LMFDB - Embedded newform 208.2.f.b.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,2,Mod(129,208)] 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, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [4] 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:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.2.f.b.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} -1.56155i q^{5} -0.438447i q^{7} -0.561553 q^{9} -5.12311i q^{11} +(0.561553 - 3.56155i) q^{13} +2.43845i q^{15} -3.56155 q^{17} -2.00000i q^{19} +0.684658i q^{21} +3.12311 q^{23} +2.56155 q^{25} +5.56155 q^{27} -5.12311 q^{29} +5.12311i q^{31} +8.00000i q^{33} -0.684658 q^{35} +9.56155i q^{37} +(-0.876894 + 5.56155i) q^{39} -8.00000i q^{41} -9.56155 q^{43} +0.876894i q^{45} +7.56155i q^{47} +6.80776 q^{49} +5.56155 q^{51} +12.2462 q^{53} -8.00000 q^{55} +3.12311i q^{57} -10.0000i q^{59} +2.87689 q^{61} +0.246211i q^{63} +(-5.56155 - 0.876894i) q^{65} +9.12311i q^{67} -4.87689 q^{69} -6.68466i q^{71} -9.36932i q^{73} -4.00000 q^{75} -2.24621 q^{77} +11.1231 q^{79} -7.00000 q^{81} -8.24621i q^{83} +5.56155i q^{85} +8.00000 q^{87} -3.12311i q^{89} +(-1.56155 - 0.246211i) q^{91} -8.00000i q^{93} -3.12311 q^{95} +6.24621i q^{97} +2.87689i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9} - 6 q^{13} - 6 q^{17} - 4 q^{23} + 2 q^{25} + 14 q^{27} - 4 q^{29} + 22 q^{35} - 20 q^{39} - 30 q^{43} - 14 q^{49} + 14 q^{51} + 16 q^{53} - 32 q^{55} + 28 q^{61} - 14 q^{65} - 36 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 - 6 * q^13 - 6 * q^17 - 4 * q^23 + 2 * q^25 + 14 * q^27 - 4 * q^29 + 22 * q^35 - 20 * q^39 - 30 * q^43 - 14 * q^49 + 14 * q^51 + 16 * q^53 - 32 * q^55 + 28 * q^61 - 14 * q^65 - 36 * q^69 - 16 * q^75 + 24 * q^77 + 28 * q^79 - 28 * q^81 + 32 * q^87 + 2 * q^91 + 4 * 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
$3$	−1.56155			−0.901563			−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155			−0.901563			−0.450781	+	0.892634i	$0.648855\pi$
$4$		0			0
$5$		−	1.56155i		−	0.698348i	−0.937058	−	0.349174i	$-0.886462\pi$
$5$		−	1.56155i		−	0.698348i	0.937058	−	0.349174i	$-0.113538\pi$
$6$		0			0
$7$		−	0.438447i		−	0.165717i	−0.996561	−	0.0828587i	$-0.973595\pi$
$7$		−	0.438447i		−	0.165717i	0.996561	−	0.0828587i	$-0.0264050\pi$
$8$		0			0
$9$	−0.561553			−0.187184
$10$		0			0
$11$		−	5.12311i		−	1.54467i	−0.635213	−	0.772337i	$-0.719088\pi$
$11$		−	5.12311i		−	1.54467i	0.635213	−	0.772337i	$-0.280912\pi$
$12$		0			0
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$13$	0.561553	−	3.56155i	0.155747	−	0.987797i
$14$		0			0
$15$			2.43845i			0.629604i
$16$		0			0
$17$	−3.56155			−0.863803			−0.431902	−	0.901921i	$-0.642157\pi$
$17$	−3.56155			−0.863803			−0.431902	+	0.901921i	$0.642157\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$			0.684658i			0.149405i
$22$		0			0
$23$	3.12311			0.651213			0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311			0.651213			0.325606	+	0.945505i	$0.394432\pi$
$24$		0			0
$25$	2.56155			0.512311
$26$		0			0
$27$	5.56155			1.07032
$28$		0			0
$29$	−5.12311			−0.951337			−0.475668	−	0.879625i	$-0.657794\pi$
$29$	−5.12311			−0.951337			−0.475668	+	0.879625i	$0.657794\pi$
$30$		0			0
$31$			5.12311i			0.920137i	0.887883	+	0.460068i	$0.152175\pi$
$31$			5.12311i			0.920137i	−0.887883	+	0.460068i	$0.847825\pi$
$32$		0			0
$33$			8.00000i			1.39262i
$34$		0			0
$35$	−0.684658			−0.115728
$36$		0			0
$37$			9.56155i			1.57191i	0.618284	+	0.785955i	$0.287828\pi$
$37$			9.56155i			1.57191i	−0.618284	+	0.785955i	$0.712172\pi$
$38$		0			0
$39$	−0.876894	+	5.56155i	−0.140415	+	0.890561i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	−9.56155			−1.45812			−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155			−1.45812			−0.729062	+	0.684448i	$0.760043\pi$
$44$		0			0
$45$			0.876894i			0.130720i
$46$		0			0
$47$			7.56155i			1.10297i	0.834186	+	0.551483i	$0.185938\pi$
$47$			7.56155i			1.10297i	−0.834186	+	0.551483i	$0.814062\pi$
$48$		0			0
$49$	6.80776			0.972538
$50$		0			0
$51$	5.56155			0.778773
$52$		0			0
$53$	12.2462			1.68215			0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462			1.68215			0.841073	+	0.540921i	$0.181924\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$			3.12311i			0.413665i
$58$		0			0
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$	2.87689			0.368349			0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689			0.368349			0.184174	+	0.982894i	$0.441039\pi$
$62$		0			0
$63$			0.246211i			0.0310197i
$64$		0			0
$65$	−5.56155	−	0.876894i	−0.689826	−	0.108765i
$66$		0			0
$67$			9.12311i			1.11456i	0.830323	+	0.557282i	$0.188156\pi$
$67$			9.12311i			1.11456i	−0.830323	+	0.557282i	$0.811844\pi$
$68$		0			0
$69$	−4.87689			−0.587109
$70$		0			0
$71$		−	6.68466i		−	0.793323i	−0.917965	−	0.396662i	$-0.870169\pi$
$71$		−	6.68466i		−	0.793323i	0.917965	−	0.396662i	$-0.129831\pi$
$72$		0			0
$73$		−	9.36932i		−	1.09660i	−0.836283	−	0.548298i	$-0.815276\pi$
$73$		−	9.36932i		−	1.09660i	0.836283	−	0.548298i	$-0.184724\pi$
$74$		0			0
$75$	−4.00000			−0.461880
$76$		0			0
$77$	−2.24621			−0.255980
$78$		0			0
$79$	11.1231			1.25145			0.625724	−	0.780045i	$-0.284804\pi$
$79$	11.1231			1.25145			0.625724	+	0.780045i	$0.284804\pi$
$80$		0			0
$81$	−7.00000			−0.777778
$82$		0			0
$83$		−	8.24621i		−	0.905139i	−0.891729	−	0.452570i	$-0.850507\pi$
$83$		−	8.24621i		−	0.905139i	0.891729	−	0.452570i	$-0.149493\pi$
$84$		0			0
$85$			5.56155i			0.603235i
$86$		0			0
$87$	8.00000			0.857690
$88$		0			0
$89$		−	3.12311i		−	0.331049i	−0.986206	−	0.165524i	$-0.947068\pi$
$89$		−	3.12311i		−	0.331049i	0.986206	−	0.165524i	$-0.0529316\pi$
$90$		0			0
$91$	−1.56155	−	0.246211i	−0.163695	−	0.0258100i
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$	−3.12311			−0.320424
$96$		0			0
$97$			6.24621i			0.634207i	0.948391	+	0.317103i	$0.102710\pi$
$97$			6.24621i			0.634207i	−0.948391	+	0.317103i	$0.897290\pi$
$98$		0			0
$99$			2.87689i			0.289139i
$100$		0			0
$101$	−11.3693			−1.13129			−0.565645	−	0.824649i	$-0.691373\pi$
$101$	−11.3693			−1.13129			−0.565645	+	0.824649i	$0.691373\pi$
$102$		0			0
$103$	−3.12311			−0.307729			−0.153864	−	0.988092i	$-0.549172\pi$
$103$	−3.12311			−0.307729			−0.153864	+	0.988092i	$0.549172\pi$
$104$		0			0
$105$	1.06913			0.104336
$106$		0			0
$107$	4.00000			0.386695			0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000			0.386695			0.193347	+	0.981130i	$0.438066\pi$
$108$		0			0
$109$		−	3.31534i		−	0.317552i	−0.987315	−	0.158776i	$-0.949245\pi$
$109$		−	3.31534i		−	0.317552i	0.987315	−	0.158776i	$-0.0507548\pi$
$110$		0			0
$111$		−	14.9309i		−	1.41718i
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$		−	4.87689i		−	0.454773i
$116$		0			0
$117$	−0.315342	+	2.00000i	−0.0291533	+	0.184900i
$118$		0			0
$119$			1.56155i			0.143147i
$120$		0			0
$121$	−15.2462			−1.38602
$122$		0			0
$123$			12.4924i			1.12640i
$124$		0			0
$125$		−	11.8078i		−	1.05612i
$126$		0			0
$127$	4.87689			0.432754			0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689			0.432754			0.216377	+	0.976310i	$0.430576\pi$
$128$		0			0
$129$	14.9309			1.31459
$130$		0			0
$131$	3.31534			0.289663			0.144831	−	0.989456i	$-0.453736\pi$
$131$	3.31534			0.289663			0.144831	+	0.989456i	$0.453736\pi$
$132$		0			0
$133$	−0.876894			−0.0760364
$134$		0			0
$135$		−	8.68466i		−	0.747456i
$136$		0			0
$137$		−	3.12311i		−	0.266825i	−0.991061	−	0.133412i	$-0.957406\pi$
$137$		−	3.12311i		−	0.266825i	0.991061	−	0.133412i	$-0.0425935\pi$
$138$		0			0
$139$	7.80776			0.662246			0.331123	−	0.943588i	$-0.392573\pi$
$139$	7.80776			0.662246			0.331123	+	0.943588i	$0.392573\pi$
$140$		0			0
$141$		−	11.8078i		−	0.994393i
$142$		0			0
$143$	−18.2462	−	2.87689i	−1.52582	−	0.240578i
$144$		0			0
$145$			8.00000i			0.664364i
$146$		0			0
$147$	−10.6307			−0.876804
$148$		0			0
$149$			7.12311i			0.583548i	0.956487	+	0.291774i	$0.0942454\pi$
$149$			7.12311i			0.583548i	−0.956487	+	0.291774i	$0.905755\pi$
$150$		0			0
$151$		−	11.5616i		−	0.940866i	−0.882436	−	0.470433i	$-0.844098\pi$
$151$		−	11.5616i		−	0.940866i	0.882436	−	0.470433i	$-0.155902\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	8.00000			0.642575
$156$		0			0
$157$	12.2462			0.977354			0.488677	−	0.872465i	$-0.337480\pi$
$157$	12.2462			0.977354			0.488677	+	0.872465i	$0.337480\pi$
$158$		0			0
$159$	−19.1231			−1.51656
$160$		0			0
$161$		−	1.36932i		−	0.107917i
$162$		0			0
$163$			20.2462i			1.58581i	0.609348	+	0.792903i	$0.291431\pi$
$163$			20.2462i			1.58581i	−0.609348	+	0.792903i	$0.708569\pi$
$164$		0			0
$165$	12.4924			0.972534
$166$		0			0
$167$			6.87689i			0.532150i	0.963952	+	0.266075i	$0.0857269\pi$
$167$			6.87689i			0.532150i	−0.963952	+	0.266075i	$0.914273\pi$
$168$		0			0
$169$	−12.3693	−	4.00000i	−0.951486	−	0.307692i
$170$		0			0
$171$			1.12311i			0.0858860i
$172$		0			0
$173$	4.24621			0.322833			0.161417	−	0.986886i	$-0.448394\pi$
$173$	4.24621			0.322833			0.161417	+	0.986886i	$0.448394\pi$
$174$		0			0
$175$		−	1.12311i		−	0.0848988i
$176$		0			0
$177$			15.6155i			1.17373i
$178$		0			0
$179$	−17.5616			−1.31261			−0.656306	−	0.754495i	$-0.727882\pi$
$179$	−17.5616			−1.31261			−0.656306	+	0.754495i	$0.727882\pi$
$180$		0			0
$181$	−8.24621			−0.612936			−0.306468	−	0.951881i	$-0.599147\pi$
$181$	−8.24621			−0.612936			−0.306468	+	0.951881i	$0.599147\pi$
$182$		0			0
$183$	−4.49242			−0.332089
$184$		0			0
$185$	14.9309			1.09774
$186$		0			0
$187$			18.2462i			1.33430i
$188$		0			0
$189$		−	2.43845i		−	0.177371i
$190$		0			0
$191$	22.2462			1.60968			0.804840	−	0.593492i	$-0.202251\pi$
$191$	22.2462			1.60968			0.804840	+	0.593492i	$0.202251\pi$
$192$		0			0
$193$			23.6155i			1.69988i	0.526877	+	0.849941i	$0.323363\pi$
$193$			23.6155i			1.69988i	−0.526877	+	0.849941i	$0.676637\pi$
$194$		0			0
$195$	8.68466	+	1.36932i	0.621921	+	0.0980588i
$196$		0			0
$197$			9.56155i			0.681232i	0.940202	+	0.340616i	$0.110636\pi$
$197$			9.56155i			0.681232i	−0.940202	+	0.340616i	$0.889364\pi$
$198$		0			0
$199$	14.2462			1.00989			0.504944	−	0.863152i	$-0.331513\pi$
$199$	14.2462			1.00989			0.504944	+	0.863152i	$0.331513\pi$
$200$		0			0
$201$		−	14.2462i		−	1.00485i
$202$		0			0
$203$			2.24621i			0.157653i
$204$		0			0
$205$	−12.4924			−0.872509
$206$		0			0
$207$	−1.75379			−0.121897
$208$		0			0
$209$	−10.2462			−0.708745
$210$		0			0
$211$	22.0540			1.51826			0.759129	−	0.650940i	$-0.225625\pi$
$211$	22.0540			1.51826			0.759129	+	0.650940i	$0.225625\pi$
$212$		0			0
$213$			10.4384i			0.715231i
$214$		0			0
$215$			14.9309i			1.01828i
$216$		0			0
$217$	2.24621			0.152483
$218$		0			0
$219$			14.6307i			0.988650i
$220$		0			0
$221$	−2.00000	+	12.6847i	−0.134535	+	0.853262i
$222$		0			0
$223$		−	27.1771i		−	1.81991i	−0.414705	−	0.909956i	$-0.636115\pi$
$223$		−	27.1771i		−	1.81991i	0.414705	−	0.909956i	$-0.363885\pi$
$224$		0			0
$225$	−1.43845			−0.0958965
$226$		0			0
$227$		−	3.36932i		−	0.223629i	−0.993729	−	0.111815i	$-0.964334\pi$
$227$		−	3.36932i		−	0.223629i	0.993729	−	0.111815i	$-0.0356663\pi$
$228$		0			0
$229$			20.6847i			1.36688i	0.730006	+	0.683440i	$0.239517\pi$
$229$			20.6847i			1.36688i	−0.730006	+	0.683440i	$0.760483\pi$
$230$		0			0
$231$	3.50758			0.230782
$232$		0			0
$233$	−3.56155			−0.233325			−0.116663	−	0.993172i	$-0.537220\pi$
$233$	−3.56155			−0.233325			−0.116663	+	0.993172i	$0.537220\pi$
$234$		0			0
$235$	11.8078			0.770254
$236$		0			0
$237$	−17.3693			−1.12826
$238$		0			0
$239$			8.93087i			0.577690i	0.957376	+	0.288845i	$0.0932712\pi$
$239$			8.93087i			0.577690i	−0.957376	+	0.288845i	$0.906729\pi$
$240$		0			0
$241$		−	27.1231i		−	1.74715i	−0.486687	−	0.873576i	$-0.661795\pi$
$241$		−	27.1231i		−	1.74715i	0.486687	−	0.873576i	$-0.338205\pi$
$242$		0			0
$243$	−5.75379			−0.369106
$244$		0			0
$245$		−	10.6307i		−	0.679169i
$246$		0			0
$247$	−7.12311	−	1.12311i	−0.453232	−	0.0714615i
$248$		0			0
$249$			12.8769i			0.816040i
$250$		0			0
$251$	−24.4924			−1.54595			−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924			−1.54595			−0.772974	+	0.634438i	$0.781232\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$		−	8.68466i		−	0.543854i
$256$		0			0
$257$	0.930870			0.0580661			0.0290330	−	0.999578i	$-0.490757\pi$
$257$	0.930870			0.0580661			0.0290330	+	0.999578i	$0.490757\pi$
$258$		0			0
$259$	4.19224			0.260493
$260$		0			0
$261$	2.87689			0.178075
$262$		0			0
$263$	4.49242			0.277015			0.138507	−	0.990361i	$-0.455770\pi$
$263$	4.49242			0.277015			0.138507	+	0.990361i	$0.455770\pi$
$264$		0			0
$265$		−	19.1231i		−	1.17472i
$266$		0			0
$267$			4.87689i			0.298461i
$268$		0			0
$269$	10.8769			0.663176			0.331588	−	0.943424i	$-0.392416\pi$
$269$	10.8769			0.663176			0.331588	+	0.943424i	$0.392416\pi$
$270$		0			0
$271$		−	14.6847i		−	0.892029i	−0.895026	−	0.446015i	$-0.852843\pi$
$271$		−	14.6847i		−	0.892029i	0.895026	−	0.446015i	$-0.147157\pi$
$272$		0			0
$273$	2.43845	+	0.384472i	0.147582	+	0.0232693i
$274$		0			0
$275$		−	13.1231i		−	0.791353i
$276$		0			0
$277$	7.36932			0.442779			0.221390	−	0.975185i	$-0.428941\pi$
$277$	7.36932			0.442779			0.221390	+	0.975185i	$0.428941\pi$
$278$		0			0
$279$		−	2.87689i		−	0.172235i
$280$		0			0
$281$			9.36932i			0.558927i	0.960156	+	0.279463i	$0.0901565\pi$
$281$			9.36932i			0.558927i	−0.960156	+	0.279463i	$0.909843\pi$
$282$		0			0
$283$	−12.0000			−0.713326			−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000			−0.713326			−0.356663	+	0.934233i	$0.616086\pi$
$284$		0			0
$285$	4.87689			0.288882
$286$		0			0
$287$	−3.50758			−0.207046
$288$		0			0
$289$	−4.31534			−0.253844
$290$		0			0
$291$		−	9.75379i		−	0.571777i
$292$		0			0
$293$		−	12.6847i		−	0.741046i	−0.928823	−	0.370523i	$-0.879179\pi$
$293$		−	12.6847i		−	0.741046i	0.928823	−	0.370523i	$-0.120821\pi$
$294$		0			0
$295$	−15.6155			−0.909171
$296$		0			0
$297$		−	28.4924i		−	1.65330i
$298$		0			0
$299$	1.75379	−	11.1231i	0.101424	−	0.643266i
$300$		0			0
$301$			4.19224i			0.241636i
$302$		0			0
$303$	17.7538			1.01993
$304$		0			0
$305$		−	4.49242i		−	0.257235i
$306$		0			0
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$		0			0
$309$	4.87689			0.277437
$310$		0			0
$311$	−19.1231			−1.08437			−0.542186	−	0.840259i	$-0.682403\pi$
$311$	−19.1231			−1.08437			−0.542186	+	0.840259i	$0.682403\pi$
$312$		0			0
$313$	4.43845			0.250876			0.125438	−	0.992101i	$-0.459966\pi$
$313$	4.43845			0.250876			0.125438	+	0.992101i	$0.459966\pi$
$314$		0			0
$315$	0.384472			0.0216625
$316$		0			0
$317$			15.1231i			0.849398i	0.905335	+	0.424699i	$0.139620\pi$
$317$			15.1231i			0.849398i	−0.905335	+	0.424699i	$0.860380\pi$
$318$		0			0
$319$			26.2462i			1.46951i
$320$		0			0
$321$	−6.24621			−0.348630
$322$		0			0
$323$			7.12311i			0.396340i
$324$		0			0
$325$	1.43845	−	9.12311i	0.0797907	−	0.506059i
$326$		0			0
$327$			5.17708i			0.286293i
$328$		0			0
$329$	3.31534			0.182781
$330$		0			0
$331$		−	0.246211i		−	0.0135330i	−0.999977	−	0.00676650i	$-0.997846\pi$
$331$		−	0.246211i		−	0.0135330i	0.999977	−	0.00676650i	$-0.00215386\pi$
$332$		0			0
$333$		−	5.36932i		−	0.294237i
$334$		0			0
$335$	14.2462			0.778354
$336$		0			0
$337$	1.31534			0.0716512			0.0358256	−	0.999358i	$-0.488594\pi$
$337$	1.31534			0.0716512			0.0358256	+	0.999358i	$0.488594\pi$
$338$		0			0
$339$	−15.6155			−0.848119
$340$		0			0
$341$	26.2462			1.42131
$342$		0			0
$343$		−	6.05398i		−	0.326884i
$344$		0			0
$345$			7.61553i			0.410006i
$346$		0			0
$347$	1.56155			0.0838285			0.0419143	−	0.999121i	$-0.486654\pi$
$347$	1.56155			0.0838285			0.0419143	+	0.999121i	$0.486654\pi$
$348$		0			0
$349$		−	10.9309i		−	0.585116i	−0.956248	−	0.292558i	$-0.905494\pi$
$349$		−	10.9309i		−	0.585116i	0.956248	−	0.292558i	$-0.0945065\pi$
$350$		0			0
$351$	3.12311	−	19.8078i	0.166699	−	1.05726i
$352$		0			0
$353$			7.61553i			0.405334i	0.979248	+	0.202667i	$0.0649608\pi$
$353$			7.61553i			0.405334i	−0.979248	+	0.202667i	$0.935039\pi$
$354$		0			0
$355$	−10.4384			−0.554015
$356$		0			0
$357$		−	2.43845i		−	0.129056i
$358$		0			0
$359$			13.1231i			0.692611i	0.938122	+	0.346306i	$0.112564\pi$
$359$			13.1231i			0.692611i	−0.938122	+	0.346306i	$0.887436\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	23.8078			1.24958
$364$		0			0
$365$	−14.6307			−0.765805
$366$		0			0
$367$	−28.4924			−1.48729			−0.743646	−	0.668573i	$-0.766905\pi$
$367$	−28.4924			−1.48729			−0.743646	+	0.668573i	$0.766905\pi$
$368$		0			0
$369$			4.49242i			0.233866i
$370$		0			0
$371$		−	5.36932i		−	0.278761i
$372$		0			0
$373$	−35.3693			−1.83135			−0.915677	−	0.401915i	$-0.868345\pi$
$373$	−35.3693			−1.83135			−0.915677	+	0.401915i	$0.868345\pi$
$374$		0			0
$375$			18.4384i			0.952157i
$376$		0			0
$377$	−2.87689	+	18.2462i	−0.148168	+	0.939728i
$378$		0			0
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$		0			0
$381$	−7.61553			−0.390155
$382$		0			0
$383$			29.8078i			1.52311i	0.648103	+	0.761553i	$0.275563\pi$
$383$			29.8078i			1.52311i	−0.648103	+	0.761553i	$0.724437\pi$
$384$		0			0
$385$			3.50758i			0.178763i
$386$		0			0
$387$	5.36932			0.272938
$388$		0			0
$389$	−11.7538			−0.595941			−0.297970	−	0.954575i	$-0.596310\pi$
$389$	−11.7538			−0.595941			−0.297970	+	0.954575i	$0.596310\pi$
$390$		0			0
$391$	−11.1231			−0.562520
$392$		0			0
$393$	−5.17708			−0.261149
$394$		0			0
$395$		−	17.3693i		−	0.873945i
$396$		0			0
$397$		−	0.876894i		−	0.0440101i	−0.999758	−	0.0220050i	$-0.992995\pi$
$397$		−	0.876894i		−	0.0440101i	0.999758	−	0.0220050i	$-0.00700499\pi$
$398$		0			0
$399$	1.36932			0.0685516
$400$		0			0
$401$		−	17.3693i		−	0.867382i	−0.901062	−	0.433691i	$-0.857211\pi$
$401$		−	17.3693i		−	0.867382i	0.901062	−	0.433691i	$-0.142789\pi$
$402$		0			0
$403$	18.2462	+	2.87689i	0.908909	+	0.143308i
$404$		0			0
$405$			10.9309i			0.543159i
$406$		0			0
$407$	48.9848			2.42809
$408$		0			0
$409$			25.3693i			1.25443i	0.778845	+	0.627216i	$0.215806\pi$
$409$			25.3693i			1.25443i	−0.778845	+	0.627216i	$0.784194\pi$
$410$		0			0
$411$			4.87689i			0.240559i
$412$		0			0
$413$	−4.38447			−0.215746
$414$		0			0
$415$	−12.8769			−0.632102
$416$		0			0
$417$	−12.1922			−0.597056
$418$		0			0
$419$	30.4384			1.48702			0.743508	−	0.668727i	$-0.233161\pi$
$419$	30.4384			1.48702			0.743508	+	0.668727i	$0.233161\pi$
$420$		0			0
$421$		−	23.8078i		−	1.16032i	−0.814503	−	0.580160i	$-0.802990\pi$
$421$		−	23.8078i		−	1.16032i	0.814503	−	0.580160i	$-0.197010\pi$
$422$		0			0
$423$		−	4.24621i		−	0.206458i
$424$		0			0
$425$	−9.12311			−0.442536
$426$		0			0
$427$		−	1.26137i		−	0.0610418i
$428$		0			0
$429$	28.4924	+	4.49242i	1.37563	+	0.216896i
$430$		0			0
$431$			37.4233i			1.80262i	0.433178	+	0.901308i	$0.357392\pi$
$431$			37.4233i			1.80262i	−0.433178	+	0.901308i	$0.642608\pi$
$432$		0			0
$433$	9.31534			0.447667			0.223833	−	0.974627i	$-0.428143\pi$
$433$	9.31534			0.447667			0.223833	+	0.974627i	$0.428143\pi$
$434$		0			0
$435$		−	12.4924i		−	0.598966i
$436$		0			0
$437$		−	6.24621i		−	0.298797i
$438$		0			0
$439$	37.8617			1.80704			0.903521	−	0.428544i	$-0.140973\pi$
$439$	37.8617			1.80704			0.903521	+	0.428544i	$0.140973\pi$
$440$		0			0
$441$	−3.82292			−0.182044
$442$		0			0
$443$	9.17708			0.436016			0.218008	−	0.975947i	$-0.430044\pi$
$443$	9.17708			0.436016			0.218008	+	0.975947i	$0.430044\pi$
$444$		0			0
$445$	−4.87689			−0.231187
$446$		0			0
$447$		−	11.1231i		−	0.526105i
$448$		0			0
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$		0			0
$451$	−40.9848			−1.92990
$452$		0			0
$453$			18.0540i			0.848250i
$454$		0			0
$455$	−0.384472	+	2.43845i	−0.0180243	+	0.114316i
$456$		0			0
$457$		−	15.6155i		−	0.730464i	−0.930917	−	0.365232i	$-0.880990\pi$
$457$		−	15.6155i		−	0.730464i	0.930917	−	0.365232i	$-0.119010\pi$
$458$		0			0
$459$	−19.8078			−0.924547
$460$		0			0
$461$			9.17708i			0.427419i	0.976897	+	0.213710i	$0.0685546\pi$
$461$			9.17708i			0.427419i	−0.976897	+	0.213710i	$0.931445\pi$
$462$		0			0
$463$			1.61553i			0.0750800i	0.999295	+	0.0375400i	$0.0119522\pi$
$463$			1.61553i			0.0750800i	−0.999295	+	0.0375400i	$0.988048\pi$
$464$		0			0
$465$	−12.4924			−0.579322
$466$		0			0
$467$	−16.4924			−0.763178			−0.381589	−	0.924332i	$-0.624623\pi$
$467$	−16.4924			−0.763178			−0.381589	+	0.924332i	$0.624623\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$	−19.1231			−0.881146
$472$		0			0
$473$			48.9848i			2.25233i
$474$		0			0
$475$		−	5.12311i		−	0.235064i
$476$		0			0
$477$	−6.87689			−0.314871
$478$		0			0
$479$			29.8078i			1.36195i	0.732306	+	0.680976i	$0.238444\pi$
$479$			29.8078i			1.36195i	−0.732306	+	0.680976i	$0.761556\pi$
$480$		0			0
$481$	34.0540	+	5.36932i	1.55273	+	0.244820i
$482$		0			0
$483$			2.13826i			0.0972942i
$484$		0			0
$485$	9.75379			0.442897
$486$		0			0
$487$			15.8617i			0.718764i	0.933191	+	0.359382i	$0.117012\pi$
$487$			15.8617i			0.718764i	−0.933191	+	0.359382i	$0.882988\pi$
$488$		0			0
$489$		−	31.6155i		−	1.42970i
$490$		0			0
$491$	−31.8078			−1.43546			−0.717732	−	0.696319i	$-0.754820\pi$
$491$	−31.8078			−1.43546			−0.717732	+	0.696319i	$0.754820\pi$
$492$		0			0
$493$	18.2462			0.821768
$494$		0			0
$495$	4.49242			0.201919
$496$		0			0
$497$	−2.93087			−0.131467
$498$		0			0
$499$			25.1231i			1.12466i	0.826911	+	0.562332i	$0.190096\pi$
$499$			25.1231i			1.12466i	−0.826911	+	0.562332i	$0.809904\pi$
$500$		0			0
$501$		−	10.7386i		−	0.479767i
$502$		0			0
$503$	−9.36932			−0.417757			−0.208879	−	0.977942i	$-0.566981\pi$
$503$	−9.36932			−0.417757			−0.208879	+	0.977942i	$0.566981\pi$
$504$		0			0
$505$			17.7538i			0.790033i
$506$		0			0
$507$	19.3153	+	6.24621i	0.857824	+	0.277404i
$508$		0			0
$509$			2.63068i			0.116603i	0.998299	+	0.0583015i	$0.0185685\pi$
$509$			2.63068i			0.116603i	−0.998299	+	0.0583015i	$0.981432\pi$
$510$		0			0
$511$	−4.10795			−0.181725
$512$		0			0
$513$		−	11.1231i		−	0.491097i
$514$		0			0
$515$			4.87689i			0.214902i
$516$		0			0
$517$	38.7386			1.70372
$518$		0			0
$519$	−6.63068			−0.291055
$520$		0			0
$521$	0.930870			0.0407821			0.0203911	−	0.999792i	$-0.493509\pi$
$521$	0.930870			0.0407821			0.0203911	+	0.999792i	$0.493509\pi$
$522$		0			0
$523$	13.7538			0.601411			0.300706	−	0.953717i	$-0.402778\pi$
$523$	13.7538			0.601411			0.300706	+	0.953717i	$0.402778\pi$
$524$		0			0
$525$			1.75379i			0.0765416i
$526$		0			0
$527$		−	18.2462i		−	0.794817i
$528$		0			0
$529$	−13.2462			−0.575922
$530$		0			0
$531$			5.61553i			0.243693i
$532$		0			0
$533$	−28.4924	−	4.49242i	−1.23414	−	0.194588i
$534$		0			0
$535$		−	6.24621i		−	0.270047i
$536$		0			0
$537$	27.4233			1.18340
$538$		0			0
$539$		−	34.8769i		−	1.50225i
$540$		0			0
$541$			25.1771i			1.08245i	0.840879	+	0.541224i	$0.182039\pi$
$541$			25.1771i			1.08245i	−0.840879	+	0.541224i	$0.817961\pi$
$542$		0			0
$543$	12.8769			0.552600
$544$		0			0
$545$	−5.17708			−0.221762
$546$		0			0
$547$	−23.8078			−1.01795			−0.508973	−	0.860782i	$-0.669975\pi$
$547$	−23.8078			−1.01795			−0.508973	+	0.860782i	$0.669975\pi$
$548$		0			0
$549$	−1.61553			−0.0689491
$550$		0			0
$551$			10.2462i			0.436503i
$552$		0			0
$553$		−	4.87689i		−	0.207387i
$554$		0			0
$555$	−23.3153			−0.989681
$556$		0			0
$557$			18.9309i			0.802127i	0.916050	+	0.401063i	$0.131359\pi$
$557$			18.9309i			0.802127i	−0.916050	+	0.401063i	$0.868641\pi$
$558$		0			0
$559$	−5.36932	+	34.0540i	−0.227098	+	1.44033i
$560$		0			0
$561$		−	28.4924i		−	1.20295i
$562$		0			0
$563$	26.9309			1.13500			0.567500	−	0.823373i	$-0.307911\pi$
$563$	26.9309			1.13500			0.567500	+	0.823373i	$0.307911\pi$
$564$		0			0
$565$		−	15.6155i		−	0.656950i
$566$		0			0
$567$			3.06913i			0.128891i
$568$		0			0
$569$	−2.19224			−0.0919033			−0.0459517	−	0.998944i	$-0.514632\pi$
$569$	−2.19224			−0.0919033			−0.0459517	+	0.998944i	$0.514632\pi$
$570$		0			0
$571$	14.0540			0.588141			0.294070	−	0.955784i	$-0.404990\pi$
$571$	14.0540			0.588141			0.294070	+	0.955784i	$0.404990\pi$
$572$		0			0
$573$	−34.7386			−1.45123
$574$		0			0
$575$	8.00000			0.333623
$576$		0			0
$577$		−	29.8617i		−	1.24316i	−0.783351	−	0.621580i	$-0.786491\pi$
$577$		−	29.8617i		−	1.24316i	0.783351	−	0.621580i	$-0.213509\pi$
$578$		0			0
$579$		−	36.8769i		−	1.53255i
$580$		0			0
$581$	−3.61553			−0.149997
$582$		0			0
$583$		−	62.7386i		−	2.59837i
$584$		0			0
$585$	3.12311	+	0.492423i	0.129125	+	0.0203592i
$586$		0			0
$587$			9.50758i			0.392420i	0.980562	+	0.196210i	$0.0628634\pi$
$587$			9.50758i			0.392420i	−0.980562	+	0.196210i	$0.937137\pi$
$588$		0			0
$589$	10.2462			0.422188
$590$		0			0
$591$		−	14.9309i		−	0.614174i
$592$		0			0
$593$		−	25.7538i		−	1.05758i	−0.848752	−	0.528791i	$-0.822646\pi$
$593$		−	25.7538i		−	1.05758i	0.848752	−	0.528791i	$-0.177354\pi$
$594$		0			0
$595$	2.43845			0.0999666
$596$		0			0
$597$	−22.2462			−0.910477
$598$		0			0
$599$	−41.3693			−1.69030			−0.845152	−	0.534526i	$-0.820490\pi$
$599$	−41.3693			−1.69030			−0.845152	+	0.534526i	$0.820490\pi$
$600$		0			0
$601$	21.8078			0.889557			0.444779	−	0.895641i	$-0.353282\pi$
$601$	21.8078			0.889557			0.444779	+	0.895641i	$0.353282\pi$
$602$		0			0
$603$		−	5.12311i		−	0.208629i
$604$		0			0
$605$			23.8078i			0.967923i
$606$		0			0
$607$	7.61553			0.309105			0.154552	−	0.987985i	$-0.450606\pi$
$607$	7.61553			0.309105			0.154552	+	0.987985i	$0.450606\pi$
$608$		0			0
$609$		−	3.50758i		−	0.142134i
$610$		0			0
$611$	26.9309	+	4.24621i	1.08951	+	0.171783i
$612$		0			0
$613$		−	18.6307i		−	0.752486i	−0.926521	−	0.376243i	$-0.877216\pi$
$613$		−	18.6307i		−	0.752486i	0.926521	−	0.376243i	$-0.122784\pi$
$614$		0			0
$615$	19.5076			0.786621
$616$		0			0
$617$		−	1.75379i		−	0.0706049i	−0.999377	−	0.0353024i	$-0.988761\pi$
$617$		−	1.75379i		−	0.0706049i	0.999377	−	0.0353024i	$-0.0112395\pi$
$618$		0			0
$619$		−	17.6155i		−	0.708028i	−0.935240	−	0.354014i	$-0.884817\pi$
$619$		−	17.6155i		−	0.708028i	0.935240	−	0.354014i	$-0.115183\pi$
$620$		0			0
$621$	17.3693			0.697007
$622$		0			0
$623$	−1.36932			−0.0548605
$624$		0			0
$625$	−5.63068			−0.225227
$626$		0			0
$627$	16.0000			0.638978
$628$		0			0
$629$		−	34.0540i		−	1.35782i
$630$		0			0
$631$		−	11.5616i		−	0.460258i	−0.973160	−	0.230129i	$-0.926085\pi$
$631$		−	11.5616i		−	0.460258i	0.973160	−	0.230129i	$-0.0739148\pi$
$632$		0			0
$633$	−34.4384			−1.36881
$634$		0			0
$635$		−	7.61553i		−	0.302213i
$636$		0			0
$637$	3.82292	−	24.2462i	0.151470	−	0.960670i
$638$		0			0
$639$			3.75379i			0.148498i
$640$		0			0
$641$	3.75379			0.148266			0.0741329	−	0.997248i	$-0.476381\pi$
$641$	3.75379			0.148266			0.0741329	+	0.997248i	$0.476381\pi$
$642$		0			0
$643$			25.1231i			0.990759i	0.868677	+	0.495379i	$0.164971\pi$
$643$			25.1231i			0.990759i	−0.868677	+	0.495379i	$0.835029\pi$
$644$		0			0
$645$		−	23.3153i		−	0.918041i
$646$		0			0
$647$	−3.12311			−0.122782			−0.0613910	−	0.998114i	$-0.519554\pi$
$647$	−3.12311			−0.122782			−0.0613910	+	0.998114i	$0.519554\pi$
$648$		0			0
$649$	−51.2311			−2.01099
$650$		0			0
$651$	−3.50758			−0.137473
$652$		0			0
$653$	13.6155			0.532817			0.266408	−	0.963860i	$-0.414163\pi$
$653$	13.6155			0.532817			0.266408	+	0.963860i	$0.414163\pi$
$654$		0			0
$655$		−	5.17708i		−	0.202285i
$656$		0			0
$657$			5.26137i			0.205265i
$658$		0			0
$659$	−6.73863			−0.262500			−0.131250	−	0.991349i	$-0.541899\pi$
$659$	−6.73863			−0.262500			−0.131250	+	0.991349i	$0.541899\pi$
$660$		0			0
$661$			32.8769i			1.27876i	0.768890	+	0.639381i	$0.220810\pi$
$661$			32.8769i			1.27876i	−0.768890	+	0.639381i	$0.779190\pi$
$662$		0			0
$663$	3.12311	−	19.8078i	0.121291	−	0.769270i
$664$		0			0
$665$			1.36932i			0.0530998i
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$			42.4384i			1.64077i
$670$		0			0
$671$		−	14.7386i		−	0.568979i
$672$		0			0
$673$	−22.3002			−0.859609			−0.429805	−	0.902922i	$-0.641418\pi$
$673$	−22.3002			−0.859609			−0.429805	+	0.902922i	$0.641418\pi$
$674$		0			0
$675$	14.2462			0.548337
$676$		0			0
$677$	−28.7386			−1.10452			−0.552258	−	0.833673i	$-0.686234\pi$
$677$	−28.7386			−1.10452			−0.552258	+	0.833673i	$0.686234\pi$
$678$		0			0
$679$	2.73863			0.105099
$680$		0			0
$681$			5.26137i			0.201616i
$682$		0			0
$683$			12.2462i			0.468588i	0.972166	+	0.234294i	$0.0752779\pi$
$683$			12.2462i			0.468588i	−0.972166	+	0.234294i	$0.924722\pi$
$684$		0			0
$685$	−4.87689			−0.186337
$686$		0			0
$687$		−	32.3002i		−	1.23233i
$688$		0			0
$689$	6.87689	−	43.6155i	0.261989	−	1.66162i
$690$		0			0
$691$			27.8617i			1.05991i	0.848026	+	0.529955i	$0.177791\pi$
$691$			27.8617i			1.05991i	−0.848026	+	0.529955i	$0.822209\pi$
$692$		0			0
$693$	1.26137			0.0479153
$694$		0			0
$695$		−	12.1922i		−	0.462478i
$696$		0			0
$697$			28.4924i			1.07923i
$698$		0			0
$699$	5.56155			0.210357
$700$		0			0
$701$	37.2311			1.40620			0.703099	−	0.711092i	$-0.251799\pi$
$701$	37.2311			1.40620			0.703099	+	0.711092i	$0.251799\pi$
$702$		0			0
$703$	19.1231			0.721242
$704$		0			0
$705$	−18.4384			−0.694432
$706$		0			0
$707$			4.98485i			0.187474i
$708$		0			0
$709$			7.12311i			0.267514i	0.991014	+	0.133757i	$0.0427041\pi$
$709$			7.12311i			0.267514i	−0.991014	+	0.133757i	$0.957296\pi$
$710$		0			0
$711$	−6.24621			−0.234251
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$	−4.49242	+	28.4924i	−0.168007	+	1.06556i
$716$		0			0
$717$		−	13.9460i		−	0.520824i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$			1.36932i			0.0509960i
$722$		0			0
$723$			42.3542i			1.57517i
$724$		0			0
$725$	−13.1231			−0.487380
$726$		0			0
$727$	−39.2311			−1.45500			−0.727500	−	0.686108i	$-0.759318\pi$
$727$	−39.2311			−1.45500			−0.727500	+	0.686108i	$0.759318\pi$
$728$		0			0
$729$	29.9848			1.11055
$730$		0			0
$731$	34.0540			1.25953
$732$		0			0
$733$			4.30019i			0.158831i	0.996842	+	0.0794155i	$0.0253054\pi$
$733$			4.30019i			0.158831i	−0.996842	+	0.0794155i	$0.974695\pi$
$734$		0			0
$735$			16.6004i			0.612314i
$736$		0			0
$737$	46.7386			1.72164
$738$		0			0
$739$		−	15.8617i		−	0.583484i	−0.956497	−	0.291742i	$-0.905765\pi$
$739$		−	15.8617i		−	0.583484i	0.956497	−	0.291742i	$-0.0942348\pi$
$740$		0			0
$741$	11.1231	+	1.75379i	0.408617	+	0.0644270i
$742$		0			0
$743$			7.94602i			0.291511i	0.989321	+	0.145756i	$0.0465613\pi$
$743$			7.94602i			0.291511i	−0.989321	+	0.145756i	$0.953439\pi$
$744$		0			0
$745$	11.1231			0.407519
$746$		0			0
$747$			4.63068i			0.169428i
$748$		0			0
$749$		−	1.75379i		−	0.0640820i
$750$		0			0
$751$	3.50758			0.127993			0.0639967	−	0.997950i	$-0.479615\pi$
$751$	3.50758			0.127993			0.0639967	+	0.997950i	$0.479615\pi$
$752$		0			0
$753$	38.2462			1.39377
$754$		0			0
$755$	−18.0540			−0.657051
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$			24.9848i			0.906892i
$760$		0			0
$761$			44.8769i			1.62679i	0.581714	+	0.813393i	$0.302382\pi$
$761$			44.8769i			1.62679i	−0.581714	+	0.813393i	$0.697618\pi$
$762$		0			0
$763$	−1.45360			−0.0526239
$764$		0			0
$765$		−	3.12311i		−	0.112916i
$766$		0			0
$767$	−35.6155	−	5.61553i	−1.28600	−	0.202765i
$768$		0			0
$769$			28.4924i			1.02746i	0.857951	+	0.513732i	$0.171737\pi$
$769$			28.4924i			1.02746i	−0.857951	+	0.513732i	$0.828263\pi$
$770$		0			0
$771$	−1.45360			−0.0523502
$772$		0			0
$773$		−	14.0540i		−	0.505486i	−0.967533	−	0.252743i	$-0.918667\pi$
$773$		−	14.0540i		−	0.505486i	0.967533	−	0.252743i	$-0.0813328\pi$
$774$		0			0
$775$			13.1231i			0.471396i
$776$		0			0
$777$	−6.54640			−0.234851
$778$		0			0
$779$	−16.0000			−0.573259
$780$		0			0
$781$	−34.2462			−1.22543
$782$		0			0
$783$	−28.4924			−1.01824
$784$		0			0
$785$		−	19.1231i		−	0.682533i
$786$		0			0
$787$		−	3.36932i		−	0.120103i	−0.998195	−	0.0600516i	$-0.980873\pi$
$787$		−	3.36932i		−	0.120103i	0.998195	−	0.0600516i	$-0.0191265\pi$
$788$		0			0
$789$	−7.01515			−0.249746
$790$		0			0
$791$		−	4.38447i		−	0.155894i
$792$		0			0
$793$	1.61553	−	10.2462i	0.0573691	−	0.363854i
$794$		0			0
$795$			29.8617i			1.05909i
$796$		0			0
$797$	12.2462			0.433783			0.216892	−	0.976196i	$-0.430408\pi$
$797$	12.2462			0.433783			0.216892	+	0.976196i	$0.430408\pi$
$798$		0			0
$799$		−	26.9309i		−	0.952746i
$800$		0			0
$801$			1.75379i			0.0619671i
$802$		0			0
$803$	−48.0000			−1.69388
$804$		0			0
$805$	−2.13826			−0.0753638
$806$		0			0
$807$	−16.9848			−0.597895
$808$		0			0
$809$	42.6847			1.50071			0.750356	−	0.661034i	$-0.229882\pi$
$809$	42.6847			1.50071			0.750356	+	0.661034i	$0.229882\pi$
$810$		0			0
$811$			23.3693i			0.820608i	0.911949	+	0.410304i	$0.134577\pi$
$811$			23.3693i			0.820608i	−0.911949	+	0.410304i	$0.865423\pi$
$812$		0			0
$813$			22.9309i			0.804221i
$814$		0			0
$815$	31.6155			1.10744
$816$		0			0
$817$			19.1231i			0.669033i
$818$		0			0
$819$	0.876894	+	0.138261i	0.0306412	+	0.00483122i
$820$		0			0
$821$		−	42.5464i		−	1.48488i	−0.669913	−	0.742440i	$-0.733669\pi$
$821$		−	42.5464i		−	1.48488i	0.669913	−	0.742440i	$-0.266331\pi$
$822$		0			0
$823$	50.3542			1.75524			0.877618	−	0.479361i	$-0.159131\pi$
$823$	50.3542			1.75524			0.877618	+	0.479361i	$0.159131\pi$
$824$		0			0
$825$			20.4924i			0.713455i
$826$		0			0
$827$			22.0000i			0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$			22.0000i			0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$		0			0
$829$	−6.87689			−0.238844			−0.119422	−	0.992844i	$-0.538104\pi$
$829$	−6.87689			−0.238844			−0.119422	+	0.992844i	$0.538104\pi$
$830$		0			0
$831$	−11.5076			−0.399193
$832$		0			0
$833$	−24.2462			−0.840081
$834$		0			0
$835$	10.7386			0.371626
$836$		0			0
$837$			28.4924i			0.984842i
$838$		0			0
$839$		−	43.8617i		−	1.51428i	−0.653255	−	0.757138i	$-0.726597\pi$
$839$		−	43.8617i		−	1.51428i	0.653255	−	0.757138i	$-0.273403\pi$
$840$		0			0
$841$	−2.75379			−0.0949582
$842$		0			0
$843$		−	14.6307i		−	0.503907i
$844$		0			0
$845$	−6.24621	+	19.3153i	−0.214876	+	0.664468i
$846$		0			0
$847$			6.68466i			0.229688i
$848$		0			0
$849$	18.7386			0.643108
$850$		0			0
$851$			29.8617i			1.02365i
$852$		0			0
$853$			4.68466i			0.160400i	0.996779	+	0.0801998i	$0.0255558\pi$
$853$			4.68466i			0.160400i	−0.996779	+	0.0801998i	$0.974444\pi$
$854$		0			0
$855$	1.75379			0.0599783
$856$		0			0
$857$	−50.4924			−1.72479			−0.862394	−	0.506237i	$-0.831036\pi$
$857$	−50.4924			−1.72479			−0.862394	+	0.506237i	$0.831036\pi$
$858$		0			0
$859$	10.2462			0.349596			0.174798	−	0.984604i	$-0.444073\pi$
$859$	10.2462			0.349596			0.174798	+	0.984604i	$0.444073\pi$
$860$		0			0
$861$	5.47727			0.186665
$862$		0			0
$863$		−	48.0540i		−	1.63578i	−0.575377	−	0.817888i	$-0.695145\pi$
$863$		−	48.0540i		−	1.63578i	0.575377	−	0.817888i	$-0.304855\pi$
$864$		0			0
$865$		−	6.63068i		−	0.225450i
$866$		0			0
$867$	6.73863			0.228856
$868$		0			0
$869$		−	56.9848i		−	1.93308i
$870$		0			0
$871$	32.4924	+	5.12311i	1.10096	+	0.173590i
$872$		0			0
$873$		−	3.50758i		−	0.118714i
$874$		0			0
$875$	−5.17708			−0.175017
$876$		0			0
$877$		−	7.42329i		−	0.250667i	−0.992115	−	0.125333i	$-0.960000\pi$
$877$		−	7.42329i		−	0.250667i	0.992115	−	0.125333i	$-0.0400000\pi$
$878$		0			0
$879$			19.8078i			0.668099i
$880$		0			0
$881$	−42.7926			−1.44172			−0.720860	−	0.693081i	$-0.756253\pi$
$881$	−42.7926			−1.44172			−0.720860	+	0.693081i	$0.756253\pi$
$882$		0			0
$883$	31.8078			1.07042			0.535208	−	0.844720i	$-0.320233\pi$
$883$	31.8078			1.07042			0.535208	+	0.844720i	$0.320233\pi$
$884$		0			0
$885$	24.3845			0.819675
$886$		0			0
$887$	5.26137			0.176659			0.0883297	−	0.996091i	$-0.471847\pi$
$887$	5.26137			0.176659			0.0883297	+	0.996091i	$0.471847\pi$
$888$		0			0
$889$		−	2.13826i		−	0.0717150i
$890$		0			0
$891$			35.8617i			1.20141i
$892$		0			0
$893$	15.1231			0.506075
$894$		0			0
$895$			27.4233i			0.916660i
$896$		0			0
$897$	−2.73863	+	17.3693i	−0.0914403	+	0.579945i
$898$		0			0
$899$		−	26.2462i		−	0.875360i
$900$		0			0
$901$	−43.6155			−1.45304
$902$		0			0
$903$		−	6.54640i		−	0.217850i
$904$		0			0
$905$			12.8769i			0.428042i
$906$		0			0
$907$	22.4384			0.745056			0.372528	−	0.928021i	$-0.378491\pi$
$907$	22.4384			0.745056			0.372528	+	0.928021i	$0.378491\pi$
$908$		0			0
$909$	6.38447			0.211760
$910$		0			0
$911$	−25.7538			−0.853261			−0.426631	−	0.904426i	$-0.640300\pi$
$911$	−25.7538			−0.853261			−0.426631	+	0.904426i	$0.640300\pi$
$912$		0			0
$913$	−42.2462			−1.39815
$914$		0			0
$915$			7.01515i			0.231914i
$916$		0			0
$917$		−	1.45360i		−	0.0480022i
$918$		0			0
$919$	−1.75379			−0.0578522			−0.0289261	−	0.999582i	$-0.509209\pi$
$919$	−1.75379			−0.0578522			−0.0289261	+	0.999582i	$0.509209\pi$
$920$		0			0
$921$			28.1080i			0.926188i
$922$		0			0
$923$	−23.8078	−	3.75379i	−0.783642	−	0.123557i
$924$		0			0
$925$			24.4924i			0.805306i
$926$		0			0
$927$	1.75379			0.0576020
$928$		0			0
$929$		−	18.7386i		−	0.614795i	−0.951581	−	0.307397i	$-0.900542\pi$
$929$		−	18.7386i		−	0.614795i	0.951581	−	0.307397i	$-0.0994581\pi$
$930$		0			0
$931$		−	13.6155i		−	0.446231i
$932$		0			0
$933$	29.8617			0.977629
$934$		0			0
$935$	28.4924			0.931802
$936$		0			0
$937$	19.7538			0.645328			0.322664	−	0.946514i	$-0.395422\pi$
$937$	19.7538			0.645328			0.322664	+	0.946514i	$0.395422\pi$
$938$		0			0
$939$	−6.93087			−0.226180
$940$		0			0
$941$			42.5464i			1.38697i	0.720470	+	0.693486i	$0.243926\pi$
$941$			42.5464i			1.38697i	−0.720470	+	0.693486i	$0.756074\pi$
$942$		0			0
$943$		−	24.9848i		−	0.813618i
$944$		0			0
$945$	−3.80776			−0.123867
$946$		0			0
$947$			15.3693i			0.499436i	0.968319	+	0.249718i	$0.0803379\pi$
$947$			15.3693i			0.499436i	−0.968319	+	0.249718i	$0.919662\pi$
$948$		0			0
$949$	−33.3693	−	5.26137i	−1.08321	−	0.170791i
$950$		0			0
$951$		−	23.6155i		−	0.765786i
$952$		0			0
$953$	20.0540			0.649612			0.324806	−	0.945781i	$-0.394701\pi$
$953$	20.0540			0.649612			0.324806	+	0.945781i	$0.394701\pi$
$954$		0			0
$955$		−	34.7386i		−	1.12412i
$956$		0			0
$957$		−	40.9848i		−	1.32485i
$958$		0			0
$959$	−1.36932			−0.0442175
$960$		0			0
$961$	4.75379			0.153348
$962$		0			0
$963$	−2.24621			−0.0723831
$964$		0			0
$965$	36.8769			1.18711
$966$		0			0
$967$			15.5616i			0.500426i	0.968191	+	0.250213i	$0.0805006\pi$
$967$			15.5616i			0.500426i	−0.968191	+	0.250213i	$0.919499\pi$
$968$		0			0
$969$		−	11.1231i		−	0.357326i
$970$		0			0
$971$	57.1771			1.83490			0.917450	−	0.397851i	$-0.130244\pi$
$971$	57.1771			1.83490			0.917450	+	0.397851i	$0.130244\pi$
$972$		0			0
$973$		−	3.42329i		−	0.109746i
$974$		0			0
$975$	−2.24621	+	14.2462i	−0.0719363	+	0.456244i
$976$		0			0
$977$		−	47.2311i		−	1.51106i	−0.655117	−	0.755528i	$-0.727381\pi$
$977$		−	47.2311i		−	1.51106i	0.655117	−	0.755528i	$-0.272619\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$			1.86174i			0.0594408i
$982$		0			0
$983$		−	10.1922i		−	0.325082i	−0.986702	−	0.162541i	$-0.948031\pi$
$983$		−	10.1922i		−	0.325082i	0.986702	−	0.162541i	$-0.0519690\pi$
$984$		0			0
$985$	14.9309			0.475737
$986$		0			0
$987$	−5.17708			−0.164788
$988$		0			0
$989$	−29.8617			−0.949548
$990$		0			0
$991$	7.61553			0.241915			0.120958	−	0.992658i	$-0.461403\pi$
$991$	7.61553			0.241915			0.120958	+	0.992658i	$0.461403\pi$
$992$		0			0
$993$			0.384472i			0.0122008i
$994$		0			0
$995$		−	22.2462i		−	0.705252i
$996$		0			0
$997$	−1.61553			−0.0511643			−0.0255821	−	0.999673i	$-0.508144\pi$
$997$	−1.61553			−0.0511643			−0.0255821	+	0.999673i	$0.508144\pi$
$998$		0			0
$999$			53.1771i			1.68245i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.f.b.129.1		4
3.2	odd	2		1872.2.c.j.1585.3		4
4.3	odd	2		104.2.f.a.25.3	✓	4
8.3	odd	2		832.2.f.i.129.2		4
8.5	even	2		832.2.f.f.129.4		4
12.11	even	2		936.2.c.d.649.3		4
13.5	odd	4		2704.2.a.t.1.1		2
13.8	odd	4		2704.2.a.s.1.1		2
13.12	even	2	inner	208.2.f.b.129.2		4
20.3	even	4		2600.2.f.b.649.3		4
20.7	even	4		2600.2.f.a.649.2		4
20.19	odd	2		2600.2.k.a.2001.1		4
39.38	odd	2		1872.2.c.j.1585.2		4
52.3	odd	6		1352.2.o.e.1161.1		8
52.7	even	12		1352.2.i.g.1329.1		4
52.11	even	12		1352.2.i.g.529.1		4
52.15	even	12		1352.2.i.h.529.1		4
52.19	even	12		1352.2.i.h.1329.1		4
52.23	odd	6		1352.2.o.e.1161.2		8
52.31	even	4		1352.2.a.e.1.2		2
52.35	odd	6		1352.2.o.e.361.1		8
52.43	odd	6		1352.2.o.e.361.2		8
52.47	even	4		1352.2.a.d.1.2		2
52.51	odd	2		104.2.f.a.25.4	yes	4
104.51	odd	2		832.2.f.i.129.1		4
104.77	even	2		832.2.f.f.129.3		4
156.155	even	2		936.2.c.d.649.2		4
260.103	even	4		2600.2.f.a.649.3		4
260.207	even	4		2600.2.f.b.649.2		4
260.259	odd	2		2600.2.k.a.2001.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.f.a.25.3	✓	4	4.3	odd	2
104.2.f.a.25.4	yes	4	52.51	odd	2
208.2.f.b.129.1		4	1.1	even	1	trivial
208.2.f.b.129.2		4	13.12	even	2	inner
832.2.f.f.129.3		4	104.77	even	2
832.2.f.f.129.4		4	8.5	even	2
832.2.f.i.129.1		4	104.51	odd	2
832.2.f.i.129.2		4	8.3	odd	2
936.2.c.d.649.2		4	156.155	even	2
936.2.c.d.649.3		4	12.11	even	2
1352.2.a.d.1.2		2	52.47	even	4
1352.2.a.e.1.2		2	52.31	even	4
1352.2.i.g.529.1		4	52.11	even	12
1352.2.i.g.1329.1		4	52.7	even	12
1352.2.i.h.529.1		4	52.15	even	12
1352.2.i.h.1329.1		4	52.19	even	12
1352.2.o.e.361.1		8	52.35	odd	6
1352.2.o.e.361.2		8	52.43	odd	6
1352.2.o.e.1161.1		8	52.3	odd	6
1352.2.o.e.1161.2		8	52.23	odd	6
1872.2.c.j.1585.2		4	39.38	odd	2
1872.2.c.j.1585.3		4	3.2	odd	2
2600.2.f.a.649.2		4	20.7	even	4
2600.2.f.a.649.3		4	260.103	even	4
2600.2.f.b.649.2		4	260.207	even	4
2600.2.f.b.649.3		4	20.3	even	4
2600.2.k.a.2001.1		4	20.19	odd	2
2600.2.k.a.2001.2		4	260.259	odd	2
2704.2.a.s.1.1		2	13.8	odd	4
2704.2.a.t.1.1		2	13.5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} -1.56155i q^{5} -0.438447i q^{7} -0.561553 q^{9} -5.12311i q^{11} +(0.561553 - 3.56155i) q^{13} +2.43845i q^{15} -3.56155 q^{17} -2.00000i q^{19} +0.684658i q^{21} +3.12311 q^{23} +2.56155 q^{25} +5.56155 q^{27} -5.12311 q^{29} +5.12311i q^{31} +8.00000i q^{33} -0.684658 q^{35} +9.56155i q^{37} +(-0.876894 + 5.56155i) q^{39} -8.00000i q^{41} -9.56155 q^{43} +0.876894i q^{45} +7.56155i q^{47} +6.80776 q^{49} +5.56155 q^{51} +12.2462 q^{53} -8.00000 q^{55} +3.12311i q^{57} -10.0000i q^{59} +2.87689 q^{61} +0.246211i q^{63} +(-5.56155 - 0.876894i) q^{65} +9.12311i q^{67} -4.87689 q^{69} -6.68466i q^{71} -9.36932i q^{73} -4.00000 q^{75} -2.24621 q^{77} +11.1231 q^{79} -7.00000 q^{81} -8.24621i q^{83} +5.56155i q^{85} +8.00000 q^{87} -3.12311i q^{89} +(-1.56155 - 0.246211i) q^{91} -8.00000i q^{93} -3.12311 q^{95} +6.24621i q^{97} +2.87689i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9} - 6 q^{13} - 6 q^{17} - 4 q^{23} + 2 q^{25} + 14 q^{27} - 4 q^{29} + 22 q^{35} - 20 q^{39} - 30 q^{43} - 14 q^{49} + 14 q^{51} + 16 q^{53} - 32 q^{55} + 28 q^{61} - 14 q^{65} - 36 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 - 6 * q^13 - 6 * q^17 - 4 * q^23 + 2 * q^25 + 14 * q^27 - 4 * q^29 + 22 * q^35 - 20 * q^39 - 30 * q^43 - 14 * q^49 + 14 * q^51 + 16 * q^53 - 32 * q^55 + 28 * q^61 - 14 * q^65 - 36 * q^69 - 16 * q^75 + 24 * q^77 + 28 * q^79 - 28 * q^81 + 32 * q^87 + 2 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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