LMFDB - Embedded newform 16.40.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,40,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 40, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 40);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 40 $
Character orbit:	$[\chi]$	$=$	16.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:	$154.143282224$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 175630027x - 142249227846 $ x^3 - 175630027*x - 142249227846
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{5}\cdot 5\cdot 13 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-812.996$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14962e9 q^{3} +5.36221e13 q^{5} -1.82084e16 q^{7} +5.86757e18 q^{9} +O(q^{10})$	$q+3.14962e9 q^{3} +5.36221e13 q^{5} -1.82084e16 q^{7} +5.86757e18 q^{9} -2.56889e20 q^{11} -2.93238e21 q^{13} +1.68890e23 q^{15} -9.89190e22 q^{17} -1.24548e25 q^{19} -5.73494e25 q^{21} +1.86286e26 q^{23} +1.05635e27 q^{25} +5.71661e27 q^{27} +1.31775e28 q^{29} -1.89026e29 q^{31} -8.09103e29 q^{33} -9.76371e29 q^{35} +3.22957e30 q^{37} -9.23588e30 q^{39} -2.27011e31 q^{41} -5.36582e31 q^{43} +3.14632e32 q^{45} -1.87430e32 q^{47} -5.78000e32 q^{49} -3.11557e32 q^{51} -3.22443e32 q^{53} -1.37749e34 q^{55} -3.92280e34 q^{57} -1.01307e34 q^{59} -8.66378e34 q^{61} -1.06839e35 q^{63} -1.57240e35 q^{65} +5.98696e33 q^{67} +5.86730e35 q^{69} -5.02476e35 q^{71} +1.14062e36 q^{73} +3.32709e36 q^{75} +4.67752e36 q^{77} -7.78268e36 q^{79} -5.77348e36 q^{81} +4.09909e37 q^{83} -5.30425e36 q^{85} +4.15041e37 q^{87} +2.04808e38 q^{89} +5.33937e37 q^{91} -5.95359e38 q^{93} -6.67854e38 q^{95} -2.29267e38 q^{97} -1.50731e39 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 1109442852 q^{3} + 17426916500490 q^{5} + 17\!\cdots\!44 q^{7}+ \cdots + 83\!\cdots\!91 q^{9}+O(q^{10}) $ 3 * q - 1109442852 * q^3 + 17426916500490 * q^5 + 17996297718635544 * q^7 + 8305879902078677391 * q^9	$ 3 q - 1109442852 q^{3} + 17426916500490 q^{5} + 17\!\cdots\!44 q^{7}+ \cdots - 10\!\cdots\!72 q^{99}+O(q^{100}) $ 3 * q - 1109442852 * q^3 + 17426916500490 * q^5 + 17996297718635544 * q^7 + 8305879902078677391 * q^9 - 667814968265751536076 * q^11 + 3156080701447967254962 * q^13 + 171209278264452417200520 * q^15 + 890025791008793741893206 * q^17 - 9227612185007697573518580 * q^19 - 190208658572592931471251744 * q^21 - 305308591470919570740188472 * q^23 + 1866127065486173095027030725 * q^25 + 11508090459893795046427628760 * q^27 - 2492682219258648277296444030 * q^29 - 306107671747822704409554746016 * q^31 - 157376353761155567959844693616 * q^33 + 1197259217905835166323668756560 * q^35 + 7599073412303579174888591086506 * q^37 - 22252083518345402015145920059992 * q^39 + 2391924206237769506232479990286 * q^41 - 162865923671547339657067136304492 * q^43 + 588908524142943445153326555470130 * q^45 - 202683231875405841765643043683056 * q^47 + 368237681722499062385273201641179 * q^49 - 785651103815434341253216161914376 * q^51 + 5869079988764517679853946632443002 * q^53 + 4993109836111426485910134681304920 * q^55 - 46129102671186186415908500969957520 * q^57 + 56082681385621280317109288669637060 * q^59 + 39577675682126428331898689976833826 * q^61 + 174786720262353444072185876495505528 * q^63 - 264857762104053085095614223637519620 * q^65 + 140659929656119892877736400727430044 * q^67 + 1584175048695702801789842685209503392 * q^69 - 3849053210874068146700295241416217896 * q^71 - 180754136492273067100959984036797778 * q^73 + 4308561205008052508725896857031567300 * q^75 + 5690389092595065468343565382905921952 * q^77 - 13807772201216229289895673856694559120 * q^79 - 18798023146176620962298765658838887237 * q^81 + 20899400147905548803677768348186808268 * q^83 - 105991789747453754784809897833488498060 * q^85 + 73247996942640959139716393654519138280 * q^87 + 205050039910109857136561853222076224510 * q^89 + 165525104998331842571199169888366982544 * q^91 - 167325068951627887866919523900021629056 * q^93 - 724803433705663618674774120451764528600 * q^95 + 173490888317728793437812209497231327206 * q^97 - 1057018158118568141770036034713090936572 * 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
$3$	3.14962e9	1.56457	0.782283	−	0.622923i	$-0.214055\pi$
$3$	3.14962e9	1.56457	0.782283	+	0.622923i	$0.214055\pi$
$4$	0	0
$5$	5.36221e13	1.25727	0.628636	−	0.777700i	$-0.283614\pi$
$5$	5.36221e13	1.25727	0.628636	+	0.777700i	$0.283614\pi$
$6$	0	0
$7$	−1.82084e16	−0.603752	−0.301876	−	0.953347i	$-0.597613\pi$
$7$	−1.82084e16	−0.603752	−0.301876	+	0.953347i	$0.597613\pi$
$8$	0	0
$9$	5.86757e18	1.44787
$10$	0	0
$11$	−2.56889e20	−1.26645	−0.633225	−	0.773968i	$-0.718269\pi$
$11$	−2.56889e20	−1.26645	−0.633225	+	0.773968i	$0.718269\pi$
$12$	0	0
$13$	−2.93238e21	−0.556319	−0.278160	−	0.960535i	$-0.589724\pi$
$13$	−2.93238e21	−0.556319	−0.278160	+	0.960535i	$0.589724\pi$
$14$	0	0
$15$	1.68890e23	1.96709
$16$	0	0
$17$	−9.89190e22	−0.100352	−0.0501759	−	0.998740i	$-0.515978\pi$
$17$	−9.89190e22	−0.100352	−0.0501759	+	0.998740i	$0.515978\pi$
$18$	0	0
$19$	−1.24548e25	−1.44425	−0.722125	−	0.691763i	$-0.756834\pi$
$19$	−1.24548e25	−1.44425	−0.722125	+	0.691763i	$0.756834\pi$
$20$	0	0
$21$	−5.73494e25	−0.944611
$22$	0	0
$23$	1.86286e26	0.520580	0.260290	−	0.965531i	$-0.416182\pi$
$23$	1.86286e26	0.520580	0.260290	+	0.965531i	$0.416182\pi$
$24$	0	0
$25$	1.05635e27	0.580732
$26$	0	0
$27$	5.71661e27	0.700721
$28$	0	0
$29$	1.31775e28	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$29$	1.31775e28	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$30$	0	0
$31$	−1.89026e29	−1.56663	−0.783317	−	0.621623i	$-0.786474\pi$
$31$	−1.89026e29	−1.56663	−0.783317	+	0.621623i	$0.786474\pi$
$32$	0	0
$33$	−8.09103e29	−1.98145
$34$	0	0
$35$	−9.76371e29	−0.759081
$36$	0	0
$37$	3.22957e30	0.849593	0.424797	−	0.905289i	$-0.360346\pi$
$37$	3.22957e30	0.849593	0.424797	+	0.905289i	$0.360346\pi$
$38$	0	0
$39$	−9.23588e30	−0.870399
$40$	0	0
$41$	−2.27011e31	−0.806793	−0.403397	−	0.915025i	$-0.632171\pi$
$41$	−2.27011e31	−0.806793	−0.403397	+	0.915025i	$0.632171\pi$
$42$	0	0
$43$	−5.36582e31	−0.753357	−0.376679	−	0.926344i	$-0.622934\pi$
$43$	−5.36582e31	−0.753357	−0.376679	+	0.926344i	$0.622934\pi$
$44$	0	0
$45$	3.14632e32	1.82036
$46$	0	0
$47$	−1.87430e32	−0.464442	−0.232221	−	0.972663i	$-0.574599\pi$
$47$	−1.87430e32	−0.464442	−0.232221	+	0.972663i	$0.574599\pi$
$48$	0	0
$49$	−5.78000e32	−0.635483
$50$	0	0
$51$	−3.11557e32	−0.157007
$52$	0	0
$53$	−3.22443e32	−0.0767490	−0.0383745	−	0.999263i	$-0.512218\pi$
$53$	−3.22443e32	−0.0767490	−0.0383745	+	0.999263i	$0.512218\pi$
$54$	0	0
$55$	−1.37749e34	−1.59227
$56$	0	0
$57$	−3.92280e34	−2.25962
$58$	0	0
$59$	−1.01307e34	−0.297870	−0.148935	−	0.988847i	$-0.547585\pi$
$59$	−1.01307e34	−0.297870	−0.148935	+	0.988847i	$0.547585\pi$
$60$	0	0
$61$	−8.66378e34	−1.32977	−0.664884	−	0.746947i	$-0.731519\pi$
$61$	−8.66378e34	−1.32977	−0.664884	+	0.746947i	$0.731519\pi$
$62$	0	0
$63$	−1.06839e35	−0.874155
$64$	0	0
$65$	−1.57240e35	−0.699445
$66$	0	0
$67$	5.98696e33	0.0147485	0.00737424	−	0.999973i	$-0.497653\pi$
$67$	5.98696e33	0.0147485	0.00737424	+	0.999973i	$0.497653\pi$
$68$	0	0
$69$	5.86730e35	0.814482
$70$	0	0
$71$	−5.02476e35	−0.399556	−0.199778	−	0.979841i	$-0.564022\pi$
$71$	−5.02476e35	−0.399556	−0.199778	+	0.979841i	$0.564022\pi$
$72$	0	0
$73$	1.14062e36	0.527651	0.263826	−	0.964570i	$-0.415016\pi$
$73$	1.14062e36	0.527651	0.263826	+	0.964570i	$0.415016\pi$
$74$	0	0
$75$	3.32709e36	0.908594
$76$	0	0
$77$	4.67752e36	0.764622
$78$	0	0
$79$	−7.78268e36	−0.771615	−0.385808	−	0.922579i	$-0.626077\pi$
$79$	−7.78268e36	−0.771615	−0.385808	+	0.922579i	$0.626077\pi$
$80$	0	0
$81$	−5.77348e36	−0.351544
$82$	0	0
$83$	4.09909e37	1.55119	0.775594	−	0.631232i	$-0.217450\pi$
$83$	4.09909e37	1.55119	0.775594	+	0.631232i	$0.217450\pi$
$84$	0	0
$85$	−5.30425e36	−0.126170
$86$	0	0
$87$	4.15041e37	0.627286
$88$	0	0
$89$	2.04808e38	1.98720	0.993600	−	0.112954i	$-0.0360312\pi$
$89$	2.04808e38	1.98720	0.993600	+	0.112954i	$0.0360312\pi$
$90$	0	0
$91$	5.33937e37	0.335879
$92$	0	0
$93$	−5.95359e38	−2.45110
$94$	0	0
$95$	−6.67854e38	−1.81581
$96$	0	0
$97$	−2.29267e38	−0.415234	−0.207617	−	0.978210i	$-0.566571\pi$
$97$	−2.29267e38	−0.415234	−0.207617	+	0.978210i	$0.566571\pi$
$98$	0	0
$99$	−1.50731e39	−1.83365
$100$	0	0
$101$	1.29569e39	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$101$	1.29569e39	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$102$	0	0
$103$	2.69807e39	1.51610	0.758049	−	0.652198i	$-0.226153\pi$
$103$	2.69807e39	1.51610	0.758049	+	0.652198i	$0.226153\pi$
$104$	0	0
$105$	−3.07520e39	−1.18763
$106$	0	0
$107$	−1.69252e39	−0.452430	−0.226215	−	0.974077i	$-0.572635\pi$
$107$	−1.69252e39	−0.452430	−0.226215	+	0.974077i	$0.572635\pi$
$108$	0	0
$109$	1.65572e39	0.308439	0.154220	−	0.988037i	$-0.450714\pi$
$109$	1.65572e39	0.308439	0.154220	+	0.988037i	$0.450714\pi$
$110$	0	0
$111$	1.01719e40	1.32924
$112$	0	0
$113$	1.65677e40	1.52839	0.764193	−	0.644988i	$-0.223137\pi$
$113$	1.65677e40	1.52839	0.764193	+	0.644988i	$0.223137\pi$
$114$	0	0
$115$	9.98905e39	0.654510
$116$	0	0
$117$	−1.72059e40	−0.805478
$118$	0	0
$119$	1.80115e39	0.0605877
$120$	0	0
$121$	2.48471e40	0.603895
$122$	0	0
$123$	−7.15000e40	−1.26228
$124$	0	0
$125$	−4.08946e40	−0.527134
$126$	0	0
$127$	−1.00910e41	−0.954473	−0.477236	−	0.878775i	$-0.658361\pi$
$127$	−1.00910e41	−0.954473	−0.477236	+	0.878775i	$0.658361\pi$
$128$	0	0
$129$	−1.69003e41	−1.17868
$130$	0	0
$131$	−3.37684e40	−0.174470	−0.0872348	−	0.996188i	$-0.527803\pi$
$131$	−3.37684e40	−0.174470	−0.0872348	+	0.996188i	$0.527803\pi$
$132$	0	0
$133$	2.26782e41	0.871969
$134$	0	0
$135$	3.06537e41	0.880997
$136$	0	0
$137$	−5.27644e41	−1.13839	−0.569193	−	0.822204i	$-0.692744\pi$
$137$	−5.27644e41	−1.13839	−0.569193	+	0.822204i	$0.692744\pi$
$138$	0	0
$139$	−6.43985e41	−1.04734	−0.523669	−	0.851922i	$-0.675437\pi$
$139$	−6.43985e41	−1.04734	−0.523669	+	0.851922i	$0.675437\pi$
$140$	0	0
$141$	−5.90335e41	−0.726650
$142$	0	0
$143$	7.53295e41	0.704551
$144$	0	0
$145$	7.06604e41	0.504081
$146$	0	0
$147$	−1.82048e42	−0.994256
$148$	0	0
$149$	1.12323e42	0.471341	0.235671	−	0.971833i	$-0.424271\pi$
$149$	1.12323e42	0.471341	0.235671	+	0.971833i	$0.424271\pi$
$150$	0	0
$151$	3.59308e42	1.16256	0.581281	−	0.813703i	$-0.302552\pi$
$151$	3.59308e42	1.16256	0.581281	+	0.813703i	$0.302552\pi$
$152$	0	0
$153$	−5.80414e41	−0.145296
$154$	0	0
$155$	−1.01360e43	−1.96968
$156$	0	0
$157$	−2.11217e42	−0.319659	−0.159829	−	0.987145i	$-0.551094\pi$
$157$	−2.11217e42	−0.319659	−0.159829	+	0.987145i	$0.551094\pi$
$158$	0	0
$159$	−1.01557e42	−0.120079
$160$	0	0
$161$	−3.39196e42	−0.314301
$162$	0	0
$163$	5.72434e42	0.416934	0.208467	−	0.978029i	$-0.433153\pi$
$163$	5.72434e42	0.416934	0.208467	+	0.978029i	$0.433153\pi$
$164$	0	0
$165$	−4.33859e43	−2.49122
$166$	0	0
$167$	−3.34941e43	−1.52054	−0.760268	−	0.649610i	$-0.774932\pi$
$167$	−3.34941e43	−1.52054	−0.760268	+	0.649610i	$0.774932\pi$
$168$	0	0
$169$	−1.91849e43	−0.690509
$170$	0	0
$171$	−7.30795e43	−2.09108
$172$	0	0
$173$	−3.41377e43	−0.778636	−0.389318	−	0.921103i	$-0.627289\pi$
$173$	−3.41377e43	−0.778636	−0.389318	+	0.921103i	$0.627289\pi$
$174$	0	0
$175$	−1.92343e43	−0.350618
$176$	0	0
$177$	−3.19080e43	−0.466037
$178$	0	0
$179$	1.44182e44	1.69151	0.845757	−	0.533569i	$-0.179149\pi$
$179$	1.44182e44	1.69151	0.845757	+	0.533569i	$0.179149\pi$
$180$	0	0
$181$	3.88244e43	0.366752	0.183376	−	0.983043i	$-0.441297\pi$
$181$	3.88244e43	0.366752	0.183376	+	0.983043i	$0.441297\pi$
$182$	0	0
$183$	−2.72877e44	−2.08051
$184$	0	0
$185$	1.73176e44	1.06817
$186$	0	0
$187$	2.54112e43	0.127091
$188$	0	0
$189$	−1.04090e44	−0.423062
$190$	0	0
$191$	1.72164e44	0.569889	0.284945	−	0.958544i	$-0.408025\pi$
$191$	1.72164e44	0.569889	0.284945	+	0.958544i	$0.408025\pi$
$192$	0	0
$193$	−6.30991e44	−1.70473	−0.852365	−	0.522947i	$-0.824832\pi$
$193$	−6.30991e44	−1.70473	−0.852365	+	0.522947i	$0.824832\pi$
$194$	0	0
$195$	−4.95248e44	−1.09433
$196$	0	0
$197$	−1.00595e45	−1.82173	−0.910865	−	0.412705i	$-0.864584\pi$
$197$	−1.00595e45	−1.82173	−0.910865	+	0.412705i	$0.864584\pi$
$198$	0	0
$199$	−1.18780e45	−1.76648	−0.883242	−	0.468918i	$-0.844644\pi$
$199$	−1.18780e45	−1.76648	−0.883242	+	0.468918i	$0.844644\pi$
$200$	0	0
$201$	1.88567e43	0.0230750
$202$	0	0
$203$	−2.39940e44	−0.242064
$204$	0	0
$205$	−1.21728e45	−1.01436
$206$	0	0
$207$	1.09305e45	0.753731
$208$	0	0
$209$	3.19950e45	1.82907
$210$	0	0
$211$	3.02479e43	0.0143611	0.00718054	−	0.999974i	$-0.497714\pi$
$211$	3.02479e43	0.0143611	0.00718054	+	0.999974i	$0.497714\pi$
$212$	0	0
$213$	−1.58261e45	−0.625132
$214$	0	0
$215$	−2.87727e45	−0.947175
$216$	0	0
$217$	3.44184e45	0.945859
$218$	0	0
$219$	3.59254e45	0.825545
$220$	0	0
$221$	2.90068e44	0.0558277
$222$	0	0
$223$	6.62720e45	1.07000	0.535002	−	0.844851i	$-0.320311\pi$
$223$	6.62720e45	1.07000	0.535002	+	0.844851i	$0.320311\pi$
$224$	0	0
$225$	6.19818e45	0.840824
$226$	0	0
$227$	5.09742e45	0.581897	0.290948	−	0.956739i	$-0.406029\pi$
$227$	5.09742e45	0.581897	0.290948	+	0.956739i	$0.406029\pi$
$228$	0	0
$229$	−5.09087e45	−0.489778	−0.244889	−	0.969551i	$-0.578752\pi$
$229$	−5.09087e45	−0.489778	−0.244889	+	0.969551i	$0.578752\pi$
$230$	0	0
$231$	1.47324e46	1.19630
$232$	0	0
$233$	2.36093e46	1.62048	0.810239	−	0.586100i	$-0.199337\pi$
$233$	2.36093e46	1.62048	0.810239	+	0.586100i	$0.199337\pi$
$234$	0	0
$235$	−1.00504e46	−0.583930
$236$	0	0
$237$	−2.45125e46	−1.20724
$238$	0	0
$239$	1.68674e46	0.705162	0.352581	−	0.935781i	$-0.385304\pi$
$239$	1.68674e46	0.705162	0.352581	+	0.935781i	$0.385304\pi$
$240$	0	0
$241$	3.12678e46	1.11113	0.555565	−	0.831473i	$-0.312502\pi$
$241$	3.12678e46	1.11113	0.555565	+	0.831473i	$0.312502\pi$
$242$	0	0
$243$	−4.13512e46	−1.25074
$244$	0	0
$245$	−3.09936e46	−0.798975
$246$	0	0
$247$	3.65222e46	0.803464
$248$	0	0
$249$	1.29106e47	2.42694
$250$	0	0
$251$	6.69718e46	1.07710	0.538548	−	0.842595i	$-0.318973\pi$
$251$	6.69718e46	1.07710	0.538548	+	0.842595i	$0.318973\pi$
$252$	0	0
$253$	−4.78548e46	−0.659288
$254$	0	0
$255$	−1.67064e46	−0.197401
$256$	0	0
$257$	−3.87410e46	−0.393074	−0.196537	−	0.980496i	$-0.562970\pi$
$257$	−3.87410e46	−0.393074	−0.196537	+	0.980496i	$0.562970\pi$
$258$	0	0
$259$	−5.88051e46	−0.512944
$260$	0	0
$261$	7.73197e46	0.580498
$262$	0	0
$263$	5.89846e45	0.0381594	0.0190797	−	0.999818i	$-0.493926\pi$
$263$	5.89846e45	0.0381594	0.0190797	+	0.999818i	$0.493926\pi$
$264$	0	0
$265$	−1.72901e46	−0.0964944
$266$	0	0
$267$	6.45068e47	3.10911
$268$	0	0
$269$	8.99725e46	0.374921	0.187461	−	0.982272i	$-0.439974\pi$
$269$	8.99725e46	0.374921	0.187461	+	0.982272i	$0.439974\pi$
$270$	0	0
$271$	−1.93454e47	−0.697711	−0.348855	−	0.937177i	$-0.613430\pi$
$271$	−1.93454e47	−0.697711	−0.348855	+	0.937177i	$0.613430\pi$
$272$	0	0
$273$	1.68170e47	0.525505
$274$	0	0
$275$	−2.71363e47	−0.735468
$276$	0	0
$277$	−5.39283e47	−1.26900	−0.634500	−	0.772923i	$-0.718794\pi$
$277$	−5.39283e47	−1.26900	−0.634500	+	0.772923i	$0.718794\pi$
$278$	0	0
$279$	−1.10912e48	−2.26828
$280$	0	0
$281$	3.42772e47	0.609862	0.304931	−	0.952374i	$-0.401367\pi$
$281$	3.42772e47	0.609862	0.304931	+	0.952374i	$0.401367\pi$
$282$	0	0
$283$	−8.61731e47	−1.33517	−0.667584	−	0.744534i	$-0.732672\pi$
$283$	−8.61731e47	−1.33517	−0.667584	+	0.744534i	$0.732672\pi$
$284$	0	0
$285$	−2.10349e48	−2.84096
$286$	0	0
$287$	4.13350e47	0.487104
$288$	0	0
$289$	−9.61861e47	−0.989929
$290$	0	0
$291$	−7.22104e47	−0.649661
$292$	0	0
$293$	1.93492e48	1.52316	0.761578	−	0.648073i	$-0.224425\pi$
$293$	1.93492e48	1.52316	0.761578	+	0.648073i	$0.224425\pi$
$294$	0	0
$295$	−5.43232e47	−0.374503
$296$	0	0
$297$	−1.46853e48	−0.887428
$298$	0	0
$299$	−5.46260e47	−0.289609
$300$	0	0
$301$	9.77027e47	0.454841
$302$	0	0
$303$	4.08094e48	1.66966
$304$	0	0
$305$	−4.64571e48	−1.67188
$306$	0	0
$307$	1.51569e48	0.480188	0.240094	−	0.970750i	$-0.422822\pi$
$307$	1.51569e48	0.480188	0.240094	+	0.970750i	$0.422822\pi$
$308$	0	0
$309$	8.49790e48	2.37204
$310$	0	0
$311$	−1.19146e48	−0.293260	−0.146630	−	0.989191i	$-0.546843\pi$
$311$	−1.19146e48	−0.293260	−0.146630	+	0.989191i	$0.546843\pi$
$312$	0	0
$313$	−4.44520e47	−0.0965555	−0.0482778	−	0.998834i	$-0.515373\pi$
$313$	−4.44520e47	−0.0965555	−0.0482778	+	0.998834i	$0.515373\pi$
$314$	0	0
$315$	−5.72893e48	−1.09905
$316$	0	0
$317$	−1.03382e48	−0.175303	−0.0876516	−	0.996151i	$-0.527936\pi$
$317$	−1.03382e48	−0.175303	−0.0876516	+	0.996151i	$0.527936\pi$
$318$	0	0
$319$	−3.38515e48	−0.507761
$320$	0	0
$321$	−5.33082e48	−0.707857
$322$	0	0
$323$	1.23202e48	0.144933
$324$	0	0
$325$	−3.09760e48	−0.323073
$326$	0	0
$327$	5.21489e48	0.482574
$328$	0	0
$329$	3.41280e48	0.280408
$330$	0	0
$331$	−6.53650e48	−0.477199	−0.238599	−	0.971118i	$-0.576688\pi$
$331$	−6.53650e48	−0.477199	−0.238599	+	0.971118i	$0.576688\pi$
$332$	0	0
$333$	1.89497e49	1.23010
$334$	0	0
$335$	3.21033e47	0.0185429
$336$	0	0
$337$	−2.75713e49	−1.41799	−0.708996	−	0.705213i	$-0.750852\pi$
$337$	−2.75713e49	−1.41799	−0.708996	+	0.705213i	$0.750852\pi$
$338$	0	0
$339$	5.21821e49	2.39126
$340$	0	0
$341$	4.85586e49	1.98406
$342$	0	0
$343$	2.70857e49	0.987427
$344$	0	0
$345$	3.14617e49	1.02402
$346$	0	0
$347$	−1.59040e48	−0.0462468	−0.0231234	−	0.999733i	$-0.507361\pi$
$347$	−1.59040e48	−0.0462468	−0.0231234	+	0.999733i	$0.507361\pi$
$348$	0	0
$349$	−4.30840e49	−1.12001	−0.560003	−	0.828491i	$-0.689200\pi$
$349$	−4.30840e49	−1.12001	−0.560003	+	0.828491i	$0.689200\pi$
$350$	0	0
$351$	−1.67633e49	−0.389825
$352$	0	0
$353$	4.31576e49	0.898361	0.449181	−	0.893441i	$-0.351716\pi$
$353$	4.31576e49	0.898361	0.449181	+	0.893441i	$0.351716\pi$
$354$	0	0
$355$	−2.69439e49	−0.502350
$356$	0	0
$357$	5.67295e48	0.0947935
$358$	0	0
$359$	2.80608e49	0.420492	0.210246	−	0.977648i	$-0.432573\pi$
$359$	2.80608e49	0.420492	0.210246	+	0.977648i	$0.432573\pi$
$360$	0	0
$361$	8.07537e49	1.08586
$362$	0	0
$363$	7.82591e49	0.944834
$364$	0	0
$365$	6.11627e49	0.663401
$366$	0	0
$367$	−8.06624e49	−0.786472	−0.393236	−	0.919438i	$-0.628644\pi$
$367$	−8.06624e49	−0.786472	−0.393236	+	0.919438i	$0.628644\pi$
$368$	0	0
$369$	−1.33200e50	−1.16813
$370$	0	0
$371$	5.87115e48	0.0463374
$372$	0	0
$373$	−2.09254e50	−1.48714	−0.743571	−	0.668657i	$-0.766870\pi$
$373$	−2.09254e50	−1.48714	−0.743571	+	0.668657i	$0.766870\pi$
$374$	0	0
$375$	−1.28803e50	−0.824736
$376$	0	0
$377$	−3.86413e49	−0.223046
$378$	0	0
$379$	−3.48223e49	−0.181298	−0.0906489	−	0.995883i	$-0.528894\pi$
$379$	−3.48223e49	−0.181298	−0.0906489	+	0.995883i	$0.528894\pi$
$380$	0	0
$381$	−3.17830e50	−1.49334
$382$	0	0
$383$	−6.27597e49	−0.266259	−0.133130	−	0.991099i	$-0.542503\pi$
$383$	−6.27597e49	−0.266259	−0.133130	+	0.991099i	$0.542503\pi$
$384$	0	0
$385$	2.50819e50	0.961338
$386$	0	0
$387$	−3.14843e50	−1.09076
$388$	0	0
$389$	−2.37857e50	−0.745245	−0.372623	−	0.927983i	$-0.621541\pi$
$389$	−2.37857e50	−0.745245	−0.372623	+	0.927983i	$0.621541\pi$
$390$	0	0
$391$	−1.84272e49	−0.0522412
$392$	0	0
$393$	−1.06358e50	−0.272969
$394$	0	0
$395$	−4.17324e50	−0.970130
$396$	0	0
$397$	−1.69815e50	−0.357734	−0.178867	−	0.983873i	$-0.557243\pi$
$397$	−1.69815e50	−0.357734	−0.178867	+	0.983873i	$0.557243\pi$
$398$	0	0
$399$	7.14277e50	1.36425
$400$	0	0
$401$	1.06858e51	1.85136	0.925681	−	0.378306i	$-0.123493\pi$
$401$	1.06858e51	1.85136	0.925681	+	0.378306i	$0.123493\pi$
$402$	0	0
$403$	5.54294e50	0.871549
$404$	0	0
$405$	−3.09586e50	−0.441986
$406$	0	0
$407$	−8.29640e50	−1.07597
$408$	0	0
$409$	1.58478e50	0.186794	0.0933972	−	0.995629i	$-0.470227\pi$
$409$	1.58478e50	0.186794	0.0933972	+	0.995629i	$0.470227\pi$
$410$	0	0
$411$	−1.66188e51	−1.78108
$412$	0	0
$413$	1.84464e50	0.179840
$414$	0	0
$415$	2.19802e51	1.95026
$416$	0	0
$417$	−2.02831e51	−1.63863
$418$	0	0
$419$	2.13326e50	0.156989	0.0784947	−	0.996915i	$-0.474989\pi$
$419$	2.13326e50	0.156989	0.0784947	+	0.996915i	$0.474989\pi$
$420$	0	0
$421$	−1.07542e51	−0.721232	−0.360616	−	0.932714i	$-0.617434\pi$
$421$	−1.07542e51	−0.721232	−0.360616	+	0.932714i	$0.617434\pi$
$422$	0	0
$423$	−1.09976e51	−0.672451
$424$	0	0
$425$	−1.04493e50	−0.0582776
$426$	0	0
$427$	1.57753e51	0.802851
$428$	0	0
$429$	2.37259e51	1.10232
$430$	0	0
$431$	−3.49435e51	−1.48272	−0.741360	−	0.671108i	$-0.765819\pi$
$431$	−3.49435e51	−1.48272	−0.741360	+	0.671108i	$0.765819\pi$
$432$	0	0
$433$	2.34196e51	0.907956	0.453978	−	0.891013i	$-0.350005\pi$
$433$	2.34196e51	0.907956	0.453978	+	0.891013i	$0.350005\pi$
$434$	0	0
$435$	2.22554e51	0.788668
$436$	0	0
$437$	−2.32016e51	−0.751847
$438$	0	0
$439$	−1.54147e51	−0.456960	−0.228480	−	0.973549i	$-0.573376\pi$
$439$	−1.54147e51	−0.456960	−0.228480	+	0.973549i	$0.573376\pi$
$440$	0	0
$441$	−3.39145e51	−0.920096
$442$	0	0
$443$	6.02650e51	1.49690	0.748449	−	0.663192i	$-0.230799\pi$
$443$	6.02650e51	1.49690	0.748449	+	0.663192i	$0.230799\pi$
$444$	0	0
$445$	1.09822e52	2.49845
$446$	0	0
$447$	3.53774e51	0.737445
$448$	0	0
$449$	8.31046e51	1.58789	0.793947	−	0.607987i	$-0.208023\pi$
$449$	8.31046e51	1.58789	0.793947	+	0.607987i	$0.208023\pi$
$450$	0	0
$451$	5.83167e51	1.02176
$452$	0	0
$453$	1.13168e52	1.81891
$454$	0	0
$455$	2.86309e51	0.422291
$456$	0	0
$457$	−7.27855e51	−0.985551	−0.492776	−	0.870156i	$-0.664018\pi$
$457$	−7.27855e51	−0.985551	−0.492776	+	0.870156i	$0.664018\pi$
$458$	0	0
$459$	−5.65481e50	−0.0703187
$460$	0	0
$461$	−1.09767e51	−0.125402	−0.0627008	−	0.998032i	$-0.519971\pi$
$461$	−1.09767e51	−0.125402	−0.0627008	+	0.998032i	$0.519971\pi$
$462$	0	0
$463$	8.46883e51	0.889189	0.444595	−	0.895732i	$-0.353348\pi$
$463$	8.46883e51	0.889189	0.444595	+	0.895732i	$0.353348\pi$
$464$	0	0
$465$	−3.19244e52	−3.08170
$466$	0	0
$467$	−1.69800e52	−1.50751	−0.753753	−	0.657158i	$-0.771758\pi$
$467$	−1.69800e52	−1.50751	−0.753753	+	0.657158i	$0.771758\pi$
$468$	0	0
$469$	−1.09013e50	−0.00890443
$470$	0	0
$471$	−6.65255e51	−0.500127
$472$	0	0
$473$	1.37842e52	0.954089
$474$	0	0
$475$	−1.31566e52	−0.838722
$476$	0	0
$477$	−1.89195e51	−0.111123
$478$	0	0
$479$	−2.99013e51	−0.161863	−0.0809314	−	0.996720i	$-0.525789\pi$
$479$	−2.99013e51	−0.161863	−0.0809314	+	0.996720i	$0.525789\pi$
$480$	0	0
$481$	−9.47031e51	−0.472645
$482$	0	0
$483$	−1.06834e52	−0.491745
$484$	0	0
$485$	−1.22938e52	−0.522061
$486$	0	0
$487$	1.46011e52	0.572232	0.286116	−	0.958195i	$-0.407636\pi$
$487$	1.46011e52	0.572232	0.286116	+	0.958195i	$0.407636\pi$
$488$	0	0
$489$	1.80295e52	0.652321
$490$	0	0
$491$	1.53063e52	0.511426	0.255713	−	0.966753i	$-0.417690\pi$
$491$	1.53063e52	0.511426	0.255713	+	0.966753i	$0.417690\pi$
$492$	0	0
$493$	−1.30350e51	−0.0402343
$494$	0	0
$495$	−8.08254e52	−2.30540
$496$	0	0
$497$	9.14926e51	0.241233
$498$	0	0
$499$	4.30786e51	0.105027	0.0525135	−	0.998620i	$-0.483277\pi$
$499$	4.30786e51	0.105027	0.0525135	+	0.998620i	$0.483277\pi$
$500$	0	0
$501$	−1.05494e53	−2.37898
$502$	0	0
$503$	7.62462e52	1.59090	0.795449	−	0.606020i	$-0.207235\pi$
$503$	7.62462e52	1.59090	0.795449	+	0.606020i	$0.207235\pi$
$504$	0	0
$505$	6.94778e52	1.34173
$506$	0	0
$507$	−6.04253e52	−1.08035
$508$	0	0
$509$	4.36698e51	0.0723078	0.0361539	−	0.999346i	$-0.488489\pi$
$509$	4.36698e51	0.0723078	0.0361539	+	0.999346i	$0.488489\pi$
$510$	0	0
$511$	−2.07689e52	−0.318571
$512$	0	0
$513$	−7.11993e52	−1.01202
$514$	0	0
$515$	1.44676e53	1.90615
$516$	0	0
$517$	4.81487e52	0.588192
$518$	0	0
$519$	−1.07521e53	−1.21823
$520$	0	0
$521$	−2.72686e51	−0.0286634	−0.0143317	−	0.999897i	$-0.504562\pi$
$521$	−2.72686e51	−0.0286634	−0.0143317	+	0.999897i	$0.504562\pi$
$522$	0	0
$523$	−3.45861e52	−0.337379	−0.168690	−	0.985669i	$-0.553954\pi$
$523$	−3.45861e52	−0.337379	−0.168690	+	0.985669i	$0.553954\pi$
$524$	0	0
$525$	−6.05808e52	−0.548566
$526$	0	0
$527$	1.86982e52	0.157215
$528$	0	0
$529$	−9.33493e52	−0.728997
$530$	0	0
$531$	−5.94428e52	−0.431277
$532$	0	0
$533$	6.65682e52	0.448835
$534$	0	0
$535$	−9.07568e52	−0.568827
$536$	0	0
$537$	4.54117e53	2.64649
$538$	0	0
$539$	1.48482e53	0.804807
$540$	0	0
$541$	−1.10485e53	−0.557129	−0.278565	−	0.960417i	$-0.589859\pi$
$541$	−1.10485e53	−0.557129	−0.278565	+	0.960417i	$0.589859\pi$
$542$	0	0
$543$	1.22282e53	0.573809
$544$	0	0
$545$	8.87832e52	0.387792
$546$	0	0
$547$	1.77885e52	0.0723413	0.0361706	−	0.999346i	$-0.488484\pi$
$547$	1.77885e52	0.0723413	0.0361706	+	0.999346i	$0.488484\pi$
$548$	0	0
$549$	−5.08354e53	−1.92533
$550$	0	0
$551$	−1.64123e53	−0.579046
$552$	0	0
$553$	1.41710e53	0.465865
$554$	0	0
$555$	5.45440e53	1.67122
$556$	0	0
$557$	3.89141e53	1.11155	0.555777	−	0.831332i	$-0.312421\pi$
$557$	3.89141e53	1.11155	0.555777	+	0.831332i	$0.312421\pi$
$558$	0	0
$559$	1.57346e53	0.419107
$560$	0	0
$561$	8.00356e52	0.198842
$562$	0	0
$563$	4.03186e53	0.934528	0.467264	−	0.884118i	$-0.345240\pi$
$563$	4.03186e53	0.934528	0.467264	+	0.884118i	$0.345240\pi$
$564$	0	0
$565$	8.88397e53	1.92160
$566$	0	0
$567$	1.05126e53	0.212246
$568$	0	0
$569$	−6.36680e52	−0.120014	−0.0600070	−	0.998198i	$-0.519112\pi$
$569$	−6.36680e52	−0.120014	−0.0600070	+	0.998198i	$0.519112\pi$
$570$	0	0
$571$	6.36424e53	1.12032	0.560160	−	0.828384i	$-0.310740\pi$
$571$	6.36424e53	1.12032	0.560160	+	0.828384i	$0.310740\pi$
$572$	0	0
$573$	5.42250e53	0.891630
$574$	0	0
$575$	1.96782e53	0.302317
$576$	0	0
$577$	−9.29606e53	−1.33466	−0.667330	−	0.744762i	$-0.732563\pi$
$577$	−9.29606e53	−1.33466	−0.667330	+	0.744762i	$0.732563\pi$
$578$	0	0
$579$	−1.98738e54	−2.66716
$580$	0	0
$581$	−7.46376e53	−0.936533
$582$	0	0
$583$	8.28319e52	0.0971988
$584$	0	0
$585$	−9.22618e53	−1.01270
$586$	0	0
$587$	−6.03454e52	−0.0619728	−0.0309864	−	0.999520i	$-0.509865\pi$
$587$	−6.03454e52	−0.0619728	−0.0309864	+	0.999520i	$0.509865\pi$
$588$	0	0
$589$	2.35428e54	2.26261
$590$	0	0
$591$	−3.16835e54	−2.85022
$592$	0	0
$593$	−1.41264e54	−1.18977	−0.594886	−	0.803810i	$-0.702803\pi$
$593$	−1.41264e54	−1.18977	−0.594886	+	0.803810i	$0.702803\pi$
$594$	0	0
$595$	9.65816e52	0.0761752
$596$	0	0
$597$	−3.74113e54	−2.76378
$598$	0	0
$599$	1.17608e54	0.813979	0.406990	−	0.913433i	$-0.366579\pi$
$599$	1.17608e54	0.813979	0.406990	+	0.913433i	$0.366579\pi$
$600$	0	0
$601$	−1.78335e53	−0.115660	−0.0578300	−	0.998326i	$-0.518418\pi$
$601$	−1.78335e53	−0.115660	−0.0578300	+	0.998326i	$0.518418\pi$
$602$	0	0
$603$	3.51289e52	0.0213539
$604$	0	0
$605$	1.33236e54	0.759260
$606$	0	0
$607$	1.82763e54	0.976582	0.488291	−	0.872681i	$-0.337620\pi$
$607$	1.82763e54	0.976582	0.488291	+	0.872681i	$0.337620\pi$
$608$	0	0
$609$	−7.55721e53	−0.378725
$610$	0	0
$611$	5.49616e53	0.258378
$612$	0	0
$613$	1.27061e54	0.560446	0.280223	−	0.959935i	$-0.409592\pi$
$613$	1.27061e54	0.560446	0.280223	+	0.959935i	$0.409592\pi$
$614$	0	0
$615$	−3.83398e54	−1.58703
$616$	0	0
$617$	1.76693e54	0.686530	0.343265	−	0.939239i	$-0.388467\pi$
$617$	1.76693e54	0.686530	0.343265	+	0.939239i	$0.388467\pi$
$618$	0	0
$619$	2.12154e54	0.773897	0.386948	−	0.922101i	$-0.373529\pi$
$619$	2.12154e54	0.773897	0.386948	+	0.922101i	$0.373529\pi$
$620$	0	0
$621$	1.06492e54	0.364781
$622$	0	0
$623$	−3.72922e54	−1.19978
$624$	0	0
$625$	−4.11434e54	−1.24348
$626$	0	0
$627$	1.00772e55	2.86170
$628$	0	0
$629$	−3.19466e53	−0.0852583
$630$	0	0
$631$	−4.38932e54	−1.10109	−0.550547	−	0.834804i	$-0.685581\pi$
$631$	−4.38932e54	−1.10109	−0.550547	+	0.834804i	$0.685581\pi$
$632$	0	0
$633$	9.52695e52	0.0224689
$634$	0	0
$635$	−5.41103e54	−1.20003
$636$	0	0
$637$	1.69491e54	0.353532
$638$	0	0
$639$	−2.94831e54	−0.578505
$640$	0	0
$641$	8.28753e54	1.53001	0.765004	−	0.644025i	$-0.222737\pi$
$641$	8.28753e54	1.53001	0.765004	+	0.644025i	$0.222737\pi$
$642$	0	0
$643$	1.96493e54	0.341376	0.170688	−	0.985325i	$-0.445401\pi$
$643$	1.96493e54	0.341376	0.170688	+	0.985325i	$0.445401\pi$
$644$	0	0
$645$	−9.06230e54	−1.48192
$646$	0	0
$647$	−8.12153e54	−1.25027	−0.625136	−	0.780516i	$-0.714956\pi$
$647$	−8.12153e54	−1.25027	−0.625136	+	0.780516i	$0.714956\pi$
$648$	0	0
$649$	2.60247e54	0.377237
$650$	0	0
$651$	1.08405e55	1.47986
$652$	0	0
$653$	−1.31229e55	−1.68741	−0.843707	−	0.536804i	$-0.819632\pi$
$653$	−1.31229e55	−1.68741	−0.843707	+	0.536804i	$0.819632\pi$
$654$	0	0
$655$	−1.81073e54	−0.219356
$656$	0	0
$657$	6.69269e54	0.763970
$658$	0	0
$659$	1.11571e55	1.20029	0.600146	−	0.799890i	$-0.295109\pi$
$659$	1.11571e55	1.20029	0.600146	+	0.799890i	$0.295109\pi$
$660$	0	0
$661$	−3.62687e54	−0.367793	−0.183897	−	0.982946i	$-0.558871\pi$
$661$	−3.62687e54	−0.367793	−0.183897	+	0.982946i	$0.558871\pi$
$662$	0	0
$663$	9.13603e53	0.0873462
$664$	0	0
$665$	1.21605e55	1.09630
$666$	0	0
$667$	2.45478e54	0.208717
$668$	0	0
$669$	2.08732e55	1.67409
$670$	0	0
$671$	2.22563e55	1.68408
$672$	0	0
$673$	−1.86876e54	−0.133432	−0.0667159	−	0.997772i	$-0.521252\pi$
$673$	−1.86876e54	−0.133432	−0.0667159	+	0.997772i	$0.521252\pi$
$674$	0	0
$675$	6.03872e54	0.406931
$676$	0	0
$677$	1.48956e55	0.947498	0.473749	−	0.880660i	$-0.342900\pi$
$677$	1.48956e55	0.947498	0.473749	+	0.880660i	$0.342900\pi$
$678$	0	0
$679$	4.17457e54	0.250698
$680$	0	0
$681$	1.60549e55	0.910416
$682$	0	0
$683$	−9.58075e54	−0.513093	−0.256546	−	0.966532i	$-0.582585\pi$
$683$	−9.58075e54	−0.513093	−0.256546	+	0.966532i	$0.582585\pi$
$684$	0	0
$685$	−2.82934e55	−1.43126
$686$	0	0
$687$	−1.60343e55	−0.766291
$688$	0	0
$689$	9.45523e53	0.0426970
$690$	0	0
$691$	−1.23084e54	−0.0525267	−0.0262634	−	0.999655i	$-0.508361\pi$
$691$	−1.23084e54	−0.0525267	−0.0262634	+	0.999655i	$0.508361\pi$
$692$	0	0
$693$	2.74457e55	1.10707
$694$	0	0
$695$	−3.45319e55	−1.31679
$696$	0	0
$697$	2.24557e54	0.0809633
$698$	0	0
$699$	7.43605e55	2.53535
$700$	0	0
$701$	−4.60890e55	−1.48627	−0.743133	−	0.669144i	$-0.766661\pi$
$701$	−4.60890e55	−1.48627	−0.743133	+	0.669144i	$0.766661\pi$
$702$	0	0
$703$	−4.02237e55	−1.22702
$704$	0	0
$705$	−3.16550e55	−0.913597
$706$	0	0
$707$	−2.35924e55	−0.644308
$708$	0	0
$709$	−4.42017e54	−0.114245	−0.0571226	−	0.998367i	$-0.518193\pi$
$709$	−4.42017e54	−0.114245	−0.0571226	+	0.998367i	$0.518193\pi$
$710$	0	0
$711$	−4.56654e55	−1.11720
$712$	0	0
$713$	−3.52128e55	−0.815558
$714$	0	0
$715$	4.03933e55	0.885812
$716$	0	0
$717$	5.31259e55	1.10327
$718$	0	0
$719$	−4.79691e55	−0.943514	−0.471757	−	0.881729i	$-0.656380\pi$
$719$	−4.79691e55	−0.943514	−0.471757	+	0.881729i	$0.656380\pi$
$720$	0	0
$721$	−4.91274e55	−0.915348
$722$	0	0
$723$	9.84817e55	1.73844
$724$	0	0
$725$	1.39200e55	0.232834
$726$	0	0
$727$	−1.02822e56	−1.62991	−0.814956	−	0.579523i	$-0.803239\pi$
$727$	−1.02822e56	−1.62991	−0.814956	+	0.579523i	$0.803239\pi$
$728$	0	0
$729$	−1.06843e56	−1.60531
$730$	0	0
$731$	5.30781e54	0.0756008
$732$	0	0
$733$	−5.77006e55	−0.779207	−0.389603	−	0.920983i	$-0.627388\pi$
$733$	−5.77006e55	−0.779207	−0.389603	+	0.920983i	$0.627388\pi$
$734$	0	0
$735$	−9.76181e55	−1.25005
$736$	0	0
$737$	−1.53798e54	−0.0186782
$738$	0	0
$739$	−1.40056e56	−1.61337	−0.806686	−	0.590980i	$-0.798741\pi$
$739$	−1.40056e56	−1.61337	−0.806686	+	0.590980i	$0.798741\pi$
$740$	0	0
$741$	1.15031e56	1.25707
$742$	0	0
$743$	−7.96262e54	−0.0825609	−0.0412804	−	0.999148i	$-0.513144\pi$
$743$	−7.96262e54	−0.0825609	−0.0412804	+	0.999148i	$0.513144\pi$
$744$	0	0
$745$	6.02299e55	0.592604
$746$	0	0
$747$	2.40517e56	2.24592
$748$	0	0
$749$	3.08181e55	0.273156
$750$	0	0
$751$	8.79022e55	0.739641	0.369821	−	0.929103i	$-0.379419\pi$
$751$	8.79022e55	0.739641	0.369821	+	0.929103i	$0.379419\pi$
$752$	0	0
$753$	2.10936e56	1.68519
$754$	0	0
$755$	1.92669e56	1.46166
$756$	0	0
$757$	−6.05512e54	−0.0436268	−0.0218134	−	0.999762i	$-0.506944\pi$
$757$	−6.05512e54	−0.0436268	−0.0218134	+	0.999762i	$0.506944\pi$
$758$	0	0
$759$	−1.50725e56	−1.03150
$760$	0	0
$761$	−2.67473e56	−1.73892	−0.869458	−	0.494008i	$-0.835532\pi$
$761$	−2.67473e56	−1.73892	−0.869458	+	0.494008i	$0.835532\pi$
$762$	0	0
$763$	−3.01479e55	−0.186221
$764$	0	0
$765$	−3.11230e55	−0.182677
$766$	0	0
$767$	2.97071e55	0.165711
$768$	0	0
$769$	3.07792e55	0.163190	0.0815951	−	0.996666i	$-0.473999\pi$
$769$	3.07792e55	0.163190	0.0815951	+	0.996666i	$0.473999\pi$
$770$	0	0
$771$	−1.22019e56	−0.614991
$772$	0	0
$773$	7.39678e55	0.354440	0.177220	−	0.984171i	$-0.443290\pi$
$773$	7.39678e55	0.354440	0.177220	+	0.984171i	$0.443290\pi$
$774$	0	0
$775$	−1.99676e56	−0.909794
$776$	0	0
$777$	−1.85214e56	−0.802535
$778$	0	0
$779$	2.82738e56	1.16521
$780$	0	0
$781$	1.29081e56	0.506018
$782$	0	0
$783$	7.53305e55	0.280942
$784$	0	0
$785$	−1.13259e56	−0.401898
$786$	0	0
$787$	3.95231e56	1.33458	0.667289	−	0.744799i	$-0.267455\pi$
$787$	3.95231e56	1.33458	0.667289	+	0.744799i	$0.267455\pi$
$788$	0	0
$789$	1.85779e55	0.0597030
$790$	0	0
$791$	−3.01671e56	−0.922767
$792$	0	0
$793$	2.54055e56	0.739776
$794$	0	0
$795$	−5.44572e55	−0.150972
$796$	0	0
$797$	−1.36161e56	−0.359432	−0.179716	−	0.983719i	$-0.557518\pi$
$797$	−1.36161e56	−0.359432	−0.179716	+	0.983719i	$0.557518\pi$
$798$	0	0
$799$	1.85404e55	0.0466076
$800$	0	0
$801$	1.20172e57	2.87721
$802$	0	0
$803$	−2.93014e56	−0.668244
$804$	0	0
$805$	−1.81884e56	−0.395162
$806$	0	0
$807$	2.83379e56	0.586590
$808$	0	0
$809$	5.42801e56	1.07064	0.535321	−	0.844649i	$-0.320191\pi$
$809$	5.42801e56	1.07064	0.535321	+	0.844649i	$0.320191\pi$
$810$	0	0
$811$	8.54286e56	1.60582	0.802910	−	0.596100i	$-0.203284\pi$
$811$	8.54286e56	1.60582	0.802910	+	0.596100i	$0.203284\pi$
$812$	0	0
$813$	−6.09306e56	−1.09162
$814$	0	0
$815$	3.06952e56	0.524199
$816$	0	0
$817$	6.68303e56	1.08804
$818$	0	0
$819$	3.13291e56	0.486309
$820$	0	0
$821$	−2.83593e56	−0.419763	−0.209882	−	0.977727i	$-0.567308\pi$
$821$	−2.83593e56	−0.419763	−0.209882	+	0.977727i	$0.567308\pi$
$822$	0	0
$823$	7.79009e56	1.09963	0.549814	−	0.835287i	$-0.314699\pi$
$823$	7.79009e56	1.09963	0.549814	+	0.835287i	$0.314699\pi$
$824$	0	0
$825$	−8.54693e56	−1.15069
$826$	0	0
$827$	−2.44580e56	−0.314097	−0.157049	−	0.987591i	$-0.550198\pi$
$827$	−2.44580e56	−0.314097	−0.157049	+	0.987591i	$0.550198\pi$
$828$	0	0
$829$	−1.26495e56	−0.154974	−0.0774871	−	0.996993i	$-0.524690\pi$
$829$	−1.26495e56	−0.154974	−0.0774871	+	0.996993i	$0.524690\pi$
$830$	0	0
$831$	−1.69854e57	−1.98543
$832$	0	0
$833$	5.71751e55	0.0637719
$834$	0	0
$835$	−1.79602e57	−1.91173
$836$	0	0
$837$	−1.08059e57	−1.09777
$838$	0	0
$839$	−1.05147e57	−1.01962	−0.509809	−	0.860287i	$-0.670284\pi$
$839$	−1.05147e57	−1.01962	−0.509809	+	0.860287i	$0.670284\pi$
$840$	0	0
$841$	−9.06598e56	−0.839253
$842$	0	0
$843$	1.07960e57	0.954170
$844$	0	0
$845$	−1.02874e57	−0.868157
$846$	0	0
$847$	−4.52425e56	−0.364603
$848$	0	0
$849$	−2.71413e57	−2.08896
$850$	0	0
$851$	6.01623e56	0.442281
$852$	0	0
$853$	2.14388e57	1.50555	0.752775	−	0.658278i	$-0.228715\pi$
$853$	2.14388e57	1.50555	0.752775	+	0.658278i	$0.228715\pi$
$854$	0	0
$855$	−3.91868e57	−2.62906
$856$	0	0
$857$	1.76630e57	1.13224	0.566119	−	0.824323i	$-0.308444\pi$
$857$	1.76630e57	1.13224	0.566119	+	0.824323i	$0.308444\pi$
$858$	0	0
$859$	1.41212e57	0.864978	0.432489	−	0.901639i	$-0.357635\pi$
$859$	1.41212e57	0.864978	0.432489	+	0.901639i	$0.357635\pi$
$860$	0	0
$861$	1.30190e57	0.762106
$862$	0	0
$863$	2.76506e57	1.54702	0.773508	−	0.633787i	$-0.218500\pi$
$863$	2.76506e57	1.54702	0.773508	+	0.633787i	$0.218500\pi$
$864$	0	0
$865$	−1.83053e57	−0.978958
$866$	0	0
$867$	−3.02950e57	−1.54881
$868$	0	0
$869$	1.99928e57	0.977212
$870$	0	0
$871$	−1.75560e55	−0.00820487
$872$	0	0
$873$	−1.34524e57	−0.601204
$874$	0	0
$875$	7.44623e56	0.318258
$876$	0	0
$877$	−7.42378e56	−0.303482	−0.151741	−	0.988420i	$-0.548488\pi$
$877$	−7.42378e56	−0.303482	−0.151741	+	0.988420i	$0.548488\pi$
$878$	0	0
$879$	6.09427e57	2.38308
$880$	0	0
$881$	6.13623e56	0.229547	0.114773	−	0.993392i	$-0.463386\pi$
$881$	6.13623e56	0.229547	0.114773	+	0.993392i	$0.463386\pi$
$882$	0	0
$883$	−2.47374e57	−0.885361	−0.442681	−	0.896679i	$-0.645972\pi$
$883$	−2.47374e57	−0.885361	−0.442681	+	0.896679i	$0.645972\pi$
$884$	0	0
$885$	−1.71097e57	−0.585936
$886$	0	0
$887$	−4.04554e57	−1.32576	−0.662882	−	0.748724i	$-0.730667\pi$
$887$	−4.04554e57	−1.32576	−0.662882	+	0.748724i	$0.730667\pi$
$888$	0	0
$889$	1.83741e57	0.576265
$890$	0	0
$891$	1.48314e57	0.445213
$892$	0	0
$893$	2.33441e57	0.670770
$894$	0	0
$895$	7.73132e57	2.12669
$896$	0	0
$897$	−1.72051e57	−0.453112
$898$	0	0
$899$	−2.49088e57	−0.628114
$900$	0	0
$901$	3.18957e55	0.00770191
$902$	0	0
$903$	3.07727e57	0.711630
$904$	0	0
$905$	2.08185e57	0.461107
$906$	0	0
$907$	−1.91109e57	−0.405452	−0.202726	−	0.979236i	$-0.564980\pi$
$907$	−1.91109e57	−0.405452	−0.202726	+	0.979236i	$0.564980\pi$
$908$	0	0
$909$	7.60256e57	1.54513
$910$	0	0
$911$	6.02986e57	1.17408	0.587041	−	0.809557i	$-0.300293\pi$
$911$	6.02986e57	1.17408	0.587041	+	0.809557i	$0.300293\pi$
$912$	0	0
$913$	−1.05301e58	−1.96450
$914$	0	0
$915$	−1.46322e58	−2.61577
$916$	0	0
$917$	6.14867e56	0.105336
$918$	0	0
$919$	−6.23642e57	−1.02396	−0.511980	−	0.858997i	$-0.671088\pi$
$919$	−6.23642e57	−1.02396	−0.511980	+	0.858997i	$0.671088\pi$
$920$	0	0
$921$	4.77384e57	0.751286
$922$	0	0
$923$	1.47345e57	0.222281
$924$	0	0
$925$	3.41154e57	0.493386
$926$	0	0
$927$	1.58311e58	2.19511
$928$	0	0
$929$	4.25561e57	0.565790	0.282895	−	0.959151i	$-0.408705\pi$
$929$	4.25561e57	0.565790	0.282895	+	0.959151i	$0.408705\pi$
$930$	0	0
$931$	7.19888e57	0.917796
$932$	0	0
$933$	−3.75266e57	−0.458825
$934$	0	0
$935$	1.36260e57	0.159787
$936$	0	0
$937$	1.14088e58	1.28327	0.641637	−	0.767008i	$-0.278255\pi$
$937$	1.14088e58	1.28327	0.641637	+	0.767008i	$0.278255\pi$
$938$	0	0
$939$	−1.40007e57	−0.151068
$940$	0	0
$941$	−2.33700e57	−0.241914	−0.120957	−	0.992658i	$-0.538596\pi$
$941$	−2.33700e57	−0.241914	−0.120957	+	0.992658i	$0.538596\pi$
$942$	0	0
$943$	−4.22890e57	−0.420000
$944$	0	0
$945$	−5.58153e57	−0.531904
$946$	0	0
$947$	4.94229e56	0.0451964	0.0225982	−	0.999745i	$-0.492806\pi$
$947$	4.94229e56	0.0451964	0.0225982	+	0.999745i	$0.492806\pi$
$948$	0	0
$949$	−3.34474e57	−0.293542
$950$	0	0
$951$	−3.25615e57	−0.274274
$952$	0	0
$953$	1.48078e56	0.0119723	0.00598617	−	0.999982i	$-0.498095\pi$
$953$	1.48078e56	0.0119723	0.00598617	+	0.999982i	$0.498095\pi$
$954$	0	0
$955$	9.23178e57	0.716506
$956$	0	0
$957$	−1.06619e58	−0.794426
$958$	0	0
$959$	9.60753e57	0.687304
$960$	0	0
$961$	2.11725e58	1.45434
$962$	0	0
$963$	−9.93101e57	−0.655059
$964$	0	0
$965$	−3.38351e58	−2.14331
$966$	0	0
$967$	6.58500e57	0.400627	0.200313	−	0.979732i	$-0.435804\pi$
$967$	6.58500e57	0.400627	0.200313	+	0.979732i	$0.435804\pi$
$968$	0	0
$969$	3.88039e57	0.226758
$970$	0	0
$971$	−4.39314e57	−0.246604	−0.123302	−	0.992369i	$-0.539348\pi$
$971$	−4.39314e57	−0.246604	−0.123302	+	0.992369i	$0.539348\pi$
$972$	0	0
$973$	1.17259e58	0.632333
$974$	0	0
$975$	−9.75628e57	−0.505469
$976$	0	0
$977$	2.41520e58	1.20229	0.601144	−	0.799141i	$-0.294712\pi$
$977$	2.41520e58	1.20229	0.601144	+	0.799141i	$0.294712\pi$
$978$	0	0
$979$	−5.26129e58	−2.51669
$980$	0	0
$981$	9.71504e57	0.446580
$982$	0	0
$983$	−7.73253e57	−0.341608	−0.170804	−	0.985305i	$-0.554636\pi$
$983$	−7.73253e57	−0.341608	−0.170804	+	0.985305i	$0.554636\pi$
$984$	0	0
$985$	−5.39411e58	−2.29041
$986$	0	0
$987$	1.07490e58	0.438717
$988$	0	0
$989$	−9.99576e57	−0.392183
$990$	0	0
$991$	2.62440e58	0.989904	0.494952	−	0.868920i	$-0.335186\pi$
$991$	2.62440e58	0.989904	0.494952	+	0.868920i	$0.335186\pi$
$992$	0	0
$993$	−2.05875e58	−0.746609
$994$	0	0
$995$	−6.36926e58	−2.22095
$996$	0	0
$997$	−2.17628e58	−0.729726	−0.364863	−	0.931061i	$-0.618884\pi$
$997$	−2.17628e58	−0.729726	−0.364863	+	0.931061i	$0.618884\pi$
$998$	0	0
$999$	1.84622e58	0.595328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.40.a.c.1.3		3
4.3	odd	2		1.40.a.a.1.2	✓	3
12.11	even	2		9.40.a.b.1.2		3
20.3	even	4		25.40.b.a.24.3		6
20.7	even	4		25.40.b.a.24.4		6
20.19	odd	2		25.40.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.40.a.a.1.2	✓	3	4.3	odd	2
9.40.a.b.1.2		3	12.11	even	2
16.40.a.c.1.3		3	1.1	even	1	trivial
25.40.a.a.1.2		3	20.19	odd	2
25.40.b.a.24.3		6	20.3	even	4
25.40.b.a.24.4		6	20.7	even	4

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 40 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+3.14962e9 q^{3} +5.36221e13 q^{5} -1.82084e16 q^{7} +5.86757e18 q^{9} +O(q^{10})\)	\(q+3.14962e9 q^{3} +5.36221e13 q^{5} -1.82084e16 q^{7} +5.86757e18 q^{9} -2.56889e20 q^{11} -2.93238e21 q^{13} +1.68890e23 q^{15} -9.89190e22 q^{17} -1.24548e25 q^{19} -5.73494e25 q^{21} +1.86286e26 q^{23} +1.05635e27 q^{25} +5.71661e27 q^{27} +1.31775e28 q^{29} -1.89026e29 q^{31} -8.09103e29 q^{33} -9.76371e29 q^{35} +3.22957e30 q^{37} -9.23588e30 q^{39} -2.27011e31 q^{41} -5.36582e31 q^{43} +3.14632e32 q^{45} -1.87430e32 q^{47} -5.78000e32 q^{49} -3.11557e32 q^{51} -3.22443e32 q^{53} -1.37749e34 q^{55} -3.92280e34 q^{57} -1.01307e34 q^{59} -8.66378e34 q^{61} -1.06839e35 q^{63} -1.57240e35 q^{65} +5.98696e33 q^{67} +5.86730e35 q^{69} -5.02476e35 q^{71} +1.14062e36 q^{73} +3.32709e36 q^{75} +4.67752e36 q^{77} -7.78268e36 q^{79} -5.77348e36 q^{81} +4.09909e37 q^{83} -5.30425e36 q^{85} +4.15041e37 q^{87} +2.04808e38 q^{89} +5.33937e37 q^{91} -5.95359e38 q^{93} -6.67854e38 q^{95} -2.29267e38 q^{97} -1.50731e39 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 1109442852 q^{3} + 17426916500490 q^{5} + 17\!\cdots\!44 q^{7}+ \cdots + 83\!\cdots\!91 q^{9}+O(q^{10}) \) 3 * q - 1109442852 * q^3 + 17426916500490 * q^5 + 17996297718635544 * q^7 + 8305879902078677391 * q^9	\( 3 q - 1109442852 q^{3} + 17426916500490 q^{5} + 17\!\cdots\!44 q^{7}+ \cdots - 10\!\cdots\!72 q^{99}+O(q^{100}) \) 3 * q - 1109442852 * q^3 + 17426916500490 * q^5 + 17996297718635544 * q^7 + 8305879902078677391 * q^9 - 667814968265751536076 * q^11 + 3156080701447967254962 * q^13 + 171209278264452417200520 * q^15 + 890025791008793741893206 * q^17 - 9227612185007697573518580 * q^19 - 190208658572592931471251744 * q^21 - 305308591470919570740188472 * q^23 + 1866127065486173095027030725 * q^25 + 11508090459893795046427628760 * q^27 - 2492682219258648277296444030 * q^29 - 306107671747822704409554746016 * q^31 - 157376353761155567959844693616 * q^33 + 1197259217905835166323668756560 * q^35 + 7599073412303579174888591086506 * q^37 - 22252083518345402015145920059992 * q^39 + 2391924206237769506232479990286 * q^41 - 162865923671547339657067136304492 * q^43 + 588908524142943445153326555470130 * q^45 - 202683231875405841765643043683056 * q^47 + 368237681722499062385273201641179 * q^49 - 785651103815434341253216161914376 * q^51 + 5869079988764517679853946632443002 * q^53 + 4993109836111426485910134681304920 * q^55 - 46129102671186186415908500969957520 * q^57 + 56082681385621280317109288669637060 * q^59 + 39577675682126428331898689976833826 * q^61 + 174786720262353444072185876495505528 * q^63 - 264857762104053085095614223637519620 * q^65 + 140659929656119892877736400727430044 * q^67 + 1584175048695702801789842685209503392 * q^69 - 3849053210874068146700295241416217896 * q^71 - 180754136492273067100959984036797778 * q^73 + 4308561205008052508725896857031567300 * q^75 + 5690389092595065468343565382905921952 * q^77 - 13807772201216229289895673856694559120 * q^79 - 18798023146176620962298765658838887237 * q^81 + 20899400147905548803677768348186808268 * q^83 - 105991789747453754784809897833488498060 * q^85 + 73247996942640959139716393654519138280 * q^87 + 205050039910109857136561853222076224510 * q^89 + 165525104998331842571199169888366982544 * q^91 - 167325068951627887866919523900021629056 * q^93 - 724803433705663618674774120451764528600 * q^95 + 173490888317728793437812209497231327206 * q^97 - 1057018158118568141770036034713090936572 * q^99

Introduction

L-functions

Modular forms

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Database

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