624648\pi$
$223$	−2541.94	−0.763323	−0.381662	+	0.924302i	$0.624648\pi$
$224$	−949.939	−0.283350
$225$	0	0
$226$	2217.52	0.652687
$227$	2993.26	0.875197	0.437598	−	0.899171i	$-0.355829\pi$
$227$	2993.26	0.875197	0.437598	+	0.899171i	$0.355829\pi$
$228$	0	0
$229$	−4305.31	−1.24237	−0.621185	−	0.783664i	$-0.713348\pi$
$229$	−4305.31	−1.24237	−0.621185	+	0.783664i	$0.713348\pi$
$230$	1011.64	0.290023
$231$	0	0
$232$	−549.636	−0.155540
$233$	5581.34	1.56930	0.784648	−	0.619942i	$-0.212844\pi$
$233$	5581.34	1.56930	0.784648	+	0.619942i	$0.212844\pi$
$234$	0	0
$235$	−641.812	−0.178158
$236$	4830.05	1.33224
$237$	0	0
$238$	166.061	0.0452274
$239$	−1409.63	−0.381512	−0.190756	−	0.981638i	$-0.561094\pi$
$239$	−1409.63	−0.381512	−0.190756	+	0.981638i	$0.561094\pi$
$240$	0	0
$241$	626.572	0.167473	0.0837366	−	0.996488i	$-0.473315\pi$
$241$	626.572	0.167473	0.0837366	+	0.996488i	$0.473315\pi$
$242$	2473.63	0.657069
$243$	0	0
$244$	−3187.42	−0.836286
$245$	3286.12	0.856909
$246$	0	0
$247$	1467.72	0.378093
$248$	−249.576	−0.0639036
$249$	0	0
$250$	2027.10	0.512821
$251$	−1705.53	−0.428892	−0.214446	−	0.976736i	$-0.568795\pi$
$251$	−1705.53	−0.428892	−0.214446	+	0.976736i	$0.568795\pi$
$252$	0	0
$253$	3978.56	0.988656
$254$	−1655.11	−0.408862
$255$	0	0
$256$	−2502.74	−0.611020
$257$	3597.38	0.873146	0.436573	−	0.899669i	$-0.356192\pi$
$257$	3597.38	0.873146	0.436573	+	0.899669i	$0.356192\pi$
$258$	0	0
$259$	921.560	0.221092
$260$	2384.55	0.568784
$261$	0	0
$262$	1410.51	0.332601
$263$	−4137.50	−0.970074	−0.485037	−	0.874494i	$-0.661194\pi$
$263$	−4137.50	−0.970074	−0.485037	+	0.874494i	$0.661194\pi$
$264$	0	0
$265$	5106.37	1.18371
$266$	−274.211	−0.0632065
$267$	0	0
$268$	−1862.15	−0.424436
$269$	−6090.99	−1.38057	−0.690287	−	0.723536i	$-0.742516\pi$
$269$	−6090.99	−1.38057	−0.690287	+	0.723536i	$0.742516\pi$
$270$	0	0
$271$	−3196.62	−0.716534	−0.358267	−	0.933619i	$-0.616632\pi$
$271$	−3196.62	−0.716534	−0.358267	+	0.933619i	$0.616632\pi$
$272$	−528.578	−0.117830
$273$	0	0
$274$	1730.32	0.381505
$275$	974.905	0.213778
$276$	0	0
$277$	−3119.36	−0.676622	−0.338311	−	0.941034i	$-0.609856\pi$
$277$	−3119.36	−0.676622	−0.338311	+	0.941034i	$0.609856\pi$
$278$	633.763	0.136729
$279$	0	0
$280$	−1028.15	−0.219442
$281$	−4948.33	−1.05051	−0.525254	−	0.850946i	$-0.676030\pi$
$281$	−4948.33	−1.05051	−0.525254	+	0.850946i	$0.676030\pi$
$282$	0	0
$283$	−4544.93	−0.954658	−0.477329	−	0.878725i	$-0.658395\pi$
$283$	−4544.93	−0.954658	−0.477329	+	0.878725i	$0.658395\pi$
$284$	1653.94	0.345575
$285$	0	0
$286$	−2887.14	−0.596924
$287$	−1102.60	−0.226774
$288$	0	0
$289$	−4353.70	−0.886160
$290$	−403.842	−0.0817739
$291$	0	0
$292$	5660.92	1.13452
$293$	−6860.11	−1.36782	−0.683911	−	0.729566i	$-0.739722\pi$
$293$	−6860.11	−1.36782	−0.683911	+	0.729566i	$0.739722\pi$
$294$	0	0
$295$	8190.27	1.61646
$296$	3489.01	0.685117
$297$	0	0
$298$	−2002.60	−0.389287
$299$	−2671.24	−0.516661
$300$	0	0
$301$	−313.323	−0.0599988
$302$	2115.12	0.403018
$303$	0	0
$304$	872.824	0.164671
$305$	−5404.89	−1.01470
$306$	0	0
$307$	6332.25	1.17720	0.588600	−	0.808424i	$-0.299679\pi$
$307$	6332.25	1.17720	0.588600	+	0.808424i	$0.299679\pi$
$308$	−1752.06	−0.324133
$309$	0	0
$310$	−183.375	−0.0335967
$311$	−7077.67	−1.29048	−0.645238	−	0.763982i	$-0.723242\pi$
$311$	−7077.67	−1.29048	−0.645238	+	0.763982i	$0.723242\pi$
$312$	0	0
$313$	1381.30	0.249443	0.124721	−	0.992192i	$-0.460196\pi$
$313$	1381.30	0.249443	0.124721	+	0.992192i	$0.460196\pi$
$314$	−4412.67	−0.793062
$315$	0	0
$316$	7884.83	1.40366
$317$	8174.93	1.44842	0.724211	−	0.689578i	$-0.242204\pi$
$317$	8174.93	1.44842	0.724211	+	0.689578i	$0.242204\pi$
$318$	0	0
$319$	−1588.23	−0.278758
$320$	−787.869	−0.137635
$321$	0	0
$322$	499.059	0.0863711
$323$	−923.549	−0.159095
$324$	0	0
$325$	−654.559	−0.111718
$326$	1300.36	0.220920
$327$	0	0
$328$	−4174.41	−0.702723
$329$	−316.618	−0.0530569
$330$	0	0
$331$	−9661.28	−1.60433	−0.802163	−	0.597105i	$-0.796318\pi$
$331$	−9661.28	−1.60433	−0.802163	+	0.597105i	$0.796318\pi$
$332$	4366.41	0.721801
$333$	0	0
$334$	−941.329	−0.154213
$335$	−3157.63	−0.514984
$336$	0	0
$337$	−4956.02	−0.801103	−0.400552	−	0.916274i	$-0.631181\pi$
$337$	−4956.02	−0.801103	−0.400552	+	0.916274i	$0.631181\pi$
$338$	−1076.45	−0.173229
$339$	0	0
$340$	−1500.46	−0.239334
$341$	−721.177	−0.114528
$342$	0	0
$343$	3376.19	0.531478
$344$	−1186.24	−0.185923
$345$	0	0
$346$	3036.62	0.471820
$347$	1015.60	0.157120	0.0785598	−	0.996909i	$-0.474968\pi$
$347$	1015.60	0.157120	0.0785598	+	0.996909i	$0.474968\pi$
$348$	0	0
$349$	12158.6	1.86485	0.932426	−	0.361360i	$-0.117687\pi$
$349$	12158.6	1.86485	0.932426	+	0.361360i	$0.117687\pi$
$350$	122.289	0.0186761
$351$	0	0
$352$	−10392.3	−1.57362
$353$	−4236.08	−0.638708	−0.319354	−	0.947635i	$-0.603466\pi$
$353$	−4236.08	−0.638708	−0.319354	+	0.947635i	$0.603466\pi$
$354$	0	0
$355$	2804.58	0.419300
$356$	−2476.12	−0.368636
$357$	0	0
$358$	−4148.70	−0.612474
$359$	517.939	0.0761443	0.0380721	−	0.999275i	$-0.487878\pi$
$359$	517.939	0.0761443	0.0380721	+	0.999275i	$0.487878\pi$
$360$	0	0
$361$	−5333.97	−0.777660
$362$	537.172	0.0779921
$363$	0	0
$364$	1176.35	0.169388
$365$	9599.18	1.37656
$366$	0	0
$367$	−4616.29	−0.656590	−0.328295	−	0.944575i	$-0.606474\pi$
$367$	−4616.29	−0.656590	−0.328295	+	0.944575i	$0.606474\pi$
$368$	−1588.53	−0.225021
$369$	0	0
$370$	2563.53	0.360194
$371$	2519.07	0.352517
$372$	0	0
$373$	4765.42	0.661512	0.330756	−	0.943716i	$-0.392696\pi$
$373$	4765.42	0.661512	0.330756	+	0.943716i	$0.392696\pi$
$374$	1816.70	0.251175
$375$	0	0
$376$	−1198.71	−0.164412
$377$	1066.35	0.145676
$378$	0	0
$379$	−2000.33	−0.271108	−0.135554	−	0.990770i	$-0.543281\pi$
$379$	−2000.33	−0.271108	−0.135554	+	0.990770i	$0.543281\pi$
$380$	2477.65	0.334476
$381$	0	0
$382$	−4783.21	−0.640655
$383$	−990.294	−0.132119	−0.0660596	−	0.997816i	$-0.521043\pi$
$383$	−990.294	−0.132119	−0.0660596	+	0.997816i	$0.521043\pi$
$384$	0	0
$385$	−2970.95	−0.393283
$386$	3039.70	0.400820
$387$	0	0
$388$	−458.930	−0.0600481
$389$	−404.411	−0.0527106	−0.0263553	−	0.999653i	$-0.508390\pi$
$389$	−404.411	−0.0527106	−0.0263553	+	0.999653i	$0.508390\pi$
$390$	0	0
$391$	1680.85	0.217402
$392$	6137.49	0.790790
$393$	0	0
$394$	5454.97	0.697507
$395$	13370.2	1.70311
$396$	0	0
$397$	2919.61	0.369096	0.184548	−	0.982824i	$-0.440918\pi$
$397$	2919.61	0.369096	0.184548	+	0.982824i	$0.440918\pi$
$398$	−2134.37	−0.268809
$399$	0	0
$400$	−389.252	−0.0486565
$401$	10186.2	1.26852	0.634258	−	0.773121i	$-0.281306\pi$
$401$	10186.2	1.26852	0.634258	+	0.773121i	$0.281306\pi$
$402$	0	0
$403$	484.203	0.0598508
$404$	−6654.38	−0.819474
$405$	0	0
$406$	−199.223	−0.0243529
$407$	10081.9	1.22786
$408$	0	0
$409$	6914.24	0.835910	0.417955	−	0.908468i	$-0.362747\pi$
$409$	6914.24	0.835910	0.417955	+	0.908468i	$0.362747\pi$
$410$	−3067.13	−0.369450
$411$	0	0
$412$	6678.48	0.798605
$413$	4040.41	0.481394
$414$	0	0
$415$	7404.09	0.875789
$416$	6977.49	0.822355
$417$	0	0
$418$	−2999.86	−0.351024
$419$	−5120.31	−0.597002	−0.298501	−	0.954409i	$-0.596487\pi$
$419$	−5120.31	−0.597002	−0.298501	+	0.954409i	$0.596487\pi$
$420$	0	0
$421$	1866.49	0.216074	0.108037	−	0.994147i	$-0.465543\pi$
$421$	1866.49	0.216074	0.108037	+	0.994147i	$0.465543\pi$
$422$	2398.38	0.276662
$423$	0	0
$424$	9537.16	1.09237
$425$	411.874	0.0470090
$426$	0	0
$427$	−2666.33	−0.302185
$428$	−6295.18	−0.710956
$429$	0	0
$430$	−871.581	−0.0977474
$431$	4090.64	0.457168	0.228584	−	0.973524i	$-0.426590\pi$
$431$	4090.64	0.457168	0.228584	+	0.973524i	$0.426590\pi$
$432$	0	0
$433$	633.052	0.0702599	0.0351299	−	0.999383i	$-0.488815\pi$
$433$	633.052	0.0702599	0.0351299	+	0.999383i	$0.488815\pi$
$434$	−90.4623	−0.0100054
$435$	0	0
$436$	−10866.7	−1.19362
$437$	−2775.53	−0.303825
$438$	0	0
$439$	−11306.5	−1.22923	−0.614614	−	0.788828i	$-0.710688\pi$
$439$	−11306.5	−1.22923	−0.614614	+	0.788828i	$0.710688\pi$
$440$	−11248.0	−1.21870
$441$	0	0
$442$	−1219.75	−0.131261
$443$	8281.30	0.888163	0.444082	−	0.895986i	$-0.353530\pi$
$443$	8281.30	0.888163	0.444082	+	0.895986i	$0.353530\pi$
$444$	0	0
$445$	−4198.74	−0.447280
$446$	−3488.26	−0.370345
$447$	0	0
$448$	−388.671	−0.0409887
$449$	−6888.40	−0.724017	−0.362008	−	0.932175i	$-0.617909\pi$
$449$	−6888.40	−0.724017	−0.362008	+	0.932175i	$0.617909\pi$
$450$	0	0
$451$	−12062.4	−1.25941
$452$	−9884.44	−1.02859
$453$	0	0
$454$	4107.59	0.424623
$455$	1994.72	0.205525
$456$	0	0
$457$	4283.60	0.438465	0.219233	−	0.975673i	$-0.429645\pi$
$457$	4283.60	0.438465	0.219233	+	0.975673i	$0.429645\pi$
$458$	−5908.09	−0.602766
$459$	0	0
$460$	−4509.29	−0.457058
$461$	−13778.3	−1.39202	−0.696009	−	0.718033i	$-0.745042\pi$
$461$	−13778.3	−1.39202	−0.696009	+	0.718033i	$0.745042\pi$
$462$	0	0
$463$	5734.53	0.575608	0.287804	−	0.957689i	$-0.407075\pi$
$463$	5734.53	0.575608	0.287804	+	0.957689i	$0.407075\pi$
$464$	634.136	0.0634461
$465$	0	0
$466$	7659.17	0.761383
$467$	8950.97	0.886941	0.443470	−	0.896289i	$-0.353747\pi$
$467$	8950.97	0.886941	0.443470	+	0.896289i	$0.353747\pi$
$468$	0	0
$469$	−1557.72	−0.153366
$470$	−880.746	−0.0864379
$471$	0	0
$472$	15296.9	1.49173
$473$	−3427.75	−0.333210
$474$	0	0
$475$	−680.114	−0.0656964
$476$	−740.203	−0.0712755
$477$	0	0
$478$	−1934.41	−0.185100
$479$	−9681.01	−0.923459	−0.461729	−	0.887021i	$-0.652771\pi$
$479$	−9681.01	−0.923459	−0.461729	+	0.887021i	$0.652771\pi$
$480$	0	0
$481$	−6769.04	−0.641666
$482$	859.833	0.0812538
$483$	0	0
$484$	−11026.0	−1.03550
$485$	−778.204	−0.0728586
$486$	0	0
$487$	8704.66	0.809950	0.404975	−	0.914328i	$-0.367280\pi$
$487$	8704.66	0.809950	0.404975	+	0.914328i	$0.367280\pi$
$488$	−10094.7	−0.936404
$489$	0	0
$490$	4509.49	0.415751
$491$	15595.7	1.43345	0.716725	−	0.697356i	$-0.245640\pi$
$491$	15595.7	1.43345	0.716725	+	0.697356i	$0.245640\pi$
$492$	0	0
$493$	−670.989	−0.0612979
$494$	2014.13	0.183441
$495$	0	0
$496$	287.945	0.0260668
$497$	1383.55	0.124871
$498$	0	0
$499$	−9696.28	−0.869870	−0.434935	−	0.900462i	$-0.643229\pi$
$499$	−9696.28	−0.869870	−0.434935	+	0.900462i	$0.643229\pi$
$500$	−9035.66	−0.808174
$501$	0	0
$502$	−2340.46	−0.208087
$503$	−20949.7	−1.85706	−0.928532	−	0.371253i	$-0.878928\pi$
$503$	−20949.7	−1.85706	−0.928532	+	0.371253i	$0.878928\pi$
$504$	0	0
$505$	−11283.8	−0.994300
$506$	5459.71	0.479671
$507$	0	0
$508$	7377.55	0.644342
$509$	−11274.7	−0.981816	−0.490908	−	0.871211i	$-0.663335\pi$
$509$	−11274.7	−0.981816	−0.490908	+	0.871211i	$0.663335\pi$
$510$	0	0
$511$	4735.46	0.409950
$512$	7613.21	0.657148
$513$	0	0
$514$	4936.62	0.423628
$515$	11324.6	0.968977
$516$	0	0
$517$	−3463.80	−0.294657
$518$	1264.64	0.107268
$519$	0	0
$520$	7551.98	0.636877
$521$	−8675.49	−0.729520	−0.364760	−	0.931102i	$-0.618849\pi$
$521$	−8675.49	−0.729520	−0.364760	+	0.931102i	$0.618849\pi$
$522$	0	0
$523$	−4226.14	−0.353339	−0.176670	−	0.984270i	$-0.556532\pi$
$523$	−4226.14	−0.353339	−0.176670	+	0.984270i	$0.556532\pi$
$524$	−6287.23	−0.524158
$525$	0	0
$526$	−5677.82	−0.470655
$527$	−304.680	−0.0251842
$528$	0	0
$529$	−7115.58	−0.584826
$530$	7007.38	0.574304
$531$	0	0
$532$	1222.27	0.0996095
$533$	8098.78	0.658156
$534$	0	0
$535$	−10674.7	−0.862630
$536$	−5897.49	−0.475248
$537$	0	0
$538$	−8358.56	−0.669820
$539$	17734.9	1.41725
$540$	0	0
$541$	13357.8	1.06154	0.530771	−	0.847515i	$-0.321902\pi$
$541$	13357.8	1.06154	0.530771	+	0.847515i	$0.321902\pi$
$542$	−4386.66	−0.347644
$543$	0	0
$544$	−4390.51	−0.346032
$545$	−18426.5	−1.44827
$546$	0	0
$547$	21671.1	1.69395	0.846974	−	0.531634i	$-0.178422\pi$
$547$	21671.1	1.69395	0.846974	+	0.531634i	$0.178422\pi$
$548$	−7712.78	−0.601229
$549$	0	0
$550$	1337.84	0.103720
$551$	1107.98	0.0856654
$552$	0	0
$553$	6595.79	0.507200
$554$	−4280.64	−0.328280
$555$	0	0
$556$	−2824.95	−0.215476
$557$	−7477.63	−0.568828	−0.284414	−	0.958702i	$-0.591799\pi$
$557$	−7477.63	−0.568828	−0.284414	+	0.958702i	$0.591799\pi$
$558$	0	0
$559$	2301.42	0.174132
$560$	1186.22	0.0895122
$561$	0	0
$562$	−6790.50	−0.509680
$563$	23304.7	1.74454	0.872269	−	0.489026i	$-0.162648\pi$
$563$	23304.7	1.74454	0.872269	+	0.489026i	$0.162648\pi$
$564$	0	0
$565$	−16761.0	−1.24803
$566$	−6236.92	−0.463176
$567$	0	0
$568$	5238.10	0.386947
$569$	14649.1	1.07930	0.539650	−	0.841890i	$-0.318557\pi$
$569$	14649.1	1.07930	0.539650	+	0.841890i	$0.318557\pi$
$570$	0	0
$571$	23164.0	1.69769	0.848846	−	0.528640i	$-0.177298\pi$
$571$	23164.0	1.69769	0.848846	+	0.528640i	$0.177298\pi$
$572$	12869.2	0.940715
$573$	0	0
$574$	−1513.07	−0.110025
$575$	1237.80	0.0897735
$576$	0	0
$577$	7865.97	0.567529	0.283765	−	0.958894i	$-0.408417\pi$
$577$	7865.97	0.567529	0.283765	+	0.958894i	$0.408417\pi$
$578$	−5974.50	−0.429942
$579$	0	0
$580$	1800.10	0.128871
$581$	3652.58	0.260817
$582$	0	0
$583$	27558.6	1.95774
$584$	17928.4	1.27034
$585$	0	0
$586$	−9413.99	−0.663632
$587$	956.182	0.0672332	0.0336166	−	0.999435i	$-0.489297\pi$
$587$	956.182	0.0672332	0.0336166	+	0.999435i	$0.489297\pi$
$588$	0	0
$589$	503.108	0.0351956
$590$	11239.4	0.784266
$591$	0	0
$592$	−4025.40	−0.279465
$593$	16966.0	1.17489	0.587444	−	0.809265i	$-0.300134\pi$
$593$	16966.0	1.17489	0.587444	+	0.809265i	$0.300134\pi$
$594$	0	0
$595$	−1255.16	−0.0864813
$596$	8926.45	0.613492
$597$	0	0
$598$	−3665.69	−0.250671
$599$	6191.41	0.422327	0.211164	−	0.977451i	$-0.432275\pi$
$599$	6191.41	0.422327	0.211164	+	0.977451i	$0.432275\pi$
$600$	0	0
$601$	−2718.54	−0.184512	−0.0922559	−	0.995735i	$-0.529408\pi$
$601$	−2718.54	−0.184512	−0.0922559	+	0.995735i	$0.529408\pi$
$602$	−429.968	−0.0291099
$603$	0	0
$604$	−9428.00	−0.635132
$605$	−18696.7	−1.25641
$606$	0	0
$607$	16825.0	1.12505	0.562524	−	0.826781i	$-0.309830\pi$
$607$	16825.0	1.12505	0.562524	+	0.826781i	$0.309830\pi$
$608$	7249.91	0.483590
$609$	0	0
$610$	−7417.03	−0.492306
$611$	2325.62	0.153985
$612$	0	0
$613$	−20175.1	−1.32930	−0.664652	−	0.747153i	$-0.731420\pi$
$613$	−20175.1	−1.32930	−0.664652	+	0.747153i	$0.731420\pi$
$614$	8689.63	0.571148
$615$	0	0
$616$	−5548.84	−0.362937
$617$	−11310.6	−0.738004	−0.369002	−	0.929429i	$-0.620301\pi$
$617$	−11310.6	−0.738004	−0.369002	+	0.929429i	$0.620301\pi$
$618$	0	0
$619$	−17059.9	−1.10775	−0.553873	−	0.832601i	$-0.686851\pi$
$619$	−17059.9	−1.10775	−0.553873	+	0.832601i	$0.686851\pi$
$620$	817.380	0.0529464
$621$	0	0
$622$	−9712.56	−0.626106
$623$	−2071.32	−0.133203
$624$	0	0
$625$	−13144.7	−0.841262
$626$	1895.53	0.121023
$627$	0	0
$628$	19669.2	1.24982
$629$	4259.34	0.270002
$630$	0	0
$631$	−13186.3	−0.831916	−0.415958	−	0.909384i	$-0.636554\pi$
$631$	−13186.3	−0.831916	−0.415958	+	0.909384i	$0.636554\pi$
$632$	24971.5	1.57170
$633$	0	0
$634$	11218.3	0.702738
$635$	12510.0	0.781805
$636$	0	0
$637$	−11907.4	−0.740638
$638$	−2179.50	−0.135246
$639$	0	0
$640$	14323.7	0.884677
$641$	−16362.0	−1.00820	−0.504102	−	0.863644i	$-0.668177\pi$
$641$	−16362.0	−1.00820	−0.504102	+	0.863644i	$0.668177\pi$
$642$	0	0
$643$	−28044.9	−1.72004	−0.860019	−	0.510262i	$-0.829548\pi$
$643$	−28044.9	−1.72004	−0.860019	+	0.510262i	$0.829548\pi$
$644$	−2224.52	−0.136115
$645$	0	0
$646$	−1267.37	−0.0771888
$647$	−21247.7	−1.29109	−0.645543	−	0.763724i	$-0.723369\pi$
$647$	−21247.7	−1.29109	−0.645543	+	0.763724i	$0.723369\pi$
$648$	0	0
$649$	44202.1	2.67347
$650$	−898.238	−0.0542028
$651$	0	0
$652$	−5796.24	−0.348157
$653$	−1259.86	−0.0755007	−0.0377504	−	0.999287i	$-0.512019\pi$
$653$	−1259.86	−0.0755007	−0.0377504	+	0.999287i	$0.512019\pi$
$654$	0	0
$655$	−10661.2	−0.635981
$656$	4816.17	0.286646
$657$	0	0
$658$	−434.489	−0.0257419
$659$	12046.7	0.712098	0.356049	−	0.934467i	$-0.384124\pi$
$659$	12046.7	0.712098	0.356049	+	0.934467i	$0.384124\pi$
$660$	0	0
$661$	13108.1	0.771324	0.385662	−	0.922640i	$-0.373973\pi$
$661$	13108.1	0.771324	0.385662	+	0.922640i	$0.373973\pi$
$662$	−13258.0	−0.778379
$663$	0	0
$664$	13828.6	0.808213
$665$	2072.60	0.120860
$666$	0	0
$667$	−2016.51	−0.117061
$668$	4195.91	0.243031
$669$	0	0
$670$	−4333.15	−0.249857
$671$	−29169.7	−1.67822
$672$	0	0
$673$	2743.65	0.157147	0.0785734	−	0.996908i	$-0.474963\pi$
$673$	2743.65	0.157147	0.0785734	+	0.996908i	$0.474963\pi$
$674$	−6801.06	−0.388675
$675$	0	0
$676$	4798.21	0.272998
$677$	−25004.0	−1.41947	−0.709735	−	0.704468i	$-0.751186\pi$
$677$	−25004.0	−1.41947	−0.709735	+	0.704468i	$0.751186\pi$
$678$	0	0
$679$	−383.903	−0.0216979
$680$	−4752.00	−0.267987
$681$	0	0
$682$	−989.657	−0.0555659
$683$	4846.23	0.271502	0.135751	−	0.990743i	$-0.456655\pi$
$683$	4846.23	0.271502	0.135751	+	0.990743i	$0.456655\pi$
$684$	0	0
$685$	−13078.5	−0.729494
$686$	4633.08	0.257860
$687$	0	0
$688$	1368.61	0.0758395
$689$	−18503.1	−1.02309
$690$	0	0
$691$	3484.58	0.191837	0.0959187	−	0.995389i	$-0.469421\pi$
$691$	3484.58	0.191837	0.0959187	+	0.995389i	$0.469421\pi$
$692$	−13535.5	−0.743560
$693$	0	0
$694$	1393.70	0.0762305
$695$	−4790.25	−0.261445
$696$	0	0
$697$	−5096.07	−0.276940
$698$	16685.0	0.904780
$699$	0	0
$700$	−545.095	−0.0294324
$701$	−15701.4	−0.845981	−0.422991	−	0.906134i	$-0.639020\pi$
$701$	−15701.4	−0.845981	−0.422991	+	0.906134i	$0.639020\pi$
$702$	0	0
$703$	−7033.32	−0.377335
$704$	−4252.05	−0.227635
$705$	0	0
$706$	−5813.10	−0.309885
$707$	−5566.50	−0.296110
$708$	0	0
$709$	15643.4	0.828634	0.414317	−	0.910133i	$-0.364020\pi$
$709$	15643.4	0.828634	0.414317	+	0.910133i	$0.364020\pi$
$710$	3848.67	0.203434
$711$	0	0
$712$	−7841.98	−0.412768
$713$	−915.649	−0.0480944
$714$	0	0
$715$	21822.2	1.14141
$716$	18492.5	0.965222
$717$	0	0
$718$	710.759	0.0369433
$719$	6964.13	0.361222	0.180611	−	0.983555i	$-0.442193\pi$
$719$	6964.13	0.361222	0.180611	+	0.983555i	$0.442193\pi$
$720$	0	0
$721$	5586.66	0.288569
$722$	−7319.71	−0.377301
$723$	0	0
$724$	−2394.41	−0.122911
$725$	−494.125	−0.0253122
$726$	0	0
$727$	−14207.2	−0.724782	−0.362391	−	0.932026i	$-0.618039\pi$
$727$	−14207.2	−0.724782	−0.362391	+	0.932026i	$0.618039\pi$
$728$	3725.53	0.189667
$729$	0	0
$730$	13172.8	0.667871
$731$	−1448.14	−0.0732716
$732$	0	0
$733$	26530.5	1.33687	0.668437	−	0.743769i	$-0.266964\pi$
$733$	26530.5	1.33687	0.668437	+	0.743769i	$0.266964\pi$
$734$	−6334.85	−0.318561
$735$	0	0
$736$	−13194.7	−0.660821
$737$	−17041.4	−0.851735
$738$	0	0
$739$	−5683.47	−0.282909	−0.141455	−	0.989945i	$-0.545178\pi$
$739$	−5683.47	−0.282909	−0.141455	+	0.989945i	$0.545178\pi$
$740$	−11426.8	−0.567643
$741$	0	0
$742$	3456.87	0.171032
$743$	15568.6	0.768715	0.384358	−	0.923184i	$-0.374423\pi$
$743$	15568.6	0.768715	0.384358	+	0.923184i	$0.374423\pi$
$744$	0	0
$745$	15136.5	0.744374
$746$	6539.50	0.320949
$747$	0	0
$748$	−8097.82	−0.395836
$749$	−5266.02	−0.256898
$750$	0	0
$751$	−8261.64	−0.401427	−0.200713	−	0.979650i	$-0.564326\pi$
$751$	−8261.64	−0.401427	−0.200713	+	0.979650i	$0.564326\pi$
$752$	1383.00	0.0670648
$753$	0	0
$754$	1463.33	0.0706783
$755$	−15987.0	−0.770630
$756$	0	0
$757$	−13381.5	−0.642481	−0.321240	−	0.946998i	$-0.604100\pi$
$757$	−13381.5	−0.642481	−0.321240	+	0.946998i	$0.604100\pi$
$758$	−2745.01	−0.131535
$759$	0	0
$760$	7846.82	0.374519
$761$	−5449.84	−0.259601	−0.129801	−	0.991540i	$-0.541434\pi$
$761$	−5449.84	−0.259601	−0.129801	+	0.991540i	$0.541434\pi$
$762$	0	0
$763$	−9090.15	−0.431305
$764$	21320.8	1.00963
$765$	0	0
$766$	−1358.96	−0.0641009
$767$	−29677.6	−1.39713
$768$	0	0
$769$	−19364.0	−0.908039	−0.454020	−	0.890992i	$-0.650010\pi$
$769$	−19364.0	−0.908039	−0.454020	+	0.890992i	$0.650010\pi$
$770$	−4076.98	−0.190811
$771$	0	0
$772$	−13549.2	−0.631668
$773$	1865.54	0.0868033	0.0434017	−	0.999058i	$-0.486180\pi$
$773$	1865.54	0.0868033	0.0434017	+	0.999058i	$0.486180\pi$
$774$	0	0
$775$	−224.370	−0.0103995
$776$	−1453.45	−0.0672369
$777$	0	0
$778$	−554.965	−0.0255739
$779$	8414.98	0.387032
$780$	0	0
$781$	15136.0	0.693483
$782$	2306.59	0.105478
$783$	0	0
$784$	−7081.05	−0.322570
$785$	33352.8	1.51645
$786$	0	0
$787$	−19207.3	−0.869970	−0.434985	−	0.900438i	$-0.643246\pi$
$787$	−19207.3	−0.869970	−0.434985	+	0.900438i	$0.643246\pi$
$788$	−24315.2	−1.09923
$789$	0	0
$790$	18347.7	0.826307
$791$	−8268.50	−0.371674
$792$	0	0
$793$	19584.7	0.877017
$794$	4006.53	0.179076
$795$	0	0
$796$	9513.78	0.423627
$797$	−186.074	−0.00826988	−0.00413494	−	0.999991i	$-0.501316\pi$
$797$	−186.074	−0.00826988	−0.00413494	+	0.999991i	$0.501316\pi$
$798$	0	0
$799$	−1463.37	−0.0647940
$800$	−3233.23	−0.142890
$801$	0	0
$802$	13978.3	0.615452
$803$	51805.9	2.27670
$804$	0	0
$805$	−3772.10	−0.165154
$806$	664.463	0.0290381
$807$	0	0
$808$	−21074.7	−0.917580
$809$	−5903.09	−0.256541	−0.128270	−	0.991739i	$-0.540943\pi$
$809$	−5903.09	−0.256541	−0.128270	+	0.991739i	$0.540943\pi$
$810$	0	0
$811$	23111.0	1.00066	0.500331	−	0.865834i	$-0.333212\pi$
$811$	23111.0	1.00066	0.500331	+	0.865834i	$0.333212\pi$
$812$	888.022	0.0383787
$813$	0	0
$814$	13835.2	0.595727
$815$	−9828.64	−0.422432
$816$	0	0
$817$	2391.27	0.102399
$818$	9488.29	0.405563
$819$	0	0
$820$	13671.5	0.582231
$821$	9644.29	0.409973	0.204987	−	0.978765i	$-0.434285\pi$
$821$	9644.29	0.409973	0.204987	+	0.978765i	$0.434285\pi$
$822$	0	0
$823$	−33573.4	−1.42199	−0.710994	−	0.703198i	$-0.751755\pi$
$823$	−33573.4	−1.42199	−0.710994	+	0.703198i	$0.751755\pi$
$824$	21151.0	0.894211
$825$	0	0
$826$	5544.59	0.233560
$827$	25916.1	1.08971	0.544855	−	0.838530i	$-0.316585\pi$
$827$	25916.1	1.08971	0.544855	+	0.838530i	$0.316585\pi$
$828$	0	0
$829$	−28650.6	−1.20033	−0.600166	−	0.799876i	$-0.704899\pi$
$829$	−28650.6	−1.20033	−0.600166	+	0.799876i	$0.704899\pi$
$830$	10160.5	0.424911
$831$	0	0
$832$	2854.86	0.118960
$833$	7492.58	0.311647
$834$	0	0
$835$	7114.97	0.294878
$836$	13371.7	0.553192
$837$	0	0
$838$	−7026.51	−0.289650
$839$	−712.960	−0.0293374	−0.0146687	−	0.999892i	$-0.504669\pi$
$839$	−712.960	−0.0293374	−0.0146687	+	0.999892i	$0.504669\pi$
$840$	0	0
$841$	−23584.0	−0.966994
$842$	2561.35	0.104834
$843$	0	0
$844$	−10690.6	−0.436002
$845$	8136.29	0.331239
$846$	0	0
$847$	−9223.44	−0.374169
$848$	−11003.4	−0.445587
$849$	0	0
$850$	565.207	0.0228076
$851$	12800.5	0.515625
$852$	0	0
$853$	−30367.2	−1.21894	−0.609469	−	0.792810i	$-0.708617\pi$
$853$	−30367.2	−1.21894	−0.609469	+	0.792810i	$0.708617\pi$
$854$	−3658.96	−0.146612
$855$	0	0
$856$	−19937.1	−0.796069
$857$	−9080.70	−0.361950	−0.180975	−	0.983488i	$-0.557925\pi$
$857$	−9080.70	−0.361950	−0.180975	+	0.983488i	$0.557925\pi$
$858$	0	0
$859$	26160.2	1.03909	0.519543	−	0.854444i	$-0.326102\pi$
$859$	26160.2	1.03909	0.519543	+	0.854444i	$0.326102\pi$
$860$	3885.01	0.154044
$861$	0	0
$862$	5613.51	0.221806
$863$	40102.0	1.58180	0.790898	−	0.611949i	$-0.209614\pi$
$863$	40102.0	1.58180	0.790898	+	0.611949i	$0.209614\pi$
$864$	0	0
$865$	−22952.1	−0.902189
$866$	868.725	0.0340883
$867$	0	0
$868$	403.229	0.0157678
$869$	72157.9	2.81679
$870$	0	0
$871$	11441.7	0.445108
$872$	−34415.2	−1.33652
$873$	0	0
$874$	−3808.80	−0.147408
$875$	−7558.48	−0.292027
$876$	0	0
$877$	25252.2	0.972299	0.486149	−	0.873876i	$-0.338401\pi$
$877$	25252.2	0.972299	0.486149	+	0.873876i	$0.338401\pi$
$878$	−15515.7	−0.596390
$879$	0	0
$880$	12977.2	0.497116
$881$	2049.26	0.0783670	0.0391835	−	0.999232i	$-0.487524\pi$
$881$	2049.26	0.0783670	0.0391835	+	0.999232i	$0.487524\pi$
$882$	0	0
$883$	39413.4	1.50211	0.751057	−	0.660237i	$-0.229544\pi$
$883$	39413.4	1.50211	0.751057	+	0.660237i	$0.229544\pi$
$884$	5436.94	0.206860
$885$	0	0
$886$	11364.3	0.430914
$887$	−36968.5	−1.39941	−0.699707	−	0.714430i	$-0.746686\pi$
$887$	−36968.5	−1.39941	−0.699707	+	0.714430i	$0.746686\pi$
$888$	0	0
$889$	6171.44	0.232827
$890$	−5761.86	−0.217009
$891$	0	0
$892$	15548.7	0.583641
$893$	2416.42	0.0905514
$894$	0	0
$895$	31357.7	1.17114
$896$	7066.15	0.263464
$897$	0	0
$898$	−9452.82	−0.351275
$899$	365.525	0.0135605
$900$	0	0
$901$	11642.9	0.430499
$902$	−16553.0	−0.611036
$903$	0	0
$904$	−31304.4	−1.15174
$905$	−4060.17	−0.149132
$906$	0	0
$907$	2710.62	0.0992334	0.0496167	−	0.998768i	$-0.484200\pi$
$907$	2710.62	0.0992334	0.0496167	+	0.998768i	$0.484200\pi$
$908$	−18309.3	−0.669180
$909$	0	0
$910$	2737.32	0.0997156
$911$	−22996.6	−0.836345	−0.418172	−	0.908368i	$-0.637329\pi$
$911$	−22996.6	−0.836345	−0.418172	+	0.908368i	$0.637329\pi$
$912$	0	0
$913$	39959.2	1.44847
$914$	5878.31	0.212732
$915$	0	0
$916$	26334.9	0.949922
$917$	−5259.38	−0.189400
$918$	0	0
$919$	−39103.8	−1.40361	−0.701804	−	0.712370i	$-0.747622\pi$
$919$	−39103.8	−1.40361	−0.701804	+	0.712370i	$0.747622\pi$
$920$	−14281.1	−0.511776
$921$	0	0
$922$	−18907.7	−0.675372
$923$	−10162.5	−0.362407
$924$	0	0
$925$	3136.64	0.111494
$926$	7869.39	0.279270
$927$	0	0
$928$	5267.30	0.186323
$929$	35954.6	1.26979	0.634894	−	0.772600i	$-0.281044\pi$
$929$	35954.6	1.26979	0.634894	+	0.772600i	$0.281044\pi$
$930$	0	0
$931$	−12372.2	−0.435536
$932$	−34140.2	−1.19989
$933$	0	0
$934$	12283.2	0.430321
$935$	−13731.4	−0.480283
$936$	0	0
$937$	−7263.94	−0.253258	−0.126629	−	0.991950i	$-0.540416\pi$
$937$	−7263.94	−0.253258	−0.126629	+	0.991950i	$0.540416\pi$
$938$	−2137.63	−0.0744094
$939$	0	0
$940$	3925.86	0.136221
$941$	−7478.91	−0.259092	−0.129546	−	0.991573i	$-0.541352\pi$
$941$	−7478.91	−0.259092	−0.129546	+	0.991573i	$0.541352\pi$
$942$	0	0
$943$	−15315.1	−0.528875
$944$	−17648.7	−0.608490
$945$	0	0
$946$	−4703.84	−0.161665
$947$	−13491.4	−0.462947	−0.231473	−	0.972841i	$-0.574355\pi$
$947$	−13491.4	−0.462947	−0.231473	+	0.972841i	$0.574355\pi$
$948$	0	0
$949$	−34782.9	−1.18978
$950$	−933.308	−0.0318742
$951$	0	0
$952$	−2344.25	−0.0798085
$953$	13981.6	0.475246	0.237623	−	0.971357i	$-0.423632\pi$
$953$	13981.6	0.475246	0.237623	+	0.971357i	$0.423632\pi$
$954$	0	0
$955$	36153.5	1.22503
$956$	8622.48	0.291706
$957$	0	0
$958$	−13285.1	−0.448039
$959$	−6451.87	−0.217249
$960$	0	0
$961$	−29625.0	−0.994429
$962$	−9289.02	−0.311320
$963$	0	0
$964$	−3832.64	−0.128051
$965$	−22975.3	−0.766427
$966$	0	0
$967$	−9081.47	−0.302007	−0.151003	−	0.988533i	$-0.548250\pi$
$967$	−9081.47	−0.302007	−0.151003	+	0.988533i	$0.548250\pi$
$968$	−34919.8	−1.15947
$969$	0	0
$970$	−1067.92	−0.0353492
$971$	9709.13	0.320887	0.160443	−	0.987045i	$-0.448708\pi$
$971$	9709.13	0.320887	0.160443	+	0.987045i	$0.448708\pi$
$972$	0	0
$973$	−2363.12	−0.0778604
$974$	11945.2	0.392967
$975$	0	0
$976$	11646.6	0.381967
$977$	−10854.9	−0.355455	−0.177727	−	0.984080i	$-0.556875\pi$
$977$	−10854.9	−0.355455	−0.177727	+	0.984080i	$0.556875\pi$
$978$	0	0
$979$	−22660.2	−0.739759
$980$	−20100.7	−0.655198
$981$	0	0
$982$	21401.7	0.695474
$983$	7510.10	0.243678	0.121839	−	0.992550i	$-0.461121\pi$
$983$	7510.10	0.243678	0.121839	+	0.992550i	$0.461121\pi$
$984$	0	0
$985$	−41231.0	−1.33373
$986$	−920.786	−0.0297402
$987$	0	0
$988$	−8977.84	−0.289092
$989$	−4352.08	−0.139927
$990$	0	0
$991$	46125.6	1.47854	0.739268	−	0.673412i	$-0.235172\pi$
$991$	46125.6	1.47854	0.739268	+	0.673412i	$0.235172\pi$
$992$	2391.75	0.0765506
$993$	0	0
$994$	1898.62	0.0605841
$995$	16132.4	0.514003
$996$	0	0
$997$	45350.1	1.44057	0.720287	−	0.693677i	$-0.244010\pi$
$997$	45350.1	1.44057	0.720287	+	0.693677i	$0.244010\pi$
$998$	−13306.0	−0.422039
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.4.a.a.1.2		2
3.2	odd	2		81.4.a.d.1.1		2
4.3	odd	2		1296.4.a.i.1.1		2
5.4	even	2		2025.4.a.n.1.1		2
9.2	odd	6		9.4.c.a.4.2	✓	4
9.4	even	3		27.4.c.a.19.1		4
9.5	odd	6		9.4.c.a.7.2	yes	4
9.7	even	3		27.4.c.a.10.1		4
12.11	even	2		1296.4.a.u.1.2		2
15.14	odd	2		2025.4.a.g.1.2		2
36.7	odd	6		432.4.i.c.145.2		4
36.11	even	6		144.4.i.c.49.2		4
36.23	even	6		144.4.i.c.97.2		4
36.31	odd	6		432.4.i.c.289.2		4
45.2	even	12		225.4.k.b.49.2		8
45.14	odd	6		225.4.e.b.151.1		4
45.23	even	12		225.4.k.b.124.2		8
45.29	odd	6		225.4.e.b.76.1		4
45.32	even	12		225.4.k.b.124.3		8
45.38	even	12		225.4.k.b.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.c.a.4.2	✓	4	9.2	odd	6
9.4.c.a.7.2	yes	4	9.5	odd	6
27.4.c.a.10.1		4	9.7	even	3
27.4.c.a.19.1		4	9.4	even	3
81.4.a.a.1.2		2	1.1	even	1	trivial
81.4.a.d.1.1		2	3.2	odd	2
144.4.i.c.49.2		4	36.11	even	6
144.4.i.c.97.2		4	36.23	even	6
225.4.e.b.76.1		4	45.29	odd	6
225.4.e.b.151.1		4	45.14	odd	6
225.4.k.b.49.2		8	45.2	even	12
225.4.k.b.49.3		8	45.38	even	12
225.4.k.b.124.2		8	45.23	even	12
225.4.k.b.124.3		8	45.32	even	12
432.4.i.c.145.2		4	36.7	odd	6
432.4.i.c.289.2		4	36.31	odd	6
1296.4.a.i.1.1		2	4.3	odd	2
1296.4.a.u.1.2		2	12.11	even	2
2025.4.a.g.1.2		2	15.14	odd	2
2025.4.a.n.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37228 q^{2} -6.11684 q^{4} -10.3723 q^{5} -5.11684 q^{7} -19.3723 q^{8} +O(q^{10})\)	\(q+1.37228 q^{2} -6.11684 q^{4} -10.3723 q^{5} -5.11684 q^{7} -19.3723 q^{8} -14.2337 q^{10} -55.9783 q^{11} +37.5842 q^{13} -7.02175 q^{14} +22.3505 q^{16} -23.6495 q^{17} +39.0516 q^{19} +63.4456 q^{20} -76.8179 q^{22} -71.0733 q^{23} -17.4158 q^{25} +51.5761 q^{26} +31.2989 q^{28} +28.3723 q^{29} +12.8832 q^{31} +185.649 q^{32} -32.4537 q^{34} +53.0733 q^{35} -180.103 q^{37} +53.5898 q^{38} +200.935 q^{40} +215.484 q^{41} +61.2337 q^{43} +342.410 q^{44} -97.5326 q^{46} +61.8776 q^{47} -316.818 q^{49} -23.8993 q^{50} -229.897 q^{52} -492.310 q^{53} +580.622 q^{55} +99.1249 q^{56} +38.9348 q^{58} -789.630 q^{59} +521.090 q^{61} +17.6793 q^{62} +75.9590 q^{64} -389.834 q^{65} +304.429 q^{67} +144.660 q^{68} +72.8316 q^{70} -270.391 q^{71} -925.464 q^{73} -247.152 q^{74} -238.873 q^{76} +286.432 q^{77} -1289.03 q^{79} -231.826 q^{80} +295.704 q^{82} -713.834 q^{83} +245.299 q^{85} +84.0298 q^{86} +1084.43 q^{88} +404.804 q^{89} -192.313 q^{91} +434.745 q^{92} +84.9135 q^{94} -405.054 q^{95} +75.0273 q^{97} -434.763 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 5 q^{4} - 15 q^{5} + 7 q^{7} - 33 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 5 * q^4 - 15 * q^5 + 7 * q^7 - 33 * q^8	\( 2 q - 3 q^{2} + 5 q^{4} - 15 q^{5} + 7 q^{7} - 33 q^{8} + 6 q^{10} - 66 q^{11} - 11 q^{13} - 60 q^{14} - 7 q^{16} - 99 q^{17} - 77 q^{19} + 12 q^{20} - 33 q^{22} - 33 q^{23} - 121 q^{25} + 264 q^{26} + 166 q^{28} + 51 q^{29} + 43 q^{31} + 423 q^{32} + 297 q^{34} - 3 q^{35} - 50 q^{37} + 561 q^{38} + 264 q^{40} - 132 q^{41} + 88 q^{43} + 231 q^{44} - 264 q^{46} - 399 q^{47} - 513 q^{49} + 429 q^{50} - 770 q^{52} - 54 q^{53} + 627 q^{55} - 66 q^{56} - 60 q^{58} - 798 q^{59} + 439 q^{61} - 114 q^{62} - 727 q^{64} - 165 q^{65} + 988 q^{67} - 693 q^{68} + 318 q^{70} - 1368 q^{71} - 455 q^{73} - 816 q^{74} - 1529 q^{76} + 165 q^{77} - 803 q^{79} - 96 q^{80} + 1815 q^{82} - 813 q^{83} + 594 q^{85} - 33 q^{86} + 1221 q^{88} + 396 q^{89} - 781 q^{91} + 858 q^{92} + 2100 q^{94} + 132 q^{95} + 736 q^{97} + 423 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 5 * q^4 - 15 * q^5 + 7 * q^7 - 33 * q^8 + 6 * q^10 - 66 * q^11 - 11 * q^13 - 60 * q^14 - 7 * q^16 - 99 * q^17 - 77 * q^19 + 12 * q^20 - 33 * q^22 - 33 * q^23 - 121 * q^25 + 264 * q^26 + 166 * q^28 + 51 * q^29 + 43 * q^31 + 423 * q^32 + 297 * q^34 - 3 * q^35 - 50 * q^37 + 561 * q^38 + 264 * q^40 - 132 * q^41 + 88 * q^43 + 231 * q^44 - 264 * q^46 - 399 * q^47 - 513 * q^49 + 429 * q^50 - 770 * q^52 - 54 * q^53 + 627 * q^55 - 66 * q^56 - 60 * q^58 - 798 * q^59 + 439 * q^61 - 114 * q^62 - 727 * q^64 - 165 * q^65 + 988 * q^67 - 693 * q^68 + 318 * q^70 - 1368 * q^71 - 455 * q^73 - 816 * q^74 - 1529 * q^76 + 165 * q^77 - 803 * q^79 - 96 * q^80 + 1815 * q^82 - 813 * q^83 + 594 * q^85 - 33 * q^86 + 1221 * q^88 + 396 * q^89 - 781 * q^91 + 858 * q^92 + 2100 * q^94 + 132 * q^95 + 736 * q^97 + 423 * q^98

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