LMFDB - Embedded newform 675.2.f.g.593.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,2,Mod(107,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.107"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.f (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-8,0,0,0,0,0,16,0,0,-16,0,0,0,0,0,8,0,0,0,0,0,-40, 0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.33973862400.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 44x^{4} + 100 $ x^8 + 44*x^4 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.3
Root			$0.880486 + 0.880486i$ of defining polynomial
Character	$\chi$	$=$	675.593
Dual form			675.2.f.g.107.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.880486 + 0.880486i) q^{2} -0.449490i q^{4} +(0.224745 - 0.224745i) q^{7} +(2.15674 - 2.15674i) q^{8} -3.91771i q^{11} +(3.22474 + 3.22474i) q^{13} +0.395769 q^{14} +2.89898 q^{16} +(-3.91771 - 3.91771i) q^{17} -3.00000i q^{19} +(3.44949 - 3.44949i) q^{22} +(-0.880486 + 0.880486i) q^{23} +5.67868i q^{26} +(-0.101021 - 0.101021i) q^{28} +7.43966 q^{29} +9.34847 q^{31} +(-1.76097 - 1.76097i) q^{32} -6.89898i q^{34} +(-1.67423 + 1.67423i) q^{37} +(2.64146 - 2.64146i) q^{38} +1.36520i q^{41} +(1.44949 + 1.44949i) q^{43} -1.76097 q^{44} -1.55051 q^{46} +(3.52194 + 3.52194i) q^{47} +6.89898i q^{49} +(1.44949 - 1.44949i) q^{52} +(-5.19397 + 5.19397i) q^{53} -0.969433i q^{56} +(6.55051 + 6.55051i) q^{58} -13.9099 q^{59} -1.00000 q^{61} +(8.23119 + 8.23119i) q^{62} -8.89898i q^{64} +(3.77526 - 3.77526i) q^{67} +(-1.76097 + 1.76097i) q^{68} -9.20063i q^{71} +(-4.67423 - 4.67423i) q^{73} -2.94828 q^{74} -1.34847 q^{76} +(-0.880486 - 0.880486i) q^{77} +5.44949i q^{79} +(-1.20204 + 1.20204i) q^{82} +(-3.03723 + 3.03723i) q^{83} +2.55251i q^{86} +(-8.44949 - 8.44949i) q^{88} -5.28291 q^{89} +1.44949 q^{91} +(0.395769 + 0.395769i) q^{92} +6.20204i q^{94} +(-10.1237 + 10.1237i) q^{97} +(-6.07445 + 6.07445i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} + 16 q^{13} - 16 q^{16} + 8 q^{22} - 40 q^{28} + 16 q^{31} + 16 q^{37} - 8 q^{43} - 32 q^{46} - 8 q^{52} + 72 q^{58} - 8 q^{61} + 40 q^{67} - 8 q^{73} + 48 q^{76} - 88 q^{82} - 48 q^{88} - 8 q^{91}+ \cdots - 32 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 16 * q^13 - 16 * q^16 + 8 * q^22 - 40 * q^28 + 16 * q^31 + 16 * q^37 - 8 * q^43 - 32 * q^46 - 8 * q^52 + 72 * q^58 - 8 * q^61 + 40 * q^67 - 8 * q^73 + 48 * q^76 - 88 * q^82 - 48 * q^88 - 8 * q^91 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	0.880486	+	0.880486i	0.622597	+	0.622597i	0.946195	−	0.323597i	$-0.104892\pi$
$2$	0.880486	+	0.880486i	0.622597	+	0.622597i	−0.323597	+	0.946195i	$0.604892\pi$
$3$		0			0
$3$		0			0
$4$		−	0.449490i		−	0.224745i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.224745	−	0.224745i	0.0849456	−	0.0849456i	−0.663357	−	0.748303i	$-0.730869\pi$
$7$	0.224745	−	0.224745i	0.0849456	−	0.0849456i	0.748303	+	0.663357i	$0.230869\pi$
$8$	2.15674	−	2.15674i	0.762523	−	0.762523i
$9$		0			0
$10$		0			0
$11$		−	3.91771i		−	1.18123i	−0.806952	−	0.590617i	$-0.798884\pi$
$11$		−	3.91771i		−	1.18123i	0.806952	−	0.590617i	$-0.201116\pi$
$12$		0			0
$13$	3.22474	+	3.22474i	0.894383	+	0.894383i	0.994932	−	0.100549i	$-0.0320599\pi$
$13$	3.22474	+	3.22474i	0.894383	+	0.894383i	−0.100549	+	0.994932i	$0.532060\pi$
$14$	0.395769			0.105774
$15$		0			0
$16$	2.89898			0.724745
$17$	−3.91771	−	3.91771i	−0.950185	−	0.950185i	0.0486320	−	0.998817i	$-0.484514\pi$
$17$	−3.91771	−	3.91771i	−0.950185	−	0.950185i	−0.998817	+	0.0486320i	$0.984514\pi$
$18$		0			0
$19$		−	3.00000i		−	0.688247i	−0.938924	−	0.344124i	$-0.888176\pi$
$19$		−	3.00000i		−	0.688247i	0.938924	−	0.344124i	$-0.111824\pi$
$20$		0			0
$21$		0			0
$22$	3.44949	−	3.44949i	0.735434	−	0.735434i
$23$	−0.880486	+	0.880486i	−0.183594	+	0.183594i	−0.792920	−	0.609326i	$-0.791440\pi$
$23$	−0.880486	+	0.880486i	−0.183594	+	0.183594i	0.609326	+	0.792920i	$0.291440\pi$
$24$		0			0
$25$		0			0
$26$			5.67868i			1.11368i
$27$		0			0
$28$	−0.101021	−	0.101021i	−0.0190911	−	0.0190911i
$29$	7.43966			1.38151			0.690755	−	0.723089i	$-0.257278\pi$
$29$	7.43966			1.38151			0.690755	+	0.723089i	$0.257278\pi$
$30$		0			0
$31$	9.34847			1.67903			0.839517	−	0.543333i	$-0.182838\pi$
$31$	9.34847			1.67903			0.839517	+	0.543333i	$0.182838\pi$
$32$	−1.76097	−	1.76097i	−0.311299	−	0.311299i
$33$		0			0
$34$		−	6.89898i		−	1.18317i
$35$		0			0
$36$		0			0
$37$	−1.67423	+	1.67423i	−0.275242	+	0.275242i	−0.831206	−	0.555964i	$-0.812349\pi$
$37$	−1.67423	+	1.67423i	−0.275242	+	0.275242i	0.555964	+	0.831206i	$0.312349\pi$
$38$	2.64146	−	2.64146i	0.428501	−	0.428501i
$39$		0			0
$40$		0			0
$41$			1.36520i			0.213209i	0.994302	+	0.106604i	$0.0339978\pi$
$41$			1.36520i			0.213209i	−0.994302	+	0.106604i	$0.966002\pi$
$42$		0			0
$43$	1.44949	+	1.44949i	0.221045	+	0.221045i	0.808938	−	0.587893i	$-0.200043\pi$
$43$	1.44949	+	1.44949i	0.221045	+	0.221045i	−0.587893	+	0.808938i	$0.700043\pi$
$44$	−1.76097			−0.265476
$45$		0			0
$46$	−1.55051			−0.228610
$47$	3.52194	+	3.52194i	0.513728	+	0.513728i	0.915667	−	0.401938i	$-0.131663\pi$
$47$	3.52194	+	3.52194i	0.513728	+	0.513728i	−0.401938	+	0.915667i	$0.631663\pi$
$48$		0			0
$49$			6.89898i			0.985568i
$50$		0			0
$51$		0			0
$52$	1.44949	−	1.44949i	0.201008	−	0.201008i
$53$	−5.19397	+	5.19397i	−0.713446	+	0.713446i	−0.967255	−	0.253808i	$-0.918317\pi$
$53$	−5.19397	+	5.19397i	−0.713446	+	0.713446i	0.253808	+	0.967255i	$0.418317\pi$
$54$		0			0
$55$		0			0
$56$		−	0.969433i		−	0.129546i
$57$		0			0
$58$	6.55051	+	6.55051i	0.860124	+	0.860124i
$59$	−13.9099			−1.81091			−0.905456	−	0.424440i	$-0.860471\pi$
$59$	−13.9099			−1.81091			−0.905456	+	0.424440i	$0.860471\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$	8.23119	+	8.23119i	1.04536	+	1.04536i
$63$		0			0
$64$		−	8.89898i		−	1.11237i
$65$		0			0
$66$		0			0
$67$	3.77526	−	3.77526i	0.461221	−	0.461221i	−0.437835	−	0.899056i	$-0.644254\pi$
$67$	3.77526	−	3.77526i	0.461221	−	0.461221i	0.899056	+	0.437835i	$0.144254\pi$
$68$	−1.76097	+	1.76097i	−0.213549	+	0.213549i
$69$		0			0
$70$		0			0
$71$		−	9.20063i		−	1.09191i	−0.837813	−	0.545957i	$-0.816166\pi$
$71$		−	9.20063i		−	1.09191i	0.837813	−	0.545957i	$-0.183834\pi$
$72$		0			0
$73$	−4.67423	−	4.67423i	−0.547078	−	0.547078i	0.378517	−	0.925594i	$-0.376434\pi$
$73$	−4.67423	−	4.67423i	−0.547078	−	0.547078i	−0.925594	+	0.378517i	$0.876434\pi$
$74$	−2.94828			−0.342731
$75$		0			0
$76$	−1.34847			−0.154680
$77$	−0.880486	−	0.880486i	−0.100341	−	0.100341i
$78$		0			0
$79$			5.44949i			0.613115i	0.951852	+	0.306558i	$0.0991773\pi$
$79$			5.44949i			0.613115i	−0.951852	+	0.306558i	$0.900823\pi$
$80$		0			0
$81$		0			0
$82$	−1.20204	+	1.20204i	−0.132743	+	0.132743i
$83$	−3.03723	+	3.03723i	−0.333379	+	0.333379i	−0.853868	−	0.520489i	$-0.825750\pi$
$83$	−3.03723	+	3.03723i	−0.333379	+	0.333379i	0.520489	+	0.853868i	$0.325750\pi$
$84$		0			0
$85$		0			0
$86$			2.55251i			0.275244i
$87$		0			0
$88$	−8.44949	−	8.44949i	−0.900719	−	0.900719i
$89$	−5.28291			−0.559988			−0.279994	−	0.960002i	$-0.590332\pi$
$89$	−5.28291			−0.559988			−0.279994	+	0.960002i	$0.590332\pi$
$90$		0			0
$91$	1.44949			0.151948
$92$	0.395769	+	0.395769i	0.0412618	+	0.0412618i
$93$		0			0
$94$			6.20204i			0.639692i
$95$		0			0
$96$		0			0
$97$	−10.1237	+	10.1237i	−1.02791	+	1.02791i	−0.0283093	+	0.999599i	$0.509012\pi$
$97$	−10.1237	+	10.1237i	−1.02791	+	1.02791i	−0.999599	+	0.0283093i	$0.990988\pi$
$98$	−6.07445	+	6.07445i	−0.613612	+	0.613612i
$99$		0			0
$100$		0			0
$101$			13.1183i			1.30532i	0.757649	+	0.652662i	$0.226348\pi$
$101$			13.1183i			1.30532i	−0.757649	+	0.652662i	$0.773652\pi$
$102$		0			0
$103$	7.57321	+	7.57321i	0.746211	+	0.746211i	0.973765	−	0.227554i	$-0.0730730\pi$
$103$	7.57321	+	7.57321i	0.746211	+	0.746211i	−0.227554	+	0.973765i	$0.573073\pi$
$104$	13.9099			1.36398
$105$		0			0
$106$	−9.14643			−0.888380
$107$	−1.27626	−	1.27626i	−0.123380	−	0.123380i	0.642721	−	0.766101i	$-0.277806\pi$
$107$	−1.27626	−	1.27626i	−0.123380	−	0.123380i	−0.766101	+	0.642721i	$0.777806\pi$
$108$		0			0
$109$			7.34847i			0.703856i	0.936027	+	0.351928i	$0.114474\pi$
$109$			7.34847i			0.703856i	−0.936027	+	0.351928i	$0.885526\pi$
$110$		0			0
$111$		0			0
$112$	0.651531	−	0.651531i	0.0615639	−	0.0615639i
$113$	4.40243	−	4.40243i	0.414146	−	0.414146i	−0.469034	−	0.883180i	$-0.655398\pi$
$113$	4.40243	−	4.40243i	0.414146	−	0.414146i	0.883180	+	0.469034i	$0.155398\pi$
$114$		0			0
$115$		0			0
$116$		−	3.34405i		−	0.310487i
$117$		0			0
$118$	−12.2474	−	12.2474i	−1.12747	−	1.12747i
$119$	−1.76097			−0.161428
$120$		0			0
$121$	−4.34847			−0.395315
$122$	−0.880486	−	0.880486i	−0.0797154	−	0.0797154i
$123$		0			0
$124$		−	4.20204i		−	0.377354i
$125$		0			0
$126$		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	$0.394745\pi$
$128$	4.31348	−	4.31348i	0.381262	−	0.381262i
$129$		0			0
$130$		0			0
$131$			7.83542i			0.684584i	0.939594	+	0.342292i	$0.111203\pi$
$131$			7.83542i			0.684584i	−0.939594	+	0.342292i	$0.888797\pi$
$132$		0			0
$133$	−0.674235	−	0.674235i	−0.0584636	−	0.0584636i
$134$	6.64812			0.574310
$135$		0			0
$136$	−16.8990			−1.44908
$137$	−10.8727	−	10.8727i	−0.928913	−	0.928913i	0.0687225	−	0.997636i	$-0.478108\pi$
$137$	−10.8727	−	10.8727i	−0.928913	−	0.928913i	−0.997636	+	0.0687225i	$0.978108\pi$
$138$		0			0
$139$			9.24745i			0.784358i	0.919889	+	0.392179i	$0.128279\pi$
$139$			9.24745i			0.784358i	−0.919889	+	0.392179i	$0.871721\pi$
$140$		0			0
$141$		0			0
$142$	8.10102	−	8.10102i	0.679823	−	0.679823i
$143$	12.6336	−	12.6336i	1.05648	−	1.05648i
$144$		0			0
$145$		0			0
$146$		−	8.23119i		−	0.681218i
$147$		0			0
$148$	0.752551	+	0.752551i	0.0618593	+	0.0618593i
$149$	9.59640			0.786167			0.393084	−	0.919503i	$-0.371408\pi$
$149$	9.59640			0.786167			0.393084	+	0.919503i	$0.371408\pi$
$150$		0			0
$151$	−5.65153			−0.459915			−0.229958	−	0.973201i	$-0.573859\pi$
$151$	−5.65153			−0.459915			−0.229958	+	0.973201i	$0.573859\pi$
$152$	−6.47022	−	6.47022i	−0.524804	−	0.524804i
$153$		0			0
$154$		−	1.55051i		−	0.124944i
$155$		0			0
$156$		0			0
$157$	4.44949	−	4.44949i	0.355108	−	0.355108i	−0.506898	−	0.862006i	$-0.669208\pi$
$157$	4.44949	−	4.44949i	0.355108	−	0.355108i	0.862006	+	0.506898i	$0.169208\pi$
$158$	−4.79820	+	4.79820i	−0.381724	+	0.381724i
$159$		0			0
$160$		0			0
$161$			0.395769i			0.0311910i
$162$		0			0
$163$	−3.32577	−	3.32577i	−0.260494	−	0.260494i	0.564761	−	0.825255i	$-0.308968\pi$
$163$	−3.32577	−	3.32577i	−0.260494	−	0.260494i	−0.825255	+	0.564761i	$0.808968\pi$
$164$	0.613644			0.0479176
$165$		0			0
$166$	−5.34847			−0.415122
$167$	5.67868	+	5.67868i	0.439430	+	0.439430i	0.891820	−	0.452390i	$-0.149429\pi$
$167$	5.67868	+	5.67868i	0.439430	+	0.439430i	−0.452390	+	0.891820i	$0.649429\pi$
$168$		0			0
$169$			7.79796i			0.599843i
$170$		0			0
$171$		0			0
$172$	0.651531	−	0.651531i	0.0496788	−	0.0496788i
$173$	0.791539	−	0.791539i	0.0601796	−	0.0601796i	−0.676376	−	0.736556i	$-0.736451\pi$
$173$	0.791539	−	0.791539i	0.0601796	−	0.0601796i	0.736556	+	0.676376i	$0.236451\pi$
$174$		0			0
$175$		0			0
$176$		−	11.3574i		−	0.856094i
$177$		0			0
$178$	−4.65153	−	4.65153i	−0.348647	−	0.348647i
$179$	6.47022			0.483607			0.241804	−	0.970325i	$-0.422261\pi$
$179$	6.47022			0.483607			0.241804	+	0.970325i	$0.422261\pi$
$180$		0			0
$181$	−8.34847			−0.620537			−0.310268	−	0.950649i	$-0.600419\pi$
$181$	−8.34847			−0.620537			−0.310268	+	0.950649i	$0.600419\pi$
$182$	1.27626	+	1.27626i	0.0946023	+	0.0946023i
$183$		0			0
$184$			3.79796i			0.279989i
$185$		0			0
$186$		0			0
$187$	−15.3485	+	15.3485i	−1.12239	+	1.12239i
$188$	1.58308	−	1.58308i	0.115458	−	0.115458i
$189$		0			0
$190$		0			0
$191$		−	10.3879i		−	0.751645i	−0.926692	−	0.375822i	$-0.877360\pi$
$191$		−	10.3879i		−	0.751645i	0.926692	−	0.375822i	$-0.122640\pi$
$192$		0			0
$193$	−13.1237	−	13.1237i	−0.944666	−	0.944666i	0.0538810	−	0.998547i	$-0.482841\pi$
$193$	−13.1237	−	13.1237i	−0.944666	−	0.944666i	−0.998547	+	0.0538810i	$0.982841\pi$
$194$	−17.8276			−1.27995
$195$		0			0
$196$	3.10102			0.221501
$197$	2.55251	+	2.55251i	0.181859	+	0.181859i	0.792165	−	0.610307i	$-0.208954\pi$
$197$	2.55251	+	2.55251i	0.181859	+	0.181859i	−0.610307	+	0.792165i	$0.708954\pi$
$198$		0			0
$199$		−	4.34847i		−	0.308255i	−0.988051	−	0.154127i	$-0.950743\pi$
$199$		−	4.34847i		−	0.308255i	0.988051	−	0.154127i	$-0.0492566\pi$
$200$		0			0
$201$		0			0
$202$	−11.5505	+	11.5505i	−0.812691	+	0.812691i
$203$	1.67202	−	1.67202i	0.117353	−	0.117353i
$204$		0			0
$205$		0			0
$206$			13.3362i			0.929178i
$207$		0			0
$208$	9.34847	+	9.34847i	0.648200	+	0.648200i
$209$	−11.7531			−0.812981
$210$		0			0
$211$	12.3485			0.850104			0.425052	−	0.905169i	$-0.360256\pi$
$211$	12.3485			0.850104			0.425052	+	0.905169i	$0.360256\pi$
$212$	2.33464	+	2.33464i	0.160343	+	0.160343i
$213$		0			0
$214$		−	2.24745i		−	0.153632i
$215$		0			0
$216$		0			0
$217$	2.10102	−	2.10102i	0.142627	−	0.142627i
$218$	−6.47022	+	6.47022i	−0.438219	+	0.438219i
$219$		0			0
$220$		0			0
$221$		−	25.2672i		−	1.69966i
$222$		0			0
$223$	6.10102	+	6.10102i	0.408555	+	0.408555i	0.881234	−	0.472680i	$-0.156713\pi$
$223$	6.10102	+	6.10102i	0.408555	+	0.408555i	−0.472680	+	0.881234i	$0.656713\pi$
$224$	−0.791539			−0.0528869
$225$		0			0
$226$	7.75255			0.515692
$227$	7.35071	+	7.35071i	0.487884	+	0.487884i	0.907638	−	0.419754i	$-0.137884\pi$
$227$	7.35071	+	7.35071i	0.487884	+	0.487884i	−0.419754	+	0.907638i	$0.637884\pi$
$228$		0			0
$229$			3.55051i			0.234624i	0.993095	+	0.117312i	$0.0374278\pi$
$229$			3.55051i			0.234624i	−0.993095	+	0.117312i	$0.962572\pi$
$230$		0			0
$231$		0			0
$232$	16.0454	−	16.0454i	1.05343	−	1.05343i
$233$	9.20063	−	9.20063i	0.602753	−	0.602753i	−0.338289	−	0.941042i	$-0.609848\pi$
$233$	9.20063	−	9.20063i	0.602753	−	0.602753i	0.941042	+	0.338289i	$0.109848\pi$
$234$		0			0
$235$		0			0
$236$			6.25235i			0.406993i
$237$		0			0
$238$	−1.55051	−	1.55051i	−0.100505	−	0.100505i
$239$	26.6324			1.72271			0.861355	−	0.508004i	$-0.169616\pi$
$239$	26.6324			1.72271			0.861355	+	0.508004i	$0.169616\pi$
$240$		0			0
$241$	−13.0000			−0.837404			−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000			−0.837404			−0.418702	+	0.908124i	$0.637515\pi$
$242$	−3.82877	−	3.82877i	−0.246122	−	0.246122i
$243$		0			0
$244$			0.449490i			0.0287756i
$245$		0			0
$246$		0			0
$247$	9.67423	−	9.67423i	0.615557	−	0.615557i
$248$	20.1622	−	20.1622i	1.28030	−	1.28030i
$249$		0			0
$250$		0			0
$251$			9.02273i			0.569510i	0.958600	+	0.284755i	$0.0919122\pi$
$251$			9.02273i			0.569510i	−0.958600	+	0.284755i	$0.908088\pi$
$252$		0			0
$253$	3.44949	+	3.44949i	0.216868	+	0.216868i
$254$	−12.3268			−0.773452
$255$		0			0
$256$	−10.2020			−0.637628
$257$	5.67868	+	5.67868i	0.354227	+	0.354227i	0.861679	−	0.507453i	$-0.169413\pi$
$257$	5.67868	+	5.67868i	0.354227	+	0.354227i	−0.507453	+	0.861679i	$0.669413\pi$
$258$		0			0
$259$			0.752551i			0.0467613i
$260$		0			0
$261$		0			0
$262$	−6.89898	+	6.89898i	−0.426220	+	0.426220i
$263$	−17.4318	+	17.4318i	−1.07489	+	1.07489i	−0.0779335	+	0.996959i	$0.524832\pi$
$263$	−17.4318	+	17.4318i	−1.07489	+	1.07489i	−0.996959	+	0.0779335i	$0.975168\pi$
$264$		0			0
$265$		0			0
$266$		−	1.18731i		−	0.0727985i
$267$		0			0
$268$	−1.69694	−	1.69694i	−0.103657	−	0.103657i
$269$	5.28291			0.322105			0.161052	−	0.986946i	$-0.448511\pi$
$269$	5.28291			0.322105			0.161052	+	0.986946i	$0.448511\pi$
$270$		0			0
$271$	19.6969			1.19650			0.598252	−	0.801308i	$-0.295862\pi$
$271$	19.6969			1.19650			0.598252	+	0.801308i	$0.295862\pi$
$272$	−11.3574	−	11.3574i	−0.688642	−	0.688642i
$273$		0			0
$274$		−	19.1464i		−	1.15668i
$275$		0			0
$276$		0			0
$277$	−2.10102	+	2.10102i	−0.126238	+	0.126238i	−0.767403	−	0.641165i	$-0.778451\pi$
$277$	−2.10102	+	2.10102i	−0.126238	+	0.126238i	0.641165	+	0.767403i	$0.278451\pi$
$278$	−8.14225	+	8.14225i	−0.488340	+	0.488340i
$279$		0			0
$280$		0			0
$281$			26.0588i			1.55454i	0.629169	+	0.777268i	$0.283395\pi$
$281$			26.0588i			1.55454i	−0.629169	+	0.777268i	$0.716605\pi$
$282$		0			0
$283$	9.89898	+	9.89898i	0.588433	+	0.588433i	0.937207	−	0.348774i	$-0.113402\pi$
$283$	9.89898	+	9.89898i	0.588433	+	0.588433i	−0.348774	+	0.937207i	$0.613402\pi$
$284$	−4.13559			−0.245402
$285$		0			0
$286$	22.2474			1.31552
$287$	0.306822	+	0.306822i	0.0181111	+	0.0181111i
$288$		0			0
$289$			13.6969i			0.805702i
$290$		0			0
$291$		0			0
$292$	−2.10102	+	2.10102i	−0.122953	+	0.122953i
$293$	−17.9165	+	17.9165i	−1.04670	+	1.04670i	−0.0478403	+	0.998855i	$0.515234\pi$
$293$	−17.9165	+	17.9165i	−1.04670	+	1.04670i	−0.998855	+	0.0478403i	$0.984766\pi$
$294$		0			0
$295$		0			0
$296$			7.22178i			0.419757i
$297$		0			0
$298$	8.44949	+	8.44949i	0.489466	+	0.489466i
$299$	−5.67868			−0.328407
$300$		0			0
$301$	0.651531			0.0375536
$302$	−4.97609	−	4.97609i	−0.286342	−	0.286342i
$303$		0			0
$304$		−	8.69694i		−	0.498804i
$305$		0			0
$306$		0			0
$307$	21.3485	−	21.3485i	1.21842	−	1.21842i	0.250237	−	0.968185i	$-0.419492\pi$
$307$	21.3485	−	21.3485i	1.21842	−	1.21842i	0.968185	−	0.250237i	$-0.0805085\pi$
$308$	−0.395769	+	0.395769i	−0.0225510	+	0.0225510i
$309$		0			0
$310$		0			0
$311$			26.0588i			1.47766i	0.673893	+	0.738829i	$0.264621\pi$
$311$			26.0588i			1.47766i	−0.673893	+	0.738829i	$0.735379\pi$
$312$		0			0
$313$	5.12372	+	5.12372i	0.289610	+	0.289610i	0.836926	−	0.547316i	$-0.184350\pi$
$313$	5.12372	+	5.12372i	0.289610	+	0.289610i	−0.547316	+	0.836926i	$0.684350\pi$
$314$	7.83542			0.442179
$315$		0			0
$316$	2.44949			0.137795
$317$	4.70925	+	4.70925i	0.264498	+	0.264498i	0.826878	−	0.562381i	$-0.190114\pi$
$317$	4.70925	+	4.70925i	0.264498	+	0.264498i	−0.562381	+	0.826878i	$0.690114\pi$
$318$		0			0
$319$		−	29.1464i		−	1.63189i
$320$		0			0
$321$		0			0
$322$	−0.348469	+	0.348469i	−0.0194194	+	0.0194194i
$323$	−11.7531	+	11.7531i	−0.653962	+	0.653962i
$324$		0			0
$325$		0			0
$326$		−	5.85658i		−	0.324366i
$327$		0			0
$328$	2.94439	+	2.94439i	0.162577	+	0.162577i
$329$	1.58308			0.0872779
$330$		0			0
$331$	24.3485			1.33831			0.669156	−	0.743122i	$-0.266656\pi$
$331$	24.3485			1.33831			0.669156	+	0.743122i	$0.266656\pi$
$332$	1.36520	+	1.36520i	0.0749252	+	0.0749252i
$333$		0			0
$334$			10.0000i			0.547176i
$335$		0			0
$336$		0			0
$337$	−16.9217	+	16.9217i	−0.921783	+	0.921783i	−0.997155	−	0.0753726i	$-0.975985\pi$
$337$	−16.9217	+	16.9217i	−0.921783	+	0.921783i	0.0753726	+	0.997155i	$0.475985\pi$
$338$	−6.86599	+	6.86599i	−0.373461	+	0.373461i
$339$		0			0
$340$		0			0
$341$		−	36.6246i		−	1.98333i
$342$		0			0
$343$	3.12372	+	3.12372i	0.168665	+	0.168665i
$344$	6.25235			0.337104
$345$		0			0
$346$	1.39388			0.0749353
$347$	−8.71591	−	8.71591i	−0.467895	−	0.467895i	0.433337	−	0.901232i	$-0.357336\pi$
$347$	−8.71591	−	8.71591i	−0.467895	−	0.467895i	−0.901232	+	0.433337i	$0.857336\pi$
$348$		0			0
$349$			0.797959i			0.0427138i	0.999772	+	0.0213569i	$0.00679862\pi$
$349$			0.797959i			0.0427138i	−0.999772	+	0.0213569i	$0.993201\pi$
$350$		0			0
$351$		0			0
$352$	−6.89898	+	6.89898i	−0.367717	+	0.367717i
$353$	17.8276	−	17.8276i	0.948867	−	0.948867i	−0.0498880	−	0.998755i	$-0.515886\pi$
$353$	17.8276	−	17.8276i	0.948867	−	0.948867i	0.998755	+	0.0498880i	$0.0158864\pi$
$354$		0			0
$355$		0			0
$356$			2.37462i			0.125854i
$357$		0			0
$358$	5.69694	+	5.69694i	0.301093	+	0.301093i
$359$	−29.9765			−1.58210			−0.791049	−	0.611752i	$-0.790465\pi$
$359$	−29.9765			−1.58210			−0.791049	+	0.611752i	$0.790465\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$	−7.35071	−	7.35071i	−0.386345	−	0.386345i
$363$		0			0
$364$		−	0.651531i		−	0.0341495i
$365$		0			0
$366$		0			0
$367$	17.9217	−	17.9217i	0.935504	−	0.935504i	−0.0625383	−	0.998043i	$-0.519920\pi$
$367$	17.9217	−	17.9217i	0.935504	−	0.935504i	0.998043	+	0.0625383i	$0.0199196\pi$
$368$	−2.55251	+	2.55251i	−0.133059	+	0.133059i
$369$		0			0
$370$		0			0
$371$			2.33464i			0.121208i
$372$		0			0
$373$	−23.4722	−	23.4722i	−1.21534	−	1.21534i	−0.969245	−	0.246100i	$-0.920851\pi$
$373$	−23.4722	−	23.4722i	−1.21534	−	1.21534i	−0.246100	−	0.969245i	$-0.579149\pi$
$374$	−27.0282			−1.39760
$375$		0			0
$376$	15.1918			0.783459
$377$	23.9910	+	23.9910i	1.23560	+	1.23560i
$378$		0			0
$379$		−	29.6969i		−	1.52543i	−0.646735	−	0.762715i	$-0.723866\pi$
$379$		−	29.6969i		−	1.52543i	0.646735	−	0.762715i	$-0.276134\pi$
$380$		0			0
$381$		0			0
$382$	9.14643	−	9.14643i	0.467972	−	0.467972i
$383$	1.76097	−	1.76097i	0.0899814	−	0.0899814i	−0.660683	−	0.750665i	$-0.729733\pi$
$383$	1.76097	−	1.76097i	0.0899814	−	0.0899814i	0.750665	+	0.660683i	$0.229733\pi$
$384$		0			0
$385$		0			0
$386$		−	23.1105i		−	1.17629i
$387$		0			0
$388$	4.55051	+	4.55051i	0.231017	+	0.231017i
$389$	6.25235			0.317007			0.158503	−	0.987358i	$-0.449333\pi$
$389$	6.25235			0.317007			0.158503	+	0.987358i	$0.449333\pi$
$390$		0			0
$391$	6.89898			0.348896
$392$	14.8793	+	14.8793i	0.751519	+	0.751519i
$393$		0			0
$394$			4.49490i			0.226450i
$395$		0			0
$396$		0			0
$397$	6.34847	−	6.34847i	0.318621	−	0.318621i	−0.529617	−	0.848237i	$-0.677664\pi$
$397$	6.34847	−	6.34847i	0.318621	−	0.318621i	0.848237	+	0.529617i	$0.177664\pi$
$398$	3.82877	−	3.82877i	0.191919	−	0.191919i
$399$		0			0
$400$		0			0
$401$			14.3056i			0.714390i	0.934030	+	0.357195i	$0.116267\pi$
$401$			14.3056i			0.714390i	−0.934030	+	0.357195i	$0.883733\pi$
$402$		0			0
$403$	30.1464	+	30.1464i	1.50170	+	1.50170i
$404$	5.89656			0.293365
$405$		0			0
$406$	2.94439			0.146127
$407$	6.55917	+	6.55917i	0.325126	+	0.325126i
$408$		0			0
$409$		−	22.1010i		−	1.09283i	−0.837516	−	0.546413i	$-0.815993\pi$
$409$		−	22.1010i		−	1.09283i	0.837516	−	0.546413i	$-0.184007\pi$
$410$		0			0
$411$		0			0
$412$	3.40408	−	3.40408i	0.167707	−	0.167707i
$413$	−3.12617	+	3.12617i	−0.153829	+	0.153829i
$414$		0			0
$415$		0			0
$416$		−	11.3574i		−	0.556841i
$417$		0			0
$418$	−10.3485	−	10.3485i	−0.506160	−	0.506160i
$419$	4.31348			0.210727			0.105364	−	0.994434i	$-0.466399\pi$
$419$	4.31348			0.210727			0.105364	+	0.994434i	$0.466399\pi$
$420$		0			0
$421$	−37.7423			−1.83945			−0.919725	−	0.392564i	$-0.871588\pi$
$421$	−37.7423			−1.83945			−0.919725	+	0.392564i	$0.871588\pi$
$422$	10.8727	+	10.8727i	0.529272	+	0.529272i
$423$		0			0
$424$			22.4041i			1.08804i
$425$		0			0
$426$		0			0
$427$	−0.224745	+	0.224745i	−0.0108762	+	0.0108762i
$428$	−0.573664	+	0.573664i	−0.0277291	+	0.0277291i
$429$		0			0
$430$		0			0
$431$			11.9310i			0.574698i	0.957826	+	0.287349i	$0.0927739\pi$
$431$			11.9310i			0.574698i	−0.957826	+	0.287349i	$0.907226\pi$
$432$		0			0
$433$	23.0000	+	23.0000i	1.10531	+	1.10531i	0.993759	+	0.111551i	$0.0355818\pi$
$433$	23.0000	+	23.0000i	1.10531	+	1.10531i	0.111551	+	0.993759i	$0.464418\pi$
$434$	3.69984			0.177598
$435$		0			0
$436$	3.30306			0.158188
$437$	2.64146	+	2.64146i	0.126358	+	0.126358i
$438$		0			0
$439$		−	0.247449i		−	0.0118101i	−0.999983	−	0.00590504i	$-0.998120\pi$
$439$		−	0.247449i		−	0.0118101i	0.999983	−	0.00590504i	$-0.00187964\pi$
$440$		0			0
$441$		0			0
$442$	22.2474	−	22.2474i	1.05820	−	1.05820i
$443$	16.6403	−	16.6403i	0.790604	−	0.790604i	−0.190989	−	0.981592i	$-0.561169\pi$
$443$	16.6403	−	16.6403i	0.790604	−	0.790604i	0.981592	+	0.190989i	$0.0611694\pi$
$444$		0			0
$445$		0			0
$446$			10.7437i			0.508730i
$447$		0			0
$448$	−2.00000	−	2.00000i	−0.0944911	−	0.0944911i
$449$	−15.8487			−0.747948			−0.373974	−	0.927439i	$-0.622005\pi$
$449$	−15.8487			−0.747948			−0.373974	+	0.927439i	$0.622005\pi$
$450$		0			0
$451$	5.34847			0.251850
$452$	−1.97885	−	1.97885i	−0.0930771	−	0.0930771i
$453$		0			0
$454$			12.9444i			0.607510i
$455$		0			0
$456$		0			0
$457$	−15.4495	+	15.4495i	−0.722697	+	0.722697i	−0.969154	−	0.246457i	$-0.920734\pi$
$457$	−15.4495	+	15.4495i	−0.722697	+	0.722697i	0.246457	+	0.969154i	$0.420734\pi$
$458$	−3.12617	+	3.12617i	−0.146077	+	0.146077i
$459$		0			0
$460$		0			0
$461$		−	8.01332i		−	0.373217i	−0.982434	−	0.186609i	$-0.940250\pi$
$461$		−	8.01332i		−	0.373217i	0.982434	−	0.186609i	$-0.0597497\pi$
$462$		0			0
$463$	−23.4722	−	23.4722i	−1.09085	−	1.09085i	−0.995438	−	0.0954070i	$-0.969585\pi$
$463$	−23.4722	−	23.4722i	−1.09085	−	1.09085i	−0.0954070	−	0.995438i	$-0.530415\pi$
$464$	21.5674			1.00124
$465$		0			0
$466$	16.2020			0.750545
$467$	−27.4240	−	27.4240i	−1.26903	−	1.26903i	−0.946589	−	0.322441i	$-0.895497\pi$
$467$	−27.4240	−	27.4240i	−1.26903	−	1.26903i	−0.322441	−	0.946589i	$-0.604503\pi$
$468$		0			0
$469$		−	1.69694i		−	0.0783573i
$470$		0			0
$471$		0			0
$472$	−30.0000	+	30.0000i	−1.38086	+	1.38086i
$473$	5.67868	−	5.67868i	0.261106	−	0.261106i
$474$		0			0
$475$		0			0
$476$			0.791539i			0.0362801i
$477$		0			0
$478$	23.4495	+	23.4495i	1.07255	+	1.07255i
$479$	38.6035			1.76384			0.881919	−	0.471402i	$-0.156252\pi$
$479$	38.6035			1.76384			0.881919	+	0.471402i	$0.156252\pi$
$480$		0			0
$481$	−10.7980			−0.492345
$482$	−11.4463	−	11.4463i	−0.521365	−	0.521365i
$483$		0			0
$484$			1.95459i			0.0888451i
$485$		0			0
$486$		0			0
$487$	−9.32577	+	9.32577i	−0.422591	+	0.422591i	−0.886095	−	0.463504i	$-0.846592\pi$
$487$	−9.32577	+	9.32577i	−0.422591	+	0.422591i	0.463504	+	0.886095i	$0.346592\pi$
$488$	−2.15674	+	2.15674i	−0.0976311	+	0.0976311i
$489$		0			0
$490$		0			0
$491$		−	22.1411i		−	0.999213i	−0.866253	−	0.499606i	$-0.833478\pi$
$491$		−	22.1411i		−	0.999213i	0.866253	−	0.499606i	$-0.166522\pi$
$492$		0			0
$493$	−29.1464	−	29.1464i	−1.31269	−	1.31269i
$494$	17.0361			0.766488
$495$		0			0
$496$	27.1010			1.21687
$497$	−2.06779	−	2.06779i	−0.0927532	−	0.0927532i
$498$		0			0
$499$		−	12.4949i		−	0.559348i	−0.960095	−	0.279674i	$-0.909774\pi$
$499$		−	12.4949i		−	0.559348i	0.960095	−	0.279674i	$-0.0902265\pi$
$500$		0			0
$501$		0			0
$502$	−7.94439	+	7.94439i	−0.354575	+	0.354575i
$503$	18.3123	−	18.3123i	0.816505	−	0.816505i	−0.169094	−	0.985600i	$-0.554084\pi$
$503$	18.3123	−	18.3123i	0.816505	−	0.816505i	0.985600	+	0.169094i	$0.0540843\pi$
$504$		0			0
$505$		0			0
$506$			6.07445i			0.270042i
$507$		0			0
$508$	3.14643	+	3.14643i	0.139600	+	0.139600i
$509$	−20.3801			−0.903332			−0.451666	−	0.892187i	$-0.649170\pi$
$509$	−20.3801			−0.903332			−0.451666	+	0.892187i	$0.649170\pi$
$510$		0			0
$511$	−2.10102			−0.0929437
$512$	−17.6097	−	17.6097i	−0.778247	−	0.778247i
$513$		0			0
$514$			10.0000i			0.441081i
$515$		0			0
$516$		0			0
$517$	13.7980	−	13.7980i	0.606834	−	0.606834i
$518$	−0.662611	+	0.662611i	−0.0291134	+	0.0291134i
$519$		0			0
$520$		0			0
$521$			24.8715i			1.08964i	0.838553	+	0.544820i	$0.183402\pi$
$521$			24.8715i			1.08964i	−0.838553	+	0.544820i	$0.816598\pi$
$522$		0			0
$523$	−19.6742	−	19.6742i	−0.860294	−	0.860294i	0.131078	−	0.991372i	$-0.458156\pi$
$523$	−19.6742	−	19.6742i	−0.860294	−	0.860294i	−0.991372	+	0.131078i	$0.958156\pi$
$524$	3.52194			0.153857
$525$		0			0
$526$	−30.6969			−1.33845
$527$	−36.6246	−	36.6246i	−1.59539	−	1.59539i
$528$		0			0
$529$			21.4495i			0.932587i
$530$		0			0
$531$		0			0
$532$	−0.303062	+	0.303062i	−0.0131394	+	0.0131394i
$533$	−4.40243	+	4.40243i	−0.190690	+	0.190690i
$534$		0			0
$535$		0			0
$536$		−	16.2845i		−	0.703383i
$537$		0			0
$538$	4.65153	+	4.65153i	0.200542	+	0.200542i
$539$	27.0282			1.16419
$540$		0			0
$541$	−17.0454			−0.732839			−0.366420	−	0.930450i	$-0.619417\pi$
$541$	−17.0454			−0.732839			−0.366420	+	0.930450i	$0.619417\pi$
$542$	17.3429	+	17.3429i	0.744940	+	0.744940i
$543$		0			0
$544$			13.7980i			0.591583i
$545$		0			0
$546$		0			0
$547$	20.9217	−	20.9217i	0.894547	−	0.894547i	−0.100401	−	0.994947i	$-0.532012\pi$
$547$	20.9217	−	20.9217i	0.894547	−	0.894547i	0.994947	+	0.100401i	$0.0320124\pi$
$548$	−4.88715	+	4.88715i	−0.208768	+	0.208768i
$549$		0			0
$550$		0			0
$551$		−	22.3190i		−	0.950820i
$552$		0			0
$553$	1.22474	+	1.22474i	0.0520814	+	0.0520814i
$554$	−3.69984			−0.157191
$555$		0			0
$556$	4.15663			0.176281
$557$	−10.3879	−	10.3879i	−0.440151	−	0.440151i	0.451912	−	0.892063i	$-0.350742\pi$
$557$	−10.3879	−	10.3879i	−0.440151	−	0.440151i	−0.892063	+	0.451912i	$0.850742\pi$
$558$		0			0
$559$			9.34847i			0.395398i
$560$		0			0
$561$		0			0
$562$	−22.9444	+	22.9444i	−0.967851	+	0.967851i
$563$	18.7970	−	18.7970i	0.792200	−	0.792200i	−0.189651	−	0.981851i	$-0.560736\pi$
$563$	18.7970	−	18.7970i	0.792200	−	0.792200i	0.981851	+	0.189651i	$0.0607358\pi$
$564$		0			0
$565$		0			0
$566$			17.4318i			0.732714i
$567$		0			0
$568$	−19.8434	−	19.8434i	−0.832609	−	0.832609i
$569$	−9.59640			−0.402302			−0.201151	−	0.979560i	$-0.564468\pi$
$569$	−9.59640			−0.402302			−0.201151	+	0.979560i	$0.564468\pi$
$570$		0			0
$571$	−5.65153			−0.236509			−0.118255	−	0.992983i	$-0.537730\pi$
$571$	−5.65153			−0.236509			−0.118255	+	0.992983i	$0.537730\pi$
$572$	−5.67868	−	5.67868i	−0.237438	−	0.237438i
$573$		0			0
$574$			0.540305i			0.0225519i
$575$		0			0
$576$		0			0
$577$	13.3258	−	13.3258i	0.554759	−	0.554759i	−0.373052	−	0.927811i	$-0.621689\pi$
$577$	13.3258	−	13.3258i	0.554759	−	0.554759i	0.927811	+	0.373052i	$0.121689\pi$
$578$	−12.0600	+	12.0600i	−0.501628	+	0.501628i
$579$		0			0
$580$		0			0
$581$			1.36520i			0.0566381i
$582$		0			0
$583$	20.3485	+	20.3485i	0.842747	+	0.842747i
$584$	−20.1622			−0.834319
$585$		0			0
$586$	−31.5505			−1.30334
$587$	12.6336	+	12.6336i	0.521445	+	0.521445i	0.918008	−	0.396562i	$-0.129797\pi$
$587$	12.6336	+	12.6336i	0.521445	+	0.521445i	−0.396562	+	0.918008i	$0.629797\pi$
$588$		0			0
$589$		−	28.0454i		−	1.15559i
$590$		0			0
$591$		0			0
$592$	−4.85357	+	4.85357i	−0.199481	+	0.199481i
$593$	6.55917	−	6.55917i	0.269353	−	0.269353i	−0.559487	−	0.828839i	$-0.689002\pi$
$593$	6.55917	−	6.55917i	0.269353	−	0.269353i	0.828839	+	0.559487i	$0.189002\pi$
$594$		0			0
$595$		0			0
$596$		−	4.31348i		−	0.176687i
$597$		0			0
$598$	−5.00000	−	5.00000i	−0.204465	−	0.204465i
$599$	−12.7226			−0.519830			−0.259915	−	0.965631i	$-0.583695\pi$
$599$	−12.7226			−0.519830			−0.259915	+	0.965631i	$0.583695\pi$
$600$		0			0
$601$	−20.6515			−0.842394			−0.421197	−	0.906969i	$-0.638390\pi$
$601$	−20.6515			−0.842394			−0.421197	+	0.906969i	$0.638390\pi$
$602$	0.573664	+	0.573664i	0.0233808	+	0.0233808i
$603$		0			0
$604$			2.54031i			0.103364i
$605$		0			0
$606$		0			0
$607$	6.77526	−	6.77526i	0.274999	−	0.274999i	−0.556110	−	0.831109i	$-0.687707\pi$
$607$	6.77526	−	6.77526i	0.274999	−	0.274999i	0.831109	+	0.556110i	$0.187707\pi$
$608$	−5.28291	+	5.28291i	−0.214250	+	0.214250i
$609$		0			0
$610$		0			0
$611$			22.7147i			0.918940i
$612$		0			0
$613$	−10.3712	−	10.3712i	−0.418888	−	0.418888i	0.465933	−	0.884820i	$-0.345719\pi$
$613$	−10.3712	−	10.3712i	−0.418888	−	0.418888i	−0.884820	+	0.465933i	$0.845719\pi$
$614$	37.5940			1.51717
$615$		0			0
$616$	−3.79796			−0.153024
$617$	25.3562	+	25.3562i	1.02080	+	1.02080i	0.999779	+	0.0210229i	$0.00669228\pi$
$617$	25.3562	+	25.3562i	1.02080	+	1.02080i	0.0210229	+	0.999779i	$0.493308\pi$
$618$		0			0
$619$			5.44949i			0.219033i	0.993985	+	0.109517i	$0.0349303\pi$
$619$			5.44949i			0.219033i	−0.993985	+	0.109517i	$0.965070\pi$
$620$		0			0
$621$		0			0
$622$	−22.9444	+	22.9444i	−0.919986	+	0.919986i
$623$	−1.18731	+	1.18731i	−0.0475685	+	0.0475685i
$624$		0			0
$625$		0			0
$626$			9.02273i			0.360621i
$627$		0			0
$628$	−2.00000	−	2.00000i	−0.0798087	−	0.0798087i
$629$	13.1183			0.523062
$630$		0			0
$631$	−13.0000			−0.517522			−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000			−0.517522			−0.258761	+	0.965941i	$0.583314\pi$
$632$	11.7531	+	11.7531i	0.467515	+	0.467515i
$633$		0			0
$634$			8.29286i			0.329351i
$635$		0			0
$636$		0			0
$637$	−22.2474	+	22.2474i	−0.881476	+	0.881476i
$638$	25.6630	−	25.6630i	1.01601	−	1.01601i
$639$		0			0
$640$		0			0
$641$		−	28.6113i		−	1.13008i	−0.825064	−	0.565039i	$-0.808861\pi$
$641$		−	28.6113i		−	1.13008i	0.825064	−	0.565039i	$-0.191139\pi$
$642$		0			0
$643$	−9.44949	−	9.44949i	−0.372651	−	0.372651i	0.495791	−	0.868442i	$-0.334878\pi$
$643$	−9.44949	−	9.44949i	−0.372651	−	0.372651i	−0.868442	+	0.495791i	$0.834878\pi$
$644$	0.177894			0.00701002
$645$		0			0
$646$	−20.6969			−0.814310
$647$	15.7598	+	15.7598i	0.619582	+	0.619582i	0.945424	−	0.325843i	$-0.105648\pi$
$647$	15.7598	+	15.7598i	0.619582	+	0.619582i	−0.325843	+	0.945424i	$0.605648\pi$
$648$		0			0
$649$			54.4949i			2.13911i
$650$		0			0
$651$		0			0
$652$	−1.49490	+	1.49490i	−0.0585447	+	0.0585447i
$653$	−28.2155	+	28.2155i	−1.10416	+	1.10416i	−0.110256	+	0.993903i	$0.535167\pi$
$653$	−28.2155	+	28.2155i	−1.10416	+	1.10416i	−0.993903	+	0.110256i	$0.964833\pi$
$654$		0			0
$655$		0			0
$656$			3.95769i			0.154522i
$657$		0			0
$658$	1.39388	+	1.39388i	0.0543390	+	0.0543390i
$659$	−17.2539			−0.672118			−0.336059	−	0.941841i	$-0.609094\pi$
$659$	−17.2539			−0.672118			−0.336059	+	0.941841i	$0.609094\pi$
$660$		0			0
$661$	−42.3939			−1.64893			−0.824465	−	0.565912i	$-0.808524\pi$
$661$	−42.3939			−1.64893			−0.824465	+	0.565912i	$0.808524\pi$
$662$	21.4385	+	21.4385i	0.833230	+	0.833230i
$663$		0			0
$664$			13.1010i			0.508418i
$665$		0			0
$666$		0			0
$667$	−6.55051	+	6.55051i	−0.253637	+	0.253637i
$668$	2.55251	−	2.55251i	0.0987596	−	0.0987596i
$669$		0			0
$670$		0			0
$671$			3.91771i			0.151242i
$672$		0			0
$673$	−21.5732	−	21.5732i	−0.831586	−	0.831586i	0.156147	−	0.987734i	$-0.450092\pi$
$673$	−21.5732	−	21.5732i	−0.831586	−	0.831586i	−0.987734	+	0.156147i	$0.950092\pi$
$674$	−29.7986			−1.14780
$675$		0			0
$676$	3.50510			0.134812
$677$	3.52194	+	3.52194i	0.135359	+	0.135359i	0.771540	−	0.636181i	$-0.219487\pi$
$677$	3.52194	+	3.52194i	0.135359	+	0.135359i	−0.636181	+	0.771540i	$0.719487\pi$
$678$		0			0
$679$			4.55051i			0.174633i
$680$		0			0
$681$		0			0
$682$	32.2474	−	32.2474i	1.23482	−	1.23482i
$683$	−27.5129	+	27.5129i	−1.05275	+	1.05275i	−0.0542243	+	0.998529i	$0.517269\pi$
$683$	−27.5129	+	27.5129i	−1.05275	+	1.05275i	−0.998529	+	0.0542243i	$0.982731\pi$
$684$		0			0
$685$		0			0
$686$			5.50079i			0.210021i
$687$		0			0
$688$	4.20204	+	4.20204i	0.160201	+	0.160201i
$689$	−33.4984			−1.27619
$690$		0			0
$691$	18.0454			0.686480			0.343240	−	0.939248i	$-0.388476\pi$
$691$	18.0454			0.686480			0.343240	+	0.939248i	$0.388476\pi$
$692$	−0.355788	−	0.355788i	−0.0135250	−	0.0135250i
$693$		0			0
$694$		−	15.3485i		−	0.582620i
$695$		0			0
$696$		0			0
$697$	5.34847	−	5.34847i	0.202588	−	0.202588i
$698$	−0.702591	+	0.702591i	−0.0265935	+	0.0265935i
$699$		0			0
$700$		0			0
$701$			1.36520i			0.0515630i	0.999668	+	0.0257815i	$0.00820741\pi$
$701$			1.36520i			0.0515630i	−0.999668	+	0.0257815i	$0.991793\pi$
$702$		0			0
$703$	5.02270	+	5.02270i	0.189435	+	0.189435i
$704$	−34.8636			−1.31397
$705$		0			0
$706$	31.3939			1.18152
$707$	2.94828	+	2.94828i	0.110881	+	0.110881i
$708$		0			0
$709$		−	35.9444i		−	1.34992i	−0.737854	−	0.674960i	$-0.764161\pi$
$709$		−	35.9444i		−	1.34992i	0.737854	−	0.674960i	$-0.235839\pi$
$710$		0			0
$711$		0			0
$712$	−11.3939	+	11.3939i	−0.427004	+	0.427004i
$713$	−8.23119	+	8.23119i	−0.308261	+	0.308261i
$714$		0			0
$715$		0			0
$716$		−	2.90830i		−	0.108688i
$717$		0			0
$718$	−26.3939	−	26.3939i	−0.985011	−	0.985011i
$719$	−24.6936			−0.920915			−0.460458	−	0.887682i	$-0.652315\pi$
$719$	−24.6936			−0.920915			−0.460458	+	0.887682i	$0.652315\pi$
$720$		0			0
$721$	3.40408			0.126775
$722$	8.80486	+	8.80486i	0.327683	+	0.327683i
$723$		0			0
$724$			3.75255i			0.139462i
$725$		0			0
$726$		0			0
$727$	−22.7980	+	22.7980i	−0.845529	+	0.845529i	−0.989572	−	0.144042i	$-0.953990\pi$
$727$	−22.7980	+	22.7980i	−0.845529	+	0.845529i	0.144042	+	0.989572i	$0.453990\pi$
$728$	3.12617	−	3.12617i	0.115864	−	0.115864i
$729$		0			0
$730$		0			0
$731$		−	11.3574i		−	0.420067i
$732$		0			0
$733$	−9.44949	−	9.44949i	−0.349025	−	0.349025i	0.510721	−	0.859746i	$-0.329378\pi$
$733$	−9.44949	−	9.44949i	−0.349025	−	0.349025i	−0.859746	+	0.510721i	$0.829378\pi$
$734$	31.5596			1.16489
$735$		0			0
$736$	3.10102			0.114305
$737$	−14.7904	−	14.7904i	−0.544810	−	0.544810i
$738$		0			0
$739$		−	49.3485i		−	1.81531i	−0.419714	−	0.907656i	$-0.637870\pi$
$739$		−	49.3485i		−	1.81531i	0.419714	−	0.907656i	$-0.362130\pi$
$740$		0			0
$741$		0			0
$742$	−2.05561	+	2.05561i	−0.0754639	+	0.0754639i
$743$	−20.5580	+	20.5580i	−0.754200	+	0.754200i	−0.975260	−	0.221060i	$-0.929048\pi$
$743$	−20.5580	+	20.5580i	−0.754200	+	0.754200i	0.221060	+	0.975260i	$0.429048\pi$
$744$		0			0
$745$		0			0
$746$		−	41.3339i		−	1.51334i
$747$		0			0
$748$	6.89898	+	6.89898i	0.252252	+	0.252252i
$749$	−0.573664			−0.0209612
$750$		0			0
$751$	8.30306			0.302983			0.151492	−	0.988459i	$-0.451592\pi$
$751$	8.30306			0.302983			0.151492	+	0.988459i	$0.451592\pi$
$752$	10.2100	+	10.2100i	0.372322	+	0.372322i
$753$		0			0
$754$			42.2474i			1.53856i
$755$		0			0
$756$		0			0
$757$	23.6742	−	23.6742i	0.860455	−	0.860455i	−0.130936	−	0.991391i	$-0.541798\pi$
$757$	23.6742	−	23.6742i	0.860455	−	0.860455i	0.991391	+	0.130936i	$0.0417983\pi$
$758$	26.1477	−	26.1477i	0.949728	−	0.949728i
$759$		0			0
$760$		0			0
$761$		−	9.20063i		−	0.333522i	−0.985997	−	0.166761i	$-0.946669\pi$
$761$		−	9.20063i		−	0.333522i	0.985997	−	0.166761i	$-0.0533309\pi$
$762$		0			0
$763$	1.65153	+	1.65153i	0.0597894	+	0.0597894i
$764$	−4.66927			−0.168928
$765$		0			0
$766$	3.10102			0.112044
$767$	−44.8558	−	44.8558i	−1.61965	−	1.61965i
$768$		0			0
$769$			44.6413i			1.60981i	0.593405	+	0.804904i	$0.297783\pi$
$769$			44.6413i			1.60981i	−0.593405	+	0.804904i	$0.702217\pi$
$770$		0			0
$771$		0			0
$772$	−5.89898	+	5.89898i	−0.212309	+	0.212309i
$773$	−27.5129	+	27.5129i	−0.989571	+	0.989571i	−0.999946	−	0.0103748i	$-0.996698\pi$
$773$	−27.5129	+	27.5129i	−0.989571	+	0.989571i	0.0103748	+	0.999946i	$0.496698\pi$
$774$		0			0
$775$		0			0
$776$			43.6685i			1.56761i
$777$		0			0
$778$	5.50510	+	5.50510i	0.197367	+	0.197367i
$779$	4.09561			0.146740
$780$		0			0
$781$	−36.0454			−1.28981
$782$	6.07445	+	6.07445i	0.217222	+	0.217222i
$783$		0			0
$784$			20.0000i			0.714286i
$785$		0			0
$786$		0			0
$787$	−2.52781	+	2.52781i	−0.0901066	+	0.0901066i	−0.750723	−	0.660617i	$-0.770295\pi$
$787$	−2.52781	+	2.52781i	−0.0901066	+	0.0901066i	0.660617	+	0.750723i	$0.270295\pi$
$788$	1.14733	−	1.14733i	0.0408718	−	0.0408718i
$789$		0			0
$790$		0			0
$791$		−	1.97885i		−	0.0703597i
$792$		0			0
$793$	−3.22474	−	3.22474i	−0.114514	−	0.114514i
$794$	11.1795			0.396745
$795$		0			0
$796$	−1.95459			−0.0692787
$797$	−10.1701	−	10.1701i	−0.360242	−	0.360242i	0.503660	−	0.863902i	$-0.331986\pi$
$797$	−10.1701	−	10.1701i	−0.360242	−	0.360242i	−0.863902	+	0.503660i	$0.831986\pi$
$798$		0			0
$799$		−	27.5959i		−	0.976273i
$800$		0			0
$801$		0			0
$802$	−12.5959	+	12.5959i	−0.444777	+	0.444777i
$803$	−18.3123	+	18.3123i	−0.646227	+	0.646227i
$804$		0			0
$805$		0			0
$806$			53.0870i			1.86991i
$807$		0			0
$808$	28.2929	+	28.2929i	0.995339	+	0.995339i
$809$	26.4146			0.928687			0.464343	−	0.885655i	$-0.346290\pi$
$809$	26.4146			0.928687			0.464343	+	0.885655i	$0.346290\pi$
$810$		0			0
$811$	−32.0454			−1.12527			−0.562633	−	0.826707i	$-0.690212\pi$
$811$	−32.0454			−1.12527			−0.562633	+	0.826707i	$0.690212\pi$
$812$	−0.751558	−	0.751558i	−0.0263745	−	0.0263745i
$813$		0			0
$814$			11.5505i			0.404845i
$815$		0			0
$816$		0			0
$817$	4.34847	−	4.34847i	0.152134	−	0.152134i
$818$	19.4596	−	19.4596i	0.680390	−	0.680390i
$819$		0			0
$820$		0			0
$821$			2.55251i			0.0890832i	0.999008	+	0.0445416i	$0.0141827\pi$
$821$			2.55251i			0.0890832i	−0.999008	+	0.0445416i	$0.985817\pi$
$822$		0			0
$823$	23.9217	+	23.9217i	0.833857	+	0.833857i	0.988042	−	0.154185i	$-0.0492750\pi$
$823$	23.9217	+	23.9217i	0.833857	+	0.833857i	−0.154185	+	0.988042i	$0.549275\pi$
$824$	32.6669			1.13801
$825$		0			0
$826$	−5.50510			−0.191547
$827$	23.6842	+	23.6842i	0.823579	+	0.823579i	0.986619	−	0.163040i	$-0.0521300\pi$
$827$	23.6842	+	23.6842i	0.823579	+	0.823579i	−0.163040	+	0.986619i	$0.552130\pi$
$828$		0			0
$829$		−	39.7423i		−	1.38031i	−0.723662	−	0.690154i	$-0.757543\pi$
$829$		−	39.7423i		−	1.38031i	0.723662	−	0.690154i	$-0.242457\pi$
$830$		0			0
$831$		0			0
$832$	28.6969	−	28.6969i	0.994887	−	0.994887i
$833$	27.0282	−	27.0282i	0.936472	−	0.936472i
$834$		0			0
$835$		0			0
$836$			5.28291i			0.182713i
$837$		0			0
$838$	3.79796	+	3.79796i	0.131198	+	0.131198i
$839$	−27.8198			−0.960445			−0.480222	−	0.877147i	$-0.659444\pi$
$839$	−27.8198			−0.960445			−0.480222	+	0.877147i	$0.659444\pi$
$840$		0			0
$841$	26.3485			0.908568
$842$	−33.2316	−	33.2316i	−1.14524	−	1.14524i
$843$		0			0
$844$		−	5.55051i		−	0.191056i
$845$		0			0
$846$		0			0
$847$	−0.977296	+	0.977296i	−0.0335803	+	0.0335803i
$848$	−15.0572	+	15.0572i	−0.517067	+	0.517067i
$849$		0			0
$850$		0			0
$851$		−	2.94828i		−	0.101066i
$852$		0			0
$853$	4.07832	+	4.07832i	0.139639	+	0.139639i	0.773471	−	0.633832i	$-0.218519\pi$
$853$	4.07832	+	4.07832i	0.139639	+	0.139639i	−0.633832	+	0.773471i	$0.718519\pi$
$854$	−0.395769			−0.0135429
$855$		0			0
$856$	−5.50510			−0.188161
$857$	−21.4385	−	21.4385i	−0.732325	−	0.732325i	0.238755	−	0.971080i	$-0.423261\pi$
$857$	−21.4385	−	21.4385i	−0.732325	−	0.732325i	−0.971080	+	0.238755i	$0.923261\pi$
$858$		0			0
$859$			18.5505i			0.632935i	0.948603	+	0.316468i	$0.102497\pi$
$859$			18.5505i			0.632935i	−0.948603	+	0.316468i	$0.897503\pi$
$860$		0			0
$861$		0			0
$862$	−10.5051	+	10.5051i	−0.357805	+	0.357805i
$863$	−19.5886	+	19.5886i	−0.666802	+	0.666802i	−0.956974	−	0.290172i	$-0.906287\pi$
$863$	−19.5886	+	19.5886i	−0.666802	+	0.666802i	0.290172	+	0.956974i	$0.406287\pi$
$864$		0			0
$865$		0			0
$866$			40.5023i			1.37633i
$867$		0			0
$868$	−0.944387	−	0.944387i	−0.0320546	−	0.0320546i
$869$	21.3495			0.724233
$870$		0			0
$871$	24.3485			0.825016
$872$	15.8487	+	15.8487i	0.536706	+	0.536706i
$873$		0			0
$874$			4.65153i			0.157340i
$875$		0			0
$876$		0			0
$877$	−8.22474	+	8.22474i	−0.277730	+	0.277730i	−0.832202	−	0.554472i	$-0.812920\pi$
$877$	−8.22474	+	8.22474i	−0.277730	+	0.277730i	0.554472	+	0.832202i	$0.312920\pi$
$878$	0.217875	−	0.217875i	0.00735293	−	0.00735293i
$879$		0			0
$880$		0			0
$881$		−	16.8582i		−	0.567966i	−0.958829	−	0.283983i	$-0.908344\pi$
$881$		−	16.8582i		−	0.567966i	0.958829	−	0.283983i	$-0.0916559\pi$
$882$		0			0
$883$	−13.9773	−	13.9773i	−0.470373	−	0.470373i	0.431662	−	0.902035i	$-0.357927\pi$
$883$	−13.9773	−	13.9773i	−0.470373	−	0.470373i	−0.902035	+	0.431662i	$0.857927\pi$
$884$	−11.3574			−0.381990
$885$		0			0
$886$	29.3031			0.984456
$887$	33.2806	+	33.2806i	1.11745	+	1.11745i	0.992114	+	0.125338i	$0.0400015\pi$
$887$	33.2806	+	33.2806i	1.11745	+	1.11745i	0.125338	+	0.992114i	$0.459999\pi$
$888$		0			0
$889$			3.14643i			0.105528i
$890$		0			0
$891$		0			0
$892$	2.74235	−	2.74235i	0.0918205	−	0.0918205i
$893$	10.5658	−	10.5658i	0.353572	−	0.353572i
$894$		0			0
$895$		0			0
$896$		−	1.93887i		−	0.0647730i
$897$		0			0
$898$	−13.9546	−	13.9546i	−0.465671	−	0.465671i
$899$	69.5494			2.31960
$900$		0			0
$901$	40.6969			1.35581
$902$	4.70925	+	4.70925i	0.156801	+	0.156801i
$903$		0			0
$904$		−	18.9898i		−	0.631591i
$905$		0			0
$906$		0			0
$907$	19.8763	−	19.8763i	0.659981	−	0.659981i	−0.295394	−	0.955375i	$-0.595451\pi$
$907$	19.8763	−	19.8763i	0.659981	−	0.659981i	0.955375	+	0.295394i	$0.0954511\pi$
$908$	3.30407	−	3.30407i	0.109649	−	0.109649i
$909$		0			0
$910$		0			0
$911$			38.9992i			1.29210i	0.763294	+	0.646051i	$0.223581\pi$
$911$			38.9992i			1.29210i	−0.763294	+	0.646051i	$0.776419\pi$
$912$		0			0
$913$	11.8990	+	11.8990i	0.393799	+	0.393799i
$914$	−27.2061			−0.899898
$915$		0			0
$916$	1.59592			0.0527306
$917$	1.76097	+	1.76097i	0.0581524	+	0.0581524i
$918$		0			0
$919$		−	13.3485i		−	0.440325i	−0.975463	−	0.220163i	$-0.929341\pi$
$919$		−	13.3485i		−	0.440325i	0.975463	−	0.220163i	$-0.0706588\pi$
$920$		0			0
$921$		0			0
$922$	7.05561	−	7.05561i	0.232364	−	0.232364i
$923$	29.6697	−	29.6697i	0.976589	−	0.976589i
$924$		0			0
$925$		0			0
$926$		−	41.3339i		−	1.35832i
$927$		0			0
$928$	−13.1010	−	13.1010i	−0.430062	−	0.430062i
$929$	15.0972			0.495323			0.247661	−	0.968847i	$-0.420338\pi$
$929$	15.0972			0.495323			0.247661	+	0.968847i	$0.420338\pi$
$930$		0			0
$931$	20.6969			0.678315
$932$	−4.13559	−	4.13559i	−0.135466	−	0.135466i
$933$		0			0
$934$		−	48.2929i		−	1.58019i
$935$		0			0
$936$		0			0
$937$	−32.7196	+	32.7196i	−1.06890	+	1.06890i	−0.0714607	+	0.997443i	$0.522766\pi$
$937$	−32.7196	+	32.7196i	−1.06890	+	1.06890i	−0.997443	+	0.0714607i	$0.977234\pi$
$938$	1.49413	−	1.49413i	0.0487851	−	0.0487851i
$939$		0			0
$940$		0			0
$941$		−	22.1411i		−	0.721778i	−0.932609	−	0.360889i	$-0.882473\pi$
$941$		−	22.1411i		−	0.721778i	0.932609	−	0.360889i	$-0.117527\pi$
$942$		0			0
$943$	−1.20204	−	1.20204i	−0.0391438	−	0.0391438i
$944$	−40.3244			−1.31245
$945$		0			0
$946$	10.0000			0.325128
$947$	4.70925	+	4.70925i	0.153030	+	0.153030i	0.779470	−	0.626440i	$-0.215489\pi$
$947$	4.70925	+	4.70925i	0.153030	+	0.153030i	−0.626440	+	0.779470i	$0.715489\pi$
$948$		0			0
$949$		−	30.1464i		−	0.978594i
$950$		0			0
$951$		0			0
$952$	−3.79796	+	3.79796i	−0.123093	+	0.123093i
$953$	10.3879	−	10.3879i	0.336498	−	0.336498i	−0.518549	−	0.855048i	$-0.673528\pi$
$953$	10.3879	−	10.3879i	0.336498	−	0.336498i	0.855048	+	0.518549i	$0.173528\pi$
$954$		0			0
$955$		0			0
$956$		−	11.9710i		−	0.387170i
$957$		0			0
$958$	33.9898	+	33.9898i	1.09816	+	1.09816i
$959$	−4.88715			−0.157814
$960$		0			0
$961$	56.3939			1.81916
$962$	−9.50745	−	9.50745i	−0.306532	−	0.306532i
$963$		0			0
$964$			5.84337i			0.188202i
$965$		0			0
$966$		0			0
$967$	42.4722	−	42.4722i	1.36581	−	1.36581i	0.499499	−	0.866314i	$-0.333517\pi$
$967$	42.4722	−	42.4722i	1.36581	−	1.36581i	0.866314	−	0.499499i	$-0.166483\pi$
$968$	−9.37852	+	9.37852i	−0.301437	+	0.301437i
$969$		0			0
$970$		0			0
$971$		−	47.3683i		−	1.52012i	−0.649852	−	0.760061i	$-0.725169\pi$
$971$		−	47.3683i		−	1.52012i	0.649852	−	0.760061i	$-0.274831\pi$
$972$		0			0
$973$	2.07832	+	2.07832i	0.0666278	+	0.0666278i
$974$	−16.4224			−0.526208
$975$		0			0
$976$	−2.89898			−0.0927941
$977$	−25.7520	−	25.7520i	−0.823878	−	0.823878i	0.162784	−	0.986662i	$-0.447953\pi$
$977$	−25.7520	−	25.7520i	−0.823878	−	0.823878i	−0.986662	+	0.162784i	$0.947953\pi$
$978$		0			0
$979$			20.6969i			0.661477i
$980$		0			0
$981$		0			0
$982$	19.4949	−	19.4949i	0.622107	−	0.622107i
$983$	4.62030	−	4.62030i	0.147365	−	0.147365i	−0.629575	−	0.776940i	$-0.716771\pi$
$983$	4.62030	−	4.62030i	0.147365	−	0.147365i	0.776940	+	0.629575i	$0.216771\pi$
$984$		0			0
$985$		0			0
$986$		−	51.3260i		−	1.63455i
$987$		0			0
$988$	−4.34847	−	4.34847i	−0.138343	−	0.138343i
$989$	−2.55251			−0.0811651
$990$		0			0
$991$	−14.9546			−0.475048			−0.237524	−	0.971382i	$-0.576336\pi$
$991$	−14.9546			−0.475048			−0.237524	+	0.971382i	$0.576336\pi$
$992$	−16.4624	−	16.4624i	−0.522681	−	0.522681i
$993$		0			0
$994$		−	3.64133i		−	0.115496i
$995$		0			0
$996$		0			0
$997$	9.89898	−	9.89898i	0.313504	−	0.313504i	−0.532762	−	0.846265i	$-0.678846\pi$
$997$	9.89898	−	9.89898i	0.313504	−	0.313504i	0.846265	+	0.532762i	$0.178846\pi$
$998$	11.0016	−	11.0016i	0.348249	−	0.348249i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.f.g.593.3		8
3.2	odd	2	inner	675.2.f.g.593.2		8
5.2	odd	4	inner	675.2.f.g.107.2		8
5.3	odd	4		135.2.f.b.107.3	yes	8
5.4	even	2		135.2.f.b.53.2	✓	8
15.2	even	4	inner	675.2.f.g.107.3		8
15.8	even	4		135.2.f.b.107.2	yes	8
15.14	odd	2		135.2.f.b.53.3	yes	8
20.3	even	4		2160.2.w.a.1457.3		8
20.19	odd	2		2160.2.w.a.593.2		8
45.4	even	6		405.2.m.b.188.3		16
45.13	odd	12		405.2.m.b.107.3		16
45.14	odd	6		405.2.m.b.188.2		16
45.23	even	12		405.2.m.b.107.2		16
45.29	odd	6		405.2.m.b.53.3		16
45.34	even	6		405.2.m.b.53.2		16
45.38	even	12		405.2.m.b.377.3		16
45.43	odd	12		405.2.m.b.377.2		16
60.23	odd	4		2160.2.w.a.1457.2		8
60.59	even	2		2160.2.w.a.593.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.f.b.53.2	✓	8	5.4	even	2
135.2.f.b.53.3	yes	8	15.14	odd	2
135.2.f.b.107.2	yes	8	15.8	even	4
135.2.f.b.107.3	yes	8	5.3	odd	4
405.2.m.b.53.2		16	45.34	even	6
405.2.m.b.53.3		16	45.29	odd	6
405.2.m.b.107.2		16	45.23	even	12
405.2.m.b.107.3		16	45.13	odd	12
405.2.m.b.188.2		16	45.14	odd	6
405.2.m.b.188.3		16	45.4	even	6
405.2.m.b.377.2		16	45.43	odd	12
405.2.m.b.377.3		16	45.38	even	12
675.2.f.g.107.2		8	5.2	odd	4	inner
675.2.f.g.107.3		8	15.2	even	4	inner
675.2.f.g.593.2		8	3.2	odd	2	inner
675.2.f.g.593.3		8	1.1	even	1	trivial
2160.2.w.a.593.2		8	20.19	odd	2
2160.2.w.a.593.3		8	60.59	even	2
2160.2.w.a.1457.2		8	60.23	odd	4
2160.2.w.a.1457.3		8	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.880486 + 0.880486i) q^{2} -0.449490i q^{4} +(0.224745 - 0.224745i) q^{7} +(2.15674 - 2.15674i) q^{8} -3.91771i q^{11} +(3.22474 + 3.22474i) q^{13} +0.395769 q^{14} +2.89898 q^{16} +(-3.91771 - 3.91771i) q^{17} -3.00000i q^{19} +(3.44949 - 3.44949i) q^{22} +(-0.880486 + 0.880486i) q^{23} +5.67868i q^{26} +(-0.101021 - 0.101021i) q^{28} +7.43966 q^{29} +9.34847 q^{31} +(-1.76097 - 1.76097i) q^{32} -6.89898i q^{34} +(-1.67423 + 1.67423i) q^{37} +(2.64146 - 2.64146i) q^{38} +1.36520i q^{41} +(1.44949 + 1.44949i) q^{43} -1.76097 q^{44} -1.55051 q^{46} +(3.52194 + 3.52194i) q^{47} +6.89898i q^{49} +(1.44949 - 1.44949i) q^{52} +(-5.19397 + 5.19397i) q^{53} -0.969433i q^{56} +(6.55051 + 6.55051i) q^{58} -13.9099 q^{59} -1.00000 q^{61} +(8.23119 + 8.23119i) q^{62} -8.89898i q^{64} +(3.77526 - 3.77526i) q^{67} +(-1.76097 + 1.76097i) q^{68} -9.20063i q^{71} +(-4.67423 - 4.67423i) q^{73} -2.94828 q^{74} -1.34847 q^{76} +(-0.880486 - 0.880486i) q^{77} +5.44949i q^{79} +(-1.20204 + 1.20204i) q^{82} +(-3.03723 + 3.03723i) q^{83} +2.55251i q^{86} +(-8.44949 - 8.44949i) q^{88} -5.28291 q^{89} +1.44949 q^{91} +(0.395769 + 0.395769i) q^{92} +6.20204i q^{94} +(-10.1237 + 10.1237i) q^{97} +(-6.07445 + 6.07445i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} + 16 q^{13} - 16 q^{16} + 8 q^{22} - 40 q^{28} + 16 q^{31} + 16 q^{37} - 8 q^{43} - 32 q^{46} - 8 q^{52} + 72 q^{58} - 8 q^{61} + 40 q^{67} - 8 q^{73} + 48 q^{76} - 88 q^{82} - 48 q^{88} - 8 q^{91}+ \cdots - 32 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 16 * q^13 - 16 * q^16 + 8 * q^22 - 40 * q^28 + 16 * q^31 + 16 * q^37 - 8 * q^43 - 32 * q^46 - 8 * q^52 + 72 * q^58 - 8 * q^61 + 40 * q^67 - 8 * q^73 + 48 * q^76 - 88 * q^82 - 48 * q^88 - 8 * q^91 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more