LMFDB - Embedded newform 16.44.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,44,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, 44, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 44);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 44 $
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:	$187.376632553$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 397496384250 $ x^2 - x - 397496384250
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-630473.$ 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)

$f(q)$	$=$	$q+3.29600e10 q^{3} +3.97045e14 q^{5} -1.80269e18 q^{7} +7.58103e20 q^{9} +3.33115e22 q^{11} -7.96520e23 q^{13} +1.30866e25 q^{15} -3.35920e26 q^{17} -6.16773e26 q^{19} -5.94168e28 q^{21} -2.43777e29 q^{23} -9.79224e29 q^{25} +1.41677e31 q^{27} -2.03214e31 q^{29} +1.01779e32 q^{31} +1.09795e33 q^{33} -7.15750e32 q^{35} -4.91735e33 q^{37} -2.62533e34 q^{39} -5.16303e34 q^{41} +9.91848e34 q^{43} +3.01001e35 q^{45} -2.80853e35 q^{47} +1.06589e36 q^{49} -1.10719e37 q^{51} +6.66611e36 q^{53} +1.32262e37 q^{55} -2.03288e37 q^{57} -4.38154e37 q^{59} -8.24137e37 q^{61} -1.36663e39 q^{63} -3.16254e38 q^{65} +2.28272e39 q^{67} -8.03489e39 q^{69} -6.57126e39 q^{71} -1.46269e40 q^{73} -3.22752e40 q^{75} -6.00505e40 q^{77} -4.38403e40 q^{79} +2.18114e41 q^{81} +2.02288e41 q^{83} -1.33375e41 q^{85} -6.69791e41 q^{87} -3.53186e41 q^{89} +1.43588e42 q^{91} +3.35464e42 q^{93} -2.44886e41 q^{95} +6.40027e42 q^{97} +2.52536e43 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22341634056 q^{3} - 47320658398020 q^{5} + 22\!\cdots\!28 q^{7} + 54\!\cdots\!14 q^{9} + 41\!\cdots\!16 q^{11} - 15\!\cdots\!96 q^{13} + 17\!\cdots\!40 q^{15} - 67\!\cdots\!48 q^{17} - 12\!\cdots\!40 q^{19}+ \cdots + 23\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q + 22341634056 * q^3 - 47320658398020 * q^5 + 222460131347038928 * q^7 + 542594646100314147114 * q^9 + 41887858358166241718616 * q^11 - 1511213342321158511669396 * q^13 + 17804997360134453991129840 * q^15 - 67349878355730307414152348 * q^17 - 1230904208507994915735409240 * q^19 - 80920530905061445582689383616 * q^21 - 38167008283484201492558062224 * q^23 - 1918631833123351317328900860850 * q^25 + 19941575760041650132553853212880 * q^27 - 72780948725084997011410245083700 * q^29 - 40207733905312243999256937298624 * q^31 + 1006881310052050425793011088926048 * q^33 - 1615658656617379304576130831538080 * q^35 + 3383801276601910515940174638928252 * q^37 - 18664419028213485589802473566212688 * q^39 - 58505417265395086228405561355069196 * q^41 + 285092737981269186332521030698108056 * q^43 + 396764787925879165058503325488849260 * q^45 - 1834986318326201344851046619753827872 * q^47 + 2983331119134682985187309352444961106 * q^49 - 13923666849435817419951839264623903344 * q^51 + 13488383575332508144313645312419699164 * q^53 + 9415153536190550355180017489429532240 * q^55 - 13807748645696734847941276681469831520 * q^57 - 6535875469255007418951566075676717000 * q^59 + 84915949989376412900726656980650069644 * q^61 - 1803064383904392926498433003649964350704 * q^63 + 1330967805107975013132566371731317160 * q^65 + 3889814457887475263640354897004803753608 * q^67 - 10218134177167059404862981043947605969472 * q^69 + 4500205056814555193905197090502997775696 * q^71 - 10215757578580250274581767012298787143756 * q^73 - 22300243542723736486434261503150942041800 * q^75 - 42682181092256892291285667845090826099776 * q^77 - 106263833346898669088358572211107427639520 * q^79 + 227547444210851249821957128354947134508562 * q^81 + 99535113533406416106304387849461181193256 * q^83 - 252718204268333675320943366175053150677320 * q^85 - 112757591652872720426564768050770607266000 * q^87 + 175134609645798670638848613045046472300820 * q^89 - 11482488188506183083757950956690750656544 * q^91 + 4862300735557253257085331576689556832720128 * q^93 + 28012559440364213825890153198462567470000 * q^95 - 3659395264822684391406294706085506900627388 * q^97 + 23405304422197580402007697540466944929273912 * q^99

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
$3$	3.29600e10	1.81920	0.909599	−	0.415488i	$-0.136389\pi$
$3$	3.29600e10	1.81920	0.909599	+	0.415488i	$0.136389\pi$
$4$	0	0
$5$	3.97045e14	0.372378	0.186189	−	0.982514i	$-0.440386\pi$
$5$	3.97045e14	0.372378	0.186189	+	0.982514i	$0.440386\pi$
$6$	0	0
$7$	−1.80269e18	−1.21987	−0.609936	−	0.792451i	$-0.708805\pi$
$7$	−1.80269e18	−1.21987	−0.609936	+	0.792451i	$0.708805\pi$
$8$	0	0
$9$	7.58103e20	2.30948
$10$	0	0
$11$	3.33115e22	1.35723	0.678613	−	0.734496i	$-0.262581\pi$
$11$	3.33115e22	1.35723	0.678613	+	0.734496i	$0.262581\pi$
$12$	0	0
$13$	−7.96520e23	−0.894159	−0.447079	−	0.894494i	$-0.647536\pi$
$13$	−7.96520e23	−0.894159	−0.447079	+	0.894494i	$0.647536\pi$
$14$	0	0
$15$	1.30866e25	0.677429
$16$	0	0
$17$	−3.35920e26	−1.17919	−0.589595	−	0.807699i	$-0.700713\pi$
$17$	−3.35920e26	−1.17919	−0.589595	+	0.807699i	$0.700713\pi$
$18$	0	0
$19$	−6.16773e26	−0.198117	−0.0990587	−	0.995082i	$-0.531583\pi$
$19$	−6.16773e26	−0.198117	−0.0990587	+	0.995082i	$0.531583\pi$
$20$	0	0
$21$	−5.94168e28	−2.21919
$22$	0	0
$23$	−2.43777e29	−1.28779	−0.643895	−	0.765114i	$-0.722683\pi$
$23$	−2.43777e29	−1.28779	−0.643895	+	0.765114i	$0.722683\pi$
$24$	0	0
$25$	−9.79224e29	−0.861334
$26$	0	0
$27$	1.41677e31	2.38220
$28$	0	0
$29$	−2.03214e31	−0.735183	−0.367592	−	0.929987i	$-0.619818\pi$
$29$	−2.03214e31	−0.735183	−0.367592	+	0.929987i	$0.619818\pi$
$30$	0	0
$31$	1.01779e32	0.877773	0.438887	−	0.898542i	$-0.355373\pi$
$31$	1.01779e32	0.877773	0.438887	+	0.898542i	$0.355373\pi$
$32$	0	0
$33$	1.09795e33	2.46906
$34$	0	0
$35$	−7.15750e32	−0.454254
$36$	0	0
$37$	−4.91735e33	−0.944918	−0.472459	−	0.881353i	$-0.656633\pi$
$37$	−4.91735e33	−0.944918	−0.472459	+	0.881353i	$0.656633\pi$
$38$	0	0
$39$	−2.62533e34	−1.62665
$40$	0	0
$41$	−5.16303e34	−1.09157	−0.545785	−	0.837925i	$-0.683769\pi$
$41$	−5.16303e34	−1.09157	−0.545785	+	0.837925i	$0.683769\pi$
$42$	0	0
$43$	9.91848e34	0.753136	0.376568	−	0.926389i	$-0.377104\pi$
$43$	9.91848e34	0.753136	0.376568	+	0.926389i	$0.377104\pi$
$44$	0	0
$45$	3.01001e35	0.860000
$46$	0	0
$47$	−2.80853e35	−0.315047	−0.157523	−	0.987515i	$-0.550351\pi$
$47$	−2.80853e35	−0.315047	−0.157523	+	0.987515i	$0.550351\pi$
$48$	0	0
$49$	1.06589e36	0.488088
$50$	0	0
$51$	−1.10719e37	−2.14518
$52$	0	0
$53$	6.66611e36	0.564861	0.282430	−	0.959288i	$-0.408859\pi$
$53$	6.66611e36	0.564861	0.282430	+	0.959288i	$0.408859\pi$
$54$	0	0
$55$	1.32262e37	0.505401
$56$	0	0
$57$	−2.03288e37	−0.360415
$58$	0	0
$59$	−4.38154e37	−0.370091	−0.185046	−	0.982730i	$-0.559243\pi$
$59$	−4.38154e37	−0.370091	−0.185046	+	0.982730i	$0.559243\pi$
$60$	0	0
$61$	−8.24137e37	−0.339945	−0.169972	−	0.985449i	$-0.554368\pi$
$61$	−8.24137e37	−0.339945	−0.169972	+	0.985449i	$0.554368\pi$
$62$	0	0
$63$	−1.36663e39	−2.81727
$64$	0	0
$65$	−3.16254e38	−0.332965
$66$	0	0
$67$	2.28272e39	1.25269	0.626345	−	0.779546i	$-0.284550\pi$
$67$	2.28272e39	1.25269	0.626345	+	0.779546i	$0.284550\pi$
$68$	0	0
$69$	−8.03489e39	−2.34274
$70$	0	0
$71$	−6.57126e39	−1.03656	−0.518280	−	0.855211i	$-0.673427\pi$
$71$	−6.57126e39	−1.03656	−0.518280	+	0.855211i	$0.673427\pi$
$72$	0	0
$73$	−1.46269e40	−1.26972	−0.634862	−	0.772625i	$-0.718943\pi$
$73$	−1.46269e40	−1.26972	−0.634862	+	0.772625i	$0.718943\pi$
$74$	0	0
$75$	−3.22752e40	−1.56694
$76$	0	0
$77$	−6.00505e40	−1.65564
$78$	0	0
$79$	−4.38403e40	−0.696452	−0.348226	−	0.937411i	$-0.613216\pi$
$79$	−4.38403e40	−0.696452	−0.348226	+	0.937411i	$0.613216\pi$
$80$	0	0
$81$	2.18114e41	2.02421
$82$	0	0
$83$	2.02288e41	1.11120	0.555599	−	0.831451i	$-0.312489\pi$
$83$	2.02288e41	1.11120	0.555599	+	0.831451i	$0.312489\pi$
$84$	0	0
$85$	−1.33375e41	−0.439104
$86$	0	0
$87$	−6.69791e41	−1.33744
$88$	0	0
$89$	−3.53186e41	−0.432632	−0.216316	−	0.976323i	$-0.569404\pi$
$89$	−3.53186e41	−0.432632	−0.216316	+	0.976323i	$0.569404\pi$
$90$	0	0
$91$	1.43588e42	1.09076
$92$	0	0
$93$	3.35464e42	1.59684
$94$	0	0
$95$	−2.44886e41	−0.0737746
$96$	0	0
$97$	6.40027e42	1.23199	0.615993	−	0.787752i	$-0.288755\pi$
$97$	6.40027e42	1.23199	0.615993	+	0.787752i	$0.288755\pi$
$98$	0	0
$99$	2.52536e43	3.13448
$100$	0	0
$101$	−1.03724e43	−0.837474	−0.418737	−	0.908108i	$-0.637527\pi$
$101$	−1.03724e43	−0.837474	−0.418737	+	0.908108i	$0.637527\pi$
$102$	0	0
$103$	−1.71018e43	−0.905820	−0.452910	−	0.891556i	$-0.649614\pi$
$103$	−1.71018e43	−0.905820	−0.452910	+	0.891556i	$0.649614\pi$
$104$	0	0
$105$	−2.35911e43	−0.826377
$106$	0	0
$107$	−2.99663e43	−0.699652	−0.349826	−	0.936815i	$-0.613759\pi$
$107$	−2.99663e43	−0.699652	−0.349826	+	0.936815i	$0.613759\pi$
$108$	0	0
$109$	1.74419e43	0.273479	0.136739	−	0.990607i	$-0.456338\pi$
$109$	1.74419e43	0.273479	0.136739	+	0.990607i	$0.456338\pi$
$110$	0	0
$111$	−1.62076e44	−1.71899
$112$	0	0
$113$	6.93854e43	0.501282	0.250641	−	0.968080i	$-0.419359\pi$
$113$	6.93854e43	0.501282	0.250641	+	0.968080i	$0.419359\pi$
$114$	0	0
$115$	−9.67905e43	−0.479545
$116$	0	0
$117$	−6.03844e44	−2.06504
$118$	0	0
$119$	6.05561e44	1.43846
$120$	0	0
$121$	5.07258e44	0.842061
$122$	0	0
$123$	−1.70173e45	−1.98578
$124$	0	0
$125$	−8.40183e44	−0.693120
$126$	0	0
$127$	2.42717e45	1.42338	0.711689	−	0.702495i	$-0.247931\pi$
$127$	2.42717e45	1.42338	0.711689	+	0.702495i	$0.247931\pi$
$128$	0	0
$129$	3.26913e45	1.37010
$130$	0	0
$131$	4.52439e45	1.36216	0.681078	−	0.732211i	$-0.261511\pi$
$131$	4.52439e45	1.36216	0.681078	+	0.732211i	$0.261511\pi$
$132$	0	0
$133$	1.11185e45	0.241678
$134$	0	0
$135$	5.62521e45	0.887080
$136$	0	0
$137$	4.20024e45	0.482816	0.241408	−	0.970424i	$-0.422391\pi$
$137$	4.20024e45	0.482816	0.241408	+	0.970424i	$0.422391\pi$
$138$	0	0
$139$	−1.45728e46	−1.22666	−0.613329	−	0.789827i	$-0.710170\pi$
$139$	−1.45728e46	−1.22666	−0.613329	+	0.789827i	$0.710170\pi$
$140$	0	0
$141$	−9.25691e45	−0.573132
$142$	0	0
$143$	−2.65333e46	−1.21358
$144$	0	0
$145$	−8.06848e45	−0.273766
$146$	0	0
$147$	3.51318e46	0.887928
$148$	0	0
$149$	6.79498e46	1.28435	0.642175	−	0.766558i	$-0.278032\pi$
$149$	6.79498e46	1.28435	0.642175	+	0.766558i	$0.278032\pi$
$150$	0	0
$151$	−1.18879e47	−1.68695	−0.843474	−	0.537170i	$-0.819494\pi$
$151$	−1.18879e47	−1.68695	−0.843474	+	0.537170i	$0.819494\pi$
$152$	0	0
$153$	−2.54662e47	−2.72331
$154$	0	0
$155$	4.04109e46	0.326864
$156$	0	0
$157$	−2.20019e47	−1.35088	−0.675441	−	0.737414i	$-0.736047\pi$
$157$	−2.20019e47	−1.35088	−0.675441	+	0.737414i	$0.736047\pi$
$158$	0	0
$159$	2.19715e47	1.02759
$160$	0	0
$161$	4.39456e47	1.57094
$162$	0	0
$163$	−3.40799e47	−0.934254	−0.467127	−	0.884190i	$-0.654711\pi$
$163$	−3.40799e47	−0.934254	−0.467127	+	0.884190i	$0.654711\pi$
$164$	0	0
$165$	4.35934e47	0.919425
$166$	0	0
$167$	1.70286e47	0.277188	0.138594	−	0.990349i	$-0.455742\pi$
$167$	1.70286e47	0.277188	0.138594	+	0.990349i	$0.455742\pi$
$168$	0	0
$169$	−1.59087e47	−0.200480
$170$	0	0
$171$	−4.67577e47	−0.457548
$172$	0	0
$173$	2.24170e47	0.170839	0.0854194	−	0.996345i	$-0.472777\pi$
$173$	2.24170e47	0.170839	0.0854194	+	0.996345i	$0.472777\pi$
$174$	0	0
$175$	1.76524e48	1.05072
$176$	0	0
$177$	−1.44416e48	−0.673269
$178$	0	0
$179$	−1.36907e48	−0.501284	−0.250642	−	0.968080i	$-0.580642\pi$
$179$	−1.36907e48	−0.501284	−0.250642	+	0.968080i	$0.580642\pi$
$180$	0	0
$181$	1.59585e48	0.460153	0.230077	−	0.973172i	$-0.426102\pi$
$181$	1.59585e48	0.460153	0.230077	+	0.973172i	$0.426102\pi$
$182$	0	0
$183$	−2.71635e48	−0.618426
$184$	0	0
$185$	−1.95241e48	−0.351867
$186$	0	0
$187$	−1.11900e49	−1.60043
$188$	0	0
$189$	−2.55400e49	−2.90598
$190$	0	0
$191$	−7.91365e48	−0.718058	−0.359029	−	0.933326i	$-0.616892\pi$
$191$	−7.91365e48	−0.718058	−0.359029	+	0.933326i	$0.616892\pi$
$192$	0	0
$193$	−2.15792e49	−1.56514	−0.782568	−	0.622565i	$-0.786091\pi$
$193$	−2.15792e49	−1.56514	−0.782568	+	0.622565i	$0.786091\pi$
$194$	0	0
$195$	−1.04237e49	−0.605729
$196$	0	0
$197$	−2.02401e48	−0.0944470	−0.0472235	−	0.998884i	$-0.515037\pi$
$197$	−2.02401e48	−0.0944470	−0.0472235	+	0.998884i	$0.515037\pi$
$198$	0	0
$199$	2.63656e48	0.0990141	0.0495070	−	0.998774i	$-0.484235\pi$
$199$	2.63656e48	0.0990141	0.0495070	+	0.998774i	$0.484235\pi$
$200$	0	0
$201$	7.52383e49	2.27889
$202$	0	0
$203$	3.66332e49	0.896830
$204$	0	0
$205$	−2.04995e49	−0.406477
$206$	0	0
$207$	−1.84808e50	−2.97412
$208$	0	0
$209$	−2.05456e49	−0.268890
$210$	0	0
$211$	1.00024e50	1.06667	0.533337	−	0.845903i	$-0.320938\pi$
$211$	1.00024e50	1.06667	0.533337	+	0.845903i	$0.320938\pi$
$212$	0	0
$213$	−2.16589e50	−1.88571
$214$	0	0
$215$	3.93808e49	0.280451
$216$	0	0
$217$	−1.83477e50	−1.07077
$218$	0	0
$219$	−4.82101e50	−2.30988
$220$	0	0
$221$	2.67567e50	1.05438
$222$	0	0
$223$	6.05545e50	1.96604	0.983020	−	0.183501i	$-0.0587431\pi$
$223$	6.05545e50	1.96604	0.983020	+	0.183501i	$0.0587431\pi$
$224$	0	0
$225$	−7.42352e50	−1.98923
$226$	0	0
$227$	−8.22900e50	−1.82302	−0.911510	−	0.411278i	$-0.865082\pi$
$227$	−8.22900e50	−1.82302	−0.911510	+	0.411278i	$0.865082\pi$
$228$	0	0
$229$	−7.00201e50	−1.28458	−0.642288	−	0.766464i	$-0.722015\pi$
$229$	−7.00201e50	−1.28458	−0.642288	+	0.766464i	$0.722015\pi$
$230$	0	0
$231$	−1.97926e51	−3.01194
$232$	0	0
$233$	3.12585e50	0.395199	0.197599	−	0.980283i	$-0.436686\pi$
$233$	3.12585e50	0.395199	0.197599	+	0.980283i	$0.436686\pi$
$234$	0	0
$235$	−1.11511e50	−0.117317
$236$	0	0
$237$	−1.44497e51	−1.26698
$238$	0	0
$239$	1.14055e50	0.0834755	0.0417378	−	0.999129i	$-0.486711\pi$
$239$	1.14055e50	0.0834755	0.0417378	+	0.999129i	$0.486711\pi$
$240$	0	0
$241$	5.69958e50	0.348719	0.174360	−	0.984682i	$-0.444214\pi$
$241$	5.69958e50	0.348719	0.174360	+	0.984682i	$0.444214\pi$
$242$	0	0
$243$	2.53840e51	1.30024
$244$	0	0
$245$	4.23207e50	0.181753
$246$	0	0
$247$	4.91272e50	0.177148
$248$	0	0
$249$	6.66741e51	2.02149
$250$	0	0
$251$	2.86476e50	0.0731312	0.0365656	−	0.999331i	$-0.488358\pi$
$251$	2.86476e50	0.0731312	0.0365656	+	0.999331i	$0.488358\pi$
$252$	0	0
$253$	−8.12060e51	−1.74782
$254$	0	0
$255$	−4.39604e51	−0.798818
$256$	0	0
$257$	9.14943e51	1.40550	0.702752	−	0.711435i	$-0.251954\pi$
$257$	9.14943e51	1.40550	0.702752	+	0.711435i	$0.251954\pi$
$258$	0	0
$259$	8.86448e51	1.15268
$260$	0	0
$261$	−1.54057e52	−1.69789
$262$	0	0
$263$	1.31330e52	1.22833	0.614163	−	0.789179i	$-0.289494\pi$
$263$	1.31330e52	1.22833	0.614163	+	0.789179i	$0.289494\pi$
$264$	0	0
$265$	2.64674e51	0.210342
$266$	0	0
$267$	−1.16410e52	−0.787043
$268$	0	0
$269$	1.10289e52	0.635124	0.317562	−	0.948237i	$-0.397136\pi$
$269$	1.10289e52	0.635124	0.317562	+	0.948237i	$0.397136\pi$
$270$	0	0
$271$	−1.58832e52	−0.780008	−0.390004	−	0.920813i	$-0.627526\pi$
$271$	−1.58832e52	−0.780008	−0.390004	+	0.920813i	$0.627526\pi$
$272$	0	0
$273$	4.73266e52	1.98431
$274$	0	0
$275$	−3.26195e52	−1.16903
$276$	0	0
$277$	−4.86772e52	−1.49283	−0.746416	−	0.665480i	$-0.768227\pi$
$277$	−4.86772e52	−1.49283	−0.746416	+	0.665480i	$0.768227\pi$
$278$	0	0
$279$	7.71590e52	2.02720
$280$	0	0
$281$	4.47786e52	1.00898	0.504492	−	0.863416i	$-0.331680\pi$
$281$	4.47786e52	1.00898	0.504492	+	0.863416i	$0.331680\pi$
$282$	0	0
$283$	−1.56790e52	−0.303327	−0.151663	−	0.988432i	$-0.548463\pi$
$283$	−1.56790e52	−0.303327	−0.151663	+	0.988432i	$0.548463\pi$
$284$	0	0
$285$	−8.07144e51	−0.134211
$286$	0	0
$287$	9.30737e52	1.33158
$288$	0	0
$289$	3.16892e52	0.390488
$290$	0	0
$291$	2.10953e53	2.24123
$292$	0	0
$293$	4.52079e52	0.414535	0.207267	−	0.978284i	$-0.433543\pi$
$293$	4.52079e52	0.414535	0.207267	+	0.978284i	$0.433543\pi$
$294$	0	0
$295$	−1.73967e52	−0.137814
$296$	0	0
$297$	4.71948e53	3.23318
$298$	0	0
$299$	1.94174e53	1.15149
$300$	0	0
$301$	−1.78800e53	−0.918729
$302$	0	0
$303$	−3.41875e53	−1.52353
$304$	0	0
$305$	−3.27219e52	−0.126588
$306$	0	0
$307$	−4.59415e52	−0.154429	−0.0772147	−	0.997014i	$-0.524603\pi$
$307$	−4.59415e52	−0.154429	−0.0772147	+	0.997014i	$0.524603\pi$
$308$	0	0
$309$	−5.63676e53	−1.64787
$310$	0	0
$311$	9.87805e52	0.251375	0.125688	−	0.992070i	$-0.459886\pi$
$311$	9.87805e52	0.251375	0.125688	+	0.992070i	$0.459886\pi$
$312$	0	0
$313$	6.64885e53	1.47416	0.737078	−	0.675808i	$-0.236205\pi$
$313$	6.64885e53	1.47416	0.737078	+	0.675808i	$0.236205\pi$
$314$	0	0
$315$	−5.42612e53	−1.04909
$316$	0	0
$317$	−8.52504e53	−1.43854	−0.719270	−	0.694730i	$-0.755524\pi$
$317$	−8.52504e53	−1.43854	−0.719270	+	0.694730i	$0.755524\pi$
$318$	0	0
$319$	−6.76936e53	−0.997810
$320$	0	0
$321$	−9.87689e53	−1.27281
$322$	0	0
$323$	2.07186e53	0.233618
$324$	0	0
$325$	7.79971e53	0.770170
$326$	0	0
$327$	5.74883e53	0.497511
$328$	0	0
$329$	5.06292e53	0.384317
$330$	0	0
$331$	−1.71691e54	−1.14405	−0.572026	−	0.820235i	$-0.693843\pi$
$331$	−1.71691e54	−1.14405	−0.572026	+	0.820235i	$0.693843\pi$
$332$	0	0
$333$	−3.72785e54	−2.18227
$334$	0	0
$335$	9.06341e53	0.466475
$336$	0	0
$337$	2.35850e54	1.06805	0.534027	−	0.845467i	$-0.320678\pi$
$337$	2.35850e54	1.06805	0.534027	+	0.845467i	$0.320678\pi$
$338$	0	0
$339$	2.28694e54	0.911930
$340$	0	0
$341$	3.39042e54	1.19134
$342$	0	0
$343$	2.01527e54	0.624467
$344$	0	0
$345$	−3.19021e54	−0.872387
$346$	0	0
$347$	−6.65761e54	−1.60781	−0.803904	−	0.594759i	$-0.797248\pi$
$347$	−6.65761e54	−1.60781	−0.803904	+	0.594759i	$0.797248\pi$
$348$	0	0
$349$	−5.12896e54	−1.09467	−0.547334	−	0.836914i	$-0.684357\pi$
$349$	−5.12896e54	−1.09467	−0.547334	+	0.836914i	$0.684357\pi$
$350$	0	0
$351$	−1.12849e55	−2.13007
$352$	0	0
$353$	−9.28186e54	−1.55052	−0.775262	−	0.631639i	$-0.782382\pi$
$353$	−9.28186e54	−1.55052	−0.775262	+	0.631639i	$0.782382\pi$
$354$	0	0
$355$	−2.60909e54	−0.385992
$356$	0	0
$357$	1.99593e55	2.61684
$358$	0	0
$359$	−5.79735e54	−0.674060	−0.337030	−	0.941494i	$-0.609422\pi$
$359$	−5.79735e54	−0.674060	−0.337030	+	0.941494i	$0.609422\pi$
$360$	0	0
$361$	−9.31140e54	−0.960749
$362$	0	0
$363$	1.67192e55	1.53188
$364$	0	0
$365$	−5.80752e54	−0.472818
$366$	0	0
$367$	1.84353e55	1.33454	0.667269	−	0.744817i	$-0.267463\pi$
$367$	1.84353e55	1.33454	0.667269	+	0.744817i	$0.267463\pi$
$368$	0	0
$369$	−3.91411e55	−2.52096
$370$	0	0
$371$	−1.20170e55	−0.689058
$372$	0	0
$373$	8.25340e54	0.421593	0.210796	−	0.977530i	$-0.432394\pi$
$373$	8.25340e54	0.421593	0.210796	+	0.977530i	$0.432394\pi$
$374$	0	0
$375$	−2.76924e55	−1.26092
$376$	0	0
$377$	1.61864e55	0.657371
$378$	0	0
$379$	−1.57598e55	−0.571225	−0.285612	−	0.958345i	$-0.592197\pi$
$379$	−1.57598e55	−0.571225	−0.285612	+	0.958345i	$0.592197\pi$
$380$	0	0
$381$	7.99994e55	2.58940
$382$	0	0
$383$	4.72750e55	1.36728	0.683640	−	0.729819i	$-0.260396\pi$
$383$	4.72750e55	1.36728	0.683640	+	0.729819i	$0.260396\pi$
$384$	0	0
$385$	−2.38427e55	−0.616525
$386$	0	0
$387$	7.51922e55	1.73935
$388$	0	0
$389$	7.53509e55	1.56017	0.780085	−	0.625673i	$-0.215176\pi$
$389$	7.53509e55	1.56017	0.780085	+	0.625673i	$0.215176\pi$
$390$	0	0
$391$	8.18896e55	1.51855
$392$	0	0
$393$	1.49124e56	2.47803
$394$	0	0
$395$	−1.74065e55	−0.259343
$396$	0	0
$397$	−1.44560e56	−1.93220	−0.966102	−	0.258160i	$-0.916884\pi$
$397$	−1.44560e56	−1.93220	−0.966102	+	0.258160i	$0.916884\pi$
$398$	0	0
$399$	3.66466e55	0.439660
$400$	0	0
$401$	−6.84820e55	−0.737858	−0.368929	−	0.929458i	$-0.620275\pi$
$401$	−6.84820e55	−0.737858	−0.368929	+	0.929458i	$0.620275\pi$
$402$	0	0
$403$	−8.10691e55	−0.784868
$404$	0	0
$405$	8.66012e55	0.753773
$406$	0	0
$407$	−1.63804e56	−1.28247
$408$	0	0
$409$	−3.28961e55	−0.231789	−0.115895	−	0.993262i	$-0.536973\pi$
$409$	−3.28961e55	−0.231789	−0.115895	+	0.993262i	$0.536973\pi$
$410$	0	0
$411$	1.38440e56	0.878338
$412$	0	0
$413$	7.89858e55	0.451464
$414$	0	0
$415$	8.03174e55	0.413786
$416$	0	0
$417$	−4.80318e56	−2.23153
$418$	0	0
$419$	1.62541e56	0.681338	0.340669	−	0.940183i	$-0.389347\pi$
$419$	1.62541e56	0.681338	0.340669	+	0.940183i	$0.389347\pi$
$420$	0	0
$421$	4.26597e56	1.61418	0.807092	−	0.590426i	$-0.201040\pi$
$421$	4.26597e56	1.61418	0.807092	+	0.590426i	$0.201040\pi$
$422$	0	0
$423$	−2.12915e56	−0.727594
$424$	0	0
$425$	3.28941e56	1.01568
$426$	0	0
$427$	1.48567e56	0.414689
$428$	0	0
$429$	−8.74537e56	−2.20773
$430$	0	0
$431$	−3.62720e56	−0.828534	−0.414267	−	0.910155i	$-0.635962\pi$
$431$	−3.62720e56	−0.828534	−0.414267	+	0.910155i	$0.635962\pi$
$432$	0	0
$433$	2.97238e56	0.614631	0.307315	−	0.951608i	$-0.400569\pi$
$433$	2.97238e56	0.614631	0.307315	+	0.951608i	$0.400569\pi$
$434$	0	0
$435$	−2.65937e56	−0.498035
$436$	0	0
$437$	1.50355e56	0.255134
$438$	0	0
$439$	−4.62182e56	−0.710928	−0.355464	−	0.934690i	$-0.615677\pi$
$439$	−4.62182e56	−0.710928	−0.355464	+	0.934690i	$0.615677\pi$
$440$	0	0
$441$	8.08057e56	1.12723
$442$	0	0
$443$	2.12702e56	0.269210	0.134605	−	0.990899i	$-0.457023\pi$
$443$	2.12702e56	0.269210	0.134605	+	0.990899i	$0.457023\pi$
$444$	0	0
$445$	−1.40231e56	−0.161103
$446$	0	0
$447$	2.23962e57	2.33649
$448$	0	0
$449$	1.02618e57	0.972581	0.486291	−	0.873797i	$-0.338350\pi$
$449$	1.02618e57	0.972581	0.486291	+	0.873797i	$0.338350\pi$
$450$	0	0
$451$	−1.71989e57	−1.48151
$452$	0	0
$453$	−3.91826e57	−3.06889
$454$	0	0
$455$	5.70109e56	0.406175
$456$	0	0
$457$	−1.48807e57	−0.964775	−0.482388	−	0.875958i	$-0.660230\pi$
$457$	−1.48807e57	−0.964775	−0.482388	+	0.875958i	$0.660230\pi$
$458$	0	0
$459$	−4.75921e57	−2.80907
$460$	0	0
$461$	−2.28032e57	−1.22582	−0.612910	−	0.790153i	$-0.710001\pi$
$461$	−2.28032e57	−1.22582	−0.612910	+	0.790153i	$0.710001\pi$
$462$	0	0
$463$	−3.76534e56	−0.184422	−0.0922111	−	0.995739i	$-0.529393\pi$
$463$	−3.76534e56	−0.184422	−0.0922111	+	0.995739i	$0.529393\pi$
$464$	0	0
$465$	1.33194e57	0.594629
$466$	0	0
$467$	3.19228e57	1.29953	0.649767	−	0.760133i	$-0.274866\pi$
$467$	3.19228e57	1.29953	0.649767	+	0.760133i	$0.274866\pi$
$468$	0	0
$469$	−4.11504e57	−1.52812
$470$	0	0
$471$	−7.25182e57	−2.45752
$472$	0	0
$473$	3.30400e57	1.02217
$474$	0	0
$475$	6.03958e56	0.170645
$476$	0	0
$477$	5.05360e57	1.30453
$478$	0	0
$479$	−4.90345e57	−1.15688	−0.578439	−	0.815725i	$-0.696338\pi$
$479$	−4.90345e57	−1.15688	−0.578439	+	0.815725i	$0.696338\pi$
$480$	0	0
$481$	3.91677e57	0.844906
$482$	0	0
$483$	1.44845e58	2.85785
$484$	0	0
$485$	2.54119e57	0.458765
$486$	0	0
$487$	1.10089e58	1.81917	0.909585	−	0.415519i	$-0.136400\pi$
$487$	1.10089e58	1.81917	0.909585	+	0.415519i	$0.136400\pi$
$488$	0	0
$489$	−1.12327e58	−1.69959
$490$	0	0
$491$	−1.04229e58	−1.44457	−0.722283	−	0.691598i	$-0.756907\pi$
$491$	−1.04229e58	−1.44457	−0.722283	+	0.691598i	$0.756907\pi$
$492$	0	0
$493$	6.82634e57	0.866921
$494$	0	0
$495$	1.00268e58	1.16721
$496$	0	0
$497$	1.18460e58	1.26447
$498$	0	0
$499$	1.18570e58	1.16095	0.580473	−	0.814280i	$-0.302868\pi$
$499$	1.18570e58	1.16095	0.580473	+	0.814280i	$0.302868\pi$
$500$	0	0
$501$	5.61261e57	0.504259
$502$	0	0
$503$	−1.35864e58	−1.12045	−0.560225	−	0.828340i	$-0.689285\pi$
$503$	−1.35864e58	−1.12045	−0.560225	+	0.828340i	$0.689285\pi$
$504$	0	0
$505$	−4.11832e57	−0.311857
$506$	0	0
$507$	−5.24352e57	−0.364713
$508$	0	0
$509$	1.54608e58	0.988097	0.494048	−	0.869434i	$-0.335517\pi$
$509$	1.54608e58	0.988097	0.494048	+	0.869434i	$0.335517\pi$
$510$	0	0
$511$	2.63678e58	1.54890
$512$	0	0
$513$	−8.73825e57	−0.471955
$514$	0	0
$515$	−6.79019e57	−0.337308
$516$	0	0
$517$	−9.35565e57	−0.427590
$518$	0	0
$519$	7.38864e57	0.310789
$520$	0	0
$521$	−4.99449e58	−1.93410	−0.967052	−	0.254579i	$-0.918063\pi$
$521$	−4.99449e58	−1.93410	−0.967052	+	0.254579i	$0.918063\pi$
$522$	0	0
$523$	4.06006e57	0.144793	0.0723963	−	0.997376i	$-0.476935\pi$
$523$	4.06006e57	0.144793	0.0723963	+	0.997376i	$0.476935\pi$
$524$	0	0
$525$	5.81823e58	1.91146
$526$	0	0
$527$	−3.41896e58	−1.03506
$528$	0	0
$529$	2.35933e58	0.658402
$530$	0	0
$531$	−3.32166e58	−0.854718
$532$	0	0
$533$	4.11246e58	0.976037
$534$	0	0
$535$	−1.18980e58	−0.260535
$536$	0	0
$537$	−4.51243e58	−0.911935
$538$	0	0
$539$	3.55066e58	0.662446
$540$	0	0
$541$	−3.01019e58	−0.518624	−0.259312	−	0.965794i	$-0.583496\pi$
$541$	−3.01019e58	−0.518624	−0.259312	+	0.965794i	$0.583496\pi$
$542$	0	0
$543$	5.25992e58	0.837110
$544$	0	0
$545$	6.92520e57	0.101837
$546$	0	0
$547$	6.26196e58	0.851104	0.425552	−	0.904934i	$-0.360080\pi$
$547$	6.26196e58	0.851104	0.425552	+	0.904934i	$0.360080\pi$
$548$	0	0
$549$	−6.24781e58	−0.785095
$550$	0	0
$551$	1.25337e58	0.145653
$552$	0	0
$553$	7.90306e58	0.849582
$554$	0	0
$555$	−6.43513e58	−0.640115
$556$	0	0
$557$	7.30619e58	0.672673	0.336337	−	0.941742i	$-0.390812\pi$
$557$	7.30619e58	0.672673	0.336337	+	0.941742i	$0.390812\pi$
$558$	0	0
$559$	−7.90026e58	−0.673423
$560$	0	0
$561$	−3.68822e59	−2.91149
$562$	0	0
$563$	1.91890e59	1.40321	0.701603	−	0.712568i	$-0.252468\pi$
$563$	1.91890e59	1.40321	0.701603	+	0.712568i	$0.252468\pi$
$564$	0	0
$565$	2.75491e58	0.186666
$566$	0	0
$567$	−3.93194e59	−2.46928
$568$	0	0
$569$	3.18324e59	1.85334	0.926671	−	0.375873i	$-0.122657\pi$
$569$	3.18324e59	1.85334	0.926671	+	0.375873i	$0.122657\pi$
$570$	0	0
$571$	−2.66967e59	−1.44139	−0.720694	−	0.693254i	$-0.756176\pi$
$571$	−2.66967e59	−1.44139	−0.720694	+	0.693254i	$0.756176\pi$
$572$	0	0
$573$	−2.60834e59	−1.30629
$574$	0	0
$575$	2.38713e59	1.10922
$576$	0	0
$577$	−3.30424e58	−0.142492	−0.0712461	−	0.997459i	$-0.522698\pi$
$577$	−3.30424e58	−0.142492	−0.0712461	+	0.997459i	$0.522698\pi$
$578$	0	0
$579$	−7.11248e59	−2.84729
$580$	0	0
$581$	−3.64664e59	−1.35552
$582$	0	0
$583$	2.22058e59	0.766643
$584$	0	0
$585$	−2.39753e59	−0.768976
$586$	0	0
$587$	1.06658e59	0.317888	0.158944	−	0.987288i	$-0.449191\pi$
$587$	1.06658e59	0.317888	0.158944	+	0.987288i	$0.449191\pi$
$588$	0	0
$589$	−6.27746e58	−0.173902
$590$	0	0
$591$	−6.67112e58	−0.171818
$592$	0	0
$593$	4.73920e58	0.113509	0.0567544	−	0.998388i	$-0.481925\pi$
$593$	4.73920e58	0.113509	0.0567544	+	0.998388i	$0.481925\pi$
$594$	0	0
$595$	2.40435e59	0.535651
$596$	0	0
$597$	8.69010e58	0.180126
$598$	0	0
$599$	1.38891e59	0.267915	0.133957	−	0.990987i	$-0.457231\pi$
$599$	1.38891e59	0.267915	0.133957	+	0.990987i	$0.457231\pi$
$600$	0	0
$601$	9.03635e59	1.62253	0.811263	−	0.584681i	$-0.198780\pi$
$601$	9.03635e59	1.62253	0.811263	+	0.584681i	$0.198780\pi$
$602$	0	0
$603$	1.73053e60	2.89306
$604$	0	0
$605$	2.01404e59	0.313565
$606$	0	0
$607$	−2.30469e59	−0.334237	−0.167119	−	0.985937i	$-0.553446\pi$
$607$	−2.30469e59	−0.334237	−0.167119	+	0.985937i	$0.553446\pi$
$608$	0	0
$609$	1.20743e60	1.63151
$610$	0	0
$611$	2.23705e59	0.281702
$612$	0	0
$613$	5.31241e59	0.623579	0.311789	−	0.950151i	$-0.399072\pi$
$613$	5.31241e59	0.623579	0.311789	+	0.950151i	$0.399072\pi$
$614$	0	0
$615$	−6.75664e59	−0.739462
$616$	0	0
$617$	3.01800e59	0.308027	0.154013	−	0.988069i	$-0.450780\pi$
$617$	3.01800e59	0.308027	0.154013	+	0.988069i	$0.450780\pi$
$618$	0	0
$619$	−1.17709e59	−0.112063	−0.0560313	−	0.998429i	$-0.517845\pi$
$619$	−1.17709e59	−0.112063	−0.0560313	+	0.998429i	$0.517845\pi$
$620$	0	0
$621$	−3.45376e60	−3.06777
$622$	0	0
$623$	6.36687e59	0.527756
$624$	0	0
$625$	7.79659e59	0.603232
$626$	0	0
$627$	−6.77184e59	−0.489164
$628$	0	0
$629$	1.65183e60	1.11424
$630$	0	0
$631$	2.18281e60	1.37526	0.687631	−	0.726061i	$-0.258651\pi$
$631$	2.18281e60	1.37526	0.687631	+	0.726061i	$0.258651\pi$
$632$	0	0
$633$	3.29679e60	1.94049
$634$	0	0
$635$	9.63695e59	0.530035
$636$	0	0
$637$	−8.49006e59	−0.436428
$638$	0	0
$639$	−4.98169e60	−2.39391
$640$	0	0
$641$	−2.64938e60	−1.19041	−0.595205	−	0.803574i	$-0.702929\pi$
$641$	−2.64938e60	−1.19041	−0.595205	+	0.803574i	$0.702929\pi$
$642$	0	0
$643$	2.29486e60	0.964316	0.482158	−	0.876084i	$-0.339853\pi$
$643$	2.29486e60	0.964316	0.482158	+	0.876084i	$0.339853\pi$
$644$	0	0
$645$	1.29799e60	0.510196
$646$	0	0
$647$	1.33978e60	0.492708	0.246354	−	0.969180i	$-0.420767\pi$
$647$	1.33978e60	0.492708	0.246354	+	0.969180i	$0.420767\pi$
$648$	0	0
$649$	−1.45956e60	−0.502297
$650$	0	0
$651$	−6.04739e60	−1.94794
$652$	0	0
$653$	1.76959e60	0.533628	0.266814	−	0.963748i	$-0.414029\pi$
$653$	1.76959e60	0.533628	0.266814	+	0.963748i	$0.414029\pi$
$654$	0	0
$655$	1.79638e60	0.507237
$656$	0	0
$657$	−1.10887e61	−2.93240
$658$	0	0
$659$	2.81423e60	0.697146	0.348573	−	0.937282i	$-0.386666\pi$
$659$	2.81423e60	0.697146	0.348573	+	0.937282i	$0.386666\pi$
$660$	0	0
$661$	2.17575e59	0.0504983	0.0252491	−	0.999681i	$-0.491962\pi$
$661$	2.17575e59	0.0504983	0.0252491	+	0.999681i	$0.491962\pi$
$662$	0	0
$663$	8.81899e60	1.91813
$664$	0	0
$665$	4.41455e59	0.0899956
$666$	0	0
$667$	4.95389e60	0.946761
$668$	0	0
$669$	1.99588e61	3.57661
$670$	0	0
$671$	−2.74533e60	−0.461382
$672$	0	0
$673$	8.95136e60	1.41112	0.705562	−	0.708648i	$-0.250695\pi$
$673$	8.95136e60	1.41112	0.705562	+	0.708648i	$0.250695\pi$
$674$	0	0
$675$	−1.38733e61	−2.05187
$676$	0	0
$677$	7.11840e59	0.0987930	0.0493965	−	0.998779i	$-0.484270\pi$
$677$	7.11840e59	0.0987930	0.0493965	+	0.998779i	$0.484270\pi$
$678$	0	0
$679$	−1.15377e61	−1.50287
$680$	0	0
$681$	−2.71228e61	−3.31643
$682$	0	0
$683$	1.26607e61	1.45349	0.726744	−	0.686908i	$-0.241032\pi$
$683$	1.26607e61	1.45349	0.726744	+	0.686908i	$0.241032\pi$
$684$	0	0
$685$	1.66768e60	0.179790
$686$	0	0
$687$	−2.30786e61	−2.33690
$688$	0	0
$689$	−5.30969e60	−0.505075
$690$	0	0
$691$	−2.15812e60	−0.192885	−0.0964423	−	0.995339i	$-0.530746\pi$
$691$	−2.15812e60	−0.192885	−0.0964423	+	0.995339i	$0.530746\pi$
$692$	0	0
$693$	−4.55245e61	−3.82367
$694$	0	0
$695$	−5.78604e60	−0.456781
$696$	0	0
$697$	1.73436e61	1.28717
$698$	0	0
$699$	1.03028e61	0.718944
$700$	0	0
$701$	1.28524e61	0.843425	0.421712	−	0.906730i	$-0.361429\pi$
$701$	1.28524e61	0.843425	0.421712	+	0.906730i	$0.361429\pi$
$702$	0	0
$703$	3.03289e60	0.187205
$704$	0	0
$705$	−3.67540e60	−0.213422
$706$	0	0
$707$	1.86983e61	1.02161
$708$	0	0
$709$	5.66834e60	0.291449	0.145724	−	0.989325i	$-0.453449\pi$
$709$	5.66834e60	0.291449	0.145724	+	0.989325i	$0.453449\pi$
$710$	0	0
$711$	−3.32354e61	−1.60844
$712$	0	0
$713$	−2.48114e61	−1.13039
$714$	0	0
$715$	−1.05349e61	−0.451909
$716$	0	0
$717$	3.75925e60	0.151858
$718$	0	0
$719$	1.32395e61	0.503734	0.251867	−	0.967762i	$-0.418955\pi$
$719$	1.32395e61	0.503734	0.251867	+	0.967762i	$0.418955\pi$
$720$	0	0
$721$	3.08294e61	1.10499
$722$	0	0
$723$	1.87858e61	0.634390
$724$	0	0
$725$	1.98992e61	0.633239
$726$	0	0
$727$	3.95559e61	1.18637	0.593187	−	0.805064i	$-0.297869\pi$
$727$	3.95559e61	1.18637	0.593187	+	0.805064i	$0.297869\pi$
$728$	0	0
$729$	1.20680e61	0.341187
$730$	0	0
$731$	−3.33181e61	−0.888090
$732$	0	0
$733$	−2.46425e61	−0.619369	−0.309684	−	0.950839i	$-0.600223\pi$
$733$	−2.46425e61	−0.619369	−0.309684	+	0.950839i	$0.600223\pi$
$734$	0	0
$735$	1.39489e61	0.330645
$736$	0	0
$737$	7.60409e61	1.70018
$738$	0	0
$739$	2.08042e61	0.438829	0.219414	−	0.975632i	$-0.429585\pi$
$739$	2.08042e61	0.438829	0.219414	+	0.975632i	$0.429585\pi$
$740$	0	0
$741$	1.61923e61	0.322268
$742$	0	0
$743$	3.49360e60	0.0656165	0.0328083	−	0.999462i	$-0.489555\pi$
$743$	3.49360e60	0.0656165	0.0328083	+	0.999462i	$0.489555\pi$
$744$	0	0
$745$	2.69791e61	0.478264
$746$	0	0
$747$	1.53355e62	2.56629
$748$	0	0
$749$	5.40201e61	0.853487
$750$	0	0
$751$	7.03445e61	1.04947	0.524737	−	0.851264i	$-0.324164\pi$
$751$	7.03445e61	1.04947	0.524737	+	0.851264i	$0.324164\pi$
$752$	0	0
$753$	9.44223e60	0.133040
$754$	0	0
$755$	−4.72004e61	−0.628183
$756$	0	0
$757$	−1.01445e62	−1.27547	−0.637736	−	0.770255i	$-0.720129\pi$
$757$	−1.01445e62	−1.27547	−0.637736	+	0.770255i	$0.720129\pi$
$758$	0	0
$759$	−2.67655e62	−3.17963
$760$	0	0
$761$	6.71650e61	0.754002	0.377001	−	0.926213i	$-0.376955\pi$
$761$	6.71650e61	0.754002	0.377001	+	0.926213i	$0.376955\pi$
$762$	0	0
$763$	−3.14424e61	−0.333609
$764$	0	0
$765$	−1.01112e62	−1.01410
$766$	0	0
$767$	3.48999e61	0.330920
$768$	0	0
$769$	−1.24682e62	−1.11786	−0.558931	−	0.829214i	$-0.688788\pi$
$769$	−1.24682e62	−1.11786	−0.558931	+	0.829214i	$0.688788\pi$
$770$	0	0
$771$	3.01565e62	2.55689
$772$	0	0
$773$	−1.19282e62	−0.956570	−0.478285	−	0.878205i	$-0.658741\pi$
$773$	−1.19282e62	−0.956570	−0.478285	+	0.878205i	$0.658741\pi$
$774$	0	0
$775$	−9.96646e61	−0.756056
$776$	0	0
$777$	2.92173e62	2.09695
$778$	0	0
$779$	3.18442e61	0.216259
$780$	0	0
$781$	−2.18899e62	−1.40685
$782$	0	0
$783$	−2.87907e62	−1.75135
$784$	0	0
$785$	−8.73574e61	−0.503039
$786$	0	0
$787$	1.29833e62	0.707826	0.353913	−	0.935278i	$-0.384851\pi$
$787$	1.29833e62	0.707826	0.353913	+	0.935278i	$0.384851\pi$
$788$	0	0
$789$	4.32862e62	2.23457
$790$	0	0
$791$	−1.25081e62	−0.611499
$792$	0	0
$793$	6.56442e61	0.303964
$794$	0	0
$795$	8.72366e61	0.382653
$796$	0	0
$797$	1.44847e62	0.601942	0.300971	−	0.953633i	$-0.402689\pi$
$797$	1.44847e62	0.601942	0.300971	+	0.953633i	$0.402689\pi$
$798$	0	0
$799$	9.43441e61	0.371500
$800$	0	0
$801$	−2.67752e62	−0.999155
$802$	0	0
$803$	−4.87243e62	−1.72330
$804$	0	0
$805$	1.74484e62	0.584983
$806$	0	0
$807$	3.63513e62	1.15542
$808$	0	0
$809$	−9.08798e61	−0.273889	−0.136944	−	0.990579i	$-0.543728\pi$
$809$	−9.08798e61	−0.273889	−0.136944	+	0.990579i	$0.543728\pi$
$810$	0	0
$811$	−1.53395e60	−0.00438393	−0.00219197	−	0.999998i	$-0.500698\pi$
$811$	−1.53395e60	−0.00438393	−0.00219197	+	0.999998i	$0.500698\pi$
$812$	0	0
$813$	−5.23511e62	−1.41899
$814$	0	0
$815$	−1.35313e62	−0.347896
$816$	0	0
$817$	−6.11744e61	−0.149209
$818$	0	0
$819$	1.08855e63	2.51909
$820$	0	0
$821$	−4.54457e62	−0.997964	−0.498982	−	0.866612i	$-0.666293\pi$
$821$	−4.54457e62	−0.997964	−0.498982	+	0.866612i	$0.666293\pi$
$822$	0	0
$823$	−9.00456e62	−1.87658	−0.938288	−	0.345856i	$-0.887588\pi$
$823$	−9.00456e62	−1.87658	−0.938288	+	0.345856i	$0.887588\pi$
$824$	0	0
$825$	−1.07514e63	−2.12669
$826$	0	0
$827$	−4.33820e62	−0.814593	−0.407296	−	0.913296i	$-0.633528\pi$
$827$	−4.33820e62	−0.814593	−0.407296	+	0.913296i	$0.633528\pi$
$828$	0	0
$829$	3.96801e62	0.707375	0.353688	−	0.935364i	$-0.384928\pi$
$829$	3.96801e62	0.707375	0.353688	+	0.935364i	$0.384928\pi$
$830$	0	0
$831$	−1.60440e63	−2.71576
$832$	0	0
$833$	−3.58055e62	−0.575548
$834$	0	0
$835$	6.76110e61	0.103219
$836$	0	0
$837$	1.44198e63	2.09103
$838$	0	0
$839$	−5.28190e62	−0.727626	−0.363813	−	0.931472i	$-0.618525\pi$
$839$	−5.28190e62	−0.727626	−0.363813	+	0.931472i	$0.618525\pi$
$840$	0	0
$841$	−3.51079e62	−0.459505
$842$	0	0
$843$	1.47590e63	1.83554
$844$	0	0
$845$	−6.31648e61	−0.0746545
$846$	0	0
$847$	−9.14432e62	−1.02721
$848$	0	0
$849$	−5.16781e62	−0.551812
$850$	0	0
$851$	1.19874e63	1.21685
$852$	0	0
$853$	2.92763e62	0.282561	0.141281	−	0.989970i	$-0.454878\pi$
$853$	2.92763e62	0.282561	0.141281	+	0.989970i	$0.454878\pi$
$854$	0	0
$855$	−1.85649e62	−0.170381
$856$	0	0
$857$	−2.84286e62	−0.248123	−0.124062	−	0.992275i	$-0.539592\pi$
$857$	−2.84286e62	−0.248123	−0.124062	+	0.992275i	$0.539592\pi$
$858$	0	0
$859$	8.07624e62	0.670434	0.335217	−	0.942141i	$-0.391190\pi$
$859$	8.07624e62	0.670434	0.335217	+	0.942141i	$0.391190\pi$
$860$	0	0
$861$	3.06771e63	2.42240
$862$	0	0
$863$	2.13069e63	1.60062	0.800310	−	0.599586i	$-0.204668\pi$
$863$	2.13069e63	1.60062	0.800310	+	0.599586i	$0.204668\pi$
$864$	0	0
$865$	8.90055e61	0.0636166
$866$	0	0
$867$	1.04448e63	0.710375
$868$	0	0
$869$	−1.46039e63	−0.945242
$870$	0	0
$871$	−1.81823e63	−1.12010
$872$	0	0
$873$	4.85206e63	2.84525
$874$	0	0
$875$	1.51459e63	0.845518
$876$	0	0
$877$	−9.77134e62	−0.519353	−0.259677	−	0.965696i	$-0.583616\pi$
$877$	−9.77134e62	−0.519353	−0.259677	+	0.965696i	$0.583616\pi$
$878$	0	0
$879$	1.49005e63	0.754120
$880$	0	0
$881$	4.67468e62	0.225304	0.112652	−	0.993634i	$-0.464065\pi$
$881$	4.67468e62	0.225304	0.112652	+	0.993634i	$0.464065\pi$
$882$	0	0
$883$	−2.00447e63	−0.920120	−0.460060	−	0.887888i	$-0.652172\pi$
$883$	−2.00447e63	−0.920120	−0.460060	+	0.887888i	$0.652172\pi$
$884$	0	0
$885$	−5.73394e62	−0.250711
$886$	0	0
$887$	−2.72376e62	−0.113452	−0.0567259	−	0.998390i	$-0.518066\pi$
$887$	−2.72376e62	−0.113452	−0.0567259	+	0.998390i	$0.518066\pi$
$888$	0	0
$889$	−4.37545e63	−1.73634
$890$	0	0
$891$	7.26573e63	2.74732
$892$	0	0
$893$	1.73222e62	0.0624163
$894$	0	0
$895$	−5.43580e62	−0.186667
$896$	0	0
$897$	6.39995e63	2.09478
$898$	0	0
$899$	−2.06829e63	−0.645324
$900$	0	0
$901$	−2.23928e63	−0.666078
$902$	0	0
$903$	−5.89324e63	−1.67135
$904$	0	0
$905$	6.33624e62	0.171351
$906$	0	0
$907$	6.10336e62	0.157403	0.0787013	−	0.996898i	$-0.474923\pi$
$907$	6.10336e62	0.157403	0.0787013	+	0.996898i	$0.474923\pi$
$908$	0	0
$909$	−7.86337e63	−1.93413
$910$	0	0
$911$	−8.12058e63	−1.90521	−0.952603	−	0.304218i	$-0.901605\pi$
$911$	−8.12058e63	−1.90521	−0.952603	+	0.304218i	$0.901605\pi$
$912$	0	0
$913$	6.73853e63	1.50815
$914$	0	0
$915$	−1.07851e63	−0.230288
$916$	0	0
$917$	−8.15609e63	−1.66166
$918$	0	0
$919$	6.01090e63	1.16857	0.584287	−	0.811547i	$-0.301374\pi$
$919$	6.01090e63	1.16857	0.584287	+	0.811547i	$0.301374\pi$
$920$	0	0
$921$	−1.51423e63	−0.280938
$922$	0	0
$923$	5.23414e63	0.926849
$924$	0	0
$925$	4.81519e63	0.813890
$926$	0	0
$927$	−1.29649e64	−2.09197
$928$	0	0
$929$	7.66144e63	1.18025	0.590123	−	0.807313i	$-0.299079\pi$
$929$	7.66144e63	1.18025	0.590123	+	0.807313i	$0.299079\pi$
$930$	0	0
$931$	−6.57414e62	−0.0966987
$932$	0	0
$933$	3.25580e63	0.457301
$934$	0	0
$935$	−4.44293e63	−0.595964
$936$	0	0
$937$	−1.34664e64	−1.72524	−0.862621	−	0.505850i	$-0.831179\pi$
$937$	−1.34664e64	−1.72524	−0.862621	+	0.505850i	$0.831179\pi$
$938$	0	0
$939$	2.19146e64	2.68178
$940$	0	0
$941$	2.22801e63	0.260459	0.130230	−	0.991484i	$-0.458429\pi$
$941$	2.22801e63	0.260459	0.130230	+	0.991484i	$0.458429\pi$
$942$	0	0
$943$	1.25863e64	1.40571
$944$	0	0
$945$	−1.01405e64	−1.08212
$946$	0	0
$947$	1.00072e64	1.02044	0.510220	−	0.860044i	$-0.329564\pi$
$947$	1.00072e64	1.02044	0.510220	+	0.860044i	$0.329564\pi$
$948$	0	0
$949$	1.16506e64	1.13534
$950$	0	0
$951$	−2.80985e64	−2.61699
$952$	0	0
$953$	−1.92139e63	−0.171048	−0.0855240	−	0.996336i	$-0.527256\pi$
$953$	−1.92139e63	−0.171048	−0.0855240	+	0.996336i	$0.527256\pi$
$954$	0	0
$955$	−3.14207e63	−0.267389
$956$	0	0
$957$	−2.23118e64	−1.81521
$958$	0	0
$959$	−7.57175e63	−0.588974
$960$	0	0
$961$	−3.08576e63	−0.229514
$962$	0	0
$963$	−2.27176e64	−1.61583
$964$	0	0
$965$	−8.56789e63	−0.582823
$966$	0	0
$967$	−1.51693e64	−0.986953	−0.493476	−	0.869759i	$-0.664274\pi$
$967$	−1.51693e64	−0.986953	−0.493476	+	0.869759i	$0.664274\pi$
$968$	0	0
$969$	6.82885e63	0.424997
$970$	0	0
$971$	−1.62846e64	−0.969536	−0.484768	−	0.874643i	$-0.661096\pi$
$971$	−1.62846e64	−0.969536	−0.484768	+	0.874643i	$0.661096\pi$
$972$	0	0
$973$	2.62702e64	1.49637
$974$	0	0
$975$	2.57078e64	1.40109
$976$	0	0
$977$	−3.85777e63	−0.201188	−0.100594	−	0.994928i	$-0.532074\pi$
$977$	−3.85777e63	−0.201188	−0.100594	+	0.994928i	$0.532074\pi$
$978$	0	0
$979$	−1.17652e64	−0.587179
$980$	0	0
$981$	1.32227e64	0.631593
$982$	0	0
$983$	3.53894e64	1.61798	0.808988	−	0.587825i	$-0.200016\pi$
$983$	3.53894e64	1.61798	0.808988	+	0.587825i	$0.200016\pi$
$984$	0	0
$985$	−8.03621e62	−0.0351700
$986$	0	0
$987$	1.66874e64	0.699148
$988$	0	0
$989$	−2.41790e64	−0.969880
$990$	0	0
$991$	1.37174e64	0.526853	0.263426	−	0.964680i	$-0.415147\pi$
$991$	1.37174e64	0.526853	0.263426	+	0.964680i	$0.415147\pi$
$992$	0	0
$993$	−5.65894e64	−2.08126
$994$	0	0
$995$	1.04683e63	0.0368707
$996$	0	0
$997$	6.70387e63	0.226141	0.113071	−	0.993587i	$-0.463931\pi$
$997$	6.70387e63	0.226141	0.113071	+	0.993587i	$0.463931\pi$
$998$	0	0
$999$	−6.96675e64	−2.25098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.44.a.b.1.2		2
4.3	odd	2		2.44.a.b.1.1	✓	2
12.11	even	2		18.44.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.44.a.b.1.1	✓	2	4.3	odd	2
16.44.a.b.1.2		2	1.1	even	1	trivial
18.44.a.c.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 44 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+3.29600e10 q^{3} +3.97045e14 q^{5} -1.80269e18 q^{7} +7.58103e20 q^{9} +3.33115e22 q^{11} -7.96520e23 q^{13} +1.30866e25 q^{15} -3.35920e26 q^{17} -6.16773e26 q^{19} -5.94168e28 q^{21} -2.43777e29 q^{23} -9.79224e29 q^{25} +1.41677e31 q^{27} -2.03214e31 q^{29} +1.01779e32 q^{31} +1.09795e33 q^{33} -7.15750e32 q^{35} -4.91735e33 q^{37} -2.62533e34 q^{39} -5.16303e34 q^{41} +9.91848e34 q^{43} +3.01001e35 q^{45} -2.80853e35 q^{47} +1.06589e36 q^{49} -1.10719e37 q^{51} +6.66611e36 q^{53} +1.32262e37 q^{55} -2.03288e37 q^{57} -4.38154e37 q^{59} -8.24137e37 q^{61} -1.36663e39 q^{63} -3.16254e38 q^{65} +2.28272e39 q^{67} -8.03489e39 q^{69} -6.57126e39 q^{71} -1.46269e40 q^{73} -3.22752e40 q^{75} -6.00505e40 q^{77} -4.38403e40 q^{79} +2.18114e41 q^{81} +2.02288e41 q^{83} -1.33375e41 q^{85} -6.69791e41 q^{87} -3.53186e41 q^{89} +1.43588e42 q^{91} +3.35464e42 q^{93} -2.44886e41 q^{95} +6.40027e42 q^{97} +2.52536e43 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22341634056 q^{3} - 47320658398020 q^{5} + 22\!\cdots\!28 q^{7} + 54\!\cdots\!14 q^{9} + 41\!\cdots\!16 q^{11} - 15\!\cdots\!96 q^{13} + 17\!\cdots\!40 q^{15} - 67\!\cdots\!48 q^{17} - 12\!\cdots\!40 q^{19}+ \cdots + 23\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q + 22341634056 * q^3 - 47320658398020 * q^5 + 222460131347038928 * q^7 + 542594646100314147114 * q^9 + 41887858358166241718616 * q^11 - 1511213342321158511669396 * q^13 + 17804997360134453991129840 * q^15 - 67349878355730307414152348 * q^17 - 1230904208507994915735409240 * q^19 - 80920530905061445582689383616 * q^21 - 38167008283484201492558062224 * q^23 - 1918631833123351317328900860850 * q^25 + 19941575760041650132553853212880 * q^27 - 72780948725084997011410245083700 * q^29 - 40207733905312243999256937298624 * q^31 + 1006881310052050425793011088926048 * q^33 - 1615658656617379304576130831538080 * q^35 + 3383801276601910515940174638928252 * q^37 - 18664419028213485589802473566212688 * q^39 - 58505417265395086228405561355069196 * q^41 + 285092737981269186332521030698108056 * q^43 + 396764787925879165058503325488849260 * q^45 - 1834986318326201344851046619753827872 * q^47 + 2983331119134682985187309352444961106 * q^49 - 13923666849435817419951839264623903344 * q^51 + 13488383575332508144313645312419699164 * q^53 + 9415153536190550355180017489429532240 * q^55 - 13807748645696734847941276681469831520 * q^57 - 6535875469255007418951566075676717000 * q^59 + 84915949989376412900726656980650069644 * q^61 - 1803064383904392926498433003649964350704 * q^63 + 1330967805107975013132566371731317160 * q^65 + 3889814457887475263640354897004803753608 * q^67 - 10218134177167059404862981043947605969472 * q^69 + 4500205056814555193905197090502997775696 * q^71 - 10215757578580250274581767012298787143756 * q^73 - 22300243542723736486434261503150942041800 * q^75 - 42682181092256892291285667845090826099776 * q^77 - 106263833346898669088358572211107427639520 * q^79 + 227547444210851249821957128354947134508562 * q^81 + 99535113533406416106304387849461181193256 * q^83 - 252718204268333675320943366175053150677320 * q^85 - 112757591652872720426564768050770607266000 * q^87 + 175134609645798670638848613045046472300820 * q^89 - 11482488188506183083757950956690750656544 * q^91 + 4862300735557253257085331576689556832720128 * q^93 + 28012559440364213825890153198462567470000 * q^95 - 3659395264822684391406294706085506900627388 * q^97 + 23405304422197580402007697540466944929273912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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