LMFDB - Embedded newform 256.5.c.j.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,5,Mod(255,256)]

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, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$26.4627105495$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.5.c.j.255.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{3} -20.7846 q^{5} -40.0000i q^{7} +69.0000 q^{9} +O(q^{10})$	$q-3.46410i q^{3} -20.7846 q^{5} -40.0000i q^{7} +69.0000 q^{9} +197.454i q^{11} +256.344 q^{13} +72.0000i q^{15} -198.000 q^{17} -252.879i q^{19} -138.564 q^{21} -504.000i q^{23} -193.000 q^{25} -519.615i q^{27} -436.477 q^{29} -1696.00i q^{31} +684.000 q^{33} +831.384i q^{35} +1752.84 q^{37} -888.000i q^{39} -18.0000 q^{41} -1548.45i q^{43} -1434.14 q^{45} -2448.00i q^{47} +801.000 q^{49} +685.892i q^{51} -3263.18 q^{53} -4104.00i q^{55} -876.000 q^{57} +5268.90i q^{59} +1004.59 q^{61} -2760.00i q^{63} -5328.00 q^{65} -1804.80i q^{67} -1745.91 q^{69} -7848.00i q^{71} +4310.00 q^{73} +668.572i q^{75} +7898.15 q^{77} -4144.00i q^{79} +3789.00 q^{81} -9093.27i q^{83} +4115.35 q^{85} +1512.00i q^{87} -3114.00 q^{89} -10253.7i q^{91} -5875.12 q^{93} +5256.00i q^{95} +4730.00 q^{97} +13624.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 276 q^{9}+O(q^{10}) $ 4 * q + 276 * q^9	$ 4 q + 276 q^{9} - 792 q^{17} - 772 q^{25} + 2736 q^{33} - 72 q^{41} + 3204 q^{49} - 3504 q^{57} - 21312 q^{65} + 17240 q^{73} + 15156 q^{81} - 12456 q^{89} + 18920 q^{97}+O(q^{100}) $ 4 * q + 276 * q^9 - 792 * q^17 - 772 * q^25 + 2736 * q^33 - 72 * q^41 + 3204 * q^49 - 3504 * q^57 - 21312 * q^65 + 17240 * q^73 + 15156 * q^81 - 12456 * q^89 + 18920 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 3.46410i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 3.46410i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	0	0
$5$	−20.7846	−0.831384	−0.415692	−	0.909505i	$-0.636461\pi$
$5$	−20.7846	−0.831384	−0.415692	+	0.909505i	$0.636461\pi$
$6$	0	0
$7$	− 40.0000i	− 0.816327i	−0.912909	−	0.408163i	$-0.866169\pi$
$7$	− 40.0000i	− 0.816327i	0.912909	−	0.408163i	$-0.133831\pi$
$8$	0	0
$9$	69.0000	0.851852
$10$	0	0
$11$	197.454i	1.63185i	0.578158	+	0.815925i	$0.303772\pi$
$11$	197.454i	1.63185i	−0.578158	+	0.815925i	$0.696228\pi$
$12$	0	0
$13$	256.344	1.51683	0.758413	−	0.651775i	$-0.225975\pi$
$13$	256.344	1.51683	0.758413	+	0.651775i	$0.225975\pi$
$14$	0	0
$15$	72.0000i	0.320000i
$16$	0	0
$17$	−198.000	−0.685121	−0.342561	−	0.939496i	$-0.611294\pi$
$17$	−198.000	−0.685121	−0.342561	+	0.939496i	$0.611294\pi$
$18$	0	0
$19$	− 252.879i	− 0.700497i	−0.936657	−	0.350249i	$-0.886097\pi$
$19$	− 252.879i	− 0.700497i	0.936657	−	0.350249i	$-0.113903\pi$
$20$	0	0
$21$	−138.564	−0.314204
$22$	0	0
$23$	− 504.000i	− 0.952741i	−0.879245	−	0.476371i	$-0.841952\pi$
$23$	− 504.000i	− 0.952741i	0.879245	−	0.476371i	$-0.158048\pi$
$24$	0	0
$25$	−193.000	−0.308800
$26$	0	0
$27$	− 519.615i	− 0.712778i
$28$	0	0
$29$	−436.477	−0.518997	−0.259499	−	0.965743i	$-0.583557\pi$
$29$	−436.477	−0.518997	−0.259499	+	0.965743i	$0.583557\pi$
$30$	0	0
$31$	− 1696.00i	− 1.76483i	−0.470473	−	0.882414i	$-0.655917\pi$
$31$	− 1696.00i	− 1.76483i	0.470473	−	0.882414i	$-0.344083\pi$
$32$	0	0
$33$	684.000	0.628099
$34$	0	0
$35$	831.384i	0.678681i
$36$	0	0
$37$	1752.84	1.28038	0.640188	−	0.768218i	$-0.278856\pi$
$37$	1752.84	1.28038	0.640188	+	0.768218i	$0.278856\pi$
$38$	0	0
$39$	− 888.000i	− 0.583826i
$40$	0	0
$41$	−18.0000	−0.0107079	−0.00535396	−	0.999986i	$-0.501704\pi$
$41$	−18.0000	−0.0107079	−0.00535396	+	0.999986i	$0.501704\pi$
$42$	0	0
$43$	− 1548.45i	− 0.837455i	−0.908112	−	0.418727i	$-0.862476\pi$
$43$	− 1548.45i	− 0.837455i	0.908112	−	0.418727i	$-0.137524\pi$
$44$	0	0
$45$	−1434.14	−0.708216
$46$	0	0
$47$	− 2448.00i	− 1.10819i	−0.832452	−	0.554097i	$-0.813064\pi$
$47$	− 2448.00i	− 1.10819i	0.832452	−	0.554097i	$-0.186936\pi$
$48$	0	0
$49$	801.000	0.333611
$50$	0	0
$51$	685.892i	0.263703i
$52$	0	0
$53$	−3263.18	−1.16169	−0.580844	−	0.814015i	$-0.697278\pi$
$53$	−3263.18	−1.16169	−0.580844	+	0.814015i	$0.697278\pi$
$54$	0	0
$55$	− 4104.00i	− 1.35669i
$56$	0	0
$57$	−876.000	−0.269621
$58$	0	0
$59$	5268.90i	1.51362i	0.653637	+	0.756808i	$0.273242\pi$
$59$	5268.90i	1.51362i	−0.653637	+	0.756808i	$0.726758\pi$
$60$	0	0
$61$	1004.59	0.269978	0.134989	−	0.990847i	$-0.456900\pi$
$61$	1004.59	0.269978	0.134989	+	0.990847i	$0.456900\pi$
$62$	0	0
$63$	− 2760.00i	− 0.695389i
$64$	0	0
$65$	−5328.00	−1.26107
$66$	0	0
$67$	− 1804.80i	− 0.402049i	−0.979586	−	0.201024i	$-0.935573\pi$
$67$	− 1804.80i	− 0.402049i	0.979586	−	0.201024i	$-0.0644270\pi$
$68$	0	0
$69$	−1745.91	−0.366710
$70$	0	0
$71$	− 7848.00i	− 1.55683i	−0.627748	−	0.778417i	$-0.716023\pi$
$71$	− 7848.00i	− 1.55683i	0.627748	−	0.778417i	$-0.283977\pi$
$72$	0	0
$73$	4310.00	0.808782	0.404391	−	0.914586i	$-0.367483\pi$
$73$	4310.00	0.808782	0.404391	+	0.914586i	$0.367483\pi$
$74$	0	0
$75$	668.572i	0.118857i
$76$	0	0
$77$	7898.15	1.33212
$78$	0	0
$79$	− 4144.00i	− 0.663996i	−0.943280	−	0.331998i	$-0.892277\pi$
$79$	− 4144.00i	− 0.663996i	0.943280	−	0.331998i	$-0.107723\pi$
$80$	0	0
$81$	3789.00	0.577503
$82$	0	0
$83$	− 9093.27i	− 1.31997i	−0.751279	−	0.659985i	$-0.770563\pi$
$83$	− 9093.27i	− 1.31997i	0.751279	−	0.659985i	$-0.229437\pi$
$84$	0	0
$85$	4115.35	0.569599
$86$	0	0
$87$	1512.00i	0.199762i
$88$	0	0
$89$	−3114.00	−0.393132	−0.196566	−	0.980491i	$-0.562979\pi$
$89$	−3114.00	−0.393132	−0.196566	+	0.980491i	$0.562979\pi$
$90$	0	0
$91$	− 10253.7i	− 1.23822i
$92$	0	0
$93$	−5875.12	−0.679283
$94$	0	0
$95$	5256.00i	0.582382i
$96$	0	0
$97$	4730.00	0.502710	0.251355	−	0.967895i	$-0.419124\pi$
$97$	4730.00	0.502710	0.251355	+	0.967895i	$0.419124\pi$
$98$	0	0
$99$	13624.3i	1.39009i
$100$	0	0
$101$	−20.7846	−0.00203751	−0.00101875	−	0.999999i	$-0.500324\pi$
$101$	−20.7846	−0.00203751	−0.00101875	+	0.999999i	$0.500324\pi$
$102$	0	0
$103$	9016.00i	0.849844i	0.905230	+	0.424922i	$0.139699\pi$
$103$	9016.00i	0.849844i	−0.905230	+	0.424922i	$0.860301\pi$
$104$	0	0
$105$	2880.00	0.261224
$106$	0	0
$107$	2359.05i	0.206049i	0.994679	+	0.103024i	$0.0328520\pi$
$107$	2359.05i	0.206049i	−0.994679	+	0.103024i	$0.967148\pi$
$108$	0	0
$109$	−14486.9	−1.21933	−0.609666	−	0.792659i	$-0.708696\pi$
$109$	−14486.9	−1.21933	−0.609666	+	0.792659i	$0.708696\pi$
$110$	0	0
$111$	− 6072.00i	− 0.492817i
$112$	0	0
$113$	−2142.00	−0.167750	−0.0838750	−	0.996476i	$-0.526730\pi$
$113$	−2142.00	−0.167750	−0.0838750	+	0.996476i	$0.526730\pi$
$114$	0	0
$115$	10475.4i	0.792094i
$116$	0	0
$117$	17687.7	1.29211
$118$	0	0
$119$	7920.00i	0.559283i
$120$	0	0
$121$	−24347.0	−1.66293
$122$	0	0
$123$	62.3538i	0.00412148i
$124$	0	0
$125$	17001.8	1.08812
$126$	0	0
$127$	5056.00i	0.313473i	0.987640	+	0.156736i	$0.0500973\pi$
$127$	5056.00i	0.313473i	−0.987640	+	0.156736i	$0.949903\pi$
$128$	0	0
$129$	−5364.00	−0.322336
$130$	0	0
$131$	− 9176.41i	− 0.534724i	−0.963596	−	0.267362i	$-0.913848\pi$
$131$	− 9176.41i	− 0.534724i	0.963596	−	0.267362i	$-0.0861520\pi$
$132$	0	0
$133$	−10115.2	−0.571834
$134$	0	0
$135$	10800.0i	0.592593i
$136$	0	0
$137$	32526.0	1.73296	0.866482	−	0.499208i	$-0.166376\pi$
$137$	32526.0	1.73296	0.866482	+	0.499208i	$0.166376\pi$
$138$	0	0
$139$	− 1105.05i	− 0.0571942i	−0.999591	−	0.0285971i	$-0.990896\pi$
$139$	− 1105.05i	− 0.0571942i	0.999591	−	0.0285971i	$-0.00910397\pi$
$140$	0	0
$141$	−8480.12	−0.426544
$142$	0	0
$143$	50616.0i	2.47523i
$144$	0	0
$145$	9072.00	0.431486
$146$	0	0
$147$	− 2774.75i	− 0.128407i
$148$	0	0
$149$	−34523.2	−1.55503	−0.777515	−	0.628864i	$-0.783520\pi$
$149$	−34523.2	−1.55503	−0.777515	+	0.628864i	$0.783520\pi$
$150$	0	0
$151$	14216.0i	0.623481i	0.950167	+	0.311741i	$0.100912\pi$
$151$	14216.0i	0.623481i	−0.950167	+	0.311741i	$0.899088\pi$
$152$	0	0
$153$	−13662.0	−0.583622
$154$	0	0
$155$	35250.7i	1.46725i
$156$	0	0
$157$	−270.200	−0.0109619	−0.00548095	−	0.999985i	$-0.501745\pi$
$157$	−270.200	−0.0109619	−0.00548095	+	0.999985i	$0.501745\pi$
$158$	0	0
$159$	11304.0i	0.447134i
$160$	0	0
$161$	−20160.0	−0.777748
$162$	0	0
$163$	− 28658.5i	− 1.07864i	−0.842099	−	0.539322i	$-0.818680\pi$
$163$	− 28658.5i	− 1.07864i	0.842099	−	0.539322i	$-0.181320\pi$
$164$	0	0
$165$	−14216.7	−0.522192
$166$	0	0
$167$	21816.0i	0.782244i	0.920339	+	0.391122i	$0.127913\pi$
$167$	21816.0i	0.782244i	−0.920339	+	0.391122i	$0.872087\pi$
$168$	0	0
$169$	37151.0	1.30076
$170$	0	0
$171$	− 17448.7i	− 0.596720i
$172$	0	0
$173$	31904.4	1.06600	0.533001	−	0.846115i	$-0.321064\pi$
$173$	31904.4	1.06600	0.533001	+	0.846115i	$0.321064\pi$
$174$	0	0
$175$	7720.00i	0.252082i
$176$	0	0
$177$	18252.0	0.582591
$178$	0	0
$179$	− 10589.8i	− 0.330506i	−0.986251	−	0.165253i	$-0.947156\pi$
$179$	− 10589.8i	− 0.330506i	0.986251	−	0.165253i	$-0.0528441\pi$
$180$	0	0
$181$	36920.4	1.12696	0.563481	−	0.826129i	$-0.309462\pi$
$181$	36920.4	1.12696	0.563481	+	0.826129i	$0.309462\pi$
$182$	0	0
$183$	− 3480.00i	− 0.103915i
$184$	0	0
$185$	−36432.0	−1.06449
$186$	0	0
$187$	− 39095.9i	− 1.11801i
$188$	0	0
$189$	−20784.6	−0.581860
$190$	0	0
$191$	6912.00i	0.189468i	0.995503	+	0.0947342i	$0.0302001\pi$
$191$	6912.00i	0.189468i	−0.995503	+	0.0947342i	$0.969800\pi$
$192$	0	0
$193$	−29446.0	−0.790518	−0.395259	−	0.918570i	$-0.629345\pi$
$193$	−29446.0	−0.790518	−0.395259	+	0.918570i	$0.629345\pi$
$194$	0	0
$195$	18456.7i	0.485384i
$196$	0	0
$197$	40550.8	1.04488	0.522440	−	0.852676i	$-0.325022\pi$
$197$	40550.8	1.04488	0.522440	+	0.852676i	$0.325022\pi$
$198$	0	0
$199$	1432.00i	0.0361607i	0.999837	+	0.0180804i	$0.00575547\pi$
$199$	1432.00i	0.0361607i	−0.999837	+	0.0180804i	$0.994245\pi$
$200$	0	0
$201$	−6252.00	−0.154749
$202$	0	0
$203$	17459.1i	0.423671i
$204$	0	0
$205$	374.123	0.00890239
$206$	0	0
$207$	− 34776.0i	− 0.811594i
$208$	0	0
$209$	49932.0	1.14311
$210$	0	0
$211$	75985.1i	1.70672i	0.521319	+	0.853362i	$0.325440\pi$
$211$	75985.1i	1.70672i	−0.521319	+	0.853362i	$0.674560\pi$
$212$	0	0
$213$	−27186.3	−0.599226
$214$	0	0
$215$	32184.0i	0.696247i
$216$	0	0
$217$	−67840.0	−1.44068
$218$	0	0
$219$	− 14930.3i	− 0.311300i
$220$	0	0
$221$	−50756.0	−1.03921
$222$	0	0
$223$	− 19808.0i	− 0.398319i	−0.979967	−	0.199159i	$-0.936179\pi$
$223$	− 19808.0i	− 0.398319i	0.979967	−	0.199159i	$-0.0638212\pi$
$224$	0	0
$225$	−13317.0	−0.263052
$226$	0	0
$227$	28152.8i	0.546348i	0.961965	+	0.273174i	$0.0880734\pi$
$227$	28152.8i	0.546348i	−0.961965	+	0.273174i	$0.911927\pi$
$228$	0	0
$229$	45372.8	0.865216	0.432608	−	0.901582i	$-0.357593\pi$
$229$	45372.8	0.865216	0.432608	+	0.901582i	$0.357593\pi$
$230$	0	0
$231$	− 27360.0i	− 0.512734i
$232$	0	0
$233$	105750.	1.94791	0.973954	−	0.226745i	$-0.0728083\pi$
$233$	105750.	1.94791	0.973954	+	0.226745i	$0.0728083\pi$
$234$	0	0
$235$	50880.7i	0.921335i
$236$	0	0
$237$	−14355.2	−0.255572
$238$	0	0
$239$	− 23184.0i	− 0.405875i	−0.979192	−	0.202938i	$-0.934951\pi$
$239$	− 23184.0i	− 0.405875i	0.979192	−	0.202938i	$-0.0650489\pi$
$240$	0	0
$241$	13306.0	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$241$	13306.0	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$242$	0	0
$243$	− 55214.3i	− 0.935059i
$244$	0	0
$245$	−16648.5	−0.277359
$246$	0	0
$247$	− 64824.0i	− 1.06253i
$248$	0	0
$249$	−31500.0	−0.508056
$250$	0	0
$251$	− 63569.7i	− 1.00903i	−0.863404	−	0.504514i	$-0.831672\pi$
$251$	− 63569.7i	− 1.00903i	0.863404	−	0.504514i	$-0.168328\pi$
$252$	0	0
$253$	99516.7	1.55473
$254$	0	0
$255$	− 14256.0i	− 0.219239i
$256$	0	0
$257$	−2142.00	−0.0324305	−0.0162152	−	0.999869i	$-0.505162\pi$
$257$	−2142.00	−0.0324305	−0.0162152	+	0.999869i	$0.505162\pi$
$258$	0	0
$259$	− 70113.4i	− 1.04521i
$260$	0	0
$261$	−30116.9	−0.442109
$262$	0	0
$263$	− 59112.0i	− 0.854602i	−0.904109	−	0.427301i	$-0.859464\pi$
$263$	− 59112.0i	− 0.854602i	0.904109	−	0.427301i	$-0.140536\pi$
$264$	0	0
$265$	67824.0	0.965810
$266$	0	0
$267$	10787.2i	0.151317i
$268$	0	0
$269$	8293.06	0.114607	0.0573034	−	0.998357i	$-0.481750\pi$
$269$	8293.06	0.114607	0.0573034	+	0.998357i	$0.481750\pi$
$270$	0	0
$271$	− 97712.0i	− 1.33048i	−0.746628	−	0.665241i	$-0.768329\pi$
$271$	− 97712.0i	− 1.33048i	0.746628	−	0.665241i	$-0.231671\pi$
$272$	0	0
$273$	−35520.0	−0.476593
$274$	0	0
$275$	− 38108.6i	− 0.503915i
$276$	0	0
$277$	−82577.3	−1.07622	−0.538110	−	0.842875i	$-0.680861\pi$
$277$	−82577.3	−1.07622	−0.538110	+	0.842875i	$0.680861\pi$
$278$	0	0
$279$	− 117024.i	− 1.50337i
$280$	0	0
$281$	−116586.	−1.47650	−0.738251	−	0.674527i	$-0.764348\pi$
$281$	−116586.	−1.47650	−0.738251	+	0.674527i	$0.764348\pi$
$282$	0	0
$283$	142032.i	1.77342i	0.462324	+	0.886711i	$0.347016\pi$
$283$	142032.i	1.77342i	−0.462324	+	0.886711i	$0.652984\pi$
$284$	0	0
$285$	18207.3	0.224159
$286$	0	0
$287$	720.000i	0.00874115i
$288$	0	0
$289$	−44317.0	−0.530609
$290$	0	0
$291$	− 16385.2i	− 0.193493i
$292$	0	0
$293$	−42088.8	−0.490266	−0.245133	−	0.969489i	$-0.578832\pi$
$293$	−42088.8	−0.490266	−0.245133	+	0.969489i	$0.578832\pi$
$294$	0	0
$295$	− 109512.i	− 1.25840i
$296$	0	0
$297$	102600.	1.16315
$298$	0	0
$299$	− 129197.i	− 1.44514i
$300$	0	0
$301$	−61938.1	−0.683636
$302$	0	0
$303$	72.0000i	0 0.000784237i
$304$	0	0
$305$	−20880.0	−0.224456
$306$	0	0
$307$	39875.3i	0.423084i	0.977369	+	0.211542i	$0.0678486\pi$
$307$	39875.3i	0.423084i	−0.977369	+	0.211542i	$0.932151\pi$
$308$	0	0
$309$	31232.3	0.327105
$310$	0	0
$311$	143784.i	1.48659i	0.668966	+	0.743293i	$0.266737\pi$
$311$	143784.i	1.48659i	−0.668966	+	0.743293i	$0.733263\pi$
$312$	0	0
$313$	−15826.0	−0.161541	−0.0807704	−	0.996733i	$-0.525738\pi$
$313$	−15826.0	−0.161541	−0.0807704	+	0.996733i	$0.525738\pi$
$314$	0	0
$315$	57365.5i	0.578136i
$316$	0	0
$317$	−113006.	−1.12456	−0.562280	−	0.826947i	$-0.690076\pi$
$317$	−113006.	−1.12456	−0.562280	+	0.826947i	$0.690076\pi$
$318$	0	0
$319$	− 86184.0i	− 0.846926i
$320$	0	0
$321$	8172.00	0.0793082
$322$	0	0
$323$	50070.1i	0.479925i
$324$	0	0
$325$	−49474.3	−0.468396
$326$	0	0
$327$	50184.0i	0.469321i
$328$	0	0
$329$	−97920.0	−0.904648
$330$	0	0
$331$	− 34776.1i	− 0.317413i	−0.987326	−	0.158707i	$-0.949268\pi$
$331$	− 34776.1i	− 0.317413i	0.987326	−	0.158707i	$-0.0507324\pi$
$332$	0	0
$333$	120946.	1.09069
$334$	0	0
$335$	37512.0i	0.334257i
$336$	0	0
$337$	158114.	1.39223	0.696114	−	0.717931i	$-0.254911\pi$
$337$	158114.	1.39223	0.696114	+	0.717931i	$0.254911\pi$
$338$	0	0
$339$	7420.11i	0.0645670i
$340$	0	0
$341$	334882.	2.87993
$342$	0	0
$343$	− 128080.i	− 1.08866i
$344$	0	0
$345$	36288.0	0.304877
$346$	0	0
$347$	− 122598.i	− 1.01818i	−0.860713	−	0.509090i	$-0.829982\pi$
$347$	− 122598.i	− 1.01818i	0.860713	−	0.509090i	$-0.170018\pi$
$348$	0	0
$349$	1780.55	0.0146185	0.00730925	−	0.999973i	$-0.497673\pi$
$349$	1780.55	0.0146185	0.00730925	+	0.999973i	$0.497673\pi$
$350$	0	0
$351$	− 133200.i	− 1.08116i
$352$	0	0
$353$	41346.0	0.331806	0.165903	−	0.986142i	$-0.446946\pi$
$353$	41346.0	0.331806	0.165903	+	0.986142i	$0.446946\pi$
$354$	0	0
$355$	163118.i	1.29433i
$356$	0	0
$357$	27435.7	0.215268
$358$	0	0
$359$	150840.i	1.17038i	0.810895	+	0.585191i	$0.198980\pi$
$359$	150840.i	1.17038i	−0.810895	+	0.585191i	$0.801020\pi$
$360$	0	0
$361$	66373.0	0.509304
$362$	0	0
$363$	84340.5i	0.640063i
$364$	0	0
$365$	−89581.7	−0.672409
$366$	0	0
$367$	79664.0i	0.591466i	0.955271	+	0.295733i	$0.0955639\pi$
$367$	79664.0i	0.591466i	−0.955271	+	0.295733i	$0.904436\pi$
$368$	0	0
$369$	−1242.00	−0.00912155
$370$	0	0
$371$	130527.i	0.948317i
$372$	0	0
$373$	−135176.	−0.971589	−0.485794	−	0.874073i	$-0.661470\pi$
$373$	−135176.	−0.971589	−0.485794	+	0.874073i	$0.661470\pi$
$374$	0	0
$375$	− 58896.0i	− 0.418816i
$376$	0	0
$377$	−111888.	−0.787229
$378$	0	0
$379$	− 16735.1i	− 0.116506i	−0.998302	−	0.0582531i	$-0.981447\pi$
$379$	− 16735.1i	− 0.116506i	0.998302	−	0.0582531i	$-0.0185530\pi$
$380$	0	0
$381$	17514.5	0.120656
$382$	0	0
$383$	157824.i	1.07591i	0.842974	+	0.537954i	$0.180803\pi$
$383$	157824.i	1.07591i	−0.842974	+	0.537954i	$0.819197\pi$
$384$	0	0
$385$	−164160.	−1.10751
$386$	0	0
$387$	− 106843.i	− 0.713387i
$388$	0	0
$389$	−50735.2	−0.335282	−0.167641	−	0.985848i	$-0.553615\pi$
$389$	−50735.2	−0.335282	−0.167641	+	0.985848i	$0.553615\pi$
$390$	0	0
$391$	99792.0i	0.652743i
$392$	0	0
$393$	−31788.0	−0.205816
$394$	0	0
$395$	86131.4i	0.552036i
$396$	0	0
$397$	−81829.0	−0.519190	−0.259595	−	0.965718i	$-0.583589\pi$
$397$	−81829.0	−0.519190	−0.259595	+	0.965718i	$0.583589\pi$
$398$	0	0
$399$	35040.0i	0.220099i
$400$	0	0
$401$	33210.0	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$401$	33210.0	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$402$	0	0
$403$	− 434759.i	− 2.67694i
$404$	0	0
$405$	−78752.9	−0.480127
$406$	0	0
$407$	346104.i	2.08938i
$408$	0	0
$409$	123950.	0.740969	0.370484	−	0.928839i	$-0.379192\pi$
$409$	123950.	0.740969	0.370484	+	0.928839i	$0.379192\pi$
$410$	0	0
$411$	− 112673.i	− 0.667018i
$412$	0	0
$413$	210756.	1.23561
$414$	0	0
$415$	189000.i	1.09740i
$416$	0	0
$417$	−3828.00	−0.0220140
$418$	0	0
$419$	24910.4i	0.141890i	0.997480	+	0.0709450i	$0.0226015\pi$
$419$	24910.4i	0.141890i	−0.997480	+	0.0709450i	$0.977399\pi$
$420$	0	0
$421$	18990.2	0.107143	0.0535717	−	0.998564i	$-0.482939\pi$
$421$	18990.2	0.107143	0.0535717	+	0.998564i	$0.482939\pi$
$422$	0	0
$423$	− 168912.i	− 0.944017i
$424$	0	0
$425$	38214.0	0.211565
$426$	0	0
$427$	− 40183.6i	− 0.220390i
$428$	0	0
$429$	175339.	0.952717
$430$	0	0
$431$	304560.i	1.63953i	0.572703	+	0.819763i	$0.305895\pi$
$431$	304560.i	1.63953i	−0.572703	+	0.819763i	$0.694105\pi$
$432$	0	0
$433$	−237062.	−1.26440	−0.632202	−	0.774803i	$-0.717849\pi$
$433$	−237062.	−1.26440	−0.632202	+	0.774803i	$0.717849\pi$
$434$	0	0
$435$	− 31426.3i	− 0.166079i
$436$	0	0
$437$	−127451.	−0.667392
$438$	0	0
$439$	8488.00i	0.0440429i	0.999757	+	0.0220215i	$0.00701022\pi$
$439$	8488.00i	0.0440429i	−0.999757	+	0.0220215i	$0.992990\pi$
$440$	0	0
$441$	55269.0	0.284187
$442$	0	0
$443$	227997.i	1.16177i	0.813985	+	0.580886i	$0.197294\pi$
$443$	227997.i	1.16177i	−0.813985	+	0.580886i	$0.802706\pi$
$444$	0	0
$445$	64723.3	0.326844
$446$	0	0
$447$	119592.i	0.598532i
$448$	0	0
$449$	49338.0	0.244731	0.122365	−	0.992485i	$-0.460952\pi$
$449$	49338.0	0.244731	0.122365	+	0.992485i	$0.460952\pi$
$450$	0	0
$451$	− 3554.17i	− 0.0174737i
$452$	0	0
$453$	49245.7	0.239978
$454$	0	0
$455$	213120.i	1.02944i
$456$	0	0
$457$	−206866.	−0.990505	−0.495253	−	0.868749i	$-0.664924\pi$
$457$	−206866.	−0.990505	−0.495253	+	0.868749i	$0.664924\pi$
$458$	0	0
$459$	102884.i	0.488339i
$460$	0	0
$461$	−194897.	−0.917073	−0.458537	−	0.888676i	$-0.651626\pi$
$461$	−194897.	−0.917073	−0.458537	+	0.888676i	$0.651626\pi$
$462$	0	0
$463$	− 137648.i	− 0.642108i	−0.947061	−	0.321054i	$-0.895963\pi$
$463$	− 137648.i	− 0.642108i	0.947061	−	0.321054i	$-0.104037\pi$
$464$	0	0
$465$	122112.	0.564745
$466$	0	0
$467$	270501.i	1.24033i	0.784473	+	0.620163i	$0.212933\pi$
$467$	270501.i	1.24033i	−0.784473	+	0.620163i	$0.787067\pi$
$468$	0	0
$469$	−72191.9	−0.328203
$470$	0	0
$471$	936.000i	0.00421924i
$472$	0	0
$473$	305748.	1.36660
$474$	0	0
$475$	48805.7i	0.216313i
$476$	0	0
$477$	−225160.	−0.989587
$478$	0	0
$479$	183456.i	0.799578i	0.916607	+	0.399789i	$0.130917\pi$
$479$	183456.i	0.799578i	−0.916607	+	0.399789i	$0.869083\pi$
$480$	0	0
$481$	449328.	1.94211
$482$	0	0
$483$	69836.3i	0.299355i
$484$	0	0
$485$	−98311.2	−0.417945
$486$	0	0
$487$	108152.i	0.456012i	0.973660	+	0.228006i	$0.0732206\pi$
$487$	108152.i	0.456012i	−0.973660	+	0.228006i	$0.926779\pi$
$488$	0	0
$489$	−99276.0	−0.415171
$490$	0	0
$491$	− 170652.i	− 0.707862i	−0.935272	−	0.353931i	$-0.884845\pi$
$491$	− 170652.i	− 0.707862i	0.935272	−	0.353931i	$-0.115155\pi$
$492$	0	0
$493$	86422.4	0.355576
$494$	0	0
$495$	− 283176.i	− 1.15570i
$496$	0	0
$497$	−313920.	−1.27088
$498$	0	0
$499$	146874.i	0.589855i	0.955520	+	0.294927i	$0.0952955\pi$
$499$	146874.i	0.589855i	−0.955520	+	0.294927i	$0.904705\pi$
$500$	0	0
$501$	75572.8	0.301086
$502$	0	0
$503$	184104.i	0.727658i	0.931466	+	0.363829i	$0.118531\pi$
$503$	184104.i	0.727658i	−0.931466	+	0.363829i	$0.881469\pi$
$504$	0	0
$505$	432.000	0.00169395
$506$	0	0
$507$	− 128695.i	− 0.500663i
$508$	0	0
$509$	−441070.	−1.70244	−0.851221	−	0.524808i	$-0.824137\pi$
$509$	−441070.	−1.70244	−0.851221	+	0.524808i	$0.824137\pi$
$510$	0	0
$511$	− 172400.i	− 0.660230i
$512$	0	0
$513$	−131400.	−0.499299
$514$	0	0
$515$	− 187394.i	− 0.706547i
$516$	0	0
$517$	483367.	1.80841
$518$	0	0
$519$	− 110520.i	− 0.410304i
$520$	0	0
$521$	−332658.	−1.22553	−0.612763	−	0.790267i	$-0.709942\pi$
$521$	−332658.	−1.22553	−0.612763	+	0.790267i	$0.709942\pi$
$522$	0	0
$523$	− 350037.i	− 1.27971i	−0.768497	−	0.639854i	$-0.778995\pi$
$523$	− 350037.i	− 1.27971i	0.768497	−	0.639854i	$-0.221005\pi$
$524$	0	0
$525$	26742.9	0.0970263
$526$	0	0
$527$	335808.i	1.20912i
$528$	0	0
$529$	25825.0	0.0922845
$530$	0	0
$531$	363554.i	1.28938i
$532$	0	0
$533$	−4614.18	−0.0162420
$534$	0	0
$535$	− 49032.0i	− 0.171306i
$536$	0	0
$537$	−36684.0	−0.127212
$538$	0	0
$539$	158160.i	0.544403i
$540$	0	0
$541$	158136.	0.540302	0.270151	−	0.962818i	$-0.412926\pi$
$541$	158136.	0.540302	0.270151	+	0.962818i	$0.412926\pi$
$542$	0	0
$543$	− 127896.i	− 0.433768i
$544$	0	0
$545$	301104.	1.01373
$546$	0	0
$547$	− 38385.7i	− 0.128291i	−0.997941	−	0.0641453i	$-0.979568\pi$
$547$	− 38385.7i	− 0.128291i	0.997941	−	0.0641453i	$-0.0204321\pi$
$548$	0	0
$549$	69316.7	0.229982
$550$	0	0
$551$	110376.i	0.363556i
$552$	0	0
$553$	−165760.	−0.542038
$554$	0	0
$555$	126204.i	0.409720i
$556$	0	0
$557$	540379.	1.74176	0.870880	−	0.491496i	$-0.163550\pi$
$557$	540379.	1.74176	0.870880	+	0.491496i	$0.163550\pi$
$558$	0	0
$559$	− 396936.i	− 1.27027i
$560$	0	0
$561$	−135432.	−0.430324
$562$	0	0
$563$	− 390782.i	− 1.23287i	−0.787405	−	0.616435i	$-0.788576\pi$
$563$	− 390782.i	− 1.23287i	0.787405	−	0.616435i	$-0.211424\pi$
$564$	0	0
$565$	44520.6	0.139465
$566$	0	0
$567$	− 151560.i	− 0.471431i
$568$	0	0
$569$	461646.	1.42589	0.712943	−	0.701222i	$-0.247362\pi$
$569$	461646.	1.42589	0.712943	+	0.701222i	$0.247362\pi$
$570$	0	0
$571$	− 215076.i	− 0.659658i	−0.944041	−	0.329829i	$-0.893009\pi$
$571$	− 215076.i	− 0.659658i	0.944041	−	0.329829i	$-0.106991\pi$
$572$	0	0
$573$	23943.9	0.0729265
$574$	0	0
$575$	97272.0i	0.294206i
$576$	0	0
$577$	−282590.	−0.848800	−0.424400	−	0.905475i	$-0.639515\pi$
$577$	−282590.	−0.848800	−0.424400	+	0.905475i	$0.639515\pi$
$578$	0	0
$579$	102004.i	0.304270i
$580$	0	0
$581$	−363731.	−1.07753
$582$	0	0
$583$	− 644328.i	− 1.89570i
$584$	0	0
$585$	−367632.	−1.07424
$586$	0	0
$587$	355032.i	1.03037i	0.857080	+	0.515183i	$0.172276\pi$
$587$	355032.i	1.03037i	−0.857080	+	0.515183i	$0.827724\pi$
$588$	0	0
$589$	−428883.	−1.23626
$590$	0	0
$591$	− 140472.i	− 0.402175i
$592$	0	0
$593$	119682.	0.340345	0.170173	−	0.985414i	$-0.445567\pi$
$593$	119682.	0.340345	0.170173	+	0.985414i	$0.445567\pi$
$594$	0	0
$595$	− 164614.i	− 0.464979i
$596$	0	0
$597$	4960.59	0.0139183
$598$	0	0
$599$	− 505080.i	− 1.40769i	−0.710354	−	0.703844i	$-0.751465\pi$
$599$	− 505080.i	− 1.40769i	0.710354	−	0.703844i	$-0.248535\pi$
$600$	0	0
$601$	576406.	1.59580	0.797902	−	0.602787i	$-0.205943\pi$
$601$	576406.	1.59580	0.797902	+	0.602787i	$0.205943\pi$
$602$	0	0
$603$	− 124531.i	− 0.342486i
$604$	0	0
$605$	506043.	1.38254
$606$	0	0
$607$	− 464672.i	− 1.26116i	−0.776126	−	0.630578i	$-0.782818\pi$
$607$	− 464672.i	− 1.26116i	0.776126	−	0.630578i	$-0.217182\pi$
$608$	0	0
$609$	60480.0	0.163071
$610$	0	0
$611$	− 627529.i	− 1.68094i
$612$	0	0
$613$	−292613.	−0.778704	−0.389352	−	0.921089i	$-0.627301\pi$
$613$	−292613.	−0.778704	−0.389352	+	0.921089i	$0.627301\pi$
$614$	0	0
$615$	− 1296.00i	− 0.00342653i
$616$	0	0
$617$	294678.	0.774065	0.387032	−	0.922066i	$-0.373500\pi$
$617$	294678.	0.774065	0.387032	+	0.922066i	$0.373500\pi$
$618$	0	0
$619$	347273.i	0.906336i	0.891425	+	0.453168i	$0.149706\pi$
$619$	347273.i	0.906336i	−0.891425	+	0.453168i	$0.850294\pi$
$620$	0	0
$621$	−261886.	−0.679093
$622$	0	0
$623$	124560.i	0.320924i
$624$	0	0
$625$	−232751.	−0.595843
$626$	0	0
$627$	− 172970.i	− 0.439982i
$628$	0	0
$629$	−347061.	−0.877213
$630$	0	0
$631$	− 434008.i	− 1.09003i	−0.838426	−	0.545016i	$-0.816524\pi$
$631$	− 434008.i	− 1.09003i	0.838426	−	0.545016i	$-0.183476\pi$
$632$	0	0
$633$	263220.	0.656918
$634$	0	0
$635$	− 105087.i	− 0.260616i
$636$	0	0
$637$	205331.	0.506030
$638$	0	0
$639$	− 541512.i	− 1.32619i
$640$	0	0
$641$	−68742.0	−0.167304	−0.0836520	−	0.996495i	$-0.526658\pi$
$641$	−68742.0	−0.167304	−0.0836520	+	0.996495i	$0.526658\pi$
$642$	0	0
$643$	610538.i	1.47669i	0.674421	+	0.738347i	$0.264393\pi$
$643$	610538.i	1.47669i	−0.674421	+	0.738347i	$0.735607\pi$
$644$	0	0
$645$	111489.	0.267985
$646$	0	0
$647$	114840.i	0.274337i	0.990548	+	0.137169i	$0.0438002\pi$
$647$	114840.i	0.274337i	−0.990548	+	0.137169i	$0.956200\pi$
$648$	0	0
$649$	−1.04036e6	−2.46999
$650$	0	0
$651$	235005.i	0.554517i
$652$	0	0
$653$	294289.	0.690157	0.345079	−	0.938574i	$-0.387852\pi$
$653$	294289.	0.690157	0.345079	+	0.938574i	$0.387852\pi$
$654$	0	0
$655$	190728.i	0.444562i
$656$	0	0
$657$	297390.	0.688963
$658$	0	0
$659$	333105.i	0.767025i	0.923536	+	0.383513i	$0.125286\pi$
$659$	333105.i	0.767025i	−0.923536	+	0.383513i	$0.874714\pi$
$660$	0	0
$661$	296589.	0.678817	0.339409	−	0.940639i	$-0.389773\pi$
$661$	296589.	0.678817	0.339409	+	0.940639i	$0.389773\pi$
$662$	0	0
$663$	175824.i	0.399992i
$664$	0	0
$665$	210240.	0.475414
$666$	0	0
$667$	219984.i	0.494470i
$668$	0	0
$669$	−68616.9	−0.153313
$670$	0	0
$671$	198360.i	0.440564i
$672$	0	0
$673$	379066.	0.836921	0.418461	−	0.908235i	$-0.362570\pi$
$673$	379066.	0.836921	0.418461	+	0.908235i	$0.362570\pi$
$674$	0	0
$675$	100286.i	0.220106i
$676$	0	0
$677$	204832.	0.446911	0.223456	−	0.974714i	$-0.428266\pi$
$677$	204832.	0.446911	0.223456	+	0.974714i	$0.428266\pi$
$678$	0	0
$679$	− 189200.i	− 0.410376i
$680$	0	0
$681$	97524.0	0.210289
$682$	0	0
$683$	13250.2i	0.0284041i	0.999899	+	0.0142020i	$0.00452080\pi$
$683$	13250.2i	0.0284041i	−0.999899	+	0.0142020i	$0.995479\pi$
$684$	0	0
$685$	−676040.	−1.44076
$686$	0	0
$687$	− 157176.i	− 0.333022i
$688$	0	0
$689$	−836496.	−1.76208
$690$	0	0
$691$	485913.i	1.01766i	0.860867	+	0.508830i	$0.169922\pi$
$691$	485913.i	1.01766i	−0.860867	+	0.508830i	$0.830078\pi$
$692$	0	0
$693$	544972.	1.13477
$694$	0	0
$695$	22968.0i	0.0475503i
$696$	0	0
$697$	3564.00	0.00733622
$698$	0	0
$699$	− 366329.i	− 0.749750i
$700$	0	0
$701$	−119823.	−0.243840	−0.121920	−	0.992540i	$-0.538905\pi$
$701$	−119823.	−0.243840	−0.121920	+	0.992540i	$0.538905\pi$
$702$	0	0
$703$	− 443256.i	− 0.896900i
$704$	0	0
$705$	176256.	0.354622
$706$	0	0
$707$	831.384i	0.00166327i
$708$	0	0
$709$	−564974.	−1.12392	−0.561961	−	0.827164i	$-0.689953\pi$
$709$	−564974.	−1.12392	−0.561961	+	0.827164i	$0.689953\pi$
$710$	0	0
$711$	− 285936.i	− 0.565626i
$712$	0	0
$713$	−854784.	−1.68142
$714$	0	0
$715$	− 1.05203e6i	− 2.05787i
$716$	0	0
$717$	−80311.7	−0.156221
$718$	0	0
$719$	− 453168.i	− 0.876600i	−0.898829	−	0.438300i	$-0.855581\pi$
$719$	− 453168.i	− 0.876600i	0.898829	−	0.438300i	$-0.144419\pi$
$720$	0	0
$721$	360640.	0.693751
$722$	0	0
$723$	− 46093.3i	− 0.0881783i
$724$	0	0
$725$	84240.0	0.160266
$726$	0	0
$727$	604808.i	1.14432i	0.820141	+	0.572162i	$0.193895\pi$
$727$	604808.i	1.14432i	−0.820141	+	0.572162i	$0.806105\pi$
$728$	0	0
$729$	115641.	0.217599
$730$	0	0
$731$	306594.i	0.573758i
$732$	0	0
$733$	−1.05230e6	−1.95854	−0.979272	−	0.202550i	$-0.935077\pi$
$733$	−1.05230e6	−1.95854	−0.979272	+	0.202550i	$0.935077\pi$
$734$	0	0
$735$	57672.0i	0.106756i
$736$	0	0
$737$	356364.	0.656083
$738$	0	0
$739$	− 502936.i	− 0.920923i	−0.887680	−	0.460462i	$-0.847684\pi$
$739$	− 502936.i	− 0.920923i	0.887680	−	0.460462i	$-0.152316\pi$
$740$	0	0
$741$	−224557.	−0.408969
$742$	0	0
$743$	499320.i	0.904485i	0.891895	+	0.452242i	$0.149376\pi$
$743$	499320.i	0.904485i	−0.891895	+	0.452242i	$0.850624\pi$
$744$	0	0
$745$	717552.	1.29283
$746$	0	0
$747$	− 627435.i	− 1.12442i
$748$	0	0
$749$	94362.1	0.168203
$750$	0	0
$751$	− 207376.i	− 0.367687i	−0.982955	−	0.183844i	$-0.941146\pi$
$751$	− 207376.i	− 0.367687i	0.982955	−	0.183844i	$-0.0588540\pi$
$752$	0	0
$753$	−220212.	−0.388375
$754$	0	0
$755$	− 295474.i	− 0.518353i
$756$	0	0
$757$	175041.	0.305456	0.152728	−	0.988268i	$-0.451194\pi$
$757$	175041.	0.305456	0.152728	+	0.988268i	$0.451194\pi$
$758$	0	0
$759$	− 344736.i	− 0.598416i
$760$	0	0
$761$	−93906.0	−0.162153	−0.0810763	−	0.996708i	$-0.525836\pi$
$761$	−93906.0	−0.162153	−0.0810763	+	0.996708i	$0.525836\pi$
$762$	0	0
$763$	579475.i	0.995372i
$764$	0	0
$765$	283959.	0.485214
$766$	0	0
$767$	1.35065e6i	2.29589i
$768$	0	0
$769$	162362.	0.274556	0.137278	−	0.990533i	$-0.456165\pi$
$769$	162362.	0.274556	0.137278	+	0.990533i	$0.456165\pi$
$770$	0	0
$771$	7420.11i	0.0124825i
$772$	0	0
$773$	504630.	0.844527	0.422264	−	0.906473i	$-0.361236\pi$
$773$	504630.	0.844527	0.422264	+	0.906473i	$0.361236\pi$
$774$	0	0
$775$	327328.i	0.544979i
$776$	0	0
$777$	−242880.	−0.402300
$778$	0	0
$779$	4551.83i	0.00750086i
$780$	0	0
$781$	1.54962e6	2.54052
$782$	0	0
$783$	226800.i	0.369930i
$784$	0	0
$785$	5616.00	0.00911355
$786$	0	0
$787$	− 38413.4i	− 0.0620203i	−0.999519	−	0.0310101i	$-0.990128\pi$
$787$	− 38413.4i	− 0.0620203i	0.999519	−	0.0310101i	$-0.00987241\pi$
$788$	0	0
$789$	−204770.	−0.328937
$790$	0	0
$791$	85680.0i	0.136939i
$792$	0	0
$793$	257520.	0.409510
$794$	0	0
$795$	− 234949.i	− 0.371740i
$796$	0	0
$797$	269098.	0.423637	0.211819	−	0.977309i	$-0.432061\pi$
$797$	269098.	0.423637	0.211819	+	0.977309i	$0.432061\pi$
$798$	0	0
$799$	484704.i	0.759247i
$800$	0	0
$801$	−214866.	−0.334890
$802$	0	0
$803$	851026.i	1.31981i
$804$	0	0
$805$	419018.	0.646607
$806$	0	0
$807$	− 28728.0i	− 0.0441122i
$808$	0	0
$809$	813294.	1.24265	0.621327	−	0.783551i	$-0.286594\pi$
$809$	813294.	1.24265	0.621327	+	0.783551i	$0.286594\pi$
$810$	0	0
$811$	1.19639e6i	1.81900i	0.415706	+	0.909499i	$0.363535\pi$
$811$	1.19639e6i	1.81900i	−0.415706	+	0.909499i	$0.636465\pi$
$812$	0	0
$813$	−338484.	−0.512103
$814$	0	0
$815$	595656.i	0.896768i
$816$	0	0
$817$	−391572.	−0.586634
$818$	0	0
$819$	− 707508.i	− 1.05478i
$820$	0	0
$821$	−681340.	−1.01083	−0.505415	−	0.862877i	$-0.668660\pi$
$821$	−681340.	−1.01083	−0.505415	+	0.862877i	$0.668660\pi$
$822$	0	0
$823$	688808.i	1.01695i	0.861078	+	0.508474i	$0.169790\pi$
$823$	688808.i	1.01695i	−0.861078	+	0.508474i	$0.830210\pi$
$824$	0	0
$825$	−132012.	−0.193957
$826$	0	0
$827$	518815.i	0.758580i	0.925278	+	0.379290i	$0.123832\pi$
$827$	518815.i	0.758580i	−0.925278	+	0.379290i	$0.876168\pi$
$828$	0	0
$829$	1.00144e6	1.45718	0.728592	−	0.684948i	$-0.240175\pi$
$829$	1.00144e6	1.45718	0.728592	+	0.684948i	$0.240175\pi$
$830$	0	0
$831$	286056.i	0.414237i
$832$	0	0
$833$	−158598.	−0.228564
$834$	0	0
$835$	− 453437.i	− 0.650345i
$836$	0	0
$837$	−881267.	−1.25793
$838$	0	0
$839$	520344.i	0.739208i	0.929189	+	0.369604i	$0.120507\pi$
$839$	520344.i	0.739208i	−0.929189	+	0.369604i	$0.879493\pi$
$840$	0	0
$841$	−516769.	−0.730642
$842$	0	0
$843$	403866.i	0.568306i
$844$	0	0
$845$	−772169.	−1.08143
$846$	0	0
$847$	973880.i	1.35750i
$848$	0	0
$849$	492012.	0.682591
$850$	0	0
$851$	− 883429.i	− 1.21987i
$852$	0	0
$853$	−271357.	−0.372943	−0.186472	−	0.982460i	$-0.559705\pi$
$853$	−271357.	−0.372943	−0.186472	+	0.982460i	$0.559705\pi$
$854$	0	0
$855$	362664.i	0.496103i
$856$	0	0
$857$	−324018.	−0.441172	−0.220586	−	0.975368i	$-0.570797\pi$
$857$	−324018.	−0.441172	−0.220586	+	0.975368i	$0.570797\pi$
$858$	0	0
$859$	− 638832.i	− 0.865766i	−0.901450	−	0.432883i	$-0.857496\pi$
$859$	− 638832.i	− 0.865766i	0.901450	−	0.432883i	$-0.142504\pi$
$860$	0	0
$861$	2494.15	0.00336447
$862$	0	0
$863$	− 399456.i	− 0.536349i	−0.963370	−	0.268174i	$-0.913580\pi$
$863$	− 399456.i	− 0.536349i	0.963370	−	0.268174i	$-0.0864204\pi$
$864$	0	0
$865$	−663120.	−0.886257
$866$	0	0
$867$	153519.i	0.204232i
$868$	0	0
$869$	818249.	1.08354
$870$	0	0
$871$	− 462648.i	− 0.609838i
$872$	0	0
$873$	326370.	0.428235
$874$	0	0
$875$	− 680072.i	− 0.888258i
$876$	0	0
$877$	−35216.1	−0.0457869	−0.0228935	−	0.999738i	$-0.507288\pi$
$877$	−35216.1	−0.0457869	−0.0228935	+	0.999738i	$0.507288\pi$
$878$	0	0
$879$	145800.i	0.188703i
$880$	0	0
$881$	−876510.	−1.12929	−0.564644	−	0.825334i	$-0.690987\pi$
$881$	−876510.	−1.12929	−0.564644	+	0.825334i	$0.690987\pi$
$882$	0	0
$883$	1.21983e6i	1.56451i	0.622958	+	0.782255i	$0.285931\pi$
$883$	1.21983e6i	1.56451i	−0.622958	+	0.782255i	$0.714069\pi$
$884$	0	0
$885$	−379361.	−0.484357
$886$	0	0
$887$	853992.i	1.08544i	0.839913	+	0.542721i	$0.182606\pi$
$887$	853992.i	1.08544i	−0.839913	+	0.542721i	$0.817394\pi$
$888$	0	0
$889$	202240.	0.255896
$890$	0	0
$891$	748152.i	0.942399i
$892$	0	0
$893$	−619049.	−0.776286
$894$	0	0
$895$	220104.i	0.274778i
$896$	0	0
$897$	−447552.	−0.556235
$898$	0	0
$899$	740265.i	0.915941i
$900$	0	0
$901$	646110.	0.795897
$902$	0	0
$903$	214560.i	0.263132i
$904$	0	0
$905$	−767376.	−0.936938
$906$	0	0
$907$	804506.i	0.977946i	0.872299	+	0.488973i	$0.162628\pi$
$907$	804506.i	0.977946i	−0.872299	+	0.488973i	$0.837372\pi$
$908$	0	0
$909$	−1434.14	−0.00173565
$910$	0	0
$911$	− 617904.i	− 0.744534i	−0.928126	−	0.372267i	$-0.878581\pi$
$911$	− 617904.i	− 0.744534i	0.928126	−	0.372267i	$-0.121419\pi$
$912$	0	0
$913$	1.79550e6	2.15399
$914$	0	0
$915$	72330.4i	0.0863931i
$916$	0	0
$917$	−367056.	−0.436510
$918$	0	0
$919$	− 930040.i	− 1.10121i	−0.834766	−	0.550606i	$-0.814397\pi$
$919$	− 930040.i	− 1.10121i	0.834766	−	0.550606i	$-0.185603\pi$
$920$	0	0
$921$	138132.	0.162845
$922$	0	0
$923$	− 2.01178e6i	− 2.36145i
$924$	0	0
$925$	−338297.	−0.395380
$926$	0	0
$927$	622104.i	0.723942i
$928$	0	0
$929$	1.63507e6	1.89454	0.947270	−	0.320436i	$-0.103829\pi$
$929$	1.63507e6	1.89454	0.947270	+	0.320436i	$0.103829\pi$
$930$	0	0
$931$	− 202556.i	− 0.233694i
$932$	0	0
$933$	498082.	0.572187
$934$	0	0
$935$	812592.i	0.929500i
$936$	0	0
$937$	1.16956e6	1.33212	0.666059	−	0.745899i	$-0.267980\pi$
$937$	1.16956e6	1.33212	0.666059	+	0.745899i	$0.267980\pi$
$938$	0	0
$939$	54822.9i	0.0621771i
$940$	0	0
$941$	−68693.1	−0.0775772	−0.0387886	−	0.999247i	$-0.512350\pi$
$941$	−68693.1	−0.0775772	−0.0387886	+	0.999247i	$0.512350\pi$
$942$	0	0
$943$	9072.00i	0.0102019i
$944$	0	0
$945$	432000.	0.483749
$946$	0	0
$947$	195178.i	0.217636i	0.994062	+	0.108818i	$0.0347066\pi$
$947$	195178.i	0.217636i	−0.994062	+	0.108818i	$0.965293\pi$
$948$	0	0
$949$	1.10484e6	1.22678
$950$	0	0
$951$	391464.i	0.432843i
$952$	0	0
$953$	−1.61311e6	−1.77614	−0.888070	−	0.459709i	$-0.847954\pi$
$953$	−1.61311e6	−1.77614	−0.888070	+	0.459709i	$0.847954\pi$
$954$	0	0
$955$	− 143663.i	− 0.157521i
$956$	0	0
$957$	−298550.	−0.325982
$958$	0	0
$959$	− 1.30104e6i	− 1.41466i
$960$	0	0
$961$	−1.95290e6	−2.11462
$962$	0	0
$963$	162775.i	0.175523i
$964$	0	0
$965$	612024.	0.657224
$966$	0	0
$967$	1.15382e6i	1.23391i	0.786998	+	0.616955i	$0.211634\pi$
$967$	1.15382e6i	1.23391i	−0.786998	+	0.616955i	$0.788366\pi$
$968$	0	0
$969$	173448.	0.184723
$970$	0	0
$971$	− 1.34490e6i	− 1.42643i	−0.700944	−	0.713216i	$-0.747238\pi$
$971$	− 1.34490e6i	− 1.42643i	0.700944	−	0.713216i	$-0.252762\pi$
$972$	0	0
$973$	−44201.9	−0.0466891
$974$	0	0
$975$	171384.i	0.180286i
$976$	0	0
$977$	803322.	0.841590	0.420795	−	0.907156i	$-0.361751\pi$
$977$	803322.	0.841590	0.420795	+	0.907156i	$0.361751\pi$
$978$	0	0
$979$	− 614871.i	− 0.641533i
$980$	0	0
$981$	−999594.	−1.03869
$982$	0	0
$983$	− 590328.i	− 0.610923i	−0.952204	−	0.305461i	$-0.901189\pi$
$983$	− 590328.i	− 0.610923i	0.952204	−	0.305461i	$-0.0988107\pi$
$984$	0	0
$985$	−842832.	−0.868697
$986$	0	0
$987$	339205.i	0.348199i
$988$	0	0
$989$	−780421.	−0.797877
$990$	0	0
$991$	182240.i	0.185565i	0.995686	+	0.0927826i	$0.0295761\pi$
$991$	182240.i	0.185565i	−0.995686	+	0.0927826i	$0.970424\pi$
$992$	0	0
$993$	−120468.	−0.122172
$994$	0	0
$995$	− 29763.6i	− 0.0300634i
$996$	0	0
$997$	−1.90123e6	−1.91269	−0.956345	−	0.292241i	$-0.905599\pi$
$997$	−1.90123e6	−1.91269	−0.956345	+	0.292241i	$0.905599\pi$
$998$	0	0
$999$	− 910800.i	− 0.912624i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.5.c.j.255.1		4
4.3	odd	2	inner	256.5.c.j.255.3		4
8.3	odd	2	inner	256.5.c.j.255.2		4
8.5	even	2	inner	256.5.c.j.255.4		4
16.3	odd	4		64.5.d.a.31.2	yes	4
16.5	even	4		64.5.d.a.31.1	✓	4
16.11	odd	4		64.5.d.a.31.3	yes	4
16.13	even	4		64.5.d.a.31.4	yes	4
48.5	odd	4		576.5.b.e.415.4		4
48.11	even	4		576.5.b.e.415.3		4
48.29	odd	4		576.5.b.e.415.2		4
48.35	even	4		576.5.b.e.415.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.5.d.a.31.1	✓	4	16.5	even	4
64.5.d.a.31.2	yes	4	16.3	odd	4
64.5.d.a.31.3	yes	4	16.11	odd	4
64.5.d.a.31.4	yes	4	16.13	even	4
256.5.c.j.255.1		4	1.1	even	1	trivial
256.5.c.j.255.2		4	8.3	odd	2	inner
256.5.c.j.255.3		4	4.3	odd	2	inner
256.5.c.j.255.4		4	8.5	even	2	inner
576.5.b.e.415.1		4	48.35	even	4
576.5.b.e.415.2		4	48.29	odd	4
576.5.b.e.415.3		4	48.11	even	4
576.5.b.e.415.4		4	48.5	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.46410i q^{3} -20.7846 q^{5} -40.0000i q^{7} +69.0000 q^{9} +O(q^{10})\)	\(q-3.46410i q^{3} -20.7846 q^{5} -40.0000i q^{7} +69.0000 q^{9} +197.454i q^{11} +256.344 q^{13} +72.0000i q^{15} -198.000 q^{17} -252.879i q^{19} -138.564 q^{21} -504.000i q^{23} -193.000 q^{25} -519.615i q^{27} -436.477 q^{29} -1696.00i q^{31} +684.000 q^{33} +831.384i q^{35} +1752.84 q^{37} -888.000i q^{39} -18.0000 q^{41} -1548.45i q^{43} -1434.14 q^{45} -2448.00i q^{47} +801.000 q^{49} +685.892i q^{51} -3263.18 q^{53} -4104.00i q^{55} -876.000 q^{57} +5268.90i q^{59} +1004.59 q^{61} -2760.00i q^{63} -5328.00 q^{65} -1804.80i q^{67} -1745.91 q^{69} -7848.00i q^{71} +4310.00 q^{73} +668.572i q^{75} +7898.15 q^{77} -4144.00i q^{79} +3789.00 q^{81} -9093.27i q^{83} +4115.35 q^{85} +1512.00i q^{87} -3114.00 q^{89} -10253.7i q^{91} -5875.12 q^{93} +5256.00i q^{95} +4730.00 q^{97} +13624.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 276 q^{9}+O(q^{10}) \) 4 * q + 276 * q^9	\( 4 q + 276 q^{9} - 792 q^{17} - 772 q^{25} + 2736 q^{33} - 72 q^{41} + 3204 q^{49} - 3504 q^{57} - 21312 q^{65} + 17240 q^{73} + 15156 q^{81} - 12456 q^{89} + 18920 q^{97}+O(q^{100}) \) 4 * q + 276 * q^9 - 792 * q^17 - 772 * q^25 + 2736 * q^33 - 72 * q^41 + 3204 * q^49 - 3504 * q^57 - 21312 * q^65 + 17240 * q^73 + 15156 * q^81 - 12456 * q^89 + 18920 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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