LMFDB - Embedded newform 684.2.j.a.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(49,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 4]))

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

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

chi := DirichletCharacter("684.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.j (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$5.46176749826$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.4
Character	$\chi$	$=$	684.49
Dual form			684.2.j.a.349.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.55715 - 0.758482i) q^{3} +(-0.482969 - 0.836526i) q^{5} +(0.302844 + 0.524541i) q^{7} +(1.84941 + 2.36214i) q^{9} +(1.73310 + 3.00181i) q^{11} -3.51890 q^{13} +(0.117562 + 1.66892i) q^{15} +(-0.224370 + 0.388620i) q^{17} +(4.35090 + 0.263993i) q^{19} +(-0.0737171 - 1.04649i) q^{21} +5.73635 q^{23} +(2.03348 - 3.52210i) q^{25} +(-1.08816 - 5.08094i) q^{27} +(-3.34350 + 5.79111i) q^{29} +(3.03763 - 5.26133i) q^{31} +(-0.421863 - 5.98878i) q^{33} +(0.292528 - 0.506674i) q^{35} +0.0149820 q^{37} +(5.47944 + 2.66902i) q^{39} +(4.60225 + 7.97133i) q^{41} +12.7725 q^{43} +(1.08278 - 2.68792i) q^{45} +(4.76208 - 8.24816i) q^{47} +(3.31657 - 5.74447i) q^{49} +(0.644138 - 0.434958i) q^{51} +(2.06576 + 3.57800i) q^{53} +(1.67406 - 2.89956i) q^{55} +(-6.57475 - 3.71116i) q^{57} +(2.24012 + 3.88000i) q^{59} +(-6.10177 + 10.5686i) q^{61} +(-0.678956 + 1.68545i) q^{63} +(1.69952 + 2.94365i) q^{65} -0.961656 q^{67} +(-8.93233 - 4.35092i) q^{69} +(-5.31499 + 9.20582i) q^{71} +(3.99363 - 6.91717i) q^{73} +(-5.83788 + 3.94206i) q^{75} +(-1.04972 + 1.81816i) q^{77} +3.70000 q^{79} +(-2.15938 + 8.73711i) q^{81} +(-3.23115 - 5.59652i) q^{83} +0.433455 q^{85} +(9.59877 - 6.48162i) q^{87} +(-6.55308 - 11.3503i) q^{89} +(-1.06568 - 1.84581i) q^{91} +(-8.72066 + 5.88867i) q^{93} +(-1.88051 - 3.76714i) q^{95} +6.25397 q^{97} +(-3.88548 + 9.64539i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 2 q^{3} - q^{7} - 2 q^{9} - q^{11} + 2 q^{13} + q^{15} + 5 q^{17} + q^{19} + 6 q^{21} + 8 q^{23} - 20 q^{25} + 7 q^{27} - 9 q^{29} + 2 q^{31} - q^{33} - 6 q^{35} + 2 q^{37} + 3 q^{39} - 25 q^{41}+ \cdots + 28 q^{99}+O(q^{100}) $ 40 * q - 2 * q^3 - q^7 - 2 * q^9 - q^11 + 2 * q^13 + q^15 + 5 * q^17 + q^19 + 6 * q^21 + 8 * q^23 - 20 * q^25 + 7 * q^27 - 9 * q^29 + 2 * q^31 - q^33 - 6 * q^35 + 2 * q^37 + 3 * q^39 - 25 * q^41 - 16 * q^43 + 44 * q^45 - 10 * q^47 - 21 * q^49 + 19 * q^51 - 8 * q^53 - 22 * q^59 - 7 * q^61 + 10 * q^63 - 6 * q^65 + 20 * q^67 + 38 * q^69 - 5 * q^71 - 10 * q^73 - 10 * q^77 + 2 * q^79 + 22 * q^81 + q^83 - 52 * q^87 - 34 * q^89 + 16 * q^91 - 8 * q^93 + 36 * q^95 + 2 * q^97 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.55715	−	0.758482i	−0.899019	−	0.437910i
$3$	−1.55715	−	0.758482i	−0.899019	−	0.437910i
$4$		0			0
$5$	−0.482969	−	0.836526i	−0.215990	−	0.374106i	0.737588	−	0.675251i	$-0.235965\pi$
$5$	−0.482969	−	0.836526i	−0.215990	−	0.374106i	−0.953578	+	0.301145i	$0.902631\pi$
$6$		0			0
$7$	0.302844	+	0.524541i	0.114464	+	0.198258i	0.917565	−	0.397585i	$-0.130152\pi$
$7$	0.302844	+	0.524541i	0.114464	+	0.198258i	−0.803101	+	0.595843i	$0.796818\pi$
$8$		0			0
$9$	1.84941	+	2.36214i	0.616470	+	0.787379i
$10$		0			0
$11$	1.73310	+	3.00181i	0.522548	+	0.905080i	0.999656	+	0.0262353i	$0.00835190\pi$
$11$	1.73310	+	3.00181i	0.522548	+	0.905080i	−0.477107	+	0.878845i	$0.658315\pi$
$12$		0			0
$13$	−3.51890			−0.975967			−0.487983	−	0.872853i	$-0.662267\pi$
$13$	−3.51890			−0.975967			−0.487983	+	0.872853i	$0.662267\pi$
$14$		0			0
$15$	0.117562	+	1.66892i	0.0303545	+	0.430912i
$16$		0			0
$17$	−0.224370	+	0.388620i	−0.0544177	+	0.0942542i	−0.891951	−	0.452132i	$-0.850664\pi$
$17$	−0.224370	+	0.388620i	−0.0544177	+	0.0942542i	0.837533	+	0.546386i	$0.183997\pi$
$18$		0			0
$19$	4.35090	+	0.263993i	0.998164	+	0.0605642i
$19$	4.35090	+	0.263993i	0.998164	+	0.0605642i
$20$		0			0
$21$	−0.0737171	−	1.04649i	−0.0160864	−	0.228363i
$22$		0			0
$23$	5.73635			1.19611			0.598055	−	0.801455i	$-0.295940\pi$
$23$	5.73635			1.19611			0.598055	+	0.801455i	$0.295940\pi$
$24$		0			0
$25$	2.03348	−	3.52210i	0.406697	−	0.704419i
$26$		0			0
$27$	−1.08816	−	5.08094i	−0.209417	−	0.977827i
$28$		0			0
$29$	−3.34350	+	5.79111i	−0.620872	+	1.07538i	0.368452	+	0.929647i	$0.379888\pi$
$29$	−3.34350	+	5.79111i	−0.620872	+	1.07538i	−0.989324	+	0.145735i	$0.953445\pi$
$30$		0			0
$31$	3.03763	−	5.26133i	0.545574	−	0.944962i	−0.452996	−	0.891512i	$-0.649645\pi$
$31$	3.03763	−	5.26133i	0.545574	−	0.944962i	0.998571	−	0.0534499i	$-0.0170217\pi$
$32$		0			0
$33$	−0.421863	−	5.98878i	−0.0734370	−	1.04251i
$34$		0			0
$35$	0.292528	−	0.506674i	0.0494463	−	0.0856435i
$36$		0			0
$37$	0.0149820			0.00246303			0.00123151	−	0.999999i	$-0.499608\pi$
$37$	0.0149820			0.00246303			0.00123151	+	0.999999i	$0.499608\pi$
$38$		0			0
$39$	5.47944	+	2.66902i	0.877412	+	0.427386i
$40$		0			0
$41$	4.60225	+	7.97133i	0.718751	+	1.24491i	0.961495	+	0.274822i	$0.0886190\pi$
$41$	4.60225	+	7.97133i	0.718751	+	1.24491i	−0.242744	+	0.970090i	$0.578048\pi$
$42$		0			0
$43$	12.7725			1.94778			0.973891	−	0.227015i	$-0.0728966\pi$
$43$	12.7725			1.94778			0.973891	+	0.227015i	$0.0728966\pi$
$44$		0			0
$45$	1.08278	−	2.68792i	0.161412	−	0.400691i
$46$		0			0
$47$	4.76208	−	8.24816i	0.694620	−	1.20312i	−0.275688	−	0.961247i	$-0.588906\pi$
$47$	4.76208	−	8.24816i	0.694620	−	1.20312i	0.970309	−	0.241871i	$-0.0777609\pi$
$48$		0			0
$49$	3.31657	−	5.74447i	0.473796	−	0.820638i
$50$		0			0
$51$	0.644138	−	0.434958i	0.0901974	−	0.0609063i
$52$		0			0
$53$	2.06576	+	3.57800i	0.283754	+	0.491476i	0.972306	−	0.233711i	$-0.0750869\pi$
$53$	2.06576	+	3.57800i	0.283754	+	0.491476i	−0.688553	+	0.725186i	$0.741754\pi$
$54$		0			0
$55$	1.67406	−	2.89956i	0.225731	−	0.390977i
$56$		0			0
$57$	−6.57475	−	3.71116i	−0.870847	−	0.491555i
$58$		0			0
$59$	2.24012	+	3.88000i	0.291639	+	0.505134i	0.974197	−	0.225697i	$-0.0724661\pi$
$59$	2.24012	+	3.88000i	0.291639	+	0.505134i	−0.682558	+	0.730831i	$0.739133\pi$
$60$		0			0
$61$	−6.10177	+	10.5686i	−0.781252	+	1.35317i	0.149961	+	0.988692i	$0.452085\pi$
$61$	−6.10177	+	10.5686i	−0.781252	+	1.35317i	−0.931213	+	0.364476i	$0.881248\pi$
$62$		0			0
$63$	−0.678956	+	1.68545i	−0.0855404	+	0.212347i
$64$		0			0
$65$	1.69952	+	2.94365i	0.210799	+	0.365115i
$66$		0			0
$67$	−0.961656			−0.117485			−0.0587425	−	0.998273i	$-0.518709\pi$
$67$	−0.961656			−0.117485			−0.0587425	+	0.998273i	$0.518709\pi$
$68$		0			0
$69$	−8.93233	−	4.35092i	−1.07533	−	0.523789i
$70$		0			0
$71$	−5.31499	+	9.20582i	−0.630773	+	1.09253i	0.356621	+	0.934249i	$0.383929\pi$
$71$	−5.31499	+	9.20582i	−0.630773	+	1.09253i	−0.987394	+	0.158281i	$0.949405\pi$
$72$		0			0
$73$	3.99363	−	6.91717i	0.467419	−	0.809594i	−0.531888	−	0.846815i	$-0.678517\pi$
$73$	3.99363	−	6.91717i	0.467419	−	0.809594i	0.999307	+	0.0372212i	$0.0118506\pi$
$74$		0			0
$75$	−5.83788	+	3.94206i	−0.674100	+	0.455189i
$76$		0			0
$77$	−1.04972	+	1.81816i	−0.119626	+	0.207199i
$78$		0			0
$79$	3.70000			0.416283			0.208141	−	0.978099i	$-0.433259\pi$
$79$	3.70000			0.416283			0.208141	+	0.978099i	$0.433259\pi$
$80$		0			0
$81$	−2.15938	+	8.73711i	−0.239931	+	0.970790i
$82$		0			0
$83$	−3.23115	−	5.59652i	−0.354665	−	0.614298i	0.632396	−	0.774646i	$-0.282072\pi$
$83$	−3.23115	−	5.59652i	−0.354665	−	0.614298i	−0.987061	+	0.160348i	$0.948738\pi$
$84$		0			0
$85$	0.433455			0.0470147
$86$		0			0
$87$	9.59877	−	6.48162i	1.02910	−	0.694902i
$88$		0			0
$89$	−6.55308	−	11.3503i	−0.694625	−	1.20313i	−0.970307	−	0.241877i	$-0.922237\pi$
$89$	−6.55308	−	11.3503i	−0.694625	−	1.20313i	0.275682	−	0.961249i	$-0.411096\pi$
$90$		0			0
$91$	−1.06568	−	1.84581i	−0.111713	−	0.193493i
$92$		0			0
$93$	−8.72066	+	5.88867i	−0.904290	+	0.610626i
$94$		0			0
$95$	−1.88051	−	3.76714i	−0.192936	−	0.386500i
$96$		0			0
$97$	6.25397			0.634995			0.317497	−	0.948259i	$-0.397158\pi$
$97$	6.25397			0.634995			0.317497	+	0.948259i	$0.397158\pi$
$98$		0			0
$99$	−3.88548	+	9.64539i	−0.390506	+	0.969398i
$100$		0			0
$101$	−3.49968	+	6.06162i	−0.348231	+	0.603154i	−0.985935	−	0.167128i	$-0.946551\pi$
$101$	−3.49968	+	6.06162i	−0.348231	+	0.603154i	0.637704	+	0.770281i	$0.279884\pi$
$102$		0			0
$103$	−6.34696	+	10.9933i	−0.625385	+	1.08320i	0.363081	+	0.931757i	$0.381725\pi$
$103$	−6.34696	+	10.9933i	−0.625385	+	1.08320i	−0.988466	+	0.151441i	$0.951609\pi$
$104$		0			0
$105$	−0.839813	+	0.567088i	−0.0819573	+	0.0553421i
$106$		0			0
$107$	2.32136			0.224415			0.112207	−	0.993685i	$-0.464208\pi$
$107$	2.32136			0.224415			0.112207	+	0.993685i	$0.464208\pi$
$108$		0			0
$109$	5.16363	−	8.94368i	0.494586	−	0.856649i	−0.505394	−	0.862889i	$-0.668653\pi$
$109$	5.16363	−	8.94368i	0.494586	−	0.856649i	0.999981	+	0.00623986i	$0.00198622\pi$
$110$		0			0
$111$	−0.0233292	−	0.0113636i	−0.00221431	−	0.00107858i
$112$		0			0
$113$	−2.18832	+	3.79029i	−0.205860	+	0.356560i	−0.950406	−	0.311011i	$-0.899333\pi$
$113$	−2.18832	+	3.79029i	−0.205860	+	0.356560i	0.744546	+	0.667571i	$0.232666\pi$
$114$		0			0
$115$	−2.77047	−	4.79860i	−0.258348	−	0.447472i
$116$		0			0
$117$	−6.50788	−	8.31212i	−0.601654	−	0.768455i
$118$		0			0
$119$	−0.271796			−0.0249155
$120$		0			0
$121$	−0.507248	+	0.878579i	−0.0461135	+	0.0798708i
$122$		0			0
$123$	−1.12026	−	15.9033i	−0.101011	−	1.43395i
$124$		0			0
$125$	−8.75812			−0.783350
$126$		0			0
$127$	1.08855	+	1.88542i	0.0965929	+	0.167304i	0.910272	−	0.414010i	$-0.135872\pi$
$127$	1.08855	+	1.88542i	0.0965929	+	0.167304i	−0.813679	+	0.581314i	$0.802539\pi$
$128$		0			0
$129$	−19.8886	−	9.68769i	−1.75109	−	0.852954i
$130$		0			0
$131$	7.54805	+	13.0736i	0.659476	+	1.14225i	0.980751	+	0.195260i	$0.0625551\pi$
$131$	7.54805	+	13.0736i	0.659476	+	1.14225i	−0.321276	+	0.946986i	$0.604112\pi$
$132$		0			0
$133$	1.17917	+	2.36217i	0.102247	+	0.204827i
$134$		0			0
$135$	−3.72479	+	3.36421i	−0.320579	+	0.289545i
$136$		0			0
$137$	−5.71756	+	9.90310i	−0.488484	+	0.846079i	−0.999912	−	0.0132471i	$-0.995783\pi$
$137$	−5.71756	+	9.90310i	−0.488484	+	0.846079i	0.511428	+	0.859326i	$0.329117\pi$
$138$		0			0
$139$	−0.379169			−0.0321607			−0.0160804	−	0.999871i	$-0.505119\pi$
$139$	−0.379169			−0.0321607			−0.0160804	+	0.999871i	$0.505119\pi$
$140$		0			0
$141$	−13.6713	+	9.23164i	−1.15133	+	0.777444i
$142$		0			0
$143$	−6.09859	−	10.5631i	−0.509990	−	0.883328i
$144$		0			0
$145$	6.45922			0.536409
$146$		0			0
$147$	−9.52147	+	6.42942i	−0.785317	+	0.530289i
$148$		0			0
$149$	11.4335	+	19.8033i	0.936665	+	1.62235i	0.771638	+	0.636062i	$0.219438\pi$
$149$	11.4335	+	19.8033i	0.936665	+	1.62235i	0.165027	+	0.986289i	$0.447229\pi$
$150$		0			0
$151$	−1.96099	−	3.39653i	−0.159583	−	0.276406i	0.775136	−	0.631795i	$-0.217682\pi$
$151$	−1.96099	−	3.39653i	−0.159583	−	0.276406i	−0.934718	+	0.355390i	$0.884348\pi$
$152$		0			0
$153$	−1.33293	+	0.188725i	−0.107761	+	0.0152575i
$154$		0			0
$155$	−5.86832			−0.471355
$156$		0			0
$157$	−6.83695	−	11.8419i	−0.545648	−	0.945090i	−0.998566	−	0.0535379i	$-0.982950\pi$
$157$	−6.83695	−	11.8419i	−0.545648	−	0.945090i	0.452918	−	0.891552i	$-0.350383\pi$
$158$		0			0
$159$	−0.502838	−	7.13830i	−0.0398777	−	0.566104i
$160$		0			0
$161$	1.73722	+	3.00895i	0.136912	+	0.237139i
$162$		0			0
$163$	−16.9820			−1.33013			−0.665067	−	0.746784i	$-0.731597\pi$
$163$	−16.9820			−1.33013			−0.665067	+	0.746784i	$0.731597\pi$
$164$		0			0
$165$	−4.80603	+	3.24529i	−0.374149	+	0.252646i
$166$		0			0
$167$	15.8316			1.22509			0.612543	−	0.790437i	$-0.290147\pi$
$167$	15.8316			1.22509			0.612543	+	0.790437i	$0.290147\pi$
$168$		0			0
$169$	−0.617360			−0.0474892
$170$		0			0
$171$	7.42300	+	10.7656i	0.567651	+	0.823269i
$172$		0			0
$173$	−4.85976			−0.369481			−0.184740	−	0.982787i	$-0.559144\pi$
$173$	−4.85976			−0.369481			−0.184740	+	0.982787i	$0.559144\pi$
$174$		0			0
$175$	2.46331			0.186209
$176$		0			0
$177$	−0.545281	−	7.74083i	−0.0409859	−	0.581836i
$178$		0			0
$179$	−14.6950			−1.09836			−0.549180	−	0.835704i	$-0.685060\pi$
$179$	−14.6950			−1.09836			−0.549180	+	0.835704i	$0.685060\pi$
$180$		0			0
$181$	−10.2340	−	17.7258i	−0.760685	−	1.31755i	−0.942498	−	0.334213i	$-0.891530\pi$
$181$	−10.2340	−	17.7258i	−0.760685	−	1.31755i	0.181812	−	0.983333i	$-0.441804\pi$
$182$		0			0
$183$	17.5174	−	11.8287i	1.29493	−	0.874405i
$184$		0			0
$185$	−0.00723584	−	0.0125328i	−0.000531989	−	0.000921433i
$186$		0			0
$187$	−1.55542			−0.113744
$188$		0			0
$189$	2.33562	−	2.10952i	0.169891	−	0.153445i
$190$		0			0
$191$	4.75649	+	8.23848i	0.344167	+	0.596115i	0.985202	−	0.171396i	$-0.0548279\pi$
$191$	4.75649	+	8.23848i	0.344167	+	0.596115i	−0.641035	+	0.767512i	$0.721495\pi$
$192$		0			0
$193$	5.00419	+	8.66751i	0.360209	+	0.623901i	0.987995	−	0.154485i	$-0.0493720\pi$
$193$	5.00419	+	8.66751i	0.360209	+	0.623901i	−0.627786	+	0.778386i	$0.716039\pi$
$194$		0			0
$195$	−0.413690	−	5.87275i	−0.0296249	−	0.420556i
$196$		0			0
$197$	−10.4934			−0.747623			−0.373812	−	0.927505i	$-0.621949\pi$
$197$	−10.4934			−0.747623			−0.373812	+	0.927505i	$0.621949\pi$
$198$		0			0
$199$	1.49793	+	2.59449i	0.106185	+	0.183919i	0.914222	−	0.405214i	$-0.132803\pi$
$199$	1.49793	+	2.59449i	0.106185	+	0.183919i	−0.808036	+	0.589132i	$0.799470\pi$
$200$		0			0
$201$	1.49744	+	0.729400i	0.105621	+	0.0514479i
$202$		0			0
$203$	−4.05024			−0.284271
$204$		0			0
$205$	4.44548	−	7.69980i	0.310486	−	0.537778i
$206$		0			0
$207$	10.6088	+	13.5500i	0.737366	+	0.941792i
$208$		0			0
$209$	6.74807	+	13.5181i	0.466774	+	0.935066i
$210$		0			0
$211$	6.00832	+	10.4067i	0.413630	+	0.716428i	0.995284	−	0.0970085i	$-0.0309274\pi$
$211$	6.00832	+	10.4067i	0.413630	+	0.716428i	−0.581654	+	0.813437i	$0.697594\pi$
$212$		0			0
$213$	15.2587	−	10.3035i	1.04551	−	0.705984i
$214$		0			0
$215$	−6.16870	−	10.6845i	−0.420702	−	0.728677i
$216$		0			0
$217$	3.67971			0.249795
$218$		0			0
$219$	−11.4652	+	7.74195i	−0.774748	+	0.523152i
$220$		0			0
$221$	0.789535	−	1.36751i	0.0531099	−	0.0919890i
$222$		0			0
$223$	−2.08522			−0.139637			−0.0698183	−	0.997560i	$-0.522242\pi$
$223$	−2.08522			−0.139637			−0.0698183	+	0.997560i	$0.522242\pi$
$224$		0			0
$225$	12.0804	−	1.71043i	0.805361	−	0.114029i
$226$		0			0
$227$	1.35735	+	2.35099i	0.0900902	+	0.156041i	0.907549	−	0.419946i	$-0.137951\pi$
$227$	1.35735	+	2.35099i	0.0900902	+	0.156041i	−0.817459	+	0.575987i	$0.804618\pi$
$228$		0			0
$229$	6.68386	−	11.5768i	0.441682	−	0.765016i	−0.556132	−	0.831094i	$-0.687715\pi$
$229$	6.68386	−	11.5768i	0.441682	−	0.765016i	0.997814	+	0.0660780i	$0.0210486\pi$
$230$		0			0
$231$	3.01361	−	2.03495i	0.198281	−	0.133890i
$232$		0			0
$233$	10.3831	−	17.9840i	0.680218	−	1.17817i	−0.294696	−	0.955591i	$-0.595219\pi$
$233$	10.3831	−	17.9840i	0.680218	−	1.17817i	0.974914	−	0.222581i	$-0.0714482\pi$
$234$		0			0
$235$	−9.19973			−0.600124
$236$		0			0
$237$	−5.76145	−	2.80639i	−0.374246	−	0.182295i
$238$		0			0
$239$	13.4560	−	23.3064i	0.870395	−	1.50757i	0.00880603	−	0.999961i	$-0.497197\pi$
$239$	13.4560	−	23.3064i	0.870395	−	1.50757i	0.861589	−	0.507607i	$-0.169470\pi$
$240$		0			0
$241$	−7.22192	+	12.5087i	−0.465205	+	0.805758i	−0.999211	−	0.0397224i	$-0.987353\pi$
$241$	−7.22192	+	12.5087i	−0.465205	+	0.805758i	0.534006	+	0.845481i	$0.320686\pi$
$242$		0			0
$243$	9.98941	−	11.9671i	0.640821	−	0.767690i
$244$		0			0
$245$	−6.40720			−0.409341
$246$		0			0
$247$	−15.3104	−	0.928966i	−0.974175	−	0.0591087i
$248$		0			0
$249$	0.786514	+	11.1654i	0.0498433	+	0.707577i
$250$		0			0
$251$	2.64721	+	4.58510i	0.167090	+	0.289409i	0.937396	−	0.348266i	$-0.113230\pi$
$251$	2.64721	+	4.58510i	0.167090	+	0.289409i	−0.770305	+	0.637675i	$0.779896\pi$
$252$		0			0
$253$	9.94164	+	17.2194i	0.625026	+	1.08258i
$254$		0			0
$255$	−0.674952	−	0.328768i	−0.0422671	−	0.0205882i
$256$		0			0
$257$	−12.8038			−0.798678			−0.399339	−	0.916803i	$-0.630760\pi$
$257$	−12.8038			−0.798678			−0.399339	+	0.916803i	$0.630760\pi$
$258$		0			0
$259$	0.00453721	+	0.00785868i	0.000281929	+	0.000488315i
$260$		0			0
$261$	−19.8629	+	2.81233i	−1.22948	+	0.174079i
$262$		0			0
$263$	−4.47388			−0.275871			−0.137936	−	0.990441i	$-0.544047\pi$
$263$	−4.47388			−0.275871			−0.137936	+	0.990441i	$0.544047\pi$
$264$		0			0
$265$	1.99539	−	3.45612i	0.122576	−	0.212308i
$266$		0			0
$267$	1.59512	+	22.6444i	0.0976200	+	1.38582i
$268$		0			0
$269$	11.7219	−	20.3030i	0.714700	−	1.23790i	−0.248375	−	0.968664i	$-0.579897\pi$
$269$	11.7219	−	20.3030i	0.714700	−	1.23790i	0.963075	−	0.269232i	$-0.0867700\pi$
$270$		0			0
$271$	−7.23016	+	12.5230i	−0.439201	+	0.760718i	−0.997628	−	0.0688354i	$-0.978072\pi$
$271$	−7.23016	+	12.5230i	−0.439201	+	0.760718i	0.558427	+	0.829554i	$0.311405\pi$
$272$		0			0
$273$	0.259403	+	3.68249i	0.0156998	+	0.222874i
$274$		0			0
$275$	14.0969			0.850074
$276$		0			0
$277$	−3.01769	−	5.22679i	−0.181315	−	0.314047i	0.761013	−	0.648736i	$-0.224702\pi$
$277$	−3.01769	−	5.22679i	−0.181315	−	0.314047i	−0.942329	+	0.334689i	$0.891369\pi$
$278$		0			0
$279$	18.0458	−	2.55505i	1.08037	−	0.152967i
$280$		0			0
$281$	5.64502	−	9.77745i	0.336753	−	0.583274i	−0.647067	−	0.762433i	$-0.724004\pi$
$281$	5.64502	−	9.77745i	0.336753	−	0.583274i	0.983820	+	0.179159i	$0.0573378\pi$
$282$		0			0
$283$	9.05358	+	15.6813i	0.538179	+	0.932154i	0.999002	+	0.0446618i	$0.0142210\pi$
$283$	9.05358	+	15.6813i	0.538179	+	0.932154i	−0.460823	+	0.887492i	$0.652446\pi$
$284$		0			0
$285$	0.0709184	+	7.29232i	0.00420085	+	0.431960i
$286$		0			0
$287$	−2.78753	+	4.82814i	−0.164543	+	0.284996i
$288$		0			0
$289$	8.39932	+	14.5480i	0.494077	+	0.855767i
$290$		0			0
$291$	−9.73835	−	4.74353i	−0.570872	−	0.278071i
$292$		0			0
$293$	3.95827	−	6.85592i	0.231245	−	0.400527i	−0.726930	−	0.686711i	$-0.759054\pi$
$293$	3.95827	−	6.85592i	0.231245	−	0.400527i	0.958175	+	0.286184i	$0.0923869\pi$
$294$		0			0
$295$	2.16382	−	3.74784i	0.125982	−	0.218208i
$296$		0			0
$297$	13.3661	−	12.0722i	0.775581	−	0.700500i
$298$		0			0
$299$	−20.1856			−1.16736
$300$		0			0
$301$	3.86807	+	6.69969i	0.222952	+	0.386163i
$302$		0			0
$303$	10.0471	−	6.78438i	0.577193	−	0.389753i
$304$		0			0
$305$	11.7879			0.674971
$306$		0			0
$307$	14.0319	−	24.3039i	0.800840	−	1.38710i	−0.118224	−	0.992987i	$-0.537720\pi$
$307$	14.0319	−	24.3039i	0.800840	−	1.38710i	0.919064	−	0.394109i	$-0.128947\pi$
$308$		0			0
$309$	18.2213	−	12.3041i	1.03658	−	0.699953i
$310$		0			0
$311$	2.02934	−	3.51492i	0.115073	−	0.199313i	−0.802736	−	0.596335i	$-0.796623\pi$
$311$	2.02934	−	3.51492i	0.115073	−	0.199313i	0.917809	+	0.397022i	$0.129956\pi$
$312$		0			0
$313$	6.11043	−	10.5836i	0.345382	−	0.598219i	−0.640041	−	0.768341i	$-0.721083\pi$
$313$	6.11043	−	10.5836i	0.345382	−	0.598219i	0.985423	+	0.170122i	$0.0544160\pi$
$314$		0			0
$315$	1.73784	−	0.246055i	0.0979161	−	0.0138637i
$316$		0			0
$317$	−10.6745	+	18.4887i	−0.599539	+	1.03843i	0.393351	+	0.919389i	$0.371316\pi$
$317$	−10.6745	+	18.4887i	−0.599539	+	1.03843i	−0.992889	+	0.119043i	$0.962017\pi$
$318$		0			0
$319$	−23.1784			−1.29774
$320$		0			0
$321$	−3.61470	−	1.76071i	−0.201753	−	0.0982734i
$322$		0			0
$323$	−1.07880	+	1.63161i	−0.0600262	+	0.0907854i
$324$		0			0
$325$	−7.15562	+	12.3939i	−0.396922	+	0.687489i
$326$		0			0
$327$	−14.8242	+	10.0101i	−0.819778	+	0.553559i
$328$		0			0
$329$	5.76867			0.318037
$330$		0			0
$331$	4.08520	+	7.07577i	0.224543	+	0.388919i	0.956182	−	0.292772i	$-0.0945778\pi$
$331$	4.08520	+	7.07577i	0.224543	+	0.388919i	−0.731639	+	0.681692i	$0.761244\pi$
$332$		0			0
$333$	0.0277079	+	0.0353895i	0.00151838	+	0.00193933i
$334$		0			0
$335$	0.464450	+	0.804451i	0.0253756	+	0.0439518i
$336$		0			0
$337$	−2.88655	−	4.99965i	−0.157240	−	0.272348i	0.776632	−	0.629954i	$-0.216926\pi$
$337$	−2.88655	−	4.99965i	−0.157240	−	0.272348i	−0.933873	+	0.357606i	$0.883593\pi$
$338$		0			0
$339$	6.28241	−	4.24223i	0.341213	−	0.230406i
$340$		0			0
$341$	21.0580			1.14036
$342$		0			0
$343$	8.25743			0.445859
$344$		0			0
$345$	0.674378	+	9.57348i	0.0363073	+	0.515419i
$346$		0			0
$347$	−13.3030	−	23.0415i	−0.714144	−	1.23693i	−0.963289	−	0.268468i	$-0.913483\pi$
$347$	−13.3030	−	23.0415i	−0.714144	−	1.23693i	0.249145	−	0.968466i	$-0.419851\pi$
$348$		0			0
$349$	−11.9520	−	20.7015i	−0.639776	−	1.10813i	−0.985482	−	0.169782i	$-0.945694\pi$
$349$	−11.9520	−	20.7015i	−0.639776	−	1.10813i	0.345705	−	0.938343i	$-0.387640\pi$
$350$		0			0
$351$	3.82913	+	17.8793i	0.204384	+	0.954326i
$352$		0			0
$353$	1.96382	+	3.40144i	0.104524	+	0.181041i	0.913544	−	0.406741i	$-0.133335\pi$
$353$	1.96382	+	3.40144i	0.104524	+	0.181041i	−0.809020	+	0.587781i	$0.800001\pi$
$354$		0			0
$355$	10.2679			0.544963
$356$		0			0
$357$	0.423227	+	0.206153i	0.0223995	+	0.0109108i
$358$		0			0
$359$	−4.95990	+	8.59080i	−0.261774	+	0.453405i	−0.966713	−	0.255862i	$-0.917641\pi$
$359$	−4.95990	+	8.59080i	−0.261774	+	0.453405i	0.704940	+	0.709267i	$0.250974\pi$
$360$		0			0
$361$	18.8606	+	2.29722i	0.992664	+	0.120906i
$362$		0			0
$363$	1.45625	−	0.983338i	0.0764331	−	0.0516118i
$364$		0			0
$365$	−7.71519			−0.403832
$366$		0			0
$367$	−11.3471	+	19.6537i	−0.592312	+	1.02591i	0.401609	+	0.915811i	$0.368451\pi$
$367$	−11.3471	+	19.6537i	−0.592312	+	1.02591i	−0.993920	+	0.110102i	$0.964882\pi$
$368$		0			0
$369$	−10.3179	+	25.6134i	−0.537130	+	1.33338i
$370$		0			0
$371$	−1.25120	+	2.16715i	−0.0649593	+	0.112513i
$372$		0			0
$373$	11.7057	−	20.2749i	0.606100	−	1.04980i	−0.385776	−	0.922592i	$-0.626066\pi$
$373$	11.7057	−	20.2749i	0.606100	−	1.04980i	0.991877	−	0.127204i	$-0.0406003\pi$
$374$		0			0
$375$	13.6377	+	6.64288i	0.704246	+	0.343037i
$376$		0			0
$377$	11.7654	−	20.3783i	0.605950	−	1.04954i
$378$		0			0
$379$	0.486258			0.0249774			0.0124887	−	0.999922i	$-0.496025\pi$
$379$	0.486258			0.0249774			0.0124887	+	0.999922i	$0.496025\pi$
$380$		0			0
$381$	−0.264970	−	3.76152i	−0.0135748	−	0.192708i
$382$		0			0
$383$	−1.66891	−	2.89064i	−0.0852775	−	0.147705i	0.820232	−	0.572031i	$-0.193844\pi$
$383$	−1.66891	−	2.89064i	−0.0852775	−	0.147705i	−0.905509	+	0.424326i	$0.860511\pi$
$384$		0			0
$385$	2.02792			0.103352
$386$		0			0
$387$	23.6215	+	30.1703i	1.20075	+	1.53364i
$388$		0			0
$389$	−10.1047	+	17.5019i	−0.512330	+	0.887382i	0.487568	+	0.873085i	$0.337884\pi$
$389$	−10.1047	+	17.5019i	−0.512330	+	0.887382i	−0.999898	+	0.0142966i	$0.995449\pi$
$390$		0			0
$391$	−1.28706	+	2.22926i	−0.0650896	+	0.112738i
$392$		0			0
$393$	−1.83732	−	26.0826i	−0.0926803	−	1.31569i
$394$		0			0
$395$	−1.78699	−	3.09515i	−0.0899130	−	0.155734i
$396$		0			0
$397$	−18.8489	+	32.6472i	−0.945997	+	1.63852i	−0.192255	+	0.981345i	$0.561580\pi$
$397$	−18.8489	+	32.6472i	−0.945997	+	1.63852i	−0.753742	+	0.657170i	$0.771753\pi$
$398$		0			0
$399$	−0.0444692	−	4.57263i	−0.00222625	−	0.228918i
$400$		0			0
$401$	−3.03437	−	5.25568i	−0.151529	−	0.262456i	0.780261	−	0.625454i	$-0.215086\pi$
$401$	−3.03437	−	5.25568i	−0.151529	−	0.262456i	−0.931790	+	0.362998i	$0.881753\pi$
$402$		0			0
$403$	−10.6891	+	18.5141i	−0.532462	+	0.922252i
$404$		0			0
$405$	8.35173	−	2.41338i	0.415001	−	0.119922i
$406$		0			0
$407$	0.0259653	+	0.0449732i	0.00128705	+	0.00222924i
$408$		0			0
$409$	13.5907			0.672019			0.336009	−	0.941859i	$-0.390923\pi$
$409$	13.5907			0.672019			0.336009	+	0.941859i	$0.390923\pi$
$410$		0			0
$411$	16.4144	−	11.0839i	0.809663	−	0.546729i
$412$		0			0
$413$	−1.35682	+	2.35007i	−0.0667645	+	0.115640i
$414$		0			0
$415$	−3.12109	+	5.40589i	−0.153208	+	0.265364i
$416$		0			0
$417$	0.590422	+	0.287593i	0.0289131	+	0.0140835i
$418$		0			0
$419$	18.5949	−	32.2073i	0.908420	−	1.57343i	0.0921599	−	0.995744i	$-0.470623\pi$
$419$	18.5949	−	32.2073i	0.908420	−	1.57343i	0.816260	−	0.577685i	$-0.196044\pi$
$420$		0			0
$421$	−19.5164			−0.951173			−0.475587	−	0.879669i	$-0.657764\pi$
$421$	−19.5164			−0.951173			−0.475587	+	0.879669i	$0.657764\pi$
$422$		0			0
$423$	28.2903	−	4.00554i	1.37552	−	0.194756i
$424$		0			0
$425$	0.912505	+	1.58050i	0.0442630	+	0.0766657i
$426$		0			0
$427$	−7.39154			−0.357702
$428$		0			0
$429$	1.48449	+	21.0739i	0.0716721	+	1.01746i
$430$		0			0
$431$	−16.0690	−	27.8322i	−0.774014	−	1.34063i	−0.935347	−	0.353732i	$-0.884912\pi$
$431$	−16.0690	−	27.8322i	−0.774014	−	1.34063i	0.161332	−	0.986900i	$-0.448421\pi$
$432$		0			0
$433$	−7.24786	−	12.5537i	−0.348310	−	0.603290i	0.637640	−	0.770335i	$-0.279911\pi$
$433$	−7.24786	−	12.5537i	−0.348310	−	0.603290i	−0.985949	+	0.167045i	$0.946578\pi$
$434$		0			0
$435$	−10.0579	−	4.89920i	−0.482242	−	0.234899i
$436$		0			0
$437$	24.9582	+	1.51436i	1.19391	+	0.0724415i
$438$		0			0
$439$	−29.8332			−1.42386			−0.711930	−	0.702251i	$-0.752179\pi$
$439$	−29.8332			−1.42386			−0.711930	+	0.702251i	$0.752179\pi$
$440$		0			0
$441$	19.7029	−	2.78968i	0.938234	−	0.132842i
$442$		0			0
$443$	8.94460	−	15.4925i	0.424971	−	0.736071i	−0.571447	−	0.820639i	$-0.693618\pi$
$443$	8.94460	−	15.4925i	0.424971	−	0.736071i	0.996418	+	0.0845679i	$0.0269510\pi$
$444$		0			0
$445$	−6.32986	+	10.9636i	−0.300064	+	0.519727i
$446$		0			0
$447$	−2.78309	−	39.5087i	−0.131635	−	1.86870i
$448$		0			0
$449$	5.87066			0.277054			0.138527	−	0.990359i	$-0.455763\pi$
$449$	5.87066			0.277054			0.138527	+	0.990359i	$0.455763\pi$
$450$		0			0
$451$	−15.9523	+	27.6302i	−0.751164	+	1.30105i
$452$		0			0
$453$	0.477335	+	6.77626i	0.0224272	+	0.318377i
$454$		0			0
$455$	−1.02938	+	1.78293i	−0.0482580	+	0.0835852i
$456$		0			0
$457$	2.64379	+	4.57918i	0.123671	+	0.214205i	0.921213	−	0.389059i	$-0.127200\pi$
$457$	2.64379	+	4.57918i	0.123671	+	0.214205i	−0.797541	+	0.603264i	$0.793866\pi$
$458$		0			0
$459$	2.21870	+	0.717128i	0.103560	+	0.0334727i
$460$		0			0
$461$	−17.3111			−0.806260			−0.403130	−	0.915143i	$-0.632078\pi$
$461$	−17.3111			−0.806260			−0.403130	+	0.915143i	$0.632078\pi$
$462$		0			0
$463$	−0.0471263	+	0.0816252i	−0.00219015	+	0.00379344i	−0.867118	−	0.498102i	$-0.834030\pi$
$463$	−0.0471263	+	0.0816252i	−0.00219015	+	0.00379344i	0.864928	+	0.501896i	$0.167364\pi$
$464$		0			0
$465$	9.13783	+	4.45102i	0.423757	+	0.206411i
$466$		0			0
$467$	35.8380			1.65839			0.829193	−	0.558962i	$-0.188800\pi$
$467$	35.8380			1.65839			0.829193	+	0.558962i	$0.188800\pi$
$468$		0			0
$469$	−0.291232	−	0.504429i	−0.0134478	−	0.0232923i
$470$		0			0
$471$	1.66422	+	23.6253i	0.0766834	+	1.08860i
$472$		0			0
$473$	22.1359	+	38.3405i	1.01781	+	1.76290i
$474$		0			0
$475$	9.77728	−	14.7874i	0.448613	−	0.678495i
$476$		0			0
$477$	−4.63129	+	11.4968i	−0.212052	+	0.526401i
$478$		0			0
$479$	9.31857	−	16.1402i	0.425776	−	0.737466i	−0.570716	−	0.821147i	$-0.693334\pi$
$479$	9.31857	−	16.1402i	0.425776	−	0.737466i	0.996493	+	0.0836811i	$0.0266677\pi$
$480$		0			0
$481$	−0.0527201			−0.00240383
$482$		0			0
$483$	−0.422867	−	6.00303i	−0.0192411	−	0.273147i
$484$		0			0
$485$	−3.02047	−	5.23161i	−0.137153	−	0.237555i
$486$		0			0
$487$	20.9594			0.949761			0.474881	−	0.880050i	$-0.342491\pi$
$487$	20.9594			0.949761			0.474881	+	0.880050i	$0.342491\pi$
$488$		0			0
$489$	26.4435	+	12.8806i	1.19582	+	0.582479i
$490$		0			0
$491$	−18.7391	−	32.4570i	−0.845683	−	1.46477i	−0.885027	−	0.465540i	$-0.845860\pi$
$491$	−18.7391	−	32.4570i	−0.845683	−	1.46477i	0.0393442	−	0.999226i	$-0.487473\pi$
$492$		0			0
$493$	−1.50036	−	2.59870i	−0.0675729	−	0.117040i
$494$		0			0
$495$	9.94518	−	1.40811i	0.447003	−	0.0632898i
$496$		0			0
$497$	−6.43845			−0.288804
$498$		0			0
$499$	3.93297	+	6.81210i	0.176064	+	0.304951i	0.940529	−	0.339714i	$-0.110330\pi$
$499$	3.93297	+	6.81210i	0.176064	+	0.304951i	−0.764465	+	0.644665i	$0.776997\pi$
$500$		0			0
$501$	−24.6521	−	12.0080i	−1.10138	−	0.536477i
$502$		0			0
$503$	−12.1397	−	21.0266i	−0.541282	−	0.937528i	−0.998831	−	0.0483438i	$-0.984606\pi$
$503$	−12.1397	−	21.0266i	−0.541282	−	0.937528i	0.457548	−	0.889185i	$-0.348728\pi$
$504$		0			0
$505$	6.76094			0.300858
$506$		0			0
$507$	0.961319	+	0.468256i	0.0426937	+	0.0207960i
$508$		0			0
$509$	−27.3018			−1.21013			−0.605064	−	0.796177i	$-0.706853\pi$
$509$	−27.3018			−1.21013			−0.605064	+	0.796177i	$0.706853\pi$
$510$		0			0
$511$	4.83779			0.214011
$512$		0			0
$513$	−3.39314	−	22.3939i	−0.149811	−	0.988715i
$514$		0			0
$515$	12.2615			0.540308
$516$		0			0
$517$	33.0126			1.45189
$518$		0			0
$519$	7.56736	+	3.68604i	0.332170	+	0.161799i
$520$		0			0
$521$	−17.3591			−0.760516			−0.380258	−	0.924880i	$-0.624165\pi$
$521$	−17.3591			−0.760516			−0.380258	+	0.924880i	$0.624165\pi$
$522$		0			0
$523$	−22.2174	−	38.4817i	−0.971501	−	1.68269i	−0.691029	−	0.722827i	$-0.742842\pi$
$523$	−22.2174	−	38.4817i	−0.971501	−	1.68269i	−0.280472	−	0.959862i	$-0.590491\pi$
$524$		0			0
$525$	−3.83574	−	1.86838i	−0.167405	−	0.0815428i
$526$		0			0
$527$	1.36311	+	2.36097i	0.0593778	+	0.102845i
$528$		0			0
$529$	9.90565			0.430681
$530$		0			0
$531$	−5.02220	+	12.4672i	−0.217945	+	0.541030i
$532$		0			0
$533$	−16.1948	−	28.0503i	−0.701477	−	1.21499i
$534$		0			0
$535$	−1.12115	−	1.94188i	−0.0484713	−	0.0839548i
$536$		0			0
$537$	22.8823	+	11.1459i	0.987445	+	0.480982i
$538$		0			0
$539$	22.9918			0.990325
$540$		0			0
$541$	3.37948	+	5.85343i	0.145295	+	0.251659i	0.929483	−	0.368865i	$-0.120253\pi$
$541$	3.37948	+	5.85343i	0.145295	+	0.251659i	−0.784188	+	0.620524i	$0.786920\pi$
$542$		0			0
$543$	2.49111	+	35.3639i	0.106904	+	1.51761i
$544$		0			0
$545$	−9.97549			−0.427303
$546$		0			0
$547$	8.22328	−	14.2431i	0.351602	−	0.608993i	−0.634928	−	0.772571i	$-0.718970\pi$
$547$	8.22328	−	14.2431i	0.351602	−	0.608993i	0.986530	+	0.163579i	$0.0523037\pi$
$548$		0			0
$549$	−36.2491	+	5.13240i	−1.54707	+	0.219046i
$550$		0			0
$551$	−16.0760	+	24.3139i	−0.684862	+	1.03581i
$552$		0			0
$553$	1.12052	+	1.94081i	0.0476495	+	0.0825314i
$554$		0			0
$555$	0.00176132	+	0.0250037i	7.47638e−5	+	0.00106135i
$556$		0			0
$557$	−10.8263	−	18.7517i	−0.458724	−	0.794533i	0.540170	−	0.841556i	$-0.318360\pi$
$557$	−10.8263	−	18.7517i	−0.458724	−	0.794533i	−0.998894	+	0.0470229i	$0.985027\pi$
$558$		0			0
$559$	−44.9450			−1.90097
$560$		0			0
$561$	2.42201	+	1.17976i	0.102258	+	0.0498094i
$562$		0			0
$563$	−5.34965	+	9.26586i	−0.225461	+	0.390509i	−0.956458	−	0.291871i	$-0.905722\pi$
$563$	−5.34965	+	9.26586i	−0.225461	+	0.390509i	0.730997	+	0.682381i	$0.239055\pi$
$564$		0			0
$565$	4.22757			0.177855
$566$		0			0
$567$	−5.23693	+	1.51330i	−0.219930	+	0.0635527i
$568$		0			0
$569$	5.83846	+	10.1125i	0.244761	+	0.423939i	0.962064	−	0.272822i	$-0.0879571\pi$
$569$	5.83846	+	10.1125i	0.244761	+	0.423939i	−0.717303	+	0.696761i	$0.754624\pi$
$570$		0			0
$571$	−19.0612	+	33.0150i	−0.797687	+	1.38163i	0.123431	+	0.992353i	$0.460610\pi$
$571$	−19.0612	+	33.0150i	−0.797687	+	1.38163i	−0.921119	+	0.389282i	$0.872723\pi$
$572$		0			0
$573$	−1.15781	−	16.4362i	−0.0483680	−	0.686633i
$574$		0			0
$575$	11.6648	−	20.2040i	0.486454	−	0.842563i
$576$		0			0
$577$	−43.5999			−1.81509			−0.907544	−	0.419957i	$-0.862045\pi$
$577$	−43.5999			−1.81509			−0.907544	+	0.419957i	$0.862045\pi$
$578$		0			0
$579$	−1.21810	−	17.2922i	−0.0506225	−	0.718638i
$580$		0			0
$581$	1.95707	−	3.38975i	0.0811930	−	0.140630i
$582$		0			0
$583$	−7.16031	+	12.4020i	−0.296550	+	0.513639i
$584$		0			0
$585$	−3.81020	+	9.45850i	−0.157532	+	0.391061i
$586$		0			0
$587$	−19.5320			−0.806172			−0.403086	−	0.915162i	$-0.632062\pi$
$587$	−19.5320			−0.806172			−0.403086	+	0.915162i	$0.632062\pi$
$588$		0			0
$589$	14.6054	−	22.0896i	0.601804	−	0.910185i
$590$		0			0
$591$	16.3397	+	7.95905i	0.672127	+	0.327392i
$592$		0			0
$593$	12.2452	+	21.2093i	0.502851	+	0.870963i	0.999995	+	0.00329492i	$0.00104881\pi$
$593$	12.2452	+	21.2093i	0.502851	+	0.870963i	−0.497144	+	0.867668i	$0.665618\pi$
$594$		0			0
$595$	0.131269	+	0.227365i	0.00538151	+	0.00932105i
$596$		0			0
$597$	−0.364620	−	5.17616i	−0.0149229	−	0.211846i
$598$		0			0
$599$	−14.3258			−0.585336			−0.292668	−	0.956214i	$-0.594543\pi$
$599$	−14.3258			−0.585336			−0.292668	+	0.956214i	$0.594543\pi$
$600$		0			0
$601$	15.5141	+	26.8712i	0.632833	+	1.09610i	0.986970	+	0.160906i	$0.0514415\pi$
$601$	15.5141	+	26.8712i	0.632833	+	1.09610i	−0.354136	+	0.935194i	$0.615225\pi$
$602$		0			0
$603$	−1.77850	−	2.27156i	−0.0724259	−	0.0925052i
$604$		0			0
$605$	0.979939			0.0398402
$606$		0			0
$607$	−10.5360	+	18.2489i	−0.427644	+	0.740702i	−0.996663	−	0.0816226i	$-0.973990\pi$
$607$	−10.5360	+	18.2489i	−0.427644	+	0.740702i	0.569019	+	0.822324i	$0.307323\pi$
$608$		0			0
$609$	6.30681	+	3.07203i	0.255565	+	0.124485i
$610$		0			0
$611$	−16.7573	+	29.0244i	−0.677926	+	1.17420i
$612$		0			0
$613$	−0.306338	+	0.530594i	−0.0123729	+	0.0214305i	−0.872146	−	0.489246i	$-0.837272\pi$
$613$	−0.306338	+	0.530594i	−0.0123729	+	0.0214305i	0.859773	+	0.510677i	$0.170605\pi$
$614$		0			0
$615$	−12.7624	+	8.61790i	−0.514631	+	0.347507i
$616$		0			0
$617$	8.77749			0.353368			0.176684	−	0.984268i	$-0.443463\pi$
$617$	8.77749			0.353368			0.176684	+	0.984268i	$0.443463\pi$
$618$		0			0
$619$	−16.8409	−	29.1692i	−0.676892	−	1.17241i	−0.975912	−	0.218164i	$-0.929993\pi$
$619$	−16.8409	−	29.1692i	−0.676892	−	1.17241i	0.299020	−	0.954247i	$-0.403340\pi$
$620$		0			0
$621$	−6.24207	−	29.1460i	−0.250485	−	1.16959i
$622$		0			0
$623$	3.96912	−	6.87472i	0.159020	−	0.275430i
$624$		0			0
$625$	−5.93752	−	10.2841i	−0.237501	−	0.411363i
$626$		0			0
$627$	−0.254486	−	26.1679i	−0.0101632	−	1.04505i
$628$		0			0
$629$	−0.00336151	+	0.00582231i	−0.000134032	+	0.000232151i
$630$		0			0
$631$	9.60714	+	16.6401i	0.382454	+	0.662430i	0.991412	−	0.130772i	$-0.0417456\pi$
$631$	9.60714	+	16.6401i	0.382454	+	0.662430i	−0.608958	+	0.793202i	$0.708412\pi$
$632$		0			0
$633$	−1.46252	−	20.7620i	−0.0581300	−	0.825215i
$634$		0			0
$635$	1.05147	−	1.82120i	0.0417262	−	0.0722720i
$636$		0			0
$637$	−11.6707	+	20.2142i	−0.462409	+	0.800916i
$638$		0			0
$639$	−31.5750	+	4.47061i	−1.24909	+	0.176855i
$640$		0			0
$641$	25.7464			1.01692			0.508460	−	0.861086i	$-0.330215\pi$
$641$	25.7464			1.01692			0.508460	+	0.861086i	$0.330215\pi$
$642$		0			0
$643$	−15.0258	−	26.0254i	−0.592559	−	1.02634i	−0.993886	−	0.110408i	$-0.964784\pi$
$643$	−15.0258	−	26.0254i	−0.592559	−	1.02634i	0.401327	−	0.915935i	$-0.368549\pi$
$644$		0			0
$645$	1.50156	+	21.3162i	0.0591239	+	0.839324i
$646$		0			0
$647$	−4.20889			−0.165469			−0.0827343	−	0.996572i	$-0.526365\pi$
$647$	−4.20889			−0.165469			−0.0827343	+	0.996572i	$0.526365\pi$
$648$		0			0
$649$	−7.76469	+	13.4488i	−0.304791	+	0.527913i
$650$		0			0
$651$	−5.72985	−	2.79100i	−0.224571	−	0.109388i
$652$		0			0
$653$	7.70804	−	13.3507i	0.301639	−	0.522454i	−0.674868	−	0.737938i	$-0.735800\pi$
$653$	7.70804	−	13.3507i	0.301639	−	0.522454i	0.976507	+	0.215484i	$0.0691330\pi$
$654$		0			0
$655$	7.29094	−	12.6283i	0.284881	−	0.493428i
$656$		0			0
$657$	23.7252	−	3.35918i	0.925606	−	0.131054i
$658$		0			0
$659$	−13.2807	+	23.0029i	−0.517343	+	0.896064i	0.482454	+	0.875921i	$0.339746\pi$
$659$	−13.2807	+	23.0029i	−0.517343	+	0.896064i	−0.999797	+	0.0201431i	$0.993588\pi$
$660$		0			0
$661$	12.6301			0.491252			0.245626	−	0.969365i	$-0.421006\pi$
$661$	12.6301			0.491252			0.245626	+	0.969365i	$0.421006\pi$
$662$		0			0
$663$	−2.26666	+	1.53057i	−0.0880297	+	0.0594425i
$664$		0			0
$665$	1.40652	−	2.12726i	0.0545425	−	0.0824916i
$666$		0			0
$667$	−19.1795	+	33.2198i	−0.742632	+	1.28628i
$668$		0			0
$669$	3.24699	+	1.58160i	0.125536	+	0.0611483i
$670$		0			0
$671$	−42.2998			−1.63297
$672$		0			0
$673$	−18.9462	−	32.8158i	−0.730322	−	1.26495i	−0.956746	−	0.290925i	$-0.906037\pi$
$673$	−18.9462	−	32.8158i	−0.730322	−	1.26495i	0.226424	−	0.974029i	$-0.427296\pi$
$674$		0			0
$675$	−20.1083	−	6.49939i	−0.773969	−	0.250162i
$676$		0			0
$677$	8.43804	+	14.6151i	0.324300	+	0.561705i	0.981371	−	0.192125i	$-0.0615378\pi$
$677$	8.43804	+	14.6151i	0.324300	+	0.561705i	−0.657070	+	0.753829i	$0.728204\pi$
$678$		0			0
$679$	1.89398	+	3.28047i	0.0726842	+	0.125893i
$680$		0			0
$681$	−0.330400	−	4.69036i	−0.0126609	−	0.179735i
$682$		0			0
$683$	−42.4985			−1.62616			−0.813080	−	0.582151i	$-0.802211\pi$
$683$	−42.4985			−1.62616			−0.813080	+	0.582151i	$0.802211\pi$
$684$		0			0
$685$	11.0456			0.422031
$686$		0			0
$687$	−19.1885	+	12.9572i	−0.732088	+	0.494346i
$688$		0			0
$689$	−7.26919	−	12.5906i	−0.276934	−	0.479664i
$690$		0			0
$691$	21.7812	+	37.7261i	0.828596	+	1.43517i	0.899140	+	0.437662i	$0.144193\pi$
$691$	21.7812	+	37.7261i	0.828596	+	1.43517i	−0.0705438	+	0.997509i	$0.522473\pi$
$692$		0			0
$693$	−6.23610	+	0.882951i	−0.236890	+	0.0335406i
$694$		0			0
$695$	0.183127	+	0.317185i	0.00694640	+	0.0120315i
$696$		0			0
$697$	−4.13043			−0.156451
$698$		0			0
$699$	−29.8085	+	20.1284i	−1.12746	+	0.761324i
$700$		0			0
$701$	−20.6791	+	35.8172i	−0.781038	+	1.35280i	0.150300	+	0.988640i	$0.451976\pi$
$701$	−20.6791	+	35.8172i	−0.781038	+	1.35280i	−0.931338	+	0.364157i	$0.881357\pi$
$702$		0			0
$703$	0.0651852	+	0.00395515i	0.00245851	+	0.000149171i
$704$		0			0
$705$	14.3253	+	6.97784i	0.539523	+	0.262801i
$706$		0			0
$707$	−4.23943			−0.159440
$708$		0			0
$709$	4.74152	−	8.21256i	0.178072	−	0.308429i	−0.763148	−	0.646223i	$-0.776347\pi$
$709$	4.74152	−	8.21256i	0.178072	−	0.308429i	0.941220	+	0.337794i	$0.109681\pi$
$710$		0			0
$711$	6.84282	+	8.73991i	0.256626	+	0.327772i
$712$		0			0
$713$	17.4249	−	30.1808i	0.652567	−	1.13028i
$714$		0			0
$715$	−5.89085	+	10.2033i	−0.220305	+	0.381580i
$716$		0			0
$717$	−38.6304	+	26.0854i	−1.44268	+	0.974177i
$718$		0			0
$719$	−5.52329	+	9.56662i	−0.205984	+	0.356775i	−0.950446	−	0.310890i	$-0.899373\pi$
$719$	−5.52329	+	9.56662i	−0.205984	+	0.356775i	0.744462	+	0.667665i	$0.232706\pi$
$720$		0			0
$721$	−7.68856			−0.286337
$722$		0			0
$723$	20.7332	−	14.0002i	0.771078	−	0.520674i
$724$		0			0
$725$	13.5979	+	23.5522i	0.505013	+	0.874708i
$726$		0			0
$727$	−41.6500			−1.54471			−0.772357	−	0.635189i	$-0.780922\pi$
$727$	−41.6500			−1.54471			−0.772357	+	0.635189i	$0.780922\pi$
$728$		0			0
$729$	−24.6318	+	11.0577i	−0.912289	+	0.409546i
$730$		0			0
$731$	−2.86576	+	4.96364i	−0.105994	+	0.183587i
$732$		0			0
$733$	−24.6939	+	42.7712i	−0.912092	+	1.57979i	−0.100987	+	0.994888i	$0.532200\pi$
$733$	−24.6939	+	42.7712i	−0.912092	+	1.57979i	−0.811104	+	0.584901i	$0.801133\pi$
$734$		0			0
$735$	9.97694	+	4.85975i	0.368005	+	0.179254i
$736$		0			0
$737$	−1.66664	−	2.88671i	−0.0613916	−	0.106333i
$738$		0			0
$739$	14.6923	−	25.4478i	0.540465	−	0.936113i	−0.458412	−	0.888740i	$-0.651582\pi$
$739$	14.6923	−	25.4478i	0.540465	−	0.936113i	0.998877	−	0.0473731i	$-0.0150850\pi$
$740$		0			0
$741$	23.1359	+	13.0592i	0.849917	+	0.479741i
$742$		0			0
$743$	−3.14346	−	5.44463i	−0.115322	−	0.199744i	0.802586	−	0.596536i	$-0.203457\pi$
$743$	−3.14346	−	5.44463i	−0.115322	−	0.199744i	−0.917909	+	0.396792i	$0.870123\pi$
$744$		0			0
$745$	11.0440	−	19.1288i	0.404621	−	0.700824i
$746$		0			0
$747$	7.24402	−	17.9827i	0.265045	−	0.657951i
$748$		0			0
$749$	0.703011	+	1.21765i	0.0256875	+	0.0444920i
$750$		0			0
$751$	−44.2112			−1.61329			−0.806644	−	0.591037i	$-0.798719\pi$
$751$	−44.2112			−1.61329			−0.806644	+	0.591037i	$0.798719\pi$
$752$		0			0
$753$	−0.644373	−	9.14753i	−0.0234823	−	0.333355i
$754$		0			0
$755$	−1.89419	+	3.28083i	−0.0689366	+	0.119402i
$756$		0			0
$757$	8.26360	−	14.3130i	0.300346	−	0.520214i	−0.675869	−	0.737022i	$-0.736231\pi$
$757$	8.26360	−	14.3130i	0.300346	−	0.520214i	0.976214	+	0.216808i	$0.0695646\pi$
$758$		0			0
$759$	−2.41995	−	34.3537i	−0.0878388	−	1.24696i
$760$		0			0
$761$	−1.48812	+	2.57749i	−0.0539442	+	0.0934341i	−0.891737	−	0.452555i	$-0.850513\pi$
$761$	−1.48812	+	2.57749i	−0.0539442	+	0.0934341i	0.837792	+	0.545989i	$0.183846\pi$
$762$		0			0
$763$	6.25510			0.226450
$764$		0			0
$765$	0.801635	+	1.02388i	0.0289832	+	0.0370184i
$766$		0			0
$767$	−7.88276	−	13.6533i	−0.284630	−	0.492993i
$768$		0			0
$769$	−7.57441			−0.273140			−0.136570	−	0.990630i	$-0.543608\pi$
$769$	−7.57441			−0.273140			−0.136570	+	0.990630i	$0.543608\pi$
$770$		0			0
$771$	19.9374	+	9.71144i	0.718026	+	0.349749i
$772$		0			0
$773$	−20.3006	−	35.1617i	−0.730162	−	1.26468i	−0.956814	−	0.290701i	$-0.906111\pi$
$773$	−20.3006	−	35.1617i	−0.730162	−	1.26468i	0.226652	−	0.973976i	$-0.427222\pi$
$774$		0			0
$775$	−12.3539	−	21.3976i	−0.443766	−	0.768626i
$776$		0			0
$777$	−0.00110443	−	0.0156785i	−3.96212e−5	−	0.000562464i
$778$		0			0
$779$	17.9195	+	35.8974i	0.642034	+	1.28616i
$780$		0			0
$781$	−36.8455			−1.31844
$782$		0			0
$783$	33.0625	+	10.6864i	1.18156	+	0.381902i
$784$		0			0
$785$	−6.60407	+	11.4386i	−0.235709	+	0.408260i
$786$		0			0
$787$	22.2753	−	38.5819i	0.794027	−	1.37530i	−0.129428	−	0.991589i	$-0.541314\pi$
$787$	22.2753	−	38.5819i	0.794027	−	1.37530i	0.923455	−	0.383706i	$-0.125353\pi$
$788$		0			0
$789$	6.96649	+	3.39336i	0.248014	+	0.120807i
$790$		0			0
$791$	−2.65088			−0.0942546
$792$		0			0
$793$	21.4715	−	37.1897i	0.762476	−	1.32065i
$794$		0			0
$795$	−5.72852	+	3.86821i	−0.203170	+	0.137191i
$796$		0			0
$797$	8.21620	−	14.2309i	0.291033	−	0.504084i	−0.683021	−	0.730398i	$-0.739334\pi$
$797$	8.21620	−	14.2309i	0.291033	−	0.504084i	0.974054	+	0.226315i	$0.0726678\pi$
$798$		0			0
$799$	2.13693	+	3.70128i	0.0755993	+	0.130942i
$800$		0			0
$801$	14.6916	−	36.4705i	0.519101	−	1.28862i
$802$		0			0
$803$	27.6854			0.976996
$804$		0			0
$805$	1.67804	−	2.90646i	0.0591433	−	0.102439i
$806$		0			0
$807$	−33.6523	+	22.7239i	−1.18462	+	0.799918i
$808$		0			0
$809$	39.0676			1.37354			0.686771	−	0.726873i	$-0.259027\pi$
$809$	39.0676			1.37354			0.686771	+	0.726873i	$0.259027\pi$
$810$		0			0
$811$	19.3327	+	33.4852i	0.678863	+	1.17583i	0.975323	+	0.220781i	$0.0708605\pi$
$811$	19.3327	+	33.4852i	0.678863	+	1.17583i	−0.296460	+	0.955045i	$0.595806\pi$
$812$		0			0
$813$	20.7569	−	14.0162i	0.727976	−	0.491569i
$814$		0			0
$815$	8.20178	+	14.2059i	0.287296	+	0.497611i
$816$		0			0
$817$	55.5717	+	3.37185i	1.94421	+	0.117966i
$818$		0			0
$819$	2.38918	−	5.93093i	0.0834846	−	0.207243i
$820$		0			0
$821$	−8.14175	+	14.1019i	−0.284149	+	0.492161i	−0.972403	−	0.233310i	$-0.925044\pi$
$821$	−8.14175	+	14.1019i	−0.284149	+	0.492161i	0.688253	+	0.725470i	$0.258378\pi$
$822$		0			0
$823$	6.59475			0.229879			0.114939	−	0.993373i	$-0.463333\pi$
$823$	6.59475			0.229879			0.114939	+	0.993373i	$0.463333\pi$
$824$		0			0
$825$	−21.9509	−	10.6922i	−0.764233	−	0.372256i
$826$		0			0
$827$	3.83447	+	6.64150i	0.133338	+	0.230948i	0.924961	−	0.380062i	$-0.124097\pi$
$827$	3.83447	+	6.64150i	0.133338	+	0.230948i	−0.791624	+	0.611009i	$0.790764\pi$
$828$		0			0
$829$	43.3013			1.50391			0.751957	−	0.659212i	$-0.229110\pi$
$829$	43.3013			1.50391			0.751957	+	0.659212i	$0.229110\pi$
$830$		0			0
$831$	0.734553	+	10.4277i	0.0254814	+	0.361734i
$832$		0			0
$833$	1.48828	+	2.57777i	0.0515658	+	0.0893145i
$834$		0			0
$835$	−7.64617	−	13.2435i	−0.264606	−	0.458312i
$836$		0			0
$837$	−30.0379	−	9.70883i	−1.03826	−	0.335586i
$838$		0			0
$839$	52.4101			1.80940			0.904700	−	0.426050i	$-0.140095\pi$
$839$	52.4101			1.80940			0.904700	+	0.426050i	$0.140095\pi$
$840$		0			0
$841$	−7.85796	−	13.6104i	−0.270964	−	0.469324i
$842$		0			0
$843$	−16.2061	+	10.9433i	−0.558169	+	0.376907i
$844$		0			0
$845$	0.298165	+	0.516437i	0.0102572	+	0.0177660i
$846$		0			0
$847$	−0.614468			−0.0211134
$848$		0			0
$849$	−2.20379	−	31.2850i	−0.0756337	−	1.07370i
$850$		0			0
$851$	0.0859420			0.00294605
$852$		0			0
$853$	47.9998			1.64348			0.821741	−	0.569861i	$-0.193003\pi$
$853$	47.9998			1.64348			0.821741	+	0.569861i	$0.193003\pi$
$854$		0			0
$855$	5.42067	−	11.4090i	0.185383	−	0.390180i
$856$		0			0
$857$	−26.2786			−0.897659			−0.448830	−	0.893617i	$-0.648159\pi$
$857$	−26.2786			−0.897659			−0.448830	+	0.893617i	$0.648159\pi$
$858$		0			0
$859$	33.0600			1.12799			0.563996	−	0.825778i	$-0.309263\pi$
$859$	33.0600			1.12799			0.563996	+	0.825778i	$0.309263\pi$
$860$		0			0
$861$	8.00265	−	5.40383i	0.272730	−	0.184162i
$862$		0			0
$863$	−16.7020			−0.568541			−0.284271	−	0.958744i	$-0.591751\pi$
$863$	−16.7020			−0.568541			−0.284271	+	0.958744i	$0.591751\pi$
$864$		0			0
$865$	2.34711	+	4.06532i	0.0798042	+	0.138225i
$866$		0			0
$867$	−2.04453	−	29.0242i	−0.0694358	−	0.985712i
$868$		0			0
$869$	6.41246	+	11.1067i	0.217528	+	0.376769i
$870$		0			0
$871$	3.38397			0.114661
$872$		0			0
$873$	11.5662	+	14.7727i	0.391455	+	0.499981i
$874$		0			0
$875$	−2.65234	−	4.59400i	−0.0896656	−	0.155305i
$876$		0			0
$877$	−8.62160	−	14.9330i	−0.291131	−	0.504253i	0.682947	−	0.730468i	$-0.260698\pi$
$877$	−8.62160	−	14.9330i	−0.291131	−	0.504253i	−0.974077	+	0.226215i	$0.927365\pi$
$878$		0			0
$879$	−11.3637	+	7.67340i	−0.383288	+	0.258817i
$880$		0			0
$881$	9.55580			0.321943			0.160972	−	0.986959i	$-0.448537\pi$
$881$	9.55580			0.321943			0.160972	+	0.986959i	$0.448537\pi$
$882$		0			0
$883$	−2.94463	−	5.10025i	−0.0990948	−	0.171637i	0.812215	−	0.583358i	$-0.198261\pi$
$883$	−2.94463	−	5.10025i	−0.0990948	−	0.171637i	−0.911310	+	0.411720i	$0.864928\pi$
$884$		0			0
$885$	−6.21205	+	4.19472i	−0.208816	+	0.141004i
$886$		0			0
$887$	2.63685			0.0885368			0.0442684	−	0.999020i	$-0.485904\pi$
$887$	2.63685			0.0885368			0.0442684	+	0.999020i	$0.485904\pi$
$888$		0			0
$889$	−0.659320	+	1.14198i	−0.0221129	+	0.0383006i
$890$		0			0
$891$	−29.9696	+	8.66022i	−1.00402	+	0.290128i
$892$		0			0
$893$	22.8968	−	34.6297i	0.766211	−	1.15884i
$894$		0			0
$895$	7.09724	+	12.2928i	0.237235	+	0.410903i
$896$		0			0
$897$	31.4319	+	15.3104i	1.04948	+	0.511200i
$898$		0			0
$899$	20.3126	+	35.1825i	0.677464	+	1.17340i
$900$		0			0
$901$	−1.85398			−0.0617649
$902$		0			0
$903$	−0.941549	−	13.3663i	−0.0313328	−	0.444801i
$904$		0			0
$905$	−9.88538	+	17.1220i	−0.328601	+	0.569154i
$906$		0			0
$907$	44.2840			1.47043			0.735213	−	0.677836i	$-0.237082\pi$
$907$	44.2840			1.47043			0.735213	+	0.677836i	$0.237082\pi$
$908$		0			0
$909$	−20.7907	+	2.94370i	−0.689584	+	0.0976362i
$910$		0			0
$911$	−27.3226	−	47.3241i	−0.905237	−	1.56792i	−0.820598	−	0.571505i	$-0.806360\pi$
$911$	−27.3226	−	47.3241i	−0.905237	−	1.56792i	−0.0846387	−	0.996412i	$-0.526974\pi$
$912$		0			0
$913$	11.1998	−	19.3986i	0.370659	−	0.642000i
$914$		0			0
$915$	−18.3554	−	8.94088i	−0.606811	−	0.295576i
$916$		0			0
$917$	−4.57176	+	7.91853i	−0.150973	+	0.261493i
$918$		0			0
$919$	3.54623			0.116979			0.0584896	−	0.998288i	$-0.481372\pi$
$919$	3.54623			0.116979			0.0584896	+	0.998288i	$0.481372\pi$
$920$		0			0
$921$	−40.2837	+	27.2018i	−1.32739	+	0.896329i
$922$		0			0
$923$	18.7029	−	32.3944i	0.615613	−	1.06627i
$924$		0			0
$925$	0.0304656	−	0.0527681i	0.00100170	−	0.00173500i
$926$		0			0
$927$	−37.7057	+	5.33864i	−1.23842	+	0.175344i
$928$		0			0
$929$	−24.0640			−0.789515			−0.394757	−	0.918785i	$-0.629171\pi$
$929$	−24.0640			−0.789515			−0.394757	+	0.918785i	$0.629171\pi$
$930$		0			0
$931$	15.9466	−	24.1180i	0.522627	−	0.790437i
$932$		0			0
$933$	−5.82599	+	3.93403i	−0.190734	+	0.128794i
$934$		0			0
$935$	0.751219	+	1.30115i	0.0245675	+	0.0425521i
$936$		0			0
$937$	21.0532	+	36.4652i	0.687778	+	1.19127i	0.972555	+	0.232673i	$0.0747471\pi$
$937$	21.0532	+	36.4652i	0.687778	+	1.19127i	−0.284777	+	0.958594i	$0.591920\pi$
$938$		0			0
$939$	−17.5423	+	11.8455i	−0.572471	+	0.386564i
$940$		0			0
$941$	44.8743			1.46286			0.731430	−	0.681917i	$-0.238853\pi$
$941$	44.8743			1.46286			0.731430	+	0.681917i	$0.238853\pi$
$942$		0			0
$943$	26.4001	+	45.7263i	0.859705	+	1.48905i
$944$		0			0
$945$	−2.89270	−	0.934975i	−0.0940994	−	0.0304147i
$946$		0			0
$947$	−6.22469			−0.202275			−0.101138	−	0.994872i	$-0.532248\pi$
$947$	−6.22469			−0.202275			−0.101138	+	0.994872i	$0.532248\pi$
$948$		0			0
$949$	−14.0532	+	24.3408i	−0.456185	+	0.790136i
$950$		0			0
$951$	30.6451	−	20.6933i	0.993736	−	0.671025i
$952$		0			0
$953$	−2.59279	+	4.49085i	−0.0839888	+	0.145473i	−0.904960	−	0.425497i	$-0.860099\pi$
$953$	−2.59279	+	4.49085i	−0.0839888	+	0.145473i	0.820971	+	0.570970i	$0.193433\pi$
$954$		0			0
$955$	4.59447	−	7.95785i	0.148674	−	0.257510i
$956$		0			0
$957$	36.0922	+	17.5804i	1.16669	+	0.568295i
$958$		0			0
$959$	−6.92611			−0.223656
$960$		0			0
$961$	−2.95438	−	5.11713i	−0.0953024	−	0.165069i
$962$		0			0
$963$	4.29315	+	5.48337i	0.138345	+	0.176699i
$964$		0			0
$965$	4.83373	−	8.37227i	0.155603	−	0.269513i
$966$		0			0
$967$	−2.12836	−	3.68642i	−0.0684433	−	0.118547i	0.829773	−	0.558101i	$-0.188470\pi$
$967$	−2.12836	−	3.68642i	−0.0684433	−	0.118547i	−0.898216	+	0.439554i	$0.855137\pi$
$968$		0			0
$969$	2.91741	−	1.72241i	0.0937206	−	0.0553317i
$970$		0			0
$971$	−22.9152	+	39.6904i	−0.735385	+	1.27372i	0.219169	+	0.975687i	$0.429665\pi$
$971$	−22.9152	+	39.6904i	−0.735385	+	1.27372i	−0.954554	+	0.298037i	$0.903668\pi$
$972$		0			0
$973$	−0.114829	−	0.198890i	−0.00368126	−	0.00637612i
$974$		0			0
$975$	20.5429	−	13.8717i	0.657899	−	0.444250i
$976$		0			0
$977$	−30.5392	+	52.8954i	−0.977035	+	1.69227i	−0.303987	+	0.952676i	$0.598318\pi$
$977$	−30.5392	+	52.8954i	−0.977035	+	1.69227i	−0.673048	+	0.739598i	$0.735015\pi$
$978$		0			0
$979$	22.7142	−	39.3422i	0.725950	−	1.25738i
$980$		0			0
$981$	30.6758	−	4.34331i	0.979404	−	0.138671i
$982$		0			0
$983$	45.0777			1.43775			0.718877	−	0.695137i	$-0.244656\pi$
$983$	45.0777			1.43775			0.718877	+	0.695137i	$0.244656\pi$
$984$		0			0
$985$	5.06798	+	8.77800i	0.161479	+	0.279690i
$986$		0			0
$987$	−8.98266	−	4.37543i	−0.285921	−	0.139272i
$988$		0			0
$989$	73.2673			2.32976
$990$		0			0
$991$	20.4508	−	35.4218i	0.649640	−	1.12521i	−0.333569	−	0.942726i	$-0.608253\pi$
$991$	20.4508	−	35.4218i	0.649640	−	1.12521i	0.983209	−	0.182484i	$-0.0584138\pi$
$992$		0			0
$993$	−0.994403	−	14.1166i	−0.0315564	−	0.447975i
$994$		0			0
$995$	1.44691	−	2.50612i	0.0458700	−	0.0794492i
$996$		0			0
$997$	−11.6326	+	20.1483i	−0.368408	+	0.638102i	−0.989317	−	0.145781i	$-0.953431\pi$
$997$	−11.6326	+	20.1483i	−0.368408	+	0.638102i	0.620908	+	0.783883i	$0.286764\pi$
$998$		0			0
$999$	−0.0163028	−	0.0761226i	−0.000515799	−	0.00240841i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.j.a.49.4	✓	40
3.2	odd	2		2052.2.j.a.1873.13		40
9.2	odd	6		2052.2.l.a.505.8		40
9.7	even	3		684.2.l.a.277.17	yes	40
19.7	even	3		684.2.l.a.121.17	yes	40
57.26	odd	6		2052.2.l.a.577.8		40
171.7	even	3	inner	684.2.j.a.349.4	yes	40
171.83	odd	6		2052.2.j.a.1261.13		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.j.a.49.4	✓	40	1.1	even	1	trivial
684.2.j.a.349.4	yes	40	171.7	even	3	inner
684.2.l.a.121.17	yes	40	19.7	even	3
684.2.l.a.277.17	yes	40	9.7	even	3
2052.2.j.a.1261.13		40	171.83	odd	6
2052.2.j.a.1873.13		40	3.2	odd	2
2052.2.l.a.505.8		40	9.2	odd	6
2052.2.l.a.577.8		40	57.26	odd	6

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.55715 - 0.758482i) q^{3} +(-0.482969 - 0.836526i) q^{5} +(0.302844 + 0.524541i) q^{7} +(1.84941 + 2.36214i) q^{9} +(1.73310 + 3.00181i) q^{11} -3.51890 q^{13} +(0.117562 + 1.66892i) q^{15} +(-0.224370 + 0.388620i) q^{17} +(4.35090 + 0.263993i) q^{19} +(-0.0737171 - 1.04649i) q^{21} +5.73635 q^{23} +(2.03348 - 3.52210i) q^{25} +(-1.08816 - 5.08094i) q^{27} +(-3.34350 + 5.79111i) q^{29} +(3.03763 - 5.26133i) q^{31} +(-0.421863 - 5.98878i) q^{33} +(0.292528 - 0.506674i) q^{35} +0.0149820 q^{37} +(5.47944 + 2.66902i) q^{39} +(4.60225 + 7.97133i) q^{41} +12.7725 q^{43} +(1.08278 - 2.68792i) q^{45} +(4.76208 - 8.24816i) q^{47} +(3.31657 - 5.74447i) q^{49} +(0.644138 - 0.434958i) q^{51} +(2.06576 + 3.57800i) q^{53} +(1.67406 - 2.89956i) q^{55} +(-6.57475 - 3.71116i) q^{57} +(2.24012 + 3.88000i) q^{59} +(-6.10177 + 10.5686i) q^{61} +(-0.678956 + 1.68545i) q^{63} +(1.69952 + 2.94365i) q^{65} -0.961656 q^{67} +(-8.93233 - 4.35092i) q^{69} +(-5.31499 + 9.20582i) q^{71} +(3.99363 - 6.91717i) q^{73} +(-5.83788 + 3.94206i) q^{75} +(-1.04972 + 1.81816i) q^{77} +3.70000 q^{79} +(-2.15938 + 8.73711i) q^{81} +(-3.23115 - 5.59652i) q^{83} +0.433455 q^{85} +(9.59877 - 6.48162i) q^{87} +(-6.55308 - 11.3503i) q^{89} +(-1.06568 - 1.84581i) q^{91} +(-8.72066 + 5.88867i) q^{93} +(-1.88051 - 3.76714i) q^{95} +6.25397 q^{97} +(-3.88548 + 9.64539i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 2 q^{3} - q^{7} - 2 q^{9} - q^{11} + 2 q^{13} + q^{15} + 5 q^{17} + q^{19} + 6 q^{21} + 8 q^{23} - 20 q^{25} + 7 q^{27} - 9 q^{29} + 2 q^{31} - q^{33} - 6 q^{35} + 2 q^{37} + 3 q^{39} - 25 q^{41}+ \cdots + 28 q^{99}+O(q^{100}) \) 40 * q - 2 * q^3 - q^7 - 2 * q^9 - q^11 + 2 * q^13 + q^15 + 5 * q^17 + q^19 + 6 * q^21 + 8 * q^23 - 20 * q^25 + 7 * q^27 - 9 * q^29 + 2 * q^31 - q^33 - 6 * q^35 + 2 * q^37 + 3 * q^39 - 25 * q^41 - 16 * q^43 + 44 * q^45 - 10 * q^47 - 21 * q^49 + 19 * q^51 - 8 * q^53 - 22 * q^59 - 7 * q^61 + 10 * q^63 - 6 * q^65 + 20 * q^67 + 38 * q^69 - 5 * q^71 - 10 * q^73 - 10 * q^77 + 2 * q^79 + 22 * q^81 + q^83 - 52 * q^87 - 34 * q^89 + 16 * q^91 - 8 * q^93 + 36 * q^95 + 2 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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