LMFDB - Embedded newform 208.8.f.a.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(129,208)] 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("208.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 449x^{4} + 37224x^{2} + 205776 $ x^6 + 449x^4 + 37224x^2 + 205776
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$10.0583i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.8.f.a.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-50.8656 q^{3} -8.88672i q^{5} +510.452i q^{7} +400.306 q^{9} -4065.46i q^{11} +(2772.33 + 7420.43i) q^{13} +452.028i q^{15} -1791.54 q^{17} -19743.0i q^{19} -25964.4i q^{21} +11749.0 q^{23} +78046.0 q^{25} +90881.2 q^{27} -183054. q^{29} -247198. i q^{31} +206792. i q^{33} +4536.25 q^{35} -203775. i q^{37} +(-141016. - 377444. i) q^{39} -419322. i q^{41} -17481.7 q^{43} -3557.41i q^{45} +1.17985e6i q^{47} +562982. q^{49} +91127.9 q^{51} +1.32587e6 q^{53} -36128.6 q^{55} +1.00424e6i q^{57} -401485. i q^{59} -3.28655e6 q^{61} +204337. i q^{63} +(65943.3 - 24636.9i) q^{65} +2.10629e6i q^{67} -597618. q^{69} +4.22986e6i q^{71} +3.29726e6i q^{73} -3.96986e6 q^{75} +2.07522e6 q^{77} -3.79387e6 q^{79} -5.49819e6 q^{81} +4.36639e6i q^{83} +15920.9i q^{85} +9.31117e6 q^{87} +8.72168e6i q^{89} +(-3.78777e6 + 1.41514e6i) q^{91} +1.25738e7i q^{93} -175451. q^{95} +9.21482e6i q^{97} -1.62743e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 56 q^{3} - 1150 q^{9} - 5018 q^{13} + 13152 q^{17} - 27264 q^{23} - 18262 q^{25} - 194560 q^{27} + 42924 q^{29} + 546720 q^{35} - 511160 q^{39} + 1005576 q^{43} + 3246846 q^{49} - 297984 q^{51} + 1705524 q^{53}+ \cdots + 22075632 q^{95}+O(q^{100}) $ 6 * q + 56 * q^3 - 1150 * q^9 - 5018 * q^13 + 13152 * q^17 - 27264 * q^23 - 18262 * q^25 - 194560 * q^27 + 42924 * q^29 + 546720 * q^35 - 511160 * q^39 + 1005576 * q^43 + 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 + 5320576 * q^55 - 8237572 * q^61 + 7139028 * q^65 - 6017400 * q^69 - 20426072 * q^75 + 4093560 * q^77 - 15147200 * q^79 - 21318442 * q^81 + 36474240 * q^87 - 21281832 * q^91 + 22075632 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−50.8656			−1.08768			−0.543838	−	0.839190i	$-0.683029\pi$
$3$	−50.8656			−1.08768			−0.543838	+	0.839190i	$0.683029\pi$
$4$		0			0
$5$		−	8.88672i		−	0.0317941i	−0.999874	−	0.0158971i	$-0.994940\pi$
$5$		−	8.88672i		−	0.0317941i	0.999874	−	0.0158971i	$-0.00506040\pi$
$6$		0			0
$7$			510.452i			0.562486i	0.959637	+	0.281243i	$0.0907467\pi$
$7$			510.452i			0.562486i	−0.959637	+	0.281243i	$0.909253\pi$
$8$		0			0
$9$	400.306			0.183039
$10$		0			0
$11$		−	4065.46i		−	0.920949i	−0.887673	−	0.460474i	$-0.847679\pi$
$11$		−	4065.46i		−	0.920949i	0.887673	−	0.460474i	$-0.152321\pi$
$12$		0			0
$13$	2772.33	+	7420.43i	0.349980	+	0.936757i
$13$	2772.33	+	7420.43i	0.349980	+	0.936757i
$14$		0			0
$15$			452.028i			0.0345817i
$16$		0			0
$17$	−1791.54			−0.0884415			−0.0442207	−	0.999022i	$-0.514080\pi$
$17$	−1791.54			−0.0884415			−0.0442207	+	0.999022i	$0.514080\pi$
$18$		0			0
$19$		−	19743.0i		−	0.660353i	−0.943919	−	0.330176i	$-0.892892\pi$
$19$		−	19743.0i		−	0.660353i	0.943919	−	0.330176i	$-0.107108\pi$
$20$		0			0
$21$		−	25964.4i		−	0.611802i
$22$		0			0
$23$	11749.0			0.201350			0.100675	−	0.994919i	$-0.467900\pi$
$23$	11749.0			0.201350			0.100675	+	0.994919i	$0.467900\pi$
$24$		0			0
$25$	78046.0			0.998989
$26$		0			0
$27$	90881.2			0.888589
$28$		0			0
$29$	−183054.			−1.39376			−0.696879	−	0.717189i	$-0.745429\pi$
$29$	−183054.			−1.39376			−0.696879	+	0.717189i	$0.745429\pi$
$30$		0			0
$31$		−	247198.i		−	1.49032i	−0.666888	−	0.745158i	$-0.732374\pi$
$31$		−	247198.i		−	1.49032i	0.666888	−	0.745158i	$-0.267626\pi$
$32$		0			0
$33$			206792.i			1.00169i
$34$		0			0
$35$	4536.25			0.0178837
$36$		0			0
$37$		−	203775.i		−	0.661372i	−0.943741	−	0.330686i	$-0.892720\pi$
$37$		−	203775.i		−	0.661372i	0.943741	−	0.330686i	$-0.107280\pi$
$38$		0			0
$39$	−141016.	−	377444.i	−0.380665	−	1.01889i
$40$		0			0
$41$		−	419322.i		−	0.950175i	−0.879938	−	0.475088i	$-0.842416\pi$
$41$		−	419322.i		−	0.950175i	0.879938	−	0.475088i	$-0.157584\pi$
$42$		0			0
$43$	−17481.7			−0.0335309			−0.0167654	−	0.999859i	$-0.505337\pi$
$43$	−17481.7			−0.0335309			−0.0167654	+	0.999859i	$0.505337\pi$
$44$		0			0
$45$		−	3557.41i		−	0.00581956i
$46$		0			0
$47$			1.17985e6i			1.65761i	0.559537	+	0.828806i	$0.310979\pi$
$47$			1.17985e6i			1.65761i	−0.559537	+	0.828806i	$0.689021\pi$
$48$		0			0
$49$	562982.			0.683610
$50$		0			0
$51$	91127.9			0.0961957
$52$		0			0
$53$	1.32587e6			1.22330			0.611651	−	0.791127i	$-0.290506\pi$
$53$	1.32587e6			1.22330			0.611651	+	0.791127i	$0.290506\pi$
$54$		0			0
$55$	−36128.6			−0.0292807
$56$		0			0
$57$			1.00424e6i			0.718250i
$58$		0			0
$59$		−	401485.i		−	0.254499i	−0.991871	−	0.127250i	$-0.959385\pi$
$59$		−	401485.i		−	0.254499i	0.991871	−	0.127250i	$-0.0406150\pi$
$60$		0			0
$61$	−3.28655e6			−1.85390			−0.926949	−	0.375188i	$-0.877578\pi$
$61$	−3.28655e6			−1.85390			−0.926949	+	0.375188i	$0.877578\pi$
$62$		0			0
$63$			204337.i			0.102957i
$64$		0			0
$65$	65943.3	−	24636.9i	0.0297834	−	0.0111273i
$66$		0			0
$67$			2.10629e6i			0.855572i	0.903880	+	0.427786i	$0.140706\pi$
$67$			2.10629e6i			0.855572i	−0.903880	+	0.427786i	$0.859294\pi$
$68$		0			0
$69$	−597618.			−0.219004
$70$		0			0
$71$			4.22986e6i			1.40256i	0.712886	+	0.701280i	$0.247388\pi$
$71$			4.22986e6i			1.40256i	−0.712886	+	0.701280i	$0.752612\pi$
$72$		0			0
$73$			3.29726e6i			0.992027i	0.868315	+	0.496014i	$0.165203\pi$
$73$			3.29726e6i			0.992027i	−0.868315	+	0.496014i	$0.834797\pi$
$74$		0			0
$75$	−3.96986e6			−1.08658
$76$		0			0
$77$	2.07522e6			0.518021
$78$		0			0
$79$	−3.79387e6			−0.865741			−0.432870	−	0.901456i	$-0.642499\pi$
$79$	−3.79387e6			−0.865741			−0.432870	+	0.901456i	$0.642499\pi$
$80$		0			0
$81$	−5.49819e6			−1.14954
$82$		0			0
$83$			4.36639e6i			0.838203i	0.907939	+	0.419102i	$0.137655\pi$
$83$			4.36639e6i			0.838203i	−0.907939	+	0.419102i	$0.862345\pi$
$84$		0			0
$85$			15920.9i			0.00281192i
$86$		0			0
$87$	9.31117e6			1.51596
$88$		0			0
$89$			8.72168e6i			1.31140i	0.755022	+	0.655700i	$0.227626\pi$
$89$			8.72168e6i			1.31140i	−0.755022	+	0.655700i	$0.772374\pi$
$90$		0			0
$91$	−3.78777e6	+	1.41514e6i	−0.526913	+	0.196859i
$92$		0			0
$93$			1.25738e7i			1.62098i
$94$		0			0
$95$	−175451.			−0.0209953
$96$		0			0
$97$			9.21482e6i			1.02515i	0.858644	+	0.512573i	$0.171308\pi$
$97$			9.21482e6i			1.02515i	−0.858644	+	0.512573i	$0.828692\pi$
$98$		0			0
$99$		−	1.62743e6i		−	0.168569i
$100$		0			0
$101$	2.06993e6			0.199908			0.0999539	−	0.994992i	$-0.468130\pi$
$101$	2.06993e6			0.199908			0.0999539	+	0.994992i	$0.468130\pi$
$102$		0			0
$103$	−1.74931e7			−1.57738			−0.788688	−	0.614793i	$-0.789239\pi$
$103$	−1.74931e7			−1.57738			−0.788688	+	0.614793i	$0.789239\pi$
$104$		0			0
$105$	−230739.			−0.0194517
$106$		0			0
$107$	1.55162e7			1.22445			0.612226	−	0.790683i	$-0.290274\pi$
$107$	1.55162e7			1.22445			0.612226	+	0.790683i	$0.290274\pi$
$108$		0			0
$109$		−	1.59529e7i		−	1.17990i	−0.807438	−	0.589952i	$-0.799147\pi$
$109$		−	1.59529e7i		−	1.17990i	0.807438	−	0.589952i	$-0.200853\pi$
$110$		0			0
$111$			1.03651e7i			0.719358i
$112$		0			0
$113$	−1.65661e7			−1.08006			−0.540028	−	0.841647i	$-0.681586\pi$
$113$	−1.65661e7			−1.08006			−0.540028	+	0.841647i	$0.681586\pi$
$114$		0			0
$115$		−	104410.i		−	0.00640176i
$116$		0			0
$117$	1.10978e6	+	2.97044e6i	0.0640599	+	0.171463i
$118$		0			0
$119$		−	914497.i		−	0.0497471i
$120$		0			0
$121$	2.95919e6			0.151853
$122$		0			0
$123$			2.13290e7i			1.03348i
$124$		0			0
$125$		−	1.38785e6i		−	0.0635561i
$126$		0			0
$127$	−1.98781e6			−0.0861118			−0.0430559	−	0.999073i	$-0.513709\pi$
$127$	−1.98781e6			−0.0861118			−0.0430559	+	0.999073i	$0.513709\pi$
$128$		0			0
$129$	889218.			0.0364707
$130$		0			0
$131$	−1.58779e7			−0.617081			−0.308541	−	0.951211i	$-0.599841\pi$
$131$	−1.58779e7			−0.617081			−0.308541	+	0.951211i	$0.599841\pi$
$132$		0			0
$133$	1.00779e7			0.371439
$134$		0			0
$135$		−	807636.i		−	0.0282519i
$136$		0			0
$137$			4.95975e7i			1.64793i	0.566643	+	0.823963i	$0.308242\pi$
$137$			4.95975e7i			1.64793i	−0.566643	+	0.823963i	$0.691758\pi$
$138$		0			0
$139$	5.71630e6			0.180536			0.0902678	−	0.995918i	$-0.471228\pi$
$139$	5.71630e6			0.180536			0.0902678	+	0.995918i	$0.471228\pi$
$140$		0			0
$141$		−	6.00135e7i		−	1.80294i
$142$		0			0
$143$	3.01675e7	−	1.12708e7i	0.862705	−	0.322313i
$144$		0			0
$145$			1.62675e6i			0.0443133i
$146$		0			0
$147$	−2.86364e7			−0.743546
$148$		0			0
$149$			1.51998e7i			0.376431i	0.982128	+	0.188216i	$0.0602704\pi$
$149$			1.51998e7i			0.376431i	−0.982128	+	0.188216i	$0.939730\pi$
$150$		0			0
$151$			6.47181e7i			1.52970i	0.644208	+	0.764850i	$0.277187\pi$
$151$			6.47181e7i			1.52970i	−0.644208	+	0.764850i	$0.722813\pi$
$152$		0			0
$153$	−717165.			−0.0161882
$154$		0			0
$155$	−2.19678e6			−0.0473833
$156$		0			0
$157$	2.86572e6			0.0590997			0.0295499	−	0.999563i	$-0.490593\pi$
$157$	2.86572e6			0.0590997			0.0295499	+	0.999563i	$0.490593\pi$
$158$		0			0
$159$	−6.74409e7			−1.33056
$160$		0			0
$161$			5.99728e6i			0.113257i
$162$		0			0
$163$		−	3.56517e7i		−	0.644797i	−0.946604	−	0.322399i	$-0.895511\pi$
$163$		−	3.56517e7i		−	0.644797i	0.946604	−	0.322399i	$-0.104489\pi$
$164$		0			0
$165$	1.83770e6			0.0318480
$166$		0			0
$167$		−	2.20447e7i		−	0.366266i	−0.983088	−	0.183133i	$-0.941376\pi$
$167$		−	2.20447e7i		−	0.366266i	0.983088	−	0.183133i	$-0.0586239\pi$
$168$		0			0
$169$	−4.73769e7	+	4.11437e7i	−0.755028	+	0.655692i
$170$		0			0
$171$		−	7.90325e6i		−	0.120870i
$172$		0			0
$173$	−1.96908e7			−0.289136			−0.144568	−	0.989495i	$-0.546179\pi$
$173$	−1.96908e7			−0.289136			−0.144568	+	0.989495i	$0.546179\pi$
$174$		0			0
$175$			3.98387e7i			0.561917i
$176$		0			0
$177$			2.04217e7i			0.276813i
$178$		0			0
$179$	7.90229e7			1.02983			0.514917	−	0.857240i	$-0.327823\pi$
$179$	7.90229e7			1.02983			0.514917	+	0.857240i	$0.327823\pi$
$180$		0			0
$181$	4.94136e7			0.619400			0.309700	−	0.950834i	$-0.399771\pi$
$181$	4.94136e7			0.619400			0.309700	+	0.950834i	$0.399771\pi$
$182$		0			0
$183$	1.67172e8			2.01644
$184$		0			0
$185$	−1.81090e6			−0.0210277
$186$		0			0
$187$			7.28345e6i			0.0814501i
$188$		0			0
$189$			4.63905e7i			0.499819i
$190$		0			0
$191$	−1.64464e8			−1.70787			−0.853935	−	0.520379i	$-0.825791\pi$
$191$	−1.64464e8			−1.70787			−0.853935	+	0.520379i	$0.825791\pi$
$192$		0			0
$193$			8.78627e7i			0.879740i	0.898062	+	0.439870i	$0.144975\pi$
$193$			8.78627e7i			0.879740i	−0.898062	+	0.439870i	$0.855025\pi$
$194$		0			0
$195$	−3.35424e6	+	1.25317e6i	−0.0323946	+	0.0121029i
$196$		0			0
$197$		−	1.98762e7i		−	0.185225i	−0.995702	−	0.0926127i	$-0.970478\pi$
$197$		−	1.98762e7i		−	0.185225i	0.995702	−	0.0926127i	$-0.0295218\pi$
$198$		0			0
$199$	2.80120e7			0.251975			0.125988	−	0.992032i	$-0.459790\pi$
$199$	2.80120e7			0.251975			0.125988	+	0.992032i	$0.459790\pi$
$200$		0			0
$201$		−	1.07138e8i		−	0.930585i
$202$		0			0
$203$		−	9.34405e7i		−	0.783969i
$204$		0			0
$205$	−3.72640e6			−0.0302100
$206$		0			0
$207$	4.70318e6			0.0368549
$208$		0			0
$209$	−8.02645e7			−0.608151
$210$		0			0
$211$	−2.39701e7			−0.175663			−0.0878315	−	0.996135i	$-0.527994\pi$
$211$	−2.39701e7			−0.175663			−0.0878315	+	0.996135i	$0.527994\pi$
$212$		0			0
$213$		−	2.15154e8i		−	1.52553i
$214$		0			0
$215$			155355.i			0.00106608i
$216$		0			0
$217$	1.26182e8			0.838281
$218$		0			0
$219$		−	1.67717e8i		−	1.07900i
$220$		0			0
$221$	−4.96675e6	−	1.32940e7i	−0.0309527	−	0.0828482i
$222$		0			0
$223$		−	1.88776e8i		−	1.13993i	−0.821668	−	0.569967i	$-0.806956\pi$
$223$		−	1.88776e8i		−	1.13993i	0.821668	−	0.569967i	$-0.193044\pi$
$224$		0			0
$225$	3.12423e7			0.182854
$226$		0			0
$227$			766382.i			0.00434865i	0.999998	+	0.00217433i	$0.000692110\pi$
$227$			766382.i			0.00434865i	−0.999998	+	0.00217433i	$0.999308\pi$
$228$		0			0
$229$		−	1.06420e8i		−	0.585597i	−0.956174	−	0.292798i	$-0.905414\pi$
$229$		−	1.06420e8i		−	0.585597i	0.956174	−	0.292798i	$-0.0945865\pi$
$230$		0			0
$231$	−1.05557e8			−0.563439
$232$		0			0
$233$	−2.05485e7			−0.106423			−0.0532113	−	0.998583i	$-0.516946\pi$
$233$	−2.05485e7			−0.106423			−0.0532113	+	0.998583i	$0.516946\pi$
$234$		0			0
$235$	1.04850e7			0.0527023
$236$		0			0
$237$	1.92977e8			0.941645
$238$		0			0
$239$			6.47473e7i			0.306781i	0.988166	+	0.153391i	$0.0490193\pi$
$239$			6.47473e7i			0.306781i	−0.988166	+	0.153391i	$0.950981\pi$
$240$		0			0
$241$		−	4.01781e8i		−	1.84897i	−0.381217	−	0.924486i	$-0.624495\pi$
$241$		−	4.01781e8i		−	1.84897i	0.381217	−	0.924486i	$-0.375505\pi$
$242$		0			0
$243$	8.09115e7			0.361733
$244$		0			0
$245$		−	5.00306e6i		−	0.0217348i
$246$		0			0
$247$	1.46502e8	−	5.47342e7i	0.618590	−	0.231110i
$248$		0			0
$249$		−	2.22099e8i		−	0.911694i
$250$		0			0
$251$	2.77091e8			1.10602			0.553011	−	0.833174i	$-0.313479\pi$
$251$	2.77091e8			1.10602			0.553011	+	0.833174i	$0.313479\pi$
$252$		0			0
$253$		−	4.77650e7i		−	0.185433i
$254$		0			0
$255$		−	809828.i		−	0.00305846i
$256$		0			0
$257$	4.67270e8			1.71713			0.858563	−	0.512708i	$-0.171358\pi$
$257$	4.67270e8			1.71713			0.858563	+	0.512708i	$0.171358\pi$
$258$		0			0
$259$	1.04017e8			0.372012
$260$		0			0
$261$	−7.32778e7			−0.255112
$262$		0			0
$263$	−7.58768e7			−0.257196			−0.128598	−	0.991697i	$-0.541048\pi$
$263$	−7.58768e7			−0.257196			−0.128598	+	0.991697i	$0.541048\pi$
$264$		0			0
$265$		−	1.17826e7i		−	0.0388938i
$266$		0			0
$267$		−	4.43633e8i		−	1.42638i
$268$		0			0
$269$	−1.93468e8			−0.606005			−0.303002	−	0.952990i	$-0.597989\pi$
$269$	−1.93468e8			−0.606005			−0.303002	+	0.952990i	$0.597989\pi$
$270$		0			0
$271$			4.04769e8i			1.23542i	0.786406	+	0.617710i	$0.211940\pi$
$271$			4.04769e8i			1.23542i	−0.786406	+	0.617710i	$0.788060\pi$
$272$		0			0
$273$	1.92667e8	−	7.19819e7i	0.573110	−	0.214118i
$274$		0			0
$275$		−	3.17293e8i		−	0.920018i
$276$		0			0
$277$	−4.66835e8			−1.31973			−0.659864	−	0.751385i	$-0.729386\pi$
$277$	−4.66835e8			−1.31973			−0.659864	+	0.751385i	$0.729386\pi$
$278$		0			0
$279$		−	9.89546e7i		−	0.272786i
$280$		0			0
$281$			2.83566e8i			0.762397i	0.924493	+	0.381199i	$0.124489\pi$
$281$			2.83566e8i			0.762397i	−0.924493	+	0.381199i	$0.875511\pi$
$282$		0			0
$283$	−3.54304e8			−0.929229			−0.464615	−	0.885513i	$-0.653807\pi$
$283$	−3.54304e8			−0.929229			−0.464615	+	0.885513i	$0.653807\pi$
$284$		0			0
$285$	8.92441e6			0.0228361
$286$		0			0
$287$	2.14044e8			0.534460
$288$		0			0
$289$	−4.07129e8			−0.992178
$290$		0			0
$291$		−	4.68717e8i		−	1.11503i
$292$		0			0
$293$			5.68237e8i			1.31975i	0.751374	+	0.659877i	$0.229392\pi$
$293$			5.68237e8i			1.31975i	−0.751374	+	0.659877i	$0.770608\pi$
$294$		0			0
$295$	−3.56788e6			−0.00809158
$296$		0			0
$297$		−	3.69474e8i		−	0.818345i
$298$		0			0
$299$	3.25720e7	+	8.71824e7i	0.0704685	+	0.188616i
$300$		0			0
$301$		−	8.92359e6i		−	0.0188606i
$302$		0			0
$303$	−1.05288e8			−0.217435
$304$		0			0
$305$			2.92067e7i			0.0589430i
$306$		0			0
$307$			6.06885e8i			1.19708i	0.801094	+	0.598538i	$0.204251\pi$
$307$			6.06885e8i			1.19708i	−0.801094	+	0.598538i	$0.795749\pi$
$308$		0			0
$309$	8.89794e8			1.71567
$310$		0			0
$311$	−5.70878e8			−1.07617			−0.538086	−	0.842890i	$-0.680853\pi$
$311$	−5.70878e8			−1.07617			−0.538086	+	0.842890i	$0.680853\pi$
$312$		0			0
$313$	1.19008e8			0.219367			0.109684	−	0.993967i	$-0.465016\pi$
$313$	1.19008e8			0.219367			0.109684	+	0.993967i	$0.465016\pi$
$314$		0			0
$315$	1.81589e6			0.00327342
$316$		0			0
$317$			7.11011e8i			1.25363i	0.779169	+	0.626814i	$0.215642\pi$
$317$			7.11011e8i			1.25363i	−0.779169	+	0.626814i	$0.784358\pi$
$318$		0			0
$319$			7.44201e8i			1.28358i
$320$		0			0
$321$	−7.89240e8			−1.33181
$322$		0			0
$323$			3.53705e7i			0.0584026i
$324$		0			0
$325$	2.16369e8	+	5.79135e8i	0.349626	+	0.935810i
$326$		0			0
$327$			8.11453e8i			1.28335i
$328$		0			0
$329$	−6.02254e8			−0.932383
$330$		0			0
$331$		−	1.03380e9i		−	1.56689i	−0.621459	−	0.783447i	$-0.713460\pi$
$331$		−	1.03380e9i		−	1.56689i	0.621459	−	0.783447i	$-0.286540\pi$
$332$		0			0
$333$		−	8.15724e7i		−	0.121057i
$334$		0			0
$335$	1.87180e7			0.0272021
$336$		0			0
$337$	8.03307e8			1.14334			0.571672	−	0.820482i	$-0.306295\pi$
$337$	8.03307e8			1.14334			0.571672	+	0.820482i	$0.306295\pi$
$338$		0			0
$339$	8.42645e8			1.17475
$340$		0			0
$341$	−1.00497e9			−1.37250
$342$		0			0
$343$			7.07754e8i			0.947007i
$344$		0			0
$345$			5.31087e6i			0.00696303i
$346$		0			0
$347$	−3.48273e8			−0.447473			−0.223737	−	0.974650i	$-0.571826\pi$
$347$	−3.48273e8			−0.447473			−0.223737	+	0.974650i	$0.571826\pi$
$348$		0			0
$349$			6.90910e8i			0.870027i	0.900424	+	0.435013i	$0.143256\pi$
$349$			6.90910e8i			0.870027i	−0.900424	+	0.435013i	$0.856744\pi$
$350$		0			0
$351$	2.51953e8	+	6.74377e8i	0.310988	+	0.832392i
$352$		0			0
$353$			3.16345e8i			0.382780i	0.981514	+	0.191390i	$0.0612996\pi$
$353$			3.16345e8i			0.382780i	−0.981514	+	0.191390i	$0.938700\pi$
$354$		0			0
$355$	3.75896e7			0.0445932
$356$		0			0
$357$			4.65164e7i			0.0541087i
$358$		0			0
$359$		−	5.14441e8i		−	0.586820i	−0.955987	−	0.293410i	$-0.905210\pi$
$359$		−	5.14441e8i		−	0.586820i	0.955987	−	0.293410i	$-0.0947901\pi$
$360$		0			0
$361$	5.04085e8			0.563934
$362$		0			0
$363$	−1.50521e8			−0.165167
$364$		0			0
$365$	2.93019e7			0.0315406
$366$		0			0
$367$	−1.47338e9			−1.55591			−0.777953	−	0.628323i	$-0.783742\pi$
$367$	−1.47338e9			−1.55591			−0.777953	+	0.628323i	$0.783742\pi$
$368$		0			0
$369$		−	1.67857e8i		−	0.173919i
$370$		0			0
$371$			6.76791e8i			0.688091i
$372$		0			0
$373$	−9.69375e8			−0.967188			−0.483594	−	0.875292i	$-0.660669\pi$
$373$	−9.69375e8			−0.967188			−0.483594	+	0.875292i	$0.660669\pi$
$374$		0			0
$375$			7.05937e7i			0.0691284i
$376$		0			0
$377$	−5.07487e8	−	1.35834e9i	−0.487787	−	1.30561i
$378$		0			0
$379$			3.46921e8i			0.327335i	0.986516	+	0.163668i	$0.0523325\pi$
$379$			3.46921e8i			0.327335i	−0.986516	+	0.163668i	$0.947668\pi$
$380$		0			0
$381$	1.01111e8			0.0936617
$382$		0			0
$383$			1.83888e9i			1.67247i	0.548373	+	0.836234i	$0.315247\pi$
$383$			1.83888e9i			1.67247i	−0.548373	+	0.836234i	$0.684753\pi$
$384$		0			0
$385$		−	1.84419e7i		−	0.0164700i
$386$		0			0
$387$	−6.99804e6			−0.00613745
$388$		0			0
$389$	1.00176e9			0.862861			0.431431	−	0.902146i	$-0.358009\pi$
$389$	1.00176e9			0.862861			0.431431	+	0.902146i	$0.358009\pi$
$390$		0			0
$391$	−2.10488e7			−0.0178077
$392$		0			0
$393$	8.07636e8			0.671184
$394$		0			0
$395$			3.37151e7i			0.0275255i
$396$		0			0
$397$		−	1.39793e9i		−	1.12129i	−0.828055	−	0.560647i	$-0.810552\pi$
$397$		−	1.39793e9i		−	1.12129i	0.828055	−	0.560647i	$-0.189448\pi$
$398$		0			0
$399$	−5.12616e8			−0.404005
$400$		0			0
$401$			1.21136e9i			0.938144i	0.883160	+	0.469072i	$0.155411\pi$
$401$			1.21136e9i			0.938144i	−0.883160	+	0.469072i	$0.844589\pi$
$402$		0			0
$403$	1.83431e9	−	6.85313e8i	1.39606	−	0.521580i
$404$		0			0
$405$			4.88609e7i			0.0365485i
$406$		0			0
$407$	−8.28441e8			−0.609089
$408$		0			0
$409$			1.45644e9i			1.05260i	0.850300	+	0.526299i	$0.176421\pi$
$409$			1.45644e9i			1.05260i	−0.850300	+	0.526299i	$0.823579\pi$
$410$		0			0
$411$		−	2.52281e9i		−	1.79241i
$412$		0			0
$413$	2.04939e8			0.143152
$414$		0			0
$415$	3.88029e7			0.0266499
$416$		0			0
$417$	−2.90763e8			−0.196364
$418$		0			0
$419$	−1.35223e9			−0.898051			−0.449026	−	0.893519i	$-0.648229\pi$
$419$	−1.35223e9			−0.898051			−0.449026	+	0.893519i	$0.648229\pi$
$420$		0			0
$421$			1.35611e9i			0.885742i	0.896585	+	0.442871i	$0.146040\pi$
$421$			1.35611e9i			0.885742i	−0.896585	+	0.442871i	$0.853960\pi$
$422$		0			0
$423$			4.72299e8i			0.303407i
$424$		0			0
$425$	−1.39823e8			−0.0883521
$426$		0			0
$427$		−	1.67763e9i		−	1.04279i
$428$		0			0
$429$	−1.53448e9	+	5.73295e8i	−0.938344	+	0.350573i
$430$		0			0
$431$			1.27281e9i			0.765764i	0.923797	+	0.382882i	$0.125068\pi$
$431$			1.27281e9i			0.765764i	−0.923797	+	0.382882i	$0.874932\pi$
$432$		0			0
$433$	−3.39003e8			−0.200676			−0.100338	−	0.994953i	$-0.531992\pi$
$433$	−3.39003e8			−0.200676			−0.100338	+	0.994953i	$0.531992\pi$
$434$		0			0
$435$		−	8.27458e7i		−	0.0481985i
$436$		0			0
$437$		−	2.31960e8i		−	0.132962i
$438$		0			0
$439$	1.40195e9			0.790870			0.395435	−	0.918494i	$-0.370594\pi$
$439$	1.40195e9			0.790870			0.395435	+	0.918494i	$0.370594\pi$
$440$		0			0
$441$	2.25365e8			0.125127
$442$		0			0
$443$	3.31716e9			1.81282			0.906408	−	0.422404i	$-0.138813\pi$
$443$	3.31716e9			1.81282			0.906408	+	0.422404i	$0.138813\pi$
$444$		0			0
$445$	7.75071e7			0.0416948
$446$		0			0
$447$		−	7.73146e8i		−	0.409435i
$448$		0			0
$449$			8.90007e8i			0.464014i	0.972714	+	0.232007i	$0.0745292\pi$
$449$			8.90007e8i			0.464014i	−0.972714	+	0.232007i	$0.925471\pi$
$450$		0			0
$451$	−1.70474e9			−0.875063
$452$		0			0
$453$		−	3.29192e9i		−	1.66382i
$454$		0			0
$455$	1.25760e7	+	3.36609e7i	0.00625895	+	0.0167527i
$456$		0			0
$457$			1.66381e9i			0.815448i	0.913105	+	0.407724i	$0.133677\pi$
$457$			1.66381e9i			0.815448i	−0.913105	+	0.407724i	$0.866323\pi$
$458$		0			0
$459$	−1.62818e8			−0.0785881
$460$		0			0
$461$		−	2.68236e7i		−	0.0127516i	−0.999980	−	0.00637580i	$-0.997971\pi$
$461$		−	2.68236e7i		−	0.0127516i	0.999980	−	0.00637580i	$-0.00202949\pi$
$462$		0			0
$463$			3.55847e9i			1.66621i	0.553115	+	0.833105i	$0.313439\pi$
$463$			3.55847e9i			1.66621i	−0.553115	+	0.833105i	$0.686561\pi$
$464$		0			0
$465$	1.11740e8			0.0515376
$466$		0			0
$467$	−2.59791e9			−1.18036			−0.590180	−	0.807272i	$-0.700943\pi$
$467$	−2.59791e9			−1.18036			−0.590180	+	0.807272i	$0.700943\pi$
$468$		0			0
$469$	−1.07516e9			−0.481247
$470$		0			0
$471$	−1.45767e8			−0.0642813
$472$		0			0
$473$			7.10713e7i			0.0308802i
$474$		0			0
$475$		−	1.54086e9i		−	0.659685i
$476$		0			0
$477$	5.30752e8			0.223912
$478$		0			0
$479$			2.90167e9i			1.20635i	0.797608	+	0.603176i	$0.206098\pi$
$479$			2.90167e9i			1.20635i	−0.797608	+	0.603176i	$0.793902\pi$
$480$		0			0
$481$	1.51210e9	−	5.64932e8i	0.619545	−	0.231467i
$482$		0			0
$483$		−	3.05055e8i		−	0.123187i
$484$		0			0
$485$	8.18896e7			0.0325936
$486$		0			0
$487$		−	1.15313e9i		−	0.452406i	−0.974080	−	0.226203i	$-0.927369\pi$
$487$		−	1.15313e9i		−	0.452406i	0.974080	−	0.226203i	$-0.0726313\pi$
$488$		0			0
$489$			1.81344e9i			0.701331i
$490$		0			0
$491$	3.78661e9			1.44366			0.721831	−	0.692069i	$-0.243301\pi$
$491$	3.78661e9			1.44366			0.721831	+	0.692069i	$0.243301\pi$
$492$		0			0
$493$	3.27950e8			0.123266
$494$		0			0
$495$	−1.44625e7			−0.00535951
$496$		0			0
$497$	−2.15914e9			−0.788920
$498$		0			0
$499$		−	1.98153e9i		−	0.713918i	−0.934120	−	0.356959i	$-0.883814\pi$
$499$		−	1.98153e9i		−	0.713918i	0.934120	−	0.356959i	$-0.116186\pi$
$500$		0			0
$501$			1.12132e9i			0.398379i
$502$		0			0
$503$	−2.32772e9			−0.815536			−0.407768	−	0.913086i	$-0.633693\pi$
$503$	−2.32772e9			−0.815536			−0.407768	+	0.913086i	$0.633693\pi$
$504$		0			0
$505$		−	1.83949e7i		−	0.00635589i
$506$		0			0
$507$	2.40985e9	−	2.09280e9i	0.821226	−	0.713181i
$508$		0			0
$509$		−	1.44356e9i		−	0.485203i	−0.970126	−	0.242601i	$-0.921999\pi$
$509$		−	1.44356e9i		−	0.485203i	0.970126	−	0.242601i	$-0.0780007\pi$
$510$		0			0
$511$	−1.68309e9			−0.558001
$512$		0			0
$513$		−	1.79427e9i		−	0.586782i
$514$		0			0
$515$			1.55456e8i			0.0501513i
$516$		0			0
$517$	4.79662e9			1.52657
$518$		0			0
$519$	1.00158e9			0.314486
$520$		0			0
$521$	2.02069e9			0.625990			0.312995	−	0.949755i	$-0.398668\pi$
$521$	2.02069e9			0.625990			0.312995	+	0.949755i	$0.398668\pi$
$522$		0			0
$523$	−3.62640e9			−1.10846			−0.554230	−	0.832364i	$-0.686987\pi$
$523$	−3.62640e9			−1.10846			−0.554230	+	0.832364i	$0.686987\pi$
$524$		0			0
$525$		−	2.02642e9i		−	0.611184i
$526$		0			0
$527$			4.42865e8i			0.131806i
$528$		0			0
$529$	−3.26679e9			−0.959458
$530$		0			0
$531$		−	1.60717e8i		−	0.0465833i
$532$		0			0
$533$	3.11155e9	−	1.16250e9i	0.890084	−	0.332542i
$534$		0			0
$535$		−	1.37888e8i		−	0.0389304i
$536$		0			0
$537$	−4.01955e9			−1.12013
$538$		0			0
$539$		−	2.28878e9i		−	0.629569i
$540$		0			0
$541$		−	7.29172e9i		−	1.97988i	−0.141481	−	0.989941i	$-0.545186\pi$
$541$		−	7.29172e9i		−	1.97988i	0.141481	−	0.989941i	$-0.454814\pi$
$542$		0			0
$543$	−2.51345e9			−0.673706
$544$		0			0
$545$	−1.41769e8			−0.0375140
$546$		0			0
$547$	2.20432e9			0.575862			0.287931	−	0.957651i	$-0.407033\pi$
$547$	2.20432e9			0.575862			0.287931	+	0.957651i	$0.407033\pi$
$548$		0			0
$549$	−1.31562e9			−0.339335
$550$		0			0
$551$			3.61405e9i			0.920372i
$552$		0			0
$553$		−	1.93659e9i		−	0.486967i
$554$		0			0
$555$	9.21122e7			0.0228713
$556$		0			0
$557$			6.48577e8i			0.159026i	0.996834	+	0.0795131i	$0.0253365\pi$
$557$			6.48577e8i			0.159026i	−0.996834	+	0.0795131i	$0.974663\pi$
$558$		0			0
$559$	−4.84651e7	−	1.29722e8i	−0.0117351	−	0.0314103i
$560$		0			0
$561$		−	3.70477e8i		−	0.0885913i
$562$		0			0
$563$	7.79259e8			0.184036			0.0920180	−	0.995757i	$-0.470668\pi$
$563$	7.79259e8			0.184036			0.0920180	+	0.995757i	$0.470668\pi$
$564$		0			0
$565$			1.47218e8i			0.0343394i
$566$		0			0
$567$		−	2.80656e9i		−	0.646598i
$568$		0			0
$569$	4.16166e9			0.947051			0.473525	−	0.880780i	$-0.342981\pi$
$569$	4.16166e9			0.947051			0.473525	+	0.880780i	$0.342981\pi$
$570$		0			0
$571$	8.48385e9			1.90707			0.953535	−	0.301282i	$-0.0974145\pi$
$571$	8.48385e9			1.90707			0.953535	+	0.301282i	$0.0974145\pi$
$572$		0			0
$573$	8.36557e9			1.85761
$574$		0			0
$575$	9.16960e8			0.201147
$576$		0			0
$577$		−	4.94714e9i		−	1.07211i	−0.844184	−	0.536054i	$-0.819914\pi$
$577$		−	4.94714e9i		−	1.07211i	0.844184	−	0.536054i	$-0.180086\pi$
$578$		0			0
$579$		−	4.46919e9i		−	0.956872i
$580$		0			0
$581$	−2.22883e9			−0.471478
$582$		0			0
$583$		−	5.39026e9i		−	1.12660i
$584$		0			0
$585$	2.63975e7	−	9.86230e6i	0.00545151	−	0.00203673i
$586$		0			0
$587$		−	6.91524e9i		−	1.41115i	−0.708634	−	0.705577i	$-0.750688\pi$
$587$		−	6.91524e9i		−	1.41115i	0.708634	−	0.705577i	$-0.249312\pi$
$588$		0			0
$589$	−4.88043e9			−0.984134
$590$		0			0
$591$			1.01101e9i			0.201465i
$592$		0			0
$593$			4.73029e9i			0.931529i	0.884909	+	0.465765i	$0.154221\pi$
$593$			4.73029e9i			0.931529i	−0.884909	+	0.465765i	$0.845779\pi$
$594$		0			0
$595$	−8.12688e6			−0.00158166
$596$		0			0
$597$	−1.42484e9			−0.274067
$598$		0			0
$599$	5.05854e9			0.961682			0.480841	−	0.876808i	$-0.340331\pi$
$599$	5.05854e9			0.961682			0.480841	+	0.876808i	$0.340331\pi$
$600$		0			0
$601$	−2.81440e9			−0.528840			−0.264420	−	0.964408i	$-0.585181\pi$
$601$	−2.81440e9			−0.528840			−0.264420	+	0.964408i	$0.585181\pi$
$602$		0			0
$603$			8.43160e8i			0.156603i
$604$		0			0
$605$		−	2.62975e7i		−	0.00482804i
$606$		0			0
$607$	3.54047e9			0.642541			0.321271	−	0.946987i	$-0.395890\pi$
$607$	3.54047e9			0.642541			0.321271	+	0.946987i	$0.395890\pi$
$608$		0			0
$609$			4.75290e9i			0.852704i
$610$		0			0
$611$	−8.75496e9	+	3.27092e9i	−1.55278	+	0.580130i
$612$		0			0
$613$		−	1.90580e8i		−	0.0334169i	−0.999860	−	0.0167085i	$-0.994681\pi$
$613$		−	1.90580e8i		−	0.0334169i	0.999860	−	0.0167085i	$-0.00531872\pi$
$614$		0			0
$615$	1.89545e8			0.0328587
$616$		0			0
$617$		−	9.31051e9i		−	1.59579i	−0.602798	−	0.797894i	$-0.705947\pi$
$617$		−	9.31051e9i		−	1.59579i	0.602798	−	0.797894i	$-0.294053\pi$
$618$		0			0
$619$		−	8.75671e9i		−	1.48396i	−0.670419	−	0.741982i	$-0.733886\pi$
$619$		−	8.75671e9i		−	1.48396i	0.670419	−	0.741982i	$-0.266114\pi$
$620$		0			0
$621$	1.06776e9			0.178918
$622$		0			0
$623$	−4.45200e9			−0.737644
$624$		0			0
$625$	6.08501e9			0.996968
$626$		0			0
$627$	4.08270e9			0.661471
$628$		0			0
$629$			3.65072e8i			0.0584927i
$630$		0			0
$631$		−	2.59032e9i		−	0.410440i	−0.978716	−	0.205220i	$-0.934209\pi$
$631$		−	2.59032e9i		−	0.410440i	0.978716	−	0.205220i	$-0.0657910\pi$
$632$		0			0
$633$	1.21925e9			0.191065
$634$		0			0
$635$			1.76652e7i			0.00273785i
$636$		0			0
$637$	1.56077e9	+	4.17757e9i	0.239250	+	0.640376i
$638$		0			0
$639$			1.69324e9i			0.256723i
$640$		0			0
$641$	4.49376e9			0.673918			0.336959	−	0.941519i	$-0.390602\pi$
$641$	4.49376e9			0.673918			0.336959	+	0.941519i	$0.390602\pi$
$642$		0			0
$643$			5.61571e9i			0.833041i	0.909126	+	0.416521i	$0.136751\pi$
$643$			5.61571e9i			0.833041i	−0.909126	+	0.416521i	$0.863249\pi$
$644$		0			0
$645$		−	7.90224e6i		−	0.00115955i
$646$		0			0
$647$	1.45087e9			0.210602			0.105301	−	0.994440i	$-0.466419\pi$
$647$	1.45087e9			0.210602			0.105301	+	0.994440i	$0.466419\pi$
$648$		0			0
$649$	−1.63222e9			−0.234381
$650$		0			0
$651$	−6.41834e9			−0.911779
$652$		0			0
$653$	−9.22610e9			−1.29665			−0.648324	−	0.761365i	$-0.724530\pi$
$653$	−9.22610e9			−1.29665			−0.648324	+	0.761365i	$0.724530\pi$
$654$		0			0
$655$			1.41102e8i			0.0196195i
$656$		0			0
$657$			1.31991e9i			0.181579i
$658$		0			0
$659$	−4.26023e9			−0.579875			−0.289937	−	0.957046i	$-0.593634\pi$
$659$	−4.26023e9			−0.579875			−0.289937	+	0.957046i	$0.593634\pi$
$660$		0			0
$661$		−	1.14507e9i		−	0.154216i	−0.997023	−	0.0771078i	$-0.975431\pi$
$661$		−	1.14507e9i		−	0.154216i	0.997023	−	0.0771078i	$-0.0245685\pi$
$662$		0			0
$663$	2.52636e8	+	6.76207e8i	0.0336665	+	0.0901120i
$664$		0			0
$665$		−	8.95592e7i		−	0.0118096i
$666$		0			0
$667$	−2.15070e9			−0.280634
$668$		0			0
$669$			9.60219e9i			1.23988i
$670$		0			0
$671$			1.33613e10i			1.70735i
$672$		0			0
$673$	−5.46085e9			−0.690570			−0.345285	−	0.938498i	$-0.612218\pi$
$673$	−5.46085e9			−0.690570			−0.345285	+	0.938498i	$0.612218\pi$
$674$		0			0
$675$	7.09292e9			0.887691
$676$		0			0
$677$	−8.92821e9			−1.10587			−0.552935	−	0.833225i	$-0.686492\pi$
$677$	−8.92821e9			−1.10587			−0.552935	+	0.833225i	$0.686492\pi$
$678$		0			0
$679$	−4.70372e9			−0.576630
$680$		0			0
$681$		−	3.89824e7i		−	0.00472992i
$682$		0			0
$683$			8.50973e9i			1.02198i	0.859586	+	0.510991i	$0.170721\pi$
$683$			8.50973e9i			1.02198i	−0.859586	+	0.510991i	$0.829279\pi$
$684$		0			0
$685$	4.40759e8			0.0523944
$686$		0			0
$687$			5.41311e9i			0.636940i
$688$		0			0
$689$	3.67573e9	+	9.83849e9i	0.428131	+	1.14594i
$690$		0			0
$691$			1.11787e9i			0.128889i	0.997921	+	0.0644447i	$0.0205276\pi$
$691$			1.11787e9i			0.128889i	−0.997921	+	0.0644447i	$0.979472\pi$
$692$		0			0
$693$	8.30724e8			0.0948179
$694$		0			0
$695$		−	5.07991e7i		−	0.00573997i
$696$		0			0
$697$			7.51233e8i			0.0840349i
$698$		0			0
$699$	1.04521e9			0.115753
$700$		0			0
$701$	3.84858e9			0.421976			0.210988	−	0.977489i	$-0.432332\pi$
$701$	3.84858e9			0.421976			0.210988	+	0.977489i	$0.432332\pi$
$702$		0			0
$703$	−4.02314e9			−0.436739
$704$		0			0
$705$	−5.33324e8			−0.0573230
$706$		0			0
$707$			1.05660e9i			0.112445i
$708$		0			0
$709$			5.36316e9i			0.565144i	0.959246	+	0.282572i	$0.0911875\pi$
$709$			5.36316e9i			0.565144i	−0.959246	+	0.282572i	$0.908812\pi$
$710$		0			0
$711$	−1.51871e9			−0.158464
$712$		0			0
$713$		−	2.90432e9i		−	0.300076i
$714$		0			0
$715$	−1.00160e8	−	2.68090e8i	−0.0102477	−	0.0274290i
$716$		0			0
$717$		−	3.29341e9i		−	0.333679i
$718$		0			0
$719$	−6.62611e9			−0.664825			−0.332413	−	0.943134i	$-0.607863\pi$
$719$	−6.62611e9			−0.664825			−0.332413	+	0.943134i	$0.607863\pi$
$720$		0			0
$721$		−	8.92936e9i		−	0.887252i
$722$		0			0
$723$			2.04368e10i			2.01108i
$724$		0			0
$725$	−1.42867e10			−1.39235
$726$		0			0
$727$	9.52174e8			0.0919064			0.0459532	−	0.998944i	$-0.485367\pi$
$727$	9.52174e8			0.0919064			0.0459532	+	0.998944i	$0.485367\pi$
$728$		0			0
$729$	7.90894e9			0.756087
$730$		0			0
$731$	3.13193e7			0.00296552
$732$		0			0
$733$			1.46510e10i			1.37405i	0.726632	+	0.687027i	$0.241084\pi$
$733$			1.46510e10i			1.37405i	−0.726632	+	0.687027i	$0.758916\pi$
$734$		0			0
$735$			2.54484e8i			0.0236404i
$736$		0			0
$737$	8.56305e9			0.787938
$738$		0			0
$739$			1.21913e10i			1.11121i	0.831447	+	0.555604i	$0.187513\pi$
$739$			1.21913e10i			1.11121i	−0.831447	+	0.555604i	$0.812487\pi$
$740$		0			0
$741$	−7.45189e9	+	2.78408e9i	−0.672826	+	0.251373i
$742$		0			0
$743$		−	3.89555e9i		−	0.348424i	−0.984708	−	0.174212i	$-0.944262\pi$
$743$		−	3.89555e9i		−	0.348424i	0.984708	−	0.174212i	$-0.0557378\pi$
$744$		0			0
$745$	1.35076e8			0.0119683
$746$		0			0
$747$			1.74789e9i			0.153424i
$748$		0			0
$749$			7.92027e9i			0.688737i
$750$		0			0
$751$	1.53353e10			1.32115			0.660574	−	0.750761i	$-0.270313\pi$
$751$	1.53353e10			1.32115			0.660574	+	0.750761i	$0.270313\pi$
$752$		0			0
$753$	−1.40944e10			−1.20299
$754$		0			0
$755$	5.75132e8			0.0486355
$756$		0			0
$757$	1.71312e10			1.43533			0.717665	−	0.696389i	$-0.245211\pi$
$757$	1.71312e10			1.43533			0.717665	+	0.696389i	$0.245211\pi$
$758$		0			0
$759$			2.42959e9i			0.201691i
$760$		0			0
$761$			9.75251e9i			0.802177i	0.916039	+	0.401088i	$0.131368\pi$
$761$			9.75251e9i			0.802177i	−0.916039	+	0.401088i	$0.868632\pi$
$762$		0			0
$763$	8.14318e9			0.663679
$764$		0			0
$765$			6.37325e6i			0.000514690i
$766$		0			0
$767$	2.97919e9	−	1.11305e9i	0.238404	−	0.0890697i
$768$		0			0
$769$		−	6.98230e9i		−	0.553677i	−0.960916	−	0.276839i	$-0.910713\pi$
$769$		−	6.98230e9i		−	0.553677i	0.960916	−	0.276839i	$-0.0892867\pi$
$770$		0			0
$771$	−2.37680e10			−1.86768
$772$		0			0
$773$			1.86799e10i			1.45461i	0.686314	+	0.727305i	$0.259227\pi$
$773$			1.86799e10i			1.45461i	−0.686314	+	0.727305i	$0.740773\pi$
$774$		0			0
$775$		−	1.92928e10i		−	1.48881i
$776$		0			0
$777$	−5.29091e9			−0.404629
$778$		0			0
$779$	−8.27868e9			−0.627451
$780$		0			0
$781$	1.71963e10			1.29169
$782$		0			0
$783$	−1.66362e10			−1.23848
$784$		0			0
$785$		−	2.54669e7i		−	0.00187902i
$786$		0			0
$787$		−	5.17067e9i		−	0.378125i	−0.981965	−	0.189062i	$-0.939455\pi$
$787$		−	5.17067e9i		−	0.378125i	0.981965	−	0.189062i	$-0.0605448\pi$
$788$		0			0
$789$	3.85952e9			0.279746
$790$		0			0
$791$		−	8.45620e9i		−	0.607516i
$792$		0			0
$793$	−9.11140e9	−	2.43876e10i	−0.648827	−	1.73665i
$794$		0			0
$795$			5.99329e8i			0.0423039i
$796$		0			0
$797$	−8.89261e9			−0.622193			−0.311097	−	0.950378i	$-0.600696\pi$
$797$	−8.89261e9			−0.622193			−0.311097	+	0.950378i	$0.600696\pi$
$798$		0			0
$799$		−	2.11374e9i		−	0.146602i
$800$		0			0
$801$			3.49134e9i			0.240037i
$802$		0			0
$803$	1.34049e10			0.913606
$804$		0			0
$805$	5.32962e7			0.00360090
$806$		0			0
$807$	9.84086e9			0.659137
$808$		0			0
$809$	−1.93280e10			−1.28342			−0.641709	−	0.766948i	$-0.721774\pi$
$809$	−1.93280e10			−1.28342			−0.641709	+	0.766948i	$0.721774\pi$
$810$		0			0
$811$			1.58565e9i			0.104384i	0.998637	+	0.0521922i	$0.0166208\pi$
$811$			1.58565e9i			0.104384i	−0.998637	+	0.0521922i	$0.983379\pi$
$812$		0			0
$813$		−	2.05888e10i		−	1.34374i
$814$		0			0
$815$	−3.16827e8			−0.0205008
$816$		0			0
$817$			3.45142e8i			0.0221422i
$818$		0			0
$819$	−1.51627e9	+	5.66489e8i	−0.0964455	+	0.0360328i
$820$		0			0
$821$			1.92167e10i			1.21193i	0.795490	+	0.605967i	$0.207214\pi$
$821$			1.92167e10i			1.21193i	−0.795490	+	0.605967i	$0.792786\pi$
$822$		0			0
$823$	−7.27465e9			−0.454897			−0.227448	−	0.973790i	$-0.573038\pi$
$823$	−7.27465e9			−0.454897			−0.227448	+	0.973790i	$0.573038\pi$
$824$		0			0
$825$			1.61393e10i			1.00068i
$826$		0			0
$827$			7.39476e9i			0.454626i	0.973822	+	0.227313i	$0.0729941\pi$
$827$			7.39476e9i			0.454626i	−0.973822	+	0.227313i	$0.927006\pi$
$828$		0			0
$829$	−1.20575e10			−0.735048			−0.367524	−	0.930014i	$-0.619794\pi$
$829$	−1.20575e10			−0.735048			−0.367524	+	0.930014i	$0.619794\pi$
$830$		0			0
$831$	2.37458e10			1.43544
$832$		0			0
$833$	−1.00861e9			−0.0604594
$834$		0			0
$835$	−1.95905e8			−0.0116451
$836$		0			0
$837$		−	2.24656e10i		−	1.32428i
$838$		0			0
$839$			1.74685e9i			0.102115i	0.998696	+	0.0510575i	$0.0162592\pi$
$839$			1.74685e9i			0.102115i	−0.998696	+	0.0510575i	$0.983741\pi$
$840$		0			0
$841$	1.62591e10			0.942562
$842$		0			0
$843$		−	1.44237e10i		−	0.829241i
$844$		0			0
$845$	3.65633e8	+	4.21026e8i	0.0208471	+	0.0240055i
$846$		0			0
$847$			1.51052e9i			0.0854153i
$848$		0			0
$849$	1.80218e10			1.01070
$850$		0			0
$851$		−	2.39415e9i		−	0.133167i
$852$		0			0
$853$		−	2.10759e9i		−	0.116269i	−0.998309	−	0.0581346i	$-0.981485\pi$
$853$		−	2.10759e9i		−	0.116269i	0.998309	−	0.0581346i	$-0.0185153\pi$
$854$		0			0
$855$	−7.02340e7			−0.00384296
$856$		0			0
$857$	2.74086e10			1.48749			0.743744	−	0.668464i	$-0.233048\pi$
$857$	2.74086e10			1.48749			0.743744	+	0.668464i	$0.233048\pi$
$858$		0			0
$859$	−2.95561e9			−0.159100			−0.0795502	−	0.996831i	$-0.525348\pi$
$859$	−2.95561e9			−0.159100			−0.0795502	+	0.996831i	$0.525348\pi$
$860$		0			0
$861$	−1.08874e10			−0.581320
$862$		0			0
$863$		−	2.24829e10i		−	1.19073i	−0.803454	−	0.595366i	$-0.797007\pi$
$863$		−	2.24829e10i		−	1.19073i	0.803454	−	0.595366i	$-0.202993\pi$
$864$		0			0
$865$			1.74987e8i			0.00919283i
$866$		0			0
$867$	2.07088e10			1.07917
$868$		0			0
$869$			1.54238e10i			0.797303i
$870$		0			0
$871$	−1.56296e10	+	5.83933e9i	−0.801463	+	0.299433i
$872$		0			0
$873$			3.68875e9i			0.187642i
$874$		0			0
$875$	7.08430e8			0.0357494
$876$		0			0
$877$		−	3.27881e10i		−	1.64141i	−0.571350	−	0.820707i	$-0.693580\pi$
$877$		−	3.27881e10i		−	1.64141i	0.571350	−	0.820707i	$-0.306420\pi$
$878$		0			0
$879$		−	2.89037e10i		−	1.43546i
$880$		0			0
$881$	−3.78138e10			−1.86310			−0.931548	−	0.363619i	$-0.881541\pi$
$881$	−3.78138e10			−1.86310			−0.931548	+	0.363619i	$0.881541\pi$
$882$		0			0
$883$	−8.05584e9			−0.393775			−0.196888	−	0.980426i	$-0.563083\pi$
$883$	−8.05584e9			−0.393775			−0.196888	+	0.980426i	$0.563083\pi$
$884$		0			0
$885$	1.81482e8			0.00880102
$886$		0			0
$887$	3.15643e10			1.51867			0.759335	−	0.650700i	$-0.225524\pi$
$887$	3.15643e10			1.51867			0.759335	+	0.650700i	$0.225524\pi$
$888$		0			0
$889$		−	1.01468e9i		−	0.0484367i
$890$		0			0
$891$			2.23527e10i			1.05866i
$892$		0			0
$893$	2.32937e10			1.09461
$894$		0			0
$895$		−	7.02255e8i		−	0.0327427i
$896$		0			0
$897$	−1.65679e9	−	4.43458e9i	−0.0766469	−	0.205154i
$898$		0			0
$899$			4.52506e10i			2.07714i
$900$		0			0
$901$	−2.37535e9			−0.108191
$902$		0			0
$903$			4.53903e8i			0.0205143i
$904$		0			0
$905$		−	4.39125e8i		−	0.0196933i
$906$		0			0
$907$	−3.74494e10			−1.66656			−0.833278	−	0.552854i	$-0.813539\pi$
$907$	−3.74494e10			−1.66656			−0.833278	+	0.552854i	$0.813539\pi$
$908$		0			0
$909$	8.28603e8			0.0365909
$910$		0			0
$911$	−4.69150e9			−0.205588			−0.102794	−	0.994703i	$-0.532778\pi$
$911$	−4.69150e9			−0.205588			−0.102794	+	0.994703i	$0.532778\pi$
$912$		0			0
$913$	1.77514e10			0.771942
$914$		0			0
$915$		−	1.48561e9i		−	0.0641109i
$916$		0			0
$917$		−	8.10488e9i		−	0.347099i
$918$		0			0
$919$	3.13816e10			1.33374			0.666870	−	0.745174i	$-0.267634\pi$
$919$	3.13816e10			1.33374			0.666870	+	0.745174i	$0.267634\pi$
$920$		0			0
$921$		−	3.08695e10i		−	1.30203i
$922$		0			0
$923$	−3.13873e10	+	1.17266e10i	−1.31386	+	0.490868i
$924$		0			0
$925$		−	1.59039e10i		−	0.660703i
$926$		0			0
$927$	−7.00257e9			−0.288721
$928$		0			0
$929$			2.99605e10i			1.22601i	0.790079	+	0.613005i	$0.210039\pi$
$929$			2.99605e10i			1.22601i	−0.790079	+	0.613005i	$0.789961\pi$
$930$		0			0
$931$		−	1.11150e10i		−	0.451424i
$932$		0			0
$933$	2.90380e10			1.17053
$934$		0			0
$935$	6.47260e7			0.00258963
$936$		0			0
$937$	3.17700e10			1.26162			0.630810	−	0.775937i	$-0.282723\pi$
$937$	3.17700e10			1.26162			0.630810	+	0.775937i	$0.282723\pi$
$938$		0			0
$939$	−6.05341e9			−0.238600
$940$		0			0
$941$			3.63923e8i			0.0142379i	0.999975	+	0.00711895i	$0.00226605\pi$
$941$			3.63923e8i			0.0142379i	−0.999975	+	0.00711895i	$0.997734\pi$
$942$		0			0
$943$		−	4.92660e9i		−	0.191318i
$944$		0			0
$945$	4.12259e8			0.0158913
$946$		0			0
$947$			8.10602e9i			0.310158i	0.987902	+	0.155079i	$0.0495632\pi$
$947$			8.10602e9i			0.310158i	−0.987902	+	0.155079i	$0.950437\pi$
$948$		0			0
$949$	−2.44671e10	+	9.14110e9i	−0.929289	+	0.347190i
$950$		0			0
$951$		−	3.61660e10i		−	1.36354i
$952$		0			0
$953$	−3.07889e10			−1.15231			−0.576155	−	0.817340i	$-0.695448\pi$
$953$	−3.07889e10			−1.15231			−0.576155	+	0.817340i	$0.695448\pi$
$954$		0			0
$955$			1.46155e9i			0.0543002i
$956$		0			0
$957$		−	3.78542e10i		−	1.39612i
$958$		0			0
$959$	−2.53171e10			−0.926936
$960$		0			0
$961$	−3.35940e10			−1.22104
$962$		0			0
$963$	6.21122e9			0.224122
$964$		0			0
$965$	7.80812e8			0.0279705
$966$		0			0
$967$		−	2.71301e10i		−	0.964846i	−0.875938	−	0.482423i	$-0.839757\pi$
$967$		−	2.71301e10i		−	0.964846i	0.875938	−	0.482423i	$-0.160243\pi$
$968$		0			0
$969$		−	1.79914e9i		−	0.0635231i
$970$		0			0
$971$	2.35980e10			0.827197			0.413598	−	0.910459i	$-0.364272\pi$
$971$	2.35980e10			0.827197			0.413598	+	0.910459i	$0.364272\pi$
$972$		0			0
$973$			2.91789e9i			0.101549i
$974$		0			0
$975$	−1.10057e10	−	2.94580e10i	−0.380280	−	1.01786i
$976$		0			0
$977$			1.46911e9i			0.0503992i	0.999682	+	0.0251996i	$0.00802212\pi$
$977$			1.46911e9i			0.0503992i	−0.999682	+	0.0251996i	$0.991978\pi$
$978$		0			0
$979$	3.54576e10			1.20773
$980$		0			0
$981$		−	6.38603e9i		−	0.215968i
$982$		0			0
$983$			3.36024e10i			1.12832i	0.825665	+	0.564161i	$0.190800\pi$
$983$			3.36024e10i			1.12832i	−0.825665	+	0.564161i	$0.809200\pi$
$984$		0			0
$985$	−1.76634e8			−0.00588908
$986$		0			0
$987$	3.06340e10			1.01413
$988$		0			0
$989$	−2.05392e8			−0.00675146
$990$		0			0
$991$	2.03355e10			0.663737			0.331869	−	0.943326i	$-0.392321\pi$
$991$	2.03355e10			0.663737			0.331869	+	0.943326i	$0.392321\pi$
$992$		0			0
$993$			5.25849e10i			1.70427i
$994$		0			0
$995$		−	2.48935e8i		−	0.00801133i
$996$		0			0
$997$	8.04557e9			0.257113			0.128556	−	0.991702i	$-0.458966\pi$
$997$	8.04557e9			0.257113			0.128556	+	0.991702i	$0.458966\pi$
$998$		0			0
$999$		−	1.85193e10i		−	0.587688i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.f.a.129.1		6
4.3	odd	2		13.8.b.a.12.2	✓	6
12.11	even	2		117.8.b.b.64.5		6
13.12	even	2	inner	208.8.f.a.129.2		6
52.31	even	4		169.8.a.d.1.2		6
52.47	even	4		169.8.a.d.1.5		6
52.51	odd	2		13.8.b.a.12.5	yes	6
156.155	even	2		117.8.b.b.64.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.2	✓	6	4.3	odd	2
13.8.b.a.12.5	yes	6	52.51	odd	2
117.8.b.b.64.2		6	156.155	even	2
117.8.b.b.64.5		6	12.11	even	2
169.8.a.d.1.2		6	52.31	even	4
169.8.a.d.1.5		6	52.47	even	4
208.8.f.a.129.1		6	1.1	even	1	trivial
208.8.f.a.129.2		6	13.12	even	2	inner

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-50.8656 q^{3} -8.88672i q^{5} +510.452i q^{7} +400.306 q^{9} -4065.46i q^{11} +(2772.33 + 7420.43i) q^{13} +452.028i q^{15} -1791.54 q^{17} -19743.0i q^{19} -25964.4i q^{21} +11749.0 q^{23} +78046.0 q^{25} +90881.2 q^{27} -183054. q^{29} -247198. i q^{31} +206792. i q^{33} +4536.25 q^{35} -203775. i q^{37} +(-141016. - 377444. i) q^{39} -419322. i q^{41} -17481.7 q^{43} -3557.41i q^{45} +1.17985e6i q^{47} +562982. q^{49} +91127.9 q^{51} +1.32587e6 q^{53} -36128.6 q^{55} +1.00424e6i q^{57} -401485. i q^{59} -3.28655e6 q^{61} +204337. i q^{63} +(65943.3 - 24636.9i) q^{65} +2.10629e6i q^{67} -597618. q^{69} +4.22986e6i q^{71} +3.29726e6i q^{73} -3.96986e6 q^{75} +2.07522e6 q^{77} -3.79387e6 q^{79} -5.49819e6 q^{81} +4.36639e6i q^{83} +15920.9i q^{85} +9.31117e6 q^{87} +8.72168e6i q^{89} +(-3.78777e6 + 1.41514e6i) q^{91} +1.25738e7i q^{93} -175451. q^{95} +9.21482e6i q^{97} -1.62743e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 56 q^{3} - 1150 q^{9} - 5018 q^{13} + 13152 q^{17} - 27264 q^{23} - 18262 q^{25} - 194560 q^{27} + 42924 q^{29} + 546720 q^{35} - 511160 q^{39} + 1005576 q^{43} + 3246846 q^{49} - 297984 q^{51} + 1705524 q^{53}+ \cdots + 22075632 q^{95}+O(q^{100}) \) 6 * q + 56 * q^3 - 1150 * q^9 - 5018 * q^13 + 13152 * q^17 - 27264 * q^23 - 18262 * q^25 - 194560 * q^27 + 42924 * q^29 + 546720 * q^35 - 511160 * q^39 + 1005576 * q^43 + 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 + 5320576 * q^55 - 8237572 * q^61 + 7139028 * q^65 - 6017400 * q^69 - 20426072 * q^75 + 4093560 * q^77 - 15147200 * q^79 - 21318442 * q^81 + 36474240 * q^87 - 21281832 * q^91 + 22075632 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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