LMFDB - Embedded newform 162.10.a.f.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,10,Mod(1,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-64,0,1024,1968] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$83.4358054585$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 953x^{2} + 954x + 195702 $ x^4 - 2x^3 - 953x^2 + 954*x + 195702
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			$26.1057$ 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)

$f(q)$	$=$	\(q-16.0000 q^{2} +256.000 q^{4} +1985.27 q^{5} +3496.33 q^{7} -4096.00 q^{8} -31764.3 q^{10} -92426.8 q^{11} -156467. q^{13} -55941.3 q^{14} +65536.0 q^{16} +606661. q^{17} +793782. q^{19} +508229. q^{20} +1.47883e6 q^{22} +107894. q^{23} +1.98817e6 q^{25} +2.50348e6 q^{26} +895062. q^{28} -3.95259e6 q^{29} -356140. q^{31} -1.04858e6 q^{32} -9.70657e6 q^{34} +6.94117e6 q^{35} +4.63798e6 q^{37} -1.27005e7 q^{38} -8.13167e6 q^{40} +2.51530e7 q^{41} -4.09013e6 q^{43} -2.36613e7 q^{44} -1.72630e6 q^{46} +4.52307e7 q^{47} -2.81293e7 q^{49} -3.18108e7 q^{50} -4.00556e7 q^{52} -41008.3 q^{53} -1.83492e8 q^{55} -1.43210e7 q^{56} +6.32414e7 q^{58} +1.44716e8 q^{59} +2.38311e7 q^{61} +5.69824e6 q^{62} +1.67772e7 q^{64} -3.10630e8 q^{65} -1.29326e8 q^{67} +1.55305e8 q^{68} -1.11059e8 q^{70} +2.92490e8 q^{71} +1.45424e8 q^{73} -7.42077e7 q^{74} +2.03208e8 q^{76} -3.23155e8 q^{77} -9.11909e7 q^{79} +1.30107e8 q^{80} -4.02448e8 q^{82} +3.33108e8 q^{83} +1.20439e9 q^{85} +6.54421e7 q^{86} +3.78580e8 q^{88} +4.73579e8 q^{89} -5.47062e8 q^{91} +2.76208e7 q^{92} -7.23691e8 q^{94} +1.57587e9 q^{95} -1.68662e9 q^{97} +4.50068e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{2} + 1024 q^{4} + 1968 q^{5} + 4496 q^{7} - 16384 q^{8} - 31488 q^{10} - 8784 q^{11} - 162556 q^{13} - 71936 q^{14} + 262144 q^{16} + 538080 q^{17} - 8224 q^{19} + 503808 q^{20} + 140544 q^{22}+ \cdots + 2038788288 q^{98}+O(q^{100}) $ 4 * q - 64 * q^2 + 1024 * q^4 + 1968 * q^5 + 4496 * q^7 - 16384 * q^8 - 31488 * q^10 - 8784 * q^11 - 162556 * q^13 - 71936 * q^14 + 262144 * q^16 + 538080 * q^17 - 8224 * q^19 + 503808 * q^20 + 140544 * q^22 + 2594736 * q^23 + 1873192 * q^25 + 2600896 * q^26 + 1150976 * q^28 + 3232656 * q^29 + 3482576 * q^31 - 4194304 * q^32 - 8609280 * q^34 + 7969296 * q^35 - 2487892 * q^37 + 131584 * q^38 - 8060928 * q^40 + 34657152 * q^41 - 41410000 * q^43 - 2248704 * q^44 - 41515776 * q^46 + 40558848 * q^47 - 127424268 * q^49 - 29971072 * q^50 - 41614336 * q^52 + 8421024 * q^53 - 229068000 * q^55 - 18415616 * q^56 - 51722496 * q^58 + 26843328 * q^59 - 111504484 * q^61 - 55721216 * q^62 + 67108864 * q^64 + 23507616 * q^65 - 208064512 * q^67 + 137748480 * q^68 - 127508736 * q^70 + 356008272 * q^71 - 37126108 * q^73 + 39806272 * q^74 - 2105344 * q^76 + 33010560 * q^77 - 645134848 * q^79 + 128974848 * q^80 - 554514432 * q^82 + 1019036256 * q^83 + 80955540 * q^85 + 662560000 * q^86 + 35979264 * q^88 + 1548192768 * q^89 - 130776608 * q^91 + 664252416 * q^92 - 648941568 * q^94 + 3445933872 * q^95 + 1112014568 * q^97 + 2038788288 * 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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	0	0
$3$	0	0
$4$	256.000	0.500000
$5$	1985.27	1.42054	0.710272	−	0.703927i	$-0.248572\pi$
$5$	1985.27	1.42054	0.710272	+	0.703927i	$0.248572\pi$
$6$	0	0
$7$	3496.33	0.550392	0.275196	−	0.961388i	$-0.411257\pi$
$7$	3496.33	0.550392	0.275196	+	0.961388i	$0.411257\pi$
$8$	−4096.00	−0.353553
$9$	0	0
$10$	−31764.3	−1.00448
$11$	−92426.8	−1.90340	−0.951702	−	0.307023i	$-0.900667\pi$
$11$	−92426.8	−1.90340	−0.951702	+	0.307023i	$0.900667\pi$
$12$	0	0
$13$	−156467.	−1.51942	−0.759711	−	0.650261i	$-0.774660\pi$
$13$	−156467.	−1.51942	−0.759711	+	0.650261i	$0.774660\pi$
$14$	−55941.3	−0.389186
$15$	0	0
$16$	65536.0	0.250000
$17$	606661.	1.76167	0.880837	−	0.473419i	$-0.156980\pi$
$17$	606661.	1.76167	0.880837	+	0.473419i	$0.156980\pi$
$18$	0	0
$19$	793782.	1.39736	0.698682	−	0.715432i	$-0.253770\pi$
$19$	793782.	1.39736	0.698682	+	0.715432i	$0.253770\pi$
$20$	508229.	0.710272
$21$	0	0
$22$	1.47883e6	1.34591
$23$	107894.	0.0803936	0.0401968	−	0.999192i	$-0.487202\pi$
$23$	107894.	0.0803936	0.0401968	+	0.999192i	$0.487202\pi$
$24$	0	0
$25$	1.98817e6	1.01795
$26$	2.50348e6	1.07439
$27$	0	0
$28$	895062.	0.275196
$29$	−3.95259e6	−1.03774	−0.518872	−	0.854852i	$-0.673648\pi$
$29$	−3.95259e6	−1.03774	−0.518872	+	0.854852i	$0.673648\pi$
$30$	0	0
$31$	−356140.	−0.0692617	−0.0346309	−	0.999400i	$-0.511026\pi$
$31$	−356140.	−0.0692617	−0.0346309	+	0.999400i	$0.511026\pi$
$32$	−1.04858e6	−0.176777
$33$	0	0
$34$	−9.70657e6	−1.24569
$35$	6.94117e6	0.781855
$36$	0	0
$37$	4.63798e6	0.406838	0.203419	−	0.979092i	$-0.434795\pi$
$37$	4.63798e6	0.406838	0.203419	+	0.979092i	$0.434795\pi$
$38$	−1.27005e7	−0.988086
$39$	0	0
$40$	−8.13167e6	−0.502238
$41$	2.51530e7	1.39015	0.695075	−	0.718937i	$-0.255371\pi$
$41$	2.51530e7	1.39015	0.695075	+	0.718937i	$0.255371\pi$
$42$	0	0
$43$	−4.09013e6	−0.182444	−0.0912219	−	0.995831i	$-0.529077\pi$
$43$	−4.09013e6	−0.182444	−0.0912219	+	0.995831i	$0.529077\pi$
$44$	−2.36613e7	−0.951702
$45$	0	0
$46$	−1.72630e6	−0.0568469
$47$	4.52307e7	1.35205	0.676025	−	0.736879i	$-0.263701\pi$
$47$	4.52307e7	1.35205	0.676025	+	0.736879i	$0.263701\pi$
$48$	0	0
$49$	−2.81293e7	−0.697069
$50$	−3.18108e7	−0.719796
$51$	0	0
$52$	−4.00556e7	−0.759711
$53$	−41008.3	−0.000713889 0	−0.000356944	−	1.00000i	$-0.500114\pi$
$53$	−41008.3	−0.000713889 0	−0.000356944		1.00000i	$0.500114\pi$
$54$	0	0
$55$	−1.83492e8	−2.70387
$56$	−1.43210e7	−0.194593
$57$	0	0
$58$	6.32414e7	0.733796
$59$	1.44716e8	1.55483	0.777417	−	0.628986i	$-0.216530\pi$
$59$	1.44716e8	1.55483	0.777417	+	0.628986i	$0.216530\pi$
$60$	0	0
$61$	2.38311e7	0.220374	0.110187	−	0.993911i	$-0.464855\pi$
$61$	2.38311e7	0.220374	0.110187	+	0.993911i	$0.464855\pi$
$62$	5.69824e6	0.0489754
$63$	0	0
$64$	1.67772e7	0.125000
$65$	−3.10630e8	−2.15841
$66$	0	0
$67$	−1.29326e8	−0.784057	−0.392029	−	0.919953i	$-0.628227\pi$
$67$	−1.29326e8	−0.784057	−0.392029	+	0.919953i	$0.628227\pi$
$68$	1.55305e8	0.880837
$69$	0	0
$70$	−1.11059e8	−0.552855
$71$	2.92490e8	1.36599	0.682996	−	0.730422i	$-0.260676\pi$
$71$	2.92490e8	1.36599	0.682996	+	0.730422i	$0.260676\pi$
$72$	0	0
$73$	1.45424e8	0.599355	0.299678	−	0.954041i	$-0.403121\pi$
$73$	1.45424e8	0.599355	0.299678	+	0.954041i	$0.403121\pi$
$74$	−7.42077e7	−0.287678
$75$	0	0
$76$	2.03208e8	0.698682
$77$	−3.23155e8	−1.04762
$78$	0	0
$79$	−9.11909e7	−0.263408	−0.131704	−	0.991289i	$-0.542045\pi$
$79$	−9.11909e7	−0.263408	−0.131704	+	0.991289i	$0.542045\pi$
$80$	1.30107e8	0.355136
$81$	0	0
$82$	−4.02448e8	−0.982985
$83$	3.33108e8	0.770430	0.385215	−	0.922827i	$-0.374127\pi$
$83$	3.33108e8	0.770430	0.385215	+	0.922827i	$0.374127\pi$
$84$	0	0
$85$	1.20439e9	2.50254
$86$	6.54421e7	0.129007
$87$	0	0
$88$	3.78580e8	0.672955
$89$	4.73579e8	0.800087	0.400044	−	0.916496i	$-0.368995\pi$
$89$	4.73579e8	0.800087	0.400044	+	0.916496i	$0.368995\pi$
$90$	0	0
$91$	−5.47062e8	−0.836277
$92$	2.76208e7	0.0401968
$93$	0	0
$94$	−7.23691e8	−0.956043
$95$	1.57587e9	1.98502
$96$	0	0
$97$	−1.68662e9	−1.93440	−0.967198	−	0.254025i	$-0.918246\pi$
$97$	−1.68662e9	−1.93440	−0.967198	+	0.254025i	$0.918246\pi$
$98$	4.50068e8	0.492902
$99$	0	0
$100$	5.08973e8	0.508973
$101$	7.73784e8	0.739901	0.369950	−	0.929052i	$-0.379375\pi$
$101$	7.73784e8	0.739901	0.369950	+	0.929052i	$0.379375\pi$
$102$	0	0
$103$	6.55861e8	0.574175	0.287087	−	0.957904i	$-0.407313\pi$
$103$	6.55861e8	0.574175	0.287087	+	0.957904i	$0.407313\pi$
$104$	6.40890e8	0.537197
$105$	0	0
$106$	656133.	0.000504796 0
$107$	−4.88739e8	−0.360454	−0.180227	−	0.983625i	$-0.557683\pi$
$107$	−4.88739e8	−0.360454	−0.180227	+	0.983625i	$0.557683\pi$
$108$	0	0
$109$	−1.55839e9	−1.05744	−0.528721	−	0.848796i	$-0.677328\pi$
$109$	−1.55839e9	−1.05744	−0.528721	+	0.848796i	$0.677328\pi$
$110$	2.93588e9	1.91192
$111$	0	0
$112$	2.29136e8	0.137598
$113$	−4.71555e8	−0.272069	−0.136035	−	0.990704i	$-0.543436\pi$
$113$	−4.71555e8	−0.272069	−0.136035	+	0.990704i	$0.543436\pi$
$114$	0	0
$115$	2.14199e8	0.114203
$116$	−1.01186e9	−0.518872
$117$	0	0
$118$	−2.31546e9	−1.09943
$119$	2.12109e9	0.969611
$120$	0	0
$121$	6.18477e9	2.62295
$122$	−3.81297e8	−0.155828
$123$	0	0
$124$	−9.11719e7	−0.0346309
$125$	6.95826e7	0.0254921
$126$	0	0
$127$	2.23394e9	0.761999	0.380999	−	0.924575i	$-0.375580\pi$
$127$	2.23394e9	0.761999	0.380999	+	0.924575i	$0.375580\pi$
$128$	−2.68435e8	−0.0883883
$129$	0	0
$130$	4.97008e9	1.52622
$131$	−6.31423e8	−0.187327	−0.0936634	−	0.995604i	$-0.529858\pi$
$131$	−6.31423e8	−0.187327	−0.0936634	+	0.995604i	$0.529858\pi$
$132$	0	0
$133$	2.77533e9	0.769098
$134$	2.06921e9	0.554412
$135$	0	0
$136$	−2.48488e9	−0.622846
$137$	3.89809e9	0.945388	0.472694	−	0.881227i	$-0.343282\pi$
$137$	3.89809e9	0.945388	0.472694	+	0.881227i	$0.343282\pi$
$138$	0	0
$139$	4.74014e9	1.07702	0.538511	−	0.842619i	$-0.318987\pi$
$139$	4.74014e9	1.07702	0.538511	+	0.842619i	$0.318987\pi$
$140$	1.77694e9	0.390928
$141$	0	0
$142$	−4.67984e9	−0.965903
$143$	1.44618e10	2.89207
$144$	0	0
$145$	−7.84695e9	−1.47416
$146$	−2.32679e9	−0.423808
$147$	0	0
$148$	1.18732e9	0.203419
$149$	8.22496e8	0.136709	0.0683543	−	0.997661i	$-0.478225\pi$
$149$	8.22496e8	0.136709	0.0683543	+	0.997661i	$0.478225\pi$
$150$	0	0
$151$	6.31122e6	0.000987909 0	0.000493954	−	1.00000i	$-0.499843\pi$
$151$	6.31122e6	0.000987909 0	0.000493954		1.00000i	$0.499843\pi$
$152$	−3.25133e9	−0.494043
$153$	0	0
$154$	5.17048e9	0.740778
$155$	−7.07035e8	−0.0983893
$156$	0	0
$157$	4.84563e9	0.636505	0.318253	−	0.948006i	$-0.396904\pi$
$157$	4.84563e9	0.636505	0.318253	+	0.948006i	$0.396904\pi$
$158$	1.45905e9	0.186258
$159$	0	0
$160$	−2.08171e9	−0.251119
$161$	3.77233e8	0.0442480
$162$	0	0
$163$	−2.47695e9	−0.274836	−0.137418	−	0.990513i	$-0.543880\pi$
$163$	−2.47695e9	−0.274836	−0.137418	+	0.990513i	$0.543880\pi$
$164$	6.43916e9	0.695075
$165$	0	0
$166$	−5.32972e9	−0.544776
$167$	1.20374e10	1.19759	0.598795	−	0.800902i	$-0.295646\pi$
$167$	1.20374e10	1.19759	0.598795	+	0.800902i	$0.295646\pi$
$168$	0	0
$169$	1.38775e10	1.30864
$170$	−1.92702e10	−1.76956
$171$	0	0
$172$	−1.04707e9	−0.0912219
$173$	5.55954e9	0.471880	0.235940	−	0.971768i	$-0.424183\pi$
$173$	5.55954e9	0.471880	0.235940	+	0.971768i	$0.424183\pi$
$174$	0	0
$175$	6.95132e9	0.560268
$176$	−6.05729e9	−0.475851
$177$	0	0
$178$	−7.57726e9	−0.565747
$179$	1.40559e10	1.02334	0.511669	−	0.859183i	$-0.329027\pi$
$179$	1.40559e10	1.02334	0.511669	+	0.859183i	$0.329027\pi$
$180$	0	0
$181$	−8.33092e9	−0.576952	−0.288476	−	0.957487i	$-0.593149\pi$
$181$	−8.33092e9	−0.576952	−0.288476	+	0.957487i	$0.593149\pi$
$182$	8.75299e9	0.591337
$183$	0	0
$184$	−4.41933e8	−0.0284234
$185$	9.20765e9	0.577931
$186$	0	0
$187$	−5.60717e10	−3.35318
$188$	1.15791e10	0.676025
$189$	0	0
$190$	−2.52139e10	−1.40362
$191$	−1.80382e10	−0.980716	−0.490358	−	0.871521i	$-0.663134\pi$
$191$	−1.80382e10	−0.980716	−0.490358	+	0.871521i	$0.663134\pi$
$192$	0	0
$193$	−2.68848e10	−1.39476	−0.697380	−	0.716702i	$-0.745651\pi$
$193$	−2.68848e10	−1.39476	−0.697380	+	0.716702i	$0.745651\pi$
$194$	2.69860e10	1.36782
$195$	0	0
$196$	−7.20109e9	−0.348535
$197$	−6.01487e9	−0.284530	−0.142265	−	0.989829i	$-0.545439\pi$
$197$	−6.01487e9	−0.284530	−0.142265	+	0.989829i	$0.545439\pi$
$198$	0	0
$199$	4.06076e10	1.83556	0.917781	−	0.397087i	$-0.129979\pi$
$199$	4.06076e10	1.83556	0.917781	+	0.397087i	$0.129979\pi$
$200$	−8.14356e9	−0.359898
$201$	0	0
$202$	−1.23805e10	−0.523189
$203$	−1.38196e10	−0.571166
$204$	0	0
$205$	4.99355e10	1.97477
$206$	−1.04938e10	−0.406003
$207$	0	0
$208$	−1.02542e10	−0.379855
$209$	−7.33667e10	−2.65975
$210$	0	0
$211$	8.50220e9	0.295298	0.147649	−	0.989040i	$-0.452829\pi$
$211$	8.50220e9	0.295298	0.147649	+	0.989040i	$0.452829\pi$
$212$	−1.04981e7	−0.000356944 0
$213$	0	0
$214$	7.81983e9	0.254880
$215$	−8.12002e9	−0.259169
$216$	0	0
$217$	−1.24518e9	−0.0381211
$218$	2.49342e10	0.747724
$219$	0	0
$220$	−4.69740e10	−1.35193
$221$	−9.49225e10	−2.67673
$222$	0	0
$223$	6.43288e10	1.74194	0.870971	−	0.491334i	$-0.163491\pi$
$223$	6.43288e10	1.74194	0.870971	+	0.491334i	$0.163491\pi$
$224$	−3.66617e9	−0.0972964
$225$	0	0
$226$	7.54489e9	0.192382
$227$	−1.54775e10	−0.386887	−0.193444	−	0.981111i	$-0.561966\pi$
$227$	−1.54775e10	−0.386887	−0.193444	+	0.981111i	$0.561966\pi$
$228$	0	0
$229$	5.55452e10	1.33471	0.667355	−	0.744740i	$-0.267427\pi$
$229$	5.55452e10	1.33471	0.667355	+	0.744740i	$0.267427\pi$
$230$	−3.42718e9	−0.0807535
$231$	0	0
$232$	1.61898e10	0.366898
$233$	7.45295e10	1.65663	0.828317	−	0.560259i	$-0.189299\pi$
$233$	7.45295e10	1.65663	0.828317	+	0.560259i	$0.189299\pi$
$234$	0	0
$235$	8.97951e10	1.92065
$236$	3.70474e10	0.777417
$237$	0	0
$238$	−3.39374e10	−0.685618
$239$	3.32399e10	0.658975	0.329487	−	0.944160i	$-0.393124\pi$
$239$	3.32399e10	0.658975	0.329487	+	0.944160i	$0.393124\pi$
$240$	0	0
$241$	−3.35131e10	−0.639937	−0.319969	−	0.947428i	$-0.603672\pi$
$241$	−3.35131e10	−0.639937	−0.319969	+	0.947428i	$0.603672\pi$
$242$	−9.89564e10	−1.85470
$243$	0	0
$244$	6.10076e9	0.110187
$245$	−5.58442e10	−0.990217
$246$	0	0
$247$	−1.24201e11	−2.12319
$248$	1.45875e9	0.0244877
$249$	0	0
$250$	−1.11332e9	−0.0180256
$251$	−1.33603e10	−0.212464	−0.106232	−	0.994341i	$-0.533879\pi$
$251$	−1.33603e10	−0.212464	−0.106232	+	0.994341i	$0.533879\pi$
$252$	0	0
$253$	−9.97229e9	−0.153022
$254$	−3.57430e10	−0.538814
$255$	0	0
$256$	4.29497e9	0.0625000
$257$	−1.24665e10	−0.178257	−0.0891284	−	0.996020i	$-0.528408\pi$
$257$	−1.24665e10	−0.178257	−0.0891284	+	0.996020i	$0.528408\pi$
$258$	0	0
$259$	1.62159e10	0.223920
$260$	−7.95212e10	−1.07920
$261$	0	0
$262$	1.01028e10	0.132460
$263$	7.58179e10	0.977172	0.488586	−	0.872516i	$-0.337513\pi$
$263$	7.58179e10	0.977172	0.488586	+	0.872516i	$0.337513\pi$
$264$	0	0
$265$	−8.14126e7	−0.00101411
$266$	−4.44052e10	−0.543834
$267$	0	0
$268$	−3.31074e10	−0.392029
$269$	−9.89798e10	−1.15255	−0.576277	−	0.817255i	$-0.695495\pi$
$269$	−9.89798e10	−1.15255	−0.576277	+	0.817255i	$0.695495\pi$
$270$	0	0
$271$	6.28601e10	0.707968	0.353984	−	0.935252i	$-0.384827\pi$
$271$	6.28601e10	0.707968	0.353984	+	0.935252i	$0.384827\pi$
$272$	3.97581e10	0.440419
$273$	0	0
$274$	−6.23695e10	−0.668490
$275$	−1.83761e11	−1.93756
$276$	0	0
$277$	1.25003e11	1.27573	0.637867	−	0.770147i	$-0.279817\pi$
$277$	1.25003e11	1.27573	0.637867	+	0.770147i	$0.279817\pi$
$278$	−7.58422e10	−0.761569
$279$	0	0
$280$	−2.84310e10	−0.276428
$281$	1.05613e10	0.101051	0.0505254	−	0.998723i	$-0.483910\pi$
$281$	1.05613e10	0.101051	0.0505254	+	0.998723i	$0.483910\pi$
$282$	0	0
$283$	−1.22575e11	−1.13596	−0.567978	−	0.823043i	$-0.692274\pi$
$283$	−1.22575e11	−1.13596	−0.567978	+	0.823043i	$0.692274\pi$
$284$	7.48774e10	0.682996
$285$	0	0
$286$	−2.31388e11	−2.04501
$287$	8.79432e10	0.765127
$288$	0	0
$289$	2.49449e11	2.10350
$290$	1.25551e11	1.04239
$291$	0	0
$292$	3.72286e10	0.299678
$293$	−8.83183e10	−0.700078	−0.350039	−	0.936735i	$-0.613832\pi$
$293$	−8.83183e10	−0.700078	−0.350039	+	0.936735i	$0.613832\pi$
$294$	0	0
$295$	2.87301e11	2.20871
$296$	−1.89972e10	−0.143839
$297$	0	0
$298$	−1.31599e10	−0.0966676
$299$	−1.68819e10	−0.122152
$300$	0	0
$301$	−1.43005e10	−0.100416
$302$	−1.00979e8	−0.000698557 0
$303$	0	0
$304$	5.20213e10	0.349341
$305$	4.73112e10	0.313050
$306$	0	0
$307$	3.87556e10	0.249007	0.124503	−	0.992219i	$-0.460266\pi$
$307$	3.87556e10	0.249007	0.124503	+	0.992219i	$0.460266\pi$
$308$	−8.27277e10	−0.523809
$309$	0	0
$310$	1.13126e10	0.0695718
$311$	2.17955e11	1.32113	0.660563	−	0.750770i	$-0.270317\pi$
$311$	2.17955e11	1.32113	0.660563	+	0.750770i	$0.270317\pi$
$312$	0	0
$313$	−7.22320e10	−0.425383	−0.212692	−	0.977119i	$-0.568223\pi$
$313$	−7.22320e10	−0.425383	−0.212692	+	0.977119i	$0.568223\pi$
$314$	−7.75301e10	−0.450077
$315$	0	0
$316$	−2.33449e10	−0.131704
$317$	−2.76799e11	−1.53957	−0.769783	−	0.638306i	$-0.779636\pi$
$317$	−2.76799e11	−1.53957	−0.769783	+	0.638306i	$0.779636\pi$
$318$	0	0
$319$	3.65325e11	1.97525
$320$	3.33073e10	0.177568
$321$	0	0
$322$	−6.03573e9	−0.0312880
$323$	4.81556e11	2.46170
$324$	0	0
$325$	−3.11084e11	−1.54669
$326$	3.96313e10	0.194338
$327$	0	0
$328$	−1.03027e11	−0.491493
$329$	1.58142e11	0.744157
$330$	0	0
$331$	−1.81639e11	−0.831732	−0.415866	−	0.909426i	$-0.636522\pi$
$331$	−1.81639e11	−0.831732	−0.415866	+	0.909426i	$0.636522\pi$
$332$	8.52755e10	0.385215
$333$	0	0
$334$	−1.92598e11	−0.846824
$335$	−2.56746e11	−1.11379
$336$	0	0
$337$	−6.20015e10	−0.261859	−0.130930	−	0.991392i	$-0.541796\pi$
$337$	−6.20015e10	−0.261859	−0.130930	+	0.991392i	$0.541796\pi$
$338$	−2.22040e11	−0.925350
$339$	0	0
$340$	3.08323e11	1.25127
$341$	3.29169e10	0.131833
$342$	0	0
$343$	−2.39439e11	−0.934052
$344$	1.67532e10	0.0645036
$345$	0	0
$346$	−8.89527e10	−0.333669
$347$	−2.55503e11	−0.946049	−0.473024	−	0.881049i	$-0.656838\pi$
$347$	−2.55503e11	−0.946049	−0.473024	+	0.881049i	$0.656838\pi$
$348$	0	0
$349$	−2.45242e11	−0.884873	−0.442437	−	0.896800i	$-0.645886\pi$
$349$	−2.45242e11	−0.884873	−0.442437	+	0.896800i	$0.645886\pi$
$350$	−1.11221e11	−0.396170
$351$	0	0
$352$	9.69166e10	0.336478
$353$	2.76510e11	0.947818	0.473909	−	0.880574i	$-0.342843\pi$
$353$	2.76510e11	0.947818	0.473909	+	0.880574i	$0.342843\pi$
$354$	0	0
$355$	5.80672e11	1.94045
$356$	1.21236e11	0.400044
$357$	0	0
$358$	−2.24894e11	−0.723609
$359$	−5.93920e11	−1.88713	−0.943567	−	0.331181i	$-0.892553\pi$
$359$	−5.93920e11	−1.88713	−0.943567	+	0.331181i	$0.892553\pi$
$360$	0	0
$361$	3.07401e11	0.952628
$362$	1.33295e11	0.407967
$363$	0	0
$364$	−1.40048e11	−0.418138
$365$	2.88707e11	0.851410
$366$	0	0
$367$	−1.74195e11	−0.501230	−0.250615	−	0.968087i	$-0.580633\pi$
$367$	−1.74195e11	−0.501230	−0.250615	+	0.968087i	$0.580633\pi$
$368$	7.07094e9	0.0200984
$369$	0	0
$370$	−1.47322e11	−0.408659
$371$	−1.43379e8	−0.000392918 0
$372$	0	0
$373$	−4.95214e10	−0.132466	−0.0662328	−	0.997804i	$-0.521098\pi$
$373$	−4.95214e10	−0.132466	−0.0662328	+	0.997804i	$0.521098\pi$
$374$	8.97148e11	2.37106
$375$	0	0
$376$	−1.85265e11	−0.478022
$377$	6.18450e11	1.57677
$378$	0	0
$379$	−6.61837e11	−1.64769	−0.823843	−	0.566819i	$-0.808174\pi$
$379$	−6.61837e11	−1.64769	−0.823843	+	0.566819i	$0.808174\pi$
$380$	4.03423e11	0.992509
$381$	0	0
$382$	2.88611e11	0.693471
$383$	2.41369e11	0.573175	0.286587	−	0.958054i	$-0.407479\pi$
$383$	2.41369e11	0.573175	0.286587	+	0.958054i	$0.407479\pi$
$384$	0	0
$385$	−6.41550e11	−1.48819
$386$	4.30157e11	0.986244
$387$	0	0
$388$	−4.31776e11	−0.967198
$389$	6.15256e11	1.36233	0.681165	−	0.732130i	$-0.261474\pi$
$389$	6.15256e11	1.36233	0.681165	+	0.732130i	$0.261474\pi$
$390$	0	0
$391$	6.54550e10	0.141627
$392$	1.15217e11	0.246451
$393$	0	0
$394$	9.62379e10	0.201193
$395$	−1.81039e11	−0.374183
$396$	0	0
$397$	−6.78022e11	−1.36989	−0.684946	−	0.728594i	$-0.740174\pi$
$397$	−6.78022e11	−1.36989	−0.684946	+	0.728594i	$0.740174\pi$
$398$	−6.49722e11	−1.29794
$399$	0	0
$400$	1.30297e11	0.254486
$401$	1.74580e11	0.337166	0.168583	−	0.985687i	$-0.446081\pi$
$401$	1.74580e11	0.337166	0.168583	+	0.985687i	$0.446081\pi$
$402$	0	0
$403$	5.57243e10	0.105238
$404$	1.98089e11	0.369950
$405$	0	0
$406$	2.21113e11	0.403875
$407$	−4.28674e11	−0.774377
$408$	0	0
$409$	1.88605e11	0.333271	0.166636	−	0.986019i	$-0.446710\pi$
$409$	1.88605e11	0.333271	0.166636	+	0.986019i	$0.446710\pi$
$410$	−7.98967e11	−1.39637
$411$	0	0
$412$	1.67900e11	0.287087
$413$	5.05977e11	0.855767
$414$	0	0
$415$	6.61309e11	1.09443
$416$	1.64068e11	0.268598
$417$	0	0
$418$	1.17387e12	1.88073
$419$	4.12173e11	0.653305	0.326653	−	0.945144i	$-0.394079\pi$
$419$	4.12173e11	0.653305	0.326653	+	0.945144i	$0.394079\pi$
$420$	0	0
$421$	9.50502e11	1.47463	0.737316	−	0.675548i	$-0.236093\pi$
$421$	9.50502e11	1.47463	0.737316	+	0.675548i	$0.236093\pi$
$422$	−1.36035e11	−0.208807
$423$	0	0
$424$	1.67970e8	0.000252398 0
$425$	1.20615e12	1.79329
$426$	0	0
$427$	8.33215e10	0.121292
$428$	−1.25117e11	−0.180227
$429$	0	0
$430$	1.29920e11	0.183260
$431$	−3.36328e11	−0.469478	−0.234739	−	0.972058i	$-0.575424\pi$
$431$	−3.36328e11	−0.469478	−0.234739	+	0.972058i	$0.575424\pi$
$432$	0	0
$433$	1.15663e12	1.58124	0.790620	−	0.612307i	$-0.209758\pi$
$433$	1.15663e12	1.58124	0.790620	+	0.612307i	$0.209758\pi$
$434$	1.99230e10	0.0269557
$435$	0	0
$436$	−3.98948e11	−0.528721
$437$	8.56442e10	0.112339
$438$	0	0
$439$	−5.91073e11	−0.759540	−0.379770	−	0.925081i	$-0.623997\pi$
$439$	−5.91073e11	−0.759540	−0.379770	+	0.925081i	$0.623997\pi$
$440$	7.51584e11	0.955962
$441$	0	0
$442$	1.51876e12	1.89273
$443$	6.18685e11	0.763226	0.381613	−	0.924322i	$-0.375369\pi$
$443$	6.18685e11	0.763226	0.381613	+	0.924322i	$0.375369\pi$
$444$	0	0
$445$	9.40182e11	1.13656
$446$	−1.02926e12	−1.23174
$447$	0	0
$448$	5.86588e10	0.0687989
$449$	8.68676e11	1.00867	0.504335	−	0.863508i	$-0.331738\pi$
$449$	8.68676e11	1.00867	0.504335	+	0.863508i	$0.331738\pi$
$450$	0	0
$451$	−2.32481e12	−2.64602
$452$	−1.20718e11	−0.136035
$453$	0	0
$454$	2.47640e11	0.273571
$455$	−1.08607e12	−1.18797
$456$	0	0
$457$	4.76007e11	0.510494	0.255247	−	0.966876i	$-0.417843\pi$
$457$	4.76007e11	0.510494	0.255247	+	0.966876i	$0.417843\pi$
$458$	−8.88723e11	−0.943782
$459$	0	0
$460$	5.48348e10	0.0571014
$461$	−2.88560e11	−0.297565	−0.148782	−	0.988870i	$-0.547535\pi$
$461$	−2.88560e11	−0.297565	−0.148782	+	0.988870i	$0.547535\pi$
$462$	0	0
$463$	−4.69763e11	−0.475077	−0.237539	−	0.971378i	$-0.576341\pi$
$463$	−4.69763e11	−0.475077	−0.237539	+	0.971378i	$0.576341\pi$
$464$	−2.59037e11	−0.259436
$465$	0	0
$466$	−1.19247e12	−1.17142
$467$	−1.29156e12	−1.25658	−0.628289	−	0.777980i	$-0.716245\pi$
$467$	−1.29156e12	−1.25658	−0.628289	+	0.777980i	$0.716245\pi$
$468$	0	0
$469$	−4.52166e11	−0.431539
$470$	−1.43672e12	−1.35810
$471$	0	0
$472$	−5.92759e11	−0.549717
$473$	3.78038e11	0.347264
$474$	0	0
$475$	1.57818e12	1.42244
$476$	5.42999e11	0.484805
$477$	0	0
$478$	−5.31838e11	−0.465965
$479$	3.29340e11	0.285848	0.142924	−	0.989734i	$-0.454350\pi$
$479$	3.29340e11	0.285848	0.142924	+	0.989734i	$0.454350\pi$
$480$	0	0
$481$	−7.25692e11	−0.618158
$482$	5.36209e11	0.452504
$483$	0	0
$484$	1.58330e12	1.31147
$485$	−3.34840e12	−2.74789
$486$	0	0
$487$	3.91063e11	0.315040	0.157520	−	0.987516i	$-0.449650\pi$
$487$	3.91063e11	0.315040	0.157520	+	0.987516i	$0.449650\pi$
$488$	−9.76121e10	−0.0779139
$489$	0	0
$490$	8.93507e11	0.700189
$491$	1.65436e12	1.28458	0.642292	−	0.766460i	$-0.277984\pi$
$491$	1.65436e12	1.28458	0.642292	+	0.766460i	$0.277984\pi$
$492$	0	0
$493$	−2.39788e12	−1.82817
$494$	1.98721e12	1.50132
$495$	0	0
$496$	−2.33400e10	−0.0173154
$497$	1.02264e12	0.751831
$498$	0	0
$499$	−2.11182e12	−1.52477	−0.762386	−	0.647123i	$-0.775972\pi$
$499$	−2.11182e12	−1.52477	−0.762386	+	0.647123i	$0.775972\pi$
$500$	1.78131e10	0.0127460
$501$	0	0
$502$	2.13765e11	0.150234
$503$	−1.34067e12	−0.933829	−0.466915	−	0.884302i	$-0.654634\pi$
$503$	−1.34067e12	−0.933829	−0.466915	+	0.884302i	$0.654634\pi$
$504$	0	0
$505$	1.53617e12	1.05106
$506$	1.59557e11	0.108203
$507$	0	0
$508$	5.71888e11	0.380999
$509$	−2.19594e12	−1.45007	−0.725037	−	0.688710i	$-0.758177\pi$
$509$	−2.19594e12	−1.45007	−0.725037	+	0.688710i	$0.758177\pi$
$510$	0	0
$511$	5.08452e11	0.329880
$512$	−6.87195e10	−0.0441942
$513$	0	0
$514$	1.99464e11	0.126047
$515$	1.30206e12	0.815641
$516$	0	0
$517$	−4.18053e12	−2.57350
$518$	−2.59455e11	−0.158335
$519$	0	0
$520$	1.27234e12	0.763112
$521$	−1.88176e12	−1.11891	−0.559454	−	0.828862i	$-0.688989\pi$
$521$	−1.88176e12	−1.11891	−0.559454	+	0.828862i	$0.688989\pi$
$522$	0	0
$523$	2.34521e12	1.37064	0.685322	−	0.728240i	$-0.259662\pi$
$523$	2.34521e12	1.37064	0.685322	+	0.728240i	$0.259662\pi$
$524$	−1.61644e11	−0.0936634
$525$	0	0
$526$	−1.21309e12	−0.690965
$527$	−2.16056e11	−0.122017
$528$	0	0
$529$	−1.78951e12	−0.993537
$530$	1.30260e9	0.000717084 0
$531$	0	0
$532$	7.10483e11	0.384549
$533$	−3.93562e12	−2.11223
$534$	0	0
$535$	−9.70280e11	−0.512041
$536$	5.29718e11	0.277206
$537$	0	0
$538$	1.58368e12	0.814979
$539$	2.59990e12	1.32680
$540$	0	0
$541$	−1.50433e12	−0.755015	−0.377507	−	0.926007i	$-0.623219\pi$
$541$	−1.50433e12	−0.755015	−0.377507	+	0.926007i	$0.623219\pi$
$542$	−1.00576e12	−0.500609
$543$	0	0
$544$	−6.36130e11	−0.311423
$545$	−3.09382e12	−1.50214
$546$	0	0
$547$	−8.62929e11	−0.412128	−0.206064	−	0.978539i	$-0.566065\pi$
$547$	−8.62929e11	−0.412128	−0.206064	+	0.978539i	$0.566065\pi$
$548$	9.97912e11	0.472694
$549$	0	0
$550$	2.94017e12	1.37006
$551$	−3.13749e12	−1.45011
$552$	0	0
$553$	−3.18834e11	−0.144978
$554$	−2.00004e12	−0.902080
$555$	0	0
$556$	1.21348e12	0.538511
$557$	−8.24319e11	−0.362866	−0.181433	−	0.983403i	$-0.558074\pi$
$557$	−8.24319e11	−0.362866	−0.181433	+	0.983403i	$0.558074\pi$
$558$	0	0
$559$	6.39971e11	0.277209
$560$	4.54896e11	0.195464
$561$	0	0
$562$	−1.68981e11	−0.0714537
$563$	4.02214e12	1.68721	0.843606	−	0.536963i	$-0.180428\pi$
$563$	4.02214e12	1.68721	0.843606	+	0.536963i	$0.180428\pi$
$564$	0	0
$565$	−9.36165e11	−0.386487
$566$	1.96120e12	0.803243
$567$	0	0
$568$	−1.19804e12	−0.482951
$569$	−3.67705e12	−1.47060	−0.735300	−	0.677741i	$-0.762959\pi$
$569$	−3.67705e12	−1.47060	−0.735300	+	0.677741i	$0.762959\pi$
$570$	0	0
$571$	2.74332e12	1.07998	0.539988	−	0.841673i	$-0.318429\pi$
$571$	2.74332e12	1.07998	0.539988	+	0.841673i	$0.318429\pi$
$572$	3.70221e12	1.44604
$573$	0	0
$574$	−1.40709e12	−0.541027
$575$	2.14512e11	0.0818363
$576$	0	0
$577$	−3.52716e12	−1.32475	−0.662375	−	0.749172i	$-0.730452\pi$
$577$	−3.52716e12	−1.32475	−0.662375	+	0.749172i	$0.730452\pi$
$578$	−3.99119e12	−1.48740
$579$	0	0
$580$	−2.00882e12	−0.737081
$581$	1.16466e12	0.424038
$582$	0	0
$583$	3.79027e9	0.00135882
$584$	−5.95658e11	−0.211904
$585$	0	0
$586$	1.41309e12	0.495030
$587$	−4.41152e11	−0.153362	−0.0766808	−	0.997056i	$-0.524432\pi$
$587$	−4.41152e11	−0.153362	−0.0766808	+	0.997056i	$0.524432\pi$
$588$	0	0
$589$	−2.82697e11	−0.0967839
$590$	−4.59682e12	−1.56179
$591$	0	0
$592$	3.03955e11	0.101709
$593$	2.03482e11	0.0675739	0.0337870	−	0.999429i	$-0.489243\pi$
$593$	2.03482e11	0.0675739	0.0337870	+	0.999429i	$0.489243\pi$
$594$	0	0
$595$	4.21094e12	1.37738
$596$	2.10559e11	0.0683543
$597$	0	0
$598$	2.70110e11	0.0863744
$599$	−2.44246e12	−0.775189	−0.387594	−	0.921830i	$-0.626694\pi$
$599$	−2.44246e12	−0.775189	−0.387594	+	0.921830i	$0.626694\pi$
$600$	0	0
$601$	3.51374e12	1.09859	0.549293	−	0.835630i	$-0.314897\pi$
$601$	3.51374e12	1.09859	0.549293	+	0.835630i	$0.314897\pi$
$602$	2.28807e11	0.0710045
$603$	0	0
$604$	1.61567e9	0.000493954 0
$605$	1.22785e13	3.72601
$606$	0	0
$607$	−5.35941e12	−1.60239	−0.801194	−	0.598404i	$-0.795802\pi$
$607$	−5.35941e12	−1.60239	−0.801194	+	0.598404i	$0.795802\pi$
$608$	−8.32340e11	−0.247022
$609$	0	0
$610$	−7.56979e11	−0.221360
$611$	−7.07712e12	−2.05433
$612$	0	0
$613$	−1.14841e12	−0.328491	−0.164245	−	0.986420i	$-0.552519\pi$
$613$	−1.14841e12	−0.328491	−0.164245	+	0.986420i	$0.552519\pi$
$614$	−6.20089e11	−0.176075
$615$	0	0
$616$	1.32364e12	0.370389
$617$	−1.00275e12	−0.278554	−0.139277	−	0.990253i	$-0.544478\pi$
$617$	−1.00275e12	−0.278554	−0.139277	+	0.990253i	$0.544478\pi$
$618$	0	0
$619$	2.16456e10	0.00592599	0.00296300	−	0.999996i	$-0.499057\pi$
$619$	2.16456e10	0.00592599	0.00296300	+	0.999996i	$0.499057\pi$
$620$	−1.81001e11	−0.0491947
$621$	0	0
$622$	−3.48728e12	−0.934178
$623$	1.65579e12	0.440361
$624$	0	0
$625$	−3.74501e12	−0.981733
$626$	1.15571e12	0.300791
$627$	0	0
$628$	1.24048e12	0.318253
$629$	2.81368e12	0.716716
$630$	0	0
$631$	4.01608e12	1.00849	0.504244	−	0.863561i	$-0.331771\pi$
$631$	4.01608e12	1.00849	0.504244	+	0.863561i	$0.331771\pi$
$632$	3.73518e11	0.0931289
$633$	0	0
$634$	4.42879e12	1.08864
$635$	4.43497e12	1.08245
$636$	0	0
$637$	4.40131e12	1.05914
$638$	−5.84520e12	−1.39671
$639$	0	0
$640$	−5.32917e11	−0.125560
$641$	−8.03352e12	−1.87951	−0.939755	−	0.341849i	$-0.888947\pi$
$641$	−8.03352e12	−1.87951	−0.939755	+	0.341849i	$0.888947\pi$
$642$	0	0
$643$	6.81517e12	1.57227	0.786135	−	0.618055i	$-0.212079\pi$
$643$	6.81517e12	1.57227	0.786135	+	0.618055i	$0.212079\pi$
$644$	9.65717e10	0.0221240
$645$	0	0
$646$	−7.70490e12	−1.74069
$647$	−1.78611e12	−0.400718	−0.200359	−	0.979723i	$-0.564211\pi$
$647$	−1.78611e12	−0.400718	−0.200359	+	0.979723i	$0.564211\pi$
$648$	0	0
$649$	−1.33757e13	−2.95948
$650$	4.97735e12	1.09367
$651$	0	0
$652$	−6.34100e11	−0.137418
$653$	4.20558e11	0.0905142	0.0452571	−	0.998975i	$-0.485589\pi$
$653$	4.20558e11	0.0905142	0.0452571	+	0.998975i	$0.485589\pi$
$654$	0	0
$655$	−1.25355e12	−0.266106
$656$	1.64843e12	0.347538
$657$	0	0
$658$	−2.53026e12	−0.526198
$659$	2.55219e12	0.527144	0.263572	−	0.964640i	$-0.415099\pi$
$659$	2.55219e12	0.527144	0.263572	+	0.964640i	$0.415099\pi$
$660$	0	0
$661$	6.36695e12	1.29725	0.648627	−	0.761106i	$-0.275343\pi$
$661$	6.36695e12	1.29725	0.648627	+	0.761106i	$0.275343\pi$
$662$	2.90623e12	0.588124
$663$	0	0
$664$	−1.36441e12	−0.272388
$665$	5.50977e12	1.09254
$666$	0	0
$667$	−4.26460e11	−0.0834281
$668$	3.08157e12	0.598795
$669$	0	0
$670$	4.10794e12	0.787567
$671$	−2.20263e12	−0.419460
$672$	0	0
$673$	1.69690e12	0.318852	0.159426	−	0.987210i	$-0.449036\pi$
$673$	1.69690e12	0.318852	0.159426	+	0.987210i	$0.449036\pi$
$674$	9.92024e11	0.185162
$675$	0	0
$676$	3.55264e12	0.654321
$677$	2.51339e12	0.459845	0.229922	−	0.973209i	$-0.426153\pi$
$677$	2.51339e12	0.459845	0.229922	+	0.973209i	$0.426153\pi$
$678$	0	0
$679$	−5.89700e12	−1.06467
$680$	−4.93316e12	−0.884780
$681$	0	0
$682$	−5.26671e11	−0.0932201
$683$	9.57413e12	1.68347	0.841737	−	0.539888i	$-0.181533\pi$
$683$	9.57413e12	1.68347	0.841737	+	0.539888i	$0.181533\pi$
$684$	0	0
$685$	7.73877e12	1.34296
$686$	3.83102e12	0.660475
$687$	0	0
$688$	−2.68051e11	−0.0456109
$689$	6.41646e9	0.00108470
$690$	0	0
$691$	−6.38711e12	−1.06574	−0.532872	−	0.846196i	$-0.678887\pi$
$691$	−6.38711e12	−1.06574	−0.532872	+	0.846196i	$0.678887\pi$
$692$	1.42324e12	0.235940
$693$	0	0
$694$	4.08805e12	0.668957
$695$	9.41046e12	1.52996
$696$	0	0
$697$	1.52593e13	2.44899
$698$	3.92388e12	0.625700
$699$	0	0
$700$	1.77954e12	0.280134
$701$	7.78314e12	1.21737	0.608687	−	0.793411i	$-0.291697\pi$
$701$	7.78314e12	1.21737	0.608687	+	0.793411i	$0.291697\pi$
$702$	0	0
$703$	3.68154e12	0.568501
$704$	−1.55067e12	−0.237926
$705$	0	0
$706$	−4.42416e12	−0.670208
$707$	2.70541e12	0.407235
$708$	0	0
$709$	−2.82895e12	−0.420453	−0.210226	−	0.977653i	$-0.567420\pi$
$709$	−2.82895e12	−0.420453	−0.210226	+	0.977653i	$0.567420\pi$
$710$	−9.29075e12	−1.37211
$711$	0	0
$712$	−1.93978e12	−0.282874
$713$	−3.84254e10	−0.00556820
$714$	0	0
$715$	2.87105e13	4.10832
$716$	3.59830e12	0.511669
$717$	0	0
$718$	9.50272e12	1.33441
$719$	−1.31870e12	−0.184021	−0.0920103	−	0.995758i	$-0.529329\pi$
$719$	−1.31870e12	−0.184021	−0.0920103	+	0.995758i	$0.529329\pi$
$720$	0	0
$721$	2.29311e12	0.316021
$722$	−4.91842e12	−0.673610
$723$	0	0
$724$	−2.13272e12	−0.288476
$725$	−7.85843e12	−1.05637
$726$	0	0
$727$	6.88647e12	0.914306	0.457153	−	0.889388i	$-0.348869\pi$
$727$	6.88647e12	0.914306	0.457153	+	0.889388i	$0.348869\pi$
$728$	2.24077e12	0.295669
$729$	0	0
$730$	−4.61931e12	−0.602038
$731$	−2.48132e12	−0.321407
$732$	0	0
$733$	−1.09729e13	−1.40396	−0.701979	−	0.712198i	$-0.747700\pi$
$733$	−1.09729e13	−1.40396	−0.701979	+	0.712198i	$0.747700\pi$
$734$	2.78711e12	0.354423
$735$	0	0
$736$	−1.13135e11	−0.0142117
$737$	1.19532e13	1.49238
$738$	0	0
$739$	−7.00379e12	−0.863839	−0.431920	−	0.901912i	$-0.642164\pi$
$739$	−7.00379e12	−0.863839	−0.431920	+	0.901912i	$0.642164\pi$
$740$	2.35716e12	0.288965
$741$	0	0
$742$	2.29406e9	0.000277835 0
$743$	1.14605e13	1.37960	0.689799	−	0.724001i	$-0.257699\pi$
$743$	1.14605e13	1.37960	0.689799	+	0.724001i	$0.257699\pi$
$744$	0	0
$745$	1.63288e12	0.194201
$746$	7.92342e11	0.0936673
$747$	0	0
$748$	−1.43544e13	−1.67659
$749$	−1.70880e12	−0.198391
$750$	0	0
$751$	7.25973e11	0.0832800	0.0416400	−	0.999133i	$-0.486742\pi$
$751$	7.25973e11	0.0832800	0.0416400	+	0.999133i	$0.486742\pi$
$752$	2.96424e12	0.338012
$753$	0	0
$754$	−9.89521e12	−1.11495
$755$	1.25295e10	0.00140337
$756$	0	0
$757$	−3.91708e12	−0.433542	−0.216771	−	0.976222i	$-0.569553\pi$
$757$	−3.91708e12	−0.433542	−0.216771	+	0.976222i	$0.569553\pi$
$758$	1.05894e13	1.16509
$759$	0	0
$760$	−6.45477e12	−0.701810
$761$	−1.62290e13	−1.75413	−0.877064	−	0.480374i	$-0.840501\pi$
$761$	−1.62290e13	−1.75413	−0.877064	+	0.480374i	$0.840501\pi$
$762$	0	0
$763$	−5.44865e12	−0.582007
$764$	−4.61778e12	−0.490358
$765$	0	0
$766$	−3.86191e12	−0.405296
$767$	−2.26434e13	−2.36245
$768$	0	0
$769$	−6.91300e12	−0.712850	−0.356425	−	0.934324i	$-0.616004\pi$
$769$	−6.91300e12	−0.712850	−0.356425	+	0.934324i	$0.616004\pi$
$770$	1.02648e13	1.05231
$771$	0	0
$772$	−6.88252e12	−0.697380
$773$	1.72391e11	0.0173663	0.00868315	−	0.999962i	$-0.497236\pi$
$773$	1.72391e11	0.0173663	0.00868315	+	0.999962i	$0.497236\pi$
$774$	0	0
$775$	−7.08069e11	−0.0705046
$776$	6.90841e12	0.683912
$777$	0	0
$778$	−9.84409e12	−0.963313
$779$	1.99660e13	1.94255
$780$	0	0
$781$	−2.70339e13	−2.60004
$782$	−1.04728e12	−0.100146
$783$	0	0
$784$	−1.84348e12	−0.174267
$785$	9.61989e12	0.904184
$786$	0	0
$787$	9.84010e12	0.914351	0.457176	−	0.889376i	$-0.348861\pi$
$787$	9.84010e12	0.914351	0.457176	+	0.889376i	$0.348861\pi$
$788$	−1.53981e12	−0.142265
$789$	0	0
$790$	2.89662e12	0.264587
$791$	−1.64872e12	−0.149745
$792$	0	0
$793$	−3.72879e12	−0.334841
$794$	1.08484e13	0.968660
$795$	0	0
$796$	1.03956e13	0.917781
$797$	2.06457e12	0.181245	0.0906226	−	0.995885i	$-0.471114\pi$
$797$	2.06457e12	0.181245	0.0906226	+	0.995885i	$0.471114\pi$
$798$	0	0
$799$	2.74397e13	2.38187
$800$	−2.08475e12	−0.179949
$801$	0	0
$802$	−2.79328e12	−0.238413
$803$	−1.34411e13	−1.14082
$804$	0	0
$805$	7.48910e11	0.0628562
$806$	−8.91588e11	−0.0744143
$807$	0	0
$808$	−3.16942e12	−0.261594
$809$	−1.49430e13	−1.22651	−0.613254	−	0.789886i	$-0.710140\pi$
$809$	−1.49430e13	−1.22651	−0.613254	+	0.789886i	$0.710140\pi$
$810$	0	0
$811$	−8.17959e12	−0.663954	−0.331977	−	0.943288i	$-0.607716\pi$
$811$	−8.17959e12	−0.663954	−0.331977	+	0.943288i	$0.607716\pi$
$812$	−3.53781e12	−0.285583
$813$	0	0
$814$	6.85878e12	0.547567
$815$	−4.91742e12	−0.390417
$816$	0	0
$817$	−3.24667e12	−0.254941
$818$	−3.01768e12	−0.235658
$819$	0	0
$820$	1.27835e13	0.987385
$821$	−4.70391e12	−0.361339	−0.180669	−	0.983544i	$-0.557826\pi$
$821$	−4.70391e12	−0.361339	−0.180669	+	0.983544i	$0.557826\pi$
$822$	0	0
$823$	−2.42535e13	−1.84279	−0.921393	−	0.388633i	$-0.872947\pi$
$823$	−2.42535e13	−1.84279	−0.921393	+	0.388633i	$0.872947\pi$
$824$	−2.68641e12	−0.203001
$825$	0	0
$826$	−8.09563e12	−0.605119
$827$	6.61385e11	0.0491676	0.0245838	−	0.999698i	$-0.492174\pi$
$827$	6.61385e11	0.0491676	0.0245838	+	0.999698i	$0.492174\pi$
$828$	0	0
$829$	1.49619e13	1.10025	0.550126	−	0.835081i	$-0.314580\pi$
$829$	1.49619e13	1.10025	0.550126	+	0.835081i	$0.314580\pi$
$830$	−1.05809e13	−0.773878
$831$	0	0
$832$	−2.62508e12	−0.189928
$833$	−1.70649e13	−1.22801
$834$	0	0
$835$	2.38975e13	1.70123
$836$	−1.87819e13	−1.32988
$837$	0	0
$838$	−6.59476e12	−0.461956
$839$	−1.06092e13	−0.739187	−0.369593	−	0.929194i	$-0.620503\pi$
$839$	−1.06092e13	−0.739187	−0.369593	+	0.929194i	$0.620503\pi$
$840$	0	0
$841$	1.11580e12	0.0769135
$842$	−1.52080e13	−1.04272
$843$	0	0
$844$	2.17656e12	0.147649
$845$	2.75506e13	1.85898
$846$	0	0
$847$	2.16240e13	1.44365
$848$	−2.68752e9	−0.000178472 0
$849$	0	0
$850$	−1.92984e13	−1.26805
$851$	5.00410e11	0.0327072
$852$	0	0
$853$	−1.04916e13	−0.678533	−0.339266	−	0.940690i	$-0.610179\pi$
$853$	−1.04916e13	−0.678533	−0.339266	+	0.940690i	$0.610179\pi$
$854$	−1.33314e12	−0.0857662
$855$	0	0
$856$	2.00188e12	0.127440
$857$	−7.29064e12	−0.461691	−0.230846	−	0.972990i	$-0.574149\pi$
$857$	−7.29064e12	−0.461691	−0.230846	+	0.972990i	$0.574149\pi$
$858$	0	0
$859$	−7.12984e12	−0.446797	−0.223399	−	0.974727i	$-0.571715\pi$
$859$	−7.12984e12	−0.446797	−0.223399	+	0.974727i	$0.571715\pi$
$860$	−2.07872e12	−0.129585
$861$	0	0
$862$	5.38125e12	0.331971
$863$	6.59591e12	0.404787	0.202393	−	0.979304i	$-0.435128\pi$
$863$	6.59591e12	0.404787	0.202393	+	0.979304i	$0.435128\pi$
$864$	0	0
$865$	1.10372e13	0.670326
$866$	−1.85060e13	−1.11811
$867$	0	0
$868$	−3.18767e11	−0.0190605
$869$	8.42848e12	0.501372
$870$	0	0
$871$	2.02352e13	1.19131
$872$	6.38316e12	0.373862
$873$	0	0
$874$	−1.37031e12	−0.0794358
$875$	2.43284e11	0.0140306
$876$	0	0
$877$	−1.74094e13	−0.993771	−0.496885	−	0.867816i	$-0.665523\pi$
$877$	−1.74094e13	−0.993771	−0.496885	+	0.867816i	$0.665523\pi$
$878$	9.45717e12	0.537076
$879$	0	0
$880$	−1.20254e13	−0.675967
$881$	1.81298e13	1.01392	0.506958	−	0.861971i	$-0.330770\pi$
$881$	1.81298e13	1.01392	0.506958	+	0.861971i	$0.330770\pi$
$882$	0	0
$883$	−1.92731e12	−0.106691	−0.0533457	−	0.998576i	$-0.516989\pi$
$883$	−1.92731e12	−0.106691	−0.0533457	+	0.998576i	$0.516989\pi$
$884$	−2.43002e13	−1.33836
$885$	0	0
$886$	−9.89896e12	−0.539682
$887$	9.81475e12	0.532382	0.266191	−	0.963920i	$-0.414235\pi$
$887$	9.81475e12	0.532382	0.266191	+	0.963920i	$0.414235\pi$
$888$	0	0
$889$	7.81059e12	0.419398
$890$	−1.50429e13	−0.803669
$891$	0	0
$892$	1.64682e13	0.870971
$893$	3.59033e13	1.88931
$894$	0	0
$895$	2.79047e13	1.45370
$896$	−9.38540e11	−0.0486482
$897$	0	0
$898$	−1.38988e13	−0.713237
$899$	1.40767e12	0.0718760
$900$	0	0
$901$	−2.48781e10	−0.00125764
$902$	3.71970e13	1.87102
$903$	0	0
$904$	1.93149e12	0.0961911
$905$	−1.65391e13	−0.819585
$906$	0	0
$907$	6.78766e12	0.333033	0.166516	−	0.986039i	$-0.446748\pi$
$907$	6.78766e12	0.333033	0.166516	+	0.986039i	$0.446748\pi$
$908$	−3.96224e12	−0.193444
$909$	0	0
$910$	1.73771e13	0.840020
$911$	−2.98654e13	−1.43660	−0.718300	−	0.695734i	$-0.755079\pi$
$911$	−2.98654e13	−1.43660	−0.718300	+	0.695734i	$0.755079\pi$
$912$	0	0
$913$	−3.07881e13	−1.46644
$914$	−7.61612e12	−0.360974
$915$	0	0
$916$	1.42196e13	0.667355
$917$	−2.20767e12	−0.103103
$918$	0	0
$919$	3.83406e13	1.77312	0.886562	−	0.462609i	$-0.153087\pi$
$919$	3.83406e13	1.77312	0.886562	+	0.462609i	$0.153087\pi$
$920$	−8.77358e11	−0.0403768
$921$	0	0
$922$	4.61696e12	0.210410
$923$	−4.57651e13	−2.07552
$924$	0	0
$925$	9.22111e12	0.414138
$926$	7.51621e12	0.335931
$927$	0	0
$928$	4.14459e12	0.183449
$929$	1.75006e13	0.770873	0.385437	−	0.922734i	$-0.374051\pi$
$929$	1.75006e13	0.770873	0.385437	+	0.922734i	$0.374051\pi$
$930$	0	0
$931$	−2.23285e13	−0.974060
$932$	1.90796e13	0.828317
$933$	0	0
$934$	2.06650e13	0.888535
$935$	−1.11318e14	−4.76334
$936$	0	0
$937$	−5.75033e12	−0.243705	−0.121853	−	0.992548i	$-0.538883\pi$
$937$	−5.75033e12	−0.243705	−0.121853	+	0.992548i	$0.538883\pi$
$938$	7.23465e12	0.305144
$939$	0	0
$940$	2.29875e13	0.960323
$941$	−3.85054e13	−1.60092	−0.800458	−	0.599389i	$-0.795410\pi$
$941$	−3.85054e13	−1.60092	−0.800458	+	0.599389i	$0.795410\pi$
$942$	0	0
$943$	2.71385e12	0.111759
$944$	9.48414e12	0.388708
$945$	0	0
$946$	−6.04861e12	−0.245553
$947$	4.22310e13	1.70630	0.853152	−	0.521662i	$-0.174688\pi$
$947$	4.22310e13	1.70630	0.853152	+	0.521662i	$0.174688\pi$
$948$	0	0
$949$	−2.27541e13	−0.910673
$950$	−2.52508e13	−1.00582
$951$	0	0
$952$	−8.68798e12	−0.342809
$953$	−4.92293e12	−0.193333	−0.0966664	−	0.995317i	$-0.530818\pi$
$953$	−4.92293e12	−0.193333	−0.0966664	+	0.995317i	$0.530818\pi$
$954$	0	0
$955$	−3.58107e13	−1.39315
$956$	8.50941e12	0.329487
$957$	0	0
$958$	−5.26944e12	−0.202125
$959$	1.36290e13	0.520333
$960$	0	0
$961$	−2.63128e13	−0.995203
$962$	1.16111e13	0.437104
$963$	0	0
$964$	−8.57934e12	−0.319969
$965$	−5.33737e13	−1.98132
$966$	0	0
$967$	−1.61089e13	−0.592445	−0.296222	−	0.955119i	$-0.595727\pi$
$967$	−1.61089e13	−0.592445	−0.296222	+	0.955119i	$0.595727\pi$
$968$	−2.53328e13	−0.927352
$969$	0	0
$970$	5.35745e13	1.94305
$971$	−2.15164e13	−0.776754	−0.388377	−	0.921501i	$-0.626964\pi$
$971$	−2.15164e13	−0.776754	−0.388377	+	0.921501i	$0.626964\pi$
$972$	0	0
$973$	1.65731e13	0.592784
$974$	−6.25700e12	−0.222767
$975$	0	0
$976$	1.56179e12	0.0550934
$977$	2.35966e13	0.828560	0.414280	−	0.910149i	$-0.364033\pi$
$977$	2.35966e13	0.828560	0.414280	+	0.910149i	$0.364033\pi$
$978$	0	0
$979$	−4.37714e13	−1.52289
$980$	−1.42961e13	−0.495109
$981$	0	0
$982$	−2.64697e13	−0.908338
$983$	−4.07118e13	−1.39069	−0.695343	−	0.718678i	$-0.744748\pi$
$983$	−4.07118e13	−1.39069	−0.695343	+	0.718678i	$0.744748\pi$
$984$	0	0
$985$	−1.19411e13	−0.404188
$986$	3.83661e13	1.29271
$987$	0	0
$988$	−3.17954e13	−1.06159
$989$	−4.41300e11	−0.0146673
$990$	0	0
$991$	2.38946e13	0.786988	0.393494	−	0.919327i	$-0.371266\pi$
$991$	2.38946e13	0.786988	0.393494	+	0.919327i	$0.371266\pi$
$992$	3.73440e11	0.0122439
$993$	0	0
$994$	−1.63623e13	−0.531625
$995$	8.06172e13	2.60750
$996$	0	0
$997$	1.97857e13	0.634195	0.317098	−	0.948393i	$-0.397292\pi$
$997$	1.97857e13	0.634195	0.317098	+	0.948393i	$0.397292\pi$
$998$	3.37891e13	1.07818
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.10.a.f.1.4	✓	4
3.2	odd	2		162.10.a.g.1.1	yes	4
9.2	odd	6		162.10.c.s.109.4		8
9.4	even	3		162.10.c.t.55.1		8
9.5	odd	6		162.10.c.s.55.4		8
9.7	even	3		162.10.c.t.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.10.a.f.1.4	✓	4	1.1	even	1	trivial
162.10.a.g.1.1	yes	4	3.2	odd	2
162.10.c.s.55.4		8	9.5	odd	6
162.10.c.s.109.4		8	9.2	odd	6
162.10.c.t.55.1		8	9.4	even	3
162.10.c.t.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0000 q^{2} +256.000 q^{4} +1985.27 q^{5} +3496.33 q^{7} -4096.00 q^{8} -31764.3 q^{10} -92426.8 q^{11} -156467. q^{13} -55941.3 q^{14} +65536.0 q^{16} +606661. q^{17} +793782. q^{19} +508229. q^{20} +1.47883e6 q^{22} +107894. q^{23} +1.98817e6 q^{25} +2.50348e6 q^{26} +895062. q^{28} -3.95259e6 q^{29} -356140. q^{31} -1.04858e6 q^{32} -9.70657e6 q^{34} +6.94117e6 q^{35} +4.63798e6 q^{37} -1.27005e7 q^{38} -8.13167e6 q^{40} +2.51530e7 q^{41} -4.09013e6 q^{43} -2.36613e7 q^{44} -1.72630e6 q^{46} +4.52307e7 q^{47} -2.81293e7 q^{49} -3.18108e7 q^{50} -4.00556e7 q^{52} -41008.3 q^{53} -1.83492e8 q^{55} -1.43210e7 q^{56} +6.32414e7 q^{58} +1.44716e8 q^{59} +2.38311e7 q^{61} +5.69824e6 q^{62} +1.67772e7 q^{64} -3.10630e8 q^{65} -1.29326e8 q^{67} +1.55305e8 q^{68} -1.11059e8 q^{70} +2.92490e8 q^{71} +1.45424e8 q^{73} -7.42077e7 q^{74} +2.03208e8 q^{76} -3.23155e8 q^{77} -9.11909e7 q^{79} +1.30107e8 q^{80} -4.02448e8 q^{82} +3.33108e8 q^{83} +1.20439e9 q^{85} +6.54421e7 q^{86} +3.78580e8 q^{88} +4.73579e8 q^{89} -5.47062e8 q^{91} +2.76208e7 q^{92} -7.23691e8 q^{94} +1.57587e9 q^{95} -1.68662e9 q^{97} +4.50068e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{2} + 1024 q^{4} + 1968 q^{5} + 4496 q^{7} - 16384 q^{8} - 31488 q^{10} - 8784 q^{11} - 162556 q^{13} - 71936 q^{14} + 262144 q^{16} + 538080 q^{17} - 8224 q^{19} + 503808 q^{20} + 140544 q^{22}+ \cdots + 2038788288 q^{98}+O(q^{100}) \) 4 * q - 64 * q^2 + 1024 * q^4 + 1968 * q^5 + 4496 * q^7 - 16384 * q^8 - 31488 * q^10 - 8784 * q^11 - 162556 * q^13 - 71936 * q^14 + 262144 * q^16 + 538080 * q^17 - 8224 * q^19 + 503808 * q^20 + 140544 * q^22 + 2594736 * q^23 + 1873192 * q^25 + 2600896 * q^26 + 1150976 * q^28 + 3232656 * q^29 + 3482576 * q^31 - 4194304 * q^32 - 8609280 * q^34 + 7969296 * q^35 - 2487892 * q^37 + 131584 * q^38 - 8060928 * q^40 + 34657152 * q^41 - 41410000 * q^43 - 2248704 * q^44 - 41515776 * q^46 + 40558848 * q^47 - 127424268 * q^49 - 29971072 * q^50 - 41614336 * q^52 + 8421024 * q^53 - 229068000 * q^55 - 18415616 * q^56 - 51722496 * q^58 + 26843328 * q^59 - 111504484 * q^61 - 55721216 * q^62 + 67108864 * q^64 + 23507616 * q^65 - 208064512 * q^67 + 137748480 * q^68 - 127508736 * q^70 + 356008272 * q^71 - 37126108 * q^73 + 39806272 * q^74 - 2105344 * q^76 + 33010560 * q^77 - 645134848 * q^79 + 128974848 * q^80 - 554514432 * q^82 + 1019036256 * q^83 + 80955540 * q^85 + 662560000 * q^86 + 35979264 * q^88 + 1548192768 * q^89 - 130776608 * q^91 + 664252416 * q^92 - 648941568 * q^94 + 3445933872 * q^95 + 1112014568 * q^97 + 2038788288 * q^98

Introduction

L-functions

Modular forms

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Database

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