LMFDB - Embedded newform 490.2.c.e.99.1

Newspace parameters

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	490.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.91266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	490.99
Dual form			490.2.c.e.99.4

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.22474 + 0.224745i) q^{5} -2.44949 q^{6} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.22474 + 0.224745i) q^{5} -2.44949 q^{6} +1.00000i q^{8} -3.00000 q^{9} +(0.224745 + 2.22474i) q^{10} -4.89898 q^{11} +2.44949i q^{12} +4.44949i q^{13} +(0.550510 + 5.44949i) q^{15} +1.00000 q^{16} -2.00000i q^{17} +3.00000i q^{18} +1.55051 q^{19} +(2.22474 - 0.224745i) q^{20} +4.89898i q^{22} -2.89898i q^{23} +2.44949 q^{24} +(4.89898 - 1.00000i) q^{25} +4.44949 q^{26} -6.89898 q^{29} +(5.44949 - 0.550510i) q^{30} -8.89898 q^{31} -1.00000i q^{32} +12.0000i q^{33} -2.00000 q^{34} +3.00000 q^{36} +2.00000i q^{37} -1.55051i q^{38} +10.8990 q^{39} +(-0.224745 - 2.22474i) q^{40} +1.10102 q^{41} +0.898979i q^{43} +4.89898 q^{44} +(6.67423 - 0.674235i) q^{45} -2.89898 q^{46} -8.89898i q^{47} -2.44949i q^{48} +(-1.00000 - 4.89898i) q^{50} -4.89898 q^{51} -4.44949i q^{52} +10.8990i q^{53} +(10.8990 - 1.10102i) q^{55} -3.79796i q^{57} +6.89898i q^{58} -1.55051 q^{59} +(-0.550510 - 5.44949i) q^{60} -3.55051 q^{61} +8.89898i q^{62} -1.00000 q^{64} +(-1.00000 - 9.89898i) q^{65} +12.0000 q^{66} -8.00000i q^{67} +2.00000i q^{68} -7.10102 q^{69} -1.10102 q^{71} -3.00000i q^{72} +2.89898i q^{73} +2.00000 q^{74} +(-2.44949 - 12.0000i) q^{75} -1.55051 q^{76} -10.8990i q^{78} -6.89898 q^{79} +(-2.22474 + 0.224745i) q^{80} -9.00000 q^{81} -1.10102i q^{82} -2.44949i q^{83} +(0.449490 + 4.44949i) q^{85} +0.898979 q^{86} +16.8990i q^{87} -4.89898i q^{88} -10.0000 q^{89} +(-0.674235 - 6.67423i) q^{90} +2.89898i q^{92} +21.7980i q^{93} -8.89898 q^{94} +(-3.44949 + 0.348469i) q^{95} -2.44949 q^{96} -15.7980i q^{97} +14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^5 - 12 * q^9	$ 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9} - 4 q^{10} + 12 q^{15} + 4 q^{16} + 16 q^{19} + 4 q^{20} + 8 q^{26} - 8 q^{29} + 12 q^{30} - 16 q^{31} - 8 q^{34} + 12 q^{36} + 24 q^{39} + 4 q^{40} + 24 q^{41} + 12 q^{45} + 8 q^{46} - 4 q^{50} + 24 q^{55} - 16 q^{59} - 12 q^{60} - 24 q^{61} - 4 q^{64} - 4 q^{65} + 48 q^{66} - 48 q^{69} - 24 q^{71} + 8 q^{74} - 16 q^{76} - 8 q^{79} - 4 q^{80} - 36 q^{81} - 8 q^{85} - 16 q^{86} - 40 q^{89} + 12 q^{90} - 16 q^{94} - 4 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^5 - 12 * q^9 - 4 * q^10 + 12 * q^15 + 4 * q^16 + 16 * q^19 + 4 * q^20 + 8 * q^26 - 8 * q^29 + 12 * q^30 - 16 * q^31 - 8 * q^34 + 12 * q^36 + 24 * q^39 + 4 * q^40 + 24 * q^41 + 12 * q^45 + 8 * q^46 - 4 * q^50 + 24 * q^55 - 16 * q^59 - 12 * q^60 - 24 * q^61 - 4 * q^64 - 4 * q^65 + 48 * q^66 - 48 * q^69 - 24 * q^71 + 8 * q^74 - 16 * q^76 - 8 * q^79 - 4 * q^80 - 36 * q^81 - 8 * q^85 - 16 * q^86 - 40 * q^89 + 12 * q^90 - 16 * q^94 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		−	2.44949i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$		−	2.44949i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$	−1.00000			−0.500000
$5$	−2.22474	+	0.224745i	−0.994936	+	0.100509i
$5$	−2.22474	+	0.224745i	−0.994936	+	0.100509i
$6$	−2.44949			−1.00000
$7$		0			0
$7$		0			0
$8$			1.00000i			0.353553i
$9$	−3.00000			−1.00000
$10$	0.224745	+	2.22474i	0.0710706	+	0.703526i
$11$	−4.89898			−1.47710			−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898			−1.47710			−0.738549	+	0.674200i	$0.764489\pi$
$12$			2.44949i			0.707107i
$13$			4.44949i			1.23407i	0.786937	+	0.617033i	$0.211666\pi$
$13$			4.44949i			1.23407i	−0.786937	+	0.617033i	$0.788334\pi$
$14$		0			0
$15$	0.550510	+	5.44949i	0.142141	+	1.40705i
$16$	1.00000			0.250000
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$			3.00000i			0.707107i
$19$	1.55051			0.355711			0.177856	−	0.984057i	$-0.443084\pi$
$19$	1.55051			0.355711			0.177856	+	0.984057i	$0.443084\pi$
$20$	2.22474	−	0.224745i	0.497468	−	0.0502545i
$21$		0			0
$22$			4.89898i			1.04447i
$23$		−	2.89898i		−	0.604479i	−0.953232	−	0.302240i	$-0.902266\pi$
$23$		−	2.89898i		−	0.604479i	0.953232	−	0.302240i	$-0.0977342\pi$
$24$	2.44949			0.500000
$25$	4.89898	−	1.00000i	0.979796	−	0.200000i
$26$	4.44949			0.872617
$27$		0			0
$28$		0			0
$29$	−6.89898			−1.28111			−0.640554	−	0.767913i	$-0.721295\pi$
$29$	−6.89898			−1.28111			−0.640554	+	0.767913i	$0.721295\pi$
$30$	5.44949	−	0.550510i	0.994936	−	0.100509i
$31$	−8.89898			−1.59830			−0.799152	−	0.601129i	$-0.794718\pi$
$31$	−8.89898			−1.59830			−0.799152	+	0.601129i	$0.794718\pi$
$32$		−	1.00000i		−	0.176777i
$33$			12.0000i			2.08893i
$34$	−2.00000			−0.342997
$35$		0			0
$36$	3.00000			0.500000
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		−	1.55051i		−	0.251526i
$39$	10.8990			1.74523
$40$	−0.224745	−	2.22474i	−0.0355353	−	0.351763i
$41$	1.10102			0.171951			0.0859753	−	0.996297i	$-0.472599\pi$
$41$	1.10102			0.171951			0.0859753	+	0.996297i	$0.472599\pi$
$42$		0			0
$43$			0.898979i			0.137093i	0.997648	+	0.0685465i	$0.0218362\pi$
$43$			0.898979i			0.137093i	−0.997648	+	0.0685465i	$0.978164\pi$
$44$	4.89898			0.738549
$45$	6.67423	−	0.674235i	0.994936	−	0.100509i
$46$	−2.89898			−0.427431
$47$		−	8.89898i		−	1.29805i	−0.760767	−	0.649025i	$-0.775177\pi$
$47$		−	8.89898i		−	1.29805i	0.760767	−	0.649025i	$-0.224823\pi$
$48$		−	2.44949i		−	0.353553i
$49$		0			0
$50$	−1.00000	−	4.89898i	−0.141421	−	0.692820i
$51$	−4.89898			−0.685994
$52$		−	4.44949i		−	0.617033i
$53$			10.8990i			1.49709i	0.663084	+	0.748545i	$0.269247\pi$
$53$			10.8990i			1.49709i	−0.663084	+	0.748545i	$0.730753\pi$
$54$		0			0
$55$	10.8990	−	1.10102i	1.46962	−	0.148462i
$56$		0			0
$57$		−	3.79796i		−	0.503052i
$58$			6.89898i			0.905880i
$59$	−1.55051			−0.201859			−0.100930	−	0.994894i	$-0.532182\pi$
$59$	−1.55051			−0.201859			−0.100930	+	0.994894i	$0.532182\pi$
$60$	−0.550510	−	5.44949i	−0.0710706	−	0.703526i
$61$	−3.55051			−0.454596			−0.227298	−	0.973825i	$-0.572989\pi$
$61$	−3.55051			−0.454596			−0.227298	+	0.973825i	$0.572989\pi$
$62$			8.89898i			1.13017i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−1.00000	−	9.89898i	−0.124035	−	1.22782i
$66$	12.0000			1.47710
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$			2.00000i			0.242536i
$69$	−7.10102			−0.854862
$70$		0			0
$71$	−1.10102			−0.130667			−0.0653335	−	0.997863i	$-0.520811\pi$
$71$	−1.10102			−0.130667			−0.0653335	+	0.997863i	$0.520811\pi$
$72$		−	3.00000i		−	0.353553i
$73$			2.89898i			0.339300i	0.985504	+	0.169650i	$0.0542637\pi$
$73$			2.89898i			0.339300i	−0.985504	+	0.169650i	$0.945736\pi$
$74$	2.00000			0.232495
$75$	−2.44949	−	12.0000i	−0.282843	−	1.38564i
$76$	−1.55051			−0.177856
$77$		0			0
$78$		−	10.8990i		−	1.23407i
$79$	−6.89898			−0.776196			−0.388098	−	0.921618i	$-0.626868\pi$
$79$	−6.89898			−0.776196			−0.388098	+	0.921618i	$0.626868\pi$
$80$	−2.22474	+	0.224745i	−0.248734	+	0.0251272i
$81$	−9.00000			−1.00000
$82$		−	1.10102i		−	0.121587i
$83$		−	2.44949i		−	0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$		−	2.44949i		−	0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$		0			0
$85$	0.449490	+	4.44949i	0.0487540	+	0.482615i
$86$	0.898979			0.0969395
$87$			16.8990i			1.81176i
$88$		−	4.89898i		−	0.522233i
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$	−0.674235	−	6.67423i	−0.0710706	−	0.703526i
$91$		0			0
$92$			2.89898i			0.302240i
$93$			21.7980i			2.26034i
$94$	−8.89898			−0.917860
$95$	−3.44949	+	0.348469i	−0.353910	+	0.0357522i
$96$	−2.44949			−0.250000
$97$		−	15.7980i		−	1.60404i	−0.597297	−	0.802020i	$-0.703759\pi$
$97$		−	15.7980i		−	1.60404i	0.597297	−	0.802020i	$-0.296241\pi$
$98$		0			0
$99$	14.6969			1.47710
$100$	−4.89898	+	1.00000i	−0.489898	+	0.100000i
$101$	−3.55051			−0.353289			−0.176644	−	0.984275i	$-0.556524\pi$
$101$	−3.55051			−0.353289			−0.176644	+	0.984275i	$0.556524\pi$
$102$			4.89898i			0.485071i
$103$			12.8990i			1.27097i	0.772111	+	0.635487i	$0.219201\pi$
$103$			12.8990i			1.27097i	−0.772111	+	0.635487i	$0.780799\pi$
$104$	−4.44949			−0.436308
$105$		0			0
$106$	10.8990			1.05860
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	−6.89898			−0.660802			−0.330401	−	0.943841i	$-0.607184\pi$
$109$	−6.89898			−0.660802			−0.330401	+	0.943841i	$0.607184\pi$
$110$	−1.10102	−	10.8990i	−0.104978	−	1.03918i
$111$	4.89898			0.464991
$112$		0			0
$113$		−	19.7980i		−	1.86244i	−0.364464	−	0.931218i	$-0.618748\pi$
$113$		−	19.7980i		−	1.86244i	0.364464	−	0.931218i	$-0.381252\pi$
$114$	−3.79796			−0.355711
$115$	0.651531	+	6.44949i	0.0607556	+	0.601418i
$116$	6.89898			0.640554
$117$		−	13.3485i		−	1.23407i
$118$			1.55051i			0.142736i
$119$		0			0
$120$	−5.44949	+	0.550510i	−0.497468	+	0.0502545i
$121$	13.0000			1.18182
$122$			3.55051i			0.321448i
$123$		−	2.69694i		−	0.243175i
$124$	8.89898			0.799152
$125$	−10.6742	+	3.32577i	−0.954733	+	0.297465i
$126$		0			0
$127$		−	14.8990i		−	1.32207i	−0.750355	−	0.661035i	$-0.770117\pi$
$127$		−	14.8990i		−	1.32207i	0.750355	−	0.661035i	$-0.229883\pi$
$128$			1.00000i			0.0883883i
$129$	2.20204			0.193879
$130$	−9.89898	+	1.00000i	−0.868198	+	0.0877058i
$131$	6.44949			0.563495			0.281747	−	0.959489i	$-0.409086\pi$
$131$	6.44949			0.563495			0.281747	+	0.959489i	$0.409086\pi$
$132$		−	12.0000i		−	1.04447i
$133$		0			0
$134$	−8.00000			−0.691095
$135$		0			0
$136$	2.00000			0.171499
$137$		−	1.79796i		−	0.153610i	−0.997046	−	0.0768050i	$-0.975528\pi$
$137$		−	1.79796i		−	0.153610i	0.997046	−	0.0768050i	$-0.0244719\pi$
$138$			7.10102i			0.604479i
$139$	1.55051			0.131513			0.0657563	−	0.997836i	$-0.479054\pi$
$139$	1.55051			0.131513			0.0657563	+	0.997836i	$0.479054\pi$
$140$		0			0
$141$	−21.7980			−1.83572
$142$			1.10102i			0.0923956i
$143$		−	21.7980i		−	1.82284i
$144$	−3.00000			−0.250000
$145$	15.3485	−	1.55051i	1.27462	−	0.128763i
$146$	2.89898			0.239921
$147$		0			0
$148$		−	2.00000i		−	0.164399i
$149$	3.79796			0.311141			0.155570	−	0.987825i	$-0.450278\pi$
$149$	3.79796			0.311141			0.155570	+	0.987825i	$0.450278\pi$
$150$	−12.0000	+	2.44949i	−0.979796	+	0.200000i
$151$	19.5959			1.59469			0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959			1.59469			0.797347	+	0.603522i	$0.206236\pi$
$152$			1.55051i			0.125763i
$153$			6.00000i			0.485071i
$154$		0			0
$155$	19.7980	−	2.00000i	1.59021	−	0.160644i
$156$	−10.8990			−0.872617
$157$		−	3.55051i		−	0.283362i	−0.989912	−	0.141681i	$-0.954749\pi$
$157$		−	3.55051i		−	0.283362i	0.989912	−	0.141681i	$-0.0452507\pi$
$158$			6.89898i			0.548853i
$159$	26.6969			2.11720
$160$	0.224745	+	2.22474i	0.0177676	+	0.175882i
$161$		0			0
$162$			9.00000i			0.707107i
$163$			7.10102i			0.556195i	0.960553	+	0.278097i	$0.0897038\pi$
$163$			7.10102i			0.556195i	−0.960553	+	0.278097i	$0.910296\pi$
$164$	−1.10102			−0.0859753
$165$	−2.69694	−	26.6969i	−0.209956	−	2.07835i
$166$	−2.44949			−0.190117
$167$			4.89898i			0.379094i	0.981872	+	0.189547i	$0.0607020\pi$
$167$			4.89898i			0.379094i	−0.981872	+	0.189547i	$0.939298\pi$
$168$		0			0
$169$	−6.79796			−0.522920
$170$	4.44949	−	0.449490i	0.341260	−	0.0344743i
$171$	−4.65153			−0.355711
$172$		−	0.898979i		−	0.0685465i
$173$		−	6.24745i		−	0.474985i	−0.971389	−	0.237492i	$-0.923675\pi$
$173$		−	6.24745i		−	0.474985i	0.971389	−	0.237492i	$-0.0763255\pi$
$174$	16.8990			1.28111
$175$		0			0
$176$	−4.89898			−0.369274
$177$			3.79796i			0.285472i
$178$			10.0000i			0.749532i
$179$	−13.7980			−1.03131			−0.515654	−	0.856797i	$-0.672451\pi$
$179$	−13.7980			−1.03131			−0.515654	+	0.856797i	$0.672451\pi$
$180$	−6.67423	+	0.674235i	−0.497468	+	0.0502545i
$181$	10.2474			0.761687			0.380843	−	0.924640i	$-0.375634\pi$
$181$	10.2474			0.761687			0.380843	+	0.924640i	$0.375634\pi$
$182$		0			0
$183$			8.69694i			0.642896i
$184$	2.89898			0.213716
$185$	−0.449490	−	4.44949i	−0.0330471	−	0.327133i
$186$	21.7980			1.59830
$187$			9.79796i			0.716498i
$188$			8.89898i			0.649025i
$189$		0			0
$190$	0.348469	+	3.44949i	0.0252806	+	0.250252i
$191$	12.6969			0.918718			0.459359	−	0.888251i	$-0.348079\pi$
$191$	12.6969			0.918718			0.459359	+	0.888251i	$0.348079\pi$
$192$			2.44949i			0.176777i
$193$			21.5959i			1.55451i	0.629187	+	0.777254i	$0.283388\pi$
$193$			21.5959i			1.55451i	−0.629187	+	0.777254i	$0.716612\pi$
$194$	−15.7980			−1.13423
$195$	−24.2474	+	2.44949i	−1.73640	+	0.175412i
$196$		0			0
$197$			18.8990i			1.34650i	0.739417	+	0.673248i	$0.235101\pi$
$197$			18.8990i			1.34650i	−0.739417	+	0.673248i	$0.764899\pi$
$198$		−	14.6969i		−	1.04447i
$199$	16.8990			1.19794			0.598968	−	0.800773i	$-0.295577\pi$
$199$	16.8990			1.19794			0.598968	+	0.800773i	$0.295577\pi$
$200$	1.00000	+	4.89898i	0.0707107	+	0.346410i
$201$	−19.5959			−1.38219
$202$			3.55051i			0.249813i
$203$		0			0
$204$	4.89898			0.342997
$205$	−2.44949	+	0.247449i	−0.171080	+	0.0172826i
$206$	12.8990			0.898714
$207$			8.69694i			0.604479i
$208$			4.44949i			0.308517i
$209$	−7.59592			−0.525421
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		−	10.8990i		−	0.748545i
$213$			2.69694i			0.184791i
$214$	−8.00000			−0.546869
$215$	−0.202041	−	2.00000i	−0.0137791	−	0.136399i
$216$		0			0
$217$		0			0
$218$			6.89898i			0.467258i
$219$	7.10102			0.479842
$220$	−10.8990	+	1.10102i	−0.734809	+	0.0742308i
$221$	8.89898			0.598610
$222$		−	4.89898i		−	0.328798i
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$	−14.6969	+	3.00000i	−0.979796	+	0.200000i
$226$	−19.7980			−1.31694
$227$		−	7.34847i		−	0.487735i	−0.969809	−	0.243868i	$-0.921584\pi$
$227$		−	7.34847i		−	0.487735i	0.969809	−	0.243868i	$-0.0784162\pi$
$228$			3.79796i			0.251526i
$229$	−19.1464			−1.26523			−0.632616	−	0.774466i	$-0.718019\pi$
$229$	−19.1464			−1.26523			−0.632616	+	0.774466i	$0.718019\pi$
$230$	6.44949	−	0.651531i	0.425267	−	0.0429607i
$231$		0			0
$232$		−	6.89898i		−	0.452940i
$233$		−	29.7980i		−	1.95213i	−0.217481	−	0.976065i	$-0.569784\pi$
$233$		−	29.7980i		−	1.95213i	0.217481	−	0.976065i	$-0.430216\pi$
$234$	−13.3485			−0.872617
$235$	2.00000	+	19.7980i	0.130466	+	1.29148i
$236$	1.55051			0.100930
$237$			16.8990i			1.09771i
$238$		0			0
$239$	6.20204			0.401177			0.200588	−	0.979676i	$-0.435715\pi$
$239$	6.20204			0.401177			0.200588	+	0.979676i	$0.435715\pi$
$240$	0.550510	+	5.44949i	0.0355353	+	0.351763i
$241$	8.69694			0.560219			0.280110	−	0.959968i	$-0.409629\pi$
$241$	8.69694			0.560219			0.280110	+	0.959968i	$0.409629\pi$
$242$		−	13.0000i		−	0.835672i
$243$			22.0454i			1.41421i
$244$	3.55051			0.227298
$245$		0			0
$246$	−2.69694			−0.171951
$247$			6.89898i			0.438972i
$248$		−	8.89898i		−	0.565086i
$249$	−6.00000			−0.380235
$250$	3.32577	+	10.6742i	0.210340	+	0.675098i
$251$	6.44949			0.407088			0.203544	−	0.979066i	$-0.434754\pi$
$251$	6.44949			0.407088			0.203544	+	0.979066i	$0.434754\pi$
$252$		0			0
$253$			14.2020i			0.892875i
$254$	−14.8990			−0.934845
$255$	10.8990	−	1.10102i	0.682521	−	0.0689486i
$256$	1.00000			0.0625000
$257$			8.69694i			0.542500i	0.962509	+	0.271250i	$0.0874370\pi$
$257$			8.69694i			0.542500i	−0.962509	+	0.271250i	$0.912563\pi$
$258$		−	2.20204i		−	0.137093i
$259$		0			0
$260$	1.00000	+	9.89898i	0.0620174	+	0.613909i
$261$	20.6969			1.28111
$262$		−	6.44949i		−	0.398451i
$263$		−	9.79796i		−	0.604168i	−0.953281	−	0.302084i	$-0.902318\pi$
$263$		−	9.79796i		−	0.604168i	0.953281	−	0.302084i	$-0.0976823\pi$
$264$	−12.0000			−0.738549
$265$	−2.44949	−	24.2474i	−0.150471	−	1.48951i
$266$		0			0
$267$			24.4949i			1.49906i
$268$			8.00000i			0.488678i
$269$	19.1464			1.16738			0.583689	−	0.811977i	$-0.301609\pi$
$269$	19.1464			1.16738			0.583689	+	0.811977i	$0.301609\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$		−	2.00000i		−	0.121268i
$273$		0			0
$274$	−1.79796			−0.108619
$275$	−24.0000	+	4.89898i	−1.44725	+	0.295420i
$276$	7.10102			0.427431
$277$		−	14.8990i		−	0.895193i	−0.894236	−	0.447596i	$-0.852280\pi$
$277$		−	14.8990i		−	0.895193i	0.894236	−	0.447596i	$-0.147720\pi$
$278$		−	1.55051i		−	0.0929934i
$279$	26.6969			1.59830
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$			21.7980i			1.29805i
$283$			3.75255i			0.223066i	0.993761	+	0.111533i	$0.0355761\pi$
$283$			3.75255i			0.223066i	−0.993761	+	0.111533i	$0.964424\pi$
$284$	1.10102			0.0653335
$285$	0.853572	+	8.44949i	0.0505612	+	0.500505i
$286$	−21.7980			−1.28894
$287$		0			0
$288$			3.00000i			0.176777i
$289$	13.0000			0.764706
$290$	−1.55051	−	15.3485i	−0.0910491	−	0.901293i
$291$	−38.6969			−2.26845
$292$		−	2.89898i		−	0.169650i
$293$			18.2474i			1.06603i	0.846107	+	0.533014i	$0.178941\pi$
$293$			18.2474i			1.06603i	−0.846107	+	0.533014i	$0.821059\pi$
$294$		0			0
$295$	3.44949	−	0.348469i	0.200837	−	0.0202887i
$296$	−2.00000			−0.116248
$297$		0			0
$298$		−	3.79796i		−	0.220010i
$299$	12.8990			0.745967
$300$	2.44949	+	12.0000i	0.141421	+	0.692820i
$301$		0			0
$302$		−	19.5959i		−	1.12762i
$303$			8.69694i			0.499626i
$304$	1.55051			0.0889279
$305$	7.89898	−	0.797959i	0.452294	−	0.0456910i
$306$	6.00000			0.342997
$307$			20.2474i			1.15558i	0.816184	+	0.577791i	$0.196085\pi$
$307$			20.2474i			1.15558i	−0.816184	+	0.577791i	$0.803915\pi$
$308$		0			0
$309$	31.5959			1.79743
$310$	−2.00000	−	19.7980i	−0.113592	−	1.12445i
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$			10.8990i			0.617033i
$313$		−	21.5959i		−	1.22067i	−0.792142	−	0.610337i	$-0.791034\pi$
$313$		−	21.5959i		−	1.22067i	0.792142	−	0.610337i	$-0.208966\pi$
$314$	−3.55051			−0.200367
$315$		0			0
$316$	6.89898			0.388098
$317$		−	22.4949i		−	1.26344i	−0.775197	−	0.631720i	$-0.782349\pi$
$317$		−	22.4949i		−	1.26344i	0.775197	−	0.631720i	$-0.217651\pi$
$318$		−	26.6969i		−	1.49709i
$319$	33.7980			1.89232
$320$	2.22474	−	0.224745i	0.124367	−	0.0125636i
$321$	−19.5959			−1.09374
$322$		0			0
$323$		−	3.10102i		−	0.172545i
$324$	9.00000			0.500000
$325$	4.44949	+	21.7980i	0.246813	+	1.20913i
$326$	7.10102			0.393289
$327$			16.8990i			0.934516i
$328$			1.10102i			0.0607937i
$329$		0			0
$330$	−26.6969	+	2.69694i	−1.46962	+	0.148462i
$331$	−18.6969			−1.02768			−0.513838	−	0.857887i	$-0.671777\pi$
$331$	−18.6969			−1.02768			−0.513838	+	0.857887i	$0.671777\pi$
$332$			2.44949i			0.134433i
$333$		−	6.00000i		−	0.328798i
$334$	4.89898			0.268060
$335$	1.79796	+	17.7980i	0.0982330	+	0.972406i
$336$		0			0
$337$			9.59592i			0.522723i	0.965241	+	0.261361i	$0.0841715\pi$
$337$			9.59592i			0.522723i	−0.965241	+	0.261361i	$0.915829\pi$
$338$			6.79796i			0.369760i
$339$	−48.4949			−2.63388
$340$	−0.449490	−	4.44949i	−0.0243770	−	0.241307i
$341$	43.5959			2.36085
$342$			4.65153i			0.251526i
$343$		0			0
$344$	−0.898979			−0.0484697
$345$	15.7980	−	1.59592i	0.850534	−	0.0859213i
$346$	−6.24745			−0.335865
$347$			28.8990i			1.55138i	0.631115	+	0.775689i	$0.282598\pi$
$347$			28.8990i			1.55138i	−0.631115	+	0.775689i	$0.717402\pi$
$348$		−	16.8990i		−	0.905880i
$349$	−8.44949			−0.452291			−0.226145	−	0.974094i	$-0.572612\pi$
$349$	−8.44949			−0.452291			−0.226145	+	0.974094i	$0.572612\pi$
$350$		0			0
$351$		0			0
$352$			4.89898i			0.261116i
$353$			22.8990i			1.21879i	0.792867	+	0.609395i	$0.208588\pi$
$353$			22.8990i			1.21879i	−0.792867	+	0.609395i	$0.791412\pi$
$354$	3.79796			0.201859
$355$	2.44949	−	0.247449i	0.130005	−	0.0131332i
$356$	10.0000			0.529999
$357$		0			0
$358$			13.7980i			0.729245i
$359$	27.5959			1.45646			0.728228	−	0.685334i	$-0.240344\pi$
$359$	27.5959			1.45646			0.728228	+	0.685334i	$0.240344\pi$
$360$	0.674235	+	6.67423i	0.0355353	+	0.351763i
$361$	−16.5959			−0.873469
$362$		−	10.2474i		−	0.538594i
$363$		−	31.8434i		−	1.67134i
$364$		0			0
$365$	−0.651531	−	6.44949i	−0.0341027	−	0.337582i
$366$	8.69694			0.454596
$367$		−	32.0000i		−	1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$		−	32.0000i		−	1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$		−	2.89898i		−	0.151120i
$369$	−3.30306			−0.171951
$370$	−4.44949	+	0.449490i	−0.231318	+	0.0233679i
$371$		0			0
$372$		−	21.7980i		−	1.13017i
$373$			4.69694i			0.243198i	0.992579	+	0.121599i	$0.0388022\pi$
$373$			4.69694i			0.243198i	−0.992579	+	0.121599i	$0.961198\pi$
$374$	9.79796			0.506640
$375$	8.14643	+	26.1464i	0.420680	+	1.35020i
$376$	8.89898			0.458930
$377$		−	30.6969i		−	1.58097i
$378$		0			0
$379$	−30.6969			−1.57680			−0.788398	−	0.615166i	$-0.789089\pi$
$379$	−30.6969			−1.57680			−0.788398	+	0.615166i	$0.789089\pi$
$380$	3.44949	−	0.348469i	0.176955	−	0.0178761i
$381$	−36.4949			−1.86969
$382$		−	12.6969i		−	0.649632i
$383$		−	7.10102i		−	0.362845i	−0.983405	−	0.181423i	$-0.941930\pi$
$383$		−	7.10102i		−	0.362845i	0.983405	−	0.181423i	$-0.0580702\pi$
$384$	2.44949			0.125000
$385$		0			0
$386$	21.5959			1.09920
$387$		−	2.69694i		−	0.137093i
$388$			15.7980i			0.802020i
$389$	−13.1010			−0.664248			−0.332124	−	0.943236i	$-0.607765\pi$
$389$	−13.1010			−0.664248			−0.332124	+	0.943236i	$0.607765\pi$
$390$	2.44949	+	24.2474i	0.124035	+	1.22782i
$391$	−5.79796			−0.293215
$392$		0			0
$393$		−	15.7980i		−	0.796902i
$394$	18.8990			0.952117
$395$	15.3485	−	1.55051i	0.772265	−	0.0780146i
$396$	−14.6969			−0.738549
$397$			2.65153i			0.133077i	0.997784	+	0.0665383i	$0.0211954\pi$
$397$			2.65153i			0.133077i	−0.997784	+	0.0665383i	$0.978805\pi$
$398$		−	16.8990i		−	0.847069i
$399$		0			0
$400$	4.89898	−	1.00000i	0.244949	−	0.0500000i
$401$	−29.3939			−1.46786			−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939			−1.46786			−0.733930	+	0.679225i	$0.762316\pi$
$402$			19.5959i			0.977356i
$403$		−	39.5959i		−	1.97241i
$404$	3.55051			0.176644
$405$	20.0227	−	2.02270i	0.994936	−	0.100509i
$406$		0			0
$407$		−	9.79796i		−	0.485667i
$408$		−	4.89898i		−	0.242536i
$409$	34.4949			1.70566			0.852831	−	0.522186i	$-0.174883\pi$
$409$	34.4949			1.70566			0.852831	+	0.522186i	$0.174883\pi$
$410$	0.247449	+	2.44949i	0.0122206	+	0.120972i
$411$	−4.40408			−0.217237
$412$		−	12.8990i		−	0.635487i
$413$		0			0
$414$	8.69694			0.427431
$415$	0.550510	+	5.44949i	0.0270235	+	0.267505i
$416$	4.44949			0.218154
$417$		−	3.79796i		−	0.185987i
$418$			7.59592i			0.371528i
$419$	−1.55051			−0.0757474			−0.0378737	−	0.999283i	$-0.512058\pi$
$419$	−1.55051			−0.0757474			−0.0378737	+	0.999283i	$0.512058\pi$
$420$		0			0
$421$	−4.20204			−0.204795			−0.102397	−	0.994744i	$-0.532651\pi$
$421$	−4.20204			−0.204795			−0.102397	+	0.994744i	$0.532651\pi$
$422$		−	12.0000i		−	0.584151i
$423$			26.6969i			1.29805i
$424$	−10.8990			−0.529301
$425$	−2.00000	−	9.79796i	−0.0970143	−	0.475271i
$426$	2.69694			0.130667
$427$		0			0
$428$			8.00000i			0.386695i
$429$	−53.3939			−2.57788
$430$	−2.00000	+	0.202041i	−0.0964486	+	0.00974328i
$431$	−1.79796			−0.0866046			−0.0433023	−	0.999062i	$-0.513788\pi$
$431$	−1.79796			−0.0866046			−0.0433023	+	0.999062i	$0.513788\pi$
$432$		0			0
$433$		−	0.202041i		−	0.00970947i	−0.999988	−	0.00485474i	$-0.998455\pi$
$433$		−	0.202041i		−	0.00970947i	0.999988	−	0.00485474i	$-0.00154532\pi$
$434$		0			0
$435$	−3.79796	−	37.5959i	−0.182098	−	1.80259i
$436$	6.89898			0.330401
$437$		−	4.49490i		−	0.215020i
$438$		−	7.10102i		−	0.339300i
$439$	−21.3939			−1.02107			−0.510537	−	0.859856i	$-0.670553\pi$
$439$	−21.3939			−1.02107			−0.510537	+	0.859856i	$0.670553\pi$
$440$	1.10102	+	10.8990i	0.0524891	+	0.519588i
$441$		0			0
$442$		−	8.89898i		−	0.423281i
$443$		−	9.79796i		−	0.465515i	−0.972535	−	0.232758i	$-0.925225\pi$
$443$		−	9.79796i		−	0.465515i	0.972535	−	0.232758i	$-0.0747749\pi$
$444$	−4.89898			−0.232495
$445$	22.2474	−	2.24745i	1.05463	−	0.106539i
$446$	−4.00000			−0.189405
$447$		−	9.30306i		−	0.440020i
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$	3.00000	+	14.6969i	0.141421	+	0.692820i
$451$	−5.39388			−0.253988
$452$			19.7980i			0.931218i
$453$		−	48.0000i		−	2.25524i
$454$	−7.34847			−0.344881
$455$		0			0
$456$	3.79796			0.177856
$457$			29.5959i			1.38444i	0.721687	+	0.692219i	$0.243367\pi$
$457$			29.5959i			1.38444i	−0.721687	+	0.692219i	$0.756633\pi$
$458$			19.1464i			0.894654i
$459$		0			0
$460$	−0.651531	−	6.44949i	−0.0303778	−	0.300709i
$461$	−17.3485			−0.807999			−0.403999	−	0.914759i	$-0.632380\pi$
$461$	−17.3485			−0.807999			−0.403999	+	0.914759i	$0.632380\pi$
$462$		0			0
$463$		−	3.59592i		−	0.167116i	−0.996503	−	0.0835582i	$-0.973372\pi$
$463$		−	3.59592i		−	0.167116i	0.996503	−	0.0835582i	$-0.0266285\pi$
$464$	−6.89898			−0.320277
$465$	−4.89898	−	48.4949i	−0.227185	−	2.24890i
$466$	−29.7980			−1.38036
$467$		−	10.4495i		−	0.483545i	−0.970333	−	0.241772i	$-0.922271\pi$
$467$		−	10.4495i		−	0.483545i	0.970333	−	0.241772i	$-0.0777287\pi$
$468$			13.3485i			0.617033i
$469$		0			0
$470$	19.7980	−	2.00000i	0.913212	−	0.0922531i
$471$	−8.69694			−0.400734
$472$		−	1.55051i		−	0.0713680i
$473$		−	4.40408i		−	0.202500i
$474$	16.8990			0.776196
$475$	7.59592	−	1.55051i	0.348525	−	0.0711423i
$476$		0			0
$477$		−	32.6969i		−	1.49709i
$478$		−	6.20204i		−	0.283675i
$479$	9.30306			0.425068			0.212534	−	0.977154i	$-0.431828\pi$
$479$	9.30306			0.425068			0.212534	+	0.977154i	$0.431828\pi$
$480$	5.44949	−	0.550510i	0.248734	−	0.0251272i
$481$	−8.89898			−0.405759
$482$		−	8.69694i		−	0.396135i
$483$		0			0
$484$	−13.0000			−0.590909
$485$	3.55051	+	35.1464i	0.161220	+	1.59592i
$486$	22.0454			1.00000
$487$		−	7.30306i		−	0.330933i	−0.986215	−	0.165467i	$-0.947087\pi$
$487$		−	7.30306i		−	0.330933i	0.986215	−	0.165467i	$-0.0529130\pi$
$488$		−	3.55051i		−	0.160724i
$489$	17.3939			0.786578
$490$		0			0
$491$	19.5959			0.884351			0.442176	−	0.896928i	$-0.354207\pi$
$491$	19.5959			0.884351			0.442176	+	0.896928i	$0.354207\pi$
$492$			2.69694i			0.121587i
$493$			13.7980i			0.621429i
$494$	6.89898			0.310400
$495$	−32.6969	+	3.30306i	−1.46962	+	0.148462i
$496$	−8.89898			−0.399576
$497$		0			0
$498$			6.00000i			0.268866i
$499$	−6.20204			−0.277641			−0.138821	−	0.990318i	$-0.544331\pi$
$499$	−6.20204			−0.277641			−0.138821	+	0.990318i	$0.544331\pi$
$500$	10.6742	−	3.32577i	0.477366	−	0.148733i
$501$	12.0000			0.536120
$502$		−	6.44949i		−	0.287855i
$503$		−	4.00000i		−	0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$		−	4.00000i		−	0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$		0			0
$505$	7.89898	−	0.797959i	0.351500	−	0.0355087i
$506$	14.2020			0.631358
$507$			16.6515i			0.739520i
$508$			14.8990i			0.661035i
$509$	−31.5505			−1.39845			−0.699226	−	0.714901i	$-0.746472\pi$
$509$	−31.5505			−1.39845			−0.699226	+	0.714901i	$0.746472\pi$
$510$	−1.10102	−	10.8990i	−0.0487540	−	0.482615i
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	8.69694			0.383606
$515$	−2.89898	−	28.6969i	−0.127744	−	1.26454i
$516$	−2.20204			−0.0969395
$517$			43.5959i			1.91735i
$518$		0			0
$519$	−15.3031			−0.671730
$520$	9.89898	−	1.00000i	0.434099	−	0.0438529i
$521$	−32.6969			−1.43248			−0.716239	−	0.697855i	$-0.754138\pi$
$521$	−32.6969			−1.43248			−0.716239	+	0.697855i	$0.754138\pi$
$522$		−	20.6969i		−	0.905880i
$523$		−	33.1464i		−	1.44939i	−0.689069	−	0.724696i	$-0.741980\pi$
$523$		−	33.1464i		−	1.44939i	0.689069	−	0.724696i	$-0.258020\pi$
$524$	−6.44949			−0.281747
$525$		0			0
$526$	−9.79796			−0.427211
$527$			17.7980i			0.775291i
$528$			12.0000i			0.522233i
$529$	14.5959			0.634605
$530$	−24.2474	+	2.44949i	−1.05324	+	0.106399i
$531$	4.65153			0.201859
$532$		0			0
$533$			4.89898i			0.212198i
$534$	24.4949			1.06000
$535$	1.79796	+	17.7980i	0.0777325	+	0.769473i
$536$	8.00000			0.345547
$537$			33.7980i			1.45849i
$538$		−	19.1464i		−	0.825461i
$539$		0			0
$540$		0			0
$541$	9.59592			0.412561			0.206280	−	0.978493i	$-0.433864\pi$
$541$	9.59592			0.412561			0.206280	+	0.978493i	$0.433864\pi$
$542$			12.0000i			0.515444i
$543$		−	25.1010i		−	1.07719i
$544$	−2.00000			−0.0857493
$545$	15.3485	−	1.55051i	0.657456	−	0.0664166i
$546$		0			0
$547$		−	18.6969i		−	0.799423i	−0.916641	−	0.399712i	$-0.869110\pi$
$547$		−	18.6969i		−	0.799423i	0.916641	−	0.399712i	$-0.130890\pi$
$548$			1.79796i			0.0768050i
$549$	10.6515			0.454596
$550$	4.89898	+	24.0000i	0.208893	+	1.02336i
$551$	−10.6969			−0.455705
$552$		−	7.10102i		−	0.302240i
$553$		0			0
$554$	−14.8990			−0.632997
$555$	−10.8990	+	1.10102i	−0.462636	+	0.0467357i
$556$	−1.55051			−0.0657563
$557$			12.6969i			0.537987i	0.963142	+	0.268993i	$0.0866909\pi$
$557$			12.6969i			0.537987i	−0.963142	+	0.268993i	$0.913309\pi$
$558$		−	26.6969i		−	1.13017i
$559$	−4.00000			−0.169182
$560$		0			0
$561$	24.0000			1.01328
$562$			18.0000i			0.759284i
$563$		−	30.0454i		−	1.26626i	−0.774044	−	0.633131i	$-0.781769\pi$
$563$		−	30.0454i		−	1.26626i	0.774044	−	0.633131i	$-0.218231\pi$
$564$	21.7980			0.917860
$565$	4.44949	+	44.0454i	0.187191	+	1.85300i
$566$	3.75255			0.157731
$567$		0			0
$568$		−	1.10102i		−	0.0461978i
$569$	−33.7980			−1.41688			−0.708442	−	0.705769i	$-0.750602\pi$
$569$	−33.7980			−1.41688			−0.708442	+	0.705769i	$0.750602\pi$
$570$	8.44949	−	0.853572i	0.353910	−	0.0357522i
$571$	−11.1010			−0.464563			−0.232282	−	0.972649i	$-0.574619\pi$
$571$	−11.1010			−0.464563			−0.232282	+	0.972649i	$0.574619\pi$
$572$			21.7980i			0.911418i
$573$		−	31.1010i		−	1.29926i
$574$		0			0
$575$	−2.89898	−	14.2020i	−0.120896	−	0.592266i
$576$	3.00000			0.125000
$577$			2.49490i			0.103864i	0.998651	+	0.0519320i	$0.0165379\pi$
$577$			2.49490i			0.103864i	−0.998651	+	0.0519320i	$0.983462\pi$
$578$		−	13.0000i		−	0.540729i
$579$	52.8990			2.19841
$580$	−15.3485	+	1.55051i	−0.637310	+	0.0643814i
$581$		0			0
$582$			38.6969i			1.60404i
$583$		−	53.3939i		−	2.21135i
$584$	−2.89898			−0.119961
$585$	3.00000	+	29.6969i	0.124035	+	1.22782i
$586$	18.2474			0.753795
$587$		−	1.14643i		−	0.0473182i	−0.999720	−	0.0236591i	$-0.992468\pi$
$587$		−	1.14643i		−	0.0473182i	0.999720	−	0.0236591i	$-0.00753162\pi$
$588$		0			0
$589$	−13.7980			−0.568535
$590$	−0.348469	−	3.44949i	−0.0143463	−	0.142013i
$591$	46.2929			1.90423
$592$			2.00000i			0.0821995i
$593$		−	10.8990i		−	0.447567i	−0.974639	−	0.223784i	$-0.928159\pi$
$593$		−	10.8990i		−	0.447567i	0.974639	−	0.223784i	$-0.0718409\pi$
$594$		0			0
$595$		0			0
$596$	−3.79796			−0.155570
$597$		−	41.3939i		−	1.69414i
$598$		−	12.8990i		−	0.527478i
$599$	13.1010			0.535293			0.267647	−	0.963517i	$-0.413754\pi$
$599$	13.1010			0.535293			0.267647	+	0.963517i	$0.413754\pi$
$600$	12.0000	−	2.44949i	0.489898	−	0.100000i
$601$	39.3939			1.60691			0.803455	−	0.595366i	$-0.202993\pi$
$601$	39.3939			1.60691			0.803455	+	0.595366i	$0.202993\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$	−19.5959			−0.797347
$605$	−28.9217	+	2.92168i	−1.17583	+	0.118783i
$606$	8.69694			0.353289
$607$		−	33.3939i		−	1.35542i	−0.735331	−	0.677708i	$-0.762973\pi$
$607$		−	33.3939i		−	1.35542i	0.735331	−	0.677708i	$-0.237027\pi$
$608$		−	1.55051i		−	0.0628815i
$609$		0			0
$610$	−0.797959	−	7.89898i	−0.0323084	−	0.319820i
$611$	39.5959			1.60188
$612$		−	6.00000i		−	0.242536i
$613$			27.7980i			1.12275i	0.827562	+	0.561374i	$0.189727\pi$
$613$			27.7980i			1.12275i	−0.827562	+	0.561374i	$0.810273\pi$
$614$	20.2474			0.817121
$615$	0.606123	+	6.00000i	0.0244412	+	0.241943i
$616$		0			0
$617$			29.5959i			1.19149i	0.803175	+	0.595743i	$0.203142\pi$
$617$			29.5959i			1.19149i	−0.803175	+	0.595743i	$0.796858\pi$
$618$		−	31.5959i		−	1.27097i
$619$	−41.5505			−1.67006			−0.835028	−	0.550207i	$-0.814549\pi$
$619$	−41.5505			−1.67006			−0.835028	+	0.550207i	$0.814549\pi$
$620$	−19.7980	+	2.00000i	−0.795105	+	0.0803219i
$621$		0			0
$622$			12.0000i			0.481156i
$623$		0			0
$624$	10.8990			0.436308
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$	−21.5959			−0.863146
$627$			18.6061i			0.743057i
$628$			3.55051i			0.141681i
$629$	4.00000			0.159490
$630$		0			0
$631$	−42.4949			−1.69170			−0.845848	−	0.533425i	$-0.820905\pi$
$631$	−42.4949			−1.69170			−0.845848	+	0.533425i	$0.820905\pi$
$632$		−	6.89898i		−	0.274427i
$633$		−	29.3939i		−	1.16830i
$634$	−22.4949			−0.893387
$635$	3.34847	+	33.1464i	0.132880	+	1.31538i
$636$	−26.6969			−1.05860
$637$		0			0
$638$		−	33.7980i		−	1.33807i
$639$	3.30306			0.130667
$640$	−0.224745	−	2.22474i	−0.00888382	−	0.0879408i
$641$	25.7980			1.01896			0.509479	−	0.860483i	$-0.329838\pi$
$641$	25.7980			1.01896			0.509479	+	0.860483i	$0.329838\pi$
$642$			19.5959i			0.773389i
$643$			25.1464i			0.991678i	0.868414	+	0.495839i	$0.165139\pi$
$643$			25.1464i			0.991678i	−0.868414	+	0.495839i	$0.834861\pi$
$644$		0			0
$645$	−4.89898	+	0.494897i	−0.192897	+	0.0194866i
$646$	−3.10102			−0.122008
$647$			46.2929i			1.81996i	0.414652	+	0.909980i	$0.363903\pi$
$647$			46.2929i			1.81996i	−0.414652	+	0.909980i	$0.636097\pi$
$648$		−	9.00000i		−	0.353553i
$649$	7.59592			0.298166
$650$	21.7980	−	4.44949i	0.854986	−	0.174523i
$651$		0			0
$652$		−	7.10102i		−	0.278097i
$653$			20.2020i			0.790567i	0.918559	+	0.395283i	$0.129354\pi$
$653$			20.2020i			0.790567i	−0.918559	+	0.395283i	$0.870646\pi$
$654$	16.8990			0.660802
$655$	−14.3485	+	1.44949i	−0.560641	+	0.0566363i
$656$	1.10102			0.0429876
$657$		−	8.69694i		−	0.339300i
$658$		0			0
$659$	−16.8990			−0.658291			−0.329145	−	0.944279i	$-0.606761\pi$
$659$	−16.8990			−0.658291			−0.329145	+	0.944279i	$0.606761\pi$
$660$	2.69694	+	26.6969i	0.104978	+	1.03918i
$661$	40.9444			1.59255			0.796276	−	0.604933i	$-0.206800\pi$
$661$	40.9444			1.59255			0.796276	+	0.604933i	$0.206800\pi$
$662$			18.6969i			0.726677i
$663$		−	21.7980i		−	0.846563i
$664$	2.44949			0.0950586
$665$		0			0
$666$	−6.00000			−0.232495
$667$			20.0000i			0.774403i
$668$		−	4.89898i		−	0.189547i
$669$	−9.79796			−0.378811
$670$	17.7980	−	1.79796i	0.687595	−	0.0694612i
$671$	17.3939			0.671483
$672$		0			0
$673$			17.7980i			0.686061i	0.939324	+	0.343030i	$0.111453\pi$
$673$			17.7980i			0.686061i	−0.939324	+	0.343030i	$0.888547\pi$
$674$	9.59592			0.369621
$675$		0			0
$676$	6.79796			0.261460
$677$			36.4495i			1.40087i	0.713717	+	0.700434i	$0.247010\pi$
$677$			36.4495i			1.40087i	−0.713717	+	0.700434i	$0.752990\pi$
$678$			48.4949i			1.86244i
$679$		0			0
$680$	−4.44949	+	0.449490i	−0.170630	+	0.0172371i
$681$	−18.0000			−0.689761
$682$		−	43.5959i		−	1.66937i
$683$		−	3.59592i		−	0.137594i	−0.997631	−	0.0687970i	$-0.978084\pi$
$683$		−	3.59592i		−	0.137594i	0.997631	−	0.0687970i	$-0.0219161\pi$
$684$	4.65153			0.177856
$685$	0.404082	+	4.00000i	0.0154392	+	0.152832i
$686$		0			0
$687$			46.8990i			1.78931i
$688$			0.898979i			0.0342733i
$689$	−48.4949			−1.84751
$690$	−1.59592	−	15.7980i	−0.0607556	−	0.601418i
$691$	−21.1464			−0.804448			−0.402224	−	0.915541i	$-0.631763\pi$
$691$	−21.1464			−0.804448			−0.402224	+	0.915541i	$0.631763\pi$
$692$			6.24745i			0.237492i
$693$		0			0
$694$	28.8990			1.09699
$695$	−3.44949	+	0.348469i	−0.130847	+	0.0132182i
$696$	−16.8990			−0.640554
$697$		−	2.20204i		−	0.0834083i
$698$			8.44949i			0.319818i
$699$	−72.9898			−2.76073
$700$		0			0
$701$	11.3031			0.426911			0.213455	−	0.976953i	$-0.431528\pi$
$701$	11.3031			0.426911			0.213455	+	0.976953i	$0.431528\pi$
$702$		0			0
$703$			3.10102i			0.116957i
$704$	4.89898			0.184637
$705$	48.4949	−	4.89898i	1.82642	−	0.184506i
$706$	22.8990			0.861814
$707$		0			0
$708$		−	3.79796i		−	0.142736i
$709$	28.2929			1.06256			0.531280	−	0.847196i	$-0.321711\pi$
$709$	28.2929			1.06256			0.531280	+	0.847196i	$0.321711\pi$
$710$	−0.247449	−	2.44949i	−0.00928658	−	0.0919277i
$711$	20.6969			0.776196
$712$		−	10.0000i		−	0.374766i
$713$			25.7980i			0.966141i
$714$		0			0
$715$	4.89898	+	48.4949i	0.183211	+	1.81361i
$716$	13.7980			0.515654
$717$		−	15.1918i		−	0.567350i
$718$		−	27.5959i		−	1.02987i
$719$	−4.49490			−0.167631			−0.0838157	−	0.996481i	$-0.526711\pi$
$719$	−4.49490			−0.167631			−0.0838157	+	0.996481i	$0.526711\pi$
$720$	6.67423	−	0.674235i	0.248734	−	0.0251272i
$721$		0			0
$722$			16.5959i			0.617636i
$723$		−	21.3031i		−	0.792269i
$724$	−10.2474			−0.380843
$725$	−33.7980	+	6.89898i	−1.25522	+	0.256222i
$726$	−31.8434			−1.18182
$727$		−	22.6969i		−	0.841783i	−0.907111	−	0.420891i	$-0.861717\pi$
$727$		−	22.6969i		−	0.841783i	0.907111	−	0.420891i	$-0.138283\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	−6.44949	+	0.651531i	−0.238706	+	0.0241142i
$731$	1.79796			0.0664999
$732$		−	8.69694i		−	0.321448i
$733$			39.6413i			1.46419i	0.681205	+	0.732093i	$0.261456\pi$
$733$			39.6413i			1.46419i	−0.681205	+	0.732093i	$0.738544\pi$
$734$	−32.0000			−1.18114
$735$		0			0
$736$	−2.89898			−0.106858
$737$			39.1918i			1.44365i
$738$			3.30306i			0.121587i
$739$	−4.49490			−0.165347			−0.0826737	−	0.996577i	$-0.526346\pi$
$739$	−4.49490			−0.165347			−0.0826737	+	0.996577i	$0.526346\pi$
$740$	0.449490	+	4.44949i	0.0165236	+	0.163566i
$741$	16.8990			0.620800
$742$		0			0
$743$			44.6969i			1.63977i	0.572527	+	0.819886i	$0.305963\pi$
$743$			44.6969i			1.63977i	−0.572527	+	0.819886i	$0.694037\pi$
$744$	−21.7980			−0.799152
$745$	−8.44949	+	0.853572i	−0.309565	+	0.0312725i
$746$	4.69694			0.171967
$747$			7.34847i			0.268866i
$748$		−	9.79796i		−	0.358249i
$749$		0			0
$750$	26.1464	−	8.14643i	0.954733	−	0.297465i
$751$	−41.7980			−1.52523			−0.762615	−	0.646853i	$-0.776085\pi$
$751$	−41.7980			−1.52523			−0.762615	+	0.646853i	$0.776085\pi$
$752$		−	8.89898i		−	0.324512i
$753$		−	15.7980i		−	0.575710i
$754$	−30.6969			−1.11792
$755$	−43.5959	+	4.40408i	−1.58662	+	0.160281i
$756$		0			0
$757$		−	51.7980i		−	1.88263i	−0.337531	−	0.941314i	$-0.609592\pi$
$757$		−	51.7980i		−	1.88263i	0.337531	−	0.941314i	$-0.390408\pi$
$758$			30.6969i			1.11496i
$759$	34.7878			1.26272
$760$	−0.348469	−	3.44949i	−0.0126403	−	0.125126i
$761$	21.1010			0.764911			0.382456	−	0.923974i	$-0.375078\pi$
$761$	21.1010			0.764911			0.382456	+	0.923974i	$0.375078\pi$
$762$			36.4949i			1.32207i
$763$		0			0
$764$	−12.6969			−0.459359
$765$	−1.34847	−	13.3485i	−0.0487540	−	0.482615i
$766$	−7.10102			−0.256570
$767$		−	6.89898i		−	0.249108i
$768$		−	2.44949i		−	0.0883883i
$769$	−40.6969			−1.46757			−0.733785	−	0.679382i	$-0.762248\pi$
$769$	−40.6969			−1.46757			−0.733785	+	0.679382i	$0.762248\pi$
$770$		0			0
$771$	21.3031			0.767211
$772$		−	21.5959i		−	0.777254i
$773$			1.34847i			0.0485011i	0.999706	+	0.0242505i	$0.00771994\pi$
$773$			1.34847i			0.0485011i	−0.999706	+	0.0242505i	$0.992280\pi$
$774$	−2.69694			−0.0969395
$775$	−43.5959	+	8.89898i	−1.56601	+	0.319661i
$776$	15.7980			0.567114
$777$		0			0
$778$			13.1010i			0.469694i
$779$	1.70714			0.0611648
$780$	24.2474	−	2.44949i	0.868198	−	0.0877058i
$781$	5.39388			0.193008
$782$			5.79796i			0.207335i
$783$		0			0
$784$		0			0
$785$	0.797959	+	7.89898i	0.0284804	+	0.281927i
$786$	−15.7980			−0.563495
$787$		−	50.4495i		−	1.79833i	−0.437610	−	0.899165i	$-0.644175\pi$
$787$		−	50.4495i		−	1.79833i	0.437610	−	0.899165i	$-0.355825\pi$
$788$		−	18.8990i		−	0.673248i
$789$	−24.0000			−0.854423
$790$	−1.55051	−	15.3485i	−0.0551647

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:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.22474 + 0.224745i) q^{5} -2.44949 q^{6} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -2.44949i q^{3} -1.00000 q^{4} +(-2.22474 + 0.224745i) q^{5} -2.44949 q^{6} +1.00000i q^{8} -3.00000 q^{9} +(0.224745 + 2.22474i) q^{10} -4.89898 q^{11} +2.44949i q^{12} +4.44949i q^{13} +(0.550510 + 5.44949i) q^{15} +1.00000 q^{16} -2.00000i q^{17} +3.00000i q^{18} +1.55051 q^{19} +(2.22474 - 0.224745i) q^{20} +4.89898i q^{22} -2.89898i q^{23} +2.44949 q^{24} +(4.89898 - 1.00000i) q^{25} +4.44949 q^{26} -6.89898 q^{29} +(5.44949 - 0.550510i) q^{30} -8.89898 q^{31} -1.00000i q^{32} +12.0000i q^{33} -2.00000 q^{34} +3.00000 q^{36} +2.00000i q^{37} -1.55051i q^{38} +10.8990 q^{39} +(-0.224745 - 2.22474i) q^{40} +1.10102 q^{41} +0.898979i q^{43} +4.89898 q^{44} +(6.67423 - 0.674235i) q^{45} -2.89898 q^{46} -8.89898i q^{47} -2.44949i q^{48} +(-1.00000 - 4.89898i) q^{50} -4.89898 q^{51} -4.44949i q^{52} +10.8990i q^{53} +(10.8990 - 1.10102i) q^{55} -3.79796i q^{57} +6.89898i q^{58} -1.55051 q^{59} +(-0.550510 - 5.44949i) q^{60} -3.55051 q^{61} +8.89898i q^{62} -1.00000 q^{64} +(-1.00000 - 9.89898i) q^{65} +12.0000 q^{66} -8.00000i q^{67} +2.00000i q^{68} -7.10102 q^{69} -1.10102 q^{71} -3.00000i q^{72} +2.89898i q^{73} +2.00000 q^{74} +(-2.44949 - 12.0000i) q^{75} -1.55051 q^{76} -10.8990i q^{78} -6.89898 q^{79} +(-2.22474 + 0.224745i) q^{80} -9.00000 q^{81} -1.10102i q^{82} -2.44949i q^{83} +(0.449490 + 4.44949i) q^{85} +0.898979 q^{86} +16.8990i q^{87} -4.89898i q^{88} -10.0000 q^{89} +(-0.674235 - 6.67423i) q^{90} +2.89898i q^{92} +21.7980i q^{93} -8.89898 q^{94} +(-3.44949 + 0.348469i) q^{95} -2.44949 q^{96} -15.7980i q^{97} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^5 - 12 * q^9	\( 4 q - 4 q^{4} - 4 q^{5} - 12 q^{9} - 4 q^{10} + 12 q^{15} + 4 q^{16} + 16 q^{19} + 4 q^{20} + 8 q^{26} - 8 q^{29} + 12 q^{30} - 16 q^{31} - 8 q^{34} + 12 q^{36} + 24 q^{39} + 4 q^{40} + 24 q^{41} + 12 q^{45} + 8 q^{46} - 4 q^{50} + 24 q^{55} - 16 q^{59} - 12 q^{60} - 24 q^{61} - 4 q^{64} - 4 q^{65} + 48 q^{66} - 48 q^{69} - 24 q^{71} + 8 q^{74} - 16 q^{76} - 8 q^{79} - 4 q^{80} - 36 q^{81} - 8 q^{85} - 16 q^{86} - 40 q^{89} + 12 q^{90} - 16 q^{94} - 4 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^5 - 12 * q^9 - 4 * q^10 + 12 * q^15 + 4 * q^16 + 16 * q^19 + 4 * q^20 + 8 * q^26 - 8 * q^29 + 12 * q^30 - 16 * q^31 - 8 * q^34 + 12 * q^36 + 24 * q^39 + 4 * q^40 + 24 * q^41 + 12 * q^45 + 8 * q^46 - 4 * q^50 + 24 * q^55 - 16 * q^59 - 12 * q^60 - 24 * q^61 - 4 * q^64 - 4 * q^65 + 48 * q^66 - 48 * q^69 - 24 * q^71 + 8 * q^74 - 16 * q^76 - 8 * q^79 - 4 * q^80 - 36 * q^81 - 8 * q^85 - 16 * q^86 - 40 * q^89 + 12 * q^90 - 16 * q^94 - 4 * q^95

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