LMFDB - Embedded newform 1350.2.e.j.451.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(451,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1350.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.e (of order $3$, 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:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1350.451
Dual form			1350.2.e.j.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.224745 - 0.389270i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.224745 - 0.389270i) q^{7} +1.00000 q^{8} +(-1.72474 + 2.98735i) q^{11} +(-1.22474 - 2.12132i) q^{13} +(0.224745 + 0.389270i) q^{14} +(-0.500000 + 0.866025i) q^{16} +5.89898 q^{17} -5.44949 q^{19} +(-1.72474 - 2.98735i) q^{22} +(3.44949 + 5.97469i) q^{23} +2.44949 q^{26} -0.449490 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-0.775255 - 1.34278i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-2.94949 + 5.10867i) q^{34} +8.00000 q^{37} +(2.72474 - 4.71940i) q^{38} +(-0.500000 - 0.866025i) q^{41} +(-1.27526 + 2.20881i) q^{43} +3.44949 q^{44} -6.89898 q^{46} +(-2.22474 + 3.85337i) q^{47} +(3.39898 + 5.88721i) q^{49} +(-1.22474 + 2.12132i) q^{52} -3.55051 q^{53} +(0.224745 - 0.389270i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(6.62372 + 11.4726i) q^{59} +(-2.22474 + 3.85337i) q^{61} +1.55051 q^{62} +1.00000 q^{64} +(-2.27526 - 3.94086i) q^{67} +(-2.94949 - 5.10867i) q^{68} +2.44949 q^{71} +14.7980 q^{73} +(-4.00000 + 6.92820i) q^{74} +(2.72474 + 4.71940i) q^{76} +(0.775255 + 1.34278i) q^{77} +(-3.67423 + 6.36396i) q^{79} +1.00000 q^{82} +(-2.00000 + 3.46410i) q^{83} +(-1.27526 - 2.20881i) q^{86} +(-1.72474 + 2.98735i) q^{88} -3.10102 q^{89} -1.10102 q^{91} +(3.44949 - 5.97469i) q^{92} +(-2.22474 - 3.85337i) q^{94} +(-6.50000 + 11.2583i) q^{97} -6.79796 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} - 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^7 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} - 4 q^{7} + 4 q^{8} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{17} - 12 q^{19} - 2 q^{22} + 4 q^{23} + 8 q^{28} - 12 q^{29} - 8 q^{31} - 2 q^{32} - 2 q^{34} + 32 q^{37} + 6 q^{38} - 2 q^{41} - 10 q^{43} + 4 q^{44} - 8 q^{46} - 4 q^{47} - 6 q^{49} - 24 q^{53} - 4 q^{56} - 12 q^{58} + 2 q^{59} - 4 q^{61} + 16 q^{62} + 4 q^{64} - 14 q^{67} - 2 q^{68} + 20 q^{73} - 16 q^{74} + 6 q^{76} + 8 q^{77} + 4 q^{82} - 8 q^{83} - 10 q^{86} - 2 q^{88} - 32 q^{89} - 24 q^{91} + 4 q^{92} - 4 q^{94} - 26 q^{97} + 12 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^7 + 4 * q^8 - 2 * q^11 - 4 * q^14 - 2 * q^16 + 4 * q^17 - 12 * q^19 - 2 * q^22 + 4 * q^23 + 8 * q^28 - 12 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^34 + 32 * q^37 + 6 * q^38 - 2 * q^41 - 10 * q^43 + 4 * q^44 - 8 * q^46 - 4 * q^47 - 6 * q^49 - 24 * q^53 - 4 * q^56 - 12 * q^58 + 2 * q^59 - 4 * q^61 + 16 * q^62 + 4 * q^64 - 14 * q^67 - 2 * q^68 + 20 * q^73 - 16 * q^74 + 6 * q^76 + 8 * q^77 + 4 * q^82 - 8 * q^83 - 10 * q^86 - 2 * q^88 - 32 * q^89 - 24 * q^91 + 4 * q^92 - 4 * q^94 - 26 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.224745	−	0.389270i	0.0849456	−	0.147130i	−0.820422	−	0.571758i	$-0.806262\pi$
$7$	0.224745	−	0.389270i	0.0849456	−	0.147130i	0.905368	+	0.424628i	$0.139595\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$		0			0
$11$	−1.72474	+	2.98735i	−0.520030	+	0.900719i	0.479699	+	0.877433i	$0.340746\pi$
$11$	−1.72474	+	2.98735i	−0.520030	+	0.900719i	−0.999729	+	0.0232854i	$0.992587\pi$
$12$		0			0
$13$	−1.22474	−	2.12132i	−0.339683	−	0.588348i	0.644690	−	0.764444i	$-0.276986\pi$
$13$	−1.22474	−	2.12132i	−0.339683	−	0.588348i	−0.984373	+	0.176096i	$0.943653\pi$
$14$	0.224745	+	0.389270i	0.0600656	+	0.104037i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	5.89898			1.43071			0.715356	−	0.698760i	$-0.246264\pi$
$17$	5.89898			1.43071			0.715356	+	0.698760i	$0.246264\pi$
$18$		0			0
$19$	−5.44949			−1.25020			−0.625099	−	0.780545i	$-0.714942\pi$
$19$	−5.44949			−1.25020			−0.625099	+	0.780545i	$0.714942\pi$
$20$		0			0
$21$		0			0
$22$	−1.72474	−	2.98735i	−0.367717	−	0.636904i
$23$	3.44949	+	5.97469i	0.719268	+	1.24581i	0.961290	+	0.275538i	$0.0888561\pi$
$23$	3.44949	+	5.97469i	0.719268	+	1.24581i	−0.242022	+	0.970271i	$0.577811\pi$
$24$		0			0
$25$		0			0
$26$	2.44949			0.480384
$27$		0			0
$28$	−0.449490			−0.0849456
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	−0.775255	−	1.34278i	−0.139240	−	0.241171i	0.787969	−	0.615715i	$-0.211133\pi$
$31$	−0.775255	−	1.34278i	−0.139240	−	0.241171i	−0.927209	+	0.374544i	$0.877799\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−2.94949	+	5.10867i	−0.505833	+	0.876129i
$35$		0			0
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$	2.72474	−	4.71940i	0.442012	−	0.765587i
$39$		0			0
$40$		0			0
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	0.824338	−	0.566099i	$-0.191548\pi$
$41$	−0.500000	−	0.866025i	−0.0780869	−	0.135250i	−0.902424	+	0.430848i	$0.858214\pi$
$42$		0			0
$43$	−1.27526	+	2.20881i	−0.194475	+	0.336840i	−0.946728	−	0.322034i	$-0.895634\pi$
$43$	−1.27526	+	2.20881i	−0.194475	+	0.336840i	0.752254	+	0.658874i	$0.228967\pi$
$44$	3.44949			0.520030
$45$		0			0
$46$	−6.89898			−1.01720
$47$	−2.22474	+	3.85337i	−0.324512	+	0.562072i	−0.981414	−	0.191905i	$-0.938534\pi$
$47$	−2.22474	+	3.85337i	−0.324512	+	0.562072i	0.656901	+	0.753977i	$0.271867\pi$
$48$		0			0
$49$	3.39898	+	5.88721i	0.485568	+	0.841029i
$50$		0			0
$51$		0			0
$52$	−1.22474	+	2.12132i	−0.169842	+	0.294174i
$53$	−3.55051			−0.487700			−0.243850	−	0.969813i	$-0.578410\pi$
$53$	−3.55051			−0.487700			−0.243850	+	0.969813i	$0.578410\pi$
$54$		0			0
$55$		0			0
$56$	0.224745	−	0.389270i	0.0300328	−	0.0520183i
$57$		0			0
$58$	−3.00000	−	5.19615i	−0.393919	−	0.682288i
$59$	6.62372	+	11.4726i	0.862335	+	1.49361i	0.869669	+	0.493636i	$0.164332\pi$
$59$	6.62372	+	11.4726i	0.862335	+	1.49361i	−0.00733331	+	0.999973i	$0.502334\pi$
$60$		0			0
$61$	−2.22474	+	3.85337i	−0.284849	+	0.493374i	−0.972573	−	0.232599i	$-0.925277\pi$
$61$	−2.22474	+	3.85337i	−0.284849	+	0.493374i	0.687723	+	0.725973i	$0.258610\pi$
$62$	1.55051			0.196915
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−2.27526	−	3.94086i	−0.277967	−	0.481452i	0.692913	−	0.721022i	$-0.256327\pi$
$67$	−2.27526	−	3.94086i	−0.277967	−	0.481452i	−0.970879	+	0.239569i	$0.922994\pi$
$68$	−2.94949	−	5.10867i	−0.357678	−	0.619517i
$69$		0			0
$70$		0			0
$71$	2.44949			0.290701			0.145350	−	0.989380i	$-0.453569\pi$
$71$	2.44949			0.290701			0.145350	+	0.989380i	$0.453569\pi$
$72$		0			0
$73$	14.7980			1.73197			0.865985	−	0.500070i	$-0.166692\pi$
$73$	14.7980			1.73197			0.865985	+	0.500070i	$0.166692\pi$
$74$	−4.00000	+	6.92820i	−0.464991	+	0.805387i
$75$		0			0
$76$	2.72474	+	4.71940i	0.312550	+	0.541352i
$77$	0.775255	+	1.34278i	0.0883485	+	0.153024i
$78$		0			0
$79$	−3.67423	+	6.36396i	−0.413384	+	0.716002i	−0.995257	−	0.0972777i	$-0.968987\pi$
$79$	−3.67423	+	6.36396i	−0.413384	+	0.716002i	0.581874	+	0.813279i	$0.302320\pi$
$80$		0			0
$81$		0			0
$82$	1.00000			0.110432
$83$	−2.00000	+	3.46410i	−0.219529	+	0.380235i	−0.954664	−	0.297686i	$-0.903785\pi$
$83$	−2.00000	+	3.46410i	−0.219529	+	0.380235i	0.735135	+	0.677920i	$0.237119\pi$
$84$		0			0
$85$		0			0
$86$	−1.27526	−	2.20881i	−0.137514	−	0.238182i
$87$		0			0
$88$	−1.72474	+	2.98735i	−0.183858	+	0.318452i
$89$	−3.10102			−0.328708			−0.164354	−	0.986401i	$-0.552554\pi$
$89$	−3.10102			−0.328708			−0.164354	+	0.986401i	$0.552554\pi$
$90$		0			0
$91$	−1.10102			−0.115418
$92$	3.44949	−	5.97469i	0.359634	−	0.622905i
$93$		0			0
$94$	−2.22474	−	3.85337i	−0.229465	−	0.397445i
$95$		0			0
$96$		0			0
$97$	−6.50000	+	11.2583i	−0.659975	+	1.14311i	0.320647	+	0.947199i	$0.396100\pi$
$97$	−6.50000	+	11.2583i	−0.659975	+	1.14311i	−0.980622	+	0.195911i	$0.937234\pi$
$98$	−6.79796			−0.686698
$99$		0			0
$100$		0			0
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	−0.993481	−	0.113998i	$-0.963634\pi$
$101$	−4.00000	+	6.92820i	−0.398015	+	0.689382i	0.595466	+	0.803380i	$0.296967\pi$
$102$		0			0
$103$	7.12372	+	12.3387i	0.701921	+	1.21576i	0.967791	+	0.251755i	$0.0810076\pi$
$103$	7.12372	+	12.3387i	0.701921	+	1.21576i	−0.265870	+	0.964009i	$0.585659\pi$
$104$	−1.22474	−	2.12132i	−0.120096	−	0.208013i
$105$		0			0
$106$	1.77526	−	3.07483i	0.172428	−	0.298654i
$107$	−16.3485			−1.58047			−0.790233	−	0.612806i	$-0.790041\pi$
$107$	−16.3485			−1.58047			−0.790233	+	0.612806i	$0.790041\pi$
$108$		0			0
$109$	−8.00000			−0.766261			−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000			−0.766261			−0.383131	+	0.923694i	$0.625154\pi$
$110$		0			0
$111$		0			0
$112$	0.224745	+	0.389270i	0.0212364	+	0.0367825i
$113$	−2.44949	−	4.24264i	−0.230429	−	0.399114i	0.727506	−	0.686102i	$-0.240679\pi$
$113$	−2.44949	−	4.24264i	−0.230429	−	0.399114i	−0.957934	+	0.286988i	$0.907346\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$	−13.2474			−1.21953
$119$	1.32577	−	2.29629i	0.121533	−	0.210501i
$120$		0			0
$121$	−0.449490	−	0.778539i	−0.0408627	−	0.0707763i
$122$	−2.22474	−	3.85337i	−0.201419	−	0.348868i
$123$		0			0
$124$	−0.775255	+	1.34278i	−0.0696200	+	0.120585i
$125$		0			0
$126$		0			0
$127$	6.89898			0.612185			0.306093	−	0.952002i	$-0.400978\pi$
$127$	6.89898			0.612185			0.306093	+	0.952002i	$0.400978\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	2.44949	+	4.24264i	0.214013	+	0.370681i	0.952967	−	0.303075i	$-0.0980132\pi$
$131$	2.44949	+	4.24264i	0.214013	+	0.370681i	−0.738954	+	0.673756i	$0.764680\pi$
$132$		0			0
$133$	−1.22474	+	2.12132i	−0.106199	+	0.183942i
$134$	4.55051			0.393104
$135$		0			0
$136$	5.89898			0.505833
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	−0.794808	−	0.606861i	$-0.792428\pi$
$137$	1.50000	−	2.59808i	0.128154	−	0.221969i	0.922961	+	0.384893i	$0.125762\pi$
$138$		0			0
$139$	−6.62372	−	11.4726i	−0.561817	−	0.973096i	−0.997338	−	0.0729170i	$-0.976769\pi$
$139$	−6.62372	−	11.4726i	−0.561817	−	0.973096i	0.435521	−	0.900179i	$-0.356564\pi$
$140$		0			0
$141$		0			0
$142$	−1.22474	+	2.12132i	−0.102778	+	0.178017i
$143$	8.44949			0.706582
$144$		0			0
$145$		0			0
$146$	−7.39898	+	12.8154i	−0.612344	+	1.06061i
$147$		0			0
$148$	−4.00000	−	6.92820i	−0.328798	−	0.569495i
$149$	−4.12372	−	7.14250i	−0.337829	−	0.585136i	0.646195	−	0.763172i	$-0.276359\pi$
$149$	−4.12372	−	7.14250i	−0.337829	−	0.585136i	−0.984024	+	0.178036i	$0.943026\pi$
$150$		0			0
$151$	−1.44949	+	2.51059i	−0.117958	+	0.204309i	−0.918958	−	0.394355i	$-0.870968\pi$
$151$	−1.44949	+	2.51059i	−0.117958	+	0.204309i	0.801000	+	0.598664i	$0.204301\pi$
$152$	−5.44949			−0.442012
$153$		0			0
$154$	−1.55051			−0.124944
$155$		0			0
$156$		0			0
$157$	−8.00000	−	13.8564i	−0.638470	−	1.10586i	−0.985769	−	0.168107i	$-0.946235\pi$
$157$	−8.00000	−	13.8564i	−0.638470	−	1.10586i	0.347299	−	0.937754i	$-0.387099\pi$
$158$	−3.67423	−	6.36396i	−0.292306	−	0.506290i
$159$		0			0
$160$		0			0
$161$	3.10102			0.244395
$162$		0			0
$163$	8.89898			0.697022			0.348511	−	0.937305i	$-0.386687\pi$
$163$	8.89898			0.697022			0.348511	+	0.937305i	$0.386687\pi$
$164$	−0.500000	+	0.866025i	−0.0390434	+	0.0676252i
$165$		0			0
$166$	−2.00000	−	3.46410i	−0.155230	−	0.268866i
$167$	0.123724	+	0.214297i	0.00957408	+	0.0165828i	0.870773	−	0.491686i	$-0.163619\pi$
$167$	0.123724	+	0.214297i	0.00957408	+	0.0165828i	−0.861199	+	0.508268i	$0.830286\pi$
$168$		0			0
$169$	3.50000	−	6.06218i	0.269231	−	0.466321i
$170$		0			0
$171$		0			0
$172$	2.55051			0.194475
$173$	−5.89898	+	10.2173i	−0.448491	+	0.776809i	−0.998288	−	0.0584890i	$-0.981372\pi$
$173$	−5.89898	+	10.2173i	−0.448491	+	0.776809i	0.549797	+	0.835298i	$0.314705\pi$
$174$		0			0
$175$		0			0
$176$	−1.72474	−	2.98735i	−0.130008	−	0.225180i
$177$		0			0
$178$	1.55051	−	2.68556i	0.116216	−	0.201291i
$179$	0.898979			0.0671929			0.0335964	−	0.999435i	$-0.489304\pi$
$179$	0.898979			0.0671929			0.0335964	+	0.999435i	$0.489304\pi$
$180$		0			0
$181$	5.55051			0.412566			0.206283	−	0.978492i	$-0.433863\pi$
$181$	5.55051			0.412566			0.206283	+	0.978492i	$0.433863\pi$
$182$	0.550510	−	0.953512i	0.0408065	−	0.0706790i
$183$		0			0
$184$	3.44949	+	5.97469i	0.254300	+	0.440460i
$185$		0			0
$186$		0			0
$187$	−10.1742	+	17.6223i	−0.744014	+	1.28867i
$188$	4.44949			0.324512
$189$		0			0
$190$		0			0
$191$	9.12372	−	15.8028i	0.660170	−	1.14345i	−0.320401	−	0.947282i	$-0.603818\pi$
$191$	9.12372	−	15.8028i	0.660170	−	1.14345i	0.980571	−	0.196165i	$-0.0628489\pi$
$192$		0			0
$193$	−6.84847	−	11.8619i	−0.492964	−	0.853838i	0.507004	−	0.861944i	$-0.330753\pi$
$193$	−6.84847	−	11.8619i	−0.492964	−	0.853838i	−0.999967	+	0.00810596i	$0.997420\pi$
$194$	−6.50000	−	11.2583i	−0.466673	−	0.808301i
$195$		0			0
$196$	3.39898	−	5.88721i	0.242784	−	0.420515i
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	15.5505			1.10235			0.551173	−	0.834391i	$-0.314180\pi$
$199$	15.5505			1.10235			0.551173	+	0.834391i	$0.314180\pi$
$200$		0			0
$201$		0			0
$202$	−4.00000	−	6.92820i	−0.281439	−	0.487467i
$203$	1.34847	+	2.33562i	0.0946440	+	0.163928i
$204$		0			0
$205$		0			0
$206$	−14.2474			−0.992667
$207$		0			0
$208$	2.44949			0.169842
$209$	9.39898	−	16.2795i	0.650141	−	1.12608i
$210$		0			0
$211$	−1.89898	−	3.28913i	−0.130731	−	0.226433i	0.793227	−	0.608925i	$-0.208399\pi$
$211$	−1.89898	−	3.28913i	−0.130731	−	0.226433i	−0.923959	+	0.382492i	$0.875066\pi$
$212$	1.77526	+	3.07483i	0.121925	+	0.211180i
$213$		0			0
$214$	8.17423	−	14.1582i	0.558779	−	0.967834i
$215$		0			0
$216$		0			0
$217$	−0.696938			−0.0473113
$218$	4.00000	−	6.92820i	0.270914	−	0.469237i
$219$		0			0
$220$		0			0
$221$	−7.22474	−	12.5136i	−0.485989	−	0.841758i
$222$		0			0
$223$	−4.55051	+	7.88171i	−0.304725	+	0.527799i	−0.977200	−	0.212321i	$-0.931898\pi$
$223$	−4.55051	+	7.88171i	−0.304725	+	0.527799i	0.672475	+	0.740120i	$0.265231\pi$
$224$	−0.449490			−0.0300328
$225$		0			0
$226$	4.89898			0.325875
$227$	−1.72474	+	2.98735i	−0.114475	+	0.198277i	−0.917570	−	0.397574i	$-0.869852\pi$
$227$	−1.72474	+	2.98735i	−0.114475	+	0.198277i	0.803095	+	0.595852i	$0.203185\pi$
$228$		0			0
$229$	9.22474	+	15.9777i	0.609588	+	1.05584i	0.991308	+	0.131560i	$0.0419986\pi$
$229$	9.22474	+	15.9777i	0.609588	+	1.05584i	−0.381720	+	0.924278i	$0.624668\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000	+	5.19615i	−0.196960	+	0.341144i
$233$	13.6969			0.897316			0.448658	−	0.893703i	$-0.351902\pi$
$233$	13.6969			0.897316			0.448658	+	0.893703i	$0.351902\pi$
$234$		0			0
$235$		0			0
$236$	6.62372	−	11.4726i	0.431168	−	0.746804i
$237$		0			0
$238$	1.32577	+	2.29629i	0.0859366	+	0.148847i
$239$	−0.348469	−	0.603566i	−0.0225406	−	0.0390415i	0.854535	−	0.519394i	$-0.173842\pi$
$239$	−0.348469	−	0.603566i	−0.0225406	−	0.0390415i	−0.877076	+	0.480352i	$0.840509\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$	0.898979			0.0577886
$243$		0			0
$244$	4.44949			0.284849
$245$		0			0
$246$		0			0
$247$	6.67423	+	11.5601i	0.424671	+	0.735552i
$248$	−0.775255	−	1.34278i	−0.0492287	−	0.0852667i
$249$		0			0
$250$		0			0
$251$	−6.55051			−0.413465			−0.206732	−	0.978398i	$-0.566283\pi$
$251$	−6.55051			−0.413465			−0.206732	+	0.978398i	$0.566283\pi$
$252$		0			0
$253$	−23.7980			−1.49616
$254$	−3.44949	+	5.97469i	−0.216440	+	0.374885i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	5.05051	+	8.74774i	0.315042	+	0.545669i	0.979446	−	0.201704i	$-0.0646480\pi$
$257$	5.05051	+	8.74774i	0.315042	+	0.545669i	−0.664404	+	0.747373i	$0.731315\pi$
$258$		0			0
$259$	1.79796	−	3.11416i	0.111720	−	0.193504i
$260$		0			0
$261$		0			0
$262$	−4.89898			−0.302660
$263$	−6.22474	+	10.7816i	−0.383834	+	0.664820i	−0.991607	−	0.129291i	$-0.958730\pi$
$263$	−6.22474	+	10.7816i	−0.383834	+	0.664820i	0.607773	+	0.794111i	$0.292063\pi$
$264$		0			0
$265$		0			0
$266$	−1.22474	−	2.12132i	−0.0750939	−	0.130066i
$267$		0			0
$268$	−2.27526	+	3.94086i	−0.138983	+	0.240726i
$269$	−16.0454			−0.978306			−0.489153	−	0.872198i	$-0.662694\pi$
$269$	−16.0454			−0.978306			−0.489153	+	0.872198i	$0.662694\pi$
$270$		0			0
$271$	−15.5959			−0.947385			−0.473692	−	0.880690i	$-0.657079\pi$
$271$	−15.5959			−0.947385			−0.473692	+	0.880690i	$0.657079\pi$
$272$	−2.94949	+	5.10867i	−0.178839	+	0.309758i
$273$		0			0
$274$	1.50000	+	2.59808i	0.0906183	+	0.156956i
$275$		0			0
$276$		0			0
$277$	14.7980	−	25.6308i	0.889123	−	1.54001i	0.0482095	−	0.998837i	$-0.484648\pi$
$277$	14.7980	−	25.6308i	0.889123	−	1.54001i	0.840914	−	0.541169i	$-0.182018\pi$
$278$	13.2474			0.794529
$279$		0			0
$280$		0			0
$281$	6.00000	−	10.3923i	0.357930	−	0.619953i	−0.629685	−	0.776851i	$-0.716816\pi$
$281$	6.00000	−	10.3923i	0.357930	−	0.619953i	0.987615	+	0.156898i	$0.0501493\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$	−1.22474	−	2.12132i	−0.0726752	−	0.125877i
$285$		0			0
$286$	−4.22474	+	7.31747i	−0.249814	+	0.432691i
$287$	−0.449490			−0.0265325
$288$		0			0
$289$	17.7980			1.04694
$290$		0			0
$291$		0			0
$292$	−7.39898	−	12.8154i	−0.432993	−	0.749965i
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	0.999549	+	0.0300351i	$0.00956192\pi$
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	−0.473763	+	0.880652i	$0.657105\pi$
$294$		0			0
$295$		0			0
$296$	8.00000			0.464991
$297$		0			0
$298$	8.24745			0.477762
$299$	8.44949	−	14.6349i	0.488647	−	0.846361i
$300$		0			0
$301$	0.573214	+	0.992836i	0.0330395	+	0.0572261i
$302$	−1.44949	−	2.51059i	−0.0834088	−	0.144468i
$303$		0			0
$304$	2.72474	−	4.71940i	0.156275	−	0.270676i
$305$		0			0
$306$		0			0
$307$	29.9444			1.70902			0.854508	−	0.519438i	$-0.173859\pi$
$307$	29.9444			1.70902			0.854508	+	0.519438i	$0.173859\pi$
$308$	0.775255	−	1.34278i	0.0441743	−	0.0765121i
$309$		0			0
$310$		0			0
$311$	6.55051	+	11.3458i	0.371445	+	0.643362i	0.989788	−	0.142546i	$-0.0455290\pi$
$311$	6.55051	+	11.3458i	0.371445	+	0.643362i	−0.618343	+	0.785909i	$0.712196\pi$
$312$		0			0
$313$	10.8485	−	18.7901i	0.613192	−	1.06208i	−0.377507	−	0.926007i	$-0.623219\pi$
$313$	10.8485	−	18.7901i	0.613192	−	1.06208i	0.990699	−	0.136073i	$-0.0434480\pi$
$314$	16.0000			0.902932
$315$		0			0
$316$	7.34847			0.413384
$317$	11.4722	−	19.8704i	0.644343	−	1.11603i	−0.340110	−	0.940386i	$-0.610464\pi$
$317$	11.4722	−	19.8704i	0.644343	−	1.11603i	0.984453	−	0.175649i	$-0.0562022\pi$
$318$		0			0
$319$	−10.3485	−	17.9241i	−0.579403	−	1.00356i
$320$		0			0
$321$		0			0
$322$	−1.55051	+	2.68556i	−0.0864066	+	0.149661i
$323$	−32.1464			−1.78868
$324$		0			0
$325$		0			0
$326$	−4.44949	+	7.70674i	−0.246434	+	0.426837i
$327$		0			0
$328$	−0.500000	−	0.866025i	−0.0276079	−	0.0478183i
$329$	1.00000	+	1.73205i	0.0551318	+	0.0954911i
$330$		0			0
$331$	12.6969	−	21.9917i	0.697887	−	1.20878i	−0.271311	−	0.962492i	$-0.587457\pi$
$331$	12.6969	−	21.9917i	0.697887	−	1.20878i	0.969198	−	0.246284i	$-0.0792095\pi$
$332$	4.00000			0.219529
$333$		0			0
$334$	−0.247449			−0.0135398
$335$		0			0
$336$		0			0
$337$	−9.29796	−	16.1045i	−0.506492	−	0.877270i	−0.999972	−	0.00751272i	$-0.997609\pi$
$337$	−9.29796	−	16.1045i	−0.506492	−	0.877270i	0.493480	−	0.869757i	$-0.335725\pi$
$338$	3.50000	+	6.06218i	0.190375	+	0.329739i
$339$		0			0
$340$		0			0
$341$	5.34847			0.289636
$342$		0			0
$343$	6.20204			0.334879
$344$	−1.27526	+	2.20881i	−0.0687571	+	0.119091i
$345$		0			0
$346$	−5.89898	−	10.2173i	−0.317131	−	0.549287i
$347$	4.62372	+	8.00853i	0.248215	+	0.429920i	0.963030	−	0.269392i	$-0.0868229\pi$
$347$	4.62372	+	8.00853i	0.248215	+	0.429920i	−0.714816	+	0.699313i	$0.753490\pi$
$348$		0			0
$349$	13.7980	−	23.8988i	0.738588	−	1.27927i	−0.214543	−	0.976714i	$-0.568826\pi$
$349$	13.7980	−	23.8988i	0.738588	−	1.27927i	0.953131	−	0.302557i	$-0.0978403\pi$
$350$		0			0
$351$		0			0
$352$	3.44949			0.183858
$353$	16.2980	−	28.2289i	0.867453	−	1.50247i	0.00286194	−	0.999996i	$-0.499089\pi$
$353$	16.2980	−	28.2289i	0.867453	−	1.50247i	0.864591	−	0.502476i	$-0.167578\pi$
$354$		0			0
$355$		0			0
$356$	1.55051	+	2.68556i	0.0821769	+	0.142335i
$357$		0			0
$358$	−0.449490	+	0.778539i	−0.0237563	+	0.0411471i
$359$	−3.55051			−0.187389			−0.0936944	−	0.995601i	$-0.529868\pi$
$359$	−3.55051			−0.187389			−0.0936944	+	0.995601i	$0.529868\pi$
$360$		0			0
$361$	10.6969			0.562997
$362$	−2.77526	+	4.80688i	−0.145864	+	0.252644i
$363$		0			0
$364$	0.550510	+	0.953512i	0.0288546	+	0.0499776i
$365$		0			0
$366$		0			0
$367$	−13.7980	+	23.8988i	−0.720248	+	1.24751i	0.240653	+	0.970611i	$0.422638\pi$
$367$	−13.7980	+	23.8988i	−0.720248	+	1.24751i	−0.960900	+	0.276894i	$0.910695\pi$
$368$	−6.89898			−0.359634
$369$		0			0
$370$		0			0
$371$	−0.797959	+	1.38211i	−0.0414280	+	0.0717553i
$372$		0			0
$373$	6.79796	+	11.7744i	0.351985	+	0.609656i	0.986597	−	0.163175i	$-0.0521735\pi$
$373$	6.79796	+	11.7744i	0.351985	+	0.609656i	−0.634612	+	0.772831i	$0.718840\pi$
$374$	−10.1742	−	17.6223i	−0.526097	−	0.911227i
$375$		0			0
$376$	−2.22474	+	3.85337i	−0.114732	+	0.198722i
$377$	14.6969			0.756931
$378$		0			0
$379$	4.14643			0.212988			0.106494	−	0.994313i	$-0.466038\pi$
$379$	4.14643			0.212988			0.106494	+	0.994313i	$0.466038\pi$
$380$		0			0
$381$		0			0
$382$	9.12372	+	15.8028i	0.466810	+	0.808539i
$383$	−8.89898	−	15.4135i	−0.454717	−	0.787592i	0.543955	−	0.839114i	$-0.316926\pi$
$383$	−8.89898	−	15.4135i	−0.454717	−	0.787592i	−0.998672	+	0.0515220i	$0.983593\pi$
$384$		0			0
$385$		0			0
$386$	13.6969			0.697156
$387$		0			0
$388$	13.0000			0.659975
$389$	8.77526	−	15.1992i	0.444923	−	0.770629i	−0.553124	−	0.833099i	$-0.686564\pi$
$389$	8.77526	−	15.1992i	0.444923	−	0.770629i	0.998047	+	0.0624697i	$0.0198977\pi$
$390$		0			0
$391$	20.3485	+	35.2446i	1.02907	+	1.78240i
$392$	3.39898	+	5.88721i	0.171674	+	0.297349i
$393$		0			0
$394$	−4.00000	+	6.92820i	−0.201517	+	0.349038i
$395$		0			0
$396$		0			0
$397$	−1.79796			−0.0902370			−0.0451185	−	0.998982i	$-0.514367\pi$
$397$	−1.79796			−0.0902370			−0.0451185	+	0.998982i	$0.514367\pi$
$398$	−7.77526	+	13.4671i	−0.389738	+	0.675047i
$399$		0			0
$400$		0			0
$401$	4.60102	+	7.96920i	0.229764	+	0.397963i	0.957738	−	0.287642i	$-0.0928712\pi$
$401$	4.60102	+	7.96920i	0.229764	+	0.397963i	−0.727974	+	0.685605i	$0.759538\pi$
$402$		0			0
$403$	−1.89898	+	3.28913i	−0.0945949	+	0.163843i
$404$	8.00000			0.398015
$405$		0			0
$406$	−2.69694			−0.133847
$407$	−13.7980	+	23.8988i	−0.683939	+	1.18462i
$408$		0			0
$409$	−7.94949	−	13.7689i	−0.393077	−	0.680829i	0.599777	−	0.800167i	$-0.295256\pi$
$409$	−7.94949	−	13.7689i	−0.393077	−	0.680829i	−0.992854	+	0.119338i	$0.961923\pi$
$410$		0			0
$411$		0			0
$412$	7.12372	−	12.3387i	0.350961	−	0.607882i
$413$	5.95459			0.293006
$414$		0			0
$415$		0			0
$416$	−1.22474	+	2.12132i	−0.0600481	+	0.104006i
$417$		0			0
$418$	9.39898	+	16.2795i	0.459719	+	0.796257i
$419$	−4.44949	−	7.70674i	−0.217372	−	0.376499i	0.736632	−	0.676294i	$-0.236415\pi$
$419$	−4.44949	−	7.70674i	−0.217372	−	0.376499i	−0.954004	+	0.299795i	$0.903082\pi$
$420$		0			0
$421$	5.77526	−	10.0030i	0.281469	−	0.487518i	−0.690278	−	0.723544i	$-0.742512\pi$
$421$	5.77526	−	10.0030i	0.281469	−	0.487518i	0.971747	+	0.236026i	$0.0758451\pi$
$422$	3.79796			0.184882
$423$		0			0
$424$	−3.55051			−0.172428
$425$		0			0
$426$		0			0
$427$	1.00000	+	1.73205i	0.0483934	+	0.0838198i
$428$	8.17423	+	14.1582i	0.395117	+	0.684362i
$429$		0			0
$430$		0			0
$431$	−38.2474			−1.84231			−0.921157	−	0.389190i	$-0.872755\pi$
$431$	−38.2474			−1.84231			−0.921157	+	0.389190i	$0.872755\pi$
$432$		0			0
$433$	23.0000			1.10531			0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000			1.10531			0.552655	+	0.833410i	$0.313615\pi$
$434$	0.348469	−	0.603566i	0.0167271	−	0.0289721i
$435$		0			0
$436$	4.00000	+	6.92820i	0.191565	+	0.331801i
$437$	−18.7980	−	32.5590i	−0.899228	−	1.55751i
$438$		0			0
$439$	−11.0227	+	19.0919i	−0.526085	+	0.911206i	0.473453	+	0.880819i	$0.343007\pi$
$439$	−11.0227	+	19.0919i	−0.526085	+	0.911206i	−0.999538	+	0.0303869i	$0.990326\pi$
$440$		0			0
$441$		0			0
$442$	14.4495			0.687292
$443$	10.6237	−	18.4008i	0.504748	−	0.874250i	−0.495237	−	0.868758i	$-0.664919\pi$
$443$	10.6237	−	18.4008i	0.504748	−	0.874250i	0.999985	−	0.00549166i	$-0.00174806\pi$
$444$		0			0
$445$		0			0
$446$	−4.55051	−	7.88171i	−0.215473	−	0.373210i
$447$		0			0
$448$	0.224745	−	0.389270i	0.0106182	−	0.0183913i
$449$	−18.7980			−0.887131			−0.443565	−	0.896242i	$-0.646287\pi$
$449$	−18.7980			−0.887131			−0.443565	+	0.896242i	$0.646287\pi$
$450$		0			0
$451$	3.44949			0.162430
$452$	−2.44949	+	4.24264i	−0.115214	+	0.199557i
$453$		0			0
$454$	−1.72474	−	2.98735i	−0.0809463	−	0.140203i
$455$		0			0
$456$		0			0
$457$	8.94949	−	15.5010i	0.418639	−	0.725105i	−0.577163	−	0.816629i	$-0.695840\pi$
$457$	8.94949	−	15.5010i	0.418639	−	0.725105i	0.995803	+	0.0915238i	$0.0291738\pi$
$458$	−18.4495			−0.862088
$459$		0			0
$460$		0			0
$461$	−1.22474	+	2.12132i	−0.0570421	+	0.0987997i	−0.893136	−	0.449786i	$-0.851500\pi$
$461$	−1.22474	+	2.12132i	−0.0570421	+	0.0987997i	0.836094	+	0.548586i	$0.184834\pi$
$462$		0			0
$463$	−12.0000	−	20.7846i	−0.557687	−	0.965943i	−0.997689	−	0.0679458i	$-0.978356\pi$
$463$	−12.0000	−	20.7846i	−0.557687	−	0.965943i	0.440002	−	0.897997i	$-0.354978\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	−6.84847	+	11.8619i	−0.317249	+	0.549492i
$467$	10.3485			0.478870			0.239435	−	0.970912i	$-0.423038\pi$
$467$	10.3485			0.478870			0.239435	+	0.970912i	$0.423038\pi$
$468$		0			0
$469$	−2.04541			−0.0944482
$470$		0			0
$471$		0			0
$472$	6.62372	+	11.4726i	0.304882	+	0.528070i
$473$	−4.39898	−	7.61926i	−0.202265	−	0.350334i
$474$		0			0
$475$		0			0
$476$	−2.65153			−0.121533
$477$		0			0
$478$	0.696938			0.0318772
$479$	8.34847	−	14.4600i	0.381451	−	0.660693i	−0.609819	−	0.792541i	$-0.708758\pi$
$479$	8.34847	−	14.4600i	0.381451	−	0.660693i	0.991270	+	0.131848i	$0.0420911\pi$
$480$		0			0
$481$	−9.79796	−	16.9706i	−0.446748	−	0.773791i
$482$	−0.500000	−	0.866025i	−0.0227744	−	0.0394464i
$483$		0			0
$484$	−0.449490	+	0.778539i	−0.0204314	+	0.0353881i
$485$		0			0
$486$		0			0
$487$	−25.1010			−1.13744			−0.568718	−	0.822533i	$-0.692560\pi$
$487$	−25.1010			−1.13744			−0.568718	+	0.822533i	$0.692560\pi$
$488$	−2.22474	+	3.85337i	−0.100709	+	0.174434i
$489$		0			0
$490$		0			0
$491$	9.27526	+	16.0652i	0.418586	+	0.725013i	0.995798	−	0.0915820i	$-0.0291924\pi$
$491$	9.27526	+	16.0652i	0.418586	+	0.725013i	−0.577211	+	0.816595i	$0.695859\pi$
$492$		0			0
$493$	−17.6969	+	30.6520i	−0.797030	+	1.38050i
$494$	−13.3485			−0.600576
$495$		0			0
$496$	1.55051			0.0696200
$497$	0.550510	−	0.953512i	0.0246938	−	0.0427708i
$498$		0			0
$499$	10.6237	+	18.4008i	0.475583	+	0.823734i	0.999609	−	0.0279682i	$-0.00890372\pi$
$499$	10.6237	+	18.4008i	0.475583	+	0.823734i	−0.524026	+	0.851703i	$0.675570\pi$
$500$		0			0
$501$		0			0
$502$	3.27526	−	5.67291i	0.146182	−	0.253194i
$503$	−14.4495			−0.644271			−0.322135	−	0.946694i	$-0.604401\pi$
$503$	−14.4495			−0.644271			−0.322135	+	0.946694i	$0.604401\pi$
$504$		0			0
$505$		0			0
$506$	11.8990	−	20.6096i	0.528974	−	0.916210i
$507$		0			0
$508$	−3.44949	−	5.97469i	−0.153046	−	0.265084i
$509$	15.7980	+	27.3629i	0.700232	+	1.21284i	0.968385	+	0.249461i	$0.0802535\pi$
$509$	15.7980	+	27.3629i	0.700232	+	1.21284i	−0.268153	+	0.963376i	$0.586413\pi$
$510$		0			0
$511$	3.32577	−	5.76039i	0.147123	−	0.254825i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−10.1010			−0.445537
$515$		0			0
$516$		0			0
$517$	−7.67423	−	13.2922i	−0.337512	−	0.584589i
$518$	1.79796	+	3.11416i	0.0789978	+	0.136828i
$519$		0			0
$520$		0			0
$521$	−21.6969			−0.950560			−0.475280	−	0.879835i	$-0.657653\pi$
$521$	−21.6969			−0.950560			−0.475280	+	0.879835i	$0.657653\pi$
$522$		0			0
$523$	−10.2020			−0.446104			−0.223052	−	0.974807i	$-0.571602\pi$
$523$	−10.2020			−0.446104			−0.223052	+	0.974807i	$0.571602\pi$
$524$	2.44949	−	4.24264i	0.107006	−	0.185341i
$525$		0			0
$526$	−6.22474	−	10.7816i	−0.271412	−	0.470099i
$527$	−4.57321	−	7.92104i	−0.199212	−	0.345046i
$528$		0			0
$529$	−12.2980	+	21.3007i	−0.534694	+	0.926117i
$530$		0			0
$531$		0			0
$532$	2.44949			0.106199
$533$	−1.22474	+	2.12132i	−0.0530496	+	0.0918846i
$534$		0			0
$535$		0			0
$536$	−2.27526	−	3.94086i	−0.0982761	−	0.170219i
$537$		0			0
$538$	8.02270	−	13.8957i	0.345883	−	0.599087i
$539$	−23.4495			−1.01004
$540$		0			0
$541$	−0.404082			−0.0173728			−0.00868642	−	0.999962i	$-0.502765\pi$
$541$	−0.404082			−0.0173728			−0.00868642	+	0.999962i	$0.502765\pi$
$542$	7.79796	−	13.5065i	0.334951	−	0.580152i
$543$		0			0
$544$	−2.94949	−	5.10867i	−0.126458	−	0.219032i
$545$		0			0
$546$		0			0
$547$	−8.62372	+	14.9367i	−0.368724	+	0.638648i	−0.989366	−	0.145445i	$-0.953539\pi$
$547$	−8.62372	+	14.9367i	−0.368724	+	0.638648i	0.620642	+	0.784094i	$0.286872\pi$
$548$	−3.00000			−0.128154
$549$		0			0
$550$		0			0
$551$	16.3485	−	28.3164i	0.696468	−	1.20632i
$552$		0			0
$553$	1.65153	+	2.86054i	0.0702302	+	0.121642i
$554$	14.7980	+	25.6308i	0.628705	+	1.08895i
$555$		0			0
$556$	−6.62372	+	11.4726i	−0.280908	+	0.486548i
$557$	−14.9444			−0.633214			−0.316607	−	0.948557i	$-0.602544\pi$
$557$	−14.9444			−0.633214			−0.316607	+	0.948557i	$0.602544\pi$
$558$		0			0
$559$	6.24745			0.264239
$560$		0			0
$561$		0			0
$562$	6.00000	+	10.3923i	0.253095	+	0.438373i
$563$	0.926786	+	1.60524i	0.0390594	+	0.0676528i	0.884894	−	0.465792i	$-0.154231\pi$
$563$	0.926786	+	1.60524i	0.0390594	+	0.0676528i	−0.845835	+	0.533445i	$0.820897\pi$
$564$		0			0
$565$		0			0
$566$	4.00000			0.168133
$567$		0			0
$568$	2.44949			0.102778
$569$	−15.7474	+	27.2754i	−0.660167	+	1.14344i	0.320404	+	0.947281i	$0.396181\pi$
$569$	−15.7474	+	27.2754i	−0.660167	+	1.14344i	−0.980571	+	0.196162i	$0.937152\pi$
$570$		0			0
$571$	−18.6237	−	32.2572i	−0.779379	−	1.34992i	−0.932300	−	0.361685i	$-0.882202\pi$
$571$	−18.6237	−	32.2572i	−0.779379	−	1.34992i	0.152922	−	0.988238i	$-0.451132\pi$
$572$	−4.22474	−	7.31747i	−0.176645	−	0.305959i
$573$		0			0
$574$	0.224745	−	0.389270i	0.00938067	−	0.0162478i
$575$		0			0
$576$		0			0
$577$	−15.6969			−0.653472			−0.326736	−	0.945116i	$-0.605949\pi$
$577$	−15.6969			−0.653472			−0.326736	+	0.945116i	$0.605949\pi$
$578$	−8.89898	+	15.4135i	−0.370149	+	0.641116i
$579$		0			0
$580$		0			0
$581$	0.898979	+	1.55708i	0.0372960	+	0.0645985i
$582$		0			0
$583$	6.12372	−	10.6066i	0.253619	−	0.439281i
$584$	14.7980			0.612344
$585$		0			0
$586$	−18.0000			−0.743573
$587$	−11.9722	+	20.7364i	−0.494145	+	0.855885i	−0.999977	−	0.00674727i	$-0.997852\pi$
$587$	−11.9722	+	20.7364i	−0.494145	+	0.855885i	0.505832	+	0.862632i	$0.331186\pi$
$588$		0			0
$589$	4.22474	+	7.31747i	0.174078	+	0.301511i
$590$		0			0
$591$		0			0
$592$	−4.00000	+	6.92820i	−0.164399	+	0.284747i
$593$	17.3939			0.714281			0.357140	−	0.934051i	$-0.383752\pi$
$593$	17.3939			0.714281			0.357140	+	0.934051i	$0.383752\pi$
$594$		0			0
$595$		0			0
$596$	−4.12372	+	7.14250i	−0.168914	+	0.292568i
$597$		0			0
$598$	8.44949	+	14.6349i	0.345525	+	0.598467i
$599$	16.8990	+	29.2699i	0.690474	+	1.19594i	0.971683	+	0.236289i	$0.0759312\pi$
$599$	16.8990	+	29.2699i	0.690474	+	1.19594i	−0.281209	+	0.959646i	$0.590736\pi$
$600$		0			0
$601$	−19.3990	+	33.6000i	−0.791301	+	1.37057i	0.133861	+	0.991000i	$0.457262\pi$
$601$	−19.3990	+	33.6000i	−0.791301	+	1.37057i	−0.925162	+	0.379573i	$0.876071\pi$
$602$	−1.14643			−0.0467249
$603$		0			0
$604$	2.89898			0.117958
$605$		0			0
$606$		0			0
$607$	13.7980	+	23.8988i	0.560042	+	0.970021i	0.997492	+	0.0707783i	$0.0225483\pi$
$607$	13.7980	+	23.8988i	0.560042	+	0.970021i	−0.437450	+	0.899243i	$0.644118\pi$
$608$	2.72474	+	4.71940i	0.110503	+	0.191397i
$609$		0			0
$610$		0			0
$611$	10.8990			0.440926
$612$		0			0
$613$	−36.9444			−1.49217			−0.746085	−	0.665851i	$-0.768069\pi$
$613$	−36.9444			−1.49217			−0.746085	+	0.665851i	$0.768069\pi$
$614$	−14.9722	+	25.9326i	−0.604229	+	1.04655i
$615$		0			0
$616$	0.775255	+	1.34278i	0.0312359	+	0.0541022i
$617$	4.15153	+	7.19066i	0.167134	+	0.289485i	0.937411	−	0.348224i	$-0.113215\pi$
$617$	4.15153	+	7.19066i	0.167134	+	0.289485i	−0.770277	+	0.637710i	$0.779882\pi$
$618$		0			0
$619$	14.2753	−	24.7255i	0.573771	−	0.993800i	−0.422403	−	0.906408i	$-0.638813\pi$
$619$	14.2753	−	24.7255i	0.573771	−	0.993800i	0.996174	−	0.0873923i	$-0.0278534\pi$
$620$		0			0
$621$		0			0
$622$	−13.1010			−0.525303
$623$	−0.696938	+	1.20713i	−0.0279222	+	0.0483628i
$624$		0			0
$625$		0			0
$626$	10.8485	+	18.7901i	0.433592	+	0.751003i
$627$		0			0
$628$	−8.00000	+	13.8564i	−0.319235	+	0.552931i
$629$	47.1918			1.88166
$630$		0			0
$631$	−11.3485			−0.451775			−0.225888	−	0.974153i	$-0.572528\pi$
$631$	−11.3485			−0.451775			−0.225888	+	0.974153i	$0.572528\pi$
$632$	−3.67423	+	6.36396i	−0.146153	+	0.253145i
$633$		0			0
$634$	11.4722	+	19.8704i	0.455619	+	0.789155i
$635$		0			0
$636$		0			0
$637$	8.32577	−	14.4206i	0.329879	−	0.571367i
$638$	20.6969			0.819400
$639$		0			0
$640$		0			0
$641$	−18.5000	+	32.0429i	−0.730706	+	1.26562i	0.225876	+	0.974156i	$0.427476\pi$
$641$	−18.5000	+	32.0429i	−0.730706	+	1.26562i	−0.956582	+	0.291464i	$0.905858\pi$
$642$		0			0
$643$	−7.62372	−	13.2047i	−0.300650	−	0.520742i	0.675633	−	0.737238i	$-0.263870\pi$
$643$	−7.62372	−	13.2047i	−0.300650	−	0.520742i	−0.976283	+	0.216496i	$0.930537\pi$
$644$	−1.55051	−	2.68556i	−0.0610987	−	0.105826i
$645$		0			0
$646$	16.0732	−	27.8396i	0.632392	−	1.09534i
$647$	−34.8990			−1.37202			−0.686010	−	0.727592i	$-0.740639\pi$
$647$	−34.8990			−1.37202			−0.686010	+	0.727592i	$0.740639\pi$
$648$		0			0
$649$	−45.6969			−1.79376
$650$		0			0
$651$		0			0
$652$	−4.44949	−	7.70674i	−0.174255	−	0.301819i
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	−0.979214	−	0.202828i	$-0.934987\pi$
$653$	−17.0000	−	29.4449i	−0.665261	−	1.15227i	0.313953	−	0.949439i	$-0.398347\pi$
$654$		0			0
$655$		0			0
$656$	1.00000			0.0390434
$657$		0			0
$658$	−2.00000			−0.0779681
$659$	−17.8990	+	31.0019i	−0.697245	+	1.20766i	0.272173	+	0.962248i	$0.412258\pi$
$659$	−17.8990	+	31.0019i	−0.697245	+	1.20766i	−0.969418	+	0.245416i	$0.921076\pi$
$660$		0			0
$661$	−2.89898	−	5.02118i	−0.112757	−	0.195301i	0.804124	−	0.594462i	$-0.202635\pi$
$661$	−2.89898	−	5.02118i	−0.112757	−	0.195301i	−0.916881	+	0.399161i	$0.869302\pi$
$662$	12.6969	+	21.9917i	0.493481	+	0.854733i
$663$		0			0
$664$	−2.00000	+	3.46410i	−0.0776151	+	0.134433i
$665$		0			0
$666$		0			0
$667$	−41.3939			−1.60278
$668$	0.123724	−	0.214297i	0.00478704	−	0.00829139i
$669$		0			0
$670$		0			0
$671$	−7.67423	−	13.2922i	−0.296261	−	0.513138i
$672$		0			0
$673$	14.4495	−	25.0273i	0.556987	−	0.964730i	−0.440759	−	0.897625i	$-0.645291\pi$
$673$	14.4495	−	25.0273i	0.556987	−	0.964730i	0.997746	−	0.0671042i	$-0.0213760\pi$
$674$	18.5959			0.716288
$675$		0			0
$676$	−7.00000			−0.269231
$677$	19.7980	−	34.2911i	0.760897	−	1.31791i	−0.181491	−	0.983393i	$-0.558092\pi$
$677$	19.7980	−	34.2911i	0.760897	−	1.31791i	0.942389	−	0.334520i	$-0.108574\pi$
$678$		0			0
$679$	2.92168	+	5.06050i	0.112124	+	0.194204i
$680$		0			0
$681$		0			0
$682$	−2.67423	+	4.63191i	−0.102402	+	0.177365i
$683$	45.4495			1.73908			0.869538	−	0.493866i	$-0.164417\pi$
$683$	45.4495			1.73908			0.869538	+	0.493866i	$0.164417\pi$
$684$		0			0
$685$		0			0
$686$	−3.10102	+	5.37113i	−0.118398	+	0.205071i
$687$		0			0
$688$	−1.27526	−	2.20881i	−0.0486186	−	0.0842100i
$689$	4.34847	+	7.53177i	0.165663	+	0.286938i
$690$		0			0
$691$	−8.79796	+	15.2385i	−0.334690	+	0.579700i	−0.983425	−	0.181314i	$-0.941965\pi$
$691$	−8.79796	+	15.2385i	−0.334690	+	0.579700i	0.648735	+	0.761014i	$0.275298\pi$
$692$	11.7980			0.448491
$693$		0			0
$694$	−9.24745			−0.351028
$695$		0			0
$696$		0			0
$697$	−2.94949	−	5.10867i	−0.111720	−	0.193505i
$698$	13.7980	+	23.8988i	0.522260	+	0.904582i
$699$		0			0
$700$		0			0
$701$	39.3939			1.48789			0.743943	−	0.668243i	$-0.232953\pi$
$701$	39.3939			1.48789			0.743943	+	0.668243i	$0.232953\pi$
$702$		0			0
$703$	−43.5959			−1.64425
$704$	−1.72474	+	2.98735i	−0.0650038	+	0.112590i
$705$		0			0
$706$	16.2980	+	28.2289i	0.613382	+	1.06241i
$707$	1.79796	+	3.11416i	0.0676192	+	0.117120i
$708$		0			0
$709$	−18.6742	+	32.3447i	−0.701326	+	1.21473i	0.266676	+	0.963786i	$0.414075\pi$
$709$	−18.6742	+	32.3447i	−0.701326	+	1.21473i	−0.968001	+	0.250945i	$0.919259\pi$
$710$		0			0
$711$		0			0
$712$	−3.10102			−0.116216
$713$	5.34847	−	9.26382i	0.200302	−	0.346933i
$714$		0			0
$715$		0			0
$716$	−0.449490	−	0.778539i	−0.0167982	−	0.0290954i
$717$		0			0
$718$	1.77526	−	3.07483i	0.0662519	−	0.114752i
$719$	−41.7980			−1.55880			−0.779400	−	0.626526i	$-0.784476\pi$
$719$	−41.7980			−1.55880			−0.779400	+	0.626526i	$0.784476\pi$
$720$		0			0
$721$	6.40408			0.238500
$722$	−5.34847	+	9.26382i	−0.199049	+	0.344764i
$723$		0			0
$724$	−2.77526	−	4.80688i	−0.103142	−	0.178646i
$725$		0			0
$726$		0			0
$727$	12.6742	−	21.9524i	0.470061	−	0.814170i	−0.529353	−	0.848402i	$-0.677565\pi$
$727$	12.6742	−	21.9524i	0.470061	−	0.814170i	0.999414	+	0.0342318i	$0.0108985\pi$
$728$	−1.10102			−0.0408065
$729$		0			0
$730$		0			0
$731$	−7.52270	+	13.0297i	−0.278237	+	0.481921i
$732$		0			0
$733$	−10.5732	−	18.3133i	−0.390531	−	0.676419i	0.601989	−	0.798504i	$-0.294375\pi$
$733$	−10.5732	−	18.3133i	−0.390531	−	0.676419i	−0.992520	+	0.122086i	$0.961042\pi$
$734$	−13.7980	−	23.8988i	−0.509292	−	0.882120i
$735$		0			0
$736$	3.44949	−	5.97469i	0.127150	−	0.220230i
$737$	15.6969			0.578204
$738$		0			0
$739$	−17.2474			−0.634458			−0.317229	−	0.948349i	$-0.602752\pi$
$739$	−17.2474			−0.634458			−0.317229	+	0.948349i	$0.602752\pi$
$740$		0			0
$741$		0			0
$742$	−0.797959	−	1.38211i	−0.0292940	−	0.0507387i
$743$	−11.4495	−	19.8311i	−0.420041	−	0.727532i	0.575902	−	0.817519i	$-0.304651\pi$
$743$	−11.4495	−	19.8311i	−0.420041	−	0.727532i	−0.995943	+	0.0899863i	$0.971318\pi$
$744$		0			0
$745$		0			0
$746$	−13.5959			−0.497782
$747$		0			0
$748$	20.3485			0.744014
$749$	−3.67423	+	6.36396i	−0.134254	+	0.232534i
$750$		0			0
$751$	26.4949	+	45.8905i	0.966813	+	1.67457i	0.704664	+	0.709541i	$0.251098\pi$
$751$	26.4949	+	45.8905i	0.966813	+	1.67457i	0.262148	+	0.965028i	$0.415569\pi$
$752$	−2.22474	−	3.85337i	−0.0811281	−	0.140518i
$753$		0			0
$754$	−7.34847	+	12.7279i	−0.267615	+	0.463524i
$755$		0			0
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$	−2.07321	+	3.59091i	−0.0753025	+	0.130428i
$759$		0			0
$760$		0			0
$761$	0.247449	+	0.428594i	0.00897001	+	0.0155365i	0.870476	−	0.492212i	$-0.163811\pi$
$761$	0.247449	+	0.428594i	0.00897001	+	0.0155365i	−0.861506	+	0.507748i	$0.830478\pi$
$762$		0			0
$763$	−1.79796	+	3.11416i	−0.0650905	+	0.112740i
$764$	−18.2474			−0.660170
$765$		0			0
$766$	17.7980			0.643066
$767$	16.2247	−	28.1021i	0.585842	−	1.01471i
$768$		0			0
$769$	−12.2474	−	21.2132i	−0.441654	−	0.764968i	0.556158	−	0.831076i	$-0.312275\pi$
$769$	−12.2474	−	21.2132i	−0.441654	−	0.764968i	−0.997812	+	0.0661088i	$0.978942\pi$
$770$		0			0
$771$		0			0
$772$	−6.84847	+	11.8619i	−0.246482	+	0.426919i
$773$	−35.3939			−1.27303			−0.636515	−	0.771265i	$-0.719625\pi$
$773$	−35.3939			−1.27303			−0.636515	+	0.771265i	$0.719625\pi$
$774$		0			0
$775$		0			0
$776$	−6.50000	+	11.2583i	−0.233336	+	0.404151i
$777$		0			0
$778$	8.77526	+	15.1992i	0.314608	+	0.544917i
$779$	2.72474	+	4.71940i	0.0976241	+	0.169090i
$780$		0			0
$781$	−4.22474	+	7.31747i	−0.151173	+	0.261840i
$782$	−40.6969			−1.45532
$783$		0			0
$784$	−6.79796			−0.242784
$785$		0			0
$786$		0			0
$787$	25.6969	+	44.5084i	0.915997	+	1.58655i	0.805435	+	0.592684i	$0.201932\pi$
$787$	25.6969	+	44.5084i	0.915997	+	1.58655i	0.110562	+	0.993869i	$0.464735\pi$
$788$	−4.00000	−	6.92820i	−0.142494	−	0.246807i
$789$		0			0
$790$		0			0
$791$	−2.20204			−0.0782956
$792$		0			0
$793$	10.8990			0.387034
$794$	0.898979	−	1.55708i	0.0319036	−	0.0552586i
$795$		0			0
$796$	−7.77526	−	13.4671i	−0.275587	−	0.477330i
$797$	1.79796	+	3.11416i	0.0636870	+	0.110309i	0.896111	−	0.443830i	$-0.146381\pi$
$797$	1.79796	+	3.11416i	0.0636870	+	0.110309i	−0.832424	+	0.554139i	$0.813047\pi$
$798$		0			0
$799$	−13.1237	+	22.7310i	−0.464284	+	0.804163i
$800$		0			0
$801$		0			0
$802$	−9.20204			−0.324935
$803$	−25.5227	+	44.2066i	−0.900677	+	1.56002i
$804$		0			0
$805$		0			0
$806$	−1.89898	−	3.28913i	−0.0668887	−	0.115855i
$807$		0			0
$808$	−4.00000	+	6.92820i	−0.140720	+	0.243733i
$809$	41.0908			1.44468			0.722338	−	0.691540i	$-0.243067\pi$
$809$	41.0908			1.44468			0.722338	+	0.691540i	$0.243067\pi$
$810$		0			0
$811$	7.24745			0.254492			0.127246	−	0.991871i	$-0.459386\pi$
$811$	7.24745			0.254492			0.127246	+	0.991871i	$0.459386\pi$
$812$	1.34847	−	2.33562i	0.0473220	−	0.0819641i
$813$		0			0
$814$	−13.7980	−	23.8988i	−0.483618	−	0.837651i
$815$		0			0
$816$		0			0
$817$	6.94949	−	12.0369i	0.243132	−	0.421117i
$818$	15.8990			0.555895
$819$		0			0
$820$		0			0
$821$	1.02270	−	1.77138i	0.0356926	−	0.0618214i	−0.847627	−	0.530592i	$-0.821970\pi$
$821$	1.02270	−	1.77138i	0.0356926	−	0.0618214i	0.883320	+	0.468771i	$0.155303\pi$
$822$		0			0
$823$	15.7980	+	27.3629i	0.550682	+	0.953810i	0.998225	+	0.0595473i	$0.0189657\pi$
$823$	15.7980	+	27.3629i	0.550682	+	0.953810i	−0.447543	+	0.894262i	$0.647701\pi$
$824$	7.12372	+	12.3387i	0.248167	+	0.429837i
$825$		0			0
$826$	−2.97730	+	5.15683i	−0.103593	+	0.179429i
$827$	−5.79796			−0.201615			−0.100807	−	0.994906i	$-0.532143\pi$
$827$	−5.79796			−0.201615			−0.100807	+	0.994906i	$0.532143\pi$
$828$		0			0
$829$	−26.4495			−0.918629			−0.459314	−	0.888274i	$-0.651905\pi$
$829$	−26.4495			−0.918629			−0.459314	+	0.888274i	$0.651905\pi$
$830$		0			0
$831$		0			0
$832$	−1.22474	−	2.12132i	−0.0424604	−	0.0735436i
$833$	20.0505	+	34.7285i	0.694709	+	1.20327i
$834$		0			0
$835$		0			0
$836$	−18.7980			−0.650141
$837$		0			0
$838$	8.89898			0.307410
$839$	20.1237	−	34.8553i	0.694748	−	1.20334i	−0.275517	−	0.961296i	$-0.588849\pi$
$839$	20.1237	−	34.8553i	0.694748	−	1.20334i	0.970266	−	0.242043i	$-0.0778175\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$	5.77526	+	10.0030i	0.199028	+	0.344727i
$843$		0			0
$844$	−1.89898	+	3.28913i	−0.0653656	+	0.113216i
$845$		0			0
$846$		0			0
$847$	−0.404082			−0.0138844
$848$	1.77526	−	3.07483i	0.0609625	−	0.105590i
$849$		0			0
$850$		0			0
$851$	27.5959	+	47.7975i	0.945976	+	1.63848i
$852$		0			0
$853$	−4.57321	+	7.92104i	−0.156584	+	0.271211i	−0.933635	−	0.358227i	$-0.883381\pi$
$853$	−4.57321	+	7.92104i	−0.156584	+	0.271211i	0.777051	+	0.629438i	$0.216715\pi$
$854$	−2.00000			−0.0684386
$855$		0			0
$856$	−16.3485			−0.558779
$857$	2.69694	−	4.67123i	0.0921257	−	0.159566i	−0.816280	−	0.577657i	$-0.803967\pi$
$857$	2.69694	−	4.67123i	0.0921257	−	0.159566i	0.908405	+	0.418091i	$0.137301\pi$
$858$		0			0
$859$	18.8712	+	32.6858i	0.643876	+	1.11523i	0.984560	+	0.175048i	$0.0560081\pi$
$859$	18.8712	+	32.6858i	0.643876	+	1.11523i	−0.340684	+	0.940178i	$0.610659\pi$
$860$		0			0
$861$		0			0
$862$	19.1237	−	33.1233i	0.651357	−	1.12818i
$863$	−26.4495			−0.900351			−0.450176	−	0.892940i	$-0.648639\pi$
$863$	−26.4495			−0.900351			−0.450176	+	0.892940i	$0.648639\pi$
$864$		0			0
$865$		0			0
$866$	−11.5000	+	19.9186i	−0.390786	+	0.676861i
$867$		0			0
$868$	0.348469	+	0.603566i	0.0118278	+	0.0204864i
$869$	−12.6742	−	21.9524i	−0.429944	−	0.744685i
$870$		0			0
$871$	−5.57321	+	9.65309i	−0.188841	+	0.327082i
$872$	−8.00000			−0.270914
$873$		0			0
$874$	37.5959			1.27170
$875$		0			0
$876$		0			0
$877$	10.4268	+	18.0597i	0.352088	+	0.609834i	0.986615	−	0.163067i	$-0.0521386\pi$
$877$	10.4268	+	18.0597i	0.352088	+	0.609834i	−0.634527	+	0.772900i	$0.718805\pi$
$878$	−11.0227	−	19.0919i	−0.371998	−	0.644320i
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−6.55051			−0.220442			−0.110221	−	0.993907i	$-0.535156\pi$
$883$	−6.55051			−0.220442			−0.110221	+	0.993907i	$0.535156\pi$
$884$	−7.22474	+	12.5136i	−0.242994	+	0.420879i
$885$		0			0
$886$	10.6237	+	18.4008i	0.356911	+	0.618188i
$887$	9.67423	+	16.7563i	0.324829	+	0.562620i	0.981478	−	0.191576i	$-0.0613599\pi$
$887$	9.67423	+	16.7563i	0.324829	+	0.562620i	−0.656649	+	0.754197i	$0.728027\pi$
$888$		0			0
$889$	1.55051	−	2.68556i	0.0520024	−	0.0900709i
$890$		0			0
$891$		0			0
$892$	9.10102			0.304725
$893$	12.1237	−	20.9989i	0.405705	−	0.702702i
$894$		0			0
$895$		0			0
$896$	0.224745	+	0.389270i	0.00750820	+	0.0130046i
$897$		0			0
$898$	9.39898	−	16.2795i	0.313648	−	0.543254i
$899$	9.30306			0.310274
$900$		0			0
$901$	−20.9444			−0.697759
$902$	−1.72474	+	2.98735i	−0.0574277	+	0.0994677i
$903$		0			0
$904$	−2.44949	−	4.24264i	−0.0814688	−	0.141108i
$905$		0			0
$906$		0			0
$907$	19.8712	−	34.4179i	0.659811	−	1.14283i	−0.320853	−	0.947129i	$-0.603969\pi$
$907$	19.8712	−	34.4179i	0.659811	−	1.14283i	0.980664	−	0.195698i	$-0.0626972\pi$
$908$	3.44949			0.114475
$909$		0			0
$910$		0			0
$911$	12.1237	−	20.9989i	0.401677	−	0.695725i	−0.592252	−	0.805753i	$-0.701761\pi$
$911$	12.1237	−	20.9989i	0.401677	−	0.695725i	0.993928	+	0.110028i	$0.0350942\pi$
$912$		0			0
$913$	−6.89898	−	11.9494i	−0.228323	−	0.395467i
$914$	8.94949	+	15.5010i	0.296023	+	0.512727i
$915$		0			0
$916$	9.22474	−	15.9777i	0.304794	−	0.527919i
$917$	2.20204			0.0727178
$918$		0			0
$919$	1.10102			0.0363193			0.0181597	−	0.999835i	$-0.494219\pi$
$919$	1.10102			0.0363193			0.0181597	+	0.999835i	$0.494219\pi$
$920$		0			0
$921$		0			0
$922$	−1.22474	−	2.12132i	−0.0403348	−	0.0698620i
$923$	−3.00000	−	5.19615i	−0.0987462	−	0.171033i
$924$		0			0
$925$		0			0
$926$	24.0000			0.788689
$927$		0			0
$928$	6.00000			0.196960
$929$	−8.20204	+	14.2064i	−0.269100	+	0.466095i	−0.968630	−	0.248508i	$-0.920060\pi$
$929$	−8.20204	+	14.2064i	−0.269100	+	0.466095i	0.699529	+	0.714604i	$0.253393\pi$
$930$		0			0
$931$	−18.5227	−	32.0823i	−0.607057	−	1.05145i
$932$	−6.84847	−	11.8619i	−0.224329	−	0.388549i
$933$		0			0
$934$	−5.17423	+	8.96204i	−0.169306	+	0.293247i
$935$		0			0
$936$		0			0
$937$	−0.404082			−0.0132008			−0.00660039	−	0.999978i	$-0.502101\pi$
$937$	−0.404082			−0.0132008			−0.00660039	+	0.999978i	$0.502101\pi$
$938$	1.02270	−	1.77138i	0.0333925	−	0.0578374i
$939$		0			0
$940$		0			0
$941$	−15.1010	−	26.1557i	−0.492279	−	0.852653i	0.507681	−	0.861545i	$-0.330503\pi$
$941$	−15.1010	−	26.1557i	−0.492279	−	0.852653i	−0.999960	+	0.00889239i	$0.997169\pi$
$942$		0			0
$943$	3.44949	−	5.97469i	0.112331	−	0.194563i
$944$	−13.2474			−0.431168
$945$		0			0
$946$	8.79796			0.286046
$947$	1.62372	−	2.81237i	0.0527640	−	0.0913898i	−0.838437	−	0.544998i	$-0.816530\pi$
$947$	1.62372	−	2.81237i	0.0527640	−	0.0913898i	0.891201	+	0.453609i	$0.149864\pi$
$948$		0			0
$949$	−18.1237	−	31.3912i	−0.588321	−	1.01900i
$950$		0			0
$951$		0			0
$952$	1.32577	−	2.29629i	0.0429683	−	0.0744233i
$953$	31.2020			1.01073			0.505367	−	0.862905i	$-0.331357\pi$
$953$	31.2020			1.01073			0.505367	+	0.862905i	$0.331357\pi$
$954$		0			0
$955$		0			0
$956$	−0.348469	+	0.603566i	−0.0112703	+	0.0195207i
$957$		0			0
$958$	8.34847	+	14.4600i	0.269727	+	0.467181i
$959$	−0.674235	−	1.16781i	−0.0217722	−	0.0377105i
$960$		0			0
$961$	14.2980	−	24.7648i	0.461224	−	0.798864i
$962$	19.5959			0.631798
$963$		0			0
$964$	1.00000			0.0322078
$965$		0			0
$966$		0			0
$967$	−0.348469	−	0.603566i	−0.0112060	−	0.0194094i	0.860368	−	0.509673i	$-0.170234\pi$
$967$	−0.348469	−	0.603566i	−0.0112060	−	0.0194094i	−0.871574	+	0.490264i	$0.836900\pi$
$968$	−0.449490	−	0.778539i	−0.0144471	−	0.0250232i
$969$		0			0
$970$		0			0
$971$	−35.3939			−1.13584			−0.567922	−	0.823083i	$-0.692252\pi$
$971$	−35.3939			−1.13584			−0.567922	+	0.823083i	$0.692252\pi$
$972$		0			0
$973$	−5.95459			−0.190895
$974$	12.5505	−	21.7381i	0.402144	−	0.696534i
$975$		0			0
$976$	−2.22474	−	3.85337i	−0.0712123	−	0.123343i
$977$	14.0505	+	24.3362i	0.449516	+	0.778584i	0.998354	−	0.0573443i	$-0.0182633\pi$
$977$	14.0505	+	24.3362i	0.449516	+	0.778584i	−0.548839	+	0.835928i	$0.684930\pi$
$978$		0			0
$979$	5.34847	−	9.26382i	0.170938	−	0.296073i
$980$		0			0
$981$		0			0
$982$	−18.5505			−0.591971
$983$	10.6969	−	18.5276i	0.341179	−	0.590940i	−0.643473	−	0.765469i	$-0.722507\pi$
$983$	10.6969	−	18.5276i	0.341179	−	0.590940i	0.984652	+	0.174529i	$0.0558403\pi$
$984$		0			0
$985$		0			0
$986$	−17.6969	−	30.6520i	−0.563585	−	0.976158i
$987$		0			0
$988$	6.67423	−	11.5601i	0.212336	−	0.367776i
$989$	−17.5959			−0.559518
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$	−0.775255	+	1.34278i	−0.0246144	+	0.0426333i
$993$		0			0
$994$	0.550510	+	0.953512i	0.0174611	+	0.0302436i
$995$		0			0
$996$		0			0
$997$	−10.4722	+	18.1384i	−0.331658	+	0.574448i	−0.982837	−	0.184476i	$-0.940941\pi$
$997$	−10.4722	+	18.1384i	−0.331658	+	0.574448i	0.651179	+	0.758924i	$0.274275\pi$
$998$	−21.2474			−0.672576
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.e.j.451.2		4
3.2	odd	2		450.2.e.n.151.1		4
5.2	odd	4		270.2.i.b.19.1		8
5.3	odd	4		270.2.i.b.19.4		8
5.4	even	2		1350.2.e.m.451.1		4
9.2	odd	6		4050.2.a.bq.1.1		2
9.4	even	3	inner	1350.2.e.j.901.2		4
9.5	odd	6		450.2.e.n.301.1		4
9.7	even	3		4050.2.a.bz.1.1		2
15.2	even	4		90.2.i.b.79.3	yes	8
15.8	even	4		90.2.i.b.79.2	yes	8
15.14	odd	2		450.2.e.k.151.2		4
20.3	even	4		2160.2.by.d.289.3		8
20.7	even	4		2160.2.by.d.289.2		8
45.2	even	12		810.2.c.f.649.2		4
45.4	even	6		1350.2.e.m.901.1		4
45.7	odd	12		810.2.c.e.649.3		4
45.13	odd	12		270.2.i.b.199.1		8
45.14	odd	6		450.2.e.k.301.2		4
45.22	odd	12		270.2.i.b.199.4		8
45.23	even	12		90.2.i.b.49.3	yes	8
45.29	odd	6		4050.2.a.bs.1.2		2
45.32	even	12		90.2.i.b.49.2	✓	8
45.34	even	6		4050.2.a.bm.1.2		2
45.38	even	12		810.2.c.f.649.4		4
45.43	odd	12		810.2.c.e.649.1		4
60.23	odd	4		720.2.by.c.529.3		8
60.47	odd	4		720.2.by.c.529.2		8
180.23	odd	12		720.2.by.c.49.2		8
180.67	even	12		2160.2.by.d.1009.3		8
180.103	even	12		2160.2.by.d.1009.2		8
180.167	odd	12		720.2.by.c.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.b.49.2	✓	8	45.32	even	12
90.2.i.b.49.3	yes	8	45.23	even	12
90.2.i.b.79.2	yes	8	15.8	even	4
90.2.i.b.79.3	yes	8	15.2	even	4
270.2.i.b.19.1		8	5.2	odd	4
270.2.i.b.19.4		8	5.3	odd	4
270.2.i.b.199.1		8	45.13	odd	12
270.2.i.b.199.4		8	45.22	odd	12
450.2.e.k.151.2		4	15.14	odd	2
450.2.e.k.301.2		4	45.14	odd	6
450.2.e.n.151.1		4	3.2	odd	2
450.2.e.n.301.1		4	9.5	odd	6
720.2.by.c.49.2		8	180.23	odd	12
720.2.by.c.49.3		8	180.167	odd	12
720.2.by.c.529.2		8	60.47	odd	4
720.2.by.c.529.3		8	60.23	odd	4
810.2.c.e.649.1		4	45.43	odd	12
810.2.c.e.649.3		4	45.7	odd	12
810.2.c.f.649.2		4	45.2	even	12
810.2.c.f.649.4		4	45.38	even	12
1350.2.e.j.451.2		4	1.1	even	1	trivial
1350.2.e.j.901.2		4	9.4	even	3	inner
1350.2.e.m.451.1		4	5.4	even	2
1350.2.e.m.901.1		4	45.4	even	6
2160.2.by.d.289.2		8	20.7	even	4
2160.2.by.d.289.3		8	20.3	even	4
2160.2.by.d.1009.2		8	180.103	even	12
2160.2.by.d.1009.3		8	180.67	even	12
4050.2.a.bm.1.2		2	45.34	even	6
4050.2.a.bq.1.1		2	9.2	odd	6
4050.2.a.bs.1.2		2	45.29	odd	6
4050.2.a.bz.1.1		2	9.7	even	3

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.224745 - 0.389270i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(0.224745 - 0.389270i) q^{7} +1.00000 q^{8} +(-1.72474 + 2.98735i) q^{11} +(-1.22474 - 2.12132i) q^{13} +(0.224745 + 0.389270i) q^{14} +(-0.500000 + 0.866025i) q^{16} +5.89898 q^{17} -5.44949 q^{19} +(-1.72474 - 2.98735i) q^{22} +(3.44949 + 5.97469i) q^{23} +2.44949 q^{26} -0.449490 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-0.775255 - 1.34278i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-2.94949 + 5.10867i) q^{34} +8.00000 q^{37} +(2.72474 - 4.71940i) q^{38} +(-0.500000 - 0.866025i) q^{41} +(-1.27526 + 2.20881i) q^{43} +3.44949 q^{44} -6.89898 q^{46} +(-2.22474 + 3.85337i) q^{47} +(3.39898 + 5.88721i) q^{49} +(-1.22474 + 2.12132i) q^{52} -3.55051 q^{53} +(0.224745 - 0.389270i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(6.62372 + 11.4726i) q^{59} +(-2.22474 + 3.85337i) q^{61} +1.55051 q^{62} +1.00000 q^{64} +(-2.27526 - 3.94086i) q^{67} +(-2.94949 - 5.10867i) q^{68} +2.44949 q^{71} +14.7980 q^{73} +(-4.00000 + 6.92820i) q^{74} +(2.72474 + 4.71940i) q^{76} +(0.775255 + 1.34278i) q^{77} +(-3.67423 + 6.36396i) q^{79} +1.00000 q^{82} +(-2.00000 + 3.46410i) q^{83} +(-1.27526 - 2.20881i) q^{86} +(-1.72474 + 2.98735i) q^{88} -3.10102 q^{89} -1.10102 q^{91} +(3.44949 - 5.97469i) q^{92} +(-2.22474 - 3.85337i) q^{94} +(-6.50000 + 11.2583i) q^{97} -6.79796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^7 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} - 4 q^{7} + 4 q^{8} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{17} - 12 q^{19} - 2 q^{22} + 4 q^{23} + 8 q^{28} - 12 q^{29} - 8 q^{31} - 2 q^{32} - 2 q^{34} + 32 q^{37} + 6 q^{38} - 2 q^{41} - 10 q^{43} + 4 q^{44} - 8 q^{46} - 4 q^{47} - 6 q^{49} - 24 q^{53} - 4 q^{56} - 12 q^{58} + 2 q^{59} - 4 q^{61} + 16 q^{62} + 4 q^{64} - 14 q^{67} - 2 q^{68} + 20 q^{73} - 16 q^{74} + 6 q^{76} + 8 q^{77} + 4 q^{82} - 8 q^{83} - 10 q^{86} - 2 q^{88} - 32 q^{89} - 24 q^{91} + 4 q^{92} - 4 q^{94} - 26 q^{97} + 12 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 - 4 * q^7 + 4 * q^8 - 2 * q^11 - 4 * q^14 - 2 * q^16 + 4 * q^17 - 12 * q^19 - 2 * q^22 + 4 * q^23 + 8 * q^28 - 12 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^34 + 32 * q^37 + 6 * q^38 - 2 * q^41 - 10 * q^43 + 4 * q^44 - 8 * q^46 - 4 * q^47 - 6 * q^49 - 24 * q^53 - 4 * q^56 - 12 * q^58 + 2 * q^59 - 4 * q^61 + 16 * q^62 + 4 * q^64 - 14 * q^67 - 2 * q^68 + 20 * q^73 - 16 * q^74 + 6 * q^76 + 8 * q^77 + 4 * q^82 - 8 * q^83 - 10 * q^86 - 2 * q^88 - 32 * q^89 - 24 * q^91 + 4 * q^92 - 4 * q^94 - 26 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more