LMFDB - Embedded newform 140.2.q.b.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.b.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 2.17945i) q^{5} +(-1.13746 - 2.38876i) q^{7} +O(q^{10})$	\(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 2.17945i) q^{5} +(-1.13746 - 2.38876i) q^{7} +(2.63746 + 4.56821i) q^{11} +2.62685i q^{13} +(-2.63746 - 2.83616i) q^{15} +(-0.362541 + 0.209313i) q^{17} +(1.63746 - 2.83616i) q^{19} +(-3.77492 - 2.59808i) q^{21} +(6.77492 + 3.91150i) q^{23} +(-4.50000 + 2.17945i) q^{25} +5.19615i q^{27} -4.27492 q^{29} +(-1.63746 - 2.83616i) q^{31} +(7.91238 + 4.56821i) q^{33} +(-4.63746 + 3.67341i) q^{35} +(-8.63746 - 4.98684i) q^{37} +(2.27492 + 3.94027i) q^{39} -3.72508 q^{41} -2.15068i q^{43} +(5.63746 + 3.25479i) q^{47} +(-4.41238 + 5.43424i) q^{49} +(-0.362541 + 0.627940i) q^{51} +(-4.91238 + 2.83616i) q^{53} +(8.63746 - 8.03231i) q^{55} -5.67232i q^{57} +(-1.63746 - 2.83616i) q^{59} +(6.77492 - 11.7345i) q^{61} +(5.72508 - 1.31342i) q^{65} +(-3.04983 + 1.76082i) q^{67} +13.5498 q^{69} -4.54983 q^{71} +(5.63746 - 3.25479i) q^{73} +(-4.86254 + 7.16629i) q^{75} +(7.91238 - 11.4964i) q^{77} +(3.63746 - 6.30026i) q^{79} +(4.50000 + 7.79423i) q^{81} -7.40437i q^{83} +(0.637459 + 0.685484i) q^{85} +(-6.41238 + 3.70219i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(6.27492 - 2.98793i) q^{91} +(-4.91238 - 2.83616i) q^{93} +(-7.00000 - 2.15068i) q^{95} +6.92820i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7}+O(q^{10}) $ 4 * q + 6 * q^3 - 2 * q^5 + 3 * q^7	$ 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{11} - 3 q^{15} - 9 q^{17} - q^{19} + 12 q^{23} - 18 q^{25} - 2 q^{29} + q^{31} + 9 q^{33} - 11 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} + 38 q^{65} + 18 q^{67} + 24 q^{69} + 12 q^{71} + 15 q^{73} - 27 q^{75} + 9 q^{77} + 7 q^{79} + 18 q^{81} - 5 q^{85} - 3 q^{87} - 14 q^{89} + 10 q^{91} + 3 q^{93} - 28 q^{95}+O(q^{100}) $ 4 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 3 * q^11 - 3 * q^15 - 9 * q^17 - q^19 + 12 * q^23 - 18 * q^25 - 2 * q^29 + q^31 + 9 * q^33 - 11 * 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 + 38 * q^65 + 18 * q^67 + 24 * q^69 + 12 * q^71 + 15 * q^73 - 27 * q^75 + 9 * q^77 + 7 * q^79 + 18 * q^81 - 5 * q^85 - 3 * q^87 - 14 * q^89 + 10 * q^91 + 3 * q^93 - 28 * q^95

Display 100 coefficients 10 coefficients

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		−	1.00000i	$-0.5\pi$
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	−0.500000	−	2.17945i	−0.223607	−	0.974679i
$5$	−0.500000	−	2.17945i	−0.223607	−	0.974679i
$6$		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	$0.118684\pi$
$13$			2.62685i			0.728557i	−0.931290	+	0.364278i	$0.881316\pi$
$14$		0			0
$15$	−2.63746	−	2.83616i	−0.680989	−	0.732294i
$16$		0			0
$17$	−0.362541	+	0.209313i	−0.0879292	+	0.0507659i	−0.543320	−	0.839526i	$-0.682833\pi$
$17$	−0.362541	+	0.209313i	−0.0879292	+	0.0507659i	0.455391	+	0.890292i	$0.349500\pi$
$18$		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	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	6.77492	+	3.91150i	1.41267	+	0.815604i	0.417029	+	0.908893i	$0.363071\pi$
$24$		0			0
$25$	−4.50000	+	2.17945i	−0.900000	+	0.435890i
$26$		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
$35$	−4.63746	+	3.67341i	−0.783874	+	0.620920i
$36$		0			0
$37$	−8.63746	−	4.98684i	−1.41999	−	0.819831i	−0.423692	−	0.905806i	$-0.639266\pi$
$37$	−8.63746	−	4.98684i	−1.41999	−	0.819831i	−0.996297	+	0.0859750i	$0.972599\pi$
$38$		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	$-0.947564\pi$
$43$		−	2.15068i		−	0.327975i	0.986462	−	0.163988i	$-0.0524357\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.63746	+	3.25479i	0.822308	+	0.474760i	0.851212	−	0.524823i	$-0.175868\pi$
$47$	5.63746	+	3.25479i	0.822308	+	0.474760i	−0.0289038	+	0.999582i	$0.509202\pi$
$48$		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	−0.797880	−	0.602816i	$-0.794045\pi$
$53$	−4.91238	+	2.83616i	−0.674767	+	0.389577i	0.123114	+	0.992393i	$0.460712\pi$
$54$		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
$65$	5.72508	−	1.31342i	0.710109	−	0.162910i
$66$		0			0
$67$	−3.04983	+	1.76082i	−0.372597	+	0.215119i	−0.674592	−	0.738191i	$-0.735681\pi$
$67$	−3.04983	+	1.76082i	−0.372597	+	0.215119i	0.301996	+	0.953309i	$0.402347\pi$
$68$		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	−0.132392	−	0.991197i	$-0.542266\pi$
$73$	5.63746	−	3.25479i	0.659815	−	0.380944i	0.792206	+	0.610253i	$0.208932\pi$
$74$		0			0
$75$	−4.86254	+	7.16629i	−0.561478	+	0.827492i
$76$		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	$-0.866795\pi$
$83$		−	7.40437i		−	0.812736i	0.913710	−	0.406368i	$-0.133205\pi$
$84$		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
$95$	−7.00000	−	2.15068i	−0.718185	−	0.220655i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		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	−0.0660098	−	0.997819i	$-0.521027\pi$
$103$	−9.77492	−	5.64355i	−0.963151	−	0.556076i	−0.897141	+	0.441743i	$0.854360\pi$
$104$		0			0
$105$	−3.77492	+	9.52628i	−0.368394	+	0.929670i
$106$		0			0
$107$	3.04983	+	1.76082i	0.294839	+	0.170225i	0.640122	−	0.768273i	$-0.278884\pi$
$107$	3.04983	+	1.76082i	0.294839	+	0.170225i	−0.345283	+	0.938499i	$0.612217\pi$
$108$		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	$0.0648477\pi$
$113$			4.30136i			0.404637i	−0.979320	+	0.202319i	$0.935152\pi$
$114$		0			0
$115$	5.13746	−	16.7213i	0.479070	−	1.55927i
$116$		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	$0.244232\pi$
$127$			15.6460i			1.38836i	−0.719802	+	0.694179i	$0.755768\pi$
$128$		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
$135$	11.3248	−	2.59808i	0.974679	−	0.223607i
$136$		0			0
$137$	18.4622	−	10.6592i	1.57733	−	0.910674i	0.582103	−	0.813115i	$-0.302230\pi$
$137$	18.4622	−	10.6592i	1.57733	−	0.910674i	0.995230	−	0.0975588i	$-0.0311034\pi$
$138$		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
$145$	2.13746	+	9.31697i	0.177506	+	0.773732i
$146$		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	−0.421743	−	0.906715i	$-0.638582\pi$
$157$	1.91238	−	1.10411i	0.152624	−	0.0881176i	0.574367	+	0.818598i	$0.305248\pi$
$158$		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	0.295123	−	0.955459i	$-0.404639\pi$
$163$	−4.91238	−	2.83616i	−0.384767	−	0.222145i	−0.679890	+	0.733314i	$0.737973\pi$
$164$		0			0
$165$	6.00000	−	19.5287i	0.467099	−	1.52031i
$166$		0			0
$167$			0.476171i			0.0368472i	0.999830	+	0.0184236i	$0.00586474\pi$
$167$			0.476171i			0.0368472i	−0.999830	+	0.0184236i	$0.994135\pi$
$168$		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	−0.360488	−	0.932764i	$-0.617390\pi$
$173$	−17.7371	−	10.2405i	−1.34853	−	0.778573i	−0.988041	+	0.154190i	$0.950723\pi$
$174$		0			0
$175$	10.3248	+	8.27040i	0.780478	+	0.625183i
$176$		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
$185$	−6.54983	+	21.3183i	−0.481553	+	1.56735i
$186$		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	0.343803	−	0.939042i	$-0.388285\pi$
$193$	18.4622	−	10.6592i	1.32894	−	0.767263i	0.985136	+	0.171778i	$0.0549513\pi$
$194$		0			0
$195$	7.45017	−	6.92820i	0.533517	−	0.496139i
$196$		0			0
$197$		−	8.60271i		−	0.612918i	−0.951884	−	0.306459i	$-0.900856\pi$
$197$		−	8.60271i		−	0.612918i	0.951884	−	0.306459i	$-0.0991442\pi$
$198$		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
$205$	1.86254	+	8.11863i	0.130086	+	0.567030i
$206$		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
$215$	−4.68729	+	1.07534i	−0.319671	+	0.0733375i
$216$		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	$-0.905715\pi$
$223$		−	8.71780i		−	0.583787i	0.956451	−	0.291893i	$-0.0942853\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.9124	−	9.76436i	1.12251	−	0.648084i	0.180472	−	0.983580i	$-0.442237\pi$
$227$	16.9124	−	9.76436i	1.12251	−	0.648084i	0.942041	+	0.335496i	$0.108904\pi$
$228$		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	0.0370231	−	0.999314i	$-0.488212\pi$
$233$	−12.3625	−	7.13752i	−0.809897	−	0.467594i	−0.846920	+	0.531720i	$0.821546\pi$
$234$		0			0
$235$	4.27492	−	13.9140i	0.278865	−	0.907646i
$236$		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
$245$	14.0498	+	6.89943i	0.897611	+	0.440788i
$246$		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
$255$	1.54983	+	0.476171i	0.0970544	+	0.0298190i
$256$		0			0
$257$	−10.0876	−	5.82409i	−0.629249	−	0.363297i	0.151212	−	0.988501i	$-0.451682\pi$
$257$	−10.0876	−	5.82409i	−0.629249	−	0.363297i	−0.780461	+	0.625204i	$0.785016\pi$
$258$		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	−0.479037	−	0.877795i	$-0.659014\pi$
$263$	0.675248	−	0.389855i	0.0416376	−	0.0240395i	0.520674	+	0.853755i	$0.325681\pi$
$264$		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
$275$	−21.8248	−	14.8087i	−1.31608	−	0.893001i
$276$		0			0
$277$	−12.3625	+	7.13752i	−0.742793	+	0.428852i	−0.823084	−	0.567920i	$-0.807748\pi$
$277$	−12.3625	+	7.13752i	−0.742793	+	0.428852i	0.0802909	+	0.996771i	$0.474415\pi$
$278$		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	−0.191666	−	0.981460i	$-0.561389\pi$
$283$	9.46221	−	5.46301i	0.562470	−	0.324742i	0.754136	+	0.656718i	$0.228056\pi$
$284$		0			0
$285$	−12.3625	+	2.83616i	−0.732294	+	0.168000i
$286$		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	$0.0648660\pi$
$293$			6.92820i			0.404750i	−0.979308	+	0.202375i	$0.935134\pi$
$294$		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
$305$	−28.9622	−	8.89834i	−1.65837	−	0.509517i
$306$		0			0
$307$		−	26.5145i		−	1.51326i	−0.653843	−	0.756631i	$-0.726844\pi$
$307$		−	26.5145i		−	1.51326i	0.653843	−	0.756631i	$-0.273156\pi$
$308$		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	$0.0623438\pi$
$313$	29.0120	+	16.7501i	1.63986	+	0.946772i	0.658977	+	0.752163i	$0.270990\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	22.1873	+	12.8098i	1.24616	+	0.719472i	0.970342	−	0.241737i	$-0.0777171\pi$
$317$	22.1873	+	12.8098i	1.24616	+	0.719472i	0.275821	+	0.961209i	$0.411050\pi$
$318$		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
$325$	−5.72508	−	11.8208i	−0.317570	−	0.655701i
$326$		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	$0.0373774\pi$
$337$			4.30136i			0.234310i	−0.993114	+	0.117155i	$0.962623\pi$
$338$		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
$345$	−6.77492	−	29.5312i	−0.364749	−	1.58991i
$346$		0			0
$347$	−10.5000	+	6.06218i	−0.563670	+	0.325435i	−0.754617	−	0.656165i	$-0.772177\pi$
$347$	−10.5000	+	6.06218i	−0.563670	+	0.325435i	0.190947	+	0.981600i	$0.438844\pi$
$348$		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	−0.299379	−	0.954134i	$-0.596779\pi$
$353$	7.08762	−	4.09204i	0.377236	−	0.217797i	0.676615	+	0.736337i	$0.263446\pi$
$354$		0			0
$355$	2.27492	+	9.91613i	0.120740	+	0.526294i
$356$		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	−0.357388	−	0.933956i	$-0.616333\pi$
$367$	5.22508	−	3.01670i	0.272747	−	0.157471i	0.630135	+	0.776485i	$0.282999\pi$
$368$		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	0.259429	−	0.965762i	$-0.416466\pi$
$373$	−8.63746	−	4.98684i	−0.447231	−	0.258209i	−0.706660	+	0.707553i	$0.749799\pi$
$374$		0			0
$375$	18.0498	+	7.01452i	0.932089	+	0.362228i
$376$		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	0.629833	−	0.776730i	$-0.283123\pi$
$383$	5.32475	+	3.07425i	0.272082	+	0.157087i	−0.357751	+	0.933817i	$0.616456\pi$
$384$		0			0
$385$	−29.0120	−	11.4964i	−1.47859	−	0.585912i
$386$		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
$395$	−15.5498	−	4.77753i	−0.782397	−	0.240383i
$396$		0			0
$397$	9.36254	+	5.40547i	0.469892	+	0.271293i	0.716195	−	0.697901i	$-0.245882\pi$
$397$	9.36254	+	5.40547i	0.469892	+	0.271293i	−0.246302	+	0.969193i	$0.579216\pi$
$398$		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
$415$	−16.1375	+	3.70219i	−0.792157	+	0.181733i
$416$		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
$425$	1.17525	−	1.73205i	0.0570079	−	0.0840168i
$426$		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	$-0.856288\pi$
$433$		−	18.1578i		−	0.872606i	0.899800	−	0.436303i	$-0.143712\pi$
$434$		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	0.728247	−	0.685315i	$-0.240335\pi$
$443$	10.5000	+	6.06218i	0.498870	+	0.288023i	−0.229377	+	0.973338i	$0.573669\pi$
$444$		0			0
$445$	14.9622	+	4.59698i	0.709277	+	0.217918i
$446$		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
$455$	−9.64950	−	12.1819i	−0.452376	−	0.571096i
$456$		0			0
$457$	−1.18729	−	0.685484i	−0.0555392	−	0.0320656i	0.471973	−	0.881613i	$-0.343542\pi$
$457$	−1.18729	−	0.685484i	−0.0555392	−	0.0320656i	−0.527512	+	0.849547i	$0.676875\pi$
$458$		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	$0.0159142\pi$
$463$			2.15068i			0.0999505i	−0.998750	+	0.0499752i	$0.984086\pi$
$464$		0			0
$465$	−3.72508	+	12.1244i	−0.172747	+	0.562254i
$466$		0			0
$467$	13.5997	+	7.85177i	0.629318	+	0.363337i	0.780488	−	0.625171i	$-0.214971\pi$
$467$	13.5997	+	7.85177i	0.629318	+	0.363337i	−0.151170	+	0.988508i	$0.548304\pi$
$468$		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
$485$	15.0997	−	3.46410i	0.685641	−	0.157297i
$486$		0			0
$487$	−2.53779	+	1.46519i	−0.114998	+	0.0663943i	−0.556396	−	0.830917i	$-0.687816\pi$
$487$	−2.53779	+	1.46519i	−0.114998	+	0.0663943i	0.441398	+	0.897312i	$0.354483\pi$
$488$		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	$0.250509\pi$
$503$			31.7682i			1.41647i	−0.705975	+	0.708236i	$0.749491\pi$
$504$		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
$515$	−7.41238	+	24.1257i	−0.326628	+	1.06311i
$516$		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	0.354400	−	0.935094i	$-0.384685\pi$
$523$	−6.36254	−	3.67341i	−0.278215	−	0.160627i	−0.632615	+	0.774467i	$0.718018\pi$
$524$		0			0
$525$	22.6495	+	3.46410i	0.988505	+	0.151186i
$526$		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
$535$	2.31271	−	7.52737i	0.0999870	−	0.325437i
$536$		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	$0.144697\pi$
$547$			20.5386i			0.878168i	−0.898446	+	0.439084i	$0.855303\pi$
$548$		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
$555$	8.63746	+	37.6498i	0.366640	+	1.59815i
$556$		0			0
$557$	−8.63746	+	4.98684i	−0.365981	+	0.211299i	−0.671701	−	0.740822i	$-0.734436\pi$
$557$	−8.63746	+	4.98684i	−0.365981	+	0.211299i	0.305720	+	0.952121i	$0.401103\pi$
$558$		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	−0.852491	−	0.522743i	$-0.824909\pi$
$563$	−19.5997	+	11.3159i	−0.826028	+	0.476907i	0.0264630	+	0.999650i	$0.491576\pi$
$564$		0			0
$565$	9.37459	−	2.15068i	0.394392	−	0.0904797i
$566$		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	−0.428372	−	0.903602i	$-0.640913\pi$
$577$	3.36254	−	1.94136i	0.139984	−	0.0808200i	0.568357	+	0.822782i	$0.307579\pi$
$578$		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	$-0.858051\pi$
$587$		−	20.8997i		−	0.862623i	0.902203	−	0.431311i	$-0.141949\pi$
$588$		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	−0.229600	−	0.973285i	$-0.573742\pi$
$593$	−28.9124	−	16.6926i	−1.18729	−	0.685482i	−0.957689	+	0.287804i	$0.907075\pi$
$594$		0			0
$595$	0.912376	−	2.30245i	0.0374038	−	0.0943911i
$596$		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
$605$	35.9622	+	11.0490i	1.46207	+	0.449206i
$606$		0			0
$607$	9.87459	+	5.70109i	0.400797	+	0.231400i	0.686828	−	0.726820i	$-0.259003\pi$
$607$	9.87459	+	5.70109i	0.400797	+	0.231400i	−0.286031	+	0.958220i	$0.592336\pi$
$608$		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	−0.905885	−	0.423524i	$-0.860793\pi$
$613$	−24.5619	+	14.1808i	−0.992045	+	0.572757i	−0.0861600	+	0.996281i	$0.527460\pi$
$614$		0			0
$615$	9.82475	+	10.5649i	0.396172	+	0.426019i
$616$		0			0
$617$			31.2920i			1.25977i	0.776689	+	0.629884i	$0.216898\pi$
$617$			31.2920i			1.25977i	−0.776689	+	0.629884i	$0.783102\pi$
$618$		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
$625$	15.5000	−	19.6150i	0.620000	−	0.784602i
$626$		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
$635$	34.0997	−	7.82300i	1.35320	−	0.310446i
$636$		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	$-0.787422\pi$
$643$		−	31.4071i		−	1.23857i	0.785164	−	0.619287i	$-0.212578\pi$
$644$		0			0
$645$	−6.09967	+	5.67232i	−0.240174	+	0.223348i
$646$		0			0
$647$	−23.3248	+	13.4666i	−0.916991	+	0.529425i	−0.882674	−	0.469986i	$-0.844259\pi$
$647$	−23.3248	+	13.4666i	−0.916991	+	0.529425i	−0.0343169	+	0.999411i	$0.510926\pi$
$648$		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	−0.0646444	−	0.997908i	$-0.520591\pi$
$653$	−24.5619	−	14.1808i	−0.961181	−	0.554938i	−0.896536	+	0.442970i	$0.853925\pi$
$654$		0			0
$655$	−22.9244	−	7.04329i	−0.895731	−	0.275204i
$656$		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
$665$	2.82475	+	19.1676i	0.109539	+	0.743289i
$666$		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	$-0.793928\pi$
$673$		−	31.2920i		−	1.20622i	0.797659	−	0.603109i	$-0.206072\pi$
$674$		0			0
$675$	−11.3248	−	23.3827i	−0.435890	−	0.900000i
$676$		0			0
$677$	40.1873	+	23.2021i	1.54452	+	0.891731i	0.998545	+	0.0539317i	$0.0171753\pi$
$677$	40.1873	+	23.2021i	1.54452	+	0.891731i	0.545979	+	0.837799i	$0.316158\pi$
$678$		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	−0.147582	−	0.989050i	$-0.547149\pi$
$683$	16.5997	−	9.58382i	0.635169	−	0.366715i	0.782751	+	0.622335i	$0.213816\pi$
$684$		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
$695$	6.54983	+	28.5501i	0.248449	+	1.08297i
$696$		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
$705$	−5.63746	−	24.5731i	−0.212319	−	0.925477i
$706$		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
$725$	19.2371	−	9.31697i	0.714449	−	0.346023i
$726$		0			0
$727$		−	3.10302i		−	0.115085i	−0.998343	−	0.0575423i	$-0.981674\pi$
$727$		−	3.10302i		−	0.115085i	0.998343	−	0.0575423i	$-0.0183264\pi$
$728$		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	−0.243721	−	0.969845i	$-0.578368\pi$
$733$	−32.6375	−	18.8432i	−1.20549	−	0.695991i	−0.961771	+	0.273854i	$0.911701\pi$
$734$		0			0
$735$	27.0498	−	1.81837i	0.997748	−	0.0670715i
$736$		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	$-0.962239\pi$
$743$		−	6.45203i		−	0.236702i	0.992972	−	0.118351i	$-0.0377608\pi$
$744$		0			0
$745$	16.1375	+	4.95807i	0.591231	+	0.181650i
$746$		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	$-0.776094\pi$
$757$		−	35.5934i		−	1.29366i	0.762633	−	0.646831i	$-0.223906\pi$
$758$		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	−0.972238	−	0.233995i	$-0.924820\pi$
$773$	−34.9124	+	20.1567i	−1.25571	+	0.724985i	−0.283473	+	0.958980i	$0.591487\pi$
$774$		0			0
$775$	13.5498	+	9.19397i	0.486724	+	0.330257i
$776$		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	−0.473027	−	0.881048i	$-0.656839\pi$
$787$	1.50000	−	0.866025i	0.0534692	−	0.0308705i	0.526496	+	0.850177i	$0.323505\pi$
$788$		0			0
$789$	0.675248	−	1.16956i	0.0240395	−	0.0416376i
$790$		0			0
$791$	10.2749	−	4.89261i	0.365334

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)

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 2.17945i) q^{5} +(-1.13746 - 2.38876i) q^{7} +O(q^{10})\)	\(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 2.17945i) q^{5} +(-1.13746 - 2.38876i) q^{7} +(2.63746 + 4.56821i) q^{11} +2.62685i q^{13} +(-2.63746 - 2.83616i) q^{15} +(-0.362541 + 0.209313i) q^{17} +(1.63746 - 2.83616i) q^{19} +(-3.77492 - 2.59808i) q^{21} +(6.77492 + 3.91150i) q^{23} +(-4.50000 + 2.17945i) q^{25} +5.19615i q^{27} -4.27492 q^{29} +(-1.63746 - 2.83616i) q^{31} +(7.91238 + 4.56821i) q^{33} +(-4.63746 + 3.67341i) q^{35} +(-8.63746 - 4.98684i) q^{37} +(2.27492 + 3.94027i) q^{39} -3.72508 q^{41} -2.15068i q^{43} +(5.63746 + 3.25479i) q^{47} +(-4.41238 + 5.43424i) q^{49} +(-0.362541 + 0.627940i) q^{51} +(-4.91238 + 2.83616i) q^{53} +(8.63746 - 8.03231i) q^{55} -5.67232i q^{57} +(-1.63746 - 2.83616i) q^{59} +(6.77492 - 11.7345i) q^{61} +(5.72508 - 1.31342i) q^{65} +(-3.04983 + 1.76082i) q^{67} +13.5498 q^{69} -4.54983 q^{71} +(5.63746 - 3.25479i) q^{73} +(-4.86254 + 7.16629i) q^{75} +(7.91238 - 11.4964i) q^{77} +(3.63746 - 6.30026i) q^{79} +(4.50000 + 7.79423i) q^{81} -7.40437i q^{83} +(0.637459 + 0.685484i) q^{85} +(-6.41238 + 3.70219i) q^{87} +(-3.50000 + 6.06218i) q^{89} +(6.27492 - 2.98793i) q^{91} +(-4.91238 - 2.83616i) q^{93} +(-7.00000 - 2.15068i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7}+O(q^{10}) \) 4 * q + 6 * q^3 - 2 * q^5 + 3 * q^7	\( 4 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{11} - 3 q^{15} - 9 q^{17} - q^{19} + 12 q^{23} - 18 q^{25} - 2 q^{29} + q^{31} + 9 q^{33} - 11 q^{35} - 27 q^{37} - 6 q^{39} - 30 q^{41} + 15 q^{47} + 5 q^{49} - 9 q^{51} + 3 q^{53} + 27 q^{55} + q^{59} + 12 q^{61} + 38 q^{65} + 18 q^{67} + 24 q^{69} + 12 q^{71} + 15 q^{73} - 27 q^{75} + 9 q^{77} + 7 q^{79} + 18 q^{81} - 5 q^{85} - 3 q^{87} - 14 q^{89} + 10 q^{91} + 3 q^{93} - 28 q^{95}+O(q^{100}) \) 4 * q + 6 * q^3 - 2 * q^5 + 3 * q^7 + 3 * q^11 - 3 * q^15 - 9 * q^17 - q^19 + 12 * q^23 - 18 * q^25 - 2 * q^29 + q^31 + 9 * q^33 - 11 * 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 + 38 * q^65 + 18 * q^67 + 24 * q^69 + 12 * q^71 + 15 * q^73 - 27 * q^75 + 9 * q^77 + 7 * q^79 + 18 * q^81 - 5 * q^85 - 3 * q^87 - 14 * q^89 + 10 * q^91 + 3 * q^93 - 28 * q^95

Introduction

L-functions

Modular forms

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