LMFDB - Embedded newform 256.10.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,10,Mod(1,256)]

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([0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$131.849174058$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 1131x^{6} + 409780x^{4} - 50526912x^{2} + 1648955392 $ x^8 - 1131x^6 + 409780x^4 - 50526912*x^2 + 1648955392
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{52}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-7.11952$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+100.481 q^{3} -2583.09 q^{5} +6967.65 q^{7} -9586.48 q^{9} +O(q^{10})$	$q+100.481 q^{3} -2583.09 q^{5} +6967.65 q^{7} -9586.48 q^{9} +25054.5 q^{11} +29701.4 q^{13} -259553. q^{15} -138527. q^{17} -489569. q^{19} +700119. q^{21} +847079. q^{23} +4.71924e6 q^{25} -2.94104e6 q^{27} -1.13646e6 q^{29} -4.35747e6 q^{31} +2.51751e6 q^{33} -1.79981e7 q^{35} -535315. q^{37} +2.98444e6 q^{39} -1.45816e7 q^{41} -3.96441e7 q^{43} +2.47628e7 q^{45} +4.48997e7 q^{47} +8.19448e6 q^{49} -1.39194e7 q^{51} +4.85666e7 q^{53} -6.47180e7 q^{55} -4.91926e7 q^{57} -4.19685e6 q^{59} +6.38659e7 q^{61} -6.67952e7 q^{63} -7.67215e7 q^{65} +5.79621e7 q^{67} +8.51157e7 q^{69} +2.74912e8 q^{71} +9.16969e7 q^{73} +4.74196e8 q^{75} +1.74571e8 q^{77} -2.02396e8 q^{79} -1.06829e8 q^{81} +6.11053e8 q^{83} +3.57829e8 q^{85} -1.14193e8 q^{87} +7.71588e8 q^{89} +2.06949e8 q^{91} -4.37845e8 q^{93} +1.26460e9 q^{95} +1.08292e9 q^{97} -2.40184e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4800 q^{7} + 39368 q^{9}+O(q^{10}) $ 8 * q + 4800 * q^7 + 39368 * q^9	$ 8 q + 4800 q^{7} + 39368 q^{9} + 163136 q^{15} - 102000 q^{17} + 3412032 q^{23} + 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} - 17590208 q^{39} + 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} + 7056832 q^{55} - 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} + 560234688 q^{71} + 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} + 540527424 q^{87} - 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) $ 8 * q + 4800 * q^7 + 39368 * q^9 + 163136 * q^15 - 102000 * q^17 + 3412032 * q^23 + 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 - 17590208 * q^39 + 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 + 7056832 * q^55 - 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 + 560234688 * q^71 + 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 + 540527424 * q^87 - 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	100.481	0.716210	0.358105	−	0.933681i	$-0.383423\pi$
$3$	100.481	0.716210	0.358105	+	0.933681i	$0.383423\pi$
$4$	0	0
$5$	−2583.09	−1.84831	−0.924155	−	0.382017i	$-0.875230\pi$
$5$	−2583.09	−1.84831	−0.924155	+	0.382017i	$0.875230\pi$
$6$	0	0
$7$	6967.65	1.09684	0.548422	−	0.836202i	$-0.315229\pi$
$7$	6967.65	1.09684	0.548422	+	0.836202i	$0.315229\pi$
$8$	0	0
$9$	−9586.48	−0.487044
$10$	0	0
$11$	25054.5	0.515962	0.257981	−	0.966150i	$-0.416943\pi$
$11$	25054.5	0.515962	0.257981	+	0.966150i	$0.416943\pi$
$12$	0	0
$13$	29701.4	0.288425	0.144212	−	0.989547i	$-0.453935\pi$
$13$	29701.4	0.288425	0.144212	+	0.989547i	$0.453935\pi$
$14$	0	0
$15$	−259553.	−1.32378
$16$	0	0
$17$	−138527.	−0.402268	−0.201134	−	0.979564i	$-0.564463\pi$
$17$	−138527.	−0.402268	−0.201134	+	0.979564i	$0.564463\pi$
$18$	0	0
$19$	−489569.	−0.861832	−0.430916	−	0.902392i	$-0.641809\pi$
$19$	−489569.	−0.861832	−0.430916	+	0.902392i	$0.641809\pi$
$20$	0	0
$21$	700119.	0.785570
$22$	0	0
$23$	847079.	0.631173	0.315587	−	0.948897i	$-0.397799\pi$
$23$	847079.	0.631173	0.315587	+	0.948897i	$0.397799\pi$
$24$	0	0
$25$	4.71924e6	2.41625
$26$	0	0
$27$	−2.94104e6	−1.06504
$28$	0	0
$29$	−1.13646e6	−0.298376	−0.149188	−	0.988809i	$-0.547666\pi$
$29$	−1.13646e6	−0.298376	−0.149188	+	0.988809i	$0.547666\pi$
$30$	0	0
$31$	−4.35747e6	−0.847436	−0.423718	−	0.905794i	$-0.639275\pi$
$31$	−4.35747e6	−0.847436	−0.423718	+	0.905794i	$0.639275\pi$
$32$	0	0
$33$	2.51751e6	0.369537
$34$	0	0
$35$	−1.79981e7	−2.02731
$36$	0	0
$37$	−535315.	−0.0469571	−0.0234786	−	0.999724i	$-0.507474\pi$
$37$	−535315.	−0.0469571	−0.0234786	+	0.999724i	$0.507474\pi$
$38$	0	0
$39$	2.98444e6	0.206572
$40$	0	0
$41$	−1.45816e7	−0.805894	−0.402947	−	0.915223i	$-0.632014\pi$
$41$	−1.45816e7	−0.805894	−0.402947	+	0.915223i	$0.632014\pi$
$42$	0	0
$43$	−3.96441e7	−1.76836	−0.884179	−	0.467148i	$-0.845281\pi$
$43$	−3.96441e7	−1.76836	−0.884179	+	0.467148i	$0.845281\pi$
$44$	0	0
$45$	2.47628e7	0.900208
$46$	0	0
$47$	4.48997e7	1.34216	0.671078	−	0.741387i	$-0.265832\pi$
$47$	4.48997e7	1.34216	0.671078	+	0.741387i	$0.265832\pi$
$48$	0	0
$49$	8.19448e6	0.203067
$50$	0	0
$51$	−1.39194e7	−0.288108
$52$	0	0
$53$	4.85666e7	0.845466	0.422733	−	0.906254i	$-0.361071\pi$
$53$	4.85666e7	0.845466	0.422733	+	0.906254i	$0.361071\pi$
$54$	0	0
$55$	−6.47180e7	−0.953658
$56$	0	0
$57$	−4.91926e7	−0.617252
$58$	0	0
$59$	−4.19685e6	−0.0450910	−0.0225455	−	0.999746i	$-0.507177\pi$
$59$	−4.19685e6	−0.0450910	−0.0225455	+	0.999746i	$0.507177\pi$
$60$	0	0
$61$	6.38659e7	0.590588	0.295294	−	0.955406i	$-0.404582\pi$
$61$	6.38659e7	0.590588	0.295294	+	0.955406i	$0.404582\pi$
$62$	0	0
$63$	−6.67952e7	−0.534211
$64$	0	0
$65$	−7.67215e7	−0.533098
$66$	0	0
$67$	5.79621e7	0.351404	0.175702	−	0.984443i	$-0.443780\pi$
$67$	5.79621e7	0.351404	0.175702	+	0.984443i	$0.443780\pi$
$68$	0	0
$69$	8.51157e7	0.452052
$70$	0	0
$71$	2.74912e8	1.28390	0.641950	−	0.766746i	$-0.278125\pi$
$71$	2.74912e8	1.28390	0.641950	+	0.766746i	$0.278125\pi$
$72$	0	0
$73$	9.16969e7	0.377921	0.188961	−	0.981985i	$-0.439488\pi$
$73$	9.16969e7	0.377921	0.188961	+	0.981985i	$0.439488\pi$
$74$	0	0
$75$	4.74196e8	1.73054
$76$	0	0
$77$	1.74571e8	0.565930
$78$	0	0
$79$	−2.02396e8	−0.584627	−0.292314	−	0.956322i	$-0.594425\pi$
$79$	−2.02396e8	−0.584627	−0.292314	+	0.956322i	$0.594425\pi$
$80$	0	0
$81$	−1.06829e8	−0.275744
$82$	0	0
$83$	6.11053e8	1.41328	0.706639	−	0.707574i	$-0.250210\pi$
$83$	6.11053e8	1.41328	0.706639	+	0.707574i	$0.250210\pi$
$84$	0	0
$85$	3.57829e8	0.743516
$86$	0	0
$87$	−1.14193e8	−0.213700
$88$	0	0
$89$	7.71588e8	1.30356	0.651779	−	0.758409i	$-0.274023\pi$
$89$	7.71588e8	1.30356	0.651779	+	0.758409i	$0.274023\pi$
$90$	0	0
$91$	2.06949e8	0.316357
$92$	0	0
$93$	−4.37845e8	−0.606942
$94$	0	0
$95$	1.26460e9	1.59293
$96$	0	0
$97$	1.08292e9	1.24200	0.621001	−	0.783810i	$-0.286726\pi$
$97$	1.08292e9	1.24200	0.621001	+	0.783810i	$0.286726\pi$
$98$	0	0
$99$	−2.40184e8	−0.251296
$100$	0	0
$101$	−6.22244e8	−0.594996	−0.297498	−	0.954722i	$-0.596152\pi$
$101$	−6.22244e8	−0.594996	−0.297498	+	0.954722i	$0.596152\pi$
$102$	0	0
$103$	6.34170e8	0.555186	0.277593	−	0.960699i	$-0.410463\pi$
$103$	6.34170e8	0.555186	0.277593	+	0.960699i	$0.410463\pi$
$104$	0	0
$105$	−1.80847e9	−1.45198
$106$	0	0
$107$	−1.54145e9	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$107$	−1.54145e9	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$108$	0	0
$109$	2.19786e9	1.49135	0.745676	−	0.666309i	$-0.232127\pi$
$109$	2.19786e9	1.49135	0.745676	+	0.666309i	$0.232127\pi$
$110$	0	0
$111$	−5.37892e7	−0.0336311
$112$	0	0
$113$	−5.43476e8	−0.313565	−0.156782	−	0.987633i	$-0.550112\pi$
$113$	−5.43476e8	−0.313565	−0.156782	+	0.987633i	$0.550112\pi$
$114$	0	0
$115$	−2.18808e9	−1.16660
$116$	0	0
$117$	−2.84732e8	−0.140475
$118$	0	0
$119$	−9.65210e8	−0.441225
$120$	0	0
$121$	−1.73022e9	−0.733783
$122$	0	0
$123$	−1.46518e9	−0.577189
$124$	0	0
$125$	−7.14513e9	−2.61767
$126$	0	0
$127$	2.41305e9	0.823093	0.411547	−	0.911389i	$-0.364989\pi$
$127$	2.41305e9	0.823093	0.411547	+	0.911389i	$0.364989\pi$
$128$	0	0
$129$	−3.98349e9	−1.26652
$130$	0	0
$131$	−1.57746e9	−0.467990	−0.233995	−	0.972238i	$-0.575180\pi$
$131$	−1.57746e9	−0.467990	−0.233995	+	0.972238i	$0.575180\pi$
$132$	0	0
$133$	−3.41114e9	−0.945295
$134$	0	0
$135$	7.59698e9	1.96852
$136$	0	0
$137$	6.39076e9	1.54992	0.774961	−	0.632009i	$-0.217769\pi$
$137$	6.39076e9	1.54992	0.774961	+	0.632009i	$0.217769\pi$
$138$	0	0
$139$	−2.45897e9	−0.558710	−0.279355	−	0.960188i	$-0.590121\pi$
$139$	−2.45897e9	−0.558710	−0.279355	+	0.960188i	$0.590121\pi$
$140$	0	0
$141$	4.51158e9	0.961265
$142$	0	0
$143$	7.44153e8	0.148816
$144$	0	0
$145$	2.93559e9	0.551492
$146$	0	0
$147$	8.23393e8	0.145438
$148$	0	0
$149$	3.84540e9	0.639150	0.319575	−	0.947561i	$-0.396460\pi$
$149$	3.84540e9	0.639150	0.319575	+	0.947561i	$0.396460\pi$
$150$	0	0
$151$	4.07627e9	0.638067	0.319034	−	0.947743i	$-0.396642\pi$
$151$	4.07627e9	0.638067	0.319034	+	0.947743i	$0.396642\pi$
$152$	0	0
$153$	1.32799e9	0.195922
$154$	0	0
$155$	1.12557e10	1.56632
$156$	0	0
$157$	−7.59409e9	−0.997533	−0.498767	−	0.866736i	$-0.666214\pi$
$157$	−7.59409e9	−0.997533	−0.498767	+	0.866736i	$0.666214\pi$
$158$	0	0
$159$	4.88004e9	0.605531
$160$	0	0
$161$	5.90214e9	0.692298
$162$	0	0
$163$	7.71277e9	0.855788	0.427894	−	0.903829i	$-0.359256\pi$
$163$	7.71277e9	0.855788	0.427894	+	0.903829i	$0.359256\pi$
$164$	0	0
$165$	−6.50295e9	−0.683019
$166$	0	0
$167$	1.01170e10	1.00653	0.503265	−	0.864132i	$-0.332132\pi$
$167$	1.01170e10	1.00653	0.503265	+	0.864132i	$0.332132\pi$
$168$	0	0
$169$	−9.72232e9	−0.916811
$170$	0	0
$171$	4.69324e9	0.419750
$172$	0	0
$173$	1.89724e10	1.61033	0.805164	−	0.593053i	$-0.202078\pi$
$173$	1.89724e10	1.61033	0.805164	+	0.593053i	$0.202078\pi$
$174$	0	0
$175$	3.28820e10	2.65025
$176$	0	0
$177$	−4.21706e8	−0.0322946
$178$	0	0
$179$	1.72674e10	1.25716	0.628578	−	0.777746i	$-0.283637\pi$
$179$	1.72674e10	1.25716	0.628578	+	0.777746i	$0.283637\pi$
$180$	0	0
$181$	1.03363e10	0.715830	0.357915	−	0.933754i	$-0.383488\pi$
$181$	1.03363e10	0.715830	0.357915	+	0.933754i	$0.383488\pi$
$182$	0	0
$183$	6.41733e9	0.422985
$184$	0	0
$185$	1.38277e9	0.0867913
$186$	0	0
$187$	−3.47073e9	−0.207555
$188$	0	0
$189$	−2.04921e10	−1.16818
$190$	0	0
$191$	9.14299e9	0.497094	0.248547	−	0.968620i	$-0.420047\pi$
$191$	9.14299e9	0.497094	0.248547	+	0.968620i	$0.420047\pi$
$192$	0	0
$193$	8.45502e8	0.0438639	0.0219319	−	0.999759i	$-0.493018\pi$
$193$	8.45502e8	0.0438639	0.0219319	+	0.999759i	$0.493018\pi$
$194$	0	0
$195$	−7.70909e9	−0.381810
$196$	0	0
$197$	−2.12407e9	−0.100478	−0.0502390	−	0.998737i	$-0.515998\pi$
$197$	−2.12407e9	−0.100478	−0.0502390	+	0.998737i	$0.515998\pi$
$198$	0	0
$199$	−2.92879e9	−0.132388	−0.0661941	−	0.997807i	$-0.521086\pi$
$199$	−2.92879e9	−0.132388	−0.0661941	+	0.997807i	$0.521086\pi$
$200$	0	0
$201$	5.82411e9	0.251679
$202$	0	0
$203$	−7.91848e9	−0.327272
$204$	0	0
$205$	3.76656e10	1.48954
$206$	0	0
$207$	−8.12051e9	−0.307409
$208$	0	0
$209$	−1.22659e10	−0.444673
$210$	0	0
$211$	−3.16571e10	−1.09951	−0.549757	−	0.835325i	$-0.685280\pi$
$211$	−3.16571e10	−1.09951	−0.549757	+	0.835325i	$0.685280\pi$
$212$	0	0
$213$	2.76236e10	0.919542
$214$	0	0
$215$	1.02404e11	3.26847
$216$	0	0
$217$	−3.03613e10	−0.929505
$218$	0	0
$219$	9.21383e9	0.270671
$220$	0	0
$221$	−4.11446e9	−0.116024
$222$	0	0
$223$	4.92453e10	1.33350	0.666750	−	0.745282i	$-0.267685\pi$
$223$	4.92453e10	1.33350	0.666750	+	0.745282i	$0.267685\pi$
$224$	0	0
$225$	−4.52409e10	−1.17682
$226$	0	0
$227$	−5.67209e10	−1.41784	−0.708919	−	0.705290i	$-0.750817\pi$
$227$	−5.67209e10	−1.41784	−0.708919	+	0.705290i	$0.750817\pi$
$228$	0	0
$229$	3.33645e9	0.0801723	0.0400862	−	0.999196i	$-0.487237\pi$
$229$	3.33645e9	0.0801723	0.0400862	+	0.999196i	$0.487237\pi$
$230$	0	0
$231$	1.75411e10	0.405324
$232$	0	0
$233$	4.60006e10	1.02250	0.511249	−	0.859433i	$-0.329183\pi$
$233$	4.60006e10	1.02250	0.511249	+	0.859433i	$0.329183\pi$
$234$	0	0
$235$	−1.15980e11	−2.48072
$236$	0	0
$237$	−2.03370e10	−0.418716
$238$	0	0
$239$	4.05955e10	0.804798	0.402399	−	0.915464i	$-0.368176\pi$
$239$	4.05955e10	0.804798	0.402399	+	0.915464i	$0.368176\pi$
$240$	0	0
$241$	3.45208e10	0.659181	0.329590	−	0.944124i	$-0.393089\pi$
$241$	3.45208e10	0.659181	0.329590	+	0.944124i	$0.393089\pi$
$242$	0	0
$243$	4.71541e10	0.867544
$244$	0	0
$245$	−2.11671e10	−0.375331
$246$	0	0
$247$	−1.45409e10	−0.248573
$248$	0	0
$249$	6.13995e10	1.01220
$250$	0	0
$251$	7.83857e10	1.24654	0.623268	−	0.782008i	$-0.285804\pi$
$251$	7.83857e10	1.24654	0.623268	+	0.782008i	$0.285804\pi$
$252$	0	0
$253$	2.12231e10	0.325661
$254$	0	0
$255$	3.59552e10	0.532513
$256$	0	0
$257$	1.06489e11	1.52267	0.761337	−	0.648356i	$-0.224543\pi$
$257$	1.06489e11	1.52267	0.761337	+	0.648356i	$0.224543\pi$
$258$	0	0
$259$	−3.72988e9	−0.0515046
$260$	0	0
$261$	1.08947e10	0.145322
$262$	0	0
$263$	−6.39865e8	−0.00824684	−0.00412342	−	0.999991i	$-0.501313\pi$
$263$	−6.39865e8	−0.00824684	−0.00412342	+	0.999991i	$0.501313\pi$
$264$	0	0
$265$	−1.25452e11	−1.56268
$266$	0	0
$267$	7.75302e10	0.933620
$268$	0	0
$269$	−9.60820e10	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$269$	−9.60820e10	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$270$	0	0
$271$	9.78435e10	1.10197	0.550986	−	0.834515i	$-0.314252\pi$
$271$	9.78435e10	1.10197	0.550986	+	0.834515i	$0.314252\pi$
$272$	0	0
$273$	2.07945e10	0.226578
$274$	0	0
$275$	1.18238e11	1.24669
$276$	0	0
$277$	−1.61396e11	−1.64716	−0.823579	−	0.567202i	$-0.808026\pi$
$277$	−1.61396e11	−1.64716	−0.823579	+	0.567202i	$0.808026\pi$
$278$	0	0
$279$	4.17728e10	0.412738
$280$	0	0
$281$	2.71451e10	0.259724	0.129862	−	0.991532i	$-0.458547\pi$
$281$	2.71451e10	0.259724	0.129862	+	0.991532i	$0.458547\pi$
$282$	0	0
$283$	2.80153e10	0.259631	0.129815	−	0.991538i	$-0.458562\pi$
$283$	2.80153e10	0.259631	0.129815	+	0.991538i	$0.458562\pi$
$284$	0	0
$285$	1.27069e11	1.14087
$286$	0	0
$287$	−1.01599e11	−0.883940
$288$	0	0
$289$	−9.93980e10	−0.838180
$290$	0	0
$291$	1.08813e11	0.889534
$292$	0	0
$293$	2.49001e10	0.197377	0.0986884	−	0.995118i	$-0.468535\pi$
$293$	2.49001e10	0.197377	0.0986884	+	0.995118i	$0.468535\pi$
$294$	0	0
$295$	1.08409e10	0.0833421
$296$	0	0
$297$	−7.36861e10	−0.549518
$298$	0	0
$299$	2.51594e10	0.182046
$300$	0	0
$301$	−2.76226e11	−1.93961
$302$	0	0
$303$	−6.25239e10	−0.426142
$304$	0	0
$305$	−1.64971e11	−1.09159
$306$	0	0
$307$	−1.71738e11	−1.10343	−0.551713	−	0.834034i	$-0.686026\pi$
$307$	−1.71738e11	−1.10343	−0.551713	+	0.834034i	$0.686026\pi$
$308$	0	0
$309$	6.37223e10	0.397629
$310$	0	0
$311$	−1.24783e11	−0.756371	−0.378185	−	0.925730i	$-0.623452\pi$
$311$	−1.24783e11	−0.756371	−0.378185	+	0.925730i	$0.623452\pi$
$312$	0	0
$313$	2.95278e10	0.173893	0.0869463	−	0.996213i	$-0.472289\pi$
$313$	2.95278e10	0.173893	0.0869463	+	0.996213i	$0.472289\pi$
$314$	0	0
$315$	1.72538e11	0.987388
$316$	0	0
$317$	1.20582e11	0.670678	0.335339	−	0.942098i	$-0.391149\pi$
$317$	1.20582e11	0.670678	0.335339	+	0.942098i	$0.391149\pi$
$318$	0	0
$319$	−2.84735e10	−0.153951
$320$	0	0
$321$	−1.54887e11	−0.814223
$322$	0	0
$323$	6.78187e10	0.346687
$324$	0	0
$325$	1.40168e11	0.696906
$326$	0	0
$327$	2.20844e11	1.06812
$328$	0	0
$329$	3.12845e11	1.47214
$330$	0	0
$331$	−4.18670e11	−1.91711	−0.958553	−	0.284913i	$-0.908035\pi$
$331$	−4.18670e11	−1.91711	−0.958553	+	0.284913i	$0.908035\pi$
$332$	0	0
$333$	5.13179e9	0.0228702
$334$	0	0
$335$	−1.49721e11	−0.649504
$336$	0	0
$337$	2.95362e11	1.24744	0.623721	−	0.781647i	$-0.285620\pi$
$337$	2.95362e11	1.24744	0.623721	+	0.781647i	$0.285620\pi$
$338$	0	0
$339$	−5.46092e10	−0.224578
$340$	0	0
$341$	−1.09174e11	−0.437245
$342$	0	0
$343$	−2.24073e11	−0.874111
$344$	0	0
$345$	−2.19862e11	−0.835533
$346$	0	0
$347$	1.55047e10	0.0574093	0.0287046	−	0.999588i	$-0.490862\pi$
$347$	1.55047e10	0.0574093	0.0287046	+	0.999588i	$0.490862\pi$
$348$	0	0
$349$	3.06969e11	1.10759	0.553797	−	0.832652i	$-0.313178\pi$
$349$	3.06969e11	1.10759	0.553797	+	0.832652i	$0.313178\pi$
$350$	0	0
$351$	−8.73531e10	−0.307182
$352$	0	0
$353$	2.99671e11	1.02721	0.513604	−	0.858028i	$-0.328310\pi$
$353$	2.99671e11	1.02721	0.513604	+	0.858028i	$0.328310\pi$
$354$	0	0
$355$	−7.10124e11	−2.37305
$356$	0	0
$357$	−9.69856e10	−0.316010
$358$	0	0
$359$	−4.04368e11	−1.28485	−0.642424	−	0.766349i	$-0.722071\pi$
$359$	−4.04368e11	−1.28485	−0.642424	+	0.766349i	$0.722071\pi$
$360$	0	0
$361$	−8.30100e10	−0.257246
$362$	0	0
$363$	−1.73855e11	−0.525542
$364$	0	0
$365$	−2.36861e11	−0.698516
$366$	0	0
$367$	−5.07965e11	−1.46163	−0.730813	−	0.682578i	$-0.760859\pi$
$367$	−5.07965e11	−1.46163	−0.730813	+	0.682578i	$0.760859\pi$
$368$	0	0
$369$	1.39786e11	0.392506
$370$	0	0
$371$	3.38395e11	0.927345
$372$	0	0
$373$	6.94617e10	0.185804	0.0929022	−	0.995675i	$-0.470386\pi$
$373$	6.94617e10	0.185804	0.0929022	+	0.995675i	$0.470386\pi$
$374$	0	0
$375$	−7.17953e11	−1.87480
$376$	0	0
$377$	−3.37546e10	−0.0860591
$378$	0	0
$379$	−4.45070e11	−1.10803	−0.554015	−	0.832507i	$-0.686905\pi$
$379$	−4.45070e11	−1.10803	−0.554015	+	0.832507i	$0.686905\pi$
$380$	0	0
$381$	2.42466e11	0.589507
$382$	0	0
$383$	4.50375e11	1.06950	0.534749	−	0.845011i	$-0.320406\pi$
$383$	4.50375e11	1.06950	0.534749	+	0.845011i	$0.320406\pi$
$384$	0	0
$385$	−4.50932e11	−1.04601
$386$	0	0
$387$	3.80047e11	0.861268
$388$	0	0
$389$	5.76192e11	1.27583	0.637916	−	0.770106i	$-0.279797\pi$
$389$	5.76192e11	1.27583	0.637916	+	0.770106i	$0.279797\pi$
$390$	0	0
$391$	−1.17344e11	−0.253901
$392$	0	0
$393$	−1.58505e11	−0.335179
$394$	0	0
$395$	5.22807e11	1.08057
$396$	0	0
$397$	3.22788e11	0.652169	0.326084	−	0.945341i	$-0.394271\pi$
$397$	3.22788e11	0.652169	0.326084	+	0.945341i	$0.394271\pi$
$398$	0	0
$399$	−3.42756e11	−0.677030
$400$	0	0
$401$	1.47838e11	0.285520	0.142760	−	0.989757i	$-0.454402\pi$
$401$	1.47838e11	0.285520	0.142760	+	0.989757i	$0.454402\pi$
$402$	0	0
$403$	−1.29423e11	−0.244421
$404$	0	0
$405$	2.75949e11	0.509661
$406$	0	0
$407$	−1.34120e10	−0.0242281
$408$	0	0
$409$	7.24078e11	1.27947	0.639735	−	0.768595i	$-0.279044\pi$
$409$	7.24078e11	1.27947	0.639735	+	0.768595i	$0.279044\pi$
$410$	0	0
$411$	6.42153e11	1.11007
$412$	0	0
$413$	−2.92422e10	−0.0494578
$414$	0	0
$415$	−1.57841e12	−2.61218
$416$	0	0
$417$	−2.47081e11	−0.400153
$418$	0	0
$419$	6.10316e11	0.967368	0.483684	−	0.875243i	$-0.339298\pi$
$419$	6.10316e11	0.967368	0.483684	+	0.875243i	$0.339298\pi$
$420$	0	0
$421$	−7.16040e11	−1.11088	−0.555441	−	0.831556i	$-0.687450\pi$
$421$	−7.16040e11	−1.11088	−0.555441	+	0.831556i	$0.687450\pi$
$422$	0	0
$423$	−4.30430e11	−0.653689
$424$	0	0
$425$	−6.53744e11	−0.971981
$426$	0	0
$427$	4.44995e11	0.647783
$428$	0	0
$429$	7.47735e10	0.106584
$430$	0	0
$431$	−4.78823e11	−0.668385	−0.334193	−	0.942505i	$-0.608464\pi$
$431$	−4.78823e11	−0.668385	−0.334193	+	0.942505i	$0.608464\pi$
$432$	0	0
$433$	−3.13312e11	−0.428334	−0.214167	−	0.976797i	$-0.568704\pi$
$433$	−3.13312e11	−0.428334	−0.214167	+	0.976797i	$0.568704\pi$
$434$	0	0
$435$	2.94972e11	0.394984
$436$	0	0
$437$	−4.14703e11	−0.543965
$438$	0	0
$439$	6.95985e11	0.894354	0.447177	−	0.894445i	$-0.352429\pi$
$439$	6.95985e11	0.894354	0.447177	+	0.894445i	$0.352429\pi$
$440$	0	0
$441$	−7.85563e10	−0.0989025
$442$	0	0
$443$	3.55482e11	0.438531	0.219266	−	0.975665i	$-0.429634\pi$
$443$	3.55482e11	0.438531	0.219266	+	0.975665i	$0.429634\pi$
$444$	0	0
$445$	−1.99308e12	−2.40938
$446$	0	0
$447$	3.86391e11	0.457765
$448$	0	0
$449$	−8.15188e11	−0.946562	−0.473281	−	0.880911i	$-0.656931\pi$
$449$	−8.15188e11	−0.946562	−0.473281	+	0.880911i	$0.656931\pi$
$450$	0	0
$451$	−3.65334e11	−0.415811
$452$	0	0
$453$	4.09589e11	0.456990
$454$	0	0
$455$	−5.34568e11	−0.584725
$456$	0	0
$457$	5.94548e11	0.637623	0.318811	−	0.947818i	$-0.396716\pi$
$457$	5.94548e11	0.637623	0.318811	+	0.947818i	$0.396716\pi$
$458$	0	0
$459$	4.07414e11	0.428429
$460$	0	0
$461$	1.42516e12	1.46964	0.734818	−	0.678265i	$-0.237268\pi$
$461$	1.42516e12	1.46964	0.734818	+	0.678265i	$0.237268\pi$
$462$	0	0
$463$	1.04996e12	1.06183	0.530917	−	0.847424i	$-0.321847\pi$
$463$	1.04996e12	1.06183	0.530917	+	0.847424i	$0.321847\pi$
$464$	0	0
$465$	1.13099e12	1.12182
$466$	0	0
$467$	1.15656e11	0.112523	0.0562616	−	0.998416i	$-0.482082\pi$
$467$	1.15656e11	0.112523	0.0562616	+	0.998416i	$0.482082\pi$
$468$	0	0
$469$	4.03859e11	0.385436
$470$	0	0
$471$	−7.63065e11	−0.714443
$472$	0	0
$473$	−9.93261e11	−0.912406
$474$	0	0
$475$	−2.31039e12	−2.08240
$476$	0	0
$477$	−4.65583e11	−0.411779
$478$	0	0
$479$	−1.37302e12	−1.19170	−0.595849	−	0.803096i	$-0.703184\pi$
$479$	−1.37302e12	−1.19170	−0.595849	+	0.803096i	$0.703184\pi$
$480$	0	0
$481$	−1.58996e10	−0.0135436
$482$	0	0
$483$	5.93056e11	0.495831
$484$	0	0
$485$	−2.79727e12	−2.29561
$486$	0	0
$487$	−1.53458e12	−1.23626	−0.618131	−	0.786075i	$-0.712110\pi$
$487$	−1.53458e12	−1.23626	−0.618131	+	0.786075i	$0.712110\pi$
$488$	0	0
$489$	7.74990e11	0.612923
$490$	0	0
$491$	−5.12475e11	−0.397929	−0.198965	−	0.980007i	$-0.563758\pi$
$491$	−5.12475e11	−0.397929	−0.198965	+	0.980007i	$0.563758\pi$
$492$	0	0
$493$	1.57431e11	0.120027
$494$	0	0
$495$	6.20418e11	0.464473
$496$	0	0
$497$	1.91549e12	1.40824
$498$	0	0
$499$	−2.45745e12	−1.77432	−0.887159	−	0.461463i	$-0.847325\pi$
$499$	−2.45745e12	−1.77432	−0.887159	+	0.461463i	$0.847325\pi$
$500$	0	0
$501$	1.01657e12	0.720887
$502$	0	0
$503$	−2.11765e11	−0.147502	−0.0737512	−	0.997277i	$-0.523497\pi$
$503$	−2.11765e11	−0.147502	−0.0737512	+	0.997277i	$0.523497\pi$
$504$	0	0
$505$	1.60731e12	1.09974
$506$	0	0
$507$	−9.76913e11	−0.656629
$508$	0	0
$509$	−2.27707e12	−1.50365	−0.751824	−	0.659363i	$-0.770826\pi$
$509$	−2.27707e12	−1.50365	−0.751824	+	0.659363i	$0.770826\pi$
$510$	0	0
$511$	6.38911e11	0.414521
$512$	0	0
$513$	1.43984e12	0.917881
$514$	0	0
$515$	−1.63812e12	−1.02616
$516$	0	0
$517$	1.12494e12	0.692501
$518$	0	0
$519$	1.90637e12	1.15333
$520$	0	0
$521$	−3.15742e12	−1.87742	−0.938712	−	0.344703i	$-0.887980\pi$
$521$	−3.15742e12	−1.87742	−0.938712	+	0.344703i	$0.887980\pi$
$522$	0	0
$523$	6.70464e11	0.391848	0.195924	−	0.980619i	$-0.437229\pi$
$523$	6.70464e11	0.391848	0.195924	+	0.980619i	$0.437229\pi$
$524$	0	0
$525$	3.30403e12	1.89814
$526$	0	0
$527$	6.03629e11	0.340896
$528$	0	0
$529$	−1.08361e12	−0.601621
$530$	0	0
$531$	4.02330e10	0.0219613
$532$	0	0
$533$	−4.33094e11	−0.232440
$534$	0	0
$535$	3.98172e12	2.10125
$536$	0	0
$537$	1.73506e12	0.900387
$538$	0	0
$539$	2.05308e11	0.104775
$540$	0	0
$541$	6.87645e11	0.345125	0.172562	−	0.984999i	$-0.444795\pi$
$541$	6.87645e11	0.345125	0.172562	+	0.984999i	$0.444795\pi$
$542$	0	0
$543$	1.03860e12	0.512685
$544$	0	0
$545$	−5.67726e12	−2.75648
$546$	0	0
$547$	2.73174e12	1.30466	0.652328	−	0.757936i	$-0.273792\pi$
$547$	2.73174e12	1.30466	0.652328	+	0.757936i	$0.273792\pi$
$548$	0	0
$549$	−6.12249e11	−0.287642
$550$	0	0
$551$	5.56377e11	0.257150
$552$	0	0
$553$	−1.41022e12	−0.641245
$554$	0	0
$555$	1.38942e11	0.0621608
$556$	0	0
$557$	−3.59464e12	−1.58236	−0.791182	−	0.611580i	$-0.790534\pi$
$557$	−3.59464e12	−1.58236	−0.791182	+	0.611580i	$0.790534\pi$
$558$	0	0
$559$	−1.17749e12	−0.510038
$560$	0	0
$561$	−3.48744e11	−0.148653
$562$	0	0
$563$	4.60263e11	0.193072	0.0965358	−	0.995330i	$-0.469224\pi$
$563$	4.60263e11	0.193072	0.0965358	+	0.995330i	$0.469224\pi$
$564$	0	0
$565$	1.40385e12	0.579565
$566$	0	0
$567$	−7.44347e11	−0.302449
$568$	0	0
$569$	1.91911e12	0.767527	0.383764	−	0.923431i	$-0.374628\pi$
$569$	1.91911e12	0.767527	0.383764	+	0.923431i	$0.374628\pi$
$570$	0	0
$571$	2.01838e12	0.794586	0.397293	−	0.917692i	$-0.369950\pi$
$571$	2.01838e12	0.794586	0.397293	+	0.917692i	$0.369950\pi$
$572$	0	0
$573$	9.18701e11	0.356023
$574$	0	0
$575$	3.99757e12	1.52507
$576$	0	0
$577$	−6.91957e11	−0.259889	−0.129945	−	0.991521i	$-0.541480\pi$
$577$	−6.91957e11	−0.259889	−0.129945	+	0.991521i	$0.541480\pi$
$578$	0	0
$579$	8.49572e10	0.0314157
$580$	0	0
$581$	4.25760e12	1.55015
$582$	0	0
$583$	1.21681e12	0.436229
$584$	0	0
$585$	7.35490e11	0.259642
$586$	0	0
$587$	1.99542e12	0.693686	0.346843	−	0.937923i	$-0.387254\pi$
$587$	1.99542e12	0.693686	0.346843	+	0.937923i	$0.387254\pi$
$588$	0	0
$589$	2.13328e12	0.730347
$590$	0	0
$591$	−2.13430e11	−0.0719633
$592$	0	0
$593$	5.19417e12	1.72492	0.862462	−	0.506121i	$-0.168921\pi$
$593$	5.19417e12	1.72492	0.862462	+	0.506121i	$0.168921\pi$
$594$	0	0
$595$	2.49323e12	0.815521
$596$	0	0
$597$	−2.94289e11	−0.0948177
$598$	0	0
$599$	4.47443e12	1.42010	0.710048	−	0.704154i	$-0.248673\pi$
$599$	4.47443e12	1.42010	0.710048	+	0.704154i	$0.248673\pi$
$600$	0	0
$601$	−3.10552e12	−0.970955	−0.485477	−	0.874249i	$-0.661354\pi$
$601$	−3.10552e12	−0.970955	−0.485477	+	0.874249i	$0.661354\pi$
$602$	0	0
$603$	−5.55652e11	−0.171149
$604$	0	0
$605$	4.46932e12	1.35626
$606$	0	0
$607$	3.79043e12	1.13329	0.566643	−	0.823964i	$-0.308242\pi$
$607$	3.79043e12	1.13329	0.566643	+	0.823964i	$0.308242\pi$
$608$	0	0
$609$	−7.95660e11	−0.234396
$610$	0	0
$611$	1.33358e12	0.387111
$612$	0	0
$613$	−6.09780e12	−1.74422	−0.872110	−	0.489310i	$-0.837249\pi$
$613$	−6.09780e12	−1.74422	−0.872110	+	0.489310i	$0.837249\pi$
$614$	0	0
$615$	3.78470e12	1.06682
$616$	0	0
$617$	2.07665e12	0.576872	0.288436	−	0.957499i	$-0.406865\pi$
$617$	2.07665e12	0.576872	0.288436	+	0.957499i	$0.406865\pi$
$618$	0	0
$619$	6.03694e12	1.65276	0.826379	−	0.563115i	$-0.190397\pi$
$619$	6.03694e12	1.65276	0.826379	+	0.563115i	$0.190397\pi$
$620$	0	0
$621$	−2.49129e12	−0.672221
$622$	0	0
$623$	5.37615e12	1.42980
$624$	0	0
$625$	9.23927e12	2.42202
$626$	0	0
$627$	−1.23249e12	−0.318479
$628$	0	0
$629$	7.41558e10	0.0188893
$630$	0	0
$631$	−7.01879e12	−1.76250	−0.881252	−	0.472648i	$-0.843298\pi$
$631$	−7.01879e12	−1.76250	−0.881252	+	0.472648i	$0.843298\pi$
$632$	0	0
$633$	−3.18095e12	−0.787482
$634$	0	0
$635$	−6.23312e12	−1.52133
$636$	0	0
$637$	2.43388e11	0.0585695
$638$	0	0
$639$	−2.63544e12	−0.625316
$640$	0	0
$641$	−6.65994e12	−1.55815	−0.779075	−	0.626931i	$-0.784311\pi$
$641$	−6.65994e12	−1.55815	−0.779075	+	0.626931i	$0.784311\pi$
$642$	0	0
$643$	3.60863e12	0.832516	0.416258	−	0.909247i	$-0.363341\pi$
$643$	3.60863e12	0.832516	0.416258	+	0.909247i	$0.363341\pi$
$644$	0	0
$645$	1.02897e13	2.34091
$646$	0	0
$647$	3.26245e12	0.731938	0.365969	−	0.930627i	$-0.380738\pi$
$647$	3.26245e12	0.731938	0.365969	+	0.930627i	$0.380738\pi$
$648$	0	0
$649$	−1.05150e11	−0.0232652
$650$	0	0
$651$	−3.05075e12	−0.665720
$652$	0	0
$653$	−4.35965e11	−0.0938302	−0.0469151	−	0.998899i	$-0.514939\pi$
$653$	−4.35965e11	−0.0938302	−0.0469151	+	0.998899i	$0.514939\pi$
$654$	0	0
$655$	4.07471e12	0.864990
$656$	0	0
$657$	−8.79050e11	−0.184064
$658$	0	0
$659$	−2.27883e12	−0.470683	−0.235341	−	0.971913i	$-0.575621\pi$
$659$	−2.27883e12	−0.470683	−0.235341	+	0.971913i	$0.575621\pi$
$660$	0	0
$661$	−6.28607e12	−1.28077	−0.640387	−	0.768052i	$-0.721226\pi$
$661$	−6.28607e12	−1.28077	−0.640387	+	0.768052i	$0.721226\pi$
$662$	0	0
$663$	−4.13427e11	−0.0830975
$664$	0	0
$665$	8.81130e12	1.74720
$666$	0	0
$667$	−9.62674e11	−0.188327
$668$	0	0
$669$	4.94824e12	0.955065
$670$	0	0
$671$	1.60012e12	0.304721
$672$	0	0
$673$	−1.30391e12	−0.245007	−0.122504	−	0.992468i	$-0.539092\pi$
$673$	−1.30391e12	−0.245007	−0.122504	+	0.992468i	$0.539092\pi$
$674$	0	0
$675$	−1.38795e13	−2.57339
$676$	0	0
$677$	−3.77662e12	−0.690962	−0.345481	−	0.938426i	$-0.612284\pi$
$677$	−3.77662e12	−0.690962	−0.345481	+	0.938426i	$0.612284\pi$
$678$	0	0
$679$	7.54538e12	1.36228
$680$	0	0
$681$	−5.69939e12	−1.01547
$682$	0	0
$683$	−8.09351e12	−1.42313	−0.711563	−	0.702622i	$-0.752013\pi$
$683$	−8.09351e12	−1.42313	−0.711563	+	0.702622i	$0.752013\pi$
$684$	0	0
$685$	−1.65079e13	−2.86474
$686$	0	0
$687$	3.35251e11	0.0574202
$688$	0	0
$689$	1.44250e12	0.243853
$690$	0	0
$691$	1.04215e12	0.173891	0.0869455	−	0.996213i	$-0.472289\pi$
$691$	1.04215e12	0.173891	0.0869455	+	0.996213i	$0.472289\pi$
$692$	0	0
$693$	−1.67352e12	−0.275633
$694$	0	0
$695$	6.35174e12	1.03267
$696$	0	0
$697$	2.01995e12	0.324185
$698$	0	0
$699$	4.62221e12	0.732322
$700$	0	0
$701$	−2.19669e12	−0.343587	−0.171794	−	0.985133i	$-0.554956\pi$
$701$	−2.19669e12	−0.343587	−0.171794	+	0.985133i	$0.554956\pi$
$702$	0	0
$703$	2.62073e11	0.0404692
$704$	0	0
$705$	−1.16538e13	−1.77672
$706$	0	0
$707$	−4.33557e12	−0.652618
$708$	0	0
$709$	7.02990e11	0.104482	0.0522410	−	0.998635i	$-0.483364\pi$
$709$	7.02990e11	0.104482	0.0522410	+	0.998635i	$0.483364\pi$
$710$	0	0
$711$	1.94026e12	0.284739
$712$	0	0
$713$	−3.69112e12	−0.534879
$714$	0	0
$715$	−1.92222e12	−0.275058
$716$	0	0
$717$	4.07909e12	0.576404
$718$	0	0
$719$	−1.18349e13	−1.65152	−0.825759	−	0.564022i	$-0.809253\pi$
$719$	−1.18349e13	−1.65152	−0.825759	+	0.564022i	$0.809253\pi$
$720$	0	0
$721$	4.41867e12	0.608952
$722$	0	0
$723$	3.46870e12	0.472111
$724$	0	0
$725$	−5.36325e12	−0.720953
$726$	0	0
$727$	−4.66170e12	−0.618927	−0.309464	−	0.950911i	$-0.600150\pi$
$727$	−4.66170e12	−0.618927	−0.309464	+	0.950911i	$0.600150\pi$
$728$	0	0
$729$	6.84083e12	0.897088
$730$	0	0
$731$	5.49179e12	0.711354
$732$	0	0
$733$	1.16839e13	1.49493	0.747466	−	0.664300i	$-0.231270\pi$
$733$	1.16839e13	1.49493	0.747466	+	0.664300i	$0.231270\pi$
$734$	0	0
$735$	−2.12690e12	−0.268815
$736$	0	0
$737$	1.45221e12	0.181311
$738$	0	0
$739$	1.25228e13	1.54455	0.772274	−	0.635290i	$-0.219119\pi$
$739$	1.25228e13	1.54455	0.772274	+	0.635290i	$0.219119\pi$
$740$	0	0
$741$	−1.46109e12	−0.178031
$742$	0	0
$743$	−1.10289e13	−1.32765	−0.663823	−	0.747890i	$-0.731067\pi$
$743$	−1.10289e13	−1.32765	−0.663823	+	0.747890i	$0.731067\pi$
$744$	0	0
$745$	−9.93301e12	−1.18135
$746$	0	0
$747$	−5.85785e12	−0.688328
$748$	0	0
$749$	−1.07403e13	−1.24695
$750$	0	0
$751$	6.45522e12	0.740511	0.370255	−	0.928930i	$-0.379270\pi$
$751$	6.45522e12	0.740511	0.370255	+	0.928930i	$0.379270\pi$
$752$	0	0
$753$	7.87631e12	0.892781
$754$	0	0
$755$	−1.05294e13	−1.17935
$756$	0	0
$757$	6.48010e12	0.717217	0.358608	−	0.933488i	$-0.383251\pi$
$757$	6.48010e12	0.717217	0.358608	+	0.933488i	$0.383251\pi$
$758$	0	0
$759$	2.13253e12	0.233242
$760$	0	0
$761$	−2.62886e12	−0.284142	−0.142071	−	0.989856i	$-0.545376\pi$
$761$	−2.62886e12	−0.284142	−0.142071	+	0.989856i	$0.545376\pi$
$762$	0	0
$763$	1.53139e13	1.63578
$764$	0	0
$765$	−3.43032e12	−0.362125
$766$	0	0
$767$	−1.24652e11	−0.0130053
$768$	0	0
$769$	−1.43517e13	−1.47991	−0.739955	−	0.672656i	$-0.765153\pi$
$769$	−1.43517e13	−1.47991	−0.739955	+	0.672656i	$0.765153\pi$
$770$	0	0
$771$	1.07002e13	1.09055
$772$	0	0
$773$	6.56407e12	0.661250	0.330625	−	0.943762i	$-0.392740\pi$
$773$	6.56407e12	0.661250	0.330625	+	0.943762i	$0.392740\pi$
$774$	0	0
$775$	−2.05640e13	−2.04762
$776$	0	0
$777$	−3.74784e11	−0.0368881
$778$	0	0
$779$	7.13870e12	0.694545
$780$	0	0
$781$	6.88778e12	0.662444
$782$	0	0
$783$	3.34238e12	0.317781
$784$	0	0
$785$	1.96162e13	1.84375
$786$	0	0
$787$	1.96637e13	1.82717	0.913584	−	0.406649i	$-0.133303\pi$
$787$	1.96637e13	1.82717	0.913584	+	0.406649i	$0.133303\pi$
$788$	0	0
$789$	−6.42945e10	−0.00590646
$790$	0	0
$791$	−3.78675e12	−0.343932
$792$	0	0
$793$	1.89691e12	0.170340
$794$	0	0
$795$	−1.26056e13	−1.11921
$796$	0	0
$797$	2.05859e12	0.180721	0.0903603	−	0.995909i	$-0.471198\pi$
$797$	2.05859e12	0.180721	0.0903603	+	0.995909i	$0.471198\pi$
$798$	0	0
$799$	−6.21983e12	−0.539906
$800$	0	0
$801$	−7.39681e12	−0.634890
$802$	0	0
$803$	2.29741e12	0.194993
$804$	0	0
$805$	−1.52458e13	−1.27958
$806$	0	0
$807$	−9.65446e12	−0.801304
$808$	0	0
$809$	9.44339e12	0.775103	0.387552	−	0.921848i	$-0.373321\pi$
$809$	9.44339e12	0.775103	0.387552	+	0.921848i	$0.373321\pi$
$810$	0	0
$811$	3.40891e12	0.276708	0.138354	−	0.990383i	$-0.455819\pi$
$811$	3.40891e12	0.276708	0.138354	+	0.990383i	$0.455819\pi$
$812$	0	0
$813$	9.83146e12	0.789242
$814$	0	0
$815$	−1.99228e13	−1.58176
$816$	0	0
$817$	1.94085e13	1.52403
$818$	0	0
$819$	−1.98391e12	−0.154080
$820$	0	0
$821$	−1.44900e12	−0.111307	−0.0556535	−	0.998450i	$-0.517724\pi$
$821$	−1.44900e12	−0.111307	−0.0556535	+	0.998450i	$0.517724\pi$
$822$	0	0
$823$	9.50599e12	0.722267	0.361134	−	0.932514i	$-0.382390\pi$
$823$	9.50599e12	0.722267	0.361134	+	0.932514i	$0.382390\pi$
$824$	0	0
$825$	1.18807e13	0.892894
$826$	0	0
$827$	1.32590e13	0.985683	0.492842	−	0.870119i	$-0.335958\pi$
$827$	1.32590e13	0.985683	0.492842	+	0.870119i	$0.335958\pi$
$828$	0	0
$829$	−1.63326e13	−1.20104	−0.600522	−	0.799608i	$-0.705040\pi$
$829$	−1.63326e13	−1.20104	−0.600522	+	0.799608i	$0.705040\pi$
$830$	0	0
$831$	−1.62173e13	−1.17971
$832$	0	0
$833$	−1.13516e12	−0.0816873
$834$	0	0
$835$	−2.61331e13	−1.86038
$836$	0	0
$837$	1.28155e13	0.902549
$838$	0	0
$839$	2.23005e12	0.155377	0.0776884	−	0.996978i	$-0.475246\pi$
$839$	2.23005e12	0.155377	0.0776884	+	0.996978i	$0.475246\pi$
$840$	0	0
$841$	−1.32156e13	−0.910972
$842$	0	0
$843$	2.72758e12	0.186017
$844$	0	0
$845$	2.51137e13	1.69455
$846$	0	0
$847$	−1.20556e13	−0.804846
$848$	0	0
$849$	2.81501e12	0.185950
$850$	0	0
$851$	−4.53454e11	−0.0296381
$852$	0	0
$853$	2.83205e13	1.83160	0.915801	−	0.401633i	$-0.131557\pi$
$853$	2.83205e13	1.83160	0.915801	+	0.401633i	$0.131557\pi$
$854$	0	0
$855$	−1.21231e13	−0.775828
$856$	0	0
$857$	6.76700e12	0.428531	0.214266	−	0.976775i	$-0.431264\pi$
$857$	6.76700e12	0.428531	0.214266	+	0.976775i	$0.431264\pi$
$858$	0	0
$859$	−2.30991e12	−0.144753	−0.0723763	−	0.997377i	$-0.523058\pi$
$859$	−2.30991e12	−0.144753	−0.0723763	+	0.997377i	$0.523058\pi$
$860$	0	0
$861$	−1.02089e13	−0.633087
$862$	0	0
$863$	1.05303e13	0.646237	0.323119	−	0.946358i	$-0.395269\pi$
$863$	1.05303e13	0.646237	0.323119	+	0.946358i	$0.395269\pi$
$864$	0	0
$865$	−4.90074e13	−2.97638
$866$	0	0
$867$	−9.98766e12	−0.600313
$868$	0	0
$869$	−5.07091e12	−0.301646
$870$	0	0
$871$	1.72156e12	0.101354
$872$	0	0
$873$	−1.03814e13	−0.604909
$874$	0	0
$875$	−4.97848e13	−2.87118
$876$	0	0
$877$	1.37761e13	0.786373	0.393186	−	0.919459i	$-0.371373\pi$
$877$	1.37761e13	0.786373	0.393186	+	0.919459i	$0.371373\pi$
$878$	0	0
$879$	2.50199e12	0.141363
$880$	0	0
$881$	−1.13133e13	−0.632698	−0.316349	−	0.948643i	$-0.602457\pi$
$881$	−1.13133e13	−0.632698	−0.316349	+	0.948643i	$0.602457\pi$
$882$	0	0
$883$	−3.91577e12	−0.216768	−0.108384	−	0.994109i	$-0.534568\pi$
$883$	−3.91577e12	−0.216768	−0.108384	+	0.994109i	$0.534568\pi$
$884$	0	0
$885$	1.08930e12	0.0596904
$886$	0	0
$887$	−1.62707e12	−0.0882573	−0.0441287	−	0.999026i	$-0.514051\pi$
$887$	−1.62707e12	−0.0882573	−0.0441287	+	0.999026i	$0.514051\pi$
$888$	0	0
$889$	1.68133e13	0.902805
$890$	0	0
$891$	−2.67654e12	−0.142274
$892$	0	0
$893$	−2.19815e13	−1.15671
$894$	0	0
$895$	−4.46034e13	−2.32362
$896$	0	0
$897$	2.52806e12	0.130383
$898$	0	0
$899$	4.95211e12	0.252855
$900$	0	0
$901$	−6.72780e12	−0.340104
$902$	0	0
$903$	−2.77556e13	−1.38917
$904$	0	0
$905$	−2.66995e13	−1.32308
$906$	0	0
$907$	7.72659e12	0.379101	0.189551	−	0.981871i	$-0.439297\pi$
$907$	7.72659e12	0.379101	0.189551	+	0.981871i	$0.439297\pi$
$908$	0	0
$909$	5.96513e12	0.289789
$910$	0	0
$911$	1.47051e13	0.707350	0.353675	−	0.935368i	$-0.384932\pi$
$911$	1.47051e13	0.707350	0.353675	+	0.935368i	$0.384932\pi$
$912$	0	0
$913$	1.53096e13	0.729198
$914$	0	0
$915$	−1.65766e13	−0.781807
$916$	0	0
$917$	−1.09911e13	−0.513312
$918$	0	0
$919$	−3.87199e13	−1.79066	−0.895332	−	0.445399i	$-0.853062\pi$
$919$	−3.87199e13	−1.79066	−0.895332	+	0.445399i	$0.853062\pi$
$920$	0	0
$921$	−1.72565e13	−0.790285
$922$	0	0
$923$	8.16529e12	0.370309
$924$	0	0
$925$	−2.52628e12	−0.113460
$926$	0	0
$927$	−6.07946e12	−0.270400
$928$	0	0
$929$	7.75866e12	0.341756	0.170878	−	0.985292i	$-0.445340\pi$
$929$	7.75866e12	0.341756	0.170878	+	0.985292i	$0.445340\pi$
$930$	0	0
$931$	−4.01176e12	−0.175010
$932$	0	0
$933$	−1.25384e13	−0.541720
$934$	0	0
$935$	8.96521e12	0.383626
$936$	0	0
$937$	1.60112e13	0.678573	0.339286	−	0.940683i	$-0.389814\pi$
$937$	1.60112e13	0.678573	0.339286	+	0.940683i	$0.389814\pi$
$938$	0	0
$939$	2.96699e12	0.124543
$940$	0	0
$941$	1.55404e13	0.646115	0.323057	−	0.946379i	$-0.395289\pi$
$941$	1.55404e13	0.646115	0.323057	+	0.946379i	$0.395289\pi$
$942$	0	0
$943$	−1.23518e13	−0.508659
$944$	0	0
$945$	5.29330e13	2.15915
$946$	0	0
$947$	1.38291e13	0.558750	0.279375	−	0.960182i	$-0.409873\pi$
$947$	1.38291e13	0.558750	0.279375	+	0.960182i	$0.409873\pi$
$948$	0	0
$949$	2.72353e12	0.109002
$950$	0	0
$951$	1.21162e13	0.480346
$952$	0	0
$953$	1.62441e12	0.0637935	0.0318967	−	0.999491i	$-0.489845\pi$
$953$	1.62441e12	0.0637935	0.0318967	+	0.999491i	$0.489845\pi$
$954$	0	0
$955$	−2.36172e13	−0.918784
$956$	0	0
$957$	−2.86105e12	−0.110261
$958$	0	0
$959$	4.45285e13	1.70002
$960$	0	0
$961$	−7.45207e12	−0.281852
$962$	0	0
$963$	1.47771e13	0.553696
$964$	0	0
$965$	−2.18401e12	−0.0810740
$966$	0	0
$967$	2.99781e13	1.10252	0.551259	−	0.834334i	$-0.314148\pi$
$967$	2.99781e13	1.10252	0.551259	+	0.834334i	$0.314148\pi$
$968$	0	0
$969$	6.81452e12	0.248301
$970$	0	0
$971$	3.61921e13	1.30655	0.653277	−	0.757119i	$-0.273394\pi$
$971$	3.61921e13	1.30655	0.653277	+	0.757119i	$0.273394\pi$
$972$	0	0
$973$	−1.71332e13	−0.612818
$974$	0	0
$975$	1.40843e13	0.499131
$976$	0	0
$977$	2.29688e13	0.806515	0.403258	−	0.915087i	$-0.367878\pi$
$977$	2.29688e13	0.806515	0.403258	+	0.915087i	$0.367878\pi$
$978$	0	0
$979$	1.93317e13	0.672586
$980$	0	0
$981$	−2.10697e13	−0.726353
$982$	0	0
$983$	−2.02509e13	−0.691758	−0.345879	−	0.938279i	$-0.612419\pi$
$983$	−2.02509e13	−0.691758	−0.345879	+	0.938279i	$0.612419\pi$
$984$	0	0
$985$	5.48667e12	0.185715
$986$	0	0
$987$	3.14351e13	1.05436
$988$	0	0
$989$	−3.35817e13	−1.11614
$990$	0	0
$991$	1.63486e12	0.0538456	0.0269228	−	0.999638i	$-0.491429\pi$
$991$	1.63486e12	0.0538456	0.0269228	+	0.999638i	$0.491429\pi$
$992$	0	0
$993$	−4.20686e13	−1.37305
$994$	0	0
$995$	7.56533e12	0.244694
$996$	0	0
$997$	−9.91675e12	−0.317864	−0.158932	−	0.987290i	$-0.550805\pi$
$997$	−9.91675e12	−0.317864	−0.158932	+	0.987290i	$0.550805\pi$
$998$	0	0
$999$	1.57438e12	0.0500110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.10.a.s.1.6		8
4.3	odd	2		256.10.a.p.1.3		8
8.3	odd	2		256.10.a.p.1.6		8
8.5	even	2	inner	256.10.a.s.1.3		8
16.3	odd	4		8.10.b.a.5.2	yes	8
16.5	even	4		32.10.b.a.17.3		8
16.11	odd	4		8.10.b.a.5.1	✓	8
16.13	even	4		32.10.b.a.17.6		8
48.5	odd	4		288.10.d.b.145.8		8
48.11	even	4		72.10.d.b.37.8		8
48.29	odd	4		288.10.d.b.145.1		8
48.35	even	4		72.10.d.b.37.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.b.a.5.1	✓	8	16.11	odd	4
8.10.b.a.5.2	yes	8	16.3	odd	4
32.10.b.a.17.3		8	16.5	even	4
32.10.b.a.17.6		8	16.13	even	4
72.10.d.b.37.7		8	48.35	even	4
72.10.d.b.37.8		8	48.11	even	4
256.10.a.p.1.3		8	4.3	odd	2
256.10.a.p.1.6		8	8.3	odd	2
256.10.a.s.1.3		8	8.5	even	2	inner
256.10.a.s.1.6		8	1.1	even	1	trivial
288.10.d.b.145.1		8	48.29	odd	4
288.10.d.b.145.8		8	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+100.481 q^{3} -2583.09 q^{5} +6967.65 q^{7} -9586.48 q^{9} +O(q^{10})\)	\(q+100.481 q^{3} -2583.09 q^{5} +6967.65 q^{7} -9586.48 q^{9} +25054.5 q^{11} +29701.4 q^{13} -259553. q^{15} -138527. q^{17} -489569. q^{19} +700119. q^{21} +847079. q^{23} +4.71924e6 q^{25} -2.94104e6 q^{27} -1.13646e6 q^{29} -4.35747e6 q^{31} +2.51751e6 q^{33} -1.79981e7 q^{35} -535315. q^{37} +2.98444e6 q^{39} -1.45816e7 q^{41} -3.96441e7 q^{43} +2.47628e7 q^{45} +4.48997e7 q^{47} +8.19448e6 q^{49} -1.39194e7 q^{51} +4.85666e7 q^{53} -6.47180e7 q^{55} -4.91926e7 q^{57} -4.19685e6 q^{59} +6.38659e7 q^{61} -6.67952e7 q^{63} -7.67215e7 q^{65} +5.79621e7 q^{67} +8.51157e7 q^{69} +2.74912e8 q^{71} +9.16969e7 q^{73} +4.74196e8 q^{75} +1.74571e8 q^{77} -2.02396e8 q^{79} -1.06829e8 q^{81} +6.11053e8 q^{83} +3.57829e8 q^{85} -1.14193e8 q^{87} +7.71588e8 q^{89} +2.06949e8 q^{91} -4.37845e8 q^{93} +1.26460e9 q^{95} +1.08292e9 q^{97} -2.40184e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4800 q^{7} + 39368 q^{9}+O(q^{10}) \) 8 * q + 4800 * q^7 + 39368 * q^9	\( 8 q + 4800 q^{7} + 39368 q^{9} + 163136 q^{15} - 102000 q^{17} + 3412032 q^{23} + 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} - 17590208 q^{39} + 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} + 7056832 q^{55} - 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} + 560234688 q^{71} + 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} + 540527424 q^{87} - 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) \) 8 * q + 4800 * q^7 + 39368 * q^9 + 163136 * q^15 - 102000 * q^17 + 3412032 * q^23 + 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 - 17590208 * q^39 + 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 + 7056832 * q^55 - 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 + 560234688 * q^71 + 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 + 540527424 * q^87 - 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

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