LMFDB - Embedded newform 1296.4.a.s.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 162)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+11.1962 q^{5} -18.3923 q^{7} +O(q^{10})$	$q+11.1962 q^{5} -18.3923 q^{7} +23.5692 q^{11} -67.7461 q^{13} +117.158 q^{17} -110.315 q^{19} -69.2154 q^{23} +0.353829 q^{25} +198.373 q^{29} +311.061 q^{31} -205.923 q^{35} -206.608 q^{37} -132.631 q^{41} +335.177 q^{43} +379.061 q^{47} -4.72312 q^{49} +190.908 q^{53} +263.885 q^{55} +337.723 q^{59} +277.469 q^{61} -758.496 q^{65} -665.069 q^{67} +528.431 q^{71} -73.8306 q^{73} -433.492 q^{77} +479.808 q^{79} +179.769 q^{83} +1311.72 q^{85} +846.458 q^{89} +1246.01 q^{91} -1235.11 q^{95} -672.985 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} - 16 q^{7}+O(q^{10}) $ 2 * q + 12 * q^5 - 16 * q^7	$ 2 q + 12 q^{5} - 16 q^{7} - 36 q^{11} + 10 q^{13} + 120 q^{17} + 8 q^{19} - 180 q^{23} - 124 q^{25} + 324 q^{29} + 248 q^{31} - 204 q^{35} - 434 q^{37} + 192 q^{41} + 608 q^{43} + 384 q^{47} - 342 q^{49} - 408 q^{53} + 216 q^{55} + 1008 q^{59} + 742 q^{61} - 696 q^{65} + 104 q^{67} + 1140 q^{71} + 850 q^{73} - 576 q^{77} + 440 q^{79} - 264 q^{83} + 1314 q^{85} + 768 q^{89} + 1432 q^{91} - 1140 q^{95} - 764 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 - 16 * q^7 - 36 * q^11 + 10 * q^13 + 120 * q^17 + 8 * q^19 - 180 * q^23 - 124 * q^25 + 324 * q^29 + 248 * q^31 - 204 * q^35 - 434 * q^37 + 192 * q^41 + 608 * q^43 + 384 * q^47 - 342 * q^49 - 408 * q^53 + 216 * q^55 + 1008 * q^59 + 742 * q^61 - 696 * q^65 + 104 * q^67 + 1140 * q^71 + 850 * q^73 - 576 * q^77 + 440 * q^79 - 264 * q^83 + 1314 * q^85 + 768 * q^89 + 1432 * q^91 - 1140 * q^95 - 764 * 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
$5$	11.1962	1.00141	0.500707	−	0.865617i	$-0.333073\pi$
$5$	11.1962	1.00141	0.500707	+	0.865617i	$0.333073\pi$
$6$	0	0
$7$	−18.3923	−0.993091	−0.496546	−	0.868011i	$-0.665398\pi$
$7$	−18.3923	−0.993091	−0.496546	+	0.868011i	$0.665398\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	23.5692	0.646035	0.323018	−	0.946393i	$-0.395303\pi$
$11$	23.5692	0.646035	0.323018	+	0.946393i	$0.395303\pi$
$12$	0	0
$13$	−67.7461	−1.44534	−0.722669	−	0.691194i	$-0.757085\pi$
$13$	−67.7461	−1.44534	−0.722669	+	0.691194i	$0.757085\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	117.158	1.67147	0.835733	−	0.549137i	$-0.185043\pi$
$17$	117.158	1.67147	0.835733	+	0.549137i	$0.185043\pi$
$18$	0	0
$19$	−110.315	−1.33200	−0.666002	−	0.745950i	$-0.731996\pi$
$19$	−110.315	−1.33200	−0.666002	+	0.745950i	$0.731996\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−69.2154	−0.627496	−0.313748	−	0.949506i	$-0.601585\pi$
$23$	−69.2154	−0.627496	−0.313748	+	0.949506i	$0.601585\pi$
$24$	0	0
$25$	0.353829	0.00283063
$26$	0	0
$27$	0	0
$28$	0	0
$29$	198.373	1.27024	0.635120	−	0.772414i	$-0.280951\pi$
$29$	198.373	1.27024	0.635120	+	0.772414i	$0.280951\pi$
$30$	0	0
$31$	311.061	1.80220	0.901101	−	0.433608i	$-0.142760\pi$
$31$	311.061	1.80220	0.901101	+	0.433608i	$0.142760\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−205.923	−0.994496
$36$	0	0
$37$	−206.608	−0.918003	−0.459001	−	0.888436i	$-0.651793\pi$
$37$	−206.608	−0.918003	−0.459001	+	0.888436i	$0.651793\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−132.631	−0.505206	−0.252603	−	0.967570i	$-0.581287\pi$
$41$	−132.631	−0.505206	−0.252603	+	0.967570i	$0.581287\pi$
$42$	0	0
$43$	335.177	1.18870	0.594349	−	0.804207i	$-0.297410\pi$
$43$	335.177	1.18870	0.594349	+	0.804207i	$0.297410\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	379.061	1.17642	0.588211	−	0.808708i	$-0.299833\pi$
$47$	379.061	1.17642	0.588211	+	0.808708i	$0.299833\pi$
$48$	0	0
$49$	−4.72312	−0.0137700
$50$	0	0
$51$	0	0
$52$	0	0
$53$	190.908	0.494777	0.247388	−	0.968916i	$-0.420428\pi$
$53$	190.908	0.494777	0.247388	+	0.968916i	$0.420428\pi$
$54$	0	0
$55$	263.885	0.646949
$56$	0	0
$57$	0	0
$58$	0	0
$59$	337.723	0.745217	0.372609	−	0.927989i	$-0.378463\pi$
$59$	337.723	0.745217	0.372609	+	0.927989i	$0.378463\pi$
$60$	0	0
$61$	277.469	0.582398	0.291199	−	0.956662i	$-0.405946\pi$
$61$	277.469	0.582398	0.291199	+	0.956662i	$0.405946\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−758.496	−1.44738
$66$	0	0
$67$	−665.069	−1.21270	−0.606352	−	0.795197i	$-0.707368\pi$
$67$	−665.069	−1.21270	−0.606352	+	0.795197i	$0.707368\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	528.431	0.883284	0.441642	−	0.897191i	$-0.354396\pi$
$71$	528.431	0.883284	0.441642	+	0.897191i	$0.354396\pi$
$72$	0	0
$73$	−73.8306	−0.118373	−0.0591865	−	0.998247i	$-0.518851\pi$
$73$	−73.8306	−0.118373	−0.0591865	+	0.998247i	$0.518851\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−433.492	−0.641572
$78$	0	0
$79$	479.808	0.683324	0.341662	−	0.939823i	$-0.389010\pi$
$79$	479.808	0.683324	0.341662	+	0.939823i	$0.389010\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	179.769	0.237738	0.118869	−	0.992910i	$-0.462073\pi$
$83$	179.769	0.237738	0.118869	+	0.992910i	$0.462073\pi$
$84$	0	0
$85$	1311.72	1.67383
$86$	0	0
$87$	0	0
$88$	0	0
$89$	846.458	1.00814	0.504069	−	0.863663i	$-0.331836\pi$
$89$	846.458	1.00814	0.504069	+	0.863663i	$0.331836\pi$
$90$	0	0
$91$	1246.01	1.43535
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1235.11	−1.33389
$96$	0	0
$97$	−672.985	−0.704446	−0.352223	−	0.935916i	$-0.614574\pi$
$97$	−672.985	−0.704446	−0.352223	+	0.935916i	$0.614574\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	206.523	0.203463	0.101732	−	0.994812i	$-0.467562\pi$
$101$	206.523	0.203463	0.101732	+	0.994812i	$0.467562\pi$
$102$	0	0
$103$	1371.77	1.31228	0.656138	−	0.754641i	$-0.272189\pi$
$103$	1371.77	1.31228	0.656138	+	0.754641i	$0.272189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1267.00	1.14472	0.572362	−	0.820001i	$-0.306027\pi$
$107$	1267.00	1.14472	0.572362	+	0.820001i	$0.306027\pi$
$108$	0	0
$109$	1725.13	1.51594	0.757970	−	0.652289i	$-0.226191\pi$
$109$	1725.13	1.51594	0.757970	+	0.652289i	$0.226191\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1740.20	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$113$	−1740.20	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$114$	0	0
$115$	−774.946	−0.628383
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2154.80	−1.65992
$120$	0	0
$121$	−775.492	−0.582639
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1395.56	−0.998580
$126$	0	0
$127$	−492.131	−0.343855	−0.171927	−	0.985110i	$-0.554999\pi$
$127$	−492.131	−0.343855	−0.171927	+	0.985110i	$0.554999\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1919.28	−1.28006	−0.640030	−	0.768350i	$-0.721078\pi$
$131$	−1919.28	−1.28006	−0.640030	+	0.768350i	$0.721078\pi$
$132$	0	0
$133$	2028.95	1.32280
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2622.86	1.63566	0.817832	−	0.575458i	$-0.195176\pi$
$137$	2622.86	1.63566	0.817832	+	0.575458i	$0.195176\pi$
$138$	0	0
$139$	628.569	0.383558	0.191779	−	0.981438i	$-0.438574\pi$
$139$	628.569	0.383558	0.191779	+	0.981438i	$0.438574\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1596.72	−0.933739
$144$	0	0
$145$	2221.02	1.27204
$146$	0	0
$147$	0	0
$148$	0	0
$149$	568.312	0.312469	0.156235	−	0.987720i	$-0.450064\pi$
$149$	568.312	0.312469	0.156235	+	0.987720i	$0.450064\pi$
$150$	0	0
$151$	−357.430	−0.192631	−0.0963155	−	0.995351i	$-0.530706\pi$
$151$	−357.430	−0.192631	−0.0963155	+	0.995351i	$0.530706\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3482.69	1.80475
$156$	0	0
$157$	−727.253	−0.369689	−0.184844	−	0.982768i	$-0.559178\pi$
$157$	−727.253	−0.369689	−0.184844	+	0.982768i	$0.559178\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1273.03	0.623161
$162$	0	0
$163$	396.554	0.190555	0.0952776	−	0.995451i	$-0.469626\pi$
$163$	396.554	0.190555	0.0952776	+	0.995451i	$0.469626\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3178.52	−1.47282	−0.736411	−	0.676534i	$-0.763481\pi$
$167$	−3178.52	−1.47282	−0.736411	+	0.676534i	$0.763481\pi$
$168$	0	0
$169$	2392.54	1.08900
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2152.65	0.946029	0.473014	−	0.881055i	$-0.343166\pi$
$173$	2152.65	0.946029	0.473014	+	0.881055i	$0.343166\pi$
$174$	0	0
$175$	−6.50773	−0.00281108
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4490.29	1.87497	0.937487	−	0.348022i	$-0.113146\pi$
$179$	4490.29	1.87497	0.937487	+	0.348022i	$0.113146\pi$
$180$	0	0
$181$	1407.32	0.577931	0.288966	−	0.957340i	$-0.406689\pi$
$181$	1407.32	0.577931	0.288966	+	0.957340i	$0.406689\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2313.21	−0.919301
$186$	0	0
$187$	2761.31	1.07983
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−772.277	−0.292565	−0.146283	−	0.989243i	$-0.546731\pi$
$191$	−772.277	−0.292565	−0.146283	+	0.989243i	$0.546731\pi$
$192$	0	0
$193$	3652.68	1.36231	0.681154	−	0.732140i	$-0.261478\pi$
$193$	3652.68	1.36231	0.681154	+	0.732140i	$0.261478\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2647.40	0.957460	0.478730	−	0.877962i	$-0.341097\pi$
$197$	2647.40	0.957460	0.478730	+	0.877962i	$0.341097\pi$
$198$	0	0
$199$	−1470.22	−0.523723	−0.261861	−	0.965106i	$-0.584336\pi$
$199$	−1470.22	−0.523723	−0.261861	+	0.965106i	$0.584336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3648.54	−1.26146
$204$	0	0
$205$	−1484.95	−0.505920
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2600.05	−0.860522
$210$	0	0
$211$	−1536.01	−0.501152	−0.250576	−	0.968097i	$-0.580620\pi$
$211$	−1536.01	−0.501152	−0.250576	+	0.968097i	$0.580620\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3752.69	1.19038
$216$	0	0
$217$	−5721.14	−1.78975
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7936.98	−2.41583
$222$	0	0
$223$	1657.53	0.497742	0.248871	−	0.968537i	$-0.419940\pi$
$223$	1657.53	0.497742	0.248871	+	0.968537i	$0.419940\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1514.26	0.442753	0.221377	−	0.975188i	$-0.428945\pi$
$227$	1514.26	0.442753	0.221377	+	0.975188i	$0.428945\pi$
$228$	0	0
$229$	4299.04	1.24056	0.620280	−	0.784380i	$-0.287019\pi$
$229$	4299.04	1.24056	0.620280	+	0.784380i	$0.287019\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1336.78	0.375860	0.187930	−	0.982182i	$-0.439822\pi$
$233$	1336.78	0.375860	0.187930	+	0.982182i	$0.439822\pi$
$234$	0	0
$235$	4244.03	1.17809
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6878.63	1.86168	0.930840	−	0.365427i	$-0.119077\pi$
$239$	6878.63	1.86168	0.930840	+	0.365427i	$0.119077\pi$
$240$	0	0
$241$	1531.29	0.409291	0.204646	−	0.978836i	$-0.434396\pi$
$241$	1531.29	0.409291	0.204646	+	0.978836i	$0.434396\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−52.8808	−0.0137895
$246$	0	0
$247$	7473.44	1.92520
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1181.60	−0.297139	−0.148570	−	0.988902i	$-0.547467\pi$
$251$	−1181.60	−0.297139	−0.148570	+	0.988902i	$0.547467\pi$
$252$	0	0
$253$	−1631.35	−0.405384
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4381.35	1.06343	0.531714	−	0.846924i	$-0.321548\pi$
$257$	4381.35	1.06343	0.531714	+	0.846924i	$0.321548\pi$
$258$	0	0
$259$	3799.99	0.911660
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2800.08	0.656502	0.328251	−	0.944590i	$-0.393541\pi$
$263$	2800.08	0.656502	0.328251	+	0.944590i	$0.393541\pi$
$264$	0	0
$265$	2137.43	0.495477
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2803.73	0.635489	0.317745	−	0.948176i	$-0.397075\pi$
$269$	2803.73	0.635489	0.317745	+	0.948176i	$0.397075\pi$
$270$	0	0
$271$	6332.36	1.41942	0.709711	−	0.704493i	$-0.248825\pi$
$271$	6332.36	1.41942	0.709711	+	0.704493i	$0.248825\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.33948	0.00182869
$276$	0	0
$277$	927.661	0.201219	0.100610	−	0.994926i	$-0.467921\pi$
$277$	927.661	0.201219	0.100610	+	0.994926i	$0.467921\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1141.53	0.242341	0.121170	−	0.992632i	$-0.461335\pi$
$281$	1141.53	0.242341	0.121170	+	0.992632i	$0.461335\pi$
$282$	0	0
$283$	−3611.05	−0.758496	−0.379248	−	0.925295i	$-0.623817\pi$
$283$	−3611.05	−0.758496	−0.379248	+	0.925295i	$0.623817\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2439.38	0.501715
$288$	0	0
$289$	8812.92	1.79380
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2324.21	0.463419	0.231710	−	0.972785i	$-0.425568\pi$
$293$	2324.21	0.463419	0.231710	+	0.972785i	$0.425568\pi$
$294$	0	0
$295$	3781.20	0.746271
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4689.08	0.906944
$300$	0	0
$301$	−6164.68	−1.18049
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3106.59	0.583222
$306$	0	0
$307$	−6968.51	−1.29548	−0.647742	−	0.761860i	$-0.724287\pi$
$307$	−6968.51	−1.29548	−0.647742	+	0.761860i	$0.724287\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6340.31	−1.15603	−0.578016	−	0.816026i	$-0.696173\pi$
$311$	−6340.31	−1.15603	−0.578016	+	0.816026i	$0.696173\pi$
$312$	0	0
$313$	2388.83	0.431389	0.215694	−	0.976461i	$-0.430799\pi$
$313$	2388.83	0.431389	0.215694	+	0.976461i	$0.430799\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5861.26	−1.03849	−0.519245	−	0.854625i	$-0.673787\pi$
$317$	−5861.26	−1.03849	−0.519245	+	0.854625i	$0.673787\pi$
$318$	0	0
$319$	4675.50	0.820620
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12924.3	−2.22640
$324$	0	0
$325$	−23.9706	−0.00409122
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6971.81	−1.16829
$330$	0	0
$331$	−4964.96	−0.824468	−0.412234	−	0.911078i	$-0.635251\pi$
$331$	−4964.96	−0.824468	−0.412234	+	0.911078i	$0.635251\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7446.21	−1.21442
$336$	0	0
$337$	−3097.66	−0.500713	−0.250357	−	0.968154i	$-0.580548\pi$
$337$	−3097.66	−0.500713	−0.250357	+	0.968154i	$0.580548\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7331.48	1.16429
$342$	0	0
$343$	6395.43	1.00677
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8044.20	−1.24448	−0.622241	−	0.782826i	$-0.713778\pi$
$347$	−8044.20	−1.24448	−0.622241	+	0.782826i	$0.713778\pi$
$348$	0	0
$349$	−1144.71	−0.175572	−0.0877862	−	0.996139i	$-0.527979\pi$
$349$	−1144.71	−0.175572	−0.0877862	+	0.996139i	$0.527979\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7801.26	1.17626	0.588129	−	0.808767i	$-0.299865\pi$
$353$	7801.26	1.17626	0.588129	+	0.808767i	$0.299865\pi$
$354$	0	0
$355$	5916.39	0.884534
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−341.307	−0.0501769	−0.0250885	−	0.999685i	$-0.507987\pi$
$359$	−341.307	−0.0501769	−0.0250885	+	0.999685i	$0.507987\pi$
$360$	0	0
$361$	5310.48	0.774235
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−826.619	−0.118540
$366$	0	0
$367$	−119.368	−0.0169781	−0.00848903	−	0.999964i	$-0.502702\pi$
$367$	−119.368	−0.0169781	−0.00848903	+	0.999964i	$0.502702\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3511.23	−0.491359
$372$	0	0
$373$	−4374.43	−0.607237	−0.303619	−	0.952794i	$-0.598195\pi$
$373$	−4374.43	−0.607237	−0.303619	+	0.952794i	$0.598195\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13439.0	−1.83593
$378$	0	0
$379$	−8949.46	−1.21294	−0.606468	−	0.795108i	$-0.707414\pi$
$379$	−8949.46	−1.21294	−0.606468	+	0.795108i	$0.707414\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−206.463	−0.0275451	−0.0137726	−	0.999905i	$-0.504384\pi$
$383$	−206.463	−0.0275451	−0.0137726	+	0.999905i	$0.504384\pi$
$384$	0	0
$385$	−4853.45	−0.642479
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2028.94	−0.264451	−0.132225	−	0.991220i	$-0.542212\pi$
$389$	−2028.94	−0.264451	−0.132225	+	0.991220i	$0.542212\pi$
$390$	0	0
$391$	−8109.11	−1.04884
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5372.00	0.684290
$396$	0	0
$397$	−6646.07	−0.840193	−0.420097	−	0.907479i	$-0.638004\pi$
$397$	−6646.07	−0.840193	−0.420097	+	0.907479i	$0.638004\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.63448	−0.000203546 0	−0.000101773	−	1.00000i	$-0.500032\pi$
$401$	−1.63448	−0.000203546 0	−0.000101773		1.00000i	$0.500032\pi$
$402$	0	0
$403$	−21073.2	−2.60479
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4869.58	−0.593062
$408$	0	0
$409$	−6203.35	−0.749966	−0.374983	−	0.927032i	$-0.622351\pi$
$409$	−6203.35	−0.749966	−0.374983	+	0.927032i	$0.622351\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6211.51	−0.740068
$414$	0	0
$415$	2012.72	0.238074
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9750.37	1.13684	0.568421	−	0.822738i	$-0.307555\pi$
$419$	9750.37	1.13684	0.568421	+	0.822738i	$0.307555\pi$
$420$	0	0
$421$	−10061.4	−1.16475	−0.582377	−	0.812919i	$-0.697877\pi$
$421$	−10061.4	−1.16475	−0.582377	+	0.812919i	$0.697877\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	41.4538	0.00473130
$426$	0	0
$427$	−5103.30	−0.578375
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6763.21	0.755853	0.377926	−	0.925836i	$-0.376637\pi$
$431$	6763.21	0.755853	0.377926	+	0.925836i	$0.376637\pi$
$432$	0	0
$433$	−10601.4	−1.17660	−0.588302	−	0.808641i	$-0.700204\pi$
$433$	−10601.4	−1.17660	−0.588302	+	0.808641i	$0.700204\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7635.52	0.835827
$438$	0	0
$439$	12568.9	1.36647	0.683237	−	0.730197i	$-0.260572\pi$
$439$	12568.9	1.36647	0.683237	+	0.730197i	$0.260572\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10255.4	1.09989	0.549944	−	0.835202i	$-0.314649\pi$
$443$	10255.4	1.09989	0.549944	+	0.835202i	$0.314649\pi$
$444$	0	0
$445$	9477.07	1.00956
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4080.23	−0.428860	−0.214430	−	0.976739i	$-0.568789\pi$
$449$	−4080.23	−0.428860	−0.214430	+	0.976739i	$0.568789\pi$
$450$	0	0
$451$	−3126.00	−0.326381
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13950.5	1.43738
$456$	0	0
$457$	−2183.20	−0.223470	−0.111735	−	0.993738i	$-0.535641\pi$
$457$	−2183.20	−0.223470	−0.111735	+	0.993738i	$0.535641\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3250.26	0.328372	0.164186	−	0.986429i	$-0.447500\pi$
$461$	3250.26	0.328372	0.164186	+	0.986429i	$0.447500\pi$
$462$	0	0
$463$	−18991.1	−1.90625	−0.953124	−	0.302580i	$-0.902152\pi$
$463$	−18991.1	−1.90625	−0.953124	+	0.302580i	$0.902152\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6906.52	−0.684359	−0.342180	−	0.939635i	$-0.611165\pi$
$467$	−6906.52	−0.684359	−0.342180	+	0.939635i	$0.611165\pi$
$468$	0	0
$469$	12232.2	1.20432
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7899.86	0.767941
$474$	0	0
$475$	−39.0328	−0.00377041
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7380.27	−0.703994	−0.351997	−	0.936001i	$-0.614497\pi$
$479$	−7380.27	−0.703994	−0.351997	+	0.936001i	$0.614497\pi$
$480$	0	0
$481$	13996.9	1.32682
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7534.84	−0.705442
$486$	0	0
$487$	8756.51	0.814774	0.407387	−	0.913256i	$-0.366440\pi$
$487$	8756.51	0.814774	0.407387	+	0.913256i	$0.366440\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11837.9	−1.08806	−0.544031	−	0.839065i	$-0.683103\pi$
$491$	−11837.9	−1.08806	−0.544031	+	0.839065i	$0.683103\pi$
$492$	0	0
$493$	23240.9	2.12316
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9719.06	−0.877182
$498$	0	0
$499$	−8578.38	−0.769581	−0.384791	−	0.923004i	$-0.625726\pi$
$499$	−8578.38	−0.769581	−0.384791	+	0.923004i	$0.625726\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3611.17	0.320107	0.160054	−	0.987108i	$-0.448833\pi$
$503$	3611.17	0.320107	0.160054	+	0.987108i	$0.448833\pi$
$504$	0	0
$505$	2312.26	0.203751
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9059.74	0.788931	0.394465	−	0.918911i	$-0.370930\pi$
$509$	9059.74	0.788931	0.394465	+	0.918911i	$0.370930\pi$
$510$	0	0
$511$	1357.92	0.117555
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15358.5	1.31413
$516$	0	0
$517$	8934.18	0.760010
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12834.1	−1.07922	−0.539609	−	0.841916i	$-0.681428\pi$
$521$	−12834.1	−1.07922	−0.539609	+	0.841916i	$0.681428\pi$
$522$	0	0
$523$	−7061.21	−0.590373	−0.295187	−	0.955440i	$-0.595382\pi$
$523$	−7061.21	−0.590373	−0.295187	+	0.955440i	$0.595382\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36443.2	3.01232
$528$	0	0
$529$	−7376.23	−0.606249
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8985.22	0.730193
$534$	0	0
$535$	14185.5	1.14634
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−111.320	−0.00889593
$540$	0	0
$541$	−21645.6	−1.72018	−0.860091	−	0.510141i	$-0.829593\pi$
$541$	−21645.6	−1.72018	−0.860091	+	0.510141i	$0.829593\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	19314.8	1.51808
$546$	0	0
$547$	−8507.05	−0.664964	−0.332482	−	0.943110i	$-0.607886\pi$
$547$	−8507.05	−0.664964	−0.332482	+	0.943110i	$0.607886\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21883.6	−1.69196
$552$	0	0
$553$	−8824.77	−0.678603
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−636.486	−0.0484179	−0.0242090	−	0.999707i	$-0.507707\pi$
$557$	−636.486	−0.0484179	−0.0242090	+	0.999707i	$0.507707\pi$
$558$	0	0
$559$	−22706.9	−1.71807
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13562.6	−1.01527	−0.507635	−	0.861572i	$-0.669480\pi$
$563$	−13562.6	−1.01527	−0.507635	+	0.861572i	$0.669480\pi$
$564$	0	0
$565$	−19483.5	−1.45076
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21116.7	−1.55582	−0.777909	−	0.628377i	$-0.783719\pi$
$569$	−21116.7	−1.55582	−0.777909	+	0.628377i	$0.783719\pi$
$570$	0	0
$571$	14527.1	1.06469	0.532347	−	0.846526i	$-0.321310\pi$
$571$	14527.1	1.06469	0.532347	+	0.846526i	$0.321310\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−24.4904	−0.00177621
$576$	0	0
$577$	−14590.9	−1.05273	−0.526366	−	0.850258i	$-0.676446\pi$
$577$	−14590.9	−1.05273	−0.526366	+	0.850258i	$0.676446\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3306.37	−0.236095
$582$	0	0
$583$	4499.54	0.319643
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18341.5	−1.28967	−0.644833	−	0.764324i	$-0.723073\pi$
$587$	−18341.5	−1.28967	−0.644833	+	0.764324i	$0.723073\pi$
$588$	0	0
$589$	−34314.9	−2.40054
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28380.4	1.96534	0.982668	−	0.185376i	$-0.0593504\pi$
$593$	28380.4	1.96534	0.982668	+	0.185376i	$0.0593504\pi$
$594$	0	0
$595$	−24125.5	−1.66227
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23881.1	1.62897	0.814487	−	0.580181i	$-0.197018\pi$
$599$	23881.1	1.62897	0.814487	+	0.580181i	$0.197018\pi$
$600$	0	0
$601$	16620.8	1.12808	0.564039	−	0.825748i	$-0.309247\pi$
$601$	16620.8	1.12808	0.564039	+	0.825748i	$0.309247\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8682.53	−0.583463
$606$	0	0
$607$	1570.78	0.105034	0.0525172	−	0.998620i	$-0.483276\pi$
$607$	1570.78	0.105034	0.0525172	+	0.998620i	$0.483276\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25680.0	−1.70033
$612$	0	0
$613$	−5766.40	−0.379939	−0.189969	−	0.981790i	$-0.560839\pi$
$613$	−5766.40	−0.379939	−0.189969	+	0.981790i	$0.560839\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6962.57	0.454299	0.227149	−	0.973860i	$-0.427059\pi$
$617$	6962.57	0.454299	0.227149	+	0.973860i	$0.427059\pi$
$618$	0	0
$619$	−2660.42	−0.172748	−0.0863741	−	0.996263i	$-0.527528\pi$
$619$	−2660.42	−0.172748	−0.0863741	+	0.996263i	$0.527528\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15568.3	−1.00117
$624$	0	0
$625$	−15669.1	−1.00282
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24205.7	−1.53441
$630$	0	0
$631$	20432.5	1.28908	0.644538	−	0.764573i	$-0.277050\pi$
$631$	20432.5	1.28908	0.644538	+	0.764573i	$0.277050\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5509.97	−0.344341
$636$	0	0
$637$	319.973	0.0199024
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26068.3	1.60630	0.803149	−	0.595779i	$-0.203156\pi$
$641$	26068.3	1.60630	0.803149	+	0.595779i	$0.203156\pi$
$642$	0	0
$643$	−21170.4	−1.29841	−0.649207	−	0.760612i	$-0.724899\pi$
$643$	−21170.4	−1.29841	−0.649207	+	0.760612i	$0.724899\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8291.45	−0.503818	−0.251909	−	0.967751i	$-0.581058\pi$
$647$	−8291.45	−0.503818	−0.251909	+	0.967751i	$0.581058\pi$
$648$	0	0
$649$	7959.87	0.481436
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24685.0	1.47932	0.739662	−	0.672979i	$-0.234985\pi$
$653$	24685.0	1.47932	0.739662	+	0.672979i	$0.234985\pi$
$654$	0	0
$655$	−21488.5	−1.28187
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29954.4	1.77065	0.885325	−	0.464973i	$-0.153936\pi$
$659$	29954.4	1.77065	0.885325	+	0.464973i	$0.153936\pi$
$660$	0	0
$661$	−4126.76	−0.242833	−0.121416	−	0.992602i	$-0.538744\pi$
$661$	−4126.76	−0.242833	−0.121416	+	0.992602i	$0.538744\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22716.5	1.32467
$666$	0	0
$667$	−13730.5	−0.797070
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6539.73	0.376250
$672$	0	0
$673$	−26329.7	−1.50808	−0.754039	−	0.656830i	$-0.771897\pi$
$673$	−26329.7	−1.50808	−0.754039	+	0.656830i	$0.771897\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6603.86	0.374900	0.187450	−	0.982274i	$-0.439978\pi$
$677$	6603.86	0.374900	0.187450	+	0.982274i	$0.439978\pi$
$678$	0	0
$679$	12377.7	0.699579
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12706.4	−0.711854	−0.355927	−	0.934514i	$-0.615835\pi$
$683$	−12706.4	−0.711854	−0.355927	+	0.934514i	$0.615835\pi$
$684$	0	0
$685$	29365.9	1.63798
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12933.3	−0.715120
$690$	0	0
$691$	15730.2	0.865999	0.432999	−	0.901394i	$-0.357455\pi$
$691$	15730.2	0.865999	0.432999	+	0.901394i	$0.357455\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7037.55	0.384100
$696$	0	0
$697$	−15538.7	−0.844434
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−652.959	−0.0351811	−0.0175905	−	0.999845i	$-0.505600\pi$
$701$	−652.959	−0.0351811	−0.0175905	+	0.999845i	$0.505600\pi$
$702$	0	0
$703$	22792.0	1.22278
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3798.43	−0.202058
$708$	0	0
$709$	−24884.4	−1.31813	−0.659065	−	0.752086i	$-0.729048\pi$
$709$	−24884.4	−1.31813	−0.659065	+	0.752086i	$0.729048\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21530.2	−1.13088
$714$	0	0
$715$	−17877.2	−0.935060
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21323.8	−1.10604	−0.553020	−	0.833168i	$-0.686525\pi$
$719$	−21323.8	−1.10604	−0.553020	+	0.833168i	$0.686525\pi$
$720$	0	0
$721$	−25230.0	−1.30321
$722$	0	0
$723$	0	0
$724$	0	0
$725$	70.1902	0.00359558
$726$	0	0
$727$	−8032.01	−0.409753	−0.204877	−	0.978788i	$-0.565679\pi$
$727$	−8032.01	−0.409753	−0.204877	+	0.978788i	$0.565679\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	39268.5	1.98687
$732$	0	0
$733$	28457.4	1.43397	0.716984	−	0.697090i	$-0.245522\pi$
$733$	28457.4	1.43397	0.716984	+	0.697090i	$0.245522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15675.2	−0.783449
$738$	0	0
$739$	11006.3	0.547868	0.273934	−	0.961748i	$-0.411675\pi$
$739$	11006.3	0.547868	0.273934	+	0.961748i	$0.411675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4652.91	−0.229742	−0.114871	−	0.993380i	$-0.536646\pi$
$743$	−4652.91	−0.229742	−0.114871	+	0.993380i	$0.536646\pi$
$744$	0	0
$745$	6362.91	0.312911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−23303.0	−1.13682
$750$	0	0
$751$	17357.7	0.843400	0.421700	−	0.906735i	$-0.361434\pi$
$751$	17357.7	0.843400	0.421700	+	0.906735i	$0.361434\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4001.85	−0.192903
$756$	0	0
$757$	119.139	0.00572019	0.00286010	−	0.999996i	$-0.499090\pi$
$757$	119.139	0.00572019	0.00286010	+	0.999996i	$0.499090\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8843.02	−0.421234	−0.210617	−	0.977569i	$-0.567547\pi$
$761$	−8843.02	−0.421234	−0.210617	+	0.977569i	$0.567547\pi$
$762$	0	0
$763$	−31729.1	−1.50547
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22879.4	−1.07709
$768$	0	0
$769$	2693.10	0.126288	0.0631442	−	0.998004i	$-0.479887\pi$
$769$	2693.10	0.126288	0.0631442	+	0.998004i	$0.479887\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18116.5	−0.842958	−0.421479	−	0.906838i	$-0.638489\pi$
$773$	−18116.5	−0.842958	−0.421479	+	0.906838i	$0.638489\pi$
$774$	0	0
$775$	110.063	0.00510137
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14631.2	0.672936
$780$	0	0
$781$	12454.7	0.570633
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8142.44	−0.370212
$786$	0	0
$787$	−31747.5	−1.43796	−0.718980	−	0.695031i	$-0.755391\pi$
$787$	−31747.5	−1.43796	−0.718980	+	0.695031i	$0.755391\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32006.2	1.43870
$792$	0	0
$793$	−18797.5	−0.841763
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28265.8	−1.25624	−0.628122	−	0.778115i	$-0.716176\pi$
$797$	−28265.8	−1.25624	−0.628122	+	0.778115i	$0.716176\pi$
$798$	0	0
$799$	44410.0	1.96635
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1740.13	−0.0764731
$804$	0	0
$805$	14253.0	0.624042
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42553.4	−1.84932	−0.924659	−	0.380795i	$-0.875650\pi$
$809$	−42553.4	−1.84932	−0.924659	+	0.380795i	$0.875650\pi$
$810$	0	0
$811$	6900.03	0.298758	0.149379	−	0.988780i	$-0.452273\pi$
$811$	6900.03	0.298758	0.149379	+	0.988780i	$0.452273\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4439.88	0.190825
$816$	0	0
$817$	−36975.2	−1.58335
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22359.2	0.950476	0.475238	−	0.879857i	$-0.342362\pi$
$821$	22359.2	0.950476	0.475238	+	0.879857i	$0.342362\pi$
$822$	0	0
$823$	−791.570	−0.0335266	−0.0167633	−	0.999859i	$-0.505336\pi$
$823$	−791.570	−0.0335266	−0.0167633	+	0.999859i	$0.505336\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23005.9	−0.967343	−0.483671	−	0.875250i	$-0.660697\pi$
$827$	−23005.9	−0.967343	−0.483671	+	0.875250i	$0.660697\pi$
$828$	0	0
$829$	−15420.2	−0.646040	−0.323020	−	0.946392i	$-0.604698\pi$
$829$	−15420.2	−0.646040	−0.323020	+	0.946392i	$0.604698\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−553.350	−0.0230161
$834$	0	0
$835$	−35587.2	−1.47491
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−46290.1	−1.90478	−0.952391	−	0.304878i	$-0.901384\pi$
$839$	−46290.1	−1.90478	−0.952391	+	0.304878i	$0.901384\pi$
$840$	0	0
$841$	14962.9	0.613509
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26787.2	1.09054
$846$	0	0
$847$	14263.1	0.578613
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14300.4	0.576043
$852$	0	0
$853$	27902.4	1.12000	0.559999	−	0.828493i	$-0.310801\pi$
$853$	27902.4	1.12000	0.559999	+	0.828493i	$0.310801\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6552.93	0.261195	0.130597	−	0.991435i	$-0.458310\pi$
$857$	6552.93	0.261195	0.130597	+	0.991435i	$0.458310\pi$
$858$	0	0
$859$	−34857.7	−1.38455	−0.692275	−	0.721634i	$-0.743392\pi$
$859$	−34857.7	−1.38455	−0.692275	+	0.721634i	$0.743392\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10885.2	0.429359	0.214680	−	0.976685i	$-0.431129\pi$
$863$	10885.2	0.429359	0.214680	+	0.976685i	$0.431129\pi$
$864$	0	0
$865$	24101.4	0.947367
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11308.7	0.441451
$870$	0	0
$871$	45055.9	1.75277
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25667.5	0.991681
$876$	0	0
$877$	−27913.8	−1.07478	−0.537390	−	0.843334i	$-0.680590\pi$
$877$	−27913.8	−1.07478	−0.537390	+	0.843334i	$0.680590\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10694.5	0.408975	0.204488	−	0.978869i	$-0.434447\pi$
$881$	10694.5	0.408975	0.204488	+	0.978869i	$0.434447\pi$
$882$	0	0
$883$	−3265.74	−0.124463	−0.0622315	−	0.998062i	$-0.519822\pi$
$883$	−3265.74	−0.124463	−0.0622315	+	0.998062i	$0.519822\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9392.88	−0.355560	−0.177780	−	0.984070i	$-0.556892\pi$
$887$	−9392.88	−0.355560	−0.177780	+	0.984070i	$0.556892\pi$
$888$	0	0
$889$	9051.41	0.341479
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−41816.3	−1.56700
$894$	0	0
$895$	50274.0	1.87762
$896$	0	0
$897$	0	0
$898$	0	0
$899$	61706.2	2.28923
$900$	0	0
$901$	22366.3	0.827002
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15756.6	0.578748
$906$	0	0
$907$	−13538.5	−0.495631	−0.247816	−	0.968807i	$-0.579713\pi$
$907$	−13538.5	−0.495631	−0.247816	+	0.968807i	$0.579713\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37806.4	−1.37495	−0.687476	−	0.726207i	$-0.741282\pi$
$911$	−37806.4	−1.37495	−0.687476	+	0.726207i	$0.741282\pi$
$912$	0	0
$913$	4237.02	0.153587
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35299.9	1.27122
$918$	0	0
$919$	−30674.2	−1.10103	−0.550515	−	0.834825i	$-0.685569\pi$
$919$	−30674.2	−1.10103	−0.550515	+	0.834825i	$0.685569\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35799.1	−1.27664
$924$	0	0
$925$	−73.1038	−0.00259853
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28117.4	0.993004	0.496502	−	0.868035i	$-0.334617\pi$
$929$	28117.4	0.993004	0.496502	+	0.868035i	$0.334617\pi$
$930$	0	0
$931$	521.033	0.0183417
$932$	0	0
$933$	0	0
$934$	0	0
$935$	30916.1	1.08135
$936$	0	0
$937$	31859.0	1.11077	0.555384	−	0.831594i	$-0.312571\pi$
$937$	31859.0	1.11077	0.555384	+	0.831594i	$0.312571\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2263.14	−0.0784018	−0.0392009	−	0.999231i	$-0.512481\pi$
$941$	−2263.14	−0.0784018	−0.0392009	+	0.999231i	$0.512481\pi$
$942$	0	0
$943$	9180.09	0.317015
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6985.33	0.239697	0.119848	−	0.992792i	$-0.461759\pi$
$947$	6985.33	0.239697	0.119848	+	0.992792i	$0.461759\pi$
$948$	0	0
$949$	5001.74	0.171089
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26436.8	0.898608	0.449304	−	0.893379i	$-0.351672\pi$
$953$	26436.8	0.898608	0.449304	+	0.893379i	$0.351672\pi$
$954$	0	0
$955$	−8646.53	−0.292979
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−48240.4	−1.62436
$960$	0	0
$961$	66968.2	2.24794
$962$	0	0
$963$	0	0
$964$	0	0
$965$	40895.9	1.36423
$966$	0	0
$967$	100.156	0.00333072	0.00166536	−	0.999999i	$-0.499470\pi$
$967$	100.156	0.00333072	0.00166536	+	0.999999i	$0.499470\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−678.145	−0.0224127	−0.0112063	−	0.999937i	$-0.503567\pi$
$971$	−678.145	−0.0224127	−0.0112063	+	0.999937i	$0.503567\pi$
$972$	0	0
$973$	−11560.8	−0.380908
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9748.25	−0.319216	−0.159608	−	0.987180i	$-0.551023\pi$
$977$	−9748.25	−0.319216	−0.159608	+	0.987180i	$0.551023\pi$
$978$	0	0
$979$	19950.3	0.651293
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3515.69	0.114072	0.0570361	−	0.998372i	$-0.481835\pi$
$983$	3515.69	0.114072	0.0570361	+	0.998372i	$0.481835\pi$
$984$	0	0
$985$	29640.7	0.958815
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23199.4	−0.745903
$990$	0	0
$991$	612.517	0.0196339	0.00981697	−	0.999952i	$-0.496875\pi$
$991$	612.517	0.0196339	0.00981697	+	0.999952i	$0.496875\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16460.8	−0.524463
$996$	0	0
$997$	22956.4	0.729223	0.364611	−	0.931160i	$-0.381202\pi$
$997$	22956.4	0.729223	0.364611	+	0.931160i	$0.381202\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.s.1.2		2
3.2	odd	2		1296.4.a.j.1.1		2
4.3	odd	2		162.4.a.h.1.2	yes	2
12.11	even	2		162.4.a.e.1.1	✓	2
36.7	odd	6		162.4.c.i.109.1		4
36.11	even	6		162.4.c.j.109.2		4
36.23	even	6		162.4.c.j.55.2		4
36.31	odd	6		162.4.c.i.55.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.1	✓	2	12.11	even	2
162.4.a.h.1.2	yes	2	4.3	odd	2
162.4.c.i.55.1		4	36.31	odd	6
162.4.c.i.109.1		4	36.7	odd	6
162.4.c.j.55.2		4	36.23	even	6
162.4.c.j.109.2		4	36.11	even	6
1296.4.a.j.1.1		2	3.2	odd	2
1296.4.a.s.1.2		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+11.1962 q^{5} -18.3923 q^{7} +O(q^{10})\)	\(q+11.1962 q^{5} -18.3923 q^{7} +23.5692 q^{11} -67.7461 q^{13} +117.158 q^{17} -110.315 q^{19} -69.2154 q^{23} +0.353829 q^{25} +198.373 q^{29} +311.061 q^{31} -205.923 q^{35} -206.608 q^{37} -132.631 q^{41} +335.177 q^{43} +379.061 q^{47} -4.72312 q^{49} +190.908 q^{53} +263.885 q^{55} +337.723 q^{59} +277.469 q^{61} -758.496 q^{65} -665.069 q^{67} +528.431 q^{71} -73.8306 q^{73} -433.492 q^{77} +479.808 q^{79} +179.769 q^{83} +1311.72 q^{85} +846.458 q^{89} +1246.01 q^{91} -1235.11 q^{95} -672.985 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} - 16 q^{7}+O(q^{10}) \) 2 * q + 12 * q^5 - 16 * q^7	\( 2 q + 12 q^{5} - 16 q^{7} - 36 q^{11} + 10 q^{13} + 120 q^{17} + 8 q^{19} - 180 q^{23} - 124 q^{25} + 324 q^{29} + 248 q^{31} - 204 q^{35} - 434 q^{37} + 192 q^{41} + 608 q^{43} + 384 q^{47} - 342 q^{49} - 408 q^{53} + 216 q^{55} + 1008 q^{59} + 742 q^{61} - 696 q^{65} + 104 q^{67} + 1140 q^{71} + 850 q^{73} - 576 q^{77} + 440 q^{79} - 264 q^{83} + 1314 q^{85} + 768 q^{89} + 1432 q^{91} - 1140 q^{95} - 764 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 - 16 * q^7 - 36 * q^11 + 10 * q^13 + 120 * q^17 + 8 * q^19 - 180 * q^23 - 124 * q^25 + 324 * q^29 + 248 * q^31 - 204 * q^35 - 434 * q^37 + 192 * q^41 + 608 * q^43 + 384 * q^47 - 342 * q^49 - 408 * q^53 + 216 * q^55 + 1008 * q^59 + 742 * q^61 - 696 * q^65 + 104 * q^67 + 1140 * q^71 + 850 * q^73 - 576 * q^77 + 440 * q^79 - 264 * q^83 + 1314 * q^85 + 768 * q^89 + 1432 * q^91 - 1140 * q^95 - 764 * q^97

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