LMFDB - Embedded newform 252.9.bg.a.29.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(29,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.29");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bg (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.9
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-71.3437 - 38.3547i) q^{3} +(330.062 - 190.561i) q^{5} +(453.746 - 785.912i) q^{7} +(3618.84 + 5472.73i) q^{9} +O(q^{10})$	\(q+(-71.3437 - 38.3547i) q^{3} +(330.062 - 190.561i) q^{5} +(453.746 - 785.912i) q^{7} +(3618.84 + 5472.73i) q^{9} +(-9675.17 - 5585.96i) q^{11} +(-10779.7 - 18671.0i) q^{13} +(-30856.7 + 935.924i) q^{15} +97980.2i q^{17} +153991. q^{19} +(-62515.3 + 38666.5i) q^{21} +(307963. - 177802. i) q^{23} +(-122685. + 212497. i) q^{25} +(-48276.4 - 529244. i) q^{27} +(489663. + 282707. i) q^{29} +(47183.3 + 81723.8i) q^{31} +(476014. + 769611. i) q^{33} -345866. i q^{35} -2.30555e6 q^{37} +(52943.5 + 1.74551e6i) q^{39} +(-3.00171e6 + 1.73304e6i) q^{41} +(-2.40865e6 + 4.17191e6i) q^{43} +(2.23733e6 + 1.11673e6i) q^{45} +(829403. + 478856. i) q^{47} +(-411772. - 713209. i) q^{49} +(3.75800e6 - 6.99026e6i) q^{51} +831652. i q^{53} -4.25787e6 q^{55} +(-1.09863e7 - 5.90629e6i) q^{57} +(-1.19496e7 + 6.89910e6i) q^{59} +(8.64840e6 - 1.49795e7i) q^{61} +(5.94311e6 - 360857. i) q^{63} +(-7.11592e6 - 4.10838e6i) q^{65} +(1.78435e7 + 3.09059e7i) q^{67} +(-2.87908e7 + 873261. i) q^{69} -1.94134e6i q^{71} -9.04455e6 q^{73} +(1.69031e7 - 1.04548e7i) q^{75} +(-8.78015e6 + 5.06922e6i) q^{77} +(1.46148e6 - 2.53137e6i) q^{79} +(-1.68548e7 + 3.96098e7i) q^{81} +(7.10733e7 + 4.10342e7i) q^{83} +(1.86712e7 + 3.23395e7i) q^{85} +(-2.40912e7 - 3.89503e7i) q^{87} -7.94663e6i q^{89} -1.95650e7 q^{91} +(-231737. - 7.64018e6i) q^{93} +(5.08267e7 - 2.93448e7i) q^{95} +(8.46479e7 - 1.46615e8i) q^{97} +(-4.44242e6 - 7.31643e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	−71.3437	−	38.3547i	−0.880786	−	0.473515i
$3$	−71.3437	−	38.3547i	−0.880786	−	0.473515i
$4$		0			0
$5$	330.062	−	190.561i	0.528099	−	0.304898i	−0.212143	−	0.977239i	$-0.568044\pi$
$5$	330.062	−	190.561i	0.528099	−	0.304898i	0.740242	+	0.672341i	$0.234711\pi$
$6$		0			0
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$7$	453.746	−	785.912i	0.188982	−	0.327327i
$8$		0			0
$9$	3618.84	+	5472.73i	0.551568	+	0.834130i
$10$		0			0
$11$	−9675.17	−	5585.96i	−0.660827	−	0.381529i	0.131765	−	0.991281i	$-0.457936\pi$
$11$	−9675.17	−	5585.96i	−0.660827	−	0.381529i	−0.792592	+	0.609752i	$0.791269\pi$
$12$		0			0
$13$	−10779.7	−	18671.0i	−0.377427	−	0.653723i	0.613260	−	0.789881i	$-0.289858\pi$
$13$	−10779.7	−	18671.0i	−0.377427	−	0.653723i	−0.990687	+	0.136158i	$0.956524\pi$
$14$		0			0
$15$	−30856.7	+	935.924i	−0.609515	+	0.0184874i
$16$		0			0
$17$			97980.2i			1.17312i	0.809906	+	0.586560i	$0.199518\pi$
$17$			97980.2i			1.17312i	−0.809906	+	0.586560i	$0.800482\pi$
$18$		0			0
$19$	153991.			1.18163			0.590816	−	0.806806i	$-0.298806\pi$
$19$	153991.			1.18163			0.590816	+	0.806806i	$0.298806\pi$
$20$		0			0
$21$	−62515.3	+	38666.5i	−0.321447	+	0.198819i
$22$		0			0
$23$	307963.	−	177802.i	1.10049	−	0.635369i	0.164142	−	0.986437i	$-0.447515\pi$
$23$	307963.	−	177802.i	1.10049	−	0.635369i	0.936350	+	0.351067i	$0.114181\pi$
$24$		0			0
$25$	−122685.	+	212497.i	−0.314075	+	0.543993i
$26$		0			0
$27$	−48276.4	−	529244.i	−0.0908406	−	0.995865i
$28$		0			0
$29$	489663.	+	282707.i	0.692318	+	0.399710i	0.804480	−	0.593980i	$-0.202444\pi$
$29$	489663.	+	282707.i	0.692318	+	0.399710i	−0.112162	+	0.993690i	$0.535777\pi$
$30$		0			0
$31$	47183.3	+	81723.8i	0.0510906	+	0.0884916i	0.890440	−	0.455101i	$-0.150397\pi$
$31$	47183.3	+	81723.8i	0.0510906	+	0.0884916i	−0.839349	+	0.543593i	$0.817064\pi$
$32$		0			0
$33$	476014.	+	769611.i	0.401388	+	0.648957i
$34$		0			0
$35$		−	345866.i		−	0.230481i
$36$		0			0
$37$	−2.30555e6			−1.23018			−0.615090	−	0.788457i	$-0.710880\pi$
$37$	−2.30555e6			−1.23018			−0.615090	+	0.788457i	$0.710880\pi$
$38$		0			0
$39$	52943.5	+	1.74551e6i	0.0228852	+	0.754507i
$40$		0			0
$41$	−3.00171e6	+	1.73304e6i	−1.06227	+	0.613300i	−0.926058	−	0.377380i	$-0.876825\pi$
$41$	−3.00171e6	+	1.73304e6i	−1.06227	+	0.613300i	−0.136208	+	0.990680i	$0.543492\pi$
$42$		0			0
$43$	−2.40865e6	+	4.17191e6i	−0.704531	+	1.22028i	0.262329	+	0.964978i	$0.415509\pi$
$43$	−2.40865e6	+	4.17191e6i	−0.704531	+	1.22028i	−0.966860	+	0.255305i	$0.917824\pi$
$44$		0			0
$45$	2.23733e6	+	1.11673e6i	0.545607	+	0.272331i
$46$		0			0
$47$	829403.	+	478856.i	0.169971	+	0.0981326i	0.582572	−	0.812779i	$-0.302046\pi$
$47$	829403.	+	478856.i	0.169971	+	0.0981326i	−0.412601	+	0.910912i	$0.635380\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	3.75800e6	−	6.99026e6i	0.555490	−	1.03327i
$52$		0			0
$53$			831652.i			0.105399i	0.998610	+	0.0526997i	$0.0167826\pi$
$53$			831652.i			0.105399i	−0.998610	+	0.0526997i	$0.983217\pi$
$54$		0			0
$55$	−4.25787e6			−0.465309
$56$		0			0
$57$	−1.09863e7	−	5.90629e6i	−1.04076	−	0.559520i
$58$		0			0
$59$	−1.19496e7	+	6.89910e6i	−0.986155	+	0.569357i	−0.904123	−	0.427273i	$-0.859474\pi$
$59$	−1.19496e7	+	6.89910e6i	−0.986155	+	0.569357i	−0.0820322	+	0.996630i	$0.526141\pi$
$60$		0			0
$61$	8.64840e6	−	1.49795e7i	0.624621	−	1.08187i	−0.363994	−	0.931401i	$-0.618587\pi$
$61$	8.64840e6	−	1.49795e7i	0.624621	−	1.08187i	0.988614	−	0.150473i	$-0.0480797\pi$
$62$		0			0
$63$	5.94311e6	−	360857.i	0.377270	−	0.0229072i
$64$		0			0
$65$	−7.11592e6	−	4.10838e6i	−0.398637	−	0.230153i
$66$		0			0
$67$	1.78435e7	+	3.09059e7i	0.885485	+	1.53371i	0.845157	+	0.534519i	$0.179507\pi$
$67$	1.78435e7	+	3.09059e7i	0.885485	+	1.53371i	0.0403286	+	0.999186i	$0.487160\pi$
$68$		0			0
$69$	−2.87908e7	+	873261.i	−1.27015	+	0.0385254i
$70$		0			0
$71$		−	1.94134e6i		−	0.0763955i	−0.999270	−	0.0381978i	$-0.987838\pi$
$71$		−	1.94134e6i		−	0.0763955i	0.999270	−	0.0381978i	$-0.0121617\pi$
$72$		0			0
$73$	−9.04455e6			−0.318490			−0.159245	−	0.987239i	$-0.550906\pi$
$73$	−9.04455e6			−0.318490			−0.159245	+	0.987239i	$0.550906\pi$
$74$		0			0
$75$	1.69031e7	−	1.04548e7i	0.534221	−	0.330423i
$76$		0			0
$77$	−8.78015e6	+	5.06922e6i	−0.249769	+	0.144204i
$78$		0			0
$79$	1.46148e6	−	2.53137e6i	0.0375220	−	0.0649900i	−0.846655	−	0.532143i	$-0.821387\pi$
$79$	1.46148e6	−	2.53137e6i	0.0375220	−	0.0649900i	0.884177	+	0.467153i	$0.154720\pi$
$80$		0			0
$81$	−1.68548e7	+	3.96098e7i	−0.391546	+	0.920159i
$82$		0			0
$83$	7.10733e7	+	4.10342e7i	1.49759	+	0.864637i	0.999996	−	0.00277050i	$-0.000881878\pi$
$83$	7.10733e7	+	4.10342e7i	1.49759	+	0.864637i	0.497599	+	0.867407i	$0.334215\pi$
$84$		0			0
$85$	1.86712e7	+	3.23395e7i	0.357682	+	0.619523i
$86$		0			0
$87$	−2.40912e7	−	3.89503e7i	−0.420516	−	0.679882i
$88$		0			0
$89$		−	7.94663e6i		−	0.126655i	−0.997993	−	0.0633276i	$-0.979829\pi$
$89$		−	7.94663e6i		−	0.126655i	0.997993	−	0.0633276i	$-0.0201713\pi$
$90$		0			0
$91$	−1.95650e7			−0.285308
$92$		0			0
$93$	−231737.	−	7.64018e6i	−0.00309787	−	0.102134i
$94$		0			0
$95$	5.08267e7	−	2.93448e7i	0.624018	−	0.360277i
$96$		0			0
$97$	8.46479e7	−	1.46615e8i	0.956158	−	1.65611i	0.224461	−	0.974483i	$-0.427938\pi$
$97$	8.46479e7	−	1.46615e8i	0.956158	−	1.65611i	0.731697	−	0.681631i	$-0.238729\pi$
$98$		0			0
$99$	−4.44242e6	−	7.31643e7i	−0.0462465	−	0.761655i
$100$		0			0
$101$	1.20108e8	+	6.93446e7i	1.15422	+	0.666388i	0.949912	−	0.312519i	$-0.101173\pi$
$101$	1.20108e8	+	6.93446e7i	1.15422	+	0.666388i	0.204307	+	0.978907i	$0.434506\pi$
$102$		0			0
$103$	1.50115e7	+	2.60007e7i	0.133375	+	0.231013i	0.924976	−	0.380026i	$-0.124085\pi$
$103$	1.50115e7	+	2.60007e7i	0.133375	+	0.231013i	−0.791600	+	0.611039i	$0.790752\pi$
$104$		0			0
$105$	−1.32656e7	+	2.46753e7i	−0.109136	+	0.203005i
$106$		0			0
$107$		−	1.19162e8i		−	0.909082i	−0.890726	−	0.454541i	$-0.849803\pi$
$107$		−	1.19162e8i		−	0.909082i	0.890726	−	0.454541i	$-0.150197\pi$
$108$		0			0
$109$	1.50377e8			1.06531			0.532655	−	0.846332i	$-0.321194\pi$
$109$	1.50377e8			1.06531			0.532655	+	0.846332i	$0.321194\pi$
$110$		0			0
$111$	1.64487e8	+	8.84288e7i	1.08352	+	0.582508i
$112$		0			0
$113$	−2.37010e7	+	1.36838e7i	−0.145363	+	0.0839253i	−0.570917	−	0.821007i	$-0.693412\pi$
$113$	−2.37010e7	+	1.36838e7i	−0.145363	+	0.0839253i	0.425555	+	0.904933i	$0.360079\pi$
$114$		0			0
$115$	6.77645e7	−	1.17371e8i	0.387445	−	0.671075i
$116$		0			0
$117$	6.31712e7	−	1.26561e8i	0.337113	−	0.675395i
$118$		0			0
$119$	7.70038e7	+	4.44581e7i	0.383994	+	0.221699i
$120$		0			0
$121$	−4.47735e7	−	7.75499e7i	−0.208872	−	0.361776i
$122$		0			0
$123$	2.80623e8	−	8.51167e6i	1.22604	−	0.0371873i
$124$		0			0
$125$			2.42392e8i			0.992838i
$126$		0			0
$127$	2.47276e8			0.950533			0.475266	−	0.879842i	$-0.342352\pi$
$127$	2.47276e8			0.950533			0.475266	+	0.879842i	$0.342352\pi$
$128$		0			0
$129$	3.31854e8	−	2.05256e8i	1.19836	−	0.741203i
$130$		0			0
$131$	−4.29940e7	+	2.48226e7i	−0.145990	+	0.0842874i	−0.571216	−	0.820800i	$-0.693528\pi$
$131$	−4.29940e7	+	2.48226e7i	−0.145990	+	0.0842874i	0.425226	+	0.905087i	$0.360195\pi$
$132$		0			0
$133$	6.98731e7	−	1.21024e8i	0.223307	−	0.386780i
$134$		0			0
$135$	−1.16787e8	−	1.65483e8i	−0.351610	−	0.498218i
$136$		0			0
$137$	−3.97520e8	−	2.29508e8i	−1.12843	−	0.651502i	−0.184894	−	0.982759i	$-0.559194\pi$
$137$	−3.97520e8	−	2.29508e8i	−1.12843	−	0.651502i	−0.943541	+	0.331257i	$0.892527\pi$
$138$		0			0
$139$	3.02203e8	+	5.23430e8i	0.809541	+	1.40217i	0.913182	+	0.407552i	$0.133618\pi$
$139$	3.02203e8	+	5.23430e8i	0.809541	+	1.40217i	−0.103641	+	0.994615i	$0.533049\pi$
$140$		0			0
$141$	−4.08063e7	−	6.59748e7i	−0.103241	−	0.166917i
$142$		0			0
$143$			2.40860e8i			0.575997i
$144$		0			0
$145$	2.15492e8			0.487483
$146$		0			0
$147$	2.02238e6	+	6.66763e7i	0.00433105	+	0.142791i
$148$		0			0
$149$	6.10324e8	−	3.52371e8i	1.23827	−	0.714915i	0.269529	−	0.962992i	$-0.413132\pi$
$149$	6.10324e8	−	3.52371e8i	1.23827	−	0.714915i	0.968740	+	0.248077i	$0.0797985\pi$
$150$		0			0
$151$	−2.92551e8	+	5.06714e8i	−0.562722	+	0.974664i	0.434535	+	0.900655i	$0.356913\pi$
$151$	−2.92551e8	+	5.06714e8i	−0.562722	+	0.974664i	−0.997258	+	0.0740088i	$0.976421\pi$
$152$		0			0
$153$	−5.36219e8	+	3.54574e8i	−0.978535	+	0.647056i
$154$		0			0
$155$	3.11468e7	+	1.79826e7i	0.0539618	+	0.0311548i
$156$		0			0
$157$	−1.02080e8	−	1.76808e8i	−0.168013	−	0.291008i	0.769708	−	0.638396i	$-0.220402\pi$
$157$	−1.02080e8	−	1.76808e8i	−0.168013	−	0.291008i	−0.937721	+	0.347389i	$0.887068\pi$
$158$		0			0
$159$	3.18977e7	−	5.93331e7i	0.0499081	−	0.0928343i
$160$		0			0
$161$		−	3.22709e8i		−	0.480294i
$162$		0			0
$163$	−3.95411e8			−0.560143			−0.280071	−	0.959979i	$-0.590358\pi$
$163$	−3.95411e8			−0.560143			−0.280071	+	0.959979i	$0.590358\pi$
$164$		0			0
$165$	3.03772e8	+	1.63309e8i	0.409838	+	0.220331i
$166$		0			0
$167$	1.10489e9	−	6.37907e8i	1.42054	−	0.820147i	0.424191	−	0.905573i	$-0.360558\pi$
$167$	1.10489e9	−	6.37907e8i	1.42054	−	0.820147i	0.996345	+	0.0854259i	$0.0272251\pi$
$168$		0			0
$169$	1.75462e8	−	3.03909e8i	0.215098	−	0.372561i
$170$		0			0
$171$	5.57270e8	+	8.42753e8i	0.651750	+	0.985634i
$172$		0			0
$173$	2.49826e8	+	1.44237e8i	0.278903	+	0.161025i	0.632927	−	0.774212i	$-0.281853\pi$
$173$	2.49826e8	+	1.44237e8i	0.278903	+	0.161025i	−0.354023	+	0.935237i	$0.615187\pi$
$174$		0			0
$175$	1.11336e8	+	1.92840e8i	0.118709	+	0.205610i
$176$		0			0
$177$	1.11714e9	−	3.38843e7i	1.13819	−	0.0345228i
$178$		0			0
$179$		−	1.46978e9i		−	1.43166i	−0.698275	−	0.715829i	$-0.746049\pi$
$179$		−	1.46978e9i		−	1.43166i	0.698275	−	0.715829i	$-0.253951\pi$
$180$		0			0
$181$	9.99701e8			0.931442			0.465721	−	0.884932i	$-0.345795\pi$
$181$	9.99701e8			0.931442			0.465721	+	0.884932i	$0.345795\pi$
$182$		0			0
$183$	−1.19154e9	+	7.36983e8i	−1.06244	+	0.657133i
$184$		0			0
$185$	−7.60975e8	+	4.39349e8i	−0.649656	+	0.375079i
$186$		0			0
$187$	5.47314e8	−	9.47975e8i	0.447579	−	0.775230i
$188$		0			0
$189$	−4.37844e8	−	2.02201e8i	−0.343141	−	0.158466i
$190$		0			0
$191$	1.74539e9	+	1.00770e9i	1.31147	+	0.757179i	0.982340	−	0.187105i	$-0.0599104\pi$
$191$	1.74539e9	+	1.00770e9i	1.31147	+	0.757179i	0.329133	+	0.944284i	$0.393244\pi$
$192$		0			0
$193$	−2.33070e8	−	4.03690e8i	−0.167980	−	0.290950i	0.769729	−	0.638370i	$-0.220391\pi$
$193$	−2.33070e8	−	4.03690e8i	−0.167980	−	0.290950i	−0.937710	+	0.347420i	$0.887058\pi$
$194$		0			0
$195$	3.50100e8	+	5.66036e8i	0.242133	+	0.391476i
$196$		0			0
$197$			1.30433e9i			0.866008i	0.901392	+	0.433004i	$0.142546\pi$
$197$			1.30433e9i			0.866008i	−0.901392	+	0.433004i	$0.857454\pi$
$198$		0			0
$199$	−8.81613e8			−0.562168			−0.281084	−	0.959683i	$-0.590694\pi$
$199$	−8.81613e8			−0.562168			−0.281084	+	0.959683i	$0.590694\pi$
$200$		0			0
$201$	−8.76369e7	−	2.88932e9i	−0.0536911	−	1.77016i
$202$		0			0
$203$	4.44366e8	−	2.56555e8i	0.261672	−	0.151076i
$204$		0			0
$205$	−6.60500e8	+	1.14402e9i	−0.373988	+	0.647766i
$206$		0			0
$207$	2.08753e9	+	1.04196e9i	1.13698	+	0.567504i
$208$		0			0
$209$	−1.48989e9	−	8.60191e8i	−0.780855	−	0.450827i
$210$		0			0
$211$	1.05517e9	+	1.82760e9i	0.532343	+	0.922045i	0.999287	+	0.0377578i	$0.0120215\pi$
$211$	1.05517e9	+	1.82760e9i	0.532343	+	0.922045i	−0.466944	+	0.884287i	$0.654645\pi$
$212$		0			0
$213$	−7.44594e7	+	1.38502e8i	−0.0361744	+	0.0672881i
$214$		0			0
$215$			1.83598e9i			0.859240i
$216$		0			0
$217$	8.56370e7			0.0386209
$218$		0			0
$219$	6.45271e8	+	3.46901e8i	0.280521	+	0.150810i
$220$		0			0
$221$	1.82938e9	−	1.05620e9i	0.766895	−	0.442767i
$222$		0			0
$223$	1.65843e9	−	2.87248e9i	0.670621	−	1.16155i	−0.307107	−	0.951675i	$-0.599361\pi$
$223$	1.65843e9	−	2.87248e9i	0.670621	−	1.16155i	0.977728	−	0.209875i	$-0.0673058\pi$
$224$		0			0
$225$	−1.60692e9	+	9.75696e7i	−0.626995	+	0.0380701i
$226$		0			0
$227$	2.10570e9	+	1.21573e9i	0.793037	+	0.457860i	0.841030	−	0.540988i	$-0.181949\pi$
$227$	2.10570e9	+	1.21573e9i	0.793037	+	0.457860i	−0.0479939	+	0.998848i	$0.515283\pi$
$228$		0			0
$229$	1.46420e9	+	2.53607e9i	0.532425	+	0.922188i	0.999283	+	0.0378554i	$0.0120526\pi$
$229$	1.46420e9	+	2.53607e9i	0.532425	+	0.922188i	−0.466858	+	0.884332i	$0.654614\pi$
$230$		0			0
$231$	8.20836e8	−	2.48970e7i	0.288276	−	0.00874379i
$232$		0			0
$233$		−	1.88046e9i		−	0.638028i	−0.947750	−	0.319014i	$-0.896648\pi$
$233$		−	1.88046e9i		−	0.638028i	0.947750	−	0.319014i	$-0.103352\pi$
$234$		0			0
$235$	3.65005e8			0.119682
$236$		0			0
$237$	−2.01357e8	+	1.24542e8i	−0.0638226	+	0.0394751i
$238$		0			0
$239$	3.41920e9	−	1.97407e9i	1.04793	−	0.605023i	0.125861	−	0.992048i	$-0.459831\pi$
$239$	3.41920e9	−	1.97407e9i	1.04793	−	0.605023i	0.922069	+	0.387025i	$0.126497\pi$
$240$		0			0
$241$	−2.09508e9	+	3.62878e9i	−0.621058	+	1.07570i	0.368231	+	0.929734i	$0.379964\pi$
$241$	−2.09508e9	+	3.62878e9i	−0.621058	+	1.07570i	−0.989289	+	0.145970i	$0.953370\pi$
$242$		0			0
$243$	2.72170e9	−	2.17945e9i	0.780576	−	0.625060i
$244$		0			0
$245$	−2.71820e8	−	1.56935e8i	−0.0754426	−	0.0435568i
$246$		0			0
$247$	−1.65998e9	−	2.87517e9i	−0.445980	−	0.772459i
$248$		0			0
$249$	−3.49678e9	−	5.65353e9i	−0.909642	−	1.47069i
$250$		0			0
$251$		−	1.08553e9i		−	0.273493i	−0.990606	−	0.136746i	$-0.956335\pi$
$251$		−	1.08553e9i		−	0.273493i	0.990606	−	0.136746i	$-0.0436645\pi$
$252$		0			0
$253$	−3.97279e9			−0.969647
$254$		0			0
$255$	−9.17020e7	−	3.02335e9i	−0.0216879	−	0.715035i
$256$		0			0
$257$	1.97499e9	−	1.14026e9i	0.452723	−	0.261380i	−0.256257	−	0.966609i	$-0.582489\pi$
$257$	1.97499e9	−	1.14026e9i	0.452723	−	0.261380i	0.708980	+	0.705229i	$0.249156\pi$
$258$		0			0
$259$	−1.04614e9	+	1.81196e9i	−0.232482	+	0.402671i
$260$		0			0
$261$	2.24832e8	+	3.70287e9i	0.0484503	+	0.797951i
$262$		0			0
$263$	−2.26917e9	−	1.31011e9i	−0.474290	−	0.273832i	0.243744	−	0.969840i	$-0.421624\pi$
$263$	−2.26917e9	−	1.31011e9i	−0.474290	−	0.273832i	−0.718034	+	0.696008i	$0.754958\pi$
$264$		0			0
$265$	1.58481e8	+	2.74496e8i	0.0321360	+	0.0556613i
$266$		0			0
$267$	−3.04790e8	+	5.66942e8i	−0.0599731	+	0.111556i
$268$		0			0
$269$		−	3.97833e9i		−	0.759787i	−0.925030	−	0.379894i	$-0.875961\pi$
$269$		−	3.97833e9i		−	0.759787i	0.925030	−	0.379894i	$-0.124039\pi$
$270$		0			0
$271$	3.82295e9			0.708796			0.354398	−	0.935095i	$-0.384686\pi$
$271$	3.82295e9			0.708796			0.354398	+	0.935095i	$0.384686\pi$
$272$		0			0
$273$	1.39584e9	+	7.50408e8i	0.251295	+	0.135097i
$274$		0			0
$275$	2.37401e9	−	1.37063e9i	0.415098	−	0.239657i
$276$		0			0
$277$	2.95196e9	−	5.11294e9i	0.501407	−	0.868463i	−0.498591	−	0.866837i	$-0.666149\pi$
$277$	2.95196e9	−	5.11294e9i	0.501407	−	0.868463i	0.999999	−	0.00162580i	$-0.000517507\pi$
$278$		0			0
$279$	−2.76504e8	+	5.53966e8i	−0.0456335	+	0.0914254i
$280$		0			0
$281$	−1.47839e9	−	8.53547e8i	−0.237117	−	0.136900i	0.376734	−	0.926322i	$-0.377047\pi$
$281$	−1.47839e9	−	8.53547e8i	−0.237117	−	0.136900i	−0.613851	+	0.789422i	$0.710380\pi$
$282$		0			0
$283$	3.89530e9	+	6.74685e9i	0.607288	+	1.05185i	0.991685	+	0.128686i	$0.0410761\pi$
$283$	3.89530e9	+	6.74685e9i	0.607288	+	1.05185i	−0.384397	+	0.923168i	$0.625591\pi$
$284$		0			0
$285$	−4.75167e9	+	1.44124e8i	−0.720223	+	0.0218453i
$286$		0			0
$287$			3.14544e9i			0.463611i
$288$		0			0
$289$	−2.62436e9			−0.376211
$290$		0			0
$291$	−1.16624e10	+	7.21337e9i	−1.62636	+	1.00593i
$292$		0			0
$293$	−6.86770e9	+	3.96507e9i	−0.931839	+	0.537998i	−0.887393	−	0.461014i	$-0.847486\pi$
$293$	−6.86770e9	+	3.96507e9i	−0.931839	+	0.537998i	−0.0444463	+	0.999012i	$0.514152\pi$
$294$		0			0
$295$	−2.62940e9	+	4.55426e9i	−0.347191	+	0.601353i
$296$		0			0
$297$	−2.48925e9	+	5.39019e9i	−0.319921	+	0.692753i
$298$		0			0
$299$	−6.63949e9	−	3.83331e9i	−0.830711	−	0.479611i
$300$		0			0
$301$	2.18583e9	+	3.78598e9i	0.266288	+	0.461224i
$302$		0			0
$303$	−5.90928e9	−	9.55402e9i	−0.701075	−	1.13348i
$304$		0			0
$305$		−	6.59219e9i		−	0.761782i
$306$		0			0
$307$	−5.37784e9			−0.605416			−0.302708	−	0.953083i	$-0.597891\pi$
$307$	−5.37784e9			−0.605416			−0.302708	+	0.953083i	$0.597891\pi$
$308$		0			0
$309$	−7.37277e7	−	2.43075e9i	−0.00808718	−	0.266628i
$310$		0			0
$311$	−6.31947e9	+	3.64855e9i	−0.675522	+	0.390013i	−0.798166	−	0.602438i	$-0.794196\pi$
$311$	−6.31947e9	+	3.64855e9i	−0.675522	+	0.390013i	0.122644	+	0.992451i	$0.460863\pi$
$312$		0			0
$313$	−2.21887e9	+	3.84320e9i	−0.231182	+	0.400419i	−0.958156	−	0.286245i	$-0.907593\pi$
$313$	−2.21887e9	+	3.84320e9i	−0.231182	+	0.400419i	0.726974	+	0.686665i	$0.240926\pi$
$314$		0			0
$315$	1.89283e9	−	1.25163e9i	0.192251	−	0.127126i
$316$		0			0
$317$	6.60084e9	+	3.81100e9i	0.653676	+	0.377400i	0.789863	−	0.613283i	$-0.210152\pi$
$317$	6.60084e9	+	3.81100e9i	0.653676	+	0.377400i	−0.136187	+	0.990683i	$0.543485\pi$
$318$		0			0
$319$	−3.15839e9	−	5.47048e9i	−0.305002	−	0.528279i
$320$		0			0
$321$	−4.57042e9	+	8.50146e9i	−0.430463	+	0.800706i
$322$		0			0
$323$			1.50881e10i			1.38620i
$324$		0			0
$325$	5.29004e9			0.474161
$326$		0			0
$327$	−1.07285e10	−	5.76767e9i	−0.938310	−	0.504440i
$328$		0			0
$329$	7.52677e8	−	4.34558e8i	0.0642429	−	0.0370906i
$330$		0			0
$331$	−1.17329e9	+	2.03219e9i	−0.0977444	+	0.169298i	−0.910751	−	0.412957i	$-0.864496\pi$
$331$	−1.17329e9	+	2.03219e9i	−0.0977444	+	0.169298i	0.813006	+	0.582255i	$0.197829\pi$
$332$		0			0
$333$	−8.34343e9	−	1.26177e10i	−0.678528	−	1.02613i
$334$		0			0
$335$	1.17789e10	+	6.80056e9i	0.935247	+	0.539965i
$336$		0			0
$337$	7.60146e9	+	1.31661e10i	0.589356	+	1.02079i	0.994317	+	0.106460i	$0.0339518\pi$
$337$	7.60146e9	+	1.31661e10i	0.589356	+	1.02079i	−0.404961	+	0.914334i	$0.632715\pi$
$338$		0			0
$339$	2.21576e9	−	6.72067e7i	0.167773	−	0.00508878i
$340$		0			0
$341$		−	1.05426e9i		−	0.0779702i
$342$		0			0
$343$	−7.47359e8			−0.0539949
$344$		0			0
$345$	−9.33631e9	+	5.77463e9i	−0.659020	+	0.407613i
$346$		0			0
$347$	−6.95047e9	+	4.01285e9i	−0.479398	+	0.276780i	−0.720166	−	0.693802i	$-0.755934\pi$
$347$	−6.95047e9	+	4.01285e9i	−0.479398	+	0.276780i	0.240768	+	0.970583i	$0.422601\pi$
$348$		0			0
$349$	−6.52738e9	+	1.13057e10i	−0.439984	+	0.762075i	−0.997688	−	0.0679653i	$-0.978349\pi$
$349$	−6.52738e9	+	1.13057e10i	−0.439984	+	0.762075i	0.557704	+	0.830040i	$0.311683\pi$
$350$		0			0
$351$	−9.36109e9	+	6.60645e9i	−0.616734	+	0.435251i
$352$		0			0
$353$	−1.91250e10	−	1.10418e10i	−1.23169	−	0.711118i	−0.264310	−	0.964438i	$-0.585144\pi$
$353$	−1.91250e10	−	1.10418e10i	−1.23169	−	0.711118i	−0.967383	+	0.253320i	$0.918478\pi$
$354$		0			0
$355$	−3.69944e8	−	6.40761e8i	−0.0232928	−	0.0403444i
$356$		0			0
$357$	−3.78855e9	−	6.12526e9i	−0.233239	−	0.377096i
$358$		0			0
$359$		−	2.11225e10i		−	1.27165i	−0.771834	−	0.635824i	$-0.780661\pi$
$359$		−	2.11225e10i		−	1.27165i	0.771834	−	0.635824i	$-0.219339\pi$
$360$		0			0
$361$	6.72980e9			0.396254
$362$		0			0
$363$	2.19901e8	+	7.24997e9i	0.0126649	+	0.417551i
$364$		0			0
$365$	−2.98526e9	+	1.72354e9i	−0.168194	+	0.0971068i
$366$		0			0
$367$	−7.90952e9	+	1.36997e10i	−0.435999	+	0.755173i	−0.997377	−	0.0723871i	$-0.976938\pi$
$367$	−7.90952e9	+	1.36997e10i	−0.435999	+	0.755173i	0.561377	+	0.827560i	$0.310272\pi$
$368$		0			0
$369$	−2.03472e10	−	1.01560e10i	−1.09748	−	0.547792i
$370$		0			0
$371$	6.53605e8	+	3.77359e8i	0.0345000	+	0.0199186i
$372$		0			0
$373$	−6.48126e9	−	1.12259e10i	−0.334830	−	0.579942i	0.648622	−	0.761111i	$-0.275346\pi$
$373$	−6.48126e9	−	1.12259e10i	−0.334830	−	0.579942i	−0.983452	+	0.181168i	$0.942012\pi$
$374$		0			0
$375$	9.29687e9	−	1.72931e10i	0.470123	−	0.874478i
$376$		0			0
$377$		−	1.21900e10i		−	0.603445i
$378$		0			0
$379$	−3.77687e10			−1.83052			−0.915261	−	0.402862i	$-0.868015\pi$
$379$	−3.77687e10			−1.83052			−0.915261	+	0.402862i	$0.868015\pi$
$380$		0			0
$381$	−1.76416e10	−	9.48419e9i	−0.837216	−	0.450091i
$382$		0			0
$383$	2.94216e10	−	1.69866e10i	1.36732	−	0.789424i	0.376737	−	0.926320i	$-0.377046\pi$
$383$	2.94216e10	−	1.69866e10i	1.36732	−	0.789424i	0.990585	+	0.136896i	$0.0437126\pi$
$384$		0			0
$385$	−1.93199e9	+	3.34631e9i	−0.0879352	+	0.152308i
$386$		0			0
$387$	−3.15482e10	+	1.91556e9i	−1.40647	+	0.0853988i
$388$		0			0
$389$	1.07345e10	+	6.19757e9i	0.468796	+	0.270659i	0.715735	−	0.698371i	$-0.246092\pi$
$389$	1.07345e10	+	6.19757e9i	0.468796	+	0.270659i	−0.246940	+	0.969031i	$0.579425\pi$
$390$		0			0
$391$	1.74211e10	+	3.01743e10i	0.745365	+	1.29101i
$392$		0			0
$393$	4.01942e9	−	1.21914e8i	0.168497	−	0.00511074i
$394$		0			0
$395$		−	1.11401e9i		−	0.0457615i
$396$		0			0
$397$	−3.23769e10			−1.30339			−0.651693	−	0.758483i	$-0.725941\pi$
$397$	−3.23769e10			−1.30339			−0.651693	+	0.758483i	$0.725941\pi$
$398$		0			0
$399$	−9.62682e9	+	5.95431e9i	−0.379832	+	0.234931i
$400$		0			0
$401$	8.25544e9	−	4.76628e9i	0.319274	−	0.184333i	−0.331795	−	0.943351i	$-0.607654\pi$
$401$	8.25544e9	−	4.76628e9i	0.319274	−	0.184333i	0.651069	+	0.759019i	$0.274321\pi$
$402$		0			0
$403$	1.01724e9	−	1.76192e9i	0.0385660	−	0.0667982i
$404$		0			0
$405$	1.98498e9	+	1.62855e10i	0.0737797	+	0.605316i
$406$		0			0
$407$	2.23066e10	+	1.28787e10i	0.812936	+	0.469349i
$408$		0			0
$409$	2.40162e10	+	4.15973e10i	0.858246	+	1.48653i	0.873601	+	0.486643i	$0.161779\pi$
$409$	2.40162e10	+	4.15973e10i	0.858246	+	1.48653i	−0.0153552	+	0.999882i	$0.504888\pi$
$410$		0			0
$411$	1.95578e10	+	3.16207e10i	0.685414	+	1.10816i
$412$		0			0
$413$			1.25218e10i			0.430393i
$414$		0			0
$415$	3.12781e10			1.05450
$416$		0			0
$417$	−1.48424e9	−	4.89343e10i	−0.0490863	−	1.61834i
$418$		0			0
$419$	1.13736e10	−	6.56658e9i	0.369015	−	0.213051i	−0.304013	−	0.952668i	$-0.598327\pi$
$419$	1.13736e10	−	6.56658e9i	0.369015	−	0.213051i	0.673028	+	0.739617i	$0.264993\pi$
$420$		0			0
$421$	1.86522e10	−	3.23065e10i	0.593746	−	1.02840i	−0.399976	−	0.916526i	$-0.630982\pi$
$421$	1.86522e10	−	3.23065e10i	0.593746	−	1.02840i	0.993722	−	0.111873i	$-0.0356851\pi$
$422$		0			0
$423$	3.80826e8	+	6.27200e9i	0.0118950	+	0.195904i
$424$		0			0
$425$	−2.08205e10	−	1.20207e10i	−0.638169	−	0.368447i
$426$		0			0
$427$	−7.84836e9	−	1.35938e10i	−0.236084	−	0.408910i
$428$		0			0
$429$	9.23810e9	−	1.71838e10i	0.272743	−	0.507330i
$430$		0			0
$431$		−	2.81087e10i		−	0.814575i	−0.913300	−	0.407288i	$-0.866475\pi$
$431$		−	2.81087e10i		−	0.814575i	0.913300	−	0.407288i	$-0.133525\pi$
$432$		0			0
$433$	−1.41184e10			−0.401637			−0.200818	−	0.979628i	$-0.564360\pi$
$433$	−1.41184e10			−0.401637			−0.200818	+	0.979628i	$0.564360\pi$
$434$		0			0
$435$	−1.53740e10	−	8.26513e9i	−0.429368	−	0.230830i
$436$		0			0
$437$	4.74236e10	−	2.73800e10i	1.30038	−	0.750773i
$438$		0			0
$439$	−1.36919e10	+	2.37150e10i	−0.368641	+	0.638506i	−0.989353	−	0.145533i	$-0.953510\pi$
$439$	−1.36919e10	+	2.37150e10i	−0.368641	+	0.638506i	0.620712	+	0.784039i	$0.286844\pi$
$440$		0			0
$441$	2.41306e9	−	4.83450e9i	0.0637991	−	0.127820i
$442$		0			0
$443$	1.94948e10	+	1.12553e10i	0.506179	+	0.292243i	0.731262	−	0.682097i	$-0.238932\pi$
$443$	1.94948e10	+	1.12553e10i	0.506179	+	0.292243i	−0.225082	+	0.974340i	$0.572265\pi$
$444$		0			0
$445$	−1.51432e9	−	2.62288e9i	−0.0386169	−	0.0668864i
$446$		0			0
$447$	−5.70578e10	+	1.73064e9i	−1.42917	+	0.0433487i
$448$		0			0
$449$		−	6.38016e10i		−	1.56981i	−0.619618	−	0.784903i	$-0.712713\pi$
$449$		−	6.38016e10i		−	1.56981i	0.619618	−	0.784903i	$-0.287287\pi$
$450$		0			0
$451$	3.87228e10			0.935966
$452$		0			0
$453$	4.03065e10	−	2.49301e10i	0.957155	−	0.592013i
$454$		0			0
$455$	−6.45765e9	+	3.72832e9i	−0.150671	+	0.0869898i
$456$		0			0
$457$	−2.97707e10	+	5.15643e10i	−0.682533	+	1.18218i	0.291672	+	0.956518i	$0.405788\pi$
$457$	−2.97707e10	+	5.15643e10i	−0.682533	+	1.18218i	−0.974205	+	0.225663i	$0.927545\pi$
$458$		0			0
$459$	5.18554e10	−	4.73013e9i	1.16827	−	0.106567i
$460$		0			0
$461$	6.17704e10	+	3.56632e10i	1.36766	+	0.789616i	0.990628	−	0.136586i	$-0.0436130\pi$
$461$	6.17704e10	+	3.56632e10i	1.36766	+	0.789616i	0.377027	+	0.926202i	$0.376946\pi$
$462$		0			0
$463$	3.12824e10	+	5.41827e10i	0.680732	+	1.17906i	0.974758	+	0.223265i	$0.0716715\pi$
$463$	3.12824e10	+	5.41827e10i	0.680732	+	1.17906i	−0.294026	+	0.955797i	$0.594995\pi$
$464$		0			0
$465$	−1.53241e9	−	2.47757e9i	−0.0327765	−	0.0529924i
$466$		0			0
$467$		−	7.25796e10i		−	1.52597i	−0.646414	−	0.762987i	$-0.723732\pi$
$467$		−	7.25796e10i		−	1.52597i	0.646414	−	0.762987i	$-0.276268\pi$
$468$		0			0
$469$	3.23857e10			0.669364
$470$		0			0
$471$	5.01359e8	+	1.65294e10i	0.0101874	+	0.335872i
$472$		0			0
$473$	4.66082e10	−	2.69093e10i	0.931147	−	0.537598i
$474$		0			0
$475$	−1.88925e10	+	3.27228e10i	−0.371121	+	0.642800i
$476$		0			0
$477$	−4.55140e9	+	3.00961e9i	−0.0879168	+	0.0581349i
$478$		0			0
$479$	1.03735e10	+	5.98912e9i	0.197052	+	0.113768i	0.595280	−	0.803518i	$-0.297041\pi$
$479$	1.03735e10	+	5.98912e9i	0.197052	+	0.113768i	−0.398227	+	0.917287i	$0.630375\pi$
$480$		0			0
$481$	2.48532e10	+	4.30469e10i	0.464303	+	0.804196i
$482$		0			0
$483$	−1.23774e10	+	2.30232e10i	−0.227426	+	0.423036i
$484$		0			0
$485$		−	6.45224e10i		−	1.16612i
$486$		0			0
$487$	−9.64760e9			−0.171516			−0.0857578	−	0.996316i	$-0.527331\pi$
$487$	−9.64760e9			−0.171516			−0.0857578	+	0.996316i	$0.527331\pi$
$488$		0			0
$489$	2.82101e10	+	1.51659e10i	0.493366	+	0.265236i
$490$		0			0
$491$	7.88890e10	−	4.55466e10i	1.35735	−	0.783664i	0.368081	−	0.929794i	$-0.380015\pi$
$491$	7.88890e10	−	4.55466e10i	1.35735	−	0.783664i	0.989265	+	0.146130i	$0.0466817\pi$
$492$		0			0
$493$	−2.76997e10	+	4.79773e10i	−0.468908	+	0.812172i
$494$		0			0
$495$	−1.54085e10	−	2.33022e10i	−0.256650	−	0.388128i
$496$		0			0
$497$	−1.52572e9	−	8.80876e8i	−0.0250063	−	0.0144374i
$498$		0			0
$499$	2.28384e10	+	3.95572e10i	0.368352	+	0.638005i	0.989308	−	0.145841i	$-0.0465887\pi$
$499$	2.28384e10	+	3.95572e10i	0.368352	+	0.638005i	−0.620956	+	0.783845i	$0.713255\pi$
$500$		0			0
$501$	−1.03293e11	+	3.13302e9i	−1.63954	+	0.0497294i
$502$		0			0
$503$			5.71568e10i			0.892885i	0.894812	+	0.446443i	$0.147309\pi$
$503$			5.71568e10i			0.892885i	−0.894812	+	0.446443i	$0.852691\pi$
$504$		0			0
$505$	5.28576e10			0.812721
$506$		0			0
$507$	−2.41744e10	+	1.49522e10i	−0.365868	+	0.226294i
$508$		0			0
$509$	−4.73158e10	+	2.73178e10i	−0.704913	+	0.406982i	−0.809175	−	0.587568i	$-0.800085\pi$
$509$	−4.73158e10	+	2.73178e10i	−0.704913	+	0.406982i	0.104262	+	0.994550i	$0.466752\pi$
$510$		0			0
$511$	−4.10393e9	+	7.10822e9i	−0.0601889	+	0.104250i
$512$		0			0
$513$	−7.43416e9	−	8.14990e10i	−0.107340	−	1.17675i
$514$		0			0
$515$	9.90945e9	+	5.72122e9i	0.140871	+	0.0813317i
$516$		0			0
$517$	−5.34974e9	−	9.26603e9i	−0.0748809	−	0.129697i
$518$		0			0
$519$	−1.22914e10	−	1.98724e10i	−0.169407	−	0.273893i
$520$		0			0
$521$			8.21044e10i			1.11433i	0.830400	+	0.557167i	$0.188112\pi$
$521$			8.21044e10i			1.11433i	−0.830400	+	0.557167i	$0.811888\pi$
$522$		0			0
$523$	−1.58314e8			−0.00211598			−0.00105799	−	0.999999i	$-0.500337\pi$
$523$	−1.58314e8			−0.00211598			−0.00105799	+	0.999999i	$0.500337\pi$
$524$		0			0
$525$	−5.46818e8	−	1.80282e10i	−0.00719789	−	0.237309i
$526$		0			0
$527$	−8.00732e9	+	4.62303e9i	−0.103811	+	0.0599355i
$528$		0			0
$529$	2.40719e10	−	4.16938e10i	0.307389	−	0.532413i
$530$		0			0
$531$	−8.10005e10	−	4.04301e10i	−1.01885	−	0.508542i
$532$		0			0
$533$	6.47150e10	+	3.73632e10i	0.801856	+	0.462952i
$534$		0			0
$535$	−2.27077e10	−	3.93308e10i	−0.277177	−	0.480085i
$536$		0			0
$537$	−5.63729e10	+	1.04859e11i	−0.677911	+	1.26098i
$538$		0			0
$539$			9.20056e9i			0.109008i
$540$		0			0
$541$	3.09192e10			0.360943			0.180472	−	0.983580i	$-0.442238\pi$
$541$	3.09192e10			0.360943			0.180472	+	0.983580i	$0.442238\pi$
$542$		0			0
$543$	−7.13223e10	−	3.83432e10i	−0.820401	−	0.441051i
$544$		0			0
$545$	4.96337e10	−	2.86560e10i	0.562589	−	0.324811i
$546$		0			0
$547$	1.26698e8	−	2.19447e8i	0.00141520	−	0.00245120i	−0.865317	−	0.501225i	$-0.832883\pi$
$547$	1.26698e8	−	2.19447e8i	0.00141520	−	0.00245120i	0.866732	+	0.498774i	$0.166216\pi$
$548$		0			0
$549$	1.13276e11	−	6.87792e9i	1.24694	−	0.0757126i
$550$		0			0
$551$	7.54040e10	+	4.35345e10i	0.818065	+	0.472310i
$552$		0			0
$553$	−1.32629e9	−	2.29720e9i	−0.0141820	−	0.0245639i
$554$		0			0
$555$	7.11418e10	−	2.15782e9i	0.749813	−	0.0227428i
$556$		0			0
$557$		−	1.49513e11i		−	1.55331i	−0.629927	−	0.776654i	$-0.716915\pi$
$557$		−	1.49513e11i		−	1.55331i	0.629927	−	0.776654i	$-0.283085\pi$
$558$		0			0
$559$	1.03858e11			1.06364
$560$		0			0
$561$	−7.54066e10	+	4.66400e10i	−0.761304	+	0.470876i
$562$		0			0
$563$	1.24232e11	−	7.17255e10i	1.23652	−	0.713904i	0.268138	−	0.963381i	$-0.413592\pi$
$563$	1.24232e11	−	7.17255e10i	1.23652	−	0.713904i	0.968381	+	0.249476i	$0.0802584\pi$
$564$		0			0
$565$	−5.21520e9	+	9.03299e9i	−0.0511773	+	0.0886416i
$566$		0			0
$567$	2.34820e10	+	3.12192e10i	0.227197	+	0.302057i
$568$		0			0
$569$	−8.79999e10	−	5.08068e10i	−0.839524	−	0.484700i	0.0175781	−	0.999845i	$-0.494404\pi$
$569$	−8.79999e10	−	5.08068e10i	−0.839524	−	0.484700i	−0.857103	+	0.515146i	$0.827738\pi$
$570$		0			0
$571$	5.68425e10	+	9.84542e10i	0.534723	+	0.926168i	0.999177	+	0.0405703i	$0.0129175\pi$
$571$	5.68425e10	+	9.84542e10i	0.534723	+	0.926168i	−0.464453	+	0.885598i	$0.653749\pi$
$572$		0			0
$573$	−8.58725e10	−	1.38837e11i	−0.796591	−	1.28791i
$574$		0			0
$575$			8.72550e10i			0.798214i
$576$		0			0
$577$	−8.29985e10			−0.748802			−0.374401	−	0.927267i	$-0.622152\pi$
$577$	−8.29985e10			−0.748802			−0.374401	+	0.927267i	$0.622152\pi$
$578$		0			0
$579$	1.14471e9	+	3.77401e10i	0.0101854	+	0.335806i
$580$		0			0
$581$	6.44985e10	−	3.72382e10i	0.566038	−	0.326802i
$582$		0			0
$583$	4.64558e9	−	8.04637e9i	0.0402129	−	0.0696508i
$584$		0			0
$585$	−3.26732e9	−	5.38111e10i	−0.0278977	−	0.459460i
$586$		0			0
$587$	−1.09685e9	−	6.33266e8i	−0.00923835	−	0.00533376i	0.495374	−	0.868680i	$-0.335031\pi$
$587$	−1.09685e9	−	6.33266e8i	−0.00923835	−	0.00533376i	−0.504612	+	0.863346i	$0.668364\pi$
$588$		0			0
$589$	7.26582e9	+	1.25848e10i	0.0603703	+	0.104564i
$590$		0			0
$591$	5.00271e10	−	9.30555e10i	0.410067	−	0.762767i
$592$		0			0
$593$			1.09759e11i			0.887610i	0.896123	+	0.443805i	$0.146372\pi$
$593$			1.09759e11i			0.887610i	−0.896123	+	0.443805i	$0.853628\pi$
$594$		0			0
$595$	3.38880e10			0.270382
$596$		0			0
$597$	6.28975e10	+	3.38140e10i	0.495149	+	0.266195i
$598$		0			0
$599$	1.48946e11	−	8.59940e10i	1.15697	−	0.667976i	0.206393	−	0.978469i	$-0.433827\pi$
$599$	1.48946e11	−	8.59940e10i	1.15697	−	0.667976i	0.950576	+	0.310493i	$0.100494\pi$
$600$		0			0
$601$	−3.50014e10	+	6.06241e10i	−0.268279	+	0.464673i	−0.968418	−	0.249334i	$-0.919788\pi$
$601$	−3.50014e10	+	6.06241e10i	−0.268279	+	0.464673i	0.700138	+	0.714007i	$0.253122\pi$
$602$		0			0
$603$	−1.04567e11	+	2.09496e11i	−0.790904	+	1.58455i
$604$		0			0
$605$	−2.95560e10	−	1.70642e10i	−0.220609	−	0.127369i
$606$		0			0
$607$	8.03179e10	+	1.39115e11i	0.591640	+	1.02475i	0.994012	+	0.109274i	$0.0348527\pi$
$607$	8.03179e10	+	1.39115e11i	0.591640	+	1.02475i	−0.402371	+	0.915477i	$0.631814\pi$
$608$		0			0
$609$	−4.15428e10	+	1.26005e9i	−0.302013	+	0.00916046i
$610$		0			0
$611$		−	2.06477e10i		−	0.148152i
$612$		0			0
$613$	6.60131e10			0.467507			0.233754	−	0.972296i	$-0.424899\pi$
$613$	6.60131e10			0.467507			0.233754	+	0.972296i	$0.424899\pi$
$614$		0			0
$615$	9.10010e10	−	5.62853e10i	0.636130	−	0.393454i
$616$		0			0
$617$	−1.70486e11	+	9.84300e10i	−1.17638	+	0.679183i	−0.955174	−	0.296044i	$-0.904332\pi$
$617$	−1.70486e11	+	9.84300e10i	−1.17638	+	0.679183i	−0.221205	+	0.975227i	$0.570999\pi$
$618$		0			0
$619$	−5.60627e10	+	9.71035e10i	−0.381866	+	0.661412i	−0.991329	−	0.131403i	$-0.958052\pi$
$619$	−5.60627e10	+	9.71035e10i	−0.381866	+	0.661412i	0.609463	+	0.792815i	$0.291385\pi$
$620$		0			0
$621$	−1.08968e11	−	1.54404e11i	−0.732712	−	1.03822i
$622$		0			0
$623$	−6.24535e9	−	3.60575e9i	−0.0414576	−	0.0239356i
$624$		0			0
$625$	−1.73346e9	−	3.00244e9i	−0.0113604	−	0.0196768i
$626$		0			0
$627$	7.33021e10	+	1.18514e11i	0.474293	+	0.766828i
$628$		0			0
$629$		−	2.25899e11i		−	1.44315i
$630$		0			0
$631$	1.32766e11			0.837471			0.418735	−	0.908108i	$-0.362473\pi$
$631$	1.32766e11			0.837471			0.418735	+	0.908108i	$0.362473\pi$
$632$		0			0
$633$	−5.18236e9	−	1.70858e11i	−0.0322784	−	1.06420i
$634$		0			0
$635$	8.16163e10	−	4.71212e10i	0.501975	−	0.289815i
$636$		0			0
$637$	−8.87754e9	+	1.53763e10i	−0.0539181	+	0.0933889i
$638$		0			0
$639$	1.06244e10	−	7.02539e9i	0.0637238	−	0.0421373i
$640$		0			0
$641$	1.40042e11	+	8.08532e10i	0.829518	+	0.478922i	0.853687	−	0.520786i	$-0.174361\pi$
$641$	1.40042e11	+	8.08532e10i	0.829518	+	0.478922i	−0.0241698	+	0.999708i	$0.507694\pi$
$642$		0			0
$643$	6.48177e10	+	1.12268e11i	0.379184	+	0.656766i	0.990944	−	0.134278i	$-0.0428714\pi$
$643$	6.48177e10	+	1.12268e11i	0.379184	+	0.656766i	−0.611760	+	0.791044i	$0.709538\pi$
$644$		0			0
$645$	7.04185e10	−	1.30986e11i	0.406863	−	0.756807i
$646$		0			0
$647$		−	3.31053e11i		−	1.88921i	−0.328206	−	0.944606i	$-0.606444\pi$
$647$		−	3.31053e11i		−	1.88921i	0.328206	−	0.944606i	$-0.393556\pi$
$648$		0			0
$649$	1.54153e11			0.868904
$650$		0			0
$651$	−6.10965e9	−	3.28458e9i	−0.0340167	−	0.0182876i
$652$		0			0
$653$	3.02907e10	−	1.74883e10i	0.166593	−	0.0961825i	−0.414385	−	0.910101i	$-0.636003\pi$
$653$	3.02907e10	−	1.74883e10i	0.166593	−	0.0961825i	0.580978	+	0.813919i	$0.302670\pi$
$654$		0			0
$655$	−9.46045e9	+	1.63860e10i	−0.0513981	+	0.0890240i
$656$		0			0
$657$	−3.27308e10	−	4.94983e10i	−0.175669	−	0.265662i
$658$		0			0
$659$	−2.21185e10	−	1.27701e10i	−0.117277	−	0.0677101i	0.440214	−	0.897893i	$-0.354903\pi$
$659$	−2.21185e10	−	1.27701e10i	−0.117277	−	0.0677101i	−0.557491	+	0.830183i	$0.688236\pi$
$660$		0			0
$661$	1.62613e10	+	2.81654e10i	0.0851822	+	0.147540i	0.905469	−	0.424413i	$-0.139519\pi$
$661$	1.62613e10	+	2.81654e10i	0.0851822	+	0.147540i	−0.820287	+	0.571953i	$0.806186\pi$
$662$		0			0
$663$	−1.71025e11	+	5.18741e9i	−0.885127	+	0.0268471i
$664$		0			0
$665$		−	5.32604e10i		−	0.272344i
$666$		0			0
$667$	2.01064e11			1.01585
$668$		0			0
$669$	−2.28492e11	+	1.41325e11i	−1.14069	+	0.705528i
$670$		0			0
$671$	−1.67349e11	+	9.66193e10i	−0.825533	+	0.476621i
$672$		0			0
$673$	−1.46350e10	+	2.53486e10i	−0.0713401	+	0.123565i	−0.899489	−	0.436944i	$-0.856061\pi$
$673$	−1.46350e10	+	2.53486e10i	−0.0713401	+	0.123565i	0.828149	+	0.560508i	$0.189394\pi$
$674$		0			0
$675$	1.18386e11	+	5.46719e10i	0.570275	+	0.263359i
$676$		0			0
$677$	4.04717e10	+	2.33664e10i	0.192662	+	0.111234i	0.593228	−	0.805034i	$-0.297853\pi$
$677$	4.04717e10	+	2.33664e10i	0.192662	+	0.111234i	−0.400566	+	0.916268i	$0.631186\pi$
$678$		0			0
$679$	−7.68174e10	−	1.33052e11i	−0.361394	−	0.625952i
$680$		0			0
$681$	−1.03600e11	−	1.67498e11i	−0.481692	−	0.778791i
$682$		0			0
$683$			3.23544e11i			1.48679i	0.668851	+	0.743396i	$0.266786\pi$
$683$			3.23544e11i			1.48679i	−0.668851	+	0.743396i	$0.733214\pi$
$684$		0			0
$685$	−1.74941e11			−0.794566
$686$		0			0
$687$	−7.19129e9	−	2.37092e11i	−0.0322835	−	1.06436i
$688$		0			0
$689$	1.55277e10	−	8.96495e9i	0.0689019	−	0.0397806i
$690$		0			0
$691$	−1.62839e11	+	2.82046e11i	−0.714245	+	1.23711i	0.249005	+	0.968502i	$0.419896\pi$
$691$	−1.62839e11	+	2.82046e11i	−0.714245	+	1.23711i	−0.963250	+	0.268606i	$0.913437\pi$
$692$		0			0
$693$	−5.95164e10	−	2.97067e10i	−0.258050	−	0.128802i
$694$		0			0
$695$	1.99491e11	+	1.15176e11i	0.855035	+	0.493655i
$696$		0			0
$697$	−1.69803e11	−	2.94108e11i	−0.719475	−	1.24617i
$698$		0			0
$699$	−7.21243e10	+	1.34159e11i	−0.302115	+	0.561966i
$700$		0			0
$701$		−	3.54750e10i		−	0.146910i	−0.997299	−	0.0734549i	$-0.976598\pi$
$701$		−	3.54750e10i		−	0.146910i	0.997299	−	0.0734549i	$-0.0234025\pi$
$702$		0			0
$703$	−3.55036e11			−1.45362
$704$		0			0
$705$	−2.60408e10	−	1.39997e10i	−0.105414	−	0.0566710i
$706$		0			0
$707$	1.08998e11	−	6.29298e10i	0.436254	−	0.251871i
$708$		0			0
$709$	−1.81412e11	+	3.14214e11i	−0.717927	+	1.24349i	0.243892	+	0.969802i	$0.421576\pi$
$709$	−1.81412e11	+	3.14214e11i	−0.717927	+	1.24349i	−0.961820	+	0.273684i	$0.911758\pi$
$710$		0			0
$711$	1.91424e10	−	1.16229e9i	0.0749060	−	0.00454818i
$712$		0			0
$713$	2.90614e10	+	1.67786e10i	0.112450	+	0.0649229i
$714$		0			0
$715$	4.58985e10	+	7.94986e10i	0.175620	+	0.304183i
$716$		0			0
$717$	−3.19653e11	+	9.69549e9i	−1.20949	+	0.0366854i
$718$		0			0
$719$			2.40396e11i			0.899523i	0.893149	+	0.449761i	$0.148491\pi$
$719$			2.40396e11i			0.899523i	−0.893149	+	0.449761i	$0.851509\pi$
$720$		0			0
$721$	2.72457e10			0.100822
$722$		0			0
$723$	2.88651e11	−	1.78535e11i	1.05638	−	0.653385i
$724$		0			0
$725$	−1.20149e11	+	6.93681e10i	−0.434879	+	0.251078i
$726$		0			0
$727$	3.03544e10	−	5.25754e10i	0.108664	−	0.188211i	−0.806565	−	0.591145i	$-0.798676\pi$
$727$	3.03544e10	−	5.25754e10i	0.108664	−	0.188211i	0.915229	+	0.402934i	$0.132009\pi$
$728$		0			0
$729$	−2.77768e11	+	5.11000e10i	−0.983496	+	0.180930i
$730$		0			0
$731$	−4.08764e11	−	2.36000e11i	−1.43154	−	0.826500i
$732$		0			0
$733$	−2.46897e11	−	4.27639e11i	−0.855265	−	1.48136i	−0.876399	−	0.481585i	$-0.840061\pi$
$733$	−2.46897e11	−	4.27639e11i	−0.855265	−	1.48136i	0.0211346	−	0.999777i	$-0.493272\pi$
$734$		0			0
$735$	1.33734e10	+	2.16219e10i	0.0458240	+	0.0740874i
$736$		0			0
$737$		−	3.98693e11i		−	1.35135i
$738$		0			0
$739$	−1.31092e11			−0.439539			−0.219770	−	0.975552i	$-0.570531\pi$
$739$	−1.31092e11			−0.439539			−0.219770	+	0.975552i	$0.570531\pi$
$740$		0			0
$741$	8.15284e9	+	2.68793e11i	0.0270418	+	0.891549i
$742$		0			0
$743$	−4.62365e10	+	2.66947e10i	−0.151715	+	0.0875930i	−0.573936	−	0.818900i	$-0.694584\pi$
$743$	−4.62365e10	+	2.66947e10i	−0.151715	+	0.0875930i	0.422220	+	0.906493i	$0.361251\pi$
$744$		0			0
$745$	1.34296e11	−	2.32608e11i	0.435952	−	0.755092i
$746$		0			0
$747$	3.26338e10	+	5.37461e11i	0.104806	+	1.72609i
$748$		0			0
$749$	−9.36509e10	−	5.40693e10i	−0.297567	−	0.171800i
$750$		0			0
$751$	−1.00303e11	−	1.73730e11i	−0.315321	−	0.546152i	0.664184	−	0.747569i	$-0.268779\pi$
$751$	−1.00303e11	−	1.73730e11i	−0.315321	−	0.546152i	−0.979506	+	0.201416i	$0.935446\pi$
$752$		0			0
$753$	−4.16350e10	+	7.74455e10i	−0.129503	+	0.240888i
$754$		0			0
$755$			2.22996e11i			0.686291i
$756$		0			0
$757$	−5.11424e11			−1.55739			−0.778695	−	0.627402i	$-0.784118\pi$
$757$	−5.11424e11			−1.55739			−0.778695	+	0.627402i	$0.784118\pi$
$758$		0			0
$759$	2.83433e11	+	1.52375e11i	0.854051	+	0.459142i
$760$		0			0
$761$	−4.59246e11	+	2.65146e11i	−1.36933	+	0.790581i	−0.990842	−	0.135027i	$-0.956888\pi$
$761$	−4.59246e11	+	2.65146e11i	−1.36933	+	0.790581i	−0.378484	+	0.925608i	$0.623554\pi$
$762$		0			0
$763$	6.82331e10	−	1.18183e11i	0.201325	−	0.348705i
$764$		0			0
$765$	−1.09417e11	+	2.19214e11i	−0.319477	+	0.640062i
$766$		0			0
$767$	2.57626e11	+	1.48740e11i	0.744403	+	0.429781i
$768$		0			0
$769$	−7.53529e10	−	1.30515e11i	−0.215474	−	0.373211i	0.737945	−	0.674861i	$-0.235796\pi$
$769$	−7.53529e10	−	1.30515e11i	−0.215474	−	0.373211i	−0.953419	+	0.301649i	$0.902463\pi$
$770$		0			0
$771$	−1.84637e11	+	5.60029e9i	−0.522519	+	0.0158487i
$772$		0			0
$773$		−	2.57744e11i		−	0.721888i	−0.932588	−	0.360944i	$-0.882455\pi$
$773$		−	2.57744e11i		−	0.721888i	0.932588	−	0.360944i	$-0.117545\pi$
$774$		0			0
$775$	−2.31548e10			−0.0641851
$776$		0			0
$777$	1.44132e11	−	8.91478e10i	0.395437	−	0.244583i
$778$		0			0
$779$	−4.62238e11	+	2.66873e11i	−1.25521	+	0.724695i
$780$		0			0
$781$	−1.08442e10	+	1.87828e10i	−0.0291471	+	0.0504843i
$782$		0			0
$783$	1.25982e11	−	2.72799e11i	0.335167	−	0.725766i
$784$		0			0
$785$	−6.73856e10	−	3.89051e10i	−0.177455	−	0.102454i
$786$		0			0
$787$	3.52866e11	+	6.11183e11i	0.919838	+	1.59321i	0.799659	+	0.600455i	$0.205014\pi$
$787$	3.52866e11	+	6.11183e11i	0.919838	+	1.59321i	0.120179	+	0.992752i	$0.461653\pi$
$788$		0			0
$789$	1.11642e11	+	1.80501e11i	0.288085	+	0.465770i
$790$		0			0
$791$			2.48359e10i			0.0634415i
$792$		0			0
$793$	−3.72908e11			−0.942994
$794$		0			0
$795$	−7.78363e8	−	2.56620e10i	−0.00194856	−	0.0642425i
$796$		0			0
$797$	2.51593e11	−	1.45257e11i	0.623541	−	0.360001i	−0.154706	−	0.987961i	$-0.549443\pi$
$797$	2.51593e11	−	1.45257e11i	0.623541	−	0.360001i	0.778246	+	0.627959i	$0.216110\pi$
$798$		0			0
$799$	−4.69184e10	+	8.12650e10i	−0.115121	+	0.199396i
$800$		0			0
$801$	4.34897e10	−	2.87576e10i	0.105647	−	0.0698589i
$802$		0			0
$803$	8.75076e10	+	5.05225e10i	0.210467	+	0.121513i
$804$		0			0
$805$	−6.14958e10	−	1.06514e11i	−0.146441	−	0.253643i
$806$		0			0
$807$	−1.52588e11	+	2.83829e11i	−0.359770	+	0.669210i
$808$		0			0
$809$		−	2.53213e11i		−	0.591142i	−0.955321	−	0.295571i	$-0.904490\pi$
$809$		−	2.53213e11i		−	0.591142i	0.955321	−	0.295571i	$-0.0955098\pi$
$810$		0			0
$811$	3.35767e11			0.776165			0.388083	−	0.921625i	$-0.373138\pi$
$811$	3.35767e11			0.776165			0.388083	+	0.921625i	$0.373138\pi$
$812$		0			0
$813$	−2.72743e11	−	1.46628e11i	−0.624297	−	0.335625i
$814$		0			0
$815$	−1.30510e11	+	7.53500e10i	−0.295811	+	0.170786i
$816$		0			0
$817$	−3.70912e11	+	6.42438e11i	−0.832496	+	1.44193i
$818$		0			0
$819$	−7.08025e10	−	1.07074e11i	−0.157367	−	0.237984i
$820$		0			0
$821$	−1.30786e11	−	7.55092e10i	−0.287864	−	0.166198i	0.349114	−	0.937080i	$-0.386483\pi$
$821$	−1.30786e11	−	7.55092e10i	−0.287864	−	0.166198i	−0.636978	+	0.770882i	$0.719816\pi$
$822$		0			0
$823$	1.63800e11	+	2.83711e11i	0.357039	+	0.618410i	0.987465	−	0.157840i	$-0.0504530\pi$
$823$	1.63800e11	+	2.83711e11i	0.357039	+	0.618410i	−0.630426	+	0.776250i	$0.717120\pi$
$824$		0			0
$825$	−2.21940e11	+	6.73174e9i	−0.479094	+	0.0145315i
$826$		0			0
$827$			4.88332e11i			1.04398i	0.852951	+	0.521991i	$0.174811\pi$
$827$			4.88332e11i			1.04398i	−0.852951	+	0.521991i	$0.825189\pi$
$828$		0			0
$829$	−8.30431e10			−0.175827			−0.0879135	−	0.996128i	$-0.528020\pi$
$829$	−8.30431e10			−0.175827			−0.0879135	+	0.996128i	$0.528020\pi$
$830$		0			0
$831$	−4.06708e11	+	2.51554e11i	−0.852862	+	0.527506i
$832$		0			0
$833$	6.98804e10	−	4.03454e10i	0.145136	−	0.0837943i
$834$		0			0
$835$	2.43121e11	−	4.21097e11i	0.500122	−	0.866237i
$836$		0			0
$837$	4.09740e10	−	2.89168e10i	0.0834846	−	0.0589180i
$838$		0			0
$839$	−4.78280e11	−	2.76135e11i	−0.965238	−	0.557280i	−0.0674567	−	0.997722i	$-0.521488\pi$
$839$	−4.78280e11	−	2.76135e11i	−0.965238	−	0.557280i	−0.897781	+	0.440442i	$0.854822\pi$
$840$		0			0
$841$	−9.02764e10	−	1.56363e11i	−0.180464	−	0.312572i
$842$		0			0
$843$	7.27360e10	+	1.17598e11i	0.144025	+	0.232858i
$844$		0			0
$845$		−	1.33745e11i		−	0.262332i
$846$		0			0
$847$	−8.12632e10			−0.157892
$848$		0			0
$849$	−1.91314e10	−	6.30748e11i	−0.0368228	−	1.21402i
$850$		0			0
$851$	−7.10025e11	+	4.09933e11i	−1.35380	+	0.781618i
$852$		0			0
$853$	5.00274e11	−	8.66500e11i	0.944956	−	1.63671i	0.189118	−	0.981954i	$-0.439437\pi$
$853$	5.00274e11	−	8.66500e11i	0.944956	−	1.63671i	0.755839	−	0.654758i	$-0.227229\pi$
$854$		0			0
$855$	3.44529e11	+	1.71966e11i	0.644706	+	0.321795i
$856$		0			0
$857$	8.82414e10	+	5.09462e10i	0.163587	+	0.0944471i	0.579559	−	0.814931i	$-0.303225\pi$
$857$	8.82414e10	+	5.09462e10i	0.163587	+	0.0944471i	−0.415971	+	0.909378i	$0.636558\pi$
$858$		0			0
$859$	2.61167e11	+	4.52355e11i	0.479674	+	0.830819i	0.999728	−	0.0233137i	$-0.00742165\pi$
$859$	2.61167e11	+	4.52355e11i	0.479674	+	0.830819i	−0.520054	+	0.854133i	$0.674088\pi$
$860$		0			0
$861$	1.20642e11	−	2.24407e11i	0.219527	−	0.408342i
$862$		0			0
$863$		−	9.63840e11i		−	1.73765i	−0.495121	−	0.868824i	$-0.664876\pi$
$863$		−	9.63840e11i		−	1.73765i	0.495121	−	0.868824i	$-0.335124\pi$
$864$		0			0
$865$	1.09944e11			0.196385
$866$		0			0
$867$	1.87231e11	+	1.00656e11i	0.331362	+	0.178141i
$868$		0			0
$869$	−2.82802e10	+	1.63276e10i	−0.0495911	+	0.0286314i
$870$		0			0
$871$	3.84695e11	−	6.66312e11i	0.668412	−	1.15772i
$872$		0			0
$873$	1.10871e12	−	6.73191e10i	1.90880	−	0.115899i
$874$		0			0
$875$	1.90499e11	+	1.09985e11i	0.324983	+	0.187629i
$876$		0			0
$877$	2.66937e11	+	4.62348e11i	0.451242	+	0.781575i	0.998463	−	0.0554134i	$-0.0176477\pi$
$877$	2.66937e11	+	4.62348e11i	0.451242	+	0.781575i	−0.547221	+	0.836988i	$0.684314\pi$
$878$		0			0
$879$	6.42046e11	−	1.94741e10i	1.07550	−	0.0326213i
$880$		0			0
$881$			5.92891e11i			0.984173i	0.870546	+	0.492086i	$0.163766\pi$
$881$			5.92891e11i			0.984173i	−0.870546	+	0.492086i	$0.836234\pi$
$882$		0			0
$883$	−5.83283e11			−0.959481			−0.479740	−	0.877410i	$-0.659269\pi$
$883$	−5.83283e11			−0.959481			−0.479740	+	0.877410i	$0.659269\pi$
$884$		0			0
$885$	3.62268e11	−	2.24068e11i	0.590551	−	0.365263i
$886$		0			0
$887$	−4.36452e11	+	2.51986e11i	−0.705086	+	0.407081i	−0.809239	−	0.587480i	$-0.800120\pi$
$887$	−4.36452e11	+	2.51986e11i	−0.705086	+	0.407081i	0.104153	+	0.994561i	$0.466787\pi$
$888$		0			0
$889$	1.12201e11	−	1.94337e11i	0.179634	−	0.311135i
$890$		0			0
$891$	3.84332e11	−	2.89082e11i	0.609811	−	0.458680i
$892$		0			0
$893$	1.27721e11	+	7.37397e10i	0.200843	+	0.115957i
$894$		0			0
$895$	−2.80082e11	−	4.85117e11i	−0.436510	−	0.756057i
$896$		0			0
$897$	3.26660e11	+	5.28138e11i	0.504575	+	0.815788i
$898$		0			0
$899$			5.33562e10i			0.0816858i
$900$		0			0
$901$	−8.14854e10			−0.123646
$902$		0			0
$903$	−1.07355e10	−	3.53942e11i	−0.0161463	−	0.532331i
$904$		0			0
$905$	3.29963e11	−	1.90504e11i	0.491893	−	0.283995i
$906$		0			0
$907$	6.72391e11	−	1.16462e12i	0.993557	−	1.72089i	0.398626	−	0.917114i	$-0.369487\pi$
$907$	6.72391e11	−	1.16462e12i	0.993557	−	1.72089i	0.594931	−	0.803777i	$-0.297180\pi$
$908$		0			0
$909$	5.51486e10	+	9.08268e11i	0.0807754	+	1.33033i
$910$		0			0
$911$	2.94833e11	+	1.70222e11i	0.428058	+	0.247139i	0.698519	−	0.715592i	$-0.253843\pi$
$911$	2.94833e11	+	1.70222e11i	0.428058	+	0.247139i	−0.270461	+	0.962731i	$0.587176\pi$
$912$		0			0
$913$	−4.58431e11	−	7.94026e11i	−0.659768	−	1.14275i
$914$		0			0
$915$	−2.52841e11	+	4.70311e11i	−0.360715	+	0.670967i
$916$		0			0
$917$			4.50527e10i			0.0637152i
$918$		0			0
$919$	6.04243e11			0.847129			0.423564	−	0.905866i	$-0.360779\pi$
$919$	6.04243e11			0.847129			0.423564	+	0.905866i	$0.360779\pi$
$920$		0			0
$921$	3.83675e11	+	2.06265e11i	0.533242	+	0.286673i
$922$		0			0
$923$	−3.62467e10	+	2.09270e10i	−0.0499415	+	0.0288337i
$924$		0			0
$925$	2.82858e11	−	4.89924e11i	0.386368	−	0.669209i
$926$		0			0
$927$	−8.79705e10	+	1.76246e11i	−0.119129	+	0.238672i
$928$		0			0
$929$	−7.30185e10	−	4.21572e10i	−0.0980325	−	0.0565991i	0.450182	−	0.892937i	$-0.351359\pi$
$929$	−7.30185e10	−	4.21572e10i	−0.0980325	−	0.0565991i	−0.548215	+	0.836338i	$0.684692\pi$
$930$		0			0
$931$	−6.34093e10	−	1.09828e11i	−0.0844023	−	0.146189i
$932$		0			0
$933$	5.90793e11	−	1.79195e10i	0.779667	−	0.0236483i
$934$		0			0
$935$		−	4.17187e11i		−	0.545864i
$936$		0			0
$937$	−1.12477e12			−1.45917			−0.729584	−	0.683891i	$-0.760286\pi$
$937$	−1.12477e12			−1.45917			−0.729584	+	0.683891i	$0.760286\pi$
$938$		0			0
$939$	3.05707e11	−	1.89084e11i	0.393227	−	0.243216i
$940$		0			0
$941$	−5.23706e10	+	3.02362e10i	−0.0667927	+	0.0385628i	−0.533024	−	0.846100i	$-0.678945\pi$
$941$	−5.23706e10	+	3.02362e10i	−0.0667927	+	0.0385628i	0.466232	+	0.884663i	$0.345611\pi$
$942$		0			0
$943$	−6.16277e11	+	1.06742e12i	−0.779344	+	1.34986i
$944$		0			0
$945$	−1.83047e11	+	1.66972e10i	−0.229528	+	0.0209370i
$946$		0			0
$947$	9.70075e11	+	5.60073e11i	1.20616	+	0.696377i	0.961918	−	0.273337i	$-0.0881274\pi$
$947$	9.70075e11	+	5.60073e11i	1.20616	+	0.696377i	0.244243	+	0.969714i	$0.421461\pi$
$948$		0			0
$949$	9.74974e10	+	1.68871e11i	0.120207	+	0.208204i
$950$		0			0
$951$	−3.24759e11	−	5.25064e11i	−0.397044	−	0.641933i
$952$		0			0
$953$			9.75269e11i			1.18237i	0.806537	+	0.591184i	$0.201339\pi$
$953$			9.75269e11i			1.18237i	−0.806537	+	0.591184i	$0.798661\pi$
$954$		0			0
$955$	7.68115e11			0.923449
$956$		0			0
$957$	1.55121e10	+	5.11423e11i	0.0184937	+	0.609723i
$958$		0			0
$959$	−3.60746e11	+	2.08277e11i	−0.426508	+	0.246245i
$960$		0			0
$961$	4.21993e11	−	7.30913e11i	0.494779	−	0.856983i
$962$		0			0
$963$	6.52141e11	−	4.31228e11i	0.758292	−	0.501420i
$964$		0			0
$965$	−1.53855e11	−	8.88283e10i	−0.177420	−	0.102434i
$966$		0			0
$967$	−8.22506e11	−	1.42462e12i	−0.940661	−	1.62927i	−0.764215	−	0.644962i	$-0.776873\pi$
$967$	−8.22506e11	−	1.42462e12i	−0.940661	−	1.62927i	−0.176446	−	0.984310i	$-0.556460\pi$
$968$		0			0
$969$	5.78700e11	−	1.07644e12i	0.656384	−	1.22094i
$970$		0			0
$971$			6.59794e11i			0.742218i	0.928589	+	0.371109i	$0.121022\pi$
$971$			6.59794e11i			0.742218i	−0.928589	+	0.371109i	$0.878978\pi$
$972$		0			0
$973$	5.48493e11			0.611956
$974$		0			0
$975$	−3.77411e11	−	2.02898e11i	−0.417634	−	0.224522i
$976$		0			0
$977$	−2.48488e10	+	1.43465e10i	−0.0272727	+	0.0157459i	−0.513574	−	0.858045i	$-0.671679\pi$
$977$	−2.48488e10	+	1.43465e10i	−0.0272727	+	0.0157459i	0.486302	+	0.873791i	$0.338346\pi$
$978$		0			0
$979$	−4.43896e10	+	7.68850e10i	−0.0483226	+	0.0836972i
$980$		0			0
$981$	5.44191e11	+	8.22973e11i	0.587591	+	0.888607i
$982$		0			0
$983$	−3.46553e11	−	2.00083e11i	−0.371155	−	0.214287i	0.302808	−	0.953052i	$-0.402076\pi$
$983$	−3.46553e11	−	2.00083e11i	−0.371155	−	0.214287i	−0.673963	+	0.738765i	$0.735409\pi$
$984$		0			0
$985$	2.48554e11	+	4.30508e11i	0.264044	+	0.457337i
$986$		0			0
$987$	−7.03661e10	+	2.13429e9i	−0.0741472	+	0.00224898i
$988$		0			0
$989$			1.71306e12i			1.79055i
$990$		0			0
$991$	7.05806e11			0.731797			0.365899	−	0.930655i	$-0.380762\pi$
$991$	7.05806e11			0.731797			0.365899	+	0.930655i	$0.380762\pi$
$992$		0			0
$993$	1.61651e11	−	9.99829e10i	0.166257	−	0.102832i
$994$		0			0
$995$	−2.90987e11	+	1.68001e11i	−0.296880	+	0.171404i
$996$		0			0
$997$	3.19259e11	−	5.52973e11i	0.323119	−	0.559659i	−0.658011	−	0.753009i	$-0.728602\pi$
$997$	3.19259e11	−	5.52973e11i	0.323119	−	0.559659i	0.981130	+	0.193350i	$0.0619352\pi$
$998$		0			0
$999$	1.11304e11	+	1.22020e12i	0.111750	+	1.22509i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.9	✓	96
9.5	odd	6	inner	252.9.bg.a.113.9	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.9	✓	96	1.1	even	1	trivial
252.9.bg.a.113.9	yes	96	9.5	odd	6	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-71.3437 - 38.3547i) q^{3} +(330.062 - 190.561i) q^{5} +(453.746 - 785.912i) q^{7} +(3618.84 + 5472.73i) q^{9} +O(q^{10})\)	\(q+(-71.3437 - 38.3547i) q^{3} +(330.062 - 190.561i) q^{5} +(453.746 - 785.912i) q^{7} +(3618.84 + 5472.73i) q^{9} +(-9675.17 - 5585.96i) q^{11} +(-10779.7 - 18671.0i) q^{13} +(-30856.7 + 935.924i) q^{15} +97980.2i q^{17} +153991. q^{19} +(-62515.3 + 38666.5i) q^{21} +(307963. - 177802. i) q^{23} +(-122685. + 212497. i) q^{25} +(-48276.4 - 529244. i) q^{27} +(489663. + 282707. i) q^{29} +(47183.3 + 81723.8i) q^{31} +(476014. + 769611. i) q^{33} -345866. i q^{35} -2.30555e6 q^{37} +(52943.5 + 1.74551e6i) q^{39} +(-3.00171e6 + 1.73304e6i) q^{41} +(-2.40865e6 + 4.17191e6i) q^{43} +(2.23733e6 + 1.11673e6i) q^{45} +(829403. + 478856. i) q^{47} +(-411772. - 713209. i) q^{49} +(3.75800e6 - 6.99026e6i) q^{51} +831652. i q^{53} -4.25787e6 q^{55} +(-1.09863e7 - 5.90629e6i) q^{57} +(-1.19496e7 + 6.89910e6i) q^{59} +(8.64840e6 - 1.49795e7i) q^{61} +(5.94311e6 - 360857. i) q^{63} +(-7.11592e6 - 4.10838e6i) q^{65} +(1.78435e7 + 3.09059e7i) q^{67} +(-2.87908e7 + 873261. i) q^{69} -1.94134e6i q^{71} -9.04455e6 q^{73} +(1.69031e7 - 1.04548e7i) q^{75} +(-8.78015e6 + 5.06922e6i) q^{77} +(1.46148e6 - 2.53137e6i) q^{79} +(-1.68548e7 + 3.96098e7i) q^{81} +(7.10733e7 + 4.10342e7i) q^{83} +(1.86712e7 + 3.23395e7i) q^{85} +(-2.40912e7 - 3.89503e7i) q^{87} -7.94663e6i q^{89} -1.95650e7 q^{91} +(-231737. - 7.64018e6i) q^{93} +(5.08267e7 - 2.93448e7i) q^{95} +(8.46479e7 - 1.46615e8i) q^{97} +(-4.44242e6 - 7.31643e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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