LMFDB - Embedded newform 880.2.f.d.351.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(351,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-8,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.170772624.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.8
Root			$0.335728 - 1.37379i$ of defining polynomial
Character	$\chi$	$=$	880.351
Dual form			880.2.f.d.351.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52434i q^{3} -1.00000 q^{5} +2.20979 q^{7} -3.37228 q^{9} +(3.22060 - 0.792287i) q^{11} +6.85407i q^{13} -2.52434i q^{15} +1.27582i q^{17} -3.22060 q^{19} +5.57825i q^{21} -1.58457i q^{23} +1.00000 q^{25} -0.939764i q^{27} +3.02661i q^{29} -0.644810i q^{31} +(2.00000 + 8.12989i) q^{33} -2.20979 q^{35} -0.372281 q^{37} -17.3020 q^{39} -5.10328i q^{41} +5.43039 q^{43} +3.37228 q^{45} +8.51278i q^{47} -2.11684 q^{49} -3.22060 q^{51} -13.1168 q^{53} +(-3.22060 + 0.792287i) q^{55} -8.12989i q^{57} -8.51278i q^{59} +10.6815i q^{61} -7.45202 q^{63} -6.85407i q^{65} -3.46410i q^{67} +4.00000 q^{69} -9.45254i q^{71} +1.75079i q^{73} +2.52434i q^{75} +(7.11684 - 1.75079i) q^{77} -12.8824 q^{79} -7.74456 q^{81} +11.8716 q^{83} -1.27582i q^{85} -7.64018 q^{87} +13.1168 q^{89} +15.1460i q^{91} +1.62772 q^{93} +3.22060 q^{95} -14.0000 q^{97} +(-10.8608 + 2.67181i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 4 q^{9} + 8 q^{25} + 16 q^{33} + 20 q^{37} + 4 q^{45} + 52 q^{49} - 36 q^{53} + 32 q^{69} - 12 q^{77} - 16 q^{81} + 36 q^{89} + 36 q^{93} - 112 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 4 * q^9 + 8 * q^25 + 16 * q^33 + 20 * q^37 + 4 * q^45 + 52 * q^49 - 36 * q^53 + 32 * q^69 - 12 * q^77 - 16 * q^81 + 36 * q^89 + 36 * q^93 - 112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			2.52434i			1.45743i	0.684819	+	0.728714i	$0.259881\pi$
$3$			2.52434i			1.45743i	−0.684819	+	0.728714i	$0.740119\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	2.20979			0.835221			0.417610	−	0.908626i	$-0.362868\pi$
$7$	2.20979			0.835221			0.417610	+	0.908626i	$0.362868\pi$
$8$		0			0
$9$	−3.37228			−1.12409
$10$		0			0
$11$	3.22060	−	0.792287i	0.971048	−	0.238884i
$11$	3.22060	−	0.792287i	0.971048	−	0.238884i
$12$		0			0
$13$			6.85407i			1.90098i	0.310761	+	0.950488i	$0.399416\pi$
$13$			6.85407i			1.90098i	−0.310761	+	0.950488i	$0.600584\pi$
$14$		0			0
$15$		−	2.52434i		−	0.651781i
$16$		0			0
$17$			1.27582i			0.309432i	0.987959	+	0.154716i	$0.0494463\pi$
$17$			1.27582i			0.309432i	−0.987959	+	0.154716i	$0.950554\pi$
$18$		0			0
$19$	−3.22060			−0.738857			−0.369428	−	0.929259i	$-0.620447\pi$
$19$	−3.22060			−0.738857			−0.369428	+	0.929259i	$0.620447\pi$
$20$		0			0
$21$			5.57825i			1.21727i
$22$		0			0
$23$		−	1.58457i		−	0.330407i	−0.986260	−	0.165203i	$-0.947172\pi$
$23$		−	1.58457i		−	0.330407i	0.986260	−	0.165203i	$-0.0528280\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	0.939764i		−	0.180858i
$28$		0			0
$29$			3.02661i			0.562027i	0.959704	+	0.281013i	$0.0906705\pi$
$29$			3.02661i			0.562027i	−0.959704	+	0.281013i	$0.909329\pi$
$30$		0			0
$31$		−	0.644810i		−	0.115811i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$		−	0.644810i		−	0.115811i	0.998322	−	0.0579057i	$-0.0184423\pi$
$32$		0			0
$33$	2.00000	+	8.12989i	0.348155	+	1.41523i
$34$		0			0
$35$	−2.20979			−0.373522
$36$		0			0
$37$	−0.372281			−0.0612027			−0.0306013	−	0.999532i	$-0.509742\pi$
$37$	−0.372281			−0.0612027			−0.0306013	+	0.999532i	$0.509742\pi$
$38$		0			0
$39$	−17.3020			−2.77053
$40$		0			0
$41$		−	5.10328i		−	0.796999i	−0.917169	−	0.398499i	$-0.869531\pi$
$41$		−	5.10328i		−	0.796999i	0.917169	−	0.398499i	$-0.130469\pi$
$42$		0			0
$43$	5.43039			0.828127			0.414063	−	0.910248i	$-0.364109\pi$
$43$	5.43039			0.828127			0.414063	+	0.910248i	$0.364109\pi$
$44$		0			0
$45$	3.37228			0.502710
$46$		0			0
$47$			8.51278i			1.24172i	0.783923	+	0.620858i	$0.213216\pi$
$47$			8.51278i			1.24172i	−0.783923	+	0.620858i	$0.786784\pi$
$48$		0			0
$49$	−2.11684			−0.302406
$50$		0			0
$51$	−3.22060			−0.450975
$52$		0			0
$53$	−13.1168			−1.80174			−0.900869	−	0.434092i	$-0.857069\pi$
$53$	−13.1168			−1.80174			−0.900869	+	0.434092i	$0.857069\pi$
$54$		0			0
$55$	−3.22060	+	0.792287i	−0.434266	+	0.106832i
$56$		0			0
$57$		−	8.12989i		−	1.07683i
$58$		0			0
$59$		−	8.51278i		−	1.10827i	−0.832427	−	0.554135i	$-0.813049\pi$
$59$		−	8.51278i		−	1.10827i	0.832427	−	0.554135i	$-0.186951\pi$
$60$		0			0
$61$			10.6815i			1.36763i	0.729656	+	0.683815i	$0.239680\pi$
$61$			10.6815i			1.36763i	−0.729656	+	0.683815i	$0.760320\pi$
$62$		0			0
$63$	−7.45202			−0.938866
$64$		0			0
$65$		−	6.85407i		−	0.850143i
$66$		0			0
$67$		−	3.46410i		−	0.423207i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$		−	3.46410i		−	0.423207i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$		−	9.45254i		−	1.12181i	−0.827880	−	0.560905i	$-0.810453\pi$
$71$		−	9.45254i		−	1.12181i	0.827880	−	0.560905i	$-0.189547\pi$
$72$		0			0
$73$			1.75079i			0.204914i	0.994737	+	0.102457i	$0.0326704\pi$
$73$			1.75079i			0.204914i	−0.994737	+	0.102457i	$0.967330\pi$
$74$		0			0
$75$			2.52434i			0.291485i
$76$		0			0
$77$	7.11684	−	1.75079i	0.811040	−	0.199520i
$78$		0			0
$79$	−12.8824			−1.44938			−0.724692	−	0.689073i	$-0.758018\pi$
$79$	−12.8824			−1.44938			−0.724692	+	0.689073i	$0.758018\pi$
$80$		0			0
$81$	−7.74456			−0.860507
$82$		0			0
$83$	11.8716			1.30308			0.651538	−	0.758616i	$-0.274124\pi$
$83$	11.8716			1.30308			0.651538	+	0.758616i	$0.274124\pi$
$84$		0			0
$85$		−	1.27582i		−	0.138382i
$86$		0			0
$87$	−7.64018			−0.819113
$88$		0			0
$89$	13.1168			1.39038			0.695191	−	0.718825i	$-0.255320\pi$
$89$	13.1168			1.39038			0.695191	+	0.718825i	$0.255320\pi$
$90$		0			0
$91$			15.1460i			1.58774i
$92$		0			0
$93$	1.62772			0.168787
$94$		0			0
$95$	3.22060			0.330427
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−10.8608	+	2.67181i	−1.09155	+	0.268527i
$100$		0			0
$101$			7.65492i			0.761693i	0.924638	+	0.380847i	$0.124367\pi$
$101$			7.65492i			0.761693i	−0.924638	+	0.380847i	$0.875633\pi$
$102$		0			0
$103$		−	11.6819i		−	1.15105i	−0.817783	−	0.575527i	$-0.804797\pi$
$103$		−	11.6819i		−	1.15105i	0.817783	−	0.575527i	$-0.195203\pi$
$104$		0			0
$105$		−	5.57825i		−	0.544381i
$106$		0			0
$107$	13.8932			1.34311			0.671554	−	0.740955i	$-0.265627\pi$
$107$	13.8932			1.34311			0.671554	+	0.740955i	$0.265627\pi$
$108$		0			0
$109$			11.1565i			1.06860i	0.845295	+	0.534299i	$0.179424\pi$
$109$			11.1565i			1.06860i	−0.845295	+	0.534299i	$0.820576\pi$
$110$		0			0
$111$		−	0.939764i		−	0.0891984i
$112$		0			0
$113$	11.4891			1.08081			0.540403	−	0.841406i	$-0.318272\pi$
$113$	11.4891			1.08081			0.540403	+	0.841406i	$0.318272\pi$
$114$		0			0
$115$			1.58457i			0.147762i
$116$		0			0
$117$		−	23.1138i		−	2.13688i
$118$		0			0
$119$			2.81929i			0.258444i
$120$		0			0
$121$	9.74456	−	5.10328i	0.885869	−	0.463935i
$122$		0			0
$123$	12.8824			1.16157
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	13.8932			1.23282			0.616412	−	0.787424i	$-0.288586\pi$
$127$	13.8932			1.23282			0.616412	+	0.787424i	$0.288586\pi$
$128$		0			0
$129$			13.7081i			1.20693i
$130$		0			0
$131$	1.19897			0.104755			0.0523773	−	0.998627i	$-0.483320\pi$
$131$	1.19897			0.104755			0.0523773	+	0.998627i	$0.483320\pi$
$132$		0			0
$133$	−7.11684			−0.617109
$134$		0			0
$135$			0.939764i			0.0808820i
$136$		0			0
$137$	11.4891			0.981582			0.490791	−	0.871277i	$-0.336708\pi$
$137$	11.4891			0.981582			0.490791	+	0.871277i	$0.336708\pi$
$138$		0			0
$139$	17.3020			1.46754			0.733768	−	0.679401i	$-0.237760\pi$
$139$	17.3020			1.46754			0.733768	+	0.679401i	$0.237760\pi$
$140$		0			0
$141$	−21.4891			−1.80971
$142$		0			0
$143$	5.43039	+	22.0742i	0.454112	+	1.84594i
$144$		0			0
$145$		−	3.02661i		−	0.251346i
$146$		0			0
$147$		−	5.34363i		−	0.440735i
$148$		0			0
$149$		−	8.12989i		−	0.666026i	−0.942922	−	0.333013i	$-0.891935\pi$
$149$		−	8.12989i		−	0.666026i	0.942922	−	0.333013i	$-0.108065\pi$
$150$		0			0
$151$	4.41957			0.359660			0.179830	−	0.983698i	$-0.442445\pi$
$151$	4.41957			0.359660			0.179830	+	0.983698i	$0.442445\pi$
$152$		0			0
$153$		−	4.30243i		−	0.347831i
$154$		0			0
$155$			0.644810i			0.0517924i
$156$		0			0
$157$	5.86141			0.467791			0.233896	−	0.972262i	$-0.424853\pi$
$157$	5.86141			0.467791			0.233896	+	0.972262i	$0.424853\pi$
$158$		0			0
$159$		−	33.1113i		−	2.62590i
$160$		0			0
$161$		−	3.50157i		−	0.275962i
$162$		0			0
$163$			6.28339i			0.492153i	0.969250	+	0.246077i	$0.0791415\pi$
$163$			6.28339i			0.492153i	−0.969250	+	0.246077i	$0.920858\pi$
$164$		0			0
$165$	−2.00000	−	8.12989i	−0.155700	−	0.632911i
$166$		0			0
$167$	23.9313			1.85186			0.925931	−	0.377692i	$-0.123282\pi$
$167$	23.9313			1.85186			0.925931	+	0.377692i	$0.123282\pi$
$168$		0			0
$169$	−33.9783			−2.61371
$170$		0			0
$171$	10.8608			0.830544
$172$		0			0
$173$		−	4.30243i		−	0.327107i	−0.986534	−	0.163554i	$-0.947704\pi$
$173$		−	4.30243i		−	0.327107i	0.986534	−	0.163554i	$-0.0522957\pi$
$174$		0			0
$175$	2.20979			0.167044
$176$		0			0
$177$	21.4891			1.61522
$178$		0			0
$179$			8.51278i			0.636275i	0.948045	+	0.318137i	$0.103057\pi$
$179$			8.51278i			0.636275i	−0.948045	+	0.318137i	$0.896943\pi$
$180$		0			0
$181$	19.4891			1.44862			0.724308	−	0.689477i	$-0.242160\pi$
$181$	19.4891			1.44862			0.724308	+	0.689477i	$0.242160\pi$
$182$		0			0
$183$	−26.9638			−1.99322
$184$		0			0
$185$	0.372281			0.0273707
$186$		0			0
$187$	1.01082	+	4.10891i	0.0739182	+	0.300473i
$188$		0			0
$189$		−	2.07668i		−	0.151056i
$190$		0			0
$191$			9.80240i			0.709277i	0.935004	+	0.354638i	$0.115396\pi$
$191$			9.80240i			0.709277i	−0.935004	+	0.354638i	$0.884604\pi$
$192$		0			0
$193$			26.1405i			1.88163i	0.338919	+	0.940815i	$0.389939\pi$
$193$			26.1405i			1.88163i	−0.338919	+	0.940815i	$0.610061\pi$
$194$		0			0
$195$	17.3020			1.23902
$196$		0			0
$197$		−	24.0638i		−	1.71447i	−0.514923	−	0.857236i	$-0.672179\pi$
$197$		−	24.0638i		−	1.71447i	0.514923	−	0.857236i	$-0.327821\pi$
$198$		0			0
$199$			22.7190i			1.61051i	0.592929	+	0.805255i	$0.297972\pi$
$199$			22.7190i			1.61051i	−0.592929	+	0.805255i	$0.702028\pi$
$200$		0			0
$201$	8.74456			0.616794
$202$		0			0
$203$			6.68815i			0.469416i
$204$		0			0
$205$			5.10328i			0.356429i
$206$		0			0
$207$			5.34363i			0.371408i
$208$		0			0
$209$	−10.3723	+	2.55164i	−0.717466	+	0.176501i
$210$		0			0
$211$	−16.4793			−1.13448			−0.567242	−	0.823551i	$-0.691989\pi$
$211$	−16.4793			−1.13448			−0.567242	+	0.823551i	$0.691989\pi$
$212$		0			0
$213$	23.8614			1.63496
$214$		0			0
$215$	−5.43039			−0.370349
$216$		0			0
$217$		−	1.42489i		−	0.0967280i
$218$		0			0
$219$	−4.41957			−0.298647
$220$		0			0
$221$	−8.74456			−0.588223
$222$		0			0
$223$		−	19.8997i		−	1.33259i	−0.745690	−	0.666293i	$-0.767880\pi$
$223$		−	19.8997i		−	1.33259i	0.745690	−	0.666293i	$-0.232120\pi$
$224$		0			0
$225$	−3.37228			−0.224819
$226$		0			0
$227$	1.01082			0.0670902			0.0335451	−	0.999437i	$-0.489320\pi$
$227$	1.01082			0.0670902			0.0335451	+	0.999437i	$0.489320\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$	4.41957	+	17.9653i	0.290787	+	1.18203i
$232$		0			0
$233$		−	17.5356i		−	1.14880i	−0.818576	−	0.574398i	$-0.805236\pi$
$233$		−	17.5356i		−	1.14880i	0.818576	−	0.574398i	$-0.194764\pi$
$234$		0			0
$235$		−	8.51278i		−	0.555312i
$236$		0			0
$237$		−	32.5196i		−	2.11237i
$238$		0			0
$239$	12.8824			0.833294			0.416647	−	0.909068i	$-0.363205\pi$
$239$	12.8824			0.833294			0.416647	+	0.909068i	$0.363205\pi$
$240$		0			0
$241$		−	21.3631i		−	1.37612i	−0.725656	−	0.688058i	$-0.758464\pi$
$241$		−	21.3631i		−	1.37612i	0.725656	−	0.688058i	$-0.241536\pi$
$242$		0			0
$243$		−	22.3692i		−	1.43498i
$244$		0			0
$245$	2.11684			0.135240
$246$		0			0
$247$		−	22.0742i		−	1.40455i
$248$		0			0
$249$			29.9679i			1.89914i
$250$		0			0
$251$		−	6.63325i		−	0.418687i	−0.977842	−	0.209344i	$-0.932867\pi$
$251$		−	6.63325i		−	0.418687i	0.977842	−	0.209344i	$-0.0671327\pi$
$252$		0			0
$253$	−1.25544	−	5.10328i	−0.0789287	−	0.320841i
$254$		0			0
$255$	3.22060			0.201682
$256$		0			0
$257$	−9.25544			−0.577338			−0.288669	−	0.957429i	$-0.593213\pi$
$257$	−9.25544			−0.577338			−0.288669	+	0.957429i	$0.593213\pi$
$258$		0			0
$259$	−0.822662			−0.0511177
$260$		0			0
$261$		−	10.2066i		−	0.631771i
$262$		0			0
$263$	−23.9313			−1.47567			−0.737835	−	0.674981i	$-0.764152\pi$
$263$	−23.9313			−1.47567			−0.737835	+	0.674981i	$0.764152\pi$
$264$		0			0
$265$	13.1168			0.805761
$266$		0			0
$267$			33.1113i			2.02638i
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	−13.2587			−0.805410			−0.402705	−	0.915330i	$-0.631930\pi$
$271$	−13.2587			−0.805410			−0.402705	+	0.915330i	$0.631930\pi$
$272$		0			0
$273$	−38.2337			−2.31401
$274$		0			0
$275$	3.22060	−	0.792287i	0.194210	−	0.0477767i
$276$		0			0
$277$		−	17.0606i		−	1.02507i	−0.858665	−	0.512537i	$-0.828706\pi$
$277$		−	17.0606i		−	1.02507i	0.858665	−	0.512537i	$-0.171294\pi$
$278$		0			0
$279$			2.17448i			0.130183i
$280$		0			0
$281$		−	32.5196i		−	1.93995i	−0.243197	−	0.969977i	$-0.578196\pi$
$281$		−	32.5196i		−	1.93995i	0.243197	−	0.969977i	$-0.421804\pi$
$282$		0			0
$283$	−18.6891			−1.11095			−0.555476	−	0.831533i	$-0.687464\pi$
$283$	−18.6891			−1.11095			−0.555476	+	0.831533i	$0.687464\pi$
$284$		0			0
$285$			8.12989i			0.481573i
$286$		0			0
$287$		−	11.2772i		−	0.665670i
$288$		0			0
$289$	15.3723			0.904252
$290$		0			0
$291$		−	35.3407i		−	2.07171i
$292$		0			0
$293$		−	10.3556i		−	0.604983i	−0.953152	−	0.302491i	$-0.902182\pi$
$293$		−	10.3556i		−	0.604983i	0.953152	−	0.302491i	$-0.0978184\pi$
$294$		0			0
$295$			8.51278i			0.495633i
$296$		0			0
$297$	−0.744563	−	3.02661i	−0.0432039	−	0.175621i
$298$		0			0
$299$	10.8608			0.628095
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	−19.3236			−1.11011
$304$		0			0
$305$		−	10.6815i		−	0.611623i
$306$		0			0
$307$	22.7324			1.29741			0.648703	−	0.761042i	$-0.275312\pi$
$307$	22.7324			1.29741			0.648703	+	0.761042i	$0.275312\pi$
$308$		0			0
$309$	29.4891			1.67758
$310$		0			0
$311$		−	5.69349i		−	0.322848i	−0.986885	−	0.161424i	$-0.948391\pi$
$311$		−	5.69349i		−	0.322848i	0.986885	−	0.161424i	$-0.0516087\pi$
$312$		0			0
$313$	13.2554			0.749242			0.374621	−	0.927178i	$-0.377773\pi$
$313$	13.2554			0.749242			0.374621	+	0.927178i	$0.377773\pi$
$314$		0			0
$315$	7.45202			0.419874
$316$		0			0
$317$	4.37228			0.245572			0.122786	−	0.992433i	$-0.460817\pi$
$317$	4.37228			0.245572			0.122786	+	0.992433i	$0.460817\pi$
$318$		0			0
$319$	2.39794	+	9.74749i	0.134259	+	0.545755i
$320$		0			0
$321$			35.0712i			1.95748i
$322$		0			0
$323$		−	4.10891i		−	0.228626i
$324$		0			0
$325$			6.85407i			0.380195i
$326$		0			0
$327$	−28.1628			−1.55740
$328$		0			0
$329$			18.8114i			1.03711i
$330$		0			0
$331$		−	25.5383i		−	1.40371i	−0.712318	−	0.701857i	$-0.752355\pi$
$331$		−	25.5383i		−	1.40371i	0.712318	−	0.701857i	$-0.247645\pi$
$332$		0			0
$333$	1.25544			0.0687975
$334$		0			0
$335$			3.46410i			0.189264i
$336$		0			0
$337$		−	3.82746i		−	0.208495i	−0.994551	−	0.104248i	$-0.966757\pi$
$337$		−	3.82746i		−	0.208495i	0.994551	−	0.104248i	$-0.0332434\pi$
$338$		0			0
$339$			29.0024i			1.57520i
$340$		0			0
$341$	−0.510875	−	2.07668i	−0.0276654	−	0.112458i
$342$		0			0
$343$	−20.1463			−1.08780
$344$		0			0
$345$	−4.00000			−0.215353
$346$		0			0
$347$	−3.03245			−0.162790			−0.0813952	−	0.996682i	$-0.525938\pi$
$347$	−3.03245			−0.162790			−0.0813952	+	0.996682i	$0.525938\pi$
$348$		0			0
$349$			26.4663i			1.41671i	0.705856	+	0.708355i	$0.250563\pi$
$349$			26.4663i			1.41671i	−0.705856	+	0.708355i	$0.749437\pi$
$350$		0			0
$351$	6.44121			0.343806
$352$		0			0
$353$	−20.2337			−1.07693			−0.538465	−	0.842648i	$-0.680996\pi$
$353$	−20.2337			−1.07693			−0.538465	+	0.842648i	$0.680996\pi$
$354$		0			0
$355$			9.45254i			0.501689i
$356$		0			0
$357$	−7.11684			−0.376663
$358$		0			0
$359$	15.2804			0.806466			0.403233	−	0.915097i	$-0.367886\pi$
$359$	15.2804			0.806466			0.403233	+	0.915097i	$0.367886\pi$
$360$		0			0
$361$	−8.62772			−0.454090
$362$		0			0
$363$	12.8824	+	24.5986i	0.676151	+	1.29109i
$364$		0			0
$365$		−	1.75079i		−	0.0916403i
$366$		0			0
$367$			12.9715i			0.677109i	0.940947	+	0.338555i	$0.109938\pi$
$367$			12.9715i			0.677109i	−0.940947	+	0.338555i	$0.890062\pi$
$368$		0			0
$369$			17.2097i			0.895902i
$370$		0			0
$371$	−28.9854			−1.50485
$372$		0			0
$373$			24.0638i			1.24598i	0.782232	+	0.622988i	$0.214081\pi$
$373$			24.0638i			1.24598i	−0.782232	+	0.622988i	$0.785919\pi$
$374$		0			0
$375$		−	2.52434i		−	0.130356i
$376$		0			0
$377$	−20.7446			−1.06840
$378$		0			0
$379$		−	16.0309i		−	0.823451i	−0.911308	−	0.411726i	$-0.864926\pi$
$379$		−	16.0309i		−	0.823451i	0.911308	−	0.411726i	$-0.135074\pi$
$380$		0			0
$381$			35.0712i			1.79675i
$382$		0			0
$383$		−	1.58457i		−	0.0809679i	−0.999180	−	0.0404840i	$-0.987110\pi$
$383$		−	1.58457i		−	0.0809679i	0.999180	−	0.0404840i	$-0.0128900\pi$
$384$		0			0
$385$	−7.11684	+	1.75079i	−0.362708	+	0.0892283i
$386$		0			0
$387$	−18.3128			−0.930892
$388$		0			0
$389$	−25.7228			−1.30420			−0.652099	−	0.758134i	$-0.726111\pi$
$389$	−25.7228			−1.30420			−0.652099	+	0.758134i	$0.726111\pi$
$390$		0			0
$391$	2.02163			0.102238
$392$		0			0
$393$			3.02661i			0.152672i
$394$		0			0
$395$	12.8824			0.648184
$396$		0			0
$397$	−7.48913			−0.375868			−0.187934	−	0.982182i	$-0.560179\pi$
$397$	−7.48913			−0.375868			−0.187934	+	0.982182i	$0.560179\pi$
$398$		0			0
$399$		−	17.9653i		−	0.899391i
$400$		0			0
$401$	19.6277			0.980161			0.490081	−	0.871677i	$-0.336967\pi$
$401$	19.6277			0.980161			0.490081	+	0.871677i	$0.336967\pi$
$402$		0			0
$403$	4.41957			0.220155
$404$		0			0
$405$	7.74456			0.384830
$406$		0			0
$407$	−1.19897	+	0.294954i	−0.0594307	+	0.0146203i
$408$		0			0
$409$			8.60485i			0.425483i	0.977109	+	0.212741i	$0.0682392\pi$
$409$			8.60485i			0.425483i	−0.977109	+	0.212741i	$0.931761\pi$
$410$		0			0
$411$			29.0024i			1.43058i
$412$		0			0
$413$		−	18.8114i		−	0.925649i
$414$		0			0
$415$	−11.8716			−0.582754
$416$		0			0
$417$			43.6761i			2.13883i
$418$		0			0
$419$			0.294954i			0.0144094i	0.999974	+	0.00720471i	$0.00229335\pi$
$419$			0.294954i			0.0144094i	−0.999974	+	0.00720471i	$0.997707\pi$
$420$		0			0
$421$	−36.2337			−1.76592			−0.882961	−	0.469446i	$-0.844454\pi$
$421$	−36.2337			−1.76592			−0.882961	+	0.469446i	$0.844454\pi$
$422$		0			0
$423$		−	28.7075i		−	1.39581i
$424$		0			0
$425$			1.27582i			0.0618864i
$426$		0			0
$427$			23.6039i			1.14227i
$428$		0			0
$429$	−55.7228	+	13.7081i	−2.69032	+	0.661835i
$430$		0			0
$431$	8.46284			0.407641			0.203820	−	0.979008i	$-0.434664\pi$
$431$	8.46284			0.407641			0.203820	+	0.979008i	$0.434664\pi$
$432$		0			0
$433$	24.2337			1.16460			0.582298	−	0.812975i	$-0.302154\pi$
$433$	24.2337			1.16460			0.582298	+	0.812975i	$0.302154\pi$
$434$		0			0
$435$	7.64018			0.366318
$436$		0			0
$437$			5.10328i			0.244123i
$438$		0			0
$439$	8.83915			0.421869			0.210935	−	0.977500i	$-0.432349\pi$
$439$	8.83915			0.421869			0.210935	+	0.977500i	$0.432349\pi$
$440$		0			0
$441$	7.13859			0.339933
$442$		0			0
$443$			6.63325i			0.315155i	0.987507	+	0.157578i	$0.0503684\pi$
$443$			6.63325i			0.315155i	−0.987507	+	0.157578i	$0.949632\pi$
$444$		0			0
$445$	−13.1168			−0.621798
$446$		0			0
$447$	20.5226			0.970685
$448$		0			0
$449$	34.4674			1.62662			0.813308	−	0.581833i	$-0.197664\pi$
$449$	34.4674			1.62662			0.813308	+	0.581833i	$0.197664\pi$
$450$		0			0
$451$	−4.04326	−	16.4356i	−0.190390	−	0.773924i
$452$		0			0
$453$			11.1565i			0.524178i
$454$		0			0
$455$		−	15.1460i		−	0.710057i
$456$		0			0
$457$			17.5356i			0.820281i	0.912022	+	0.410140i	$0.134520\pi$
$457$			17.5356i			0.820281i	−0.912022	+	0.410140i	$0.865480\pi$
$458$		0			0
$459$	1.19897			0.0559631
$460$		0			0
$461$			5.57825i			0.259805i	0.991527	+	0.129902i	$0.0414664\pi$
$461$			5.57825i			0.259805i	−0.991527	+	0.129902i	$0.958534\pi$
$462$		0			0
$463$		−	19.8997i		−	0.924820i	−0.886666	−	0.462410i	$-0.846985\pi$
$463$		−	19.8997i		−	0.924820i	0.886666	−	0.462410i	$-0.153015\pi$
$464$		0			0
$465$	−1.62772			−0.0754836
$466$		0			0
$467$			24.5986i			1.13829i	0.822239	+	0.569143i	$0.192725\pi$
$467$			24.5986i			1.13829i	−0.822239	+	0.569143i	$0.807275\pi$
$468$		0			0
$469$		−	7.65492i		−	0.353472i
$470$		0			0
$471$			14.7962i			0.681772i
$472$		0			0
$473$	17.4891	−	4.30243i	0.804151	−	0.197826i
$474$		0			0
$475$	−3.22060			−0.147771
$476$		0			0
$477$	44.2337			2.02532
$478$		0			0
$479$	10.4845			0.479048			0.239524	−	0.970890i	$-0.423009\pi$
$479$	10.4845			0.479048			0.239524	+	0.970890i	$0.423009\pi$
$480$		0			0
$481$		−	2.55164i		−	0.116345i
$482$		0			0
$483$	8.83915			0.402195
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$			16.0309i			0.726429i	0.931706	+	0.363214i	$0.118321\pi$
$487$			16.0309i			0.726429i	−0.931706	+	0.363214i	$0.881679\pi$
$488$		0			0
$489$	−15.8614			−0.717278
$490$		0			0
$491$	−14.0814			−0.635484			−0.317742	−	0.948177i	$-0.602925\pi$
$491$	−14.0814			−0.635484			−0.317742	+	0.948177i	$0.602925\pi$
$492$		0			0
$493$	−3.86141			−0.173909
$494$		0			0
$495$	10.8608	−	2.67181i	0.488156	−	0.120089i
$496$		0			0
$497$		−	20.8881i		−	0.936959i
$498$		0			0
$499$		−	35.0458i		−	1.56886i	−0.620215	−	0.784432i	$-0.712954\pi$
$499$		−	35.0458i		−	1.56886i	0.620215	−	0.784432i	$-0.287046\pi$
$500$		0			0
$501$			60.4108i			2.69896i
$502$		0			0
$503$	20.7107			0.923446			0.461723	−	0.887024i	$-0.347231\pi$
$503$	20.7107			0.923446			0.461723	+	0.887024i	$0.347231\pi$
$504$		0			0
$505$		−	7.65492i		−	0.340640i
$506$		0			0
$507$		−	85.7726i		−	3.80929i
$508$		0			0
$509$	−14.7446			−0.653541			−0.326771	−	0.945104i	$-0.605960\pi$
$509$	−14.7446			−0.653541			−0.326771	+	0.945104i	$0.605960\pi$
$510$		0			0
$511$			3.86886i			0.171148i
$512$		0			0
$513$			3.02661i			0.133628i
$514$		0			0
$515$			11.6819i			0.514767i
$516$		0			0
$517$	6.74456	+	27.4163i	0.296626	+	1.20577i
$518$		0			0
$519$	10.8608			0.476735
$520$		0			0
$521$	−0.510875			−0.0223818			−0.0111909	−	0.999937i	$-0.503562\pi$
$521$	−0.510875			−0.0223818			−0.0111909	+	0.999937i	$0.503562\pi$
$522$		0			0
$523$	9.47365			0.414254			0.207127	−	0.978314i	$-0.433589\pi$
$523$	9.47365			0.414254			0.207127	+	0.978314i	$0.433589\pi$
$524$		0			0
$525$			5.57825i			0.243455i
$526$		0			0
$527$	0.822662			0.0358357
$528$		0			0
$529$	20.4891			0.890832
$530$		0			0
$531$			28.7075i			1.24580i
$532$		0			0
$533$	34.9783			1.51508
$534$		0			0
$535$	−13.8932			−0.600657
$536$		0			0
$537$	−21.4891			−0.927324
$538$		0			0
$539$	−6.81751	+	1.67715i	−0.293651	+	0.0722399i
$540$		0			0
$541$		−	33.9444i		−	1.45939i	−0.683775	−	0.729693i	$-0.739663\pi$
$541$		−	33.9444i		−	1.45939i	0.683775	−	0.729693i	$-0.260337\pi$
$542$		0			0
$543$			49.1971i			2.11125i
$544$		0			0
$545$		−	11.1565i		−	0.477892i
$546$		0			0
$547$	−24.7540			−1.05840			−0.529202	−	0.848496i	$-0.677509\pi$
$547$	−24.7540			−1.05840			−0.529202	+	0.848496i	$0.677509\pi$
$548$		0			0
$549$		−	36.0211i		−	1.53734i
$550$		0			0
$551$		−	9.74749i		−	0.415257i
$552$		0			0
$553$	−28.4674			−1.21056
$554$		0			0
$555$			0.939764i			0.0398908i
$556$		0			0
$557$			26.6154i			1.12773i	0.825867	+	0.563866i	$0.190686\pi$
$557$			26.6154i			1.12773i	−0.825867	+	0.563866i	$0.809314\pi$
$558$		0			0
$559$			37.2203i			1.57425i
$560$		0			0
$561$	−10.3723	+	2.55164i	−0.437918	+	0.107730i
$562$		0			0
$563$	−3.40876			−0.143662			−0.0718310	−	0.997417i	$-0.522884\pi$
$563$	−3.40876			−0.143662			−0.0718310	+	0.997417i	$0.522884\pi$
$564$		0			0
$565$	−11.4891			−0.483351
$566$		0			0
$567$	−17.1138			−0.718713
$568$		0			0
$569$		−	12.7582i		−	0.534852i	−0.963578	−	0.267426i	$-0.913827\pi$
$569$		−	12.7582i		−	0.534852i	0.963578	−	0.267426i	$-0.0861731\pi$
$570$		0			0
$571$	−18.1246			−0.758493			−0.379246	−	0.925296i	$-0.623817\pi$
$571$	−18.1246			−0.758493			−0.379246	+	0.925296i	$0.623817\pi$
$572$		0			0
$573$	−24.7446			−1.03372
$574$		0			0
$575$		−	1.58457i		−	0.0660813i
$576$		0			0
$577$	19.4891			0.811343			0.405671	−	0.914019i	$-0.367038\pi$
$577$	19.4891			0.811343			0.405671	+	0.914019i	$0.367038\pi$
$578$		0			0
$579$	−65.9873			−2.74234
$580$		0			0
$581$	26.2337			1.08836
$582$		0			0
$583$	−42.2441	+	10.3923i	−1.74957	+	0.430405i
$584$		0			0
$585$			23.1138i			0.955640i
$586$		0			0
$587$		−	27.7677i		−	1.14610i	−0.819522	−	0.573048i	$-0.805761\pi$
$587$		−	27.7677i		−	1.14610i	0.819522	−	0.573048i	$-0.194239\pi$
$588$		0			0
$589$			2.07668i			0.0855680i
$590$		0			0
$591$	60.7451			2.49872
$592$		0			0
$593$		−	10.3556i		−	0.425255i	−0.977133	−	0.212628i	$-0.931798\pi$
$593$		−	10.3556i		−	0.425255i	0.977133	−	0.212628i	$-0.0682021\pi$
$594$		0			0
$595$		−	2.81929i		−	0.115580i
$596$		0			0
$597$	−57.3505			−2.34720
$598$		0			0
$599$		−	32.1167i		−	1.31225i	−0.754651	−	0.656126i	$-0.772194\pi$
$599$		−	32.1167i		−	1.31225i	0.754651	−	0.656126i	$-0.227806\pi$
$600$		0			0
$601$		−	21.3631i		−	0.871417i	−0.900088	−	0.435709i	$-0.856498\pi$
$601$		−	21.3631i		−	0.871417i	0.900088	−	0.435709i	$-0.143502\pi$
$602$		0			0
$603$			11.6819i			0.475725i
$604$		0			0
$605$	−9.74456	+	5.10328i	−0.396173	+	0.207478i
$606$		0			0
$607$	12.6943			0.515244			0.257622	−	0.966246i	$-0.417061\pi$
$607$	12.6943			0.515244			0.257622	+	0.966246i	$0.417061\pi$
$608$		0			0
$609$	−16.8832			−0.684140
$610$		0			0
$611$	−58.3472			−2.36047
$612$		0			0
$613$			29.1671i			1.17805i	0.808116	+	0.589023i	$0.200487\pi$
$613$			29.1671i			1.17805i	−0.808116	+	0.589023i	$0.799513\pi$
$614$		0			0
$615$	−12.8824			−0.519469
$616$		0			0
$617$	21.2554			0.855712			0.427856	−	0.903847i	$-0.359269\pi$
$617$	21.2554			0.855712			0.427856	+	0.903847i	$0.359269\pi$
$618$		0			0
$619$		−	9.10268i		−	0.365868i	−0.983125	−	0.182934i	$-0.941441\pi$
$619$		−	9.10268i		−	0.365868i	0.983125	−	0.182934i	$-0.0585594\pi$
$620$		0			0
$621$	−1.48913			−0.0597565
$622$		0			0
$623$	28.9854			1.16128
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−6.44121	−	26.1831i	−0.257237	−	1.04565i
$628$		0			0
$629$		−	0.474964i		−	0.0189381i
$630$		0			0
$631$		−	13.2116i		−	0.525945i	−0.964803	−	0.262973i	$-0.915297\pi$
$631$		−	13.2116i		−	0.525945i	0.964803	−	0.262973i	$-0.0847029\pi$
$632$		0			0
$633$		−	41.5994i		−	1.65343i
$634$		0			0
$635$	−13.8932			−0.551336
$636$		0			0
$637$		−	14.5090i		−	0.574867i
$638$		0			0
$639$			31.8766i			1.26102i
$640$		0			0
$641$	−15.3505			−0.606310			−0.303155	−	0.952941i	$-0.598040\pi$
$641$	−15.3505			−0.606310			−0.303155	+	0.952941i	$0.598040\pi$
$642$		0			0
$643$			8.86263i			0.349508i	0.984612	+	0.174754i	$0.0559130\pi$
$643$			8.86263i			0.349508i	−0.984612	+	0.174754i	$0.944087\pi$
$644$		0			0
$645$		−	13.7081i		−	0.539757i
$646$		0			0
$647$			6.63325i			0.260780i	0.991463	+	0.130390i	$0.0416229\pi$
$647$			6.63325i			0.260780i	−0.991463	+	0.130390i	$0.958377\pi$
$648$		0			0
$649$	−6.74456	−	27.4163i	−0.264747	−	1.07618i
$650$		0			0
$651$	3.59691			0.140974
$652$		0			0
$653$	36.0951			1.41251			0.706255	−	0.707957i	$-0.250383\pi$
$653$	36.0951			1.41251			0.706255	+	0.707957i	$0.250383\pi$
$654$		0			0
$655$	−1.19897			−0.0468476
$656$		0			0
$657$		−	5.90414i		−	0.230342i
$658$		0			0
$659$	−37.8246			−1.47344			−0.736718	−	0.676200i	$-0.763625\pi$
$659$	−37.8246			−1.47344			−0.736718	+	0.676200i	$0.763625\pi$
$660$		0			0
$661$	−30.7446			−1.19582			−0.597912	−	0.801561i	$-0.704003\pi$
$661$	−30.7446			−1.19582			−0.597912	+	0.801561i	$0.704003\pi$
$662$		0			0
$663$		−	22.0742i		−	0.857292i
$664$		0			0
$665$	7.11684			0.275979
$666$		0			0
$667$	4.79588			0.185697
$668$		0			0
$669$	50.2337			1.94215
$670$		0			0
$671$	8.46284	+	34.4010i	0.326704	+	1.32803i
$672$		0			0
$673$			36.3470i			1.40107i	0.713616	+	0.700537i	$0.247056\pi$
$673$			36.3470i			1.40107i	−0.713616	+	0.700537i	$0.752944\pi$
$674$		0			0
$675$		−	0.939764i		−	0.0361715i
$676$		0			0
$677$			25.6655i			0.986405i	0.869915	+	0.493202i	$0.164174\pi$
$677$			25.6655i			0.986405i	−0.869915	+	0.493202i	$0.835826\pi$
$678$		0			0
$679$	−30.9370			−1.18725
$680$		0			0
$681$			2.55164i			0.0977791i
$682$		0			0
$683$		−	39.7446i		−	1.52078i	−0.649464	−	0.760392i	$-0.725007\pi$
$683$		−	39.7446i		−	1.52078i	0.649464	−	0.760392i	$-0.274993\pi$
$684$		0			0
$685$	−11.4891			−0.438977
$686$		0			0
$687$		−	5.04868i		−	0.192619i
$688$		0			0
$689$		−	89.9037i		−	3.42506i
$690$		0			0
$691$			31.1769i			1.18603i	0.805193	+	0.593013i	$0.202062\pi$
$691$			31.1769i			1.18603i	−0.805193	+	0.593013i	$0.797938\pi$
$692$		0			0
$693$	−24.0000	+	5.90414i	−0.911685	+	0.224280i
$694$		0			0
$695$	−17.3020			−0.656302
$696$		0			0
$697$	6.51087			0.246617
$698$		0			0
$699$	44.2658			1.67429
$700$		0			0
$701$		−	26.9413i		−	1.01756i	−0.860897	−	0.508780i	$-0.830097\pi$
$701$		−	26.9413i		−	1.01756i	0.860897	−	0.508780i	$-0.169903\pi$
$702$		0			0
$703$	1.19897			0.0452200
$704$		0			0
$705$	21.4891			0.809327
$706$		0			0
$707$			16.9157i			0.636182i
$708$		0			0
$709$	30.7446			1.15464			0.577318	−	0.816519i	$-0.304099\pi$
$709$	30.7446			1.15464			0.577318	+	0.816519i	$0.304099\pi$
$710$		0			0
$711$	43.4431			1.62924
$712$		0			0
$713$	−1.02175			−0.0382648
$714$		0			0
$715$	−5.43039	−	22.0742i	−0.203085	−	0.825529i
$716$		0			0
$717$			32.5196i			1.21447i
$718$		0			0
$719$			13.9113i			0.518804i	0.965769	+	0.259402i	$0.0835255\pi$
$719$			13.9113i			0.518804i	−0.965769	+	0.259402i	$0.916474\pi$
$720$		0			0
$721$		−	25.8146i		−	0.961384i
$722$		0			0
$723$	53.9276			2.00559
$724$		0			0
$725$			3.02661i			0.112405i
$726$		0			0
$727$			4.75372i			0.176306i	0.996107	+	0.0881529i	$0.0280964\pi$
$727$			4.75372i			0.176306i	−0.996107	+	0.0881529i	$0.971904\pi$
$728$		0			0
$729$	33.2337			1.23088
$730$		0			0
$731$			6.92820i			0.256249i
$732$		0			0
$733$			2.70071i			0.0997531i	0.998755	+	0.0498766i	$0.0158828\pi$
$733$			2.70071i			0.0997531i	−0.998755	+	0.0498766i	$0.984117\pi$
$734$		0			0
$735$			5.34363i			0.197103i
$736$		0			0
$737$	−2.74456	−	11.1565i	−0.101097	−	0.410955i
$738$		0			0
$739$	−4.41957			−0.162577			−0.0812883	−	0.996691i	$-0.525903\pi$
$739$	−4.41957			−0.162577			−0.0812883	+	0.996691i	$0.525903\pi$
$740$		0			0
$741$	55.7228			2.04703
$742$		0			0
$743$	28.3509			1.04009			0.520047	−	0.854138i	$-0.325914\pi$
$743$	28.3509			1.04009			0.520047	+	0.854138i	$0.325914\pi$
$744$		0			0
$745$			8.12989i			0.297856i
$746$		0			0
$747$	−40.0344			−1.46478
$748$		0			0
$749$	30.7011			1.12179
$750$		0			0
$751$		−	51.7215i		−	1.88734i	−0.330883	−	0.943672i	$-0.607347\pi$
$751$		−	51.7215i		−	1.88734i	0.330883	−	0.943672i	$-0.392653\pi$
$752$		0			0
$753$	16.7446			0.610206
$754$		0			0
$755$	−4.41957			−0.160845
$756$		0			0
$757$	30.4674			1.10736			0.553678	−	0.832731i	$-0.313224\pi$
$757$	30.4674			1.10736			0.553678	+	0.832731i	$0.313224\pi$
$758$		0			0
$759$	12.8824	−	3.16915i	0.467602	−	0.115033i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			24.6535i			0.892516i
$764$		0			0
$765$			4.30243i			0.155555i
$766$		0			0
$767$	58.3472			2.10679
$768$		0			0
$769$		−	18.8114i		−	0.678357i	−0.940722	−	0.339179i	$-0.889851\pi$
$769$		−	18.8114i		−	0.678357i	0.940722	−	0.339179i	$-0.110149\pi$
$770$		0			0
$771$		−	23.3639i		−	0.841429i
$772$		0			0
$773$	−43.6277			−1.56918			−0.784590	−	0.620015i	$-0.787127\pi$
$773$	−43.6277			−1.56918			−0.784590	+	0.620015i	$0.787127\pi$
$774$		0			0
$775$		−	0.644810i		−	0.0231623i
$776$		0			0
$777$		−	2.07668i		−	0.0745004i
$778$		0			0
$779$			16.4356i			0.588868i
$780$		0			0
$781$	−7.48913	−	30.4429i	−0.267982	−	1.08933i
$782$		0			0
$783$	2.84429			0.101647
$784$		0			0
$785$	−5.86141			−0.209203
$786$		0			0
$787$	−27.1519			−0.967862			−0.483931	−	0.875106i	$-0.660791\pi$
$787$	−27.1519			−0.967862			−0.483931	+	0.875106i	$0.660791\pi$
$788$		0			0
$789$		−	60.4108i		−	2.15068i
$790$		0			0
$791$	25.3885			0.902712
$792$		0			0
$793$	−73.2119			−2.59983
$794$		0			0
$795$			33.1113i			1.17434i
$796$		0			0
$797$	−23.4891			−0.832028			−0.416014	−	0.909358i	$-0.636573\pi$
$797$	−23.4891			−0.832028			−0.416014	+	0.909358i	$0.636573\pi$
$798$		0			0
$799$	−10.8608			−0.384227
$800$		0			0
$801$	−44.2337			−1.56292
$802$		0			0
$803$	1.38712	+	5.63858i	0.0489506	+	0.198981i
$804$		0			0
$805$			3.50157i			0.123414i
$806$		0			0
$807$			15.1460i			0.533165i
$808$		0			0
$809$			22.9648i			0.807398i	0.914892	+	0.403699i	$0.132276\pi$
$809$			22.9648i			0.807398i	−0.914892	+	0.403699i	$0.867724\pi$
$810$		0			0
$811$	37.4483			1.31499			0.657493	−	0.753461i	$-0.271617\pi$
$811$	37.4483			1.31499			0.657493	+	0.753461i	$0.271617\pi$
$812$		0			0
$813$		−	33.4695i		−	1.17383i
$814$		0			0
$815$		−	6.28339i		−	0.220098i
$816$		0			0
$817$	−17.4891			−0.611867
$818$		0			0
$819$		−	51.0767i		−	1.78476i
$820$		0			0
$821$			24.8646i			0.867782i	0.900965	+	0.433891i	$0.142860\pi$
$821$			24.8646i			0.867782i	−0.900965	+	0.433891i	$0.857140\pi$
$822$		0			0
$823$			7.81306i			0.272346i	0.990685	+	0.136173i	$0.0434803\pi$
$823$			7.81306i			0.272346i	−0.990685	+	0.136173i	$0.956520\pi$
$824$		0			0
$825$	2.00000	+	8.12989i	0.0696311	+	0.283046i
$826$		0			0
$827$	−9.47365			−0.329431			−0.164716	−	0.986341i	$-0.552671\pi$
$827$	−9.47365			−0.329431			−0.164716	+	0.986341i	$0.552671\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$	43.0668			1.49397
$832$		0			0
$833$		−	2.70071i		−	0.0935742i
$834$		0			0
$835$	−23.9313			−0.828178
$836$		0			0
$837$	−0.605969			−0.0209454
$838$		0			0
$839$		−	34.9360i		−	1.20612i	−0.797694	−	0.603062i	$-0.793947\pi$
$839$		−	34.9360i		−	1.20612i	0.797694	−	0.603062i	$-0.206053\pi$
$840$		0			0
$841$	19.8397			0.684126
$842$		0			0
$843$	82.0903			2.82734
$844$		0			0
$845$	33.9783			1.16889
$846$		0			0
$847$	21.5334	−	11.2772i	0.739896	−	0.387488i
$848$		0			0
$849$		−	47.1776i		−	1.61913i
$850$		0			0
$851$			0.589907i			0.0202218i
$852$		0			0
$853$			11.0074i			0.376887i	0.982084	+	0.188443i	$0.0603442\pi$
$853$			11.0074i			0.376887i	−0.982084	+	0.188443i	$0.939656\pi$
$854$		0			0
$855$	−10.8608			−0.371431
$856$		0			0
$857$			7.32903i			0.250355i	0.992134	+	0.125177i	$0.0399500\pi$
$857$			7.32903i			0.250355i	−0.992134	+	0.125177i	$0.960050\pi$
$858$		0			0
$859$			46.3229i			1.58052i	0.612773	+	0.790259i	$0.290054\pi$
$859$			46.3229i			1.58052i	−0.612773	+	0.790259i	$0.709946\pi$
$860$		0			0
$861$	28.4674			0.970166
$862$		0			0
$863$		−	21.0796i		−	0.717557i	−0.933423	−	0.358778i	$-0.883193\pi$
$863$		−	21.0796i		−	0.717557i	0.933423	−	0.358778i	$-0.116807\pi$
$864$		0			0
$865$			4.30243i			0.146287i
$866$		0			0
$867$			38.8048i			1.31788i
$868$		0			0
$869$	−41.4891	+	10.2066i	−1.40742	+	0.346234i
$870$		0			0
$871$	23.7432			0.804507
$872$		0			0
$873$	47.2119			1.59788
$874$		0			0
$875$	−2.20979			−0.0747044
$876$		0			0
$877$			26.6154i			0.898739i	0.893346	+	0.449369i	$0.148351\pi$
$877$			26.6154i			0.898739i	−0.893346	+	0.449369i	$0.851649\pi$
$878$		0			0
$879$	26.1411			0.881718
$880$		0			0
$881$	−23.4891			−0.791369			−0.395684	−	0.918387i	$-0.629493\pi$
$881$	−23.4891			−0.791369			−0.395684	+	0.918387i	$0.629493\pi$
$882$		0			0
$883$		−	4.99377i		−	0.168054i	−0.996463	−	0.0840269i	$-0.973222\pi$
$883$		−	4.99377i		−	0.168054i	0.996463	−	0.0840269i	$-0.0267782\pi$
$884$		0			0
$885$	−21.4891			−0.722349
$886$		0			0
$887$	−46.4756			−1.56050			−0.780248	−	0.625470i	$-0.784907\pi$
$887$	−46.4756			−1.56050			−0.780248	+	0.625470i	$0.784907\pi$
$888$		0			0
$889$	30.7011			1.02968
$890$		0			0
$891$	−24.9422	+	6.13592i	−0.835594	+	0.205561i
$892$		0			0
$893$		−	27.4163i		−	0.917451i
$894$		0			0
$895$		−	8.51278i		−	0.284551i
$896$		0			0
$897$			27.4163i			0.915403i
$898$		0			0
$899$	1.95159			0.0650890
$900$		0			0
$901$		−	16.7347i		−	0.557515i
$902$		0			0
$903$			30.2921i			1.00806i
$904$		0			0
$905$	−19.4891			−0.647840
$906$		0			0
$907$			50.4319i			1.67456i	0.546773	+	0.837281i	$0.315856\pi$
$907$			50.4319i			1.67456i	−0.546773	+	0.837281i	$0.684144\pi$
$908$		0			0
$909$		−	25.8146i		−	0.856215i
$910$		0			0
$911$			8.27273i			0.274088i	0.990565	+	0.137044i	$0.0437601\pi$
$911$			8.27273i			0.274088i	−0.990565	+	0.137044i	$0.956240\pi$
$912$		0			0
$913$	38.2337	−	9.40571i	1.26535	−	0.311284i
$914$		0			0
$915$	26.9638			0.891395
$916$		0			0
$917$	2.64947			0.0874931
$918$		0			0
$919$	12.8824			0.424952			0.212476	−	0.977166i	$-0.431847\pi$
$919$	12.8824			0.424952			0.212476	+	0.977166i	$0.431847\pi$
$920$		0			0
$921$			57.3842i			1.89087i
$922$		0			0
$923$	64.7884			2.13253
$924$		0			0
$925$	−0.372281			−0.0122405
$926$		0			0
$927$			39.3947i			1.29389i
$928$		0			0
$929$	−6.60597			−0.216735			−0.108367	−	0.994111i	$-0.534562\pi$
$929$	−6.60597			−0.216735			−0.108367	+	0.994111i	$0.534562\pi$
$930$		0			0
$931$	6.81751			0.223435
$932$		0			0
$933$	14.3723			0.470527
$934$		0			0
$935$	−1.01082	−	4.10891i	−0.0330572	−	0.134376i
$936$		0			0
$937$			8.45578i			0.276238i	0.990416	+	0.138119i	$0.0441057\pi$
$937$			8.45578i			0.276238i	−0.990416	+	0.138119i	$0.955894\pi$
$938$		0			0
$939$			33.4612i			1.09197i
$940$		0			0
$941$		−	32.0446i		−	1.04462i	−0.852755	−	0.522312i	$-0.825070\pi$
$941$		−	32.0446i		−	1.04462i	0.852755	−	0.522312i	$-0.174930\pi$
$942$		0			0
$943$	−8.08653			−0.263334
$944$		0			0
$945$			2.07668i			0.0675543i
$946$		0			0
$947$		−	23.3089i		−	0.757439i	−0.925511	−	0.378720i	$-0.876365\pi$
$947$		−	23.3089i		−	0.757439i	0.925511	−	0.378720i	$-0.123635\pi$
$948$		0			0
$949$	−12.0000			−0.389536
$950$		0			0
$951$			11.0371i			0.357903i
$952$		0			0
$953$			7.98082i			0.258524i	0.991610	+	0.129262i	$0.0412608\pi$
$953$			7.98082i			0.258524i	−0.991610	+	0.129262i	$0.958739\pi$
$954$		0			0
$955$		−	9.80240i		−	0.317198i
$956$		0			0
$957$	−24.6060	+	6.05321i	−0.795398	+	0.195673i
$958$		0			0
$959$	25.3885			0.819838
$960$		0			0
$961$	30.5842			0.986588
$962$		0			0
$963$	−46.8519			−1.50978
$964$		0			0
$965$		−	26.1405i		−	0.841491i
$966$		0			0
$967$	41.6096			1.33808			0.669038	−	0.743228i	$-0.266706\pi$
$967$	41.6096			1.33808			0.669038	+	0.743228i	$0.266706\pi$
$968$		0			0
$969$	10.3723			0.333206
$970$		0			0
$971$		−	18.0202i		−	0.578296i	−0.957284	−	0.289148i	$-0.906628\pi$
$971$		−	18.0202i		−	0.578296i	0.957284	−	0.289148i	$-0.0933720\pi$
$972$		0			0
$973$	38.2337			1.22572
$974$		0			0
$975$	−17.3020			−0.554107
$976$		0			0
$977$	−34.4674			−1.10271			−0.551355	−	0.834271i	$-0.685889\pi$
$977$	−34.4674			−1.10271			−0.551355	+	0.834271i	$0.685889\pi$
$978$		0			0
$979$	42.2441	−	10.3923i	1.35013	−	0.332140i
$980$		0			0
$981$		−	37.6228i		−	1.20120i
$982$		0			0
$983$			36.9253i			1.17773i	0.808230	+	0.588867i	$0.200426\pi$
$983$			36.9253i			1.17773i	−0.808230	+	0.588867i	$0.799574\pi$
$984$		0			0
$985$			24.0638i			0.766736i
$986$		0			0
$987$	−47.4864			−1.51151
$988$		0			0
$989$		−	8.60485i		−	0.273618i
$990$		0			0
$991$			17.3205i			0.550204i	0.961415	+	0.275102i	$0.0887116\pi$
$991$			17.3205i			0.550204i	−0.961415	+	0.275102i	$0.911288\pi$
$992$		0			0
$993$	64.4674			2.04581
$994$		0			0
$995$		−	22.7190i		−	0.720242i
$996$		0			0
$997$			19.6123i			0.621127i	0.950553	+	0.310564i	$0.100518\pi$
$997$			19.6123i			0.621127i	−0.950553	+	0.310564i	$0.899482\pi$
$998$		0			0
$999$			0.349857i			0.0110690i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.f.d.351.8	yes	8
4.3	odd	2	inner	880.2.f.d.351.1	✓	8
8.3	odd	2		3520.2.f.j.2111.7		8
8.5	even	2		3520.2.f.j.2111.2		8
11.10	odd	2	inner	880.2.f.d.351.7	yes	8
44.43	even	2	inner	880.2.f.d.351.2	yes	8
88.21	odd	2		3520.2.f.j.2111.1		8
88.43	even	2		3520.2.f.j.2111.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.2.f.d.351.1	✓	8	4.3	odd	2	inner
880.2.f.d.351.2	yes	8	44.43	even	2	inner
880.2.f.d.351.7	yes	8	11.10	odd	2	inner
880.2.f.d.351.8	yes	8	1.1	even	1	trivial
3520.2.f.j.2111.1		8	88.21	odd	2
3520.2.f.j.2111.2		8	8.5	even	2
3520.2.f.j.2111.7		8	8.3	odd	2
3520.2.f.j.2111.8		8	88.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.52434i q^{3} -1.00000 q^{5} +2.20979 q^{7} -3.37228 q^{9} +(3.22060 - 0.792287i) q^{11} +6.85407i q^{13} -2.52434i q^{15} +1.27582i q^{17} -3.22060 q^{19} +5.57825i q^{21} -1.58457i q^{23} +1.00000 q^{25} -0.939764i q^{27} +3.02661i q^{29} -0.644810i q^{31} +(2.00000 + 8.12989i) q^{33} -2.20979 q^{35} -0.372281 q^{37} -17.3020 q^{39} -5.10328i q^{41} +5.43039 q^{43} +3.37228 q^{45} +8.51278i q^{47} -2.11684 q^{49} -3.22060 q^{51} -13.1168 q^{53} +(-3.22060 + 0.792287i) q^{55} -8.12989i q^{57} -8.51278i q^{59} +10.6815i q^{61} -7.45202 q^{63} -6.85407i q^{65} -3.46410i q^{67} +4.00000 q^{69} -9.45254i q^{71} +1.75079i q^{73} +2.52434i q^{75} +(7.11684 - 1.75079i) q^{77} -12.8824 q^{79} -7.74456 q^{81} +11.8716 q^{83} -1.27582i q^{85} -7.64018 q^{87} +13.1168 q^{89} +15.1460i q^{91} +1.62772 q^{93} +3.22060 q^{95} -14.0000 q^{97} +(-10.8608 + 2.67181i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 4 q^{9} + 8 q^{25} + 16 q^{33} + 20 q^{37} + 4 q^{45} + 52 q^{49} - 36 q^{53} + 32 q^{69} - 12 q^{77} - 16 q^{81} + 36 q^{89} + 36 q^{93} - 112 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 4 * q^9 + 8 * q^25 + 16 * q^33 + 20 * q^37 + 4 * q^45 + 52 * q^49 - 36 * q^53 + 32 * q^69 - 12 * q^77 - 16 * q^81 + 36 * q^89 + 36 * q^93 - 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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