LMFDB - Embedded newform 72.10.d.b.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,10,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, 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:	$37.0825802038$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 $ x^8 - x^7 + 59x^6 - 313x^5 - 315x^4 - 92091x^3 + 1261649x^2 - 16074123x + 251007534
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.4
Root			$-2.43481 + 11.2224i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.10.d.b.37.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.86961 + 22.4447i) q^{2} +(-495.531 - 128.815i) q^{4} -1417.55i q^{5} +5087.57 q^{7} +(4313.20 - 10752.4i) q^{8} +O(q^{10})$	\(q+(-2.86961 + 22.4447i) q^{2} +(-495.531 - 128.815i) q^{4} -1417.55i q^{5} +5087.57 q^{7} +(4313.20 - 10752.4i) q^{8} +(31816.6 + 4067.83i) q^{10} +14811.2i q^{11} +64089.8i q^{13} +(-14599.3 + 114189. i) q^{14} +(228957. + 127664. i) q^{16} -251342. q^{17} -511568. i q^{19} +(-182602. + 702441. i) q^{20} +(-332432. - 42502.3i) q^{22} -1.96905e6 q^{23} -56329.6 q^{25} +(-1.43848e6 - 183913. i) q^{26} +(-2.52105e6 - 655356. i) q^{28} -2.16116e6i q^{29} -3.03997e6 q^{31} +(-3.52240e6 + 4.77253e6i) q^{32} +(721253. - 5.64129e6i) q^{34} -7.21189e6i q^{35} +8.74315e6i q^{37} +(1.14820e7 + 1.46800e6i) q^{38} +(-1.52421e7 - 6.11419e6i) q^{40} +1.48418e7 q^{41} +153307. i q^{43} +(1.90790e6 - 7.33938e6i) q^{44} +(5.65042e6 - 4.41949e7i) q^{46} -5.17595e7 q^{47} -1.44703e7 q^{49} +(161644. - 1.26430e6i) q^{50} +(8.25574e6 - 3.17584e7i) q^{52} -4.26277e7i q^{53} +2.09956e7 q^{55} +(2.19437e7 - 5.47035e7i) q^{56} +(4.85067e7 + 6.20170e6i) q^{58} -1.15231e7i q^{59} -1.95800e8i q^{61} +(8.72353e6 - 6.82312e7i) q^{62} +(-9.70103e7 - 9.27545e7i) q^{64} +9.08506e7 q^{65} -1.69718e8i q^{67} +(1.24547e8 + 3.23766e7i) q^{68} +(1.61869e8 + 2.06953e7i) q^{70} -2.23552e8 q^{71} -4.10130e8 q^{73} +(-1.96238e8 - 2.50895e7i) q^{74} +(-6.58977e7 + 2.53497e8i) q^{76} +7.53528e7i q^{77} +2.51240e7 q^{79} +(1.80970e8 - 3.24559e8i) q^{80} +(-4.25902e7 + 3.33120e8i) q^{82} +4.73174e8i q^{83} +3.56290e8i q^{85} +(-3.44094e6 - 439932. i) q^{86} +(1.59255e8 + 6.38836e7i) q^{88} -8.45560e8 q^{89} +3.26061e8i q^{91} +(9.75726e8 + 2.53644e8i) q^{92} +(1.48530e8 - 1.16173e9i) q^{94} -7.25174e8 q^{95} -7.97179e8 q^{97} +(4.15241e7 - 3.24781e8i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 18 q^{2} - 428 q^{4} + 4800 q^{7} + 3384 q^{8}+O(q^{10}) $ 8 * q + 18 * q^2 - 428 * q^4 + 4800 * q^7 + 3384 * q^8	$ 8 q + 18 q^{2} - 428 q^{4} + 4800 q^{7} + 3384 q^{8} + 26392 q^{10} + 72336 q^{14} + 185616 q^{16} + 102000 q^{17} - 1245264 q^{20} - 2373124 q^{22} - 3412032 q^{23} - 2423384 q^{25} - 4551240 q^{26} + 7509920 q^{28} + 803584 q^{31} + 14113248 q^{32} + 27757244 q^{34} + 63661140 q^{38} - 93063648 q^{40} + 2180784 q^{41} - 114013320 q^{44} - 131840944 q^{46} - 7432320 q^{47} + 24436680 q^{49} - 231784902 q^{50} + 219270896 q^{52} + 7056832 q^{55} + 358503360 q^{56} + 375425192 q^{58} + 344291904 q^{62} - 316815296 q^{64} + 146501760 q^{65} - 79875048 q^{68} - 56202048 q^{70} - 560234688 q^{71} - 523987120 q^{73} + 65773608 q^{74} - 87532760 q^{76} - 248943744 q^{79} - 890441280 q^{80} - 1051981172 q^{82} - 1492810428 q^{86} + 1544767952 q^{88} - 744827856 q^{89} + 2959012128 q^{92} + 3068552352 q^{94} + 1465245504 q^{95} - 9932784 q^{97} + 3062604162 q^{98}+O(q^{100}) $ 8 * q + 18 * q^2 - 428 * q^4 + 4800 * q^7 + 3384 * q^8 + 26392 * q^10 + 72336 * q^14 + 185616 * q^16 + 102000 * q^17 - 1245264 * q^20 - 2373124 * q^22 - 3412032 * q^23 - 2423384 * q^25 - 4551240 * q^26 + 7509920 * q^28 + 803584 * q^31 + 14113248 * q^32 + 27757244 * q^34 + 63661140 * q^38 - 93063648 * q^40 + 2180784 * q^41 - 114013320 * q^44 - 131840944 * q^46 - 7432320 * q^47 + 24436680 * q^49 - 231784902 * q^50 + 219270896 * q^52 + 7056832 * q^55 + 358503360 * q^56 + 375425192 * q^58 + 344291904 * q^62 - 316815296 * q^64 + 146501760 * q^65 - 79875048 * q^68 - 56202048 * q^70 - 560234688 * q^71 - 523987120 * q^73 + 65773608 * q^74 - 87532760 * q^76 - 248943744 * q^79 - 890441280 * q^80 - 1051981172 * q^82 - 1492810428 * q^86 + 1544767952 * q^88 - 744827856 * q^89 + 2959012128 * q^92 + 3068552352 * q^94 + 1465245504 * q^95 - 9932784 * q^97 + 3062604162 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	−2.86961	+	22.4447i	−0.126820	+	0.991926i
$2$	−2.86961	+	22.4447i	−0.126820	+	0.991926i
$3$		0			0
$3$		0			0
$4$	−495.531	−	128.815i	−0.967833	−	0.251592i
$5$		−	1417.55i		−	1.01432i	−0.861853	−	0.507159i	$-0.830696\pi$
$5$		−	1417.55i		−	1.01432i	0.861853	−	0.507159i	$-0.169304\pi$
$6$		0			0
$7$	5087.57			0.800883			0.400441	−	0.916322i	$-0.368857\pi$
$7$	5087.57			0.800883			0.400441	+	0.916322i	$0.368857\pi$
$8$	4313.20	−	10752.4i	0.372302	−	0.928112i
$9$		0			0
$10$	31816.6	+	4067.83i	1.00613	+	0.128636i
$11$			14811.2i			0.305016i	0.988302	+	0.152508i	$0.0487349\pi$
$11$			14811.2i			0.305016i	−0.988302	+	0.152508i	$0.951265\pi$
$12$		0			0
$13$			64089.8i			0.622363i	0.950351	+	0.311181i	$0.100725\pi$
$13$			64089.8i			0.622363i	−0.950351	+	0.311181i	$0.899275\pi$
$14$	−14599.3	+	114189.i	−0.101568	+	0.794416i
$15$		0			0
$16$	228957.	+	127664.i	0.873403	+	0.486999i
$17$	−251342.			−0.729868			−0.364934	−	0.931033i	$-0.618908\pi$
$17$	−251342.			−0.729868			−0.364934	+	0.931033i	$0.618908\pi$
$18$		0			0
$19$		−	511568.i		−	0.900558i	−0.892888	−	0.450279i	$-0.851324\pi$
$19$		−	511568.i		−	0.900558i	0.892888	−	0.450279i	$-0.148676\pi$
$20$	−182602.	+	702441.i	−0.255195	+	0.981691i
$21$		0			0
$22$	−332432.	−	42502.3i	−0.302553	−	0.0386821i
$23$	−1.96905e6			−1.46718			−0.733588	−	0.679594i	$-0.762156\pi$
$23$	−1.96905e6			−1.46718			−0.733588	+	0.679594i	$0.762156\pi$
$24$		0			0
$25$	−56329.6			−0.0288408
$26$	−1.43848e6	−	183913.i	−0.617337	−	0.0789281i
$27$		0			0
$28$	−2.52105e6	−	655356.i	−0.775121	−	0.201496i
$29$		−	2.16116e6i		−	0.567409i	−0.958912	−	0.283705i	$-0.908436\pi$
$29$		−	2.16116e6i		−	0.567409i	0.958912	−	0.283705i	$-0.0915636\pi$
$30$		0			0
$31$	−3.03997e6			−0.591210			−0.295605	−	0.955310i	$-0.595521\pi$
$31$	−3.03997e6			−0.591210			−0.295605	+	0.955310i	$0.595521\pi$
$32$	−3.52240e6	+	4.77253e6i	−0.593832	+	0.804589i
$33$		0			0
$34$	721253.	−	5.64129e6i	0.0925620	−	0.723975i
$35$		−	7.21189e6i		−	0.812350i
$36$		0			0
$37$			8.74315e6i			0.766938i	0.923554	+	0.383469i	$0.125271\pi$
$37$			8.74315e6i			0.766938i	−0.923554	+	0.383469i	$0.874729\pi$
$38$	1.14820e7	+	1.46800e6i	0.893287	+	0.114209i
$39$		0			0
$40$	−1.52421e7	−	6.11419e6i	−0.941400	−	0.377632i
$41$	1.48418e7			0.820274			0.410137	−	0.912024i	$-0.365481\pi$
$41$	1.48418e7			0.820274			0.410137	+	0.912024i	$0.365481\pi$
$42$		0			0
$43$			153307.i			0.00683840i	0.999994	+	0.00341920i	$0.00108837\pi$
$43$			153307.i			0.00683840i	−0.999994	+	0.00341920i	$0.998912\pi$
$44$	1.90790e6	−	7.33938e6i	0.0767396	−	0.295204i
$45$		0			0
$46$	5.65042e6	−	4.41949e7i	0.186068	−	1.45533i
$47$	−5.17595e7			−1.54721			−0.773605	−	0.633668i	$-0.781549\pi$
$47$	−5.17595e7			−1.54721			−0.773605	+	0.633668i	$0.781549\pi$
$48$		0			0
$49$	−1.44703e7			−0.358587
$50$	161644.	−	1.26430e6i	0.00365759	−	0.0286079i
$51$		0			0
$52$	8.25574e6	−	3.17584e7i	0.156582	−	0.602343i
$53$		−	4.26277e7i		−	0.742080i	−0.928617	−	0.371040i	$-0.879001\pi$
$53$		−	4.26277e7i		−	0.742080i	0.928617	−	0.371040i	$-0.120999\pi$
$54$		0			0
$55$	2.09956e7			0.309383
$56$	2.19437e7	−	5.47035e7i	0.298170	−	0.743309i
$57$		0			0
$58$	4.85067e7	+	6.20170e6i	0.562828	+	0.0719589i
$59$		−	1.15231e7i		−	0.123804i	−0.998082	−	0.0619021i	$-0.980283\pi$
$59$		−	1.15231e7i		−	0.123804i	0.998082	−	0.0619021i	$-0.0197167\pi$
$60$		0			0
$61$		−	1.95800e8i		−	1.81062i	−0.424748	−	0.905312i	$-0.639637\pi$
$61$		−	1.95800e8i		−	1.81062i	0.424748	−	0.905312i	$-0.360363\pi$
$62$	8.72353e6	−	6.82312e7i	0.0749773	−	0.586436i
$63$		0			0
$64$	−9.70103e7	−	9.27545e7i	−0.722783	−	0.691075i
$65$	9.08506e7			0.631274
$66$		0			0
$67$		−	1.69718e8i		−	1.02894i	−0.857508	−	0.514470i	$-0.827989\pi$
$67$		−	1.69718e8i		−	1.02894i	0.857508	−	0.514470i	$-0.172011\pi$
$68$	1.24547e8	+	3.23766e7i	0.706390	+	0.183629i
$69$		0			0
$70$	1.61869e8	+	2.06953e7i	0.805790	+	0.103022i
$71$	−2.23552e8			−1.04404			−0.522018	−	0.852934i	$-0.674821\pi$
$71$	−2.23552e8			−1.04404			−0.522018	+	0.852934i	$0.674821\pi$
$72$		0			0
$73$	−4.10130e8			−1.69032			−0.845159	−	0.534514i	$-0.820495\pi$
$73$	−4.10130e8			−1.69032			−0.845159	+	0.534514i	$0.820495\pi$
$74$	−1.96238e8	−	2.50895e7i	−0.760746	−	0.0972632i
$75$		0			0
$76$	−6.58977e7	+	2.53497e8i	−0.226574	+	0.871590i
$77$			7.53528e7i			0.244282i
$78$		0			0
$79$	2.51240e7			0.0725716			0.0362858	−	0.999341i	$-0.488447\pi$
$79$	2.51240e7			0.0725716			0.0362858	+	0.999341i	$0.488447\pi$
$80$	1.80970e8	−	3.24559e8i	0.493972	−	0.885908i
$81$		0			0
$82$	−4.25902e7	+	3.33120e8i	−0.104027	+	0.813651i
$83$			4.73174e8i			1.09438i	0.837008	+	0.547191i	$0.184303\pi$
$83$			4.73174e8i			1.09438i	−0.837008	+	0.547191i	$0.815697\pi$
$84$		0			0
$85$			3.56290e8i			0.740318i
$86$	−3.44094e6	−	439932.i	−0.00678318	−	0.000867247i
$87$		0			0
$88$	1.59255e8	+	6.38836e7i	0.283089	+	0.113558i
$89$	−8.45560e8			−1.42853			−0.714265	−	0.699875i	$-0.753239\pi$
$89$	−8.45560e8			−1.42853			−0.714265	+	0.699875i	$0.753239\pi$
$90$		0			0
$91$			3.26061e8i			0.498439i
$92$	9.75726e8	+	2.53644e8i	1.41998	+	0.369130i
$93$		0			0
$94$	1.48530e8	−	1.16173e9i	0.196217	−	1.53472i
$95$	−7.25174e8			−0.913452
$96$		0			0
$97$	−7.97179e8			−0.914288			−0.457144	−	0.889393i	$-0.651128\pi$
$97$	−7.97179e8			−0.914288			−0.457144	+	0.889393i	$0.651128\pi$
$98$	4.15241e7	−	3.24781e8i	0.0454760	−	0.355692i
$99$		0			0
$100$	2.79130e7	+	7.25611e6i	0.0279130	+	0.00725611i
$101$			8.00801e8i			0.765735i	0.923803	+	0.382868i	$0.125063\pi$
$101$			8.00801e8i			0.765735i	−0.923803	+	0.382868i	$0.874937\pi$
$102$		0			0
$103$	8.20324e8			0.718155			0.359077	−	0.933308i	$-0.383091\pi$
$103$	8.20324e8			0.718155			0.359077	+	0.933308i	$0.383091\pi$
$104$	6.89118e8	+	2.76432e8i	0.577622	+	0.231707i
$105$		0			0
$106$	9.56767e8	+	1.22325e8i	0.736088	+	0.0941107i
$107$		−	5.88826e8i		−	0.434270i	−0.976142	−	0.217135i	$-0.930329\pi$
$107$		−	5.88826e8i		−	0.434270i	0.976142	−	0.217135i	$-0.0696712\pi$
$108$		0			0
$109$			8.89549e8i			0.603602i	0.953371	+	0.301801i	$0.0975878\pi$
$109$			8.89549e8i			0.603602i	−0.953371	+	0.301801i	$0.902412\pi$
$110$	−6.02492e7	+	4.71240e8i	−0.0392360	+	0.306885i
$111$		0			0
$112$	1.16484e9	+	6.49498e8i	0.699493	+	0.390029i
$113$	−3.64198e8			−0.210128			−0.105064	−	0.994465i	$-0.533505\pi$
$113$	−3.64198e8			−0.210128			−0.105064	+	0.994465i	$0.533505\pi$
$114$		0			0
$115$			2.79124e9i			1.48818i
$116$	−2.78391e8	+	1.07092e9i	−0.142756	+	0.549158i
$117$		0			0
$118$	2.58633e8	+	3.30669e7i	0.122805	+	0.0157009i
$119$	−1.27872e9			−0.584539
$120$		0			0
$121$	2.13858e9			0.906965
$122$	4.39467e9	+	5.61870e8i	1.79600	+	0.229624i
$123$		0			0
$124$	1.50640e9	+	3.91594e8i	0.572192	+	0.148744i
$125$		−	2.68881e9i		−	0.985064i
$126$		0			0
$127$	−3.86422e9			−1.31809			−0.659046	−	0.752103i	$-0.729040\pi$
$127$	−3.86422e9			−1.31809			−0.659046	+	0.752103i	$0.729040\pi$
$128$	2.36023e9	−	1.91120e9i	0.777159	−	0.629305i
$129$		0			0
$130$	−2.60706e8	+	2.03912e9i	−0.0800582	+	0.626176i
$131$			3.05723e9i			0.907000i	0.891256	+	0.453500i	$0.149825\pi$
$131$			3.05723e9i			0.907000i	−0.891256	+	0.453500i	$0.850175\pi$
$132$		0			0
$133$		−	2.60263e9i		−	0.721242i
$134$	3.80926e9	+	4.87024e8i	1.02063	+	0.130490i
$135$		0			0
$136$	−1.08409e9	+	2.70252e9i	−0.271731	+	0.677399i
$137$	−3.15497e9			−0.765160			−0.382580	−	0.923922i	$-0.624964\pi$
$137$	−3.15497e9			−0.765160			−0.382580	+	0.923922i	$0.624964\pi$
$138$		0			0
$139$			1.46170e9i			0.332117i	0.986116	+	0.166058i	$0.0531040\pi$
$139$			1.46170e9i			0.332117i	−0.986116	+	0.166058i	$0.946896\pi$
$140$	−9.29002e8	+	3.57371e9i	−0.204381	+	0.786219i
$141$		0			0
$142$	6.41507e8	−	5.01756e9i	0.132405	−	1.03561i
$143$	−9.49244e8			−0.189830
$144$		0			0
$145$	−3.06356e9			−0.575534
$146$	1.17691e9	−	9.20525e9i	0.214367	−	1.67667i
$147$		0			0
$148$	1.12625e9	−	4.33250e9i	0.192956	−	0.742268i
$149$			2.14316e9i			0.356218i	0.984011	+	0.178109i	$0.0569980\pi$
$149$			2.14316e9i			0.356218i	−0.984011	+	0.178109i	$0.943002\pi$
$150$		0			0
$151$	−4.26252e9			−0.667222			−0.333611	−	0.942711i	$-0.608267\pi$
$151$	−4.26252e9			−0.667222			−0.333611	+	0.942711i	$0.608267\pi$
$152$	−5.50058e9	−	2.20650e9i	−0.835819	−	0.335279i
$153$		0			0
$154$	−1.69127e9	−	2.16233e8i	−0.242309	−	0.0309798i
$155$			4.30931e9i			0.599674i
$156$		0			0
$157$		−	1.08699e10i		−	1.42783i	−0.700232	−	0.713915i	$-0.746920\pi$
$157$		−	1.08699e10i		−	1.42783i	0.700232	−	0.713915i	$-0.253080\pi$
$158$	−7.20961e7	+	5.63901e8i	−0.00920354	+	0.0719856i
$159$		0			0
$160$	6.76532e9	+	4.99318e9i	0.816109	+	0.602334i
$161$	−1.00177e10			−1.17504
$162$		0			0
$163$			4.36394e9i			0.484211i	0.970250	+	0.242105i	$0.0778380\pi$
$163$			4.36394e9i			0.484211i	−0.970250	+	0.242105i	$0.922162\pi$
$164$	−7.35456e9	−	1.91185e9i	−0.793888	−	0.206375i
$165$		0			0
$166$	−1.06203e10	−	1.35783e9i	−1.08555	−	0.138790i
$167$	7.71676e8			0.0767734			0.0383867	−	0.999263i	$-0.487778\pi$
$167$	7.71676e8			0.0767734			0.0383867	+	0.999263i	$0.487778\pi$
$168$		0			0
$169$	6.49700e9			0.612665
$170$	−7.99683e9	−	1.02241e9i	−0.734341	−	0.0938872i
$171$		0			0
$172$	1.97483e7	−	7.59684e7i	0.00172049	−	0.00661843i
$173$			1.47368e9i			0.125082i	0.998042	+	0.0625411i	$0.0199204\pi$
$173$			1.47368e9i			0.125082i	−0.998042	+	0.0625411i	$0.980080\pi$
$174$		0			0
$175$	−2.86581e8			−0.0230981
$176$	−1.89085e9	+	3.39112e9i	−0.148542	+	0.266401i
$177$		0			0
$178$	2.42643e9	−	1.89784e10i	0.181166	−	1.41700i
$179$		−	1.18045e9i		−	0.0859429i	−0.999076	−	0.0429714i	$-0.986318\pi$
$179$		−	1.18045e9i		−	0.0859429i	0.999076	−	0.0429714i	$-0.0136825\pi$
$180$		0			0
$181$		−	1.33366e10i		−	0.923616i	−0.886980	−	0.461808i	$-0.847201\pi$
$181$		−	1.33366e10i		−	0.923616i	0.886980	−	0.461808i	$-0.152799\pi$
$182$	−7.31834e9	−	9.35668e8i	−0.494415	−	0.0632122i
$183$		0			0
$184$	−8.49293e9	+	2.11720e10i	−0.546232	+	1.36170i
$185$	1.23939e10			0.777919
$186$		0			0
$187$		−	3.72266e9i		−	0.222621i
$188$	2.56484e10	+	6.66741e9i	1.49744	+	0.389266i
$189$		0			0
$190$	2.08097e9	−	1.62763e10i	0.115844	−	0.906077i
$191$	2.67293e10			1.45324			0.726619	−	0.687040i	$-0.241090\pi$
$191$	2.67293e10			1.45324			0.726619	+	0.687040i	$0.241090\pi$
$192$		0			0
$193$	−1.13253e10			−0.587548			−0.293774	−	0.955875i	$-0.594911\pi$
$193$	−1.13253e10			−0.587548			−0.293774	+	0.955875i	$0.594911\pi$
$194$	2.28759e9	−	1.78924e10i	0.115950	−	0.906905i
$195$		0			0
$196$	7.17047e9	+	1.86399e9i	0.347052	+	0.0902177i
$197$		−	3.97475e10i		−	1.88024i	−0.340850	−	0.940118i	$-0.610715\pi$
$197$		−	3.97475e10i		−	1.88024i	0.340850	−	0.940118i	$-0.389285\pi$
$198$		0			0
$199$	2.07651e10			0.938630			0.469315	−	0.883031i	$-0.344501\pi$
$199$	2.07651e10			0.938630			0.469315	+	0.883031i	$0.344501\pi$
$200$	−2.42961e8	+	6.05678e8i	−0.0107375	+	0.0267674i
$201$		0			0
$202$	−1.79738e10	−	2.29799e9i	−0.759552	−	0.0971107i
$203$		−	1.09951e10i		−	0.454428i
$204$		0			0
$205$		−	2.10390e10i		−	0.832018i
$206$	−2.35401e9	+	1.84120e10i	−0.0910765	+	0.712356i
$207$		0			0
$208$	−8.18194e9	+	1.46738e10i	−0.303090	+	0.543573i
$209$	7.57691e9			0.274684
$210$		0			0
$211$			5.11435e10i			1.77631i	0.459541	+	0.888157i	$0.348014\pi$
$211$			5.11435e10i			1.77631i	−0.459541	+	0.888157i	$0.651986\pi$
$212$	−5.49110e9	+	2.11233e10i	−0.186702	+	0.718210i
$213$		0			0
$214$	1.32160e10	+	1.68970e9i	0.430763	+	0.0550742i
$215$	2.17321e8			0.00693631
$216$		0			0
$217$	−1.54660e10			−0.473490
$218$	−1.99657e10	−	2.55266e9i	−0.598728	−	0.0765489i
$219$		0			0
$220$	−1.04040e10	−	2.70455e9i	−0.299431	−	0.0778383i
$221$		−	1.61084e10i		−	0.454243i
$222$		0			0
$223$	4.79011e10			1.29710			0.648550	−	0.761172i	$-0.275376\pi$
$223$	4.79011e10			1.29710			0.648550	+	0.761172i	$0.275376\pi$
$224$	−1.79204e10	+	2.42806e10i	−0.475590	+	0.644382i
$225$		0			0
$226$	1.04511e9	−	8.17432e9i	0.0266485	−	0.208432i
$227$			4.51527e10i			1.12867i	0.825545	+	0.564336i	$0.190868\pi$
$227$			4.51527e10i			1.12867i	−0.825545	+	0.564336i	$0.809132\pi$
$228$		0			0
$229$		−	1.30792e10i		−	0.314282i	−0.987576	−	0.157141i	$-0.949772\pi$
$229$		−	1.30792e10i		−	0.314282i	0.987576	−	0.157141i	$-0.0502278\pi$
$230$	−6.26485e10	−	8.00977e9i	−1.47617	−	0.188732i
$231$		0			0
$232$	−2.32377e10	−	9.32154e9i	−0.526619	−	0.211247i
$233$	7.90446e10			1.75700			0.878498	−	0.477746i	$-0.158546\pi$
$233$	7.90446e10			1.75700			0.878498	+	0.477746i	$0.158546\pi$
$234$		0			0
$235$			7.33717e10i			1.56936i
$236$	−1.48435e9	+	5.71005e9i	−0.0311482	+	0.119822i
$237$		0			0
$238$	3.66942e9	−	2.87004e10i	0.0741313	−	0.579819i
$239$	−1.26518e10			−0.250819			−0.125410	−	0.992105i	$-0.540025\pi$
$239$	−1.26518e10			−0.250819			−0.125410	+	0.992105i	$0.540025\pi$
$240$		0			0
$241$	5.85645e10			1.11830			0.559149	−	0.829067i	$-0.311128\pi$
$241$	5.85645e10			1.11830			0.559149	+	0.829067i	$0.311128\pi$
$242$	−6.13689e9	+	4.79998e10i	−0.115021	+	0.899642i
$243$		0			0
$244$	−2.52220e10	+	9.70248e10i	−0.455539	+	1.75238i
$245$			2.05124e10i			0.363721i
$246$		0			0
$247$	3.27862e10			0.560474
$248$	−1.31120e10	+	3.26869e10i	−0.220108	+	0.548709i
$249$		0			0
$250$	6.03495e10	+	7.71583e9i	0.977111	+	0.124926i
$251$			6.88194e10i			1.09441i	0.837000	+	0.547204i	$0.184308\pi$
$251$			6.88194e10i			1.09441i	−0.837000	+	0.547204i	$0.815692\pi$
$252$		0			0
$253$		−	2.91640e10i		−	0.447512i
$254$	1.10888e10	−	8.67314e10i	0.167161	−	1.30745i
$255$		0			0
$256$	3.61234e10	+	5.84591e10i	0.525664	+	0.850692i
$257$	1.19348e10			0.170654			0.0853271	−	0.996353i	$-0.472806\pi$
$257$	1.19348e10			0.170654			0.0853271	+	0.996353i	$0.472806\pi$
$258$		0			0
$259$			4.44814e10i			0.614227i
$260$	−4.50192e10	−	1.17029e10i	−0.610968	−	0.158824i
$261$		0			0
$262$	−6.86187e10	−	8.77307e9i	−0.899677	−	0.115026i
$263$	−3.19598e10			−0.411911			−0.205956	−	0.978561i	$-0.566030\pi$
$263$	−3.19598e10			−0.411911			−0.205956	+	0.978561i	$0.566030\pi$
$264$		0			0
$265$	−6.04270e10			−0.752705
$266$	5.84154e10	+	7.46855e9i	0.715418	+	0.0914680i
$267$		0			0
$268$	−2.18622e10	+	8.41003e10i	−0.258874	+	0.995843i
$269$		−	7.08498e10i		−	0.824999i	−0.910958	−	0.412500i	$-0.864656\pi$
$269$		−	7.08498e10i		−	0.824999i	0.910958	−	0.412500i	$-0.135344\pi$
$270$		0			0
$271$	−8.70264e10			−0.980143			−0.490071	−	0.871682i	$-0.663029\pi$
$271$	−8.70264e10			−0.980143			−0.490071	+	0.871682i	$0.663029\pi$
$272$	−5.75465e10	−	3.20872e10i	−0.637469	−	0.355445i
$273$		0			0
$274$	9.05354e9	−	7.08124e10i	0.0970378	−	0.758982i
$275$		−	8.34307e8i		−	0.00879688i
$276$		0			0
$277$		−	1.55080e11i		−	1.58270i	−0.611365	−	0.791349i	$-0.709379\pi$
$277$		−	1.55080e11i		−	1.58270i	0.611365	−	0.791349i	$-0.290621\pi$
$278$	−3.28074e10	−	4.19450e9i	−0.329435	−	0.0421191i
$279$		0			0
$280$	−7.75451e10	−	3.11064e10i	−0.753951	−	0.302439i
$281$	2.79959e10			0.267865			0.133933	−	0.990990i	$-0.457239\pi$
$281$	2.79959e10			0.267865			0.133933	+	0.990990i	$0.457239\pi$
$282$		0			0
$283$		−	9.14271e10i		−	0.847298i	−0.905826	−	0.423649i	$-0.860749\pi$
$283$		−	9.14271e10i		−	0.847298i	0.905826	−	0.423649i	$-0.139251\pi$
$284$	1.10777e11	+	2.87969e10i	1.01045	+	0.262672i
$285$		0			0
$286$	2.72396e9	−	2.13055e10i	0.0240743	−	0.188298i
$287$	7.55086e10			0.656943
$288$		0			0
$289$	−5.54153e10			−0.467293
$290$	8.79123e9	−	6.87608e10i	0.0729892	−	0.570887i
$291$		0			0
$292$	2.03232e11	+	5.28310e10i	1.63595	+	0.425271i
$293$			9.37104e10i			0.742819i	0.928469	+	0.371410i	$0.121125\pi$
$293$			9.37104e10i			0.742819i	−0.928469	+	0.371410i	$0.878875\pi$
$294$		0			0
$295$	−1.63346e10			−0.125577
$296$	9.40098e10	+	3.77110e10i	0.711804	+	0.285532i
$297$		0			0
$298$	−4.81026e10	−	6.15004e9i	−0.353342	−	0.0451757i
$299$		−	1.26196e11i		−	0.913116i
$300$		0			0
$301$			7.79960e8i			0.00547676i
$302$	1.22318e10	−	9.56711e10i	0.0846172	−	0.661835i
$303$		0			0
$304$	6.53087e10	−	1.17127e11i	0.438571	−	0.786550i
$305$	−2.77557e11			−1.83655
$306$		0			0
$307$			1.03026e11i			0.661951i	0.943639	+	0.330976i	$0.107378\pi$
$307$			1.03026e11i			0.661951i	−0.943639	+	0.330976i	$0.892622\pi$
$308$	9.70659e9	−	3.73396e10i	0.0614594	−	0.236424i
$309$		0			0
$310$	−9.67213e10	−	1.23661e10i	−0.594833	−	0.0760508i
$311$	1.43254e11			0.868333			0.434167	−	0.900833i	$-0.357043\pi$
$311$	1.43254e11			0.868333			0.434167	+	0.900833i	$0.357043\pi$
$312$		0			0
$313$	2.53850e11			1.49495			0.747477	−	0.664287i	$-0.231265\pi$
$313$	2.53850e11			1.49495			0.747477	+	0.664287i	$0.231265\pi$
$314$	2.43972e11	+	3.11924e10i	1.41630	+	0.181078i
$315$		0			0
$316$	−1.24497e10	−	3.23635e9i	−0.0702372	−	0.0182585i
$317$		−	2.17416e11i		−	1.20927i	−0.796502	−	0.604636i	$-0.793318\pi$
$317$		−	2.17416e11i		−	1.20927i	0.796502	−	0.604636i	$-0.206682\pi$
$318$		0			0
$319$	3.20093e10			0.173069
$320$	−1.31484e11	+	1.37517e11i	−0.700970	+	0.733132i
$321$		0			0
$322$	2.87469e10	−	2.24844e11i	0.149018	−	1.16555i
$323$			1.28578e11i			0.657289i
$324$		0			0
$325$		−	3.61015e9i		−	0.0179494i
$326$	−9.79474e10	−	1.25228e10i	−0.480301	−	0.0614077i
$327$		0			0
$328$	6.40157e10	−	1.59585e11i	0.305389	−	0.761306i
$329$	−2.63330e11			−1.23913
$330$		0			0
$331$		−	2.98163e10i		−	0.136530i	−0.997667	−	0.0682650i	$-0.978254\pi$
$331$		−	2.98163e10i		−	0.136530i	0.997667	−	0.0682650i	$-0.0217463\pi$
$332$	6.09520e10	−	2.34472e11i	0.275338	−	1.05918i
$333$		0			0
$334$	−2.21441e9	+	1.73201e10i	−0.00973642	+	0.0761535i
$335$	−2.40584e11			−1.04367
$336$		0			0
$337$	−3.56448e11			−1.50543			−0.752716	−	0.658346i	$-0.771257\pi$
$337$	−3.56448e11			−1.50543			−0.752716	+	0.658346i	$0.771257\pi$
$338$	−1.86439e10	+	1.45823e11i	−0.0776982	+	0.607718i
$339$		0			0
$340$	4.58956e10	−	1.76553e11i	0.186258	−	0.716505i
$341$		−	4.50255e10i		−	0.180328i
$342$		0			0
$343$	−2.78920e11			−1.08807
$344$	1.64842e9	+	6.61245e8i	0.00634680	+	0.00254595i
$345$		0			0
$346$	−3.30763e10	−	4.22889e9i	−0.124072	−	0.0158629i
$347$			3.73598e10i			0.138332i	0.997605	+	0.0691659i	$0.0220338\pi$
$347$			3.73598e10i			0.138332i	−0.997605	+	0.0691659i	$0.977966\pi$
$348$		0			0
$349$			1.15752e11i			0.417653i	0.977953	+	0.208827i	$0.0669644\pi$
$349$			1.15752e11i			0.417653i	−0.977953	+	0.208827i	$0.933036\pi$
$350$	8.22375e8	−	6.43222e9i	0.00292930	−	0.0229116i
$351$		0			0
$352$	−7.06868e10	−	5.21708e10i	−0.245412	−	0.181128i
$353$	−1.94182e11			−0.665614			−0.332807	−	0.942995i	$-0.607996\pi$
$353$	−1.94182e11			−0.665614			−0.332807	+	0.942995i	$0.607996\pi$
$354$		0			0
$355$			3.16897e11i			1.05899i
$356$	4.19001e11	+	1.08921e11i	1.38258	+	0.359407i
$357$		0			0
$358$	2.64949e10	+	3.38744e9i	0.0852490	+	0.0108993i
$359$	−2.22958e11			−0.708432			−0.354216	−	0.935164i	$-0.615252\pi$
$359$	−2.22958e11			−0.708432			−0.354216	+	0.935164i	$0.615252\pi$
$360$		0			0
$361$	6.09863e10			0.188995
$362$	2.99336e11	+	3.82708e10i	0.916158	+	0.117133i
$363$		0			0
$364$	4.20016e10	−	1.61573e11i	0.125404	−	0.482406i
$365$			5.81381e11i			1.71452i
$366$		0			0
$367$	−2.95151e11			−0.849272			−0.424636	−	0.905364i	$-0.639598\pi$
$367$	−2.95151e11			−0.849272			−0.424636	+	0.905364i	$0.639598\pi$
$368$	−4.50829e11	−	2.51377e11i	−1.28144	−	0.714513i
$369$		0			0
$370$	−3.55656e10	+	2.78177e11i	−0.0986558	+	0.771638i
$371$		−	2.16871e11i		−	0.594319i
$372$		0			0
$373$			1.80321e10i			0.0482343i	0.999709	+	0.0241172i	$0.00767748\pi$
$373$			1.80321e10i			0.0482343i	−0.999709	+	0.0241172i	$0.992323\pi$
$374$	8.35541e10	+	1.06826e10i	0.220824	+	0.0282328i
$375$		0			0
$376$	−2.23249e11	+	5.56538e11i	−0.576029	+	1.43598i
$377$	1.38508e11			0.353134
$378$		0			0
$379$		−	6.09778e11i		−	1.51808i	−0.651042	−	0.759042i	$-0.725668\pi$
$379$		−	6.09778e11i		−	1.51808i	0.651042	−	0.759042i	$-0.274332\pi$
$380$	3.59346e11	+	9.34135e10i	0.884070	+	0.229818i
$381$		0			0
$382$	−7.67026e10	+	5.99931e11i	−0.184300	+	1.44150i
$383$	−1.33302e11			−0.316549			−0.158274	−	0.987395i	$-0.550593\pi$
$383$	−1.33302e11			−0.316549			−0.158274	+	0.987395i	$0.550593\pi$
$384$		0			0
$385$	1.06816e11			0.247779
$386$	3.24993e10	−	2.54194e11i	0.0745129	−	0.582804i
$387$		0			0
$388$	3.95026e11	+	1.02689e11i	0.884878	+	0.230028i
$389$			2.05551e10i			0.0455142i	0.999741	+	0.0227571i	$0.00724444\pi$
$389$			2.05551e10i			0.0455142i	−0.999741	+	0.0227571i	$0.992756\pi$
$390$		0			0
$391$	4.94905e11			1.07084
$392$	−6.24132e10	+	1.55590e11i	−0.133503	+	0.332809i
$393$		0			0
$394$	8.92122e11	+	1.14060e11i	1.86505	+	0.238452i
$395$		−	3.56146e10i		−	0.0736107i
$396$		0			0
$397$			5.93486e11i			1.19909i	0.800339	+	0.599547i	$0.204653\pi$
$397$			5.93486e11i			1.19909i	−0.800339	+	0.599547i	$0.795347\pi$
$398$	−5.95877e10	+	4.66066e11i	−0.119037	+	0.931051i
$399$		0			0
$400$	−1.28971e10	−	7.19125e9i	−0.0251896	−	0.0140454i
$401$	1.00640e12			1.94367			0.971836	−	0.235660i	$-0.0757252\pi$
$401$	1.00640e12			1.94367			0.971836	+	0.235660i	$0.0757252\pi$
$402$		0			0
$403$		−	1.94831e11i		−	0.367947i
$404$	1.03155e11	−	3.96822e11i	0.192653	−	0.741104i
$405$		0			0
$406$	2.46781e11	+	3.15516e10i	0.450759	+	0.0576307i
$407$	−1.29496e11			−0.233928
$408$		0			0
$409$	−3.40532e11			−0.601732			−0.300866	−	0.953666i	$-0.597276\pi$
$409$	−3.40532e11			−0.601732			−0.300866	+	0.953666i	$0.597276\pi$
$410$	4.72215e11	+	6.03738e10i	0.825301	+	0.105517i
$411$		0			0
$412$	−4.06496e11	−	1.05670e11i	−0.695054	−	0.180682i
$413$		−	5.86246e10i		−	0.0991527i
$414$		0			0
$415$	6.70749e11			1.11005
$416$	−3.05871e11	−	2.25750e11i	−0.500746	−	0.369579i
$417$		0			0
$418$	−2.17428e10	+	1.70062e11i	−0.0348355	+	0.272466i
$419$			8.47876e11i			1.34391i	0.740593	+	0.671953i	$0.234544\pi$
$419$			8.47876e11i			1.34391i	−0.740593	+	0.671953i	$0.765456\pi$
$420$		0			0
$421$			5.27094e11i			0.817747i	0.912591	+	0.408873i	$0.134078\pi$
$421$			5.27094e11i			0.817747i	−0.912591	+	0.408873i	$0.865922\pi$
$422$	−1.14790e12	−	1.46762e11i	−1.76197	−	0.225272i
$423$		0			0
$424$	−4.58350e11	−	1.83862e11i	−0.688733	−	0.276278i
$425$	1.41580e10			0.0210499
$426$		0			0
$427$		−	9.96145e11i		−	1.45010i
$428$	−7.58497e10	+	2.91781e11i	−0.109259	+	0.420301i
$429$		0			0
$430$	−6.23627e8	+	4.87771e9i	−0.000879664	+	0.00688030i
$431$	−1.12919e11			−0.157622			−0.0788112	−	0.996890i	$-0.525112\pi$
$431$	−1.12919e11			−0.157622			−0.0788112	+	0.996890i	$0.525112\pi$
$432$		0			0
$433$	9.18037e11			1.25506			0.627530	−	0.778592i	$-0.284066\pi$
$433$	9.18037e11			1.25506			0.627530	+	0.778592i	$0.284066\pi$
$434$	4.43815e10	−	3.47131e11i	0.0600480	−	0.469666i
$435$		0			0
$436$	1.14588e11	−	4.40799e11i	0.151862	−	0.584186i
$437$			1.00730e12i			1.32128i
$438$		0			0
$439$	−9.32700e11			−1.19854			−0.599269	−	0.800548i	$-0.704542\pi$
$439$	−9.32700e11			−1.19854			−0.599269	+	0.800548i	$0.704542\pi$
$440$	9.05583e10	−	2.25753e11i	0.115184	−	0.287142i
$441$		0			0
$442$	3.61549e11	+	4.62249e10i	0.450575	+	0.0576071i
$443$		−	1.58499e12i		−	1.95528i	−0.210283	−	0.977641i	$-0.567439\pi$
$443$		−	1.58499e12i		−	1.95528i	0.210283	−	0.977641i	$-0.432561\pi$
$444$		0			0
$445$			1.19863e12i			1.44898i
$446$	−1.37457e11	+	1.07513e12i	−0.164498	+	1.28663i
$447$		0			0
$448$	−4.93546e11	−	4.71895e11i	−0.578864	−	0.553470i
$449$	5.33305e11			0.619251			0.309626	−	0.950859i	$-0.399796\pi$
$449$	5.33305e11			0.619251			0.309626	+	0.950859i	$0.399796\pi$
$450$		0			0
$451$			2.19824e11i			0.250196i
$452$	1.80471e11	+	4.69143e10i	0.203369	+	0.0528667i
$453$		0			0
$454$	−1.01344e12	−	1.29571e11i	−1.11956	−	0.143138i
$455$	4.62208e11			0.505576
$456$		0			0
$457$	−6.95059e11			−0.745416			−0.372708	−	0.927949i	$-0.621571\pi$
$457$	−6.95059e11			−0.745416			−0.372708	+	0.927949i	$0.621571\pi$
$458$	2.93558e11	+	3.75321e10i	0.311745	+	0.0398573i
$459$		0			0
$460$	3.59554e11	−	1.38314e12i	0.374415	−	1.44031i
$461$		−	1.51250e12i		−	1.55970i	−0.625966	−	0.779851i	$-0.715295\pi$
$461$		−	1.51250e12i		−	1.55970i	0.625966	−	0.779851i	$-0.284705\pi$
$462$		0			0
$463$	1.10015e12			1.11260			0.556300	−	0.830981i	$-0.312220\pi$
$463$	1.10015e12			1.11260			0.556300	+	0.830981i	$0.312220\pi$
$464$	2.75902e11	−	4.94814e11i	0.276328	−	0.495577i
$465$		0			0
$466$	−2.26827e11	+	1.77413e12i	−0.222823	+	1.74281i
$467$		−	7.29651e11i		−	0.709887i	−0.934888	−	0.354944i	$-0.884500\pi$
$467$		−	7.29651e11i		−	0.709887i	0.934888	−	0.354944i	$-0.115500\pi$
$468$		0			0
$469$		−	8.63450e11i		−	0.824061i
$470$	−1.64681e12	−	2.10548e11i	−1.55669	−	0.199027i
$471$		0			0
$472$	−1.23901e11	−	4.97015e10i	−0.114904	−	0.0460925i
$473$	−2.27066e9			−0.00208582
$474$		0			0
$475$			2.88164e10i			0.0259728i
$476$	6.33644e11	+	1.64718e11i	0.565736	+	0.147065i
$477$		0			0
$478$	3.63057e10	−	2.83965e11i	0.0318089	−	0.248794i
$479$	9.75429e11			0.846614			0.423307	−	0.905986i	$-0.360869\pi$
$479$	9.75429e11			0.846614			0.423307	+	0.905986i	$0.360869\pi$
$480$		0			0
$481$	−5.60346e11			−0.477314
$482$	−1.68057e11	+	1.31446e12i	−0.141823	+	1.10927i
$483$		0			0
$484$	−1.05973e12	−	2.75481e11i	−0.877791	−	0.228186i
$485$			1.13004e12i			0.927378i
$486$		0			0
$487$	7.09068e11			0.571225			0.285613	−	0.958345i	$-0.407803\pi$
$487$	7.09068e11			0.571225			0.285613	+	0.958345i	$0.407803\pi$
$488$	−2.10532e12	−	8.44525e11i	−1.68046	−	0.674098i
$489$		0			0
$490$	−4.60394e11	−	5.88626e10i	−0.360784	−	0.0461272i
$491$			8.30763e10i			0.0645075i	0.999480	+	0.0322538i	$0.0102685\pi$
$491$			8.30763e10i			0.0645075i	−0.999480	+	0.0322538i	$0.989732\pi$
$492$		0			0
$493$			5.43190e11i			0.414134i
$494$	−9.40838e10	+	7.35878e11i	−0.0710794	+	0.555948i
$495$		0			0
$496$	−6.96023e11	−	3.88094e11i	−0.516364	−	0.287918i
$497$	−1.13734e12			−0.836151
$498$		0			0
$499$		−	1.54674e12i		−	1.11677i	−0.829581	−	0.558386i	$-0.811421\pi$
$499$		−	1.54674e12i		−	1.11677i	0.829581	−	0.558386i	$-0.188579\pi$
$500$	−3.46359e11	+	1.33239e12i	−0.247835	+	0.953378i
$501$		0			0
$502$	−1.54463e12	−	1.97485e11i	−1.08557	−	0.138793i
$503$	9.81095e11			0.683369			0.341684	−	0.939815i	$-0.389003\pi$
$503$	9.81095e11			0.683369			0.341684	+	0.939815i	$0.389003\pi$
$504$		0			0
$505$	1.13518e12			0.776699
$506$	6.54577e11	+	8.36893e10i	0.443898	+	0.0567535i
$507$		0			0
$508$	1.91484e12	+	4.97771e11i	1.27569	+	0.331622i
$509$		−	1.02396e12i		−	0.676167i	−0.941116	−	0.338083i	$-0.890221\pi$
$509$		−	1.02396e12i		−	0.676167i	0.941116	−	0.338083i	$-0.109779\pi$
$510$		0			0
$511$	−2.08656e12			−1.35375
$512$	−1.41576e12	+	6.43024e11i	−0.910488	+	0.413535i
$513$		0			0
$514$	−3.42483e10	+	2.67874e11i	−0.0216424	+	0.169276i
$515$		−	1.16285e12i		−	0.728437i
$516$		0			0
$517$		−	7.66618e11i		−	0.471923i
$518$	−9.98372e11	−	1.27644e11i	−0.609268	−	0.0778964i
$519$		0			0
$520$	3.91857e11	−	9.76861e11i	0.235024	−	0.585892i
$521$	−1.53658e11			−0.0913664			−0.0456832	−	0.998956i	$-0.514546\pi$
$521$	−1.53658e11			−0.0913664			−0.0456832	+	0.998956i	$0.514546\pi$
$522$		0			0
$523$		−	2.08535e12i		−	1.21877i	−0.792875	−	0.609385i	$-0.791417\pi$
$523$		−	2.08535e12i		−	1.21877i	0.792875	−	0.609385i	$-0.208583\pi$
$524$	3.93818e11	−	1.51495e12i	0.228194	−	0.877825i
$525$		0			0
$526$	9.17123e10	−	7.17329e11i	0.0522386	−	0.408585i
$527$	7.64071e11			0.431505
$528$		0			0
$529$	2.07602e12			1.15261
$530$	1.73402e11	−	1.35627e12i	0.0954582	−	0.746628i
$531$		0			0
$532$	−3.35259e11	+	1.28968e12i	−0.181459	+	0.698042i
$533$			9.51207e11i			0.510508i
$534$		0			0
$535$	−8.34691e11			−0.440488
$536$	−1.82487e12	−	7.32027e11i	−0.954972	−	0.383076i
$537$		0			0
$538$	1.59020e12	+	2.03312e11i	0.818338	+	0.104627i
$539$		−	2.14322e11i		−	0.109375i
$540$		0			0
$541$			4.46995e11i			0.224344i	0.993689	+	0.112172i	$0.0357808\pi$
$541$			4.46995e11i			0.224344i	−0.993689	+	0.112172i	$0.964219\pi$
$542$	2.49732e11	−	1.95328e12i	0.124302	−	0.972229i
$543$		0			0
$544$	8.85325e11	−	1.19954e12i	0.433419	−	0.587244i
$545$	1.26098e12			0.612244
$546$		0			0
$547$		−	3.46729e11i		−	0.165595i	−0.996566	−	0.0827974i	$-0.973615\pi$
$547$		−	3.46729e11i		−	0.165595i	0.996566	−	0.0827974i	$-0.0263854\pi$
$548$	1.56338e12	+	4.06408e11i	0.740548	+	0.192508i
$549$		0			0
$550$	1.87258e10	+	2.39414e9i	0.00872585	+	0.00111562i
$551$	−1.10558e12			−0.510985
$552$		0			0
$553$	1.27820e11			0.0581213
$554$	3.48073e12	+	4.45020e11i	1.56992	+	0.200718i
$555$		0			0
$556$	1.88289e11	−	7.24315e11i	0.0835580	−	0.321434i
$557$			5.59793e11i			0.246422i	0.992380	+	0.123211i	$0.0393192\pi$
$557$			5.59793e11i			0.246422i	−0.992380	+	0.123211i	$0.960681\pi$
$558$		0			0
$559$	−9.82542e9			−0.00425596
$560$	9.20698e11	−	1.65121e12i	0.395613	−	0.709508i
$561$		0			0
$562$	−8.03374e10	+	6.28360e11i	−0.0339707	+	0.265702i
$563$			2.30313e12i			0.966119i	0.875587	+	0.483060i	$0.160475\pi$
$563$			2.30313e12i			0.966119i	−0.875587	+	0.483060i	$0.839525\pi$
$564$		0			0
$565$			5.16270e11i			0.213137i
$566$	2.05206e12	+	2.62360e11i	0.840457	+	0.107454i
$567$		0			0
$568$	−9.64225e11	+	2.40372e12i	−0.388697	+	0.968983i
$569$	8.14910e11			0.325915			0.162958	−	0.986633i	$-0.447897\pi$
$569$	8.14910e11			0.325915			0.162958	+	0.986633i	$0.447897\pi$
$570$		0			0
$571$			3.63502e12i			1.43102i	0.698605	+	0.715508i	$0.253805\pi$
$571$			3.63502e12i			1.43102i	−0.698605	+	0.715508i	$0.746195\pi$
$572$	4.70379e11	+	1.22277e11i	0.183724	+	0.0477599i
$573$		0			0
$574$	−2.16680e11	+	1.69477e12i	−0.0833136	+	0.651639i
$575$	1.10916e11			0.0423145
$576$		0			0
$577$	2.45038e11			0.0920326			0.0460163	−	0.998941i	$-0.485347\pi$
$577$	2.45038e11			0.0920326			0.0460163	+	0.998941i	$0.485347\pi$
$578$	1.59020e11	−	1.24378e12i	0.0592621	−	0.463520i
$579$		0			0
$580$	1.51809e12	+	3.94634e11i	0.557021	+	0.144800i
$581$			2.40730e12i			0.876472i
$582$		0			0
$583$	6.31366e11			0.226346
$584$	−1.76897e12	+	4.40988e12i	−0.629309	+	1.56880i
$585$		0			0
$586$	−2.10330e12	−	2.68912e11i	−0.736822	−	0.0942045i
$587$			2.81390e12i			0.978223i	0.872221	+	0.489112i	$0.162679\pi$
$587$			2.81390e12i			0.978223i	−0.872221	+	0.489112i	$0.837321\pi$
$588$		0			0
$589$			1.55515e12i			0.532419i
$590$	4.68740e10	−	3.66626e11i	0.0159257	−	0.124563i
$591$		0			0
$592$	−1.11618e12	+	2.00181e12i	−0.373498	+	0.669846i
$593$	−7.42491e11			−0.246573			−0.123286	−	0.992371i	$-0.539343\pi$
$593$	−7.42491e11			−0.246573			−0.123286	+	0.992371i	$0.539343\pi$
$594$		0			0
$595$			1.81265e12i			0.592908i
$596$	2.76072e11	−	1.06200e12i	0.0896218	−	0.344760i
$597$		0			0
$598$	2.83244e12	+	3.62134e11i	0.905743	+	0.115801i
$599$	−8.31869e11			−0.264018			−0.132009	−	0.991248i	$-0.542143\pi$
$599$	−8.31869e11			−0.264018			−0.132009	+	0.991248i	$0.542143\pi$
$600$		0			0
$601$	−1.71774e12			−0.537060			−0.268530	−	0.963271i	$-0.586538\pi$
$601$	−1.71774e12			−0.537060			−0.268530	+	0.963271i	$0.586538\pi$
$602$	−1.75060e10	−	2.23818e9i	−0.00543253	−	0.000694563i
$603$		0			0
$604$	2.11221e12	+	5.49078e11i	0.645760	+	0.167868i
$605$		−	3.03155e12i		−	0.919951i
$606$		0			0
$607$	−2.97709e12			−0.890107			−0.445053	−	0.895504i	$-0.646815\pi$
$607$	−2.97709e12			−0.890107			−0.445053	+	0.895504i	$0.646815\pi$
$608$	2.44147e12	+	1.80194e12i	0.724580	+	0.534780i
$609$		0			0
$610$	7.96480e11	−	6.22968e12i	0.232911	−	1.82172i
$611$		−	3.31725e12i		−	0.962926i
$612$		0			0
$613$		−	3.26103e12i		−	0.932788i	−0.884577	−	0.466394i	$-0.845553\pi$
$613$		−	3.26103e12i		−	0.932788i	0.884577	−	0.466394i	$-0.154447\pi$
$614$	−2.31240e12	−	2.95646e11i	−0.656607	−	0.0839488i
$615$		0			0
$616$	8.10223e11	+	3.25012e11i	0.226721	+	0.0909465i
$617$	−4.07434e12			−1.13181			−0.565906	−	0.824470i	$-0.691473\pi$
$617$	−4.07434e12			−1.13181			−0.565906	+	0.824470i	$0.691473\pi$
$618$		0			0
$619$			3.60019e12i			0.985638i	0.870132	+	0.492819i	$0.164034\pi$
$619$			3.60019e12i			0.985638i	−0.870132	+	0.492819i	$0.835966\pi$
$620$	5.55106e11	−	2.13540e12i	0.150873	−	0.580385i
$621$		0			0
$622$	−4.11085e11	+	3.21531e12i	−0.110122	+	0.861322i
$623$	−4.30184e12			−1.14409
$624$		0			0
$625$	−3.92154e12			−1.02801
$626$	−7.28452e11	+	5.69760e12i	−0.189590	+	1.48288i
$627$		0			0
$628$	−1.40021e12	+	5.38636e12i	−0.359231	+	1.38190i
$629$		−	2.19752e12i		−	0.559763i
$630$		0			0
$631$	4.14069e12			1.03978			0.519890	−	0.854233i	$-0.325973\pi$
$631$	4.14069e12			1.03978			0.519890	+	0.854233i	$0.325973\pi$
$632$	1.08365e11	−	2.70143e11i	0.0270185	−	0.0673546i
$633$		0			0
$634$	4.87984e12	+	6.23899e11i	1.19951	+	0.153360i
$635$			5.47774e12i			1.33696i
$636$		0			0
$637$		−	9.27396e11i		−	0.223171i
$638$	−9.18544e10	+	7.18441e11i	−0.0219486	+	0.171671i
$639$		0			0
$640$	−2.70922e12	−	3.34575e12i	−0.638315	−	0.788286i
$641$	−1.23747e12			−0.289516			−0.144758	−	0.989467i	$-0.546240\pi$
$641$	−1.23747e12			−0.289516			−0.144758	+	0.989467i	$0.546240\pi$
$642$		0			0
$643$			2.36693e12i			0.546054i	0.962006	+	0.273027i	$0.0880248\pi$
$643$			2.36693e12i			0.546054i	−0.962006	+	0.273027i	$0.911975\pi$
$644$	4.96407e12	+	1.29043e12i	1.13724	+	0.295630i
$645$		0			0
$646$	−2.88590e12	−	3.68970e11i	−0.651982	−	0.0833574i
$647$	−5.75807e12			−1.29184			−0.645918	−	0.763406i	$-0.723525\pi$
$647$	−5.75807e12			−1.29184			−0.645918	+	0.763406i	$0.723525\pi$
$648$		0			0
$649$	1.70671e11			0.0377622
$650$	8.10288e10	+	1.03597e10i	0.0178045	+	0.00227635i
$651$		0			0
$652$	5.62142e11	−	2.16247e12i	0.121824	−	0.468635i
$653$		−	1.19202e12i		−	0.256552i	−0.991739	−	0.128276i	$-0.959056\pi$
$653$		−	1.19202e12i		−	0.256552i	0.991739	−	0.128276i	$-0.0409443\pi$
$654$		0			0
$655$	4.33379e12			0.919987
$656$	3.39814e12	+	1.89476e12i	0.716429	+	0.399472i
$657$		0			0
$658$	7.55654e11	−	5.91036e12i	0.157147	−	1.22913i
$659$		−	6.40718e11i		−	0.132337i	−0.997808	−	0.0661687i	$-0.978922\pi$
$659$		−	6.40718e11i		−	0.132337i	0.997808	−	0.0661687i	$-0.0210776\pi$
$660$		0			0
$661$		−	6.37585e12i		−	1.29907i	−0.760333	−	0.649533i	$-0.774964\pi$
$661$		−	6.37585e12i		−	1.29907i	0.760333	−	0.649533i	$-0.225036\pi$
$662$	6.69219e11	+	8.55613e10i	0.135428	+	0.0173148i
$663$		0			0
$664$	5.08775e12	+	2.04089e12i	1.01571	+	0.407440i
$665$	−3.68937e12			−0.731568
$666$		0			0
$667$			4.25545e12i			0.832490i
$668$	−3.82389e11	−	9.94037e10i	−0.0743039	−	0.0193156i
$669$		0			0
$670$	6.90382e11	−	5.39983e12i	0.132359	−	1.03525i
$671$	2.90002e12			0.552268
$672$		0			0
$673$	−5.29863e12			−0.995624			−0.497812	−	0.867285i	$-0.665863\pi$
$673$	−5.29863e12			−0.995624			−0.497812	+	0.867285i	$0.665863\pi$
$674$	1.02287e12	−	8.00036e12i	0.190919	−	1.49328i
$675$		0			0
$676$	−3.21946e12	−	8.36913e11i	−0.592957	−	0.154142i
$677$			2.91606e12i			0.533516i	0.963764	+	0.266758i	$0.0859525\pi$
$677$			2.91606e12i			0.533516i	−0.963764	+	0.266758i	$0.914048\pi$
$678$		0			0
$679$	−4.05570e12			−0.732237
$680$	3.83097e12	+	1.53675e12i	0.687098	+	0.275622i
$681$		0			0
$682$	1.01058e12	+	1.29206e11i	0.178872	+	0.0228692i
$683$		−	1.41684e12i		−	0.249131i	−0.992211	−	0.124565i	$-0.960246\pi$
$683$		−	1.41684e12i		−	0.249131i	0.992211	−	0.124565i	$-0.0397536\pi$
$684$		0			0
$685$			4.47233e12i			0.776116i
$686$	8.00393e11	−	6.26028e12i	0.137989	−	1.07928i
$687$		0			0
$688$	−1.95718e10	+	3.51008e10i	−0.00333029	+	0.00597268i
$689$	2.73200e12			0.461843
$690$		0			0
$691$		−	1.87083e12i		−	0.312164i	−0.987744	−	0.156082i	$-0.950114\pi$
$691$		−	1.87083e12i		−	0.312164i	0.987744	−	0.156082i	$-0.0498865\pi$
$692$	1.89832e11	−	7.30253e11i	0.0314697	−	0.121059i
$693$		0			0
$694$	−8.38530e11	−	1.07208e11i	−0.137215	−	0.0175433i
$695$	2.07203e12			0.336872
$696$		0			0
$697$	−3.73036e12			−0.598692
$698$	−2.59803e12	−	3.32165e11i	−0.414281	−	0.0529668i
$699$		0			0
$700$	1.42009e11	+	3.69160e10i	0.0223551	+	0.00581130i
$701$		−	8.31406e12i		−	1.30042i	−0.759757	−	0.650208i	$-0.774682\pi$
$701$		−	8.31406e12i		−	1.30042i	0.759757	−	0.650208i	$-0.225318\pi$
$702$		0			0
$703$	4.47271e12			0.690672
$704$	1.37380e12	−	1.43684e12i	0.210789	−	0.220460i
$705$		0			0
$706$	5.57227e11	−	4.35836e12i	0.0844133	−	0.660240i
$707$			4.07413e12i			0.613264i
$708$		0			0
$709$		−	1.00136e13i		−	1.48827i	−0.668030	−	0.744134i	$-0.732862\pi$
$709$		−	1.00136e13i		−	1.48827i	0.668030	−	0.744134i	$-0.267138\pi$
$710$	−7.11265e12	−	9.09370e11i	−1.05043	−	0.134301i
$711$		0			0
$712$	−3.64707e12	+	9.09180e12i	−0.531844	+	1.32584i
$713$	5.98586e12			0.867409
$714$		0			0
$715$			1.34560e12i			0.192548i
$716$	−1.52060e11	+	5.84951e11i	−0.0216226	+	0.0831784i
$717$		0			0
$718$	6.39803e11	−	5.00423e12i	0.0898435	−	0.702712i
$719$	3.50640e12			0.489307			0.244654	−	0.969611i	$-0.421326\pi$
$719$	3.50640e12			0.489307			0.244654	+	0.969611i	$0.421326\pi$
$720$		0			0
$721$	4.17346e12			0.575158
$722$	−1.75007e11	+	1.36882e12i	−0.0239683	+	0.187469i
$723$		0			0
$724$	−1.71796e12	+	6.60869e12i	−0.232375	+	0.893906i
$725$			1.21737e11i			0.0163645i
$726$		0			0
$727$	−2.99547e11			−0.0397703			−0.0198852	−	0.999802i	$-0.506330\pi$
$727$	−2.99547e11			−0.0397703			−0.0198852	+	0.999802i	$0.506330\pi$
$728$	3.50594e12	+	1.40637e12i	0.462608	+	0.185570i
$729$		0			0
$730$	−1.30489e13	−	1.66834e12i	−1.70068	−	0.217436i
$731$		−	3.85325e10i		−	0.00499113i
$732$		0			0
$733$		−	6.88627e10i		−	0.00881082i	−0.999990	−	0.00440541i	$-0.998598\pi$
$733$		−	6.88627e10i		−	0.00881082i	0.999990	−	0.00440541i	$-0.00140229\pi$
$734$	8.46968e11	−	6.62457e12i	0.107705	−	0.842414i
$735$		0			0
$736$	6.93579e12	−	9.39738e12i	0.871256	−	1.18047i
$737$	2.51372e12			0.313843
$738$		0			0
$739$		−	9.78319e12i		−	1.20665i	−0.797496	−	0.603324i	$-0.793843\pi$
$739$		−	9.78319e12i		−	1.20665i	0.797496	−	0.603324i	$-0.206157\pi$
$740$	−6.14154e12	−	1.59652e12i	−0.752896	−	0.195718i
$741$		0			0
$742$	4.86762e12	+	6.22337e11i	0.589520	+	0.0753716i
$743$	1.30634e12			0.157256			0.0786279	−	0.996904i	$-0.474946\pi$
$743$	1.30634e12			0.157256			0.0786279	+	0.996904i	$0.474946\pi$
$744$		0			0
$745$	3.03804e12			0.361319
$746$	−4.04725e11	−	5.17451e10i	−0.0478449	−	0.00611709i
$747$		0			0
$748$	−4.79536e11	+	1.84469e12i	−0.0560098	+	0.215460i
$749$		−	2.99569e12i		−	0.347799i
$750$		0			0
$751$	−1.05656e13			−1.21204			−0.606019	−	0.795451i	$-0.707234\pi$
$751$	−1.05656e13			−1.21204			−0.606019	+	0.795451i	$0.707234\pi$
$752$	−1.18507e13	−	6.60781e12i	−1.35134	−	0.753490i
$753$		0			0
$754$	−3.97465e11	+	3.10878e12i	−0.0447846	+	0.350283i
$755$			6.04235e12i			0.676775i
$756$		0			0
$757$		−	1.53287e13i		−	1.69658i	−0.529535	−	0.848288i	$-0.677634\pi$
$757$		−	1.53287e13i		−	1.69658i	0.529535	−	0.848288i	$-0.322366\pi$
$758$	1.36863e13	+	1.74983e12i	1.50583	+	0.192524i
$759$		0			0
$760$	−3.12782e12	+	7.79736e12i	−0.340080	+	0.847786i
$761$	3.80613e12			0.411389			0.205695	−	0.978616i	$-0.434055\pi$
$761$	3.80613e12			0.411389			0.205695	+	0.978616i	$0.434055\pi$
$762$		0			0
$763$			4.52564e12i			0.483414i
$764$	−1.32452e13	−	3.44314e12i	−1.40649	−	0.365624i
$765$		0			0
$766$	3.82524e11	−	2.99192e12i	0.0401448	−	0.313993i
$767$	7.38513e11			0.0770511
$768$		0			0
$769$	3.96740e12			0.409107			0.204554	−	0.978855i	$-0.434426\pi$
$769$	3.96740e12			0.409107			0.204554	+	0.978855i	$0.434426\pi$
$770$	−3.06522e11	+	2.39747e12i	−0.0314234	+	0.245779i
$771$		0			0
$772$	5.61205e12	+	1.45888e12i	0.568648	+	0.147823i
$773$			7.06875e12i			0.712090i	0.934469	+	0.356045i	$0.115875\pi$
$773$			7.06875e12i			0.712090i	−0.934469	+	0.356045i	$0.884125\pi$
$774$		0			0
$775$	1.71240e11			0.0170509
$776$	−3.43839e12	+	8.57158e12i	−0.340391	+	0.848561i
$777$		0			0
$778$	−4.61354e11	−	5.89853e10i	−0.0451467	−	0.00577212i
$779$		−	7.59258e12i		−	0.738704i
$780$		0			0
$781$		−	3.31106e12i		−	0.318448i
$782$	−1.42019e12	+	1.11080e13i	−0.135805	+	1.06220i
$783$		0			0
$784$	−3.31307e12	−	1.84733e12i	−0.313191	−	0.174631i
$785$	−1.54086e13			−1.44827
$786$		0			0
$787$			8.40522e12i			0.781021i	0.920598	+	0.390511i	$0.127702\pi$
$787$			8.40522e12i			0.781021i	−0.920598	+	0.390511i	$0.872298\pi$
$788$	−5.12009e12	+	1.96961e13i	−0.473053	+	1.81975i
$789$		0			0
$790$	7.99359e11	+	1.02200e11i	0.0730163	+	0.00933532i
$791$	−1.85288e12			−0.168288
$792$		0			0
$793$	1.25488e13			1.12686
$794$	−1.33206e13	−	1.70308e12i	−1.18941	−	0.152069i
$795$		0			0
$796$	−1.02897e13	−	2.67486e12i	−0.908438	−	0.236152i
$797$			2.76521e11i			0.0242754i	0.999926	+	0.0121377i	$0.00386364\pi$
$797$			2.76521e11i			0.0242754i	−0.999926	+	0.0121377i	$0.996136\pi$
$798$		0			0
$799$	1.30093e13			1.12926
$800$	1.98415e11	−	2.68835e11i	0.0171266	−	0.0232050i
$801$		0			0
$802$	−2.88799e12	+	2.25885e13i	−0.246497	+	1.92798i
$803$		−	6.07450e12i		−	0.515574i
$804$		0			0
$805$			1.42006e13i			1.19186i
$806$	4.37292e12	+	5.59089e11i	0.364976	+	0.0466631i
$807$		0			0
$808$	8.61053e12	+	3.45402e12i	0.710688	+	0.285084i
$809$	1.10211e13			0.904601			0.452300	−	0.891866i	$-0.350604\pi$
$809$	1.10211e13			0.904601			0.452300	+	0.891866i	$0.350604\pi$
$810$		0			0
$811$			2.02649e13i			1.64494i	0.568808	+	0.822470i	$0.307405\pi$
$811$			2.02649e13i			1.64494i	−0.568808	+	0.822470i	$0.692595\pi$
$812$	−1.41633e12	+	5.44839e12i	−0.114331	+	0.439811i
$813$		0			0
$814$	3.71604e11	−	2.90651e12i	0.0296668	−	0.232039i
$815$	6.18611e12			0.491144
$816$		0			0
$817$	7.84270e10			0.00615838
$818$	9.77195e11	−	7.64314e12i	0.0763117	−	0.596873i
$819$		0			0
$820$	−2.71015e12	+	1.04255e13i	−0.209329	+	0.805255i
$821$		−	4.75674e12i		−	0.365397i	−0.983169	−	0.182699i	$-0.941517\pi$
$821$		−	4.75674e12i		−	0.365397i	0.983169	−	0.182699i	$-0.0584833\pi$
$822$		0			0
$823$	−1.58721e13			−1.20596			−0.602981	−	0.797755i	$-0.706021\pi$
$823$	−1.58721e13			−1.20596			−0.602981	+	0.797755i	$0.706021\pi$
$824$	3.53823e12	−	8.82045e12i	0.267370	−	0.666528i
$825$		0			0
$826$	1.31581e12	+	1.68230e11i	0.0983521	+	0.0125746i
$827$			1.83601e13i			1.36490i	0.730932	+	0.682451i	$0.239086\pi$
$827$			1.83601e13i			1.36490i	−0.730932	+	0.682451i	$0.760914\pi$
$828$		0			0
$829$			6.16980e12i			0.453707i	0.973929	+	0.226854i	$0.0728439\pi$
$829$			6.16980e12i			0.453707i	−0.973929	+	0.226854i	$0.927156\pi$
$830$	−1.92479e12	+	1.50548e13i	−0.140777	+	1.10109i
$831$		0			0
$832$	5.94461e12	−	6.21736e12i	0.430099	−	0.449833i
$833$	3.63698e12			0.261721
$834$		0			0
$835$		−	1.09389e12i		−	0.0778727i
$836$	−3.75459e12	−	9.76022e11i	−0.265849	−	0.0691085i
$837$		0			0
$838$	−1.90303e13	−	2.43307e12i	−1.33306	−	0.170434i
$839$	2.48397e13			1.73068			0.865342	−	0.501183i	$-0.167101\pi$
$839$	2.48397e13			1.73068			0.865342	+	0.501183i	$0.167101\pi$
$840$		0			0
$841$	9.83652e12			0.678047
$842$	−1.18305e13	−	1.51256e12i	−0.811144	−	0.103707i
$843$		0			0
$844$	6.58807e12	−	2.53432e13i	0.446907	−	1.71918i
$845$		−	9.20984e12i		−	0.621437i
$846$		0			0
$847$	1.08802e13			0.726373
$848$	5.44202e12	−	9.75993e12i	0.361392	−	0.648135i
$849$		0			0
$850$	−4.06279e10	+	3.17772e11i	−0.00266956	+	0.0208800i
$851$		−	1.72157e13i		−	1.12523i
$852$		0			0
$853$			2.52887e13i			1.63552i	0.575560	+	0.817760i	$0.304784\pi$
$853$			2.52887e13i			1.63552i	−0.575560	+	0.817760i	$0.695216\pi$
$854$	2.23582e13	+	2.85855e12i	1.43839	+	0.183901i
$855$		0			0
$856$	−6.33129e12	−	2.53972e12i	−0.403051	−	0.161679i
$857$	1.49919e13			0.949387			0.474693	−	0.880151i	$-0.342559\pi$
$857$	1.49919e13			0.949387			0.474693	+	0.880151i	$0.342559\pi$
$858$		0			0
$859$		−	1.48492e13i		−	0.930540i	−0.885169	−	0.465270i	$-0.845957\pi$
$859$		−	1.48492e13i		−	0.930540i	0.885169	−	0.465270i	$-0.154043\pi$
$860$	−1.07689e11	−	2.79943e10i	−0.00671319	−	0.00174512i
$861$		0			0
$862$	3.24033e11	−	2.53443e12i	0.0199897	−	0.156350i
$863$	2.31037e13			1.41786			0.708928	−	0.705281i	$-0.249179\pi$
$863$	2.31037e13			1.41786			0.708928	+	0.705281i	$0.249179\pi$
$864$		0			0
$865$	2.08902e12			0.126873
$866$	−2.63441e12	+	2.06051e13i	−0.159167	+	1.24493i
$867$		0			0
$868$	7.66390e12	+	1.99226e12i	0.458259	+	0.119126i
$869$			3.72116e11i			0.0221355i
$870$		0			0
$871$	1.08772e13			0.640374
$872$	9.56478e12	+	3.83681e12i	0.560210	+	0.224722i
$873$		0			0
$874$	−2.26087e13	−	2.89057e12i	−1.31061	−	0.167565i
$875$		−	1.36795e13i		−	0.788921i
$876$		0			0
$877$			1.71535e11i			0.00979164i	0.999988	+	0.00489582i	$0.00155839\pi$
$877$			1.71535e11i			0.00979164i	−0.999988	+	0.00489582i	$0.998442\pi$
$878$	2.67649e12	−	2.09342e13i	0.151999	−	1.18886i
$879$		0			0
$880$	4.80709e12	+	2.68038e12i	0.270216	+	0.150669i
$881$	−1.01574e13			−0.568058			−0.284029	−	0.958816i	$-0.591671\pi$
$881$	−1.01574e13			−0.568058			−0.284029	+	0.958816i	$0.591671\pi$
$882$		0			0
$883$		−	1.93867e13i		−	1.07320i	−0.843837	−	0.536600i	$-0.819708\pi$
$883$		−	1.93867e13i		−	1.07320i	0.843837	−	0.536600i	$-0.180292\pi$
$884$	−2.07501e12	+	7.98222e12i	−0.114284	+	0.439631i
$885$		0			0
$886$	3.55746e13	+	4.54830e12i	1.93949	+	0.247969i
$887$	−2.56537e13			−1.39153			−0.695766	−	0.718268i	$-0.744935\pi$
$887$	−2.56537e13			−1.39153			−0.695766	+	0.718268i	$0.744935\pi$
$888$		0			0
$889$	−1.96595e13			−1.05564
$890$	−2.69028e13	−	3.43959e12i	−1.43728	−	0.183760i
$891$		0			0
$892$	−2.37364e13	−	6.17039e12i	−1.25538	−	0.326340i
$893$			2.64785e13i			1.39335i
$894$		0			0
$895$	−1.67335e12			−0.0871734
$896$	1.20078e13	−	9.72335e12i	0.622413	−	0.503999i
$897$		0			0
$898$	−1.53038e12	+	1.19699e13i	−0.0785335	+	0.614251i
$899$			6.56987e12i			0.335458i
$900$		0			0
$901$			1.07141e13i			0.541621i
$902$	−4.93389e12	−	6.30810e11i	−0.248176	−	0.0317299i
$903$		0			0
$904$	−1.57086e12	+	3.91600e12i	−0.0782312	+	0.195023i
$905$	−1.89053e13			−0.936840
$906$		0			0
$907$		−	3.07269e13i		−	1.50760i	−0.657105	−	0.753799i	$-0.728219\pi$
$907$		−	3.07269e13i		−	1.50760i	0.657105	−	0.753799i	$-0.271781\pi$
$908$	5.81636e12	−	2.23746e13i	0.283965	−	1.09237i
$909$		0			0
$910$	−1.32636e12	+	1.03741e13i	−0.0641172	+	0.501494i
$911$	−2.80997e13			−1.35166			−0.675832	−	0.737056i	$-0.736215\pi$
$911$	−2.80997e13			−1.35166			−0.675832	+	0.737056i	$0.736215\pi$
$912$		0			0
$913$	−7.00825e12			−0.333804
$914$	1.99455e12	−	1.56004e13i	0.0945337	−	0.739397i
$915$		0			0
$916$	−1.68479e12	+	6.48112e12i	−0.0790710	+	0.304173i
$917$			1.55539e13i			0.726401i
$918$		0			0
$919$	1.57307e13			0.727494			0.363747	−	0.931498i	$-0.381497\pi$
$919$	1.57307e13			0.727494			0.363747	+	0.931498i	$0.381497\pi$
$920$	3.00125e13	+	1.20392e13i	1.38120	+	0.554053i
$921$		0			0
$922$	3.39476e13	+	4.34029e12i	1.54711	+	0.197802i
$923$		−	1.43274e13i		−	0.649770i
$924$		0			0
$925$		−	4.92498e11i		−	0.0221191i
$926$	−3.15702e12	+	2.46927e13i	−0.141100	+	1.10362i
$927$		0			0
$928$	1.03142e13	+	7.61247e12i	0.456532	+	0.336946i
$929$	6.66185e12			0.293443			0.146722	−	0.989178i	$-0.453128\pi$
$929$	6.66185e12			0.293443			0.146722	+	0.989178i	$0.453128\pi$
$930$		0			0
$931$			7.40252e12i			0.322928i
$932$	−3.91690e13	−	1.01822e13i	−1.70048	−	0.442047i
$933$		0			0
$934$	1.63768e13	+	2.09382e12i	0.704155	+	0.0900280i
$935$	−5.27707e12			−0.225809
$936$		0			0
$937$	2.89231e12			0.122579			0.0612897	−	0.998120i	$-0.480479\pi$
$937$	2.89231e12			0.122579			0.0612897	+	0.998120i	$0.480479\pi$
$938$	1.93799e13	+	2.47777e12i	0.817407	+	0.104507i
$939$		0			0
$940$	9.45140e12	−	3.63579e13i	0.394840	−	1.51888i
$941$		−	7.76858e12i		−	0.322990i	−0.986874	−	0.161495i	$-0.948369\pi$
$941$		−	7.76858e12i		−	0.322990i	0.986874	−	0.161495i	$-0.0516315\pi$
$942$		0			0
$943$	−2.92243e13			−1.20349
$944$	1.47108e12	−	2.63830e12i	0.0602925	−	0.108131i
$945$		0			0
$946$	6.51591e9	−	5.09643e10i	0.000264524	−	0.00206898i
$947$			9.80623e12i			0.396212i	0.980181	+	0.198106i	$0.0634790\pi$
$947$			9.80623e12i			0.396212i	−0.980181	+	0.198106i	$0.936521\pi$
$948$		0			0
$949$		−	2.62851e13i		−	1.05199i
$950$	−6.46776e11	−	8.26919e10i	−0.0257631	−	0.00329387i
$951$		0			0
$952$	−5.51537e12	+	1.37493e13i	−0.217625	+	0.542517i
$953$	−2.10091e12			−0.0825067			−0.0412534	−	0.999149i	$-0.513135\pi$
$953$	−2.10091e12			−0.0825067			−0.0412534	+	0.999149i	$0.513135\pi$
$954$		0			0
$955$		−	3.78901e13i		−	1.47405i
$956$	6.26934e12	+	1.62974e12i	0.242751	+	0.0631042i
$957$		0			0
$958$	−2.79910e12	+	2.18932e13i	−0.107368	+	0.839779i
$959$	−1.60511e13			−0.612804
$960$		0			0
$961$	−1.71982e13			−0.650471
$962$	1.60798e12	−	1.25768e13i	0.0605330	−	0.473460i
$963$		0			0
$964$	−2.90205e13	−	7.54400e12i	−1.08233	−	0.281355i
$965$			1.60543e13i			0.595960i
$966$		0			0
$967$	−3.69001e13			−1.35709			−0.678544	−	0.734560i	$-0.737389\pi$
$967$	−3.69001e13			−1.35709			−0.678544	+	0.734560i	$0.737389\pi$
$968$	9.22412e12	−	2.29948e13i	0.337665	−	0.841765i
$969$		0			0
$970$	−2.53635e13	−	3.24278e12i	−0.919890	−	0.117610i
$971$			1.52779e13i			0.551542i	0.961223	+	0.275771i	$0.0889331\pi$
$971$			1.52779e13i			0.551542i	−0.961223	+	0.275771i	$0.911067\pi$
$972$		0			0
$973$			7.43648e12i			0.265986i
$974$	−2.03475e12	+	1.59148e13i	−0.0724429	+	0.566613i
$975$		0			0
$976$	2.49966e13	−	4.48298e13i	0.881771	−	1.58140i
$977$	1.67353e13			0.587637			0.293818	−	0.955861i	$-0.405074\pi$
$977$	1.67353e13			0.587637			0.293818	+	0.955861i	$0.405074\pi$
$978$		0			0
$979$		−	1.25237e13i		−	0.435724i
$980$	2.64231e12	−	1.01645e13i	0.0915095	−	0.352021i
$981$		0			0
$982$	−1.86462e12	−	2.38397e11i	−0.0639867	−	0.00818085i
$983$	−4.18461e13			−1.42943			−0.714717	−	0.699414i	$-0.753444\pi$
$983$	−4.18461e13			−1.42943			−0.714717	+	0.699414i	$0.753444\pi$
$984$		0			0
$985$	−5.63442e13			−1.90716
$986$	−1.21918e13	−	1.55875e12i	−0.410790	−	0.0525205i
$987$		0			0
$988$	−1.62466e13	−	4.22337e12i	−0.542445	−	0.141011i
$989$		−	3.01870e11i		−	0.0100331i
$990$		0			0
$991$	4.79950e13			1.58075			0.790377	−	0.612620i	$-0.209885\pi$
$991$	4.79950e13			1.58075			0.790377	+	0.612620i	$0.209885\pi$
$992$	1.07080e13	−	1.45084e13i	0.351079	−	0.475681i
$993$		0			0
$994$	3.26371e12	−	2.55272e13i	0.106041	−	0.829400i
$995$		−	2.94356e13i		−	0.952069i
$996$		0			0
$997$			3.46659e13i			1.11115i	0.831465	+	0.555576i	$0.187502\pi$
$997$			3.46659e13i			1.11115i	−0.831465	+	0.555576i	$0.812498\pi$
$998$	3.47161e13	+	4.43854e12i	1.10776	+	0.141629i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.10.d.b.37.4		8
3.2	odd	2		8.10.b.a.5.5	✓	8
4.3	odd	2		288.10.d.b.145.2		8
8.3	odd	2		288.10.d.b.145.7		8
8.5	even	2	inner	72.10.d.b.37.3		8
12.11	even	2		32.10.b.a.17.1		8
24.5	odd	2		8.10.b.a.5.6	yes	8
24.11	even	2		32.10.b.a.17.8		8
48.5	odd	4		256.10.a.p.1.8		8
48.11	even	4		256.10.a.s.1.1		8
48.29	odd	4		256.10.a.p.1.1		8
48.35	even	4		256.10.a.s.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.b.a.5.5	✓	8	3.2	odd	2
8.10.b.a.5.6	yes	8	24.5	odd	2
32.10.b.a.17.1		8	12.11	even	2
32.10.b.a.17.8		8	24.11	even	2
72.10.d.b.37.3		8	8.5	even	2	inner
72.10.d.b.37.4		8	1.1	even	1	trivial
256.10.a.p.1.1		8	48.29	odd	4
256.10.a.p.1.8		8	48.5	odd	4
256.10.a.s.1.1		8	48.11	even	4
256.10.a.s.1.8		8	48.35	even	4
288.10.d.b.145.2		8	4.3	odd	2
288.10.d.b.145.7		8	8.3	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.86961 + 22.4447i) q^{2} +(-495.531 - 128.815i) q^{4} -1417.55i q^{5} +5087.57 q^{7} +(4313.20 - 10752.4i) q^{8} +O(q^{10})\)	\(q+(-2.86961 + 22.4447i) q^{2} +(-495.531 - 128.815i) q^{4} -1417.55i q^{5} +5087.57 q^{7} +(4313.20 - 10752.4i) q^{8} +(31816.6 + 4067.83i) q^{10} +14811.2i q^{11} +64089.8i q^{13} +(-14599.3 + 114189. i) q^{14} +(228957. + 127664. i) q^{16} -251342. q^{17} -511568. i q^{19} +(-182602. + 702441. i) q^{20} +(-332432. - 42502.3i) q^{22} -1.96905e6 q^{23} -56329.6 q^{25} +(-1.43848e6 - 183913. i) q^{26} +(-2.52105e6 - 655356. i) q^{28} -2.16116e6i q^{29} -3.03997e6 q^{31} +(-3.52240e6 + 4.77253e6i) q^{32} +(721253. - 5.64129e6i) q^{34} -7.21189e6i q^{35} +8.74315e6i q^{37} +(1.14820e7 + 1.46800e6i) q^{38} +(-1.52421e7 - 6.11419e6i) q^{40} +1.48418e7 q^{41} +153307. i q^{43} +(1.90790e6 - 7.33938e6i) q^{44} +(5.65042e6 - 4.41949e7i) q^{46} -5.17595e7 q^{47} -1.44703e7 q^{49} +(161644. - 1.26430e6i) q^{50} +(8.25574e6 - 3.17584e7i) q^{52} -4.26277e7i q^{53} +2.09956e7 q^{55} +(2.19437e7 - 5.47035e7i) q^{56} +(4.85067e7 + 6.20170e6i) q^{58} -1.15231e7i q^{59} -1.95800e8i q^{61} +(8.72353e6 - 6.82312e7i) q^{62} +(-9.70103e7 - 9.27545e7i) q^{64} +9.08506e7 q^{65} -1.69718e8i q^{67} +(1.24547e8 + 3.23766e7i) q^{68} +(1.61869e8 + 2.06953e7i) q^{70} -2.23552e8 q^{71} -4.10130e8 q^{73} +(-1.96238e8 - 2.50895e7i) q^{74} +(-6.58977e7 + 2.53497e8i) q^{76} +7.53528e7i q^{77} +2.51240e7 q^{79} +(1.80970e8 - 3.24559e8i) q^{80} +(-4.25902e7 + 3.33120e8i) q^{82} +4.73174e8i q^{83} +3.56290e8i q^{85} +(-3.44094e6 - 439932. i) q^{86} +(1.59255e8 + 6.38836e7i) q^{88} -8.45560e8 q^{89} +3.26061e8i q^{91} +(9.75726e8 + 2.53644e8i) q^{92} +(1.48530e8 - 1.16173e9i) q^{94} -7.25174e8 q^{95} -7.97179e8 q^{97} +(4.15241e7 - 3.24781e8i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 18 q^{2} - 428 q^{4} + 4800 q^{7} + 3384 q^{8}+O(q^{10}) \) 8 * q + 18 * q^2 - 428 * q^4 + 4800 * q^7 + 3384 * q^8	\( 8 q + 18 q^{2} - 428 q^{4} + 4800 q^{7} + 3384 q^{8} + 26392 q^{10} + 72336 q^{14} + 185616 q^{16} + 102000 q^{17} - 1245264 q^{20} - 2373124 q^{22} - 3412032 q^{23} - 2423384 q^{25} - 4551240 q^{26} + 7509920 q^{28} + 803584 q^{31} + 14113248 q^{32} + 27757244 q^{34} + 63661140 q^{38} - 93063648 q^{40} + 2180784 q^{41} - 114013320 q^{44} - 131840944 q^{46} - 7432320 q^{47} + 24436680 q^{49} - 231784902 q^{50} + 219270896 q^{52} + 7056832 q^{55} + 358503360 q^{56} + 375425192 q^{58} + 344291904 q^{62} - 316815296 q^{64} + 146501760 q^{65} - 79875048 q^{68} - 56202048 q^{70} - 560234688 q^{71} - 523987120 q^{73} + 65773608 q^{74} - 87532760 q^{76} - 248943744 q^{79} - 890441280 q^{80} - 1051981172 q^{82} - 1492810428 q^{86} + 1544767952 q^{88} - 744827856 q^{89} + 2959012128 q^{92} + 3068552352 q^{94} + 1465245504 q^{95} - 9932784 q^{97} + 3062604162 q^{98}+O(q^{100}) \) 8 * q + 18 * q^2 - 428 * q^4 + 4800 * q^7 + 3384 * q^8 + 26392 * q^10 + 72336 * q^14 + 185616 * q^16 + 102000 * q^17 - 1245264 * q^20 - 2373124 * q^22 - 3412032 * q^23 - 2423384 * q^25 - 4551240 * q^26 + 7509920 * q^28 + 803584 * q^31 + 14113248 * q^32 + 27757244 * q^34 + 63661140 * q^38 - 93063648 * q^40 + 2180784 * q^41 - 114013320 * q^44 - 131840944 * q^46 - 7432320 * q^47 + 24436680 * q^49 - 231784902 * q^50 + 219270896 * q^52 + 7056832 * q^55 + 358503360 * q^56 + 375425192 * q^58 + 344291904 * q^62 - 316815296 * q^64 + 146501760 * q^65 - 79875048 * q^68 - 56202048 * q^70 - 560234688 * q^71 - 523987120 * q^73 + 65773608 * q^74 - 87532760 * q^76 - 248943744 * q^79 - 890441280 * q^80 - 1051981172 * q^82 - 1492810428 * q^86 + 1544767952 * q^88 - 744827856 * q^89 + 2959012128 * q^92 + 3068552352 * q^94 + 1465245504 * q^95 - 9932784 * q^97 + 3062604162 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more