LMFDB - Embedded newform 162.4.a.g.1.1

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(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.62348$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -10.8704 q^{5} +24.8704 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -10.8704 q^{5} +24.8704 q^{7} +8.00000 q^{8} -21.7409 q^{10} +42.7409 q^{11} +15.1296 q^{13} +49.7409 q^{14} +16.0000 q^{16} +13.8704 q^{17} +143.352 q^{19} -43.4817 q^{20} +85.4817 q^{22} -19.1296 q^{23} -6.83384 q^{25} +30.2591 q^{26} +99.4817 q^{28} -226.093 q^{29} +59.3887 q^{31} +32.0000 q^{32} +27.7409 q^{34} -270.352 q^{35} -84.1860 q^{37} +286.704 q^{38} -86.9634 q^{40} -203.259 q^{41} +325.890 q^{43} +170.963 q^{44} -38.2591 q^{46} -10.9436 q^{47} +275.538 q^{49} -13.6677 q^{50} +60.5183 q^{52} +140.186 q^{53} -464.611 q^{55} +198.963 q^{56} -452.186 q^{58} -114.741 q^{59} -755.056 q^{61} +118.777 q^{62} +64.0000 q^{64} -164.465 q^{65} -767.445 q^{67} +55.4817 q^{68} -540.704 q^{70} -335.854 q^{71} +167.279 q^{73} -168.372 q^{74} +573.409 q^{76} +1062.98 q^{77} -25.3155 q^{79} -173.927 q^{80} -406.518 q^{82} -287.721 q^{83} -150.777 q^{85} +651.780 q^{86} +341.927 q^{88} +860.817 q^{89} +376.279 q^{91} -76.5183 q^{92} -21.8872 q^{94} -1558.30 q^{95} -402.149 q^{97} +551.076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26} + 76 q^{28} - 237 q^{29} + 211 q^{31} + 64 q^{32} - 6 q^{34} - 387 q^{35} + 262 q^{37} + 266 q^{38} + 72 q^{40} - 468 q^{41} - 86 q^{43} + 96 q^{44} - 138 q^{46} - 483 q^{47} - 33 q^{49} + 526 q^{50} + 244 q^{52} - 150 q^{53} - 837 q^{55} + 152 q^{56} - 474 q^{58} - 168 q^{59} - 1049 q^{61} + 422 q^{62} + 128 q^{64} + 747 q^{65} - 1166 q^{67} - 12 q^{68} - 774 q^{70} + 312 q^{71} - 311 q^{73} + 524 q^{74} + 532 q^{76} + 1173 q^{77} + 349 q^{79} + 144 q^{80} - 936 q^{82} - 1221 q^{83} - 486 q^{85} - 172 q^{86} + 192 q^{88} + 492 q^{89} + 107 q^{91} - 276 q^{92} - 966 q^{94} - 1764 q^{95} - 128 q^{97} - 66 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8 + 18 * q^10 + 24 * q^11 + 61 * q^13 + 38 * q^14 + 32 * q^16 - 3 * q^17 + 133 * q^19 + 36 * q^20 + 48 * q^22 - 69 * q^23 + 263 * q^25 + 122 * q^26 + 76 * q^28 - 237 * q^29 + 211 * q^31 + 64 * q^32 - 6 * q^34 - 387 * q^35 + 262 * q^37 + 266 * q^38 + 72 * q^40 - 468 * q^41 - 86 * q^43 + 96 * q^44 - 138 * q^46 - 483 * q^47 - 33 * q^49 + 526 * q^50 + 244 * q^52 - 150 * q^53 - 837 * q^55 + 152 * q^56 - 474 * q^58 - 168 * q^59 - 1049 * q^61 + 422 * q^62 + 128 * q^64 + 747 * q^65 - 1166 * q^67 - 12 * q^68 - 774 * q^70 + 312 * q^71 - 311 * q^73 + 524 * q^74 + 532 * q^76 + 1173 * q^77 + 349 * q^79 + 144 * q^80 - 936 * q^82 - 1221 * q^83 - 486 * q^85 - 172 * q^86 + 192 * q^88 + 492 * q^89 + 107 * q^91 - 276 * q^92 - 966 * q^94 - 1764 * q^95 - 128 * q^97 - 66 * 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
$5$	−10.8704	−0.972280	−0.486140	−	0.873881i	$-0.661596\pi$
$5$	−10.8704	−0.972280	−0.486140	+	0.873881i	$0.661596\pi$
$6$	0	0
$7$	24.8704	1.34288	0.671438	−	0.741060i	$-0.265677\pi$
$7$	24.8704	1.34288	0.671438	+	0.741060i	$0.265677\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−21.7409	−0.687506
$11$	42.7409	1.17153	0.585766	−	0.810480i	$-0.300794\pi$
$11$	42.7409	1.17153	0.585766	+	0.810480i	$0.300794\pi$
$12$	0	0
$13$	15.1296	0.322784	0.161392	−	0.986890i	$-0.448402\pi$
$13$	15.1296	0.322784	0.161392	+	0.986890i	$0.448402\pi$
$14$	49.7409	0.949557
$15$	0	0
$16$	16.0000	0.250000
$17$	13.8704	0.197887	0.0989433	−	0.995093i	$-0.468454\pi$
$17$	13.8704	0.197887	0.0989433	+	0.995093i	$0.468454\pi$
$18$	0	0
$19$	143.352	1.73091	0.865454	−	0.500989i	$-0.167030\pi$
$19$	143.352	1.73091	0.865454	+	0.500989i	$0.167030\pi$
$20$	−43.4817	−0.486140
$21$	0	0
$22$	85.4817	0.828398
$23$	−19.1296	−0.173426	−0.0867129	−	0.996233i	$-0.527636\pi$
$23$	−19.1296	−0.173426	−0.0867129	+	0.996233i	$0.527636\pi$
$24$	0	0
$25$	−6.83384	−0.0546707
$26$	30.2591	0.228243
$27$	0	0
$28$	99.4817	0.671438
$29$	−226.093	−1.44774	−0.723869	−	0.689937i	$-0.757638\pi$
$29$	−226.093	−1.44774	−0.723869	+	0.689937i	$0.757638\pi$
$30$	0	0
$31$	59.3887	0.344082	0.172041	−	0.985090i	$-0.444964\pi$
$31$	59.3887	0.344082	0.172041	+	0.985090i	$0.444964\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	27.7409	0.139927
$35$	−270.352	−1.30565
$36$	0	0
$37$	−84.1860	−0.374056	−0.187028	−	0.982355i	$-0.559886\pi$
$37$	−84.1860	−0.374056	−0.187028	+	0.982355i	$0.559886\pi$
$38$	286.704	1.22394
$39$	0	0
$40$	−86.9634	−0.343753
$41$	−203.259	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$41$	−203.259	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$42$	0	0
$43$	325.890	1.15576	0.577881	−	0.816121i	$-0.303880\pi$
$43$	325.890	1.15576	0.577881	+	0.816121i	$0.303880\pi$
$44$	170.963	0.585766
$45$	0	0
$46$	−38.2591	−0.122631
$47$	−10.9436	−0.0339636	−0.0169818	−	0.999856i	$-0.505406\pi$
$47$	−10.9436	−0.0339636	−0.0169818	+	0.999856i	$0.505406\pi$
$48$	0	0
$49$	275.538	0.803318
$50$	−13.6677	−0.0386580
$51$	0	0
$52$	60.5183	0.161392
$53$	140.186	0.363321	0.181661	−	0.983361i	$-0.441853\pi$
$53$	140.186	0.363321	0.181661	+	0.983361i	$0.441853\pi$
$54$	0	0
$55$	−464.611	−1.13906
$56$	198.963	0.474779
$57$	0	0
$58$	−452.186	−1.02371
$59$	−114.741	−0.253186	−0.126593	−	0.991955i	$-0.540404\pi$
$59$	−114.741	−0.253186	−0.126593	+	0.991955i	$0.540404\pi$
$60$	0	0
$61$	−755.056	−1.58484	−0.792419	−	0.609978i	$-0.791178\pi$
$61$	−755.056	−1.58484	−0.792419	+	0.609978i	$0.791178\pi$
$62$	118.777	0.243302
$63$	0	0
$64$	64.0000	0.125000
$65$	−164.465	−0.313836
$66$	0	0
$67$	−767.445	−1.39938	−0.699689	−	0.714447i	$-0.746678\pi$
$67$	−767.445	−1.39938	−0.699689	+	0.714447i	$0.746678\pi$
$68$	55.4817	0.0989433
$69$	0	0
$70$	−540.704	−0.923236
$71$	−335.854	−0.561387	−0.280694	−	0.959797i	$-0.590564\pi$
$71$	−335.854	−0.561387	−0.280694	+	0.959797i	$0.590564\pi$
$72$	0	0
$73$	167.279	0.268199	0.134099	−	0.990968i	$-0.457186\pi$
$73$	167.279	0.268199	0.134099	+	0.990968i	$0.457186\pi$
$74$	−168.372	−0.264498
$75$	0	0
$76$	573.409	0.865454
$77$	1062.98	1.57322
$78$	0	0
$79$	−25.3155	−0.0360534	−0.0180267	−	0.999838i	$-0.505738\pi$
$79$	−25.3155	−0.0360534	−0.0180267	+	0.999838i	$0.505738\pi$
$80$	−173.927	−0.243070
$81$	0	0
$82$	−406.518	−0.547469
$83$	−287.721	−0.380500	−0.190250	−	0.981736i	$-0.560930\pi$
$83$	−287.721	−0.380500	−0.190250	+	0.981736i	$0.560930\pi$
$84$	0	0
$85$	−150.777	−0.192401
$86$	651.780	0.817248
$87$	0	0
$88$	341.927	0.414199
$89$	860.817	1.02524	0.512620	−	0.858615i	$-0.328675\pi$
$89$	860.817	1.02524	0.512620	+	0.858615i	$0.328675\pi$
$90$	0	0
$91$	376.279	0.433459
$92$	−76.5183	−0.0867129
$93$	0	0
$94$	−21.8872	−0.0240159
$95$	−1558.30	−1.68293
$96$	0	0
$97$	−402.149	−0.420949	−0.210475	−	0.977599i	$-0.567501\pi$
$97$	−402.149	−0.420949	−0.210475	+	0.977599i	$0.567501\pi$
$98$	551.076	0.568032
$99$	0	0
$100$	−27.3353	−0.0273353
$101$	1331.43	1.31170	0.655852	−	0.754890i	$-0.272310\pi$
$101$	1331.43	1.31170	0.655852	+	0.754890i	$0.272310\pi$
$102$	0	0
$103$	−518.505	−0.496017	−0.248009	−	0.968758i	$-0.579776\pi$
$103$	−518.505	−0.496017	−0.248009	+	0.968758i	$0.579776\pi$
$104$	121.037	0.114121
$105$	0	0
$106$	280.372	0.256907
$107$	−1471.87	−1.32982	−0.664912	−	0.746922i	$-0.731531\pi$
$107$	−1471.87	−1.32982	−0.664912	+	0.746922i	$0.731531\pi$
$108$	0	0
$109$	−643.668	−0.565616	−0.282808	−	0.959176i	$-0.591266\pi$
$109$	−643.668	−0.565616	−0.282808	+	0.959176i	$0.591266\pi$
$110$	−929.223	−0.805435
$111$	0	0
$112$	397.927	0.335719
$113$	−1023.73	−0.852249	−0.426125	−	0.904665i	$-0.640121\pi$
$113$	−1023.73	−0.852249	−0.426125	+	0.904665i	$0.640121\pi$
$114$	0	0
$115$	207.947	0.168618
$116$	−904.372	−0.723869
$117$	0	0
$118$	−229.482	−0.179030
$119$	344.963	0.265737
$120$	0	0
$121$	495.780	0.372487
$122$	−1510.11	−1.12065
$123$	0	0
$124$	237.555	0.172041
$125$	1433.09	1.02544
$126$	0	0
$127$	−31.4481	−0.0219730	−0.0109865	−	0.999940i	$-0.503497\pi$
$127$	−31.4481	−0.0219730	−0.0109865	+	0.999940i	$0.503497\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−328.930	−0.221916
$131$	1937.36	1.29212	0.646059	−	0.763287i	$-0.276416\pi$
$131$	1937.36	1.29212	0.646059	+	0.763287i	$0.276416\pi$
$132$	0	0
$133$	3565.23	2.32439
$134$	−1534.89	−0.989510
$135$	0	0
$136$	110.963	0.0699635
$137$	−1158.59	−0.722521	−0.361261	−	0.932465i	$-0.617654\pi$
$137$	−1158.59	−0.722521	−0.361261	+	0.932465i	$0.617654\pi$
$138$	0	0
$139$	2311.60	1.41055	0.705277	−	0.708931i	$-0.250822\pi$
$139$	2311.60	1.41055	0.705277	+	0.708931i	$0.250822\pi$
$140$	−1081.41	−0.652826
$141$	0	0
$142$	−671.707	−0.396961
$143$	646.651	0.378151
$144$	0	0
$145$	2457.73	1.40761
$146$	334.558	0.189645
$147$	0	0
$148$	−336.744	−0.187028
$149$	2950.21	1.62209	0.811043	−	0.584987i	$-0.198900\pi$
$149$	2950.21	1.62209	0.811043	+	0.584987i	$0.198900\pi$
$150$	0	0
$151$	−1726.40	−0.930412	−0.465206	−	0.885202i	$-0.654020\pi$
$151$	−1726.40	−0.930412	−0.465206	+	0.885202i	$0.654020\pi$
$152$	1146.82	0.611968
$153$	0	0
$154$	2125.97	1.11244
$155$	−645.581	−0.334544
$156$	0	0
$157$	1283.39	0.652392	0.326196	−	0.945302i	$-0.394233\pi$
$157$	1283.39	0.652392	0.326196	+	0.945302i	$0.394233\pi$
$158$	−50.6311	−0.0254936
$159$	0	0
$160$	−347.854	−0.171877
$161$	−475.761	−0.232889
$162$	0	0
$163$	1033.93	0.496831	0.248415	−	0.968654i	$-0.420090\pi$
$163$	1033.93	0.496831	0.248415	+	0.968654i	$0.420090\pi$
$164$	−813.037	−0.387119
$165$	0	0
$166$	−575.442	−0.269054
$167$	282.617	0.130956	0.0654778	−	0.997854i	$-0.479143\pi$
$167$	282.617	0.130956	0.0654778	+	0.997854i	$0.479143\pi$
$168$	0	0
$169$	−1968.10	−0.895811
$170$	−301.555	−0.136048
$171$	0	0
$172$	1303.56	0.577881
$173$	3532.72	1.55253	0.776266	−	0.630405i	$-0.217111\pi$
$173$	3532.72	1.55253	0.776266	+	0.630405i	$0.217111\pi$
$174$	0	0
$175$	−169.960	−0.0734160
$176$	683.854	0.292883
$177$	0	0
$178$	1721.63	0.724955
$179$	4052.74	1.69227	0.846135	−	0.532969i	$-0.178924\pi$
$179$	4052.74	1.69227	0.846135	+	0.532969i	$0.178924\pi$
$180$	0	0
$181$	−2830.97	−1.16257	−0.581283	−	0.813702i	$-0.697449\pi$
$181$	−2830.97	−1.16257	−0.581283	+	0.813702i	$0.697449\pi$
$182$	752.558	0.306502
$183$	0	0
$184$	−153.037	−0.0613153
$185$	915.137	0.363688
$186$	0	0
$187$	592.834	0.231831
$188$	−43.7744	−0.0169818
$189$	0	0
$190$	−3116.60	−1.19001
$191$	−4509.24	−1.70826	−0.854129	−	0.520061i	$-0.825909\pi$
$191$	−4509.24	−1.70826	−0.854129	+	0.520061i	$0.825909\pi$
$192$	0	0
$193$	−3221.12	−1.20136	−0.600678	−	0.799491i	$-0.705103\pi$
$193$	−3221.12	−1.20136	−0.600678	+	0.799491i	$0.705103\pi$
$194$	−804.299	−0.297656
$195$	0	0
$196$	1102.15	0.401659
$197$	−3784.20	−1.36859	−0.684297	−	0.729204i	$-0.739891\pi$
$197$	−3784.20	−1.36859	−0.684297	+	0.729204i	$0.739891\pi$
$198$	0	0
$199$	−2926.27	−1.04240	−0.521200	−	0.853435i	$-0.674515\pi$
$199$	−2926.27	−1.04240	−0.521200	+	0.853435i	$0.674515\pi$
$200$	−54.6707	−0.0193290
$201$	0	0
$202$	2662.86	0.927515
$203$	−5623.03	−1.94413
$204$	0	0
$205$	2209.51	0.752776
$206$	−1037.01	−0.350737
$207$	0	0
$208$	242.073	0.0806959
$209$	6126.99	2.02781
$210$	0	0
$211$	313.474	0.102277	0.0511385	−	0.998692i	$-0.483715\pi$
$211$	313.474	0.102277	0.0511385	+	0.998692i	$0.483715\pi$
$212$	560.744	0.181661
$213$	0	0
$214$	−2943.74	−0.940327
$215$	−3542.57	−1.12373
$216$	0	0
$217$	1477.02	0.462059
$218$	−1287.34	−0.399951
$219$	0	0
$220$	−1858.45	−0.569529
$221$	209.854	0.0638746
$222$	0	0
$223$	−710.386	−0.213323	−0.106661	−	0.994295i	$-0.534016\pi$
$223$	−710.386	−0.213323	−0.106661	+	0.994295i	$0.534016\pi$
$224$	795.854	0.237389
$225$	0	0
$226$	−2047.45	−0.602631
$227$	−32.6616	−0.00954991	−0.00477496	−	0.999989i	$-0.501520\pi$
$227$	−32.6616	−0.00954991	−0.00477496	+	0.999989i	$0.501520\pi$
$228$	0	0
$229$	−5503.85	−1.58823	−0.794114	−	0.607768i	$-0.792065\pi$
$229$	−5503.85	−1.58823	−0.794114	+	0.607768i	$0.792065\pi$
$230$	415.893	0.119231
$231$	0	0
$232$	−1808.74	−0.511853
$233$	6788.81	1.90880	0.954399	−	0.298534i	$-0.0964977\pi$
$233$	6788.81	1.90880	0.954399	+	0.298534i	$0.0964977\pi$
$234$	0	0
$235$	118.962	0.0330221
$236$	−458.963	−0.126593
$237$	0	0
$238$	689.927	0.187905
$239$	429.389	0.116213	0.0581064	−	0.998310i	$-0.481494\pi$
$239$	429.389	0.116213	0.0581064	+	0.998310i	$0.481494\pi$
$240$	0	0
$241$	−4843.65	−1.29463	−0.647317	−	0.762221i	$-0.724109\pi$
$241$	−4843.65	−1.29463	−0.647317	+	0.762221i	$0.724109\pi$
$242$	991.561	0.263388
$243$	0	0
$244$	−3020.23	−0.792419
$245$	−2995.22	−0.781050
$246$	0	0
$247$	2168.86	0.558709
$248$	475.110	0.121651
$249$	0	0
$250$	2866.18	0.725093
$251$	−2400.87	−0.603752	−0.301876	−	0.953347i	$-0.597613\pi$
$251$	−2400.87	−0.603752	−0.301876	+	0.953347i	$0.597613\pi$
$252$	0	0
$253$	−817.614	−0.203174
$254$	−62.8963	−0.0155373
$255$	0	0
$256$	256.000	0.0625000
$257$	−2980.99	−0.723538	−0.361769	−	0.932268i	$-0.617827\pi$
$257$	−2980.99	−0.723538	−0.361769	+	0.932268i	$0.617827\pi$
$258$	0	0
$259$	−2093.74	−0.502312
$260$	−657.860	−0.156918
$261$	0	0
$262$	3874.71	0.913666
$263$	−5560.82	−1.30378	−0.651891	−	0.758313i	$-0.726024\pi$
$263$	−5560.82	−1.30378	−0.651891	+	0.758313i	$0.726024\pi$
$264$	0	0
$265$	−1523.88	−0.353250
$266$	7130.46	1.64360
$267$	0	0
$268$	−3069.78	−0.699689
$269$	−6288.50	−1.42534	−0.712671	−	0.701499i	$-0.752515\pi$
$269$	−6288.50	−1.42534	−0.712671	+	0.701499i	$0.752515\pi$
$270$	0	0
$271$	6854.90	1.53655	0.768275	−	0.640119i	$-0.221115\pi$
$271$	6854.90	1.53655	0.768275	+	0.640119i	$0.221115\pi$
$272$	221.927	0.0494717
$273$	0	0
$274$	−2317.19	−0.510900
$275$	−292.084	−0.0640485
$276$	0	0
$277$	898.172	0.194823	0.0974114	−	0.995244i	$-0.468944\pi$
$277$	898.172	0.194823	0.0974114	+	0.995244i	$0.468944\pi$
$278$	4623.20	0.997413
$279$	0	0
$280$	−2162.82	−0.461618
$281$	−238.770	−0.0506897	−0.0253448	−	0.999679i	$-0.508068\pi$
$281$	−238.770	−0.0506897	−0.0253448	+	0.999679i	$0.508068\pi$
$282$	0	0
$283$	2071.16	0.435045	0.217523	−	0.976055i	$-0.430202\pi$
$283$	2071.16	0.435045	0.217523	+	0.976055i	$0.430202\pi$
$284$	−1343.41	−0.280694
$285$	0	0
$286$	1293.30	0.267393
$287$	−5055.14	−1.03971
$288$	0	0
$289$	−4720.61	−0.960841
$290$	4915.45	0.995329
$291$	0	0
$292$	669.116	0.134099
$293$	6577.76	1.31153	0.655763	−	0.754967i	$-0.272347\pi$
$293$	6577.76	1.31153	0.655763	+	0.754967i	$0.272347\pi$
$294$	0	0
$295$	1247.28	0.246168
$296$	−673.488	−0.132249
$297$	0	0
$298$	5900.42	1.14699
$299$	−289.422	−0.0559790
$300$	0	0
$301$	8105.03	1.55205
$302$	−3452.80	−0.657901
$303$	0	0
$304$	2293.63	0.432727
$305$	8207.78	1.54091
$306$	0	0
$307$	−5237.30	−0.973644	−0.486822	−	0.873501i	$-0.661844\pi$
$307$	−5237.30	−0.973644	−0.486822	+	0.873501i	$0.661844\pi$
$308$	4251.93	0.786612
$309$	0	0
$310$	−1291.16	−0.236558
$311$	−5704.99	−1.04019	−0.520097	−	0.854107i	$-0.674104\pi$
$311$	−5704.99	−1.04019	−0.520097	+	0.854107i	$0.674104\pi$
$312$	0	0
$313$	5077.48	0.916920	0.458460	−	0.888715i	$-0.348401\pi$
$313$	5077.48	0.916920	0.458460	+	0.888715i	$0.348401\pi$
$314$	2566.78	0.461311
$315$	0	0
$316$	−101.262	−0.0180267
$317$	−2868.91	−0.508309	−0.254155	−	0.967164i	$-0.581797\pi$
$317$	−2868.91	−0.508309	−0.254155	+	0.967164i	$0.581797\pi$
$318$	0	0
$319$	−9663.41	−1.69607
$320$	−695.707	−0.121535
$321$	0	0
$322$	−951.521	−0.164678
$323$	1988.36	0.342523
$324$	0	0
$325$	−103.393	−0.0176468
$326$	2067.85	0.351312
$327$	0	0
$328$	−1626.07	−0.273734
$329$	−272.172	−0.0456089
$330$	0	0
$331$	2031.35	0.337320	0.168660	−	0.985674i	$-0.446056\pi$
$331$	2031.35	0.337320	0.168660	+	0.985674i	$0.446056\pi$
$332$	−1150.88	−0.190250
$333$	0	0
$334$	565.235	0.0925996
$335$	8342.46	1.36059
$336$	0	0
$337$	9798.28	1.58382	0.791909	−	0.610640i	$-0.209088\pi$
$337$	9798.28	1.58382	0.791909	+	0.610640i	$0.209088\pi$
$338$	−3936.19	−0.633434
$339$	0	0
$340$	−603.110	−0.0962006
$341$	2538.32	0.403103
$342$	0	0
$343$	−1677.81	−0.264120
$344$	2607.12	0.408624
$345$	0	0
$346$	7065.45	1.09781
$347$	4556.56	0.704925	0.352463	−	0.935826i	$-0.385344\pi$
$347$	4556.56	0.704925	0.352463	+	0.935826i	$0.385344\pi$
$348$	0	0
$349$	3348.89	0.513644	0.256822	−	0.966459i	$-0.417325\pi$
$349$	3348.89	0.513644	0.256822	+	0.966459i	$0.417325\pi$
$350$	−339.921	−0.0519129
$351$	0	0
$352$	1367.71	0.207100
$353$	−1862.27	−0.280789	−0.140395	−	0.990096i	$-0.544837\pi$
$353$	−1862.27	−0.280789	−0.140395	+	0.990096i	$0.544837\pi$
$354$	0	0
$355$	3650.87	0.545826
$356$	3443.27	0.512620
$357$	0	0
$358$	8105.49	1.19662
$359$	−6179.46	−0.908466	−0.454233	−	0.890883i	$-0.650087\pi$
$359$	−6179.46	−0.908466	−0.454233	+	0.890883i	$0.650087\pi$
$360$	0	0
$361$	13690.8	1.99604
$362$	−5661.94	−0.822058
$363$	0	0
$364$	1505.12	0.216729
$365$	−1818.39	−0.260765
$366$	0	0
$367$	6873.95	0.977704	0.488852	−	0.872367i	$-0.337416\pi$
$367$	6873.95	0.977704	0.488852	+	0.872367i	$0.337416\pi$
$368$	−306.073	−0.0433564
$369$	0	0
$370$	1830.27	0.257166
$371$	3486.48	0.487896
$372$	0	0
$373$	1270.03	0.176299	0.0881494	−	0.996107i	$-0.471905\pi$
$373$	1270.03	0.176299	0.0881494	+	0.996107i	$0.471905\pi$
$374$	1185.67	0.163929
$375$	0	0
$376$	−87.5489	−0.0120079
$377$	−3420.69	−0.467306
$378$	0	0
$379$	2490.54	0.337548	0.168774	−	0.985655i	$-0.446019\pi$
$379$	2490.54	0.337548	0.168774	+	0.985655i	$0.446019\pi$
$380$	−6233.20	−0.841464
$381$	0	0
$382$	−9018.48	−1.20792
$383$	−312.425	−0.0416820	−0.0208410	−	0.999783i	$-0.506634\pi$
$383$	−312.425	−0.0416820	−0.0208410	+	0.999783i	$0.506634\pi$
$384$	0	0
$385$	−11555.1	−1.52961
$386$	−6442.25	−0.849487
$387$	0	0
$388$	−1608.60	−0.210475
$389$	−9643.17	−1.25688	−0.628442	−	0.777856i	$-0.716307\pi$
$389$	−9643.17	−1.25688	−0.628442	+	0.777856i	$0.716307\pi$
$390$	0	0
$391$	−265.335	−0.0343186
$392$	2204.30	0.284016
$393$	0	0
$394$	−7568.40	−0.967742
$395$	275.191	0.0350540
$396$	0	0
$397$	−2260.32	−0.285749	−0.142874	−	0.989741i	$-0.545634\pi$
$397$	−2260.32	−0.285749	−0.142874	+	0.989741i	$0.545634\pi$
$398$	−5852.53	−0.737088
$399$	0	0
$400$	−109.341	−0.0136677
$401$	−10169.3	−1.26641	−0.633204	−	0.773985i	$-0.718261\pi$
$401$	−10169.3	−1.26641	−0.633204	+	0.773985i	$0.718261\pi$
$402$	0	0
$403$	898.526	0.111064
$404$	5325.71	0.655852
$405$	0	0
$406$	−11246.1	−1.37471
$407$	−3598.18	−0.438219
$408$	0	0
$409$	−949.833	−0.114832	−0.0574159	−	0.998350i	$-0.518286\pi$
$409$	−949.833	−0.114832	−0.0574159	+	0.998350i	$0.518286\pi$
$410$	4419.03	0.532293
$411$	0	0
$412$	−2074.02	−0.248009
$413$	−2853.65	−0.339998
$414$	0	0
$415$	3127.65	0.369953
$416$	484.146	0.0570606
$417$	0	0
$418$	12254.0	1.43388
$419$	11799.3	1.37573	0.687866	−	0.725837i	$-0.258547\pi$
$419$	11799.3	1.37573	0.687866	+	0.725837i	$0.258547\pi$
$420$	0	0
$421$	−6412.93	−0.742392	−0.371196	−	0.928555i	$-0.621052\pi$
$421$	−6412.93	−0.742392	−0.371196	+	0.928555i	$0.621052\pi$
$422$	626.948	0.0723207
$423$	0	0
$424$	1121.49	0.128453
$425$	−94.7882	−0.0108186
$426$	0	0
$427$	−18778.6	−2.12824
$428$	−5887.48	−0.664912
$429$	0	0
$430$	−7085.13	−0.794594
$431$	12042.7	1.34589	0.672945	−	0.739693i	$-0.265029\pi$
$431$	12042.7	1.34589	0.672945	+	0.739693i	$0.265029\pi$
$432$	0	0
$433$	7279.83	0.807959	0.403980	−	0.914768i	$-0.367627\pi$
$433$	7279.83	0.807959	0.403980	+	0.914768i	$0.367627\pi$
$434$	2954.05	0.326725
$435$	0	0
$436$	−2574.67	−0.282808
$437$	−2742.27	−0.300184
$438$	0	0
$439$	−3598.70	−0.391245	−0.195623	−	0.980679i	$-0.562673\pi$
$439$	−3598.70	−0.391245	−0.195623	+	0.980679i	$0.562673\pi$
$440$	−3716.89	−0.402718
$441$	0	0
$442$	419.707	0.0451662
$443$	14786.4	1.58583	0.792913	−	0.609335i	$-0.208563\pi$
$443$	14786.4	1.58583	0.792913	+	0.609335i	$0.208563\pi$
$444$	0	0
$445$	−9357.45	−0.996822
$446$	−1420.77	−0.150842
$447$	0	0
$448$	1591.71	0.167860
$449$	−114.489	−0.0120336	−0.00601681	−	0.999982i	$-0.501915\pi$
$449$	−114.489	−0.0120336	−0.00601681	+	0.999982i	$0.501915\pi$
$450$	0	0
$451$	−8687.47	−0.907044
$452$	−4094.91	−0.426125
$453$	0	0
$454$	−65.3233	−0.00675281
$455$	−4090.31	−0.421444
$456$	0	0
$457$	6311.14	0.646001	0.323001	−	0.946399i	$-0.395308\pi$
$457$	6311.14	0.646001	0.323001	+	0.946399i	$0.395308\pi$
$458$	−11007.7	−1.12305
$459$	0	0
$460$	831.786	0.0843092
$461$	13744.8	1.38863	0.694317	−	0.719670i	$-0.255707\pi$
$461$	13744.8	1.38863	0.694317	+	0.719670i	$0.255707\pi$
$462$	0	0
$463$	15648.6	1.57074	0.785369	−	0.619028i	$-0.212473\pi$
$463$	15648.6	1.57074	0.785369	+	0.619028i	$0.212473\pi$
$464$	−3617.49	−0.361935
$465$	0	0
$466$	13577.6	1.34972
$467$	7395.79	0.732840	0.366420	−	0.930450i	$-0.380583\pi$
$467$	7395.79	0.732840	0.366420	+	0.930450i	$0.380583\pi$
$468$	0	0
$469$	−19086.7	−1.87919
$470$	237.923	0.0233502
$471$	0	0
$472$	−917.927	−0.0895148
$473$	13928.8	1.35401
$474$	0	0
$475$	−979.645	−0.0946299
$476$	1379.85	0.132869
$477$	0	0
$478$	858.777	0.0821748
$479$	6261.07	0.597235	0.298617	−	0.954373i	$-0.403475\pi$
$479$	6261.07	0.597235	0.298617	+	0.954373i	$0.403475\pi$
$480$	0	0
$481$	−1273.70	−0.120739
$482$	−9687.30	−0.915445
$483$	0	0
$484$	1983.12	0.186244
$485$	4371.54	0.409281
$486$	0	0
$487$	−10314.7	−0.959763	−0.479881	−	0.877333i	$-0.659320\pi$
$487$	−10314.7	−0.959763	−0.479881	+	0.877333i	$0.659320\pi$
$488$	−6040.45	−0.560325
$489$	0	0
$490$	−5990.43	−0.552286
$491$	2760.90	0.253763	0.126881	−	0.991918i	$-0.459503\pi$
$491$	2760.90	0.253763	0.126881	+	0.991918i	$0.459503\pi$
$492$	0	0
$493$	−3136.01	−0.286488
$494$	4337.71	0.395067
$495$	0	0
$496$	950.220	0.0860204
$497$	−8352.82	−0.753874
$498$	0	0
$499$	−9585.99	−0.859976	−0.429988	−	0.902835i	$-0.641482\pi$
$499$	−9585.99	−0.859976	−0.429988	+	0.902835i	$0.641482\pi$
$500$	5732.36	0.512718
$501$	0	0
$502$	−4801.75	−0.426917
$503$	−8829.60	−0.782689	−0.391344	−	0.920244i	$-0.627990\pi$
$503$	−8829.60	−0.782689	−0.391344	+	0.920244i	$0.627990\pi$
$504$	0	0
$505$	−14473.2	−1.27534
$506$	−1635.23	−0.143666
$507$	0	0
$508$	−125.793	−0.0109865
$509$	−4741.74	−0.412916	−0.206458	−	0.978456i	$-0.566194\pi$
$509$	−4741.74	−0.412916	−0.206458	+	0.978456i	$0.566194\pi$
$510$	0	0
$511$	4160.30	0.360158
$512$	512.000	0.0441942
$513$	0	0
$514$	−5961.99	−0.511619
$515$	5636.37	0.482268
$516$	0	0
$517$	−467.739	−0.0397894
$518$	−4187.48	−0.355188
$519$	0	0
$520$	−1315.72	−0.110958
$521$	2753.22	0.231518	0.115759	−	0.993277i	$-0.463070\pi$
$521$	2753.22	0.231518	0.115759	+	0.993277i	$0.463070\pi$
$522$	0	0
$523$	17115.3	1.43098	0.715489	−	0.698624i	$-0.246204\pi$
$523$	17115.3	1.43098	0.715489	+	0.698624i	$0.246204\pi$
$524$	7749.42	0.646059
$525$	0	0
$526$	−11121.6	−0.921913
$527$	823.747	0.0680891
$528$	0	0
$529$	−11801.1	−0.969924
$530$	−3047.76	−0.249786
$531$	0	0
$532$	14260.9	1.16220
$533$	−3075.22	−0.249911
$534$	0	0
$535$	15999.9	1.29296
$536$	−6139.56	−0.494755
$537$	0	0
$538$	−12577.0	−1.00787
$539$	11776.7	0.941113
$540$	0	0
$541$	17880.1	1.42093	0.710467	−	0.703731i	$-0.248484\pi$
$541$	17880.1	1.42093	0.710467	+	0.703731i	$0.248484\pi$
$542$	13709.8	1.08651
$543$	0	0
$544$	443.854	0.0349817
$545$	6996.94	0.549938
$546$	0	0
$547$	−13069.5	−1.02159	−0.510795	−	0.859703i	$-0.670649\pi$
$547$	−13069.5	−1.02159	−0.510795	+	0.859703i	$0.670649\pi$
$548$	−4634.38	−0.361261
$549$	0	0
$550$	−584.168	−0.0452891
$551$	−32410.9	−2.50590
$552$	0	0
$553$	−629.608	−0.0484153
$554$	1796.34	0.137761
$555$	0	0
$556$	9246.39	0.705277
$557$	−9507.62	−0.723251	−0.361626	−	0.932323i	$-0.617778\pi$
$557$	−9507.62	−0.723251	−0.361626	+	0.932323i	$0.617778\pi$
$558$	0	0
$559$	4930.58	0.373061
$560$	−4325.63	−0.326413
$561$	0	0
$562$	−477.539	−0.0358430
$563$	20444.6	1.53044	0.765221	−	0.643768i	$-0.222630\pi$
$563$	20444.6	1.53044	0.765221	+	0.643768i	$0.222630\pi$
$564$	0	0
$565$	11128.3	0.828625
$566$	4142.33	0.307624
$567$	0	0
$568$	−2686.83	−0.198480
$569$	−2646.06	−0.194954	−0.0974770	−	0.995238i	$-0.531077\pi$
$569$	−2646.06	−0.194954	−0.0974770	+	0.995238i	$0.531077\pi$
$570$	0	0
$571$	1757.03	0.128773	0.0643864	−	0.997925i	$-0.479491\pi$
$571$	1757.03	0.128773	0.0643864	+	0.997925i	$0.479491\pi$
$572$	2586.60	0.189076
$573$	0	0
$574$	−10110.3	−0.735183
$575$	130.728	0.00948130
$576$	0	0
$577$	−7515.43	−0.542238	−0.271119	−	0.962546i	$-0.587394\pi$
$577$	−7515.43	−0.542238	−0.271119	+	0.962546i	$0.587394\pi$
$578$	−9441.22	−0.679417
$579$	0	0
$580$	9830.91	0.703804
$581$	−7155.75	−0.510964
$582$	0	0
$583$	5991.67	0.425642
$584$	1338.23	0.0948226
$585$	0	0
$586$	13155.5	0.927388
$587$	4952.49	0.348230	0.174115	−	0.984725i	$-0.444294\pi$
$587$	4952.49	0.348230	0.174115	+	0.984725i	$0.444294\pi$
$588$	0	0
$589$	8513.50	0.595573
$590$	2494.56	0.174067
$591$	0	0
$592$	−1346.98	−0.0935141
$593$	−17115.3	−1.18523	−0.592613	−	0.805487i	$-0.701904\pi$
$593$	−17115.3	−1.18523	−0.592613	+	0.805487i	$0.701904\pi$
$594$	0	0
$595$	−3749.90	−0.258371
$596$	11800.8	0.811043
$597$	0	0
$598$	−578.845	−0.0395831
$599$	−8414.13	−0.573943	−0.286972	−	0.957939i	$-0.592649\pi$
$599$	−8414.13	−0.573943	−0.286972	+	0.957939i	$0.592649\pi$
$600$	0	0
$601$	28094.2	1.90680	0.953399	−	0.301712i	$-0.0975582\pi$
$601$	28094.2	1.90680	0.953399	+	0.301712i	$0.0975582\pi$
$602$	16210.1	1.09746
$603$	0	0
$604$	−6905.59	−0.465206
$605$	−5389.34	−0.362162
$606$	0	0
$607$	1430.85	0.0956775	0.0478388	−	0.998855i	$-0.484767\pi$
$607$	1430.85	0.0956775	0.0478388	+	0.998855i	$0.484767\pi$
$608$	4587.27	0.305984
$609$	0	0
$610$	16415.6	1.08959
$611$	−165.572	−0.0109629
$612$	0	0
$613$	14438.1	0.951306	0.475653	−	0.879633i	$-0.342212\pi$
$613$	14438.1	0.951306	0.475653	+	0.879633i	$0.342212\pi$
$614$	−10474.6	−0.688470
$615$	0	0
$616$	8503.87	0.556218
$617$	−25444.9	−1.66025	−0.830125	−	0.557577i	$-0.811731\pi$
$617$	−25444.9	−1.66025	−0.830125	+	0.557577i	$0.811731\pi$
$618$	0	0
$619$	−3479.46	−0.225931	−0.112966	−	0.993599i	$-0.536035\pi$
$619$	−3479.46	−0.225931	−0.112966	+	0.993599i	$0.536035\pi$
$620$	−2582.32	−0.167272
$621$	0	0
$622$	−11410.0	−0.735528
$623$	21408.9	1.37677
$624$	0	0
$625$	−14724.1	−0.942340
$626$	10155.0	0.648361
$627$	0	0
$628$	5133.55	0.326196
$629$	−1167.70	−0.0740208
$630$	0	0
$631$	11151.7	0.703552	0.351776	−	0.936084i	$-0.385578\pi$
$631$	11151.7	0.703552	0.351776	+	0.936084i	$0.385578\pi$
$632$	−202.524	−0.0127468
$633$	0	0
$634$	−5737.82	−0.359429
$635$	341.855	0.0213639
$636$	0	0
$637$	4168.77	0.259298
$638$	−19326.8	−1.19930
$639$	0	0
$640$	−1391.41	−0.0859383
$641$	1092.78	0.0673356	0.0336678	−	0.999433i	$-0.489281\pi$
$641$	1092.78	0.0673356	0.0336678	+	0.999433i	$0.489281\pi$
$642$	0	0
$643$	31694.0	1.94384	0.971922	−	0.235305i	$-0.0756089\pi$
$643$	31694.0	1.94384	0.971922	+	0.235305i	$0.0756089\pi$
$644$	−1903.04	−0.116445
$645$	0	0
$646$	3976.71	0.242201
$647$	13719.9	0.833672	0.416836	−	0.908982i	$-0.363139\pi$
$647$	13719.9	0.833672	0.416836	+	0.908982i	$0.363139\pi$
$648$	0	0
$649$	−4904.12	−0.296616
$650$	−206.786	−0.0124782
$651$	0	0
$652$	4135.71	0.248415
$653$	−13285.2	−0.796155	−0.398078	−	0.917352i	$-0.630323\pi$
$653$	−13285.2	−0.796155	−0.398078	+	0.917352i	$0.630323\pi$
$654$	0	0
$655$	−21059.9	−1.25630
$656$	−3252.15	−0.193559
$657$	0	0
$658$	−544.344	−0.0322504
$659$	2593.98	0.153334	0.0766670	−	0.997057i	$-0.475572\pi$
$659$	2593.98	0.153334	0.0766670	+	0.997057i	$0.475572\pi$
$660$	0	0
$661$	−15875.3	−0.934159	−0.467079	−	0.884215i	$-0.654694\pi$
$661$	−15875.3	−0.934159	−0.467079	+	0.884215i	$0.654694\pi$
$662$	4062.70	0.238522
$663$	0	0
$664$	−2301.77	−0.134527
$665$	−38755.6	−2.25996
$666$	0	0
$667$	4325.06	0.251075
$668$	1130.47	0.0654778
$669$	0	0
$670$	16684.9	0.962081
$671$	−32271.8	−1.85669
$672$	0	0
$673$	21434.7	1.22771	0.613853	−	0.789420i	$-0.289619\pi$
$673$	21434.7	1.22771	0.613853	+	0.789420i	$0.289619\pi$
$674$	19596.6	1.11993
$675$	0	0
$676$	−7872.38	−0.447905
$677$	20333.0	1.15430	0.577150	−	0.816638i	$-0.304165\pi$
$677$	20333.0	1.15430	0.577150	+	0.816638i	$0.304165\pi$
$678$	0	0
$679$	−10001.6	−0.565283
$680$	−1206.22	−0.0680241
$681$	0	0
$682$	5076.65	0.285037
$683$	27149.0	1.52097	0.760487	−	0.649353i	$-0.224960\pi$
$683$	27149.0	1.52097	0.760487	+	0.649353i	$0.224960\pi$
$684$	0	0
$685$	12594.4	0.702493
$686$	−3355.61	−0.186761
$687$	0	0
$688$	5214.24	0.288941
$689$	2120.95	0.117274
$690$	0	0
$691$	22710.2	1.25027	0.625134	−	0.780517i	$-0.285044\pi$
$691$	22710.2	1.25027	0.625134	+	0.780517i	$0.285044\pi$
$692$	14130.9	0.776266
$693$	0	0
$694$	9113.12	0.498457
$695$	−25128.0	−1.37146
$696$	0	0
$697$	−2819.29	−0.153211
$698$	6697.78	0.363201
$699$	0	0
$700$	−679.842	−0.0367080
$701$	20079.5	1.08187	0.540936	−	0.841064i	$-0.318070\pi$
$701$	20079.5	1.08187	0.540936	+	0.841064i	$0.318070\pi$
$702$	0	0
$703$	−12068.2	−0.647457
$704$	2735.41	0.146441
$705$	0	0
$706$	−3724.54	−0.198548
$707$	33113.2	1.76146
$708$	0	0
$709$	−28983.7	−1.53527	−0.767634	−	0.640889i	$-0.778566\pi$
$709$	−28983.7	−1.53527	−0.767634	+	0.640889i	$0.778566\pi$
$710$	7301.74	0.385957
$711$	0	0
$712$	6886.54	0.362477
$713$	−1136.08	−0.0596726
$714$	0	0
$715$	−7029.37	−0.367669
$716$	16211.0	0.846135
$717$	0	0
$718$	−12358.9	−0.642383
$719$	−14496.2	−0.751901	−0.375951	−	0.926640i	$-0.622684\pi$
$719$	−14496.2	−0.751901	−0.375951	+	0.926640i	$0.622684\pi$
$720$	0	0
$721$	−12895.4	−0.666090
$722$	27381.7	1.41141
$723$	0	0
$724$	−11323.9	−0.581283
$725$	1545.08	0.0791488
$726$	0	0
$727$	5000.50	0.255101	0.127550	−	0.991832i	$-0.459289\pi$
$727$	5000.50	0.255101	0.127550	+	0.991832i	$0.459289\pi$
$728$	3010.23	0.153251
$729$	0	0
$730$	−3636.79	−0.184388
$731$	4520.24	0.228710
$732$	0	0
$733$	17515.6	0.882609	0.441305	−	0.897357i	$-0.354516\pi$
$733$	17515.6	0.882609	0.441305	+	0.897357i	$0.354516\pi$
$734$	13747.9	0.691341
$735$	0	0
$736$	−612.146	−0.0306576
$737$	−32801.3	−1.63942
$738$	0	0
$739$	−20169.2	−1.00397	−0.501985	−	0.864876i	$-0.667397\pi$
$739$	−20169.2	−1.00397	−0.501985	+	0.864876i	$0.667397\pi$
$740$	3660.55	0.181844
$741$	0	0
$742$	6972.97	0.344994
$743$	36702.6	1.81223	0.906116	−	0.423029i	$-0.139033\pi$
$743$	36702.6	1.81223	0.906116	+	0.423029i	$0.139033\pi$
$744$	0	0
$745$	−32070.1	−1.57712
$746$	2540.05	0.124662
$747$	0	0
$748$	2371.34	0.115915
$749$	−36606.0	−1.78579
$750$	0	0
$751$	−33320.1	−1.61900	−0.809499	−	0.587122i	$-0.800261\pi$
$751$	−33320.1	−1.61900	−0.809499	+	0.587122i	$0.800261\pi$
$752$	−175.098	−0.00849090
$753$	0	0
$754$	−6841.38	−0.330436
$755$	18766.7	0.904622
$756$	0	0
$757$	26515.6	1.27309	0.636543	−	0.771241i	$-0.280364\pi$
$757$	26515.6	1.27309	0.636543	+	0.771241i	$0.280364\pi$
$758$	4981.08	0.238682
$759$	0	0
$760$	−12466.4	−0.595005
$761$	5684.35	0.270772	0.135386	−	0.990793i	$-0.456773\pi$
$761$	5684.35	0.270772	0.135386	+	0.990793i	$0.456773\pi$
$762$	0	0
$763$	−16008.3	−0.759553
$764$	−18037.0	−0.854129
$765$	0	0
$766$	−624.851	−0.0294736
$767$	−1735.98	−0.0817244
$768$	0	0
$769$	−398.377	−0.0186812	−0.00934060	−	0.999956i	$-0.502973\pi$
$769$	−398.377	−0.0186812	−0.00934060	+	0.999956i	$0.502973\pi$
$770$	−23110.2	−1.08160
$771$	0	0
$772$	−12884.5	−0.600678
$773$	8437.17	0.392579	0.196290	−	0.980546i	$-0.437111\pi$
$773$	8437.17	0.392579	0.196290	+	0.980546i	$0.437111\pi$
$774$	0	0
$775$	−405.853	−0.0188112
$776$	−3217.20	−0.148828
$777$	0	0
$778$	−19286.3	−0.888751
$779$	−29137.6	−1.34013
$780$	0	0
$781$	−14354.7	−0.657683
$782$	−530.671	−0.0242669
$783$	0	0
$784$	4408.61	0.200830
$785$	−13951.0	−0.634308
$786$	0	0
$787$	16277.9	0.737286	0.368643	−	0.929571i	$-0.379823\pi$
$787$	16277.9	0.737286	0.368643	+	0.929571i	$0.379823\pi$
$788$	−15136.8	−0.684297
$789$	0	0
$790$	550.381	0.0247870
$791$	−25460.5	−1.14447
$792$	0	0
$793$	−11423.7	−0.511560
$794$	−4520.64	−0.202055
$795$	0	0
$796$	−11705.1	−0.521200
$797$	−5112.11	−0.227202	−0.113601	−	0.993526i	$-0.536239\pi$
$797$	−5112.11	−0.227202	−0.113601	+	0.993526i	$0.536239\pi$
$798$	0	0
$799$	−151.793	−0.00672094
$800$	−218.683	−0.00966450
$801$	0	0
$802$	−20338.6	−0.895485
$803$	7149.64	0.314204
$804$	0	0
$805$	5171.72	0.226434
$806$	1797.05	0.0785341
$807$	0	0
$808$	10651.4	0.463757
$809$	13141.2	0.571100	0.285550	−	0.958364i	$-0.407824\pi$
$809$	13141.2	0.571100	0.285550	+	0.958364i	$0.407824\pi$
$810$	0	0
$811$	−18614.2	−0.805957	−0.402979	−	0.915209i	$-0.632025\pi$
$811$	−18614.2	−0.805957	−0.402979	+	0.915209i	$0.632025\pi$
$812$	−22492.1	−0.972067
$813$	0	0
$814$	−7196.36	−0.309868
$815$	−11239.2	−0.483059
$816$	0	0
$817$	46717.1	2.00052
$818$	−1899.67	−0.0811983
$819$	0	0
$820$	8838.05	0.376388
$821$	11320.0	0.481208	0.240604	−	0.970623i	$-0.422655\pi$
$821$	11320.0	0.481208	0.240604	+	0.970623i	$0.422655\pi$
$822$	0	0
$823$	10866.6	0.460249	0.230125	−	0.973161i	$-0.426087\pi$
$823$	10866.6	0.460249	0.230125	+	0.973161i	$0.426087\pi$
$824$	−4148.04	−0.175369
$825$	0	0
$826$	−5707.31	−0.240415
$827$	13059.3	0.549114	0.274557	−	0.961571i	$-0.411469\pi$
$827$	13059.3	0.549114	0.274557	+	0.961571i	$0.411469\pi$
$828$	0	0
$829$	−21203.7	−0.888341	−0.444171	−	0.895942i	$-0.646502\pi$
$829$	−21203.7	−0.888341	−0.444171	+	0.895942i	$0.646502\pi$
$830$	6255.30	0.261596
$831$	0	0
$832$	968.293	0.0403480
$833$	3821.83	0.158966
$834$	0	0
$835$	−3072.17	−0.127326
$836$	24508.0	1.01391
$837$	0	0
$838$	23598.5	0.972790
$839$	14480.6	0.595858	0.297929	−	0.954588i	$-0.403704\pi$
$839$	14480.6	0.595858	0.297929	+	0.954588i	$0.403704\pi$
$840$	0	0
$841$	26729.0	1.09595
$842$	−12825.9	−0.524950
$843$	0	0
$844$	1253.90	0.0511385
$845$	21394.0	0.870979
$846$	0	0
$847$	12330.3	0.500204
$848$	2242.98	0.0908303
$849$	0	0
$850$	−189.576	−0.00764990
$851$	1610.44	0.0648710
$852$	0	0
$853$	−6467.23	−0.259594	−0.129797	−	0.991541i	$-0.541433\pi$
$853$	−6467.23	−0.259594	−0.129797	+	0.991541i	$0.541433\pi$
$854$	−37557.1	−1.50489
$855$	0	0
$856$	−11775.0	−0.470164
$857$	−3894.64	−0.155237	−0.0776186	−	0.996983i	$-0.524732\pi$
$857$	−3894.64	−0.155237	−0.0776186	+	0.996983i	$0.524732\pi$
$858$	0	0
$859$	−37653.8	−1.49561	−0.747805	−	0.663918i	$-0.768892\pi$
$859$	−37653.8	−1.49561	−0.747805	+	0.663918i	$0.768892\pi$
$860$	−14170.3	−0.561863
$861$	0	0
$862$	24085.5	0.951688
$863$	−47067.5	−1.85654	−0.928271	−	0.371905i	$-0.878705\pi$
$863$	−47067.5	−1.85654	−0.928271	+	0.371905i	$0.878705\pi$
$864$	0	0
$865$	−38402.2	−1.50950
$866$	14559.7	0.571313
$867$	0	0
$868$	5908.09	0.231030
$869$	−1082.01	−0.0422377
$870$	0	0
$871$	−11611.1	−0.451697
$872$	−5149.34	−0.199976
$873$	0	0
$874$	−5484.53	−0.212262
$875$	35641.6	1.37703
$876$	0	0
$877$	−7443.54	−0.286603	−0.143301	−	0.989679i	$-0.545772\pi$
$877$	−7443.54	−0.286603	−0.143301	+	0.989679i	$0.545772\pi$
$878$	−7197.40	−0.276652
$879$	0	0
$880$	−7433.78	−0.284764
$881$	13781.9	0.527040	0.263520	−	0.964654i	$-0.415116\pi$
$881$	13781.9	0.527040	0.263520	+	0.964654i	$0.415116\pi$
$882$	0	0
$883$	12230.3	0.466119	0.233060	−	0.972462i	$-0.425126\pi$
$883$	12230.3	0.466119	0.233060	+	0.972462i	$0.425126\pi$
$884$	839.415	0.0319373
$885$	0	0
$886$	29572.7	1.12135
$887$	1674.87	0.0634011	0.0317005	−	0.999497i	$-0.489908\pi$
$887$	1674.87	0.0634011	0.0317005	+	0.999497i	$0.489908\pi$
$888$	0	0
$889$	−782.128	−0.0295070
$890$	−18714.9	−0.704859
$891$	0	0
$892$	−2841.54	−0.106661
$893$	−1568.79	−0.0587878
$894$	0	0
$895$	−44055.1	−1.64536
$896$	3183.41	0.118695
$897$	0	0
$898$	−228.979	−0.00850905
$899$	−13427.4	−0.498140
$900$	0	0
$901$	1944.44	0.0718964
$902$	−17374.9	−0.641377
$903$	0	0
$904$	−8189.82	−0.301316
$905$	30773.8	1.13034
$906$	0	0
$907$	17088.1	0.625580	0.312790	−	0.949822i	$-0.398736\pi$
$907$	17088.1	0.625580	0.312790	+	0.949822i	$0.398736\pi$
$908$	−130.647	−0.00477496
$909$	0	0
$910$	−8180.63	−0.298006
$911$	−21288.4	−0.774222	−0.387111	−	0.922033i	$-0.626527\pi$
$911$	−21288.4	−0.774222	−0.387111	+	0.922033i	$0.626527\pi$
$912$	0	0
$913$	−12297.4	−0.445768
$914$	12622.3	0.456792
$915$	0	0
$916$	−22015.4	−0.794114
$917$	48182.8	1.73516
$918$	0	0
$919$	−10413.4	−0.373784	−0.186892	−	0.982380i	$-0.559841\pi$
$919$	−10413.4	−0.373784	−0.186892	+	0.982380i	$0.559841\pi$
$920$	1663.57	0.0596156
$921$	0	0
$922$	27489.6	0.981912
$923$	−5081.32	−0.181207
$924$	0	0
$925$	575.313	0.0204499
$926$	31297.2	1.11068
$927$	0	0
$928$	−7234.98	−0.255926
$929$	6823.43	0.240979	0.120490	−	0.992715i	$-0.461554\pi$
$929$	6823.43	0.240979	0.120490	+	0.992715i	$0.461554\pi$
$930$	0	0
$931$	39499.0	1.39047
$932$	27155.2	0.954399
$933$	0	0
$934$	14791.6	0.518196
$935$	−6444.36	−0.225404
$936$	0	0
$937$	41049.8	1.43120	0.715602	−	0.698508i	$-0.246152\pi$
$937$	41049.8	1.43120	0.715602	+	0.698508i	$0.246152\pi$
$938$	−38173.4	−1.32879
$939$	0	0
$940$	475.847	0.0165111
$941$	−2416.12	−0.0837017	−0.0418508	−	0.999124i	$-0.513325\pi$
$941$	−2416.12	−0.0837017	−0.0418508	+	0.999124i	$0.513325\pi$
$942$	0	0
$943$	3888.26	0.134273
$944$	−1835.85	−0.0632966
$945$	0	0
$946$	27857.7	0.957432
$947$	2394.07	0.0821509	0.0410755	−	0.999156i	$-0.486922\pi$
$947$	2394.07	0.0821509	0.0410755	+	0.999156i	$0.486922\pi$
$948$	0	0
$949$	2530.86	0.0865702
$950$	−1959.29	−0.0669134
$951$	0	0
$952$	2759.71	0.0939523
$953$	50651.3	1.72168	0.860838	−	0.508879i	$-0.169940\pi$
$953$	50651.3	1.72168	0.860838	+	0.508879i	$0.169940\pi$
$954$	0	0
$955$	49017.4	1.66091
$956$	1717.55	0.0581064
$957$	0	0
$958$	12522.1	0.422309
$959$	−28814.7	−0.970257
$960$	0	0
$961$	−26264.0	−0.881608
$962$	−2547.40	−0.0853756
$963$	0	0
$964$	−19374.6	−0.647317
$965$	35015.0	1.16805
$966$	0	0
$967$	25034.9	0.832542	0.416271	−	0.909241i	$-0.363337\pi$
$967$	25034.9	0.832542	0.416271	+	0.909241i	$0.363337\pi$
$968$	3966.24	0.131694
$969$	0	0
$970$	8743.07	0.289405
$971$	−49553.7	−1.63775	−0.818875	−	0.573972i	$-0.805402\pi$
$971$	−49553.7	−1.63775	−0.818875	+	0.573972i	$0.805402\pi$
$972$	0	0
$973$	57490.4	1.89420
$974$	−20629.4	−0.678655
$975$	0	0
$976$	−12080.9	−0.396209
$977$	35176.2	1.15188	0.575939	−	0.817492i	$-0.304636\pi$
$977$	35176.2	1.15188	0.575939	+	0.817492i	$0.304636\pi$
$978$	0	0
$979$	36792.1	1.20110
$980$	−11980.9	−0.390525
$981$	0	0
$982$	5521.79	0.179437
$983$	−33349.3	−1.08207	−0.541036	−	0.840999i	$-0.681968\pi$
$983$	−33349.3	−1.08207	−0.541036	+	0.840999i	$0.681968\pi$
$984$	0	0
$985$	41135.8	1.33066
$986$	−6272.01	−0.202578
$987$	0	0
$988$	8675.43	0.279354
$989$	−6234.14	−0.200439
$990$	0	0
$991$	23066.3	0.739378	0.369689	−	0.929156i	$-0.379464\pi$
$991$	23066.3	0.739378	0.369689	+	0.929156i	$0.379464\pi$
$992$	1900.44	0.0608256
$993$	0	0
$994$	−16705.6	−0.533069
$995$	31809.7	1.01350
$996$	0	0
$997$	−55769.9	−1.77156	−0.885782	−	0.464102i	$-0.846377\pi$
$997$	−55769.9	−1.77156	−0.885782	+	0.464102i	$0.846377\pi$
$998$	−19172.0	−0.608095
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.a.g.1.1		2
3.2	odd	2		162.4.a.f.1.2		2
4.3	odd	2		1296.4.a.r.1.1		2
9.2	odd	6		18.4.c.b.13.2	yes	4
9.4	even	3		54.4.c.b.19.2		4
9.5	odd	6		18.4.c.b.7.2	✓	4
9.7	even	3		54.4.c.b.37.2		4
12.11	even	2		1296.4.a.l.1.2		2
36.7	odd	6		432.4.i.b.145.2		4
36.11	even	6		144.4.i.b.49.1		4
36.23	even	6		144.4.i.b.97.1		4
36.31	odd	6		432.4.i.b.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.b.7.2	✓	4	9.5	odd	6
18.4.c.b.13.2	yes	4	9.2	odd	6
54.4.c.b.19.2		4	9.4	even	3
54.4.c.b.37.2		4	9.7	even	3
144.4.i.b.49.1		4	36.11	even	6
144.4.i.b.97.1		4	36.23	even	6
162.4.a.f.1.2		2	3.2	odd	2
162.4.a.g.1.1		2	1.1	even	1	trivial
432.4.i.b.145.2		4	36.7	odd	6
432.4.i.b.289.2		4	36.31	odd	6
1296.4.a.l.1.2		2	12.11	even	2
1296.4.a.r.1.1		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -10.8704 q^{5} +24.8704 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -10.8704 q^{5} +24.8704 q^{7} +8.00000 q^{8} -21.7409 q^{10} +42.7409 q^{11} +15.1296 q^{13} +49.7409 q^{14} +16.0000 q^{16} +13.8704 q^{17} +143.352 q^{19} -43.4817 q^{20} +85.4817 q^{22} -19.1296 q^{23} -6.83384 q^{25} +30.2591 q^{26} +99.4817 q^{28} -226.093 q^{29} +59.3887 q^{31} +32.0000 q^{32} +27.7409 q^{34} -270.352 q^{35} -84.1860 q^{37} +286.704 q^{38} -86.9634 q^{40} -203.259 q^{41} +325.890 q^{43} +170.963 q^{44} -38.2591 q^{46} -10.9436 q^{47} +275.538 q^{49} -13.6677 q^{50} +60.5183 q^{52} +140.186 q^{53} -464.611 q^{55} +198.963 q^{56} -452.186 q^{58} -114.741 q^{59} -755.056 q^{61} +118.777 q^{62} +64.0000 q^{64} -164.465 q^{65} -767.445 q^{67} +55.4817 q^{68} -540.704 q^{70} -335.854 q^{71} +167.279 q^{73} -168.372 q^{74} +573.409 q^{76} +1062.98 q^{77} -25.3155 q^{79} -173.927 q^{80} -406.518 q^{82} -287.721 q^{83} -150.777 q^{85} +651.780 q^{86} +341.927 q^{88} +860.817 q^{89} +376.279 q^{91} -76.5183 q^{92} -21.8872 q^{94} -1558.30 q^{95} -402.149 q^{97} +551.076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 9 q^{5} + 19 q^{7} + 16 q^{8} + 18 q^{10} + 24 q^{11} + 61 q^{13} + 38 q^{14} + 32 q^{16} - 3 q^{17} + 133 q^{19} + 36 q^{20} + 48 q^{22} - 69 q^{23} + 263 q^{25} + 122 q^{26} + 76 q^{28} - 237 q^{29} + 211 q^{31} + 64 q^{32} - 6 q^{34} - 387 q^{35} + 262 q^{37} + 266 q^{38} + 72 q^{40} - 468 q^{41} - 86 q^{43} + 96 q^{44} - 138 q^{46} - 483 q^{47} - 33 q^{49} + 526 q^{50} + 244 q^{52} - 150 q^{53} - 837 q^{55} + 152 q^{56} - 474 q^{58} - 168 q^{59} - 1049 q^{61} + 422 q^{62} + 128 q^{64} + 747 q^{65} - 1166 q^{67} - 12 q^{68} - 774 q^{70} + 312 q^{71} - 311 q^{73} + 524 q^{74} + 532 q^{76} + 1173 q^{77} + 349 q^{79} + 144 q^{80} - 936 q^{82} - 1221 q^{83} - 486 q^{85} - 172 q^{86} + 192 q^{88} + 492 q^{89} + 107 q^{91} - 276 q^{92} - 966 q^{94} - 1764 q^{95} - 128 q^{97} - 66 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 9 * q^5 + 19 * q^7 + 16 * q^8 + 18 * q^10 + 24 * q^11 + 61 * q^13 + 38 * q^14 + 32 * q^16 - 3 * q^17 + 133 * q^19 + 36 * q^20 + 48 * q^22 - 69 * q^23 + 263 * q^25 + 122 * q^26 + 76 * q^28 - 237 * q^29 + 211 * q^31 + 64 * q^32 - 6 * q^34 - 387 * q^35 + 262 * q^37 + 266 * q^38 + 72 * q^40 - 468 * q^41 - 86 * q^43 + 96 * q^44 - 138 * q^46 - 483 * q^47 - 33 * q^49 + 526 * q^50 + 244 * q^52 - 150 * q^53 - 837 * q^55 + 152 * q^56 - 474 * q^58 - 168 * q^59 - 1049 * q^61 + 422 * q^62 + 128 * q^64 + 747 * q^65 - 1166 * q^67 - 12 * q^68 - 774 * q^70 + 312 * q^71 - 311 * q^73 + 524 * q^74 + 532 * q^76 + 1173 * q^77 + 349 * q^79 + 144 * q^80 - 936 * q^82 - 1221 * q^83 - 486 * q^85 - 172 * q^86 + 192 * q^88 + 492 * q^89 + 107 * q^91 - 276 * q^92 - 966 * q^94 - 1764 * q^95 - 128 * q^97 - 66 * q^98

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