LMFDB - Embedded newform 16.22.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,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, 22, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 22 $
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:	$44.7163750859$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{358549}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 89637 $ x^2 - x - 89637
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$299.895$ 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)

$f(q)$	$=$	$q-62251.6 q^{3} -1.24528e6 q^{5} +2.63007e8 q^{7} -6.58509e9 q^{9} +O(q^{10})$	$q-62251.6 q^{3} -1.24528e6 q^{5} +2.63007e8 q^{7} -6.58509e9 q^{9} +3.37055e10 q^{11} +2.50845e11 q^{13} +7.75208e10 q^{15} +2.34664e11 q^{17} +3.24706e13 q^{19} -1.63726e13 q^{21} +9.63098e13 q^{23} -4.75286e14 q^{25} +1.06111e15 q^{27} -3.86879e15 q^{29} +2.26841e15 q^{31} -2.09822e15 q^{33} -3.27518e14 q^{35} +4.75366e15 q^{37} -1.56155e16 q^{39} +3.56442e16 q^{41} -2.20776e17 q^{43} +8.20030e15 q^{45} -4.79937e17 q^{47} -4.89373e17 q^{49} -1.46082e16 q^{51} -1.86349e18 q^{53} -4.19729e16 q^{55} -2.02135e18 q^{57} -1.23147e18 q^{59} -6.27167e18 q^{61} -1.73193e18 q^{63} -3.12373e17 q^{65} -1.72809e19 q^{67} -5.99544e18 q^{69} -3.13094e19 q^{71} +5.49389e19 q^{73} +2.95873e19 q^{75} +8.86480e18 q^{77} +8.34613e19 q^{79} +2.82681e18 q^{81} +1.22864e20 q^{83} -2.92223e17 q^{85} +2.40839e20 q^{87} +1.21225e20 q^{89} +6.59742e19 q^{91} -1.41212e20 q^{93} -4.04350e19 q^{95} +3.73762e20 q^{97} -2.21954e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) $ 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9	$ 2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9} - 53806403320 q^{11} - 490366676932 q^{13} + 639834721680 q^{15} - 6593864672092 q^{17} - 19302397925320 q^{19} - 135055584824256 q^{21} + 409737865776272 q^{23} - 940878149007650 q^{25} + 22\!\cdots\!56 q^{27}+ \cdots - 17\!\cdots\!04 q^{99}+O(q^{100}) $ 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9 - 53806403320 * q^11 - 490366676932 * q^13 + 639834721680 * q^15 - 6593864672092 * q^17 - 19302397925320 * q^19 - 135055584824256 * q^21 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2267939450073456 * q^27 - 2404787522145060 * q^29 + 8689907170559168 * q^31 - 16772541164782752 * q^33 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 139904623071296688 * q^39 + 68178038573558676 * q^41 - 264529652266004024 * q^43 + 67413140092548540 * q^45 - 426494411558622432 * q^47 - 546967577640131534 * q^49 - 1159640456555021520 * q^51 - 3055980275589518132 * q^53 - 335437371694825040 * q^55 - 10702822880368260192 * q^57 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 - 14229493501717343376 * q^63 - 2797969869756700760 * q^65 - 16674123174011538088 * q^67 + 46561318352256083136 * q^69 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 48484754640473875800 * q^75 + 70803926825553273024 * q^77 + 85280702218715897824 * q^79 + 20489906394726573618 * q^81 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 486328037598631978704 * q^87 - 59742932430695979660 * q^89 + 590588534777956016544 * q^91 + 935567489996727394560 * q^93 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97 - 1767190682286719892504 * 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
$3$	−62251.6	−0.608664	−0.304332	−	0.952566i	$-0.598433\pi$
$3$	−62251.6	−0.608664	−0.304332	+	0.952566i	$0.598433\pi$
$4$	0	0
$5$	−1.24528e6	−0.0570273	−0.0285136	−	0.999593i	$-0.509077\pi$
$5$	−1.24528e6	−0.0570273	−0.0285136	+	0.999593i	$0.509077\pi$
$6$	0	0
$7$	2.63007e8	0.351916	0.175958	−	0.984398i	$-0.443698\pi$
$7$	2.63007e8	0.351916	0.175958	+	0.984398i	$0.443698\pi$
$8$	0	0
$9$	−6.58509e9	−0.629529
$10$	0	0
$11$	3.37055e10	0.391812	0.195906	−	0.980623i	$-0.437235\pi$
$11$	3.37055e10	0.391812	0.195906	+	0.980623i	$0.437235\pi$
$12$	0	0
$13$	2.50845e11	0.504662	0.252331	−	0.967641i	$-0.418803\pi$
$13$	2.50845e11	0.504662	0.252331	+	0.967641i	$0.418803\pi$
$14$	0	0
$15$	7.75208e10	0.0347104
$16$	0	0
$17$	2.34664e11	0.0282314	0.0141157	−	0.999900i	$-0.495507\pi$
$17$	2.34664e11	0.0282314	0.0141157	+	0.999900i	$0.495507\pi$
$18$	0	0
$19$	3.24706e13	1.21500	0.607501	−	0.794319i	$-0.292172\pi$
$19$	3.24706e13	1.21500	0.607501	+	0.794319i	$0.292172\pi$
$20$	0	0
$21$	−1.63726e13	−0.214198
$22$	0	0
$23$	9.63098e13	0.484761	0.242381	−	0.970181i	$-0.422072\pi$
$23$	9.63098e13	0.484761	0.242381	+	0.970181i	$0.422072\pi$
$24$	0	0
$25$	−4.75286e14	−0.996748
$26$	0	0
$27$	1.06111e15	0.991835
$28$	0	0
$29$	−3.86879e15	−1.70764	−0.853820	−	0.520568i	$-0.825720\pi$
$29$	−3.86879e15	−1.70764	−0.853820	+	0.520568i	$0.825720\pi$
$30$	0	0
$31$	2.26841e15	0.497077	0.248539	−	0.968622i	$-0.420050\pi$
$31$	2.26841e15	0.497077	0.248539	+	0.968622i	$0.420050\pi$
$32$	0	0
$33$	−2.09822e15	−0.238482
$34$	0	0
$35$	−3.27518e14	−0.0200688
$36$	0	0
$37$	4.75366e15	0.162521	0.0812606	−	0.996693i	$-0.474105\pi$
$37$	4.75366e15	0.162521	0.0812606	+	0.996693i	$0.474105\pi$
$38$	0	0
$39$	−1.56155e16	−0.307170
$40$	0	0
$41$	3.56442e16	0.414723	0.207361	−	0.978264i	$-0.433512\pi$
$41$	3.56442e16	0.414723	0.207361	+	0.978264i	$0.433512\pi$
$42$	0	0
$43$	−2.20776e17	−1.55788	−0.778939	−	0.627100i	$-0.784242\pi$
$43$	−2.20776e17	−1.55788	−0.778939	+	0.627100i	$0.784242\pi$
$44$	0	0
$45$	8.20030e15	0.0359003
$46$	0	0
$47$	−4.79937e17	−1.33093	−0.665467	−	0.746427i	$-0.731768\pi$
$47$	−4.79937e17	−1.33093	−0.665467	+	0.746427i	$0.731768\pi$
$48$	0	0
$49$	−4.89373e17	−0.876155
$50$	0	0
$51$	−1.46082e16	−0.0171834
$52$	0	0
$53$	−1.86349e18	−1.46362	−0.731812	−	0.681506i	$-0.761325\pi$
$53$	−1.86349e18	−1.46362	−0.731812	+	0.681506i	$0.761325\pi$
$54$	0	0
$55$	−4.19729e16	−0.0223440
$56$	0	0
$57$	−2.02135e18	−0.739528
$58$	0	0
$59$	−1.23147e18	−0.313674	−0.156837	−	0.987625i	$-0.550130\pi$
$59$	−1.23147e18	−0.313674	−0.156837	+	0.987625i	$0.550130\pi$
$60$	0	0
$61$	−6.27167e18	−1.12569	−0.562846	−	0.826562i	$-0.690294\pi$
$61$	−6.27167e18	−1.12569	−0.562846	+	0.826562i	$0.690294\pi$
$62$	0	0
$63$	−1.73193e18	−0.221541
$64$	0	0
$65$	−3.12373e17	−0.0287795
$66$	0	0
$67$	−1.72809e19	−1.15820	−0.579098	−	0.815258i	$-0.696595\pi$
$67$	−1.72809e19	−1.15820	−0.579098	+	0.815258i	$0.696595\pi$
$68$	0	0
$69$	−5.99544e18	−0.295057
$70$	0	0
$71$	−3.13094e19	−1.14146	−0.570732	−	0.821137i	$-0.693340\pi$
$71$	−3.13094e19	−1.14146	−0.570732	+	0.821137i	$0.693340\pi$
$72$	0	0
$73$	5.49389e19	1.49620	0.748100	−	0.663586i	$-0.230966\pi$
$73$	5.49389e19	1.49620	0.748100	+	0.663586i	$0.230966\pi$
$74$	0	0
$75$	2.95873e19	0.606684
$76$	0	0
$77$	8.86480e18	0.137885
$78$	0	0
$79$	8.34613e19	0.991747	0.495873	−	0.868395i	$-0.334848\pi$
$79$	8.34613e19	0.991747	0.495873	+	0.868395i	$0.334848\pi$
$80$	0	0
$81$	2.82681e18	0.0258347
$82$	0	0
$83$	1.22864e20	0.869171	0.434585	−	0.900631i	$-0.356895\pi$
$83$	1.22864e20	0.869171	0.434585	+	0.900631i	$0.356895\pi$
$84$	0	0
$85$	−2.92223e17	−0.00160996
$86$	0	0
$87$	2.40839e20	1.03938
$88$	0	0
$89$	1.21225e20	0.412096	0.206048	−	0.978542i	$-0.433940\pi$
$89$	1.21225e20	0.412096	0.206048	+	0.978542i	$0.433940\pi$
$90$	0	0
$91$	6.59742e19	0.177598
$92$	0	0
$93$	−1.41212e20	−0.302553
$94$	0	0
$95$	−4.04350e19	−0.0692883
$96$	0	0
$97$	3.73762e20	0.514627	0.257313	−	0.966328i	$-0.417163\pi$
$97$	3.73762e20	0.514627	0.257313	+	0.966328i	$0.417163\pi$
$98$	0	0
$99$	−2.21954e20	−0.246657
$100$	0	0
$101$	1.12796e21	1.01606	0.508031	−	0.861339i	$-0.330374\pi$
$101$	1.12796e21	1.01606	0.508031	+	0.861339i	$0.330374\pi$
$102$	0	0
$103$	−1.15466e21	−0.846573	−0.423287	−	0.905996i	$-0.639124\pi$
$103$	−1.15466e21	−0.846573	−0.423287	+	0.905996i	$0.639124\pi$
$104$	0	0
$105$	2.03885e19	0.0122151
$106$	0	0
$107$	−1.23814e21	−0.608469	−0.304235	−	0.952597i	$-0.598401\pi$
$107$	−1.23814e21	−0.608469	−0.304235	+	0.952597i	$0.598401\pi$
$108$	0	0
$109$	−1.33848e21	−0.541544	−0.270772	−	0.962643i	$-0.587279\pi$
$109$	−1.33848e21	−0.541544	−0.270772	+	0.962643i	$0.587279\pi$
$110$	0	0
$111$	−2.95923e20	−0.0989208
$112$	0	0
$113$	−6.49910e21	−1.80107	−0.900533	−	0.434787i	$-0.856824\pi$
$113$	−6.49910e21	−1.80107	−0.900533	+	0.434787i	$0.856824\pi$
$114$	0	0
$115$	−1.19933e20	−0.0276446
$116$	0	0
$117$	−1.65184e21	−0.317699
$118$	0	0
$119$	6.17183e19	0.00993507
$120$	0	0
$121$	−6.26419e21	−0.846483
$122$	0	0
$123$	−2.21891e21	−0.252427
$124$	0	0
$125$	1.18566e21	0.113869
$126$	0	0
$127$	5.02960e21	0.408878	0.204439	−	0.978879i	$-0.434463\pi$
$127$	5.02960e21	0.408878	0.204439	+	0.978879i	$0.434463\pi$
$128$	0	0
$129$	1.37437e22	0.948223
$130$	0	0
$131$	2.11255e21	0.124011	0.0620053	−	0.998076i	$-0.480250\pi$
$131$	2.11255e21	0.124011	0.0620053	+	0.998076i	$0.480250\pi$
$132$	0	0
$133$	8.54000e21	0.427578
$134$	0	0
$135$	−1.32138e21	−0.0565616
$136$	0	0
$137$	2.68746e22	0.985770	0.492885	−	0.870095i	$-0.335942\pi$
$137$	2.68746e22	0.985770	0.492885	+	0.870095i	$0.335942\pi$
$138$	0	0
$139$	−9.03627e21	−0.284665	−0.142332	−	0.989819i	$-0.545460\pi$
$139$	−9.03627e21	−0.284665	−0.142332	+	0.989819i	$0.545460\pi$
$140$	0	0
$141$	2.98768e22	0.810092
$142$	0	0
$143$	8.45488e21	0.197733
$144$	0	0
$145$	4.81774e21	0.0973821
$146$	0	0
$147$	3.04643e22	0.533284
$148$	0	0
$149$	−5.86168e22	−0.890362	−0.445181	−	0.895441i	$-0.646861\pi$
$149$	−5.86168e22	−0.890362	−0.445181	+	0.895441i	$0.646861\pi$
$150$	0	0
$151$	8.08836e22	1.06808	0.534038	−	0.845460i	$-0.320674\pi$
$151$	8.08836e22	1.06808	0.534038	+	0.845460i	$0.320674\pi$
$152$	0	0
$153$	−1.54528e21	−0.0177725
$154$	0	0
$155$	−2.82481e21	−0.0283470
$156$	0	0
$157$	−5.42012e22	−0.475404	−0.237702	−	0.971338i	$-0.576394\pi$
$157$	−5.42012e22	−0.475404	−0.237702	+	0.971338i	$0.576394\pi$
$158$	0	0
$159$	1.16005e23	0.890855
$160$	0	0
$161$	2.53302e22	0.170595
$162$	0	0
$163$	−1.61447e23	−0.955126	−0.477563	−	0.878597i	$-0.658480\pi$
$163$	−1.61447e23	−0.955126	−0.477563	+	0.878597i	$0.658480\pi$
$164$	0	0
$165$	2.61288e21	0.0136000
$166$	0	0
$167$	−2.19835e23	−1.00826	−0.504132	−	0.863627i	$-0.668188\pi$
$167$	−2.19835e23	−1.00826	−0.504132	+	0.863627i	$0.668188\pi$
$168$	0	0
$169$	−1.84141e23	−0.745316
$170$	0	0
$171$	−2.13822e23	−0.764879
$172$	0	0
$173$	−3.84695e22	−0.121796	−0.0608979	−	0.998144i	$-0.519396\pi$
$173$	−3.84695e22	−0.121796	−0.0608979	+	0.998144i	$0.519396\pi$
$174$	0	0
$175$	−1.25004e23	−0.350771
$176$	0	0
$177$	7.66611e22	0.190922
$178$	0	0
$179$	−6.03667e23	−1.33610	−0.668052	−	0.744115i	$-0.732872\pi$
$179$	−6.03667e23	−1.33610	−0.668052	+	0.744115i	$0.732872\pi$
$180$	0	0
$181$	−5.64803e21	−0.0111243	−0.00556214	−	0.999985i	$-0.501770\pi$
$181$	−5.64803e21	−0.0111243	−0.00556214	+	0.999985i	$0.501770\pi$
$182$	0	0
$183$	3.90421e23	0.685168
$184$	0	0
$185$	−5.91965e21	−0.00926815
$186$	0	0
$187$	7.90947e21	0.0110614
$188$	0	0
$189$	2.79079e23	0.349042
$190$	0	0
$191$	1.34983e24	1.51157	0.755783	−	0.654822i	$-0.227256\pi$
$191$	1.34983e24	1.51157	0.755783	+	0.654822i	$0.227256\pi$
$192$	0	0
$193$	−1.12919e24	−1.13348	−0.566741	−	0.823896i	$-0.691796\pi$
$193$	−1.12919e24	−1.13348	−0.566741	+	0.823896i	$0.691796\pi$
$194$	0	0
$195$	1.94457e22	0.0175170
$196$	0	0
$197$	5.95420e23	0.481868	0.240934	−	0.970541i	$-0.422546\pi$
$197$	5.95420e23	0.481868	0.240934	+	0.970541i	$0.422546\pi$
$198$	0	0
$199$	2.68672e24	1.95554	0.977768	−	0.209690i	$-0.0672454\pi$
$199$	2.68672e24	1.95554	0.977768	+	0.209690i	$0.0672454\pi$
$200$	0	0
$201$	1.07577e24	0.704952
$202$	0	0
$203$	−1.01752e24	−0.600945
$204$	0	0
$205$	−4.43870e22	−0.0236505
$206$	0	0
$207$	−6.34208e23	−0.305171
$208$	0	0
$209$	1.09444e24	0.476053
$210$	0	0
$211$	−1.31634e24	−0.518085	−0.259043	−	0.965866i	$-0.583407\pi$
$211$	−1.31634e24	−0.518085	−0.259043	+	0.965866i	$0.583407\pi$
$212$	0	0
$213$	1.94906e24	0.694767
$214$	0	0
$215$	2.74928e23	0.0888415
$216$	0	0
$217$	5.96609e23	0.174929
$218$	0	0
$219$	−3.42003e24	−0.910683
$220$	0	0
$221$	5.88643e22	0.0142473
$222$	0	0
$223$	7.81247e24	1.72023	0.860116	−	0.510098i	$-0.170391\pi$
$223$	7.81247e24	1.72023	0.860116	+	0.510098i	$0.170391\pi$
$224$	0	0
$225$	3.12980e24	0.627481
$226$	0	0
$227$	−9.76737e24	−1.78446	−0.892228	−	0.451584i	$-0.850859\pi$
$227$	−9.76737e24	−1.78446	−0.892228	+	0.451584i	$0.850859\pi$
$228$	0	0
$229$	−9.30771e24	−1.55085	−0.775426	−	0.631439i	$-0.782465\pi$
$229$	−9.30771e24	−1.55085	−0.775426	+	0.631439i	$0.782465\pi$
$230$	0	0
$231$	−5.51848e23	−0.0839255
$232$	0	0
$233$	−1.31126e24	−0.182159	−0.0910797	−	0.995844i	$-0.529032\pi$
$233$	−1.31126e24	−0.182159	−0.0910797	+	0.995844i	$0.529032\pi$
$234$	0	0
$235$	5.97657e23	0.0758996
$236$	0	0
$237$	−5.19560e24	−0.603640
$238$	0	0
$239$	8.42576e24	0.896254	0.448127	−	0.893970i	$-0.352091\pi$
$239$	8.42576e24	0.896254	0.448127	+	0.893970i	$0.352091\pi$
$240$	0	0
$241$	−1.44027e25	−1.40367	−0.701834	−	0.712341i	$-0.747635\pi$
$241$	−1.44027e25	−1.40367	−0.701834	+	0.712341i	$0.747635\pi$
$242$	0	0
$243$	−1.12755e25	−1.00756
$244$	0	0
$245$	6.09407e23	0.0499648
$246$	0	0
$247$	8.14509e24	0.613166
$248$	0	0
$249$	−7.64849e24	−0.529033
$250$	0	0
$251$	−1.42742e25	−0.907772	−0.453886	−	0.891060i	$-0.649963\pi$
$251$	−1.42742e25	−0.907772	−0.453886	+	0.891060i	$0.649963\pi$
$252$	0	0
$253$	3.24617e24	0.189935
$254$	0	0
$255$	1.81913e22	0.000979924 0
$256$	0	0
$257$	3.27132e25	1.62340	0.811700	−	0.584075i	$-0.198543\pi$
$257$	3.27132e25	1.62340	0.811700	+	0.584075i	$0.198543\pi$
$258$	0	0
$259$	1.25025e24	0.0571938
$260$	0	0
$261$	2.54763e25	1.07501
$262$	0	0
$263$	1.57524e24	0.0613494	0.0306747	−	0.999529i	$-0.490234\pi$
$263$	1.57524e24	0.0613494	0.0306747	+	0.999529i	$0.490234\pi$
$264$	0	0
$265$	2.32057e24	0.0834665
$266$	0	0
$267$	−7.54647e24	−0.250828
$268$	0	0
$269$	2.47417e25	0.760378	0.380189	−	0.924909i	$-0.375859\pi$
$269$	2.47417e25	0.760378	0.380189	+	0.924909i	$0.375859\pi$
$270$	0	0
$271$	6.32727e25	1.79903	0.899515	−	0.436890i	$-0.143920\pi$
$271$	6.32727e25	1.79903	0.899515	+	0.436890i	$0.143920\pi$
$272$	0	0
$273$	−4.10700e24	−0.108098
$274$	0	0
$275$	−1.60198e25	−0.390538
$276$	0	0
$277$	−1.52097e25	−0.343624	−0.171812	−	0.985130i	$-0.554962\pi$
$277$	−1.52097e25	−0.343624	−0.171812	+	0.985130i	$0.554962\pi$
$278$	0	0
$279$	−1.49377e25	−0.312924
$280$	0	0
$281$	−4.87649e24	−0.0947743	−0.0473872	−	0.998877i	$-0.515089\pi$
$281$	−4.87649e24	−0.0947743	−0.0473872	+	0.998877i	$0.515089\pi$
$282$	0	0
$283$	6.39728e25	1.15408	0.577042	−	0.816714i	$-0.304207\pi$
$283$	6.39728e25	1.15408	0.577042	+	0.816714i	$0.304207\pi$
$284$	0	0
$285$	2.51714e24	0.0421733
$286$	0	0
$287$	9.37468e24	0.145947
$288$	0	0
$289$	−6.90369e25	−0.999203
$290$	0	0
$291$	−2.32673e25	−0.313234
$292$	0	0
$293$	−2.68070e25	−0.335845	−0.167922	−	0.985800i	$-0.553706\pi$
$293$	−2.68070e25	−0.335845	−0.167922	+	0.985800i	$0.553706\pi$
$294$	0	0
$295$	1.53353e24	0.0178880
$296$	0	0
$297$	3.57652e25	0.388613
$298$	0	0
$299$	2.41589e25	0.244641
$300$	0	0
$301$	−5.80657e25	−0.548241
$302$	0	0
$303$	−7.02175e25	−0.618440
$304$	0	0
$305$	7.81000e24	0.0641952
$306$	0	0
$307$	−5.13119e25	−0.393791	−0.196895	−	0.980425i	$-0.563086\pi$
$307$	−5.13119e25	−0.393791	−0.196895	+	0.980425i	$0.563086\pi$
$308$	0	0
$309$	7.18796e25	0.515278
$310$	0	0
$311$	−2.73650e26	−1.83321	−0.916603	−	0.399798i	$-0.869080\pi$
$311$	−2.73650e26	−1.83321	−0.916603	+	0.399798i	$0.869080\pi$
$312$	0	0
$313$	4.66082e25	0.291909	0.145954	−	0.989291i	$-0.453375\pi$
$313$	4.66082e25	0.291909	0.145954	+	0.989291i	$0.453375\pi$
$314$	0	0
$315$	2.15674e24	0.0126339
$316$	0	0
$317$	−1.24217e26	−0.680860	−0.340430	−	0.940270i	$-0.610573\pi$
$317$	−1.24217e26	−0.680860	−0.340430	+	0.940270i	$0.610573\pi$
$318$	0	0
$319$	−1.30400e26	−0.669074
$320$	0	0
$321$	7.70760e25	0.370353
$322$	0	0
$323$	7.61966e24	0.0343012
$324$	0	0
$325$	−1.19223e26	−0.503021
$326$	0	0
$327$	8.33226e25	0.329618
$328$	0	0
$329$	−1.26227e26	−0.468377
$330$	0	0
$331$	3.57554e26	1.24494	0.622469	−	0.782644i	$-0.286130\pi$
$331$	3.57554e26	1.24494	0.622469	+	0.782644i	$0.286130\pi$
$332$	0	0
$333$	−3.13033e25	−0.102312
$334$	0	0
$335$	2.15196e25	0.0660488
$336$	0	0
$337$	5.04047e25	0.145331	0.0726653	−	0.997356i	$-0.476850\pi$
$337$	5.04047e25	0.145331	0.0726653	+	0.997356i	$0.476850\pi$
$338$	0	0
$339$	4.04580e26	1.09624
$340$	0	0
$341$	7.64580e25	0.194761
$342$	0	0
$343$	−2.75610e26	−0.660248
$344$	0	0
$345$	7.46601e24	0.0168263
$346$	0	0
$347$	7.30981e26	1.55041	0.775205	−	0.631709i	$-0.217646\pi$
$347$	7.30981e26	1.55041	0.775205	+	0.631709i	$0.217646\pi$
$348$	0	0
$349$	6.72114e26	1.34207	0.671036	−	0.741425i	$-0.265850\pi$
$349$	6.72114e26	1.34207	0.671036	+	0.741425i	$0.265850\pi$
$350$	0	0
$351$	2.66174e26	0.500542
$352$	0	0
$353$	5.30749e25	0.0940275	0.0470138	−	0.998894i	$-0.485030\pi$
$353$	5.30749e25	0.0940275	0.0470138	+	0.998894i	$0.485030\pi$
$354$	0	0
$355$	3.89890e25	0.0650945
$356$	0	0
$357$	−3.84206e24	−0.00604711
$358$	0	0
$359$	−8.31857e26	−1.23469	−0.617343	−	0.786694i	$-0.711791\pi$
$359$	−8.31857e26	−1.23469	−0.617343	+	0.786694i	$0.711791\pi$
$360$	0	0
$361$	3.40128e26	0.476231
$362$	0	0
$363$	3.89956e26	0.515224
$364$	0	0
$365$	−6.84144e25	−0.0853243
$366$	0	0
$367$	−1.02517e27	−1.20727	−0.603635	−	0.797261i	$-0.706281\pi$
$367$	−1.02517e27	−1.20727	−0.603635	+	0.797261i	$0.706281\pi$
$368$	0	0
$369$	−2.34720e26	−0.261080
$370$	0	0
$371$	−4.90111e26	−0.515072
$372$	0	0
$373$	1.01508e27	1.00822	0.504112	−	0.863638i	$-0.331820\pi$
$373$	1.01508e27	1.00822	0.504112	+	0.863638i	$0.331820\pi$
$374$	0	0
$375$	−7.38094e25	−0.0693080
$376$	0	0
$377$	−9.70468e26	−0.861782
$378$	0	0
$379$	−9.76777e26	−0.820510	−0.410255	−	0.911971i	$-0.634560\pi$
$379$	−9.76777e26	−0.820510	−0.410255	+	0.911971i	$0.634560\pi$
$380$	0	0
$381$	−3.13100e26	−0.248869
$382$	0	0
$383$	3.96291e26	0.298145	0.149072	−	0.988826i	$-0.452371\pi$
$383$	3.96291e26	0.298145	0.149072	+	0.988826i	$0.452371\pi$
$384$	0	0
$385$	−1.10392e25	−0.00786320
$386$	0	0
$387$	1.45383e27	0.980728
$388$	0	0
$389$	−2.89702e27	−1.85132	−0.925658	−	0.378361i	$-0.876488\pi$
$389$	−2.89702e27	−1.85132	−0.925658	+	0.378361i	$0.876488\pi$
$390$	0	0
$391$	2.26004e25	0.0136855
$392$	0	0
$393$	−1.31510e26	−0.0754807
$394$	0	0
$395$	−1.03933e26	−0.0565566
$396$	0	0
$397$	−5.79178e26	−0.298890	−0.149445	−	0.988770i	$-0.547749\pi$
$397$	−5.79178e26	−0.298890	−0.149445	+	0.988770i	$0.547749\pi$
$398$	0	0
$399$	−5.31629e26	−0.260251
$400$	0	0
$401$	−1.05664e27	−0.490807	−0.245403	−	0.969421i	$-0.578920\pi$
$401$	−1.05664e27	−0.490807	−0.245403	+	0.969421i	$0.578920\pi$
$402$	0	0
$403$	5.69020e26	0.250856
$404$	0	0
$405$	−3.52017e24	−0.00147328
$406$	0	0
$407$	1.60225e26	0.0636778
$408$	0	0
$409$	2.30402e27	0.869746	0.434873	−	0.900492i	$-0.356793\pi$
$409$	2.30402e27	0.869746	0.434873	+	0.900492i	$0.356793\pi$
$410$	0	0
$411$	−1.67299e27	−0.600002
$412$	0	0
$413$	−3.23886e26	−0.110387
$414$	0	0
$415$	−1.53000e26	−0.0495664
$416$	0	0
$417$	5.62522e26	0.173265
$418$	0	0
$419$	−2.38736e27	−0.699311	−0.349655	−	0.936878i	$-0.613701\pi$
$419$	−2.38736e27	−0.699311	−0.349655	+	0.936878i	$0.613701\pi$
$420$	0	0
$421$	4.44803e27	1.23938	0.619691	−	0.784846i	$-0.287258\pi$
$421$	4.44803e27	1.23938	0.619691	+	0.784846i	$0.287258\pi$
$422$	0	0
$423$	3.16043e27	0.837861
$424$	0	0
$425$	−1.11532e26	−0.0281396
$426$	0	0
$427$	−1.64949e27	−0.396149
$428$	0	0
$429$	−5.26330e26	−0.120353
$430$	0	0
$431$	2.79963e27	0.609662	0.304831	−	0.952406i	$-0.401400\pi$
$431$	2.79963e27	0.609662	0.304831	+	0.952406i	$0.401400\pi$
$432$	0	0
$433$	6.23965e27	1.29431	0.647154	−	0.762360i	$-0.275959\pi$
$433$	6.23965e27	1.29431	0.647154	+	0.762360i	$0.275959\pi$
$434$	0	0
$435$	−2.99912e26	−0.0592729
$436$	0	0
$437$	3.12723e27	0.588986
$438$	0	0
$439$	7.65214e27	1.37374	0.686871	−	0.726779i	$-0.258984\pi$
$439$	7.65214e27	1.37374	0.686871	+	0.726779i	$0.258984\pi$
$440$	0	0
$441$	3.22257e27	0.551565
$442$	0	0
$443$	−2.50862e27	−0.409446	−0.204723	−	0.978820i	$-0.565629\pi$
$443$	−2.50862e27	−0.409446	−0.204723	+	0.978820i	$0.565629\pi$
$444$	0	0
$445$	−1.50960e26	−0.0235007
$446$	0	0
$447$	3.64899e27	0.541931
$448$	0	0
$449$	−9.77491e27	−1.38524	−0.692622	−	0.721301i	$-0.743544\pi$
$449$	−9.77491e27	−1.38524	−0.692622	+	0.721301i	$0.743544\pi$
$450$	0	0
$451$	1.20141e27	0.162493
$452$	0	0
$453$	−5.03513e27	−0.650099
$454$	0	0
$455$	−8.21564e25	−0.0101280
$456$	0	0
$457$	−6.63390e27	−0.780996	−0.390498	−	0.920604i	$-0.627697\pi$
$457$	−6.63390e27	−0.780996	−0.390498	+	0.920604i	$0.627697\pi$
$458$	0	0
$459$	2.49003e26	0.0280009
$460$	0	0
$461$	−1.50903e27	−0.162121	−0.0810605	−	0.996709i	$-0.525831\pi$
$461$	−1.50903e27	−0.162121	−0.0810605	+	0.996709i	$0.525831\pi$
$462$	0	0
$463$	5.89041e27	0.604707	0.302354	−	0.953196i	$-0.402228\pi$
$463$	5.89041e27	0.604707	0.302354	+	0.953196i	$0.402228\pi$
$464$	0	0
$465$	1.75849e26	0.0172538
$466$	0	0
$467$	−1.38550e28	−1.29951	−0.649756	−	0.760142i	$-0.725129\pi$
$467$	−1.38550e28	−1.29951	−0.649756	+	0.760142i	$0.725129\pi$
$468$	0	0
$469$	−4.54501e27	−0.407587
$470$	0	0
$471$	3.37411e27	0.289361
$472$	0	0
$473$	−7.44138e27	−0.610395
$474$	0	0
$475$	−1.54328e28	−1.21105
$476$	0	0
$477$	1.22712e28	0.921393
$478$	0	0
$479$	−2.10471e28	−1.51241	−0.756204	−	0.654336i	$-0.772948\pi$
$479$	−2.10471e28	−1.51241	−0.756204	+	0.654336i	$0.772948\pi$
$480$	0	0
$481$	1.19243e27	0.0820183
$482$	0	0
$483$	−1.57684e27	−0.103835
$484$	0	0
$485$	−4.65439e26	−0.0293478
$486$	0	0
$487$	1.93093e28	1.16604	0.583018	−	0.812459i	$-0.301872\pi$
$487$	1.93093e28	1.16604	0.583018	+	0.812459i	$0.301872\pi$
$488$	0	0
$489$	1.00504e28	0.581351
$490$	0	0
$491$	−2.38236e28	−1.32023	−0.660117	−	0.751163i	$-0.729493\pi$
$491$	−2.38236e28	−1.32023	−0.660117	+	0.751163i	$0.729493\pi$
$492$	0	0
$493$	−9.07865e26	−0.0482091
$494$	0	0
$495$	2.76395e26	0.0140662
$496$	0	0
$497$	−8.23460e27	−0.401699
$498$	0	0
$499$	2.22172e28	1.03905	0.519523	−	0.854457i	$-0.326110\pi$
$499$	2.22172e28	1.03905	0.519523	+	0.854457i	$0.326110\pi$
$500$	0	0
$501$	1.36851e28	0.613694
$502$	0	0
$503$	−8.37368e27	−0.360124	−0.180062	−	0.983655i	$-0.557630\pi$
$503$	−8.37368e27	−0.360124	−0.180062	+	0.983655i	$0.557630\pi$
$504$	0	0
$505$	−1.40463e27	−0.0579433
$506$	0	0
$507$	1.14631e28	0.453647
$508$	0	0
$509$	−3.38122e28	−1.28392	−0.641958	−	0.766740i	$-0.721878\pi$
$509$	−3.38122e28	−1.28392	−0.641958	+	0.766740i	$0.721878\pi$
$510$	0	0
$511$	1.44493e28	0.526536
$512$	0	0
$513$	3.44547e28	1.20508
$514$	0	0
$515$	1.43788e27	0.0482778
$516$	0	0
$517$	−1.61765e28	−0.521477
$518$	0	0
$519$	2.39479e27	0.0741327
$520$	0	0
$521$	8.19578e27	0.243665	0.121833	−	0.992551i	$-0.461123\pi$
$521$	8.19578e27	0.243665	0.121833	+	0.992551i	$0.461123\pi$
$522$	0	0
$523$	6.28168e28	1.79394	0.896970	−	0.442092i	$-0.145764\pi$
$523$	6.28168e28	1.79394	0.896970	+	0.442092i	$0.145764\pi$
$524$	0	0
$525$	7.78169e27	0.213502
$526$	0	0
$527$	5.32314e26	0.0140332
$528$	0	0
$529$	−3.01960e28	−0.765006
$530$	0	0
$531$	8.10935e27	0.197467
$532$	0	0
$533$	8.94117e27	0.209295
$534$	0	0
$535$	1.54183e27	0.0346994
$536$	0	0
$537$	3.75792e28	0.813238
$538$	0	0
$539$	−1.64946e28	−0.343288
$540$	0	0
$541$	4.40803e28	0.882416	0.441208	−	0.897405i	$-0.354550\pi$
$541$	4.40803e28	0.882416	0.441208	+	0.897405i	$0.354550\pi$
$542$	0	0
$543$	3.51599e26	0.00677094
$544$	0	0
$545$	1.66679e27	0.0308828
$546$	0	0
$547$	3.93832e28	0.702174	0.351087	−	0.936343i	$-0.385812\pi$
$547$	3.93832e28	0.702174	0.351087	+	0.936343i	$0.385812\pi$
$548$	0	0
$549$	4.12995e28	0.708656
$550$	0	0
$551$	−1.25622e29	−2.07479
$552$	0	0
$553$	2.19509e28	0.349011
$554$	0	0
$555$	3.68508e26	0.00564118
$556$	0	0
$557$	−5.79761e28	−0.854612	−0.427306	−	0.904107i	$-0.640537\pi$
$557$	−5.79761e28	−0.854612	−0.427306	+	0.904107i	$0.640537\pi$
$558$	0	0
$559$	−5.53806e28	−0.786202
$560$	0	0
$561$	−4.92377e26	−0.00673268
$562$	0	0
$563$	−3.32225e28	−0.437618	−0.218809	−	0.975768i	$-0.570217\pi$
$563$	−3.32225e28	−0.437618	−0.218809	+	0.975768i	$0.570217\pi$
$564$	0	0
$565$	8.09322e27	0.102710
$566$	0	0
$567$	7.43471e26	0.00909163
$568$	0	0
$569$	−1.08751e29	−1.28160	−0.640801	−	0.767707i	$-0.721398\pi$
$569$	−1.08751e29	−1.28160	−0.640801	+	0.767707i	$0.721398\pi$
$570$	0	0
$571$	1.31209e28	0.149033	0.0745166	−	0.997220i	$-0.476259\pi$
$571$	1.31209e28	0.149033	0.0745166	+	0.997220i	$0.476259\pi$
$572$	0	0
$573$	−8.40288e28	−0.920036
$574$	0	0
$575$	−4.57747e28	−0.483185
$576$	0	0
$577$	−1.17664e29	−1.19756	−0.598778	−	0.800915i	$-0.704347\pi$
$577$	−1.17664e29	−1.19756	−0.598778	+	0.800915i	$0.704347\pi$
$578$	0	0
$579$	7.02938e28	0.689910
$580$	0	0
$581$	3.23142e28	0.305875
$582$	0	0
$583$	−6.28098e28	−0.573466
$584$	0	0
$585$	2.05701e27	0.0181175
$586$	0	0
$587$	1.41639e29	1.20360	0.601799	−	0.798647i	$-0.294451\pi$
$587$	1.41639e29	1.20360	0.601799	+	0.798647i	$0.294451\pi$
$588$	0	0
$589$	7.36566e28	0.603950
$590$	0	0
$591$	−3.70659e28	−0.293296
$592$	0	0
$593$	−6.70941e28	−0.512401	−0.256201	−	0.966624i	$-0.582471\pi$
$593$	−6.70941e28	−0.512401	−0.256201	+	0.966624i	$0.582471\pi$
$594$	0	0
$595$	−7.68567e25	−0.000566570 0
$596$	0	0
$597$	−1.67253e29	−1.19026
$598$	0	0
$599$	−2.42035e28	−0.166302	−0.0831510	−	0.996537i	$-0.526498\pi$
$599$	−2.42035e28	−0.166302	−0.0831510	+	0.996537i	$0.526498\pi$
$600$	0	0
$601$	2.08024e29	1.38016	0.690082	−	0.723731i	$-0.257575\pi$
$601$	2.08024e29	1.38016	0.690082	+	0.723731i	$0.257575\pi$
$602$	0	0
$603$	1.13797e29	0.729117
$604$	0	0
$605$	7.80068e27	0.0482726
$606$	0	0
$607$	1.15837e29	0.692417	0.346208	−	0.938158i	$-0.387469\pi$
$607$	1.15837e29	0.692417	0.346208	+	0.938158i	$0.387469\pi$
$608$	0	0
$609$	6.33423e28	0.365773
$610$	0	0
$611$	−1.20390e29	−0.671672
$612$	0	0
$613$	2.36853e29	1.27686	0.638430	−	0.769680i	$-0.279584\pi$
$613$	2.36853e29	1.27686	0.638430	+	0.769680i	$0.279584\pi$
$614$	0	0
$615$	2.76316e27	0.0143952
$616$	0	0
$617$	−4.18061e28	−0.210497	−0.105248	−	0.994446i	$-0.533564\pi$
$617$	−4.18061e28	−0.210497	−0.105248	+	0.994446i	$0.533564\pi$
$618$	0	0
$619$	4.89038e27	0.0238008	0.0119004	−	0.999929i	$-0.496212\pi$
$619$	4.89038e27	0.0238008	0.0119004	+	0.999929i	$0.496212\pi$
$620$	0	0
$621$	1.02195e29	0.480803
$622$	0	0
$623$	3.18831e28	0.145023
$624$	0	0
$625$	2.25158e29	0.990254
$626$	0	0
$627$	−6.81305e28	−0.289756
$628$	0	0
$629$	1.11551e27	0.00458820
$630$	0	0
$631$	−2.10752e29	−0.838424	−0.419212	−	0.907888i	$-0.637694\pi$
$631$	−2.10752e29	−0.838424	−0.419212	+	0.907888i	$0.637694\pi$
$632$	0	0
$633$	8.19441e28	0.315340
$634$	0	0
$635$	−6.26327e27	−0.0233172
$636$	0	0
$637$	−1.22757e29	−0.442163
$638$	0	0
$639$	2.06175e29	0.718584
$640$	0	0
$641$	6.89760e28	0.232642	0.116321	−	0.993212i	$-0.462890\pi$
$641$	6.89760e28	0.232642	0.116321	+	0.993212i	$0.462890\pi$
$642$	0	0
$643$	−8.28400e28	−0.270411	−0.135206	−	0.990818i	$-0.543170\pi$
$643$	−8.28400e28	−0.270411	−0.135206	+	0.990818i	$0.543170\pi$
$644$	0	0
$645$	−1.71147e28	−0.0540746
$646$	0	0
$647$	5.69599e29	1.74211	0.871053	−	0.491189i	$-0.163438\pi$
$647$	5.69599e29	1.74211	0.871053	+	0.491189i	$0.163438\pi$
$648$	0	0
$649$	−4.15074e28	−0.122901
$650$	0	0
$651$	−3.71398e28	−0.106473
$652$	0	0
$653$	−3.86586e29	−1.07314	−0.536571	−	0.843855i	$-0.680281\pi$
$653$	−3.86586e29	−1.07314	−0.536571	+	0.843855i	$0.680281\pi$
$654$	0	0
$655$	−2.63072e27	−0.00707199
$656$	0	0
$657$	−3.61778e29	−0.941901
$658$	0	0
$659$	3.00496e29	0.757777	0.378889	−	0.925442i	$-0.376306\pi$
$659$	3.00496e29	0.757777	0.378889	+	0.925442i	$0.376306\pi$
$660$	0	0
$661$	2.82911e29	0.691089	0.345545	−	0.938402i	$-0.387694\pi$
$661$	2.82911e29	0.691089	0.345545	+	0.938402i	$0.387694\pi$
$662$	0	0
$663$	−3.66440e27	−0.00867182
$664$	0	0
$665$	−1.06347e28	−0.0243836
$666$	0	0
$667$	−3.72602e29	−0.827798
$668$	0	0
$669$	−4.86339e29	−1.04704
$670$	0	0
$671$	−2.11390e29	−0.441060
$672$	0	0
$673$	−4.94836e29	−1.00070	−0.500349	−	0.865824i	$-0.666795\pi$
$673$	−4.94836e29	−1.00070	−0.500349	+	0.865824i	$0.666795\pi$
$674$	0	0
$675$	−5.04329e29	−0.988609
$676$	0	0
$677$	9.03975e28	0.171781	0.0858906	−	0.996305i	$-0.472626\pi$
$677$	9.03975e28	0.171781	0.0858906	+	0.996305i	$0.472626\pi$
$678$	0	0
$679$	9.83022e28	0.181105
$680$	0	0
$681$	6.08034e29	1.08613
$682$	0	0
$683$	−6.74953e29	−1.16911	−0.584556	−	0.811354i	$-0.698731\pi$
$683$	−6.74953e29	−1.16911	−0.584556	+	0.811354i	$0.698731\pi$
$684$	0	0
$685$	−3.34664e28	−0.0562158
$686$	0	0
$687$	5.79420e29	0.943947
$688$	0	0
$689$	−4.67447e29	−0.738636
$690$	0	0
$691$	−1.18142e30	−1.81087	−0.905433	−	0.424489i	$-0.860454\pi$
$691$	−1.18142e30	−1.81087	−0.905433	+	0.424489i	$0.860454\pi$
$692$	0	0
$693$	−5.83755e28	−0.0868024
$694$	0	0
$695$	1.12527e28	0.0162337
$696$	0	0
$697$	8.36439e27	0.0117082
$698$	0	0
$699$	8.16280e28	0.110874
$700$	0	0
$701$	8.90277e29	1.17351	0.586754	−	0.809766i	$-0.300406\pi$
$701$	8.90277e29	1.17351	0.586754	+	0.809766i	$0.300406\pi$
$702$	0	0
$703$	1.54354e29	0.197464
$704$	0	0
$705$	−3.72051e28	−0.0461973
$706$	0	0
$707$	2.96662e29	0.357568
$708$	0	0
$709$	4.82848e29	0.564970	0.282485	−	0.959272i	$-0.408841\pi$
$709$	4.82848e29	0.564970	0.282485	+	0.959272i	$0.408841\pi$
$710$	0	0
$711$	−5.49600e29	−0.624333
$712$	0	0
$713$	2.18470e29	0.240964
$714$	0	0
$715$	−1.05287e28	−0.0112762
$716$	0	0
$717$	−5.24517e29	−0.545517
$718$	0	0
$719$	9.70706e29	0.980470	0.490235	−	0.871590i	$-0.336911\pi$
$719$	9.70706e29	0.980470	0.490235	+	0.871590i	$0.336911\pi$
$720$	0	0
$721$	−3.03685e29	−0.297922
$722$	0	0
$723$	8.96589e29	0.854361
$724$	0	0
$725$	1.83878e30	1.70209
$726$	0	0
$727$	1.65557e30	1.48880	0.744401	−	0.667733i	$-0.232735\pi$
$727$	1.65557e30	1.48880	0.744401	+	0.667733i	$0.232735\pi$
$728$	0	0
$729$	6.72350e29	0.587430
$730$	0	0
$731$	−5.18081e28	−0.0439810
$732$	0	0
$733$	−1.70639e30	−1.40762	−0.703811	−	0.710387i	$-0.748520\pi$
$733$	−1.70639e30	−1.40762	−0.703811	+	0.710387i	$0.748520\pi$
$734$	0	0
$735$	−3.79366e28	−0.0304117
$736$	0	0
$737$	−5.82463e29	−0.453795
$738$	0	0
$739$	1.51635e30	1.14824	0.574118	−	0.818772i	$-0.305345\pi$
$739$	1.51635e30	1.14824	0.574118	+	0.818772i	$0.305345\pi$
$740$	0	0
$741$	−5.07045e29	−0.373212
$742$	0	0
$743$	1.01870e30	0.728890	0.364445	−	0.931225i	$-0.381259\pi$
$743$	1.01870e30	0.728890	0.364445	+	0.931225i	$0.381259\pi$
$744$	0	0
$745$	7.29945e28	0.0507749
$746$	0	0
$747$	−8.09071e29	−0.547168
$748$	0	0
$749$	−3.25639e29	−0.214130
$750$	0	0
$751$	−2.00722e30	−1.28344	−0.641721	−	0.766938i	$-0.721779\pi$
$751$	−2.00722e30	−1.28344	−0.641721	+	0.766938i	$0.721779\pi$
$752$	0	0
$753$	8.88590e29	0.552528
$754$	0	0
$755$	−1.00723e29	−0.0609095
$756$	0	0
$757$	−6.30787e29	−0.371002	−0.185501	−	0.982644i	$-0.559391\pi$
$757$	−6.30787e29	−0.371002	−0.185501	+	0.982644i	$0.559391\pi$
$758$	0	0
$759$	−2.02079e29	−0.115607
$760$	0	0
$761$	2.91644e30	1.62299	0.811493	−	0.584363i	$-0.198655\pi$
$761$	2.91644e30	1.62299	0.811493	+	0.584363i	$0.198655\pi$
$762$	0	0
$763$	−3.52030e29	−0.190578
$764$	0	0
$765$	1.92431e27	0.00101352
$766$	0	0
$767$	−3.08909e29	−0.158299
$768$	0	0
$769$	−2.22144e30	−1.10766	−0.553831	−	0.832629i	$-0.686835\pi$
$769$	−2.22144e30	−1.10766	−0.553831	+	0.832629i	$0.686835\pi$
$770$	0	0
$771$	−2.03645e30	−0.988104
$772$	0	0
$773$	1.99610e30	0.942535	0.471267	−	0.881990i	$-0.343797\pi$
$773$	1.99610e30	0.942535	0.471267	+	0.881990i	$0.343797\pi$
$774$	0	0
$775$	−1.07814e30	−0.495461
$776$	0	0
$777$	−7.78299e28	−0.0348118
$778$	0	0
$779$	1.15739e30	0.503889
$780$	0	0
$781$	−1.05530e30	−0.447239
$782$	0	0
$783$	−4.10520e30	−1.69370
$784$	0	0
$785$	6.74958e28	0.0271110
$786$	0	0
$787$	3.43376e30	1.34287	0.671437	−	0.741062i	$-0.265677\pi$
$787$	3.43376e30	1.34287	0.671437	+	0.741062i	$0.265677\pi$
$788$	0	0
$789$	−9.80610e28	−0.0373412
$790$	0	0
$791$	−1.70931e30	−0.633823
$792$	0	0
$793$	−1.57322e30	−0.568095
$794$	0	0
$795$	−1.44459e29	−0.0508030
$796$	0	0
$797$	−2.08927e30	−0.715620	−0.357810	−	0.933794i	$-0.616476\pi$
$797$	−2.08927e30	−0.715620	−0.357810	+	0.933794i	$0.616476\pi$
$798$	0	0
$799$	−1.12624e29	−0.0375741
$800$	0	0
$801$	−7.98280e29	−0.259426
$802$	0	0
$803$	1.85174e30	0.586230
$804$	0	0
$805$	−3.15432e28	−0.00972857
$806$	0	0
$807$	−1.54021e30	−0.462815
$808$	0	0
$809$	1.63457e30	0.478567	0.239284	−	0.970950i	$-0.423087\pi$
$809$	1.63457e30	0.478567	0.239284	+	0.970950i	$0.423087\pi$
$810$	0	0
$811$	6.21829e28	0.0177399	0.00886997	−	0.999961i	$-0.497177\pi$
$811$	6.21829e28	0.0177399	0.00886997	+	0.999961i	$0.497177\pi$
$812$	0	0
$813$	−3.93882e30	−1.09500
$814$	0	0
$815$	2.01047e29	0.0544683
$816$	0	0
$817$	−7.16872e30	−1.89282
$818$	0	0
$819$	−4.34446e29	−0.111803
$820$	0	0
$821$	5.54995e30	1.39215	0.696074	−	0.717970i	$-0.254928\pi$
$821$	5.54995e30	1.39215	0.696074	+	0.717970i	$0.254928\pi$
$822$	0	0
$823$	8.10790e29	0.198249	0.0991243	−	0.995075i	$-0.468396\pi$
$823$	8.10790e29	0.198249	0.0991243	+	0.995075i	$0.468396\pi$
$824$	0	0
$825$	9.97257e29	0.237706
$826$	0	0
$827$	8.17222e30	1.89903	0.949516	−	0.313719i	$-0.101575\pi$
$827$	8.17222e30	1.89903	0.949516	+	0.313719i	$0.101575\pi$
$828$	0	0
$829$	−3.53326e29	−0.0800484	−0.0400242	−	0.999199i	$-0.512744\pi$
$829$	−3.53326e29	−0.0800484	−0.0400242	+	0.999199i	$0.512744\pi$
$830$	0	0
$831$	9.46829e29	0.209151
$832$	0	0
$833$	−1.14838e29	−0.0247351
$834$	0	0
$835$	2.73757e29	0.0574986
$836$	0	0
$837$	2.40702e30	0.493019
$838$	0	0
$839$	−2.12371e30	−0.424223	−0.212112	−	0.977245i	$-0.568034\pi$
$839$	−2.12371e30	−0.424223	−0.212112	+	0.977245i	$0.568034\pi$
$840$	0	0
$841$	9.83471e30	1.91604
$842$	0	0
$843$	3.03569e29	0.0576857
$844$	0	0
$845$	2.29308e29	0.0425034
$846$	0	0
$847$	−1.64753e30	−0.297891
$848$	0	0
$849$	−3.98241e30	−0.702450
$850$	0	0
$851$	4.57824e29	0.0787840
$852$	0	0
$853$	9.31841e30	1.56451	0.782253	−	0.622961i	$-0.214071\pi$
$853$	9.31841e30	1.56451	0.782253	+	0.622961i	$0.214071\pi$
$854$	0	0
$855$	2.66268e29	0.0436190
$856$	0	0
$857$	8.07467e30	1.29070	0.645352	−	0.763886i	$-0.276711\pi$
$857$	8.07467e30	1.29070	0.645352	+	0.763886i	$0.276711\pi$
$858$	0	0
$859$	−2.78850e30	−0.434953	−0.217476	−	0.976066i	$-0.569782\pi$
$859$	−2.78850e30	−0.434953	−0.217476	+	0.976066i	$0.569782\pi$
$860$	0	0
$861$	−5.83589e29	−0.0888329
$862$	0	0
$863$	7.81319e29	0.116069	0.0580344	−	0.998315i	$-0.481517\pi$
$863$	7.81319e29	0.116069	0.0580344	+	0.998315i	$0.481517\pi$
$864$	0	0
$865$	4.79054e28	0.00694569
$866$	0	0
$867$	4.29766e30	0.608179
$868$	0	0
$869$	2.81311e30	0.388579
$870$	0	0
$871$	−4.33484e30	−0.584498
$872$	0	0
$873$	−2.46126e30	−0.323972
$874$	0	0
$875$	3.11838e29	0.0400723
$876$	0	0
$877$	−7.95009e30	−0.997416	−0.498708	−	0.866770i	$-0.666192\pi$
$877$	−7.95009e30	−0.997416	−0.498708	+	0.866770i	$0.666192\pi$
$878$	0	0
$879$	1.66878e30	0.204417
$880$	0	0
$881$	−9.87210e30	−1.18076	−0.590381	−	0.807125i	$-0.701023\pi$
$881$	−9.87210e30	−1.18076	−0.590381	+	0.807125i	$0.701023\pi$
$882$	0	0
$883$	−2.79595e30	−0.326544	−0.163272	−	0.986581i	$-0.552205\pi$
$883$	−2.79595e30	−0.326544	−0.163272	+	0.986581i	$0.552205\pi$
$884$	0	0
$885$	−9.54647e28	−0.0108878
$886$	0	0
$887$	2.90087e30	0.323095	0.161547	−	0.986865i	$-0.448352\pi$
$887$	2.90087e30	0.323095	0.161547	+	0.986865i	$0.448352\pi$
$888$	0	0
$889$	1.32282e30	0.143891
$890$	0	0
$891$	9.52791e28	0.0101224
$892$	0	0
$893$	−1.55838e31	−1.61709
$894$	0	0
$895$	7.51735e29	0.0761944
$896$	0	0
$897$	−1.50393e30	−0.148904
$898$	0	0
$899$	−8.77601e30	−0.848829
$900$	0	0
$901$	−4.37293e29	−0.0413202
$902$	0	0
$903$	3.61468e30	0.333694
$904$	0	0
$905$	7.03339e27	0.000634387 0
$906$	0	0
$907$	−5.28748e30	−0.465985	−0.232993	−	0.972479i	$-0.574852\pi$
$907$	−5.28748e30	−0.465985	−0.232993	+	0.972479i	$0.574852\pi$
$908$	0	0
$909$	−7.42773e30	−0.639640
$910$	0	0
$911$	1.11142e31	0.935269	0.467634	−	0.883922i	$-0.345106\pi$
$911$	1.11142e31	0.935269	0.467634	+	0.883922i	$0.345106\pi$
$912$	0	0
$913$	4.14120e30	0.340552
$914$	0	0
$915$	−4.86185e29	−0.0390733
$916$	0	0
$917$	5.55617e29	0.0436413
$918$	0	0
$919$	−6.90233e30	−0.529886	−0.264943	−	0.964264i	$-0.585353\pi$
$919$	−6.90233e30	−0.529886	−0.264943	+	0.964264i	$0.585353\pi$
$920$	0	0
$921$	3.19425e30	0.239686
$922$	0	0
$923$	−7.85381e30	−0.576053
$924$	0	0
$925$	−2.25935e30	−0.161993
$926$	0	0
$927$	7.60356e30	0.532942
$928$	0	0
$929$	−2.04813e31	−1.40344	−0.701718	−	0.712455i	$-0.747583\pi$
$929$	−2.04813e31	−1.40344	−0.701718	+	0.712455i	$0.747583\pi$
$930$	0	0
$931$	−1.58902e31	−1.06453
$932$	0	0
$933$	1.70352e31	1.11581
$934$	0	0
$935$	−9.84952e27	−0.000630802 0
$936$	0	0
$937$	2.29135e31	1.43492	0.717458	−	0.696601i	$-0.245305\pi$
$937$	2.29135e31	1.43492	0.717458	+	0.696601i	$0.245305\pi$
$938$	0	0
$939$	−2.90144e30	−0.177674
$940$	0	0
$941$	−7.14878e30	−0.428096	−0.214048	−	0.976823i	$-0.568665\pi$
$941$	−7.14878e30	−0.428096	−0.214048	+	0.976823i	$0.568665\pi$
$942$	0	0
$943$	3.43288e30	0.201042
$944$	0	0
$945$	−3.47532e29	−0.0199049
$946$	0	0
$947$	−2.01049e31	−1.12623	−0.563116	−	0.826378i	$-0.690397\pi$
$947$	−2.01049e31	−1.12623	−0.563116	+	0.826378i	$0.690397\pi$
$948$	0	0
$949$	1.37812e31	0.755076
$950$	0	0
$951$	7.73269e30	0.414415
$952$	0	0
$953$	1.77316e31	0.929550	0.464775	−	0.885429i	$-0.346135\pi$
$953$	1.77316e31	0.929550	0.464775	+	0.885429i	$0.346135\pi$
$954$	0	0
$955$	−1.68091e30	−0.0862005
$956$	0	0
$957$	8.11759e30	0.407241
$958$	0	0
$959$	7.06821e30	0.346908
$960$	0	0
$961$	−1.56798e31	−0.752914
$962$	0	0
$963$	8.15324e30	0.383049
$964$	0	0
$965$	1.40616e30	0.0646394
$966$	0	0
$967$	1.43980e31	0.647624	0.323812	−	0.946121i	$-0.395035\pi$
$967$	1.43980e31	0.647624	0.323812	+	0.946121i	$0.395035\pi$
$968$	0	0
$969$	−4.74336e29	−0.0208779
$970$	0	0
$971$	1.69355e31	0.729452	0.364726	−	0.931115i	$-0.381163\pi$
$971$	1.69355e31	0.729452	0.364726	+	0.931115i	$0.381163\pi$
$972$	0	0
$973$	−2.37660e30	−0.100178
$974$	0	0
$975$	7.42185e30	0.306171
$976$	0	0
$977$	−6.75065e30	−0.272554	−0.136277	−	0.990671i	$-0.543514\pi$
$977$	−6.75065e30	−0.272554	−0.136277	+	0.990671i	$0.543514\pi$
$978$	0	0
$979$	4.08596e30	0.161464
$980$	0	0
$981$	8.81402e30	0.340918
$982$	0	0
$983$	1.83872e31	0.696153	0.348076	−	0.937466i	$-0.386835\pi$
$983$	1.83872e31	0.696153	0.348076	+	0.937466i	$0.386835\pi$
$984$	0	0
$985$	−7.41466e29	−0.0274796
$986$	0	0
$987$	7.85783e30	0.285084
$988$	0	0
$989$	−2.12629e31	−0.755199
$990$	0	0
$991$	4.23711e31	1.47332	0.736659	−	0.676264i	$-0.236402\pi$
$991$	4.23711e31	1.47332	0.736659	+	0.676264i	$0.236402\pi$
$992$	0	0
$993$	−2.22583e31	−0.757749
$994$	0	0
$995$	−3.34573e30	−0.111519
$996$	0	0
$997$	5.70834e31	1.86299	0.931495	−	0.363753i	$-0.118505\pi$
$997$	5.70834e31	1.86299	0.931495	+	0.363753i	$0.118505\pi$
$998$	0	0
$999$	5.04414e30	0.161194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.a.e.1.1		2
4.3	odd	2		8.22.a.a.1.2	✓	2
8.3	odd	2		64.22.a.k.1.1		2
8.5	even	2		64.22.a.h.1.2		2
12.11	even	2		72.22.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.a.1.2	✓	2	4.3	odd	2
16.22.a.e.1.1		2	1.1	even	1	trivial
64.22.a.h.1.2		2	8.5	even	2
64.22.a.k.1.1		2	8.3	odd	2
72.22.a.b.1.2		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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-62251.6 q^{3} -1.24528e6 q^{5} +2.63007e8 q^{7} -6.58509e9 q^{9} +O(q^{10})\)	\(q-62251.6 q^{3} -1.24528e6 q^{5} +2.63007e8 q^{7} -6.58509e9 q^{9} +3.37055e10 q^{11} +2.50845e11 q^{13} +7.75208e10 q^{15} +2.34664e11 q^{17} +3.24706e13 q^{19} -1.63726e13 q^{21} +9.63098e13 q^{23} -4.75286e14 q^{25} +1.06111e15 q^{27} -3.86879e15 q^{29} +2.26841e15 q^{31} -2.09822e15 q^{33} -3.27518e14 q^{35} +4.75366e15 q^{37} -1.56155e16 q^{39} +3.56442e16 q^{41} -2.20776e17 q^{43} +8.20030e15 q^{45} -4.79937e17 q^{47} -4.89373e17 q^{49} -1.46082e16 q^{51} -1.86349e18 q^{53} -4.19729e16 q^{55} -2.02135e18 q^{57} -1.23147e18 q^{59} -6.27167e18 q^{61} -1.73193e18 q^{63} -3.12373e17 q^{65} -1.72809e19 q^{67} -5.99544e18 q^{69} -3.13094e19 q^{71} +5.49389e19 q^{73} +2.95873e19 q^{75} +8.86480e18 q^{77} +8.34613e19 q^{79} +2.82681e18 q^{81} +1.22864e20 q^{83} -2.92223e17 q^{85} +2.40839e20 q^{87} +1.21225e20 q^{89} +6.59742e19 q^{91} -1.41212e20 q^{93} -4.04350e19 q^{95} +3.73762e20 q^{97} -2.21954e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) \) 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9	\( 2 q + 105432 q^{3} + 2108140 q^{5} - 444771792 q^{7} + 11072347578 q^{9} - 53806403320 q^{11} - 490366676932 q^{13} + 639834721680 q^{15} - 6593864672092 q^{17} - 19302397925320 q^{19} - 135055584824256 q^{21} + 409737865776272 q^{23} - 940878149007650 q^{25} + 22\!\cdots\!56 q^{27}+ \cdots - 17\!\cdots\!04 q^{99}+O(q^{100}) \) 2 * q + 105432 * q^3 + 2108140 * q^5 - 444771792 * q^7 + 11072347578 * q^9 - 53806403320 * q^11 - 490366676932 * q^13 + 639834721680 * q^15 - 6593864672092 * q^17 - 19302397925320 * q^19 - 135055584824256 * q^21 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2267939450073456 * q^27 - 2404787522145060 * q^29 + 8689907170559168 * q^31 - 16772541164782752 * q^33 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 139904623071296688 * q^39 + 68178038573558676 * q^41 - 264529652266004024 * q^43 + 67413140092548540 * q^45 - 426494411558622432 * q^47 - 546967577640131534 * q^49 - 1159640456555021520 * q^51 - 3055980275589518132 * q^53 - 335437371694825040 * q^55 - 10702822880368260192 * q^57 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 - 14229493501717343376 * q^63 - 2797969869756700760 * q^65 - 16674123174011538088 * q^67 + 46561318352256083136 * q^69 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 48484754640473875800 * q^75 + 70803926825553273024 * q^77 + 85280702218715897824 * q^79 + 20489906394726573618 * q^81 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 486328037598631978704 * q^87 - 59742932430695979660 * q^89 + 590588534777956016544 * q^91 + 935567489996727394560 * q^93 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97 - 1767190682286719892504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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