LMFDB - Embedded newform 1296.4.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
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:	$\Z[a_1, \ldots, a_{5}]$
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	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3723 q^{5} +5.11684 q^{7} +O(q^{10})$	$q+10.3723 q^{5} +5.11684 q^{7} -55.9783 q^{11} +37.5842 q^{13} +23.6495 q^{17} -39.0516 q^{19} -71.0733 q^{23} -17.4158 q^{25} -28.3723 q^{29} -12.8832 q^{31} +53.0733 q^{35} -180.103 q^{37} -215.484 q^{41} -61.2337 q^{43} +61.8776 q^{47} -316.818 q^{49} +492.310 q^{53} -580.622 q^{55} -789.630 q^{59} +521.090 q^{61} +389.834 q^{65} -304.429 q^{67} -270.391 q^{71} -925.464 q^{73} -286.432 q^{77} +1289.03 q^{79} -713.834 q^{83} +245.299 q^{85} -404.804 q^{89} +192.313 q^{91} -405.054 q^{95} +75.0273 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 15 q^{5} - 7 q^{7}+O(q^{10}) $ 2 * q + 15 * q^5 - 7 * q^7	$ 2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97}+O(q^{100}) $ 2 * q + 15 * q^5 - 7 * q^7 - 66 * q^11 - 11 * q^13 + 99 * q^17 + 77 * q^19 - 33 * q^23 - 121 * q^25 - 51 * q^29 - 43 * q^31 - 3 * q^35 - 50 * q^37 + 132 * q^41 - 88 * q^43 - 399 * q^47 - 513 * q^49 + 54 * q^53 - 627 * q^55 - 798 * q^59 + 439 * q^61 + 165 * q^65 - 988 * q^67 - 1368 * q^71 - 455 * q^73 - 165 * q^77 + 803 * q^79 - 813 * q^83 + 594 * q^85 - 396 * q^89 + 781 * q^91 + 132 * q^95 + 736 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	10.3723	0.927725	0.463863	−	0.885907i	$-0.346463\pi$
$5$	10.3723	0.927725	0.463863	+	0.885907i	$0.346463\pi$
$6$	0	0
$7$	5.11684	0.276284	0.138142	−	0.990412i	$-0.455887\pi$
$7$	5.11684	0.276284	0.138142	+	0.990412i	$0.455887\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−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	$0.368690\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.6495	0.337402	0.168701	−	0.985667i	$-0.446043\pi$
$17$	23.6495	0.337402	0.168701	+	0.985667i	$0.446043\pi$
$18$	0	0
$19$	−39.0516	−0.471529	−0.235764	−	0.971810i	$-0.575759\pi$
$19$	−39.0516	−0.471529	−0.235764	+	0.971810i	$0.575759\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−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
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−28.3723	−0.181676	−0.0908379	−	0.995866i	$-0.528955\pi$
$29$	−28.3723	−0.181676	−0.0908379	+	0.995866i	$0.528955\pi$
$30$	0	0
$31$	−12.8832	−0.0746414	−0.0373207	−	0.999303i	$-0.511882\pi$
$31$	−12.8832	−0.0746414	−0.0373207	+	0.999303i	$0.511882\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	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	$0.631031\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−215.484	−0.820802	−0.410401	−	0.911905i	$-0.634611\pi$
$41$	−215.484	−0.820802	−0.410401	+	0.911905i	$0.634611\pi$
$42$	0	0
$43$	−61.2337	−0.217164	−0.108582	−	0.994087i	$-0.534631\pi$
$43$	−61.2337	−0.217164	−0.108582	+	0.994087i	$0.534631\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	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
$50$	0	0
$51$	0	0
$52$	0	0
$53$	492.310	1.27592	0.637962	−	0.770068i	$-0.279778\pi$
$53$	492.310	1.27592	0.637962	+	0.770068i	$0.279778\pi$
$54$	0	0
$55$	−580.622	−1.42347
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−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	$0.315818\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	389.834	0.743892
$66$	0	0
$67$	−304.429	−0.555104	−0.277552	−	0.960711i	$-0.589523\pi$
$67$	−304.429	−0.555104	−0.277552	+	0.960711i	$0.589523\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−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	$0.766075\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−286.432	−0.423921
$78$	0	0
$79$	1289.03	1.83579	0.917897	−	0.396818i	$-0.129886\pi$
$79$	1289.03	1.83579	0.917897	+	0.396818i	$0.129886\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−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
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−404.804	−0.482125	−0.241063	−	0.970510i	$-0.577496\pi$
$89$	−404.804	−0.482125	−0.241063	+	0.970510i	$0.577496\pi$
$90$	0	0
$91$	192.313	0.221537
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−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	$0.487498\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1087.88	−1.07176	−0.535881	−	0.844294i	$-0.680020\pi$
$101$	−1087.88	−1.07176	−0.535881	+	0.844294i	$0.680020\pi$
$102$	0	0
$103$	1091.82	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$103$	1091.82	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	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	$0.214940\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1615.94	−1.34526	−0.672631	−	0.739978i	$-0.734836\pi$
$113$	−1615.94	−1.34526	−0.672631	+	0.739978i	$0.734836\pi$
$114$	0	0
$115$	−737.193	−0.597770
$116$	0	0
$117$	0	0
$118$	0	0
$119$	121.011	0.0932187
$120$	0	0
$121$	1802.56	1.35429
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1477.18	−1.05698
$126$	0	0
$127$	1206.10	0.842711	0.421356	−	0.906895i	$-0.361554\pi$
$127$	1206.10	0.842711	0.421356	+	0.906895i	$0.361554\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	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
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1260.91	−0.786326	−0.393163	−	0.919469i	$-0.628619\pi$
$137$	−1260.91	−0.786326	−0.393163	+	0.919469i	$0.628619\pi$
$138$	0	0
$139$	−461.832	−0.281813	−0.140907	−	0.990023i	$-0.545002\pi$
$139$	−461.832	−0.281813	−0.140907	+	0.990023i	$0.545002\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2103.90	−1.23033
$144$	0	0
$145$	−294.285	−0.168545
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1459.32	0.802365	0.401182	−	0.915998i	$-0.368599\pi$
$149$	1459.32	0.802365	0.401182	+	0.915998i	$0.368599\pi$
$150$	0	0
$151$	−1541.32	−0.830666	−0.415333	−	0.909669i	$-0.636335\pi$
$151$	−1541.32	−0.830666	−0.415333	+	0.909669i	$0.636335\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−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	$0.804527\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−363.671	−0.178021
$162$	0	0
$163$	−947.587	−0.455342	−0.227671	−	0.973738i	$-0.573111\pi$
$163$	−947.587	−0.455342	−0.227671	+	0.973738i	$0.573111\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−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
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2212.83	−0.972475	−0.486237	−	0.873827i	$-0.661631\pi$
$173$	−2212.83	−0.972475	−0.486237	+	0.873827i	$0.661631\pi$
$174$	0	0
$175$	−89.1138	−0.0384936
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−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	$0.474388\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1868.08	−0.742400
$186$	0	0
$187$	−1323.86	−0.517700
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−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	$0.364457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3975.11	−1.43764	−0.718820	−	0.695196i	$-0.755318\pi$
$197$	−3975.11	−1.43764	−0.718820	+	0.695196i	$0.755318\pi$
$198$	0	0
$199$	1555.34	0.554046	0.277023	−	0.960863i	$-0.410652\pi$
$199$	1555.34	0.554046	0.277023	+	0.960863i	$0.410652\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−145.177	−0.0501941
$204$	0	0
$205$	−2235.06	−0.761479
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2186.04	0.723500
$210$	0	0
$211$	−1747.73	−0.570231	−0.285115	−	0.958493i	$-0.592032\pi$
$211$	−1747.73	−0.570231	−0.285115	+	0.958493i	$0.592032\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−635.133	−0.201468
$216$	0	0
$217$	−65.9211	−0.0206222
$218$	0	0
$219$	0	0
$220$	0	0
$221$	888.847	0.270544
$222$	0	0
$223$	2541.94	0.763323	0.381662	−	0.924302i	$-0.375352\pi$
$223$	2541.94	0.763323	0.381662	+	0.924302i	$0.375352\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	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	$0.713348\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5581.34	−1.56930	−0.784648	−	0.619942i	$-0.787156\pi$
$233$	−5581.34	−1.56930	−0.784648	+	0.619942i	$0.787156\pi$
$234$	0	0
$235$	641.812	0.178158
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−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	$0.473315\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3286.12	−0.856909
$246$	0	0
$247$	−1467.72	−0.378093
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−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
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3597.38	−0.873146	−0.436573	−	0.899669i	$-0.643808\pi$
$257$	−3597.38	−0.873146	−0.436573	+	0.899669i	$0.643808\pi$
$258$	0	0
$259$	−921.560	−0.221092
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−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
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6090.99	1.38057	0.690287	−	0.723536i	$-0.257484\pi$
$269$	6090.99	1.38057	0.690287	+	0.723536i	$0.257484\pi$
$270$	0	0
$271$	3196.62	0.716534	0.358267	−	0.933619i	$-0.383368\pi$
$271$	3196.62	0.716534	0.358267	+	0.933619i	$0.383368\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	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	$0.609856\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4948.33	1.05051	0.525254	−	0.850946i	$-0.323970\pi$
$281$	4948.33	1.05051	0.525254	+	0.850946i	$0.323970\pi$
$282$	0	0
$283$	4544.93	0.954658	0.477329	−	0.878725i	$-0.341605\pi$
$283$	4544.93	0.954658	0.477329	+	0.878725i	$0.341605\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1102.60	−0.226774
$288$	0	0
$289$	−4353.70	−0.886160
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6860.11	1.36782	0.683911	−	0.729566i	$-0.260278\pi$
$293$	6860.11	1.36782	0.683911	+	0.729566i	$0.260278\pi$
$294$	0	0
$295$	−8190.27	−1.61646
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2671.24	−0.516661
$300$	0	0
$301$	−313.323	−0.0599988
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5404.89	1.01470
$306$	0	0
$307$	−6332.25	−1.17720	−0.588600	−	0.808424i	$-0.700321\pi$
$307$	−6332.25	−1.17720	−0.588600	+	0.808424i	$0.700321\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−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	$0.460196\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8174.93	−1.44842	−0.724211	−	0.689578i	$-0.757796\pi$
$317$	−8174.93	−1.44842	−0.724211	+	0.689578i	$0.757796\pi$
$318$	0	0
$319$	1588.23	0.278758
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−923.549	−0.159095
$324$	0	0
$325$	−654.559	−0.111718
$326$	0	0
$327$	0	0
$328$	0	0
$329$	316.618	0.0530569
$330$	0	0
$331$	9661.28	1.60433	0.802163	−	0.597105i	$-0.203682\pi$
$331$	9661.28	1.60433	0.802163	+	0.597105i	$0.203682\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−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	$0.631181\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	721.177	0.114528
$342$	0	0
$343$	−3376.19	−0.531478
$344$	0	0
$345$	0	0
$346$	0	0
$347$	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	$0.117687\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4236.08	0.638708	0.319354	−	0.947635i	$-0.396534\pi$
$353$	4236.08	0.638708	0.319354	+	0.947635i	$0.396534\pi$
$354$	0	0
$355$	−2804.58	−0.419300
$356$	0	0
$357$	0	0
$358$	0	0
$359$	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
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9599.18	−1.37656
$366$	0	0
$367$	4616.29	0.656590	0.328295	−	0.944575i	$-0.393526\pi$
$367$	4616.29	0.656590	0.328295	+	0.944575i	$0.393526\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	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	$0.392696\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1066.35	−0.145676
$378$	0	0
$379$	2000.33	0.271108	0.135554	−	0.990770i	$-0.456719\pi$
$379$	2000.33	0.271108	0.135554	+	0.990770i	$0.456719\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−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
$386$	0	0
$387$	0	0
$388$	0	0
$389$	404.411	0.0527106	0.0263553	−	0.999653i	$-0.491610\pi$
$389$	404.411	0.0527106	0.0263553	+	0.999653i	$0.491610\pi$
$390$	0	0
$391$	−1680.85	−0.217402
$392$	0	0
$393$	0	0
$394$	0	0
$395$	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	$0.440918\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10186.2	−1.26852	−0.634258	−	0.773121i	$-0.718694\pi$
$401$	−10186.2	−1.26852	−0.634258	+	0.773121i	$0.718694\pi$
$402$	0	0
$403$	−484.203	−0.0598508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	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	$0.362747\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4040.41	−0.481394
$414$	0	0
$415$	−7404.09	−0.875789
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−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	$0.465543\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−411.874	−0.0470090
$426$	0	0
$427$	2666.33	0.302185
$428$	0	0
$429$	0	0
$430$	0	0
$431$	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	$0.488815\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2775.53	0.303825
$438$	0	0
$439$	11306.5	1.22923	0.614614	−	0.788828i	$-0.289312\pi$
$439$	11306.5	1.22923	0.614614	+	0.788828i	$0.289312\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	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
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6888.40	0.724017	0.362008	−	0.932175i	$-0.382091\pi$
$449$	6888.40	0.724017	0.362008	+	0.932175i	$0.382091\pi$
$450$	0	0
$451$	12062.4	1.25941
$452$	0	0
$453$	0	0
$454$	0	0
$455$	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	$0.429645\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13778.3	1.39202	0.696009	−	0.718033i	$-0.254958\pi$
$461$	13778.3	1.39202	0.696009	+	0.718033i	$0.254958\pi$
$462$	0	0
$463$	−5734.53	−0.575608	−0.287804	−	0.957689i	$-0.592925\pi$
$463$	−5734.53	−0.575608	−0.287804	+	0.957689i	$0.592925\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	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
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3427.75	0.333210
$474$	0	0
$475$	680.114	0.0656964
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−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
$482$	0	0
$483$	0	0
$484$	0	0
$485$	778.204	0.0728586
$486$	0	0
$487$	−8704.66	−0.809950	−0.404975	−	0.914328i	$-0.632720\pi$
$487$	−8704.66	−0.809950	−0.404975	+	0.914328i	$0.632720\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	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
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1383.55	−0.124871
$498$	0	0
$499$	9696.28	0.869870	0.434935	−	0.900462i	$-0.356771\pi$
$499$	9696.28	0.869870	0.434935	+	0.900462i	$0.356771\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−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
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11274.7	0.981816	0.490908	−	0.871211i	$-0.336665\pi$
$509$	11274.7	0.981816	0.490908	+	0.871211i	$0.336665\pi$
$510$	0	0
$511$	−4735.46	−0.409950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11324.6	0.968977
$516$	0	0
$517$	−3463.80	−0.294657
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8675.49	0.729520	0.364760	−	0.931102i	$-0.381151\pi$
$521$	8675.49	0.729520	0.364760	+	0.931102i	$0.381151\pi$
$522$	0	0
$523$	4226.14	0.353339	0.176670	−	0.984270i	$-0.443468\pi$
$523$	4226.14	0.353339	0.176670	+	0.984270i	$0.443468\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−304.680	−0.0251842
$528$	0	0
$529$	−7115.58	−0.584826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8098.78	−0.658156
$534$	0	0
$535$	10674.7	0.862630
$536$	0	0
$537$	0	0
$538$	0	0
$539$	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	$0.321902\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18426.5	1.44827
$546$	0	0
$547$	−21671.1	−1.69395	−0.846974	−	0.531634i	$-0.821578\pi$
$547$	−21671.1	−1.69395	−0.846974	+	0.531634i	$0.821578\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1107.98	0.0856654
$552$	0	0
$553$	6595.79	0.507200
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7477.63	0.568828	0.284414	−	0.958702i	$-0.408201\pi$
$557$	7477.63	0.568828	0.284414	+	0.958702i	$0.408201\pi$
$558$	0	0
$559$	−2301.42	−0.174132
$560$	0	0
$561$	0	0
$562$	0	0
$563$	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
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14649.1	−1.07930	−0.539650	−	0.841890i	$-0.681443\pi$
$569$	−14649.1	−1.07930	−0.539650	+	0.841890i	$0.681443\pi$
$570$	0	0
$571$	−23164.0	−1.69769	−0.848846	−	0.528640i	$-0.822702\pi$
$571$	−23164.0	−1.69769	−0.848846	+	0.528640i	$0.822702\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	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	$0.408417\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3652.58	−0.260817
$582$	0	0
$583$	−27558.6	−1.95774
$584$	0	0
$585$	0	0
$586$	0	0
$587$	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
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16966.0	−1.17489	−0.587444	−	0.809265i	$-0.699866\pi$
$593$	−16966.0	−1.17489	−0.587444	+	0.809265i	$0.699866\pi$
$594$	0	0
$595$	1255.16	0.0864813
$596$	0	0
$597$	0	0
$598$	0	0
$599$	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	$0.529408\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18696.7	1.25641
$606$	0	0
$607$	−16825.0	−1.12505	−0.562524	−	0.826781i	$-0.690170\pi$
$607$	−16825.0	−1.12505	−0.562524	+	0.826781i	$0.690170\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	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	$0.731420\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11310.6	0.738004	0.369002	−	0.929429i	$-0.379699\pi$
$617$	11310.6	0.738004	0.369002	+	0.929429i	$0.379699\pi$
$618$	0	0
$619$	17059.9	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$619$	17059.9	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2071.32	−0.133203
$624$	0	0
$625$	−13144.7	−0.841262
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4259.34	−0.270002
$630$	0	0
$631$	13186.3	0.831916	0.415958	−	0.909384i	$-0.363446\pi$
$631$	13186.3	0.831916	0.415958	+	0.909384i	$0.363446\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12510.0	0.781805
$636$	0	0
$637$	−11907.4	−0.740638
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16362.0	1.00820	0.504102	−	0.863644i	$-0.331823\pi$
$641$	16362.0	1.00820	0.504102	+	0.863644i	$0.331823\pi$
$642$	0	0
$643$	28044.9	1.72004	0.860019	−	0.510262i	$-0.170452\pi$
$643$	28044.9	1.72004	0.860019	+	0.510262i	$0.170452\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−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
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1259.86	0.0755007	0.0377504	−	0.999287i	$-0.487981\pi$
$653$	1259.86	0.0755007	0.0377504	+	0.999287i	$0.487981\pi$
$654$	0	0
$655$	10661.2	0.635981
$656$	0	0
$657$	0	0
$658$	0	0
$659$	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	$0.373973\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2072.60	−0.120860
$666$	0	0
$667$	2016.51	0.117061
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−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	$0.474963\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25004.0	1.41947	0.709735	−	0.704468i	$-0.248814\pi$
$677$	25004.0	1.41947	0.709735	+	0.704468i	$0.248814\pi$
$678$	0	0
$679$	383.903	0.0216979
$680$	0	0
$681$	0	0
$682$	0	0
$683$	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
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18503.1	1.02309
$690$	0	0
$691$	−3484.58	−0.191837	−0.0959187	−	0.995389i	$-0.530579\pi$
$691$	−3484.58	−0.191837	−0.0959187	+	0.995389i	$0.530579\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4790.25	−0.261445
$696$	0	0
$697$	−5096.07	−0.276940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15701.4	0.845981	0.422991	−	0.906134i	$-0.360980\pi$
$701$	15701.4	0.845981	0.422991	+	0.906134i	$0.360980\pi$
$702$	0	0
$703$	7033.32	0.377335
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−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	$0.364020\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	915.649	0.0480944
$714$	0	0
$715$	−21822.2	−1.14141
$716$	0	0
$717$	0	0
$718$	0	0
$719$	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
$722$	0	0
$723$	0	0
$724$	0	0
$725$	494.125	0.0253122
$726$	0	0
$727$	14207.2	0.724782	0.362391	−	0.932026i	$-0.381961\pi$
$727$	14207.2	0.724782	0.362391	+	0.932026i	$0.381961\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−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	$0.266964\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17041.4	0.851735
$738$	0	0
$739$	5683.47	0.282909	0.141455	−	0.989945i	$-0.454822\pi$
$739$	5683.47	0.282909	0.141455	+	0.989945i	$0.454822\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	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
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5266.02	0.256898
$750$	0	0
$751$	8261.64	0.401427	0.200713	−	0.979650i	$-0.435674\pi$
$751$	8261.64	0.401427	0.200713	+	0.979650i	$0.435674\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−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	$0.604100\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5449.84	0.259601	0.129801	−	0.991540i	$-0.458566\pi$
$761$	5449.84	0.259601	0.129801	+	0.991540i	$0.458566\pi$
$762$	0	0
$763$	9090.15	0.431305
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−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	$0.650010\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1865.54	−0.0868033	−0.0434017	−	0.999058i	$-0.513820\pi$
$773$	−1865.54	−0.0868033	−0.0434017	+	0.999058i	$0.513820\pi$
$774$	0	0
$775$	224.370	0.0103995
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8414.98	0.387032
$780$	0	0
$781$	15136.0	0.693483
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33352.8	−1.51645
$786$	0	0
$787$	19207.3	0.869970	0.434985	−	0.900438i	$-0.356754\pi$
$787$	19207.3	0.869970	0.434985	+	0.900438i	$0.356754\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8268.50	−0.371674
$792$	0	0
$793$	19584.7	0.877017
$794$	0	0
$795$	0	0
$796$	0	0
$797$	186.074	0.00826988	0.00413494	−	0.999991i	$-0.498684\pi$
$797$	186.074	0.00826988	0.00413494	+	0.999991i	$0.498684\pi$
$798$	0	0
$799$	1463.37	0.0647940
$800$	0	0
$801$	0	0
$802$	0	0
$803$	51805.9	2.27670
$804$	0	0
$805$	−3772.10	−0.165154
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5903.09	0.256541	0.128270	−	0.991739i	$-0.459057\pi$
$809$	5903.09	0.256541	0.128270	+	0.991739i	$0.459057\pi$
$810$	0	0
$811$	−23111.0	−1.00066	−0.500331	−	0.865834i	$-0.666788\pi$
$811$	−23111.0	−1.00066	−0.500331	+	0.865834i	$0.666788\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9828.64	−0.422432
$816$	0	0
$817$	2391.27	0.102399
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9644.29	−0.409973	−0.204987	−	0.978765i	$-0.565715\pi$
$821$	−9644.29	−0.409973	−0.204987	+	0.978765i	$0.565715\pi$
$822$	0	0
$823$	33573.4	1.42199	0.710994	−	0.703198i	$-0.248245\pi$
$823$	33573.4	1.42199	0.710994	+	0.703198i	$0.248245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	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	$0.704899\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7492.58	−0.311647
$834$	0	0
$835$	−7114.97	−0.294878
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−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
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8136.29	−0.331239
$846$	0	0
$847$	9223.44	0.374169
$848$	0	0
$849$	0	0
$850$	0	0
$851$	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	$0.708617\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9080.70	0.361950	0.180975	−	0.983488i	$-0.442075\pi$
$857$	9080.70	0.361950	0.180975	+	0.983488i	$0.442075\pi$
$858$	0	0
$859$	−26160.2	−1.03909	−0.519543	−	0.854444i	$-0.673898\pi$
$859$	−26160.2	−1.03909	−0.519543	+	0.854444i	$0.673898\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	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
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−72157.9	−2.81679
$870$	0	0
$871$	−11441.7	−0.445108
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−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	$0.338401\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2049.26	−0.0783670	−0.0391835	−	0.999232i	$-0.512476\pi$
$881$	−2049.26	−0.0783670	−0.0391835	+	0.999232i	$0.512476\pi$
$882$	0	0
$883$	−39413.4	−1.50211	−0.751057	−	0.660237i	$-0.770456\pi$
$883$	−39413.4	−1.50211	−0.751057	+	0.660237i	$0.770456\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−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
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2416.42	−0.0905514
$894$	0	0
$895$	−31357.7	−1.17114
$896$	0	0
$897$	0	0
$898$	0	0
$899$	365.525	0.0135605
$900$	0	0
$901$	11642.9	0.430499
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4060.17	0.149132
$906$	0	0
$907$	−2710.62	−0.0992334	−0.0496167	−	0.998768i	$-0.515800\pi$
$907$	−2710.62	−0.0992334	−0.0496167	+	0.998768i	$0.515800\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−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
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5259.38	0.189400
$918$	0	0
$919$	39103.8	1.40361	0.701804	−	0.712370i	$-0.252378\pi$
$919$	39103.8	1.40361	0.701804	+	0.712370i	$0.252378\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10162.5	−0.362407
$924$	0	0
$925$	3136.64	0.111494
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35954.6	−1.26979	−0.634894	−	0.772600i	$-0.718956\pi$
$929$	−35954.6	−1.26979	−0.634894	+	0.772600i	$0.718956\pi$
$930$	0	0
$931$	12372.2	0.435536
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−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	$0.540416\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7478.91	0.259092	0.129546	−	0.991573i	$-0.458648\pi$
$941$	7478.91	0.259092	0.129546	+	0.991573i	$0.458648\pi$
$942$	0	0
$943$	15315.1	0.528875
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−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
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13981.6	−0.475246	−0.237623	−	0.971357i	$-0.576368\pi$
$953$	−13981.6	−0.475246	−0.237623	+	0.971357i	$0.576368\pi$
$954$	0	0
$955$	−36153.5	−1.22503
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6451.87	−0.217249
$960$	0	0
$961$	−29625.0	−0.994429
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22975.3	0.766427
$966$	0	0
$967$	9081.47	0.302007	0.151003	−	0.988533i	$-0.451750\pi$
$967$	9081.47	0.302007	0.151003	+	0.988533i	$0.451750\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	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
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10854.9	0.355455	0.177727	−	0.984080i	$-0.443125\pi$
$977$	10854.9	0.355455	0.177727	+	0.984080i	$0.443125\pi$
$978$	0	0
$979$	22660.2	0.739759
$980$	0	0
$981$	0	0
$982$	0	0
$983$	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
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4352.08	0.139927
$990$	0	0
$991$	−46125.6	−1.47854	−0.739268	−	0.673412i	$-0.764828\pi$
$991$	−46125.6	−1.47854	−0.739268	+	0.673412i	$0.764828\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	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	$0.244010\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.c.a.4.2	✓	4	36.7	odd	6
9.4.c.a.7.2	yes	4	36.31	odd	6
27.4.c.a.10.1		4	36.11	even	6
27.4.c.a.19.1		4	36.23	even	6
81.4.a.a.1.2		2	12.11	even	2
81.4.a.d.1.1		2	4.3	odd	2
144.4.i.c.49.2		4	9.7	even	3
144.4.i.c.97.2		4	9.4	even	3
225.4.e.b.76.1		4	180.79	odd	6
225.4.e.b.151.1		4	180.139	odd	6
225.4.k.b.49.2		8	180.7	even	12
225.4.k.b.49.3		8	180.43	even	12
225.4.k.b.124.2		8	180.103	even	12
225.4.k.b.124.3		8	180.67	even	12
432.4.i.c.145.2		4	9.2	odd	6
432.4.i.c.289.2		4	9.5	odd	6
1296.4.a.i.1.1		2	3.2	odd	2
1296.4.a.u.1.2		2	1.1	even	1	trivial
2025.4.a.g.1.2		2	20.19	odd	2
2025.4.a.n.1.1		2	60.59	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3723 q^{5} +5.11684 q^{7} +O(q^{10})\)	\(q+10.3723 q^{5} +5.11684 q^{7} -55.9783 q^{11} +37.5842 q^{13} +23.6495 q^{17} -39.0516 q^{19} -71.0733 q^{23} -17.4158 q^{25} -28.3723 q^{29} -12.8832 q^{31} +53.0733 q^{35} -180.103 q^{37} -215.484 q^{41} -61.2337 q^{43} +61.8776 q^{47} -316.818 q^{49} +492.310 q^{53} -580.622 q^{55} -789.630 q^{59} +521.090 q^{61} +389.834 q^{65} -304.429 q^{67} -270.391 q^{71} -925.464 q^{73} -286.432 q^{77} +1289.03 q^{79} -713.834 q^{83} +245.299 q^{85} -404.804 q^{89} +192.313 q^{91} -405.054 q^{95} +75.0273 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 15 q^{5} - 7 q^{7}+O(q^{10}) \) 2 * q + 15 * q^5 - 7 * q^7	\( 2 q + 15 q^{5} - 7 q^{7} - 66 q^{11} - 11 q^{13} + 99 q^{17} + 77 q^{19} - 33 q^{23} - 121 q^{25} - 51 q^{29} - 43 q^{31} - 3 q^{35} - 50 q^{37} + 132 q^{41} - 88 q^{43} - 399 q^{47} - 513 q^{49} + 54 q^{53} - 627 q^{55} - 798 q^{59} + 439 q^{61} + 165 q^{65} - 988 q^{67} - 1368 q^{71} - 455 q^{73} - 165 q^{77} + 803 q^{79} - 813 q^{83} + 594 q^{85} - 396 q^{89} + 781 q^{91} + 132 q^{95} + 736 q^{97}+O(q^{100}) \) 2 * q + 15 * q^5 - 7 * q^7 - 66 * q^11 - 11 * q^13 + 99 * q^17 + 77 * q^19 - 33 * q^23 - 121 * q^25 - 51 * q^29 - 43 * q^31 - 3 * q^35 - 50 * q^37 + 132 * q^41 - 88 * q^43 - 399 * q^47 - 513 * q^49 + 54 * q^53 - 627 * q^55 - 798 * q^59 + 439 * q^61 + 165 * q^65 - 988 * q^67 - 1368 * q^71 - 455 * q^73 - 165 * q^77 + 803 * q^79 - 813 * q^83 + 594 * q^85 - 396 * q^89 + 781 * q^91 + 132 * q^95 + 736 * q^97

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