LMFDB - Embedded newform 27.16.a.c.1.5

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			1.5
Root			$0.841091$ 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)

$f(q)$	$=$	\(q+342.118 q^{2} +84276.6 q^{4} -267526. q^{5} +2.26179e6 q^{7} +1.76220e7 q^{8} -9.15253e7 q^{10} +6.31585e7 q^{11} +2.11169e7 q^{13} +7.73799e8 q^{14} +3.26722e9 q^{16} +2.97747e9 q^{17} -8.21682e8 q^{19} -2.25462e10 q^{20} +2.16077e10 q^{22} +7.25431e9 q^{23} +4.10525e10 q^{25} +7.22448e9 q^{26} +1.90616e11 q^{28} +1.08710e11 q^{29} -2.13478e11 q^{31} +5.40338e11 q^{32} +1.01865e12 q^{34} -6.05087e11 q^{35} -6.01338e11 q^{37} -2.81112e11 q^{38} -4.71434e12 q^{40} -2.96711e11 q^{41} -7.47193e11 q^{43} +5.32278e12 q^{44} +2.48183e12 q^{46} +2.99462e12 q^{47} +3.68134e11 q^{49} +1.40448e13 q^{50} +1.77966e12 q^{52} -6.43802e12 q^{53} -1.68965e13 q^{55} +3.98573e13 q^{56} +3.71918e13 q^{58} -1.71672e13 q^{59} +1.10169e13 q^{61} -7.30348e13 q^{62} +7.77987e13 q^{64} -5.64933e12 q^{65} +8.02005e13 q^{67} +2.50931e14 q^{68} -2.07011e14 q^{70} +4.45943e13 q^{71} -1.21952e14 q^{73} -2.05728e14 q^{74} -6.92485e13 q^{76} +1.42851e14 q^{77} -2.36661e13 q^{79} -8.74067e14 q^{80} -1.01510e14 q^{82} +1.36525e14 q^{83} -7.96551e14 q^{85} -2.55628e14 q^{86} +1.11298e15 q^{88} -7.14258e14 q^{89} +4.77621e13 q^{91} +6.11368e14 q^{92} +1.02451e15 q^{94} +2.19821e14 q^{95} -4.61589e14 q^{97} +1.25945e14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	342.118	1.88995	0.944976	−	0.327140i	$-0.106085\pi$
$2$	342.118	1.88995	0.944976	+	0.327140i	$0.106085\pi$
$3$	0	0
$3$	0	0
$4$	84276.6	2.57192
$5$	−267526.	−1.53141	−0.765704	−	0.643194i	$-0.777609\pi$
$5$	−267526.	−1.53141	−0.765704	+	0.643194i	$0.777609\pi$
$6$	0	0
$7$	2.26179e6	1.03805	0.519024	−	0.854760i	$-0.326296\pi$
$7$	2.26179e6	1.03805	0.519024	+	0.854760i	$0.326296\pi$
$8$	1.76220e7	2.97085
$9$	0	0
$10$	−9.15253e7	−2.89429
$11$	6.31585e7	0.977208	0.488604	−	0.872506i	$-0.337506\pi$
$11$	6.31585e7	0.977208	0.488604	+	0.872506i	$0.337506\pi$
$12$	0	0
$13$	2.11169e7	0.0933374	0.0466687	−	0.998910i	$-0.485139\pi$
$13$	2.11169e7	0.0933374	0.0466687	+	0.998910i	$0.485139\pi$
$14$	7.73799e8	1.96186
$15$	0	0
$16$	3.26722e9	3.04284
$17$	2.97747e9	1.75987	0.879935	−	0.475093i	$-0.157586\pi$
$17$	2.97747e9	1.75987	0.879935	+	0.475093i	$0.157586\pi$
$18$	0	0
$19$	−8.21682e8	−0.210888	−0.105444	−	0.994425i	$-0.533626\pi$
$19$	−8.21682e8	−0.210888	−0.105444	+	0.994425i	$0.533626\pi$
$20$	−2.25462e10	−3.93865
$21$	0	0
$22$	2.16077e10	1.84688
$23$	7.25431e9	0.444260	0.222130	−	0.975017i	$-0.428699\pi$
$23$	7.25431e9	0.444260	0.222130	+	0.975017i	$0.428699\pi$
$24$	0	0
$25$	4.10525e10	1.34521
$26$	7.22448e9	0.176403
$27$	0	0
$28$	1.90616e11	2.66977
$29$	1.08710e11	1.17027	0.585135	−	0.810936i	$-0.301041\pi$
$29$	1.08710e11	1.17027	0.585135	+	0.810936i	$0.301041\pi$
$30$	0	0
$31$	−2.13478e11	−1.39361	−0.696805	−	0.717260i	$-0.745396\pi$
$31$	−2.13478e11	−1.39361	−0.696805	+	0.717260i	$0.745396\pi$
$32$	5.40338e11	2.77997
$33$	0	0
$34$	1.01865e12	3.32607
$35$	−6.05087e11	−1.58967
$36$	0	0
$37$	−6.01338e11	−1.04137	−0.520686	−	0.853748i	$-0.674324\pi$
$37$	−6.01338e11	−1.04137	−0.520686	+	0.853748i	$0.674324\pi$
$38$	−2.81112e11	−0.398568
$39$	0	0
$40$	−4.71434e12	−4.54958
$41$	−2.96711e11	−0.237933	−0.118966	−	0.992898i	$-0.537958\pi$
$41$	−2.96711e11	−0.237933	−0.118966	+	0.992898i	$0.537958\pi$
$42$	0	0
$43$	−7.47193e11	−0.419198	−0.209599	−	0.977787i	$-0.567216\pi$
$43$	−7.47193e11	−0.419198	−0.209599	+	0.977787i	$0.567216\pi$
$44$	5.32278e12	2.51330
$45$	0	0
$46$	2.48183e12	0.839630
$47$	2.99462e12	0.862200	0.431100	−	0.902304i	$-0.358126\pi$
$47$	2.99462e12	0.862200	0.431100	+	0.902304i	$0.358126\pi$
$48$	0	0
$49$	3.68134e11	0.0775418
$50$	1.40448e13	2.54238
$51$	0	0
$52$	1.77966e12	0.240056
$53$	−6.43802e12	−0.752806	−0.376403	−	0.926456i	$-0.622839\pi$
$53$	−6.43802e12	−0.752806	−0.376403	+	0.926456i	$0.622839\pi$
$54$	0	0
$55$	−1.68965e13	−1.49650
$56$	3.98573e13	3.08388
$57$	0	0
$58$	3.71918e13	2.21175
$59$	−1.71672e13	−0.898069	−0.449034	−	0.893514i	$-0.648232\pi$
$59$	−1.71672e13	−0.898069	−0.449034	+	0.893514i	$0.648232\pi$
$60$	0	0
$61$	1.10169e13	0.448833	0.224417	−	0.974493i	$-0.427952\pi$
$61$	1.10169e13	0.448833	0.224417	+	0.974493i	$0.427952\pi$
$62$	−7.30348e13	−2.63386
$63$	0	0
$64$	7.77987e13	2.21117
$65$	−5.64933e12	−0.142938
$66$	0	0
$67$	8.02005e13	1.61665	0.808325	−	0.588736i	$-0.200374\pi$
$67$	8.02005e13	1.61665	0.808325	+	0.588736i	$0.200374\pi$
$68$	2.50931e14	4.52624
$69$	0	0
$70$	−2.07011e14	−3.00440
$71$	4.45943e13	0.581891	0.290946	−	0.956740i	$-0.406030\pi$
$71$	4.45943e13	0.581891	0.290946	+	0.956740i	$0.406030\pi$
$72$	0	0
$73$	−1.21952e14	−1.29202	−0.646010	−	0.763329i	$-0.723563\pi$
$73$	−1.21952e14	−1.29202	−0.646010	+	0.763329i	$0.723563\pi$
$74$	−2.05728e14	−1.96814
$75$	0	0
$76$	−6.92485e13	−0.542386
$77$	1.42851e14	1.01439
$78$	0	0
$79$	−2.36661e13	−0.138651	−0.0693256	−	0.997594i	$-0.522085\pi$
$79$	−2.36661e13	−0.138651	−0.0693256	+	0.997594i	$0.522085\pi$
$80$	−8.74067e14	−4.65983
$81$	0	0
$82$	−1.01510e14	−0.449682
$83$	1.36525e14	0.552240	0.276120	−	0.961123i	$-0.410951\pi$
$83$	1.36525e14	0.552240	0.276120	+	0.961123i	$0.410951\pi$
$84$	0	0
$85$	−7.96551e14	−2.69508
$86$	−2.55628e14	−0.792264
$87$	0	0
$88$	1.11298e15	2.90313
$89$	−7.14258e14	−1.71171	−0.855854	−	0.517217i	$-0.826968\pi$
$89$	−7.14258e14	−1.71171	−0.855854	+	0.517217i	$0.826968\pi$
$90$	0	0
$91$	4.77621e13	0.0968886
$92$	6.11368e14	1.14260
$93$	0	0
$94$	1.02451e15	1.62952
$95$	2.19821e14	0.322955
$96$	0	0
$97$	−4.61589e14	−0.580053	−0.290027	−	0.957019i	$-0.593664\pi$
$97$	−4.61589e14	−0.580053	−0.290027	+	0.957019i	$0.593664\pi$
$98$	1.25945e14	0.146550
$99$	0	0
$100$	3.45976e15	3.45976
$101$	4.74964e14	0.440809	0.220405	−	0.975409i	$-0.429262\pi$
$101$	4.74964e14	0.440809	0.220405	+	0.975409i	$0.429262\pi$
$102$	0	0
$103$	−2.42447e15	−1.94240	−0.971198	−	0.238272i	$-0.923419\pi$
$103$	−2.42447e15	−1.94240	−0.971198	+	0.238272i	$0.923419\pi$
$104$	3.72123e14	0.277291
$105$	0	0
$106$	−2.20256e15	−1.42277
$107$	3.30056e14	0.198705	0.0993526	−	0.995052i	$-0.468323\pi$
$107$	3.30056e14	0.198705	0.0993526	+	0.995052i	$0.468323\pi$
$108$	0	0
$109$	−1.53691e15	−0.805285	−0.402643	−	0.915357i	$-0.631908\pi$
$109$	−1.53691e15	−0.805285	−0.402643	+	0.915357i	$0.631908\pi$
$110$	−5.78061e15	−2.82832
$111$	0	0
$112$	7.38977e15	3.15861
$113$	−1.04775e15	−0.418956	−0.209478	−	0.977813i	$-0.567176\pi$
$113$	−1.04775e15	−0.418956	−0.209478	+	0.977813i	$0.567176\pi$
$114$	0	0
$115$	−1.94072e15	−0.680343
$116$	9.16174e15	3.00984
$117$	0	0
$118$	−5.87321e15	−1.69731
$119$	6.73442e15	1.82683
$120$	0	0
$121$	−1.88249e14	−0.0450653
$122$	3.76907e15	0.848274
$123$	0	0
$124$	−1.79912e16	−3.58425
$125$	−2.81836e15	−0.528655
$126$	0	0
$127$	5.91825e14	0.0985520	0.0492760	−	0.998785i	$-0.484309\pi$
$127$	5.91825e14	0.0985520	0.0492760	+	0.998785i	$0.484309\pi$
$128$	8.91054e15	1.39904
$129$	0	0
$130$	−1.93274e15	−0.270145
$131$	−5.91456e15	−0.780526	−0.390263	−	0.920703i	$-0.627616\pi$
$131$	−5.91456e15	−0.780526	−0.390263	+	0.920703i	$0.627616\pi$
$132$	0	0
$133$	−1.85847e15	−0.218912
$134$	2.74380e16	3.05539
$135$	0	0
$136$	5.24690e16	5.22831
$137$	−9.76125e15	−0.920664	−0.460332	−	0.887747i	$-0.652270\pi$
$137$	−9.76125e15	−0.920664	−0.460332	+	0.887747i	$0.652270\pi$
$138$	0	0
$139$	1.45594e16	1.23177	0.615887	−	0.787834i	$-0.288798\pi$
$139$	1.45594e16	1.23177	0.615887	+	0.787834i	$0.288798\pi$
$140$	−5.09947e16	−4.08851
$141$	0	0
$142$	1.52565e16	1.09975
$143$	1.33371e15	0.0912100
$144$	0	0
$145$	−2.90828e16	−1.79216
$146$	−4.17221e16	−2.44185
$147$	0	0
$148$	−5.06787e16	−2.67832
$149$	3.05473e16	1.53488	0.767442	−	0.641119i	$-0.221529\pi$
$149$	3.05473e16	1.53488	0.767442	+	0.641119i	$0.221529\pi$
$150$	0	0
$151$	3.84044e16	1.74604	0.873019	−	0.487686i	$-0.162159\pi$
$151$	3.84044e16	1.74604	0.873019	+	0.487686i	$0.162159\pi$
$152$	−1.44797e16	−0.626515
$153$	0	0
$154$	4.88720e16	1.91714
$155$	5.71110e16	2.13419
$156$	0	0
$157$	−2.70042e15	−0.0916607	−0.0458304	−	0.998949i	$-0.514593\pi$
$157$	−2.70042e15	−0.0916607	−0.0458304	+	0.998949i	$0.514593\pi$
$158$	−8.09660e15	−0.262044
$159$	0	0
$160$	−1.44554e17	−4.25727
$161$	1.64077e16	0.461163
$162$	0	0
$163$	1.64078e16	0.420381	0.210191	−	0.977660i	$-0.432592\pi$
$163$	1.64078e16	0.420381	0.210191	+	0.977660i	$0.432592\pi$
$164$	−2.50058e16	−0.611944
$165$	0	0
$166$	4.67077e16	1.04371
$167$	−8.89147e15	−0.189933	−0.0949664	−	0.995480i	$-0.530274\pi$
$167$	−8.89147e15	−0.189933	−0.0949664	+	0.995480i	$0.530274\pi$
$168$	0	0
$169$	−5.07400e16	−0.991288
$170$	−2.72514e17	−5.09357
$171$	0	0
$172$	−6.29708e16	−1.07814
$173$	−2.94391e16	−0.482591	−0.241295	−	0.970452i	$-0.577572\pi$
$173$	−2.94391e16	−0.482591	−0.241295	+	0.970452i	$0.577572\pi$
$174$	0	0
$175$	9.28522e16	1.39639
$176$	2.06353e17	2.97349
$177$	0	0
$178$	−2.44360e17	−3.23505
$179$	−2.47532e16	−0.314220	−0.157110	−	0.987581i	$-0.550218\pi$
$179$	−2.47532e16	−0.314220	−0.157110	+	0.987581i	$0.550218\pi$
$180$	0	0
$181$	1.40133e17	1.63663	0.818317	−	0.574767i	$-0.194907\pi$
$181$	1.40133e17	1.63663	0.818317	+	0.574767i	$0.194907\pi$
$182$	1.63403e16	0.183115
$183$	0	0
$184$	1.27835e17	1.31983
$185$	1.60873e17	1.59476
$186$	0	0
$187$	1.88053e17	1.71976
$188$	2.52377e17	2.21751
$189$	0	0
$190$	7.52047e16	0.610370
$191$	−1.10161e17	−0.859563	−0.429782	−	0.902933i	$-0.641409\pi$
$191$	−1.10161e17	−0.859563	−0.429782	+	0.902933i	$0.641409\pi$
$192$	0	0
$193$	−1.39680e17	−1.00799	−0.503993	−	0.863708i	$-0.668136\pi$
$193$	−1.39680e17	−1.00799	−0.503993	+	0.863708i	$0.668136\pi$
$194$	−1.57918e17	−1.09627
$195$	0	0
$196$	3.10251e16	0.199431
$197$	2.11422e17	1.30814	0.654068	−	0.756435i	$-0.273061\pi$
$197$	2.11422e17	1.30814	0.654068	+	0.756435i	$0.273061\pi$
$198$	0	0
$199$	1.36285e17	0.781720	0.390860	−	0.920450i	$-0.372178\pi$
$199$	1.36285e17	0.781720	0.390860	+	0.920450i	$0.372178\pi$
$200$	7.23427e17	3.99641
$201$	0	0
$202$	1.62494e17	0.833108
$203$	2.45880e17	1.21480
$204$	0	0
$205$	7.93778e16	0.364372
$206$	−8.29455e17	−3.67104
$207$	0	0
$208$	6.89938e16	0.284011
$209$	−5.18962e16	−0.206081
$210$	0	0
$211$	−1.97728e17	−0.731055	−0.365528	−	0.930801i	$-0.619111\pi$
$211$	−1.97728e17	−0.731055	−0.365528	+	0.930801i	$0.619111\pi$
$212$	−5.42574e17	−1.93615
$213$	0	0
$214$	1.12918e17	0.375543
$215$	1.99893e17	0.641963
$216$	0	0
$217$	−4.82844e17	−1.44663
$218$	−5.25804e17	−1.52195
$219$	0	0
$220$	−1.42398e18	−3.84888
$221$	6.28751e16	0.164262
$222$	0	0
$223$	−4.17003e17	−1.01825	−0.509123	−	0.860694i	$-0.670030\pi$
$223$	−4.17003e17	−1.01825	−0.509123	+	0.860694i	$0.670030\pi$
$224$	1.22213e18	2.88574
$225$	0	0
$226$	−3.58453e17	−0.791806
$227$	−1.68451e17	−0.359980	−0.179990	−	0.983668i	$-0.557607\pi$
$227$	−1.68451e17	−0.359980	−0.179990	+	0.983668i	$0.557607\pi$
$228$	0	0
$229$	3.24883e16	0.0650071	0.0325036	−	0.999472i	$-0.489652\pi$
$229$	3.24883e16	0.0650071	0.0325036	+	0.999472i	$0.489652\pi$
$230$	−6.63953e17	−1.28582
$231$	0	0
$232$	1.91569e18	3.47669
$233$	7.09468e17	1.24670	0.623352	−	0.781941i	$-0.285770\pi$
$233$	7.09468e17	1.24670	0.623352	+	0.781941i	$0.285770\pi$
$234$	0	0
$235$	−8.01139e17	−1.32038
$236$	−1.44679e18	−2.30976
$237$	0	0
$238$	2.30396e18	3.45262
$239$	8.34668e17	1.21208	0.606038	−	0.795436i	$-0.292758\pi$
$239$	8.34668e17	1.21208	0.606038	+	0.795436i	$0.292758\pi$
$240$	0	0
$241$	−3.88898e17	−0.530527	−0.265264	−	0.964176i	$-0.585459\pi$
$241$	−3.88898e17	−0.530527	−0.265264	+	0.964176i	$0.585459\pi$
$242$	−6.44032e16	−0.0851711
$243$	0	0
$244$	9.28466e17	1.15436
$245$	−9.84854e16	−0.118748
$246$	0	0
$247$	−1.73514e16	−0.0196837
$248$	−3.76192e18	−4.14020
$249$	0	0
$250$	−9.64213e17	−0.999132
$251$	−4.39686e17	−0.442171	−0.221085	−	0.975254i	$-0.570960\pi$
$251$	−4.39686e17	−0.442171	−0.221085	+	0.975254i	$0.570960\pi$
$252$	0	0
$253$	4.58171e17	0.434134
$254$	2.02474e17	0.186259
$255$	0	0
$256$	4.99146e17	0.432940
$257$	1.31961e18	1.11160	0.555798	−	0.831318i	$-0.312413\pi$
$257$	1.31961e18	1.11160	0.555798	+	0.831318i	$0.312413\pi$
$258$	0	0
$259$	−1.36010e18	−1.08099
$260$	−4.76106e17	−0.367624
$261$	0	0
$262$	−2.02347e18	−1.47516
$263$	1.49133e17	0.105659	0.0528295	−	0.998604i	$-0.483176\pi$
$263$	1.49133e17	0.105659	0.0528295	+	0.998604i	$0.483176\pi$
$264$	0	0
$265$	1.72234e18	1.15285
$266$	−6.35816e17	−0.413732
$267$	0	0
$268$	6.75902e18	4.15789
$269$	−2.00604e18	−1.20004	−0.600021	−	0.799984i	$-0.704841\pi$
$269$	−2.00604e18	−1.20004	−0.600021	+	0.799984i	$0.704841\pi$
$270$	0	0
$271$	−4.00215e17	−0.226477	−0.113238	−	0.993568i	$-0.536122\pi$
$271$	−4.00215e17	−0.226477	−0.113238	+	0.993568i	$0.536122\pi$
$272$	9.72807e18	5.35500
$273$	0	0
$274$	−3.33950e18	−1.74001
$275$	2.59282e18	1.31455
$276$	0	0
$277$	−1.13848e17	−0.0546671	−0.0273335	−	0.999626i	$-0.508702\pi$
$277$	−1.13848e17	−0.0546671	−0.0273335	+	0.999626i	$0.508702\pi$
$278$	4.98102e18	2.32799
$279$	0	0
$280$	−1.06628e19	−4.72267
$281$	−4.35169e18	−1.87655	−0.938277	−	0.345885i	$-0.887579\pi$
$281$	−4.35169e18	−1.87655	−0.938277	+	0.345885i	$0.887579\pi$
$282$	0	0
$283$	−2.59033e18	−1.05915	−0.529574	−	0.848264i	$-0.677648\pi$
$283$	−2.59033e18	−1.05915	−0.529574	+	0.848264i	$0.677648\pi$
$284$	3.75825e18	1.49658
$285$	0	0
$286$	4.56288e17	0.172383
$287$	−6.71098e17	−0.246986
$288$	0	0
$289$	6.00292e18	2.09714
$290$	−9.94976e18	−3.38710
$291$	0	0
$292$	−1.02777e19	−3.32297
$293$	2.81651e18	0.887574	0.443787	−	0.896132i	$-0.353635\pi$
$293$	2.81651e18	0.887574	0.443787	+	0.896132i	$0.353635\pi$
$294$	0	0
$295$	4.59267e18	1.37531
$296$	−1.05968e19	−3.09376
$297$	0	0
$298$	1.04508e19	2.90085
$299$	1.53189e17	0.0414661
$300$	0	0
$301$	−1.68999e18	−0.435147
$302$	1.31388e19	3.29993
$303$	0	0
$304$	−2.68462e18	−0.641698
$305$	−2.94730e18	−0.687347
$306$	0	0
$307$	−2.47472e18	−0.549526	−0.274763	−	0.961512i	$-0.588599\pi$
$307$	−2.47472e18	−0.549526	−0.274763	+	0.961512i	$0.588599\pi$
$308$	1.20390e19	2.60892
$309$	0	0
$310$	1.95387e19	4.03351
$311$	−1.67434e18	−0.337396	−0.168698	−	0.985668i	$-0.553956\pi$
$311$	−1.67434e18	−0.337396	−0.168698	+	0.985668i	$0.553956\pi$
$312$	0	0
$313$	−4.86543e18	−0.934413	−0.467206	−	0.884148i	$-0.654739\pi$
$313$	−4.86543e18	−0.934413	−0.467206	+	0.884148i	$0.654739\pi$
$314$	−9.23860e17	−0.173234
$315$	0	0
$316$	−1.99450e18	−0.356600
$317$	2.16540e18	0.378088	0.189044	−	0.981969i	$-0.439461\pi$
$317$	2.16540e18	0.378088	0.189044	+	0.981969i	$0.439461\pi$
$318$	0	0
$319$	6.86599e18	1.14360
$320$	−2.08132e19	−3.38621
$321$	0	0
$322$	5.61337e18	0.871575
$323$	−2.44653e18	−0.371135
$324$	0	0
$325$	8.66903e17	0.125558
$326$	5.61340e18	0.794501
$327$	0	0
$328$	−5.22864e18	−0.706862
$329$	6.77321e18	0.895004
$330$	0	0
$331$	9.68114e18	1.22241	0.611204	−	0.791473i	$-0.290685\pi$
$331$	9.68114e18	1.22241	0.611204	+	0.791473i	$0.290685\pi$
$332$	1.15059e19	1.42031
$333$	0	0
$334$	−3.04193e18	−0.358964
$335$	−2.14557e19	−2.47575
$336$	0	0
$337$	1.55152e19	1.71211	0.856057	−	0.516881i	$-0.172907\pi$
$337$	1.55152e19	1.71211	0.856057	+	0.516881i	$0.172907\pi$
$338$	−1.73590e19	−1.87349
$339$	0	0
$340$	−6.71306e19	−6.93152
$341$	−1.34830e19	−1.36185
$342$	0	0
$343$	−9.90535e18	−0.957555
$344$	−1.31670e19	−1.24537
$345$	0	0
$346$	−1.00716e19	−0.912074
$347$	3.82882e18	0.339308	0.169654	−	0.985504i	$-0.445735\pi$
$347$	3.82882e18	0.339308	0.169654	+	0.985504i	$0.445735\pi$
$348$	0	0
$349$	1.06883e19	0.907227	0.453614	−	0.891198i	$-0.350135\pi$
$349$	1.06883e19	0.907227	0.453614	+	0.891198i	$0.350135\pi$
$350$	3.17664e19	2.63911
$351$	0	0
$352$	3.41269e19	2.71661
$353$	−1.41220e19	−1.10049	−0.550246	−	0.835002i	$-0.685466\pi$
$353$	−1.41220e19	−1.10049	−0.550246	+	0.835002i	$0.685466\pi$
$354$	0	0
$355$	−1.19301e19	−0.891112
$356$	−6.01952e19	−4.40237
$357$	0	0
$358$	−8.46852e18	−0.593861
$359$	1.73731e19	1.19308	0.596539	−	0.802584i	$-0.296542\pi$
$359$	1.73731e19	1.19308	0.596539	+	0.802584i	$0.296542\pi$
$360$	0	0
$361$	−1.45060e19	−0.955526
$362$	4.79420e19	3.09316
$363$	0	0
$364$	4.02523e18	0.249190
$365$	3.26254e19	1.97861
$366$	0	0
$367$	3.08931e18	0.179832	0.0899158	−	0.995949i	$-0.471340\pi$
$367$	3.08931e18	0.179832	0.0899158	+	0.995949i	$0.471340\pi$
$368$	2.37014e19	1.35181
$369$	0	0
$370$	5.50377e19	3.01403
$371$	−1.45614e19	−0.781448
$372$	0	0
$373$	−2.56873e19	−1.32404	−0.662022	−	0.749485i	$-0.730301\pi$
$373$	−2.56873e19	−1.32404	−0.662022	+	0.749485i	$0.730301\pi$
$374$	6.43362e19	3.25026
$375$	0	0
$376$	5.27712e19	2.56146
$377$	2.29563e18	0.109230
$378$	0	0
$379$	1.09687e19	0.501603	0.250801	−	0.968039i	$-0.419306\pi$
$379$	1.09687e19	0.501603	0.250801	+	0.968039i	$0.419306\pi$
$380$	1.85258e19	0.830614
$381$	0	0
$382$	−3.76881e19	−1.62453
$383$	−4.05439e19	−1.71370	−0.856849	−	0.515567i	$-0.827581\pi$
$383$	−4.05439e19	−1.71370	−0.856849	+	0.515567i	$0.827581\pi$
$384$	0	0
$385$	−3.82164e19	−1.55344
$386$	−4.77871e19	−1.90505
$387$	0	0
$388$	−3.89012e19	−1.49185
$389$	−3.73547e19	−1.40515	−0.702576	−	0.711608i	$-0.747967\pi$
$389$	−3.73547e19	−1.40515	−0.702576	+	0.711608i	$0.747967\pi$
$390$	0	0
$391$	2.15995e19	0.781840
$392$	6.48726e18	0.230365
$393$	0	0
$394$	7.23312e19	2.47231
$395$	6.33130e18	0.212332
$396$	0	0
$397$	−1.08495e19	−0.350333	−0.175166	−	0.984539i	$-0.556046\pi$
$397$	−1.08495e19	−0.350333	−0.175166	+	0.984539i	$0.556046\pi$
$398$	4.66256e19	1.47741
$399$	0	0
$400$	1.34128e20	4.09325
$401$	7.51940e18	0.225217	0.112608	−	0.993639i	$-0.464079\pi$
$401$	7.51940e18	0.225217	0.112608	+	0.993639i	$0.464079\pi$
$402$	0	0
$403$	−4.50801e18	−0.130076
$404$	4.00283e19	1.13372
$405$	0	0
$406$	8.41200e19	2.29590
$407$	−3.79796e19	−1.01764
$408$	0	0
$409$	−5.13896e19	−1.32724	−0.663621	−	0.748069i	$-0.730981\pi$
$409$	−5.13896e19	−1.32724	−0.663621	+	0.748069i	$0.730981\pi$
$410$	2.71566e19	0.688646
$411$	0	0
$412$	−2.04326e20	−4.99568
$413$	−3.88286e19	−0.932238
$414$	0	0
$415$	−3.65241e19	−0.845704
$416$	1.14103e19	0.259475
$417$	0	0
$418$	−1.77546e19	−0.389484
$419$	3.43832e19	0.740867	0.370434	−	0.928859i	$-0.379209\pi$
$419$	3.43832e19	0.740867	0.370434	+	0.928859i	$0.379209\pi$
$420$	0	0
$421$	−6.51594e19	−1.35476	−0.677378	−	0.735635i	$-0.736884\pi$
$421$	−6.51594e19	−1.35476	−0.677378	+	0.735635i	$0.736884\pi$
$422$	−6.76463e19	−1.38166
$423$	0	0
$424$	−1.13451e20	−2.23647
$425$	1.22233e20	2.36739
$426$	0	0
$427$	2.49179e19	0.465910
$428$	2.78160e19	0.511053
$429$	0	0
$430$	6.83871e19	1.21328
$431$	6.34723e19	1.10664	0.553318	−	0.832970i	$-0.313361\pi$
$431$	6.34723e19	1.10664	0.553318	+	0.832970i	$0.313361\pi$
$432$	0	0
$433$	−2.16773e19	−0.365044	−0.182522	−	0.983202i	$-0.558426\pi$
$433$	−2.16773e19	−0.365044	−0.182522	+	0.983202i	$0.558426\pi$
$434$	−1.65189e20	−2.73407
$435$	0	0
$436$	−1.29525e20	−2.07113
$437$	−5.96073e18	−0.0936891
$438$	0	0
$439$	1.00725e20	1.52986	0.764931	−	0.644113i	$-0.222773\pi$
$439$	1.00725e20	1.52986	0.764931	+	0.644113i	$0.222773\pi$
$440$	−2.97751e20	−4.44588
$441$	0	0
$442$	2.15107e19	0.310447
$443$	1.00834e20	1.43079	0.715397	−	0.698718i	$-0.246246\pi$
$443$	1.00834e20	1.43079	0.715397	+	0.698718i	$0.246246\pi$
$444$	0	0
$445$	1.91082e20	2.62132
$446$	−1.42664e20	−1.92444
$447$	0	0
$448$	1.75964e20	2.29530
$449$	−1.32180e20	−1.69558	−0.847791	−	0.530331i	$-0.822068\pi$
$449$	−1.32180e20	−1.69558	−0.847791	+	0.530331i	$0.822068\pi$
$450$	0	0
$451$	−1.87398e19	−0.232510
$452$	−8.83006e19	−1.07752
$453$	0	0
$454$	−5.76300e19	−0.680346
$455$	−1.27776e19	−0.148376
$456$	0	0
$457$	4.41888e19	0.496525	0.248263	−	0.968693i	$-0.420140\pi$
$457$	4.41888e19	0.496525	0.248263	+	0.968693i	$0.420140\pi$
$458$	1.11148e19	0.122860
$459$	0	0
$460$	−1.63557e20	−1.74979
$461$	1.25702e20	1.32308	0.661539	−	0.749911i	$-0.269904\pi$
$461$	1.25702e20	1.32308	0.661539	+	0.749911i	$0.269904\pi$
$462$	0	0
$463$	2.69447e19	0.274547	0.137273	−	0.990533i	$-0.456166\pi$
$463$	2.69447e19	0.274547	0.137273	+	0.990533i	$0.456166\pi$
$464$	3.55181e20	3.56094
$465$	0	0
$466$	2.42722e20	2.35621
$467$	1.11580e20	1.06588	0.532942	−	0.846152i	$-0.321086\pi$
$467$	1.11580e20	1.06588	0.532942	+	0.846152i	$0.321086\pi$
$468$	0	0
$469$	1.81397e20	1.67816
$470$	−2.74084e20	−2.49545
$471$	0	0
$472$	−3.02521e20	−2.66802
$473$	−4.71916e19	−0.409644
$474$	0	0
$475$	−3.37321e19	−0.283688
$476$	5.67554e20	4.69845
$477$	0	0
$478$	2.85555e20	2.29076
$479$	−1.46973e20	−1.16071	−0.580353	−	0.814365i	$-0.697085\pi$
$479$	−1.46973e20	−1.16071	−0.580353	+	0.814365i	$0.697085\pi$
$480$	0	0
$481$	−1.26984e19	−0.0971990
$482$	−1.33049e20	−1.00267
$483$	0	0
$484$	−1.58650e19	−0.115904
$485$	1.23487e20	0.888298
$486$	0	0
$487$	−2.67872e20	−1.86836	−0.934178	−	0.356806i	$-0.883866\pi$
$487$	−2.67872e20	−1.86836	−0.934178	+	0.356806i	$0.883866\pi$
$488$	1.94140e20	1.33342
$489$	0	0
$490$	−3.36936e19	−0.224428
$491$	−2.42923e20	−1.59352	−0.796759	−	0.604298i	$-0.793454\pi$
$491$	−2.42923e20	−1.59352	−0.796759	+	0.604298i	$0.793454\pi$
$492$	0	0
$493$	3.23682e20	2.05952
$494$	−5.93622e18	−0.0372013
$495$	0	0
$496$	−6.97482e20	−4.24053
$497$	1.00863e20	0.604030
$498$	0	0
$499$	−1.84613e19	−0.107277	−0.0536387	−	0.998560i	$-0.517082\pi$
$499$	−1.84613e19	−0.107277	−0.0536387	+	0.998560i	$0.517082\pi$
$500$	−2.37522e20	−1.35966
$501$	0	0
$502$	−1.50425e20	−0.835682
$503$	6.79894e19	0.372119	0.186059	−	0.982538i	$-0.440428\pi$
$503$	6.79894e19	0.372119	0.186059	+	0.982538i	$0.440428\pi$
$504$	0	0
$505$	−1.27065e20	−0.675058
$506$	1.56749e20	0.820493
$507$	0	0
$508$	4.98770e19	0.253468
$509$	2.02967e20	1.01635	0.508175	−	0.861254i	$-0.330320\pi$
$509$	2.02967e20	1.01635	0.508175	+	0.861254i	$0.330320\pi$
$510$	0	0
$511$	−2.75831e20	−1.34118
$512$	−1.21214e20	−0.580801
$513$	0	0
$514$	4.51461e20	2.10086
$515$	6.48609e20	2.97460
$516$	0	0
$517$	1.89136e20	0.842549
$518$	−4.65315e20	−2.04302
$519$	0	0
$520$	−9.95524e19	−0.424646
$521$	6.67657e18	0.0280718	0.0140359	−	0.999901i	$-0.495532\pi$
$521$	6.67657e18	0.0280718	0.0140359	+	0.999901i	$0.495532\pi$
$522$	0	0
$523$	3.75218e20	1.53293	0.766463	−	0.642289i	$-0.222015\pi$
$523$	3.75218e20	1.53293	0.766463	+	0.642289i	$0.222015\pi$
$524$	−4.98458e20	−2.00745
$525$	0	0
$526$	5.10211e19	0.199690
$527$	−6.35626e20	−2.45257
$528$	0	0
$529$	−2.14010e20	−0.802633
$530$	5.89242e20	2.17884
$531$	0	0
$532$	−1.56626e20	−0.563022
$533$	−6.26562e18	−0.0222080
$534$	0	0
$535$	−8.82985e19	−0.304299
$536$	1.41329e21	4.80282
$537$	0	0
$538$	−6.86300e20	−2.26802
$539$	2.32508e19	0.0757744
$540$	0	0
$541$	−4.47985e20	−1.41999	−0.709993	−	0.704209i	$-0.751302\pi$
$541$	−4.47985e20	−1.41999	−0.709993	+	0.704209i	$0.751302\pi$
$542$	−1.36921e20	−0.428030
$543$	0	0
$544$	1.60884e21	4.89239
$545$	4.11163e20	1.23322
$546$	0	0
$547$	6.31361e19	0.184235	0.0921177	−	0.995748i	$-0.470636\pi$
$547$	6.31361e19	0.184235	0.0921177	+	0.995748i	$0.470636\pi$
$548$	−8.22645e20	−2.36787
$549$	0	0
$550$	8.87048e20	2.48443
$551$	−8.93253e19	−0.246796
$552$	0	0
$553$	−5.35278e19	−0.143927
$554$	−3.89493e19	−0.103318
$555$	0	0
$556$	1.22701e21	3.16802
$557$	−1.38388e20	−0.352521	−0.176260	−	0.984344i	$-0.556400\pi$
$557$	−1.38388e20	−0.352521	−0.176260	+	0.984344i	$0.556400\pi$
$558$	0	0
$559$	−1.57784e19	−0.0391269
$560$	−1.97696e21	−4.83712
$561$	0	0
$562$	−1.48879e21	−3.54660
$563$	−8.29296e20	−1.94938	−0.974690	−	0.223559i	$-0.928233\pi$
$563$	−8.29296e20	−1.94938	−0.974690	+	0.223559i	$0.928233\pi$
$564$	0	0
$565$	2.80300e20	0.641592
$566$	−8.86197e20	−2.00174
$567$	0	0
$568$	7.85841e20	1.72871
$569$	5.56188e19	0.120748	0.0603739	−	0.998176i	$-0.480771\pi$
$569$	5.56188e19	0.120748	0.0603739	+	0.998176i	$0.480771\pi$
$570$	0	0
$571$	3.85265e20	0.814682	0.407341	−	0.913276i	$-0.366456\pi$
$571$	3.85265e20	0.814682	0.407341	+	0.913276i	$0.366456\pi$
$572$	1.12401e20	0.234585
$573$	0	0
$574$	−2.29594e20	−0.466791
$575$	2.97808e20	0.597622
$576$	0	0
$577$	−8.58771e20	−1.67903	−0.839515	−	0.543336i	$-0.817161\pi$
$577$	−8.58771e20	−1.67903	−0.839515	+	0.543336i	$0.817161\pi$
$578$	2.05370e21	3.96350
$579$	0	0
$580$	−2.45100e21	−4.60929
$581$	3.08792e20	0.573251
$582$	0	0
$583$	−4.06616e20	−0.735648
$584$	−2.14905e21	−3.83839
$585$	0	0
$586$	9.63580e20	1.67747
$587$	−5.92204e20	−1.01786	−0.508928	−	0.860809i	$-0.669958\pi$
$587$	−5.92204e20	−1.01786	−0.508928	+	0.860809i	$0.669958\pi$
$588$	0	0
$589$	1.75411e20	0.293896
$590$	1.57124e21	2.59927
$591$	0	0
$592$	−1.96471e21	−3.16873
$593$	4.12318e20	0.656631	0.328316	−	0.944568i	$-0.393519\pi$
$593$	4.12318e20	0.656631	0.328316	+	0.944568i	$0.393519\pi$
$594$	0	0
$595$	−1.80163e21	−2.79762
$596$	2.57442e21	3.94759
$597$	0	0
$598$	5.24086e19	0.0783689
$599$	−2.48379e19	−0.0366787	−0.0183394	−	0.999832i	$-0.505838\pi$
$599$	−2.48379e19	−0.0366787	−0.0183394	+	0.999832i	$0.505838\pi$
$600$	0	0
$601$	2.06154e20	0.296915	0.148458	−	0.988919i	$-0.452569\pi$
$601$	2.06154e20	0.296915	0.148458	+	0.988919i	$0.452569\pi$
$602$	−5.78177e20	−0.822407
$603$	0	0
$604$	3.23659e21	4.49066
$605$	5.03614e19	0.0690133
$606$	0	0
$607$	9.06191e20	1.21145	0.605724	−	0.795675i	$-0.292884\pi$
$607$	9.06191e20	1.21145	0.605724	+	0.795675i	$0.292884\pi$
$608$	−4.43986e20	−0.586262
$609$	0	0
$610$	−1.00832e21	−1.29905
$611$	6.32373e19	0.0804756
$612$	0	0
$613$	−5.48960e20	−0.681690	−0.340845	−	0.940120i	$-0.610713\pi$
$613$	−5.48960e20	−0.681690	−0.340845	+	0.940120i	$0.610713\pi$
$614$	−8.46646e20	−1.03858
$615$	0	0
$616$	2.51733e21	3.01359
$617$	−3.07414e20	−0.363567	−0.181784	−	0.983339i	$-0.558187\pi$
$617$	−3.07414e20	−0.363567	−0.181784	+	0.983339i	$0.558187\pi$
$618$	0	0
$619$	2.80264e20	0.323510	0.161755	−	0.986831i	$-0.448285\pi$
$619$	2.80264e20	0.323510	0.161755	+	0.986831i	$0.448285\pi$
$620$	4.81312e21	5.48895
$621$	0	0
$622$	−5.72820e20	−0.637662
$623$	−1.61550e21	−1.77683
$624$	0	0
$625$	−4.98837e20	−0.535623
$626$	−1.66455e21	−1.76599
$627$	0	0
$628$	−2.27582e20	−0.235744
$629$	−1.79047e21	−1.83268
$630$	0	0
$631$	1.45234e21	1.45161	0.725803	−	0.687903i	$-0.241468\pi$
$631$	1.45234e21	1.45161	0.725803	+	0.687903i	$0.241468\pi$
$632$	−4.17044e20	−0.411912
$633$	0	0
$634$	7.40820e20	0.714568
$635$	−1.58329e20	−0.150923
$636$	0	0
$637$	7.77387e18	0.00723755
$638$	2.34898e21	2.16134
$639$	0	0
$640$	−2.38380e21	−2.14250
$641$	1.55508e20	0.138139	0.0690696	−	0.997612i	$-0.477997\pi$
$641$	1.55508e20	0.138139	0.0690696	+	0.997612i	$0.477997\pi$
$642$	0	0
$643$	1.36048e21	1.18062	0.590310	−	0.807176i	$-0.299005\pi$
$643$	1.36048e21	1.18062	0.590310	+	0.807176i	$0.299005\pi$
$644$	1.38279e21	1.18607
$645$	0	0
$646$	−8.37003e20	−0.701428
$647$	−4.49528e20	−0.372370	−0.186185	−	0.982515i	$-0.559612\pi$
$647$	−4.49528e20	−0.372370	−0.186185	+	0.982515i	$0.559612\pi$
$648$	0	0
$649$	−1.08426e21	−0.877600
$650$	2.96583e20	0.237299
$651$	0	0
$652$	1.38279e21	1.08119
$653$	2.16710e20	0.167506	0.0837531	−	0.996487i	$-0.473309\pi$
$653$	2.16710e20	0.167506	0.0837531	+	0.996487i	$0.473309\pi$
$654$	0	0
$655$	1.58230e21	1.19530
$656$	−9.69421e20	−0.723991
$657$	0	0
$658$	2.31723e21	1.69152
$659$	5.81906e20	0.419964	0.209982	−	0.977705i	$-0.432659\pi$
$659$	5.81906e20	0.419964	0.209982	+	0.977705i	$0.432659\pi$
$660$	0	0
$661$	1.80020e21	1.27002	0.635009	−	0.772505i	$-0.280996\pi$
$661$	1.80020e21	1.27002	0.635009	+	0.772505i	$0.280996\pi$
$662$	3.31209e21	2.31029
$663$	0	0
$664$	2.40585e21	1.64062
$665$	4.97189e20	0.335243
$666$	0	0
$667$	7.88619e20	0.519904
$668$	−7.49343e20	−0.488491
$669$	0	0
$670$	−7.34038e21	−4.67905
$671$	6.95811e20	0.438603
$672$	0	0
$673$	−2.41671e20	−0.148975	−0.0744873	−	0.997222i	$-0.523732\pi$
$673$	−2.41671e20	−0.148975	−0.0744873	+	0.997222i	$0.523732\pi$
$674$	5.30802e21	3.23581
$675$	0	0
$676$	−4.27619e21	−2.54951
$677$	2.85210e21	1.68170	0.840851	−	0.541267i	$-0.182055\pi$
$677$	2.85210e21	1.68170	0.840851	+	0.541267i	$0.182055\pi$
$678$	0	0
$679$	−1.04402e21	−0.602122
$680$	−1.40368e22	−8.00667
$681$	0	0
$682$	−4.61277e21	−2.57382
$683$	−2.69275e21	−1.48607	−0.743037	−	0.669250i	$-0.766615\pi$
$683$	−2.69275e21	−1.48607	−0.743037	+	0.669250i	$0.766615\pi$
$684$	0	0
$685$	2.61139e21	1.40991
$686$	−3.38880e21	−1.80973
$687$	0	0
$688$	−2.44125e21	−1.27555
$689$	−1.35951e20	−0.0702650
$690$	0	0
$691$	3.91482e20	0.197982	0.0989911	−	0.995088i	$-0.468438\pi$
$691$	3.91482e20	0.197982	0.0989911	+	0.995088i	$0.468438\pi$
$692$	−2.48103e21	−1.24118
$693$	0	0
$694$	1.30991e21	0.641276
$695$	−3.89500e21	−1.88635
$696$	0	0
$697$	−8.83448e20	−0.418731
$698$	3.65664e21	1.71462
$699$	0	0
$700$	7.82526e21	3.59140
$701$	7.18120e20	0.326071	0.163035	−	0.986620i	$-0.447872\pi$
$701$	7.18120e20	0.326071	0.163035	+	0.986620i	$0.447872\pi$
$702$	0	0
$703$	4.94108e20	0.219613
$704$	4.91365e21	2.16077
$705$	0	0
$706$	−4.83140e21	−2.07988
$707$	1.07427e21	0.457581
$708$	0	0
$709$	4.44129e21	1.85209	0.926047	−	0.377409i	$-0.123185\pi$
$709$	4.44129e21	1.85209	0.926047	+	0.377409i	$0.123185\pi$
$710$	−4.08151e21	−1.68416
$711$	0	0
$712$	−1.25866e22	−5.08522
$713$	−1.54864e21	−0.619125
$714$	0	0
$715$	−3.56803e20	−0.139680
$716$	−2.08612e21	−0.808149
$717$	0	0
$718$	5.94364e21	2.25486
$719$	1.63687e21	0.614536	0.307268	−	0.951623i	$-0.400585\pi$
$719$	1.63687e21	0.614536	0.307268	+	0.951623i	$0.400585\pi$
$720$	0	0
$721$	−5.48365e21	−2.01630
$722$	−4.96275e21	−1.80590
$723$	0	0
$724$	1.18099e22	4.20929
$725$	4.46283e21	1.57426
$726$	0	0
$727$	−3.83870e21	−1.32641	−0.663203	−	0.748440i	$-0.730803\pi$
$727$	−3.83870e21	−1.32641	−0.663203	+	0.748440i	$0.730803\pi$
$728$	8.41663e20	0.287841
$729$	0	0
$730$	1.11617e22	3.73947
$731$	−2.22475e21	−0.737734
$732$	0	0
$733$	4.44283e21	1.44338	0.721689	−	0.692217i	$-0.243366\pi$
$733$	4.44283e21	1.44338	0.721689	+	0.692217i	$0.243366\pi$
$734$	1.05691e21	0.339873
$735$	0	0
$736$	3.91978e21	1.23503
$737$	5.06535e21	1.57980
$738$	0	0
$739$	−6.10620e21	−1.86611	−0.933056	−	0.359732i	$-0.882868\pi$
$739$	−6.10620e21	−1.86611	−0.933056	+	0.359732i	$0.882868\pi$
$740$	1.35579e22	4.10160
$741$	0	0
$742$	−4.98173e21	−1.47690
$743$	3.14913e21	0.924220	0.462110	−	0.886823i	$-0.347093\pi$
$743$	3.14913e21	0.924220	0.462110	+	0.886823i	$0.347093\pi$
$744$	0	0
$745$	−8.17218e21	−2.35053
$746$	−8.78808e21	−2.50238
$747$	0	0
$748$	1.58484e22	4.42308
$749$	7.46517e20	0.206265
$750$	0	0
$751$	4.62975e21	1.25389	0.626944	−	0.779065i	$-0.284306\pi$
$751$	4.62975e21	1.25389	0.626944	+	0.779065i	$0.284306\pi$
$752$	9.78410e21	2.62354
$753$	0	0
$754$	7.85376e20	0.206439
$755$	−1.02742e22	−2.67389
$756$	0	0
$757$	2.12624e21	0.542493	0.271246	−	0.962510i	$-0.412564\pi$
$757$	2.12624e21	0.542493	0.271246	+	0.962510i	$0.412564\pi$
$758$	3.75258e21	0.948005
$759$	0	0
$760$	3.87369e21	0.959450
$761$	5.97478e21	1.46533	0.732667	−	0.680587i	$-0.238275\pi$
$761$	5.97478e21	1.46533	0.732667	+	0.680587i	$0.238275\pi$
$762$	0	0
$763$	−3.47617e21	−0.835924
$764$	−9.28399e21	−2.21072
$765$	0	0
$766$	−1.38708e22	−3.23881
$767$	−3.62519e20	−0.0838234
$768$	0	0
$769$	1.20804e21	0.273926	0.136963	−	0.990576i	$-0.456266\pi$
$769$	1.20804e21	0.273926	0.136963	+	0.990576i	$0.456266\pi$
$770$	−1.30745e22	−2.93593
$771$	0	0
$772$	−1.17718e22	−2.59246
$773$	5.61608e21	1.22486	0.612430	−	0.790524i	$-0.290192\pi$
$773$	5.61608e21	1.22486	0.612430	+	0.790524i	$0.290192\pi$
$774$	0	0
$775$	−8.76383e21	−1.87470
$776$	−8.13413e21	−1.72325
$777$	0	0
$778$	−1.27797e22	−2.65567
$779$	2.43802e20	0.0501772
$780$	0	0
$781$	2.81651e21	0.568628
$782$	7.38957e21	1.47764
$783$	0	0
$784$	1.20278e21	0.235947
$785$	7.22431e20	0.140370
$786$	0	0
$787$	−3.95370e21	−0.753692	−0.376846	−	0.926276i	$-0.622991\pi$
$787$	−3.95370e21	−0.753692	−0.376846	+	0.926276i	$0.622991\pi$
$788$	1.78179e22	3.36442
$789$	0	0
$790$	2.16605e21	0.401296
$791$	−2.36979e21	−0.434896
$792$	0	0
$793$	2.32643e20	0.0418930
$794$	−3.71181e21	−0.662112
$795$	0	0
$796$	1.14857e22	2.01052
$797$	2.43101e21	0.421550	0.210775	−	0.977535i	$-0.432401\pi$
$797$	2.43101e21	0.421550	0.210775	+	0.977535i	$0.432401\pi$
$798$	0	0
$799$	8.91640e21	1.51736
$800$	2.21822e22	3.73964
$801$	0	0
$802$	2.57252e21	0.425649
$803$	−7.70234e21	−1.26257
$804$	0	0
$805$	−4.38949e21	−0.706228
$806$	−1.54227e21	−0.245837
$807$	0	0
$808$	8.36982e21	1.30958
$809$	8.93451e21	1.38502	0.692511	−	0.721407i	$-0.256504\pi$
$809$	8.93451e21	1.38502	0.692511	+	0.721407i	$0.256504\pi$
$810$	0	0
$811$	4.25917e21	0.648140	0.324070	−	0.946033i	$-0.394949\pi$
$811$	4.25917e21	0.648140	0.324070	+	0.946033i	$0.394949\pi$
$812$	2.07219e22	3.12435
$813$	0	0
$814$	−1.29935e22	−1.92328
$815$	−4.38951e21	−0.643775
$816$	0	0
$817$	6.13955e20	0.0884038
$818$	−1.75813e22	−2.50842
$819$	0	0
$820$	6.68969e21	0.937135
$821$	−2.11501e21	−0.293589	−0.146794	−	0.989167i	$-0.546896\pi$
$821$	−2.11501e21	−0.293589	−0.146794	+	0.989167i	$0.546896\pi$
$822$	0	0
$823$	−9.06755e21	−1.23592	−0.617961	−	0.786209i	$-0.712041\pi$
$823$	−9.06755e21	−1.23592	−0.617961	+	0.786209i	$0.712041\pi$
$824$	−4.27240e22	−5.77056
$825$	0	0
$826$	−1.32840e22	−1.76188
$827$	1.78356e21	0.234421	0.117210	−	0.993107i	$-0.462605\pi$
$827$	1.78356e21	0.234421	0.117210	+	0.993107i	$0.462605\pi$
$828$	0	0
$829$	9.78631e21	1.26316	0.631582	−	0.775309i	$-0.282406\pi$
$829$	9.78631e21	1.26316	0.631582	+	0.775309i	$0.282406\pi$
$830$	−1.24955e22	−1.59834
$831$	0	0
$832$	1.64287e21	0.206385
$833$	1.09611e21	0.136463
$834$	0	0
$835$	2.37870e21	0.290865
$836$	−4.37363e21	−0.530024
$837$	0	0
$838$	1.17631e22	1.40020
$839$	1.42376e22	1.67966	0.839831	−	0.542848i	$-0.182654\pi$
$839$	1.42376e22	1.67966	0.839831	+	0.542848i	$0.182654\pi$
$840$	0	0
$841$	3.18876e21	0.369532
$842$	−2.22922e22	−2.56042
$843$	0	0
$844$	−1.66639e22	−1.88021
$845$	1.35743e22	1.51807
$846$	0	0
$847$	−4.25779e20	−0.0467799
$848$	−2.10344e22	−2.29067
$849$	0	0
$850$	4.18180e22	4.47426
$851$	−4.36229e21	−0.462640
$852$	0	0
$853$	−3.21853e21	−0.335382	−0.167691	−	0.985840i	$-0.553631\pi$
$853$	−3.21853e21	−0.335382	−0.167691	+	0.985840i	$0.553631\pi$
$854$	8.52485e21	0.880548
$855$	0	0
$856$	5.81624e21	0.590323
$857$	6.54708e21	0.658706	0.329353	−	0.944207i	$-0.393169\pi$
$857$	6.54708e21	0.658706	0.329353	+	0.944207i	$0.393169\pi$
$858$	0	0
$859$	6.45559e21	0.638244	0.319122	−	0.947714i	$-0.396612\pi$
$859$	6.45559e21	0.638244	0.319122	+	0.947714i	$0.396612\pi$
$860$	1.68463e22	1.65108
$861$	0	0
$862$	2.17150e22	2.09149
$863$	−8.19889e21	−0.782842	−0.391421	−	0.920212i	$-0.628016\pi$
$863$	−8.19889e21	−0.782842	−0.391421	+	0.920212i	$0.628016\pi$
$864$	0	0
$865$	7.87573e21	0.739043
$866$	−7.41619e21	−0.689916
$867$	0	0
$868$	−4.06924e22	−3.72062
$869$	−1.49472e21	−0.135491
$870$	0	0
$871$	1.69359e21	0.150894
$872$	−2.70834e22	−2.39238
$873$	0	0
$874$	−2.03927e21	−0.177068
$875$	−6.37455e21	−0.548769
$876$	0	0
$877$	−6.54760e21	−0.554096	−0.277048	−	0.960856i	$-0.589356\pi$
$877$	−6.54760e21	−0.554096	−0.277048	+	0.960856i	$0.589356\pi$
$878$	3.44597e22	2.89136
$879$	0	0
$880$	−5.52048e22	−4.55362
$881$	1.09863e22	0.898527	0.449263	−	0.893399i	$-0.351687\pi$
$881$	1.09863e22	0.898527	0.449263	+	0.893399i	$0.351687\pi$
$882$	0	0
$883$	7.48310e21	0.601695	0.300847	−	0.953672i	$-0.402731\pi$
$883$	7.48310e21	0.601695	0.300847	+	0.953672i	$0.402731\pi$
$884$	5.29890e21	0.422468
$885$	0	0
$886$	3.44970e22	2.70413
$887$	−5.52940e21	−0.429784	−0.214892	−	0.976638i	$-0.568940\pi$
$887$	−5.52940e21	−0.429784	−0.214892	+	0.976638i	$0.568940\pi$
$888$	0	0
$889$	1.33858e21	0.102302
$890$	6.53727e22	4.95417
$891$	0	0
$892$	−3.51436e22	−2.61884
$893$	−2.46063e21	−0.181828
$894$	0	0
$895$	6.62213e21	0.481199
$896$	2.01538e22	1.45227
$897$	0	0
$898$	−4.52212e22	−3.20457
$899$	−2.32073e22	−1.63090
$900$	0	0
$901$	−1.91690e22	−1.32484
$902$	−6.41123e21	−0.439432
$903$	0	0
$904$	−1.84634e22	−1.24465
$905$	−3.74892e22	−2.50635
$906$	0	0
$907$	2.91236e22	1.91510	0.957549	−	0.288270i	$-0.0930801\pi$
$907$	2.91236e22	1.91510	0.957549	+	0.288270i	$0.0930801\pi$
$908$	−1.41964e22	−0.925840
$909$	0	0
$910$	−4.37144e21	−0.280423
$911$	9.29157e21	0.591155	0.295577	−	0.955319i	$-0.404488\pi$
$911$	9.29157e21	0.591155	0.295577	+	0.955319i	$0.404488\pi$
$912$	0	0
$913$	8.62274e21	0.539653
$914$	1.51178e22	0.938408
$915$	0	0
$916$	2.73800e21	0.167193
$917$	−1.33775e22	−0.810223
$918$	0	0
$919$	7.85960e21	0.468311	0.234155	−	0.972199i	$-0.424768\pi$
$919$	7.85960e21	0.468311	0.234155	+	0.972199i	$0.424768\pi$
$920$	−3.41993e22	−2.02120
$921$	0	0
$922$	4.30049e22	2.50055
$923$	9.41695e20	0.0543122
$924$	0	0
$925$	−2.46864e22	−1.40086
$926$	9.21828e21	0.518880
$927$	0	0
$928$	5.87403e22	3.25332
$929$	2.86294e22	1.57288	0.786439	−	0.617668i	$-0.211922\pi$
$929$	2.86294e22	1.57288	0.786439	+	0.617668i	$0.211922\pi$
$930$	0	0
$931$	−3.02489e20	−0.0163526
$932$	5.97915e22	3.20642
$933$	0	0
$934$	3.81735e22	2.01447
$935$	−5.03090e22	−2.63365
$936$	0	0
$937$	−1.16685e22	−0.601132	−0.300566	−	0.953761i	$-0.597176\pi$
$937$	−1.16685e22	−0.601132	−0.300566	+	0.953761i	$0.597176\pi$
$938$	6.20590e22	3.17164
$939$	0	0
$940$	−6.75172e22	−3.39591
$941$	1.21221e22	0.604859	0.302430	−	0.953172i	$-0.402202\pi$
$941$	1.21221e22	0.604859	0.302430	+	0.953172i	$0.402202\pi$
$942$	0	0
$943$	−2.15243e21	−0.105704
$944$	−5.60891e22	−2.73268
$945$	0	0
$946$	−1.61451e22	−0.774206
$947$	2.10131e21	0.0999692	0.0499846	−	0.998750i	$-0.484083\pi$
$947$	2.10131e21	0.0999692	0.0499846	+	0.998750i	$0.484083\pi$
$948$	0	0
$949$	−2.57526e21	−0.120594
$950$	−1.15403e22	−0.536157
$951$	0	0
$952$	1.18674e23	5.42723
$953$	3.46487e22	1.57214	0.786069	−	0.618139i	$-0.212113\pi$
$953$	3.46487e22	1.57214	0.786069	+	0.618139i	$0.212113\pi$
$954$	0	0
$955$	2.94709e22	1.31634
$956$	7.03430e22	3.11736
$957$	0	0
$958$	−5.02822e22	−2.19368
$959$	−2.20779e22	−0.955693
$960$	0	0
$961$	2.21078e22	0.942150
$962$	−4.34435e21	−0.183701
$963$	0	0
$964$	−3.27750e22	−1.36447
$965$	3.73680e22	1.54364
$966$	0	0
$967$	1.55510e22	0.632498	0.316249	−	0.948676i	$-0.397576\pi$
$967$	1.55510e22	0.632498	0.316249	+	0.948676i	$0.397576\pi$
$968$	−3.31732e21	−0.133882
$969$	0	0
$970$	4.22471e22	1.67884
$971$	2.17968e22	0.859505	0.429752	−	0.902947i	$-0.358601\pi$
$971$	2.17968e22	0.859505	0.429752	+	0.902947i	$0.358601\pi$
$972$	0	0
$973$	3.29302e22	1.27864
$974$	−9.16437e22	−3.53110
$975$	0	0
$976$	3.59946e22	1.36573
$977$	−3.15962e22	−1.18967	−0.594833	−	0.803849i	$-0.702782\pi$
$977$	−3.15962e22	−1.18967	−0.594833	+	0.803849i	$0.702782\pi$
$978$	0	0
$979$	−4.51115e22	−1.67270
$980$	−8.30001e21	−0.305410
$981$	0	0
$982$	−8.31081e22	−3.01167
$983$	−3.14050e22	−1.12940	−0.564700	−	0.825296i	$-0.691008\pi$
$983$	−3.14050e22	−1.12940	−0.564700	+	0.825296i	$0.691008\pi$
$984$	0	0
$985$	−5.65608e22	−2.00329
$986$	1.10737e23	3.89240
$987$	0	0
$988$	−1.46232e21	−0.0506249
$989$	−5.42037e21	−0.186233
$990$	0	0
$991$	−2.12502e22	−0.719137	−0.359568	−	0.933119i	$-0.617076\pi$
$991$	−2.12502e22	−0.719137	−0.359568	+	0.933119i	$0.617076\pi$
$992$	−1.15350e23	−3.87420
$993$	0	0
$994$	3.45070e22	1.14159
$995$	−3.64599e22	−1.19713
$996$	0	0
$997$	−3.98547e22	−1.28904	−0.644518	−	0.764589i	$-0.722942\pi$
$997$	−3.98547e22	−1.28904	−0.644518	+	0.764589i	$0.722942\pi$
$998$	−6.31594e21	−0.202749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.16.a.c.1.5	yes	5
3.2	odd	2		27.16.a.b.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.16.a.b.1.1	✓	5	3.2	odd	2
27.16.a.c.1.5	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)

\(f(q)\)	\(=\)	\(q+342.118 q^{2} +84276.6 q^{4} -267526. q^{5} +2.26179e6 q^{7} +1.76220e7 q^{8} -9.15253e7 q^{10} +6.31585e7 q^{11} +2.11169e7 q^{13} +7.73799e8 q^{14} +3.26722e9 q^{16} +2.97747e9 q^{17} -8.21682e8 q^{19} -2.25462e10 q^{20} +2.16077e10 q^{22} +7.25431e9 q^{23} +4.10525e10 q^{25} +7.22448e9 q^{26} +1.90616e11 q^{28} +1.08710e11 q^{29} -2.13478e11 q^{31} +5.40338e11 q^{32} +1.01865e12 q^{34} -6.05087e11 q^{35} -6.01338e11 q^{37} -2.81112e11 q^{38} -4.71434e12 q^{40} -2.96711e11 q^{41} -7.47193e11 q^{43} +5.32278e12 q^{44} +2.48183e12 q^{46} +2.99462e12 q^{47} +3.68134e11 q^{49} +1.40448e13 q^{50} +1.77966e12 q^{52} -6.43802e12 q^{53} -1.68965e13 q^{55} +3.98573e13 q^{56} +3.71918e13 q^{58} -1.71672e13 q^{59} +1.10169e13 q^{61} -7.30348e13 q^{62} +7.77987e13 q^{64} -5.64933e12 q^{65} +8.02005e13 q^{67} +2.50931e14 q^{68} -2.07011e14 q^{70} +4.45943e13 q^{71} -1.21952e14 q^{73} -2.05728e14 q^{74} -6.92485e13 q^{76} +1.42851e14 q^{77} -2.36661e13 q^{79} -8.74067e14 q^{80} -1.01510e14 q^{82} +1.36525e14 q^{83} -7.96551e14 q^{85} -2.55628e14 q^{86} +1.11298e15 q^{88} -7.14258e14 q^{89} +4.77621e13 q^{91} +6.11368e14 q^{92} +1.02451e15 q^{94} +2.19821e14 q^{95} -4.61589e14 q^{97} +1.25945e14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 273 q^{2} + 95405 q^{4} - 16725 q^{5} + 2153539 q^{7} + 13315731 q^{8} - 30467925 q^{10} + 115943493 q^{11} + 65408428 q^{13} + 885109227 q^{14} + 1597063649 q^{16} + 1975444272 q^{17} - 263500334 q^{19}+ \cdots - 930960542438550 q^{98}+O(q^{100}) \) 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

Related objects

Downloads

Learn more