LMFDB - Embedded newform 16.28.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,28,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, 28, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 28 $
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:	$73.8968919741$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1059289}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 264822 $ x^2 - x - 264822
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-514.109$ 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)

$f(q)$	$=$	$q+3.00840e6 q^{3} +9.02501e8 q^{5} -3.76979e11 q^{7} +1.42486e12 q^{9} +1.19981e14 q^{11} +8.45735e14 q^{13} +2.71508e15 q^{15} -5.80089e16 q^{17} +2.28933e16 q^{19} -1.13410e18 q^{21} +4.11252e18 q^{23} -6.63607e18 q^{25} -1.86543e19 q^{27} -5.09330e19 q^{29} -2.28760e20 q^{31} +3.60949e20 q^{33} -3.40224e20 q^{35} -1.37094e21 q^{37} +2.54431e21 q^{39} +1.26146e21 q^{41} +1.57776e22 q^{43} +1.28593e21 q^{45} -1.74091e22 q^{47} +7.64006e22 q^{49} -1.74514e23 q^{51} -2.42990e23 q^{53} +1.08283e23 q^{55} +6.88720e22 q^{57} +4.33841e23 q^{59} -1.61997e24 q^{61} -5.37140e23 q^{63} +7.63276e23 q^{65} -3.04796e24 q^{67} +1.23721e25 q^{69} +1.33723e24 q^{71} -2.26071e25 q^{73} -1.99639e25 q^{75} -4.52302e25 q^{77} -2.50702e25 q^{79} -6.69849e25 q^{81} -2.43633e25 q^{83} -5.23531e25 q^{85} -1.53227e26 q^{87} +3.08582e26 q^{89} -3.18824e26 q^{91} -6.88202e26 q^{93} +2.06612e25 q^{95} -2.50167e26 q^{97} +1.70955e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 483720 q^{3} + 145079100 q^{5} - 60475251760 q^{7} + 173252390058 q^{9} + 24840277565400 q^{11} - 79026950880020 q^{13} + 46\!\cdots\!00 q^{15} - 57\!\cdots\!40 q^{17} + 35\!\cdots\!72 q^{19} - 19\!\cdots\!88 q^{21}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) $ 2 * q + 483720 * q^3 + 145079100 * q^5 - 60475251760 * q^7 + 173252390058 * q^9 + 24840277565400 * q^11 - 79026950880020 * q^13 + 4627325283375600 * q^15 - 57199393642550940 * q^17 + 355562290954316072 * q^19 - 1933171089411511488 * q^21 + 6588852862675601520 * q^23 - 13512966285544371250 * q^25 + 3757770863228771280 * q^27 + 50363331391885736268 * q^29 - 289592255497320916672 * q^31 + 601148166645981667680 * q^33 - 579950211055058474400 * q^35 - 1026380758945616198660 * q^37 + 4879030784008287487152 * q^39 - 9667004963486656782348 * q^41 + 13487141643503459284760 * q^43 + 2233924267093408305900 * q^45 - 46961105511875608404000 * q^47 + 110862765316290827124306 * q^49 - 176557790235135981293424 * q^51 + 4224695398452731028060 * q^53 + 180343991690673418674000 * q^55 - 771009892031359579225440 * q^57 + 1004029738899231296722104 * q^59 - 1726777676727260480588276 * q^61 - 933277108270869338684400 * q^63 + 1463710693249729982514600 * q^65 - 9033075655753326333646840 * q^67 + 6120160785644798981928384 * q^69 - 13203608118408237669065136 * q^71 - 18212101269545484387315020 * q^73 - 2602008494342196687165000 * q^75 - 75342425018265756076461120 * q^77 - 134061090639694576525024 * q^79 - 114023903391905644807269102 * q^81 - 113710918401726608059277400 * q^83 - 52966281741728089323184200 * q^85 - 408967478461534854560014800 * q^87 + 30590274319231503302052372 * q^89 - 611514423095313234144615968 * q^91 - 534620231259281913597438720 * q^93 - 231309527733675980899160400 * q^95 - 1444066674910179813599054780 * q^97 + 290033083248945697236886200 * 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
$3$	3.00840e6	1.08943	0.544714	−	0.838622i	$-0.316638\pi$
$3$	3.00840e6	1.08943	0.544714	+	0.838622i	$0.316638\pi$
$4$	0	0
$5$	9.02501e8	0.330638	0.165319	−	0.986240i	$-0.447135\pi$
$5$	9.02501e8	0.330638	0.165319	+	0.986240i	$0.447135\pi$
$6$	0	0
$7$	−3.76979e11	−1.47060	−0.735298	−	0.677744i	$-0.762958\pi$
$7$	−3.76979e11	−1.47060	−0.735298	+	0.677744i	$0.762958\pi$
$8$	0	0
$9$	1.42486e12	0.186852
$10$	0	0
$11$	1.19981e14	1.04788	0.523938	−	0.851756i	$-0.324462\pi$
$11$	1.19981e14	1.04788	0.523938	+	0.851756i	$0.324462\pi$
$12$	0	0
$13$	8.45735e14	0.774460	0.387230	−	0.921983i	$-0.373432\pi$
$13$	8.45735e14	0.774460	0.387230	+	0.921983i	$0.373432\pi$
$14$	0	0
$15$	2.71508e15	0.360206
$16$	0	0
$17$	−5.80089e16	−1.42048	−0.710239	−	0.703961i	$-0.751413\pi$
$17$	−5.80089e16	−1.42048	−0.710239	+	0.703961i	$0.751413\pi$
$18$	0	0
$19$	2.28933e16	0.124892	0.0624459	−	0.998048i	$-0.480110\pi$
$19$	2.28933e16	0.124892	0.0624459	+	0.998048i	$0.480110\pi$
$20$	0	0
$21$	−1.13410e18	−1.60211
$22$	0	0
$23$	4.11252e18	1.70131	0.850653	−	0.525728i	$-0.176207\pi$
$23$	4.11252e18	1.70131	0.850653	+	0.525728i	$0.176207\pi$
$24$	0	0
$25$	−6.63607e18	−0.890679
$26$	0	0
$27$	−1.86543e19	−0.885866
$28$	0	0
$29$	−5.09330e19	−0.921779	−0.460889	−	0.887458i	$-0.652470\pi$
$29$	−5.09330e19	−0.921779	−0.460889	+	0.887458i	$0.652470\pi$
$30$	0	0
$31$	−2.28760e20	−1.68266	−0.841332	−	0.540518i	$-0.818228\pi$
$31$	−2.28760e20	−1.68266	−0.841332	+	0.540518i	$0.818228\pi$
$32$	0	0
$33$	3.60949e20	1.14158
$34$	0	0
$35$	−3.40224e20	−0.486235
$36$	0	0
$37$	−1.37094e21	−0.925328	−0.462664	−	0.886534i	$-0.653106\pi$
$37$	−1.37094e21	−0.925328	−0.462664	+	0.886534i	$0.653106\pi$
$38$	0	0
$39$	2.54431e21	0.843718
$40$	0	0
$41$	1.26146e21	0.212957	0.106479	−	0.994315i	$-0.466042\pi$
$41$	1.26146e21	0.212957	0.106479	+	0.994315i	$0.466042\pi$
$42$	0	0
$43$	1.57776e22	1.40029	0.700144	−	0.714002i	$-0.253119\pi$
$43$	1.57776e22	1.40029	0.700144	+	0.714002i	$0.253119\pi$
$44$	0	0
$45$	1.28593e21	0.0617802
$46$	0	0
$47$	−1.74091e22	−0.465004	−0.232502	−	0.972596i	$-0.574691\pi$
$47$	−1.74091e22	−0.465004	−0.232502	+	0.972596i	$0.574691\pi$
$48$	0	0
$49$	7.64006e22	1.16265
$50$	0	0
$51$	−1.74514e23	−1.54751
$52$	0	0
$53$	−2.42990e23	−1.28193	−0.640964	−	0.767571i	$-0.721465\pi$
$53$	−2.42990e23	−1.28193	−0.640964	+	0.767571i	$0.721465\pi$
$54$	0	0
$55$	1.08283e23	0.346467
$56$	0	0
$57$	6.88720e22	0.136061
$58$	0	0
$59$	4.33841e23	0.538057	0.269029	−	0.963132i	$-0.413297\pi$
$59$	4.33841e23	0.538057	0.269029	+	0.963132i	$0.413297\pi$
$60$	0	0
$61$	−1.61997e24	−1.28102	−0.640508	−	0.767951i	$-0.721276\pi$
$61$	−1.61997e24	−1.28102	−0.640508	+	0.767951i	$0.721276\pi$
$62$	0	0
$63$	−5.37140e23	−0.274783
$64$	0	0
$65$	7.63276e23	0.256066
$66$	0	0
$67$	−3.04796e24	−0.679203	−0.339601	−	0.940569i	$-0.610292\pi$
$67$	−3.04796e24	−0.679203	−0.339601	+	0.940569i	$0.610292\pi$
$68$	0	0
$69$	1.23721e25	1.85345
$70$	0	0
$71$	1.33723e24	0.136212	0.0681062	−	0.997678i	$-0.478304\pi$
$71$	1.33723e24	0.136212	0.0681062	+	0.997678i	$0.478304\pi$
$72$	0	0
$73$	−2.26071e25	−1.58265	−0.791326	−	0.611394i	$-0.790609\pi$
$73$	−2.26071e25	−1.58265	−0.791326	+	0.611394i	$0.790609\pi$
$74$	0	0
$75$	−1.99639e25	−0.970330
$76$	0	0
$77$	−4.52302e25	−1.54100
$78$	0	0
$79$	−2.50702e25	−0.604217	−0.302108	−	0.953274i	$-0.597690\pi$
$79$	−2.50702e25	−0.604217	−0.302108	+	0.953274i	$0.597690\pi$
$80$	0	0
$81$	−6.69849e25	−1.15194
$82$	0	0
$83$	−2.43633e25	−0.301427	−0.150714	−	0.988577i	$-0.548157\pi$
$83$	−2.43633e25	−0.301427	−0.150714	+	0.988577i	$0.548157\pi$
$84$	0	0
$85$	−5.23531e25	−0.469664
$86$	0	0
$87$	−1.53227e26	−1.00421
$88$	0	0
$89$	3.08582e26	1.48801	0.744004	−	0.668175i	$-0.232924\pi$
$89$	3.08582e26	1.48801	0.744004	+	0.668175i	$0.232924\pi$
$90$	0	0
$91$	−3.18824e26	−1.13892
$92$	0	0
$93$	−6.88202e26	−1.83314
$94$	0	0
$95$	2.06612e25	0.0412939
$96$	0	0
$97$	−2.50167e26	−0.377408	−0.188704	−	0.982034i	$-0.560429\pi$
$97$	−2.50167e26	−0.377408	−0.188704	+	0.982034i	$0.560429\pi$
$98$	0	0
$99$	1.70955e26	0.195797
$100$	0	0
$101$	−1.27747e27	−1.11690	−0.558449	−	0.829539i	$-0.688603\pi$
$101$	−1.27747e27	−1.11690	−0.558449	+	0.829539i	$0.688603\pi$
$102$	0	0
$103$	−1.21454e27	−0.814910	−0.407455	−	0.913225i	$-0.633584\pi$
$103$	−1.21454e27	−0.814910	−0.407455	+	0.913225i	$0.633584\pi$
$104$	0	0
$105$	−1.02353e27	−0.529717
$106$	0	0
$107$	−1.12588e27	−0.451661	−0.225830	−	0.974167i	$-0.572510\pi$
$107$	−1.12588e27	−0.451661	−0.225830	+	0.974167i	$0.572510\pi$
$108$	0	0
$109$	−8.86361e26	−0.276919	−0.138460	−	0.990368i	$-0.544215\pi$
$109$	−8.86361e26	−0.276919	−0.138460	+	0.990368i	$0.544215\pi$
$110$	0	0
$111$	−4.12434e27	−1.00808
$112$	0	0
$113$	9.53256e27	1.83084	0.915421	−	0.402499i	$-0.131858\pi$
$113$	9.53256e27	1.83084	0.915421	+	0.402499i	$0.131858\pi$
$114$	0	0
$115$	3.71155e27	0.562516
$116$	0	0
$117$	1.20505e27	0.144709
$118$	0	0
$119$	2.18681e28	2.08895
$120$	0	0
$121$	1.28536e27	0.0980445
$122$	0	0
$123$	3.79499e27	0.232002
$124$	0	0
$125$	−1.27132e28	−0.625130
$126$	0	0
$127$	1.16152e27	0.0460972	0.0230486	−	0.999734i	$-0.492663\pi$
$127$	1.16152e27	0.0460972	0.0230486	+	0.999734i	$0.492663\pi$
$128$	0	0
$129$	4.74654e28	1.52551
$130$	0	0
$131$	2.58218e28	0.674255	0.337127	−	0.941459i	$-0.390545\pi$
$131$	2.58218e28	0.674255	0.337127	+	0.941459i	$0.390545\pi$
$132$	0	0
$133$	−8.63027e27	−0.183665
$134$	0	0
$135$	−1.68355e28	−0.292901
$136$	0	0
$137$	1.02664e29	1.46451	0.732253	−	0.681033i	$-0.238469\pi$
$137$	1.02664e29	1.46451	0.732253	+	0.681033i	$0.238469\pi$
$138$	0	0
$139$	−7.13626e28	−0.837088	−0.418544	−	0.908197i	$-0.637459\pi$
$139$	−7.13626e28	−0.837088	−0.418544	+	0.908197i	$0.637459\pi$
$140$	0	0
$141$	−5.23736e28	−0.506588
$142$	0	0
$143$	1.01472e29	0.811538
$144$	0	0
$145$	−4.59671e28	−0.304775
$146$	0	0
$147$	2.29843e29	1.26663
$148$	0	0
$149$	−4.81525e28	−0.221108	−0.110554	−	0.993870i	$-0.535262\pi$
$149$	−4.81525e28	−0.221108	−0.110554	+	0.993870i	$0.535262\pi$
$150$	0	0
$151$	2.75466e28	0.105652	0.0528261	−	0.998604i	$-0.483177\pi$
$151$	2.75466e28	0.105652	0.0528261	+	0.998604i	$0.483177\pi$
$152$	0	0
$153$	−8.26544e28	−0.265419
$154$	0	0
$155$	−2.06456e29	−0.556353
$156$	0	0
$157$	5.34419e29	1.21126	0.605629	−	0.795747i	$-0.292921\pi$
$157$	5.34419e29	1.21126	0.605629	+	0.795747i	$0.292921\pi$
$158$	0	0
$159$	−7.31010e29	−1.39657
$160$	0	0
$161$	−1.55033e30	−2.50193
$162$	0	0
$163$	−9.07370e29	−1.23951	−0.619757	−	0.784794i	$-0.712769\pi$
$163$	−9.07370e29	−1.23951	−0.619757	+	0.784794i	$0.712769\pi$
$164$	0	0
$165$	3.25757e29	0.377451
$166$	0	0
$167$	−8.15269e29	−0.802839	−0.401420	−	0.915894i	$-0.631483\pi$
$167$	−8.15269e29	−0.802839	−0.401420	+	0.915894i	$0.631483\pi$
$168$	0	0
$169$	−4.77266e29	−0.400212
$170$	0	0
$171$	3.26196e28	0.0233362
$172$	0	0
$173$	−1.82838e30	−1.11801	−0.559003	−	0.829166i	$-0.688816\pi$
$173$	−1.82838e30	−1.11801	−0.559003	+	0.829166i	$0.688816\pi$
$174$	0	0
$175$	2.50166e30	1.30983
$176$	0	0
$177$	1.30517e30	0.586174
$178$	0	0
$179$	3.64275e30	1.40577	0.702885	−	0.711304i	$-0.251895\pi$
$179$	3.64275e30	1.40577	0.702885	+	0.711304i	$0.251895\pi$
$180$	0	0
$181$	9.83969e29	0.326829	0.163414	−	0.986558i	$-0.447749\pi$
$181$	9.83969e29	0.326829	0.163414	+	0.986558i	$0.447749\pi$
$182$	0	0
$183$	−4.87352e30	−1.39557
$184$	0	0
$185$	−1.23728e30	−0.305948
$186$	0	0
$187$	−6.95995e30	−1.48849
$188$	0	0
$189$	7.03227e30	1.30275
$190$	0	0
$191$	1.23099e31	1.97836	0.989178	−	0.146721i	$-0.0468720\pi$
$191$	1.23099e31	1.97836	0.989178	+	0.146721i	$0.0468720\pi$
$192$	0	0
$193$	4.96173e30	0.692803	0.346402	−	0.938086i	$-0.387403\pi$
$193$	4.96173e30	0.692803	0.346402	+	0.938086i	$0.387403\pi$
$194$	0	0
$195$	2.29624e30	0.278965
$196$	0	0
$197$	1.42433e29	0.0150771	0.00753853	−	0.999972i	$-0.497600\pi$
$197$	1.42433e29	0.0150771	0.00753853	+	0.999972i	$0.497600\pi$
$198$	0	0
$199$	−8.43469e30	−0.779027	−0.389514	−	0.921021i	$-0.627357\pi$
$199$	−8.43469e30	−0.779027	−0.389514	+	0.921021i	$0.627357\pi$
$200$	0	0
$201$	−9.16947e30	−0.739942
$202$	0	0
$203$	1.92007e31	1.35556
$204$	0	0
$205$	1.13847e30	0.0704118
$206$	0	0
$207$	5.85975e30	0.317892
$208$	0	0
$209$	2.74675e30	0.130871
$210$	0	0
$211$	1.64953e31	0.691111	0.345555	−	0.938398i	$-0.387691\pi$
$211$	1.64953e31	0.691111	0.345555	+	0.938398i	$0.387691\pi$
$212$	0	0
$213$	4.02290e30	0.148394
$214$	0	0
$215$	1.42393e31	0.462988
$216$	0	0
$217$	8.62377e31	2.47452
$218$	0	0
$219$	−6.80110e31	−1.72419
$220$	0	0
$221$	−4.90602e31	−1.10010
$222$	0	0
$223$	−2.32346e31	−0.461339	−0.230670	−	0.973032i	$-0.574092\pi$
$223$	−2.32346e31	−0.461339	−0.230670	+	0.973032i	$0.574092\pi$
$224$	0	0
$225$	−9.45545e30	−0.166425
$226$	0	0
$227$	6.90063e31	1.07780	0.538902	−	0.842369i	$-0.318839\pi$
$227$	6.90063e31	1.07780	0.538902	+	0.842369i	$0.318839\pi$
$228$	0	0
$229$	−9.88419e31	−1.37139	−0.685697	−	0.727887i	$-0.740502\pi$
$229$	−9.88419e31	−1.37139	−0.685697	+	0.727887i	$0.740502\pi$
$230$	0	0
$231$	−1.36070e32	−1.67881
$232$	0	0
$233$	−8.98268e31	−0.986509	−0.493254	−	0.869885i	$-0.664193\pi$
$233$	−8.98268e31	−0.986509	−0.493254	+	0.869885i	$0.664193\pi$
$234$	0	0
$235$	−1.57118e31	−0.153748
$236$	0	0
$237$	−7.54213e31	−0.658250
$238$	0	0
$239$	4.92530e31	0.383761	0.191880	−	0.981418i	$-0.438541\pi$
$239$	4.92530e31	0.383761	0.191880	+	0.981418i	$0.438541\pi$
$240$	0	0
$241$	−2.98565e31	−0.207878	−0.103939	−	0.994584i	$-0.533145\pi$
$241$	−2.98565e31	−0.207878	−0.103939	+	0.994584i	$0.533145\pi$
$242$	0	0
$243$	−5.92670e31	−0.369087
$244$	0	0
$245$	6.89516e31	0.384417
$246$	0	0
$247$	1.93616e31	0.0967237
$248$	0	0
$249$	−7.32946e31	−0.328383
$250$	0	0
$251$	−2.50757e32	−1.00846	−0.504228	−	0.863570i	$-0.668223\pi$
$251$	−2.50757e32	−1.00846	−0.504228	+	0.863570i	$0.668223\pi$
$252$	0	0
$253$	4.93423e32	1.78276
$254$	0	0
$255$	−1.57499e32	−0.511664
$256$	0	0
$257$	−5.26158e30	−0.0153822	−0.00769111	−	0.999970i	$-0.502448\pi$
$257$	−5.26158e30	−0.0153822	−0.00769111	+	0.999970i	$0.502448\pi$
$258$	0	0
$259$	5.16816e32	1.36078
$260$	0	0
$261$	−7.25722e31	−0.172236
$262$	0	0
$263$	2.18683e32	0.468179	0.234090	−	0.972215i	$-0.424789\pi$
$263$	2.18683e32	0.468179	0.234090	+	0.972215i	$0.424789\pi$
$264$	0	0
$265$	−2.19299e32	−0.423854
$266$	0	0
$267$	9.28337e32	1.62108
$268$	0	0
$269$	5.35651e32	0.845720	0.422860	−	0.906195i	$-0.361026\pi$
$269$	5.35651e32	0.845720	0.422860	+	0.906195i	$0.361026\pi$
$270$	0	0
$271$	−3.03296e32	−0.433292	−0.216646	−	0.976250i	$-0.569512\pi$
$271$	−3.03296e32	−0.433292	−0.216646	+	0.976250i	$0.569512\pi$
$272$	0	0
$273$	−9.59149e32	−1.24077
$274$	0	0
$275$	−7.96200e32	−0.933321
$276$	0	0
$277$	−6.54386e31	−0.0695596	−0.0347798	−	0.999395i	$-0.511073\pi$
$277$	−6.54386e31	−0.0695596	−0.0347798	+	0.999395i	$0.511073\pi$
$278$	0	0
$279$	−3.25950e32	−0.314409
$280$	0	0
$281$	2.00731e33	1.75824	0.879121	−	0.476598i	$-0.158130\pi$
$281$	2.00731e33	1.75824	0.879121	+	0.476598i	$0.158130\pi$
$282$	0	0
$283$	2.05262e33	1.63378	0.816888	−	0.576796i	$-0.195697\pi$
$283$	2.05262e33	1.63378	0.816888	+	0.576796i	$0.195697\pi$
$284$	0	0
$285$	6.21570e31	0.0449867
$286$	0	0
$287$	−4.75545e32	−0.313174
$288$	0	0
$289$	1.69733e33	1.01776
$290$	0	0
$291$	−7.52601e32	−0.411158
$292$	0	0
$293$	2.23798e33	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$293$	2.23798e33	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$294$	0	0
$295$	3.91542e32	0.177902
$296$	0	0
$297$	−2.23815e33	−0.928278
$298$	0	0
$299$	3.47810e33	1.31759
$300$	0	0
$301$	−5.94783e33	−2.05926
$302$	0	0
$303$	−3.84315e33	−1.21678
$304$	0	0
$305$	−1.46203e33	−0.423553
$306$	0	0
$307$	1.56124e32	0.0414097	0.0207049	−	0.999786i	$-0.493409\pi$
$307$	1.56124e32	0.0414097	0.0207049	+	0.999786i	$0.493409\pi$
$308$	0	0
$309$	−3.65382e33	−0.887785
$310$	0	0
$311$	−5.69789e33	−1.26896	−0.634481	−	0.772939i	$-0.718786\pi$
$311$	−5.69789e33	−1.26896	−0.634481	+	0.772939i	$0.718786\pi$
$312$	0	0
$313$	3.46371e32	0.0707445	0.0353723	−	0.999374i	$-0.488738\pi$
$313$	3.46371e32	0.0707445	0.0353723	+	0.999374i	$0.488738\pi$
$314$	0	0
$315$	−4.84769e32	−0.0908537
$316$	0	0
$317$	6.10550e33	1.05056	0.525281	−	0.850929i	$-0.323960\pi$
$317$	6.10550e33	1.05056	0.525281	+	0.850929i	$0.323960\pi$
$318$	0	0
$319$	−6.11098e33	−0.965910
$320$	0	0
$321$	−3.38711e33	−0.492052
$322$	0	0
$323$	−1.32801e33	−0.177406
$324$	0	0
$325$	−5.61236e33	−0.689795
$326$	0	0
$327$	−2.66653e33	−0.301683
$328$	0	0
$329$	6.56288e33	0.683832
$330$	0	0
$331$	4.78257e33	0.459181	0.229590	−	0.973287i	$-0.426261\pi$
$331$	4.78257e33	0.459181	0.229590	+	0.973287i	$0.426261\pi$
$332$	0	0
$333$	−1.95339e33	−0.172899
$334$	0	0
$335$	−2.75079e33	−0.224570
$336$	0	0
$337$	1.28316e34	0.966666	0.483333	−	0.875437i	$-0.339426\pi$
$337$	1.28316e34	0.966666	0.483333	+	0.875437i	$0.339426\pi$
$338$	0	0
$339$	2.86777e34	1.99457
$340$	0	0
$341$	−2.74468e34	−1.76322
$342$	0	0
$343$	−4.02925e33	−0.239196
$344$	0	0
$345$	1.11658e34	0.612820
$346$	0	0
$347$	3.03623e34	1.54130	0.770649	−	0.637260i	$-0.219932\pi$
$347$	3.03623e34	1.54130	0.770649	+	0.637260i	$0.219932\pi$
$348$	0	0
$349$	2.98480e34	1.40208	0.701039	−	0.713123i	$-0.252720\pi$
$349$	2.98480e34	1.40208	0.701039	+	0.713123i	$0.252720\pi$
$350$	0	0
$351$	−1.57766e34	−0.686068
$352$	0	0
$353$	−1.86056e34	−0.749351	−0.374675	−	0.927156i	$-0.622246\pi$
$353$	−1.86056e34	−0.749351	−0.374675	+	0.927156i	$0.622246\pi$
$354$	0	0
$355$	1.20685e33	0.0450370
$356$	0	0
$357$	6.57881e34	2.27576
$358$	0	0
$359$	−5.56754e34	−1.78603	−0.893014	−	0.450029i	$-0.851413\pi$
$359$	−5.56754e34	−1.78603	−0.893014	+	0.450029i	$0.851413\pi$
$360$	0	0
$361$	−3.30765e34	−0.984402
$362$	0	0
$363$	3.86688e33	0.106812
$364$	0	0
$365$	−2.04029e34	−0.523285
$366$	0	0
$367$	1.68674e34	0.401843	0.200921	−	0.979607i	$-0.435606\pi$
$367$	1.68674e34	0.401843	0.200921	+	0.979607i	$0.435606\pi$
$368$	0	0
$369$	1.79740e33	0.0397915
$370$	0	0
$371$	9.16020e34	1.88520
$372$	0	0
$373$	7.35219e34	1.40717	0.703586	−	0.710610i	$-0.251581\pi$
$373$	7.35219e34	1.40717	0.703586	+	0.710610i	$0.251581\pi$
$374$	0	0
$375$	−3.82464e34	−0.681033
$376$	0	0
$377$	−4.30758e34	−0.713881
$378$	0	0
$379$	4.43586e34	0.684461	0.342231	−	0.939616i	$-0.388818\pi$
$379$	4.43586e34	0.684461	0.342231	+	0.939616i	$0.388818\pi$
$380$	0	0
$381$	3.49430e33	0.0502196
$382$	0	0
$383$	3.92129e34	0.525104	0.262552	−	0.964918i	$-0.415436\pi$
$383$	3.92129e34	0.525104	0.262552	+	0.964918i	$0.415436\pi$
$384$	0	0
$385$	−4.08203e34	−0.509514
$386$	0	0
$387$	2.24808e34	0.261646
$388$	0	0
$389$	−1.27767e35	−1.38708	−0.693539	−	0.720420i	$-0.743949\pi$
$389$	−1.27767e35	−1.38708	−0.693539	+	0.720420i	$0.743949\pi$
$390$	0	0
$391$	−2.38563e35	−2.41667
$392$	0	0
$393$	7.76822e34	0.734551
$394$	0	0
$395$	−2.26259e34	−0.199777
$396$	0	0
$397$	−1.02565e35	−0.845914	−0.422957	−	0.906150i	$-0.639008\pi$
$397$	−1.02565e35	−0.845914	−0.422957	+	0.906150i	$0.639008\pi$
$398$	0	0
$399$	−2.59633e34	−0.200090
$400$	0	0
$401$	2.02460e35	1.45844	0.729221	−	0.684278i	$-0.239883\pi$
$401$	2.02460e35	1.45844	0.729221	+	0.684278i	$0.239883\pi$
$402$	0	0
$403$	−1.93470e35	−1.30316
$404$	0	0
$405$	−6.04539e34	−0.380874
$406$	0	0
$407$	−1.64486e35	−0.969629
$408$	0	0
$409$	−1.85908e35	−1.02573	−0.512866	−	0.858469i	$-0.671416\pi$
$409$	−1.85908e35	−1.02573	−0.512866	+	0.858469i	$0.671416\pi$
$410$	0	0
$411$	3.08854e35	1.59547
$412$	0	0
$413$	−1.63549e35	−0.791265
$414$	0	0
$415$	−2.19879e34	−0.0996633
$416$	0	0
$417$	−2.14687e35	−0.911946
$418$	0	0
$419$	−3.36370e34	−0.133945	−0.0669727	−	0.997755i	$-0.521334\pi$
$419$	−3.36370e34	−0.133945	−0.0669727	+	0.997755i	$0.521334\pi$
$420$	0	0
$421$	−6.56415e33	−0.0245115	−0.0122558	−	0.999925i	$-0.503901\pi$
$421$	−6.56415e33	−0.0245115	−0.0122558	+	0.999925i	$0.503901\pi$
$422$	0	0
$423$	−2.48055e34	−0.0868867
$424$	0	0
$425$	3.84952e35	1.26519
$426$	0	0
$427$	6.10695e35	1.88386
$428$	0	0
$429$	3.05267e35	0.884112
$430$	0	0
$431$	1.92022e35	0.522285	0.261143	−	0.965300i	$-0.415901\pi$
$431$	1.92022e35	0.522285	0.261143	+	0.965300i	$0.415901\pi$
$432$	0	0
$433$	2.53533e35	0.647810	0.323905	−	0.946090i	$-0.395004\pi$
$433$	2.53533e35	0.647810	0.323905	+	0.946090i	$0.395004\pi$
$434$	0	0
$435$	−1.38287e35	−0.332030
$436$	0	0
$437$	9.41490e34	0.212479
$438$	0	0
$439$	1.34689e35	0.285799	0.142900	−	0.989737i	$-0.454357\pi$
$439$	1.34689e35	0.285799	0.142900	+	0.989737i	$0.454357\pi$
$440$	0	0
$441$	1.08860e35	0.217244
$442$	0	0
$443$	−4.59219e35	−0.862124	−0.431062	−	0.902322i	$-0.641861\pi$
$443$	−4.59219e35	−0.862124	−0.431062	+	0.902322i	$0.641861\pi$
$444$	0	0
$445$	2.78495e35	0.491992
$446$	0	0
$447$	−1.44862e35	−0.240881
$448$	0	0
$449$	3.59402e35	0.562672	0.281336	−	0.959609i	$-0.409222\pi$
$449$	3.59402e35	0.562672	0.281336	+	0.959609i	$0.409222\pi$
$450$	0	0
$451$	1.51351e35	0.223153
$452$	0	0
$453$	8.28710e34	0.115100
$454$	0	0
$455$	−2.87739e35	−0.376569
$456$	0	0
$457$	−1.11556e36	−1.37602	−0.688008	−	0.725704i	$-0.741514\pi$
$457$	−1.11556e36	−1.37602	−0.688008	+	0.725704i	$0.741514\pi$
$458$	0	0
$459$	1.08212e36	1.25835
$460$	0	0
$461$	−7.21038e35	−0.790670	−0.395335	−	0.918537i	$-0.629372\pi$
$461$	−7.21038e35	−0.790670	−0.395335	+	0.918537i	$0.629372\pi$
$462$	0	0
$463$	1.02072e36	1.05575	0.527876	−	0.849321i	$-0.322988\pi$
$463$	1.02072e36	1.05575	0.527876	+	0.849321i	$0.322988\pi$
$464$	0	0
$465$	−6.21102e35	−0.606106
$466$	0	0
$467$	−1.40942e36	−1.29796	−0.648980	−	0.760805i	$-0.724804\pi$
$467$	−1.40942e36	−1.29796	−0.648980	+	0.760805i	$0.724804\pi$
$468$	0	0
$469$	1.14902e36	0.998832
$470$	0	0
$471$	1.60775e36	1.31958
$472$	0	0
$473$	1.89301e36	1.46733
$474$	0	0
$475$	−1.51921e35	−0.111238
$476$	0	0
$477$	−3.46225e35	−0.239531
$478$	0	0
$479$	8.04487e34	0.0526005	0.0263002	−	0.999654i	$-0.491627\pi$
$479$	8.04487e34	0.0526005	0.0263002	+	0.999654i	$0.491627\pi$
$480$	0	0
$481$	−1.15945e36	−0.716629
$482$	0	0
$483$	−4.66402e36	−2.72567
$484$	0	0
$485$	−2.25776e35	−0.124785
$486$	0	0
$487$	−1.44084e36	−0.753310	−0.376655	−	0.926354i	$-0.622926\pi$
$487$	−1.44084e36	−0.753310	−0.376655	+	0.926354i	$0.622926\pi$
$488$	0	0
$489$	−2.72973e36	−1.35036
$490$	0	0
$491$	2.03452e36	0.952494	0.476247	−	0.879311i	$-0.341997\pi$
$491$	2.03452e36	0.952494	0.476247	+	0.879311i	$0.341997\pi$
$492$	0	0
$493$	2.95457e36	1.30937
$494$	0	0
$495$	1.54287e35	0.0647380
$496$	0	0
$497$	−5.04105e35	−0.200313
$498$	0	0
$499$	−2.09784e36	−0.789615	−0.394808	−	0.918764i	$-0.629189\pi$
$499$	−2.09784e36	−0.789615	−0.394808	+	0.918764i	$0.629189\pi$
$500$	0	0
$501$	−2.45265e36	−0.874635
$502$	0	0
$503$	−3.01251e35	−0.101803	−0.0509016	−	0.998704i	$-0.516209\pi$
$503$	−3.01251e35	−0.101803	−0.0509016	+	0.998704i	$0.516209\pi$
$504$	0	0
$505$	−1.15292e36	−0.369289
$506$	0	0
$507$	−1.43581e36	−0.436002
$508$	0	0
$509$	−3.84477e36	−1.10708	−0.553541	−	0.832822i	$-0.686724\pi$
$509$	−3.84477e36	−1.10708	−0.553541	+	0.832822i	$0.686724\pi$
$510$	0	0
$511$	8.52239e36	2.32744
$512$	0	0
$513$	−4.27058e35	−0.110637
$514$	0	0
$515$	−1.09612e36	−0.269440
$516$	0	0
$517$	−2.08876e36	−0.487266
$518$	0	0
$519$	−5.50049e36	−1.21799
$520$	0	0
$521$	5.89950e35	0.124024	0.0620120	−	0.998075i	$-0.480248\pi$
$521$	5.89950e35	0.124024	0.0620120	+	0.998075i	$0.480248\pi$
$522$	0	0
$523$	−8.51690e36	−1.70023	−0.850117	−	0.526594i	$-0.823469\pi$
$523$	−8.51690e36	−1.70023	−0.850117	+	0.526594i	$0.823469\pi$
$524$	0	0
$525$	7.52598e36	1.42696
$526$	0	0
$527$	1.32701e37	2.39019
$528$	0	0
$529$	1.10696e37	1.89444
$530$	0	0
$531$	6.18161e35	0.100537
$532$	0	0
$533$	1.06686e36	0.164927
$534$	0	0
$535$	−1.01611e36	−0.149336
$536$	0	0
$537$	1.09589e37	1.53148
$538$	0	0
$539$	9.16660e36	1.21832
$540$	0	0
$541$	−8.17889e36	−1.03403	−0.517013	−	0.855978i	$-0.672956\pi$
$541$	−8.17889e36	−1.03403	−0.517013	+	0.855978i	$0.672956\pi$
$542$	0	0
$543$	2.96017e36	0.356056
$544$	0	0
$545$	−7.99941e35	−0.0915599
$546$	0	0
$547$	2.25930e36	0.246119	0.123059	−	0.992399i	$-0.460729\pi$
$547$	2.25930e36	0.246119	0.123059	+	0.992399i	$0.460729\pi$
$548$	0	0
$549$	−2.30823e36	−0.239360
$550$	0	0
$551$	−1.16602e36	−0.115123
$552$	0	0
$553$	9.45095e36	0.888559
$554$	0	0
$555$	−3.72222e36	−0.333309
$556$	0	0
$557$	−1.44303e37	−1.23092	−0.615462	−	0.788167i	$-0.711030\pi$
$557$	−1.44303e37	−1.23092	−0.615462	+	0.788167i	$0.711030\pi$
$558$	0	0
$559$	1.33437e37	1.08447
$560$	0	0
$561$	−2.09383e37	−1.62160
$562$	0	0
$563$	−2.56319e37	−1.89198	−0.945992	−	0.324191i	$-0.894908\pi$
$563$	−2.56319e37	−1.89198	−0.945992	+	0.324191i	$0.894908\pi$
$564$	0	0
$565$	8.60314e36	0.605345
$566$	0	0
$567$	2.52519e37	1.69404
$568$	0	0
$569$	−1.26520e37	−0.809366	−0.404683	−	0.914457i	$-0.632618\pi$
$569$	−1.26520e37	−0.809366	−0.404683	+	0.914457i	$0.632618\pi$
$570$	0	0
$571$	−1.18072e37	−0.720376	−0.360188	−	0.932880i	$-0.617287\pi$
$571$	−1.18072e37	−0.720376	−0.360188	+	0.932880i	$0.617287\pi$
$572$	0	0
$573$	3.70331e37	2.15527
$574$	0	0
$575$	−2.72910e37	−1.51532
$576$	0	0
$577$	−7.88254e36	−0.417631	−0.208816	−	0.977955i	$-0.566961\pi$
$577$	−7.88254e36	−0.417631	−0.208816	+	0.977955i	$0.566961\pi$
$578$	0	0
$579$	1.49269e37	0.754758
$580$	0	0
$581$	9.18446e36	0.443278
$582$	0	0
$583$	−2.91541e37	−1.34330
$584$	0	0
$585$	1.08756e36	0.0478463
$586$	0	0
$587$	1.23694e36	0.0519677	0.0259838	−	0.999662i	$-0.491728\pi$
$587$	1.23694e36	0.0519677	0.0259838	+	0.999662i	$0.491728\pi$
$588$	0	0
$589$	−5.23707e36	−0.210151
$590$	0	0
$591$	4.28495e35	0.0164254
$592$	0	0
$593$	−1.16613e37	−0.427081	−0.213540	−	0.976934i	$-0.568499\pi$
$593$	−1.16613e37	−0.427081	−0.213540	+	0.976934i	$0.568499\pi$
$594$	0	0
$595$	1.97360e37	0.690686
$596$	0	0
$597$	−2.53749e37	−0.848693
$598$	0	0
$599$	3.54285e35	0.0113264	0.00566319	−	0.999984i	$-0.498197\pi$
$599$	3.54285e35	0.0113264	0.00566319	+	0.999984i	$0.498197\pi$
$600$	0	0
$601$	−6.27911e37	−1.91908	−0.959539	−	0.281575i	$-0.909143\pi$
$601$	−6.27911e37	−1.91908	−0.959539	+	0.281575i	$0.909143\pi$
$602$	0	0
$603$	−4.34290e36	−0.126910
$604$	0	0
$605$	1.16004e36	0.0324172
$606$	0	0
$607$	−6.16722e36	−0.164832	−0.0824161	−	0.996598i	$-0.526264\pi$
$607$	−6.16722e36	−0.164832	−0.0824161	+	0.996598i	$0.526264\pi$
$608$	0	0
$609$	5.77633e37	1.47679
$610$	0	0
$611$	−1.47235e37	−0.360127
$612$	0	0
$613$	−1.58191e37	−0.370225	−0.185112	−	0.982717i	$-0.559265\pi$
$613$	−1.58191e37	−0.370225	−0.185112	+	0.982717i	$0.559265\pi$
$614$	0	0
$615$	3.42498e36	0.0767085
$616$	0	0
$617$	−6.89426e35	−0.0147788	−0.00738938	−	0.999973i	$-0.502352\pi$
$617$	−6.89426e35	−0.0147788	−0.00738938	+	0.999973i	$0.502352\pi$
$618$	0	0
$619$	5.16416e37	1.05968	0.529842	−	0.848097i	$-0.322251\pi$
$619$	5.16416e37	1.05968	0.529842	+	0.848097i	$0.322251\pi$
$620$	0	0
$621$	−7.67162e37	−1.50713
$622$	0	0
$623$	−1.16329e38	−2.18826
$624$	0	0
$625$	3.79689e37	0.683987
$626$	0	0
$627$	8.26331e36	0.142575
$628$	0	0
$629$	7.95269e37	1.31441
$630$	0	0
$631$	2.90903e37	0.460630	0.230315	−	0.973116i	$-0.426024\pi$
$631$	2.90903e37	0.460630	0.230315	+	0.973116i	$0.426024\pi$
$632$	0	0
$633$	4.96245e37	0.752915
$634$	0	0
$635$	1.04827e36	0.0152415
$636$	0	0
$637$	6.46147e37	0.900428
$638$	0	0
$639$	1.90535e36	0.0254515
$640$	0	0
$641$	−3.34459e37	−0.428311	−0.214155	−	0.976800i	$-0.568700\pi$
$641$	−3.34459e37	−0.428311	−0.214155	+	0.976800i	$0.568700\pi$
$642$	0	0
$643$	5.66150e37	0.695158	0.347579	−	0.937651i	$-0.387004\pi$
$643$	5.66150e37	0.695158	0.347579	+	0.937651i	$0.387004\pi$
$644$	0	0
$645$	4.28375e37	0.504392
$646$	0	0
$647$	6.66099e37	0.752196	0.376098	−	0.926580i	$-0.377266\pi$
$647$	6.66099e37	0.752196	0.376098	+	0.926580i	$0.377266\pi$
$648$	0	0
$649$	5.20525e37	0.563817
$650$	0	0
$651$	2.59437e38	2.69581
$652$	0	0
$653$	5.83600e36	0.0581818	0.0290909	−	0.999577i	$-0.490739\pi$
$653$	5.83600e36	0.0581818	0.0290909	+	0.999577i	$0.490739\pi$
$654$	0	0
$655$	2.33042e37	0.222934
$656$	0	0
$657$	−3.22118e37	−0.295721
$658$	0	0
$659$	6.15562e37	0.542399	0.271200	−	0.962523i	$-0.412580\pi$
$659$	6.15562e37	0.542399	0.271200	+	0.962523i	$0.412580\pi$
$660$	0	0
$661$	−9.28033e37	−0.784953	−0.392476	−	0.919762i	$-0.628382\pi$
$661$	−9.28033e37	−0.784953	−0.392476	+	0.919762i	$0.628382\pi$
$662$	0	0
$663$	−1.47593e38	−1.19848
$664$	0	0
$665$	−7.78883e36	−0.0607267
$666$	0	0
$667$	−2.09463e38	−1.56823
$668$	0	0
$669$	−6.98990e37	−0.502595
$670$	0	0
$671$	−1.94365e38	−1.34235
$672$	0	0
$673$	1.13009e38	0.749736	0.374868	−	0.927078i	$-0.377688\pi$
$673$	1.13009e38	0.749736	0.374868	+	0.927078i	$0.377688\pi$
$674$	0	0
$675$	1.23791e38	0.789022
$676$	0	0
$677$	−1.47748e38	−0.904844	−0.452422	−	0.891804i	$-0.649440\pi$
$677$	−1.47748e38	−0.904844	−0.452422	+	0.891804i	$0.649440\pi$
$678$	0	0
$679$	9.43075e37	0.555015
$680$	0	0
$681$	2.07598e38	1.17419
$682$	0	0
$683$	−9.59633e37	−0.521706	−0.260853	−	0.965379i	$-0.584004\pi$
$683$	−9.59633e37	−0.521706	−0.260853	+	0.965379i	$0.584004\pi$
$684$	0	0
$685$	9.26544e37	0.484221
$686$	0	0
$687$	−2.97356e38	−1.49403
$688$	0	0
$689$	−2.05505e38	−0.992803
$690$	0	0
$691$	1.85005e38	0.859467	0.429733	−	0.902956i	$-0.358608\pi$
$691$	1.85005e38	0.859467	0.429733	+	0.902956i	$0.358608\pi$
$692$	0	0
$693$	−6.44464e37	−0.287939
$694$	0	0
$695$	−6.44048e37	−0.276773
$696$	0	0
$697$	−7.31762e37	−0.302501
$698$	0	0
$699$	−2.70235e38	−1.07473
$700$	0	0
$701$	−2.12622e38	−0.813608	−0.406804	−	0.913515i	$-0.633357\pi$
$701$	−2.12622e38	−0.813608	−0.406804	+	0.913515i	$0.633357\pi$
$702$	0	0
$703$	−3.13853e37	−0.115566
$704$	0	0
$705$	−4.72672e37	−0.167497
$706$	0	0
$707$	4.81580e38	1.64251
$708$	0	0
$709$	2.32594e38	0.763614	0.381807	−	0.924242i	$-0.375302\pi$
$709$	2.32594e38	0.763614	0.381807	+	0.924242i	$0.375302\pi$
$710$	0	0
$711$	−3.57215e37	−0.112899
$712$	0	0
$713$	−9.40781e38	−2.86273
$714$	0	0
$715$	9.15784e37	0.268325
$716$	0	0
$717$	1.48172e38	0.418080
$718$	0	0
$719$	1.04280e38	0.283375	0.141688	−	0.989911i	$-0.454747\pi$
$719$	1.04280e38	0.283375	0.141688	+	0.989911i	$0.454747\pi$
$720$	0	0
$721$	4.57856e38	1.19840
$722$	0	0
$723$	−8.98203e37	−0.226468
$724$	0	0
$725$	3.37995e38	0.821009
$726$	0	0
$727$	−8.31544e37	−0.194613	−0.0973063	−	0.995254i	$-0.531023\pi$
$727$	−8.31544e37	−0.194613	−0.0973063	+	0.995254i	$0.531023\pi$
$728$	0	0
$729$	3.32501e38	0.749845
$730$	0	0
$731$	−9.15243e38	−1.98908
$732$	0	0
$733$	1.51696e38	0.317738	0.158869	−	0.987300i	$-0.449215\pi$
$733$	1.51696e38	0.317738	0.158869	+	0.987300i	$0.449215\pi$
$734$	0	0
$735$	2.07434e38	0.418794
$736$	0	0
$737$	−3.65696e38	−0.711720
$738$	0	0
$739$	2.89991e38	0.544107	0.272053	−	0.962282i	$-0.412297\pi$
$739$	2.89991e38	0.544107	0.272053	+	0.962282i	$0.412297\pi$
$740$	0	0
$741$	5.82474e37	0.105373
$742$	0	0
$743$	4.53556e37	0.0791193	0.0395596	−	0.999217i	$-0.487404\pi$
$743$	4.53556e37	0.0791193	0.0395596	+	0.999217i	$0.487404\pi$
$744$	0	0
$745$	−4.34577e37	−0.0731066
$746$	0	0
$747$	−3.47142e37	−0.0563222
$748$	0	0
$749$	4.24434e38	0.664211
$750$	0	0
$751$	−8.41113e37	−0.126974	−0.0634870	−	0.997983i	$-0.520222\pi$
$751$	−8.41113e37	−0.126974	−0.0634870	+	0.997983i	$0.520222\pi$
$752$	0	0
$753$	−7.54376e38	−1.09864
$754$	0	0
$755$	2.48608e37	0.0349326
$756$	0	0
$757$	−1.31808e38	−0.178709	−0.0893544	−	0.996000i	$-0.528480\pi$
$757$	−1.31808e38	−0.178709	−0.0893544	+	0.996000i	$0.528480\pi$
$758$	0	0
$759$	1.48441e39	1.94218
$760$	0	0
$761$	1.16038e39	1.46524	0.732618	−	0.680640i	$-0.238298\pi$
$761$	1.16038e39	1.46524	0.732618	+	0.680640i	$0.238298\pi$
$762$	0	0
$763$	3.34139e38	0.407236
$764$	0	0
$765$	−7.45956e37	−0.0877574
$766$	0	0
$767$	3.66914e38	0.416704
$768$	0	0
$769$	7.10302e38	0.778821	0.389410	−	0.921064i	$-0.372679\pi$
$769$	7.10302e38	0.778821	0.389410	+	0.921064i	$0.372679\pi$
$770$	0	0
$771$	−1.58289e37	−0.0167578
$772$	0	0
$773$	−1.17473e39	−1.20092	−0.600461	−	0.799654i	$-0.705016\pi$
$773$	−1.17473e39	−1.20092	−0.600461	+	0.799654i	$0.705016\pi$
$774$	0	0
$775$	1.51807e39	1.49871
$776$	0	0
$777$	1.55479e39	1.48247
$778$	0	0
$779$	2.88790e37	0.0265966
$780$	0	0
$781$	1.60441e38	0.142734
$782$	0	0
$783$	9.50120e38	0.816572
$784$	0	0
$785$	4.82314e38	0.400488
$786$	0	0
$787$	−2.33439e39	−1.87290	−0.936449	−	0.350805i	$-0.885908\pi$
$787$	−2.33439e39	−1.87290	−0.936449	+	0.350805i	$0.885908\pi$
$788$	0	0
$789$	6.57885e38	0.510047
$790$	0	0
$791$	−3.59357e39	−2.69243
$792$	0	0
$793$	−1.37007e39	−0.992096
$794$	0	0
$795$	−6.59737e38	−0.461758
$796$	0	0
$797$	−1.54886e39	−1.04791	−0.523957	−	0.851745i	$-0.675545\pi$
$797$	−1.54886e39	−1.04791	−0.523957	+	0.851745i	$0.675545\pi$
$798$	0	0
$799$	1.00989e39	0.660527
$800$	0	0
$801$	4.39685e38	0.278037
$802$	0	0
$803$	−2.71241e39	−1.65842
$804$	0	0
$805$	−1.39918e39	−0.827234
$806$	0	0
$807$	1.61145e39	0.921350
$808$	0	0
$809$	−8.27333e38	−0.457484	−0.228742	−	0.973487i	$-0.573461\pi$
$809$	−8.27333e38	−0.457484	−0.228742	+	0.973487i	$0.573461\pi$
$810$	0	0
$811$	2.60639e39	1.39399	0.696993	−	0.717078i	$-0.254521\pi$
$811$	2.60639e39	1.39399	0.696993	+	0.717078i	$0.254521\pi$
$812$	0	0
$813$	−9.12434e38	−0.472040
$814$	0	0
$815$	−8.18902e38	−0.409830
$816$	0	0
$817$	3.61201e38	0.174885
$818$	0	0
$819$	−4.54278e38	−0.212809
$820$	0	0
$821$	−1.98954e39	−0.901822	−0.450911	−	0.892569i	$-0.648901\pi$
$821$	−1.98954e39	−0.901822	−0.450911	+	0.892569i	$0.648901\pi$
$822$	0	0
$823$	2.42547e39	1.06389	0.531947	−	0.846778i	$-0.321460\pi$
$823$	2.42547e39	1.06389	0.531947	+	0.846778i	$0.321460\pi$
$824$	0	0
$825$	−2.39529e39	−1.01679
$826$	0	0
$827$	−4.67442e39	−1.92045	−0.960226	−	0.279226i	$-0.909922\pi$
$827$	−4.67442e39	−1.92045	−0.960226	+	0.279226i	$0.909922\pi$
$828$	0	0
$829$	3.96570e39	1.57701	0.788504	−	0.615030i	$-0.210856\pi$
$829$	3.96570e39	1.57701	0.788504	+	0.615030i	$0.210856\pi$
$830$	0	0
$831$	−1.96865e38	−0.0757801
$832$	0	0
$833$	−4.43192e39	−1.65152
$834$	0	0
$835$	−7.35781e38	−0.265449
$836$	0	0
$837$	4.26736e39	1.49062
$838$	0	0
$839$	5.00134e38	0.169161	0.0845804	−	0.996417i	$-0.473045\pi$
$839$	5.00134e38	0.169161	0.0845804	+	0.996417i	$0.473045\pi$
$840$	0	0
$841$	−4.58959e38	−0.150324
$842$	0	0
$843$	6.03878e39	1.91548
$844$	0	0
$845$	−4.30733e38	−0.132325
$846$	0	0
$847$	−4.84554e38	−0.144184
$848$	0	0
$849$	6.17511e39	1.77988
$850$	0	0
$851$	−5.63803e39	−1.57427
$852$	0	0
$853$	1.96601e39	0.531831	0.265915	−	0.963996i	$-0.414326\pi$
$853$	1.96601e39	0.531831	0.265915	+	0.963996i	$0.414326\pi$
$854$	0	0
$855$	2.94392e37	0.00771584
$856$	0	0
$857$	1.84272e39	0.467969	0.233985	−	0.972240i	$-0.424823\pi$
$857$	1.84272e39	0.467969	0.233985	+	0.972240i	$0.424823\pi$
$858$	0	0
$859$	−2.19919e39	−0.541197	−0.270598	−	0.962692i	$-0.587222\pi$
$859$	−2.19919e39	−0.541197	−0.270598	+	0.962692i	$0.587222\pi$
$860$	0	0
$861$	−1.43063e39	−0.341181
$862$	0	0
$863$	2.64850e39	0.612144	0.306072	−	0.952008i	$-0.400985\pi$
$863$	2.64850e39	0.612144	0.306072	+	0.952008i	$0.400985\pi$
$864$	0	0
$865$	−1.65011e39	−0.369655
$866$	0	0
$867$	5.10623e39	1.10877
$868$	0	0
$869$	−3.00794e39	−0.633144
$870$	0	0
$871$	−2.57776e39	−0.526015
$872$	0	0
$873$	−3.56451e38	−0.0705193
$874$	0	0
$875$	4.79261e39	0.919313
$876$	0	0
$877$	8.30033e38	0.154384	0.0771918	−	0.997016i	$-0.475405\pi$
$877$	8.30033e38	0.154384	0.0771918	+	0.997016i	$0.475405\pi$
$878$	0	0
$879$	6.73273e39	1.21434
$880$	0	0
$881$	−4.48294e39	−0.784130	−0.392065	−	0.919938i	$-0.628239\pi$
$881$	−4.48294e39	−0.784130	−0.392065	+	0.919938i	$0.628239\pi$
$882$	0	0
$883$	1.55764e38	0.0264239	0.0132119	−	0.999913i	$-0.495794\pi$
$883$	1.55764e38	0.0264239	0.0132119	+	0.999913i	$0.495794\pi$
$884$	0	0
$885$	1.17791e39	0.193811
$886$	0	0
$887$	1.61984e37	0.00258525	0.00129262	−	0.999999i	$-0.499589\pi$
$887$	1.61984e37	0.00258525	0.00129262	+	0.999999i	$0.499589\pi$
$888$	0	0
$889$	−4.37867e38	−0.0677904
$890$	0	0
$891$	−8.03689e39	−1.20709
$892$	0	0
$893$	−3.98552e38	−0.0580751
$894$	0	0
$895$	3.28759e39	0.464800
$896$	0	0
$897$	1.04635e40	1.43542
$898$	0	0
$899$	1.16515e40	1.55104
$900$	0	0
$901$	1.40956e40	1.82095
$902$	0	0
$903$	−1.78934e40	−2.24341
$904$	0	0
$905$	8.88032e38	0.108062
$906$	0	0
$907$	1.48316e40	1.75182	0.875912	−	0.482472i	$-0.160261\pi$
$907$	1.48316e40	1.75182	0.875912	+	0.482472i	$0.160261\pi$
$908$	0	0
$909$	−1.82022e39	−0.208694
$910$	0	0
$911$	7.22078e38	0.0803685	0.0401843	−	0.999192i	$-0.487206\pi$
$911$	7.22078e38	0.0803685	0.0401843	+	0.999192i	$0.487206\pi$
$912$	0	0
$913$	−2.92313e39	−0.315858
$914$	0	0
$915$	−4.39836e39	−0.461430
$916$	0	0
$917$	−9.73427e39	−0.991556
$918$	0	0
$919$	−1.77181e40	−1.75250	−0.876248	−	0.481861i	$-0.839961\pi$
$919$	−1.77181e40	−1.75250	−0.876248	+	0.481861i	$0.839961\pi$
$920$	0	0
$921$	4.69684e38	0.0451129
$922$	0	0
$923$	1.13094e39	0.105491
$924$	0	0
$925$	9.09767e39	0.824170
$926$	0	0
$927$	−1.73055e39	−0.152267
$928$	0	0
$929$	−8.24704e39	−0.704832	−0.352416	−	0.935843i	$-0.614640\pi$
$929$	−8.24704e39	−0.704832	−0.352416	+	0.935843i	$0.614640\pi$
$930$	0	0
$931$	1.74906e39	0.145206
$932$	0	0
$933$	−1.71415e40	−1.38244
$934$	0	0
$935$	−6.28136e39	−0.492149
$936$	0	0
$937$	−1.37250e40	−1.04478	−0.522392	−	0.852705i	$-0.674960\pi$
$937$	−1.37250e40	−1.04478	−0.522392	+	0.852705i	$0.674960\pi$
$938$	0	0
$939$	1.04202e39	0.0770710
$940$	0	0
$941$	−1.35387e40	−0.973012	−0.486506	−	0.873677i	$-0.661729\pi$
$941$	−1.35387e40	−0.973012	−0.486506	+	0.873677i	$0.661729\pi$
$942$	0	0
$943$	5.18780e39	0.362306
$944$	0	0
$945$	6.34663e39	0.430739
$946$	0	0
$947$	−1.23594e40	−0.815218	−0.407609	−	0.913156i	$-0.633637\pi$
$947$	−1.23594e40	−0.815218	−0.407609	+	0.913156i	$0.633637\pi$
$948$	0	0
$949$	−1.91196e40	−1.22570
$950$	0	0
$951$	1.83678e40	1.14451
$952$	0	0
$953$	1.59950e40	0.968794	0.484397	−	0.874848i	$-0.339039\pi$
$953$	1.59950e40	0.968794	0.484397	+	0.874848i	$0.339039\pi$
$954$	0	0
$955$	1.11097e40	0.654119
$956$	0	0
$957$	−1.83843e40	−1.05229
$958$	0	0
$959$	−3.87022e40	−2.15370
$960$	0	0
$961$	3.38485e40	1.83136
$962$	0	0
$963$	−1.60422e39	−0.0843936
$964$	0	0
$965$	4.47797e39	0.229067
$966$	0	0
$967$	−2.71147e40	−1.34880	−0.674399	−	0.738367i	$-0.735597\pi$
$967$	−2.71147e40	−1.34880	−0.674399	+	0.738367i	$0.735597\pi$
$968$	0	0
$969$	−3.99519e39	−0.193271
$970$	0	0
$971$	6.75635e39	0.317872	0.158936	−	0.987289i	$-0.449194\pi$
$971$	6.75635e39	0.317872	0.158936	+	0.987289i	$0.449194\pi$
$972$	0	0
$973$	2.69022e40	1.23102
$974$	0	0
$975$	−1.68842e40	−0.751481
$976$	0	0
$977$	−2.46931e40	−1.06905	−0.534526	−	0.845152i	$-0.679510\pi$
$977$	−2.46931e40	−1.06905	−0.534526	+	0.845152i	$0.679510\pi$
$978$	0	0
$979$	3.70239e40	1.55925
$980$	0	0
$981$	−1.26294e39	−0.0517428
$982$	0	0
$983$	9.26756e39	0.369397	0.184698	−	0.982795i	$-0.440869\pi$
$983$	9.26756e39	0.369397	0.184698	+	0.982795i	$0.440869\pi$
$984$	0	0
$985$	1.28546e38	0.00498504
$986$	0	0
$987$	1.97437e40	0.744986
$988$	0	0
$989$	6.48858e40	2.38232
$990$	0	0
$991$	5.13676e40	1.83525	0.917625	−	0.397448i	$-0.130104\pi$
$991$	5.13676e40	1.83525	0.917625	+	0.397448i	$0.130104\pi$
$992$	0	0
$993$	1.43879e40	0.500244
$994$	0	0
$995$	−7.61231e39	−0.257576
$996$	0	0
$997$	3.94653e38	0.0129966	0.00649832	−	0.999979i	$-0.497932\pi$
$997$	3.94653e38	0.0129966	0.00649832	+	0.999979i	$0.497932\pi$
$998$	0	0
$999$	2.55740e40	0.819717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.28.a.c.1.2		2
4.3	odd	2		4.28.a.a.1.1	✓	2
12.11	even	2		36.28.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.28.a.a.1.1	✓	2	4.3	odd	2
16.28.a.c.1.2		2	1.1	even	1	trivial
36.28.a.a.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:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00840e6 q^{3} +9.02501e8 q^{5} -3.76979e11 q^{7} +1.42486e12 q^{9} +1.19981e14 q^{11} +8.45735e14 q^{13} +2.71508e15 q^{15} -5.80089e16 q^{17} +2.28933e16 q^{19} -1.13410e18 q^{21} +4.11252e18 q^{23} -6.63607e18 q^{25} -1.86543e19 q^{27} -5.09330e19 q^{29} -2.28760e20 q^{31} +3.60949e20 q^{33} -3.40224e20 q^{35} -1.37094e21 q^{37} +2.54431e21 q^{39} +1.26146e21 q^{41} +1.57776e22 q^{43} +1.28593e21 q^{45} -1.74091e22 q^{47} +7.64006e22 q^{49} -1.74514e23 q^{51} -2.42990e23 q^{53} +1.08283e23 q^{55} +6.88720e22 q^{57} +4.33841e23 q^{59} -1.61997e24 q^{61} -5.37140e23 q^{63} +7.63276e23 q^{65} -3.04796e24 q^{67} +1.23721e25 q^{69} +1.33723e24 q^{71} -2.26071e25 q^{73} -1.99639e25 q^{75} -4.52302e25 q^{77} -2.50702e25 q^{79} -6.69849e25 q^{81} -2.43633e25 q^{83} -5.23531e25 q^{85} -1.53227e26 q^{87} +3.08582e26 q^{89} -3.18824e26 q^{91} -6.88202e26 q^{93} +2.06612e25 q^{95} -2.50167e26 q^{97} +1.70955e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 483720 q^{3} + 145079100 q^{5} - 60475251760 q^{7} + 173252390058 q^{9} + 24840277565400 q^{11} - 79026950880020 q^{13} + 46\!\cdots\!00 q^{15} - 57\!\cdots\!40 q^{17} + 35\!\cdots\!72 q^{19} - 19\!\cdots\!88 q^{21}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) \) 2 * q + 483720 * q^3 + 145079100 * q^5 - 60475251760 * q^7 + 173252390058 * q^9 + 24840277565400 * q^11 - 79026950880020 * q^13 + 4627325283375600 * q^15 - 57199393642550940 * q^17 + 355562290954316072 * q^19 - 1933171089411511488 * q^21 + 6588852862675601520 * q^23 - 13512966285544371250 * q^25 + 3757770863228771280 * q^27 + 50363331391885736268 * q^29 - 289592255497320916672 * q^31 + 601148166645981667680 * q^33 - 579950211055058474400 * q^35 - 1026380758945616198660 * q^37 + 4879030784008287487152 * q^39 - 9667004963486656782348 * q^41 + 13487141643503459284760 * q^43 + 2233924267093408305900 * q^45 - 46961105511875608404000 * q^47 + 110862765316290827124306 * q^49 - 176557790235135981293424 * q^51 + 4224695398452731028060 * q^53 + 180343991690673418674000 * q^55 - 771009892031359579225440 * q^57 + 1004029738899231296722104 * q^59 - 1726777676727260480588276 * q^61 - 933277108270869338684400 * q^63 + 1463710693249729982514600 * q^65 - 9033075655753326333646840 * q^67 + 6120160785644798981928384 * q^69 - 13203608118408237669065136 * q^71 - 18212101269545484387315020 * q^73 - 2602008494342196687165000 * q^75 - 75342425018265756076461120 * q^77 - 134061090639694576525024 * q^79 - 114023903391905644807269102 * q^81 - 113710918401726608059277400 * q^83 - 52966281741728089323184200 * q^85 - 408967478461534854560014800 * q^87 + 30590274319231503302052372 * q^89 - 611514423095313234144615968 * q^91 - 534620231259281913597438720 * q^93 - 231309527733675980899160400 * q^95 - 1444066674910179813599054780 * q^97 + 290033083248945697236886200 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more