LMFDB - Embedded newform 162.8.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,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, 8, names="a")

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
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:	$50.6063741284$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.43103376.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 383x^{2} + 384x + 18612 $ x^4 - 2x^3 - 383x^2 + 384*x + 18612
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$18.5928$ 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)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} +280.425 q^{5} -164.323 q^{7} -512.000 q^{8} -2243.40 q^{10} +2037.94 q^{11} -1678.33 q^{13} +1314.59 q^{14} +4096.00 q^{16} -31685.7 q^{17} -12641.8 q^{19} +17947.2 q^{20} -16303.5 q^{22} -49466.9 q^{23} +513.276 q^{25} +13426.7 q^{26} -10516.7 q^{28} +7552.79 q^{29} +156035. q^{31} -32768.0 q^{32} +253485. q^{34} -46080.4 q^{35} +541440. q^{37} +101134. q^{38} -143578. q^{40} -537583. q^{41} +200498. q^{43} +130428. q^{44} +395735. q^{46} +410871. q^{47} -796541. q^{49} -4106.21 q^{50} -107413. q^{52} -1.36778e6 q^{53} +571489. q^{55} +84133.6 q^{56} -60422.3 q^{58} -798889. q^{59} +569439. q^{61} -1.24828e6 q^{62} +262144. q^{64} -470647. q^{65} -4.80453e6 q^{67} -2.02788e6 q^{68} +368643. q^{70} -2.45714e6 q^{71} +1.60344e6 q^{73} -4.33152e6 q^{74} -809075. q^{76} -334881. q^{77} +5.58998e6 q^{79} +1.14862e6 q^{80} +4.30067e6 q^{82} -9.82905e6 q^{83} -8.88546e6 q^{85} -1.60398e6 q^{86} -1.04342e6 q^{88} +117249. q^{89} +275789. q^{91} -3.16588e6 q^{92} -3.28697e6 q^{94} -3.54508e6 q^{95} -7.79245e6 q^{97} +6.37233e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) $ 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	280.425	1.00328	0.501640	−	0.865077i	$-0.332730\pi$
$5$	280.425	1.00328	0.501640	+	0.865077i	$0.332730\pi$
$6$	0	0
$7$	−164.323	−0.181074	−0.0905370	−	0.995893i	$-0.528858\pi$
$7$	−164.323	−0.181074	−0.0905370	+	0.995893i	$0.528858\pi$
$8$	−512.000	−0.353553
$9$	0	0
$10$	−2243.40	−0.709426
$11$	2037.94	0.461654	0.230827	−	0.972995i	$-0.425857\pi$
$11$	2037.94	0.461654	0.230827	+	0.972995i	$0.425857\pi$
$12$	0	0
$13$	−1678.33	−0.211873	−0.105937	−	0.994373i	$-0.533784\pi$
$13$	−1678.33	−0.211873	−0.105937	+	0.994373i	$0.533784\pi$
$14$	1314.59	0.128039
$15$	0	0
$16$	4096.00	0.250000
$17$	−31685.7	−1.56420	−0.782099	−	0.623154i	$-0.785851\pi$
$17$	−31685.7	−1.56420	−0.782099	+	0.623154i	$0.785851\pi$
$18$	0	0
$19$	−12641.8	−0.422835	−0.211418	−	0.977396i	$-0.567808\pi$
$19$	−12641.8	−0.422835	−0.211418	+	0.977396i	$0.567808\pi$
$20$	17947.2	0.501640
$21$	0	0
$22$	−16303.5	−0.326439
$23$	−49466.9	−0.847749	−0.423874	−	0.905721i	$-0.639330\pi$
$23$	−49466.9	−0.847749	−0.423874	+	0.905721i	$0.639330\pi$
$24$	0	0
$25$	513.276	0.00656993
$26$	13426.7	0.149817
$27$	0	0
$28$	−10516.7	−0.0905370
$29$	7552.79	0.0575062	0.0287531	−	0.999587i	$-0.490846\pi$
$29$	7552.79	0.0575062	0.0287531	+	0.999587i	$0.490846\pi$
$30$	0	0
$31$	156035.	0.940712	0.470356	−	0.882477i	$-0.344125\pi$
$31$	156035.	0.940712	0.470356	+	0.882477i	$0.344125\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	253485.	1.10606
$35$	−46080.4	−0.181668
$36$	0	0
$37$	541440.	1.75729	0.878647	−	0.477472i	$-0.158447\pi$
$37$	541440.	1.75729	0.878647	+	0.477472i	$0.158447\pi$
$38$	101134.	0.298990
$39$	0	0
$40$	−143578.	−0.354713
$41$	−537583.	−1.21815	−0.609077	−	0.793111i	$-0.708460\pi$
$41$	−537583.	−1.21815	−0.609077	+	0.793111i	$0.708460\pi$
$42$	0	0
$43$	200498.	0.384565	0.192282	−	0.981340i	$-0.438411\pi$
$43$	200498.	0.384565	0.192282	+	0.981340i	$0.438411\pi$
$44$	130428.	0.230827
$45$	0	0
$46$	395735.	0.599449
$47$	410871.	0.577249	0.288624	−	0.957442i	$-0.406802\pi$
$47$	410871.	0.577249	0.288624	+	0.957442i	$0.406802\pi$
$48$	0	0
$49$	−796541.	−0.967212
$50$	−4106.21	−0.00464564
$51$	0	0
$52$	−107413.	−0.105937
$53$	−1.36778e6	−1.26198	−0.630988	−	0.775792i	$-0.717350\pi$
$53$	−1.36778e6	−1.26198	−0.630988	+	0.775792i	$0.717350\pi$
$54$	0	0
$55$	571489.	0.463168
$56$	84133.6	0.0640193
$57$	0	0
$58$	−60422.3	−0.0406630
$59$	−798889.	−0.506412	−0.253206	−	0.967412i	$-0.581485\pi$
$59$	−798889.	−0.506412	−0.253206	+	0.967412i	$0.581485\pi$
$60$	0	0
$61$	569439.	0.321213	0.160606	−	0.987019i	$-0.448655\pi$
$61$	569439.	0.321213	0.160606	+	0.987019i	$0.448655\pi$
$62$	−1.24828e6	−0.665184
$63$	0	0
$64$	262144.	0.125000
$65$	−470647.	−0.212568
$66$	0	0
$67$	−4.80453e6	−1.95159	−0.975795	−	0.218686i	$-0.929823\pi$
$67$	−4.80453e6	−1.95159	−0.975795	+	0.218686i	$0.929823\pi$
$68$	−2.02788e6	−0.782099
$69$	0	0
$70$	368643.	0.128459
$71$	−2.45714e6	−0.814751	−0.407376	−	0.913261i	$-0.633556\pi$
$71$	−2.45714e6	−0.814751	−0.407376	+	0.913261i	$0.633556\pi$
$72$	0	0
$73$	1.60344e6	0.482418	0.241209	−	0.970473i	$-0.422456\pi$
$73$	1.60344e6	0.482418	0.241209	+	0.970473i	$0.422456\pi$
$74$	−4.33152e6	−1.24259
$75$	0	0
$76$	−809075.	−0.211418
$77$	−334881.	−0.0835936
$78$	0	0
$79$	5.58998e6	1.27560	0.637801	−	0.770201i	$-0.279844\pi$
$79$	5.58998e6	1.27560	0.637801	+	0.770201i	$0.279844\pi$
$80$	1.14862e6	0.250820
$81$	0	0
$82$	4.30067e6	0.861365
$83$	−9.82905e6	−1.88685	−0.943427	−	0.331580i	$-0.892418\pi$
$83$	−9.82905e6	−1.88685	−0.943427	+	0.331580i	$0.892418\pi$
$84$	0	0
$85$	−8.88546e6	−1.56933
$86$	−1.60398e6	−0.271928
$87$	0	0
$88$	−1.04342e6	−0.163219
$89$	117249.	0.0176296	0.00881480	−	0.999961i	$-0.497194\pi$
$89$	117249.	0.0176296	0.00881480	+	0.999961i	$0.497194\pi$
$90$	0	0
$91$	275789.	0.0383648
$92$	−3.16588e6	−0.423874
$93$	0	0
$94$	−3.28697e6	−0.408177
$95$	−3.54508e6	−0.424222
$96$	0	0
$97$	−7.79245e6	−0.866908	−0.433454	−	0.901176i	$-0.642705\pi$
$97$	−7.79245e6	−0.866908	−0.433454	+	0.901176i	$0.642705\pi$
$98$	6.37233e6	0.683922
$99$	0	0
$100$	32849.7	0.00328497
$101$	−1.15485e6	−0.111532	−0.0557661	−	0.998444i	$-0.517760\pi$
$101$	−1.15485e6	−0.111532	−0.0557661	+	0.998444i	$0.517760\pi$
$102$	0	0
$103$	−1.69867e7	−1.53171	−0.765857	−	0.643011i	$-0.777685\pi$
$103$	−1.69867e7	−1.53171	−0.765857	+	0.643011i	$0.777685\pi$
$104$	859307.	0.0749086
$105$	0	0
$106$	1.09423e7	0.892352
$107$	−2.24189e7	−1.76918	−0.884589	−	0.466372i	$-0.845561\pi$
$107$	−2.24189e7	−1.76918	−0.884589	+	0.466372i	$0.845561\pi$
$108$	0	0
$109$	1.75728e7	1.29971	0.649857	−	0.760056i	$-0.274829\pi$
$109$	1.75728e7	1.29971	0.649857	+	0.760056i	$0.274829\pi$
$110$	−4.57191e6	−0.327509
$111$	0	0
$112$	−673069.	−0.0452685
$113$	6.95639e6	0.453533	0.226767	−	0.973949i	$-0.427184\pi$
$113$	6.95639e6	0.453533	0.226767	+	0.973949i	$0.427184\pi$
$114$	0	0
$115$	−1.38718e7	−0.850529
$116$	483379.	0.0287531
$117$	0	0
$118$	6.39111e6	0.358088
$119$	5.20670e6	0.283236
$120$	0	0
$121$	−1.53340e7	−0.786875
$122$	−4.55551e6	−0.227132
$123$	0	0
$124$	9.98626e6	0.470356
$125$	−2.17643e7	−0.996688
$126$	0	0
$127$	−3.25050e7	−1.40811	−0.704057	−	0.710144i	$-0.748630\pi$
$127$	−3.25050e7	−1.40811	−0.704057	+	0.710144i	$0.748630\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	3.76517e6	0.150308
$131$	−1.63139e7	−0.634029	−0.317015	−	0.948421i	$-0.602680\pi$
$131$	−1.63139e7	−0.634029	−0.317015	+	0.948421i	$0.602680\pi$
$132$	0	0
$133$	2.07734e6	0.0765645
$134$	3.84362e7	1.37998
$135$	0	0
$136$	1.62231e7	0.553028
$137$	3.61594e7	1.20143	0.600716	−	0.799462i	$-0.294882\pi$
$137$	3.61594e7	1.20143	0.600716	+	0.799462i	$0.294882\pi$
$138$	0	0
$139$	5.35183e7	1.69025	0.845125	−	0.534569i	$-0.179526\pi$
$139$	5.35183e7	1.69025	0.845125	+	0.534569i	$0.179526\pi$
$140$	−2.94915e6	−0.0908339
$141$	0	0
$142$	1.96571e7	0.576116
$143$	−3.42034e6	−0.0978122
$144$	0	0
$145$	2.11799e6	0.0576948
$146$	−1.28276e7	−0.341121
$147$	0	0
$148$	3.46522e7	0.878647
$149$	5.44218e6	0.134779	0.0673893	−	0.997727i	$-0.478533\pi$
$149$	5.44218e6	0.134779	0.0673893	+	0.997727i	$0.478533\pi$
$150$	0	0
$151$	2.19361e7	0.518490	0.259245	−	0.965812i	$-0.416526\pi$
$151$	2.19361e7	0.518490	0.259245	+	0.965812i	$0.416526\pi$
$152$	6.47260e6	0.149495
$153$	0	0
$154$	2.67905e6	0.0591096
$155$	4.37562e7	0.943798
$156$	0	0
$157$	4.99067e6	0.102922	0.0514612	−	0.998675i	$-0.483612\pi$
$157$	4.99067e6	0.102922	0.0514612	+	0.998675i	$0.483612\pi$
$158$	−4.47198e7	−0.901987
$159$	0	0
$160$	−9.18897e6	−0.177356
$161$	8.12857e6	0.153505
$162$	0	0
$163$	−4.47642e7	−0.809606	−0.404803	−	0.914404i	$-0.632660\pi$
$163$	−4.47642e7	−0.809606	−0.404803	+	0.914404i	$0.632660\pi$
$164$	−3.44053e7	−0.609077
$165$	0	0
$166$	7.86324e7	1.33421
$167$	6.14781e7	1.02144	0.510720	−	0.859747i	$-0.329379\pi$
$167$	6.14781e7	1.02144	0.510720	+	0.859747i	$0.329379\pi$
$168$	0	0
$169$	−5.99317e7	−0.955110
$170$	7.10837e7	1.10968
$171$	0	0
$172$	1.28318e7	0.192282
$173$	1.02491e8	1.50496	0.752479	−	0.658616i	$-0.228858\pi$
$173$	1.02491e8	1.50496	0.752479	+	0.658616i	$0.228858\pi$
$174$	0	0
$175$	−84343.2	−0.00118964
$176$	8.34740e6	0.115414
$177$	0	0
$178$	−937988.	−0.0124660
$179$	−6.08971e7	−0.793617	−0.396809	−	0.917901i	$-0.629882\pi$
$179$	−6.08971e7	−0.793617	−0.396809	+	0.917901i	$0.629882\pi$
$180$	0	0
$181$	−9.62195e7	−1.20611	−0.603057	−	0.797698i	$-0.706051\pi$
$181$	−9.62195e7	−1.20611	−0.603057	+	0.797698i	$0.706051\pi$
$182$	−2.20631e6	−0.0271280
$183$	0	0
$184$	2.53271e7	0.299725
$185$	1.51833e8	1.76306
$186$	0	0
$187$	−6.45735e7	−0.722119
$188$	2.62957e7	0.288624
$189$	0	0
$190$	2.83606e7	0.299970
$191$	7.39207e7	0.767626	0.383813	−	0.923411i	$-0.374611\pi$
$191$	7.39207e7	0.767626	0.383813	+	0.923411i	$0.374611\pi$
$192$	0	0
$193$	8.14695e7	0.815726	0.407863	−	0.913043i	$-0.366274\pi$
$193$	8.14695e7	0.815726	0.407863	+	0.913043i	$0.366274\pi$
$194$	6.23396e7	0.612997
$195$	0	0
$196$	−5.09786e7	−0.483606
$197$	2.44033e7	0.227414	0.113707	−	0.993514i	$-0.463728\pi$
$197$	2.44033e7	0.227414	0.113707	+	0.993514i	$0.463728\pi$
$198$	0	0
$199$	−1.52134e8	−1.36848	−0.684241	−	0.729256i	$-0.739867\pi$
$199$	−1.52134e8	−1.36848	−0.684241	+	0.729256i	$0.739867\pi$
$200$	−262797.	−0.00232282
$201$	0	0
$202$	9.23879e6	0.0788652
$203$	−1.24110e6	−0.0104129
$204$	0	0
$205$	−1.50752e8	−1.22215
$206$	1.35893e8	1.08309
$207$	0	0
$208$	−6.87445e6	−0.0529684
$209$	−2.57632e7	−0.195204
$210$	0	0
$211$	−5.87824e7	−0.430783	−0.215392	−	0.976528i	$-0.569103\pi$
$211$	−5.87824e7	−0.430783	−0.215392	+	0.976528i	$0.569103\pi$
$212$	−8.75380e7	−0.630988
$213$	0	0
$214$	1.79351e8	1.25100
$215$	5.62246e7	0.385826
$216$	0	0
$217$	−2.56402e7	−0.170339
$218$	−1.40582e8	−0.919037
$219$	0	0
$220$	3.65753e7	0.231584
$221$	5.31791e7	0.331412
$222$	0	0
$223$	−3.31815e7	−0.200368	−0.100184	−	0.994969i	$-0.531943\pi$
$223$	−3.31815e7	−0.200368	−0.100184	+	0.994969i	$0.531943\pi$
$224$	5.38455e6	0.0320097
$225$	0	0
$226$	−5.56511e7	−0.320697
$227$	−1.38592e8	−0.786407	−0.393203	−	0.919452i	$-0.628633\pi$
$227$	−1.38592e8	−0.786407	−0.393203	+	0.919452i	$0.628633\pi$
$228$	0	0
$229$	6.70368e7	0.368884	0.184442	−	0.982843i	$-0.440952\pi$
$229$	6.70368e7	0.368884	0.184442	+	0.982843i	$0.440952\pi$
$230$	1.10974e8	0.601415
$231$	0	0
$232$	−3.86703e6	−0.0203315
$233$	−1.30899e8	−0.677940	−0.338970	−	0.940797i	$-0.610078\pi$
$233$	−1.30899e8	−0.677940	−0.338970	+	0.940797i	$0.610078\pi$
$234$	0	0
$235$	1.15219e8	0.579142
$236$	−5.11289e7	−0.253206
$237$	0	0
$238$	−4.16536e7	−0.200278
$239$	−1.92741e7	−0.0913234	−0.0456617	−	0.998957i	$-0.514540\pi$
$239$	−1.92741e7	−0.0913234	−0.0456617	+	0.998957i	$0.514540\pi$
$240$	0	0
$241$	6.84990e7	0.315228	0.157614	−	0.987501i	$-0.449620\pi$
$241$	6.84990e7	0.315228	0.157614	+	0.987501i	$0.449620\pi$
$242$	1.22672e8	0.556405
$243$	0	0
$244$	3.64441e7	0.160606
$245$	−2.23370e8	−0.970384
$246$	0	0
$247$	2.12171e7	0.0895875
$248$	−7.98901e7	−0.332592
$249$	0	0
$250$	1.74114e8	0.704765
$251$	−2.45365e7	−0.0979387	−0.0489693	−	0.998800i	$-0.515594\pi$
$251$	−2.45365e7	−0.0979387	−0.0489693	+	0.998800i	$0.515594\pi$
$252$	0	0
$253$	−1.00811e8	−0.391367
$254$	2.60040e8	0.995687
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−3.32725e8	−1.22270	−0.611349	−	0.791361i	$-0.709373\pi$
$257$	−3.32725e8	−1.22270	−0.611349	+	0.791361i	$0.709373\pi$
$258$	0	0
$259$	−8.89713e7	−0.318200
$260$	−3.01214e7	−0.106284
$261$	0	0
$262$	1.30512e8	0.448326
$263$	3.70551e8	1.25604	0.628019	−	0.778198i	$-0.283866\pi$
$263$	3.70551e8	1.25604	0.628019	+	0.778198i	$0.283866\pi$
$264$	0	0
$265$	−3.83560e8	−1.26612
$266$	−1.66187e7	−0.0541393
$267$	0	0
$268$	−3.07490e8	−0.975795
$269$	−4.30980e8	−1.34997	−0.674985	−	0.737831i	$-0.735850\pi$
$269$	−4.30980e8	−1.34997	−0.674985	+	0.737831i	$0.735850\pi$
$270$	0	0
$271$	3.41000e8	1.04079	0.520394	−	0.853927i	$-0.325785\pi$
$271$	3.41000e8	1.04079	0.520394	+	0.853927i	$0.325785\pi$
$272$	−1.29785e8	−0.391050
$273$	0	0
$274$	−2.89275e8	−0.849541
$275$	1.04602e6	0.00303304
$276$	0	0
$277$	6.82082e8	1.92822	0.964112	−	0.265496i	$-0.0855356\pi$
$277$	6.82082e8	1.92822	0.964112	+	0.265496i	$0.0855356\pi$
$278$	−4.28147e8	−1.19519
$279$	0	0
$280$	2.35932e7	0.0642293
$281$	−6.74238e7	−0.181276	−0.0906382	−	0.995884i	$-0.528891\pi$
$281$	−6.74238e7	−0.181276	−0.0906382	+	0.995884i	$0.528891\pi$
$282$	0	0
$283$	−5.15722e8	−1.35258	−0.676290	−	0.736636i	$-0.736413\pi$
$283$	−5.15722e8	−1.35258	−0.676290	+	0.736636i	$0.736413\pi$
$284$	−1.57257e8	−0.407376
$285$	0	0
$286$	2.73627e7	0.0691637
$287$	8.83375e7	0.220576
$288$	0	0
$289$	5.93644e8	1.44672
$290$	−1.69439e7	−0.0407964
$291$	0	0
$292$	1.02620e8	0.241209
$293$	−8.06018e7	−0.187201	−0.0936005	−	0.995610i	$-0.529838\pi$
$293$	−8.06018e7	−0.187201	−0.0936005	+	0.995610i	$0.529838\pi$
$294$	0	0
$295$	−2.24029e8	−0.508073
$296$	−2.77217e8	−0.621297
$297$	0	0
$298$	−4.35374e7	−0.0953029
$299$	8.30219e7	0.179615
$300$	0	0
$301$	−3.29464e7	−0.0696347
$302$	−1.75489e8	−0.366628
$303$	0	0
$304$	−5.17808e7	−0.105709
$305$	1.59685e8	0.322266
$306$	0	0
$307$	3.87201e8	0.763752	0.381876	−	0.924214i	$-0.375278\pi$
$307$	3.87201e8	0.763752	0.381876	+	0.924214i	$0.375278\pi$
$308$	−2.14324e7	−0.0417968
$309$	0	0
$310$	−3.50050e8	−0.667366
$311$	−6.30776e8	−1.18909	−0.594543	−	0.804064i	$-0.702667\pi$
$311$	−6.30776e8	−1.18909	−0.594543	+	0.804064i	$0.702667\pi$
$312$	0	0
$313$	4.74951e8	0.875475	0.437737	−	0.899103i	$-0.355780\pi$
$313$	4.74951e8	0.875475	0.437737	+	0.899103i	$0.355780\pi$
$314$	−3.99254e7	−0.0727772
$315$	0	0
$316$	3.57759e8	0.637801
$317$	−2.90309e8	−0.511863	−0.255931	−	0.966695i	$-0.582382\pi$
$317$	−2.90309e8	−0.511863	−0.255931	+	0.966695i	$0.582382\pi$
$318$	0	0
$319$	1.53921e7	0.0265480
$320$	7.35118e7	0.125410
$321$	0	0
$322$	−6.50285e7	−0.108545
$323$	4.00564e8	0.661398
$324$	0	0
$325$	−861448.	−0.00139199
$326$	3.58113e8	0.572478
$327$	0	0
$328$	2.75243e8	0.430683
$329$	−6.75157e7	−0.104525
$330$	0	0
$331$	1.04983e9	1.59118	0.795589	−	0.605836i	$-0.207161\pi$
$331$	1.04983e9	1.59118	0.795589	+	0.605836i	$0.207161\pi$
$332$	−6.29059e8	−0.943427
$333$	0	0
$334$	−4.91825e8	−0.722267
$335$	−1.34731e9	−1.95799
$336$	0	0
$337$	5.09260e8	0.724827	0.362414	−	0.932017i	$-0.381953\pi$
$337$	5.09260e8	0.724827	0.362414	+	0.932017i	$0.381953\pi$
$338$	4.79454e8	0.675365
$339$	0	0
$340$	−5.68670e8	−0.784664
$341$	3.17990e8	0.434284
$342$	0	0
$343$	2.66218e8	0.356211
$344$	−1.02655e8	−0.135964
$345$	0	0
$346$	−8.19928e8	−1.06417
$347$	−4.69074e8	−0.602682	−0.301341	−	0.953516i	$-0.597434\pi$
$347$	−4.69074e8	−0.602682	−0.301341	+	0.953516i	$0.597434\pi$
$348$	0	0
$349$	−2.18782e8	−0.275500	−0.137750	−	0.990467i	$-0.543987\pi$
$349$	−2.18782e8	−0.275500	−0.137750	+	0.990467i	$0.543987\pi$
$350$	674746.	0.000841205 0
$351$	0	0
$352$	−6.67792e7	−0.0816097
$353$	−4.43032e8	−0.536072	−0.268036	−	0.963409i	$-0.586375\pi$
$353$	−4.43032e8	−0.536072	−0.268036	+	0.963409i	$0.586375\pi$
$354$	0	0
$355$	−6.89043e8	−0.817423
$356$	7.50391e6	0.00881480
$357$	0	0
$358$	4.87177e8	0.561172
$359$	4.34672e8	0.495828	0.247914	−	0.968782i	$-0.420255\pi$
$359$	4.34672e8	0.495828	0.247914	+	0.968782i	$0.420255\pi$
$360$	0	0
$361$	−7.34057e8	−0.821210
$362$	7.69756e8	0.852851
$363$	0	0
$364$	1.76505e7	0.0191824
$365$	4.49646e8	0.484001
$366$	0	0
$367$	−1.64735e9	−1.73962	−0.869809	−	0.493389i	$-0.835758\pi$
$367$	−1.64735e9	−1.73962	−0.869809	+	0.493389i	$0.835758\pi$
$368$	−2.02616e8	−0.211937
$369$	0	0
$370$	−1.21467e9	−1.24667
$371$	2.24758e8	0.228511
$372$	0	0
$373$	5.70827e8	0.569539	0.284770	−	0.958596i	$-0.408083\pi$
$373$	5.70827e8	0.569539	0.284770	+	0.958596i	$0.408083\pi$
$374$	5.16588e8	0.510615
$375$	0	0
$376$	−2.10366e8	−0.204088
$377$	−1.26761e7	−0.0121840
$378$	0	0
$379$	−1.71431e8	−0.161753	−0.0808763	−	0.996724i	$-0.525772\pi$
$379$	−1.71431e8	−0.161753	−0.0808763	+	0.996724i	$0.525772\pi$
$380$	−2.26885e8	−0.212111
$381$	0	0
$382$	−5.91366e8	−0.542793
$383$	6.66865e8	0.606515	0.303258	−	0.952909i	$-0.401926\pi$
$383$	6.66865e8	0.606515	0.303258	+	0.952909i	$0.401926\pi$
$384$	0	0
$385$	−9.39090e7	−0.0838677
$386$	−6.51756e8	−0.576806
$387$	0	0
$388$	−4.98717e8	−0.433454
$389$	−5.08814e8	−0.438263	−0.219132	−	0.975695i	$-0.570322\pi$
$389$	−5.08814e8	−0.438263	−0.219132	+	0.975695i	$0.570322\pi$
$390$	0	0
$391$	1.56739e9	1.32605
$392$	4.07829e8	0.341961
$393$	0	0
$394$	−1.95227e8	−0.160806
$395$	1.56757e9	1.27979
$396$	0	0
$397$	9.18742e8	0.736931	0.368466	−	0.929641i	$-0.379883\pi$
$397$	9.18742e8	0.736931	0.368466	+	0.929641i	$0.379883\pi$
$398$	1.21707e9	0.967663
$399$	0	0
$400$	2.10238e6	0.00164248
$401$	1.97725e9	1.53129	0.765643	−	0.643266i	$-0.222421\pi$
$401$	1.97725e9	1.53129	0.765643	+	0.643266i	$0.222421\pi$
$402$	0	0
$403$	−2.61879e8	−0.199312
$404$	−7.39103e7	−0.0557661
$405$	0	0
$406$	9.92880e6	0.00736301
$407$	1.10342e9	0.811262
$408$	0	0
$409$	2.48921e9	1.79899	0.899497	−	0.436927i	$-0.143933\pi$
$409$	2.48921e9	1.79899	0.899497	+	0.436927i	$0.143933\pi$
$410$	1.20602e9	0.864190
$411$	0	0
$412$	−1.08715e9	−0.765857
$413$	1.31276e8	0.0916981
$414$	0	0
$415$	−2.75631e9	−1.89304
$416$	5.49956e7	0.0374543
$417$	0	0
$418$	2.06106e8	0.138030
$419$	−1.79159e9	−1.18984	−0.594921	−	0.803784i	$-0.702817\pi$
$419$	−1.79159e9	−1.18984	−0.594921	+	0.803784i	$0.702817\pi$
$420$	0	0
$421$	−1.39662e7	−0.00912199	−0.00456099	−	0.999990i	$-0.501452\pi$
$421$	−1.39662e7	−0.00912199	−0.00456099	+	0.999990i	$0.501452\pi$
$422$	4.70259e8	0.304610
$423$	0	0
$424$	7.00304e8	0.446176
$425$	−1.62635e7	−0.0102767
$426$	0	0
$427$	−9.35722e7	−0.0581633
$428$	−1.43481e9	−0.884589
$429$	0	0
$430$	−4.49796e8	−0.272820
$431$	2.57849e9	1.55129	0.775647	−	0.631166i	$-0.217423\pi$
$431$	2.57849e9	1.55129	0.775647	+	0.631166i	$0.217423\pi$
$432$	0	0
$433$	1.82232e9	1.07874	0.539372	−	0.842068i	$-0.318662\pi$
$433$	1.82232e9	1.07874	0.539372	+	0.842068i	$0.318662\pi$
$434$	2.05122e8	0.120448
$435$	0	0
$436$	1.12466e9	0.649857
$437$	6.25350e8	0.358458
$438$	0	0
$439$	1.33131e9	0.751023	0.375512	−	0.926818i	$-0.377467\pi$
$439$	1.33131e9	0.751023	0.375512	+	0.926818i	$0.377467\pi$
$440$	−2.92603e8	−0.163755
$441$	0	0
$442$	−4.25433e8	−0.234344
$443$	3.71360e7	0.0202947	0.0101473	−	0.999949i	$-0.496770\pi$
$443$	3.71360e7	0.0202947	0.0101473	+	0.999949i	$0.496770\pi$
$444$	0	0
$445$	3.28794e7	0.0176874
$446$	2.65452e8	0.141682
$447$	0	0
$448$	−4.30764e7	−0.0226343
$449$	1.95772e9	1.02068	0.510338	−	0.859974i	$-0.329520\pi$
$449$	1.95772e9	1.02068	0.510338	+	0.859974i	$0.329520\pi$
$450$	0	0
$451$	−1.09556e9	−0.562366
$452$	4.45209e8	0.226767
$453$	0	0
$454$	1.10873e9	0.556073
$455$	7.73383e7	0.0384906
$456$	0	0
$457$	9.99179e8	0.489707	0.244854	−	0.969560i	$-0.421260\pi$
$457$	9.99179e8	0.489707	0.244854	+	0.969560i	$0.421260\pi$
$458$	−5.36295e8	−0.260840
$459$	0	0
$460$	−8.87793e8	−0.425265
$461$	1.64807e9	0.783468	0.391734	−	0.920079i	$-0.371875\pi$
$461$	1.64807e9	0.783468	0.391734	+	0.920079i	$0.371875\pi$
$462$	0	0
$463$	3.47062e9	1.62508	0.812538	−	0.582909i	$-0.198086\pi$
$463$	3.47062e9	1.62508	0.812538	+	0.582909i	$0.198086\pi$
$464$	3.09362e7	0.0143765
$465$	0	0
$466$	1.04719e9	0.479376
$467$	−2.49926e9	−1.13554	−0.567769	−	0.823188i	$-0.692193\pi$
$467$	−2.49926e9	−1.13554	−0.567769	+	0.823188i	$0.692193\pi$
$468$	0	0
$469$	7.89496e8	0.353382
$470$	−9.21749e8	−0.409515
$471$	0	0
$472$	4.09031e8	0.179044
$473$	4.08602e8	0.177536
$474$	0	0
$475$	−6.48873e6	−0.00277800
$476$	3.33229e8	0.141618
$477$	0	0
$478$	1.54193e8	0.0645754
$479$	4.39611e9	1.82765	0.913827	−	0.406103i	$-0.133112\pi$
$479$	4.39611e9	1.82765	0.913827	+	0.406103i	$0.133112\pi$
$480$	0	0
$481$	−9.08717e8	−0.372324
$482$	−5.47992e8	−0.222900
$483$	0	0
$484$	−9.81375e8	−0.393438
$485$	−2.18520e9	−0.869752
$486$	0	0
$487$	−2.82989e9	−1.11025	−0.555123	−	0.831769i	$-0.687329\pi$
$487$	−2.82989e9	−1.11025	−0.555123	+	0.831769i	$0.687329\pi$
$488$	−2.91553e8	−0.113566
$489$	0	0
$490$	1.78696e9	0.686165
$491$	−2.45842e9	−0.937283	−0.468641	−	0.883389i	$-0.655256\pi$
$491$	−2.45842e9	−0.937283	−0.468641	+	0.883389i	$0.655256\pi$
$492$	0	0
$493$	−2.39315e8	−0.0899511
$494$	−1.69737e8	−0.0633480
$495$	0	0
$496$	6.39121e8	0.235178
$497$	4.03765e8	0.147530
$498$	0	0
$499$	−3.00604e9	−1.08304	−0.541518	−	0.840689i	$-0.682150\pi$
$499$	−3.00604e9	−1.08304	−0.541518	+	0.840689i	$0.682150\pi$
$500$	−1.39291e9	−0.498344
$501$	0	0
$502$	1.96292e8	0.0692531
$503$	2.43279e8	0.0852348	0.0426174	−	0.999091i	$-0.486430\pi$
$503$	2.43279e8	0.0852348	0.0426174	+	0.999091i	$0.486430\pi$
$504$	0	0
$505$	−3.23849e8	−0.111898
$506$	8.06484e8	0.276738
$507$	0	0
$508$	−2.08032e9	−0.704057
$509$	4.00432e9	1.34591	0.672956	−	0.739682i	$-0.265024\pi$
$509$	4.00432e9	1.34591	0.672956	+	0.739682i	$0.265024\pi$
$510$	0	0
$511$	−2.63483e8	−0.0873535
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	2.66180e9	0.864578
$515$	−4.76349e9	−1.53674
$516$	0	0
$517$	8.37330e8	0.266489
$518$	7.11770e8	0.225002
$519$	0	0
$520$	2.40971e8	0.0751542
$521$	−5.72614e9	−1.77390	−0.886951	−	0.461864i	$-0.847181\pi$
$521$	−5.72614e9	−1.77390	−0.886951	+	0.461864i	$0.847181\pi$
$522$	0	0
$523$	1.53753e9	0.469966	0.234983	−	0.971999i	$-0.424497\pi$
$523$	1.53753e9	0.469966	0.234983	+	0.971999i	$0.424497\pi$
$524$	−1.04409e9	−0.317015
$525$	0	0
$526$	−2.96441e9	−0.888153
$527$	−4.94409e9	−1.47146
$528$	0	0
$529$	−9.57851e8	−0.281322
$530$	3.06848e9	0.895279
$531$	0	0
$532$	1.32950e8	0.0382822
$533$	9.02244e8	0.258095
$534$	0	0
$535$	−6.28683e9	−1.77498
$536$	2.45992e9	0.689991
$537$	0	0
$538$	3.44784e9	0.954573
$539$	−1.62330e9	−0.446518
$540$	0	0
$541$	3.90611e9	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$541$	3.90611e9	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$542$	−2.72800e9	−0.735948
$543$	0	0
$544$	1.03828e9	0.276514
$545$	4.92785e9	1.30398
$546$	0	0
$547$	−5.00854e9	−1.30844	−0.654222	−	0.756303i	$-0.727004\pi$
$547$	−5.00854e9	−1.30844	−0.654222	+	0.756303i	$0.727004\pi$
$548$	2.31420e9	0.600716
$549$	0	0
$550$	−8.36820e6	−0.00214468
$551$	−9.54808e7	−0.0243156
$552$	0	0
$553$	−9.18564e8	−0.230979
$554$	−5.45666e9	−1.36346
$555$	0	0
$556$	3.42517e9	0.845125
$557$	1.45387e9	0.356479	0.178239	−	0.983987i	$-0.442960\pi$
$557$	1.45387e9	0.356479	0.178239	+	0.983987i	$0.442960\pi$
$558$	0	0
$559$	−3.36502e8	−0.0814790
$560$	−1.88745e8	−0.0454170
$561$	0	0
$562$	5.39390e8	0.128182
$563$	−5.91199e9	−1.39622	−0.698111	−	0.715989i	$-0.745976\pi$
$563$	−5.91199e9	−1.39622	−0.698111	+	0.715989i	$0.745976\pi$
$564$	0	0
$565$	1.95075e9	0.455021
$566$	4.12577e9	0.956418
$567$	0	0
$568$	1.25805e9	0.288058
$569$	−7.62555e9	−1.73531	−0.867657	−	0.497163i	$-0.834375\pi$
$569$	−7.62555e9	−1.73531	−0.867657	+	0.497163i	$0.834375\pi$
$570$	0	0
$571$	−7.37293e9	−1.65735	−0.828675	−	0.559731i	$-0.810905\pi$
$571$	−7.37293e9	−1.65735	−0.828675	+	0.559731i	$0.810905\pi$
$572$	−2.18902e8	−0.0489061
$573$	0	0
$574$	−7.06700e8	−0.155971
$575$	−2.53902e7	−0.00556965
$576$	0	0
$577$	1.95790e9	0.424302	0.212151	−	0.977237i	$-0.431953\pi$
$577$	1.95790e9	0.424302	0.212151	+	0.977237i	$0.431953\pi$
$578$	−4.74915e9	−1.02298
$579$	0	0
$580$	1.35552e8	0.0288474
$581$	1.61514e9	0.341660
$582$	0	0
$583$	−2.78746e9	−0.582597
$584$	−8.20964e8	−0.170561
$585$	0	0
$586$	6.44815e8	0.132371
$587$	2.19857e9	0.448649	0.224324	−	0.974515i	$-0.427983\pi$
$587$	2.19857e9	0.448649	0.224324	+	0.974515i	$0.427983\pi$
$588$	0	0
$589$	−1.97257e9	−0.397766
$590$	1.79223e9	0.359262
$591$	0	0
$592$	2.21774e9	0.439323
$593$	−6.54769e9	−1.28943	−0.644713	−	0.764425i	$-0.723023\pi$
$593$	−6.54769e9	−1.28943	−0.644713	+	0.764425i	$0.723023\pi$
$594$	0	0
$595$	1.46009e9	0.284165
$596$	3.48299e8	0.0673893
$597$	0	0
$598$	−6.64176e8	−0.127007
$599$	5.27283e9	1.00242	0.501210	−	0.865326i	$-0.332888\pi$
$599$	5.27283e9	1.00242	0.501210	+	0.865326i	$0.332888\pi$
$600$	0	0
$601$	5.19270e9	0.975737	0.487869	−	0.872917i	$-0.337774\pi$
$601$	5.19270e9	0.975737	0.487869	+	0.872917i	$0.337774\pi$
$602$	2.63571e8	0.0492392
$603$	0	0
$604$	1.40391e9	0.259245
$605$	−4.30003e9	−0.789456
$606$	0	0
$607$	−2.43708e9	−0.442293	−0.221147	−	0.975241i	$-0.570980\pi$
$607$	−2.43708e9	−0.442293	−0.221147	+	0.975241i	$0.570980\pi$
$608$	4.14246e8	0.0747474
$609$	0	0
$610$	−1.27748e9	−0.227877
$611$	−6.89579e8	−0.122304
$612$	0	0
$613$	−3.31215e9	−0.580763	−0.290382	−	0.956911i	$-0.593782\pi$
$613$	−3.31215e9	−0.580763	−0.290382	+	0.956911i	$0.593782\pi$
$614$	−3.09761e9	−0.540054
$615$	0	0
$616$	1.71459e8	0.0295548
$617$	−3.27130e9	−0.560690	−0.280345	−	0.959899i	$-0.590449\pi$
$617$	−3.27130e9	−0.560690	−0.280345	+	0.959899i	$0.590449\pi$
$618$	0	0
$619$	−1.43605e9	−0.243361	−0.121681	−	0.992569i	$-0.538828\pi$
$619$	−1.43605e9	−0.243361	−0.121681	+	0.992569i	$0.538828\pi$
$620$	2.80040e9	0.471899
$621$	0	0
$622$	5.04620e9	0.840811
$623$	−1.92667e7	−0.00319226
$624$	0	0
$625$	−6.14335e9	−1.00653
$626$	−3.79961e9	−0.619054
$627$	0	0
$628$	3.19403e8	0.0514612
$629$	−1.71559e10	−2.74876
$630$	0	0
$631$	−7.26152e9	−1.15060	−0.575301	−	0.817942i	$-0.695115\pi$
$631$	−7.26152e9	−1.15060	−0.575301	+	0.817942i	$0.695115\pi$
$632$	−2.86207e9	−0.450994
$633$	0	0
$634$	2.32247e9	0.361942
$635$	−9.11523e9	−1.41273
$636$	0	0
$637$	1.33686e9	0.204927
$638$	−1.23137e8	−0.0187722
$639$	0	0
$640$	−5.88094e8	−0.0886782
$641$	8.95695e9	1.34325	0.671625	−	0.740891i	$-0.265596\pi$
$641$	8.95695e9	1.34325	0.671625	+	0.740891i	$0.265596\pi$
$642$	0	0
$643$	−2.59547e9	−0.385015	−0.192508	−	0.981295i	$-0.561662\pi$
$643$	−2.59547e9	−0.385015	−0.192508	+	0.981295i	$0.561662\pi$
$644$	5.20228e8	0.0767527
$645$	0	0
$646$	−3.20451e9	−0.467679
$647$	9.11067e9	1.32247	0.661234	−	0.750180i	$-0.270033\pi$
$647$	9.11067e9	1.32247	0.661234	+	0.750180i	$0.270033\pi$
$648$	0	0
$649$	−1.62809e9	−0.233787
$650$	6.89158e6	0.000984288 0
$651$	0	0
$652$	−2.86491e9	−0.404803
$653$	1.02334e10	1.43821	0.719107	−	0.694899i	$-0.244551\pi$
$653$	1.02334e10	1.43821	0.719107	+	0.694899i	$0.244551\pi$
$654$	0	0
$655$	−4.57484e9	−0.636109
$656$	−2.20194e9	−0.304539
$657$	0	0
$658$	5.40126e8	0.0739102
$659$	1.07582e10	1.46434	0.732168	−	0.681124i	$-0.238508\pi$
$659$	1.07582e10	1.46434	0.732168	+	0.681124i	$0.238508\pi$
$660$	0	0
$661$	6.05822e9	0.815906	0.407953	−	0.913003i	$-0.366243\pi$
$661$	6.05822e9	0.815906	0.407953	+	0.913003i	$0.366243\pi$
$662$	−8.39860e9	−1.12513
$663$	0	0
$664$	5.03247e9	0.667104
$665$	5.82539e8	0.0768156
$666$	0	0
$667$	−3.73613e8	−0.0487508
$668$	3.93460e9	0.510720
$669$	0	0
$670$	1.07785e10	1.38451
$671$	1.16048e9	0.148289
$672$	0	0
$673$	4.23487e9	0.535534	0.267767	−	0.963484i	$-0.413714\pi$
$673$	4.23487e9	0.535534	0.267767	+	0.963484i	$0.413714\pi$
$674$	−4.07408e9	−0.512530
$675$	0	0
$676$	−3.83563e9	−0.477555
$677$	−1.98516e9	−0.245887	−0.122944	−	0.992414i	$-0.539233\pi$
$677$	−1.98516e9	−0.245887	−0.122944	+	0.992414i	$0.539233\pi$
$678$	0	0
$679$	1.28048e9	0.156975
$680$	4.54936e9	0.554841
$681$	0	0
$682$	−2.54392e9	−0.307085
$683$	2.07576e9	0.249290	0.124645	−	0.992201i	$-0.460221\pi$
$683$	2.07576e9	0.249290	0.124645	+	0.992201i	$0.460221\pi$
$684$	0	0
$685$	1.01400e10	1.20537
$686$	−2.12974e9	−0.251879
$687$	0	0
$688$	8.21238e8	0.0961412
$689$	2.29559e9	0.267379
$690$	0	0
$691$	1.45644e9	0.167927	0.0839634	−	0.996469i	$-0.473242\pi$
$691$	1.45644e9	0.167927	0.0839634	+	0.996469i	$0.473242\pi$
$692$	6.55943e9	0.752479
$693$	0	0
$694$	3.75259e9	0.426160
$695$	1.50079e10	1.69579
$696$	0	0
$697$	1.70337e10	1.90544
$698$	1.75025e9	0.194808
$699$	0	0
$700$	−5.39797e6	−0.000594822 0
$701$	6.55669e9	0.718905	0.359452	−	0.933163i	$-0.382964\pi$
$701$	6.55669e9	0.718905	0.359452	+	0.933163i	$0.382964\pi$
$702$	0	0
$703$	−6.84477e9	−0.743046
$704$	5.34233e8	0.0577068
$705$	0	0
$706$	3.54425e9	0.379060
$707$	1.89769e8	0.0201956
$708$	0	0
$709$	6.40302e9	0.674719	0.337360	−	0.941376i	$-0.390466\pi$
$709$	6.40302e9	0.674719	0.337360	+	0.941376i	$0.390466\pi$
$710$	5.51234e9	0.578005
$711$	0	0
$712$	−6.00313e7	−0.00623301
$713$	−7.71858e9	−0.797488
$714$	0	0
$715$	−9.59149e8	−0.0981330
$716$	−3.89742e9	−0.396809
$717$	0	0
$718$	−3.47738e9	−0.350603
$719$	1.33984e10	1.34432	0.672158	−	0.740408i	$-0.265368\pi$
$719$	1.33984e10	1.34432	0.672158	+	0.740408i	$0.265368\pi$
$720$	0	0
$721$	2.79130e9	0.277354
$722$	5.87245e9	0.580683
$723$	0	0
$724$	−6.15805e9	−0.603057
$725$	3.87666e6	0.000377812 0
$726$	0	0
$727$	−1.44080e10	−1.39070	−0.695351	−	0.718670i	$-0.744751\pi$
$727$	−1.44080e10	−1.39070	−0.695351	+	0.718670i	$0.744751\pi$
$728$	−1.41204e8	−0.0135640
$729$	0	0
$730$	−3.59717e9	−0.342240
$731$	−6.35290e9	−0.601536
$732$	0	0
$733$	1.12128e10	1.05159	0.525797	−	0.850610i	$-0.323767\pi$
$733$	1.12128e10	1.05159	0.525797	+	0.850610i	$0.323767\pi$
$734$	1.31788e10	1.23010
$735$	0	0
$736$	1.62093e9	0.149862
$737$	−9.79133e9	−0.900960
$738$	0	0
$739$	7.79784e9	0.710753	0.355377	−	0.934723i	$-0.384353\pi$
$739$	7.79784e9	0.710753	0.355377	+	0.934723i	$0.384353\pi$
$740$	9.71734e9	0.881528
$741$	0	0
$742$	−1.79807e9	−0.161582
$743$	−1.56159e10	−1.39671	−0.698353	−	0.715753i	$-0.746084\pi$
$743$	−1.56159e10	−1.39671	−0.698353	+	0.715753i	$0.746084\pi$
$744$	0	0
$745$	1.52612e9	0.135221
$746$	−4.56662e9	−0.402725
$747$	0	0
$748$	−4.13270e9	−0.361059
$749$	3.68395e9	0.320352
$750$	0	0
$751$	1.27457e10	1.09805	0.549025	−	0.835806i	$-0.314999\pi$
$751$	1.27457e10	1.09805	0.549025	+	0.835806i	$0.314999\pi$
$752$	1.68293e9	0.144312
$753$	0	0
$754$	1.01409e8	0.00861541
$755$	6.15144e9	0.520190
$756$	0	0
$757$	−2.31819e10	−1.94229	−0.971144	−	0.238492i	$-0.923347\pi$
$757$	−2.31819e10	−1.94229	−0.971144	+	0.238492i	$0.923347\pi$
$758$	1.37144e9	0.114376
$759$	0	0
$760$	1.81508e9	0.149985
$761$	2.22033e10	1.82630	0.913148	−	0.407629i	$-0.133644\pi$
$761$	2.22033e10	1.82630	0.913148	+	0.407629i	$0.133644\pi$
$762$	0	0
$763$	−2.88762e9	−0.235345
$764$	4.73093e9	0.383813
$765$	0	0
$766$	−5.33492e9	−0.428871
$767$	1.34080e9	0.107295
$768$	0	0
$769$	2.19911e10	1.74383	0.871916	−	0.489655i	$-0.162877\pi$
$769$	2.19911e10	1.74383	0.871916	+	0.489655i	$0.162877\pi$
$770$	7.51272e8	0.0593034
$771$	0	0
$772$	5.21405e9	0.407863
$773$	1.12771e10	0.878151	0.439075	−	0.898450i	$-0.355306\pi$
$773$	1.12771e10	0.878151	0.439075	+	0.898450i	$0.355306\pi$
$774$	0	0
$775$	8.00891e7	0.00618042
$776$	3.98974e9	0.306498
$777$	0	0
$778$	4.07051e9	0.309899
$779$	6.79602e9	0.515079
$780$	0	0
$781$	−5.00749e9	−0.376133
$782$	−1.25391e10	−0.937657
$783$	0	0
$784$	−3.26263e9	−0.241803
$785$	1.39951e9	0.103260
$786$	0	0
$787$	−1.83418e10	−1.34131	−0.670657	−	0.741768i	$-0.733988\pi$
$787$	−1.83418e10	−1.34131	−0.670657	+	0.741768i	$0.733988\pi$
$788$	1.56181e9	0.113707
$789$	0	0
$790$	−1.25406e10	−0.904946
$791$	−1.14310e9	−0.0821231
$792$	0	0
$793$	−9.55709e8	−0.0680565
$794$	−7.34994e9	−0.521089
$795$	0	0
$796$	−9.73655e9	−0.684241
$797$	−1.57035e9	−0.109874	−0.0549368	−	0.998490i	$-0.517496\pi$
$797$	−1.57035e9	−0.109874	−0.0549368	+	0.998490i	$0.517496\pi$
$798$	0	0
$799$	−1.30187e10	−0.902932
$800$	−1.68190e7	−0.00116141
$801$	0	0
$802$	−1.58180e10	−1.08278
$803$	3.26772e9	0.222710
$804$	0	0
$805$	2.27945e9	0.154009
$806$	2.09503e9	0.140935
$807$	0	0
$808$	5.91283e8	0.0394326
$809$	3.66110e9	0.243104	0.121552	−	0.992585i	$-0.461213\pi$
$809$	3.66110e9	0.243104	0.121552	+	0.992585i	$0.461213\pi$
$810$	0	0
$811$	−2.32307e10	−1.52929	−0.764645	−	0.644451i	$-0.777086\pi$
$811$	−2.32307e10	−1.52929	−0.764645	+	0.644451i	$0.777086\pi$
$812$	−7.94304e7	−0.00520644
$813$	0	0
$814$	−8.82737e9	−0.573649
$815$	−1.25530e10	−0.812261
$816$	0	0
$817$	−2.53465e9	−0.162608
$818$	−1.99137e10	−1.27208
$819$	0	0
$820$	−9.64812e9	−0.611075
$821$	−2.53061e10	−1.59597	−0.797984	−	0.602678i	$-0.794100\pi$
$821$	−2.53061e10	−1.59597	−0.797984	+	0.602678i	$0.794100\pi$
$822$	0	0
$823$	8.28677e9	0.518186	0.259093	−	0.965852i	$-0.416576\pi$
$823$	8.28677e9	0.518186	0.259093	+	0.965852i	$0.416576\pi$
$824$	8.69717e9	0.541543
$825$	0	0
$826$	−1.05021e9	−0.0648404
$827$	3.06921e10	1.88694	0.943470	−	0.331459i	$-0.107541\pi$
$827$	3.06921e10	1.88694	0.943470	+	0.331459i	$0.107541\pi$
$828$	0	0
$829$	1.83150e10	1.11652	0.558260	−	0.829666i	$-0.311469\pi$
$829$	1.83150e10	1.11652	0.558260	+	0.829666i	$0.311469\pi$
$830$	2.20505e10	1.33858
$831$	0	0
$832$	−4.39965e8	−0.0264842
$833$	2.52389e10	1.51291
$834$	0	0
$835$	1.72400e10	1.02479
$836$	−1.64885e9	−0.0976018
$837$	0	0
$838$	1.43327e10	0.841346
$839$	−3.00211e10	−1.75493	−0.877466	−	0.479639i	$-0.840767\pi$
$839$	−3.00211e10	−1.75493	−0.877466	+	0.479639i	$0.840767\pi$
$840$	0	0
$841$	−1.71928e10	−0.996693
$842$	1.11729e8	0.00645022
$843$	0	0
$844$	−3.76207e9	−0.215392
$845$	−1.68064e10	−0.958242
$846$	0	0
$847$	2.51973e9	0.142483
$848$	−5.60243e9	−0.315494
$849$	0	0
$850$	1.30108e8	0.00726671
$851$	−2.67834e10	−1.48974
$852$	0	0
$853$	2.78651e10	1.53723	0.768615	−	0.639712i	$-0.220946\pi$
$853$	2.78651e10	1.53723	0.768615	+	0.639712i	$0.220946\pi$
$854$	7.48578e8	0.0411277
$855$	0	0
$856$	1.14785e10	0.625499
$857$	3.41062e9	0.185098	0.0925488	−	0.995708i	$-0.470499\pi$
$857$	3.41062e9	0.185098	0.0925488	+	0.995708i	$0.470499\pi$
$858$	0	0
$859$	9.01531e9	0.485294	0.242647	−	0.970115i	$-0.421984\pi$
$859$	9.01531e9	0.485294	0.242647	+	0.970115i	$0.421984\pi$
$860$	3.59837e9	0.192913
$861$	0	0
$862$	−2.06279e10	−1.09693
$863$	2.43096e9	0.128748	0.0643739	−	0.997926i	$-0.479495\pi$
$863$	2.43096e9	0.128748	0.0643739	+	0.997926i	$0.479495\pi$
$864$	0	0
$865$	2.87411e10	1.50989
$866$	−1.45786e10	−0.762787
$867$	0	0
$868$	−1.64098e9	−0.0851693
$869$	1.13920e10	0.588887
$870$	0	0
$871$	8.06359e9	0.413490
$872$	−8.99727e9	−0.459518
$873$	0	0
$874$	−5.00280e9	−0.253468
$875$	3.57638e9	0.180474
$876$	0	0
$877$	−9.98233e9	−0.499728	−0.249864	−	0.968281i	$-0.580386\pi$
$877$	−9.98233e9	−0.499728	−0.249864	+	0.968281i	$0.580386\pi$
$878$	−1.06505e10	−0.531054
$879$	0	0
$880$	2.34082e9	0.115792
$881$	2.89122e10	1.42451	0.712256	−	0.701920i	$-0.247673\pi$
$881$	2.89122e10	1.42451	0.712256	+	0.701920i	$0.247673\pi$
$882$	0	0
$883$	−8.55345e9	−0.418099	−0.209049	−	0.977905i	$-0.567037\pi$
$883$	−8.55345e9	−0.418099	−0.209049	+	0.977905i	$0.567037\pi$
$884$	3.40346e9	0.165706
$885$	0	0
$886$	−2.97088e8	−0.0143505
$887$	−1.78318e10	−0.857953	−0.428977	−	0.903316i	$-0.641126\pi$
$887$	−1.78318e10	−0.857953	−0.428977	+	0.903316i	$0.641126\pi$
$888$	0	0
$889$	5.34134e9	0.254973
$890$	−2.63036e8	−0.0125069
$891$	0	0
$892$	−2.12362e9	−0.100184
$893$	−5.19415e9	−0.244081
$894$	0	0
$895$	−1.70771e10	−0.796220
$896$	3.44611e8	0.0160048
$897$	0	0
$898$	−1.56618e10	−0.721727
$899$	1.17850e9	0.0540968
$900$	0	0
$901$	4.33391e10	1.97398
$902$	8.76450e9	0.397653
$903$	0	0
$904$	−3.56167e9	−0.160348
$905$	−2.69824e10	−1.21007
$906$	0	0
$907$	−6.42631e8	−0.0285980	−0.0142990	−	0.999898i	$-0.504552\pi$
$907$	−6.42631e8	−0.0285980	−0.0142990	+	0.999898i	$0.504552\pi$
$908$	−8.86988e9	−0.393203
$909$	0	0
$910$	−6.18706e8	−0.0272170
$911$	9.22976e9	0.404460	0.202230	−	0.979338i	$-0.435181\pi$
$911$	9.22976e9	0.404460	0.202230	+	0.979338i	$0.435181\pi$
$912$	0	0
$913$	−2.00310e10	−0.871074
$914$	−7.99343e9	−0.346275
$915$	0	0
$916$	4.29036e9	0.184442
$917$	2.68076e9	0.114806
$918$	0	0
$919$	1.03928e10	0.441701	0.220850	−	0.975308i	$-0.429117\pi$
$919$	1.03928e10	0.441701	0.220850	+	0.975308i	$0.429117\pi$
$920$	7.10234e9	0.300707
$921$	0	0
$922$	−1.31845e10	−0.553996
$923$	4.12389e9	0.172624
$924$	0	0
$925$	2.77908e8	0.0115453
$926$	−2.77650e10	−1.14910
$927$	0	0
$928$	−2.47490e8	−0.0101658
$929$	−1.67610e10	−0.685876	−0.342938	−	0.939358i	$-0.611422\pi$
$929$	−1.67610e10	−0.685876	−0.342938	+	0.939358i	$0.611422\pi$
$930$	0	0
$931$	1.00697e10	0.408971
$932$	−8.37755e9	−0.338970
$933$	0	0
$934$	1.99940e10	0.802946
$935$	−1.81080e10	−0.724487
$936$	0	0
$937$	2.27134e9	0.0901973	0.0450987	−	0.998983i	$-0.485640\pi$
$937$	2.27134e9	0.0901973	0.0450987	+	0.998983i	$0.485640\pi$
$938$	−6.31597e9	−0.249879
$939$	0	0
$940$	7.37399e9	0.289571
$941$	1.86944e9	0.0731388	0.0365694	−	0.999331i	$-0.488357\pi$
$941$	1.86944e9	0.0731388	0.0365694	+	0.999331i	$0.488357\pi$
$942$	0	0
$943$	2.65926e10	1.03269
$944$	−3.27225e9	−0.126603
$945$	0	0
$946$	−3.26881e9	−0.125537
$947$	4.99793e9	0.191234	0.0956171	−	0.995418i	$-0.469518\pi$
$947$	4.99793e9	0.191234	0.0956171	+	0.995418i	$0.469518\pi$
$948$	0	0
$949$	−2.69111e9	−0.102212
$950$	5.19098e7	0.00196434
$951$	0	0
$952$	−2.66583e9	−0.100139
$953$	−4.30036e10	−1.60946	−0.804730	−	0.593641i	$-0.797690\pi$
$953$	−4.30036e10	−1.60946	−0.804730	+	0.593641i	$0.797690\pi$
$954$	0	0
$955$	2.07292e10	0.770143
$956$	−1.23354e9	−0.0456617
$957$	0	0
$958$	−3.51689e10	−1.29235
$959$	−5.94184e9	−0.217548
$960$	0	0
$961$	−3.16560e9	−0.115060
$962$	7.26973e9	0.263273
$963$	0	0
$964$	4.38394e9	0.157614
$965$	2.28461e10	0.818401
$966$	0	0
$967$	−4.60361e10	−1.63721	−0.818607	−	0.574354i	$-0.805253\pi$
$967$	−4.60361e10	−1.63721	−0.818607	+	0.574354i	$0.805253\pi$
$968$	7.85100e9	0.278202
$969$	0	0
$970$	1.74816e10	0.615007
$971$	−3.33640e10	−1.16953	−0.584765	−	0.811203i	$-0.698813\pi$
$971$	−3.33640e10	−1.16953	−0.584765	+	0.811203i	$0.698813\pi$
$972$	0	0
$973$	−8.79431e9	−0.306060
$974$	2.26392e10	0.785062
$975$	0	0
$976$	2.33242e9	0.0803032
$977$	−2.78640e10	−0.955899	−0.477950	−	0.878387i	$-0.658620\pi$
$977$	−2.78640e10	−0.955899	−0.477950	+	0.878387i	$0.658620\pi$
$978$	0	0
$979$	2.38945e8	0.00813878
$980$	−1.42957e10	−0.485192
$981$	0	0
$982$	1.96673e10	0.662759
$983$	−3.09396e10	−1.03891	−0.519455	−	0.854498i	$-0.673865\pi$
$983$	−3.09396e10	−1.03891	−0.519455	+	0.854498i	$0.673865\pi$
$984$	0	0
$985$	6.84330e9	0.228160
$986$	1.91452e9	0.0636050
$987$	0	0
$988$	1.35790e9	0.0447938
$989$	−9.91799e9	−0.326014
$990$	0	0
$991$	−1.80515e10	−0.589192	−0.294596	−	0.955622i	$-0.595185\pi$
$991$	−1.80515e10	−0.589192	−0.294596	+	0.955622i	$0.595185\pi$
$992$	−5.11296e9	−0.166296
$993$	0	0
$994$	−3.23012e9	−0.104320
$995$	−4.26621e10	−1.37297
$996$	0	0
$997$	2.62841e9	0.0839963	0.0419982	−	0.999118i	$-0.486628\pi$
$997$	2.62841e9	0.0839963	0.0419982	+	0.999118i	$0.486628\pi$
$998$	2.40483e10	0.765823
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.a.g.1.4	✓	4
3.2	odd	2		162.8.a.j.1.1	yes	4
9.2	odd	6		162.8.c.q.109.4		8
9.4	even	3		162.8.c.r.55.1		8
9.5	odd	6		162.8.c.q.55.4		8
9.7	even	3		162.8.c.r.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.g.1.4	✓	4	1.1	even	1	trivial
162.8.a.j.1.1	yes	4	3.2	odd	2
162.8.c.q.55.4		8	9.5	odd	6
162.8.c.q.109.4		8	9.2	odd	6
162.8.c.r.55.1		8	9.4	even	3
162.8.c.r.109.1		8	9.7	even	3

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} +280.425 q^{5} -164.323 q^{7} -512.000 q^{8} -2243.40 q^{10} +2037.94 q^{11} -1678.33 q^{13} +1314.59 q^{14} +4096.00 q^{16} -31685.7 q^{17} -12641.8 q^{19} +17947.2 q^{20} -16303.5 q^{22} -49466.9 q^{23} +513.276 q^{25} +13426.7 q^{26} -10516.7 q^{28} +7552.79 q^{29} +156035. q^{31} -32768.0 q^{32} +253485. q^{34} -46080.4 q^{35} +541440. q^{37} +101134. q^{38} -143578. q^{40} -537583. q^{41} +200498. q^{43} +130428. q^{44} +395735. q^{46} +410871. q^{47} -796541. q^{49} -4106.21 q^{50} -107413. q^{52} -1.36778e6 q^{53} +571489. q^{55} +84133.6 q^{56} -60422.3 q^{58} -798889. q^{59} +569439. q^{61} -1.24828e6 q^{62} +262144. q^{64} -470647. q^{65} -4.80453e6 q^{67} -2.02788e6 q^{68} +368643. q^{70} -2.45714e6 q^{71} +1.60344e6 q^{73} -4.33152e6 q^{74} -809075. q^{76} -334881. q^{77} +5.58998e6 q^{79} +1.14862e6 q^{80} +4.30067e6 q^{82} -9.82905e6 q^{83} -8.88546e6 q^{85} -1.60398e6 q^{86} -1.04342e6 q^{88} +117249. q^{89} +275789. q^{91} -3.16588e6 q^{92} -3.28697e6 q^{94} -3.54508e6 q^{95} -7.79245e6 q^{97} +6.37233e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 528 q^{5} + 560 q^{7} - 2048 q^{8} + 4224 q^{10} - 2160 q^{11} + 13460 q^{13} - 4480 q^{14} + 16384 q^{16} - 22560 q^{17} + 36704 q^{19} - 33792 q^{20} + 17280 q^{22} - 62640 q^{23}+ \cdots - 11595168 q^{98}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 528 * q^5 + 560 * q^7 - 2048 * q^8 + 4224 * q^10 - 2160 * q^11 + 13460 * q^13 - 4480 * q^14 + 16384 * q^16 - 22560 * q^17 + 36704 * q^19 - 33792 * q^20 + 17280 * q^22 - 62640 * q^23 + 94696 * q^25 - 107680 * q^26 + 35840 * q^28 - 68400 * q^29 + 227504 * q^31 - 131072 * q^32 + 180480 * q^34 + 63024 * q^35 + 523580 * q^37 - 293632 * q^38 + 270336 * q^40 - 67200 * q^41 + 562640 * q^43 - 138240 * q^44 + 501120 * q^46 - 515328 * q^47 + 1449396 * q^49 - 757568 * q^50 + 861440 * q^52 - 2498016 * q^53 + 1322784 * q^55 - 286720 * q^56 + 547200 * q^58 - 4155840 * q^59 + 2130764 * q^61 - 1820032 * q^62 + 1048576 * q^64 - 8966496 * q^65 - 1205440 * q^67 - 1443840 * q^68 - 504192 * q^70 - 12486480 * q^71 - 4820860 * q^73 - 4188640 * q^74 + 2349056 * q^76 - 18686592 * q^77 - 11471680 * q^79 - 2162688 * q^80 + 537600 * q^82 - 16811232 * q^83 - 21006396 * q^85 - 4501120 * q^86 + 1105920 * q^88 - 16857600 * q^89 - 23200544 * q^91 - 4008960 * q^92 + 4122624 * q^94 - 14090160 * q^95 - 27078040 * q^97 - 11595168 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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