LMFDB - Embedded newform 252.9.bg.a.29.18

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			29.18
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-26.4497 + 76.5599i) q^{3} +(151.995 - 87.7545i) q^{5} +(-453.746 + 785.912i) q^{7} +(-5161.83 - 4049.97i) q^{9} +O(q^{10})$	\(q+(-26.4497 + 76.5599i) q^{3} +(151.995 - 87.7545i) q^{5} +(-453.746 + 785.912i) q^{7} +(-5161.83 - 4049.97i) q^{9} +(4585.15 + 2647.24i) q^{11} +(-10189.5 - 17648.7i) q^{13} +(2698.24 + 13957.8i) q^{15} +77603.4i q^{17} -114107. q^{19} +(-48167.9 - 55525.9i) q^{21} +(254810. - 147115. i) q^{23} +(-179911. + 311615. i) q^{25} +(446594. - 288068. i) q^{27} +(-232731. - 134367. i) q^{29} +(-730557. - 1.26536e6i) q^{31} +(-323948. + 281020. i) q^{33} +159273. i q^{35} +1.45042e6 q^{37} +(1.62069e6 - 313303. i) q^{39} +(4.03869e6 - 2.33174e6i) q^{41} +(-1.48102e6 + 2.56521e6i) q^{43} +(-1.13998e6 - 162602. i) q^{45} +(-3.01280e6 - 1.73944e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-5.94131e6 - 2.05259e6i) q^{51} +230423. i q^{53} +929227. q^{55} +(3.01810e6 - 8.73604e6i) q^{57} +(-2.24458e6 + 1.29591e6i) q^{59} +(8.11563e6 - 1.40567e7i) q^{61} +(5.52508e6 - 2.21908e6i) q^{63} +(-3.09750e6 - 1.78834e6i) q^{65} +(-1.91512e7 - 3.31709e7i) q^{67} +(4.52343e6 + 2.33994e7i) q^{69} +2.05780e7i q^{71} +1.51131e7 q^{73} +(-1.90986e7 - 2.20161e7i) q^{75} +(-4.16099e6 + 2.40235e6i) q^{77} +(4.91246e6 - 8.50863e6i) q^{79} +(1.02422e7 + 4.18105e7i) q^{81} +(2.60363e7 + 1.50321e7i) q^{83} +(6.81005e6 + 1.17953e7i) q^{85} +(1.64428e7 - 1.42639e7i) q^{87} -9.44654e7i q^{89} +1.84938e7 q^{91} +(1.16199e8 - 2.24629e7i) q^{93} +(-1.73438e7 + 1.00134e7i) q^{95} +(-4.17520e7 + 7.23166e7i) q^{97} +(-1.29465e7 - 3.22343e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−26.4497	+	76.5599i	−0.326539	+	0.945184i
$3$	−26.4497	+	76.5599i	−0.326539	+	0.945184i
$4$		0			0
$5$	151.995	−	87.7545i	0.243192	−	0.140407i	−0.373451	−	0.927650i	$-0.621826\pi$
$5$	151.995	−	87.7545i	0.243192	−	0.140407i	0.616643	+	0.787243i	$0.288492\pi$
$6$		0			0
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$8$		0			0
$9$	−5161.83	−	4049.97i	−0.786744	−	0.617279i
$10$		0			0
$11$	4585.15	+	2647.24i	0.313172	+	0.180810i	0.648345	−	0.761347i	$-0.275461\pi$
$11$	4585.15	+	2647.24i	0.313172	+	0.180810i	−0.335173	+	0.942157i	$0.608795\pi$
$12$		0			0
$13$	−10189.5	−	17648.7i	−0.356762	−	0.617930i	0.630656	−	0.776063i	$-0.282786\pi$
$13$	−10189.5	−	17648.7i	−0.356762	−	0.617930i	−0.987418	+	0.158133i	$0.949453\pi$
$14$		0			0
$15$	2698.24	+	13957.8i	0.0532987	+	0.275710i
$16$		0			0
$17$			77603.4i			0.929149i	0.885534	+	0.464574i	$0.153793\pi$
$17$			77603.4i			0.929149i	−0.885534	+	0.464574i	$0.846207\pi$
$18$		0			0
$19$	−114107.			−0.875587			−0.437793	−	0.899076i	$-0.644240\pi$
$19$	−114107.			−0.875587			−0.437793	+	0.899076i	$0.644240\pi$
$20$		0			0
$21$	−48167.9	−	55525.9i	−0.247674	−	0.285508i
$22$		0			0
$23$	254810.	−	147115.i	0.910553	−	0.525708i	0.0299436	−	0.999552i	$-0.490467\pi$
$23$	254810.	−	147115.i	0.910553	−	0.525708i	0.880609	+	0.473844i	$0.157134\pi$
$24$		0			0
$25$	−179911.	+	311615.i	−0.460572	+	0.797734i
$26$		0			0
$27$	446594.	−	288068.i	0.840345	−	0.542052i
$28$		0			0
$29$	−232731.	−	134367.i	−0.329050	−	0.189977i	0.326369	−	0.945242i	$-0.394175\pi$
$29$	−232731.	−	134367.i	−0.329050	−	0.189977i	−0.655419	+	0.755265i	$0.727508\pi$
$30$		0			0
$31$	−730557.	−	1.26536e6i	−0.791057	−	1.37015i	−0.925313	−	0.379204i	$-0.876198\pi$
$31$	−730557.	−	1.26536e6i	−0.791057	−	1.37015i	0.134257	−	0.990947i	$-0.457135\pi$
$32$		0			0
$33$	−323948.	+	281020.i	−0.273161	+	0.236963i
$34$		0			0
$35$			159273.i			0.106138i
$36$		0			0
$37$	1.45042e6			0.773904			0.386952	−	0.922100i	$-0.373528\pi$
$37$	1.45042e6			0.773904			0.386952	+	0.922100i	$0.373528\pi$
$38$		0			0
$39$	1.62069e6	−	313303.i	0.700554	−	0.135427i
$40$		0			0
$41$	4.03869e6	−	2.33174e6i	1.42924	−	0.825172i	0.432179	−	0.901788i	$-0.357745\pi$
$41$	4.03869e6	−	2.33174e6i	1.42924	−	0.825172i	0.997061	+	0.0766158i	$0.0244115\pi$
$42$		0			0
$43$	−1.48102e6	+	2.56521e6i	−0.433200	+	0.750324i	−0.997147	−	0.0754868i	$-0.975949\pi$
$43$	−1.48102e6	+	2.56521e6i	−0.433200	+	0.750324i	0.563947	+	0.825811i	$0.309282\pi$
$44$		0			0
$45$	−1.13998e6	−	162602.i	−0.278001	−	0.0396531i
$46$		0			0
$47$	−3.01280e6	−	1.73944e6i	−0.617417	−	0.356466i	0.158446	−	0.987368i	$-0.449352\pi$
$47$	−3.01280e6	−	1.73944e6i	−0.617417	−	0.356466i	−0.775863	+	0.630902i	$0.782685\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	−5.94131e6	−	2.05259e6i	−0.878216	−	0.303404i
$52$		0			0
$53$			230423.i			0.0292027i	0.999893	+	0.0146013i	$0.00464792\pi$
$53$			230423.i			0.0292027i	−0.999893	+	0.0146013i	$0.995352\pi$
$54$		0			0
$55$	929227.			0.101548
$56$		0			0
$57$	3.01810e6	−	8.73604e6i	0.285914	−	0.827590i
$58$		0			0
$59$	−2.24458e6	+	1.29591e6i	−0.185237	+	0.106946i	−0.589751	−	0.807585i	$-0.700774\pi$
$59$	−2.24458e6	+	1.29591e6i	−0.185237	+	0.106946i	0.404514	+	0.914532i	$0.367441\pi$
$60$		0			0
$61$	8.11563e6	−	1.40567e7i	0.586142	−	1.01523i	−0.408590	−	0.912718i	$-0.633979\pi$
$61$	8.11563e6	−	1.40567e7i	0.586142	−	1.01523i	0.994732	−	0.102510i	$-0.0326873\pi$
$62$		0			0
$63$	5.52508e6	−	2.21908e6i	0.350733	−	0.140868i
$64$		0			0
$65$	−3.09750e6	−	1.78834e6i	−0.173524	−	0.100184i
$66$		0			0
$67$	−1.91512e7	−	3.31709e7i	−0.950381	−	1.64611i	−0.744601	−	0.667510i	$-0.767360\pi$
$67$	−1.91512e7	−	3.31709e7i	−0.950381	−	1.64611i	−0.205780	−	0.978598i	$-0.565973\pi$
$68$		0			0
$69$	4.52343e6	+	2.33994e7i	0.199559	+	1.03230i
$70$		0			0
$71$			2.05780e7i			0.809784i	0.914365	+	0.404892i	$0.132691\pi$
$71$			2.05780e7i			0.809784i	−0.914365	+	0.404892i	$0.867309\pi$
$72$		0			0
$73$	1.51131e7			0.532184			0.266092	−	0.963948i	$-0.414268\pi$
$73$	1.51131e7			0.532184			0.266092	+	0.963948i	$0.414268\pi$
$74$		0			0
$75$	−1.90986e7	−	2.20161e7i	−0.603610	−	0.695816i
$76$		0			0
$77$	−4.16099e6	+	2.40235e6i	−0.118368	+	0.0683397i
$78$		0			0
$79$	4.91246e6	−	8.50863e6i	0.126122	−	0.218450i	−0.796049	−	0.605232i	$-0.793080\pi$
$79$	4.91246e6	−	8.50863e6i	0.126122	−	0.218450i	0.922171	+	0.386783i	$0.126414\pi$
$80$		0			0
$81$	1.02422e7	+	4.18105e7i	0.237932	+	0.971282i
$82$		0			0
$83$	2.60363e7	+	1.50321e7i	0.548615	+	0.316743i	0.748563	−	0.663064i	$-0.230744\pi$
$83$	2.60363e7	+	1.50321e7i	0.548615	+	0.316743i	−0.199948	+	0.979806i	$0.564077\pi$
$84$		0			0
$85$	6.81005e6	+	1.17953e7i	0.130459	+	0.225962i
$86$		0			0
$87$	1.64428e7	−	1.42639e7i	0.287011	−	0.248978i
$88$		0			0
$89$		−	9.44654e7i		−	1.50561i	−0.658244	−	0.752805i	$-0.728700\pi$
$89$		−	9.44654e7i		−	1.50561i	0.658244	−	0.752805i	$-0.271300\pi$
$90$		0			0
$91$	1.84938e7			0.269687
$92$		0			0
$93$	1.16199e8	−	2.24629e7i	1.55335	−	0.300286i
$94$		0			0
$95$	−1.73438e7	+	1.00134e7i	−0.212936	+	0.122939i
$96$		0			0
$97$	−4.17520e7	+	7.23166e7i	−0.471618	+	0.816866i	−0.999473	−	0.0324683i	$-0.989663\pi$
$97$	−4.17520e7	+	7.23166e7i	−0.471618	+	0.816866i	0.527855	+	0.849335i	$0.322997\pi$
$98$		0			0
$99$	−1.29465e7	−	3.22343e7i	−0.134776	−	0.335565i
$100$		0			0
$101$	1.04874e8	+	6.05493e7i	1.00782	+	0.581867i	0.910553	−	0.413391i	$-0.135656\pi$
$101$	1.04874e8	+	6.05493e7i	1.00782	+	0.581867i	0.0972693	+	0.995258i	$0.468989\pi$
$102$		0			0
$103$	8.58348e7	+	1.48670e8i	0.762631	+	1.32092i	0.941490	+	0.337041i	$0.109426\pi$
$103$	8.58348e7	+	1.48670e8i	0.762631	+	1.32092i	−0.178859	+	0.983875i	$0.557240\pi$
$104$		0			0
$105$	−1.21939e7	−	4.21272e6i	−0.100320	−	0.0346582i
$106$		0			0
$107$			2.19162e8i			1.67198i	0.548746	+	0.835989i	$0.315106\pi$
$107$			2.19162e8i			1.67198i	−0.548746	+	0.835989i	$0.684894\pi$
$108$		0			0
$109$	3.26136e7			0.231043			0.115522	−	0.993305i	$-0.463146\pi$
$109$	3.26136e7			0.231043			0.115522	+	0.993305i	$0.463146\pi$
$110$		0			0
$111$	−3.83632e7	+	1.11044e8i	−0.252710	+	0.731482i
$112$		0			0
$113$	6.62730e7	−	3.82627e7i	0.406465	−	0.234672i	−0.282805	−	0.959177i	$-0.591265\pi$
$113$	6.62730e7	−	3.82627e7i	0.406465	−	0.234672i	0.689270	+	0.724505i	$0.257932\pi$
$114$		0			0
$115$	2.58199e7	−	4.47214e7i	0.147626	−	0.255696i
$116$		0			0
$117$	−1.88803e7	+	1.32367e8i	−0.100755	+	0.706375i
$118$		0			0
$119$	−6.09894e7	−	3.52123e7i	−0.304135	−	0.175593i
$120$		0			0
$121$	−9.31637e7	−	1.61364e8i	−0.434616	−	0.752776i
$122$		0			0
$123$	7.16955e7	+	3.70875e8i	0.313236	+	1.62035i
$124$		0			0
$125$			1.31710e8i			0.539484i
$126$		0			0
$127$	3.44302e8			1.32350			0.661750	−	0.749724i	$-0.269814\pi$
$127$	3.44302e8			1.32350			0.661750	+	0.749724i	$0.269814\pi$
$128$		0			0
$129$	−1.57219e8	−	1.81236e8i	−0.567737	−	0.654464i
$130$		0			0
$131$	2.36310e8	−	1.36434e8i	0.802411	−	0.463272i	−0.0419024	−	0.999122i	$-0.513342\pi$
$131$	2.36310e8	−	1.36434e8i	0.802411	−	0.463272i	0.844314	+	0.535849i	$0.180009\pi$
$132$		0			0
$133$	5.17758e7	−	8.96783e7i	0.165470	−	0.286603i
$134$		0			0
$135$	4.26008e7	−	8.29756e7i	0.128258	−	0.249813i
$136$		0			0
$137$	3.14816e8	+	1.81759e8i	0.893666	+	0.515958i	0.875140	−	0.483870i	$-0.160769\pi$
$137$	3.14816e8	+	1.81759e8i	0.893666	+	0.515958i	0.0185260	+	0.999828i	$0.494103\pi$
$138$		0			0
$139$	2.39977e8	+	4.15652e8i	0.642850	+	1.11345i	0.984794	+	0.173728i	$0.0555814\pi$
$139$	2.39977e8	+	4.15652e8i	0.642850	+	1.11345i	−0.341944	+	0.939720i	$0.611085\pi$
$140$		0			0
$141$	2.12859e8	−	1.84652e8i	0.538537	−	0.467172i
$142$		0			0
$143$		−	1.07896e8i		−	0.258024i
$144$		0			0
$145$	−4.71653e7			−0.106697
$146$		0			0
$147$	6.54944e7	−	1.26610e7i	0.140260	−	0.0271143i
$148$		0			0
$149$	7.86359e8	−	4.54005e8i	1.59542	−	0.921118i	0.603069	−	0.797689i	$-0.293944\pi$
$149$	7.86359e8	−	4.54005e8i	1.59542	−	0.921118i	0.992353	−	0.123429i	$-0.0393890\pi$
$150$		0			0
$151$	2.31177e8	−	4.00410e8i	0.444669	−	0.770189i	−0.553360	−	0.832942i	$-0.686655\pi$
$151$	2.31177e8	−	4.00410e8i	0.444669	−	0.770189i	0.998029	+	0.0627532i	$0.0199881\pi$
$152$		0			0
$153$	3.14292e8	−	4.00576e8i	0.573544	−	0.731002i
$154$		0			0
$155$	−2.22082e8	−	1.28219e8i	−0.384758	−	0.222140i
$156$		0			0
$157$	3.37697e8	+	5.84909e8i	0.555813	+	0.962697i	0.997840	+	0.0656949i	$0.0209264\pi$
$157$	3.37697e8	+	5.84909e8i	0.555813	+	0.962697i	−0.442026	+	0.897002i	$0.645740\pi$
$158$		0			0
$159$	−1.76412e7	−	6.09462e6i	−0.0276019	−	0.00953582i
$160$		0			0
$161$			2.67011e8i			0.397398i
$162$		0			0
$163$	−6.85487e8			−0.971066			−0.485533	−	0.874218i	$-0.661374\pi$
$163$	−6.85487e8			−0.971066			−0.485533	+	0.874218i	$0.661374\pi$
$164$		0			0
$165$	−2.45778e7	+	7.11415e7i	−0.0331594	+	0.0959814i
$166$		0			0
$167$	−1.62236e8	+	9.36667e7i	−0.208584	+	0.120426i	−0.600653	−	0.799510i	$-0.705093\pi$
$167$	−1.62236e8	+	9.36667e7i	−0.208584	+	0.120426i	0.392069	+	0.919936i	$0.371759\pi$
$168$		0			0
$169$	2.00214e8	−	3.46781e8i	0.245442	−	0.425118i
$170$		0			0
$171$	5.89002e8	+	4.62131e8i	0.688863	+	0.540482i
$172$		0			0
$173$	−4.23635e8	−	2.44586e8i	−0.472941	−	0.273053i	0.244529	−	0.969642i	$-0.421367\pi$
$173$	−4.23635e8	−	2.44586e8i	−0.472941	−	0.273053i	−0.717470	+	0.696589i	$0.754700\pi$
$174$		0			0
$175$	−1.63268e8	−	2.82788e8i	−0.174080	−	0.301515i
$176$		0			0
$177$	−3.98461e7	−	2.06121e8i	−0.0405969	−	0.210005i
$178$		0			0
$179$		−	5.28810e8i		−	0.515095i	−0.966266	−	0.257548i	$-0.917086\pi$
$179$		−	5.28810e8i		−	0.515095i	0.966266	−	0.257548i	$-0.0829144\pi$
$180$		0			0
$181$	1.39250e9			1.29742			0.648711	−	0.761035i	$-0.275308\pi$
$181$	1.39250e9			1.29742			0.648711	+	0.761035i	$0.275308\pi$
$182$		0			0
$183$	8.61522e8	+	9.93127e8i	0.768178	+	0.885524i
$184$		0			0
$185$	2.20457e8	−	1.27281e8i	0.188208	−	0.108662i
$186$		0			0
$187$	−2.05435e8	+	3.55823e8i	−0.167999	+	0.290983i
$188$		0			0
$189$	2.37560e7	+	4.81693e8i	0.0186177	+	0.377506i
$190$		0			0
$191$	−4.02183e7	−	2.32200e7i	−0.0302197	−	0.0174474i	0.484814	−	0.874617i	$-0.338887\pi$
$191$	−4.02183e7	−	2.32200e7i	−0.0302197	−	0.0174474i	−0.515034	+	0.857170i	$0.672221\pi$
$192$		0			0
$193$	7.37662e7	+	1.27767e8i	0.0531653	+	0.0920850i	0.891383	−	0.453250i	$-0.149736\pi$
$193$	7.37662e7	+	1.27767e8i	0.0531653	+	0.0920850i	−0.838218	+	0.545335i	$0.816402\pi$
$194$		0			0
$195$	2.18843e8	−	1.89843e8i	0.151354	−	0.131298i
$196$		0			0
$197$		−	7.05344e7i		−	0.0468313i	−0.999726	−	0.0234156i	$-0.992546\pi$
$197$		−	7.05344e7i		−	0.0468313i	0.999726	−	0.0234156i	$-0.00745411\pi$
$198$		0			0
$199$	2.07303e9			1.32188			0.660940	−	0.750438i	$-0.270158\pi$
$199$	2.07303e9			1.32188			0.660940	+	0.750438i	$0.270158\pi$
$200$		0			0
$201$	3.04611e9	−	5.88856e8i	1.86621	−	0.360765i
$202$		0			0
$203$	2.11202e8	−	1.21937e8i	0.124369	−	0.0718047i
$204$		0			0
$205$	4.09241e8	−	7.08826e8i	0.231720	−	0.401351i
$206$		0			0
$207$	−1.91109e9	−	2.72592e8i	−1.04088	−	0.148468i
$208$		0			0
$209$	−5.23199e8	−	3.02069e8i	−0.274209	−	0.158315i
$210$		0			0
$211$	5.09923e7	+	8.83213e7i	0.0257262	+	0.0445590i	0.878602	−	0.477555i	$-0.158477\pi$
$211$	5.09923e7	+	8.83213e7i	0.0257262	+	0.0445590i	−0.852876	+	0.522114i	$0.825144\pi$
$212$		0			0
$213$	−1.57545e9	−	5.44281e8i	−0.765394	−	0.264426i
$214$		0			0
$215$			5.19866e8i			0.243297i
$216$		0			0
$217$	1.32595e9			0.597983
$218$		0			0
$219$	−3.99736e8	+	1.15706e9i	−0.173779	+	0.503011i
$220$		0			0
$221$	1.36960e9	−	7.90739e8i	0.574149	−	0.331485i
$222$		0			0
$223$	−3.30199e8	+	5.71921e8i	−0.133523	+	0.231269i	−0.925032	−	0.379889i	$-0.875962\pi$
$223$	−3.30199e8	+	5.71921e8i	−0.133523	+	0.231269i	0.791509	+	0.611157i	$0.209296\pi$
$224$		0			0
$225$	2.19070e9	−	8.79868e8i	0.854776	−	0.343311i
$226$		0			0
$227$	4.02256e9	+	2.32242e9i	1.51495	+	0.874658i	0.999846	+	0.0175311i	$0.00558060\pi$
$227$	4.02256e9	+	2.32242e9i	1.51495	+	0.874658i	0.515106	+	0.857127i	$0.327753\pi$
$228$		0			0
$229$	5.48661e7	+	9.50308e7i	0.0199509	+	0.0345559i	0.875828	−	0.482623i	$-0.160316\pi$
$229$	5.48661e7	+	9.50308e7i	0.0199509	+	0.0345559i	−0.855878	+	0.517178i	$0.826982\pi$
$230$		0			0
$231$	−7.38666e7	−	3.82106e8i	−0.0259418	−	0.134195i
$232$		0			0
$233$			1.17177e9i			0.397574i	0.980043	+	0.198787i	$0.0637003\pi$
$233$			1.17177e9i			0.397574i	−0.980043	+	0.198787i	$0.936300\pi$
$234$		0			0
$235$	−6.10575e8			−0.200201
$236$		0			0
$237$	5.21486e8	+	6.01148e8i	0.165291	+	0.190541i
$238$		0			0
$239$	−1.84753e9	+	1.06667e9i	−0.566238	+	0.326918i	−0.755645	−	0.654981i	$-0.772677\pi$
$239$	−1.84753e9	+	1.06667e9i	−0.566238	+	0.326918i	0.189408	+	0.981899i	$0.439343\pi$
$240$		0			0
$241$	−1.35053e9	+	2.33918e9i	−0.400345	+	0.693419i	−0.993767	−	0.111473i	$-0.964443\pi$
$241$	−1.35053e9	+	2.33918e9i	−0.400345	+	0.693419i	0.593422	+	0.804891i	$0.297777\pi$
$242$		0			0
$243$	−3.47191e9	−	3.21732e8i	−0.995734	−	0.0922719i
$244$		0			0
$245$	−1.25175e8	−	7.22696e7i	−0.0347418	−	0.0200582i
$246$		0			0
$247$	1.16269e9	+	2.01385e9i	0.312376	+	0.541051i
$248$		0			0
$249$	−1.83951e9	+	1.59574e9i	−0.478525	+	0.415113i
$250$		0			0
$251$		−	4.18873e9i		−	1.05533i	−0.849453	−	0.527664i	$-0.823068\pi$
$251$		−	4.18873e9i		−	1.05533i	0.849453	−	0.527664i	$-0.176932\pi$
$252$		0			0
$253$	1.55779e9			0.380212
$254$		0			0
$255$	−1.08317e9	+	2.09393e8i	−0.256175	+	0.0495224i
$256$		0			0
$257$	−6.14502e9	+	3.54783e9i	−1.40861	+	0.813261i	−0.995254	−	0.0973088i	$-0.968977\pi$
$257$	−6.14502e9	+	3.54783e9i	−1.40861	+	0.813261i	−0.413355	+	0.910570i	$0.635643\pi$
$258$		0			0
$259$	−6.58123e8	+	1.13990e9i	−0.146254	+	0.253320i
$260$		0			0
$261$	6.57134e8	+	1.63613e9i	0.141609	+	0.352579i
$262$		0			0
$263$	−5.18413e9	−	2.99306e9i	−1.08356	−	0.625594i	−0.151706	−	0.988426i	$-0.548477\pi$
$263$	−5.18413e9	−	2.99306e9i	−1.08356	−	0.625594i	−0.931855	+	0.362832i	$0.881810\pi$
$264$		0			0
$265$	2.02207e7	+	3.50232e7i	0.00410026	+	0.00710187i
$266$		0			0
$267$	7.23226e9	+	2.49858e9i	1.42308	+	0.491641i
$268$		0			0
$269$		−	4.12840e9i		−	0.788447i	−0.919015	−	0.394223i	$-0.871014\pi$
$269$		−	4.12840e9i		−	0.788447i	0.919015	−	0.394223i	$-0.128986\pi$
$270$		0			0
$271$	−2.31738e9			−0.429655			−0.214828	−	0.976652i	$-0.568919\pi$
$271$	−2.31738e9			−0.429655			−0.214828	+	0.976652i	$0.568919\pi$
$272$		0			0
$273$	−4.89154e8	+	1.41588e9i	−0.0880633	+	0.254903i
$274$		0			0
$275$	−1.64984e9	+	9.52533e8i	−0.288476	+	0.166552i
$276$		0			0
$277$	1.47372e9	−	2.55256e9i	0.250320	−	0.433567i	−0.713294	−	0.700865i	$-0.752797\pi$
$277$	1.47372e9	−	2.55256e9i	0.250320	−	0.433567i	0.963614	+	0.267298i	$0.0861308\pi$
$278$		0			0
$279$	−1.35367e9	+	9.49032e9i	−0.223406	+	1.56626i
$280$		0			0
$281$	1.94232e9	+	1.12140e9i	0.311526	+	0.179860i	0.647609	−	0.761973i	$-0.275769\pi$
$281$	1.94232e9	+	1.12140e9i	0.311526	+	0.179860i	−0.336083	+	0.941832i	$0.609102\pi$
$282$		0			0
$283$	−3.11268e9	−	5.39131e9i	−0.485275	−	0.840521i	0.514582	−	0.857441i	$-0.327947\pi$
$283$	−3.11268e9	−	5.39131e9i	−0.485275	−	0.840521i	−0.999857	+	0.0169201i	$0.994614\pi$
$284$		0			0
$285$	−3.07890e8	−	1.59269e9i	−0.0466676	−	0.241408i
$286$		0			0
$287$			4.23207e9i			0.623771i
$288$		0			0
$289$	9.53466e8			0.136683
$290$		0			0
$291$	−4.43222e9	−	5.10928e9i	−0.618087	−	0.712505i
$292$		0			0
$293$	1.15799e10	−	6.68566e9i	1.57121	−	0.907138i	0.575188	−	0.818021i	$-0.304929\pi$
$293$	1.15799e10	−	6.68566e9i	1.57121	−	0.907138i	0.996021	−	0.0891174i	$-0.0284046\pi$
$294$		0			0
$295$	−2.27443e8	+	3.93944e8i	−0.0300321	+	0.0520171i
$296$		0			0
$297$	2.81028e9	−	1.38597e8i	0.361181	−	0.0178126i
$298$		0			0
$299$	−5.19276e9	−	2.99804e9i	−0.649701	−	0.375105i
$300$		0			0
$301$	−1.34402e9	−	2.32791e9i	−0.163734	−	0.283596i
$302$		0			0
$303$	−7.40954e9	+	6.42766e9i	−0.879065	+	0.762575i
$304$		0			0
$305$		−	2.84873e9i		−	0.329194i
$306$		0			0
$307$	7.04877e9			0.793524			0.396762	−	0.917922i	$-0.370134\pi$
$307$	7.04877e9			0.793524			0.396762	+	0.917922i	$0.370134\pi$
$308$		0			0
$309$	−1.36525e10	+	2.63922e9i	−1.49754	+	0.289495i
$310$		0			0
$311$	3.82080e8	−	2.20594e8i	0.0408426	−	0.0235805i	−0.479440	−	0.877575i	$-0.659160\pi$
$311$	3.82080e8	−	2.20594e8i	0.0408426	−	0.0235805i	0.520282	+	0.853994i	$0.325827\pi$
$312$		0			0
$313$	3.84042e9	−	6.65180e9i	0.400130	−	0.693045i	−0.593611	−	0.804752i	$-0.702298\pi$
$313$	3.84042e9	−	6.65180e9i	0.400130	−	0.693045i	0.993741	+	0.111706i	$0.0356316\pi$
$314$		0			0
$315$	6.45051e8	−	8.22140e8i	0.0655167	−	0.0835033i
$316$		0			0
$317$	−1.69445e10	−	9.78291e9i	−1.67800	−	0.968793i	−0.962935	−	0.269732i	$-0.913065\pi$
$317$	−1.69445e10	−	9.78291e9i	−1.67800	−	0.968793i	−0.715063	−	0.699061i	$-0.753602\pi$
$318$		0			0
$319$	−7.11404e8	−	1.23219e9i	−0.0686995	−	0.118991i
$320$		0			0
$321$	−1.67790e10	−	5.79678e9i	−1.58033	−	0.545967i
$322$		0			0
$323$		−	8.85512e9i		−	0.813550i
$324$		0			0
$325$	7.33279e9			0.657258
$326$		0			0
$327$	−8.62621e8	+	2.49690e9i	−0.0754447	+	0.218378i
$328$		0			0
$329$	2.73409e9	−	1.57853e9i	0.233362	−	0.134731i
$330$		0			0
$331$	−1.56542e9	+	2.71138e9i	−0.130412	+	0.225881i	−0.923836	−	0.382790i	$-0.874963\pi$
$331$	−1.56542e9	+	2.71138e9i	−0.130412	+	0.225881i	0.793423	+	0.608670i	$0.208297\pi$
$332$		0			0
$333$	−7.48682e9	−	5.87416e9i	−0.608864	−	0.477715i
$334$		0			0
$335$	−5.82179e9	−	3.36121e9i	−0.462251	−	0.266880i
$336$		0			0
$337$	2.45702e9	+	4.25568e9i	0.190497	+	0.329951i	0.945415	−	0.325868i	$-0.105657\pi$
$337$	2.45702e9	+	4.25568e9i	0.190497	+	0.329951i	−0.754918	+	0.655819i	$0.772323\pi$
$338$		0			0
$339$	1.17649e9	+	6.08589e9i	0.0890818	+	0.460813i
$340$		0			0
$341$		−	7.73583e9i		−	0.572123i
$342$		0			0
$343$	7.47359e8			0.0539949
$344$		0			0
$345$	2.74094e9	+	3.15964e9i	0.193474	+	0.223029i
$346$		0			0
$347$	−2.41145e10	+	1.39225e10i	−1.66326	+	0.960285i	−0.692122	+	0.721781i	$0.743324\pi$
$347$	−2.41145e10	+	1.39225e10i	−1.66326	+	0.960285i	−0.971141	+	0.238505i	$0.923343\pi$
$348$		0			0
$349$	−4.68506e9	+	8.11476e9i	−0.315801	+	0.546983i	−0.979607	−	0.200921i	$-0.935606\pi$
$349$	−4.68506e9	+	8.11476e9i	−0.315801	+	0.546983i	0.663807	+	0.747904i	$0.268940\pi$
$350$		0			0
$351$	−9.63459e9	−	4.94653e9i	−0.634753	−	0.325891i
$352$		0			0
$353$	−1.09025e10	−	6.29454e9i	−0.702143	−	0.405383i	0.106002	−	0.994366i	$-0.466195\pi$
$353$	−1.09025e10	−	6.29454e9i	−0.702143	−	0.405383i	−0.808145	+	0.588983i	$0.799528\pi$
$354$		0			0
$355$	1.80581e9	+	3.12775e9i	0.113699	+	0.196933i
$356$		0			0
$357$	4.30900e9	−	3.73799e9i	0.265279	−	0.230126i
$358$		0			0
$359$		−	7.36435e9i		−	0.443360i	−0.975119	−	0.221680i	$-0.928846\pi$
$359$		−	7.36435e9i		−	0.443360i	0.975119	−	0.221680i	$-0.0711541\pi$
$360$		0			0
$361$	−3.96308e9			−0.233348
$362$		0			0
$363$	1.48182e10	−	2.86457e9i	0.853431	−	0.164980i
$364$		0			0
$365$	2.29711e9	−	1.32624e9i	0.129423	−	0.0747224i
$366$		0			0
$367$	1.05039e10	−	1.81933e10i	0.579010	−	1.00287i	−0.416583	−	0.909098i	$-0.636773\pi$
$367$	1.05039e10	−	1.81933e10i	0.579010	−	1.00287i	0.995593	−	0.0937771i	$-0.0298941\pi$
$368$		0			0
$369$	−3.02905e10	−	4.32054e9i	−1.63381	−	0.233041i
$370$		0			0
$371$	−1.81092e8	−	1.04554e8i	−0.00955882	−	0.00551879i
$372$		0			0
$373$	−1.57611e10	−	2.72990e10i	−0.814238	−	1.41030i	−0.909874	−	0.414885i	$-0.863822\pi$
$373$	−1.57611e10	−	2.72990e10i	−0.814238	−	1.41030i	0.0956357	−	0.995416i	$-0.469512\pi$
$374$		0			0
$375$	−1.00837e10	−	3.48369e9i	−0.509912	−	0.176163i
$376$		0			0
$377$			5.47653e9i			0.271107i
$378$		0			0
$379$	3.35850e9			0.162775			0.0813876	−	0.996683i	$-0.474065\pi$
$379$	3.35850e9			0.162775			0.0813876	+	0.996683i	$0.474065\pi$
$380$		0			0
$381$	−9.10667e9	+	2.63597e10i	−0.432175	+	1.25095i
$382$		0			0
$383$	1.99420e10	−	1.15135e10i	0.926772	−	0.535072i	0.0409826	−	0.999160i	$-0.486951\pi$
$383$	1.99420e10	−	1.15135e10i	0.926772	−	0.535072i	0.885789	+	0.464088i	$0.153618\pi$
$384$		0			0
$385$	−4.21633e8	+	7.30290e8i	−0.0191908	+	0.0332394i
$386$		0			0
$387$	1.80338e10	−	7.24307e9i	0.803977	−	0.322908i
$388$		0			0
$389$	−2.07863e10	−	1.20010e10i	−0.907775	−	0.524104i	−0.0280606	−	0.999606i	$-0.508933\pi$
$389$	−2.07863e10	−	1.20010e10i	−0.907775	−	0.524104i	−0.879715	+	0.475502i	$0.842266\pi$
$390$		0			0
$391$	1.14166e10	+	1.97741e10i	0.488461	+	0.846039i
$392$		0			0
$393$	4.19502e9	+	2.17005e10i	0.175859	+	0.909703i
$394$		0			0
$395$		−	1.72436e9i		−	0.0708336i
$396$		0			0
$397$	−2.72783e8			−0.0109813			−0.00549067	−	0.999985i	$-0.501748\pi$
$397$	−2.72783e8			−0.0109813			−0.00549067	+	0.999985i	$0.501748\pi$
$398$		0			0
$399$	5.49631e9	+	6.33591e9i	0.216860	+	0.249987i
$400$		0			0
$401$	9.45397e9	−	5.45825e9i	0.365626	−	0.211094i	−0.305920	−	0.952057i	$-0.598964\pi$
$401$	9.45397e9	−	5.45825e9i	0.365626	−	0.211094i	0.671546	+	0.740963i	$0.265631\pi$
$402$		0			0
$403$	−1.48880e10	+	2.57868e10i	−0.564438	+	0.977635i
$404$		0			0
$405$	5.22582e9	+	5.45619e9i	0.194238	+	0.202801i
$406$		0			0
$407$	6.65039e9	+	3.83961e9i	0.242365	+	0.139929i
$408$		0			0
$409$	2.26467e10	+	3.92253e10i	0.809305	+	1.40176i	0.913346	+	0.407184i	$0.133489\pi$
$409$	2.26467e10	+	3.92253e10i	0.809305	+	1.40176i	−0.104042	+	0.994573i	$0.533178\pi$
$410$		0			0
$411$	−2.22423e10	+	1.92948e10i	−0.779492	+	0.676198i
$412$		0			0
$413$		−	2.35205e9i		−	0.0808439i
$414$		0			0
$415$	5.27653e9			0.177892
$416$		0			0
$417$	−3.81695e10	+	7.37872e9i	−1.26233	+	0.244026i
$418$		0			0
$419$	−1.64502e10	+	9.49753e9i	−0.533722	+	0.308144i	−0.742531	−	0.669812i	$-0.766375\pi$
$419$	−1.64502e10	+	9.49753e9i	−0.533722	+	0.308144i	0.208809	+	0.977956i	$0.433041\pi$
$420$		0			0
$421$	1.90372e10	−	3.29734e10i	0.606003	−	1.04963i	−0.385889	−	0.922545i	$-0.626105\pi$
$421$	1.90372e10	−	3.29734e10i	0.606003	−	1.04963i	0.991892	−	0.127083i	$-0.0405614\pi$
$422$		0			0
$423$	8.50687e9	+	2.11804e10i	0.265710	+	0.661566i
$424$		0			0
$425$	−2.41824e10	−	1.39617e10i	−0.741213	−	0.427940i
$426$		0			0
$427$	7.36488e9	+	1.27563e10i	0.221541	+	0.383720i
$428$		0			0
$429$	8.26049e9	+	2.85381e9i	0.243880	+	0.0842551i
$430$		0			0
$431$		−	5.45707e10i		−	1.58143i	−0.612184	−	0.790715i	$-0.709709\pi$
$431$		−	5.45707e10i		−	1.58143i	0.612184	−	0.790715i	$-0.290291\pi$
$432$		0			0
$433$	4.32171e10			1.22943			0.614716	−	0.788749i	$-0.289271\pi$
$433$	4.32171e10			1.22943			0.614716	+	0.788749i	$0.289271\pi$
$434$		0			0
$435$	1.24751e9	−	3.61097e9i	0.0348407	−	0.100848i
$436$		0			0
$437$	−2.90757e10	+	1.67869e10i	−0.797268	+	0.460303i
$438$		0			0
$439$	2.31220e10	−	4.00485e10i	0.622540	−	1.07827i	−0.366471	−	0.930430i	$-0.619434\pi$
$439$	2.31220e10	−	4.00485e10i	0.622540	−	1.07827i	0.989011	−	0.147842i	$-0.0472326\pi$
$440$		0			0
$441$	−7.62982e8	+	5.34912e9i	−0.0201725	+	0.141426i
$442$		0			0
$443$	−3.86389e10	−	2.23082e10i	−1.00325	−	0.579227i	−0.0940422	−	0.995568i	$-0.529979\pi$
$443$	−3.86389e10	−	2.23082e10i	−1.00325	−	0.579227i	−0.909208	+	0.416341i	$0.863312\pi$
$444$		0			0
$445$	−8.28976e9	−	1.43583e10i	−0.211398	−	0.366153i
$446$		0			0
$447$	1.39596e10	+	7.22118e10i	0.349657	+	1.80875i
$448$		0			0
$449$		−	2.25839e10i		−	0.555665i	−0.960630	−	0.277833i	$-0.910384\pi$
$449$		−	2.25839e10i		−	0.555665i	0.960630	−	0.277833i	$-0.0896161\pi$
$450$		0			0
$451$	2.46907e10			0.596797
$452$		0			0
$453$	2.45408e10	+	2.82896e10i	0.582768	+	0.671790i
$454$		0			0
$455$	2.81096e9	−	1.62291e9i	0.0655857	−	0.0378659i
$456$		0			0
$457$	−1.27122e10	+	2.20181e10i	−0.291444	+	0.504795i	−0.974151	−	0.225897i	$-0.927469\pi$
$457$	−1.27122e10	+	2.20181e10i	−0.291444	+	0.504795i	0.682708	+	0.730692i	$0.260802\pi$
$458$		0			0
$459$	2.23551e10	+	3.46572e10i	0.503646	+	0.780806i
$460$		0			0
$461$	3.23124e10	+	1.86556e10i	0.715427	+	0.413052i	0.813067	−	0.582170i	$-0.197796\pi$
$461$	3.23124e10	+	1.86556e10i	0.715427	+	0.413052i	−0.0976401	+	0.995222i	$0.531129\pi$
$462$		0			0
$463$	2.98756e10	+	5.17461e10i	0.650119	+	1.12604i	0.983094	+	0.183103i	$0.0586143\pi$
$463$	2.98756e10	+	5.17461e10i	0.650119	+	1.12604i	−0.332975	+	0.942936i	$0.608052\pi$
$464$		0			0
$465$	1.56905e10	−	1.36112e10i	0.335602	−	0.291129i
$466$		0			0
$467$		−	1.51518e10i		−	0.318565i	−0.987233	−	0.159282i	$-0.949082\pi$
$467$		−	1.51518e10i		−	0.318565i	0.987233	−	0.159282i	$-0.0509181\pi$
$468$		0			0
$469$	3.47592e10			0.718420
$470$		0			0
$471$	−5.37125e10	+	1.03834e10i	−1.09142	+	0.210987i
$472$		0			0
$473$	−1.35814e10	+	7.84124e9i	−0.271332	+	0.156654i
$474$		0			0
$475$	2.05291e10	−	3.55575e10i	0.403270	−	0.698485i
$476$		0			0
$477$	9.33207e8	−	1.18940e9i	0.0180262	−	0.0229750i
$478$		0			0
$479$	2.42241e10	+	1.39858e10i	0.460157	+	0.265672i	0.712110	−	0.702068i	$-0.247740\pi$
$479$	2.42241e10	+	1.39858e10i	0.460157	+	0.265672i	−0.251953	+	0.967739i	$0.581073\pi$
$480$		0			0
$481$	−1.47790e10	−	2.55980e10i	−0.276100	−	0.478219i
$482$		0			0
$483$	−2.04423e10	−	7.06235e9i	−0.375614	−	0.129766i
$484$		0			0
$485$			1.46557e10i			0.264874i
$486$		0			0
$487$	3.80481e10			0.676422			0.338211	−	0.941070i	$-0.390178\pi$
$487$	3.80481e10			0.676422			0.338211	+	0.941070i	$0.390178\pi$
$488$		0			0
$489$	1.81309e10	−	5.24808e10i	0.317091	−	0.917836i
$490$		0			0
$491$	3.98522e8	−	2.30087e8i	0.00685687	−	0.00395882i	−0.496568	−	0.867998i	$-0.665407\pi$
$491$	3.98522e8	−	2.30087e8i	0.00685687	−	0.00395882i	0.503425	+	0.864039i	$0.332073\pi$
$492$		0			0
$493$	1.04274e10	−	1.80607e10i	0.176517	−	0.305737i
$494$		0			0
$495$	−4.79651e9	−	3.76334e9i	−0.0798922	−	0.0626834i
$496$		0			0
$497$	−1.61725e10	−	9.33718e9i	−0.265064	−	0.153035i
$498$		0			0
$499$	4.74964e10	+	8.22662e10i	0.766053	+	1.32684i	0.939688	+	0.342032i	$0.111115\pi$
$499$	4.74964e10	+	8.22662e10i	0.766053	+	1.32684i	−0.173635	+	0.984810i	$0.555551\pi$
$500$		0			0
$501$	−2.88003e9	−	1.48982e10i	−0.0457137	−	0.236473i
$502$		0			0
$503$		−	4.39958e10i		−	0.687289i	−0.939100	−	0.343644i	$-0.888339\pi$
$503$		−	4.39958e10i		−	0.687289i	0.939100	−	0.343644i	$-0.111661\pi$
$504$		0			0
$505$	2.12539e10			0.326793
$506$		0			0
$507$	2.12539e10	+	2.45006e10i	0.321668	+	0.370805i
$508$		0			0
$509$	6.21327e9	−	3.58723e9i	0.0925655	−	0.0534427i	−0.453003	−	0.891509i	$-0.649647\pi$
$509$	6.21327e9	−	3.58723e9i	0.0925655	−	0.0534427i	0.545568	+	0.838066i	$0.316314\pi$
$510$		0			0
$511$	−6.85750e9	+	1.18775e10i	−0.100573	+	0.174198i
$512$		0			0
$513$	−5.09596e10	+	3.28707e10i	−0.735795	+	0.474613i
$514$		0			0
$515$	2.60930e10	+	1.50648e10i	0.370932	+	0.214158i
$516$		0			0
$517$	−9.20942e9	−	1.59512e10i	−0.128905	−	0.223270i
$518$		0			0
$519$	2.99305e10	−	2.59642e10i	0.412519	−	0.357854i
$520$		0			0
$521$		−	9.87279e10i		−	1.33995i	−0.742383	−	0.669975i	$-0.766305\pi$
$521$		−	9.87279e10i		−	1.33995i	0.742383	−	0.669975i	$-0.233695\pi$
$522$		0			0
$523$	8.57689e10			1.14636			0.573182	−	0.819428i	$-0.305709\pi$
$523$	8.57689e10			1.14636			0.573182	+	0.819428i	$0.305709\pi$
$524$		0			0
$525$	2.59686e10	−	5.02010e9i	0.341831	−	0.0660808i
$526$		0			0
$527$	9.81965e10	−	5.66938e10i	1.27307	−	0.735009i
$528$		0			0
$529$	4.12991e9	−	7.15321e9i	0.0527373	−	0.0913436i
$530$		0			0
$531$	1.68345e10	+	2.40122e9i	0.211750	+	0.0302033i
$532$		0			0
$533$	−8.23043e10	−	4.75184e10i	−1.01980	−	0.588780i
$534$		0			0
$535$	1.92325e10	+	3.33116e10i	0.234758	+	0.406612i
$536$		0			0
$537$	4.04856e10	+	1.39869e10i	0.486860	+	0.168199i
$538$		0			0
$539$		−	4.36023e9i		−	0.0516599i
$540$		0			0
$541$	1.32377e11			1.54533			0.772667	−	0.634811i	$-0.218922\pi$
$541$	1.32377e11			1.54533			0.772667	+	0.634811i	$0.218922\pi$
$542$		0			0
$543$	−3.68312e10	+	1.06610e11i	−0.423659	+	1.22630i
$544$		0			0
$545$	4.95712e9	−	2.86199e9i	0.0561879	−	0.0324401i
$546$		0			0
$547$	4.03528e10	−	6.98932e10i	0.450739	−	0.780702i	−0.547693	−	0.836679i	$-0.684494\pi$
$547$	4.03528e10	−	6.98932e10i	0.450739	−	0.780702i	0.998432	+	0.0559769i	$0.0178273\pi$
$548$		0			0
$549$	−9.88206e10	+	3.96901e10i	−1.08782	+	0.436911i
$550$		0			0
$551$	2.65563e10	+	1.53323e10i	0.288112	+	0.166342i
$552$		0			0
$553$	4.45802e9	+	7.72152e9i	0.0476696	+	0.0825662i
$554$		0			0
$555$	3.91359e9	+	2.02447e10i	0.0412481	+	0.213373i
$556$		0			0
$557$		−	1.12556e11i		−	1.16935i	−0.811266	−	0.584677i	$-0.801221\pi$
$557$		−	1.12556e11i		−	1.16935i	0.811266	−	0.584677i	$-0.198779\pi$
$558$		0			0
$559$	6.03635e10			0.618197
$560$		0			0
$561$	−2.18081e10	−	2.51395e10i	−0.220174	−	0.253807i
$562$		0			0
$563$	1.14159e11	−	6.59097e10i	1.13626	−	0.656018i	0.190756	−	0.981637i	$-0.438906\pi$
$563$	1.14159e11	−	6.59097e10i	1.13626	−	0.656018i	0.945501	+	0.325619i	$0.105573\pi$
$564$		0			0
$565$	6.71545e9	−	1.16315e10i	0.0658994	−	0.114141i
$566$		0			0
$567$	−3.75067e10	−	1.09219e10i	−0.362892	−	0.105673i
$568$		0			0
$569$	−3.93006e10	−	2.26902e10i	−0.374930	−	0.216466i	0.300680	−	0.953725i	$-0.402786\pi$
$569$	−3.93006e10	−	2.26902e10i	−0.374930	−	0.216466i	−0.675610	+	0.737259i	$0.736120\pi$
$570$		0			0
$571$	−3.75464e10	−	6.50323e10i	−0.353203	−	0.611765i	0.633606	−	0.773656i	$-0.281574\pi$
$571$	−3.75464e10	−	6.50323e10i	−0.353203	−	0.611765i	−0.986809	+	0.161891i	$0.948241\pi$
$572$		0			0
$573$	2.84148e9	−	2.46494e9i	0.0263589	−	0.0228659i
$574$		0			0
$575$			1.05870e11i			0.968504i
$576$		0			0
$577$	−4.26783e10			−0.385039			−0.192519	−	0.981293i	$-0.561666\pi$
$577$	−4.26783e10			−0.385039			−0.192519	+	0.981293i	$0.561666\pi$
$578$		0			0
$579$	−1.17329e10	+	2.26814e9i	−0.104398	+	0.0201816i
$580$		0			0
$581$	−2.36278e10	+	1.36415e10i	−0.207357	+	0.119718i
$582$		0			0
$583$	−6.09985e8	+	1.05652e9i	−0.00528013	+	0.00914545i
$584$		0			0
$585$	8.74604e9	+	2.17759e10i	0.0746772	+	0.185932i
$586$		0			0
$587$	−1.11663e11	−	6.44686e10i	−0.940495	−	0.542995i	−0.0503793	−	0.998730i	$-0.516043\pi$
$587$	−1.11663e11	−	6.44686e10i	−0.940495	−	0.542995i	−0.890115	+	0.455735i	$0.849376\pi$
$588$		0			0
$589$	8.33620e10	+	1.44387e11i	0.692639	+	1.19969i
$590$		0			0
$591$	5.40010e9	+	1.86561e9i	0.0442641	+	0.0152922i
$592$		0			0
$593$		−	1.79018e11i		−	1.44770i	−0.689957	−	0.723851i	$-0.742370\pi$
$593$		−	1.79018e11i		−	1.44770i	0.689957	−	0.723851i	$-0.257630\pi$
$594$		0			0
$595$	−1.23601e10			−0.0986178
$596$		0			0
$597$	−5.48309e10	+	1.58711e11i	−0.431646	+	1.24942i
$598$		0			0
$599$	−1.70973e11	+	9.87113e10i	−1.32807	+	0.766761i	−0.985001	−	0.172550i	$-0.944799\pi$
$599$	−1.70973e11	+	9.87113e10i	−1.32807	+	0.766761i	−0.343068	+	0.939311i	$0.611466\pi$
$600$		0			0
$601$	6.71568e10	−	1.16319e11i	0.514745	−	0.891564i	−0.485109	−	0.874454i	$-0.661220\pi$
$601$	6.71568e10	−	1.16319e11i	0.514745	−	0.891564i	0.999854	−	0.0171105i	$-0.00544670\pi$
$602$		0			0
$603$	−3.54858e10	+	2.48785e11i	−0.268402	+	1.88172i
$604$		0			0
$605$	−2.83209e10	−	1.63511e10i	−0.211390	−	0.122046i
$606$		0			0
$607$	8.16170e10	+	1.41365e11i	0.601209	+	1.04132i	0.992638	+	0.121117i	$0.0386475\pi$
$607$	8.16170e10	+	1.41365e11i	0.601209	+	1.04132i	−0.391429	+	0.920208i	$0.628019\pi$
$608$		0			0
$609$	3.74929e9	+	1.93948e10i	0.0272571	+	0.140999i
$610$		0			0
$611$			7.08960e10i			0.508694i
$612$		0			0
$613$	1.51830e11			1.07527			0.537633	−	0.843179i	$-0.319319\pi$
$613$	1.51830e11			1.07527			0.537633	+	0.843179i	$0.319319\pi$
$614$		0			0
$615$	4.34433e10	+	5.00797e10i	0.303685	+	0.350075i
$616$		0			0
$617$	1.85080e11	−	1.06856e11i	1.27708	−	0.737323i	0.300770	−	0.953697i	$-0.402756\pi$
$617$	1.85080e11	−	1.06856e11i	1.27708	−	0.737323i	0.976311	+	0.216374i	$0.0694230\pi$
$618$		0			0
$619$	−5.05022e10	+	8.74724e10i	−0.343991	+	0.595811i	−0.985170	−	0.171581i	$-0.945112\pi$
$619$	−5.05022e10	+	8.74724e10i	−0.343991	+	0.595811i	0.641179	+	0.767392i	$0.278446\pi$
$620$		0			0
$621$	7.14175e10	−	1.39103e11i	0.480218	−	0.935342i
$622$		0			0
$623$	7.42414e10	+	4.28633e10i	0.492827	+	0.284534i
$624$		0			0
$625$	−5.87195e10	−	1.01705e11i	−0.384824	−	0.666535i
$626$		0			0
$627$	3.69648e10	−	3.20664e10i	0.239176	−	0.207482i
$628$		0			0
$629$			1.12558e11i			0.719072i
$630$		0			0
$631$	6.45734e10			0.407320			0.203660	−	0.979042i	$-0.434716\pi$
$631$	6.45734e10			0.407320			0.203660	+	0.979042i	$0.434716\pi$
$632$		0			0
$633$	−8.11060e9	+	1.56789e9i	−0.0505171	+	0.00976567i
$634$		0			0
$635$	5.23322e10	−	3.02140e10i	0.321865	−	0.185829i
$636$		0			0
$637$	−8.39147e9	+	1.45345e10i	−0.0509660	+	0.0882757i
$638$		0			0
$639$	8.33402e10	−	1.06220e11i	0.499863	−	0.637093i
$640$		0			0
$641$	−1.80435e10	−	1.04174e10i	−0.106878	−	0.0617062i	0.445608	−	0.895228i	$-0.352988\pi$
$641$	−1.80435e10	−	1.04174e10i	−0.106878	−	0.0617062i	−0.552486	+	0.833522i	$0.686321\pi$
$642$		0			0
$643$	1.24834e10	+	2.16220e10i	0.0730282	+	0.126489i	0.900227	−	0.435421i	$-0.143400\pi$
$643$	1.24834e10	+	2.16220e10i	0.0730282	+	0.126489i	−0.827199	+	0.561909i	$0.810067\pi$
$644$		0			0
$645$	−3.98009e10	−	1.37503e10i	−0.229961	−	0.0794462i
$646$		0			0
$647$			1.80439e11i			1.02971i	0.857278	+	0.514853i	$0.172154\pi$
$647$			1.80439e11i			1.02971i	−0.857278	+	0.514853i	$0.827846\pi$
$648$		0			0
$649$	−1.37223e10			−0.0773478
$650$		0			0
$651$	−3.50710e10	+	1.01515e11i	−0.195265	+	0.565203i
$652$		0			0
$653$	−8.35648e10	+	4.82462e10i	−0.459590	+	0.265345i	−0.711872	−	0.702309i	$-0.752152\pi$
$653$	−8.35648e10	+	4.82462e10i	−0.459590	+	0.265345i	0.252282	+	0.967654i	$0.418819\pi$
$654$		0			0
$655$	2.39453e10	−	4.14745e10i	0.130093	−	0.225329i
$656$		0			0
$657$	−7.80111e10	−	6.12075e10i	−0.418692	−	0.328506i
$658$		0			0
$659$	−2.72109e11	−	1.57102e11i	−1.44279	−	0.832993i	−0.444751	−	0.895654i	$-0.646708\pi$
$659$	−2.72109e11	−	1.57102e11i	−1.44279	−	0.832993i	−0.998035	+	0.0626612i	$0.980041\pi$
$660$		0			0
$661$	−1.10345e11	−	1.91123e11i	−0.578024	−	1.00117i	−0.995706	−	0.0925733i	$-0.970491\pi$
$661$	−1.10345e11	−	1.91123e11i	−0.578024	−	1.00117i	0.417682	−	0.908593i	$-0.362843\pi$
$662$		0			0
$663$	2.43134e10	+	1.25771e11i	0.125832	+	0.650919i
$664$		0			0
$665$		−	1.81742e10i		−	0.0929329i
$666$		0			0
$667$	−7.90696e10			−0.399490
$668$		0			0
$669$	−3.50526e10	−	4.04071e10i	−0.174991	−	0.201722i
$670$		0			0
$671$	7.44227e10	−	4.29680e10i	0.367126	−	0.211960i
$672$		0			0
$673$	8.24643e10	−	1.42832e11i	0.401981	−	0.696252i	−0.591984	−	0.805950i	$-0.701655\pi$
$673$	8.24643e10	−	1.42832e11i	0.401981	−	0.696252i	0.993965	+	0.109698i	$0.0349883\pi$
$674$		0			0
$675$	9.41927e9	+	1.90992e11i	0.0453735	+	0.920025i
$676$		0			0
$677$	2.07045e10	+	1.19538e10i	0.0985622	+	0.0569049i	0.548471	−	0.836170i	$-0.315210\pi$
$677$	2.07045e10	+	1.19538e10i	0.0985622	+	0.0569049i	−0.449909	+	0.893075i	$0.648543\pi$
$678$		0			0
$679$	−3.78896e10	−	6.56268e10i	−0.178255	−	0.308746i
$680$		0			0
$681$	−2.84200e11	+	2.46539e11i	−1.32140	+	1.14630i
$682$		0			0
$683$			3.73875e11i			1.71808i	0.511908	+	0.859040i	$0.328939\pi$
$683$			3.73875e11i			1.71808i	−0.511908	+	0.859040i	$0.671061\pi$
$684$		0			0
$685$	6.38008e10			0.289777
$686$		0			0
$687$	−8.72674e9	+	1.68700e9i	−0.0391765	+	0.00757337i
$688$		0			0
$689$	4.06667e9	−	2.34789e9i	0.0180452	−	0.0104184i
$690$		0			0
$691$	−4.81237e10	+	8.33527e10i	−0.211080	+	0.365601i	−0.952053	−	0.305934i	$-0.901031\pi$
$691$	−4.81237e10	+	8.33527e10i	−0.211080	+	0.365601i	0.740973	+	0.671535i	$0.234365\pi$
$692$		0			0
$693$	3.12077e10	+	4.45137e9i	0.135310	+	0.0193002i
$694$		0			0
$695$	7.29505e10	+	4.21180e10i	0.312672	+	0.180521i
$696$		0			0
$697$	1.80951e11	+	3.13416e11i	0.766707	+	1.32798i
$698$		0			0
$699$	−8.97105e10	−	3.09929e10i	−0.375781	−	0.129824i
$700$		0			0
$701$		−	1.85660e11i		−	0.768860i	−0.923154	−	0.384430i	$-0.874398\pi$
$701$		−	1.85660e11i		−	0.768860i	0.923154	−	0.384430i	$-0.125602\pi$
$702$		0			0
$703$	−1.65504e11			−0.677620
$704$		0			0
$705$	1.61495e10	−	4.67455e10i	0.0653737	−	0.189227i
$706$		0			0
$707$	−9.51728e10	+	5.49480e10i	−0.380921	+	0.219925i
$708$		0			0
$709$	2.95355e10	−	5.11569e10i	0.116885	−	0.202451i	−0.801647	−	0.597798i	$-0.796042\pi$
$709$	2.95355e10	−	5.11569e10i	0.116885	−	0.202451i	0.918532	+	0.395347i	$0.129376\pi$
$710$		0			0
$711$	−5.98169e10	+	2.40248e10i	−0.234070	+	0.0940114i
$712$		0			0
$713$	−3.72307e11	−	2.14951e11i	−1.44060	−	0.831729i
$714$		0			0
$715$	−9.46834e9	−	1.63996e10i	−0.0362284	−	0.0627495i
$716$		0			0
$717$	−3.27976e10	−	1.69659e11i	−0.124098	−	0.641950i
$718$		0			0
$719$		−	2.65867e11i		−	0.994831i	−0.867512	−	0.497416i	$-0.834282\pi$
$719$		−	2.65867e11i		−	0.994831i	0.867512	−	0.497416i	$-0.165718\pi$
$720$		0			0
$721$	−1.55789e11			−0.576495
$722$		0			0
$723$	−1.43366e11	−	1.65267e11i	−0.524679	−	0.604828i
$724$		0			0
$725$	8.37417e10	−	4.83483e10i	0.303103	−	0.174996i
$726$		0			0
$727$	−1.86582e11	+	3.23170e11i	−0.667932	+	1.15689i	0.310550	+	0.950557i	$0.399487\pi$
$727$	−1.86582e11	+	3.23170e11i	−0.667932	+	1.15689i	−0.978481	+	0.206335i	$0.933846\pi$
$728$		0			0
$729$	1.16463e11	−	2.57299e11i	0.412360	−	0.911021i
$730$		0			0
$731$	−1.99069e11	−	1.14933e11i	−0.697163	−	0.402507i
$732$		0			0
$733$	−7.05761e10	−	1.22241e11i	−0.244479	−	0.423450i	0.717506	−	0.696552i	$-0.245284\pi$
$733$	−7.05761e10	−	1.22241e11i	−0.244479	−	0.423450i	−0.961985	+	0.273102i	$0.911950\pi$
$734$		0			0
$735$	8.84378e9	−	7.67184e9i	0.0303032	−	0.0262876i
$736$		0			0
$737$		−	2.02791e11i		−	0.687353i
$738$		0			0
$739$	−2.86877e11			−0.961875			−0.480937	−	0.876755i	$-0.659704\pi$
$739$	−2.86877e11			−0.961875			−0.480937	+	0.876755i	$0.659704\pi$
$740$		0			0
$741$	−1.84933e11	+	3.57501e10i	−0.613396	+	0.118578i
$742$		0			0
$743$	−2.37516e10	+	1.37130e10i	−0.0779361	+	0.0449964i	−0.538462	−	0.842650i	$-0.680994\pi$
$743$	−2.37516e10	+	1.37130e10i	−0.0779361	+	0.0449964i	0.460526	+	0.887646i	$0.347661\pi$
$744$		0			0
$745$	7.96818e10	−	1.38013e11i	0.258663	−	0.448017i
$746$		0			0
$747$	−7.35156e10	−	1.83039e11i	−0.236101	−	0.587844i
$748$		0			0
$749$	−1.72242e11	−	9.94441e10i	−0.547284	−	0.315974i
$750$		0			0
$751$	4.28712e10	+	7.42551e10i	0.134774	+	0.233435i	0.925511	−	0.378721i	$-0.123636\pi$
$751$	4.28712e10	+	7.42551e10i	0.134774	+	0.233435i	−0.790737	+	0.612156i	$0.790303\pi$
$752$		0			0
$753$	3.20689e11	+	1.10791e11i	0.997479	+	0.344606i
$754$		0			0
$755$		−	8.11472e10i		−	0.249739i
$756$		0			0
$757$	−3.84864e11			−1.17199			−0.585995	−	0.810314i	$-0.699296\pi$
$757$	−3.84864e11			−1.17199			−0.585995	+	0.810314i	$0.699296\pi$
$758$		0			0
$759$	−4.12030e10	+	1.19264e11i	−0.124154	+	0.359371i
$760$		0			0
$761$	1.41084e10	−	8.14546e9i	0.0420666	−	0.0242872i	−0.478819	−	0.877914i	$-0.658935\pi$
$761$	1.41084e10	−	8.14546e9i	0.0420666	−	0.0242872i	0.520886	+	0.853626i	$0.325602\pi$
$762$		0			0
$763$	−1.47983e10	+	2.56314e10i	−0.0436631	+	0.0756266i
$764$		0			0
$765$	1.26185e10	−	8.84660e10i	0.0368436	−	0.258304i
$766$		0			0
$767$	4.57422e10	+	2.64093e10i	0.132171	+	0.0763088i
$768$		0			0
$769$	−1.98141e11	−	3.43190e11i	−0.566590	−	0.981363i	−0.996900	−	0.0786813i	$-0.974929\pi$
$769$	−1.98141e11	−	3.43190e11i	−0.566590	−	0.981363i	0.430310	−	0.902681i	$-0.358404\pi$
$770$		0			0
$771$	−1.09087e11	−	5.64301e11i	−0.308715	−	1.59696i
$772$		0			0
$773$		−	6.53340e10i		−	0.182988i	−0.995806	−	0.0914938i	$-0.970836\pi$
$773$		−	6.53340e10i		−	0.182988i	0.995806	−	0.0914938i	$-0.0291642\pi$
$774$		0			0
$775$	5.25741e11			1.45735
$776$		0			0
$777$	−6.98637e10	−	8.05359e10i	−0.191676	−	0.220956i
$778$		0			0
$779$	−4.60844e11	+	2.66069e11i	−1.25142	+	0.722510i
$780$		0			0
$781$	−5.44747e10	+	9.43530e10i	−0.146417	+	0.253601i
$782$		0			0
$783$	−1.42643e11	+	7.03483e9i	−0.379493	+	0.0187157i
$784$		0			0
$785$	1.02657e11	+	5.92689e10i	0.270339	+	0.156080i
$786$		0			0
$787$	−5.18896e10	−	8.98755e10i	−0.135264	−	0.234284i	0.790434	−	0.612547i	$-0.209855\pi$
$787$	−5.18896e10	−	8.98755e10i	−0.135264	−	0.234284i	−0.925698	+	0.378263i	$0.876522\pi$
$788$		0			0
$789$	3.66267e11	−	3.17731e11i	0.945126	−	0.819883i
$790$		0			0
$791$			6.94463e10i			0.177396i
$792$		0			0
$793$	−3.30776e11			−0.836453
$794$		0			0
$795$	−3.21620e9	+	6.21738e8i	−0.00805146	+	0.00155646i
$796$		0			0
$797$	1.83517e11	−	1.05953e11i	0.454823	−	0.262592i	−0.255042	−	0.966930i	$-0.582089\pi$
$797$	1.83517e11	−	1.05953e11i	0.454823	−	0.262592i	0.709865	+	0.704338i	$0.248756\pi$
$798$		0			0
$799$	1.34987e11	−	2.33803e11i	0.331210	−	0.573672i
$800$		0			0
$801$	−3.82582e11	+	4.87614e11i	−0.929382	+	1.18453i
$802$		0			0
$803$	6.92957e10	+	4.00079e10i	0.166665	+	0.0962240i
$804$		0			0
$805$	2.34314e10	+	4.05844e10i	0.0557975	+	0.0966441i
$806$		0			0
$807$	3.16070e11	+	1.09195e11i	0.745227	+	0.257459i
$808$		0			0
$809$			1.15554e11i			0.269769i	0.990861	+	0.134884i	$0.0430663\pi$
$809$			1.15554e11i			0.269769i	−0.990861	+	0.134884i	$0.956934\pi$
$810$		0			0
$811$	−2.31485e11			−0.535105			−0.267552	−	0.963543i	$-0.586215\pi$
$811$	−2.31485e11			−0.535105			−0.267552	+	0.963543i	$0.586215\pi$
$812$		0			0
$813$	6.12940e10	−	1.77418e11i	0.140299	−	0.406103i
$814$		0			0
$815$	−1.04191e11	+	6.01545e10i	−0.236156	+	0.136345i
$816$		0			0
$817$	1.68996e11	−	2.92709e11i	0.379304	−	0.656974i
$818$		0			0
$819$	−9.54616e10	−	7.48992e10i	−0.212174	−	0.166472i
$820$		0			0
$821$	1.98266e11	+	1.14469e11i	0.436390	+	0.251950i	0.702065	−	0.712113i	$-0.252262\pi$
$821$	1.98266e11	+	1.14469e11i	0.436390	+	0.251950i	−0.265675	+	0.964063i	$0.585595\pi$
$822$		0			0
$823$	3.08518e11	+	5.34368e11i	0.672482	+	1.16477i	0.977198	+	0.212330i	$0.0681051\pi$
$823$	3.08518e11	+	5.34368e11i	0.672482	+	1.16477i	−0.304716	+	0.952443i	$0.598562\pi$
$824$		0			0
$825$	−2.92882e10	−	1.51505e11i	−0.0632232	−	0.327049i
$826$		0			0
$827$			5.47691e11i			1.17088i	0.810715	+	0.585442i	$0.199079\pi$
$827$			5.47691e11i			1.17088i	−0.810715	+	0.585442i	$0.800921\pi$
$828$		0			0
$829$	−2.35777e11			−0.499211			−0.249605	−	0.968348i	$-0.580301\pi$
$829$	−2.35777e11			−0.499211			−0.249605	+	0.968348i	$0.580301\pi$
$830$		0			0
$831$	1.56444e11	+	1.80342e11i	0.328061	+	0.378175i
$832$		0			0
$833$	5.53475e10	−	3.19549e10i	0.114952	−	0.0663678i
$834$		0			0
$835$	−1.64393e10	+	2.84738e10i	−0.0338173	+	0.0585732i
$836$		0			0
$837$	−6.90773e11	−	3.54653e11i	−1.40745	−	0.722606i
$838$		0			0
$839$	2.76270e11	+	1.59504e11i	0.557552	+	0.321903i	0.752162	−	0.658978i	$-0.229011\pi$
$839$	2.76270e11	+	1.59504e11i	0.557552	+	0.321903i	−0.194610	+	0.980881i	$0.562344\pi$
$840$		0			0
$841$	−2.14014e11	−	3.70683e11i	−0.427817	−	0.741001i
$842$		0			0
$843$	−1.37228e11	+	1.19043e11i	−0.271726	+	0.235718i
$844$		0			0
$845$		−	7.02788e10i		−	0.137847i
$846$		0			0
$847$	1.69091e11			0.328539
$848$		0			0
$849$	4.95087e11	−	9.57075e10i	0.952908	−	0.184211i
$850$		0			0
$851$	3.69582e11	−	2.13378e11i	0.704680	−	0.406847i
$852$		0			0
$853$	−2.06299e11	+	3.57321e11i	−0.389674	+	0.674935i	−0.992406	−	0.123009i	$-0.960746\pi$
$853$	−2.06299e11	+	3.57321e11i	−0.389674	+	0.674935i	0.602732	+	0.797944i	$0.294079\pi$
$854$		0			0
$855$	1.30080e11	+	1.85541e10i	0.243414	+	0.0347197i
$856$		0			0
$857$	−6.30153e11	−	3.63819e11i	−1.16821	−	0.674469i	−0.214955	−	0.976624i	$-0.568961\pi$
$857$	−6.30153e11	−	3.63819e11i	−1.16821	−	0.674469i	−0.953259	+	0.302155i	$0.902294\pi$
$858$		0			0
$859$	−4.79374e11	−	8.30300e11i	−0.880444	−	1.52497i	−0.850848	−	0.525412i	$-0.823911\pi$
$859$	−4.79374e11	−	8.30300e11i	−0.880444	−	1.52497i	−0.0295957	−	0.999562i	$-0.509422\pi$
$860$		0			0
$861$	−3.24007e11	−	1.11937e11i	−0.589578	−	0.203686i
$862$		0			0
$863$			6.69243e11i			1.20654i	0.797538	+	0.603268i	$0.206135\pi$
$863$			6.69243e11i			1.20654i	−0.797538	+	0.603268i	$0.793865\pi$
$864$		0			0
$865$	−8.58539e10			−0.153354
$866$		0			0
$867$	−2.52189e10	+	7.29972e10i	−0.0446323	+	0.129190i
$868$		0			0
$869$	4.50487e10	−	2.60089e10i	0.0789956	−	0.0456081i
$870$		0			0
$871$	−3.90282e11	+	6.75989e11i	−0.678120	+	1.17454i
$872$		0			0
$873$	5.08397e11	−	2.04191e11i	0.875277	−	0.351545i
$874$		0			0
$875$	−1.03512e11	−	5.97630e10i	−0.176588	−	0.101953i
$876$		0			0
$877$	1.36073e11	+	2.35686e11i	0.230025	+	0.398415i	0.957815	−	0.287385i	$-0.0927859\pi$
$877$	1.36073e11	+	2.35686e11i	0.230025	+	0.398415i	−0.727790	+	0.685800i	$0.759453\pi$
$878$		0			0
$879$	2.05568e11	+	1.06339e12i	0.344350	+	1.78130i
$880$		0			0
$881$			5.36764e11i			0.891004i	0.895281	+	0.445502i	$0.146975\pi$
$881$			5.36764e11i			0.891004i	−0.895281	+	0.445502i	$0.853025\pi$
$882$		0			0
$883$	4.00106e11			0.658161			0.329081	−	0.944302i	$-0.393261\pi$
$883$	4.00106e11			0.658161			0.329081	+	0.944302i	$0.393261\pi$
$884$		0			0
$885$	−2.41445e10	−	2.78327e10i	−0.0393590	−	0.0453714i
$886$		0			0
$887$	4.72684e11	−	2.72904e11i	0.763618	−	0.440875i	−0.0669754	−	0.997755i	$-0.521335\pi$
$887$	4.72684e11	−	2.72904e11i	0.763618	−	0.440875i	0.830593	+	0.556880i	$0.188002\pi$
$888$		0			0
$889$	−1.56226e11	+	2.70591e11i	−0.250118	+	0.433217i
$890$		0			0
$891$	−6.37202e10	+	2.18821e11i	−0.101104	+	0.347198i
$892$		0			0
$893$	3.43782e11	+	1.98483e11i	0.540602	+	0.312117i
$894$		0			0
$895$	−4.64054e10	−	8.03766e10i	−0.0723231	−	0.125267i
$896$		0			0
$897$	3.66877e11	−	3.18260e11i	0.566696	−	0.491600i
$898$		0			0
$899$			3.92652e11i			0.601131i
$900$		0			0
$901$	−1.78816e10			−0.0271336
$902$		0			0
$903$	2.13773e11	−	4.13254e10i	0.321516	−	0.0621536i
$904$		0			0
$905$	2.11653e11	−	1.22198e11i	0.315523	−	0.182167i
$906$		0			0
$907$	−2.65476e11	+	4.59818e11i	−0.392280	+	0.679449i	−0.992750	−	0.120198i	$-0.961647\pi$
$907$	−2.65476e11	+	4.59818e11i	−0.392280	+	0.679449i	0.600470	+	0.799647i	$0.294980\pi$
$908$		0			0
$909$	−2.96121e11	−	7.37283e11i	−0.433724	−	1.07989i
$910$		0			0
$911$	1.35769e11	+	7.83863e10i	0.197119	+	0.113806i	0.595311	−	0.803496i	$-0.297029\pi$
$911$	1.35769e11	+	7.83863e10i	0.197119	+	0.113806i	−0.398192	+	0.917302i	$0.630362\pi$
$912$		0			0
$913$	7.95870e10	+	1.37849e11i	0.114540	+	0.198390i
$914$		0			0
$915$	2.18098e11	+	7.53481e10i	0.311149	+	0.107495i
$916$		0			0
$917$			2.47625e11i			0.350201i
$918$		0			0
$919$	−1.03308e12			−1.44834			−0.724170	−	0.689621i	$-0.757777\pi$
$919$	−1.03308e12			−1.44834			−0.724170	+	0.689621i	$0.757777\pi$
$920$		0			0
$921$	−1.86438e11	+	5.39653e11i	−0.259117	+	0.750026i
$922$		0			0
$923$	3.63174e11	−	2.09679e11i	0.500390	−	0.288900i
$924$		0			0
$925$	−2.60946e11	+	4.51972e11i	−0.356438	+	0.617369i
$926$		0			0
$927$	1.59046e11	−	1.11504e12i	0.215379	−	1.50998i
$928$		0			0
$929$	1.23776e12	+	7.14621e11i	1.66178	+	0.959429i	0.971865	+	0.235538i	$0.0756851\pi$
$929$	1.23776e12	+	7.14621e11i	1.66178	+	0.959429i	0.689914	+	0.723891i	$0.257648\pi$
$930$		0			0
$931$	4.69862e10	+	8.13824e10i	0.0625419	+	0.108326i
$932$		0			0
$933$	6.78276e9	+	3.50867e10i	0.00895117	+	0.0463037i
$934$		0			0
$935$			7.21112e10i			0.0943531i
$936$		0			0
$937$	7.63306e11			0.990239			0.495120	−	0.868825i	$-0.335124\pi$
$937$	7.63306e11			0.990239			0.495120	+	0.868825i	$0.335124\pi$
$938$		0			0
$939$	4.07683e11	+	4.69960e11i	0.524397	+	0.604503i
$940$		0			0
$941$	1.60716e11	−	9.27897e10i	0.204975	−	0.118343i	−0.393999	−	0.919111i	$-0.628909\pi$
$941$	1.60716e11	−	9.27897e10i	0.204975	−	0.118343i	0.598974	+	0.800768i	$0.295575\pi$
$942$		0			0
$943$	6.86066e11	−	1.18830e12i	0.867599	−	1.50272i
$944$		0			0
$945$	4.58815e10	+	7.11304e10i	0.0575322	+	0.0891924i
$946$		0			0
$947$	−8.23181e10	−	4.75264e10i	−0.102352	−	0.0590928i	0.447950	−	0.894058i	$-0.352154\pi$
$947$	−8.23181e10	−	4.75264e10i	−0.102352	−	0.0590928i	−0.550302	+	0.834966i	$0.685487\pi$
$948$		0			0
$949$	−1.53994e11	−	2.66726e11i	−0.189863	−	0.328852i
$950$		0			0
$951$	1.19715e12	−	1.03851e12i	1.46362	−	1.26967i
$952$		0			0
$953$		−	2.67458e11i		−	0.324253i	−0.986770	−	0.162126i	$-0.948165\pi$
$953$		−	2.67458e11i		−	0.324253i	0.986770	−	0.162126i	$-0.0518352\pi$
$954$		0			0
$955$	−8.15065e9			−0.00979893
$956$		0			0
$957$	1.13153e11	−	2.18740e10i	0.134901	−	0.0260784i
$958$		0			0
$959$	−2.85694e11	+	1.64945e11i	−0.337774	+	0.195014i
$960$		0			0
$961$	−6.40983e11	+	1.11021e12i	−0.751541	+	1.30171i
$962$		0			0
$963$	8.87601e11	−	1.13128e12i	1.03208	−	1.31542i
$964$		0			0
$965$	2.24242e10	+	1.29466e10i	0.0258588	+	0.0149296i
$966$		0			0
$967$	−1.56014e11	−	2.70225e11i	−0.178426	−	0.309043i	0.762915	−	0.646498i	$-0.223767\pi$
$967$	−1.56014e11	−	2.70225e11i	−0.178426	−	0.309043i	−0.941342	+	0.337455i	$0.890434\pi$
$968$		0			0
$969$	6.77947e11	+	2.34215e11i	0.768954	+	0.265656i
$970$		0			0
$971$			9.71177e11i			1.09250i	0.837622	+	0.546250i	$0.183945\pi$
$971$			9.71177e11i			1.09250i	−0.837622	+	0.546250i	$0.816055\pi$
$972$		0			0
$973$	−4.35554e11			−0.485949
$974$		0			0
$975$	−1.93950e11	+	5.61397e11i	−0.214621	+	0.621229i
$976$		0			0
$977$	−1.06521e12	+	6.14997e11i	−1.16911	+	0.674985i	−0.953470	−	0.301488i	$-0.902517\pi$
$977$	−1.06521e12	+	6.14997e11i	−1.16911	+	0.674985i	−0.215639	+	0.976473i	$0.569183\pi$
$978$		0			0
$979$	2.50072e11	−	4.33138e11i	0.272229	−	0.471515i
$980$		0			0
$981$	−1.68346e11	−	1.32084e11i	−0.181772	−	0.142618i
$982$		0			0
$983$	1.14407e12	+	6.60528e11i	1.22529	+	0.707419i	0.966040	−	0.258392i	$-0.0831926\pi$
$983$	1.14407e12	+	6.60528e11i	1.22529	+	0.707419i	0.259246	+	0.965811i	$0.416526\pi$
$984$		0			0
$985$	−6.18970e9	−	1.07209e10i	−0.00657544	−	0.0113890i
$986$		0			0
$987$	4.85361e10	+	2.51073e11i	0.0511442	+	0.264565i
$988$		0			0
$989$			8.71521e11i			0.910946i
$990$		0			0
$991$	−3.62020e11			−0.375351			−0.187676	−	0.982231i	$-0.560095\pi$
$991$	−3.62020e11			−0.375351			−0.187676	+	0.982231i	$0.560095\pi$
$992$		0			0
$993$	−1.66178e11	−	1.91563e11i	−0.170914	−	0.197022i
$994$		0			0
$995$	3.15090e11	−	1.81917e11i	0.321471	−	0.185601i
$996$		0			0
$997$	2.78647e11	−	4.82631e11i	0.282016	−	0.488467i	−0.689865	−	0.723938i	$-0.742330\pi$
$997$	2.78647e11	−	4.82631e11i	0.282016	−	0.488467i	0.971881	+	0.235471i	$0.0756634\pi$
$998$		0			0
$999$	6.47749e11	−	4.17820e11i	0.650347	−	0.419496i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.18	✓	96
9.5	odd	6	inner	252.9.bg.a.113.18	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.18	✓	96	1.1	even	1	trivial
252.9.bg.a.113.18	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)

\(f(q)\)	\(=\)	\(q+(-26.4497 + 76.5599i) q^{3} +(151.995 - 87.7545i) q^{5} +(-453.746 + 785.912i) q^{7} +(-5161.83 - 4049.97i) q^{9} +O(q^{10})\)	\(q+(-26.4497 + 76.5599i) q^{3} +(151.995 - 87.7545i) q^{5} +(-453.746 + 785.912i) q^{7} +(-5161.83 - 4049.97i) q^{9} +(4585.15 + 2647.24i) q^{11} +(-10189.5 - 17648.7i) q^{13} +(2698.24 + 13957.8i) q^{15} +77603.4i q^{17} -114107. q^{19} +(-48167.9 - 55525.9i) q^{21} +(254810. - 147115. i) q^{23} +(-179911. + 311615. i) q^{25} +(446594. - 288068. i) q^{27} +(-232731. - 134367. i) q^{29} +(-730557. - 1.26536e6i) q^{31} +(-323948. + 281020. i) q^{33} +159273. i q^{35} +1.45042e6 q^{37} +(1.62069e6 - 313303. i) q^{39} +(4.03869e6 - 2.33174e6i) q^{41} +(-1.48102e6 + 2.56521e6i) q^{43} +(-1.13998e6 - 162602. i) q^{45} +(-3.01280e6 - 1.73944e6i) q^{47} +(-411772. - 713209. i) q^{49} +(-5.94131e6 - 2.05259e6i) q^{51} +230423. i q^{53} +929227. q^{55} +(3.01810e6 - 8.73604e6i) q^{57} +(-2.24458e6 + 1.29591e6i) q^{59} +(8.11563e6 - 1.40567e7i) q^{61} +(5.52508e6 - 2.21908e6i) q^{63} +(-3.09750e6 - 1.78834e6i) q^{65} +(-1.91512e7 - 3.31709e7i) q^{67} +(4.52343e6 + 2.33994e7i) q^{69} +2.05780e7i q^{71} +1.51131e7 q^{73} +(-1.90986e7 - 2.20161e7i) q^{75} +(-4.16099e6 + 2.40235e6i) q^{77} +(4.91246e6 - 8.50863e6i) q^{79} +(1.02422e7 + 4.18105e7i) q^{81} +(2.60363e7 + 1.50321e7i) q^{83} +(6.81005e6 + 1.17953e7i) q^{85} +(1.64428e7 - 1.42639e7i) q^{87} -9.44654e7i q^{89} +1.84938e7 q^{91} +(1.16199e8 - 2.24629e7i) q^{93} +(-1.73438e7 + 1.00134e7i) q^{95} +(-4.17520e7 + 7.23166e7i) q^{97} +(-1.29465e7 - 3.22343e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 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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