LMFDB - Embedded newform 84.9.m.b.61.5

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			61.5
Root			$221.993 + 384.503i$ of defining polynomial
Character	$\chi$	$=$	84.61
Dual form			84.9.m.b.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.5000 - 23.3827i) q^{3} +(632.551 + 365.204i) q^{5} +(984.437 + 2189.91i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})$	\(q+(40.5000 - 23.3827i) q^{3} +(632.551 + 365.204i) q^{5} +(984.437 + 2189.91i) q^{7} +(1093.50 - 1894.00i) q^{9} +(6353.95 + 11005.4i) q^{11} +21176.4i q^{13} +34157.8 q^{15} +(-82551.9 + 47661.3i) q^{17} +(-133336. - 76981.8i) q^{19} +(91075.6 + 65672.4i) q^{21} +(-47111.5 + 81599.5i) q^{23} +(71435.0 + 123729. i) q^{25} -102276. i q^{27} +541664. q^{29} +(-185730. + 107231. i) q^{31} +(514670. + 297145. i) q^{33} +(-177055. + 1.74475e6i) q^{35} +(-370376. + 641510. i) q^{37} +(495160. + 857643. i) q^{39} -782599. i q^{41} -221324. q^{43} +(1.38339e6 - 798700. i) q^{45} +(7.13230e6 + 4.11784e6i) q^{47} +(-3.82657e6 + 4.31165e6i) q^{49} +(-2.22890e6 + 3.86057e6i) q^{51} +(659083. + 1.14157e6i) q^{53} +9.28194e6i q^{55} -7.20016e6 q^{57} +(-7.52865e6 + 4.34667e6i) q^{59} +(1.31192e7 + 7.57439e6i) q^{61} +(5.22416e6 + 530140. i) q^{63} +(-7.73369e6 + 1.33951e7i) q^{65} +(1.99100e7 + 3.44851e7i) q^{67} +4.40638e6i q^{69} -9.34713e6 q^{71} +(2.94520e7 - 1.70041e7i) q^{73} +(5.78623e6 + 3.34068e6i) q^{75} +(-1.78456e7 + 2.47486e7i) q^{77} +(2.26450e7 - 3.92223e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} -6.58893e7i q^{83} -6.96244e7 q^{85} +(2.19374e7 - 1.26656e7i) q^{87} +(-1.09902e7 - 6.34517e6i) q^{89} +(-4.63742e7 + 2.08468e7i) q^{91} +(-5.01470e6 + 8.68571e6i) q^{93} +(-5.62280e7 - 9.73898e7i) q^{95} -6.15780e6i q^{97} +2.77922e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	632.551	+	365.204i	1.01208	+	0.584326i	0.911801	−	0.410633i	$-0.134692\pi$
$5$	632.551	+	365.204i	1.01208	+	0.584326i	0.100282	+	0.994959i	$0.468026\pi$
$6$		0			0
$7$	984.437	+	2189.91i	0.410011	+	0.912080i
$7$	984.437	+	2189.91i	0.410011	+	0.912080i
$8$		0			0
$9$	1093.50	−	1894.00i	0.166667	−	0.288675i
$10$		0			0
$11$	6353.95	+	11005.4i	0.433983	+	0.751681i	0.997212	−	0.0746198i	$-0.0237743\pi$
$11$	6353.95	+	11005.4i	0.433983	+	0.751681i	−0.563229	+	0.826301i	$0.690441\pi$
$12$		0			0
$13$			21176.4i			0.741444i	0.928744	+	0.370722i	$0.120890\pi$
$13$			21176.4i			0.741444i	−0.928744	+	0.370722i	$0.879110\pi$
$14$		0			0
$15$	34157.8			0.674721
$16$		0			0
$17$	−82551.9	+	47661.3i	−0.988397	+	0.570651i	−0.904795	−	0.425848i	$-0.859976\pi$
$17$	−82551.9	+	47661.3i	−0.988397	+	0.570651i	−0.0836020	+	0.996499i	$0.526642\pi$
$18$		0			0
$19$	−133336.	−	76981.8i	−1.02314	−	0.590709i	−0.108126	−	0.994137i	$-0.534485\pi$
$19$	−133336.	−	76981.8i	−1.02314	−	0.590709i	−0.915011	+	0.403428i	$0.867818\pi$
$20$		0			0
$21$	91075.6	+	65672.4i	0.468301	+	0.337680i
$22$		0			0
$23$	−47111.5	+	81599.5i	−0.168351	+	0.291593i	−0.937840	−	0.347067i	$-0.887178\pi$
$23$	−47111.5	+	81599.5i	−0.168351	+	0.291593i	0.769489	+	0.638660i	$0.220511\pi$
$24$		0			0
$25$	71435.0	+	123729.i	0.182874	+	0.316746i
$26$		0			0
$27$		−	102276.i		−	0.192450i
$28$		0			0
$29$	541664.			0.765840			0.382920	−	0.923782i	$-0.374919\pi$
$29$	541664.			0.765840			0.382920	+	0.923782i	$0.374919\pi$
$30$		0			0
$31$	−185730.	+	107231.i	−0.201110	+	0.116111i	−0.597173	−	0.802112i	$-0.703710\pi$
$31$	−185730.	+	107231.i	−0.201110	+	0.116111i	0.396063	+	0.918223i	$0.370376\pi$
$32$		0			0
$33$	514670.	+	297145.i	0.433983	+	0.250560i
$34$		0			0
$35$	−177055.	+	1.74475e6i	−0.117987	+	1.16268i
$36$		0			0
$37$	−370376.	+	641510.i	−0.197622	+	0.342292i	−0.947757	−	0.318993i	$-0.896655\pi$
$37$	−370376.	+	641510.i	−0.197622	+	0.342292i	0.750135	+	0.661285i	$0.229989\pi$
$38$		0			0
$39$	495160.	+	857643.i	0.214036	+	0.370722i
$40$		0			0
$41$		−	782599.i		−	0.276952i	−0.990366	−	0.138476i	$-0.955780\pi$
$41$		−	782599.i		−	0.276952i	0.990366	−	0.138476i	$-0.0442203\pi$
$42$		0			0
$43$	−221324.			−0.0647373			−0.0323687	−	0.999476i	$-0.510305\pi$
$43$	−221324.			−0.0647373			−0.0323687	+	0.999476i	$0.510305\pi$
$44$		0			0
$45$	1.38339e6	−	798700.i	0.337361	−	0.194775i
$46$		0			0
$47$	7.13230e6	+	4.11784e6i	1.46163	+	0.843874i	0.999087	−	0.0427209i	$-0.0136026\pi$
$47$	7.13230e6	+	4.11784e6i	1.46163	+	0.843874i	0.462546	+	0.886595i	$0.346936\pi$
$48$		0			0
$49$	−3.82657e6	+	4.31165e6i	−0.663782	+	0.747926i
$50$		0			0
$51$	−2.22890e6	+	3.86057e6i	−0.329466	+	0.570651i
$52$		0			0
$53$	659083.	+	1.14157e6i	0.0835289	+	0.144676i	0.904763	−	0.425915i	$-0.140048\pi$
$53$	659083.	+	1.14157e6i	0.0835289	+	0.144676i	−0.821235	+	0.570591i	$0.806714\pi$
$54$		0			0
$55$			9.28194e6i			1.01435i
$56$		0			0
$57$	−7.20016e6			−0.682092
$58$		0			0
$59$	−7.52865e6	+	4.34667e6i	−0.621311	+	0.358714i	−0.777379	−	0.629032i	$-0.783451\pi$
$59$	−7.52865e6	+	4.34667e6i	−0.621311	+	0.358714i	0.156068	+	0.987746i	$0.450118\pi$
$60$		0			0
$61$	1.31192e7	+	7.57439e6i	0.947521	+	0.547052i	0.892310	−	0.451423i	$-0.149083\pi$
$61$	1.31192e7	+	7.57439e6i	0.947521	+	0.547052i	0.0552112	+	0.998475i	$0.482417\pi$
$62$		0			0
$63$	5.22416e6	+	530140.i	0.331630	+	0.0336534i
$64$		0			0
$65$	−7.73369e6	+	1.33951e7i	−0.433245	+	0.750402i
$66$		0			0
$67$	1.99100e7	+	3.44851e7i	0.988033	+	1.71132i	0.627596	+	0.778539i	$0.284039\pi$
$67$	1.99100e7	+	3.44851e7i	0.988033	+	1.71132i	0.360436	+	0.932784i	$0.382628\pi$
$68$		0			0
$69$			4.40638e6i			0.194395i
$70$		0			0
$71$	−9.34713e6			−0.367828			−0.183914	−	0.982942i	$-0.558877\pi$
$71$	−9.34713e6			−0.367828			−0.183914	+	0.982942i	$0.558877\pi$
$72$		0			0
$73$	2.94520e7	−	1.70041e7i	1.03711	−	0.598774i	0.118094	−	0.993002i	$-0.462321\pi$
$73$	2.94520e7	−	1.70041e7i	1.03711	−	0.598774i	0.919012	+	0.394229i	$0.128988\pi$
$74$		0			0
$75$	5.78623e6	+	3.34068e6i	0.182874	+	0.105582i
$76$		0			0
$77$	−1.78456e7	+	2.47486e7i	−0.507656	+	0.704025i
$78$		0			0
$79$	2.26450e7	−	3.92223e7i	0.581385	−	1.00699i	−0.413931	−	0.910308i	$-0.635844\pi$
$79$	2.26450e7	−	3.92223e7i	0.581385	−	1.00699i	0.995316	−	0.0966799i	$-0.0308223\pi$
$80$		0			0
$81$	−2.39148e6	−	4.14217e6i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	6.58893e7i		−	1.38836i	−0.719801	−	0.694180i	$-0.755767\pi$
$83$		−	6.58893e7i		−	1.38836i	0.719801	−	0.694180i	$-0.244233\pi$
$84$		0			0
$85$	−6.96244e7			−1.33378
$86$		0			0
$87$	2.19374e7	−	1.26656e7i	0.382920	−	0.221079i
$88$		0			0
$89$	−1.09902e7	−	6.34517e6i	−0.175164	−	0.101131i	0.409855	−	0.912151i	$-0.365579\pi$
$89$	−1.09902e7	−	6.34517e6i	−0.175164	−	0.101131i	−0.585018	+	0.811020i	$0.698913\pi$
$90$		0			0
$91$	−4.63742e7	+	2.08468e7i	−0.676256	+	0.304000i
$92$		0			0
$93$	−5.01470e6	+	8.68571e6i	−0.0670368	+	0.116111i
$94$		0			0
$95$	−5.62280e7	−	9.73898e7i	−0.690333	−	1.19569i
$96$		0			0
$97$		−	6.15780e6i		−	0.0695566i	−0.999395	−	0.0347783i	$-0.988927\pi$
$97$		−	6.15780e6i		−	0.0695566i	0.999395	−	0.0347783i	$-0.0110725\pi$
$98$		0			0
$99$	2.77922e7			0.289322
$100$		0			0
$101$	1.15329e8	−	6.65855e7i	1.10829	−	0.639874i	0.169907	−	0.985460i	$-0.445653\pi$
$101$	1.15329e8	−	6.65855e7i	1.10829	−	0.639874i	0.938387	+	0.345587i	$0.112320\pi$
$102$		0			0
$103$	4.81925e7	+	2.78240e7i	0.428184	+	0.247212i	0.698573	−	0.715539i	$-0.253819\pi$
$103$	4.81925e7	+	2.78240e7i	0.428184	+	0.247212i	−0.270389	+	0.962751i	$0.587152\pi$
$104$		0			0
$105$	3.36262e7	+	7.48023e7i	0.276643	+	0.615400i
$106$		0			0
$107$	7.23667e6	−	1.25343e7i	0.0552082	−	0.0956234i	−0.837101	−	0.547049i	$-0.815751\pi$
$107$	7.23667e6	−	1.25343e7i	0.0552082	−	0.0956234i	0.892309	+	0.451426i	$0.149084\pi$
$108$		0			0
$109$	−9.01793e7	−	1.56195e8i	−0.638853	−	1.10653i	−0.985685	−	0.168598i	$-0.946076\pi$
$109$	−9.01793e7	−	1.56195e8i	−0.638853	−	1.10653i	0.346832	−	0.937927i	$-0.387257\pi$
$110$		0			0
$111$			3.46415e7i			0.228195i
$112$		0			0
$113$	−2.21791e7			−0.136028			−0.0680142	−	0.997684i	$-0.521666\pi$
$113$	−2.21791e7			−0.136028			−0.0680142	+	0.997684i	$0.521666\pi$
$114$		0			0
$115$	−5.96009e7	+	3.44106e7i	−0.340770	+	0.196744i
$116$		0			0
$117$	4.01080e7	+	2.31564e7i	0.214036	+	0.123574i
$118$		0			0
$119$	−1.85641e8	−	1.33861e8i	−0.925733	−	0.667524i
$120$		0			0
$121$	2.64341e7	−	4.57852e7i	0.123317	−	0.213591i
$122$		0			0
$123$	−1.82993e7	−	3.16953e7i	−0.0799490	−	0.138476i
$124$		0			0
$125$		−	1.80962e8i		−	0.741221i
$126$		0			0
$127$	1.77711e8			0.683123			0.341561	−	0.939859i	$-0.389044\pi$
$127$	1.77711e8			0.683123			0.341561	+	0.939859i	$0.389044\pi$
$128$		0			0
$129$	−8.96362e6	+	5.17515e6i	−0.0323687	+	0.0186881i
$130$		0			0
$131$	−4.83164e8	−	2.78955e8i	−1.64062	−	0.947215i	−0.980612	−	0.195960i	$-0.937218\pi$
$131$	−4.83164e8	−	2.78955e8i	−1.64062	−	0.947215i	−0.660012	−	0.751255i	$-0.729449\pi$
$132$		0			0
$133$	3.73216e7	−	3.67778e8i	0.119276	−	1.17538i
$134$		0			0
$135$	3.73515e7	−	6.46947e7i	0.112454	−	0.194775i
$136$		0			0
$137$	−1.96914e8	−	3.41064e8i	−0.558977	−	0.968176i	−0.997582	−	0.0694962i	$-0.977861\pi$
$137$	−1.96914e8	−	3.41064e8i	−0.558977	−	0.968176i	0.438606	−	0.898680i	$-0.355472\pi$
$138$		0			0
$139$			4.85677e8i			1.30103i	0.759492	+	0.650516i	$0.225447\pi$
$139$			4.85677e8i			1.30103i	−0.759492	+	0.650516i	$0.774553\pi$
$140$		0			0
$141$	3.85144e8			0.974422
$142$		0			0
$143$	−2.33054e8	+	1.34554e8i	−0.557329	+	0.321774i
$144$		0			0
$145$	3.42630e8	+	1.97818e8i	0.775093	+	0.447500i
$146$		0			0
$147$	−5.41582e7	+	2.64097e8i	−0.115983	+	0.565580i
$148$		0			0
$149$	1.60563e8	−	2.78103e8i	0.325762	−	0.564236i	−0.655905	−	0.754844i	$-0.727713\pi$
$149$	1.60563e8	−	2.78103e8i	0.325762	−	0.564236i	0.981666	+	0.190608i	$0.0610459\pi$
$150$		0			0
$151$	4.12702e8	+	7.14822e8i	0.793833	+	1.37496i	0.923577	+	0.383412i	$0.125251\pi$
$151$	4.12702e8	+	7.14822e8i	0.793833	+	1.37496i	−0.129744	+	0.991547i	$0.541416\pi$
$152$		0			0
$153$			2.08471e8i			0.380434i
$154$		0			0
$155$	−1.56645e8			−0.271387
$156$		0			0
$157$	9.15549e8	−	5.28592e8i	1.50689	−	0.870006i	0.506926	−	0.861989i	$-0.330782\pi$
$157$	9.15549e8	−	5.28592e8i	1.50689	−	0.870006i	0.999968	−	0.00801653i	$-0.00255177\pi$
$158$		0			0
$159$	5.33857e7	+	3.08223e7i	0.0835289	+	0.0482254i
$160$		0			0
$161$	−2.25074e8	−	2.28402e7i	−0.334982	−	0.0339935i
$162$		0			0
$163$	−5.67831e8	+	9.83512e8i	−0.804394	+	1.39325i	0.112306	+	0.993674i	$0.464176\pi$
$163$	−5.67831e8	+	9.83512e8i	−0.804394	+	1.39325i	−0.916699	+	0.399577i	$0.869157\pi$
$164$		0			0
$165$	2.17037e8	+	3.75919e8i	0.292818	+	0.507175i
$166$		0			0
$167$			1.00076e9i			1.28666i	0.765591	+	0.643328i	$0.222447\pi$
$167$			1.00076e9i			1.28666i	−0.765591	+	0.643328i	$0.777553\pi$
$168$		0			0
$169$	3.67292e8			0.450261
$170$		0			0
$171$	−2.91606e8	+	1.68359e8i	−0.341046	+	0.196903i
$172$		0			0
$173$	−5.46649e8	−	3.15608e8i	−0.610273	−	0.352341i	0.162799	−	0.986659i	$-0.447948\pi$
$173$	−5.46649e8	−	3.15608e8i	−0.610273	−	0.352341i	−0.773072	+	0.634318i	$0.781281\pi$
$174$		0			0
$175$	−2.00632e8	+	2.78239e8i	−0.213918	+	0.296665i
$176$		0			0
$177$	−2.03274e8	+	3.52080e8i	−0.207104	+	0.358714i
$178$		0			0
$179$	−5.77342e8	−	9.99986e8i	−0.562369	−	0.974052i	−0.997289	−	0.0735826i	$-0.976557\pi$
$179$	−5.77342e8	−	9.99986e8i	−0.562369	−	0.974052i	0.434920	−	0.900469i	$-0.356777\pi$
$180$		0			0
$181$			1.42212e9i			1.32502i	0.749052	+	0.662511i	$0.230509\pi$
$181$			1.42212e9i			1.32502i	−0.749052	+	0.662511i	$0.769491\pi$
$182$		0			0
$183$	7.08438e8			0.631681
$184$		0			0
$185$	−4.68564e8	+	2.70525e8i	−0.400020	+	0.230952i
$186$		0			0
$187$	−1.04906e9	−	6.05676e8i	−0.857895	−	0.495306i
$188$		0			0
$189$	2.23974e8	−	1.00684e8i	0.175530	−	0.0789067i
$190$		0			0
$191$	3.05844e8	−	5.29737e8i	0.229809	−	0.398040i	−0.727943	−	0.685638i	$-0.759523\pi$
$191$	3.05844e8	−	5.29737e8i	0.229809	−	0.398040i	0.957751	+	0.287598i	$0.0928566\pi$
$192$		0			0
$193$	9.49542e8	+	1.64465e9i	0.684360	+	1.18535i	0.973637	+	0.228101i	$0.0732518\pi$
$193$	9.49542e8	+	1.64465e9i	0.684360	+	1.18535i	−0.289277	+	0.957245i	$0.593415\pi$
$194$		0			0
$195$			7.23338e8i			0.500268i
$196$		0			0
$197$	1.00843e9			0.669544			0.334772	−	0.942299i	$-0.391341\pi$
$197$	1.00843e9			0.669544			0.334772	+	0.942299i	$0.391341\pi$
$198$		0			0
$199$	2.76255e8	−	1.59496e8i	0.176156	−	0.101704i	−0.409329	−	0.912387i	$-0.634237\pi$
$199$	2.76255e8	−	1.59496e8i	0.176156	−	0.101704i	0.585485	+	0.810683i	$0.300904\pi$
$200$		0			0
$201$	1.61271e9	+	9.31097e8i	0.988033	+	0.570441i
$202$		0			0
$203$	5.33234e8	+	1.18619e9i	0.314003	+	0.698507i
$204$		0			0
$205$	2.85808e8	−	4.95034e8i	0.161830	−	0.280298i
$206$		0			0
$207$	1.03033e8	+	1.78458e8i	0.0561170	+	0.0971975i
$208$		0			0
$209$		−	1.95655e9i		−	1.02543i
$210$		0			0
$211$	−1.39969e9			−0.706160			−0.353080	−	0.935593i	$-0.614866\pi$
$211$	−1.39969e9			−0.706160			−0.353080	+	0.935593i	$0.614866\pi$
$212$		0			0
$213$	−3.78559e8	+	2.18561e8i	−0.183914	+	0.106183i
$214$		0			0
$215$	−1.39999e8	−	8.08283e7i	−0.0655195	−	0.0378277i
$216$		0			0
$217$	−4.17665e8	−	3.01168e8i	−0.188360	−	0.135822i
$218$		0			0
$219$	7.95204e8	−	1.37733e9i	0.345702	−	0.598774i
$220$		0			0
$221$	−1.00929e9	−	1.74815e9i	−0.423106	−	0.732840i
$222$		0			0
$223$		−	3.09118e9i		−	1.24999i	−0.780630	−	0.624993i	$-0.785102\pi$
$223$		−	3.09118e9i		−	1.24999i	0.780630	−	0.624993i	$-0.214898\pi$
$224$		0			0
$225$	3.12457e8			0.121916
$226$		0			0
$227$	−3.27805e9	+	1.89258e9i	−1.23456	+	0.712774i	−0.967977	−	0.251038i	$-0.919228\pi$
$227$	−3.27805e9	+	1.89258e9i	−1.23456	+	0.712774i	−0.266583	+	0.963812i	$0.585895\pi$
$228$		0			0
$229$	−2.74391e9	−	1.58420e9i	−0.997763	−	0.576059i	−0.0901773	−	0.995926i	$-0.528743\pi$
$229$	−2.74391e9	−	1.58420e9i	−0.997763	−	0.576059i	−0.907586	+	0.419867i	$0.862077\pi$
$230$		0			0
$231$	−1.44059e8	+	1.41960e9i	−0.0505932	+	0.498560i
$232$		0			0
$233$	2.76314e9	−	4.78590e9i	0.937518	−	1.62383i	0.167437	−	0.985883i	$-0.446451\pi$
$233$	2.76314e9	−	4.78590e9i	0.937518	−	1.62383i	0.770081	−	0.637946i	$-0.220216\pi$
$234$		0			0
$235$	3.00770e9	+	5.20949e9i	0.986195	+	1.70814i
$236$		0			0
$237$		−	2.11800e9i		−	0.671326i
$238$		0			0
$239$	−3.93530e9			−1.20611			−0.603054	−	0.797701i	$-0.706049\pi$
$239$	−3.93530e9			−1.20611			−0.603054	+	0.797701i	$0.706049\pi$
$240$		0			0
$241$	5.76301e9	−	3.32727e9i	1.70837	−	0.986326i	0.771780	−	0.635889i	$-0.219367\pi$
$241$	5.76301e9	−	3.32727e9i	1.70837	−	0.986326i	0.936586	−	0.350437i	$-0.113967\pi$
$242$		0			0
$243$	−1.93710e8	−	1.11839e8i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−3.99513e9	+	1.32986e9i	−1.10883	+	0.369098i
$246$		0			0
$247$	1.63019e9	−	2.82358e9i	0.437977	−	0.758599i
$248$		0			0
$249$	−1.54067e9	−	2.66852e9i	−0.400785	−	0.694180i
$250$		0			0
$251$		−	4.25680e9i		−	1.07248i	−0.844066	−	0.536239i	$-0.819845\pi$
$251$		−	4.25680e9i		−	1.07248i	0.844066	−	0.536239i	$-0.180155\pi$
$252$		0			0
$253$	−1.19738e9			−0.292246
$254$		0			0
$255$	−2.81979e9	+	1.62801e9i	−0.666892	+	0.385031i
$256$		0			0
$257$	−2.63398e9	−	1.52073e9i	−0.603782	−	0.348593i	0.166746	−	0.986000i	$-0.446674\pi$
$257$	−2.63398e9	−	1.52073e9i	−0.603782	−	0.348593i	−0.770528	+	0.637406i	$0.780007\pi$
$258$		0			0
$259$	−1.76946e9	−	1.79562e8i	−0.393225	−	0.0399039i
$260$		0			0
$261$	5.92309e8	−	1.02591e9i	0.127640	−	0.221079i
$262$		0			0
$263$	4.20815e9	+	7.28873e9i	0.879566	+	1.52345i	0.851818	+	0.523838i	$0.175500\pi$
$263$	4.20815e9	+	7.28873e9i	0.879566	+	1.52345i	0.0277486	+	0.999615i	$0.491166\pi$
$264$		0			0
$265$			9.62798e8i			0.195232i
$266$		0			0
$267$	−5.93468e8			−0.116776
$268$		0			0
$269$	6.40038e9	−	3.69526e9i	1.22235	−	0.705726i	0.256934	−	0.966429i	$-0.417288\pi$
$269$	6.40038e9	−	3.69526e9i	1.22235	−	0.705726i	0.965419	+	0.260703i	$0.0839542\pi$
$270$		0			0
$271$	−3.72934e9	−	2.15314e9i	−0.691441	−	0.399204i	0.112711	−	0.993628i	$-0.464047\pi$
$271$	−3.72934e9	−	2.15314e9i	−0.691441	−	0.399204i	−0.804152	+	0.594424i	$0.797380\pi$
$272$		0			0
$273$	−1.39070e9	+	1.92865e9i	−0.250371	+	0.347218i
$274$		0			0
$275$	−9.07789e8	+	1.57234e9i	−0.158728	+	0.274925i
$276$		0			0
$277$	−2.54312e7	−	4.40481e7i	−0.00431964	−	0.00748183i	0.863858	−	0.503736i	$-0.168042\pi$
$277$	−2.54312e7	−	4.40481e7i	−0.00431964	−	0.00748183i	−0.868177	+	0.496254i	$0.834708\pi$
$278$		0			0
$279$			4.69028e8i			0.0774074i
$280$		0			0
$281$	1.03421e10			1.65877			0.829383	−	0.558680i	$-0.188692\pi$
$281$	1.03421e10			1.65877			0.829383	+	0.558680i	$0.188692\pi$
$282$		0			0
$283$	−2.55984e9	+	1.47792e9i	−0.399086	+	0.230412i	−0.686090	−	0.727517i	$-0.740674\pi$
$283$	−2.55984e9	+	1.47792e9i	−0.399086	+	0.230412i	0.287003	+	0.957930i	$0.407341\pi$
$284$		0			0
$285$	−4.55447e9	−	2.62953e9i	−0.690333	−	0.398564i
$286$		0			0
$287$	1.71382e9	−	7.70419e8i	0.252602	−	0.113553i
$288$		0			0
$289$	1.05533e9	−	1.82788e9i	0.151285	−	0.262034i
$290$		0			0
$291$	−1.43986e8	−	2.49391e8i	−0.0200793	−	0.0347783i
$292$		0			0
$293$			4.22563e9i			0.573351i	0.958028	+	0.286676i	$0.0925502\pi$
$293$			4.22563e9i			0.573351i	−0.958028	+	0.286676i	$0.907450\pi$
$294$		0			0
$295$	−6.34968e9			−0.838424
$296$		0			0
$297$	1.12558e9	−	6.49856e8i	0.144661	−	0.0835201i
$298$		0			0
$299$	−1.72798e9	−	9.97651e8i	−0.216199	−	0.124823i
$300$		0			0
$301$	−2.17880e8	−	4.84679e8i	−0.0265430	−	0.0590456i
$302$		0			0
$303$	3.11390e9	−	5.39343e9i	0.369431	−	0.639874i
$304$		0			0
$305$	5.53239e9	+	9.58238e9i	0.639313	+	1.10732i
$306$		0			0
$307$		−	2.96070e9i		−	0.333304i	−0.986016	−	0.166652i	$-0.946704\pi$
$307$		−	2.96070e9i		−	0.333304i	0.986016	−	0.166652i	$-0.0532956\pi$
$308$		0			0
$309$	2.60240e9			0.285456
$310$		0			0
$311$	−7.74956e9	+	4.47421e9i	−0.828391	+	0.478272i	−0.853302	−	0.521418i	$-0.825403\pi$
$311$	−7.74956e9	+	4.47421e9i	−0.828391	+	0.478272i	0.0249101	+	0.999690i	$0.492070\pi$
$312$		0			0
$313$	−5.04664e9	−	2.91368e9i	−0.525806	−	0.303574i	0.213501	−	0.976943i	$-0.431513\pi$
$313$	−5.04664e9	−	2.91368e9i	−0.525806	−	0.303574i	−0.739307	+	0.673369i	$0.764847\pi$
$314$		0			0
$315$	3.11094e9	+	2.24322e9i	0.315972	+	0.227840i
$316$		0			0
$317$	2.50614e9	−	4.34076e9i	0.248181	−	0.429861i	−0.714840	−	0.699288i	$-0.753501\pi$
$317$	2.50614e9	−	4.34076e9i	0.248181	−	0.429861i	0.963021	+	0.269426i	$0.0868340\pi$
$318$		0			0
$319$	3.44171e9	+	5.96121e9i	0.332362	+	0.575667i
$320$		0			0
$321$		−	6.76851e8i		−	0.0637489i
$322$		0			0
$323$	1.46762e10			1.34835
$324$		0			0
$325$	−2.62013e9	+	1.51273e9i	−0.234850	+	0.135590i
$326$		0			0
$327$	−7.30452e9	−	4.21727e9i	−0.638853	−	0.368842i
$328$		0			0
$329$	−1.99637e9	+	1.96728e10i	−0.170395	+	1.67912i
$330$		0			0
$331$	4.78357e9	−	8.28539e9i	0.398511	−	0.690242i	−0.595031	−	0.803703i	$-0.702860\pi$
$331$	4.78357e9	−	8.28539e9i	0.398511	−	0.690242i	0.993542	+	0.113461i	$0.0361937\pi$
$332$		0			0
$333$	8.10012e8	+	1.40298e9i	0.0658741	+	0.114097i
$334$		0			0
$335$			2.90848e10i			2.30933i
$336$		0			0
$337$	2.32884e9			0.180560			0.0902798	−	0.995916i	$-0.471224\pi$
$337$	2.32884e9			0.180560			0.0902798	+	0.995916i	$0.471224\pi$
$338$		0			0
$339$	−8.98252e8	+	5.18606e8i	−0.0680142	+	0.0392680i
$340$		0			0
$341$	−2.36023e9	−	1.36268e9i	−0.174557	−	0.100781i
$342$		0			0
$343$	−1.32091e10	−	4.13528e9i	−0.954327	−	0.298764i
$344$		0			0
$345$	−1.60922e9	+	2.78726e9i	−0.113590	+	0.196744i
$346$		0			0
$347$	1.04233e10	+	1.80536e10i	0.718928	+	1.24522i	0.961425	+	0.275068i	$0.0887002\pi$
$347$	1.04233e10	+	1.80536e10i	0.718928	+	1.24522i	−0.242497	+	0.970152i	$0.577966\pi$
$348$		0			0
$349$			2.91490e10i			1.96482i	0.186745	+	0.982409i	$0.440206\pi$
$349$			2.91490e10i			1.96482i	−0.186745	+	0.982409i	$0.559794\pi$
$350$		0			0
$351$	2.16583e9			0.142691
$352$		0			0
$353$	9.78829e8	−	5.65127e8i	0.0630388	−	0.0363955i	−0.468149	−	0.883649i	$-0.655079\pi$
$353$	9.78829e8	−	5.65127e8i	0.0630388	−	0.0363955i	0.531188	+	0.847254i	$0.321746\pi$
$354$		0			0
$355$	−5.91254e9	−	3.41361e9i	−0.372272	−	0.214932i
$356$		0			0
$357$	−1.06485e10	−	1.08059e9i	−0.655564	−	0.0665258i
$358$		0			0
$359$	5.49405e9	−	9.51597e9i	0.330761	−	0.572896i	−0.651900	−	0.758305i	$-0.726028\pi$
$359$	5.49405e9	−	9.51597e9i	0.330761	−	0.572896i	0.982661	+	0.185409i	$0.0593611\pi$
$360$		0			0
$361$	3.36060e9	+	5.82073e9i	0.197874	+	0.342727i
$362$		0			0
$363$		−	2.47240e9i		−	0.142394i
$364$		0			0
$365$	2.48399e10			1.39952
$366$		0			0
$367$	−2.47539e9	+	1.42917e9i	−0.136452	+	0.0787805i	−0.566672	−	0.823944i	$-0.691769\pi$
$367$	−2.47539e9	+	1.42917e9i	−0.136452	+	0.0787805i	0.430220	+	0.902724i	$0.358436\pi$
$368$		0			0
$369$	−1.48224e9	−	8.55772e8i	−0.0799490	−	0.0461586i
$370$		0			0
$371$	−1.85109e9	+	2.56713e9i	−0.0977086	+	0.135504i
$372$		0			0
$373$	8.91785e9	−	1.54462e10i	0.460707	−	0.797968i	−0.538289	−	0.842760i	$-0.680929\pi$
$373$	8.91785e9	−	1.54462e10i	0.460707	−	0.797968i	0.998996	+	0.0447919i	$0.0142625\pi$
$374$		0			0
$375$	−4.23138e9	−	7.32897e9i	−0.213972	−	0.370610i
$376$		0			0
$377$			1.14705e10i			0.567827i
$378$		0			0
$379$	3.25934e10			1.57969			0.789847	−	0.613303i	$-0.210160\pi$
$379$	3.25934e10			1.57969			0.789847	+	0.613303i	$0.210160\pi$
$380$		0			0
$381$	7.19729e9	−	4.15536e9i	0.341561	−	0.197201i
$382$		0			0
$383$	−1.90349e10	−	1.09898e10i	−0.884619	−	0.510735i	−0.0124407	−	0.999923i	$-0.503960\pi$
$383$	−1.90349e10	−	1.09898e10i	−0.884619	−	0.510735i	−0.872179	+	0.489187i	$0.837293\pi$
$384$		0			0
$385$	−2.03266e10	+	9.13749e9i	−0.925170	+	0.415895i
$386$		0			0
$387$	−2.42018e8	+	4.19187e8i	−0.0107896	+	0.0186881i
$388$		0			0
$389$	−1.09555e10	−	1.89755e10i	−0.478448	−	0.828695i	0.521247	−	0.853406i	$-0.325467\pi$
$389$	−1.09555e10	−	1.89755e10i	−0.478448	−	0.828695i	−0.999695	+	0.0247103i	$0.992134\pi$
$390$		0			0
$391$		−	8.98159e9i		−	0.384279i
$392$		0			0
$393$	−2.60908e10			−1.09375
$394$		0			0
$395$	2.86482e10	−	1.65401e10i	1.17682	−	0.679437i
$396$		0			0
$397$	−4.10594e10	−	2.37057e10i	−1.65292	−	0.954312i	−0.975865	−	0.218376i	$-0.929924\pi$
$397$	−4.10594e10	−	2.37057e10i	−1.65292	−	0.954312i	−0.677052	−	0.735936i	$-0.736743\pi$
$398$		0			0
$399$	−7.08810e9	−	1.57677e10i	−0.279665	−	0.622123i
$400$		0			0
$401$	6.47336e8	−	1.12122e9i	0.0250353	−	0.0433623i	−0.853236	−	0.521524i	$-0.825364\pi$
$401$	6.47336e8	−	1.12122e9i	0.0250353	−	0.0433623i	0.878272	+	0.478162i	$0.158697\pi$
$402$		0			0
$403$	−2.27076e9	−	3.93308e9i	−0.0860898	−	0.149112i
$404$		0			0
$405$		−	3.49352e9i		−	0.129850i
$406$		0			0
$407$	−9.41340e9			−0.343059
$408$		0			0
$409$	1.05517e10	−	6.09204e9i	0.377077	−	0.217705i	−0.299469	−	0.954106i	$-0.596809\pi$
$409$	1.05517e10	−	6.09204e9i	0.377077	−	0.217705i	0.676546	+	0.736401i	$0.263476\pi$
$410$		0			0
$411$	−1.59500e10	−	9.20874e9i	−0.558977	−	0.322725i
$412$		0			0
$413$	−1.69303e10	−	1.22080e10i	−0.581921	−	0.419609i
$414$		0			0
$415$	2.40630e10	−	4.16783e10i	0.811255	−	1.40514i
$416$		0			0
$417$	1.13564e10	+	1.96699e10i	0.375576	+	0.650516i
$418$		0			0
$419$		−	2.83249e9i		−	0.0918992i	−0.998944	−	0.0459496i	$-0.985369\pi$
$419$		−	2.83249e9i		−	0.0918992i	0.998944	−	0.0459496i	$-0.0146314\pi$
$420$		0			0
$421$	1.47344e10			0.469033			0.234517	−	0.972112i	$-0.424649\pi$
$421$	1.47344e10			0.469033			0.234517	+	0.972112i	$0.424649\pi$
$422$		0			0
$423$	1.55983e10	−	9.00571e9i	0.487211	−	0.281291i
$424$		0			0
$425$	−1.17942e10	−	6.80938e9i	−0.361503	−	0.208714i
$426$		0			0
$427$	−3.67214e9	+	3.61864e10i	−0.110461	+	1.08851i
$428$		0			0
$429$	−6.29245e9	+	1.08988e10i	−0.185776	+	0.321774i
$430$		0			0
$431$	1.99404e10	+	3.45377e10i	0.577862	+	1.00089i	0.995724	+	0.0923761i	$0.0294462\pi$
$431$	1.99404e10	+	3.45377e10i	0.577862	+	1.00089i	−0.417862	+	0.908510i	$0.637220\pi$
$432$		0			0
$433$		−	3.77901e10i		−	1.07505i	−0.843249	−	0.537523i	$-0.819360\pi$
$433$		−	3.77901e10i		−	1.07505i	0.843249	−	0.537523i	$-0.180640\pi$
$434$		0			0
$435$	1.85020e10			0.516729
$436$		0			0
$437$	1.25634e10	−	7.25345e9i	0.344493	−	0.198893i
$438$		0			0
$439$	7.02545e9	+	4.05614e9i	0.189154	+	0.109208i	0.591587	−	0.806242i	$-0.298502\pi$
$439$	7.02545e9	+	4.05614e9i	0.189154	+	0.109208i	−0.402432	+	0.915450i	$0.631835\pi$
$440$		0			0
$441$	3.98190e9	+	1.19623e10i	0.105277	+	0.316272i
$442$		0			0
$443$	2.85877e10	−	4.95154e10i	0.742274	−	1.28566i	−0.209183	−	0.977876i	$-0.567081\pi$
$443$	2.85877e10	−	4.95154e10i	0.742274	−	1.28566i	0.951457	−	0.307780i	$-0.0995861\pi$
$444$		0			0
$445$	−4.63456e9	−	8.02729e9i	−0.118187	−	0.204705i
$446$		0			0
$447$		−	1.50176e10i		−	0.376157i
$448$		0			0
$449$	−6.27811e10			−1.54470			−0.772349	−	0.635199i	$-0.780918\pi$
$449$	−6.27811e10			−1.54470			−0.772349	+	0.635199i	$0.780918\pi$
$450$		0			0
$451$	8.61279e9	−	4.97259e9i	0.208179	−	0.120192i
$452$		0			0
$453$	3.34289e10	+	1.93002e10i	0.793833	+	0.458320i
$454$		0			0
$455$	−3.69474e10	−	3.74937e9i	−0.862062	−	0.0874809i
$456$		0			0
$457$	−8.34859e9	+	1.44602e10i	−0.191403	+	0.331519i	−0.945715	−	0.324996i	$-0.894637\pi$
$457$	−8.34859e9	+	1.44602e10i	−0.191403	+	0.331519i	0.754313	+	0.656515i	$0.227970\pi$
$458$		0			0
$459$	4.87461e9	+	8.44306e9i	0.109822	+	0.190217i
$460$		0			0
$461$		−	1.39906e10i		−	0.309766i	−0.987933	−	0.154883i	$-0.950500\pi$
$461$		−	1.39906e10i		−	0.309766i	0.987933	−	0.154883i	$-0.0495001\pi$
$462$		0			0
$463$	−2.56476e10			−0.558114			−0.279057	−	0.960274i	$-0.590022\pi$
$463$	−2.56476e10			−0.558114			−0.279057	+	0.960274i	$0.590022\pi$
$464$		0			0
$465$	−6.34411e9	+	3.66277e9i	−0.135693	+	0.0783426i
$466$		0			0
$467$	−3.36726e10	−	1.94409e10i	−0.707960	−	0.408741i	0.102345	−	0.994749i	$-0.467365\pi$
$467$	−3.36726e10	−	1.94409e10i	−0.707960	−	0.408741i	−0.810305	+	0.586008i	$0.800699\pi$
$468$		0			0
$469$	−5.59189e10	+	7.75493e10i	−1.15576	+	1.60283i
$470$		0			0
$471$	2.47198e10	−	4.28160e10i	0.502298	−	0.870006i
$472$		0			0
$473$	−1.40628e9	−	2.43575e9i	−0.0280949	−	0.0486618i
$474$		0			0
$475$		−	2.19968e10i		−	0.432100i
$476$		0			0
$477$	2.88283e9			0.0556859
$478$		0			0
$479$	6.40797e9	−	3.69965e9i	0.121725	−	0.0702778i	−0.437901	−	0.899023i	$-0.644278\pi$
$479$	6.40797e9	−	3.69965e9i	0.121725	−	0.0702778i	0.559626	+	0.828745i	$0.310945\pi$
$480$		0			0
$481$	−1.35849e10	−	7.84322e9i	−0.253790	−	0.146526i
$482$		0			0
$483$	−9.64954e9	+	4.33780e9i	−0.177304	+	0.0797041i
$484$		0			0
$485$	2.24885e9	−	3.89512e9i	0.0406437	−	0.0703970i
$486$		0			0
$487$	−1.11183e10	−	1.92575e10i	−0.197662	−	0.342361i	0.750108	−	0.661316i	$-0.230002\pi$
$487$	−1.11183e10	−	1.92575e10i	−0.197662	−	0.342361i	−0.947770	+	0.318954i	$0.896668\pi$
$488$		0			0
$489$			5.31097e10i			0.928834i
$490$		0			0
$491$	3.62576e10			0.623839			0.311920	−	0.950109i	$-0.399028\pi$
$491$	3.62576e10			0.623839			0.311920	+	0.950109i	$0.399028\pi$
$492$		0			0
$493$	−4.47154e10	+	2.58164e10i	−0.756953	+	0.437027i
$494$		0			0
$495$	1.75800e10	+	1.01498e10i	0.292818	+	0.169058i
$496$		0			0
$497$	−9.20166e9	−	2.04693e10i	−0.150814	−	0.335489i
$498$		0			0
$499$	3.67859e10	−	6.37151e10i	0.593307	−	1.02764i	−0.400476	−	0.916307i	$-0.631155\pi$
$499$	3.67859e10	−	6.37151e10i	0.593307	−	1.02764i	0.993783	−	0.111331i	$-0.0355114\pi$
$500$		0			0
$501$	2.34004e10	+	4.05306e10i	0.371425	+	0.643328i
$502$		0			0
$503$			5.49729e10i			0.858770i	0.903122	+	0.429385i	$0.141270\pi$
$503$			5.49729e10i			0.858770i	−0.903122	+	0.429385i	$0.858730\pi$
$504$		0			0
$505$	9.72691e10			1.49558
$506$		0			0
$507$	1.48753e10	−	8.58827e9i	0.225131	−	0.129979i
$508$		0			0
$509$	−5.64437e10	−	3.25878e10i	−0.840900	−	0.485494i	0.0166697	−	0.999861i	$-0.494694\pi$
$509$	−5.64437e10	−	3.25878e10i	−0.840900	−	0.485494i	−0.857570	+	0.514367i	$0.828027\pi$
$510$		0			0
$511$	6.62311e10	+	4.77576e10i	0.971355	+	0.700421i
$512$		0			0
$513$	−7.87338e9	+	1.36371e10i	−0.113682	+	0.196903i
$514$		0			0
$515$	2.03228e10	+	3.52002e10i	0.288905	+	0.500398i
$516$		0			0
$517$			1.04658e11i			1.46491i
$518$		0			0
$519$	−2.95190e10			−0.406849
$520$		0			0
$521$	6.65862e9	−	3.84435e9i	0.0903718	−	0.0521762i	−0.454133	−	0.890934i	$-0.650051\pi$
$521$	6.65862e9	−	3.84435e9i	0.0903718	−	0.0521762i	0.544505	+	0.838758i	$0.316718\pi$
$522$		0			0
$523$	1.07896e11	+	6.22937e10i	1.44211	+	0.832602i	0.997990	−	0.0633652i	$-0.0201833\pi$
$523$	1.07896e11	+	6.22937e10i	1.44211	+	0.832602i	0.444119	+	0.895968i	$0.353517\pi$
$524$		0			0
$525$	−1.61960e9	+	1.59600e10i	−0.0213192	+	0.210085i
$526$		0			0
$527$	1.02215e10	−	1.77042e10i	0.132518	−	0.229528i
$528$		0			0
$529$	3.47165e10	+	6.01307e10i	0.443316	+	0.767846i
$530$		0			0
$531$			1.90123e10i			0.239143i
$532$		0			0
$533$	1.65726e10			0.205344
$534$		0			0
$535$	9.15513e9	−	5.28572e9i	0.111750	−	0.0645191i
$536$		0			0
$537$	−4.67647e10	−	2.69996e10i	−0.562369	−	0.324684i
$538$		0			0
$539$	−7.17651e10	−	1.47168e10i	−0.850272	−	0.174365i
$540$		0			0
$541$	−4.97811e10	+	8.62233e10i	−0.581132	+	1.00655i	0.414213	+	0.910180i	$0.364057\pi$
$541$	−4.97811e10	+	8.62233e10i	−0.581132	+	1.00655i	−0.995346	+	0.0963708i	$0.969277\pi$
$542$		0			0
$543$	3.32531e10	+	5.75960e10i	0.382501	+	0.662511i
$544$		0			0
$545$		−	1.31735e11i		−	1.49319i
$546$		0			0
$547$	−2.34938e10			−0.262424			−0.131212	−	0.991354i	$-0.541887\pi$
$547$	−2.34938e10			−0.262424			−0.131212	+	0.991354i	$0.541887\pi$
$548$		0			0
$549$	2.86918e10	−	1.65652e10i	0.315840	−	0.182351i
$550$		0			0
$551$	−7.22235e10	−	4.16982e10i	−0.783559	−	0.452388i
$552$		0			0
$553$	1.08186e11	+	1.09785e10i	1.15683	+	0.117393i
$554$		0			0
$555$	−1.26512e10	+	2.19126e10i	−0.133340	+	0.230952i
$556$		0			0
$557$	1.24108e10	+	2.14962e10i	0.128938	+	0.223327i	0.923265	−	0.384163i	$-0.125510\pi$
$557$	1.24108e10	+	2.14962e10i	0.128938	+	0.223327i	−0.794328	+	0.607490i	$0.792177\pi$
$558$		0			0
$559$		−	4.68684e9i		−	0.0479991i
$560$		0			0
$561$	−5.66493e10			−0.571930
$562$		0			0
$563$	−1.47632e11	+	8.52355e10i	−1.46942	+	0.848373i	−0.999412	−	0.0342868i	$-0.989084\pi$
$563$	−1.47632e11	+	8.52355e10i	−1.46942	+	0.848373i	−0.470013	+	0.882660i	$0.655751\pi$
$564$		0			0
$565$	−1.40294e10	−	8.09988e9i	−0.137672	−	0.0794849i
$566$		0			0
$567$	6.71670e9	−	9.31483e9i	0.0649866	−	0.0901245i
$568$		0			0
$569$	−2.03247e10	+	3.52033e10i	−0.193898	+	0.335842i	−0.946539	−	0.322590i	$-0.895447\pi$
$569$	−2.03247e10	+	3.52033e10i	−0.193898	+	0.335842i	0.752640	+	0.658432i	$0.228780\pi$
$570$		0			0
$571$	4.24414e9	+	7.35107e9i	0.0399250	+	0.0691522i	0.885297	−	0.465025i	$-0.153955\pi$
$571$	4.24414e9	+	7.35107e9i	0.0399250	+	0.0691522i	−0.845372	+	0.534177i	$0.820621\pi$
$572$		0			0
$573$		−	2.86058e10i		−	0.265360i
$574$		0			0
$575$	−1.34616e10			−0.123148
$576$		0			0
$577$	−5.94782e10	+	3.43397e10i	−0.536604	+	0.309809i	−0.743702	−	0.668512i	$-0.766932\pi$
$577$	−5.94782e10	+	3.43397e10i	−0.536604	+	0.309809i	0.207097	+	0.978320i	$0.433598\pi$
$578$		0			0
$579$	7.69129e10	+	4.44057e10i	0.684360	+	0.395116i
$580$		0			0
$581$	1.44291e11	−	6.48638e10i	1.26630	−	0.569243i
$582$		0			0
$583$	−8.37556e9	+	1.45069e10i	−0.0725003	+	0.125574i
$584$		0			0
$585$	1.69136e10	+	2.92952e10i	0.144415	+	0.250134i
$586$		0			0
$587$			1.43383e11i			1.20766i	0.797114	+	0.603829i	$0.206359\pi$
$587$			1.43383e11i			1.20766i	−0.797114	+	0.603829i	$0.793641\pi$
$588$		0			0
$589$	3.30193e10			0.274351
$590$		0			0
$591$	4.08413e10	−	2.35797e10i	0.334772	−	0.193281i
$592$		0			0
$593$	−1.30721e11	−	7.54720e10i	−1.05713	−	0.610333i	−0.132491	−	0.991184i	$-0.542298\pi$
$593$	−1.30721e11	−	7.54720e10i	−1.05713	−	0.610333i	−0.924636	+	0.380851i	$0.875631\pi$
$594$		0			0
$595$	−6.85408e10	−	1.52471e11i	−0.546867	−	1.21652i
$596$		0			0
$597$	7.45888e9	−	1.29192e10i	0.0587187	−	0.101704i
$598$		0			0
$599$	−8.29322e10	−	1.43643e11i	−0.644193	−	1.11577i	−0.984487	−	0.175456i	$-0.943860\pi$
$599$	−8.29322e10	−	1.43643e11i	−0.644193	−	1.11577i	0.340294	−	0.940319i	$-0.389473\pi$
$600$		0			0
$601$		−	7.04035e10i		−	0.539630i	−0.962912	−	0.269815i	$-0.913037\pi$
$601$		−	7.04035e10i		−	0.539630i	0.962912	−	0.269815i	$-0.0869625\pi$
$602$		0			0
$603$	8.70862e10			0.658688
$604$		0			0
$605$	3.34418e10	−	1.93076e10i	0.249614	−	0.144115i
$606$		0			0
$607$	−5.15559e10	−	2.97658e10i	−0.379773	−	0.219262i	0.297947	−	0.954583i	$-0.403698\pi$
$607$	−5.15559e10	−	2.97658e10i	−0.379773	−	0.219262i	−0.677719	+	0.735321i	$0.737032\pi$
$608$		0			0
$609$	4.93323e10	+	3.55724e10i	0.358643	+	0.258609i
$610$		0			0
$611$	−8.72009e10	+	1.51036e11i	−0.625685	+	1.08372i
$612$		0			0
$613$	1.14654e11	+	1.98587e11i	0.811986	+	1.40640i	0.911472	+	0.411363i	$0.134947\pi$
$613$	1.14654e11	+	1.98587e11i	0.811986	+	1.40640i	−0.0994852	+	0.995039i	$0.531720\pi$
$614$		0			0
$615$		−	2.67318e10i		−	0.186865i
$616$		0			0
$617$	1.93904e11			1.33797			0.668986	−	0.743275i	$-0.266728\pi$
$617$	1.93904e11			1.33797			0.668986	+	0.743275i	$0.266728\pi$
$618$		0			0
$619$	1.77090e10	−	1.02243e10i	0.120623	−	0.0696418i	−0.438474	−	0.898744i	$-0.644481\pi$
$619$	1.77090e10	−	1.02243e10i	0.120623	−	0.0696418i	0.559098	+	0.829102i	$0.311148\pi$
$620$		0			0
$621$	8.34566e9	+	4.81837e9i	0.0561170	+	0.0323992i
$622$		0			0
$623$	3.07621e9	−	3.03138e10i	0.0204203	−	0.201228i
$624$		0			0
$625$	9.39923e10	−	1.62799e11i	0.615988	−	1.06692i
$626$		0			0
$627$	−4.57495e10	−	7.92404e10i	−0.296016	−	0.512716i
$628$		0			0
$629$		−	7.06105e10i		−	0.451094i
$630$		0			0
$631$	−1.43159e11			−0.903030			−0.451515	−	0.892263i	$-0.649116\pi$
$631$	−1.43159e11			−0.903030			−0.451515	+	0.892263i	$0.649116\pi$
$632$		0			0
$633$	−5.66876e10	+	3.27286e10i	−0.353080	+	0.203851i
$634$		0			0
$635$	1.12411e11	+	6.49006e10i	0.691377	+	0.399166i
$636$		0			0
$637$	−9.13050e10	−	8.10328e10i	−0.554545	−	0.492157i
$638$		0			0
$639$	−1.02211e10	+	1.77034e10i	−0.0613047	+	0.106183i
$640$		0			0
$641$	−1.32450e11	−	2.29411e11i	−0.784551	−	1.35888i	−0.929267	−	0.369409i	$-0.879560\pi$
$641$	−1.32450e11	−	2.29411e11i	−0.784551	−	1.35888i	0.144716	−	0.989473i	$-0.453773\pi$
$642$		0			0
$643$			1.18901e9i			0.00695572i	0.999994	+	0.00347786i	$0.00110704\pi$
$643$			1.18901e9i			0.00695572i	−0.999994	+	0.00347786i	$0.998893\pi$
$644$		0			0
$645$	−7.55994e9			−0.0436797
$646$		0			0
$647$	−8.24205e10	+	4.75855e10i	−0.470347	+	0.271555i	−0.716385	−	0.697705i	$-0.754204\pi$
$647$	−8.24205e10	+	4.75855e10i	−0.470347	+	0.271555i	0.246038	+	0.969260i	$0.420871\pi$
$648$		0			0
$649$	−9.56733e10	−	5.52370e10i	−0.539277	−	0.311352i
$650$		0			0
$651$	−2.39575e10	−	2.43118e9i	−0.133388	−	0.0135361i
$652$		0			0
$653$	1.66756e11	−	2.88829e11i	0.917124	−	1.58850i	0.113362	−	0.993554i	$-0.463838\pi$
$653$	1.66756e11	−	2.88829e11i	0.917124	−	1.58850i	0.803762	−	0.594951i	$-0.202829\pi$
$654$		0			0
$655$	−2.03751e11	−	3.52906e11i	−1.10696	−	1.91732i
$656$		0			0
$657$		−	7.43760e10i		−	0.399183i
$658$		0			0
$659$	1.17025e11			0.620493			0.310246	−	0.950656i	$-0.399588\pi$
$659$	1.17025e11			0.620493			0.310246	+	0.950656i	$0.399588\pi$
$660$		0			0
$661$	2.68769e10	−	1.55174e10i	0.140791	−	0.0812855i	−0.427950	−	0.903802i	$-0.640764\pi$
$661$	2.68769e10	−	1.55174e10i	0.140791	−	0.0812855i	0.568741	+	0.822517i	$0.307431\pi$
$662$		0			0
$663$	−8.17528e10	−	4.72000e10i	−0.423106	−	0.244280i
$664$		0			0
$665$	1.57922e11	−	2.19008e11i	0.807523	−	1.11989i
$666$		0			0
$667$	−2.55186e10	+	4.41995e10i	−0.128930	+	0.223313i
$668$		0			0
$669$	−7.22801e10	−	1.25193e11i	−0.360840	−	0.624993i
$670$		0			0
$671$			1.92509e11i			0.949645i
$672$		0			0
$673$	−2.12193e11			−1.03436			−0.517179	−	0.855877i	$-0.673018\pi$
$673$	−2.12193e11			−1.03436			−0.517179	+	0.855877i	$0.673018\pi$
$674$		0			0
$675$	1.26545e10	−	7.30608e9i	0.0609579	−	0.0351940i
$676$		0			0
$677$	9.30441e10	+	5.37191e10i	0.442929	+	0.255725i	0.704839	−	0.709367i	$-0.251019\pi$
$677$	9.30441e10	+	5.37191e10i	0.442929	+	0.255725i	−0.261910	+	0.965092i	$0.584352\pi$
$678$		0			0
$679$	1.34850e10	−	6.06196e9i	0.0634412	−	0.0285190i
$680$		0			0
$681$	−8.85074e10	+	1.53299e11i	−0.411520	+	0.712774i
$682$		0			0
$683$	5.88995e10	+	1.02017e11i	0.270663	+	0.468802i	0.969032	−	0.246936i	$-0.0794238\pi$
$683$	5.88995e10	+	1.02017e11i	0.270663	+	0.468802i	−0.698369	+	0.715738i	$0.746091\pi$
$684$		0			0
$685$		−	2.87654e11i		−	1.30650i
$686$		0			0
$687$	−1.48171e11			−0.665175
$688$		0			0
$689$	−2.41742e10	+	1.39570e10i	−0.107269	+	0.0619320i
$690$		0			0
$691$	2.96618e11	+	1.71253e11i	1.30103	+	0.751148i	0.980580	−	0.196119i	$-0.0628339\pi$
$691$	2.96618e11	+	1.71253e11i	1.30103	+	0.751148i	0.320446	+	0.947267i	$0.396167\pi$
$692$		0			0
$693$	2.73596e10	+	6.08622e10i	0.118625	+	0.263885i
$694$		0			0
$695$	−1.77371e11	+	3.07215e11i	−0.760227	+	1.31675i
$696$		0			0
$697$	3.72997e10	+	6.46050e10i	0.158043	+	0.273738i
$698$		0			0
$699$		−	2.58439e11i		−	1.08255i
$700$		0			0
$701$	−1.15900e11			−0.479967			−0.239984	−	0.970777i	$-0.577142\pi$
$701$	−1.15900e11			−0.479967			−0.239984	+	0.970777i	$0.577142\pi$
$702$		0			0
$703$	9.87691e10	−	5.70244e10i	0.404390	−	0.233474i
$704$		0			0
$705$	2.43624e11	+	1.40656e11i	0.986195	+	0.569380i
$706$		0			0
$707$	2.59351e11	+	1.87011e11i	1.03803	+	0.748498i
$708$		0			0
$709$	−7.71407e10	+	1.33612e11i	−0.305280	+	0.528761i	−0.977324	−	0.211751i	$-0.932083\pi$
$709$	−7.71407e10	+	1.33612e11i	−0.305280	+	0.528761i	0.672044	+	0.740512i	$0.265417\pi$
$710$		0			0
$711$	−4.95246e10	−	8.57791e10i	−0.193795	−	0.335663i
$712$		0			0
$713$		−	2.02073e10i		−	0.0781897i
$714$		0			0
$715$	−1.96558e11			−0.752084
$716$		0			0
$717$	−1.59380e11	+	9.20178e10i	−0.603054	+	0.348173i
$718$		0			0
$719$	1.82639e11	+	1.05447e11i	0.683404	+	0.394564i	0.801136	−	0.598482i	$-0.204229\pi$
$719$	1.82639e11	+	1.05447e11i	0.683404	+	0.394564i	−0.117732	+	0.993045i	$0.537562\pi$
$720$		0			0
$721$	−1.34893e10	+	1.32928e11i	−0.0499172	+	0.491898i
$722$		0			0
$723$	1.55601e11	−	2.69509e11i	0.569456	−	0.986326i
$724$		0			0
$725$	3.86938e10	+	6.70195e10i	0.140052	+	0.242577i
$726$		0			0
$727$		−	2.28475e10i		−	0.0817900i	−0.999163	−	0.0408950i	$-0.986979\pi$
$727$		−	2.28475e10i		−	0.0817900i	0.999163	−	0.0408950i	$-0.0130209\pi$
$728$		0			0
$729$	−1.04604e10			−0.0370370
$730$		0			0
$731$	1.82707e10	−	1.05486e10i	0.0639861	−	0.0369424i
$732$		0			0
$733$	3.90685e11	+	2.25562e11i	1.35335	+	0.781358i	0.988717	−	0.149793i	$-0.0478609\pi$
$733$	3.90685e11	+	2.25562e11i	1.35335	+	0.781358i	0.364634	+	0.931151i	$0.381194\pi$
$734$		0			0
$735$	−1.30707e11	+	1.47276e11i	−0.447868	+	0.504642i
$736$		0			0
$737$	−2.53014e11	+	4.38233e11i	−0.857579	+	1.48537i
$738$		0			0
$739$	8.06607e10	+	1.39708e11i	0.270448	+	0.468430i	0.968977	−	0.247152i	$-0.0794948\pi$
$739$	8.06607e10	+	1.39708e11i	0.270448	+	0.468430i	−0.698528	+	0.715582i	$0.746161\pi$
$740$		0			0
$741$		−	1.52473e11i		−	0.505733i
$742$		0			0
$743$	−2.11855e11			−0.695159			−0.347579	−	0.937651i	$-0.612996\pi$
$743$	−2.11855e11			−0.695159			−0.347579	+	0.937651i	$0.612996\pi$
$744$		0			0
$745$	2.03129e11	−	1.17276e11i	0.659395	−	0.380702i
$746$		0			0
$747$	−1.24794e11	−	7.20499e10i	−0.400785	−	0.231393i
$748$		0			0
$749$	3.45729e10	+	3.50841e9i	0.109852	+	0.0111477i
$750$		0			0
$751$	−2.80633e11	+	4.86071e11i	−0.882224	+	1.52806i	−0.0333617	+	0.999443i	$0.510621\pi$
$751$	−2.80633e11	+	4.86071e11i	−0.882224	+	1.52806i	−0.848862	+	0.528614i	$0.822712\pi$
$752$		0			0
$753$	−9.95354e10	−	1.72400e11i	−0.309598	−	0.536239i
$754$		0			0
$755$			6.02882e11i			1.85543i
$756$		0			0
$757$	−2.12454e11			−0.646967			−0.323483	−	0.946234i	$-0.604854\pi$
$757$	−2.12454e11			−0.646967			−0.323483	+	0.946234i	$0.604854\pi$
$758$		0			0
$759$	−4.84938e10	+	2.79979e10i	−0.146123	+	0.0843642i
$760$		0			0
$761$	3.11683e11	+	1.79950e11i	0.929338	+	0.536554i	0.886602	−	0.462533i	$-0.153059\pi$
$761$	3.11683e11	+	1.79950e11i	0.929338	+	0.536554i	0.0427362	+	0.999086i	$0.486393\pi$
$762$		0			0
$763$	2.53277e11	−	3.51248e11i	0.747304	−	1.03637i
$764$		0			0
$765$	−7.61343e10	+	1.31868e11i	−0.222297	+	0.385031i
$766$		0			0
$767$	−9.20467e10	−	1.59429e11i	−0.265966	−	0.460667i
$768$		0			0
$769$		−	1.74232e11i		−	0.498222i	−0.968475	−	0.249111i	$-0.919862\pi$
$769$		−	1.74232e11i		−	0.498222i	0.968475	−	0.249111i	$-0.0801385\pi$
$770$		0			0
$771$	−1.42235e11			−0.402521
$772$		0			0
$773$	2.44377e11	−	1.41091e11i	0.684450	−	0.395168i	−0.117079	−	0.993123i	$-0.537353\pi$
$773$	2.44377e11	−	1.41091e11i	0.684450	−	0.395168i	0.801530	+	0.597955i	$0.204020\pi$
$774$		0			0
$775$	−2.65352e10	−	1.53201e10i	−0.0735555	−	0.0424673i
$776$		0			0
$777$	−7.58617e10	+	3.41024e10i	−0.208132	+	0.0935623i
$778$		0			0
$779$	−6.02458e10	+	1.04349e11i	−0.163598	+	0.283360i
$780$		0			0
$781$	−5.93912e10	−	1.02869e11i	−0.159631	−	0.276490i
$782$		0			0
$783$		−	5.53991e10i		−	0.147386i
$784$		0			0
$785$	7.72175e11			2.03347
$786$		0			0
$787$	5.01252e11	−	2.89398e11i	1.30664	−	0.754391i	0.325110	−	0.945676i	$-0.394599\pi$
$787$	5.01252e11	−	2.89398e11i	1.30664	−	0.754391i	0.981535	+	0.191285i	$0.0612654\pi$
$788$		0			0
$789$	3.40860e11	+	1.96796e11i	0.879566	+	0.507818i
$790$		0			0
$791$	−2.18339e10	−	4.85701e10i	−0.0557732	−	0.124069i
$792$		0			0
$793$	−1.60398e11	+	2.77818e11i	−0.405608	+	0.702534i
$794$		0			0
$795$	2.25128e10	+	3.89933e10i	0.0563587	+	0.0976162i
$796$		0			0
$797$			1.44708e11i			0.358640i	0.983791	+	0.179320i	$0.0573898\pi$
$797$			1.44708e11i			0.358640i	−0.983791	+	0.179320i	$0.942610\pi$
$798$		0			0
$799$	−7.85047e11			−1.92623
$800$		0			0
$801$	−2.40355e10	+	1.38769e10i	−0.0583879	+	0.0337103i
$802$		0			0
$803$	3.74273e11	+	2.16087e11i	0.900174	+	0.519716i
$804$		0			0
$805$	−1.34029e11	−	9.66453e10i	−0.319166	−	0.230143i
$806$		0			0
$807$	1.72810e11	−	2.99316e11i	0.407451	−	0.705726i
$808$		0			0
$809$	−1.04878e11	−	1.81654e11i	−0.244844	−	0.424083i	0.717244	−	0.696823i	$-0.245404\pi$
$809$	−1.04878e11	−	1.81654e11i	−0.244844	−	0.424083i	−0.962088	+	0.272740i	$0.912070\pi$
$810$		0			0
$811$			5.06101e10i			0.116991i	0.998288	+	0.0584957i	$0.0186304\pi$
$811$			5.06101e10i			0.116991i	−0.998288	+	0.0584957i	$0.981370\pi$
$812$		0			0
$813$	−2.01384e11			−0.460961
$814$		0			0
$815$	−7.18365e11	+	4.14748e11i	−1.62823	+	0.940056i
$816$		0			0
$817$	2.95105e10	+	1.70379e10i	0.0662352	+	0.0382409i
$818$		0			0
$819$	−1.12264e10	+	1.10629e11i	−0.0249521	+	0.245885i
$820$		0			0
$821$	−3.52706e10	+	6.10905e10i	−0.0776319	+	0.134462i	−0.902228	−	0.431260i	$-0.858069\pi$
$821$	−3.52706e10	+	6.10905e10i	−0.0776319	+	0.134462i	0.824596	+	0.565722i	$0.191403\pi$
$822$		0			0
$823$	−1.72986e11	−	2.99621e11i	−0.377061	−	0.653089i	0.613572	−	0.789639i	$-0.289732\pi$
$823$	−1.72986e11	−	2.99621e11i	−0.377061	−	0.653089i	−0.990633	+	0.136550i	$0.956399\pi$
$824$		0			0
$825$			8.49062e10i			0.183284i
$826$		0			0
$827$	7.36661e11			1.57487			0.787436	−	0.616396i	$-0.211408\pi$
$827$	7.36661e11			1.57487			0.787436	+	0.616396i	$0.211408\pi$
$828$		0			0
$829$	−5.55082e10	+	3.20477e10i	−0.117527	+	0.0678545i	−0.557611	−	0.830102i	$-0.688282\pi$
$829$	−5.55082e10	+	3.20477e10i	−0.117527	+	0.0678545i	0.440084	+	0.897957i	$0.354949\pi$
$830$		0			0
$831$	−2.05992e9	−	1.18930e9i	−0.00431964	−	0.00249394i
$832$		0			0
$833$	1.10392e11	−	5.38314e11i	0.229275	−	1.11804i
$834$		0			0
$835$	−3.65480e11	+	6.33029e11i	−0.751826	+	1.30220i
$836$		0			0
$837$	1.09671e10	+	1.89957e10i	0.0223456	+	0.0387037i
$838$		0			0
$839$			3.78990e11i			0.764857i	0.923985	+	0.382428i	$0.124912\pi$
$839$			3.78990e11i			0.764857i	−0.923985	+	0.382428i	$0.875088\pi$
$840$		0			0
$841$	−2.06847e11			−0.413489
$842$		0			0
$843$	4.18857e11	−	2.41827e11i	0.829383	−	0.478845i
$844$		0			0
$845$	2.32331e11	+	1.34136e11i	0.455701	+	0.263099i
$846$		0			0
$847$	1.26288e11	+	1.28155e10i	0.245374	+	0.0249002i
$848$		0			0
$849$	−6.91156e10	+	1.19712e11i	−0.133029	+	0.230412i
$850$		0			0
$851$	−3.48980e10	−	6.04450e10i	−0.0665398	−	0.115250i
$852$		0			0
$853$		−	3.34365e11i		−	0.631575i	−0.948830	−	0.315787i	$-0.897731\pi$
$853$		−	3.34365e11i		−	0.631575i	0.948830	−	0.315787i	$-0.102269\pi$
$854$		0			0
$855$	−2.45941e11			−0.460222
$856$		0			0
$857$	−2.45660e11	+	1.41832e11i	−0.455419	+	0.262937i	−0.710116	−	0.704084i	$-0.751358\pi$
$857$	−2.45660e11	+	1.41832e11i	−0.455419	+	0.262937i	0.254697	+	0.967021i	$0.418024\pi$
$858$		0			0
$859$	4.58194e11	+	2.64539e11i	0.841544	+	0.485866i	0.857789	−	0.514002i	$-0.171838\pi$
$859$	4.58194e11	+	2.64539e11i	0.841544	+	0.485866i	−0.0162447	+	0.999868i	$0.505171\pi$
$860$		0			0
$861$	5.13951e10	−	7.12756e10i	0.0935211	−	0.129697i
$862$		0			0
$863$	8.74072e10	−	1.51394e11i	0.157581	−	0.272938i	−0.776415	−	0.630222i	$-0.782964\pi$
$863$	8.74072e10	−	1.51394e11i	0.157581	−	0.272938i	0.933996	+	0.357284i	$0.116297\pi$
$864$		0			0
$865$	−2.30522e11	−	3.99276e11i	−0.411764	−	0.713197i
$866$		0			0
$867$		−	9.87057e10i		−	0.174689i
$868$		0			0
$869$	5.75541e11			1.00925
$870$		0			0
$871$	−7.30269e11	+	4.21621e11i	−1.26885	+	0.732571i
$872$		0			0
$873$	−1.16629e10	−	6.73355e9i	−0.0200793	−	0.0115928i
$874$		0			0
$875$	3.96290e11	−	1.78146e11i	0.676053	−	0.303909i
$876$		0			0
$877$	2.49143e11	−	4.31528e11i	0.421162	−	0.729475i	−0.574891	−	0.818230i	$-0.694956\pi$
$877$	2.49143e11	−	4.31528e11i	0.421162	−	0.729475i	0.996053	+	0.0887553i	$0.0282889\pi$
$878$		0			0
$879$	9.88065e10	+	1.71138e11i	0.165512	+	0.286676i
$880$		0			0
$881$		−	2.65089e11i		−	0.440035i	−0.975496	−	0.220018i	$-0.929389\pi$
$881$		−	2.65089e11i		−	0.440035i	0.975496	−	0.220018i	$-0.0706115\pi$
$882$		0			0
$883$	2.64089e11			0.434418			0.217209	−	0.976125i	$-0.430305\pi$
$883$	2.64089e11			0.434418			0.217209	+	0.976125i	$0.430305\pi$
$884$		0			0
$885$	−2.57162e11	+	1.48473e11i	−0.419212	+	0.242032i
$886$		0			0
$887$	−1.00397e12	−	5.79640e11i	−1.62190	−	0.936406i	−0.986411	−	0.164297i	$-0.947464\pi$
$887$	−1.00397e12	−	5.79640e11i	−1.62190	−	0.936406i	−0.635491	−	0.772108i	$-0.719202\pi$
$888$		0			0
$889$	1.74945e11	+	3.89170e11i	0.280088	+	0.623063i
$890$		0			0
$891$	3.03907e10	−	5.26383e10i	0.0482204	−	0.0835201i
$892$		0			0
$893$	−6.33997e11	−	1.09811e12i	−0.996968	−	1.72680i
$894$		0			0
$895$		−	8.43390e11i		−	1.31443i
$896$		0			0
$897$	−9.33110e10			−0.144133
$898$		0			0
$899$	−1.00603e11	+	5.80832e10i	−0.154018	+	0.0889225i
$900$		0			0
$901$	−1.08817e11	−	6.28256e10i	−0.165119	−	0.0953317i
$902$		0			0
$903$	−2.01572e10	−	1.45349e10i	−0.0303165	−	0.0218605i
$904$		0			0
$905$	−5.19365e11	+	8.99566e11i	−0.774244	+	1.34103i
$906$		0			0
$907$	−2.24197e9	−	3.88320e9i	−0.00331284	−	0.00573801i	0.864364	−	0.502866i	$-0.167721\pi$
$907$	−2.24197e9	−	3.88320e9i	−0.00331284	−	0.00573801i	−0.867677	+	0.497128i	$0.834388\pi$
$908$		0			0
$909$		−	2.91245e11i		−	0.426582i
$910$		0			0
$911$	−8.14100e11			−1.18196			−0.590982	−	0.806685i	$-0.701260\pi$
$911$	−8.14100e11			−1.18196			−0.590982	+	0.806685i	$0.701260\pi$
$912$		0			0
$913$	7.25135e11	−	4.18657e11i	1.04360	−	0.602525i
$914$		0			0
$915$	4.48124e11	+	2.58724e11i	0.639313	+	0.369107i
$916$		0			0
$917$	1.35240e11	−	1.33270e12i	0.191262	−	1.88475i
$918$		0			0
$919$	5.26562e11	−	9.12032e11i	0.738222	−	1.27864i	−0.215073	−	0.976598i	$-0.568999\pi$
$919$	5.26562e11	−	9.12032e11i	0.738222	−	1.27864i	0.953295	−	0.302041i	$-0.0976679\pi$
$920$		0			0
$921$	−6.92290e10	−	1.19908e11i	−0.0962166	−	0.166652i
$922$		0			0
$923$		−	1.97938e11i		−	0.272724i
$924$		0			0
$925$	−1.05831e11			−0.144560
$926$		0			0
$927$	1.05397e11	−	6.08510e10i	0.142728	−	0.0824041i
$928$		0			0
$929$	−8.96093e11	−	5.17360e11i	−1.20307	−	0.694592i	−0.241832	−	0.970318i	$-0.577748\pi$
$929$	−8.96093e11	−	5.17360e11i	−1.20307	−	0.694592i	−0.961236	+	0.275726i	$0.911082\pi$
$930$		0			0
$931$	8.42139e11	−	2.80323e11i	1.12095	−	0.373130i
$932$		0			0
$933$	−2.09238e11	+	3.62411e11i	−0.276130	+	0.478272i
$934$		0			0
$935$	−4.42390e11	−	7.66242e11i	−0.578840	−	1.00258i
$936$		0			0
$937$			8.31714e11i			1.07899i	0.841990	+	0.539493i	$0.181384\pi$
$937$			8.31714e11i			1.07899i	−0.841990	+	0.539493i	$0.818616\pi$
$938$		0			0
$939$	−2.72519e11			−0.350537
$940$		0			0
$941$	−1.14487e12	+	6.60992e11i	−1.46015	+	0.843020i	−0.999018	−	0.0443103i	$-0.985891\pi$
$941$	−1.14487e12	+	6.60992e11i	−1.46015	+	0.843020i	−0.461135	+	0.887330i	$0.652558\pi$
$942$		0			0
$943$	6.38597e10	+	3.68694e10i	0.0807570	+	0.0466251i
$944$		0			0
$945$	1.78446e11	+	1.81084e10i	0.223758	+	0.0227067i
$946$		0			0
$947$	2.20176e11	−	3.81357e11i	0.273760	−	0.474167i	−0.696061	−	0.717982i	$-0.745066\pi$
$947$	2.20176e11	−	3.81357e11i	0.273760	−	0.474167i	0.969822	+	0.243815i	$0.0783991\pi$
$948$		0			0
$949$	3.60086e11	+	6.23687e11i	0.443957	+	0.768956i
$950$		0			0
$951$		−	2.34401e11i		−	0.286574i
$952$		0			0
$953$	7.16416e11			0.868547			0.434274	−	0.900781i	$-0.357005\pi$
$953$	7.16416e11			0.868547			0.434274	+	0.900781i	$0.357005\pi$
$954$		0			0
$955$	3.86924e11	−	2.23391e11i	0.465171	−	0.268566i
$956$		0			0
$957$	2.78778e11	+	1.60953e11i	0.332362	+	0.191889i
$958$		0			0
$959$	5.53050e11	−	7.66979e11i	0.653868	−	0.906795i
$960$		0			0
$961$	−4.03449e11	+	6.98793e11i	−0.473036	+	0.819323i
$962$		0			0
$963$	−1.58266e10	−	2.74125e10i	−0.0184027	−	0.0318745i
$964$		0			0
$965$			1.38710e12i			1.59956i
$966$		0			0
$967$	1.18405e12			1.35414			0.677070	−	0.735919i	$-0.263249\pi$
$967$	1.18405e12			1.35414			0.677070	+	0.735919i	$0.263249\pi$
$968$		0			0
$969$	5.94387e11	−	3.43169e11i	0.674177	−	0.389236i
$970$		0			0
$971$	−1.17425e12	−	6.77952e11i	−1.32094	−	0.762645i	−0.337061	−	0.941483i	$-0.609433\pi$
$971$	−1.17425e12	−	6.77952e11i	−1.32094	−	0.762645i	−0.983879	+	0.178838i	$0.942766\pi$
$972$		0			0
$973$	−1.06359e12	+	4.78118e11i	−1.18665	+	0.533438i
$974$		0			0
$975$	−7.07436e10	+	1.22531e11i	−0.0782832	+	0.135590i
$976$		0			0
$977$	2.61277e11	+	4.52545e11i	0.286763	+	0.496688i	0.973035	−	0.230657i	$-0.0740874\pi$
$977$	2.61277e11	+	4.52545e11i	0.286763	+	0.496688i	−0.686272	+	0.727345i	$0.740754\pi$
$978$		0			0
$979$		−	1.61268e11i		−	0.175556i
$980$		0			0
$981$	−3.94444e11			−0.425902
$982$		0			0
$983$	1.35501e12	−	7.82318e11i	1.45121	−	0.837855i	0.452658	−	0.891684i	$-0.350476\pi$
$983$	1.35501e12	−	7.82318e11i	1.45121	−	0.837855i	0.998550	+	0.0538289i	$0.0171426\pi$
$984$		0			0
$985$	6.37882e11	+	3.68281e11i	0.677634	+	0.391232i
$986$		0			0
$987$	3.79150e11	+	8.43430e11i	0.399524	+	0.888751i
$988$		0			0
$989$	1.04269e10	−	1.80599e10i	0.0108986	−	0.0188769i
$990$		0			0
$991$	−1.30845e11	−	2.26630e11i	−0.135663	−	0.234975i	0.790187	−	0.612865i	$-0.209983\pi$
$991$	−1.30845e11	−	2.26630e11i	−0.135663	−	0.234975i	−0.925851	+	0.377890i	$0.876650\pi$
$992$		0			0
$993$		−	4.47411e11i		−	0.460161i
$994$		0			0
$995$	2.32994e11			0.237713
$996$		0			0
$997$	−1.55089e12	+	8.95405e11i	−1.56964	+	0.906231i	−0.573427	+	0.819256i	$0.694387\pi$
$997$	−1.55089e12	+	8.95405e11i	−1.56964	+	0.906231i	−0.996211	+	0.0869745i	$0.972280\pi$
$998$		0			0
$999$	6.56110e10	+	3.78805e10i	0.0658741	+	0.0380324i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.m.b.61.5	✓	12
3.2	odd	2		252.9.z.d.145.2		12
7.3	odd	6	inner	84.9.m.b.73.5	yes	12
21.17	even	6		252.9.z.d.73.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.m.b.61.5	✓	12	1.1	even	1	trivial
84.9.m.b.73.5	yes	12	7.3	odd	6	inner
252.9.z.d.73.2		12	21.17	even	6
252.9.z.d.145.2		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(40.5000 - 23.3827i) q^{3} +(632.551 + 365.204i) q^{5} +(984.437 + 2189.91i) q^{7} +(1093.50 - 1894.00i) q^{9} +O(q^{10})\)	\(q+(40.5000 - 23.3827i) q^{3} +(632.551 + 365.204i) q^{5} +(984.437 + 2189.91i) q^{7} +(1093.50 - 1894.00i) q^{9} +(6353.95 + 11005.4i) q^{11} +21176.4i q^{13} +34157.8 q^{15} +(-82551.9 + 47661.3i) q^{17} +(-133336. - 76981.8i) q^{19} +(91075.6 + 65672.4i) q^{21} +(-47111.5 + 81599.5i) q^{23} +(71435.0 + 123729. i) q^{25} -102276. i q^{27} +541664. q^{29} +(-185730. + 107231. i) q^{31} +(514670. + 297145. i) q^{33} +(-177055. + 1.74475e6i) q^{35} +(-370376. + 641510. i) q^{37} +(495160. + 857643. i) q^{39} -782599. i q^{41} -221324. q^{43} +(1.38339e6 - 798700. i) q^{45} +(7.13230e6 + 4.11784e6i) q^{47} +(-3.82657e6 + 4.31165e6i) q^{49} +(-2.22890e6 + 3.86057e6i) q^{51} +(659083. + 1.14157e6i) q^{53} +9.28194e6i q^{55} -7.20016e6 q^{57} +(-7.52865e6 + 4.34667e6i) q^{59} +(1.31192e7 + 7.57439e6i) q^{61} +(5.22416e6 + 530140. i) q^{63} +(-7.73369e6 + 1.33951e7i) q^{65} +(1.99100e7 + 3.44851e7i) q^{67} +4.40638e6i q^{69} -9.34713e6 q^{71} +(2.94520e7 - 1.70041e7i) q^{73} +(5.78623e6 + 3.34068e6i) q^{75} +(-1.78456e7 + 2.47486e7i) q^{77} +(2.26450e7 - 3.92223e7i) q^{79} +(-2.39148e6 - 4.14217e6i) q^{81} -6.58893e7i q^{83} -6.96244e7 q^{85} +(2.19374e7 - 1.26656e7i) q^{87} +(-1.09902e7 - 6.34517e6i) q^{89} +(-4.63742e7 + 2.08468e7i) q^{91} +(-5.01470e6 + 8.68571e6i) q^{93} +(-5.62280e7 - 9.73898e7i) q^{95} -6.15780e6i q^{97} +2.77922e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9}+O(q^{10}) \) 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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