LMFDB - Embedded newform 81.12.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,12,Mod(1,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	81.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.2357976253$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 14790 x^{8} + 93060 x^{7} + 72223254 x^{6} - 592709562 x^{5} - 132941711592 x^{4} + \cdots - 18\!\cdots\!74 $ x^10 - 3x^9 - 14790x^8 + 93060x^7 + 72223254x^6 - 592709562x^5 - 132941711592x^4 + 1226670836724x^3 + 71629903044513x^2 - 424311619946427*x - 1872029164523274
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{30} $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$35.9210$ of defining polynomial
Character	$\chi$	$=$	81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-38.9210 q^{2} -533.155 q^{4} +3382.94 q^{5} +32029.1 q^{7} +100461. q^{8} -131667. q^{10} -672105. q^{11} -1.69588e6 q^{13} -1.24660e6 q^{14} -2.81815e6 q^{16} +5.41320e6 q^{17} +8.84256e6 q^{19} -1.80363e6 q^{20} +2.61590e7 q^{22} +5.22598e6 q^{23} -3.73839e7 q^{25} +6.60055e7 q^{26} -1.70765e7 q^{28} -7.23255e7 q^{29} +2.06025e8 q^{31} -9.60593e7 q^{32} -2.10687e8 q^{34} +1.08352e8 q^{35} +7.22103e8 q^{37} -3.44161e8 q^{38} +3.39854e8 q^{40} -4.29048e8 q^{41} +1.37098e9 q^{43} +3.58337e8 q^{44} -2.03400e8 q^{46} +7.40317e7 q^{47} -9.51466e8 q^{49} +1.45502e9 q^{50} +9.04170e8 q^{52} -5.54718e9 q^{53} -2.27369e9 q^{55} +3.21768e9 q^{56} +2.81498e9 q^{58} -3.61264e9 q^{59} +6.16072e9 q^{61} -8.01870e9 q^{62} +9.51029e9 q^{64} -5.73707e9 q^{65} -6.23162e9 q^{67} -2.88608e9 q^{68} -4.21718e9 q^{70} +1.73266e10 q^{71} -1.89972e10 q^{73} -2.81050e10 q^{74} -4.71446e9 q^{76} -2.15269e10 q^{77} -3.58485e10 q^{79} -9.53361e9 q^{80} +1.66990e10 q^{82} -2.49712e10 q^{83} +1.83125e10 q^{85} -5.33598e10 q^{86} -6.75205e10 q^{88} -7.94245e10 q^{89} -5.43176e10 q^{91} -2.78626e9 q^{92} -2.88139e9 q^{94} +2.99138e10 q^{95} +6.65970e10 q^{97} +3.70320e10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 33 q^{2} + 9217 q^{4} - 7230 q^{5} - 8512 q^{7} + 14559 q^{8} + 2046 q^{10} - 112776 q^{11} - 279706 q^{13} - 3901584 q^{14} + 7342081 q^{16} - 13882896 q^{17} + 3514700 q^{19} - 34163508 q^{20}+ \cdots - 655061802039 q^{98}+O(q^{100}) $ 10 * q - 33 * q^2 + 9217 * q^4 - 7230 * q^5 - 8512 * q^7 + 14559 * q^8 + 2046 * q^10 - 112776 * q^11 - 279706 * q^13 - 3901584 * q^14 + 7342081 * q^16 - 13882896 * q^17 + 3514700 * q^19 - 34163508 * q^20 - 2274591 * q^22 - 69371616 * q^23 + 45286204 * q^25 - 240964572 * q^26 - 30672898 * q^28 - 25437246 * q^29 - 114575368 * q^31 + 80396559 * q^32 + 243855063 * q^34 + 89073732 * q^35 - 67109164 * q^37 + 489799995 * q^38 - 107425416 * q^40 + 331873026 * q^41 + 1118847584 * q^43 - 139238637 * q^44 + 1441268796 * q^46 - 1469650704 * q^47 + 3553434720 * q^49 - 6643771701 * q^50 - 3632448874 * q^52 - 7457130972 * q^53 + 2490724992 * q^55 - 27669139026 * q^56 + 1387480560 * q^58 - 26505032592 * q^59 - 990409066 * q^61 - 45522498090 * q^62 - 3758354783 * q^64 - 39045315390 * q^65 - 6557215720 * q^67 - 77299152993 * q^68 + 785437278 * q^70 - 61026859872 * q^71 - 6306446968 * q^73 - 109519086216 * q^74 - 14574055597 * q^76 - 88616208018 * q^77 + 7621233248 * q^79 - 199583341536 * q^80 - 29584238667 * q^82 - 99007044180 * q^83 - 12911595156 * q^85 - 214357830519 * q^86 + 54423523605 * q^88 - 238298852412 * q^89 + 69105826108 * q^91 - 461776423998 * q^92 - 13393667064 * q^94 - 418952909328 * q^95 - 123483551938 * q^97 - 655061802039 * 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 $

$2$	−38.9210	−0.860041	−0.430020	−	0.902819i	$-0.641494\pi$
$2$	−38.9210	−0.860041	−0.430020	+	0.902819i	$0.641494\pi$
$3$	0	0
$3$	0	0
$4$	−533.155	−0.260330
$5$	3382.94	0.484126	0.242063	−	0.970260i	$-0.422176\pi$
$5$	3382.94	0.484126	0.242063	+	0.970260i	$0.422176\pi$
$6$	0	0
$7$	32029.1	0.720286	0.360143	−	0.932897i	$-0.382728\pi$
$7$	32029.1	0.720286	0.360143	+	0.932897i	$0.382728\pi$
$8$	100461.	1.08394
$9$	0	0
$10$	−131667.	−0.416368
$11$	−672105.	−1.25828	−0.629140	−	0.777292i	$-0.716593\pi$
$11$	−672105.	−1.25828	−0.629140	+	0.777292i	$0.716593\pi$
$12$	0	0
$13$	−1.69588e6	−1.26680	−0.633400	−	0.773825i	$-0.718341\pi$
$13$	−1.69588e6	−1.26680	−0.633400	+	0.773825i	$0.718341\pi$
$14$	−1.24660e6	−0.619475
$15$	0	0
$16$	−2.81815e6	−0.671899
$17$	5.41320e6	0.924667	0.462333	−	0.886706i	$-0.347012\pi$
$17$	5.41320e6	0.924667	0.462333	+	0.886706i	$0.347012\pi$
$18$	0	0
$19$	8.84256e6	0.819282	0.409641	−	0.912247i	$-0.365654\pi$
$19$	8.84256e6	0.819282	0.409641	+	0.912247i	$0.365654\pi$
$20$	−1.80363e6	−0.126033
$21$	0	0
$22$	2.61590e7	1.08217
$23$	5.22598e6	0.169303	0.0846515	−	0.996411i	$-0.473022\pi$
$23$	5.22598e6	0.169303	0.0846515	+	0.996411i	$0.473022\pi$
$24$	0	0
$25$	−3.73839e7	−0.765622
$26$	6.60055e7	1.08950
$27$	0	0
$28$	−1.70765e7	−0.187512
$29$	−7.23255e7	−0.654791	−0.327395	−	0.944887i	$-0.606171\pi$
$29$	−7.23255e7	−0.654791	−0.327395	+	0.944887i	$0.606171\pi$
$30$	0	0
$31$	2.06025e8	1.29250	0.646250	−	0.763126i	$-0.276336\pi$
$31$	2.06025e8	1.29250	0.646250	+	0.763126i	$0.276336\pi$
$32$	−9.60593e7	−0.506075
$33$	0	0
$34$	−2.10687e8	−0.795251
$35$	1.08352e8	0.348709
$36$	0	0
$37$	7.22103e8	1.71194	0.855972	−	0.517021i	$-0.172959\pi$
$37$	7.22103e8	1.71194	0.855972	+	0.517021i	$0.172959\pi$
$38$	−3.44161e8	−0.704616
$39$	0	0
$40$	3.39854e8	0.524762
$41$	−4.29048e8	−0.578356	−0.289178	−	0.957275i	$-0.593382\pi$
$41$	−4.29048e8	−0.578356	−0.289178	+	0.957275i	$0.593382\pi$
$42$	0	0
$43$	1.37098e9	1.42218	0.711088	−	0.703103i	$-0.248203\pi$
$43$	1.37098e9	1.42218	0.711088	+	0.703103i	$0.248203\pi$
$44$	3.58337e8	0.327568
$45$	0	0
$46$	−2.03400e8	−0.145608
$47$	7.40317e7	0.0470847	0.0235423	−	0.999723i	$-0.492506\pi$
$47$	7.40317e7	0.0470847	0.0235423	+	0.999723i	$0.492506\pi$
$48$	0	0
$49$	−9.51466e8	−0.481188
$50$	1.45502e9	0.658466
$51$	0	0
$52$	9.04170e8	0.329786
$53$	−5.54718e9	−1.82203	−0.911014	−	0.412377i	$-0.864699\pi$
$53$	−5.54718e9	−1.82203	−0.911014	+	0.412377i	$0.864699\pi$
$54$	0	0
$55$	−2.27369e9	−0.609167
$56$	3.21768e9	0.780743
$57$	0	0
$58$	2.81498e9	0.563147
$59$	−3.61264e9	−0.657868	−0.328934	−	0.944353i	$-0.606689\pi$
$59$	−3.61264e9	−0.657868	−0.328934	+	0.944353i	$0.606689\pi$
$60$	0	0
$61$	6.16072e9	0.933936	0.466968	−	0.884274i	$-0.345346\pi$
$61$	6.16072e9	0.933936	0.466968	+	0.884274i	$0.345346\pi$
$62$	−8.01870e9	−1.11160
$63$	0	0
$64$	9.51029e9	1.10714
$65$	−5.73707e9	−0.613291
$66$	0	0
$67$	−6.23162e9	−0.563884	−0.281942	−	0.959432i	$-0.590978\pi$
$67$	−6.23162e9	−0.563884	−0.281942	+	0.959432i	$0.590978\pi$
$68$	−2.88608e9	−0.240718
$69$	0	0
$70$	−4.21718e9	−0.299904
$71$	1.73266e10	1.13970	0.569852	−	0.821747i	$-0.307001\pi$
$71$	1.73266e10	1.13970	0.569852	+	0.821747i	$0.307001\pi$
$72$	0	0
$73$	−1.89972e10	−1.07254	−0.536270	−	0.844046i	$-0.680167\pi$
$73$	−1.89972e10	−1.07254	−0.536270	+	0.844046i	$0.680167\pi$
$74$	−2.81050e10	−1.47234
$75$	0	0
$76$	−4.71446e9	−0.213283
$77$	−2.15269e10	−0.906322
$78$	0	0
$79$	−3.58485e10	−1.31076	−0.655379	−	0.755300i	$-0.727491\pi$
$79$	−3.58485e10	−1.31076	−0.655379	+	0.755300i	$0.727491\pi$
$80$	−9.53361e9	−0.325284
$81$	0	0
$82$	1.66990e10	0.497410
$83$	−2.49712e10	−0.695842	−0.347921	−	0.937524i	$-0.613112\pi$
$83$	−2.49712e10	−0.695842	−0.347921	+	0.937524i	$0.613112\pi$
$84$	0	0
$85$	1.83125e10	0.447656
$86$	−5.33598e10	−1.22313
$87$	0	0
$88$	−6.75205e10	−1.36389
$89$	−7.94245e10	−1.50768	−0.753841	−	0.657057i	$-0.771801\pi$
$89$	−7.94245e10	−1.50768	−0.753841	+	0.657057i	$0.771801\pi$
$90$	0	0
$91$	−5.43176e10	−0.912458
$92$	−2.78626e9	−0.0440746
$93$	0	0
$94$	−2.88139e9	−0.0404947
$95$	2.99138e10	0.396636
$96$	0	0
$97$	6.65970e10	0.787427	0.393713	−	0.919233i	$-0.371190\pi$
$97$	6.65970e10	0.787427	0.393713	+	0.919233i	$0.371190\pi$
$98$	3.70320e10	0.413841
$99$	0	0
$100$	1.99314e10	0.199314
$101$	−1.32098e11	−1.25063	−0.625314	−	0.780373i	$-0.715029\pi$
$101$	−1.32098e11	−1.25063	−0.625314	+	0.780373i	$0.715029\pi$
$102$	0	0
$103$	−5.75738e10	−0.489351	−0.244675	−	0.969605i	$-0.578681\pi$
$103$	−5.75738e10	−0.489351	−0.244675	+	0.969605i	$0.578681\pi$
$104$	−1.70371e11	−1.37313
$105$	0	0
$106$	2.15902e11	1.56702
$107$	−2.21267e11	−1.52513	−0.762563	−	0.646914i	$-0.776059\pi$
$107$	−2.21267e11	−1.52513	−0.762563	+	0.646914i	$0.776059\pi$
$108$	0	0
$109$	5.26853e10	0.327977	0.163989	−	0.986462i	$-0.447564\pi$
$109$	5.26853e10	0.327977	0.163989	+	0.986462i	$0.447564\pi$
$110$	8.84943e10	0.523908
$111$	0	0
$112$	−9.02626e10	−0.483959
$113$	2.15858e11	1.10214	0.551070	−	0.834459i	$-0.314220\pi$
$113$	2.15858e11	1.10214	0.551070	+	0.834459i	$0.314220\pi$
$114$	0	0
$115$	1.76792e10	0.0819641
$116$	3.85607e10	0.170462
$117$	0	0
$118$	1.40608e11	0.565794
$119$	1.73380e11	0.666025
$120$	0	0
$121$	1.66414e11	0.583270
$122$	−2.39781e11	−0.803223
$123$	0	0
$124$	−1.09843e11	−0.336476
$125$	−2.91650e11	−0.854784
$126$	0	0
$127$	−3.30771e11	−0.888398	−0.444199	−	0.895928i	$-0.646512\pi$
$127$	−3.30771e11	−0.888398	−0.444199	+	0.895928i	$0.646512\pi$
$128$	−1.73421e11	−0.446114
$129$	0	0
$130$	2.23292e11	0.527455
$131$	1.14849e11	0.260098	0.130049	−	0.991508i	$-0.458487\pi$
$131$	1.14849e11	0.260098	0.130049	+	0.991508i	$0.458487\pi$
$132$	0	0
$133$	2.83219e11	0.590117
$134$	2.42541e11	0.484963
$135$	0	0
$136$	5.43817e11	1.00228
$137$	−6.05101e11	−1.07118	−0.535592	−	0.844477i	$-0.679912\pi$
$137$	−6.05101e11	−1.07118	−0.535592	+	0.844477i	$0.679912\pi$
$138$	0	0
$139$	5.38498e11	0.880243	0.440121	−	0.897938i	$-0.354935\pi$
$139$	5.38498e11	0.880243	0.440121	+	0.897938i	$0.354935\pi$
$140$	−5.77686e10	−0.0907795
$141$	0	0
$142$	−6.74368e11	−0.980192
$143$	1.13981e12	1.59399
$144$	0	0
$145$	−2.44673e11	−0.317001
$146$	7.39389e11	0.922428
$147$	0	0
$148$	−3.84993e11	−0.445670
$149$	1.61927e11	0.180632	0.0903159	−	0.995913i	$-0.471212\pi$
$149$	1.61927e11	0.180632	0.0903159	+	0.995913i	$0.471212\pi$
$150$	0	0
$151$	−6.46712e11	−0.670406	−0.335203	−	0.942146i	$-0.608805\pi$
$151$	−6.46712e11	−0.670406	−0.335203	+	0.942146i	$0.608805\pi$
$152$	8.88334e11	0.888048
$153$	0	0
$154$	8.37849e11	0.779474
$155$	6.96969e11	0.625733
$156$	0	0
$157$	−1.44114e12	−1.20576	−0.602878	−	0.797833i	$-0.705979\pi$
$157$	−1.44114e12	−1.20576	−0.602878	+	0.797833i	$0.705979\pi$
$158$	1.39526e12	1.12731
$159$	0	0
$160$	−3.24963e11	−0.245004
$161$	1.67383e11	0.121947
$162$	0	0
$163$	−1.78980e12	−1.21835	−0.609175	−	0.793035i	$-0.708499\pi$
$163$	−1.78980e12	−1.21835	−0.609175	+	0.793035i	$0.708499\pi$
$164$	2.28749e11	0.150563
$165$	0	0
$166$	9.71906e11	0.598452
$167$	−2.84170e11	−0.169292	−0.0846461	−	0.996411i	$-0.526976\pi$
$167$	−2.84170e11	−0.169292	−0.0846461	+	0.996411i	$0.526976\pi$
$168$	0	0
$169$	1.08386e12	0.604781
$170$	−7.12742e11	−0.385002
$171$	0	0
$172$	−7.30943e11	−0.370235
$173$	1.50737e12	0.739549	0.369775	−	0.929121i	$-0.379435\pi$
$173$	1.50737e12	0.739549	0.369775	+	0.929121i	$0.379435\pi$
$174$	0	0
$175$	−1.19737e12	−0.551467
$176$	1.89409e12	0.845437
$177$	0	0
$178$	3.09128e12	1.29667
$179$	−1.54494e12	−0.628377	−0.314189	−	0.949361i	$-0.601732\pi$
$179$	−1.54494e12	−0.628377	−0.314189	+	0.949361i	$0.601732\pi$
$180$	0	0
$181$	−3.80037e11	−0.145410	−0.0727048	−	0.997354i	$-0.523163\pi$
$181$	−3.80037e11	−0.145410	−0.0727048	+	0.997354i	$0.523163\pi$
$182$	2.11410e12	0.784751
$183$	0	0
$184$	5.25008e11	0.183514
$185$	2.44283e12	0.828798
$186$	0	0
$187$	−3.63824e12	−1.16349
$188$	−3.94704e10	−0.0122575
$189$	0	0
$190$	−1.16428e12	−0.341123
$191$	2.33028e12	0.663322	0.331661	−	0.943399i	$-0.392391\pi$
$191$	2.33028e12	0.663322	0.331661	+	0.943399i	$0.392391\pi$
$192$	0	0
$193$	−4.99261e12	−1.34203	−0.671015	−	0.741444i	$-0.734141\pi$
$193$	−4.99261e12	−1.34203	−0.671015	+	0.741444i	$0.734141\pi$
$194$	−2.59202e12	−0.677219
$195$	0	0
$196$	5.07279e11	0.125268
$197$	2.51169e12	0.603117	0.301559	−	0.953448i	$-0.402493\pi$
$197$	2.51169e12	0.603117	0.301559	+	0.953448i	$0.402493\pi$
$198$	0	0
$199$	2.69518e12	0.612204	0.306102	−	0.951999i	$-0.400975\pi$
$199$	2.69518e12	0.612204	0.306102	+	0.951999i	$0.400975\pi$
$200$	−3.75563e12	−0.829884
$201$	0	0
$202$	5.14138e12	1.07559
$203$	−2.31652e12	−0.471637
$204$	0	0
$205$	−1.45144e12	−0.279997
$206$	2.24083e12	0.420861
$207$	0	0
$208$	4.77925e12	0.851161
$209$	−5.94313e12	−1.03089
$210$	0	0
$211$	6.06584e12	0.998475	0.499238	−	0.866465i	$-0.333613\pi$
$211$	6.06584e12	0.998475	0.499238	+	0.866465i	$0.333613\pi$
$212$	2.95751e12	0.474328
$213$	0	0
$214$	8.61193e12	1.31167
$215$	4.63792e12	0.688513
$216$	0	0
$217$	6.59878e12	0.930969
$218$	−2.05056e12	−0.282074
$219$	0	0
$220$	1.21223e12	0.158584
$221$	−9.18017e12	−1.17137
$222$	0	0
$223$	−5.57666e12	−0.677170	−0.338585	−	0.940936i	$-0.609948\pi$
$223$	−5.57666e12	−0.677170	−0.338585	+	0.940936i	$0.609948\pi$
$224$	−3.07669e12	−0.364519
$225$	0	0
$226$	−8.40140e12	−0.947885
$227$	−4.01440e12	−0.442057	−0.221028	−	0.975267i	$-0.570941\pi$
$227$	−4.01440e12	−0.442057	−0.221028	+	0.975267i	$0.570941\pi$
$228$	0	0
$229$	1.02421e13	1.07472	0.537358	−	0.843354i	$-0.319422\pi$
$229$	1.02421e13	1.07472	0.537358	+	0.843354i	$0.319422\pi$
$230$	−6.88091e11	−0.0704924
$231$	0	0
$232$	−7.26591e12	−0.709751
$233$	−1.36342e13	−1.30069	−0.650343	−	0.759641i	$-0.725375\pi$
$233$	−1.36342e13	−1.30069	−0.650343	+	0.759641i	$0.725375\pi$
$234$	0	0
$235$	2.50444e11	0.0227949
$236$	1.92610e12	0.171263
$237$	0	0
$238$	−6.74812e12	−0.572808
$239$	−9.51347e12	−0.789133	−0.394567	−	0.918867i	$-0.629105\pi$
$239$	−9.51347e12	−0.789133	−0.394567	+	0.918867i	$0.629105\pi$
$240$	0	0
$241$	−7.21921e12	−0.572000	−0.286000	−	0.958230i	$-0.592326\pi$
$241$	−7.21921e12	−0.572000	−0.286000	+	0.958230i	$0.592326\pi$
$242$	−6.47699e12	−0.501636
$243$	0	0
$244$	−3.28462e12	−0.243131
$245$	−3.21875e12	−0.232956
$246$	0	0
$247$	−1.49960e13	−1.03787
$248$	2.06975e13	1.40099
$249$	0	0
$250$	1.13513e13	0.735149
$251$	−7.98786e12	−0.506087	−0.253043	−	0.967455i	$-0.581432\pi$
$251$	−7.98786e12	−0.506087	−0.253043	+	0.967455i	$0.581432\pi$
$252$	0	0
$253$	−3.51241e12	−0.213031
$254$	1.28740e13	0.764058
$255$	0	0
$256$	−1.27274e13	−0.723468
$257$	−2.75365e13	−1.53206	−0.766030	−	0.642805i	$-0.777771\pi$
$257$	−2.75365e13	−1.53206	−0.766030	+	0.642805i	$0.777771\pi$
$258$	0	0
$259$	2.31283e13	1.23309
$260$	3.05875e12	0.159658
$261$	0	0
$262$	−4.47006e12	−0.223695
$263$	−2.98881e13	−1.46467	−0.732337	−	0.680942i	$-0.761571\pi$
$263$	−2.98881e13	−1.46467	−0.732337	+	0.680942i	$0.761571\pi$
$264$	0	0
$265$	−1.87657e13	−0.882091
$266$	−1.10232e13	−0.507525
$267$	0	0
$268$	3.32242e12	0.146796
$269$	−9.80257e12	−0.424329	−0.212164	−	0.977234i	$-0.568051\pi$
$269$	−9.80257e12	−0.424329	−0.212164	+	0.977234i	$0.568051\pi$
$270$	0	0
$271$	−1.00339e13	−0.417003	−0.208501	−	0.978022i	$-0.566859\pi$
$271$	−1.00339e13	−0.417003	−0.208501	+	0.978022i	$0.566859\pi$
$272$	−1.52552e13	−0.621282
$273$	0	0
$274$	2.35511e13	0.921263
$275$	2.51259e13	0.963367
$276$	0	0
$277$	−8.52869e12	−0.314227	−0.157113	−	0.987581i	$-0.550219\pi$
$277$	−8.52869e12	−0.314227	−0.157113	+	0.987581i	$0.550219\pi$
$278$	−2.09589e13	−0.757045
$279$	0	0
$280$	1.08852e13	0.377978
$281$	3.78916e13	1.29021	0.645103	−	0.764096i	$-0.276815\pi$
$281$	3.78916e13	1.29021	0.645103	+	0.764096i	$0.276815\pi$
$282$	0	0
$283$	−4.11334e12	−0.134700	−0.0673502	−	0.997729i	$-0.521454\pi$
$283$	−4.11334e12	−0.134700	−0.0673502	+	0.997729i	$0.521454\pi$
$284$	−9.23776e12	−0.296699
$285$	0	0
$286$	−4.43627e13	−1.37090
$287$	−1.37420e13	−0.416582
$288$	0	0
$289$	−4.96913e12	−0.144991
$290$	9.52290e12	0.272634
$291$	0	0
$292$	1.01285e13	0.279214
$293$	−8.98760e12	−0.243149	−0.121574	−	0.992582i	$-0.538794\pi$
$293$	−8.98760e12	−0.243149	−0.121574	+	0.992582i	$0.538794\pi$
$294$	0	0
$295$	−1.22213e13	−0.318491
$296$	7.25433e13	1.85564
$297$	0	0
$298$	−6.30235e12	−0.155351
$299$	−8.86266e12	−0.214473
$300$	0	0
$301$	4.39111e13	1.02437
$302$	2.51707e13	0.576576
$303$	0	0
$304$	−2.49196e13	−0.550474
$305$	2.08413e13	0.452143
$306$	0	0
$307$	1.27887e13	0.267649	0.133825	−	0.991005i	$-0.457274\pi$
$307$	1.27887e13	0.267649	0.133825	+	0.991005i	$0.457274\pi$
$308$	1.14772e13	0.235943
$309$	0	0
$310$	−2.71267e13	−0.538156
$311$	1.38679e13	0.270289	0.135145	−	0.990826i	$-0.456850\pi$
$311$	1.38679e13	0.270289	0.135145	+	0.990826i	$0.456850\pi$
$312$	0	0
$313$	6.42210e13	1.20832	0.604162	−	0.796861i	$-0.293508\pi$
$313$	6.42210e13	1.20832	0.604162	+	0.796861i	$0.293508\pi$
$314$	5.60908e13	1.03700
$315$	0	0
$316$	1.91128e13	0.341229
$317$	−5.48988e13	−0.963246	−0.481623	−	0.876379i	$-0.659953\pi$
$317$	−5.48988e13	−0.963246	−0.481623	+	0.876379i	$0.659953\pi$
$318$	0	0
$319$	4.86104e13	0.823911
$320$	3.21727e13	0.535997
$321$	0	0
$322$	−6.51473e12	−0.104879
$323$	4.78666e13	0.757563
$324$	0	0
$325$	6.33987e13	0.969889
$326$	6.96608e13	1.04783
$327$	0	0
$328$	−4.31027e13	−0.626900
$329$	2.37117e12	0.0339144
$330$	0	0
$331$	3.53734e13	0.489354	0.244677	−	0.969605i	$-0.421318\pi$
$331$	3.53734e13	0.489354	0.244677	+	0.969605i	$0.421318\pi$
$332$	1.33136e13	0.181148
$333$	0	0
$334$	1.10602e13	0.145598
$335$	−2.10812e13	−0.272991
$336$	0	0
$337$	2.16449e13	0.271263	0.135631	−	0.990759i	$-0.456694\pi$
$337$	2.16449e13	0.271263	0.135631	+	0.990759i	$0.456694\pi$
$338$	−4.21851e13	−0.520136
$339$	0	0
$340$	−9.76342e12	−0.116538
$341$	−1.38470e14	−1.62633
$342$	0	0
$343$	−9.38065e13	−1.06688
$344$	1.37730e14	1.54155
$345$	0	0
$346$	−5.86685e13	−0.636042
$347$	1.27155e14	1.35682	0.678411	−	0.734683i	$-0.262669\pi$
$347$	1.27155e14	1.35682	0.678411	+	0.734683i	$0.262669\pi$
$348$	0	0
$349$	7.61293e13	0.787068	0.393534	−	0.919310i	$-0.371252\pi$
$349$	7.61293e13	0.787068	0.393534	+	0.919310i	$0.371252\pi$
$350$	4.66029e13	0.474284
$351$	0	0
$352$	6.45620e13	0.636784
$353$	1.01111e14	0.981830	0.490915	−	0.871208i	$-0.336663\pi$
$353$	1.01111e14	0.981830	0.490915	+	0.871208i	$0.336663\pi$
$354$	0	0
$355$	5.86147e13	0.551761
$356$	4.23456e13	0.392494
$357$	0	0
$358$	6.01307e13	0.540430
$359$	−1.24969e14	−1.10607	−0.553034	−	0.833159i	$-0.686530\pi$
$359$	−1.24969e14	−1.10607	−0.553034	+	0.833159i	$0.686530\pi$
$360$	0	0
$361$	−3.82993e13	−0.328777
$362$	1.47914e13	0.125058
$363$	0	0
$364$	2.89597e13	0.237540
$365$	−6.42663e13	−0.519245
$366$	0	0
$367$	−7.01276e13	−0.549826	−0.274913	−	0.961469i	$-0.588649\pi$
$367$	−7.01276e13	−0.549826	−0.274913	+	0.961469i	$0.588649\pi$
$368$	−1.47276e13	−0.113754
$369$	0	0
$370$	−9.50773e13	−0.712800
$371$	−1.77671e14	−1.31238
$372$	0	0
$373$	−1.39872e14	−1.00307	−0.501535	−	0.865138i	$-0.667231\pi$
$373$	−1.39872e14	−1.00307	−0.501535	+	0.865138i	$0.667231\pi$
$374$	1.41604e14	1.00065
$375$	0	0
$376$	7.43731e12	0.0510367
$377$	1.22656e14	0.829489
$378$	0	0
$379$	−1.23834e14	−0.813435	−0.406717	−	0.913554i	$-0.633327\pi$
$379$	−1.23834e14	−0.813435	−0.406717	+	0.913554i	$0.633327\pi$
$380$	−1.59487e13	−0.103256
$381$	0	0
$382$	−9.06968e13	−0.570484
$383$	−1.44623e14	−0.896697	−0.448348	−	0.893859i	$-0.647988\pi$
$383$	−1.44623e14	−0.896697	−0.448348	+	0.893859i	$0.647988\pi$
$384$	0	0
$385$	−7.28241e13	−0.438774
$386$	1.94317e14	1.15420
$387$	0	0
$388$	−3.55066e13	−0.204991
$389$	1.27056e14	0.723225	0.361613	−	0.932328i	$-0.382226\pi$
$389$	1.27056e14	0.723225	0.361613	+	0.932328i	$0.382226\pi$
$390$	0	0
$391$	2.82893e13	0.156549
$392$	−9.55854e13	−0.521576
$393$	0	0
$394$	−9.77575e13	−0.518706
$395$	−1.21273e14	−0.634573
$396$	0	0
$397$	3.66597e14	1.86570	0.932850	−	0.360266i	$-0.117314\pi$
$397$	3.66597e14	1.86570	0.932850	+	0.360266i	$0.117314\pi$
$398$	−1.04899e14	−0.526521
$399$	0	0
$400$	1.05353e14	0.514420
$401$	−1.09287e14	−0.526350	−0.263175	−	0.964748i	$-0.584770\pi$
$401$	−1.09287e14	−0.526350	−0.263175	+	0.964748i	$0.584770\pi$
$402$	0	0
$403$	−3.49394e14	−1.63734
$404$	7.04287e13	0.325576
$405$	0	0
$406$	9.01612e13	0.405627
$407$	−4.85329e14	−2.15411
$408$	0	0
$409$	−1.49332e14	−0.645169	−0.322584	−	0.946541i	$-0.604552\pi$
$409$	−1.49332e14	−0.645169	−0.322584	+	0.946541i	$0.604552\pi$
$410$	5.64916e13	0.240809
$411$	0	0
$412$	3.06958e13	0.127393
$413$	−1.15710e14	−0.473853
$414$	0	0
$415$	−8.44761e13	−0.336875
$416$	1.62906e14	0.641095
$417$	0	0
$418$	2.31313e14	0.886605
$419$	−4.63339e14	−1.75276	−0.876379	−	0.481623i	$-0.840048\pi$
$419$	−4.63339e14	−1.75276	−0.876379	+	0.481623i	$0.840048\pi$
$420$	0	0
$421$	1.92962e14	0.711083	0.355541	−	0.934661i	$-0.384296\pi$
$421$	1.92962e14	0.711083	0.355541	+	0.934661i	$0.384296\pi$
$422$	−2.36088e14	−0.858729
$423$	0	0
$424$	−5.57276e14	−1.97496
$425$	−2.02366e14	−0.707945
$426$	0	0
$427$	1.97322e14	0.672701
$428$	1.17970e14	0.397036
$429$	0	0
$430$	−1.80513e14	−0.592149
$431$	1.26743e14	0.410485	0.205243	−	0.978711i	$-0.434202\pi$
$431$	1.26743e14	0.410485	0.205243	+	0.978711i	$0.434202\pi$
$432$	0	0
$433$	1.11003e14	0.350471	0.175235	−	0.984527i	$-0.443931\pi$
$433$	1.11003e14	0.350471	0.175235	+	0.984527i	$0.443931\pi$
$434$	−2.56831e14	−0.800672
$435$	0	0
$436$	−2.80894e13	−0.0853822
$437$	4.62111e13	0.138707
$438$	0	0
$439$	3.79614e14	1.11119	0.555594	−	0.831454i	$-0.312491\pi$
$439$	3.79614e14	1.11119	0.555594	+	0.831454i	$0.312491\pi$
$440$	−2.28417e14	−0.660297
$441$	0	0
$442$	3.57301e14	1.00742
$443$	5.50584e14	1.53321	0.766607	−	0.642117i	$-0.221943\pi$
$443$	5.50584e14	1.53321	0.766607	+	0.642117i	$0.221943\pi$
$444$	0	0
$445$	−2.68688e14	−0.729908
$446$	2.17049e14	0.582394
$447$	0	0
$448$	3.04606e14	0.797460
$449$	−2.83340e14	−0.732744	−0.366372	−	0.930468i	$-0.619400\pi$
$449$	−2.83340e14	−0.732744	−0.366372	+	0.930468i	$0.619400\pi$
$450$	0	0
$451$	2.88366e14	0.727734
$452$	−1.15086e14	−0.286920
$453$	0	0
$454$	1.56244e14	0.380187
$455$	−1.83753e14	−0.441745
$456$	0	0
$457$	1.27497e14	0.299200	0.149600	−	0.988747i	$-0.452201\pi$
$457$	1.27497e14	0.299200	0.149600	+	0.988747i	$0.452201\pi$
$458$	−3.98633e14	−0.924300
$459$	0	0
$460$	−9.42574e12	−0.0213377
$461$	3.07348e14	0.687504	0.343752	−	0.939060i	$-0.388302\pi$
$461$	3.07348e14	0.687504	0.343752	+	0.939060i	$0.388302\pi$
$462$	0	0
$463$	4.36939e14	0.954390	0.477195	−	0.878797i	$-0.341654\pi$
$463$	4.36939e14	0.954390	0.477195	+	0.878797i	$0.341654\pi$
$464$	2.03824e14	0.439953
$465$	0	0
$466$	5.30657e14	1.11864
$467$	6.49256e13	0.135261	0.0676306	−	0.997710i	$-0.478456\pi$
$467$	6.49256e13	0.135261	0.0676306	+	0.997710i	$0.478456\pi$
$468$	0	0
$469$	−1.99593e14	−0.406157
$470$	−9.74755e12	−0.0196046
$471$	0	0
$472$	−3.62930e14	−0.713086
$473$	−9.21440e14	−1.78950
$474$	0	0
$475$	−3.30569e14	−0.627260
$476$	−9.24384e13	−0.173386
$477$	0	0
$478$	3.70274e14	0.678687
$479$	−1.01170e15	−1.83318	−0.916590	−	0.399828i	$-0.869070\pi$
$479$	−1.01170e15	−1.83318	−0.916590	+	0.399828i	$0.869070\pi$
$480$	0	0
$481$	−1.22460e15	−2.16869
$482$	2.80979e14	0.491943
$483$	0	0
$484$	−8.87244e13	−0.151843
$485$	2.25293e14	0.381214
$486$	0	0
$487$	−3.00339e14	−0.496824	−0.248412	−	0.968654i	$-0.579909\pi$
$487$	−3.00339e14	−0.496824	−0.248412	+	0.968654i	$0.579909\pi$
$488$	6.18913e14	1.01233
$489$	0	0
$490$	1.25277e14	0.200351
$491$	9.16459e14	1.44932	0.724661	−	0.689106i	$-0.241997\pi$
$491$	9.16459e14	1.44932	0.724661	+	0.689106i	$0.241997\pi$
$492$	0	0
$493$	−3.91513e14	−0.605463
$494$	5.83658e14	0.892607
$495$	0	0
$496$	−5.80608e14	−0.868429
$497$	5.54954e14	0.820913
$498$	0	0
$499$	1.18257e15	1.71109	0.855547	−	0.517726i	$-0.173221\pi$
$499$	1.18257e15	1.71109	0.855547	+	0.517726i	$0.173221\pi$
$500$	1.55495e14	0.222526
$501$	0	0
$502$	3.10896e14	0.435255
$503$	5.42478e14	0.751205	0.375603	−	0.926781i	$-0.377436\pi$
$503$	5.42478e14	0.751205	0.375603	+	0.926781i	$0.377436\pi$
$504$	0	0
$505$	−4.46878e14	−0.605462
$506$	1.36707e14	0.183215
$507$	0	0
$508$	1.76353e14	0.231276
$509$	3.73891e14	0.485062	0.242531	−	0.970144i	$-0.422022\pi$
$509$	3.73891e14	0.485062	0.242531	+	0.970144i	$0.422022\pi$
$510$	0	0
$511$	−6.08462e14	−0.772536
$512$	8.50527e14	1.06833
$513$	0	0
$514$	1.07175e15	1.31763
$515$	−1.94768e14	−0.236907
$516$	0	0
$517$	−4.97571e13	−0.0592457
$518$	−9.00176e14	−1.06051
$519$	0	0
$520$	−5.76353e14	−0.664768
$521$	−9.41376e14	−1.07437	−0.537187	−	0.843463i	$-0.680513\pi$
$521$	−9.41376e14	−1.07437	−0.537187	+	0.843463i	$0.680513\pi$
$522$	0	0
$523$	6.38561e13	0.0713581	0.0356791	−	0.999363i	$-0.488641\pi$
$523$	6.38561e13	0.0713581	0.0356791	+	0.999363i	$0.488641\pi$
$524$	−6.12326e13	−0.0677112
$525$	0	0
$526$	1.16327e15	1.25968
$527$	1.11525e15	1.19513
$528$	0	0
$529$	−9.25499e14	−0.971336
$530$	7.30382e14	0.758634
$531$	0	0
$532$	−1.51000e14	−0.153625
$533$	7.27617e14	0.732661
$534$	0	0
$535$	−7.48532e14	−0.738354
$536$	−6.26036e14	−0.611213
$537$	0	0
$538$	3.81526e14	0.364940
$539$	6.39485e14	0.605469
$540$	0	0
$541$	7.25322e14	0.672893	0.336447	−	0.941703i	$-0.390775\pi$
$541$	7.25322e14	0.672893	0.336447	+	0.941703i	$0.390775\pi$
$542$	3.90530e14	0.358639
$543$	0	0
$544$	−5.19989e14	−0.467951
$545$	1.78231e14	0.158782
$546$	0	0
$547$	−1.11494e14	−0.0973464	−0.0486732	−	0.998815i	$-0.515499\pi$
$547$	−1.11494e14	−0.0973464	−0.0486732	+	0.998815i	$0.515499\pi$
$548$	3.22613e14	0.278861
$549$	0	0
$550$	−9.77925e14	−0.828535
$551$	−6.39543e14	−0.536458
$552$	0	0
$553$	−1.14820e15	−0.944121
$554$	3.31945e14	0.270248
$555$	0	0
$556$	−2.87103e14	−0.229153
$557$	2.27003e14	0.179402	0.0897012	−	0.995969i	$-0.471409\pi$
$557$	2.27003e14	0.179402	0.0897012	+	0.995969i	$0.471409\pi$
$558$	0	0
$559$	−2.32502e15	−1.80161
$560$	−3.05353e14	−0.234297
$561$	0	0
$562$	−1.47478e15	−1.10963
$563$	1.40970e15	1.05034	0.525170	−	0.850997i	$-0.324002\pi$
$563$	1.40970e15	1.05034	0.525170	+	0.850997i	$0.324002\pi$
$564$	0	0
$565$	7.30233e14	0.533575
$566$	1.60095e14	0.115848
$567$	0	0
$568$	1.74065e15	1.23537
$569$	4.03462e14	0.283586	0.141793	−	0.989896i	$-0.454713\pi$
$569$	4.03462e14	0.283586	0.141793	+	0.989896i	$0.454713\pi$
$570$	0	0
$571$	1.38965e14	0.0958091	0.0479046	−	0.998852i	$-0.484746\pi$
$571$	1.38965e14	0.0958091	0.0479046	+	0.998852i	$0.484746\pi$
$572$	−6.07698e14	−0.414963
$573$	0	0
$574$	5.34853e14	0.358277
$575$	−1.95367e14	−0.129622
$576$	0	0
$577$	−1.49055e14	−0.0970240	−0.0485120	−	0.998823i	$-0.515448\pi$
$577$	−1.49055e14	−0.0970240	−0.0485120	+	0.998823i	$0.515448\pi$
$578$	1.93403e14	0.124698
$579$	0	0
$580$	1.30449e14	0.0825249
$581$	−7.99805e14	−0.501205
$582$	0	0
$583$	3.72829e15	2.29262
$584$	−1.90848e15	−1.16256
$585$	0	0
$586$	3.49806e14	0.209118
$587$	2.53175e15	1.49938	0.749690	−	0.661789i	$-0.230202\pi$
$587$	2.53175e15	1.49938	0.749690	+	0.661789i	$0.230202\pi$
$588$	0	0
$589$	1.82179e15	1.05892
$590$	4.75667e14	0.273916
$591$	0	0
$592$	−2.03499e15	−1.15025
$593$	−7.38349e14	−0.413486	−0.206743	−	0.978395i	$-0.566286\pi$
$593$	−7.38349e14	−0.413486	−0.206743	+	0.978395i	$0.566286\pi$
$594$	0	0
$595$	5.86533e14	0.322440
$596$	−8.63321e13	−0.0470238
$597$	0	0
$598$	3.44944e14	0.184456
$599$	2.47110e15	1.30931	0.654656	−	0.755927i	$-0.272814\pi$
$599$	2.47110e15	1.30931	0.654656	+	0.755927i	$0.272814\pi$
$600$	0	0
$601$	2.10961e15	1.09747	0.548735	−	0.835996i	$-0.315110\pi$
$601$	2.10961e15	1.09747	0.548735	+	0.835996i	$0.315110\pi$
$602$	−1.70906e15	−0.881003
$603$	0	0
$604$	3.44798e14	0.174527
$605$	5.62967e14	0.282377
$606$	0	0
$607$	2.87834e15	1.41777	0.708883	−	0.705326i	$-0.249200\pi$
$607$	2.87834e15	1.41777	0.708883	+	0.705326i	$0.249200\pi$
$608$	−8.49411e14	−0.414618
$609$	0	0
$610$	−8.11165e14	−0.388862
$611$	−1.25549e14	−0.0596468
$612$	0	0
$613$	1.06157e15	0.495352	0.247676	−	0.968843i	$-0.420333\pi$
$613$	1.06157e15	0.495352	0.247676	+	0.968843i	$0.420333\pi$
$614$	−4.97750e14	−0.230189
$615$	0	0
$616$	−2.16262e15	−0.982394
$617$	−1.87568e15	−0.844480	−0.422240	−	0.906484i	$-0.638756\pi$
$617$	−1.87568e15	−0.844480	−0.422240	+	0.906484i	$0.638756\pi$
$618$	0	0
$619$	−2.79678e15	−1.23697	−0.618485	−	0.785797i	$-0.712253\pi$
$619$	−2.79678e15	−1.23697	−0.618485	+	0.785797i	$0.712253\pi$
$620$	−3.71593e14	−0.162897
$621$	0	0
$622$	−5.39752e14	−0.232460
$623$	−2.54389e15	−1.08596
$624$	0	0
$625$	8.38752e14	0.351798
$626$	−2.49955e15	−1.03921
$627$	0	0
$628$	7.68354e14	0.313894
$629$	3.90889e15	1.58298
$630$	0	0
$631$	−1.21754e14	−0.0484530	−0.0242265	−	0.999706i	$-0.507712\pi$
$631$	−1.21754e14	−0.0484530	−0.0242265	+	0.999706i	$0.507712\pi$
$632$	−3.60139e15	−1.42078
$633$	0	0
$634$	2.13672e15	0.828431
$635$	−1.11898e15	−0.430097
$636$	0	0
$637$	1.61358e15	0.609569
$638$	−1.89196e15	−0.708597
$639$	0	0
$640$	−5.86671e14	−0.215975
$641$	−2.53322e14	−0.0924600	−0.0462300	−	0.998931i	$-0.514721\pi$
$641$	−2.53322e14	−0.0924600	−0.0462300	+	0.998931i	$0.514721\pi$
$642$	0	0
$643$	−5.12828e15	−1.83997	−0.919986	−	0.391951i	$-0.871800\pi$
$643$	−5.12828e15	−1.83997	−0.919986	+	0.391951i	$0.871800\pi$
$644$	−8.92413e13	−0.0317463
$645$	0	0
$646$	−1.86302e15	−0.651535
$647$	1.76264e14	0.0611210	0.0305605	−	0.999533i	$-0.490271\pi$
$647$	1.76264e14	0.0611210	0.0305605	+	0.999533i	$0.490271\pi$
$648$	0	0
$649$	2.42808e15	0.827783
$650$	−2.46754e15	−0.834144
$651$	0	0
$652$	9.54241e14	0.317173
$653$	−1.28765e15	−0.424399	−0.212199	−	0.977226i	$-0.568063\pi$
$653$	−1.28765e15	−0.424399	−0.212199	+	0.977226i	$0.568063\pi$
$654$	0	0
$655$	3.88528e14	0.125920
$656$	1.20912e15	0.388596
$657$	0	0
$658$	−9.22882e13	−0.0291678
$659$	−5.39386e15	−1.69056	−0.845279	−	0.534325i	$-0.820566\pi$
$659$	−5.39386e15	−1.69056	−0.845279	+	0.534325i	$0.820566\pi$
$660$	0	0
$661$	1.62130e15	0.499753	0.249876	−	0.968278i	$-0.419610\pi$
$661$	1.62130e15	0.499753	0.249876	+	0.968278i	$0.419610\pi$
$662$	−1.37677e15	−0.420865
$663$	0	0
$664$	−2.50864e15	−0.754247
$665$	9.58112e14	0.285691
$666$	0	0
$667$	−3.77972e14	−0.110858
$668$	1.51507e14	0.0440718
$669$	0	0
$670$	8.20500e14	0.234783
$671$	−4.14065e15	−1.17515
$672$	0	0
$673$	−5.93173e15	−1.65615	−0.828073	−	0.560620i	$-0.810563\pi$
$673$	−5.93173e15	−1.65615	−0.828073	+	0.560620i	$0.810563\pi$
$674$	−8.42439e14	−0.233297
$675$	0	0
$676$	−5.77868e14	−0.157443
$677$	6.75109e15	1.82447	0.912234	−	0.409670i	$-0.134356\pi$
$677$	6.75109e15	1.82447	0.912234	+	0.409670i	$0.134356\pi$
$678$	0	0
$679$	2.13304e15	0.567173
$680$	1.83970e15	0.485230
$681$	0	0
$682$	5.38941e15	1.39871
$683$	−4.47959e15	−1.15325	−0.576626	−	0.817008i	$-0.695631\pi$
$683$	−4.47959e15	−1.15325	−0.576626	+	0.817008i	$0.695631\pi$
$684$	0	0
$685$	−2.04702e15	−0.518589
$686$	3.65104e15	0.917560
$687$	0	0
$688$	−3.86361e15	−0.955558
$689$	9.40737e15	2.30814
$690$	0	0
$691$	−3.76180e15	−0.908379	−0.454189	−	0.890905i	$-0.650071\pi$
$691$	−3.76180e15	−0.908379	−0.454189	+	0.890905i	$0.650071\pi$
$692$	−8.03664e14	−0.192527
$693$	0	0
$694$	−4.94902e15	−1.16692
$695$	1.82170e15	0.426149
$696$	0	0
$697$	−2.32253e15	−0.534786
$698$	−2.96303e15	−0.676910
$699$	0	0
$700$	6.38385e14	0.143563
$701$	−5.60965e15	−1.25166	−0.625831	−	0.779959i	$-0.715240\pi$
$701$	−5.60965e15	−1.25166	−0.625831	+	0.779959i	$0.715240\pi$
$702$	0	0
$703$	6.38524e15	1.40257
$704$	−6.39192e15	−1.39310
$705$	0	0
$706$	−3.93533e15	−0.844414
$707$	−4.23097e15	−0.900810
$708$	0	0
$709$	4.92456e15	1.03232	0.516159	−	0.856493i	$-0.327361\pi$
$709$	4.92456e15	1.03232	0.516159	+	0.856493i	$0.327361\pi$
$710$	−2.28134e15	−0.474537
$711$	0	0
$712$	−7.97908e15	−1.63423
$713$	1.07668e15	0.218824
$714$	0	0
$715$	3.85591e15	0.771692
$716$	8.23694e14	0.163585
$717$	0	0
$718$	4.86390e15	0.951263
$719$	9.38020e15	1.82055	0.910276	−	0.414002i	$-0.135869\pi$
$719$	9.38020e15	1.82055	0.910276	+	0.414002i	$0.135869\pi$
$720$	0	0
$721$	−1.84403e15	−0.352472
$722$	1.49065e15	0.282762
$723$	0	0
$724$	2.02619e14	0.0378545
$725$	2.70381e15	0.501322
$726$	0	0
$727$	1.46931e15	0.268332	0.134166	−	0.990959i	$-0.457164\pi$
$727$	1.46931e15	0.268332	0.134166	+	0.990959i	$0.457164\pi$
$728$	−5.45681e15	−0.989045
$729$	0	0
$730$	2.50131e15	0.446572
$731$	7.42137e15	1.31504
$732$	0	0
$733$	−8.64113e15	−1.50834	−0.754169	−	0.656681i	$-0.771960\pi$
$733$	−8.64113e15	−1.50834	−0.754169	+	0.656681i	$0.771960\pi$
$734$	2.72944e15	0.472873
$735$	0	0
$736$	−5.02004e14	−0.0856800
$737$	4.18830e15	0.709524
$738$	0	0
$739$	1.31994e15	0.220298	0.110149	−	0.993915i	$-0.464867\pi$
$739$	1.31994e15	0.220298	0.110149	+	0.993915i	$0.464867\pi$
$740$	−1.30241e15	−0.215761
$741$	0	0
$742$	6.91513e15	1.12870
$743$	4.26310e15	0.690695	0.345348	−	0.938475i	$-0.387761\pi$
$743$	4.26310e15	0.690695	0.345348	+	0.938475i	$0.387761\pi$
$744$	0	0
$745$	5.47788e14	0.0874486
$746$	5.44394e15	0.862681
$747$	0	0
$748$	1.93975e15	0.302891
$749$	−7.08697e15	−1.09853
$750$	0	0
$751$	−6.11106e15	−0.933462	−0.466731	−	0.884399i	$-0.654568\pi$
$751$	−6.11106e15	−0.933462	−0.466731	+	0.884399i	$0.654568\pi$
$752$	−2.08632e14	−0.0316361
$753$	0	0
$754$	−4.77388e15	−0.713394
$755$	−2.18779e15	−0.324561
$756$	0	0
$757$	9.63454e15	1.40865	0.704326	−	0.709877i	$-0.251250\pi$
$757$	9.63454e15	1.40865	0.704326	+	0.709877i	$0.251250\pi$
$758$	4.81973e15	0.699587
$759$	0	0
$760$	3.00518e15	0.429928
$761$	−2.84480e15	−0.404051	−0.202026	−	0.979380i	$-0.564752\pi$
$761$	−2.84480e15	−0.404051	−0.202026	+	0.979380i	$0.564752\pi$
$762$	0	0
$763$	1.68746e15	0.236237
$764$	−1.24240e15	−0.172682
$765$	0	0
$766$	5.62889e15	0.771196
$767$	6.12663e15	0.833387
$768$	0	0
$769$	1.40069e16	1.87822	0.939110	−	0.343616i	$-0.111652\pi$
$769$	1.40069e16	1.87822	0.939110	+	0.343616i	$0.111652\pi$
$770$	2.83439e15	0.377364
$771$	0	0
$772$	2.66184e15	0.349370
$773$	1.33026e16	1.73360	0.866798	−	0.498659i	$-0.166174\pi$
$773$	1.33026e16	1.73360	0.866798	+	0.498659i	$0.166174\pi$
$774$	0	0
$775$	−7.70201e15	−0.989566
$776$	6.69041e15	0.853520
$777$	0	0
$778$	−4.94516e15	−0.622003
$779$	−3.79389e15	−0.473836
$780$	0	0
$781$	−1.16453e16	−1.43407
$782$	−1.10105e15	−0.134638
$783$	0	0
$784$	2.68137e15	0.323309
$785$	−4.87530e15	−0.583738
$786$	0	0
$787$	8.40292e15	0.992131	0.496065	−	0.868285i	$-0.334778\pi$
$787$	8.40292e15	0.992131	0.496065	+	0.868285i	$0.334778\pi$
$788$	−1.33912e15	−0.157009
$789$	0	0
$790$	4.72008e15	0.545758
$791$	6.91372e15	0.793855
$792$	0	0
$793$	−1.04479e16	−1.18311
$794$	−1.42683e16	−1.60458
$795$	0	0
$796$	−1.43695e15	−0.159375
$797$	1.01373e16	1.11661	0.558306	−	0.829635i	$-0.311451\pi$
$797$	1.01373e16	1.11661	0.558306	+	0.829635i	$0.311451\pi$
$798$	0	0
$799$	4.00749e14	0.0435376
$800$	3.59107e15	0.387462
$801$	0	0
$802$	4.25356e15	0.452682
$803$	1.27681e16	1.34956
$804$	0	0
$805$	5.66247e14	0.0590376
$806$	1.35988e16	1.40818
$807$	0	0
$808$	−1.32707e16	−1.35560
$809$	1.09699e14	0.0111297	0.00556487	−	0.999985i	$-0.498229\pi$
$809$	1.09699e14	0.0111297	0.00556487	+	0.999985i	$0.498229\pi$
$810$	0	0
$811$	5.29565e15	0.530035	0.265018	−	0.964244i	$-0.414622\pi$
$811$	5.29565e15	0.530035	0.265018	+	0.964244i	$0.414622\pi$
$812$	1.23506e15	0.122781
$813$	0	0
$814$	1.88895e16	1.85262
$815$	−6.05477e15	−0.589836
$816$	0	0
$817$	1.21229e16	1.16516
$818$	5.81214e15	0.554872
$819$	0	0
$820$	7.73845e14	0.0728916
$821$	−4.44266e15	−0.415676	−0.207838	−	0.978163i	$-0.566643\pi$
$821$	−4.44266e15	−0.415676	−0.207838	+	0.978163i	$0.566643\pi$
$822$	0	0
$823$	−1.11414e16	−1.02858	−0.514291	−	0.857616i	$-0.671945\pi$
$823$	−1.11414e16	−1.02858	−0.514291	+	0.857616i	$0.671945\pi$
$824$	−5.78393e15	−0.530424
$825$	0	0
$826$	4.50353e15	0.407533
$827$	−1.64160e16	−1.47567	−0.737833	−	0.674983i	$-0.764151\pi$
$827$	−1.64160e16	−1.47567	−0.737833	+	0.674983i	$0.764151\pi$
$828$	0	0
$829$	−1.29613e16	−1.14973	−0.574867	−	0.818247i	$-0.694946\pi$
$829$	−1.29613e16	−1.14973	−0.574867	+	0.818247i	$0.694946\pi$
$830$	3.28789e15	0.289727
$831$	0	0
$832$	−1.61284e16	−1.40253
$833$	−5.15048e15	−0.444938
$834$	0	0
$835$	−9.61327e14	−0.0819588
$836$	3.16861e15	0.268371
$837$	0	0
$838$	1.80336e16	1.50744
$839$	−4.89379e15	−0.406401	−0.203201	−	0.979137i	$-0.565134\pi$
$839$	−4.89379e15	−0.406401	−0.203201	+	0.979137i	$0.565134\pi$
$840$	0	0
$841$	−6.96953e15	−0.571249
$842$	−7.51027e15	−0.611560
$843$	0	0
$844$	−3.23403e15	−0.259933
$845$	3.66664e15	0.292790
$846$	0	0
$847$	5.33008e15	0.420122
$848$	1.56328e16	1.22422
$849$	0	0
$850$	7.87631e15	0.608862
$851$	3.77370e15	0.289838
$852$	0	0
$853$	7.07831e15	0.536673	0.268337	−	0.963325i	$-0.413526\pi$
$853$	7.07831e15	0.536673	0.268337	+	0.963325i	$0.413526\pi$
$854$	−7.67997e15	−0.578551
$855$	0	0
$856$	−2.22287e16	−1.65314
$857$	−2.31170e15	−0.170819	−0.0854095	−	0.996346i	$-0.527220\pi$
$857$	−2.31170e15	−0.170819	−0.0854095	+	0.996346i	$0.527220\pi$
$858$	0	0
$859$	−1.97815e16	−1.44310	−0.721551	−	0.692361i	$-0.756570\pi$
$859$	−1.97815e16	−1.44310	−0.721551	+	0.692361i	$0.756570\pi$
$860$	−2.47273e15	−0.179240
$861$	0	0
$862$	−4.93295e15	−0.353034
$863$	−1.59925e16	−1.13725	−0.568627	−	0.822595i	$-0.692525\pi$
$863$	−1.59925e16	−1.13725	−0.568627	+	0.822595i	$0.692525\pi$
$864$	0	0
$865$	5.09934e15	0.358035
$866$	−4.32035e15	−0.301419
$867$	0	0
$868$	−3.51818e15	−0.242359
$869$	2.40940e16	1.64930
$870$	0	0
$871$	1.05681e16	0.714327
$872$	5.29282e15	0.355506
$873$	0	0
$874$	−1.79858e15	−0.119294
$875$	−9.34127e15	−0.615689
$876$	0	0
$877$	1.40825e16	0.916602	0.458301	−	0.888797i	$-0.348458\pi$
$877$	1.40825e16	0.916602	0.458301	+	0.888797i	$0.348458\pi$
$878$	−1.47750e16	−0.955667
$879$	0	0
$880$	6.40759e15	0.409298
$881$	−2.05758e16	−1.30614	−0.653068	−	0.757299i	$-0.726519\pi$
$881$	−2.05758e16	−1.30614	−0.653068	+	0.757299i	$0.726519\pi$
$882$	0	0
$883$	−1.09044e16	−0.683627	−0.341814	−	0.939768i	$-0.611041\pi$
$883$	−1.09044e16	−0.683627	−0.341814	+	0.939768i	$0.611041\pi$
$884$	4.89446e15	0.304942
$885$	0	0
$886$	−2.14293e16	−1.31863
$887$	−6.08971e15	−0.372406	−0.186203	−	0.982511i	$-0.559618\pi$
$887$	−6.08971e15	−0.372406	−0.186203	+	0.982511i	$0.559618\pi$
$888$	0	0
$889$	−1.05943e16	−0.639901
$890$	1.04576e16	0.627751
$891$	0	0
$892$	2.97323e15	0.176287
$893$	6.54630e14	0.0385756
$894$	0	0
$895$	−5.22644e15	−0.304214
$896$	−5.55450e15	−0.321330
$897$	0	0
$898$	1.10279e16	0.630190
$899$	−1.49009e16	−0.846317
$900$	0	0
$901$	−3.00280e16	−1.68477
$902$	−1.12235e16	−0.625881
$903$	0	0
$904$	2.16853e16	1.19465
$905$	−1.28564e15	−0.0703966
$906$	0	0
$907$	−1.82430e16	−0.986859	−0.493429	−	0.869786i	$-0.664257\pi$
$907$	−1.82430e16	−0.986859	−0.493429	+	0.869786i	$0.664257\pi$
$908$	2.14030e15	0.115081
$909$	0	0
$910$	7.15185e15	0.379919
$911$	−1.23507e16	−0.652140	−0.326070	−	0.945346i	$-0.605724\pi$
$911$	−1.23507e16	−0.652140	−0.326070	+	0.945346i	$0.605724\pi$
$912$	0	0
$913$	1.67833e16	0.875564
$914$	−4.96232e15	−0.257324
$915$	0	0
$916$	−5.46063e15	−0.279781
$917$	3.67852e15	0.187345
$918$	0	0
$919$	−8.70126e14	−0.0437871	−0.0218936	−	0.999760i	$-0.506969\pi$
$919$	−8.70126e14	−0.0437871	−0.0218936	+	0.999760i	$0.506969\pi$
$920$	1.77607e15	0.0888437
$921$	0	0
$922$	−1.19623e16	−0.591282
$923$	−2.93839e16	−1.44378
$924$	0	0
$925$	−2.69950e16	−1.31070
$926$	−1.70061e16	−0.820814
$927$	0	0
$928$	6.94754e15	0.331373
$929$	−1.98416e16	−0.940784	−0.470392	−	0.882458i	$-0.655887\pi$
$929$	−1.98416e16	−0.940784	−0.470392	+	0.882458i	$0.655887\pi$
$930$	0	0
$931$	−8.41340e15	−0.394229
$932$	7.26915e15	0.338607
$933$	0	0
$934$	−2.52697e15	−0.116330
$935$	−1.23079e16	−0.563276
$936$	0	0
$937$	2.15194e16	0.973336	0.486668	−	0.873587i	$-0.338212\pi$
$937$	2.15194e16	0.973336	0.486668	+	0.873587i	$0.338212\pi$
$938$	7.76836e15	0.349312
$939$	0	0
$940$	−1.33526e14	−0.00593420
$941$	−2.01951e16	−0.892285	−0.446142	−	0.894962i	$-0.647202\pi$
$941$	−2.01951e16	−0.892285	−0.446142	+	0.894962i	$0.647202\pi$
$942$	0	0
$943$	−2.24220e15	−0.0979174
$944$	1.01810e16	0.442021
$945$	0	0
$946$	3.58634e16	1.53904
$947$	−2.57658e16	−1.09931	−0.549654	−	0.835392i	$-0.685240\pi$
$947$	−2.57658e16	−1.09931	−0.549654	+	0.835392i	$0.685240\pi$
$948$	0	0
$949$	3.22170e16	1.35869
$950$	1.28661e16	0.539469
$951$	0	0
$952$	1.74179e16	0.721927
$953$	5.27859e15	0.217524	0.108762	−	0.994068i	$-0.465311\pi$
$953$	5.27859e15	0.217524	0.108762	+	0.994068i	$0.465311\pi$
$954$	0	0
$955$	7.88318e15	0.321131
$956$	5.07216e15	0.205435
$957$	0	0
$958$	3.93763e16	1.57661
$959$	−1.93808e16	−0.771560
$960$	0	0
$961$	1.70378e16	0.670555
$962$	4.76628e16	1.86516
$963$	0	0
$964$	3.84896e15	0.148909
$965$	−1.68897e16	−0.649712
$966$	0	0
$967$	−1.11683e16	−0.424759	−0.212379	−	0.977187i	$-0.568121\pi$
$967$	−1.11683e16	−0.424759	−0.212379	+	0.977187i	$0.568121\pi$
$968$	1.67181e16	0.632227
$969$	0	0
$970$	−8.76865e15	−0.327860
$971$	3.16576e16	1.17699	0.588494	−	0.808502i	$-0.299721\pi$
$971$	3.16576e16	1.17699	0.588494	+	0.808502i	$0.299721\pi$
$972$	0	0
$973$	1.72476e16	0.634027
$974$	1.16895e16	0.427289
$975$	0	0
$976$	−1.73618e16	−0.627511
$977$	1.19881e16	0.430854	0.215427	−	0.976520i	$-0.430886\pi$
$977$	1.19881e16	0.430854	0.215427	+	0.976520i	$0.430886\pi$
$978$	0	0
$979$	5.33816e16	1.89709
$980$	1.71609e15	0.0606453
$981$	0	0
$982$	−3.56695e16	−1.24648
$983$	1.72000e16	0.597701	0.298850	−	0.954300i	$-0.403397\pi$
$983$	1.72000e16	0.597701	0.298850	+	0.954300i	$0.403397\pi$
$984$	0	0
$985$	8.49689e15	0.291985
$986$	1.52381e16	0.520723
$987$	0	0
$988$	7.99518e15	0.270187
$989$	7.16470e15	0.240779
$990$	0	0
$991$	5.83721e15	0.193999	0.0969996	−	0.995284i	$-0.469075\pi$
$991$	5.83721e15	0.193999	0.0969996	+	0.995284i	$0.469075\pi$
$992$	−1.97906e16	−0.654102
$993$	0	0
$994$	−2.15994e16	−0.706019
$995$	9.11763e15	0.296384
$996$	0	0
$997$	−2.93240e16	−0.942756	−0.471378	−	0.881931i	$-0.656243\pi$
$997$	−2.93240e16	−0.942756	−0.471378	+	0.881931i	$0.656243\pi$
$998$	−4.60268e16	−1.47161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.12.a.c.1.4		10
3.2	odd	2		81.12.a.e.1.7		10
9.2	odd	6		9.12.c.a.4.4	✓	20
9.4	even	3		27.12.c.a.19.7		20
9.5	odd	6		9.12.c.a.7.4	yes	20
9.7	even	3		27.12.c.a.10.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.12.c.a.4.4	✓	20	9.2	odd	6
9.12.c.a.7.4	yes	20	9.5	odd	6
27.12.c.a.10.7		20	9.7	even	3
27.12.c.a.19.7		20	9.4	even	3
81.12.a.c.1.4		10	1.1	even	1	trivial
81.12.a.e.1.7		10	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	81.a (trivial)

\(f(q)\)	\(=\)	\(q-38.9210 q^{2} -533.155 q^{4} +3382.94 q^{5} +32029.1 q^{7} +100461. q^{8} -131667. q^{10} -672105. q^{11} -1.69588e6 q^{13} -1.24660e6 q^{14} -2.81815e6 q^{16} +5.41320e6 q^{17} +8.84256e6 q^{19} -1.80363e6 q^{20} +2.61590e7 q^{22} +5.22598e6 q^{23} -3.73839e7 q^{25} +6.60055e7 q^{26} -1.70765e7 q^{28} -7.23255e7 q^{29} +2.06025e8 q^{31} -9.60593e7 q^{32} -2.10687e8 q^{34} +1.08352e8 q^{35} +7.22103e8 q^{37} -3.44161e8 q^{38} +3.39854e8 q^{40} -4.29048e8 q^{41} +1.37098e9 q^{43} +3.58337e8 q^{44} -2.03400e8 q^{46} +7.40317e7 q^{47} -9.51466e8 q^{49} +1.45502e9 q^{50} +9.04170e8 q^{52} -5.54718e9 q^{53} -2.27369e9 q^{55} +3.21768e9 q^{56} +2.81498e9 q^{58} -3.61264e9 q^{59} +6.16072e9 q^{61} -8.01870e9 q^{62} +9.51029e9 q^{64} -5.73707e9 q^{65} -6.23162e9 q^{67} -2.88608e9 q^{68} -4.21718e9 q^{70} +1.73266e10 q^{71} -1.89972e10 q^{73} -2.81050e10 q^{74} -4.71446e9 q^{76} -2.15269e10 q^{77} -3.58485e10 q^{79} -9.53361e9 q^{80} +1.66990e10 q^{82} -2.49712e10 q^{83} +1.83125e10 q^{85} -5.33598e10 q^{86} -6.75205e10 q^{88} -7.94245e10 q^{89} -5.43176e10 q^{91} -2.78626e9 q^{92} -2.88139e9 q^{94} +2.99138e10 q^{95} +6.65970e10 q^{97} +3.70320e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 33 q^{2} + 9217 q^{4} - 7230 q^{5} - 8512 q^{7} + 14559 q^{8} + 2046 q^{10} - 112776 q^{11} - 279706 q^{13} - 3901584 q^{14} + 7342081 q^{16} - 13882896 q^{17} + 3514700 q^{19} - 34163508 q^{20}+ \cdots - 655061802039 q^{98}+O(q^{100}) \) 10 * q - 33 * q^2 + 9217 * q^4 - 7230 * q^5 - 8512 * q^7 + 14559 * q^8 + 2046 * q^10 - 112776 * q^11 - 279706 * q^13 - 3901584 * q^14 + 7342081 * q^16 - 13882896 * q^17 + 3514700 * q^19 - 34163508 * q^20 - 2274591 * q^22 - 69371616 * q^23 + 45286204 * q^25 - 240964572 * q^26 - 30672898 * q^28 - 25437246 * q^29 - 114575368 * q^31 + 80396559 * q^32 + 243855063 * q^34 + 89073732 * q^35 - 67109164 * q^37 + 489799995 * q^38 - 107425416 * q^40 + 331873026 * q^41 + 1118847584 * q^43 - 139238637 * q^44 + 1441268796 * q^46 - 1469650704 * q^47 + 3553434720 * q^49 - 6643771701 * q^50 - 3632448874 * q^52 - 7457130972 * q^53 + 2490724992 * q^55 - 27669139026 * q^56 + 1387480560 * q^58 - 26505032592 * q^59 - 990409066 * q^61 - 45522498090 * q^62 - 3758354783 * q^64 - 39045315390 * q^65 - 6557215720 * q^67 - 77299152993 * q^68 + 785437278 * q^70 - 61026859872 * q^71 - 6306446968 * q^73 - 109519086216 * q^74 - 14574055597 * q^76 - 88616208018 * q^77 + 7621233248 * q^79 - 199583341536 * q^80 - 29584238667 * q^82 - 99007044180 * q^83 - 12911595156 * q^85 - 214357830519 * q^86 + 54423523605 * q^88 - 238298852412 * q^89 + 69105826108 * q^91 - 461776423998 * q^92 - 13393667064 * q^94 - 418952909328 * q^95 - 123483551938 * q^97 - 655061802039 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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