LMFDB - Embedded newform 1280.2.o.f.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(127,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

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

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

chi := DirichletCharacter("1280.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1280.o (of order $4$, 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:	$10.2208514587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.127
Dual form			1280.2.o.f.383.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 1.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 1.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +1.00000i q^{9} -2.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-1.00000 + 3.00000i) q^{15} +(1.00000 + 1.00000i) q^{17} +4.00000i q^{19} +6.00000i q^{21} +(1.00000 + 1.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} -10.0000i q^{31} +(2.00000 - 2.00000i) q^{33} +(3.00000 - 9.00000i) q^{35} +(-1.00000 + 1.00000i) q^{37} -6.00000 q^{39} +10.0000 q^{41} +(5.00000 - 5.00000i) q^{43} +(1.00000 + 2.00000i) q^{45} +(-3.00000 + 3.00000i) q^{47} -11.0000i q^{49} -2.00000 q^{51} +(5.00000 + 5.00000i) q^{53} +(-4.00000 + 2.00000i) q^{55} +(-4.00000 - 4.00000i) q^{57} +12.0000i q^{59} +2.00000i q^{61} +(3.00000 + 3.00000i) q^{63} +(9.00000 + 3.00000i) q^{65} +(-1.00000 - 1.00000i) q^{67} -2.00000 q^{69} +2.00000i q^{71} +(-1.00000 + 1.00000i) q^{73} +(1.00000 + 7.00000i) q^{75} +(-6.00000 + 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} +(3.00000 + 1.00000i) q^{85} -16.0000i q^{89} +18.0000 q^{91} +(10.0000 + 10.0000i) q^{93} +(4.00000 + 8.00000i) q^{95} +(-3.00000 - 3.00000i) q^{97} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7	$ 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7} - 4 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{33} + 6 q^{35} - 2 q^{37} - 12 q^{39} + 20 q^{41} + 10 q^{43} + 2 q^{45} - 6 q^{47} - 4 q^{51} + 10 q^{53} - 8 q^{55} - 8 q^{57} + 6 q^{63} + 18 q^{65} - 2 q^{67} - 4 q^{69} - 2 q^{73} + 2 q^{75} - 12 q^{77} + 16 q^{79} + 10 q^{81} - 10 q^{83} + 6 q^{85} + 36 q^{91} + 20 q^{93} + 8 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7 - 4 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^33 + 6 * q^35 - 2 * q^37 - 12 * q^39 + 20 * q^41 + 10 * q^43 + 2 * q^45 - 6 * q^47 - 4 * q^51 + 10 * q^53 - 8 * q^55 - 8 * q^57 + 6 * q^63 + 18 * q^65 - 2 * q^67 - 4 * q^69 - 2 * q^73 + 2 * q^75 - 12 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 + 6 * q^85 + 36 * q^91 + 20 * q^93 + 8 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$		0			0
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.144370	−	0.989524i	$-0.453885\pi$
$7$	3.00000	−	3.00000i	1.13389	−	1.13389i	0.989524	−	0.144370i	$-0.0461154\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	0.987797	−	0.155747i	$-0.0497784\pi$
$13$	3.00000	+	3.00000i	0.832050	+	0.832050i	−0.155747	+	0.987797i	$0.549778\pi$
$14$		0			0
$15$	−1.00000	+	3.00000i	−0.258199	+	0.774597i
$16$		0			0
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	0.817898	−	0.575363i	$-0.195139\pi$
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$			6.00000i			1.30931i
$22$		0			0
$23$	1.00000	+	1.00000i	0.208514	+	0.208514i	0.803636	−	0.595121i	$-0.202896\pi$
$23$	1.00000	+	1.00000i	0.208514	+	0.208514i	−0.595121	+	0.803636i	$0.702896\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$		0			0
$33$	2.00000	−	2.00000i	0.348155	−	0.348155i
$34$		0			0
$35$	3.00000	−	9.00000i	0.507093	−	1.52128i
$36$		0			0
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	−0.784512	−	0.620113i	$-0.787087\pi$
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	0.620113	+	0.784512i	$0.287087\pi$
$38$		0			0
$39$	−6.00000			−0.960769
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$	5.00000	−	5.00000i	0.762493	−	0.762493i	−0.214280	−	0.976772i	$-0.568740\pi$
$43$	5.00000	−	5.00000i	0.762493	−	0.762493i	0.976772	+	0.214280i	$0.0687403\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$	−3.00000	+	3.00000i	−0.437595	+	0.437595i	−0.891202	−	0.453607i	$-0.850137\pi$
$47$	−3.00000	+	3.00000i	−0.437595	+	0.437595i	0.453607	+	0.891202i	$0.350137\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	5.00000	+	5.00000i	0.686803	+	0.686803i	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	5.00000	+	5.00000i	0.686803	+	0.686803i	−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$	−4.00000	−	4.00000i	−0.529813	−	0.529813i
$58$		0			0
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$		0			0
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$		0			0
$63$	3.00000	+	3.00000i	0.377964	+	0.377964i
$64$		0			0
$65$	9.00000	+	3.00000i	1.11631	+	0.372104i
$66$		0			0
$67$	−1.00000	−	1.00000i	−0.122169	−	0.122169i	0.643379	−	0.765548i	$-0.277532\pi$
$67$	−1.00000	−	1.00000i	−0.122169	−	0.122169i	−0.765548	+	0.643379i	$0.777532\pi$
$68$		0			0
$69$	−2.00000			−0.240772
$70$		0			0
$71$			2.00000i			0.237356i	0.992933	+	0.118678i	$0.0378657\pi$
$71$			2.00000i			0.237356i	−0.992933	+	0.118678i	$0.962134\pi$
$72$		0			0
$73$	−1.00000	+	1.00000i	−0.117041	+	0.117041i	−0.763202	−	0.646160i	$-0.776374\pi$
$73$	−1.00000	+	1.00000i	−0.117041	+	0.117041i	0.646160	+	0.763202i	$0.276374\pi$
$74$		0			0
$75$	1.00000	+	7.00000i	0.115470	+	0.808290i
$76$		0			0
$77$	−6.00000	+	6.00000i	−0.683763	+	0.683763i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	−0.926100	−	0.377279i	$-0.876860\pi$
$83$	−5.00000	+	5.00000i	−0.548821	+	0.548821i	0.377279	+	0.926100i	$0.376860\pi$
$84$		0			0
$85$	3.00000	+	1.00000i	0.325396	+	0.108465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.0000i		−	1.69600i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$		−	16.0000i		−	1.69600i	0.529999	−	0.847998i	$-0.322192\pi$
$90$		0			0
$91$	18.0000			1.88691
$92$		0			0
$93$	10.0000	+	10.0000i	1.03695	+	1.03695i
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	0.538208	−	0.842812i	$-0.319101\pi$
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	−0.842812	+	0.538208i	$0.819101\pi$
$98$		0			0
$99$		−	2.00000i		−	0.201008i
$100$		0			0
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		0			0
$103$	9.00000	+	9.00000i	0.886796	+	0.886796i	0.994214	−	0.107418i	$-0.0342582\pi$
$103$	9.00000	+	9.00000i	0.886796	+	0.886796i	−0.107418	+	0.994214i	$0.534258\pi$
$104$		0			0
$105$	6.00000	+	12.0000i	0.585540	+	1.17108i
$106$		0			0
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	0.547068	−	0.837088i	$-0.315744\pi$
$107$	−3.00000	−	3.00000i	−0.290021	−	0.290021i	−0.837088	+	0.547068i	$0.815744\pi$
$108$		0			0
$109$	4.00000			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$110$		0			0
$111$		−	2.00000i		−	0.189832i
$112$		0			0
$113$	−3.00000	+	3.00000i	−0.282216	+	0.282216i	−0.833992	−	0.551776i	$-0.813950\pi$
$113$	−3.00000	+	3.00000i	−0.282216	+	0.282216i	0.551776	+	0.833992i	$0.313950\pi$
$114$		0			0
$115$	3.00000	+	1.00000i	0.279751	+	0.0932505i
$116$		0			0
$117$	−3.00000	+	3.00000i	−0.277350	+	0.277350i
$118$		0			0
$119$	6.00000			0.550019
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−10.0000	+	10.0000i	−0.901670	+	0.901670i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$	−7.00000	+	7.00000i	−0.621150	+	0.621150i	−0.945825	−	0.324676i	$-0.894745\pi$
$127$	−7.00000	+	7.00000i	−0.621150	+	0.621150i	0.324676	+	0.945825i	$0.394745\pi$
$128$		0			0
$129$			10.0000i			0.880451i
$130$		0			0
$131$	10.0000			0.873704			0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000			0.873704			0.436852	+	0.899533i	$0.356093\pi$
$132$		0			0
$133$	12.0000	+	12.0000i	1.04053	+	1.04053i
$134$		0			0
$135$	−12.0000	−	4.00000i	−1.03280	−	0.344265i
$136$		0			0
$137$	11.0000	+	11.0000i	0.939793	+	0.939793i	0.998288	−	0.0584943i	$-0.0186300\pi$
$137$	11.0000	+	11.0000i	0.939793	+	0.939793i	−0.0584943	+	0.998288i	$0.518630\pi$
$138$		0			0
$139$		−	12.0000i		−	1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$		−	12.0000i		−	1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$		0			0
$141$		−	6.00000i		−	0.505291i
$142$		0			0
$143$	−6.00000	−	6.00000i	−0.501745	−	0.501745i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	11.0000	+	11.0000i	0.907265	+	0.907265i
$148$		0			0
$149$	−4.00000			−0.327693			−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000			−0.327693			−0.163846	+	0.986486i	$0.552390\pi$
$150$		0			0
$151$		−	6.00000i		−	0.488273i	−0.969741	−	0.244137i	$-0.921495\pi$
$151$		−	6.00000i		−	0.488273i	0.969741	−	0.244137i	$-0.0785045\pi$
$152$		0			0
$153$	−1.00000	+	1.00000i	−0.0808452	+	0.0808452i
$154$		0			0
$155$	−10.0000	−	20.0000i	−0.803219	−	1.60644i
$156$		0			0
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	−0.666076	−	0.745884i	$-0.732027\pi$
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	0.745884	+	0.666076i	$0.232027\pi$
$158$		0			0
$159$	−10.0000			−0.793052
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	−0.745184	−	0.666858i	$-0.767639\pi$
$163$	−1.00000	+	1.00000i	−0.0783260	+	0.0783260i	0.666858	+	0.745184i	$0.267639\pi$
$164$		0			0
$165$	2.00000	−	6.00000i	0.155700	−	0.467099i
$166$		0			0
$167$	−1.00000	+	1.00000i	−0.0773823	+	0.0773823i	−0.744739	−	0.667356i	$-0.767426\pi$
$167$	−1.00000	+	1.00000i	−0.0773823	+	0.0773823i	0.667356	+	0.744739i	$0.267426\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	0.491011	−	0.871154i	$-0.336628\pi$
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	−0.871154	+	0.491011i	$0.836628\pi$
$174$		0			0
$175$	−3.00000	−	21.0000i	−0.226779	−	1.58745i
$176$		0			0
$177$	−12.0000	−	12.0000i	−0.901975	−	0.901975i
$178$		0			0
$179$		−	12.0000i		−	0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$		−	12.0000i		−	0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$		0			0
$181$		−	22.0000i		−	1.63525i	−0.575753	−	0.817624i	$-0.695291\pi$
$181$		−	22.0000i		−	1.63525i	0.575753	−	0.817624i	$-0.304709\pi$
$182$		0			0
$183$	−2.00000	−	2.00000i	−0.147844	−	0.147844i
$184$		0			0
$185$	−1.00000	+	3.00000i	−0.0735215	+	0.220564i
$186$		0			0
$187$	−2.00000	−	2.00000i	−0.146254	−	0.146254i
$188$		0			0
$189$	−24.0000			−1.74574
$190$		0			0
$191$			14.0000i			1.01300i	0.862239	+	0.506502i	$0.169062\pi$
$191$			14.0000i			1.01300i	−0.862239	+	0.506502i	$0.830938\pi$
$192$		0			0
$193$	−15.0000	+	15.0000i	−1.07972	+	1.07972i	−0.0831899	+	0.996534i	$0.526511\pi$
$193$	−15.0000	+	15.0000i	−1.07972	+	1.07972i	−0.996534	+	0.0831899i	$0.973489\pi$
$194$		0			0
$195$	−12.0000	+	6.00000i	−0.859338	+	0.429669i
$196$		0			0
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$	2.00000			0.141069
$202$		0			0
$203$		0			0
$204$		0			0
$205$	20.0000	−	10.0000i	1.39686	−	0.698430i
$206$		0			0
$207$	−1.00000	+	1.00000i	−0.0695048	+	0.0695048i
$208$		0			0
$209$		−	8.00000i		−	0.553372i
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$	−2.00000	−	2.00000i	−0.137038	−	0.137038i
$214$		0			0
$215$	5.00000	−	15.0000i	0.340997	−	1.02299i
$216$		0			0
$217$	−30.0000	−	30.0000i	−2.03653	−	2.03653i
$218$		0			0
$219$		−	2.00000i		−	0.135147i
$220$		0			0
$221$			6.00000i			0.403604i
$222$		0			0
$223$	−1.00000	−	1.00000i	−0.0669650	−	0.0669650i	0.672831	−	0.739796i	$-0.265078\pi$
$223$	−1.00000	−	1.00000i	−0.0669650	−	0.0669650i	−0.739796	+	0.672831i	$0.765078\pi$
$224$		0			0
$225$	4.00000	+	3.00000i	0.266667	+	0.200000i
$226$		0			0
$227$	−5.00000	−	5.00000i	−0.331862	−	0.331862i	0.521431	−	0.853293i	$-0.325398\pi$
$227$	−5.00000	−	5.00000i	−0.331862	−	0.331862i	−0.853293	+	0.521431i	$0.825398\pi$
$228$		0			0
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$		0			0
$231$		−	12.0000i		−	0.789542i
$232$		0			0
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.524097	+	0.851658i	$0.675597\pi$
$233$	−21.0000	+	21.0000i	−1.37576	+	1.37576i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	−3.00000	+	9.00000i	−0.195698	+	0.587095i
$236$		0			0
$237$	−8.00000	+	8.00000i	−0.519656	+	0.519656i
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$		0			0
$245$	−11.0000	−	22.0000i	−0.702764	−	1.40553i
$246$		0			0
$247$	−12.0000	+	12.0000i	−0.763542	+	0.763542i
$248$		0			0
$249$		−	10.0000i		−	0.633724i
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$	−2.00000	−	2.00000i	−0.125739	−	0.125739i
$254$		0			0
$255$	−4.00000	+	2.00000i	−0.250490	+	0.125245i
$256$		0			0
$257$	5.00000	+	5.00000i	0.311891	+	0.311891i	0.845642	−	0.533751i	$-0.179218\pi$
$257$	5.00000	+	5.00000i	0.311891	+	0.311891i	−0.533751	+	0.845642i	$0.679218\pi$
$258$		0			0
$259$			6.00000i			0.372822i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0000	−	11.0000i	−0.678289	−	0.678289i	0.281324	−	0.959613i	$-0.409226\pi$
$263$	−11.0000	−	11.0000i	−0.678289	−	0.678289i	−0.959613	+	0.281324i	$0.909226\pi$
$264$		0			0
$265$	15.0000	+	5.00000i	0.921443	+	0.307148i
$266$		0			0
$267$	16.0000	+	16.0000i	0.979184	+	0.979184i
$268$		0			0
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$		0			0
$271$			14.0000i			0.850439i	0.905090	+	0.425220i	$0.139803\pi$
$271$			14.0000i			0.850439i	−0.905090	+	0.425220i	$0.860197\pi$
$272$		0			0
$273$	−18.0000	+	18.0000i	−1.08941	+	1.08941i
$274$		0			0
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$	11.0000	−	11.0000i	0.660926	−	0.660926i	−0.294672	−	0.955598i	$-0.595211\pi$
$277$	11.0000	−	11.0000i	0.660926	−	0.660926i	0.955598	+	0.294672i	$0.0952105\pi$
$278$		0			0
$279$	10.0000			0.598684
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	−7.00000	+	7.00000i	−0.416107	+	0.416107i	−0.883859	−	0.467753i	$-0.845064\pi$
$283$	−7.00000	+	7.00000i	−0.416107	+	0.416107i	0.467753	+	0.883859i	$0.345064\pi$
$284$		0			0
$285$	−12.0000	−	4.00000i	−0.710819	−	0.236940i
$286$		0			0
$287$	30.0000	−	30.0000i	1.77084	−	1.77084i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$	6.00000			0.351726
$292$		0			0
$293$	−11.0000	−	11.0000i	−0.642627	−	0.642627i	0.308574	−	0.951200i	$-0.400148\pi$
$293$	−11.0000	−	11.0000i	−0.642627	−	0.642627i	−0.951200	+	0.308574i	$0.900148\pi$
$294$		0			0
$295$	12.0000	+	24.0000i	0.698667	+	1.39733i
$296$		0			0
$297$	8.00000	+	8.00000i	0.464207	+	0.464207i
$298$		0			0
$299$			6.00000i			0.346989i
$300$		0			0
$301$		−	30.0000i		−	1.72917i
$302$		0			0
$303$	6.00000	+	6.00000i	0.344691	+	0.344691i
$304$		0			0
$305$	2.00000	+	4.00000i	0.114520	+	0.229039i
$306$		0			0
$307$	−17.0000	−	17.0000i	−0.970241	−	0.970241i	0.0293286	−	0.999570i	$-0.490663\pi$
$307$	−17.0000	−	17.0000i	−0.970241	−	0.970241i	−0.999570	+	0.0293286i	$0.990663\pi$
$308$		0			0
$309$	−18.0000			−1.02398
$310$		0			0
$311$			18.0000i			1.02069i	0.859971	+	0.510343i	$0.170482\pi$
$311$			18.0000i			1.02069i	−0.859971	+	0.510343i	$0.829518\pi$
$312$		0			0
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$		0			0
$315$	9.00000	+	3.00000i	0.507093	+	0.169031i
$316$		0			0
$317$	13.0000	−	13.0000i	0.730153	−	0.730153i	−0.240497	−	0.970650i	$-0.577310\pi$
$317$	13.0000	−	13.0000i	0.730153	−	0.730153i	0.970650	+	0.240497i	$0.0773105\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	6.00000			0.334887
$322$		0			0
$323$	−4.00000	+	4.00000i	−0.222566	+	0.222566i
$324$		0			0
$325$	21.0000	−	3.00000i	1.16487	−	0.166410i
$326$		0			0
$327$	−4.00000	+	4.00000i	−0.221201	+	0.221201i
$328$		0			0
$329$			18.0000i			0.992372i
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$		0			0
$333$	−1.00000	−	1.00000i	−0.0547997	−	0.0547997i
$334$		0			0
$335$	−3.00000	−	1.00000i	−0.163908	−	0.0546358i
$336$		0			0
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	0.168585	−	0.985687i	$-0.446080\pi$
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	−0.985687	+	0.168585i	$0.946080\pi$
$338$		0			0
$339$		−	6.00000i		−	0.325875i
$340$		0			0
$341$			20.0000i			1.08306i
$342$		0			0
$343$	−12.0000	−	12.0000i	−0.647939	−	0.647939i
$344$		0			0
$345$	−4.00000	+	2.00000i	−0.215353	+	0.107676i
$346$		0			0
$347$	9.00000	+	9.00000i	0.483145	+	0.483145i	0.906135	−	0.422989i	$-0.139019\pi$
$347$	9.00000	+	9.00000i	0.483145	+	0.483145i	−0.422989	+	0.906135i	$0.639019\pi$
$348$		0			0
$349$	8.00000			0.428230			0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000			0.428230			0.214115	+	0.976808i	$0.431313\pi$
$350$		0			0
$351$		−	24.0000i		−	1.28103i
$352$		0			0
$353$	−15.0000	+	15.0000i	−0.798369	+	0.798369i	−0.982838	−	0.184469i	$-0.940943\pi$
$353$	−15.0000	+	15.0000i	−0.798369	+	0.798369i	0.184469	+	0.982838i	$0.440943\pi$
$354$		0			0
$355$	2.00000	+	4.00000i	0.106149	+	0.212298i
$356$		0			0
$357$	−6.00000	+	6.00000i	−0.317554	+	0.317554i
$358$		0			0
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	7.00000	−	7.00000i	0.367405	−	0.367405i
$364$		0			0
$365$	−1.00000	+	3.00000i	−0.0523424	+	0.157027i
$366$		0			0
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	−0.980335	−	0.197341i	$-0.936769\pi$
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	0.197341	+	0.980335i	$0.436769\pi$
$368$		0			0
$369$			10.0000i			0.520579i
$370$		0			0
$371$	30.0000			1.55752
$372$		0			0
$373$	9.00000	+	9.00000i	0.466002	+	0.466002i	0.900617	−	0.434614i	$-0.143115\pi$
$373$	9.00000	+	9.00000i	0.466002	+	0.466002i	−0.434614	+	0.900617i	$0.643115\pi$
$374$		0			0
$375$	9.00000	+	13.0000i	0.464758	+	0.671317i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		0			0
$381$		−	14.0000i		−	0.717242i
$382$		0			0
$383$	−1.00000	−	1.00000i	−0.0510976	−	0.0510976i	0.681096	−	0.732194i	$-0.261504\pi$
$383$	−1.00000	−	1.00000i	−0.0510976	−	0.0510976i	−0.732194	+	0.681096i	$0.761504\pi$
$384$		0			0
$385$	−6.00000	+	18.0000i	−0.305788	+	0.917365i
$386$		0			0
$387$	5.00000	+	5.00000i	0.254164	+	0.254164i
$388$		0			0
$389$	−4.00000			−0.202808			−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000			−0.202808			−0.101404	+	0.994845i	$0.532333\pi$
$390$		0			0
$391$			2.00000i			0.101144i
$392$		0			0
$393$	−10.0000	+	10.0000i	−0.504433	+	0.504433i
$394$		0			0
$395$	16.0000	−	8.00000i	0.805047	−	0.402524i
$396$		0			0
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	−0.975006	−	0.222178i	$-0.928683\pi$
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	0.222178	+	0.975006i	$0.428683\pi$
$398$		0			0
$399$	−24.0000			−1.20150
$400$		0			0
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		0			0
$403$	30.0000	−	30.0000i	1.49441	−	1.49441i
$404$		0			0
$405$	10.0000	−	5.00000i	0.496904	−	0.248452i
$406$		0			0
$407$	2.00000	−	2.00000i	0.0991363	−	0.0991363i
$408$		0			0
$409$		−	20.0000i		−	0.988936i	−0.869196	−	0.494468i	$-0.835363\pi$
$409$		−	20.0000i		−	0.988936i	0.869196	−	0.494468i	$-0.164637\pi$
$410$		0			0
$411$	−22.0000			−1.08518
$412$		0			0
$413$	36.0000	+	36.0000i	1.77144	+	1.77144i
$414$		0			0
$415$	−5.00000	+	15.0000i	−0.245440	+	0.736321i
$416$		0			0
$417$	12.0000	+	12.0000i	0.587643	+	0.587643i
$418$		0			0
$419$			28.0000i			1.36789i	0.729534	+	0.683945i	$0.239737\pi$
$419$			28.0000i			1.36789i	−0.729534	+	0.683945i	$0.760263\pi$
$420$		0			0
$421$		−	34.0000i		−	1.65706i	−0.559946	−	0.828529i	$-0.689178\pi$
$421$		−	34.0000i		−	1.65706i	0.559946	−	0.828529i	$-0.310822\pi$
$422$		0			0
$423$	−3.00000	−	3.00000i	−0.145865	−	0.145865i
$424$		0			0
$425$	7.00000	−	1.00000i	0.339550	−	0.0485071i
$426$		0			0
$427$	6.00000	+	6.00000i	0.290360	+	0.290360i
$428$		0			0
$429$	12.0000			0.579365
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$	21.0000	−	21.0000i	1.00920	−	1.00920i	0.00923827	−	0.999957i	$-0.497059\pi$
$433$	21.0000	−	21.0000i	1.00920	−	1.00920i	0.999957	−	0.00923827i	$-0.00294067\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.00000	+	4.00000i	−0.191346	+	0.191346i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$	11.0000			0.523810
$442$		0			0
$443$	25.0000	−	25.0000i	1.18779	−	1.18779i	0.210108	−	0.977678i	$-0.432619\pi$
$443$	25.0000	−	25.0000i	1.18779	−	1.18779i	0.977678	−	0.210108i	$-0.0673814\pi$
$444$		0			0
$445$	−16.0000	−	32.0000i	−0.758473	−	1.51695i
$446$		0			0
$447$	4.00000	−	4.00000i	0.189194	−	0.189194i
$448$		0			0
$449$		−	12.0000i		−	0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$	6.00000	+	6.00000i	0.281905	+	0.281905i
$454$		0			0
$455$	36.0000	−	18.0000i	1.68771	−	0.843853i
$456$		0			0
$457$	−9.00000	−	9.00000i	−0.421002	−	0.421002i	0.464546	−	0.885549i	$-0.346217\pi$
$457$	−9.00000	−	9.00000i	−0.421002	−	0.421002i	−0.885549	+	0.464546i	$0.846217\pi$
$458$		0			0
$459$		−	8.00000i		−	0.373408i
$460$		0			0
$461$		−	2.00000i		−	0.0931493i	−0.998915	−	0.0465746i	$-0.985169\pi$
$461$		−	2.00000i		−	0.0931493i	0.998915	−	0.0465746i	$-0.0148305\pi$
$462$		0			0
$463$	11.0000	+	11.0000i	0.511213	+	0.511213i	0.914898	−	0.403685i	$-0.132271\pi$
$463$	11.0000	+	11.0000i	0.511213	+	0.511213i	−0.403685	+	0.914898i	$0.632271\pi$
$464$		0			0
$465$	30.0000	+	10.0000i	1.39122	+	0.463739i
$466$		0			0
$467$	−13.0000	−	13.0000i	−0.601568	−	0.601568i	0.339160	−	0.940729i	$-0.389857\pi$
$467$	−13.0000	−	13.0000i	−0.601568	−	0.601568i	−0.940729	+	0.339160i	$0.889857\pi$
$468$		0			0
$469$	−6.00000			−0.277054
$470$		0			0
$471$			2.00000i			0.0921551i
$472$		0			0
$473$	−10.0000	+	10.0000i	−0.459800	+	0.459800i
$474$		0			0
$475$	16.0000	+	12.0000i	0.734130	+	0.550598i
$476$		0			0
$477$	−5.00000	+	5.00000i	−0.228934	+	0.228934i
$478$		0			0
$479$	−40.0000			−1.82765			−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000			−1.82765			−0.913823	+	0.406112i	$0.866884\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$	−6.00000	+	6.00000i	−0.273009	+	0.273009i
$484$		0			0
$485$	−9.00000	−	3.00000i	−0.408669	−	0.136223i
$486$		0			0
$487$	19.0000	−	19.0000i	0.860972	−	0.860972i	−0.130479	−	0.991451i	$-0.541651\pi$
$487$	19.0000	−	19.0000i	0.860972	−	0.860972i	0.991451	+	0.130479i	$0.0416515\pi$
$488$		0			0
$489$		−	2.00000i		−	0.0904431i
$490$		0			0
$491$	−10.0000			−0.451294			−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000			−0.451294			−0.225647	+	0.974209i	$0.572450\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−2.00000	−	4.00000i	−0.0898933	−	0.179787i
$496$		0			0
$497$	6.00000	+	6.00000i	0.269137	+	0.269137i
$498$		0			0
$499$			28.0000i			1.25345i	0.779240	+	0.626726i	$0.215605\pi$
$499$			28.0000i			1.25345i	−0.779240	+	0.626726i	$0.784395\pi$
$500$		0			0
$501$		−	2.00000i		−	0.0893534i
$502$		0			0
$503$	17.0000	+	17.0000i	0.757993	+	0.757993i	0.975957	−	0.217964i	$-0.0699416\pi$
$503$	17.0000	+	17.0000i	0.757993	+	0.757993i	−0.217964	+	0.975957i	$0.569942\pi$
$504$		0			0
$505$	−6.00000	−	12.0000i	−0.266996	−	0.533993i
$506$		0			0
$507$	−5.00000	−	5.00000i	−0.222058	−	0.222058i
$508$		0			0
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$			6.00000i			0.265424i
$512$		0			0
$513$	16.0000	−	16.0000i	0.706417	−	0.706417i
$514$		0			0
$515$	27.0000	+	9.00000i	1.18976	+	0.396587i
$516$		0			0
$517$	6.00000	−	6.00000i	0.263880	−	0.263880i
$518$		0			0
$519$	10.0000			0.438951
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$	−15.0000	+	15.0000i	−0.655904	+	0.655904i	−0.954408	−	0.298504i	$-0.903512\pi$
$523$	−15.0000	+	15.0000i	−0.655904	+	0.655904i	0.298504	+	0.954408i	$0.403512\pi$
$524$		0			0
$525$	24.0000	+	18.0000i	1.04745	+	0.785584i
$526$		0			0
$527$	10.0000	−	10.0000i	0.435607	−	0.435607i
$528$		0			0
$529$		−	21.0000i		−	0.913043i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$	30.0000	+	30.0000i	1.29944	+	1.29944i
$534$		0			0
$535$	−9.00000	−	3.00000i	−0.389104	−	0.129701i
$536$		0			0
$537$	12.0000	+	12.0000i	0.517838	+	0.517838i
$538$		0			0
$539$			22.0000i			0.947607i
$540$		0			0
$541$			30.0000i			1.28980i	0.764267	+	0.644900i	$0.223101\pi$
$541$			30.0000i			1.28980i	−0.764267	+	0.644900i	$0.776899\pi$
$542$		0			0
$543$	22.0000	+	22.0000i	0.944110	+	0.944110i
$544$		0			0
$545$	8.00000	−	4.00000i	0.342682	−	0.171341i
$546$		0			0
$547$	11.0000	+	11.0000i	0.470326	+	0.470326i	0.902020	−	0.431694i	$-0.142084\pi$
$547$	11.0000	+	11.0000i	0.470326	+	0.470326i	−0.431694	+	0.902020i	$0.642084\pi$
$548$		0			0
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$	24.0000	−	24.0000i	1.02058	−	1.02058i
$554$		0			0
$555$	−2.00000	−	4.00000i	−0.0848953	−	0.169791i
$556$		0			0
$557$	−27.0000	+	27.0000i	−1.14403	+	1.14403i	−0.156320	+	0.987706i	$0.549963\pi$
$557$	−27.0000	+	27.0000i	−1.14403	+	1.14403i	−0.987706	+	0.156320i	$0.950037\pi$
$558$		0			0
$559$	30.0000			1.26886
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$	−33.0000	+	33.0000i	−1.39078	+	1.39078i	−0.567213	+	0.823571i	$0.691978\pi$
$563$	−33.0000	+	33.0000i	−1.39078	+	1.39078i	−0.823571	+	0.567213i	$0.808022\pi$
$564$		0			0
$565$	−3.00000	+	9.00000i	−0.126211	+	0.378633i
$566$		0			0
$567$	15.0000	−	15.0000i	0.629941	−	0.629941i
$568$		0			0
$569$		−	12.0000i		−	0.503066i	−0.967849	−	0.251533i	$-0.919065\pi$
$569$		−	12.0000i		−	0.503066i	0.967849	−	0.251533i	$-0.0809347\pi$
$570$		0			0
$571$	−34.0000			−1.42286			−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000			−1.42286			−0.711428	+	0.702759i	$0.751951\pi$
$572$		0			0
$573$	−14.0000	−	14.0000i	−0.584858	−	0.584858i
$574$		0			0
$575$	7.00000	−	1.00000i	0.291920	−	0.0417029i
$576$		0			0
$577$	−19.0000	−	19.0000i	−0.790980	−	0.790980i	0.190673	−	0.981654i	$-0.438933\pi$
$577$	−19.0000	−	19.0000i	−0.790980	−	0.790980i	−0.981654	+	0.190673i	$0.938933\pi$
$578$		0			0
$579$		−	30.0000i		−	1.24676i
$580$		0			0
$581$			30.0000i			1.24461i
$582$		0			0
$583$	−10.0000	−	10.0000i	−0.414158	−	0.414158i
$584$		0			0
$585$	−3.00000	+	9.00000i	−0.124035	+	0.372104i
$586$		0			0
$587$	−23.0000	−	23.0000i	−0.949312	−	0.949312i	0.0494643	−	0.998776i	$-0.484249\pi$
$587$	−23.0000	−	23.0000i	−0.949312	−	0.949312i	−0.998776	+	0.0494643i	$0.984249\pi$
$588$		0			0
$589$	40.0000			1.64817
$590$		0			0
$591$		−	26.0000i		−	1.06950i
$592$		0			0
$593$	−7.00000	+	7.00000i	−0.287456	+	0.287456i	−0.836073	−	0.548618i	$-0.815154\pi$
$593$	−7.00000	+	7.00000i	−0.287456	+	0.287456i	0.548618	+	0.836073i	$0.315154\pi$
$594$		0			0
$595$	12.0000	−	6.00000i	0.491952	−	0.245976i
$596$		0			0
$597$	−16.0000	+	16.0000i	−0.654836	+	0.654836i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	1.00000	−	1.00000i	0.0407231	−	0.0407231i
$604$		0			0
$605$	−14.0000	+	7.00000i	−0.569181	+	0.284590i
$606$		0			0
$607$	5.00000	−	5.00000i	0.202944	−	0.202944i	−0.598316	−	0.801260i	$-0.704163\pi$
$607$	5.00000	−	5.00000i	0.202944	−	0.202944i	0.801260	+	0.598316i	$0.204163\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.0000			−0.728202
$612$		0			0
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	0.336013	−	0.941857i	$-0.390921\pi$
$613$	−15.0000	−	15.0000i	−0.605844	−	0.605844i	−0.941857	+	0.336013i	$0.890921\pi$
$614$		0			0
$615$	−10.0000	+	30.0000i	−0.403239	+	1.20972i
$616$		0			0
$617$	−13.0000	−	13.0000i	−0.523360	−	0.523360i	0.395224	−	0.918585i	$-0.370667\pi$
$617$	−13.0000	−	13.0000i	−0.523360	−	0.523360i	−0.918585	+	0.395224i	$0.870667\pi$
$618$		0			0
$619$			12.0000i			0.482321i	0.970485	+	0.241160i	$0.0775280\pi$
$619$			12.0000i			0.482321i	−0.970485	+	0.241160i	$0.922472\pi$
$620$		0			0
$621$		−	8.00000i		−	0.321029i
$622$		0			0
$623$	−48.0000	−	48.0000i	−1.92308	−	1.92308i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	8.00000	+	8.00000i	0.319489	+	0.319489i
$628$		0			0
$629$	−2.00000			−0.0797452
$630$		0			0
$631$		−	14.0000i		−	0.557331i	−0.960388	−	0.278666i	$-0.910108\pi$
$631$		−	14.0000i		−	0.557331i	0.960388	−	0.278666i	$-0.0898921\pi$
$632$		0			0
$633$	14.0000	−	14.0000i	0.556450	−	0.556450i
$634$		0			0
$635$	−7.00000	+	21.0000i	−0.277787	+	0.833360i
$636$		0			0
$637$	33.0000	−	33.0000i	1.30751	−	1.30751i
$638$		0			0
$639$	−2.00000			−0.0791188
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	27.0000	−	27.0000i	1.06478	−	1.06478i	0.0670247	−	0.997751i	$-0.478649\pi$
$643$	27.0000	−	27.0000i	1.06478	−	1.06478i	0.997751	−	0.0670247i	$-0.0213506\pi$
$644$		0			0
$645$	10.0000	+	20.0000i	0.393750	+	0.787499i
$646$		0			0
$647$	−29.0000	+	29.0000i	−1.14011	+	1.14011i	−0.151678	+	0.988430i	$0.548468\pi$
$647$	−29.0000	+	29.0000i	−1.14011	+	1.14011i	−0.988430	+	0.151678i	$0.951532\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$		0			0
$651$	60.0000			2.35159
$652$		0			0
$653$	−1.00000	−	1.00000i	−0.0391330	−	0.0391330i	0.687270	−	0.726403i	$-0.258809\pi$
$653$	−1.00000	−	1.00000i	−0.0391330	−	0.0391330i	−0.726403	+	0.687270i	$0.758809\pi$
$654$		0			0
$655$	20.0000	−	10.0000i	0.781465	−	0.390732i
$656$		0			0
$657$	−1.00000	−	1.00000i	−0.0390137	−	0.0390137i
$658$		0			0
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$	−6.00000	−	6.00000i	−0.233021	−	0.233021i
$664$		0			0
$665$	36.0000	+	12.0000i	1.39602	+	0.465340i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.00000			0.0773245
$670$		0			0
$671$		−	4.00000i		−	0.154418i
$672$		0			0
$673$	−3.00000	+	3.00000i	−0.115642	+	0.115642i	−0.762560	−	0.646918i	$-0.776058\pi$
$673$	−3.00000	+	3.00000i	−0.115642	+	0.115642i	0.646918	+	0.762560i	$0.276058\pi$
$674$		0			0
$675$	−28.0000	+	4.00000i	−1.07772	+	0.153960i
$676$		0			0
$677$	3.00000	−	3.00000i	0.115299	−	0.115299i	−0.647103	−	0.762402i	$-0.724020\pi$
$677$	3.00000	−	3.00000i	0.115299	−	0.115299i	0.762402	+	0.647103i	$0.224020\pi$
$678$		0			0
$679$	−18.0000			−0.690777
$680$		0			0
$681$	10.0000			0.383201
$682$		0			0
$683$	−11.0000	+	11.0000i	−0.420903	+	0.420903i	−0.885515	−	0.464611i	$-0.846194\pi$
$683$	−11.0000	+	11.0000i	−0.420903	+	0.420903i	0.464611	+	0.885515i	$0.346194\pi$
$684$		0			0
$685$	33.0000	+	11.0000i	1.26087	+	0.420288i
$686$		0			0
$687$	8.00000	−	8.00000i	0.305219	−	0.305219i
$688$		0			0
$689$			30.0000i			1.14291i
$690$		0			0
$691$	−14.0000			−0.532585			−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000			−0.532585			−0.266293	+	0.963892i	$0.585799\pi$
$692$		0			0
$693$	−6.00000	−	6.00000i	−0.227921	−	0.227921i
$694$		0			0
$695$	−12.0000	−	24.0000i	−0.455186	−	0.910372i
$696$		0			0
$697$	10.0000	+	10.0000i	0.378777	+	0.378777i
$698$		0			0
$699$		−	42.0000i		−	1.58859i
$700$		0			0
$701$			34.0000i			1.28416i	0.766637	+	0.642081i	$0.221929\pi$
$701$			34.0000i			1.28416i	−0.766637	+	0.642081i	$0.778071\pi$
$702$		0			0
$703$	−4.00000	−	4.00000i	−0.150863	−	0.150863i
$704$		0			0
$705$	−6.00000	−	12.0000i	−0.225973	−	0.451946i
$706$		0			0
$707$	−18.0000	−	18.0000i	−0.676960	−	0.676960i
$708$		0			0
$709$	48.0000			1.80268			0.901339	−	0.433114i	$-0.142585\pi$
$709$	48.0000			1.80268			0.901339	+	0.433114i	$0.142585\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$		0			0
$713$	10.0000	−	10.0000i	0.374503	−	0.374503i
$714$		0			0
$715$	−18.0000	−	6.00000i	−0.673162	−	0.224387i
$716$		0			0
$717$	−16.0000	+	16.0000i	−0.597531	+	0.597531i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	54.0000			2.01107
$722$		0			0
$723$	2.00000	−	2.00000i	0.0743808	−	0.0743808i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	3.00000	−	3.00000i	0.111264	−	0.111264i	−0.649283	−	0.760547i	$-0.724931\pi$
$727$	3.00000	−	3.00000i	0.111264	−	0.111264i	0.760547	+	0.649283i	$0.224931\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$	10.0000			0.369863
$732$		0			0
$733$	27.0000	+	27.0000i	0.997268	+	0.997268i	0.999996	−	0.00272852i	$-0.000868517\pi$
$733$	27.0000	+	27.0000i	0.997268	+	0.997268i	−0.00272852	+	0.999996i	$0.500869\pi$
$734$		0			0
$735$	33.0000	+	11.0000i	1.21722	+	0.405741i
$736$		0			0
$737$	2.00000	+	2.00000i	0.0736709	+	0.0736709i
$738$		0			0
$739$		−	28.0000i		−	1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$		−	28.0000i		−	1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$		0			0
$741$		−	24.0000i		−	0.881662i
$742$		0			0
$743$	21.0000	+	21.0000i	0.770415	+	0.770415i	0.978179	−	0.207764i	$-0.0666185\pi$
$743$	21.0000	+	21.0000i	0.770415	+	0.770415i	−0.207764	+	0.978179i	$0.566619\pi$
$744$		0			0
$745$	−8.00000	+	4.00000i	−0.293097	+	0.146549i
$746$		0			0
$747$	−5.00000	−	5.00000i	−0.182940	−	0.182940i
$748$		0			0
$749$	−18.0000			−0.657706
$750$		0			0
$751$		−	2.00000i		−	0.0729810i	−0.999334	−	0.0364905i	$-0.988382\pi$
$751$		−	2.00000i		−	0.0729810i	0.999334	−	0.0364905i	$-0.0116179\pi$
$752$		0			0
$753$	−6.00000	+	6.00000i	−0.218652	+	0.218652i
$754$		0			0
$755$	−6.00000	−	12.0000i	−0.218362	−	0.436725i
$756$		0			0
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	−0.271790	−	0.962357i	$-0.587616\pi$
$757$	19.0000	−	19.0000i	0.690567	−	0.690567i	0.962357	+	0.271790i	$0.0876156\pi$
$758$		0			0
$759$	4.00000			0.145191
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$	12.0000	−	12.0000i	0.434429	−	0.434429i
$764$		0			0
$765$	−1.00000	+	3.00000i	−0.0361551	+	0.108465i
$766$		0			0
$767$	−36.0000	+	36.0000i	−1.29988	+	1.29988i
$768$		0			0
$769$		−	8.00000i		−	0.288487i	−0.989542	−	0.144244i	$-0.953925\pi$
$769$		−	8.00000i		−	0.288487i	0.989542	−	0.144244i	$-0.0460749\pi$
$770$		0			0
$771$	−10.0000			−0.360141
$772$		0			0
$773$	17.0000	+	17.0000i	0.611448	+	0.611448i	0.943323	−	0.331876i	$-0.107681\pi$
$773$	17.0000	+	17.0000i	0.611448	+	0.611448i	−0.331876	+	0.943323i	$0.607681\pi$
$774$		0			0
$775$	−40.0000	−	30.0000i	−1.43684	−	1.07763i
$776$		0			0
$777$	−6.00000	−	6.00000i	−0.215249	−	0.215249i
$778$		0			0
$779$			40.0000i			1.43315i
$780$		0			0
$781$		−	4.00000i		−	0.143131i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.00000	−	3.00000i	0.0356915	−	0.107075i
$786$		0			0
$787$	31.0000	+	31.0000i	1.10503	+	1.10503i	0.993794	+	0.111237i	$0.0354812\pi$
$787$	31.0000	+	31.0000i	1.10503	+	1.10503i	0.111237	+	0.993794i	$0.464519\pi$
$788$		0			0
$789$	22.0000			0.783221

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1280.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 1.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 1.00000i) q^{5} +(3.00000 - 3.00000i) q^{7} +1.00000i q^{9} -2.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-1.00000 + 3.00000i) q^{15} +(1.00000 + 1.00000i) q^{17} +4.00000i q^{19} +6.00000i q^{21} +(1.00000 + 1.00000i) q^{23} +(3.00000 - 4.00000i) q^{25} +(-4.00000 - 4.00000i) q^{27} -10.0000i q^{31} +(2.00000 - 2.00000i) q^{33} +(3.00000 - 9.00000i) q^{35} +(-1.00000 + 1.00000i) q^{37} -6.00000 q^{39} +10.0000 q^{41} +(5.00000 - 5.00000i) q^{43} +(1.00000 + 2.00000i) q^{45} +(-3.00000 + 3.00000i) q^{47} -11.0000i q^{49} -2.00000 q^{51} +(5.00000 + 5.00000i) q^{53} +(-4.00000 + 2.00000i) q^{55} +(-4.00000 - 4.00000i) q^{57} +12.0000i q^{59} +2.00000i q^{61} +(3.00000 + 3.00000i) q^{63} +(9.00000 + 3.00000i) q^{65} +(-1.00000 - 1.00000i) q^{67} -2.00000 q^{69} +2.00000i q^{71} +(-1.00000 + 1.00000i) q^{73} +(1.00000 + 7.00000i) q^{75} +(-6.00000 + 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} +(3.00000 + 1.00000i) q^{85} -16.0000i q^{89} +18.0000 q^{91} +(10.0000 + 10.0000i) q^{93} +(4.00000 + 8.00000i) q^{95} +(-3.00000 - 3.00000i) q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7	\( 2 q - 2 q^{3} + 4 q^{5} + 6 q^{7} - 4 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{23} + 6 q^{25} - 8 q^{27} + 4 q^{33} + 6 q^{35} - 2 q^{37} - 12 q^{39} + 20 q^{41} + 10 q^{43} + 2 q^{45} - 6 q^{47} - 4 q^{51} + 10 q^{53} - 8 q^{55} - 8 q^{57} + 6 q^{63} + 18 q^{65} - 2 q^{67} - 4 q^{69} - 2 q^{73} + 2 q^{75} - 12 q^{77} + 16 q^{79} + 10 q^{81} - 10 q^{83} + 6 q^{85} + 36 q^{91} + 20 q^{93} + 8 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 6 * q^7 - 4 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^23 + 6 * q^25 - 8 * q^27 + 4 * q^33 + 6 * q^35 - 2 * q^37 - 12 * q^39 + 20 * q^41 + 10 * q^43 + 2 * q^45 - 6 * q^47 - 4 * q^51 + 10 * q^53 - 8 * q^55 - 8 * q^57 + 6 * q^63 + 18 * q^65 - 2 * q^67 - 4 * q^69 - 2 * q^73 + 2 * q^75 - 12 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 + 6 * q^85 + 36 * q^91 + 20 * q^93 + 8 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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