LMFDB - Embedded newform 16.34.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,34,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, 34, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 34);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 34 $
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:	$110.372526210$
Analytic rank:	$0$
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 - 589050 $ x^2 - x - 589050
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-766.996$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.55221e7 q^{3} +2.04307e10 q^{5} +1.22168e14 q^{7} -5.31812e15 q^{9} +O(q^{10})$	$q+1.55221e7 q^{3} +2.04307e10 q^{5} +1.22168e14 q^{7} -5.31812e15 q^{9} -2.15763e17 q^{11} -1.07762e18 q^{13} +3.17129e17 q^{15} +2.54532e20 q^{17} +1.22020e21 q^{19} +1.89631e21 q^{21} +5.11280e21 q^{23} -1.15998e23 q^{25} -1.68837e23 q^{27} -1.64733e23 q^{29} +6.75706e24 q^{31} -3.34910e24 q^{33} +2.49599e24 q^{35} -7.46520e25 q^{37} -1.67269e25 q^{39} +4.96453e26 q^{41} +1.99347e26 q^{43} -1.08653e26 q^{45} -2.16452e27 q^{47} +7.19411e27 q^{49} +3.95088e27 q^{51} -3.60439e28 q^{53} -4.40819e27 q^{55} +1.89401e28 q^{57} +1.87520e29 q^{59} +4.18340e27 q^{61} -6.49706e29 q^{63} -2.20165e28 q^{65} -4.85975e29 q^{67} +7.93615e28 q^{69} +3.42819e30 q^{71} +7.01467e30 q^{73} -1.80053e30 q^{75} -2.63594e31 q^{77} +2.95630e30 q^{79} +2.69431e31 q^{81} +1.23020e31 q^{83} +5.20028e30 q^{85} -2.55701e30 q^{87} +7.05623e31 q^{89} -1.31651e32 q^{91} +1.04884e32 q^{93} +2.49297e31 q^{95} -7.71791e32 q^{97} +1.14745e33 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 37919880 q^{3} - 181061536500 q^{5} + 67153080066800 q^{7} - 80\!\cdots\!54 q^{9}+O(q^{10}) $ 2 * q - 37919880 * q^3 - 181061536500 * q^5 + 67153080066800 * q^7 - 8021136954970854 * q^9	$ 2 q - 37919880 q^{3} - 181061536500 q^{5} + 67153080066800 q^{7} - 80\!\cdots\!54 q^{9}+ \cdots + 92\!\cdots\!28 q^{99}+O(q^{100}) $ 2 * q - 37919880 * q^3 - 181061536500 * q^5 + 67153080066800 * q^7 - 8021136954970854 * q^9 - 133871815441914264 * q^11 - 2981610478259443940 * q^13 + 11085280069139874000 * q^15 - 79361149261175525340 * q^17 + 1360696443041697171800 * q^19 + 4836438843082801141824 * q^21 - 26163854053674631579440 * q^23 - 191814088084783724781250 * q^25 + 272704742315639906405040 * q^27 - 1674519971630399929109700 * q^29 + 6238301225723632337095616 * q^31 - 7725507376265263259943840 * q^33 + 13581143663546216842308000 * q^35 - 104802741516758180098173620 * q^37 + 85026272438019668836783248 * q^39 + 277566613022002182732188244 * q^41 - 1567661189367072113768291800 * q^43 + 435982958561646097146265500 * q^45 - 5421061461431251260476972640 * q^47 + 2489799365888888933444851314 * q^49 + 21794809591439435388668624496 * q^51 - 26867414785542903092975171220 * q^53 - 20908570805489145209222682000 * q^55 + 11431882183748158902465090720 * q^57 + 305786409635298433026663995400 * q^59 - 5745893983413264859067362436 * q^61 - 500999494400792409792776544720 * q^63 + 361623311364964256200313721000 * q^65 - 1598906337224495341750209565960 * q^67 + 1750848530317427976275742443712 * q^69 + 2669761937956104715736810084976 * q^71 + 946314321191998870965166536340 * q^73 + 2251234781189208709982677125000 * q^75 - 30864620292716415864215760033600 * q^77 + 8535199972765636469730106191200 * q^79 + 18372400450705101408694430349042 * q^81 - 29072605803747945444403983322920 * q^83 + 72477213859794605306963311287000 * q^85 + 78129025791401437906177732680720 * q^87 + 132719467655604316264938597726900 * q^89 - 26902102147452358410548887211744 * q^91 + 132607664882695458799488635592960 * q^93 - 3378716387067841884487537110000 * q^95 - 367787330930535727150839877856060 * q^97 + 926100695332764780030656940070728 * 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
$3$	1.55221e7	0.208185	0.104093	−	0.994568i	$-0.466806\pi$
$3$	1.55221e7	0.208185	0.104093	+	0.994568i	$0.466806\pi$
$4$	0	0
$5$	2.04307e10	0.0598796	0.0299398	−	0.999552i	$-0.490468\pi$
$5$	2.04307e10	0.0598796	0.0299398	+	0.999552i	$0.490468\pi$
$6$	0	0
$7$	1.22168e14	1.38944	0.694722	−	0.719278i	$-0.255527\pi$
$7$	1.22168e14	1.38944	0.694722	+	0.719278i	$0.255527\pi$
$8$	0	0
$9$	−5.31812e15	−0.956659
$10$	0	0
$11$	−2.15763e17	−1.41578	−0.707892	−	0.706321i	$-0.750354\pi$
$11$	−2.15763e17	−1.41578	−0.707892	+	0.706321i	$0.750354\pi$
$12$	0	0
$13$	−1.07762e18	−0.449158	−0.224579	−	0.974456i	$-0.572101\pi$
$13$	−1.07762e18	−0.449158	−0.224579	+	0.974456i	$0.572101\pi$
$14$	0	0
$15$	3.17129e17	0.0124661
$16$	0	0
$17$	2.54532e20	1.26863	0.634316	−	0.773074i	$-0.281282\pi$
$17$	2.54532e20	1.26863	0.634316	+	0.773074i	$0.281282\pi$
$18$	0	0
$19$	1.22020e21	0.970505	0.485252	−	0.874374i	$-0.338728\pi$
$19$	1.22020e21	0.970505	0.485252	+	0.874374i	$0.338728\pi$
$20$	0	0
$21$	1.89631e21	0.289262
$22$	0	0
$23$	5.11280e21	0.173840	0.0869199	−	0.996215i	$-0.472298\pi$
$23$	5.11280e21	0.173840	0.0869199	+	0.996215i	$0.472298\pi$
$24$	0	0
$25$	−1.15998e23	−0.996414
$26$	0	0
$27$	−1.68837e23	−0.407348
$28$	0	0
$29$	−1.64733e23	−0.122240	−0.0611201	−	0.998130i	$-0.519467\pi$
$29$	−1.64733e23	−0.122240	−0.0611201	+	0.998130i	$0.519467\pi$
$30$	0	0
$31$	6.75706e24	1.66836	0.834181	−	0.551491i	$-0.185941\pi$
$31$	6.75706e24	1.66836	0.834181	+	0.551491i	$0.185941\pi$
$32$	0	0
$33$	−3.34910e24	−0.294746
$34$	0	0
$35$	2.49599e24	0.0831994
$36$	0	0
$37$	−7.46520e25	−0.994748	−0.497374	−	0.867536i	$-0.665702\pi$
$37$	−7.46520e25	−0.994748	−0.497374	+	0.867536i	$0.665702\pi$
$38$	0	0
$39$	−1.67269e25	−0.0935082
$40$	0	0
$41$	4.96453e26	1.21603	0.608016	−	0.793925i	$-0.291966\pi$
$41$	4.96453e26	1.21603	0.608016	+	0.793925i	$0.291966\pi$
$42$	0	0
$43$	1.99347e26	0.222525	0.111263	−	0.993791i	$-0.464511\pi$
$43$	1.99347e26	0.222525	0.111263	+	0.993791i	$0.464511\pi$
$44$	0	0
$45$	−1.08653e26	−0.0572844
$46$	0	0
$47$	−2.16452e27	−0.556862	−0.278431	−	0.960456i	$-0.589814\pi$
$47$	−2.16452e27	−0.556862	−0.278431	+	0.960456i	$0.589814\pi$
$48$	0	0
$49$	7.19411e27	0.930555
$50$	0	0
$51$	3.95088e27	0.264111
$52$	0	0
$53$	−3.60439e28	−1.27726	−0.638631	−	0.769513i	$-0.720499\pi$
$53$	−3.60439e28	−1.27726	−0.638631	+	0.769513i	$0.720499\pi$
$54$	0	0
$55$	−4.40819e27	−0.0847766
$56$	0	0
$57$	1.89401e28	0.202045
$58$	0	0
$59$	1.87520e29	1.13237	0.566186	−	0.824278i	$-0.308418\pi$
$59$	1.87520e29	1.13237	0.566186	+	0.824278i	$0.308418\pi$
$60$	0	0
$61$	4.18340e27	0.0145743	0.00728715	−	0.999973i	$-0.497680\pi$
$61$	4.18340e27	0.0145743	0.00728715	+	0.999973i	$0.497680\pi$
$62$	0	0
$63$	−6.49706e29	−1.32922
$64$	0	0
$65$	−2.20165e28	−0.0268954
$66$	0	0
$67$	−4.85975e29	−0.360064	−0.180032	−	0.983661i	$-0.557620\pi$
$67$	−4.85975e29	−0.360064	−0.180032	+	0.983661i	$0.557620\pi$
$68$	0	0
$69$	7.93615e28	0.0361909
$70$	0	0
$71$	3.42819e30	0.975667	0.487834	−	0.872937i	$-0.337787\pi$
$71$	3.42819e30	0.975667	0.487834	+	0.872937i	$0.337787\pi$
$72$	0	0
$73$	7.01467e30	1.26235	0.631175	−	0.775640i	$-0.282573\pi$
$73$	7.01467e30	1.26235	0.631175	+	0.775640i	$0.282573\pi$
$74$	0	0
$75$	−1.80053e30	−0.207439
$76$	0	0
$77$	−2.63594e31	−1.96715
$78$	0	0
$79$	2.95630e30	0.144511	0.0722555	−	0.997386i	$-0.476980\pi$
$79$	2.95630e30	0.144511	0.0722555	+	0.997386i	$0.476980\pi$
$80$	0	0
$81$	2.69431e31	0.871855
$82$	0	0
$83$	1.23020e31	0.266187	0.133094	−	0.991103i	$-0.457509\pi$
$83$	1.23020e31	0.266187	0.133094	+	0.991103i	$0.457509\pi$
$84$	0	0
$85$	5.20028e30	0.0759652
$86$	0	0
$87$	−2.55701e30	−0.0254486
$88$	0	0
$89$	7.05623e31	0.482656	0.241328	−	0.970444i	$-0.422417\pi$
$89$	7.05623e31	0.482656	0.241328	+	0.970444i	$0.422417\pi$
$90$	0	0
$91$	−1.31651e32	−0.624080
$92$	0	0
$93$	1.04884e32	0.347329
$94$	0	0
$95$	2.49297e31	0.0581135
$96$	0	0
$97$	−7.71791e32	−1.27575	−0.637875	−	0.770140i	$-0.720187\pi$
$97$	−7.71791e32	−1.27575	−0.637875	+	0.770140i	$0.720187\pi$
$98$	0	0
$99$	1.14745e33	1.35442
$100$	0	0
$101$	1.88601e33	1.60045	0.800224	−	0.599701i	$-0.204714\pi$
$101$	1.88601e33	1.60045	0.800224	+	0.599701i	$0.204714\pi$
$102$	0	0
$103$	4.05317e32	0.248874	0.124437	−	0.992227i	$-0.460287\pi$
$103$	4.05317e32	0.248874	0.124437	+	0.992227i	$0.460287\pi$
$104$	0	0
$105$	3.87431e31	0.0173209
$106$	0	0
$107$	1.13905e33	0.373003	0.186501	−	0.982455i	$-0.440285\pi$
$107$	1.13905e33	0.373003	0.186501	+	0.982455i	$0.440285\pi$
$108$	0	0
$109$	5.47789e32	0.132153	0.0660764	−	0.997815i	$-0.478952\pi$
$109$	5.47789e32	0.132153	0.0660764	+	0.997815i	$0.478952\pi$
$110$	0	0
$111$	−1.15876e33	−0.207092
$112$	0	0
$113$	7.05689e33	0.939332	0.469666	−	0.882844i	$-0.344374\pi$
$113$	7.05689e33	0.939332	0.469666	+	0.882844i	$0.344374\pi$
$114$	0	0
$115$	1.04458e32	0.0104095
$116$	0	0
$117$	5.73090e33	0.429691
$118$	0	0
$119$	3.10958e34	1.76269
$120$	0	0
$121$	2.33284e34	1.00444
$122$	0	0
$123$	7.70601e33	0.253160
$124$	0	0
$125$	−4.74838e33	−0.119545
$126$	0	0
$127$	1.88228e34	0.364689	0.182344	−	0.983235i	$-0.441631\pi$
$127$	1.88228e34	0.364689	0.182344	+	0.983235i	$0.441631\pi$
$128$	0	0
$129$	3.09428e33	0.0463265
$130$	0	0
$131$	1.26378e35	1.46790	0.733949	−	0.679204i	$-0.237675\pi$
$131$	1.26378e35	1.46790	0.733949	+	0.679204i	$0.237675\pi$
$132$	0	0
$133$	1.49070e35	1.34846
$134$	0	0
$135$	−3.44947e33	−0.0243918
$136$	0	0
$137$	2.95759e35	1.64077	0.820387	−	0.571809i	$-0.193758\pi$
$137$	2.95759e35	1.64077	0.820387	+	0.571809i	$0.193758\pi$
$138$	0	0
$139$	8.22498e34	0.359245	0.179622	−	0.983736i	$-0.442512\pi$
$139$	8.22498e34	0.359245	0.179622	+	0.983736i	$0.442512\pi$
$140$	0	0
$141$	−3.35980e34	−0.115931
$142$	0	0
$143$	2.32510e35	0.635911
$144$	0	0
$145$	−3.36562e33	−0.00731970
$146$	0	0
$147$	1.11668e35	0.193728
$148$	0	0
$149$	5.29045e35	0.734378	0.367189	−	0.930146i	$-0.380320\pi$
$149$	5.29045e35	0.734378	0.367189	+	0.930146i	$0.380320\pi$
$150$	0	0
$151$	5.18728e35	0.577856	0.288928	−	0.957351i	$-0.406701\pi$
$151$	5.18728e35	0.577856	0.288928	+	0.957351i	$0.406701\pi$
$152$	0	0
$153$	−1.35363e36	−1.21365
$154$	0	0
$155$	1.38052e35	0.0999009
$156$	0	0
$157$	1.31075e36	0.767669	0.383835	−	0.923402i	$-0.374603\pi$
$157$	1.31075e36	0.767669	0.383835	+	0.923402i	$0.374603\pi$
$158$	0	0
$159$	−5.59478e35	−0.265908
$160$	0	0
$161$	6.24622e35	0.241541
$162$	0	0
$163$	−1.61518e36	−0.509478	−0.254739	−	0.967010i	$-0.581990\pi$
$163$	−1.61518e36	−0.509478	−0.254739	+	0.967010i	$0.581990\pi$
$164$	0	0
$165$	−6.84245e34	−0.0176493
$166$	0	0
$167$	6.45682e35	0.136520	0.0682601	−	0.997668i	$-0.478255\pi$
$167$	6.45682e35	0.136520	0.0682601	+	0.997668i	$0.478255\pi$
$168$	0	0
$169$	−4.59487e36	−0.798257
$170$	0	0
$171$	−6.48919e36	−0.928442
$172$	0	0
$173$	−6.58049e36	−0.777137	−0.388569	−	0.921420i	$-0.627030\pi$
$173$	−6.58049e36	−0.777137	−0.388569	+	0.921420i	$0.627030\pi$
$174$	0	0
$175$	−1.41713e37	−1.38446
$176$	0	0
$177$	2.91071e36	0.235743
$178$	0	0
$179$	2.08823e37	1.40508	0.702542	−	0.711642i	$-0.252048\pi$
$179$	2.08823e37	1.40508	0.702542	+	0.711642i	$0.252048\pi$
$180$	0	0
$181$	−2.42658e37	−1.35925	−0.679624	−	0.733561i	$-0.737857\pi$
$181$	−2.42658e37	−1.35925	−0.679624	+	0.733561i	$0.737857\pi$
$182$	0	0
$183$	6.49353e34	0.00303416
$184$	0	0
$185$	−1.52520e36	−0.0595652
$186$	0	0
$187$	−5.49185e37	−1.79611
$188$	0	0
$189$	−2.06265e37	−0.565987
$190$	0	0
$191$	−4.06968e37	−0.938664	−0.469332	−	0.883022i	$-0.655505\pi$
$191$	−4.06968e37	−0.938664	−0.469332	+	0.883022i	$0.655505\pi$
$192$	0	0
$193$	−5.23242e37	−1.01627	−0.508133	−	0.861279i	$-0.669664\pi$
$193$	−5.23242e37	−1.01627	−0.508133	+	0.861279i	$0.669664\pi$
$194$	0	0
$195$	−3.41743e35	−0.00559924
$196$	0	0
$197$	1.35006e38	1.86922	0.934611	−	0.355672i	$-0.115748\pi$
$197$	1.35006e38	1.86922	0.934611	+	0.355672i	$0.115748\pi$
$198$	0	0
$199$	9.91562e37	1.16210	0.581051	−	0.813867i	$-0.302642\pi$
$199$	9.91562e37	1.16210	0.581051	+	0.813867i	$0.302642\pi$
$200$	0	0
$201$	−7.54336e36	−0.0749601
$202$	0	0
$203$	−2.01252e37	−0.169846
$204$	0	0
$205$	1.01429e37	0.0728155
$206$	0	0
$207$	−2.71905e37	−0.166305
$208$	0	0
$209$	−2.63274e38	−1.37403
$210$	0	0
$211$	2.07128e38	0.923802	0.461901	−	0.886931i	$-0.347168\pi$
$211$	2.07128e38	0.923802	0.461901	+	0.886931i	$0.347168\pi$
$212$	0	0
$213$	5.32127e37	0.203120
$214$	0	0
$215$	4.07280e36	0.0133247
$216$	0	0
$217$	8.25499e38	2.31809
$218$	0	0
$219$	1.08883e38	0.262803
$220$	0	0
$221$	−2.74288e38	−0.569816
$222$	0	0
$223$	4.51334e38	0.808106	0.404053	−	0.914736i	$-0.367601\pi$
$223$	4.51334e38	0.808106	0.404053	+	0.914736i	$0.367601\pi$
$224$	0	0
$225$	6.16891e38	0.953229
$226$	0	0
$227$	2.43217e38	0.324764	0.162382	−	0.986728i	$-0.448082\pi$
$227$	2.43217e38	0.324764	0.162382	+	0.986728i	$0.448082\pi$
$228$	0	0
$229$	3.41418e38	0.394458	0.197229	−	0.980357i	$-0.436806\pi$
$229$	3.41418e38	0.394458	0.197229	+	0.980357i	$0.436806\pi$
$230$	0	0
$231$	−4.09153e38	−0.409533
$232$	0	0
$233$	−6.99608e37	−0.0607410	−0.0303705	−	0.999539i	$-0.509669\pi$
$233$	−6.99608e37	−0.0607410	−0.0303705	+	0.999539i	$0.509669\pi$
$234$	0	0
$235$	−4.42228e37	−0.0333447
$236$	0	0
$237$	4.58880e37	0.0300851
$238$	0	0
$239$	1.63523e39	0.933282	0.466641	−	0.884447i	$-0.345464\pi$
$239$	1.63523e39	0.933282	0.466641	+	0.884447i	$0.345464\pi$
$240$	0	0
$241$	−3.64560e38	−0.181338	−0.0906689	−	0.995881i	$-0.528900\pi$
$241$	−3.64560e38	−0.181338	−0.0906689	+	0.995881i	$0.528900\pi$
$242$	0	0
$243$	1.35679e39	0.588856
$244$	0	0
$245$	1.46981e38	0.0557213
$246$	0	0
$247$	−1.31491e39	−0.435910
$248$	0	0
$249$	1.90953e38	0.0554164
$250$	0	0
$251$	−1.94693e39	−0.495147	−0.247573	−	0.968869i	$-0.579633\pi$
$251$	−1.94693e39	−0.495147	−0.247573	+	0.968869i	$0.579633\pi$
$252$	0	0
$253$	−1.10315e39	−0.246120
$254$	0	0
$255$	8.07194e37	0.0158149
$256$	0	0
$257$	−5.64464e39	−0.972164	−0.486082	−	0.873913i	$-0.661574\pi$
$257$	−5.64464e39	−0.972164	−0.486082	+	0.873913i	$0.661574\pi$
$258$	0	0
$259$	−9.12012e39	−1.38215
$260$	0	0
$261$	8.76071e38	0.116942
$262$	0	0
$263$	−9.75930e39	−1.14855	−0.574274	−	0.818663i	$-0.694715\pi$
$263$	−9.75930e39	−1.14855	−0.574274	+	0.818663i	$0.694715\pi$
$264$	0	0
$265$	−7.36404e38	−0.0764820
$266$	0	0
$267$	1.09528e39	0.100482
$268$	0	0
$269$	−1.08142e40	−0.877168	−0.438584	−	0.898690i	$-0.644520\pi$
$269$	−1.08142e40	−0.877168	−0.438584	+	0.898690i	$0.644520\pi$
$270$	0	0
$271$	6.19121e39	0.444408	0.222204	−	0.975000i	$-0.428675\pi$
$271$	6.19121e39	0.444408	0.222204	+	0.975000i	$0.428675\pi$
$272$	0	0
$273$	−2.04350e39	−0.129924
$274$	0	0
$275$	2.50280e40	1.41071
$276$	0	0
$277$	−5.15662e39	−0.257898	−0.128949	−	0.991651i	$-0.541160\pi$
$277$	−5.15662e39	−0.257898	−0.128949	+	0.991651i	$0.541160\pi$
$278$	0	0
$279$	−3.59349e40	−1.59605
$280$	0	0
$281$	−1.28386e40	−0.506832	−0.253416	−	0.967357i	$-0.581554\pi$
$281$	−1.28386e40	−0.506832	−0.253416	+	0.967357i	$0.581554\pi$
$282$	0	0
$283$	−3.83806e40	−1.34783	−0.673915	−	0.738809i	$-0.735389\pi$
$283$	−3.83806e40	−1.34783	−0.673915	+	0.738809i	$0.735389\pi$
$284$	0	0
$285$	3.86961e38	0.0120984
$286$	0	0
$287$	6.06509e40	1.68961
$288$	0	0
$289$	2.45321e40	0.609426
$290$	0	0
$291$	−1.19798e40	−0.265593
$292$	0	0
$293$	1.90344e40	0.376897	0.188449	−	0.982083i	$-0.439654\pi$
$293$	1.90344e40	0.376897	0.188449	+	0.982083i	$0.439654\pi$
$294$	0	0
$295$	3.83117e39	0.0678060
$296$	0	0
$297$	3.64287e40	0.576717
$298$	0	0
$299$	−5.50964e39	−0.0780816
$300$	0	0
$301$	2.43538e40	0.309186
$302$	0	0
$303$	2.92749e40	0.333190
$304$	0	0
$305$	8.54701e37	0.000872703 0
$306$	0	0
$307$	1.48955e41	1.36544	0.682718	−	0.730682i	$-0.260798\pi$
$307$	1.48955e41	1.36544	0.682718	+	0.730682i	$0.260798\pi$
$308$	0	0
$309$	6.29138e39	0.0518121
$310$	0	0
$311$	7.37928e39	0.0546345	0.0273173	−	0.999627i	$-0.491304\pi$
$311$	7.37928e39	0.0546345	0.0273173	+	0.999627i	$0.491304\pi$
$312$	0	0
$313$	−2.63324e40	−0.175391	−0.0876957	−	0.996147i	$-0.527950\pi$
$313$	−2.63324e40	−0.175391	−0.0876957	+	0.996147i	$0.527950\pi$
$314$	0	0
$315$	−1.32740e40	−0.0795934
$316$	0	0
$317$	1.79226e41	0.968109	0.484054	−	0.875038i	$-0.339164\pi$
$317$	1.79226e41	0.968109	0.484054	+	0.875038i	$0.339164\pi$
$318$	0	0
$319$	3.55433e40	0.173066
$320$	0	0
$321$	1.76805e40	0.0776538
$322$	0	0
$323$	3.10581e41	1.23121
$324$	0	0
$325$	1.25001e41	0.447548
$326$	0	0
$327$	8.50285e39	0.0275123
$328$	0	0
$329$	−2.64436e41	−0.773728
$330$	0	0
$331$	−6.16181e41	−1.63135	−0.815674	−	0.578512i	$-0.803634\pi$
$331$	−6.16181e41	−1.63135	−0.815674	+	0.578512i	$0.803634\pi$
$332$	0	0
$333$	3.97009e41	0.951635
$334$	0	0
$335$	−9.92883e39	−0.0215605
$336$	0	0
$337$	4.51357e41	0.888437	0.444218	−	0.895919i	$-0.353481\pi$
$337$	4.51357e41	0.888437	0.444218	+	0.895919i	$0.353481\pi$
$338$	0	0
$339$	1.09538e41	0.195555
$340$	0	0
$341$	−1.45792e42	−2.36204
$342$	0	0
$343$	−6.55899e40	−0.0964903
$344$	0	0
$345$	1.62141e39	0.00216710
$346$	0	0
$347$	−8.17900e41	−0.993720	−0.496860	−	0.867831i	$-0.665514\pi$
$347$	−8.17900e41	−0.993720	−0.496860	+	0.867831i	$0.665514\pi$
$348$	0	0
$349$	−1.22969e42	−1.35886	−0.679432	−	0.733739i	$-0.737774\pi$
$349$	−1.22969e42	−1.35886	−0.679432	+	0.733739i	$0.737774\pi$
$350$	0	0
$351$	1.81942e41	0.182964
$352$	0	0
$353$	1.62570e41	0.148853	0.0744266	−	0.997226i	$-0.476287\pi$
$353$	1.62570e41	0.148853	0.0744266	+	0.997226i	$0.476287\pi$
$354$	0	0
$355$	7.00404e40	0.0584226
$356$	0	0
$357$	4.82673e41	0.366967
$358$	0	0
$359$	−3.64036e41	−0.252398	−0.126199	−	0.992005i	$-0.540278\pi$
$359$	−3.64036e41	−0.252398	−0.126199	+	0.992005i	$0.540278\pi$
$360$	0	0
$361$	−9.18755e40	−0.0581207
$362$	0	0
$363$	3.62106e41	0.209111
$364$	0	0
$365$	1.43315e41	0.0755891
$366$	0	0
$367$	−1.35581e42	−0.653442	−0.326721	−	0.945121i	$-0.605944\pi$
$367$	−1.35581e42	−0.653442	−0.326721	+	0.945121i	$0.605944\pi$
$368$	0	0
$369$	−2.64020e42	−1.16333
$370$	0	0
$371$	−4.40343e42	−1.77469
$372$	0	0
$373$	1.93457e42	0.713493	0.356746	−	0.934201i	$-0.383886\pi$
$373$	1.93457e42	0.713493	0.356746	+	0.934201i	$0.383886\pi$
$374$	0	0
$375$	−7.37049e40	−0.0248874
$376$	0	0
$377$	1.77519e41	0.0549052
$378$	0	0
$379$	5.45789e42	1.54696	0.773478	−	0.633823i	$-0.218515\pi$
$379$	5.45789e42	1.54696	0.773478	+	0.633823i	$0.218515\pi$
$380$	0	0
$381$	2.92169e41	0.0759229
$382$	0	0
$383$	5.06942e42	1.20831	0.604155	−	0.796867i	$-0.293511\pi$
$383$	5.06942e42	1.20831	0.604155	+	0.796867i	$0.293511\pi$
$384$	0	0
$385$	−5.38542e41	−0.117792
$386$	0	0
$387$	−1.06015e42	−0.212881
$388$	0	0
$389$	7.51028e42	1.38512	0.692560	−	0.721360i	$-0.256483\pi$
$389$	7.51028e42	1.38512	0.692560	+	0.721360i	$0.256483\pi$
$390$	0	0
$391$	1.30137e42	0.220539
$392$	0	0
$393$	1.96166e42	0.305595
$394$	0	0
$395$	6.03993e40	0.00865327
$396$	0	0
$397$	−1.26926e43	−1.67304	−0.836519	−	0.547938i	$-0.815413\pi$
$397$	−1.26926e43	−1.67304	−0.836519	+	0.547938i	$0.815413\pi$
$398$	0	0
$399$	2.31389e42	0.280730
$400$	0	0
$401$	5.29553e42	0.591598	0.295799	−	0.955250i	$-0.404414\pi$
$401$	5.29553e42	0.591598	0.295799	+	0.955250i	$0.404414\pi$
$402$	0	0
$403$	−7.28153e42	−0.749358
$404$	0	0
$405$	5.50467e41	0.0522064
$406$	0	0
$407$	1.61071e43	1.40835
$408$	0	0
$409$	1.80907e43	1.45888	0.729442	−	0.684042i	$-0.239780\pi$
$409$	1.80907e43	1.45888	0.729442	+	0.684042i	$0.239780\pi$
$410$	0	0
$411$	4.59081e42	0.341585
$412$	0	0
$413$	2.29090e43	1.57337
$414$	0	0
$415$	2.51339e41	0.0159392
$416$	0	0
$417$	1.27669e42	0.0747896
$418$	0	0
$419$	−9.59306e42	−0.519309	−0.259654	−	0.965702i	$-0.583609\pi$
$419$	−9.59306e42	−0.519309	−0.259654	+	0.965702i	$0.583609\pi$
$420$	0	0
$421$	−1.95406e43	−0.977875	−0.488938	−	0.872319i	$-0.662615\pi$
$421$	−1.95406e43	−0.977875	−0.488938	+	0.872319i	$0.662615\pi$
$422$	0	0
$423$	1.15112e43	0.532727
$424$	0	0
$425$	−2.95252e43	−1.26408
$426$	0	0
$427$	5.11080e41	0.0202502
$428$	0	0
$429$	3.60904e42	0.132387
$430$	0	0
$431$	−4.01382e43	−1.36359	−0.681794	−	0.731544i	$-0.738800\pi$
$431$	−4.01382e43	−1.36359	−0.681794	+	0.731544i	$0.738800\pi$
$432$	0	0
$433$	−1.24146e43	−0.390735	−0.195367	−	0.980730i	$-0.562590\pi$
$433$	−1.24146e43	−0.390735	−0.195367	+	0.980730i	$0.562590\pi$
$434$	0	0
$435$	−5.22416e40	−0.00152386
$436$	0	0
$437$	6.23865e42	0.168712
$438$	0	0
$439$	6.07044e43	1.52249	0.761247	−	0.648462i	$-0.224587\pi$
$439$	6.07044e43	1.52249	0.761247	+	0.648462i	$0.224587\pi$
$440$	0	0
$441$	−3.82592e43	−0.890223
$442$	0	0
$443$	−6.61294e43	−1.42802	−0.714009	−	0.700137i	$-0.753122\pi$
$443$	−6.61294e43	−1.42802	−0.714009	+	0.700137i	$0.753122\pi$
$444$	0	0
$445$	1.44164e42	0.0289013
$446$	0	0
$447$	8.21191e42	0.152887
$448$	0	0
$449$	−1.54504e43	−0.267225	−0.133612	−	0.991034i	$-0.542658\pi$
$449$	−1.54504e43	−0.267225	−0.133612	+	0.991034i	$0.542658\pi$
$450$	0	0
$451$	−1.07116e44	−1.72164
$452$	0	0
$453$	8.05175e42	0.120301
$454$	0	0
$455$	−2.68972e42	−0.0373697
$456$	0	0
$457$	7.12331e43	0.920586	0.460293	−	0.887767i	$-0.347744\pi$
$457$	7.12331e43	0.920586	0.460293	+	0.887767i	$0.347744\pi$
$458$	0	0
$459$	−4.29745e43	−0.516774
$460$	0	0
$461$	−1.34004e44	−1.49986	−0.749932	−	0.661515i	$-0.769914\pi$
$461$	−1.34004e44	−1.49986	−0.749932	+	0.661515i	$0.769914\pi$
$462$	0	0
$463$	5.53073e43	0.576363	0.288182	−	0.957576i	$-0.406949\pi$
$463$	5.53073e43	0.576363	0.288182	+	0.957576i	$0.406949\pi$
$464$	0	0
$465$	2.14286e42	0.0207979
$466$	0	0
$467$	3.51798e43	0.318101	0.159050	−	0.987270i	$-0.449157\pi$
$467$	3.51798e43	0.318101	0.159050	+	0.987270i	$0.449157\pi$
$468$	0	0
$469$	−5.93707e43	−0.500288
$470$	0	0
$471$	2.03456e43	0.159818
$472$	0	0
$473$	−4.30116e43	−0.315048
$474$	0	0
$475$	−1.41541e44	−0.967025
$476$	0	0
$477$	1.91686e44	1.22190
$478$	0	0
$479$	2.90520e44	1.72838	0.864192	−	0.503162i	$-0.167830\pi$
$479$	2.90520e44	1.72838	0.864192	+	0.503162i	$0.167830\pi$
$480$	0	0
$481$	8.04464e43	0.446799
$482$	0	0
$483$	9.69546e42	0.0502853
$484$	0	0
$485$	−1.57683e43	−0.0763915
$486$	0	0
$487$	1.45706e44	0.659551	0.329776	−	0.944059i	$-0.393027\pi$
$487$	1.45706e44	0.659551	0.329776	+	0.944059i	$0.393027\pi$
$488$	0	0
$489$	−2.50710e43	−0.106066
$490$	0	0
$491$	−1.43984e44	−0.569469	−0.284735	−	0.958606i	$-0.591905\pi$
$491$	−1.43984e44	−0.569469	−0.284735	+	0.958606i	$0.591905\pi$
$492$	0	0
$493$	−4.19299e43	−0.155078
$494$	0	0
$495$	2.34433e43	0.0811023
$496$	0	0
$497$	4.18816e44	1.35564
$498$	0	0
$499$	−1.76176e44	−0.533687	−0.266843	−	0.963740i	$-0.585981\pi$
$499$	−1.76176e44	−0.533687	−0.266843	+	0.963740i	$0.585981\pi$
$500$	0	0
$501$	1.00224e43	0.0284215
$502$	0	0
$503$	2.43829e44	0.647460	0.323730	−	0.946149i	$-0.395063\pi$
$503$	2.43829e44	0.647460	0.323730	+	0.946149i	$0.395063\pi$
$504$	0	0
$505$	3.85326e43	0.0958343
$506$	0	0
$507$	−7.13222e43	−0.166185
$508$	0	0
$509$	3.49786e44	0.763763	0.381882	−	0.924211i	$-0.375276\pi$
$509$	3.49786e44	0.763763	0.381882	+	0.924211i	$0.375276\pi$
$510$	0	0
$511$	8.56970e44	1.75396
$512$	0	0
$513$	−2.06015e44	−0.395333
$514$	0	0
$515$	8.28092e42	0.0149025
$516$	0	0
$517$	4.67023e44	0.788396
$518$	0	0
$519$	−1.02143e44	−0.161789
$520$	0	0
$521$	−3.31981e44	−0.493505	−0.246752	−	0.969079i	$-0.579363\pi$
$521$	−3.31981e44	−0.493505	−0.246752	+	0.969079i	$0.579363\pi$
$522$	0	0
$523$	−1.02645e45	−1.43238	−0.716190	−	0.697905i	$-0.754116\pi$
$523$	−1.02645e45	−1.43238	−0.716190	+	0.697905i	$0.754116\pi$
$524$	0	0
$525$	−2.19968e44	−0.288225
$526$	0	0
$527$	1.71989e45	2.11654
$528$	0	0
$529$	−8.38864e44	−0.969780
$530$	0	0
$531$	−9.97254e44	−1.08329
$532$	0	0
$533$	−5.34987e44	−0.546191
$534$	0	0
$535$	2.32717e43	0.0223353
$536$	0	0
$537$	3.24137e44	0.292518
$538$	0	0
$539$	−1.55222e45	−1.31746
$540$	0	0
$541$	4.08746e44	0.326361	0.163180	−	0.986596i	$-0.447825\pi$
$541$	4.08746e44	0.326361	0.163180	+	0.986596i	$0.447825\pi$
$542$	0	0
$543$	−3.76657e44	−0.282976
$544$	0	0
$545$	1.11917e43	0.00791326
$546$	0	0
$547$	−7.54748e44	−0.502355	−0.251178	−	0.967941i	$-0.580818\pi$
$547$	−7.54748e44	−0.502355	−0.251178	+	0.967941i	$0.580818\pi$
$548$	0	0
$549$	−2.22479e43	−0.0139426
$550$	0	0
$551$	−2.01008e44	−0.118635
$552$	0	0
$553$	3.61166e44	0.200790
$554$	0	0
$555$	−2.36743e43	−0.0124006
$556$	0	0
$557$	−3.77046e45	−1.86116	−0.930581	−	0.366087i	$-0.880697\pi$
$557$	−3.77046e45	−1.86116	−0.930581	+	0.366087i	$0.880697\pi$
$558$	0	0
$559$	−2.14819e44	−0.0999491
$560$	0	0
$561$	−8.52453e44	−0.373924
$562$	0	0
$563$	−1.99523e45	−0.825284	−0.412642	−	0.910893i	$-0.635394\pi$
$563$	−1.99523e45	−0.825284	−0.412642	+	0.910893i	$0.635394\pi$
$564$	0	0
$565$	1.44177e44	0.0562468
$566$	0	0
$567$	3.29159e45	1.21139
$568$	0	0
$569$	2.38895e45	0.829574	0.414787	−	0.909919i	$-0.363856\pi$
$569$	2.38895e45	0.829574	0.414787	+	0.909919i	$0.363856\pi$
$570$	0	0
$571$	−9.25875e44	−0.303429	−0.151714	−	0.988424i	$-0.548479\pi$
$571$	−9.25875e44	−0.303429	−0.151714	+	0.988424i	$0.548479\pi$
$572$	0	0
$573$	−6.31701e44	−0.195416
$574$	0	0
$575$	−5.93074e44	−0.173217
$576$	0	0
$577$	−3.73598e45	−1.03040	−0.515198	−	0.857071i	$-0.672281\pi$
$577$	−3.73598e45	−1.03040	−0.515198	+	0.857071i	$0.672281\pi$
$578$	0	0
$579$	−8.12183e44	−0.211572
$580$	0	0
$581$	1.50292e45	0.369853
$582$	0	0
$583$	7.77694e45	1.80833
$584$	0	0
$585$	1.17087e44	0.0257298
$586$	0	0
$587$	−6.35230e45	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$587$	−6.35230e45	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$588$	0	0
$589$	8.24499e45	1.61915
$590$	0	0
$591$	2.09558e45	0.389145
$592$	0	0
$593$	7.20521e45	1.26545	0.632724	−	0.774378i	$-0.281937\pi$
$593$	7.20521e45	1.26545	0.632724	+	0.774378i	$0.281937\pi$
$594$	0	0
$595$	6.35310e44	0.105549
$596$	0	0
$597$	1.53912e45	0.241933
$598$	0	0
$599$	6.37291e45	0.947971	0.473985	−	0.880533i	$-0.342815\pi$
$599$	6.37291e45	0.947971	0.473985	+	0.880533i	$0.342815\pi$
$600$	0	0
$601$	−8.13523e45	−1.14536	−0.572679	−	0.819780i	$-0.694096\pi$
$601$	−8.13523e45	−1.14536	−0.572679	+	0.819780i	$0.694096\pi$
$602$	0	0
$603$	2.58447e45	0.344458
$604$	0	0
$605$	4.76616e44	0.0601458
$606$	0	0
$607$	−8.31211e45	−0.993339	−0.496670	−	0.867940i	$-0.665444\pi$
$607$	−8.31211e45	−0.993339	−0.496670	+	0.867940i	$0.665444\pi$
$608$	0	0
$609$	−3.12386e44	−0.0353595
$610$	0	0
$611$	2.33253e45	0.250119
$612$	0	0
$613$	−2.99559e44	−0.0304358	−0.0152179	−	0.999884i	$-0.504844\pi$
$613$	−2.99559e44	−0.0304358	−0.0152179	+	0.999884i	$0.504844\pi$
$614$	0	0
$615$	1.57440e44	0.0151591
$616$	0	0
$617$	9.03503e45	0.824564	0.412282	−	0.911056i	$-0.364732\pi$
$617$	9.03503e45	0.824564	0.412282	+	0.911056i	$0.364732\pi$
$618$	0	0
$619$	1.15706e46	1.00106	0.500528	−	0.865720i	$-0.333139\pi$
$619$	1.15706e46	1.00106	0.500528	+	0.865720i	$0.333139\pi$
$620$	0	0
$621$	−8.63230e44	−0.0708133
$622$	0	0
$623$	8.62048e45	0.670623
$624$	0	0
$625$	1.34069e46	0.989256
$626$	0	0
$627$	−4.08658e45	−0.286052
$628$	0	0
$629$	−1.90013e46	−1.26197
$630$	0	0
$631$	1.43742e46	0.905939	0.452970	−	0.891526i	$-0.350365\pi$
$631$	1.43742e46	0.905939	0.452970	+	0.891526i	$0.350365\pi$
$632$	0	0
$633$	3.21507e45	0.192322
$634$	0	0
$635$	3.84563e44	0.0218374
$636$	0	0
$637$	−7.75250e45	−0.417966
$638$	0	0
$639$	−1.82315e46	−0.933381
$640$	0	0
$641$	1.34213e46	0.652586	0.326293	−	0.945269i	$-0.394200\pi$
$641$	1.34213e46	0.652586	0.326293	+	0.945269i	$0.394200\pi$
$642$	0	0
$643$	−3.48572e46	−1.60995	−0.804973	−	0.593312i	$-0.797820\pi$
$643$	−3.48572e46	−1.60995	−0.804973	+	0.593312i	$0.797820\pi$
$644$	0	0
$645$	6.32185e43	0.00277402
$646$	0	0
$647$	−3.47978e46	−1.45088	−0.725440	−	0.688285i	$-0.758364\pi$
$647$	−3.47978e46	−1.45088	−0.725440	+	0.688285i	$0.758364\pi$
$648$	0	0
$649$	−4.04598e46	−1.60319
$650$	0	0
$651$	1.28135e46	0.482594
$652$	0	0
$653$	1.19815e46	0.428986	0.214493	−	0.976726i	$-0.431190\pi$
$653$	1.19815e46	0.428986	0.214493	+	0.976726i	$0.431190\pi$
$654$	0	0
$655$	2.58200e45	0.0878972
$656$	0	0
$657$	−3.73049e46	−1.20764
$658$	0	0
$659$	2.00047e46	0.615918	0.307959	−	0.951400i	$-0.400354\pi$
$659$	2.00047e46	0.615918	0.307959	+	0.951400i	$0.400354\pi$
$660$	0	0
$661$	3.80588e46	1.11463	0.557316	−	0.830301i	$-0.311831\pi$
$661$	3.80588e46	1.11463	0.557316	+	0.830301i	$0.311831\pi$
$662$	0	0
$663$	−4.25754e45	−0.118628
$664$	0	0
$665$	3.04561e45	0.0807454
$666$	0	0
$667$	−8.42247e44	−0.0212502
$668$	0	0
$669$	7.00567e45	0.168236
$670$	0	0
$671$	−9.02622e44	−0.0206341
$672$	0	0
$673$	−5.38328e46	−1.17165	−0.585825	−	0.810437i	$-0.699230\pi$
$673$	−5.38328e46	−1.17165	−0.585825	+	0.810437i	$0.699230\pi$
$674$	0	0
$675$	1.95847e46	0.405887
$676$	0	0
$677$	−9.38585e46	−1.85251	−0.926255	−	0.376898i	$-0.876991\pi$
$677$	−9.38585e46	−1.85251	−0.926255	+	0.376898i	$0.876991\pi$
$678$	0	0
$679$	−9.42884e46	−1.77258
$680$	0	0
$681$	3.77525e45	0.0676111
$682$	0	0
$683$	−1.22152e46	−0.208430	−0.104215	−	0.994555i	$-0.533233\pi$
$683$	−1.22152e46	−0.208430	−0.104215	+	0.994555i	$0.533233\pi$
$684$	0	0
$685$	6.04258e45	0.0982489
$686$	0	0
$687$	5.29953e45	0.0821205
$688$	0	0
$689$	3.88416e46	0.573693
$690$	0	0
$691$	1.01396e47	1.42768	0.713841	−	0.700308i	$-0.246954\pi$
$691$	1.01396e47	1.42768	0.713841	+	0.700308i	$0.246954\pi$
$692$	0	0
$693$	1.40182e47	1.88189
$694$	0	0
$695$	1.68042e45	0.0215115
$696$	0	0
$697$	1.26363e47	1.54270
$698$	0	0
$699$	−1.08594e45	−0.0126454
$700$	0	0
$701$	−1.07227e47	−1.19112	−0.595560	−	0.803311i	$-0.703070\pi$
$701$	−1.07227e47	−1.19112	−0.595560	+	0.803311i	$0.703070\pi$
$702$	0	0
$703$	−9.10906e46	−0.965408
$704$	0	0
$705$	−6.86432e44	−0.00694188
$706$	0	0
$707$	2.30411e47	2.22373
$708$	0	0
$709$	−3.54939e45	−0.0326957	−0.0163478	−	0.999866i	$-0.505204\pi$
$709$	−3.54939e45	−0.0326957	−0.0163478	+	0.999866i	$0.505204\pi$
$710$	0	0
$711$	−1.57219e46	−0.138248
$712$	0	0
$713$	3.45475e46	0.290028
$714$	0	0
$715$	4.75035e45	0.0380781
$716$	0	0
$717$	2.53822e46	0.194296
$718$	0	0
$719$	−6.94896e46	−0.508035	−0.254017	−	0.967200i	$-0.581752\pi$
$719$	−6.94896e46	−0.508035	−0.254017	+	0.967200i	$0.581752\pi$
$720$	0	0
$721$	4.95169e46	0.345797
$722$	0	0
$723$	−5.65875e45	−0.0377519
$724$	0	0
$725$	1.91087e46	0.121802
$726$	0	0
$727$	1.73359e47	1.05592	0.527958	−	0.849270i	$-0.322958\pi$
$727$	1.73359e47	1.05592	0.527958	+	0.849270i	$0.322958\pi$
$728$	0	0
$729$	−1.28718e47	−0.749264
$730$	0	0
$731$	5.07401e46	0.282303
$732$	0	0
$733$	4.59301e46	0.244277	0.122138	−	0.992513i	$-0.461025\pi$
$733$	4.59301e46	0.244277	0.122138	+	0.992513i	$0.461025\pi$
$734$	0	0
$735$	2.28146e45	0.0116004
$736$	0	0
$737$	1.04855e47	0.509773
$738$	0	0
$739$	3.24514e47	1.50869	0.754346	−	0.656477i	$-0.227954\pi$
$739$	3.24514e47	1.50869	0.754346	+	0.656477i	$0.227954\pi$
$740$	0	0
$741$	−2.04102e46	−0.0907502
$742$	0	0
$743$	−3.63726e47	−1.54689	−0.773443	−	0.633866i	$-0.781467\pi$
$743$	−3.63726e47	−1.54689	−0.773443	+	0.633866i	$0.781467\pi$
$744$	0	0
$745$	1.08088e46	0.0439743
$746$	0	0
$747$	−6.54236e46	−0.254651
$748$	0	0
$749$	1.39156e47	0.518267
$750$	0	0
$751$	−2.28361e47	−0.813885	−0.406943	−	0.913454i	$-0.633405\pi$
$751$	−2.28361e47	−0.813885	−0.406943	+	0.913454i	$0.633405\pi$
$752$	0	0
$753$	−3.02205e46	−0.103082
$754$	0	0
$755$	1.05980e46	0.0346018
$756$	0	0
$757$	−1.23646e47	−0.386453	−0.193226	−	0.981154i	$-0.561895\pi$
$757$	−1.23646e47	−0.386453	−0.193226	+	0.981154i	$0.561895\pi$
$758$	0	0
$759$	−1.71233e46	−0.0512386
$760$	0	0
$761$	−1.65073e47	−0.472966	−0.236483	−	0.971636i	$-0.575995\pi$
$761$	−1.65073e47	−0.472966	−0.236483	+	0.971636i	$0.575995\pi$
$762$	0	0
$763$	6.69225e46	0.183619
$764$	0	0
$765$	−2.76557e46	−0.0726728
$766$	0	0
$767$	−2.02075e47	−0.508614
$768$	0	0
$769$	−2.15810e47	−0.520339	−0.260170	−	0.965563i	$-0.583778\pi$
$769$	−2.15810e47	−0.520339	−0.260170	+	0.965563i	$0.583778\pi$
$770$	0	0
$771$	−8.76167e46	−0.202390
$772$	0	0
$773$	5.97126e46	0.132161	0.0660806	−	0.997814i	$-0.478951\pi$
$773$	5.97126e46	0.132161	0.0660806	+	0.997814i	$0.478951\pi$
$774$	0	0
$775$	−7.83805e47	−1.66238
$776$	0	0
$777$	−1.41564e47	−0.287743
$778$	0	0
$779$	6.05774e47	1.18016
$780$	0	0
$781$	−7.39675e47	−1.38133
$782$	0	0
$783$	2.78131e46	0.0497943
$784$	0	0
$785$	2.67795e46	0.0459677
$786$	0	0
$787$	−2.40999e47	−0.396673	−0.198336	−	0.980134i	$-0.563554\pi$
$787$	−2.40999e47	−0.396673	−0.198336	+	0.980134i	$0.563554\pi$
$788$	0	0
$789$	−1.51485e47	−0.239111
$790$	0	0
$791$	8.62128e47	1.30515
$792$	0	0
$793$	−4.50811e45	−0.00654616
$794$	0	0
$795$	−1.14306e46	−0.0159225
$796$	0	0
$797$	5.30320e47	0.708722	0.354361	−	0.935109i	$-0.384698\pi$
$797$	5.30320e47	0.708722	0.354361	+	0.935109i	$0.384698\pi$
$798$	0	0
$799$	−5.50941e47	−0.706452
$800$	0	0
$801$	−3.75259e47	−0.461737
$802$	0	0
$803$	−1.51350e48	−1.78722
$804$	0	0
$805$	1.27615e46	0.0144634
$806$	0	0
$807$	−1.67860e47	−0.182614
$808$	0	0
$809$	6.52623e47	0.681572	0.340786	−	0.940141i	$-0.389307\pi$
$809$	6.52623e47	0.681572	0.340786	+	0.940141i	$0.389307\pi$
$810$	0	0
$811$	4.54256e47	0.455466	0.227733	−	0.973724i	$-0.426869\pi$
$811$	4.54256e47	0.455466	0.227733	+	0.973724i	$0.426869\pi$
$812$	0	0
$813$	9.61007e46	0.0925194
$814$	0	0
$815$	−3.29994e46	−0.0305074
$816$	0	0
$817$	2.43243e47	0.215962
$818$	0	0
$819$	7.00135e47	0.597032
$820$	0	0
$821$	−3.88660e47	−0.318352	−0.159176	−	0.987250i	$-0.550884\pi$
$821$	−3.88660e47	−0.318352	−0.159176	+	0.987250i	$0.550884\pi$
$822$	0	0
$823$	2.16262e48	1.70170	0.850849	−	0.525410i	$-0.176088\pi$
$823$	2.16262e48	1.70170	0.850849	+	0.525410i	$0.176088\pi$
$824$	0	0
$825$	3.88488e47	0.293689
$826$	0	0
$827$	5.70292e47	0.414244	0.207122	−	0.978315i	$-0.433590\pi$
$827$	5.70292e47	0.414244	0.207122	+	0.978315i	$0.433590\pi$
$828$	0	0
$829$	1.81254e48	1.26514	0.632569	−	0.774504i	$-0.282001\pi$
$829$	1.81254e48	1.26514	0.632569	+	0.774504i	$0.282001\pi$
$830$	0	0
$831$	−8.00417e46	−0.0536907
$832$	0	0
$833$	1.83113e48	1.18053
$834$	0	0
$835$	1.31918e46	0.00817478
$836$	0	0
$837$	−1.14084e48	−0.679603
$838$	0	0
$839$	1.88483e47	0.107945	0.0539723	−	0.998542i	$-0.482812\pi$
$839$	1.88483e47	0.107945	0.0539723	+	0.998542i	$0.482812\pi$
$840$	0	0
$841$	−1.78894e48	−0.985057
$842$	0	0
$843$	−1.99283e47	−0.105515
$844$	0	0
$845$	−9.38766e46	−0.0477993
$846$	0	0
$847$	2.84999e48	1.39562
$848$	0	0
$849$	−5.95748e47	−0.280599
$850$	0	0
$851$	−3.81681e47	−0.172927
$852$	0	0
$853$	3.65000e48	1.59087	0.795434	−	0.606040i	$-0.207243\pi$
$853$	3.65000e48	1.59087	0.795434	+	0.606040i	$0.207243\pi$
$854$	0	0
$855$	−1.32579e47	−0.0555948
$856$	0	0
$857$	−1.58574e48	−0.639807	−0.319903	−	0.947450i	$-0.603650\pi$
$857$	−1.58574e48	−0.639807	−0.319903	+	0.947450i	$0.603650\pi$
$858$	0	0
$859$	−2.88906e48	−1.12168	−0.560840	−	0.827924i	$-0.689522\pi$
$859$	−2.88906e48	−1.12168	−0.560840	+	0.827924i	$0.689522\pi$
$860$	0	0
$861$	9.41431e47	0.351752
$862$	0	0
$863$	−3.38580e46	−0.0121754	−0.00608770	−	0.999981i	$-0.501938\pi$
$863$	−3.38580e46	−0.0121754	−0.00608770	+	0.999981i	$0.501938\pi$
$864$	0	0
$865$	−1.34444e47	−0.0465347
$866$	0	0
$867$	3.80791e47	0.126874
$868$	0	0
$869$	−6.37858e47	−0.204596
$870$	0	0
$871$	5.23695e47	0.161726
$872$	0	0
$873$	4.10448e48	1.22046
$874$	0	0
$875$	−5.80101e47	−0.166100
$876$	0	0
$877$	−4.24252e48	−1.16985	−0.584925	−	0.811087i	$-0.698876\pi$
$877$	−4.24252e48	−1.16985	−0.584925	+	0.811087i	$0.698876\pi$
$878$	0	0
$879$	2.95454e47	0.0784646
$880$	0	0
$881$	−2.17405e48	−0.556119	−0.278059	−	0.960564i	$-0.589691\pi$
$881$	−2.17405e48	−0.556119	−0.278059	+	0.960564i	$0.589691\pi$
$882$	0	0
$883$	−4.77705e48	−1.17708	−0.588542	−	0.808466i	$-0.700298\pi$
$883$	−4.77705e48	−1.17708	−0.588542	+	0.808466i	$0.700298\pi$
$884$	0	0
$885$	5.94679e46	0.0141162
$886$	0	0
$887$	5.12275e48	1.17156	0.585779	−	0.810471i	$-0.300789\pi$
$887$	5.12275e48	1.17156	0.585779	+	0.810471i	$0.300789\pi$
$888$	0	0
$889$	2.29955e48	0.506714
$890$	0	0
$891$	−5.81331e48	−1.23436
$892$	0	0
$893$	−2.64116e48	−0.540437
$894$	0	0
$895$	4.26640e47	0.0841359
$896$	0	0
$897$	−8.55214e46	−0.0162555
$898$	0	0
$899$	−1.11311e48	−0.203941
$900$	0	0
$901$	−9.17434e48	−1.62038
$902$	0	0
$903$	3.78023e47	0.0643681
$904$	0	0
$905$	−4.95769e47	−0.0813912
$906$	0	0
$907$	−7.71888e48	−1.22190	−0.610948	−	0.791671i	$-0.709211\pi$
$907$	−7.71888e48	−1.22190	−0.610948	+	0.791671i	$0.709211\pi$
$908$	0	0
$909$	−1.00300e49	−1.53108
$910$	0	0
$911$	1.06274e49	1.56450	0.782250	−	0.622964i	$-0.214072\pi$
$911$	1.06274e49	1.56450	0.782250	+	0.622964i	$0.214072\pi$
$912$	0	0
$913$	−2.65432e48	−0.376864
$914$	0	0
$915$	1.32668e45	0.000181684 0
$916$	0	0
$917$	1.54394e49	2.03956
$918$	0	0
$919$	−8.89051e48	−1.13298	−0.566488	−	0.824070i	$-0.691698\pi$
$919$	−8.89051e48	−1.13298	−0.566488	+	0.824070i	$0.691698\pi$
$920$	0	0
$921$	2.31210e48	0.284264
$922$	0	0
$923$	−3.69428e48	−0.438229
$924$	0	0
$925$	8.65948e48	0.991182
$926$	0	0
$927$	−2.15552e48	−0.238088
$928$	0	0
$929$	−7.35883e48	−0.784422	−0.392211	−	0.919875i	$-0.628290\pi$
$929$	−7.35883e48	−0.784422	−0.392211	+	0.919875i	$0.628290\pi$
$930$	0	0
$931$	8.77828e48	0.903108
$932$	0	0
$933$	1.14542e47	0.0113741
$934$	0	0
$935$	−1.12203e48	−0.107550
$936$	0	0
$937$	−7.92924e48	−0.733717	−0.366858	−	0.930277i	$-0.619567\pi$
$937$	−7.92924e48	−0.733717	−0.366858	+	0.930277i	$0.619567\pi$
$938$	0	0
$939$	−4.08734e47	−0.0365140
$940$	0	0
$941$	1.44230e49	1.24402	0.622008	−	0.783011i	$-0.286317\pi$
$941$	1.44230e49	1.24402	0.622008	+	0.783011i	$0.286317\pi$
$942$	0	0
$943$	2.53827e48	0.211395
$944$	0	0
$945$	−4.21416e47	−0.0338911
$946$	0	0
$947$	−9.83625e48	−0.763932	−0.381966	−	0.924176i	$-0.624753\pi$
$947$	−9.83625e48	−0.763932	−0.381966	+	0.924176i	$0.624753\pi$
$948$	0	0
$949$	−7.55913e48	−0.566995
$950$	0	0
$951$	2.78198e48	0.201546
$952$	0	0
$953$	−8.19274e48	−0.573319	−0.286660	−	0.958033i	$-0.592545\pi$
$953$	−8.19274e48	−0.573319	−0.286660	+	0.958033i	$0.592545\pi$
$954$	0	0
$955$	−8.31467e47	−0.0562068
$956$	0	0
$957$	5.51707e47	0.0360298
$958$	0	0
$959$	3.61324e49	2.27976
$960$	0	0
$961$	2.92544e49	1.78343
$962$	0	0
$963$	−6.05763e48	−0.356837
$964$	0	0
$965$	−1.06902e48	−0.0608536
$966$	0	0
$967$	1.65015e49	0.907791	0.453895	−	0.891055i	$-0.350034\pi$
$967$	1.65015e49	0.907791	0.453895	+	0.891055i	$0.350034\pi$
$968$	0	0
$969$	4.82088e48	0.256321
$970$	0	0
$971$	3.28602e49	1.68870	0.844352	−	0.535789i	$-0.179986\pi$
$971$	3.28602e49	1.68870	0.844352	+	0.535789i	$0.179986\pi$
$972$	0	0
$973$	1.00483e49	0.499151
$974$	0	0
$975$	1.94029e48	0.0931730
$976$	0	0
$977$	1.17177e49	0.543977	0.271989	−	0.962300i	$-0.412319\pi$
$977$	1.17177e49	0.543977	0.271989	+	0.962300i	$0.412319\pi$
$978$	0	0
$979$	−1.52247e49	−0.683336
$980$	0	0
$981$	−2.91321e48	−0.126425
$982$	0	0
$983$	−3.61572e49	−1.51726	−0.758632	−	0.651519i	$-0.774132\pi$
$983$	−3.61572e49	−1.51726	−0.758632	+	0.651519i	$0.774132\pi$
$984$	0	0
$985$	2.75828e48	0.111928
$986$	0	0
$987$	−4.10461e48	−0.161079
$988$	0	0
$989$	1.01922e48	0.0386838
$990$	0	0
$991$	−2.27623e49	−0.835603	−0.417802	−	0.908538i	$-0.637199\pi$
$991$	−2.27623e49	−0.835603	−0.417802	+	0.908538i	$0.637199\pi$
$992$	0	0
$993$	−9.56445e48	−0.339623
$994$	0	0
$995$	2.02584e48	0.0695863
$996$	0	0
$997$	−5.01160e49	−1.66535	−0.832676	−	0.553761i	$-0.813192\pi$
$997$	−5.01160e49	−1.66535	−0.832676	+	0.553761i	$0.813192\pi$
$998$	0	0
$999$	1.26040e49	0.405209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.34.a.b.1.2		2
4.3	odd	2		1.34.a.a.1.2	✓	2
12.11	even	2		9.34.a.b.1.1		2
20.3	even	4		25.34.b.a.24.2		4
20.7	even	4		25.34.b.a.24.3		4
20.19	odd	2		25.34.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.34.a.a.1.2	✓	2	4.3	odd	2
9.34.a.b.1.1		2	12.11	even	2
16.34.a.b.1.2		2	1.1	even	1	trivial
25.34.a.a.1.1		2	20.19	odd	2
25.34.b.a.24.2		4	20.3	even	4
25.34.b.a.24.3		4	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+1.55221e7 q^{3} +2.04307e10 q^{5} +1.22168e14 q^{7} -5.31812e15 q^{9} +O(q^{10})\)	\(q+1.55221e7 q^{3} +2.04307e10 q^{5} +1.22168e14 q^{7} -5.31812e15 q^{9} -2.15763e17 q^{11} -1.07762e18 q^{13} +3.17129e17 q^{15} +2.54532e20 q^{17} +1.22020e21 q^{19} +1.89631e21 q^{21} +5.11280e21 q^{23} -1.15998e23 q^{25} -1.68837e23 q^{27} -1.64733e23 q^{29} +6.75706e24 q^{31} -3.34910e24 q^{33} +2.49599e24 q^{35} -7.46520e25 q^{37} -1.67269e25 q^{39} +4.96453e26 q^{41} +1.99347e26 q^{43} -1.08653e26 q^{45} -2.16452e27 q^{47} +7.19411e27 q^{49} +3.95088e27 q^{51} -3.60439e28 q^{53} -4.40819e27 q^{55} +1.89401e28 q^{57} +1.87520e29 q^{59} +4.18340e27 q^{61} -6.49706e29 q^{63} -2.20165e28 q^{65} -4.85975e29 q^{67} +7.93615e28 q^{69} +3.42819e30 q^{71} +7.01467e30 q^{73} -1.80053e30 q^{75} -2.63594e31 q^{77} +2.95630e30 q^{79} +2.69431e31 q^{81} +1.23020e31 q^{83} +5.20028e30 q^{85} -2.55701e30 q^{87} +7.05623e31 q^{89} -1.31651e32 q^{91} +1.04884e32 q^{93} +2.49297e31 q^{95} -7.71791e32 q^{97} +1.14745e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 37919880 q^{3} - 181061536500 q^{5} + 67153080066800 q^{7} - 80\!\cdots\!54 q^{9}+O(q^{10}) \) 2 * q - 37919880 * q^3 - 181061536500 * q^5 + 67153080066800 * q^7 - 8021136954970854 * q^9	\( 2 q - 37919880 q^{3} - 181061536500 q^{5} + 67153080066800 q^{7} - 80\!\cdots\!54 q^{9}+ \cdots + 92\!\cdots\!28 q^{99}+O(q^{100}) \) 2 * q - 37919880 * q^3 - 181061536500 * q^5 + 67153080066800 * q^7 - 8021136954970854 * q^9 - 133871815441914264 * q^11 - 2981610478259443940 * q^13 + 11085280069139874000 * q^15 - 79361149261175525340 * q^17 + 1360696443041697171800 * q^19 + 4836438843082801141824 * q^21 - 26163854053674631579440 * q^23 - 191814088084783724781250 * q^25 + 272704742315639906405040 * q^27 - 1674519971630399929109700 * q^29 + 6238301225723632337095616 * q^31 - 7725507376265263259943840 * q^33 + 13581143663546216842308000 * q^35 - 104802741516758180098173620 * q^37 + 85026272438019668836783248 * q^39 + 277566613022002182732188244 * q^41 - 1567661189367072113768291800 * q^43 + 435982958561646097146265500 * q^45 - 5421061461431251260476972640 * q^47 + 2489799365888888933444851314 * q^49 + 21794809591439435388668624496 * q^51 - 26867414785542903092975171220 * q^53 - 20908570805489145209222682000 * q^55 + 11431882183748158902465090720 * q^57 + 305786409635298433026663995400 * q^59 - 5745893983413264859067362436 * q^61 - 500999494400792409792776544720 * q^63 + 361623311364964256200313721000 * q^65 - 1598906337224495341750209565960 * q^67 + 1750848530317427976275742443712 * q^69 + 2669761937956104715736810084976 * q^71 + 946314321191998870965166536340 * q^73 + 2251234781189208709982677125000 * q^75 - 30864620292716415864215760033600 * q^77 + 8535199972765636469730106191200 * q^79 + 18372400450705101408694430349042 * q^81 - 29072605803747945444403983322920 * q^83 + 72477213859794605306963311287000 * q^85 + 78129025791401437906177732680720 * q^87 + 132719467655604316264938597726900 * q^89 - 26902102147452358410548887211744 * q^91 + 132607664882695458799488635592960 * q^93 - 3378716387067841884487537110000 * q^95 - 367787330930535727150839877856060 * q^97 + 926100695332764780030656940070728 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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