LMFDB - Embedded newform 256.9.d.f.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(127,256)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	256.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$104.288924176$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 81 $ x^4 - 17*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$-2.95804 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.127
Dual form			256.9.d.f.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+141.986 q^{3} -510.000i q^{5} +2555.75i q^{7} +13599.0 q^{9} -19168.1 q^{11} +27710.0i q^{13} -72412.8i q^{15} +50370.0 q^{17} +108619. q^{19} +362880. i q^{21} -176347. i q^{23} +130525. q^{25} +999297. q^{27} -54978.0i q^{29} +1.17564e6i q^{31} -2.72160e6 q^{33} +1.30343e6 q^{35} +793730. i q^{37} +3.93443e6i q^{39} +75582.0 q^{41} -499648. q^{43} -6.93549e6i q^{45} +2.86755e6i q^{47} -767039. q^{49} +7.15183e6 q^{51} +1.11662e7i q^{53} +9.77573e6i q^{55} +1.54224e7 q^{57} +2.18325e7 q^{59} +2.38266e7i q^{61} +3.47556e7i q^{63} +1.41321e7 q^{65} +7.49473e6 q^{67} -2.50387e7i q^{69} +1.00824e7i q^{71} -6.51661e6 q^{73} +1.85327e7 q^{75} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -2.56887e7i q^{85} -7.80610e6i q^{87} -8.67958e7 q^{89} -7.08197e7 q^{91} +1.66925e8i q^{93} -5.53958e7i q^{95} -4.66703e7 q^{97} -2.60667e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100}) $ 4 * q + 54396 * q^9 + 201480 * q^17 + 522100 * q^25 - 10886400 * q^33 + 302328 * q^41 - 3068156 * q^49 + 61689600 * q^57 + 56528400 * q^65 - 26066440 * q^73 + 210652164 * q^81 - 347183112 * q^89 - 186681080 * 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
$3$	141.986	1.75291	0.876456	−	0.481481i	$-0.159901\pi$
$3$	141.986	1.75291	0.876456	+	0.481481i	$0.159901\pi$
$4$	0	0
$5$	− 510.000i	− 0.816000i	−0.912982	−	0.408000i	$-0.866226\pi$
$5$	− 510.000i	− 0.816000i	0.912982	−	0.408000i	$-0.133774\pi$
$6$	0	0
$7$	2555.75i	1.06445i	0.846603	+	0.532225i	$0.178644\pi$
$7$	2555.75i	1.06445i	−0.846603	+	0.532225i	$0.821356\pi$
$8$	0	0
$9$	13599.0	2.07270
$10$	0	0
$11$	−19168.1	−1.30921	−0.654603	−	0.755972i	$-0.727164\pi$
$11$	−19168.1	−1.30921	−0.654603	+	0.755972i	$0.727164\pi$
$12$	0	0
$13$	27710.0i	0.970204i	0.874458	+	0.485102i	$0.161218\pi$
$13$	27710.0i	0.970204i	−0.874458	+	0.485102i	$0.838782\pi$
$14$	0	0
$15$	− 72412.8i	− 1.43038i
$16$	0	0
$17$	50370.0	0.603082	0.301541	−	0.953453i	$-0.402499\pi$
$17$	50370.0	0.603082	0.301541	+	0.953453i	$0.402499\pi$
$18$	0	0
$19$	108619.	0.833474	0.416737	−	0.909027i	$-0.363174\pi$
$19$	108619.	0.833474	0.416737	+	0.909027i	$0.363174\pi$
$20$	0	0
$21$	362880.i	1.86589i
$22$	0	0
$23$	− 176347.i	− 0.630167i	−0.949064	−	0.315083i	$-0.897968\pi$
$23$	− 176347.i	− 0.630167i	0.949064	−	0.315083i	$-0.102032\pi$
$24$	0	0
$25$	130525.	0.334144
$26$	0	0
$27$	999297.	1.88035
$28$	0	0
$29$	− 54978.0i	− 0.0777315i	−0.999244	−	0.0388657i	$-0.987626\pi$
$29$	− 54978.0i	− 0.0777315i	0.999244	−	0.0388657i	$-0.0123745\pi$
$30$	0	0
$31$	1.17564e6i	1.27300i	0.771276	+	0.636501i	$0.219619\pi$
$31$	1.17564e6i	1.27300i	−0.771276	+	0.636501i	$0.780381\pi$
$32$	0	0
$33$	−2.72160e6	−2.29493
$34$	0	0
$35$	1.30343e6	0.868592
$36$	0	0
$37$	793730.i	0.423512i	0.977323	+	0.211756i	$0.0679182\pi$
$37$	793730.i	0.423512i	−0.977323	+	0.211756i	$0.932082\pi$
$38$	0	0
$39$	3.93443e6i	1.70068i
$40$	0	0
$41$	75582.0	0.0267475	0.0133737	−	0.999911i	$-0.495743\pi$
$41$	75582.0	0.0267475	0.0133737	+	0.999911i	$0.495743\pi$
$42$	0	0
$43$	−499648.	−0.146147	−0.0730736	−	0.997327i	$-0.523281\pi$
$43$	−499648.	−0.146147	−0.0730736	+	0.997327i	$0.523281\pi$
$44$	0	0
$45$	− 6.93549e6i	− 1.69133i
$46$	0	0
$47$	2.86755e6i	0.587651i	0.955859	+	0.293825i	$0.0949284\pi$
$47$	2.86755e6i	0.587651i	−0.955859	+	0.293825i	$0.905072\pi$
$48$	0	0
$49$	−767039.	−0.133056
$50$	0	0
$51$	7.15183e6	1.05715
$52$	0	0
$53$	1.11662e7i	1.41515i	0.706639	+	0.707575i	$0.250211\pi$
$53$	1.11662e7i	1.41515i	−0.706639	+	0.707575i	$0.749789\pi$
$54$	0	0
$55$	9.77573e6i	1.06831i
$56$	0	0
$57$	1.54224e7	1.46101
$58$	0	0
$59$	2.18325e7	1.80175	0.900875	−	0.434078i	$-0.142926\pi$
$59$	2.18325e7	1.80175	0.900875	+	0.434078i	$0.142926\pi$
$60$	0	0
$61$	2.38266e7i	1.72085i	0.509577	+	0.860425i	$0.329802\pi$
$61$	2.38266e7i	1.72085i	−0.509577	+	0.860425i	$0.670198\pi$
$62$	0	0
$63$	3.47556e7i	2.20629i
$64$	0	0
$65$	1.41321e7	0.791687
$66$	0	0
$67$	7.49473e6	0.371926	0.185963	−	0.982557i	$-0.440460\pi$
$67$	7.49473e6	0.371926	0.185963	+	0.982557i	$0.440460\pi$
$68$	0	0
$69$	− 2.50387e7i	− 1.10463i
$70$	0	0
$71$	1.00824e7i	0.396763i	0.980125	+	0.198382i	$0.0635685\pi$
$71$	1.00824e7i	0.396763i	−0.980125	+	0.198382i	$0.936431\pi$
$72$	0	0
$73$	−6.51661e6	−0.229472	−0.114736	−	0.993396i	$-0.536602\pi$
$73$	−6.51661e6	−0.229472	−0.114736	+	0.993396i	$0.536602\pi$
$74$	0	0
$75$	1.85327e7	0.585725
$76$	0	0
$77$	− 4.89888e7i	− 1.39359i
$78$	0	0
$79$	− 4.87892e7i	− 1.25261i	−0.779579	−	0.626304i	$-0.784567\pi$
$79$	− 4.87892e7i	− 1.25261i	0.779579	−	0.626304i	$-0.215433\pi$
$80$	0	0
$81$	5.26630e7	1.22339
$82$	0	0
$83$	7.34483e7	1.54764	0.773819	−	0.633407i	$-0.218344\pi$
$83$	7.34483e7	1.54764	0.773819	+	0.633407i	$0.218344\pi$
$84$	0	0
$85$	− 2.56887e7i	− 0.492115i
$86$	0	0
$87$	− 7.80610e6i	− 0.136256i
$88$	0	0
$89$	−8.67958e7	−1.38337	−0.691685	−	0.722199i	$-0.743131\pi$
$89$	−8.67958e7	−1.38337	−0.691685	+	0.722199i	$0.743131\pi$
$90$	0	0
$91$	−7.08197e7	−1.03273
$92$	0	0
$93$	1.66925e8i	2.23146i
$94$	0	0
$95$	− 5.53958e7i	− 0.680115i
$96$	0	0
$97$	−4.66703e7	−0.527173	−0.263587	−	0.964636i	$-0.584905\pi$
$97$	−4.66703e7	−0.527173	−0.263587	+	0.964636i	$0.584905\pi$
$98$	0	0
$99$	−2.60667e8	−2.71360
$100$	0	0
$101$	6.59910e7i	0.634161i	0.948399	+	0.317080i	$0.102703\pi$
$101$	6.59910e7i	0.634161i	−0.948399	+	0.317080i	$0.897297\pi$
$102$	0	0
$103$	− 1.64884e8i	− 1.46497i	−0.680782	−	0.732486i	$-0.738360\pi$
$103$	− 1.64884e8i	− 1.46497i	0.680782	−	0.732486i	$-0.261640\pi$
$104$	0	0
$105$	1.85069e8	1.52257
$106$	0	0
$107$	−1.27326e8	−0.971364	−0.485682	−	0.874136i	$-0.661429\pi$
$107$	−1.27326e8	−0.971364	−0.485682	+	0.874136i	$0.661429\pi$
$108$	0	0
$109$	1.56119e8i	1.10598i	0.833186	+	0.552992i	$0.186514\pi$
$109$	1.56119e8i	1.10598i	−0.833186	+	0.552992i	$0.813486\pi$
$110$	0	0
$111$	1.12698e8i	0.742380i
$112$	0	0
$113$	2.36346e8	1.44955	0.724776	−	0.688984i	$-0.241943\pi$
$113$	2.36346e8	1.44955	0.724776	+	0.688984i	$0.241943\pi$
$114$	0	0
$115$	−8.99367e7	−0.514216
$116$	0	0
$117$	3.76828e8i	2.01094i
$118$	0	0
$119$	1.28733e8i	0.641951i
$120$	0	0
$121$	1.53057e8	0.714023
$122$	0	0
$123$	1.07316e7	0.0468860
$124$	0	0
$125$	− 2.65786e8i	− 1.08866i
$126$	0	0
$127$	− 3.67741e8i	− 1.41360i	−0.707412	−	0.706802i	$-0.750137\pi$
$127$	− 3.67741e8i	− 1.41360i	0.707412	−	0.706802i	$-0.249863\pi$
$128$	0	0
$129$	−7.09430e7	−0.256183
$130$	0	0
$131$	2.16350e8	0.734636	0.367318	−	0.930095i	$-0.380276\pi$
$131$	2.16350e8	0.734636	0.367318	+	0.930095i	$0.380276\pi$
$132$	0	0
$133$	2.77603e8i	0.887193i
$134$	0	0
$135$	− 5.09641e8i	− 1.53437i
$136$	0	0
$137$	3.86442e8	1.09699	0.548494	−	0.836155i	$-0.315201\pi$
$137$	3.86442e8	1.09699	0.548494	+	0.836155i	$0.315201\pi$
$138$	0	0
$139$	−3.51077e8	−0.940465	−0.470232	−	0.882543i	$-0.655830\pi$
$139$	−3.51077e8	−0.940465	−0.470232	+	0.882543i	$0.655830\pi$
$140$	0	0
$141$	4.07151e8i	1.03010i
$142$	0	0
$143$	− 5.31148e8i	− 1.27020i
$144$	0	0
$145$	−2.80388e7	−0.0634289
$146$	0	0
$147$	−1.08909e8	−0.233235
$148$	0	0
$149$	− 4.54099e8i	− 0.921308i	−0.887580	−	0.460654i	$-0.847615\pi$
$149$	− 4.54099e8i	− 0.921308i	0.887580	−	0.460654i	$-0.152385\pi$
$150$	0	0
$151$	6.60188e8i	1.26987i	0.772565	+	0.634936i	$0.218973\pi$
$151$	6.60188e8i	1.26987i	−0.772565	+	0.634936i	$0.781027\pi$
$152$	0	0
$153$	6.84982e8	1.25001
$154$	0	0
$155$	5.99578e8	1.03877
$156$	0	0
$157$	− 4.35318e8i	− 0.716486i	−0.933628	−	0.358243i	$-0.883376\pi$
$157$	− 4.35318e8i	− 0.716486i	0.933628	−	0.358243i	$-0.116624\pi$
$158$	0	0
$159$	1.58544e9i	2.48063i
$160$	0	0
$161$	4.50697e8	0.670782
$162$	0	0
$163$	2.44065e8	0.345744	0.172872	−	0.984944i	$-0.444695\pi$
$163$	2.44065e8	0.345744	0.172872	+	0.984944i	$0.444695\pi$
$164$	0	0
$165$	1.38802e9i	1.87266i
$166$	0	0
$167$	− 6.71351e8i	− 0.863145i	−0.902078	−	0.431573i	$-0.857959\pi$
$167$	− 6.71351e8i	− 0.863145i	0.902078	−	0.431573i	$-0.142041\pi$
$168$	0	0
$169$	4.78866e7	0.0587040
$170$	0	0
$171$	1.47711e9	1.72754
$172$	0	0
$173$	1.76764e9i	1.97337i	0.162644	+	0.986685i	$0.447998\pi$
$173$	1.76764e9i	1.97337i	−0.162644	+	0.986685i	$0.552002\pi$
$174$	0	0
$175$	3.33589e8i	0.355680i
$176$	0	0
$177$	3.09990e9	3.15831
$178$	0	0
$179$	7.56967e8	0.737335	0.368668	−	0.929561i	$-0.379814\pi$
$179$	7.56967e8	0.737335	0.368668	+	0.929561i	$0.379814\pi$
$180$	0	0
$181$	6.27094e8i	0.584277i	0.956376	+	0.292138i	$0.0943668\pi$
$181$	6.27094e8i	0.584277i	−0.956376	+	0.292138i	$0.905633\pi$
$182$	0	0
$183$	3.38304e9i	3.01650i
$184$	0	0
$185$	4.04802e8	0.345586
$186$	0	0
$187$	−9.65497e8	−0.789559
$188$	0	0
$189$	2.55395e9i	2.00154i
$190$	0	0
$191$	1.07924e9i	0.810933i	0.914110	+	0.405466i	$0.132891\pi$
$191$	1.07924e9i	0.810933i	−0.914110	+	0.405466i	$0.867109\pi$
$192$	0	0
$193$	−2.96757e7	−0.0213881	−0.0106940	−	0.999943i	$-0.503404\pi$
$193$	−2.96757e7	−0.0213881	−0.0106940	+	0.999943i	$0.503404\pi$
$194$	0	0
$195$	2.00656e9	1.38776
$196$	0	0
$197$	− 1.12484e9i	− 0.746837i	−0.927663	−	0.373419i	$-0.878186\pi$
$197$	− 1.12484e9i	− 0.746837i	0.927663	−	0.373419i	$-0.121814\pi$
$198$	0	0
$199$	− 1.04718e9i	− 0.667742i	−0.942619	−	0.333871i	$-0.891645\pi$
$199$	− 1.04718e9i	− 0.667742i	0.942619	−	0.333871i	$-0.108355\pi$
$200$	0	0
$201$	1.06415e9	0.651954
$202$	0	0
$203$	1.40510e8	0.0827413
$204$	0	0
$205$	− 3.85468e7i	− 0.0218259i
$206$	0	0
$207$	− 2.39814e9i	− 1.30615i
$208$	0	0
$209$	−2.08202e9	−1.09119
$210$	0	0
$211$	−2.77676e9	−1.40090	−0.700452	−	0.713699i	$-0.747018\pi$
$211$	−2.77676e9	−1.40090	−0.700452	+	0.713699i	$0.747018\pi$
$212$	0	0
$213$	1.43156e9i	0.695491i
$214$	0	0
$215$	2.54821e8i	0.119256i
$216$	0	0
$217$	−3.00465e9	−1.35505
$218$	0	0
$219$	−9.25267e8	−0.402245
$220$	0	0
$221$	1.39575e9i	0.585113i
$222$	0	0
$223$	2.27822e9i	0.921248i	0.887595	+	0.460624i	$0.152374\pi$
$223$	2.27822e9i	0.921248i	−0.887595	+	0.460624i	$0.847626\pi$
$224$	0	0
$225$	1.77501e9	0.692581
$226$	0	0
$227$	−1.48863e9	−0.560639	−0.280319	−	0.959907i	$-0.590440\pi$
$227$	−1.48863e9	−0.560639	−0.280319	+	0.959907i	$0.590440\pi$
$228$	0	0
$229$	1.69447e9i	0.616157i	0.951361	+	0.308079i	$0.0996860\pi$
$229$	1.69447e9i	0.616157i	−0.951361	+	0.308079i	$0.900314\pi$
$230$	0	0
$231$	− 6.95572e9i	− 2.44284i
$232$	0	0
$233$	−5.21423e8	−0.176916	−0.0884580	−	0.996080i	$-0.528194\pi$
$233$	−5.21423e8	−0.176916	−0.0884580	+	0.996080i	$0.528194\pi$
$234$	0	0
$235$	1.46245e9	0.479523
$236$	0	0
$237$	− 6.92738e9i	− 2.19571i
$238$	0	0
$239$	− 4.40690e9i	− 1.35065i	−0.737522	−	0.675323i	$-0.764004\pi$
$239$	− 4.40690e9i	− 1.35065i	0.737522	−	0.675323i	$-0.235996\pi$
$240$	0	0
$241$	−1.62148e9	−0.480666	−0.240333	−	0.970691i	$-0.577257\pi$
$241$	−1.62148e9	−0.480666	−0.240333	+	0.970691i	$0.577257\pi$
$242$	0	0
$243$	9.21023e8	0.264147
$244$	0	0
$245$	3.91190e8i	0.108573i
$246$	0	0
$247$	3.00984e9i	0.808640i
$248$	0	0
$249$	1.04286e10	2.71287
$250$	0	0
$251$	1.31321e8	0.0330855	0.0165428	−	0.999863i	$-0.494734\pi$
$251$	1.31321e8	0.0330855	0.0165428	+	0.999863i	$0.494734\pi$
$252$	0	0
$253$	3.38023e9i	0.825019i
$254$	0	0
$255$	− 3.64743e9i	− 0.862634i
$256$	0	0
$257$	5.27789e9	1.20984	0.604920	−	0.796287i	$-0.293205\pi$
$257$	5.27789e9	1.20984	0.604920	+	0.796287i	$0.293205\pi$
$258$	0	0
$259$	−2.02857e9	−0.450808
$260$	0	0
$261$	− 7.47646e8i	− 0.161114i
$262$	0	0
$263$	− 7.38745e9i	− 1.54409i	−0.635570	−	0.772044i	$-0.719235\pi$
$263$	− 7.38745e9i	− 1.54409i	0.635570	−	0.772044i	$-0.280765\pi$
$264$	0	0
$265$	5.69477e9	1.15476
$266$	0	0
$267$	−1.23238e10	−2.42493
$268$	0	0
$269$	− 3.46450e8i	− 0.0661655i	−0.999453	−	0.0330828i	$-0.989468\pi$
$269$	− 3.46450e8i	− 0.0661655i	0.999453	−	0.0330828i	$-0.0105325\pi$
$270$	0	0
$271$	6.51715e6i	0.00120832i	1.00000		0.000604158i	$0.000192310\pi$
$271$	6.51715e6i	0.00120832i	−1.00000		0.000604158i	$0.999808\pi$
$272$	0	0
$273$	−1.00554e10	−1.81029
$274$	0	0
$275$	−2.50192e9	−0.437464
$276$	0	0
$277$	2.15061e9i	0.365293i	0.983179	+	0.182647i	$0.0584664\pi$
$277$	2.15061e9i	0.365293i	−0.983179	+	0.182647i	$0.941534\pi$
$278$	0	0
$279$	1.59876e10i	2.63855i
$280$	0	0
$281$	−1.04256e10	−1.67215	−0.836074	−	0.548616i	$-0.815155\pi$
$281$	−1.04256e10	−1.67215	−0.836074	+	0.548616i	$0.815155\pi$
$282$	0	0
$283$	−1.28042e9	−0.199622	−0.0998108	−	0.995006i	$-0.531824\pi$
$283$	−1.28042e9	−0.199622	−0.0998108	+	0.995006i	$0.531824\pi$
$284$	0	0
$285$	− 7.86542e9i	− 1.19218i
$286$	0	0
$287$	1.93168e8i	0.0284714i
$288$	0	0
$289$	−4.43862e9	−0.636292
$290$	0	0
$291$	−6.62652e9	−0.924089
$292$	0	0
$293$	− 2.13786e9i	− 0.290074i	−0.989426	−	0.145037i	$-0.953670\pi$
$293$	− 2.13786e9i	− 0.290074i	0.989426	−	0.145037i	$-0.0463301\pi$
$294$	0	0
$295$	− 1.11346e10i	− 1.47023i
$296$	0	0
$297$	−1.91546e10	−2.46177
$298$	0	0
$299$	4.88656e9	0.611390
$300$	0	0
$301$	− 1.27697e9i	− 0.155567i
$302$	0	0
$303$	9.36980e9i	1.11163i
$304$	0	0
$305$	1.21516e10	1.40421
$306$	0	0
$307$	5.45140e9	0.613698	0.306849	−	0.951758i	$-0.400725\pi$
$307$	5.45140e9	0.613698	0.306849	+	0.951758i	$0.400725\pi$
$308$	0	0
$309$	− 2.34112e10i	− 2.56797i
$310$	0	0
$311$	− 1.07550e10i	− 1.14965i	−0.818275	−	0.574827i	$-0.805069\pi$
$311$	− 1.07550e10i	− 1.14965i	0.818275	−	0.574827i	$-0.194931\pi$
$312$	0	0
$313$	2.99804e8	0.0312364	0.0156182	−	0.999878i	$-0.495028\pi$
$313$	2.99804e8	0.0312364	0.0156182	+	0.999878i	$0.495028\pi$
$314$	0	0
$315$	1.77254e10	1.80033
$316$	0	0
$317$	− 5.31172e9i	− 0.526015i	−0.964794	−	0.263007i	$-0.915286\pi$
$317$	− 5.31172e9i	− 0.526015i	0.964794	−	0.263007i	$-0.0847144\pi$
$318$	0	0
$319$	1.05382e9i	0.101767i
$320$	0	0
$321$	−1.80785e10	−1.70272
$322$	0	0
$323$	5.47115e9	0.502653
$324$	0	0
$325$	3.61685e9i	0.324188i
$326$	0	0
$327$	2.21667e10i	1.93869i
$328$	0	0
$329$	−7.32872e9	−0.625525
$330$	0	0
$331$	1.01004e10	0.841446	0.420723	−	0.907189i	$-0.361776\pi$
$331$	1.01004e10	0.841446	0.420723	+	0.907189i	$0.361776\pi$
$332$	0	0
$333$	1.07939e10i	0.877815i
$334$	0	0
$335$	− 3.82231e9i	− 0.303492i
$336$	0	0
$337$	−1.84359e10	−1.42937	−0.714684	−	0.699448i	$-0.753429\pi$
$337$	−1.84359e10	−1.42937	−0.714684	+	0.699448i	$0.753429\pi$
$338$	0	0
$339$	3.35578e10	2.54094
$340$	0	0
$341$	− 2.25348e10i	− 1.66662i
$342$	0	0
$343$	1.27730e10i	0.922820i
$344$	0	0
$345$	−1.27697e10	−0.901376
$346$	0	0
$347$	−1.27822e10	−0.881636	−0.440818	−	0.897597i	$-0.645312\pi$
$347$	−1.27822e10	−0.881636	−0.440818	+	0.897597i	$0.645312\pi$
$348$	0	0
$349$	6.39381e8i	0.0430981i	0.999768	+	0.0215490i	$0.00685980\pi$
$349$	6.39381e8i	0.0430981i	−0.999768	+	0.0215490i	$0.993140\pi$
$350$	0	0
$351$	2.76905e10i	1.82433i
$352$	0	0
$353$	2.59837e10	1.67341	0.836705	−	0.547653i	$-0.184479\pi$
$353$	2.59837e10	1.67341	0.836705	+	0.547653i	$0.184479\pi$
$354$	0	0
$355$	5.14203e9	0.323759
$356$	0	0
$357$	1.82783e10i	1.12528i
$358$	0	0
$359$	2.22541e10i	1.33978i	0.742461	+	0.669889i	$0.233658\pi$
$359$	2.22541e10i	1.33978i	−0.742461	+	0.669889i	$0.766342\pi$
$360$	0	0
$361$	−5.18543e9	−0.305320
$362$	0	0
$363$	2.17320e10	1.25162
$364$	0	0
$365$	3.32347e9i	0.187249i
$366$	0	0
$367$	− 1.86229e10i	− 1.02656i	−0.858221	−	0.513280i	$-0.828430\pi$
$367$	− 1.86229e10i	− 1.02656i	0.858221	−	0.513280i	$-0.171570\pi$
$368$	0	0
$369$	1.02784e9	0.0554396
$370$	0	0
$371$	−2.85380e10	−1.50636
$372$	0	0
$373$	1.19680e10i	0.618283i	0.951016	+	0.309141i	$0.100042\pi$
$373$	1.19680e10i	0.618283i	−0.951016	+	0.309141i	$0.899958\pi$
$374$	0	0
$375$	− 3.77379e10i	− 1.90833i
$376$	0	0
$377$	1.52344e9	0.0754154
$378$	0	0
$379$	−2.30787e10	−1.11855	−0.559275	−	0.828982i	$-0.688920\pi$
$379$	−2.30787e10	−1.11855	−0.559275	+	0.828982i	$0.688920\pi$
$380$	0	0
$381$	− 5.22141e10i	− 2.47792i
$382$	0	0
$383$	− 1.43419e10i	− 0.666518i	−0.942835	−	0.333259i	$-0.891852\pi$
$383$	− 1.43419e10i	− 0.666518i	0.942835	−	0.333259i	$-0.108148\pi$
$384$	0	0
$385$	−2.49843e10	−1.13717
$386$	0	0
$387$	−6.79472e9	−0.302920
$388$	0	0
$389$	− 2.73457e10i	− 1.19424i	−0.802152	−	0.597119i	$-0.796312\pi$
$389$	− 2.73457e10i	− 1.19424i	0.802152	−	0.597119i	$-0.203688\pi$
$390$	0	0
$391$	− 8.88257e9i	− 0.380042i
$392$	0	0
$393$	3.07187e10	1.28775
$394$	0	0
$395$	−2.48825e10	−1.02213
$396$	0	0
$397$	− 3.99456e10i	− 1.60808i	−0.594576	−	0.804039i	$-0.702680\pi$
$397$	− 3.99456e10i	− 1.60808i	0.594576	−	0.804039i	$-0.297320\pi$
$398$	0	0
$399$	3.94157e10i	1.55517i
$400$	0	0
$401$	−2.12767e10	−0.822863	−0.411431	−	0.911441i	$-0.634971\pi$
$401$	−2.12767e10	−0.822863	−0.411431	+	0.911441i	$0.634971\pi$
$402$	0	0
$403$	−3.25771e10	−1.23507
$404$	0	0
$405$	− 2.68582e10i	− 0.998288i
$406$	0	0
$407$	− 1.52143e10i	− 0.554465i
$408$	0	0
$409$	−1.14283e10	−0.408404	−0.204202	−	0.978929i	$-0.565460\pi$
$409$	−1.14283e10	−0.408404	−0.204202	+	0.978929i	$0.565460\pi$
$410$	0	0
$411$	5.48693e10	1.92292
$412$	0	0
$413$	5.57982e10i	1.91788i
$414$	0	0
$415$	− 3.74586e10i	− 1.26287i
$416$	0	0
$417$	−4.98479e10	−1.64855
$418$	0	0
$419$	1.10009e10	0.356922	0.178461	−	0.983947i	$-0.442888\pi$
$419$	1.10009e10	0.356922	0.178461	+	0.983947i	$0.442888\pi$
$420$	0	0
$421$	2.28766e10i	0.728220i	0.931356	+	0.364110i	$0.118627\pi$
$421$	2.28766e10i	0.728220i	−0.931356	+	0.364110i	$0.881373\pi$
$422$	0	0
$423$	3.89958e10i	1.21802i
$424$	0	0
$425$	6.57454e9	0.201516
$426$	0	0
$427$	−6.08948e10	−1.83176
$428$	0	0
$429$	− 7.54155e10i	− 2.22655i
$430$	0	0
$431$	− 9.55108e8i	− 0.0276786i	−0.999904	−	0.0138393i	$-0.995595\pi$
$431$	− 9.55108e8i	− 0.0276786i	0.999904	−	0.0138393i	$-0.00440532\pi$
$432$	0	0
$433$	3.82225e10	1.08735	0.543673	−	0.839297i	$-0.317033\pi$
$433$	3.82225e10	1.08735	0.543673	+	0.839297i	$0.317033\pi$
$434$	0	0
$435$	−3.98111e9	−0.111185
$436$	0	0
$437$	− 1.91546e10i	− 0.525228i
$438$	0	0
$439$	− 6.40288e10i	− 1.72392i	−0.506976	−	0.861960i	$-0.669237\pi$
$439$	− 6.40288e10i	− 1.72392i	0.506976	−	0.861960i	$-0.330763\pi$
$440$	0	0
$441$	−1.04310e10	−0.275785
$442$	0	0
$443$	−7.47659e10	−1.94128	−0.970641	−	0.240533i	$-0.922678\pi$
$443$	−7.47659e10	−1.94128	−0.970641	+	0.240533i	$0.922678\pi$
$444$	0	0
$445$	4.42658e10i	1.12883i
$446$	0	0
$447$	− 6.44756e10i	− 1.61497i
$448$	0	0
$449$	2.51987e10	0.620001	0.310000	−	0.950736i	$-0.399671\pi$
$449$	2.51987e10	0.620001	0.310000	+	0.950736i	$0.399671\pi$
$450$	0	0
$451$	−1.44876e9	−0.0350180
$452$	0	0
$453$	9.37373e10i	2.22597i
$454$	0	0
$455$	3.61181e10i	0.842711i
$456$	0	0
$457$	−4.66828e9	−0.107027	−0.0535133	−	0.998567i	$-0.517042\pi$
$457$	−4.66828e9	−0.107027	−0.0535133	+	0.998567i	$0.517042\pi$
$458$	0	0
$459$	5.03346e10	1.13401
$460$	0	0
$461$	3.88096e10i	0.859281i	0.903000	+	0.429641i	$0.141360\pi$
$461$	3.88096e10i	0.859281i	−0.903000	+	0.429641i	$0.858640\pi$
$462$	0	0
$463$	− 3.23432e10i	− 0.703817i	−0.936034	−	0.351908i	$-0.885533\pi$
$463$	− 3.23432e10i	− 0.703817i	0.936034	−	0.351908i	$-0.114467\pi$
$464$	0	0
$465$	8.51316e10	1.82087
$466$	0	0
$467$	−2.31902e10	−0.487570	−0.243785	−	0.969829i	$-0.578389\pi$
$467$	−2.31902e10	−0.487570	−0.243785	+	0.969829i	$0.578389\pi$
$468$	0	0
$469$	1.91546e10i	0.395897i
$470$	0	0
$471$	− 6.18090e10i	− 1.25594i
$472$	0	0
$473$	9.57731e9	0.191337
$474$	0	0
$475$	1.41775e10	0.278500
$476$	0	0
$477$	1.51849e11i	2.93318i
$478$	0	0
$479$	− 4.83542e9i	− 0.0918528i	−0.998945	−	0.0459264i	$-0.985376\pi$
$479$	− 4.83542e9i	− 0.0918528i	0.998945	−	0.0459264i	$-0.0146240\pi$
$480$	0	0
$481$	−2.19943e10	−0.410893
$482$	0	0
$483$	6.39926e10	1.17582
$484$	0	0
$485$	2.38018e10i	0.430173i
$486$	0	0
$487$	− 3.03878e10i	− 0.540236i	−0.962827	−	0.270118i	$-0.912937\pi$
$487$	− 3.03878e10i	− 0.540236i	0.962827	−	0.270118i	$-0.0870628\pi$
$488$	0	0
$489$	3.46538e10	0.606059
$490$	0	0
$491$	5.56483e10	0.957472	0.478736	−	0.877959i	$-0.341095\pi$
$491$	5.56483e10	0.957472	0.478736	+	0.877959i	$0.341095\pi$
$492$	0	0
$493$	− 2.76924e9i	− 0.0468784i
$494$	0	0
$495$	1.32940e11i	2.21429i
$496$	0	0
$497$	−2.57681e10	−0.422335
$498$	0	0
$499$	−7.88458e10	−1.27168	−0.635838	−	0.771822i	$-0.719345\pi$
$499$	−7.88458e10	−1.27168	−0.635838	+	0.771822i	$0.719345\pi$
$500$	0	0
$501$	− 9.53224e10i	− 1.51302i
$502$	0	0
$503$	− 4.41092e10i	− 0.689061i	−0.938775	−	0.344530i	$-0.888038\pi$
$503$	− 4.41092e10i	− 0.689061i	0.938775	−	0.344530i	$-0.111962\pi$
$504$	0	0
$505$	3.36554e10	0.517475
$506$	0	0
$507$	6.79923e9	0.102903
$508$	0	0
$509$	− 1.05927e10i	− 0.157811i	−0.996882	−	0.0789055i	$-0.974857\pi$
$509$	− 1.05927e10i	− 0.157811i	0.996882	−	0.0789055i	$-0.0251425\pi$
$510$	0	0
$511$	− 1.66548e10i	− 0.244262i
$512$	0	0
$513$	1.08543e11	1.56723
$514$	0	0
$515$	−8.40908e10	−1.19542
$516$	0	0
$517$	− 5.49654e10i	− 0.769356i
$518$	0	0
$519$	2.50979e11i	3.45914i
$520$	0	0
$521$	−1.24958e11	−1.69595	−0.847973	−	0.530039i	$-0.822177\pi$
$521$	−1.24958e11	−1.69595	−0.847973	+	0.530039i	$0.822177\pi$
$522$	0	0
$523$	−2.80408e10	−0.374786	−0.187393	−	0.982285i	$-0.560004\pi$
$523$	−2.80408e10	−0.374786	−0.187393	+	0.982285i	$0.560004\pi$
$524$	0	0
$525$	4.73649e10i	0.623476i
$526$	0	0
$527$	5.92172e10i	0.767724i
$528$	0	0
$529$	4.72129e10	0.602890
$530$	0	0
$531$	2.96900e11	3.73449
$532$	0	0
$533$	2.09438e9i	0.0259505i
$534$	0	0
$535$	6.49363e10i	0.792633i
$536$	0	0
$537$	1.07479e11	1.29248
$538$	0	0
$539$	1.47027e10	0.174197
$540$	0	0
$541$	− 1.44659e11i	− 1.68871i	−0.535782	−	0.844356i	$-0.679983\pi$
$541$	− 1.44659e11i	− 1.68871i	0.535782	−	0.844356i	$-0.320017\pi$
$542$	0	0
$543$	8.90386e10i	1.02419i
$544$	0	0
$545$	7.96205e10	0.902483
$546$	0	0
$547$	1.03774e10	0.115915	0.0579573	−	0.998319i	$-0.481541\pi$
$547$	1.03774e10	0.115915	0.0579573	+	0.998319i	$0.481541\pi$
$548$	0	0
$549$	3.24018e11i	3.56681i
$550$	0	0
$551$	− 5.97167e9i	− 0.0647872i
$552$	0	0
$553$	1.24693e11	1.33334
$554$	0	0
$555$	5.74762e10	0.605782
$556$	0	0
$557$	5.47312e9i	0.0568610i	0.999596	+	0.0284305i	$0.00905093\pi$
$557$	5.47312e9i	0.0568610i	−0.999596	+	0.0284305i	$0.990949\pi$
$558$	0	0
$559$	− 1.38453e10i	− 0.141793i
$560$	0	0
$561$	−1.37087e11	−1.38403
$562$	0	0
$563$	−4.36118e10	−0.434081	−0.217040	−	0.976163i	$-0.569640\pi$
$563$	−4.36118e10	−0.434081	−0.217040	+	0.976163i	$0.569640\pi$
$564$	0	0
$565$	− 1.20536e11i	− 1.18283i
$566$	0	0
$567$	1.34593e11i	1.30224i
$568$	0	0
$569$	−1.27822e10	−0.121943	−0.0609716	−	0.998140i	$-0.519420\pi$
$569$	−1.27822e10	−0.121943	−0.0609716	+	0.998140i	$0.519420\pi$
$570$	0	0
$571$	7.59455e10	0.714427	0.357213	−	0.934023i	$-0.383727\pi$
$571$	7.59455e10	0.714427	0.357213	+	0.934023i	$0.383727\pi$
$572$	0	0
$573$	1.53237e11i	1.42149i
$574$	0	0
$575$	− 2.30176e10i	− 0.210566i
$576$	0	0
$577$	2.13827e10	0.192912	0.0964560	−	0.995337i	$-0.469249\pi$
$577$	2.13827e10	0.192912	0.0964560	+	0.995337i	$0.469249\pi$
$578$	0	0
$579$	−4.21353e9	−0.0374914
$580$	0	0
$581$	1.87715e11i	1.64738i
$582$	0	0
$583$	− 2.14035e11i	− 1.85272i
$584$	0	0
$585$	1.92182e11	1.64093
$586$	0	0
$587$	−6.07298e10	−0.511504	−0.255752	−	0.966742i	$-0.582323\pi$
$587$	−6.07298e10	−0.511504	−0.255752	+	0.966742i	$0.582323\pi$
$588$	0	0
$589$	1.27697e11i	1.06101i
$590$	0	0
$591$	− 1.59711e11i	− 1.30914i
$592$	0	0
$593$	1.15978e11	0.937899	0.468949	−	0.883225i	$-0.344633\pi$
$593$	1.15978e11	0.937899	0.468949	+	0.883225i	$0.344633\pi$
$594$	0	0
$595$	6.56538e10	0.523832
$596$	0	0
$597$	− 1.48685e11i	− 1.17049i
$598$	0	0
$599$	− 2.40647e11i	− 1.86927i	−0.355607	−	0.934636i	$-0.615726\pi$
$599$	− 2.40647e11i	− 1.86927i	0.355607	−	0.934636i	$-0.384274\pi$
$600$	0	0
$601$	1.92942e11	1.47887	0.739434	−	0.673229i	$-0.235093\pi$
$601$	1.92942e11	1.47887	0.739434	+	0.673229i	$0.235093\pi$
$602$	0	0
$603$	1.01921e11	0.770892
$604$	0	0
$605$	− 7.80591e10i	− 0.582643i
$606$	0	0
$607$	1.62042e11i	1.19364i	0.802376	+	0.596819i	$0.203569\pi$
$607$	1.62042e11i	1.19364i	−0.802376	+	0.596819i	$0.796431\pi$
$608$	0	0
$609$	1.99504e10	0.145038
$610$	0	0
$611$	−7.94597e10	−0.570141
$612$	0	0
$613$	1.76424e11i	1.24944i	0.780847	+	0.624722i	$0.214788\pi$
$613$	1.76424e11i	1.24944i	−0.780847	+	0.624722i	$0.785212\pi$
$614$	0	0
$615$	− 5.47311e9i	− 0.0382590i
$616$	0	0
$617$	9.84986e10	0.679656	0.339828	−	0.940488i	$-0.389631\pi$
$617$	9.84986e10	0.679656	0.339828	+	0.940488i	$0.389631\pi$
$618$	0	0
$619$	1.28596e10	0.0875923	0.0437961	−	0.999040i	$-0.486055\pi$
$619$	1.28596e10	0.0875923	0.0437961	+	0.999040i	$0.486055\pi$
$620$	0	0
$621$	− 1.76223e11i	− 1.18494i
$622$	0	0
$623$	− 2.21828e11i	− 1.47253i
$624$	0	0
$625$	−8.45648e10	−0.554204
$626$	0	0
$627$	−2.95618e11	−1.91276
$628$	0	0
$629$	3.99802e10i	0.255413i
$630$	0	0
$631$	1.73463e11i	1.09418i	0.837073	+	0.547091i	$0.184265\pi$
$631$	1.73463e11i	1.09418i	−0.837073	+	0.547091i	$0.815735\pi$
$632$	0	0
$633$	−3.94261e11	−2.45566
$634$	0	0
$635$	−1.87548e11	−1.15350
$636$	0	0
$637$	− 2.12547e10i	− 0.129091i
$638$	0	0
$639$	1.37111e11i	0.822372i
$640$	0	0
$641$	−1.13903e11	−0.674690	−0.337345	−	0.941381i	$-0.609529\pi$
$641$	−1.13903e11	−0.674690	−0.337345	+	0.941381i	$0.609529\pi$
$642$	0	0
$643$	−7.04067e10	−0.411879	−0.205940	−	0.978565i	$-0.566025\pi$
$643$	−7.04067e10	−0.411879	−0.205940	+	0.978565i	$0.566025\pi$
$644$	0	0
$645$	3.61810e10i	0.209046i
$646$	0	0
$647$	− 1.99175e11i	− 1.13663i	−0.822812	−	0.568314i	$-0.807596\pi$
$647$	− 1.99175e11i	− 1.13663i	0.822812	−	0.568314i	$-0.192404\pi$
$648$	0	0
$649$	−4.18487e11	−2.35886
$650$	0	0
$651$	−4.26617e11	−2.37528
$652$	0	0
$653$	− 6.49972e9i	− 0.0357472i	−0.999840	−	0.0178736i	$-0.994310\pi$
$653$	− 6.49972e9i	− 0.0357472i	0.999840	−	0.0178736i	$-0.00568964\pi$
$654$	0	0
$655$	− 1.10339e11i	− 0.599463i
$656$	0	0
$657$	−8.86194e10	−0.475628
$658$	0	0
$659$	2.20982e11	1.17170	0.585848	−	0.810421i	$-0.300761\pi$
$659$	2.20982e11	1.17170	0.585848	+	0.810421i	$0.300761\pi$
$660$	0	0
$661$	− 2.69549e11i	− 1.41199i	−0.708217	−	0.705995i	$-0.750500\pi$
$661$	− 2.69549e11i	− 1.41199i	0.708217	−	0.705995i	$-0.249500\pi$
$662$	0	0
$663$	1.98177e11i	1.02565i
$664$	0	0
$665$	1.41578e11	0.723949
$666$	0	0
$667$	−9.69518e9	−0.0489838
$668$	0	0
$669$	3.23476e11i	1.61487i
$670$	0	0
$671$	− 4.56711e11i	− 2.25295i
$672$	0	0
$673$	9.44470e10	0.460392	0.230196	−	0.973144i	$-0.426063\pi$
$673$	9.44470e10	0.460392	0.230196	+	0.973144i	$0.426063\pi$
$674$	0	0
$675$	1.30433e11	0.628309
$676$	0	0
$677$	8.02735e10i	0.382136i	0.981577	+	0.191068i	$0.0611950\pi$
$677$	8.02735e10i	0.382136i	−0.981577	+	0.191068i	$0.938805\pi$
$678$	0	0
$679$	− 1.19277e11i	− 0.561150i
$680$	0	0
$681$	−2.11364e11	−0.982751
$682$	0	0
$683$	3.00783e11	1.38220	0.691099	−	0.722760i	$-0.257127\pi$
$683$	3.00783e11	1.38220	0.691099	+	0.722760i	$0.257127\pi$
$684$	0	0
$685$	− 1.97085e11i	− 0.895142i
$686$	0	0
$687$	2.40591e11i	1.08007i
$688$	0	0
$689$	−3.09416e11	−1.37298
$690$	0	0
$691$	−3.06208e11	−1.34309	−0.671544	−	0.740964i	$-0.734369\pi$
$691$	−3.06208e11	−1.34309	−0.671544	+	0.740964i	$0.734369\pi$
$692$	0	0
$693$	− 6.66199e11i	− 2.88849i
$694$	0	0
$695$	1.79049e11i	0.767419i
$696$	0	0
$697$	3.80707e9	0.0161309
$698$	0	0
$699$	−7.40348e10	−0.310118
$700$	0	0
$701$	2.73603e11i	1.13305i	0.824045	+	0.566524i	$0.191712\pi$
$701$	2.73603e11i	1.13305i	−0.824045	+	0.566524i	$0.808288\pi$
$702$	0	0
$703$	8.62143e10i	0.352987i
$704$	0	0
$705$	2.07647e11	0.840562
$706$	0	0
$707$	−1.68656e11	−0.675033
$708$	0	0
$709$	− 1.76662e11i	− 0.699129i	−0.936912	−	0.349564i	$-0.886330\pi$
$709$	− 1.76662e11i	− 0.699129i	0.936912	−	0.349564i	$-0.113670\pi$
$710$	0	0
$711$	− 6.63484e11i	− 2.59628i
$712$	0	0
$713$	2.07321e11	0.802203
$714$	0	0
$715$	−2.70885e11	−1.03648
$716$	0	0
$717$	− 6.25718e11i	− 2.36756i
$718$	0	0
$719$	2.25510e11i	0.843821i	0.906637	+	0.421911i	$0.138640\pi$
$719$	2.25510e11i	0.843821i	−0.906637	+	0.421911i	$0.861360\pi$
$720$	0	0
$721$	4.21402e11	1.55939
$722$	0	0
$723$	−2.30227e11	−0.842565
$724$	0	0
$725$	− 7.17600e9i	− 0.0259735i
$726$	0	0
$727$	2.87080e11i	1.02770i	0.857881	+	0.513849i	$0.171781\pi$
$727$	2.87080e11i	1.02770i	−0.857881	+	0.513849i	$0.828219\pi$
$728$	0	0
$729$	−2.14750e11	−0.760366
$730$	0	0
$731$	−2.51673e10	−0.0881388
$732$	0	0
$733$	2.94176e11i	1.01904i	0.860459	+	0.509520i	$0.170177\pi$
$733$	2.94176e11i	1.01904i	−0.860459	+	0.509520i	$0.829823\pi$
$734$	0	0
$735$	5.55435e10i	0.190320i
$736$	0	0
$737$	−1.43660e11	−0.486928
$738$	0	0
$739$	9.69888e10	0.325195	0.162598	−	0.986692i	$-0.448013\pi$
$739$	9.69888e10	0.325195	0.162598	+	0.986692i	$0.448013\pi$
$740$	0	0
$741$	4.27355e11i	1.41748i
$742$	0	0
$743$	− 1.01567e11i	− 0.333271i	−0.986019	−	0.166635i	$-0.946710\pi$
$743$	− 1.01567e11i	− 0.333271i	0.986019	−	0.166635i	$-0.0532902\pi$
$744$	0	0
$745$	−2.31590e11	−0.751788
$746$	0	0
$747$	9.98824e11	3.20779
$748$	0	0
$749$	− 3.25413e11i	− 1.03397i
$750$	0	0
$751$	− 4.17899e11i	− 1.31375i	−0.754001	−	0.656873i	$-0.771879\pi$
$751$	− 4.17899e11i	− 1.31375i	0.754001	−	0.656873i	$-0.228121\pi$
$752$	0	0
$753$	1.86457e10	0.0579960
$754$	0	0
$755$	3.36696e11	1.03621
$756$	0	0
$757$	1.82006e11i	0.554244i	0.960835	+	0.277122i	$0.0893806\pi$
$757$	1.82006e11i	0.554244i	−0.960835	+	0.277122i	$0.910619\pi$
$758$	0	0
$759$	4.79945e11i	1.44619i
$760$	0	0
$761$	4.27419e11	1.27443	0.637213	−	0.770687i	$-0.280087\pi$
$761$	4.27419e11	1.27443	0.637213	+	0.770687i	$0.280087\pi$
$762$	0	0
$763$	−3.99000e11	−1.17727
$764$	0	0
$765$	− 3.49341e11i	− 1.02001i
$766$	0	0
$767$	6.04978e11i	1.74807i
$768$	0	0
$769$	−5.09969e11	−1.45827	−0.729136	−	0.684368i	$-0.760078\pi$
$769$	−5.09969e11	−1.45827	−0.729136	+	0.684368i	$0.760078\pi$
$770$	0	0
$771$	7.49386e11	2.12074
$772$	0	0
$773$	− 1.49408e11i	− 0.418462i	−0.977866	−	0.209231i	$-0.932904\pi$
$773$	− 1.49408e11i	− 0.418462i	0.977866	−	0.209231i	$-0.0670961\pi$
$774$	0	0
$775$	1.53451e11i	0.425366i
$776$	0	0
$777$	−2.88029e11	−0.790227
$778$	0	0
$779$	8.20966e9	0.0222933
$780$	0	0
$781$	− 1.93261e11i	− 0.519445i
$782$	0	0
$783$	− 5.49393e10i	− 0.146163i
$784$	0	0
$785$	−2.22012e11	−0.584652
$786$	0	0
$787$	7.33252e11	1.91141	0.955706	−	0.294323i	$-0.0950943\pi$
$787$	7.33252e11	1.91141	0.955706	+	0.294323i	$0.0950943\pi$
$788$	0	0
$789$	− 1.04891e12i	− 2.70665i
$790$	0	0
$791$	6.04040e11i	1.54298i
$792$	0	0
$793$	−6.60236e11	−1.66958
$794$	0	0
$795$	8.08577e11	2.02420
$796$	0	0
$797$	− 3.02703e11i	− 0.750212i	−0.926982	−	0.375106i	$-0.877606\pi$
$797$	− 3.02703e11i	− 0.750212i	0.926982	−	0.375106i	$-0.122394\pi$
$798$	0	0
$799$	1.44438e11i	0.354401i
$800$	0	0
$801$	−1.18034e12	−2.86732
$802$	0	0
$803$	1.24911e11	0.300427
$804$	0	0
$805$	− 2.29855e11i	− 0.547358i
$806$	0	0
$807$	− 4.91911e10i	− 0.115982i
$808$	0	0
$809$	5.84316e11	1.36412	0.682062	−	0.731295i	$-0.261084\pi$
$809$	5.84316e11	1.36412	0.682062	+	0.731295i	$0.261084\pi$
$810$	0	0
$811$	1.21470e11	0.280793	0.140396	−	0.990095i	$-0.455162\pi$
$811$	1.21470e11	0.280793	0.140396	+	0.990095i	$0.455162\pi$
$812$	0	0
$813$	9.25344e8i	0.00211807i
$814$	0	0
$815$	− 1.24473e11i	− 0.282127i
$816$	0	0
$817$	−5.42714e10	−0.121810
$818$	0	0
$819$	−9.63078e11	−2.14055
$820$	0	0
$821$	4.52470e11i	0.995903i	0.867205	+	0.497952i	$0.165914\pi$
$821$	4.52470e11i	0.995903i	−0.867205	+	0.497952i	$0.834086\pi$
$822$	0	0
$823$	− 3.06704e11i	− 0.668528i	−0.942479	−	0.334264i	$-0.891512\pi$
$823$	− 3.06704e11i	− 0.668528i	0.942479	−	0.334264i	$-0.108488\pi$
$824$	0	0
$825$	−3.55237e11	−0.766835
$826$	0	0
$827$	−2.93276e11	−0.626982	−0.313491	−	0.949591i	$-0.601499\pi$
$827$	−2.93276e11	−0.626982	−0.313491	+	0.949591i	$0.601499\pi$
$828$	0	0
$829$	− 3.35532e11i	− 0.710421i	−0.934786	−	0.355210i	$-0.884409\pi$
$829$	− 3.35532e11i	− 0.710421i	0.934786	−	0.355210i	$-0.115591\pi$
$830$	0	0
$831$	3.05356e11i	0.640327i
$832$	0	0
$833$	−3.86358e10	−0.0802434
$834$	0	0
$835$	−3.42389e11	−0.704326
$836$	0	0
$837$	1.17482e12i	2.39369i
$838$	0	0
$839$	3.42844e11i	0.691908i	0.938252	+	0.345954i	$0.112445\pi$
$839$	3.42844e11i	0.691908i	−0.938252	+	0.345954i	$0.887555\pi$
$840$	0	0
$841$	4.97224e11	0.993958
$842$	0	0
$843$	−1.48029e12	−2.93113
$844$	0	0
$845$	− 2.44222e10i	− 0.0479024i
$846$	0	0
$847$	3.91175e11i	0.760042i
$848$	0	0
$849$	−1.81802e11	−0.349919
$850$	0	0
$851$	1.39972e11	0.266883
$852$	0	0
$853$	− 5.08662e11i	− 0.960801i	−0.877049	−	0.480400i	$-0.840491\pi$
$853$	− 5.08662e11i	− 0.960801i	0.877049	−	0.480400i	$-0.159509\pi$
$854$	0	0
$855$	− 7.53328e11i	− 1.40968i
$856$	0	0
$857$	−6.06764e11	−1.12486	−0.562428	−	0.826846i	$-0.690133\pi$
$857$	−6.06764e11	−1.12486	−0.562428	+	0.826846i	$0.690133\pi$
$858$	0	0
$859$	9.49431e11	1.74378	0.871888	−	0.489705i	$-0.162895\pi$
$859$	9.49431e11	1.74378	0.871888	+	0.489705i	$0.162895\pi$
$860$	0	0
$861$	2.74272e10i	0.0499078i
$862$	0	0
$863$	− 2.99836e10i	− 0.0540556i	−0.999635	−	0.0270278i	$-0.991396\pi$
$863$	− 2.99836e10i	− 0.0540556i	0.999635	−	0.0270278i	$-0.00860426\pi$
$864$	0	0
$865$	9.01494e11	1.61027
$866$	0	0
$867$	−6.30222e11	−1.11536
$868$	0	0
$869$	9.35196e11i	1.63992i
$870$	0	0
$871$	2.07679e11i	0.360844i
$872$	0	0
$873$	−6.34669e11	−1.09267
$874$	0	0
$875$	6.79283e11	1.15883
$876$	0	0
$877$	8.80195e11i	1.48792i	0.668223	+	0.743961i	$0.267055\pi$
$877$	8.80195e11i	1.48792i	−0.668223	+	0.743961i	$0.732945\pi$
$878$	0	0
$879$	− 3.03546e11i	− 0.508474i
$880$	0	0
$881$	1.04085e12	1.72776	0.863879	−	0.503699i	$-0.168028\pi$
$881$	1.04085e12	1.72776	0.863879	+	0.503699i	$0.168028\pi$
$882$	0	0
$883$	−7.60446e11	−1.25091	−0.625454	−	0.780261i	$-0.715086\pi$
$883$	−7.60446e11	−1.25091	−0.625454	+	0.780261i	$0.715086\pi$
$884$	0	0
$885$	− 1.58095e12i	− 2.57718i
$886$	0	0
$887$	− 3.34097e11i	− 0.539732i	−0.962898	−	0.269866i	$-0.913021\pi$
$887$	− 3.34097e11i	− 0.539732i	0.962898	−	0.269866i	$-0.0869794\pi$
$888$	0	0
$889$	9.39853e11	1.50471
$890$	0	0
$891$	−1.00945e12	−1.60167
$892$	0	0
$893$	3.11471e11i	0.489792i
$894$	0	0
$895$	− 3.86053e11i	− 0.601666i
$896$	0	0
$897$	6.93823e11	1.07171
$898$	0	0
$899$	6.46345e10	0.0989523
$900$	0	0
$901$	5.62442e11i	0.853451i
$902$	0	0
$903$	− 1.81312e11i	− 0.272695i
$904$	0	0
$905$	3.19818e11	0.476770
$906$	0	0
$907$	−7.65213e11	−1.13071	−0.565357	−	0.824846i	$-0.691262\pi$
$907$	−7.65213e11	−1.13071	−0.565357	+	0.824846i	$0.691262\pi$
$908$	0	0
$909$	8.97412e11i	1.31443i
$910$	0	0
$911$	− 3.83541e11i	− 0.556851i	−0.960458	−	0.278425i	$-0.910188\pi$
$911$	− 3.83541e11i	− 0.556851i	0.960458	−	0.278425i	$-0.0898125\pi$
$912$	0	0
$913$	−1.40786e12	−2.02618
$914$	0	0
$915$	1.72535e12	2.46146
$916$	0	0
$917$	5.52937e11i	0.781984i
$918$	0	0
$919$	− 6.82775e11i	− 0.957229i	−0.878025	−	0.478615i	$-0.841139\pi$
$919$	− 6.82775e11i	− 0.957229i	0.878025	−	0.478615i	$-0.158861\pi$
$920$	0	0
$921$	7.74022e11	1.07576
$922$	0	0
$923$	−2.79384e11	−0.384941
$924$	0	0
$925$	1.03602e11i	0.141514i
$926$	0	0
$927$	− 2.24226e12i	− 3.03645i
$928$	0	0
$929$	2.94973e11	0.396021	0.198011	−	0.980200i	$-0.436552\pi$
$929$	2.94973e11	0.396021	0.198011	+	0.980200i	$0.436552\pi$
$930$	0	0
$931$	−8.33152e10	−0.110898
$932$	0	0
$933$	− 1.52705e12i	− 2.01524i
$934$	0	0
$935$	4.92404e11i	0.644280i
$936$	0	0
$937$	−1.03941e12	−1.34843	−0.674217	−	0.738533i	$-0.735519\pi$
$937$	−1.03941e12	−1.34843	−0.674217	+	0.738533i	$0.735519\pi$
$938$	0	0
$939$	4.25680e10	0.0547546
$940$	0	0
$941$	− 1.26891e11i	− 0.161836i	−0.996721	−	0.0809178i	$-0.974215\pi$
$941$	− 1.26891e11i	− 0.161836i	0.996721	−	0.0809178i	$-0.0257851\pi$
$942$	0	0
$943$	− 1.33286e10i	− 0.0168554i
$944$	0	0
$945$	1.30251e12	1.63326
$946$	0	0
$947$	−6.13064e11	−0.762265	−0.381133	−	0.924520i	$-0.624466\pi$
$947$	−6.13064e11	−0.762265	−0.381133	+	0.924520i	$0.624466\pi$
$948$	0	0
$949$	− 1.80575e11i	− 0.222635i
$950$	0	0
$951$	− 7.54189e11i	− 0.922058i
$952$	0	0
$953$	−6.58227e11	−0.798002	−0.399001	−	0.916951i	$-0.630643\pi$
$953$	−6.58227e11	−0.798002	−0.399001	+	0.916951i	$0.630643\pi$
$954$	0	0
$955$	5.50413e11	0.661721
$956$	0	0
$957$	1.49628e11i	0.178388i
$958$	0	0
$959$	9.87647e11i	1.16769i
$960$	0	0
$961$	−5.29246e11	−0.620532
$962$	0	0
$963$	−1.73151e12	−2.01335
$964$	0	0
$965$	1.51346e10i	0.0174527i
$966$	0	0
$967$	5.41485e11i	0.619271i	0.950855	+	0.309635i	$0.100207\pi$
$967$	5.41485e11i	0.619271i	−0.950855	+	0.309635i	$0.899793\pi$
$968$	0	0
$969$	7.76826e11	0.881107
$970$	0	0
$971$	3.54981e11	0.399327	0.199663	−	0.979865i	$-0.436015\pi$
$971$	3.54981e11	0.399327	0.199663	+	0.979865i	$0.436015\pi$
$972$	0	0
$973$	− 8.97263e11i	− 1.00108i
$974$	0	0
$975$	5.13541e11i	0.568273i
$976$	0	0
$977$	6.02238e11	0.660982	0.330491	−	0.943809i	$-0.392786\pi$
$977$	6.02238e11	0.660982	0.330491	+	0.943809i	$0.392786\pi$
$978$	0	0
$979$	1.66371e12	1.81112
$980$	0	0
$981$	2.12306e12i	2.29238i
$982$	0	0
$983$	− 1.37465e12i	− 1.47223i	−0.676854	−	0.736117i	$-0.736657\pi$
$983$	− 1.37465e12i	− 1.47223i	0.676854	−	0.736117i	$-0.263343\pi$
$984$	0	0
$985$	−5.73668e11	−0.609419
$986$	0	0
$987$	−1.04058e12	−1.09649
$988$	0	0
$989$	8.81113e10i	0.0920972i
$990$	0	0
$991$	1.01081e12i	1.04803i	0.851709	+	0.524015i	$0.175567\pi$
$991$	1.01081e12i	1.04803i	−0.851709	+	0.524015i	$0.824433\pi$
$992$	0	0
$993$	1.43411e12	1.47498
$994$	0	0
$995$	−5.34061e11	−0.544877
$996$	0	0
$997$	− 3.28556e11i	− 0.332528i	−0.986081	−	0.166264i	$-0.946830\pi$
$997$	− 3.28556e11i	− 0.332528i	0.986081	−	0.166264i	$-0.0531704\pi$
$998$	0	0
$999$	7.93172e11i	0.796353i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.9.d.f.127.3		4
4.3	odd	2	inner	256.9.d.f.127.1		4
8.3	odd	2	inner	256.9.d.f.127.4		4
8.5	even	2	inner	256.9.d.f.127.2		4
16.3	odd	4		16.9.c.a.15.1	✓	2
16.5	even	4		64.9.c.d.63.1		2
16.11	odd	4		64.9.c.d.63.2		2
16.13	even	4		16.9.c.a.15.2	yes	2
48.29	odd	4		144.9.g.g.127.1		2
48.35	even	4		144.9.g.g.127.2		2
80.3	even	4		400.9.h.b.399.3		4
80.13	odd	4		400.9.h.b.399.2		4
80.19	odd	4		400.9.b.c.351.2		2
80.29	even	4		400.9.b.c.351.1		2
80.67	even	4		400.9.h.b.399.1		4
80.77	odd	4		400.9.h.b.399.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.9.c.a.15.1	✓	2	16.3	odd	4
16.9.c.a.15.2	yes	2	16.13	even	4
64.9.c.d.63.1		2	16.5	even	4
64.9.c.d.63.2		2	16.11	odd	4
144.9.g.g.127.1		2	48.29	odd	4
144.9.g.g.127.2		2	48.35	even	4
256.9.d.f.127.1		4	4.3	odd	2	inner
256.9.d.f.127.2		4	8.5	even	2	inner
256.9.d.f.127.3		4	1.1	even	1	trivial
256.9.d.f.127.4		4	8.3	odd	2	inner
400.9.b.c.351.1		2	80.29	even	4
400.9.b.c.351.2		2	80.19	odd	4
400.9.h.b.399.1		4	80.67	even	4
400.9.h.b.399.2		4	80.13	odd	4
400.9.h.b.399.3		4	80.3	even	4
400.9.h.b.399.4		4	80.77	odd	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+141.986 q^{3} -510.000i q^{5} +2555.75i q^{7} +13599.0 q^{9} -19168.1 q^{11} +27710.0i q^{13} -72412.8i q^{15} +50370.0 q^{17} +108619. q^{19} +362880. i q^{21} -176347. i q^{23} +130525. q^{25} +999297. q^{27} -54978.0i q^{29} +1.17564e6i q^{31} -2.72160e6 q^{33} +1.30343e6 q^{35} +793730. i q^{37} +3.93443e6i q^{39} +75582.0 q^{41} -499648. q^{43} -6.93549e6i q^{45} +2.86755e6i q^{47} -767039. q^{49} +7.15183e6 q^{51} +1.11662e7i q^{53} +9.77573e6i q^{55} +1.54224e7 q^{57} +2.18325e7 q^{59} +2.38266e7i q^{61} +3.47556e7i q^{63} +1.41321e7 q^{65} +7.49473e6 q^{67} -2.50387e7i q^{69} +1.00824e7i q^{71} -6.51661e6 q^{73} +1.85327e7 q^{75} -4.89888e7i q^{77} -4.87892e7i q^{79} +5.26630e7 q^{81} +7.34483e7 q^{83} -2.56887e7i q^{85} -7.80610e6i q^{87} -8.67958e7 q^{89} -7.08197e7 q^{91} +1.66925e8i q^{93} -5.53958e7i q^{95} -4.66703e7 q^{97} -2.60667e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 54396 q^{9} + 201480 q^{17} + 522100 q^{25} - 10886400 q^{33} + 302328 q^{41} - 3068156 q^{49} + 61689600 q^{57} + 56528400 q^{65} - 26066440 q^{73} + 210652164 q^{81} - 347183112 q^{89} - 186681080 q^{97}+O(q^{100}) \) 4 * q + 54396 * q^9 + 201480 * q^17 + 522100 * q^25 - 10886400 * q^33 + 302328 * q^41 - 3068156 * q^49 + 61689600 * q^57 + 56528400 * q^65 - 26066440 * q^73 + 210652164 * q^81 - 347183112 * q^89 - 186681080 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more