LMFDB - Embedded newform 1520.2.q.k.961.1

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.4601315889.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 $ x^8 - x^7 + 6x^6 - 3x^5 + 26x^4 - 14x^3 + 31x^2 + 12x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$-0.245959 + 0.426014i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.k.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.37901 + 2.38851i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.11474 q^{7} +(-2.30333 - 3.98948i) q^{9} +O(q^{10})$	\(q+(-1.37901 + 2.38851i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.11474 q^{7} +(-2.30333 - 3.98948i) q^{9} +6.09857 q^{11} +(-1.17028 - 2.02698i) q^{13} +(-1.37901 - 2.38851i) q^{15} +(3.80333 - 6.58756i) q^{17} +(-1.46155 + 4.10657i) q^{19} +(5.67425 - 9.82810i) q^{21} +(-2.98977 - 5.17843i) q^{23} +(-0.500000 - 0.866025i) q^{25} +4.43118 q^{27} +(-0.969629 - 1.67945i) q^{29} +1.43118 q^{31} +(-8.40998 + 14.5665i) q^{33} +(2.05737 - 3.56347i) q^{35} +8.91733 q^{37} +6.45529 q^{39} +(-1.70065 + 2.94561i) q^{41} +(-0.655338 + 1.13508i) q^{43} +4.60665 q^{45} +(0.192567 + 0.333536i) q^{47} +9.93105 q^{49} +(10.4896 + 18.1686i) q^{51} +(-0.597970 - 1.03572i) q^{53} +(-3.04929 + 5.28152i) q^{55} +(-7.79309 - 9.15391i) q^{57} +(5.48566 - 9.50145i) q^{59} +(-0.0825396 - 0.142963i) q^{61} +(9.47758 + 16.4157i) q^{63} +2.34056 q^{65} +(-0.848637 - 1.46988i) q^{67} +16.4917 q^{69} +(4.75679 - 8.23901i) q^{71} +(-5.97360 + 10.3466i) q^{73} +2.75802 q^{75} -25.0940 q^{77} +(-3.87563 + 6.71280i) q^{79} +(0.799351 - 1.38452i) q^{81} +3.25570 q^{83} +(3.80333 + 6.58756i) q^{85} +5.34851 q^{87} +(9.26077 + 16.0401i) q^{89} +(4.81538 + 8.34049i) q^{91} +(-1.97360 + 3.41838i) q^{93} +(-2.82562 - 3.31902i) q^{95} +(0.203531 - 0.352527i) q^{97} +(-14.0470 - 24.3301i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9	$ 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 8 q^{11} + q^{13} - q^{15} + 13 q^{17} - q^{19} + 12 q^{21} - 8 q^{23} - 4 q^{25} + 20 q^{27} - 3 q^{29} - 4 q^{31} - 15 q^{33} + 2 q^{35} + 20 q^{37} + 2 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{45} - 10 q^{47} + 12 q^{49} + 16 q^{51} - 11 q^{53} - 4 q^{55} - 29 q^{57} - q^{59} + 25 q^{63} - 2 q^{65} + 8 q^{67} + 22 q^{69} + 4 q^{71} - 20 q^{73} + 2 q^{75} - 36 q^{77} + 3 q^{79} + 12 q^{81} + 30 q^{83} + 13 q^{85} - 24 q^{87} + 17 q^{89} + 4 q^{91} + 12 q^{93} - 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) $ 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 8 * q^11 + q^13 - q^15 + 13 * q^17 - q^19 + 12 * q^21 - 8 * q^23 - 4 * q^25 + 20 * q^27 - 3 * q^29 - 4 * q^31 - 15 * q^33 + 2 * q^35 + 20 * q^37 + 2 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^45 - 10 * q^47 + 12 * q^49 + 16 * q^51 - 11 * q^53 - 4 * q^55 - 29 * q^57 - q^59 + 25 * q^63 - 2 * q^65 + 8 * q^67 + 22 * q^69 + 4 * q^71 - 20 * q^73 + 2 * q^75 - 36 * q^77 + 3 * q^79 + 12 * q^81 + 30 * q^83 + 13 * q^85 - 24 * q^87 + 17 * q^89 + 4 * q^91 + 12 * q^93 - 4 * q^95 + 11 * q^97 - 30 * 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
$3$	−1.37901	+	2.38851i	−0.796171	+	1.37901i	0.125922	+	0.992040i	$0.459811\pi$
$3$	−1.37901	+	2.38851i	−0.796171	+	1.37901i	−0.922093	+	0.386968i	$0.873522\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−4.11474			−1.55522			−0.777612	−	0.628745i	$-0.783569\pi$
$7$	−4.11474			−1.55522			−0.777612	+	0.628745i	$0.783569\pi$
$8$		0			0
$9$	−2.30333	−	3.98948i	−0.767776	−	1.32983i
$10$		0			0
$11$	6.09857			1.83879			0.919394	−	0.393337i	$-0.128679\pi$
$11$	6.09857			1.83879			0.919394	+	0.393337i	$0.128679\pi$
$12$		0			0
$13$	−1.17028	−	2.02698i	−0.324577	−	0.562183i	0.656850	−	0.754021i	$-0.271889\pi$
$13$	−1.17028	−	2.02698i	−0.324577	−	0.562183i	−0.981427	+	0.191838i	$0.938555\pi$
$14$		0			0
$15$	−1.37901	−	2.38851i	−0.356058	−	0.616711i
$16$		0			0
$17$	3.80333	−	6.58756i	0.922442	−	1.59772i	0.126818	−	0.991926i	$-0.459523\pi$
$17$	3.80333	−	6.58756i	0.922442	−	1.59772i	0.795624	−	0.605791i	$-0.207143\pi$
$18$		0			0
$19$	−1.46155	+	4.10657i	−0.335302	+	0.942111i
$19$	−1.46155	+	4.10657i	−0.335302	+	0.942111i
$20$		0			0
$21$	5.67425	−	9.82810i	1.23822	−	2.14467i
$22$		0			0
$23$	−2.98977	−	5.17843i	−0.623410	−	1.07978i	−0.988846	−	0.148941i	$-0.952414\pi$
$23$	−2.98977	−	5.17843i	−0.623410	−	1.07978i	0.365436	−	0.930836i	$-0.380920\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	4.43118			0.852780
$28$		0			0
$29$	−0.969629	−	1.67945i	−0.180056	−	0.311866i	0.761844	−	0.647761i	$-0.224294\pi$
$29$	−0.969629	−	1.67945i	−0.180056	−	0.311866i	−0.941899	+	0.335895i	$0.890961\pi$
$30$		0			0
$31$	1.43118			0.257047			0.128523	−	0.991706i	$-0.458976\pi$
$31$	1.43118			0.257047			0.128523	+	0.991706i	$0.458976\pi$
$32$		0			0
$33$	−8.40998	+	14.5665i	−1.46399	+	2.53570i
$34$		0			0
$35$	2.05737	−	3.56347i	0.347759	−	0.602336i
$36$		0			0
$37$	8.91733			1.46600			0.733000	−	0.680229i	$-0.238120\pi$
$37$	8.91733			1.46600			0.733000	+	0.680229i	$0.238120\pi$
$38$		0			0
$39$	6.45529			1.03367
$40$		0			0
$41$	−1.70065	+	2.94561i	−0.265597	+	0.460027i	−0.967720	−	0.252029i	$-0.918902\pi$
$41$	−1.70065	+	2.94561i	−0.265597	+	0.460027i	0.702123	+	0.712056i	$0.252236\pi$
$42$		0			0
$43$	−0.655338	+	1.13508i	−0.0999381	+	0.173098i	−0.911659	−	0.410948i	$-0.865198\pi$
$43$	−0.655338	+	1.13508i	−0.0999381	+	0.173098i	0.811721	+	0.584046i	$0.198531\pi$
$44$		0			0
$45$	4.60665			0.686719
$46$		0			0
$47$	0.192567	+	0.333536i	0.0280888	+	0.0486512i	0.879728	−	0.475477i	$-0.157725\pi$
$47$	0.192567	+	0.333536i	0.0280888	+	0.0486512i	−0.851639	+	0.524128i	$0.824391\pi$
$48$		0			0
$49$	9.93105			1.41872
$50$		0			0
$51$	10.4896	+	18.1686i	1.46884	+	2.54411i
$52$		0			0
$53$	−0.597970	−	1.03572i	−0.0821376	−	0.142266i	0.822030	−	0.569444i	$-0.192841\pi$
$53$	−0.597970	−	1.03572i	−0.0821376	−	0.142266i	−0.904168	+	0.427177i	$0.859508\pi$
$54$		0			0
$55$	−3.04929	+	5.28152i	−0.411166	+	0.712160i
$56$		0			0
$57$	−7.79309	−	9.15391i	−1.03222	−	1.21247i
$58$		0			0
$59$	5.48566	−	9.50145i	0.714172	−	1.23698i	−0.249106	−	0.968476i	$-0.580137\pi$
$59$	5.48566	−	9.50145i	0.714172	−	1.23698i	0.963278	−	0.268506i	$-0.0865301\pi$
$60$		0			0
$61$	−0.0825396	−	0.142963i	−0.0105681	−	0.0183045i	0.860693	−	0.509124i	$-0.170031\pi$
$61$	−0.0825396	−	0.142963i	−0.0105681	−	0.0183045i	−0.871261	+	0.490820i	$0.836697\pi$
$62$		0			0
$63$	9.47758	+	16.4157i	1.19406	+	2.06818i
$64$		0			0
$65$	2.34056			0.290310
$66$		0			0
$67$	−0.848637	−	1.46988i	−0.103678	−	0.179575i	0.809520	−	0.587093i	$-0.199728\pi$
$67$	−0.848637	−	1.46988i	−0.103678	−	0.179575i	−0.913197	+	0.407518i	$0.866394\pi$
$68$		0			0
$69$	16.4917			1.98536
$70$		0			0
$71$	4.75679	−	8.23901i	0.564527	−	0.977790i	−0.432566	−	0.901602i	$-0.642392\pi$
$71$	4.75679	−	8.23901i	0.564527	−	0.977790i	0.997093	−	0.0761880i	$-0.0242749\pi$
$72$		0			0
$73$	−5.97360	+	10.3466i	−0.699158	+	1.21098i	0.269601	+	0.962972i	$0.413108\pi$
$73$	−5.97360	+	10.3466i	−0.699158	+	1.21098i	−0.968759	+	0.248004i	$0.920225\pi$
$74$		0			0
$75$	2.75802			0.318468
$76$		0			0
$77$	−25.0940			−2.85973
$78$		0			0
$79$	−3.87563	+	6.71280i	−0.436043	+	0.755249i	−0.997380	−	0.0723387i	$-0.976954\pi$
$79$	−3.87563	+	6.71280i	−0.436043	+	0.755249i	0.561337	+	0.827587i	$0.310287\pi$
$80$		0			0
$81$	0.799351	−	1.38452i	0.0888168	−	0.153835i
$82$		0			0
$83$	3.25570			0.357360			0.178680	−	0.983907i	$-0.442817\pi$
$83$	3.25570			0.357360			0.178680	+	0.983907i	$0.442817\pi$
$84$		0			0
$85$	3.80333	+	6.58756i	0.412529	+	0.714521i
$86$		0			0
$87$	5.34851			0.573420
$88$		0			0
$89$	9.26077	+	16.0401i	0.981639	+	1.70025i	0.656010	+	0.754752i	$0.272243\pi$
$89$	9.26077	+	16.0401i	0.981639	+	1.70025i	0.325629	+	0.945498i	$0.394424\pi$
$90$		0			0
$91$	4.81538	+	8.34049i	0.504789	+	0.874321i
$92$		0			0
$93$	−1.97360	+	3.41838i	−0.204653	+	0.354470i
$94$		0			0
$95$	−2.82562	−	3.31902i	−0.289902	−	0.340524i
$96$		0			0
$97$	0.203531	−	0.352527i	0.0206655	−	0.0357937i	−0.855508	−	0.517790i	$-0.826755\pi$
$97$	0.203531	−	0.352527i	0.0206655	−	0.0357937i	0.876173	+	0.481996i	$0.160088\pi$
$98$		0			0
$99$	−14.0470	−	24.3301i	−1.41178	−	2.44527i
$100$		0			0
$101$	−1.02626	−	1.77754i	−0.102117	−	0.176872i	0.810440	−	0.585822i	$-0.199228\pi$
$101$	−1.02626	−	1.77754i	−0.102117	−	0.176872i	−0.912557	+	0.408950i	$0.865895\pi$
$102$		0			0
$103$	0.0204641			0.00201639			0.00100820	−	0.999999i	$-0.499679\pi$
$103$	0.0204641			0.00201639			0.00100820	+	0.999999i	$0.499679\pi$
$104$		0			0
$105$	5.67425	+	9.82810i	0.553750	+	0.959124i
$106$		0			0
$107$	−0.463503			−0.0448086			−0.0224043	−	0.999749i	$-0.507132\pi$
$107$	−0.463503			−0.0448086			−0.0224043	+	0.999749i	$0.507132\pi$
$108$		0			0
$109$	−7.36589	+	12.7581i	−0.705525	+	1.22200i	0.260977	+	0.965345i	$0.415955\pi$
$109$	−7.36589	+	12.7581i	−0.705525	+	1.22200i	−0.966502	+	0.256660i	$0.917378\pi$
$110$		0			0
$111$	−12.2971	+	21.2992i	−1.16719	+	2.02163i
$112$		0			0
$113$	8.36613			0.787020			0.393510	−	0.919320i	$-0.371261\pi$
$113$	8.36613			0.787020			0.393510	+	0.919320i	$0.371261\pi$
$114$		0			0
$115$	5.97954			0.557595
$116$		0			0
$117$	−5.39107	+	9.33760i	−0.498404	+	0.863261i
$118$		0			0
$119$	−15.6497	+	27.1060i	−1.43460	+	2.48481i
$120$		0			0
$121$	26.1926			2.38114
$122$		0			0
$123$	−4.69042	−	8.12404i	−0.422921	−	0.732520i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−9.23377	−	15.9934i	−0.819365	−	1.41918i	−0.906151	−	0.422954i	$-0.860993\pi$
$127$	−9.23377	−	15.9934i	−0.819365	−	1.41918i	0.0867862	−	0.996227i	$-0.472340\pi$
$128$		0			0
$129$	−1.80743	−	3.13057i	−0.159136	−	0.275631i
$130$		0			0
$131$	8.13978	−	14.0985i	0.711176	−	1.23179i	−0.253241	−	0.967403i	$-0.581496\pi$
$131$	8.13978	−	14.0985i	0.711176	−	1.23179i	0.964416	−	0.264389i	$-0.0851702\pi$
$132$		0			0
$133$	6.01388	−	16.8974i	0.521470	−	1.46519i
$134$		0			0
$135$	−2.21559	+	3.83751i	−0.190687	+	0.330280i
$136$		0			0
$137$	−2.02822	−	3.51298i	−0.173283	−	0.300134i	0.766283	−	0.642503i	$-0.222104\pi$
$137$	−2.02822	−	3.51298i	−0.173283	−	0.300134i	−0.939566	+	0.342369i	$0.888771\pi$
$138$		0			0
$139$	−5.42021	−	9.38808i	−0.459736	−	0.796287i	0.539210	−	0.842171i	$-0.318723\pi$
$139$	−5.42021	−	9.38808i	−0.459736	−	0.796287i	−0.998947	+	0.0458843i	$0.985389\pi$
$140$		0			0
$141$	−1.06221			−0.0894539
$142$		0			0
$143$	−7.13702	−	12.3617i	−0.596828	−	1.03374i
$144$		0			0
$145$	1.93926			0.161047
$146$		0			0
$147$	−13.6950	+	23.7204i	−1.12954	+	1.95643i
$148$		0			0
$149$	11.4721	−	19.8702i	0.939827	−	1.62783i	0.174037	−	0.984739i	$-0.444319\pi$
$149$	11.4721	−	19.8702i	0.939827	−	1.62783i	0.765791	−	0.643090i	$-0.222348\pi$
$150$		0			0
$151$	16.3025			1.32668			0.663338	−	0.748320i	$-0.269139\pi$
$151$	16.3025			1.32668			0.663338	+	0.748320i	$0.269139\pi$
$152$		0			0
$153$	−35.0412			−2.83291
$154$		0			0
$155$	−0.715589	+	1.23944i	−0.0574775	+	0.0995539i
$156$		0			0
$157$	4.46890	−	7.74036i	0.356657	−	0.617748i	−0.630743	−	0.775992i	$-0.717250\pi$
$157$	4.46890	−	7.74036i	0.356657	−	0.617748i	0.987400	+	0.158244i	$0.0505832\pi$
$158$		0			0
$159$	3.29842			0.261582
$160$		0			0
$161$	12.3021	+	21.3079i	0.969542	+	1.67930i
$162$		0			0
$163$	5.65488			0.442925			0.221462	−	0.975169i	$-0.428917\pi$
$163$	5.65488			0.442925			0.221462	+	0.975169i	$0.428917\pi$
$164$		0			0
$165$	−8.40998	−	14.5665i	−0.654716	−	1.13400i
$166$		0			0
$167$	6.63414	+	11.4907i	0.513365	+	0.889175i	0.999880	+	0.0155023i	$0.00493473\pi$
$167$	6.63414	+	11.4907i	0.513365	+	0.889175i	−0.486515	+	0.873672i	$0.661732\pi$
$168$		0			0
$169$	3.76090	−	6.51407i	0.289300	−	0.501082i
$170$		0			0
$171$	19.7495	−	3.62795i	1.51028	−	0.277436i
$172$		0			0
$173$	0.642354	−	1.11259i	0.0488373	−	0.0845886i	−0.840573	−	0.541698i	$-0.817782\pi$
$173$	0.642354	−	1.11259i	0.0488373	−	0.0845886i	0.889411	+	0.457109i	$0.151115\pi$
$174$		0			0
$175$	2.05737	+	3.56347i	0.155522	+	0.269373i
$176$		0			0
$177$	15.1295	+	26.2051i	1.13721	+	1.96970i
$178$		0			0
$179$	−4.27160			−0.319275			−0.159637	−	0.987176i	$-0.551032\pi$
$179$	−4.27160			−0.319275			−0.159637	+	0.987176i	$0.551032\pi$
$180$		0			0
$181$	0.869999	+	1.50688i	0.0646665	+	0.112006i	0.896546	−	0.442951i	$-0.146068\pi$
$181$	0.869999	+	1.50688i	0.0646665	+	0.112006i	−0.831879	+	0.554956i	$0.812735\pi$
$182$		0			0
$183$	0.455291			0.0336561
$184$		0			0
$185$	−4.45866	+	7.72263i	−0.327808	+	0.567779i
$186$		0			0
$187$	23.1949	−	40.1747i	1.69618	−	2.93786i
$188$		0			0
$189$	−18.2331			−1.32626
$190$		0			0
$191$	−22.4806			−1.62664			−0.813320	−	0.581817i	$-0.802342\pi$
$191$	−22.4806			−1.62664			−0.813320	+	0.581817i	$0.802342\pi$
$192$		0			0
$193$	4.08559	−	7.07645i	0.294087	−	0.509374i	−0.680685	−	0.732576i	$-0.738318\pi$
$193$	4.08559	−	7.07645i	0.294087	−	0.509374i	0.974772	+	0.223202i	$0.0716511\pi$
$194$		0			0
$195$	−3.22765	+	5.59045i	−0.231137	+	0.400340i
$196$		0			0
$197$	14.6952			1.04699			0.523493	−	0.852030i	$-0.324628\pi$
$197$	14.6952			1.04699			0.523493	+	0.852030i	$0.324628\pi$
$198$		0			0
$199$	−7.80137	−	13.5124i	−0.553025	−	0.957867i	−0.998054	−	0.0623508i	$-0.980140\pi$
$199$	−7.80137	−	13.5124i	−0.553025	−	0.957867i	0.445030	−	0.895516i	$-0.353193\pi$
$200$		0			0
$201$	4.68111			0.330180
$202$		0			0
$203$	3.98977	+	6.91048i	0.280027	+	0.485021i
$204$		0			0
$205$	−1.70065	−	2.94561i	−0.118778	−	0.205730i
$206$		0			0
$207$	−13.7728	+	23.8552i	−0.957278	+	1.65805i
$208$		0			0
$209$	−8.91335	+	25.0442i	−0.616550	+	1.73234i
$210$		0			0
$211$	7.14968	−	12.3836i	0.492204	−	0.852523i	−0.507755	−	0.861501i	$-0.669525\pi$
$211$	7.14968	−	12.3836i	0.492204	−	0.852523i	0.999960	+	0.00897821i	$0.00285789\pi$
$212$		0			0
$213$	13.1193	+	22.7233i	0.898920	+	1.55698i
$214$		0			0
$215$	−0.655338	−	1.13508i	−0.0446937	−	0.0774117i
$216$		0			0
$217$	−5.88891			−0.399766
$218$		0			0
$219$	−16.4753	−	28.5361i	−1.11330	−	1.92829i
$220$		0			0
$221$	−17.8038			−1.19761
$222$		0			0
$223$	0.145237	−	0.251557i	0.00972576	−	0.0168455i	−0.861122	−	0.508399i	$-0.830237\pi$
$223$	0.145237	−	0.251557i	0.00972576	−	0.0168455i	0.870847	+	0.491554i	$0.163571\pi$
$224$		0			0
$225$	−2.30333	+	3.98948i	−0.153555	+	0.265965i
$226$		0			0
$227$	22.6811			1.50540			0.752699	−	0.658365i	$-0.228752\pi$
$227$	22.6811			1.50540			0.752699	+	0.658365i	$0.228752\pi$
$228$		0			0
$229$	7.08241			0.468019			0.234009	−	0.972234i	$-0.424815\pi$
$229$	7.08241			0.468019			0.234009	+	0.972234i	$0.424815\pi$
$230$		0			0
$231$	34.6048	−	59.9373i	2.27683	−	3.94359i
$232$		0			0
$233$	4.98884	−	8.64093i	0.326830	−	0.566086i	−0.655051	−	0.755585i	$-0.727353\pi$
$233$	4.98884	−	8.64093i	0.326830	−	0.566086i	0.981881	+	0.189499i	$0.0606863\pi$
$234$		0			0
$235$	−0.385134			−0.0251234
$236$		0			0
$237$	−10.6891	−	18.5140i	−0.694329	−	1.20261i
$238$		0			0
$239$	28.3785			1.83565			0.917825	−	0.396986i	$-0.129944\pi$
$239$	28.3785			1.83565			0.917825	+	0.396986i	$0.129944\pi$
$240$		0			0
$241$	−2.87994	−	4.98819i	−0.185513	−	0.321318i	0.758236	−	0.651980i	$-0.226061\pi$
$241$	−2.87994	−	4.98819i	−0.185513	−	0.321318i	−0.943749	+	0.330662i	$0.892728\pi$
$242$		0			0
$243$	8.85139	+	15.3311i	0.567817	+	0.983488i
$244$		0			0
$245$	−4.96552	+	8.60054i	−0.317236	+	0.549468i
$246$		0			0
$247$	10.0343	−	1.84329i	0.638470	−	0.117286i
$248$		0			0
$249$	−4.48964	+	7.77628i	−0.284519	+	0.492802i
$250$		0			0
$251$	−3.29127	−	5.70065i	−0.207743	−	0.359822i	0.743260	−	0.669003i	$-0.233278\pi$
$251$	−3.29127	−	5.70065i	−0.207743	−	0.359822i	−0.951003	+	0.309181i	$0.899945\pi$
$252$		0			0
$253$	−18.2333	−	31.5810i	−1.14632	−	1.98548i
$254$		0			0
$255$	−20.9793			−1.31377
$256$		0			0
$257$	11.4694	+	19.8655i	0.715440	+	1.23918i	0.962790	+	0.270251i	$0.0871068\pi$
$257$	11.4694	+	19.8655i	0.715440	+	1.23918i	−0.247350	+	0.968926i	$0.579560\pi$
$258$		0			0
$259$	−36.6925			−2.27996
$260$		0			0
$261$	−4.46675	+	7.73663i	−0.276485	+	0.478885i
$262$		0			0
$263$	−3.04458	+	5.27336i	−0.187737	+	0.325170i	−0.944495	−	0.328525i	$-0.893449\pi$
$263$	−3.04458	+	5.27336i	−0.187737	+	0.325170i	0.756759	+	0.653694i	$0.226782\pi$
$264$		0			0
$265$	1.19594			0.0734661
$266$		0			0
$267$	−51.0827			−3.12621
$268$		0			0
$269$	−3.06208	+	5.30367i	−0.186698	+	0.323370i	−0.944147	−	0.329523i	$-0.893112\pi$
$269$	−3.06208	+	5.30367i	−0.186698	+	0.323370i	0.757449	+	0.652894i	$0.226445\pi$
$270$		0			0
$271$	−4.74920	+	8.22586i	−0.288493	+	0.499685i	−0.973450	−	0.228898i	$-0.926488\pi$
$271$	−4.74920	+	8.22586i	−0.288493	+	0.499685i	0.684957	+	0.728584i	$0.259821\pi$
$272$		0			0
$273$	−26.5618			−1.60759
$274$		0			0
$275$	−3.04929	−	5.28152i	−0.183879	−	0.318488i
$276$		0			0
$277$	−8.79770			−0.528603			−0.264301	−	0.964440i	$-0.585141\pi$
$277$	−8.79770			−0.528603			−0.264301	+	0.964440i	$0.585141\pi$
$278$		0			0
$279$	−3.29647	−	5.70965i	−0.197354	−	0.341828i
$280$		0			0
$281$	−12.9565	−	22.4413i	−0.772921	−	1.33874i	−0.935956	−	0.352118i	$-0.885462\pi$
$281$	−12.9565	−	22.4413i	−0.772921	−	1.33874i	0.163035	−	0.986620i	$-0.447872\pi$
$282$		0			0
$283$	−10.3636	+	17.9502i	−0.616051	+	1.06703i	0.374148	+	0.927369i	$0.377935\pi$
$283$	−10.3636	+	17.9502i	−0.616051	+	1.06703i	−0.990199	+	0.139662i	$0.955398\pi$
$284$		0			0
$285$	11.8241	−	2.17206i	0.700397	−	0.128662i
$286$		0			0
$287$	6.99772	−	12.1204i	0.413062	−	0.715445i
$288$		0			0
$289$	−20.4306	−	35.3868i	−1.20180	−	2.08158i
$290$		0			0
$291$	0.561343	+	0.972275i	0.0329065	+	0.0569958i
$292$		0			0
$293$	−14.7175			−0.859804			−0.429902	−	0.902876i	$-0.641452\pi$
$293$	−14.7175			−0.859804			−0.429902	+	0.902876i	$0.641452\pi$
$294$		0			0
$295$	5.48566	+	9.50145i	0.319388	+	0.553196i
$296$		0			0
$297$	27.0239			1.56808
$298$		0			0
$299$	−6.99772	+	12.1204i	−0.404689	+	0.700941i
$300$		0			0
$301$	2.69654	−	4.67055i	0.155426	−	0.269206i
$302$		0			0
$303$	5.66091			0.325211
$304$		0			0
$305$	0.165079			0.00945241
$306$		0			0
$307$	0.213929	−	0.370536i	0.0122096	−	0.0211476i	−0.859856	−	0.510537i	$-0.829447\pi$
$307$	0.213929	−	0.370536i	0.0122096	−	0.0211476i	0.872066	+	0.489389i	$0.162780\pi$
$308$		0			0
$309$	−0.0282202	+	0.0488788i	−0.00160539	+	0.00278062i
$310$		0			0
$311$	9.35307			0.530364			0.265182	−	0.964198i	$-0.414568\pi$
$311$	9.35307			0.530364			0.265182	+	0.964198i	$0.414568\pi$
$312$		0			0
$313$	−13.2718	−	22.9874i	−0.750165	−	1.29932i	−0.947743	−	0.319036i	$-0.896641\pi$
$313$	−13.2718	−	22.9874i	−0.750165	−	1.29932i	0.197578	−	0.980287i	$-0.436692\pi$
$314$		0			0
$315$	−18.9552			−1.06800
$316$		0			0
$317$	2.47556	+	4.28780i	0.139041	+	0.240827i	0.927134	−	0.374730i	$-0.122265\pi$
$317$	2.47556	+	4.28780i	0.139041	+	0.240827i	−0.788093	+	0.615557i	$0.788931\pi$
$318$		0			0
$319$	−5.91335	−	10.2422i	−0.331084	−	0.573455i
$320$		0			0
$321$	0.639175	−	1.10708i	0.0356753	−	0.0617914i
$322$		0			0
$323$	21.4935	+	25.2466i	1.19593	+	1.40476i
$324$		0			0
$325$	−1.17028	+	2.02698i	−0.0649153	+	0.112437i
$326$		0			0
$327$	−20.3153	−	35.1871i	−1.12344	−	1.94585i
$328$		0			0
$329$	−0.792362	−	1.37241i	−0.0436844	−	0.0756635i
$330$		0			0
$331$	−3.47145			−0.190808			−0.0954042	−	0.995439i	$-0.530414\pi$
$331$	−3.47145			−0.190808			−0.0954042	+	0.995439i	$0.530414\pi$
$332$		0			0
$333$	−20.5395	−	35.5755i	−1.12556	−	1.94953i
$334$		0			0
$335$	1.69727			0.0927320
$336$		0			0
$337$	−5.16753	+	8.95042i	−0.281493	+	0.487560i	−0.971753	−	0.236001i	$-0.924163\pi$
$337$	−5.16753	+	8.95042i	−0.281493	+	0.487560i	0.690260	+	0.723562i	$0.257496\pi$
$338$		0			0
$339$	−11.5370	+	19.9826i	−0.626602	+	1.08531i
$340$		0			0
$341$	8.72814			0.472655
$342$		0			0
$343$	−12.0605			−0.651205
$344$		0			0
$345$	−8.24583	+	14.2822i	−0.443940	+	0.768927i
$346$		0			0
$347$	0.775839	−	1.34379i	0.0416492	−	0.0721386i	−0.844449	−	0.535635i	$-0.820072\pi$
$347$	0.775839	−	1.34379i	0.0416492	−	0.0721386i	0.886099	+	0.463497i	$0.153405\pi$
$348$		0			0
$349$	8.03451			0.430078			0.215039	−	0.976606i	$-0.431012\pi$
$349$	8.03451			0.430078			0.215039	+	0.976606i	$0.431012\pi$
$350$		0			0
$351$	−5.18571	−	8.98191i	−0.276793	−	0.479419i
$352$		0			0
$353$	−17.2875			−0.920121			−0.460061	−	0.887888i	$-0.652172\pi$
$353$	−17.2875			−0.920121			−0.460061	+	0.887888i	$0.652172\pi$
$354$		0			0
$355$	4.75679	+	8.23901i	0.252464	+	0.437281i
$356$		0			0
$357$	−43.1621	−	74.7589i	−2.28438	−	3.95666i
$358$		0			0
$359$	−8.40203	+	14.5527i	−0.443442	+	0.768064i	−0.997942	−	0.0641193i	$-0.979576\pi$
$359$	−8.40203	+	14.5527i	−0.443442	+	0.768064i	0.554500	+	0.832184i	$0.312910\pi$
$360$		0			0
$361$	−14.7278	−	12.0039i	−0.775145	−	0.631783i
$362$		0			0
$363$	−36.1198	+	62.5613i	−1.89580	+	3.28362i
$364$		0			0
$365$	−5.97360	−	10.3466i	−0.312673	−	0.541565i
$366$		0			0
$367$	−16.0589	−	27.8149i	−0.838269	−	1.45193i	−0.891340	−	0.453334i	$-0.850234\pi$
$367$	−16.0589	−	27.8149i	−0.838269	−	1.45193i	0.0530711	−	0.998591i	$-0.483099\pi$
$368$		0			0
$369$	15.6686			0.815675
$370$		0			0
$371$	2.46049	+	4.26169i	0.127742	+	0.221256i
$372$		0			0
$373$	28.4233			1.47170			0.735851	−	0.677143i	$-0.236782\pi$
$373$	28.4233			1.47170			0.735851	+	0.677143i	$0.236782\pi$
$374$		0			0
$375$	−1.37901	+	2.38851i	−0.0712117	+	0.123342i
$376$		0			0
$377$	−2.26947	+	3.93084i	−0.116884	+	0.202449i
$378$		0			0
$379$	31.0504			1.59495			0.797476	−	0.603350i	$-0.206168\pi$
$379$	31.0504			1.59495			0.797476	+	0.603350i	$0.206168\pi$
$380$		0			0
$381$	50.9338			2.60942
$382$		0			0
$383$	14.6428	−	25.3621i	0.748213	−	1.29594i	−0.200465	−	0.979701i	$-0.564245\pi$
$383$	14.6428	−	25.3621i	0.748213	−	1.29594i	0.948678	−	0.316243i	$-0.102421\pi$
$384$		0			0
$385$	12.5470	−	21.7321i	0.639455	−	1.10757i
$386$		0			0
$387$	6.03783			0.306920
$388$		0			0
$389$	5.66415	+	9.81060i	0.287184	+	0.497417i	0.973136	−	0.230229i	$-0.0739476\pi$
$389$	5.66415	+	9.81060i	0.287184	+	0.497417i	−0.685953	+	0.727646i	$0.740614\pi$
$390$		0			0
$391$	−45.4843			−2.30024
$392$		0			0
$393$	22.4496	+	38.8839i	1.13243	+	1.96143i
$394$		0			0
$395$	−3.87563	−	6.71280i	−0.195004	−	0.337757i
$396$		0			0
$397$	−1.84526	+	3.19609i	−0.0926111	+	0.160407i	−0.908609	−	0.417648i	$-0.862855\pi$
$397$	−1.84526	+	3.19609i	−0.0926111	+	0.160407i	0.815998	+	0.578055i	$0.196188\pi$
$398$		0			0
$399$	32.0665	+	37.6659i	1.60533	+	1.88565i
$400$		0			0
$401$	6.63776	−	11.4969i	0.331474	−	0.574129i	−0.651327	−	0.758797i	$-0.725787\pi$
$401$	6.63776	−	11.4969i	0.331474	−	0.574129i	0.982801	+	0.184668i	$0.0591208\pi$
$402$		0			0
$403$	−1.67487	−	2.90097i	−0.0834315	−	0.144508i
$404$		0			0
$405$	0.799351	+	1.38452i	0.0397201	+	0.0687972i
$406$		0			0
$407$	54.3830			2.69566
$408$		0			0
$409$	2.01100	+	3.48315i	0.0994375	+	0.172231i	0.911452	−	0.411406i	$-0.134962\pi$
$409$	2.01100	+	3.48315i	0.0994375	+	0.172231i	−0.812014	+	0.583637i	$0.801629\pi$
$410$		0			0
$411$	11.1877			0.551850
$412$		0			0
$413$	−22.5720	+	39.0959i	−1.11070	+	1.92378i
$414$		0			0
$415$	−1.62785	+	2.81952i	−0.0799080	+	0.138405i
$416$		0			0
$417$	29.8981			1.46411
$418$		0			0
$419$	−4.41991			−0.215927			−0.107963	−	0.994155i	$-0.534433\pi$
$419$	−4.41991			−0.215927			−0.107963	+	0.994155i	$0.534433\pi$
$420$		0			0
$421$	6.58578	−	11.4069i	0.320971	−	0.555939i	−0.659717	−	0.751514i	$-0.729324\pi$
$421$	6.58578	−	11.4069i	0.320971	−	0.555939i	0.980689	+	0.195575i	$0.0626573\pi$
$422$		0			0
$423$	0.887090	−	1.53648i	0.0431318	−	0.0747064i
$424$		0			0
$425$	−7.60665			−0.368977
$426$		0			0
$427$	0.339629	+	0.588254i	0.0164358	+	0.0284676i
$428$		0			0
$429$	39.3681			1.90071
$430$		0			0
$431$	−14.4124	−	24.9630i	−0.694221	−	1.20243i	−0.970443	−	0.241332i	$-0.922416\pi$
$431$	−14.4124	−	24.9630i	−0.694221	−	1.20243i	0.276221	−	0.961094i	$-0.410918\pi$
$432$		0			0
$433$	6.30575	+	10.9219i	0.303035	+	0.524872i	0.976822	−	0.214054i	$-0.0686667\pi$
$433$	6.30575	+	10.9219i	0.303035	+	0.524872i	−0.673787	+	0.738926i	$0.735333\pi$
$434$		0			0
$435$	−2.67425	+	4.63194i	−0.128221	+	0.222085i
$436$		0			0
$437$	25.6352	−	4.70915i	1.22630	−	0.225269i
$438$		0			0
$439$	−4.78563	+	8.28896i	−0.228406	+	0.395611i	−0.957336	−	0.288978i	$-0.906685\pi$
$439$	−4.78563	+	8.28896i	−0.228406	+	0.395611i	0.728930	+	0.684588i	$0.240018\pi$
$440$		0			0
$441$	−22.8744	−	39.6197i	−1.08926	−	1.88665i
$442$		0			0
$443$	0.885541	+	1.53380i	0.0420733	+	0.0728731i	0.886295	−	0.463121i	$-0.153270\pi$
$443$	0.885541	+	1.53380i	0.0420733	+	0.0728731i	−0.844222	+	0.535994i	$0.819937\pi$
$444$		0			0
$445$	−18.5215			−0.878005
$446$		0			0
$447$	31.6401	+	54.8023i	1.49653	+	2.59206i
$448$		0			0
$449$	5.20350			0.245568			0.122784	−	0.992433i	$-0.460818\pi$
$449$	5.20350			0.245568			0.122784	+	0.992433i	$0.460818\pi$
$450$		0			0
$451$	−10.3715	+	17.9640i	−0.488376	+	0.845892i
$452$		0			0
$453$	−22.4812	+	38.9386i	−1.05626	+	1.82950i
$454$		0			0
$455$	−9.63077			−0.451497
$456$		0			0
$457$	6.45311			0.301864			0.150932	−	0.988544i	$-0.451773\pi$
$457$	6.45311			0.301864			0.150932	+	0.988544i	$0.451773\pi$
$458$		0			0
$459$	16.8532	−	29.1906i	0.786641	−	1.36250i
$460$		0			0
$461$	−1.12464	+	1.94794i	−0.0523798	+	0.0907245i	−0.891026	−	0.453951i	$-0.850014\pi$
$461$	−1.12464	+	1.94794i	−0.0523798	+	0.0907245i	0.838647	+	0.544676i	$0.183347\pi$
$462$		0			0
$463$	−33.8252			−1.57199			−0.785995	−	0.618232i	$-0.787849\pi$
$463$	−33.8252			−1.57199			−0.785995	+	0.618232i	$0.787849\pi$
$464$		0			0
$465$	−1.97360	−	3.41838i	−0.0915237	−	0.158524i
$466$		0			0
$467$	−7.01926			−0.324813			−0.162406	−	0.986724i	$-0.551926\pi$
$467$	−7.01926			−0.324813			−0.162406	+	0.986724i	$0.551926\pi$
$468$		0			0
$469$	3.49192	+	6.04818i	0.161242	+	0.279279i
$470$		0			0
$471$	12.3253	+	21.3480i	0.567919	+	0.983665i
$472$		0			0
$473$	−3.99663	+	6.92236i	−0.183765	+	0.318290i
$474$		0			0
$475$	4.28716	−	0.787545i	0.196709	−	0.0361351i
$476$		0			0
$477$	−2.75464	+	4.77118i	−0.126126	+	0.218457i
$478$		0			0
$479$	11.0213	+	19.0895i	0.503578	+	0.872223i	0.999991	+	0.00413643i	$0.00131667\pi$
$479$	11.0213	+	19.0895i	0.503578	+	0.872223i	−0.496413	+	0.868086i	$0.665350\pi$
$480$		0			0
$481$	−10.4358	−	18.0753i	−0.475829	−	0.824161i
$482$		0			0
$483$	−67.8588			−3.08768
$484$		0			0
$485$	0.203531	+	0.352527i	0.00924189	+	0.0160074i
$486$		0			0
$487$	18.1596			0.822892			0.411446	−	0.911434i	$-0.365024\pi$
$487$	18.1596			0.822892			0.411446	+	0.911434i	$0.365024\pi$
$488$		0			0
$489$	−7.79813	+	13.5068i	−0.352644	+	0.610797i
$490$		0			0
$491$	−17.0204	+	29.4802i	−0.768121	+	1.33042i	0.170460	+	0.985365i	$0.445475\pi$
$491$	−17.0204	+	29.4802i	−0.768121	+	1.33042i	−0.938581	+	0.345059i	$0.887859\pi$
$492$		0			0
$493$	−14.7513			−0.664364
$494$		0			0
$495$	28.0940			1.26273
$496$		0			0
$497$	−19.5729	+	33.9013i	−0.877967	+	1.52068i
$498$		0			0
$499$	10.1552	−	17.5893i	0.454610	−	0.787407i	−0.544056	−	0.839049i	$-0.683112\pi$
$499$	10.1552	−	17.5893i	0.454610	−	0.787407i	0.998666	+	0.0516420i	$0.0164455\pi$
$500$		0			0
$501$	−36.5941			−1.63491
$502$		0			0
$503$	0.231947	+	0.401743i	0.0103420	+	0.0179128i	0.871150	−	0.491017i	$-0.163375\pi$
$503$	0.231947	+	0.401743i	0.0103420	+	0.0179128i	−0.860808	+	0.508930i	$0.830041\pi$
$504$		0			0
$505$	2.05253			0.0913363
$506$		0			0
$507$	10.3726	+	17.9659i	0.460664	+	0.797894i
$508$		0			0
$509$	6.08533	+	10.5401i	0.269727	+	0.467181i	0.968791	−	0.247878i	$-0.0797331\pi$
$509$	6.08533	+	10.5401i	0.269727	+	0.467181i	−0.699064	+	0.715059i	$0.746400\pi$
$510$		0			0
$511$	24.5798	−	42.5735i	1.08735	−	1.88334i
$512$		0			0
$513$	−6.47638	+	18.1969i	−0.285939	+	0.803414i
$514$		0			0
$515$	−0.0102321	+	0.0177225i	−0.000450879	+	0.000780945i
$516$		0			0
$517$	1.17438	+	2.03409i	0.0516494	+	0.0894593i
$518$		0			0
$519$	1.77162	+	3.06854i	0.0777656	+	0.134694i
$520$		0			0
$521$	−41.4580			−1.81631			−0.908155	−	0.418635i	$-0.862509\pi$
$521$	−41.4580			−1.81631			−0.908155	+	0.418635i	$0.862509\pi$
$522$		0			0
$523$	−1.09675	−	1.89962i	−0.0479574	−	0.0830647i	0.841050	−	0.540957i	$-0.181938\pi$
$523$	−1.09675	−	1.89962i	−0.0479574	−	0.0830647i	−0.889008	+	0.457892i	$0.848605\pi$
$524$		0			0
$525$	−11.3485			−0.495289
$526$		0			0
$527$	5.44323	−	9.42796i	0.237111	−	0.410688i
$528$		0			0
$529$	−6.37742	+	11.0460i	−0.277279	+	0.480262i
$530$		0			0
$531$	−50.5411			−2.19330
$532$		0			0
$533$	7.96093			0.344826
$534$		0			0
$535$	0.231752	−	0.401406i	0.0100195	−	0.0173543i
$536$		0			0
$537$	5.89057	−	10.2028i	0.254197	−	0.440282i
$538$		0			0
$539$	60.5652			2.60873
$540$		0			0
$541$	−3.83490	−	6.64224i	−0.164875	−	0.285572i	0.771736	−	0.635943i	$-0.219389\pi$
$541$	−3.83490	−	6.64224i	−0.164875	−	0.285572i	−0.936611	+	0.350371i	$0.886055\pi$
$542$		0			0
$543$	−4.79894			−0.205942
$544$		0			0
$545$	−7.36589	−	12.7581i	−0.315520	−	0.546497i
$546$		0			0
$547$	19.0123	+	32.9303i	0.812909	+	1.40800i	0.910820	+	0.412804i	$0.135451\pi$
$547$	19.0123	+	32.9303i	0.812909	+	1.40800i	−0.0979112	+	0.995195i	$0.531216\pi$
$548$		0			0
$549$	−0.380231	+	0.658580i	−0.0162279	+	0.0281075i
$550$		0			0
$551$	8.31392	−	1.52725i	0.354185	−	0.0650632i
$552$		0			0
$553$	15.9472	−	27.6214i	0.678144	−	1.17458i
$554$		0			0
$555$	−12.2971	−	21.2992i	−0.521982	−	0.904099i
$556$		0			0
$557$	−11.6565	−	20.1897i	−0.493904	−	0.855466i	0.506072	−	0.862491i	$-0.331097\pi$
$557$	−11.6565	−	20.1897i	−0.493904	−	0.855466i	−0.999975	+	0.00702537i	$0.997764\pi$
$558$		0			0
$559$	3.06771			0.129750
$560$		0			0
$561$	63.9718	+	110.802i	2.70089	+	4.67808i
$562$		0			0
$563$	18.1507			0.764961			0.382481	−	0.923964i	$-0.375070\pi$
$563$	18.1507			0.764961			0.382481	+	0.923964i	$0.375070\pi$
$564$		0			0
$565$	−4.18307	+	7.24529i	−0.175983	+	0.304811i
$566$		0			0
$567$	−3.28912	+	5.69692i	−0.138130	+	0.239248i
$568$		0			0
$569$	−41.6104			−1.74440			−0.872200	−	0.489149i	$-0.837307\pi$
$569$	−41.6104			−1.74440			−0.872200	+	0.489149i	$0.837307\pi$
$570$		0			0
$571$	25.5273			1.06828			0.534142	−	0.845395i	$-0.320635\pi$
$571$	25.5273			1.06828			0.534142	+	0.845395i	$0.320635\pi$
$572$		0			0
$573$	31.0009	−	53.6952i	1.29508	−	2.24315i
$574$		0			0
$575$	−2.98977	+	5.17843i	−0.124682	+	0.215955i
$576$		0			0
$577$	−12.7680			−0.531540			−0.265770	−	0.964036i	$-0.585626\pi$
$577$	−12.7680			−0.531540			−0.265770	+	0.964036i	$0.585626\pi$
$578$		0			0
$579$	11.2681	+	19.5170i	0.468287	+	0.811097i
$580$		0			0
$581$	−13.3963			−0.555774
$582$		0			0
$583$	−3.64677	−	6.31638i	−0.151034	−	0.261598i
$584$		0			0
$585$	−5.39107	−	9.33760i	−0.222893	−	0.386062i
$586$		0			0
$587$	−12.1686	+	21.0767i	−0.502253	+	0.869927i	0.497744	+	0.867324i	$0.334162\pi$
$587$	−12.1686	+	21.0767i	−0.502253	+	0.869927i	−0.999997	+	0.00260307i	$0.999171\pi$
$588$		0			0
$589$	−2.09173	+	5.87722i	−0.0861884	+	0.242167i
$590$		0			0
$591$	−20.2647	+	35.0996i	−0.833580	+	1.44380i
$592$		0			0
$593$	10.4025	+	18.0177i	0.427180	+	0.739897i	0.996621	−	0.0821351i	$-0.0261739\pi$
$593$	10.4025	+	18.0177i	0.427180	+	0.739897i	−0.569442	+	0.822032i	$0.692841\pi$
$594$		0			0
$595$	−15.6497	−	27.1060i	−0.641574	−	1.11124i
$596$		0			0
$597$	43.0326			1.76121
$598$		0			0
$599$	−15.7162	−	27.2212i	−0.642146	−	1.11223i	−0.984953	−	0.172824i	$-0.944711\pi$
$599$	−15.7162	−	27.2212i	−0.642146	−	1.11223i	0.342807	−	0.939406i	$-0.388622\pi$
$600$		0			0
$601$	4.41844			0.180232			0.0901160	−	0.995931i	$-0.471276\pi$
$601$	4.41844			0.180232			0.0901160	+	0.995931i	$0.471276\pi$
$602$		0			0
$603$	−3.90938	+	6.77124i	−0.159202	+	0.275746i
$604$		0			0
$605$	−13.0963	+	22.6834i	−0.532440	+	0.922213i
$606$		0			0
$607$	26.2340			1.06481			0.532403	−	0.846491i	$-0.321289\pi$
$607$	26.2340			1.06481			0.532403	+	0.846491i	$0.321289\pi$
$608$		0			0
$609$	−22.0077			−0.891797
$610$		0			0
$611$	0.450714	−	0.780659i	0.0182339	−	0.0315821i
$612$		0			0
$613$	−21.6066	+	37.4237i	−0.872681	+	1.51153i	−0.0134676	+	0.999909i	$0.504287\pi$
$613$	−21.6066	+	37.4237i	−0.872681	+	1.51153i	−0.859213	+	0.511618i	$0.829046\pi$
$614$		0			0
$615$	9.38083			0.378272
$616$		0			0
$617$	8.62914	+	14.9461i	0.347396	+	0.601708i	0.985786	−	0.168005i	$-0.0537326\pi$
$617$	8.62914	+	14.9461i	0.347396	+	0.601708i	−0.638390	+	0.769713i	$0.720399\pi$
$618$		0			0
$619$	12.4548			0.500599			0.250300	−	0.968168i	$-0.419471\pi$
$619$	12.4548			0.500599			0.250300	+	0.968168i	$0.419471\pi$
$620$		0			0
$621$	−13.2482	−	22.9465i	−0.531632	−	0.920813i
$622$		0			0
$623$	−38.1056	−	66.0009i	−1.52667	−	2.64427i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−47.5268	−	55.8258i	−1.89804	−	2.22947i
$628$		0			0
$629$	33.9155	−	58.7434i	1.35230	−	2.34225i
$630$		0			0
$631$	−3.02335	−	5.23659i	−0.120358	−	0.208465i	0.799551	−	0.600598i	$-0.205071\pi$
$631$	−3.02335	−	5.23659i	−0.120358	−	0.208465i	−0.919909	+	0.392133i	$0.871737\pi$
$632$		0			0
$633$	19.7189	+	34.1542i	0.783758	+	1.35751i
$634$		0			0
$635$	18.4675			0.732862
$636$		0			0
$637$	−11.6221	−	20.1300i	−0.460484	−	0.797581i
$638$		0			0
$639$	−43.8258			−1.73372
$640$		0			0
$641$	14.8639	−	25.7449i	0.587087	−	1.01686i	−0.407525	−	0.913194i	$-0.633608\pi$
$641$	14.8639	−	25.7449i	0.587087	−	1.01686i	0.994612	−	0.103670i	$-0.0330587\pi$
$642$		0			0
$643$	14.7949	−	25.6256i	0.583455	−	1.01057i	−0.411611	−	0.911359i	$-0.635034\pi$
$643$	14.7949	−	25.6256i	0.583455	−	1.01057i	0.995066	−	0.0992137i	$-0.0316328\pi$
$644$		0			0
$645$	3.61487			0.142335
$646$		0			0
$647$	35.2808			1.38703			0.693515	−	0.720442i	$-0.256061\pi$
$647$	35.2808			1.38703			0.693515	+	0.720442i	$0.256061\pi$
$648$		0			0
$649$	33.4547	−	57.9452i	1.31321	−	2.27455i
$650$		0			0
$651$	8.12086	−	14.0657i	0.318282	−	0.551280i
$652$		0			0
$653$	6.40217			0.250536			0.125268	−	0.992123i	$-0.460021\pi$
$653$	6.40217			0.250536			0.125268	+	0.992123i	$0.460021\pi$
$654$		0			0
$655$	8.13978	+	14.0985i	0.318047	+	0.550874i
$656$		0			0
$657$	55.0367			2.14718
$658$		0			0
$659$	−20.0539	−	34.7344i	−0.781189	−	1.35306i	−0.931250	−	0.364382i	$-0.881280\pi$
$659$	−20.0539	−	34.7344i	−0.781189	−	1.35306i	0.150061	−	0.988677i	$-0.452053\pi$
$660$		0			0
$661$	−11.9089	−	20.6268i	−0.463202	−	0.802289i	0.535916	−	0.844271i	$-0.319966\pi$
$661$	−11.9089	−	20.6268i	−0.463202	−	0.802289i	−0.999118	+	0.0419816i	$0.986633\pi$
$662$		0			0
$663$	24.5516	−	42.5246i	0.953504	−	1.65152i
$664$		0			0
$665$	11.6267	+	13.6569i	0.450863	+	0.529591i
$666$		0			0
$667$	−5.79793	+	10.0423i	−0.224497	+	0.388840i
$668$		0			0
$669$	0.400565	+	0.693799i	0.0154867	+	0.0268238i
$670$		0			0
$671$	−0.503374	−	0.871869i	−0.0194325	−	0.0336581i
$672$		0			0
$673$	−2.45529			−0.0946445			−0.0473223	−	0.998880i	$-0.515069\pi$
$673$	−2.45529			−0.0946445			−0.0473223	+	0.998880i	$0.515069\pi$
$674$		0			0
$675$	−2.21559	−	3.83751i	−0.0852780	−	0.147706i
$676$		0			0
$677$	11.1525			0.428625			0.214313	−	0.976765i	$-0.431249\pi$
$677$	11.1525			0.428625			0.214313	+	0.976765i	$0.431249\pi$
$678$		0			0
$679$	−0.837478	+	1.45055i	−0.0321395	+	0.0556672i
$680$		0			0
$681$	−31.2774	+	54.1741i	−1.19855	+	2.07596i
$682$		0			0
$683$	18.0171			0.689407			0.344703	−	0.938712i	$-0.387979\pi$
$683$	18.0171			0.689407			0.344703	+	0.938712i	$0.387979\pi$
$684$		0			0
$685$	4.05644			0.154989
$686$		0			0
$687$	−9.76670	+	16.9164i	−0.372623	+	0.645402i
$688$		0			0
$689$	−1.39958	+	2.42415i	−0.0533199	+	0.0923527i
$690$		0			0
$691$	−0.184631			−0.00702370			−0.00351185	−	0.999994i	$-0.501118\pi$
$691$	−0.184631			−0.00702370			−0.00351185	+	0.999994i	$0.501118\pi$
$692$		0			0
$693$	57.7997	+	100.112i	2.19563	+	3.80294i
$694$		0			0
$695$	10.8404			0.411201
$696$		0			0
$697$	12.9362	+	22.4062i	0.489995	+	0.848697i
$698$		0			0
$699$	13.7593	+	23.8318i	0.520425	+	0.901402i
$700$		0			0
$701$	16.8233	−	29.1387i	0.635405	−	1.10055i	−0.351024	−	0.936367i	$-0.614166\pi$
$701$	16.8233	−	29.1387i	0.635405	−	1.10055i	0.986429	−	0.164188i	$-0.0525003\pi$
$702$		0			0
$703$	−13.0331	+	36.6196i	−0.491553	+	1.38113i
$704$		0			0
$705$	0.531103	−	0.919897i	0.0200025	−	0.0346453i
$706$		0			0
$707$	4.22281	+	7.31412i	0.158815	+	0.275076i
$708$		0			0
$709$	−11.5040	−	19.9255i	−0.432042	−	0.748319i	0.565007	−	0.825086i	$-0.308874\pi$
$709$	−11.5040	−	19.9255i	−0.432042	−	0.748319i	−0.997049	+	0.0767672i	$0.975540\pi$
$710$		0			0
$711$	35.7074			1.33913
$712$		0			0
$713$	−4.27889	−	7.41125i	−0.160246	−	0.277553i
$714$		0			0
$715$	14.2740			0.533819
$716$		0			0
$717$	−39.1341	+	67.7823i	−1.46149	+	2.53138i
$718$		0			0
$719$	−11.4576	+	19.8452i	−0.427297	+	0.740099i	−0.996632	−	0.0820060i	$-0.973867\pi$
$719$	−11.4576	+	19.8452i	−0.427297	+	0.740099i	0.569335	+	0.822105i	$0.307201\pi$
$720$		0			0
$721$	−0.0842045			−0.00313594
$722$		0			0
$723$	15.8858			0.590800
$724$		0			0
$725$	−0.969629	+	1.67945i	−0.0360111	+	0.0623731i
$726$		0			0
$727$	7.94740	−	13.7653i	0.294753	−	0.510527i	−0.680174	−	0.733050i	$-0.738096\pi$
$727$	7.94740	−	13.7653i	0.294753	−	0.510527i	0.974927	+	0.222523i	$0.0714293\pi$
$728$		0			0
$729$	−44.0284			−1.63068
$730$		0			0
$731$	4.98493	+	8.63415i	0.184374	+	0.319346i
$732$		0			0
$733$	18.5670			0.685787			0.342894	−	0.939374i	$-0.388593\pi$
$733$	18.5670			0.685787			0.342894	+	0.939374i	$0.388593\pi$
$734$		0			0
$735$	−13.6950	−	23.7204i	−0.505147	−	0.874941i
$736$		0			0
$737$	−5.17548	−	8.96419i	−0.190641	−	0.330200i
$738$		0			0
$739$	1.70308	−	2.94981i	0.0626486	−	0.108511i	−0.833000	−	0.553273i	$-0.813379\pi$
$739$	1.70308	−	2.94981i	0.0626486	−	0.108511i	0.895649	+	0.444763i	$0.146712\pi$
$740$		0			0
$741$	−9.43472	+	26.5091i	−0.346593	+	0.973835i
$742$		0			0
$743$	−2.08221	+	3.60650i	−0.0763890	+	0.132310i	−0.901689	−	0.432384i	$-0.857672\pi$
$743$	−2.08221	+	3.60650i	−0.0763890	+	0.132310i	0.825300	+	0.564694i	$0.191006\pi$
$744$		0			0
$745$	11.4721	+	19.8702i	0.420304	+	0.727987i
$746$		0			0
$747$	−7.49894	−	12.9885i	−0.274372	−	0.475226i
$748$		0			0
$749$	1.90719			0.0696873
$750$		0			0
$751$	8.92844	+	15.4645i	0.325803	+	0.564308i	0.981675	−	0.190565i	$-0.0610319\pi$
$751$	8.92844	+	15.4645i	0.325803	+	0.564308i	−0.655871	+	0.754873i	$0.727699\pi$
$752$		0			0
$753$	18.1548			0.661596
$754$		0			0
$755$	−8.15123	+	14.1183i	−0.296654	+	0.513819i
$756$		0			0
$757$	−17.7593	+	30.7600i	−0.645473	+	1.11799i	0.338719	+	0.940888i	$0.390006\pi$
$757$	−17.7593	+	30.7600i	−0.645473	+	1.11799i	−0.984192	+	0.177104i	$0.943327\pi$
$758$		0			0
$759$	100.576			3.65066
$760$		0			0
$761$	−46.6528			−1.69116			−0.845581	−	0.533848i	$-0.820746\pi$
$761$	−46.6528			−1.69116			−0.845581	+	0.533848i	$0.820746\pi$
$762$		0			0
$763$	30.3087	−	52.4962i	1.09725	−	1.90049i
$764$		0			0
$765$	17.5206	−	30.3466i	0.633459	−	1.09718i
$766$		0			0
$767$	−25.6790			−0.927215
$768$		0			0
$769$	17.5585	+	30.4121i	0.633174	+	1.09669i	0.986899	+	0.161340i	$0.0515817\pi$
$769$	17.5585	+	30.4121i	0.633174	+	1.09669i	−0.353725	+	0.935350i	$0.615085\pi$
$770$		0			0
$771$	−63.2654			−2.27845
$772$		0			0
$773$	−5.99581	−	10.3850i	−0.215654	−	0.373524i	0.737820	−	0.674997i	$-0.235855\pi$
$773$	−5.99581	−	10.3850i	−0.215654	−	0.373524i	−0.953475	+	0.301473i	$0.902522\pi$
$774$		0			0
$775$	−0.715589	−	1.23944i	−0.0257047	−	0.0445218i
$776$		0			0
$777$	50.5992	−	87.6404i	1.81524	−	3.14408i
$778$		0			0
$779$	−9.61076	−	11.2890i	−0.344341	−	0.404469i
$780$		0			0
$781$	29.0096	−	50.2462i	1.03805	−	1.79795i
$782$		0			0
$783$	−4.29660	−	7.44193i	−0.153548	−	0.265953i
$784$		0			0
$785$	4.46890	+	7.74036i	0.159502	+	0.276265i
$786$		0			0
$787$	−16.9481			−0.604134			−0.302067	−	0.953287i	$-0.597677\pi$
$787$	−16.9481			−0.604134			−0.302067	+	0.953287i	$0.597677\pi$
$788$		0			0
$789$	−8.39700	−	14.5440i	−0.298941	−	0.517781i
$790$		0			0
$791$	−34.4244			−1.22399

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)

\(f(q)\)	\(=\)	\(q+(-1.37901 + 2.38851i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.11474 q^{7} +(-2.30333 - 3.98948i) q^{9} +O(q^{10})\)	\(q+(-1.37901 + 2.38851i) q^{3} +(-0.500000 + 0.866025i) q^{5} -4.11474 q^{7} +(-2.30333 - 3.98948i) q^{9} +6.09857 q^{11} +(-1.17028 - 2.02698i) q^{13} +(-1.37901 - 2.38851i) q^{15} +(3.80333 - 6.58756i) q^{17} +(-1.46155 + 4.10657i) q^{19} +(5.67425 - 9.82810i) q^{21} +(-2.98977 - 5.17843i) q^{23} +(-0.500000 - 0.866025i) q^{25} +4.43118 q^{27} +(-0.969629 - 1.67945i) q^{29} +1.43118 q^{31} +(-8.40998 + 14.5665i) q^{33} +(2.05737 - 3.56347i) q^{35} +8.91733 q^{37} +6.45529 q^{39} +(-1.70065 + 2.94561i) q^{41} +(-0.655338 + 1.13508i) q^{43} +4.60665 q^{45} +(0.192567 + 0.333536i) q^{47} +9.93105 q^{49} +(10.4896 + 18.1686i) q^{51} +(-0.597970 - 1.03572i) q^{53} +(-3.04929 + 5.28152i) q^{55} +(-7.79309 - 9.15391i) q^{57} +(5.48566 - 9.50145i) q^{59} +(-0.0825396 - 0.142963i) q^{61} +(9.47758 + 16.4157i) q^{63} +2.34056 q^{65} +(-0.848637 - 1.46988i) q^{67} +16.4917 q^{69} +(4.75679 - 8.23901i) q^{71} +(-5.97360 + 10.3466i) q^{73} +2.75802 q^{75} -25.0940 q^{77} +(-3.87563 + 6.71280i) q^{79} +(0.799351 - 1.38452i) q^{81} +3.25570 q^{83} +(3.80333 + 6.58756i) q^{85} +5.34851 q^{87} +(9.26077 + 16.0401i) q^{89} +(4.81538 + 8.34049i) q^{91} +(-1.97360 + 3.41838i) q^{93} +(-2.82562 - 3.31902i) q^{95} +(0.203531 - 0.352527i) q^{97} +(-14.0470 - 24.3301i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9	\( 8 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 8 q^{11} + q^{13} - q^{15} + 13 q^{17} - q^{19} + 12 q^{21} - 8 q^{23} - 4 q^{25} + 20 q^{27} - 3 q^{29} - 4 q^{31} - 15 q^{33} + 2 q^{35} + 20 q^{37} + 2 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{45} - 10 q^{47} + 12 q^{49} + 16 q^{51} - 11 q^{53} - 4 q^{55} - 29 q^{57} - q^{59} + 25 q^{63} - 2 q^{65} + 8 q^{67} + 22 q^{69} + 4 q^{71} - 20 q^{73} + 2 q^{75} - 36 q^{77} + 3 q^{79} + 12 q^{81} + 30 q^{83} + 13 q^{85} - 24 q^{87} + 17 q^{89} + 4 q^{91} + 12 q^{93} - 4 q^{95} + 11 q^{97} - 30 q^{99}+O(q^{100}) \) 8 * q - q^3 - 4 * q^5 - 4 * q^7 - q^9 + 8 * q^11 + q^13 - q^15 + 13 * q^17 - q^19 + 12 * q^21 - 8 * q^23 - 4 * q^25 + 20 * q^27 - 3 * q^29 - 4 * q^31 - 15 * q^33 + 2 * q^35 + 20 * q^37 + 2 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^45 - 10 * q^47 + 12 * q^49 + 16 * q^51 - 11 * q^53 - 4 * q^55 - 29 * q^57 - q^59 + 25 * q^63 - 2 * q^65 + 8 * q^67 + 22 * q^69 + 4 * q^71 - 20 * q^73 + 2 * q^75 - 36 * q^77 + 3 * q^79 + 12 * q^81 + 30 * q^83 + 13 * q^85 - 24 * q^87 + 17 * q^89 + 4 * q^91 + 12 * q^93 - 4 * q^95 + 11 * q^97 - 30 * q^99

Introduction

L-functions

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