LMFDB - Embedded newform 162.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,4,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, 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("162.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.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:	$9.55830942093$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +11.1962 q^{5} +18.3923 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +11.1962 q^{5} +18.3923 q^{7} +8.00000 q^{8} +22.3923 q^{10} -23.5692 q^{11} -67.7461 q^{13} +36.7846 q^{14} +16.0000 q^{16} +117.158 q^{17} +110.315 q^{19} +44.7846 q^{20} -47.1384 q^{22} +69.2154 q^{23} +0.353829 q^{25} -135.492 q^{26} +73.5692 q^{28} +198.373 q^{29} -311.061 q^{31} +32.0000 q^{32} +234.315 q^{34} +205.923 q^{35} -206.608 q^{37} +220.631 q^{38} +89.5692 q^{40} -132.631 q^{41} -335.177 q^{43} -94.2769 q^{44} +138.431 q^{46} -379.061 q^{47} -4.72312 q^{49} +0.707658 q^{50} -270.985 q^{52} +190.908 q^{53} -263.885 q^{55} +147.138 q^{56} +396.746 q^{58} -337.723 q^{59} +277.469 q^{61} -622.123 q^{62} +64.0000 q^{64} -758.496 q^{65} +665.069 q^{67} +468.631 q^{68} +411.846 q^{70} -528.431 q^{71} -73.8306 q^{73} -413.215 q^{74} +441.261 q^{76} -433.492 q^{77} -479.808 q^{79} +179.138 q^{80} -265.261 q^{82} -179.769 q^{83} +1311.72 q^{85} -670.354 q^{86} -188.554 q^{88} +846.458 q^{89} -1246.01 q^{91} +276.862 q^{92} -758.123 q^{94} +1235.11 q^{95} -672.985 q^{97} -9.44624 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 + 24 * q^10 + 36 * q^11 + 10 * q^13 + 32 * q^14 + 32 * q^16 + 120 * q^17 - 8 * q^19 + 48 * q^20 + 72 * q^22 + 180 * q^23 - 124 * q^25 + 20 * q^26 + 64 * q^28 + 324 * q^29 - 248 * q^31 + 64 * q^32 + 240 * q^34 + 204 * q^35 - 434 * q^37 - 16 * q^38 + 96 * q^40 + 192 * q^41 - 608 * q^43 + 144 * q^44 + 360 * q^46 - 384 * q^47 - 342 * q^49 - 248 * q^50 + 40 * q^52 - 408 * q^53 - 216 * q^55 + 128 * q^56 + 648 * q^58 - 1008 * q^59 + 742 * q^61 - 496 * q^62 + 128 * q^64 - 696 * q^65 - 104 * q^67 + 480 * q^68 + 408 * q^70 - 1140 * q^71 + 850 * q^73 - 868 * q^74 - 32 * q^76 - 576 * q^77 - 440 * q^79 + 192 * q^80 + 384 * q^82 + 264 * q^83 + 1314 * q^85 - 1216 * q^86 + 288 * q^88 + 768 * q^89 - 1432 * q^91 + 720 * q^92 - 768 * q^94 + 1140 * q^95 - 764 * q^97 - 684 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	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	$-0.334602\pi$
$7$	18.3923	0.993091	0.496546	+	0.868011i	$0.334602\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	22.3923	0.708107
$11$	−23.5692	−0.646035	−0.323018	−	0.946393i	$-0.604697\pi$
$11$	−23.5692	−0.646035	−0.323018	+	0.946393i	$0.604697\pi$
$12$	0	0
$13$	−67.7461	−1.44534	−0.722669	−	0.691194i	$-0.757085\pi$
$13$	−67.7461	−1.44534	−0.722669	+	0.691194i	$0.757085\pi$
$14$	36.7846	0.702221
$15$	0	0
$16$	16.0000	0.250000
$17$	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	$-0.268004\pi$
$19$	110.315	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$20$	44.7846	0.500707
$21$	0	0
$22$	−47.1384	−0.456816
$23$	69.2154	0.627496	0.313748	−	0.949506i	$-0.398415\pi$
$23$	69.2154	0.627496	0.313748	+	0.949506i	$0.398415\pi$
$24$	0	0
$25$	0.353829	0.00283063
$26$	−135.492	−1.02201
$27$	0	0
$28$	73.5692	0.496546
$29$	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	$-0.857240\pi$
$31$	−311.061	−1.80220	−0.901101	+	0.433608i	$0.857240\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	234.315	1.18190
$35$	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	$0.651793\pi$
$38$	220.631	0.941869
$39$	0	0
$40$	89.5692	0.354053
$41$	−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	$-0.702590\pi$
$43$	−335.177	−1.18870	−0.594349	+	0.804207i	$0.702590\pi$
$44$	−94.2769	−0.323018
$45$	0	0
$46$	138.431	0.443707
$47$	−379.061	−1.17642	−0.588211	−	0.808708i	$-0.700167\pi$
$47$	−379.061	−1.17642	−0.588211	+	0.808708i	$0.700167\pi$
$48$	0	0
$49$	−4.72312	−0.0137700
$50$	0.707658	0.00200156
$51$	0	0
$52$	−270.985	−0.722669
$53$	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
$56$	147.138	0.351111
$57$	0	0
$58$	396.746	0.898195
$59$	−337.723	−0.745217	−0.372609	−	0.927989i	$-0.621537\pi$
$59$	−337.723	−0.745217	−0.372609	+	0.927989i	$0.621537\pi$
$60$	0	0
$61$	277.469	0.582398	0.291199	−	0.956662i	$-0.405946\pi$
$61$	277.469	0.582398	0.291199	+	0.956662i	$0.405946\pi$
$62$	−622.123	−1.27435
$63$	0	0
$64$	64.0000	0.125000
$65$	−758.496	−1.44738
$66$	0	0
$67$	665.069	1.21270	0.606352	−	0.795197i	$-0.292632\pi$
$67$	665.069	1.21270	0.606352	+	0.795197i	$0.292632\pi$
$68$	468.631	0.835733
$69$	0	0
$70$	411.846	0.703215
$71$	−528.431	−0.883284	−0.441642	−	0.897191i	$-0.645604\pi$
$71$	−528.431	−0.883284	−0.441642	+	0.897191i	$0.645604\pi$
$72$	0	0
$73$	−73.8306	−0.118373	−0.0591865	−	0.998247i	$-0.518851\pi$
$73$	−73.8306	−0.118373	−0.0591865	+	0.998247i	$0.518851\pi$
$74$	−413.215	−0.649126
$75$	0	0
$76$	441.261	0.666002
$77$	−433.492	−0.641572
$78$	0	0
$79$	−479.808	−0.683324	−0.341662	−	0.939823i	$-0.610990\pi$
$79$	−479.808	−0.683324	−0.341662	+	0.939823i	$0.610990\pi$
$80$	179.138	0.250354
$81$	0	0
$82$	−265.261	−0.357234
$83$	−179.769	−0.237738	−0.118869	−	0.992910i	$-0.537927\pi$
$83$	−179.769	−0.237738	−0.118869	+	0.992910i	$0.537927\pi$
$84$	0	0
$85$	1311.72	1.67383
$86$	−670.354	−0.840536
$87$	0	0
$88$	−188.554	−0.228408
$89$	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
$92$	276.862	0.313748
$93$	0	0
$94$	−758.123	−0.831855
$95$	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	$0.614574\pi$
$98$	−9.44624	−0.00973689
$99$	0	0
$100$	1.41532	0.00141532
$101$	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	$-0.727811\pi$
$103$	−1371.77	−1.31228	−0.656138	+	0.754641i	$0.727811\pi$
$104$	−541.969	−0.511004
$105$	0	0
$106$	381.815	0.349860
$107$	−1267.00	−1.14472	−0.572362	−	0.820001i	$-0.693973\pi$
$107$	−1267.00	−1.14472	−0.572362	+	0.820001i	$0.693973\pi$
$108$	0	0
$109$	1725.13	1.51594	0.757970	−	0.652289i	$-0.226191\pi$
$109$	1725.13	1.51594	0.757970	+	0.652289i	$0.226191\pi$
$110$	−527.769	−0.457462
$111$	0	0
$112$	294.277	0.248273
$113$	−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
$116$	793.492	0.635120
$117$	0	0
$118$	−675.446	−0.526948
$119$	2154.80	1.65992
$120$	0	0
$121$	−775.492	−0.582639
$122$	554.939	0.411818
$123$	0	0
$124$	−1244.25	−0.901101
$125$	−1395.56	−0.998580
$126$	0	0
$127$	492.131	0.343855	0.171927	−	0.985110i	$-0.445001\pi$
$127$	492.131	0.343855	0.171927	+	0.985110i	$0.445001\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−1516.99	−1.02345
$131$	1919.28	1.28006	0.640030	−	0.768350i	$-0.278922\pi$
$131$	1919.28	1.28006	0.640030	+	0.768350i	$0.278922\pi$
$132$	0	0
$133$	2028.95	1.32280
$134$	1330.14	0.857511
$135$	0	0
$136$	937.261	0.590952
$137$	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	$-0.561426\pi$
$139$	−628.569	−0.383558	−0.191779	+	0.981438i	$0.561426\pi$
$140$	823.692	0.497248
$141$	0	0
$142$	−1056.86	−0.624576
$143$	1596.72	0.933739
$144$	0	0
$145$	2221.02	1.27204
$146$	−147.661	−0.0837023
$147$	0	0
$148$	−826.431	−0.459001
$149$	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	$-0.469294\pi$
$151$	357.430	0.192631	0.0963155	+	0.995351i	$0.469294\pi$
$152$	882.523	0.470935
$153$	0	0
$154$	−866.985	−0.453660
$155$	−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	$0.559178\pi$
$158$	−959.615	−0.483183
$159$	0	0
$160$	358.277	0.177027
$161$	1273.03	0.623161
$162$	0	0
$163$	−396.554	−0.190555	−0.0952776	−	0.995451i	$-0.530374\pi$
$163$	−396.554	−0.190555	−0.0952776	+	0.995451i	$0.530374\pi$
$164$	−530.523	−0.252603
$165$	0	0
$166$	−359.538	−0.168106
$167$	3178.52	1.47282	0.736411	−	0.676534i	$-0.236519\pi$
$167$	3178.52	1.47282	0.736411	+	0.676534i	$0.236519\pi$
$168$	0	0
$169$	2392.54	1.08900
$170$	2623.43	1.18358
$171$	0	0
$172$	−1340.71	−0.594349
$173$	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
$176$	−377.108	−0.161509
$177$	0	0
$178$	1692.92	0.712862
$179$	−4490.29	−1.87497	−0.937487	−	0.348022i	$-0.886854\pi$
$179$	−4490.29	−1.87497	−0.937487	+	0.348022i	$0.886854\pi$
$180$	0	0
$181$	1407.32	0.577931	0.288966	−	0.957340i	$-0.406689\pi$
$181$	1407.32	0.577931	0.288966	+	0.957340i	$0.406689\pi$
$182$	−2492.02	−1.01495
$183$	0	0
$184$	553.723	0.221853
$185$	−2313.21	−0.919301
$186$	0	0
$187$	−2761.31	−1.07983
$188$	−1516.25	−0.588211
$189$	0	0
$190$	2470.22	0.943201
$191$	772.277	0.292565	0.146283	−	0.989243i	$-0.453269\pi$
$191$	772.277	0.292565	0.146283	+	0.989243i	$0.453269\pi$
$192$	0	0
$193$	3652.68	1.36231	0.681154	−	0.732140i	$-0.261478\pi$
$193$	3652.68	1.36231	0.681154	+	0.732140i	$0.261478\pi$
$194$	−1345.97	−0.498118
$195$	0	0
$196$	−18.8925	−0.00688502
$197$	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	$-0.415664\pi$
$199$	1470.22	0.523723	0.261861	+	0.965106i	$0.415664\pi$
$200$	2.83063	0.00100078
$201$	0	0
$202$	413.046	0.143870
$203$	3648.54	1.26146
$204$	0	0
$205$	−1484.95	−0.505920
$206$	−2743.54	−0.927919
$207$	0	0
$208$	−1083.94	−0.361335
$209$	−2600.05	−0.860522
$210$	0	0
$211$	1536.01	0.501152	0.250576	−	0.968097i	$-0.419380\pi$
$211$	1536.01	0.501152	0.250576	+	0.968097i	$0.419380\pi$
$212$	763.630	0.247388
$213$	0	0
$214$	−2534.00	−0.809442
$215$	−3752.69	−1.19038
$216$	0	0
$217$	−5721.14	−1.78975
$218$	3450.26	1.07193
$219$	0	0
$220$	−1055.54	−0.323474
$221$	−7936.98	−2.41583
$222$	0	0
$223$	−1657.53	−0.497742	−0.248871	−	0.968537i	$-0.580060\pi$
$223$	−1657.53	−0.497742	−0.248871	+	0.968537i	$0.580060\pi$
$224$	588.554	0.175555
$225$	0	0
$226$	−3480.39	−1.02439
$227$	−1514.26	−0.442753	−0.221377	−	0.975188i	$-0.571055\pi$
$227$	−1514.26	−0.442753	−0.221377	+	0.975188i	$0.571055\pi$
$228$	0	0
$229$	4299.04	1.24056	0.620280	−	0.784380i	$-0.287019\pi$
$229$	4299.04	1.24056	0.620280	+	0.784380i	$0.287019\pi$
$230$	1549.89	0.444334
$231$	0	0
$232$	1586.98	0.449098
$233$	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
$236$	−1350.89	−0.372609
$237$	0	0
$238$	4309.60	1.17374
$239$	−6878.63	−1.86168	−0.930840	−	0.365427i	$-0.880923\pi$
$239$	−6878.63	−1.86168	−0.930840	+	0.365427i	$0.880923\pi$
$240$	0	0
$241$	1531.29	0.409291	0.204646	−	0.978836i	$-0.434396\pi$
$241$	1531.29	0.409291	0.204646	+	0.978836i	$0.434396\pi$
$242$	−1550.98	−0.411988
$243$	0	0
$244$	1109.88	0.291199
$245$	−52.8808	−0.0137895
$246$	0	0
$247$	−7473.44	−1.92520
$248$	−2488.49	−0.637175
$249$	0	0
$250$	−2791.12	−0.706102
$251$	1181.60	0.297139	0.148570	−	0.988902i	$-0.452533\pi$
$251$	1181.60	0.297139	0.148570	+	0.988902i	$0.452533\pi$
$252$	0	0
$253$	−1631.35	−0.405384
$254$	984.261	0.243142
$255$	0	0
$256$	256.000	0.0625000
$257$	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
$260$	−3033.98	−0.723691
$261$	0	0
$262$	3838.55	0.905140
$263$	−2800.08	−0.656502	−0.328251	−	0.944590i	$-0.606459\pi$
$263$	−2800.08	−0.656502	−0.328251	+	0.944590i	$0.606459\pi$
$264$	0	0
$265$	2137.43	0.495477
$266$	4057.91	0.935362
$267$	0	0
$268$	2660.28	0.606352
$269$	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	$-0.751175\pi$
$271$	−6332.36	−1.41942	−0.709711	+	0.704493i	$0.751175\pi$
$272$	1874.52	0.417866
$273$	0	0
$274$	5245.71	1.15659
$275$	−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	$0.467921\pi$
$278$	−1257.14	−0.271216
$279$	0	0
$280$	1647.38	0.351607
$281$	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	$-0.376183\pi$
$283$	3611.05	0.758496	0.379248	+	0.925295i	$0.376183\pi$
$284$	−2113.72	−0.441642
$285$	0	0
$286$	3193.45	0.660253
$287$	−2439.38	−0.501715
$288$	0	0
$289$	8812.92	1.79380
$290$	4442.03	0.899465
$291$	0	0
$292$	−295.323	−0.0591865
$293$	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
$296$	−1652.86	−0.324563
$297$	0	0
$298$	1136.62	0.220949
$299$	−4689.08	−0.906944
$300$	0	0
$301$	−6164.68	−1.18049
$302$	714.861	0.136211
$303$	0	0
$304$	1765.05	0.333001
$305$	3106.59	0.583222
$306$	0	0
$307$	6968.51	1.29548	0.647742	−	0.761860i	$-0.275713\pi$
$307$	6968.51	1.29548	0.647742	+	0.761860i	$0.275713\pi$
$308$	−1733.97	−0.320786
$309$	0	0
$310$	−6965.38	−1.27615
$311$	6340.31	1.15603	0.578016	−	0.816026i	$-0.303827\pi$
$311$	6340.31	1.15603	0.578016	+	0.816026i	$0.303827\pi$
$312$	0	0
$313$	2388.83	0.431389	0.215694	−	0.976461i	$-0.430799\pi$
$313$	2388.83	0.431389	0.215694	+	0.976461i	$0.430799\pi$
$314$	−1454.51	−0.261409
$315$	0	0
$316$	−1919.23	−0.341662
$317$	−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
$320$	716.554	0.125177
$321$	0	0
$322$	2546.06	0.440641
$323$	12924.3	2.22640
$324$	0	0
$325$	−23.9706	−0.00409122
$326$	−793.108	−0.134743
$327$	0	0
$328$	−1061.05	−0.178617
$329$	−6971.81	−1.16829
$330$	0	0
$331$	4964.96	0.824468	0.412234	−	0.911078i	$-0.364749\pi$
$331$	4964.96	0.824468	0.412234	+	0.911078i	$0.364749\pi$
$332$	−719.077	−0.118869
$333$	0	0
$334$	6357.04	1.04144
$335$	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	$0.580548\pi$
$338$	4785.08	0.770041
$339$	0	0
$340$	5246.86	0.836915
$341$	7331.48	1.16429
$342$	0	0
$343$	−6395.43	−1.00677
$344$	−2681.42	−0.420268
$345$	0	0
$346$	4305.30	0.668943
$347$	8044.20	1.24448	0.622241	−	0.782826i	$-0.286222\pi$
$347$	8044.20	1.24448	0.622241	+	0.782826i	$0.286222\pi$
$348$	0	0
$349$	−1144.71	−0.175572	−0.0877862	−	0.996139i	$-0.527979\pi$
$349$	−1144.71	−0.175572	−0.0877862	+	0.996139i	$0.527979\pi$
$350$	13.0155	0.00198773
$351$	0	0
$352$	−754.215	−0.114204
$353$	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
$356$	3385.83	0.504069
$357$	0	0
$358$	−8980.58	−1.32581
$359$	341.307	0.0501769	0.0250885	−	0.999685i	$-0.492013\pi$
$359$	341.307	0.0501769	0.0250885	+	0.999685i	$0.492013\pi$
$360$	0	0
$361$	5310.48	0.774235
$362$	2814.65	0.408659
$363$	0	0
$364$	−4984.03	−0.717676
$365$	−826.619	−0.118540
$366$	0	0
$367$	119.368	0.0169781	0.00848903	−	0.999964i	$-0.497298\pi$
$367$	119.368	0.0169781	0.00848903	+	0.999964i	$0.497298\pi$
$368$	1107.45	0.156874
$369$	0	0
$370$	−4626.42	−0.650044
$371$	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	$0.598195\pi$
$374$	−5522.63	−0.763552
$375$	0	0
$376$	−3032.49	−0.415928
$377$	−13439.0	−1.83593
$378$	0	0
$379$	8949.46	1.21294	0.606468	−	0.795108i	$-0.292586\pi$
$379$	8949.46	1.21294	0.606468	+	0.795108i	$0.292586\pi$
$380$	4940.43	0.666944
$381$	0	0
$382$	1544.55	0.206875
$383$	206.463	0.0275451	0.0137726	−	0.999905i	$-0.495616\pi$
$383$	206.463	0.0275451	0.0137726	+	0.999905i	$0.495616\pi$
$384$	0	0
$385$	−4853.45	−0.642479
$386$	7305.35	0.963297
$387$	0	0
$388$	−2691.94	−0.352223
$389$	−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
$392$	−37.7850	−0.00486844
$393$	0	0
$394$	5294.81	0.677027
$395$	−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	$0.638004\pi$
$398$	2940.43	0.370328
$399$	0	0
$400$	5.66127	0.000707658 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
$404$	826.091	0.101732
$405$	0	0
$406$	7297.08	0.891990
$407$	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	$0.622351\pi$
$410$	−2969.91	−0.357740
$411$	0	0
$412$	−5487.08	−0.656138
$413$	−6211.51	−0.740068
$414$	0	0
$415$	−2012.72	−0.238074
$416$	−2167.88	−0.255502
$417$	0	0
$418$	−5200.09	−0.608481
$419$	−9750.37	−1.13684	−0.568421	−	0.822738i	$-0.692445\pi$
$419$	−9750.37	−1.13684	−0.568421	+	0.822738i	$0.692445\pi$
$420$	0	0
$421$	−10061.4	−1.16475	−0.582377	−	0.812919i	$-0.697877\pi$
$421$	−10061.4	−1.16475	−0.582377	+	0.812919i	$0.697877\pi$
$422$	3072.01	0.354368
$423$	0	0
$424$	1527.26	0.174930
$425$	41.4538	0.00473130
$426$	0	0
$427$	5103.30	0.578375
$428$	−5068.00	−0.572362
$429$	0	0
$430$	−7505.38	−0.841725
$431$	−6763.21	−0.755853	−0.377926	−	0.925836i	$-0.623363\pi$
$431$	−6763.21	−0.755853	−0.377926	+	0.925836i	$0.623363\pi$
$432$	0	0
$433$	−10601.4	−1.17660	−0.588302	−	0.808641i	$-0.700204\pi$
$433$	−10601.4	−1.17660	−0.588302	+	0.808641i	$0.700204\pi$
$434$	−11442.3	−1.26555
$435$	0	0
$436$	6900.52	0.757970
$437$	7635.52	0.835827
$438$	0	0
$439$	−12568.9	−1.36647	−0.683237	−	0.730197i	$-0.739428\pi$
$439$	−12568.9	−1.36647	−0.683237	+	0.730197i	$0.739428\pi$
$440$	−2111.08	−0.228731
$441$	0	0
$442$	−15874.0	−1.70825
$443$	−10255.4	−1.09989	−0.549944	−	0.835202i	$-0.685351\pi$
$443$	−10255.4	−1.09989	−0.549944	+	0.835202i	$0.685351\pi$
$444$	0	0
$445$	9477.07	1.00956
$446$	−3315.06	−0.351957
$447$	0	0
$448$	1177.11	0.124136
$449$	−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
$452$	−6960.78	−0.724353
$453$	0	0
$454$	−3028.52	−0.313074
$455$	−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	$0.535641\pi$
$458$	8598.08	0.877209
$459$	0	0
$460$	3099.78	0.314192
$461$	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	$-0.0978479\pi$
$463$	18991.1	1.90625	0.953124	+	0.302580i	$0.0978479\pi$
$464$	3173.97	0.317560
$465$	0	0
$466$	2673.56	0.265773
$467$	6906.52	0.684359	0.342180	−	0.939635i	$-0.388835\pi$
$467$	6906.52	0.684359	0.342180	+	0.939635i	$0.388835\pi$
$468$	0	0
$469$	12232.2	1.20432
$470$	−8488.06	−0.833032
$471$	0	0
$472$	−2701.78	−0.263474
$473$	7899.86	0.767941
$474$	0	0
$475$	39.0328	0.00377041
$476$	8619.20	0.829959
$477$	0	0
$478$	−13757.3	−1.31641
$479$	7380.27	0.703994	0.351997	−	0.936001i	$-0.385503\pi$
$479$	7380.27	0.703994	0.351997	+	0.936001i	$0.385503\pi$
$480$	0	0
$481$	13996.9	1.32682
$482$	3062.58	0.289413
$483$	0	0
$484$	−3101.97	−0.291319
$485$	−7534.84	−0.705442
$486$	0	0
$487$	−8756.51	−0.814774	−0.407387	−	0.913256i	$-0.633560\pi$
$487$	−8756.51	−0.814774	−0.407387	+	0.913256i	$0.633560\pi$
$488$	2219.75	0.205909
$489$	0	0
$490$	−105.762	−0.00975066
$491$	11837.9	1.08806	0.544031	−	0.839065i	$-0.316897\pi$
$491$	11837.9	1.08806	0.544031	+	0.839065i	$0.316897\pi$
$492$	0	0
$493$	23240.9	2.12316
$494$	−14946.9	−1.36132
$495$	0	0
$496$	−4976.98	−0.450551
$497$	−9719.06	−0.877182
$498$	0	0
$499$	8578.38	0.769581	0.384791	−	0.923004i	$-0.374274\pi$
$499$	8578.38	0.769581	0.384791	+	0.923004i	$0.374274\pi$
$500$	−5582.23	−0.499290
$501$	0	0
$502$	2363.20	0.210109
$503$	−3611.17	−0.320107	−0.160054	−	0.987108i	$-0.551167\pi$
$503$	−3611.17	−0.320107	−0.160054	+	0.987108i	$0.551167\pi$
$504$	0	0
$505$	2312.26	0.203751
$506$	−3262.71	−0.286650
$507$	0	0
$508$	1968.52	0.171927
$509$	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
$512$	512.000	0.0441942
$513$	0	0
$514$	8762.70	0.751957
$515$	−15358.5	−1.31413
$516$	0	0
$517$	8934.18	0.760010
$518$	−7599.98	−0.644641
$519$	0	0
$520$	−6067.97	−0.511727
$521$	−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	$-0.404618\pi$
$523$	7061.21	0.590373	0.295187	+	0.955440i	$0.404618\pi$
$524$	7677.11	0.640030
$525$	0	0
$526$	−5600.15	−0.464217
$527$	−36443.2	−3.01232
$528$	0	0
$529$	−7376.23	−0.606249
$530$	4274.86	0.350355
$531$	0	0
$532$	8115.81	0.661401
$533$	8985.22	0.730193
$534$	0	0
$535$	−14185.5	−1.14634
$536$	5320.55	0.428755
$537$	0	0
$538$	5607.47	0.449359
$539$	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	$0.829593\pi$
$542$	−12664.7	−1.00368
$543$	0	0
$544$	3749.05	0.295476
$545$	19314.8	1.51808
$546$	0	0
$547$	8507.05	0.664964	0.332482	−	0.943110i	$-0.392114\pi$
$547$	8507.05	0.664964	0.332482	+	0.943110i	$0.392114\pi$
$548$	10491.4	0.817832
$549$	0	0
$550$	−16.6790	−0.00129308
$551$	21883.6	1.69196
$552$	0	0
$553$	−8824.77	−0.678603
$554$	1855.32	0.142284
$555$	0	0
$556$	−2514.28	−0.191779
$557$	−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
$560$	3294.77	0.248624
$561$	0	0
$562$	2283.05	0.171361
$563$	13562.6	1.01527	0.507635	−	0.861572i	$-0.330520\pi$
$563$	13562.6	1.01527	0.507635	+	0.861572i	$0.330520\pi$
$564$	0	0
$565$	−19483.5	−1.45076
$566$	7222.09	0.536338
$567$	0	0
$568$	−4227.45	−0.312288
$569$	−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	$-0.678690\pi$
$571$	−14527.1	−1.06469	−0.532347	+	0.846526i	$0.678690\pi$
$572$	6386.89	0.466870
$573$	0	0
$574$	−4878.77	−0.354766
$575$	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	$0.676446\pi$
$578$	17625.8	1.26841
$579$	0	0
$580$	8884.06	0.636018
$581$	−3306.37	−0.236095
$582$	0	0
$583$	−4499.54	−0.319643
$584$	−590.645	−0.0418511
$585$	0	0
$586$	4648.42	0.327687
$587$	18341.5	1.28967	0.644833	−	0.764324i	$-0.276927\pi$
$587$	18341.5	1.28967	0.644833	+	0.764324i	$0.276927\pi$
$588$	0	0
$589$	−34314.9	−2.40054
$590$	−7562.40	−0.527693
$591$	0	0
$592$	−3305.72	−0.229501
$593$	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
$596$	2273.25	0.156235
$597$	0	0
$598$	−9378.15	−0.641306
$599$	−23881.1	−1.62897	−0.814487	−	0.580181i	$-0.802982\pi$
$599$	−23881.1	−1.62897	−0.814487	+	0.580181i	$0.802982\pi$
$600$	0	0
$601$	16620.8	1.12808	0.564039	−	0.825748i	$-0.309247\pi$
$601$	16620.8	1.12808	0.564039	+	0.825748i	$0.309247\pi$
$602$	−12329.4	−0.834729
$603$	0	0
$604$	1429.72	0.0963155
$605$	−8682.53	−0.583463
$606$	0	0
$607$	−1570.78	−0.105034	−0.0525172	−	0.998620i	$-0.516724\pi$
$607$	−1570.78	−0.105034	−0.0525172	+	0.998620i	$0.516724\pi$
$608$	3530.09	0.235467
$609$	0	0
$610$	6213.18	0.412400
$611$	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	$0.560839\pi$
$614$	13937.0	0.916046
$615$	0	0
$616$	−3467.94	−0.226830
$617$	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	$-0.472472\pi$
$619$	2660.42	0.172748	0.0863741	+	0.996263i	$0.472472\pi$
$620$	−13930.8	−0.902376
$621$	0	0
$622$	12680.6	0.817438
$623$	15568.3	1.00117
$624$	0	0
$625$	−15669.1	−1.00282
$626$	4777.66	0.305038
$627$	0	0
$628$	−2909.01	−0.184844
$629$	−24205.7	−1.53441
$630$	0	0
$631$	−20432.5	−1.28908	−0.644538	−	0.764573i	$-0.722950\pi$
$631$	−20432.5	−1.28908	−0.644538	+	0.764573i	$0.722950\pi$
$632$	−3838.46	−0.241591
$633$	0	0
$634$	−11722.5	−0.734323
$635$	5509.97	0.344341
$636$	0	0
$637$	319.973	0.0199024
$638$	−9351.00	−0.580266
$639$	0	0
$640$	1433.11	0.0885134
$641$	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	$-0.275101\pi$
$643$	21170.4	1.29841	0.649207	+	0.760612i	$0.275101\pi$
$644$	5092.12	0.311580
$645$	0	0
$646$	25848.6	1.57430
$647$	8291.45	0.503818	0.251909	−	0.967751i	$-0.418942\pi$
$647$	8291.45	0.503818	0.251909	+	0.967751i	$0.418942\pi$
$648$	0	0
$649$	7959.87	0.481436
$650$	−47.9411	−0.00289293
$651$	0	0
$652$	−1586.22	−0.0952776
$653$	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
$656$	−2122.09	−0.126301
$657$	0	0
$658$	−13943.6	−0.826108
$659$	−29954.4	−1.77065	−0.885325	−	0.464973i	$-0.846064\pi$
$659$	−29954.4	−1.77065	−0.885325	+	0.464973i	$0.846064\pi$
$660$	0	0
$661$	−4126.76	−0.242833	−0.121416	−	0.992602i	$-0.538744\pi$
$661$	−4126.76	−0.242833	−0.121416	+	0.992602i	$0.538744\pi$
$662$	9929.92	0.582987
$663$	0	0
$664$	−1438.15	−0.0840530
$665$	22716.5	1.32467
$666$	0	0
$667$	13730.5	0.797070
$668$	12714.1	0.736411
$669$	0	0
$670$	14892.4	0.858723
$671$	−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	$0.771897\pi$
$674$	−6195.32	−0.354058
$675$	0	0
$676$	9570.15	0.544501
$677$	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
$680$	10493.7	0.591788
$681$	0	0
$682$	14663.0	0.823275
$683$	12706.4	0.711854	0.355927	−	0.934514i	$-0.384165\pi$
$683$	12706.4	0.711854	0.355927	+	0.934514i	$0.384165\pi$
$684$	0	0
$685$	29365.9	1.63798
$686$	−12790.9	−0.711891
$687$	0	0
$688$	−5362.83	−0.297174
$689$	−12933.3	−0.715120
$690$	0	0
$691$	−15730.2	−0.865999	−0.432999	−	0.901394i	$-0.642545\pi$
$691$	−15730.2	−0.865999	−0.432999	+	0.901394i	$0.642545\pi$
$692$	8610.60	0.473014
$693$	0	0
$694$	16088.4	0.879982
$695$	−7037.55	−0.384100
$696$	0	0
$697$	−15538.7	−0.844434
$698$	−2289.41	−0.124148
$699$	0	0
$700$	26.0309	0.00140554
$701$	−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
$704$	−1508.43	−0.0807544
$705$	0	0
$706$	15602.5	0.831740
$707$	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	$0.729048\pi$
$710$	−11832.8	−0.625460
$711$	0	0
$712$	6771.66	0.356431
$713$	−21530.2	−1.13088
$714$	0	0
$715$	17877.2	0.935060
$716$	−17961.2	−0.937487
$717$	0	0
$718$	682.615	0.0354804
$719$	21323.8	1.10604	0.553020	−	0.833168i	$-0.313475\pi$
$719$	21323.8	1.10604	0.553020	+	0.833168i	$0.313475\pi$
$720$	0	0
$721$	−25230.0	−1.30321
$722$	10621.0	0.547467
$723$	0	0
$724$	5629.29	0.288966
$725$	70.1902	0.00359558
$726$	0	0
$727$	8032.01	0.409753	0.204877	−	0.978788i	$-0.434321\pi$
$727$	8032.01	0.409753	0.204877	+	0.978788i	$0.434321\pi$
$728$	−9968.06	−0.507474
$729$	0	0
$730$	−1653.24	−0.0838207
$731$	−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	$0.245522\pi$
$734$	238.736	0.0120053
$735$	0	0
$736$	2214.89	0.110927
$737$	−15675.2	−0.783449
$738$	0	0
$739$	−11006.3	−0.547868	−0.273934	−	0.961748i	$-0.588325\pi$
$739$	−11006.3	−0.547868	−0.273934	+	0.961748i	$0.588325\pi$
$740$	−9252.84	−0.459650
$741$	0	0
$742$	7022.46	0.347443
$743$	4652.91	0.229742	0.114871	−	0.993380i	$-0.463354\pi$
$743$	4652.91	0.229742	0.114871	+	0.993380i	$0.463354\pi$
$744$	0	0
$745$	6362.91	0.312911
$746$	−8748.86	−0.429381
$747$	0	0
$748$	−11045.3	−0.539913
$749$	−23303.0	−1.13682
$750$	0	0
$751$	−17357.7	−0.843400	−0.421700	−	0.906735i	$-0.638566\pi$
$751$	−17357.7	−0.843400	−0.421700	+	0.906735i	$0.638566\pi$
$752$	−6064.98	−0.294105
$753$	0	0
$754$	−26878.0	−1.29820
$755$	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	$0.499090\pi$
$758$	17898.9	0.857675
$759$	0	0
$760$	9880.86	0.471601
$761$	−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
$764$	3089.11	0.146283
$765$	0	0
$766$	412.926	0.0194773
$767$	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	$0.479887\pi$
$770$	−9706.89	−0.454301
$771$	0	0
$772$	14610.7	0.681154
$773$	−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
$776$	−5383.88	−0.249059
$777$	0	0
$778$	−4057.88	−0.186995
$779$	−14631.2	−0.672936
$780$	0	0
$781$	12454.7	0.570633
$782$	16218.2	0.741640
$783$	0	0
$784$	−75.5700	−0.00344251
$785$	−8142.44	−0.370212
$786$	0	0
$787$	31747.5	1.43796	0.718980	−	0.695031i	$-0.244609\pi$
$787$	31747.5	1.43796	0.718980	+	0.695031i	$0.244609\pi$
$788$	10589.6	0.478730
$789$	0	0
$790$	−10744.0	−0.483866
$791$	−32006.2	−1.43870
$792$	0	0
$793$	−18797.5	−0.841763
$794$	−13292.1	−0.594106
$795$	0	0
$796$	5880.86	0.261861
$797$	−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
$800$	11.3225	0.000500390 0
$801$	0	0
$802$	−3.26896	−0.000143929 0
$803$	1740.13	0.0764731
$804$	0	0
$805$	14253.0	0.624042
$806$	42146.4	1.84187
$807$	0	0
$808$	1652.18	0.0719351
$809$	−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	$-0.547727\pi$
$811$	−6900.03	−0.298758	−0.149379	+	0.988780i	$0.547727\pi$
$812$	14594.2	0.630732
$813$	0	0
$814$	9739.16	0.419358
$815$	−4439.88	−0.190825
$816$	0	0
$817$	−36975.2	−1.58335
$818$	−12406.7	−0.530306
$819$	0	0
$820$	−5939.81	−0.252960
$821$	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	$-0.494664\pi$
$823$	791.570	0.0335266	0.0167633	+	0.999859i	$0.494664\pi$
$824$	−10974.2	−0.463960
$825$	0	0
$826$	−12423.0	−0.523307
$827$	23005.9	0.967343	0.483671	−	0.875250i	$-0.339303\pi$
$827$	23005.9	0.967343	0.483671	+	0.875250i	$0.339303\pi$
$828$	0	0
$829$	−15420.2	−0.646040	−0.323020	−	0.946392i	$-0.604698\pi$
$829$	−15420.2	−0.646040	−0.323020	+	0.946392i	$0.604698\pi$
$830$	−4025.45	−0.168344
$831$	0	0
$832$	−4335.75	−0.180667
$833$	−553.350	−0.0230161
$834$	0	0
$835$	35587.2	1.47491
$836$	−10400.2	−0.430261
$837$	0	0
$838$	−19500.7	−0.803869
$839$	46290.1	1.90478	0.952391	−	0.304878i	$-0.0986158\pi$
$839$	46290.1	1.90478	0.952391	+	0.304878i	$0.0986158\pi$
$840$	0	0
$841$	14962.9	0.613509
$842$	−20122.8	−0.823605
$843$	0	0
$844$	6144.03	0.250576
$845$	26787.2	1.09054
$846$	0	0
$847$	−14263.1	−0.578613
$848$	3054.52	0.123694
$849$	0	0
$850$	82.9076	0.00334554
$851$	−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	$0.310801\pi$
$854$	10206.6	0.408973
$855$	0	0
$856$	−10136.0	−0.404721
$857$	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	$-0.256608\pi$
$859$	34857.7	1.38455	0.692275	+	0.721634i	$0.256608\pi$
$860$	−15010.8	−0.595189
$861$	0	0
$862$	−13526.4	−0.534469
$863$	−10885.2	−0.429359	−0.214680	−	0.976685i	$-0.568871\pi$
$863$	−10885.2	−0.429359	−0.214680	+	0.976685i	$0.568871\pi$
$864$	0	0
$865$	24101.4	0.947367
$866$	−21202.8	−0.831985
$867$	0	0
$868$	−22884.6	−0.894876
$869$	11308.7	0.441451
$870$	0	0
$871$	−45055.9	−1.75277
$872$	13801.0	0.535966
$873$	0	0
$874$	15271.0	0.591019
$875$	−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	$0.680590\pi$
$878$	−25137.8	−0.966243
$879$	0	0
$880$	−4222.15	−0.161737
$881$	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	$-0.480178\pi$
$883$	3265.74	0.124463	0.0622315	+	0.998062i	$0.480178\pi$
$884$	−31747.9	−1.20792
$885$	0	0
$886$	−20510.9	−0.777738
$887$	9392.88	0.355560	0.177780	−	0.984070i	$-0.443108\pi$
$887$	9392.88	0.355560	0.177780	+	0.984070i	$0.443108\pi$
$888$	0	0
$889$	9051.41	0.341479
$890$	18954.1	0.713870
$891$	0	0
$892$	−6630.12	−0.248871
$893$	−41816.3	−1.56700
$894$	0	0
$895$	−50274.0	−1.87762
$896$	2354.22	0.0877777
$897$	0	0
$898$	−8160.46	−0.303250
$899$	−61706.2	−2.28923
$900$	0	0
$901$	22366.3	0.827002
$902$	6252.00	0.230786
$903$	0	0
$904$	−13921.6	−0.512195
$905$	15756.6	0.578748
$906$	0	0
$907$	13538.5	0.495631	0.247816	−	0.968807i	$-0.420287\pi$
$907$	13538.5	0.495631	0.247816	+	0.968807i	$0.420287\pi$
$908$	−6057.04	−0.221377
$909$	0	0
$910$	−27901.0	−1.01638
$911$	37806.4	1.37495	0.687476	−	0.726207i	$-0.258718\pi$
$911$	37806.4	1.37495	0.687476	+	0.726207i	$0.258718\pi$
$912$	0	0
$913$	4237.02	0.153587
$914$	−4366.40	−0.158017
$915$	0	0
$916$	17196.2	0.620280
$917$	35299.9	1.27122
$918$	0	0
$919$	30674.2	1.10103	0.550515	−	0.834825i	$-0.314431\pi$
$919$	30674.2	1.10103	0.550515	+	0.834825i	$0.314431\pi$
$920$	6199.57	0.222167
$921$	0	0
$922$	6500.52	0.232194
$923$	35799.1	1.27664
$924$	0	0
$925$	−73.1038	−0.00259853
$926$	37982.3	1.34792
$927$	0	0
$928$	6347.94	0.224549
$929$	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
$932$	5347.12	0.187930
$933$	0	0
$934$	13813.0	0.483915
$935$	−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	$0.312571\pi$
$938$	24464.3	0.851586
$939$	0	0
$940$	−16976.1	−0.589043
$941$	−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
$944$	−5403.57	−0.186304
$945$	0	0
$946$	15799.7	0.543016
$947$	−6985.33	−0.239697	−0.119848	−	0.992792i	$-0.538241\pi$
$947$	−6985.33	−0.239697	−0.119848	+	0.992792i	$0.538241\pi$
$948$	0	0
$949$	5001.74	0.171089
$950$	78.0656	0.00266609
$951$	0	0
$952$	17238.4	0.586869
$953$	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
$956$	−27514.5	−0.930840
$957$	0	0
$958$	14760.5	0.497799
$959$	48240.4	1.62436
$960$	0	0
$961$	66968.2	2.24794
$962$	27993.7	0.938206
$963$	0	0
$964$	6125.17	0.204646
$965$	40895.9	1.36423
$966$	0	0
$967$	−100.156	−0.00333072	−0.00166536	−	0.999999i	$-0.500530\pi$
$967$	−100.156	−0.00333072	−0.00166536	+	0.999999i	$0.500530\pi$
$968$	−6203.94	−0.205994
$969$	0	0
$970$	−15069.7	−0.498823
$971$	678.145	0.0224127	0.0112063	−	0.999937i	$-0.496433\pi$
$971$	678.145	0.0224127	0.0112063	+	0.999937i	$0.496433\pi$
$972$	0	0
$973$	−11560.8	−0.380908
$974$	−17513.0	−0.576133
$975$	0	0
$976$	4439.51	0.145600
$977$	−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
$980$	−211.523	−0.00689476
$981$	0	0
$982$	23675.9	0.769376
$983$	−3515.69	−0.114072	−0.0570361	−	0.998372i	$-0.518165\pi$
$983$	−3515.69	−0.114072	−0.0570361	+	0.998372i	$0.518165\pi$
$984$	0	0
$985$	29640.7	0.958815
$986$	46481.9	1.50130
$987$	0	0
$988$	−29893.8	−0.962598
$989$	−23199.4	−0.745903
$990$	0	0
$991$	−612.517	−0.0196339	−0.00981697	−	0.999952i	$-0.503125\pi$
$991$	−612.517	−0.0196339	−0.00981697	+	0.999952i	$0.503125\pi$
$992$	−9953.97	−0.318587
$993$	0	0
$994$	−19438.1	−0.620261
$995$	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	$0.381202\pi$
$998$	17156.8	0.544176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.h.1.2	yes	2
3.2	odd	2		162.4.a.e.1.1	✓	2
4.3	odd	2		1296.4.a.s.1.2		2
9.2	odd	6		162.4.c.j.109.2		4
9.4	even	3		162.4.c.i.55.1		4
9.5	odd	6		162.4.c.j.55.2		4
9.7	even	3		162.4.c.i.109.1		4
12.11	even	2		1296.4.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.1	✓	2	3.2	odd	2
162.4.a.h.1.2	yes	2	1.1	even	1	trivial
162.4.c.i.55.1		4	9.4	even	3
162.4.c.i.109.1		4	9.7	even	3
162.4.c.j.55.2		4	9.5	odd	6
162.4.c.j.109.2		4	9.2	odd	6
1296.4.a.j.1.1		2	12.11	even	2
1296.4.a.s.1.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +11.1962 q^{5} +18.3923 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +11.1962 q^{5} +18.3923 q^{7} +8.00000 q^{8} +22.3923 q^{10} -23.5692 q^{11} -67.7461 q^{13} +36.7846 q^{14} +16.0000 q^{16} +117.158 q^{17} +110.315 q^{19} +44.7846 q^{20} -47.1384 q^{22} +69.2154 q^{23} +0.353829 q^{25} -135.492 q^{26} +73.5692 q^{28} +198.373 q^{29} -311.061 q^{31} +32.0000 q^{32} +234.315 q^{34} +205.923 q^{35} -206.608 q^{37} +220.631 q^{38} +89.5692 q^{40} -132.631 q^{41} -335.177 q^{43} -94.2769 q^{44} +138.431 q^{46} -379.061 q^{47} -4.72312 q^{49} +0.707658 q^{50} -270.985 q^{52} +190.908 q^{53} -263.885 q^{55} +147.138 q^{56} +396.746 q^{58} -337.723 q^{59} +277.469 q^{61} -622.123 q^{62} +64.0000 q^{64} -758.496 q^{65} +665.069 q^{67} +468.631 q^{68} +411.846 q^{70} -528.431 q^{71} -73.8306 q^{73} -413.215 q^{74} +441.261 q^{76} -433.492 q^{77} -479.808 q^{79} +179.138 q^{80} -265.261 q^{82} -179.769 q^{83} +1311.72 q^{85} -670.354 q^{86} -188.554 q^{88} +846.458 q^{89} -1246.01 q^{91} +276.862 q^{92} -758.123 q^{94} +1235.11 q^{95} -672.985 q^{97} -9.44624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{7} + 16 q^{8} + 24 q^{10} + 36 q^{11} + 10 q^{13} + 32 q^{14} + 32 q^{16} + 120 q^{17} - 8 q^{19} + 48 q^{20} + 72 q^{22} + 180 q^{23} - 124 q^{25} + 20 q^{26} + 64 q^{28} + 324 q^{29} - 248 q^{31} + 64 q^{32} + 240 q^{34} + 204 q^{35} - 434 q^{37} - 16 q^{38} + 96 q^{40} + 192 q^{41} - 608 q^{43} + 144 q^{44} + 360 q^{46} - 384 q^{47} - 342 q^{49} - 248 q^{50} + 40 q^{52} - 408 q^{53} - 216 q^{55} + 128 q^{56} + 648 q^{58} - 1008 q^{59} + 742 q^{61} - 496 q^{62} + 128 q^{64} - 696 q^{65} - 104 q^{67} + 480 q^{68} + 408 q^{70} - 1140 q^{71} + 850 q^{73} - 868 q^{74} - 32 q^{76} - 576 q^{77} - 440 q^{79} + 192 q^{80} + 384 q^{82} + 264 q^{83} + 1314 q^{85} - 1216 q^{86} + 288 q^{88} + 768 q^{89} - 1432 q^{91} + 720 q^{92} - 768 q^{94} + 1140 q^{95} - 764 q^{97} - 684 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^7 + 16 * q^8 + 24 * q^10 + 36 * q^11 + 10 * q^13 + 32 * q^14 + 32 * q^16 + 120 * q^17 - 8 * q^19 + 48 * q^20 + 72 * q^22 + 180 * q^23 - 124 * q^25 + 20 * q^26 + 64 * q^28 + 324 * q^29 - 248 * q^31 + 64 * q^32 + 240 * q^34 + 204 * q^35 - 434 * q^37 - 16 * q^38 + 96 * q^40 + 192 * q^41 - 608 * q^43 + 144 * q^44 + 360 * q^46 - 384 * q^47 - 342 * q^49 - 248 * q^50 + 40 * q^52 - 408 * q^53 - 216 * q^55 + 128 * q^56 + 648 * q^58 - 1008 * q^59 + 742 * q^61 - 496 * q^62 + 128 * q^64 - 696 * q^65 - 104 * q^67 + 480 * q^68 + 408 * q^70 - 1140 * q^71 + 850 * q^73 - 868 * q^74 - 32 * q^76 - 576 * q^77 - 440 * q^79 + 192 * q^80 + 384 * q^82 + 264 * q^83 + 1314 * q^85 - 1216 * q^86 + 288 * q^88 + 768 * q^89 - 1432 * q^91 + 720 * q^92 - 768 * q^94 + 1140 * q^95 - 764 * q^97 - 684 * q^98

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