LMFDB - Embedded newform 256.8.b.l.129.6

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			129.6
Root			$-7.53231i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.l.129.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+62.2585i q^{3} +203.009i q^{5} +534.535 q^{7} -1689.12 q^{9} -1579.58i q^{11} -1771.71i q^{13} -12639.0 q^{15} +28985.6 q^{17} -39214.3i q^{19} +33279.3i q^{21} +85265.7 q^{23} +36912.4 q^{25} +30997.2i q^{27} -246207. i q^{29} +54648.0 q^{31} +98342.3 q^{33} +108515. i q^{35} -290027. i q^{37} +110304. q^{39} +725641. q^{41} -668406. i q^{43} -342907. i q^{45} -429632. q^{47} -537815. q^{49} +1.80460e6i q^{51} -725328. i q^{53} +320669. q^{55} +2.44142e6 q^{57} -165981. i q^{59} -2.66752e6i q^{61} -902895. q^{63} +359673. q^{65} +1.16279e6i q^{67} +5.30852e6i q^{69} +3.87428e6 q^{71} -579226. q^{73} +2.29811e6i q^{75} -844341. i q^{77} -1.92515e6 q^{79} -5.62395e6 q^{81} +4.86286e6i q^{83} +5.88434e6i q^{85} +1.53285e7 q^{87} -4.58660e6 q^{89} -947042. i q^{91} +3.40230e6i q^{93} +7.96085e6 q^{95} -4.18281e6 q^{97} +2.66810e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	62.2585i	1.33130i	0.746266	+	0.665648i	$0.231845\pi$
$3$	62.2585i	1.33130i	−0.746266	+	0.665648i	$0.768155\pi$
$4$	0	0
$5$	203.009i	0.726307i	0.931729	+	0.363153i	$0.118300\pi$
$5$	203.009i	0.726307i	−0.931729	+	0.363153i	$0.881700\pi$
$6$	0	0
$7$	534.535	0.589024	0.294512	−	0.955648i	$-0.404843\pi$
$7$	534.535	0.589024	0.294512	+	0.955648i	$0.404843\pi$
$8$	0	0
$9$	−1689.12	−0.772346
$10$	0	0
$11$	− 1579.58i	− 0.357822i	−0.983865	−	0.178911i	$-0.942743\pi$
$11$	− 1579.58i	− 0.357822i	0.983865	−	0.178911i	$-0.0572574\pi$
$12$	0	0
$13$	− 1771.71i	− 0.223661i	−0.993727	−	0.111831i	$-0.964329\pi$
$13$	− 1771.71i	− 0.223661i	0.993727	−	0.111831i	$-0.0356714\pi$
$14$	0	0
$15$	−12639.0	−0.966929
$16$	0	0
$17$	28985.6	1.43091	0.715453	−	0.698661i	$-0.246220\pi$
$17$	28985.6	1.43091	0.715453	+	0.698661i	$0.246220\pi$
$18$	0	0
$19$	− 39214.3i	− 1.31162i	−0.754928	−	0.655808i	$-0.772328\pi$
$19$	− 39214.3i	− 1.31162i	0.754928	−	0.655808i	$-0.227672\pi$
$20$	0	0
$21$	33279.3i	0.784165i
$22$	0	0
$23$	85265.7	1.46126	0.730629	−	0.682774i	$-0.239227\pi$
$23$	85265.7	1.46126	0.730629	+	0.682774i	$0.239227\pi$
$24$	0	0
$25$	36912.4	0.472478
$26$	0	0
$27$	30997.2i	0.303074i
$28$	0	0
$29$	− 246207.i	− 1.87459i	−0.348531	−	0.937297i	$-0.613319\pi$
$29$	− 246207.i	− 1.87459i	0.348531	−	0.937297i	$-0.386681\pi$
$30$	0	0
$31$	54648.0	0.329464	0.164732	−	0.986338i	$-0.447324\pi$
$31$	54648.0	0.329464	0.164732	+	0.986338i	$0.447324\pi$
$32$	0	0
$33$	98342.3	0.476367
$34$	0	0
$35$	108515.i	0.427812i
$36$	0	0
$37$	− 290027.i	− 0.941310i	−0.882317	−	0.470655i	$-0.844018\pi$
$37$	− 290027.i	− 0.941310i	0.882317	−	0.470655i	$-0.155982\pi$
$38$	0	0
$39$	110304.	0.297759
$40$	0	0
$41$	725641.	1.64429	0.822145	−	0.569278i	$-0.192777\pi$
$41$	725641.	1.64429	0.822145	+	0.569278i	$0.192777\pi$
$42$	0	0
$43$	− 668406.i	− 1.28204i	−0.767525	−	0.641019i	$-0.778512\pi$
$43$	− 668406.i	− 1.28204i	0.767525	−	0.641019i	$-0.221488\pi$
$44$	0	0
$45$	− 342907.i	− 0.560961i
$46$	0	0
$47$	−429632.	−0.603607	−0.301803	−	0.953370i	$-0.597589\pi$
$47$	−429632.	−0.603607	−0.301803	+	0.953370i	$0.597589\pi$
$48$	0	0
$49$	−537815.	−0.653051
$50$	0	0
$51$	1.80460e6i	1.90496i
$52$	0	0
$53$	− 725328.i	− 0.669220i	−0.942357	−	0.334610i	$-0.891395\pi$
$53$	− 725328.i	− 0.669220i	0.942357	−	0.334610i	$-0.108605\pi$
$54$	0	0
$55$	320669.	0.259889
$56$	0	0
$57$	2.44142e6	1.74615
$58$	0	0
$59$	− 165981.i	− 0.105215i	−0.998615	−	0.0526073i	$-0.983247\pi$
$59$	− 165981.i	− 0.105215i	0.998615	−	0.0526073i	$-0.0167532\pi$
$60$	0	0
$61$	− 2.66752e6i	− 1.50471i	−0.658758	−	0.752355i	$-0.728918\pi$
$61$	− 2.66752e6i	− 1.50471i	0.658758	−	0.752355i	$-0.271082\pi$
$62$	0	0
$63$	−902895.	−0.454931
$64$	0	0
$65$	359673.	0.162447
$66$	0	0
$67$	1.16279e6i	0.472322i	0.971714	+	0.236161i	$0.0758892\pi$
$67$	1.16279e6i	0.472322i	−0.971714	+	0.236161i	$0.924111\pi$
$68$	0	0
$69$	5.30852e6i	1.94537i
$70$	0	0
$71$	3.87428e6	1.28466	0.642328	−	0.766430i	$-0.277969\pi$
$71$	3.87428e6	1.28466	0.642328	+	0.766430i	$0.277969\pi$
$72$	0	0
$73$	−579226.	−0.174268	−0.0871341	−	0.996197i	$-0.527771\pi$
$73$	−579226.	−0.174268	−0.0871341	+	0.996197i	$0.527771\pi$
$74$	0	0
$75$	2.29811e6i	0.629008i
$76$	0	0
$77$	− 844341.i	− 0.210766i
$78$	0	0
$79$	−1.92515e6	−0.439308	−0.219654	−	0.975578i	$-0.570493\pi$
$79$	−1.92515e6	−0.439308	−0.219654	+	0.975578i	$0.570493\pi$
$80$	0	0
$81$	−5.62395e6	−1.17583
$82$	0	0
$83$	4.86286e6i	0.933509i	0.884387	+	0.466755i	$0.154577\pi$
$83$	4.86286e6i	0.933509i	−0.884387	+	0.466755i	$0.845423\pi$
$84$	0	0
$85$	5.88434e6i	1.03928i
$86$	0	0
$87$	1.53285e7	2.49564
$88$	0	0
$89$	−4.58660e6	−0.689645	−0.344823	−	0.938668i	$-0.612061\pi$
$89$	−4.58660e6	−0.689645	−0.344823	+	0.938668i	$0.612061\pi$
$90$	0	0
$91$	− 947042.i	− 0.131742i
$92$	0	0
$93$	3.40230e6i	0.438614i
$94$	0	0
$95$	7.96085e6	0.952636
$96$	0	0
$97$	−4.18281e6	−0.465336	−0.232668	−	0.972556i	$-0.574746\pi$
$97$	−4.18281e6	−0.465336	−0.232668	+	0.972556i	$0.574746\pi$
$98$	0	0
$99$	2.66810e6i	0.276363i
$100$	0	0
$101$	− 4.85215e6i	− 0.468608i	−0.972163	−	0.234304i	$-0.924719\pi$
$101$	− 4.85215e6i	− 0.468608i	0.972163	−	0.234304i	$-0.0752811\pi$
$102$	0	0
$103$	−1.31137e7	−1.18249	−0.591243	−	0.806494i	$-0.701363\pi$
$103$	−1.31137e7	−1.18249	−0.591243	+	0.806494i	$0.701363\pi$
$104$	0	0
$105$	−6.75600e6	−0.569544
$106$	0	0
$107$	− 2.04501e6i	− 0.161381i	−0.996739	−	0.0806905i	$-0.974287\pi$
$107$	− 2.04501e6i	− 0.161381i	0.996739	−	0.0806905i	$-0.0257126\pi$
$108$	0	0
$109$	9.15280e6i	0.676957i	0.940974	+	0.338479i	$0.109912\pi$
$109$	9.15280e6i	0.676957i	−0.940974	+	0.338479i	$0.890088\pi$
$110$	0	0
$111$	1.80567e7	1.25316
$112$	0	0
$113$	−1.75532e7	−1.14441	−0.572207	−	0.820110i	$-0.693912\pi$
$113$	−1.75532e7	−1.14441	−0.572207	+	0.820110i	$0.693912\pi$
$114$	0	0
$115$	1.73097e7i	1.06132i
$116$	0	0
$117$	2.99264e6i	0.172744i
$118$	0	0
$119$	1.54938e7	0.842838
$120$	0	0
$121$	1.69921e7	0.871963
$122$	0	0
$123$	4.51773e7i	2.18903i
$124$	0	0
$125$	2.33536e7i	1.06947i
$126$	0	0
$127$	3.81077e7	1.65082	0.825410	−	0.564534i	$-0.190944\pi$
$127$	3.81077e7	1.65082	0.825410	+	0.564534i	$0.190944\pi$
$128$	0	0
$129$	4.16140e7	1.70677
$130$	0	0
$131$	2.06882e7i	0.804033i	0.915633	+	0.402016i	$0.131690\pi$
$131$	2.06882e7i	0.804033i	−0.915633	+	0.402016i	$0.868310\pi$
$132$	0	0
$133$	− 2.09614e7i	− 0.772573i
$134$	0	0
$135$	−6.29270e6	−0.220125
$136$	0	0
$137$	−1.98669e7	−0.660099	−0.330049	−	0.943964i	$-0.607065\pi$
$137$	−1.98669e7	−0.660099	−0.330049	+	0.943964i	$0.607065\pi$
$138$	0	0
$139$	5.73872e7i	1.81244i	0.422808	+	0.906219i	$0.361045\pi$
$139$	5.73872e7i	1.81244i	−0.422808	+	0.906219i	$0.638955\pi$
$140$	0	0
$141$	− 2.67482e7i	− 0.803578i
$142$	0	0
$143$	−2.79856e6	−0.0800310
$144$	0	0
$145$	4.99822e7	1.36153
$146$	0	0
$147$	− 3.34836e7i	− 0.869403i
$148$	0	0
$149$	− 5.61302e7i	− 1.39009i	−0.718964	−	0.695047i	$-0.755383\pi$
$149$	− 5.61302e7i	− 1.39009i	0.718964	−	0.695047i	$-0.244617\pi$
$150$	0	0
$151$	5.71149e7	1.34999	0.674994	−	0.737823i	$-0.264146\pi$
$151$	5.71149e7	1.34999	0.674994	+	0.737823i	$0.264146\pi$
$152$	0	0
$153$	−4.89602e7	−1.10516
$154$	0	0
$155$	1.10940e7i	0.239292i
$156$	0	0
$157$	− 2.12835e7i	− 0.438928i	−0.975621	−	0.219464i	$-0.929569\pi$
$157$	− 2.12835e7i	− 0.438928i	0.975621	−	0.219464i	$-0.0704309\pi$
$158$	0	0
$159$	4.51578e7	0.890929
$160$	0	0
$161$	4.55775e7	0.860716
$162$	0	0
$163$	2.30446e7i	0.416785i	0.978045	+	0.208393i	$0.0668232\pi$
$163$	2.30446e7i	0.416785i	−0.978045	+	0.208393i	$0.933177\pi$
$164$	0	0
$165$	1.99644e7i	0.345989i
$166$	0	0
$167$	−6.81864e7	−1.13290	−0.566448	−	0.824097i	$-0.691683\pi$
$167$	−6.81864e7	−1.13290	−0.566448	+	0.824097i	$0.691683\pi$
$168$	0	0
$169$	5.96096e7	0.949976
$170$	0	0
$171$	6.62377e7i	1.01302i
$172$	0	0
$173$	− 1.37015e7i	− 0.201191i	−0.994927	−	0.100595i	$-0.967925\pi$
$173$	− 1.37015e7i	− 0.201191i	0.994927	−	0.100595i	$-0.0320747\pi$
$174$	0	0
$175$	1.97309e7	0.278301
$176$	0	0
$177$	1.03337e7	0.140072
$178$	0	0
$179$	− 3.41349e6i	− 0.0444849i	−0.999753	−	0.0222424i	$-0.992919\pi$
$179$	− 3.41349e6i	− 0.0444849i	0.999753	−	0.0222424i	$-0.00708057\pi$
$180$	0	0
$181$	− 6.65355e7i	− 0.834024i	−0.908901	−	0.417012i	$-0.863077\pi$
$181$	− 6.65355e7i	− 0.834024i	0.908901	−	0.417012i	$-0.136923\pi$
$182$	0	0
$183$	1.66076e8	2.00321
$184$	0	0
$185$	5.88781e7	0.683680
$186$	0	0
$187$	− 4.57851e7i	− 0.512010i
$188$	0	0
$189$	1.65691e7i	0.178518i
$190$	0	0
$191$	1.69543e8	1.76061	0.880303	−	0.474413i	$-0.157340\pi$
$191$	1.69543e8	1.76061	0.880303	+	0.474413i	$0.157340\pi$
$192$	0	0
$193$	−1.77620e8	−1.77845	−0.889226	−	0.457468i	$-0.848757\pi$
$193$	−1.77620e8	−1.77845	−0.889226	+	0.457468i	$0.848757\pi$
$194$	0	0
$195$	2.23927e7i	0.216265i
$196$	0	0
$197$	7.47827e6i	0.0696898i	0.999393	+	0.0348449i	$0.0110937\pi$
$197$	7.47827e6i	0.0696898i	−0.999393	+	0.0348449i	$0.988906\pi$
$198$	0	0
$199$	−1.63422e8	−1.47003	−0.735014	−	0.678052i	$-0.762824\pi$
$199$	−1.63422e8	−1.47003	−0.735014	+	0.678052i	$0.762824\pi$
$200$	0	0
$201$	−7.23933e7	−0.628799
$202$	0	0
$203$	− 1.31606e8i	− 1.10418i
$204$	0	0
$205$	1.47312e8i	1.19426i
$206$	0	0
$207$	−1.44024e8	−1.12860
$208$	0	0
$209$	−6.19421e7	−0.469325
$210$	0	0
$211$	2.12526e8i	1.55749i	0.627343	+	0.778743i	$0.284143\pi$
$211$	2.12526e8i	1.55749i	−0.627343	+	0.778743i	$0.715857\pi$
$212$	0	0
$213$	2.41207e8i	1.71026i
$214$	0	0
$215$	1.35692e8	0.931153
$216$	0	0
$217$	2.92113e7	0.194062
$218$	0	0
$219$	− 3.60618e7i	− 0.232002i
$220$	0	0
$221$	− 5.13541e7i	− 0.320039i
$222$	0	0
$223$	−2.42620e8	−1.46507	−0.732536	−	0.680728i	$-0.761664\pi$
$223$	−2.42620e8	−1.46507	−0.732536	+	0.680728i	$0.761664\pi$
$224$	0	0
$225$	−6.23495e7	−0.364917
$226$	0	0
$227$	1.63844e7i	0.0929695i	0.998919	+	0.0464847i	$0.0148019\pi$
$227$	1.63844e7i	0.0929695i	−0.998919	+	0.0464847i	$0.985198\pi$
$228$	0	0
$229$	2.30425e8i	1.26796i	0.773350	+	0.633979i	$0.218579\pi$
$229$	2.30425e8i	1.26796i	−0.773350	+	0.633979i	$0.781421\pi$
$230$	0	0
$231$	5.25674e7	0.280591
$232$	0	0
$233$	9.52807e7	0.493468	0.246734	−	0.969083i	$-0.420643\pi$
$233$	9.52807e7	0.493468	0.246734	+	0.969083i	$0.420643\pi$
$234$	0	0
$235$	− 8.72191e7i	− 0.438404i
$236$	0	0
$237$	− 1.19857e8i	− 0.584848i
$238$	0	0
$239$	−1.85841e8	−0.880540	−0.440270	−	0.897865i	$-0.645117\pi$
$239$	−1.85841e8	−0.880540	−0.440270	+	0.897865i	$0.645117\pi$
$240$	0	0
$241$	−105251.	−0.000484356 0	−0.000242178	−	1.00000i	$-0.500077\pi$
$241$	−105251.	−0.000484356 0	−0.000242178		1.00000i	$0.500077\pi$
$242$	0	0
$243$	− 2.82348e8i	− 1.26230i
$244$	0	0
$245$	− 1.09181e8i	− 0.474315i
$246$	0	0
$247$	−6.94764e7	−0.293358
$248$	0	0
$249$	−3.02755e8	−1.24278
$250$	0	0
$251$	3.64315e7i	0.145418i	0.997353	+	0.0727092i	$0.0231645\pi$
$251$	3.64315e7i	0.145418i	−0.997353	+	0.0727092i	$0.976836\pi$
$252$	0	0
$253$	− 1.34684e8i	− 0.522871i
$254$	0	0
$255$	−3.66350e8	−1.38358
$256$	0	0
$257$	−2.82041e8	−1.03645	−0.518223	−	0.855246i	$-0.673406\pi$
$257$	−2.82041e8	−1.03645	−0.518223	+	0.855246i	$0.673406\pi$
$258$	0	0
$259$	− 1.55030e8i	− 0.554454i
$260$	0	0
$261$	4.15874e8i	1.44784i
$262$	0	0
$263$	−2.38965e8	−0.810008	−0.405004	−	0.914315i	$-0.632730\pi$
$263$	−2.38965e8	−0.810008	−0.405004	+	0.914315i	$0.632730\pi$
$264$	0	0
$265$	1.47248e8	0.486059
$266$	0	0
$267$	− 2.85555e8i	− 0.918121i
$268$	0	0
$269$	− 3.57893e8i	− 1.12104i	−0.828142	−	0.560519i	$-0.810602\pi$
$269$	− 3.57893e8i	− 1.12104i	0.828142	−	0.560519i	$-0.189398\pi$
$270$	0	0
$271$	−1.58788e8	−0.484646	−0.242323	−	0.970196i	$-0.577909\pi$
$271$	−1.58788e8	−0.484646	−0.242323	+	0.970196i	$0.577909\pi$
$272$	0	0
$273$	5.89614e7	0.175387
$274$	0	0
$275$	− 5.83060e7i	− 0.169063i
$276$	0	0
$277$	− 4.11223e8i	− 1.16251i	−0.813720	−	0.581257i	$-0.802561\pi$
$277$	− 4.11223e8i	− 1.16251i	0.813720	−	0.581257i	$-0.197439\pi$
$278$	0	0
$279$	−9.23071e7	−0.254460
$280$	0	0
$281$	3.05056e8	0.820177	0.410088	−	0.912046i	$-0.365498\pi$
$281$	3.05056e8	0.820177	0.410088	+	0.912046i	$0.365498\pi$
$282$	0	0
$283$	− 4.22725e8i	− 1.10868i	−0.832291	−	0.554340i	$-0.812971\pi$
$283$	− 4.22725e8i	− 1.10868i	0.832291	−	0.554340i	$-0.187029\pi$
$284$	0	0
$285$	4.95631e8i	1.26824i
$286$	0	0
$287$	3.87880e8	0.968526
$288$	0	0
$289$	4.29827e8	1.04749
$290$	0	0
$291$	− 2.60415e8i	− 0.619500i
$292$	0	0
$293$	− 4.27521e8i	− 0.992934i	−0.868056	−	0.496467i	$-0.834630\pi$
$293$	− 4.27521e8i	− 0.992934i	0.868056	−	0.496467i	$-0.165370\pi$
$294$	0	0
$295$	3.36956e7	0.0764181
$296$	0	0
$297$	4.89625e7	0.108447
$298$	0	0
$299$	− 1.51066e8i	− 0.326827i
$300$	0	0
$301$	− 3.57286e8i	− 0.755151i
$302$	0	0
$303$	3.02088e8	0.623855
$304$	0	0
$305$	5.41530e8	1.09288
$306$	0	0
$307$	2.51344e8i	0.495775i	0.968789	+	0.247888i	$0.0797363\pi$
$307$	2.51344e8i	0.495775i	−0.968789	+	0.247888i	$0.920264\pi$
$308$	0	0
$309$	− 8.16441e8i	− 1.57424i
$310$	0	0
$311$	−8.95523e8	−1.68817	−0.844083	−	0.536212i	$-0.819855\pi$
$311$	−8.95523e8	−1.68817	−0.844083	+	0.536212i	$0.819855\pi$
$312$	0	0
$313$	5.23000e8	0.964043	0.482021	−	0.876159i	$-0.339903\pi$
$313$	5.23000e8	0.964043	0.482021	+	0.876159i	$0.339903\pi$
$314$	0	0
$315$	− 1.83296e8i	− 0.330419i
$316$	0	0
$317$	7.06111e8i	1.24499i	0.782624	+	0.622495i	$0.213881\pi$
$317$	7.06111e8i	1.24499i	−0.782624	+	0.622495i	$0.786119\pi$
$318$	0	0
$319$	−3.88904e8	−0.670772
$320$	0	0
$321$	1.27319e8	0.214846
$322$	0	0
$323$	− 1.13665e9i	− 1.87680i
$324$	0	0
$325$	− 6.53981e7i	− 0.105675i
$326$	0	0
$327$	−5.69840e8	−0.901230
$328$	0	0
$329$	−2.29653e8	−0.355539
$330$	0	0
$331$	1.98152e8i	0.300331i	0.988661	+	0.150166i	$0.0479807\pi$
$331$	1.98152e8i	0.300331i	−0.988661	+	0.150166i	$0.952019\pi$
$332$	0	0
$333$	4.89891e8i	0.727018i
$334$	0	0
$335$	−2.36056e8	−0.343050
$336$	0	0
$337$	−2.39645e8	−0.341086	−0.170543	−	0.985350i	$-0.554552\pi$
$337$	−2.39645e8	−0.341086	−0.170543	+	0.985350i	$0.554552\pi$
$338$	0	0
$339$	− 1.09284e9i	− 1.52355i
$340$	0	0
$341$	− 8.63209e7i	− 0.117890i
$342$	0	0
$343$	−7.27694e8	−0.973686
$344$	0	0
$345$	−1.07768e9	−1.41293
$346$	0	0
$347$	2.75125e8i	0.353490i	0.984257	+	0.176745i	$0.0565568\pi$
$347$	2.75125e8i	0.353490i	−0.984257	+	0.176745i	$0.943443\pi$
$348$	0	0
$349$	− 1.32765e9i	− 1.67184i	−0.548854	−	0.835918i	$-0.684936\pi$
$349$	− 1.32765e9i	− 1.67184i	0.548854	−	0.835918i	$-0.315064\pi$
$350$	0	0
$351$	5.49180e7	0.0677860
$352$	0	0
$353$	1.24065e9	1.50120	0.750598	−	0.660759i	$-0.229765\pi$
$353$	1.24065e9	1.50120	0.750598	+	0.660759i	$0.229765\pi$
$354$	0	0
$355$	7.86513e8i	0.933054i
$356$	0	0
$357$	9.64622e8i	1.12207i
$358$	0	0
$359$	1.28760e9	1.46876	0.734381	−	0.678737i	$-0.237472\pi$
$359$	1.28760e9	1.46876	0.734381	+	0.678737i	$0.237472\pi$
$360$	0	0
$361$	−6.43889e8	−0.720338
$362$	0	0
$363$	1.05790e9i	1.16084i
$364$	0	0
$365$	− 1.17588e8i	− 0.126572i
$366$	0	0
$367$	4.96596e8	0.524412	0.262206	−	0.965012i	$-0.415550\pi$
$367$	4.96596e8	0.524412	0.262206	+	0.965012i	$0.415550\pi$
$368$	0	0
$369$	−1.22570e9	−1.26996
$370$	0	0
$371$	− 3.87713e8i	− 0.394187i
$372$	0	0
$373$	− 5.27113e8i	− 0.525923i	−0.964806	−	0.262962i	$-0.915301\pi$
$373$	− 5.27113e8i	− 0.525923i	0.964806	−	0.262962i	$-0.0846993\pi$
$374$	0	0
$375$	−1.45396e9	−1.42378
$376$	0	0
$377$	−4.36208e8	−0.419275
$378$	0	0
$379$	1.16631e9i	1.10047i	0.835011	+	0.550233i	$0.185461\pi$
$379$	1.16631e9i	1.10047i	−0.835011	+	0.550233i	$0.814539\pi$
$380$	0	0
$381$	2.37253e9i	2.19773i
$382$	0	0
$383$	−8.29099e8	−0.754068	−0.377034	−	0.926199i	$-0.623056\pi$
$383$	−8.29099e8	−0.754068	−0.377034	+	0.926199i	$0.623056\pi$
$384$	0	0
$385$	1.71409e8	0.153081
$386$	0	0
$387$	1.12902e9i	0.990177i
$388$	0	0
$389$	1.52571e9i	1.31416i	0.753820	+	0.657081i	$0.228209\pi$
$389$	1.52571e9i	1.31416i	−0.753820	+	0.657081i	$0.771791\pi$
$390$	0	0
$391$	2.47148e9	2.09092
$392$	0	0
$393$	−1.28802e9	−1.07040
$394$	0	0
$395$	− 3.90822e8i	− 0.319072i
$396$	0	0
$397$	− 4.58821e7i	− 0.0368024i	−0.999831	−	0.0184012i	$-0.994142\pi$
$397$	− 4.58821e7i	− 0.0368024i	0.999831	−	0.0184012i	$-0.00585762\pi$
$398$	0	0
$399$	1.30503e9	1.02852
$400$	0	0
$401$	3.83043e8	0.296648	0.148324	−	0.988939i	$-0.452612\pi$
$401$	3.83043e8	0.296648	0.148324	+	0.988939i	$0.452612\pi$
$402$	0	0
$403$	− 9.68204e7i	− 0.0736884i
$404$	0	0
$405$	− 1.14171e9i	− 0.854012i
$406$	0	0
$407$	−4.58121e8	−0.336822
$408$	0	0
$409$	1.39912e9	1.01117	0.505585	−	0.862777i	$-0.331277\pi$
$409$	1.39912e9	1.01117	0.505585	+	0.862777i	$0.331277\pi$
$410$	0	0
$411$	− 1.23689e9i	− 0.878786i
$412$	0	0
$413$	− 8.87226e7i	− 0.0619740i
$414$	0	0
$415$	−9.87205e8	−0.678014
$416$	0	0
$417$	−3.57284e9	−2.41289
$418$	0	0
$419$	1.40633e9i	0.933984i	0.884261	+	0.466992i	$0.154662\pi$
$419$	1.40633e9i	0.933984i	−0.884261	+	0.466992i	$0.845338\pi$
$420$	0	0
$421$	2.69826e9i	1.76237i	0.472772	+	0.881185i	$0.343253\pi$
$421$	2.69826e9i	1.76237i	−0.472772	+	0.881185i	$0.656747\pi$
$422$	0	0
$423$	7.25700e8	0.466193
$424$	0	0
$425$	1.06993e9	0.676072
$426$	0	0
$427$	− 1.42588e9i	− 0.886310i
$428$	0	0
$429$	− 1.74234e8i	− 0.106545i
$430$	0	0
$431$	1.53153e9	0.921415	0.460707	−	0.887552i	$-0.347596\pi$
$431$	1.53153e9	0.921415	0.460707	+	0.887552i	$0.347596\pi$
$432$	0	0
$433$	−1.09800e9	−0.649975	−0.324987	−	0.945718i	$-0.605360\pi$
$433$	−1.09800e9	−0.649975	−0.324987	+	0.945718i	$0.605360\pi$
$434$	0	0
$435$	3.11182e9i	1.81260i
$436$	0	0
$437$	− 3.34364e9i	− 1.91661i
$438$	0	0
$439$	2.20920e9	1.24626	0.623132	−	0.782117i	$-0.285860\pi$
$439$	2.20920e9	1.24626	0.623132	+	0.782117i	$0.285860\pi$
$440$	0	0
$441$	9.08436e8	0.504381
$442$	0	0
$443$	8.92969e8i	0.488004i	0.969775	+	0.244002i	$0.0784604\pi$
$443$	8.92969e8i	0.488004i	−0.969775	+	0.244002i	$0.921540\pi$
$444$	0	0
$445$	− 9.31120e8i	− 0.500894i
$446$	0	0
$447$	3.49458e9	1.85063
$448$	0	0
$449$	2.50310e9	1.30502	0.652509	−	0.757781i	$-0.273717\pi$
$449$	2.50310e9	1.30502	0.652509	+	0.757781i	$0.273717\pi$
$450$	0	0
$451$	− 1.14621e9i	− 0.588363i
$452$	0	0
$453$	3.55589e9i	1.79723i
$454$	0	0
$455$	1.92258e8	0.0956851
$456$	0	0
$457$	3.69849e9	1.81267	0.906333	−	0.422563i	$-0.138870\pi$
$457$	3.69849e9	1.81267	0.906333	+	0.422563i	$0.138870\pi$
$458$	0	0
$459$	8.98471e8i	0.433670i
$460$	0	0
$461$	− 1.58606e9i	− 0.753990i	−0.926215	−	0.376995i	$-0.876957\pi$
$461$	− 1.58606e9i	− 0.753990i	0.926215	−	0.376995i	$-0.123043\pi$
$462$	0	0
$463$	−1.21797e9	−0.570298	−0.285149	−	0.958483i	$-0.592043\pi$
$463$	−1.21797e9	−0.570298	−0.285149	+	0.958483i	$0.592043\pi$
$464$	0	0
$465$	−6.90698e8	−0.318568
$466$	0	0
$467$	1.27823e9i	0.580766i	0.956911	+	0.290383i	$0.0937826\pi$
$467$	1.27823e9i	0.580766i	−0.956911	+	0.290383i	$0.906217\pi$
$468$	0	0
$469$	6.21549e8i	0.278209i
$470$	0	0
$471$	1.32508e9	0.584343
$472$	0	0
$473$	−1.05580e9	−0.458741
$474$	0	0
$475$	− 1.44749e9i	− 0.619710i
$476$	0	0
$477$	1.22517e9i	0.516870i
$478$	0	0
$479$	−2.51593e9	−1.04598	−0.522991	−	0.852338i	$-0.675184\pi$
$479$	−2.51593e9	−1.04598	−0.522991	+	0.852338i	$0.675184\pi$
$480$	0	0
$481$	−5.13845e8	−0.210535
$482$	0	0
$483$	2.83759e9i	1.14587i
$484$	0	0
$485$	− 8.49147e8i	− 0.337977i
$486$	0	0
$487$	3.49252e9	1.37021	0.685106	−	0.728443i	$-0.259756\pi$
$487$	3.49252e9	1.37021	0.685106	+	0.728443i	$0.259756\pi$
$488$	0	0
$489$	−1.43472e9	−0.554864
$490$	0	0
$491$	4.34800e9i	1.65769i	0.559475	+	0.828847i	$0.311003\pi$
$491$	4.34800e9i	1.65769i	−0.559475	+	0.828847i	$0.688997\pi$
$492$	0	0
$493$	− 7.13646e9i	− 2.68237i
$494$	0	0
$495$	−5.41649e8	−0.200724
$496$	0	0
$497$	2.07094e9	0.756693
$498$	0	0
$499$	1.73130e9i	0.623763i	0.950121	+	0.311882i	$0.100959\pi$
$499$	1.73130e9i	0.623763i	−0.950121	+	0.311882i	$0.899041\pi$
$500$	0	0
$501$	− 4.24519e9i	− 1.50822i
$502$	0	0
$503$	1.79236e9	0.627967	0.313984	−	0.949428i	$-0.398336\pi$
$503$	1.79236e9	0.627967	0.313984	+	0.949428i	$0.398336\pi$
$504$	0	0
$505$	9.85030e8	0.340353
$506$	0	0
$507$	3.71120e9i	1.26470i
$508$	0	0
$509$	− 1.51265e9i	− 0.508423i	−0.967149	−	0.254211i	$-0.918184\pi$
$509$	− 1.51265e9i	− 0.508423i	0.967149	−	0.254211i	$-0.0818159\pi$
$510$	0	0
$511$	−3.09617e8	−0.102648
$512$	0	0
$513$	1.21553e9	0.397517
$514$	0	0
$515$	− 2.66220e9i	− 0.858847i
$516$	0	0
$517$	6.78638e8i	0.215984i
$518$	0	0
$519$	8.53036e8	0.267844
$520$	0	0
$521$	3.48250e8	0.107884	0.0539422	−	0.998544i	$-0.482821\pi$
$521$	3.48250e8	0.107884	0.0539422	+	0.998544i	$0.482821\pi$
$522$	0	0
$523$	− 2.73487e9i	− 0.835950i	−0.908458	−	0.417975i	$-0.862740\pi$
$523$	− 2.73487e9i	− 0.835950i	0.908458	−	0.417975i	$-0.137260\pi$
$524$	0	0
$525$	1.22842e9i	0.370501i
$526$	0	0
$527$	1.58400e9	0.471432
$528$	0	0
$529$	3.86542e9	1.13528
$530$	0	0
$531$	2.80362e8i	0.0812622i
$532$	0	0
$533$	− 1.28563e9i	− 0.367764i
$534$	0	0
$535$	4.15156e8	0.117212
$536$	0	0
$537$	2.12519e8	0.0592225
$538$	0	0
$539$	8.49523e8i	0.233676i
$540$	0	0
$541$	2.93847e9i	0.797866i	0.916980	+	0.398933i	$0.130619\pi$
$541$	2.93847e9i	0.797866i	−0.916980	+	0.398933i	$0.869381\pi$
$542$	0	0
$543$	4.14240e9	1.11033
$544$	0	0
$545$	−1.85810e9	−0.491679
$546$	0	0
$547$	5.18061e9i	1.35340i	0.736260	+	0.676699i	$0.236590\pi$
$547$	5.18061e9i	1.35340i	−0.736260	+	0.676699i	$0.763410\pi$
$548$	0	0
$549$	4.50576e9i	1.16216i
$550$	0	0
$551$	−9.65483e9	−2.45875
$552$	0	0
$553$	−1.02906e9	−0.258763
$554$	0	0
$555$	3.66566e9i	0.910180i
$556$	0	0
$557$	2.69427e9i	0.660615i	0.943873	+	0.330307i	$0.107152\pi$
$557$	2.69427e9i	0.660615i	−0.943873	+	0.330307i	$0.892848\pi$
$558$	0	0
$559$	−1.18422e9	−0.286742
$560$	0	0
$561$	2.85051e9	0.681636
$562$	0	0
$563$	− 5.45741e9i	− 1.28886i	−0.764661	−	0.644432i	$-0.777094\pi$
$563$	− 5.45741e9i	− 1.28886i	0.764661	−	0.644432i	$-0.222906\pi$
$564$	0	0
$565$	− 3.56347e9i	− 0.831195i
$566$	0	0
$567$	−3.00620e9	−0.692590
$568$	0	0
$569$	−3.17326e9	−0.722125	−0.361063	−	0.932542i	$-0.617586\pi$
$569$	−3.17326e9	−0.722125	−0.361063	+	0.932542i	$0.617586\pi$
$570$	0	0
$571$	− 2.68017e8i	− 0.0602471i	−0.999546	−	0.0301236i	$-0.990410\pi$
$571$	− 2.68017e8i	− 0.0602471i	0.999546	−	0.0301236i	$-0.00959007\pi$
$572$	0	0
$573$	1.05555e10i	2.34388i
$574$	0	0
$575$	3.14736e9	0.690413
$576$	0	0
$577$	−8.47388e9	−1.83640	−0.918199	−	0.396119i	$-0.870357\pi$
$577$	−8.47388e9	−1.83640	−0.918199	+	0.396119i	$0.870357\pi$
$578$	0	0
$579$	− 1.10584e10i	− 2.36764i
$580$	0	0
$581$	2.59937e9i	0.549859i
$582$	0	0
$583$	−1.14571e9	−0.239462
$584$	0	0
$585$	−6.07532e8	−0.125465
$586$	0	0
$587$	− 8.85508e9i	− 1.80700i	−0.428584	−	0.903502i	$-0.640987\pi$
$587$	− 8.85508e9i	− 1.80700i	0.428584	−	0.903502i	$-0.359013\pi$
$588$	0	0
$589$	− 2.14298e9i	− 0.432131i
$590$	0	0
$591$	−4.65586e8	−0.0927777
$592$	0	0
$593$	−3.89886e9	−0.767796	−0.383898	−	0.923375i	$-0.625419\pi$
$593$	−3.89886e9	−0.767796	−0.383898	+	0.923375i	$0.625419\pi$
$594$	0	0
$595$	3.14538e9i	0.612159i
$596$	0	0
$597$	− 1.01744e10i	− 1.95704i
$598$	0	0
$599$	1.95793e9	0.372224	0.186112	−	0.982529i	$-0.440411\pi$
$599$	1.95793e9	0.372224	0.186112	+	0.982529i	$0.440411\pi$
$600$	0	0
$601$	4.25172e8	0.0798920	0.0399460	−	0.999202i	$-0.487281\pi$
$601$	4.25172e8	0.0798920	0.0399460	+	0.999202i	$0.487281\pi$
$602$	0	0
$603$	− 1.96409e9i	− 0.364796i
$604$	0	0
$605$	3.44955e9i	0.633313i
$606$	0	0
$607$	4.98918e9	0.905459	0.452729	−	0.891648i	$-0.350450\pi$
$607$	4.98918e9	0.905459	0.452729	+	0.891648i	$0.350450\pi$
$608$	0	0
$609$	8.19361e9	1.46999
$610$	0	0
$611$	7.61183e8i	0.135004i
$612$	0	0
$613$	9.35346e9i	1.64006i	0.572318	+	0.820032i	$0.306044\pi$
$613$	9.35346e9i	1.64006i	−0.572318	+	0.820032i	$0.693956\pi$
$614$	0	0
$615$	−9.17140e9	−1.58991
$616$	0	0
$617$	−4.29392e9	−0.735963	−0.367982	−	0.929833i	$-0.619951\pi$
$617$	−4.29392e9	−0.735963	−0.367982	+	0.929833i	$0.619951\pi$
$618$	0	0
$619$	2.13759e9i	0.362249i	0.983460	+	0.181124i	$0.0579737\pi$
$619$	2.13759e9i	0.362249i	−0.983460	+	0.181124i	$0.942026\pi$
$620$	0	0
$621$	2.64300e9i	0.442870i
$622$	0	0
$623$	−2.45170e9	−0.406217
$624$	0	0
$625$	−1.85721e9	−0.304286
$626$	0	0
$627$	− 3.85642e9i	− 0.624811i
$628$	0	0
$629$	− 8.40662e9i	− 1.34693i
$630$	0	0
$631$	1.18683e10	1.88056	0.940279	−	0.340405i	$-0.110564\pi$
$631$	1.18683e10	1.88056	0.940279	+	0.340405i	$0.110564\pi$
$632$	0	0
$633$	−1.32316e10	−2.07347
$634$	0	0
$635$	7.73620e9i	1.19900i
$636$	0	0
$637$	9.52854e8i	0.146062i
$638$	0	0
$639$	−6.54413e9	−0.992199
$640$	0	0
$641$	3.07897e9	0.461746	0.230873	−	0.972984i	$-0.425842\pi$
$641$	3.07897e9	0.461746	0.230873	+	0.972984i	$0.425842\pi$
$642$	0	0
$643$	− 1.68440e9i	− 0.249866i	−0.992165	−	0.124933i	$-0.960128\pi$
$643$	− 1.68440e9i	− 0.249866i	0.992165	−	0.124933i	$-0.0398716\pi$
$644$	0	0
$645$	8.44800e9i	1.23964i
$646$	0	0
$647$	−1.22431e10	−1.77717	−0.888583	−	0.458717i	$-0.848309\pi$
$647$	−1.22431e10	−1.77717	−0.888583	+	0.458717i	$0.848309\pi$
$648$	0	0
$649$	−2.62180e8	−0.0376481
$650$	0	0
$651$	1.81865e9i	0.258354i
$652$	0	0
$653$	8.42587e8i	0.118418i	0.998246	+	0.0592092i	$0.0188579\pi$
$653$	8.42587e8i	0.118418i	−0.998246	+	0.0592092i	$0.981142\pi$
$654$	0	0
$655$	−4.19989e9	−0.583974
$656$	0	0
$657$	9.78384e8	0.134595
$658$	0	0
$659$	− 1.45717e10i	− 1.98341i	−0.128537	−	0.991705i	$-0.541028\pi$
$659$	− 1.45717e10i	− 1.98341i	0.128537	−	0.991705i	$-0.458972\pi$
$660$	0	0
$661$	3.88432e9i	0.523130i	0.965186	+	0.261565i	$0.0842386\pi$
$661$	3.88432e9i	0.523130i	−0.965186	+	0.261565i	$0.915761\pi$
$662$	0	0
$663$	3.19723e9	0.426066
$664$	0	0
$665$	4.25535e9	0.561125
$666$	0	0
$667$	− 2.09930e10i	− 2.73927i
$668$	0	0
$669$	− 1.51051e10i	− 1.95044i
$670$	0	0
$671$	−4.21356e9	−0.538418
$672$	0	0
$673$	−1.67232e9	−0.211479	−0.105739	−	0.994394i	$-0.533721\pi$
$673$	−1.67232e9	−0.211479	−0.105739	+	0.994394i	$0.533721\pi$
$674$	0	0
$675$	1.14418e9i	0.143196i
$676$	0	0
$677$	2.25476e8i	0.0279280i	0.999902	+	0.0139640i	$0.00444503\pi$
$677$	2.25476e8i	0.0279280i	−0.999902	+	0.0139640i	$0.995555\pi$
$678$	0	0
$679$	−2.23586e9	−0.274094
$680$	0	0
$681$	−1.02007e9	−0.123770
$682$	0	0
$683$	4.61244e8i	0.0553934i	0.999616	+	0.0276967i	$0.00881727\pi$
$683$	4.61244e8i	0.0553934i	−0.999616	+	0.0276967i	$0.991183\pi$
$684$	0	0
$685$	− 4.03316e9i	− 0.479434i
$686$	0	0
$687$	−1.43459e10	−1.68803
$688$	0	0
$689$	−1.28507e9	−0.149679
$690$	0	0
$691$	3.53967e9i	0.408122i	0.978958	+	0.204061i	$0.0654141\pi$
$691$	3.53967e9i	0.408122i	−0.978958	+	0.204061i	$0.934586\pi$
$692$	0	0
$693$	1.42619e9i	0.162784i
$694$	0	0
$695$	−1.16501e10	−1.31639
$696$	0	0
$697$	2.10331e10	2.35282
$698$	0	0
$699$	5.93204e9i	0.656952i
$700$	0	0
$701$	7.06039e8i	0.0774134i	0.999251	+	0.0387067i	$0.0123238\pi$
$701$	7.06039e8i	0.0774134i	−0.999251	+	0.0387067i	$0.987676\pi$
$702$	0	0
$703$	−1.13732e10	−1.23464
$704$	0	0
$705$	5.43013e9	0.583644
$706$	0	0
$707$	− 2.59364e9i	− 0.276021i
$708$	0	0
$709$	− 1.28048e9i	− 0.134931i	−0.997722	−	0.0674655i	$-0.978509\pi$
$709$	− 1.28048e9i	− 0.134931i	0.997722	−	0.0674655i	$-0.0214912\pi$
$710$	0	0
$711$	3.25181e9	0.339298
$712$	0	0
$713$	4.65960e9	0.481432
$714$	0	0
$715$	− 5.68133e8i	− 0.0581271i
$716$	0	0
$717$	− 1.15702e10i	− 1.17226i
$718$	0	0
$719$	1.73237e9	0.173816	0.0869079	−	0.996216i	$-0.472301\pi$
$719$	1.73237e9	0.173816	0.0869079	+	0.996216i	$0.472301\pi$
$720$	0	0
$721$	−7.00975e9	−0.696512
$722$	0	0
$723$	− 6.55274e6i	0 0.000644821i
$724$	0	0
$725$	− 9.08808e9i	− 0.885705i
$726$	0	0
$727$	−3.24560e9	−0.313274	−0.156637	−	0.987656i	$-0.550065\pi$
$727$	−3.24560e9	−0.313274	−0.156637	+	0.987656i	$0.550065\pi$
$728$	0	0
$729$	5.27898e9	0.504665
$730$	0	0
$731$	− 1.93741e10i	− 1.83447i
$732$	0	0
$733$	7.80499e9i	0.731996i	0.930616	+	0.365998i	$0.119272\pi$
$733$	7.80499e9i	0.731996i	−0.930616	+	0.365998i	$0.880728\pi$
$734$	0	0
$735$	6.79747e9	0.631454
$736$	0	0
$737$	1.83671e9	0.169007
$738$	0	0
$739$	2.03879e10i	1.85831i	0.369694	+	0.929154i	$0.379463\pi$
$739$	2.03879e10i	1.85831i	−0.369694	+	0.929154i	$0.620537\pi$
$740$	0	0
$741$	− 4.32550e9i	− 0.390546i
$742$	0	0
$743$	6.44304e9	0.576275	0.288137	−	0.957589i	$-0.406964\pi$
$743$	6.44304e9	0.576275	0.288137	+	0.957589i	$0.406964\pi$
$744$	0	0
$745$	1.13949e10	1.00964
$746$	0	0
$747$	− 8.21397e9i	− 0.720993i
$748$	0	0
$749$	− 1.09313e9i	− 0.0950573i
$750$	0	0
$751$	4.02921e8	0.0347120	0.0173560	−	0.999849i	$-0.494475\pi$
$751$	4.02921e8	0.0347120	0.0173560	+	0.999849i	$0.494475\pi$
$752$	0	0
$753$	−2.26817e9	−0.193595
$754$	0	0
$755$	1.15948e10i	0.980505i
$756$	0	0
$757$	3.77299e9i	0.316119i	0.987430	+	0.158059i	$0.0505237\pi$
$757$	3.77299e9i	0.316119i	−0.987430	+	0.158059i	$0.949476\pi$
$758$	0	0
$759$	8.38523e9	0.696095
$760$	0	0
$761$	−2.52431e9	−0.207633	−0.103817	−	0.994596i	$-0.533106\pi$
$761$	−2.52431e9	−0.207633	−0.103817	+	0.994596i	$0.533106\pi$
$762$	0	0
$763$	4.89249e9i	0.398744i
$764$	0	0
$765$	− 9.93936e9i	− 0.802682i
$766$	0	0
$767$	−2.94070e8	−0.0235325
$768$	0	0
$769$	−5.59413e9	−0.443599	−0.221799	−	0.975092i	$-0.571193\pi$
$769$	−5.59413e9	−0.443599	−0.221799	+	0.975092i	$0.571193\pi$
$770$	0	0
$771$	− 1.75595e10i	− 1.37982i
$772$	0	0
$773$	9.30520e9i	0.724599i	0.932062	+	0.362299i	$0.118008\pi$
$773$	9.30520e9i	0.724599i	−0.932062	+	0.362299i	$0.881992\pi$
$774$	0	0
$775$	2.01719e9	0.155665
$776$	0	0
$777$	9.65192e9	0.738142
$778$	0	0
$779$	− 2.84555e10i	− 2.15668i
$780$	0	0
$781$	− 6.11974e9i	− 0.459678i
$782$	0	0
$783$	7.63172e9	0.568141
$784$	0	0
$785$	4.32073e9	0.318797
$786$	0	0
$787$	7.70711e9i	0.563612i	0.959471	+	0.281806i	$0.0909334\pi$
$787$	7.70711e9i	0.563612i	−0.959471	+	0.281806i	$0.909067\pi$
$788$	0	0
$789$	− 1.48776e10i	− 1.07836i
$790$	0	0
$791$	−9.38282e9	−0.674087
$792$	0	0
$793$	−4.72607e9	−0.336546
$794$	0	0
$795$	9.16745e9i	0.647088i
$796$	0	0
$797$	1.88557e10i	1.31928i	0.751581	+	0.659641i	$0.229292\pi$
$797$	1.88557e10i	1.31928i	−0.751581	+	0.659641i	$0.770708\pi$
$798$	0	0
$799$	−1.24531e10	−0.863704
$800$	0	0
$801$	7.74732e9	0.532645
$802$	0	0
$803$	9.14934e8i	0.0623570i
$804$	0	0
$805$	9.25264e9i	0.625144i
$806$	0	0
$807$	2.22819e10	1.49243
$808$	0	0
$809$	−7.30677e9	−0.485183	−0.242591	−	0.970129i	$-0.577997\pi$
$809$	−7.30677e9	−0.485183	−0.242591	+	0.970129i	$0.577997\pi$
$810$	0	0
$811$	1.71333e10i	1.12789i	0.825812	+	0.563945i	$0.190717\pi$
$811$	1.71333e10i	1.12789i	−0.825812	+	0.563945i	$0.809283\pi$
$812$	0	0
$813$	− 9.88588e9i	− 0.645206i
$814$	0	0
$815$	−4.67826e9	−0.302714
$816$	0	0
$817$	−2.62111e10	−1.68154
$818$	0	0
$819$	1.59967e9i	0.101750i
$820$	0	0
$821$	1.36575e10i	0.861331i	0.902512	+	0.430666i	$0.141721\pi$
$821$	1.36575e10i	0.861331i	−0.902512	+	0.430666i	$0.858279\pi$
$822$	0	0
$823$	7.25803e9	0.453858	0.226929	−	0.973911i	$-0.427132\pi$
$823$	7.25803e9	0.453858	0.226929	+	0.973911i	$0.427132\pi$
$824$	0	0
$825$	3.63005e9	0.225073
$826$	0	0
$827$	2.25663e10i	1.38736i	0.720281	+	0.693682i	$0.244013\pi$
$827$	2.25663e10i	1.38736i	−0.720281	+	0.693682i	$0.755987\pi$
$828$	0	0
$829$	− 1.61901e10i	− 0.986979i	−0.869752	−	0.493490i	$-0.835721\pi$
$829$	− 1.61901e10i	− 0.986979i	0.869752	−	0.493490i	$-0.164279\pi$
$830$	0	0
$831$	2.56021e10	1.54765
$832$	0	0
$833$	−1.55889e10	−0.934454
$834$	0	0
$835$	− 1.38425e10i	− 0.822831i
$836$	0	0
$837$	1.69393e9i	0.0998520i
$838$	0	0
$839$	8.41622e9	0.491983	0.245992	−	0.969272i	$-0.420886\pi$
$839$	8.41622e9	0.491983	0.245992	+	0.969272i	$0.420886\pi$
$840$	0	0
$841$	−4.33680e10	−2.51410
$842$	0	0
$843$	1.89923e10i	1.09190i
$844$	0	0
$845$	1.21013e10i	0.689974i
$846$	0	0
$847$	9.08287e9	0.513607
$848$	0	0
$849$	2.63183e10	1.47598
$850$	0	0
$851$	− 2.47294e10i	− 1.37550i
$852$	0	0
$853$	1.31544e10i	0.725687i	0.931850	+	0.362844i	$0.118194\pi$
$853$	1.31544e10i	0.725687i	−0.931850	+	0.362844i	$0.881806\pi$
$854$	0	0
$855$	−1.34469e10	−0.735765
$856$	0	0
$857$	−3.19572e10	−1.73435	−0.867174	−	0.498006i	$-0.834066\pi$
$857$	−3.19572e10	−1.73435	−0.867174	+	0.498006i	$0.834066\pi$
$858$	0	0
$859$	1.50626e9i	0.0810821i	0.999178	+	0.0405410i	$0.0129082\pi$
$859$	1.50626e9i	0.0810821i	−0.999178	+	0.0405410i	$0.987092\pi$
$860$	0	0
$861$	2.41489e10i	1.28939i
$862$	0	0
$863$	−1.34154e10	−0.710504	−0.355252	−	0.934771i	$-0.615605\pi$
$863$	−1.34154e10	−0.710504	−0.355252	+	0.934771i	$0.615605\pi$
$864$	0	0
$865$	2.78153e9	0.146126
$866$	0	0
$867$	2.67604e10i	1.39452i
$868$	0	0
$869$	3.04092e9i	0.157194i
$870$	0	0
$871$	2.06012e9	0.105640
$872$	0	0
$873$	7.06527e9	0.359401
$874$	0	0
$875$	1.24833e10i	0.629944i
$876$	0	0
$877$	1.79535e10i	0.898775i	0.893337	+	0.449387i	$0.148358\pi$
$877$	1.79535e10i	0.898775i	−0.893337	+	0.449387i	$0.851642\pi$
$878$	0	0
$879$	2.66168e10	1.32189
$880$	0	0
$881$	7.94700e9	0.391550	0.195775	−	0.980649i	$-0.437278\pi$
$881$	7.94700e9	0.391550	0.195775	+	0.980649i	$0.437278\pi$
$882$	0	0
$883$	1.90736e10i	0.932329i	0.884698	+	0.466165i	$0.154365\pi$
$883$	1.90736e10i	0.932329i	−0.884698	+	0.466165i	$0.845635\pi$
$884$	0	0
$885$	2.09784e9i	0.101735i
$886$	0	0
$887$	−1.73840e10	−0.836406	−0.418203	−	0.908354i	$-0.637340\pi$
$887$	−1.73840e10	−0.836406	−0.418203	+	0.908354i	$0.637340\pi$
$888$	0	0
$889$	2.03699e10	0.972372
$890$	0	0
$891$	8.88347e9i	0.420737i
$892$	0	0
$893$	1.68477e10i	0.791700i
$894$	0	0
$895$	6.92968e8	0.0323097
$896$	0	0
$897$	9.40516e9	0.435104
$898$	0	0
$899$	− 1.34547e10i	− 0.617612i
$900$	0	0
$901$	− 2.10241e10i	− 0.957591i
$902$	0	0
$903$	2.22441e10	1.00533
$904$	0	0
$905$	1.35073e10	0.605757
$906$	0	0
$907$	1.37564e10i	0.612180i	0.952003	+	0.306090i	$0.0990208\pi$
$907$	1.37564e10i	0.612180i	−0.952003	+	0.306090i	$0.900979\pi$
$908$	0	0
$909$	8.19587e9i	0.361928i
$910$	0	0
$911$	6.97126e9	0.305490	0.152745	−	0.988266i	$-0.451189\pi$
$911$	6.97126e9	0.305490	0.152745	+	0.988266i	$0.451189\pi$
$912$	0	0
$913$	7.68128e9	0.334030
$914$	0	0
$915$	3.37148e10i	1.45495i
$916$	0	0
$917$	1.10586e10i	0.473594i
$918$	0	0
$919$	−4.22655e9	−0.179631	−0.0898156	−	0.995958i	$-0.528628\pi$
$919$	−4.22655e9	−0.179631	−0.0898156	+	0.995958i	$0.528628\pi$
$920$	0	0
$921$	−1.56483e10	−0.660023
$922$	0	0
$923$	− 6.86410e9i	− 0.287328i
$924$	0	0
$925$	− 1.07056e10i	− 0.444749i
$926$	0	0
$927$	2.21507e10	0.913289
$928$	0	0
$929$	−1.34132e9	−0.0548881	−0.0274441	−	0.999623i	$-0.508737\pi$
$929$	−1.34132e9	−0.0548881	−0.0274441	+	0.999623i	$0.508737\pi$
$930$	0	0
$931$	2.10901e10i	0.856552i
$932$	0	0
$933$	− 5.57539e10i	− 2.24745i
$934$	0	0
$935$	9.29478e9	0.371876
$936$	0	0
$937$	−1.84420e10	−0.732353	−0.366176	−	0.930545i	$-0.619333\pi$
$937$	−1.84420e10	−0.732353	−0.366176	+	0.930545i	$0.619333\pi$
$938$	0	0
$939$	3.25612e10i	1.28343i
$940$	0	0
$941$	2.73596e10i	1.07040i	0.844725	+	0.535201i	$0.179764\pi$
$941$	2.73596e10i	1.07040i	−0.844725	+	0.535201i	$0.820236\pi$
$942$	0	0
$943$	6.18723e10	2.40273
$944$	0	0
$945$	−3.36367e9	−0.129659
$946$	0	0
$947$	− 9.65273e9i	− 0.369339i	−0.982801	−	0.184670i	$-0.940879\pi$
$947$	− 9.65273e9i	− 0.369339i	0.982801	−	0.184670i	$-0.0591215\pi$
$948$	0	0
$949$	1.02622e9i	0.0389771i
$950$	0	0
$951$	−4.39614e10	−1.65745
$952$	0	0
$953$	−2.34107e10	−0.876170	−0.438085	−	0.898934i	$-0.644343\pi$
$953$	−2.34107e10	−0.876170	−0.438085	+	0.898934i	$0.644343\pi$
$954$	0	0
$955$	3.44187e10i	1.27874i
$956$	0	0
$957$	− 2.42126e10i	− 0.892995i
$958$	0	0
$959$	−1.06196e10	−0.388814
$960$	0	0
$961$	−2.45262e10	−0.891453
$962$	0	0
$963$	3.45427e9i	0.124642i
$964$	0	0
$965$	− 3.60585e10i	− 1.29170i
$966$	0	0
$967$	4.12588e10	1.46732	0.733658	−	0.679519i	$-0.237812\pi$
$967$	4.12588e10	1.46732	0.733658	+	0.679519i	$0.237812\pi$
$968$	0	0
$969$	7.07661e10	2.49857
$970$	0	0
$971$	− 2.00506e10i	− 0.702846i	−0.936217	−	0.351423i	$-0.885698\pi$
$971$	− 2.00506e10i	− 0.702846i	0.936217	−	0.351423i	$-0.114302\pi$
$972$	0	0
$973$	3.06755e10i	1.06757i
$974$	0	0
$975$	4.07159e9	0.140685
$976$	0	0
$977$	1.24784e10	0.428084	0.214042	−	0.976824i	$-0.431337\pi$
$977$	1.24784e10	0.428084	0.214042	+	0.976824i	$0.431337\pi$
$978$	0	0
$979$	7.24490e9i	0.246770i
$980$	0	0
$981$	− 1.54602e10i	− 0.522846i
$982$	0	0
$983$	−7.76481e9	−0.260732	−0.130366	−	0.991466i	$-0.541615\pi$
$983$	−7.76481e9	−0.260732	−0.130366	+	0.991466i	$0.541615\pi$
$984$	0	0
$985$	−1.51816e9	−0.0506162
$986$	0	0
$987$	− 1.42979e10i	− 0.473327i
$988$	0	0
$989$	− 5.69921e10i	− 1.87339i
$990$	0	0
$991$	6.53111e9	0.213171	0.106586	−	0.994304i	$-0.466008\pi$
$991$	6.53111e9	0.213171	0.106586	+	0.994304i	$0.466008\pi$
$992$	0	0
$993$	−1.23366e10	−0.399829
$994$	0	0
$995$	− 3.31762e10i	− 1.06769i
$996$	0	0
$997$	4.43782e10i	1.41820i	0.705110	+	0.709098i	$0.250898\pi$
$997$	4.43782e10i	1.41820i	−0.705110	+	0.709098i	$0.749102\pi$
$998$	0	0
$999$	8.99002e9	0.285287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.l.129.6		6
4.3	odd	2		256.8.b.k.129.1		6
8.3	odd	2		256.8.b.k.129.6		6
8.5	even	2	inner	256.8.b.l.129.1		6
16.3	odd	4		128.8.a.c.1.3	yes	3
16.5	even	4		128.8.a.d.1.3	yes	3
16.11	odd	4		128.8.a.b.1.1	yes	3
16.13	even	4		128.8.a.a.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.a.a.1.1	✓	3	16.13	even	4
128.8.a.b.1.1	yes	3	16.11	odd	4
128.8.a.c.1.3	yes	3	16.3	odd	4
128.8.a.d.1.3	yes	3	16.5	even	4
256.8.b.k.129.1		6	4.3	odd	2
256.8.b.k.129.6		6	8.3	odd	2
256.8.b.l.129.1		6	8.5	even	2	inner
256.8.b.l.129.6		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)

\(f(q)\)	\(=\)	\(q+62.2585i q^{3} +203.009i q^{5} +534.535 q^{7} -1689.12 q^{9} -1579.58i q^{11} -1771.71i q^{13} -12639.0 q^{15} +28985.6 q^{17} -39214.3i q^{19} +33279.3i q^{21} +85265.7 q^{23} +36912.4 q^{25} +30997.2i q^{27} -246207. i q^{29} +54648.0 q^{31} +98342.3 q^{33} +108515. i q^{35} -290027. i q^{37} +110304. q^{39} +725641. q^{41} -668406. i q^{43} -342907. i q^{45} -429632. q^{47} -537815. q^{49} +1.80460e6i q^{51} -725328. i q^{53} +320669. q^{55} +2.44142e6 q^{57} -165981. i q^{59} -2.66752e6i q^{61} -902895. q^{63} +359673. q^{65} +1.16279e6i q^{67} +5.30852e6i q^{69} +3.87428e6 q^{71} -579226. q^{73} +2.29811e6i q^{75} -844341. i q^{77} -1.92515e6 q^{79} -5.62395e6 q^{81} +4.86286e6i q^{83} +5.88434e6i q^{85} +1.53285e7 q^{87} -4.58660e6 q^{89} -947042. i q^{91} +3.40230e6i q^{93} +7.96085e6 q^{95} -4.18281e6 q^{97} +2.66810e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 106 q^{9} - 8232 q^{15} + 1076 q^{17} + 86632 q^{23} + 81206 q^{25} - 313856 q^{31} - 367000 q^{33} + 1243368 q^{39} + 1819396 q^{41} - 1705904 q^{47} - 3273066 q^{49} + 7411352 q^{55}+ \cdots - 10501388 q^{97}+O(q^{100}) \) 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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