LMFDB - Embedded newform 273.2.c.c.64.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(64,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("273.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.c (of order $2$, degree $1$, 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:	$2.17991597518$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.2
Root			$-2.54814i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.c.64.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54814i q^{2} -1.00000 q^{3} -4.49301 q^{4} -3.49301i q^{5} +2.54814i q^{6} -1.00000i q^{7} +6.35254i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.54814i q^{2} -1.00000 q^{3} -4.49301 q^{4} -3.49301i q^{5} +2.54814i q^{6} -1.00000i q^{7} +6.35254i q^{8} +1.00000 q^{9} -8.90068 q^{10} +0.708126i q^{11} +4.49301 q^{12} +(-3.25627 + 1.54814i) q^{13} -2.54814 q^{14} +3.49301i q^{15} +7.20114 q^{16} +7.09628 q^{17} -2.54814i q^{18} +0.311392i q^{19} +15.6942i q^{20} +1.00000i q^{21} +1.80440 q^{22} -7.88116 q^{23} -6.35254i q^{24} -7.20114 q^{25} +(3.94487 + 8.29742i) q^{26} -1.00000 q^{27} +4.49301i q^{28} +5.29742 q^{29} +8.90068 q^{30} -7.29742i q^{31} -5.64442i q^{32} -0.708126i q^{33} -18.0823i q^{34} -3.49301 q^{35} -4.49301 q^{36} -1.41625i q^{37} +0.793469 q^{38} +(3.25627 - 1.54814i) q^{39} +22.1895 q^{40} -11.8044i q^{41} +2.54814 q^{42} -3.29742 q^{43} -3.18162i q^{44} -3.49301i q^{45} +20.0823i q^{46} -6.11580i q^{47} -7.20114 q^{48} -1.00000 q^{49} +18.3495i q^{50} -7.09628 q^{51} +(14.6304 - 6.95581i) q^{52} -3.72764 q^{53} +2.54814i q^{54} +2.47349 q^{55} +6.35254 q^{56} -0.311392i q^{57} -13.4986i q^{58} +2.19560i q^{59} -15.6942i q^{60} +2.51253 q^{61} -18.5948 q^{62} -1.00000i q^{63} +0.0195182 q^{64} +(5.40767 + 11.3742i) q^{65} -1.80440 q^{66} +9.17858i q^{67} -31.8837 q^{68} +7.88116 q^{69} +8.90068i q^{70} -0.708126i q^{71} +6.35254i q^{72} -5.21511i q^{73} -3.60881 q^{74} +7.20114 q^{75} -1.39909i q^{76} +0.708126 q^{77} +(-3.94487 - 8.29742i) q^{78} -2.78489 q^{79} -25.1537i q^{80} +1.00000 q^{81} -30.0793 q^{82} +6.11580i q^{83} -4.49301i q^{84} -24.7874i q^{85} +8.40228i q^{86} -5.29742 q^{87} -4.49840 q^{88} -11.1018i q^{89} -8.90068 q^{90} +(1.54814 + 3.25627i) q^{91} +35.4102 q^{92} +7.29742i q^{93} -15.5839 q^{94} +1.08770 q^{95} +5.64442i q^{96} +7.79886i q^{97} +2.54814i q^{98} +0.708126i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$.

$n$	$92$	$106$	$157$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.54814i		−	1.80181i	−0.434020	−	0.900903i	$-0.642905\pi$
$2$		−	2.54814i		−	1.80181i	0.434020	−	0.900903i	$-0.357095\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−4.49301			−2.24651
$5$		−	3.49301i		−	1.56212i	−0.624454	−	0.781061i	$-0.714679\pi$
$5$		−	3.49301i		−	1.56212i	0.624454	−	0.781061i	$-0.285321\pi$
$6$			2.54814i			1.04027i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			6.35254i			2.24596i
$9$	1.00000			0.333333
$10$	−8.90068			−2.81464
$11$			0.708126i			0.213508i	0.994285	+	0.106754i	$0.0340458\pi$
$11$			0.708126i			0.213508i	−0.994285	+	0.106754i	$0.965954\pi$
$12$	4.49301			1.29702
$13$	−3.25627	+	1.54814i	−0.903126	+	0.429377i
$13$	−3.25627	+	1.54814i	−0.903126	+	0.429377i
$14$	−2.54814			−0.681019
$15$			3.49301i			0.901892i
$16$	7.20114			1.80028
$17$	7.09628			1.72110			0.860550	−	0.509366i	$-0.170120\pi$
$17$	7.09628			1.72110			0.860550	+	0.509366i	$0.170120\pi$
$18$		−	2.54814i		−	0.600602i
$19$			0.311392i			0.0714382i	0.999362	+	0.0357191i	$0.0113722\pi$
$19$			0.311392i			0.0714382i	−0.999362	+	0.0357191i	$0.988628\pi$
$20$			15.6942i			3.50932i
$21$			1.00000i			0.218218i
$22$	1.80440			0.384700
$23$	−7.88116			−1.64334			−0.821668	−	0.569966i	$-0.806956\pi$
$23$	−7.88116			−1.64334			−0.821668	+	0.569966i	$0.806956\pi$
$24$		−	6.35254i		−	1.29671i
$25$	−7.20114			−1.44023
$26$	3.94487	+	8.29742i	0.773653	+	1.62726i
$27$	−1.00000			−0.192450
$28$			4.49301i			0.849100i
$29$	5.29742			0.983706			0.491853	−	0.870678i	$-0.336320\pi$
$29$	5.29742			0.983706			0.491853	+	0.870678i	$0.336320\pi$
$30$	8.90068			1.62503
$31$		−	7.29742i		−	1.31065i	−0.755345	−	0.655327i	$-0.772531\pi$
$31$		−	7.29742i		−	1.31065i	0.755345	−	0.655327i	$-0.227469\pi$
$32$		−	5.64442i		−	0.997801i
$33$		−	0.708126i		−	0.123269i
$34$		−	18.0823i		−	3.10109i
$35$	−3.49301			−0.590427
$36$	−4.49301			−0.748835
$37$		−	1.41625i		−	0.232831i	−0.993201	−	0.116415i	$-0.962860\pi$
$37$		−	1.41625i		−	0.232831i	0.993201	−	0.116415i	$-0.0371403\pi$
$38$	0.793469			0.128718
$39$	3.25627	−	1.54814i	0.521420	−	0.247901i
$40$	22.1895			3.50847
$41$		−	11.8044i		−	1.84354i	−0.387739	−	0.921769i	$-0.626744\pi$
$41$		−	11.8044i		−	1.84354i	0.387739	−	0.921769i	$-0.373256\pi$
$42$	2.54814			0.393186
$43$	−3.29742			−0.502851			−0.251426	−	0.967877i	$-0.580899\pi$
$43$	−3.29742			−0.502851			−0.251426	+	0.967877i	$0.580899\pi$
$44$		−	3.18162i		−	0.479647i
$45$		−	3.49301i		−	0.520708i
$46$			20.0823i			2.96097i
$47$		−	6.11580i		−	0.892081i	−0.895013	−	0.446040i	$-0.852834\pi$
$47$		−	6.11580i		−	0.892081i	0.895013	−	0.446040i	$-0.147166\pi$
$48$	−7.20114			−1.03939
$49$	−1.00000			−0.142857
$50$			18.3495i			2.59501i
$51$	−7.09628			−0.993678
$52$	14.6304	−	6.95581i	2.02888	−	0.964597i
$53$	−3.72764			−0.512031			−0.256016	−	0.966673i	$-0.582410\pi$
$53$	−3.72764			−0.512031			−0.256016	+	0.966673i	$0.582410\pi$
$54$			2.54814i			0.346758i
$55$	2.47349			0.333526
$56$	6.35254			0.848894
$57$		−	0.311392i		−	0.0412448i
$58$		−	13.4986i		−	1.77245i
$59$			2.19560i			0.285842i	0.989734	+	0.142921i	$0.0456495\pi$
$59$			2.19560i			0.285842i	−0.989734	+	0.142921i	$0.954350\pi$
$60$		−	15.6942i		−	2.02611i
$61$	2.51253			0.321697			0.160848	−	0.986979i	$-0.448577\pi$
$61$	2.51253			0.321697			0.160848	+	0.986979i	$0.448577\pi$
$62$	−18.5948			−2.36155
$63$		−	1.00000i		−	0.125988i
$64$	0.0195182			0.00243977
$65$	5.40767	+	11.3742i	0.670739	+	1.41079i
$66$	−1.80440			−0.222107
$67$			9.17858i			1.12134i	0.828039	+	0.560671i	$0.189457\pi$
$67$			9.17858i			1.12134i	−0.828039	+	0.560671i	$0.810543\pi$
$68$	−31.8837			−3.86646
$69$	7.88116			0.948781
$70$			8.90068i			1.06384i
$71$		−	0.708126i		−	0.0840391i	−0.999117	−	0.0420196i	$-0.986621\pi$
$71$		−	0.708126i		−	0.0840391i	0.999117	−	0.0420196i	$-0.0133792\pi$
$72$			6.35254i			0.748654i
$73$		−	5.21511i		−	0.610383i	−0.952291	−	0.305191i	$-0.901280\pi$
$73$		−	5.21511i		−	0.610383i	0.952291	−	0.305191i	$-0.0987205\pi$
$74$	−3.60881			−0.419516
$75$	7.20114			0.831516
$76$		−	1.39909i		−	0.160486i
$77$	0.708126			0.0806985
$78$	−3.94487	−	8.29742i	−0.446669	−	0.939498i
$79$	−2.78489			−0.313324			−0.156662	−	0.987652i	$-0.550073\pi$
$79$	−2.78489			−0.313324			−0.156662	+	0.987652i	$0.550073\pi$
$80$		−	25.1537i		−	2.81227i
$81$	1.00000			0.111111
$82$	−30.0793			−3.32170
$83$			6.11580i			0.671296i	0.941988	+	0.335648i	$0.108955\pi$
$83$			6.11580i			0.671296i	−0.941988	+	0.335648i	$0.891045\pi$
$84$		−	4.49301i		−	0.490228i
$85$		−	24.7874i		−	2.68857i
$86$			8.40228i			0.906041i
$87$	−5.29742			−0.567943
$88$	−4.49840			−0.479531
$89$		−	11.1018i		−	1.17679i	−0.808573	−	0.588395i	$-0.799760\pi$
$89$		−	11.1018i		−	1.17679i	0.808573	−	0.588395i	$-0.200240\pi$
$90$	−8.90068			−0.938214
$91$	1.54814	+	3.25627i	0.162289	+	0.341349i
$92$	35.4102			3.69177
$93$			7.29742i			0.756707i
$94$	−15.5839			−1.60736
$95$	1.08770			0.111595
$96$			5.64442i			0.576081i
$97$			7.79886i			0.791854i	0.918282	+	0.395927i	$0.129577\pi$
$97$			7.79886i			0.791854i	−0.918282	+	0.395927i	$0.870423\pi$
$98$			2.54814i			0.257401i
$99$			0.708126i			0.0711694i
$100$	32.3548			3.23548
$101$	17.6911			1.76033			0.880166	−	0.474666i	$-0.157431\pi$
$101$	17.6911			1.76033			0.880166	+	0.474666i	$0.157431\pi$
$102$			18.0823i			1.79041i
$103$	5.52651			0.544543			0.272271	−	0.962221i	$-0.412225\pi$
$103$	5.52651			0.544543			0.272271	+	0.962221i	$0.412225\pi$
$104$	−9.83462	−	20.6856i	−0.964364	−	2.02839i
$105$	3.49301			0.340883
$106$			9.49856i			0.922581i
$107$	19.1786			1.85406			0.927032	−	0.374983i	$-0.122351\pi$
$107$	19.1786			1.85406			0.927032	+	0.374983i	$0.122351\pi$
$108$	4.49301			0.432340
$109$		−	3.20653i		−	0.307130i	−0.988139	−	0.153565i	$-0.950925\pi$
$109$		−	3.20653i		−	0.307130i	0.988139	−	0.153565i	$-0.0490754\pi$
$110$		−	6.30281i		−	0.600949i
$111$			1.41625i			0.134425i
$112$		−	7.20114i		−	0.680444i
$113$	−5.29742			−0.498339			−0.249170	−	0.968460i	$-0.580158\pi$
$113$	−5.29742			−0.498339			−0.249170	+	0.968460i	$0.580158\pi$
$114$	−0.793469			−0.0743152
$115$			27.5290i			2.56709i
$116$	−23.8014			−2.20990
$117$	−3.25627	+	1.54814i	−0.301042	+	0.143126i
$118$	5.59468			0.515032
$119$		−	7.09628i		−	0.650515i
$120$	−22.1895			−2.02562
$121$	10.4986			0.954414
$122$		−	6.40228i		−	0.579635i
$123$			11.8044i			1.06437i
$124$			32.7874i			2.94439i
$125$			7.68861i			0.687690i
$126$	−2.54814			−0.227006
$127$	−14.3028			−1.26917			−0.634585	−	0.772853i	$-0.718829\pi$
$127$	−14.3028			−1.26917			−0.634585	+	0.772853i	$0.718829\pi$
$128$		−	11.3386i		−	1.00220i
$129$	3.29742			0.290321
$130$	28.9830	−	13.7795i	2.54198	−	1.20854i
$131$	12.4023			1.08359			0.541796	−	0.840510i	$-0.317745\pi$
$131$	12.4023			1.08359			0.541796	+	0.840510i	$0.317745\pi$
$132$			3.18162i			0.276925i
$133$	0.311392			0.0270011
$134$	23.3883			2.02044
$135$			3.49301i			0.300631i
$136$			45.0794i			3.86553i
$137$		−	5.58390i		−	0.477065i	−0.971135	−	0.238532i	$-0.923334\pi$
$137$		−	5.58390i		−	0.477065i	0.971135	−	0.238532i	$-0.0766663\pi$
$138$		−	20.0823i		−	1.70952i
$139$	−1.52651			−0.129476			−0.0647382	−	0.997902i	$-0.520621\pi$
$139$	−1.52651			−0.129476			−0.0647382	+	0.997902i	$0.520621\pi$
$140$	15.6942			1.32640
$141$			6.11580i			0.515043i
$142$	−1.80440			−0.151422
$143$	−1.09628	−	2.30585i	−0.0916754	−	0.192825i
$144$	7.20114			0.600095
$145$		−	18.5039i		−	1.53667i
$146$	−13.2888			−1.09979
$147$	1.00000			0.0824786
$148$			6.36324i			0.523055i
$149$			15.7765i			1.29246i	0.763144	+	0.646229i	$0.223655\pi$
$149$			15.7765i			1.29246i	−0.763144	+	0.646229i	$0.776345\pi$
$150$		−	18.3495i		−	1.49823i
$151$			6.83251i			0.556021i	0.960578	+	0.278011i	$0.0896751\pi$
$151$			6.83251i			0.556021i	−0.960578	+	0.278011i	$0.910325\pi$
$152$	−1.97813			−0.160447
$153$	7.09628			0.573700
$154$		−	1.80440i		−	0.145403i
$155$	−25.4900			−2.04740
$156$	−14.6304	+	6.95581i	−1.17137	+	0.556910i
$157$	−12.1926			−0.973072			−0.486536	−	0.873661i	$-0.661740\pi$
$157$	−12.1926			−0.973072			−0.486536	+	0.873661i	$0.661740\pi$
$158$			7.09628i			0.564550i
$159$	3.72764			0.295621
$160$	−19.7160			−1.55869
$161$			7.88116i			0.621123i
$162$		−	2.54814i		−	0.200201i
$163$			10.0390i			0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$163$			10.0390i			0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$164$			53.0373i			4.14152i
$165$	−2.47349			−0.192561
$166$	15.5839			1.20955
$167$		−	24.7106i		−	1.91217i	−0.293096	−	0.956083i	$-0.594686\pi$
$167$		−	24.7106i		−	1.91217i	0.293096	−	0.956083i	$-0.405314\pi$
$168$	−6.35254			−0.490109
$169$	8.20653	−	10.0823i	0.631272	−	0.775562i
$170$	−63.1617			−4.84428
$171$			0.311392i			0.0238127i
$172$	14.8153			1.12966
$173$	10.4735			0.796285			0.398143	−	0.917324i	$-0.369655\pi$
$173$	10.4735			0.796285			0.398143	+	0.917324i	$0.369655\pi$
$174$			13.4986i			1.02332i
$175$			7.20114i			0.544355i
$176$			5.09932i			0.384375i
$177$		−	2.19560i		−	0.165031i
$178$	−28.2890			−2.12035
$179$	11.2584			0.841491			0.420745	−	0.907179i	$-0.361769\pi$
$179$	11.2584			0.841491			0.420745	+	0.907179i	$0.361769\pi$
$180$			15.6942i			1.16977i
$181$	−8.59484			−0.638849			−0.319425	−	0.947612i	$-0.603490\pi$
$181$	−8.59484			−0.638849			−0.319425	+	0.947612i	$0.603490\pi$
$182$	8.29742	−	3.94487i	0.615046	−	0.292414i
$183$	−2.51253			−0.185732
$184$		−	50.0654i		−	3.69087i
$185$	−4.94699			−0.363710
$186$	18.5948			1.36344
$187$			5.02506i			0.367469i
$188$			27.4784i			2.00406i
$189$			1.00000i			0.0727393i
$190$		−	2.77160i		−	0.201073i
$191$	4.98603			0.360776			0.180388	−	0.983596i	$-0.442265\pi$
$191$	4.98603			0.360776			0.180388	+	0.983596i	$0.442265\pi$
$192$	−0.0195182			−0.00140860
$193$		−	6.98603i		−	0.502865i	−0.967875	−	0.251433i	$-0.919098\pi$
$193$		−	6.98603i		−	0.502865i	0.967875	−	0.251433i	$-0.0809017\pi$
$194$	19.8726			1.42677
$195$	−5.40767	−	11.3742i	−0.387251	−	0.814522i
$196$	4.49301			0.320929
$197$		−	21.8044i		−	1.55350i	−0.629809	−	0.776750i	$-0.716867\pi$
$197$		−	21.8044i		−	1.55350i	0.629809	−	0.776750i	$-0.283133\pi$
$198$	1.80440			0.128233
$199$	0.622783			0.0441479			0.0220740	−	0.999756i	$-0.492973\pi$
$199$	0.622783			0.0441479			0.0220740	+	0.999756i	$0.492973\pi$
$200$		−	45.7456i		−	3.23470i
$201$		−	9.17858i		−	0.647407i
$202$		−	45.0794i		−	3.17178i
$203$		−	5.29742i		−	0.371806i
$204$	31.8837			2.23230
$205$	−41.2329			−2.87983
$206$		−	14.0823i		−	0.981161i
$207$	−7.88116			−0.547779
$208$	−23.4488	+	11.1484i	−1.62588	+	0.773000i
$209$	−0.220505			−0.0152526
$210$		−	8.90068i		−	0.614205i
$211$	16.3547			1.12590			0.562951	−	0.826491i	$-0.309666\pi$
$211$	16.3547			1.12590			0.562951	+	0.826491i	$0.309666\pi$
$212$	16.7484			1.15028
$213$			0.708126i			0.0485200i
$214$		−	48.8697i		−	3.34066i
$215$			11.5179i			0.785516i
$216$		−	6.35254i		−	0.432236i
$217$	−7.29742			−0.495381
$218$	−8.17069			−0.553389
$219$			5.21511i			0.352405i
$220$	−11.1134			−0.749268
$221$	−23.1074	+	10.9860i	−1.55437	+	0.739000i
$222$	3.60881			0.242207
$223$			8.71367i			0.583511i	0.956493	+	0.291755i	$0.0942393\pi$
$223$			8.71367i			0.583511i	−0.956493	+	0.291755i	$0.905761\pi$
$224$	−5.64442			−0.377133
$225$	−7.20114			−0.480076
$226$			13.4986i			0.897911i
$227$			12.9440i			0.859120i	0.903038	+	0.429560i	$0.141331\pi$
$227$			12.9440i			0.859120i	−0.903038	+	0.429560i	$0.858669\pi$
$228$			1.39909i			0.0926568i
$229$			19.6521i			1.29865i	0.760513	+	0.649323i	$0.224948\pi$
$229$			19.6521i			1.29865i	−0.760513	+	0.649323i	$0.775052\pi$
$230$	70.1477			4.62541
$231$	−0.708126			−0.0465913
$232$			33.6521i			2.20937i
$233$	3.72764			0.244206			0.122103	−	0.992517i	$-0.461036\pi$
$233$	3.72764			0.244206			0.122103	+	0.992517i	$0.461036\pi$
$234$	3.94487	+	8.29742i	0.257884	+	0.542419i
$235$	−21.3626			−1.39354
$236$		−	9.86484i		−	0.642146i
$237$	2.78489			0.180898
$238$	−18.0823			−1.17210
$239$			8.86165i			0.573212i	0.958048	+	0.286606i	$0.0925271\pi$
$239$			8.86165i			0.573212i	−0.958048	+	0.286606i	$0.907473\pi$
$240$			25.1537i			1.62366i
$241$			10.0025i			0.644318i	0.946686	+	0.322159i	$0.104409\pi$
$241$			10.0025i			0.644318i	−0.946686	+	0.322159i	$0.895591\pi$
$242$		−	26.7518i		−	1.71967i
$243$	−1.00000			−0.0641500
$244$	−11.2888			−0.722694
$245$			3.49301i			0.223160i
$246$	30.0793			1.91778
$247$	−0.482078	−	1.01397i	−0.0306739	−	0.0645176i
$248$	46.3572			2.94368
$249$		−	6.11580i		−	0.387573i
$250$	19.5916			1.23908
$251$	9.24557			0.583575			0.291788	−	0.956483i	$-0.405750\pi$
$251$	9.24557			0.583575			0.291788	+	0.956483i	$0.405750\pi$
$252$			4.49301i			0.283033i
$253$		−	5.58086i		−	0.350866i
$254$			36.4455i			2.28680i
$255$			24.7874i			1.55225i
$256$	−28.8532			−1.80333
$257$	−14.5516			−0.907702			−0.453851	−	0.891078i	$-0.649950\pi$
$257$	−14.5516			−0.907702			−0.453851	+	0.891078i	$0.649950\pi$
$258$		−	8.40228i		−	0.523103i
$259$	−1.41625			−0.0880017
$260$	−24.2967	−	51.1043i	−1.50682	−	3.16936i
$261$	5.29742			0.327902
$262$		−	31.6027i		−	1.95242i
$263$	−14.7137			−0.907284			−0.453642	−	0.891184i	$-0.649875\pi$
$263$	−14.7137			−0.907284			−0.453642	+	0.891184i	$0.649875\pi$
$264$	4.49840			0.276858
$265$			13.0207i			0.799856i
$266$		−	0.793469i		−	0.0486507i
$267$			11.1018i			0.679420i
$268$		−	41.2395i		−	2.51910i
$269$	−12.0433			−0.734291			−0.367145	−	0.930164i	$-0.619665\pi$
$269$	−12.0433			−0.734291			−0.367145	+	0.930164i	$0.619665\pi$
$270$	8.90068			0.541678
$271$		−	7.38830i		−	0.448808i	−0.974496	−	0.224404i	$-0.927957\pi$
$271$		−	7.38830i		−	0.448808i	0.974496	−	0.224404i	$-0.0720434\pi$
$272$	51.1013			3.09847
$273$	−1.54814	−	3.25627i	−0.0936976	−	0.197078i
$274$	−14.2286			−0.859578
$275$		−	5.09932i		−	0.307500i
$276$	−35.4102			−2.13144
$277$	13.6997			0.823135			0.411567	−	0.911379i	$-0.364981\pi$
$277$	13.6997			0.823135			0.411567	+	0.911379i	$0.364981\pi$
$278$			3.88975i			0.233292i
$279$		−	7.29742i		−	0.436885i
$280$		−	22.1895i		−	1.32608i
$281$			4.62582i			0.275953i	0.990435	+	0.137977i	$0.0440599\pi$
$281$			4.62582i			0.275953i	−0.990435	+	0.137977i	$0.955940\pi$
$282$	15.5839			0.928008
$283$	17.9721			1.06833			0.534164	−	0.845381i	$-0.320627\pi$
$283$	17.9721			1.06833			0.534164	+	0.845381i	$0.320627\pi$
$284$			3.18162i			0.188794i
$285$	−1.08770			−0.0644295
$286$	−5.87562	+	2.79347i	−0.347433	+	0.165181i
$287$	−11.8044			−0.696792
$288$		−	5.64442i		−	0.332600i
$289$	33.3572			1.96219
$290$	−47.1506			−2.76878
$291$		−	7.79886i		−	0.457177i
$292$			23.4316i			1.37123i
$293$		−	6.87023i		−	0.401363i	−0.979657	−	0.200682i	$-0.935684\pi$
$293$		−	6.87023i		−	0.401363i	0.979657	−	0.200682i	$-0.0643156\pi$
$294$		−	2.54814i		−	0.148610i
$295$	7.66924			0.446521
$296$	8.99681			0.522929
$297$		−	0.708126i		−	0.0410897i
$298$	40.2006			2.32876
$299$	25.6632	−	12.2011i	1.48414	−	0.705610i
$300$	−32.3548			−1.86801
$301$			3.29742i			0.190060i
$302$	17.4102			1.00184
$303$	−17.6911			−1.01633
$304$			2.24238i			0.128609i
$305$		−	8.77630i		−	0.502530i
$306$		−	18.0823i		−	1.03370i
$307$			11.9202i			0.680322i	0.940367	+	0.340161i	$0.110482\pi$
$307$			11.9202i			0.680322i	−0.940367	+	0.340161i	$0.889518\pi$
$308$	−3.18162			−0.181290
$309$	−5.52651			−0.314392
$310$			64.9520i			3.68903i
$311$	−24.7874			−1.40556			−0.702782	−	0.711405i	$-0.748059\pi$
$311$	−24.7874			−1.40556			−0.702782	+	0.711405i	$0.748059\pi$
$312$	9.83462	+	20.6856i	0.556776	+	1.17109i
$313$	−6.73304			−0.380574			−0.190287	−	0.981729i	$-0.560942\pi$
$313$	−6.73304			−0.380574			−0.190287	+	0.981729i	$0.560942\pi$
$314$			31.0683i			1.75329i
$315$	−3.49301			−0.196809
$316$	12.5125			0.703885
$317$		−	19.1537i		−	1.07578i	−0.843016	−	0.537889i	$-0.819222\pi$
$317$		−	19.1537i		−	1.07578i	0.843016	−	0.537889i	$-0.180778\pi$
$318$		−	9.49856i		−	0.532653i
$319$			3.75124i			0.210029i
$320$		−	0.0681772i		−	0.00381122i
$321$	−19.1786			−1.07044
$322$	20.0823			1.11914
$323$			2.20972i			0.122952i
$324$	−4.49301			−0.249612
$325$	23.4488	−	11.1484i	1.30071	−	0.618400i
$326$	25.5809			1.41679
$327$			3.20653i			0.177322i
$328$	74.9880			4.14052
$329$	−6.11580			−0.337175
$330$			6.30281i			0.346958i
$331$		−	23.5247i		−	1.29303i	−0.762900	−	0.646516i	$-0.776225\pi$
$331$		−	23.5247i		−	1.29303i	0.762900	−	0.646516i	$-0.223775\pi$
$332$		−	27.4784i		−	1.50807i
$333$		−	1.41625i		−	0.0776102i
$334$	−62.9661			−3.44535
$335$	32.0609			1.75167
$336$			7.20114i			0.392854i
$337$	29.7258			1.61927			0.809634	−	0.586935i	$-0.199666\pi$
$337$	29.7258			1.61927			0.809634	+	0.586935i	$0.199666\pi$
$338$	−25.6911	−	20.9114i	−1.39741	−	1.13743i
$339$	5.29742			0.287716
$340$			111.370i			6.03989i
$341$	5.16749			0.279836
$342$	0.793469			0.0429059
$343$			1.00000i			0.0539949i
$344$		−	20.9470i		−	1.12939i
$345$		−	27.5290i		−	1.48211i
$346$		−	26.6879i		−	1.43475i
$347$	−32.2036			−1.72878			−0.864391	−	0.502820i	$-0.832296\pi$
$347$	−32.2036			−1.72878			−0.864391	+	0.502820i	$0.832296\pi$
$348$	23.8014			1.27589
$349$		−	12.5752i		−	0.673133i	−0.941660	−	0.336567i	$-0.890734\pi$
$349$		−	12.5752i		−	0.673133i	0.941660	−	0.336567i	$-0.109266\pi$
$350$	18.3495			0.980822
$351$	3.25627	−	1.54814i	0.173807	−	0.0826336i
$352$	3.99696			0.213039
$353$			15.4132i			0.820363i	0.912004	+	0.410181i	$0.134535\pi$
$353$			15.4132i			0.820363i	−0.912004	+	0.410181i	$0.865465\pi$
$354$	−5.59468			−0.297354
$355$	−2.47349			−0.131279
$356$			49.8806i			2.64367i
$357$			7.09628i			0.375575i
$358$		−	28.6879i		−	1.51620i
$359$			24.6910i			1.30314i	0.758589	+	0.651570i	$0.225889\pi$
$359$			24.6910i			1.30314i	−0.758589	+	0.651570i	$0.774111\pi$
$360$	22.1895			1.16949
$361$	18.9030			0.994897
$362$			21.9008i			1.15108i
$363$	−10.4986			−0.551031
$364$	−6.95581	−	14.6304i	−0.364583	−	0.766844i
$365$	−18.2165			−0.953493
$366$			6.40228i			0.334652i
$367$	10.8758			0.567711			0.283855	−	0.958867i	$-0.408386\pi$
$367$	10.8758			0.567711			0.283855	+	0.958867i	$0.408386\pi$
$368$	−56.7534			−2.95847
$369$		−	11.8044i		−	0.614513i
$370$			12.6056i			0.655335i
$371$			3.72764i			0.193530i
$372$		−	32.7874i		−	1.69995i
$373$	−1.71906			−0.0890096			−0.0445048	−	0.999009i	$-0.514171\pi$
$373$	−1.71906			−0.0890096			−0.0445048	+	0.999009i	$0.514171\pi$
$374$	12.8046			0.662108
$375$		−	7.68861i		−	0.397038i
$376$	38.8509			2.00358
$377$	−17.2498	+	8.20114i	−0.888410	+	0.422380i
$378$	2.54814			0.131062
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	−4.88703			−0.250699
$381$	14.3028			0.732755
$382$		−	12.7051i		−	0.650049i
$383$		−	36.5528i		−	1.86776i	−0.357588	−	0.933879i	$-0.616401\pi$
$383$		−	36.5528i		−	1.86776i	0.357588	−	0.933879i	$-0.383599\pi$
$384$			11.3386i			0.578619i
$385$		−	2.47349i		−	0.126061i
$386$	−17.8014			−0.906066
$387$	−3.29742			−0.167617
$388$		−	35.0404i		−	1.77891i
$389$	36.1367			1.83220			0.916101	−	0.400948i	$-0.131319\pi$
$389$	36.1367			1.83220			0.916101	+	0.400948i	$0.131319\pi$
$390$	−28.9830	+	13.7795i	−1.46761	+	0.697752i
$391$	−55.9269			−2.82835
$392$		−	6.35254i		−	0.320852i
$393$	−12.4023			−0.625612
$394$	−55.5607			−2.79911
$395$			9.72764i			0.489451i
$396$		−	3.18162i		−	0.159882i
$397$			24.3657i			1.22288i	0.791290	+	0.611441i	$0.209410\pi$
$397$			24.3657i			1.22288i	−0.791290	+	0.611441i	$0.790590\pi$
$398$		−	1.58694i		−	0.0795461i
$399$	−0.311392			−0.0155891
$400$	−51.8564			−2.59282
$401$		−	3.76537i		−	0.188034i	−0.995571	−	0.0940168i	$-0.970029\pi$
$401$		−	3.76537i		−	0.188034i	0.995571	−	0.0940168i	$-0.0299707\pi$
$402$	−23.3883			−1.16650
$403$	11.2974	+	23.7623i	0.562764	+	1.18369i
$404$	−79.4864			−3.95460
$405$		−	3.49301i		−	0.173569i
$406$	−13.4986			−0.669922
$407$	1.00289			0.0497112
$408$		−	45.0794i		−	2.23176i
$409$			31.3235i			1.54885i	0.632667	+	0.774424i	$0.281960\pi$
$409$			31.3235i			1.54885i	−0.632667	+	0.774424i	$0.718040\pi$
$410$			105.067i			5.18890i
$411$			5.58390i			0.275433i
$412$	−24.8307			−1.22332
$413$	2.19560			0.108038
$414$			20.0823i			0.986991i
$415$	21.3626			1.04865
$416$	8.73834	+	18.3797i	0.428433	+	0.901140i
$417$	1.52651			0.0747533
$418$			0.561877i			0.0274823i
$419$	−26.3572			−1.28763			−0.643816	−	0.765180i	$-0.722650\pi$
$419$	−26.3572			−1.28763			−0.643816	+	0.765180i	$0.722650\pi$
$420$	−15.6942			−0.765796
$421$		−	22.5168i		−	1.09740i	−0.836019	−	0.548700i	$-0.815123\pi$
$421$		−	22.5168i		−	1.09740i	0.836019	−	0.548700i	$-0.184877\pi$
$422$		−	41.6739i		−	2.02866i
$423$		−	6.11580i		−	0.297360i
$424$		−	23.6800i		−	1.15000i
$425$	−51.1013			−2.47878
$426$	1.80440			0.0874237
$427$		−	2.51253i		−	0.121590i
$428$	−86.1696			−4.16517
$429$	1.09628	+	2.30585i	0.0529288	+	0.111327i
$430$	29.3493			1.41535
$431$		−	16.5374i		−	0.796580i	−0.917259	−	0.398290i	$-0.869604\pi$
$431$		−	16.5374i		−	0.796580i	0.917259	−	0.398290i	$-0.130396\pi$
$432$	−7.20114			−0.346465
$433$	25.2720			1.21449			0.607247	−	0.794513i	$-0.292274\pi$
$433$	25.2720			1.21449			0.607247	+	0.794513i	$0.292274\pi$
$434$			18.5948i			0.892581i
$435$			18.5039i			0.887196i
$436$			14.4070i			0.689969i
$437$		−	2.45413i		−	0.117397i
$438$	13.2888			0.634965
$439$	5.52651			0.263766			0.131883	−	0.991265i	$-0.457898\pi$
$439$	5.52651			0.263766			0.131883	+	0.991265i	$0.457898\pi$
$440$			15.7130i			0.749087i
$441$	−1.00000			−0.0476190
$442$	27.9939	+	58.8808i	1.33154	+	2.80067i
$443$	−7.25838			−0.344856			−0.172428	−	0.985022i	$-0.555161\pi$
$443$	−7.25838			−0.344856			−0.172428	+	0.985022i	$0.555161\pi$
$444$		−	6.36324i		−	0.301986i
$445$	−38.7788			−1.83829
$446$	22.2036			1.05137
$447$		−	15.7765i		−	0.746201i
$448$		−	0.0195182i		−	0.000922146i
$449$			24.1787i			1.14107i	0.821275	+	0.570533i	$0.193263\pi$
$449$			24.1787i			1.14107i	−0.821275	+	0.570533i	$0.806737\pi$
$450$			18.3495i			0.865004i
$451$	8.35901			0.393610
$452$	23.8014			1.11952
$453$		−	6.83251i		−	0.321019i
$454$	32.9830			1.54797
$455$	11.3742	−	5.40767i	0.533230	−	0.253515i
$456$	1.97813			0.0926344
$457$			33.3991i			1.56234i	0.624316	+	0.781172i	$0.285378\pi$
$457$			33.3991i			1.56234i	−0.624316	+	0.781172i	$0.714622\pi$
$458$	50.0762			2.33991
$459$	−7.09628			−0.331226
$460$		−	123.688i		−	5.76699i
$461$		−	10.3882i		−	0.483824i	−0.970298	−	0.241912i	$-0.922225\pi$
$461$		−	10.3882i		−	0.483824i	0.970298	−	0.241912i	$-0.0777746\pi$
$462$			1.80440i			0.0839485i
$463$			23.9330i			1.11226i	0.831095	+	0.556131i	$0.187715\pi$
$463$			23.9330i			1.11226i	−0.831095	+	0.556131i	$0.812285\pi$
$464$	38.1474			1.77095
$465$	25.4900			1.18207
$466$		−	9.49856i		−	0.440012i
$467$	8.84329			0.409219			0.204609	−	0.978844i	$-0.434408\pi$
$467$	8.84329			0.409219			0.204609	+	0.978844i	$0.434408\pi$
$468$	14.6304	−	6.95581i	0.676292	−	0.321532i
$469$	9.17858			0.423828
$470$			54.4348i			2.51089i
$471$	12.1926			0.561803
$472$	−13.9476			−0.641991
$473$		−	2.33499i		−	0.107363i
$474$		−	7.09628i		−	0.325943i
$475$		−	2.24238i		−	0.102887i
$476$			31.8837i			1.46139i
$477$	−3.72764			−0.170677
$478$	22.5807			1.03282
$479$		−	13.9623i		−	0.637953i	−0.947763	−	0.318976i	$-0.896661\pi$
$479$		−	13.9623i		−	0.637953i	0.947763	−	0.318976i	$-0.103339\pi$
$480$	19.7160			0.899909
$481$	2.19256	+	4.61170i	0.0999720	+	0.210275i
$482$	25.4878			1.16094
$483$		−	7.88116i		−	0.358605i
$484$	−47.1702			−2.14410
$485$	27.2415			1.23697
$486$			2.54814i			0.115586i
$487$		−	2.41306i		−	0.109346i	−0.998504	−	0.0546731i	$-0.982588\pi$
$487$		−	2.41306i		−	0.109346i	0.998504	−	0.0546731i	$-0.0174117\pi$
$488$			15.9610i			0.722519i
$489$		−	10.0390i		−	0.453981i
$490$	8.90068			0.402092
$491$	9.15063			0.412962			0.206481	−	0.978451i	$-0.433799\pi$
$491$	9.15063			0.412962			0.206481	+	0.978451i	$0.433799\pi$
$492$		−	53.0373i		−	2.39111i
$493$	37.5919			1.69306
$494$	−2.58375	+	1.22840i	−0.116248	+	0.0552684i
$495$	2.47349			0.111175
$496$		−	52.5497i		−	2.35955i
$497$	−0.708126			−0.0317638
$498$	−15.5839			−0.698331
$499$			5.34927i			0.239466i	0.992806	+	0.119733i	$0.0382039\pi$
$499$			5.34927i			0.239466i	−0.992806	+	0.119733i	$0.961796\pi$
$500$		−	34.5450i		−	1.54490i
$501$			24.7106i			1.10399i
$502$		−	23.5590i		−	1.05149i
$503$	35.1617			1.56778			0.783892	−	0.620897i	$-0.213232\pi$
$503$	35.1617			1.56778			0.783892	+	0.620897i	$0.213232\pi$
$504$	6.35254			0.282965
$505$		−	61.7953i		−	2.74985i
$506$	−14.2208			−0.632192
$507$	−8.20653	+	10.0823i	−0.364465	+	0.447771i
$508$	64.2627			2.85120
$509$			18.4790i			0.819069i	0.912295	+	0.409534i	$0.134309\pi$
$509$			18.4790i			0.819069i	−0.912295	+	0.409534i	$0.865691\pi$
$510$	63.1617			2.79685
$511$	−5.21511			−0.230703
$512$			50.8449i			2.24705i
$513$		−	0.311392i		−	0.0137483i
$514$			37.0794i			1.63550i
$515$		−	19.3042i		−	0.850643i
$516$	−14.8153			−0.652209
$517$	4.33076			0.190466
$518$			3.60881i			0.158562i
$519$	−10.4735			−0.459735
$520$	−72.2550	+	34.3525i	−3.16859	+	1.50646i
$521$	12.1385			0.531798			0.265899	−	0.964001i	$-0.414331\pi$
$521$	12.1385			0.531798			0.265899	+	0.964001i	$0.414331\pi$
$522$		−	13.4986i		−	0.590816i
$523$	−11.4205			−0.499383			−0.249691	−	0.968325i	$-0.580329\pi$
$523$	−11.4205			−0.499383			−0.249691	+	0.968325i	$0.580329\pi$
$524$	−55.7236			−2.43430
$525$		−	7.20114i		−	0.314283i
$526$			37.4925i			1.63475i
$527$		−	51.7845i		−	2.25577i
$528$		−	5.09932i		−	0.221919i
$529$	39.1128			1.70055
$530$	33.1786			1.44119
$531$			2.19560i			0.0952807i
$532$	−1.39909			−0.0606581
$533$	18.2749	+	38.4383i	0.791572	+	1.66495i
$534$	28.2890			1.22418
$535$		−	66.9910i		−	2.89628i
$536$	−58.3073			−2.51849
$537$	−11.2584			−0.485835
$538$			30.6879i			1.32305i
$539$		−	0.708126i		−	0.0305012i
$540$		−	15.6942i		−	0.675369i
$541$			16.3851i			0.704451i	0.935915	+	0.352226i	$0.114575\pi$
$541$			16.3851i			0.704451i	−0.935915	+	0.352226i	$0.885425\pi$
$542$	−18.8264			−0.808664
$543$	8.59484			0.368840
$544$		−	40.0544i		−	1.71732i
$545$	−11.2005			−0.479775
$546$	−8.29742	+	3.94487i	−0.355097	+	0.168825i
$547$	−19.4620			−0.832136			−0.416068	−	0.909333i	$-0.636592\pi$
$547$	−19.4620			−0.832136			−0.416068	+	0.909333i	$0.636592\pi$
$548$			25.0885i			1.07173i
$549$	2.51253			0.107232
$550$	−12.9938			−0.554056
$551$			1.64957i			0.0702741i
$552$			50.0654i			2.13093i
$553$			2.78489i			0.118425i
$554$		−	34.9087i		−	1.48313i
$555$	4.94699			0.209988
$556$	6.85861			0.290870
$557$			38.5248i			1.63235i	0.577806	+	0.816174i	$0.303909\pi$
$557$			38.5248i			1.63235i	−0.577806	+	0.816174i	$0.696091\pi$
$558$	−18.5948			−0.787182
$559$	10.7373	−	5.10486i	0.454138	−	0.215913i
$560$	−25.1537			−1.06294
$561$		−	5.02506i		−	0.212158i
$562$	11.7872			0.497215
$563$	−13.7903			−0.581191			−0.290595	−	0.956846i	$-0.593853\pi$
$563$	−13.7903			−0.581191			−0.290595	+	0.956846i	$0.593853\pi$
$564$		−	27.4784i		−	1.15705i
$565$			18.5039i			0.778467i
$566$		−	45.7953i		−	1.92492i
$567$		−	1.00000i		−	0.0419961i
$568$	4.49840			0.188749
$569$	−5.29742			−0.222079			−0.111040	−	0.993816i	$-0.535418\pi$
$569$	−5.29742			−0.222079			−0.111040	+	0.993816i	$0.535418\pi$
$570$			2.77160i			0.116090i
$571$	−27.7928			−1.16309			−0.581546	−	0.813514i	$-0.697552\pi$
$571$	−27.7928			−1.16309			−0.581546	+	0.813514i	$0.697552\pi$
$572$	4.92559	+	10.3602i	0.205949	+	0.433182i
$573$	−4.98603			−0.208294
$574$			30.0793i			1.25548i
$575$	56.7534			2.36678
$576$	0.0195182			0.000813256
$577$		−	18.7333i		−	0.779879i	−0.920840	−	0.389940i	$-0.872496\pi$
$577$		−	18.7333i		−	0.779879i	0.920840	−	0.389940i	$-0.127504\pi$
$578$		−	84.9987i		−	3.53548i
$579$			6.98603i			0.290329i
$580$			83.1385i			3.45214i
$581$	6.11580			0.253726
$582$	−19.8726			−0.823745
$583$		−	2.63964i		−	0.109323i
$584$	33.1292			1.37090
$585$	5.40767	+	11.3742i	0.223580	+	0.470264i
$586$	−17.5063			−0.723179
$587$		−	8.93721i		−	0.368878i	−0.982844	−	0.184439i	$-0.940953\pi$
$587$		−	8.93721i		−	0.368878i	0.982844	−	0.184439i	$-0.0590469\pi$
$588$	−4.49301			−0.185289
$589$	2.27236			0.0936308
$590$		−	19.5423i		−	0.804544i
$591$			21.8044i			0.896913i
$592$		−	10.1986i		−	0.419161i
$593$			10.9655i			0.450298i	0.974324	+	0.225149i	$0.0722869\pi$
$593$			10.9655i			0.450298i	−0.974324	+	0.225149i	$0.927713\pi$
$594$	−1.80440			−0.0740356
$595$	−24.7874			−1.01618
$596$		−	70.8838i		−	2.90351i
$597$	−0.622783			−0.0254888
$598$	−31.0902	−	65.3933i	−1.27137	−	2.67413i
$599$	32.2663			1.31836			0.659182	−	0.751983i	$-0.270903\pi$
$599$	32.2663			1.31836			0.659182	+	0.751983i	$0.270903\pi$
$600$			45.7456i			1.86755i
$601$	−19.2999			−0.787260			−0.393630	−	0.919269i	$-0.628781\pi$
$601$	−19.2999			−0.787260			−0.393630	+	0.919269i	$0.628781\pi$
$602$	8.40228			0.342451
$603$			9.17858i			0.373781i
$604$		−	30.6985i		−	1.24911i
$605$		−	36.6716i		−	1.49091i
$606$			45.0794i			1.83123i
$607$	31.6632			1.28517			0.642584	−	0.766215i	$-0.277862\pi$
$607$	31.6632			1.28517			0.642584	+	0.766215i	$0.277862\pi$
$608$	1.75762			0.0712811
$609$			5.29742i			0.214662i
$610$	−22.3632			−0.905461
$611$	9.46810	+	19.9147i	0.383038	+	0.805661i
$612$	−31.8837			−1.28882
$613$		−	3.23478i		−	0.130652i	−0.997864	−	0.0653259i	$-0.979191\pi$
$613$		−	3.23478i		−	0.130652i	0.997864	−	0.0653259i	$-0.0208087\pi$
$614$	30.3743			1.22581
$615$	41.2329			1.66267
$616$			4.49840i			0.181246i
$617$			11.1365i			0.448339i	0.974550	+	0.224169i	$0.0719669\pi$
$617$			11.1365i			0.448339i	−0.974550	+	0.224169i	$0.928033\pi$
$618$			14.0823i			0.566473i
$619$		−	6.82643i		−	0.274377i	−0.990545	−	0.137189i	$-0.956193\pi$
$619$		−	6.82643i		−	0.274377i	0.990545	−	0.137189i	$-0.0438067\pi$
$620$	114.527			4.59951
$621$	7.88116			0.316260
$622$			63.1617i			2.53255i
$623$	−11.1018			−0.444785
$624$	23.4488	−	11.1484i	0.938704	−	0.446292i
$625$	−9.14929			−0.365972
$626$			17.1567i			0.685720i
$627$	0.220505			0.00880611
$628$	54.7813			2.18601
$629$		−	10.0501i		−	0.400725i
$630$			8.90068i			0.354612i
$631$			18.9689i			0.755138i	0.925981	+	0.377569i	$0.123240\pi$
$631$			18.9689i			0.755138i	−0.925981	+	0.377569i	$0.876760\pi$
$632$		−	17.6911i		−	0.703715i
$633$	−16.3547			−0.650039
$634$	−48.8062			−1.93834
$635$			49.9599i			1.98260i
$636$	−16.7484			−0.664115
$637$	3.25627	−	1.54814i	0.129018	−	0.0613395i
$638$	9.55868			0.378432
$639$		−	0.708126i		−	0.0280130i
$640$	−39.6058			−1.56556
$641$	−8.45413			−0.333918			−0.166959	−	0.985964i	$-0.553395\pi$
$641$	−8.45413			−0.333918			−0.166959	+	0.985964i	$0.553395\pi$
$642$			48.8697i			1.92873i
$643$			37.4102i			1.47531i	0.675176	+	0.737657i	$0.264068\pi$
$643$			37.4102i			1.47531i	−0.675176	+	0.737657i	$0.735932\pi$
$644$		−	35.4102i		−	1.39536i
$645$		−	11.5179i		−	0.453518i
$646$	5.63068			0.221536
$647$	−17.2627			−0.678668			−0.339334	−	0.940666i	$-0.610202\pi$
$647$	−17.2627			−0.678668			−0.339334	+	0.940666i	$0.610202\pi$
$648$			6.35254i			0.249551i
$649$	−1.55476			−0.0610296
$650$	−28.4076	−	59.7509i	−1.11424	−	2.34362i
$651$	7.29742			0.286008
$652$		−	45.1055i		−	1.76647i
$653$	27.2456			1.06620			0.533101	−	0.846052i	$-0.321027\pi$
$653$	27.2456			1.06620			0.533101	+	0.846052i	$0.321027\pi$
$654$	8.17069			0.319499
$655$		−	43.3213i		−	1.69270i
$656$		−	85.0052i		−	3.31889i
$657$		−	5.21511i		−	0.203461i
$658$			15.5839i			0.607524i
$659$	−36.0458			−1.40414			−0.702072	−	0.712106i	$-0.747742\pi$
$659$	−36.0458			−1.40414			−0.702072	+	0.712106i	$0.747742\pi$
$660$	11.1134			0.432590
$661$			25.3626i			0.986489i	0.869891	+	0.493245i	$0.164189\pi$
$661$			25.3626i			0.986489i	−0.869891	+	0.493245i	$0.835811\pi$
$662$	−59.9441			−2.32979
$663$	23.1074	−	10.9860i	0.897416	−	0.426662i
$664$	−38.8509			−1.50771
$665$		−	1.08770i		−	0.0421790i
$666$	−3.60881			−0.139839
$667$	−41.7498			−1.61656
$668$			111.025i			4.29569i
$669$		−	8.71367i		−	0.336890i
$670$		−	81.6957i		−	3.15618i
$671$			1.77919i			0.0686849i
$672$	5.64442			0.217738
$673$	−40.7355			−1.57024			−0.785120	−	0.619344i	$-0.787399\pi$
$673$	−40.7355			−1.57024			−0.785120	+	0.619344i	$0.787399\pi$
$674$		−	75.7455i		−	2.91761i
$675$	7.20114			0.277172
$676$	−36.8720	+	45.2999i	−1.41816	+	1.74230i
$677$	−38.7333			−1.48864			−0.744322	−	0.667821i	$-0.767227\pi$
$677$	−38.7333			−1.48864			−0.744322	+	0.667821i	$0.767227\pi$
$678$		−	13.4986i		−	0.518409i
$679$	7.79886			0.299293
$680$	157.463			6.03843
$681$		−	12.9440i		−	0.496013i
$682$		−	13.1675i		−	0.504209i
$683$			32.0682i			1.22705i	0.789673	+	0.613527i	$0.210250\pi$
$683$			32.0682i			1.22705i	−0.789673	+	0.613527i	$0.789750\pi$
$684$		−	1.39909i		−	0.0534954i
$685$	−19.5046			−0.745234
$686$	2.54814			0.0972884
$687$		−	19.6521i		−	0.749774i
$688$	−23.7452			−0.905276
$689$	12.1382	−	5.77091i	0.462429	−	0.219854i
$690$	−70.1477			−2.67048
$691$		−	14.4259i		−	0.548786i	−0.961618	−	0.274393i	$-0.911523\pi$
$691$		−	14.4259i		−	0.548786i	0.961618	−	0.274393i	$-0.0884769\pi$
$692$	−47.0575			−1.78886
$693$	0.708126			0.0268995
$694$			82.0594i			3.11493i
$695$			5.33210i			0.202258i
$696$		−	33.6521i		−	1.27558i
$697$		−	83.7673i		−	3.17291i
$698$	−32.0433			−1.21286
$699$	−3.72764			−0.140992
$700$		−	32.3548i		−	1.22290i
$701$	11.0490			0.417314			0.208657	−	0.977989i	$-0.433091\pi$
$701$	11.0490			0.417314			0.208657	+	0.977989i	$0.433091\pi$
$702$	−3.94487	−	8.29742i	−0.148890	−	0.313166i
$703$	0.441009			0.0166330
$704$			0.0138213i			0.000520911i
$705$	21.3626			0.804560
$706$	39.2750			1.47813
$707$		−	17.6911i		−	0.665343i
$708$			9.86484i			0.370743i
$709$		−	19.8293i		−	0.744706i	−0.928091	−	0.372353i	$-0.878551\pi$
$709$		−	19.8293i		−	0.744706i	0.928091	−	0.372353i	$-0.121449\pi$
$710$			6.30281i			0.236540i
$711$	−2.78489			−0.104441
$712$	70.5248			2.64303
$713$			57.5122i			2.15385i
$714$	18.0823			0.676713
$715$	−8.05436	+	3.82931i	−0.301216	+	0.143208i
$716$	−50.5841			−1.89041
$717$		−	8.86165i		−	0.330944i
$718$	62.9160			2.34800
$719$	−3.21261			−0.119810			−0.0599050	−	0.998204i	$-0.519080\pi$
$719$	−3.21261			−0.119810			−0.0599050	+	0.998204i	$0.519080\pi$
$720$		−	25.1537i		−	0.937422i
$721$		−	5.52651i		−	0.205818i
$722$		−	48.1676i		−	1.79261i
$723$		−	10.0025i		−	0.371997i
$724$	38.6167			1.43518
$725$	−38.1474			−1.41676
$726$			26.7518i			0.992852i
$727$	−13.3493			−0.495097			−0.247548	−	0.968876i	$-0.579625\pi$
$727$	−13.3493			−0.495097			−0.247548	+	0.968876i	$0.579625\pi$
$728$	−20.6856	+	9.83462i	−0.766658	+	0.364495i
$729$	1.00000			0.0370370
$730$			46.4181i			1.71801i
$731$	−23.3994			−0.865458
$732$	11.2888			0.417247
$733$		−	17.4185i		−	0.643365i	−0.946848	−	0.321683i	$-0.895752\pi$
$733$		−	17.4185i		−	0.643365i	0.946848	−	0.321683i	$-0.104248\pi$
$734$		−	27.7130i		−	1.02290i
$735$		−	3.49301i		−	0.128842i
$736$			44.4846i			1.63972i
$737$	−6.49960			−0.239416
$738$	−30.0793			−1.10723
$739$		−	1.80137i		−	0.0662643i	−0.999451	−	0.0331322i	$-0.989452\pi$
$739$		−	1.80137i		−	0.0662643i	0.999451	−	0.0331322i	$-0.0105482\pi$
$740$	22.2269			0.817077
$741$	0.482078	+	1.01397i	0.0177096	+	0.0372493i
$742$	9.49856			0.348703
$743$		−	3.86484i		−	0.141787i	−0.997484	−	0.0708936i	$-0.977415\pi$
$743$		−	3.86484i		−	0.141787i	0.997484	−	0.0708936i	$-0.0225851\pi$
$744$	−46.3572			−1.69954
$745$	55.1074			2.01898
$746$			4.38041i			0.160378i
$747$			6.11580i			0.223765i
$748$		−	22.5777i		−	0.825521i
$749$		−	19.1786i		−	0.700770i
$750$	−19.5916			−0.715386
$751$	29.8923			1.09078			0.545392	−	0.838181i	$-0.316381\pi$
$751$	29.8923			1.09078			0.545392	+	0.838181i	$0.316381\pi$
$752$		−	44.0407i		−	1.60600i
$753$	−9.24557			−0.336927
$754$	20.8976	+	43.9549i	0.761047	+	1.60074i
$755$	23.8660			0.868574
$756$		−	4.49301i		−	0.163409i
$757$	−38.8933			−1.41360			−0.706800	−	0.707413i	$-0.749862\pi$
$757$	−38.8933			−1.41360			−0.706800	+	0.707413i	$0.749862\pi$
$758$	−30.5777			−1.11063
$759$			5.58086i			0.202572i
$760$			6.90963i			0.250639i
$761$		−	8.33630i		−	0.302191i	−0.988519	−	0.151095i	$-0.951720\pi$
$761$		−	8.33630i		−	0.302191i	0.988519	−	0.151095i	$-0.0482800\pi$
$762$		−	36.4455i		−	1.32028i
$763$	−3.20653			−0.116084
$764$	−22.4023			−0.810486
$765$		−	24.7874i		−	0.896190i
$766$	−93.1415			−3.36534
$767$	−3.39909	−	7.14944i	−0.122734	−	0.258151i
$768$	28.8532			1.04115
$769$		−	17.0811i		−	0.615962i	−0.951393	−	0.307981i	$-0.900347\pi$
$769$		−	17.0811i		−	0.615962i	0.951393	−	0.307981i	$-0.0996533\pi$
$770$	−6.30281			−0.227137
$771$	14.5516			0.524062
$772$			31.3883i			1.12969i
$773$			16.0532i			0.577392i	0.957421	+	0.288696i	$0.0932217\pi$
$773$			16.0532i			0.577392i	−0.957421	+	0.288696i	$0.906778\pi$
$774$			8.40228i			0.302014i
$775$			52.5497i			1.88764i
$776$	−49.5426			−1.77848
$777$	1.41625			0.0508078
$778$		−	92.0812i		−	3.30127i
$779$	3.67579			0.131699
$780$	24.2967	+	51.1043i	0.869963	+	1.82983i
$781$	0.501443			0.0179430
$782$			142.510i			5.09613i
$783$	−5.29742			−0.189314
$784$	−7.20114			−0.257184

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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.54814i q^{2} -1.00000 q^{3} -4.49301 q^{4} -3.49301i q^{5} +2.54814i q^{6} -1.00000i q^{7} +6.35254i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.54814i q^{2} -1.00000 q^{3} -4.49301 q^{4} -3.49301i q^{5} +2.54814i q^{6} -1.00000i q^{7} +6.35254i q^{8} +1.00000 q^{9} -8.90068 q^{10} +0.708126i q^{11} +4.49301 q^{12} +(-3.25627 + 1.54814i) q^{13} -2.54814 q^{14} +3.49301i q^{15} +7.20114 q^{16} +7.09628 q^{17} -2.54814i q^{18} +0.311392i q^{19} +15.6942i q^{20} +1.00000i q^{21} +1.80440 q^{22} -7.88116 q^{23} -6.35254i q^{24} -7.20114 q^{25} +(3.94487 + 8.29742i) q^{26} -1.00000 q^{27} +4.49301i q^{28} +5.29742 q^{29} +8.90068 q^{30} -7.29742i q^{31} -5.64442i q^{32} -0.708126i q^{33} -18.0823i q^{34} -3.49301 q^{35} -4.49301 q^{36} -1.41625i q^{37} +0.793469 q^{38} +(3.25627 - 1.54814i) q^{39} +22.1895 q^{40} -11.8044i q^{41} +2.54814 q^{42} -3.29742 q^{43} -3.18162i q^{44} -3.49301i q^{45} +20.0823i q^{46} -6.11580i q^{47} -7.20114 q^{48} -1.00000 q^{49} +18.3495i q^{50} -7.09628 q^{51} +(14.6304 - 6.95581i) q^{52} -3.72764 q^{53} +2.54814i q^{54} +2.47349 q^{55} +6.35254 q^{56} -0.311392i q^{57} -13.4986i q^{58} +2.19560i q^{59} -15.6942i q^{60} +2.51253 q^{61} -18.5948 q^{62} -1.00000i q^{63} +0.0195182 q^{64} +(5.40767 + 11.3742i) q^{65} -1.80440 q^{66} +9.17858i q^{67} -31.8837 q^{68} +7.88116 q^{69} +8.90068i q^{70} -0.708126i q^{71} +6.35254i q^{72} -5.21511i q^{73} -3.60881 q^{74} +7.20114 q^{75} -1.39909i q^{76} +0.708126 q^{77} +(-3.94487 - 8.29742i) q^{78} -2.78489 q^{79} -25.1537i q^{80} +1.00000 q^{81} -30.0793 q^{82} +6.11580i q^{83} -4.49301i q^{84} -24.7874i q^{85} +8.40228i q^{86} -5.29742 q^{87} -4.49840 q^{88} -11.1018i q^{89} -8.90068 q^{90} +(1.54814 + 3.25627i) q^{91} +35.4102 q^{92} +7.29742i q^{93} -15.5839 q^{94} +1.08770 q^{95} +5.64442i q^{96} +7.79886i q^{97} +2.54814i q^{98} +0.708126i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * q^95

Introduction

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