LMFDB - Embedded newform 520.2.f.b.129.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(129,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("520.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.f (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:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 $ x^10 + 11x^8 + 36x^6 + 42x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.7
Root			$2.50630i$ of defining polynomial
Character	$\chi$	$=$	520.129
Dual form			520.2.f.b.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.949078i q^{3} +(0.108880 + 2.23342i) q^{5} +3.85660 q^{7} +2.09925 q^{9} +O(q^{10})$	$q+0.949078i q^{3} +(0.108880 + 2.23342i) q^{5} +3.85660 q^{7} +2.09925 q^{9} +0.589684i q^{11} +(-3.32782 - 1.38767i) q^{13} +(-2.11969 + 0.103336i) q^{15} -3.66022i q^{17} +5.94139i q^{19} +3.66022i q^{21} -3.51775i q^{23} +(-4.97629 + 0.486349i) q^{25} +4.83959i q^{27} +5.33862 q^{29} +8.71673i q^{31} -0.559657 q^{33} +(0.419907 + 8.61340i) q^{35} +1.85660 q^{37} +(1.31701 - 3.15836i) q^{39} -4.63292i q^{41} +6.30078i q^{43} +(0.228566 + 4.68850i) q^{45} -3.85660 q^{47} +7.87339 q^{49} +3.47383 q^{51} -10.7912i q^{53} +(-1.31701 + 0.0642049i) q^{55} -5.63884 q^{57} -10.3127i q^{59} -13.4874 q^{61} +8.09598 q^{63} +(2.73692 - 7.58349i) q^{65} -4.09598 q^{67} +3.33862 q^{69} -5.34145i q^{71} -6.09598 q^{73} +(-0.461583 - 4.72289i) q^{75} +2.27418i q^{77} +11.1128 q^{79} +1.70460 q^{81} +8.77977 q^{83} +(8.17479 - 0.398525i) q^{85} +5.06677i q^{87} -0.413343i q^{89} +(-12.8341 - 5.35170i) q^{91} -8.27286 q^{93} +(-13.2696 + 0.646899i) q^{95} +8.45713 q^{97} +1.23790i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{5} - 2 q^{7} - 8 q^{9}+O(q^{10}) $ 10 * q + 3 * q^5 - 2 * q^7 - 8 * q^9	$ 10 q + 3 q^{5} - 2 q^{7} - 8 q^{9} + 2 q^{13} - 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} - 3 q^{35} - 22 q^{37} - 12 q^{39} - 4 q^{45} + 2 q^{47} + 12 q^{49} - 30 q^{51} + 12 q^{55} - 12 q^{57} - 16 q^{61} + 24 q^{63} - 5 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{73} + 21 q^{75} + 28 q^{79} + 22 q^{81} - 4 q^{83} - 25 q^{85} + 2 q^{91} + 12 q^{93} - 10 q^{95} + 72 q^{97}+O(q^{100}) $ 10 * q + 3 * q^5 - 2 * q^7 - 8 * q^9 + 2 * q^13 - 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 - 3 * q^35 - 22 * q^37 - 12 * q^39 - 4 * q^45 + 2 * q^47 + 12 * q^49 - 30 * q^51 + 12 * q^55 - 12 * q^57 - 16 * q^61 + 24 * q^63 - 5 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^73 + 21 * q^75 + 28 * q^79 + 22 * q^81 - 4 * q^83 - 25 * q^85 + 2 * q^91 + 12 * q^93 - 10 * q^95 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			0.949078i			0.547951i	0.961737	+	0.273975i	$0.0883387\pi$
$3$			0.949078i			0.547951i	−0.961737	+	0.273975i	$0.911661\pi$
$4$		0			0
$5$	0.108880	+	2.23342i	0.0486926	+	0.998814i
$5$	0.108880	+	2.23342i	0.0486926	+	0.998814i
$6$		0			0
$7$	3.85660			1.45766			0.728830	−	0.684695i	$-0.240065\pi$
$7$	3.85660			1.45766			0.728830	+	0.684695i	$0.240065\pi$
$8$		0			0
$9$	2.09925			0.699750
$10$		0			0
$11$			0.589684i			0.177797i	0.996041	+	0.0888983i	$0.0283346\pi$
$11$			0.589684i			0.177797i	−0.996041	+	0.0888983i	$0.971665\pi$
$12$		0			0
$13$	−3.32782	−	1.38767i	−0.922970	−	0.384871i
$13$	−3.32782	−	1.38767i	−0.922970	−	0.384871i
$14$		0			0
$15$	−2.11969	+	0.103336i	−0.547301	+	0.0266812i
$16$		0			0
$17$		−	3.66022i		−	0.887734i	−0.896093	−	0.443867i	$-0.853606\pi$
$17$		−	3.66022i		−	0.887734i	0.896093	−	0.443867i	$-0.146394\pi$
$18$		0			0
$19$			5.94139i			1.36305i	0.731796	+	0.681524i	$0.238683\pi$
$19$			5.94139i			1.36305i	−0.731796	+	0.681524i	$0.761317\pi$
$20$		0			0
$21$			3.66022i			0.798725i
$22$		0			0
$23$		−	3.51775i		−	0.733502i	−0.930319	−	0.366751i	$-0.880470\pi$
$23$		−	3.51775i		−	0.733502i	0.930319	−	0.366751i	$-0.119530\pi$
$24$		0			0
$25$	−4.97629	+	0.486349i	−0.995258	+	0.0972697i
$26$		0			0
$27$			4.83959i			0.931379i
$28$		0			0
$29$	5.33862			0.991357			0.495679	−	0.868506i	$-0.334919\pi$
$29$	5.33862			0.991357			0.495679	+	0.868506i	$0.334919\pi$
$30$		0			0
$31$			8.71673i			1.56557i	0.622291	+	0.782786i	$0.286202\pi$
$31$			8.71673i			1.56557i	−0.622291	+	0.782786i	$0.713798\pi$
$32$		0			0
$33$	−0.559657			−0.0974237
$34$		0			0
$35$	0.419907	+	8.61340i	0.0709773	+	1.45593i
$36$		0			0
$37$	1.85660			0.305224			0.152612	−	0.988286i	$-0.451232\pi$
$37$	1.85660			0.305224			0.152612	+	0.988286i	$0.451232\pi$
$38$		0			0
$39$	1.31701	−	3.15836i	0.210890	−	0.505742i
$40$		0			0
$41$		−	4.63292i		−	0.723540i	−0.932267	−	0.361770i	$-0.882173\pi$
$41$		−	4.63292i		−	0.723540i	0.932267	−	0.361770i	$-0.117827\pi$
$42$		0			0
$43$			6.30078i			0.960860i	0.877033	+	0.480430i	$0.159519\pi$
$43$			6.30078i			0.960860i	−0.877033	+	0.480430i	$0.840481\pi$
$44$		0			0
$45$	0.228566	+	4.68850i	0.0340727	+	0.698920i
$46$		0			0
$47$	−3.85660			−0.562543			−0.281272	−	0.959628i	$-0.590756\pi$
$47$	−3.85660			−0.562543			−0.281272	+	0.959628i	$0.590756\pi$
$48$		0			0
$49$	7.87339			1.12477
$50$		0			0
$51$	3.47383			0.486434
$52$		0			0
$53$		−	10.7912i		−	1.48229i	−0.671345	−	0.741145i	$-0.734283\pi$
$53$		−	10.7912i		−	1.48229i	0.671345	−	0.741145i	$-0.265717\pi$
$54$		0			0
$55$	−1.31701	+	0.0642049i	−0.177586	+	0.00865738i
$56$		0			0
$57$	−5.63884			−0.746883
$58$		0			0
$59$		−	10.3127i		−	1.34260i	−0.741185	−	0.671300i	$-0.765736\pi$
$59$		−	10.3127i		−	1.34260i	0.741185	−	0.671300i	$-0.234264\pi$
$60$		0			0
$61$	−13.4874			−1.72688			−0.863439	−	0.504453i	$-0.831694\pi$
$61$	−13.4874			−1.72688			−0.863439	+	0.504453i	$0.831694\pi$
$62$		0			0
$63$	8.09598			1.02000
$64$		0			0
$65$	2.73692	−	7.58349i	0.339473	−	0.940616i
$66$		0			0
$67$	−4.09598			−0.500403			−0.250202	−	0.968194i	$-0.580497\pi$
$67$	−4.09598			−0.500403			−0.250202	+	0.968194i	$0.580497\pi$
$68$		0			0
$69$	3.33862			0.401923
$70$		0			0
$71$		−	5.34145i		−	0.633913i	−0.948440	−	0.316957i	$-0.897339\pi$
$71$		−	5.34145i		−	0.633913i	0.948440	−	0.316957i	$-0.102661\pi$
$72$		0			0
$73$	−6.09598			−0.713480			−0.356740	−	0.934204i	$-0.616112\pi$
$73$	−6.09598			−0.713480			−0.356740	+	0.934204i	$0.616112\pi$
$74$		0			0
$75$	−0.461583	−	4.72289i	−0.0532990	−	0.545352i
$76$		0			0
$77$			2.27418i			0.259167i
$78$		0			0
$79$	11.1128			1.25028			0.625142	−	0.780511i	$-0.285041\pi$
$79$	11.1128			1.25028			0.625142	+	0.780511i	$0.285041\pi$
$80$		0			0
$81$	1.70460			0.189400
$82$		0			0
$83$	8.77977			0.963705			0.481852	−	0.876252i	$-0.339964\pi$
$83$	8.77977			0.963705			0.481852	+	0.876252i	$0.339964\pi$
$84$		0			0
$85$	8.17479	−	0.398525i	0.886681	−	0.0432261i
$86$		0			0
$87$			5.06677i			0.543215i
$88$		0			0
$89$		−	0.413343i		−	0.0438142i	−0.999760	−	0.0219071i	$-0.993026\pi$
$89$		−	0.413343i		−	0.0438142i	0.999760	−	0.0219071i	$-0.00697381\pi$
$90$		0			0
$91$	−12.8341	−	5.35170i	−1.34538	−	0.561011i
$92$		0			0
$93$	−8.27286			−0.857856
$94$		0			0
$95$	−13.2696	+	0.646899i	−1.36143	+	0.0663704i
$96$		0			0
$97$	8.45713			0.858692			0.429346	−	0.903140i	$-0.358744\pi$
$97$	8.45713			0.858692			0.429346	+	0.903140i	$0.358744\pi$
$98$		0			0
$99$			1.23790i			0.124413i
$100$		0			0
$101$	−11.4762			−1.14192			−0.570962	−	0.820977i	$-0.693430\pi$
$101$	−11.4762			−1.14192			−0.570962	+	0.820977i	$0.693430\pi$
$102$		0			0
$103$		−	9.28280i		−	0.914661i	−0.889297	−	0.457331i	$-0.848806\pi$
$103$		−	9.28280i		−	0.914661i	0.889297	−	0.457331i	$-0.151194\pi$
$104$		0			0
$105$	−8.17479	+	0.398525i	−0.797778	+	0.0388920i
$106$		0			0
$107$			10.8788i			1.05169i	0.850580	+	0.525846i	$0.176251\pi$
$107$			10.8788i			1.05169i	−0.850580	+	0.525846i	$0.823749\pi$
$108$		0			0
$109$		−	13.8138i		−	1.32313i	−0.749890	−	0.661563i	$-0.769894\pi$
$109$		−	13.8138i		−	1.32313i	0.749890	−	0.661563i	$-0.230106\pi$
$110$		0			0
$111$			1.76206i			0.167248i
$112$		0			0
$113$			9.43809i			0.887861i	0.896061	+	0.443931i	$0.146416\pi$
$113$			9.43809i			0.887861i	−0.896061	+	0.443931i	$0.853584\pi$
$114$		0			0
$115$	7.85660	−	0.383013i	0.732632	−	0.0357161i
$116$		0			0
$117$	−6.98592	−	2.91307i	−0.645849	−	0.269314i
$118$		0			0
$119$		−	14.1160i		−	1.29401i
$120$		0			0
$121$	10.6523			0.968388
$122$		0			0
$123$	4.39700			0.396464
$124$		0			0
$125$	−1.62804	−	11.0612i	−0.145616	−	0.989341i
$126$		0			0
$127$		−	20.1619i		−	1.78908i	−0.446992	−	0.894538i	$-0.647505\pi$
$127$		−	20.1619i		−	1.78908i	0.446992	−	0.894538i	$-0.352495\pi$
$128$		0			0
$129$	−5.97994			−0.526504
$130$		0			0
$131$	6.18960			0.540788			0.270394	−	0.962750i	$-0.412846\pi$
$131$	6.18960			0.540788			0.270394	+	0.962750i	$0.412846\pi$
$132$		0			0
$133$			22.9136i			1.98686i
$134$		0			0
$135$	−10.8088	+	0.526935i	−0.930274	+	0.0453513i
$136$		0			0
$137$	2.50691			0.214179			0.107090	−	0.994249i	$-0.465847\pi$
$137$	2.50691			0.214179			0.107090	+	0.994249i	$0.465847\pi$
$138$		0			0
$139$	9.71556			0.824063			0.412032	−	0.911170i	$-0.364819\pi$
$139$	9.71556			0.824063			0.412032	+	0.911170i	$0.364819\pi$
$140$		0			0
$141$		−	3.66022i		−	0.308246i
$142$		0			0
$143$	0.818289	−	1.96236i	0.0684288	−	0.164101i
$144$		0			0
$145$	0.581269	+	11.9234i	0.0482718	+	0.990181i
$146$		0			0
$147$			7.47247i			0.616319i
$148$		0			0
$149$		−	17.8973i		−	1.46621i	−0.680118	−	0.733103i	$-0.738071\pi$
$149$		−	17.8973i		−	1.46621i	0.680118	−	0.733103i	$-0.261929\pi$
$150$		0			0
$151$		−	1.40398i		−	0.114254i	−0.998367	−	0.0571272i	$-0.981806\pi$
$151$		−	1.40398i		−	0.114254i	0.998367	−	0.0571272i	$-0.0181940\pi$
$152$		0			0
$153$		−	7.68372i		−	0.621192i
$154$		0			0
$155$	−19.4681	+	0.949078i	−1.56371	+	0.0762318i
$156$		0			0
$157$		−	18.6161i		−	1.48573i	−0.669443	−	0.742863i	$-0.733467\pi$
$157$		−	18.6161i		−	1.48573i	0.669443	−	0.742863i	$-0.266533\pi$
$158$		0			0
$159$	10.2417			0.812222
$160$		0			0
$161$		−	13.5666i		−	1.06920i
$162$		0			0
$163$	−2.74619			−0.215098			−0.107549	−	0.994200i	$-0.534300\pi$
$163$	−2.74619			−0.215098			−0.107549	+	0.994200i	$0.534300\pi$
$164$		0			0
$165$	−0.0609354	−	1.24995i	−0.00474382	−	0.0973082i
$166$		0			0
$167$	−15.3304			−1.18630			−0.593152	−	0.805090i	$-0.702117\pi$
$167$	−15.3304			−1.18630			−0.593152	+	0.805090i	$0.702117\pi$
$168$		0			0
$169$	9.14873	+	9.23584i	0.703748	+	0.710449i
$170$		0			0
$171$			12.4725i			0.953793i
$172$		0			0
$173$			12.5456i			0.953825i	0.878951	+	0.476913i	$0.158244\pi$
$173$			12.5456i			0.953825i	−0.878951	+	0.476913i	$0.841756\pi$
$174$		0			0
$175$	−19.1916	+	1.87565i	−1.45075	+	0.141786i
$176$		0			0
$177$	9.78757			0.735679
$178$		0			0
$179$	8.82362			0.659508			0.329754	−	0.944067i	$-0.393034\pi$
$179$	8.82362			0.659508			0.329754	+	0.944067i	$0.393034\pi$
$180$		0			0
$181$	7.88929			0.586406			0.293203	−	0.956050i	$-0.405279\pi$
$181$	7.88929			0.586406			0.293203	+	0.956050i	$0.405279\pi$
$182$		0			0
$183$		−	12.8006i		−	0.946244i
$184$		0			0
$185$	0.202147	+	4.14657i	0.0148621	+	0.304862i
$186$		0			0
$187$	2.15837			0.157836
$188$		0			0
$189$			18.6644i			1.35763i
$190$		0			0
$191$	−16.2975			−1.17924			−0.589621	−	0.807680i	$-0.700723\pi$
$191$	−16.2975			−1.17924			−0.589621	+	0.807680i	$0.700723\pi$
$192$		0			0
$193$	−18.8476			−1.35668			−0.678339	−	0.734749i	$-0.737300\pi$
$193$	−18.8476			−1.35668			−0.678339	+	0.734749i	$0.737300\pi$
$194$		0			0
$195$	7.19732	+	2.59755i	0.515411	+	0.186014i
$196$		0			0
$197$	26.1727			1.86473			0.932364	−	0.361522i	$-0.117743\pi$
$197$	26.1727			1.86473			0.932364	+	0.361522i	$0.117743\pi$
$198$		0			0
$199$	−3.02941			−0.214749			−0.107375	−	0.994219i	$-0.534244\pi$
$199$	−3.02941			−0.214749			−0.107375	+	0.994219i	$0.534244\pi$
$200$		0			0
$201$		−	3.88740i		−	0.274196i
$202$		0			0
$203$	20.5890			1.44506
$204$		0			0
$205$	10.3472	−	0.504432i	0.722682	−	0.0352311i
$206$		0			0
$207$		−	7.38464i		−	0.513268i
$208$		0			0
$209$	−3.50354			−0.242345
$210$		0			0
$211$	−8.82362			−0.607443			−0.303722	−	0.952761i	$-0.598229\pi$
$211$	−8.82362			−0.607443			−0.303722	+	0.952761i	$0.598229\pi$
$212$		0			0
$213$	5.06945			0.347353
$214$		0			0
$215$	−14.0723	+	0.686029i	−0.959721	+	0.0467868i
$216$		0			0
$217$			33.6170i			2.28207i
$218$		0			0
$219$		−	5.78556i		−	0.390952i
$220$		0			0
$221$	−5.07919	+	12.1805i	−0.341663	+	0.819352i
$222$		0			0
$223$	−3.27236			−0.219133			−0.109567	−	0.993979i	$-0.534946\pi$
$223$	−3.27236			−0.219133			−0.109567	+	0.993979i	$0.534946\pi$
$224$		0			0
$225$	−10.4465	+	1.02097i	−0.696432	+	0.0680645i
$226$		0			0
$227$	−2.12956			−0.141344			−0.0706718	−	0.997500i	$-0.522514\pi$
$227$	−2.12956			−0.141344			−0.0706718	+	0.997500i	$0.522514\pi$
$228$		0			0
$229$			2.62136i			0.173225i	0.996242	+	0.0866123i	$0.0276041\pi$
$229$			2.62136i			0.173225i	−0.996242	+	0.0866123i	$0.972396\pi$
$230$		0			0
$231$	−2.15837			−0.142011
$232$		0			0
$233$			10.3197i			0.676066i	0.941134	+	0.338033i	$0.109762\pi$
$233$			10.3197i			0.676066i	−0.941134	+	0.338033i	$0.890238\pi$
$234$		0			0
$235$	−0.419907	−	8.61340i	−0.0273917	−	0.561876i
$236$		0			0
$237$			10.5469i			0.685094i
$238$		0			0
$239$			1.76463i			0.114145i	0.998370	+	0.0570723i	$0.0181766\pi$
$239$			1.76463i			0.114145i	−0.998370	+	0.0570723i	$0.981823\pi$
$240$		0			0
$241$			22.9136i			1.47599i	0.674804	+	0.737997i	$0.264228\pi$
$241$			22.9136i			1.47599i	−0.674804	+	0.737997i	$0.735772\pi$
$242$		0			0
$243$			16.1366i			1.03516i
$244$		0			0
$245$	0.857255	+	17.5846i	0.0547680	+	1.12344i
$246$		0			0
$247$	8.24470	−	19.7719i	0.524598	−	1.25805i
$248$		0			0
$249$			8.33269i			0.528063i
$250$		0			0
$251$	8.79239			0.554971			0.277486	−	0.960730i	$-0.410499\pi$
$251$	8.79239			0.554971			0.277486	+	0.960730i	$0.410499\pi$
$252$		0			0
$253$	2.07436			0.130414
$254$		0			0
$255$	0.378231	+	7.75852i	0.0236858	+	0.485857i
$256$		0			0
$257$		−	24.3907i		−	1.52145i	−0.649074	−	0.760725i	$-0.724843\pi$
$257$		−	24.3907i		−	1.52145i	0.649074	−	0.760725i	$-0.275157\pi$
$258$		0			0
$259$	7.16019			0.444912
$260$		0			0
$261$	11.2071			0.693702
$262$		0			0
$263$		−	4.14545i		−	0.255619i	−0.991799	−	0.127810i	$-0.959205\pi$
$263$		−	4.14545i		−	0.255619i	0.991799	−	0.127810i	$-0.0407947\pi$
$264$		0			0
$265$	24.1013	−	1.17495i	1.48053	−	0.0721766i
$266$		0			0
$267$	0.392295			0.0240080
$268$		0			0
$269$	10.2081			0.622402			0.311201	−	0.950344i	$-0.399269\pi$
$269$	10.2081			0.622402			0.311201	+	0.950344i	$0.399269\pi$
$270$		0			0
$271$			3.82991i			0.232650i	0.993211	+	0.116325i	$0.0371115\pi$
$271$			3.82991i			0.232650i	−0.993211	+	0.116325i	$0.962889\pi$
$272$		0			0
$273$	5.07919	−	12.1805i	0.307406	−	0.737200i
$274$		0			0
$275$	−0.286792	−	2.93444i	−0.0172942	−	0.176953i
$276$		0			0
$277$			21.9080i			1.31632i	0.752877	+	0.658162i	$0.228666\pi$
$277$			21.9080i			1.31632i	−0.752877	+	0.658162i	$0.771334\pi$
$278$		0			0
$279$			18.2986i			1.09551i
$280$		0			0
$281$		−	12.0345i		−	0.717920i	−0.933353	−	0.358960i	$-0.883131\pi$
$281$		−	12.0345i		−	0.717920i	0.933353	−	0.358960i	$-0.116869\pi$
$282$		0			0
$283$		−	14.1180i		−	0.839226i	−0.907703	−	0.419613i	$-0.862166\pi$
$283$		−	14.1180i		−	0.839226i	0.907703	−	0.419613i	$-0.137834\pi$
$284$		0			0
$285$	−0.613957	−	12.5939i	−0.0363677	−	0.745997i
$286$		0			0
$287$		−	17.8673i		−	1.05467i
$288$		0			0
$289$	3.60279			0.211929
$290$		0			0
$291$			8.02648i			0.470521i
$292$		0			0
$293$	−27.3166			−1.59585			−0.797926	−	0.602755i	$-0.794070\pi$
$293$	−27.3166			−1.59585			−0.797926	+	0.602755i	$0.794070\pi$
$294$		0			0
$295$	23.0326	−	1.12285i	1.34101	−	0.0653748i
$296$		0			0
$297$	−2.85383			−0.165596
$298$		0			0
$299$	−4.88149	+	11.7064i	−0.282304	+	0.677001i
$300$		0			0
$301$			24.2996i			1.40061i
$302$		0			0
$303$		−	10.8918i		−	0.625718i
$304$		0			0
$305$	−1.46850	−	30.1229i	−0.0840862	−	1.72483i
$306$		0			0
$307$	−25.0772			−1.43123			−0.715616	−	0.698493i	$-0.753854\pi$
$307$	−25.0772			−1.43123			−0.715616	+	0.698493i	$0.753854\pi$
$308$		0			0
$309$	8.81010			0.501189
$310$		0			0
$311$	28.3022			1.60487			0.802434	−	0.596741i	$-0.203538\pi$
$311$	28.3022			1.60487			0.802434	+	0.596741i	$0.203538\pi$
$312$		0			0
$313$			24.0428i			1.35898i	0.733685	+	0.679490i	$0.237799\pi$
$313$			24.0428i			1.35898i	−0.733685	+	0.679490i	$0.762201\pi$
$314$		0			0
$315$	0.881490	+	18.0817i	0.0496663	+	1.01879i
$316$		0			0
$317$	−32.6514			−1.83388			−0.916941	−	0.399022i	$-0.869350\pi$
$317$	−32.6514			−1.83388			−0.916941	+	0.399022i	$0.869350\pi$
$318$		0			0
$319$			3.14810i			0.176260i
$320$		0			0
$321$	−10.3248			−0.576275
$322$		0			0
$323$	21.7468			1.21002
$324$		0			0
$325$	17.2351	+	5.28698i	0.956030	+	0.293269i
$326$		0			0
$327$	13.1104			0.725007
$328$		0			0
$329$	−14.8734			−0.819997
$330$		0			0
$331$			20.3679i			1.11952i	0.828654	+	0.559761i	$0.189107\pi$
$331$			20.3679i			1.11952i	−0.828654	+	0.559761i	$0.810893\pi$
$332$		0			0
$333$	3.89748			0.213580
$334$		0			0
$335$	−0.445970	−	9.14802i	−0.0243659	−	0.499810i
$336$		0			0
$337$		−	12.1805i		−	0.663516i	−0.943364	−	0.331758i	$-0.892358\pi$
$337$		−	12.1805i		−	0.663516i	0.943364	−	0.331758i	$-0.107642\pi$
$338$		0			0
$339$	−8.95749			−0.486504
$340$		0			0
$341$	−5.14012			−0.278353
$342$		0			0
$343$	3.36833			0.181873
$344$		0			0
$345$	0.363509	+	7.45653i	0.0195707	+	0.401446i
$346$		0			0
$347$			20.9841i			1.12649i	0.826291	+	0.563243i	$0.190447\pi$
$347$			20.9841i			1.12649i	−0.826291	+	0.563243i	$0.809553\pi$
$348$		0			0
$349$		−	14.8415i		−	0.794445i	−0.917722	−	0.397222i	$-0.869974\pi$
$349$		−	14.8415i		−	0.794445i	0.917722	−	0.397222i	$-0.130026\pi$
$350$		0			0
$351$	6.71577	−	16.1053i	0.358461	−	0.859635i
$352$		0			0
$353$	−3.93760			−0.209577			−0.104789	−	0.994495i	$-0.533417\pi$
$353$	−3.93760			−0.209577			−0.104789	+	0.994495i	$0.533417\pi$
$354$		0			0
$355$	11.9297	−	0.581577i	0.633161	−	0.0308669i
$356$		0			0
$357$	13.3972			0.709055
$358$		0			0
$359$			18.2781i			0.964681i	0.875984	+	0.482340i	$0.160213\pi$
$359$			18.2781i			0.964681i	−0.875984	+	0.482340i	$0.839787\pi$
$360$		0			0
$361$	−16.3001			−0.857900
$362$		0			0
$363$			10.1098i			0.530629i
$364$		0			0
$365$	−0.663730	−	13.6148i	−0.0347412	−	0.712634i
$366$		0			0
$367$			25.0296i			1.30654i	0.757126	+	0.653268i	$0.226603\pi$
$367$			25.0296i			1.30654i	−0.757126	+	0.653268i	$0.773397\pi$
$368$		0			0
$369$		−	9.72565i		−	0.506297i
$370$		0			0
$371$		−	41.6175i		−	2.16067i
$372$		0			0
$373$			21.3882i			1.10744i	0.832704	+	0.553719i	$0.186792\pi$
$373$			21.3882i			1.10744i	−0.832704	+	0.553719i	$0.813208\pi$
$374$		0			0
$375$	10.4979	−	1.54514i	0.542110	−	0.0797904i
$376$		0			0
$377$	−17.7660	−	7.40826i	−0.914993	−	0.381545i
$378$		0			0
$379$		−	20.4784i		−	1.05191i	−0.850514	−	0.525953i	$-0.823709\pi$
$379$		−	20.4784i		−	1.05191i	0.850514	−	0.525953i	$-0.176291\pi$
$380$		0			0
$381$	19.1352			0.980325
$382$		0			0
$383$	−12.7276			−0.650352			−0.325176	−	0.945653i	$-0.605424\pi$
$383$	−12.7276			−0.650352			−0.325176	+	0.945653i	$0.605424\pi$
$384$		0			0
$385$	−5.07919	+	0.247613i	−0.258859	+	0.0126195i
$386$		0			0
$387$			13.2269i			0.672362i
$388$		0			0
$389$	−15.2680			−0.774120			−0.387060	−	0.922054i	$-0.626509\pi$
$389$	−15.2680			−0.774120			−0.387060	+	0.922054i	$0.626509\pi$
$390$		0			0
$391$	−12.8757			−0.651154
$392$		0			0
$393$			5.87442i			0.296325i
$394$		0			0
$395$	1.20996	+	24.8194i	0.0608796	+	1.24880i
$396$		0			0
$397$	−2.96702			−0.148910			−0.0744552	−	0.997224i	$-0.523722\pi$
$397$	−2.96702			−0.148910			−0.0744552	+	0.997224i	$0.523722\pi$
$398$		0			0
$399$	−21.7468			−1.08870
$400$		0			0
$401$			15.8408i			0.791050i	0.918455	+	0.395525i	$0.129437\pi$
$401$			15.8408i			0.791050i	−0.918455	+	0.395525i	$0.870563\pi$
$402$		0			0
$403$	12.0960	−	29.0077i	0.602544	−	1.44498i
$404$		0			0
$405$	0.185597	+	3.80709i	0.00922240	+	0.189176i
$406$		0			0
$407$			1.09481i			0.0542677i
$408$		0			0
$409$			19.5460i			0.966487i	0.875486	+	0.483243i	$0.160541\pi$
$409$			19.5460i			0.966487i	−0.875486	+	0.483243i	$0.839459\pi$
$410$		0			0
$411$			2.37925i			0.117360i
$412$		0			0
$413$		−	39.7720i		−	1.95705i
$414$		0			0
$415$	0.955942	+	19.6089i	0.0469253	+	0.962562i
$416$		0			0
$417$			9.22083i			0.451546i
$418$		0			0
$419$	−39.9411			−1.95125			−0.975625	−	0.219444i	$-0.929576\pi$
$419$	−39.9411			−1.95125			−0.975625	+	0.219444i	$0.929576\pi$
$420$		0			0
$421$		−	11.4551i		−	0.558287i	−0.960249	−	0.279144i	$-0.909949\pi$
$421$		−	11.4551i		−	0.558287i	0.960249	−	0.279144i	$-0.0900506\pi$
$422$		0			0
$423$	−8.09598			−0.393640
$424$		0			0
$425$	1.78014	+	18.2143i	0.0863496	+	0.883524i
$426$		0			0
$427$	−52.0154			−2.51720
$428$		0			0
$429$	1.86243	+	0.776620i	0.0899192	+	0.0374956i
$430$		0			0
$431$		−	35.4522i		−	1.70767i	−0.520542	−	0.853836i	$-0.674270\pi$
$431$		−	35.4522i		−	1.70767i	0.520542	−	0.853836i	$-0.325730\pi$
$432$		0			0
$433$		−	28.9479i		−	1.39115i	−0.718455	−	0.695574i	$-0.755150\pi$
$433$		−	28.9479i		−	1.39115i	0.718455	−	0.695574i	$-0.244850\pi$
$434$		0			0
$435$	−11.3162	+	0.551670i	−0.542571	+	0.0264506i
$436$		0			0
$437$	20.9003			0.999799
$438$		0			0
$439$	−33.6681			−1.60689			−0.803446	−	0.595377i	$-0.797003\pi$
$439$	−33.6681			−1.60689			−0.803446	+	0.595377i	$0.797003\pi$
$440$		0			0
$441$	16.5282			0.787058
$442$		0			0
$443$		−	15.1485i		−	0.719727i	−0.933005	−	0.359863i	$-0.882823\pi$
$443$		−	15.1485i		−	0.719727i	0.933005	−	0.359863i	$-0.117177\pi$
$444$		0			0
$445$	0.923166	−	0.0450048i	0.0437623	−	0.00213343i
$446$		0			0
$447$	16.9860			0.803408
$448$		0			0
$449$		−	10.5311i		−	0.496996i	−0.968632	−	0.248498i	$-0.920063\pi$
$449$		−	10.5311i		−	0.496996i	0.968632	−	0.248498i	$-0.0799369\pi$
$450$		0			0
$451$	2.73196			0.128643
$452$		0			0
$453$	1.33249			0.0626057
$454$		0			0
$455$	10.5552	−	29.2465i	0.494836	−	1.37110i
$456$		0			0
$457$	15.1073			0.706692			0.353346	−	0.935493i	$-0.385044\pi$
$457$	15.1073			0.706692			0.353346	+	0.935493i	$0.385044\pi$
$458$		0			0
$459$	17.7140			0.826817
$460$		0			0
$461$		−	14.9226i		−	0.695016i	−0.937677	−	0.347508i	$-0.887028\pi$
$461$		−	14.9226i		−	0.695016i	0.937677	−	0.347508i	$-0.112972\pi$
$462$		0			0
$463$	−0.0959767			−0.00446041			−0.00223021	−	0.999998i	$-0.500710\pi$
$463$	−0.0959767			−0.00446041			−0.00223021	+	0.999998i	$0.500710\pi$
$464$		0			0
$465$	−0.900750	−	18.4767i	−0.0417713	−	0.856839i
$466$		0			0
$467$			7.55364i			0.349541i	0.984609	+	0.174770i	$0.0559183\pi$
$467$			7.55364i			0.349541i	−0.984609	+	0.174770i	$0.944082\pi$
$468$		0			0
$469$	−15.7966			−0.729417
$470$		0			0
$471$	17.6681			0.814105
$472$		0			0
$473$	−3.71547			−0.170838
$474$		0			0
$475$	−2.88959	−	29.5661i	−0.132583	−	1.35658i
$476$		0			0
$477$		−	22.6535i		−	1.03723i
$478$		0			0
$479$			2.15464i			0.0984479i	0.998788	+	0.0492240i	$0.0156748\pi$
$479$			2.15464i			0.0984479i	−0.998788	+	0.0492240i	$0.984325\pi$
$480$		0			0
$481$	−6.17844	−	2.57636i	−0.281712	−	0.117472i
$482$		0			0
$483$	12.8757			0.585867
$484$		0			0
$485$	0.920813	+	18.8883i	0.0418120	+	0.857673i
$486$		0			0
$487$	41.6555			1.88759			0.943796	−	0.330529i	$-0.107227\pi$
$487$	41.6555			1.88759			0.943796	+	0.330529i	$0.107227\pi$
$488$		0			0
$489$		−	2.60635i		−	0.117863i
$490$		0			0
$491$	13.2206			0.596638			0.298319	−	0.954466i	$-0.403574\pi$
$491$	13.2206			0.596638			0.298319	+	0.954466i	$0.403574\pi$
$492$		0			0
$493$		−	19.5405i		−	0.880061i
$494$		0			0
$495$	−2.76473	+	0.134782i	−0.124266	+	0.00605800i
$496$		0			0
$497$		−	20.5999i		−	0.924030i
$498$		0			0
$499$			21.3534i			0.955911i	0.878384	+	0.477956i	$0.158622\pi$
$499$			21.3534i			0.955911i	−0.878384	+	0.477956i	$0.841378\pi$
$500$		0			0
$501$		−	14.5498i		−	0.650037i
$502$		0			0
$503$			39.7750i			1.77348i	0.462268	+	0.886740i	$0.347036\pi$
$503$			39.7750i			1.77348i	−0.462268	+	0.886740i	$0.652964\pi$
$504$		0			0
$505$	−1.24953	−	25.6311i	−0.0556033	−	1.14057i
$506$		0			0
$507$	−8.76554	+	8.68286i	−0.389291	+	0.385619i
$508$		0			0
$509$		−	15.9107i		−	0.705229i	−0.935769	−	0.352615i	$-0.885293\pi$
$509$		−	15.9107i		−	0.705229i	0.935769	−	0.352615i	$-0.114707\pi$
$510$		0			0
$511$	−23.5098			−1.04001
$512$		0			0
$513$	−28.7539			−1.26951
$514$		0			0
$515$	20.7324	−	1.01071i	0.913577	−	0.0445373i
$516$		0			0
$517$		−	2.27418i		−	0.100018i
$518$		0			0
$519$	−11.9068			−0.522649
$520$		0			0
$521$	13.7669			0.603137			0.301568	−	0.953445i	$-0.402490\pi$
$521$	13.7669			0.603137			0.301568	+	0.953445i	$0.402490\pi$
$522$		0			0
$523$			6.04142i			0.264173i	0.991238	+	0.132086i	$0.0421677\pi$
$523$			6.04142i			0.264173i	−0.991238	+	0.132086i	$0.957832\pi$
$524$		0			0
$525$	−1.78014	−	18.2143i	−0.0776918	−	0.794938i
$526$		0			0
$527$	31.9052			1.38981
$528$		0			0
$529$	10.6254			0.461975
$530$		0			0
$531$		−	21.6490i		−	0.939485i
$532$		0			0
$533$	−6.42897	+	15.4175i	−0.278470	+	0.667806i
$534$		0			0
$535$	−24.2968	+	1.18448i	−1.05044	+	0.0512096i
$536$		0			0
$537$			8.37431i			0.361378i
$538$		0			0
$539$			4.64282i			0.199980i
$540$		0			0
$541$		−	20.4075i		−	0.877388i	−0.898637	−	0.438694i	$-0.855441\pi$
$541$		−	20.4075i		−	0.877388i	0.898637	−	0.438694i	$-0.144559\pi$
$542$		0			0
$543$			7.48756i			0.321322i
$544$		0			0
$545$	30.8520	−	1.50405i	1.32156	−	0.0644265i
$546$		0			0
$547$		−	10.8898i		−	0.465616i	−0.972523	−	0.232808i	$-0.925209\pi$
$547$		−	10.8898i		−	0.465616i	0.972523	−	0.232808i	$-0.0747914\pi$
$548$		0			0
$549$	−28.3133			−1.20838
$550$		0			0
$551$			31.7188i			1.35127i
$552$		0			0
$553$	42.8575			1.82249
$554$		0			0
$555$	−3.93542	+	0.191853i	−0.167049	+	0.00814372i
$556$		0			0
$557$	10.2405			0.433904			0.216952	−	0.976182i	$-0.430388\pi$
$557$	10.2405			0.433904			0.216952	+	0.976182i	$0.430388\pi$
$558$		0			0
$559$	8.74343	−	20.9679i	0.369808	−	0.886846i
$560$		0			0
$561$			2.04847i			0.0864863i
$562$		0			0
$563$			25.5180i			1.07545i	0.843119	+	0.537727i	$0.180717\pi$
$563$			25.5180i			1.07545i	−0.843119	+	0.537727i	$0.819283\pi$
$564$		0			0
$565$	−21.0792	+	1.02762i	−0.886808	+	0.0432323i
$566$		0			0
$567$	6.57398			0.276081
$568$		0			0
$569$	−8.25524			−0.346078			−0.173039	−	0.984915i	$-0.555359\pi$
$569$	−8.25524			−0.346078			−0.173039	+	0.984915i	$0.555359\pi$
$570$		0			0
$571$	−20.2662			−0.848115			−0.424058	−	0.905635i	$-0.639395\pi$
$571$	−20.2662			−0.848115			−0.424058	+	0.905635i	$0.639395\pi$
$572$		0			0
$573$		−	15.4676i		−	0.646167i
$574$		0			0
$575$	1.71085	+	17.5054i	0.0713476	+	0.730024i
$576$		0			0
$577$	−4.26745			−0.177656			−0.0888280	−	0.996047i	$-0.528312\pi$
$577$	−4.26745			−0.177656			−0.0888280	+	0.996047i	$0.528312\pi$
$578$		0			0
$579$		−	17.8878i		−	0.743393i
$580$		0			0
$581$	33.8601			1.40475
$582$		0			0
$583$	6.36342			0.263546
$584$		0			0
$585$	5.74547	−	15.9196i	0.237546	−	0.658196i
$586$		0			0
$587$	−1.84276			−0.0760590			−0.0380295	−	0.999277i	$-0.512108\pi$
$587$	−1.84276			−0.0760590			−0.0380295	+	0.999277i	$0.512108\pi$
$588$		0			0
$589$	−51.7895			−2.13395
$590$		0			0
$591$			24.8399i			1.02178i
$592$		0			0
$593$	−36.3238			−1.49164			−0.745819	−	0.666148i	$-0.767942\pi$
$593$	−36.3238			−1.49164			−0.745819	+	0.666148i	$0.767942\pi$
$594$		0			0
$595$	31.5269	−	1.53695i	1.29248	−	0.0630089i
$596$		0			0
$597$		−	2.87515i		−	0.117672i
$598$		0			0
$599$	−17.2727			−0.705745			−0.352873	−	0.935671i	$-0.614795\pi$
$599$	−17.2727			−0.705745			−0.352873	+	0.935671i	$0.614795\pi$
$600$		0			0
$601$	−32.1781			−1.31257			−0.656286	−	0.754512i	$-0.727874\pi$
$601$	−32.1781			−1.31257			−0.656286	+	0.754512i	$0.727874\pi$
$602$		0			0
$603$	−8.59848			−0.350157
$604$		0			0
$605$	1.15982	+	23.7910i	0.0471534	+	0.967240i
$606$		0			0
$607$		−	6.88535i		−	0.279468i	−0.990189	−	0.139734i	$-0.955375\pi$
$607$		−	6.88535i		−	0.279468i	0.990189	−	0.139734i	$-0.0446247\pi$
$608$		0			0
$609$			19.5405i			0.791822i
$610$		0			0
$611$	12.8341	+	5.35170i	0.519211	+	0.216507i
$612$		0			0
$613$	−23.1565			−0.935283			−0.467642	−	0.883918i	$-0.654896\pi$
$613$	−23.1565			−0.935283			−0.467642	+	0.883918i	$0.654896\pi$
$614$		0			0
$615$	0.478746	+	9.82033i	0.0193049	+	0.395994i
$616$		0			0
$617$	34.3256			1.38190			0.690949	−	0.722904i	$-0.257193\pi$
$617$	34.3256			1.38190			0.690949	+	0.722904i	$0.257193\pi$
$618$		0			0
$619$			35.0066i			1.40704i	0.710678	+	0.703518i	$0.248388\pi$
$619$			35.0066i			1.40704i	−0.710678	+	0.703518i	$0.751612\pi$
$620$		0			0
$621$	17.0245			0.683169
$622$		0			0
$623$		−	1.59410i		−	0.0638662i
$624$		0			0
$625$	24.5269	−	4.84042i	0.981077	−	0.193617i
$626$		0			0
$627$		−	3.32514i		−	0.132793i
$628$		0			0
$629$		−	6.79558i		−	0.270957i
$630$		0			0
$631$		−	4.95144i		−	0.197114i	−0.995131	−	0.0985570i	$-0.968577\pi$
$631$		−	4.95144i		−	0.197114i	0.995131	−	0.0985570i	$-0.0314227\pi$
$632$		0			0
$633$		−	8.37431i		−	0.332849i
$634$		0			0
$635$	45.0298	−	2.19522i	1.78695	−	0.0871148i
$636$		0			0
$637$	−26.2012	−	10.9257i	−1.03813	−	0.432892i
$638$		0			0
$639$		−	11.2130i		−	0.443581i
$640$		0			0
$641$	−1.02941			−0.0406594			−0.0203297	−	0.999793i	$-0.506472\pi$
$641$	−1.02941			−0.0406594			−0.0203297	+	0.999793i	$0.506472\pi$
$642$		0			0
$643$	−8.30102			−0.327360			−0.163680	−	0.986513i	$-0.552337\pi$
$643$	−8.30102			−0.327360			−0.163680	+	0.986513i	$0.552337\pi$
$644$		0			0
$645$	−0.651096	−	13.3557i	−0.0256369	−	0.525880i
$646$		0			0
$647$			6.13808i			0.241313i	0.992694	+	0.120656i	$0.0385000\pi$
$647$			6.13808i			0.241313i	−0.992694	+	0.120656i	$0.961500\pi$
$648$		0			0
$649$	6.08125			0.238710
$650$		0			0
$651$	−31.9052			−1.25046
$652$		0			0
$653$		−	3.78421i		−	0.148088i	−0.997255	−	0.0740438i	$-0.976410\pi$
$653$		−	3.78421i		−	0.148088i	0.997255	−	0.0740438i	$-0.0235905\pi$
$654$		0			0
$655$	0.673924	+	13.8239i	0.0263324	+	0.540146i
$656$		0			0
$657$	−12.7970			−0.499258
$658$		0			0
$659$	−25.6634			−0.999705			−0.499853	−	0.866110i	$-0.666613\pi$
$659$	−25.6634			−0.999705			−0.499853	+	0.866110i	$0.666613\pi$
$660$		0			0
$661$			10.8080i			0.420384i	0.977660	+	0.210192i	$0.0674089\pi$
$661$			10.8080i			0.420384i	−0.977660	+	0.210192i	$0.932591\pi$
$662$		0			0
$663$	−11.5603	−	4.82055i	−0.448964	−	0.187215i
$664$		0			0
$665$	−51.1756	+	2.49483i	−1.98450	+	0.0967454i
$666$		0			0
$667$		−	18.7800i		−	0.727163i
$668$		0			0
$669$		−	3.10572i		−	0.120074i
$670$		0			0
$671$		−	7.95328i		−	0.307033i
$672$		0			0
$673$			15.8803i			0.612141i	0.952009	+	0.306071i	$0.0990143\pi$
$673$			15.8803i			0.612141i	−0.952009	+	0.306071i	$0.900986\pi$
$674$		0			0
$675$	−2.35373	−	24.0832i	−0.0905950	−	0.926963i
$676$		0			0
$677$			0.340888i			0.0131014i	0.999979	+	0.00655069i	$0.00208516\pi$
$677$			0.340888i			0.0131014i	−0.999979	+	0.00655069i	$0.997915\pi$
$678$		0			0
$679$	32.6158			1.25168
$680$		0			0
$681$		−	2.02112i		−	0.0774493i
$682$		0			0
$683$	17.9483			0.686771			0.343386	−	0.939194i	$-0.388426\pi$
$683$	17.9483			0.686771			0.343386	+	0.939194i	$0.388426\pi$
$684$		0			0
$685$	0.272952	+	5.59896i	0.0104290	+	0.213925i
$686$		0			0
$687$	−2.48788			−0.0949185
$688$		0			0
$689$	−14.9747	+	35.9112i	−0.570491	+	1.36811i
$690$		0			0
$691$			2.24959i			0.0855783i	0.999084	+	0.0427891i	$0.0136244\pi$
$691$			2.24959i			0.0855783i	−0.999084	+	0.0427891i	$0.986376\pi$
$692$		0			0
$693$			4.77407i			0.181352i
$694$		0			0
$695$	1.05783	+	21.6989i	0.0401258	+	0.823086i
$696$		0			0
$697$	−16.9575			−0.642311
$698$		0			0
$699$	−9.79421			−0.370451
$700$		0			0
$701$	−6.89790			−0.260530			−0.130265	−	0.991479i	$-0.541583\pi$
$701$	−6.89790			−0.260530			−0.130265	+	0.991479i	$0.541583\pi$
$702$		0			0
$703$			11.0308i			0.416035i
$704$		0			0
$705$	8.17479	−	0.398525i	0.307880	−	0.0150093i
$706$		0			0
$707$	−44.2591			−1.66454
$708$		0			0
$709$			35.4512i			1.33140i	0.746220	+	0.665699i	$0.231867\pi$
$709$			35.4512i			1.33140i	−0.746220	+	0.665699i	$0.768133\pi$
$710$		0			0
$711$	23.3285			0.874886
$712$		0			0
$713$	30.6633			1.14835
$714$		0			0
$715$	4.47186	+	1.61392i	0.167238	+	0.0603571i
$716$		0			0
$717$	−1.67478			−0.0625457
$718$		0			0
$719$	24.2319			0.903698			0.451849	−	0.892095i	$-0.350765\pi$
$719$	24.2319			0.903698			0.451849	+	0.892095i	$0.350765\pi$
$720$		0			0
$721$		−	35.8001i		−	1.33326i
$722$		0			0
$723$	−21.7468			−0.808772
$724$		0			0
$725$	−26.5665	+	2.59643i	−0.986656	+	0.0964291i
$726$		0			0
$727$			13.4972i			0.500585i	0.968170	+	0.250292i	$0.0805267\pi$
$727$			13.4972i			0.500585i	−0.968170	+	0.250292i	$0.919473\pi$
$728$		0			0
$729$	−10.2011			−0.377817
$730$		0			0
$731$	23.0622			0.852988
$732$		0			0
$733$	38.3706			1.41725			0.708625	−	0.705585i	$-0.249316\pi$
$733$	38.3706			1.41725			0.708625	+	0.705585i	$0.249316\pi$
$734$		0			0
$735$	−16.6891	+	0.813602i	−0.615588	+	0.0300102i
$736$		0			0
$737$		−	2.41533i		−	0.0889700i
$738$		0			0
$739$		−	9.36168i		−	0.344375i	−0.985064	−	0.172187i	$-0.944917\pi$
$739$		−	9.36168i		−	0.344375i	0.985064	−	0.172187i	$-0.0550835\pi$
$740$		0			0
$741$	18.7650	+	7.82487i	0.689351	+	0.287454i
$742$		0			0
$743$	−11.4643			−0.420585			−0.210292	−	0.977639i	$-0.567442\pi$
$743$	−11.4643			−0.420585			−0.210292	+	0.977639i	$0.567442\pi$
$744$		0			0
$745$	39.9721	−	1.94866i	1.46447	−	0.0713934i
$746$		0			0
$747$	18.4309			0.674353
$748$		0			0
$749$			41.9551i			1.53301i
$750$		0			0
$751$	6.56686			0.239628			0.119814	−	0.992796i	$-0.461770\pi$
$751$	6.56686			0.239628			0.119814	+	0.992796i	$0.461770\pi$
$752$		0			0
$753$			8.34467i			0.304097i
$754$		0			0
$755$	3.13567	−	0.152865i	0.114119	−	0.00556334i
$756$		0			0
$757$			34.6035i			1.25768i	0.777533	+	0.628842i	$0.216471\pi$
$757$			34.6035i			1.25768i	−0.777533	+	0.628842i	$0.783529\pi$
$758$		0			0
$759$			1.96873i			0.0714605i
$760$		0			0
$761$			40.4330i			1.46570i	0.680393	+	0.732848i	$0.261809\pi$
$761$			40.4330i			1.46570i	−0.680393	+	0.732848i	$0.738191\pi$
$762$		0			0
$763$		−	53.2745i		−	1.92867i
$764$		0			0
$765$	17.1609	−	0.836603i	0.620455	−	0.0302475i
$766$		0			0
$767$	−14.3107	+	34.3188i	−0.516728	+	1.23918i
$768$		0			0
$769$			9.52918i			0.343631i	0.985129	+	0.171816i	$0.0549633\pi$
$769$			9.52918i			0.343631i	−0.985129	+	0.171816i	$0.945037\pi$
$770$		0			0
$771$	23.1487			0.833680
$772$		0			0
$773$	4.70473			0.169217			0.0846087	−	0.996414i	$-0.473036\pi$
$773$	4.70473			0.169217			0.0846087	+	0.996414i	$0.473036\pi$
$774$		0			0
$775$	−4.23937	−	43.3770i	−0.152283	−	1.55815i
$776$		0			0
$777$			6.79558i			0.243790i
$778$		0			0
$779$	27.5260			0.986220
$780$		0			0
$781$	3.14977			0.112708
$782$		0			0
$783$			25.8367i			0.923330i
$784$		0			0
$785$	41.5775	−	2.02692i	1.48396	−	0.0723439i
$786$		0			0
$787$	24.8675			0.886430			0.443215	−	0.896415i	$-0.353838\pi$
$787$	24.8675			0.886430			0.443215	+	0.896415i	$0.353838\pi$
$788$		0			0
$789$	3.93436			0.140067
$790$		0			0
$791$			36.3990i			1.29420i
$792$		0			0
$793$	44.8834	+	18.7160i	1.59386	+	0.664626i
$794$		0			0
$795$	1.11512	+	22.8740i	0.0395492	+	0.811258i
$796$		0			0
$797$		−	19.3616i		−	0.685824i	−0.939368	−	0.342912i	$-0.888587\pi$
$797$		−	19.3616i		−	0.685824i	0.939368	−	0.342912i	$-0.111413\pi$
$798$		0			0
$799$			14.1160i			0.499389i
$800$		0			0
$801$		−	0.867710i		−	0.0306590i
$802$		0			0
$803$		−	3.59470i		−	0.126854i
$804$		0			0
$805$	30.2998	−	1.47713i	1.06793	−	0.0520620i
$806$		0			0
$807$			9.68833i			0.341045i
$808$		0			0
$809$	−3.50590			−0.123261			−0.0616304	−	0.998099i	$-0.519630\pi$
$809$	−3.50590			−0.123261			−0.0616304	+	0.998099i	$0.519630\pi$
$810$		0			0
$811$			7.32489i			0.257212i	0.991696	+	0.128606i	$0.0410502\pi$
$811$			7.32489i			0.257212i	−0.991696	+	0.128606i	$0.958950\pi$
$812$		0			0
$813$	−3.63488			−0.127481
$814$		0			0
$815$	−0.299005	−	6.13339i	−0.0104737	−	0.214843i
$816$		0			0
$817$	−37.4354			−1.30970
$818$		0			0
$819$	−26.9419	−	11.2346i	−0.941427	−	0.392568i
$820$		0			0
$821$			30.8633i			1.07714i	0.842582	+	0.538569i	$0.181035\pi$
$821$			30.8633i			1.07714i	−0.842582	+	0.538569i	$0.818965\pi$
$822$		0			0
$823$			49.4309i			1.72305i	0.507714	+	0.861526i	$0.330491\pi$
$823$			49.4309i			1.72305i	−0.507714	+	0.861526i	$0.669509\pi$
$824$		0			0
$825$	2.78501	−	0.272188i	0.0969617	−	0.00947638i
$826$		0			0
$827$	−31.1108			−1.08183			−0.540914	−	0.841078i	$-0.681922\pi$
$827$	−31.1108			−1.08183			−0.540914	+	0.841078i	$0.681922\pi$
$828$		0			0
$829$	0.440726			0.0153070			0.00765352	−	0.999971i	$-0.497564\pi$
$829$	0.440726			0.0153070			0.00765352	+	0.999971i	$0.497564\pi$
$830$		0			0
$831$	−20.7924			−0.721280
$832$		0			0
$833$		−	28.8183i		−	0.998497i
$834$		0			0
$835$	−1.66918	−	34.2392i	−0.0577643	−	1.18490i
$836$		0			0
$837$	−42.1854			−1.45814
$838$		0			0
$839$			21.0334i			0.726153i	0.931759	+	0.363076i	$0.118274\pi$
$839$			21.0334i			0.726153i	−0.931759	+	0.363076i	$0.881726\pi$
$840$		0			0
$841$	−0.499104			−0.0172105
$842$		0			0
$843$	11.4217			0.393385
$844$		0			0
$845$	−19.6314	+	21.4385i	−0.675339	+	0.737507i
$846$		0			0
$847$	41.0816			1.41158
$848$		0			0
$849$	13.3991			0.459855
$850$		0			0
$851$		−	6.53107i		−	0.223882i
$852$		0			0
$853$	31.6034			1.08208			0.541040	−	0.840997i	$-0.318031\pi$
$853$	31.6034			1.08208			0.541040	+	0.840997i	$0.318031\pi$
$854$		0			0
$855$	−27.8562	+	1.35800i	−0.952662	+	0.0464427i
$856$		0			0
$857$			29.1751i			0.996602i	0.867004	+	0.498301i	$0.166043\pi$
$857$			29.1751i			0.996602i	−0.867004	+	0.498301i	$0.833957\pi$
$858$		0			0
$859$	34.7221			1.18470			0.592351	−	0.805680i	$-0.298200\pi$
$859$	34.7221			1.18470			0.592351	+	0.805680i	$0.298200\pi$
$860$		0			0
$861$	16.9575			0.577910
$862$		0			0
$863$	−23.2447			−0.791258			−0.395629	−	0.918410i	$-0.629473\pi$
$863$	−23.2447			−0.791258			−0.395629	+	0.918410i	$0.629473\pi$
$864$		0			0
$865$	−28.0196	+	1.36597i	−0.952694	+	0.0464443i
$866$		0			0
$867$			3.41933i			0.116127i
$868$		0			0
$869$			6.55302i			0.222296i
$870$		0			0
$871$	13.6307	+	5.68388i	0.461857	+	0.192591i
$872$		0			0
$873$	17.7536			0.600870
$874$		0			0
$875$	−6.27870	−	42.6586i	−0.212259	−	1.44212i
$876$		0			0
$877$	14.3892			0.485890			0.242945	−	0.970040i	$-0.421887\pi$
$877$	14.3892			0.485890			0.242945	+	0.970040i	$0.421887\pi$
$878$		0			0
$879$		−	25.9256i		−	0.874449i
$880$		0			0
$881$	17.9750			0.605593			0.302797	−	0.953055i	$-0.402080\pi$
$881$	17.9750			0.605593			0.302797	+	0.953055i	$0.402080\pi$
$882$		0			0
$883$		−	31.2125i		−	1.05038i	−0.850984	−	0.525191i	$-0.823994\pi$
$883$		−	31.2125i		−	1.05038i	0.850984	−	0.525191i	$-0.176006\pi$
$884$		0			0
$885$	1.06567	+	21.8597i	0.0358221	+	0.734806i
$886$		0			0
$887$		−	53.2656i		−	1.78848i	−0.447585	−	0.894242i	$-0.647716\pi$
$887$		−	53.2656i		−	1.78848i	0.447585	−	0.894242i	$-0.352284\pi$
$888$		0			0
$889$		−	77.7563i		−	2.60786i
$890$		0			0
$891$			1.00518i			0.0336747i
$892$		0			0
$893$		−	22.9136i		−	0.766774i
$894$		0			0
$895$	0.960716	+	19.7068i	0.0321132	+	0.658726i
$896$		0			0
$897$	−11.1103	−	4.63292i	−0.370963	−	0.154689i
$898$		0			0
$899$			46.5354i			1.55204i
$900$		0			0
$901$	−39.4983			−1.31588
$902$		0			0
$903$	−23.0622			−0.767464
$904$		0			0
$905$	0.858986	+	17.6201i	0.0285537	+	0.585711i
$906$		0			0
$907$		−	41.5858i		−	1.38084i	−0.723411	−	0.690418i	$-0.757427\pi$
$907$		−	41.5858i		−	1.38084i	0.723411	−	0.690418i	$-0.242573\pi$
$908$		0			0
$909$	−24.0914			−0.799061
$910$		0			0
$911$	34.6225			1.14710			0.573548	−	0.819172i	$-0.305567\pi$
$911$	34.6225			1.14710			0.573548	+	0.819172i	$0.305567\pi$
$912$		0			0
$913$			5.17729i			0.171343i
$914$		0			0
$915$	28.5890	−	1.39372i	0.945122	−	0.0460751i
$916$		0			0
$917$	23.8708			0.788284
$918$		0			0
$919$	−47.3494			−1.56191			−0.780956	−	0.624586i	$-0.785268\pi$
$919$	−47.3494			−1.56191			−0.780956	+	0.624586i	$0.785268\pi$
$920$		0			0
$921$		−	23.8003i		−	0.784245i
$922$		0			0
$923$	−7.41218	+	17.7754i	−0.243975	+	0.585083i
$924$		0			0
$925$	−9.23900	+	0.902957i	−0.303776	+	0.0296890i
$926$		0			0
$927$		−	19.4869i		−	0.640034i
$928$		0			0
$929$		−	9.83607i		−	0.322711i	−0.986896	−	0.161355i	$-0.948413\pi$
$929$		−	9.83607i		−	0.322711i	0.986896	−	0.161355i	$-0.0515866\pi$
$930$		0			0
$931$			46.7789i			1.53312i
$932$		0			0
$933$			26.8610i			0.879389i
$934$		0			0
$935$	0.235004	+	4.82055i	0.00768545	+	0.157649i
$936$		0			0
$937$		−	1.96076i		−	0.0640554i	−0.999487	−	0.0320277i	$-0.989804\pi$
$937$		−	1.96076i		−	0.0640554i	0.999487	−	0.0320277i	$-0.0101965\pi$
$938$		0			0
$939$	−22.8185			−0.744654
$940$		0			0
$941$		−	21.2254i		−	0.691927i	−0.938248	−	0.345964i	$-0.887552\pi$
$941$		−	21.2254i		−	0.691927i	0.938248	−	0.345964i	$-0.112448\pi$
$942$		0			0
$943$	−16.2975			−0.530718
$944$		0			0
$945$	−41.6853	+	2.03218i	−1.35602	+	0.0661067i
$946$		0			0
$947$	−53.5500			−1.74014			−0.870071	−	0.492927i	$-0.835927\pi$
$947$	−53.5500			−1.74014			−0.870071	+	0.492927i	$0.835927\pi$
$948$		0			0
$949$	20.2863	+	8.45922i	0.658521	+	0.274598i
$950$		0			0
$951$		−	30.9887i		−	1.00488i
$952$		0			0
$953$		−	1.13504i		−	0.0367676i	−0.999831	−	0.0183838i	$-0.994148\pi$
$953$		−	1.13504i		−	0.0367676i	0.999831	−	0.0183838i	$-0.00585207\pi$
$954$		0			0
$955$	−1.77447	−	36.3990i	−0.0574204	−	1.17784i
$956$		0			0
$957$	−2.98780			−0.0965817
$958$		0			0
$959$	9.66814			0.312201
$960$		0			0
$961$	−44.9815			−1.45102
$962$		0			0
$963$			22.8373i			0.735921i
$964$		0			0
$965$	−2.05213	−	42.0945i	−0.0660603	−	1.35507i
$966$		0			0
$967$	21.3213			0.685647			0.342823	−	0.939400i	$-0.388617\pi$
$967$	21.3213			0.685647			0.342823	+	0.939400i	$0.388617\pi$
$968$		0			0
$969$			20.6394i			0.663033i
$970$		0			0
$971$	37.2206			1.19447			0.597233	−	0.802068i	$-0.296267\pi$
$971$	37.2206			1.19447			0.597233	+	0.802068i	$0.296267\pi$
$972$		0			0
$973$	37.4691			1.20120
$974$		0			0
$975$	−5.01776	+	16.3574i	−0.160697	+	0.523857i
$976$		0			0
$977$	22.7798			0.728789			0.364395	−	0.931245i	$-0.381276\pi$
$977$	22.7798			0.728789			0.364395	+	0.931245i	$0.381276\pi$
$978$		0			0
$979$	0.243742			0.00779002
$980$		0			0
$981$		−	28.9987i		−	0.925857i
$982$		0			0
$983$	2.48902			0.0793873			0.0396937	−	0.999212i	$-0.487362\pi$
$983$	2.48902			0.0793873			0.0396937	+	0.999212i	$0.487362\pi$
$984$		0			0
$985$	2.84968	+	58.4545i	0.0907985	+	1.86252i
$986$		0			0
$987$		−	14.1160i		−	0.449318i
$988$		0			0
$989$	22.1646			0.704793
$990$		0			0
$991$	−35.9068			−1.14062			−0.570308	−	0.821431i	$-0.693176\pi$
$991$	−35.9068			−1.14062			−0.570308	+	0.821431i	$0.693176\pi$
$992$		0			0
$993$	−19.3307			−0.613443
$994$		0			0
$995$	−0.329843	−	6.76594i	−0.0104567	−	0.214495i
$996$		0			0
$997$			15.0609i			0.476983i	0.971145	+	0.238491i	$0.0766529\pi$
$997$			15.0609i			0.476983i	−0.971145	+	0.238491i	$0.923347\pi$
$998$		0			0
$999$			8.98520i			0.284279i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.f.b.129.7	yes	10
4.3	odd	2		1040.2.f.g.129.4		10
5.2	odd	4		2600.2.k.f.2001.14		20
5.3	odd	4		2600.2.k.f.2001.7		20
5.4	even	2		520.2.f.a.129.4	✓	10
13.12	even	2		520.2.f.a.129.7	yes	10
20.19	odd	2		1040.2.f.f.129.7		10
52.51	odd	2		1040.2.f.f.129.4		10
65.12	odd	4		2600.2.k.f.2001.13		20
65.38	odd	4		2600.2.k.f.2001.8		20
65.64	even	2	inner	520.2.f.b.129.4	yes	10
260.259	odd	2		1040.2.f.g.129.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.4	✓	10	5.4	even	2
520.2.f.a.129.7	yes	10	13.12	even	2
520.2.f.b.129.4	yes	10	65.64	even	2	inner
520.2.f.b.129.7	yes	10	1.1	even	1	trivial
1040.2.f.f.129.4		10	52.51	odd	2
1040.2.f.f.129.7		10	20.19	odd	2
1040.2.f.g.129.4		10	4.3	odd	2
1040.2.f.g.129.7		10	260.259	odd	2
2600.2.k.f.2001.7		20	5.3	odd	4
2600.2.k.f.2001.8		20	65.38	odd	4
2600.2.k.f.2001.13		20	65.12	odd	4
2600.2.k.f.2001.14		20	5.2	odd	4

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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.949078i q^{3} +(0.108880 + 2.23342i) q^{5} +3.85660 q^{7} +2.09925 q^{9} +O(q^{10})\)	\(q+0.949078i q^{3} +(0.108880 + 2.23342i) q^{5} +3.85660 q^{7} +2.09925 q^{9} +0.589684i q^{11} +(-3.32782 - 1.38767i) q^{13} +(-2.11969 + 0.103336i) q^{15} -3.66022i q^{17} +5.94139i q^{19} +3.66022i q^{21} -3.51775i q^{23} +(-4.97629 + 0.486349i) q^{25} +4.83959i q^{27} +5.33862 q^{29} +8.71673i q^{31} -0.559657 q^{33} +(0.419907 + 8.61340i) q^{35} +1.85660 q^{37} +(1.31701 - 3.15836i) q^{39} -4.63292i q^{41} +6.30078i q^{43} +(0.228566 + 4.68850i) q^{45} -3.85660 q^{47} +7.87339 q^{49} +3.47383 q^{51} -10.7912i q^{53} +(-1.31701 + 0.0642049i) q^{55} -5.63884 q^{57} -10.3127i q^{59} -13.4874 q^{61} +8.09598 q^{63} +(2.73692 - 7.58349i) q^{65} -4.09598 q^{67} +3.33862 q^{69} -5.34145i q^{71} -6.09598 q^{73} +(-0.461583 - 4.72289i) q^{75} +2.27418i q^{77} +11.1128 q^{79} +1.70460 q^{81} +8.77977 q^{83} +(8.17479 - 0.398525i) q^{85} +5.06677i q^{87} -0.413343i q^{89} +(-12.8341 - 5.35170i) q^{91} -8.27286 q^{93} +(-13.2696 + 0.646899i) q^{95} +8.45713 q^{97} +1.23790i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{5} - 2 q^{7} - 8 q^{9}+O(q^{10}) \) 10 * q + 3 * q^5 - 2 * q^7 - 8 * q^9	\( 10 q + 3 q^{5} - 2 q^{7} - 8 q^{9} + 2 q^{13} - 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} - 3 q^{35} - 22 q^{37} - 12 q^{39} - 4 q^{45} + 2 q^{47} + 12 q^{49} - 30 q^{51} + 12 q^{55} - 12 q^{57} - 16 q^{61} + 24 q^{63} - 5 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{73} + 21 q^{75} + 28 q^{79} + 22 q^{81} - 4 q^{83} - 25 q^{85} + 2 q^{91} + 12 q^{93} - 10 q^{95} + 72 q^{97}+O(q^{100}) \) 10 * q + 3 * q^5 - 2 * q^7 - 8 * q^9 + 2 * q^13 - 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 - 3 * q^35 - 22 * q^37 - 12 * q^39 - 4 * q^45 + 2 * q^47 + 12 * q^49 - 30 * q^51 + 12 * q^55 - 12 * q^57 - 16 * q^61 + 24 * q^63 - 5 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^73 + 21 * q^75 + 28 * q^79 + 22 * q^81 - 4 * q^83 - 25 * q^85 + 2 * q^91 + 12 * q^93 - 10 * q^95 + 72 * q^97

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