LMFDB - Embedded newform 16.34.a.b.1.1

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			1.1
Root			$767.996$ 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)

$f(q)$	$=$	$q-5.34420e7 q^{3} -2.01492e11 q^{5} -5.50153e13 q^{7} -2.70301e15 q^{9} +O(q^{10})$	$q-5.34420e7 q^{3} -2.01492e11 q^{5} -5.50153e13 q^{7} -2.70301e15 q^{9} +8.18909e16 q^{11} -1.90399e18 q^{13} +1.07682e19 q^{15} -3.33893e20 q^{17} +1.40494e20 q^{19} +2.94013e21 q^{21} -3.12767e22 q^{23} -7.58162e22 q^{25} +4.41542e23 q^{27} -1.50979e24 q^{29} -5.18762e23 q^{31} -4.37641e24 q^{33} +1.10852e25 q^{35} -3.01507e25 q^{37} +1.01753e26 q^{39} -2.18887e26 q^{41} -1.76701e27 q^{43} +5.44636e26 q^{45} -3.25654e27 q^{47} -4.70431e27 q^{49} +1.78439e28 q^{51} +9.17652e27 q^{53} -1.65004e28 q^{55} -7.50826e27 q^{57} +1.18267e29 q^{59} -9.92930e27 q^{61} +1.48707e29 q^{63} +3.83640e29 q^{65} -1.11293e30 q^{67} +1.67149e30 q^{69} -7.58425e29 q^{71} -6.06835e30 q^{73} +4.05177e30 q^{75} -4.50525e30 q^{77} +5.57890e30 q^{79} -8.57067e30 q^{81} -4.13746e31 q^{83} +6.72769e31 q^{85} +8.06860e31 q^{87} +6.21572e31 q^{89} +1.04749e32 q^{91} +2.77237e31 q^{93} -2.83084e31 q^{95} +4.04003e32 q^{97} -2.21352e32 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.34420e7	−0.716774	−0.358387	−	0.933573i	$-0.616673\pi$
$3$	−5.34420e7	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$4$	0	0
$5$	−2.01492e11	−0.590545	−0.295273	−	0.955413i	$-0.595411\pi$
$5$	−2.01492e11	−0.590545	−0.295273	+	0.955413i	$0.595411\pi$
$6$	0	0
$7$	−5.50153e13	−0.625699	−0.312850	−	0.949803i	$-0.601284\pi$
$7$	−5.50153e13	−0.625699	−0.312850	+	0.949803i	$0.601284\pi$
$8$	0	0
$9$	−2.70301e15	−0.486236
$10$	0	0
$11$	8.18909e16	0.537349	0.268674	−	0.963231i	$-0.413414\pi$
$11$	8.18909e16	0.537349	0.268674	+	0.963231i	$0.413414\pi$
$12$	0	0
$13$	−1.90399e18	−0.793597	−0.396798	−	0.917906i	$-0.629879\pi$
$13$	−1.90399e18	−0.793597	−0.396798	+	0.917906i	$0.629879\pi$
$14$	0	0
$15$	1.07682e19	0.423287
$16$	0	0
$17$	−3.33893e20	−1.66418	−0.832090	−	0.554640i	$-0.812856\pi$
$17$	−3.33893e20	−1.66418	−0.832090	+	0.554640i	$0.812856\pi$
$18$	0	0
$19$	1.40494e20	0.111743	0.0558717	−	0.998438i	$-0.482206\pi$
$19$	1.40494e20	0.111743	0.0558717	+	0.998438i	$0.482206\pi$
$20$	0	0
$21$	2.94013e21	0.448485
$22$	0	0
$23$	−3.12767e22	−1.06344	−0.531718	−	0.846922i	$-0.678453\pi$
$23$	−3.12767e22	−1.06344	−0.531718	+	0.846922i	$0.678453\pi$
$24$	0	0
$25$	−7.58162e22	−0.651256
$26$	0	0
$27$	4.41542e23	1.06529
$28$	0	0
$29$	−1.50979e24	−1.12034	−0.560168	−	0.828379i	$-0.689264\pi$
$29$	−1.50979e24	−1.12034	−0.560168	+	0.828379i	$0.689264\pi$
$30$	0	0
$31$	−5.18762e23	−0.128086	−0.0640428	−	0.997947i	$-0.520399\pi$
$31$	−5.18762e23	−0.128086	−0.0640428	+	0.997947i	$0.520399\pi$
$32$	0	0
$33$	−4.37641e24	−0.385157
$34$	0	0
$35$	1.10852e25	0.369504
$36$	0	0
$37$	−3.01507e25	−0.401762	−0.200881	−	0.979616i	$-0.564380\pi$
$37$	−3.01507e25	−0.401762	−0.200881	+	0.979616i	$0.564380\pi$
$38$	0	0
$39$	1.01753e26	0.568829
$40$	0	0
$41$	−2.18887e26	−0.536149	−0.268075	−	0.963398i	$-0.586387\pi$
$41$	−2.18887e26	−0.536149	−0.268075	+	0.963398i	$0.586387\pi$
$42$	0	0
$43$	−1.76701e27	−1.97246	−0.986232	−	0.165369i	$-0.947118\pi$
$43$	−1.76701e27	−1.97246	−0.986232	+	0.165369i	$0.947118\pi$
$44$	0	0
$45$	5.44636e26	0.287144
$46$	0	0
$47$	−3.25654e27	−0.837802	−0.418901	−	0.908032i	$-0.637585\pi$
$47$	−3.25654e27	−0.837802	−0.418901	+	0.908032i	$0.637585\pi$
$48$	0	0
$49$	−4.70431e27	−0.608500
$50$	0	0
$51$	1.78439e28	1.19284
$52$	0	0
$53$	9.17652e27	0.325182	0.162591	−	0.986694i	$-0.448015\pi$
$53$	9.17652e27	0.325182	0.162591	+	0.986694i	$0.448015\pi$
$54$	0	0
$55$	−1.65004e28	−0.317329
$56$	0	0
$57$	−7.50826e27	−0.0800948
$58$	0	0
$59$	1.18267e29	0.714174	0.357087	−	0.934071i	$-0.383770\pi$
$59$	1.18267e29	0.714174	0.357087	+	0.934071i	$0.383770\pi$
$60$	0	0
$61$	−9.92930e27	−0.0345920	−0.0172960	−	0.999850i	$-0.505506\pi$
$61$	−9.92930e27	−0.0345920	−0.0172960	+	0.999850i	$0.505506\pi$
$62$	0	0
$63$	1.48707e29	0.304237
$64$	0	0
$65$	3.83640e29	0.468655
$66$	0	0
$67$	−1.11293e30	−0.824583	−0.412291	−	0.911052i	$-0.635271\pi$
$67$	−1.11293e30	−0.824583	−0.412291	+	0.911052i	$0.635271\pi$
$68$	0	0
$69$	1.67149e30	0.762242
$70$	0	0
$71$	−7.58425e29	−0.215849	−0.107924	−	0.994159i	$-0.534420\pi$
$71$	−7.58425e29	−0.215849	−0.107924	+	0.994159i	$0.534420\pi$
$72$	0	0
$73$	−6.06835e30	−1.09205	−0.546026	−	0.837768i	$-0.683860\pi$
$73$	−6.06835e30	−1.09205	−0.546026	+	0.837768i	$0.683860\pi$
$74$	0	0
$75$	4.05177e30	0.466803
$76$	0	0
$77$	−4.50525e30	−0.336219
$78$	0	0
$79$	5.57890e30	0.272711	0.136355	−	0.990660i	$-0.456461\pi$
$79$	5.57890e30	0.272711	0.136355	+	0.990660i	$0.456461\pi$
$80$	0	0
$81$	−8.57067e30	−0.277339
$82$	0	0
$83$	−4.13746e31	−0.895252	−0.447626	−	0.894221i	$-0.647730\pi$
$83$	−4.13746e31	−0.895252	−0.447626	+	0.894221i	$0.647730\pi$
$84$	0	0
$85$	6.72769e31	0.982775
$86$	0	0
$87$	8.06860e31	0.803028
$88$	0	0
$89$	6.21572e31	0.425164	0.212582	−	0.977143i	$-0.431813\pi$
$89$	6.21572e31	0.425164	0.212582	+	0.977143i	$0.431813\pi$
$90$	0	0
$91$	1.04749e32	0.496553
$92$	0	0
$93$	2.77237e31	0.0918084
$94$	0	0
$95$	−2.83084e31	−0.0659896
$96$	0	0
$97$	4.04003e32	0.667808	0.333904	−	0.942607i	$-0.391634\pi$
$97$	4.04003e32	0.667808	0.333904	+	0.942607i	$0.391634\pi$
$98$	0	0
$99$	−2.21352e32	−0.261278
$100$	0	0
$101$	−1.20576e33	−1.02319	−0.511596	−	0.859226i	$-0.670946\pi$
$101$	−1.20576e33	−1.02319	−0.511596	+	0.859226i	$0.670946\pi$
$102$	0	0
$103$	1.69378e33	1.04002	0.520011	−	0.854159i	$-0.325928\pi$
$103$	1.69378e33	1.04002	0.520011	+	0.854159i	$0.325928\pi$
$104$	0	0
$105$	−5.92413e32	−0.264851
$106$	0	0
$107$	−7.07121e32	−0.231559	−0.115780	−	0.993275i	$-0.536937\pi$
$107$	−7.07121e32	−0.231559	−0.115780	+	0.993275i	$0.536937\pi$
$108$	0	0
$109$	4.42814e33	1.06828	0.534139	−	0.845397i	$-0.320636\pi$
$109$	4.42814e33	1.06828	0.534139	+	0.845397i	$0.320636\pi$
$110$	0	0
$111$	1.61131e33	0.287973
$112$	0	0
$113$	1.36496e34	1.81687	0.908437	−	0.418023i	$-0.137277\pi$
$113$	1.36496e34	1.81687	0.908437	+	0.418023i	$0.137277\pi$
$114$	0	0
$115$	6.30200e33	0.628007
$116$	0	0
$117$	5.14652e33	0.385875
$118$	0	0
$119$	1.83692e34	1.04128
$120$	0	0
$121$	−1.65190e34	−0.711256
$122$	0	0
$123$	1.16977e34	0.384298
$124$	0	0
$125$	3.87332e34	0.975142
$126$	0	0
$127$	7.12185e34	1.37985	0.689925	−	0.723881i	$-0.257644\pi$
$127$	7.12185e34	1.37985	0.689925	+	0.723881i	$0.257644\pi$
$128$	0	0
$129$	9.44324e34	1.41381
$130$	0	0
$131$	2.95217e34	0.342898	0.171449	−	0.985193i	$-0.445155\pi$
$131$	2.95217e34	0.342898	0.171449	+	0.985193i	$0.445155\pi$
$132$	0	0
$133$	−7.72929e33	−0.0699178
$134$	0	0
$135$	−8.89673e34	−0.629105
$136$	0	0
$137$	8.90834e34	0.494205	0.247103	−	0.968989i	$-0.420521\pi$
$137$	8.90834e34	0.494205	0.247103	+	0.968989i	$0.420521\pi$
$138$	0	0
$139$	3.61707e35	1.57984	0.789920	−	0.613210i	$-0.210122\pi$
$139$	3.61707e35	1.57984	0.789920	+	0.613210i	$0.210122\pi$
$140$	0	0
$141$	1.74036e35	0.600515
$142$	0	0
$143$	−1.55920e35	−0.426438
$144$	0	0
$145$	3.04210e35	0.661610
$146$	0	0
$147$	2.51408e35	0.436157
$148$	0	0
$149$	1.73500e35	0.240839	0.120419	−	0.992723i	$-0.461576\pi$
$149$	1.73500e35	0.240839	0.120419	+	0.992723i	$0.461576\pi$
$150$	0	0
$151$	1.35744e36	1.51217	0.756085	−	0.654473i	$-0.227109\pi$
$151$	1.35744e36	1.51217	0.756085	+	0.654473i	$0.227109\pi$
$152$	0	0
$153$	9.02518e35	0.809184
$154$	0	0
$155$	1.04527e35	0.0756404
$156$	0	0
$157$	−2.78049e36	−1.62846	−0.814229	−	0.580543i	$-0.802840\pi$
$157$	−2.78049e36	−1.62846	−0.814229	+	0.580543i	$0.802840\pi$
$158$	0	0
$159$	−4.90411e35	−0.233082
$160$	0	0
$161$	1.72069e36	0.665390
$162$	0	0
$163$	−2.81381e36	−0.887565	−0.443782	−	0.896135i	$-0.646364\pi$
$163$	−2.81381e36	−0.887565	−0.443782	+	0.896135i	$0.646364\pi$
$164$	0	0
$165$	8.81813e35	0.227453
$166$	0	0
$167$	−6.77893e36	−1.43331	−0.716653	−	0.697430i	$-0.754327\pi$
$167$	−6.77893e36	−1.43331	−0.716653	+	0.697430i	$0.754327\pi$
$168$	0	0
$169$	−2.13094e36	−0.370204
$170$	0	0
$171$	−3.79756e35	−0.0543336
$172$	0	0
$173$	−1.42570e37	−1.68370	−0.841852	−	0.539708i	$-0.818535\pi$
$173$	−1.42570e37	−1.68370	−0.841852	+	0.539708i	$0.818535\pi$
$174$	0	0
$175$	4.17105e36	0.407490
$176$	0	0
$177$	−6.32040e36	−0.511901
$178$	0	0
$179$	−7.37934e36	−0.496527	−0.248263	−	0.968693i	$-0.579860\pi$
$179$	−7.37934e36	−0.496527	−0.248263	+	0.968693i	$0.579860\pi$
$180$	0	0
$181$	2.39313e36	0.134051	0.0670254	−	0.997751i	$-0.478649\pi$
$181$	2.39313e36	0.134051	0.0670254	+	0.997751i	$0.478649\pi$
$182$	0	0
$183$	5.30642e35	0.0247947
$184$	0	0
$185$	6.07513e36	0.237259
$186$	0	0
$187$	−2.73428e37	−0.894245
$188$	0	0
$189$	−2.42915e37	−0.666554
$190$	0	0
$191$	6.98026e35	0.0160998	0.00804991	−	0.999968i	$-0.497438\pi$
$191$	6.98026e35	0.0160998	0.00804991	+	0.999968i	$0.497438\pi$
$192$	0	0
$193$	6.67098e36	0.129567	0.0647835	−	0.997899i	$-0.479364\pi$
$193$	6.67098e36	0.129567	0.0647835	+	0.997899i	$0.479364\pi$
$194$	0	0
$195$	−2.05025e37	−0.335920
$196$	0	0
$197$	−8.89563e37	−1.23164	−0.615821	−	0.787886i	$-0.711175\pi$
$197$	−8.89563e37	−1.23164	−0.615821	+	0.787886i	$0.711175\pi$
$198$	0	0
$199$	−1.05106e38	−1.23183	−0.615916	−	0.787812i	$-0.711214\pi$
$199$	−1.05106e38	−1.23183	−0.615916	+	0.787812i	$0.711214\pi$
$200$	0	0
$201$	5.94773e37	0.591039
$202$	0	0
$203$	8.30613e37	0.700994
$204$	0	0
$205$	4.41040e37	0.316621
$206$	0	0
$207$	8.45412e37	0.517080
$208$	0	0
$209$	1.15051e37	0.0600452
$210$	0	0
$211$	5.59870e37	0.249705	0.124852	−	0.992175i	$-0.460154\pi$
$211$	5.59870e37	0.249705	0.124852	+	0.992175i	$0.460154\pi$
$212$	0	0
$213$	4.05318e37	0.154715
$214$	0	0
$215$	3.56038e38	1.16483
$216$	0	0
$217$	2.85398e37	0.0801431
$218$	0	0
$219$	3.24305e38	0.782755
$220$	0	0
$221$	6.35730e38	1.32069
$222$	0	0
$223$	−4.65676e38	−0.833784	−0.416892	−	0.908956i	$-0.636881\pi$
$223$	−4.65676e38	−0.833784	−0.416892	+	0.908956i	$0.636881\pi$
$224$	0	0
$225$	2.04932e38	0.316664
$226$	0	0
$227$	2.61802e38	0.349580	0.174790	−	0.984606i	$-0.444075\pi$
$227$	2.61802e38	0.349580	0.174790	+	0.984606i	$0.444075\pi$
$228$	0	0
$229$	9.20963e38	1.06404	0.532018	−	0.846733i	$-0.321434\pi$
$229$	9.20963e38	1.06404	0.532018	+	0.846733i	$0.321434\pi$
$230$	0	0
$231$	2.40770e38	0.240993
$232$	0	0
$233$	7.13755e37	0.0619693	0.0309847	−	0.999520i	$-0.490136\pi$
$233$	7.13755e37	0.0619693	0.0309847	+	0.999520i	$0.490136\pi$
$234$	0	0
$235$	6.56167e38	0.494760
$236$	0	0
$237$	−2.98148e38	−0.195472
$238$	0	0
$239$	1.83178e39	1.04546	0.522731	−	0.852498i	$-0.324913\pi$
$239$	1.83178e39	1.04546	0.522731	+	0.852498i	$0.324913\pi$
$240$	0	0
$241$	−3.05313e39	−1.51867	−0.759336	−	0.650698i	$-0.774476\pi$
$241$	−3.05313e39	−1.51867	−0.759336	+	0.650698i	$0.774476\pi$
$242$	0	0
$243$	−1.99652e39	−0.866505
$244$	0	0
$245$	9.47883e38	0.359347
$246$	0	0
$247$	−2.67499e38	−0.0886793
$248$	0	0
$249$	2.21114e39	0.641693
$250$	0	0
$251$	−3.03407e39	−0.771631	−0.385815	−	0.922576i	$-0.626080\pi$
$251$	−3.03407e39	−0.771631	−0.385815	+	0.922576i	$0.626080\pi$
$252$	0	0
$253$	−2.56127e39	−0.571435
$254$	0	0
$255$	−3.59541e39	−0.704427
$256$	0	0
$257$	−2.91186e39	−0.501503	−0.250752	−	0.968051i	$-0.580678\pi$
$257$	−2.91186e39	−0.501503	−0.250752	+	0.968051i	$0.580678\pi$
$258$	0	0
$259$	1.65875e39	0.251382
$260$	0	0
$261$	4.08097e39	0.544748
$262$	0	0
$263$	5.62907e39	0.662471	0.331236	−	0.943548i	$-0.392534\pi$
$263$	5.62907e39	0.662471	0.331236	+	0.943548i	$0.392534\pi$
$264$	0	0
$265$	−1.84900e39	−0.192035
$266$	0	0
$267$	−3.32180e39	−0.304746
$268$	0	0
$269$	−9.77080e39	−0.792533	−0.396266	−	0.918136i	$-0.629694\pi$
$269$	−9.77080e39	−0.792533	−0.396266	+	0.918136i	$0.629694\pi$
$270$	0	0
$271$	−2.38779e40	−1.71397	−0.856985	−	0.515342i	$-0.827665\pi$
$271$	−2.38779e40	−1.71397	−0.856985	+	0.515342i	$0.827665\pi$
$272$	0	0
$273$	−5.59798e39	−0.355916
$274$	0	0
$275$	−6.20865e39	−0.349952
$276$	0	0
$277$	−1.39835e40	−0.699358	−0.349679	−	0.936870i	$-0.613709\pi$
$277$	−1.39835e40	−0.699358	−0.349679	+	0.936870i	$0.613709\pi$
$278$	0	0
$279$	1.40222e39	0.0622798
$280$	0	0
$281$	−2.89927e40	−1.14455	−0.572274	−	0.820062i	$-0.693939\pi$
$281$	−2.89927e40	−1.14455	−0.572274	+	0.820062i	$0.693939\pi$
$282$	0	0
$283$	1.50945e40	0.530081	0.265040	−	0.964237i	$-0.414615\pi$
$283$	1.50945e40	0.530081	0.265040	+	0.964237i	$0.414615\pi$
$284$	0	0
$285$	1.51286e39	0.0472996
$286$	0	0
$287$	1.20421e40	0.335468
$288$	0	0
$289$	7.12303e40	1.76950
$290$	0	0
$291$	−2.15907e40	−0.478667
$292$	0	0
$293$	−1.40305e40	−0.277816	−0.138908	−	0.990305i	$-0.544359\pi$
$293$	−1.40305e40	−0.277816	−0.138908	+	0.990305i	$0.544359\pi$
$294$	0	0
$295$	−2.38298e40	−0.421752
$296$	0	0
$297$	3.61582e40	0.572435
$298$	0	0
$299$	5.95505e40	0.843939
$300$	0	0
$301$	9.72124e40	1.23417
$302$	0	0
$303$	6.44381e40	0.733398
$304$	0	0
$305$	2.00068e39	0.0204282
$306$	0	0
$307$	2.92194e40	0.267848	0.133924	−	0.990992i	$-0.457242\pi$
$307$	2.92194e40	0.267848	0.133924	+	0.990992i	$0.457242\pi$
$308$	0	0
$309$	−9.05190e40	−0.745461
$310$	0	0
$311$	2.24927e41	1.66531	0.832653	−	0.553794i	$-0.186821\pi$
$311$	2.24927e41	1.66531	0.832653	+	0.553794i	$0.186821\pi$
$312$	0	0
$313$	1.17836e41	0.784868	0.392434	−	0.919780i	$-0.371633\pi$
$313$	1.17836e41	0.784868	0.392434	+	0.919780i	$0.371633\pi$
$314$	0	0
$315$	−2.99633e40	−0.179666
$316$	0	0
$317$	−9.02704e40	−0.487604	−0.243802	−	0.969825i	$-0.578395\pi$
$317$	−9.02704e40	−0.487604	−0.243802	+	0.969825i	$0.578395\pi$
$318$	0	0
$319$	−1.23638e41	−0.602012
$320$	0	0
$321$	3.77900e40	0.165975
$322$	0	0
$323$	−4.69099e40	−0.185961
$324$	0	0
$325$	1.44353e41	0.516835
$326$	0	0
$327$	−2.36649e41	−0.765714
$328$	0	0
$329$	1.79159e41	0.524212
$330$	0	0
$331$	2.91394e41	0.771469	0.385734	−	0.922610i	$-0.373948\pi$
$331$	2.91394e41	0.771469	0.385734	+	0.922610i	$0.373948\pi$
$332$	0	0
$333$	8.14977e40	0.195351
$334$	0	0
$335$	2.24247e41	0.486954
$336$	0	0
$337$	−5.69474e41	−1.12094	−0.560468	−	0.828176i	$-0.689379\pi$
$337$	−5.69474e41	−1.12094	−0.560468	+	0.828176i	$0.689379\pi$
$338$	0	0
$339$	−7.29460e41	−1.30229
$340$	0	0
$341$	−4.24819e40	−0.0688266
$342$	0	0
$343$	6.84132e41	1.00644
$344$	0	0
$345$	−3.36792e41	−0.450139
$346$	0	0
$347$	−4.85020e41	−0.589283	−0.294641	−	0.955608i	$-0.595200\pi$
$347$	−4.85020e41	−0.589283	−0.294641	+	0.955608i	$0.595200\pi$
$348$	0	0
$349$	5.38533e41	0.595104	0.297552	−	0.954706i	$-0.403830\pi$
$349$	5.38533e41	0.595104	0.297552	+	0.954706i	$0.403830\pi$
$350$	0	0
$351$	−8.40692e41	−0.845414
$352$	0	0
$353$	−9.77578e41	−0.895094	−0.447547	−	0.894260i	$-0.647702\pi$
$353$	−9.77578e41	−0.895094	−0.447547	+	0.894260i	$0.647702\pi$
$354$	0	0
$355$	1.52817e41	0.127469
$356$	0	0
$357$	−9.81689e41	−0.746360
$358$	0	0
$359$	−1.18143e42	−0.819124	−0.409562	−	0.912282i	$-0.634318\pi$
$359$	−1.18143e42	−0.819124	−0.409562	+	0.912282i	$0.634318\pi$
$360$	0	0
$361$	−1.56103e42	−0.987513
$362$	0	0
$363$	8.82811e41	0.509810
$364$	0	0
$365$	1.22273e42	0.644907
$366$	0	0
$367$	−7.16875e41	−0.345504	−0.172752	−	0.984965i	$-0.555266\pi$
$367$	−7.16875e41	−0.345504	−0.172752	+	0.984965i	$0.555266\pi$
$368$	0	0
$369$	5.91654e41	0.260695
$370$	0	0
$371$	−5.04849e41	−0.203466
$372$	0	0
$373$	−6.80691e41	−0.251047	−0.125523	−	0.992091i	$-0.540061\pi$
$373$	−6.80691e41	−0.251047	−0.125523	+	0.992091i	$0.540061\pi$
$374$	0	0
$375$	−2.06998e42	−0.698956
$376$	0	0
$377$	2.87462e42	0.889096
$378$	0	0
$379$	−5.14200e42	−1.45742	−0.728711	−	0.684821i	$-0.759880\pi$
$379$	−5.14200e42	−1.45742	−0.728711	+	0.684821i	$0.759880\pi$
$380$	0	0
$381$	−3.80606e42	−0.989040
$382$	0	0
$383$	6.82496e42	1.62675	0.813375	−	0.581740i	$-0.197628\pi$
$383$	6.82496e42	1.62675	0.813375	+	0.581740i	$0.197628\pi$
$384$	0	0
$385$	9.07773e41	0.198552
$386$	0	0
$387$	4.77624e42	0.959082
$388$	0	0
$389$	3.03602e42	0.559933	0.279966	−	0.960010i	$-0.409677\pi$
$389$	3.03602e42	0.559933	0.279966	+	0.960010i	$0.409677\pi$
$390$	0	0
$391$	1.04431e43	1.76975
$392$	0	0
$393$	−1.57770e42	−0.245781
$394$	0	0
$395$	−1.12411e42	−0.161048
$396$	0	0
$397$	8.54761e42	1.12668	0.563341	−	0.826225i	$-0.309516\pi$
$397$	8.54761e42	1.12668	0.563341	+	0.826225i	$0.309516\pi$
$398$	0	0
$399$	4.13069e41	0.0501152
$400$	0	0
$401$	4.64638e42	0.519078	0.259539	−	0.965733i	$-0.416429\pi$
$401$	4.64638e42	0.519078	0.259539	+	0.965733i	$0.416429\pi$
$402$	0	0
$403$	9.87719e41	0.101648
$404$	0	0
$405$	1.72692e42	0.163782
$406$	0	0
$407$	−2.46907e42	−0.215886
$408$	0	0
$409$	1.73477e43	1.39897	0.699483	−	0.714649i	$-0.253414\pi$
$409$	1.73477e43	1.39897	0.699483	+	0.714649i	$0.253414\pi$
$410$	0	0
$411$	−4.76080e42	−0.354233
$412$	0	0
$413$	−6.50647e42	−0.446858
$414$	0	0
$415$	8.33667e42	0.528687
$416$	0	0
$417$	−1.93304e43	−1.13239
$418$	0	0
$419$	−1.59552e43	−0.863717	−0.431858	−	0.901941i	$-0.642142\pi$
$419$	−1.59552e43	−0.863717	−0.431858	+	0.901941i	$0.642142\pi$
$420$	0	0
$421$	−2.07154e43	−1.03667	−0.518333	−	0.855179i	$-0.673447\pi$
$421$	−2.07154e43	−1.03667	−0.518333	+	0.855179i	$0.673447\pi$
$422$	0	0
$423$	8.80247e42	0.407369
$424$	0	0
$425$	2.53145e43	1.08381
$426$	0	0
$427$	5.46263e41	0.0216442
$428$	0	0
$429$	8.33266e42	0.305660
$430$	0	0
$431$	2.15255e43	0.731272	0.365636	−	0.930758i	$-0.380851\pi$
$431$	2.15255e43	0.731272	0.365636	+	0.930758i	$0.380851\pi$
$432$	0	0
$433$	−2.62411e43	−0.825908	−0.412954	−	0.910752i	$-0.635503\pi$
$433$	−2.62411e43	−0.825908	−0.412954	+	0.910752i	$0.635503\pi$
$434$	0	0
$435$	−1.62576e43	−0.474225
$436$	0	0
$437$	−4.39417e42	−0.118832
$438$	0	0
$439$	−1.50091e43	−0.376434	−0.188217	−	0.982127i	$-0.560271\pi$
$439$	−1.50091e43	−0.376434	−0.188217	+	0.982127i	$0.560271\pi$
$440$	0	0
$441$	1.27158e43	0.295875
$442$	0	0
$443$	−1.03686e43	−0.223902	−0.111951	−	0.993714i	$-0.535710\pi$
$443$	−1.03686e43	−0.223902	−0.111951	+	0.993714i	$0.535710\pi$
$444$	0	0
$445$	−1.25242e43	−0.251079
$446$	0	0
$447$	−9.27219e42	−0.172627
$448$	0	0
$449$	−3.05007e42	−0.0527528	−0.0263764	−	0.999652i	$-0.508397\pi$
$449$	−3.05007e42	−0.0527528	−0.0263764	+	0.999652i	$0.508397\pi$
$450$	0	0
$451$	−1.79248e43	−0.288099
$452$	0	0
$453$	−7.25443e43	−1.08388
$454$	0	0
$455$	−2.11061e43	−0.293237
$456$	0	0
$457$	−8.15124e43	−1.05343	−0.526716	−	0.850042i	$-0.676577\pi$
$457$	−8.15124e43	−1.05343	−0.526716	+	0.850042i	$0.676577\pi$
$458$	0	0
$459$	−1.47428e44	−1.77284
$460$	0	0
$461$	9.49311e43	1.06253	0.531267	−	0.847204i	$-0.321716\pi$
$461$	9.49311e43	1.06253	0.531267	+	0.847204i	$0.321716\pi$
$462$	0	0
$463$	6.84730e43	0.713564	0.356782	−	0.934188i	$-0.383874\pi$
$463$	6.84730e43	0.713564	0.356782	+	0.934188i	$0.383874\pi$
$464$	0	0
$465$	−5.58611e42	−0.0542170
$466$	0	0
$467$	−1.13867e44	−1.02961	−0.514803	−	0.857308i	$-0.672135\pi$
$467$	−1.13867e44	−1.02961	−0.514803	+	0.857308i	$0.672135\pi$
$468$	0	0
$469$	6.12282e43	0.515941
$470$	0	0
$471$	1.48595e44	1.16724
$472$	0	0
$473$	−1.44702e44	−1.05990
$474$	0	0
$475$	−1.06517e43	−0.0727736
$476$	0	0
$477$	−2.48042e43	−0.158115
$478$	0	0
$479$	2.43710e44	1.44990	0.724948	−	0.688804i	$-0.241864\pi$
$479$	2.43710e44	1.44990	0.724948	+	0.688804i	$0.241864\pi$
$480$	0	0
$481$	5.74067e43	0.318837
$482$	0	0
$483$	−9.19573e43	−0.476934
$484$	0	0
$485$	−8.14036e43	−0.394371
$486$	0	0
$487$	1.55812e44	0.705300	0.352650	−	0.935755i	$-0.385281\pi$
$487$	1.55812e44	0.705300	0.352650	+	0.935755i	$0.385281\pi$
$488$	0	0
$489$	1.50376e44	0.636183
$490$	0	0
$491$	2.64492e43	0.104609	0.0523044	−	0.998631i	$-0.483343\pi$
$491$	2.64492e43	0.104609	0.0523044	+	0.998631i	$0.483343\pi$
$492$	0	0
$493$	5.04108e44	1.86444
$494$	0	0
$495$	4.46007e43	0.154297
$496$	0	0
$497$	4.17250e43	0.135057
$498$	0	0
$499$	−3.31565e44	−1.00441	−0.502203	−	0.864750i	$-0.667477\pi$
$499$	−3.31565e44	−1.00441	−0.502203	+	0.864750i	$0.667477\pi$
$500$	0	0
$501$	3.62279e44	1.02736
$502$	0	0
$503$	4.39485e44	1.16700	0.583502	−	0.812112i	$-0.301682\pi$
$503$	4.39485e44	1.16700	0.583502	+	0.812112i	$0.301682\pi$
$504$	0	0
$505$	2.42951e44	0.604242
$506$	0	0
$507$	1.13882e44	0.265352
$508$	0	0
$509$	−7.37330e44	−1.60997	−0.804986	−	0.593294i	$-0.797827\pi$
$509$	−7.37330e44	−1.60997	−0.804986	+	0.593294i	$0.797827\pi$
$510$	0	0
$511$	3.33852e44	0.683296
$512$	0	0
$513$	6.20338e43	0.119040
$514$	0	0
$515$	−3.41284e44	−0.614181
$516$	0	0
$517$	−2.66681e44	−0.450192
$518$	0	0
$519$	7.61920e44	1.20684
$520$	0	0
$521$	1.67201e43	0.0248552	0.0124276	−	0.999923i	$-0.496044\pi$
$521$	1.67201e43	0.0248552	0.0124276	+	0.999923i	$0.496044\pi$
$522$	0	0
$523$	−1.06261e45	−1.48284	−0.741420	−	0.671041i	$-0.765847\pi$
$523$	−1.06261e45	−1.48284	−0.741420	+	0.671041i	$0.765847\pi$
$524$	0	0
$525$	−2.22909e44	−0.292078
$526$	0	0
$527$	1.73211e44	0.213158
$528$	0	0
$529$	1.13224e44	0.130894
$530$	0	0
$531$	−3.19676e44	−0.347257
$532$	0	0
$533$	4.16759e44	0.425486
$534$	0	0
$535$	1.42480e44	0.136746
$536$	0	0
$537$	3.94367e44	0.355897
$538$	0	0
$539$	−3.85240e44	−0.326977
$540$	0	0
$541$	−1.73107e45	−1.38217	−0.691084	−	0.722774i	$-0.742867\pi$
$541$	−1.73107e45	−1.38217	−0.691084	+	0.722774i	$0.742867\pi$
$542$	0	0
$543$	−1.27894e44	−0.0960841
$544$	0	0
$545$	−8.92237e44	−0.630867
$546$	0	0
$547$	−1.88809e45	−1.25670	−0.628350	−	0.777931i	$-0.716269\pi$
$547$	−1.88809e45	−1.25670	−0.628350	+	0.777931i	$0.716269\pi$
$548$	0	0
$549$	2.68390e43	0.0168199
$550$	0	0
$551$	−2.12115e44	−0.125190
$552$	0	0
$553$	−3.06925e44	−0.170635
$554$	0	0
$555$	−3.24667e44	−0.170061
$556$	0	0
$557$	1.82254e45	0.899636	0.449818	−	0.893120i	$-0.351489\pi$
$557$	1.82254e45	0.899636	0.449818	+	0.893120i	$0.351489\pi$
$558$	0	0
$559$	3.36437e45	1.56534
$560$	0	0
$561$	1.46125e45	0.640972
$562$	0	0
$563$	−3.54966e45	−1.46824	−0.734122	−	0.679018i	$-0.762406\pi$
$563$	−3.54966e45	−1.46824	−0.734122	+	0.679018i	$0.762406\pi$
$564$	0	0
$565$	−2.75028e45	−1.07295
$566$	0	0
$567$	4.71517e44	0.173531
$568$	0	0
$569$	−3.23817e45	−1.12447	−0.562234	−	0.826978i	$-0.690058\pi$
$569$	−3.23817e45	−1.12447	−0.562234	+	0.826978i	$0.690058\pi$
$570$	0	0
$571$	−1.75221e45	−0.574237	−0.287118	−	0.957895i	$-0.592697\pi$
$571$	−1.75221e45	−0.574237	−0.287118	+	0.957895i	$0.592697\pi$
$572$	0	0
$573$	−3.73039e43	−0.0115399
$574$	0	0
$575$	2.37128e45	0.692569
$576$	0	0
$577$	−2.22828e45	−0.614566	−0.307283	−	0.951618i	$-0.599420\pi$
$577$	−2.22828e45	−0.614566	−0.307283	+	0.951618i	$0.599420\pi$
$578$	0	0
$579$	−3.56511e44	−0.0928702
$580$	0	0
$581$	2.27624e45	0.560159
$582$	0	0
$583$	7.51473e44	0.174736
$584$	0	0
$585$	−1.03698e45	−0.227877
$586$	0	0
$587$	−1.91072e45	−0.396889	−0.198444	−	0.980112i	$-0.563589\pi$
$587$	−1.91072e45	−0.396889	−0.198444	+	0.980112i	$0.563589\pi$
$588$	0	0
$589$	−7.28827e43	−0.0143127
$590$	0	0
$591$	4.75400e45	0.882808
$592$	0	0
$593$	8.87275e45	1.55832	0.779158	−	0.626827i	$-0.215647\pi$
$593$	8.87275e45	1.55832	0.779158	+	0.626827i	$0.215647\pi$
$594$	0	0
$595$	−3.70126e45	−0.614921
$596$	0	0
$597$	5.61707e45	0.882945
$598$	0	0
$599$	1.96939e45	0.292947	0.146474	−	0.989215i	$-0.453208\pi$
$599$	1.96939e45	0.292947	0.146474	+	0.989215i	$0.453208\pi$
$600$	0	0
$601$	6.49113e45	0.913885	0.456942	−	0.889496i	$-0.348945\pi$
$601$	6.49113e45	0.913885	0.456942	+	0.889496i	$0.348945\pi$
$602$	0	0
$603$	3.00827e45	0.400941
$604$	0	0
$605$	3.32846e45	0.420029
$606$	0	0
$607$	1.08656e46	1.29849	0.649247	−	0.760577i	$-0.275084\pi$
$607$	1.08656e46	1.29849	0.649247	+	0.760577i	$0.275084\pi$
$608$	0	0
$609$	−4.43896e45	−0.502454
$610$	0	0
$611$	6.20043e45	0.664877
$612$	0	0
$613$	8.70542e45	0.884488	0.442244	−	0.896895i	$-0.354182\pi$
$613$	8.70542e45	0.884488	0.442244	+	0.896895i	$0.354182\pi$
$614$	0	0
$615$	−2.35701e45	−0.226945
$616$	0	0
$617$	−1.12685e46	−1.02840	−0.514199	−	0.857671i	$-0.671911\pi$
$617$	−1.12685e46	−1.02840	−0.514199	+	0.857671i	$0.671911\pi$
$618$	0	0
$619$	1.07685e46	0.931663	0.465831	−	0.884874i	$-0.345755\pi$
$619$	1.07685e46	0.931663	0.465831	+	0.884874i	$0.345755\pi$
$620$	0	0
$621$	−1.38099e46	−1.13287
$622$	0	0
$623$	−3.41959e45	−0.266025
$624$	0	0
$625$	1.02173e45	0.0753905
$626$	0	0
$627$	−6.14858e44	−0.0430388
$628$	0	0
$629$	1.00671e46	0.668605
$630$	0	0
$631$	−2.47581e46	−1.56039	−0.780196	−	0.625535i	$-0.784881\pi$
$631$	−2.47581e46	−1.56039	−0.780196	+	0.625535i	$0.784881\pi$
$632$	0	0
$633$	−2.99206e45	−0.178982
$634$	0	0
$635$	−1.43500e46	−0.814864
$636$	0	0
$637$	8.95698e45	0.482904
$638$	0	0
$639$	2.05003e45	0.104953
$640$	0	0
$641$	−6.66844e45	−0.324240	−0.162120	−	0.986771i	$-0.551833\pi$
$641$	−6.66844e45	−0.324240	−0.162120	+	0.986771i	$0.551833\pi$
$642$	0	0
$643$	8.88748e42	0.000410485 0	0.000205243	−	1.00000i	$-0.499935\pi$
$643$	8.88748e42	0.000410485 0	0.000205243		1.00000i	$0.499935\pi$
$644$	0	0
$645$	−1.90274e46	−0.834919
$646$	0	0
$647$	1.36920e46	0.570884	0.285442	−	0.958396i	$-0.407860\pi$
$647$	1.36920e46	0.570884	0.285442	+	0.958396i	$0.407860\pi$
$648$	0	0
$649$	9.68495e45	0.383760
$650$	0	0
$651$	−1.52523e45	−0.0574444
$652$	0	0
$653$	−1.32140e46	−0.473115	−0.236557	−	0.971618i	$-0.576019\pi$
$653$	−1.32140e46	−0.473115	−0.236557	+	0.971618i	$0.576019\pi$
$654$	0	0
$655$	−5.94840e45	−0.202497
$656$	0	0
$657$	1.64028e46	0.530995
$658$	0	0
$659$	−3.14029e46	−0.966852	−0.483426	−	0.875385i	$-0.660608\pi$
$659$	−3.14029e46	−0.966852	−0.483426	+	0.875385i	$0.660608\pi$
$660$	0	0
$661$	−5.62366e46	−1.64701	−0.823503	−	0.567313i	$-0.807983\pi$
$661$	−5.62366e46	−1.64701	−0.823503	+	0.567313i	$0.807983\pi$
$662$	0	0
$663$	−3.39747e46	−0.946635
$664$	0	0
$665$	1.55739e45	0.0412896
$666$	0	0
$667$	4.72211e46	1.19141
$668$	0	0
$669$	2.48866e46	0.597634
$670$	0	0
$671$	−8.13119e44	−0.0185880
$672$	0	0
$673$	1.86687e45	0.0406317	0.0203159	−	0.999794i	$-0.493533\pi$
$673$	1.86687e45	0.0406317	0.0203159	+	0.999794i	$0.493533\pi$
$674$	0	0
$675$	−3.34760e46	−0.693779
$676$	0	0
$677$	−3.19535e46	−0.630674	−0.315337	−	0.948980i	$-0.602118\pi$
$677$	−3.19535e46	−0.630674	−0.315337	+	0.948980i	$0.602118\pi$
$678$	0	0
$679$	−2.22264e46	−0.417847
$680$	0	0
$681$	−1.39912e46	−0.250570
$682$	0	0
$683$	9.67204e46	1.65035	0.825176	−	0.564876i	$-0.191076\pi$
$683$	9.67204e46	1.65035	0.825176	+	0.564876i	$0.191076\pi$
$684$	0	0
$685$	−1.79496e46	−0.291851
$686$	0	0
$687$	−4.92181e46	−0.762673
$688$	0	0
$689$	−1.74720e46	−0.258063
$690$	0	0
$691$	4.21417e46	0.593368	0.296684	−	0.954976i	$-0.404119\pi$
$691$	4.21417e46	0.593368	0.296684	+	0.954976i	$0.404119\pi$
$692$	0	0
$693$	1.21777e46	0.163481
$694$	0	0
$695$	−7.28813e46	−0.932968
$696$	0	0
$697$	7.30848e46	0.892249
$698$	0	0
$699$	−3.81445e45	−0.0444180
$700$	0	0
$701$	1.07382e47	1.19284	0.596422	−	0.802671i	$-0.296589\pi$
$701$	1.07382e47	1.19284	0.596422	+	0.802671i	$0.296589\pi$
$702$	0	0
$703$	−4.23598e45	−0.0448943
$704$	0	0
$705$	−3.50669e46	−0.354631
$706$	0	0
$707$	6.63351e46	0.640211
$708$	0	0
$709$	9.99645e46	0.920837	0.460419	−	0.887702i	$-0.347699\pi$
$709$	9.99645e46	0.920837	0.460419	+	0.887702i	$0.347699\pi$
$710$	0	0
$711$	−1.50798e46	−0.132602
$712$	0	0
$713$	1.62251e46	0.136211
$714$	0	0
$715$	3.14166e46	0.251831
$716$	0	0
$717$	−9.78940e46	−0.749359
$718$	0	0
$719$	7.93598e46	0.580195	0.290097	−	0.956997i	$-0.406312\pi$
$719$	7.93598e46	0.580195	0.290097	+	0.956997i	$0.406312\pi$
$720$	0	0
$721$	−9.31838e46	−0.650741
$722$	0	0
$723$	1.63165e47	1.08854
$724$	0	0
$725$	1.14466e47	0.729626
$726$	0	0
$727$	−1.33485e47	−0.813046	−0.406523	−	0.913641i	$-0.633259\pi$
$727$	−1.33485e47	−0.813046	−0.406523	+	0.913641i	$0.633259\pi$
$728$	0	0
$729$	1.54343e47	0.898427
$730$	0	0
$731$	5.89992e47	3.28254
$732$	0	0
$733$	8.41792e46	0.447702	0.223851	−	0.974623i	$-0.428137\pi$
$733$	8.41792e46	0.447702	0.223851	+	0.974623i	$0.428137\pi$
$734$	0	0
$735$	−5.06568e46	−0.257571
$736$	0	0
$737$	−9.11389e46	−0.443088
$738$	0	0
$739$	−2.39720e47	−1.11448	−0.557239	−	0.830352i	$-0.688139\pi$
$739$	−2.39720e47	−1.11448	−0.557239	+	0.830352i	$0.688139\pi$
$740$	0	0
$741$	1.42957e46	0.0635630
$742$	0	0
$743$	−4.51636e47	−1.92076	−0.960380	−	0.278695i	$-0.910098\pi$
$743$	−4.51636e47	−1.92076	−0.960380	+	0.278695i	$0.910098\pi$
$744$	0	0
$745$	−3.49589e46	−0.142226
$746$	0	0
$747$	1.11836e47	0.435304
$748$	0	0
$749$	3.89025e46	0.144886
$750$	0	0
$751$	2.31769e46	0.0826030	0.0413015	−	0.999147i	$-0.486850\pi$
$751$	2.31769e46	0.0826030	0.0413015	+	0.999147i	$0.486850\pi$
$752$	0	0
$753$	1.62147e47	0.553085
$754$	0	0
$755$	−2.73514e47	−0.893006
$756$	0	0
$757$	−6.14297e46	−0.191998	−0.0959989	−	0.995381i	$-0.530605\pi$
$757$	−6.14297e46	−0.191998	−0.0959989	+	0.995381i	$0.530605\pi$
$758$	0	0
$759$	1.36880e47	0.409590
$760$	0	0
$761$	−4.18126e47	−1.19801	−0.599004	−	0.800746i	$-0.704437\pi$
$761$	−4.18126e47	−1.19801	−0.599004	+	0.800746i	$0.704437\pi$
$762$	0	0
$763$	−2.43616e47	−0.668421
$764$	0	0
$765$	−1.81850e47	−0.477860
$766$	0	0
$767$	−2.25179e47	−0.566766
$768$	0	0
$769$	−2.07493e47	−0.500286	−0.250143	−	0.968209i	$-0.580478\pi$
$769$	−2.07493e47	−0.500286	−0.250143	+	0.968209i	$0.580478\pi$
$770$	0	0
$771$	1.55616e47	0.359464
$772$	0	0
$773$	−3.82656e47	−0.846930	−0.423465	−	0.905912i	$-0.639186\pi$
$773$	−3.82656e47	−0.846930	−0.423465	+	0.905912i	$0.639186\pi$
$774$	0	0
$775$	3.93306e46	0.0834165
$776$	0	0
$777$	−8.86469e46	−0.180184
$778$	0	0
$779$	−3.07522e46	−0.0599112
$780$	0	0
$781$	−6.21081e46	−0.115986
$782$	0	0
$783$	−6.66634e47	−1.19349
$784$	0	0
$785$	5.60247e47	0.961679
$786$	0	0
$787$	6.26036e47	1.03043	0.515213	−	0.857062i	$-0.327713\pi$
$787$	6.26036e47	1.03043	0.515213	+	0.857062i	$0.327713\pi$
$788$	0	0
$789$	−3.00829e47	−0.474842
$790$	0	0
$791$	−7.50935e47	−1.13682
$792$	0	0
$793$	1.89053e46	0.0274521
$794$	0	0
$795$	9.88141e46	0.137645
$796$	0	0
$797$	1.56391e47	0.209002	0.104501	−	0.994525i	$-0.466675\pi$
$797$	1.56391e47	0.209002	0.104501	+	0.994525i	$0.466675\pi$
$798$	0	0
$799$	1.08734e48	1.39425
$800$	0	0
$801$	−1.68012e47	−0.206730
$802$	0	0
$803$	−4.96943e47	−0.586813
$804$	0	0
$805$	−3.46706e47	−0.392943
$806$	0	0
$807$	5.22171e47	0.568067
$808$	0	0
$809$	4.01144e47	0.418938	0.209469	−	0.977815i	$-0.432826\pi$
$809$	4.01144e47	0.418938	0.209469	+	0.977815i	$0.432826\pi$
$810$	0	0
$811$	−2.34209e47	−0.234833	−0.117417	−	0.993083i	$-0.537461\pi$
$811$	−2.34209e47	−0.234833	−0.117417	+	0.993083i	$0.537461\pi$
$812$	0	0
$813$	1.27608e48	1.22853
$814$	0	0
$815$	5.66962e47	0.524147
$816$	0	0
$817$	−2.48253e47	−0.220410
$818$	0	0
$819$	−2.83137e47	−0.241442
$820$	0	0
$821$	4.03504e47	0.330510	0.165255	−	0.986251i	$-0.447155\pi$
$821$	4.03504e47	0.330510	0.165255	+	0.986251i	$0.447155\pi$
$822$	0	0
$823$	−1.82505e48	−1.43608	−0.718039	−	0.696002i	$-0.754960\pi$
$823$	−1.82505e48	−1.43608	−0.718039	+	0.696002i	$0.754960\pi$
$824$	0	0
$825$	3.31803e47	0.250836
$826$	0	0
$827$	−7.52706e47	−0.546744	−0.273372	−	0.961908i	$-0.588139\pi$
$827$	−7.52706e47	−0.546744	−0.273372	+	0.961908i	$0.588139\pi$
$828$	0	0
$829$	−2.19354e48	−1.53107	−0.765537	−	0.643392i	$-0.777526\pi$
$829$	−2.19354e48	−1.53107	−0.765537	+	0.643392i	$0.777526\pi$
$830$	0	0
$831$	7.47306e47	0.501281
$832$	0	0
$833$	1.57074e48	1.01265
$834$	0	0
$835$	1.36590e48	0.846433
$836$	0	0
$837$	−2.29055e47	−0.136449
$838$	0	0
$839$	2.86394e48	1.64018	0.820091	−	0.572233i	$-0.193923\pi$
$839$	2.86394e48	1.64018	0.820091	+	0.572233i	$0.193923\pi$
$840$	0	0
$841$	4.63381e47	0.255155
$842$	0	0
$843$	1.54943e48	0.820382
$844$	0	0
$845$	4.29368e47	0.218622
$846$	0	0
$847$	9.08800e47	0.445033
$848$	0	0
$849$	−8.06680e47	−0.379948
$850$	0	0
$851$	9.43013e47	0.427248
$852$	0	0
$853$	4.28657e48	1.86832	0.934158	−	0.356859i	$-0.116152\pi$
$853$	4.28657e48	1.86832	0.934158	+	0.356859i	$0.116152\pi$
$854$	0	0
$855$	7.65179e46	0.0320865
$856$	0	0
$857$	−8.64522e47	−0.348813	−0.174406	−	0.984674i	$-0.555801\pi$
$857$	−8.64522e47	−0.348813	−0.174406	+	0.984674i	$0.555801\pi$
$858$	0	0
$859$	−2.35883e48	−0.915817	−0.457909	−	0.888999i	$-0.651401\pi$
$859$	−2.35883e48	−0.915817	−0.457909	+	0.888999i	$0.651401\pi$
$860$	0	0
$861$	−6.43555e47	−0.240455
$862$	0	0
$863$	6.27316e46	0.0225584	0.0112792	−	0.999936i	$-0.496410\pi$
$863$	6.27316e46	0.0225584	0.0112792	+	0.999936i	$0.496410\pi$
$864$	0	0
$865$	2.87267e48	0.994304
$866$	0	0
$867$	−3.80669e48	−1.26833
$868$	0	0
$869$	4.56861e47	0.146541
$870$	0	0
$871$	2.11901e48	0.654386
$872$	0	0
$873$	−1.09203e48	−0.324712
$874$	0	0
$875$	−2.13092e48	−0.610145
$876$	0	0
$877$	−2.00783e48	−0.553647	−0.276823	−	0.960921i	$-0.589282\pi$
$877$	−2.00783e48	−0.553647	−0.276823	+	0.960921i	$0.589282\pi$
$878$	0	0
$879$	7.49818e47	0.199131
$880$	0	0
$881$	−4.20296e48	−1.07511	−0.537555	−	0.843229i	$-0.680652\pi$
$881$	−4.20296e48	−1.07511	−0.537555	+	0.843229i	$0.680652\pi$
$882$	0	0
$883$	−7.94997e48	−1.95891	−0.979453	−	0.201672i	$-0.935362\pi$
$883$	−7.94997e48	−1.95891	−0.979453	+	0.201672i	$0.935362\pi$
$884$	0	0
$885$	1.27351e48	0.302301
$886$	0	0
$887$	7.43922e48	1.70133	0.850664	−	0.525710i	$-0.176200\pi$
$887$	7.43922e48	1.70133	0.850664	+	0.525710i	$0.176200\pi$
$888$	0	0
$889$	−3.91811e48	−0.863371
$890$	0	0
$891$	−7.01859e47	−0.149028
$892$	0	0
$893$	−4.57523e47	−0.0936189
$894$	0	0
$895$	1.48688e48	0.293222
$896$	0	0
$897$	−3.18250e48	−0.604913
$898$	0	0
$899$	7.83220e47	0.143499
$900$	0	0
$901$	−3.06398e48	−0.541161
$902$	0	0
$903$	−5.19523e48	−0.884620
$904$	0	0
$905$	−4.82197e47	−0.0791631
$906$	0	0
$907$	−8.96944e48	−1.41986	−0.709929	−	0.704273i	$-0.751273\pi$
$907$	−8.96944e48	−1.41986	−0.709929	+	0.704273i	$0.751273\pi$
$908$	0	0
$909$	3.25918e48	0.497513
$910$	0	0
$911$	−2.25421e48	−0.331850	−0.165925	−	0.986138i	$-0.553061\pi$
$911$	−2.25421e48	−0.331850	−0.165925	+	0.986138i	$0.553061\pi$
$912$	0	0
$913$	−3.38820e48	−0.481063
$914$	0	0
$915$	−1.06920e47	−0.0146424
$916$	0	0
$917$	−1.62415e48	−0.214551
$918$	0	0
$919$	7.04277e48	0.897507	0.448754	−	0.893656i	$-0.351868\pi$
$919$	7.04277e48	0.897507	0.448754	+	0.893656i	$0.351868\pi$
$920$	0	0
$921$	−1.56155e48	−0.191987
$922$	0	0
$923$	1.44404e48	0.171297
$924$	0	0
$925$	2.28591e48	0.261650
$926$	0	0
$927$	−4.57831e48	−0.505696
$928$	0	0
$929$	1.56732e49	1.67070	0.835348	−	0.549721i	$-0.185266\pi$
$929$	1.56732e49	1.67070	0.835348	+	0.549721i	$0.185266\pi$
$930$	0	0
$931$	−6.60926e47	−0.0679959
$932$	0	0
$933$	−1.20205e49	−1.19365
$934$	0	0
$935$	5.50937e48	0.528093
$936$	0	0
$937$	4.17295e48	0.386136	0.193068	−	0.981185i	$-0.438156\pi$
$937$	4.17295e48	0.386136	0.193068	+	0.981185i	$0.438156\pi$
$938$	0	0
$939$	−6.29740e48	−0.562573
$940$	0	0
$941$	−5.21719e48	−0.449995	−0.224998	−	0.974359i	$-0.572237\pi$
$941$	−5.21719e48	−0.449995	−0.224998	+	0.974359i	$0.572237\pi$
$942$	0	0
$943$	6.84604e48	0.570160
$944$	0	0
$945$	4.89456e48	0.393630
$946$	0	0
$947$	1.05918e49	0.822616	0.411308	−	0.911496i	$-0.365072\pi$
$947$	1.05918e49	0.822616	0.411308	+	0.911496i	$0.365072\pi$
$948$	0	0
$949$	1.15541e49	0.866650
$950$	0	0
$951$	4.82423e48	0.349502
$952$	0	0
$953$	1.49599e49	1.04688	0.523438	−	0.852064i	$-0.324649\pi$
$953$	1.49599e49	1.04688	0.523438	+	0.852064i	$0.324649\pi$
$954$	0	0
$955$	−1.40647e47	−0.00950767
$956$	0	0
$957$	6.60745e48	0.431506
$958$	0	0
$959$	−4.90095e48	−0.309224
$960$	0	0
$961$	−1.61344e49	−0.983594
$962$	0	0
$963$	1.91136e48	0.112592
$964$	0	0
$965$	−1.34415e48	−0.0765152
$966$	0	0
$967$	2.39619e49	1.31821	0.659106	−	0.752050i	$-0.270935\pi$
$967$	2.39619e49	1.31821	0.659106	+	0.752050i	$0.270935\pi$
$968$	0	0
$969$	2.50696e48	0.133292
$970$	0	0
$971$	1.35152e47	0.00694552	0.00347276	−	0.999994i	$-0.498895\pi$
$971$	1.35152e47	0.00694552	0.00347276	+	0.999994i	$0.498895\pi$
$972$	0	0
$973$	−1.98994e49	−0.988505
$974$	0	0
$975$	−7.71454e48	−0.370454
$976$	0	0
$977$	1.84460e48	0.0856333	0.0428166	−	0.999083i	$-0.486367\pi$
$977$	1.84460e48	0.0856333	0.0428166	+	0.999083i	$0.486367\pi$
$978$	0	0
$979$	5.09011e48	0.228461
$980$	0	0
$981$	−1.19693e49	−0.519435
$982$	0	0
$983$	6.69809e48	0.281072	0.140536	−	0.990076i	$-0.455117\pi$
$983$	6.69809e48	0.281072	0.140536	+	0.990076i	$0.455117\pi$
$984$	0	0
$985$	1.79240e49	0.727340
$986$	0	0
$987$	−9.57464e48	−0.375742
$988$	0	0
$989$	5.52661e49	2.09759
$990$	0	0
$991$	−1.17168e49	−0.430123	−0.215062	−	0.976600i	$-0.568995\pi$
$991$	−1.17168e49	−0.430123	−0.215062	+	0.976600i	$0.568995\pi$
$992$	0	0
$993$	−1.55727e49	−0.552969
$994$	0	0
$995$	2.11780e49	0.727453
$996$	0	0
$997$	2.44007e49	0.810834	0.405417	−	0.914132i	$-0.367126\pi$
$997$	2.44007e49	0.810834	0.405417	+	0.914132i	$0.367126\pi$
$998$	0	0
$999$	−1.33128e49	−0.427995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.34.a.b.1.1		2
4.3	odd	2		1.34.a.a.1.1	✓	2
12.11	even	2		9.34.a.b.1.2		2
20.3	even	4		25.34.b.a.24.4		4
20.7	even	4		25.34.b.a.24.1		4
20.19	odd	2		25.34.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.34.a.a.1.1	✓	2	4.3	odd	2
9.34.a.b.1.2		2	12.11	even	2
16.34.a.b.1.1		2	1.1	even	1	trivial
25.34.a.a.1.2		2	20.19	odd	2
25.34.b.a.24.1		4	20.7	even	4
25.34.b.a.24.4		4	20.3	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)

\(f(q)\)	\(=\)	\(q-5.34420e7 q^{3} -2.01492e11 q^{5} -5.50153e13 q^{7} -2.70301e15 q^{9} +O(q^{10})\)	\(q-5.34420e7 q^{3} -2.01492e11 q^{5} -5.50153e13 q^{7} -2.70301e15 q^{9} +8.18909e16 q^{11} -1.90399e18 q^{13} +1.07682e19 q^{15} -3.33893e20 q^{17} +1.40494e20 q^{19} +2.94013e21 q^{21} -3.12767e22 q^{23} -7.58162e22 q^{25} +4.41542e23 q^{27} -1.50979e24 q^{29} -5.18762e23 q^{31} -4.37641e24 q^{33} +1.10852e25 q^{35} -3.01507e25 q^{37} +1.01753e26 q^{39} -2.18887e26 q^{41} -1.76701e27 q^{43} +5.44636e26 q^{45} -3.25654e27 q^{47} -4.70431e27 q^{49} +1.78439e28 q^{51} +9.17652e27 q^{53} -1.65004e28 q^{55} -7.50826e27 q^{57} +1.18267e29 q^{59} -9.92930e27 q^{61} +1.48707e29 q^{63} +3.83640e29 q^{65} -1.11293e30 q^{67} +1.67149e30 q^{69} -7.58425e29 q^{71} -6.06835e30 q^{73} +4.05177e30 q^{75} -4.50525e30 q^{77} +5.57890e30 q^{79} -8.57067e30 q^{81} -4.13746e31 q^{83} +6.72769e31 q^{85} +8.06860e31 q^{87} +6.21572e31 q^{89} +1.04749e32 q^{91} +2.77237e31 q^{93} -2.83084e31 q^{95} +4.04003e32 q^{97} -2.21352e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 37919880 q^{3} - 181061536500 q^{5} + 67153080066800 q^{7} - 80\!\cdots\!54 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more