LMFDB - Embedded newform 300.3.f.b.199.13

Newspace parameters

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	300.f (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.17440793081$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
Defining polynomial:	$x^{16} + 5 x^{14} + 12 x^{12} + 25 x^{10} + 53 x^{8} + 100 x^{6} + 192 x^{4} + 320 x^{2} + 256$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.13
Root			$0.120653 + 1.40906i$ of defining polynomial
Character	$\chi$	$=$	300.199
Dual form			300.3.f.b.199.14

$q$-expansion

$f(q)$	$=$	$q+(1.08539 - 1.67986i) q^{2} +1.73205 q^{3} +(-1.64388 - 3.64660i) q^{4} +(1.87994 - 2.90961i) q^{6} -0.596540 q^{7} +(-7.91002 - 1.19648i) q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+(1.08539 - 1.67986i) q^{2} +1.73205 q^{3} +(-1.64388 - 3.64660i) q^{4} +(1.87994 - 2.90961i) q^{6} -0.596540 q^{7} +(-7.91002 - 1.19648i) q^{8} +3.00000 q^{9} -9.27963i q^{11} +(-2.84728 - 6.31609i) q^{12} -23.5117i q^{13} +(-0.647476 + 1.00210i) q^{14} +(-10.5953 + 11.9891i) q^{16} -3.97751i q^{17} +(3.25616 - 5.03959i) q^{18} -7.04756i q^{19} -1.03324 q^{21} +(-15.5885 - 10.0720i) q^{22} +32.0793 q^{23} +(-13.7006 - 2.07237i) q^{24} +(-39.4964 - 25.5192i) q^{26} +5.19615 q^{27} +(0.980637 + 2.17534i) q^{28} -35.6734 q^{29} +59.2585i q^{31} +(8.64000 + 30.8115i) q^{32} -16.0728i q^{33} +(-6.68167 - 4.31713i) q^{34} +(-4.93163 - 10.9398i) q^{36} +5.38761i q^{37} +(-11.8389 - 7.64932i) q^{38} -40.7234i q^{39} +40.0791 q^{41} +(-1.12146 + 1.73570i) q^{42} +36.1157 q^{43} +(-33.8391 + 15.2545i) q^{44} +(34.8184 - 53.8888i) q^{46} +74.0131 q^{47} +(-18.3517 + 20.7657i) q^{48} -48.6441 q^{49} -6.88925i q^{51} +(-85.7376 + 38.6503i) q^{52} -2.55123i q^{53} +(5.63983 - 8.72882i) q^{54} +(4.71864 + 0.713748i) q^{56} -12.2067i q^{57} +(-38.7194 + 59.9265i) q^{58} +36.4026i q^{59} -8.73223 q^{61} +(99.5461 + 64.3183i) q^{62} -1.78962 q^{63} +(61.1369 + 18.9284i) q^{64} +(-27.0001 - 17.4452i) q^{66} -69.7379 q^{67} +(-14.5044 + 6.53853i) q^{68} +55.5630 q^{69} +59.2170i q^{71} +(-23.7301 - 3.58944i) q^{72} -83.0019i q^{73} +(9.05044 + 5.84763i) q^{74} +(-25.6996 + 11.5853i) q^{76} +5.53566i q^{77} +(-68.4098 - 44.2006i) q^{78} +65.8705i q^{79} +9.00000 q^{81} +(43.5013 - 67.3274i) q^{82} +129.909 q^{83} +(1.69851 + 3.76780i) q^{84} +(39.1995 - 60.6695i) q^{86} -61.7882 q^{87} +(-11.1029 + 73.4020i) q^{88} +130.466 q^{89} +14.0256i q^{91} +(-52.7344 - 116.980i) q^{92} +102.639i q^{93} +(80.3327 - 124.332i) q^{94} +(14.9649 + 53.3671i) q^{96} -93.1113i q^{97} +(-52.7977 + 81.7155i) q^{98} -27.8389i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 20q^{4} - 12q^{6} + 48q^{9} + O(q^{10}) $	$ 16q - 20q^{4} - 12q^{6} + 48q^{9} + 40q^{14} + 68q^{16} - 96q^{21} - 36q^{24} - 72q^{26} - 128q^{29} + 184q^{34} - 60q^{36} - 32q^{41} - 344q^{44} + 304q^{46} + 112q^{49} - 36q^{54} + 232q^{56} - 352q^{61} + 220q^{64} + 216q^{66} + 192q^{69} - 264q^{74} - 48q^{76} + 144q^{81} + 72q^{84} - 400q^{86} - 160q^{89} + 192q^{94} - 348q^{96} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.08539	−	1.67986i	0.542693	−	0.839931i
$2$	1.08539	−	1.67986i	0.542693	−	0.839931i
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−1.64388	−	3.64660i	−0.410969	−	0.911649i
$5$		0			0
$5$		0			0
$6$	1.87994	−	2.90961i	0.313324	−	0.484935i
$7$	−0.596540			−0.0852199			−0.0426100	−	0.999092i	$-0.513567\pi$
$7$	−0.596540			−0.0852199			−0.0426100	+	0.999092i	$0.513567\pi$
$8$	−7.91002	−	1.19648i	−0.988753	−	0.149560i
$9$	3.00000			0.333333
$10$		0			0
$11$		−	9.27963i		−	0.843602i	−0.906688	−	0.421801i	$-0.861398\pi$
$11$		−	9.27963i		−	0.843602i	0.906688	−	0.421801i	$-0.138602\pi$
$12$	−2.84728	−	6.31609i	−0.237273	−	0.526341i
$13$		−	23.5117i		−	1.80859i	−0.426907	−	0.904295i	$-0.640397\pi$
$13$		−	23.5117i		−	1.80859i	0.426907	−	0.904295i	$-0.359603\pi$
$14$	−0.647476	+	1.00210i	−0.0462483	+	0.0715789i
$15$		0			0
$16$	−10.5953	+	11.9891i	−0.662209	+	0.749319i
$17$		−	3.97751i		−	0.233971i	−0.993134	−	0.116986i	$-0.962677\pi$
$17$		−	3.97751i		−	0.233971i	0.993134	−	0.116986i	$-0.0373231\pi$
$18$	3.25616	−	5.03959i	0.180898	−	0.279977i
$19$		−	7.04756i		−	0.370924i	−0.982651	−	0.185462i	$-0.940622\pi$
$19$		−	7.04756i		−	0.370924i	0.982651	−	0.185462i	$-0.0593782\pi$
$20$		0			0
$21$	−1.03324			−0.0492018
$22$	−15.5885	−	10.0720i	−0.708568	−	0.457817i
$23$	32.0793			1.39475			0.697376	−	0.716705i	$-0.254351\pi$
$23$	32.0793			1.39475			0.697376	+	0.716705i	$0.254351\pi$
$24$	−13.7006	−	2.07237i	−0.570857	−	0.0863486i
$25$		0			0
$26$	−39.4964	−	25.5192i	−1.51909	−	0.981509i
$27$	5.19615			0.192450
$28$	0.980637	+	2.17534i	0.0350227	+	0.0776907i
$29$	−35.6734			−1.23012			−0.615059	−	0.788481i	$-0.710868\pi$
$29$	−35.6734			−1.23012			−0.615059	+	0.788481i	$0.710868\pi$
$30$		0			0
$31$			59.2585i			1.91156i	0.294076	+	0.955782i	$0.404988\pi$
$31$			59.2585i			1.91156i	−0.294076	+	0.955782i	$0.595012\pi$
$32$	8.64000	+	30.8115i	0.270000	+	0.962860i
$33$		−	16.0728i		−	0.487054i
$34$	−6.68167	−	4.31713i	−0.196520	−	0.126974i
$35$		0			0
$36$	−4.93163	−	10.9398i	−0.136990	−	0.303883i
$37$			5.38761i			0.145611i	0.997346	+	0.0728055i	$0.0231952\pi$
$37$			5.38761i			0.145611i	−0.997346	+	0.0728055i	$0.976805\pi$
$38$	−11.8389	−	7.64932i	−0.311551	−	0.201298i
$39$		−	40.7234i		−	1.04419i
$40$		0			0
$41$	40.0791			0.977539			0.488769	−	0.872413i	$-0.337446\pi$
$41$	40.0791			0.977539			0.488769	+	0.872413i	$0.337446\pi$
$42$	−1.12146	+	1.73570i	−0.0267014	+	0.0413261i
$43$	36.1157			0.839901			0.419950	−	0.907547i	$-0.362047\pi$
$43$	36.1157			0.839901			0.419950	+	0.907547i	$0.362047\pi$
$44$	−33.8391	+	15.2545i	−0.769070	+	0.346694i
$45$		0			0
$46$	34.8184	−	53.8888i	0.756922	−	1.17150i
$47$	74.0131			1.57475			0.787373	−	0.616477i	$-0.211441\pi$
$47$	74.0131			1.57475			0.787373	+	0.616477i	$0.211441\pi$
$48$	−18.3517	+	20.7657i	−0.382327	+	0.432620i
$49$	−48.6441			−0.992738
$50$		0			0
$51$		−	6.88925i		−	0.135083i
$52$	−85.7376	+	38.6503i	−1.64880	+	0.743275i
$53$		−	2.55123i		−	0.0481364i	−0.999710	−	0.0240682i	$-0.992338\pi$
$53$		−	2.55123i		−	0.0481364i	0.999710	−	0.0240682i	$-0.00766189\pi$
$54$	5.63983	−	8.72882i	0.104441	−	0.161645i
$55$		0			0
$56$	4.71864	+	0.713748i	0.0842614	+	0.0127455i
$57$		−	12.2067i		−	0.214153i
$58$	−38.7194	+	59.9265i	−0.667576	+	1.03321i
$59$			36.4026i			0.616993i	0.951225	+	0.308497i	$0.0998259\pi$
$59$			36.4026i			0.616993i	−0.951225	+	0.308497i	$0.900174\pi$
$60$		0			0
$61$	−8.73223			−0.143151			−0.0715757	−	0.997435i	$-0.522803\pi$
$61$	−8.73223			−0.143151			−0.0715757	+	0.997435i	$0.522803\pi$
$62$	99.5461	+	64.3183i	1.60558	+	1.03739i
$63$	−1.78962			−0.0284066
$64$	61.1369	+	18.9284i	0.955264	+	0.295756i
$65$		0			0
$66$	−27.0001	−	17.4452i	−0.409092	−	0.264321i
$67$	−69.7379			−1.04086			−0.520432	−	0.853903i	$-0.674229\pi$
$67$	−69.7379			−1.04086			−0.520432	+	0.853903i	$0.674229\pi$
$68$	−14.5044	+	6.53853i	−0.213300	+	0.0961548i
$69$	55.5630			0.805261
$70$		0			0
$71$			59.2170i			0.834043i	0.908897	+	0.417021i	$0.136926\pi$
$71$			59.2170i			0.834043i	−0.908897	+	0.417021i	$0.863074\pi$
$72$	−23.7301	−	3.58944i	−0.329584	−	0.0498534i
$73$		−	83.0019i		−	1.13701i	−0.822679	−	0.568506i	$-0.807522\pi$
$73$		−	83.0019i		−	1.13701i	0.822679	−	0.568506i	$-0.192478\pi$
$74$	9.05044	+	5.84763i	0.122303	+	0.0790220i
$75$		0			0
$76$	−25.6996	+	11.5853i	−0.338153	+	0.152438i
$77$			5.53566i			0.0718917i
$78$	−68.4098	−	44.2006i	−0.877048	−	0.566675i
$79$			65.8705i			0.833804i	0.908951	+	0.416902i	$0.136884\pi$
$79$			65.8705i			0.833804i	−0.908951	+	0.416902i	$0.863116\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	43.5013	−	67.3274i	0.530503	−	0.821065i
$83$	129.909			1.56517			0.782586	−	0.622542i	$-0.213900\pi$
$83$	129.909			1.56517			0.782586	+	0.622542i	$0.213900\pi$
$84$	1.69851	+	3.76780i	0.0202204	+	0.0448547i
$85$		0			0
$86$	39.1995	−	60.6695i	0.455808	−	0.705459i
$87$	−61.7882			−0.710209
$88$	−11.1029	+	73.4020i	−0.126169	+	0.834114i
$89$	130.466			1.46591			0.732956	−	0.680277i	$-0.238140\pi$
$89$	130.466			1.46591			0.732956	+	0.680277i	$0.238140\pi$
$90$		0			0
$91$			14.0256i			0.154128i
$92$	−52.7344	−	116.980i	−0.573200	−	1.27152i
$93$			102.639i			1.10364i
$94$	80.3327	−	124.332i	0.854603	−	1.32268i
$95$		0			0
$96$	14.9649	+	53.3671i	0.155885	+	0.555908i
$97$		−	93.1113i		−	0.959911i	−0.877293	−	0.479955i	$-0.840653\pi$
$97$		−	93.1113i		−	0.959911i	0.877293	−	0.479955i	$-0.159347\pi$
$98$	−52.7977	+	81.7155i	−0.538752	+	0.833831i
$99$		−	27.8389i		−	0.281201i
$100$		0			0
$101$	−3.66081			−0.0362457			−0.0181228	−	0.999836i	$-0.505769\pi$
$101$	−3.66081			−0.0362457			−0.0181228	+	0.999836i	$0.505769\pi$
$102$	−11.5730	−	7.47749i	−0.113461	−	0.0733087i
$103$	−151.417			−1.47007			−0.735033	−	0.678032i	$-0.762833\pi$
$103$	−151.417			−1.47007			−0.735033	+	0.678032i	$0.762833\pi$
$104$	−28.1313	+	185.978i	−0.270493	+	1.78825i
$105$		0			0
$106$	−4.28571	−	2.76907i	−0.0404313	−	0.0261233i
$107$	82.8092			0.773918			0.386959	−	0.922097i	$-0.373526\pi$
$107$	82.8092			0.773918			0.386959	+	0.922097i	$0.373526\pi$
$108$	−8.54183	−	18.9483i	−0.0790910	−	0.175447i
$109$	7.36835			0.0675996			0.0337998	−	0.999429i	$-0.489239\pi$
$109$	7.36835			0.0675996			0.0337998	+	0.999429i	$0.489239\pi$
$110$		0			0
$111$			9.33161i			0.0840685i
$112$	6.32054	−	7.15197i	0.0564334	−	0.0638569i
$113$			65.0370i			0.575549i	0.957698	+	0.287774i	$0.0929153\pi$
$113$			65.0370i			0.575549i	−0.957698	+	0.287774i	$0.907085\pi$
$114$	−20.5056	−	13.2490i	−0.179874	−	0.116219i
$115$		0			0
$116$	58.6427	+	130.087i	0.505540	+	1.12144i
$117$		−	70.5350i		−	0.602864i
$118$	61.1514	+	39.5109i	0.518232	+	0.334838i
$119$			2.37274i			0.0199390i
$120$		0			0
$121$	34.8885			0.288335
$122$	−9.47784	+	14.6689i	−0.0776872	+	0.120237i
$123$	69.4190			0.564382
$124$	216.092	−	97.4136i	1.74268	−	0.785593i
$125$		0			0
$126$	−1.94243	+	3.00631i	−0.0154161	+	0.0238596i
$127$	139.469			1.09818			0.549091	−	0.835763i	$-0.314974\pi$
$127$	139.469			1.09818			0.549091	+	0.835763i	$0.314974\pi$
$128$	98.1542	−	82.1569i	0.766829	−	0.641851i
$129$	62.5543			0.484917
$130$		0			0
$131$		−	63.4856i		−	0.484623i	−0.970198	−	0.242312i	$-0.922094\pi$
$131$		−	63.4856i		−	0.484623i	0.970198	−	0.242312i	$-0.0779056\pi$
$132$	−58.6110	+	26.4217i	−0.444023	+	0.200164i
$133$			4.20415i			0.0316101i
$134$	−75.6925	+	117.150i	−0.564870	+	0.874254i
$135$		0			0
$136$	−4.75901	+	31.4622i	−0.0349927	+	0.231340i
$137$			138.157i			1.00845i	0.863573	+	0.504223i	$0.168221\pi$
$137$			138.157i			1.00845i	−0.863573	+	0.504223i	$0.831779\pi$
$138$	60.3073	−	93.3382i	0.437009	−	0.676363i
$139$		−	29.9578i		−	0.215523i	−0.994177	−	0.107762i	$-0.965632\pi$
$139$		−	29.9578i		−	0.215523i	0.994177	−	0.107762i	$-0.0343684\pi$
$140$		0			0
$141$	128.194			0.909180
$142$	99.4765	+	64.2733i	0.700539	+	0.452629i
$143$	−218.180			−1.52573
$144$	−31.7860	+	35.9673i	−0.220736	+	0.249773i
$145$		0			0
$146$	−139.432	−	90.0891i	−0.955012	−	0.617048i
$147$	−84.2541			−0.573157
$148$	19.6464	−	8.85655i	0.132746	−	0.0598416i
$149$	47.3823			0.318002			0.159001	−	0.987278i	$-0.449173\pi$
$149$	47.3823			0.318002			0.159001	+	0.987278i	$0.449173\pi$
$150$		0			0
$151$		−	109.604i		−	0.725852i	−0.931818	−	0.362926i	$-0.881778\pi$
$151$		−	109.604i		−	0.725852i	0.931818	−	0.362926i	$-0.118222\pi$
$152$	−8.43227	+	55.7463i	−0.0554755	+	0.366752i
$153$		−	11.9325i		−	0.0779904i
$154$	9.29915	+	6.00833i	0.0603841	+	0.0390151i
$155$		0			0
$156$	−148.502	+	66.9442i	−0.951936	+	0.429130i
$157$			177.588i			1.13113i	0.824703	+	0.565566i	$0.191342\pi$
$157$			177.588i			1.13113i	−0.824703	+	0.565566i	$0.808658\pi$
$158$	110.653	+	71.4950i	0.700338	+	0.452500i
$159$		−	4.41886i		−	0.0277916i
$160$		0			0
$161$	−19.1366			−0.118861
$162$	9.76847	−	15.1188i	0.0602992	−	0.0933257i
$163$	−96.8778			−0.594342			−0.297171	−	0.954824i	$-0.596043\pi$
$163$	−96.8778			−0.594342			−0.297171	+	0.954824i	$0.596043\pi$
$164$	−65.8850	−	146.152i	−0.401738	−	0.891173i
$165$		0			0
$166$	141.002	−	218.230i	0.849408	−	1.31464i
$167$	−152.605			−0.913801			−0.456901	−	0.889518i	$-0.651040\pi$
$167$	−152.605			−0.913801			−0.456901	+	0.889518i	$0.651040\pi$
$168$	8.17292	+	1.23625i	0.0486484	+	0.00735862i
$169$	−383.799			−2.27100
$170$		0			0
$171$		−	21.1427i		−	0.123641i
$172$	−59.3698	−	131.700i	−0.345173	−	0.765695i
$173$			155.773i			0.900422i	0.892922	+	0.450211i	$0.148651\pi$
$173$			155.773i			0.900422i	−0.892922	+	0.450211i	$0.851349\pi$
$174$	−67.0640	+	103.796i	−0.385425	+	0.596527i
$175$		0			0
$176$	111.254	+	98.3209i	0.632127	+	0.558641i
$177$			63.0512i			0.356221i
$178$	141.606	−	219.165i	0.795540	−	1.23126i
$179$		−	126.001i		−	0.703915i	−0.936016	−	0.351957i	$-0.885516\pi$
$179$		−	126.001i		−	0.703915i	0.936016	−	0.351957i	$-0.114484\pi$
$180$		0			0
$181$	−346.725			−1.91561			−0.957803	−	0.287424i	$-0.907201\pi$
$181$	−346.725			−1.91561			−0.957803	+	0.287424i	$0.907201\pi$
$182$	23.5612	+	15.2232i	0.129457	+	0.0836442i
$183$	−15.1247			−0.0826485
$184$	−253.748	−	38.3823i	−1.37906	−	0.208599i
$185$		0			0
$186$	172.419	+	111.403i	0.926983	+	0.598939i
$187$	−36.9098			−0.197379
$188$	−121.668	−	269.896i	−0.647171	−	1.43562i
$189$	−3.09971			−0.0164006
$190$		0			0
$191$		−	133.159i		−	0.697167i	−0.937278	−	0.348584i	$-0.886663\pi$
$191$		−	133.159i		−	0.697167i	0.937278	−	0.348584i	$-0.113337\pi$
$192$	105.892	+	32.7849i	0.551522	+	0.170755i
$193$		−	136.246i		−	0.705940i	−0.935635	−	0.352970i	$-0.885172\pi$
$193$		−	136.246i		−	0.705940i	0.935635	−	0.352970i	$-0.114828\pi$
$194$	−156.414	−	101.062i	−0.806259	−	0.520937i
$195$		0			0
$196$	79.9649	+	177.386i	0.407984	+	0.905029i
$197$		−	74.8945i		−	0.380175i	−0.981767	−	0.190087i	$-0.939123\pi$
$197$		−	74.8945i		−	0.380175i	0.981767	−	0.190087i	$-0.0608772\pi$
$198$	−46.7655	−	30.2159i	−0.236189	−	0.152606i
$199$			251.605i			1.26434i	0.774828	+	0.632172i	$0.217836\pi$
$199$			251.605i			1.26434i	−0.774828	+	0.632172i	$0.782164\pi$
$200$		0			0
$201$	−120.790			−0.600943
$202$	−3.97339	+	6.14966i	−0.0196703	+	0.0304439i
$203$	21.2806			0.104831
$204$	−25.1223	+	11.3251i	−0.123149	+	0.0555150i
$205$		0			0
$206$	−164.346	+	254.359i	−0.797794	+	1.23475i
$207$	96.2379			0.464917
$208$	281.884	+	249.114i	1.35521	+	1.19767i
$209$	−65.3987			−0.312913
$210$		0			0
$211$		−	228.203i		−	1.08153i	−0.841173	−	0.540766i	$-0.818135\pi$
$211$		−	228.203i		−	1.08153i	0.841173	−	0.540766i	$-0.181865\pi$
$212$	−9.30331	+	4.19390i	−0.0438835	+	0.0197826i
$213$			102.567i			0.481535i
$214$	89.8799	−	139.108i	0.420000	−	0.650038i
$215$		0			0
$216$	−41.1017	−	6.21710i	−0.190286	−	0.0287829i
$217$		−	35.3500i		−	0.162903i
$218$	7.99751	−	12.3778i	0.0366858	−	0.0567790i
$219$		−	143.763i		−	0.656454i
$220$		0			0
$221$	−93.5179			−0.423158
$222$	15.6758	+	10.1284i	0.0706118	+	0.0456234i
$223$	−85.9549			−0.385448			−0.192724	−	0.981253i	$-0.561732\pi$
$223$	−85.9549			−0.385448			−0.192724	+	0.981253i	$0.561732\pi$
$224$	−5.15410	−	18.3803i	−0.0230094	−	0.0820549i
$225$		0			0
$226$	109.253	+	70.5902i	0.483421	+	0.312346i
$227$	−282.357			−1.24386			−0.621932	−	0.783071i	$-0.713652\pi$
$227$	−282.357			−1.24386			−0.621932	+	0.783071i	$0.713652\pi$
$228$	−44.5130	+	20.0663i	−0.195233	+	0.0880103i
$229$	−138.263			−0.603768			−0.301884	−	0.953345i	$-0.597615\pi$
$229$	−138.263			−0.603768			−0.301884	+	0.953345i	$0.597615\pi$
$230$		0			0
$231$			9.58805i			0.0415067i
$232$	282.178	+	42.6826i	1.21628	+	0.183977i
$233$		−	0.522939i		−	0.00224438i	−0.999999	−	0.00112219i	$-0.999643\pi$
$233$		−	0.522939i		−	0.00224438i	0.999999	−	0.00112219i	$-0.000357204\pi$
$234$	−118.489	−	76.5577i	−0.506364	−	0.327170i
$235$		0			0
$236$	132.746	−	59.8413i	0.562482	−	0.253565i
$237$			114.091i			0.481397i
$238$	3.98588	+	2.57534i	0.0167474	+	0.0108208i
$239$			73.6928i			0.308338i	0.988044	+	0.154169i	$0.0492700\pi$
$239$			73.6928i			0.308338i	−0.988044	+	0.154169i	$0.950730\pi$
$240$		0			0
$241$	31.3705			0.130168			0.0650840	−	0.997880i	$-0.479268\pi$
$241$	31.3705			0.130168			0.0650840	+	0.997880i	$0.479268\pi$
$242$	37.8675	−	58.6080i	0.156477	−	0.242182i
$243$	15.5885			0.0641500
$244$	14.3547	+	31.8429i	0.0588307	+	0.130504i
$245$		0			0
$246$	75.3464	−	116.614i	0.306286	−	0.474042i
$247$	−165.700			−0.670850
$248$	70.9016	−	468.736i	0.285894	−	1.89006i
$249$	225.010			0.903653
$250$		0			0
$251$		−	78.7478i		−	0.313736i	−0.987620	−	0.156868i	$-0.949860\pi$
$251$		−	78.7478i		−	0.313736i	0.987620	−	0.156868i	$-0.0501398\pi$
$252$	2.94191	+	6.52602i	0.0116742	+	0.0258969i
$253$		−	297.684i		−	1.17662i
$254$	151.378	−	234.289i	0.595975	−	0.922397i
$255$		0			0
$256$	−31.4772	−	254.057i	−0.122958	−	0.992412i
$257$		−	243.954i		−	0.949236i	−0.880192	−	0.474618i	$-0.842586\pi$
$257$		−	243.954i		−	0.949236i	0.880192	−	0.474618i	$-0.157414\pi$
$258$	67.8956	−	105.083i	0.263161	−	0.407297i
$259$		−	3.21392i		−	0.0124090i
$260$		0			0
$261$	−107.020			−0.410039
$262$	−106.647	−	68.9064i	−0.407050	−	0.263002i
$263$	102.737			0.390635			0.195317	−	0.980740i	$-0.437426\pi$
$263$	102.737			0.390635			0.195317	+	0.980740i	$0.437426\pi$
$264$	−19.2308	+	127.136i	−0.0728439	+	0.481576i
$265$		0			0
$266$	7.06239	+	4.56312i	0.0265503	+	0.0171546i
$267$	225.974			0.846344
$268$	114.640	+	254.306i	0.427763	+	0.948903i
$269$	123.646			0.459651			0.229825	−	0.973232i	$-0.426184\pi$
$269$	123.646			0.459651			0.229825	+	0.973232i	$0.426184\pi$
$270$		0			0
$271$			332.371i			1.22646i	0.789904	+	0.613230i	$0.210130\pi$
$271$			332.371i			1.22646i	−0.789904	+	0.613230i	$0.789870\pi$
$272$	47.6868	+	42.1431i	0.175319	+	0.154938i
$273$			24.2931i			0.0889858i
$274$	232.085	+	149.954i	0.847026	+	0.547277i
$275$		0			0
$276$	−91.3386	−	202.616i	−0.330937	−	0.734115i
$277$			125.916i			0.454571i	0.973828	+	0.227286i	$0.0729851\pi$
$277$			125.916i			0.454571i	−0.973828	+	0.227286i	$0.927015\pi$
$278$	−50.3249	−	32.5157i	−0.181025	−	0.116963i
$279$			177.775i			0.637188i
$280$		0			0
$281$	52.5628			0.187056			0.0935281	−	0.995617i	$-0.470186\pi$
$281$	52.5628			0.187056			0.0935281	+	0.995617i	$0.470186\pi$
$282$	139.140	−	215.349i	0.493405	−	0.763649i
$283$	−199.288			−0.704199			−0.352100	−	0.935963i	$-0.614532\pi$
$283$	−199.288			−0.704199			−0.352100	+	0.935963i	$0.614532\pi$
$284$	215.941	−	97.3454i	0.760355	−	0.342766i
$285$		0			0
$286$	−236.809	+	366.512i	−0.828004	+	1.28151i
$287$	−23.9088			−0.0833058
$288$	25.9200	+	92.4346i	0.0900001	+	0.320953i
$289$	273.179			0.945258
$290$		0			0
$291$		−	161.274i		−	0.554205i
$292$	−302.674	+	136.445i	−1.03656	+	0.467276i
$293$			102.161i			0.348672i	0.984686	+	0.174336i	$0.0557779\pi$
$293$			102.161i			0.348672i	−0.984686	+	0.174336i	$0.944222\pi$
$294$	−91.4482	+	141.535i	−0.311048	+	0.481413i
$295$		0			0
$296$	6.44617	−	42.6161i	0.0217776	−	0.143973i
$297$		−	48.2184i		−	0.162351i
$298$	51.4281	−	79.5957i	0.172577	−	0.267100i
$299$		−	754.238i		−	2.52254i
$300$		0			0
$301$	−21.5445			−0.0715763
$302$	−184.119	−	118.962i	−0.609666	−	0.393915i
$303$	−6.34071			−0.0209264
$304$	84.4939	+	74.6713i	0.277941	+	0.245629i
$305$		0			0
$306$	−20.0450	−	12.9514i	−0.0655065	−	0.0423248i
$307$	328.391			1.06968			0.534839	−	0.844954i	$-0.320372\pi$
$307$	328.391			1.06968			0.534839	+	0.844954i	$0.320372\pi$
$308$	20.1863	−	9.09994i	0.0655401	−	0.0295453i
$309$	−262.262			−0.848743
$310$		0			0
$311$			95.4377i			0.306874i	0.988158	+	0.153437i	$0.0490342\pi$
$311$			95.4377i			0.306874i	−0.988158	+	0.153437i	$0.950966\pi$
$312$	−48.7248	+	322.123i	−0.156169	+	1.03245i
$313$			550.408i			1.75849i	0.476368	+	0.879246i	$0.341953\pi$
$313$			550.408i			1.75849i	−0.476368	+	0.879246i	$0.658047\pi$
$314$	298.323	+	192.751i	0.950073	+	0.613858i
$315$		0			0
$316$	240.203	−	108.283i	0.760137	−	0.342668i
$317$		−	439.394i		−	1.38610i	−0.720889	−	0.693051i	$-0.756266\pi$
$317$		−	439.394i		−	1.38610i	0.720889	−	0.693051i	$-0.243734\pi$
$318$	−7.42307	−	4.79617i	−0.0233430	−	0.0150823i
$319$			331.036i			1.03773i
$320$		0			0
$321$	143.430			0.446822
$322$	−20.7706	+	32.1468i	−0.0645048	+	0.0998348i
$323$	−28.0317			−0.0867855
$324$	−14.7949	−	32.8194i	−0.0456632	−	0.101294i
$325$		0			0
$326$	−105.150	+	162.741i	−0.322545	+	0.499206i
$327$	12.7624			0.0390286
$328$	−317.026	−	47.9539i	−0.966544	−	0.146201i
$329$	−44.1517			−0.134200
$330$		0			0
$331$			479.922i			1.44992i	0.688794	+	0.724958i	$0.258141\pi$
$331$			479.922i			1.44992i	−0.688794	+	0.724958i	$0.741859\pi$
$332$	−213.555	−	473.727i	−0.643237	−	1.42689i
$333$			16.1628i			0.0485370i
$334$	−165.635	+	256.355i	−0.495913	+	0.767530i
$335$		0			0
$336$	10.9475	−	12.3876i	0.0325819	−	0.0368678i
$337$		−	58.8437i		−	0.174610i	−0.996182	−	0.0873052i	$-0.972174\pi$
$337$		−	58.8437i		−	0.174610i	0.996182	−	0.0873052i	$-0.0278255\pi$
$338$	−416.570	+	644.730i	−1.23246	+	1.90748i
$339$			112.647i			0.332293i
$340$		0			0
$341$	549.897			1.61260
$342$	−35.5168	−	22.9480i	−0.103850	−	0.0670993i
$343$	58.2486			0.169821
$344$	−285.676	−	43.2118i	−0.830454	−	0.125616i
$345$		0			0
$346$	261.677	+	169.074i	0.756293	+	0.488653i
$347$	−12.1484			−0.0350099			−0.0175049	−	0.999847i	$-0.505572\pi$
$347$	−12.1484			−0.0350099			−0.0175049	+	0.999847i	$0.505572\pi$
$348$	101.572	+	225.317i	0.291874	+	0.647462i
$349$	30.9277			0.0886180			0.0443090	−	0.999018i	$-0.485891\pi$
$349$	30.9277			0.0886180			0.0443090	+	0.999018i	$0.485891\pi$
$350$		0			0
$351$		−	122.170i		−	0.348063i
$352$	285.919	−	80.1760i	0.812271	−	0.227773i
$353$		−	288.065i		−	0.816048i	−0.912971	−	0.408024i	$-0.866218\pi$
$353$		−	288.065i		−	0.816048i	0.912971	−	0.408024i	$-0.133782\pi$
$354$	105.917	+	68.4348i	0.299201	+	0.193319i
$355$		0			0
$356$	−214.470	−	475.757i	−0.602444	−	1.33640i
$357$			4.10971i			0.0115118i
$358$	−211.664	−	136.759i	−0.591240	−	0.382010i
$359$			663.911i			1.84933i	0.380776	+	0.924667i	$0.375657\pi$
$359$			663.911i			1.84933i	−0.380776	+	0.924667i	$0.624343\pi$
$360$		0			0
$361$	311.332			0.862415
$362$	−376.330	+	582.450i	−1.03959	+	1.60898i
$363$	60.4287			0.166470
$364$	51.1459	−	23.0564i	0.140511	−	0.0633418i
$365$		0			0
$366$	−16.4161	+	25.4074i	−0.0448527	+	0.0694190i
$367$	−6.08529			−0.0165812			−0.00829059	−	0.999966i	$-0.502639\pi$
$367$	−6.08529			−0.0165812			−0.00829059	+	0.999966i	$0.502639\pi$
$368$	−339.891	+	384.602i	−0.923618	+	1.04511i
$369$	120.237			0.325846
$370$		0			0
$371$			1.52191i			0.00410218i
$372$	374.282	−	168.725i	1.00613	−	0.453562i
$373$		−	204.741i		−	0.548903i	−0.961601	−	0.274451i	$-0.911504\pi$
$373$		−	204.741i		−	0.548903i	0.961601	−	0.274451i	$-0.0884963\pi$
$374$	−40.0614	+	62.0034i	−0.107116	+	0.165784i
$375$		0			0
$376$	−585.445	−	88.5552i	−1.55703	−	0.235519i
$377$			838.742i			2.22478i
$378$	−3.36438	+	5.20709i	−0.00890048	+	0.0137754i
$379$		−	402.331i		−	1.06156i	−0.847510	−	0.530780i	$-0.821899\pi$
$379$		−	402.331i		−	1.06156i	0.847510	−	0.530780i	$-0.178101\pi$
$380$		0			0
$381$	241.568			0.634036
$382$	−223.689	−	144.529i	−0.585573	−	0.378348i
$383$	−331.751			−0.866191			−0.433096	−	0.901348i	$-0.642579\pi$
$383$	−331.751			−0.866191			−0.433096	+	0.901348i	$0.642579\pi$
$384$	170.008	−	142.300i	0.442729	−	0.370573i
$385$		0			0
$386$	−228.875	−	147.880i	−0.592941	−	0.383109i
$387$	108.347			0.279967
$388$	−339.540	+	153.063i	−0.875102	+	0.394493i
$389$	623.310			1.60234			0.801169	−	0.598438i	$-0.204212\pi$
$389$	623.310			1.60234			0.801169	+	0.598438i	$0.204212\pi$
$390$		0			0
$391$		−	127.596i		−	0.326332i
$392$	384.776	+	58.2018i	0.981572	+	0.148474i
$393$		−	109.960i		−	0.279797i
$394$	−125.812	−	81.2894i	−0.319321	−	0.206318i
$395$		0			0
$396$	−101.517	+	45.7636i	−0.256357	+	0.115565i
$397$			355.449i			0.895338i	0.894199	+	0.447669i	$0.147746\pi$
$397$			355.449i			0.895338i	−0.894199	+	0.447669i	$0.852254\pi$
$398$	422.661	+	273.088i	1.06196	+	0.686151i
$399$			7.28180i			0.0182501i
$400$		0			0
$401$	−542.927			−1.35393			−0.676966	−	0.736014i	$-0.736706\pi$
$401$	−542.927			−1.35393			−0.676966	+	0.736014i	$0.736706\pi$
$402$	−131.103	+	202.910i	−0.326128	+	0.504751i
$403$	1393.27			3.45724
$404$	6.01792	+	13.3495i	0.0148958	+	0.0330433i
$405$		0			0
$406$	23.0977	−	35.7485i	0.0568908	−	0.0880505i
$407$	49.9950			0.122838
$408$	−8.24285	+	54.4941i	−0.0202031	+	0.133564i
$409$	108.497			0.265273			0.132636	−	0.991165i	$-0.457656\pi$
$409$	108.497			0.265273			0.132636	+	0.991165i	$0.457656\pi$
$410$		0			0
$411$			239.295i			0.582227i
$412$	248.910	+	552.156i	0.604151	+	1.34018i
$413$		−	21.7156i		−	0.0525801i
$414$	104.455	−	161.666i	0.252307	−	0.390499i
$415$		0			0
$416$	724.431	−	203.141i	1.74142	−	0.488320i
$417$		−	51.8884i		−	0.124433i
$418$	−70.9828	+	109.861i	−0.169815	+	0.262825i
$419$		−	172.176i		−	0.410921i	−0.978665	−	0.205460i	$-0.934131\pi$
$419$		−	172.176i		−	0.410921i	0.978665	−	0.205460i	$-0.0658691\pi$
$420$		0			0
$421$	478.522			1.13663			0.568316	−	0.822810i	$-0.307595\pi$
$421$	478.522			1.13663			0.568316	+	0.822810i	$0.307595\pi$
$422$	−383.350	−	247.688i	−0.908412	−	0.586940i
$423$	222.039			0.524915
$424$	−3.05250	+	20.1803i	−0.00719929	+	0.0475950i
$425$		0			0
$426$	172.298	+	111.325i	0.404456	+	0.261326i
$427$	5.20912			0.0121993
$428$	−136.128	−	301.972i	−0.318056	−	0.705541i
$429$	−377.898			−0.880882
$430$		0			0
$431$			290.722i			0.674530i	0.941410	+	0.337265i	$0.109502\pi$
$431$			290.722i			0.674530i	−0.941410	+	0.337265i	$0.890498\pi$
$432$	−55.0550	+	62.2972i	−0.127442	+	0.144207i
$433$			53.7726i			0.124186i	0.998070	+	0.0620931i	$0.0197776\pi$
$433$			53.7726i			0.124186i	−0.998070	+	0.0620931i	$0.980222\pi$
$434$	−59.3832	−	38.3684i	−0.136828	−	0.0884065i
$435$		0			0
$436$	−12.1127	−	26.8694i	−0.0277813	−	0.0616271i
$437$		−	226.081i		−	0.517347i
$438$	−241.503	−	156.039i	−0.551376	−	0.356253i
$439$			328.657i			0.748650i	0.927298	+	0.374325i	$0.122126\pi$
$439$			328.657i			0.748650i	−0.927298	+	0.374325i	$0.877874\pi$
$440$		0			0
$441$	−145.932			−0.330913
$442$	−101.503	+	157.097i	−0.229645	+	0.355424i
$443$	428.910			0.968194			0.484097	−	0.875014i	$-0.339148\pi$
$443$	428.910			0.968194			0.484097	+	0.875014i	$0.339148\pi$
$444$	34.0286	−	15.3400i	0.0766410	−	0.0345496i
$445$		0			0
$446$	−93.2942	+	144.392i	−0.209180	+	0.323750i
$447$	82.0685			0.183599
$448$	−36.4706	−	11.2915i	−0.0814075	−	0.0252043i
$449$	−409.229			−0.911423			−0.455711	−	0.890128i	$-0.650615\pi$
$449$	−409.229			−0.911423			−0.455711	+	0.890128i	$0.650615\pi$
$450$		0			0
$451$		−	371.919i		−	0.824654i
$452$	237.164	−	106.913i	0.524698	−	0.236533i
$453$		−	189.839i		−	0.419071i
$454$	−306.466	+	474.321i	−0.675036	+	1.04476i
$455$		0			0
$456$	−14.6051	+	96.5555i	−0.0320288	+	0.211745i
$457$		−	768.561i		−	1.68175i	−0.541228	−	0.840876i	$-0.682040\pi$
$457$		−	768.561i		−	1.68175i	0.541228	−	0.840876i	$-0.317960\pi$
$458$	−150.068	+	232.262i	−0.327660	+	0.507123i
$459$		−	20.6677i		−	0.0450278i
$460$		0			0
$461$	−316.563			−0.686687			−0.343343	−	0.939210i	$-0.611559\pi$
$461$	−316.563			−0.686687			−0.343343	+	0.939210i	$0.611559\pi$
$462$	16.1066	+	10.4067i	0.0348628	+	0.0225254i
$463$	−491.208			−1.06093			−0.530463	−	0.847708i	$-0.677982\pi$
$463$	−491.208			−1.06093			−0.530463	+	0.847708i	$0.677982\pi$
$464$	377.972	−	427.692i	0.814596	−	0.921751i
$465$		0			0
$466$	−0.878466	−	0.567591i	−0.00188512	−	0.00121801i
$467$	410.393			0.878785			0.439393	−	0.898295i	$-0.355194\pi$
$467$	410.393			0.878785			0.439393	+	0.898295i	$0.355194\pi$
$468$	−257.213	+	115.951i	−0.549600	+	0.247758i
$469$	41.6014			0.0887024
$470$		0			0
$471$			307.591i			0.653060i
$472$	43.5550	−	287.945i	0.0922776	−	0.610054i
$473$		−	335.141i		−	0.708542i
$474$	191.657	+	123.833i	0.404341	+	0.261251i
$475$		0			0
$476$	8.65243	−	3.90049i	0.0181774	−	0.00819431i
$477$		−	7.65369i		−	0.0160455i
$478$	123.794	+	79.9851i	0.258983	+	0.167333i
$479$		−	198.918i		−	0.415277i	−0.978206	−	0.207638i	$-0.933422\pi$
$479$		−	198.918i		−	0.415277i	0.978206	−	0.207638i	$-0.0665778\pi$
$480$		0			0
$481$	126.672			0.263351
$482$	34.0491	−	52.6981i	0.0706413	−	0.109332i
$483$	−33.1455			−0.0686242
$484$	−57.3524	−	127.224i	−0.118497	−	0.262860i
$485$		0			0
$486$	16.9195	−	26.1865i	0.0348138	−	0.0538816i
$487$	−204.762			−0.420456			−0.210228	−	0.977652i	$-0.567421\pi$
$487$	−204.762			−0.420456			−0.210228	+	0.977652i	$0.567421\pi$
$488$	69.0721	+	10.4479i	0.141541	+	0.0214097i
$489$	−167.797			−0.343144
$490$		0			0
$491$			788.598i			1.60611i	0.595908	+	0.803053i	$0.296792\pi$
$491$			788.598i			1.60611i	−0.595908	+	0.803053i	$0.703208\pi$
$492$	−114.116	−	253.143i	−0.231944	−	0.514519i
$493$			141.891i			0.287812i
$494$	−179.848	+	278.353i	−0.364066	+	0.563468i
$495$		0			0
$496$	−710.456	−	627.864i	−1.43237	−	1.26586i
$497$		−	35.3253i		−	0.0710771i
$498$	244.222	−	377.985i	0.490406	−	0.759006i
$499$		−	740.385i		−	1.48374i	−0.670545	−	0.741869i	$-0.733940\pi$
$499$		−	740.385i		−	1.48374i	0.670545	−	0.741869i	$-0.266060\pi$
$500$		0			0
$501$	−264.319			−0.527583
$502$	−132.285	−	85.4717i	−0.263517	−	0.170262i
$503$	−70.8800			−0.140914			−0.0704572	−	0.997515i	$-0.522446\pi$
$503$	−70.8800			−0.140914			−0.0704572	+	0.997515i	$0.522446\pi$
$504$	14.1559	+	2.14124i	0.0280871	+	0.00424850i
$505$		0			0
$506$	−500.068	−	323.102i	−0.988277	−	0.638541i
$507$	−664.760			−1.31116
$508$	−229.270	−	508.588i	−0.451318	−	1.00116i
$509$	−522.642			−1.02680			−0.513400	−	0.858149i	$-0.671614\pi$
$509$	−522.642			−1.02680			−0.513400	+	0.858149i	$0.671614\pi$
$510$		0			0
$511$			49.5139i			0.0968961i
$512$	−460.946	−	222.873i	−0.900286	−	0.435299i
$513$		−	36.6202i		−	0.0713844i
$514$	−409.809	−	264.784i	−0.797293	−	0.515144i
$515$		0			0
$516$	−102.831	−	228.110i	−0.199286	−	0.442074i
$517$		−	686.813i		−	1.32846i
$518$	−5.39894	−	3.48834i	−0.0104227	−	0.00673425i
$519$			269.807i			0.519859i
$520$		0			0
$521$	304.082			0.583650			0.291825	−	0.956472i	$-0.405738\pi$
$521$	304.082			0.583650			0.291825	+	0.956472i	$0.405738\pi$
$522$	−116.158	+	179.779i	−0.222525	+	0.344405i
$523$	174.416			0.333491			0.166746	−	0.986000i	$-0.446674\pi$
$523$	174.416			0.333491			0.166746	+	0.986000i	$0.446674\pi$
$524$	−231.507	+	104.362i	−0.441806	+	0.199165i
$525$		0			0
$526$	111.509	−	172.584i	0.211995	−	0.328106i
$527$	235.701			0.447251
$528$	192.698	+	170.297i	0.364959	+	0.322532i
$529$	500.081			0.945333
$530$		0			0
$531$			109.208i			0.205664i
$532$	15.3308	−	6.91109i	0.0288174	−	0.0129908i
$533$		−	942.327i		−	1.76797i
$534$	245.269	−	379.605i	0.459305	−	0.710871i
$535$		0			0
$536$	551.628	+	83.4401i	1.02916	+	0.155672i
$537$		−	218.240i		−	0.406405i
$538$	134.204	−	207.708i	0.249449	−	0.386075i
$539$			451.399i			0.837476i
$540$		0			0
$541$	−262.199			−0.484655			−0.242328	−	0.970194i	$-0.577911\pi$
$541$	−262.199			−0.484655			−0.242328	+	0.970194i	$0.577911\pi$
$542$	558.337	+	360.750i	1.03014	+	0.665591i
$543$	−600.545			−1.10598
$544$	122.553	−	34.3657i	0.225281	−	0.0631722i
$545$		0			0
$546$	40.8091	+	26.3674i	0.0747420	+	0.0482920i
$547$	146.179			0.267237			0.133619	−	0.991033i	$-0.457340\pi$
$547$	146.179			0.267237			0.133619	+	0.991033i	$0.457340\pi$
$548$	503.804	−	227.113i	0.919350	−	0.414440i
$549$	−26.1967			−0.0477171
$550$		0			0
$551$			251.411i			0.456281i
$552$	−439.504	−	66.4800i	−0.796203	−	0.120435i
$553$		−	39.2944i		−	0.0710568i
$554$	211.522	+	136.668i	0.381809	+	0.246693i
$555$		0			0
$556$	−109.244	+	49.2468i	−0.196482	+	0.0885734i
$557$		−	187.700i		−	0.336984i	−0.985703	−	0.168492i	$-0.946110\pi$
$557$		−	187.700i		−	0.336984i	0.985703	−	0.168492i	$-0.0538898\pi$
$558$	298.638	+	192.955i	0.535194	+	0.345797i
$559$		−	849.142i		−	1.51904i
$560$		0			0
$561$	−63.9296			−0.113957
$562$	57.0509	−	88.2982i	0.101514	−	0.157114i
$563$	447.848			0.795467			0.397734	−	0.917501i	$-0.369797\pi$
$563$	447.848			0.795467			0.397734	+	0.917501i	$0.369797\pi$
$564$	−210.736	−	467.473i	−0.373645	−	0.828853i
$565$		0			0
$566$	−216.305	+	334.777i	−0.382164	+	0.591479i
$567$	−5.36886			−0.00946888
$568$	70.8520	−	468.408i	0.124740	−	0.824662i
$569$	−1078.91			−1.89615			−0.948077	−	0.318042i	$-0.896975\pi$
$569$	−1078.91			−1.89615			−0.948077	+	0.318042i	$0.896975\pi$
$570$		0			0
$571$		−	936.324i		−	1.63980i	−0.572509	−	0.819899i	$-0.694030\pi$
$571$		−	936.324i		−	1.63980i	0.572509	−	0.819899i	$-0.305970\pi$
$572$	358.660	+	795.613i	0.627028	+	1.39093i
$573$		−	230.638i		−	0.402510i
$574$	−25.9502	+	40.1634i	−0.0452095	+	0.0699711i
$575$		0			0
$576$	183.411	+	56.7851i	0.318421	+	0.0985853i
$577$			544.832i			0.944250i	0.881532	+	0.472125i	$0.156513\pi$
$577$			544.832i			0.944250i	−0.881532	+	0.472125i	$0.843487\pi$
$578$	296.505	−	458.904i	0.512985	−	0.793951i
$579$		−	235.986i		−	0.407575i
$580$		0			0
$581$	−77.4960			−0.133384
$582$	−270.917	−	175.044i	−0.465494	−	0.300763i
$583$	−23.6745			−0.0406080
$584$	−99.3101	+	656.547i	−0.170052	+	1.12422i
$585$		0			0
$586$	171.616	+	110.884i	0.292861	+	0.189222i
$587$	−337.889			−0.575619			−0.287810	−	0.957688i	$-0.592927\pi$
$587$	−337.889			−0.575619			−0.287810	+	0.957688i	$0.592927\pi$
$588$	138.503	+	307.241i	0.235550	+	0.522519i
$589$	417.628			0.709045
$590$		0			0
$591$		−	129.721i		−	0.219494i
$592$	−64.5926	−	57.0836i	−0.109109	−	0.0964249i
$593$			567.269i			0.956608i	0.878194	+	0.478304i	$0.158748\pi$
$593$			567.269i			0.956608i	−0.878194	+	0.478304i	$0.841252\pi$
$594$	−81.0002	−	52.3355i	−0.136364	−	0.0881069i
$595$		0			0
$596$	−77.8906	−	172.784i	−0.130689	−	0.289906i
$597$			435.792i			0.729969i
$598$	−1267.02	−	818.640i	−2.11876	−	1.36896i
$599$			762.966i			1.27373i	0.770974	+	0.636867i	$0.219770\pi$
$599$			762.966i			1.27373i	−0.770974	+	0.636867i	$0.780230\pi$
$600$		0			0
$601$	−790.102			−1.31464			−0.657322	−	0.753609i	$-0.728311\pi$
$601$	−790.102			−1.31464			−0.657322	+	0.753609i	$0.728311\pi$
$602$	−23.3841	+	36.1917i	−0.0388440	+	0.0601192i
$603$	−209.214			−0.346955
$604$	−399.680	+	180.175i	−0.661722	+	0.298302i
$605$		0			0
$606$	−6.88212	+	10.6515i	−0.0113566	+	0.0175768i
$607$	522.994			0.861605			0.430802	−	0.902446i	$-0.358231\pi$
$607$	522.994			0.861605			0.430802	+	0.902446i	$0.358231\pi$
$608$	217.146	−	60.8909i	0.357148	−	0.100150i
$609$	36.8591			0.0605240
$610$		0			0
$611$		−	1740.17i		−	2.84807i
$612$	−43.5131	+	19.6156i	−0.0710999	+	0.0320516i
$613$			1026.91i			1.67522i	0.546270	+	0.837609i	$0.316047\pi$
$613$			1026.91i			1.67522i	−0.546270	+	0.837609i	$0.683953\pi$
$614$	356.431	−	551.652i	0.580506	−	0.898455i
$615$		0			0
$616$	6.62332	−	43.7872i	0.0107521	−	0.0710831i
$617$			479.223i			0.776698i	0.921512	+	0.388349i	$0.126954\pi$
$617$			479.223i			0.776698i	−0.921512	+	0.388349i	$0.873046\pi$
$618$	−284.655	+	440.563i	−0.460607	+	0.712886i
$619$		−	507.654i		−	0.820119i	−0.912059	−	0.410059i	$-0.865508\pi$
$619$		−	507.654i		−	0.820119i	0.912059	−	0.410059i	$-0.134492\pi$
$620$		0			0
$621$	166.689			0.268420
$622$	160.322	+	103.587i	0.257753	+	0.166538i
$623$	−77.8282			−0.124925
$624$	488.237	+	431.479i	0.782432	+	0.691473i
$625$		0			0
$626$	924.609	+	597.405i	1.47701	+	0.954321i
$627$	−113.274			−0.180660
$628$	647.591	−	291.932i	1.03120	−	0.464860i
$629$	21.4293			0.0340688
$630$		0			0
$631$			460.186i			0.729297i	0.931145	+	0.364648i	$0.118811\pi$
$631$			460.186i			0.729297i	−0.931145	+	0.364648i	$0.881189\pi$
$632$	78.8128	−	521.037i	0.124704	−	0.824426i
$633$		−	395.259i		−	0.624423i
$634$	−738.122	−	476.912i	−1.16423	−	0.752228i
$635$		0			0
$636$	−16.1138	+	7.26405i	−0.0253362	+	0.0114215i
$637$			1143.71i			1.79546i
$638$	556.095	+	359.302i	0.871622	+	0.563169i
$639$			177.651i			0.278014i
$640$		0			0
$641$	250.774			0.391223			0.195612	−	0.980681i	$-0.437331\pi$
$641$	250.774			0.391223			0.195612	+	0.980681i	$0.437331\pi$
$642$	155.677	−	240.942i	0.242487	−	0.375299i
$643$	−590.355			−0.918126			−0.459063	−	0.888404i	$-0.651815\pi$
$643$	−590.355			−0.918126			−0.459063	+	0.888404i	$0.651815\pi$
$644$	31.4581	+	69.7834i	0.0488480	+	0.108359i
$645$		0			0
$646$	−30.4252	+	47.0894i	−0.0470979	+	0.0728939i
$647$	319.341			0.493572			0.246786	−	0.969070i	$-0.420626\pi$
$647$	319.341			0.493572			0.246786	+	0.969070i	$0.420626\pi$
$648$	−71.1902	−	10.7683i	−0.109861	−	0.0166178i
$649$	337.803			0.520497
$650$		0			0
$651$		−	61.2280i		−	0.0940523i
$652$	159.255	+	353.274i	0.244256	+	0.541832i
$653$			88.5949i			0.135674i	0.997696	+	0.0678369i	$0.0216097\pi$
$653$			88.5949i			0.135674i	−0.997696	+	0.0678369i	$0.978390\pi$
$654$	13.8521	−	21.4390i	0.0211806	−	0.0327814i
$655$		0			0
$656$	−424.652	+	480.512i	−0.647335	+	0.732488i
$657$		−	249.006i		−	0.379004i
$658$	−47.9216	+	74.1688i	−0.0728292	+	0.112719i
$659$			758.423i			1.15087i	0.817847	+	0.575435i	$0.195167\pi$
$659$			758.423i			1.15087i	−0.817847	+	0.575435i	$0.804833\pi$
$660$		0			0
$661$	527.327			0.797771			0.398885	−	0.917001i	$-0.369397\pi$
$661$	527.327			0.797771			0.398885	+	0.917001i	$0.369397\pi$
$662$	806.203	+	520.900i	1.21783	+	0.786859i
$663$	−161.978			−0.244310
$664$	−1027.59	−	155.434i	−1.54757	−	0.234087i
$665$		0			0
$666$	27.1513	+	17.5429i	0.0407677	+	0.0263407i
$667$	−1144.38			−1.71571
$668$	250.863	+	556.488i	0.375544	+	0.833066i
$669$	−148.878			−0.222538
$670$		0			0
$671$			81.0318i			0.120763i
$672$	−8.92717	−	31.8356i	−0.0132845	−	0.0473744i
$673$			120.657i			0.179283i	0.995974	+	0.0896415i	$0.0285721\pi$
$673$			120.657i			0.179283i	−0.995974	+	0.0896415i	$0.971428\pi$
$674$	−98.8493	−	63.8681i	−0.146661	−	0.0947598i
$675$		0			0
$676$	630.918	+	1399.56i	0.933311	+	2.07036i
$677$			219.196i			0.323776i	0.986809	+	0.161888i	$0.0517583\pi$
$677$			219.196i			0.323776i	−0.986809	+	0.161888i	$0.948242\pi$
$678$	189.232	+	122.266i	0.279103	+	0.180333i
$679$			55.5446i			0.0818035i
$680$		0			0
$681$	−489.057			−0.718145
$682$	596.850	−	923.751i	0.875146	−	1.35447i
$683$	−205.502			−0.300881			−0.150441	−	0.988619i	$-0.548069\pi$
$683$	−205.502			−0.300881			−0.150441	+	0.988619i	$0.548069\pi$
$684$	−77.0988	+	34.7559i	−0.112718	+	0.0508128i
$685$		0			0
$686$	63.2222	−	97.8496i	0.0921606	−	0.142638i
$687$	−239.478			−0.348585
$688$	−382.659	+	432.995i	−0.556190	+	0.629354i
$689$	−59.9837			−0.0870591
$690$		0			0
$691$			109.536i			0.158519i	0.996854	+	0.0792593i	$0.0252555\pi$
$691$			109.536i			0.158519i	−0.996854	+	0.0792593i	$0.974744\pi$
$692$	568.042	−	256.071i	0.820869	−	0.370045i
$693$			16.6070i			0.0239639i
$694$	−13.1857	+	20.4077i	−0.0189996	+	0.0294059i
$695$		0			0
$696$	488.746	+	73.9284i	0.702221	+	0.106219i
$697$		−	159.415i		−	0.228716i
$698$	33.5685	−	51.9543i	0.0480924	−	0.0744331i
$699$		−	0.905758i		−	0.00129579i
$700$		0			0
$701$	168.847			0.240865			0.120433	−	0.992721i	$-0.461572\pi$
$701$	168.847			0.240865			0.120433	+	0.992721i	$0.461572\pi$
$702$	−205.229	−	132.602i	−0.292349	−	0.188892i
$703$	37.9695			0.0540106
$704$	175.648	−	567.327i	0.249500	−	0.805863i
$705$		0			0
$706$	−483.909	−	312.661i	−0.685424	−	0.442863i
$707$	2.18382			0.00308885
$708$	229.922	−	103.648i	0.324749	−	0.146396i
$709$	554.846			0.782576			0.391288	−	0.920268i	$-0.372030\pi$
$709$	554.846			0.782576			0.391288	+	0.920268i	$0.372030\pi$
$710$		0			0
$711$			197.612i			0.277935i
$712$	−1031.99	−	156.100i	−1.44942	−	0.219242i
$713$			1900.97i			2.66616i
$714$	6.90374	+	4.46062i	0.00966911	+	0.00624737i
$715$		0			0
$716$	−459.474	+	207.130i	−0.641723	+	0.289287i
$717$			127.640i			0.178019i
$718$	1115.28	+	720.600i	1.55331	+	1.00362i
$719$			377.485i			0.525014i	0.964930	+	0.262507i	$0.0845494\pi$
$719$			377.485i			0.525014i	−0.964930	+	0.262507i	$0.915451\pi$
$720$		0			0
$721$	90.3261			0.125279
$722$	337.915	−	522.995i	0.468027	−	0.724369i
$723$	54.3353			0.0751525
$724$	569.972	+	1264.37i	0.787255	+	1.74636i
$725$		0			0
$726$	65.5885	−	101.512i	0.0903423	−	0.139824i
$727$	−173.183			−0.238216			−0.119108	−	0.992881i	$-0.538003\pi$
$727$	−173.183			−0.238216			−0.119108	+	0.992881i	$0.538003\pi$
$728$	16.7814	−	110.943i	0.0230514	−	0.152394i
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	143.651i		−	0.196513i
$732$	24.8631	+	55.1536i	0.0339659	+	0.0753464i
$733$			278.722i			0.380249i	0.981760	+	0.190124i	$0.0608891\pi$
$733$			278.722i			0.380249i	−0.981760	+	0.190124i	$0.939111\pi$
$734$	−6.60489	+	10.2225i	−0.00899849	+	0.0139270i
$735$		0			0
$736$	277.165	+	988.412i	0.376583	+	1.34295i
$737$			647.142i			0.878075i
$738$	130.504	−	201.982i	0.176834	−	0.273688i
$739$			521.363i			0.705498i	0.935718	+	0.352749i	$0.114753\pi$
$739$			521.363i			0.705498i	−0.935718	+	0.352749i	$0.885247\pi$
$740$		0			0
$741$	−287.001			−0.387315
$742$	2.55660	+	1.65186i	0.00344555	+	0.00222622i
$743$	−1277.93			−1.71996			−0.859981	−	0.510326i	$-0.829525\pi$
$743$	−1277.93			−1.71996			−0.859981	+	0.510326i	$0.829525\pi$
$744$	122.805	−	811.874i	0.165061	−	1.09123i
$745$		0			0
$746$	−343.936	−	222.223i	−0.461041	−	0.297886i
$747$	389.728			0.521724
$748$	60.6751	+	134.595i	0.0811164	+	0.179940i
$749$	−49.3989			−0.0659532
$750$		0			0
$751$		−	1165.31i		−	1.55168i	−0.630930	−	0.775840i	$-0.717327\pi$
$751$		−	1165.31i		−	1.55168i	0.630930	−	0.775840i	$-0.282673\pi$
$752$	−784.194	+	887.350i	−1.04281	+	1.17999i
$753$		−	136.395i		−	0.181136i
$754$	1408.97	+	910.359i	1.86866	+	1.20737i
$755$		0			0
$756$	5.09554	+	11.3034i	0.00674013	+	0.0149516i
$757$		−	1063.75i		−	1.40522i	−0.711574	−	0.702611i	$-0.752017\pi$
$757$		−	1063.75i		−	1.40522i	0.711574	−	0.702611i	$-0.247983\pi$
$758$	−675.861	−	436.685i	−0.891638	−	0.576101i
$759$		−	515.604i		−	0.679320i
$760$		0			0
$761$	−677.847			−0.890732			−0.445366	−	0.895349i	$-0.646926\pi$
$761$	−677.847			−0.890732			−0.445366	+	0.895349i	$0.646926\pi$
$762$	262.194	−	405.800i	0.344087	−	0.532546i
$763$	−4.39551			−0.00576083
$764$	−485.577	+	218.897i	−0.635572	+	0.286514i
$765$		0			0
$766$	−360.078	+	557.296i	−0.470076	+	0.727541i
$767$	855.887			1.11589
$768$	−54.5201	−	440.040i	−0.0709898	−	0.572969i
$769$	1289.59			1.67697			0.838486	−	0.544922i	$-0.183441\pi$
$769$	1289.59			1.67697			0.838486	+	0.544922i	$0.183441\pi$
$770$		0			0
$771$		−	422.540i		−	0.548042i
$772$	−496.836	+	223.972i	−0.643570	+	0.290119i
$773$		−	750.339i		−	0.970684i	−0.874324	−	0.485342i	$-0.838695\pi$
$773$		−	750.339i		−	0.970684i	0.874324	−	0.485342i	$-0.161305\pi$
$774$	117.599	−	182.008i	0.151936	−	0.235153i
$775$		0			0
$776$	−111.406	+	736.513i	−0.143564	+	0.949114i
$777$		−	5.56667i		−	0.00716432i
$778$	676.532	−	1047.07i	0.869578	−	1.34585i
$779$		−	282.460i		−	0.362593i
$780$		0			0
$781$	549.512			0.703600
$782$	−214.343	−	138.491i	−0.274096	−	0.177098i
$783$	−185.365			−0.236736
$784$	515.402	−	583.200i	0.657400	−	0.743877i
$785$		0			0
$786$	−184.718	−	119.349i	−0.235011	−	0.151844i
$787$	−825.185

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	300.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.08539 - 1.67986i) q^{2} +1.73205 q^{3} +(-1.64388 - 3.64660i) q^{4} +(1.87994 - 2.90961i) q^{6} -0.596540 q^{7} +(-7.91002 - 1.19648i) q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(1.08539 - 1.67986i) q^{2} +1.73205 q^{3} +(-1.64388 - 3.64660i) q^{4} +(1.87994 - 2.90961i) q^{6} -0.596540 q^{7} +(-7.91002 - 1.19648i) q^{8} +3.00000 q^{9} -9.27963i q^{11} +(-2.84728 - 6.31609i) q^{12} -23.5117i q^{13} +(-0.647476 + 1.00210i) q^{14} +(-10.5953 + 11.9891i) q^{16} -3.97751i q^{17} +(3.25616 - 5.03959i) q^{18} -7.04756i q^{19} -1.03324 q^{21} +(-15.5885 - 10.0720i) q^{22} +32.0793 q^{23} +(-13.7006 - 2.07237i) q^{24} +(-39.4964 - 25.5192i) q^{26} +5.19615 q^{27} +(0.980637 + 2.17534i) q^{28} -35.6734 q^{29} +59.2585i q^{31} +(8.64000 + 30.8115i) q^{32} -16.0728i q^{33} +(-6.68167 - 4.31713i) q^{34} +(-4.93163 - 10.9398i) q^{36} +5.38761i q^{37} +(-11.8389 - 7.64932i) q^{38} -40.7234i q^{39} +40.0791 q^{41} +(-1.12146 + 1.73570i) q^{42} +36.1157 q^{43} +(-33.8391 + 15.2545i) q^{44} +(34.8184 - 53.8888i) q^{46} +74.0131 q^{47} +(-18.3517 + 20.7657i) q^{48} -48.6441 q^{49} -6.88925i q^{51} +(-85.7376 + 38.6503i) q^{52} -2.55123i q^{53} +(5.63983 - 8.72882i) q^{54} +(4.71864 + 0.713748i) q^{56} -12.2067i q^{57} +(-38.7194 + 59.9265i) q^{58} +36.4026i q^{59} -8.73223 q^{61} +(99.5461 + 64.3183i) q^{62} -1.78962 q^{63} +(61.1369 + 18.9284i) q^{64} +(-27.0001 - 17.4452i) q^{66} -69.7379 q^{67} +(-14.5044 + 6.53853i) q^{68} +55.5630 q^{69} +59.2170i q^{71} +(-23.7301 - 3.58944i) q^{72} -83.0019i q^{73} +(9.05044 + 5.84763i) q^{74} +(-25.6996 + 11.5853i) q^{76} +5.53566i q^{77} +(-68.4098 - 44.2006i) q^{78} +65.8705i q^{79} +9.00000 q^{81} +(43.5013 - 67.3274i) q^{82} +129.909 q^{83} +(1.69851 + 3.76780i) q^{84} +(39.1995 - 60.6695i) q^{86} -61.7882 q^{87} +(-11.1029 + 73.4020i) q^{88} +130.466 q^{89} +14.0256i q^{91} +(-52.7344 - 116.980i) q^{92} +102.639i q^{93} +(80.3327 - 124.332i) q^{94} +(14.9649 + 53.3671i) q^{96} -93.1113i q^{97} +(-52.7977 + 81.7155i) q^{98} -27.8389i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 20q^{4} - 12q^{6} + 48q^{9} + O(q^{10}) \)	\( 16q - 20q^{4} - 12q^{6} + 48q^{9} + 40q^{14} + 68q^{16} - 96q^{21} - 36q^{24} - 72q^{26} - 128q^{29} + 184q^{34} - 60q^{36} - 32q^{41} - 344q^{44} + 304q^{46} + 112q^{49} - 36q^{54} + 232q^{56} - 352q^{61} + 220q^{64} + 216q^{66} + 192q^{69} - 264q^{74} - 48q^{76} + 144q^{81} + 72q^{84} - 400q^{86} - 160q^{89} + 192q^{94} - 348q^{96} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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