LMFDB - Embedded newform 208.10.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(1,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-141] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$107.127453922$
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} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-245.871$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-263.871 q^{3} +2083.78 q^{5} -3949.77 q^{7} +49944.8 q^{9} +32770.7 q^{11} +28561.0 q^{13} -549849. q^{15} +579245. q^{17} +909955. q^{19} +1.04223e6 q^{21} +1.86994e6 q^{23} +2.38902e6 q^{25} -7.98520e6 q^{27} +4.60746e6 q^{29} -2.51506e6 q^{31} -8.64723e6 q^{33} -8.23044e6 q^{35} -1.29721e7 q^{37} -7.53641e6 q^{39} -3.17314e7 q^{41} +2.64551e7 q^{43} +1.04074e8 q^{45} +2.69503e7 q^{47} -2.47530e7 q^{49} -1.52846e8 q^{51} +1.30165e7 q^{53} +6.82869e7 q^{55} -2.40111e8 q^{57} +1.03319e7 q^{59} -1.91110e7 q^{61} -1.97270e8 q^{63} +5.95148e7 q^{65} -1.36252e8 q^{67} -4.93422e8 q^{69} -1.54745e8 q^{71} +3.56526e8 q^{73} -6.30391e8 q^{75} -1.29437e8 q^{77} +1.79298e8 q^{79} +1.12400e9 q^{81} -5.94120e6 q^{83} +1.20702e9 q^{85} -1.21577e9 q^{87} +1.96414e7 q^{89} -1.12809e8 q^{91} +6.63650e8 q^{93} +1.89615e9 q^{95} +9.74665e7 q^{97} +1.63673e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) $ 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

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
$3$	−263.871	−1.88081	−0.940406	−	0.340052i	$-0.889555\pi$
$3$	−263.871	−1.88081	−0.940406	+	0.340052i	$0.889555\pi$
$4$	0	0
$5$	2083.78	1.49103	0.745516	−	0.666488i	$-0.232203\pi$
$5$	2083.78	1.49103	0.745516	+	0.666488i	$0.232203\pi$
$6$	0	0
$7$	−3949.77	−0.621771	−0.310885	−	0.950447i	$-0.600626\pi$
$7$	−3949.77	−0.621771	−0.310885	+	0.950447i	$0.600626\pi$
$8$	0	0
$9$	49944.8	2.53746
$10$	0	0
$11$	32770.7	0.674868	0.337434	−	0.941349i	$-0.390441\pi$
$11$	32770.7	0.674868	0.337434	+	0.941349i	$0.390441\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−549849.	−2.80435
$16$	0	0
$17$	579245.	1.68206	0.841031	−	0.540987i	$-0.181949\pi$
$17$	579245.	1.68206	0.841031	+	0.540987i	$0.181949\pi$
$18$	0	0
$19$	909955.	1.60188	0.800938	−	0.598747i	$-0.204335\pi$
$19$	909955.	1.60188	0.800938	+	0.598747i	$0.204335\pi$
$20$	0	0
$21$	1.04223e6	1.16943
$22$	0	0
$23$	1.86994e6	1.39332	0.696662	−	0.717399i	$-0.254668\pi$
$23$	1.86994e6	1.39332	0.696662	+	0.717399i	$0.254668\pi$
$24$	0	0
$25$	2.38902e6	1.22318
$26$	0	0
$27$	−7.98520e6	−2.89167
$28$	0	0
$29$	4.60746e6	1.20968	0.604840	−	0.796347i	$-0.293237\pi$
$29$	4.60746e6	1.20968	0.604840	+	0.796347i	$0.293237\pi$
$30$	0	0
$31$	−2.51506e6	−0.489126	−0.244563	−	0.969633i	$-0.578644\pi$
$31$	−2.51506e6	−0.489126	−0.244563	+	0.969633i	$0.578644\pi$
$32$	0	0
$33$	−8.64723e6	−1.26930
$34$	0	0
$35$	−8.23044e6	−0.927080
$36$	0	0
$37$	−1.29721e7	−1.13790	−0.568948	−	0.822374i	$-0.692649\pi$
$37$	−1.29721e7	−1.13790	−0.568948	+	0.822374i	$0.692649\pi$
$38$	0	0
$39$	−7.53641e6	−0.521644
$40$	0	0
$41$	−3.17314e7	−1.75373	−0.876864	−	0.480739i	$-0.840368\pi$
$41$	−3.17314e7	−1.75373	−0.876864	+	0.480739i	$0.840368\pi$
$42$	0	0
$43$	2.64551e7	1.18005	0.590026	−	0.807384i	$-0.299117\pi$
$43$	2.64551e7	1.18005	0.590026	+	0.807384i	$0.299117\pi$
$44$	0	0
$45$	1.04074e8	3.78343
$46$	0	0
$47$	2.69503e7	0.805608	0.402804	−	0.915286i	$-0.368036\pi$
$47$	2.69503e7	0.805608	0.402804	+	0.915286i	$0.368036\pi$
$48$	0	0
$49$	−2.47530e7	−0.613401
$50$	0	0
$51$	−1.52846e8	−3.16364
$52$	0	0
$53$	1.30165e7	0.226596	0.113298	−	0.993561i	$-0.463858\pi$
$53$	1.30165e7	0.226596	0.113298	+	0.993561i	$0.463858\pi$
$54$	0	0
$55$	6.82869e7	1.00625
$56$	0	0
$57$	−2.40111e8	−3.01283
$58$	0	0
$59$	1.03319e7	0.111006	0.0555032	−	0.998459i	$-0.482324\pi$
$59$	1.03319e7	0.111006	0.0555032	+	0.998459i	$0.482324\pi$
$60$	0	0
$61$	−1.91110e7	−0.176725	−0.0883627	−	0.996088i	$-0.528163\pi$
$61$	−1.91110e7	−0.176725	−0.0883627	+	0.996088i	$0.528163\pi$
$62$	0	0
$63$	−1.97270e8	−1.57772
$64$	0	0
$65$	5.95148e7	0.413538
$66$	0	0
$67$	−1.36252e8	−0.826047	−0.413024	−	0.910720i	$-0.635527\pi$
$67$	−1.36252e8	−0.826047	−0.413024	+	0.910720i	$0.635527\pi$
$68$	0	0
$69$	−4.93422e8	−2.62058
$70$	0	0
$71$	−1.54745e8	−0.722692	−0.361346	−	0.932432i	$-0.617683\pi$
$71$	−1.54745e8	−0.722692	−0.361346	+	0.932432i	$0.617683\pi$
$72$	0	0
$73$	3.56526e8	1.46939	0.734697	−	0.678395i	$-0.237324\pi$
$73$	3.56526e8	1.46939	0.734697	+	0.678395i	$0.237324\pi$
$74$	0	0
$75$	−6.30391e8	−2.30056
$76$	0	0
$77$	−1.29437e8	−0.419613
$78$	0	0
$79$	1.79298e8	0.517909	0.258954	−	0.965890i	$-0.416622\pi$
$79$	1.79298e8	0.517909	0.258954	+	0.965890i	$0.416622\pi$
$80$	0	0
$81$	1.12400e9	2.90123
$82$	0	0
$83$	−5.94120e6	−0.0137411	−0.00687057	−	0.999976i	$-0.502187\pi$
$83$	−5.94120e6	−0.0137411	−0.00687057	+	0.999976i	$0.502187\pi$
$84$	0	0
$85$	1.20702e9	2.50801
$86$	0	0
$87$	−1.21577e9	−2.27518
$88$	0	0
$89$	1.96414e7	0.0331831	0.0165915	−	0.999862i	$-0.494719\pi$
$89$	1.96414e7	0.0331831	0.0165915	+	0.999862i	$0.494719\pi$
$90$	0	0
$91$	−1.12809e8	−0.172448
$92$	0	0
$93$	6.63650e8	0.919954
$94$	0	0
$95$	1.89615e9	2.38845
$96$	0	0
$97$	9.74665e7	0.111785	0.0558924	−	0.998437i	$-0.482200\pi$
$97$	9.74665e7	0.111785	0.0558924	+	0.998437i	$0.482200\pi$
$98$	0	0
$99$	1.63673e9	1.71245
$100$	0	0
$101$	1.47099e9	1.40658	0.703289	−	0.710904i	$-0.251714\pi$
$101$	1.47099e9	1.40658	0.703289	+	0.710904i	$0.251714\pi$
$102$	0	0
$103$	−4.40667e8	−0.385783	−0.192892	−	0.981220i	$-0.561787\pi$
$103$	−4.40667e8	−0.385783	−0.192892	+	0.981220i	$0.561787\pi$
$104$	0	0
$105$	2.17177e9	1.74366
$106$	0	0
$107$	−4.82772e8	−0.356053	−0.178027	−	0.984026i	$-0.556971\pi$
$107$	−4.82772e8	−0.356053	−0.178027	+	0.984026i	$0.556971\pi$
$108$	0	0
$109$	−5.63126e8	−0.382108	−0.191054	−	0.981579i	$-0.561191\pi$
$109$	−5.63126e8	−0.382108	−0.191054	+	0.981579i	$0.561191\pi$
$110$	0	0
$111$	3.42296e9	2.14017
$112$	0	0
$113$	1.35162e9	0.779832	0.389916	−	0.920851i	$-0.372504\pi$
$113$	1.35162e9	0.779832	0.389916	+	0.920851i	$0.372504\pi$
$114$	0	0
$115$	3.89654e9	2.07749
$116$	0	0
$117$	1.42647e9	0.703764
$118$	0	0
$119$	−2.28788e9	−1.04586
$120$	0	0
$121$	−1.28403e9	−0.544554
$122$	0	0
$123$	8.37299e9	3.29843
$124$	0	0
$125$	9.08299e8	0.332762
$126$	0	0
$127$	2.68968e9	0.917452	0.458726	−	0.888578i	$-0.348306\pi$
$127$	2.68968e9	0.917452	0.458726	+	0.888578i	$0.348306\pi$
$128$	0	0
$129$	−6.98072e9	−2.21946
$130$	0	0
$131$	−6.37867e9	−1.89238	−0.946192	−	0.323606i	$-0.895105\pi$
$131$	−6.37867e9	−1.89238	−0.946192	+	0.323606i	$0.895105\pi$
$132$	0	0
$133$	−3.59411e9	−0.995999
$134$	0	0
$135$	−1.66394e10	−4.31157
$136$	0	0
$137$	−2.03661e9	−0.493930	−0.246965	−	0.969024i	$-0.579433\pi$
$137$	−2.03661e9	−0.493930	−0.246965	+	0.969024i	$0.579433\pi$
$138$	0	0
$139$	4.59266e9	1.04351	0.521756	−	0.853095i	$-0.325277\pi$
$139$	4.59266e9	1.04351	0.521756	+	0.853095i	$0.325277\pi$
$140$	0	0
$141$	−7.11140e9	−1.51520
$142$	0	0
$143$	9.35964e8	0.187175
$144$	0	0
$145$	9.60094e9	1.80367
$146$	0	0
$147$	6.53158e9	1.15369
$148$	0	0
$149$	6.68745e9	1.11153	0.555766	−	0.831339i	$-0.312425\pi$
$149$	6.68745e9	1.11153	0.555766	+	0.831339i	$0.312425\pi$
$150$	0	0
$151$	8.31397e8	0.130140	0.0650702	−	0.997881i	$-0.479273\pi$
$151$	8.31397e8	0.130140	0.0650702	+	0.997881i	$0.479273\pi$
$152$	0	0
$153$	2.89302e10	4.26816
$154$	0	0
$155$	−5.24083e9	−0.729302
$156$	0	0
$157$	−2.56685e9	−0.337173	−0.168586	−	0.985687i	$-0.553920\pi$
$157$	−2.56685e9	−0.337173	−0.168586	+	0.985687i	$0.553920\pi$
$158$	0	0
$159$	−3.43467e9	−0.426185
$160$	0	0
$161$	−7.38582e9	−0.866328
$162$	0	0
$163$	5.82044e9	0.645820	0.322910	−	0.946430i	$-0.395339\pi$
$163$	5.82044e9	0.645820	0.322910	+	0.946430i	$0.395339\pi$
$164$	0	0
$165$	−1.80189e10	−1.89257
$166$	0	0
$167$	1.38231e10	1.37525	0.687624	−	0.726067i	$-0.258654\pi$
$167$	1.38231e10	1.37525	0.687624	+	0.726067i	$0.258654\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	4.54475e10	4.06469
$172$	0	0
$173$	−8.78199e9	−0.745393	−0.372697	−	0.927953i	$-0.621567\pi$
$173$	−8.78199e9	−0.745393	−0.372697	+	0.927953i	$0.621567\pi$
$174$	0	0
$175$	−9.43605e9	−0.760535
$176$	0	0
$177$	−2.72630e9	−0.208782
$178$	0	0
$179$	1.53174e9	0.111518	0.0557592	−	0.998444i	$-0.482242\pi$
$179$	1.53174e9	0.111518	0.0557592	+	0.998444i	$0.482242\pi$
$180$	0	0
$181$	−2.50217e10	−1.73286	−0.866429	−	0.499301i	$-0.833590\pi$
$181$	−2.50217e10	−1.73286	−0.866429	+	0.499301i	$0.833590\pi$
$182$	0	0
$183$	5.04283e9	0.332388
$184$	0	0
$185$	−2.70310e10	−1.69664
$186$	0	0
$187$	1.89823e10	1.13517
$188$	0	0
$189$	3.15397e10	1.79796
$190$	0	0
$191$	−2.67017e10	−1.45174	−0.725870	−	0.687832i	$-0.758563\pi$
$191$	−2.67017e10	−1.45174	−0.725870	+	0.687832i	$0.758563\pi$
$192$	0	0
$193$	3.49608e10	1.81373	0.906866	−	0.421420i	$-0.138468\pi$
$193$	3.49608e10	1.81373	0.906866	+	0.421420i	$0.138468\pi$
$194$	0	0
$195$	−1.57042e10	−0.777787
$196$	0	0
$197$	−1.40965e10	−0.666828	−0.333414	−	0.942781i	$-0.608201\pi$
$197$	−1.40965e10	−0.666828	−0.333414	+	0.942781i	$0.608201\pi$
$198$	0	0
$199$	3.22673e9	0.145856	0.0729279	−	0.997337i	$-0.476766\pi$
$199$	3.22673e9	0.145856	0.0729279	+	0.997337i	$0.476766\pi$
$200$	0	0
$201$	3.59528e10	1.55364
$202$	0	0
$203$	−1.81984e10	−0.752144
$204$	0	0
$205$	−6.61213e10	−2.61486
$206$	0	0
$207$	9.33937e10	3.53550
$208$	0	0
$209$	2.98199e10	1.08105
$210$	0	0
$211$	−3.15217e10	−1.09481	−0.547405	−	0.836868i	$-0.684384\pi$
$211$	−3.15217e10	−1.09481	−0.547405	+	0.836868i	$0.684384\pi$
$212$	0	0
$213$	4.08326e10	1.35925
$214$	0	0
$215$	5.51266e10	1.75950
$216$	0	0
$217$	9.93389e9	0.304124
$218$	0	0
$219$	−9.40768e10	−2.76366
$220$	0	0
$221$	1.65438e10	0.466520
$222$	0	0
$223$	6.47827e10	1.75423	0.877116	−	0.480278i	$-0.159464\pi$
$223$	6.47827e10	1.75423	0.877116	+	0.480278i	$0.159464\pi$
$224$	0	0
$225$	1.19319e11	3.10376
$226$	0	0
$227$	−4.85549e10	−1.21372	−0.606858	−	0.794810i	$-0.707570\pi$
$227$	−4.85549e10	−1.21372	−0.606858	+	0.794810i	$0.707570\pi$
$228$	0	0
$229$	2.84348e9	0.0683268	0.0341634	−	0.999416i	$-0.489123\pi$
$229$	2.84348e9	0.0683268	0.0341634	+	0.999416i	$0.489123\pi$
$230$	0	0
$231$	3.41545e10	0.789213
$232$	0	0
$233$	1.80472e10	0.401151	0.200576	−	0.979678i	$-0.435719\pi$
$233$	1.80472e10	0.401151	0.200576	+	0.979678i	$0.435719\pi$
$234$	0	0
$235$	5.61585e10	1.20119
$236$	0	0
$237$	−4.73115e10	−0.974089
$238$	0	0
$239$	−1.19720e9	−0.0237342	−0.0118671	−	0.999930i	$-0.503778\pi$
$239$	−1.19720e9	−0.0237342	−0.0118671	+	0.999930i	$0.503778\pi$
$240$	0	0
$241$	3.46413e10	0.661482	0.330741	−	0.943722i	$-0.392701\pi$
$241$	3.46413e10	0.661482	0.330741	+	0.943722i	$0.392701\pi$
$242$	0	0
$243$	−1.39417e11	−2.56501
$244$	0	0
$245$	−5.15797e10	−0.914601
$246$	0	0
$247$	2.59892e10	0.444280
$248$	0	0
$249$	1.56771e9	0.0258445
$250$	0	0
$251$	3.03233e10	0.482219	0.241109	−	0.970498i	$-0.422489\pi$
$251$	3.03233e10	0.482219	0.241109	+	0.970498i	$0.422489\pi$
$252$	0	0
$253$	6.12792e10	0.940310
$254$	0	0
$255$	−3.18497e11	−4.71709
$256$	0	0
$257$	−3.62984e10	−0.519025	−0.259512	−	0.965740i	$-0.583562\pi$
$257$	−3.62984e10	−0.519025	−0.259512	+	0.965740i	$0.583562\pi$
$258$	0	0
$259$	5.12367e10	0.707510
$260$	0	0
$261$	2.30119e11	3.06951
$262$	0	0
$263$	3.15096e10	0.406108	0.203054	−	0.979167i	$-0.434913\pi$
$263$	3.15096e10	0.406108	0.203054	+	0.979167i	$0.434913\pi$
$264$	0	0
$265$	2.71235e10	0.337862
$266$	0	0
$267$	−5.18278e9	−0.0624112
$268$	0	0
$269$	−1.02821e11	−1.19729	−0.598643	−	0.801016i	$-0.704293\pi$
$269$	−1.02821e11	−1.19729	−0.598643	+	0.801016i	$0.704293\pi$
$270$	0	0
$271$	1.33069e11	1.49870	0.749349	−	0.662176i	$-0.230367\pi$
$271$	1.33069e11	1.49870	0.749349	+	0.662176i	$0.230367\pi$
$272$	0	0
$273$	2.97671e10	0.324343
$274$	0	0
$275$	7.82897e10	0.825482
$276$	0	0
$277$	6.79776e10	0.693756	0.346878	−	0.937910i	$-0.387242\pi$
$277$	6.79776e10	0.693756	0.346878	+	0.937910i	$0.387242\pi$
$278$	0	0
$279$	−1.25614e11	−1.24114
$280$	0	0
$281$	−1.29384e11	−1.23794	−0.618972	−	0.785413i	$-0.712451\pi$
$281$	−1.29384e11	−1.23794	−0.618972	+	0.785413i	$0.712451\pi$
$282$	0	0
$283$	−9.73902e10	−0.902560	−0.451280	−	0.892382i	$-0.649032\pi$
$283$	−9.73902e10	−0.902560	−0.451280	+	0.892382i	$0.649032\pi$
$284$	0	0
$285$	−5.00338e11	−4.49222
$286$	0	0
$287$	1.25332e11	1.09042
$288$	0	0
$289$	2.16937e11	1.82933
$290$	0	0
$291$	−2.57186e10	−0.210246
$292$	0	0
$293$	−1.65663e11	−1.31317	−0.656586	−	0.754251i	$-0.728000\pi$
$293$	−1.65663e11	−1.31317	−0.656586	+	0.754251i	$0.728000\pi$
$294$	0	0
$295$	2.15295e10	0.165514
$296$	0	0
$297$	−2.61680e11	−1.95149
$298$	0	0
$299$	5.34073e10	0.386439
$300$	0	0
$301$	−1.04491e11	−0.733722
$302$	0	0
$303$	−3.88152e11	−2.64551
$304$	0	0
$305$	−3.98231e10	−0.263503
$306$	0	0
$307$	−8.62672e9	−0.0554272	−0.0277136	−	0.999616i	$-0.508823\pi$
$307$	−8.62672e9	−0.0554272	−0.0277136	+	0.999616i	$0.508823\pi$
$308$	0	0
$309$	1.16279e11	0.725586
$310$	0	0
$311$	2.80863e10	0.170244	0.0851221	−	0.996371i	$-0.472872\pi$
$311$	2.80863e10	0.170244	0.0851221	+	0.996371i	$0.472872\pi$
$312$	0	0
$313$	−1.34760e11	−0.793619	−0.396810	−	0.917901i	$-0.629883\pi$
$313$	−1.34760e11	−0.793619	−0.396810	+	0.917901i	$0.629883\pi$
$314$	0	0
$315$	−4.11068e11	−2.35243
$316$	0	0
$317$	1.74096e11	0.968325	0.484162	−	0.874978i	$-0.339124\pi$
$317$	1.74096e11	0.968325	0.484162	+	0.874978i	$0.339124\pi$
$318$	0	0
$319$	1.50990e11	0.816374
$320$	0	0
$321$	1.27389e11	0.669670
$322$	0	0
$323$	5.27087e11	2.69445
$324$	0	0
$325$	6.82327e10	0.339248
$326$	0	0
$327$	1.48593e11	0.718674
$328$	0	0
$329$	−1.06447e11	−0.500903
$330$	0	0
$331$	4.58632e10	0.210009	0.105005	−	0.994472i	$-0.466514\pi$
$331$	4.58632e10	0.210009	0.105005	+	0.994472i	$0.466514\pi$
$332$	0	0
$333$	−6.47888e11	−2.88736
$334$	0	0
$335$	−2.83918e11	−1.23166
$336$	0	0
$337$	3.52298e11	1.48791	0.743953	−	0.668232i	$-0.232949\pi$
$337$	3.52298e11	1.48791	0.743953	+	0.668232i	$0.232949\pi$
$338$	0	0
$339$	−3.56652e11	−1.46672
$340$	0	0
$341$	−8.24202e10	−0.330095
$342$	0	0
$343$	2.57156e11	1.00317
$344$	0	0
$345$	−1.02818e12	−3.90737
$346$	0	0
$347$	−4.35462e11	−1.61238	−0.806190	−	0.591656i	$-0.798474\pi$
$347$	−4.35462e11	−1.61238	−0.806190	+	0.591656i	$0.798474\pi$
$348$	0	0
$349$	−2.00400e11	−0.723076	−0.361538	−	0.932357i	$-0.617748\pi$
$349$	−2.00400e11	−0.723076	−0.361538	+	0.932357i	$0.617748\pi$
$350$	0	0
$351$	−2.28065e11	−0.802005
$352$	0	0
$353$	1.94480e11	0.666636	0.333318	−	0.942814i	$-0.391832\pi$
$353$	1.94480e11	0.666636	0.333318	+	0.942814i	$0.391832\pi$
$354$	0	0
$355$	−3.22454e11	−1.07756
$356$	0	0
$357$	6.03705e11	1.96706
$358$	0	0
$359$	4.20992e10	0.133767	0.0668835	−	0.997761i	$-0.478694\pi$
$359$	4.20992e10	0.133767	0.0668835	+	0.997761i	$0.478694\pi$
$360$	0	0
$361$	5.05331e11	1.56601
$362$	0	0
$363$	3.38818e11	1.02420
$364$	0	0
$365$	7.42922e11	2.19091
$366$	0	0
$367$	−2.57309e11	−0.740386	−0.370193	−	0.928955i	$-0.620708\pi$
$367$	−2.57309e11	−0.740386	−0.370193	+	0.928955i	$0.620708\pi$
$368$	0	0
$369$	−1.58482e12	−4.45001
$370$	0	0
$371$	−5.14121e10	−0.140891
$372$	0	0
$373$	4.45944e11	1.19286	0.596432	−	0.802664i	$-0.296585\pi$
$373$	4.45944e11	1.19286	0.596432	+	0.802664i	$0.296585\pi$
$374$	0	0
$375$	−2.39674e11	−0.625863
$376$	0	0
$377$	1.31594e11	0.335505
$378$	0	0
$379$	−2.95912e11	−0.736691	−0.368346	−	0.929689i	$-0.620076\pi$
$379$	−2.95912e11	−0.736691	−0.368346	+	0.929689i	$0.620076\pi$
$380$	0	0
$381$	−7.09727e11	−1.72556
$382$	0	0
$383$	1.88160e11	0.446820	0.223410	−	0.974725i	$-0.428281\pi$
$383$	1.88160e11	0.446820	0.223410	+	0.974725i	$0.428281\pi$
$384$	0	0
$385$	−2.69717e11	−0.625656
$386$	0	0
$387$	1.32129e12	2.99433
$388$	0	0
$389$	−8.74755e11	−1.93693	−0.968463	−	0.249157i	$-0.919846\pi$
$389$	−8.74755e11	−1.93693	−0.968463	+	0.249157i	$0.919846\pi$
$390$	0	0
$391$	1.08315e12	2.34366
$392$	0	0
$393$	1.68314e12	3.55922
$394$	0	0
$395$	3.73617e11	0.772218
$396$	0	0
$397$	2.86756e11	0.579369	0.289684	−	0.957122i	$-0.406450\pi$
$397$	2.86756e11	0.579369	0.289684	+	0.957122i	$0.406450\pi$
$398$	0	0
$399$	9.48381e11	1.87329
$400$	0	0
$401$	−8.12312e11	−1.56882	−0.784410	−	0.620243i	$-0.787034\pi$
$401$	−8.12312e11	−1.56882	−0.784410	+	0.620243i	$0.787034\pi$
$402$	0	0
$403$	−7.18326e10	−0.135659
$404$	0	0
$405$	2.34216e12	4.32583
$406$	0	0
$407$	−4.25105e11	−0.767929
$408$	0	0
$409$	−2.34879e10	−0.0415039	−0.0207519	−	0.999785i	$-0.506606\pi$
$409$	−2.34879e10	−0.0415039	−0.0207519	+	0.999785i	$0.506606\pi$
$410$	0	0
$411$	5.37402e11	0.928991
$412$	0	0
$413$	−4.08088e10	−0.0690205
$414$	0	0
$415$	−1.23802e10	−0.0204885
$416$	0	0
$417$	−1.21187e12	−1.96265
$418$	0	0
$419$	1.08173e12	1.71457	0.857285	−	0.514842i	$-0.172150\pi$
$419$	1.08173e12	1.71457	0.857285	+	0.514842i	$0.172150\pi$
$420$	0	0
$421$	−5.68516e11	−0.882010	−0.441005	−	0.897505i	$-0.645378\pi$
$421$	−5.68516e11	−0.882010	−0.441005	+	0.897505i	$0.645378\pi$
$422$	0	0
$423$	1.34603e12	2.04419
$424$	0	0
$425$	1.38382e12	2.05746
$426$	0	0
$427$	7.54840e10	0.109883
$428$	0	0
$429$	−2.46973e11	−0.352040
$430$	0	0
$431$	−5.19620e11	−0.725334	−0.362667	−	0.931919i	$-0.618134\pi$
$431$	−5.19620e11	−0.725334	−0.362667	+	0.931919i	$0.618134\pi$
$432$	0	0
$433$	7.75751e11	1.06054	0.530270	−	0.847829i	$-0.322091\pi$
$433$	7.75751e11	1.06054	0.530270	+	0.847829i	$0.322091\pi$
$434$	0	0
$435$	−2.53341e12	−3.39237
$436$	0	0
$437$	1.70156e12	2.23193
$438$	0	0
$439$	−4.73969e11	−0.609059	−0.304530	−	0.952503i	$-0.598499\pi$
$439$	−4.73969e11	−0.609059	−0.304530	+	0.952503i	$0.598499\pi$
$440$	0	0
$441$	−1.23628e12	−1.55648
$442$	0	0
$443$	−3.20337e11	−0.395175	−0.197588	−	0.980285i	$-0.563311\pi$
$443$	−3.20337e11	−0.395175	−0.197588	+	0.980285i	$0.563311\pi$
$444$	0	0
$445$	4.09283e10	0.0494770
$446$	0	0
$447$	−1.76462e12	−2.09058
$448$	0	0
$449$	2.49131e11	0.289281	0.144640	−	0.989484i	$-0.453797\pi$
$449$	2.49131e11	0.289281	0.144640	+	0.989484i	$0.453797\pi$
$450$	0	0
$451$	−1.03986e12	−1.18353
$452$	0	0
$453$	−2.19381e11	−0.244770
$454$	0	0
$455$	−2.35070e11	−0.257126
$456$	0	0
$457$	−1.41374e12	−1.51616	−0.758082	−	0.652159i	$-0.773863\pi$
$457$	−1.41374e12	−1.51616	−0.758082	+	0.652159i	$0.773863\pi$
$458$	0	0
$459$	−4.62538e12	−4.86397
$460$	0	0
$461$	4.34543e11	0.448103	0.224052	−	0.974577i	$-0.428072\pi$
$461$	4.34543e11	0.448103	0.224052	+	0.974577i	$0.428072\pi$
$462$	0	0
$463$	3.63064e11	0.367171	0.183586	−	0.983004i	$-0.441230\pi$
$463$	3.63064e11	0.367171	0.183586	+	0.983004i	$0.441230\pi$
$464$	0	0
$465$	1.38290e12	1.37168
$466$	0	0
$467$	−1.59471e12	−1.55152	−0.775759	−	0.631029i	$-0.782633\pi$
$467$	−1.59471e12	−1.55152	−0.775759	+	0.631029i	$0.782633\pi$
$468$	0	0
$469$	5.38162e11	0.513612
$470$	0	0
$471$	6.77317e11	0.634159
$472$	0	0
$473$	8.66952e11	0.796379
$474$	0	0
$475$	2.17390e12	1.95938
$476$	0	0
$477$	6.50106e11	0.574978
$478$	0	0
$479$	1.56758e11	0.136057	0.0680284	−	0.997683i	$-0.478329\pi$
$479$	1.56758e11	0.136057	0.0680284	+	0.997683i	$0.478329\pi$
$480$	0	0
$481$	−3.70496e11	−0.315595
$482$	0	0
$483$	1.94890e12	1.62940
$484$	0	0
$485$	2.03099e11	0.166675
$486$	0	0
$487$	−1.81741e12	−1.46411	−0.732053	−	0.681247i	$-0.761438\pi$
$487$	−1.81741e12	−1.46411	−0.732053	+	0.681247i	$0.761438\pi$
$488$	0	0
$489$	−1.53584e12	−1.21467
$490$	0	0
$491$	1.94840e12	1.51290	0.756452	−	0.654049i	$-0.226931\pi$
$491$	1.94840e12	1.51290	0.756452	+	0.654049i	$0.226931\pi$
$492$	0	0
$493$	2.66885e12	2.03476
$494$	0	0
$495$	3.41058e12	2.55331
$496$	0	0
$497$	6.11205e11	0.449349
$498$	0	0
$499$	−1.10980e11	−0.0801292	−0.0400646	−	0.999197i	$-0.512756\pi$
$499$	−1.10980e11	−0.0801292	−0.0400646	+	0.999197i	$0.512756\pi$
$500$	0	0
$501$	−3.64751e12	−2.58659
$502$	0	0
$503$	6.74950e11	0.470128	0.235064	−	0.971980i	$-0.424470\pi$
$503$	6.74950e11	0.470128	0.235064	+	0.971980i	$0.424470\pi$
$504$	0	0
$505$	3.06522e12	2.09725
$506$	0	0
$507$	−2.15247e11	−0.144678
$508$	0	0
$509$	−8.30695e11	−0.548544	−0.274272	−	0.961652i	$-0.588437\pi$
$509$	−8.30695e11	−0.548544	−0.274272	+	0.961652i	$0.588437\pi$
$510$	0	0
$511$	−1.40819e12	−0.913627
$512$	0	0
$513$	−7.26617e12	−4.63210
$514$	0	0
$515$	−9.18254e11	−0.575215
$516$	0	0
$517$	8.83181e11	0.543678
$518$	0	0
$519$	2.31731e12	1.40195
$520$	0	0
$521$	2.67978e12	1.59341	0.796707	−	0.604366i	$-0.206573\pi$
$521$	2.67978e12	1.59341	0.796707	+	0.604366i	$0.206573\pi$
$522$	0	0
$523$	−4.75935e11	−0.278157	−0.139079	−	0.990281i	$-0.544414\pi$
$523$	−4.75935e11	−0.278157	−0.139079	+	0.990281i	$0.544414\pi$
$524$	0	0
$525$	2.48990e12	1.43042
$526$	0	0
$527$	−1.45683e12	−0.822740
$528$	0	0
$529$	1.69552e12	0.941354
$530$	0	0
$531$	5.16027e11	0.281674
$532$	0	0
$533$	−9.06281e11	−0.486396
$534$	0	0
$535$	−1.00599e12	−0.530887
$536$	0	0
$537$	−4.04181e11	−0.209745
$538$	0	0
$539$	−8.11172e11	−0.413965
$540$	0	0
$541$	−9.61621e11	−0.482632	−0.241316	−	0.970447i	$-0.577579\pi$
$541$	−9.61621e11	−0.482632	−0.241316	+	0.970447i	$0.577579\pi$
$542$	0	0
$543$	6.60249e12	3.25918
$544$	0	0
$545$	−1.17343e12	−0.569736
$546$	0	0
$547$	1.17774e12	0.562478	0.281239	−	0.959638i	$-0.409255\pi$
$547$	1.17774e12	0.562478	0.281239	+	0.959638i	$0.409255\pi$
$548$	0	0
$549$	−9.54495e11	−0.448433
$550$	0	0
$551$	4.19259e12	1.93776
$552$	0	0
$553$	−7.08185e11	−0.322020
$554$	0	0
$555$	7.13269e12	3.19106
$556$	0	0
$557$	1.32941e12	0.585209	0.292604	−	0.956234i	$-0.405478\pi$
$557$	1.32941e12	0.585209	0.292604	+	0.956234i	$0.405478\pi$
$558$	0	0
$559$	7.55584e11	0.327288
$560$	0	0
$561$	−5.00886e12	−2.13504
$562$	0	0
$563$	1.68082e12	0.705070	0.352535	−	0.935799i	$-0.385320\pi$
$563$	1.68082e12	0.705070	0.352535	+	0.935799i	$0.385320\pi$
$564$	0	0
$565$	2.81647e12	1.16275
$566$	0	0
$567$	−4.43953e12	−1.80390
$568$	0	0
$569$	4.32025e12	1.72784	0.863920	−	0.503629i	$-0.168002\pi$
$569$	4.32025e12	1.72784	0.863920	+	0.503629i	$0.168002\pi$
$570$	0	0
$571$	−3.55664e12	−1.40016	−0.700080	−	0.714065i	$-0.746852\pi$
$571$	−3.55664e12	−1.40016	−0.700080	+	0.714065i	$0.746852\pi$
$572$	0	0
$573$	7.04580e12	2.73045
$574$	0	0
$575$	4.46731e12	1.70428
$576$	0	0
$577$	−2.69342e11	−0.101161	−0.0505805	−	0.998720i	$-0.516107\pi$
$577$	−2.69342e11	−0.101161	−0.0505805	+	0.998720i	$0.516107\pi$
$578$	0	0
$579$	−9.22512e12	−3.41129
$580$	0	0
$581$	2.34664e10	0.00854384
$582$	0	0
$583$	4.26560e11	0.152923
$584$	0	0
$585$	2.97246e12	1.04933
$586$	0	0
$587$	4.42348e12	1.53777	0.768887	−	0.639384i	$-0.220811\pi$
$587$	4.42348e12	1.53777	0.768887	+	0.639384i	$0.220811\pi$
$588$	0	0
$589$	−2.28859e12	−0.783519
$590$	0	0
$591$	3.71966e12	1.25418
$592$	0	0
$593$	−1.58173e12	−0.525274	−0.262637	−	0.964895i	$-0.584592\pi$
$593$	−1.58173e12	−0.525274	−0.262637	+	0.964895i	$0.584592\pi$
$594$	0	0
$595$	−4.76744e12	−1.55941
$596$	0	0
$597$	−8.51439e11	−0.274327
$598$	0	0
$599$	1.14211e12	0.362483	0.181242	−	0.983439i	$-0.441988\pi$
$599$	1.14211e12	0.362483	0.181242	+	0.983439i	$0.441988\pi$
$600$	0	0
$601$	1.17993e12	0.368912	0.184456	−	0.982841i	$-0.440948\pi$
$601$	1.17993e12	0.368912	0.184456	+	0.982841i	$0.440948\pi$
$602$	0	0
$603$	−6.80505e12	−2.09606
$604$	0	0
$605$	−2.67563e12	−0.811947
$606$	0	0
$607$	2.74076e11	0.0819449	0.0409725	−	0.999160i	$-0.486954\pi$
$607$	2.74076e11	0.0819449	0.0409725	+	0.999160i	$0.486954\pi$
$608$	0	0
$609$	4.80203e12	1.41464
$610$	0	0
$611$	7.69728e11	0.223435
$612$	0	0
$613$	6.77624e12	1.93828	0.969140	−	0.246510i	$-0.0792837\pi$
$613$	6.77624e12	1.93828	0.969140	+	0.246510i	$0.0792837\pi$
$614$	0	0
$615$	1.74475e13	4.91807
$616$	0	0
$617$	1.71098e12	0.475294	0.237647	−	0.971352i	$-0.423624\pi$
$617$	1.71098e12	0.475294	0.237647	+	0.971352i	$0.423624\pi$
$618$	0	0
$619$	−2.18576e12	−0.598405	−0.299202	−	0.954190i	$-0.596721\pi$
$619$	−2.18576e12	−0.598405	−0.299202	+	0.954190i	$0.596721\pi$
$620$	0	0
$621$	−1.49318e13	−4.02904
$622$	0	0
$623$	−7.75788e10	−0.0206323
$624$	0	0
$625$	−2.77335e12	−0.727017
$626$	0	0
$627$	−7.86859e12	−2.03326
$628$	0	0
$629$	−7.51402e12	−1.91401
$630$	0	0
$631$	1.23842e12	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$631$	1.23842e12	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$632$	0	0
$633$	8.31765e12	2.05913
$634$	0	0
$635$	5.60469e12	1.36795
$636$	0	0
$637$	−7.06969e11	−0.170127
$638$	0	0
$639$	−7.72869e12	−1.83380
$640$	0	0
$641$	−3.62933e12	−0.849114	−0.424557	−	0.905401i	$-0.639570\pi$
$641$	−3.62933e12	−0.849114	−0.424557	+	0.905401i	$0.639570\pi$
$642$	0	0
$643$	−5.17653e12	−1.19423	−0.597117	−	0.802154i	$-0.703687\pi$
$643$	−5.17653e12	−1.19423	−0.597117	+	0.802154i	$0.703687\pi$
$644$	0	0
$645$	−1.45463e13	−3.30928
$646$	0	0
$647$	4.97055e12	1.11515	0.557577	−	0.830125i	$-0.311731\pi$
$647$	4.97055e12	1.11515	0.557577	+	0.830125i	$0.311731\pi$
$648$	0	0
$649$	3.38585e11	0.0749146
$650$	0	0
$651$	−2.62126e12	−0.572000
$652$	0	0
$653$	−6.97363e12	−1.50089	−0.750446	−	0.660932i	$-0.770161\pi$
$653$	−6.97363e12	−1.50089	−0.750446	+	0.660932i	$0.770161\pi$
$654$	0	0
$655$	−1.32917e13	−2.82160
$656$	0	0
$657$	1.78066e13	3.72853
$658$	0	0
$659$	−4.81270e12	−0.994041	−0.497021	−	0.867739i	$-0.665573\pi$
$659$	−4.81270e12	−0.994041	−0.497021	+	0.867739i	$0.665573\pi$
$660$	0	0
$661$	−5.92381e11	−0.120697	−0.0603483	−	0.998177i	$-0.519221\pi$
$661$	−5.92381e11	−0.120697	−0.0603483	+	0.998177i	$0.519221\pi$
$662$	0	0
$663$	−4.36543e12	−0.877437
$664$	0	0
$665$	−7.48934e12	−1.48507
$666$	0	0
$667$	8.61568e12	1.68548
$668$	0	0
$669$	−1.70943e13	−3.29938
$670$	0	0
$671$	−6.26281e11	−0.119266
$672$	0	0
$673$	1.89498e12	0.356071	0.178035	−	0.984024i	$-0.443026\pi$
$673$	1.89498e12	0.356071	0.178035	+	0.984024i	$0.443026\pi$
$674$	0	0
$675$	−1.90768e13	−3.53702
$676$	0	0
$677$	8.77240e12	1.60498	0.802490	−	0.596666i	$-0.203508\pi$
$677$	8.77240e12	1.60498	0.802490	+	0.596666i	$0.203508\pi$
$678$	0	0
$679$	−3.84970e11	−0.0695045
$680$	0	0
$681$	1.28122e13	2.28277
$682$	0	0
$683$	−4.82174e12	−0.847834	−0.423917	−	0.905701i	$-0.639345\pi$
$683$	−4.82174e12	−0.847834	−0.423917	+	0.905701i	$0.639345\pi$
$684$	0	0
$685$	−4.24385e12	−0.736466
$686$	0	0
$687$	−7.50312e11	−0.128510
$688$	0	0
$689$	3.71764e11	0.0628465
$690$	0	0
$691$	4.81233e12	0.802979	0.401490	−	0.915864i	$-0.368493\pi$
$691$	4.81233e12	0.802979	0.401490	+	0.915864i	$0.368493\pi$
$692$	0	0
$693$	−6.46468e12	−1.06475
$694$	0	0
$695$	9.57009e12	1.55591
$696$	0	0
$697$	−1.83803e13	−2.94988
$698$	0	0
$699$	−4.76213e12	−0.754490
$700$	0	0
$701$	6.29438e12	0.984515	0.492257	−	0.870450i	$-0.336172\pi$
$701$	6.29438e12	0.984515	0.492257	+	0.870450i	$0.336172\pi$
$702$	0	0
$703$	−1.18040e13	−1.82277
$704$	0	0
$705$	−1.48186e13	−2.25921
$706$	0	0
$707$	−5.81007e12	−0.874569
$708$	0	0
$709$	1.16594e13	1.73288	0.866439	−	0.499283i	$-0.166403\pi$
$709$	1.16594e13	1.73288	0.866439	+	0.499283i	$0.166403\pi$
$710$	0	0
$711$	8.95499e12	1.31417
$712$	0	0
$713$	−4.70301e12	−0.681511
$714$	0	0
$715$	1.95034e12	0.279083
$716$	0	0
$717$	3.15905e11	0.0446396
$718$	0	0
$719$	−1.07333e13	−1.49779	−0.748896	−	0.662688i	$-0.769416\pi$
$719$	−1.07333e13	−1.49779	−0.748896	+	0.662688i	$0.769416\pi$
$720$	0	0
$721$	1.74053e12	0.239869
$722$	0	0
$723$	−9.14083e12	−1.24412
$724$	0	0
$725$	1.10073e13	1.47965
$726$	0	0
$727$	−1.21526e13	−1.61348	−0.806741	−	0.590905i	$-0.798771\pi$
$727$	−1.21526e13	−1.61348	−0.806741	+	0.590905i	$0.798771\pi$
$728$	0	0
$729$	1.46645e13	1.92306
$730$	0	0
$731$	1.53240e13	1.98492
$732$	0	0
$733$	−8.72020e11	−0.111573	−0.0557864	−	0.998443i	$-0.517767\pi$
$733$	−8.72020e11	−0.111573	−0.0557864	+	0.998443i	$0.517767\pi$
$734$	0	0
$735$	1.36104e13	1.72019
$736$	0	0
$737$	−4.46506e12	−0.557473
$738$	0	0
$739$	1.55060e13	1.91250	0.956249	−	0.292555i	$-0.0945055\pi$
$739$	1.55060e13	1.91250	0.956249	+	0.292555i	$0.0945055\pi$
$740$	0	0
$741$	−6.85780e12	−0.835608
$742$	0	0
$743$	1.23691e13	1.48898	0.744489	−	0.667635i	$-0.232693\pi$
$743$	1.23691e13	1.48898	0.744489	+	0.667635i	$0.232693\pi$
$744$	0	0
$745$	1.39352e13	1.65733
$746$	0	0
$747$	−2.96732e11	−0.0348676
$748$	0	0
$749$	1.90684e12	0.221383
$750$	0	0
$751$	−2.33911e12	−0.268331	−0.134166	−	0.990959i	$-0.542835\pi$
$751$	−2.33911e12	−0.268331	−0.134166	+	0.990959i	$0.542835\pi$
$752$	0	0
$753$	−8.00142e12	−0.906964
$754$	0	0
$755$	1.73245e12	0.194044
$756$	0	0
$757$	5.52717e12	0.611747	0.305873	−	0.952072i	$-0.401052\pi$
$757$	5.52717e12	0.611747	0.305873	+	0.952072i	$0.401052\pi$
$758$	0	0
$759$	−1.61698e13	−1.76855
$760$	0	0
$761$	1.59834e12	0.172758	0.0863792	−	0.996262i	$-0.472470\pi$
$761$	1.59834e12	0.172758	0.0863792	+	0.996262i	$0.472470\pi$
$762$	0	0
$763$	2.22422e12	0.237584
$764$	0	0
$765$	6.02843e13	6.36396
$766$	0	0
$767$	2.95091e11	0.0307876
$768$	0	0
$769$	1.00249e13	1.03374	0.516871	−	0.856063i	$-0.327097\pi$
$769$	1.00249e13	1.03374	0.516871	+	0.856063i	$0.327097\pi$
$770$	0	0
$771$	9.57808e12	0.976189
$772$	0	0
$773$	−1.50388e12	−0.151498	−0.0757488	−	0.997127i	$-0.524135\pi$
$773$	−1.50388e12	−0.151498	−0.0757488	+	0.997127i	$0.524135\pi$
$774$	0	0
$775$	−6.00851e12	−0.598287
$776$	0	0
$777$	−1.35199e13	−1.33069
$778$	0	0
$779$	−2.88742e13	−2.80925
$780$	0	0
$781$	−5.07109e12	−0.487721
$782$	0	0
$783$	−3.67915e13	−3.49800
$784$	0	0
$785$	−5.34876e12	−0.502735
$786$	0	0
$787$	5.75786e11	0.0535026	0.0267513	−	0.999642i	$-0.491484\pi$
$787$	5.75786e11	0.0535026	0.0267513	+	0.999642i	$0.491484\pi$
$788$	0	0
$789$	−8.31446e12	−0.763814
$790$	0	0
$791$	−5.33857e12	−0.484876
$792$	0	0
$793$	−5.45829e11	−0.0490148
$794$	0	0
$795$	−7.15711e12	−0.635456
$796$	0	0
$797$	−6.01490e11	−0.0528039	−0.0264020	−	0.999651i	$-0.508405\pi$
$797$	−6.01490e11	−0.0528039	−0.0264020	+	0.999651i	$0.508405\pi$
$798$	0	0
$799$	1.56108e13	1.35508
$800$	0	0
$801$	9.80984e11	0.0842007
$802$	0	0
$803$	1.16836e13	0.991647
$804$	0	0
$805$	−1.53904e13	−1.29172
$806$	0	0
$807$	2.71315e13	2.25187
$808$	0	0
$809$	−5.58387e12	−0.458318	−0.229159	−	0.973389i	$-0.573598\pi$
$809$	−5.58387e12	−0.458318	−0.229159	+	0.973389i	$0.573598\pi$
$810$	0	0
$811$	−3.37482e12	−0.273941	−0.136970	−	0.990575i	$-0.543737\pi$
$811$	−3.37482e12	−0.273941	−0.136970	+	0.990575i	$0.543737\pi$
$812$	0	0
$813$	−3.51129e13	−2.81877
$814$	0	0
$815$	1.21285e13	0.962938
$816$	0	0
$817$	2.40730e13	1.89030
$818$	0	0
$819$	−5.63423e12	−0.437580
$820$	0	0
$821$	−2.07928e13	−1.59724	−0.798619	−	0.601837i	$-0.794436\pi$
$821$	−2.07928e13	−1.59724	−0.798619	+	0.601837i	$0.794436\pi$
$822$	0	0
$823$	6.58431e12	0.500278	0.250139	−	0.968210i	$-0.419524\pi$
$823$	6.58431e12	0.500278	0.250139	+	0.968210i	$0.419524\pi$
$824$	0	0
$825$	−2.06584e13	−1.55258
$826$	0	0
$827$	1.95004e13	1.44967	0.724836	−	0.688922i	$-0.241916\pi$
$827$	1.95004e13	1.44967	0.724836	+	0.688922i	$0.241916\pi$
$828$	0	0
$829$	−1.71623e13	−1.26206	−0.631028	−	0.775760i	$-0.717367\pi$
$829$	−1.71623e13	−1.26206	−0.631028	+	0.775760i	$0.717367\pi$
$830$	0	0
$831$	−1.79373e13	−1.30482
$832$	0	0
$833$	−1.43380e13	−1.03178
$834$	0	0
$835$	2.88043e13	2.05054
$836$	0	0
$837$	2.00832e13	1.41439
$838$	0	0
$839$	2.54842e12	0.177559	0.0887794	−	0.996051i	$-0.471703\pi$
$839$	2.54842e12	0.177559	0.0887794	+	0.996051i	$0.471703\pi$
$840$	0	0
$841$	6.72156e12	0.463328
$842$	0	0
$843$	3.41406e13	2.32834
$844$	0	0
$845$	1.69980e12	0.114695
$846$	0	0
$847$	5.07161e12	0.338587
$848$	0	0
$849$	2.56984e13	1.69755
$850$	0	0
$851$	−2.42570e13	−1.58546
$852$	0	0
$853$	2.70146e13	1.74714	0.873571	−	0.486696i	$-0.161798\pi$
$853$	2.70146e13	1.74714	0.873571	+	0.486696i	$0.161798\pi$
$854$	0	0
$855$	9.47026e13	6.06058
$856$	0	0
$857$	5.73061e12	0.362900	0.181450	−	0.983400i	$-0.441921\pi$
$857$	5.73061e12	0.362900	0.181450	+	0.983400i	$0.441921\pi$
$858$	0	0
$859$	5.52068e11	0.0345958	0.0172979	−	0.999850i	$-0.494494\pi$
$859$	5.52068e11	0.0345958	0.0172979	+	0.999850i	$0.494494\pi$
$860$	0	0
$861$	−3.30714e13	−2.05087
$862$	0	0
$863$	1.84299e13	1.13103	0.565515	−	0.824738i	$-0.308678\pi$
$863$	1.84299e13	1.13103	0.565515	+	0.824738i	$0.308678\pi$
$864$	0	0
$865$	−1.82997e13	−1.11140
$866$	0	0
$867$	−5.72432e13	−3.44063
$868$	0	0
$869$	5.87572e12	0.349520
$870$	0	0
$871$	−3.89148e12	−0.229104
$872$	0	0
$873$	4.86794e12	0.283649
$874$	0	0
$875$	−3.58757e12	−0.206902
$876$	0	0
$877$	3.17878e13	1.81452	0.907262	−	0.420566i	$-0.138169\pi$
$877$	3.17878e13	1.81452	0.907262	+	0.420566i	$0.138169\pi$
$878$	0	0
$879$	4.37137e13	2.46983
$880$	0	0
$881$	−6.66509e12	−0.372747	−0.186374	−	0.982479i	$-0.559674\pi$
$881$	−6.66509e12	−0.372747	−0.186374	+	0.982479i	$0.559674\pi$
$882$	0	0
$883$	−2.99853e13	−1.65992	−0.829958	−	0.557827i	$-0.811635\pi$
$883$	−2.99853e13	−1.65992	−0.829958	+	0.557827i	$0.811635\pi$
$884$	0	0
$885$	−5.68100e12	−0.311301
$886$	0	0
$887$	−3.32734e13	−1.80485	−0.902426	−	0.430846i	$-0.858215\pi$
$887$	−3.32734e13	−1.80485	−0.902426	+	0.430846i	$0.858215\pi$
$888$	0	0
$889$	−1.06236e13	−0.570445
$890$	0	0
$891$	3.68342e13	1.95795
$892$	0	0
$893$	2.45236e13	1.29048
$894$	0	0
$895$	3.19181e12	0.166277
$896$	0	0
$897$	−1.40926e13	−0.726819
$898$	0	0
$899$	−1.15880e13	−0.591686
$900$	0	0
$901$	7.53974e12	0.381149
$902$	0	0
$903$	2.75722e13	1.37999
$904$	0	0
$905$	−5.21397e13	−2.58375
$906$	0	0
$907$	−3.58423e13	−1.75858	−0.879291	−	0.476285i	$-0.841983\pi$
$907$	−3.58423e13	−1.75858	−0.879291	+	0.476285i	$0.841983\pi$
$908$	0	0
$909$	7.34683e13	3.56913
$910$	0	0
$911$	2.69078e13	1.29433	0.647165	−	0.762350i	$-0.275954\pi$
$911$	2.69078e13	1.29433	0.647165	+	0.762350i	$0.275954\pi$
$912$	0	0
$913$	−1.94697e11	−0.00927345
$914$	0	0
$915$	1.05082e13	0.495600
$916$	0	0
$917$	2.51943e13	1.17663
$918$	0	0
$919$	1.20738e12	0.0558371	0.0279185	−	0.999610i	$-0.491112\pi$
$919$	1.20738e12	0.0558371	0.0279185	+	0.999610i	$0.491112\pi$
$920$	0	0
$921$	2.27634e12	0.104248
$922$	0	0
$923$	−4.41966e12	−0.200439
$924$	0	0
$925$	−3.09905e13	−1.39185
$926$	0	0
$927$	−2.20090e13	−0.978908
$928$	0	0
$929$	8.86037e12	0.390285	0.195142	−	0.980775i	$-0.437483\pi$
$929$	8.86037e12	0.390285	0.195142	+	0.980775i	$0.437483\pi$
$930$	0	0
$931$	−2.25241e13	−0.982593
$932$	0	0
$933$	−7.41114e12	−0.320197
$934$	0	0
$935$	3.95548e13	1.69257
$936$	0	0
$937$	7.06949e12	0.299612	0.149806	−	0.988715i	$-0.452135\pi$
$937$	7.06949e12	0.299612	0.149806	+	0.988715i	$0.452135\pi$
$938$	0	0
$939$	3.55593e13	1.49265
$940$	0	0
$941$	−2.86185e11	−0.0118985	−0.00594926	−	0.999982i	$-0.501894\pi$
$941$	−2.86185e11	−0.0118985	−0.00594926	+	0.999982i	$0.501894\pi$
$942$	0	0
$943$	−5.93358e13	−2.44351
$944$	0	0
$945$	6.57217e13	2.68081
$946$	0	0
$947$	−2.95493e13	−1.19391	−0.596956	−	0.802274i	$-0.703623\pi$
$947$	−2.95493e13	−1.19391	−0.596956	+	0.802274i	$0.703623\pi$
$948$	0	0
$949$	1.01827e13	0.407537
$950$	0	0
$951$	−4.59387e13	−1.82124
$952$	0	0
$953$	−1.03130e13	−0.405011	−0.202506	−	0.979281i	$-0.564908\pi$
$953$	−1.03130e13	−0.405011	−0.202506	+	0.979281i	$0.564908\pi$
$954$	0	0
$955$	−5.56405e13	−2.16459
$956$	0	0
$957$	−3.98418e13	−1.53545
$958$	0	0
$959$	8.04414e12	0.307111
$960$	0	0
$961$	−2.01141e13	−0.760756
$962$	0	0
$963$	−2.41119e13	−0.903470
$964$	0	0
$965$	7.28505e13	2.70433
$966$	0	0
$967$	−2.97329e13	−1.09350	−0.546749	−	0.837297i	$-0.684135\pi$
$967$	−2.97329e13	−1.09350	−0.546749	+	0.837297i	$0.684135\pi$
$968$	0	0
$969$	−1.39083e14	−5.06776
$970$	0	0
$971$	4.08487e13	1.47466	0.737330	−	0.675533i	$-0.236086\pi$
$971$	4.08487e13	1.47466	0.737330	+	0.675533i	$0.236086\pi$
$972$	0	0
$973$	−1.81399e13	−0.648825
$974$	0	0
$975$	−1.80046e13	−0.638062
$976$	0	0
$977$	4.86216e13	1.70727	0.853637	−	0.520868i	$-0.174391\pi$
$977$	4.86216e13	1.70727	0.853637	+	0.520868i	$0.174391\pi$
$978$	0	0
$979$	6.43661e11	0.0223942
$980$	0	0
$981$	−2.81252e13	−0.969584
$982$	0	0
$983$	−2.27031e13	−0.775521	−0.387760	−	0.921760i	$-0.626751\pi$
$983$	−2.27031e13	−0.775521	−0.387760	+	0.921760i	$0.626751\pi$
$984$	0	0
$985$	−2.93740e13	−0.994262
$986$	0	0
$987$	2.80884e13	0.942105
$988$	0	0
$989$	4.94694e13	1.64420
$990$	0	0
$991$	1.87834e13	0.618647	0.309323	−	0.950957i	$-0.399897\pi$
$991$	1.87834e13	0.618647	0.309323	+	0.950957i	$0.399897\pi$
$992$	0	0
$993$	−1.21020e13	−0.394988
$994$	0	0
$995$	6.72379e12	0.217476
$996$	0	0
$997$	4.05828e13	1.30081	0.650405	−	0.759587i	$-0.274599\pi$
$997$	4.05828e13	1.30081	0.650405	+	0.759587i	$0.274599\pi$
$998$	0	0
$999$	1.03585e14	3.29042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.m.1.2		8
4.3	odd	2		104.10.a.d.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.7	✓	8	4.3	odd	2
208.10.a.m.1.2		8	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-263.871 q^{3} +2083.78 q^{5} -3949.77 q^{7} +49944.8 q^{9} +32770.7 q^{11} +28561.0 q^{13} -549849. q^{15} +579245. q^{17} +909955. q^{19} +1.04223e6 q^{21} +1.86994e6 q^{23} +2.38902e6 q^{25} -7.98520e6 q^{27} +4.60746e6 q^{29} -2.51506e6 q^{31} -8.64723e6 q^{33} -8.23044e6 q^{35} -1.29721e7 q^{37} -7.53641e6 q^{39} -3.17314e7 q^{41} +2.64551e7 q^{43} +1.04074e8 q^{45} +2.69503e7 q^{47} -2.47530e7 q^{49} -1.52846e8 q^{51} +1.30165e7 q^{53} +6.82869e7 q^{55} -2.40111e8 q^{57} +1.03319e7 q^{59} -1.91110e7 q^{61} -1.97270e8 q^{63} +5.95148e7 q^{65} -1.36252e8 q^{67} -4.93422e8 q^{69} -1.54745e8 q^{71} +3.56526e8 q^{73} -6.30391e8 q^{75} -1.29437e8 q^{77} +1.79298e8 q^{79} +1.12400e9 q^{81} -5.94120e6 q^{83} +1.20702e9 q^{85} -1.21577e9 q^{87} +1.96414e7 q^{89} -1.12809e8 q^{91} +6.63650e8 q^{93} +1.89615e9 q^{95} +9.74665e7 q^{97} +1.63673e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 141 q^{3} + 2051 q^{5} + 2417 q^{7} + 115741 q^{9} + 53118 q^{11} + 228488 q^{13} + 464555 q^{15} + 433095 q^{17} + 434954 q^{19} + 906875 q^{21} + 1124296 q^{23} + 5966065 q^{25} - 7820643 q^{27}+ \cdots + 641626736 q^{99}+O(q^{100}) \) 8 * q - 141 * q^3 + 2051 * q^5 + 2417 * q^7 + 115741 * q^9 + 53118 * q^11 + 228488 * q^13 + 464555 * q^15 + 433095 * q^17 + 434954 * q^19 + 906875 * q^21 + 1124296 * q^23 + 5966065 * q^25 - 7820643 * q^27 - 1913432 * q^29 - 5220148 * q^31 - 1626882 * q^33 - 950707 * q^35 - 419409 * q^37 - 4027101 * q^39 + 12148046 * q^41 - 32143187 * q^43 + 18893544 * q^45 - 57898411 * q^47 + 104302637 * q^49 - 198521477 * q^51 + 273106866 * q^53 - 141214662 * q^55 + 253032058 * q^57 - 50734502 * q^59 + 139520018 * q^61 + 20344158 * q^63 + 58578611 * q^65 - 528144786 * q^67 + 483506576 * q^69 - 334814973 * q^71 + 630864344 * q^73 - 2265796264 * q^75 + 939664146 * q^77 - 865037608 * q^79 + 3194281040 * q^81 - 1520229952 * q^83 + 2130158385 * q^85 - 1167867006 * q^87 + 894700240 * q^89 + 69031937 * q^91 + 3803209548 * q^93 + 2695926542 * q^95 + 980232164 * q^97 + 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more