LMFDB - Embedded newform 140.2.q.a.9.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(9,140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("140.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	140.q (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.11790562830$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			9.1
Root			$-1.63746 + 1.52274i$ of defining polynomial
Character	$\chi$	$=$	140.9
Dual form			140.2.q.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.50000 + 0.866025i) q^{3} +(-1.63746 + 1.52274i) q^{5} +(-2.63746 + 0.209313i) q^{7} +(-1.13746 - 1.97014i) q^{11} +6.09095i q^{13} +(1.13746 - 3.70219i) q^{15} +(4.13746 - 2.38876i) q^{17} +(-2.13746 + 3.70219i) q^{19} +(3.77492 - 2.59808i) q^{21} +(0.774917 + 0.447399i) q^{23} +(0.362541 - 4.98684i) q^{25} -5.19615i q^{27} +3.27492 q^{29} +(2.13746 + 3.70219i) q^{31} +(3.41238 + 1.97014i) q^{33} +(4.00000 - 4.35890i) q^{35} +(4.86254 + 2.80739i) q^{37} +(-5.27492 - 9.13642i) q^{39} -11.2749 q^{41} +6.50958i q^{43} +(-1.86254 - 1.07534i) q^{47} +(6.91238 - 1.10411i) q^{49} +(-4.13746 + 7.16629i) q^{51} +(-6.41238 + 3.70219i) q^{53} +(4.86254 + 1.49397i) q^{55} -7.40437i q^{57} +(2.13746 + 3.70219i) q^{59} +(-0.774917 + 1.34220i) q^{61} +(-9.27492 - 9.97368i) q^{65} +(-12.0498 + 6.95698i) q^{67} -1.54983 q^{69} +10.5498 q^{71} +(-1.86254 + 1.07534i) q^{73} +(3.77492 + 7.79423i) q^{75} +(3.41238 + 4.95807i) q^{77} +(-0.137459 + 0.238085i) q^{79} +(4.50000 + 7.79423i) q^{81} -5.67232i q^{83} +(-3.13746 + 10.2118i) q^{85} +(-4.91238 + 2.83616i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(-1.27492 - 16.0646i) q^{91} +(-6.41238 - 3.70219i) q^{93} +(-2.13746 - 9.31697i) q^{95} -6.92820i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} + q^{5} - 3 q^{7} + 3 q^{11} - 3 q^{15} + 9 q^{17} - q^{19} - 12 q^{23} + 9 q^{25} - 2 q^{29} + q^{31} - 9 q^{33} + 16 q^{35} + 27 q^{37} - 6 q^{39} - 30 q^{41} - 15 q^{47} + 5 q^{49} - 9 q^{51}+ \cdots - q^{95}+O(q^{100}) $ 4 * q - 6 * q^3 + q^5 - 3 * q^7 + 3 * q^11 - 3 * q^15 + 9 * q^17 - q^19 - 12 * q^23 + 9 * q^25 - 2 * q^29 + q^31 - 9 * q^33 + 16 * q^35 + 27 * q^37 - 6 * q^39 - 30 * q^41 - 15 * q^47 + 5 * q^49 - 9 * q^51 - 3 * q^53 + 27 * q^55 + q^59 + 12 * q^61 - 22 * q^65 - 18 * q^67 + 24 * q^69 + 12 * q^71 - 15 * q^73 - 9 * q^77 + 7 * q^79 + 18 * q^81 - 5 * q^85 + 3 * q^87 - 14 * q^89 + 10 * q^91 - 3 * q^93 - q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$4$		0			0
$5$	−1.63746	+	1.52274i	−0.732294	+	0.680989i
$5$	−1.63746	+	1.52274i	−0.732294	+	0.680989i
$6$		0			0
$7$	−2.63746	+	0.209313i	−0.996866	+	0.0791130i
$7$	−2.63746	+	0.209313i	−0.996866	+	0.0791130i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.13746	−	1.97014i	−0.342957	−	0.594018i	0.642024	−	0.766685i	$-0.278095\pi$
$11$	−1.13746	−	1.97014i	−0.342957	−	0.594018i	−0.984980	+	0.172666i	$0.944762\pi$
$12$		0			0
$13$			6.09095i			1.68933i	0.535299	+	0.844663i	$0.320199\pi$
$13$			6.09095i			1.68933i	−0.535299	+	0.844663i	$0.679801\pi$
$14$		0			0
$15$	1.13746	−	3.70219i	0.293691	−	0.955901i
$16$		0			0
$17$	4.13746	−	2.38876i	1.00348	−	0.579360i	0.0942047	−	0.995553i	$-0.469969\pi$
$17$	4.13746	−	2.38876i	1.00348	−	0.579360i	0.909276	+	0.416193i	$0.136636\pi$
$18$		0			0
$19$	−2.13746	+	3.70219i	−0.490367	+	0.849340i	−0.999939	−	0.0110882i	$-0.996470\pi$
$19$	−2.13746	+	3.70219i	−0.490367	+	0.849340i	0.509572	+	0.860428i	$0.329804\pi$
$20$		0			0
$21$	3.77492	−	2.59808i	0.823754	−	0.566947i
$22$		0			0
$23$	0.774917	+	0.447399i	0.161581	+	0.0932891i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	0.774917	+	0.447399i	0.161581	+	0.0932891i	−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	0.362541	−	4.98684i	0.0725083	−	0.997368i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	3.27492			0.608137			0.304068	−	0.952650i	$-0.401655\pi$
$29$	3.27492			0.608137			0.304068	+	0.952650i	$0.401655\pi$
$30$		0			0
$31$	2.13746	+	3.70219i	0.383899	+	0.664932i	0.991616	−	0.129221i	$-0.0412478\pi$
$31$	2.13746	+	3.70219i	0.383899	+	0.664932i	−0.607717	+	0.794154i	$0.707914\pi$
$32$		0			0
$33$	3.41238	+	1.97014i	0.594018	+	0.342957i
$34$		0			0
$35$	4.00000	−	4.35890i	0.676123	−	0.736788i
$36$		0			0
$37$	4.86254	+	2.80739i	0.799397	+	0.461532i	0.843260	−	0.537506i	$-0.180633\pi$
$37$	4.86254	+	2.80739i	0.799397	+	0.461532i	−0.0438633	+	0.999038i	$0.513967\pi$
$38$		0			0
$39$	−5.27492	−	9.13642i	−0.844663	−	1.46300i
$40$		0			0
$41$	−11.2749			−1.76085			−0.880423	−	0.474189i	$-0.842741\pi$
$41$	−11.2749			−1.76085			−0.880423	+	0.474189i	$0.842741\pi$
$42$		0			0
$43$			6.50958i			0.992701i	0.868122	+	0.496351i	$0.165327\pi$
$43$			6.50958i			0.992701i	−0.868122	+	0.496351i	$0.834673\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.86254	−	1.07534i	−0.271680	−	0.156854i	0.357971	−	0.933733i	$-0.383469\pi$
$47$	−1.86254	−	1.07534i	−0.271680	−	0.156854i	−0.629651	+	0.776878i	$0.716802\pi$
$48$		0			0
$49$	6.91238	−	1.10411i	0.987482	−	0.157730i
$50$		0			0
$51$	−4.13746	+	7.16629i	−0.579360	+	1.00348i
$52$		0			0
$53$	−6.41238	+	3.70219i	−0.880808	+	0.508534i	−0.870925	−	0.491417i	$-0.836479\pi$
$53$	−6.41238	+	3.70219i	−0.880808	+	0.508534i	−0.00988297	+	0.999951i	$0.503146\pi$
$54$		0			0
$55$	4.86254	+	1.49397i	0.655665	+	0.201446i
$56$		0			0
$57$		−	7.40437i		−	0.980733i
$58$		0			0
$59$	2.13746	+	3.70219i	0.278273	+	0.481984i	0.970956	−	0.239259i	$-0.0769045\pi$
$59$	2.13746	+	3.70219i	0.278273	+	0.481984i	−0.692682	+	0.721243i	$0.743571\pi$
$60$		0			0
$61$	−0.774917	+	1.34220i	−0.0992180	+	0.171851i	−0.911361	−	0.411608i	$-0.864967\pi$
$61$	−0.774917	+	1.34220i	−0.0992180	+	0.171851i	0.812143	+	0.583458i	$0.198301\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.27492	−	9.97368i	−1.15041	−	1.23708i
$66$		0			0
$67$	−12.0498	+	6.95698i	−1.47212	+	0.849930i	−0.999509	−	0.0313404i	$-0.990022\pi$
$67$	−12.0498	+	6.95698i	−1.47212	+	0.849930i	−0.472613	+	0.881270i	$0.656689\pi$
$68$		0			0
$69$	−1.54983			−0.186578
$70$		0			0
$71$	10.5498			1.25204			0.626018	−	0.779809i	$-0.284684\pi$
$71$	10.5498			1.25204			0.626018	+	0.779809i	$0.284684\pi$
$72$		0			0
$73$	−1.86254	+	1.07534i	−0.217994	+	0.125859i	−0.605021	−	0.796209i	$-0.706835\pi$
$73$	−1.86254	+	1.07534i	−0.217994	+	0.125859i	0.387027	+	0.922068i	$0.373502\pi$
$74$		0			0
$75$	3.77492	+	7.79423i	0.435890	+	0.900000i
$76$		0			0
$77$	3.41238	+	4.95807i	0.388876	+	0.565024i
$78$		0			0
$79$	−0.137459	+	0.238085i	−0.0154653	+	0.0267867i	−0.873654	−	0.486547i	$-0.838256\pi$
$79$	−0.137459	+	0.238085i	−0.0154653	+	0.0267867i	0.858189	+	0.513334i	$0.171590\pi$
$80$		0			0
$81$	4.50000	+	7.79423i	0.500000	+	0.866025i
$82$		0			0
$83$		−	5.67232i		−	0.622618i	−0.950309	−	0.311309i	$-0.899233\pi$
$83$		−	5.67232i		−	0.622618i	0.950309	−	0.311309i	$-0.100767\pi$
$84$		0			0
$85$	−3.13746	+	10.2118i	−0.340305	+	1.10762i
$86$		0			0
$87$	−4.91238	+	2.83616i	−0.526662	+	0.304068i
$88$		0			0
$89$	−3.50000	+	6.06218i	−0.370999	+	0.642590i	−0.989720	−	0.143022i	$-0.954318\pi$
$89$	−3.50000	+	6.06218i	−0.370999	+	0.642590i	0.618720	+	0.785611i	$0.287651\pi$
$90$		0			0
$91$	−1.27492	−	16.0646i	−0.133648	−	1.68403i
$92$		0			0
$93$	−6.41238	−	3.70219i	−0.664932	−	0.383899i
$94$		0			0
$95$	−2.13746	−	9.31697i	−0.219299	−	0.955901i
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.774917	−	1.34220i	−0.0771071	−	0.133553i	0.824894	−	0.565288i	$-0.191235\pi$
$101$	−0.774917	−	1.34220i	−0.0771071	−	0.133553i	−0.902001	+	0.431735i	$0.857902\pi$
$102$		0			0
$103$	2.22508	+	1.28465i	0.219244	+	0.126581i	0.605600	−	0.795769i	$-0.292933\pi$
$103$	2.22508	+	1.28465i	0.219244	+	0.126581i	−0.386356	+	0.922350i	$0.626266\pi$
$104$		0			0
$105$	−2.22508	+	10.0025i	−0.217146	+	0.976139i
$106$		0			0
$107$	12.0498	+	6.95698i	1.16490	+	0.672556i	0.952474	−	0.304621i	$-0.0985297\pi$
$107$	12.0498	+	6.95698i	1.16490	+	0.672556i	0.212428	+	0.977177i	$0.431863\pi$
$108$		0			0
$109$	1.77492	+	3.07425i	0.170006	+	0.294459i	0.938422	−	0.345492i	$-0.112288\pi$
$109$	1.77492	+	3.07425i	0.170006	+	0.294459i	−0.768416	+	0.639951i	$0.778955\pi$
$110$		0			0
$111$	−9.72508			−0.923064
$112$		0			0
$113$		−	13.0192i		−	1.22474i	−0.790572	−	0.612369i	$-0.790217\pi$
$113$		−	13.0192i		−	1.22474i	0.790572	−	0.612369i	$-0.209783\pi$
$114$		0			0
$115$	−1.95017	+	0.447399i	−0.181854	+	0.0417201i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.4124	+	7.16629i	−0.954501	+	0.656933i
$120$		0			0
$121$	2.91238	−	5.04438i	0.264761	−	0.458580i
$122$		0			0
$123$	16.9124	−	9.76436i	1.52494	−	0.880423i
$124$		0			0
$125$	7.00000	+	8.71780i	0.626099	+	0.779744i
$126$		0			0
$127$			1.78959i			0.158801i	0.996843	+	0.0794004i	$0.0253006\pi$
$127$			1.78959i			0.158801i	−0.996843	+	0.0794004i	$0.974699\pi$
$128$		0			0
$129$	−5.63746	−	9.76436i	−0.496351	−	0.859704i
$130$		0			0
$131$	9.13746	−	15.8265i	0.798343	−	1.38277i	−0.122351	−	0.992487i	$-0.539043\pi$
$131$	9.13746	−	15.8265i	0.798343	−	1.38277i	0.920694	−	0.390285i	$-0.127623\pi$
$132$		0			0
$133$	4.86254	−	10.2118i	0.421636	−	0.885472i
$134$		0			0
$135$	7.91238	+	8.50848i	0.680989	+	0.732294i
$136$		0			0
$137$	7.96221	−	4.59698i	0.680258	−	0.392747i	−0.119695	−	0.992811i	$-0.538192\pi$
$137$	7.96221	−	4.59698i	0.680258	−	0.392747i	0.799952	+	0.600064i	$0.204858\pi$
$138$		0			0
$139$	17.0997			1.45037			0.725187	−	0.688551i	$-0.241753\pi$
$139$	17.0997			1.45037			0.725187	+	0.688551i	$0.241753\pi$
$140$		0			0
$141$	3.72508			0.313709
$142$		0			0
$143$	12.0000	−	6.92820i	1.00349	−	0.579365i
$144$		0			0
$145$	−5.36254	+	4.98684i	−0.445335	+	0.414134i
$146$		0			0
$147$	−9.41238	+	7.64246i	−0.776320	+	0.630339i
$148$		0			0
$149$	3.77492	−	6.53835i	0.309253	−	0.535642i	−0.668946	−	0.743311i	$-0.733254\pi$
$149$	3.77492	−	6.53835i	0.309253	−	0.535642i	0.978199	+	0.207669i	$0.0665876\pi$
$150$		0			0
$151$	10.1375	+	17.5586i	0.824975	+	1.42890i	0.901939	+	0.431864i	$0.142144\pi$
$151$	10.1375	+	17.5586i	0.824975	+	1.42890i	−0.0769640	+	0.997034i	$0.524523\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.13746	−	2.80739i	−0.733938	−	0.225495i
$156$		0			0
$157$	9.41238	−	5.43424i	0.751189	−	0.433699i	−0.0749341	−	0.997188i	$-0.523875\pi$
$157$	9.41238	−	5.43424i	0.751189	−	0.433699i	0.826123	+	0.563489i	$0.190541\pi$
$158$		0			0
$159$	6.41238	−	11.1066i	0.508534	−	0.880808i
$160$		0			0
$161$	−2.13746	−	1.01779i	−0.168455	−	0.0802135i
$162$		0			0
$163$	−6.41238	−	3.70219i	−0.502256	−	0.289978i	0.227389	−	0.973804i	$-0.426981\pi$
$163$	−6.41238	−	3.70219i	−0.502256	−	0.289978i	−0.729645	+	0.683826i	$0.760315\pi$
$164$		0			0
$165$	−8.58762	+	1.97014i	−0.668546	+	0.153375i
$166$		0			0
$167$			12.6005i			0.975058i	0.873107	+	0.487529i	$0.162102\pi$
$167$			12.6005i			0.975058i	−0.873107	+	0.487529i	$0.837898\pi$
$168$		0			0
$169$	−24.0997			−1.85382
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.2371	−	9.37451i	−1.23449	−	0.712731i	−0.266524	−	0.963828i	$-0.585875\pi$
$173$	−16.2371	−	9.37451i	−1.23449	−	0.712731i	−0.967962	+	0.251097i	$0.919209\pi$
$174$		0			0
$175$	0.0876242	+	13.2285i	0.00662376	+	0.999978i
$176$		0			0
$177$	−6.41238	−	3.70219i	−0.481984	−	0.278273i
$178$		0			0
$179$	0.137459	+	0.238085i	0.0102741	+	0.0177953i	0.871117	−	0.491076i	$-0.163396\pi$
$179$	0.137459	+	0.238085i	0.0102741	+	0.0177953i	−0.860843	+	0.508871i	$0.830063\pi$
$180$		0			0
$181$	−16.7251			−1.24317			−0.621583	−	0.783348i	$-0.713510\pi$
$181$	−16.7251			−1.24317			−0.621583	+	0.783348i	$0.713510\pi$
$182$		0			0
$183$		−	2.68439i		−	0.198436i
$184$		0			0
$185$	−12.2371	+	2.80739i	−0.899692	+	0.206403i
$186$		0			0
$187$	−9.41238	−	5.43424i	−0.688301	−	0.397391i
$188$		0			0
$189$	1.08762	+	13.7046i	0.0791130	+	0.996866i
$190$		0			0
$191$	−11.4124	+	19.7668i	−0.825771	+	1.43028i	0.0755585	+	0.997141i	$0.475926\pi$
$191$	−11.4124	+	19.7668i	−0.825771	+	1.43028i	−0.901329	+	0.433135i	$0.857407\pi$
$192$		0			0
$193$	7.96221	−	4.59698i	0.573132	−	0.330898i	−0.185267	−	0.982688i	$-0.559315\pi$
$193$	7.96221	−	4.59698i	0.573132	−	0.330898i	0.758399	+	0.651790i	$0.225982\pi$
$194$		0			0
$195$	22.5498	+	6.92820i	1.61483	+	0.496139i
$196$		0			0
$197$			26.0383i			1.85515i	0.373634	+	0.927576i	$0.378112\pi$
$197$			26.0383i			1.85515i	−0.373634	+	0.927576i	$0.621888\pi$
$198$		0			0
$199$	4.86254	+	8.42217i	0.344696	+	0.597032i	0.985299	−	0.170841i	$-0.0546485\pi$
$199$	4.86254	+	8.42217i	0.344696	+	0.597032i	−0.640602	+	0.767873i	$0.721315\pi$
$200$		0			0
$201$	12.0498	−	20.8709i	0.849930	−	1.47212i
$202$		0			0
$203$	−8.63746	+	0.685484i	−0.606231	+	0.0481115i
$204$		0			0
$205$	18.4622	−	17.1687i	1.28946	−	1.19912i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.72508			0.672698
$210$		0			0
$211$	−19.6495			−1.35273			−0.676364	−	0.736568i	$-0.736445\pi$
$211$	−19.6495			−1.35273			−0.676364	+	0.736568i	$0.736445\pi$
$212$		0			0
$213$	−15.8248	+	9.13642i	−1.08429	+	0.626018i
$214$		0			0
$215$	−9.91238	−	10.6592i	−0.676018	−	0.726949i
$216$		0			0
$217$	−6.41238	−	9.31697i	−0.435300	−	0.632477i
$218$		0			0
$219$	1.86254	−	3.22602i	0.125859	−	0.217994i
$220$		0			0
$221$	14.5498	+	25.2011i	0.978728	+	1.69521i
$222$		0			0
$223$		−	8.71780i		−	0.583787i	−0.956451	−	0.291893i	$-0.905715\pi$
$223$		−	8.71780i		−	0.583787i	0.956451	−	0.291893i	$-0.0942853\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.58762	+	3.22602i	−0.370864	+	0.214118i	−0.673836	−	0.738881i	$-0.735354\pi$
$227$	−5.58762	+	3.22602i	−0.370864	+	0.214118i	0.302972	+	0.952999i	$0.402021\pi$
$228$		0			0
$229$	−2.13746	+	3.70219i	−0.141247	+	0.244647i	−0.927967	−	0.372663i	$-0.878445\pi$
$229$	−2.13746	+	3.70219i	−0.141247	+	0.244647i	0.786719	+	0.617311i	$0.211778\pi$
$230$		0			0
$231$	−9.41238	−	4.48190i	−0.619289	−	0.294887i
$232$		0			0
$233$	16.1375	+	9.31697i	1.05720	+	0.610375i	0.924656	−	0.380802i	$-0.124352\pi$
$233$	16.1375	+	9.31697i	1.05720	+	0.610375i	0.132544	+	0.991177i	$0.457686\pi$
$234$		0			0
$235$	4.68729	−	1.07534i	0.305765	−	0.0701474i
$236$		0			0
$237$		−	0.476171i		−	0.0309306i
$238$		0			0
$239$	14.5498			0.941151			0.470575	−	0.882360i	$-0.344046\pi$
$239$	14.5498			0.941151			0.470575	+	0.882360i	$0.344046\pi$
$240$		0			0
$241$	6.41238	+	11.1066i	0.413057	+	0.715436i	0.995222	−	0.0976343i	$-0.0311275\pi$
$241$	6.41238	+	11.1066i	0.413057	+	0.715436i	−0.582165	+	0.813071i	$0.697794\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.63746	+	12.3337i	−0.615715	+	0.787969i
$246$		0			0
$247$	−22.5498	−	13.0192i	−1.43481	−	0.828389i
$248$		0			0
$249$	4.91238	+	8.50848i	0.311309	+	0.539203i
$250$		0			0
$251$	−5.45017			−0.344011			−0.172006	−	0.985096i	$-0.555025\pi$
$251$	−5.45017			−0.344011			−0.172006	+	0.985096i	$0.555025\pi$
$252$		0			0
$253$		−	2.03559i		−	0.127976i
$254$		0			0
$255$	−4.13746	−	18.0348i	−0.259098	−	1.12938i
$256$		0			0
$257$	21.4124	+	12.3624i	1.33567	+	0.771148i	0.986162	−	0.165786i	$-0.0530162\pi$
$257$	21.4124	+	12.3624i	1.33567	+	0.771148i	0.349506	+	0.936934i	$0.386350\pi$
$258$		0			0
$259$	−13.4124	−	6.38658i	−0.833404	−	0.396843i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−23.3248	+	13.4666i	−1.43827	+	0.830383i	−0.997729	−	0.0673516i	$-0.978545\pi$
$263$	−23.3248	+	13.4666i	−1.43827	+	0.830383i	−0.440536	+	0.897735i	$0.645212\pi$
$264$		0			0
$265$	4.86254	−	15.8265i	0.298704	−	0.972217i
$266$		0			0
$267$		−	12.1244i		−	0.741999i
$268$		0			0
$269$	−14.7749	−	25.5909i	−0.900843	−	1.56031i	−0.826403	−	0.563080i	$-0.809616\pi$
$269$	−14.7749	−	25.5909i	−0.900843	−	1.56031i	−0.0744400	−	0.997225i	$-0.523717\pi$
$270$		0			0
$271$	6.41238	−	11.1066i	0.389524	−	0.674676i	−0.602861	−	0.797846i	$-0.705973\pi$
$271$	6.41238	−	11.1066i	0.389524	−	0.674676i	0.992386	+	0.123170i	$0.0393062\pi$
$272$		0			0
$273$	15.8248	+	22.9928i	0.957758	+	1.39159i
$274$		0			0
$275$	−10.2371	+	4.95807i	−0.617322	+	0.298983i
$276$		0			0
$277$	16.1375	−	9.31697i	0.969606	−	0.559802i	0.0704898	−	0.997512i	$-0.477544\pi$
$277$	16.1375	−	9.31697i	0.969606	−	0.559802i	0.899116	+	0.437710i	$0.144210\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	16.9622	−	9.79314i	1.00830	−	0.582142i	0.0976056	−	0.995225i	$-0.468882\pi$
$283$	16.9622	−	9.79314i	1.00830	−	0.582142i	0.910693	+	0.413084i	$0.135548\pi$
$284$		0			0
$285$	11.2749	+	12.1244i	0.667868	+	0.718185i
$286$		0			0
$287$	29.7371	−	2.35999i	1.75533	−	0.139306i
$288$		0			0
$289$	2.91238	−	5.04438i	0.171316	−	0.296728i
$290$		0			0
$291$	6.00000	+	10.3923i	0.351726	+	0.609208i
$292$		0			0
$293$		−	6.92820i		−	0.404750i	−0.979308	−	0.202375i	$-0.935134\pi$
$293$		−	6.92820i		−	0.404750i	0.979308	−	0.202375i	$-0.0648660\pi$
$294$		0			0
$295$	−9.13746	−	2.80739i	−0.532003	−	0.163453i
$296$		0			0
$297$	−10.2371	+	5.91041i	−0.594018	+	0.342957i
$298$		0			0
$299$	−2.72508	+	4.71998i	−0.157596	+	0.272964i
$300$		0			0
$301$	−1.36254	−	17.1687i	−0.0785356	−	0.989590i
$302$		0			0
$303$	2.32475	+	1.34220i	0.133553	+	0.0771071i
$304$		0			0
$305$	−0.774917	−	3.37779i	−0.0443716	−	0.193411i
$306$		0			0
$307$		−	3.99782i		−	0.228167i	−0.993471	−	0.114084i	$-0.963607\pi$
$307$		−	3.99782i		−	0.228167i	0.993471	−	0.114084i	$-0.0363932\pi$
$308$		0			0
$309$	−4.45017			−0.253161
$310$		0			0
$311$	−6.41238	−	11.1066i	−0.363612	−	0.629795i	0.624940	−	0.780673i	$-0.285123\pi$
$311$	−6.41238	−	11.1066i	−0.363612	−	0.629795i	−0.988552	+	0.150878i	$0.951790\pi$
$312$		0			0
$313$	12.5120	+	7.22383i	0.707223	+	0.408315i	0.810032	−	0.586386i	$-0.199450\pi$
$313$	12.5120	+	7.22383i	0.707223	+	0.408315i	−0.102809	+	0.994701i	$0.532783\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.31271	−	1.91259i	−0.186060	−	0.107422i	0.404077	−	0.914725i	$-0.367593\pi$
$317$	−3.31271	−	1.91259i	−0.186060	−	0.107422i	−0.590137	+	0.807303i	$0.700926\pi$
$318$		0			0
$319$	−3.72508	−	6.45203i	−0.208565	−	0.361244i
$320$		0			0
$321$	−24.0997			−1.34511
$322$		0			0
$323$			20.4235i			1.13640i
$324$		0			0
$325$	30.3746	+	2.20822i	1.68488	+	0.122490i
$326$		0			0
$327$	−5.32475	−	3.07425i	−0.294459	−	0.170006i
$328$		0			0
$329$	5.13746	+	2.44631i	0.283237	+	0.134869i
$330$		0			0
$331$	2.41238	−	4.17836i	0.132596	−	0.229663i	−0.792080	−	0.610417i	$-0.791002\pi$
$331$	2.41238	−	4.17836i	0.132596	−	0.229663i	0.924677	+	0.380753i	$0.124335\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.13746	−	29.7405i	0.499233	−	1.62490i
$336$		0			0
$337$		−	13.0192i		−	0.709198i	−0.935018	−	0.354599i	$-0.884617\pi$
$337$		−	13.0192i		−	0.709198i	0.935018	−	0.354599i	$-0.115383\pi$
$338$		0			0
$339$	11.2749	+	19.5287i	0.612369	+	1.06065i
$340$		0			0
$341$	4.86254	−	8.42217i	0.263321	−	0.456086i
$342$		0			0
$343$	−18.0000	+	4.35890i	−0.971909	+	0.235358i
$344$		0			0
$345$	2.53779	−	2.35999i	0.136630	−	0.127058i
$346$		0			0
$347$	10.5000	−	6.06218i	0.563670	−	0.325435i	−0.190947	−	0.981600i	$-0.561156\pi$
$347$	10.5000	−	6.06218i	0.563670	−	0.325435i	0.754617	+	0.656165i	$0.227823\pi$
$348$		0			0
$349$	11.2749			0.603532			0.301766	−	0.953382i	$-0.402424\pi$
$349$	11.2749			0.603532			0.301766	+	0.953382i	$0.402424\pi$
$350$		0			0
$351$	31.6495			1.68933
$352$		0			0
$353$	−18.4124	+	10.6304i	−0.979992	+	0.565799i	−0.902268	−	0.431176i	$-0.858099\pi$
$353$	−18.4124	+	10.6304i	−0.979992	+	0.565799i	−0.0777242	+	0.996975i	$0.524765\pi$
$354$		0			0
$355$	−17.2749	+	16.0646i	−0.916857	+	0.852622i
$356$		0			0
$357$	9.41238	−	19.7668i	0.498156	−	1.04617i
$358$		0			0
$359$	−0.687293	+	1.19043i	−0.0362739	+	0.0628283i	−0.883592	−	0.468257i	$-0.844882\pi$
$359$	−0.687293	+	1.19043i	−0.0362739	+	0.0628283i	0.847318	+	0.531085i	$0.178216\pi$
$360$		0			0
$361$	0.362541	+	0.627940i	0.0190811	+	0.0330495i
$362$		0			0
$363$			10.0888i			0.529523i
$364$		0			0
$365$	1.41238	−	4.59698i	0.0739271	−	0.240617i
$366$		0			0
$367$	−12.7749	+	7.37560i	−0.666845	+	0.385003i	−0.794880	−	0.606766i	$-0.792466\pi$
$367$	−12.7749	+	7.37560i	−0.666845	+	0.385003i	0.128035	+	0.991770i	$0.459133\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.1375	−	11.1066i	0.837815	−	0.576624i
$372$		0			0
$373$	4.86254	+	2.80739i	0.251773	+	0.145361i	0.620576	−	0.784146i	$-0.286899\pi$
$373$	4.86254	+	2.80739i	0.251773	+	0.145361i	−0.368803	+	0.929508i	$0.620232\pi$
$374$		0			0
$375$	−18.0498	−	7.01452i	−0.932089	−	0.362228i
$376$		0			0
$377$			19.9474i			1.02734i
$378$		0			0
$379$	−23.6495			−1.21479			−0.607397	−	0.794399i	$-0.707786\pi$
$379$	−23.6495			−1.21479			−0.607397	+	0.794399i	$0.707786\pi$
$380$		0			0
$381$	−1.54983	−	2.68439i	−0.0794004	−	0.137526i
$382$		0			0
$383$	17.3248	+	10.0025i	0.885253	+	0.511101i	0.872387	−	0.488816i	$-0.162571\pi$
$383$	17.3248	+	10.0025i	0.885253	+	0.511101i	0.0128665	+	0.999917i	$0.495904\pi$
$384$		0			0
$385$	−13.1375	−	2.92248i	−0.669547	−	0.148943i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.68729	−	4.65453i	−0.136251	−	0.235994i	0.789824	−	0.613334i	$-0.210172\pi$
$389$	−2.68729	−	4.65453i	−0.136251	−	0.235994i	−0.926075	+	0.377340i	$0.876839\pi$
$390$		0			0
$391$	4.27492			0.216192
$392$		0			0
$393$			31.6531i			1.59669i
$394$		0			0
$395$	−0.137459	−	0.599168i	−0.00691629	−	0.0301474i
$396$		0			0
$397$	−13.1375	−	7.58492i	−0.659350	−	0.380676i	0.132679	−	0.991159i	$-0.457642\pi$
$397$	−13.1375	−	7.58492i	−0.659350	−	0.380676i	−0.792029	+	0.610483i	$0.790975\pi$
$398$		0			0
$399$	1.54983	+	19.5287i	0.0775888	+	0.977659i
$400$		0			0
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	−0.901046	−	0.433724i	$-0.857199\pi$
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	0.826139	+	0.563466i	$0.190532\pi$
$402$		0			0
$403$	−22.5498	+	13.0192i	−1.12329	+	0.648530i
$404$		0			0
$405$	−19.2371	−	5.91041i	−0.955901	−	0.293691i
$406$		0			0
$407$		−	12.7732i		−	0.633142i
$408$		0			0
$409$	5.04983	+	8.74657i	0.249698	+	0.432490i	0.963442	−	0.267917i	$-0.0863352\pi$
$409$	5.04983	+	8.74657i	0.249698	+	0.432490i	−0.713744	+	0.700407i	$0.753002\pi$
$410$		0			0
$411$	−7.96221	+	13.7910i	−0.392747	+	0.680258i
$412$		0			0
$413$	−6.41238	−	9.31697i	−0.315532	−	0.458458i
$414$		0			0
$415$	8.63746	+	9.28819i	0.423996	+	0.455940i
$416$		0			0
$417$	−25.6495	+	14.8087i	−1.25606	+	0.725187i
$418$		0			0
$419$	−17.0997			−0.835373			−0.417687	−	0.908591i	$-0.637159\pi$
$419$	−17.0997			−0.835373			−0.417687	+	0.908591i	$0.637159\pi$
$420$		0			0
$421$	3.27492			0.159610			0.0798048	−	0.996811i	$-0.474570\pi$
$421$	3.27492			0.159610			0.0798048	+	0.996811i	$0.474570\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.4124	−	21.4989i	−0.505074	−	1.04285i
$426$		0			0
$427$	1.76287	−	3.70219i	0.0853114	−	0.179161i
$428$		0			0
$429$	−12.0000	+	20.7846i	−0.579365	+	1.00349i
$430$		0			0
$431$	−9.68729	−	16.7789i	−0.466620	−	0.808210i	0.532653	−	0.846334i	$-0.321195\pi$
$431$	−9.68729	−	16.7789i	−0.466620	−	0.808210i	−0.999273	+	0.0381236i	$0.987862\pi$
$432$		0			0
$433$			26.8756i			1.29156i	0.763525	+	0.645778i	$0.223467\pi$
$433$			26.8756i			1.29156i	−0.763525	+	0.645778i	$0.776533\pi$
$434$		0			0
$435$	3.72508	−	12.1244i	0.178604	−	0.581318i
$436$		0			0
$437$	−3.31271	+	1.91259i	−0.158468	+	0.0914917i
$438$		0			0
$439$	0.587624	−	1.01779i	0.0280458	−	0.0485767i	−0.851662	−	0.524092i	$-0.824405\pi$
$439$	0.587624	−	1.01779i	0.0280458	−	0.0485767i	0.879708	+	0.475515i	$0.157738\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.5000	−	6.06218i	−0.498870	−	0.288023i	0.229377	−	0.973338i	$-0.426331\pi$
$443$	−10.5000	−	6.06218i	−0.498870	−	0.288023i	−0.728247	+	0.685315i	$0.759665\pi$
$444$		0			0
$445$	−3.50000	−	15.2561i	−0.165916	−	0.723211i
$446$		0			0
$447$			13.0767i			0.618507i
$448$		0			0
$449$	25.8248			1.21875			0.609373	−	0.792884i	$-0.291421\pi$
$449$	25.8248			1.21875			0.609373	+	0.792884i	$0.291421\pi$
$450$		0			0
$451$	12.8248	+	22.2131i	0.603894	+	1.04598i
$452$		0			0
$453$	−30.4124	−	17.5586i	−1.42890	−	0.824975i
$454$		0			0
$455$	26.5498	+	24.3638i	1.24468	+	1.14219i
$456$		0			0
$457$	−17.6873	−	10.2118i	−0.827377	−	0.477686i	0.0255769	−	0.999673i	$-0.491858\pi$
$457$	−17.6873	−	10.2118i	−0.827377	−	0.477686i	−0.852954	+	0.521987i	$0.825191\pi$
$458$		0			0
$459$	−12.4124	−	21.4989i	−0.579360	−	1.00348i
$460$		0			0
$461$	−14.0000			−0.652045			−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000			−0.652045			−0.326023	+	0.945362i	$0.605709\pi$
$462$		0			0
$463$		−	6.50958i		−	0.302526i	−0.988494	−	0.151263i	$-0.951666\pi$
$463$		−	6.50958i		−	0.302526i	0.988494	−	0.151263i	$-0.0483340\pi$
$464$		0			0
$465$	16.1375	−	3.70219i	0.748357	−	0.171685i
$466$		0			0
$467$	16.5997	+	9.58382i	0.768141	+	0.443486i	0.832211	−	0.554459i	$-0.187075\pi$
$467$	16.5997	+	9.58382i	0.768141	+	0.443486i	−0.0640700	+	0.997945i	$0.520408\pi$
$468$		0			0
$469$	30.3248	−	20.8709i	1.40027	−	0.963730i
$470$		0			0
$471$	−9.41238	+	16.3027i	−0.433699	+	0.751189i
$472$		0			0
$473$	12.8248	−	7.40437i	0.589683	−	0.340453i
$474$		0			0
$475$	17.6873	+	12.0014i	0.811549	+	0.550660i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.41238	−	11.1066i	−0.292989	−	0.507472i	0.681526	−	0.731794i	$-0.261317\pi$
$479$	−6.41238	−	11.1066i	−0.292989	−	0.507472i	−0.974515	+	0.224322i	$0.927983\pi$
$480$		0			0
$481$	−17.0997	+	29.6175i	−0.779678	+	1.35044i
$482$		0			0
$483$	4.08762	−	0.324401i	0.185993	−	0.0147608i
$484$		0			0
$485$	10.5498	+	11.3446i	0.479043	+	0.515134i
$486$		0			0
$487$	28.9622	−	16.7213i	1.31240	−	0.757716i	0.329909	−	0.944013i	$-0.392982\pi$
$487$	28.9622	−	16.7213i	1.31240	−	0.757716i	0.982494	+	0.186296i	$0.0596485\pi$
$488$		0			0
$489$	12.8248			0.579955
$490$		0			0
$491$	−13.4502			−0.606997			−0.303499	−	0.952832i	$-0.598155\pi$
$491$	−13.4502			−0.606997			−0.303499	+	0.952832i	$0.598155\pi$
$492$		0			0
$493$	13.5498	−	7.82300i	0.610254	−	0.352330i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−27.8248	+	2.20822i	−1.24811	+	0.0990523i
$498$		0			0
$499$	19.6873	−	34.0994i	0.881324	−	1.52650i	0.0314548	−	0.999505i	$-0.489986\pi$
$499$	19.6873	−	34.0994i	0.881324	−	1.52650i	0.849869	−	0.526993i	$-0.176681\pi$
$500$		0			0
$501$	−10.9124	−	18.9008i	−0.487529	−	0.844425i
$502$		0			0
$503$			16.1797i			0.721418i	0.932678	+	0.360709i	$0.117465\pi$
$503$			16.1797i			0.721418i	−0.932678	+	0.360709i	$0.882535\pi$
$504$		0			0
$505$	3.31271	+	1.01779i	0.147414	+	0.0452913i
$506$		0			0
$507$	36.1495	−	20.8709i	1.60546	−	0.926910i
$508$		0			0
$509$	−14.7749	+	25.5909i	−0.654887	+	1.13430i	0.327036	+	0.945012i	$0.393950\pi$
$509$	−14.7749	+	25.5909i	−0.654887	+	1.13430i	−0.981922	+	0.189285i	$0.939383\pi$
$510$		0			0
$511$	4.68729	−	3.22602i	0.207354	−	0.142711i
$512$		0			0
$513$	19.2371	+	11.1066i	0.849340	+	0.490367i
$514$		0			0
$515$	−5.59967	+	1.28465i	−0.246751	+	0.0566085i
$516$		0			0
$517$			4.89261i			0.215177i
$518$		0			0
$519$	32.4743			1.42546
$520$		0			0
$521$	6.41238	+	11.1066i	0.280931	+	0.486587i	0.971614	−	0.236570i	$-0.0760234\pi$
$521$	6.41238	+	11.1066i	0.280931	+	0.486587i	−0.690683	+	0.723158i	$0.742690\pi$
$522$		0			0
$523$	10.1375	+	5.85286i	0.443280	+	0.255928i	0.704988	−	0.709219i	$-0.250952\pi$
$523$	10.1375	+	5.85286i	0.443280	+	0.255928i	−0.261708	+	0.965147i	$0.584286\pi$
$524$		0			0
$525$	−11.5876	−	19.7668i	−0.505725	−	0.862695i
$526$		0			0
$527$	17.6873	+	10.2118i	0.770471	+	0.444831i
$528$		0			0
$529$	−11.0997	−	19.2252i	−0.482594	−	0.835878i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	68.6750i		−	2.97464i
$534$		0			0
$535$	−30.3248	+	6.95698i	−1.31105	+	0.300776i
$536$		0			0
$537$	−0.412376	−	0.238085i	−0.0177953	−	0.0102741i
$538$		0			0
$539$	−10.0378	−	12.3624i	−0.432358	−	0.532488i
$540$		0			0
$541$	1.22508	−	2.12191i	0.0526704	−	0.0912278i	−0.838488	−	0.544920i	$-0.816560\pi$
$541$	1.22508	−	2.12191i	0.0526704	−	0.0912278i	0.891159	+	0.453692i	$0.149893\pi$
$542$		0			0
$543$	25.0876	−	14.4843i	1.07661	−	0.621583i
$544$		0			0
$545$	−7.58762	−	2.33122i	−0.325018	−	0.0998585i
$546$		0			0
$547$			36.1271i			1.54468i	0.635208	+	0.772341i	$0.280914\pi$
$547$			36.1271i			1.54468i	−0.635208	+	0.772341i	$0.719086\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.00000	+	12.1244i	−0.298210	+	0.516515i
$552$		0			0
$553$	0.312707	−	0.656712i	0.0132977	−	0.0279262i
$554$		0			0
$555$	15.9244	−	14.8087i	0.675954	−	0.628596i
$556$		0			0
$557$	4.86254	−	2.80739i	0.206032	−	0.118953i	−0.393434	−	0.919353i	$-0.628713\pi$
$557$	4.86254	−	2.80739i	0.206032	−	0.118953i	0.599466	+	0.800400i	$0.295380\pi$
$558$		0			0
$559$	−39.6495			−1.67700
$560$		0			0
$561$	18.8248			0.794782
$562$		0			0
$563$	−10.5997	+	6.11972i	−0.446723	+	0.257916i	−0.706445	−	0.707768i	$-0.749702\pi$
$563$	−10.5997	+	6.11972i	−0.446723	+	0.257916i	0.259722	+	0.965683i	$0.416369\pi$
$564$		0			0
$565$	19.8248	+	21.3183i	0.834034	+	0.896869i
$566$		0			0
$567$	−13.5000	−	19.6150i	−0.566947	−	0.823754i
$568$		0			0
$569$	−14.6873	+	25.4391i	−0.615723	+	1.06646i	0.374534	+	0.927213i	$0.377803\pi$
$569$	−14.6873	+	25.4391i	−0.615723	+	1.06646i	−0.990257	+	0.139251i	$0.955531\pi$
$570$		0			0
$571$	0.137459	+	0.238085i	0.00575246	+	0.00996356i	0.868887	−	0.495010i	$-0.164836\pi$
$571$	0.137459	+	0.238085i	0.00575246	+	0.00996356i	−0.863135	+	0.504974i	$0.831502\pi$
$572$		0			0
$573$		−	39.5336i		−	1.65154i
$574$		0			0
$575$	2.51204	−	3.70219i	0.104760	−	0.154392i
$576$		0			0
$577$	−7.13746	+	4.12081i	−0.297136	+	0.171552i	−0.641156	−	0.767411i	$-0.721545\pi$
$577$	−7.13746	+	4.12081i	−0.297136	+	0.171552i	0.344019	+	0.938963i	$0.388211\pi$
$578$		0			0
$579$	−7.96221	+	13.7910i	−0.330898	+	0.573132i
$580$		0			0
$581$	1.18729	+	14.9605i	0.0492572	+	0.620667i
$582$		0			0
$583$	14.5876	+	8.42217i	0.604158	+	0.348811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	13.9715i		−	0.576665i	−0.957530	−	0.288333i	$-0.906899\pi$
$587$		−	13.9715i		−	0.576665i	0.957530	−	0.288333i	$-0.0931009\pi$
$588$		0			0
$589$	−18.2749			−0.753005
$590$		0			0
$591$	−22.5498	−	39.0575i	−0.927576	−	1.60661i
$592$		0			0
$593$	17.5876	+	10.1542i	0.722237	+	0.416984i	0.815576	−	0.578651i	$-0.196421\pi$
$593$	17.5876	+	10.1542i	0.722237	+	0.416984i	−0.0933384	+	0.995634i	$0.529754\pi$
$594$		0			0
$595$	6.13746	−	27.5898i	0.251611	−	1.13107i
$596$		0			0
$597$	−14.5876	−	8.42217i	−0.597032	−	0.344696i
$598$		0			0
$599$	−1.13746	−	1.97014i	−0.0464753	−	0.0804976i	0.841852	−	0.539709i	$-0.181466\pi$
$599$	−1.13746	−	1.97014i	−0.0464753	−	0.0804976i	−0.888327	+	0.459211i	$0.848132\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.91238	+	12.6948i	0.118405	+	0.516115i
$606$		0			0
$607$	27.8746	+	16.0934i	1.13139	+	0.653211i	0.944285	−	0.329128i	$-0.106755\pi$
$607$	27.8746	+	16.0934i	1.13139	+	0.653211i	0.187109	+	0.982339i	$0.440088\pi$
$608$		0			0
$609$	12.3625	−	8.50848i	0.500955	−	0.344781i
$610$		0			0
$611$	6.54983	−	11.3446i	0.264978	−	0.458955i
$612$		0			0
$613$	−32.0619	+	18.5109i	−1.29497	+	0.747650i	−0.979530	−	0.201297i	$-0.935484\pi$
$613$	−32.0619	+	18.5109i	−1.29497	+	0.747650i	−0.315437	+	0.948947i	$0.602151\pi$
$614$		0			0
$615$	−12.8248	+	41.7419i	−0.517144	+	1.68319i
$616$		0			0
$617$			3.57919i			0.144093i	0.997401	+	0.0720464i	$0.0229530\pi$
$617$			3.57919i			0.144093i	−0.997401	+	0.0720464i	$0.977047\pi$
$618$		0			0
$619$	21.9622	+	38.0397i	0.882736	+	1.52894i	0.848287	+	0.529537i	$0.177634\pi$
$619$	21.9622	+	38.0397i	0.882736	+	1.52894i	0.0344487	+	0.999406i	$0.489032\pi$
$620$		0			0
$621$	2.32475	−	4.02659i	0.0932891	−	0.161581i
$622$		0			0
$623$	7.96221	−	16.7213i	0.318999	−	0.669926i
$624$		0			0
$625$	−24.7371	−	3.61587i	−0.989485	−	0.144635i
$626$		0			0
$627$	−14.5876	+	8.42217i	−0.582574	+	0.336349i
$628$		0			0
$629$	26.8248			1.06957
$630$		0			0
$631$	2.90033			0.115460			0.0577302	−	0.998332i	$-0.481614\pi$
$631$	2.90033			0.115460			0.0577302	+	0.998332i	$0.481614\pi$
$632$		0			0
$633$	29.4743	−	17.0170i	1.17150	−	0.676364i
$634$		0			0
$635$	−2.72508	−	2.93039i	−0.108142	−	0.116289i
$636$		0			0
$637$	6.72508	+	42.1029i	0.266457	+	1.66818i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−14.0498	−	24.3350i	−0.554935	−	0.961176i	−0.997909	−	0.0646411i	$-0.979410\pi$
$641$	−14.0498	−	24.3350i	−0.554935	−	0.961176i	0.442973	−	0.896535i	$-0.353924\pi$
$642$		0			0
$643$		−	38.3353i		−	1.51180i	−0.654689	−	0.755898i	$-0.727200\pi$
$643$		−	38.3353i		−	1.51180i	0.654689	−	0.755898i	$-0.272800\pi$
$644$		0			0
$645$	24.0997	+	7.40437i	0.948924	+	0.291547i
$646$		0			0
$647$	0.675248	−	0.389855i	0.0265468	−	0.0153268i	−0.486668	−	0.873587i	$-0.661788\pi$
$647$	0.675248	−	0.389855i	0.0265468	−	0.0153268i	0.513215	+	0.858260i	$0.328454\pi$
$648$		0			0
$649$	4.86254	−	8.42217i	0.190871	−	0.330599i
$650$		0			0
$651$	17.6873	+	8.42217i	0.693220	+	0.330091i
$652$		0			0
$653$	−32.0619	−	18.5109i	−1.25468	−	0.724389i	−0.282643	−	0.959225i	$-0.591211\pi$
$653$	−32.0619	−	18.5109i	−1.25468	−	0.724389i	−0.972035	+	0.234836i	$0.924545\pi$
$654$		0			0
$655$	9.13746	+	39.8293i	0.357030	+	1.55626i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	25.4502			0.991398			0.495699	−	0.868494i	$-0.334912\pi$
$659$	25.4502			0.991398			0.495699	+	0.868494i	$0.334912\pi$
$660$		0			0
$661$	7.77492	+	13.4666i	0.302409	+	0.523788i	0.976681	−	0.214695i	$-0.0688758\pi$
$661$	7.77492	+	13.4666i	0.302409	+	0.523788i	−0.674272	+	0.738483i	$0.735542\pi$
$662$		0			0
$663$	−43.6495	−	25.2011i	−1.69521	−	0.978728i
$664$		0			0
$665$	7.58762	+	24.1257i	0.294235	+	0.935555i
$666$		0			0
$667$	2.53779	+	1.46519i	0.0982636	+	0.0567325i
$668$		0			0
$669$	7.54983	+	13.0767i	0.291893	+	0.505574i
$670$		0			0
$671$	3.52575			0.136110
$672$		0			0
$673$		−	3.57919i		−	0.137968i	−0.997618	−	0.0689838i	$-0.978024\pi$
$673$		−	3.57919i		−	0.137968i	0.997618	−	0.0689838i	$-0.0219757\pi$
$674$		0			0
$675$	−25.9124	−	1.88382i	−0.997368	−	0.0725083i
$676$		0			0
$677$	−21.3127	−	12.3049i	−0.819114	−	0.472916i	0.0309969	−	0.999519i	$-0.490132\pi$
$677$	−21.3127	−	12.3049i	−0.819114	−	0.472916i	−0.850111	+	0.526604i	$0.823465\pi$
$678$		0			0
$679$	1.45017	+	18.2728i	0.0556522	+	0.701248i
$680$		0			0
$681$	5.58762	−	9.67805i	0.214118	−	0.370864i
$682$		0			0
$683$	13.5997	−	7.85177i	0.520377	−	0.300440i	−0.216712	−	0.976236i	$-0.569533\pi$
$683$	13.5997	−	7.85177i	0.520377	−	0.300440i	0.737089	+	0.675796i	$0.236200\pi$
$684$		0			0
$685$	−6.03779	+	19.6517i	−0.230692	+	0.750854i
$686$		0			0
$687$		−	7.40437i		−	0.282494i
$688$		0			0
$689$	−22.5498	−	39.0575i	−0.859080	−	1.48797i
$690$		0			0
$691$	3.68729	−	6.38658i	0.140271	−	0.242957i	−0.787327	−	0.616535i	$-0.788536\pi$
$691$	3.68729	−	6.38658i	0.140271	−	0.242957i	0.927599	+	0.373578i	$0.121869\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−28.0000	+	26.0383i	−1.06210	+	0.987689i
$696$		0			0
$697$	−46.6495	+	26.9331i	−1.76698	+	1.02016i
$698$		0			0
$699$	−32.2749			−1.22075
$700$		0			0
$701$	−13.8248			−0.522154			−0.261077	−	0.965318i	$-0.584078\pi$
$701$	−13.8248			−0.522154			−0.261077	+	0.965318i	$0.584078\pi$
$702$		0			0
$703$	−20.7870	+	12.0014i	−0.783995	+	0.452640i
$704$		0			0
$705$	−6.09967	+	5.67232i	−0.229727	+	0.213632i
$706$		0			0
$707$	2.32475	+	3.37779i	0.0874313	+	0.127035i
$708$		0			0
$709$	−12.7749	+	22.1268i	−0.479772	+	0.830990i	−0.999731	−	0.0232018i	$-0.992614\pi$
$709$	−12.7749	+	22.1268i	−0.479772	+	0.830990i	0.519959	+	0.854191i	$0.325947\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.82518i			0.143254i
$714$		0			0
$715$	−9.09967	+	29.6175i	−0.340308	+	1.10763i
$716$		0			0
$717$	−21.8248	+	12.6005i	−0.815060	+	0.470575i
$718$		0			0
$719$	3.68729	−	6.38658i	0.137513	−	0.238179i	−0.789042	−	0.614340i	$-0.789423\pi$
$719$	3.68729	−	6.38658i	0.137513	−	0.238179i	0.926555	+	0.376160i	$0.122756\pi$
$720$		0			0
$721$	−6.13746	−	2.92248i	−0.228571	−	0.108839i
$722$		0			0
$723$	−19.2371	−	11.1066i	−0.715436	−	0.413057i
$724$		0			0
$725$	1.18729	−	16.3315i	0.0440950	−	0.606536i
$726$		0			0
$727$		−	18.6915i		−	0.693228i	−0.938008	−	0.346614i	$-0.887331\pi$
$727$		−	18.6915i		−	0.693228i	0.938008	−	0.346614i	$-0.112669\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	15.5498	+	26.9331i	0.575131	+	0.996157i
$732$		0			0
$733$	28.8625	+	16.6638i	1.06606	+	0.615491i	0.927103	−	0.374807i	$-0.122291\pi$
$733$	28.8625	+	16.6638i	1.06606	+	0.615491i	0.138959	+	0.990298i	$0.455624\pi$
$734$		0			0
$735$	3.77492	−	26.8468i	0.139240	−	0.990259i
$736$		0			0
$737$	27.4124	+	15.8265i	1.00975	+	0.582978i
$738$		0			0
$739$	15.9622	+	27.6474i	0.587179	+	1.01702i	0.994600	+	0.103784i	$0.0330950\pi$
$739$	15.9622	+	27.6474i	0.587179	+	1.01702i	−0.407420	+	0.913241i	$0.633572\pi$
$740$		0			0
$741$	45.0997			1.65678
$742$		0			0
$743$			19.5287i			0.716440i	0.933637	+	0.358220i	$0.116616\pi$
$743$			19.5287i			0.716440i	−0.933637	+	0.358220i	$0.883384\pi$
$744$		0			0
$745$	3.77492	+	16.4545i	0.138302	+	0.602846i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−33.2371	−	15.8265i	−1.21446	−	0.578289i
$750$		0			0
$751$	11.1375	−	19.2906i	0.406412	−	0.703926i	−0.588073	−	0.808808i	$-0.700113\pi$
$751$	11.1375	−	19.2906i	0.406412	−	0.703926i	0.994485	+	0.104882i	$0.0334466\pi$
$752$		0			0
$753$	8.17525	−	4.71998i	0.297923	−	0.172006i
$754$		0			0
$755$	−43.3368	−	13.3148i	−1.57719	−	0.484575i
$756$		0			0
$757$			9.43996i			0.343101i	0.985175	+	0.171551i	$0.0548777\pi$
$757$			9.43996i			0.343101i	−0.985175	+	0.171551i	$0.945122\pi$
$758$		0			0
$759$	1.76287	+	3.05338i	0.0639882	+	0.110831i
$760$		0			0
$761$	14.9622	−	25.9153i	0.542380	−	0.939429i	−0.456387	−	0.889781i	$-0.650857\pi$
$761$	14.9622	−	25.9153i	0.542380	−	0.939429i	0.998767	−	0.0496479i	$-0.0158099\pi$
$762$		0			0
$763$	−5.32475	−	7.73668i	−0.192769	−	0.280087i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.5498	+	13.0192i	−0.814227	+	0.470094i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	−42.8248			−1.54230
$772$		0			0
$773$	23.5876	−	13.6183i	0.848388	−	0.489817i	−0.0117187	−	0.999931i	$-0.503730\pi$
$773$	23.5876	−	13.6183i	0.848388	−	0.489817i	0.860107	+	0.510114i	$0.170397\pi$
$774$		0			0
$775$	19.2371	−	9.31697i	0.691018	−	0.334675i
$776$		0			0
$777$	25.6495	−	2.03559i	0.920171	−	0.0730264i
$778$		0			0
$779$	24.0997	−	41.7419i	0.863460	−	1.49556i
$780$		0			0
$781$	−12.0000	−	20.7846i	−0.429394	−	0.743732i
$782$		0			0
$783$		−	17.0170i		−	0.608137i
$784$		0			0
$785$	−7.13746	+	23.2309i	−0.254747	+	0.829147i
$786$		0			0
$787$	−1.50000	+	0.866025i	−0.0534692	+	0.0308705i	−0.526496	−	0.850177i	$-0.676495\pi$
$787$	−1.50000	+	0.866025i	−0.0534692	+	0.0308705i	0.473027	+	0.881048i	$0.343161\pi$
$788$		0			0
$789$	23.3248	−	40.3997i	0.830383	−	1.43827i
$790$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{3} +(-1.63746 + 1.52274i) q^{5} +(-2.63746 + 0.209313i) q^{7} +(-1.13746 - 1.97014i) q^{11} +6.09095i q^{13} +(1.13746 - 3.70219i) q^{15} +(4.13746 - 2.38876i) q^{17} +(-2.13746 + 3.70219i) q^{19} +(3.77492 - 2.59808i) q^{21} +(0.774917 + 0.447399i) q^{23} +(0.362541 - 4.98684i) q^{25} -5.19615i q^{27} +3.27492 q^{29} +(2.13746 + 3.70219i) q^{31} +(3.41238 + 1.97014i) q^{33} +(4.00000 - 4.35890i) q^{35} +(4.86254 + 2.80739i) q^{37} +(-5.27492 - 9.13642i) q^{39} -11.2749 q^{41} +6.50958i q^{43} +(-1.86254 - 1.07534i) q^{47} +(6.91238 - 1.10411i) q^{49} +(-4.13746 + 7.16629i) q^{51} +(-6.41238 + 3.70219i) q^{53} +(4.86254 + 1.49397i) q^{55} -7.40437i q^{57} +(2.13746 + 3.70219i) q^{59} +(-0.774917 + 1.34220i) q^{61} +(-9.27492 - 9.97368i) q^{65} +(-12.0498 + 6.95698i) q^{67} -1.54983 q^{69} +10.5498 q^{71} +(-1.86254 + 1.07534i) q^{73} +(3.77492 + 7.79423i) q^{75} +(3.41238 + 4.95807i) q^{77} +(-0.137459 + 0.238085i) q^{79} +(4.50000 + 7.79423i) q^{81} -5.67232i q^{83} +(-3.13746 + 10.2118i) q^{85} +(-4.91238 + 2.83616i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(-1.27492 - 16.0646i) q^{91} +(-6.41238 - 3.70219i) q^{93} +(-2.13746 - 9.31697i) q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + q^{5} - 3 q^{7} + 3 q^{11} - 3 q^{15} + 9 q^{17} - q^{19} - 12 q^{23} + 9 q^{25} - 2 q^{29} + q^{31} - 9 q^{33} + 16 q^{35} + 27 q^{37} - 6 q^{39} - 30 q^{41} - 15 q^{47} + 5 q^{49} - 9 q^{51}+ \cdots - q^{95}+O(q^{100}) \) 4 * q - 6 * q^3 + q^5 - 3 * q^7 + 3 * q^11 - 3 * q^15 + 9 * q^17 - q^19 - 12 * q^23 + 9 * q^25 - 2 * q^29 + q^31 - 9 * q^33 + 16 * q^35 + 27 * q^37 - 6 * q^39 - 30 * q^41 - 15 * q^47 + 5 * q^49 - 9 * q^51 - 3 * q^53 + 27 * q^55 + q^59 + 12 * q^61 - 22 * q^65 - 18 * q^67 + 24 * q^69 + 12 * q^71 - 15 * q^73 - 9 * q^77 + 7 * q^79 + 18 * q^81 - 5 * q^85 + 3 * q^87 - 14 * q^89 + 10 * q^91 - 3 * q^93 - q^95

Introduction

L-functions

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