LMFDB - Embedded newform 272.10.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(1,272)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$140.089747437$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 $ x^7 - x^6 - 2986x^5 + 8252x^4 + 2252056x^3 - 10388768x^2 - 243559296*x - 675998208
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-5.44491$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-106.475 q^{3} +1303.94 q^{5} -9199.27 q^{7} -8346.12 q^{9} +O(q^{10})$	$q-106.475 q^{3} +1303.94 q^{5} -9199.27 q^{7} -8346.12 q^{9} -62238.4 q^{11} +141901. q^{13} -138837. q^{15} +83521.0 q^{17} +941163. q^{19} +979491. q^{21} -568165. q^{23} -252865. q^{25} +2.98439e6 q^{27} -1.83597e6 q^{29} +7.34903e6 q^{31} +6.62682e6 q^{33} -1.19953e7 q^{35} -8.01674e6 q^{37} -1.51089e7 q^{39} +1.95674e7 q^{41} -3.46966e7 q^{43} -1.08828e7 q^{45} +5.63645e7 q^{47} +4.42730e7 q^{49} -8.89288e6 q^{51} -3.28783e7 q^{53} -8.11551e7 q^{55} -1.00210e8 q^{57} -1.04852e8 q^{59} +5.69264e7 q^{61} +7.67782e7 q^{63} +1.85030e8 q^{65} +1.58325e7 q^{67} +6.04952e7 q^{69} +9.81097e7 q^{71} -6.27365e7 q^{73} +2.69238e7 q^{75} +5.72548e8 q^{77} +1.38282e8 q^{79} -1.53486e8 q^{81} +6.69421e8 q^{83} +1.08906e8 q^{85} +1.95485e8 q^{87} -4.21417e7 q^{89} -1.30538e9 q^{91} -7.82486e8 q^{93} +1.22722e9 q^{95} -4.11288e8 q^{97} +5.19449e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−106.475	−0.758929	−0.379465	−	0.925206i	$-0.623892\pi$
$3$	−106.475	−0.758929	−0.379465	+	0.925206i	$0.623892\pi$
$4$	0	0
$5$	1303.94	0.933024	0.466512	−	0.884515i	$-0.345511\pi$
$5$	1303.94	0.933024	0.466512	+	0.884515i	$0.345511\pi$
$6$	0	0
$7$	−9199.27	−1.44815	−0.724073	−	0.689723i	$-0.757732\pi$
$7$	−9199.27	−1.44815	−0.724073	+	0.689723i	$0.757732\pi$
$8$	0	0
$9$	−8346.12	−0.424027
$10$	0	0
$11$	−62238.4	−1.28171	−0.640857	−	0.767660i	$-0.721421\pi$
$11$	−62238.4	−1.28171	−0.640857	+	0.767660i	$0.721421\pi$
$12$	0	0
$13$	141901.	1.37797	0.688985	−	0.724775i	$-0.258056\pi$
$13$	141901.	1.37797	0.688985	+	0.724775i	$0.258056\pi$
$14$	0	0
$15$	−138837.	−0.708099
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	941163.	1.65681	0.828407	−	0.560127i	$-0.189247\pi$
$19$	941163.	1.65681	0.828407	+	0.560127i	$0.189247\pi$
$20$	0	0
$21$	979491.	1.09904
$22$	0	0
$23$	−568165.	−0.423349	−0.211675	−	0.977340i	$-0.567892\pi$
$23$	−568165.	−0.423349	−0.211675	+	0.977340i	$0.567892\pi$
$24$	0	0
$25$	−252865.	−0.129467
$26$	0	0
$27$	2.98439e6	1.08074
$28$	0	0
$29$	−1.83597e6	−0.482032	−0.241016	−	0.970521i	$-0.577481\pi$
$29$	−1.83597e6	−0.482032	−0.241016	+	0.970521i	$0.577481\pi$
$30$	0	0
$31$	7.34903e6	1.42923	0.714615	−	0.699518i	$-0.246602\pi$
$31$	7.34903e6	1.42923	0.714615	+	0.699518i	$0.246602\pi$
$32$	0	0
$33$	6.62682e6	0.972730
$34$	0	0
$35$	−1.19953e7	−1.35115
$36$	0	0
$37$	−8.01674e6	−0.703218	−0.351609	−	0.936147i	$-0.614365\pi$
$37$	−8.01674e6	−0.703218	−0.351609	+	0.936147i	$0.614365\pi$
$38$	0	0
$39$	−1.51089e7	−1.04578
$40$	0	0
$41$	1.95674e7	1.08145	0.540724	−	0.841200i	$-0.318150\pi$
$41$	1.95674e7	1.08145	0.540724	+	0.841200i	$0.318150\pi$
$42$	0	0
$43$	−3.46966e7	−1.54767	−0.773835	−	0.633387i	$-0.781664\pi$
$43$	−3.46966e7	−1.54767	−0.773835	+	0.633387i	$0.781664\pi$
$44$	0	0
$45$	−1.08828e7	−0.395627
$46$	0	0
$47$	5.63645e7	1.68486	0.842432	−	0.538802i	$-0.181123\pi$
$47$	5.63645e7	1.68486	0.842432	+	0.538802i	$0.181123\pi$
$48$	0	0
$49$	4.42730e7	1.09713
$50$	0	0
$51$	−8.89288e6	−0.184067
$52$	0	0
$53$	−3.28783e7	−0.572359	−0.286179	−	0.958176i	$-0.592385\pi$
$53$	−3.28783e7	−0.572359	−0.286179	+	0.958176i	$0.592385\pi$
$54$	0	0
$55$	−8.11551e7	−1.19587
$56$	0	0
$57$	−1.00210e8	−1.25740
$58$	0	0
$59$	−1.04852e8	−1.12653	−0.563267	−	0.826275i	$-0.690456\pi$
$59$	−1.04852e8	−1.12653	−0.563267	+	0.826275i	$0.690456\pi$
$60$	0	0
$61$	5.69264e7	0.526416	0.263208	−	0.964739i	$-0.415219\pi$
$61$	5.69264e7	0.526416	0.263208	+	0.964739i	$0.415219\pi$
$62$	0	0
$63$	7.67782e7	0.614052
$64$	0	0
$65$	1.85030e8	1.28568
$66$	0	0
$67$	1.58325e7	0.0959872	0.0479936	−	0.998848i	$-0.484717\pi$
$67$	1.58325e7	0.0959872	0.0479936	+	0.998848i	$0.484717\pi$
$68$	0	0
$69$	6.04952e7	0.321292
$70$	0	0
$71$	9.81097e7	0.458194	0.229097	−	0.973404i	$-0.426423\pi$
$71$	9.81097e7	0.458194	0.229097	+	0.973404i	$0.426423\pi$
$72$	0	0
$73$	−6.27365e7	−0.258564	−0.129282	−	0.991608i	$-0.541267\pi$
$73$	−6.27365e7	−0.258564	−0.129282	+	0.991608i	$0.541267\pi$
$74$	0	0
$75$	2.69238e7	0.0982563
$76$	0	0
$77$	5.72548e8	1.85611
$78$	0	0
$79$	1.38282e8	0.399433	0.199717	−	0.979854i	$-0.435998\pi$
$79$	1.38282e8	0.399433	0.199717	+	0.979854i	$0.435998\pi$
$80$	0	0
$81$	−1.53486e8	−0.396175
$82$	0	0
$83$	6.69421e8	1.54827	0.774137	−	0.633018i	$-0.218184\pi$
$83$	6.69421e8	1.54827	0.774137	+	0.633018i	$0.218184\pi$
$84$	0	0
$85$	1.08906e8	0.226291
$86$	0	0
$87$	1.95485e8	0.365828
$88$	0	0
$89$	−4.21417e7	−0.0711963	−0.0355981	−	0.999366i	$-0.511334\pi$
$89$	−4.21417e7	−0.0711963	−0.0355981	+	0.999366i	$0.511334\pi$
$90$	0	0
$91$	−1.30538e9	−1.99550
$92$	0	0
$93$	−7.82486e8	−1.08468
$94$	0	0
$95$	1.22722e9	1.54585
$96$	0	0
$97$	−4.11288e8	−0.471707	−0.235854	−	0.971789i	$-0.575789\pi$
$97$	−4.11288e8	−0.471707	−0.235854	+	0.971789i	$0.575789\pi$
$98$	0	0
$99$	5.19449e8	0.543481
$100$	0	0
$101$	1.81470e7	0.0173524	0.00867620	−	0.999962i	$-0.497238\pi$
$101$	1.81470e7	0.0173524	0.00867620	+	0.999962i	$0.497238\pi$
$102$	0	0
$103$	−7.17681e8	−0.628296	−0.314148	−	0.949374i	$-0.601719\pi$
$103$	−7.17681e8	−0.628296	−0.314148	+	0.949374i	$0.601719\pi$
$104$	0	0
$105$	1.27720e9	1.02543
$106$	0	0
$107$	7.18789e8	0.530120	0.265060	−	0.964232i	$-0.414608\pi$
$107$	7.18789e8	0.530120	0.265060	+	0.964232i	$0.414608\pi$
$108$	0	0
$109$	3.41476e8	0.231708	0.115854	−	0.993266i	$-0.463040\pi$
$109$	3.41476e8	0.231708	0.115854	+	0.993266i	$0.463040\pi$
$110$	0	0
$111$	8.53581e8	0.533693
$112$	0	0
$113$	−1.70781e9	−0.985339	−0.492670	−	0.870216i	$-0.663979\pi$
$113$	−1.70781e9	−0.985339	−0.492670	+	0.870216i	$0.663979\pi$
$114$	0	0
$115$	−7.40853e8	−0.394995
$116$	0	0
$117$	−1.18432e9	−0.584296
$118$	0	0
$119$	−7.68332e8	−0.351227
$120$	0	0
$121$	1.51567e9	0.642792
$122$	0	0
$123$	−2.08343e9	−0.820743
$124$	0	0
$125$	−2.87648e9	−1.05382
$126$	0	0
$127$	−1.94335e9	−0.662880	−0.331440	−	0.943476i	$-0.607534\pi$
$127$	−1.94335e9	−0.662880	−0.331440	+	0.943476i	$0.607534\pi$
$128$	0	0
$129$	3.69431e9	1.17457
$130$	0	0
$131$	−3.30692e9	−0.981075	−0.490538	−	0.871420i	$-0.663200\pi$
$131$	−3.30692e9	−0.981075	−0.490538	+	0.871420i	$0.663200\pi$
$132$	0	0
$133$	−8.65802e9	−2.39931
$134$	0	0
$135$	3.89147e9	1.00835
$136$	0	0
$137$	−4.83938e9	−1.17367	−0.586836	−	0.809706i	$-0.699627\pi$
$137$	−4.83938e9	−1.17367	−0.586836	+	0.809706i	$0.699627\pi$
$138$	0	0
$139$	−3.57524e9	−0.812342	−0.406171	−	0.913797i	$-0.633136\pi$
$139$	−3.57524e9	−0.812342	−0.406171	+	0.913797i	$0.633136\pi$
$140$	0	0
$141$	−6.00139e9	−1.27869
$142$	0	0
$143$	−8.83168e9	−1.76616
$144$	0	0
$145$	−2.39400e9	−0.449747
$146$	0	0
$147$	−4.71396e9	−0.832641
$148$	0	0
$149$	−2.20680e9	−0.366796	−0.183398	−	0.983039i	$-0.558710\pi$
$149$	−2.20680e9	−0.366796	−0.183398	+	0.983039i	$0.558710\pi$
$150$	0	0
$151$	−7.78978e9	−1.21935	−0.609676	−	0.792651i	$-0.708700\pi$
$151$	−7.78978e9	−1.21935	−0.609676	+	0.792651i	$0.708700\pi$
$152$	0	0
$153$	−6.97076e8	−0.102842
$154$	0	0
$155$	9.58269e9	1.33351
$156$	0	0
$157$	5.99611e9	0.787627	0.393814	−	0.919190i	$-0.371155\pi$
$157$	5.99611e9	0.787627	0.393814	+	0.919190i	$0.371155\pi$
$158$	0	0
$159$	3.50071e9	0.434380
$160$	0	0
$161$	5.22670e9	0.613072
$162$	0	0
$163$	−2.20096e9	−0.244212	−0.122106	−	0.992517i	$-0.538965\pi$
$163$	−2.20096e9	−0.244212	−0.122106	+	0.992517i	$0.538965\pi$
$164$	0	0
$165$	8.64098e9	0.907580
$166$	0	0
$167$	1.54378e9	0.153589	0.0767946	−	0.997047i	$-0.475531\pi$
$167$	1.54378e9	0.153589	0.0767946	+	0.997047i	$0.475531\pi$
$168$	0	0
$169$	9.53135e9	0.898802
$170$	0	0
$171$	−7.85506e9	−0.702533
$172$	0	0
$173$	1.36568e10	1.15916	0.579578	−	0.814917i	$-0.303217\pi$
$173$	1.36568e10	1.15916	0.579578	+	0.814917i	$0.303217\pi$
$174$	0	0
$175$	2.32618e9	0.187487
$176$	0	0
$177$	1.11641e10	0.854959
$178$	0	0
$179$	−7.48481e9	−0.544932	−0.272466	−	0.962165i	$-0.587839\pi$
$179$	−7.48481e9	−0.544932	−0.272466	+	0.962165i	$0.587839\pi$
$180$	0	0
$181$	−1.98188e10	−1.37254	−0.686269	−	0.727348i	$-0.740753\pi$
$181$	−1.98188e10	−1.37254	−0.686269	+	0.727348i	$0.740753\pi$
$182$	0	0
$183$	−6.06123e9	−0.399513
$184$	0	0
$185$	−1.04534e10	−0.656119
$186$	0	0
$187$	−5.19821e9	−0.310861
$188$	0	0
$189$	−2.74543e10	−1.56506
$190$	0	0
$191$	3.54665e10	1.92827	0.964137	−	0.265405i	$-0.0855056\pi$
$191$	3.54665e10	1.92827	0.964137	+	0.265405i	$0.0855056\pi$
$192$	0	0
$193$	−1.16540e10	−0.604599	−0.302299	−	0.953213i	$-0.597754\pi$
$193$	−1.16540e10	−0.604599	−0.302299	+	0.953213i	$0.597754\pi$
$194$	0	0
$195$	−1.97011e10	−0.975739
$196$	0	0
$197$	4.99943e9	0.236495	0.118248	−	0.992984i	$-0.462272\pi$
$197$	4.99943e9	0.236495	0.118248	+	0.992984i	$0.462272\pi$
$198$	0	0
$199$	1.90482e9	0.0861026	0.0430513	−	0.999073i	$-0.486292\pi$
$199$	1.90482e9	0.0861026	0.0430513	+	0.999073i	$0.486292\pi$
$200$	0	0
$201$	−1.68576e9	−0.0728475
$202$	0	0
$203$	1.68896e10	0.698052
$204$	0	0
$205$	2.55147e10	1.00902
$206$	0	0
$207$	4.74197e9	0.179511
$208$	0	0
$209$	−5.85765e10	−2.12356
$210$	0	0
$211$	−4.45582e10	−1.54759	−0.773797	−	0.633434i	$-0.781645\pi$
$211$	−4.45582e10	−1.54759	−0.773797	+	0.633434i	$0.781645\pi$
$212$	0	0
$213$	−1.04462e10	−0.347737
$214$	0	0
$215$	−4.52422e10	−1.44401
$216$	0	0
$217$	−6.76057e10	−2.06973
$218$	0	0
$219$	6.67986e9	0.196232
$220$	0	0
$221$	1.18517e10	0.334207
$222$	0	0
$223$	1.00748e10	0.272812	0.136406	−	0.990653i	$-0.456445\pi$
$223$	1.00748e10	0.272812	0.136406	+	0.990653i	$0.456445\pi$
$224$	0	0
$225$	2.11044e9	0.0548975
$226$	0	0
$227$	−7.51701e10	−1.87901	−0.939505	−	0.342536i	$-0.888714\pi$
$227$	−7.51701e10	−1.87901	−0.939505	+	0.342536i	$0.888714\pi$
$228$	0	0
$229$	8.19379e10	1.96891	0.984453	−	0.175647i	$-0.0562016\pi$
$229$	8.19379e10	1.96891	0.984453	+	0.175647i	$0.0562016\pi$
$230$	0	0
$231$	−6.09619e10	−1.40866
$232$	0	0
$233$	−4.97304e10	−1.10540	−0.552701	−	0.833380i	$-0.686403\pi$
$233$	−4.97304e10	−1.10540	−0.552701	+	0.833380i	$0.686403\pi$
$234$	0	0
$235$	7.34959e10	1.57202
$236$	0	0
$237$	−1.47236e10	−0.303142
$238$	0	0
$239$	4.06399e10	0.805680	0.402840	−	0.915270i	$-0.368023\pi$
$239$	4.06399e10	0.805680	0.402840	+	0.915270i	$0.368023\pi$
$240$	0	0
$241$	−3.05939e10	−0.584196	−0.292098	−	0.956388i	$-0.594353\pi$
$241$	−3.05939e10	−0.584196	−0.292098	+	0.956388i	$0.594353\pi$
$242$	0	0
$243$	−4.23994e10	−0.780067
$244$	0	0
$245$	5.77293e10	1.02364
$246$	0	0
$247$	1.33552e11	2.28304
$248$	0	0
$249$	−7.12765e10	−1.17503
$250$	0	0
$251$	−2.05450e10	−0.326719	−0.163359	−	0.986567i	$-0.552233\pi$
$251$	−2.05450e10	−0.326719	−0.163359	+	0.986567i	$0.552233\pi$
$252$	0	0
$253$	3.53617e10	0.542613
$254$	0	0
$255$	−1.15958e10	−0.171739
$256$	0	0
$257$	−8.98169e10	−1.28428	−0.642139	−	0.766588i	$-0.721953\pi$
$257$	−8.98169e10	−1.28428	−0.642139	+	0.766588i	$0.721953\pi$
$258$	0	0
$259$	7.37482e10	1.01836
$260$	0	0
$261$	1.53233e10	0.204394
$262$	0	0
$263$	−1.43373e11	−1.84785	−0.923923	−	0.382577i	$-0.875037\pi$
$263$	−1.43373e11	−1.84785	−0.923923	+	0.382577i	$0.875037\pi$
$264$	0	0
$265$	−4.28714e10	−0.534024
$266$	0	0
$267$	4.48703e9	0.0540329
$268$	0	0
$269$	−1.67468e11	−1.95005	−0.975025	−	0.222094i	$-0.928711\pi$
$269$	−1.67468e11	−1.95005	−0.975025	+	0.222094i	$0.928711\pi$
$270$	0	0
$271$	−4.52996e10	−0.510190	−0.255095	−	0.966916i	$-0.582107\pi$
$271$	−4.52996e10	−0.510190	−0.255095	+	0.966916i	$0.582107\pi$
$272$	0	0
$273$	1.38991e11	1.51444
$274$	0	0
$275$	1.57379e10	0.165940
$276$	0	0
$277$	1.93194e9	0.0197167	0.00985836	−	0.999951i	$-0.496862\pi$
$277$	1.93194e9	0.0197167	0.00985836	+	0.999951i	$0.496862\pi$
$278$	0	0
$279$	−6.13358e10	−0.606032
$280$	0	0
$281$	1.16694e11	1.11653	0.558265	−	0.829662i	$-0.311467\pi$
$281$	1.16694e11	1.11653	0.558265	+	0.829662i	$0.311467\pi$
$282$	0	0
$283$	1.65132e8	0.00153036	0.000765180	−	1.00000i	$-0.499756\pi$
$283$	1.65132e8	0.00153036	0.000765180		1.00000i	$0.499756\pi$
$284$	0	0
$285$	−1.30668e11	−1.17319
$286$	0	0
$287$	−1.80006e11	−1.56609
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	4.37918e10	0.357993
$292$	0	0
$293$	6.45700e10	0.511831	0.255915	−	0.966699i	$-0.417623\pi$
$293$	6.45700e10	0.511831	0.255915	+	0.966699i	$0.417623\pi$
$294$	0	0
$295$	−1.36721e11	−1.05108
$296$	0	0
$297$	−1.85744e11	−1.38519
$298$	0	0
$299$	−8.06230e10	−0.583363
$300$	0	0
$301$	3.19183e11	2.24125
$302$	0	0
$303$	−1.93220e9	−0.0131692
$304$	0	0
$305$	7.42286e10	0.491159
$306$	0	0
$307$	−8.54538e10	−0.549046	−0.274523	−	0.961581i	$-0.588520\pi$
$307$	−8.54538e10	−0.549046	−0.274523	+	0.961581i	$0.588520\pi$
$308$	0	0
$309$	7.64150e10	0.476832
$310$	0	0
$311$	7.31980e10	0.443688	0.221844	−	0.975082i	$-0.428792\pi$
$311$	7.31980e10	0.443688	0.221844	+	0.975082i	$0.428792\pi$
$312$	0	0
$313$	1.60425e10	0.0944764	0.0472382	−	0.998884i	$-0.484958\pi$
$313$	1.60425e10	0.0944764	0.0472382	+	0.998884i	$0.484958\pi$
$314$	0	0
$315$	1.00114e11	0.572925
$316$	0	0
$317$	8.53822e10	0.474899	0.237449	−	0.971400i	$-0.423689\pi$
$317$	8.53822e10	0.474899	0.237449	+	0.971400i	$0.423689\pi$
$318$	0	0
$319$	1.14268e11	0.617827
$320$	0	0
$321$	−7.65329e10	−0.402324
$322$	0	0
$323$	7.86069e10	0.401836
$324$	0	0
$325$	−3.58818e10	−0.178402
$326$	0	0
$327$	−3.63586e10	−0.175850
$328$	0	0
$329$	−5.18512e11	−2.43993
$330$	0	0
$331$	−5.68636e10	−0.260380	−0.130190	−	0.991489i	$-0.541559\pi$
$331$	−5.68636e10	−0.260380	−0.130190	+	0.991489i	$0.541559\pi$
$332$	0	0
$333$	6.69087e10	0.298183
$334$	0	0
$335$	2.06447e10	0.0895583
$336$	0	0
$337$	3.07106e11	1.29704	0.648521	−	0.761197i	$-0.275388\pi$
$337$	3.07106e11	1.29704	0.648521	+	0.761197i	$0.275388\pi$
$338$	0	0
$339$	1.81838e11	0.747803
$340$	0	0
$341$	−4.57392e11	−1.83187
$342$	0	0
$343$	−3.60556e10	−0.140653
$344$	0	0
$345$	7.88821e10	0.299773
$346$	0	0
$347$	−3.34687e10	−0.123924	−0.0619621	−	0.998079i	$-0.519736\pi$
$347$	−3.34687e10	−0.123924	−0.0619621	+	0.998079i	$0.519736\pi$
$348$	0	0
$349$	−3.80974e11	−1.37462	−0.687308	−	0.726366i	$-0.741208\pi$
$349$	−3.80974e11	−1.37462	−0.687308	+	0.726366i	$0.741208\pi$
$350$	0	0
$351$	4.23488e11	1.48922
$352$	0	0
$353$	2.09902e11	0.719499	0.359750	−	0.933049i	$-0.382862\pi$
$353$	2.09902e11	0.719499	0.359750	+	0.933049i	$0.382862\pi$
$354$	0	0
$355$	1.27929e11	0.427506
$356$	0	0
$357$	8.18080e10	0.266556
$358$	0	0
$359$	−3.14148e11	−0.998181	−0.499091	−	0.866550i	$-0.666332\pi$
$359$	−3.14148e11	−0.998181	−0.499091	+	0.866550i	$0.666332\pi$
$360$	0	0
$361$	5.63101e11	1.74503
$362$	0	0
$363$	−1.61381e11	−0.487834
$364$	0	0
$365$	−8.18047e10	−0.241246
$366$	0	0
$367$	−2.40379e11	−0.691671	−0.345835	−	0.938295i	$-0.612404\pi$
$367$	−2.40379e11	−0.691671	−0.345835	+	0.938295i	$0.612404\pi$
$368$	0	0
$369$	−1.63312e11	−0.458563
$370$	0	0
$371$	3.02457e11	0.828859
$372$	0	0
$373$	2.93908e11	0.786179	0.393089	−	0.919500i	$-0.371406\pi$
$373$	2.93908e11	0.786179	0.393089	+	0.919500i	$0.371406\pi$
$374$	0	0
$375$	3.06273e11	0.799774
$376$	0	0
$377$	−2.60526e11	−0.664225
$378$	0	0
$379$	−3.76034e11	−0.936161	−0.468081	−	0.883686i	$-0.655054\pi$
$379$	−3.76034e11	−0.936161	−0.468081	+	0.883686i	$0.655054\pi$
$380$	0	0
$381$	2.06918e11	0.503079
$382$	0	0
$383$	−3.73836e11	−0.887741	−0.443870	−	0.896091i	$-0.646395\pi$
$383$	−3.73836e11	−0.887741	−0.443870	+	0.896091i	$0.646395\pi$
$384$	0	0
$385$	7.46568e11	1.73179
$386$	0	0
$387$	2.89582e11	0.656253
$388$	0	0
$389$	−4.62613e10	−0.102434	−0.0512171	−	0.998688i	$-0.516310\pi$
$389$	−4.62613e10	−0.102434	−0.0512171	+	0.998688i	$0.516310\pi$
$390$	0	0
$391$	−4.74537e10	−0.102677
$392$	0	0
$393$	3.52103e11	0.744567
$394$	0	0
$395$	1.80312e11	0.372681
$396$	0	0
$397$	−3.36827e11	−0.680534	−0.340267	−	0.940329i	$-0.610517\pi$
$397$	−3.36827e11	−0.680534	−0.340267	+	0.940329i	$0.610517\pi$
$398$	0	0
$399$	9.21861e11	1.82091
$400$	0	0
$401$	−9.81350e11	−1.89528	−0.947642	−	0.319335i	$-0.896541\pi$
$401$	−9.81350e11	−1.89528	−0.947642	+	0.319335i	$0.896541\pi$
$402$	0	0
$403$	1.04283e12	1.96944
$404$	0	0
$405$	−2.00137e11	−0.369640
$406$	0	0
$407$	4.98949e11	0.901325
$408$	0	0
$409$	8.17703e11	1.44491	0.722455	−	0.691418i	$-0.243014\pi$
$409$	8.17703e11	1.44491	0.722455	+	0.691418i	$0.243014\pi$
$410$	0	0
$411$	5.15272e11	0.890734
$412$	0	0
$413$	9.64566e11	1.63139
$414$	0	0
$415$	8.72885e11	1.44458
$416$	0	0
$417$	3.80673e11	0.616510
$418$	0	0
$419$	4.18196e11	0.662852	0.331426	−	0.943481i	$-0.392470\pi$
$419$	4.18196e11	0.662852	0.331426	+	0.943481i	$0.392470\pi$
$420$	0	0
$421$	3.11784e11	0.483709	0.241855	−	0.970313i	$-0.422244\pi$
$421$	3.11784e11	0.483709	0.241855	+	0.970313i	$0.422244\pi$
$422$	0	0
$423$	−4.70424e11	−0.714427
$424$	0	0
$425$	−2.11196e10	−0.0314004
$426$	0	0
$427$	−5.23681e11	−0.762328
$428$	0	0
$429$	9.40351e11	1.34039
$430$	0	0
$431$	−8.66540e11	−1.20960	−0.604799	−	0.796378i	$-0.706747\pi$
$431$	−8.66540e11	−1.20960	−0.604799	+	0.796378i	$0.706747\pi$
$432$	0	0
$433$	2.95119e11	0.403461	0.201730	−	0.979441i	$-0.435344\pi$
$433$	2.95119e11	0.403461	0.201730	+	0.979441i	$0.435344\pi$
$434$	0	0
$435$	2.54901e11	0.341326
$436$	0	0
$437$	−5.34736e11	−0.701411
$438$	0	0
$439$	−5.48786e11	−0.705201	−0.352600	−	0.935774i	$-0.614703\pi$
$439$	−5.48786e11	−0.705201	−0.352600	+	0.935774i	$0.614703\pi$
$440$	0	0
$441$	−3.69508e11	−0.465211
$442$	0	0
$443$	1.03597e12	1.27800	0.638999	−	0.769207i	$-0.279349\pi$
$443$	1.03597e12	1.27800	0.638999	+	0.769207i	$0.279349\pi$
$444$	0	0
$445$	−5.49503e10	−0.0664278
$446$	0	0
$447$	2.34968e11	0.278372
$448$	0	0
$449$	−1.04924e12	−1.21833	−0.609167	−	0.793042i	$-0.708496\pi$
$449$	−1.04924e12	−1.21833	−0.609167	+	0.793042i	$0.708496\pi$
$450$	0	0
$451$	−1.21784e12	−1.38611
$452$	0	0
$453$	8.29415e11	0.925401
$454$	0	0
$455$	−1.70214e12	−1.86185
$456$	0	0
$457$	1.13929e11	0.122183	0.0610917	−	0.998132i	$-0.480542\pi$
$457$	1.13929e11	0.122183	0.0610917	+	0.998132i	$0.480542\pi$
$458$	0	0
$459$	2.49260e11	0.262117
$460$	0	0
$461$	1.04783e12	1.08053	0.540264	−	0.841496i	$-0.318325\pi$
$461$	1.04783e12	1.08053	0.540264	+	0.841496i	$0.318325\pi$
$462$	0	0
$463$	−1.26218e11	−0.127646	−0.0638231	−	0.997961i	$-0.520329\pi$
$463$	−1.26218e11	−0.127646	−0.0638231	+	0.997961i	$0.520329\pi$
$464$	0	0
$465$	−1.02032e12	−1.01204
$466$	0	0
$467$	1.20765e12	1.17494	0.587470	−	0.809246i	$-0.300124\pi$
$467$	1.20765e12	1.17494	0.587470	+	0.809246i	$0.300124\pi$
$468$	0	0
$469$	−1.45648e11	−0.139003
$470$	0	0
$471$	−6.38434e11	−0.597753
$472$	0	0
$473$	2.15946e12	1.98367
$474$	0	0
$475$	−2.37988e11	−0.214503
$476$	0	0
$477$	2.74406e11	0.242695
$478$	0	0
$479$	1.58750e12	1.37786	0.688930	−	0.724828i	$-0.258081\pi$
$479$	1.58750e12	1.37786	0.688930	+	0.724828i	$0.258081\pi$
$480$	0	0
$481$	−1.13758e12	−0.969014
$482$	0	0
$483$	−5.56512e11	−0.465278
$484$	0	0
$485$	−5.36294e11	−0.440114
$486$	0	0
$487$	−9.39880e11	−0.757167	−0.378584	−	0.925567i	$-0.623589\pi$
$487$	−9.39880e11	−0.757167	−0.378584	+	0.925567i	$0.623589\pi$
$488$	0	0
$489$	2.34347e11	0.185340
$490$	0	0
$491$	−9.66438e11	−0.750425	−0.375213	−	0.926939i	$-0.622430\pi$
$491$	−9.66438e11	−0.750425	−0.375213	+	0.926939i	$0.622430\pi$
$492$	0	0
$493$	−1.53342e11	−0.116910
$494$	0	0
$495$	6.77330e11	0.507081
$496$	0	0
$497$	−9.02538e11	−0.663532
$498$	0	0
$499$	1.22995e12	0.888045	0.444022	−	0.896016i	$-0.353551\pi$
$499$	1.22995e12	0.888045	0.444022	+	0.896016i	$0.353551\pi$
$500$	0	0
$501$	−1.64373e11	−0.116563
$502$	0	0
$503$	1.98521e12	1.38277	0.691386	−	0.722485i	$-0.257000\pi$
$503$	1.98521e12	1.38277	0.691386	+	0.722485i	$0.257000\pi$
$504$	0	0
$505$	2.36626e10	0.0161902
$506$	0	0
$507$	−1.01485e12	−0.682127
$508$	0	0
$509$	2.25974e12	1.49221	0.746104	−	0.665830i	$-0.231922\pi$
$509$	2.25974e12	1.49221	0.746104	+	0.665830i	$0.231922\pi$
$510$	0	0
$511$	5.77130e11	0.374438
$512$	0	0
$513$	2.80880e12	1.79058
$514$	0	0
$515$	−9.35813e11	−0.586215
$516$	0	0
$517$	−3.50803e12	−2.15952
$518$	0	0
$519$	−1.45411e12	−0.879717
$520$	0	0
$521$	4.78832e11	0.284717	0.142358	−	0.989815i	$-0.454531\pi$
$521$	4.78832e11	0.284717	0.142358	+	0.989815i	$0.454531\pi$
$522$	0	0
$523$	9.37326e11	0.547814	0.273907	−	0.961756i	$-0.411684\pi$
$523$	9.37326e11	0.547814	0.273907	+	0.961756i	$0.411684\pi$
$524$	0	0
$525$	−2.47679e11	−0.142290
$526$	0	0
$527$	6.13798e11	0.346639
$528$	0	0
$529$	−1.47834e12	−0.820775
$530$	0	0
$531$	8.75110e11	0.477680
$532$	0	0
$533$	2.77663e12	1.49020
$534$	0	0
$535$	9.37257e11	0.494614
$536$	0	0
$537$	7.96944e11	0.413565
$538$	0	0
$539$	−2.75548e12	−1.40620
$540$	0	0
$541$	−3.44109e12	−1.72706	−0.863532	−	0.504293i	$-0.831753\pi$
$541$	−3.44109e12	−1.72706	−0.863532	+	0.504293i	$0.831753\pi$
$542$	0	0
$543$	2.11021e12	1.04166
$544$	0	0
$545$	4.45264e11	0.216189
$546$	0	0
$547$	−8.05238e11	−0.384575	−0.192288	−	0.981339i	$-0.561591\pi$
$547$	−8.05238e11	−0.384575	−0.192288	+	0.981339i	$0.561591\pi$
$548$	0	0
$549$	−4.75114e11	−0.223215
$550$	0	0
$551$	−1.72795e12	−0.798637
$552$	0	0
$553$	−1.27210e12	−0.578438
$554$	0	0
$555$	1.11302e12	0.497948
$556$	0	0
$557$	−2.13503e11	−0.0939842	−0.0469921	−	0.998895i	$-0.514964\pi$
$557$	−2.13503e11	−0.0939842	−0.0469921	+	0.998895i	$0.514964\pi$
$558$	0	0
$559$	−4.92347e12	−2.13264
$560$	0	0
$561$	5.53479e11	0.235922
$562$	0	0
$563$	1.94290e12	0.815010	0.407505	−	0.913203i	$-0.366399\pi$
$563$	1.94290e12	0.815010	0.407505	+	0.913203i	$0.366399\pi$
$564$	0	0
$565$	−2.22688e12	−0.919345
$566$	0	0
$567$	1.41196e12	0.573719
$568$	0	0
$569$	−9.66618e11	−0.386589	−0.193295	−	0.981141i	$-0.561917\pi$
$569$	−9.66618e11	−0.386589	−0.193295	+	0.981141i	$0.561917\pi$
$570$	0	0
$571$	−3.96159e12	−1.55958	−0.779788	−	0.626043i	$-0.784673\pi$
$571$	−3.96159e12	−1.55958	−0.779788	+	0.626043i	$0.784673\pi$
$572$	0	0
$573$	−3.77629e12	−1.46342
$574$	0	0
$575$	1.43669e11	0.0548098
$576$	0	0
$577$	2.21488e12	0.831876	0.415938	−	0.909393i	$-0.363453\pi$
$577$	2.21488e12	0.831876	0.415938	+	0.909393i	$0.363453\pi$
$578$	0	0
$579$	1.24086e12	0.458847
$580$	0	0
$581$	−6.15819e12	−2.24213
$582$	0	0
$583$	2.04629e12	0.733600
$584$	0	0
$585$	−1.54428e12	−0.545162
$586$	0	0
$587$	6.70503e11	0.233093	0.116546	−	0.993185i	$-0.462818\pi$
$587$	6.70503e11	0.233093	0.116546	+	0.993185i	$0.462818\pi$
$588$	0	0
$589$	6.91664e12	2.36797
$590$	0	0
$591$	−5.32314e11	−0.179483
$592$	0	0
$593$	−4.14332e12	−1.37595	−0.687975	−	0.725735i	$-0.741500\pi$
$593$	−4.14332e12	−1.37595	−0.687975	+	0.725735i	$0.741500\pi$
$594$	0	0
$595$	−1.00186e12	−0.327703
$596$	0	0
$597$	−2.02816e11	−0.0653458
$598$	0	0
$599$	−2.11940e12	−0.672653	−0.336327	−	0.941745i	$-0.609185\pi$
$599$	−2.11940e12	−0.672653	−0.336327	+	0.941745i	$0.609185\pi$
$600$	0	0
$601$	4.44856e12	1.39086	0.695431	−	0.718593i	$-0.255213\pi$
$601$	4.44856e12	1.39086	0.695431	+	0.718593i	$0.255213\pi$
$602$	0	0
$603$	−1.32140e11	−0.0407011
$604$	0	0
$605$	1.97634e12	0.599740
$606$	0	0
$607$	−8.59588e11	−0.257005	−0.128502	−	0.991709i	$-0.541017\pi$
$607$	−8.59588e11	−0.257005	−0.128502	+	0.991709i	$0.541017\pi$
$608$	0	0
$609$	−1.79832e12	−0.529772
$610$	0	0
$611$	7.99816e12	2.32169
$612$	0	0
$613$	−5.43289e12	−1.55403	−0.777013	−	0.629484i	$-0.783266\pi$
$613$	−5.43289e12	−1.55403	−0.777013	+	0.629484i	$0.783266\pi$
$614$	0	0
$615$	−2.71667e12	−0.765772
$616$	0	0
$617$	−1.22774e12	−0.341053	−0.170527	−	0.985353i	$-0.554547\pi$
$617$	−1.22774e12	−0.341053	−0.170527	+	0.985353i	$0.554547\pi$
$618$	0	0
$619$	−4.33143e12	−1.18583	−0.592917	−	0.805264i	$-0.702024\pi$
$619$	−4.33143e12	−1.18583	−0.592917	+	0.805264i	$0.702024\pi$
$620$	0	0
$621$	−1.69563e12	−0.457529
$622$	0	0
$623$	3.87673e11	0.103103
$624$	0	0
$625$	−3.25688e12	−0.853771
$626$	0	0
$627$	6.23692e12	1.61163
$628$	0	0
$629$	−6.69566e11	−0.170555
$630$	0	0
$631$	−1.79967e12	−0.451919	−0.225959	−	0.974137i	$-0.572552\pi$
$631$	−1.79967e12	−0.451919	−0.225959	+	0.974137i	$0.572552\pi$
$632$	0	0
$633$	4.74433e12	1.17451
$634$	0	0
$635$	−2.53402e12	−0.618483
$636$	0	0
$637$	6.28238e12	1.51181
$638$	0	0
$639$	−8.18835e11	−0.194286
$640$	0	0
$641$	5.02365e12	1.17532	0.587662	−	0.809106i	$-0.300048\pi$
$641$	5.02365e12	1.17532	0.587662	+	0.809106i	$0.300048\pi$
$642$	0	0
$643$	−4.60458e12	−1.06229	−0.531143	−	0.847283i	$-0.678237\pi$
$643$	−4.60458e12	−1.06229	−0.531143	+	0.847283i	$0.678237\pi$
$644$	0	0
$645$	4.81716e12	1.09590
$646$	0	0
$647$	−4.15507e12	−0.932201	−0.466100	−	0.884732i	$-0.654341\pi$
$647$	−4.15507e12	−0.932201	−0.466100	+	0.884732i	$0.654341\pi$
$648$	0	0
$649$	6.52584e12	1.44390
$650$	0	0
$651$	7.19830e12	1.57078
$652$	0	0
$653$	3.43743e11	0.0739817	0.0369909	−	0.999316i	$-0.488223\pi$
$653$	3.43743e11	0.0739817	0.0369909	+	0.999316i	$0.488223\pi$
$654$	0	0
$655$	−4.31202e12	−0.915367
$656$	0	0
$657$	5.23606e11	0.109638
$658$	0	0
$659$	−2.39727e12	−0.495145	−0.247572	−	0.968869i	$-0.579633\pi$
$659$	−2.39727e12	−0.495145	−0.247572	+	0.968869i	$0.579633\pi$
$660$	0	0
$661$	−2.09899e12	−0.427665	−0.213833	−	0.976870i	$-0.568595\pi$
$661$	−2.09899e12	−0.427665	−0.213833	+	0.976870i	$0.568595\pi$
$662$	0	0
$663$	−1.26191e12	−0.253639
$664$	0	0
$665$	−1.12895e13	−2.23861
$666$	0	0
$667$	1.04314e12	0.204068
$668$	0	0
$669$	−1.07271e12	−0.207045
$670$	0	0
$671$	−3.54301e12	−0.674715
$672$	0	0
$673$	−1.61276e12	−0.303042	−0.151521	−	0.988454i	$-0.548417\pi$
$673$	−1.61276e12	−0.303042	−0.151521	+	0.988454i	$0.548417\pi$
$674$	0	0
$675$	−7.54650e11	−0.139920
$676$	0	0
$677$	2.64830e12	0.484527	0.242264	−	0.970210i	$-0.422110\pi$
$677$	2.64830e12	0.484527	0.242264	+	0.970210i	$0.422110\pi$
$678$	0	0
$679$	3.78355e12	0.683101
$680$	0	0
$681$	8.00373e12	1.42604
$682$	0	0
$683$	9.69821e12	1.70529	0.852645	−	0.522491i	$-0.174997\pi$
$683$	9.69821e12	1.70529	0.852645	+	0.522491i	$0.174997\pi$
$684$	0	0
$685$	−6.31026e12	−1.09506
$686$	0	0
$687$	−8.72432e12	−1.49426
$688$	0	0
$689$	−4.66546e12	−0.788693
$690$	0	0
$691$	7.85953e12	1.31143	0.655715	−	0.755008i	$-0.272367\pi$
$691$	7.85953e12	1.31143	0.655715	+	0.755008i	$0.272367\pi$
$692$	0	0
$693$	−4.77855e12	−0.787040
$694$	0	0
$695$	−4.66190e12	−0.757934
$696$	0	0
$697$	1.63429e12	0.262290
$698$	0	0
$699$	5.29503e12	0.838922
$700$	0	0
$701$	1.35808e12	0.212419	0.106210	−	0.994344i	$-0.466129\pi$
$701$	1.35808e12	0.212419	0.106210	+	0.994344i	$0.466129\pi$
$702$	0	0
$703$	−7.54506e12	−1.16510
$704$	0	0
$705$	−7.82546e12	−1.19305
$706$	0	0
$707$	−1.66940e11	−0.0251288
$708$	0	0
$709$	8.38924e12	1.24685	0.623426	−	0.781883i	$-0.285740\pi$
$709$	8.38924e12	1.24685	0.623426	+	0.781883i	$0.285740\pi$
$710$	0	0
$711$	−1.15412e12	−0.169370
$712$	0	0
$713$	−4.17546e12	−0.605064
$714$	0	0
$715$	−1.15160e13	−1.64787
$716$	0	0
$717$	−4.32713e12	−0.611454
$718$	0	0
$719$	−1.05644e12	−0.147423	−0.0737117	−	0.997280i	$-0.523484\pi$
$719$	−1.05644e12	−0.147423	−0.0737117	+	0.997280i	$0.523484\pi$
$720$	0	0
$721$	6.60214e12	0.909864
$722$	0	0
$723$	3.25748e12	0.443363
$724$	0	0
$725$	4.64254e11	0.0624072
$726$	0	0
$727$	−1.20219e12	−0.159614	−0.0798068	−	0.996810i	$-0.525430\pi$
$727$	−1.20219e12	−0.159614	−0.0798068	+	0.996810i	$0.525430\pi$
$728$	0	0
$729$	7.53554e12	0.988190
$730$	0	0
$731$	−2.89789e12	−0.375365
$732$	0	0
$733$	−1.38044e13	−1.76624	−0.883121	−	0.469146i	$-0.844562\pi$
$733$	−1.38044e13	−1.76624	−0.883121	+	0.469146i	$0.844562\pi$
$734$	0	0
$735$	−6.14672e12	−0.776874
$736$	0	0
$737$	−9.85390e11	−0.123028
$738$	0	0
$739$	1.03973e13	1.28240	0.641198	−	0.767375i	$-0.278438\pi$
$739$	1.03973e13	1.28240	0.641198	+	0.767375i	$0.278438\pi$
$740$	0	0
$741$	−1.42199e13	−1.73267
$742$	0	0
$743$	−7.06440e12	−0.850405	−0.425202	−	0.905098i	$-0.639797\pi$
$743$	−7.06440e12	−0.850405	−0.425202	+	0.905098i	$0.639797\pi$
$744$	0	0
$745$	−2.87753e12	−0.342229
$746$	0	0
$747$	−5.58707e12	−0.656510
$748$	0	0
$749$	−6.61233e12	−0.767691
$750$	0	0
$751$	−5.52292e12	−0.633562	−0.316781	−	0.948499i	$-0.602602\pi$
$751$	−5.52292e12	−0.633562	−0.316781	+	0.948499i	$0.602602\pi$
$752$	0	0
$753$	2.18752e12	0.247956
$754$	0	0
$755$	−1.01574e13	−1.13768
$756$	0	0
$757$	−5.93646e12	−0.657046	−0.328523	−	0.944496i	$-0.606551\pi$
$757$	−5.93646e12	−0.657046	−0.328523	+	0.944496i	$0.606551\pi$
$758$	0	0
$759$	−3.76513e12	−0.411805
$760$	0	0
$761$	−1.16527e13	−1.25949	−0.629745	−	0.776802i	$-0.716840\pi$
$761$	−1.16527e13	−1.25949	−0.629745	+	0.776802i	$0.716840\pi$
$762$	0	0
$763$	−3.14133e12	−0.335547
$764$	0	0
$765$	−9.08945e11	−0.0959536
$766$	0	0
$767$	−1.48786e13	−1.55233
$768$	0	0
$769$	−3.13586e12	−0.323361	−0.161681	−	0.986843i	$-0.551691\pi$
$769$	−3.13586e12	−0.323361	−0.161681	+	0.986843i	$0.551691\pi$
$770$	0	0
$771$	9.56324e12	0.974676
$772$	0	0
$773$	1.41981e13	1.43029	0.715143	−	0.698979i	$-0.246362\pi$
$773$	1.41981e13	1.43029	0.715143	+	0.698979i	$0.246362\pi$
$774$	0	0
$775$	−1.85831e12	−0.185038
$776$	0	0
$777$	−7.85232e12	−0.772865
$778$	0	0
$779$	1.84161e13	1.79176
$780$	0	0
$781$	−6.10619e12	−0.587274
$782$	0	0
$783$	−5.47927e12	−0.520949
$784$	0	0
$785$	7.81856e12	0.734875
$786$	0	0
$787$	7.60742e12	0.706889	0.353444	−	0.935456i	$-0.385010\pi$
$787$	7.60742e12	0.706889	0.353444	+	0.935456i	$0.385010\pi$
$788$	0	0
$789$	1.52656e13	1.40238
$790$	0	0
$791$	1.57106e13	1.42691
$792$	0	0
$793$	8.07790e12	0.725386
$794$	0	0
$795$	4.56472e12	0.405286
$796$	0	0
$797$	8.82832e11	0.0775025	0.0387513	−	0.999249i	$-0.487662\pi$
$797$	8.82832e11	0.0775025	0.0387513	+	0.999249i	$0.487662\pi$
$798$	0	0
$799$	4.70762e12	0.408640
$800$	0	0
$801$	3.51720e11	0.0301891
$802$	0	0
$803$	3.90462e12	0.331405
$804$	0	0
$805$	6.81530e12	0.572010
$806$	0	0
$807$	1.78311e13	1.47995
$808$	0	0
$809$	1.36683e13	1.12188	0.560940	−	0.827856i	$-0.310440\pi$
$809$	1.36683e13	1.12188	0.560940	+	0.827856i	$0.310440\pi$
$810$	0	0
$811$	1.63530e13	1.32741	0.663703	−	0.747996i	$-0.268984\pi$
$811$	1.63530e13	1.32741	0.663703	+	0.747996i	$0.268984\pi$
$812$	0	0
$813$	4.82326e12	0.387198
$814$	0	0
$815$	−2.86992e12	−0.227856
$816$	0	0
$817$	−3.26551e13	−2.56420
$818$	0	0
$819$	1.08949e13	0.846146
$820$	0	0
$821$	−4.83235e12	−0.371206	−0.185603	−	0.982625i	$-0.559424\pi$
$821$	−4.83235e12	−0.371206	−0.185603	+	0.982625i	$0.559424\pi$
$822$	0	0
$823$	1.61695e13	1.22856	0.614282	−	0.789087i	$-0.289446\pi$
$823$	1.61695e13	1.22856	0.614282	+	0.789087i	$0.289446\pi$
$824$	0	0
$825$	−1.67569e12	−0.125937
$826$	0	0
$827$	−1.37247e13	−1.02030	−0.510149	−	0.860086i	$-0.670410\pi$
$827$	−1.37247e13	−1.02030	−0.510149	+	0.860086i	$0.670410\pi$
$828$	0	0
$829$	2.58304e13	1.89949	0.949743	−	0.313031i	$-0.101344\pi$
$829$	2.58304e13	1.89949	0.949743	+	0.313031i	$0.101344\pi$
$830$	0	0
$831$	−2.05703e11	−0.0149636
$832$	0	0
$833$	3.69773e12	0.266092
$834$	0	0
$835$	2.01299e12	0.143302
$836$	0	0
$837$	2.19324e13	1.54462
$838$	0	0
$839$	5.42716e11	0.0378132	0.0189066	−	0.999821i	$-0.493981\pi$
$839$	5.42716e11	0.0378132	0.0189066	+	0.999821i	$0.493981\pi$
$840$	0	0
$841$	−1.11363e13	−0.767645
$842$	0	0
$843$	−1.24250e13	−0.847368
$844$	0	0
$845$	1.24283e13	0.838604
$846$	0	0
$847$	−1.39431e13	−0.930857
$848$	0	0
$849$	−1.75824e10	−0.00116143
$850$	0	0
$851$	4.55483e12	0.297707
$852$	0	0
$853$	−1.71048e13	−1.10623	−0.553116	−	0.833104i	$-0.686561\pi$
$853$	−1.71048e13	−1.10623	−0.553116	+	0.833104i	$0.686561\pi$
$854$	0	0
$855$	−1.02425e13	−0.655480
$856$	0	0
$857$	−6.06810e12	−0.384272	−0.192136	−	0.981368i	$-0.561542\pi$
$857$	−6.06810e12	−0.384272	−0.192136	+	0.981368i	$0.561542\pi$
$858$	0	0
$859$	1.87986e13	1.17803	0.589016	−	0.808122i	$-0.299516\pi$
$859$	1.87986e13	1.17803	0.589016	+	0.808122i	$0.299516\pi$
$860$	0	0
$861$	1.91661e13	1.18855
$862$	0	0
$863$	6.53337e12	0.400949	0.200474	−	0.979699i	$-0.435752\pi$
$863$	6.53337e12	0.400949	0.200474	+	0.979699i	$0.435752\pi$
$864$	0	0
$865$	1.78077e13	1.08152
$866$	0	0
$867$	−7.42742e11	−0.0446429
$868$	0	0
$869$	−8.60646e12	−0.511960
$870$	0	0
$871$	2.24665e12	0.132268
$872$	0	0
$873$	3.43265e12	0.200017
$874$	0	0
$875$	2.64615e13	1.52608
$876$	0	0
$877$	−1.60851e13	−0.918178	−0.459089	−	0.888390i	$-0.651824\pi$
$877$	−1.60851e13	−0.918178	−0.459089	+	0.888390i	$0.651824\pi$
$878$	0	0
$879$	−6.87507e12	−0.388443
$880$	0	0
$881$	3.30698e13	1.84944	0.924720	−	0.380647i	$-0.124299\pi$
$881$	3.30698e13	1.84944	0.924720	+	0.380647i	$0.124299\pi$
$882$	0	0
$883$	7.25465e12	0.401599	0.200800	−	0.979632i	$-0.435646\pi$
$883$	7.25465e12	0.401599	0.200800	+	0.979632i	$0.435646\pi$
$884$	0	0
$885$	1.45574e13	0.797697
$886$	0	0
$887$	1.13226e13	0.614170	0.307085	−	0.951682i	$-0.400646\pi$
$887$	1.13226e13	0.614170	0.307085	+	0.951682i	$0.400646\pi$
$888$	0	0
$889$	1.78774e13	0.959947
$890$	0	0
$891$	9.55274e12	0.507783
$892$	0	0
$893$	5.30482e13	2.79151
$894$	0	0
$895$	−9.75974e12	−0.508434
$896$	0	0
$897$	8.58432e12	0.442731
$898$	0	0
$899$	−1.34926e13	−0.688934
$900$	0	0
$901$	−2.74603e12	−0.138817
$902$	0	0
$903$	−3.39850e13	−1.70095
$904$	0	0
$905$	−2.58426e13	−1.28061
$906$	0	0
$907$	−3.43776e13	−1.68672	−0.843360	−	0.537348i	$-0.819426\pi$
$907$	−3.43776e13	−1.68672	−0.843360	+	0.537348i	$0.819426\pi$
$908$	0	0
$909$	−1.51457e11	−0.00735788
$910$	0	0
$911$	1.18306e13	0.569082	0.284541	−	0.958664i	$-0.408159\pi$
$911$	1.18306e13	0.569082	0.284541	+	0.958664i	$0.408159\pi$
$912$	0	0
$913$	−4.16637e13	−1.98445
$914$	0	0
$915$	−7.90348e12	−0.372755
$916$	0	0
$917$	3.04212e13	1.42074
$918$	0	0
$919$	2.68999e13	1.24403	0.622015	−	0.783005i	$-0.286314\pi$
$919$	2.68999e13	1.24403	0.622015	+	0.783005i	$0.286314\pi$
$920$	0	0
$921$	9.09868e12	0.416687
$922$	0	0
$923$	1.39218e13	0.631378
$924$	0	0
$925$	2.02716e12	0.0910436
$926$	0	0
$927$	5.98985e12	0.266414
$928$	0	0
$929$	−9.83481e12	−0.433207	−0.216603	−	0.976260i	$-0.569498\pi$
$929$	−9.83481e12	−0.433207	−0.216603	+	0.976260i	$0.569498\pi$
$930$	0	0
$931$	4.16681e13	1.81773
$932$	0	0
$933$	−7.79374e12	−0.336728
$934$	0	0
$935$	−6.77816e12	−0.290041
$936$	0	0
$937$	9.23144e12	0.391238	0.195619	−	0.980680i	$-0.437328\pi$
$937$	9.23144e12	0.391238	0.195619	+	0.980680i	$0.437328\pi$
$938$	0	0
$939$	−1.70812e12	−0.0717009
$940$	0	0
$941$	−1.00553e13	−0.418062	−0.209031	−	0.977909i	$-0.567031\pi$
$941$	−1.00553e13	−0.418062	−0.209031	+	0.977909i	$0.567031\pi$
$942$	0	0
$943$	−1.11175e13	−0.457830
$944$	0	0
$945$	−3.57987e13	−1.46024
$946$	0	0
$947$	−3.28956e13	−1.32912	−0.664558	−	0.747237i	$-0.731380\pi$
$947$	−3.28956e13	−1.32912	−0.664558	+	0.747237i	$0.731380\pi$
$948$	0	0
$949$	−8.90236e12	−0.356293
$950$	0	0
$951$	−9.09106e12	−0.360414
$952$	0	0
$953$	−2.79353e13	−1.09707	−0.548537	−	0.836126i	$-0.684815\pi$
$953$	−2.79353e13	−1.09707	−0.548537	+	0.836126i	$0.684815\pi$
$954$	0	0
$955$	4.62463e13	1.79913
$956$	0	0
$957$	−1.21667e13	−0.468887
$958$	0	0
$959$	4.45187e13	1.69965
$960$	0	0
$961$	2.75686e13	1.04270
$962$	0	0
$963$	−5.99909e12	−0.224785
$964$	0	0
$965$	−1.51961e13	−0.564105
$966$	0	0
$967$	−3.57620e13	−1.31523	−0.657617	−	0.753352i	$-0.728435\pi$
$967$	−3.57620e13	−1.31523	−0.657617	+	0.753352i	$0.728435\pi$
$968$	0	0
$969$	−8.36965e12	−0.304965
$970$	0	0
$971$	5.49737e12	0.198458	0.0992290	−	0.995065i	$-0.468362\pi$
$971$	5.49737e12	0.198458	0.0992290	+	0.995065i	$0.468362\pi$
$972$	0	0
$973$	3.28896e13	1.17639
$974$	0	0
$975$	3.82051e12	0.135394
$976$	0	0
$977$	−1.41757e13	−0.497759	−0.248879	−	0.968534i	$-0.580062\pi$
$977$	−1.41757e13	−0.497759	−0.248879	+	0.968534i	$0.580062\pi$
$978$	0	0
$979$	2.62283e12	0.0912533
$980$	0	0
$981$	−2.85000e12	−0.0982503
$982$	0	0
$983$	−4.48632e13	−1.53250	−0.766248	−	0.642545i	$-0.777879\pi$
$983$	−4.48632e13	−1.53250	−0.766248	+	0.642545i	$0.777879\pi$
$984$	0	0
$985$	6.51896e12	0.220656
$986$	0	0
$987$	5.52085e13	1.85173
$988$	0	0
$989$	1.97134e13	0.655205
$990$	0	0
$991$	−2.93025e13	−0.965101	−0.482550	−	0.875868i	$-0.660289\pi$
$991$	−2.93025e13	−0.965101	−0.482550	+	0.875868i	$0.660289\pi$
$992$	0	0
$993$	6.05454e12	0.197610
$994$	0	0
$995$	2.48378e12	0.0803357
$996$	0	0
$997$	1.15625e13	0.370616	0.185308	−	0.982680i	$-0.440672\pi$
$997$	1.15625e13	0.370616	0.185308	+	0.982680i	$0.440672\pi$
$998$	0	0
$999$	−2.39251e13	−0.759993

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.g.1.4		7
4.3	odd	2		17.10.a.b.1.5	✓	7
12.11	even	2		153.10.a.f.1.3		7
68.67	odd	2		289.10.a.b.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.b.1.5	✓	7	4.3	odd	2
153.10.a.f.1.3		7	12.11	even	2
272.10.a.g.1.4		7	1.1	even	1	trivial
289.10.a.b.1.5		7	68.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q-106.475 q^{3} +1303.94 q^{5} -9199.27 q^{7} -8346.12 q^{9} +O(q^{10})\)	\(q-106.475 q^{3} +1303.94 q^{5} -9199.27 q^{7} -8346.12 q^{9} -62238.4 q^{11} +141901. q^{13} -138837. q^{15} +83521.0 q^{17} +941163. q^{19} +979491. q^{21} -568165. q^{23} -252865. q^{25} +2.98439e6 q^{27} -1.83597e6 q^{29} +7.34903e6 q^{31} +6.62682e6 q^{33} -1.19953e7 q^{35} -8.01674e6 q^{37} -1.51089e7 q^{39} +1.95674e7 q^{41} -3.46966e7 q^{43} -1.08828e7 q^{45} +5.63645e7 q^{47} +4.42730e7 q^{49} -8.89288e6 q^{51} -3.28783e7 q^{53} -8.11551e7 q^{55} -1.00210e8 q^{57} -1.04852e8 q^{59} +5.69264e7 q^{61} +7.67782e7 q^{63} +1.85030e8 q^{65} +1.58325e7 q^{67} +6.04952e7 q^{69} +9.81097e7 q^{71} -6.27365e7 q^{73} +2.69238e7 q^{75} +5.72548e8 q^{77} +1.38282e8 q^{79} -1.53486e8 q^{81} +6.69421e8 q^{83} +1.08906e8 q^{85} +1.95485e8 q^{87} -4.21417e7 q^{89} -1.30538e9 q^{91} -7.82486e8 q^{93} +1.22722e9 q^{95} -4.11288e8 q^{97} +5.19449e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * q^99

Introduction

L-functions

Modular forms

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Database

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