LMFDB - Embedded newform 60.6.i.a.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(17,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("60.17");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.i (of order $4$, 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:	$9.62302918878$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} + \cdots + 48\!\cdots\!00 $ x^20 - 2x^19 + 2x^18 - 382x^17 + 117610x^16 - 661518x^15 + 1160778x^14 - 161859774x^13 + 12872525097x^12 - 60314430972x^11 + 238998177000x^10 - 23647959230160x^9 + 244333990200544x^8 + 369253960052800x^7 + 17753999958800000x^6 - 195554595639827200x^5 + 1195920778734496000x^4 - 5202205084237824000x^3 + 36839424090977280000x^2 - 600638984600463360000*x + 4896482487496796160000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{14}\cdot 5^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			53.2
Root			$2.77873 - 6.78762i$ of defining polynomial
Character	$\chi$	$=$	60.53
Dual form			60.6.i.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-12.0403 - 9.90111i) q^{3} +(-51.4858 - 21.7765i) q^{5} +(56.0116 - 56.0116i) q^{7} +(46.9362 + 238.424i) q^{9} +O(q^{10})$	\(q+(-12.0403 - 9.90111i) q^{3} +(-51.4858 - 21.7765i) q^{5} +(56.0116 - 56.0116i) q^{7} +(46.9362 + 238.424i) q^{9} +616.203i q^{11} +(-323.806 - 323.806i) q^{13} +(404.292 + 771.961i) q^{15} +(1378.31 + 1378.31i) q^{17} +256.440i q^{19} +(-1228.97 + 119.818i) q^{21} +(-2641.63 + 2641.63i) q^{23} +(2176.57 + 2242.36i) q^{25} +(1795.54 - 3335.41i) q^{27} -381.009 q^{29} -7358.84 q^{31} +(6101.09 - 7419.24i) q^{33} +(-4103.54 + 1664.07i) q^{35} +(6067.48 - 6067.48i) q^{37} +(692.674 + 7104.76i) q^{39} -1462.80i q^{41} +(3450.11 + 3450.11i) q^{43} +(2775.49 - 13297.5i) q^{45} +(-6963.19 - 6963.19i) q^{47} +10532.4i q^{49} +(-2948.43 - 30242.0i) q^{51} +(-5262.26 + 5262.26i) q^{53} +(13418.7 - 31725.7i) q^{55} +(2539.04 - 3087.61i) q^{57} -33815.5 q^{59} -31177.0 q^{61} +(15983.5 + 10725.5i) q^{63} +(9620.07 + 23722.8i) q^{65} +(-15756.9 + 15756.9i) q^{67} +(57961.0 - 5650.87i) q^{69} +75916.6i q^{71} +(51080.6 + 51080.6i) q^{73} +(-4004.69 - 48549.0i) q^{75} +(34514.5 + 34514.5i) q^{77} +42068.3i q^{79} +(-54643.0 + 22381.4i) q^{81} +(-48969.9 + 48969.9i) q^{83} +(-40948.7 - 100978. i) q^{85} +(4587.46 + 3772.42i) q^{87} +21879.1 q^{89} -36273.8 q^{91} +(88602.4 + 72860.7i) q^{93} +(5584.36 - 13203.0i) q^{95} +(63524.7 - 63524.7i) q^{97} +(-146917. + 28922.2i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) $ 20 * q + 2 * q^3 + 76 * q^7	$ 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 76 * q^7 + 1068 * q^13 - 130 * q^15 + 2180 * q^21 + 4060 * q^25 + 1454 * q^27 - 4720 * q^31 - 460 * q^33 - 612 * q^37 - 24012 * q^43 - 18860 * q^45 - 31700 * q^51 + 19200 * q^55 + 33476 * q^57 + 59880 * q^61 + 67208 * q^63 - 80804 * q^67 - 56956 * q^73 - 102470 * q^75 - 9980 * q^81 + 239260 * q^85 + 71540 * q^87 + 218520 * q^91 + 307928 * q^93 - 151164 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−12.0403	−	9.90111i	−0.772384	−	0.635156i
$3$	−12.0403	−	9.90111i	−0.772384	−	0.635156i
$4$		0			0
$5$	−51.4858	−	21.7765i	−0.921006	−	0.389549i
$5$	−51.4858	−	21.7765i	−0.921006	−	0.389549i
$6$		0			0
$7$	56.0116	−	56.0116i	0.432049	−	0.432049i	−0.457276	−	0.889325i	$-0.651175\pi$
$7$	56.0116	−	56.0116i	0.432049	−	0.432049i	0.889325	+	0.457276i	$0.151175\pi$
$8$		0			0
$9$	46.9362	+	238.424i	0.193153	+	0.981169i
$10$		0			0
$11$			616.203i			1.53547i	0.640767	+	0.767736i	$0.278617\pi$
$11$			616.203i			1.53547i	−0.640767	+	0.767736i	$0.721383\pi$
$12$		0			0
$13$	−323.806	−	323.806i	−0.531407	−	0.531407i	0.389584	−	0.920991i	$-0.372619\pi$
$13$	−323.806	−	323.806i	−0.531407	−	0.531407i	−0.920991	+	0.389584i	$0.872619\pi$
$14$		0			0
$15$	404.292	+	771.961i	0.463945	+	0.885864i
$16$		0			0
$17$	1378.31	+	1378.31i	1.15671	+	1.15671i	0.985179	+	0.171532i	$0.0548717\pi$
$17$	1378.31	+	1378.31i	1.15671	+	1.15671i	0.171532	+	0.985179i	$0.445128\pi$
$18$		0			0
$19$			256.440i			0.162968i	0.996675	+	0.0814839i	$0.0259659\pi$
$19$			256.440i			0.162968i	−0.996675	+	0.0814839i	$0.974034\pi$
$20$		0			0
$21$	−1228.97	+	119.818i	−0.608126	+	0.0592889i
$22$		0			0
$23$	−2641.63	+	2641.63i	−1.04124	+	1.04124i	−0.0421311	+	0.999112i	$0.513415\pi$
$23$	−2641.63	+	2641.63i	−1.04124	+	1.04124i	−0.999112	+	0.0421311i	$0.986585\pi$
$24$		0			0
$25$	2176.57	+	2242.36i	0.696503	+	0.717554i
$26$		0			0
$27$	1795.54	−	3335.41i	0.474007	−	0.880521i
$28$		0			0
$29$	−381.009			−0.0841280			−0.0420640	−	0.999115i	$-0.513393\pi$
$29$	−381.009			−0.0841280			−0.0420640	+	0.999115i	$0.513393\pi$
$30$		0			0
$31$	−7358.84			−1.37532			−0.687662	−	0.726031i	$-0.741363\pi$
$31$	−7358.84			−1.37532			−0.687662	+	0.726031i	$0.741363\pi$
$32$		0			0
$33$	6101.09	−	7419.24i	0.975264	−	1.18597i
$34$		0			0
$35$	−4103.54	+	1664.07i	−0.566224	+	0.229615i
$36$		0			0
$37$	6067.48	−	6067.48i	0.728625	−	0.728625i	−0.241721	−	0.970346i	$-0.577712\pi$
$37$	6067.48	−	6067.48i	0.728625	−	0.728625i	0.970346	+	0.241721i	$0.0777119\pi$
$38$		0			0
$39$	692.674	+	7104.76i	0.0729235	+	0.747976i
$40$		0			0
$41$		−	1462.80i		−	0.135902i	−0.997689	−	0.0679508i	$-0.978354\pi$
$41$		−	1462.80i		−	0.135902i	0.997689	−	0.0679508i	$-0.0216461\pi$
$42$		0			0
$43$	3450.11	+	3450.11i	0.284552	+	0.284552i	0.834921	−	0.550369i	$-0.185513\pi$
$43$	3450.11	+	3450.11i	0.284552	+	0.284552i	−0.550369	+	0.834921i	$0.685513\pi$
$44$		0			0
$45$	2775.49	−	13297.5i	0.204319	−	0.978904i
$46$		0			0
$47$	−6963.19	−	6963.19i	−0.459794	−	0.459794i	0.438794	−	0.898588i	$-0.355406\pi$
$47$	−6963.19	−	6963.19i	−0.459794	−	0.459794i	−0.898588	+	0.438794i	$0.855406\pi$
$48$		0			0
$49$			10532.4i			0.626668i
$50$		0			0
$51$	−2948.43	−	30242.0i	−0.158732	−	1.62812i
$52$		0			0
$53$	−5262.26	+	5262.26i	−0.257325	+	0.257325i	−0.823965	−	0.566640i	$-0.808243\pi$
$53$	−5262.26	+	5262.26i	−0.257325	+	0.257325i	0.566640	+	0.823965i	$0.308243\pi$
$54$		0			0
$55$	13418.7	−	31725.7i	0.598142	−	1.41418i
$56$		0			0
$57$	2539.04	−	3087.61i	0.103510	−	0.125874i
$58$		0			0
$59$	−33815.5			−1.26470			−0.632348	−	0.774684i	$-0.717909\pi$
$59$	−33815.5			−1.26470			−0.632348	+	0.774684i	$0.717909\pi$
$60$		0			0
$61$	−31177.0			−1.07278			−0.536388	−	0.843971i	$-0.680212\pi$
$61$	−31177.0			−1.07278			−0.536388	+	0.843971i	$0.680212\pi$
$62$		0			0
$63$	15983.5	+	10725.5i	0.507364	+	0.340461i
$64$		0			0
$65$	9620.07	+	23722.8i	0.282420	+	0.696438i
$66$		0			0
$67$	−15756.9	+	15756.9i	−0.428830	+	0.428830i	−0.888230	−	0.459400i	$-0.848065\pi$
$67$	−15756.9	+	15756.9i	−0.428830	+	0.428830i	0.459400	+	0.888230i	$0.348065\pi$
$68$		0			0
$69$	57961.0	−	5650.87i	1.46559	−	0.142887i
$70$		0			0
$71$			75916.6i			1.78727i	0.448791	+	0.893637i	$0.351855\pi$
$71$			75916.6i			1.78727i	−0.448791	+	0.893637i	$0.648145\pi$
$72$		0			0
$73$	51080.6	+	51080.6i	1.12189	+	1.12189i	0.991458	+	0.130428i	$0.0416352\pi$
$73$	51080.6	+	51080.6i	1.12189	+	1.12189i	0.130428	+	0.991458i	$0.458365\pi$
$74$		0			0
$75$	−4004.69	−	48549.0i	−0.0822083	−	0.996615i
$76$		0			0
$77$	34514.5	+	34514.5i	0.663399	+	0.663399i
$78$		0			0
$79$			42068.3i			0.758380i	0.925319	+	0.379190i	$0.123797\pi$
$79$			42068.3i			0.758380i	−0.925319	+	0.379190i	$0.876203\pi$
$80$		0			0
$81$	−54643.0	+	22381.4i	−0.925384	+	0.379031i
$82$		0			0
$83$	−48969.9	+	48969.9i	−0.780251	+	0.780251i	−0.979873	−	0.199622i	$-0.936029\pi$
$83$	−48969.9	+	48969.9i	−0.780251	+	0.780251i	0.199622	+	0.979873i	$0.436029\pi$
$84$		0			0
$85$	−40948.7	−	100978.i	−0.614741	−	1.51593i
$86$		0			0
$87$	4587.46	+	3772.42i	0.0649791	+	0.0534345i
$88$		0			0
$89$	21879.1			0.292788			0.146394	−	0.989226i	$-0.453233\pi$
$89$	21879.1			0.292788			0.146394	+	0.989226i	$0.453233\pi$
$90$		0			0
$91$	−36273.8			−0.459187
$92$		0			0
$93$	88602.4	+	72860.7i	1.06228	+	0.873546i
$94$		0			0
$95$	5584.36	−	13203.0i	0.0634840	−	0.150094i
$96$		0			0
$97$	63524.7	−	63524.7i	0.685509	−	0.685509i	−0.275727	−	0.961236i	$-0.588919\pi$
$97$	63524.7	−	63524.7i	0.685509	−	0.685509i	0.961236	+	0.275727i	$0.0889186\pi$
$98$		0			0
$99$	−146917.	+	28922.2i	−1.50656	+	0.296581i
$100$		0			0
$101$		−	155948.i		−	1.52117i	−0.649240	−	0.760584i	$-0.724913\pi$
$101$		−	155948.i		−	1.52117i	0.649240	−	0.760584i	$-0.275087\pi$
$102$		0			0
$103$	−23215.6	−	23215.6i	−0.215619	−	0.215619i	0.591030	−	0.806649i	$-0.298721\pi$
$103$	−23215.6	−	23215.6i	−0.215619	−	0.215619i	−0.806649	+	0.591030i	$0.798721\pi$
$104$		0			0
$105$	65883.8	+	20593.7i	0.583183	+	0.182290i
$106$		0			0
$107$	−52970.3	−	52970.3i	−0.447273	−	0.447273i	0.447174	−	0.894447i	$-0.352431\pi$
$107$	−52970.3	−	52970.3i	−0.447273	−	0.447273i	−0.894447	+	0.447174i	$0.852431\pi$
$108$		0			0
$109$			164337.i			1.32485i	0.749127	+	0.662427i	$0.230473\pi$
$109$			164337.i			1.32485i	−0.749127	+	0.662427i	$0.769527\pi$
$110$		0			0
$111$	−133129.	+	12979.3i	−1.02557	+	0.0999872i
$112$		0			0
$113$	83175.2	−	83175.2i	0.612770	−	0.612770i	−0.330897	−	0.943667i	$-0.607351\pi$
$113$	83175.2	−	83175.2i	0.612770	−	0.612770i	0.943667	+	0.330897i	$0.107351\pi$
$114$		0			0
$115$	193532.	−	78481.0i	1.36461	−	0.553375i
$116$		0			0
$117$	62005.0	−	92401.5i	0.418757	−	0.624043i
$118$		0			0
$119$	154403.			0.999511
$120$		0			0
$121$	−218655.			−1.35767
$122$		0			0
$123$	−14483.3	+	17612.5i	−0.0863188	+	0.104968i
$124$		0			0
$125$	−63231.9	−	162848.i	−0.361960	−	0.932193i
$126$		0			0
$127$	24604.9	−	24604.9i	0.135367	−	0.135367i	−0.636177	−	0.771543i	$-0.719485\pi$
$127$	24604.9	−	24604.9i	0.135367	−	0.135367i	0.771543	+	0.636177i	$0.219485\pi$
$128$		0			0
$129$	−7380.34	−	75700.2i	−0.0390483	−	0.400518i
$130$		0			0
$131$		−	212325.i		−	1.08099i	−0.841347	−	0.540496i	$-0.818237\pi$
$131$		−	212325.i		−	1.08099i	0.841347	−	0.540496i	$-0.181763\pi$
$132$		0			0
$133$	14363.6	+	14363.6i	0.0704101	+	0.0704101i
$134$		0			0
$135$	−165078.	+	132626.i	−0.779570	+	0.626315i
$136$		0			0
$137$	−26773.9	−	26773.9i	−0.121874	−	0.121874i	0.643539	−	0.765413i	$-0.277465\pi$
$137$	−26773.9	−	26773.9i	−0.121874	−	0.121874i	−0.765413	+	0.643539i	$0.777465\pi$
$138$		0			0
$139$			180163.i			0.790913i	0.918485	+	0.395457i	$0.129414\pi$
$139$			180163.i			0.790913i	−0.918485	+	0.395457i	$0.870586\pi$
$140$		0			0
$141$	14895.4	+	152782.i	0.0630963	+	0.647179i
$142$		0			0
$143$	199530.	−	199530.i	0.815960	−	0.815960i
$144$		0			0
$145$	19616.6	+	8297.04i	0.0774824	+	0.0327720i
$146$		0			0
$147$	104282.	−	126813.i	0.398032	−	0.484028i
$148$		0			0
$149$	−300303.			−1.10814			−0.554070	−	0.832470i	$-0.686926\pi$
$149$	−300303.			−1.10814			−0.554070	+	0.832470i	$0.686926\pi$
$150$		0			0
$151$	−11066.3			−0.0394968			−0.0197484	−	0.999805i	$-0.506287\pi$
$151$	−11066.3			−0.0394968			−0.0197484	+	0.999805i	$0.506287\pi$
$152$		0			0
$153$	−263930.	+	393315.i	−0.911506	+	1.35835i
$154$		0			0
$155$	378876.	+	160250.i	1.26668	+	0.535757i
$156$		0			0
$157$	44512.4	−	44512.4i	0.144123	−	0.144123i	−0.631364	−	0.775487i	$-0.717505\pi$
$157$	44512.4	−	44512.4i	0.144123	−	0.144123i	0.775487	+	0.631364i	$0.217505\pi$
$158$		0			0
$159$	115461.	−	11256.8i	0.362195	−	0.0353120i
$160$		0			0
$161$			295924.i			0.899736i
$162$		0			0
$163$	−264018.	−	264018.i	−0.778330	−	0.778330i	0.201216	−	0.979547i	$-0.435511\pi$
$163$	−264018.	−	264018.i	−0.778330	−	0.778330i	−0.979547	+	0.201216i	$0.935511\pi$
$164$		0			0
$165$	−475684.	+	249125.i	−1.36022	+	0.712374i
$166$		0			0
$167$	303339.	+	303339.i	0.841660	+	0.841660i	0.989075	−	0.147414i	$-0.0470951\pi$
$167$	303339.	+	303339.i	0.841660	+	0.841660i	−0.147414	+	0.989075i	$0.547095\pi$
$168$		0			0
$169$		−	161592.i		−	0.435213i
$170$		0			0
$171$	−61141.5	+	12036.3i	−0.159899	+	0.0314777i
$172$		0			0
$173$	−125811.	+	125811.i	−0.319596	+	0.319596i	−0.848612	−	0.529016i	$-0.822561\pi$
$173$	−125811.	+	125811.i	−0.319596	+	0.319596i	0.529016	+	0.848612i	$0.322561\pi$
$174$		0			0
$175$	247511.	+	3684.74i	0.610942	+	0.00909519i
$176$		0			0
$177$	407148.	+	334811.i	0.976831	+	0.803280i
$178$		0			0
$179$	444862.			1.03775			0.518875	−	0.854850i	$-0.326351\pi$
$179$	444862.			1.03775			0.518875	+	0.854850i	$0.326351\pi$
$180$		0			0
$181$	741760.			1.68293			0.841467	−	0.540308i	$-0.181692\pi$
$181$	741760.			1.68293			0.841467	+	0.540308i	$0.181692\pi$
$182$		0			0
$183$	375379.	+	308686.i	0.828595	+	0.681381i
$184$		0			0
$185$	−444517.	+	180261.i	−0.954903	+	0.387232i
$186$		0			0
$187$	−849318.	+	849318.i	−1.77610	+	1.77610i
$188$		0			0
$189$	−86250.6	−	287392.i	−0.175634	−	0.585222i
$190$		0			0
$191$			381675.i			0.757025i	0.925596	+	0.378513i	$0.123564\pi$
$191$			381675.i			0.757025i	−0.925596	+	0.378513i	$0.876436\pi$
$192$		0			0
$193$	−351676.	−	351676.i	−0.679594	−	0.679594i	0.280314	−	0.959908i	$-0.409561\pi$
$193$	−351676.	−	351676.i	−0.679594	−	0.679594i	−0.959908	+	0.280314i	$0.909561\pi$
$194$		0			0
$195$	119054.	−	380878.i	0.224211	−	0.717298i
$196$		0			0
$197$	−766300.	−	766300.i	−1.40680	−	1.40680i	−0.775694	−	0.631109i	$-0.782600\pi$
$197$	−766300.	−	766300.i	−1.40680	−	1.40680i	−0.631109	−	0.775694i	$-0.717400\pi$
$198$		0			0
$199$		−	536580.i		−	0.960509i	−0.877129	−	0.480255i	$-0.840544\pi$
$199$		−	536580.i		−	0.960509i	0.877129	−	0.480255i	$-0.159456\pi$
$200$		0			0
$201$	345729.	−	33706.7i	0.603595	−	0.0588472i
$202$		0			0
$203$	−21340.9	+	21340.9i	−0.0363474	+	0.0363474i
$204$		0			0
$205$	−31854.6	+	75313.3i	−0.0529404	+	0.125166i
$206$		0			0
$207$	−753816.	−	505840.i	−1.22275	−	0.820516i
$208$		0			0
$209$	−158019.			−0.250232
$210$		0			0
$211$	1.10664e6			1.71120			0.855602	−	0.517634i	$-0.173187\pi$
$211$	1.10664e6			1.71120			0.855602	+	0.517634i	$0.173187\pi$
$212$		0			0
$213$	751659.	−	914057.i	1.13520	−	1.38046i
$214$		0			0
$215$	−102500.	−	252763.i	−0.151227	−	0.372921i
$216$		0			0
$217$	−412180.	+	412180.i	−0.594207	+	0.594207i
$218$		0			0
$219$	−109270.	−	1.12078e6i	−0.153953	−	1.57910i
$220$		0			0
$221$		−	892612.i		−	1.22937i
$222$		0			0
$223$	−501399.	−	501399.i	−0.675183	−	0.675183i	0.283723	−	0.958906i	$-0.408430\pi$
$223$	−501399.	−	501399.i	−0.675183	−	0.675183i	−0.958906	+	0.283723i	$0.908430\pi$
$224$		0			0
$225$	−432472.	+	624194.i	−0.569510	+	0.821984i
$226$		0			0
$227$	686298.	+	686298.i	0.883992	+	0.883992i	0.993938	−	0.109946i	$-0.0350677\pi$
$227$	686298.	+	686298.i	0.883992	+	0.883992i	−0.109946	+	0.993938i	$0.535068\pi$
$228$		0			0
$229$			1.02915e6i			1.29685i	0.761279	+	0.648424i	$0.224572\pi$
$229$			1.02915e6i			1.29685i	−0.761279	+	0.648424i	$0.775428\pi$
$230$		0			0
$231$	−73832.1	−	757295.i	−0.0910364	−	0.933760i
$232$		0			0
$233$	−768347.	+	768347.i	−0.927187	+	0.927187i	−0.997523	−	0.0703361i	$-0.977593\pi$
$233$	−768347.	+	768347.i	−0.927187	+	0.927187i	0.0703361	+	0.997523i	$0.477593\pi$
$234$		0			0
$235$	206872.	+	510139.i	0.244361	+	0.602586i
$236$		0			0
$237$	416522.	−	506513.i	0.481690	−	0.585760i
$238$		0			0
$239$	−518074.			−0.586675			−0.293337	−	0.956009i	$-0.594766\pi$
$239$	−518074.			−0.586675			−0.293337	+	0.956009i	$0.594766\pi$
$240$		0			0
$241$	1.07418e6			1.19133			0.595666	−	0.803232i	$-0.296888\pi$
$241$	1.07418e6			1.19133			0.595666	+	0.803232i	$0.296888\pi$
$242$		0			0
$243$	879517.	+	271548.i	0.955495	+	0.295006i
$244$		0			0
$245$	229358.	−	542269.i	0.244118	−	0.577164i
$246$		0			0
$247$	83037.0	−	83037.0i	0.0866022	−	0.0866022i
$248$		0			0
$249$	1.07447e6	−	104755.i	1.09823	−	0.107072i
$250$		0			0
$251$			914663.i			0.916382i	0.888854	+	0.458191i	$0.151503\pi$
$251$			914663.i			0.916382i	−0.888854	+	0.458191i	$0.848497\pi$
$252$		0			0
$253$	−1.62778e6	−	1.62778e6i	−1.59880	−	1.59880i
$254$		0			0
$255$	−506762.	+	1.62124e6i	−0.488038	+	1.56134i
$256$		0			0
$257$	101079.	+	101079.i	0.0954619	+	0.0954619i	0.753225	−	0.657763i	$-0.228497\pi$
$257$	101079.	+	101079.i	0.0954619	+	0.0954619i	−0.657763	+	0.753225i	$0.728497\pi$
$258$		0			0
$259$		−	679698.i		−	0.629603i
$260$		0			0
$261$	−17883.1	−	90841.8i	−0.0162496	−	0.0825438i
$262$		0			0
$263$	−8443.27	+	8443.27i	−0.00752699	+	0.00752699i	−0.710860	−	0.703333i	$-0.751694\pi$
$263$	−8443.27	+	8443.27i	−0.00752699	+	0.00752699i	0.703333	+	0.710860i	$0.251694\pi$
$264$		0			0
$265$	385525.	−	156338.i	0.337239	−	0.136757i
$266$		0			0
$267$	−263430.	−	216627.i	−0.226145	−	0.185966i
$268$		0			0
$269$	−630649.			−0.531382			−0.265691	−	0.964058i	$-0.585600\pi$
$269$	−630649.			−0.531382			−0.265691	+	0.964058i	$0.585600\pi$
$270$		0			0
$271$	−458865.			−0.379544			−0.189772	−	0.981828i	$-0.560775\pi$
$271$	−458865.			−0.379544			−0.189772	+	0.981828i	$0.560775\pi$
$272$		0			0
$273$	436747.	+	359151.i	0.354669	+	0.291656i
$274$		0			0
$275$	−1.38175e6	+	1.34121e6i	−1.10178	+	1.06946i
$276$		0			0
$277$	358085.	−	358085.i	0.280405	−	0.280405i	−0.552865	−	0.833271i	$-0.686466\pi$
$277$	358085.	−	358085.i	0.280405	−	0.280405i	0.833271	+	0.552865i	$0.186466\pi$
$278$		0			0
$279$	−345396.	−	1.75452e6i	−0.265648	−	1.34943i
$280$		0			0
$281$			2.32732e6i			1.75829i	0.476556	+	0.879144i	$0.341885\pi$
$281$			2.32732e6i			1.75829i	−0.476556	+	0.879144i	$0.658115\pi$
$282$		0			0
$283$	−12977.6	−	12977.6i	−0.00963229	−	0.00963229i	0.702274	−	0.711907i	$-0.252168\pi$
$283$	−12977.6	−	12977.6i	−0.00963229	−	0.00963229i	−0.711907	+	0.702274i	$0.752168\pi$
$284$		0			0
$285$	−197962.	+	103677.i	−0.144367	+	0.0756081i
$286$		0			0
$287$	−81933.6	−	81933.6i	−0.0587161	−	0.0587161i
$288$		0			0
$289$			2.37962e6i			1.67596i
$290$		0			0
$291$	−1.39382e6	+	135890.i	−0.964881	+	0.0940705i
$292$		0			0
$293$	779591.	−	779591.i	0.530515	−	0.530515i	−0.390211	−	0.920726i	$-0.627598\pi$
$293$	779591.	−	779591.i	0.530515	−	0.530515i	0.920726	+	0.390211i	$0.127598\pi$
$294$		0			0
$295$	1.74102e6	+	736383.i	1.16479	+	0.492661i
$296$		0			0
$297$	2.05529e6	+	1.10641e6i	1.35201	+	0.727825i
$298$		0			0
$299$	1.71075e6			1.10665
$300$		0			0
$301$	386492.			0.245881
$302$		0			0
$303$	−1.54406e6	+	1.87766e6i	−0.966179	+	1.17492i
$304$		0			0
$305$	1.60517e6	+	678924.i	0.988033	+	0.417899i
$306$		0			0
$307$	−1.17490e6	+	1.17490e6i	−0.711470	+	0.711470i	−0.966843	−	0.255373i	$-0.917802\pi$
$307$	−1.17490e6	+	1.17490e6i	−0.711470	+	0.711470i	0.255373	+	0.966843i	$0.417802\pi$
$308$		0			0
$309$	49662.0	+	509383.i	0.0295888	+	0.303493i
$310$		0			0
$311$		−	1.66861e6i		−	0.978258i	−0.872211	−	0.489129i	$-0.837315\pi$
$311$		−	1.66861e6i		−	0.978258i	0.872211	−	0.489129i	$-0.162685\pi$
$312$		0			0
$313$	−1.86555e6	−	1.86555e6i	−1.07633	−	1.07633i	−0.996835	−	0.0794979i	$-0.974668\pi$
$313$	−1.86555e6	−	1.86555e6i	−1.07633	−	1.07633i	−0.0794979	−	0.996835i	$-0.525332\pi$
$314$		0			0
$315$	−589357.	−	900276.i	−0.334659	−	0.511210i
$316$		0			0
$317$	449760.	+	449760.i	0.251381	+	0.251381i	0.821537	−	0.570156i	$-0.193117\pi$
$317$	449760.	+	449760.i	0.251381	+	0.251381i	−0.570156	+	0.821537i	$0.693117\pi$
$318$		0			0
$319$		−	234779.i		−	0.129176i
$320$		0			0
$321$	113312.	+	1.16224e6i	0.0613781	+	0.629555i
$322$		0			0
$323$	−353454.	+	353454.i	−0.188507	+	0.188507i
$324$		0			0
$325$	21301.7	−	1.43088e6i	0.0111868	−	0.751440i
$326$		0			0
$327$	1.62711e6	−	1.97866e6i	0.841489	−	1.02330i
$328$		0			0
$329$	−780039.			−0.397307
$330$		0			0
$331$	−918132.			−0.460612			−0.230306	−	0.973118i	$-0.573973\pi$
$331$	−918132.			−0.460612			−0.230306	+	0.973118i	$0.573973\pi$
$332$		0			0
$333$	1.73142e6	+	1.16185e6i	0.855640	+	0.574168i
$334$		0			0
$335$	1.15439e6	−	468128.i	0.562005	−	0.227904i
$336$		0			0
$337$	−284329.	+	284329.i	−0.136379	+	0.136379i	−0.772001	−	0.635622i	$-0.780744\pi$
$337$	−284329.	+	284329.i	−0.136379	+	0.136379i	0.635622	+	0.772001i	$0.280744\pi$
$338$		0			0
$339$	−1.82498e6	+	177925.i	−0.862499	+	0.0840888i
$340$		0			0
$341$		−	4.53454e6i		−	2.11177i
$342$		0			0
$343$	1.53132e6	+	1.53132e6i	0.702800	+	0.702800i
$344$		0			0
$345$	−3.10722e6	−	971245.i	−1.40548	−	0.439320i
$346$		0			0
$347$	−2.36134e6	−	2.36134e6i	−1.05277	−	1.05277i	−0.998528	−	0.0542447i	$-0.982725\pi$
$347$	−2.36134e6	−	2.36134e6i	−1.05277	−	1.05277i	−0.0542447	−	0.998528i	$-0.517275\pi$
$348$		0			0
$349$			2.67825e6i			1.17703i	0.808487	+	0.588514i	$0.200287\pi$
$349$			2.67825e6i			1.17703i	−0.808487	+	0.588514i	$0.799713\pi$
$350$		0			0
$351$	−1.66143e6	+	498620.i	−0.719806	+	0.216024i
$352$		0			0
$353$	2.48203e6	−	2.48203e6i	1.06016	−	1.06016i	0.0620850	−	0.998071i	$-0.480225\pi$
$353$	2.48203e6	−	2.48203e6i	1.06016	−	1.06016i	0.998071	−	0.0620850i	$-0.0197750\pi$
$354$		0			0
$355$	1.65320e6	−	3.90863e6i	0.696231	−	1.64609i
$356$		0			0
$357$	−1.85905e6	−	1.52876e6i	−0.772006	−	0.634846i
$358$		0			0
$359$	−1.03055e6			−0.422020			−0.211010	−	0.977484i	$-0.567675\pi$
$359$	−1.03055e6			−0.422020			−0.211010	+	0.977484i	$0.567675\pi$
$360$		0			0
$361$	2.41034e6			0.973441
$362$		0			0
$363$	2.63266e6	+	2.16492e6i	1.04864	+	0.862334i
$364$		0			0
$365$	−1.51757e6	−	3.74228e6i	−0.596233	−	1.47029i
$366$		0			0
$367$	−781513.	+	781513.i	−0.302880	+	0.302880i	−0.842140	−	0.539259i	$-0.818704\pi$
$367$	−781513.	+	781513.i	−0.302880	+	0.302880i	0.539259	+	0.842140i	$0.318704\pi$
$368$		0			0
$369$	348766.	−	68658.1i	0.133342	−	0.0262498i
$370$		0			0
$371$			589495.i			0.222354i
$372$		0			0
$373$	1.19110e6	+	1.19110e6i	0.443279	+	0.443279i	0.893112	−	0.449833i	$-0.148517\pi$
$373$	1.19110e6	+	1.19110e6i	0.443279	+	0.443279i	−0.449833	+	0.893112i	$0.648517\pi$
$374$		0			0
$375$	−851042.	+	2.58679e6i	−0.312516	+	0.949912i
$376$		0			0
$377$	123373.	+	123373.i	0.0447062	+	0.0447062i
$378$		0			0
$379$		−	2.99522e6i		−	1.07110i	−0.844503	−	0.535551i	$-0.820104\pi$
$379$		−	2.99522e6i		−	1.07110i	0.844503	−	0.535551i	$-0.179896\pi$
$380$		0			0
$381$	−539866.	+	52633.9i	−0.190534	+	0.0185760i
$382$		0			0
$383$	−72587.9	+	72587.9i	−0.0252853	+	0.0252853i	−0.719636	−	0.694351i	$-0.755691\pi$
$383$	−72587.9	+	72587.9i	−0.0252853	+	0.0252853i	0.694351	+	0.719636i	$0.255691\pi$
$384$		0			0
$385$	−1.02540e6	−	2.52861e6i	−0.352567	−	0.869420i
$386$		0			0
$387$	−660654.	+	984524.i	−0.224232	+	0.334156i
$388$		0			0
$389$	5.47068e6			1.83302			0.916510	−	0.400011i	$-0.130994\pi$
$389$	5.47068e6			1.83302			0.916510	+	0.400011i	$0.130994\pi$
$390$		0			0
$391$	−7.28197e6			−2.40883
$392$		0			0
$393$	−2.10225e6	+	2.55645e6i	−0.686599	+	0.834941i
$394$		0			0
$395$	916098.	−	2.16592e6i	0.295426	−	0.698472i
$396$		0			0
$397$	957388.	−	957388.i	0.304868	−	0.304868i	−0.538047	−	0.842915i	$-0.680838\pi$
$397$	957388.	−	957388.i	0.304868	−	0.304868i	0.842915	+	0.538047i	$0.180838\pi$
$398$		0			0
$399$	−30726.1	−	315158.i	−0.00966218	−	0.0991050i
$400$		0			0
$401$			1.26667e6i			0.393370i	0.980467	+	0.196685i	$0.0630177\pi$
$401$			1.26667e6i			0.393370i	−0.980467	+	0.196685i	$0.936982\pi$
$402$		0			0
$403$	2.38284e6	+	2.38284e6i	0.730857	+	0.730857i
$404$		0			0
$405$	3.30073e6	+	37606.9i	0.999935	+	0.0113928i
$406$		0			0
$407$	3.73880e6	+	3.73880e6i	1.11878	+	1.11878i
$408$		0			0
$409$			4.60839e6i			1.36220i	0.732190	+	0.681101i	$0.238498\pi$
$409$			4.60839e6i			1.36220i	−0.732190	+	0.681101i	$0.761502\pi$
$410$		0			0
$411$	57273.8	+	587457.i	0.0167244	+	0.171542i
$412$		0			0
$413$	−1.89406e6	+	1.89406e6i	−0.546410	+	0.546410i
$414$		0			0
$415$	3.58765e6	−	1.45486e6i	1.02256	−	0.414669i
$416$		0			0
$417$	1.78381e6	−	2.16921e6i	0.502353	−	0.610888i
$418$		0			0
$419$	−2.24477e6			−0.624651			−0.312325	−	0.949975i	$-0.601108\pi$
$419$	−2.24477e6			−0.624651			−0.312325	+	0.949975i	$0.601108\pi$
$420$		0			0
$421$	−688460.			−0.189310			−0.0946550	−	0.995510i	$-0.530175\pi$
$421$	−688460.			−0.189310			−0.0946550	+	0.995510i	$0.530175\pi$
$422$		0			0
$423$	1.33337e6	−	1.98702e6i	0.362325	−	0.539947i
$424$		0			0
$425$	−90672.6	+	6.09065e6i	−0.0243503	+	1.63565i
$426$		0			0
$427$	−1.74627e6	+	1.74627e6i	−0.463492	+	0.463492i
$428$		0			0
$429$	−4.37797e6	+	426828.i	−1.14850	+	0.111972i
$430$		0			0
$431$		−	937065.i		−	0.242983i	−0.992592	−	0.121492i	$-0.961232\pi$
$431$		−	937065.i		−	0.242983i	0.992592	−	0.121492i	$-0.0387678\pi$
$432$		0			0
$433$	−1.41482e6	−	1.41482e6i	−0.362645	−	0.362645i	0.502141	−	0.864786i	$-0.332546\pi$
$433$	−1.41482e6	−	1.41482e6i	−0.362645	−	0.362645i	−0.864786	+	0.502141i	$0.832546\pi$
$434$		0			0
$435$	−154039.	−	294124.i	−0.0390308	−	0.0745260i
$436$		0			0
$437$	−677419.	−	677419.i	−0.169689	−	0.169689i
$438$		0			0
$439$			4.73461e6i			1.17253i	0.810120	+	0.586264i	$0.199402\pi$
$439$			4.73461e6i			1.17253i	−0.810120	+	0.586264i	$0.800598\pi$
$440$		0			0
$441$	−2.51118e6	+	494350.i	−0.614867	+	0.121043i
$442$		0			0
$443$	−1.73457e6	+	1.73457e6i	−0.419934	+	0.419934i	−0.885181	−	0.465247i	$-0.845966\pi$
$443$	−1.73457e6	+	1.73457e6i	−0.419934	+	0.419934i	0.465247	+	0.885181i	$0.345966\pi$
$444$		0			0
$445$	−1.12646e6	−	476449.i	−0.269660	−	0.114055i
$446$		0			0
$447$	3.61573e6	+	2.97334e6i	0.855909	+	0.703842i
$448$		0			0
$449$	−4.89094e6			−1.14492			−0.572462	−	0.819931i	$-0.694012\pi$
$449$	−4.89094e6			−1.14492			−0.572462	+	0.819931i	$0.694012\pi$
$450$		0			0
$451$	901379.			0.208673
$452$		0			0
$453$	133242.	+	109569.i	0.0305067	+	0.0250866i
$454$		0			0
$455$	1.86759e6	+	789916.i	0.422914	+	0.178876i
$456$		0			0
$457$	5.42002e6	−	5.42002e6i	1.21398	−	1.21398i	0.244268	−	0.969708i	$-0.421452\pi$
$457$	5.42002e6	−	5.42002e6i	1.21398	−	1.21398i	0.969708	−	0.244268i	$-0.0785477\pi$
$458$		0			0
$459$	7.07204e6	−	2.12242e6i	1.56680	−	0.470218i
$460$		0			0
$461$			5.42757e6i			1.18947i	0.803922	+	0.594735i	$0.202743\pi$
$461$			5.42757e6i			1.18947i	−0.803922	+	0.594735i	$0.797257\pi$
$462$		0			0
$463$	−2.02984e6	−	2.02984e6i	−0.440058	−	0.440058i	0.451974	−	0.892031i	$-0.350720\pi$
$463$	−2.02984e6	−	2.02984e6i	−0.440058	−	0.440058i	−0.892031	+	0.451974i	$0.850720\pi$
$464$		0			0
$465$	−2.97512e6	−	5.68074e6i	−0.638075	−	1.21835i
$466$		0			0
$467$	−2.16522e6	−	2.16522e6i	−0.459421	−	0.459421i	0.439044	−	0.898465i	$-0.355317\pi$
$467$	−2.16522e6	−	2.16522e6i	−0.459421	−	0.459421i	−0.898465	+	0.439044i	$0.855317\pi$
$468$		0			0
$469$			1.76514e6i			0.370551i
$470$		0			0
$471$	−976663.	+	95219.2i	−0.202858	+	0.0197775i
$472$		0			0
$473$	−2.12597e6	+	2.12597e6i	−0.436922	+	0.436922i
$474$		0			0
$475$	−575030.	+	558160.i	−0.116938	+	0.113508i
$476$		0			0
$477$	−1.50164e6	−	1.00766e6i	−0.302183	−	0.202776i
$478$		0			0
$479$	5.20850e6			1.03723			0.518614	−	0.855009i	$-0.326448\pi$
$479$	5.20850e6			1.03723			0.518614	+	0.855009i	$0.326448\pi$
$480$		0			0
$481$	−3.92938e6			−0.774392
$482$		0			0
$483$	2.92997e6	−	3.56300e6i	0.571473	−	0.694941i
$484$		0			0
$485$	−4.65396e6	+	1.88727e6i	−0.898397	+	0.364318i
$486$		0			0
$487$	446510.	−	446510.i	0.0853118	−	0.0853118i	−0.663163	−	0.748475i	$-0.730786\pi$
$487$	446510.	−	446510.i	0.0853118	−	0.0853118i	0.748475	+	0.663163i	$0.230786\pi$
$488$		0			0
$489$	564777.	+	5.79291e6i	0.106808	+	1.09553i
$490$		0			0
$491$		−	303583.i		−	0.0568295i	−0.999596	−	0.0284148i	$-0.990954\pi$
$491$		−	303583.i		−	0.0568295i	0.999596	−	0.0284148i	$-0.00904592\pi$
$492$		0			0
$493$	−525149.	−	525149.i	−0.0973118	−	0.0973118i
$494$		0			0
$495$	8.19398e6	+	1.71026e6i	1.50308	+	0.313725i
$496$		0			0
$497$	4.25221e6	+	4.25221e6i	0.772190	+	0.772190i
$498$		0			0
$499$		−	6.40003e6i		−	1.15062i	−0.817937	−	0.575308i	$-0.804882\pi$
$499$		−	6.40003e6i		−	1.15062i	0.817937	−	0.575308i	$-0.195118\pi$
$500$		0			0
$501$	−648891.	−	6.65567e6i	−0.115499	−	1.18467i
$502$		0			0
$503$	5.22988e6	−	5.22988e6i	0.921662	−	0.921662i	−0.0754852	−	0.997147i	$-0.524051\pi$
$503$	5.22988e6	−	5.22988e6i	0.921662	−	0.921662i	0.997147	+	0.0754852i	$0.0240505\pi$
$504$		0			0
$505$	−3.39600e6	+	8.02912e6i	−0.592570	+	1.40100i
$506$		0			0
$507$	−1.59994e6	+	1.94561e6i	−0.276429	+	0.336152i
$508$		0			0
$509$	−7.48590e6			−1.28071			−0.640353	−	0.768080i	$-0.721212\pi$
$509$	−7.48590e6			−1.28071			−0.640353	+	0.768080i	$0.721212\pi$
$510$		0			0
$511$	5.72221e6			0.969419
$512$		0			0
$513$	855333.	+	460448.i	0.143497	+	0.0772480i
$514$		0			0
$515$	689721.	+	1.70083e6i	0.114592	+	0.282581i
$516$		0			0
$517$	4.29074e6	−	4.29074e6i	0.706001	−	0.706001i
$518$		0			0
$519$	2.76046e6	−	269129.i	0.449845	−	0.0438573i
$520$		0			0
$521$		−	1.59522e6i		−	0.257470i	−0.991679	−	0.128735i	$-0.958908\pi$
$521$		−	1.59522e6i		−	0.257470i	0.991679	−	0.128735i	$-0.0410917\pi$
$522$		0			0
$523$	1.71677e6	+	1.71677e6i	0.274447	+	0.274447i	0.830888	−	0.556440i	$-0.187833\pi$
$523$	1.71677e6	+	1.71677e6i	0.274447	+	0.274447i	−0.556440	+	0.830888i	$0.687833\pi$
$524$		0			0
$525$	−2.94362e6	−	2.49500e6i	−0.466104	−	0.395068i
$526$		0			0
$527$	−1.01428e7	−	1.01428e7i	−1.59085	−	1.59085i
$528$		0			0
$529$		−	7.52006e6i		−	1.16837i
$530$		0			0
$531$	−1.58717e6	−	8.06243e6i	−0.244280	−	1.24088i
$532$		0			0
$533$	−473663.	+	473663.i	−0.0722190	+	0.0722190i
$534$		0			0
$535$	1.57371e6	+	3.88072e6i	0.237706	+	0.586176i
$536$		0			0
$537$	−5.35626e6	−	4.40462e6i	−0.801541	−	0.659133i
$538$		0			0
$539$	−6.49009e6			−0.962230
$540$		0			0
$541$	−1.05415e6			−0.154849			−0.0774245	−	0.996998i	$-0.524670\pi$
$541$	−1.05415e6			−0.154849			−0.0774245	+	0.996998i	$0.524670\pi$
$542$		0			0
$543$	−8.93099e6	−	7.34425e6i	−1.29987	−	1.06893i
$544$		0			0
$545$	3.57867e6	−	8.46100e6i	0.516096	−	1.22020i
$546$		0			0
$547$	−7.50469e6	+	7.50469e6i	−1.07242	+	1.07242i	−0.0752551	+	0.997164i	$0.523977\pi$
$547$	−7.50469e6	+	7.50469e6i	−1.07242	+	1.07242i	−0.997164	+	0.0752551i	$0.976023\pi$
$548$		0			0
$549$	−1.46333e6	−	7.43333e6i	−0.207210	−	1.05257i
$550$		0			0
$551$		−	97706.1i		−	0.0137102i
$552$		0			0
$553$	2.35631e6	+	2.35631e6i	0.327657	+	0.327657i
$554$		0			0
$555$	7.13689e6	+	2.23082e6i	0.983504	+	0.307421i
$556$		0			0
$557$	4.27281e6	+	4.27281e6i	0.583546	+	0.583546i	0.935876	−	0.352330i	$-0.114610\pi$
$557$	4.27281e6	+	4.27281e6i	0.583546	+	0.583546i	−0.352330	+	0.935876i	$0.614610\pi$
$558$		0			0
$559$		−	2.23434e6i		−	0.302426i
$560$		0			0
$561$	1.86352e7	−	1.81683e6i	2.49993	−	0.243729i
$562$		0			0
$563$	−6.78859e6	+	6.78859e6i	−0.902628	+	0.902628i	−0.995663	−	0.0930348i	$-0.970343\pi$
$563$	−6.78859e6	+	6.78859e6i	−0.902628	+	0.902628i	0.0930348	+	0.995663i	$0.470343\pi$
$564$		0			0
$565$	−6.09360e6	+	2.47108e6i	−0.803069	+	0.325661i
$566$		0			0
$567$	−1.80702e6	+	4.31426e6i	−0.236051	+	0.563571i
$568$		0			0
$569$	8.49421e6			1.09987			0.549936	−	0.835207i	$-0.314652\pi$
$569$	8.49421e6			1.09987			0.549936	+	0.835207i	$0.314652\pi$
$570$		0			0
$571$	1.25808e7			1.61480			0.807398	−	0.590007i	$-0.200875\pi$
$571$	1.25808e7			1.61480			0.807398	+	0.590007i	$0.200875\pi$
$572$		0			0
$573$	3.77901e6	−	4.59547e6i	0.480829	−	0.584714i
$574$		0			0
$575$	−1.16732e7	−	173780.i	−1.47238	−	0.0219195i
$576$		0			0
$577$	−4.93393e6	+	4.93393e6i	−0.616955	+	0.616955i	−0.944749	−	0.327794i	$-0.893695\pi$
$577$	−4.93393e6	+	4.93393e6i	−0.616955	+	0.616955i	0.327794	+	0.944749i	$0.393695\pi$
$578$		0			0
$579$	752292.	+	7.71626e6i	0.0932589	+	0.956556i
$580$		0			0
$581$			5.48577e6i			0.674213i
$582$		0			0
$583$	−3.24262e6	−	3.24262e6i	−0.395115	−	0.395115i
$584$		0			0
$585$	−5.20455e6	+	3.40711e6i	−0.628773	+	0.411620i
$586$		0			0
$587$	1.05978e7	+	1.05978e7i	1.26946	+	1.26946i	0.946368	+	0.323091i	$0.104722\pi$
$587$	1.05978e7	+	1.05978e7i	1.26946	+	1.26946i	0.323091	+	0.946368i	$0.395278\pi$
$588$		0			0
$589$		−	1.88710e6i		−	0.224134i
$590$		0			0
$591$	1.63924e6	+	1.68137e7i	0.193052	+	1.98013i
$592$		0			0
$593$	410012.	−	410012.i	0.0478807	−	0.0478807i	−0.682761	−	0.730642i	$-0.739221\pi$
$593$	410012.	−	410012.i	0.0478807	−	0.0478807i	0.730642	+	0.682761i	$0.239221\pi$
$594$		0			0
$595$	−7.94954e6	−	3.36235e6i	−0.920555	−	0.389359i
$596$		0			0
$597$	−5.31273e6	+	6.46056e6i	−0.610073	+	0.741882i
$598$		0			0
$599$	3.14687e6			0.358354			0.179177	−	0.983817i	$-0.442657\pi$
$599$	3.14687e6			0.358354			0.179177	+	0.983817i	$0.442657\pi$
$600$		0			0
$601$	4.12306e6			0.465622			0.232811	−	0.972522i	$-0.425208\pi$
$601$	4.12306e6			0.465622			0.232811	+	0.972522i	$0.425208\pi$
$602$		0			0
$603$	−4.49641e6	−	3.01726e6i	−0.503584	−	0.337925i
$604$		0			0
$605$	1.12576e7	+	4.76152e6i	1.25042	+	0.528880i
$606$		0			0
$607$	138597.	−	138597.i	0.0152680	−	0.0152680i	−0.699432	−	0.714700i	$-0.746563\pi$
$607$	138597.	−	138597.i	0.0152680	−	0.0152680i	0.714700	+	0.699432i	$0.246563\pi$
$608$		0			0
$609$	468250.	−	45651.7i	0.0511604	−	0.00498786i
$610$		0			0
$611$			4.50945e6i			0.488676i
$612$		0			0
$613$	2.87301e6	+	2.87301e6i	0.308806	+	0.308806i	0.844446	−	0.535640i	$-0.179930\pi$
$613$	2.87301e6	+	2.87301e6i	0.308806	+	0.308806i	−0.535640	+	0.844446i	$0.679930\pi$
$614$		0			0
$615$	1.12922e6	−	591397.i	0.120390	−	0.0630509i
$616$		0			0
$617$	−6.00454e6	−	6.00454e6i	−0.634990	−	0.634990i	0.314325	−	0.949315i	$-0.398222\pi$
$617$	−6.00454e6	−	6.00454e6i	−0.634990	−	0.634990i	−0.949315	+	0.314325i	$0.898222\pi$
$618$		0			0
$619$		−	1.13901e7i		−	1.19482i	−0.801937	−	0.597409i	$-0.796197\pi$
$619$		−	1.13901e7i		−	1.19482i	0.801937	−	0.597409i	$-0.203803\pi$
$620$		0			0
$621$	4.06777e6	+	1.35541e7i	0.423279	+	1.41039i
$622$		0			0
$623$	1.22548e6	−	1.22548e6i	0.126499	−	0.126499i
$624$		0			0
$625$	−290701.	+	9.76130e6i	−0.0297677	+	0.999557i
$626$		0			0
$627$	1.90259e6	+	1.56456e6i	0.193275	+	0.158937i
$628$		0			0
$629$	1.67257e7			1.68562
$630$		0			0
$631$	−2.66614e6			−0.266569			−0.133284	−	0.991078i	$-0.542552\pi$
$631$	−2.66614e6			−0.266569			−0.133284	+	0.991078i	$0.542552\pi$
$632$		0			0
$633$	−1.33243e7	−	1.09570e7i	−1.32171	−	1.08688i
$634$		0			0
$635$	−1.80261e6	+	730995.i	−0.177406	+	0.0719416i
$636$		0			0
$637$	3.41046e6	−	3.41046e6i	0.333015	−	0.333015i
$638$		0			0
$639$	−1.81003e7	+	3.56324e6i	−1.75362	+	0.345217i
$640$		0			0
$641$		−	6.50816e6i		−	0.625624i	−0.949815	−	0.312812i	$-0.898729\pi$
$641$		−	6.50816e6i		−	0.625624i	0.949815	−	0.312812i	$-0.101271\pi$
$642$		0			0
$643$	6.81712e6	+	6.81712e6i	0.650239	+	0.650239i	0.953051	−	0.302811i	$-0.0979252\pi$
$643$	6.81712e6	+	6.81712e6i	0.650239	+	0.650239i	−0.302811	+	0.953051i	$0.597925\pi$
$644$		0			0
$645$	−1.26850e6	+	4.05820e6i	−0.120058	+	0.384091i
$646$		0			0
$647$	−4.91694e6	−	4.91694e6i	−0.461779	−	0.461779i	0.437459	−	0.899238i	$-0.355878\pi$
$647$	−4.91694e6	−	4.91694e6i	−0.461779	−	0.461779i	−0.899238	+	0.437459i	$0.855878\pi$
$648$		0			0
$649$		−	2.08372e7i		−	1.94190i
$650$		0			0
$651$	9.04381e6	−	881721.i	0.836371	−	0.0815415i
$652$		0			0
$653$	251758.	−	251758.i	0.0231047	−	0.0231047i	−0.695460	−	0.718565i	$-0.744799\pi$
$653$	251758.	−	251758.i	0.0231047	−	0.0231047i	0.718565	+	0.695460i	$0.244799\pi$
$654$		0			0
$655$	−4.62368e6	+	1.09317e7i	−0.421100	+	0.995600i
$656$		0			0
$657$	−9.78131e6	+	1.45764e7i	−0.884064	+	1.31745i
$658$		0			0
$659$	−1.12808e7			−1.01187			−0.505936	−	0.862571i	$-0.668853\pi$
$659$	−1.12808e7			−1.01187			−0.505936	+	0.862571i	$0.668853\pi$
$660$		0			0
$661$	1.01880e6			0.0906950			0.0453475	−	0.998971i	$-0.485560\pi$
$661$	1.01880e6			0.0906950			0.0453475	+	0.998971i	$0.485560\pi$
$662$		0			0
$663$	−8.83784e6	+	1.07473e7i	−0.780841	+	0.949544i
$664$		0			0
$665$	−426733.	−	1.05231e6i	−0.0374199	−	0.0922763i
$666$		0			0
$667$	1.00649e6	−	1.00649e6i	0.0875977	−	0.0875977i
$668$		0			0
$669$	1.07257e6	+	1.10014e7i	0.0926536	+	0.950348i
$670$		0			0
$671$		−	1.92113e7i		−	1.64722i
$672$		0			0
$673$	1.76092e6	+	1.76092e6i	0.149865	+	0.149865i	0.778058	−	0.628193i	$-0.216205\pi$
$673$	1.76092e6	+	1.76092e6i	0.149865	+	0.149865i	−0.628193	+	0.778058i	$0.716205\pi$
$674$		0			0
$675$	1.13873e7	−	3.23352e6i	0.961969	−	0.273159i
$676$		0			0
$677$	5.23077e6	+	5.23077e6i	0.438626	+	0.438626i	0.891549	−	0.452924i	$-0.149619\pi$
$677$	5.23077e6	+	5.23077e6i	0.438626	+	0.438626i	−0.452924	+	0.891549i	$0.649619\pi$
$678$		0			0
$679$		−	7.11624e6i		−	0.592347i
$680$		0			0
$681$	−1.46810e6	−	1.50583e7i	−0.121308	−	1.24425i
$682$		0			0
$683$	8.63973e6	−	8.63973e6i	0.708677	−	0.708677i	−0.257580	−	0.966257i	$-0.582925\pi$
$683$	8.63973e6	−	8.63973e6i	0.708677	−	0.708677i	0.966257	+	0.257580i	$0.0829250\pi$
$684$		0			0
$685$	795435.	+	1.96152e6i	0.0647707	+	0.159722i
$686$		0			0
$687$	1.01897e7	−	1.23912e7i	0.823701	−	1.00166i
$688$		0			0
$689$	3.40791e6			0.273489
$690$		0			0
$691$	−9.88818e6			−0.787809			−0.393905	−	0.919151i	$-0.628876\pi$
$691$	−9.88818e6			−0.787809			−0.393905	+	0.919151i	$0.628876\pi$
$692$		0			0
$693$	−6.60910e6	+	9.84906e6i	−0.522769	+	0.779043i
$694$		0			0
$695$	3.92331e6	−	9.27584e6i	0.308100	−	0.728435i
$696$		0			0
$697$	2.01619e6	−	2.01619e6i	0.157199	−	0.157199i
$698$		0			0
$699$	1.68586e7	−	1.64362e6i	1.30505	−	0.127235i
$700$		0			0
$701$			1.61068e6i			0.123798i	0.998082	+	0.0618991i	$0.0197157\pi$
$701$			1.61068e6i			0.123798i	−0.998082	+	0.0618991i	$0.980284\pi$
$702$		0			0
$703$	1.55594e6	+	1.55594e6i	0.118742	+	0.118742i
$704$		0			0
$705$	2.56015e6	−	8.19047e6i	0.193996	−	0.620635i
$706$		0			0
$707$	−8.73491e6	−	8.73491e6i	−0.657219	−	0.657219i
$708$		0			0
$709$			1.99678e7i			1.49182i	0.666049	+	0.745908i	$0.267984\pi$
$709$			1.99678e7i			1.49182i	−0.666049	+	0.745908i	$0.732016\pi$
$710$		0			0
$711$	−1.00301e7	+	1.97452e6i	−0.744099	+	0.146483i
$712$		0			0
$713$	1.94393e7	−	1.94393e7i	1.43205	−	1.43205i
$714$		0			0
$715$	−1.46180e7	+	5.92791e6i	−1.06936	+	0.433647i
$716$		0			0
$717$	6.23776e6	+	5.12951e6i	0.453138	+	0.372630i
$718$		0			0
$719$	−8.21981e6			−0.592979			−0.296490	−	0.955036i	$-0.595816\pi$
$719$	−8.21981e6			−0.592979			−0.296490	+	0.955036i	$0.595816\pi$
$720$		0			0
$721$	−2.60069e6			−0.186316
$722$		0			0
$723$	−1.29334e7	−	1.06355e7i	−0.920166	−	0.756682i
$724$		0			0
$725$	−829294.	−	854359.i	−0.0585954	−	0.0603664i
$726$		0			0
$727$	1.42367e7	−	1.42367e7i	0.999020	−	0.999020i	−0.000979495	−	1.00000i	$-0.500312\pi$
$727$	1.42367e7	−	1.42367e7i	0.999020	−	0.999020i	1.00000		0.000979495i	$0.000311783\pi$
$728$		0			0
$729$	−7.90100e6	−	1.19777e7i	−0.550634	−	0.834747i
$730$		0			0
$731$			9.51064e6i			0.658289i
$732$		0			0
$733$	−4.76397e6	−	4.76397e6i	−0.327499	−	0.327499i	0.524136	−	0.851635i	$-0.324388\pi$
$733$	−4.76397e6	−	4.76397e6i	−0.327499	−	0.327499i	−0.851635	+	0.524136i	$0.824388\pi$
$734$		0			0
$735$	−8.13060e6	+	4.25816e6i	−0.555142	+	0.290739i
$736$		0			0
$737$	−9.70947e6	−	9.70947e6i	−0.658456	−	0.658456i
$738$		0			0
$739$			1.49992e7i			1.01031i	0.863028	+	0.505157i	$0.168565\pi$
$739$			1.49992e7i			1.01031i	−0.863028	+	0.505157i	$0.831435\pi$
$740$		0			0
$741$	−1.82195e6	+	177629.i	−0.121896	+	0.0118842i
$742$		0			0
$743$	6.28423e6	−	6.28423e6i	0.417619	−	0.417619i	−0.466764	−	0.884382i	$-0.654580\pi$
$743$	6.28423e6	−	6.28423e6i	0.417619	−	0.417619i	0.884382	+	0.466764i	$0.154580\pi$
$744$		0			0
$745$	1.54614e7	+	6.53955e6i	1.02060	+	0.431675i
$746$		0			0
$747$	−1.39741e7	−	9.37715e6i	−0.916266	−	0.614850i
$748$		0			0
$749$	−5.93390e6			−0.386488
$750$		0			0
$751$	5.31885e6			0.344126			0.172063	−	0.985086i	$-0.444957\pi$
$751$	5.31885e6			0.344126			0.172063	+	0.985086i	$0.444957\pi$
$752$		0			0
$753$	9.05618e6	−	1.10128e7i	0.582046	−	0.707799i
$754$		0			0
$755$	569759.	+	240986.i	0.0363768	+	0.0153859i
$756$		0			0
$757$	1.38183e7	−	1.38183e7i	0.876423	−	0.876423i	−0.116740	−	0.993163i	$-0.537244\pi$
$757$	1.38183e7	−	1.38183e7i	0.876423	−	0.876423i	0.993163	+	0.116740i	$0.0372444\pi$
$758$		0			0
$759$	3.48208e6	+	3.57157e7i	0.219399	+	2.25037i
$760$		0			0
$761$		−	6.28201e6i		−	0.393221i	−0.980482	−	0.196611i	$-0.937007\pi$
$761$		−	6.28201e6i		−	0.393221i	0.980482	−	0.196611i	$-0.0629935\pi$
$762$		0			0
$763$	9.20475e6	+	9.20475e6i	0.572401	+	0.572401i
$764$		0			0
$765$	2.21536e7	−	1.45027e7i	1.36865	−	0.895972i
$766$		0			0
$767$	1.09497e7	+	1.09497e7i	0.672068	+	0.672068i
$768$		0			0
$769$			77953.3i			0.00475356i	0.999997	+	0.00237678i	$0.000756553\pi$
$769$			77953.3i			0.00475356i	−0.999997	+	0.00237678i	$0.999243\pi$
$770$		0			0
$771$	−216225.	−	2.21782e6i	−0.0131000	−	0.134366i
$772$		0			0
$773$	−1.35614e7	+	1.35614e7i	−0.816313	+	0.816313i	−0.985572	−	0.169259i	$-0.945863\pi$
$773$	−1.35614e7	+	1.35614e7i	−0.816313	+	0.816313i	0.169259	+	0.985572i	$0.445863\pi$
$774$		0			0
$775$	−1.60170e7	−	1.65011e7i	−0.957917	−	0.986870i
$776$		0			0
$777$	−6.72977e6	+	8.18375e6i	−0.399896	+	0.486295i
$778$		0			0
$779$	375120.			0.0221476
$780$		0			0
$781$	−4.67800e7			−2.74431
$782$		0			0
$783$	−684117.	+	1.27082e6i	−0.0398773	+	0.0740765i
$784$		0			0
$785$	−3.26108e6	+	1.32243e6i	−0.188880	+	0.0765948i
$786$		0			0
$787$	2.89852e6	−	2.89852e6i	0.166817	−	0.166817i	−0.618762	−	0.785579i	$-0.712365\pi$
$787$	2.89852e6	−	2.89852e6i	0.166817	−	0.166817i	0.785579	+	0.618762i	$0.212365\pi$
$788$		0			0
$789$	185257.	−	18061.5i	0.0105945	−	0.00103291i
$790$		0			0
$791$		−	9.31755e6i		−	0.529494i
$792$		0			0
$793$	1.00953e7	+	1.00953e7i	0.570081	+	0.570081i
$794$		0			0
$795$	−6.18974e6	−	1.93477e6i	−0.347340	−	0.108570i
$796$		0			0
$797$	1.24492e7	+	1.24492e7i	0.694219	+	0.694219i	0.963157	−	0.268939i	$-0.0866729\pi$
$797$	1.24492e7	+	1.24492e7i	0.694219	+	0.694219i	−0.268939	+	0.963157i	$0.586673\pi$
$798$		0			0
$799$		−	1.91949e7i		−	1.06370i
$800$		0			0
$801$	1.02692e6	+	5.21649e6i	0.0565529	+	0.287275i
$802$		0			0
$803$	−3.14760e7	+	3.14760e7i	−1.72262	+	1.72262i
$804$		0			0
$805$	6.44417e6	−	1.52359e7i	0.350491	−	0.828662i
$806$		0			0
$807$	7.59319e6	+	6.24413e6i	0.410431	+	0.337511i
$808$		0			0
$809$	−4.88806e6			−0.262582			−0.131291	−	0.991344i	$-0.541912\pi$
$809$	−4.88806e6			−0.262582			−0.131291	+	0.991344i	$0.541912\pi$
$810$		0			0
$811$	6.56543e6			0.350518			0.175259	−	0.984522i	$-0.443924\pi$
$811$	6.56543e6			0.350518			0.175259	+	0.984522i	$0.443924\pi$
$812$		0			0
$813$	5.52486e6	+	4.54327e6i	0.293153	+	0.241070i
$814$		0			0
$815$	7.84378e6	+	1.93425e7i	0.413649	+	1.02004i
$816$		0			0
$817$	−884747.	+	884747.i	−0.0463728	+	0.0463728i
$818$		0			0
$819$	−1.70255e6	−	8.64855e6i	−0.0886934	−	0.450540i
$820$		0			0
$821$			1.76992e7i			0.916424i	0.888843	+	0.458212i	$0.151510\pi$
$821$			1.76992e7i			0.916424i	−0.888843	+	0.458212i	$0.848490\pi$
$822$		0			0
$823$	1.74735e7	+	1.74735e7i	0.899249	+	0.899249i	0.995370	−	0.0961203i	$-0.0306434\pi$
$823$	1.74735e7	+	1.74735e7i	0.899249	+	0.899249i	−0.0961203	+	0.995370i	$0.530643\pi$
$824$		0			0
$825$	2.99160e7	−	2.46770e6i	1.53027	−	0.126229i
$826$		0			0
$827$	1.27114e7	+	1.27114e7i	0.646294	+	0.646294i	0.952095	−	0.305801i	$-0.0989243\pi$
$827$	1.27114e7	+	1.27114e7i	0.646294	+	0.646294i	−0.305801	+	0.952095i	$0.598924\pi$
$828$		0			0
$829$			5.67658e6i			0.286880i	0.989659	+	0.143440i	$0.0458164\pi$
$829$			5.67658e6i			0.286880i	−0.989659	+	0.143440i	$0.954184\pi$
$830$		0			0
$831$	−7.85687e6	+	766001.i	−0.394682	+	0.0384793i
$832$		0			0
$833$	−1.45169e7	+	1.45169e7i	−0.724873	+	0.724873i
$834$		0			0
$835$	−9.01199e6	−	2.22233e7i	−0.447306	−	1.10304i
$836$		0			0
$837$	−1.32131e7	+	2.45447e7i	−0.651914	+	1.21100i
$838$		0			0
$839$	3.51049e7			1.72172			0.860861	−	0.508841i	$-0.169926\pi$
$839$	3.51049e7			1.72172			0.860861	+	0.508841i	$0.169926\pi$
$840$		0			0
$841$	−2.03660e7			−0.992922
$842$		0			0
$843$	2.30430e7	−	2.80215e7i	1.11679	−	1.35807i
$844$		0			0
$845$	−3.51890e6	+	8.31968e6i	−0.169537	+	0.400834i
$846$		0			0
$847$	−1.22472e7	+	1.22472e7i	−0.586581	+	0.586581i
$848$		0			0
$849$	27761.3	+	284747.i	0.00132181	+	0.0135578i
$850$		0			0
$851$			3.20561e7i			1.51735i
$852$		0			0
$853$	1.52150e7	+	1.52150e7i	0.715977	+	0.715977i	0.967779	−	0.251801i	$-0.0810230\pi$
$853$	1.52150e7	+	1.52150e7i	0.715977	+	0.715977i	−0.251801	+	0.967779i	$0.581023\pi$
$854$		0			0
$855$	3.41002e6	+	711746.i	0.159530	+	0.0332974i
$856$		0			0
$857$	−2.09661e7	−	2.09661e7i	−0.975136	−	0.975136i	0.0245623	−	0.999698i	$-0.492181\pi$
$857$	−2.09661e7	−	2.09661e7i	−0.975136	−	0.975136i	−0.999698	+	0.0245623i	$0.992181\pi$
$858$		0			0
$859$			2.09600e7i			0.969188i	0.874739	+	0.484594i	$0.161033\pi$
$859$			2.09600e7i			0.969188i	−0.874739	+	0.484594i	$0.838967\pi$
$860$		0			0
$861$	175269.	+	1.79774e6i	0.00805745	+	0.0826453i
$862$		0			0
$863$	−733482.	+	733482.i	−0.0335245	+	0.0335245i	−0.723670	−	0.690146i	$-0.757546\pi$
$863$	−733482.	+	733482.i	−0.0335245	+	0.0335245i	0.690146	+	0.723670i	$0.257546\pi$
$864$		0			0
$865$	9.21716e6	−	3.73774e6i	0.418848	−	0.169851i
$866$		0			0
$867$	2.35609e7	−	2.86513e7i	1.06450	−	1.29448i
$868$		0			0
$869$	−2.59226e7			−1.16447
$870$		0			0
$871$	1.02044e7			0.455766
$872$		0			0
$873$	1.81274e7	+	1.21642e7i	0.805008	+	0.540192i
$874$		0			0
$875$	−1.26631e7	−	5.57963e6i	−0.559138	−	0.246369i
$876$		0			0
$877$	3.51575e6	−	3.51575e6i	0.154354	−	0.154354i	−0.625705	−	0.780060i	$-0.715189\pi$
$877$	3.51575e6	−	3.51575e6i	0.154354	−	0.154354i	0.780060	+	0.625705i	$0.215189\pi$
$878$		0			0
$879$	−1.71053e7	+	1.66767e6i	−0.746721	+	0.0728011i
$880$		0			0
$881$		−	878251.i		−	0.0381223i	−0.999818	−	0.0190611i	$-0.993932\pi$
$881$		−	878251.i		−	0.0381223i	0.999818	−	0.0190611i	$-0.00606772\pi$
$882$		0			0
$883$	1.79607e7	+	1.79607e7i	0.775214	+	0.775214i	0.979013	−	0.203798i	$-0.0653287\pi$
$883$	1.79607e7	+	1.79607e7i	0.775214	+	0.775214i	−0.203798	+	0.979013i	$0.565329\pi$
$884$		0			0
$885$	−1.36713e7	−	2.61043e7i	−0.586749	−	1.12035i
$886$		0			0
$887$	1.40908e7	+	1.40908e7i	0.601349	+	0.601349i	0.940670	−	0.339321i	$-0.110197\pi$
$887$	1.40908e7	+	1.40908e7i	0.601349	+	0.601349i	−0.339321	+	0.940670i	$0.610197\pi$
$888$		0			0
$889$		−	2.75632e6i		−	0.116970i
$890$		0			0
$891$	−1.37915e7	−	3.36712e7i	−0.581991	−	1.42090i
$892$		0			0
$893$	1.78564e6	−	1.78564e6i	0.0749317	−	0.0749317i
$894$		0			0
$895$	−2.29041e7	−	9.68752e6i	−0.955773	−	0.404255i
$896$		0			0
$897$	−2.05979e7	−	1.69383e7i	−0.854756	−	0.702894i
$898$		0			0
$899$	2.80379e6			0.115703
$900$		0			0
$901$	−1.45060e7			−0.595302
$902$		0			0
$903$	−4.65347e6	−	3.82670e6i	−0.189914	−	0.156173i
$904$		0			0
$905$	−3.81901e7	−	1.61529e7i	−1.54999	−	0.655586i
$906$		0			0
$907$	5.21671e6	−	5.21671e6i	0.210561	−	0.210561i	−0.593945	−	0.804506i	$-0.702430\pi$
$907$	5.21671e6	−	5.21671e6i	0.210561	−	0.210561i	0.804506	+	0.593945i	$0.202430\pi$
$908$		0			0
$909$	3.71818e7	−	7.31961e6i	1.49252	−	0.293818i
$910$		0			0
$911$		−	2.36471e6i		−	0.0944021i	−0.998885	−	0.0472010i	$-0.984970\pi$
$911$		−	2.36471e6i		−	0.0944021i	0.998885	−	0.0472010i	$-0.0150301\pi$
$912$		0			0
$913$	−3.01754e7	−	3.01754e7i	−1.19805	−	1.19805i
$914$		0			0
$915$	−1.26046e7	−	2.40674e7i	−0.497709	−	0.950334i
$916$		0			0
$917$	−1.18927e7	−	1.18927e7i	−0.467041	−	0.467041i
$918$		0			0
$919$			1.72028e7i			0.671908i	0.941878	+	0.335954i	$0.109059\pi$
$919$			1.72028e7i			0.671908i	−0.941878	+	0.335954i	$0.890941\pi$
$920$		0			0
$921$	2.57790e7	−	2.51331e6i	1.00142	−	0.0976330i
$922$		0			0
$923$	2.45823e7	−	2.45823e7i	0.949770	−	0.949770i
$924$		0			0
$925$	2.68117e7	+	399151.i	1.03032	+	0.0153385i
$926$		0			0
$927$	4.44551e6	−	6.62482e6i	0.169911	−	0.253206i
$928$		0			0
$929$	−3.23733e7			−1.23069			−0.615344	−	0.788258i	$-0.710983\pi$
$929$	−3.23733e7			−1.23069			−0.615344	+	0.788258i	$0.710983\pi$
$930$		0			0
$931$	−2.70093e6			−0.102127
$932$		0			0
$933$	−1.65211e7	+	2.00905e7i	−0.621347	+	0.755591i
$934$		0			0
$935$	6.22230e7	−	2.52327e7i	2.32767	−	0.943918i
$936$		0			0
$937$	−1.53693e7	+	1.53693e7i	−0.571882	+	0.571882i	−0.932654	−	0.360772i	$-0.882513\pi$
$937$	−1.53693e7	+	1.53693e7i	−0.571882	+	0.571882i	0.360772	+	0.932654i	$0.382513\pi$
$938$		0			0
$939$	3.99072e6	+	4.09328e7i	0.147702	+	1.51498i
$940$		0			0
$941$		−	3.45303e7i		−	1.27124i	−0.772003	−	0.635619i	$-0.780745\pi$
$941$		−	3.45303e7i		−	1.27124i	0.772003	−	0.635619i	$-0.219255\pi$
$942$		0			0
$943$	3.86417e6	+	3.86417e6i	0.141507	+	0.141507i
$944$		0			0
$945$	−1.81771e6	+	1.66749e7i	−0.0662133	+	0.607411i
$946$		0			0
$947$	−2.20443e7	−	2.20443e7i	−0.798769	−	0.798769i	0.184132	−	0.982901i	$-0.441053\pi$
$947$	−2.20443e7	−	2.20443e7i	−0.798769	−	0.798769i	−0.982901	+	0.184132i	$0.941053\pi$
$948$		0			0
$949$		−	3.30805e7i		−	1.19236i
$950$		0			0
$951$	−962109.	−	9.86835e6i	−0.0344964	−	0.353829i
$952$		0			0
$953$	−1.46652e7	+	1.46652e7i	−0.523064	+	0.523064i	−0.918495	−	0.395431i	$-0.870595\pi$
$953$	−1.46652e7	+	1.46652e7i	−0.523064	+	0.523064i	0.395431	+	0.918495i	$0.370595\pi$
$954$		0			0
$955$	8.31153e6	−	1.96508e7i	0.294899	−	0.697225i
$956$		0			0
$957$	−2.32457e6	+	2.82680e6i	−0.0820471	+	0.0997736i
$958$		0			0
$959$	−2.99930e6			−0.105311
$960$		0			0
$961$	2.55234e7			0.891517
$962$		0			0
$963$	1.01432e7	−	1.51156e7i	0.352458	−	0.525243i
$964$		0			0
$965$	1.04481e7	+	2.57646e7i	0.361175	+	0.890646i
$966$		0			0
$967$	1.51230e6	−	1.51230e6i	0.0520081	−	0.0520081i	−0.680624	−	0.732633i	$-0.738291\pi$
$967$	1.51230e6	−	1.51230e6i	0.0520081	−	0.0520081i	0.732633	+	0.680624i	$0.238291\pi$
$968$		0			0
$969$	7.75527e6	−	756095.i	0.265331	−	0.0258683i
$970$		0			0
$971$			1.56209e6i			0.0531688i	0.999647	+	0.0265844i	$0.00846307\pi$
$971$			1.56209e6i			0.0531688i	−0.999647	+	0.0265844i	$0.991537\pi$
$972$		0			0
$973$	1.00912e7	+	1.00912e7i	0.341713	+	0.341713i
$974$		0			0
$975$	−1.44237e7	+	1.70172e7i	−0.485922	+	0.573294i
$976$		0			0
$977$	2.74017e7	+	2.74017e7i	0.918421	+	0.918421i	0.996915	−	0.0784935i	$-0.0250110\pi$
$977$	2.74017e7	+	2.74017e7i	0.918421	+	0.918421i	−0.0784935	+	0.996915i	$0.525011\pi$
$978$		0			0
$979$			1.34819e7i			0.449568i
$980$		0			0
$981$	−3.91818e7	+	7.71333e6i	−1.29990	+	0.255899i
$982$		0			0
$983$	1.94825e7	−	1.94825e7i	0.643074	−	0.643074i	−0.308236	−	0.951310i	$-0.599739\pi$
$983$	1.94825e7	−	1.94825e7i	0.643074	−	0.643074i	0.951310	+	0.308236i	$0.0997385\pi$
$984$		0			0
$985$	2.27663e7	+	5.61409e7i	0.747655	+	1.84369i
$986$		0			0
$987$	9.39188e6	+	7.72325e6i	0.306874	+	0.252352i
$988$		0			0
$989$	−1.82278e7			−0.592576
$990$		0			0
$991$	1.01093e7			0.326991			0.163495	−	0.986544i	$-0.447723\pi$
$991$	1.01093e7			0.326991			0.163495	+	0.986544i	$0.447723\pi$
$992$		0			0
$993$	1.10546e7	+	9.09053e6i	0.355769	+	0.292561i
$994$		0			0
$995$	−1.16848e7	+	2.76262e7i	−0.374166	+	0.884634i
$996$		0			0
$997$	−2.66908e6	+	2.66908e6i	−0.0850401	+	0.0850401i	−0.748347	−	0.663307i	$-0.769152\pi$
$997$	−2.66908e6	+	2.66908e6i	−0.0850401	+	0.0850401i	0.663307	+	0.748347i	$0.269152\pi$
$998$		0			0
$999$	−9.34314e6	−	3.11319e7i	−0.296196	−	0.986943i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.i.a.53.2	yes	20
3.2	odd	2	inner	60.6.i.a.53.4	yes	20
5.2	odd	4	inner	60.6.i.a.17.4	yes	20
5.3	odd	4		300.6.i.d.257.7		20
5.4	even	2		300.6.i.d.293.9		20
15.2	even	4	inner	60.6.i.a.17.2	✓	20
15.8	even	4		300.6.i.d.257.9		20
15.14	odd	2		300.6.i.d.293.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.i.a.17.2	✓	20	15.2	even	4	inner
60.6.i.a.17.4	yes	20	5.2	odd	4	inner
60.6.i.a.53.2	yes	20	1.1	even	1	trivial
60.6.i.a.53.4	yes	20	3.2	odd	2	inner
300.6.i.d.257.7		20	5.3	odd	4
300.6.i.d.257.9		20	15.8	even	4
300.6.i.d.293.7		20	15.14	odd	2
300.6.i.d.293.9		20	5.4	even	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-12.0403 - 9.90111i) q^{3} +(-51.4858 - 21.7765i) q^{5} +(56.0116 - 56.0116i) q^{7} +(46.9362 + 238.424i) q^{9} +O(q^{10})\)	\(q+(-12.0403 - 9.90111i) q^{3} +(-51.4858 - 21.7765i) q^{5} +(56.0116 - 56.0116i) q^{7} +(46.9362 + 238.424i) q^{9} +616.203i q^{11} +(-323.806 - 323.806i) q^{13} +(404.292 + 771.961i) q^{15} +(1378.31 + 1378.31i) q^{17} +256.440i q^{19} +(-1228.97 + 119.818i) q^{21} +(-2641.63 + 2641.63i) q^{23} +(2176.57 + 2242.36i) q^{25} +(1795.54 - 3335.41i) q^{27} -381.009 q^{29} -7358.84 q^{31} +(6101.09 - 7419.24i) q^{33} +(-4103.54 + 1664.07i) q^{35} +(6067.48 - 6067.48i) q^{37} +(692.674 + 7104.76i) q^{39} -1462.80i q^{41} +(3450.11 + 3450.11i) q^{43} +(2775.49 - 13297.5i) q^{45} +(-6963.19 - 6963.19i) q^{47} +10532.4i q^{49} +(-2948.43 - 30242.0i) q^{51} +(-5262.26 + 5262.26i) q^{53} +(13418.7 - 31725.7i) q^{55} +(2539.04 - 3087.61i) q^{57} -33815.5 q^{59} -31177.0 q^{61} +(15983.5 + 10725.5i) q^{63} +(9620.07 + 23722.8i) q^{65} +(-15756.9 + 15756.9i) q^{67} +(57961.0 - 5650.87i) q^{69} +75916.6i q^{71} +(51080.6 + 51080.6i) q^{73} +(-4004.69 - 48549.0i) q^{75} +(34514.5 + 34514.5i) q^{77} +42068.3i q^{79} +(-54643.0 + 22381.4i) q^{81} +(-48969.9 + 48969.9i) q^{83} +(-40948.7 - 100978. i) q^{85} +(4587.46 + 3772.42i) q^{87} +21879.1 q^{89} -36273.8 q^{91} +(88602.4 + 72860.7i) q^{93} +(5584.36 - 13203.0i) q^{95} +(63524.7 - 63524.7i) q^{97} +(-146917. + 28922.2i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) \) 20 * q + 2 * q^3 + 76 * q^7	\( 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 76 * q^7 + 1068 * q^13 - 130 * q^15 + 2180 * q^21 + 4060 * q^25 + 1454 * q^27 - 4720 * q^31 - 460 * q^33 - 612 * q^37 - 24012 * q^43 - 18860 * q^45 - 31700 * q^51 + 19200 * q^55 + 33476 * q^57 + 59880 * q^61 + 67208 * q^63 - 80804 * q^67 - 56956 * q^73 - 102470 * q^75 - 9980 * q^81 + 239260 * q^85 + 71540 * q^87 + 218520 * q^91 + 307928 * q^93 - 151164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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