LMFDB - Embedded newform 405.2.e.b.136.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(136,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.136"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-2,-1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.2.e.b.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +2.00000 q^{10} +(-1.00000 + 1.73205i) q^{11} +(2.50000 + 4.33013i) q^{13} +(3.00000 + 5.19615i) q^{14} +(2.00000 - 3.46410i) q^{16} +8.00000 q^{17} +1.00000 q^{19} +(-1.00000 + 1.73205i) q^{20} +(-2.00000 - 3.46410i) q^{22} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} -10.0000 q^{26} -6.00000 q^{28} +(1.00000 - 1.73205i) q^{29} +(4.00000 + 6.92820i) q^{32} +(-8.00000 + 13.8564i) q^{34} -3.00000 q^{35} +5.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(-5.00000 - 8.66025i) q^{41} +(-2.00000 + 3.46410i) q^{43} +4.00000 q^{44} -12.0000 q^{46} +(2.00000 - 3.46410i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(-1.00000 - 1.73205i) q^{50} +(5.00000 - 8.66025i) q^{52} +2.00000 q^{53} +2.00000 q^{55} +(2.00000 + 3.46410i) q^{58} +(-4.00000 - 6.92820i) q^{59} +(-3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +(2.50000 - 4.33013i) q^{65} +(4.50000 + 7.79423i) q^{67} +(-8.00000 - 13.8564i) q^{68} +(3.00000 - 5.19615i) q^{70} -2.00000 q^{71} -5.00000 q^{73} +(-5.00000 + 8.66025i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.00000 + 5.19615i) q^{77} +(1.50000 - 2.59808i) q^{79} -4.00000 q^{80} +20.0000 q^{82} +(3.00000 - 5.19615i) q^{83} +(-4.00000 - 6.92820i) q^{85} +(-4.00000 - 6.92820i) q^{86} +12.0000 q^{89} +15.0000 q^{91} +(6.00000 - 10.3923i) q^{92} +(4.00000 + 6.92820i) q^{94} +(-0.500000 - 0.866025i) q^{95} +(6.50000 - 11.2583i) q^{97} +4.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} - q^{5} + 3 q^{7} + 4 q^{10} - 2 q^{11} + 5 q^{13} + 6 q^{14} + 4 q^{16} + 16 q^{17} + 2 q^{19} - 2 q^{20} - 4 q^{22} + 6 q^{23} - q^{25} - 20 q^{26} - 12 q^{28} + 2 q^{29} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 - q^5 + 3 * q^7 + 4 * q^10 - 2 * q^11 + 5 * q^13 + 6 * q^14 + 4 * q^16 + 16 * q^17 + 2 * q^19 - 2 * q^20 - 4 * q^22 + 6 * q^23 - q^25 - 20 * q^26 - 12 * q^28 + 2 * q^29 + 8 * q^32 - 16 * q^34 - 6 * q^35 + 10 * q^37 - 2 * q^38 - 10 * q^41 - 4 * q^43 + 8 * q^44 - 24 * q^46 + 4 * q^47 - 2 * q^49 - 2 * q^50 + 10 * q^52 + 4 * q^53 + 4 * q^55 + 4 * q^58 - 8 * q^59 - 7 * q^61 - 16 * q^64 + 5 * q^65 + 9 * q^67 - 16 * q^68 + 6 * q^70 - 4 * q^71 - 10 * q^73 - 10 * q^74 - 2 * q^76 + 6 * q^77 + 3 * q^79 - 8 * q^80 + 40 * q^82 + 6 * q^83 - 8 * q^85 - 8 * q^86 + 24 * q^89 + 30 * q^91 + 12 * q^92 + 8 * q^94 - q^95 + 13 * q^97 + 8 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.00000	+	1.73205i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$2$	−1.00000	+	1.73205i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	−0.429919	−	0.902867i	$-0.641458\pi$
$7$	1.50000	−	2.59808i	0.566947	−	0.981981i	0.996866	−	0.0791130i	$-0.0252088\pi$
$8$		0			0
$9$		0			0
$10$	2.00000			0.632456
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.00000	+	5.19615i	0.801784	+	1.38873i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	8.00000			1.94029			0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000			1.94029			0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$	1.00000			0.229416			0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$	−1.00000	+	1.73205i	−0.223607	+	0.387298i
$21$		0			0
$22$	−2.00000	−	3.46410i	−0.426401	−	0.738549i
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	−10.0000			−1.96116
$27$		0			0
$28$	−6.00000			−1.13389
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	4.00000	+	6.92820i	0.707107	+	1.22474i
$33$		0			0
$34$	−8.00000	+	13.8564i	−1.37199	+	2.37635i
$35$	−3.00000			−0.507093
$36$		0			0
$37$	5.00000			0.821995			0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000			0.821995			0.410997	+	0.911636i	$0.365181\pi$
$38$	−1.00000	+	1.73205i	−0.162221	+	0.280976i
$39$		0			0
$40$		0			0
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	−0.931436	−	0.363905i	$-0.881443\pi$
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	0.150567	−	0.988600i	$-0.451890\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$	4.00000			0.603023
$45$		0			0
$46$	−12.0000			−1.76930
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	−0.682489	−	0.730896i	$-0.739102\pi$
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	0.974219	+	0.225605i	$0.0724358\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$	−1.00000	−	1.73205i	−0.141421	−	0.244949i
$51$		0			0
$52$	5.00000	−	8.66025i	0.693375	−	1.20096i
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$		0			0
$58$	2.00000	+	3.46410i	0.262613	+	0.454859i
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.50000	−	4.33013i	0.310087	−	0.537086i
$66$		0			0
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	0.998290	+	0.0584478i	$0.0186151\pi$
$67$	4.50000	+	7.79423i	0.549762	+	0.952217i	−0.448528	+	0.893769i	$0.648052\pi$
$68$	−8.00000	−	13.8564i	−0.970143	−	1.68034i
$69$		0			0
$70$	3.00000	−	5.19615i	0.358569	−	0.621059i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−5.00000			−0.585206			−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000			−0.585206			−0.292603	+	0.956234i	$0.594521\pi$
$74$	−5.00000	+	8.66025i	−0.581238	+	1.00673i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	3.00000	+	5.19615i	0.341882	+	0.592157i
$78$		0			0
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	−0.769222	−	0.638982i	$-0.779356\pi$
$79$	1.50000	−	2.59808i	0.168763	−	0.292306i	0.937985	+	0.346675i	$0.112689\pi$
$80$	−4.00000			−0.447214
$81$		0			0
$82$	20.0000			2.20863
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$	−4.00000	−	6.92820i	−0.433861	−	0.751469i
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$		0			0
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		0			0
$91$	15.0000			1.57243
$92$	6.00000	−	10.3923i	0.625543	−	1.08347i
$93$		0			0
$94$	4.00000	+	6.92820i	0.412568	+	0.714590i
$95$	−0.500000	−	0.866025i	−0.0512989	−	0.0888523i
$96$		0			0
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	−0.320647	−	0.947199i	$-0.603900\pi$
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	0.980622	−	0.195911i	$-0.0627665\pi$
$98$	4.00000			0.404061
$99$		0			0
$100$	2.00000			0.200000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−8.50000	−	14.7224i	−0.837530	−	1.45064i	−0.891954	−	0.452126i	$-0.850666\pi$
$103$	−8.50000	−	14.7224i	−0.837530	−	1.45064i	0.0544240	−	0.998518i	$-0.482668\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000	+	3.46410i	−0.194257	+	0.336463i
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	−2.00000	+	3.46410i	−0.190693	+	0.330289i
$111$		0			0
$112$	−6.00000	−	10.3923i	−0.566947	−	0.981981i
$113$	5.00000	+	8.66025i	0.470360	+	0.814688i	0.999425	−	0.0338931i	$-0.0107906\pi$
$113$	5.00000	+	8.66025i	0.470360	+	0.814688i	−0.529065	+	0.848581i	$0.677457\pi$
$114$		0			0
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$	−4.00000			−0.371391
$117$		0			0
$118$	16.0000			1.47292
$119$	12.0000	−	20.7846i	1.10004	−	1.90532i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	−7.00000	−	12.1244i	−0.633750	−	1.09769i
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$	5.00000	+	8.66025i	0.438529	+	0.759555i
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	1.50000	−	2.59808i	0.130066	−	0.225282i
$134$	−18.0000			−1.55496
$135$		0			0
$136$		0			0
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	−0.965250	−	0.261329i	$-0.915839\pi$
$137$	−3.00000	+	5.19615i	−0.256307	+	0.443937i	0.708942	+	0.705266i	$0.249173\pi$
$138$		0			0
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	0.998179	+	0.0603135i	$0.0192101\pi$
$139$	6.50000	+	11.2583i	0.551323	+	0.954919i	−0.446857	+	0.894606i	$0.647457\pi$
$140$	3.00000	+	5.19615i	0.253546	+	0.439155i
$141$		0			0
$142$	2.00000	−	3.46410i	0.167836	−	0.290701i
$143$	−10.0000			−0.836242
$144$		0			0
$145$	−2.00000			−0.166091
$146$	5.00000	−	8.66025i	0.413803	−	0.716728i
$147$		0			0
$148$	−5.00000	−	8.66025i	−0.410997	−	0.711868i
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	0.772399	−	0.635138i	$-0.219057\pi$
$149$	−2.00000	−	3.46410i	−0.163846	−	0.283790i	−0.936245	+	0.351348i	$0.885723\pi$
$150$		0			0
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	−0.885653	−	0.464348i	$-0.846289\pi$
$151$	−0.500000	+	0.866025i	−0.0406894	+	0.0704761i	0.844963	+	0.534824i	$0.179622\pi$
$152$		0			0
$153$		0			0
$154$	−12.0000			−0.966988
$155$		0			0
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$	3.00000	+	5.19615i	0.238667	+	0.413384i
$159$		0			0
$160$	4.00000	−	6.92820i	0.316228	−	0.547723i
$161$	18.0000			1.41860
$162$		0			0
$163$	−19.0000			−1.48819			−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000			−1.48819			−0.744097	+	0.668071i	$0.767120\pi$
$164$	−10.0000	+	17.3205i	−0.780869	+	1.35250i
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	0.534875	−	0.844931i	$-0.320359\pi$
$167$	−6.00000	−	10.3923i	−0.464294	−	0.804181i	−0.999169	+	0.0407502i	$0.987025\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$	16.0000			1.22714
$171$		0			0
$172$	8.00000			0.609994
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	−0.998755	−	0.0498898i	$-0.984113\pi$
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	0.542583	+	0.840002i	$0.317446\pi$
$174$		0			0
$175$	1.50000	+	2.59808i	0.113389	+	0.196396i
$176$	4.00000	+	6.92820i	0.301511	+	0.522233i
$177$		0			0
$178$	−12.0000	+	20.7846i	−0.899438	+	1.55787i
$179$	−22.0000			−1.64436			−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000			−1.64436			−0.822179	+	0.569230i	$0.807242\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$	−15.0000	+	25.9808i	−1.11187	+	1.92582i
$183$		0			0
$184$		0			0
$185$	−2.50000	−	4.33013i	−0.183804	−	0.318357i
$186$		0			0
$187$	−8.00000	+	13.8564i	−0.585018	+	1.01328i
$188$	−8.00000			−0.583460
$189$		0			0
$190$	2.00000			0.145095
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$	13.0000	+	22.5167i	0.933346	+	1.61660i
$195$		0			0
$196$	−2.00000	+	3.46410i	−0.142857	+	0.247436i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−17.0000			−1.20510			−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000			−1.20510			−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.00000	−	5.19615i	−0.210559	−	0.364698i
$204$		0			0
$205$	−5.00000	+	8.66025i	−0.349215	+	0.604858i
$206$	34.0000			2.36889
$207$		0			0
$208$	20.0000			1.38675
$209$	−1.00000	+	1.73205i	−0.0691714	+	0.119808i
$210$		0			0
$211$	−11.5000	−	19.9186i	−0.791693	−	1.37125i	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−11.5000	−	19.9186i	−0.791693	−	1.37125i	0.133226	−	0.991086i	$-0.457467\pi$
$212$	−2.00000	−	3.46410i	−0.137361	−	0.237915i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$	4.00000			0.272798
$216$		0			0
$217$		0			0
$218$	10.0000	−	17.3205i	0.677285	−	1.17309i
$219$		0			0
$220$	−2.00000	−	3.46410i	−0.134840	−	0.233550i
$221$	20.0000	+	34.6410i	1.34535	+	2.33021i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$	24.0000			1.60357
$225$		0			0
$226$	−20.0000			−1.33038
$227$	−5.00000	+	8.66025i	−0.331862	+	0.574801i	−0.982877	−	0.184263i	$-0.941010\pi$
$227$	−5.00000	+	8.66025i	−0.331862	+	0.574801i	0.651015	+	0.759065i	$0.274343\pi$
$228$		0			0
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	0.749714	−	0.661762i	$-0.230191\pi$
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	−0.947960	+	0.318390i	$0.896858\pi$
$230$	6.00000	+	10.3923i	0.395628	+	0.685248i
$231$		0			0
$232$		0			0
$233$	24.0000			1.57229			0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000			1.57229			0.786146	+	0.618041i	$0.212073\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$	−8.00000	+	13.8564i	−0.520756	+	0.901975i
$237$		0			0
$238$	24.0000	+	41.5692i	1.55569	+	2.69453i
$239$	−13.0000	−	22.5167i	−0.840900	−	1.45648i	−0.889135	−	0.457646i	$-0.848693\pi$
$239$	−13.0000	−	22.5167i	−0.840900	−	1.45648i	0.0482346	−	0.998836i	$-0.484640\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$	−14.0000			−0.899954
$243$		0			0
$244$	14.0000			0.896258
$245$	−1.00000	+	1.73205i	−0.0638877	+	0.110657i
$246$		0			0
$247$	2.50000	+	4.33013i	0.159071	+	0.275519i
$248$		0			0
$249$		0			0
$250$	−1.00000	+	1.73205i	−0.0632456	+	0.109545i
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$	8.00000	−	13.8564i	0.501965	−	0.869428i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	7.50000	−	12.9904i	0.466027	−	0.807183i
$260$	−10.0000			−0.620174
$261$		0			0
$262$	−24.0000			−1.48272
$263$	−14.0000	+	24.2487i	−0.863277	+	1.49524i	0.00547092	+	0.999985i	$0.498259\pi$
$263$	−14.0000	+	24.2487i	−0.863277	+	1.49524i	−0.868748	+	0.495255i	$0.835075\pi$
$264$		0			0
$265$	−1.00000	−	1.73205i	−0.0614295	−	0.106399i
$266$	3.00000	+	5.19615i	0.183942	+	0.318597i
$267$		0			0
$268$	9.00000	−	15.5885i	0.549762	−	0.952217i
$269$	−16.0000			−0.975537			−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000			−0.975537			−0.487769	+	0.872973i	$0.662189\pi$
$270$		0			0
$271$	13.0000			0.789694			0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000			0.789694			0.394847	+	0.918747i	$0.370798\pi$
$272$	16.0000	−	27.7128i	0.970143	−	1.68034i
$273$		0			0
$274$	−6.00000	−	10.3923i	−0.362473	−	0.627822i
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	15.0000	−	25.9808i	0.901263	−	1.56103i	0.0754058	−	0.997153i	$-0.475975\pi$
$277$	15.0000	−	25.9808i	0.901263	−	1.56103i	0.825857	−	0.563880i	$-0.190692\pi$
$278$	−26.0000			−1.55938
$279$		0			0
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	0.630738	−	0.775996i	$-0.282752\pi$
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	−0.987401	+	0.158237i	$0.949419\pi$
$284$	2.00000	+	3.46410i	0.118678	+	0.205557i
$285$		0			0
$286$	10.0000	−	17.3205i	0.591312	−	1.02418i
$287$	−30.0000			−1.77084
$288$		0			0
$289$	47.0000			2.76471
$290$	2.00000	−	3.46410i	0.117444	−	0.203419i
$291$		0			0
$292$	5.00000	+	8.66025i	0.292603	+	0.506803i
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	0.999549	+	0.0300351i	$0.00956192\pi$
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	−0.473763	+	0.880652i	$0.657105\pi$
$294$		0			0
$295$	−4.00000	+	6.92820i	−0.232889	+	0.403376i
$296$		0			0
$297$		0			0
$298$	8.00000			0.463428
$299$	−15.0000	+	25.9808i	−0.867472	+	1.50251i
$300$		0			0
$301$	6.00000	+	10.3923i	0.345834	+	0.599002i
$302$	−1.00000	−	1.73205i	−0.0575435	−	0.0996683i
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	7.00000			0.400819
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$	6.00000	−	10.3923i	0.341882	−	0.592157i
$309$		0			0
$310$		0			0
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	−0.974841	−	0.222900i	$-0.928448\pi$
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	0.294384	−	0.955687i	$-0.404886\pi$
$312$		0			0
$313$	3.50000	−	6.06218i	0.197832	−	0.342655i	−0.749993	−	0.661445i	$-0.769943\pi$
$313$	3.50000	−	6.06218i	0.197832	−	0.342655i	0.947825	+	0.318791i	$0.103277\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	−6.00000			−0.337526
$317$	10.0000	−	17.3205i	0.561656	−	0.972817i	−0.435696	−	0.900094i	$-0.643498\pi$
$317$	10.0000	−	17.3205i	0.561656	−	0.972817i	0.997352	−	0.0727229i	$-0.0231689\pi$
$318$		0			0
$319$	2.00000	+	3.46410i	0.111979	+	0.193952i
$320$	4.00000	+	6.92820i	0.223607	+	0.387298i
$321$		0			0
$322$	−18.0000	+	31.1769i	−1.00310	+	1.73742i
$323$	8.00000			0.445132
$324$		0			0
$325$	−5.00000			−0.277350
$326$	19.0000	−	32.9090i	1.05231	−	1.82266i
$327$		0			0
$328$		0			0
$329$	−6.00000	−	10.3923i	−0.330791	−	0.572946i
$330$		0			0
$331$	10.5000	−	18.1865i	0.577132	−	0.999622i	−0.418674	−	0.908137i	$-0.637505\pi$
$331$	10.5000	−	18.1865i	0.577132	−	0.999622i	0.995806	−	0.0914858i	$-0.0291616\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$	24.0000			1.31322
$335$	4.50000	−	7.79423i	0.245861	−	0.425844i
$336$		0			0
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	0.754811	−	0.655942i	$-0.227729\pi$
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	−0.945468	+	0.325714i	$0.894395\pi$
$338$	−12.0000	−	20.7846i	−0.652714	−	1.13053i
$339$		0			0
$340$	−8.00000	+	13.8564i	−0.433861	+	0.751469i
$341$		0			0
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$	−12.0000	−	20.7846i	−0.645124	−	1.11739i
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	0.968452	−	0.249198i	$-0.0801671\pi$
$347$	5.00000	+	8.66025i	0.268414	+	0.464907i	−0.700038	+	0.714105i	$0.746834\pi$
$348$		0			0
$349$	−9.50000	+	16.4545i	−0.508523	+	0.880788i	0.491428	+	0.870918i	$0.336475\pi$
$349$	−9.50000	+	16.4545i	−0.508523	+	0.880788i	−0.999951	+	0.00987003i	$0.996858\pi$
$350$	−6.00000			−0.320713
$351$		0			0
$352$	−16.0000			−0.852803
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	−0.980352	−	0.197256i	$-0.936797\pi$
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	0.661004	+	0.750382i	$0.270130\pi$
$354$		0			0
$355$	1.00000	+	1.73205i	0.0530745	+	0.0919277i
$356$	−12.0000	−	20.7846i	−0.635999	−	1.10158i
$357$		0			0
$358$	22.0000	−	38.1051i	1.16274	−	2.01392i
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$	−5.00000	+	8.66025i	−0.262794	+	0.455173i
$363$		0			0
$364$	−15.0000	−	25.9808i	−0.786214	−	1.36176i
$365$	2.50000	+	4.33013i	0.130856	+	0.226649i
$366$		0			0
$367$	−10.5000	+	18.1865i	−0.548096	+	0.949329i	0.450310	+	0.892873i	$0.351314\pi$
$367$	−10.5000	+	18.1865i	−0.548096	+	0.949329i	−0.998405	+	0.0564568i	$0.982020\pi$
$368$	24.0000			1.25109
$369$		0			0
$370$	10.0000			0.519875
$371$	3.00000	−	5.19615i	0.155752	−	0.269771i
$372$		0			0
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$	−16.0000	−	27.7128i	−0.827340	−	1.43300i
$375$		0			0
$376$		0			0
$377$	10.0000			0.515026
$378$		0			0
$379$	−11.0000			−0.565032			−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000			−0.565032			−0.282516	+	0.959263i	$0.591169\pi$
$380$	−1.00000	+	1.73205i	−0.0512989	+	0.0888523i
$381$		0			0
$382$	−4.00000	−	6.92820i	−0.204658	−	0.354478i
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	−0.990702	−	0.136047i	$-0.956560\pi$
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	0.377531	−	0.925997i	$-0.376773\pi$
$384$		0			0
$385$	3.00000	−	5.19615i	0.152894	−	0.264820i
$386$	10.0000			0.508987
$387$		0			0
$388$	−26.0000			−1.31995
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	−0.383076	−	0.923717i	$-0.625135\pi$
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	0.991500	−	0.130105i	$-0.0415314\pi$
$390$		0			0
$391$	24.0000	+	41.5692i	1.21373	+	2.10225i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.00000			−0.150946
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$	17.0000	−	29.4449i	0.852133	−	1.47594i
$399$		0			0
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	12.0000			0.595550
$407$	−5.00000	+	8.66025i	−0.247841	+	0.429273i
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$	−10.0000	−	17.3205i	−0.493865	−	0.855399i
$411$		0			0
$412$	−17.0000	+	29.4449i	−0.837530	+	1.45064i
$413$	−24.0000			−1.18096
$414$		0			0
$415$	−6.00000			−0.294528
$416$	−20.0000	+	34.6410i	−0.980581	+	1.69842i
$417$		0			0
$418$	−2.00000	−	3.46410i	−0.0978232	−	0.169435i
$419$	16.0000	+	27.7128i	0.781651	+	1.35386i	0.930979	+	0.365072i	$0.118956\pi$
$419$	16.0000	+	27.7128i	0.781651	+	1.35386i	−0.149328	+	0.988788i	$0.547711\pi$
$420$		0			0
$421$	−0.500000	+	0.866025i	−0.0243685	+	0.0422075i	−0.877952	−	0.478748i	$-0.841091\pi$
$421$	−0.500000	+	0.866025i	−0.0243685	+	0.0422075i	0.853584	+	0.520955i	$0.174424\pi$
$422$	46.0000			2.23924
$423$		0			0
$424$		0			0
$425$	−4.00000	+	6.92820i	−0.194029	+	0.336067i
$426$		0			0
$427$	10.5000	+	18.1865i	0.508131	+	0.880108i
$428$	6.00000	+	10.3923i	0.290021	+	0.502331i
$429$		0			0
$430$	−4.00000	+	6.92820i	−0.192897	+	0.334108i
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$		0			0
$433$	−38.0000			−1.82616			−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000			−1.82616			−0.913082	+	0.407777i	$0.866304\pi$
$434$		0			0
$435$		0			0
$436$	10.0000	+	17.3205i	0.478913	+	0.829502i
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	−0.177325	−	0.984152i	$-0.556744\pi$
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	0.940963	−	0.338508i	$-0.109922\pi$
$440$		0			0
$441$		0			0
$442$	−80.0000			−3.80521
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	−6.00000	−	10.3923i	−0.284427	−	0.492642i
$446$	−8.00000	−	13.8564i	−0.378811	−	0.656120i
$447$		0			0
$448$	−12.0000	+	20.7846i	−0.566947	+	0.981981i
$449$	−4.00000			−0.188772			−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000			−0.188772			−0.0943858	+	0.995536i	$0.530089\pi$
$450$		0			0
$451$	20.0000			0.941763
$452$	10.0000	−	17.3205i	0.470360	−	0.814688i
$453$		0			0
$454$	−10.0000	−	17.3205i	−0.469323	−	0.812892i
$455$	−7.50000	−	12.9904i	−0.351605	−	0.608998i
$456$		0			0
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	0.485299	+	0.874348i	$0.338711\pi$
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	−0.999857	+	0.0168929i	$0.994623\pi$
$458$	12.0000			0.560723
$459$		0			0
$460$	−12.0000			−0.559503
$461$	−6.00000	+	10.3923i	−0.279448	+	0.484018i	−0.971248	−	0.238071i	$-0.923485\pi$
$461$	−6.00000	+	10.3923i	−0.279448	+	0.484018i	0.691800	+	0.722089i	$0.256818\pi$
$462$		0			0
$463$	−16.5000	−	28.5788i	−0.766820	−	1.32817i	−0.939279	−	0.343155i	$-0.888505\pi$
$463$	−16.5000	−	28.5788i	−0.766820	−	1.32817i	0.172459	−	0.985017i	$-0.444829\pi$
$464$	−4.00000	−	6.92820i	−0.185695	−	0.321634i
$465$		0			0
$466$	−24.0000	+	41.5692i	−1.11178	+	1.92566i
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	27.0000			1.24674
$470$	4.00000	−	6.92820i	0.184506	−	0.319574i
$471$		0			0
$472$		0			0
$473$	−4.00000	−	6.92820i	−0.183920	−	0.318559i
$474$		0			0
$475$	−0.500000	+	0.866025i	−0.0229416	+	0.0397360i
$476$	−48.0000			−2.20008
$477$		0			0
$478$	52.0000			2.37842
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	−0.789314	−	0.613990i	$-0.789564\pi$
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	0.926388	+	0.376571i	$0.122897\pi$
$480$		0			0
$481$	12.5000	+	21.6506i	0.569951	+	0.987184i
$482$	−1.00000	−	1.73205i	−0.0455488	−	0.0788928i
$483$		0			0
$484$	7.00000	−	12.1244i	0.318182	−	0.551107i
$485$	−13.0000			−0.590300
$486$		0			0
$487$	5.00000			0.226572			0.113286	−	0.993562i	$-0.463862\pi$
$487$	5.00000			0.226572			0.113286	+	0.993562i	$0.463862\pi$
$488$		0			0
$489$		0			0
$490$	−2.00000	−	3.46410i	−0.0903508	−	0.156492i
$491$	−11.0000	−	19.0526i	−0.496423	−	0.859830i	0.503568	−	0.863955i	$-0.332020\pi$
$491$	−11.0000	−	19.0526i	−0.496423	−	0.859830i	−0.999991	+	0.00412539i	$0.998687\pi$
$492$		0			0
$493$	8.00000	−	13.8564i	0.360302	−	0.624061i
$494$	−10.0000			−0.449921
$495$		0			0
$496$		0			0
$497$	−3.00000	+	5.19615i	−0.134568	+	0.233079i
$498$		0			0
$499$	−8.00000	−	13.8564i	−0.358129	−	0.620298i	0.629519	−	0.776985i	$-0.283252\pi$
$499$	−8.00000	−	13.8564i	−0.358129	−	0.620298i	−0.987648	+	0.156687i	$0.949919\pi$
$500$	−1.00000	−	1.73205i	−0.0447214	−	0.0774597i
$501$		0			0
$502$	6.00000	−	10.3923i	0.267793	−	0.463831i
$503$	−26.0000			−1.15928			−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000			−1.15928			−0.579641	+	0.814872i	$0.696807\pi$
$504$		0			0
$505$		0			0
$506$	12.0000	−	20.7846i	0.533465	−	0.923989i
$507$		0			0
$508$	8.00000	+	13.8564i	0.354943	+	0.614779i
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	0.946111	+	0.323843i	$0.104975\pi$
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	−0.192599	+	0.981278i	$0.561692\pi$
$510$		0			0
$511$	−7.50000	+	12.9904i	−0.331780	+	0.574661i
$512$	32.0000			1.41421
$513$		0			0
$514$		0			0
$515$	−8.50000	+	14.7224i	−0.374555	+	0.648748i
$516$		0			0
$517$	4.00000	+	6.92820i	0.175920	+	0.304702i
$518$	15.0000	+	25.9808i	0.659062	+	1.14153i
$519$		0			0
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$	19.0000			0.830812			0.415406	−	0.909636i	$-0.363640\pi$
$523$	19.0000			0.830812			0.415406	+	0.909636i	$0.363640\pi$
$524$	12.0000	−	20.7846i	0.524222	−	0.907980i
$525$		0			0
$526$	−28.0000	−	48.4974i	−1.22086	−	2.11459i
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	4.00000			0.173749
$531$		0			0
$532$	−6.00000			−0.260133
$533$	25.0000	−	43.3013i	1.08287	−	1.87559i
$534$		0			0
$535$	3.00000	+	5.19615i	0.129701	+	0.224649i
$536$		0			0
$537$		0			0
$538$	16.0000	−	27.7128i	0.689809	−	1.19478i
$539$	4.00000			0.172292
$540$		0			0
$541$	33.0000			1.41878			0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000			1.41878			0.709390	+	0.704816i	$0.248970\pi$
$542$	−13.0000	+	22.5167i	−0.558398	+	0.967173i
$543$		0			0
$544$	32.0000	+	55.4256i	1.37199	+	2.37635i
$545$	5.00000	+	8.66025i	0.214176	+	0.370965i
$546$		0			0
$547$	−18.5000	+	32.0429i	−0.791003	+	1.37006i	0.134344	+	0.990935i	$0.457107\pi$
$547$	−18.5000	+	32.0429i	−0.791003	+	1.37006i	−0.925347	+	0.379122i	$0.876226\pi$
$548$	12.0000			0.512615
$549$		0			0
$550$	4.00000			0.170561
$551$	1.00000	−	1.73205i	0.0426014	−	0.0737878i
$552$		0			0
$553$	−4.50000	−	7.79423i	−0.191359	−	0.331444i
$554$	30.0000	+	51.9615i	1.27458	+	2.20763i
$555$		0			0
$556$	13.0000	−	22.5167i	0.551323	−	0.954919i
$557$	−12.0000			−0.508456			−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000			−0.508456			−0.254228	+	0.967144i	$0.581821\pi$
$558$		0			0
$559$	−20.0000			−0.845910
$560$	−6.00000	+	10.3923i	−0.253546	+	0.439155i
$561$		0			0
$562$		0			0
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	0.943560	+	0.331202i	$0.107454\pi$
$563$	18.0000	+	31.1769i	0.758610	+	1.31395i	−0.184950	+	0.982748i	$0.559212\pi$
$564$		0			0
$565$	5.00000	−	8.66025i	0.210352	−	0.364340i
$566$	24.0000			1.00880
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	0.808962	−	0.587861i	$-0.200030\pi$
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	−0.913584	+	0.406651i	$0.866697\pi$
$572$	10.0000	+	17.3205i	0.418121	+	0.724207i
$573$		0			0
$574$	30.0000	−	51.9615i	1.25218	−	2.16883i
$575$	−6.00000			−0.250217
$576$		0			0
$577$	17.0000			0.707719			0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000			0.707719			0.353860	+	0.935299i	$0.384869\pi$
$578$	−47.0000	+	81.4064i	−1.95494	+	3.38606i
$579$		0			0
$580$	2.00000	+	3.46410i	0.0830455	+	0.143839i
$581$	−9.00000	−	15.5885i	−0.373383	−	0.646718i
$582$		0			0
$583$	−2.00000	+	3.46410i	−0.0828315	+	0.143468i
$584$		0			0
$585$		0			0
$586$	−36.0000			−1.48715
$587$	−9.00000	+	15.5885i	−0.371470	+	0.643404i	−0.989792	−	0.142520i	$-0.954479\pi$
$587$	−9.00000	+	15.5885i	−0.371470	+	0.643404i	0.618322	+	0.785925i	$0.287813\pi$
$588$		0			0
$589$		0			0
$590$	−8.00000	−	13.8564i	−0.329355	−	0.570459i
$591$		0			0
$592$	10.0000	−	17.3205i	0.410997	−	0.711868i
$593$	22.0000			0.903432			0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000			0.903432			0.451716	+	0.892162i	$0.350812\pi$
$594$		0			0
$595$	−24.0000			−0.983904
$596$	−4.00000	+	6.92820i	−0.163846	+	0.283790i
$597$		0			0
$598$	−30.0000	−	51.9615i	−1.22679	−	2.12486i
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	0.745613	−	0.666379i	$-0.232157\pi$
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	−0.949908	+	0.312531i	$0.898823\pi$
$600$		0			0
$601$	17.0000	−	29.4449i	0.693444	−	1.20108i	−0.277258	−	0.960796i	$-0.589426\pi$
$601$	17.0000	−	29.4449i	0.693444	−	1.20108i	0.970702	−	0.240285i	$-0.0772411\pi$
$602$	−24.0000			−0.978167
$603$		0			0
$604$	2.00000			0.0813788
$605$	3.50000	−	6.06218i	0.142295	−	0.246463i
$606$		0			0
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	0.855700	−	0.517472i	$-0.173127\pi$
$607$	−0.500000	−	0.866025i	−0.0202944	−	0.0351509i	−0.875994	+	0.482322i	$0.839794\pi$
$608$	4.00000	+	6.92820i	0.162221	+	0.280976i
$609$		0			0
$610$	−7.00000	+	12.1244i	−0.283422	+	0.490901i
$611$	20.0000			0.809113
$612$		0			0
$613$	1.00000			0.0403896			0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000			0.0403896			0.0201948	+	0.999796i	$0.493571\pi$
$614$	−16.0000	+	27.7128i	−0.645707	+	1.11840i
$615$		0			0
$616$		0			0
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	0.988343	−	0.152242i	$-0.0486493\pi$
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	−0.626017	+	0.779809i	$0.715316\pi$
$618$		0			0
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	−0.609488	−	0.792796i	$-0.708625\pi$
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	0.991325	+	0.131434i	$0.0419582\pi$
$620$		0			0
$621$		0			0
$622$	48.0000			1.92462
$623$	18.0000	−	31.1769i	0.721155	−	1.24908i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	7.00000	+	12.1244i	0.279776	+	0.484587i
$627$		0			0
$628$	−2.00000	+	3.46410i	−0.0798087	+	0.138233i
$629$	40.0000			1.59490
$630$		0			0
$631$	−11.0000			−0.437903			−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000			−0.437903			−0.218952	+	0.975736i	$0.570264\pi$
$632$		0			0
$633$		0			0
$634$	20.0000	+	34.6410i	0.794301	+	1.37577i
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	5.00000	−	8.66025i	0.198107	−	0.343132i
$638$	−8.00000			−0.316723
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	6.00000	+	10.3923i	0.236617	+	0.409832i	0.959741	−	0.280885i	$-0.0906280\pi$
$643$	6.00000	+	10.3923i	0.236617	+	0.409832i	−0.723124	+	0.690718i	$0.757295\pi$
$644$	−18.0000	−	31.1769i	−0.709299	−	1.22854i
$645$		0			0
$646$	−8.00000	+	13.8564i	−0.314756	+	0.545173i
$647$	38.0000			1.49393			0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000			1.49393			0.746967	+	0.664861i	$0.231509\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$	5.00000	−	8.66025i	0.196116	−	0.339683i
$651$		0			0
$652$	19.0000	+	32.9090i	0.744097	+	1.28881i
$653$	−2.00000	−	3.46410i	−0.0782660	−	0.135561i	0.824236	−	0.566247i	$-0.191605\pi$
$653$	−2.00000	−	3.46410i	−0.0782660	−	0.135561i	−0.902502	+	0.430686i	$0.858272\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$	−40.0000			−1.56174
$657$		0			0
$658$	24.0000			0.935617
$659$	−16.0000	+	27.7128i	−0.623272	+	1.07954i	0.365601	+	0.930772i	$0.380864\pi$
$659$	−16.0000	+	27.7128i	−0.623272	+	1.07954i	−0.988872	+	0.148766i	$0.952470\pi$
$660$		0			0
$661$	−3.50000	−	6.06218i	−0.136134	−	0.235791i	0.789896	−	0.613241i	$-0.210135\pi$
$661$	−3.50000	−	6.06218i	−0.136134	−	0.235791i	−0.926030	+	0.377450i	$0.876801\pi$
$662$	21.0000	+	36.3731i	0.816188	+	1.41368i
$663$		0			0
$664$		0			0
$665$	−3.00000			−0.116335
$666$		0			0
$667$	12.0000			0.464642
$668$	−12.0000	+	20.7846i	−0.464294	+	0.804181i
$669$		0			0
$670$	9.00000	+	15.5885i	0.347700	+	0.602235i
$671$	−7.00000	−	12.1244i	−0.270232	−	0.468056i
$672$		0			0
$673$	−16.5000	+	28.5788i	−0.636028	+	1.10163i	0.350268	+	0.936650i	$0.386091\pi$
$673$	−16.5000	+	28.5788i	−0.636028	+	1.10163i	−0.986296	+	0.164984i	$0.947243\pi$
$674$	14.0000			0.539260
$675$		0			0
$676$	24.0000			0.923077
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	−0.727386	−	0.686229i	$-0.759265\pi$
$677$	6.00000	−	10.3923i	0.230599	−	0.399409i	0.957984	+	0.286820i	$0.0925982\pi$
$678$		0			0
$679$	−19.5000	−	33.7750i	−0.748341	−	1.29617i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$	−15.0000	+	25.9808i	−0.572703	+	0.991950i
$687$		0			0
$688$	8.00000	+	13.8564i	0.304997	+	0.528271i
$689$	5.00000	+	8.66025i	0.190485	+	0.329929i
$690$		0			0
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	−0.610704	−	0.791859i	$-0.709113\pi$
$691$	10.0000	−	17.3205i	0.380418	−	0.658903i	0.991122	+	0.132956i	$0.0424468\pi$
$692$	24.0000			0.912343
$693$		0			0
$694$	−20.0000			−0.759190
$695$	6.50000	−	11.2583i	0.246559	−	0.427053i
$696$		0			0
$697$	−40.0000	−	69.2820i	−1.51511	−	2.62424i
$698$	−19.0000	−	32.9090i	−0.719161	−	1.24562i
$699$		0			0
$700$	3.00000	−	5.19615i	0.113389	−	0.196396i
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$	5.00000			0.188579
$704$	8.00000	−	13.8564i	0.301511	−	0.522233i
$705$		0			0
$706$	−12.0000	−	20.7846i	−0.451626	−	0.782239i
$707$		0			0
$708$		0			0
$709$	−12.5000	+	21.6506i	−0.469447	+	0.813107i	−0.999390	−	0.0349269i	$-0.988880\pi$
$709$	−12.5000	+	21.6506i	−0.469447	+	0.813107i	0.529943	+	0.848034i	$0.322213\pi$
$710$	−4.00000			−0.150117
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	5.00000	+	8.66025i	0.186989	+	0.323875i
$716$	22.0000	+	38.1051i	0.822179	+	1.42406i
$717$		0			0
$718$	−18.0000	+	31.1769i	−0.671754	+	1.16351i
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	−51.0000			−1.89934
$722$	18.0000	−	31.1769i	0.669891	−	1.16028i
$723$		0			0
$724$	−5.00000	−	8.66025i	−0.185824	−	0.321856i
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	−0.678676	−	0.734438i	$-0.737446\pi$
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	0.975380	+	0.220532i	$0.0707793\pi$
$728$		0			0
$729$		0			0
$730$	−10.0000			−0.370117
$731$	−16.0000	+	27.7128i	−0.591781	+	1.02500i
$732$		0			0
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	0.994471	−	0.105010i	$-0.0334875\pi$
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	−0.588177	+	0.808732i	$0.700154\pi$
$734$	−21.0000	−	36.3731i	−0.775124	−	1.34255i
$735$		0			0
$736$	−24.0000	+	41.5692i	−0.884652	+	1.53226i
$737$	−18.0000			−0.663039
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	−5.00000	+	8.66025i	−0.183804	+	0.318357i
$741$		0			0
$742$	6.00000	+	10.3923i	0.220267	+	0.381514i
$743$	−16.0000	−	27.7128i	−0.586983	−	1.01668i	−0.994625	−	0.103543i	$-0.966982\pi$
$743$	−16.0000	−	27.7128i	−0.586983	−	1.01668i	0.407642	−	0.913142i	$-0.366351\pi$
$744$		0			0
$745$	−2.00000	+	3.46410i	−0.0732743	+	0.126915i
$746$	−22.0000			−0.805477
$747$		0			0
$748$	32.0000			1.17004
$749$	−9.00000	+	15.5885i	−0.328853	+	0.569590i
$750$		0			0
$751$	−12.5000	−	21.6506i	−0.456131	−	0.790043i	0.542621	−	0.839978i	$-0.317432\pi$
$751$	−12.5000	−	21.6506i	−0.456131	−	0.790043i	−0.998752	+	0.0499348i	$0.984099\pi$
$752$	−8.00000	−	13.8564i	−0.291730	−	0.505291i
$753$		0			0
$754$	−10.0000	+	17.3205i	−0.364179	+	0.630776i
$755$	1.00000			0.0363937
$756$		0			0
$757$	−53.0000			−1.92632			−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000			−1.92632			−0.963159	+	0.268933i	$0.913329\pi$
$758$	11.0000	−	19.0526i	0.399538	−	0.692020i
$759$		0			0
$760$		0			0
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	0.655515	−	0.755182i	$-0.272452\pi$
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	−0.981764	+	0.190101i	$0.939118\pi$
$762$		0			0
$763$	−15.0000	+	25.9808i	−0.543036	+	0.940567i
$764$	8.00000			0.289430
$765$		0			0
$766$	48.0000			1.73431
$767$	20.0000	−	34.6410i	0.722158	−	1.25081i
$768$		0			0
$769$	18.5000	+	32.0429i	0.667127	+	1.15550i	0.978704	+	0.205277i	$0.0658095\pi$
$769$	18.5000	+	32.0429i	0.667127	+	1.15550i	−0.311577	+	0.950221i	$0.600857\pi$
$770$	6.00000	+	10.3923i	0.216225	+	0.374513i
$771$		0			0
$772$	−5.00000	+	8.66025i	−0.179954	+	0.311689i
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	24.0000	+	41.5692i	0.860442	+	1.49033i
$779$	−5.00000	−	8.66025i	−0.179144	−	0.310286i
$780$		0			0
$781$	2.00000	−	3.46410i	0.0715656	−	0.123955i
$782$	−96.0000			−3.43295
$783$		0			0
$784$	−8.00000			−0.285714
$785$	−1.00000	+	1.73205i	−0.0356915	+	0.0618195i
$786$		0			0
$787$	20.5000	+	35.5070i	0.730746	+	1.26569i	0.956565	+	0.291520i	$0.0941610\pi$
$787$	20.5000	+	35.5070i	0.730746	+	1.26569i	−0.225819	+	0.974169i	$0.572506\pi$
$788$		0			0
$789$		0			0
$790$	3.00000	−	5.19615i	0.106735	−	0.184871i
$791$	30.0000			1.06668
$792$		0			0
$793$	−35.0000			−1.24289
$794$	2.00000	−	3.46410i	0.0709773	−	0.122936i
$795$		0			0
$796$	17.0000	+	29.4449i	0.602549	+	1.04365i
$797$	−8.00000	−	13.8564i	−0.283375	−	0.490819i	0.688839	−	0.724914i	$-0.258121\pi$
$797$	−8.00000	−	13.8564i	−0.283375	−	0.490819i	−0.972214	+	0.234095i	$0.924787\pi$
$798$		0			0
$799$	16.0000	−	27.7128i	0.566039	−	0.980409i
$800$	−8.00000			−0.282843
$801$		0			0
$802$	−12.0000			−0.423735
$803$	5.00000	−	8.66025i	0.176446	−	0.305614i
$804$		0			0
$805$	−9.00000	−	15.5885i	−0.317208	−	0.549421i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.0000			0.562530			0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000			0.562530			0.281265	+	0.959630i	$0.409246\pi$
$810$		0			0
$811$	−12.0000			−0.421377			−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000			−0.421377			−0.210688	+	0.977553i	$0.567571\pi$
$812$	−6.00000	+	10.3923i	−0.210559	+	0.364698i
$813$		0			0
$814$	−10.0000	−	17.3205i	−0.350500	−	0.607083i
$815$	9.50000	+	16.4545i	0.332770	+	0.576375i
$816$		0			0
$817$	−2.00000	+	3.46410i	−0.0699711	+	0.121194i
$818$	−14.0000			−0.489499
$819$		0			0
$820$	20.0000			0.698430
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	−0.0542853	−	0.998525i	$-0.517288\pi$
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	0.891891	−	0.452250i	$-0.149379\pi$
$822$		0			0
$823$	−5.50000	−	9.52628i	−0.191718	−	0.332065i	0.754102	−	0.656758i	$-0.228073\pi$
$823$	−5.50000	−	9.52628i	−0.191718	−	0.332065i	−0.945820	+	0.324692i	$0.894739\pi$
$824$		0			0
$825$		0			0
$826$	24.0000	−	41.5692i	0.835067	−	1.44638i
$827$	−28.0000			−0.973655			−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000			−0.973655			−0.486828	+	0.873498i	$0.661846\pi$
$828$		0			0
$829$	−51.0000			−1.77130			−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000			−1.77130			−0.885652	+	0.464350i	$0.846288\pi$
$830$	6.00000	−	10.3923i	0.208263	−	0.360722i
$831$		0			0
$832$	−20.0000	−	34.6410i	−0.693375	−	1.20096i
$833$	−8.00000	−	13.8564i	−0.277184	−	0.480096i
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$	4.00000			0.138343
$837$		0			0
$838$	−64.0000			−2.21084
$839$	−2.00000	+	3.46410i	−0.0690477	+	0.119594i	−0.898482	−	0.439010i	$-0.855329\pi$
$839$	−2.00000	+	3.46410i	−0.0690477	+	0.119594i	0.829435	+	0.558604i	$0.188663\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$	−1.00000	−	1.73205i	−0.0344623	−	0.0596904i
$843$		0			0
$844$	−23.0000	+	39.8372i	−0.791693	+	1.37125i
$845$	12.0000			0.412813
$846$		0			0
$847$	21.0000			0.721569
$848$	4.00000	−	6.92820i	0.137361	−	0.237915i
$849$		0			0
$850$	−8.00000	−	13.8564i	−0.274398	−	0.475271i
$851$	15.0000	+	25.9808i	0.514193	+	0.890609i
$852$		0			0
$853$	−10.5000	+	18.1865i	−0.359513	+	0.622695i	−0.987880	−	0.155222i	$-0.950391\pi$
$853$	−10.5000	+	18.1865i	−0.359513	+	0.622695i	0.628366	+	0.777918i	$0.283724\pi$
$854$	−42.0000			−1.43721
$855$		0			0
$856$		0			0
$857$	−13.0000	+	22.5167i	−0.444072	+	0.769154i	−0.997987	−	0.0634184i	$-0.979800\pi$
$857$	−13.0000	+	22.5167i	−0.444072	+	0.769154i	0.553915	+	0.832573i	$0.313133\pi$
$858$		0			0
$859$	−3.50000	−	6.06218i	−0.119418	−	0.206839i	0.800119	−	0.599841i	$-0.204770\pi$
$859$	−3.50000	−	6.06218i	−0.119418	−	0.206839i	−0.919537	+	0.393003i	$0.871436\pi$
$860$	−4.00000	−	6.92820i	−0.136399	−	0.236250i
$861$		0			0
$862$	30.0000	−	51.9615i	1.02180	−	1.76982i
$863$	4.00000			0.136162			0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000			0.136162			0.0680808	+	0.997680i	$0.478312\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$	38.0000	−	65.8179i	1.29129	−	2.23658i
$867$		0			0
$868$		0			0
$869$	3.00000	+	5.19615i	0.101768	+	0.176267i
$870$		0			0
$871$	−22.5000	+	38.9711i	−0.762383	+	1.32049i
$872$		0			0
$873$		0			0
$874$	−12.0000			−0.405906
$875$	1.50000	−	2.59808i	0.0507093	−	0.0878310i
$876$		0			0
$877$	−4.50000	−	7.79423i	−0.151954	−	0.263192i	0.779992	−	0.625790i	$-0.215223\pi$
$877$	−4.50000	−	7.79423i	−0.151954	−	0.263192i	−0.931946	+	0.362598i	$0.881890\pi$
$878$	32.0000	+	55.4256i	1.07995	+	1.87052i
$879$		0			0
$880$	4.00000	−	6.92820i	0.134840	−	0.233550i
$881$	38.0000			1.28025			0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000			1.28025			0.640126	+	0.768270i	$0.278882\pi$
$882$		0			0
$883$	−19.0000			−0.639401			−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000			−0.639401			−0.319700	+	0.947519i	$0.603582\pi$
$884$	40.0000	−	69.2820i	1.34535	−	2.33021i
$885$		0			0
$886$	−12.0000	−	20.7846i	−0.403148	−	0.698273i
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	−0.915722	−	0.401813i	$-0.868380\pi$
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	0.109881	−	0.993945i	$-0.464953\pi$
$888$		0			0
$889$	−12.0000	+	20.7846i	−0.402467	+	0.697093i
$890$	24.0000			0.804482
$891$		0			0
$892$	16.0000			0.535720
$893$	2.00000	−	3.46410i	0.0669274	−	0.115922i
$894$		0			0
$895$	11.0000	+	19.0526i	0.367689	+	0.636857i
$896$		0			0
$897$		0			0
$898$	4.00000	−	6.92820i	0.133482	−	0.231197i
$899$		0			0
$900$		0			0
$901$	16.0000			0.533037
$902$	−20.0000	+	34.6410i	−0.665927	+	1.15342i
$903$		0			0
$904$		0			0
$905$	−2.50000	−	4.33013i	−0.0831028	−	0.143938i
$906$		0			0
$907$	10.5000	−	18.1865i	0.348647	−	0.603874i	−0.637363	−	0.770564i	$-0.719975\pi$
$907$	10.5000	−	18.1865i	0.348647	−	0.603874i	0.986009	+	0.166690i	$0.0533080\pi$
$908$	20.0000			0.663723
$909$		0			0
$910$	30.0000			0.994490
$911$	20.0000	−	34.6410i	0.662630	−	1.14771i	−0.317293	−	0.948328i	$-0.602774\pi$
$911$	20.0000	−	34.6410i	0.662630	−	1.14771i	0.979922	−	0.199380i	$-0.0638929\pi$
$912$		0			0
$913$	6.00000	+	10.3923i	0.198571	+	0.343935i
$914$	−22.0000	−	38.1051i	−0.727695	−	1.26041i
$915$		0			0
$916$	−6.00000	+	10.3923i	−0.198246	+	0.343371i
$917$	36.0000			1.18882
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$	−12.0000	−	20.7846i	−0.395199	−	0.684505i
$923$	−5.00000	−	8.66025i	−0.164577	−	0.285056i
$924$		0			0
$925$	−2.50000	+	4.33013i	−0.0821995	+	0.142374i
$926$	66.0000			2.16889
$927$		0			0
$928$	16.0000			0.525226
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	0.439932	+	0.898031i	$0.355003\pi$
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	−0.997684	+	0.0680235i	$0.978331\pi$
$930$		0			0
$931$	−1.00000	−	1.73205i	−0.0327737	−	0.0567657i
$932$	−24.0000	−	41.5692i	−0.786146	−	1.36165i
$933$		0			0
$934$	−28.0000	+	48.4974i	−0.916188	+	1.58688i
$935$	16.0000			0.523256
$936$		0			0
$937$	33.0000			1.07806			0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000			1.07806			0.539032	+	0.842286i	$0.318790\pi$
$938$	−27.0000	+	46.7654i	−0.881581	+	1.52694i
$939$		0			0
$940$	4.00000	+	6.92820i	0.130466	+	0.225973i
$941$	−11.0000	−	19.0526i	−0.358590	−	0.621096i	0.629136	−	0.777295i	$-0.283409\pi$
$941$	−11.0000	−	19.0526i	−0.358590	−	0.621096i	−0.987725	+	0.156200i	$0.950076\pi$
$942$		0			0
$943$	30.0000	−	51.9615i	0.976934	−	1.69210i
$944$	−32.0000			−1.04151
$945$		0			0
$946$	16.0000			0.520205
$947$	24.0000	−	41.5692i	0.779895	−	1.35082i	−0.152106	−	0.988364i	$-0.548606\pi$
$947$	24.0000	−	41.5692i	0.779895	−	1.35082i	0.932002	−	0.362454i	$-0.118061\pi$
$948$		0			0
$949$	−12.5000	−	21.6506i	−0.405767	−	0.702809i
$950$	−1.00000	−	1.73205i	−0.0324443	−	0.0561951i
$951$		0			0
$952$		0			0
$953$	−40.0000			−1.29573			−0.647864	−	0.761756i	$-0.724337\pi$
$953$	−40.0000			−1.29573			−0.647864	+	0.761756i	$0.724337\pi$
$954$		0			0
$955$	4.00000			0.129437
$956$	−26.0000	+	45.0333i	−0.840900	+	1.45648i
$957$		0			0
$958$	6.00000	+	10.3923i	0.193851	+	0.335760i
$959$	9.00000	+	15.5885i	0.290625	+	0.503378i
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$	−50.0000			−1.61206
$963$		0			0
$964$	2.00000			0.0644157
$965$	−2.50000	+	4.33013i	−0.0804778	+	0.139392i
$966$		0			0
$967$	6.50000	+	11.2583i	0.209026	+	0.362043i	0.951408	−	0.307933i	$-0.0996374\pi$
$967$	6.50000	+	11.2583i	0.209026	+	0.362043i	−0.742382	+	0.669977i	$0.766304\pi$
$968$		0			0
$969$		0			0
$970$	13.0000	−	22.5167i	0.417405	−	0.722966i
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$	39.0000			1.25028
$974$	−5.00000	+	8.66025i	−0.160210	+	0.277492i
$975$		0			0
$976$	14.0000	+	24.2487i	0.448129	+	0.776182i
$977$	−31.0000	−	53.6936i	−0.991778	−	1.71781i	−0.606715	−	0.794919i	$-0.707513\pi$
$977$	−31.0000	−	53.6936i	−0.991778	−	1.71781i	−0.385063	−	0.922890i	$-0.625820\pi$
$978$		0			0
$979$	−12.0000	+	20.7846i	−0.383522	+	0.664279i
$980$	4.00000			0.127775
$981$		0			0
$982$	44.0000			1.40410
$983$	27.0000	−	46.7654i	0.861166	−	1.49158i	−0.00963785	−	0.999954i	$-0.503068\pi$
$983$	27.0000	−	46.7654i	0.861166	−	1.49158i	0.870804	−	0.491630i	$-0.163599\pi$
$984$		0			0
$985$		0			0
$986$	16.0000	+	27.7128i	0.509544	+	0.882556i
$987$		0			0
$988$	5.00000	−	8.66025i	0.159071	−	0.275519i
$989$	−24.0000			−0.763156
$990$		0			0
$991$	−25.0000			−0.794151			−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000			−0.794151			−0.397076	+	0.917786i	$0.629975\pi$
$992$		0			0
$993$		0			0
$994$	−6.00000	−	10.3923i	−0.190308	−	0.329624i
$995$	8.50000	+	14.7224i	0.269468	+	0.466732i
$996$		0			0
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	−0.314188	−	0.949361i	$-0.601732\pi$
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	0.979265	−	0.202586i	$-0.0649345\pi$
$998$	32.0000			1.01294
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.b.136.1		2
3.2	odd	2		405.2.e.h.136.1		2
9.2	odd	6		135.2.a.a.1.1	✓	1
9.4	even	3	inner	405.2.e.b.271.1		2
9.5	odd	6		405.2.e.h.271.1		2
9.7	even	3		135.2.a.b.1.1	yes	1
36.7	odd	6		2160.2.a.v.1.1		1
36.11	even	6		2160.2.a.j.1.1		1
45.2	even	12		675.2.b.a.649.1		2
45.7	odd	12		675.2.b.b.649.2		2
45.29	odd	6		675.2.a.i.1.1		1
45.34	even	6		675.2.a.a.1.1		1
45.38	even	12		675.2.b.a.649.2		2
45.43	odd	12		675.2.b.b.649.1		2
63.20	even	6		6615.2.a.a.1.1		1
63.34	odd	6		6615.2.a.j.1.1		1
72.11	even	6		8640.2.a.ce.1.1		1
72.29	odd	6		8640.2.a.bh.1.1		1
72.43	odd	6		8640.2.a.bb.1.1		1
72.61	even	6		8640.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.a.a.1.1	✓	1	9.2	odd	6
135.2.a.b.1.1	yes	1	9.7	even	3
405.2.e.b.136.1		2	1.1	even	1	trivial
405.2.e.b.271.1		2	9.4	even	3	inner
405.2.e.h.136.1		2	3.2	odd	2
405.2.e.h.271.1		2	9.5	odd	6
675.2.a.a.1.1		1	45.34	even	6
675.2.a.i.1.1		1	45.29	odd	6
675.2.b.a.649.1		2	45.2	even	12
675.2.b.a.649.2		2	45.38	even	12
675.2.b.b.649.1		2	45.43	odd	12
675.2.b.b.649.2		2	45.7	odd	12
2160.2.a.j.1.1		1	36.11	even	6
2160.2.a.v.1.1		1	36.7	odd	6
6615.2.a.a.1.1		1	63.20	even	6
6615.2.a.j.1.1		1	63.34	odd	6
8640.2.a.c.1.1		1	72.61	even	6
8640.2.a.bb.1.1		1	72.43	odd	6
8640.2.a.bh.1.1		1	72.29	odd	6
8640.2.a.ce.1.1		1	72.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(1.50000 - 2.59808i) q^{7} +2.00000 q^{10} +(-1.00000 + 1.73205i) q^{11} +(2.50000 + 4.33013i) q^{13} +(3.00000 + 5.19615i) q^{14} +(2.00000 - 3.46410i) q^{16} +8.00000 q^{17} +1.00000 q^{19} +(-1.00000 + 1.73205i) q^{20} +(-2.00000 - 3.46410i) q^{22} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} -10.0000 q^{26} -6.00000 q^{28} +(1.00000 - 1.73205i) q^{29} +(4.00000 + 6.92820i) q^{32} +(-8.00000 + 13.8564i) q^{34} -3.00000 q^{35} +5.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(-5.00000 - 8.66025i) q^{41} +(-2.00000 + 3.46410i) q^{43} +4.00000 q^{44} -12.0000 q^{46} +(2.00000 - 3.46410i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(-1.00000 - 1.73205i) q^{50} +(5.00000 - 8.66025i) q^{52} +2.00000 q^{53} +2.00000 q^{55} +(2.00000 + 3.46410i) q^{58} +(-4.00000 - 6.92820i) q^{59} +(-3.50000 + 6.06218i) q^{61} -8.00000 q^{64} +(2.50000 - 4.33013i) q^{65} +(4.50000 + 7.79423i) q^{67} +(-8.00000 - 13.8564i) q^{68} +(3.00000 - 5.19615i) q^{70} -2.00000 q^{71} -5.00000 q^{73} +(-5.00000 + 8.66025i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(3.00000 + 5.19615i) q^{77} +(1.50000 - 2.59808i) q^{79} -4.00000 q^{80} +20.0000 q^{82} +(3.00000 - 5.19615i) q^{83} +(-4.00000 - 6.92820i) q^{85} +(-4.00000 - 6.92820i) q^{86} +12.0000 q^{89} +15.0000 q^{91} +(6.00000 - 10.3923i) q^{92} +(4.00000 + 6.92820i) q^{94} +(-0.500000 - 0.866025i) q^{95} +(6.50000 - 11.2583i) q^{97} +4.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} - q^{5} + 3 q^{7} + 4 q^{10} - 2 q^{11} + 5 q^{13} + 6 q^{14} + 4 q^{16} + 16 q^{17} + 2 q^{19} - 2 q^{20} - 4 q^{22} + 6 q^{23} - q^{25} - 20 q^{26} - 12 q^{28} + 2 q^{29} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 - q^5 + 3 * q^7 + 4 * q^10 - 2 * q^11 + 5 * q^13 + 6 * q^14 + 4 * q^16 + 16 * q^17 + 2 * q^19 - 2 * q^20 - 4 * q^22 + 6 * q^23 - q^25 - 20 * q^26 - 12 * q^28 + 2 * q^29 + 8 * q^32 - 16 * q^34 - 6 * q^35 + 10 * q^37 - 2 * q^38 - 10 * q^41 - 4 * q^43 + 8 * q^44 - 24 * q^46 + 4 * q^47 - 2 * q^49 - 2 * q^50 + 10 * q^52 + 4 * q^53 + 4 * q^55 + 4 * q^58 - 8 * q^59 - 7 * q^61 - 16 * q^64 + 5 * q^65 + 9 * q^67 - 16 * q^68 + 6 * q^70 - 4 * q^71 - 10 * q^73 - 10 * q^74 - 2 * q^76 + 6 * q^77 + 3 * q^79 - 8 * q^80 + 40 * q^82 + 6 * q^83 - 8 * q^85 - 8 * q^86 + 24 * q^89 + 30 * q^91 + 12 * q^92 + 8 * q^94 - q^95 + 13 * q^97 + 8 * q^98

Introduction

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