LMFDB - Embedded newform 256.8.b.l.129.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,8,Mod(129,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.50765497344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 101x^{4} + 2512x^{2} + 36 $ x^6 + 101x^4 + 2512x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$-0.119748i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.l.129.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.95798i q^{3} -362.486i q^{5} -715.055 q^{7} +2178.25 q^{9} +7298.99i q^{11} -10398.4i q^{13} +1072.23 q^{15} -11225.0 q^{17} +26695.5i q^{19} -2115.12i q^{21} -23531.8 q^{23} -53270.9 q^{25} +12912.3i q^{27} +44148.6i q^{29} -311489. q^{31} -21590.3 q^{33} +259197. i q^{35} +139795. i q^{37} +30758.3 q^{39} +753482. q^{41} -125011. i q^{43} -789585. i q^{45} +798879. q^{47} -312239. q^{49} -33203.5i q^{51} +1.14579e6i q^{53} +2.64578e6 q^{55} -78964.7 q^{57} -2.01433e6i q^{59} +5678.84i q^{61} -1.55757e6 q^{63} -3.76927e6 q^{65} +3.80769e6i q^{67} -69606.5i q^{69} +2.78448e6 q^{71} +1.85336e6 q^{73} -157574. i q^{75} -5.21918e6i q^{77} -2.26678e6 q^{79} +4.72564e6 q^{81} +2.79311e6i q^{83} +4.06892e6i q^{85} -130591. q^{87} +5.86158e6 q^{89} +7.43544e6i q^{91} -921379. i q^{93} +9.67673e6 q^{95} -1.46999e7 q^{97} +1.58990e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{7} + 106 q^{9} - 8232 q^{15} + 1076 q^{17} + 86632 q^{23} + 81206 q^{25} - 313856 q^{31} - 367000 q^{33} + 1243368 q^{39} + 1819396 q^{41} - 1705904 q^{47} - 3273066 q^{49} + 7411352 q^{55}+ \cdots - 10501388 q^{97}+O(q^{100}) $ 6 * q + 24 * q^7 + 106 * q^9 - 8232 * q^15 + 1076 * q^17 + 86632 * q^23 + 81206 * q^25 - 313856 * q^31 - 367000 * q^33 + 1243368 * q^39 + 1819396 * q^41 - 1705904 * q^47 - 3273066 * q^49 + 7411352 * q^55 + 6490536 * q^57 - 5088856 * q^63 - 9549544 * q^65 + 19882936 * q^71 + 11588940 * q^73 - 2009008 * q^79 - 12889762 * q^81 + 36376264 * q^87 + 9662700 * q^89 + 30260056 * q^95 - 10501388 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$3$	2.95798i	0.0632515i	0.999500	+	0.0316258i	$0.0100685\pi$
$3$	2.95798i	0.0632515i	−0.999500	+	0.0316258i	$0.989932\pi$
$4$	0	0
$5$	− 362.486i	− 1.29687i	−0.761271	−	0.648434i	$-0.775424\pi$
$5$	− 362.486i	− 1.29687i	0.761271	−	0.648434i	$-0.224576\pi$
$6$	0	0
$7$	−715.055	−0.787946	−0.393973	−	0.919122i	$-0.628900\pi$
$7$	−715.055	−0.787946	−0.393973	+	0.919122i	$0.628900\pi$
$8$	0	0
$9$	2178.25	0.995999
$10$	0	0
$11$	7298.99i	1.65344i	0.562614	+	0.826720i	$0.309796\pi$
$11$	7298.99i	1.65344i	−0.562614	+	0.826720i	$0.690204\pi$
$12$	0	0
$13$	− 10398.4i	− 1.31270i	−0.754457	−	0.656349i	$-0.772100\pi$
$13$	− 10398.4i	− 1.31270i	0.754457	−	0.656349i	$-0.227900\pi$
$14$	0	0
$15$	1072.23	0.0820289
$16$	0	0
$17$	−11225.0	−0.554137	−0.277068	−	0.960850i	$-0.589363\pi$
$17$	−11225.0	−0.554137	−0.277068	+	0.960850i	$0.589363\pi$
$18$	0	0
$19$	26695.5i	0.892895i	0.894810	+	0.446447i	$0.147311\pi$
$19$	26695.5i	0.892895i	−0.894810	+	0.446447i	$0.852689\pi$
$20$	0	0
$21$	− 2115.12i	− 0.0498388i
$22$	0	0
$23$	−23531.8	−0.403280	−0.201640	−	0.979460i	$-0.564627\pi$
$23$	−23531.8	−0.403280	−0.201640	+	0.979460i	$0.564627\pi$
$24$	0	0
$25$	−53270.9	−0.681867
$26$	0	0
$27$	12912.3i	0.126250i
$28$	0	0
$29$	44148.6i	0.336143i	0.985775	+	0.168071i	$0.0537539\pi$
$29$	44148.6i	0.336143i	−0.985775	+	0.168071i	$0.946246\pi$
$30$	0	0
$31$	−311489.	−1.87792	−0.938960	−	0.344025i	$-0.888209\pi$
$31$	−311489.	−1.87792	−0.938960	+	0.344025i	$0.888209\pi$
$32$	0	0
$33$	−21590.3	−0.104583
$34$	0	0
$35$	259197.i	1.02186i
$36$	0	0
$37$	139795.i	0.453716i	0.973928	+	0.226858i	$0.0728454\pi$
$37$	139795.i	0.453716i	−0.973928	+	0.226858i	$0.927155\pi$
$38$	0	0
$39$	30758.3	0.0830302
$40$	0	0
$41$	753482.	1.70738	0.853688	−	0.520785i	$-0.174361\pi$
$41$	753482.	1.70738	0.853688	+	0.520785i	$0.174361\pi$
$42$	0	0
$43$	− 125011.i	− 0.239777i	−0.992787	−	0.119889i	$-0.961746\pi$
$43$	− 125011.i	− 0.239777i	0.992787	−	0.119889i	$-0.0382537\pi$
$44$	0	0
$45$	− 789585.i	− 1.29168i
$46$	0	0
$47$	798879.	1.12238	0.561188	−	0.827688i	$-0.310344\pi$
$47$	798879.	1.12238	0.561188	+	0.827688i	$0.310344\pi$
$48$	0	0
$49$	−312239.	−0.379141
$50$	0	0
$51$	− 33203.5i	− 0.0350500i
$52$	0	0
$53$	1.14579e6i	1.05716i	0.848884	+	0.528580i	$0.177275\pi$
$53$	1.14579e6i	1.05716i	−0.848884	+	0.528580i	$0.822725\pi$
$54$	0	0
$55$	2.64578e6	2.14429
$56$	0	0
$57$	−78964.7	−0.0564769
$58$	0	0
$59$	− 2.01433e6i	− 1.27687i	−0.769674	−	0.638437i	$-0.779581\pi$
$59$	− 2.01433e6i	− 1.27687i	0.769674	−	0.638437i	$-0.220419\pi$
$60$	0	0
$61$	5678.84i	0.00320336i	0.999999	+	0.00160168i	$0.000509831\pi$
$61$	5678.84i	0.00320336i	−0.999999	+	0.00160168i	$0.999490\pi$
$62$	0	0
$63$	−1.55757e6	−0.784794
$64$	0	0
$65$	−3.76927e6	−1.70240
$66$	0	0
$67$	3.80769e6i	1.54668i	0.633993	+	0.773339i	$0.281415\pi$
$67$	3.80769e6i	1.54668i	−0.633993	+	0.773339i	$0.718585\pi$
$68$	0	0
$69$	− 69606.5i	− 0.0255081i
$70$	0	0
$71$	2.78448e6	0.923293	0.461647	−	0.887064i	$-0.347259\pi$
$71$	2.78448e6	0.923293	0.461647	+	0.887064i	$0.347259\pi$
$72$	0	0
$73$	1.85336e6	0.557609	0.278805	−	0.960348i	$-0.410062\pi$
$73$	1.85336e6	0.557609	0.278805	+	0.960348i	$0.410062\pi$
$74$	0	0
$75$	− 157574.i	− 0.0431291i
$76$	0	0
$77$	− 5.21918e6i	− 1.30282i
$78$	0	0
$79$	−2.26678e6	−0.517267	−0.258633	−	0.965976i	$-0.583272\pi$
$79$	−2.26678e6	−0.517267	−0.258633	+	0.965976i	$0.583272\pi$
$80$	0	0
$81$	4.72564e6	0.988014
$82$	0	0
$83$	2.79311e6i	0.536186i	0.963393	+	0.268093i	$0.0863934\pi$
$83$	2.79311e6i	0.536186i	−0.963393	+	0.268093i	$0.913607\pi$
$84$	0	0
$85$	4.06892e6i	0.718642i
$86$	0	0
$87$	−130591.	−0.0212615
$88$	0	0
$89$	5.86158e6	0.881353	0.440676	−	0.897666i	$-0.354739\pi$
$89$	5.86158e6	0.881353	0.440676	+	0.897666i	$0.354739\pi$
$90$	0	0
$91$	7.43544e6i	1.03434i
$92$	0	0
$93$	− 921379.i	− 0.118781i
$94$	0	0
$95$	9.67673e6	1.15797
$96$	0	0
$97$	−1.46999e7	−1.63536	−0.817682	−	0.575670i	$-0.804741\pi$
$97$	−1.46999e7	−1.63536	−0.817682	+	0.575670i	$0.804741\pi$
$98$	0	0
$99$	1.58990e7i	1.64682i
$100$	0	0
$101$	7.80039e6i	0.753340i	0.926347	+	0.376670i	$0.122931\pi$
$101$	7.80039e6i	0.753340i	−0.926347	+	0.376670i	$0.877069\pi$
$102$	0	0
$103$	1.20366e7	1.08536	0.542678	−	0.839941i	$-0.317410\pi$
$103$	1.20366e7	1.08536	0.542678	+	0.839941i	$0.317410\pi$
$104$	0	0
$105$	−766701.	−0.0646343
$106$	0	0
$107$	1.89549e7i	1.49582i	0.663802	+	0.747908i	$0.268942\pi$
$107$	1.89549e7i	1.49582i	−0.663802	+	0.747908i	$0.731058\pi$
$108$	0	0
$109$	1.75553e7i	1.29842i	0.760608	+	0.649212i	$0.224901\pi$
$109$	1.75553e7i	1.29842i	−0.760608	+	0.649212i	$0.775099\pi$
$110$	0	0
$111$	−413510.	−0.0286982
$112$	0	0
$113$	2.26566e7	1.47713	0.738566	−	0.674181i	$-0.235503\pi$
$113$	2.26566e7	1.47713	0.738566	+	0.674181i	$0.235503\pi$
$114$	0	0
$115$	8.52993e6i	0.523001i
$116$	0	0
$117$	− 2.26503e7i	− 1.30745i
$118$	0	0
$119$	8.02653e6	0.436630
$120$	0	0
$121$	−3.37881e7	−1.73386
$122$	0	0
$123$	2.22878e6i	0.107994i
$124$	0	0
$125$	− 9.00926e6i	− 0.412576i
$126$	0	0
$127$	4.34969e7	1.88428	0.942140	−	0.335220i	$-0.108810\pi$
$127$	4.34969e7	1.88428	0.942140	+	0.335220i	$0.108810\pi$
$128$	0	0
$129$	369779.	0.0151663
$130$	0	0
$131$	− 2.44242e7i	− 0.949230i	−0.880194	−	0.474615i	$-0.842587\pi$
$131$	− 2.44242e7i	− 0.949230i	0.880194	−	0.474615i	$-0.157413\pi$
$132$	0	0
$133$	− 1.90887e7i	− 0.703553i
$134$	0	0
$135$	4.68053e6	0.163730
$136$	0	0
$137$	2.33908e7	0.777184	0.388592	−	0.921410i	$-0.372962\pi$
$137$	2.33908e7	0.777184	0.388592	+	0.921410i	$0.372962\pi$
$138$	0	0
$139$	8.57032e6i	0.270673i	0.990800	+	0.135337i	$0.0432116\pi$
$139$	8.57032e6i	0.270673i	−0.990800	+	0.135337i	$0.956788\pi$
$140$	0	0
$141$	2.36307e6i	0.0709920i
$142$	0	0
$143$	7.58979e7	2.17047
$144$	0	0
$145$	1.60032e7	0.435933
$146$	0	0
$147$	− 923596.i	− 0.0239812i
$148$	0	0
$149$	− 837746.i	− 0.0207472i	−0.999946	−	0.0103736i	$-0.996698\pi$
$149$	− 837746.i	− 0.0207472i	0.999946	−	0.0103736i	$-0.00330209\pi$
$150$	0	0
$151$	3.95926e7	0.935825	0.467912	−	0.883775i	$-0.345006\pi$
$151$	3.95926e7	0.935825	0.467912	+	0.883775i	$0.345006\pi$
$152$	0	0
$153$	−2.44510e7	−0.551920
$154$	0	0
$155$	1.12910e8i	2.43542i
$156$	0	0
$157$	3.58442e7i	0.739213i	0.929188	+	0.369607i	$0.120508\pi$
$157$	3.58442e7i	0.739213i	−0.929188	+	0.369607i	$0.879492\pi$
$158$	0	0
$159$	−3.38923e6	−0.0668669
$160$	0	0
$161$	1.68265e7	0.317763
$162$	0	0
$163$	2.00430e7i	0.362498i	0.983437	+	0.181249i	$0.0580139\pi$
$163$	2.00430e7i	0.362498i	−0.983437	+	0.181249i	$0.941986\pi$
$164$	0	0
$165$	7.82617e6i	0.135630i
$166$	0	0
$167$	−4.04947e7	−0.672807	−0.336403	−	0.941718i	$-0.609211\pi$
$167$	−4.04947e7	−0.672807	−0.336403	+	0.941718i	$0.609211\pi$
$168$	0	0
$169$	−4.53784e7	−0.723178
$170$	0	0
$171$	5.81494e7i	0.889322i
$172$	0	0
$173$	5.72293e7i	0.840345i	0.907444	+	0.420172i	$0.138030\pi$
$173$	5.72293e7i	0.840345i	−0.907444	+	0.420172i	$0.861970\pi$
$174$	0	0
$175$	3.80916e7	0.537275
$176$	0	0
$177$	5.95834e6	0.0807643
$178$	0	0
$179$	1.52231e8i	1.98389i	0.126660	+	0.991946i	$0.459574\pi$
$179$	1.52231e8i	1.98389i	−0.126660	+	0.991946i	$0.540426\pi$
$180$	0	0
$181$	− 2.20505e7i	− 0.276403i	−0.990404	−	0.138201i	$-0.955868\pi$
$181$	− 2.20505e7i	− 0.276403i	0.990404	−	0.138201i	$-0.0441321\pi$
$182$	0	0
$183$	−16797.9	−0.000202617 0
$184$	0	0
$185$	5.06735e7	0.588410
$186$	0	0
$187$	− 8.19315e7i	− 0.916231i
$188$	0	0
$189$	− 9.23303e6i	− 0.0994782i
$190$	0	0
$191$	−1.06826e8	−1.10933	−0.554666	−	0.832073i	$-0.687154\pi$
$191$	−1.06826e8	−1.10933	−0.554666	+	0.832073i	$0.687154\pi$
$192$	0	0
$193$	6.77720e7	0.678577	0.339289	−	0.940682i	$-0.389814\pi$
$193$	6.77720e7	0.678577	0.339289	+	0.940682i	$0.389814\pi$
$194$	0	0
$195$	− 1.11494e7i	− 0.107679i
$196$	0	0
$197$	4.58559e6i	0.0427330i	0.999772	+	0.0213665i	$0.00680168\pi$
$197$	4.58559e6i	0.0427330i	−0.999772	+	0.0213665i	$0.993198\pi$
$198$	0	0
$199$	9.30771e7	0.837254	0.418627	−	0.908158i	$-0.362512\pi$
$199$	9.30771e7	0.837254	0.418627	+	0.908158i	$0.362512\pi$
$200$	0	0
$201$	−1.12631e7	−0.0978297
$202$	0	0
$203$	− 3.15687e7i	− 0.264862i
$204$	0	0
$205$	− 2.73126e8i	− 2.21424i
$206$	0	0
$207$	−5.12581e7	−0.401667
$208$	0	0
$209$	−1.94850e8	−1.47635
$210$	0	0
$211$	− 1.64300e8i	− 1.20407i	−0.798472	−	0.602033i	$-0.794358\pi$
$211$	− 1.64300e8i	− 1.20407i	0.798472	−	0.602033i	$-0.205642\pi$
$212$	0	0
$213$	8.23643e6i	0.0583997i
$214$	0	0
$215$	−4.53146e7	−0.310959
$216$	0	0
$217$	2.22732e8	1.47970
$218$	0	0
$219$	5.48220e6i	0.0352696i
$220$	0	0
$221$	1.16723e8i	0.727414i
$222$	0	0
$223$	1.50678e8	0.909880	0.454940	−	0.890522i	$-0.349661\pi$
$223$	1.50678e8	0.909880	0.454940	+	0.890522i	$0.349661\pi$
$224$	0	0
$225$	−1.16037e8	−0.679139
$226$	0	0
$227$	1.08452e8i	0.615384i	0.951486	+	0.307692i	$0.0995567\pi$
$227$	1.08452e8i	0.615384i	−0.951486	+	0.307692i	$0.900443\pi$
$228$	0	0
$229$	2.21842e8i	1.22073i	0.792120	+	0.610366i	$0.208978\pi$
$229$	2.21842e8i	1.22073i	−0.792120	+	0.610366i	$0.791022\pi$
$230$	0	0
$231$	1.54382e7	0.0824054
$232$	0	0
$233$	−1.54008e8	−0.797620	−0.398810	−	0.917034i	$-0.630577\pi$
$233$	−1.54008e8	−0.797620	−0.398810	+	0.917034i	$0.630577\pi$
$234$	0	0
$235$	− 2.89582e8i	− 1.45557i
$236$	0	0
$237$	− 6.70509e6i	− 0.0327179i
$238$	0	0
$239$	−3.75515e8	−1.77924	−0.889620	−	0.456701i	$-0.849031\pi$
$239$	−3.75515e8	−1.77924	−0.889620	+	0.456701i	$0.849031\pi$
$240$	0	0
$241$	−2.33310e8	−1.07368	−0.536839	−	0.843684i	$-0.680382\pi$
$241$	−2.33310e8	−1.07368	−0.536839	+	0.843684i	$0.680382\pi$
$242$	0	0
$243$	4.22176e7i	0.188743i
$244$	0	0
$245$	1.13182e8i	0.491696i
$246$	0	0
$247$	2.77591e8	1.17210
$248$	0	0
$249$	−8.26197e6	−0.0339145
$250$	0	0
$251$	− 3.97050e8i	− 1.58485i	−0.609972	−	0.792423i	$-0.708819\pi$
$251$	− 3.97050e8i	− 1.58485i	0.609972	−	0.792423i	$-0.291181\pi$
$252$	0	0
$253$	− 1.71758e8i	− 0.666800i
$254$	0	0
$255$	−1.20358e7	−0.0454552
$256$	0	0
$257$	5.13203e8	1.88592	0.942960	−	0.332907i	$-0.108029\pi$
$257$	5.13203e8	1.88592	0.942960	+	0.332907i	$0.108029\pi$
$258$	0	0
$259$	− 9.99609e7i	− 0.357504i
$260$	0	0
$261$	9.61666e7i	0.334798i
$262$	0	0
$263$	7.43951e6	0.0252173	0.0126087	−	0.999921i	$-0.495986\pi$
$263$	7.43951e6	0.0252173	0.0126087	+	0.999921i	$0.495986\pi$
$264$	0	0
$265$	4.15333e8	1.37100
$266$	0	0
$267$	1.73384e7i	0.0557469i
$268$	0	0
$269$	9.30769e7i	0.291547i	0.989318	+	0.145774i	$0.0465671\pi$
$269$	9.30769e7i	0.291547i	−0.989318	+	0.145774i	$0.953433\pi$
$270$	0	0
$271$	−2.06914e8	−0.631534	−0.315767	−	0.948837i	$-0.602262\pi$
$271$	−2.06914e8	−0.631534	−0.315767	+	0.948837i	$0.602262\pi$
$272$	0	0
$273$	−2.19939e7	−0.0654233
$274$	0	0
$275$	− 3.88824e8i	− 1.12743i
$276$	0	0
$277$	− 5.45157e8i	− 1.54114i	−0.637354	−	0.770571i	$-0.719971\pi$
$277$	− 5.45157e8i	− 1.54114i	0.637354	−	0.770571i	$-0.280029\pi$
$278$	0	0
$279$	−6.78502e8	−1.87041
$280$	0	0
$281$	−1.34356e8	−0.361230	−0.180615	−	0.983554i	$-0.557809\pi$
$281$	−1.34356e8	−0.361230	−0.180615	+	0.983554i	$0.557809\pi$
$282$	0	0
$283$	− 2.03505e8i	− 0.533731i	−0.963734	−	0.266865i	$-0.914012\pi$
$283$	− 2.03505e8i	− 0.533731i	0.963734	−	0.266865i	$-0.0859879\pi$
$284$	0	0
$285$	2.86236e7i	0.0732431i
$286$	0	0
$287$	−5.38781e8	−1.34532
$288$	0	0
$289$	−2.84337e8	−0.692933
$290$	0	0
$291$	− 4.34821e7i	− 0.103439i
$292$	0	0
$293$	1.57765e8i	0.366415i	0.983074	+	0.183207i	$0.0586480\pi$
$293$	1.57765e8i	0.366415i	−0.983074	+	0.183207i	$0.941352\pi$
$294$	0	0
$295$	−7.30165e8	−1.65594
$296$	0	0
$297$	−9.42469e7	−0.208747
$298$	0	0
$299$	2.44693e8i	0.529386i
$300$	0	0
$301$	8.93896e7i	0.188931i
$302$	0	0
$303$	−2.30734e7	−0.0476499
$304$	0	0
$305$	2.05850e6	0.00415433
$306$	0	0
$307$	− 3.79087e8i	− 0.747747i	−0.927480	−	0.373873i	$-0.878029\pi$
$307$	− 3.79087e8i	− 0.747747i	0.927480	−	0.373873i	$-0.121971\pi$
$308$	0	0
$309$	3.56039e7i	0.0686504i
$310$	0	0
$311$	−6.18149e8	−1.16528	−0.582642	−	0.812729i	$-0.697981\pi$
$311$	−6.18149e8	−1.16528	−0.582642	+	0.812729i	$0.697981\pi$
$312$	0	0
$313$	−4.64450e8	−0.856119	−0.428060	−	0.903751i	$-0.640803\pi$
$313$	−4.64450e8	−0.856119	−0.428060	+	0.903751i	$0.640803\pi$
$314$	0	0
$315$	5.64597e8i	1.01777i
$316$	0	0
$317$	− 1.44567e8i	− 0.254895i	−0.991845	−	0.127447i	$-0.959322\pi$
$317$	− 1.44567e8i	− 0.254895i	0.991845	−	0.127447i	$-0.0406784\pi$
$318$	0	0
$319$	−3.22240e8	−0.555792
$320$	0	0
$321$	−5.60682e7	−0.0946127
$322$	0	0
$323$	− 2.99658e8i	− 0.494785i
$324$	0	0
$325$	5.53932e8i	0.895086i
$326$	0	0
$327$	−5.19283e7	−0.0821273
$328$	0	0
$329$	−5.71243e8	−0.884372
$330$	0	0
$331$	− 4.95770e8i	− 0.751418i	−0.926738	−	0.375709i	$-0.877399\pi$
$331$	− 4.95770e8i	− 0.751418i	0.926738	−	0.375709i	$-0.122601\pi$
$332$	0	0
$333$	3.04508e8i	0.451901i
$334$	0	0
$335$	1.38023e9	2.00584
$336$	0	0
$337$	1.05972e9	1.50830	0.754148	−	0.656705i	$-0.228050\pi$
$337$	1.05972e9	1.50830	0.754148	+	0.656705i	$0.228050\pi$
$338$	0	0
$339$	6.70177e7i	0.0934308i
$340$	0	0
$341$	− 2.27356e9i	− 3.10503i
$342$	0	0
$343$	8.12147e8	1.08669
$344$	0	0
$345$	−2.52314e7	−0.0330806
$346$	0	0
$347$	− 3.45456e8i	− 0.443853i	−0.975063	−	0.221927i	$-0.928765\pi$
$347$	− 3.45456e8i	− 0.443853i	0.975063	−	0.221927i	$-0.0712345\pi$
$348$	0	0
$349$	1.46126e9i	1.84009i	0.391811	+	0.920046i	$0.371849\pi$
$349$	1.46126e9i	1.84009i	−0.391811	+	0.920046i	$0.628151\pi$
$350$	0	0
$351$	1.34268e8	0.165728
$352$	0	0
$353$	−4.48645e7	−0.0542864	−0.0271432	−	0.999632i	$-0.508641\pi$
$353$	−4.48645e7	−0.0542864	−0.0271432	+	0.999632i	$0.508641\pi$
$354$	0	0
$355$	− 1.00933e9i	− 1.19739i
$356$	0	0
$357$	2.37423e7i	0.0276175i
$358$	0	0
$359$	−3.28983e8	−0.375269	−0.187634	−	0.982239i	$-0.560082\pi$
$359$	−3.28983e8	−0.375269	−0.187634	+	0.982239i	$0.560082\pi$
$360$	0	0
$361$	1.81223e8	0.202739
$362$	0	0
$363$	− 9.99445e7i	− 0.109670i
$364$	0	0
$365$	− 6.71817e8i	− 0.723146i
$366$	0	0
$367$	−6.26178e8	−0.661251	−0.330626	−	0.943762i	$-0.607260\pi$
$367$	−6.26178e8	−0.661251	−0.330626	+	0.943762i	$0.607260\pi$
$368$	0	0
$369$	1.64127e9	1.70055
$370$	0	0
$371$	− 8.19305e8i	− 0.832985i
$372$	0	0
$373$	1.10119e9i	1.09870i	0.835591	+	0.549352i	$0.185125\pi$
$373$	1.10119e9i	1.09870i	−0.835591	+	0.549352i	$0.814875\pi$
$374$	0	0
$375$	2.66492e7	0.0260961
$376$	0	0
$377$	4.59075e8	0.441254
$378$	0	0
$379$	2.96469e8i	0.279732i	0.990170	+	0.139866i	$0.0446672\pi$
$379$	2.96469e8i	0.279732i	−0.990170	+	0.139866i	$0.955333\pi$
$380$	0	0
$381$	1.28663e8i	0.119184i
$382$	0	0
$383$	7.05233e8	0.641412	0.320706	−	0.947179i	$-0.396080\pi$
$383$	7.05233e8	0.641412	0.320706	+	0.947179i	$0.396080\pi$
$384$	0	0
$385$	−1.89188e9	−1.68959
$386$	0	0
$387$	− 2.72305e8i	− 0.238818i
$388$	0	0
$389$	1.28994e9i	1.11108i	0.831491	+	0.555539i	$0.187488\pi$
$389$	1.28994e9i	1.11108i	−0.831491	+	0.555539i	$0.812512\pi$
$390$	0	0
$391$	2.64145e8	0.223472
$392$	0	0
$393$	7.22464e7	0.0600402
$394$	0	0
$395$	8.21675e8i	0.670827i
$396$	0	0
$397$	− 1.73147e9i	− 1.38883i	−0.719576	−	0.694414i	$-0.755664\pi$
$397$	− 1.73147e9i	− 1.38883i	0.719576	−	0.694414i	$-0.244336\pi$
$398$	0	0
$399$	5.64641e7	0.0445008
$400$	0	0
$401$	−7.16737e8	−0.555079	−0.277539	−	0.960714i	$-0.589519\pi$
$401$	−7.16737e8	−0.555079	−0.277539	+	0.960714i	$0.589519\pi$
$402$	0	0
$403$	3.23899e9i	2.46514i
$404$	0	0
$405$	− 1.71298e9i	− 1.28132i
$406$	0	0
$407$	−1.02036e9	−0.750192
$408$	0	0
$409$	−6.14164e8	−0.443867	−0.221934	−	0.975062i	$-0.571237\pi$
$409$	−6.14164e8	−0.443867	−0.221934	+	0.975062i	$0.571237\pi$
$410$	0	0
$411$	6.91896e7i	0.0491580i
$412$	0	0
$413$	1.44036e9i	1.00611i
$414$	0	0
$415$	1.01246e9	0.695362
$416$	0	0
$417$	−2.53508e7	−0.0171205
$418$	0	0
$419$	4.45049e8i	0.295569i	0.989020	+	0.147785i	$0.0472142\pi$
$419$	4.45049e8i	0.295569i	−0.989020	+	0.147785i	$0.952786\pi$
$420$	0	0
$421$	2.16226e9i	1.41228i	0.708072	+	0.706140i	$0.249565\pi$
$421$	2.16226e9i	1.41228i	−0.708072	+	0.706140i	$0.750435\pi$
$422$	0	0
$423$	1.74016e9	1.11789
$424$	0	0
$425$	5.97968e8	0.377848
$426$	0	0
$427$	− 4.06069e6i	− 0.00252407i
$428$	0	0
$429$	2.24504e8i	0.137285i
$430$	0	0
$431$	−1.00597e9	−0.605221	−0.302611	−	0.953114i	$-0.597858\pi$
$431$	−1.00597e9	−0.605221	−0.302611	+	0.953114i	$0.597858\pi$
$432$	0	0
$433$	−1.27827e9	−0.756684	−0.378342	−	0.925666i	$-0.623506\pi$
$433$	−1.27827e9	−0.756684	−0.378342	+	0.925666i	$0.623506\pi$
$434$	0	0
$435$	4.73372e7i	0.0275734i
$436$	0	0
$437$	− 6.28192e8i	− 0.360087i
$438$	0	0
$439$	2.69631e9	1.52105	0.760526	−	0.649307i	$-0.224941\pi$
$439$	2.69631e9	1.52105	0.760526	+	0.649307i	$0.224941\pi$
$440$	0	0
$441$	−6.80134e8	−0.377624
$442$	0	0
$443$	2.14940e9i	1.17464i	0.809355	+	0.587319i	$0.199817\pi$
$443$	2.14940e9i	1.17464i	−0.809355	+	0.587319i	$0.800183\pi$
$444$	0	0
$445$	− 2.12474e9i	− 1.14300i
$446$	0	0
$447$	2.47804e6	0.00131229
$448$	0	0
$449$	2.72637e8	0.142142	0.0710710	−	0.997471i	$-0.477358\pi$
$449$	2.72637e8	0.142142	0.0710710	+	0.997471i	$0.477358\pi$
$450$	0	0
$451$	5.49966e9i	2.82304i
$452$	0	0
$453$	1.17114e8i	0.0591923i
$454$	0	0
$455$	2.69524e9	1.34140
$456$	0	0
$457$	2.75201e9	1.34879	0.674393	−	0.738372i	$-0.264405\pi$
$457$	2.75201e9	1.34879	0.674393	+	0.738372i	$0.264405\pi$
$458$	0	0
$459$	− 1.44941e8i	− 0.0699597i
$460$	0	0
$461$	− 2.98450e9i	− 1.41879i	−0.704810	−	0.709396i	$-0.748968\pi$
$461$	− 2.98450e9i	− 1.41879i	0.704810	−	0.709396i	$-0.251032\pi$
$462$	0	0
$463$	−2.35761e9	−1.10392	−0.551961	−	0.833870i	$-0.686120\pi$
$463$	−2.35761e9	−1.10392	−0.551961	+	0.833870i	$0.686120\pi$
$464$	0	0
$465$	−3.33987e8	−0.154044
$466$	0	0
$467$	1.25311e9i	0.569353i	0.958624	+	0.284677i	$0.0918862\pi$
$467$	1.25311e9i	0.569353i	−0.958624	+	0.284677i	$0.908114\pi$
$468$	0	0
$469$	− 2.72271e9i	− 1.21870i
$470$	0	0
$471$	−1.06026e8	−0.0467564
$472$	0	0
$473$	9.12452e8	0.396457
$474$	0	0
$475$	− 1.42209e9i	− 0.608836i
$476$	0	0
$477$	2.49582e9i	1.05293i
$478$	0	0
$479$	9.16863e8	0.381180	0.190590	−	0.981670i	$-0.438960\pi$
$479$	9.16863e8	0.381180	0.190590	+	0.981670i	$0.438960\pi$
$480$	0	0
$481$	1.45364e9	0.595593
$482$	0	0
$483$	4.97725e7i	0.0200990i
$484$	0	0
$485$	5.32852e9i	2.12085i
$486$	0	0
$487$	1.04379e8	0.0409507	0.0204754	−	0.999790i	$-0.493482\pi$
$487$	1.04379e8	0.0409507	0.0204754	+	0.999790i	$0.493482\pi$
$488$	0	0
$489$	−5.92867e7	−0.0229285
$490$	0	0
$491$	1.13088e9i	0.431152i	0.976487	+	0.215576i	$0.0691629\pi$
$491$	1.13088e9i	0.431152i	−0.976487	+	0.215576i	$0.930837\pi$
$492$	0	0
$493$	− 4.95570e8i	− 0.186269i
$494$	0	0
$495$	5.76317e9	2.13572
$496$	0	0
$497$	−1.99106e9	−0.727505
$498$	0	0
$499$	2.91325e8i	0.104960i	0.998622	+	0.0524802i	$0.0167126\pi$
$499$	2.91325e8i	0.104960i	−0.998622	+	0.0524802i	$0.983287\pi$
$500$	0	0
$501$	− 1.19782e8i	− 0.0425560i
$502$	0	0
$503$	3.54979e7	0.0124370	0.00621848	−	0.999981i	$-0.498021\pi$
$503$	3.54979e7	0.0124370	0.00621848	+	0.999981i	$0.498021\pi$
$504$	0	0
$505$	2.82753e9	0.976983
$506$	0	0
$507$	− 1.34228e8i	− 0.0457421i
$508$	0	0
$509$	5.24184e8i	0.176186i	0.996112	+	0.0880929i	$0.0280773\pi$
$509$	5.24184e8i	0.176186i	−0.996112	+	0.0880929i	$0.971923\pi$
$510$	0	0
$511$	−1.32526e9	−0.439366
$512$	0	0
$513$	−3.44701e8	−0.112728
$514$	0	0
$515$	− 4.36308e9i	− 1.40756i
$516$	0	0
$517$	5.83101e9i	1.85578i
$518$	0	0
$519$	−1.69283e8	−0.0531531
$520$	0	0
$521$	−5.69562e9	−1.76445	−0.882224	−	0.470830i	$-0.843954\pi$
$521$	−5.69562e9	−1.76445	−0.882224	+	0.470830i	$0.843954\pi$
$522$	0	0
$523$	1.65242e9i	0.505086i	0.967586	+	0.252543i	$0.0812669\pi$
$523$	1.65242e9i	0.505086i	−0.967586	+	0.252543i	$0.918733\pi$
$524$	0	0
$525$	1.12674e8i	0.0339834i
$526$	0	0
$527$	3.49648e9	1.04062
$528$	0	0
$529$	−2.85108e9	−0.837365
$530$	0	0
$531$	− 4.38771e9i	− 1.27177i
$532$	0	0
$533$	− 7.83501e9i	− 2.24127i
$534$	0	0
$535$	6.87088e9	1.93988
$536$	0	0
$537$	−4.50297e8	−0.125484
$538$	0	0
$539$	− 2.27903e9i	− 0.626887i
$540$	0	0
$541$	− 9.74487e8i	− 0.264598i	−0.991210	−	0.132299i	$-0.957764\pi$
$541$	− 9.74487e8i	− 0.264598i	0.991210	−	0.132299i	$-0.0422358\pi$
$542$	0	0
$543$	6.52248e7	0.0174829
$544$	0	0
$545$	6.36356e9	1.68388
$546$	0	0
$547$	5.69600e9i	1.48804i	0.668158	+	0.744020i	$0.267083\pi$
$547$	5.69600e9i	1.48804i	−0.668158	+	0.744020i	$0.732917\pi$
$548$	0	0
$549$	1.23699e7i	0.00319054i
$550$	0	0
$551$	−1.17857e9	−0.300140
$552$	0	0
$553$	1.62087e9	0.407578
$554$	0	0
$555$	1.49891e8i	0.0372178i
$556$	0	0
$557$	5.93761e9i	1.45586i	0.685653	+	0.727929i	$0.259517\pi$
$557$	5.93761e9i	1.45586i	−0.685653	+	0.727929i	$0.740483\pi$
$558$	0	0
$559$	−1.29991e9	−0.314755
$560$	0	0
$561$	2.42352e8	0.0579530
$562$	0	0
$563$	1.27832e9i	0.301898i	0.988542	+	0.150949i	$0.0482329\pi$
$563$	1.27832e9i	0.301898i	−0.988542	+	0.150949i	$0.951767\pi$
$564$	0	0
$565$	− 8.21268e9i	− 1.91565i
$566$	0	0
$567$	−3.37909e9	−0.778502
$568$	0	0
$569$	9.00139e8	0.204841	0.102420	−	0.994741i	$-0.467341\pi$
$569$	9.00139e8	0.204841	0.102420	+	0.994741i	$0.467341\pi$
$570$	0	0
$571$	4.35919e9i	0.979894i	0.871752	+	0.489947i	$0.162984\pi$
$571$	4.35919e9i	0.979894i	−0.871752	+	0.489947i	$0.837016\pi$
$572$	0	0
$573$	− 3.15990e8i	− 0.0701669i
$574$	0	0
$575$	1.25356e9	0.274984
$576$	0	0
$577$	5.38461e8	0.116691	0.0583457	−	0.998296i	$-0.481417\pi$
$577$	5.38461e8	0.116691	0.0583457	+	0.998296i	$0.481417\pi$
$578$	0	0
$579$	2.00468e8i	0.0429210i
$580$	0	0
$581$	− 1.99723e9i	− 0.422485i
$582$	0	0
$583$	−8.36313e9	−1.74795
$584$	0	0
$585$	−8.21042e9	−1.69559
$586$	0	0
$587$	7.48508e8i	0.152744i	0.997079	+	0.0763718i	$0.0243336\pi$
$587$	7.48508e8i	0.152744i	−0.997079	+	0.0763718i	$0.975666\pi$
$588$	0	0
$589$	− 8.31536e9i	− 1.67679i
$590$	0	0
$591$	−1.35641e7	−0.00270292
$592$	0	0
$593$	6.19788e9	1.22054	0.610269	−	0.792194i	$-0.291061\pi$
$593$	6.19788e9	1.22054	0.610269	+	0.792194i	$0.291061\pi$
$594$	0	0
$595$	− 2.90950e9i	− 0.566251i
$596$	0	0
$597$	2.75320e8i	0.0529576i
$598$	0	0
$599$	−8.22711e9	−1.56406	−0.782030	−	0.623241i	$-0.785816\pi$
$599$	−8.22711e9	−1.56406	−0.782030	+	0.623241i	$0.785816\pi$
$600$	0	0
$601$	−1.54211e9	−0.289772	−0.144886	−	0.989448i	$-0.546281\pi$
$601$	−1.54211e9	−0.289772	−0.144886	+	0.989448i	$0.546281\pi$
$602$	0	0
$603$	8.29410e9i	1.54049i
$604$	0	0
$605$	1.22477e10i	2.24859i
$606$	0	0
$607$	−5.19592e9	−0.942979	−0.471490	−	0.881872i	$-0.656283\pi$
$607$	−5.19592e9	−0.942979	−0.471490	+	0.881872i	$0.656283\pi$
$608$	0	0
$609$	9.33795e7	0.0167529
$610$	0	0
$611$	− 8.30707e9i	− 1.47334i
$612$	0	0
$613$	9.51568e9i	1.66851i	0.551380	+	0.834254i	$0.314101\pi$
$613$	9.51568e9i	1.66851i	−0.551380	+	0.834254i	$0.685899\pi$
$614$	0	0
$615$	8.07902e8	0.140054
$616$	0	0
$617$	−1.68458e9	−0.288731	−0.144365	−	0.989524i	$-0.546114\pi$
$617$	−1.68458e9	−0.288731	−0.144365	+	0.989524i	$0.546114\pi$
$618$	0	0
$619$	− 1.82919e9i	− 0.309985i	−0.987916	−	0.154993i	$-0.950465\pi$
$619$	− 1.82919e9i	− 0.309985i	0.987916	−	0.154993i	$-0.0495354\pi$
$620$	0	0
$621$	− 3.03850e8i	− 0.0509141i
$622$	0	0
$623$	−4.19136e9	−0.694458
$624$	0	0
$625$	−7.42752e9	−1.21692
$626$	0	0
$627$	− 5.76363e8i	− 0.0933812i
$628$	0	0
$629$	− 1.56920e9i	− 0.251421i
$630$	0	0
$631$	−7.37467e9	−1.16853	−0.584265	−	0.811563i	$-0.698617\pi$
$631$	−7.37467e9	−1.16853	−0.584265	+	0.811563i	$0.698617\pi$
$632$	0	0
$633$	4.85997e8	0.0761589
$634$	0	0
$635$	− 1.57670e10i	− 2.44366i
$636$	0	0
$637$	3.24679e9i	0.497698i
$638$	0	0
$639$	6.06529e9	0.919599
$640$	0	0
$641$	−7.89218e8	−0.118357	−0.0591785	−	0.998247i	$-0.518848\pi$
$641$	−7.89218e8	−0.118357	−0.0591785	+	0.998247i	$0.518848\pi$
$642$	0	0
$643$	− 7.69347e9i	− 1.14126i	−0.821208	−	0.570629i	$-0.806700\pi$
$643$	− 7.69347e9i	− 1.14126i	0.821208	−	0.570629i	$-0.193300\pi$
$644$	0	0
$645$	− 1.34040e8i	− 0.0196686i
$646$	0	0
$647$	−2.01740e9	−0.292837	−0.146419	−	0.989223i	$-0.546775\pi$
$647$	−2.01740e9	−0.292837	−0.146419	+	0.989223i	$0.546775\pi$
$648$	0	0
$649$	1.47026e10	2.11124
$650$	0	0
$651$	6.58837e8i	0.0935933i
$652$	0	0
$653$	− 2.41614e9i	− 0.339567i	−0.985481	−	0.169784i	$-0.945693\pi$
$653$	− 2.41614e9i	− 0.339567i	0.985481	−	0.169784i	$-0.0543069\pi$
$654$	0	0
$655$	−8.85343e9	−1.23103
$656$	0	0
$657$	4.03708e9	0.555378
$658$	0	0
$659$	− 6.27602e9i	− 0.854251i	−0.904192	−	0.427126i	$-0.859526\pi$
$659$	− 6.27602e9i	− 0.854251i	0.904192	−	0.427126i	$-0.140474\pi$
$660$	0	0
$661$	− 2.24996e9i	− 0.303018i	−0.988456	−	0.151509i	$-0.951587\pi$
$661$	− 2.24996e9i	− 0.303018i	0.988456	−	0.151509i	$-0.0484133\pi$
$662$	0	0
$663$	−3.45263e8	−0.0460101
$664$	0	0
$665$	−6.91940e9	−0.912415
$666$	0	0
$667$	− 1.03889e9i	− 0.135560i
$668$	0	0
$669$	4.45704e8i	0.0575513i
$670$	0	0
$671$	−4.14498e7	−0.00529656
$672$	0	0
$673$	−3.16874e9	−0.400714	−0.200357	−	0.979723i	$-0.564210\pi$
$673$	−3.16874e9	−0.400714	−0.200357	+	0.979723i	$0.564210\pi$
$674$	0	0
$675$	− 6.87851e8i	− 0.0860857i
$676$	0	0
$677$	2.37662e9i	0.294373i	0.989109	+	0.147187i	$0.0470218\pi$
$677$	2.37662e9i	0.294373i	−0.989109	+	0.147187i	$0.952978\pi$
$678$	0	0
$679$	1.05113e10	1.28858
$680$	0	0
$681$	−3.20798e8	−0.0389240
$682$	0	0
$683$	− 5.19062e9i	− 0.623371i	−0.950185	−	0.311685i	$-0.899107\pi$
$683$	− 5.19062e9i	− 0.623371i	0.950185	−	0.311685i	$-0.100893\pi$
$684$	0	0
$685$	− 8.47884e9i	− 1.00790i
$686$	0	0
$687$	−6.56205e8	−0.0772131
$688$	0	0
$689$	1.19144e10	1.38773
$690$	0	0
$691$	1.24469e10i	1.43512i	0.696496	+	0.717560i	$0.254741\pi$
$691$	1.24469e10i	1.43512i	−0.696496	+	0.717560i	$0.745259\pi$
$692$	0	0
$693$	− 1.13687e10i	− 1.29761i
$694$	0	0
$695$	3.10662e9	0.351027
$696$	0	0
$697$	−8.45786e9	−0.946119
$698$	0	0
$699$	− 4.55551e8i	− 0.0504507i
$700$	0	0
$701$	5.62046e8i	0.0616253i	0.999525	+	0.0308126i	$0.00980952\pi$
$701$	5.62046e8i	0.0616253i	−0.999525	+	0.0308126i	$0.990190\pi$
$702$	0	0
$703$	−3.73188e9	−0.405121
$704$	0	0
$705$	8.56579e8	0.0920673
$706$	0	0
$707$	− 5.57771e9i	− 0.593592i
$708$	0	0
$709$	1.37748e10i	1.45152i	0.687949	+	0.725759i	$0.258511\pi$
$709$	1.37748e10i	1.45152i	−0.687949	+	0.725759i	$0.741489\pi$
$710$	0	0
$711$	−4.93762e9	−0.515197
$712$	0	0
$713$	7.32989e9	0.757328
$714$	0	0
$715$	− 2.75119e10i	− 2.81481i
$716$	0	0
$717$	− 1.11077e9i	− 0.112540i
$718$	0	0
$719$	−1.36301e10	−1.36757	−0.683785	−	0.729684i	$-0.739667\pi$
$719$	−1.36301e10	−1.36757	−0.683785	+	0.729684i	$0.739667\pi$
$720$	0	0
$721$	−8.60681e9	−0.855203
$722$	0	0
$723$	− 6.90128e8i	− 0.0679118i
$724$	0	0
$725$	− 2.35183e9i	− 0.229205i
$726$	0	0
$727$	1.17899e10	1.13799	0.568995	−	0.822341i	$-0.307333\pi$
$727$	1.17899e10	1.13799	0.568995	+	0.822341i	$0.307333\pi$
$728$	0	0
$729$	1.02101e10	0.976075
$730$	0	0
$731$	1.40325e9i	0.132869i
$732$	0	0
$733$	− 9.79375e9i	− 0.918513i	−0.888304	−	0.459256i	$-0.848116\pi$
$733$	− 9.79375e9i	− 0.918513i	0.888304	−	0.459256i	$-0.151884\pi$
$734$	0	0
$735$	−3.34790e8	−0.0311005
$736$	0	0
$737$	−2.77923e10	−2.55734
$738$	0	0
$739$	6.85928e9i	0.625206i	0.949884	+	0.312603i	$0.101201\pi$
$739$	6.85928e9i	0.625206i	−0.949884	+	0.312603i	$0.898799\pi$
$740$	0	0
$741$	8.21107e8i	0.0741372i
$742$	0	0
$743$	−1.71545e10	−1.53433	−0.767164	−	0.641451i	$-0.778333\pi$
$743$	−1.71545e10	−1.53433	−0.767164	+	0.641451i	$0.778333\pi$
$744$	0	0
$745$	−3.03671e8	−0.0269064
$746$	0	0
$747$	6.08410e9i	0.534040i
$748$	0	0
$749$	− 1.35538e10i	− 1.17862i
$750$	0	0
$751$	1.15572e10	0.995665	0.497832	−	0.867273i	$-0.334130\pi$
$751$	1.15572e10	0.995665	0.497832	+	0.867273i	$0.334130\pi$
$752$	0	0
$753$	1.17447e9	0.100244
$754$	0	0
$755$	− 1.43517e10i	− 1.21364i
$756$	0	0
$757$	1.28367e10i	1.07552i	0.843099	+	0.537758i	$0.180729\pi$
$757$	1.28367e10i	1.07552i	−0.843099	+	0.537758i	$0.819271\pi$
$758$	0	0
$759$	5.08057e8	0.0421761
$760$	0	0
$761$	9.25308e9	0.761097	0.380549	−	0.924761i	$-0.375735\pi$
$761$	9.25308e9	0.761097	0.380549	+	0.924761i	$0.375735\pi$
$762$	0	0
$763$	− 1.25530e10i	− 1.02309i
$764$	0	0
$765$	8.86312e9i	0.715767i
$766$	0	0
$767$	−2.09458e10	−1.67615
$768$	0	0
$769$	−8.83050e9	−0.700234	−0.350117	−	0.936706i	$-0.613858\pi$
$769$	−8.83050e9	−0.700234	−0.350117	+	0.936706i	$0.613858\pi$
$770$	0	0
$771$	1.51804e9i	0.119287i
$772$	0	0
$773$	1.19769e10i	0.932645i	0.884615	+	0.466323i	$0.154421\pi$
$773$	1.19769e10i	0.932645i	−0.884615	+	0.466323i	$0.845579\pi$
$774$	0	0
$775$	1.65933e10	1.28049
$776$	0	0
$777$	2.95682e8	0.0226127
$778$	0	0
$779$	2.01146e10i	1.52451i
$780$	0	0
$781$	2.03239e10i	1.52661i
$782$	0	0
$783$	−5.70061e8	−0.0424380
$784$	0	0
$785$	1.29930e10	0.958662
$786$	0	0
$787$	1.01486e10i	0.742155i	0.928602	+	0.371077i	$0.121012\pi$
$787$	1.01486e10i	0.742155i	−0.928602	+	0.371077i	$0.878988\pi$
$788$	0	0
$789$	2.20059e7i	0.00159503i
$790$	0	0
$791$	−1.62007e10	−1.16390
$792$	0	0
$793$	5.90509e7	0.00420505
$794$	0	0
$795$	1.22855e9i	0.0867176i
$796$	0	0
$797$	− 2.13861e10i	− 1.49633i	−0.663512	−	0.748166i	$-0.730935\pi$
$797$	− 2.13861e10i	− 1.49633i	0.663512	−	0.748166i	$-0.269065\pi$
$798$	0	0
$799$	−8.96745e9	−0.621950
$800$	0	0
$801$	1.27680e10	0.877827
$802$	0	0
$803$	1.35277e10i	0.921973i
$804$	0	0
$805$	− 6.09937e9i	− 0.412097i
$806$	0	0
$807$	−2.75320e8	−0.0184408
$808$	0	0
$809$	−1.11340e10	−0.739318	−0.369659	−	0.929168i	$-0.620525\pi$
$809$	−1.11340e10	−0.739318	−0.369659	+	0.929168i	$0.620525\pi$
$810$	0	0
$811$	6.29809e9i	0.414606i	0.978277	+	0.207303i	$0.0664686\pi$
$811$	6.29809e9i	0.414606i	−0.978277	+	0.207303i	$0.933531\pi$
$812$	0	0
$813$	− 6.12047e8i	− 0.0399455i
$814$	0	0
$815$	7.26528e9	0.470112
$816$	0	0
$817$	3.33722e9	0.214096
$818$	0	0
$819$	1.61962e10i	1.03020i
$820$	0	0
$821$	− 1.63187e10i	− 1.02916i	−0.857442	−	0.514581i	$-0.827948\pi$
$821$	− 1.63187e10i	− 1.02916i	0.857442	−	0.514581i	$-0.172052\pi$
$822$	0	0
$823$	8.61829e9	0.538917	0.269458	−	0.963012i	$-0.413155\pi$
$823$	8.61829e9	0.538917	0.269458	+	0.963012i	$0.413155\pi$
$824$	0	0
$825$	1.15013e9	0.0713114
$826$	0	0
$827$	7.60314e9i	0.467437i	0.972304	+	0.233719i	$0.0750895\pi$
$827$	7.60314e9i	0.467437i	−0.972304	+	0.233719i	$0.924911\pi$
$828$	0	0
$829$	5.02101e9i	0.306091i	0.988219	+	0.153045i	$0.0489081\pi$
$829$	5.02101e9i	0.306091i	−0.988219	+	0.153045i	$0.951092\pi$
$830$	0	0
$831$	1.61256e9	0.0974796
$832$	0	0
$833$	3.50489e9	0.210096
$834$	0	0
$835$	1.46787e10i	0.872542i
$836$	0	0
$837$	− 4.02205e9i	− 0.237087i
$838$	0	0
$839$	−1.34694e10	−0.787376	−0.393688	−	0.919244i	$-0.628801\pi$
$839$	−1.34694e10	−0.787376	−0.393688	+	0.919244i	$0.628801\pi$
$840$	0	0
$841$	1.53008e10	0.887008
$842$	0	0
$843$	− 3.97421e8i	− 0.0228483i
$844$	0	0
$845$	1.64490e10i	0.937867i
$846$	0	0
$847$	2.41604e10	1.36619
$848$	0	0
$849$	6.01963e8	0.0337593
$850$	0	0
$851$	− 3.28961e9i	− 0.182975i
$852$	0	0
$853$	1.08663e10i	0.599461i	0.954024	+	0.299731i	$0.0968968\pi$
$853$	1.08663e10i	0.599461i	−0.954024	+	0.299731i	$0.903103\pi$
$854$	0	0
$855$	2.10783e10	1.15333
$856$	0	0
$857$	5.18222e9	0.281244	0.140622	−	0.990063i	$-0.455090\pi$
$857$	5.18222e9	0.281244	0.140622	+	0.990063i	$0.455090\pi$
$858$	0	0
$859$	6.90203e9i	0.371536i	0.982594	+	0.185768i	$0.0594773\pi$
$859$	6.90203e9i	0.371536i	−0.982594	+	0.185768i	$0.940523\pi$
$860$	0	0
$861$	− 1.59370e9i	− 0.0850935i
$862$	0	0
$863$	1.60877e10	0.852033	0.426017	−	0.904715i	$-0.359916\pi$
$863$	1.60877e10	0.852033	0.426017	+	0.904715i	$0.359916\pi$
$864$	0	0
$865$	2.07448e10	1.08982
$866$	0	0
$867$	− 8.41064e8i	− 0.0438290i
$868$	0	0
$869$	− 1.65452e10i	− 0.855270i
$870$	0	0
$871$	3.95939e10	2.03032
$872$	0	0
$873$	−3.20201e10	−1.62882
$874$	0	0
$875$	6.44212e9i	0.325088i
$876$	0	0
$877$	− 1.08720e10i	− 0.544266i	−0.962260	−	0.272133i	$-0.912271\pi$
$877$	− 1.08720e10i	− 0.544266i	0.962260	−	0.272133i	$-0.0877291\pi$
$878$	0	0
$879$	−4.66664e8	−0.0231763
$880$	0	0
$881$	−2.06281e10	−1.01635	−0.508176	−	0.861253i	$-0.669680\pi$
$881$	−2.06281e10	−1.01635	−0.508176	+	0.861253i	$0.669680\pi$
$882$	0	0
$883$	− 3.77865e10i	− 1.84703i	−0.383562	−	0.923515i	$-0.625303\pi$
$883$	− 3.77865e10i	− 1.84703i	0.383562	−	0.923515i	$-0.374697\pi$
$884$	0	0
$885$	− 2.15981e9i	− 0.104741i
$886$	0	0
$887$	−4.65710e9	−0.224070	−0.112035	−	0.993704i	$-0.535737\pi$
$887$	−4.65710e9	−0.224070	−0.112035	+	0.993704i	$0.535737\pi$
$888$	0	0
$889$	−3.11027e10	−1.48471
$890$	0	0
$891$	3.44924e10i	1.63362i
$892$	0	0
$893$	2.13265e10i	1.00216i
$894$	0	0
$895$	5.51816e10	2.57285
$896$	0	0
$897$	−7.23797e8	−0.0334844
$898$	0	0
$899$	− 1.37518e10i	− 0.631249i
$900$	0	0
$901$	− 1.28616e10i	− 0.585811i
$902$	0	0
$903$	−2.64413e8	−0.0119502
$904$	0	0
$905$	−7.99298e9	−0.358458
$906$	0	0
$907$	2.02909e10i	0.902975i	0.892277	+	0.451488i	$0.149106\pi$
$907$	2.02909e10i	0.902975i	−0.892277	+	0.451488i	$0.850894\pi$
$908$	0	0
$909$	1.69912e10i	0.750327i
$910$	0	0
$911$	1.59071e10	0.697069	0.348535	−	0.937296i	$-0.386679\pi$
$911$	1.59071e10	0.697069	0.348535	+	0.937296i	$0.386679\pi$
$912$	0	0
$913$	−2.03869e10	−0.886551
$914$	0	0
$915$	6.08900e6i	0 0.000262768i
$916$	0	0
$917$	1.74647e10i	0.747942i
$918$	0	0
$919$	−1.82106e10	−0.773963	−0.386981	−	0.922088i	$-0.626482\pi$
$919$	−1.82106e10	−0.773963	−0.386981	+	0.922088i	$0.626482\pi$
$920$	0	0
$921$	1.12133e9	0.0472961
$922$	0	0
$923$	− 2.89541e10i	− 1.21201i
$924$	0	0
$925$	− 7.44698e9i	− 0.309374i
$926$	0	0
$927$	2.62187e10	1.08101
$928$	0	0
$929$	9.72491e9	0.397952	0.198976	−	0.980004i	$-0.436238\pi$
$929$	9.72491e9	0.397952	0.198976	+	0.980004i	$0.436238\pi$
$930$	0	0
$931$	− 8.33536e9i	− 0.338533i
$932$	0	0
$933$	− 1.82847e9i	− 0.0737060i
$934$	0	0
$935$	−2.96990e10	−1.18823
$936$	0	0
$937$	3.22623e10	1.28117	0.640585	−	0.767887i	$-0.278692\pi$
$937$	3.22623e10	1.28117	0.640585	+	0.767887i	$0.278692\pi$
$938$	0	0
$939$	− 1.37384e9i	− 0.0541508i
$940$	0	0
$941$	− 2.31558e9i	− 0.0905933i	−0.998974	−	0.0452966i	$-0.985577\pi$
$941$	− 2.31558e9i	− 0.0905933i	0.998974	−	0.0452966i	$-0.0144233\pi$
$942$	0	0
$943$	−1.77307e10	−0.688551
$944$	0	0
$945$	−3.34684e9	−0.129010
$946$	0	0
$947$	2.20741e10i	0.844612i	0.906453	+	0.422306i	$0.138779\pi$
$947$	2.20741e10i	0.844612i	−0.906453	+	0.422306i	$0.861221\pi$
$948$	0	0
$949$	− 1.92720e10i	− 0.731973i
$950$	0	0
$951$	4.27626e8	0.0161225
$952$	0	0
$953$	−1.01706e9	−0.0380647	−0.0190323	−	0.999819i	$-0.506059\pi$
$953$	−1.01706e9	−0.0380647	−0.0190323	+	0.999819i	$0.506059\pi$
$954$	0	0
$955$	3.87230e10i	1.43866i
$956$	0	0
$957$	− 9.53180e8i	− 0.0351547i
$958$	0	0
$959$	−1.67257e10	−0.612379
$960$	0	0
$961$	6.95130e10	2.52659
$962$	0	0
$963$	4.12885e10i	1.48983i
$964$	0	0
$965$	− 2.45664e10i	− 0.880026i
$966$	0	0
$967$	3.62288e10	1.28843	0.644216	−	0.764844i	$-0.277184\pi$
$967$	3.62288e10	1.28843	0.644216	+	0.764844i	$0.277184\pi$
$968$	0	0
$969$	8.86382e8	0.0312959
$970$	0	0
$971$	5.55229e9i	0.194628i	0.995254	+	0.0973139i	$0.0310251\pi$
$971$	5.55229e9i	0.194628i	−0.995254	+	0.0973139i	$0.968975\pi$
$972$	0	0
$973$	− 6.12825e9i	− 0.213276i
$974$	0	0
$975$	−1.63852e9	−0.0566156
$976$	0	0
$977$	1.96135e10	0.672861	0.336430	−	0.941708i	$-0.390780\pi$
$977$	1.96135e10	0.672861	0.336430	+	0.941708i	$0.390780\pi$
$978$	0	0
$979$	4.27836e10i	1.45726i
$980$	0	0
$981$	3.82399e10i	1.29323i
$982$	0	0
$983$	2.57104e10	0.863319	0.431659	−	0.902037i	$-0.357928\pi$
$983$	2.57104e10	0.863319	0.431659	+	0.902037i	$0.357928\pi$
$984$	0	0
$985$	1.66221e9	0.0554190
$986$	0	0
$987$	− 1.68973e9i	− 0.0559379i
$988$	0	0
$989$	2.94172e9i	0.0966974i
$990$	0	0
$991$	2.56956e9	0.0838688	0.0419344	−	0.999120i	$-0.486648\pi$
$991$	2.56956e9	0.0838688	0.0419344	+	0.999120i	$0.486648\pi$
$992$	0	0
$993$	1.46648e9	0.0475284
$994$	0	0
$995$	− 3.37391e10i	− 1.08581i
$996$	0	0
$997$	− 5.32099e10i	− 1.70043i	−0.526433	−	0.850216i	$-0.676471\pi$
$997$	− 5.32099e10i	− 1.70043i	0.526433	−	0.850216i	$-0.323529\pi$
$998$	0	0
$999$	−1.80507e9	−0.0572816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.l.129.4		6
4.3	odd	2		256.8.b.k.129.3		6
8.3	odd	2		256.8.b.k.129.4		6
8.5	even	2	inner	256.8.b.l.129.3		6
16.3	odd	4		128.8.a.c.1.2	yes	3
16.5	even	4		128.8.a.d.1.2	yes	3
16.11	odd	4		128.8.a.b.1.2	yes	3
16.13	even	4		128.8.a.a.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.a.a.1.2	✓	3	16.13	even	4
128.8.a.b.1.2	yes	3	16.11	odd	4
128.8.a.c.1.2	yes	3	16.3	odd	4
128.8.a.d.1.2	yes	3	16.5	even	4
256.8.b.k.129.3		6	4.3	odd	2
256.8.b.k.129.4		6	8.3	odd	2
256.8.b.l.129.3		6	8.5	even	2	inner
256.8.b.l.129.4		6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.95798i q^{3} -362.486i q^{5} -715.055 q^{7} +2178.25 q^{9} +7298.99i q^{11} -10398.4i q^{13} +1072.23 q^{15} -11225.0 q^{17} +26695.5i q^{19} -2115.12i q^{21} -23531.8 q^{23} -53270.9 q^{25} +12912.3i q^{27} +44148.6i q^{29} -311489. q^{31} -21590.3 q^{33} +259197. i q^{35} +139795. i q^{37} +30758.3 q^{39} +753482. q^{41} -125011. i q^{43} -789585. i q^{45} +798879. q^{47} -312239. q^{49} -33203.5i q^{51} +1.14579e6i q^{53} +2.64578e6 q^{55} -78964.7 q^{57} -2.01433e6i q^{59} +5678.84i q^{61} -1.55757e6 q^{63} -3.76927e6 q^{65} +3.80769e6i q^{67} -69606.5i q^{69} +2.78448e6 q^{71} +1.85336e6 q^{73} -157574. i q^{75} -5.21918e6i q^{77} -2.26678e6 q^{79} +4.72564e6 q^{81} +2.79311e6i q^{83} +4.06892e6i q^{85} -130591. q^{87} +5.86158e6 q^{89} +7.43544e6i q^{91} -921379. i q^{93} +9.67673e6 q^{95} -1.46999e7 q^{97} +1.58990e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 106 q^{9} - 8232 q^{15} + 1076 q^{17} + 86632 q^{23} + 81206 q^{25} - 313856 q^{31} - 367000 q^{33} + 1243368 q^{39} + 1819396 q^{41} - 1705904 q^{47} - 3273066 q^{49} + 7411352 q^{55}+ \cdots - 10501388 q^{97}+O(q^{100}) \) 6 * q + 24 * q^7 + 106 * q^9 - 8232 * q^15 + 1076 * q^17 + 86632 * q^23 + 81206 * q^25 - 313856 * q^31 - 367000 * q^33 + 1243368 * q^39 + 1819396 * q^41 - 1705904 * q^47 - 3273066 * q^49 + 7411352 * q^55 + 6490536 * q^57 - 5088856 * q^63 - 9549544 * q^65 + 19882936 * q^71 + 11588940 * q^73 - 2009008 * q^79 - 12889762 * q^81 + 36376264 * q^87 + 9662700 * q^89 + 30260056 * q^95 - 10501388 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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