LMFDB - Embedded newform 27.16.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [27,16,Mod(1,27)] 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("27.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	27.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,273] 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:	$38.5272463770$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3470x^{3} - 72162x^{2} - 343807x + 342288 $ x^5 - 2x^4 - 3470x^3 - 72162x^2 - 343807x + 342288
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{14}\cdot 5\cdot 71 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$68.7647$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+225.004 q^{2} +17858.9 q^{4} +338163. q^{5} +626121. q^{7} -3.35462e6 q^{8} +7.60882e7 q^{10} +5.98997e7 q^{11} +1.31085e8 q^{13} +1.40880e8 q^{14} -1.34000e9 q^{16} -9.05163e8 q^{17} +3.66261e9 q^{19} +6.03922e9 q^{20} +1.34777e10 q^{22} -2.86522e10 q^{23} +8.38370e10 q^{25} +2.94948e10 q^{26} +1.11818e10 q^{28} +1.63064e11 q^{29} -1.17149e10 q^{31} -1.91582e11 q^{32} -2.03665e11 q^{34} +2.11731e11 q^{35} +6.27662e11 q^{37} +8.24102e11 q^{38} -1.13441e12 q^{40} +5.13971e11 q^{41} +3.28498e11 q^{43} +1.06974e12 q^{44} -6.44687e12 q^{46} +1.43735e12 q^{47} -4.35553e12 q^{49} +1.88637e13 q^{50} +2.34104e12 q^{52} -2.56371e12 q^{53} +2.02559e13 q^{55} -2.10040e12 q^{56} +3.66900e13 q^{58} +2.85622e13 q^{59} -1.68793e13 q^{61} -2.63589e12 q^{62} +8.02439e11 q^{64} +4.43283e13 q^{65} -4.75907e13 q^{67} -1.61652e13 q^{68} +4.76404e13 q^{70} -1.04997e14 q^{71} -8.48244e13 q^{73} +1.41227e14 q^{74} +6.54101e13 q^{76} +3.75044e13 q^{77} -2.07977e14 q^{79} -4.53140e14 q^{80} +1.15646e14 q^{82} -3.60410e14 q^{83} -3.06093e14 q^{85} +7.39134e13 q^{86} -2.00940e14 q^{88} -3.31163e14 q^{89} +8.20753e13 q^{91} -5.11696e14 q^{92} +3.23410e14 q^{94} +1.23856e15 q^{95} +1.15557e15 q^{97} -9.80013e14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 273 q^{2} + 95405 q^{4} - 16725 q^{5} + 2153539 q^{7} + 13315731 q^{8} - 30467925 q^{10} + 115943493 q^{11} + 65408428 q^{13} + 885109227 q^{14} + 1597063649 q^{16} + 1975444272 q^{17} - 263500334 q^{19}+ \cdots - 930960542438550 q^{98}+O(q^{100}) $ 5 * q + 273 * q^2 + 95405 * q^4 - 16725 * q^5 + 2153539 * q^7 + 13315731 * q^8 - 30467925 * q^10 + 115943493 * q^11 + 65408428 * q^13 + 885109227 * q^14 + 1597063649 * q^16 + 1975444272 * q^17 - 263500334 * q^19 - 15752098605 * q^20 + 15874285977 * q^22 - 10719254334 * q^23 + 65878470860 * q^25 + 51424383876 * q^26 + 121134553795 * q^28 + 286370793390 * q^29 - 300406709951 * q^31 + 625606682931 * q^32 + 822965721624 * q^34 + 244477120485 * q^35 + 112184696098 * q^37 + 1764566464506 * q^38 - 4498792795755 * q^40 + 2894246258082 * q^41 + 1139498878582 * q^43 + 12717942305061 * q^44 - 457487589726 * q^46 + 15070114266798 * q^47 - 2056340352918 * q^49 + 42624872351220 * q^50 + 485820206164 * q^52 + 23527241658009 * q^53 - 4705777789605 * q^55 + 68598287921541 * q^56 + 47931112265094 * q^58 + 32938855380780 * q^59 - 20198107823912 * q^61 + 4988384282481 * q^62 + 53228069199833 * q^64 + 54515008918020 * q^65 + 63021911290318 * q^67 + 176777881834968 * q^68 - 178477863934095 * q^70 + 2034300387888 * q^71 - 67077230062931 * q^73 - 255364190895342 * q^74 + 14768524959274 * q^76 - 217101976083597 * q^77 - 107382054455360 * q^79 - 1363798750436985 * q^80 + 136128058396938 * q^82 - 754356450391395 * q^83 - 762873649735680 * q^85 - 1276712187741210 * q^86 + 312847358643387 * q^88 - 1353618746250426 * q^89 + 569165803886180 * q^91 - 872785068972294 * q^92 + 554616946407222 * q^94 + 190785462235950 * q^95 - 702219406917653 * q^97 - 930960542438550 * 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 $

$2$	225.004	1.24298	0.621492	−	0.783420i	$-0.286527\pi$
$2$	225.004	1.24298	0.621492	+	0.783420i	$0.286527\pi$
$3$	0	0
$3$	0	0
$4$	17858.9	0.545010
$5$	338163.	1.93576	0.967880	−	0.251411i	$-0.0808947\pi$
$5$	338163.	1.93576	0.967880	+	0.251411i	$0.0808947\pi$
$6$	0	0
$7$	626121.	0.287358	0.143679	−	0.989624i	$-0.454107\pi$
$7$	626121.	0.287358	0.143679	+	0.989624i	$0.454107\pi$
$8$	−3.35462e6	−0.565546
$9$	0	0
$10$	7.60882e7	2.40612
$11$	5.98997e7	0.926785	0.463393	−	0.886153i	$-0.346632\pi$
$11$	5.98997e7	0.926785	0.463393	+	0.886153i	$0.346632\pi$
$12$	0	0
$13$	1.31085e8	0.579401	0.289700	−	0.957117i	$-0.406444\pi$
$13$	1.31085e8	0.579401	0.289700	+	0.957117i	$0.406444\pi$
$14$	1.40880e8	0.357181
$15$	0	0
$16$	−1.34000e9	−1.24797
$17$	−9.05163e8	−0.535007	−0.267504	−	0.963557i	$-0.586199\pi$
$17$	−9.05163e8	−0.535007	−0.267504	+	0.963557i	$0.586199\pi$
$18$	0	0
$19$	3.66261e9	0.940023	0.470012	−	0.882660i	$-0.344250\pi$
$19$	3.66261e9	0.940023	0.470012	+	0.882660i	$0.344250\pi$
$20$	6.03922e9	1.05501
$21$	0	0
$22$	1.34777e10	1.15198
$23$	−2.86522e10	−1.75469	−0.877343	−	0.479863i	$-0.840686\pi$
$23$	−2.86522e10	−1.75469	−0.877343	+	0.479863i	$0.840686\pi$
$24$	0	0
$25$	8.38370e10	2.74717
$26$	2.94948e10	0.720186
$27$	0	0
$28$	1.11818e10	0.156613
$29$	1.63064e11	1.75538	0.877692	−	0.479225i	$-0.159082\pi$
$29$	1.63064e11	1.75538	0.877692	+	0.479225i	$0.159082\pi$
$30$	0	0
$31$	−1.17149e10	−0.0764759	−0.0382380	−	0.999269i	$-0.512174\pi$
$31$	−1.17149e10	−0.0764759	−0.0382380	+	0.999269i	$0.512174\pi$
$32$	−1.91582e11	−0.985666
$33$	0	0
$34$	−2.03665e11	−0.665006
$35$	2.11731e11	0.556256
$36$	0	0
$37$	6.27662e11	1.08696	0.543479	−	0.839423i	$-0.317107\pi$
$37$	6.27662e11	1.08696	0.543479	+	0.839423i	$0.317107\pi$
$38$	8.24102e11	1.16843
$39$	0	0
$40$	−1.13441e12	−1.09476
$41$	5.13971e11	0.412154	0.206077	−	0.978536i	$-0.433930\pi$
$41$	5.13971e11	0.412154	0.206077	+	0.978536i	$0.433930\pi$
$42$	0	0
$43$	3.28498e11	0.184297	0.0921487	−	0.995745i	$-0.470626\pi$
$43$	3.28498e11	0.184297	0.0921487	+	0.995745i	$0.470626\pi$
$44$	1.06974e12	0.505107
$45$	0	0
$46$	−6.44687e12	−2.18105
$47$	1.43735e12	0.413837	0.206918	−	0.978358i	$-0.433657\pi$
$47$	1.43735e12	0.413837	0.206918	+	0.978358i	$0.433657\pi$
$48$	0	0
$49$	−4.35553e12	−0.917426
$50$	1.88637e13	3.41469
$51$	0	0
$52$	2.34104e12	0.315779
$53$	−2.56371e12	−0.299778	−0.149889	−	0.988703i	$-0.547892\pi$
$53$	−2.56371e12	−0.299778	−0.149889	+	0.988703i	$0.547892\pi$
$54$	0	0
$55$	2.02559e13	1.79403
$56$	−2.10040e12	−0.162514
$57$	0	0
$58$	3.66900e13	2.18191
$59$	2.85622e13	1.49418	0.747088	−	0.664725i	$-0.231451\pi$
$59$	2.85622e13	1.49418	0.747088	+	0.664725i	$0.231451\pi$
$60$	0	0
$61$	−1.68793e13	−0.687670	−0.343835	−	0.939030i	$-0.611726\pi$
$61$	−1.68793e13	−0.687670	−0.343835	+	0.939030i	$0.611726\pi$
$62$	−2.63589e12	−0.0950584
$63$	0	0
$64$	8.02439e11	0.0228067
$65$	4.43283e13	1.12158
$66$	0	0
$67$	−4.75907e13	−0.959316	−0.479658	−	0.877456i	$-0.659239\pi$
$67$	−4.75907e13	−0.959316	−0.479658	+	0.877456i	$0.659239\pi$
$68$	−1.61652e13	−0.291584
$69$	0	0
$70$	4.76404e13	0.691417
$71$	−1.04997e14	−1.37006	−0.685029	−	0.728515i	$-0.740211\pi$
$71$	−1.04997e14	−1.37006	−0.685029	+	0.728515i	$0.740211\pi$
$72$	0	0
$73$	−8.48244e13	−0.898668	−0.449334	−	0.893364i	$-0.648339\pi$
$73$	−8.48244e13	−0.898668	−0.449334	+	0.893364i	$0.648339\pi$
$74$	1.41227e14	1.35107
$75$	0	0
$76$	6.54101e13	0.512322
$77$	3.75044e13	0.266319
$78$	0	0
$79$	−2.07977e14	−1.21846	−0.609232	−	0.792992i	$-0.708522\pi$
$79$	−2.07977e14	−1.21846	−0.609232	+	0.792992i	$0.708522\pi$
$80$	−4.53140e14	−2.41578
$81$	0	0
$82$	1.15646e14	0.512301
$83$	−3.60410e14	−1.45784	−0.728922	−	0.684597i	$-0.759978\pi$
$83$	−3.60410e14	−1.45784	−0.728922	+	0.684597i	$0.759978\pi$
$84$	0	0
$85$	−3.06093e14	−1.03565
$86$	7.39134e13	0.229079
$87$	0	0
$88$	−2.00940e14	−0.524140
$89$	−3.31163e14	−0.793629	−0.396814	−	0.917899i	$-0.629884\pi$
$89$	−3.31163e14	−0.793629	−0.396814	+	0.917899i	$0.629884\pi$
$90$	0	0
$91$	8.20753e13	0.166495
$92$	−5.11696e14	−0.956321
$93$	0	0
$94$	3.23410e14	0.514393
$95$	1.23856e15	1.81966
$96$	0	0
$97$	1.15557e15	1.45214	0.726069	−	0.687622i	$-0.241345\pi$
$97$	1.15557e15	1.45214	0.726069	+	0.687622i	$0.241345\pi$
$98$	−9.80013e14	−1.14035
$99$	0	0
$100$	1.49723e15	1.49723
$101$	−8.00845e14	−0.743256	−0.371628	−	0.928382i	$-0.621200\pi$
$101$	−8.00845e14	−0.743256	−0.371628	+	0.928382i	$0.621200\pi$
$102$	0	0
$103$	8.15198e14	0.653106	0.326553	−	0.945179i	$-0.394113\pi$
$103$	8.15198e14	0.653106	0.326553	+	0.945179i	$0.394113\pi$
$104$	−4.39741e14	−0.327678
$105$	0	0
$106$	−5.76845e14	−0.372619
$107$	1.29732e15	0.781032	0.390516	−	0.920596i	$-0.372297\pi$
$107$	1.29732e15	0.781032	0.390516	+	0.920596i	$0.372297\pi$
$108$	0	0
$109$	−1.88622e15	−0.988313	−0.494157	−	0.869373i	$-0.664523\pi$
$109$	−1.88622e15	−0.988313	−0.494157	+	0.869373i	$0.664523\pi$
$110$	4.55766e15	2.22996
$111$	0	0
$112$	−8.39003e14	−0.358615
$113$	−2.50679e15	−1.00237	−0.501186	−	0.865339i	$-0.667103\pi$
$113$	−2.50679e15	−1.00237	−0.501186	+	0.865339i	$0.667103\pi$
$114$	0	0
$115$	−9.68913e15	−3.39665
$116$	2.91213e15	0.956701
$117$	0	0
$118$	6.42661e15	1.85724
$119$	−5.66741e14	−0.153739
$120$	0	0
$121$	−5.89280e14	−0.141069
$122$	−3.79791e15	−0.854763
$123$	0	0
$124$	−2.09214e14	−0.0416801
$125$	1.80307e16	3.38210
$126$	0	0
$127$	−3.35126e15	−0.558059	−0.279030	−	0.960283i	$-0.590013\pi$
$127$	−3.35126e15	−0.558059	−0.279030	+	0.960283i	$0.590013\pi$
$128$	6.45831e15	1.01401
$129$	0	0
$130$	9.97405e15	1.39411
$131$	−2.20452e14	−0.0290924	−0.0145462	−	0.999894i	$-0.504630\pi$
$131$	−2.20452e14	−0.0290924	−0.0145462	+	0.999894i	$0.504630\pi$
$132$	0	0
$133$	2.29324e15	0.270123
$134$	−1.07081e16	−1.19241
$135$	0	0
$136$	3.03647e15	0.302571
$137$	−1.42851e16	−1.34735	−0.673674	−	0.739029i	$-0.735285\pi$
$137$	−1.42851e16	−1.34735	−0.673674	+	0.739029i	$0.735285\pi$
$138$	0	0
$139$	−1.32392e16	−1.12009	−0.560043	−	0.828463i	$-0.689215\pi$
$139$	−1.32392e16	−1.12009	−0.560043	+	0.828463i	$0.689215\pi$
$140$	3.78128e15	0.303165
$141$	0	0
$142$	−2.36248e16	−1.70296
$143$	7.85197e15	0.536980
$144$	0	0
$145$	5.51422e16	3.39800
$146$	−1.90858e16	−1.11703
$147$	0	0
$148$	1.12093e16	0.592402
$149$	2.69433e14	0.0135380	0.00676898	−	0.999977i	$-0.497845\pi$
$149$	2.69433e14	0.0135380	0.00676898	+	0.999977i	$0.497845\pi$
$150$	0	0
$151$	−5.52936e15	−0.251390	−0.125695	−	0.992069i	$-0.540116\pi$
$151$	−5.52936e15	−0.251390	−0.125695	+	0.992069i	$0.540116\pi$
$152$	−1.22866e16	−0.531626
$153$	0	0
$154$	8.43865e15	0.331030
$155$	−3.96154e15	−0.148039
$156$	0	0
$157$	3.21545e16	1.09142	0.545712	−	0.837973i	$-0.316259\pi$
$157$	3.21545e16	1.09142	0.545712	+	0.837973i	$0.316259\pi$
$158$	−4.67958e16	−1.51453
$159$	0	0
$160$	−6.47860e16	−1.90801
$161$	−1.79398e16	−0.504223
$162$	0	0
$163$	−2.62998e16	−0.673823	−0.336912	−	0.941536i	$-0.609382\pi$
$163$	−2.62998e16	−0.673823	−0.336912	+	0.941536i	$0.609382\pi$
$164$	9.17894e15	0.224628
$165$	0	0
$166$	−8.10937e16	−1.81208
$167$	4.18840e16	0.894695	0.447347	−	0.894360i	$-0.352369\pi$
$167$	4.18840e16	0.894695	0.447347	+	0.894360i	$0.352369\pi$
$168$	0	0
$169$	−3.40025e16	−0.664295
$170$	−6.88722e16	−1.28729
$171$	0	0
$172$	5.86660e15	0.100444
$173$	5.00325e16	0.820174	0.410087	−	0.912046i	$-0.365498\pi$
$173$	5.00325e16	0.820174	0.410087	+	0.912046i	$0.365498\pi$
$174$	0	0
$175$	5.24921e16	0.789420
$176$	−8.02657e16	−1.15660
$177$	0	0
$178$	−7.45132e16	−0.986468
$179$	−3.46993e16	−0.440477	−0.220238	−	0.975446i	$-0.570684\pi$
$179$	−3.46993e16	−0.440477	−0.220238	+	0.975446i	$0.570684\pi$
$180$	0	0
$181$	1.03066e17	1.20372	0.601862	−	0.798600i	$-0.294426\pi$
$181$	1.03066e17	1.20372	0.601862	+	0.798600i	$0.294426\pi$
$182$	1.84673e16	0.206951
$183$	0	0
$184$	9.61172e16	0.992356
$185$	2.12252e17	2.10409
$186$	0	0
$187$	−5.42189e16	−0.495837
$188$	2.56695e16	0.225545
$189$	0	0
$190$	2.78681e17	2.26181
$191$	1.48673e17	1.16006	0.580031	−	0.814594i	$-0.303040\pi$
$191$	1.48673e17	1.16006	0.580031	+	0.814594i	$0.303040\pi$
$192$	0	0
$193$	1.26887e17	0.915669	0.457834	−	0.889037i	$-0.348625\pi$
$193$	1.26887e17	0.915669	0.457834	+	0.889037i	$0.348625\pi$
$194$	2.60008e17	1.80498
$195$	0	0
$196$	−7.77849e16	−0.500006
$197$	−8.38907e16	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$197$	−8.38907e16	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$198$	0	0
$199$	−1.72506e17	−0.989480	−0.494740	−	0.869041i	$-0.664737\pi$
$199$	−1.72506e17	−0.989480	−0.494740	+	0.869041i	$0.664737\pi$
$200$	−2.81241e17	−1.55365
$201$	0	0
$202$	−1.80193e17	−0.923855
$203$	1.02098e17	0.504423
$204$	0	0
$205$	1.73806e17	0.797831
$206$	1.83423e17	0.811801
$207$	0	0
$208$	−1.75655e17	−0.723077
$209$	2.19389e17	0.871200
$210$	0	0
$211$	−3.34019e17	−1.23496	−0.617479	−	0.786587i	$-0.711846\pi$
$211$	−3.34019e17	−1.23496	−0.617479	+	0.786587i	$0.711846\pi$
$212$	−4.57849e16	−0.163382
$213$	0	0
$214$	2.91903e17	0.970811
$215$	1.11086e17	0.356756
$216$	0	0
$217$	−7.33493e15	−0.0219760
$218$	−4.24408e17	−1.22846
$219$	0	0
$220$	3.61747e17	0.977766
$221$	−1.18654e17	−0.309984
$222$	0	0
$223$	−4.22651e16	−0.103204	−0.0516018	−	0.998668i	$-0.516433\pi$
$223$	−4.22651e16	−0.103204	−0.0516018	+	0.998668i	$0.516433\pi$
$224$	−1.19954e17	−0.283239
$225$	0	0
$226$	−5.64038e17	−1.24593
$227$	4.59947e17	0.982911	0.491456	−	0.870903i	$-0.336465\pi$
$227$	4.59947e17	0.982911	0.491456	+	0.870903i	$0.336465\pi$
$228$	0	0
$229$	−4.27605e17	−0.855611	−0.427806	−	0.903871i	$-0.640713\pi$
$229$	−4.27605e17	−0.855611	−0.427806	+	0.903871i	$0.640713\pi$
$230$	−2.18010e18	−4.22199
$231$	0	0
$232$	−5.47016e17	−0.992750
$233$	1.04449e18	1.83541	0.917705	−	0.397263i	$-0.130040\pi$
$233$	1.04449e18	1.83541	0.917705	+	0.397263i	$0.130040\pi$
$234$	0	0
$235$	4.86060e17	0.801089
$236$	5.10089e17	0.814340
$237$	0	0
$238$	−1.27519e17	−0.191095
$239$	−1.50944e17	−0.219196	−0.109598	−	0.993976i	$-0.534956\pi$
$239$	−1.50944e17	−0.219196	−0.109598	+	0.993976i	$0.534956\pi$
$240$	0	0
$241$	8.22907e16	0.112259	0.0561297	−	0.998423i	$-0.482124\pi$
$241$	8.22907e16	0.112259	0.0561297	+	0.998423i	$0.482124\pi$
$242$	−1.32590e17	−0.175346
$243$	0	0
$244$	−3.01445e17	−0.374787
$245$	−1.47288e18	−1.77592
$246$	0	0
$247$	4.80115e17	0.544650
$248$	3.92989e16	0.0432507
$249$	0	0
$250$	4.05697e18	4.20390
$251$	−1.34443e18	−1.35203	−0.676013	−	0.736890i	$-0.736294\pi$
$251$	−1.34443e18	−1.35203	−0.676013	+	0.736890i	$0.736294\pi$
$252$	0	0
$253$	−1.71626e18	−1.62622
$254$	−7.54048e17	−0.693659
$255$	0	0
$256$	1.42685e18	1.23760
$257$	9.49432e17	0.799771	0.399885	−	0.916565i	$-0.369050\pi$
$257$	9.49432e17	0.799771	0.399885	+	0.916565i	$0.369050\pi$
$258$	0	0
$259$	3.92992e17	0.312346
$260$	7.91653e17	0.611273
$261$	0	0
$262$	−4.96026e16	−0.0361614
$263$	−6.09598e16	−0.0431892	−0.0215946	−	0.999767i	$-0.506874\pi$
$263$	−6.09598e16	−0.0431892	−0.0215946	+	0.999767i	$0.506874\pi$
$264$	0	0
$265$	−8.66952e17	−0.580298
$266$	5.15988e17	0.335759
$267$	0	0
$268$	−8.49917e17	−0.522836
$269$	2.05359e18	1.22849	0.614246	−	0.789114i	$-0.289460\pi$
$269$	2.05359e18	1.22849	0.614246	+	0.789114i	$0.289460\pi$
$270$	0	0
$271$	−1.31009e18	−0.741364	−0.370682	−	0.928760i	$-0.620876\pi$
$271$	−1.31009e18	−0.741364	−0.370682	+	0.928760i	$0.620876\pi$
$272$	1.21292e18	0.667675
$273$	0	0
$274$	−3.21421e18	−1.67473
$275$	5.02180e18	2.54604
$276$	0	0
$277$	−2.38123e18	−1.14341	−0.571707	−	0.820458i	$-0.693719\pi$
$277$	−2.38123e18	−1.14341	−0.571707	+	0.820458i	$0.693719\pi$
$278$	−2.97888e18	−1.39225
$279$	0	0
$280$	−7.10277e17	−0.314588
$281$	−2.97286e18	−1.28197	−0.640984	−	0.767554i	$-0.721474\pi$
$281$	−2.97286e18	−1.28197	−0.640984	+	0.767554i	$0.721474\pi$
$282$	0	0
$283$	9.79557e17	0.400527	0.200263	−	0.979742i	$-0.435820\pi$
$283$	9.79557e17	0.400527	0.200263	+	0.979742i	$0.435820\pi$
$284$	−1.87513e18	−0.746695
$285$	0	0
$286$	1.76673e18	0.667458
$287$	3.21808e17	0.118436
$288$	0	0
$289$	−2.04310e18	−0.713767
$290$	1.24072e19	4.22366
$291$	0	0
$292$	−1.51487e18	−0.489783
$293$	2.17680e18	0.685981	0.342991	−	0.939339i	$-0.388560\pi$
$293$	2.17680e18	0.685981	0.342991	+	0.939339i	$0.388560\pi$
$294$	0	0
$295$	9.65869e18	2.89237
$296$	−2.10556e18	−0.614725
$297$	0	0
$298$	6.06235e16	0.0168275
$299$	−3.75589e18	−1.01667
$300$	0	0
$301$	2.05680e17	0.0529593
$302$	−1.24413e18	−0.312474
$303$	0	0
$304$	−4.90790e18	−1.17312
$305$	−5.70795e18	−1.33116
$306$	0	0
$307$	3.35005e18	0.743898	0.371949	−	0.928253i	$-0.378690\pi$
$307$	3.35005e18	0.743898	0.371949	+	0.928253i	$0.378690\pi$
$308$	6.69787e17	0.145146
$309$	0	0
$310$	−8.91363e17	−0.184010
$311$	−1.34255e18	−0.270537	−0.135268	−	0.990809i	$-0.543190\pi$
$311$	−1.34255e18	−0.270537	−0.135268	+	0.990809i	$0.543190\pi$
$312$	0	0
$313$	1.14612e18	0.220113	0.110057	−	0.993925i	$-0.464897\pi$
$313$	1.14612e18	0.220113	0.110057	+	0.993925i	$0.464897\pi$
$314$	7.23489e18	1.35662
$315$	0	0
$316$	−3.71424e18	−0.664075
$317$	−9.70031e17	−0.169372	−0.0846859	−	0.996408i	$-0.526989\pi$
$317$	−9.70031e17	−0.169372	−0.0846859	+	0.996408i	$0.526989\pi$
$318$	0	0
$319$	9.76746e18	1.62686
$320$	2.71355e17	0.0441483
$321$	0	0
$322$	−4.03652e18	−0.626741
$323$	−3.31526e18	−0.502919
$324$	0	0
$325$	1.09898e19	1.59171
$326$	−5.91757e18	−0.837552
$327$	0	0
$328$	−1.72417e18	−0.233092
$329$	8.99956e17	0.118919
$330$	0	0
$331$	−1.40805e19	−1.77790	−0.888952	−	0.458000i	$-0.848566\pi$
$331$	−1.40805e19	−1.77790	−0.888952	+	0.458000i	$0.848566\pi$
$332$	−6.43651e18	−0.794538
$333$	0	0
$334$	9.42408e18	1.11209
$335$	−1.60935e19	−1.85701
$336$	0	0
$337$	1.64277e18	0.181281	0.0906404	−	0.995884i	$-0.471109\pi$
$337$	1.64277e18	0.181281	0.0906404	+	0.995884i	$0.471109\pi$
$338$	−7.65071e18	−0.825708
$339$	0	0
$340$	−5.46648e18	−0.564437
$341$	−7.01717e17	−0.0708768
$342$	0	0
$343$	−5.69964e18	−0.550987
$344$	−1.10198e18	−0.104229
$345$	0	0
$346$	1.12575e19	1.01946
$347$	5.79681e18	0.513710	0.256855	−	0.966450i	$-0.417314\pi$
$347$	5.79681e18	0.513710	0.256855	+	0.966450i	$0.417314\pi$
$348$	0	0
$349$	−2.17877e18	−0.184935	−0.0924677	−	0.995716i	$-0.529475\pi$
$349$	−2.17877e18	−0.184935	−0.0924677	+	0.995716i	$0.529475\pi$
$350$	1.18109e19	0.981237
$351$	0	0
$352$	−1.14757e19	−0.913501
$353$	−4.24598e18	−0.330878	−0.165439	−	0.986220i	$-0.552904\pi$
$353$	−4.24598e18	−0.330878	−0.165439	+	0.986220i	$0.552904\pi$
$354$	0	0
$355$	−3.55061e19	−2.65211
$356$	−5.91421e18	−0.432535
$357$	0	0
$358$	−7.80749e18	−0.547505
$359$	−3.84070e18	−0.263756	−0.131878	−	0.991266i	$-0.542101\pi$
$359$	−3.84070e18	−0.263756	−0.131878	+	0.991266i	$0.542101\pi$
$360$	0	0
$361$	−1.76642e18	−0.116356
$362$	2.31903e19	1.49621
$363$	0	0
$364$	1.46577e18	0.0907416
$365$	−2.86845e19	−1.73961
$366$	0	0
$367$	−1.60333e19	−0.933316	−0.466658	−	0.884438i	$-0.654542\pi$
$367$	−1.60333e19	−0.933316	−0.466658	+	0.884438i	$0.654542\pi$
$368$	3.83940e19	2.18980
$369$	0	0
$370$	4.77576e19	2.61535
$371$	−1.60519e18	−0.0861435
$372$	0	0
$373$	1.86268e19	0.960113	0.480057	−	0.877237i	$-0.340616\pi$
$373$	1.86268e19	0.960113	0.480057	+	0.877237i	$0.340616\pi$
$374$	−1.21995e19	−0.616318
$375$	0	0
$376$	−4.82176e18	−0.234044
$377$	2.13753e19	1.01707
$378$	0	0
$379$	2.06427e19	0.943999	0.472000	−	0.881599i	$-0.343532\pi$
$379$	2.06427e19	0.943999	0.472000	+	0.881599i	$0.343532\pi$
$380$	2.21193e19	0.991732
$381$	0	0
$382$	3.34520e19	1.44194
$383$	1.29313e19	0.546576	0.273288	−	0.961932i	$-0.411889\pi$
$383$	1.29313e19	0.546576	0.273288	+	0.961932i	$0.411889\pi$
$384$	0	0
$385$	1.26826e19	0.515530
$386$	2.85502e19	1.13816
$387$	0	0
$388$	2.06372e19	0.791429
$389$	2.53083e19	0.952009	0.476005	−	0.879443i	$-0.342085\pi$
$389$	2.53083e19	0.952009	0.476005	+	0.879443i	$0.342085\pi$
$390$	0	0
$391$	2.59349e19	0.938770
$392$	1.46111e19	0.518846
$393$	0	0
$394$	−1.88758e19	−0.645183
$395$	−7.03304e19	−2.35866
$396$	0	0
$397$	−2.82018e19	−0.910641	−0.455321	−	0.890328i	$-0.650475\pi$
$397$	−2.82018e19	−0.910641	−0.455321	+	0.890328i	$0.650475\pi$
$398$	−3.88146e19	−1.22991
$399$	0	0
$400$	−1.12342e20	−3.42840
$401$	1.86496e19	0.558582	0.279291	−	0.960207i	$-0.409901\pi$
$401$	1.86496e19	0.558582	0.279291	+	0.960207i	$0.409901\pi$
$402$	0	0
$403$	−1.53565e18	−0.0443102
$404$	−1.43022e19	−0.405081
$405$	0	0
$406$	2.29724e19	0.626990
$407$	3.75967e19	1.00738
$408$	0	0
$409$	4.72343e19	1.21992	0.609961	−	0.792431i	$-0.291185\pi$
$409$	4.72343e19	1.21992	0.609961	+	0.792431i	$0.291185\pi$
$410$	3.91071e19	0.991692
$411$	0	0
$412$	1.45585e19	0.355949
$413$	1.78834e19	0.429363
$414$	0	0
$415$	−1.21877e20	−2.82204
$416$	−2.51136e19	−0.571096
$417$	0	0
$418$	4.93634e19	1.08289
$419$	7.96907e19	1.71713	0.858563	−	0.512708i	$-0.171358\pi$
$419$	7.96907e19	1.71713	0.858563	+	0.512708i	$0.171358\pi$
$420$	0	0
$421$	−2.96485e19	−0.616435	−0.308218	−	0.951316i	$-0.599733\pi$
$421$	−2.96485e19	−0.616435	−0.308218	+	0.951316i	$0.599733\pi$
$422$	−7.51556e19	−1.53503
$423$	0	0
$424$	8.60025e18	0.169538
$425$	−7.58861e19	−1.46976
$426$	0	0
$427$	−1.05685e19	−0.197607
$428$	2.31687e19	0.425670
$429$	0	0
$430$	2.49948e19	0.443442
$431$	−5.84458e19	−1.01900	−0.509499	−	0.860471i	$-0.670169\pi$
$431$	−5.84458e19	−1.01900	−0.509499	+	0.860471i	$0.670169\pi$
$432$	0	0
$433$	6.25471e19	1.05329	0.526645	−	0.850085i	$-0.323450\pi$
$433$	6.25471e19	1.05329	0.526645	+	0.850085i	$0.323450\pi$
$434$	−1.65039e18	−0.0273158
$435$	0	0
$436$	−3.36858e19	−0.538640
$437$	−1.04942e20	−1.64945
$438$	0	0
$439$	1.17832e19	0.178970	0.0894851	−	0.995988i	$-0.471478\pi$
$439$	1.17832e19	0.178970	0.0894851	+	0.995988i	$0.471478\pi$
$440$	−6.79507e19	−1.01461
$441$	0	0
$442$	−2.66976e19	−0.385305
$443$	−2.22694e18	−0.0315996	−0.0157998	−	0.999875i	$-0.505029\pi$
$443$	−2.22694e18	−0.0315996	−0.0157998	+	0.999875i	$0.505029\pi$
$444$	0	0
$445$	−1.11987e20	−1.53628
$446$	−9.50981e18	−0.128280
$447$	0	0
$448$	5.02424e17	0.00655368
$449$	7.64814e19	0.981088	0.490544	−	0.871416i	$-0.336798\pi$
$449$	7.64814e19	0.981088	0.490544	+	0.871416i	$0.336798\pi$
$450$	0	0
$451$	3.07867e19	0.381978
$452$	−4.47684e19	−0.546303
$453$	0	0
$454$	1.03490e20	1.22174
$455$	2.77549e19	0.322295
$456$	0	0
$457$	5.42282e19	0.609332	0.304666	−	0.952459i	$-0.401455\pi$
$457$	5.42282e19	0.609332	0.304666	+	0.952459i	$0.401455\pi$
$458$	−9.62129e19	−1.06351
$459$	0	0
$460$	−1.73037e20	−1.85121
$461$	1.29183e20	1.35972	0.679859	−	0.733342i	$-0.262041\pi$
$461$	1.29183e20	1.35972	0.679859	+	0.733342i	$0.262041\pi$
$462$	0	0
$463$	9.06888e19	0.924052	0.462026	−	0.886866i	$-0.347123\pi$
$463$	9.06888e19	0.924052	0.462026	+	0.886866i	$0.347123\pi$
$464$	−2.18506e20	−2.19067
$465$	0	0
$466$	2.35014e20	2.28138
$467$	6.21059e19	0.593275	0.296638	−	0.954990i	$-0.404135\pi$
$467$	6.21059e19	0.593275	0.296638	+	0.954990i	$0.404135\pi$
$468$	0	0
$469$	−2.97976e19	−0.275667
$470$	1.09366e20	0.995741
$471$	0	0
$472$	−9.58152e19	−0.845025
$473$	1.96769e19	0.170804
$474$	0	0
$475$	3.07062e20	2.58240
$476$	−1.01214e19	−0.0837890
$477$	0	0
$478$	−3.39630e19	−0.272457
$479$	−5.08603e19	−0.401663	−0.200832	−	0.979626i	$-0.564364\pi$
$479$	−5.08603e19	−0.401663	−0.200832	+	0.979626i	$0.564364\pi$
$480$	0	0
$481$	8.22773e19	0.629784
$482$	1.85158e19	0.139537
$483$	0	0
$484$	−1.05239e19	−0.0768839
$485$	3.90771e20	2.81099
$486$	0	0
$487$	−4.66077e19	−0.325080	−0.162540	−	0.986702i	$-0.551969\pi$
$487$	−4.66077e19	−0.325080	−0.162540	+	0.986702i	$0.551969\pi$
$488$	5.66235e19	0.388909
$489$	0	0
$490$	−3.31405e20	−2.20744
$491$	−4.08772e19	−0.268145	−0.134073	−	0.990972i	$-0.542806\pi$
$491$	−4.08772e19	−0.268145	−0.134073	+	0.990972i	$0.542806\pi$
$492$	0	0
$493$	−1.47599e20	−0.939143
$494$	1.08028e20	0.676992
$495$	0	0
$496$	1.56980e19	0.0954400
$497$	−6.57408e19	−0.393697
$498$	0	0
$499$	9.22038e19	0.535790	0.267895	−	0.963448i	$-0.413672\pi$
$499$	9.22038e19	0.535790	0.267895	+	0.963448i	$0.413672\pi$
$500$	3.22007e20	1.84328
$501$	0	0
$502$	−3.02502e20	−1.68055
$503$	−8.69650e19	−0.475976	−0.237988	−	0.971268i	$-0.576488\pi$
$503$	−8.69650e19	−0.475976	−0.237988	+	0.971268i	$0.576488\pi$
$504$	0	0
$505$	−2.70817e20	−1.43877
$506$	−3.86165e20	−2.02136
$507$	0	0
$508$	−5.98497e19	−0.304148
$509$	−1.28666e20	−0.644290	−0.322145	−	0.946690i	$-0.604404\pi$
$509$	−1.28666e20	−0.644290	−0.322145	+	0.946690i	$0.604404\pi$
$510$	0	0
$511$	−5.31104e19	−0.258239
$512$	1.09422e20	0.524299
$513$	0	0
$514$	2.13626e20	0.994102
$515$	2.75670e20	1.26426
$516$	0	0
$517$	8.60969e19	0.383538
$518$	8.84249e19	0.388241
$519$	0	0
$520$	−1.48704e20	−0.634306
$521$	3.32471e20	1.39788	0.698942	−	0.715178i	$-0.253654\pi$
$521$	3.32471e20	1.39788	0.698942	+	0.715178i	$0.253654\pi$
$522$	0	0
$523$	2.81814e18	0.0115133	0.00575665	−	0.999983i	$-0.498168\pi$
$523$	2.81814e18	0.0115133	0.00575665	+	0.999983i	$0.498168\pi$
$524$	−3.93702e18	−0.0158556
$525$	0	0
$526$	−1.37162e19	−0.0536835
$527$	1.06039e19	0.0409152
$528$	0	0
$529$	5.54315e20	2.07892
$530$	−1.95068e20	−0.721301
$531$	0	0
$532$	4.09546e19	0.147220
$533$	6.73740e19	0.238802
$534$	0	0
$535$	4.38706e20	1.51189
$536$	1.59649e20	0.542537
$537$	0	0
$538$	4.62067e20	1.52700
$539$	−2.60895e20	−0.850257
$540$	0	0
$541$	−5.67492e20	−1.79879	−0.899394	−	0.437139i	$-0.855992\pi$
$541$	−5.67492e20	−1.79879	−0.899394	+	0.437139i	$0.855992\pi$
$542$	−2.94776e20	−0.921503
$543$	0	0
$544$	1.73413e20	0.527339
$545$	−6.37852e20	−1.91314
$546$	0	0
$547$	−5.68054e20	−1.65762	−0.828809	−	0.559532i	$-0.810981\pi$
$547$	−5.68054e20	−1.65762	−0.828809	+	0.559532i	$0.810981\pi$
$548$	−2.55116e20	−0.734317
$549$	0	0
$550$	1.12993e21	3.16468
$551$	5.97238e20	1.65010
$552$	0	0
$553$	−1.30219e20	−0.350135
$554$	−5.35787e20	−1.42125
$555$	0	0
$556$	−2.36438e20	−0.610458
$557$	6.55817e19	0.167059	0.0835293	−	0.996505i	$-0.473381\pi$
$557$	6.55817e19	0.167059	0.0835293	+	0.996505i	$0.473381\pi$
$558$	0	0
$559$	4.30613e19	0.106782
$560$	−2.83720e20	−0.694193
$561$	0	0
$562$	−6.68906e20	−1.59347
$563$	1.55041e20	0.364447	0.182224	−	0.983257i	$-0.441671\pi$
$563$	1.55041e20	0.364447	0.182224	+	0.983257i	$0.441671\pi$
$564$	0	0
$565$	−8.47704e20	−1.94035
$566$	2.20404e20	0.497848
$567$	0	0
$568$	3.52225e20	0.774831
$569$	4.88149e20	1.05977	0.529884	−	0.848070i	$-0.322235\pi$
$569$	4.88149e20	1.05977	0.529884	+	0.848070i	$0.322235\pi$
$570$	0	0
$571$	2.11425e19	0.0447081	0.0223540	−	0.999750i	$-0.492884\pi$
$571$	2.11425e19	0.0447081	0.0223540	+	0.999750i	$0.492884\pi$
$572$	1.40227e20	0.292659
$573$	0	0
$574$	7.24081e19	0.147214
$575$	−2.40211e21	−4.82042
$576$	0	0
$577$	−9.69561e19	−0.189564	−0.0947821	−	0.995498i	$-0.530215\pi$
$577$	−9.69561e19	−0.189564	−0.0947821	+	0.995498i	$0.530215\pi$
$578$	−4.59707e20	−0.887201
$579$	0	0
$580$	9.84777e20	1.85194
$581$	−2.25660e20	−0.418923
$582$	0	0
$583$	−1.53565e20	−0.277830
$584$	2.84553e20	0.508238
$585$	0	0
$586$	4.89790e20	0.852664
$587$	−6.70360e20	−1.15218	−0.576092	−	0.817385i	$-0.695423\pi$
$587$	−6.70360e20	−1.15218	−0.576092	+	0.817385i	$0.695423\pi$
$588$	0	0
$589$	−4.29070e19	−0.0718892
$590$	2.17325e21	3.59517
$591$	0	0
$592$	−8.41068e20	−1.35650
$593$	9.09711e20	1.44875	0.724375	−	0.689406i	$-0.242129\pi$
$593$	9.09711e20	1.44875	0.724375	+	0.689406i	$0.242129\pi$
$594$	0	0
$595$	−1.91651e20	−0.297601
$596$	4.81177e18	0.00737832
$597$	0	0
$598$	−8.45090e20	−1.26370
$599$	5.15025e20	0.760549	0.380274	−	0.924874i	$-0.375830\pi$
$599$	5.15025e20	0.760549	0.380274	+	0.924874i	$0.375830\pi$
$600$	0	0
$601$	6.88926e20	0.992234	0.496117	−	0.868256i	$-0.334759\pi$
$601$	6.88926e20	0.992234	0.496117	+	0.868256i	$0.334759\pi$
$602$	4.62787e19	0.0658276
$603$	0	0
$604$	−9.87482e19	−0.137010
$605$	−1.99273e20	−0.273076
$606$	0	0
$607$	−1.37057e21	−1.83226	−0.916128	−	0.400885i	$-0.868703\pi$
$607$	−1.37057e21	−1.83226	−0.916128	+	0.400885i	$0.868703\pi$
$608$	−7.01690e20	−0.926549
$609$	0	0
$610$	−1.28431e21	−1.65462
$611$	1.88416e20	0.239777
$612$	0	0
$613$	3.73313e20	0.463575	0.231787	−	0.972766i	$-0.425543\pi$
$613$	3.73313e20	0.463575	0.231787	+	0.972766i	$0.425543\pi$
$614$	7.53775e20	0.924654
$615$	0	0
$616$	−1.25813e20	−0.150616
$617$	6.54641e20	0.774221	0.387110	−	0.922033i	$-0.373473\pi$
$617$	6.54641e20	0.774221	0.387110	+	0.922033i	$0.373473\pi$
$618$	0	0
$619$	2.49440e20	0.287930	0.143965	−	0.989583i	$-0.454015\pi$
$619$	2.49440e20	0.287930	0.143965	+	0.989583i	$0.454015\pi$
$620$	−7.07487e19	−0.0806827
$621$	0	0
$622$	−3.02079e20	−0.336273
$623$	−2.07348e20	−0.228055
$624$	0	0
$625$	3.53881e21	3.79977
$626$	2.57881e20	0.273598
$627$	0	0
$628$	5.74242e20	0.594837
$629$	−5.68136e20	−0.581531
$630$	0	0
$631$	−4.10195e20	−0.409987	−0.204993	−	0.978763i	$-0.565717\pi$
$631$	−4.10195e20	−0.409987	−0.204993	+	0.978763i	$0.565717\pi$
$632$	6.97684e20	0.689098
$633$	0	0
$634$	−2.18261e20	−0.210527
$635$	−1.13327e21	−1.08027
$636$	0	0
$637$	−5.70947e20	−0.531557
$638$	2.19772e21	2.02217
$639$	0	0
$640$	2.18396e21	1.96289
$641$	1.62071e20	0.143969	0.0719846	−	0.997406i	$-0.477067\pi$
$641$	1.62071e20	0.143969	0.0719846	+	0.997406i	$0.477067\pi$
$642$	0	0
$643$	1.31227e21	1.13878	0.569392	−	0.822066i	$-0.307179\pi$
$643$	1.31227e21	1.13878	0.569392	+	0.822066i	$0.307179\pi$
$644$	−3.20384e20	−0.274806
$645$	0	0
$646$	−7.45947e20	−0.625121
$647$	2.02095e21	1.67407	0.837034	−	0.547150i	$-0.184287\pi$
$647$	2.02095e21	1.67407	0.837034	+	0.547150i	$0.184287\pi$
$648$	0	0
$649$	1.71087e21	1.38478
$650$	2.47275e21	1.97847
$651$	0	0
$652$	−4.69686e20	−0.367240
$653$	−5.85047e20	−0.452212	−0.226106	−	0.974103i	$-0.572600\pi$
$653$	−5.85047e20	−0.452212	−0.226106	+	0.974103i	$0.572600\pi$
$654$	0	0
$655$	−7.45488e19	−0.0563159
$656$	−6.88722e20	−0.514357
$657$	0	0
$658$	2.02494e20	0.147815
$659$	−2.12486e21	−1.53352	−0.766761	−	0.641933i	$-0.778133\pi$
$659$	−2.12486e21	−1.53352	−0.766761	+	0.641933i	$0.778133\pi$
$660$	0	0
$661$	−9.22767e20	−0.651000	−0.325500	−	0.945542i	$-0.605533\pi$
$661$	−9.22767e20	−0.651000	−0.325500	+	0.945542i	$0.605533\pi$
$662$	−3.16817e21	−2.20991
$663$	0	0
$664$	1.20904e21	0.824477
$665$	7.75489e20	0.522893
$666$	0	0
$667$	−4.67214e21	−3.08015
$668$	7.48002e20	0.487617
$669$	0	0
$670$	−3.62109e21	−2.30823
$671$	−1.01106e21	−0.637322
$672$	0	0
$673$	2.24077e21	1.38129	0.690644	−	0.723194i	$-0.257327\pi$
$673$	2.24077e21	1.38129	0.690644	+	0.723194i	$0.257327\pi$
$674$	3.69629e20	0.225329
$675$	0	0
$676$	−6.07247e20	−0.362047
$677$	−2.92445e20	−0.172437	−0.0862184	−	0.996276i	$-0.527478\pi$
$677$	−2.92445e20	−0.172437	−0.0862184	+	0.996276i	$0.527478\pi$
$678$	0	0
$679$	7.23526e20	0.417283
$680$	1.02682e21	0.585705
$681$	0	0
$682$	−1.57889e20	−0.0880987
$683$	−7.11906e20	−0.392887	−0.196443	−	0.980515i	$-0.562939\pi$
$683$	−7.11906e20	−0.392887	−0.196443	+	0.980515i	$0.562939\pi$
$684$	0	0
$685$	−4.83071e21	−2.60814
$686$	−1.28244e21	−0.684868
$687$	0	0
$688$	−4.40188e20	−0.229998
$689$	−3.36065e20	−0.173692
$690$	0	0
$691$	1.66163e21	0.840327	0.420163	−	0.907448i	$-0.361973\pi$
$691$	1.66163e21	0.840327	0.420163	+	0.907448i	$0.361973\pi$
$692$	8.93524e20	0.447003
$693$	0	0
$694$	1.30431e21	0.638533
$695$	−4.47702e21	−2.16822
$696$	0	0
$697$	−4.65227e20	−0.220505
$698$	−4.90232e20	−0.229872
$699$	0	0
$700$	9.37449e20	0.430242
$701$	6.27673e20	0.285002	0.142501	−	0.989795i	$-0.454486\pi$
$701$	6.27673e20	0.285002	0.142501	+	0.989795i	$0.454486\pi$
$702$	0	0
$703$	2.29888e21	1.02177
$704$	4.80658e19	0.0211369
$705$	0	0
$706$	−9.55363e20	−0.411276
$707$	−5.01426e20	−0.213580
$708$	0	0
$709$	9.17275e20	0.382519	0.191259	−	0.981540i	$-0.438743\pi$
$709$	9.17275e20	0.382519	0.191259	+	0.981540i	$0.438743\pi$
$710$	−7.98903e21	−3.29653
$711$	0	0
$712$	1.11093e21	0.448833
$713$	3.35657e20	0.134191
$714$	0	0
$715$	2.65525e21	1.03947
$716$	−6.19690e20	−0.240064
$717$	0	0
$718$	−8.64173e20	−0.327844
$719$	3.88381e20	0.145811	0.0729057	−	0.997339i	$-0.476773\pi$
$719$	3.88381e20	0.145811	0.0729057	+	0.997339i	$0.476773\pi$
$720$	0	0
$721$	5.10413e20	0.187675
$722$	−3.97451e20	−0.144629
$723$	0	0
$724$	1.84065e21	0.656041
$725$	1.36708e22	4.82234
$726$	0	0
$727$	3.48981e20	0.120585	0.0602926	−	0.998181i	$-0.480797\pi$
$727$	3.48981e20	0.120585	0.0602926	+	0.998181i	$0.480797\pi$
$728$	−2.75331e20	−0.0941607
$729$	0	0
$730$	−6.45414e21	−2.16230
$731$	−2.97344e20	−0.0986005
$732$	0	0
$733$	−2.99287e21	−0.972317	−0.486159	−	0.873871i	$-0.661602\pi$
$733$	−2.99287e21	−0.972317	−0.486159	+	0.873871i	$0.661602\pi$
$734$	−3.60757e21	−1.16010
$735$	0	0
$736$	5.48925e21	1.72954
$737$	−2.85067e21	−0.889080
$738$	0	0
$739$	2.27132e21	0.694137	0.347069	−	0.937840i	$-0.387177\pi$
$739$	2.27132e21	0.694137	0.347069	+	0.937840i	$0.387177\pi$
$740$	3.79059e21	1.14675
$741$	0	0
$742$	−3.61175e20	−0.107075
$743$	−1.24464e21	−0.365282	−0.182641	−	0.983180i	$-0.558465\pi$
$743$	−1.24464e21	−0.365282	−0.182641	+	0.983180i	$0.558465\pi$
$744$	0	0
$745$	9.11123e19	0.0262063
$746$	4.19111e21	1.19341
$747$	0	0
$748$	−9.68289e20	−0.270236
$749$	8.12280e20	0.224436
$750$	0	0
$751$	5.58741e21	1.51325	0.756625	−	0.653849i	$-0.226847\pi$
$751$	5.58741e21	1.51325	0.756625	+	0.653849i	$0.226847\pi$
$752$	−1.92605e21	−0.516458
$753$	0	0
$754$	4.80952e21	1.26420
$755$	−1.86983e21	−0.486631
$756$	0	0
$757$	−5.61350e21	−1.43224	−0.716118	−	0.697979i	$-0.754083\pi$
$757$	−5.61350e21	−1.43224	−0.716118	+	0.697979i	$0.754083\pi$
$758$	4.64468e21	1.17338
$759$	0	0
$760$	−4.15490e21	−1.02910
$761$	−1.63802e21	−0.401729	−0.200864	−	0.979619i	$-0.564375\pi$
$761$	−1.63802e21	−0.401729	−0.200864	+	0.979619i	$0.564375\pi$
$762$	0	0
$763$	−1.18100e21	−0.284000
$764$	2.65513e21	0.632245
$765$	0	0
$766$	2.90959e21	0.679385
$767$	3.74409e21	0.865727
$768$	0	0
$769$	4.65164e20	0.105477	0.0527386	−	0.998608i	$-0.483205\pi$
$769$	4.65164e20	0.105477	0.0527386	+	0.998608i	$0.483205\pi$
$770$	2.85364e21	0.640795
$771$	0	0
$772$	2.26607e21	0.499048
$773$	−1.70986e21	−0.372918	−0.186459	−	0.982463i	$-0.559701\pi$
$773$	−1.70986e21	−0.372918	−0.186459	+	0.982463i	$0.559701\pi$
$774$	0	0
$775$	−9.82139e20	−0.210092
$776$	−3.87649e21	−0.821251
$777$	0	0
$778$	5.69448e21	1.18333
$779$	1.88247e21	0.387434
$780$	0	0
$781$	−6.28928e21	−1.26975
$782$	5.83547e21	1.16688
$783$	0	0
$784$	5.83642e21	1.14492
$785$	1.08735e22	2.11274
$786$	0	0
$787$	−8.35194e21	−1.59212	−0.796062	−	0.605215i	$-0.793087\pi$
$787$	−8.35194e21	−1.59212	−0.796062	+	0.605215i	$0.793087\pi$
$788$	−1.49819e21	−0.282893
$789$	0	0
$790$	−1.58246e22	−2.93177
$791$	−1.56955e21	−0.288040
$792$	0	0
$793$	−2.21263e21	−0.398437
$794$	−6.34551e21	−1.13191
$795$	0	0
$796$	−3.08077e21	−0.539276
$797$	9.77322e21	1.69473	0.847364	−	0.531012i	$-0.178188\pi$
$797$	9.77322e21	1.69473	0.847364	+	0.531012i	$0.178188\pi$
$798$	0	0
$799$	−1.30104e21	−0.221406
$800$	−1.60617e22	−2.70779
$801$	0	0
$802$	4.19624e21	0.694308
$803$	−5.08095e21	−0.832873
$804$	0	0
$805$	−6.06657e21	−0.976055
$806$	−3.45527e20	−0.0550769
$807$	0	0
$808$	2.68653e21	0.420345
$809$	1.08324e22	1.67924	0.839620	−	0.543174i	$-0.182778\pi$
$809$	1.08324e22	1.67924	0.839620	+	0.543174i	$0.182778\pi$
$810$	0	0
$811$	−6.79008e21	−1.03328	−0.516641	−	0.856202i	$-0.672818\pi$
$811$	−6.79008e21	−1.03328	−0.516641	+	0.856202i	$0.672818\pi$
$812$	1.82335e21	0.274916
$813$	0	0
$814$	8.45942e21	1.25215
$815$	−8.89365e21	−1.30436
$816$	0	0
$817$	1.20316e21	0.173244
$818$	1.06279e22	1.51634
$819$	0	0
$820$	3.10398e21	0.434826
$821$	5.96458e21	0.827954	0.413977	−	0.910287i	$-0.364139\pi$
$821$	5.96458e21	0.827954	0.413977	+	0.910287i	$0.364139\pi$
$822$	0	0
$823$	3.81758e21	0.520343	0.260171	−	0.965562i	$-0.416221\pi$
$823$	3.81758e21	0.520343	0.260171	+	0.965562i	$0.416221\pi$
$824$	−2.73468e21	−0.369362
$825$	0	0
$826$	4.02384e21	0.533691
$827$	−3.40375e20	−0.0447369	−0.0223685	−	0.999750i	$-0.507121\pi$
$827$	−3.40375e20	−0.0447369	−0.0223685	+	0.999750i	$0.507121\pi$
$828$	0	0
$829$	7.04924e21	0.909879	0.454939	−	0.890522i	$-0.349661\pi$
$829$	7.04924e21	0.909879	0.454939	+	0.890522i	$0.349661\pi$
$830$	−2.74229e22	−3.50774
$831$	0	0
$832$	1.05188e20	0.0132142
$833$	3.94247e21	0.490829
$834$	0	0
$835$	1.41637e22	1.73192
$836$	3.91804e21	0.474812
$837$	0	0
$838$	1.79307e22	2.13436
$839$	−1.16619e22	−1.37580	−0.687898	−	0.725808i	$-0.741466\pi$
$839$	−1.16619e22	−1.37580	−0.687898	+	0.725808i	$0.741466\pi$
$840$	0	0
$841$	1.79606e22	2.08137
$842$	−6.67104e21	−0.766219
$843$	0	0
$844$	−5.96520e21	−0.673064
$845$	−1.14984e22	−1.28592
$846$	0	0
$847$	−3.68960e20	−0.0405372
$848$	3.43537e21	0.374115
$849$	0	0
$850$	−1.70747e22	−1.82688
$851$	−1.79839e22	−1.90727
$852$	0	0
$853$	−1.44889e22	−1.50979	−0.754897	−	0.655843i	$-0.772313\pi$
$853$	−1.44889e22	−1.50979	−0.754897	+	0.655843i	$0.772313\pi$
$854$	−2.37795e21	−0.245623
$855$	0	0
$856$	−4.35201e21	−0.441710
$857$	8.36197e20	0.0841303	0.0420652	−	0.999115i	$-0.486606\pi$
$857$	8.36197e20	0.0841303	0.0420652	+	0.999115i	$0.486606\pi$
$858$	0	0
$859$	−2.41500e21	−0.238763	−0.119382	−	0.992848i	$-0.538091\pi$
$859$	−2.41500e21	−0.238763	−0.119382	+	0.992848i	$0.538091\pi$
$860$	1.98387e21	0.194435
$861$	0	0
$862$	−1.31505e22	−1.26660
$863$	1.69939e22	1.62260	0.811300	−	0.584631i	$-0.198761\pi$
$863$	1.69939e22	1.62260	0.811300	+	0.584631i	$0.198761\pi$
$864$	0	0
$865$	1.69192e22	1.58766
$866$	1.40734e22	1.30922
$867$	0	0
$868$	−1.30994e20	−0.0119771
$869$	−1.24578e22	−1.12926
$870$	0	0
$871$	−6.23845e21	−0.555828
$872$	6.32756e21	0.558937
$873$	0	0
$874$	−2.36124e22	−2.05024
$875$	1.12894e22	0.971873
$876$	0	0
$877$	1.53851e22	1.30197	0.650987	−	0.759088i	$-0.274355\pi$
$877$	1.53851e22	1.30197	0.650987	+	0.759088i	$0.274355\pi$
$878$	2.65128e21	0.222457
$879$	0	0
$880$	−2.71429e22	−2.23891
$881$	6.57993e21	0.538148	0.269074	−	0.963120i	$-0.413282\pi$
$881$	6.57993e21	0.538148	0.269074	+	0.963120i	$0.413282\pi$
$882$	0	0
$883$	2.13856e22	1.71956	0.859779	−	0.510666i	$-0.170601\pi$
$883$	2.13856e22	1.71956	0.859779	+	0.510666i	$0.170601\pi$
$884$	−2.11902e21	−0.168944
$885$	0	0
$886$	−5.01071e20	−0.0392777
$887$	−1.81183e22	−1.40828	−0.704141	−	0.710060i	$-0.748668\pi$
$887$	−1.81183e22	−1.40828	−0.704141	+	0.710060i	$0.748668\pi$
$888$	0	0
$889$	−2.09829e21	−0.160363
$890$	−2.51976e22	−1.90957
$891$	0	0
$892$	−7.54806e20	−0.0562470
$893$	5.26446e21	0.389016
$894$	0	0
$895$	−1.17340e22	−0.852657
$896$	4.04368e21	0.291385
$897$	0	0
$898$	1.72086e22	1.21948
$899$	−1.91027e21	−0.134245
$900$	0	0
$901$	2.32057e21	0.160383
$902$	6.92713e21	0.474793
$903$	0	0
$904$	8.40931e21	0.566888
$905$	3.48532e22	2.33012
$906$	0	0
$907$	−1.36542e22	−0.897865	−0.448933	−	0.893566i	$-0.648196\pi$
$907$	−1.36542e22	−0.897865	−0.448933	+	0.893566i	$0.648196\pi$
$908$	8.21414e21	0.535696
$909$	0	0
$910$	6.24496e21	0.400608
$911$	−3.43297e21	−0.218415	−0.109207	−	0.994019i	$-0.534831\pi$
$911$	−3.43297e21	−0.218415	−0.109207	+	0.994019i	$0.534831\pi$
$912$	0	0
$913$	−2.15884e22	−1.35111
$914$	1.22016e22	0.757390
$915$	0	0
$916$	−7.63654e21	−0.466316
$917$	−1.38030e20	−0.00835992
$918$	0	0
$919$	−2.48854e22	−1.48279	−0.741393	−	0.671071i	$-0.765834\pi$
$919$	−2.48854e22	−1.48279	−0.741393	+	0.671071i	$0.765834\pi$
$920$	3.25033e22	1.92096
$921$	0	0
$922$	2.90668e22	1.69011
$923$	−1.37636e22	−0.793813
$924$	0	0
$925$	5.26212e22	2.98606
$926$	2.04054e22	1.14858
$927$	0	0
$928$	−3.12401e22	−1.73022
$929$	−2.33577e22	−1.28325	−0.641626	−	0.767017i	$-0.721740\pi$
$929$	−2.33577e22	−1.28325	−0.641626	+	0.767017i	$0.721740\pi$
$930$	0	0
$931$	−1.59526e22	−0.862401
$932$	1.86533e22	1.00032
$933$	0	0
$934$	1.39741e22	0.737432
$935$	−1.83349e22	−0.959822
$936$	0	0
$937$	4.75651e21	0.245042	0.122521	−	0.992466i	$-0.460902\pi$
$937$	4.75651e21	0.245042	0.122521	+	0.992466i	$0.460902\pi$
$938$	−6.70458e21	−0.342649
$939$	0	0
$940$	8.68048e21	0.436601
$941$	3.10932e22	1.55147	0.775735	−	0.631059i	$-0.217379\pi$
$941$	3.10932e22	1.55147	0.775735	+	0.631059i	$0.217379\pi$
$942$	0	0
$943$	−1.47264e22	−0.723201
$944$	−3.82734e22	−1.86469
$945$	0	0
$946$	4.42739e21	0.212307
$947$	−3.43792e22	−1.63557	−0.817787	−	0.575521i	$-0.804799\pi$
$947$	−3.43792e22	−1.63557	−0.817787	+	0.575521i	$0.804799\pi$
$948$	0	0
$949$	−1.11192e22	−0.520689
$950$	6.90902e22	3.20989
$951$	0	0
$952$	1.90120e21	0.0869462
$953$	−5.10344e21	−0.231561	−0.115781	−	0.993275i	$-0.536937\pi$
$953$	−5.10344e21	−0.231561	−0.115781	+	0.993275i	$0.536937\pi$
$954$	0	0
$955$	5.02757e22	2.24560
$956$	−2.69569e21	−0.119464
$957$	0	0
$958$	−1.14438e22	−0.499261
$959$	−8.94421e21	−0.387171
$960$	0	0
$961$	−2.33280e22	−0.994151
$962$	1.85127e22	0.782812
$963$	0	0
$964$	1.46962e21	0.0611825
$965$	4.29087e22	1.77252
$966$	0	0
$967$	1.73952e22	0.707506	0.353753	−	0.935339i	$-0.384905\pi$
$967$	1.73952e22	0.707506	0.353753	+	0.935339i	$0.384905\pi$
$968$	1.97681e21	0.0797809
$969$	0	0
$970$	8.79252e22	3.49402
$971$	−1.01158e22	−0.398891	−0.199445	−	0.979909i	$-0.563914\pi$
$971$	−1.01158e22	−0.398891	−0.199445	+	0.979909i	$0.563914\pi$
$972$	0	0
$973$	−8.28935e21	−0.321865
$974$	−1.04869e22	−0.404069
$975$	0	0
$976$	2.26183e22	0.858194
$977$	−3.41880e22	−1.28725	−0.643627	−	0.765339i	$-0.722571\pi$
$977$	−3.41880e22	−1.28725	−0.643627	+	0.765339i	$0.722571\pi$
$978$	0	0
$979$	−1.98366e22	−0.735524
$980$	−2.63040e22	−0.967891
$981$	0	0
$982$	−9.19754e21	−0.333300
$983$	8.96590e21	0.322435	0.161218	−	0.986919i	$-0.448458\pi$
$983$	8.96590e21	0.322435	0.161218	+	0.986919i	$0.448458\pi$
$984$	0	0
$985$	−2.83688e22	−1.00478
$986$	−3.32104e22	−1.16734
$987$	0	0
$988$	8.57431e21	0.296840
$989$	−9.41220e21	−0.323384
$990$	0	0
$991$	−3.60018e22	−1.21835	−0.609174	−	0.793036i	$-0.708499\pi$
$991$	−3.60018e22	−1.21835	−0.609174	+	0.793036i	$0.708499\pi$
$992$	2.24436e21	0.0753797
$993$	0	0
$994$	−1.47920e22	−0.489359
$995$	−5.83353e22	−1.91540
$996$	0	0
$997$	2.32465e22	0.751872	0.375936	−	0.926646i	$-0.377321\pi$
$997$	2.32465e22	0.751872	0.375936	+	0.926646i	$0.377321\pi$
$998$	2.07462e22	0.665979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.16.a.c.1.4	yes	5
3.2	odd	2		27.16.a.b.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.16.a.b.1.2	✓	5	3.2	odd	2
27.16.a.c.1.4	yes	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q+225.004 q^{2} +17858.9 q^{4} +338163. q^{5} +626121. q^{7} -3.35462e6 q^{8} +7.60882e7 q^{10} +5.98997e7 q^{11} +1.31085e8 q^{13} +1.40880e8 q^{14} -1.34000e9 q^{16} -9.05163e8 q^{17} +3.66261e9 q^{19} +6.03922e9 q^{20} +1.34777e10 q^{22} -2.86522e10 q^{23} +8.38370e10 q^{25} +2.94948e10 q^{26} +1.11818e10 q^{28} +1.63064e11 q^{29} -1.17149e10 q^{31} -1.91582e11 q^{32} -2.03665e11 q^{34} +2.11731e11 q^{35} +6.27662e11 q^{37} +8.24102e11 q^{38} -1.13441e12 q^{40} +5.13971e11 q^{41} +3.28498e11 q^{43} +1.06974e12 q^{44} -6.44687e12 q^{46} +1.43735e12 q^{47} -4.35553e12 q^{49} +1.88637e13 q^{50} +2.34104e12 q^{52} -2.56371e12 q^{53} +2.02559e13 q^{55} -2.10040e12 q^{56} +3.66900e13 q^{58} +2.85622e13 q^{59} -1.68793e13 q^{61} -2.63589e12 q^{62} +8.02439e11 q^{64} +4.43283e13 q^{65} -4.75907e13 q^{67} -1.61652e13 q^{68} +4.76404e13 q^{70} -1.04997e14 q^{71} -8.48244e13 q^{73} +1.41227e14 q^{74} +6.54101e13 q^{76} +3.75044e13 q^{77} -2.07977e14 q^{79} -4.53140e14 q^{80} +1.15646e14 q^{82} -3.60410e14 q^{83} -3.06093e14 q^{85} +7.39134e13 q^{86} -2.00940e14 q^{88} -3.31163e14 q^{89} +8.20753e13 q^{91} -5.11696e14 q^{92} +3.23410e14 q^{94} +1.23856e15 q^{95} +1.15557e15 q^{97} -9.80013e14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 273 q^{2} + 95405 q^{4} - 16725 q^{5} + 2153539 q^{7} + 13315731 q^{8} - 30467925 q^{10} + 115943493 q^{11} + 65408428 q^{13} + 885109227 q^{14} + 1597063649 q^{16} + 1975444272 q^{17} - 263500334 q^{19}+ \cdots - 930960542438550 q^{98}+O(q^{100}) \) 5 * q + 273 * q^2 + 95405 * q^4 - 16725 * q^5 + 2153539 * q^7 + 13315731 * q^8 - 30467925 * q^10 + 115943493 * q^11 + 65408428 * q^13 + 885109227 * q^14 + 1597063649 * q^16 + 1975444272 * q^17 - 263500334 * q^19 - 15752098605 * q^20 + 15874285977 * q^22 - 10719254334 * q^23 + 65878470860 * q^25 + 51424383876 * q^26 + 121134553795 * q^28 + 286370793390 * q^29 - 300406709951 * q^31 + 625606682931 * q^32 + 822965721624 * q^34 + 244477120485 * q^35 + 112184696098 * q^37 + 1764566464506 * q^38 - 4498792795755 * q^40 + 2894246258082 * q^41 + 1139498878582 * q^43 + 12717942305061 * q^44 - 457487589726 * q^46 + 15070114266798 * q^47 - 2056340352918 * q^49 + 42624872351220 * q^50 + 485820206164 * q^52 + 23527241658009 * q^53 - 4705777789605 * q^55 + 68598287921541 * q^56 + 47931112265094 * q^58 + 32938855380780 * q^59 - 20198107823912 * q^61 + 4988384282481 * q^62 + 53228069199833 * q^64 + 54515008918020 * q^65 + 63021911290318 * q^67 + 176777881834968 * q^68 - 178477863934095 * q^70 + 2034300387888 * q^71 - 67077230062931 * q^73 - 255364190895342 * q^74 + 14768524959274 * q^76 - 217101976083597 * q^77 - 107382054455360 * q^79 - 1363798750436985 * q^80 + 136128058396938 * q^82 - 754356450391395 * q^83 - 762873649735680 * q^85 - 1276712187741210 * q^86 + 312847358643387 * q^88 - 1353618746250426 * q^89 + 569165803886180 * q^91 - 872785068972294 * q^92 + 554616946407222 * q^94 + 190785462235950 * q^95 - 702219406917653 * q^97 - 930960542438550 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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