LMFDB - Embedded newform 140.2.q.a.9.2

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			9.2
Root			$2.13746 - 0.656712i$ of defining polynomial
Character	$\chi$	$=$	140.9
Dual form			140.2.q.a.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.50000 + 0.866025i) q^{3} +(2.13746 - 0.656712i) q^{5} +(1.13746 + 2.38876i) q^{7} +(2.63746 + 4.56821i) q^{11} -2.62685i q^{13} +(-2.63746 + 2.83616i) q^{15} +(0.362541 - 0.209313i) q^{17} +(1.63746 - 2.83616i) q^{19} +(-3.77492 - 2.59808i) q^{21} +(-6.77492 - 3.91150i) q^{23} +(4.13746 - 2.80739i) q^{25} -5.19615i q^{27} -4.27492 q^{29} +(-1.63746 - 2.83616i) q^{31} +(-7.91238 - 4.56821i) q^{33} +(4.00000 + 4.35890i) q^{35} +(8.63746 + 4.98684i) q^{37} +(2.27492 + 3.94027i) q^{39} -3.72508 q^{41} +2.15068i q^{43} +(-5.63746 - 3.25479i) q^{47} +(-4.41238 + 5.43424i) q^{49} +(-0.362541 + 0.627940i) q^{51} +(4.91238 - 2.83616i) q^{53} +(8.63746 + 8.03231i) q^{55} +5.67232i q^{57} +(-1.63746 - 2.83616i) q^{59} +(6.77492 - 11.7345i) q^{61} +(-1.72508 - 5.61478i) q^{65} +(3.04983 - 1.76082i) q^{67} +13.5498 q^{69} -4.54983 q^{71} +(-5.63746 + 3.25479i) q^{73} +(-3.77492 + 7.79423i) q^{75} +(-7.91238 + 11.4964i) q^{77} +(3.63746 - 6.30026i) q^{79} +(4.50000 + 7.79423i) q^{81} +7.40437i q^{83} +(0.637459 - 0.685484i) q^{85} +(6.41238 - 3.70219i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(6.27492 - 2.98793i) q^{91} +(4.91238 + 2.83616i) q^{93} +(1.63746 - 7.13752i) q^{95} -6.92820i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$6$		0			0
$7$	1.13746	+	2.38876i	0.429919	+	0.902867i
$7$	1.13746	+	2.38876i	0.429919	+	0.902867i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.63746	+	4.56821i	0.795224	+	1.37737i	0.922697	+	0.385526i	$0.125980\pi$
$11$	2.63746	+	4.56821i	0.795224	+	1.37737i	−0.127473	+	0.991842i	$0.540687\pi$
$12$		0			0
$13$		−	2.62685i		−	0.728557i	−0.931290	−	0.364278i	$-0.881316\pi$
$13$		−	2.62685i		−	0.728557i	0.931290	−	0.364278i	$-0.118684\pi$
$14$		0			0
$15$	−2.63746	+	2.83616i	−0.680989	+	0.732294i
$16$		0			0
$17$	0.362541	−	0.209313i	0.0879292	−	0.0507659i	−0.455391	−	0.890292i	$-0.650500\pi$
$17$	0.362541	−	0.209313i	0.0879292	−	0.0507659i	0.543320	+	0.839526i	$0.317167\pi$
$18$		0			0
$19$	1.63746	−	2.83616i	0.375659	−	0.650660i	−0.614767	−	0.788709i	$-0.710750\pi$
$19$	1.63746	−	2.83616i	0.375659	−	0.650660i	0.990425	+	0.138049i	$0.0440831\pi$
$20$		0			0
$21$	−3.77492	−	2.59808i	−0.823754	−	0.566947i
$22$		0			0
$23$	−6.77492	−	3.91150i	−1.41267	−	0.815604i	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−6.77492	−	3.91150i	−1.41267	−	0.815604i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	4.13746	−	2.80739i	0.827492	−	0.561478i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	−4.27492			−0.793832			−0.396916	−	0.917855i	$-0.629920\pi$
$29$	−4.27492			−0.793832			−0.396916	+	0.917855i	$0.629920\pi$
$30$		0			0
$31$	−1.63746	−	2.83616i	−0.294096	−	0.509390i	0.680678	−	0.732583i	$-0.261685\pi$
$31$	−1.63746	−	2.83616i	−0.294096	−	0.509390i	−0.974774	+	0.223193i	$0.928352\pi$
$32$		0			0
$33$	−7.91238	−	4.56821i	−1.37737	−	0.795224i
$34$		0			0
$35$	4.00000	+	4.35890i	0.676123	+	0.736788i
$36$		0			0
$37$	8.63746	+	4.98684i	1.41999	+	0.819831i	0.996297	−	0.0859750i	$-0.0274005\pi$
$37$	8.63746	+	4.98684i	1.41999	+	0.819831i	0.423692	+	0.905806i	$0.360734\pi$
$38$		0			0
$39$	2.27492	+	3.94027i	0.364278	+	0.630949i
$40$		0			0
$41$	−3.72508			−0.581760			−0.290880	−	0.956760i	$-0.593948\pi$
$41$	−3.72508			−0.581760			−0.290880	+	0.956760i	$0.593948\pi$
$42$		0			0
$43$			2.15068i			0.327975i	0.986462	+	0.163988i	$0.0524357\pi$
$43$			2.15068i			0.327975i	−0.986462	+	0.163988i	$0.947564\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.63746	−	3.25479i	−0.822308	−	0.474760i	0.0289038	−	0.999582i	$-0.490798\pi$
$47$	−5.63746	−	3.25479i	−0.822308	−	0.474760i	−0.851212	+	0.524823i	$0.824132\pi$
$48$		0			0
$49$	−4.41238	+	5.43424i	−0.630339	+	0.776320i
$50$		0			0
$51$	−0.362541	+	0.627940i	−0.0507659	+	0.0879292i
$52$		0			0
$53$	4.91238	−	2.83616i	0.674767	−	0.389577i	−0.123114	−	0.992393i	$-0.539288\pi$
$53$	4.91238	−	2.83616i	0.674767	−	0.389577i	0.797880	+	0.602816i	$0.205955\pi$
$54$		0			0
$55$	8.63746	+	8.03231i	1.16467	+	1.08308i
$56$		0			0
$57$			5.67232i			0.751318i
$58$		0			0
$59$	−1.63746	−	2.83616i	−0.213179	−	0.369237i	0.739529	−	0.673125i	$-0.235048\pi$
$59$	−1.63746	−	2.83616i	−0.213179	−	0.369237i	−0.952708	+	0.303888i	$0.901715\pi$
$60$		0			0
$61$	6.77492	−	11.7345i	0.867439	−	1.50245i	0.00283468	−	0.999996i	$-0.499098\pi$
$61$	6.77492	−	11.7345i	0.867439	−	1.50245i	0.864605	−	0.502453i	$-0.167569\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.72508	−	5.61478i	−0.213970	−	0.696428i
$66$		0			0
$67$	3.04983	−	1.76082i	0.372597	−	0.215119i	−0.301996	−	0.953309i	$-0.597653\pi$
$67$	3.04983	−	1.76082i	0.372597	−	0.215119i	0.674592	+	0.738191i	$0.264319\pi$
$68$		0			0
$69$	13.5498			1.63121
$70$		0			0
$71$	−4.54983			−0.539966			−0.269983	−	0.962865i	$-0.587018\pi$
$71$	−4.54983			−0.539966			−0.269983	+	0.962865i	$0.587018\pi$
$72$		0			0
$73$	−5.63746	+	3.25479i	−0.659815	+	0.380944i	−0.792206	−	0.610253i	$-0.791068\pi$
$73$	−5.63746	+	3.25479i	−0.659815	+	0.380944i	0.132392	+	0.991197i	$0.457734\pi$
$74$		0			0
$75$	−3.77492	+	7.79423i	−0.435890	+	0.900000i
$76$		0			0
$77$	−7.91238	+	11.4964i	−0.901699	+	1.31014i
$78$		0			0
$79$	3.63746	−	6.30026i	0.409246	−	0.708835i	−0.585559	−	0.810630i	$-0.699125\pi$
$79$	3.63746	−	6.30026i	0.409246	−	0.708835i	0.994805	+	0.101795i	$0.0324584\pi$
$80$		0			0
$81$	4.50000	+	7.79423i	0.500000	+	0.866025i
$82$		0			0
$83$			7.40437i			0.812736i	0.913710	+	0.406368i	$0.133205\pi$
$83$			7.40437i			0.812736i	−0.913710	+	0.406368i	$0.866795\pi$
$84$		0			0
$85$	0.637459	−	0.685484i	0.0691421	−	0.0743512i
$86$		0			0
$87$	6.41238	−	3.70219i	0.687479	−	0.396916i
$88$		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
$91$	6.27492	−	2.98793i	0.657790	−	0.313220i
$92$		0			0
$93$	4.91238	+	2.83616i	0.509390	+	0.294096i
$94$		0			0
$95$	1.63746	−	7.13752i	0.168000	−	0.732294i
$96$		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
$101$	6.77492	+	11.7345i	0.674129	+	1.16763i	0.976723	+	0.214507i	$0.0688144\pi$
$101$	6.77492	+	11.7345i	0.674129	+	1.16763i	−0.302593	+	0.953120i	$0.597852\pi$
$102$		0			0
$103$	9.77492	+	5.64355i	0.963151	+	0.556076i	0.897141	−	0.441743i	$-0.145640\pi$
$103$	9.77492	+	5.64355i	0.963151	+	0.556076i	0.0660098	+	0.997819i	$0.478973\pi$
$104$		0			0
$105$	−9.77492	−	3.07425i	−0.953934	−	0.300016i
$106$		0			0
$107$	−3.04983	−	1.76082i	−0.294839	−	0.170225i	0.345283	−	0.938499i	$-0.387783\pi$
$107$	−3.04983	−	1.76082i	−0.294839	−	0.170225i	−0.640122	+	0.768273i	$0.721116\pi$
$108$		0			0
$109$	−5.77492	−	10.0025i	−0.553137	−	0.958061i	−0.998046	−	0.0624852i	$-0.980097\pi$
$109$	−5.77492	−	10.0025i	−0.553137	−	0.958061i	0.444909	−	0.895576i	$-0.353236\pi$
$110$		0			0
$111$	−17.2749			−1.63966
$112$		0			0
$113$		−	4.30136i		−	0.404637i	−0.979320	−	0.202319i	$-0.935152\pi$
$113$		−	4.30136i		−	0.404637i	0.979320	−	0.202319i	$-0.0648477\pi$
$114$		0			0
$115$	−17.0498	−	3.91150i	−1.58991	−	0.364749i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.912376	+	0.627940i	0.0836374	+	0.0575632i
$120$		0			0
$121$	−8.41238	+	14.5707i	−0.764761	+	1.32461i
$122$		0			0
$123$	5.58762	−	3.22602i	0.503819	−	0.290880i
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		−	15.6460i		−	1.38836i	−0.719802	−	0.694179i	$-0.755768\pi$
$127$		−	15.6460i		−	1.38836i	0.719802	−	0.694179i	$-0.244232\pi$
$128$		0			0
$129$	−1.86254	−	3.22602i	−0.163988	−	0.284035i
$130$		0			0
$131$	5.36254	−	9.28819i	0.468527	−	0.811513i	−0.530826	−	0.847481i	$-0.678118\pi$
$131$	5.36254	−	9.28819i	0.468527	−	0.811513i	0.999353	+	0.0359678i	$0.0114514\pi$
$132$		0			0
$133$	8.63746	+	0.685484i	0.748963	+	0.0594390i
$134$		0			0
$135$	−3.41238	−	11.1066i	−0.293691	−	0.955901i
$136$		0			0
$137$	−18.4622	+	10.6592i	−1.57733	+	0.910674i	−0.582103	+	0.813115i	$0.697770\pi$
$137$	−18.4622	+	10.6592i	−1.57733	+	0.910674i	−0.995230	+	0.0975588i	$0.968897\pi$
$138$		0			0
$139$	−13.0997			−1.11110			−0.555550	−	0.831483i	$-0.687492\pi$
$139$	−13.0997			−1.11110			−0.555550	+	0.831483i	$0.687492\pi$
$140$		0			0
$141$	11.2749			0.949519
$142$		0			0
$143$	12.0000	−	6.92820i	1.00349	−	0.579365i
$144$		0			0
$145$	−9.13746	+	2.80739i	−0.758825	+	0.233141i
$146$		0			0
$147$	1.91238	−	11.9726i	0.157730	−	0.987482i
$148$		0			0
$149$	−3.77492	+	6.53835i	−0.309253	+	0.535642i	−0.978199	−	0.207669i	$-0.933412\pi$
$149$	−3.77492	+	6.53835i	−0.309253	+	0.535642i	0.668946	+	0.743311i	$0.266746\pi$
$150$		0			0
$151$	6.36254	+	11.0202i	0.517776	+	0.896815i	0.999787	+	0.0206494i	$0.00657337\pi$
$151$	6.36254	+	11.0202i	0.517776	+	0.896815i	−0.482011	+	0.876165i	$0.660093\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.36254	−	4.98684i	−0.430730	−	0.400553i
$156$		0			0
$157$	−1.91238	+	1.10411i	−0.152624	+	0.0881176i	−0.574367	−	0.818598i	$-0.694752\pi$
$157$	−1.91238	+	1.10411i	−0.152624	+	0.0881176i	0.421743	+	0.906715i	$0.361418\pi$
$158$		0			0
$159$	−4.91238	+	8.50848i	−0.389577	+	0.674767i
$160$		0			0
$161$	1.63746	−	20.6328i	0.129050	−	1.62610i
$162$		0			0
$163$	4.91238	+	2.83616i	0.384767	+	0.222145i	0.679890	−	0.733314i	$-0.262027\pi$
$163$	4.91238	+	2.83616i	0.384767	+	0.222145i	−0.295123	+	0.955459i	$0.595361\pi$
$164$		0			0
$165$	−19.9124	−	4.56821i	−1.55018	−	0.355635i
$166$		0			0
$167$		−	0.476171i		−	0.0368472i	−0.999830	−	0.0184236i	$-0.994135\pi$
$167$		−	0.476171i		−	0.0368472i	0.999830	−	0.0184236i	$-0.00586474\pi$
$168$		0			0
$169$	6.09967			0.469205
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.7371	+	10.2405i	1.34853	+	0.778573i	0.988041	−	0.154190i	$-0.0492769\pi$
$173$	17.7371	+	10.2405i	1.34853	+	0.778573i	0.360488	+	0.932764i	$0.382610\pi$
$174$		0			0
$175$	11.4124	+	6.69012i	0.862695	+	0.505725i
$176$		0			0
$177$	4.91238	+	2.83616i	0.369237	+	0.213179i
$178$		0			0
$179$	−3.63746	−	6.30026i	−0.271876	−	0.470904i	0.697466	−	0.716618i	$-0.254311\pi$
$179$	−3.63746	−	6.30026i	−0.271876	−	0.470904i	−0.969342	+	0.245714i	$0.920978\pi$
$180$		0			0
$181$	−24.2749			−1.80434			−0.902170	−	0.431380i	$-0.858027\pi$
$181$	−24.2749			−1.80434			−0.902170	+	0.431380i	$0.858027\pi$
$182$		0			0
$183$			23.4690i			1.73488i
$184$		0			0
$185$	21.7371	+	4.98684i	1.59815	+	0.366640i
$186$		0			0
$187$	1.91238	+	1.10411i	0.139847	+	0.0807406i
$188$		0			0
$189$	12.4124	−	5.91041i	0.902867	−	0.429919i
$190$		0			0
$191$	−0.0876242	+	0.151770i	−0.00634026	+	0.0109817i	−0.869178	−	0.494499i	$-0.835352\pi$
$191$	−0.0876242	+	0.151770i	−0.00634026	+	0.0109817i	0.862838	+	0.505481i	$0.168685\pi$
$192$		0			0
$193$	−18.4622	+	10.6592i	−1.32894	+	0.767263i	−0.985136	−	0.171778i	$-0.945049\pi$
$193$	−18.4622	+	10.6592i	−1.32894	+	0.767263i	−0.343803	+	0.939042i	$0.611715\pi$
$194$		0			0
$195$	7.45017	+	6.92820i	0.533517	+	0.496139i
$196$		0			0
$197$			8.60271i			0.612918i	0.951884	+	0.306459i	$0.0991442\pi$
$197$			8.60271i			0.612918i	−0.951884	+	0.306459i	$0.900856\pi$
$198$		0			0
$199$	8.63746	+	14.9605i	0.612293	+	1.06052i	0.990853	+	0.134946i	$0.0430861\pi$
$199$	8.63746	+	14.9605i	0.612293	+	1.06052i	−0.378560	+	0.925577i	$0.623581\pi$
$200$		0			0
$201$	−3.04983	+	5.28247i	−0.215119	+	0.372597i
$202$		0			0
$203$	−4.86254	−	10.2118i	−0.341284	−	0.716725i
$204$		0			0
$205$	−7.96221	+	2.44631i	−0.556105	+	0.170857i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.2749			1.19493
$210$		0			0
$211$	25.6495			1.76578			0.882892	−	0.469576i	$-0.155593\pi$
$211$	25.6495			1.76578			0.882892	+	0.469576i	$0.155593\pi$
$212$		0			0
$213$	6.82475	−	3.94027i	0.467624	−	0.269983i
$214$		0			0
$215$	1.41238	+	4.59698i	0.0963232	+	0.313512i
$216$		0			0
$217$	4.91238	−	7.13752i	0.333474	−	0.484526i
$218$		0			0
$219$	5.63746	−	9.76436i	0.380944	−	0.659815i
$220$		0			0
$221$	−0.549834	−	0.952341i	−0.0369859	−	0.0640614i
$222$		0			0
$223$			8.71780i			0.583787i	0.956451	+	0.291893i	$0.0942853\pi$
$223$			8.71780i			0.583787i	−0.956451	+	0.291893i	$0.905715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−16.9124	+	9.76436i	−1.12251	+	0.648084i	−0.942041	−	0.335496i	$-0.891096\pi$
$227$	−16.9124	+	9.76436i	−1.12251	+	0.648084i	−0.180472	+	0.983580i	$0.557763\pi$
$228$		0			0
$229$	1.63746	−	2.83616i	0.108206	−	0.187419i	−0.806837	−	0.590774i	$-0.798823\pi$
$229$	1.63746	−	2.83616i	0.108206	−	0.187419i	0.915044	+	0.403355i	$0.132156\pi$
$230$		0			0
$231$	1.91238	−	24.0969i	0.125825	−	1.58546i
$232$		0			0
$233$	12.3625	+	7.13752i	0.809897	+	0.467594i	0.846920	−	0.531720i	$-0.178454\pi$
$233$	12.3625	+	7.13752i	0.809897	+	0.467594i	−0.0370231	+	0.999314i	$0.511788\pi$
$234$		0			0
$235$	−14.1873	−	3.25479i	−0.925477	−	0.212319i
$236$		0			0
$237$			12.6005i			0.818492i
$238$		0			0
$239$	−0.549834			−0.0355658			−0.0177829	−	0.999842i	$-0.505661\pi$
$239$	−0.549834			−0.0355658			−0.0177829	+	0.999842i	$0.505661\pi$
$240$		0			0
$241$	−4.91238	−	8.50848i	−0.316434	−	0.548080i	0.663307	−	0.748347i	$-0.269152\pi$
$241$	−4.91238	−	8.50848i	−0.316434	−	0.548080i	−0.979741	+	0.200267i	$0.935819\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5.86254	+	14.5131i	−0.374544	+	0.927209i
$246$		0			0
$247$	−7.45017	−	4.30136i	−0.474043	−	0.273689i
$248$		0			0
$249$	−6.41238	−	11.1066i	−0.406368	−	0.703850i
$250$		0			0
$251$	−20.5498			−1.29709			−0.648547	−	0.761175i	$-0.724623\pi$
$251$	−20.5498			−1.29709			−0.648547	+	0.761175i	$0.724623\pi$
$252$		0			0
$253$		−	41.2657i		−	2.59435i
$254$		0			0
$255$	−0.362541	+	1.58028i	−0.0227032	+	0.0989611i
$256$		0			0
$257$	10.0876	+	5.82409i	0.629249	+	0.363297i	0.780461	−	0.625204i	$-0.214984\pi$
$257$	10.0876	+	5.82409i	0.629249	+	0.363297i	−0.151212	+	0.988501i	$0.548318\pi$
$258$		0			0
$259$	−2.08762	+	26.3052i	−0.129719	+	1.63452i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.675248	+	0.389855i	−0.0416376	+	0.0240395i	−0.520674	−	0.853755i	$-0.674319\pi$
$263$	−0.675248	+	0.389855i	−0.0416376	+	0.0240395i	0.479037	+	0.877795i	$0.340986\pi$
$264$		0			0
$265$	8.63746	−	9.28819i	0.530595	−	0.570569i
$266$		0			0
$267$		−	12.1244i		−	0.741999i
$268$		0			0
$269$	−7.22508	−	12.5142i	−0.440521	−	0.763005i	0.557207	−	0.830374i	$-0.311873\pi$
$269$	−7.22508	−	12.5142i	−0.440521	−	0.763005i	−0.997728	+	0.0673687i	$0.978540\pi$
$270$		0			0
$271$	−4.91238	+	8.50848i	−0.298406	+	0.516854i	−0.975771	−	0.218793i	$-0.929788\pi$
$271$	−4.91238	+	8.50848i	−0.298406	+	0.516854i	0.677366	+	0.735646i	$0.263121\pi$
$272$		0			0
$273$	−6.82475	+	9.91613i	−0.413053	+	0.600152i
$274$		0			0
$275$	23.7371	+	11.4964i	1.43140	+	0.693260i
$276$		0			0
$277$	12.3625	−	7.13752i	0.742793	−	0.428852i	−0.0802909	−	0.996771i	$-0.525585\pi$
$277$	12.3625	−	7.13752i	0.742793	−	0.428852i	0.823084	+	0.567920i	$0.192252\pi$
$278$		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
$283$	−9.46221	+	5.46301i	−0.562470	+	0.324742i	−0.754136	−	0.656718i	$-0.771944\pi$
$283$	−9.46221	+	5.46301i	−0.562470	+	0.324742i	0.191666	+	0.981460i	$0.438611\pi$
$284$		0			0
$285$	3.72508	+	12.1244i	0.220655	+	0.718185i
$286$		0			0
$287$	−4.23713	−	8.89834i	−0.250110	−	0.525252i
$288$		0			0
$289$	−8.41238	+	14.5707i	−0.494846	+	0.857098i
$290$		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
$295$	−5.36254	−	4.98684i	−0.312219	−	0.290345i
$296$		0			0
$297$	23.7371	−	13.7046i	1.37737	−	0.795224i
$298$		0			0
$299$	−10.2749	+	17.7967i	−0.594214	+	1.02921i
$300$		0			0
$301$	−5.13746	+	2.44631i	−0.296118	+	0.141003i
$302$		0			0
$303$	−20.3248	−	11.7345i	−1.16763	−	0.674129i
$304$		0			0
$305$	6.77492	−	29.5312i	0.387931	−	1.69095i
$306$		0			0
$307$			26.5145i			1.51326i	0.653843	+	0.756631i	$0.273156\pi$
$307$			26.5145i			1.51326i	−0.653843	+	0.756631i	$0.726844\pi$
$308$		0			0
$309$	−19.5498			−1.11215
$310$		0			0
$311$	4.91238	+	8.50848i	0.278555	+	0.482472i	0.971026	−	0.238974i	$-0.0768111\pi$
$311$	4.91238	+	8.50848i	0.278555	+	0.482472i	−0.692471	+	0.721446i	$0.743478\pi$
$312$		0			0
$313$	−29.0120	−	16.7501i	−1.63986	−	0.946772i	−0.980881	−	0.194609i	$-0.937656\pi$
$313$	−29.0120	−	16.7501i	−1.63986	−	0.946772i	−0.658977	−	0.752163i	$-0.729010\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.1873	−	12.8098i	−1.24616	−	0.719472i	−0.275821	−	0.961209i	$-0.588950\pi$
$317$	−22.1873	−	12.8098i	−1.24616	−	0.719472i	−0.970342	+	0.241737i	$0.922283\pi$
$318$		0			0
$319$	−11.2749	−	19.5287i	−0.631274	−	1.09340i
$320$		0			0
$321$	6.09967			0.340450
$322$		0			0
$323$		−	1.37097i		−	0.0762827i
$324$		0			0
$325$	−7.37459	−	10.8685i	−0.409068	−	0.602875i
$326$		0			0
$327$	17.3248	+	10.0025i	0.958061	+	0.553137i
$328$		0			0
$329$	1.36254	−	17.1687i	0.0751193	−	0.946543i
$330$		0			0
$331$	−8.91238	+	15.4367i	−0.489868	+	0.848477i	−0.999932	−	0.0116596i	$-0.996289\pi$
$331$	−8.91238	+	15.4367i	−0.489868	+	0.848477i	0.510064	+	0.860137i	$0.329622\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.36254	−	5.76655i	0.292987	−	0.315060i
$336$		0			0
$337$		−	4.30136i		−	0.234310i	−0.993114	−	0.117155i	$-0.962623\pi$
$337$		−	4.30136i		−	0.234310i	0.993114	−	0.117155i	$-0.0373774\pi$
$338$		0			0
$339$	3.72508	+	6.45203i	0.202319	+	0.350426i
$340$		0			0
$341$	8.63746	−	14.9605i	0.467745	−	0.810157i
$342$		0			0
$343$	−18.0000	−	4.35890i	−0.971909	−	0.235358i
$344$		0			0
$345$	28.9622	−	8.89834i	1.55927	−	0.479070i
$346$		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
$349$	3.72508			0.199399			0.0996996	−	0.995018i	$-0.468212\pi$
$349$	3.72508			0.199399			0.0996996	+	0.995018i	$0.468212\pi$
$350$		0			0
$351$	−13.6495			−0.728557
$352$		0			0
$353$	−7.08762	+	4.09204i	−0.377236	+	0.217797i	−0.676615	−	0.736337i	$-0.736554\pi$
$353$	−7.08762	+	4.09204i	−0.377236	+	0.217797i	0.299379	+	0.954134i	$0.403221\pi$
$354$		0			0
$355$	−9.72508	+	2.98793i	−0.516154	+	0.158583i
$356$		0			0
$357$	−1.91238	−	0.151770i	−0.101214	−	0.00803249i
$358$		0			0
$359$	18.1873	−	31.5013i	0.959889	−	1.66258i	0.237127	−	0.971479i	$-0.423794\pi$
$359$	18.1873	−	31.5013i	0.959889	−	1.66258i	0.722762	−	0.691097i	$-0.242872\pi$
$360$		0			0
$361$	4.13746	+	7.16629i	0.217761	+	0.377173i
$362$		0			0
$363$		−	29.1413i		−	1.52952i
$364$		0			0
$365$	−9.91238	+	10.6592i	−0.518837	+	0.557926i
$366$		0			0
$367$	−5.22508	+	3.01670i	−0.272747	+	0.157471i	−0.630135	−	0.776485i	$-0.717001\pi$
$367$	−5.22508	+	3.01670i	−0.272747	+	0.157471i	0.357388	+	0.933956i	$0.383667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.3625	+	8.50848i	0.641831	+	0.441739i
$372$		0			0
$373$	8.63746	+	4.98684i	0.447231	+	0.258209i	0.706660	−	0.707553i	$-0.250201\pi$
$373$	8.63746	+	4.98684i	0.447231	+	0.258209i	−0.259429	+	0.965762i	$0.583534\pi$
$374$		0			0
$375$	−2.95017	+	19.1389i	−0.152346	+	0.988327i
$376$		0			0
$377$			11.2296i			0.578352i
$378$		0			0
$379$	21.6495			1.11206			0.556030	−	0.831162i	$-0.312324\pi$
$379$	21.6495			1.11206			0.556030	+	0.831162i	$0.312324\pi$
$380$		0			0
$381$	13.5498	+	23.4690i	0.694179	+	1.20235i
$382$		0			0
$383$	−5.32475	−	3.07425i	−0.272082	−	0.157087i	0.357751	−	0.933817i	$-0.383544\pi$
$383$	−5.32475	−	3.07425i	−0.272082	−	0.157087i	−0.629833	+	0.776730i	$0.716877\pi$
$384$		0			0
$385$	−9.36254	+	29.7693i	−0.477159	+	1.51718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.1873	+	28.0372i	0.820728	+	1.42154i	0.905141	+	0.425112i	$0.139765\pi$
$389$	16.1873	+	28.0372i	0.820728	+	1.42154i	−0.0844123	+	0.996431i	$0.526901\pi$
$390$		0			0
$391$	−3.27492			−0.165620
$392$		0			0
$393$			18.5764i			0.937055i
$394$		0			0
$395$	3.63746	−	15.8553i	0.183020	−	0.797767i
$396$		0			0
$397$	−9.36254	−	5.40547i	−0.469892	−	0.271293i	0.246302	−	0.969193i	$-0.420784\pi$
$397$	−9.36254	−	5.40547i	−0.469892	−	0.271293i	−0.716195	+	0.697901i	$0.754118\pi$
$398$		0			0
$399$	−13.5498	+	6.45203i	−0.678340	+	0.323006i
$400$		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
$403$	−7.45017	+	4.30136i	−0.371119	+	0.214266i
$404$		0			0
$405$	14.7371	+	13.7046i	0.732294	+	0.680989i
$406$		0			0
$407$			52.6103i			2.60780i
$408$		0			0
$409$	−10.0498	−	17.4068i	−0.496932	−	0.860712i	0.503061	−	0.864251i	$-0.332207\pi$
$409$	−10.0498	−	17.4068i	−0.496932	−	0.860712i	−0.999994	+	0.00353862i	$0.998874\pi$
$410$		0			0
$411$	18.4622	−	31.9775i	0.910674	−	1.57733i
$412$		0			0
$413$	4.91238	−	7.13752i	0.241722	−	0.351214i
$414$		0			0
$415$	4.86254	+	15.8265i	0.238693	+	0.776894i
$416$		0			0
$417$	19.6495	−	11.3446i	0.962240	−	0.555550i
$418$		0			0
$419$	13.0997			0.639961			0.319980	−	0.947424i	$-0.396324\pi$
$419$	13.0997			0.639961			0.319980	+	0.947424i	$0.396324\pi$
$420$		0			0
$421$	−4.27492			−0.208347			−0.104173	−	0.994559i	$-0.533220\pi$
$421$	−4.27492			−0.208347			−0.104173	+	0.994559i	$0.533220\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.912376	−	1.88382i	0.0442567	−	0.0913787i
$426$		0			0
$427$	35.7371	+	2.83616i	1.72944	+	0.137251i
$428$		0			0
$429$	−12.0000	+	20.7846i	−0.579365	+	1.00349i
$430$		0			0
$431$	9.18729	+	15.9129i	0.442536	+	0.766495i	0.997877	−	0.0651276i	$-0.0207454\pi$
$431$	9.18729	+	15.9129i	0.442536	+	0.766495i	−0.555341	+	0.831623i	$0.687412\pi$
$432$		0			0
$433$			18.1578i			0.872606i	0.899800	+	0.436303i	$0.143712\pi$
$433$			18.1578i			0.872606i	−0.899800	+	0.436303i	$0.856288\pi$
$434$		0			0
$435$	11.2749	−	12.1244i	0.540591	−	0.581318i
$436$		0			0
$437$	−22.1873	+	12.8098i	−1.06136	+	0.612778i
$438$		0			0
$439$	11.9124	−	20.6328i	0.568547	−	0.984752i	−0.428163	−	0.903701i	$-0.640839\pi$
$439$	11.9124	−	20.6328i	0.568547	−	0.984752i	0.996710	−	0.0810504i	$-0.0258275\pi$
$440$		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
$449$	3.17525			0.149849			0.0749246	−	0.997189i	$-0.476128\pi$
$449$	3.17525			0.149849			0.0749246	+	0.997189i	$0.476128\pi$
$450$		0			0
$451$	−9.82475	−	17.0170i	−0.462629	−	0.801298i
$452$		0			0
$453$	−19.0876	−	11.0202i	−0.896815	−	0.517776i
$454$		0			0
$455$	11.4502	−	10.5074i	0.536792	−	0.492594i
$456$		0			0
$457$	1.18729	+	0.685484i	0.0555392	+	0.0320656i	0.527512	−	0.849547i	$-0.323125\pi$
$457$	1.18729	+	0.685484i	0.0555392	+	0.0320656i	−0.471973	+	0.881613i	$0.656458\pi$
$458$		0			0
$459$	−1.08762	−	1.88382i	−0.0507659	−	0.0879292i
$460$		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
$463$		−	2.15068i		−	0.0999505i	−0.998750	−	0.0499752i	$-0.984086\pi$
$463$		−	2.15068i		−	0.0999505i	0.998750	−	0.0499752i	$-0.0159142\pi$
$464$		0			0
$465$	12.3625	+	2.83616i	0.573299	+	0.131524i
$466$		0			0
$467$	−13.5997	−	7.85177i	−0.629318	−	0.363337i	0.151170	−	0.988508i	$-0.451696\pi$
$467$	−13.5997	−	7.85177i	−0.629318	−	0.363337i	−0.780488	+	0.625171i	$0.785029\pi$
$468$		0			0
$469$	7.67525	+	5.28247i	0.354410	+	0.243922i
$470$		0			0
$471$	1.91238	−	3.31233i	0.0881176	−	0.152624i
$472$		0			0
$473$	−9.82475	+	5.67232i	−0.451743	+	0.260814i
$474$		0			0
$475$	−1.18729	−	16.3315i	−0.0544767	−	0.749340i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.91238	+	8.50848i	0.224452	+	0.388763i	0.956155	−	0.292861i	$-0.0946074\pi$
$479$	4.91238	+	8.50848i	0.224452	+	0.388763i	−0.731703	+	0.681624i	$0.761274\pi$
$480$		0			0
$481$	13.0997	−	22.6893i	0.597293	−	1.03454i
$482$		0			0
$483$	15.4124	+	32.3673i	0.701287	+	1.47277i
$484$		0			0
$485$	−4.54983	−	14.8087i	−0.206597	−	0.672431i
$486$		0			0
$487$	2.53779	−	1.46519i	0.114998	−	0.0663943i	−0.441398	−	0.897312i	$-0.645517\pi$
$487$	2.53779	−	1.46519i	0.114998	−	0.0663943i	0.556396	+	0.830917i	$0.312184\pi$
$488$		0			0
$489$	−9.82475			−0.444291
$490$		0			0
$491$	−28.5498			−1.28844			−0.644218	−	0.764842i	$-0.722817\pi$
$491$	−28.5498			−1.28844			−0.644218	+	0.764842i	$0.722817\pi$
$492$		0			0
$493$	−1.54983	+	0.894797i	−0.0698010	+	0.0402996i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−5.17525	−	10.8685i	−0.232142	−	0.487518i
$498$		0			0
$499$	0.812707	−	1.40765i	0.0363818	−	0.0630151i	−0.847261	−	0.531177i	$-0.821750\pi$
$499$	0.812707	−	1.40765i	0.0363818	−	0.0630151i	0.883643	+	0.468161i	$0.155083\pi$
$500$		0			0
$501$	0.412376	+	0.714256i	0.0184236	+	0.0319106i
$502$		0			0
$503$		−	31.7682i		−	1.41647i	−0.705975	−	0.708236i	$-0.749491\pi$
$503$		−	31.7682i		−	1.41647i	0.705975	−	0.708236i	$-0.250509\pi$
$504$		0			0
$505$	22.1873	+	20.6328i	0.987322	+	0.918149i
$506$		0			0
$507$	−9.14950	+	5.28247i	−0.406344	+	0.234603i
$508$		0			0
$509$	−7.22508	+	12.5142i	−0.320246	+	0.554683i	−0.980539	−	0.196326i	$-0.937099\pi$
$509$	−7.22508	+	12.5142i	−0.320246	+	0.554683i	0.660293	+	0.751008i	$0.270432\pi$
$510$		0			0
$511$	−14.1873	−	9.76436i	−0.627609	−	0.431950i
$512$		0			0
$513$	−14.7371	−	8.50848i	−0.650660	−	0.375659i
$514$		0			0
$515$	24.5997	+	5.64355i	1.08399	+	0.248685i
$516$		0			0
$517$		−	34.3375i		−	1.51016i
$518$		0			0
$519$	−35.4743			−1.55715
$520$		0			0
$521$	−4.91238	−	8.50848i	−0.215215	−	0.372763i	0.738124	−	0.674665i	$-0.235712\pi$
$521$	−4.91238	−	8.50848i	−0.215215	−	0.372763i	−0.953339	+	0.301902i	$0.902379\pi$
$522$		0			0
$523$	6.36254	+	3.67341i	0.278215	+	0.160627i	0.632615	−	0.774467i	$-0.281982\pi$
$523$	6.36254	+	3.67341i	0.278215	+	0.160627i	−0.354400	+	0.935094i	$0.615315\pi$
$524$		0			0
$525$	−22.9124	−	0.151770i	−0.999978	−	0.00662376i
$526$		0			0
$527$	−1.18729	−	0.685484i	−0.0517193	−	0.0298602i
$528$		0			0
$529$	19.0997	+	33.0816i	0.830420	+	1.43833i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9.78523i			0.423845i
$534$		0			0
$535$	−7.67525	−	1.76082i	−0.331830	−	0.0761270i
$536$		0			0
$537$	10.9124	+	6.30026i	0.470904	+	0.271876i
$538$		0			0
$539$	−36.4622	−	5.82409i	−1.57054	−	0.250861i
$540$		0			0
$541$	8.77492	−	15.1986i	0.377263	−	0.653439i	−0.613400	−	0.789773i	$-0.710199\pi$
$541$	8.77492	−	15.1986i	0.377263	−	0.653439i	0.990663	+	0.136334i	$0.0435319\pi$
$542$		0			0
$543$	36.4124	−	21.0227i	1.56260	−	0.902170i
$544$		0			0
$545$	−18.9124	−	17.5874i	−0.810117	−	0.753360i
$546$		0			0
$547$		−	20.5386i		−	0.878168i	−0.898446	−	0.439084i	$-0.855303\pi$
$547$		−	20.5386i		−	0.878168i	0.898446	−	0.439084i	$-0.144697\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.00000	+	12.1244i	−0.298210	+	0.516515i
$552$		0			0
$553$	19.1873	+	1.52274i	0.815927	+	0.0647534i
$554$		0			0
$555$	−36.9244	+	11.3446i	−1.56735	+	0.481553i
$556$		0			0
$557$	8.63746	−	4.98684i	0.365981	−	0.211299i	−0.305720	−	0.952121i	$-0.598897\pi$
$557$	8.63746	−	4.98684i	0.365981	−	0.211299i	0.671701	+	0.740822i	$0.265564\pi$
$558$		0			0
$559$	5.64950			0.238949
$560$		0			0
$561$	−3.82475			−0.161481
$562$		0			0
$563$	19.5997	−	11.3159i	0.826028	−	0.476907i	−0.0264630	−	0.999650i	$-0.508424\pi$
$563$	19.5997	−	11.3159i	0.826028	−	0.476907i	0.852491	+	0.522743i	$0.175091\pi$
$564$		0			0
$565$	−2.82475	−	9.19397i	−0.118838	−	0.386793i
$566$		0			0
$567$	−13.5000	+	19.6150i	−0.566947	+	0.823754i
$568$		0			0
$569$	4.18729	−	7.25260i	0.175540	−	0.304045i	−0.764808	−	0.644259i	$-0.777166\pi$
$569$	4.18729	−	7.25260i	0.175540	−	0.304045i	0.940348	+	0.340214i	$0.110499\pi$
$570$		0			0
$571$	−3.63746	−	6.30026i	−0.152223	−	0.263658i	0.779821	−	0.626002i	$-0.215310\pi$
$571$	−3.63746	−	6.30026i	−0.152223	−	0.263658i	−0.932044	+	0.362344i	$0.881976\pi$
$572$		0			0
$573$		−	0.303539i		−	0.0126805i
$574$		0			0
$575$	−39.0120	+	2.83616i	−1.62691	+	0.118276i
$576$		0			0
$577$	−3.36254	+	1.94136i	−0.139984	+	0.0808200i	−0.568357	−	0.822782i	$-0.692421\pi$
$577$	−3.36254	+	1.94136i	−0.139984	+	0.0808200i	0.428372	+	0.903602i	$0.359087\pi$
$578$		0			0
$579$	18.4622	−	31.9775i	0.767263	−	1.32894i
$580$		0			0
$581$	−17.6873	+	8.42217i	−0.733793	+	0.349410i
$582$		0			0
$583$	25.9124	+	14.9605i	1.07318	+	0.619601i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			20.8997i			0.862623i	0.902203	+	0.431311i	$0.141949\pi$
$587$			20.8997i			0.862623i	−0.902203	+	0.431311i	$0.858051\pi$
$588$		0			0
$589$	−10.7251			−0.441919
$590$		0			0
$591$	−7.45017	−	12.9041i	−0.306459	−	0.530802i
$592$		0			0
$593$	28.9124	+	16.6926i	1.18729	+	0.685482i	0.957689	−	0.287804i	$-0.0929250\pi$
$593$	28.9124	+	16.6926i	1.18729	+	0.685482i	0.229600	+	0.973285i	$0.426258\pi$
$594$		0			0
$595$	2.36254	+	0.743028i	0.0968548	+	0.0304612i
$596$		0			0
$597$	−25.9124	−	14.9605i	−1.06052	−	0.612293i
$598$		0			0
$599$	2.63746	+	4.56821i	0.107764	+	0.186652i	0.914864	−	0.403762i	$-0.132298\pi$
$599$	2.63746	+	4.56821i	0.107764	+	0.186652i	−0.807100	+	0.590414i	$0.798964\pi$
$600$		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
$605$	−8.41238	+	36.6687i	−0.342012	+	1.49079i
$606$		0			0
$607$	−9.87459	−	5.70109i	−0.400797	−	0.231400i	0.286031	−	0.958220i	$-0.407664\pi$
$607$	−9.87459	−	5.70109i	−0.400797	−	0.231400i	−0.686828	+	0.726820i	$0.740997\pi$
$608$		0			0
$609$	16.1375	+	11.1066i	0.653923	+	0.450061i
$610$		0			0
$611$	−8.54983	+	14.8087i	−0.345889	+	0.599098i
$612$		0			0
$613$	24.5619	−	14.1808i	0.992045	−	0.572757i	0.0861600	−	0.996281i	$-0.472540\pi$
$613$	24.5619	−	14.1808i	0.992045	−	0.572757i	0.905885	+	0.423524i	$0.139207\pi$
$614$		0			0
$615$	9.82475	−	10.5649i	0.396172	−	0.426019i
$616$		0			0
$617$		−	31.2920i		−	1.25977i	−0.776689	−	0.629884i	$-0.783102\pi$
$617$		−	31.2920i		−	1.25977i	0.776689	−	0.629884i	$-0.216898\pi$
$618$		0			0
$619$	−4.46221	−	7.72877i	−0.179351	−	0.310646i	0.762307	−	0.647215i	$-0.224067\pi$
$619$	−4.46221	−	7.72877i	−0.179351	−	0.310646i	−0.941659	+	0.336570i	$0.890733\pi$
$620$		0			0
$621$	−20.3248	+	35.2035i	−0.815604	+	1.41267i
$622$		0			0
$623$	−18.4622	−	1.46519i	−0.739673	−	0.0587017i
$624$		0			0
$625$	9.23713	−	23.2309i	0.369485	−	0.929237i
$626$		0			0
$627$	−25.9124	+	14.9605i	−1.03484	+	0.597466i
$628$		0			0
$629$	4.17525			0.166478
$630$		0			0
$631$	33.0997			1.31768			0.658839	−	0.752284i	$-0.271048\pi$
$631$	33.0997			1.31768			0.658839	+	0.752284i	$0.271048\pi$
$632$		0			0
$633$	−38.4743	+	22.2131i	−1.52921	+	0.882892i
$634$		0			0
$635$	−10.2749	−	33.4427i	−0.407748	−	1.32713i
$636$		0			0
$637$	14.2749	+	11.5906i	0.565593	+	0.459238i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.04983	+	1.81837i	0.0414660	+	0.0718212i	0.886014	−	0.463659i	$-0.153464\pi$
$641$	1.04983	+	1.81837i	0.0414660	+	0.0718212i	−0.844548	+	0.535481i	$0.820131\pi$
$642$		0			0
$643$			31.4071i			1.23857i	0.785164	+	0.619287i	$0.212578\pi$
$643$			31.4071i			1.23857i	−0.785164	+	0.619287i	$0.787422\pi$
$644$		0			0
$645$	−6.09967	−	5.67232i	−0.240174	−	0.223348i
$646$		0			0
$647$	23.3248	−	13.4666i	0.916991	−	0.529425i	0.0343169	−	0.999411i	$-0.489074\pi$
$647$	23.3248	−	13.4666i	0.916991	−	0.529425i	0.882674	+	0.469986i	$0.155741\pi$
$648$		0			0
$649$	8.63746	−	14.9605i	0.339050	−	0.587252i
$650$		0			0
$651$	−1.18729	+	14.9605i	−0.0465337	+	0.586349i
$652$		0			0
$653$	24.5619	+	14.1808i	0.961181	+	0.554938i	0.896536	−	0.442970i	$-0.146075\pi$
$653$	24.5619	+	14.1808i	0.961181	+	0.554938i	0.0646444	+	0.997908i	$0.479409\pi$
$654$		0			0
$655$	5.36254	−	23.3748i	0.209532	−	0.913328i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.5498			1.57960			0.789799	−	0.613366i	$-0.210185\pi$
$659$	40.5498			1.57960			0.789799	+	0.613366i	$0.210185\pi$
$660$		0			0
$661$	0.225083	+	0.389855i	0.00875471	+	0.0151636i	0.870370	−	0.492399i	$-0.163880\pi$
$661$	0.225083	+	0.389855i	0.00875471	+	0.0151636i	−0.861615	+	0.507563i	$0.830547\pi$
$662$		0			0
$663$	1.64950	+	0.952341i	0.0640614	+	0.0369859i
$664$		0			0
$665$	18.9124	−	4.20713i	0.733390	−	0.163145i
$666$		0			0
$667$	28.9622	+	16.7213i	1.12142	+	0.647453i
$668$		0			0
$669$	−7.54983	−	13.0767i	−0.291893	−	0.505574i
$670$		0			0
$671$	71.4743			2.75923
$672$		0			0
$673$			31.2920i			1.20622i	0.797659	+	0.603109i	$0.206072\pi$
$673$			31.2920i			1.20622i	−0.797659	+	0.603109i	$0.793928\pi$
$674$		0			0
$675$	−14.5876	−	21.4989i	−0.561478	−	0.827492i
$676$		0			0
$677$	−40.1873	−	23.2021i	−1.54452	−	0.891731i	−0.998545	−	0.0539317i	$-0.982825\pi$
$677$	−40.1873	−	23.2021i	−1.54452	−	0.891731i	−0.545979	−	0.837799i	$-0.683842\pi$
$678$		0			0
$679$	16.5498	−	7.88054i	0.635124	−	0.302428i
$680$		0			0
$681$	16.9124	−	29.2931i	0.648084	−	1.12251i
$682$		0			0
$683$	−16.5997	+	9.58382i	−0.635169	+	0.366715i	−0.782751	−	0.622335i	$-0.786184\pi$
$683$	−16.5997	+	9.58382i	−0.635169	+	0.366715i	0.147582	+	0.989050i	$0.452851\pi$
$684$		0			0
$685$	−32.4622	+	34.9079i	−1.24032	+	1.33376i
$686$		0			0
$687$			5.67232i			0.216413i
$688$		0			0
$689$	−7.45017	−	12.9041i	−0.283829	−	0.491606i
$690$		0			0
$691$	−15.1873	+	26.3052i	−0.577752	+	1.00070i	0.417985	+	0.908454i	$0.362737\pi$
$691$	−15.1873	+	26.3052i	−0.577752	+	1.00070i	−0.995737	+	0.0922416i	$0.970597\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−28.0000	+	8.60271i	−1.06210	+	0.326319i
$696$		0			0
$697$	−1.35050	+	0.779710i	−0.0511537	+	0.0295336i
$698$		0			0
$699$	−24.7251			−0.935189
$700$		0			0
$701$	8.82475			0.333306			0.166653	−	0.986016i	$-0.446704\pi$
$701$	8.82475			0.333306			0.166653	+	0.986016i	$0.446704\pi$
$702$		0			0
$703$	28.2870	−	16.3315i	1.06686	−	0.615954i
$704$		0			0
$705$	24.0997	−	7.40437i	0.907646	−	0.278865i
$706$		0			0
$707$	−20.3248	+	29.5312i	−0.764391	+	1.11063i
$708$		0			0
$709$	−5.22508	+	9.05011i	−0.196232	+	0.339884i	−0.947304	−	0.320337i	$-0.896204\pi$
$709$	−5.22508	+	9.05011i	−0.196232	+	0.339884i	0.751072	+	0.660221i	$0.229537\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			25.6197i			0.959465i
$714$		0			0
$715$	21.0997	−	22.6893i	0.789083	−	0.848531i
$716$		0			0
$717$	0.824752	−	0.476171i	0.0308009	−	0.0177829i
$718$		0			0
$719$	−15.1873	+	26.3052i	−0.566390	+	0.981017i	0.430528	+	0.902577i	$0.358327\pi$
$719$	−15.1873	+	26.3052i	−0.566390	+	0.981017i	−0.996919	+	0.0784400i	$0.975006\pi$
$720$		0			0
$721$	−2.36254	+	29.7693i	−0.0879856	+	1.10867i
$722$		0			0
$723$	14.7371	+	8.50848i	0.548080	+	0.316434i
$724$		0			0
$725$	−17.6873	+	12.0014i	−0.656890	+	0.445719i
$726$		0			0
$727$			3.10302i			0.115085i	0.998343	+	0.0575423i	$0.0183264\pi$
$727$			3.10302i			0.115085i	−0.998343	+	0.0575423i	$0.981674\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	0.450166	+	0.779710i	0.0166500	+	0.0288386i
$732$		0			0
$733$	32.6375	+	18.8432i	1.20549	+	0.695991i	0.961771	−	0.273854i	$-0.0882986\pi$
$733$	32.6375	+	18.8432i	1.20549	+	0.695991i	0.243721	+	0.969845i	$0.421632\pi$
$734$		0			0
$735$	−3.77492	−	26.8468i	−0.139240	−	0.990259i
$736$		0			0
$737$	16.0876	+	9.28819i	0.592595	+	0.342135i
$738$		0			0
$739$	−10.4622	−	18.1211i	−0.384859	−	0.666595i	0.606891	−	0.794785i	$-0.292416\pi$
$739$	−10.4622	−	18.1211i	−0.384859	−	0.666595i	−0.991750	+	0.128190i	$0.959083\pi$
$740$		0			0
$741$	14.9003			0.547377
$742$		0			0
$743$			6.45203i			0.236702i	0.992972	+	0.118351i	$0.0377608\pi$
$743$			6.45203i			0.236702i	−0.992972	+	0.118351i	$0.962239\pi$
$744$		0			0
$745$	−3.77492	+	16.4545i	−0.138302	+	0.602846i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.737127	−	9.28819i	0.0269341	−	0.339383i
$750$		0			0
$751$	7.36254	−	12.7523i	0.268663	−	0.465338i	−0.699854	−	0.714286i	$-0.746752\pi$
$751$	7.36254	−	12.7523i	0.268663	−	0.465338i	0.968517	+	0.248948i	$0.0800849\pi$
$752$		0			0
$753$	30.8248	−	17.7967i	1.12332	−	0.648547i
$754$		0			0
$755$	20.8368	+	19.3770i	0.758329	+	0.705200i
$756$		0			0
$757$			35.5934i			1.29366i	0.762633	+	0.646831i	$0.223906\pi$
$757$			35.5934i			1.29366i	−0.762633	+	0.646831i	$0.776094\pi$
$758$		0			0
$759$	35.7371	+	61.8985i	1.29718	+	2.24677i
$760$		0			0
$761$	−11.4622	+	19.8531i	−0.415505	+	0.719675i	−0.995481	−	0.0949578i	$-0.969728\pi$
$761$	−11.4622	+	19.8531i	−0.415505	+	0.719675i	0.579977	+	0.814633i	$0.303062\pi$
$762$		0			0
$763$	17.3248	−	25.1723i	0.627198	−	0.911298i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.45017	+	4.30136i	−0.269010	+	0.155313i
$768$		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
$771$	−20.1752			−0.726594
$772$		0			0
$773$	34.9124	−	20.1567i	1.25571	−	0.724985i	0.283473	−	0.958980i	$-0.408513\pi$
$773$	34.9124	−	20.1567i	1.25571	−	0.724985i	0.972238	+	0.233995i	$0.0751800\pi$
$774$		0			0
$775$	−14.7371	−	7.13752i	−0.529373	−	0.256387i
$776$		0			0
$777$	−19.6495	−	41.2657i	−0.704922	−	1.48040i
$778$		0			0
$779$	−6.09967	+	10.5649i	−0.218543	+	0.378528i
$780$		0			0
$781$	−12.0000	−	20.7846i	−0.429394	−	0.743732i
$782$		0			0
$783$			22.2131i			0.793832i
$784$		0			0
$785$	−3.36254	+	3.61587i	−0.120014	+	0.129056i
$786$		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
$789$	0.675248	−	1.16956i	0.0240395	−	0.0416376i
$790$		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)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{3} +(2.13746 - 0.656712i) q^{5} +(1.13746 + 2.38876i) q^{7} +(2.63746 + 4.56821i) q^{11} -2.62685i q^{13} +(-2.63746 + 2.83616i) q^{15} +(0.362541 - 0.209313i) q^{17} +(1.63746 - 2.83616i) q^{19} +(-3.77492 - 2.59808i) q^{21} +(-6.77492 - 3.91150i) q^{23} +(4.13746 - 2.80739i) q^{25} -5.19615i q^{27} -4.27492 q^{29} +(-1.63746 - 2.83616i) q^{31} +(-7.91238 - 4.56821i) q^{33} +(4.00000 + 4.35890i) q^{35} +(8.63746 + 4.98684i) q^{37} +(2.27492 + 3.94027i) q^{39} -3.72508 q^{41} +2.15068i q^{43} +(-5.63746 - 3.25479i) q^{47} +(-4.41238 + 5.43424i) q^{49} +(-0.362541 + 0.627940i) q^{51} +(4.91238 - 2.83616i) q^{53} +(8.63746 + 8.03231i) q^{55} +5.67232i q^{57} +(-1.63746 - 2.83616i) q^{59} +(6.77492 - 11.7345i) q^{61} +(-1.72508 - 5.61478i) q^{65} +(3.04983 - 1.76082i) q^{67} +13.5498 q^{69} -4.54983 q^{71} +(-5.63746 + 3.25479i) q^{73} +(-3.77492 + 7.79423i) q^{75} +(-7.91238 + 11.4964i) q^{77} +(3.63746 - 6.30026i) q^{79} +(4.50000 + 7.79423i) q^{81} +7.40437i q^{83} +(0.637459 - 0.685484i) q^{85} +(6.41238 - 3.70219i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(6.27492 - 2.98793i) q^{91} +(4.91238 + 2.83616i) q^{93} +(1.63746 - 7.13752i) q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + q^{5} - 3 q^{7} + 3 q^{11} - 3 q^{15} + 9 q^{17} - q^{19} - 12 q^{23} + 9 q^{25} - 2 q^{29} + q^{31} - 9 q^{33} + 16 q^{35} + 27 q^{37} - 6 q^{39} - 30 q^{41} - 15 q^{47} + 5 q^{49} - 9 q^{51}+ \cdots - q^{95}+O(q^{100}) \) 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

Modular forms

Varieties

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Groups

Database

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