LMFDB - Embedded newform 144.3.m.c.19.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,3,Mod(19,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.19"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.m (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$3.92371580679$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			19.4
Root			$-0.455024 + 1.94755i$ of defining polynomial
Character	$\chi$	$=$	144.19
Dual form			144.3.m.c.91.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.455024 - 1.94755i) q^{2} +(-3.58591 + 1.77236i) q^{4} +(3.40572 - 3.40572i) q^{5} +12.1303 q^{7} +(5.08344 + 6.17727i) q^{8} +(-8.18251 - 5.08314i) q^{10} +(-9.81086 - 9.81086i) q^{11} +(-7.76859 - 7.76859i) q^{13} +(-5.51959 - 23.6244i) q^{14} +(9.71745 - 12.7111i) q^{16} -9.73087 q^{17} +(11.2823 - 11.2823i) q^{19} +(-6.17643 + 18.2488i) q^{20} +(-14.6430 + 23.5713i) q^{22} +20.2635 q^{23} +1.80207i q^{25} +(-11.5948 + 18.6646i) q^{26} +(-43.4982 + 21.4994i) q^{28} +(16.4069 + 16.4069i) q^{29} -26.3542i q^{31} +(-29.1771 - 13.1414i) q^{32} +(4.42778 + 18.9514i) q^{34} +(41.3125 - 41.3125i) q^{35} +(-23.7263 + 23.7263i) q^{37} +(-27.1066 - 16.8392i) q^{38} +(38.3509 + 3.72526i) q^{40} +24.7452i q^{41} +(29.8844 + 29.8844i) q^{43} +(52.5692 + 17.7924i) q^{44} +(-9.22036 - 39.4641i) q^{46} +31.3325i q^{47} +98.1448 q^{49} +(3.50963 - 0.819987i) q^{50} +(41.6262 + 14.0887i) q^{52} +(-36.8742 + 36.8742i) q^{53} -66.8262 q^{55} +(61.6638 + 74.9322i) q^{56} +(24.4877 - 39.4188i) q^{58} +(14.1325 + 14.1325i) q^{59} +(-42.5199 - 42.5199i) q^{61} +(-51.3260 + 11.9918i) q^{62} +(-12.3172 + 62.8036i) q^{64} -52.9153 q^{65} +(48.7789 - 48.7789i) q^{67} +(34.8940 - 17.2467i) q^{68} +(-99.2565 - 61.6601i) q^{70} -7.73935 q^{71} +85.4163i q^{73} +(57.0041 + 35.4121i) q^{74} +(-20.4610 + 60.4537i) q^{76} +(-119.009 - 119.009i) q^{77} +105.294i q^{79} +(-10.1954 - 76.3854i) q^{80} +(48.1926 - 11.2597i) q^{82} +(62.1229 - 62.1229i) q^{83} +(-33.1407 + 33.1407i) q^{85} +(44.6033 - 71.7996i) q^{86} +(10.7313 - 110.477i) q^{88} +127.172i q^{89} +(-94.2355 - 94.2355i) q^{91} +(-72.6629 + 35.9142i) q^{92} +(61.0215 - 14.2570i) q^{94} -76.8489i q^{95} -147.348 q^{97} +(-44.6582 - 191.142i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} + 12 q^{8} - 56 q^{10} - 32 q^{11} + 44 q^{14} + 32 q^{16} - 32 q^{19} - 80 q^{20} + 32 q^{22} + 128 q^{23} + 100 q^{26} - 120 q^{28} - 32 q^{29} - 160 q^{32} + 96 q^{34} - 96 q^{35} - 96 q^{37}+ \cdots + 440 q^{98}+O(q^{100}) $ 16 * q + 12 * q^4 + 12 * q^8 - 56 * q^10 - 32 * q^11 + 44 * q^14 + 32 * q^16 - 32 * q^19 - 80 * q^20 + 32 * q^22 + 128 * q^23 + 100 * q^26 - 120 * q^28 - 32 * q^29 - 160 * q^32 + 96 * q^34 - 96 * q^35 - 96 * q^37 - 168 * q^38 + 48 * q^40 + 160 * q^43 - 88 * q^44 + 136 * q^46 + 112 * q^49 + 236 * q^50 - 48 * q^52 + 160 * q^53 - 256 * q^55 + 224 * q^56 + 144 * q^58 + 128 * q^59 - 32 * q^61 + 276 * q^62 - 408 * q^64 + 32 * q^65 + 320 * q^67 + 448 * q^68 - 384 * q^70 - 512 * q^71 - 348 * q^74 + 72 * q^76 - 224 * q^77 - 552 * q^80 - 40 * q^82 + 160 * q^83 + 160 * q^85 - 528 * q^86 + 480 * q^88 - 480 * q^91 - 496 * q^92 + 312 * q^94 + 440 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

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 $

$2$	−0.455024	−	1.94755i	−0.227512	−	0.973775i
$2$	−0.455024	−	1.94755i	−0.227512	−	0.973775i
$3$		0			0
$3$		0			0
$4$	−3.58591	+	1.77236i	−0.896477	+	0.443091i
$5$	3.40572	−	3.40572i	0.681145	−	0.681145i	−0.279113	−	0.960258i	$-0.590040\pi$
$5$	3.40572	−	3.40572i	0.681145	−	0.681145i	0.960258	+	0.279113i	$0.0900405\pi$
$6$		0			0
$7$	12.1303			1.73290			0.866452	−	0.499261i	$-0.166395\pi$
$7$	12.1303			1.73290			0.866452	+	0.499261i	$0.166395\pi$
$8$	5.08344	+	6.17727i	0.635430	+	0.772158i
$9$		0			0
$10$	−8.18251	−	5.08314i	−0.818251	−	0.508314i
$11$	−9.81086	−	9.81086i	−0.891896	−	0.891896i	0.102805	−	0.994702i	$-0.467218\pi$
$11$	−9.81086	−	9.81086i	−0.891896	−	0.891896i	−0.994702	+	0.102805i	$0.967218\pi$
$12$		0			0
$13$	−7.76859	−	7.76859i	−0.597584	−	0.597584i	0.342085	−	0.939669i	$-0.388867\pi$
$13$	−7.76859	−	7.76859i	−0.597584	−	0.597584i	−0.939669	+	0.342085i	$0.888867\pi$
$14$	−5.51959	−	23.6244i	−0.394256	−	1.68746i
$15$		0			0
$16$	9.71745	−	12.7111i	0.607341	−	0.794442i
$17$	−9.73087			−0.572404			−0.286202	−	0.958169i	$-0.592393\pi$
$17$	−9.73087			−0.572404			−0.286202	+	0.958169i	$0.592393\pi$
$18$		0			0
$19$	11.2823	−	11.2823i	0.593806	−	0.593806i	−0.344851	−	0.938657i	$-0.612071\pi$
$19$	11.2823	−	11.2823i	0.593806	−	0.593806i	0.938657	+	0.344851i	$0.112071\pi$
$20$	−6.17643	+	18.2488i	−0.308821	+	0.912440i
$21$		0			0
$22$	−14.6430	+	23.5713i	−0.665589	+	1.07142i
$23$	20.2635			0.881020			0.440510	−	0.897748i	$-0.354798\pi$
$23$	20.2635			0.881020			0.440510	+	0.897748i	$0.354798\pi$
$24$		0			0
$25$			1.80207i			0.0720830i
$26$	−11.5948	+	18.6646i	−0.445955	+	0.717870i
$27$		0			0
$28$	−43.4982	+	21.4994i	−1.55351	+	0.767834i
$29$	16.4069	+	16.4069i	0.565754	+	0.565754i	0.930936	−	0.365182i	$-0.118993\pi$
$29$	16.4069	+	16.4069i	0.565754	+	0.565754i	−0.365182	+	0.930936i	$0.618993\pi$
$30$		0			0
$31$		−	26.3542i		−	0.850134i	−0.905162	−	0.425067i	$-0.860251\pi$
$31$		−	26.3542i		−	0.850134i	0.905162	−	0.425067i	$-0.139749\pi$
$32$	−29.1771	−	13.1414i	−0.911785	−	0.410668i
$33$		0			0
$34$	4.42778	+	18.9514i	0.130229	+	0.557393i
$35$	41.3125	−	41.3125i	1.18036	−	1.18036i
$36$		0			0
$37$	−23.7263	+	23.7263i	−0.641250	+	0.641250i	−0.950863	−	0.309613i	$-0.899801\pi$
$37$	−23.7263	+	23.7263i	−0.641250	+	0.641250i	0.309613	+	0.950863i	$0.399801\pi$
$38$	−27.1066	−	16.8392i	−0.713332	−	0.443136i
$39$		0			0
$40$	38.3509	+	3.72526i	0.958772	+	0.0931315i
$41$			24.7452i			0.603542i	0.953380	+	0.301771i	$0.0975779\pi$
$41$			24.7452i			0.603542i	−0.953380	+	0.301771i	$0.902422\pi$
$42$		0			0
$43$	29.8844	+	29.8844i	0.694987	+	0.694987i	0.963325	−	0.268338i	$-0.0864744\pi$
$43$	29.8844	+	29.8844i	0.694987	+	0.694987i	−0.268338	+	0.963325i	$0.586474\pi$
$44$	52.5692	+	17.7924i	1.19476	+	0.404373i
$45$		0			0
$46$	−9.22036	−	39.4641i	−0.200443	−	0.857915i
$47$			31.3325i			0.666648i	0.942812	+	0.333324i	$0.108170\pi$
$47$			31.3325i			0.666648i	−0.942812	+	0.333324i	$0.891830\pi$
$48$		0			0
$49$	98.1448			2.00295
$50$	3.50963	−	0.819987i	0.0701926	−	0.0163997i
$51$		0			0
$52$	41.6262	+	14.0887i	0.800504	+	0.270936i
$53$	−36.8742	+	36.8742i	−0.695739	+	0.695739i	−0.963489	−	0.267750i	$-0.913720\pi$
$53$	−36.8742	+	36.8742i	−0.695739	+	0.695739i	0.267750	+	0.963489i	$0.413720\pi$
$54$		0			0
$55$	−66.8262			−1.21502
$56$	61.6638	+	74.9322i	1.10114	+	1.33808i
$57$		0			0
$58$	24.4877	−	39.4188i	0.422202	−	0.679634i
$59$	14.1325	+	14.1325i	0.239534	+	0.239534i	0.816657	−	0.577123i	$-0.195825\pi$
$59$	14.1325	+	14.1325i	0.239534	+	0.239534i	−0.577123	+	0.816657i	$0.695825\pi$
$60$		0			0
$61$	−42.5199	−	42.5199i	−0.697048	−	0.697048i	0.266725	−	0.963773i	$-0.414059\pi$
$61$	−42.5199	−	42.5199i	−0.697048	−	0.697048i	−0.963773	+	0.266725i	$0.914059\pi$
$62$	−51.3260	+	11.9918i	−0.827839	+	0.193416i
$63$		0			0
$64$	−12.3172	+	62.8036i	−0.192457	+	0.981305i
$65$	−52.9153			−0.814082
$66$		0			0
$67$	48.7789	−	48.7789i	0.728044	−	0.728044i	−0.242186	−	0.970230i	$-0.577864\pi$
$67$	48.7789	−	48.7789i	0.728044	−	0.728044i	0.970230	+	0.242186i	$0.0778644\pi$
$68$	34.8940	−	17.2467i	0.513147	−	0.253627i
$69$		0			0
$70$	−99.2565	−	61.6601i	−1.41795	−	0.880858i
$71$	−7.73935			−0.109005			−0.0545025	−	0.998514i	$-0.517357\pi$
$71$	−7.73935			−0.109005			−0.0545025	+	0.998514i	$0.517357\pi$
$72$		0			0
$73$			85.4163i			1.17009i	0.811002	+	0.585043i	$0.198923\pi$
$73$			85.4163i			1.17009i	−0.811002	+	0.585043i	$0.801077\pi$
$74$	57.0041	+	35.4121i	0.770326	+	0.478541i
$75$		0			0
$76$	−20.4610	+	60.4537i	−0.269223	+	0.795443i
$77$	−119.009	−	119.009i	−1.54557	−	1.54557i
$78$		0			0
$79$			105.294i			1.33283i	0.745581	+	0.666416i	$0.232172\pi$
$79$			105.294i			1.33283i	−0.745581	+	0.666416i	$0.767828\pi$
$80$	−10.1954	−	76.3854i	−0.127443	−	0.954817i
$81$		0			0
$82$	48.1926	−	11.2597i	0.587715	−	0.137313i
$83$	62.1229	−	62.1229i	0.748469	−	0.748469i	−0.225723	−	0.974192i	$-0.572474\pi$
$83$	62.1229	−	62.1229i	0.748469	−	0.748469i	0.974192	+	0.225723i	$0.0724743\pi$
$84$		0			0
$85$	−33.1407	+	33.1407i	−0.389890	+	0.389890i
$86$	44.6033	−	71.7996i	0.518643	−	0.834879i
$87$		0			0
$88$	10.7313	−	110.477i	0.121947	−	1.25542i
$89$			127.172i			1.42890i	0.699685	+	0.714451i	$0.253324\pi$
$89$			127.172i			1.42890i	−0.699685	+	0.714451i	$0.746676\pi$
$90$		0			0
$91$	−94.2355	−	94.2355i	−1.03555	−	1.03555i
$92$	−72.6629	+	35.9142i	−0.789814	+	0.390372i
$93$		0			0
$94$	61.0215	−	14.2570i	0.649165	−	0.151670i
$95$		−	76.8489i		−	0.808936i
$96$		0			0
$97$	−147.348			−1.51905			−0.759525	−	0.650478i	$-0.774569\pi$
$97$	−147.348			−1.51905			−0.759525	+	0.650478i	$0.774569\pi$
$98$	−44.6582	−	191.142i	−0.455696	−	1.95043i
$99$		0			0
$100$	−3.19393	−	6.46207i	−0.0319393	−	0.0646207i
$101$	−12.7690	+	12.7690i	−0.126426	+	0.126426i	−0.767489	−	0.641063i	$-0.778494\pi$
$101$	−12.7690	+	12.7690i	−0.126426	+	0.126426i	0.641063	+	0.767489i	$0.278494\pi$
$102$		0			0
$103$	−17.7621			−0.172448			−0.0862240	−	0.996276i	$-0.527480\pi$
$103$	−17.7621			−0.172448			−0.0862240	+	0.996276i	$0.527480\pi$
$104$	8.49746	−	87.4798i	0.0817063	−	0.841152i
$105$		0			0
$106$	88.5929	+	55.0357i	0.835782	+	0.519204i
$107$	15.8889	+	15.8889i	0.148494	+	0.148494i	0.777445	−	0.628951i	$-0.216515\pi$
$107$	15.8889	+	15.8889i	0.148494	+	0.148494i	−0.628951	+	0.777445i	$0.716515\pi$
$108$		0			0
$109$	−79.3257	−	79.3257i	−0.727758	−	0.727758i	0.242414	−	0.970173i	$-0.422061\pi$
$109$	−79.3257	−	79.3257i	−0.727758	−	0.727758i	−0.970173	+	0.242414i	$0.922061\pi$
$110$	30.4075	+	130.147i	0.276432	+	1.18316i
$111$		0			0
$112$	117.876	−	154.189i	1.05246	−	1.37669i
$113$	167.538			1.48263			0.741317	−	0.671155i	$-0.234201\pi$
$113$	167.538			1.48263			0.741317	+	0.671155i	$0.234201\pi$
$114$		0			0
$115$	69.0118	−	69.0118i	0.600102	−	0.600102i
$116$	−87.9125	−	29.7546i	−0.757866	−	0.256505i
$117$		0			0
$118$	21.0932	−	33.9545i	0.178756	−	0.287750i
$119$	−118.039			−0.991921
$120$		0			0
$121$			71.5059i			0.590958i
$122$	−63.4621	+	102.157i	−0.520181	+	0.837355i
$123$		0			0
$124$	46.7092	+	94.5035i	0.376687	+	0.762125i
$125$	91.2805	+	91.2805i	0.730244	+	0.730244i
$126$		0			0
$127$			198.247i			1.56100i	0.625156	+	0.780500i	$0.285035\pi$
$127$			198.247i			1.56100i	−0.625156	+	0.780500i	$0.714965\pi$
$128$	127.918	−	4.58871i	0.999357	−	0.0358493i
$129$		0			0
$130$	24.0778	+	103.055i	0.185213	+	0.792733i
$131$	−134.339	+	134.339i	−1.02549	+	1.02549i	−0.0258197	+	0.999667i	$0.508220\pi$
$131$	−134.339	+	134.339i	−1.02549	+	1.02549i	−0.999667	+	0.0258197i	$0.991780\pi$
$132$		0			0
$133$	136.858	−	136.858i	1.02901	−	1.02901i
$134$	−117.195	−	72.8039i	−0.874590	−	0.543313i
$135$		0			0
$136$	−49.4663	−	60.1102i	−0.363723	−	0.441987i
$137$		−	255.937i		−	1.86816i	−0.357069	−	0.934078i	$-0.616224\pi$
$137$		−	255.937i		−	1.86816i	0.357069	−	0.934078i	$-0.383776\pi$
$138$		0			0
$139$	−21.7231	−	21.7231i	−0.156281	−	0.156281i	0.624635	−	0.780917i	$-0.285248\pi$
$139$	−21.7231	−	21.7231i	−0.156281	−	0.156281i	−0.780917	+	0.624635i	$0.785248\pi$
$140$	−74.9220	+	221.364i	−0.535157	+	1.58117i
$141$		0			0
$142$	3.52159	+	15.0728i	0.0247999	+	0.106146i
$143$			152.433i			1.06597i
$144$		0			0
$145$	111.755			0.770722
$146$	166.353	−	38.8665i	1.13940	−	0.266209i
$147$		0			0
$148$	43.0286	−	127.132i	0.290734	−	0.858998i
$149$	34.2444	−	34.2444i	0.229828	−	0.229828i	−0.582793	−	0.812621i	$-0.698040\pi$
$149$	34.2444	−	34.2444i	0.229828	−	0.229828i	0.812621	+	0.582793i	$0.198040\pi$
$150$		0			0
$151$	−14.4645			−0.0957913			−0.0478956	−	0.998852i	$-0.515251\pi$
$151$	−14.4645			−0.0957913			−0.0478956	+	0.998852i	$0.515251\pi$
$152$	127.047	+	12.3409i	0.835835	+	0.0811898i
$153$		0			0
$154$	−177.624	+	285.928i	−1.15340	+	1.85667i
$155$	−89.7550	−	89.7550i	−0.579064	−	0.579064i
$156$		0			0
$157$	31.4652	+	31.4652i	0.200415	+	0.200415i	0.800178	−	0.599763i	$-0.204738\pi$
$157$	31.4652	+	31.4652i	0.200415	+	0.200415i	−0.599763	+	0.800178i	$0.704738\pi$
$158$	205.065	−	47.9111i	1.29788	−	0.303235i
$159$		0			0
$160$	−144.125	+	54.6133i	−0.900782	+	0.341333i
$161$	245.802			1.52672
$162$		0			0
$163$	31.4002	−	31.4002i	0.192640	−	0.192640i	−0.604196	−	0.796836i	$-0.706506\pi$
$163$	31.4002	−	31.4002i	0.192640	−	0.192640i	0.796836	+	0.604196i	$0.206506\pi$
$164$	−43.8576	−	88.7341i	−0.267424	−	0.541062i
$165$		0			0
$166$	−149.255	−	92.7201i	−0.899126	−	0.558555i
$167$	−36.4796			−0.218441			−0.109220	−	0.994018i	$-0.534835\pi$
$167$	−36.4796			−0.218441			−0.109220	+	0.994018i	$0.534835\pi$
$168$		0			0
$169$		−	48.2981i		−	0.285788i
$170$	79.6229	+	49.4633i	0.468370	+	0.290961i
$171$		0			0
$172$	−160.129	−	54.1967i	−0.930982	−	0.315097i
$173$	−97.6419	−	97.6419i	−0.564404	−	0.564404i	0.366151	−	0.930555i	$-0.380675\pi$
$173$	−97.6419	−	97.6419i	−0.564404	−	0.564404i	−0.930555	+	0.366151i	$0.880675\pi$
$174$		0			0
$175$			21.8598i			0.124913i
$176$	−220.043	+	29.3700i	−1.25024	+	0.166875i
$177$		0			0
$178$	247.674	−	57.8664i	1.39143	−	0.325092i
$179$	−89.7427	+	89.7427i	−0.501356	+	0.501356i	−0.911859	−	0.410503i	$-0.865353\pi$
$179$	−89.7427	+	89.7427i	−0.501356	+	0.501356i	0.410503	+	0.911859i	$0.365353\pi$
$180$		0			0
$181$	−115.497	+	115.497i	−0.638108	+	0.638108i	−0.950088	−	0.311981i	$-0.899008\pi$
$181$	−115.497	+	115.497i	−0.638108	+	0.638108i	0.311981	+	0.950088i	$0.399008\pi$
$182$	−140.649	+	226.408i	−0.772796	+	1.24400i
$183$		0			0
$184$	103.008	+	125.173i	0.559827	+	0.680287i
$185$			161.610i			0.873569i
$186$		0			0
$187$	95.4682	+	95.4682i	0.510525	+	0.510525i
$188$	−55.5325	−	112.355i	−0.295386	−	0.597634i
$189$		0			0
$190$	−149.667	+	34.9681i	−0.787722	+	0.184043i
$191$		−	62.6278i		−	0.327894i	−0.986469	−	0.163947i	$-0.947577\pi$
$191$		−	62.6278i		−	0.327894i	0.986469	−	0.163947i	$-0.0524227\pi$
$192$		0			0
$193$	223.342			1.15721			0.578607	−	0.815607i	$-0.303597\pi$
$193$	223.342			1.15721			0.578607	+	0.815607i	$0.303597\pi$
$194$	67.0468	+	286.967i	0.345602	+	1.47921i
$195$		0			0
$196$	−351.938	+	173.948i	−1.79560	+	0.887491i
$197$	−29.0959	+	29.0959i	−0.147695	+	0.147695i	−0.777087	−	0.629393i	$-0.783304\pi$
$197$	−29.0959	+	29.0959i	−0.147695	+	0.147695i	0.629393	+	0.777087i	$0.283304\pi$
$198$		0			0
$199$	11.6967			0.0587776			0.0293888	−	0.999568i	$-0.490644\pi$
$199$	11.6967			0.0587776			0.0293888	+	0.999568i	$0.490644\pi$
$200$	−11.1319	+	9.16074i	−0.0556595	+	0.0458037i
$201$		0			0
$202$	30.6786	+	19.0581i	0.151874	+	0.0943472i
$203$	199.021	+	199.021i	0.980398	+	0.980398i
$204$		0			0
$205$	84.2755	+	84.2755i	0.411100	+	0.411100i
$206$	8.08220	+	34.5927i	0.0392340	+	0.167926i
$207$		0			0
$208$	−174.238	+	23.2562i	−0.837682	+	0.111809i
$209$	−221.378			−1.05923
$210$		0			0
$211$	−0.215765	+	0.215765i	−0.00102258	+	0.00102258i	−0.707618	−	0.706595i	$-0.750230\pi$
$211$	−0.215765	+	0.215765i	−0.00102258	+	0.00102258i	0.706595	+	0.707618i	$0.250230\pi$
$212$	66.8728	−	197.582i	0.315438	−	0.931989i
$213$		0			0
$214$	23.7146	−	38.1742i	0.110816	−	0.178384i
$215$	203.556			0.946773
$216$		0			0
$217$		−	319.684i		−	1.47320i
$218$	−118.396	+	190.586i	−0.543099	+	0.874247i
$219$		0			0
$220$	239.632	−	118.440i	1.08924	−	0.538365i
$221$	75.5951	+	75.5951i	0.342059	+	0.342059i
$222$		0			0
$223$		−	371.347i		−	1.66523i	−0.553850	−	0.832617i	$-0.686842\pi$
$223$		−	371.347i		−	1.66523i	0.553850	−	0.832617i	$-0.313158\pi$
$224$	−353.928	−	159.409i	−1.58004	−	0.711648i
$225$		0			0
$226$	−76.2337	−	326.288i	−0.337317	−	1.44375i
$227$	209.823	−	209.823i	0.924330	−	0.924330i	−0.0730018	−	0.997332i	$-0.523258\pi$
$227$	209.823	−	209.823i	0.924330	−	0.924330i	0.997332	+	0.0730018i	$0.0232579\pi$
$228$		0			0
$229$	152.751	−	152.751i	0.667037	−	0.667037i	−0.289992	−	0.957029i	$-0.593653\pi$
$229$	152.751	−	152.751i	0.667037	−	0.667037i	0.957029	+	0.289992i	$0.0936527\pi$
$230$	−165.806	−	103.002i	−0.720895	−	0.447834i
$231$		0			0
$232$	−17.9462	+	184.753i	−0.0773544	+	0.796350i
$233$		−	272.899i		−	1.17124i	−0.810586	−	0.585619i	$-0.800851\pi$
$233$		−	272.899i		−	1.17124i	0.810586	−	0.585619i	$-0.199149\pi$
$234$		0			0
$235$	106.710	+	106.710i	0.454084	+	0.454084i
$236$	−75.7259	−	25.6299i	−0.320873	−	0.108601i
$237$		0			0
$238$	53.7104	+	229.886i	0.225674	+	0.965908i
$239$			104.650i			0.437866i	0.975740	+	0.218933i	$0.0702576\pi$
$239$			104.650i			0.437866i	−0.975740	+	0.218933i	$0.929742\pi$
$240$		0			0
$241$	148.875			0.617737			0.308869	−	0.951105i	$-0.400050\pi$
$241$	148.875			0.617737			0.308869	+	0.951105i	$0.400050\pi$
$242$	139.261	−	32.5369i	0.575460	−	0.134450i
$243$		0			0
$244$	227.833	+	77.1117i	0.933743	+	0.316031i
$245$	334.254	−	334.254i	1.36430	−	1.36430i
$246$		0			0
$247$	−175.295			−0.709698
$248$	162.797	−	133.970i	0.656438	−	0.540201i
$249$		0			0
$250$	136.239	−	219.308i	0.544954	−	0.877233i
$251$	−143.712	−	143.712i	−0.572558	−	0.572558i	0.360284	−	0.932843i	$-0.382680\pi$
$251$	−143.712	−	143.712i	−0.572558	−	0.572558i	−0.932843	+	0.360284i	$0.882680\pi$
$252$		0			0
$253$	−198.802	−	198.802i	−0.785778	−	0.785778i
$254$	386.096	−	90.2071i	1.52006	−	0.355146i
$255$		0			0
$256$	−67.1424	−	247.038i	−0.262275	−	0.964993i
$257$	−134.023			−0.521489			−0.260745	−	0.965408i	$-0.583968\pi$
$257$	−134.023			−0.521489			−0.260745	+	0.965408i	$0.583968\pi$
$258$		0			0
$259$	−287.807	+	287.807i	−1.11122	+	1.11122i
$260$	189.749	−	93.7853i	0.729806	−	0.360713i
$261$		0			0
$262$	322.759	+	200.504i	1.23190	+	0.765283i
$263$	−290.386			−1.10413			−0.552066	−	0.833801i	$-0.686160\pi$
$263$	−290.386			−1.10413			−0.552066	+	0.833801i	$0.686160\pi$
$264$		0			0
$265$			251.166i			0.947798i
$266$	−328.812	−	204.264i	−1.23613	−	0.767911i
$267$		0			0
$268$	−88.4627	+	261.371i	−0.330085	+	0.975264i
$269$	74.2628	+	74.2628i	0.276070	+	0.276070i	0.831538	−	0.555468i	$-0.187461\pi$
$269$	74.2628	+	74.2628i	0.276070	+	0.276070i	−0.555468	+	0.831538i	$0.687461\pi$
$270$		0			0
$271$			70.8329i			0.261376i	0.991424	+	0.130688i	$0.0417186\pi$
$271$			70.8329i			0.261376i	−0.991424	+	0.130688i	$0.958281\pi$
$272$	−94.5593	+	123.690i	−0.347644	+	0.454742i
$273$		0			0
$274$	−498.451	+	116.458i	−1.81916	+	0.425028i
$275$	17.6799	−	17.6799i	0.0642906	−	0.0642906i
$276$		0			0
$277$	−96.6953	+	96.6953i	−0.349081	+	0.349081i	−0.859767	−	0.510686i	$-0.829391\pi$
$277$	−96.6953	+	96.6953i	−0.349081	+	0.349081i	0.510686	+	0.859767i	$0.329391\pi$
$278$	−32.4223	+	52.1914i	−0.116627	+	0.187739i
$279$		0			0
$280$	465.209	+	45.1886i	1.66146	+	0.161388i
$281$			138.151i			0.491640i	0.969316	+	0.245820i	$0.0790572\pi$
$281$			138.151i			0.491640i	−0.969316	+	0.245820i	$0.920943\pi$
$282$		0			0
$283$	−295.011	−	295.011i	−1.04244	−	1.04244i	−0.999059	−	0.0433821i	$-0.986187\pi$
$283$	−295.011	−	295.011i	−1.04244	−	1.04244i	−0.0433821	−	0.999059i	$-0.513813\pi$
$284$	27.7526	−	13.7170i	0.0977204	−	0.0482991i
$285$		0			0
$286$	296.871	−	69.3607i	1.03801	−	0.242520i
$287$			300.168i			1.04588i
$288$		0			0
$289$	−194.310			−0.672353
$290$	−50.8510	−	217.648i	−0.175348	−	0.750510i
$291$		0			0
$292$	−151.389	−	306.295i	−0.518455	−	1.04896i
$293$	33.4759	−	33.4759i	0.114252	−	0.114252i	−0.647669	−	0.761922i	$-0.724256\pi$
$293$	33.4759	−	33.4759i	0.114252	−	0.114252i	0.761922	+	0.647669i	$0.224256\pi$
$294$		0			0
$295$	96.2630			0.326315
$296$	−267.174	−	25.9523i	−0.902616	−	0.0876768i
$297$		0			0
$298$	−82.2748	−	51.1107i	−0.276090	−	0.171513i
$299$	−157.418	−	157.418i	−0.526483	−	0.526483i
$300$		0			0
$301$	362.508	+	362.508i	1.20434	+	1.20434i
$302$	6.58169	+	28.1703i	0.0217937	+	0.0932792i
$303$		0			0
$304$	−33.7749	−	253.046i	−0.111102	−	0.832387i
$305$	−289.622			−0.949582
$306$		0			0
$307$	−92.6638	+	92.6638i	−0.301836	+	0.301836i	−0.841732	−	0.539896i	$-0.818464\pi$
$307$	−92.6638	+	92.6638i	−0.301836	+	0.301836i	0.539896	+	0.841732i	$0.318464\pi$
$308$	637.682	+	215.828i	2.07040	+	0.700739i
$309$		0			0
$310$	−133.962	+	215.643i	−0.432135	+	0.695623i
$311$	18.5610			0.0596817			0.0298408	−	0.999555i	$-0.490500\pi$
$311$	18.5610			0.0596817			0.0298408	+	0.999555i	$0.490500\pi$
$312$		0			0
$313$		−	55.1534i		−	0.176209i	−0.996111	−	0.0881045i	$-0.971919\pi$
$313$		−	55.1534i		−	0.176209i	0.996111	−	0.0881045i	$-0.0280809\pi$
$314$	46.9626	−	75.5975i	0.149563	−	0.240756i
$315$		0			0
$316$	−186.619	−	377.573i	−0.590566	−	1.19485i
$317$	−62.2977	−	62.2977i	−0.196523	−	0.196523i	0.601985	−	0.798507i	$-0.294377\pi$
$317$	−62.2977	−	62.2977i	−0.196523	−	0.196523i	−0.798507	+	0.601985i	$0.794377\pi$
$318$		0			0
$319$		−	321.931i		−	1.00919i
$320$	171.943	+	255.841i	0.537320	+	0.799502i
$321$		0			0
$322$	−111.846	−	478.712i	−0.347348	−	1.48668i
$323$	−109.787	+	109.787i	−0.339897	+	0.339897i
$324$		0			0
$325$	13.9996	−	13.9996i	0.0430756	−	0.0430756i
$326$	−75.4414	−	46.8657i	−0.231415	−	0.143760i
$327$		0			0
$328$	−152.858	+	125.791i	−0.466030	+	0.383509i
$329$			380.073i			1.15524i
$330$		0			0
$331$	373.767	+	373.767i	1.12921	+	1.12921i	0.990307	+	0.138899i	$0.0443564\pi$
$331$	373.767	+	373.767i	1.12921	+	1.12921i	0.138899	+	0.990307i	$0.455644\pi$
$332$	−112.663	+	332.871i	−0.339345	+	1.00262i
$333$		0			0
$334$	16.5991	+	71.0459i	0.0496979	+	0.212712i
$335$		−	332.255i		−	0.991807i
$336$		0			0
$337$	−519.936			−1.54284			−0.771419	−	0.636328i	$-0.780453\pi$
$337$	−519.936			−1.54284			−0.771419	+	0.636328i	$0.780453\pi$
$338$	−94.0630	+	21.9768i	−0.278293	+	0.0650201i
$339$		0			0
$340$	60.1020	−	177.577i	0.176771	−	0.522284i
$341$	−258.557	+	258.557i	−0.758231	+	0.758231i
$342$		0			0
$343$	596.142			1.73802
$344$	−32.6883	+	336.520i	−0.0950240	+	0.978255i
$345$		0			0
$346$	−145.733	+	234.592i	−0.421194	+	0.678011i
$347$	−122.160	−	122.160i	−0.352045	−	0.352045i	0.508825	−	0.860870i	$-0.330080\pi$
$347$	−122.160	−	122.160i	−0.352045	−	0.352045i	−0.860870	+	0.508825i	$0.830080\pi$
$348$		0			0
$349$	−279.483	−	279.483i	−0.800810	−	0.800810i	0.182412	−	0.983222i	$-0.441609\pi$
$349$	−279.483	−	279.483i	−0.800810	−	0.800810i	−0.983222	+	0.182412i	$0.941609\pi$
$350$	42.5730	−	9.94671i	0.121637	−	0.0284192i
$351$		0			0
$352$	157.324	+	415.181i	0.446944	+	1.17949i
$353$	212.266			0.601320			0.300660	−	0.953731i	$-0.402793\pi$
$353$	212.266			0.601320			0.300660	+	0.953731i	$0.402793\pi$
$354$		0			0
$355$	−26.3581	+	26.3581i	−0.0742482	+	0.0742482i
$356$	−225.396	−	456.028i	−0.633134	−	1.28098i
$357$		0			0
$358$	215.614	+	133.943i	0.602272	+	0.374144i
$359$	435.033			1.21179			0.605895	−	0.795545i	$-0.292815\pi$
$359$	435.033			1.21179			0.605895	+	0.795545i	$0.292815\pi$
$360$		0			0
$361$			106.419i			0.294789i
$362$	277.491	+	172.383i	0.766551	+	0.476196i
$363$		0			0
$364$	504.939	+	170.900i	1.38720	+	0.469505i
$365$	290.905	+	290.905i	0.796999	+	0.796999i
$366$		0			0
$367$			125.535i			0.342058i	0.985266	+	0.171029i	$0.0547091\pi$
$367$			125.535i			0.342058i	−0.985266	+	0.171029i	$0.945291\pi$
$368$	196.909	−	257.570i	0.535079	−	0.699919i
$369$		0			0
$370$	314.744	−	73.5365i	0.850660	−	0.198747i
$371$	−447.295	+	447.295i	−1.20565	+	1.20565i
$372$		0			0
$373$	−302.389	+	302.389i	−0.810694	+	0.810694i	−0.984738	−	0.174044i	$-0.944317\pi$
$373$	−302.389	+	302.389i	−0.810694	+	0.810694i	0.174044	+	0.984738i	$0.444317\pi$
$374$	142.489	−	229.370i	0.380986	−	0.613287i
$375$		0			0
$376$	−193.549	+	159.277i	−0.514758	+	0.423608i
$377$		−	254.917i		−	0.676171i
$378$		0			0
$379$	189.784	+	189.784i	0.500751	+	0.500751i	0.911671	−	0.410921i	$-0.134793\pi$
$379$	189.784	+	189.784i	0.500751	+	0.500751i	−0.410921	+	0.911671i	$0.634793\pi$
$380$	136.204	+	275.573i	0.358432	+	0.725192i
$381$		0			0
$382$	−121.971	+	28.4972i	−0.319296	+	0.0745999i
$383$		−	639.916i		−	1.67080i	−0.549644	−	0.835399i	$-0.685237\pi$
$383$		−	639.916i		−	1.67080i	0.549644	−	0.835399i	$-0.314763\pi$
$384$		0			0
$385$	−810.623			−2.10551
$386$	−101.626	−	434.970i	−0.263280	−	1.12687i
$387$		0			0
$388$	528.376	−	261.154i	1.36179	−	0.673078i
$389$	−499.333	+	499.333i	−1.28363	+	1.28363i	−0.345046	+	0.938586i	$0.612137\pi$
$389$	−499.333	+	499.333i	−1.28363	+	1.28363i	−0.938586	+	0.345046i	$0.887863\pi$
$390$		0			0
$391$	−197.181			−0.504300
$392$	498.913	+	606.266i	1.27274	+	1.54660i
$393$		0			0
$394$	69.9050	+	43.4264i	0.177424	+	0.110219i
$395$	358.601	+	358.601i	0.907851	+	0.907851i
$396$		0			0
$397$	492.518	+	492.518i	1.24060	+	1.24060i	0.959753	+	0.280846i	$0.0906151\pi$
$397$	492.518	+	492.518i	1.24060	+	1.24060i	0.280846	+	0.959753i	$0.409385\pi$
$398$	−5.32230	−	22.7800i	−0.0133726	−	0.0572361i
$399$		0			0
$400$	22.9063	+	17.5116i	0.0572657	+	0.0437789i
$401$	−705.045			−1.75822			−0.879109	−	0.476621i	$-0.841862\pi$
$401$	−705.045			−1.75822			−0.879109	+	0.476621i	$0.841862\pi$
$402$		0			0
$403$	−204.735	+	204.735i	−0.508026	+	0.508026i
$404$	23.1572	−	68.4200i	0.0573198	−	0.169356i
$405$		0			0
$406$	297.044	−	478.162i	0.731635	−	1.17774i
$407$	465.550			1.14386
$408$		0			0
$409$		−	279.815i		−	0.684144i	−0.939674	−	0.342072i	$-0.888871\pi$
$409$		−	279.815i		−	0.684144i	0.939674	−	0.342072i	$-0.111129\pi$
$410$	125.783	−	202.478i	0.306789	−	0.493849i
$411$		0			0
$412$	63.6934	−	31.4810i	0.154596	−	0.0764102i
$413$	171.432	+	171.432i	0.415090	+	0.415090i
$414$		0			0
$415$		−	423.147i		−	1.01963i
$416$	124.575	+	328.755i	0.299459	+	0.790276i
$417$		0			0
$418$	100.732	+	431.146i	0.240987	+	1.03145i
$419$	573.583	−	573.583i	1.36893	−	1.36893i	0.506968	−	0.861965i	$-0.330766\pi$
$419$	573.583	−	573.583i	1.36893	−	1.36893i	0.861965	−	0.506968i	$-0.169234\pi$
$420$		0			0
$421$	−213.341	+	213.341i	−0.506749	+	0.506749i	−0.913527	−	0.406778i	$-0.866652\pi$
$421$	−213.341	+	213.341i	−0.506749	+	0.506749i	0.406778	+	0.913527i	$0.366652\pi$
$422$	0.518392	+	0.322035i	0.00122842	+	0.000763117i
$423$		0			0
$424$	−415.229	−	40.3338i	−0.979314	−	0.0951269i
$425$		−	17.5358i		−	0.0412606i
$426$		0			0
$427$	−515.781	−	515.781i	−1.20792	−	1.20792i
$428$	−85.1369	−	28.8152i	−0.198918	−	0.0673251i
$429$		0			0
$430$	−92.6230	−	396.436i	−0.215402	−	0.921944i
$431$			166.900i			0.387239i	0.981077	+	0.193619i	$0.0620227\pi$
$431$			166.900i			0.387239i	−0.981077	+	0.193619i	$0.937977\pi$
$432$		0			0
$433$	233.153			0.538459			0.269230	−	0.963076i	$-0.413231\pi$
$433$	233.153			0.538459			0.269230	+	0.963076i	$0.413231\pi$
$434$	−622.602	+	145.464i	−1.43457	+	0.335171i
$435$		0			0
$436$	425.048	+	143.860i	0.974882	+	0.329955i
$437$	228.619	−	228.619i	0.523155	−	0.523155i
$438$		0			0
$439$	440.480			1.00337			0.501686	−	0.865050i	$-0.332713\pi$
$439$	440.480			1.00337			0.501686	+	0.865050i	$0.332713\pi$
$440$	−339.707	−	412.803i	−0.772061	−	0.938189i
$441$		0			0
$442$	112.828	−	181.623i	0.255266	−	0.410912i
$443$	312.524	+	312.524i	0.705473	+	0.705473i	0.965580	−	0.260107i	$-0.0837579\pi$
$443$	312.524	+	312.524i	0.705473	+	0.705473i	−0.260107	+	0.965580i	$0.583758\pi$
$444$		0			0
$445$	433.114	+	433.114i	0.973290	+	0.973290i
$446$	−723.217	+	168.972i	−1.62156	+	0.378861i
$447$		0			0
$448$	−149.412	+	761.827i	−0.333509	+	1.70051i
$449$	734.338			1.63550			0.817748	−	0.575576i	$-0.195222\pi$
$449$	734.338			1.63550			0.817748	+	0.575576i	$0.195222\pi$
$450$		0			0
$451$	242.772	−	242.772i	0.538297	−	0.538297i
$452$	−600.774	+	296.938i	−1.32915	+	0.656942i
$453$		0			0
$454$	−504.115	−	313.166i	−1.11039	−	0.689794i
$455$	−641.880			−1.41073
$456$		0			0
$457$		−	692.749i		−	1.51586i	−0.652335	−	0.757931i	$-0.726211\pi$
$457$		−	692.749i		−	1.51586i	0.652335	−	0.757931i	$-0.273789\pi$
$458$	−366.997	−	227.986i	−0.801303	−	0.497785i
$459$		0			0
$460$	−125.156	+	369.784i	−0.272078	+	0.803878i
$461$	−298.447	−	298.447i	−0.647391	−	0.647391i	0.304971	−	0.952362i	$-0.401353\pi$
$461$	−298.447	−	298.447i	−0.647391	−	0.647391i	−0.952362	+	0.304971i	$0.901353\pi$
$462$		0			0
$463$		−	281.830i		−	0.608705i	−0.952560	−	0.304352i	$-0.901560\pi$
$463$		−	281.830i		−	0.608705i	0.952560	−	0.304352i	$-0.0984400\pi$
$464$	367.982	−	49.1159i	0.793065	−	0.105853i
$465$		0			0
$466$	−531.484	+	124.175i	−1.14052	+	0.266471i
$467$	−198.116	+	198.116i	−0.424232	+	0.424232i	−0.886658	−	0.462426i	$-0.846979\pi$
$467$	−198.116	+	198.116i	−0.424232	+	0.424232i	0.462426	+	0.886658i	$0.346979\pi$
$468$		0			0
$469$	591.704	−	591.704i	1.26163	−	1.26163i
$470$	159.267	−	256.378i	0.338866	−	0.545485i
$471$		0			0
$472$	−15.4585	+	159.142i	−0.0327510	+	0.337166i
$473$		−	586.384i		−	1.23971i
$474$		0			0
$475$	20.3316	+	20.3316i	0.0428033	+	0.0428033i
$476$	423.276	−	209.207i	0.889234	−	0.439512i
$477$		0			0
$478$	203.811	−	47.6182i	0.426383	−	0.0996197i
$479$		−	917.713i		−	1.91589i	−0.286945	−	0.957947i	$-0.592640\pi$
$479$		−	917.713i		−	1.91589i	0.286945	−	0.957947i	$-0.407360\pi$
$480$		0			0
$481$	368.639			0.766401
$482$	−67.7416	−	289.941i	−0.140543	−	0.601537i
$483$		0			0
$484$	−126.735	−	256.413i	−0.261848	−	0.529780i
$485$	−501.826	+	501.826i	−1.03469	+	1.03469i
$486$		0			0
$487$	−426.183			−0.875119			−0.437559	−	0.899190i	$-0.644157\pi$
$487$	−426.183			−0.875119			−0.437559	+	0.899190i	$0.644157\pi$
$488$	46.5093	−	478.805i	0.0953059	−	0.981157i
$489$		0			0
$490$	−803.070	−	498.883i	−1.63892	−	1.01813i
$491$	266.299	+	266.299i	0.542361	+	0.542361i	0.924220	−	0.381859i	$-0.124716\pi$
$491$	266.299	+	266.299i	0.542361	+	0.542361i	−0.381859	+	0.924220i	$0.624716\pi$
$492$		0			0
$493$	−159.653	−	159.653i	−0.323840	−	0.323840i
$494$	79.7636	+	341.396i	0.161465	+	0.691086i
$495$		0			0
$496$	−334.989	−	256.095i	−0.675382	−	0.516321i
$497$	−93.8809			−0.188895
$498$		0			0
$499$	−264.104	+	264.104i	−0.529266	+	0.529266i	−0.920353	−	0.391088i	$-0.872099\pi$
$499$	−264.104	+	264.104i	−0.529266	+	0.529266i	0.391088	+	0.920353i	$0.372099\pi$
$500$	−489.106	−	165.541i	−0.978211	−	0.331082i
$501$		0			0
$502$	−214.494	+	345.279i	−0.427279	+	0.687807i
$503$	574.766			1.14268			0.571338	−	0.820715i	$-0.306425\pi$
$503$	574.766			1.14268			0.571338	+	0.820715i	$0.306425\pi$
$504$		0			0
$505$			86.9756i			0.172229i
$506$	−296.717	+	477.636i	−0.586398	+	0.943946i
$507$		0			0
$508$	−351.366	−	710.895i	−0.691665	−	1.39940i
$509$	−170.592	−	170.592i	−0.335152	−	0.335152i	0.519387	−	0.854539i	$-0.326160\pi$
$509$	−170.592	−	170.592i	−0.335152	−	0.335152i	−0.854539	+	0.519387i	$0.826160\pi$
$510$		0			0
$511$			1036.13i			2.02765i
$512$	−450.568	+	243.172i	−0.880016	+	0.474944i
$513$		0			0
$514$	60.9835	+	261.016i	0.118645	+	0.507813i
$515$	−60.4930	+	60.4930i	−0.117462	+	0.117462i
$516$		0			0
$517$	307.398	−	307.398i	0.594581	−	0.594581i
$518$	691.478	+	429.560i	1.33490	+	0.829266i
$519$		0			0
$520$	−268.992	−	326.872i	−0.517293	−	0.628600i
$521$			37.1210i			0.0712496i	0.999365	+	0.0356248i	$0.0113421\pi$
$521$			37.1210i			0.0712496i	−0.999365	+	0.0356248i	$0.988658\pi$
$522$		0			0
$523$	−199.555	−	199.555i	−0.381558	−	0.381558i	0.490105	−	0.871663i	$-0.336958\pi$
$523$	−199.555	−	199.555i	−0.381558	−	0.381558i	−0.871663	+	0.490105i	$0.836958\pi$
$524$	243.629	−	719.823i	0.464941	−	1.37371i
$525$		0			0
$526$	132.133	+	565.542i	0.251203	+	1.07518i
$527$			256.449i			0.486620i
$528$		0			0
$529$	−118.392			−0.223804
$530$	489.159	−	114.287i	0.922942	−	0.215635i
$531$		0			0
$532$	−248.198	+	733.323i	−0.466538	+	1.37843i
$533$	192.236	−	192.236i	0.360667	−	0.360667i
$534$		0			0
$535$	108.226			0.202292
$536$	549.286	+	53.3555i	1.02479	+	0.0995439i
$537$		0			0
$538$	110.839	−	178.422i	0.206021	−	0.331639i
$539$	−962.884	−	962.884i	−1.78643	−	1.78643i
$540$		0			0
$541$	278.121	+	278.121i	0.514086	+	0.514086i	0.915776	−	0.401690i	$-0.131577\pi$
$541$	278.121	+	278.121i	0.514086	+	0.514086i	−0.401690	+	0.915776i	$0.631577\pi$
$542$	137.951	−	32.2307i	0.254521	−	0.0594661i
$543$		0			0
$544$	283.919	+	127.877i	0.521910	+	0.235068i
$545$	−540.323			−0.991418
$546$		0			0
$547$	724.938	−	724.938i	1.32530	−	1.32530i	0.415876	−	0.909421i	$-0.363475\pi$
$547$	724.938	−	724.938i	1.32530	−	1.32530i	0.909421	−	0.415876i	$-0.136525\pi$
$548$	453.614	+	917.767i	0.827763	+	1.67476i
$549$		0			0
$550$	−42.4773	−	26.3877i	−0.0772314	−	0.0479777i
$551$	370.215			0.671897
$552$		0			0
$553$			1277.25i			2.30967i
$554$	232.318	+	144.320i	0.419346	+	0.260506i
$555$		0			0
$556$	116.398	+	39.3958i	0.209350	+	0.0708558i
$557$	−268.298	−	268.298i	−0.481685	−	0.481685i	0.423985	−	0.905669i	$-0.360631\pi$
$557$	−268.298	−	268.298i	−0.481685	−	0.481685i	−0.905669	+	0.423985i	$0.860631\pi$
$558$		0			0
$559$		−	464.320i		−	0.830625i
$560$	−123.674	−	926.579i	−0.220846	−	1.65461i
$561$		0			0
$562$	269.056	−	62.8619i	0.478747	−	0.111854i
$563$	78.4662	−	78.4662i	0.139372	−	0.139372i	−0.633979	−	0.773350i	$-0.718579\pi$
$563$	78.4662	−	78.4662i	0.139372	−	0.139372i	0.773350	+	0.633979i	$0.218579\pi$
$564$		0			0
$565$	570.587	−	570.587i	1.00989	−	1.00989i
$566$	−440.311	+	708.785i	−0.777935	+	1.25227i
$567$		0			0
$568$	−39.3426	−	47.8080i	−0.0692651	−	0.0841691i
$569$			801.999i			1.40949i	0.709461	+	0.704744i	$0.248938\pi$
$569$			801.999i			1.40949i	−0.709461	+	0.704744i	$0.751062\pi$
$570$		0			0
$571$	79.9964	+	79.9964i	0.140099	+	0.140099i	0.773678	−	0.633579i	$-0.218415\pi$
$571$	79.9964	+	79.9964i	0.140099	+	0.140099i	−0.633579	+	0.773678i	$0.718415\pi$
$572$	−270.167	−	546.611i	−0.472320	−	0.955613i
$573$		0			0
$574$	584.592	−	136.583i	1.01845	−	0.237950i
$575$			36.5163i			0.0635066i
$576$		0			0
$577$	−237.186			−0.411068			−0.205534	−	0.978650i	$-0.565893\pi$
$577$	−237.186			−0.411068			−0.205534	+	0.978650i	$0.565893\pi$
$578$	88.4158	+	378.429i	0.152968	+	0.654721i
$579$		0			0
$580$	−400.742	+	198.070i	−0.690934	+	0.341500i
$581$	753.571	−	753.571i	1.29702	−	1.29702i
$582$		0			0
$583$	723.534			1.24105
$584$	−527.639	+	434.209i	−0.903492	+	0.743509i
$585$		0			0
$586$	−80.4283	−	49.9637i	−0.137250	−	0.0852622i
$587$	267.958	+	267.958i	0.456487	+	0.456487i	0.897500	−	0.441014i	$-0.145381\pi$
$587$	267.958	+	267.958i	0.456487	+	0.456487i	−0.441014	+	0.897500i	$0.645381\pi$
$588$		0			0
$589$	−297.336	−	297.336i	−0.504815	−	0.504815i
$590$	−43.8020	−	187.477i	−0.0742407	−	0.317758i
$591$		0			0
$592$	71.0273	+	532.145i	0.119979	+	0.898893i
$593$	607.086			1.02375			0.511877	−	0.859059i	$-0.328950\pi$
$593$	607.086			1.02375			0.511877	+	0.859059i	$0.328950\pi$
$594$		0			0
$595$	−402.007	+	402.007i	−0.675642	+	0.675642i
$596$	−62.1037	+	183.491i	−0.104201	+	0.307871i
$597$		0			0
$598$	−234.951	+	378.210i	−0.392895	+	0.632457i
$599$	−575.392			−0.960587			−0.480294	−	0.877108i	$-0.659470\pi$
$599$	−575.392			−0.960587			−0.480294	+	0.877108i	$0.659470\pi$
$600$		0			0
$601$		−	310.094i		−	0.515963i	−0.966150	−	0.257981i	$-0.916943\pi$
$601$		−	310.094i		−	0.515963i	0.966150	−	0.257981i	$-0.0830573\pi$
$602$	541.052	−	870.952i	0.898758	−	1.44676i
$603$		0			0
$604$	51.8683	−	25.6363i	0.0858746	−	0.0424443i
$605$	243.529	+	243.529i	0.402528	+	0.402528i
$606$		0			0
$607$		−	556.510i		−	0.916820i	−0.888741	−	0.458410i	$-0.848419\pi$
$607$		−	556.510i		−	0.916820i	0.888741	−	0.458410i	$-0.151581\pi$
$608$	−477.451	+	180.920i	−0.785281	+	0.297566i
$609$		0			0
$610$	131.785	+	564.054i	0.216041	+	0.924679i
$611$	243.409	−	243.409i	0.398378	−	0.398378i
$612$		0			0
$613$	−326.241	+	326.241i	−0.532204	+	0.532204i	−0.921228	−	0.389024i	$-0.872812\pi$
$613$	−326.241	+	326.241i	−0.532204	+	0.532204i	0.389024	+	0.921228i	$0.372812\pi$
$614$	222.632	+	138.303i	0.362592	+	0.225249i
$615$		0			0
$616$	130.175	−	1340.12i	0.211323	−	2.17553i
$617$			502.068i			0.813725i	0.913490	+	0.406862i	$0.133377\pi$
$617$			502.068i			0.813725i	−0.913490	+	0.406862i	$0.866623\pi$
$618$		0			0
$619$	304.429	+	304.429i	0.491808	+	0.491808i	0.908876	−	0.417067i	$-0.136942\pi$
$619$	304.429	+	304.429i	0.491808	+	0.491808i	−0.417067	+	0.908876i	$0.636942\pi$
$620$	480.932	+	162.774i	0.775696	+	0.262539i
$621$		0			0
$622$	−8.44570	−	36.1485i	−0.0135783	−	0.0581166i
$623$			1542.64i			2.47615i
$624$		0			0
$625$	576.701			0.922721
$626$	−107.414	+	25.0961i	−0.171588	+	0.0400896i
$627$		0			0
$628$	−168.599	−	57.0635i	−0.268470	−	0.0908654i
$629$	230.877	−	230.877i	0.367054	−	0.367054i
$630$		0			0
$631$	8.60592			0.0136385			0.00681927	−	0.999977i	$-0.497829\pi$
$631$	8.60592			0.0136385			0.00681927	+	0.999977i	$0.497829\pi$
$632$	−650.427	+	535.254i	−1.02916	+	0.846921i
$633$		0			0
$634$	−92.9809	+	149.675i	−0.146658	+	0.236080i
$635$	675.174	+	675.174i	1.06327	+	1.06327i
$636$		0			0
$637$	−762.446	−	762.446i	−1.19693	−	1.19693i
$638$	−626.977	+	146.486i	−0.982723	+	0.229603i
$639$		0			0
$640$	420.025	−	451.280i	0.656289	−	0.705126i
$641$	445.780			0.695445			0.347722	−	0.937598i	$-0.386955\pi$
$641$	445.780			0.695445			0.347722	+	0.937598i	$0.386955\pi$
$642$		0			0
$643$	−118.001	+	118.001i	−0.183517	+	0.183517i	−0.792886	−	0.609369i	$-0.791423\pi$
$643$	−118.001	+	118.001i	−0.183517	+	0.183517i	0.609369	+	0.792886i	$0.291423\pi$
$644$	−881.424	+	435.651i	−1.36867	+	0.676477i
$645$		0			0
$646$	263.771	+	163.860i	0.408314	+	0.253653i
$647$	−1081.35			−1.67132			−0.835662	−	0.549243i	$-0.814916\pi$
$647$	−1081.35			−1.67132			−0.835662	+	0.549243i	$0.814916\pi$
$648$		0			0
$649$		−	277.305i		−	0.427280i
$650$	−33.6350	−	20.8947i	−0.0517462	−	0.0321458i
$651$		0			0
$652$	−56.9457	+	168.251i	−0.0873400	+	0.258054i
$653$	586.227	+	586.227i	0.897744	+	0.897744i	0.995236	−	0.0974927i	$-0.0310823\pi$
$653$	586.227	+	586.227i	0.897744	+	0.897744i	−0.0974927	+	0.995236i	$0.531082\pi$
$654$		0			0
$655$			915.041i			1.39701i
$656$	314.538	+	240.461i	0.479479	+	0.366556i
$657$		0			0
$658$	740.211	−	172.942i	1.12494	−	0.262830i
$659$	−469.999	+	469.999i	−0.713201	+	0.713201i	−0.967204	−	0.254003i	$-0.918253\pi$
$659$	−469.999	+	469.999i	−0.713201	+	0.713201i	0.254003	+	0.967204i	$0.418253\pi$
$660$		0			0
$661$	−884.745	+	884.745i	−1.33849	+	1.33849i	−0.440976	+	0.897519i	$0.645368\pi$
$661$	−884.745	+	884.745i	−1.33849	+	1.33849i	−0.897519	+	0.440976i	$0.854632\pi$
$662$	557.857	−	898.003i	0.842685	−	1.35650i
$663$		0			0
$664$	699.548	+	67.9515i	1.05354	+	0.102337i
$665$		−	932.202i		−	1.40181i
$666$		0			0
$667$	332.460	+	332.460i	0.498441	+	0.498441i
$668$	130.813	−	64.6552i	0.195827	−	0.0967892i
$669$		0			0
$670$	−647.084	+	151.184i	−0.965797	+	0.225648i
$671$			834.314i			1.24339i
$672$		0			0
$673$	684.329			1.01683			0.508417	−	0.861111i	$-0.330231\pi$
$673$	684.329			1.01683			0.508417	+	0.861111i	$0.330231\pi$
$674$	236.584	+	1012.60i	0.351014	+	1.50238i
$675$		0			0
$676$	85.6018	+	173.192i	0.126630	+	0.256202i
$677$	383.762	−	383.762i	0.566857	−	0.566857i	−0.364390	−	0.931246i	$-0.618722\pi$
$677$	383.762	−	383.762i	0.566857	−	0.566857i	0.931246	+	0.364390i	$0.118722\pi$
$678$		0			0
$679$	−1787.38			−2.63237
$680$	−373.188	−	36.2500i	−0.548805	−	0.0533089i
$681$		0			0
$682$	621.202	+	385.903i	0.910854	+	0.565840i
$683$	−903.626	−	903.626i	−1.32302	−	1.32302i	−0.911315	−	0.411709i	$-0.864932\pi$
$683$	−903.626	−	903.626i	−1.32302	−	1.32302i	−0.411709	−	0.911315i	$-0.635068\pi$
$684$		0			0
$685$	−871.652	−	871.652i	−1.27248	−	1.27248i
$686$	−271.259	−	1161.02i	−0.395421	−	1.69244i
$687$		0			0
$688$	670.263	−	89.4625i	0.974220	−	0.130033i
$689$	572.920			0.831524
$690$		0			0
$691$	−63.6870	+	63.6870i	−0.0921665	+	0.0921665i	−0.751687	−	0.659520i	$-0.770759\pi$
$691$	−63.6870	+	63.6870i	−0.0921665	+	0.0921665i	0.659520	+	0.751687i	$0.270759\pi$
$692$	523.192	+	177.078i	0.756057	+	0.255893i
$693$		0			0
$694$	−182.326	+	293.497i	−0.262718	+	0.422907i
$695$	−147.966			−0.212901
$696$		0			0
$697$		−	240.793i		−	0.345470i
$698$	−417.135	+	671.478i	−0.597615	+	0.962003i
$699$		0			0
$700$	−38.7434	−	78.3870i	−0.0553478	−	0.111981i
$701$	−218.312	−	218.312i	−0.311430	−	0.311430i	0.534033	−	0.845463i	$-0.320676\pi$
$701$	−218.312	−	218.312i	−0.311430	−	0.311430i	−0.845463	+	0.534033i	$0.820676\pi$
$702$		0			0
$703$			535.374i			0.761557i
$704$	736.999	−	495.314i	1.04687	−	0.703571i
$705$		0			0
$706$	−96.5861	−	413.399i	−0.136808	−	0.585551i
$707$	−154.893	+	154.893i	−0.219084	+	0.219084i
$708$		0			0
$709$	−822.199	+	822.199i	−1.15966	+	1.15966i	−0.175112	+	0.984548i	$0.556029\pi$
$709$	−822.199	+	822.199i	−1.15966	+	1.15966i	−0.984548	+	0.175112i	$0.943971\pi$
$710$	63.3273	+	39.3402i	0.0891934	+	0.0554087i
$711$		0			0
$712$	−785.577	+	646.473i	−1.10334	+	0.907968i
$713$		−	534.026i		−	0.748985i
$714$		0			0
$715$	519.145	+	519.145i	0.726077	+	0.726077i
$716$	162.752	−	480.866i	0.227307	−	0.671600i
$717$		0			0
$718$	−197.950	−	847.248i	−0.275697	−	1.18001i
$719$		−	340.913i		−	0.474149i	−0.971491	−	0.237074i	$-0.923811\pi$
$719$		−	340.913i		−	0.474149i	0.971491	−	0.237074i	$-0.0761885\pi$
$720$		0			0
$721$	−215.461			−0.298836
$722$	207.256	−	48.4231i	0.287058	−	0.0670680i
$723$		0			0
$724$	209.460	−	618.867i	0.289309	−	0.854788i
$725$	−29.5664	+	29.5664i	−0.0407813	+	0.0407813i
$726$		0			0
$727$	−803.090			−1.10466			−0.552331	−	0.833625i	$-0.686262\pi$
$727$	−803.090			−1.10466			−0.552331	+	0.833625i	$0.686262\pi$
$728$	103.077	−	1061.16i	0.141589	−	1.45763i
$729$		0			0
$730$	434.183	−	698.920i	0.594771	−	0.957424i
$731$	−290.802	−	290.802i	−0.397813	−	0.397813i
$732$		0			0
$733$	481.592	+	481.592i	0.657015	+	0.657015i	0.954673	−	0.297658i	$-0.0962054\pi$
$733$	481.592	+	481.592i	0.657015	+	0.657015i	−0.297658	+	0.954673i	$0.596205\pi$
$734$	244.486	−	57.1215i	0.333087	−	0.0778222i
$735$		0			0
$736$	−591.229	−	266.290i	−0.803301	−	0.361807i
$737$	−957.127			−1.29868
$738$		0			0
$739$	173.622	−	173.622i	0.234941	−	0.234941i	−0.579810	−	0.814752i	$-0.696873\pi$
$739$	173.622	−	173.622i	0.234941	−	0.234941i	0.814752	+	0.579810i	$0.196873\pi$
$740$	−286.432	−	579.519i	−0.387071	−	0.783134i
$741$		0			0
$742$	1074.66	+	667.600i	1.44833	+	0.899731i
$743$	−1316.22			−1.77149			−0.885744	−	0.464173i	$-0.846351\pi$
$743$	−1316.22			−1.77149			−0.885744	+	0.464173i	$0.846351\pi$
$744$		0			0
$745$		−	233.254i		−	0.313093i
$746$	726.512	+	451.323i	0.973876	+	0.604991i
$747$		0			0
$748$	−511.545	−	173.136i	−0.683883	−	0.231465i
$749$	192.737	+	192.737i	0.257326	+	0.257326i
$750$		0			0
$751$		−	322.977i		−	0.430062i	−0.976607	−	0.215031i	$-0.931015\pi$
$751$		−	322.977i		−	0.430062i	0.976607	−	0.215031i	$-0.0689853\pi$
$752$	398.269	+	304.472i	0.529613	+	0.404882i
$753$		0			0
$754$	−496.463	+	115.993i	−0.658439	+	0.153837i
$755$	−49.2621	+	49.2621i	−0.0652478	+	0.0652478i
$756$		0			0
$757$	−80.2744	+	80.2744i	−0.106043	+	0.106043i	−0.758138	−	0.652095i	$-0.773890\pi$
$757$	−80.2744	+	80.2744i	−0.106043	+	0.106043i	0.652095	+	0.758138i	$0.273890\pi$
$758$	283.258	−	455.971i	0.373692	−	0.601545i
$759$		0			0
$760$	474.716	−	390.657i	0.624627	−	0.514023i
$761$			596.664i			0.784053i	0.919954	+	0.392027i	$0.128226\pi$
$761$			596.664i			0.784053i	−0.919954	+	0.392027i	$0.871774\pi$
$762$		0			0
$763$	−962.246	−	962.246i	−1.26113	−	1.26113i
$764$	110.999	+	224.578i	0.145287	+	0.293950i
$765$		0			0
$766$	−1246.27	+	291.177i	−1.62698	+	0.380127i
$767$		−	219.580i		−	0.286284i
$768$		0			0
$769$	1515.31			1.97050			0.985249	−	0.171129i	$-0.0547416\pi$
$769$	1515.31			1.97050			0.985249	+	0.171129i	$0.0547416\pi$
$770$	368.853	+	1578.73i	0.479030	+	2.05030i
$771$		0			0
$772$	−800.884	+	395.844i	−1.03741	+	0.512751i
$773$	−607.901	+	607.901i	−0.786418	+	0.786418i	−0.980905	−	0.194487i	$-0.937696\pi$
$773$	−607.901	+	607.901i	−0.786418	+	0.786418i	0.194487	+	0.980905i	$0.437696\pi$
$774$		0			0
$775$	47.4922			0.0612802
$776$	−749.034	−	910.207i	−0.965250	−	1.17295i
$777$		0			0
$778$	1199.68	+	745.267i	1.54201	+	0.957927i
$779$	279.184	+	279.184i	0.358387	+	0.358387i
$780$		0			0
$781$	75.9297	+	75.9297i	0.0972211	+	0.0972211i
$782$	89.7221	+	384.020i	0.114734	+	0.491074i
$783$		0			0
$784$	953.717	−	1247.52i	1.21648	−	1.59123i
$785$	214.324			0.273024
$786$		0			0
$787$	356.009	−	356.009i	0.452362	−	0.452362i	−0.443776	−	0.896138i	$-0.646361\pi$
$787$	356.009	−	356.009i	0.452362	−	0.452362i	0.896138	+	0.443776i	$0.146361\pi$
$788$	52.7666	−	155.904i	0.0669627	−	0.197847i
$789$		0			0
$790$	535.222	−	861.566i	0.677496	−	1.09059i
$791$	2032.29			2.56926
$792$		0			0
$793$			660.640i			0.833089i
$794$	735.096	−	1183.31i	0.925814	−	1.49032i
$795$		0			0
$796$	−41.9434	+	20.7309i	−0.0526927	+	0.0260438i
$797$	−971.380	−	971.380i	−1.21880	−	1.21880i	−0.968054	−	0.250742i	$-0.919326\pi$
$797$	−971.380	−	971.380i	−1.21880	−	1.21880i	−0.250742	−	0.968054i	$-0.580674\pi$
$798$		0			0
$799$		−	304.892i		−	0.381592i
$800$	23.6818	−	52.5793i	0.0296022	−	0.0657242i
$801$		0			0
$802$	320.813	+	1373.11i	0.400016	+	1.71211i
$803$	838.008	−	838.008i	1.04360	−	1.04360i
$804$		0			0
$805$	837.135	−	837.135i	1.03992	−	1.03992i
$806$	491.890	+	305.572i	0.610285	+	0.379121i
$807$		0			0
$808$	−143.788	−	13.9671i	−0.177956	−	0.0172860i
$809$		−	678.276i		−	0.838412i	−0.907891	−	0.419206i	$-0.862308\pi$
$809$		−	678.276i		−	0.838412i	0.907891	−	0.419206i	$-0.137692\pi$
$810$		0			0
$811$	204.625	+	204.625i	0.252312	+	0.252312i	0.821918	−	0.569606i	$-0.192904\pi$
$811$	204.625	+	204.625i	0.252312	+	0.252312i	−0.569606	+	0.821918i	$0.692904\pi$
$812$	−1066.41	−	360.933i	−1.31331	−	0.444498i
$813$		0			0
$814$	−211.836	−	906.682i	−0.260241	−	1.11386i
$815$		−	213.881i		−	0.262431i
$816$		0			0
$817$	674.331			0.825375
$818$	−544.954	+	127.323i	−0.666203	+	0.155651i
$819$		0			0
$820$	−451.571	−	152.837i	−0.550696	−	0.186387i
$821$	−326.524	+	326.524i	−0.397715	+	0.397715i	−0.877426	−	0.479711i	$-0.840741\pi$
$821$	−326.524	+	326.524i	−0.397715	+	0.397715i	0.479711	+	0.877426i	$0.340741\pi$
$822$		0			0
$823$	804.270			0.977241			0.488621	−	0.872496i	$-0.337500\pi$
$823$	804.270			0.977241			0.488621	+	0.872496i	$0.337500\pi$
$824$	−90.2929	−	109.722i	−0.109579	−	0.133157i
$825$		0			0
$826$	255.867	−	411.879i	0.309767	−	0.498642i
$827$	−848.530	−	848.530i	−1.02603	−	1.02603i	−0.999652	−	0.0263821i	$-0.991601\pi$
$827$	−848.530	−	848.530i	−1.02603	−	1.02603i	−0.0263821	−	0.999652i	$-0.508399\pi$
$828$		0			0
$829$	49.5139	+	49.5139i	0.0597273	+	0.0597273i	0.736340	−	0.676612i	$-0.236553\pi$
$829$	49.5139	+	49.5139i	0.0597273	+	0.0597273i	−0.676612	+	0.736340i	$0.736553\pi$
$830$	−824.101	+	192.542i	−0.992892	+	0.231978i
$831$		0			0
$832$	583.582	−	392.207i	0.701421	−	0.471403i
$833$	−955.034			−1.14650
$834$		0			0
$835$	−124.240	+	124.240i	−0.148790	+	0.148790i
$836$	793.842	−	392.363i	0.949572	−	0.469334i
$837$		0			0
$838$	−1378.08	−	856.088i	−1.64448	−	1.02158i
$839$	−866.213			−1.03244			−0.516218	−	0.856457i	$-0.672660\pi$
$839$	−866.213			−1.03244			−0.516218	+	0.856457i	$0.672660\pi$
$840$		0			0
$841$		−	302.629i		−	0.359844i
$842$	512.568	+	318.417i	0.608751	+	0.378168i
$843$		0			0
$844$	0.391299	−	1.15613i	0.000463625	−	0.00136982i
$845$	−164.490	−	164.490i	−0.194663	−	0.194663i
$846$		0			0
$847$			867.390i			1.02407i
$848$	110.387	+	827.033i	0.130173	+	0.975274i
$849$		0			0
$850$	−34.1518	+	7.97919i	−0.0401786	+	0.00938728i
$851$	−480.776	+	480.776i	−0.564954	+	0.564954i
$852$		0			0
$853$	313.947	−	313.947i	0.368050	−	0.368050i	−0.498715	−	0.866766i	$-0.666195\pi$
$853$	313.947	−	313.947i	0.368050	−	0.368050i	0.866766	+	0.498715i	$0.166195\pi$
$854$	−769.816	+	1239.20i	−0.901424	+	1.45106i
$855$		0			0
$856$	−17.3796	+	178.920i	−0.0203033	+	0.209019i
$857$			473.297i			0.552272i	0.961119	+	0.276136i	$0.0890540\pi$
$857$			473.297i			0.552272i	−0.961119	+	0.276136i	$0.910946\pi$
$858$		0			0
$859$	−595.383	−	595.383i	−0.693112	−	0.693112i	0.269803	−	0.962915i	$-0.413041\pi$
$859$	−595.383	−	595.383i	−0.693112	−	0.693112i	−0.962915	+	0.269803i	$0.913041\pi$
$860$	−729.934	+	360.776i	−0.848760	+	0.419507i
$861$		0			0
$862$	325.046	−	75.9435i	0.377084	−	0.0881015i
$863$			742.134i			0.859947i	0.902842	+	0.429973i	$0.141477\pi$
$863$			742.134i			0.859947i	−0.902842	+	0.429973i	$0.858523\pi$
$864$		0			0
$865$	−665.083			−0.768882
$866$	−106.090	−	454.077i	−0.122506	−	0.524338i
$867$		0			0
$868$	566.597	+	1146.36i	0.652762	+	1.32069i
$869$	1033.02	−	1033.02i	1.18875	−	1.18875i
$870$		0			0
$871$	−757.887			−0.870134
$872$	86.7682	−	893.263i	0.0995048	−	1.02438i
$873$		0			0
$874$	−549.274	−	341.220i	−0.628459	−	0.390411i
$875$	1107.26	+	1107.26i	1.26544	+	1.26544i
$876$		0			0
$877$	−791.224	−	791.224i	−0.902194	−	0.902194i	0.0934320	−	0.995626i	$-0.470216\pi$
$877$	−791.224	−	791.224i	−0.902194	−	0.902194i	−0.995626	+	0.0934320i	$0.970216\pi$
$878$	−200.429	−	857.858i	−0.228279	−	0.977059i
$879$		0			0
$880$	−649.380	+	849.432i	−0.737932	+	0.965264i
$881$	1524.92			1.73090			0.865450	−	0.500995i	$-0.167033\pi$
$881$	1524.92			1.73090			0.865450	+	0.500995i	$0.167033\pi$
$882$		0			0
$883$	−314.328	+	314.328i	−0.355978	+	0.355978i	−0.862328	−	0.506350i	$-0.830994\pi$
$883$	−314.328	+	314.328i	−0.355978	+	0.355978i	0.506350	+	0.862328i	$0.330994\pi$
$884$	−405.059	−	137.095i	−0.458212	−	0.155085i
$885$		0			0
$886$	466.451	−	750.863i	0.526468	−	0.847475i
$887$	1520.80			1.71454			0.857271	−	0.514866i	$-0.172158\pi$
$887$	1520.80			1.71454			0.857271	+	0.514866i	$0.172158\pi$
$888$		0			0
$889$			2404.80i			2.70506i
$890$	646.434	−	1040.59i	0.726330	−	1.16920i
$891$		0			0
$892$	658.162	+	1331.62i	0.737850	+	1.49284i
$893$	353.503	+	353.503i	0.395860	+	0.395860i
$894$		0			0
$895$			611.278i			0.682992i
$896$	1551.68	−	55.6625i	1.73179	−	0.0621234i
$897$		0			0
$898$	−334.141	−	1430.16i	−0.372095	−	1.59261i
$899$	432.389	−	432.389i	0.480967	−	0.480967i
$900$		0			0
$901$	358.818	−	358.818i	0.398244	−	0.398244i
$902$	−583.278	−	362.344i	−0.646649	−	0.401711i
$903$		0			0
$904$	851.668	+	1034.92i	0.942111	+	1.14483i
$905$			786.705i			0.869288i
$906$		0			0
$907$	−216.886	−	216.886i	−0.239125	−	0.239125i	0.577363	−	0.816488i	$-0.304082\pi$
$907$	−216.886	−	216.886i	−0.239125	−	0.239125i	−0.816488	+	0.577363i	$0.804082\pi$
$908$	−380.523	+	1124.29i	−0.419078	+	1.23820i
$909$		0			0
$910$	292.071	+	1250.09i	0.320957	+	1.37373i
$911$			799.632i			0.877752i	0.898548	+	0.438876i	$0.144623\pi$
$911$			799.632i			0.877752i	−0.898548	+	0.438876i	$0.855377\pi$
$912$		0			0
$913$	−1218.96			−1.33511
$914$	−1349.16	+	315.217i	−1.47611	+	0.344877i
$915$		0			0
$916$	−277.021	+	818.483i	−0.302425	+	0.893541i
$917$	−1629.57	+	1629.57i	−1.77707	+	1.77707i
$918$		0			0
$919$	−640.590			−0.697051			−0.348525	−	0.937299i	$-0.613317\pi$
$919$	−640.590			−0.697051			−0.348525	+	0.937299i	$0.613317\pi$
$920$	777.121	+	75.4867i	0.844697	+	0.0820507i
$921$		0			0
$922$	−445.440	+	717.042i	−0.483124	+	0.777702i
$923$	60.1238	+	60.1238i	0.0651396	+	0.0651396i
$924$		0			0
$925$	−42.7565	−	42.7565i	−0.0462232	−	0.0462232i
$926$	−548.879	+	128.240i	−0.592742	+	0.138488i
$927$		0			0
$928$	−263.096	−	694.315i	−0.283509	−	0.748184i
$929$	−118.633			−0.127699			−0.0638496	−	0.997960i	$-0.520338\pi$
$929$	−118.633			−0.127699			−0.0638496	+	0.997960i	$0.520338\pi$
$930$		0			0
$931$	1107.30	−	1107.30i	1.18937	−	1.18937i
$932$	483.676	+	978.589i	0.518966	+	1.04999i
$933$		0			0
$934$	475.989	+	295.694i	0.509624	+	0.316589i
$935$	650.277			0.695483
$936$		0			0
$937$		−	731.334i		−	0.780506i	−0.920708	−	0.390253i	$-0.872388\pi$
$937$		−	731.334i		−	0.780506i	0.920708	−	0.390253i	$-0.127612\pi$
$938$	−1421.61	−	883.135i	−1.51558	−	0.941508i
$939$		0			0
$940$	−571.780	−	193.523i	−0.608276	−	0.205875i
$941$	−980.281	−	980.281i	−1.04174	−	1.04174i	−0.999090	−	0.0426536i	$-0.986419\pi$
$941$	−980.281	−	980.281i	−1.04174	−	1.04174i	−0.0426536	−	0.999090i	$-0.513581\pi$
$942$		0			0
$943$			501.424i			0.531733i
$944$	316.972	−	42.3074i	0.335775	−	0.0448172i
$945$		0			0
$946$	−1142.01	+	266.819i	−1.20720	+	0.282049i
$947$	240.008	−	240.008i	0.253441	−	0.253441i	−0.568939	−	0.822380i	$-0.692646\pi$
$947$	240.008	−	240.008i	0.253441	−	0.253441i	0.822380	+	0.568939i	$0.192646\pi$
$948$		0			0
$949$	663.564	−	663.564i	0.699225	−	0.699225i
$950$	30.3454	−	48.8481i	0.0319425	−	0.0514191i
$951$		0			0
$952$	−600.043	−	729.156i	−0.630297	−	0.765920i
$953$		−	780.049i		−	0.818519i	−0.912418	−	0.409259i	$-0.865787\pi$
$953$		−	780.049i		−	0.818519i	0.912418	−	0.409259i	$-0.134213\pi$
$954$		0			0
$955$	−213.293	−	213.293i	−0.223344	−	0.223344i
$956$	−185.478	−	375.265i	−0.194014	−	0.392536i
$957$		0			0
$958$	−1787.29	+	417.582i	−1.86565	+	0.435889i
$959$		−	3104.60i		−	3.23733i
$960$		0			0
$961$	266.459			0.277272
$962$	−167.740	−	717.943i	−0.174365	−	0.746303i
$963$		0			0
$964$	−533.851	+	263.860i	−0.553787	+	0.273714i
$965$	760.642	−	760.642i	0.788230	−	0.788230i
$966$		0			0
$967$	1783.10			1.84395			0.921975	−	0.387249i	$-0.126575\pi$
$967$	1783.10			1.84395			0.921975	+	0.387249i	$0.126575\pi$
$968$	−441.711	+	363.496i	−0.456313	+	0.375513i
$969$		0			0
$970$	1205.67	+	748.989i	1.24296	+	0.772154i
$971$	159.340	+	159.340i	0.164099	+	0.164099i	0.784380	−	0.620281i	$-0.212981\pi$
$971$	159.340	+	159.340i	0.164099	+	0.164099i	−0.620281	+	0.784380i	$0.712981\pi$
$972$		0			0
$973$	−263.509	−	263.509i	−0.270821	−	0.270821i
$974$	193.923	+	830.013i	0.199100	+	0.852169i
$975$		0			0
$976$	−953.659	+	127.288i	−0.977110	+	0.130418i
$977$	970.922			0.993779			0.496889	−	0.867814i	$-0.334475\pi$
$977$	970.922			0.993779			0.496889	+	0.867814i	$0.334475\pi$
$978$		0			0
$979$	1247.67	−	1247.67i	1.27443	−	1.27443i
$980$	−606.184	+	1791.02i	−0.618555	+	1.82758i
$981$		0			0
$982$	397.459	−	639.804i	0.404744	−	0.651531i
$983$	−1266.90			−1.28881			−0.644406	−	0.764684i	$-0.722895\pi$
$983$	−1266.90			−1.28881			−0.644406	+	0.764684i	$0.722895\pi$
$984$		0			0
$985$			198.185i			0.201203i
$986$	−238.287	+	383.579i	−0.241670	+	0.389025i
$987$		0			0
$988$	628.593	−	310.687i	0.636227	−	0.314461i
$989$	605.562	+	605.562i	0.612297	+	0.612297i
$990$		0			0
$991$			222.422i			0.224442i	0.993683	+	0.112221i	$0.0357964\pi$
$991$			222.422i			0.224442i	−0.993683	+	0.112221i	$0.964204\pi$
$992$	−346.330	+	768.938i	−0.349123	+	0.775139i
$993$		0			0
$994$	42.7180	+	182.838i	0.0429759	+	0.183941i
$995$	39.8359	−	39.8359i	0.0400360	−	0.0400360i
$996$		0			0
$997$	−441.746	+	441.746i	−0.443075	+	0.443075i	−0.893044	−	0.449969i	$-0.851435\pi$
$997$	−441.746	+	441.746i	−0.443075	+	0.443075i	0.449969	+	0.893044i	$0.351435\pi$
$998$	634.528	+	394.181i	0.635800	+	0.394971i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.m.c.19.4		16
3.2	odd	2		48.3.l.a.19.5	✓	16
4.3	odd	2		576.3.m.c.559.6		16
8.3	odd	2		1152.3.m.c.991.3		16
8.5	even	2		1152.3.m.f.991.3		16
12.11	even	2		192.3.l.a.175.6		16
16.3	odd	4		1152.3.m.f.415.3		16
16.5	even	4		576.3.m.c.271.6		16
16.11	odd	4	inner	144.3.m.c.91.4		16
16.13	even	4		1152.3.m.c.415.3		16
24.5	odd	2		384.3.l.a.223.7		16
24.11	even	2		384.3.l.b.223.3		16
48.5	odd	4		192.3.l.a.79.6		16
48.11	even	4		48.3.l.a.43.5	yes	16
48.29	odd	4		384.3.l.b.31.3		16
48.35	even	4		384.3.l.a.31.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.5	✓	16	3.2	odd	2
48.3.l.a.43.5	yes	16	48.11	even	4
144.3.m.c.19.4		16	1.1	even	1	trivial
144.3.m.c.91.4		16	16.11	odd	4	inner
192.3.l.a.79.6		16	48.5	odd	4
192.3.l.a.175.6		16	12.11	even	2
384.3.l.a.31.7		16	48.35	even	4
384.3.l.a.223.7		16	24.5	odd	2
384.3.l.b.31.3		16	48.29	odd	4
384.3.l.b.223.3		16	24.11	even	2
576.3.m.c.271.6		16	16.5	even	4
576.3.m.c.559.6		16	4.3	odd	2
1152.3.m.c.415.3		16	16.13	even	4
1152.3.m.c.991.3		16	8.3	odd	2
1152.3.m.f.415.3		16	16.3	odd	4
1152.3.m.f.991.3		16	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.455024 - 1.94755i) q^{2} +(-3.58591 + 1.77236i) q^{4} +(3.40572 - 3.40572i) q^{5} +12.1303 q^{7} +(5.08344 + 6.17727i) q^{8} +(-8.18251 - 5.08314i) q^{10} +(-9.81086 - 9.81086i) q^{11} +(-7.76859 - 7.76859i) q^{13} +(-5.51959 - 23.6244i) q^{14} +(9.71745 - 12.7111i) q^{16} -9.73087 q^{17} +(11.2823 - 11.2823i) q^{19} +(-6.17643 + 18.2488i) q^{20} +(-14.6430 + 23.5713i) q^{22} +20.2635 q^{23} +1.80207i q^{25} +(-11.5948 + 18.6646i) q^{26} +(-43.4982 + 21.4994i) q^{28} +(16.4069 + 16.4069i) q^{29} -26.3542i q^{31} +(-29.1771 - 13.1414i) q^{32} +(4.42778 + 18.9514i) q^{34} +(41.3125 - 41.3125i) q^{35} +(-23.7263 + 23.7263i) q^{37} +(-27.1066 - 16.8392i) q^{38} +(38.3509 + 3.72526i) q^{40} +24.7452i q^{41} +(29.8844 + 29.8844i) q^{43} +(52.5692 + 17.7924i) q^{44} +(-9.22036 - 39.4641i) q^{46} +31.3325i q^{47} +98.1448 q^{49} +(3.50963 - 0.819987i) q^{50} +(41.6262 + 14.0887i) q^{52} +(-36.8742 + 36.8742i) q^{53} -66.8262 q^{55} +(61.6638 + 74.9322i) q^{56} +(24.4877 - 39.4188i) q^{58} +(14.1325 + 14.1325i) q^{59} +(-42.5199 - 42.5199i) q^{61} +(-51.3260 + 11.9918i) q^{62} +(-12.3172 + 62.8036i) q^{64} -52.9153 q^{65} +(48.7789 - 48.7789i) q^{67} +(34.8940 - 17.2467i) q^{68} +(-99.2565 - 61.6601i) q^{70} -7.73935 q^{71} +85.4163i q^{73} +(57.0041 + 35.4121i) q^{74} +(-20.4610 + 60.4537i) q^{76} +(-119.009 - 119.009i) q^{77} +105.294i q^{79} +(-10.1954 - 76.3854i) q^{80} +(48.1926 - 11.2597i) q^{82} +(62.1229 - 62.1229i) q^{83} +(-33.1407 + 33.1407i) q^{85} +(44.6033 - 71.7996i) q^{86} +(10.7313 - 110.477i) q^{88} +127.172i q^{89} +(-94.2355 - 94.2355i) q^{91} +(-72.6629 + 35.9142i) q^{92} +(61.0215 - 14.2570i) q^{94} -76.8489i q^{95} -147.348 q^{97} +(-44.6582 - 191.142i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} + 12 q^{8} - 56 q^{10} - 32 q^{11} + 44 q^{14} + 32 q^{16} - 32 q^{19} - 80 q^{20} + 32 q^{22} + 128 q^{23} + 100 q^{26} - 120 q^{28} - 32 q^{29} - 160 q^{32} + 96 q^{34} - 96 q^{35} - 96 q^{37}+ \cdots + 440 q^{98}+O(q^{100}) \) 16 * q + 12 * q^4 + 12 * q^8 - 56 * q^10 - 32 * q^11 + 44 * q^14 + 32 * q^16 - 32 * q^19 - 80 * q^20 + 32 * q^22 + 128 * q^23 + 100 * q^26 - 120 * q^28 - 32 * q^29 - 160 * q^32 + 96 * q^34 - 96 * q^35 - 96 * q^37 - 168 * q^38 + 48 * q^40 + 160 * q^43 - 88 * q^44 + 136 * q^46 + 112 * q^49 + 236 * q^50 - 48 * q^52 + 160 * q^53 - 256 * q^55 + 224 * q^56 + 144 * q^58 + 128 * q^59 - 32 * q^61 + 276 * q^62 - 408 * q^64 + 32 * q^65 + 320 * q^67 + 448 * q^68 - 384 * q^70 - 512 * q^71 - 348 * q^74 + 72 * q^76 - 224 * q^77 - 552 * q^80 - 40 * q^82 + 160 * q^83 + 160 * q^85 - 528 * q^86 + 480 * q^88 - 480 * q^91 - 496 * q^92 + 312 * q^94 + 440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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