LMFDB - Embedded newform 456.2.bg.c.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(25,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))

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

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

chi := DirichletCharacter("456.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.bg (of order $9$, degree $6$, 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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$3$ over $\Q(\zeta_{9})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 3 x^{17} + 30 x^{16} - 51 x^{15} + 501 x^{14} - 768 x^{13} + 4499 x^{12} - 2946 x^{11} + \cdots + 64 $ x^18 - 3x^17 + 30x^16 - 51x^15 + 501x^14 - 768x^13 + 4499x^12 - 2946x^11 + 21141x^10 - 7991x^9 + 67482x^8 + 18777x^7 + 57847x^6 - 1446x^5 + 22860x^4 + 3088x^3 + 3312x^2 - 384*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			289.2
Root			$-0.214696 - 0.371865i$ of defining polynomial
Character	$\chi$	$=$	456.289
Dual form			456.2.bg.c.385.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.173648 - 0.984808i) q^{3} +(-0.328934 + 0.276008i) q^{5} +(-1.95470 + 3.38563i) q^{7} +(-0.939693 + 0.342020i) q^{9} +O(q^{10})$	\(q+(-0.173648 - 0.984808i) q^{3} +(-0.328934 + 0.276008i) q^{5} +(-1.95470 + 3.38563i) q^{7} +(-0.939693 + 0.342020i) q^{9} +(2.16709 + 3.75351i) q^{11} +(-0.0235730 + 0.133689i) q^{13} +(0.328934 + 0.276008i) q^{15} +(-5.59735 - 2.03727i) q^{17} +(-1.47658 + 4.10118i) q^{19} +(3.67363 + 1.33709i) q^{21} +(6.07703 + 5.09923i) q^{23} +(-0.836224 + 4.74246i) q^{25} +(0.500000 + 0.866025i) q^{27} +(7.20815 - 2.62355i) q^{29} +(2.60402 - 4.51030i) q^{31} +(3.32017 - 2.78596i) q^{33} +(-0.291497 - 1.65316i) q^{35} -5.10285 q^{37} +0.135751 q^{39} +(-0.555615 - 3.15105i) q^{41} +(-7.94070 + 6.66304i) q^{43} +(0.214696 - 0.371865i) q^{45} +(3.56422 - 1.29727i) q^{47} +(-4.14167 - 7.17358i) q^{49} +(-1.03435 + 5.86608i) q^{51} +(5.49087 + 4.60739i) q^{53} +(-1.74883 - 0.636521i) q^{55} +(4.29528 + 0.741988i) q^{57} +(0.0603913 + 0.0219807i) q^{59} +(4.56070 + 3.82688i) q^{61} +(0.678859 - 3.85000i) q^{63} +(-0.0291453 - 0.0504812i) q^{65} +(-15.0223 + 5.46767i) q^{67} +(3.96650 - 6.87018i) q^{69} +(-5.18052 + 4.34697i) q^{71} +(-1.71264 - 9.71289i) q^{73} +4.81562 q^{75} -16.9440 q^{77} +(-1.77304 - 10.0554i) q^{79} +(0.766044 - 0.642788i) q^{81} +(3.33003 - 5.76778i) q^{83} +(2.40346 - 0.874788i) q^{85} +(-3.83538 - 6.64307i) q^{87} +(2.18923 - 12.4157i) q^{89} +(-0.406544 - 0.341131i) q^{91} +(-4.89396 - 1.78126i) q^{93} +(-0.646262 - 1.75657i) q^{95} +(7.87968 + 2.86797i) q^{97} +(-3.32017 - 2.78596i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 3 q^{5}+O(q^{10}) $ 18 * q - 3 * q^5	$ 18 q - 3 q^{5} - 3 q^{11} + 18 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{19} - 3 q^{21} + 12 q^{23} - 3 q^{25} + 9 q^{27} + 3 q^{29} + 21 q^{31} + 6 q^{33} + 12 q^{35} - 30 q^{37} + 6 q^{39} + 33 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47} - 21 q^{49} + 12 q^{51} + 30 q^{53} - 24 q^{55} - 21 q^{57} - 9 q^{59} + 48 q^{61} - 6 q^{63} - 9 q^{65} - 18 q^{67} - 6 q^{69} - 51 q^{71} - 12 q^{75} + 30 q^{77} - 39 q^{79} - 30 q^{83} - 87 q^{85} - 21 q^{87} + 45 q^{89} + 72 q^{91} - 27 q^{93} - 78 q^{95} + 39 q^{97} - 6 q^{99}+O(q^{100}) $ 18 * q - 3 * q^5 - 3 * q^11 + 18 * q^13 + 3 * q^15 - 3 * q^17 - 3 * q^19 - 3 * q^21 + 12 * q^23 - 3 * q^25 + 9 * q^27 + 3 * q^29 + 21 * q^31 + 6 * q^33 + 12 * q^35 - 30 * q^37 + 6 * q^39 + 33 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 - 21 * q^49 + 12 * q^51 + 30 * q^53 - 24 * q^55 - 21 * q^57 - 9 * q^59 + 48 * q^61 - 6 * q^63 - 9 * q^65 - 18 * q^67 - 6 * q^69 - 51 * q^71 - 12 * q^75 + 30 * q^77 - 39 * q^79 - 30 * q^83 - 87 * q^85 - 21 * q^87 + 45 * q^89 + 72 * q^91 - 27 * q^93 - 78 * q^95 + 39 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/456\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$343$
$\chi(n)$	$e\left(\frac{1}{9}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.173648	−	0.984808i	−0.100256	−	0.568579i
$3$	−0.173648	−	0.984808i	−0.100256	−	0.568579i
$4$		0			0
$5$	−0.328934	+	0.276008i	−0.147104	+	0.123435i	−0.713370	−	0.700788i	$-0.752832\pi$
$5$	−0.328934	+	0.276008i	−0.147104	+	0.123435i	0.566266	+	0.824223i	$0.308387\pi$
$6$		0			0
$7$	−1.95470	+	3.38563i	−0.738806	+	1.27965i	0.214228	+	0.976784i	$0.431276\pi$
$7$	−1.95470	+	3.38563i	−0.738806	+	1.27965i	−0.953033	+	0.302865i	$0.902057\pi$
$8$		0			0
$9$	−0.939693	+	0.342020i	−0.313231	+	0.114007i
$10$		0			0
$11$	2.16709	+	3.75351i	0.653402	+	1.13172i	0.982292	+	0.187357i	$0.0599921\pi$
$11$	2.16709	+	3.75351i	0.653402	+	1.13172i	−0.328890	+	0.944368i	$0.606675\pi$
$12$		0			0
$13$	−0.0235730	+	0.133689i	−0.00653797	+	0.0370787i	−0.987902	−	0.155078i	$-0.950437\pi$
$13$	−0.0235730	+	0.133689i	−0.00653797	+	0.0370787i	0.981364	+	0.192157i	$0.0615482\pi$
$14$		0			0
$15$	0.328934	+	0.276008i	0.0849304	+	0.0712650i
$16$		0			0
$17$	−5.59735	−	2.03727i	−1.35756	−	0.494110i	−0.442259	−	0.896888i	$-0.645823\pi$
$17$	−5.59735	−	2.03727i	−1.35756	−	0.494110i	−0.915298	+	0.402777i	$0.868045\pi$
$18$		0			0
$19$	−1.47658	+	4.10118i	−0.338751	+	0.940876i
$19$	−1.47658	+	4.10118i	−0.338751	+	0.940876i
$20$		0			0
$21$	3.67363	+	1.33709i	0.801651	+	0.291777i
$22$		0			0
$23$	6.07703	+	5.09923i	1.26715	+	1.06326i	0.994882	+	0.101041i	$0.0322174\pi$
$23$	6.07703	+	5.09923i	1.26715	+	1.06326i	0.272266	+	0.962222i	$0.412227\pi$
$24$		0			0
$25$	−0.836224	+	4.74246i	−0.167245	+	0.948492i
$26$		0			0
$27$	0.500000	+	0.866025i	0.0962250	+	0.166667i
$28$		0			0
$29$	7.20815	−	2.62355i	1.33852	−	0.487181i	0.429174	−	0.903222i	$-0.358805\pi$
$29$	7.20815	−	2.62355i	1.33852	−	0.487181i	0.909346	+	0.416041i	$0.136583\pi$
$30$		0			0
$31$	2.60402	−	4.51030i	0.467696	−	0.810074i	−0.531622	−	0.846982i	$-0.678417\pi$
$31$	2.60402	−	4.51030i	0.467696	−	0.810074i	0.999319	+	0.0369076i	$0.0117507\pi$
$32$		0			0
$33$	3.32017	−	2.78596i	0.577968	−	0.484973i
$34$		0			0
$35$	−0.291497	−	1.65316i	−0.0492720	−	0.279435i
$36$		0			0
$37$	−5.10285			−0.838903			−0.419451	−	0.907778i	$-0.637778\pi$
$37$	−5.10285			−0.838903			−0.419451	+	0.907778i	$0.637778\pi$
$38$		0			0
$39$	0.135751			0.0217376
$40$		0			0
$41$	−0.555615	−	3.15105i	−0.0867726	−	0.492112i	−0.996960	−	0.0779151i	$-0.975174\pi$
$41$	−0.555615	−	3.15105i	−0.0867726	−	0.492112i	0.910187	−	0.414197i	$-0.135937\pi$
$42$		0			0
$43$	−7.94070	+	6.66304i	−1.21095	+	1.01610i	−0.211697	+	0.977335i	$0.567899\pi$
$43$	−7.94070	+	6.66304i	−1.21095	+	1.01610i	−0.999248	+	0.0387688i	$0.987656\pi$
$44$		0			0
$45$	0.214696	−	0.371865i	0.0320050	−	0.0554344i
$46$		0			0
$47$	3.56422	−	1.29727i	0.519895	−	0.189226i	−0.0687255	−	0.997636i	$-0.521893\pi$
$47$	3.56422	−	1.29727i	0.519895	−	0.189226i	0.588621	+	0.808409i	$0.299671\pi$
$48$		0			0
$49$	−4.14167	−	7.17358i	−0.591667	−	1.02480i
$50$		0			0
$51$	−1.03435	+	5.86608i	−0.144838	+	0.821416i
$52$		0			0
$53$	5.49087	+	4.60739i	0.754229	+	0.632873i	0.936618	−	0.350353i	$-0.113939\pi$
$53$	5.49087	+	4.60739i	0.754229	+	0.632873i	−0.182389	+	0.983226i	$0.558383\pi$
$54$		0			0
$55$	−1.74883	−	0.636521i	−0.235812	−	0.0858285i
$56$		0			0
$57$	4.29528	+	0.741988i	0.568924	+	0.0982787i
$58$		0			0
$59$	0.0603913	+	0.0219807i	0.00786228	+	0.00286164i	0.345948	−	0.938254i	$-0.387557\pi$
$59$	0.0603913	+	0.0219807i	0.00786228	+	0.00286164i	−0.338086	+	0.941115i	$0.609779\pi$
$60$		0			0
$61$	4.56070	+	3.82688i	0.583937	+	0.489982i	0.886238	−	0.463231i	$-0.153310\pi$
$61$	4.56070	+	3.82688i	0.583937	+	0.489982i	−0.302300	+	0.953213i	$0.597754\pi$
$62$		0			0
$63$	0.678859	−	3.85000i	0.0855282	−	0.485054i
$64$		0			0
$65$	−0.0291453	−	0.0504812i	−0.00361503	−	0.00626142i
$66$		0			0
$67$	−15.0223	+	5.46767i	−1.83527	+	0.667983i	−0.843962	+	0.536403i	$0.819783\pi$
$67$	−15.0223	+	5.46767i	−1.83527	+	0.667983i	−0.991306	+	0.131580i	$0.957995\pi$
$68$		0			0
$69$	3.96650	−	6.87018i	0.477510	−	0.827072i
$70$		0			0
$71$	−5.18052	+	4.34697i	−0.614815	+	0.515891i	−0.896169	−	0.443713i	$-0.853661\pi$
$71$	−5.18052	+	4.34697i	−0.614815	+	0.515891i	0.281354	+	0.959604i	$0.409216\pi$
$72$		0			0
$73$	−1.71264	−	9.71289i	−0.200450	−	1.13681i	−0.904441	−	0.426599i	$-0.859712\pi$
$73$	−1.71264	−	9.71289i	−0.200450	−	1.13681i	0.703991	−	0.710209i	$-0.251399\pi$
$74$		0			0
$75$	4.81562			0.556060
$76$		0			0
$77$	−16.9440			−1.93095
$78$		0			0
$79$	−1.77304	−	10.0554i	−0.199483	−	1.13132i	−0.905888	−	0.423517i	$-0.860795\pi$
$79$	−1.77304	−	10.0554i	−0.199483	−	1.13132i	0.706405	−	0.707808i	$-0.250316\pi$
$80$		0			0
$81$	0.766044	−	0.642788i	0.0851160	−	0.0714208i
$82$		0			0
$83$	3.33003	−	5.76778i	0.365518	−	0.633096i	−0.623341	−	0.781950i	$-0.714225\pi$
$83$	3.33003	−	5.76778i	0.365518	−	0.633096i	0.988859	+	0.148854i	$0.0475585\pi$
$84$		0			0
$85$	2.40346	−	0.874788i	0.260692	−	0.0948841i
$86$		0			0
$87$	−3.83538	−	6.64307i	−0.411196	−	0.712212i
$88$		0			0
$89$	2.18923	−	12.4157i	0.232057	−	1.31606i	−0.616665	−	0.787225i	$-0.711517\pi$
$89$	2.18923	−	12.4157i	0.232057	−	1.31606i	0.848723	−	0.528838i	$-0.177372\pi$
$90$		0			0
$91$	−0.406544	−	0.341131i	−0.0426174	−	0.0357602i
$92$		0			0
$93$	−4.89396	−	1.78126i	−0.507480	−	0.184708i
$94$		0			0
$95$	−0.646262	−	1.75657i	−0.0663051	−	0.180220i
$96$		0			0
$97$	7.87968	+	2.86797i	0.800060	+	0.291198i	0.709511	−	0.704694i	$-0.248916\pi$
$97$	7.87968	+	2.86797i	0.800060	+	0.291198i	0.0905488	+	0.995892i	$0.471138\pi$
$98$		0			0
$99$	−3.32017	−	2.78596i	−0.333690	−	0.279999i
$100$		0			0
$101$	−0.211201	+	1.19778i	−0.0210153	+	0.119184i	−0.993511	−	0.113738i	$-0.963718\pi$
$101$	−0.211201	+	1.19778i	−0.0210153	+	0.119184i	0.972495	+	0.232922i	$0.0748286\pi$
$102$		0			0
$103$	7.27755	+	12.6051i	0.717078	+	1.24202i	0.962153	+	0.272511i	$0.0878542\pi$
$103$	7.27755	+	12.6051i	0.717078	+	1.24202i	−0.245075	+	0.969504i	$0.578812\pi$
$104$		0			0
$105$	−1.57743	+	0.574137i	−0.153941	+	0.0560300i
$106$		0			0
$107$	7.67812	−	13.2989i	0.742272	−	1.28565i	−0.209186	−	0.977876i	$-0.567082\pi$
$107$	7.67812	−	13.2989i	0.742272	−	1.28565i	0.951458	−	0.307777i	$-0.0995852\pi$
$108$		0			0
$109$	−3.41338	+	2.86417i	−0.326942	+	0.274337i	−0.791453	−	0.611230i	$-0.790675\pi$
$109$	−3.41338	+	2.86417i	−0.326942	+	0.274337i	0.464510	+	0.885568i	$0.346230\pi$
$110$		0			0
$111$	0.886100	+	5.02532i	0.0841049	+	0.476982i
$112$		0			0
$113$	−2.89325			−0.272174			−0.136087	−	0.990697i	$-0.543453\pi$
$113$	−2.89325			−0.272174			−0.136087	+	0.990697i	$0.543453\pi$
$114$		0			0
$115$	−3.40637			−0.317646
$116$		0			0
$117$	−0.0235730	−	0.133689i	−0.00217932	−	0.0123596i
$118$		0			0
$119$	17.8386	−	14.9683i	1.63526	−	1.37214i
$120$		0			0
$121$	−3.89254	+	6.74208i	−0.353868	+	0.612917i
$122$		0			0
$123$	−3.00670	+	1.09435i	−0.271105	+	0.0986741i
$124$		0			0
$125$	−2.10738	−	3.65009i	−0.188490	−	0.326474i
$126$		0			0
$127$	2.85972	−	16.2183i	0.253759	−	1.43914i	−0.545478	−	0.838125i	$-0.683652\pi$
$127$	2.85972	−	16.2183i	0.253759	−	1.43914i	0.799237	−	0.601016i	$-0.205237\pi$
$128$		0			0
$129$	7.94070	+	6.66304i	0.699140	+	0.586648i
$130$		0			0
$131$	−8.92144	−	3.24714i	−0.779469	−	0.283704i	−0.0785180	−	0.996913i	$-0.525019\pi$
$131$	−8.92144	−	3.24714i	−0.779469	−	0.283704i	−0.700951	+	0.713209i	$0.747241\pi$
$132$		0			0
$133$	−10.9988	−	13.0157i	−0.953719	−	1.12861i
$134$		0			0
$135$	−0.403497	−	0.146861i	−0.0347275	−	0.0126398i
$136$		0			0
$137$	9.06007	+	7.60230i	0.774054	+	0.649508i	0.941744	−	0.336332i	$-0.109186\pi$
$137$	9.06007	+	7.60230i	0.774054	+	0.649508i	−0.167690	+	0.985840i	$0.553631\pi$
$138$		0			0
$139$	−1.32623	+	7.52142i	−0.112489	+	0.637959i	0.875473	+	0.483266i	$0.160550\pi$
$139$	−1.32623	+	7.52142i	−0.112489	+	0.637959i	−0.987963	+	0.154692i	$0.950561\pi$
$140$		0			0
$141$	−1.89648	−	3.28481i	−0.159713	−	0.276631i
$142$		0			0
$143$	−0.552888	+	0.201235i	−0.0462348	+	0.0168281i
$144$		0			0
$145$	−1.64688	+	2.85248i	−0.136766	+	0.236886i
$146$		0			0
$147$	−6.34541	+	5.32443i	−0.523360	+	0.439152i
$148$		0			0
$149$	1.37145	+	7.77789i	0.112354	+	0.637190i	0.988026	+	0.154285i	$0.0493073\pi$
$149$	1.37145	+	7.77789i	0.112354	+	0.637190i	−0.875673	+	0.482905i	$0.839582\pi$
$150$		0			0
$151$	8.96656			0.729689			0.364844	−	0.931069i	$-0.381122\pi$
$151$	8.96656			0.729689			0.364844	+	0.931069i	$0.381122\pi$
$152$		0			0
$153$	5.95658			0.481561
$154$		0			0
$155$	0.388329	+	2.20232i	0.0311913	+	0.176895i
$156$		0			0
$157$	−1.78309	+	1.49619i	−0.142306	+	0.119409i	−0.711162	−	0.703028i	$-0.751831\pi$
$157$	−1.78309	+	1.49619i	−0.142306	+	0.119409i	0.568856	+	0.822437i	$0.307386\pi$
$158$		0			0
$159$	3.58391	−	6.20751i	0.284223	−	0.492288i
$160$		0			0
$161$	−29.1429	+	10.6071i	−2.29678	+	0.835959i
$162$		0			0
$163$	3.06566	+	5.30988i	0.240121	+	0.415902i	0.960749	−	0.277421i	$-0.0894795\pi$
$163$	3.06566	+	5.30988i	0.240121	+	0.415902i	−0.720628	+	0.693322i	$0.756146\pi$
$164$		0			0
$165$	−0.323170	+	1.83279i	−0.0251588	+	0.142683i
$166$		0			0
$167$	2.83124	+	2.37569i	0.219088	+	0.183836i	0.745725	−	0.666253i	$-0.232103\pi$
$167$	2.83124	+	2.37569i	0.219088	+	0.183836i	−0.526638	+	0.850090i	$0.676548\pi$
$168$		0			0
$169$	12.1987	+	4.43996i	0.938361	+	0.341535i
$170$		0			0
$171$	−0.0151526	−	4.35887i	−0.00115875	−	0.333331i
$172$		0			0
$173$	4.14540	+	1.50880i	0.315169	+	0.114712i	0.494761	−	0.869029i	$-0.335256\pi$
$173$	4.14540	+	1.50880i	0.315169	+	0.114712i	−0.179592	+	0.983741i	$0.557478\pi$
$174$		0			0
$175$	−14.4217	−	12.1012i	−1.09018	−	0.914766i
$176$		0			0
$177$	0.0111599	−	0.0632908i	0.000838827	−	0.00475723i
$178$		0			0
$179$	9.66948	+	16.7480i	0.722731	+	1.25181i	0.959901	+	0.280339i	$0.0904469\pi$
$179$	9.66948	+	16.7480i	0.722731	+	1.25181i	−0.237170	+	0.971468i	$0.576220\pi$
$180$		0			0
$181$	11.3381	−	4.12673i	0.842754	−	0.306737i	0.115671	−	0.993288i	$-0.463098\pi$
$181$	11.3381	−	4.12673i	0.842754	−	0.306737i	0.727082	+	0.686550i	$0.240876\pi$
$182$		0			0
$183$	2.97678	−	5.15594i	0.220050	−	0.381138i
$184$		0			0
$185$	1.67850	−	1.40843i	0.123406	−	0.103550i
$186$		0			0
$187$	−4.48305	−	25.4246i	−0.327833	−	1.85923i
$188$		0			0
$189$	−3.90939			−0.284366
$190$		0			0
$191$	19.9344			1.44240			0.721201	−	0.692726i	$-0.243590\pi$
$191$	19.9344			1.44240			0.721201	+	0.692726i	$0.243590\pi$
$192$		0			0
$193$	−0.121315	−	0.688010i	−0.00873242	−	0.0495240i	0.980130	−	0.198355i	$-0.0635600\pi$
$193$	−0.121315	−	0.688010i	−0.00873242	−	0.0495240i	−0.988863	+	0.148831i	$0.952449\pi$
$194$		0			0
$195$	−0.0446533	+	0.0374685i	−0.00319769	+	0.00268318i
$196$		0			0
$197$	−1.21735	+	2.10851i	−0.0867326	+	0.150225i	−0.906128	−	0.423003i	$-0.860976\pi$
$197$	−1.21735	+	2.10851i	−0.0867326	+	0.150225i	0.819396	+	0.573228i	$0.194309\pi$
$198$		0			0
$199$	0.704515	−	0.256422i	0.0499417	−	0.0181773i	−0.316928	−	0.948449i	$-0.602652\pi$
$199$	0.704515	−	0.256422i	0.0499417	−	0.0181773i	0.366870	+	0.930272i	$0.380429\pi$
$200$		0			0
$201$	7.99321	+	13.8446i	0.563797	+	0.976525i
$202$		0			0
$203$	−5.20736	+	29.5324i	−0.365485	+	2.07277i
$204$		0			0
$205$	1.05248	+	0.883133i	0.0735082	+	0.0616807i
$206$		0			0
$207$	−7.45458	−	2.71324i	−0.518129	−	0.188584i
$208$		0			0
$209$	−18.5937	+	3.34526i	−1.28615	+	0.231396i
$210$		0			0
$211$	13.8375	+	5.03644i	0.952614	+	0.346723i	0.771135	−	0.636672i	$-0.219689\pi$
$211$	13.8375	+	5.03644i	0.952614	+	0.346723i	0.181479	+	0.983395i	$0.441912\pi$
$212$		0			0
$213$	5.18052	+	4.34697i	0.354963	+	0.297850i
$214$		0			0
$215$	0.772912	−	4.38340i	0.0527121	−	0.298945i
$216$		0			0
$217$	10.1801	+	17.6325i	0.691073	+	1.19697i
$218$		0			0
$219$	−9.26793	+	3.37325i	−0.626269	+	0.227943i
$220$		0			0
$221$	0.404307	−	0.700280i	0.0271966	−	0.0471059i
$222$		0			0
$223$	−15.9571	+	13.3896i	−1.06857	+	0.896633i	−0.994922	−	0.100652i	$-0.967907\pi$
$223$	−15.9571	+	13.3896i	−1.06857	+	0.896633i	−0.0736436	+	0.997285i	$0.523463\pi$
$224$		0			0
$225$	−0.836224	−	4.74246i	−0.0557483	−	0.316164i
$226$		0			0
$227$	−10.7521			−0.713642			−0.356821	−	0.934173i	$-0.616139\pi$
$227$	−10.7521			−0.713642			−0.356821	+	0.934173i	$0.616139\pi$
$228$		0			0
$229$	11.9356			0.788727			0.394364	−	0.918954i	$-0.370965\pi$
$229$	11.9356			0.788727			0.394364	+	0.918954i	$0.370965\pi$
$230$		0			0
$231$	2.94229	+	16.6866i	0.193589	+	1.09790i
$232$		0			0
$233$	13.3725	−	11.2209i	0.876061	−	0.735103i	−0.0893042	−	0.996004i	$-0.528464\pi$
$233$	13.3725	−	11.2209i	0.876061	−	0.735103i	0.965365	+	0.260902i	$0.0840199\pi$
$234$		0			0
$235$	−0.814336	+	1.41047i	−0.0531214	+	0.0920090i
$236$		0			0
$237$	−9.59479	+	3.49222i	−0.623248	+	0.226844i
$238$		0			0
$239$	−4.01877	−	6.96071i	−0.259952	−	0.450251i	0.706277	−	0.707936i	$-0.250374\pi$
$239$	−4.01877	−	6.96071i	−0.259952	−	0.450251i	−0.966229	+	0.257685i	$0.917040\pi$
$240$		0			0
$241$	−1.29513	+	7.34503i	−0.0834265	+	0.473135i	0.914259	+	0.405131i	$0.132774\pi$
$241$	−1.29513	+	7.34503i	−0.0834265	+	0.473135i	−0.997685	+	0.0680039i	$0.978337\pi$
$242$		0			0
$243$	−0.766044	−	0.642788i	−0.0491418	−	0.0412348i
$244$		0			0
$245$	3.34231	+	1.21650i	0.213532	+	0.0777193i
$246$		0			0
$247$	−0.513476	−	0.294080i	−0.0326717	−	0.0187119i
$248$		0			0
$249$	−6.25841	−	2.27787i	−0.396610	−	0.144354i
$250$		0			0
$251$	−3.93230	−	3.29959i	−0.248205	−	0.208268i	0.510194	−	0.860059i	$-0.329574\pi$
$251$	−3.93230	−	3.29959i	−0.248205	−	0.208268i	−0.758399	+	0.651791i	$0.774018\pi$
$252$		0			0
$253$	−5.97055	+	33.8607i	−0.375365	+	2.12880i
$254$		0			0
$255$	−1.27885	−	2.21504i	−0.0800850	−	0.138711i
$256$		0			0
$257$	−18.3124	+	6.66517i	−1.14230	+	0.415762i	−0.842742	−	0.538318i	$-0.819060\pi$
$257$	−18.3124	+	6.66517i	−1.14230	+	0.415762i	−0.299554	+	0.954079i	$0.596838\pi$
$258$		0			0
$259$	9.97451	−	17.2764i	0.619786	−	1.07350i
$260$		0			0
$261$	−5.87614	+	4.93066i	−0.363724	+	0.305200i
$262$		0			0
$263$	−3.12670	−	17.7324i	−0.192801	−	1.09343i	−0.915516	−	0.402282i	$-0.868217\pi$
$263$	−3.12670	−	17.7324i	−0.192801	−	1.09343i	0.722715	−	0.691146i	$-0.242894\pi$
$264$		0			0
$265$	−3.07781			−0.189068
$266$		0			0
$267$	−12.6072			−0.771551
$268$		0			0
$269$	3.46795	+	19.6677i	0.211444	+	1.19916i	0.886971	+	0.461825i	$0.152805\pi$
$269$	3.46795	+	19.6677i	0.211444	+	1.19916i	−0.675526	+	0.737336i	$0.736084\pi$
$270$		0			0
$271$	20.1744	−	16.9283i	1.22551	−	1.02832i	0.226989	−	0.973897i	$-0.427112\pi$
$271$	20.1744	−	16.9283i	1.22551	−	1.02832i	0.998518	−	0.0544246i	$-0.0173324\pi$
$272$		0			0
$273$	−0.265353	+	0.459604i	−0.0160599	+	0.0278165i
$274$		0			0
$275$	−19.6130	+	7.13856i	−1.18271	+	0.430471i
$276$		0			0
$277$	5.46245	+	9.46124i	0.328207	+	0.568471i	0.982156	−	0.188068i	$-0.0602224\pi$
$277$	5.46245	+	9.46124i	0.328207	+	0.568471i	−0.653949	+	0.756538i	$0.726889\pi$
$278$		0			0
$279$	−0.904368	+	5.12893i	−0.0541431	+	0.307061i
$280$		0			0
$281$	−8.39555	−	7.04470i	−0.500837	−	0.420252i	0.357054	−	0.934084i	$-0.383781\pi$
$281$	−8.39555	−	7.04470i	−0.500837	−	0.420252i	−0.857891	+	0.513832i	$0.828226\pi$
$282$		0			0
$283$	21.6993	+	7.89791i	1.28989	+	0.469482i	0.893692	−	0.448682i	$-0.148106\pi$
$283$	21.6993	+	7.89791i	1.28989	+	0.469482i	0.396200	+	0.918164i	$0.370329\pi$
$284$		0			0
$285$	−1.61766	+	0.941469i	−0.0958218	+	0.0557678i
$286$		0			0
$287$	11.7544	+	4.27824i	0.693838	+	0.252536i
$288$		0			0
$289$	14.1571	+	11.8792i	0.832771	+	0.698778i
$290$		0			0
$291$	1.45611	−	8.25799i	0.0853584	−	0.484092i
$292$		0			0
$293$	4.49967	+	7.79366i	0.262874	+	0.455310i	0.967004	−	0.254760i	$-0.0819965\pi$
$293$	4.49967	+	7.79366i	0.262874	+	0.455310i	−0.704131	+	0.710070i	$0.748663\pi$
$294$		0			0
$295$	−0.0259316	+	0.00943833i	−0.00150980	+	0.000549521i
$296$		0			0
$297$	−2.16709	+	3.75351i	−0.125747	+	0.217801i
$298$		0			0
$299$	−0.824965	+	0.692228i	−0.0477090	+	0.0400326i
$300$		0			0
$301$	−7.03695	−	39.9085i	−0.405603	−	2.30029i
$302$		0			0
$303$	1.21626			0.0698723
$304$		0			0
$305$	−2.55642			−0.146380
$306$		0			0
$307$	2.55591	+	14.4953i	0.145873	+	0.827290i	0.966662	+	0.256057i	$0.0824235\pi$
$307$	2.55591	+	14.4953i	0.145873	+	0.827290i	−0.820788	+	0.571233i	$0.806465\pi$
$308$		0			0
$309$	11.1498	−	9.35583i	0.634293	−	0.532235i
$310$		0			0
$311$	−3.49563	+	6.05461i	−0.198219	+	0.343325i	−0.947951	−	0.318416i	$-0.896849\pi$
$311$	−3.49563	+	6.05461i	−0.198219	+	0.343325i	0.749732	+	0.661742i	$0.230182\pi$
$312$		0			0
$313$	18.2331	−	6.63630i	1.03059	−	0.375106i	0.229288	−	0.973359i	$-0.426360\pi$
$313$	18.2331	−	6.63630i	1.03059	−	0.375106i	0.801307	+	0.598253i	$0.204138\pi$
$314$		0			0
$315$	0.839332	+	1.45377i	0.0472910	+	0.0819104i
$316$		0			0
$317$	−0.901677	+	5.11367i	−0.0506432	+	0.287212i	−0.999603	−	0.0281860i	$-0.991027\pi$
$317$	−0.901677	+	5.11367i	−0.0506432	+	0.287212i	0.948959	+	0.315398i	$0.102138\pi$
$318$		0			0
$319$	25.4682	+	21.3704i	1.42595	+	1.19651i
$320$		0			0
$321$	−14.4302	−	5.25215i	−0.805412	−	0.293146i
$322$		0			0
$323$	16.6202	−	19.9476i	0.924771	−	1.10991i
$324$		0			0
$325$	−0.614303	−	0.223588i	−0.0340754	−	0.0124024i
$326$		0			0
$327$	3.41338	+	2.86417i	0.188760	+	0.158389i
$328$		0			0
$329$	−2.57489	+	14.6029i	−0.141958	+	0.805085i
$330$		0			0
$331$	−8.53569	−	14.7842i	−0.469164	−	0.812616i	0.530215	−	0.847863i	$-0.322111\pi$
$331$	−8.53569	−	14.7842i	−0.469164	−	0.812616i	−0.999379	+	0.0352477i	$0.988778\pi$
$332$		0			0
$333$	4.79511	−	1.74528i	0.262770	−	0.0956405i
$334$		0			0
$335$	3.43222	−	5.94479i	0.187522	−	0.324798i
$336$		0			0
$337$	16.5345	−	13.8741i	0.900691	−	0.755770i	−0.0696340	−	0.997573i	$-0.522183\pi$
$337$	16.5345	−	13.8741i	0.900691	−	0.755770i	0.970325	+	0.241803i	$0.0777387\pi$
$338$		0			0
$339$	0.502407	+	2.84929i	0.0272870	+	0.154752i
$340$		0			0
$341$	22.5726			1.22237
$342$		0			0
$343$	5.01709			0.270897
$344$		0			0
$345$	0.591510	+	3.35462i	0.0318458	+	0.180607i
$346$		0			0
$347$	−15.6056	+	13.0947i	−0.837754	+	0.702959i	−0.957057	−	0.289899i	$-0.906378\pi$
$347$	−15.6056	+	13.0947i	−0.837754	+	0.702959i	0.119303	+	0.992858i	$0.461934\pi$
$348$		0			0
$349$	14.6724	−	25.4133i	0.785394	−	1.36034i	−0.143370	−	0.989669i	$-0.545794\pi$
$349$	14.6724	−	25.4133i	0.785394	−	1.36034i	0.928764	−	0.370673i	$-0.120873\pi$
$350$		0			0
$351$	−0.127565	+	0.0464297i	−0.00680890	+	0.00247824i
$352$		0			0
$353$	−14.5079	−	25.1284i	−0.772178	−	1.33745i	−0.936367	−	0.351022i	$-0.885834\pi$
$353$	−14.5079	−	25.1284i	−0.772178	−	1.33745i	0.164190	−	0.986429i	$-0.447499\pi$
$354$		0			0
$355$	0.504248	−	2.85973i	0.0267627	−	0.151779i
$356$		0			0
$357$	−17.8386	−	14.9683i	−0.944117	−	0.792208i
$358$		0			0
$359$	−23.3857	−	8.51170i	−1.23425	−	0.449230i	−0.359199	−	0.933261i	$-0.616950\pi$
$359$	−23.3857	−	8.51170i	−1.23425	−	0.449230i	−0.875051	+	0.484030i	$0.839172\pi$
$360$		0			0
$361$	−14.6394	−	12.1115i	−0.770495	−	0.637446i
$362$		0			0
$363$	7.31559	+	2.66266i	0.383969	+	0.139753i
$364$		0			0
$365$	3.24419	+	2.72219i	0.169808	+	0.142486i
$366$		0			0
$367$	−5.51049	+	31.2515i	−0.287645	+	1.63132i	0.408036	+	0.912966i	$0.366214\pi$
$367$	−5.51049	+	31.2515i	−0.287645	+	1.63132i	−0.695682	+	0.718350i	$0.744897\pi$
$368$		0			0
$369$	1.59983	+	2.77099i	0.0832839	+	0.144252i
$370$		0			0
$371$	−26.3319	+	9.58402i	−1.36708	+	0.497578i
$372$		0			0
$373$	−4.38380	+	7.59297i	−0.226985	+	0.393149i	−0.956913	−	0.290375i	$-0.906220\pi$
$373$	−4.38380	+	7.59297i	−0.226985	+	0.393149i	0.729928	+	0.683524i	$0.239553\pi$
$374$		0			0
$375$	−3.22869	+	2.70919i	−0.166729	+	0.139902i
$376$		0			0
$377$	0.180823	+	1.02550i	0.00931284	+	0.0528157i
$378$		0			0
$379$	−5.64917			−0.290178			−0.145089	−	0.989419i	$-0.546347\pi$
$379$	−5.64917			−0.290178			−0.145089	+	0.989419i	$0.546347\pi$
$380$		0			0
$381$	−16.4685			−0.843706
$382$		0			0
$383$	−0.874156	−	4.95758i	−0.0446673	−	0.253321i	0.954295	−	0.298866i	$-0.0966085\pi$
$383$	−0.874156	−	4.95758i	−0.0446673	−	0.253321i	−0.998962	+	0.0455457i	$0.985497\pi$
$384$		0			0
$385$	5.57345	−	4.67668i	0.284050	−	0.238346i
$386$		0			0
$387$	5.18292	−	8.97709i	0.263463	−	0.456331i
$388$		0			0
$389$	20.8075	−	7.57333i	1.05498	−	0.383983i	0.244443	−	0.969664i	$-0.421395\pi$
$389$	20.8075	−	7.57333i	1.05498	−	0.383983i	0.810542	+	0.585681i	$0.199173\pi$
$390$		0			0
$391$	−23.6267	−	40.9227i	−1.19486	−	2.06955i
$392$		0			0
$393$	−1.64861	+	9.34976i	−0.0831616	+	0.471633i
$394$		0			0
$395$	3.35860	+	2.81820i	0.168989	+	0.141799i
$396$		0			0
$397$	−32.6035	−	11.8667i	−1.63632	−	0.595573i	−0.649933	−	0.759991i	$-0.725203\pi$
$397$	−32.6035	−	11.8667i	−1.63632	−	0.595573i	−0.986391	+	0.164418i	$0.947425\pi$
$398$		0			0
$399$	−10.9081	+	13.0919i	−0.546086	+	0.655414i
$400$		0			0
$401$	23.7032	+	8.62724i	1.18368	+	0.430824i	0.857500	−	0.514484i	$-0.172017\pi$
$401$	23.7032	+	8.62724i	1.18368	+	0.430824i	0.326179	+	0.945308i	$0.394239\pi$
$402$		0			0
$403$	0.541593	+	0.454451i	0.0269787	+	0.0226378i
$404$		0			0
$405$	−0.0745633	+	0.422869i	−0.00370508	+	0.0210125i
$406$		0			0
$407$	−11.0583	−	19.1536i	−0.548140	−	0.949407i
$408$		0			0
$409$	13.2428	−	4.81999i	0.654814	−	0.238333i	0.00681846	−	0.999977i	$-0.497830\pi$
$409$	13.2428	−	4.81999i	0.654814	−	0.238333i	0.647996	+	0.761644i	$0.275607\pi$
$410$		0			0
$411$	5.91354	−	10.2425i	0.291693	−	0.505228i
$412$		0			0
$413$	−0.192465	+	0.161497i	−0.00947059	+	0.00794677i
$414$		0			0
$415$	0.496596	+	2.81633i	0.0243769	+	0.138248i
$416$		0			0
$417$	7.63745			0.374008
$418$		0			0
$419$	14.0323			0.685524			0.342762	−	0.939422i	$-0.388638\pi$
$419$	14.0323			0.685524			0.342762	+	0.939422i	$0.388638\pi$
$420$		0			0
$421$	0.0176863	+	0.100304i	0.000861978	+	0.00488852i	0.985236	−	0.171203i	$-0.0547655\pi$
$421$	0.0176863	+	0.100304i	0.000861978	+	0.00488852i	−0.984374	+	0.176092i	$0.943654\pi$
$422$		0			0
$423$	−2.90558	+	2.43807i	−0.141274	+	0.118543i
$424$		0			0
$425$	14.3423	−	24.8416i	0.695704	−	1.20499i
$426$		0			0
$427$	−21.8712	+	7.96046i	−1.05842	+	0.385234i
$428$		0			0
$429$	0.294185	+	0.509544i	0.0142034	+	0.0246010i
$430$		0			0
$431$	−3.60275	+	20.4322i	−0.173539	+	0.984186i	0.766279	+	0.642508i	$0.222106\pi$
$431$	−3.60275	+	20.4322i	−0.173539	+	0.984186i	−0.939817	+	0.341678i	$0.889005\pi$
$432$		0			0
$433$	−11.7634	−	9.87070i	−0.565315	−	0.474356i	0.314772	−	0.949167i	$-0.398072\pi$
$433$	−11.7634	−	9.87070i	−0.565315	−	0.474356i	−0.880088	+	0.474812i	$0.842516\pi$
$434$		0			0
$435$	3.09513	+	1.12653i	0.148400	+	0.0540132i
$436$		0			0
$437$	−29.8861	+	17.3936i	−1.42965	+	0.832047i
$438$		0			0
$439$	24.8082	+	9.02945i	1.18403	+	0.430952i	0.857624	−	0.514277i	$-0.171940\pi$
$439$	24.8082	+	9.02945i	1.18403	+	0.430952i	0.326407	+	0.945229i	$0.394162\pi$
$440$		0			0
$441$	6.34541	+	5.32443i	0.302162	+	0.253544i
$442$		0			0
$443$	−3.05430	+	17.3218i	−0.145114	+	0.822983i	0.822161	+	0.569255i	$0.192768\pi$
$443$	−3.05430	+	17.3218i	−0.145114	+	0.822983i	−0.967275	+	0.253729i	$0.918343\pi$
$444$		0			0
$445$	2.70673	+	4.68819i	0.128311	+	0.222242i
$446$		0			0
$447$	7.42158	−	2.70123i	0.351029	−	0.127764i
$448$		0			0
$449$	8.82244	−	15.2809i	0.416357	−	0.721151i	−0.579213	−	0.815176i	$-0.696640\pi$
$449$	8.82244	−	15.2809i	0.416357	−	0.721151i	0.995570	+	0.0940252i	$0.0299734\pi$
$450$		0			0
$451$	10.6234	−	8.91411i	0.500238	−	0.419749i
$452$		0			0
$453$	−1.55703	−	8.83034i	−0.0731555	−	0.414886i
$454$		0			0
$455$	0.227881			0.0106832
$456$		0			0
$457$	−27.7425			−1.29774			−0.648869	−	0.760900i	$-0.724758\pi$
$457$	−27.7425			−1.29774			−0.648869	+	0.760900i	$0.724758\pi$
$458$		0			0
$459$	−1.03435	−	5.86608i	−0.0482792	−	0.273805i
$460$		0			0
$461$	29.7697	−	24.9798i	1.38651	−	1.16342i	0.419785	−	0.907624i	$-0.362106\pi$
$461$	29.7697	−	24.9798i	1.38651	−	1.16342i	0.966730	−	0.255800i	$-0.0823389\pi$
$462$		0			0
$463$	6.26163	−	10.8455i	0.291003	−	0.504031i	−0.683044	−	0.730377i	$-0.739344\pi$
$463$	6.26163	−	10.8455i	0.291003	−	0.504031i	0.974047	+	0.226345i	$0.0726778\pi$
$464$		0			0
$465$	2.10143	−	0.764859i	0.0974516	−	0.0354695i
$466$		0			0
$467$	13.5781	+	23.5180i	0.628320	+	1.08828i	0.987889	+	0.155163i	$0.0495903\pi$
$467$	13.5781	+	23.5180i	0.628320	+	1.08828i	−0.359569	+	0.933118i	$0.617076\pi$
$468$		0			0
$469$	10.8525	−	61.5477i	0.501123	−	2.84201i
$470$		0			0
$471$	1.78309	+	1.49619i	0.0821603	+	0.0689407i
$472$		0			0
$473$	−42.2180	−	15.3661i	−1.94118	−	0.706533i
$474$		0			0
$475$	−18.2149	−	10.4321i	−0.835759	−	0.478660i
$476$		0			0
$477$	−6.73555	−	2.45154i	−0.308399	−	0.112248i
$478$		0			0
$479$	−20.1063	−	16.8712i	−0.918682	−	0.770866i	0.0550684	−	0.998483i	$-0.482462\pi$
$479$	−20.1063	−	16.8712i	−0.918682	−	0.770866i	−0.973751	+	0.227616i	$0.926907\pi$
$480$		0			0
$481$	0.120289	−	0.682195i	0.00548472	−	0.0311054i
$482$		0			0
$483$	15.5066	+	26.8582i	0.705575	+	1.22209i
$484$		0			0
$485$	−3.38348	+	1.23148i	−0.153636	+	0.0559188i
$486$		0			0
$487$	1.30424	−	2.25902i	0.0591009	−	0.102366i	−0.834961	−	0.550309i	$-0.814510\pi$
$487$	1.30424	−	2.25902i	0.0591009	−	0.102366i	0.894062	+	0.447943i	$0.147843\pi$
$488$		0			0
$489$	4.69686	−	3.94114i	0.212400	−	0.178224i
$490$		0			0
$491$	−3.68911	−	20.9220i	−0.166487	−	0.944196i	−0.947518	−	0.319703i	$-0.896417\pi$
$491$	−3.68911	−	20.9220i	−0.166487	−	0.944196i	0.781030	−	0.624493i	$-0.214694\pi$
$492$		0			0
$493$	−45.6914			−2.05784
$494$		0			0
$495$	1.86106			0.0836486
$496$		0			0
$497$	−4.59091	−	26.0363i	−0.205930	−	1.16789i
$498$		0			0
$499$	−23.6115	+	19.8124i	−1.05699	+	0.886923i	−0.993812	−	0.111078i	$-0.964570\pi$
$499$	−23.6115	+	19.8124i	−1.05699	+	0.886923i	−0.0631826	+	0.998002i	$0.520125\pi$
$500$		0			0
$501$	1.84796	−	3.20076i	0.0825607	−	0.142999i
$502$		0			0
$503$	21.3001	−	7.75260i	0.949725	−	0.345672i	0.179726	−	0.983717i	$-0.442479\pi$
$503$	21.3001	−	7.75260i	0.949725	−	0.345672i	0.769999	+	0.638045i	$0.220257\pi$
$504$		0			0
$505$	−0.261127	−	0.452285i	−0.0116200	−	0.0201264i
$506$		0			0
$507$	2.25423	−	12.7844i	0.100114	−	0.567773i
$508$		0			0
$509$	6.51595	+	5.46753i	0.288814	+	0.242344i	0.775670	−	0.631138i	$-0.217412\pi$
$509$	6.51595	+	5.46753i	0.288814	+	0.242344i	−0.486856	+	0.873482i	$0.661856\pi$
$510$		0			0
$511$	36.2320	+	13.1874i	1.60281	+	0.583374i
$512$		0			0
$513$	−4.29002	+	0.771833i	−0.189409	+	0.0340772i
$514$		0			0
$515$	−5.87294	−	2.13757i	−0.258793	−	0.0941928i
$516$		0			0
$517$	12.5933	+	10.5670i	0.553853	+	0.464738i
$518$		0			0
$519$	0.766039	−	4.34443i	0.0336254	−	0.190699i
$520$		0			0
$521$	4.60021	+	7.96780i	0.201539	+	0.349076i	0.949024	−	0.315202i	$-0.102072\pi$
$521$	4.60021	+	7.96780i	0.201539	+	0.349076i	−0.747486	+	0.664278i	$0.768739\pi$
$522$		0			0
$523$	34.2194	−	12.4549i	1.49631	−	0.544613i	0.541209	−	0.840888i	$-0.317967\pi$
$523$	34.2194	−	12.4549i	1.49631	−	0.544613i	0.955103	+	0.296275i	$0.0957445\pi$
$524$		0			0
$525$	−9.41308	+	16.3039i	−0.410820	+	0.711562i
$526$		0			0
$527$	−23.7643	+	19.9406i	−1.03519	+	0.868628i
$528$		0			0
$529$	6.93419	+	39.3258i	0.301487	+	1.70982i
$530$		0			0
$531$	−0.0642671			−0.00278896
$532$		0			0
$533$	0.434359			0.0188142
$534$		0			0
$535$	1.14501	+	6.49368i	0.0495032	+	0.280746i
$536$		0			0
$537$	14.8145	−	12.4308i	0.639293	−	0.536431i
$538$		0			0
$539$	17.9507	−	31.0916i	0.773193	−	1.33921i
$540$		0			0
$541$	−38.1517	+	13.8861i	−1.64027	+	0.597009i	−0.987086	−	0.160190i	$-0.948789\pi$
$541$	−38.1517	+	13.8861i	−1.64027	+	0.597009i	−0.653183	+	0.757200i	$0.726567\pi$
$542$		0			0
$543$	−6.03287	−	10.4492i	−0.258895	−	0.448420i
$544$		0			0
$545$	0.332243	−	1.88424i	0.0142317	−	0.0807121i
$546$		0			0
$547$	10.3502	+	8.68488i	0.442544	+	0.371339i	0.836661	−	0.547722i	$-0.184505\pi$
$547$	10.3502	+	8.68488i	0.442544	+	0.371339i	−0.394116	+	0.919061i	$0.628949\pi$
$548$		0			0
$549$	−5.59452	−	2.03624i	−0.238768	−	0.0869046i
$550$		0			0
$551$	0.116232	+	33.4358i	0.00495165	+	1.42441i
$552$		0			0
$553$	37.5098	+	13.6524i	1.59508	+	0.580561i
$554$		0			0
$555$	−1.67850	−	1.40843i	−0.0712483	−	0.0597844i
$556$		0			0
$557$	4.90730	−	27.8307i	0.207929	−	1.17922i	−0.684835	−	0.728698i	$-0.740126\pi$
$557$	4.90730	−	27.8307i	0.207929	−	1.17922i	0.892764	−	0.450525i	$-0.148763\pi$
$558$		0			0
$559$	−0.703590	−	1.21865i	−0.0297587	−	0.0515435i
$560$		0			0
$561$	−24.2599	+	8.82988i	−1.02425	+	0.372798i
$562$		0			0
$563$	10.0442	−	17.3971i	0.423314	−	0.733201i	−0.572947	−	0.819592i	$-0.694200\pi$
$563$	10.0442	−	17.3971i	0.423314	−	0.733201i	0.996261	+	0.0863910i	$0.0275334\pi$
$564$		0			0
$565$	0.951687	−	0.798560i	0.0400378	−	0.0335957i
$566$		0			0
$567$	0.678859	+	3.85000i	0.0285094	+	0.161685i
$568$		0			0
$569$	33.9493			1.42323			0.711614	−	0.702571i	$-0.247965\pi$
$569$	33.9493			1.42323			0.711614	+	0.702571i	$0.247965\pi$
$570$		0			0
$571$	−9.86417			−0.412803			−0.206401	−	0.978467i	$-0.566175\pi$
$571$	−9.86417			−0.412803			−0.206401	+	0.978467i	$0.566175\pi$
$572$		0			0
$573$	−3.46157	−	19.6315i	−0.144609	−	0.820119i
$574$		0			0
$575$	−29.2647	+	24.5560i	−1.22042	+	1.02405i
$576$		0			0
$577$	−16.5868	+	28.7292i	−0.690519	+	1.19601i	0.281149	+	0.959664i	$0.409285\pi$
$577$	−16.5868	+	28.7292i	−0.690519	+	1.19601i	−0.971668	+	0.236350i	$0.924049\pi$
$578$		0			0
$579$	−0.656491	+	0.238943i	−0.0272828	+	0.00993014i
$580$		0			0
$581$	13.0184	+	22.5485i	0.540094	+	0.935470i
$582$		0			0
$583$	−5.39466	+	30.5946i	−0.223424	+	1.26710i
$584$		0			0
$585$	0.0446533	+	0.0374685i	0.00184618	+	0.00154913i
$586$		0			0
$587$	0.919917	+	0.334822i	0.0379690	+	0.0138196i	0.360935	−	0.932591i	$-0.382458\pi$
$587$	0.919917	+	0.334822i	0.0379690	+	0.0138196i	−0.322966	+	0.946411i	$0.604680\pi$
$588$		0			0
$589$	14.6525	+	17.3394i	0.603746	+	0.714458i
$590$		0			0
$591$	2.28787	+	0.832717i	0.0941104	+	0.0342534i
$592$		0			0
$593$	5.84026	+	4.90056i	0.239831	+	0.201242i	0.754779	−	0.655980i	$-0.227744\pi$
$593$	5.84026	+	4.90056i	0.239831	+	0.201242i	−0.514948	+	0.857222i	$0.672189\pi$
$594$		0			0
$595$	−1.73632	+	9.84718i	−0.0711823	+	0.403695i
$596$		0			0
$597$	−0.374865	−	0.649284i	−0.0153422	−	0.0265734i
$598$		0			0
$599$	−34.2157	+	12.4535i	−1.39802	+	0.508836i	−0.927589	−	0.373603i	$-0.878122\pi$
$599$	−34.2157	+	12.4535i	−1.39802	+	0.508836i	−0.470427	+	0.882439i	$0.655900\pi$
$600$		0			0
$601$	10.5601	−	18.2906i	0.430754	−	0.746088i	−0.566184	−	0.824279i	$-0.691581\pi$
$601$	10.5601	−	18.2906i	0.430754	−	0.746088i	0.996938	+	0.0781907i	$0.0249143\pi$
$602$		0			0
$603$	12.2463	−	10.2759i	0.498708	−	0.418466i
$604$		0			0
$605$	−0.580481	−	3.29207i	−0.0235999	−	0.133842i
$606$		0			0
$607$	−46.6970			−1.89537			−0.947686	−	0.319205i	$-0.896584\pi$
$607$	−46.6970			−1.89537			−0.947686	+	0.319205i	$0.896584\pi$
$608$		0			0
$609$	29.9880			1.21517
$610$		0			0
$611$	0.0894116	+	0.507078i	0.00361721	+	0.0205142i
$612$		0			0
$613$	1.60940	−	1.35044i	0.0650029	−	0.0545439i	−0.609707	−	0.792627i	$-0.708713\pi$
$613$	1.60940	−	1.35044i	0.0650029	−	0.0545439i	0.674710	+	0.738083i	$0.264269\pi$
$614$		0			0
$615$	0.686956	−	1.18984i	0.0277007	−	0.0479791i
$616$		0			0
$617$	8.04441	−	2.92793i	0.323856	−	0.117874i	−0.174976	−	0.984573i	$-0.555985\pi$
$617$	8.04441	−	2.92793i	0.323856	−	0.117874i	0.498832	+	0.866699i	$0.333763\pi$
$618$		0			0
$619$	4.58356	+	7.93896i	0.184229	+	0.319094i	0.943316	−	0.331895i	$-0.107688\pi$
$619$	4.58356	+	7.93896i	0.184229	+	0.319094i	−0.759087	+	0.650989i	$0.774355\pi$
$620$		0			0
$621$	−1.37755	+	7.81248i	−0.0552792	+	0.313504i
$622$		0			0
$623$	37.7558	+	31.6809i	1.51265	+	1.26927i
$624$		0			0
$625$	−20.9254	−	7.61622i	−0.837015	−	0.304649i
$626$		0			0
$627$	6.52320	+	17.7303i	0.260512	+	0.708081i
$628$		0			0
$629$	28.5624	+	10.3959i	1.13886	+	0.414510i
$630$		0			0
$631$	−25.4431	−	21.3493i	−1.01288	−	0.849904i	−0.0241601	−	0.999708i	$-0.507691\pi$
$631$	−25.4431	−	21.3493i	−1.01288	−	0.849904i	−0.988716	+	0.149804i	$0.952136\pi$
$632$		0			0
$633$	2.55707	−	14.5019i	0.101634	−	0.576397i
$634$		0			0
$635$	3.53572	+	6.12405i	0.140311	+	0.243026i
$636$		0			0
$637$	1.05666	−	0.384593i	0.0418665	−	0.0152381i
$638$		0			0
$639$	3.38134	−	5.85666i	0.133764	−	0.231686i
$640$		0			0
$641$	−22.2427	+	18.6638i	−0.878532	+	0.737176i	−0.965877	−	0.259002i	$-0.916606\pi$
$641$	−22.2427	+	18.6638i	−0.878532	+	0.737176i	0.0873446	+	0.996178i	$0.472162\pi$
$642$		0			0
$643$	−3.48780	−	19.7803i	−0.137545	−	0.780058i	−0.973053	−	0.230580i	$-0.925938\pi$
$643$	−3.48780	−	19.7803i	−0.137545	−	0.780058i	0.835508	−	0.549478i	$-0.185173\pi$
$644$		0			0
$645$	−4.45102			−0.175259
$646$		0			0
$647$	−5.00534			−0.196780			−0.0983902	−	0.995148i	$-0.531369\pi$
$647$	−5.00534			−0.196780			−0.0983902	+	0.995148i	$0.531369\pi$
$648$		0			0
$649$	0.0483688	+	0.274313i	0.00189864	+	0.0107677i
$650$		0			0
$651$	15.5969	−	13.0873i	0.611290	−	0.512934i
$652$		0			0
$653$	−2.89930	+	5.02173i	−0.113458	+	0.196515i	−0.917162	−	0.398514i	$-0.869526\pi$
$653$	−2.89930	+	5.02173i	−0.113458	+	0.196515i	0.803704	+	0.595029i	$0.202859\pi$
$654$		0			0
$655$	3.83080	−	1.39430i	0.149682	−	0.0544797i
$656$		0			0
$657$	4.93136	+	8.54137i	0.192391	+	0.333231i
$658$		0			0
$659$	2.65900	−	15.0800i	0.103580	−	0.587432i	−0.888198	−	0.459461i	$-0.848043\pi$
$659$	2.65900	−	15.0800i	0.103580	−	0.587432i	0.991778	−	0.127971i	$-0.0408464\pi$
$660$		0			0
$661$	20.8445	+	17.4907i	0.810759	+	0.680308i	0.950789	−	0.309840i	$-0.100275\pi$
$661$	20.8445	+	17.4907i	0.810759	+	0.680308i	−0.140030	+	0.990147i	$0.544720\pi$
$662$		0			0
$663$	−0.759848	−	0.276562i	−0.0295101	−	0.0107408i
$664$		0			0
$665$	7.21034	+	1.24555i	0.279605	+	0.0483003i
$666$		0			0
$667$	57.1822	+	20.8126i	2.21410	+	0.805868i
$668$		0			0
$669$	15.9571	+	13.3896i	0.616936	+	0.517671i
$670$		0			0
$671$	−4.48079	+	25.4118i	−0.172979	+	0.981011i
$672$		0			0
$673$	−13.9591	−	24.1779i	−0.538085	−	0.931991i	−0.999007	−	0.0445503i	$-0.985814\pi$
$673$	−13.9591	−	24.1779i	−0.538085	−	0.931991i	0.460922	−	0.887441i	$-0.347519\pi$
$674$		0			0
$675$	−4.52520	+	1.64704i	−0.174175	+	0.0633946i
$676$		0			0
$677$	15.4697	−	26.7943i	0.594548	−	1.02979i	−0.399062	−	0.916924i	$-0.630664\pi$
$677$	15.4697	−	26.7943i	0.594548	−	1.02979i	0.993610	−	0.112864i	$-0.0360024\pi$
$678$		0			0
$679$	−25.1123	+	21.0717i	−0.963720	+	0.808657i
$680$		0			0
$681$	1.86708	+	10.5888i	0.0715468	+	0.405762i
$682$		0			0
$683$	−42.8497			−1.63960			−0.819798	−	0.572653i	$-0.805914\pi$
$683$	−42.8497			−1.63960			−0.819798	+	0.572653i	$0.805914\pi$
$684$		0			0
$685$	−5.07846			−0.194038
$686$		0			0
$687$	−2.07260	−	11.7543i	−0.0790745	−	0.448454i
$688$		0			0
$689$	−0.745393	+	0.625459i	−0.0283972	+	0.0238281i
$690$		0			0
$691$	−4.60525	+	7.97652i	−0.175192	+	0.303441i	−0.940228	−	0.340547i	$-0.889388\pi$
$691$	−4.60525	+	7.97652i	−0.175192	+	0.303441i	0.765036	+	0.643988i	$0.222721\pi$
$692$		0			0
$693$	15.9221	−	5.79519i	0.604832	−	0.220141i
$694$		0			0
$695$	−1.63973	−	2.84010i	−0.0621986	−	0.107731i
$696$		0			0
$697$	−3.30957	+	18.7695i	−0.125359	+	0.710945i
$698$		0			0
$699$	−13.3725	−	11.2209i	−0.505794	−	0.424412i
$700$		0			0
$701$	32.5177	+	11.8355i	1.22818	+	0.447020i	0.872973	−	0.487769i	$-0.162189\pi$
$701$	32.5177	+	11.8355i	1.22818	+	0.447020i	0.355204	+	0.934789i	$0.384411\pi$
$702$		0			0
$703$	7.53478	−	20.9277i	0.284180	−	0.789303i
$704$		0			0
$705$	1.53045	+	0.557039i	0.0576401	+	0.0209793i
$706$		0			0
$707$	−3.64242	−	3.05635i	−0.136987	−	0.114946i
$708$		0			0
$709$	−4.26706	+	24.1997i	−0.160253	+	0.908839i	0.793572	+	0.608476i	$0.208219\pi$
$709$	−4.26706	+	24.1997i	−0.160253	+	0.908839i	−0.953825	+	0.300363i	$0.902892\pi$
$710$		0			0
$711$	5.10528	+	8.84260i	0.191463	+	0.331623i
$712$		0			0
$713$	38.8238	−	14.1307i	1.45396	−	0.529199i
$714$		0			0
$715$	0.126321	−	0.218794i	0.00472414	−	0.00818245i
$716$		0			0
$717$	−6.15711	+	5.16643i	−0.229941	+	0.192944i
$718$		0			0
$719$	−3.52850	−	20.0111i	−0.131591	−	0.746289i	−0.977173	−	0.212444i	$-0.931858\pi$
$719$	−3.52850	−	20.0111i	−0.131591	−	0.746289i	0.845582	−	0.533845i	$-0.179253\pi$
$720$		0			0
$721$	−56.9016			−2.11912
$722$		0			0
$723$	7.45834			0.277379
$724$		0			0
$725$	6.41447	+	36.3783i	0.238227	+	1.35105i
$726$		0			0
$727$	−22.8678	+	19.1884i	−0.848121	+	0.711658i	−0.959375	−	0.282134i	$-0.908958\pi$
$727$	−22.8678	+	19.1884i	−0.848121	+	0.711658i	0.111254	+	0.993792i	$0.464513\pi$
$728$		0			0
$729$	−0.500000	+	0.866025i	−0.0185185	+	0.0320750i
$730$		0			0
$731$	58.0213	−	21.1180i	2.14599	−	0.781078i
$732$		0			0
$733$	6.35559	+	11.0082i	0.234749	+	0.406597i	0.959200	−	0.282729i	$-0.0912398\pi$
$733$	6.35559	+	11.0082i	0.234749	+	0.406597i	−0.724451	+	0.689327i	$0.757906\pi$
$734$		0			0
$735$	0.617633	−	3.50277i	0.0227817	−	0.129202i
$736$		0			0
$737$	−53.0776	−	44.5374i	−1.95514	−	1.64056i
$738$		0			0
$739$	−14.6777	−	5.34223i	−0.539926	−	0.196517i	0.0576389	−	0.998337i	$-0.481643\pi$
$739$	−14.6777	−	5.34223i	−0.539926	−	0.196517i	−0.597565	+	0.801820i	$0.703865\pi$
$740$		0			0
$741$	−0.200448	+	0.556742i	−0.00736365	+	0.0204524i
$742$		0			0
$743$	−11.4945	−	4.18366i	−0.421692	−	0.153483i	0.122453	−	0.992474i	$-0.460924\pi$
$743$	−11.4945	−	4.18366i	−0.421692	−	0.153483i	−0.544146	+	0.838991i	$0.683146\pi$
$744$		0			0
$745$	−2.59788	−	2.17988i	−0.0951790	−	0.0798646i
$746$		0			0
$747$	−1.15651	+	6.55888i	−0.0423144	+	0.239977i
$748$		0			0
$749$	30.0168	+	51.9906i	1.09679	+	1.89970i
$750$		0			0
$751$	−13.9015	+	5.05974i	−0.507274	+	0.184633i	−0.582963	−	0.812499i	$-0.698107\pi$
$751$	−13.9015	+	5.05974i	−0.507274	+	0.184633i	0.0756887	+	0.997131i	$0.475884\pi$
$752$		0			0
$753$	−2.56663	+	4.44553i	−0.0935331	+	0.162004i
$754$		0			0
$755$	−2.94941	+	2.47485i	−0.107340	+	0.0900689i
$756$		0			0
$757$	7.14819	+	40.5394i	0.259805	+	1.47343i	0.783430	+	0.621480i	$0.213468\pi$
$757$	7.14819	+	40.5394i	0.259805	+	1.47343i	−0.523624	+	0.851949i	$0.675420\pi$
$758$		0			0
$759$	34.3830			1.24802
$760$		0			0
$761$	−14.5028			−0.525726			−0.262863	−	0.964833i	$-0.584667\pi$
$761$	−14.5028			−0.525726			−0.262863	+	0.964833i	$0.584667\pi$
$762$		0			0
$763$	−3.02489	−	17.1550i	−0.109508	−	0.621053i
$764$		0			0
$765$	−1.95932	+	1.64406i	−0.0708393	+	0.0594413i
$766$		0			0
$767$	−0.00436218	+	0.00755551i	−0.000157509	+	0.000272814i
$768$		0			0
$769$	−33.8302	+	12.3132i	−1.21995	+	0.444025i	−0.870144	−	0.492798i	$-0.835974\pi$
$769$	−33.8302	+	12.3132i	−1.21995	+	0.444025i	−0.349804	+	0.936823i	$0.613752\pi$
$770$		0			0
$771$	9.74382	+	16.8768i	0.350915	+	0.607803i
$772$		0			0
$773$	−1.70404	+	9.66407i	−0.0612900	+	0.347593i	0.938706	+	0.344719i	$0.112026\pi$
$773$	−1.70404	+	9.66407i	−0.0612900	+	0.347593i	−0.999996	+	0.00287358i	$0.999085\pi$
$774$		0			0
$775$	19.2124	+	16.1211i	0.690129	+	0.579087i
$776$		0			0
$777$	−18.7459	−	6.82297i	−0.672507	−	0.244773i
$778$		0			0
$779$	13.7435	+	2.37411i	0.492410	+	0.0850613i
$780$		0			0
$781$	−27.5430	−	10.0248i	−0.985567	−	0.358717i
$782$		0			0
$783$	5.87614	+	4.93066i	0.209996	+	0.176208i
$784$		0			0
$785$	0.173558	−	0.984294i	0.00619453	−	0.0351309i
$786$		0			0
$787$	−3.60527	−	6.24451i	−0.128514	−	0.222593i	0.794587	−	0.607150i	$-0.207687\pi$
$787$	−3.60527	−	6.24451i	−0.128514	−	0.222593i	−0.923101	+	0.384558i	$0.874354\pi$
$788$		0			0
$789$	−16.9201	+	6.15841i	−0.602371	+	0.219245i
$790$		0			0
$791$	5.65542	−	9.79547i	0.201083	−	0.348287i
$792$		0			0
$793$	−0.619121	+	0.519504i	−0.0219856	+	0.0184481i
$794$		0			0
$795$	0.534456	+	3.03105i	0.0189552	+	0.107500i
$796$		0			0
$797$	−38.8912			−1.37760			−0.688798	−	0.724953i	$-0.741861\pi$
$797$	−38.8912			−1.37760			−0.688798	+	0.724953i	$0.741861\pi$
$798$		0			0
$799$	−22.5931			−0.799286
$800$		0			0
$801$	2.18923	+	12.4157i	0.0773525	+	0.438688i
$802$		0			0
$803$	32.7459	−	27.4771i	1.15558	−	0.969646i
$804$		0			0
$805$	6.65842	−	11.5327i	0.234678	−	0.406475i
$806$		0			0
$807$	18.7667	−	6.83052i	0.660619	−	0.240446i
$808$		0			0
$809$	12.6692	+	21.9436i	0.445424	+	0.771497i	0.998082	−	0.0619116i	$-0.0197197\pi$
$809$	12.6692	+	21.9436i	0.445424	+	0.771497i	−0.552658	+	0.833408i	$0.686386\pi$
$810$		0			0
$811$	−2.87067	+	16.2804i	−0.100803	+	0.571682i	0.892011	+	0.452014i	$0.149294\pi$
$811$	−2.87067	+	16.2804i	−0.100803	+	0.571682i	−0.992814	+	0.119668i	$0.961817\pi$
$812$		0			0
$813$	−20.1744	−	16.9283i	−0.707546	−	0.593702i
$814$		0			0
$815$	−2.47397	−	0.900452i	−0.0866594	−	0.0315414i
$816$		0			0
$817$	−15.6012	−	42.4048i	−0.545818	−	1.48356i
$818$		0			0
$819$	0.498700	+	0.181512i	0.0174260	+	0.00634254i
$820$		0			0
$821$	25.5824	+	21.4662i	0.892832	+	0.749175i	0.968776	−	0.247937i	$-0.0797527\pi$
$821$	25.5824	+	21.4662i	0.892832	+	0.749175i	−0.0759445	+	0.997112i	$0.524197\pi$
$822$		0			0
$823$	−2.36134	+	13.3918i	−0.0823112	+	0.466810i	0.915593	+	0.402106i	$0.131722\pi$
$823$	−2.36134	+	13.3918i	−0.0823112	+	0.466810i	−0.997904	+	0.0647043i	$0.979390\pi$
$824$		0			0
$825$	10.4359	+	18.0755i	0.363331	+	0.629307i
$826$		0			0
$827$	−27.4874	+	10.0046i	−0.955831	+	0.347894i	−0.772398	−	0.635138i	$-0.780943\pi$
$827$	−27.4874	+	10.0046i	−0.955831	+	0.347894i	−0.183433	+	0.983032i	$0.558721\pi$
$828$		0			0
$829$	−23.4167	+	40.5589i	−0.813295	+	1.40867i	0.0972508	+	0.995260i	$0.468995\pi$
$829$	−23.4167	+	40.5589i	−0.813295	+	1.40867i	−0.910546	+	0.413408i	$0.864338\pi$
$830$		0			0
$831$	8.36895	−	7.02239i	0.290316	−	0.243604i
$832$		0			0
$833$	8.56786	+	48.5908i	0.296859	+	1.68357i
$834$		0			0
$835$	−1.58700			−0.0549204
$836$		0			0
$837$	5.20805			0.180016
$838$		0			0
$839$	−3.07186	−	17.4214i	−0.106052	−	0.601453i	−0.990795	−	0.135373i	$-0.956777\pi$
$839$	−3.07186	−	17.4214i	−0.106052	−	0.601453i	0.884742	−	0.466080i	$-0.154334\pi$
$840$		0			0
$841$	22.8591	−	19.1811i	0.788245	−	0.661416i
$842$		0			0
$843$	−5.47981	+	9.49130i	−0.188735	+	0.326898i
$844$		0			0
$845$	−5.23803	+	1.90649i	−0.180194	+	0.0655851i
$846$		0			0
$847$	−15.2175	−	26.3574i	−0.522879	−	0.905652i
$848$		0			0
$849$	4.00988	−	22.7411i	0.137619	−	0.780474i
$850$		0			0
$851$	−31.0101	−	26.0206i	−1.06301	−	0.891974i
$852$		0			0
$853$	10.9217	+	3.97519i	0.373953	+	0.136108i	0.522158	−	0.852849i	$-0.325127\pi$
$853$	10.9217	+	3.97519i	0.373953	+	0.136108i	−0.148205	+	0.988957i	$0.547349\pi$
$854$		0			0
$855$	1.20807	+	1.42960i	0.0413151	+	0.0488912i
$856$		0			0
$857$	−45.8227	−	16.6781i	−1.56527	−	0.569713i	−0.593337	−	0.804954i	$-0.702190\pi$
$857$	−45.8227	−	16.6781i	−1.56527	−	0.569713i	−0.971937	+	0.235241i	$0.924412\pi$
$858$		0			0
$859$	−28.1582	−	23.6276i	−0.960747	−	0.806162i	0.0203276	−	0.999793i	$-0.493529\pi$
$859$	−28.1582	−	23.6276i	−0.960747	−	0.806162i	−0.981074	+	0.193631i	$0.937974\pi$
$860$		0			0
$861$	2.17212	−	12.3187i	0.0740256	−	0.419820i
$862$		0			0
$863$	1.63689	+	2.83518i	0.0557205	+	0.0965107i	0.892540	−	0.450968i	$-0.148921\pi$
$863$	1.63689	+	2.83518i	0.0557205	+	0.0965107i	−0.836820	+	0.547479i	$0.815588\pi$
$864$		0			0
$865$	−1.78001	+	0.647869i	−0.0605220	+	0.0220282i
$866$		0			0
$867$	9.24039	−	16.0048i	0.313820	−	0.543553i
$868$		0			0
$869$	33.9008	−	28.4462i	1.15001	−	0.964970i
$870$		0			0
$871$	−0.376848	−	2.13721i	−0.0127690	−	0.0724165i
$872$		0			0
$873$	−8.38538			−0.283802
$874$		0			0
$875$	16.4771			0.557029
$876$		0			0
$877$	−5.17156	−	29.3294i	−0.174631	−	0.990384i	−0.938569	−	0.345093i	$-0.887847\pi$
$877$	−5.17156	−	29.3294i	−0.174631	−	0.990384i	0.763937	−	0.645291i	$-0.223264\pi$
$878$		0			0
$879$	6.89390	−	5.78466i	0.232525	−	0.195112i
$880$		0			0
$881$	2.54296	−	4.40453i	0.0856744	−	0.148392i	−0.820004	−	0.572358i	$-0.806029\pi$
$881$	2.54296	−	4.40453i	0.0856744	−	0.148392i	0.905678	+	0.423965i	$0.139362\pi$
$882$		0			0
$883$	26.5651	−	9.66890i	0.893986	−	0.325384i	0.146146	−	0.989263i	$-0.453313\pi$
$883$	26.5651	−	9.66890i	0.893986	−	0.325384i	0.747840	+	0.663879i	$0.231091\pi$
$884$		0			0
$885$	0.0137979	+	0.0238987i	0.000463812	+	0.000803346i
$886$		0			0
$887$	3.00988	−	17.0699i	0.101062	−	0.573150i	−0.891659	−	0.452709i	$-0.850458\pi$
$887$	3.00988	−	17.0699i	0.101062	−	0.573150i	0.992720	−	0.120442i	$-0.0384311\pi$
$888$		0			0
$889$	49.3193	+	41.3838i	1.65412	+	1.38797i
$890$		0			0
$891$	4.07279	+	1.48238i	0.136444	+	0.0496615i
$892$		0			0
$893$	0.0574734	+	16.5331i	0.00192328	+	0.553258i
$894$		0			0
$895$	−7.80322	−	2.84014i	−0.260833	−	0.0949354i
$896$		0			0
$897$	0.824965	+	0.692228i	0.0275448	+	0.0231128i
$898$		0			0
$899$	6.93718	−	39.3427i	0.231368	−	1.31215i
$900$		0			0
$901$	−21.3478	−	36.9755i	−0.711199	−	1.23183i
$902$		0			0
$903$	−38.0803	+	13.8601i	−1.26723	+	0.461235i
$904$		0			0
$905$	−2.59047	+	4.48683i	−0.0861102	+	0.149147i
$906$		0			0
$907$	23.4639	−	19.6885i	0.779106	−	0.653747i	−0.163917	−	0.986474i	$-0.552413\pi$
$907$	23.4639	−	19.6885i	0.779106	−	0.653747i	0.943023	+	0.332727i	$0.107969\pi$
$908$		0			0
$909$	−0.211201	−	1.19778i	−0.00700511	−	0.0397279i
$910$		0			0
$911$	40.9437			1.35652			0.678262	−	0.734820i	$-0.262734\pi$
$911$	40.9437			1.35652			0.678262	+	0.734820i	$0.262734\pi$
$912$		0			0
$913$	28.8659			0.955321
$914$		0			0
$915$	0.443917	+	2.51758i	0.0146755	+	0.0832287i
$916$		0			0
$917$	28.4323	−	23.8575i	0.938917	−	0.787845i
$918$		0			0
$919$	−0.628356	+	1.08835i	−0.0207276	+	0.0359012i	−0.876203	−	0.481942i	$-0.839932\pi$
$919$	−0.628356	+	1.08835i	−0.0207276	+	0.0359012i	0.855476	+	0.517843i	$0.173265\pi$
$920$		0			0
$921$	13.8312	−	5.03416i	0.455755	−	0.165881i
$922$		0			0
$923$	−0.459022	−	0.795050i	−0.0151089	−	0.0261694i
$924$		0			0
$925$	4.26712	−	24.2001i	0.140302	−	0.795693i
$926$		0			0
$927$	−11.1498	−	9.35583i	−0.366209	−	0.307286i
$928$		0			0
$929$	20.6540	+	7.51744i	0.677635	+	0.246639i	0.657832	−	0.753165i	$-0.271474\pi$
$929$	20.6540	+	7.51744i	0.677635	+	0.246639i	0.0198035	+	0.999804i	$0.493696\pi$
$930$		0			0
$931$	35.5357	−	6.39335i	1.16464	−	0.209534i
$932$		0			0
$933$	6.56964	+	2.39115i	0.215080	+	0.0782828i
$934$		0			0
$935$	8.49204	+	7.12566i	0.277719	+	0.233034i
$936$		0			0
$937$	−8.28843	+	47.0060i	−0.270771	+	1.53562i	0.481310	+	0.876550i	$0.340161\pi$
$937$	−8.28843	+	47.0060i	−0.270771	+	1.53562i	−0.752082	+	0.659070i	$0.770950\pi$
$938$		0			0
$939$	−9.70162	−	16.8037i	−0.316600	−	0.548368i
$940$		0			0
$941$	−1.92400	+	0.700280i	−0.0627208	+	0.0228285i	−0.373190	−	0.927755i	$-0.621736\pi$
$941$	−1.92400	+	0.700280i	−0.0627208	+	0.0228285i	0.310469	+	0.950583i	$0.399514\pi$
$942$		0			0
$943$	12.6915	−	21.9822i	0.413291	−	0.715840i
$944$		0			0
$945$	1.28593	−	1.07902i	0.0418314	−	0.0351007i
$946$		0			0
$947$	−8.91185	−	50.5416i	−0.289596	−	1.64238i	−0.688390	−	0.725341i	$-0.741682\pi$
$947$	−8.91185	−	50.5416i	−0.289596	−	1.64238i	0.398793	−	0.917041i	$-0.369429\pi$
$948$		0			0
$949$	1.33888			0.0434619
$950$		0			0
$951$	5.19255			0.168380
$952$		0			0
$953$	−0.270252	−	1.53267i	−0.00875432	−	0.0496482i	0.980118	−	0.198417i	$-0.0635801\pi$
$953$	−0.270252	−	1.53267i	−0.00875432	−	0.0496482i	−0.988872	+	0.148769i	$0.952469\pi$
$954$		0			0
$955$	−6.55710	+	5.50206i	−0.212183	+	0.178042i
$956$		0			0
$957$	16.6232	−	28.7922i	0.537352	−	0.930720i
$958$		0			0
$959$	−43.4483	+	15.8139i	−1.40302	+	0.510657i
$960$		0			0
$961$	1.93812	+	3.35692i	0.0625200	+	0.108288i
$962$		0			0
$963$	−2.66658	+	15.1229i	−0.0859295	+	0.487330i
$964$		0			0
$965$	0.229801	+	0.192826i	0.00739755	+	0.00620728i
$966$		0			0
$967$	14.8643	+	5.41015i	0.478002	+	0.173979i	0.569774	−	0.821801i	$-0.307031\pi$
$967$	14.8643	+	5.41015i	0.478002	+	0.173979i	−0.0917716	+	0.995780i	$0.529253\pi$
$968$		0			0
$969$	−22.5306	−	12.9038i	−0.723786	−	0.414530i
$970$		0			0
$971$	17.8360	+	6.49179i	0.572386	+	0.208331i	0.611965	−	0.790885i	$-0.290379\pi$
$971$	17.8360	+	6.49179i	0.572386	+	0.208331i	−0.0395789	+	0.999216i	$0.512602\pi$
$972$		0			0
$973$	−22.8724	−	19.1922i	−0.733255	−	0.615274i
$974$		0			0
$975$	−0.113519	+	0.643796i	−0.00363551	+	0.0206180i
$976$		0			0
$977$	1.12329	+	1.94559i	0.0359372	+	0.0622451i	0.883435	−	0.468554i	$-0.155225\pi$
$977$	1.12329	+	1.94559i	0.0359372	+	0.0622451i	−0.847497	+	0.530800i	$0.821892\pi$
$978$		0			0
$979$	51.3467	−	18.6887i	1.64105	−	0.597293i
$980$		0			0
$981$	2.22793	−	3.85888i	0.0711322	−	0.123205i
$982$		0			0
$983$	−12.1151	+	10.1658i	−0.386412	+	0.324238i	−0.815214	−	0.579160i	$-0.803381\pi$
$983$	−12.1151	+	10.1658i	−0.386412	+	0.324238i	0.428801	+	0.903399i	$0.358936\pi$
$984$		0			0
$985$	−0.181539	−	1.02956i	−0.00578432	−	0.0328045i
$986$		0			0
$987$	14.8282			0.471987
$988$		0			0
$989$	−82.2323			−2.61483
$990$		0			0
$991$	4.20236	+	23.8328i	0.133492	+	0.757072i	0.975898	+	0.218228i	$0.0700277\pi$
$991$	4.20236	+	23.8328i	0.133492	+	0.757072i	−0.842405	+	0.538844i	$0.818861\pi$
$992$		0			0
$993$	−13.0774	+	10.9733i	−0.415000	+	0.348226i
$994$		0			0
$995$	−0.160964	+	0.278798i	−0.00510291	+	0.00883849i
$996$		0			0
$997$	−42.3811	+	15.4255i	−1.34222	+	0.488529i	−0.910512	−	0.413483i	$-0.864312\pi$
$997$	−42.3811	+	15.4255i	−1.34222	+	0.488529i	−0.431711	+	0.902012i	$0.642090\pi$
$998$		0			0
$999$	−2.55142	−	4.41919i	−0.0807234	−	0.139817i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bg.c.289.2	✓	18
4.3	odd	2		912.2.bo.k.289.2		18
19.5	even	9	inner	456.2.bg.c.385.2	yes	18
19.9	even	9		8664.2.a.bo.1.4		9
19.10	odd	18		8664.2.a.bq.1.4		9
76.43	odd	18		912.2.bo.k.385.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bg.c.289.2	✓	18	1.1	even	1	trivial
456.2.bg.c.385.2	yes	18	19.5	even	9	inner
912.2.bo.k.289.2		18	4.3	odd	2
912.2.bo.k.385.2		18	76.43	odd	18
8664.2.a.bo.1.4		9	19.9	even	9
8664.2.a.bq.1.4		9	19.10	odd	18

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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.bg (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.173648 - 0.984808i) q^{3} +(-0.328934 + 0.276008i) q^{5} +(-1.95470 + 3.38563i) q^{7} +(-0.939693 + 0.342020i) q^{9} +O(q^{10})\)	\(q+(-0.173648 - 0.984808i) q^{3} +(-0.328934 + 0.276008i) q^{5} +(-1.95470 + 3.38563i) q^{7} +(-0.939693 + 0.342020i) q^{9} +(2.16709 + 3.75351i) q^{11} +(-0.0235730 + 0.133689i) q^{13} +(0.328934 + 0.276008i) q^{15} +(-5.59735 - 2.03727i) q^{17} +(-1.47658 + 4.10118i) q^{19} +(3.67363 + 1.33709i) q^{21} +(6.07703 + 5.09923i) q^{23} +(-0.836224 + 4.74246i) q^{25} +(0.500000 + 0.866025i) q^{27} +(7.20815 - 2.62355i) q^{29} +(2.60402 - 4.51030i) q^{31} +(3.32017 - 2.78596i) q^{33} +(-0.291497 - 1.65316i) q^{35} -5.10285 q^{37} +0.135751 q^{39} +(-0.555615 - 3.15105i) q^{41} +(-7.94070 + 6.66304i) q^{43} +(0.214696 - 0.371865i) q^{45} +(3.56422 - 1.29727i) q^{47} +(-4.14167 - 7.17358i) q^{49} +(-1.03435 + 5.86608i) q^{51} +(5.49087 + 4.60739i) q^{53} +(-1.74883 - 0.636521i) q^{55} +(4.29528 + 0.741988i) q^{57} +(0.0603913 + 0.0219807i) q^{59} +(4.56070 + 3.82688i) q^{61} +(0.678859 - 3.85000i) q^{63} +(-0.0291453 - 0.0504812i) q^{65} +(-15.0223 + 5.46767i) q^{67} +(3.96650 - 6.87018i) q^{69} +(-5.18052 + 4.34697i) q^{71} +(-1.71264 - 9.71289i) q^{73} +4.81562 q^{75} -16.9440 q^{77} +(-1.77304 - 10.0554i) q^{79} +(0.766044 - 0.642788i) q^{81} +(3.33003 - 5.76778i) q^{83} +(2.40346 - 0.874788i) q^{85} +(-3.83538 - 6.64307i) q^{87} +(2.18923 - 12.4157i) q^{89} +(-0.406544 - 0.341131i) q^{91} +(-4.89396 - 1.78126i) q^{93} +(-0.646262 - 1.75657i) q^{95} +(7.87968 + 2.86797i) q^{97} +(-3.32017 - 2.78596i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 3 q^{5}+O(q^{10}) \) 18 * q - 3 * q^5	\( 18 q - 3 q^{5} - 3 q^{11} + 18 q^{13} + 3 q^{15} - 3 q^{17} - 3 q^{19} - 3 q^{21} + 12 q^{23} - 3 q^{25} + 9 q^{27} + 3 q^{29} + 21 q^{31} + 6 q^{33} + 12 q^{35} - 30 q^{37} + 6 q^{39} + 33 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47} - 21 q^{49} + 12 q^{51} + 30 q^{53} - 24 q^{55} - 21 q^{57} - 9 q^{59} + 48 q^{61} - 6 q^{63} - 9 q^{65} - 18 q^{67} - 6 q^{69} - 51 q^{71} - 12 q^{75} + 30 q^{77} - 39 q^{79} - 30 q^{83} - 87 q^{85} - 21 q^{87} + 45 q^{89} + 72 q^{91} - 27 q^{93} - 78 q^{95} + 39 q^{97} - 6 q^{99}+O(q^{100}) \) 18 * q - 3 * q^5 - 3 * q^11 + 18 * q^13 + 3 * q^15 - 3 * q^17 - 3 * q^19 - 3 * q^21 + 12 * q^23 - 3 * q^25 + 9 * q^27 + 3 * q^29 + 21 * q^31 + 6 * q^33 + 12 * q^35 - 30 * q^37 + 6 * q^39 + 33 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 - 21 * q^49 + 12 * q^51 + 30 * q^53 - 24 * q^55 - 21 * q^57 - 9 * q^59 + 48 * q^61 - 6 * q^63 - 9 * q^65 - 18 * q^67 - 6 * q^69 - 51 * q^71 - 12 * q^75 + 30 * q^77 - 39 * q^79 - 30 * q^83 - 87 * q^85 - 21 * q^87 + 45 * q^89 + 72 * q^91 - 27 * q^93 - 78 * q^95 + 39 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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