LMFDB - Embedded newform 800.2.n.e.543.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(543,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("800.543");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	800.n (of order $4$, degree $2$, 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:	$6.38803216170$
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			543.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.543
Dual form			800.2.n.e.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(-1.00000 + 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +2.00000i q^{11} +(-3.00000 - 3.00000i) q^{13} +(-1.00000 + 1.00000i) q^{17} +4.00000 q^{19} -6.00000 q^{21} +(1.00000 - 1.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} +10.0000i q^{31} +(-2.00000 - 2.00000i) q^{33} +(1.00000 - 1.00000i) q^{37} +6.00000 q^{39} -10.0000 q^{41} +(-5.00000 + 5.00000i) q^{43} +(3.00000 + 3.00000i) q^{47} +11.0000i q^{49} -2.00000i q^{51} +(5.00000 + 5.00000i) q^{53} +(-4.00000 + 4.00000i) q^{57} -12.0000 q^{59} +2.00000 q^{61} +(-3.00000 + 3.00000i) q^{63} +(1.00000 + 1.00000i) q^{67} +2.00000i q^{69} +2.00000i q^{71} +(-1.00000 - 1.00000i) q^{73} +(-6.00000 + 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} -16.0000i q^{89} -18.0000i q^{91} +(-10.0000 - 10.0000i) q^{93} +(3.00000 - 3.00000i) q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^7	$ 2 q - 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 8 q^{19} - 12 q^{21} + 2 q^{23} - 8 q^{27} - 4 q^{33} + 2 q^{37} + 12 q^{39} - 20 q^{41} - 10 q^{43} + 6 q^{47} + 10 q^{53} - 8 q^{57} - 24 q^{59} + 4 q^{61} - 6 q^{63} + 2 q^{67} - 2 q^{73} - 12 q^{77} + 16 q^{79} + 10 q^{81} - 10 q^{83} - 20 q^{93} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^7 - 6 * q^13 - 2 * q^17 + 8 * q^19 - 12 * q^21 + 2 * q^23 - 8 * q^27 - 4 * q^33 + 2 * q^37 + 12 * q^39 - 20 * q^41 - 10 * q^43 + 6 * q^47 + 10 * q^53 - 8 * q^57 - 24 * q^59 + 4 * q^61 - 6 * q^63 + 2 * q^67 - 2 * q^73 - 12 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 - 20 * q^93 + 6 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.989524	+	0.144370i	$0.0461154\pi$
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.144370	+	0.989524i	$0.453885\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	0.575363	+	0.817898i	$0.304861\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−6.00000			−1.30931
$22$		0			0
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	0.803636	+	0.595121i	$0.202896\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$	−2.00000	−	2.00000i	−0.348155	−	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	−0.976772	−	0.214280i	$-0.931260\pi$
$43$	−5.00000	+	5.00000i	−0.762493	+	0.762493i	0.214280	+	0.976772i	$0.431260\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	0.891202	−	0.453607i	$-0.149863\pi$
$47$	3.00000	+	3.00000i	0.437595	+	0.437595i	−0.453607	+	0.891202i	$0.649863\pi$
$48$		0			0
$49$			11.0000i			1.57143i
$50$		0			0
$51$		−	2.00000i		−	0.280056i
$52$		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
$55$		0			0
$56$		0			0
$57$	−4.00000	+	4.00000i	−0.529813	+	0.529813i
$58$		0			0
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$	−3.00000	+	3.00000i	−0.377964	+	0.377964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.00000	+	1.00000i	0.122169	+	0.122169i	0.765548	−	0.643379i	$-0.222468\pi$
$67$	1.00000	+	1.00000i	0.122169	+	0.122169i	−0.643379	+	0.765548i	$0.722468\pi$
$68$		0			0
$69$			2.00000i			0.240772i
$70$		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	0.646160	−	0.763202i	$-0.276374\pi$
$73$	−1.00000	−	1.00000i	−0.117041	−	0.117041i	−0.763202	+	0.646160i	$0.776374\pi$
$74$		0			0
$75$		0			0
$76$		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
$85$		0			0
$86$		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
$91$		−	18.0000i		−	1.88691i
$92$		0			0
$93$	−10.0000	−	10.0000i	−1.03695	−	1.03695i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	−0.538208	−	0.842812i	$-0.680899\pi$
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	0.842812	+	0.538208i	$0.180899\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	9.00000	−	9.00000i	0.886796	−	0.886796i	−0.107418	−	0.994214i	$-0.534258\pi$
$103$	9.00000	−	9.00000i	0.886796	−	0.886796i	0.994214	+	0.107418i	$0.0342582\pi$
$104$		0			0
$105$		0			0
$106$		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
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$			2.00000i			0.189832i
$112$		0			0
$113$	3.00000	+	3.00000i	0.282216	+	0.282216i	0.833992	−	0.551776i	$-0.186050\pi$
$113$	3.00000	+	3.00000i	0.282216	+	0.282216i	−0.551776	+	0.833992i	$0.686050\pi$
$114$		0			0
$115$		0			0
$116$		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
$125$		0			0
$126$		0			0
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	0.945825	−	0.324676i	$-0.105255\pi$
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	−0.324676	+	0.945825i	$0.605255\pi$
$128$		0			0
$129$		−	10.0000i		−	0.880451i
$130$		0			0
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$		0			0
$133$	12.0000	+	12.0000i	1.04053	+	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.0000	−	11.0000i	0.939793	−	0.939793i	−0.0584943	−	0.998288i	$-0.518630\pi$
$137$	11.0000	−	11.0000i	0.939793	−	0.939793i	0.998288	+	0.0584943i	$0.0186300\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		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
$149$			4.00000i			0.327693i	0.986486	+	0.163846i	$0.0523901\pi$
$149$			4.00000i			0.327693i	−0.986486	+	0.163846i	$0.947610\pi$
$150$		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
$155$		0			0
$156$		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
$165$		0			0
$166$		0			0
$167$	−1.00000	−	1.00000i	−0.0773823	−	0.0773823i	0.667356	−	0.744739i	$-0.267426\pi$
$167$	−1.00000	−	1.00000i	−0.0773823	−	0.0773823i	−0.744739	+	0.667356i	$0.767426\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$			4.00000i			0.305888i
$172$		0			0
$173$	5.00000	+	5.00000i	0.380143	+	0.380143i	0.871154	−	0.491011i	$-0.163372\pi$
$173$	5.00000	+	5.00000i	0.380143	+	0.380143i	−0.491011	+	0.871154i	$0.663372\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	12.0000	−	12.0000i	0.901975	−	0.901975i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	−2.00000	+	2.00000i	−0.147844	+	0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.00000	−	2.00000i	−0.146254	−	0.146254i
$188$		0			0
$189$		−	24.0000i		−	1.74574i
$190$		0			0
$191$		−	14.0000i		−	1.01300i	−0.862239	−	0.506502i	$-0.830938\pi$
$191$		−	14.0000i		−	1.01300i	0.862239	−	0.506502i	$-0.169062\pi$
$192$		0			0
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.996534	+	0.0831899i	$0.0265108\pi$
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.0831899	+	0.996534i	$0.473489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	0.997459	+	0.0712470i	$0.0226979\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$	−2.00000			−0.141069
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.00000	+	1.00000i	0.0695048	+	0.0695048i
$208$		0			0
$209$			8.00000i			0.553372i
$210$		0			0
$211$		−	14.0000i		−	0.963800i	−0.876226	−	0.481900i	$-0.839947\pi$
$211$		−	14.0000i		−	0.963800i	0.876226	−	0.481900i	$-0.160053\pi$
$212$		0			0
$213$	−2.00000	−	2.00000i	−0.137038	−	0.137038i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−30.0000	+	30.0000i	−2.03653	+	2.03653i
$218$		0			0
$219$	2.00000			0.135147
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$	1.00000	−	1.00000i	0.0669650	−	0.0669650i	−0.672831	−	0.739796i	$-0.734922\pi$
$223$	1.00000	−	1.00000i	0.0669650	−	0.0669650i	0.739796	+	0.672831i	$0.234922\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.00000	+	5.00000i	0.331862	+	0.331862i	0.853293	−	0.521431i	$-0.174602\pi$
$227$	5.00000	+	5.00000i	0.331862	+	0.331862i	−0.521431	+	0.853293i	$0.674602\pi$
$228$		0			0
$229$			8.00000i			0.528655i	0.964433	+	0.264327i	$0.0851500\pi$
$229$			8.00000i			0.528655i	−0.964433	+	0.264327i	$0.914850\pi$
$230$		0			0
$231$		−	12.0000i		−	0.789542i
$232$		0			0
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.524097	−	0.851658i	$-0.675597\pi$
$234$		0			0
$235$		0			0
$236$		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
$245$		0			0
$246$		0			0
$247$	−12.0000	−	12.0000i	−0.763542	−	0.763542i
$248$		0			0
$249$		−	10.0000i		−	0.633724i
$250$		0			0
$251$		−	6.00000i		−	0.378717i	−0.981908	−	0.189358i	$-0.939359\pi$
$251$		−	6.00000i		−	0.378717i	0.981908	−	0.189358i	$-0.0606408\pi$
$252$		0			0
$253$	2.00000	+	2.00000i	0.125739	+	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.00000	+	5.00000i	−0.311891	+	0.311891i	−0.845642	−	0.533751i	$-0.820782\pi$
$257$	−5.00000	+	5.00000i	−0.311891	+	0.311891i	0.533751	+	0.845642i	$0.320782\pi$
$258$		0			0
$259$	6.00000			0.372822
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.0000	+	11.0000i	−0.678289	+	0.678289i	−0.959613	−	0.281324i	$-0.909226\pi$
$263$	−11.0000	+	11.0000i	−0.678289	+	0.678289i	0.281324	+	0.959613i	$0.409226\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	16.0000	+	16.0000i	0.979184	+	0.979184i
$268$		0			0
$269$		−	20.0000i		−	1.21942i	−0.792624	−	0.609711i	$-0.791286\pi$
$269$		−	20.0000i		−	1.21942i	0.792624	−	0.609711i	$-0.208714\pi$
$270$		0			0
$271$		−	14.0000i		−	0.850439i	−0.905090	−	0.425220i	$-0.860197\pi$
$271$		−	14.0000i		−	0.850439i	0.905090	−	0.425220i	$-0.139803\pi$
$272$		0			0
$273$	18.0000	+	18.0000i	1.08941	+	1.08941i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	−0.955598	−	0.294672i	$-0.904789\pi$
$277$	−11.0000	+	11.0000i	−0.660926	+	0.660926i	0.294672	+	0.955598i	$0.404789\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	7.00000	−	7.00000i	0.416107	−	0.416107i	−0.467753	−	0.883859i	$-0.654936\pi$
$283$	7.00000	−	7.00000i	0.416107	−	0.416107i	0.883859	+	0.467753i	$0.154936\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.0000	−	30.0000i	−1.77084	−	1.77084i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$			6.00000i			0.351726i
$292$		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
$295$		0			0
$296$		0			0
$297$	8.00000	−	8.00000i	0.464207	−	0.464207i
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	−30.0000			−1.72917
$302$		0			0
$303$	−6.00000	+	6.00000i	−0.344691	+	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	17.0000	+	17.0000i	0.970241	+	0.970241i	0.999570	−	0.0293286i	$-0.00933691\pi$
$307$	17.0000	+	17.0000i	0.970241	+	0.970241i	−0.0293286	+	0.999570i	$0.509337\pi$
$308$		0			0
$309$			18.0000i			1.02398i
$310$		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	0.405420	−	0.914130i	$-0.367125\pi$
$313$	−9.00000	−	9.00000i	−0.508710	−	0.508710i	−0.914130	+	0.405420i	$0.867125\pi$
$314$		0			0
$315$		0			0
$316$		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
$325$		0			0
$326$		0			0
$327$	−4.00000	−	4.00000i	−0.221201	−	0.221201i
$328$		0			0
$329$			18.0000i			0.992372i
$330$		0			0
$331$			26.0000i			1.42909i	0.699590	+	0.714545i	$0.253366\pi$
$331$			26.0000i			1.42909i	−0.699590	+	0.714545i	$0.746634\pi$
$332$		0			0
$333$	1.00000	+	1.00000i	0.0547997	+	0.0547997i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.0000	−	15.0000i	0.817102	−	0.817102i	−0.168585	−	0.985687i	$-0.553920\pi$
$337$	15.0000	−	15.0000i	0.817102	−	0.817102i	0.985687	+	0.168585i	$0.0539198\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−20.0000			−1.08306
$342$		0			0
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		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
$349$			8.00000i			0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$			8.00000i			0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$		0			0
$351$			24.0000i			1.28103i
$352$		0			0
$353$	15.0000	+	15.0000i	0.798369	+	0.798369i	0.982838	−	0.184469i	$-0.0590565\pi$
$353$	15.0000	+	15.0000i	0.798369	+	0.798369i	−0.184469	+	0.982838i	$0.559057\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	6.00000	−	6.00000i	0.317554	−	0.317554i
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−7.00000	+	7.00000i	−0.367405	+	0.367405i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.0000	+	15.0000i	0.782994	+	0.782994i	0.980335	−	0.197341i	$-0.0632307\pi$
$367$	15.0000	+	15.0000i	0.782994	+	0.782994i	−0.197341	+	0.980335i	$0.563231\pi$
$368$		0			0
$369$		−	10.0000i		−	0.520579i
$370$		0			0
$371$			30.0000i			1.55752i
$372$		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
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	−14.0000			−0.717242
$382$		0			0
$383$	1.00000	−	1.00000i	0.0510976	−	0.0510976i	−0.681096	−	0.732194i	$-0.738496\pi$
$383$	1.00000	−	1.00000i	0.0510976	−	0.0510976i	0.732194	+	0.681096i	$0.238496\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.00000	−	5.00000i	−0.254164	−	0.254164i
$388$		0			0
$389$			4.00000i			0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$			4.00000i			0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$		0			0
$391$			2.00000i			0.101144i
$392$		0			0
$393$	−10.0000	−	10.0000i	−0.504433	−	0.504433i
$394$		0			0
$395$		0			0
$396$		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
$405$		0			0
$406$		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
$411$			22.0000i			1.08518i
$412$		0			0
$413$	−36.0000	−	36.0000i	−1.77144	−	1.77144i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−12.0000	+	12.0000i	−0.587643	+	0.587643i
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$	−3.00000	+	3.00000i	−0.145865	+	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00000	+	6.00000i	0.290360	+	0.290360i
$428$		0			0
$429$			12.0000i			0.579365i
$430$		0			0
$431$		−	30.0000i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$		0			0
$433$	−21.0000	−	21.0000i	−1.00920	−	1.00920i	−0.999957	−	0.00923827i	$-0.997059\pi$
$433$	−21.0000	−	21.0000i	−1.00920	−	1.00920i	−0.00923827	−	0.999957i	$-0.502941\pi$
$434$		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	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	−11.0000			−0.523810
$442$		0			0
$443$	−25.0000	+	25.0000i	−1.18779	+	1.18779i	−0.210108	+	0.977678i	$0.567381\pi$
$443$	−25.0000	+	25.0000i	−1.18779	+	1.18779i	−0.977678	+	0.210108i	$0.932619\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−4.00000	−	4.00000i	−0.189194	−	0.189194i
$448$		0			0
$449$			12.0000i			0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		0			0
$451$		−	20.0000i		−	0.941763i
$452$		0			0
$453$	6.00000	+	6.00000i	0.281905	+	0.281905i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.00000	+	9.00000i	−0.421002	+	0.421002i	−0.885549	−	0.464546i	$-0.846217\pi$
$457$	−9.00000	+	9.00000i	−0.421002	+	0.421002i	0.464546	+	0.885549i	$0.346217\pi$
$458$		0			0
$459$	8.00000			0.373408
$460$		0			0
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	−11.0000	+	11.0000i	−0.511213	+	0.511213i	−0.914898	−	0.403685i	$-0.867729\pi$
$463$	−11.0000	+	11.0000i	−0.511213	+	0.511213i	0.403685	+	0.914898i	$0.367729\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.0000	+	13.0000i	0.601568	+	0.601568i	0.940729	−	0.339160i	$-0.110143\pi$
$467$	13.0000	+	13.0000i	0.601568	+	0.601568i	−0.339160	+	0.940729i	$0.610143\pi$
$468$		0			0
$469$			6.00000i			0.277054i
$470$		0			0
$471$			2.00000i			0.0921551i
$472$		0			0
$473$	−10.0000	−	10.0000i	−0.459800	−	0.459800i
$474$		0			0
$475$		0			0
$476$		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
$485$		0			0
$486$		0			0
$487$	19.0000	+	19.0000i	0.860972	+	0.860972i	0.991451	−	0.130479i	$-0.0416515\pi$
$487$	19.0000	+	19.0000i	0.860972	+	0.860972i	−0.130479	+	0.991451i	$0.541651\pi$
$488$		0			0
$489$		−	2.00000i		−	0.0904431i
$490$		0			0
$491$			10.0000i			0.451294i	0.974209	+	0.225647i	$0.0724495\pi$
$491$			10.0000i			0.451294i	−0.974209	+	0.225647i	$0.927550\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	+	6.00000i	−0.269137	+	0.269137i
$498$		0			0
$499$	28.0000			1.25345			0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000			1.25345			0.626726	+	0.779240i	$0.284395\pi$
$500$		0			0
$501$	2.00000			0.0893534
$502$		0			0
$503$	17.0000	−	17.0000i	0.757993	−	0.757993i	−0.217964	−	0.975957i	$-0.569942\pi$
$503$	17.0000	−	17.0000i	0.757993	−	0.757993i	0.975957	+	0.217964i	$0.0699416\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.00000	−	5.00000i	−0.222058	−	0.222058i
$508$		0			0
$509$		−	24.0000i		−	1.06378i	−0.846813	−	0.531891i	$-0.821482\pi$
$509$		−	24.0000i		−	1.06378i	0.846813	−	0.531891i	$-0.178518\pi$
$510$		0			0
$511$		−	6.00000i		−	0.265424i
$512$		0			0
$513$	−16.0000	−	16.0000i	−0.706417	−	0.706417i
$514$		0			0
$515$		0			0
$516$		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	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		0			0
$523$	15.0000	−	15.0000i	0.655904	−	0.655904i	−0.298504	−	0.954408i	$-0.596488\pi$
$523$	15.0000	−	15.0000i	0.655904	−	0.655904i	0.954408	+	0.298504i	$0.0964877\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	−	10.0000i	−0.435607	−	0.435607i
$528$		0			0
$529$			21.0000i			0.913043i
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$	30.0000	+	30.0000i	1.29944	+	1.29944i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	12.0000	−	12.0000i	0.517838	−	0.517838i
$538$		0			0
$539$	−22.0000			−0.947607
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$	−22.0000	+	22.0000i	−0.944110	+	0.944110i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−11.0000	−	11.0000i	−0.470326	−	0.470326i	0.431694	−	0.902020i	$-0.357916\pi$
$547$	−11.0000	−	11.0000i	−0.470326	−	0.470326i	−0.902020	+	0.431694i	$0.857916\pi$
$548$		0			0
$549$			2.00000i			0.0853579i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	24.0000	+	24.0000i	1.02058	+	1.02058i
$554$		0			0
$555$		0			0
$556$		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
$565$		0			0
$566$		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
$571$			34.0000i			1.42286i	0.702759	+	0.711428i	$0.251951\pi$
$571$			34.0000i			1.42286i	−0.702759	+	0.711428i	$0.748049\pi$
$572$		0			0
$573$	14.0000	+	14.0000i	0.584858	+	0.584858i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	19.0000	−	19.0000i	0.790980	−	0.790980i	−0.190673	−	0.981654i	$-0.561067\pi$
$577$	19.0000	−	19.0000i	0.790980	−	0.790980i	0.981654	+	0.190673i	$0.0610671\pi$
$578$		0			0
$579$	−30.0000			−1.24676
$580$		0			0
$581$	−30.0000			−1.24461
$582$		0			0
$583$	−10.0000	+	10.0000i	−0.414158	+	0.414158i
$584$		0			0
$585$		0			0
$586$		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
$589$			40.0000i			1.64817i
$590$		0			0
$591$			26.0000i			1.06950i
$592$		0			0
$593$	7.00000	+	7.00000i	0.287456	+	0.287456i	0.836073	−	0.548618i	$-0.184846\pi$
$593$	7.00000	+	7.00000i	0.287456	+	0.287456i	−0.548618	+	0.836073i	$0.684846\pi$
$594$		0			0
$595$		0			0
$596$		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	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$	−1.00000	+	1.00000i	−0.0407231	+	0.0407231i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	0.598316	−	0.801260i	$-0.295837\pi$
$607$	−5.00000	−	5.00000i	−0.202944	−	0.202944i	−0.801260	+	0.598316i	$0.795837\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	18.0000i		−	0.728202i
$612$		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
$615$		0			0
$616$		0			0
$617$	−13.0000	+	13.0000i	−0.523360	+	0.523360i	−0.918585	−	0.395224i	$-0.870667\pi$
$617$	−13.0000	+	13.0000i	−0.523360	+	0.523360i	0.395224	+	0.918585i	$0.370667\pi$
$618$		0			0
$619$	−12.0000			−0.482321			−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000			−0.482321			−0.241160	+	0.970485i	$0.577528\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$	48.0000	−	48.0000i	1.92308	−	1.92308i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.00000	−	8.00000i	−0.319489	−	0.319489i
$628$		0			0
$629$			2.00000i			0.0797452i
$630$		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
$635$		0			0
$636$		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
$645$		0			0
$646$		0			0
$647$	−29.0000	−	29.0000i	−1.14011	−	1.14011i	−0.988430	−	0.151678i	$-0.951532\pi$
$647$	−29.0000	−	29.0000i	−1.14011	−	1.14011i	−0.151678	−	0.988430i	$-0.548468\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$		0			0
$651$		−	60.0000i		−	2.35159i
$652$		0			0
$653$	1.00000	+	1.00000i	0.0391330	+	0.0391330i	0.726403	−	0.687270i	$-0.241191\pi$
$653$	1.00000	+	1.00000i	0.0391330	+	0.0391330i	−0.687270	+	0.726403i	$0.741191\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.00000	−	1.00000i	0.0390137	−	0.0390137i
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$		0			0
$663$	−6.00000	+	6.00000i	−0.233021	+	0.233021i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			2.00000i			0.0773245i
$670$		0			0
$671$			4.00000i			0.154418i
$672$		0			0
$673$	3.00000	+	3.00000i	0.115642	+	0.115642i	0.762560	−	0.646918i	$-0.223942\pi$
$673$	3.00000	+	3.00000i	0.115642	+	0.115642i	−0.646918	+	0.762560i	$0.723942\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.00000	+	3.00000i	−0.115299	+	0.115299i	−0.762402	−	0.647103i	$-0.775980\pi$
$677$	−3.00000	+	3.00000i	−0.115299	+	0.115299i	0.647103	+	0.762402i	$0.275980\pi$
$678$		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	−0.464611	−	0.885515i	$-0.653806\pi$
$683$	11.0000	−	11.0000i	0.420903	−	0.420903i	0.885515	+	0.464611i	$0.153806\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8.00000	−	8.00000i	−0.305219	−	0.305219i
$688$		0			0
$689$		−	30.0000i		−	1.14291i
$690$		0			0
$691$		−	14.0000i		−	0.532585i	−0.963892	−	0.266293i	$-0.914201\pi$
$691$		−	14.0000i		−	0.532585i	0.963892	−	0.266293i	$-0.0857987\pi$
$692$		0			0
$693$	−6.00000	−	6.00000i	−0.227921	−	0.227921i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	10.0000	−	10.0000i	0.378777	−	0.378777i
$698$		0			0
$699$	42.0000			1.58859
$700$		0			0
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$	4.00000	−	4.00000i	0.150863	−	0.150863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18.0000	+	18.0000i	0.676960	+	0.676960i
$708$		0			0
$709$		−	48.0000i		−	1.80268i	−0.433114	−	0.901339i	$-0.642585\pi$
$709$		−	48.0000i		−	1.80268i	0.433114	−	0.901339i	$-0.357415\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$		0			0
$713$	10.0000	+	10.0000i	0.374503	+	0.374503i
$714$		0			0
$715$		0			0
$716$		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	0.760547	−	0.649283i	$-0.224931\pi$
$727$	3.00000	+	3.00000i	0.111264	+	0.111264i	−0.649283	+	0.760547i	$0.724931\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$		−	10.0000i		−	0.369863i
$732$		0			0
$733$	−27.0000	−	27.0000i	−0.997268	−	0.997268i	0.00272852	−	0.999996i	$-0.499131\pi$
$733$	−27.0000	−	27.0000i	−0.997268	−	0.997268i	−0.999996	+	0.00272852i	$0.999131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.00000	+	2.00000i	−0.0736709	+	0.0736709i
$738$		0			0
$739$	−28.0000			−1.03000			−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000			−1.03000			−0.514998	+	0.857191i	$0.672207\pi$
$740$		0			0
$741$	24.0000			0.881662
$742$		0			0
$743$	21.0000	−	21.0000i	0.770415	−	0.770415i	−0.207764	−	0.978179i	$-0.566619\pi$
$743$	21.0000	−	21.0000i	0.770415	−	0.770415i	0.978179	+	0.207764i	$0.0666185\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.00000	−	5.00000i	−0.182940	−	0.182940i
$748$		0			0
$749$		−	18.0000i		−	0.657706i
$750$		0			0
$751$			2.00000i			0.0729810i	0.999334	+	0.0364905i	$0.0116179\pi$
$751$			2.00000i			0.0729810i	−0.999334	+	0.0364905i	$0.988382\pi$
$752$		0			0
$753$	6.00000	+	6.00000i	0.218652	+	0.218652i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−19.0000	+	19.0000i	−0.690567	+	0.690567i	−0.962357	−	0.271790i	$-0.912384\pi$
$757$	−19.0000	+	19.0000i	−0.690567	+	0.690567i	0.271790	+	0.962357i	$0.412384\pi$
$758$		0			0
$759$	−4.00000			−0.145191
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	−12.0000	+	12.0000i	−0.434429	+	0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	36.0000	+	36.0000i	1.29988	+	1.29988i
$768$		0			0
$769$			8.00000i			0.288487i	0.989542	+	0.144244i	$0.0460749\pi$
$769$			8.00000i			0.288487i	−0.989542	+	0.144244i	$0.953925\pi$
$770$		0			0
$771$		−	10.0000i		−	0.360141i
$772$		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
$775$		0			0
$776$		0			0
$777$	−6.00000	+	6.00000i	−0.215249	+	0.215249i
$778$		0			0
$779$	−40.0000			−1.43315
$780$		0			0
$781$	−4.00000			−0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−31.0000	−	31.0000i	−1.10503	−	1.10503i	−0.993794	−	0.111237i	$-0.964519\pi$
$787$	−31.0000	−	31.0000i	−1.10503	−	1.10503i	−0.111237	−	0.993794i	$-0.535481\pi$
$788$		0			0
$789$		−	22.0000i		−	0.783221i
$790$		0			0
$791$			18.0000i			0.640006i
$792$		0			0
$793$	−6.00000	−	6.00000i	−0.213066	−	0.213066i
$794$		0			0
$795$		0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{3} +(3.00000 + 3.00000i) q^{7} +1.00000i q^{9} +2.00000i q^{11} +(-3.00000 - 3.00000i) q^{13} +(-1.00000 + 1.00000i) q^{17} +4.00000 q^{19} -6.00000 q^{21} +(1.00000 - 1.00000i) q^{23} +(-4.00000 - 4.00000i) q^{27} +10.0000i q^{31} +(-2.00000 - 2.00000i) q^{33} +(1.00000 - 1.00000i) q^{37} +6.00000 q^{39} -10.0000 q^{41} +(-5.00000 + 5.00000i) q^{43} +(3.00000 + 3.00000i) q^{47} +11.0000i q^{49} -2.00000i q^{51} +(5.00000 + 5.00000i) q^{53} +(-4.00000 + 4.00000i) q^{57} -12.0000 q^{59} +2.00000 q^{61} +(-3.00000 + 3.00000i) q^{63} +(1.00000 + 1.00000i) q^{67} +2.00000i q^{69} +2.00000i q^{71} +(-1.00000 - 1.00000i) q^{73} +(-6.00000 + 6.00000i) q^{77} +8.00000 q^{79} +5.00000 q^{81} +(-5.00000 + 5.00000i) q^{83} -16.0000i q^{89} -18.0000i q^{91} +(-10.0000 - 10.0000i) q^{93} +(3.00000 - 3.00000i) q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^7	\( 2 q - 2 q^{3} + 6 q^{7} - 6 q^{13} - 2 q^{17} + 8 q^{19} - 12 q^{21} + 2 q^{23} - 8 q^{27} - 4 q^{33} + 2 q^{37} + 12 q^{39} - 20 q^{41} - 10 q^{43} + 6 q^{47} + 10 q^{53} - 8 q^{57} - 24 q^{59} + 4 q^{61} - 6 q^{63} + 2 q^{67} - 2 q^{73} - 12 q^{77} + 16 q^{79} + 10 q^{81} - 10 q^{83} - 20 q^{93} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^7 - 6 * q^13 - 2 * q^17 + 8 * q^19 - 12 * q^21 + 2 * q^23 - 8 * q^27 - 4 * q^33 + 2 * q^37 + 12 * q^39 - 20 * q^41 - 10 * q^43 + 6 * q^47 + 10 * q^53 - 8 * q^57 - 24 * q^59 + 4 * q^61 - 6 * q^63 + 2 * q^67 - 2 * q^73 - 12 * q^77 + 16 * q^79 + 10 * q^81 - 10 * q^83 - 20 * q^93 + 6 * q^97 - 4 * q^99

Introduction

L-functions

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