LMFDB - Embedded newform 84.9.m.b.73.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(61,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("84.61");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.m (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:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + \cdots + 45\!\cdots\!96 $ x^12 - 3x^11 + 148097x^10 + 46071824x^9 + 21578502553x^8 + 3561445462121x^7 + 576413321817541x^6 + 47217566733462528x^5 + 5214056955297543333x^4 + 358752845334081085965x^3 + 30962072851910211245661x^2 + 1221542968331193193318500*x + 45396580558961892385326096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{10}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			73.2
Root			$-72.3408 + 125.298i$ of defining polynomial
Character	$\chi$	$=$	84.73
Dual form			84.9.m.b.61.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.5000 + 23.3827i) q^{3} +(-225.043 + 129.928i) q^{5} +(597.275 + 2325.52i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})$	\(q+(40.5000 + 23.3827i) q^{3} +(-225.043 + 129.928i) q^{5} +(597.275 + 2325.52i) q^{7} +(1093.50 + 1894.00i) q^{9} +(9723.49 - 16841.6i) q^{11} +46381.7i q^{13} -12152.3 q^{15} +(-105886. - 61133.6i) q^{17} +(18446.7 - 10650.2i) q^{19} +(-30187.4 + 108150. i) q^{21} +(152199. + 263616. i) q^{23} +(-161550. + 279812. i) q^{25} +102276. i q^{27} +88119.4 q^{29} +(-1.54998e6 - 894884. i) q^{31} +(787602. - 454722. i) q^{33} +(-436564. - 445739. i) q^{35} +(755092. + 1.30786e6i) q^{37} +(-1.08453e6 + 1.87846e6i) q^{39} +188022. i q^{41} -123402. q^{43} +(-492168. - 284153. i) q^{45} +(-5.58909e6 + 3.22686e6i) q^{47} +(-5.05133e6 + 2.77796e6i) q^{49} +(-2.85893e6 - 4.95182e6i) q^{51} +(-6.88508e6 + 1.19253e7i) q^{53} +5.05343e6i q^{55} +996123. q^{57} +(1.00305e7 + 5.79109e6i) q^{59} +(-1.92663e7 + 1.11234e7i) q^{61} +(-3.75142e6 + 3.67420e6i) q^{63} +(-6.02630e6 - 1.04379e7i) q^{65} +(5.75873e6 - 9.97442e6i) q^{67} +1.42353e7i q^{69} -3.11345e7 q^{71} +(-1.08437e7 - 6.26059e6i) q^{73} +(-1.30855e7 + 7.55493e6i) q^{75} +(4.49731e7 + 1.25532e7i) q^{77} +(2.13222e7 + 3.69312e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} -7.19982e7i q^{83} +3.17720e7 q^{85} +(3.56884e6 + 2.06047e6i) q^{87} +(4.37683e7 - 2.52696e7i) q^{89} +(-1.07862e8 + 2.77026e7i) q^{91} +(-4.18496e7 - 7.24856e7i) q^{93} +(-2.76753e6 + 4.79351e6i) q^{95} -1.16642e8i q^{97} +4.25305e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) $ 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9	$ 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99}+O(q^{100}) $ 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9 - 17919 * q^11 + 15390 * q^15 - 205782 * q^17 + 74313 * q^19 - 39609 * q^21 - 62832 * q^23 + 878679 * q^25 - 575454 * q^29 + 1442952 * q^31 - 1451439 * q^33 - 3989514 * q^35 - 2058621 * q^37 - 930933 * q^39 + 7721322 * q^43 + 623295 * q^45 + 12088194 * q^47 - 16964694 * q^49 - 5556114 * q^51 - 5506743 * q^53 + 4012902 * q^57 + 7511901 * q^59 - 37215576 * q^61 - 3641355 * q^63 + 5047122 * q^65 - 36824553 * q^67 - 30011556 * q^71 + 95080185 * q^73 + 71172999 * q^75 - 38333727 * q^77 + 8514456 * q^79 - 28697814 * q^81 + 20121540 * q^85 - 23305887 * q^87 + 83038554 * q^89 - 198538635 * q^91 + 38959704 * q^93 - 221605224 * q^95 - 78377706 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	40.5000	+	23.3827i	0.500000	+	0.288675i
$3$	40.5000	+	23.3827i	0.500000	+	0.288675i
$4$		0			0
$5$	−225.043	+	129.928i	−0.360068	+	0.207885i	−0.669111	−	0.743163i	$-0.733325\pi$
$5$	−225.043	+	129.928i	−0.360068	+	0.207885i	0.309042	+	0.951048i	$0.399991\pi$
$6$		0			0
$7$	597.275	+	2325.52i	0.248761	+	0.968565i
$7$	597.275	+	2325.52i	0.248761	+	0.968565i
$8$		0			0
$9$	1093.50	+	1894.00i	0.166667	+	0.288675i
$10$		0			0
$11$	9723.49	−	16841.6i	0.664127	−	1.15030i	−0.315394	−	0.948961i	$-0.602137\pi$
$11$	9723.49	−	16841.6i	0.664127	−	1.15030i	0.979521	−	0.201341i	$-0.0645300\pi$
$12$		0			0
$13$			46381.7i			1.62395i	0.583690	+	0.811976i	$0.301608\pi$
$13$			46381.7i			1.62395i	−0.583690	+	0.811976i	$0.698392\pi$
$14$		0			0
$15$	−12152.3			−0.240045
$16$		0			0
$17$	−105886.	−	61133.6i	−1.26778	−	0.731955i	−0.293215	−	0.956047i	$-0.594725\pi$
$17$	−105886.	−	61133.6i	−1.26778	−	0.731955i	−0.974568	+	0.224092i	$0.928058\pi$
$18$		0			0
$19$	18446.7	−	10650.2i	0.141548	−	0.0817230i	−0.427553	−	0.903990i	$-0.640624\pi$
$19$	18446.7	−	10650.2i	0.141548	−	0.0817230i	0.569102	+	0.822267i	$0.307291\pi$
$20$		0			0
$21$	−30187.4	+	108150.i	−0.155220	+	0.556094i
$22$		0			0
$23$	152199.	+	263616.i	0.543876	+	0.942020i	0.998677	+	0.0514271i	$0.0163770\pi$
$23$	152199.	+	263616.i	0.543876	+	0.942020i	−0.454801	+	0.890593i	$0.650290\pi$
$24$		0			0
$25$	−161550.	+	279812.i	−0.413567	+	0.716320i
$26$		0			0
$27$			102276.i			0.192450i
$28$		0			0
$29$	88119.4			0.124589			0.0622945	−	0.998058i	$-0.480158\pi$
$29$	88119.4			0.124589			0.0622945	+	0.998058i	$0.480158\pi$
$30$		0			0
$31$	−1.54998e6	−	894884.i	−1.67834	−	0.968991i	−0.962718	−	0.270505i	$-0.912809\pi$
$31$	−1.54998e6	−	894884.i	−1.67834	−	0.968991i	−0.715624	−	0.698486i	$-0.753857\pi$
$32$		0			0
$33$	787602.	−	454722.i	0.664127	−	0.383434i
$34$		0			0
$35$	−436564.	−	445739.i	−0.290921	−	0.297036i
$36$		0			0
$37$	755092.	+	1.30786e6i	0.402896	+	0.697836i	0.994074	−	0.108705i	$-0.0346703\pi$
$37$	755092.	+	1.30786e6i	0.402896	+	0.697836i	−0.591178	+	0.806541i	$0.701337\pi$
$38$		0			0
$39$	−1.08453e6	+	1.87846e6i	−0.468795	+	0.811976i
$40$		0			0
$41$			188022.i			0.0665385i	0.999446	+	0.0332692i	$0.0105919\pi$
$41$			188022.i			0.0665385i	−0.999446	+	0.0332692i	$0.989408\pi$
$42$		0			0
$43$	−123402.			−0.0360953			−0.0180476	−	0.999837i	$-0.505745\pi$
$43$	−123402.			−0.0360953			−0.0180476	+	0.999837i	$0.505745\pi$
$44$		0			0
$45$	−492168.	−	284153.i	−0.120023	−	0.0692952i
$46$		0			0
$47$	−5.58909e6	+	3.22686e6i	−1.14538	+	0.661285i	−0.947757	−	0.318993i	$-0.896655\pi$
$47$	−5.58909e6	+	3.22686e6i	−1.14538	+	0.661285i	−0.197623	+	0.980278i	$0.563322\pi$
$48$		0			0
$49$	−5.05133e6	+	2.77796e6i	−0.876236	+	0.481882i
$50$		0			0
$51$	−2.85893e6	−	4.95182e6i	−0.422594	−	0.731955i
$52$		0			0
$53$	−6.88508e6	+	1.19253e7i	−0.872580	+	1.51135i	−0.0132616	+	0.999912i	$0.504221\pi$
$53$	−6.88508e6	+	1.19253e7i	−0.872580	+	1.51135i	−0.859318	+	0.511441i	$0.829112\pi$
$54$		0			0
$55$			5.05343e6i			0.552250i
$56$		0			0
$57$	996123.			0.0943656
$58$		0			0
$59$	1.00305e7	+	5.79109e6i	0.827776	+	0.477917i	0.853090	−	0.521763i	$-0.174725\pi$
$59$	1.00305e7	+	5.79109e6i	0.827776	+	0.477917i	−0.0253148	+	0.999680i	$0.508059\pi$
$60$		0			0
$61$	−1.92663e7	+	1.11234e7i	−1.39149	+	0.803375i	−0.993480	−	0.114007i	$-0.963631\pi$
$61$	−1.92663e7	+	1.11234e7i	−1.39149	+	0.803375i	−0.398007	+	0.917383i	$0.630298\pi$
$62$		0			0
$63$	−3.75142e6	+	3.67420e6i	−0.238140	+	0.233239i
$64$		0			0
$65$	−6.02630e6	−	1.04379e7i	−0.337596	−	0.584734i
$66$		0			0
$67$	5.75873e6	−	9.97442e6i	0.285777	−	0.494981i	−0.687020	−	0.726638i	$-0.741082\pi$
$67$	5.75873e6	−	9.97442e6i	0.285777	−	0.494981i	0.972797	+	0.231658i	$0.0744149\pi$
$68$		0			0
$69$			1.42353e7i			0.628013i
$70$		0			0
$71$	−3.11345e7			−1.22520			−0.612602	−	0.790392i	$-0.709877\pi$
$71$	−3.11345e7			−1.22520			−0.612602	+	0.790392i	$0.709877\pi$
$72$		0			0
$73$	−1.08437e7	−	6.26059e6i	−0.381843	−	0.220457i	0.296777	−	0.954947i	$-0.404088\pi$
$73$	−1.08437e7	−	6.26059e6i	−0.381843	−	0.220457i	−0.678620	+	0.734490i	$0.737422\pi$
$74$		0			0
$75$	−1.30855e7	+	7.55493e6i	−0.413567	+	0.238773i
$76$		0			0
$77$	4.49731e7	+	1.25532e7i	1.27935	+	0.357100i
$78$		0			0
$79$	2.13222e7	+	3.69312e7i	0.547424	+	0.948167i	0.998450	+	0.0556558i	$0.0177249\pi$
$79$	2.13222e7	+	3.69312e7i	0.547424	+	0.948167i	−0.451026	+	0.892511i	$0.648942\pi$
$80$		0			0
$81$	−2.39148e6	+	4.14217e6i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		−	7.19982e7i		−	1.51708i	−0.651625	−	0.758541i	$-0.725912\pi$
$83$		−	7.19982e7i		−	1.51708i	0.651625	−	0.758541i	$-0.274088\pi$
$84$		0			0
$85$	3.17720e7			0.608651
$86$		0			0
$87$	3.56884e6	+	2.06047e6i	0.0622945	+	0.0359657i
$88$		0			0
$89$	4.37683e7	−	2.52696e7i	0.697589	−	0.402753i	−0.108860	−	0.994057i	$-0.534720\pi$
$89$	4.37683e7	−	2.52696e7i	0.697589	−	0.402753i	0.806449	+	0.591304i	$0.201387\pi$
$90$		0			0
$91$	−1.07862e8	+	2.77026e7i	−1.57290	+	0.403976i
$92$		0			0
$93$	−4.18496e7	−	7.24856e7i	−0.559447	−	0.968991i
$94$		0			0
$95$	−2.76753e6	+	4.79351e6i	−0.0339781	+	0.0588517i
$96$		0			0
$97$		−	1.16642e8i		−	1.31755i	−0.752339	−	0.658776i	$-0.771075\pi$
$97$		−	1.16642e8i		−	1.31755i	0.752339	−	0.658776i	$-0.228925\pi$
$98$		0			0
$99$	4.25305e7			0.442751
$100$		0			0
$101$	9.23786e7	+	5.33348e7i	0.887740	+	0.512537i	0.873203	−	0.487357i	$-0.162039\pi$
$101$	9.23786e7	+	5.33348e7i	0.887740	+	0.512537i	0.0145374	+	0.999894i	$0.495372\pi$
$102$		0			0
$103$	1.77725e8	−	1.02609e8i	1.57906	−	0.911671i	0.584070	−	0.811703i	$-0.301459\pi$
$103$	1.77725e8	−	1.02609e8i	1.57906	−	0.911671i	0.994991	−	0.0999675i	$-0.0318739\pi$
$104$		0			0
$105$	−7.25827e6	−	2.82605e7i	−0.0597139	−	0.232500i
$106$		0			0
$107$	8.90703e7	+	1.54274e8i	0.679513	+	1.17695i	0.975128	+	0.221644i	$0.0711423\pi$
$107$	8.90703e7	+	1.54274e8i	0.679513	+	1.17695i	−0.295615	+	0.955307i	$0.595524\pi$
$108$		0			0
$109$	−5.48168e7	+	9.49454e7i	−0.388336	+	0.672617i	−0.992226	−	0.124450i	$-0.960283\pi$
$109$	−5.48168e7	+	9.49454e7i	−0.388336	+	0.672617i	0.603890	+	0.797068i	$0.293617\pi$
$110$		0			0
$111$			7.06243e7i			0.465224i
$112$		0			0
$113$	2.74592e8			1.68413			0.842063	−	0.539379i	$-0.181341\pi$
$113$	2.74592e8			1.68413			0.842063	+	0.539379i	$0.181341\pi$
$114$		0			0
$115$	−6.85024e7	−	3.95499e7i	−0.391665	−	0.226128i
$116$		0			0
$117$	−8.78469e7	+	5.07184e7i	−0.468795	+	0.270659i
$118$		0			0
$119$	7.89243e7	−	2.82755e8i	0.393571	−	1.41001i
$120$		0			0
$121$	−8.19129e7	−	1.41877e8i	−0.382130	−	0.661868i
$122$		0			0
$123$	−4.39645e6	+	7.61488e6i	−0.0192080	+	0.0332692i
$124$		0			0
$125$		−	1.85466e8i		−	0.759669i
$126$		0			0
$127$	−3.97605e7			−0.152840			−0.0764199	−	0.997076i	$-0.524349\pi$
$127$	−3.97605e7			−0.152840			−0.0764199	+	0.997076i	$0.524349\pi$
$128$		0			0
$129$	−4.99780e6	−	2.88548e6i	−0.0180476	−	0.0104198i
$130$		0			0
$131$	3.31807e7	−	1.91569e7i	0.112668	−	0.0650490i	−0.442607	−	0.896716i	$-0.645946\pi$
$131$	3.31807e7	−	1.91569e7i	0.112668	−	0.0650490i	0.555275	+	0.831667i	$0.312613\pi$
$132$		0			0
$133$	3.57851e7	+	3.65372e7i	0.114366	+	0.116769i
$134$		0			0
$135$	−1.32885e7	−	2.30164e7i	−0.0400076	−	0.0692952i
$136$		0			0
$137$	2.07018e8	−	3.58565e8i	0.587659	−	1.01786i	−0.406879	−	0.913482i	$-0.633383\pi$
$137$	2.07018e8	−	3.58565e8i	0.587659	−	1.01786i	0.994538	−	0.104374i	$-0.0332838\pi$
$138$		0			0
$139$			2.15646e8i			0.577674i	0.957378	+	0.288837i	$0.0932686\pi$
$139$			2.15646e8i			0.577674i	−0.957378	+	0.288837i	$0.906731\pi$
$140$		0			0
$141$	−3.01811e8			−0.763587
$142$		0			0
$143$	7.81141e8	+	4.50992e8i	1.86804	+	1.07851i
$144$		0			0
$145$	−1.98306e7	+	1.14492e7i	−0.0448605	+	0.0259002i
$146$		0			0
$147$	−2.69535e8	−	5.60637e6i	−0.577225	−	0.0120064i
$148$		0			0
$149$	−2.59668e8	−	4.49759e8i	−0.526834	−	0.912503i	−0.999511	−	0.0312676i	$-0.990046\pi$
$149$	−2.59668e8	−	4.49759e8i	−0.526834	−	0.912503i	0.472677	−	0.881236i	$-0.343288\pi$
$150$		0			0
$151$	−1.65131e8	+	2.86015e8i	−0.317630	+	0.550151i	−0.979993	−	0.199032i	$-0.936220\pi$
$151$	−1.65131e8	+	2.86015e8i	−0.317630	+	0.550151i	0.662363	+	0.749183i	$0.269554\pi$
$152$		0			0
$153$		−	2.67398e8i		−	0.487970i
$154$		0			0
$155$	4.65083e8			0.805757
$156$		0			0
$157$	−2.58909e8	−	1.49481e8i	−0.426137	−	0.246030i	0.271563	−	0.962421i	$-0.412460\pi$
$157$	−2.58909e8	−	1.49481e8i	−0.426137	−	0.246030i	−0.697700	+	0.716390i	$0.745793\pi$
$158$		0			0
$159$	−5.57691e8	+	3.21983e8i	−0.872580	+	0.503784i
$160$		0			0
$161$	−5.22141e8	+	5.11393e8i	−0.777113	+	0.761117i
$162$		0			0
$163$	2.92142e8	+	5.06004e8i	0.413850	+	0.716810i	0.995307	−	0.0967676i	$-0.0308504\pi$
$163$	2.92142e8	+	5.06004e8i	0.413850	+	0.716810i	−0.581457	+	0.813577i	$0.697517\pi$
$164$		0			0
$165$	−1.18163e8	+	2.04664e8i	−0.159421	+	0.276125i
$166$		0			0
$167$			7.46657e8i			0.959965i	0.877278	+	0.479983i	$0.159357\pi$
$167$			7.46657e8i			0.959965i	−0.877278	+	0.479983i	$0.840643\pi$
$168$		0			0
$169$	−1.33553e9			−1.63722
$170$		0			0
$171$	4.03430e7	+	2.32920e7i	0.0471828	+	0.0272410i
$172$		0			0
$173$	9.08205e8	−	5.24352e8i	1.01391	−	0.585381i	0.101576	−	0.994828i	$-0.467612\pi$
$173$	9.08205e8	−	5.24352e8i	1.01391	−	0.585381i	0.912334	+	0.409447i	$0.134278\pi$
$174$		0			0
$175$	−7.47200e8	−	2.08563e8i	−0.796681	−	0.222374i
$176$		0			0
$177$	2.70822e8	+	4.69078e8i	0.275925	+	0.477917i
$178$		0			0
$179$	−1.24237e8	+	2.15185e8i	−0.121015	+	0.209604i	−0.920168	−	0.391523i	$-0.871948\pi$
$179$	−1.24237e8	+	2.15185e8i	−0.121015	+	0.209604i	0.799153	+	0.601127i	$0.205282\pi$
$180$		0			0
$181$		−	4.91957e8i		−	0.458366i	−0.973383	−	0.229183i	$-0.926394\pi$
$181$		−	4.91957e8i		−	0.458366i	0.973383	−	0.229183i	$-0.0736055\pi$
$182$		0			0
$183$	−1.04038e9			−0.927658
$184$		0			0
$185$	−3.39856e8	−	1.96216e8i	−0.290140	−	0.167512i
$186$		0			0
$187$	−2.05917e9	+	1.18886e9i	−1.68394	+	0.972222i
$188$		0			0
$189$	−2.37845e8	+	6.10868e7i	−0.186400	+	0.0478741i
$190$		0			0
$191$	9.53669e8	+	1.65180e9i	0.716579	+	1.24115i	0.962347	+	0.271823i	$0.0876265\pi$
$191$	9.53669e8	+	1.65180e9i	0.716579	+	1.24115i	−0.245768	+	0.969329i	$0.579040\pi$
$192$		0			0
$193$	−4.70293e8	+	8.14571e8i	−0.338953	+	0.587083i	−0.984236	−	0.176860i	$-0.943406\pi$
$193$	−4.70293e8	+	8.14571e8i	−0.338953	+	0.587083i	0.645283	+	0.763943i	$0.276739\pi$
$194$		0			0
$195$		−	5.63645e8i		−	0.389822i
$196$		0			0
$197$	1.22807e9			0.815379			0.407689	−	0.913121i	$-0.366335\pi$
$197$	1.22807e9			0.815379			0.407689	+	0.913121i	$0.366335\pi$
$198$		0			0
$199$	9.98896e7	+	5.76713e7i	0.0636954	+	0.0367745i	0.531509	−	0.847052i	$-0.321625\pi$
$199$	9.98896e7	+	5.76713e7i	0.0636954	+	0.0367745i	−0.467814	+	0.883827i	$0.654958\pi$
$200$		0			0
$201$	4.66457e8	−	2.69309e8i	0.285777	−	0.164994i
$202$		0			0
$203$	5.26315e7	+	2.04924e8i	0.0309929	+	0.120672i
$204$		0			0
$205$	−2.44294e7	−	4.23129e7i	−0.0138324	−	0.0239584i
$206$		0			0
$207$	−3.32859e8	+	5.76528e8i	−0.181292	+	0.314007i
$208$		0			0
$209$		−	4.14229e8i		−	0.217098i
$210$		0			0
$211$	2.33973e9			1.18042			0.590209	−	0.807250i	$-0.299045\pi$
$211$	2.33973e9			1.18042			0.590209	+	0.807250i	$0.299045\pi$
$212$		0			0
$213$	−1.26095e9	−	7.28008e8i	−0.612602	−	0.353686i
$214$		0			0
$215$	2.77708e7	−	1.60335e7i	0.0129968	−	0.00750368i
$216$		0			0
$217$	1.15531e9	−	4.13902e9i	0.521025	−	1.86663i
$218$		0			0
$219$	−2.92779e8	−	5.07108e8i	−0.127281	−	0.220457i
$220$		0			0
$221$	2.83548e9	−	4.91120e9i	1.18866	−	2.05882i
$222$		0			0
$223$			1.14133e9i			0.461523i	0.973010	+	0.230762i	$0.0741218\pi$
$223$			1.14133e9i			0.461523i	−0.973010	+	0.230762i	$0.925878\pi$
$224$		0			0
$225$	−7.06618e8			−0.275712
$226$		0			0
$227$	−1.82252e8	−	1.05223e8i	−0.0686387	−	0.0396285i	0.465288	−	0.885159i	$-0.345951\pi$
$227$	−1.82252e8	−	1.05223e8i	−0.0686387	−	0.0396285i	−0.533927	+	0.845531i	$0.679284\pi$
$228$		0			0
$229$	−3.18730e9	+	1.84019e9i	−1.15899	+	0.669146i	−0.951063	−	0.308997i	$-0.900007\pi$
$229$	−3.18730e9	+	1.84019e9i	−1.15899	+	0.669146i	−0.207932	+	0.978143i	$0.566673\pi$
$230$		0			0
$231$	1.52788e9	+	1.55999e9i	0.536590	+	0.547867i
$232$		0			0
$233$	1.42196e8	+	2.46291e8i	0.0482464	+	0.0835652i	0.889140	−	0.457635i	$-0.151303\pi$
$233$	1.42196e8	+	2.46291e8i	0.0482464	+	0.0835652i	−0.840894	+	0.541200i	$0.817970\pi$
$234$		0			0
$235$	8.38522e8	−	1.45236e9i	0.274943	−	0.476216i
$236$		0			0
$237$			1.99428e9i			0.632111i
$238$		0			0
$239$	2.62607e9			0.804850			0.402425	−	0.915453i	$-0.368167\pi$
$239$	2.62607e9			0.804850			0.402425	+	0.915453i	$0.368167\pi$
$240$		0			0
$241$	2.05781e8	+	1.18808e8i	0.0610011	+	0.0352190i	0.530190	−	0.847879i	$-0.322120\pi$
$241$	2.05781e8	+	1.18808e8i	0.0610011	+	0.0352190i	−0.469189	+	0.883098i	$0.655454\pi$
$242$		0			0
$243$	−1.93710e8	+	1.11839e8i	−0.0555556	+	0.0320750i
$244$		0			0
$245$	7.75828e8	−	1.28147e9i	0.215328	−	0.355667i
$246$		0			0
$247$	4.93976e8	+	8.55591e8i	0.132714	+	0.229868i
$248$		0			0
$249$	1.68351e9	−	2.91593e9i	0.437944	−	0.758541i
$250$		0			0
$251$		−	3.49203e9i		−	0.879797i	−0.898047	−	0.439899i	$-0.855014\pi$
$251$		−	3.49203e9i		−	0.879797i	0.898047	−	0.439899i	$-0.144986\pi$
$252$		0			0
$253$	5.91961e9			1.44481
$254$		0			0
$255$	1.28676e9	+	7.42914e8i	0.304325	+	0.175702i
$256$		0			0
$257$	−3.36384e9	+	1.94211e9i	−0.771086	+	0.445187i	−0.833262	−	0.552879i	$-0.813529\pi$
$257$	−3.36384e9	+	1.94211e9i	−0.771086	+	0.445187i	0.0621759	+	0.998065i	$0.480196\pi$
$258$		0			0
$259$	−2.59046e9	+	2.53714e9i	−0.575675	+	0.563825i
$260$		0			0
$261$	9.63585e7	+	1.66898e8i	0.0207648	+	0.0359657i
$262$		0			0
$263$	−9.40863e8	+	1.62962e9i	−0.196654	+	0.340615i	−0.947442	−	0.319929i	$-0.896341\pi$
$263$	−9.40863e8	+	1.62962e9i	−0.196654	+	0.340615i	0.750787	+	0.660544i	$0.229674\pi$
$264$		0			0
$265$		−	3.57827e9i		−	0.725587i
$266$		0			0
$267$	2.36349e9			0.465059
$268$		0			0
$269$	2.78854e9	+	1.60997e9i	0.532559	+	0.307473i	0.742058	−	0.670336i	$-0.233850\pi$
$269$	2.78854e9	+	1.60997e9i	0.532559	+	0.307473i	−0.209499	+	0.977809i	$0.567183\pi$
$270$		0			0
$271$	4.42388e9	−	2.55413e9i	0.820212	−	0.473550i	−0.0302773	−	0.999542i	$-0.509639\pi$
$271$	4.42388e9	−	2.55413e9i	0.820212	−	0.473550i	0.850490	+	0.525992i	$0.176306\pi$
$272$		0			0
$273$	−5.01617e9	−	1.40014e9i	−0.903070	−	0.252070i
$274$		0			0
$275$	3.14165e9	+	5.44150e9i	0.549323	+	0.951455i
$276$		0			0
$277$	5.20091e9	−	9.00825e9i	0.883406	−	1.53010i	0.0358772	−	0.999356i	$-0.488577\pi$
$277$	5.20091e9	−	9.00825e9i	0.883406	−	1.53010i	0.847529	−	0.530749i	$-0.178089\pi$
$278$		0			0
$279$		−	3.91422e9i		−	0.645994i
$280$		0			0
$281$	−9.79146e9			−1.57044			−0.785222	−	0.619215i	$-0.787451\pi$
$281$	−9.79146e9			−1.57044			−0.785222	+	0.619215i	$0.787451\pi$
$282$		0			0
$283$	−9.03261e8	−	5.21498e8i	−0.140821	−	0.0813031i	0.427934	−	0.903810i	$-0.359242\pi$
$283$	−9.03261e8	−	5.21498e8i	−0.140821	−	0.0813031i	−0.568755	+	0.822507i	$0.692575\pi$
$284$		0			0
$285$	−2.24170e8	+	1.29425e8i	−0.0339781	+	0.0196172i
$286$		0			0
$287$	−4.37249e8	+	1.12301e8i	−0.0644468	+	0.0165522i
$288$		0			0
$289$	3.98675e9	+	6.90526e9i	0.571515	+	0.989893i
$290$		0			0
$291$	2.72740e9	−	4.72400e9i	0.380344	−	0.658776i
$292$		0			0
$293$		−	2.83834e9i		−	0.385118i	−0.981285	−	0.192559i	$-0.938321\pi$
$293$		−	2.83834e9i		−	0.385118i	0.981285	−	0.192559i	$-0.0616787\pi$
$294$		0			0
$295$	−3.00971e9			−0.397408
$296$		0			0
$297$	1.72249e9	+	9.94478e8i	0.221376	+	0.127811i
$298$		0			0
$299$	−1.22270e10	+	7.05924e9i	−1.52980	+	0.883228i
$300$		0			0
$301$	−7.37052e7	−	2.86975e8i	−0.00897909	−	0.0349606i
$302$		0			0
$303$	2.49422e9	+	4.32012e9i	0.295913	+	0.512537i
$304$		0			0
$305$	2.89049e9	−	5.00648e9i	0.334020	−	0.578540i
$306$		0			0
$307$			1.01655e10i			1.14439i	0.820116	+	0.572197i	$0.193909\pi$
$307$			1.01655e10i			1.14439i	−0.820116	+	0.572197i	$0.806091\pi$
$308$		0			0
$309$	9.59713e9			1.05271
$310$		0			0
$311$	−9.35774e9	−	5.40269e9i	−1.00030	−	0.577522i	−0.0919614	−	0.995763i	$-0.529314\pi$
$311$	−9.35774e9	−	5.40269e9i	−1.00030	−	0.577522i	−0.908336	+	0.418240i	$0.862647\pi$
$312$		0			0
$313$	−7.81261e9	+	4.51061e9i	−0.813989	+	0.469957i	−0.848339	−	0.529453i	$-0.822397\pi$
$313$	−7.81261e9	+	4.51061e9i	−0.813989	+	0.469957i	0.0343502	+	0.999410i	$0.489064\pi$
$314$		0			0
$315$	3.66846e8	−	1.31427e9i	0.0372599	−	0.133488i
$316$		0			0
$317$	2.10486e9	+	3.64573e9i	0.208443	+	0.361034i	0.951224	−	0.308500i	$-0.0998271\pi$
$317$	2.10486e9	+	3.64573e9i	0.208443	+	0.361034i	−0.742781	+	0.669534i	$0.766494\pi$
$318$		0			0
$319$	8.56828e8	−	1.48407e9i	0.0827429	−	0.143315i
$320$		0			0
$321$			8.33081e9i			0.784634i
$322$		0			0
$323$	−2.60435e9			−0.239270
$324$		0			0
$325$	−1.29782e10	−	7.49295e9i	−1.16327	−	0.671614i
$326$		0			0
$327$	−4.44016e9	+	2.56353e9i	−0.388336	+	0.224206i
$328$		0			0
$329$	−1.08424e10	−	1.10702e10i	−0.925424	−	0.944873i
$330$		0			0
$331$	5.98155e9	+	1.03603e10i	0.498312	+	0.863102i	0.999998	−	0.00194777i	$-0.000619995\pi$
$331$	5.98155e9	+	1.03603e10i	0.498312	+	0.863102i	−0.501686	+	0.865050i	$0.667287\pi$
$332$		0			0
$333$	−1.65139e9	+	2.86028e9i	−0.134299	+	0.232612i
$334$		0			0
$335$			2.99289e9i			0.237636i
$336$		0			0
$337$	8.47484e9			0.657070			0.328535	−	0.944492i	$-0.393445\pi$
$337$	8.47484e9			0.657070			0.328535	+	0.944492i	$0.393445\pi$
$338$		0			0
$339$	1.11210e10	+	6.42071e9i	0.842063	+	0.486165i
$340$		0			0
$341$	−3.01425e10	+	1.74028e10i	−2.22927	+	1.28707i
$342$		0			0
$343$	−9.47723e9	−	1.00878e10i	−0.684708	−	0.728818i
$344$		0			0
$345$	−1.84956e9	−	3.20354e9i	−0.130555	−	0.226128i
$346$		0			0
$347$	−7.92809e9	+	1.37319e10i	−0.546828	+	0.947134i	0.451661	+	0.892190i	$0.350832\pi$
$347$	−7.92809e9	+	1.37319e10i	−0.546828	+	0.947134i	−0.998489	+	0.0549447i	$0.982502\pi$
$348$		0			0
$349$		−	7.90260e9i		−	0.532682i	−0.963879	−	0.266341i	$-0.914185\pi$
$349$		−	7.90260e9i		−	0.532682i	0.963879	−	0.266341i	$-0.0858148\pi$
$350$		0			0
$351$	−4.74373e9			−0.312530
$352$		0			0
$353$	8.28745e9	+	4.78476e9i	0.533731	+	0.308150i	0.742534	−	0.669808i	$-0.233624\pi$
$353$	8.28745e9	+	4.78476e9i	0.533731	+	0.308150i	−0.208804	+	0.977958i	$0.566957\pi$
$354$		0			0
$355$	7.00658e9	−	4.04525e9i	0.441157	−	0.254702i
$356$		0			0
$357$	9.80801e9	−	9.60612e9i	0.603821	−	0.591392i
$358$		0			0
$359$	5.12021e9	+	8.86847e9i	0.308255	+	0.533914i	0.977981	−	0.208695i	$-0.0669216\pi$
$359$	5.12021e9	+	8.86847e9i	0.308255	+	0.533914i	−0.669726	+	0.742609i	$0.733588\pi$
$360$		0			0
$361$	−8.26493e9	+	1.43153e10i	−0.486643	+	0.842890i
$362$		0			0
$363$		−	7.66138e9i		−	0.441246i
$364$		0			0
$365$	3.25371e9			0.183319
$366$		0			0
$367$	2.80339e10	+	1.61854e10i	1.54532	+	0.892193i	0.998489	+	0.0549514i	$0.0175004\pi$
$367$	2.80339e10	+	1.61854e10i	1.54532	+	0.892193i	0.546834	+	0.837241i	$0.315833\pi$
$368$		0			0
$369$	−3.56113e8	+	2.05602e8i	−0.0192080	+	0.0110897i
$370$		0			0
$371$	−3.18449e10	−	8.88873e9i	−1.68091	−	0.469185i
$372$		0			0
$373$	1.25975e10	+	2.18195e10i	0.650803	+	1.12722i	0.982928	+	0.183988i	$0.0589008\pi$
$373$	1.25975e10	+	2.18195e10i	0.650803	+	1.12722i	−0.332126	+	0.943235i	$0.607766\pi$
$374$		0			0
$375$	4.33670e9	−	7.51138e9i	0.219298	−	0.379835i
$376$		0			0
$377$			4.08713e9i			0.202327i
$378$		0			0
$379$	−6.03658e9			−0.292573			−0.146287	−	0.989242i	$-0.546732\pi$
$379$	−6.03658e9			−0.292573			−0.146287	+	0.989242i	$0.546732\pi$
$380$		0			0
$381$	−1.61030e9	−	9.29707e8i	−0.0764199	−	0.0441211i
$382$		0			0
$383$	−9.57427e9	+	5.52771e9i	−0.444949	+	0.256892i	−0.705695	−	0.708516i	$-0.749365\pi$
$383$	−9.57427e9	+	5.52771e9i	−0.444949	+	0.256892i	0.260745	+	0.965408i	$0.416032\pi$
$384$		0			0
$385$	−1.17519e10	+	3.01829e9i	−0.534890	+	0.137378i
$386$		0			0
$387$	−1.34941e8	−	2.33724e8i	−0.00601588	−	0.0104198i
$388$		0			0
$389$	1.58412e10	−	2.74378e10i	0.691814	−	1.19826i	−0.279428	−	0.960167i	$-0.590145\pi$
$389$	1.58412e10	−	2.74378e10i	0.691814	−	1.19826i	0.971243	−	0.238091i	$-0.0765217\pi$
$390$		0			0
$391$		−	3.72178e10i		−	1.59237i
$392$		0			0
$393$	1.79176e9			0.0751121
$394$		0			0
$395$	−9.59682e9	−	5.54072e9i	−0.394220	−	0.227603i
$396$		0			0
$397$	−1.81623e10	+	1.04860e10i	−0.731155	+	0.422132i	−0.818844	−	0.574015i	$-0.805385\pi$
$397$	−1.81623e10	+	1.04860e10i	−0.731155	+	0.422132i	0.0876897	+	0.996148i	$0.472052\pi$
$398$		0			0
$399$	5.94960e8	+	2.31651e9i	0.0234745	+	0.0913992i
$400$		0			0
$401$	−5.84375e9	−	1.01217e10i	−0.226003	−	0.391449i	0.730617	−	0.682788i	$-0.239233\pi$
$401$	−5.84375e9	−	1.01217e10i	−0.226003	−	0.391449i	−0.956620	+	0.291339i	$0.905899\pi$
$402$		0			0
$403$	4.15062e10	−	7.18909e10i	1.57360	−	2.72555i
$404$		0			0
$405$		−	1.24289e9i		−	0.0461968i
$406$		0			0
$407$	2.93685e10			1.07030
$408$		0			0
$409$	2.98804e10	+	1.72515e10i	1.06781	+	0.616499i	0.927582	−	0.373620i	$-0.121884\pi$
$409$	2.98804e10	+	1.72515e10i	1.06781	+	0.616499i	0.140226	+	0.990119i	$0.455217\pi$
$410$		0			0
$411$	1.67684e10	−	9.68127e9i	0.587659	−	0.339285i
$412$		0			0
$413$	−7.47637e9	+	2.67849e10i	−0.256975	+	0.920642i
$414$		0			0
$415$	9.35461e9	+	1.62027e10i	0.315379	+	0.546253i
$416$		0			0
$417$	−5.04239e9	+	8.73368e9i	−0.166760	+	0.288837i
$418$		0			0
$419$		−	1.71261e10i		−	0.555652i	−0.960631	−	0.277826i	$-0.910386\pi$
$419$		−	1.71261e10i		−	0.555652i	0.960631	−	0.277826i	$-0.0896139\pi$
$420$		0			0
$421$	3.64524e10			1.16037			0.580187	−	0.814484i	$-0.302980\pi$
$421$	3.64524e10			1.16037			0.580187	+	0.814484i	$0.302980\pi$
$422$		0			0
$423$	−1.22233e10	−	7.05715e9i	−0.381793	−	0.220428i
$424$		0			0
$425$	3.42119e10	−	1.97522e10i	1.04863	−	0.605425i
$426$		0			0
$427$	−3.73750e10	−	3.81605e10i	−1.12427	−	1.14790i
$428$		0			0
$429$	2.10908e10	+	3.65303e10i	0.622679	+	1.07851i
$430$		0			0
$431$	1.09904e10	−	1.90360e10i	0.318497	−	0.551654i	−0.661677	−	0.749789i	$-0.730155\pi$
$431$	1.09904e10	−	1.90360e10i	0.318497	−	0.551654i	0.980175	+	0.198135i	$0.0634884\pi$
$432$		0			0
$433$		−	1.35544e10i		−	0.385593i	−0.981239	−	0.192796i	$-0.938244\pi$
$433$		−	1.35544e10i		−	0.385593i	0.981239	−	0.192796i	$-0.0617557\pi$
$434$		0			0
$435$	−1.07085e9			−0.0299070
$436$		0			0
$437$	5.61514e9	+	3.24190e9i	0.153969	+	0.0888943i
$438$		0			0
$439$	−4.13004e10	+	2.38448e10i	−1.11198	+	0.642001i	−0.939341	−	0.342985i	$-0.888562\pi$
$439$	−4.13004e10	+	2.38448e10i	−1.11198	+	0.642001i	−0.172637	+	0.984986i	$0.555229\pi$
$440$		0			0
$441$	−1.07851e10	−	6.52950e9i	−0.285147	−	0.172634i
$442$		0			0
$443$	9.20913e9	+	1.59507e10i	0.239113	+	0.414156i	0.960460	−	0.278418i	$-0.0898100\pi$
$443$	9.20913e9	+	1.59507e10i	0.239113	+	0.414156i	−0.721347	+	0.692574i	$0.756477\pi$
$444$		0			0
$445$	−6.56648e9	+	1.13735e10i	−0.167453	+	0.290037i
$446$		0			0
$447$		−	2.42870e10i		−	0.608336i
$448$		0			0
$449$	4.72530e10			1.16264			0.581319	−	0.813676i	$-0.302537\pi$
$449$	4.72530e10			1.16264			0.581319	+	0.813676i	$0.302537\pi$
$450$		0			0
$451$	3.16658e9	+	1.82823e9i	0.0765393	+	0.0441900i
$452$		0			0
$453$	−1.33756e10	+	7.72242e9i	−0.317630	+	0.183384i
$454$		0			0
$455$	2.06741e10	−	2.02486e10i	0.482372	−	0.472443i
$456$		0			0
$457$	9.30524e9	+	1.61172e10i	0.213335	+	0.369508i	0.952756	−	0.303736i	$-0.0982340\pi$
$457$	9.30524e9	+	1.61172e10i	0.213335	+	0.369508i	−0.739421	+	0.673243i	$0.764901\pi$
$458$		0			0
$459$	6.25249e9	−	1.08296e10i	0.140865	−	0.243985i
$460$		0			0
$461$		−	9.20893e9i		−	0.203895i	−0.994790	−	0.101947i	$-0.967493\pi$
$461$		−	9.20893e9i		−	0.203895i	0.994790	−	0.101947i	$-0.0325073\pi$
$462$		0			0
$463$	−1.42010e10			−0.309025			−0.154513	−	0.987991i	$-0.549381\pi$
$463$	−1.42010e10			−0.309025			−0.154513	+	0.987991i	$0.549381\pi$
$464$		0			0
$465$	1.88359e10	+	1.08749e10i	0.402878	+	0.232602i
$466$		0			0
$467$	−2.95507e10	+	1.70611e10i	−0.621299	+	0.358707i	−0.777374	−	0.629038i	$-0.783449\pi$
$467$	−2.95507e10	+	1.70611e10i	−0.621299	+	0.358707i	0.156076	+	0.987745i	$0.450116\pi$
$468$		0			0
$469$	2.66353e10	+	7.43460e9i	0.550511	+	0.153662i
$470$		0			0
$471$	−6.99056e9	−	1.21080e10i	−0.142046	−	0.246030i
$472$		0			0
$473$	−1.19990e9	+	2.07829e9i	−0.0239718	+	0.0415204i
$474$		0			0
$475$			6.88216e9i			0.135192i
$476$		0			0
$477$	−3.01153e10			−0.581720
$478$		0			0
$479$	7.59507e9	+	4.38502e9i	0.144275	+	0.0832970i	0.570400	−	0.821367i	$-0.306788\pi$
$479$	7.59507e9	+	4.38502e9i	0.144275	+	0.0832970i	−0.426125	+	0.904664i	$0.640122\pi$
$480$		0			0
$481$	−6.06607e10	+	3.50225e10i	−1.13325	+	0.654284i
$482$		0			0
$483$	−3.31044e10	+	8.50236e9i	−0.608272	+	0.156225i
$484$		0			0
$485$	1.51551e10	+	2.62494e10i	0.273900	+	0.474408i
$486$		0			0
$487$	−6.41038e9	+	1.11031e10i	−0.113964	+	0.197392i	−0.917365	−	0.398047i	$-0.869688\pi$
$487$	−6.41038e9	+	1.11031e10i	−0.113964	+	0.197392i	0.803401	+	0.595438i	$0.203022\pi$
$488$		0			0
$489$			2.73242e10i			0.477873i
$490$		0			0
$491$	6.34320e10			1.09140			0.545698	−	0.837982i	$-0.316265\pi$
$491$	6.34320e10			1.09140			0.545698	+	0.837982i	$0.316265\pi$
$492$		0			0
$493$	−9.33065e9	−	5.38705e9i	−0.157952	−	0.0911934i
$494$		0			0
$495$	−9.57118e9	+	5.52592e9i	−0.159421	+	0.0920416i
$496$		0			0
$497$	−1.85958e10	−	7.24040e10i	−0.304783	−	1.18669i
$498$		0			0
$499$	−3.36380e10	−	5.82628e10i	−0.542536	−	0.939700i	−0.998758	−	0.0498336i	$-0.984131\pi$
$499$	−3.36380e10	−	5.82628e10i	−0.542536	−	0.939700i	0.456222	−	0.889866i	$-0.349202\pi$
$500$		0			0
$501$	−1.74589e10	+	3.02396e10i	−0.277118	+	0.479983i
$502$		0			0
$503$		−	1.04697e11i		−	1.63554i	−0.575542	−	0.817772i	$-0.695209\pi$
$503$		−	1.04697e11i		−	1.63554i	0.575542	−	0.817772i	$-0.304791\pi$
$504$		0			0
$505$	−2.77188e10			−0.426196
$506$		0			0
$507$	−5.40891e10	−	3.12283e10i	−0.818611	−	0.472625i
$508$		0			0
$509$	−4.94469e10	+	2.85482e10i	−0.736662	+	0.425312i	−0.820854	−	0.571137i	$-0.806502\pi$
$509$	−4.94469e10	+	2.85482e10i	−0.736662	+	0.425312i	0.0841923	+	0.996450i	$0.473169\pi$
$510$		0			0
$511$	8.08251e9	−	2.89565e10i	0.118539	−	0.424680i
$512$		0			0
$513$	1.08926e9	+	1.88666e9i	0.0157276	+	0.0272410i
$514$		0			0
$515$	−2.66637e10	+	4.61830e10i	−0.379046	+	0.656527i
$516$		0			0
$517$			1.25505e11i			1.75671i
$518$		0			0
$519$	4.90430e10			0.675940
$520$		0			0
$521$	7.48609e10	+	4.32210e10i	1.01602	+	0.586602i	0.912950	−	0.408072i	$-0.133799\pi$
$521$	7.48609e10	+	4.32210e10i	1.01602	+	0.586602i	0.103074	+	0.994674i	$0.467132\pi$
$522$		0			0
$523$	1.27623e10	−	7.36833e9i	0.170578	−	0.0984832i	−0.412280	−	0.911057i	$-0.635268\pi$
$523$	1.27623e10	−	7.36833e9i	0.170578	−	0.0984832i	0.582858	+	0.812574i	$0.301934\pi$
$524$		0			0
$525$	−2.53848e10	−	2.59183e10i	−0.334147	−	0.341169i
$526$		0			0
$527$	1.09415e11	+	1.89512e11i	1.41852	+	2.45694i
$528$		0			0
$529$	−7.17338e9	+	1.24247e10i	−0.0916012	+	0.158658i
$530$		0			0
$531$			2.53302e10i			0.318611i
$532$		0			0
$533$	−8.72077e9			−0.108055
$534$		0			0
$535$	−4.00892e10	−	2.31455e10i	−0.489342	−	0.282522i
$536$		0			0
$537$	−1.00632e10	+	5.80999e9i	−0.121015	+	0.0698679i
$538$		0			0
$539$	−2.33136e9	+	1.12084e11i	−0.0276220	+	1.32797i
$540$		0			0
$541$	−5.99660e10	−	1.03864e11i	−0.700029	−	1.21249i	−0.968456	−	0.249185i	$-0.919837\pi$
$541$	−5.99660e10	−	1.03864e11i	−0.700029	−	1.21249i	0.268427	−	0.963300i	$-0.413496\pi$
$542$		0			0
$543$	1.15033e10	−	1.99243e10i	0.132319	−	0.229183i
$544$		0			0
$545$		−	2.84890e10i		−	0.322917i
$546$		0			0
$547$	−7.72241e10			−0.862588			−0.431294	−	0.902211i	$-0.641943\pi$
$547$	−7.72241e10			−0.862588			−0.431294	+	0.902211i	$0.641943\pi$
$548$		0			0
$549$	−4.21354e10	−	2.43269e10i	−0.463829	−	0.267792i
$550$		0			0
$551$	1.62551e9	−	9.38491e8i	0.0176354	−	0.0101818i
$552$		0			0
$553$	−7.31491e10	+	7.16434e10i	−0.782183	+	0.766083i
$554$		0			0
$555$	−9.17610e9	−	1.58935e10i	−0.0967133	−	0.167512i
$556$		0			0
$557$	8.89933e10	−	1.54141e11i	0.924564	−	1.60139i	0.132302	−	0.991209i	$-0.457763\pi$
$557$	8.89933e10	−	1.54141e11i	0.924564	−	1.60139i	0.792262	−	0.610182i	$-0.208904\pi$
$558$		0			0
$559$		−	5.72362e9i		−	0.0586170i
$560$		0			0
$561$	−1.11195e11			−1.12263
$562$		0			0
$563$	−1.36022e11	−	7.85323e10i	−1.35386	−	0.781654i	−0.365076	−	0.930978i	$-0.618957\pi$
$563$	−1.36022e11	−	7.85323e10i	−1.35386	−	0.781654i	−0.988788	+	0.149323i	$0.952290\pi$
$564$		0			0
$565$	−6.17950e10	+	3.56774e10i	−0.606400	+	0.350105i
$566$		0			0
$567$	−1.10611e10	−	3.08744e9i	−0.107020	−	0.0298721i
$568$		0			0
$569$	−2.78668e8	−	4.82667e8i	−0.00265851	−	0.00460467i	0.864693	−	0.502301i	$-0.167513\pi$
$569$	−2.78668e8	−	4.82667e8i	−0.00265851	−	0.00460467i	−0.867352	+	0.497696i	$0.834180\pi$
$570$		0			0
$571$	1.85718e10	−	3.21672e10i	0.174706	−	0.302600i	−0.765353	−	0.643610i	$-0.777436\pi$
$571$	1.85718e10	−	3.21672e10i	0.174706	−	0.302600i	0.940060	+	0.341010i	$0.110769\pi$
$572$		0			0
$573$			8.91974e10i			0.827434i
$574$		0			0
$575$	−9.83506e10			−0.899716
$576$		0			0
$577$	3.46036e10	+	1.99784e10i	0.312189	+	0.180243i	0.647906	−	0.761721i	$-0.275645\pi$
$577$	3.46036e10	+	1.99784e10i	0.312189	+	0.180243i	−0.335716	+	0.941963i	$0.608978\pi$
$578$		0			0
$579$	−3.80937e10	+	2.19934e10i	−0.338953	+	0.195694i
$580$		0			0
$581$	1.67434e11	−	4.30027e10i	1.46939	−	0.377391i
$582$		0			0
$583$	1.33894e11	+	2.31911e11i	1.15901	+	2.00746i
$584$		0			0
$585$	1.31795e10	−	2.28276e10i	0.112532	−	0.194911i
$586$		0			0
$587$			1.12252e11i			0.945460i	0.881207	+	0.472730i	$0.156731\pi$
$587$			1.12252e11i			0.945460i	−0.881207	+	0.472730i	$0.843269\pi$
$588$		0			0
$589$	−3.81229e10			−0.316756
$590$		0			0
$591$	4.97370e10	+	2.87156e10i	0.407689	+	0.235380i
$592$		0			0
$593$	4.84487e10	−	2.79719e10i	0.391799	−	0.226205i	−0.291140	−	0.956680i	$-0.594035\pi$
$593$	4.84487e10	−	2.79719e10i	0.391799	−	0.226205i	0.682939	+	0.730475i	$0.260701\pi$
$594$		0			0
$595$	1.89766e10	+	7.38865e10i	0.151409	+	0.589518i
$596$		0			0
$597$	2.69702e9	+	4.67137e9i	0.0212318	+	0.0367745i
$598$		0			0
$599$	1.46361e10	−	2.53505e10i	0.113689	−	0.196916i	−0.803566	−	0.595216i	$-0.797067\pi$
$599$	1.46361e10	−	2.53505e10i	0.113689	−	0.196916i	0.917255	+	0.398300i	$0.130400\pi$
$600$		0			0
$601$			3.04406e10i			0.233322i	0.993172	+	0.116661i	$0.0372191\pi$
$601$			3.04406e10i			0.233322i	−0.993172	+	0.116661i	$0.962781\pi$
$602$		0			0
$603$	2.51887e10			0.190518
$604$		0			0
$605$	3.68678e10	+	2.12856e10i	0.275186	+	0.158879i
$606$		0			0
$607$	1.87228e11	−	1.08096e11i	1.37917	−	0.796262i	0.387107	−	0.922035i	$-0.373474\pi$
$607$	1.87228e11	−	1.08096e11i	1.37917	−	0.796262i	0.992059	+	0.125772i	$0.0401409\pi$
$608$		0			0
$609$	−2.66009e9	+	9.53008e9i	−0.0193387	+	0.0692831i
$610$		0			0
$611$	−1.49667e11	−	2.59232e11i	−1.07390	−	1.86004i
$612$		0			0
$613$	3.15010e10	−	5.45613e10i	0.223091	−	0.386405i	−0.732654	−	0.680601i	$-0.761719\pi$
$613$	3.15010e10	−	5.45613e10i	0.223091	−	0.386405i	0.955745	+	0.294196i	$0.0950519\pi$
$614$		0			0
$615$		−	2.28490e9i		−	0.0159723i
$616$		0			0
$617$	2.31396e11			1.59667			0.798336	−	0.602213i	$-0.205714\pi$
$617$	2.31396e11			1.59667			0.798336	+	0.602213i	$0.205714\pi$
$618$		0			0
$619$	−8.62110e10	−	4.97739e10i	−0.587219	−	0.339031i	0.176778	−	0.984251i	$-0.443432\pi$
$619$	−8.62110e10	−	4.97739e10i	−0.587219	−	0.339031i	−0.763997	+	0.645220i	$0.776766\pi$
$620$		0			0
$621$	−2.69615e10	+	1.55663e10i	−0.181292	+	0.104669i
$622$		0			0
$623$	8.49068e10	+	8.66913e10i	0.563625	+	0.575471i
$624$		0			0
$625$	−3.90080e10	−	6.75639e10i	−0.255643	−	0.442787i
$626$		0			0
$627$	9.68579e9	−	1.67763e10i	0.0626708	−	0.108549i
$628$		0			0
$629$		−	1.84646e11i		−	1.17961i
$630$		0			0
$631$	−3.88355e10			−0.244969			−0.122485	−	0.992470i	$-0.539086\pi$
$631$	−3.88355e10			−0.244969			−0.122485	+	0.992470i	$0.539086\pi$
$632$		0			0
$633$	9.47590e10	+	5.47092e10i	0.590209	+	0.340757i
$634$		0			0
$635$	8.94780e9	−	5.16602e9i	0.0550328	−	0.0317732i
$636$		0			0
$637$	−1.28846e11	−	2.34289e11i	−0.782554	−	1.42297i
$638$		0			0
$639$	−3.40455e10	−	5.89686e10i	−0.204201	−	0.353686i
$640$		0			0
$641$	−1.01203e11	+	1.75289e11i	−0.599462	+	1.03830i	0.393439	+	0.919351i	$0.371285\pi$
$641$	−1.01203e11	+	1.75289e11i	−0.599462	+	1.03830i	−0.992901	+	0.118948i	$0.962048\pi$
$642$		0			0
$643$		−	1.46591e11i		−	0.857558i	−0.903409	−	0.428779i	$-0.858944\pi$
$643$		−	1.46591e11i		−	0.857558i	0.903409	−	0.428779i	$-0.141056\pi$
$644$		0			0
$645$	1.49962e9			0.00866450
$646$		0			0
$647$	2.40431e11	+	1.38813e11i	1.37206	+	0.792158i	0.991187	−	0.132470i	$-0.0422907\pi$
$647$	2.40431e11	+	1.38813e11i	1.37206	+	0.792158i	0.380871	+	0.924628i	$0.375624\pi$
$648$		0			0
$649$	1.95062e11	−	1.12619e11i	1.09950	−	0.634795i
$650$		0			0
$651$	1.43571e11	−	1.40616e11i	0.799362	−	0.782908i
$652$		0			0
$653$	−1.35533e11	−	2.34750e11i	−0.745406	−	1.29108i	−0.950005	−	0.312236i	$-0.898922\pi$
$653$	−1.35533e11	−	2.34750e11i	−0.745406	−	1.29108i	0.204598	−	0.978846i	$-0.434411\pi$
$654$		0			0
$655$	−4.97805e9	+	8.62224e9i	−0.0270455	+	0.0468441i
$656$		0			0
$657$		−	2.73838e10i		−	0.146971i
$658$		0			0
$659$	8.76828e10			0.464914			0.232457	−	0.972607i	$-0.425324\pi$
$659$	8.76828e10			0.464914			0.232457	+	0.972607i	$0.425324\pi$
$660$		0			0
$661$	−3.77894e10	−	2.18177e10i	−0.197954	−	0.114289i	0.397747	−	0.917495i	$-0.369792\pi$
$661$	−3.77894e10	−	2.18177e10i	−0.197954	−	0.114289i	−0.595701	+	0.803206i	$0.703126\pi$
$662$		0			0
$663$	2.29674e11	−	1.32602e11i	1.18866	−	0.686273i
$664$		0			0
$665$	−1.28004e10	−	3.57292e9i	−0.0654541	−	0.0182699i
$666$		0			0
$667$	1.34117e10	+	2.32297e10i	0.0677609	+	0.117365i
$668$		0			0
$669$	−2.66875e10	+	4.62241e10i	−0.133230	+	0.230762i
$670$		0			0
$671$			4.32633e11i			2.13417i
$672$		0			0
$673$	1.88626e11			0.919480			0.459740	−	0.888054i	$-0.347943\pi$
$673$	1.88626e11			0.919480			0.459740	+	0.888054i	$0.347943\pi$
$674$		0			0
$675$	−2.86180e10	−	1.65226e10i	−0.137856	−	0.0795911i
$676$		0			0
$677$	3.41068e10	−	1.96916e10i	0.162363	−	0.0937401i	−0.416617	−	0.909082i	$-0.636784\pi$
$677$	3.41068e10	−	1.96916e10i	0.162363	−	0.0937401i	0.578980	+	0.815342i	$0.303451\pi$
$678$		0			0
$679$	2.71253e11	−	6.96673e10i	1.27613	−	0.327755i
$680$		0			0
$681$	−4.92080e9	−	8.52308e9i	−0.0228796	−	0.0396285i
$682$		0			0
$683$	−1.05360e11	+	1.82490e11i	−0.484167	+	0.838601i	−0.999835	−	0.0181874i	$-0.994210\pi$
$683$	−1.05360e11	+	1.82490e11i	−0.484167	+	0.838601i	0.515668	+	0.856788i	$0.327544\pi$
$684$		0			0
$685$			1.07590e11i			0.488663i
$686$		0			0
$687$	−1.72114e11			−0.772663
$688$		0			0
$689$	−5.53116e11	−	3.19342e11i	−2.45437	−	1.41703i
$690$		0			0
$691$	−3.03078e10	+	1.74982e10i	−0.132936	+	0.0767505i	−0.564993	−	0.825096i	$-0.691121\pi$
$691$	−3.03078e10	+	1.74982e10i	−0.132936	+	0.0767505i	0.432057	+	0.901846i	$0.357788\pi$
$692$		0			0
$693$	2.54024e10	+	9.89058e10i	0.110139	+	0.428834i
$694$		0			0
$695$	−2.80186e10	−	4.85296e10i	−0.120090	−	0.208002i
$696$		0			0
$697$	1.14944e10	−	1.99090e10i	0.0487031	−	0.0843563i
$698$		0			0
$699$			1.32997e10i			0.0557101i
$700$		0			0
$701$	8.58032e10			0.355330			0.177665	−	0.984091i	$-0.443146\pi$
$701$	8.58032e10			0.355330			0.177665	+	0.984091i	$0.443146\pi$
$702$		0			0
$703$	2.78580e10	+	1.60838e10i	0.114059	+	0.0658517i
$704$		0			0
$705$	6.79203e10	−	3.92138e10i	0.274943	−	0.158739i
$706$		0			0
$707$	−6.88559e10	+	2.46684e11i	−0.275590	+	0.987333i
$708$		0			0
$709$	9.74057e10	+	1.68712e11i	0.385478	+	0.667668i	0.991835	−	0.127525i	$-0.0407032\pi$
$709$	9.74057e10	+	1.68712e11i	0.385478	+	0.667668i	−0.606357	+	0.795192i	$0.707370\pi$
$710$		0			0
$711$	−4.66317e10	+	8.07685e10i	−0.182475	+	0.316056i
$712$		0			0
$713$		−	5.44801e11i		−	2.10804i
$714$		0			0
$715$	−2.34387e11			−0.896827
$716$		0			0
$717$	1.06356e11	+	6.14046e10i	0.402425	+	0.232340i
$718$		0			0
$719$	−2.89871e11	+	1.67357e11i	−1.08465	+	0.626222i	−0.932147	−	0.362081i	$-0.882066\pi$
$719$	−2.89871e11	+	1.67357e11i	−1.08465	+	0.626222i	−0.152502	+	0.988303i	$0.548733\pi$
$720$		0			0
$721$	3.44771e11	+	3.52017e11i	1.27582	+	1.30263i
$722$		0			0
$723$	5.55609e9	+	9.62343e9i	0.0203337	+	0.0352190i
$724$		0			0
$725$	−1.42357e10	+	2.46569e10i	−0.0515259	+	0.0892455i
$726$		0			0
$727$			9.86180e10i			0.353035i	0.984297	+	0.176518i	$0.0564833\pi$
$727$			9.86180e10i			0.353035i	−0.984297	+	0.176518i	$0.943517\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	1.30667e10	+	7.54404e9i	0.0457609	+	0.0264201i
$732$		0			0
$733$	2.61192e11	−	1.50799e11i	0.904782	−	0.522376i	0.0260337	−	0.999661i	$-0.491712\pi$
$733$	2.61192e11	−	1.50799e11i	0.904782	−	0.522376i	0.878749	+	0.477285i	$0.158379\pi$
$734$		0			0
$735$	6.13852e10	−	3.37586e10i	0.210336	−	0.115674i
$736$		0			0
$737$	−1.11990e11	−	1.93972e11i	−0.379585	−	0.657460i
$738$		0			0
$739$	−1.82846e11	+	3.16698e11i	−0.613065	+	1.06186i	0.377655	+	0.925946i	$0.376730\pi$
$739$	−1.82846e11	+	3.16698e11i	−0.613065	+	1.06186i	−0.990721	+	0.135914i	$0.956603\pi$
$740$		0			0
$741$			4.62019e10i			0.153245i
$742$		0			0
$743$	−3.52082e11			−1.15528			−0.577642	−	0.816290i	$-0.696027\pi$
$743$	−3.52082e11			−1.15528			−0.577642	+	0.816290i	$0.696027\pi$
$744$		0			0
$745$	1.16873e11	+	6.74766e10i	0.379392	+	0.219042i
$746$		0			0
$747$	1.36364e11	−	7.87300e10i	0.437944	−	0.252847i
$748$		0			0
$749$	−3.05569e11	+	2.99279e11i	−0.970917	+	0.950932i
$750$		0			0
$751$	−1.44039e11	−	2.49484e11i	−0.452816	−	0.784300i	0.545744	−	0.837952i	$-0.316247\pi$
$751$	−1.44039e11	−	2.49484e11i	−0.452816	−	0.784300i	−0.998560	+	0.0536517i	$0.982914\pi$
$752$		0			0
$753$	8.16530e10	−	1.41427e11i	0.253976	−	0.439899i
$754$		0			0
$755$		−	8.58209e10i		−	0.264122i
$756$		0			0
$757$	−5.34615e11			−1.62801			−0.814006	−	0.580857i	$-0.802718\pi$
$757$	−5.34615e11			−1.62801			−0.814006	+	0.580857i	$0.802718\pi$
$758$		0			0
$759$	2.39744e11	+	1.38416e11i	0.722405	+	0.417081i
$760$		0			0
$761$	2.20516e10	−	1.27315e10i	0.0657507	−	0.0379612i	−0.466764	−	0.884382i	$-0.654580\pi$
$761$	2.20516e10	−	1.27315e10i	0.0657507	−	0.0379612i	0.532515	+	0.846421i	$0.321247\pi$
$762$		0			0
$763$	−2.53539e11	−	7.07692e10i	−0.748076	−	0.208807i
$764$		0			0
$765$	3.47426e10	+	6.01760e10i	0.101442	+	0.175702i
$766$		0			0
$767$	−2.68601e11	+	4.65230e11i	−0.776114	+	1.34427i
$768$		0			0
$769$			1.61134e11i			0.460768i	0.973100	+	0.230384i	$0.0739982\pi$
$769$			1.61134e11i			0.460768i	−0.973100	+	0.230384i	$0.926002\pi$
$770$		0			0
$771$	−1.81647e11			−0.514057
$772$		0			0
$773$	−2.00134e11	−	1.15547e11i	−0.560535	−	0.323625i	0.192825	−	0.981233i	$-0.438235\pi$
$773$	−2.00134e11	−	1.15547e11i	−0.560535	−	0.323625i	−0.753360	+	0.657608i	$0.771568\pi$
$774$		0			0
$775$	5.00799e11	−	2.89136e11i	1.38821	−	0.801486i
$776$		0			0
$777$	−1.64239e11	+	4.21821e10i	−0.450600	+	0.115730i
$778$		0			0
$779$	2.00247e9	+	3.46839e9i	0.00543772	+	0.00941841i
$780$		0			0
$781$	−3.02736e11	+	5.24353e11i	−0.813691	+	1.40935i
$782$		0			0
$783$			9.01249e9i			0.0239772i
$784$		0			0
$785$	7.76875e10			0.204585
$786$		0			0
$787$	8.19226e9	+	4.72980e9i	0.0213553	+	0.0123295i	0.510640	−	0.859795i	$-0.329409\pi$
$787$	8.19226e9	+	4.72980e9i	0.0213553	+	0.0123295i	−0.489284	+	0.872124i	$0.662742\pi$
$788$		0			0
$789$	−7.62099e10	+	4.39998e10i	−0.196654	+	0.113538i
$790$		0			0
$791$	1.64007e11	+	6.38571e11i	0.418945	+	1.63119i
$792$		0			0
$793$	−5.15923e11	−	8.93604e11i	−1.30464	−	2.25971i
$794$		0			0
$795$	8.36695e10	−	1.44920e11i	0.209459	−	0.362793i
$796$		0			0
$797$			1.06562e11i			0.264100i	0.991243	+	0.132050i	$0.0421559\pi$
$797$			1.06562e11i			0.264100i	−0.991243	+	0.132050i	$0.957844\pi$
$798$		0			0
$799$	7.89079e11			1.93612
$800$		0			0
$801$	9.57212e10	+	5.52647e10i	0.232530	+	0.134251i
$802$		0			0
$803$	−2.10876e11	+	1.21750e11i	−0.507184	+	0.292823i
$804$		0			0
$805$	5.10594e10	−	1.82926e11i	0.121588	−	0.435604i
$806$		0			0
$807$	7.52906e10	+	1.30407e11i	0.177520	+	0.307473i
$808$		0			0
$809$	−1.71049e11	+	2.96266e11i	−0.399326	+	0.691652i	−0.993643	−	0.112578i	$-0.964089\pi$
$809$	−1.71049e11	+	2.96266e11i	−0.399326	+	0.691652i	0.594317	+	0.804231i	$0.297422\pi$
$810$		0			0
$811$			2.89997e11i			0.670362i	0.942154	+	0.335181i	$0.108797\pi$
$811$			2.89997e11i			0.670362i	−0.942154	+	0.335181i	$0.891203\pi$
$812$		0			0
$813$	2.38890e11			0.546808
$814$		0			0
$815$	−1.31489e11	−	7.59150e10i	−0.298029	−	0.172067i
$816$		0			0
$817$	−2.27637e9	+	1.31426e9i	−0.00510923	+	0.00294981i
$818$		0			0
$819$	−1.70416e11	−	1.73997e11i	−0.378768	−	0.386729i
$820$		0			0
$821$	−3.60897e11	−	6.25091e11i	−0.794347	−	1.37585i	−0.923253	−	0.384193i	$-0.874480\pi$
$821$	−3.60897e11	−	6.25091e11i	−0.794347	−	1.37585i	0.128906	−	0.991657i	$-0.458853\pi$
$822$		0			0
$823$	−7.88089e10	+	1.36501e11i	−0.171781	+	0.297534i	−0.939043	−	0.343801i	$-0.888286\pi$
$823$	−7.88089e10	+	1.36501e11i	−0.171781	+	0.297534i	0.767261	+	0.641335i	$0.221619\pi$
$824$		0			0
$825$			2.93841e11i			0.634303i
$826$		0			0
$827$	−6.81570e11			−1.45710			−0.728549	−	0.684994i	$-0.759805\pi$
$827$	−6.81570e11			−1.45710			−0.728549	+	0.684994i	$0.759805\pi$
$828$		0			0
$829$	4.06194e11	+	2.34516e11i	0.860034	+	0.496541i	0.864024	−	0.503451i	$-0.167937\pi$
$829$	4.06194e11	+	2.34516e11i	0.860034	+	0.496541i	−0.00398941	+	0.999992i	$0.501270\pi$
$830$		0			0
$831$	4.21274e11	−	2.43223e11i	0.883406	−	0.510035i
$832$		0			0
$833$	7.04693e11	+	1.46578e10i	1.46359	+	0.0304430i
$834$		0			0
$835$	−9.70120e10	−	1.68030e11i	−0.199563	−	0.345653i
$836$		0			0
$837$	9.15250e10	−	1.58526e11i	0.186482	−	0.322997i
$838$		0			0
$839$		−	9.20167e11i		−	1.85703i	−0.371294	−	0.928515i	$-0.621086\pi$
$839$		−	9.20167e11i		−	1.85703i	0.371294	−	0.928515i	$-0.378914\pi$
$840$		0			0
$841$	−4.92481e11			−0.984478
$842$		0			0
$843$	−3.96554e11	−	2.28951e11i	−0.785222	−	0.453348i
$844$		0			0
$845$	3.00552e11	−	1.73524e11i	0.589512	−	0.340355i
$846$		0			0
$847$	2.81015e11	−	2.75230e11i	0.546004	−	0.534765i
$848$		0			0
$849$	−2.43881e10	−	4.22413e10i	−0.0469404	−	0.0813031i
$850$		0			0
$851$	−2.29848e11	+	3.98108e11i	−0.438251	+	0.759072i
$852$		0			0
$853$			5.25940e11i			0.993437i	0.867912	+	0.496718i	$0.165462\pi$
$853$			5.25940e11i			0.993437i	−0.867912	+	0.496718i	$0.834538\pi$
$854$		0			0
$855$	−1.21052e10			−0.0226520
$856$		0			0
$857$	5.85568e10	+	3.38078e10i	0.108556	+	0.0626748i	0.553295	−	0.832985i	$-0.313370\pi$
$857$	5.85568e10	+	3.38078e10i	0.108556	+	0.0626748i	−0.444739	+	0.895660i	$0.646704\pi$
$858$		0			0
$859$	−2.39418e11	+	1.38228e11i	−0.439728	+	0.253877i	−0.703482	−	0.710713i	$-0.748373\pi$
$859$	−2.39418e11	+	1.38228e11i	−0.439728	+	0.253877i	0.263754	+	0.964590i	$0.415039\pi$
$860$		0			0
$861$	−2.03345e10	−	5.67588e9i	−0.0370016	−	0.0103281i
$862$		0			0
$863$	2.25121e11	+	3.89921e11i	0.405857	+	0.702964i	0.994421	−	0.105486i	$-0.0336399\pi$
$863$	2.25121e11	+	3.89921e11i	0.405857	+	0.702964i	−0.588564	+	0.808450i	$0.700307\pi$
$864$		0			0
$865$	−1.36256e11	+	2.36003e11i	−0.243384	+	0.421554i
$866$		0			0
$867$			3.72884e11i			0.659929i
$868$		0			0
$869$	8.29305e11			1.45424
$870$		0			0
$871$	4.62631e11	+	2.67100e11i	0.803825	+	0.464089i
$872$		0			0
$873$	2.20919e11	−	1.27548e11i	0.380344	−	0.219592i
$874$		0			0
$875$	4.31306e11	−	1.10774e11i	0.735789	−	0.188976i
$876$		0			0
$877$	−9.03817e10	−	1.56546e11i	−0.152786	−	0.264632i	0.779465	−	0.626446i	$-0.215491\pi$
$877$	−9.03817e10	−	1.56546e11i	−0.152786	−	0.264632i	−0.932250	+	0.361814i	$0.882158\pi$
$878$		0			0
$879$	6.63680e10	−	1.14953e11i	0.111174	−	0.192559i
$880$		0			0
$881$			2.07020e11i			0.343644i	0.985128	+	0.171822i	$0.0549655\pi$
$881$			2.07020e11i			0.343644i	−0.985128	+	0.171822i	$0.945035\pi$
$882$		0			0
$883$	−1.01138e11			−0.166369			−0.0831845	−	0.996534i	$-0.526509\pi$
$883$	−1.01138e11			−0.166369			−0.0831845	+	0.996534i	$0.526509\pi$
$884$		0			0
$885$	−1.21893e11	−	7.03750e10i	−0.198704	−	0.114722i
$886$		0			0
$887$	−1.35333e11	+	7.81346e10i	−0.218630	+	0.126226i	−0.605316	−	0.795985i	$-0.706953\pi$
$887$	−1.35333e11	+	7.81346e10i	−0.218630	+	0.126226i	0.386686	+	0.922212i	$0.373620\pi$
$888$		0			0
$889$	−2.37479e10	−	9.24640e10i	−0.0380206	−	0.148035i
$890$		0			0
$891$	4.65071e10	+	8.05527e10i	0.0737919	+	0.127811i
$892$		0			0
$893$	−6.87336e10	+	1.19050e11i	−0.108084	+	0.187208i
$894$		0			0
$895$		−	6.45676e10i		−	0.100629i
$896$		0			0
$897$	−6.60256e11			−1.01986
$898$		0			0
$899$	−1.36584e11	−	7.88566e10i	−0.209103	−	0.120726i
$900$		0			0
$901$	1.45807e12	−	8.41819e11i	2.21248	−	1.27738i
$902$		0			0
$903$	3.72520e9	−	1.33459e10i	0.00560271	−	0.0200723i
$904$		0			0
$905$	6.39192e10	+	1.10711e11i	0.0952877	+	0.165043i
$906$		0			0
$907$	6.32401e11	−	1.09535e12i	0.934466	−	1.61854i	0.158882	−	0.987298i	$-0.449211\pi$
$907$	6.32401e11	−	1.09535e12i	0.934466	−	1.61854i	0.775584	−	0.631245i	$-0.217456\pi$
$908$		0			0
$909$			2.33286e11i			0.341691i
$910$		0			0
$911$	1.10755e12			1.60801			0.804005	−	0.594623i	$-0.202699\pi$
$911$	1.10755e12			1.60801			0.804005	+	0.594623i	$0.202699\pi$
$912$		0			0
$913$	−1.21256e12	−	7.00074e11i	−1.74510	−	1.00754i
$914$		0			0
$915$	2.34130e11	−	1.35175e11i	0.334020	−	0.192847i
$916$		0			0
$917$	6.43679e10	+	6.57207e10i	0.0910316	+	0.0929447i
$918$		0			0
$919$	3.08017e10	+	5.33501e10i	0.0431830	+	0.0747951i	0.886809	−	0.462136i	$-0.152917\pi$
$919$	3.08017e10	+	5.33501e10i	0.0431830	+	0.0747951i	−0.843626	+	0.536931i	$0.819583\pi$
$920$		0			0
$921$	−2.37697e11	+	4.11703e11i	−0.330358	+	0.572197i
$922$		0			0
$923$		−	1.44407e12i		−	1.98967i
$924$		0			0
$925$	−4.87940e11			−0.666498
$926$		0			0
$927$	3.88684e11	+	2.24407e11i	0.526353	+	0.303890i
$928$		0			0
$929$	3.05667e11	−	1.76477e11i	0.410379	−	0.236933i	−0.280573	−	0.959833i	$-0.590525\pi$
$929$	3.05667e11	−	1.76477e11i	0.410379	−	0.236933i	0.690953	+	0.722900i	$0.257191\pi$
$930$		0			0
$931$	−6.35946e10	+	1.05042e11i	−0.0846489	+	0.139818i
$932$		0			0
$933$	−2.52659e11	−	4.37618e11i	−0.333433	−	0.577522i
$934$		0			0
$935$	3.08934e11	−	5.35090e11i	0.404222	−	0.700132i
$936$		0			0
$937$			1.00546e12i			1.30439i	0.758051	+	0.652195i	$0.226152\pi$
$937$			1.00546e12i			1.30439i	−0.758051	+	0.652195i	$0.773848\pi$
$938$		0			0
$939$	−4.21881e11			−0.542659
$940$		0			0
$941$	−1.52508e11	−	8.80503e10i	−0.194506	−	0.112298i	0.399584	−	0.916696i	$-0.369154\pi$
$941$	−1.52508e11	−	8.80503e10i	−0.194506	−	0.112298i	−0.594090	+	0.804398i	$0.702488\pi$
$942$		0			0
$943$	−4.95655e10	+	2.86167e10i	−0.0626806	+	0.0361886i
$944$		0			0
$945$	4.55884e10	−	4.46500e10i	0.0571645	−	0.0559879i
$946$		0			0
$947$	−2.75622e11	−	4.77392e11i	−0.342700	−	0.593574i	0.642233	−	0.766510i	$-0.278008\pi$
$947$	−2.75622e11	−	4.77392e11i	−0.342700	−	0.593574i	−0.984933	+	0.172935i	$0.944675\pi$
$948$		0			0
$949$	2.90377e11	−	5.02947e11i	0.358012	−	0.620094i
$950$		0			0
$951$			1.96870e11i			0.240689i
$952$		0			0
$953$	8.34459e11			1.01166			0.505829	−	0.862634i	$-0.331187\pi$
$953$	8.34459e11			1.01166			0.505829	+	0.862634i	$0.331187\pi$
$954$		0			0
$955$	−4.29232e11	−	2.47817e11i	−0.516035	−	0.297933i
$956$		0			0
$957$	6.94030e10	−	4.00699e10i	0.0827429	−	0.0477716i
$958$		0			0
$959$	9.57499e11	+	2.67263e11i	1.13205	+	0.315983i
$960$		0			0
$961$	1.17519e12	+	2.03549e12i	1.37789	+	2.38657i
$962$		0			0
$963$	−1.94797e11	+	3.37398e11i	−0.226504	+	0.392317i
$964$		0			0
$965$		−	2.44418e11i		−	0.281853i
$966$		0			0
$967$	−7.68610e11			−0.879022			−0.439511	−	0.898237i	$-0.644848\pi$
$967$	−7.68610e11			−0.879022			−0.439511	+	0.898237i	$0.644848\pi$
$968$		0			0
$969$	−1.05476e11	−	6.08966e10i	−0.119635	−	0.0690713i
$970$		0			0
$971$	4.08759e11	−	2.35997e11i	0.459823	−	0.265479i	−0.252147	−	0.967689i	$-0.581137\pi$
$971$	4.08759e11	−	2.35997e11i	0.459823	−	0.265479i	0.711970	+	0.702210i	$0.247803\pi$
$972$		0			0
$973$	−5.01491e11	+	1.28800e11i	−0.559515	+	0.143703i
$974$		0			0
$975$	−3.50411e11	−	6.06929e11i	−0.387756	−	0.671614i
$976$		0			0
$977$	7.48912e11	−	1.29715e12i	0.821964	−	1.42368i	−0.0822542	−	0.996611i	$-0.526212\pi$
$977$	7.48912e11	−	1.29715e12i	0.821964	−	1.42368i	0.904218	−	0.427071i	$-0.140455\pi$
$978$		0			0
$979$		−	9.82835e11i		−	1.06992i
$980$		0			0
$981$	−2.39768e11			−0.258890
$982$		0			0
$983$	5.25935e11	+	3.03649e11i	0.563272	+	0.325205i	0.754458	−	0.656349i	$-0.227900\pi$
$983$	5.25935e11	+	3.03649e11i	0.563272	+	0.325205i	−0.191186	+	0.981554i	$0.561233\pi$
$984$		0			0
$985$	−2.76369e11	+	1.59562e11i	−0.293592	+	0.169505i
$986$		0			0
$987$	−1.80264e11	−	7.01868e11i	−0.189951	−	0.739583i
$988$		0			0
$989$	−1.87817e10	−	3.25309e10i	−0.0196313	−	0.0340025i
$990$		0			0
$991$	−4.64617e11	+	8.04740e11i	−0.481726	+	0.834374i	−0.999780	−	0.0209739i	$-0.993323\pi$
$991$	−4.64617e11	+	8.04740e11i	−0.481726	+	0.834374i	0.518054	+	0.855348i	$0.326657\pi$
$992$		0			0
$993$			5.59459e11i			0.575401i
$994$		0			0
$995$	−2.99725e10			−0.0305796
$996$		0			0
$997$	1.58181e12	+	9.13259e11i	1.60094	+	0.924300i	0.991301	+	0.131618i	$0.0420172\pi$
$997$	1.58181e12	+	9.13259e11i	1.60094	+	0.924300i	0.609635	+	0.792682i	$0.291316\pi$
$998$		0			0
$999$	−1.33762e11	+	7.72277e10i	−0.134299	+	0.0775374i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.b.73.2	yes	12
3.2	odd	2		252.9.z.d.73.5		12
7.5	odd	6	inner	84.9.m.b.61.2	✓	12
21.5	even	6		252.9.z.d.145.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.b.61.2	✓	12	7.5	odd	6	inner
84.9.m.b.73.2	yes	12	1.1	even	1	trivial
252.9.z.d.73.5		12	3.2	odd	2
252.9.z.d.145.5		12	21.5	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(40.5000 + 23.3827i) q^{3} +(-225.043 + 129.928i) q^{5} +(597.275 + 2325.52i) q^{7} +(1093.50 + 1894.00i) q^{9} +O(q^{10})\)	\(q+(40.5000 + 23.3827i) q^{3} +(-225.043 + 129.928i) q^{5} +(597.275 + 2325.52i) q^{7} +(1093.50 + 1894.00i) q^{9} +(9723.49 - 16841.6i) q^{11} +46381.7i q^{13} -12152.3 q^{15} +(-105886. - 61133.6i) q^{17} +(18446.7 - 10650.2i) q^{19} +(-30187.4 + 108150. i) q^{21} +(152199. + 263616. i) q^{23} +(-161550. + 279812. i) q^{25} +102276. i q^{27} +88119.4 q^{29} +(-1.54998e6 - 894884. i) q^{31} +(787602. - 454722. i) q^{33} +(-436564. - 445739. i) q^{35} +(755092. + 1.30786e6i) q^{37} +(-1.08453e6 + 1.87846e6i) q^{39} +188022. i q^{41} -123402. q^{43} +(-492168. - 284153. i) q^{45} +(-5.58909e6 + 3.22686e6i) q^{47} +(-5.05133e6 + 2.77796e6i) q^{49} +(-2.85893e6 - 4.95182e6i) q^{51} +(-6.88508e6 + 1.19253e7i) q^{53} +5.05343e6i q^{55} +996123. q^{57} +(1.00305e7 + 5.79109e6i) q^{59} +(-1.92663e7 + 1.11234e7i) q^{61} +(-3.75142e6 + 3.67420e6i) q^{63} +(-6.02630e6 - 1.04379e7i) q^{65} +(5.75873e6 - 9.97442e6i) q^{67} +1.42353e7i q^{69} -3.11345e7 q^{71} +(-1.08437e7 - 6.26059e6i) q^{73} +(-1.30855e7 + 7.55493e6i) q^{75} +(4.49731e7 + 1.25532e7i) q^{77} +(2.13222e7 + 3.69312e7i) q^{79} +(-2.39148e6 + 4.14217e6i) q^{81} -7.19982e7i q^{83} +3.17720e7 q^{85} +(3.56884e6 + 2.06047e6i) q^{87} +(4.37683e7 - 2.52696e7i) q^{89} +(-1.07862e8 + 2.77026e7i) q^{91} +(-4.18496e7 - 7.24856e7i) q^{93} +(-2.76753e6 + 4.79351e6i) q^{95} -1.16642e8i q^{97} +4.25305e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) \) 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9	\( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99}+O(q^{100}) \) 12 * q + 486 * q^3 + 285 * q^5 + 198 * q^7 + 13122 * q^9 - 17919 * q^11 + 15390 * q^15 - 205782 * q^17 + 74313 * q^19 - 39609 * q^21 - 62832 * q^23 + 878679 * q^25 - 575454 * q^29 + 1442952 * q^31 - 1451439 * q^33 - 3989514 * q^35 - 2058621 * q^37 - 930933 * q^39 + 7721322 * q^43 + 623295 * q^45 + 12088194 * q^47 - 16964694 * q^49 - 5556114 * q^51 - 5506743 * q^53 + 4012902 * q^57 + 7511901 * q^59 - 37215576 * q^61 - 3641355 * q^63 + 5047122 * q^65 - 36824553 * q^67 - 30011556 * q^71 + 95080185 * q^73 + 71172999 * q^75 - 38333727 * q^77 + 8514456 * q^79 - 28697814 * q^81 + 20121540 * q^85 - 23305887 * q^87 + 83038554 * q^89 - 198538635 * q^91 + 38959704 * q^93 - 221605224 * q^95 - 78377706 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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