LMFDB - Embedded newform 1520.2.q.h.961.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, not 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 4x + 16 $ x^4 - x^3 + 5x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.2
Root			$-0.780776 + 1.35234i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.h.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.780776 - 1.35234i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.56155 q^{7} +(0.280776 + 0.486319i) q^{9} +O(q^{10})$	\(q+(0.780776 - 1.35234i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.56155 q^{7} +(0.280776 + 0.486319i) q^{9} -1.00000 q^{11} +(1.00000 + 1.73205i) q^{13} +(0.780776 + 1.35234i) q^{15} +(1.56155 - 2.70469i) q^{17} +(2.50000 - 3.57071i) q^{19} +(-3.56155 + 6.16879i) q^{21} +(3.84233 + 6.65511i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.56155 q^{27} +(-1.00000 - 1.73205i) q^{29} +6.24621 q^{31} +(-0.780776 + 1.35234i) q^{33} +(2.28078 - 3.95042i) q^{35} +7.68466 q^{37} +3.12311 q^{39} +(-1.06155 + 1.83866i) q^{41} +(-2.56155 + 4.43674i) q^{43} -0.561553 q^{45} +(5.56155 + 9.63289i) q^{47} +13.8078 q^{49} +(-2.43845 - 4.22351i) q^{51} +(1.71922 + 2.97778i) q^{53} +(0.500000 - 0.866025i) q^{55} +(-2.87689 - 6.16879i) q^{57} +(5.21922 - 9.03996i) q^{59} +(1.56155 + 2.70469i) q^{61} +(-1.28078 - 2.21837i) q^{63} -2.00000 q^{65} +(6.78078 + 11.7446i) q^{67} +12.0000 q^{69} +(-0.123106 + 0.213225i) q^{71} +(-5.34233 + 9.25319i) q^{73} -1.56155 q^{75} +4.56155 q^{77} +(-1.43845 + 2.49146i) q^{79} +(3.50000 - 6.06218i) q^{81} -9.80776 q^{83} +(1.56155 + 2.70469i) q^{85} -3.12311 q^{87} +(-4.84233 - 8.38716i) q^{89} +(-4.56155 - 7.90084i) q^{91} +(4.87689 - 8.44703i) q^{93} +(1.84233 + 3.95042i) q^{95} +(5.34233 - 9.25319i) q^{97} +(-0.280776 - 0.486319i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9	$ 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - 2 q^{17} + 10 q^{19} - 6 q^{21} + 3 q^{23} - 2 q^{25} + 14 q^{27} - 4 q^{29} - 8 q^{31} + q^{33} + 5 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{43} + 6 q^{45} + 14 q^{47} + 14 q^{49} - 18 q^{51} + 11 q^{53} + 2 q^{55} - 28 q^{57} + 25 q^{59} - 2 q^{61} - q^{63} - 8 q^{65} + 23 q^{67} + 48 q^{69} + 16 q^{71} - 9 q^{73} + 2 q^{75} + 10 q^{77} - 14 q^{79} + 14 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87} - 7 q^{89} - 10 q^{91} + 36 q^{93} - 5 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9 - 4 * q^11 + 4 * q^13 - q^15 - 2 * q^17 + 10 * q^19 - 6 * q^21 + 3 * q^23 - 2 * q^25 + 14 * q^27 - 4 * q^29 - 8 * q^31 + q^33 + 5 * q^35 + 6 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^43 + 6 * q^45 + 14 * q^47 + 14 * q^49 - 18 * q^51 + 11 * q^53 + 2 * q^55 - 28 * q^57 + 25 * q^59 - 2 * q^61 - q^63 - 8 * q^65 + 23 * q^67 + 48 * q^69 + 16 * q^71 - 9 * q^73 + 2 * q^75 + 10 * q^77 - 14 * q^79 + 14 * q^81 + 2 * q^83 - 2 * q^85 + 4 * q^87 - 7 * q^89 - 10 * q^91 + 36 * q^93 - 5 * q^95 + 9 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.780776	−	1.35234i	0.450781	−	0.780776i	−0.547653	−	0.836705i	$-0.684479\pi$
$3$	0.780776	−	1.35234i	0.450781	−	0.780776i	0.998435	+	0.0559290i	$0.0178120\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−4.56155			−1.72410			−0.862052	−	0.506819i	$-0.830821\pi$
$7$	−4.56155			−1.72410			−0.862052	+	0.506819i	$0.830821\pi$
$8$		0			0
$9$	0.280776	+	0.486319i	0.0935921	+	0.162106i
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$	0.780776	+	1.35234i	0.201596	+	0.349174i
$16$		0			0
$17$	1.56155	−	2.70469i	0.378732	−	0.655983i	−0.612146	−	0.790745i	$-0.709693\pi$
$17$	1.56155	−	2.70469i	0.378732	−	0.655983i	0.990878	+	0.134761i	$0.0430268\pi$
$18$		0			0
$19$	2.50000	−	3.57071i	0.573539	−	0.819178i
$19$	2.50000	−	3.57071i	0.573539	−	0.819178i
$20$		0			0
$21$	−3.56155	+	6.16879i	−0.777195	+	1.34614i
$22$		0			0
$23$	3.84233	+	6.65511i	0.801181	+	1.38769i	0.918839	+	0.394632i	$0.129128\pi$
$23$	3.84233	+	6.65511i	0.801181	+	1.38769i	−0.117658	+	0.993054i	$0.537539\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	5.56155			1.07032
$28$		0			0
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	0.758115	−	0.652121i	$-0.226120\pi$
$29$	−1.00000	−	1.73205i	−0.185695	−	0.321634i	−0.943811	+	0.330487i	$0.892787\pi$
$30$		0			0
$31$	6.24621			1.12185			0.560926	−	0.827866i	$-0.310445\pi$
$31$	6.24621			1.12185			0.560926	+	0.827866i	$0.310445\pi$
$32$		0			0
$33$	−0.780776	+	1.35234i	−0.135916	+	0.235413i
$34$		0			0
$35$	2.28078	−	3.95042i	0.385522	−	0.667743i
$36$		0			0
$37$	7.68466			1.26335			0.631675	−	0.775233i	$-0.282368\pi$
$37$	7.68466			1.26335			0.631675	+	0.775233i	$0.282368\pi$
$38$		0			0
$39$	3.12311			0.500097
$40$		0			0
$41$	−1.06155	+	1.83866i	−0.165787	+	0.287151i	−0.936934	−	0.349505i	$-0.886350\pi$
$41$	−1.06155	+	1.83866i	−0.165787	+	0.287151i	0.771148	+	0.636656i	$0.219683\pi$
$42$		0			0
$43$	−2.56155	+	4.43674i	−0.390633	+	0.676596i	−0.992533	−	0.121975i	$-0.961077\pi$
$43$	−2.56155	+	4.43674i	−0.390633	+	0.676596i	0.601900	+	0.798571i	$0.294411\pi$
$44$		0			0
$45$	−0.561553			−0.0837114
$46$		0			0
$47$	5.56155	+	9.63289i	0.811236	+	1.40510i	0.911999	+	0.410191i	$0.134538\pi$
$47$	5.56155	+	9.63289i	0.811236	+	1.40510i	−0.100764	+	0.994910i	$0.532129\pi$
$48$		0			0
$49$	13.8078			1.97254
$50$		0			0
$51$	−2.43845	−	4.22351i	−0.341451	−	0.591410i
$52$		0			0
$53$	1.71922	+	2.97778i	0.236154	+	0.409030i	0.959607	−	0.281343i	$-0.0907798\pi$
$53$	1.71922	+	2.97778i	0.236154	+	0.409030i	−0.723454	+	0.690373i	$0.757447\pi$
$54$		0			0
$55$	0.500000	−	0.866025i	0.0674200	−	0.116775i
$56$		0			0
$57$	−2.87689	−	6.16879i	−0.381054	−	0.817076i
$58$		0			0
$59$	5.21922	−	9.03996i	0.679485	−	1.17690i	−0.295651	−	0.955296i	$-0.595537\pi$
$59$	5.21922	−	9.03996i	0.679485	−	1.17690i	0.975136	−	0.221607i	$-0.0711301\pi$
$60$		0			0
$61$	1.56155	+	2.70469i	0.199936	+	0.346300i	0.948508	−	0.316754i	$-0.102593\pi$
$61$	1.56155	+	2.70469i	0.199936	+	0.346300i	−0.748571	+	0.663054i	$0.769260\pi$
$62$		0			0
$63$	−1.28078	−	2.21837i	−0.161363	−	0.279488i
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	6.78078	+	11.7446i	0.828404	+	1.43484i	0.899290	+	0.437353i	$0.144084\pi$
$67$	6.78078	+	11.7446i	0.828404	+	1.43484i	−0.0708863	+	0.997484i	$0.522583\pi$
$68$		0			0
$69$	12.0000			1.44463
$70$		0			0
$71$	−0.123106	+	0.213225i	−0.0146099	+	0.0253052i	−0.873238	−	0.487294i	$-0.837984\pi$
$71$	−0.123106	+	0.213225i	−0.0146099	+	0.0253052i	0.858628	+	0.512599i	$0.171317\pi$
$72$		0			0
$73$	−5.34233	+	9.25319i	−0.625272	+	1.08300i	0.363216	+	0.931705i	$0.381679\pi$
$73$	−5.34233	+	9.25319i	−0.625272	+	1.08300i	−0.988488	+	0.151298i	$0.951655\pi$
$74$		0			0
$75$	−1.56155			−0.180313
$76$		0			0
$77$	4.56155			0.519837
$78$		0			0
$79$	−1.43845	+	2.49146i	−0.161838	+	0.280312i	−0.935528	−	0.353253i	$-0.885076\pi$
$79$	−1.43845	+	2.49146i	−0.161838	+	0.280312i	0.773690	+	0.633564i	$0.218409\pi$
$80$		0			0
$81$	3.50000	−	6.06218i	0.388889	−	0.673575i
$82$		0			0
$83$	−9.80776			−1.07654			−0.538271	−	0.842772i	$-0.680922\pi$
$83$	−9.80776			−1.07654			−0.538271	+	0.842772i	$0.680922\pi$
$84$		0			0
$85$	1.56155	+	2.70469i	0.169374	+	0.293365i
$86$		0			0
$87$	−3.12311			−0.334832
$88$		0			0
$89$	−4.84233	−	8.38716i	−0.513286	−	0.889037i	−0.999881	−	0.0154098i	$-0.995095\pi$
$89$	−4.84233	−	8.38716i	−0.513286	−	0.889037i	0.486595	−	0.873627i	$-0.338239\pi$
$90$		0			0
$91$	−4.56155	−	7.90084i	−0.478181	−	0.828233i
$92$		0			0
$93$	4.87689	−	8.44703i	0.505710	−	0.875916i
$94$		0			0
$95$	1.84233	+	3.95042i	0.189019	+	0.405305i
$96$		0			0
$97$	5.34233	−	9.25319i	0.542431	−	0.939519i	−0.456332	−	0.889809i	$-0.650837\pi$
$97$	5.34233	−	9.25319i	0.542431	−	0.939519i	0.998764	−	0.0497093i	$-0.0158295\pi$
$98$		0			0
$99$	−0.280776	−	0.486319i	−0.0282191	−	0.0488769i
$100$		0			0
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	0.502477	−	0.864590i	$-0.332422\pi$
$101$	−5.00000	−	8.66025i	−0.497519	−	0.861727i	−0.999996	+	0.00286291i	$0.999089\pi$
$102$		0			0
$103$	−14.8078			−1.45905			−0.729526	−	0.683953i	$-0.760259\pi$
$103$	−14.8078			−1.45905			−0.729526	+	0.683953i	$0.760259\pi$
$104$		0			0
$105$	−3.56155	−	6.16879i	−0.347572	−	0.602012i
$106$		0			0
$107$	14.2462			1.37723			0.688617	−	0.725126i	$-0.258218\pi$
$107$	14.2462			1.37723			0.688617	+	0.725126i	$0.258218\pi$
$108$		0			0
$109$	−1.12311	+	1.94528i	−0.107574	+	0.186324i	−0.914787	−	0.403937i	$-0.867642\pi$
$109$	−1.12311	+	1.94528i	−0.107574	+	0.186324i	0.807213	+	0.590260i	$0.200975\pi$
$110$		0			0
$111$	6.00000	−	10.3923i	0.569495	−	0.986394i
$112$		0			0
$113$	3.80776			0.358204			0.179102	−	0.983830i	$-0.442681\pi$
$113$	3.80776			0.358204			0.179102	+	0.983830i	$0.442681\pi$
$114$		0			0
$115$	−7.68466			−0.716598
$116$		0			0
$117$	−0.561553	+	0.972638i	−0.0519156	+	0.0899204i
$118$		0			0
$119$	−7.12311	+	12.3376i	−0.652974	+	1.13098i
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$	1.65767	+	2.87117i	0.149467	+	0.258885i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−5.71922	−	9.90599i	−0.507499	−	0.879014i	−0.999962	−	0.00868089i	$-0.997237\pi$
$127$	−5.71922	−	9.90599i	−0.507499	−	0.879014i	0.492463	−	0.870333i	$-0.336097\pi$
$128$		0			0
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−3.93845	+	6.82159i	−0.344104	+	0.596005i	−0.985191	−	0.171463i	$-0.945151\pi$
$131$	−3.93845	+	6.82159i	−0.344104	+	0.596005i	0.641087	+	0.767468i	$0.278484\pi$
$132$		0			0
$133$	−11.4039	+	16.2880i	−0.988842	+	1.41235i
$134$		0			0
$135$	−2.78078	+	4.81645i	−0.239331	+	0.414534i
$136$		0			0
$137$	4.78078	+	8.28055i	0.408449	+	0.707455i	0.994716	−	0.102663i	$-0.0327363\pi$
$137$	4.78078	+	8.28055i	0.408449	+	0.707455i	−0.586267	+	0.810118i	$0.699403\pi$
$138$		0			0
$139$	−5.90388	−	10.2258i	−0.500761	−	0.867343i	−1.00000		0.000878648i	$-0.999720\pi$
$139$	−5.90388	−	10.2258i	−0.500761	−	0.867343i	0.499239	−	0.866464i	$-0.333613\pi$
$140$		0			0
$141$	17.3693			1.46276
$142$		0			0
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$		0			0
$145$	2.00000			0.166091
$146$		0			0
$147$	10.7808	−	18.6729i	0.889183	−	1.54011i
$148$		0			0
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	0.326411	+	0.945228i	$0.394160\pi$
$149$	−8.00000	+	13.8564i	−0.655386	+	1.13516i	−0.981797	+	0.189933i	$0.939173\pi$
$150$		0			0
$151$	12.4924			1.01662			0.508309	−	0.861174i	$-0.330271\pi$
$151$	12.4924			1.01662			0.508309	+	0.861174i	$0.330271\pi$
$152$		0			0
$153$	1.75379			0.141785
$154$		0			0
$155$	−3.12311	+	5.40938i	−0.250854	+	0.434492i
$156$		0			0
$157$	−3.28078	+	5.68247i	−0.261834	+	0.453511i	−0.966729	−	0.255801i	$-0.917661\pi$
$157$	−3.28078	+	5.68247i	−0.261834	+	0.453511i	0.704895	+	0.709312i	$0.250994\pi$
$158$		0			0
$159$	5.36932			0.425815
$160$		0			0
$161$	−17.5270	−	30.3576i	−1.38132	−	2.39252i
$162$		0			0
$163$	20.0540			1.57075			0.785374	−	0.619021i	$-0.212471\pi$
$163$	20.0540			1.57075			0.785374	+	0.619021i	$0.212471\pi$
$164$		0			0
$165$	−0.780776	−	1.35234i	−0.0607834	−	0.105280i
$166$		0			0
$167$	−5.84233	−	10.1192i	−0.452093	−	0.783048i	0.546423	−	0.837509i	$-0.315989\pi$
$167$	−5.84233	−	10.1192i	−0.452093	−	0.783048i	−0.998516	+	0.0544614i	$0.982656\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$	2.43845	+	0.213225i	0.186473	+	0.0163057i
$172$		0			0
$173$	−7.40388	+	12.8239i	−0.562907	+	0.974983i	0.434334	+	0.900752i	$0.356984\pi$
$173$	−7.40388	+	12.8239i	−0.562907	+	0.974983i	−0.997241	+	0.0742313i	$0.976350\pi$
$174$		0			0
$175$	2.28078	+	3.95042i	0.172410	+	0.298624i
$176$		0			0
$177$	−8.15009	−	14.1164i	−0.612599	−	1.06105i
$178$		0			0
$179$	21.4924			1.60642			0.803210	−	0.595697i	$-0.203124\pi$
$179$	21.4924			1.60642			0.803210	+	0.595697i	$0.203124\pi$
$180$		0			0
$181$	1.68466	+	2.91791i	0.125220	+	0.216887i	0.921819	−	0.387621i	$-0.126703\pi$
$181$	1.68466	+	2.91791i	0.125220	+	0.216887i	−0.796599	+	0.604508i	$0.793370\pi$
$182$		0			0
$183$	4.87689			0.360510
$184$		0			0
$185$	−3.84233	+	6.65511i	−0.282494	+	0.489293i
$186$		0			0
$187$	−1.56155	+	2.70469i	−0.114192	+	0.197786i
$188$		0			0
$189$	−25.3693			−1.84535
$190$		0			0
$191$	−19.3693			−1.40151			−0.700757	−	0.713400i	$-0.747154\pi$
$191$	−19.3693			−1.40151			−0.700757	+	0.713400i	$0.747154\pi$
$192$		0			0
$193$	7.80776	−	13.5234i	0.562015	−	0.973439i	−0.435305	−	0.900283i	$-0.643360\pi$
$193$	7.80776	−	13.5234i	0.562015	−	0.973439i	0.997321	−	0.0731559i	$-0.0233071\pi$
$194$		0			0
$195$	−1.56155	+	2.70469i	−0.111825	+	0.193687i
$196$		0			0
$197$	18.5616			1.32246			0.661228	−	0.750185i	$-0.270036\pi$
$197$	18.5616			1.32246			0.661228	+	0.750185i	$0.270036\pi$
$198$		0			0
$199$	−5.56155	−	9.63289i	−0.394248	−	0.682858i	0.598757	−	0.800931i	$-0.295662\pi$
$199$	−5.56155	−	9.63289i	−0.394248	−	0.682858i	−0.993005	+	0.118073i	$0.962328\pi$
$200$		0			0
$201$	21.1771			1.49372
$202$		0			0
$203$	4.56155	+	7.90084i	0.320158	+	0.554530i
$204$		0			0
$205$	−1.06155	−	1.83866i	−0.0741421	−	0.128418i
$206$		0			0
$207$	−2.15767	+	3.73720i	−0.149968	+	0.259753i
$208$		0			0
$209$	−2.50000	+	3.57071i	−0.172929	+	0.246991i
$210$		0			0
$211$	−7.84233	+	13.5833i	−0.539888	+	0.935114i	0.459021	+	0.888425i	$0.348200\pi$
$211$	−7.84233	+	13.5833i	−0.539888	+	0.935114i	−0.998909	+	0.0466885i	$0.985133\pi$
$212$		0			0
$213$	0.192236	+	0.332962i	0.0131718	+	0.0228142i
$214$		0			0
$215$	−2.56155	−	4.43674i	−0.174696	−	0.302583i
$216$		0			0
$217$	−28.4924			−1.93419
$218$		0			0
$219$	8.34233	+	14.4493i	0.563722	+	0.976396i
$220$		0			0
$221$	6.24621			0.420166
$222$		0			0
$223$	11.8423	−	20.5115i	0.793021	−	1.37355i	−0.131067	−	0.991373i	$-0.541840\pi$
$223$	11.8423	−	20.5115i	0.793021	−	1.37355i	0.924088	−	0.382179i	$-0.124826\pi$
$224$		0			0
$225$	0.280776	−	0.486319i	0.0187184	−	0.0324213i
$226$		0			0
$227$	8.93087			0.592763			0.296381	−	0.955070i	$-0.404220\pi$
$227$	8.93087			0.592763			0.296381	+	0.955070i	$0.404220\pi$
$228$		0			0
$229$	21.1231			1.39585			0.697927	−	0.716169i	$-0.254106\pi$
$229$	21.1231			1.39585			0.697927	+	0.716169i	$0.254106\pi$
$230$		0			0
$231$	3.56155	−	6.16879i	0.234333	−	0.405877i
$232$		0			0
$233$	−7.34233	+	12.7173i	−0.481012	+	0.833137i	−0.999763	−	0.0217884i	$-0.993064\pi$
$233$	−7.34233	+	12.7173i	−0.481012	+	0.833137i	0.518751	+	0.854926i	$0.326397\pi$
$234$		0			0
$235$	−11.1231			−0.725591
$236$		0			0
$237$	2.24621	+	3.89055i	0.145907	+	0.252719i
$238$		0			0
$239$	−1.75379			−0.113443			−0.0567216	−	0.998390i	$-0.518065\pi$
$239$	−1.75379			−0.113443			−0.0567216	+	0.998390i	$0.518065\pi$
$240$		0			0
$241$	2.21922	+	3.84381i	0.142953	+	0.247601i	0.928607	−	0.371064i	$-0.121007\pi$
$241$	2.21922	+	3.84381i	0.142953	+	0.247601i	−0.785655	+	0.618665i	$0.787674\pi$
$242$		0			0
$243$	2.87689	+	4.98293i	0.184553	+	0.319655i
$244$		0			0
$245$	−6.90388	+	11.9579i	−0.441073	+	0.763961i
$246$		0			0
$247$	8.68466	+	0.759413i	0.552592	+	0.0483203i
$248$		0			0
$249$	−7.65767	+	13.2635i	−0.485285	+	0.840539i
$250$		0			0
$251$	−1.46543	−	2.53821i	−0.0924974	−	0.160210i	0.816064	−	0.577962i	$-0.196152\pi$
$251$	−1.46543	−	2.53821i	−0.0924974	−	0.160210i	−0.908561	+	0.417751i	$0.862818\pi$
$252$		0			0
$253$	−3.84233	−	6.65511i	−0.241565	−	0.418403i
$254$		0			0
$255$	4.87689			0.305403
$256$		0			0
$257$	12.0270	+	20.8314i	0.750223	+	1.29942i	0.947714	+	0.319120i	$0.103387\pi$
$257$	12.0270	+	20.8314i	0.750223	+	1.29942i	−0.197492	+	0.980305i	$0.563280\pi$
$258$		0			0
$259$	−35.0540			−2.17815
$260$		0			0
$261$	0.561553	−	0.972638i	0.0347592	−	0.0602048i
$262$		0			0
$263$	−11.4039	+	19.7521i	−0.703193	+	1.21797i	0.264146	+	0.964483i	$0.414910\pi$
$263$	−11.4039	+	19.7521i	−0.703193	+	1.21797i	−0.967340	+	0.253484i	$0.918423\pi$
$264$		0			0
$265$	−3.43845			−0.211222
$266$		0			0
$267$	−15.1231			−0.925519
$268$		0			0
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	−0.920534	−	0.390664i	$-0.872246\pi$
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	0.798591	+	0.601874i	$0.205579\pi$
$270$		0			0
$271$	4.12311	−	7.14143i	0.250461	−	0.433811i	−0.713192	−	0.700969i	$-0.752751\pi$
$271$	4.12311	−	7.14143i	0.250461	−	0.433811i	0.963653	+	0.267158i	$0.0860845\pi$
$272$		0			0
$273$	−14.2462			−0.862220
$274$		0			0
$275$	0.500000	+	0.866025i	0.0301511	+	0.0522233i
$276$		0			0
$277$	−12.2462			−0.735804			−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462			−0.735804			−0.367902	+	0.929865i	$0.619924\pi$
$278$		0			0
$279$	1.75379	+	3.03765i	0.104997	+	0.181859i
$280$		0			0
$281$	8.18466	+	14.1762i	0.488256	+	0.845684i	0.999909	−	0.0135084i	$-0.00429998\pi$
$281$	8.18466	+	14.1762i	0.488256	+	0.845684i	−0.511653	+	0.859192i	$0.670967\pi$
$282$		0			0
$283$	−11.7808	+	20.4049i	−0.700294	+	1.21295i	0.268069	+	0.963400i	$0.413615\pi$
$283$	−11.7808	+	20.4049i	−0.700294	+	1.21295i	−0.968363	+	0.249546i	$0.919719\pi$
$284$		0			0
$285$	6.78078	+	0.592932i	0.401659	+	0.0351222i
$286$		0			0
$287$	4.84233	−	8.38716i	0.285834	−	0.495078i
$288$		0			0
$289$	3.62311	+	6.27540i	0.213124	+	0.369141i
$290$		0			0
$291$	−8.34233	−	14.4493i	−0.489036	−	0.847035i
$292$		0			0
$293$	19.4384			1.13561			0.567803	−	0.823164i	$-0.307793\pi$
$293$	19.4384			1.13561			0.567803	+	0.823164i	$0.307793\pi$
$294$		0			0
$295$	5.21922	+	9.03996i	0.303875	+	0.526327i
$296$		0			0
$297$	−5.56155			−0.322714
$298$		0			0
$299$	−7.68466	+	13.3102i	−0.444415	+	0.769750i
$300$		0			0
$301$	11.6847	−	20.2384i	0.673493	−	1.16652i
$302$		0			0
$303$	−15.6155			−0.897089
$304$		0			0
$305$	−3.12311			−0.178829
$306$		0			0
$307$	2.46543	−	4.27026i	0.140710	−	0.243717i	−0.787054	−	0.616884i	$-0.788395\pi$
$307$	2.46543	−	4.27026i	0.140710	−	0.243717i	0.927764	+	0.373167i	$0.121728\pi$
$308$		0			0
$309$	−11.5616	+	20.0252i	−0.657714	+	1.13919i
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$	−8.34233	−	14.4493i	−0.471536	−	0.816725i	0.527933	−	0.849286i	$-0.322967\pi$
$313$	−8.34233	−	14.4493i	−0.471536	−	0.816725i	−0.999470	+	0.0325609i	$0.989634\pi$
$314$		0			0
$315$	2.56155			0.144327
$316$		0			0
$317$	−3.15767	−	5.46925i	−0.177352	−	0.307183i	0.763620	−	0.645665i	$-0.223420\pi$
$317$	−3.15767	−	5.46925i	−0.177352	−	0.307183i	−0.940973	+	0.338482i	$0.890087\pi$
$318$		0			0
$319$	1.00000	+	1.73205i	0.0559893	+	0.0969762i
$320$		0			0
$321$	11.1231	−	19.2658i	0.620831	−	1.07531i
$322$		0			0
$323$	−5.75379	−	12.3376i	−0.320149	−	0.686481i
$324$		0			0
$325$	1.00000	−	1.73205i	0.0554700	−	0.0960769i
$326$		0			0
$327$	1.75379	+	3.03765i	0.0969847	+	0.167982i
$328$		0			0
$329$	−25.3693	−	43.9409i	−1.39866	−	2.42254i
$330$		0			0
$331$	9.49242			0.521751			0.260875	−	0.965372i	$-0.415989\pi$
$331$	9.49242			0.521751			0.260875	+	0.965372i	$0.415989\pi$
$332$		0			0
$333$	2.15767	+	3.73720i	0.118240	+	0.204797i
$334$		0			0
$335$	−13.5616			−0.740947
$336$		0			0
$337$	8.02699	−	13.9032i	0.437258	−	0.757353i	−0.560219	−	0.828345i	$-0.689283\pi$
$337$	8.02699	−	13.9032i	0.437258	−	0.757353i	0.997477	+	0.0709917i	$0.0226164\pi$
$338$		0			0
$339$	2.97301	−	5.14941i	0.161472	−	0.279677i
$340$		0			0
$341$	−6.24621			−0.338251
$342$		0			0
$343$	−31.0540			−1.67676
$344$		0			0
$345$	−6.00000	+	10.3923i	−0.323029	+	0.559503i
$346$		0			0
$347$	−5.34233	+	9.25319i	−0.286791	+	0.496737i	−0.973042	−	0.230628i	$-0.925922\pi$
$347$	−5.34233	+	9.25319i	−0.286791	+	0.496737i	0.686251	+	0.727365i	$0.259255\pi$
$348$		0			0
$349$	2.24621			0.120237			0.0601185	−	0.998191i	$-0.480852\pi$
$349$	2.24621			0.120237			0.0601185	+	0.998191i	$0.480852\pi$
$350$		0			0
$351$	5.56155	+	9.63289i	0.296854	+	0.514166i
$352$		0			0
$353$	−21.1771			−1.12714			−0.563571	−	0.826068i	$-0.690573\pi$
$353$	−21.1771			−1.12714			−0.563571	+	0.826068i	$0.690573\pi$
$354$		0			0
$355$	−0.123106	−	0.213225i	−0.00653377	−	0.0113168i
$356$		0			0
$357$	11.1231	+	19.2658i	0.588697	+	1.01965i
$358$		0			0
$359$	3.43845	−	5.95557i	0.181474	−	0.314323i	−0.760908	−	0.648859i	$-0.775246\pi$
$359$	3.43845	−	5.95557i	0.181474	−	0.314323i	0.942383	+	0.334536i	$0.108580\pi$
$360$		0			0
$361$	−6.50000	−	17.8536i	−0.342105	−	0.939662i
$362$		0			0
$363$	−7.80776	+	13.5234i	−0.409801	+	0.709797i
$364$		0			0
$365$	−5.34233	−	9.25319i	−0.279630	−	0.484334i
$366$		0			0
$367$	−17.1231	−	29.6581i	−0.893819	−	1.54814i	−0.835260	−	0.549856i	$-0.814683\pi$
$367$	−17.1231	−	29.6581i	−0.893819	−	1.54814i	−0.0585592	−	0.998284i	$-0.518651\pi$
$368$		0			0
$369$	−1.19224			−0.0620653
$370$		0			0
$371$	−7.84233	−	13.5833i	−0.407153	−	0.705210i
$372$		0			0
$373$	−15.1922			−0.786624			−0.393312	−	0.919405i	$-0.628671\pi$
$373$	−15.1922			−0.786624			−0.393312	+	0.919405i	$0.628671\pi$
$374$		0			0
$375$	0.780776	−	1.35234i	0.0403191	−	0.0698348i
$376$		0			0
$377$	2.00000	−	3.46410i	0.103005	−	0.178410i
$378$		0			0
$379$	−32.4924			−1.66902			−0.834512	−	0.550990i	$-0.814250\pi$
$379$	−32.4924			−1.66902			−0.834512	+	0.550990i	$0.814250\pi$
$380$		0			0
$381$	−17.8617			−0.915085
$382$		0			0
$383$	−1.56155	+	2.70469i	−0.0797916	+	0.138203i	−0.903160	−	0.429304i	$-0.858759\pi$
$383$	−1.56155	+	2.70469i	−0.0797916	+	0.138203i	0.823368	+	0.567507i	$0.192092\pi$
$384$		0			0
$385$	−2.28078	+	3.95042i	−0.116239	+	0.201332i
$386$		0			0
$387$	−2.87689			−0.146241
$388$		0			0
$389$	5.43845	+	9.41967i	0.275740	+	0.477596i	0.970322	−	0.241818i	$-0.0777437\pi$
$389$	5.43845	+	9.41967i	0.275740	+	0.477596i	−0.694581	+	0.719414i	$0.744410\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	6.15009	+	10.6523i	0.310231	+	0.537336i
$394$		0			0
$395$	−1.43845	−	2.49146i	−0.0723761	−	0.125359i
$396$		0			0
$397$	12.6501	−	21.9106i	0.634890	−	1.09966i	−0.351648	−	0.936132i	$-0.614379\pi$
$397$	12.6501	−	21.9106i	0.634890	−	1.09966i	0.986538	−	0.163530i	$-0.0522879\pi$
$398$		0			0
$399$	13.1231	+	28.1393i	0.656977	+	1.40873i
$400$		0			0
$401$	0.465435	−	0.806157i	0.0232427	−	0.0402575i	−0.854170	−	0.519994i	$-0.825934\pi$
$401$	0.465435	−	0.806157i	0.0232427	−	0.0402575i	0.877413	+	0.479736i	$0.159268\pi$
$402$		0			0
$403$	6.24621	+	10.8188i	0.311146	+	0.538921i
$404$		0			0
$405$	3.50000	+	6.06218i	0.173916	+	0.301232i
$406$		0			0
$407$	−7.68466			−0.380914
$408$		0			0
$409$	−10.5000	−	18.1865i	−0.519192	−	0.899266i	−0.999751	−	0.0223042i	$-0.992900\pi$
$409$	−10.5000	−	18.1865i	−0.519192	−	0.899266i	0.480560	−	0.876962i	$-0.340434\pi$
$410$		0			0
$411$	14.9309			0.736485
$412$		0			0
$413$	−23.8078	+	41.2363i	−1.17150	+	2.02910i
$414$		0			0
$415$	4.90388	−	8.49377i	0.240722	−	0.416943i
$416$		0			0
$417$	−18.4384			−0.902935
$418$		0			0
$419$	13.4384			0.656511			0.328256	−	0.944589i	$-0.393539\pi$
$419$	13.4384			0.656511			0.328256	+	0.944589i	$0.393539\pi$
$420$		0			0
$421$	0.246211	−	0.426450i	0.0119996	−	0.0207839i	−0.859963	−	0.510356i	$-0.829514\pi$
$421$	0.246211	−	0.426450i	0.0119996	−	0.0207839i	0.871963	+	0.489572i	$0.162847\pi$
$422$		0			0
$423$	−3.12311	+	5.40938i	−0.151851	+	0.263013i
$424$		0			0
$425$	−3.12311			−0.151493
$426$		0			0
$427$	−7.12311	−	12.3376i	−0.344711	−	0.597057i
$428$		0			0
$429$	−3.12311			−0.150785
$430$		0			0
$431$	8.68466	+	15.0423i	0.418325	+	0.724561i	0.995771	−	0.0918683i	$-0.0292839\pi$
$431$	8.68466	+	15.0423i	0.418325	+	0.724561i	−0.577446	+	0.816429i	$0.695951\pi$
$432$		0			0
$433$	−6.36932	−	11.0320i	−0.306090	−	0.530163i	0.671414	−	0.741083i	$-0.265687\pi$
$433$	−6.36932	−	11.0320i	−0.306090	−	0.530163i	−0.977503	+	0.210920i	$0.932354\pi$
$434$		0			0
$435$	1.56155	−	2.70469i	0.0748707	−	0.129680i
$436$		0			0
$437$	33.3693	+	2.91791i	1.59627	+	0.139583i
$438$		0			0
$439$	8.24621	−	14.2829i	0.393570	−	0.681684i	−0.599347	−	0.800489i	$-0.704573\pi$
$439$	8.24621	−	14.2829i	0.393570	−	0.681684i	0.992918	+	0.118806i	$0.0379065\pi$
$440$		0			0
$441$	3.87689	+	6.71498i	0.184614	+	0.319761i
$442$		0			0
$443$	2.53457	+	4.39000i	0.120421	+	0.208575i	0.919934	−	0.392074i	$-0.128242\pi$
$443$	2.53457	+	4.39000i	0.120421	+	0.208575i	−0.799513	+	0.600649i	$0.794909\pi$
$444$		0			0
$445$	9.68466			0.459097
$446$		0			0
$447$	12.4924	+	21.6375i	0.590871	+	1.02342i
$448$		0			0
$449$	29.0000			1.36859			0.684297	−	0.729203i	$-0.260109\pi$
$449$	29.0000			1.36859			0.684297	+	0.729203i	$0.260109\pi$
$450$		0			0
$451$	1.06155	−	1.83866i	0.0499866	−	0.0865793i
$452$		0			0
$453$	9.75379	−	16.8941i	0.458273	−	0.793752i
$454$		0			0
$455$	9.12311			0.427698
$456$		0			0
$457$	−30.0540			−1.40587			−0.702933	−	0.711256i	$-0.748127\pi$
$457$	−30.0540			−1.40587			−0.702933	+	0.711256i	$0.748127\pi$
$458$		0			0
$459$	8.68466	−	15.0423i	0.405365	−	0.702113i
$460$		0			0
$461$	−13.4924	+	23.3696i	−0.628405	+	1.08843i	0.359467	+	0.933158i	$0.382958\pi$
$461$	−13.4924	+	23.3696i	−0.628405	+	1.08843i	−0.987872	+	0.155271i	$0.950375\pi$
$462$		0			0
$463$	15.4384			0.717485			0.358743	−	0.933436i	$-0.383206\pi$
$463$	15.4384			0.717485			0.358743	+	0.933436i	$0.383206\pi$
$464$		0			0
$465$	4.87689	+	8.44703i	0.226161	+	0.391722i
$466$		0			0
$467$	−12.4384			−0.575583			−0.287791	−	0.957693i	$-0.592921\pi$
$467$	−12.4384			−0.575583			−0.287791	+	0.957693i	$0.592921\pi$
$468$		0			0
$469$	−30.9309	−	53.5738i	−1.42825	−	2.47381i
$470$		0			0
$471$	5.12311	+	8.87348i	0.236060	+	0.408868i
$472$		0			0
$473$	2.56155	−	4.43674i	0.117780	−	0.204002i
$474$		0			0
$475$	−4.34233	−	0.379706i	−0.199240	−	0.0174221i
$476$		0			0
$477$	−0.965435	+	1.67218i	−0.0442042	+	0.0765640i
$478$		0			0
$479$	2.31534	+	4.01029i	0.105791	+	0.183235i	0.914061	−	0.405577i	$-0.132929\pi$
$479$	2.31534	+	4.01029i	0.105791	+	0.183235i	−0.808270	+	0.588812i	$0.799596\pi$
$480$		0			0
$481$	7.68466	+	13.3102i	0.350390	+	0.606894i
$482$		0			0
$483$	−54.7386			−2.49069
$484$		0			0
$485$	5.34233	+	9.25319i	0.242583	+	0.420166i
$486$		0			0
$487$	−21.9309			−0.993783			−0.496891	−	0.867813i	$-0.665525\pi$
$487$	−21.9309			−0.993783			−0.496891	+	0.867813i	$0.665525\pi$
$488$		0			0
$489$	15.6577	−	27.1199i	0.708064	−	1.22640i
$490$		0			0
$491$	18.0885	−	31.3303i	0.816324	−	1.41392i	−0.0920486	−	0.995755i	$-0.529342\pi$
$491$	18.0885	−	31.3303i	0.816324	−	1.41392i	0.908373	−	0.418161i	$-0.137325\pi$
$492$		0			0
$493$	−6.24621			−0.281315
$494$		0			0
$495$	0.561553			0.0252399
$496$		0			0
$497$	0.561553	−	0.972638i	0.0251891	−	0.0436288i
$498$		0			0
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	−0.334227	−	0.942493i	$-0.608475\pi$
$499$	14.5000	−	25.1147i	0.649109	−	1.12429i	0.983336	−	0.181797i	$-0.0581915\pi$
$500$		0			0
$501$	−18.2462			−0.815181
$502$		0			0
$503$	−17.2116	−	29.8114i	−0.767429	−	1.32923i	−0.938953	−	0.344047i	$-0.888202\pi$
$503$	−17.2116	−	29.8114i	−0.767429	−	1.32923i	0.171523	−	0.985180i	$-0.445131\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$	−7.02699	−	12.1711i	−0.312079	−	0.540538i
$508$		0			0
$509$	4.56155	+	7.90084i	0.202187	+	0.350199i	0.949233	−	0.314574i	$-0.101862\pi$
$509$	4.56155	+	7.90084i	0.202187	+	0.350199i	−0.747046	+	0.664773i	$0.768528\pi$
$510$		0			0
$511$	24.3693	−	42.2089i	1.07804	−	1.86721i
$512$		0			0
$513$	13.9039	−	19.8587i	0.613871	−	0.876784i
$514$		0			0
$515$	7.40388	−	12.8239i	0.326254	−	0.565089i
$516$		0			0
$517$	−5.56155	−	9.63289i	−0.244597	−	0.423654i
$518$		0			0
$519$	11.5616	+	20.0252i	0.507496	+	0.879009i
$520$		0			0
$521$	−2.19224			−0.0960436			−0.0480218	−	0.998846i	$-0.515292\pi$
$521$	−2.19224			−0.0960436			−0.0480218	+	0.998846i	$0.515292\pi$
$522$		0			0
$523$	19.6847	+	34.0948i	0.860750	+	1.49086i	0.871206	+	0.490917i	$0.163338\pi$
$523$	19.6847	+	34.0948i	0.860750	+	1.49086i	−0.0104561	+	0.999945i	$0.503328\pi$
$524$		0			0
$525$	7.12311			0.310878
$526$		0			0
$527$	9.75379	−	16.8941i	0.424882	−	0.735917i
$528$		0			0
$529$	−18.0270	+	31.2237i	−0.783782	+	1.35755i
$530$		0			0
$531$	5.86174			0.254378
$532$		0			0
$533$	−4.24621			−0.183924
$534$		0			0
$535$	−7.12311	+	12.3376i	−0.307959	+	0.533400i
$536$		0			0
$537$	16.7808	−	29.0652i	0.724144	−	1.25425i
$538$		0			0
$539$	−13.8078			−0.594743
$540$		0			0
$541$	−2.24621	−	3.89055i	−0.0965722	−	0.167268i	0.813692	−	0.581297i	$-0.197455\pi$
$541$	−2.24621	−	3.89055i	−0.0965722	−	0.167268i	−0.910264	+	0.414029i	$0.864121\pi$
$542$		0			0
$543$	5.26137			0.225787
$544$		0			0
$545$	−1.12311	−	1.94528i	−0.0481086	−	0.0833265i
$546$		0			0
$547$	−6.56155	−	11.3649i	−0.280552	−	0.485930i	0.690969	−	0.722884i	$-0.257184\pi$
$547$	−6.56155	−	11.3649i	−0.280552	−	0.485930i	−0.971521	+	0.236955i	$0.923851\pi$
$548$		0			0
$549$	−0.876894	+	1.51883i	−0.0374249	+	0.0648219i
$550$		0			0
$551$	−8.68466	−	0.759413i	−0.369979	−	0.0323521i
$552$		0			0
$553$	6.56155	−	11.3649i	0.279026	−	0.483287i
$554$		0			0
$555$	6.00000	+	10.3923i	0.254686	+	0.441129i
$556$		0			0
$557$	18.3348	+	31.7567i	0.776868	+	1.34558i	0.933738	+	0.357956i	$0.116526\pi$
$557$	18.3348	+	31.7567i	0.776868	+	1.34558i	−0.156870	+	0.987619i	$0.550140\pi$
$558$		0			0
$559$	−10.2462			−0.433369
$560$		0			0
$561$	2.43845	+	4.22351i	0.102951	+	0.178317i
$562$		0			0
$563$	32.3002			1.36129			0.680645	−	0.732613i	$-0.261700\pi$
$563$	32.3002			1.36129			0.680645	+	0.732613i	$0.261700\pi$
$564$		0			0
$565$	−1.90388	+	3.29762i	−0.0800969	+	0.138732i
$566$		0			0
$567$	−15.9654	+	27.6529i	−0.670485	+	1.16131i
$568$		0			0
$569$	34.1771			1.43278			0.716389	−	0.697701i	$-0.245794\pi$
$569$	34.1771			1.43278			0.716389	+	0.697701i	$0.245794\pi$
$570$		0			0
$571$	−15.8078			−0.661534			−0.330767	−	0.943712i	$-0.607307\pi$
$571$	−15.8078			−0.661534			−0.330767	+	0.943712i	$0.607307\pi$
$572$		0			0
$573$	−15.1231	+	26.1940i	−0.631777	+	1.09427i
$574$		0			0
$575$	3.84233	−	6.65511i	0.160236	−	0.277537i
$576$		0			0
$577$	−28.6847			−1.19416			−0.597079	−	0.802182i	$-0.703672\pi$
$577$	−28.6847			−1.19416			−0.597079	+	0.802182i	$0.703672\pi$
$578$		0			0
$579$	−12.1922	−	21.1176i	−0.506692	−	0.877616i
$580$		0			0
$581$	44.7386			1.85607
$582$		0			0
$583$	−1.71922	−	2.97778i	−0.0712030	−	0.123327i
$584$		0			0
$585$	−0.561553	−	0.972638i	−0.0232174	−	0.0402136i
$586$		0			0
$587$	−11.3693	+	19.6922i	−0.469262	+	0.812786i	−0.999383	−	0.0351368i	$-0.988813\pi$
$587$	−11.3693	+	19.6922i	−0.469262	+	0.812786i	0.530121	+	0.847922i	$0.322147\pi$
$588$		0			0
$589$	15.6155	−	22.3034i	0.643427	−	0.918997i
$590$		0			0
$591$	14.4924	−	25.1016i	0.596139	−	1.03254i
$592$		0			0
$593$	22.2732	+	38.5783i	0.914651	+	1.58422i	0.807412	+	0.589988i	$0.200868\pi$
$593$	22.2732	+	38.5783i	0.914651	+	1.58422i	0.107239	+	0.994233i	$0.465799\pi$
$594$		0			0
$595$	−7.12311	−	12.3376i	−0.292019	−	0.505792i
$596$		0			0
$597$	−17.3693			−0.710879
$598$		0			0
$599$	0.246211	+	0.426450i	0.0100599	+	0.0174243i	0.871012	−	0.491263i	$-0.163464\pi$
$599$	0.246211	+	0.426450i	0.0100599	+	0.0174243i	−0.860952	+	0.508687i	$0.830131\pi$
$600$		0			0
$601$	−36.3693			−1.48354			−0.741768	−	0.670657i	$-0.766012\pi$
$601$	−36.3693			−1.48354			−0.741768	+	0.670657i	$0.766012\pi$
$602$		0			0
$603$	−3.80776	+	6.59524i	−0.155064	+	0.268579i
$604$		0			0
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$		0			0
$607$	−30.8078			−1.25045			−0.625224	−	0.780445i	$-0.714993\pi$
$607$	−30.8078			−1.25045			−0.625224	+	0.780445i	$0.714993\pi$
$608$		0			0
$609$	14.2462			0.577286
$610$		0			0
$611$	−11.1231	+	19.2658i	−0.449993	+	0.779410i
$612$		0			0
$613$	−9.15767	+	15.8616i	−0.369875	+	0.640642i	−0.989546	−	0.144220i	$-0.953933\pi$
$613$	−9.15767	+	15.8616i	−0.369875	+	0.640642i	0.619671	+	0.784862i	$0.287266\pi$
$614$		0			0
$615$	−3.31534			−0.133687
$616$		0			0
$617$	−13.9039	−	24.0822i	−0.559749	−	0.969514i	−0.997517	−	0.0704260i	$-0.977564\pi$
$617$	−13.9039	−	24.0822i	−0.559749	−	0.969514i	0.437768	−	0.899088i	$-0.355769\pi$
$618$		0			0
$619$	11.0540			0.444297			0.222148	−	0.975013i	$-0.428693\pi$
$619$	11.0540			0.444297			0.222148	+	0.975013i	$0.428693\pi$
$620$		0			0
$621$	21.3693	+	37.0127i	0.857521	+	1.48527i
$622$		0			0
$623$	22.0885	+	38.2585i	0.884959	+	1.53279i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	2.87689	+	6.16879i	0.114892	+	0.246358i
$628$		0			0
$629$	12.0000	−	20.7846i	0.478471	−	0.828737i
$630$		0			0
$631$	−6.12311	−	10.6055i	−0.243757	−	0.422199i	0.718024	−	0.696018i	$-0.245047\pi$
$631$	−6.12311	−	10.6055i	−0.243757	−	0.422199i	−0.961781	+	0.273818i	$0.911713\pi$
$632$		0			0
$633$	12.2462	+	21.2111i	0.486743	+	0.843064i
$634$		0			0
$635$	11.4384			0.453921
$636$		0			0
$637$	13.8078	+	23.9157i	0.547084	+	0.947576i
$638$		0			0
$639$	−0.138261			−0.00546951
$640$		0			0
$641$	−5.78078	+	10.0126i	−0.228327	+	0.395474i	−0.957312	−	0.289055i	$-0.906659\pi$
$641$	−5.78078	+	10.0126i	−0.228327	+	0.395474i	0.728985	+	0.684529i	$0.239992\pi$
$642$		0			0
$643$	−9.21922	+	15.9682i	−0.363571	+	0.629723i	−0.988546	−	0.150922i	$-0.951776\pi$
$643$	−9.21922	+	15.9682i	−0.363571	+	0.629723i	0.624975	+	0.780645i	$0.285109\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	42.1771			1.65815			0.829076	−	0.559136i	$-0.188867\pi$
$647$	42.1771			1.65815			0.829076	+	0.559136i	$0.188867\pi$
$648$		0			0
$649$	−5.21922	+	9.03996i	−0.204872	+	0.354849i
$650$		0			0
$651$	−22.2462	+	38.5316i	−0.871898	+	1.51017i
$652$		0			0
$653$	−33.9309			−1.32782			−0.663909	−	0.747814i	$-0.731104\pi$
$653$	−33.9309			−1.32782			−0.663909	+	0.747814i	$0.731104\pi$
$654$		0			0
$655$	−3.93845	−	6.82159i	−0.153888	−	0.266542i
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−4.96543	−	8.60039i	−0.193426	−	0.335023i	0.752957	−	0.658069i	$-0.228627\pi$
$659$	−4.96543	−	8.60039i	−0.193426	−	0.335023i	−0.946383	+	0.323046i	$0.895293\pi$
$660$		0			0
$661$	−8.93087	−	15.4687i	−0.347371	−	0.601663i	0.638411	−	0.769696i	$-0.279592\pi$
$661$	−8.93087	−	15.4687i	−0.347371	−	0.601663i	−0.985782	+	0.168032i	$0.946259\pi$
$662$		0			0
$663$	4.87689	−	8.44703i	0.189403	−	0.328055i
$664$		0			0
$665$	−8.40388	−	18.0201i	−0.325889	−	0.698788i
$666$		0			0
$667$	7.68466	−	13.3102i	0.297551	−	0.515374i
$668$		0			0
$669$	−18.4924	−	32.0298i	−0.714958	−	1.23834i
$670$		0			0
$671$	−1.56155	−	2.70469i	−0.0602831	−	0.104413i
$672$		0			0
$673$	−28.1080			−1.08348			−0.541741	−	0.840546i	$-0.682235\pi$
$673$	−28.1080			−1.08348			−0.541741	+	0.840546i	$0.682235\pi$
$674$		0			0
$675$	−2.78078	−	4.81645i	−0.107032	−	0.185385i
$676$		0			0
$677$	22.1771			0.852334			0.426167	−	0.904644i	$-0.359864\pi$
$677$	22.1771			0.852334			0.426167	+	0.904644i	$0.359864\pi$
$678$		0			0
$679$	−24.3693	+	42.2089i	−0.935209	+	1.61983i
$680$		0			0
$681$	6.97301	−	12.0776i	0.267206	−	0.462815i
$682$		0			0
$683$	−28.0000			−1.07139			−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000			−1.07139			−0.535695	+	0.844411i	$0.679950\pi$
$684$		0			0
$685$	−9.56155			−0.365328
$686$		0			0
$687$	16.4924	−	28.5657i	0.629225	−	1.08985i
$688$		0			0
$689$	−3.43845	+	5.95557i	−0.130994	+	0.226889i
$690$		0			0
$691$	49.9309			1.89946			0.949730	−	0.313070i	$-0.101358\pi$
$691$	49.9309			1.89946			0.949730	+	0.313070i	$0.101358\pi$
$692$		0			0
$693$	1.28078	+	2.21837i	0.0486527	+	0.0842689i
$694$		0			0
$695$	11.8078			0.447894
$696$		0			0
$697$	3.31534	+	5.74234i	0.125578	+	0.217507i
$698$		0			0
$699$	11.4654	+	19.8587i	0.433663	+	0.751126i
$700$		0			0
$701$	13.2462	−	22.9431i	0.500302	−	0.866549i	−0.499697	−	0.866200i	$-0.666555\pi$
$701$	13.2462	−	22.9431i	0.500302	−	0.866549i	1.00000		0.000349325i	$-0.000111194\pi$
$702$		0			0
$703$	19.2116	−	27.4397i	0.724581	−	1.03491i
$704$		0			0
$705$	−8.68466	+	15.0423i	−0.327083	+	0.566525i
$706$		0			0
$707$	22.8078	+	39.5042i	0.857774	+	1.48571i
$708$		0			0
$709$	20.0000	+	34.6410i	0.751116	+	1.30097i	0.947282	+	0.320400i	$0.103817\pi$
$709$	20.0000	+	34.6410i	0.751116	+	1.30097i	−0.196167	+	0.980571i	$0.562849\pi$
$710$		0			0
$711$	−1.61553			−0.0605870
$712$		0			0
$713$	24.0000	+	41.5692i	0.898807	+	1.55678i
$714$		0			0
$715$	2.00000			0.0747958
$716$		0			0
$717$	−1.36932	+	2.37173i	−0.0511381	+	0.0885737i
$718$		0			0
$719$	11.5616	−	20.0252i	0.431173	−	0.746814i	−0.565801	−	0.824542i	$-0.691433\pi$
$719$	11.5616	−	20.0252i	0.431173	−	0.746814i	0.996975	+	0.0777277i	$0.0247665\pi$
$720$		0			0
$721$	67.5464			2.51556
$722$		0			0
$723$	6.93087			0.257762
$724$		0			0
$725$	−1.00000	+	1.73205i	−0.0371391	+	0.0643268i
$726$		0			0
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	−0.930618	−	0.365991i	$-0.880730\pi$
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	0.782267	+	0.622944i	$0.214063\pi$
$728$		0			0
$729$	29.9848			1.11055
$730$		0			0
$731$	8.00000	+	13.8564i	0.295891	+	0.512498i
$732$		0			0
$733$	1.93087			0.0713183			0.0356591	−	0.999364i	$-0.488647\pi$
$733$	1.93087			0.0713183			0.0356591	+	0.999364i	$0.488647\pi$
$734$		0			0
$735$	10.7808	+	18.6729i	0.397655	+	0.688759i
$736$		0			0
$737$	−6.78078	−	11.7446i	−0.249773	−	0.432620i
$738$		0			0
$739$	17.6231	−	30.5241i	0.648276	−	1.12285i	−0.335258	−	0.942126i	$-0.608823\pi$
$739$	17.6231	−	30.5241i	0.648276	−	1.12285i	0.983534	−	0.180721i	$-0.0578432\pi$
$740$		0			0
$741$	7.80776	−	11.1517i	0.286825	−	0.409669i
$742$		0			0
$743$	−11.2808	+	19.5389i	−0.413852	+	0.716812i	−0.995307	−	0.0967662i	$-0.969150\pi$
$743$	−11.2808	+	19.5389i	−0.413852	+	0.716812i	0.581456	+	0.813578i	$0.302483\pi$
$744$		0			0
$745$	−8.00000	−	13.8564i	−0.293097	−	0.507659i
$746$		0			0
$747$	−2.75379	−	4.76970i	−0.100756	−	0.174514i
$748$		0			0
$749$	−64.9848			−2.37449
$750$		0			0
$751$	−21.5616	−	37.3457i	−0.786792	−	1.36276i	−0.927923	−	0.372773i	$-0.878407\pi$
$751$	−21.5616	−	37.3457i	−0.786792	−	1.36276i	0.141130	−	0.989991i	$-0.454926\pi$
$752$		0			0
$753$	−4.57671			−0.166785
$754$		0			0
$755$	−6.24621	+	10.8188i	−0.227323	+	0.393735i
$756$		0			0
$757$	7.71922	−	13.3701i	0.280560	−	0.485944i	−0.690963	−	0.722890i	$-0.742813\pi$
$757$	7.71922	−	13.3701i	0.280560	−	0.485944i	0.971523	+	0.236946i	$0.0761465\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	13.9848			0.506950			0.253475	−	0.967342i	$-0.418426\pi$
$761$	13.9848			0.506950			0.253475	+	0.967342i	$0.418426\pi$
$762$		0			0
$763$	5.12311	−	8.87348i	0.185469	−	0.321242i
$764$		0			0
$765$	−0.876894	+	1.51883i	−0.0317042	+	0.0549133i
$766$		0			0
$767$	20.8769			0.753821
$768$		0			0
$769$	3.24621	+	5.62260i	0.117061	+	0.202756i	0.918602	−	0.395184i	$-0.129319\pi$
$769$	3.24621	+	5.62260i	0.117061	+	0.202756i	−0.801541	+	0.597940i	$0.795986\pi$
$770$		0			0
$771$	37.5616			1.35275
$772$		0			0
$773$	−19.6501	−	34.0350i	−0.706765	−	1.22415i	−0.966051	−	0.258352i	$-0.916821\pi$
$773$	−19.6501	−	34.0350i	−0.706765	−	1.22415i	0.259286	−	0.965801i	$-0.416513\pi$
$774$		0			0
$775$	−3.12311	−	5.40938i	−0.112185	−	0.194311i
$776$		0			0
$777$	−27.3693	+	47.4050i	−0.981869	+	1.70065i
$778$		0			0
$779$	3.91146	+	8.38716i	0.140143	+	0.300501i
$780$		0			0
$781$	0.123106	−	0.213225i	0.00440507	−	0.00762980i
$782$		0			0
$783$	−5.56155	−	9.63289i	−0.198754	−	0.344251i
$784$		0			0
$785$	−3.28078	−	5.68247i	−0.117096	−	0.202816i
$786$		0			0
$787$	−4.43845			−0.158214			−0.0791068	−	0.996866i	$-0.525207\pi$
$787$	−4.43845			−0.158214			−0.0791068	+	0.996866i	$0.525207\pi$
$788$		0			0
$789$	17.8078	+	30.8440i	0.633973	+	1.09807i
$790$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.780776 - 1.35234i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.56155 q^{7} +(0.280776 + 0.486319i) q^{9} +O(q^{10})\)	\(q+(0.780776 - 1.35234i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.56155 q^{7} +(0.280776 + 0.486319i) q^{9} -1.00000 q^{11} +(1.00000 + 1.73205i) q^{13} +(0.780776 + 1.35234i) q^{15} +(1.56155 - 2.70469i) q^{17} +(2.50000 - 3.57071i) q^{19} +(-3.56155 + 6.16879i) q^{21} +(3.84233 + 6.65511i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.56155 q^{27} +(-1.00000 - 1.73205i) q^{29} +6.24621 q^{31} +(-0.780776 + 1.35234i) q^{33} +(2.28078 - 3.95042i) q^{35} +7.68466 q^{37} +3.12311 q^{39} +(-1.06155 + 1.83866i) q^{41} +(-2.56155 + 4.43674i) q^{43} -0.561553 q^{45} +(5.56155 + 9.63289i) q^{47} +13.8078 q^{49} +(-2.43845 - 4.22351i) q^{51} +(1.71922 + 2.97778i) q^{53} +(0.500000 - 0.866025i) q^{55} +(-2.87689 - 6.16879i) q^{57} +(5.21922 - 9.03996i) q^{59} +(1.56155 + 2.70469i) q^{61} +(-1.28078 - 2.21837i) q^{63} -2.00000 q^{65} +(6.78078 + 11.7446i) q^{67} +12.0000 q^{69} +(-0.123106 + 0.213225i) q^{71} +(-5.34233 + 9.25319i) q^{73} -1.56155 q^{75} +4.56155 q^{77} +(-1.43845 + 2.49146i) q^{79} +(3.50000 - 6.06218i) q^{81} -9.80776 q^{83} +(1.56155 + 2.70469i) q^{85} -3.12311 q^{87} +(-4.84233 - 8.38716i) q^{89} +(-4.56155 - 7.90084i) q^{91} +(4.87689 - 8.44703i) q^{93} +(1.84233 + 3.95042i) q^{95} +(5.34233 - 9.25319i) q^{97} +(-0.280776 - 0.486319i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9	\( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - 2 q^{17} + 10 q^{19} - 6 q^{21} + 3 q^{23} - 2 q^{25} + 14 q^{27} - 4 q^{29} - 8 q^{31} + q^{33} + 5 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{43} + 6 q^{45} + 14 q^{47} + 14 q^{49} - 18 q^{51} + 11 q^{53} + 2 q^{55} - 28 q^{57} + 25 q^{59} - 2 q^{61} - q^{63} - 8 q^{65} + 23 q^{67} + 48 q^{69} + 16 q^{71} - 9 q^{73} + 2 q^{75} + 10 q^{77} - 14 q^{79} + 14 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87} - 7 q^{89} - 10 q^{91} + 36 q^{93} - 5 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q - q^3 - 2 * q^5 - 10 * q^7 - 3 * q^9 - 4 * q^11 + 4 * q^13 - q^15 - 2 * q^17 + 10 * q^19 - 6 * q^21 + 3 * q^23 - 2 * q^25 + 14 * q^27 - 4 * q^29 - 8 * q^31 + q^33 + 5 * q^35 + 6 * q^37 - 4 * q^39 + 4 * q^41 - 2 * q^43 + 6 * q^45 + 14 * q^47 + 14 * q^49 - 18 * q^51 + 11 * q^53 + 2 * q^55 - 28 * q^57 + 25 * q^59 - 2 * q^61 - q^63 - 8 * q^65 + 23 * q^67 + 48 * q^69 + 16 * q^71 - 9 * q^73 + 2 * q^75 + 10 * q^77 - 14 * q^79 + 14 * q^81 + 2 * q^83 - 2 * q^85 + 4 * q^87 - 7 * q^89 - 10 * q^91 + 36 * q^93 - 5 * q^95 + 9 * q^97 + 3 * q^99

Introduction

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