LMFDB - Embedded newform 162.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,6,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, 6, names="a")

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
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:	$25.9821788097$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 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			$-2.44949$ 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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -56.3939 q^{5} -22.3031 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} -56.3939 q^{5} -22.3031 q^{7} +64.0000 q^{8} -225.576 q^{10} +592.968 q^{11} -259.061 q^{13} -89.2122 q^{14} +256.000 q^{16} -1627.51 q^{17} -2642.42 q^{19} -902.302 q^{20} +2371.87 q^{22} -3136.42 q^{23} +55.2694 q^{25} -1036.24 q^{26} -356.849 q^{28} -124.580 q^{29} -8752.17 q^{31} +1024.00 q^{32} -6510.05 q^{34} +1257.76 q^{35} +9741.67 q^{37} -10569.7 q^{38} -3609.21 q^{40} +7471.24 q^{41} -12504.9 q^{43} +9487.49 q^{44} -12545.7 q^{46} +8817.46 q^{47} -16309.6 q^{49} +221.077 q^{50} -4144.98 q^{52} +10782.1 q^{53} -33439.8 q^{55} -1427.40 q^{56} -498.319 q^{58} +15553.7 q^{59} -11930.4 q^{61} -35008.7 q^{62} +4096.00 q^{64} +14609.5 q^{65} +6179.58 q^{67} -26040.2 q^{68} +5031.02 q^{70} +10498.9 q^{71} -38380.3 q^{73} +38966.7 q^{74} -42278.7 q^{76} -13225.0 q^{77} +17853.7 q^{79} -14436.8 q^{80} +29885.0 q^{82} +43220.6 q^{83} +91781.7 q^{85} -50019.7 q^{86} +37950.0 q^{88} +94095.6 q^{89} +5777.86 q^{91} -50182.8 q^{92} +35269.8 q^{94} +149016. q^{95} -81486.7 q^{97} -65238.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 - 54 * q^5 - 74 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8} - 216 q^{10} - 78 q^{11} - 1106 q^{13} - 296 q^{14} + 512 q^{16} - 492 q^{17} - 1640 q^{19} - 864 q^{20} - 312 q^{22} - 5538 q^{23} - 3064 q^{25} - 4424 q^{26} - 1184 q^{28} - 3894 q^{29} - 4718 q^{31} + 2048 q^{32} - 1968 q^{34} + 1134 q^{35} - 4796 q^{37} - 6560 q^{38} - 3456 q^{40} + 15354 q^{41} - 32858 q^{43} - 1248 q^{44} - 22152 q^{46} + 24954 q^{47} - 30444 q^{49} - 12256 q^{50} - 17696 q^{52} + 16332 q^{53} - 35046 q^{55} - 4736 q^{56} - 15576 q^{58} + 21966 q^{59} - 3050 q^{61} - 18872 q^{62} + 8192 q^{64} + 12582 q^{65} - 36758 q^{67} - 7872 q^{68} + 4536 q^{70} + 73848 q^{71} - 51188 q^{73} - 19184 q^{74} - 26240 q^{76} + 21462 q^{77} + 14926 q^{79} - 13824 q^{80} + 61416 q^{82} + 90762 q^{83} + 94500 q^{85} - 131432 q^{86} - 4992 q^{88} + 9300 q^{89} + 49562 q^{91} - 88608 q^{92} + 99816 q^{94} + 151416 q^{95} + 30262 q^{97} - 121776 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 54 * q^5 - 74 * q^7 + 128 * q^8 - 216 * q^10 - 78 * q^11 - 1106 * q^13 - 296 * q^14 + 512 * q^16 - 492 * q^17 - 1640 * q^19 - 864 * q^20 - 312 * q^22 - 5538 * q^23 - 3064 * q^25 - 4424 * q^26 - 1184 * q^28 - 3894 * q^29 - 4718 * q^31 + 2048 * q^32 - 1968 * q^34 + 1134 * q^35 - 4796 * q^37 - 6560 * q^38 - 3456 * q^40 + 15354 * q^41 - 32858 * q^43 - 1248 * q^44 - 22152 * q^46 + 24954 * q^47 - 30444 * q^49 - 12256 * q^50 - 17696 * q^52 + 16332 * q^53 - 35046 * q^55 - 4736 * q^56 - 15576 * q^58 + 21966 * q^59 - 3050 * q^61 - 18872 * q^62 + 8192 * q^64 + 12582 * q^65 - 36758 * q^67 - 7872 * q^68 + 4536 * q^70 + 73848 * q^71 - 51188 * q^73 - 19184 * q^74 - 26240 * q^76 + 21462 * q^77 + 14926 * q^79 - 13824 * q^80 + 61416 * q^82 + 90762 * q^83 + 94500 * q^85 - 131432 * q^86 - 4992 * q^88 + 9300 * q^89 + 49562 * q^91 - 88608 * q^92 + 99816 * q^94 + 151416 * q^95 + 30262 * q^97 - 121776 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−56.3939	−1.00880	−0.504402	−	0.863469i	$-0.668287\pi$
$5$	−56.3939	−1.00880	−0.504402	+	0.863469i	$0.668287\pi$
$6$	0	0
$7$	−22.3031	−0.172036	−0.0860180	−	0.996294i	$-0.527414\pi$
$7$	−22.3031	−0.172036	−0.0860180	+	0.996294i	$0.527414\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−225.576	−0.713332
$11$	592.968	1.47758	0.738788	−	0.673938i	$-0.235398\pi$
$11$	592.968	1.47758	0.738788	+	0.673938i	$0.235398\pi$
$12$	0	0
$13$	−259.061	−0.425152	−0.212576	−	0.977145i	$-0.568185\pi$
$13$	−259.061	−0.425152	−0.212576	+	0.977145i	$0.568185\pi$
$14$	−89.2122	−0.121648
$15$	0	0
$16$	256.000	0.250000
$17$	−1627.51	−1.36585	−0.682923	−	0.730490i	$-0.739292\pi$
$17$	−1627.51	−1.36585	−0.682923	+	0.730490i	$0.739292\pi$
$18$	0	0
$19$	−2642.42	−1.67926	−0.839630	−	0.543159i	$-0.817228\pi$
$19$	−2642.42	−1.67926	−0.839630	+	0.543159i	$0.817228\pi$
$20$	−902.302	−0.504402
$21$	0	0
$22$	2371.87	1.04480
$23$	−3136.42	−1.23628	−0.618138	−	0.786070i	$-0.712113\pi$
$23$	−3136.42	−1.23628	−0.618138	+	0.786070i	$0.712113\pi$
$24$	0	0
$25$	55.2694	0.0176862
$26$	−1036.24	−0.300628
$27$	0	0
$28$	−356.849	−0.0860180
$29$	−124.580	−0.0275076	−0.0137538	−	0.999905i	$-0.504378\pi$
$29$	−124.580	−0.0275076	−0.0137538	+	0.999905i	$0.504378\pi$
$30$	0	0
$31$	−8752.17	−1.63573	−0.817865	−	0.575411i	$-0.804842\pi$
$31$	−8752.17	−1.63573	−0.817865	+	0.575411i	$0.804842\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−6510.05	−0.965799
$35$	1257.76	0.173551
$36$	0	0
$37$	9741.67	1.16985	0.584924	−	0.811088i	$-0.301124\pi$
$37$	9741.67	1.16985	0.584924	+	0.811088i	$0.301124\pi$
$38$	−10569.7	−1.18742
$39$	0	0
$40$	−3609.21	−0.356666
$41$	7471.24	0.694118	0.347059	−	0.937843i	$-0.387180\pi$
$41$	7471.24	0.694118	0.347059	+	0.937843i	$0.387180\pi$
$42$	0	0
$43$	−12504.9	−1.03136	−0.515679	−	0.856782i	$-0.672460\pi$
$43$	−12504.9	−1.03136	−0.515679	+	0.856782i	$0.672460\pi$
$44$	9487.49	0.738788
$45$	0	0
$46$	−12545.7	−0.874178
$47$	8817.46	0.582236	0.291118	−	0.956687i	$-0.405973\pi$
$47$	8817.46	0.582236	0.291118	+	0.956687i	$0.405973\pi$
$48$	0	0
$49$	−16309.6	−0.970404
$50$	221.077	0.0125060
$51$	0	0
$52$	−4144.98	−0.212576
$53$	10782.1	0.527244	0.263622	−	0.964626i	$-0.415083\pi$
$53$	10782.1	0.527244	0.263622	+	0.964626i	$0.415083\pi$
$54$	0	0
$55$	−33439.8	−1.49058
$56$	−1427.40	−0.0608239
$57$	0	0
$58$	−498.319	−0.0194508
$59$	15553.7	0.581708	0.290854	−	0.956767i	$-0.406061\pi$
$59$	15553.7	0.581708	0.290854	+	0.956767i	$0.406061\pi$
$60$	0	0
$61$	−11930.4	−0.410517	−0.205259	−	0.978708i	$-0.565804\pi$
$61$	−11930.4	−0.410517	−0.205259	+	0.978708i	$0.565804\pi$
$62$	−35008.7	−1.15664
$63$	0	0
$64$	4096.00	0.125000
$65$	14609.5	0.428895
$66$	0	0
$67$	6179.58	0.168179	0.0840896	−	0.996458i	$-0.473202\pi$
$67$	6179.58	0.168179	0.0840896	+	0.996458i	$0.473202\pi$
$68$	−26040.2	−0.682923
$69$	0	0
$70$	5031.02	0.122719
$71$	10498.9	0.247171	0.123586	−	0.992334i	$-0.460561\pi$
$71$	10498.9	0.247171	0.123586	+	0.992334i	$0.460561\pi$
$72$	0	0
$73$	−38380.3	−0.842950	−0.421475	−	0.906840i	$-0.638487\pi$
$73$	−38380.3	−0.842950	−0.421475	+	0.906840i	$0.638487\pi$
$74$	38966.7	0.827207
$75$	0	0
$76$	−42278.7	−0.839630
$77$	−13225.0	−0.254196
$78$	0	0
$79$	17853.7	0.321856	0.160928	−	0.986966i	$-0.448551\pi$
$79$	17853.7	0.321856	0.160928	+	0.986966i	$0.448551\pi$
$80$	−14436.8	−0.252201
$81$	0	0
$82$	29885.0	0.490815
$83$	43220.6	0.688644	0.344322	−	0.938852i	$-0.388109\pi$
$83$	43220.6	0.688644	0.344322	+	0.938852i	$0.388109\pi$
$84$	0	0
$85$	91781.7	1.37787
$86$	−50019.7	−0.729281
$87$	0	0
$88$	37950.0	0.522402
$89$	94095.6	1.25920	0.629599	−	0.776920i	$-0.283219\pi$
$89$	94095.6	1.25920	0.629599	+	0.776920i	$0.283219\pi$
$90$	0	0
$91$	5777.86	0.0731414
$92$	−50182.8	−0.618138
$93$	0	0
$94$	35269.8	0.411703
$95$	149016.	1.69404
$96$	0	0
$97$	−81486.7	−0.879341	−0.439670	−	0.898159i	$-0.644905\pi$
$97$	−81486.7	−0.879341	−0.439670	+	0.898159i	$0.644905\pi$
$98$	−65238.3	−0.686179
$99$	0	0
$100$	884.310	0.00884310
$101$	−132868.	−1.29603	−0.648017	−	0.761626i	$-0.724402\pi$
$101$	−132868.	−1.29603	−0.648017	+	0.761626i	$0.724402\pi$
$102$	0	0
$103$	−124111.	−1.15270	−0.576352	−	0.817202i	$-0.695524\pi$
$103$	−124111.	−1.15270	−0.576352	+	0.817202i	$0.695524\pi$
$104$	−16579.9	−0.150314
$105$	0	0
$106$	43128.2	0.372818
$107$	−102868.	−0.868605	−0.434303	−	0.900767i	$-0.643005\pi$
$107$	−102868.	−0.868605	−0.434303	+	0.900767i	$0.643005\pi$
$108$	0	0
$109$	229354.	1.84901	0.924506	−	0.381167i	$-0.124478\pi$
$109$	229354.	1.84901	0.924506	+	0.381167i	$0.124478\pi$
$110$	−133759.	−1.05400
$111$	0	0
$112$	−5709.58	−0.0430090
$113$	−74933.7	−0.552054	−0.276027	−	0.961150i	$-0.589018\pi$
$113$	−74933.7	−0.552054	−0.276027	+	0.961150i	$0.589018\pi$
$114$	0	0
$115$	176875.	1.24716
$116$	−1993.27	−0.0137538
$117$	0	0
$118$	62215.0	0.411330
$119$	36298.5	0.234975
$120$	0	0
$121$	190560.	1.18323
$122$	−47721.7	−0.290280
$123$	0	0
$124$	−140035.	−0.817865
$125$	173114.	0.990962
$126$	0	0
$127$	34578.3	0.190237	0.0951184	−	0.995466i	$-0.469677\pi$
$127$	34578.3	0.190237	0.0951184	+	0.995466i	$0.469677\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	58437.9	0.303275
$131$	−79623.7	−0.405382	−0.202691	−	0.979243i	$-0.564969\pi$
$131$	−79623.7	−0.405382	−0.202691	+	0.979243i	$0.564969\pi$
$132$	0	0
$133$	58934.1	0.288893
$134$	24718.3	0.118921
$135$	0	0
$136$	−104161.	−0.482900
$137$	271041.	1.23377	0.616883	−	0.787055i	$-0.288395\pi$
$137$	271041.	1.23377	0.616883	+	0.787055i	$0.288395\pi$
$138$	0	0
$139$	33168.8	0.145611	0.0728053	−	0.997346i	$-0.476805\pi$
$139$	33168.8	0.145611	0.0728053	+	0.997346i	$0.476805\pi$
$140$	20124.1	0.0867753
$141$	0	0
$142$	41995.6	0.174777
$143$	−153615.	−0.628194
$144$	0	0
$145$	7025.53	0.0277497
$146$	−153521.	−0.596055
$147$	0	0
$148$	155867.	0.584924
$149$	36634.8	0.135185	0.0675925	−	0.997713i	$-0.478468\pi$
$149$	36634.8	0.135185	0.0675925	+	0.997713i	$0.478468\pi$
$150$	0	0
$151$	494971.	1.76659	0.883297	−	0.468813i	$-0.155318\pi$
$151$	494971.	1.76659	0.883297	+	0.468813i	$0.155318\pi$
$152$	−169115.	−0.593708
$153$	0	0
$154$	−52900.0	−0.179744
$155$	493569.	1.65013
$156$	0	0
$157$	−57831.7	−0.187248	−0.0936239	−	0.995608i	$-0.529845\pi$
$157$	−57831.7	−0.187248	−0.0936239	+	0.995608i	$0.529845\pi$
$158$	71414.9	0.227586
$159$	0	0
$160$	−57747.3	−0.178333
$161$	69951.8	0.212684
$162$	0	0
$163$	−135023.	−0.398050	−0.199025	−	0.979994i	$-0.563777\pi$
$163$	−135023.	−0.398050	−0.199025	+	0.979994i	$0.563777\pi$
$164$	119540.	0.347059
$165$	0	0
$166$	172882.	0.486945
$167$	406029.	1.12659	0.563294	−	0.826256i	$-0.309534\pi$
$167$	406029.	1.12659	0.563294	+	0.826256i	$0.309534\pi$
$168$	0	0
$169$	−304180.	−0.819246
$170$	367127.	0.974303
$171$	0	0
$172$	−200079.	−0.515679
$173$	−228242.	−0.579802	−0.289901	−	0.957057i	$-0.593622\pi$
$173$	−228242.	−0.579802	−0.289901	+	0.957057i	$0.593622\pi$
$174$	0	0
$175$	−1232.68	−0.00304266
$176$	151800.	0.369394
$177$	0	0
$178$	376382.	0.890388
$179$	19370.0	0.0451853	0.0225927	−	0.999745i	$-0.492808\pi$
$179$	19370.0	0.0451853	0.0225927	+	0.999745i	$0.492808\pi$
$180$	0	0
$181$	−137790.	−0.312623	−0.156311	−	0.987708i	$-0.549960\pi$
$181$	−137790.	−0.312623	−0.156311	+	0.987708i	$0.549960\pi$
$182$	23111.4	0.0517188
$183$	0	0
$184$	−200731.	−0.437089
$185$	−549371.	−1.18015
$186$	0	0
$187$	−965063.	−2.01814
$188$	141079.	0.291118
$189$	0	0
$190$	596065.	1.19787
$191$	−336523.	−0.667469	−0.333735	−	0.942667i	$-0.608309\pi$
$191$	−336523.	−0.667469	−0.333735	+	0.942667i	$0.608309\pi$
$192$	0	0
$193$	−740873.	−1.43169	−0.715847	−	0.698257i	$-0.753959\pi$
$193$	−740873.	−1.43169	−0.715847	+	0.698257i	$0.753959\pi$
$194$	−325947.	−0.621788
$195$	0	0
$196$	−260953.	−0.485202
$197$	−219128.	−0.402284	−0.201142	−	0.979562i	$-0.564465\pi$
$197$	−219128.	−0.402284	−0.201142	+	0.979562i	$0.564465\pi$
$198$	0	0
$199$	−215378.	−0.385540	−0.192770	−	0.981244i	$-0.561747\pi$
$199$	−215378.	−0.385540	−0.192770	+	0.981244i	$0.561747\pi$
$200$	3537.24	0.00625301
$201$	0	0
$202$	−531472.	−0.916435
$203$	2778.51	0.00473229
$204$	0	0
$205$	−421332.	−0.700229
$206$	−496444.	−0.815084
$207$	0	0
$208$	−66319.7	−0.106288
$209$	−1.56687e6	−2.48123
$210$	0	0
$211$	4571.15	0.00706837	0.00353419	−	0.999994i	$-0.498875\pi$
$211$	4571.15	0.00706837	0.00353419	+	0.999994i	$0.498875\pi$
$212$	172513.	0.263622
$213$	0	0
$214$	−411474.	−0.614197
$215$	705201.	1.04044
$216$	0	0
$217$	195200.	0.281404
$218$	917416.	1.30745
$219$	0	0
$220$	−535037.	−0.745292
$221$	421625.	0.580692
$222$	0	0
$223$	945528.	1.27325	0.636623	−	0.771175i	$-0.280331\pi$
$223$	945528.	1.27325	0.636623	+	0.771175i	$0.280331\pi$
$224$	−22838.3	−0.0304120
$225$	0	0
$226$	−299735.	−0.390361
$227$	−183890.	−0.236861	−0.118430	−	0.992962i	$-0.537786\pi$
$227$	−183890.	−0.236861	−0.118430	+	0.992962i	$0.537786\pi$
$228$	0	0
$229$	558434.	0.703693	0.351846	−	0.936058i	$-0.385554\pi$
$229$	558434.	0.703693	0.351846	+	0.936058i	$0.385554\pi$
$230$	707500.	0.881875
$231$	0	0
$232$	−7973.10	−0.00972539
$233$	−1.04850e6	−1.26526	−0.632630	−	0.774454i	$-0.718025\pi$
$233$	−1.04850e6	−1.26526	−0.632630	+	0.774454i	$0.718025\pi$
$234$	0	0
$235$	−497251.	−0.587362
$236$	248860.	0.290854
$237$	0	0
$238$	145194.	0.166152
$239$	−713803.	−0.808321	−0.404160	−	0.914688i	$-0.632436\pi$
$239$	−713803.	−0.808321	−0.404160	+	0.914688i	$0.632436\pi$
$240$	0	0
$241$	−770937.	−0.855020	−0.427510	−	0.904011i	$-0.640609\pi$
$241$	−770937.	−0.855020	−0.427510	+	0.904011i	$0.640609\pi$
$242$	762242.	0.836670
$243$	0	0
$244$	−190887.	−0.205259
$245$	919760.	0.978947
$246$	0	0
$247$	684549.	0.713940
$248$	−560139.	−0.578318
$249$	0	0
$250$	692456.	0.700716
$251$	−450848.	−0.451696	−0.225848	−	0.974163i	$-0.572515\pi$
$251$	−450848.	−0.451696	−0.225848	+	0.974163i	$0.572515\pi$
$252$	0	0
$253$	−1.85980e6	−1.82669
$254$	138313.	0.134518
$255$	0	0
$256$	65536.0	0.0625000
$257$	−885607.	−0.836389	−0.418194	−	0.908358i	$-0.637337\pi$
$257$	−885607.	−0.836389	−0.418194	+	0.908358i	$0.637337\pi$
$258$	0	0
$259$	−217269.	−0.201256
$260$	233751.	0.214448
$261$	0	0
$262$	−318495.	−0.286648
$263$	287968.	0.256718	0.128359	−	0.991728i	$-0.459029\pi$
$263$	287968.	0.256718	0.128359	+	0.991728i	$0.459029\pi$
$264$	0	0
$265$	−608042.	−0.531886
$266$	235736.	0.204278
$267$	0	0
$268$	98873.3	0.0840896
$269$	−898860.	−0.757376	−0.378688	−	0.925524i	$-0.623625\pi$
$269$	−898860.	−0.757376	−0.378688	+	0.925524i	$0.623625\pi$
$270$	0	0
$271$	1.89319e6	1.56592	0.782962	−	0.622069i	$-0.213708\pi$
$271$	1.89319e6	1.56592	0.782962	+	0.622069i	$0.213708\pi$
$272$	−416643.	−0.341462
$273$	0	0
$274$	1.08416e6	0.872405
$275$	32773.0	0.0261327
$276$	0	0
$277$	1.73212e6	1.35637	0.678187	−	0.734889i	$-0.262766\pi$
$277$	1.73212e6	1.35637	0.678187	+	0.734889i	$0.262766\pi$
$278$	132675.	0.102962
$279$	0	0
$280$	80496.4	0.0613594
$281$	293257.	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$281$	293257.	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$282$	0	0
$283$	−798164.	−0.592415	−0.296208	−	0.955124i	$-0.595722\pi$
$283$	−798164.	−0.592415	−0.296208	+	0.955124i	$0.595722\pi$
$284$	167982.	0.123586
$285$	0	0
$286$	−614460.	−0.444200
$287$	−166632.	−0.119413
$288$	0	0
$289$	1.22894e6	0.865537
$290$	28102.1	0.0196220
$291$	0	0
$292$	−614085.	−0.421475
$293$	−1.39073e6	−0.946398	−0.473199	−	0.880955i	$-0.656901\pi$
$293$	−1.39073e6	−0.946398	−0.473199	+	0.880955i	$0.656901\pi$
$294$	0	0
$295$	−877136.	−0.586830
$296$	623467.	0.413603
$297$	0	0
$298$	146539.	0.0955902
$299$	812526.	0.525605
$300$	0	0
$301$	278898.	0.177431
$302$	1.97988e6	1.24917
$303$	0	0
$304$	−676460.	−0.419815
$305$	672803.	0.414132
$306$	0	0
$307$	598019.	0.362134	0.181067	−	0.983471i	$-0.442045\pi$
$307$	598019.	0.362134	0.181067	+	0.983471i	$0.442045\pi$
$308$	−211600.	−0.127098
$309$	0	0
$310$	1.97427e6	1.16682
$311$	2.97803e6	1.74594	0.872968	−	0.487777i	$-0.162192\pi$
$311$	2.97803e6	1.74594	0.872968	+	0.487777i	$0.162192\pi$
$312$	0	0
$313$	2.50373e6	1.44453	0.722266	−	0.691615i	$-0.243100\pi$
$313$	2.50373e6	1.44453	0.722266	+	0.691615i	$0.243100\pi$
$314$	−231327.	−0.132404
$315$	0	0
$316$	285660.	0.160928
$317$	−2.08635e6	−1.16611	−0.583053	−	0.812434i	$-0.698142\pi$
$317$	−2.08635e6	−1.16611	−0.583053	+	0.812434i	$0.698142\pi$
$318$	0	0
$319$	−73871.8	−0.0406445
$320$	−230989.	−0.126101
$321$	0	0
$322$	279807.	0.150390
$323$	4.30057e6	2.29361
$324$	0	0
$325$	−14318.1	−0.00751932
$326$	−540090.	−0.281464
$327$	0	0
$328$	478160.	0.245408
$329$	−196656.	−0.100166
$330$	0	0
$331$	−1.89513e6	−0.950755	−0.475378	−	0.879782i	$-0.657689\pi$
$331$	−1.89513e6	−0.950755	−0.475378	+	0.879782i	$0.657689\pi$
$332$	691529.	0.344322
$333$	0	0
$334$	1.62411e6	0.796619
$335$	−348491.	−0.169660
$336$	0	0
$337$	1.33312e6	0.639434	0.319717	−	0.947513i	$-0.396412\pi$
$337$	1.33312e6	0.639434	0.319717	+	0.947513i	$0.396412\pi$
$338$	−1.21672e6	−0.579294
$339$	0	0
$340$	1.46851e6	0.688936
$341$	−5.18976e6	−2.41691
$342$	0	0
$343$	738601.	0.338980
$344$	−800315.	−0.364640
$345$	0	0
$346$	−912967.	−0.409982
$347$	−368752.	−0.164404	−0.0822018	−	0.996616i	$-0.526195\pi$
$347$	−368752.	−0.164404	−0.0822018	+	0.996616i	$0.526195\pi$
$348$	0	0
$349$	2.73539e6	1.20214	0.601071	−	0.799195i	$-0.294741\pi$
$349$	2.73539e6	1.20214	0.601071	+	0.799195i	$0.294741\pi$
$350$	−4930.70	−0.00215149
$351$	0	0
$352$	607200.	0.261201
$353$	3.29025e6	1.40537	0.702687	−	0.711499i	$-0.251983\pi$
$353$	3.29025e6	1.40537	0.702687	+	0.711499i	$0.251983\pi$
$354$	0	0
$355$	−592074.	−0.249348
$356$	1.50553e6	0.629599
$357$	0	0
$358$	77480.1	0.0319508
$359$	1.85769e6	0.760740	0.380370	−	0.924834i	$-0.375797\pi$
$359$	1.85769e6	0.760740	0.380370	+	0.924834i	$0.375797\pi$
$360$	0	0
$361$	4.50629e6	1.81991
$362$	−551159.	−0.221058
$363$	0	0
$364$	92445.7	0.0365707
$365$	2.16442e6	0.850371
$366$	0	0
$367$	−2.27639e6	−0.882231	−0.441115	−	0.897450i	$-0.645417\pi$
$367$	−2.27639e6	−0.882231	−0.441115	+	0.897450i	$0.645417\pi$
$368$	−802924.	−0.309069
$369$	0	0
$370$	−2.19748e6	−0.834490
$371$	−240473.	−0.0907050
$372$	0	0
$373$	528964.	0.196858	0.0984292	−	0.995144i	$-0.468618\pi$
$373$	528964.	0.196858	0.0984292	+	0.995144i	$0.468618\pi$
$374$	−3.86025e6	−1.42704
$375$	0	0
$376$	564318.	0.205851
$377$	32273.8	0.0116949
$378$	0	0
$379$	−3.23027e6	−1.15516	−0.577578	−	0.816336i	$-0.696002\pi$
$379$	−3.23027e6	−1.15516	−0.577578	+	0.816336i	$0.696002\pi$
$380$	2.38426e6	0.847022
$381$	0	0
$382$	−1.34609e6	−0.471972
$383$	−3.92903e6	−1.36864	−0.684319	−	0.729183i	$-0.739900\pi$
$383$	−3.92903e6	−1.36864	−0.684319	+	0.729183i	$0.739900\pi$
$384$	0	0
$385$	745810.	0.256434
$386$	−2.96349e6	−1.01236
$387$	0	0
$388$	−1.30379e6	−0.439670
$389$	−4.69206e6	−1.57213	−0.786066	−	0.618142i	$-0.787886\pi$
$389$	−4.69206e6	−1.57213	−0.786066	+	0.618142i	$0.787886\pi$
$390$	0	0
$391$	5.10457e6	1.68856
$392$	−1.04381e6	−0.343089
$393$	0	0
$394$	−876513.	−0.284458
$395$	−1.00684e6	−0.324690
$396$	0	0
$397$	−4.97999e6	−1.58581	−0.792907	−	0.609343i	$-0.791433\pi$
$397$	−4.97999e6	−1.58581	−0.792907	+	0.609343i	$0.791433\pi$
$398$	−861513.	−0.272618
$399$	0	0
$400$	14149.0	0.00442155
$401$	−3.48426e6	−1.08205	−0.541027	−	0.841005i	$-0.681965\pi$
$401$	−3.48426e6	−1.08205	−0.541027	+	0.841005i	$0.681965\pi$
$402$	0	0
$403$	2.26735e6	0.695433
$404$	−2.12589e6	−0.648017
$405$	0	0
$406$	11114.0	0.00334623
$407$	5.77650e6	1.72854
$408$	0	0
$409$	−1.29484e6	−0.382744	−0.191372	−	0.981518i	$-0.561294\pi$
$409$	−1.29484e6	−0.382744	−0.191372	+	0.981518i	$0.561294\pi$
$410$	−1.68533e6	−0.495137
$411$	0	0
$412$	−1.98578e6	−0.576352
$413$	−346896.	−0.100075
$414$	0	0
$415$	−2.43737e6	−0.694708
$416$	−265279.	−0.0751569
$417$	0	0
$418$	−6.26749e6	−1.75450
$419$	6.37479e6	1.77391	0.886953	−	0.461859i	$-0.152818\pi$
$419$	6.37479e6	1.77391	0.886953	+	0.461859i	$0.152818\pi$
$420$	0	0
$421$	−4.41604e6	−1.21430	−0.607152	−	0.794586i	$-0.707688\pi$
$421$	−4.41604e6	−1.21430	−0.607152	+	0.794586i	$0.707688\pi$
$422$	18284.6	0.00499809
$423$	0	0
$424$	690052.	0.186409
$425$	−89951.5	−0.0241566
$426$	0	0
$427$	266085.	0.0706238
$428$	−1.64589e6	−0.434303
$429$	0	0
$430$	2.82080e6	0.735701
$431$	6.65197e6	1.72487	0.862436	−	0.506166i	$-0.168938\pi$
$431$	6.65197e6	1.72487	0.862436	+	0.506166i	$0.168938\pi$
$432$	0	0
$433$	−3.41878e6	−0.876297	−0.438149	−	0.898903i	$-0.644366\pi$
$433$	−3.41878e6	−0.876297	−0.438149	+	0.898903i	$0.644366\pi$
$434$	780801.	0.198983
$435$	0	0
$436$	3.66966e6	0.924506
$437$	8.28775e6	2.07603
$438$	0	0
$439$	3964.68	0.000981854 0	0.000490927	−	1.00000i	$-0.499844\pi$
$439$	3964.68	0.000981854 0	0.000490927		1.00000i	$0.499844\pi$
$440$	−2.14015e6	−0.527001
$441$	0	0
$442$	1.68650e6	0.410611
$443$	−3.82834e6	−0.926833	−0.463417	−	0.886141i	$-0.653377\pi$
$443$	−3.82834e6	−0.926833	−0.463417	+	0.886141i	$0.653377\pi$
$444$	0	0
$445$	−5.30641e6	−1.27028
$446$	3.78211e6	0.900321
$447$	0	0
$448$	−91353.3	−0.0215045
$449$	−1.62273e6	−0.379866	−0.189933	−	0.981797i	$-0.560827\pi$
$449$	−1.62273e6	−0.379866	−0.189933	+	0.981797i	$0.560827\pi$
$450$	0	0
$451$	4.43021e6	1.02561
$452$	−1.19894e6	−0.276027
$453$	0	0
$454$	−735559.	−0.167486
$455$	−325836.	−0.0737854
$456$	0	0
$457$	−7.52754e6	−1.68602	−0.843010	−	0.537898i	$-0.819219\pi$
$457$	−7.52754e6	−1.68602	−0.843010	+	0.537898i	$0.819219\pi$
$458$	2.23374e6	0.497586
$459$	0	0
$460$	2.83000e6	0.623580
$461$	−720594.	−0.157920	−0.0789602	−	0.996878i	$-0.525160\pi$
$461$	−720594.	−0.157920	−0.0789602	+	0.996878i	$0.525160\pi$
$462$	0	0
$463$	−5.45577e6	−1.18278	−0.591389	−	0.806386i	$-0.701420\pi$
$463$	−5.45577e6	−1.18278	−0.591389	+	0.806386i	$0.701420\pi$
$464$	−31892.4	−0.00687689
$465$	0	0
$466$	−4.19401e6	−0.894674
$467$	3.70082e6	0.785245	0.392623	−	0.919700i	$-0.371568\pi$
$467$	3.70082e6	0.785245	0.392623	+	0.919700i	$0.371568\pi$
$468$	0	0
$469$	−137824.	−0.0289329
$470$	−1.98900e6	−0.415328
$471$	0	0
$472$	995440.	0.205665
$473$	−7.41502e6	−1.52391
$474$	0	0
$475$	−146045.	−0.0296997
$476$	580776.	0.117487
$477$	0	0
$478$	−2.85521e6	−0.571569
$479$	6.01603e6	1.19804	0.599020	−	0.800734i	$-0.295557\pi$
$479$	6.01603e6	1.19804	0.599020	+	0.800734i	$0.295557\pi$
$480$	0	0
$481$	−2.52369e6	−0.497363
$482$	−3.08375e6	−0.604590
$483$	0	0
$484$	3.04897e6	0.591615
$485$	4.59535e6	0.887083
$486$	0	0
$487$	1.27159e6	0.242954	0.121477	−	0.992594i	$-0.461237\pi$
$487$	1.27159e6	0.242954	0.121477	+	0.992594i	$0.461237\pi$
$488$	−763548.	−0.145140
$489$	0	0
$490$	3.67904e6	0.692220
$491$	3.92029e6	0.733863	0.366931	−	0.930248i	$-0.380408\pi$
$491$	3.92029e6	0.733863	0.366931	+	0.930248i	$0.380408\pi$
$492$	0	0
$493$	202755.	0.0375711
$494$	2.73819e6	0.504832
$495$	0	0
$496$	−2.24056e6	−0.408932
$497$	−234158.	−0.0425224
$498$	0	0
$499$	−4.39562e6	−0.790257	−0.395129	−	0.918626i	$-0.629300\pi$
$499$	−4.39562e6	−0.790257	−0.395129	+	0.918626i	$0.629300\pi$
$500$	2.76982e6	0.495481
$501$	0	0
$502$	−1.80339e6	−0.319397
$503$	−6.81583e6	−1.20115	−0.600577	−	0.799567i	$-0.705062\pi$
$503$	−6.81583e6	−1.20115	−0.600577	+	0.799567i	$0.705062\pi$
$504$	0	0
$505$	7.49294e6	1.30745
$506$	−7.43920e6	−1.29167
$507$	0	0
$508$	553253.	0.0951184
$509$	−4.66602e6	−0.798274	−0.399137	−	0.916891i	$-0.630690\pi$
$509$	−4.66602e6	−0.798274	−0.399137	+	0.916891i	$0.630690\pi$
$510$	0	0
$511$	855999.	0.145018
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.54243e6	−0.591416
$515$	6.99911e6	1.16285
$516$	0	0
$517$	5.22848e6	0.860298
$518$	−869076.	−0.142309
$519$	0	0
$520$	935006.	0.151637
$521$	619313.	0.0999576	0.0499788	−	0.998750i	$-0.484085\pi$
$521$	619313.	0.0999576	0.0499788	+	0.998750i	$0.484085\pi$
$522$	0	0
$523$	−4.63344e6	−0.740713	−0.370356	−	0.928890i	$-0.620764\pi$
$523$	−4.63344e6	−0.740713	−0.370356	+	0.928890i	$0.620764\pi$
$524$	−1.27398e6	−0.202691
$525$	0	0
$526$	1.15187e6	0.181527
$527$	1.42443e7	2.23416
$528$	0	0
$529$	3.40081e6	0.528376
$530$	−2.43217e6	−0.376100
$531$	0	0
$532$	942945.	0.144447
$533$	−1.93551e6	−0.295105
$534$	0	0
$535$	5.80115e6	0.876253
$536$	395493.	0.0594603
$537$	0	0
$538$	−3.59544e6	−0.535545
$539$	−9.67106e6	−1.43384
$540$	0	0
$541$	−9.72857e6	−1.42908	−0.714539	−	0.699596i	$-0.753364\pi$
$541$	−9.72857e6	−1.42908	−0.714539	+	0.699596i	$0.753364\pi$
$542$	7.57276e6	1.10728
$543$	0	0
$544$	−1.66657e6	−0.241450
$545$	−1.29342e7	−1.86529
$546$	0	0
$547$	9.27795e6	1.32582	0.662909	−	0.748700i	$-0.269322\pi$
$547$	9.27795e6	1.32582	0.662909	+	0.748700i	$0.269322\pi$
$548$	4.33665e6	0.616883
$549$	0	0
$550$	131092.	0.0184786
$551$	329192.	0.0461923
$552$	0	0
$553$	−398193.	−0.0553708
$554$	6.92849e6	0.959101
$555$	0	0
$556$	530701.	0.0728053
$557$	−2.83520e6	−0.387209	−0.193604	−	0.981080i	$-0.562018\pi$
$557$	−2.83520e6	−0.387209	−0.193604	+	0.981080i	$0.562018\pi$
$558$	0	0
$559$	3.23954e6	0.438484
$560$	321986.	0.0433877
$561$	0	0
$562$	1.17303e6	0.156663
$563$	−9.71416e6	−1.29162	−0.645809	−	0.763499i	$-0.723480\pi$
$563$	−9.71416e6	−1.29162	−0.645809	+	0.763499i	$0.723480\pi$
$564$	0	0
$565$	4.22580e6	0.556914
$566$	−3.19266e6	−0.418901
$567$	0	0
$568$	671930.	0.0873883
$569$	1.80297e6	0.233458	0.116729	−	0.993164i	$-0.462759\pi$
$569$	1.80297e6	0.233458	0.116729	+	0.993164i	$0.462759\pi$
$570$	0	0
$571$	−6.49691e6	−0.833905	−0.416953	−	0.908928i	$-0.636902\pi$
$571$	−6.49691e6	−0.833905	−0.416953	+	0.908928i	$0.636902\pi$
$572$	−2.45784e6	−0.314097
$573$	0	0
$574$	−666526.	−0.0844379
$575$	−173348.	−0.0218650
$576$	0	0
$577$	3.93067e6	0.491504	0.245752	−	0.969333i	$-0.420965\pi$
$577$	3.93067e6	0.491504	0.245752	+	0.969333i	$0.420965\pi$
$578$	4.91576e6	0.612027
$579$	0	0
$580$	112408.	0.0138749
$581$	−963951.	−0.118472
$582$	0	0
$583$	6.39342e6	0.779043
$584$	−2.45634e6	−0.298028
$585$	0	0
$586$	−5.56292e6	−0.669205
$587$	−2.15461e6	−0.258092	−0.129046	−	0.991639i	$-0.541191\pi$
$587$	−2.15461e6	−0.258092	−0.129046	+	0.991639i	$0.541191\pi$
$588$	0	0
$589$	2.31269e7	2.74681
$590$	−3.50854e6	−0.414951
$591$	0	0
$592$	2.49387e6	0.292462
$593$	1.38401e7	1.61623	0.808113	−	0.589027i	$-0.200489\pi$
$593$	1.38401e7	1.61623	0.808113	+	0.589027i	$0.200489\pi$
$594$	0	0
$595$	−2.04701e6	−0.237044
$596$	586157.	0.0675925
$597$	0	0
$598$	3.25010e6	0.371659
$599$	166025.	0.0189063	0.00945313	−	0.999955i	$-0.496991\pi$
$599$	166025.	0.0189063	0.00945313	+	0.999955i	$0.496991\pi$
$600$	0	0
$601$	−1.46582e7	−1.65536	−0.827682	−	0.561198i	$-0.810341\pi$
$601$	−1.46582e7	−1.65536	−0.827682	+	0.561198i	$0.810341\pi$
$602$	1.11559e6	0.125463
$603$	0	0
$604$	7.91953e6	0.883297
$605$	−1.07464e7	−1.19365
$606$	0	0
$607$	−2.91095e6	−0.320673	−0.160337	−	0.987062i	$-0.551258\pi$
$607$	−2.91095e6	−0.320673	−0.160337	+	0.987062i	$0.551258\pi$
$608$	−2.70584e6	−0.296854
$609$	0	0
$610$	2.69121e6	0.292835
$611$	−2.28426e6	−0.247539
$612$	0	0
$613$	1.46967e7	1.57968	0.789838	−	0.613315i	$-0.210164\pi$
$613$	1.46967e7	1.57968	0.789838	+	0.613315i	$0.210164\pi$
$614$	2.39207e6	0.256067
$615$	0	0
$616$	−846401.	−0.0898720
$617$	2.98185e6	0.315335	0.157668	−	0.987492i	$-0.449603\pi$
$617$	2.98185e6	0.315335	0.157668	+	0.987492i	$0.449603\pi$
$618$	0	0
$619$	1.75899e6	0.184517	0.0922583	−	0.995735i	$-0.470591\pi$
$619$	1.75899e6	0.184517	0.0922583	+	0.995735i	$0.470591\pi$
$620$	7.89710e6	0.825065
$621$	0	0
$622$	1.19121e7	1.23456
$623$	−2.09862e6	−0.216627
$624$	0	0
$625$	−9.93529e6	−1.01737
$626$	1.00149e7	1.02144
$627$	0	0
$628$	−925307.	−0.0936239
$629$	−1.58547e7	−1.59783
$630$	0	0
$631$	4.55795e6	0.455718	0.227859	−	0.973694i	$-0.426827\pi$
$631$	4.55795e6	0.455718	0.227859	+	0.973694i	$0.426827\pi$
$632$	1.14264e6	0.113793
$633$	0	0
$634$	−8.34539e6	−0.824562
$635$	−1.95001e6	−0.191912
$636$	0	0
$637$	4.22518e6	0.412569
$638$	−295487.	−0.0287400
$639$	0	0
$640$	−923957.	−0.0891665
$641$	−1.72544e7	−1.65865	−0.829326	−	0.558765i	$-0.811275\pi$
$641$	−1.72544e7	−1.65865	−0.829326	+	0.558765i	$0.811275\pi$
$642$	0	0
$643$	1.33621e7	1.27453	0.637263	−	0.770647i	$-0.280067\pi$
$643$	1.33621e7	1.27453	0.637263	+	0.770647i	$0.280067\pi$
$644$	1.11923e6	0.106342
$645$	0	0
$646$	1.72023e7	1.62183
$647$	−357748.	−0.0335982	−0.0167991	−	0.999859i	$-0.505348\pi$
$647$	−357748.	−0.0335982	−0.0167991	+	0.999859i	$0.505348\pi$
$648$	0	0
$649$	9.22288e6	0.859518
$650$	−57272.6	−0.00531696
$651$	0	0
$652$	−2.16036e6	−0.199025
$653$	1.22418e7	1.12347	0.561737	−	0.827316i	$-0.310133\pi$
$653$	1.22418e7	1.12347	0.561737	+	0.827316i	$0.310133\pi$
$654$	0	0
$655$	4.49029e6	0.408951
$656$	1.91264e6	0.173529
$657$	0	0
$658$	−786626.	−0.0708277
$659$	−1.05104e7	−0.942766	−0.471383	−	0.881929i	$-0.656245\pi$
$659$	−1.05104e7	−0.942766	−0.471383	+	0.881929i	$0.656245\pi$
$660$	0	0
$661$	3.02422e6	0.269222	0.134611	−	0.990899i	$-0.457022\pi$
$661$	3.02422e6	0.269222	0.134611	+	0.990899i	$0.457022\pi$
$662$	−7.58052e6	−0.672286
$663$	0	0
$664$	2.76612e6	0.243473
$665$	−3.32352e6	−0.291437
$666$	0	0
$667$	390734.	0.0340069
$668$	6.49646e6	0.563294
$669$	0	0
$670$	−1.39396e6	−0.119968
$671$	−7.07437e6	−0.606571
$672$	0	0
$673$	−1.22963e7	−1.04649	−0.523245	−	0.852182i	$-0.675279\pi$
$673$	−1.22963e7	−1.04649	−0.523245	+	0.852182i	$0.675279\pi$
$674$	5.33249e6	0.452148
$675$	0	0
$676$	−4.86688e6	−0.409623
$677$	−1.21122e6	−0.101567	−0.0507833	−	0.998710i	$-0.516172\pi$
$677$	−1.21122e6	−0.101567	−0.0507833	+	0.998710i	$0.516172\pi$
$678$	0	0
$679$	1.81740e6	0.151278
$680$	5.87403e6	0.487151
$681$	0	0
$682$	−2.07590e7	−1.70902
$683$	5.89207e6	0.483299	0.241650	−	0.970364i	$-0.422312\pi$
$683$	5.89207e6	0.483299	0.241650	+	0.970364i	$0.422312\pi$
$684$	0	0
$685$	−1.52850e7	−1.24463
$686$	2.95440e6	0.239695
$687$	0	0
$688$	−3.20126e6	−0.257840
$689$	−2.79321e6	−0.224159
$690$	0	0
$691$	−1.24001e7	−0.987940	−0.493970	−	0.869479i	$-0.664455\pi$
$691$	−1.24001e7	−0.987940	−0.493970	+	0.869479i	$0.664455\pi$
$692$	−3.65187e6	−0.289901
$693$	0	0
$694$	−1.47501e6	−0.116251
$695$	−1.87052e6	−0.146893
$696$	0	0
$697$	−1.21595e7	−0.948059
$698$	1.09416e7	0.850043
$699$	0	0
$700$	−19722.8	−0.00152133
$701$	1.29116e7	0.992396	0.496198	−	0.868209i	$-0.334729\pi$
$701$	1.29116e7	0.992396	0.496198	+	0.868209i	$0.334729\pi$
$702$	0	0
$703$	−2.57416e7	−1.96448
$704$	2.42880e6	0.184697
$705$	0	0
$706$	1.31610e7	0.993750
$707$	2.96336e6	0.222965
$708$	0	0
$709$	1.87352e7	1.39973	0.699864	−	0.714276i	$-0.253244\pi$
$709$	1.87352e7	1.39973	0.699864	+	0.714276i	$0.253244\pi$
$710$	−2.36830e6	−0.176315
$711$	0	0
$712$	6.02212e6	0.445194
$713$	2.74505e7	2.02221
$714$	0	0
$715$	8.66295e6	0.633725
$716$	309920.	0.0225927
$717$	0	0
$718$	7.43075e6	0.537925
$719$	1.08129e7	0.780044	0.390022	−	0.920806i	$-0.372467\pi$
$719$	1.08129e7	0.780044	0.390022	+	0.920806i	$0.372467\pi$
$720$	0	0
$721$	2.76806e6	0.198306
$722$	1.80251e7	1.28687
$723$	0	0
$724$	−2.20464e6	−0.156311
$725$	−6885.44	−0.000486504 0
$726$	0	0
$727$	−5.61421e6	−0.393960	−0.196980	−	0.980407i	$-0.563113\pi$
$727$	−5.61421e6	−0.393960	−0.196980	+	0.980407i	$0.563113\pi$
$728$	369783.	0.0258594
$729$	0	0
$730$	8.65766e6	0.601303
$731$	2.03519e7	1.40868
$732$	0	0
$733$	−1.76823e7	−1.21557	−0.607785	−	0.794102i	$-0.707942\pi$
$733$	−1.76823e7	−1.21557	−0.607785	+	0.794102i	$0.707942\pi$
$734$	−9.10557e6	−0.623831
$735$	0	0
$736$	−3.21170e6	−0.218545
$737$	3.66430e6	0.248498
$738$	0	0
$739$	9.10643e6	0.613390	0.306695	−	0.951808i	$-0.400777\pi$
$739$	9.10643e6	0.613390	0.306695	+	0.951808i	$0.400777\pi$
$740$	−8.78993e6	−0.590073
$741$	0	0
$742$	−961891.	−0.0641381
$743$	331033.	0.0219988	0.0109994	−	0.999940i	$-0.496499\pi$
$743$	331033.	0.0219988	0.0109994	+	0.999940i	$0.496499\pi$
$744$	0	0
$745$	−2.06598e6	−0.136375
$746$	2.11585e6	0.139200
$747$	0	0
$748$	−1.54410e7	−1.00907
$749$	2.29428e6	0.149431
$750$	0	0
$751$	−2.80825e7	−1.81692	−0.908460	−	0.417972i	$-0.862741\pi$
$751$	−2.80825e7	−1.81692	−0.908460	+	0.417972i	$0.862741\pi$
$752$	2.25727e6	0.145559
$753$	0	0
$754$	129095.	0.00826954
$755$	−2.79133e7	−1.78215
$756$	0	0
$757$	−1.85991e7	−1.17965	−0.589824	−	0.807532i	$-0.700803\pi$
$757$	−1.85991e7	−1.17965	−0.589824	+	0.807532i	$0.700803\pi$
$758$	−1.29211e7	−0.816818
$759$	0	0
$760$	9.53705e6	0.598935
$761$	−1.03323e7	−0.646747	−0.323374	−	0.946271i	$-0.604817\pi$
$761$	−1.03323e7	−0.646747	−0.323374	+	0.946271i	$0.604817\pi$
$762$	0	0
$763$	−5.11529e6	−0.318097
$764$	−5.38437e6	−0.333735
$765$	0	0
$766$	−1.57161e7	−0.967773
$767$	−4.02937e6	−0.247314
$768$	0	0
$769$	2.00522e7	1.22278	0.611388	−	0.791331i	$-0.290611\pi$
$769$	2.00522e7	1.22278	0.611388	+	0.791331i	$0.290611\pi$
$770$	2.98324e6	0.181326
$771$	0	0
$772$	−1.18540e7	−0.715847
$773$	−2.68448e7	−1.61589	−0.807943	−	0.589260i	$-0.799419\pi$
$773$	−2.68448e7	−1.61589	−0.807943	+	0.589260i	$0.799419\pi$
$774$	0	0
$775$	−483727.	−0.0289298
$776$	−5.21515e6	−0.310894
$777$	0	0
$778$	−1.87682e7	−1.11167
$779$	−1.97422e7	−1.16560
$780$	0	0
$781$	6.22552e6	0.365214
$782$	2.04183e7	1.19399
$783$	0	0
$784$	−4.17525e6	−0.242601
$785$	3.26135e6	0.188896
$786$	0	0
$787$	7.67121e6	0.441496	0.220748	−	0.975331i	$-0.429150\pi$
$787$	7.67121e6	0.441496	0.220748	+	0.975331i	$0.429150\pi$
$788$	−3.50605e6	−0.201142
$789$	0	0
$790$	−4.02737e6	−0.229590
$791$	1.67125e6	0.0949731
$792$	0	0
$793$	3.09071e6	0.174532
$794$	−1.99200e7	−1.12134
$795$	0	0
$796$	−3.44605e6	−0.192770
$797$	8.83287e6	0.492556	0.246278	−	0.969199i	$-0.420792\pi$
$797$	8.83287e6	0.492556	0.246278	+	0.969199i	$0.420792\pi$
$798$	0	0
$799$	−1.43505e7	−0.795245
$800$	56595.8	0.00312651
$801$	0	0
$802$	−1.39370e7	−0.765128
$803$	−2.27583e7	−1.24552
$804$	0	0
$805$	−3.94486e6	−0.214556
$806$	9.06939e6	0.491746
$807$	0	0
$808$	−8.50355e6	−0.458218
$809$	2.96255e6	0.159146	0.0795729	−	0.996829i	$-0.474644\pi$
$809$	2.96255e6	0.159146	0.0795729	+	0.996829i	$0.474644\pi$
$810$	0	0
$811$	1.86775e7	0.997161	0.498581	−	0.866843i	$-0.333855\pi$
$811$	1.86775e7	0.997161	0.498581	+	0.866843i	$0.333855\pi$
$812$	44456.1	0.00236615
$813$	0	0
$814$	2.31060e7	1.22226
$815$	7.61445e6	0.401554
$816$	0	0
$817$	3.30432e7	1.73192
$818$	−5.17937e6	−0.270641
$819$	0	0
$820$	−6.74132e6	−0.350115
$821$	2.72527e7	1.41108	0.705540	−	0.708670i	$-0.250704\pi$
$821$	2.72527e7	1.41108	0.705540	+	0.708670i	$0.250704\pi$
$822$	0	0
$823$	−1.75672e7	−0.904071	−0.452035	−	0.892000i	$-0.649302\pi$
$823$	−1.75672e7	−0.904071	−0.452035	+	0.892000i	$0.649302\pi$
$824$	−7.94311e6	−0.407542
$825$	0	0
$826$	−1.38758e6	−0.0707635
$827$	1.92954e7	0.981046	0.490523	−	0.871428i	$-0.336806\pi$
$827$	1.92954e7	0.981046	0.490523	+	0.871428i	$0.336806\pi$
$828$	0	0
$829$	4.98059e6	0.251707	0.125853	−	0.992049i	$-0.459833\pi$
$829$	4.98059e6	0.251707	0.125853	+	0.992049i	$0.459833\pi$
$830$	−9.74950e6	−0.491232
$831$	0	0
$832$	−1.06111e6	−0.0531440
$833$	2.65440e7	1.32542
$834$	0	0
$835$	−2.28975e7	−1.13651
$836$	−2.50699e7	−1.24062
$837$	0	0
$838$	2.54992e7	1.25434
$839$	−2.94710e7	−1.44541	−0.722703	−	0.691159i	$-0.757101\pi$
$839$	−2.94710e7	−1.44541	−0.722703	+	0.691159i	$0.757101\pi$
$840$	0	0
$841$	−2.04956e7	−0.999243
$842$	−1.76641e7	−0.858643
$843$	0	0
$844$	73138.4	0.00353419
$845$	1.71539e7	0.826459
$846$	0	0
$847$	−4.25008e6	−0.203558
$848$	2.76021e6	0.131811
$849$	0	0
$850$	−359806.	−0.0170813
$851$	−3.05540e7	−1.44625
$852$	0	0
$853$	−2.73093e7	−1.28510	−0.642551	−	0.766243i	$-0.722124\pi$
$853$	−2.73093e7	−1.28510	−0.642551	+	0.766243i	$0.722124\pi$
$854$	1.06434e6	0.0499386
$855$	0	0
$856$	−6.58358e6	−0.307098
$857$	3.08855e6	0.143649	0.0718244	−	0.997417i	$-0.477118\pi$
$857$	3.08855e6	0.143649	0.0718244	+	0.997417i	$0.477118\pi$
$858$	0	0
$859$	−2.49115e6	−0.115190	−0.0575952	−	0.998340i	$-0.518343\pi$
$859$	−2.49115e6	−0.115190	−0.0575952	+	0.998340i	$0.518343\pi$
$860$	1.12832e7	0.520220
$861$	0	0
$862$	2.66079e7	1.21967
$863$	−6.26889e6	−0.286526	−0.143263	−	0.989685i	$-0.545759\pi$
$863$	−6.26889e6	−0.286526	−0.143263	+	0.989685i	$0.545759\pi$
$864$	0	0
$865$	1.28714e7	0.584907
$866$	−1.36751e7	−0.619636
$867$	0	0
$868$	3.12320e6	0.140702
$869$	1.05867e7	0.475566
$870$	0	0
$871$	−1.60089e6	−0.0715017
$872$	1.46786e7	0.653725
$873$	0	0
$874$	3.31510e7	1.46797
$875$	−3.86097e6	−0.170481
$876$	0	0
$877$	2.61600e6	0.114852	0.0574261	−	0.998350i	$-0.481711\pi$
$877$	2.61600e6	0.114852	0.0574261	+	0.998350i	$0.481711\pi$
$878$	15858.7	0.000694275 0
$879$	0	0
$880$	−8.56058e6	−0.372646
$881$	1.63208e7	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$881$	1.63208e7	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$882$	0	0
$883$	−2.80814e6	−0.121204	−0.0606020	−	0.998162i	$-0.519302\pi$
$883$	−2.80814e6	−0.121204	−0.0606020	+	0.998162i	$0.519302\pi$
$884$	6.74601e6	0.290346
$885$	0	0
$886$	−1.53134e7	−0.655370
$887$	2.61982e7	1.11805	0.559027	−	0.829150i	$-0.311175\pi$
$887$	2.61982e7	1.11805	0.559027	+	0.829150i	$0.311175\pi$
$888$	0	0
$889$	−771202.	−0.0327276
$890$	−2.12257e7	−0.898227
$891$	0	0
$892$	1.51285e7	0.636623
$893$	−2.32994e7	−0.977725
$894$	0	0
$895$	−1.09235e6	−0.0455832
$896$	−365413.	−0.0152060
$897$	0	0
$898$	−6.49092e6	−0.268606
$899$	1.09034e6	0.0449949
$900$	0	0
$901$	−1.75479e7	−0.720135
$902$	1.77208e7	0.725217
$903$	0	0
$904$	−4.79576e6	−0.195180
$905$	7.77050e6	0.315375
$906$	0	0
$907$	2.79242e7	1.12710	0.563550	−	0.826082i	$-0.309435\pi$
$907$	2.79242e7	1.12710	0.563550	+	0.826082i	$0.309435\pi$
$908$	−2.94224e6	−0.118430
$909$	0	0
$910$	−1.30334e6	−0.0521741
$911$	4.56090e7	1.82077	0.910384	−	0.413763i	$-0.135786\pi$
$911$	4.56090e7	1.82077	0.910384	+	0.413763i	$0.135786\pi$
$912$	0	0
$913$	2.56284e7	1.01752
$914$	−3.01102e7	−1.19220
$915$	0	0
$916$	8.93494e6	0.351846
$917$	1.77585e6	0.0697402
$918$	0	0
$919$	6.63615e6	0.259196	0.129598	−	0.991567i	$-0.458631\pi$
$919$	6.63615e6	0.259196	0.129598	+	0.991567i	$0.458631\pi$
$920$	1.13200e7	0.440938
$921$	0	0
$922$	−2.88237e6	−0.111667
$923$	−2.71986e6	−0.105085
$924$	0	0
$925$	538416.	0.0206901
$926$	−2.18231e7	−0.836351
$927$	0	0
$928$	−127570.	−0.00486270
$929$	2.05335e7	0.780590	0.390295	−	0.920690i	$-0.372373\pi$
$929$	2.05335e7	0.780590	0.390295	+	0.920690i	$0.372373\pi$
$930$	0	0
$931$	4.30967e7	1.62956
$932$	−1.67760e7	−0.632630
$933$	0	0
$934$	1.48033e7	0.555252
$935$	5.44237e7	2.03591
$936$	0	0
$937$	−3.95168e7	−1.47039	−0.735196	−	0.677855i	$-0.762910\pi$
$937$	−3.95168e7	−1.47039	−0.735196	+	0.677855i	$0.762910\pi$
$938$	−551295.	−0.0204586
$939$	0	0
$940$	−7.95601e6	−0.293681
$941$	9.82637e6	0.361759	0.180879	−	0.983505i	$-0.442106\pi$
$941$	9.82637e6	0.361759	0.180879	+	0.983505i	$0.442106\pi$
$942$	0	0
$943$	−2.34330e7	−0.858121
$944$	3.98176e6	0.145427
$945$	0	0
$946$	−2.96601e7	−1.07757
$947$	−1.90818e7	−0.691422	−0.345711	−	0.938341i	$-0.612362\pi$
$947$	−1.90818e7	−0.691422	−0.345711	+	0.938341i	$0.612362\pi$
$948$	0	0
$949$	9.94286e6	0.358382
$950$	−584179.	−0.0210009
$951$	0	0
$952$	2.32310e6	0.0830761
$953$	−3.72070e7	−1.32707	−0.663533	−	0.748147i	$-0.730944\pi$
$953$	−3.72070e7	−1.32707	−0.663533	+	0.748147i	$0.730944\pi$
$954$	0	0
$955$	1.89778e7	0.673346
$956$	−1.14209e7	−0.404160
$957$	0	0
$958$	2.40641e7	0.847142
$959$	−6.04504e6	−0.212252
$960$	0	0
$961$	4.79713e7	1.67561
$962$	−1.00948e7	−0.351689
$963$	0	0
$964$	−1.23350e7	−0.427510
$965$	4.17807e7	1.44430
$966$	0	0
$967$	5.38164e7	1.85075	0.925377	−	0.379048i	$-0.123748\pi$
$967$	5.38164e7	1.85075	0.925377	+	0.379048i	$0.123748\pi$
$968$	1.21959e7	0.418335
$969$	0	0
$970$	1.83814e7	0.627262
$971$	−3.16679e7	−1.07788	−0.538940	−	0.842344i	$-0.681175\pi$
$971$	−3.16679e7	−1.07788	−0.538940	+	0.842344i	$0.681175\pi$
$972$	0	0
$973$	−739766.	−0.0250503
$974$	5.08635e6	0.171794
$975$	0	0
$976$	−3.05419e6	−0.102629
$977$	6.06230e6	0.203189	0.101595	−	0.994826i	$-0.467606\pi$
$977$	6.06230e6	0.203189	0.101595	+	0.994826i	$0.467606\pi$
$978$	0	0
$979$	5.57957e7	1.86056
$980$	1.47162e7	0.489474
$981$	0	0
$982$	1.56812e7	0.518919
$983$	−2.27002e7	−0.749283	−0.374642	−	0.927170i	$-0.622234\pi$
$983$	−2.27002e7	−0.749283	−0.374642	+	0.927170i	$0.622234\pi$
$984$	0	0
$985$	1.23575e7	0.405826
$986$	811019.	0.0265668
$987$	0	0
$988$	1.09528e7	0.356970
$989$	3.92207e7	1.27504
$990$	0	0
$991$	4.31783e7	1.39663	0.698315	−	0.715790i	$-0.253933\pi$
$991$	4.31783e7	1.39663	0.698315	+	0.715790i	$0.253933\pi$
$992$	−8.96222e6	−0.289159
$993$	0	0
$994$	−936631.	−0.0300679
$995$	1.21460e7	0.388934
$996$	0	0
$997$	−2.65605e7	−0.846251	−0.423125	−	0.906071i	$-0.639067\pi$
$997$	−2.65605e7	−0.846251	−0.423125	+	0.906071i	$0.639067\pi$
$998$	−1.75825e7	−0.558796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.6.a.f.1.1		2
3.2	odd	2		162.6.a.e.1.2		2
9.2	odd	6		18.6.c.a.13.1	yes	4
9.4	even	3		54.6.c.a.19.2		4
9.5	odd	6		18.6.c.a.7.1	✓	4
9.7	even	3		54.6.c.a.37.2		4
36.7	odd	6		432.6.i.a.145.2		4
36.11	even	6		144.6.i.a.49.2		4
36.23	even	6		144.6.i.a.97.2		4
36.31	odd	6		432.6.i.a.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.c.a.7.1	✓	4	9.5	odd	6
18.6.c.a.13.1	yes	4	9.2	odd	6
54.6.c.a.19.2		4	9.4	even	3
54.6.c.a.37.2		4	9.7	even	3
144.6.i.a.49.2		4	36.11	even	6
144.6.i.a.97.2		4	36.23	even	6
162.6.a.e.1.2		2	3.2	odd	2
162.6.a.f.1.1		2	1.1	even	1	trivial
432.6.i.a.145.2		4	36.7	odd	6
432.6.i.a.289.2		4	36.31	odd	6

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -56.3939 q^{5} -22.3031 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -56.3939 q^{5} -22.3031 q^{7} +64.0000 q^{8} -225.576 q^{10} +592.968 q^{11} -259.061 q^{13} -89.2122 q^{14} +256.000 q^{16} -1627.51 q^{17} -2642.42 q^{19} -902.302 q^{20} +2371.87 q^{22} -3136.42 q^{23} +55.2694 q^{25} -1036.24 q^{26} -356.849 q^{28} -124.580 q^{29} -8752.17 q^{31} +1024.00 q^{32} -6510.05 q^{34} +1257.76 q^{35} +9741.67 q^{37} -10569.7 q^{38} -3609.21 q^{40} +7471.24 q^{41} -12504.9 q^{43} +9487.49 q^{44} -12545.7 q^{46} +8817.46 q^{47} -16309.6 q^{49} +221.077 q^{50} -4144.98 q^{52} +10782.1 q^{53} -33439.8 q^{55} -1427.40 q^{56} -498.319 q^{58} +15553.7 q^{59} -11930.4 q^{61} -35008.7 q^{62} +4096.00 q^{64} +14609.5 q^{65} +6179.58 q^{67} -26040.2 q^{68} +5031.02 q^{70} +10498.9 q^{71} -38380.3 q^{73} +38966.7 q^{74} -42278.7 q^{76} -13225.0 q^{77} +17853.7 q^{79} -14436.8 q^{80} +29885.0 q^{82} +43220.6 q^{83} +91781.7 q^{85} -50019.7 q^{86} +37950.0 q^{88} +94095.6 q^{89} +5777.86 q^{91} -50182.8 q^{92} +35269.8 q^{94} +149016. q^{95} -81486.7 q^{97} -65238.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 - 54 * q^5 - 74 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} - 54 q^{5} - 74 q^{7} + 128 q^{8} - 216 q^{10} - 78 q^{11} - 1106 q^{13} - 296 q^{14} + 512 q^{16} - 492 q^{17} - 1640 q^{19} - 864 q^{20} - 312 q^{22} - 5538 q^{23} - 3064 q^{25} - 4424 q^{26} - 1184 q^{28} - 3894 q^{29} - 4718 q^{31} + 2048 q^{32} - 1968 q^{34} + 1134 q^{35} - 4796 q^{37} - 6560 q^{38} - 3456 q^{40} + 15354 q^{41} - 32858 q^{43} - 1248 q^{44} - 22152 q^{46} + 24954 q^{47} - 30444 q^{49} - 12256 q^{50} - 17696 q^{52} + 16332 q^{53} - 35046 q^{55} - 4736 q^{56} - 15576 q^{58} + 21966 q^{59} - 3050 q^{61} - 18872 q^{62} + 8192 q^{64} + 12582 q^{65} - 36758 q^{67} - 7872 q^{68} + 4536 q^{70} + 73848 q^{71} - 51188 q^{73} - 19184 q^{74} - 26240 q^{76} + 21462 q^{77} + 14926 q^{79} - 13824 q^{80} + 61416 q^{82} + 90762 q^{83} + 94500 q^{85} - 131432 q^{86} - 4992 q^{88} + 9300 q^{89} + 49562 q^{91} - 88608 q^{92} + 99816 q^{94} + 151416 q^{95} + 30262 q^{97} - 121776 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 54 * q^5 - 74 * q^7 + 128 * q^8 - 216 * q^10 - 78 * q^11 - 1106 * q^13 - 296 * q^14 + 512 * q^16 - 492 * q^17 - 1640 * q^19 - 864 * q^20 - 312 * q^22 - 5538 * q^23 - 3064 * q^25 - 4424 * q^26 - 1184 * q^28 - 3894 * q^29 - 4718 * q^31 + 2048 * q^32 - 1968 * q^34 + 1134 * q^35 - 4796 * q^37 - 6560 * q^38 - 3456 * q^40 + 15354 * q^41 - 32858 * q^43 - 1248 * q^44 - 22152 * q^46 + 24954 * q^47 - 30444 * q^49 - 12256 * q^50 - 17696 * q^52 + 16332 * q^53 - 35046 * q^55 - 4736 * q^56 - 15576 * q^58 + 21966 * q^59 - 3050 * q^61 - 18872 * q^62 + 8192 * q^64 + 12582 * q^65 - 36758 * q^67 - 7872 * q^68 + 4536 * q^70 + 73848 * q^71 - 51188 * q^73 - 19184 * q^74 - 26240 * q^76 + 21462 * q^77 + 14926 * q^79 - 13824 * q^80 + 61416 * q^82 + 90762 * q^83 + 94500 * q^85 - 131432 * q^86 - 4992 * q^88 + 9300 * q^89 + 49562 * q^91 - 88608 * q^92 + 99816 * q^94 + 151416 * q^95 + 30262 * q^97 - 121776 * q^98

Introduction

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