LMFDB - Embedded newform 300.3.f.a.199.2

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	300.199
Dual form			300.3.f.a.199.1

$q$-expansion

$f(q)$	$=$	$q+(-1.73205 + 1.00000i) q^{2} -1.73205 q^{3} +(2.00000 - 3.46410i) q^{4} +(3.00000 - 1.73205i) q^{6} -6.92820 q^{7} +8.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+(-1.73205 + 1.00000i) q^{2} -1.73205 q^{3} +(2.00000 - 3.46410i) q^{4} +(3.00000 - 1.73205i) q^{6} -6.92820 q^{7} +8.00000i q^{8} +3.00000 q^{9} +6.92820i q^{11} +(-3.46410 + 6.00000i) q^{12} +2.00000i q^{13} +(12.0000 - 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} -10.0000i q^{17} +(-5.19615 + 3.00000i) q^{18} -20.7846i q^{19} +12.0000 q^{21} +(-6.92820 - 12.0000i) q^{22} +27.7128 q^{23} -13.8564i q^{24} +(-2.00000 - 3.46410i) q^{26} -5.19615 q^{27} +(-13.8564 + 24.0000i) q^{28} +26.0000 q^{29} +6.92820i q^{31} +(27.7128 + 16.0000i) q^{32} -12.0000i q^{33} +(10.0000 + 17.3205i) q^{34} +(6.00000 - 10.3923i) q^{36} -26.0000i q^{37} +(20.7846 + 36.0000i) q^{38} -3.46410i q^{39} +58.0000 q^{41} +(-20.7846 + 12.0000i) q^{42} +48.4974 q^{43} +(24.0000 + 13.8564i) q^{44} +(-48.0000 + 27.7128i) q^{46} +69.2820 q^{47} +(13.8564 + 24.0000i) q^{48} -1.00000 q^{49} +17.3205i q^{51} +(6.92820 + 4.00000i) q^{52} -74.0000i q^{53} +(9.00000 - 5.19615i) q^{54} -55.4256i q^{56} +36.0000i q^{57} +(-45.0333 + 26.0000i) q^{58} +90.0666i q^{59} +26.0000 q^{61} +(-6.92820 - 12.0000i) q^{62} -20.7846 q^{63} -64.0000 q^{64} +(12.0000 + 20.7846i) q^{66} +6.92820 q^{67} +(-34.6410 - 20.0000i) q^{68} -48.0000 q^{69} +24.0000i q^{72} -46.0000i q^{73} +(26.0000 + 45.0333i) q^{74} +(-72.0000 - 41.5692i) q^{76} -48.0000i q^{77} +(3.46410 + 6.00000i) q^{78} -117.779i q^{79} +9.00000 q^{81} +(-100.459 + 58.0000i) q^{82} -48.4974 q^{83} +(24.0000 - 41.5692i) q^{84} +(-84.0000 + 48.4974i) q^{86} -45.0333 q^{87} -55.4256 q^{88} -82.0000 q^{89} -13.8564i q^{91} +(55.4256 - 96.0000i) q^{92} -12.0000i q^{93} +(-120.000 + 69.2820i) q^{94} +(-48.0000 - 27.7128i) q^{96} -2.00000i q^{97} +(1.73205 - 1.00000i) q^{98} +20.7846i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q + 8q^{4} + 12q^{6} + 12q^{9} + O(q^{10}) $	$ 4q + 8q^{4} + 12q^{6} + 12q^{9} + 48q^{14} - 32q^{16} + 48q^{21} - 8q^{26} + 104q^{29} + 40q^{34} + 24q^{36} + 232q^{41} + 96q^{44} - 192q^{46} - 4q^{49} + 36q^{54} + 104q^{61} - 256q^{64} + 48q^{66} - 192q^{69} + 104q^{74} - 288q^{76} + 36q^{81} + 96q^{84} - 336q^{86} - 328q^{89} - 480q^{94} - 192q^{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 $

$2$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$2$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	2.00000	−	3.46410i	0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$	3.00000	−	1.73205i	0.500000	−	0.288675i
$7$	−6.92820			−0.989743			−0.494872	−	0.868966i	$-0.664785\pi$
$7$	−6.92820			−0.989743			−0.494872	+	0.868966i	$0.664785\pi$
$8$			8.00000i			1.00000i
$9$	3.00000			0.333333
$10$		0			0
$11$			6.92820i			0.629837i	0.949119	+	0.314918i	$0.101977\pi$
$11$			6.92820i			0.629837i	−0.949119	+	0.314918i	$0.898023\pi$
$12$	−3.46410	+	6.00000i	−0.288675	+	0.500000i
$13$			2.00000i			0.153846i	0.997037	+	0.0769231i	$0.0245096\pi$
$13$			2.00000i			0.153846i	−0.997037	+	0.0769231i	$0.975490\pi$
$14$	12.0000	−	6.92820i	0.857143	−	0.494872i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$		−	10.0000i		−	0.588235i	−0.955769	−	0.294118i	$-0.904974\pi$
$17$		−	10.0000i		−	0.588235i	0.955769	−	0.294118i	$-0.0950258\pi$
$18$	−5.19615	+	3.00000i	−0.288675	+	0.166667i
$19$		−	20.7846i		−	1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$		−	20.7846i		−	1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$		0			0
$21$	12.0000			0.571429
$22$	−6.92820	−	12.0000i	−0.314918	−	0.545455i
$23$	27.7128			1.20490			0.602452	−	0.798155i	$-0.294190\pi$
$23$	27.7128			1.20490			0.602452	+	0.798155i	$0.294190\pi$
$24$		−	13.8564i		−	0.577350i
$25$		0			0
$26$	−2.00000	−	3.46410i	−0.0769231	−	0.133235i
$27$	−5.19615			−0.192450
$28$	−13.8564	+	24.0000i	−0.494872	+	0.857143i
$29$	26.0000			0.896552			0.448276	−	0.893895i	$-0.352038\pi$
$29$	26.0000			0.896552			0.448276	+	0.893895i	$0.352038\pi$
$30$		0			0
$31$			6.92820i			0.223490i	0.993737	+	0.111745i	$0.0356441\pi$
$31$			6.92820i			0.223490i	−0.993737	+	0.111745i	$0.964356\pi$
$32$	27.7128	+	16.0000i	0.866025	+	0.500000i
$33$		−	12.0000i		−	0.363636i
$34$	10.0000	+	17.3205i	0.294118	+	0.509427i
$35$		0			0
$36$	6.00000	−	10.3923i	0.166667	−	0.288675i
$37$		−	26.0000i		−	0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$		−	26.0000i		−	0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	20.7846	+	36.0000i	0.546963	+	0.947368i
$39$		−	3.46410i		−	0.0888231i
$40$		0			0
$41$	58.0000			1.41463			0.707317	−	0.706896i	$-0.249905\pi$
$41$	58.0000			1.41463			0.707317	+	0.706896i	$0.249905\pi$
$42$	−20.7846	+	12.0000i	−0.494872	+	0.285714i
$43$	48.4974			1.12785			0.563924	−	0.825827i	$-0.309291\pi$
$43$	48.4974			1.12785			0.563924	+	0.825827i	$0.309291\pi$
$44$	24.0000	+	13.8564i	0.545455	+	0.314918i
$45$		0			0
$46$	−48.0000	+	27.7128i	−1.04348	+	0.602452i
$47$	69.2820			1.47409			0.737043	−	0.675846i	$-0.236222\pi$
$47$	69.2820			1.47409			0.737043	+	0.675846i	$0.236222\pi$
$48$	13.8564	+	24.0000i	0.288675	+	0.500000i
$49$	−1.00000			−0.0204082
$50$		0			0
$51$			17.3205i			0.339618i
$52$	6.92820	+	4.00000i	0.133235	+	0.0769231i
$53$		−	74.0000i		−	1.39623i	−0.715987	−	0.698113i	$-0.754023\pi$
$53$		−	74.0000i		−	1.39623i	0.715987	−	0.698113i	$-0.245977\pi$
$54$	9.00000	−	5.19615i	0.166667	−	0.0962250i
$55$		0			0
$56$		−	55.4256i		−	0.989743i
$57$			36.0000i			0.631579i
$58$	−45.0333	+	26.0000i	−0.776437	+	0.448276i
$59$			90.0666i			1.52655i	0.646072	+	0.763277i	$0.276411\pi$
$59$			90.0666i			1.52655i	−0.646072	+	0.763277i	$0.723589\pi$
$60$		0			0
$61$	26.0000			0.426230			0.213115	−	0.977027i	$-0.431639\pi$
$61$	26.0000			0.426230			0.213115	+	0.977027i	$0.431639\pi$
$62$	−6.92820	−	12.0000i	−0.111745	−	0.193548i
$63$	−20.7846			−0.329914
$64$	−64.0000			−1.00000
$65$		0			0
$66$	12.0000	+	20.7846i	0.181818	+	0.314918i
$67$	6.92820			0.103406			0.0517030	−	0.998663i	$-0.483535\pi$
$67$	6.92820			0.103406			0.0517030	+	0.998663i	$0.483535\pi$
$68$	−34.6410	−	20.0000i	−0.509427	−	0.294118i
$69$	−48.0000			−0.695652
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$			24.0000i			0.333333i
$73$		−	46.0000i		−	0.630137i	−0.949069	−	0.315068i	$-0.897973\pi$
$73$		−	46.0000i		−	0.630137i	0.949069	−	0.315068i	$-0.102027\pi$
$74$	26.0000	+	45.0333i	0.351351	+	0.608558i
$75$		0			0
$76$	−72.0000	−	41.5692i	−0.947368	−	0.546963i
$77$		−	48.0000i		−	0.623377i
$78$	3.46410	+	6.00000i	0.0444116	+	0.0769231i
$79$		−	117.779i		−	1.49088i	−0.666573	−	0.745440i	$-0.732240\pi$
$79$		−	117.779i		−	1.49088i	0.666573	−	0.745440i	$-0.267760\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	−100.459	+	58.0000i	−1.22511	+	0.707317i
$83$	−48.4974			−0.584306			−0.292153	−	0.956372i	$-0.594372\pi$
$83$	−48.4974			−0.584306			−0.292153	+	0.956372i	$0.594372\pi$
$84$	24.0000	−	41.5692i	0.285714	−	0.494872i
$85$		0			0
$86$	−84.0000	+	48.4974i	−0.976744	+	0.563924i
$87$	−45.0333			−0.517624
$88$	−55.4256			−0.629837
$89$	−82.0000			−0.921348			−0.460674	−	0.887569i	$-0.652392\pi$
$89$	−82.0000			−0.921348			−0.460674	+	0.887569i	$0.652392\pi$
$90$		0			0
$91$		−	13.8564i		−	0.152268i
$92$	55.4256	−	96.0000i	0.602452	−	1.04348i
$93$		−	12.0000i		−	0.129032i
$94$	−120.000	+	69.2820i	−1.27660	+	0.737043i
$95$		0			0
$96$	−48.0000	−	27.7128i	−0.500000	−	0.288675i
$97$		−	2.00000i		−	0.0206186i	−0.999947	−	0.0103093i	$-0.996718\pi$
$97$		−	2.00000i		−	0.0206186i	0.999947	−	0.0103093i	$-0.00328160\pi$
$98$	1.73205	−	1.00000i	0.0176740	−	0.0102041i
$99$			20.7846i			0.209946i
$100$		0			0
$101$	−74.0000			−0.732673			−0.366337	−	0.930482i	$-0.619388\pi$
$101$	−74.0000			−0.732673			−0.366337	+	0.930482i	$0.619388\pi$
$102$	−17.3205	−	30.0000i	−0.169809	−	0.294118i
$103$	76.2102			0.739905			0.369953	−	0.929051i	$-0.379374\pi$
$103$	76.2102			0.739905			0.369953	+	0.929051i	$0.379374\pi$
$104$	−16.0000			−0.153846
$105$		0			0
$106$	74.0000	+	128.172i	0.698113	+	1.20917i
$107$	20.7846			0.194249			0.0971243	−	0.995272i	$-0.469036\pi$
$107$	20.7846			0.194249			0.0971243	+	0.995272i	$0.469036\pi$
$108$	−10.3923	+	18.0000i	−0.0962250	+	0.166667i
$109$	46.0000			0.422018			0.211009	−	0.977484i	$-0.432325\pi$
$109$	46.0000			0.422018			0.211009	+	0.977484i	$0.432325\pi$
$110$		0			0
$111$			45.0333i			0.405706i
$112$	55.4256	+	96.0000i	0.494872	+	0.857143i
$113$		−	110.000i		−	0.973451i	−0.873555	−	0.486726i	$-0.838191\pi$
$113$		−	110.000i		−	0.973451i	0.873555	−	0.486726i	$-0.161809\pi$
$114$	−36.0000	−	62.3538i	−0.315789	−	0.546963i
$115$		0			0
$116$	52.0000	−	90.0666i	0.448276	−	0.776437i
$117$			6.00000i			0.0512821i
$118$	−90.0666	−	156.000i	−0.763277	−	1.32203i
$119$			69.2820i			0.582202i
$120$		0			0
$121$	73.0000			0.603306
$122$	−45.0333	+	26.0000i	−0.369126	+	0.213115i
$123$	−100.459			−0.816739
$124$	24.0000	+	13.8564i	0.193548	+	0.111745i
$125$		0			0
$126$	36.0000	−	20.7846i	0.285714	−	0.164957i
$127$	−145.492			−1.14561			−0.572804	−	0.819692i	$-0.694144\pi$
$127$	−145.492			−1.14561			−0.572804	+	0.819692i	$0.694144\pi$
$128$	110.851	−	64.0000i	0.866025	−	0.500000i
$129$	−84.0000			−0.651163
$130$		0			0
$131$			117.779i			0.899080i	0.893260	+	0.449540i	$0.148412\pi$
$131$			117.779i			0.899080i	−0.893260	+	0.449540i	$0.851588\pi$
$132$	−41.5692	−	24.0000i	−0.314918	−	0.181818i
$133$			144.000i			1.08271i
$134$	−12.0000	+	6.92820i	−0.0895522	+	0.0517030i
$135$		0			0
$136$	80.0000			0.588235
$137$		−	10.0000i		−	0.0729927i	−0.999334	−	0.0364964i	$-0.988380\pi$
$137$		−	10.0000i		−	0.0729927i	0.999334	−	0.0364964i	$-0.0116197\pi$
$138$	83.1384	−	48.0000i	0.602452	−	0.347826i
$139$		−	48.4974i		−	0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$		−	48.4974i		−	0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$		0			0
$141$	−120.000			−0.851064
$142$		0			0
$143$	−13.8564			−0.0968979
$144$	−24.0000	−	41.5692i	−0.166667	−	0.288675i
$145$		0			0
$146$	46.0000	+	79.6743i	0.315068	+	0.545715i
$147$	1.73205			0.0117827
$148$	−90.0666	−	52.0000i	−0.608558	−	0.351351i
$149$	2.00000			0.0134228			0.00671141	−	0.999977i	$-0.497864\pi$
$149$	2.00000			0.0134228			0.00671141	+	0.999977i	$0.497864\pi$
$150$		0			0
$151$		−	90.0666i		−	0.596468i	−0.954493	−	0.298234i	$-0.903602\pi$
$151$		−	90.0666i		−	0.596468i	0.954493	−	0.298234i	$-0.0963975\pi$
$152$	166.277			1.09393
$153$		−	30.0000i		−	0.196078i
$154$	48.0000	+	83.1384i	0.311688	+	0.539860i
$155$		0			0
$156$	−12.0000	−	6.92820i	−0.0769231	−	0.0444116i
$157$			214.000i			1.36306i	0.731791	+	0.681529i	$0.238685\pi$
$157$			214.000i			1.36306i	−0.731791	+	0.681529i	$0.761315\pi$
$158$	117.779	+	204.000i	0.745440	+	1.29114i
$159$			128.172i			0.806112i
$160$		0			0
$161$	−192.000			−1.19255
$162$	−15.5885	+	9.00000i	−0.0962250	+	0.0555556i
$163$	−20.7846			−0.127513			−0.0637565	−	0.997965i	$-0.520308\pi$
$163$	−20.7846			−0.127513			−0.0637565	+	0.997965i	$0.520308\pi$
$164$	116.000	−	200.918i	0.707317	−	1.22511i
$165$		0			0
$166$	84.0000	−	48.4974i	0.506024	−	0.292153i
$167$	96.9948			0.580807			0.290404	−	0.956904i	$-0.406210\pi$
$167$	96.9948			0.580807			0.290404	+	0.956904i	$0.406210\pi$
$168$			96.0000i			0.571429i
$169$	165.000			0.976331
$170$		0			0
$171$		−	62.3538i		−	0.364642i
$172$	96.9948	−	168.000i	0.563924	−	0.976744i
$173$			334.000i			1.93064i	0.261077	+	0.965318i	$0.415922\pi$
$173$			334.000i			1.93064i	−0.261077	+	0.965318i	$0.584078\pi$
$174$	78.0000	−	45.0333i	0.448276	−	0.258812i
$175$		0			0
$176$	96.0000	−	55.4256i	0.545455	−	0.314918i
$177$		−	156.000i		−	0.881356i
$178$	142.028	−	82.0000i	0.797911	−	0.460674i
$179$			187.061i			1.04504i	0.852628	+	0.522518i	$0.175007\pi$
$179$			187.061i			1.04504i	−0.852628	+	0.522518i	$0.824993\pi$
$180$		0			0
$181$	2.00000			0.0110497			0.00552486	−	0.999985i	$-0.498241\pi$
$181$	2.00000			0.0110497			0.00552486	+	0.999985i	$0.498241\pi$
$182$	13.8564	+	24.0000i	0.0761341	+	0.131868i
$183$	−45.0333			−0.246084
$184$			221.703i			1.20490i
$185$		0			0
$186$	12.0000	+	20.7846i	0.0645161	+	0.111745i
$187$	69.2820			0.370492
$188$	138.564	−	240.000i	0.737043	−	1.27660i
$189$	36.0000			0.190476
$190$		0			0
$191$		−	221.703i		−	1.16075i	−0.814351	−	0.580373i	$-0.802907\pi$
$191$		−	221.703i		−	1.16075i	0.814351	−	0.580373i	$-0.197093\pi$
$192$	110.851			0.577350
$193$			290.000i			1.50259i	0.659966	+	0.751295i	$0.270571\pi$
$193$			290.000i			1.50259i	−0.659966	+	0.751295i	$0.729429\pi$
$194$	2.00000	+	3.46410i	0.0103093	+	0.0178562i
$195$		0			0
$196$	−2.00000	+	3.46410i	−0.0102041	+	0.0176740i
$197$			26.0000i			0.131980i	0.997820	+	0.0659898i	$0.0210205\pi$
$197$			26.0000i			0.131980i	−0.997820	+	0.0659898i	$0.978980\pi$
$198$	−20.7846	−	36.0000i	−0.104973	−	0.181818i
$199$		−	394.908i		−	1.98446i	−0.124416	−	0.992230i	$-0.539706\pi$
$199$		−	394.908i		−	1.98446i	0.124416	−	0.992230i	$-0.460294\pi$
$200$		0			0
$201$	−12.0000			−0.0597015
$202$	128.172	−	74.0000i	0.634514	−	0.366337i
$203$	−180.133			−0.887356
$204$	60.0000	+	34.6410i	0.294118	+	0.169809i
$205$		0			0
$206$	−132.000	+	76.2102i	−0.640777	+	0.369953i
$207$	83.1384			0.401635
$208$	27.7128	−	16.0000i	0.133235	−	0.0769231i
$209$	144.000			0.688995
$210$		0			0
$211$		−	242.487i		−	1.14923i	−0.818425	−	0.574614i	$-0.805152\pi$
$211$		−	242.487i		−	1.14923i	0.818425	−	0.574614i	$-0.194848\pi$
$212$	−256.344	−	148.000i	−1.20917	−	0.698113i
$213$		0			0
$214$	−36.0000	+	20.7846i	−0.168224	+	0.0971243i
$215$		0			0
$216$		−	41.5692i		−	0.192450i
$217$		−	48.0000i		−	0.221198i
$218$	−79.6743	+	46.0000i	−0.365479	+	0.211009i
$219$			79.6743i			0.363810i
$220$		0			0
$221$	20.0000			0.0904977
$222$	−45.0333	−	78.0000i	−0.202853	−	0.351351i
$223$	339.482			1.52234			0.761170	−	0.648552i	$-0.224625\pi$
$223$	339.482			1.52234			0.761170	+	0.648552i	$0.224625\pi$
$224$	−192.000	−	110.851i	−0.857143	−	0.494872i
$225$		0			0
$226$	110.000	+	190.526i	0.486726	+	0.843034i
$227$	−284.056			−1.25135			−0.625675	−	0.780084i	$-0.715176\pi$
$227$	−284.056			−1.25135			−0.625675	+	0.780084i	$0.715176\pi$
$228$	124.708	+	72.0000i	0.546963	+	0.315789i
$229$	142.000			0.620087			0.310044	−	0.950722i	$-0.399656\pi$
$229$	142.000			0.620087			0.310044	+	0.950722i	$0.399656\pi$
$230$		0			0
$231$			83.1384i			0.359907i
$232$			208.000i			0.896552i
$233$			82.0000i			0.351931i	0.984396	+	0.175966i	$0.0563048\pi$
$233$			82.0000i			0.351931i	−0.984396	+	0.175966i	$0.943695\pi$
$234$	−6.00000	−	10.3923i	−0.0256410	−	0.0444116i
$235$		0			0
$236$	312.000	+	180.133i	1.32203	+	0.763277i
$237$			204.000i			0.860759i
$238$	−69.2820	−	120.000i	−0.291101	−	0.504202i
$239$		−	387.979i		−	1.62334i	−0.584113	−	0.811672i	$-0.698558\pi$
$239$		−	387.979i		−	1.62334i	0.584113	−	0.811672i	$-0.301442\pi$
$240$		0			0
$241$	−46.0000			−0.190871			−0.0954357	−	0.995436i	$-0.530424\pi$
$241$	−46.0000			−0.190871			−0.0954357	+	0.995436i	$0.530424\pi$
$242$	−126.440	+	73.0000i	−0.522478	+	0.301653i
$243$	−15.5885			−0.0641500
$244$	52.0000	−	90.0666i	0.213115	−	0.369126i
$245$		0			0
$246$	174.000	−	100.459i	0.707317	−	0.408370i
$247$	41.5692			0.168296
$248$	−55.4256			−0.223490
$249$	84.0000			0.337349
$250$		0			0
$251$			145.492i			0.579650i	0.957080	+	0.289825i	$0.0935972\pi$
$251$			145.492i			0.579650i	−0.957080	+	0.289825i	$0.906403\pi$
$252$	−41.5692	+	72.0000i	−0.164957	+	0.285714i
$253$			192.000i			0.758893i
$254$	252.000	−	145.492i	0.992126	−	0.572804i
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$			254.000i			0.988327i	0.869369	+	0.494163i	$0.164526\pi$
$257$			254.000i			0.988327i	−0.869369	+	0.494163i	$0.835474\pi$
$258$	145.492	−	84.0000i	0.563924	−	0.325581i
$259$			180.133i			0.695495i
$260$		0			0
$261$	78.0000			0.298851
$262$	−117.779	−	204.000i	−0.449540	−	0.778626i
$263$	−152.420			−0.579546			−0.289773	−	0.957095i	$-0.593580\pi$
$263$	−152.420			−0.579546			−0.289773	+	0.957095i	$0.593580\pi$
$264$	96.0000			0.363636
$265$		0			0
$266$	−144.000	−	249.415i	−0.541353	−	0.937652i
$267$	142.028			0.531941
$268$	13.8564	−	24.0000i	0.0517030	−	0.0895522i
$269$	−262.000			−0.973978			−0.486989	−	0.873408i	$-0.661905\pi$
$269$	−262.000			−0.973978			−0.486989	+	0.873408i	$0.661905\pi$
$270$		0			0
$271$			20.7846i			0.0766960i	0.999264	+	0.0383480i	$0.0122095\pi$
$271$			20.7846i			0.0766960i	−0.999264	+	0.0383480i	$0.987790\pi$
$272$	−138.564	+	80.0000i	−0.509427	+	0.294118i
$273$			24.0000i			0.0879121i
$274$	10.0000	+	17.3205i	0.0364964	+	0.0632135i
$275$		0			0
$276$	−96.0000	+	166.277i	−0.347826	+	0.602452i
$277$		−	290.000i		−	1.04693i	−0.852047	−	0.523466i	$-0.824639\pi$
$277$		−	290.000i		−	1.04693i	0.852047	−	0.523466i	$-0.175361\pi$
$278$	48.4974	+	84.0000i	0.174451	+	0.302158i
$279$			20.7846i			0.0744968i
$280$		0			0
$281$	226.000			0.804270			0.402135	−	0.915580i	$-0.368268\pi$
$281$	226.000			0.804270			0.402135	+	0.915580i	$0.368268\pi$
$282$	207.846	−	120.000i	0.737043	−	0.425532i
$283$	−297.913			−1.05270			−0.526348	−	0.850269i	$-0.676439\pi$
$283$	−297.913			−1.05270			−0.526348	+	0.850269i	$0.676439\pi$
$284$		0			0
$285$		0			0
$286$	24.0000	−	13.8564i	0.0839161	−	0.0484490i
$287$	−401.836			−1.40012
$288$	83.1384	+	48.0000i	0.288675	+	0.166667i
$289$	189.000			0.653979
$290$		0			0
$291$			3.46410i			0.0119041i
$292$	−159.349	−	92.0000i	−0.545715	−	0.315068i
$293$		−	362.000i		−	1.23549i	−0.786377	−	0.617747i	$-0.788045\pi$
$293$		−	362.000i		−	1.23549i	0.786377	−	0.617747i	$-0.211955\pi$
$294$	−3.00000	+	1.73205i	−0.0102041	+	0.00589133i
$295$		0			0
$296$	208.000			0.702703
$297$		−	36.0000i		−	0.121212i
$298$	−3.46410	+	2.00000i	−0.0116245	+	0.00671141i
$299$			55.4256i			0.185370i
$300$		0			0
$301$	−336.000			−1.11628
$302$	90.0666	+	156.000i	0.298234	+	0.516556i
$303$	128.172			0.423009
$304$	−288.000	+	166.277i	−0.947368	+	0.546963i
$305$		0			0
$306$	30.0000	+	51.9615i	0.0980392	+	0.169809i
$307$	145.492			0.473916			0.236958	−	0.971520i	$-0.423850\pi$
$307$	145.492			0.473916			0.236958	+	0.971520i	$0.423850\pi$
$308$	−166.277	−	96.0000i	−0.539860	−	0.311688i
$309$	−132.000			−0.427184
$310$		0			0
$311$		−	235.559i		−	0.757424i	−0.925515	−	0.378712i	$-0.876367\pi$
$311$		−	235.559i		−	0.757424i	0.925515	−	0.378712i	$-0.123633\pi$
$312$	27.7128			0.0888231
$313$		−	478.000i		−	1.52716i	−0.645715	−	0.763578i	$-0.723441\pi$
$313$		−	478.000i		−	1.52716i	0.645715	−	0.763578i	$-0.276559\pi$
$314$	−214.000	−	370.659i	−0.681529	−	1.18044i
$315$		0			0
$316$	−408.000	−	235.559i	−1.29114	−	0.745440i
$317$			170.000i			0.536278i	0.963380	+	0.268139i	$0.0864086\pi$
$317$			170.000i			0.536278i	−0.963380	+	0.268139i	$0.913591\pi$
$318$	−128.172	−	222.000i	−0.403056	−	0.698113i
$319$			180.133i			0.564681i
$320$		0			0
$321$	−36.0000			−0.112150
$322$	332.554	−	192.000i	1.03278	−	0.596273i
$323$	−207.846			−0.643486
$324$	18.0000	−	31.1769i	0.0555556	−	0.0962250i
$325$		0			0
$326$	36.0000	−	20.7846i	0.110429	−	0.0637565i
$327$	−79.6743			−0.243652
$328$			464.000i			1.41463i
$329$	−480.000			−1.45897
$330$		0			0
$331$			408.764i			1.23494i	0.786596	+	0.617468i	$0.211842\pi$
$331$			408.764i			1.23494i	−0.786596	+	0.617468i	$0.788158\pi$
$332$	−96.9948	+	168.000i	−0.292153	+	0.506024i
$333$		−	78.0000i		−	0.234234i
$334$	−168.000	+	96.9948i	−0.502994	+	0.290404i
$335$		0			0
$336$	−96.0000	−	166.277i	−0.285714	−	0.494872i
$337$		−	338.000i		−	1.00297i	−0.865167	−	0.501484i	$-0.832788\pi$
$337$		−	338.000i		−	1.00297i	0.865167	−	0.501484i	$-0.167212\pi$
$338$	−285.788	+	165.000i	−0.845528	+	0.488166i
$339$			190.526i			0.562022i
$340$		0			0
$341$	−48.0000			−0.140762
$342$	62.3538	+	108.000i	0.182321	+	0.315789i
$343$	346.410			1.00994
$344$			387.979i			1.12785i
$345$		0			0
$346$	−334.000	−	578.505i	−0.965318	−	1.67198i
$347$	200.918			0.579014			0.289507	−	0.957176i	$-0.406509\pi$
$347$	200.918			0.579014			0.289507	+	0.957176i	$0.406509\pi$
$348$	−90.0666	+	156.000i	−0.258812	+	0.448276i
$349$	−506.000			−1.44986			−0.724928	−	0.688824i	$-0.758127\pi$
$349$	−506.000			−1.44986			−0.724928	+	0.688824i	$0.758127\pi$
$350$		0			0
$351$		−	10.3923i		−	0.0296077i
$352$	−110.851	+	192.000i	−0.314918	+	0.545455i
$353$			178.000i			0.504249i	0.967695	+	0.252125i	$0.0811293\pi$
$353$			178.000i			0.504249i	−0.967695	+	0.252125i	$0.918871\pi$
$354$	156.000	+	270.200i	0.440678	+	0.763277i
$355$		0			0
$356$	−164.000	+	284.056i	−0.460674	+	0.797911i
$357$		−	120.000i		−	0.336134i
$358$	−187.061	−	324.000i	−0.522518	−	0.905028i
$359$		−	166.277i		−	0.463167i	−0.972815	−	0.231583i	$-0.925609\pi$
$359$		−	166.277i		−	0.463167i	0.972815	−	0.231583i	$-0.0743906\pi$
$360$		0			0
$361$	−71.0000			−0.196676
$362$	−3.46410	+	2.00000i	−0.00956934	+	0.00552486i
$363$	−126.440			−0.348319
$364$	−48.0000	−	27.7128i	−0.131868	−	0.0761341i
$365$		0			0
$366$	78.0000	−	45.0333i	0.213115	−	0.123042i
$367$	200.918			0.547460			0.273730	−	0.961807i	$-0.411742\pi$
$367$	200.918			0.547460			0.273730	+	0.961807i	$0.411742\pi$
$368$	−221.703	−	384.000i	−0.602452	−	1.04348i
$369$	174.000			0.471545
$370$		0			0
$371$			512.687i			1.38191i
$372$	−41.5692	−	24.0000i	−0.111745	−	0.0645161i
$373$		−	310.000i		−	0.831099i	−0.909571	−	0.415550i	$-0.863589\pi$
$373$		−	310.000i		−	0.831099i	0.909571	−	0.415550i	$-0.136411\pi$
$374$	−120.000	+	69.2820i	−0.320856	+	0.185246i
$375$		0			0
$376$			554.256i			1.47409i
$377$			52.0000i			0.137931i
$378$	−62.3538	+	36.0000i	−0.164957	+	0.0952381i
$379$			436.477i			1.15165i	0.817572	+	0.575827i	$0.195320\pi$
$379$			436.477i			1.15165i	−0.817572	+	0.575827i	$0.804680\pi$
$380$		0			0
$381$	252.000			0.661417
$382$	221.703	+	384.000i	0.580373	+	1.00524i
$383$	609.682			1.59186			0.795929	−	0.605390i	$-0.206983\pi$
$383$	609.682			1.59186			0.795929	+	0.605390i	$0.206983\pi$
$384$	−192.000	+	110.851i	−0.500000	+	0.288675i
$385$		0			0
$386$	−290.000	−	502.295i	−0.751295	−	1.30128i
$387$	145.492			0.375949
$388$	−6.92820	−	4.00000i	−0.0178562	−	0.0103093i
$389$	578.000			1.48586			0.742931	−	0.669368i	$-0.233435\pi$
$389$	578.000			1.48586			0.742931	+	0.669368i	$0.233435\pi$
$390$		0			0
$391$		−	277.128i		−	0.708768i
$392$		−	8.00000i		−	0.0204082i
$393$		−	204.000i		−	0.519084i
$394$	−26.0000	−	45.0333i	−0.0659898	−	0.114298i
$395$		0			0
$396$	72.0000	+	41.5692i	0.181818	+	0.104973i
$397$		−	26.0000i		−	0.0654912i	−0.999464	−	0.0327456i	$-0.989575\pi$
$397$		−	26.0000i		−	0.0654912i	0.999464	−	0.0327456i	$-0.0104251\pi$
$398$	394.908	+	684.000i	0.992230	+	1.71859i
$399$		−	249.415i		−	0.625101i
$400$		0			0
$401$	250.000			0.623441			0.311721	−	0.950174i	$-0.399095\pi$
$401$	250.000			0.623441			0.311721	+	0.950174i	$0.399095\pi$
$402$	20.7846	−	12.0000i	0.0517030	−	0.0298507i
$403$	−13.8564			−0.0343831
$404$	−148.000	+	256.344i	−0.366337	+	0.634514i
$405$		0			0
$406$	312.000	−	180.133i	0.768473	−	0.443678i
$407$	180.133			0.442588
$408$	−138.564			−0.339618
$409$	−290.000			−0.709046			−0.354523	−	0.935047i	$-0.615357\pi$
$409$	−290.000			−0.709046			−0.354523	+	0.935047i	$0.615357\pi$
$410$		0			0
$411$			17.3205i			0.0421424i
$412$	152.420	−	264.000i	0.369953	−	0.640777i
$413$		−	624.000i		−	1.51090i
$414$	−144.000	+	83.1384i	−0.347826	+	0.200817i
$415$		0			0
$416$	−32.0000	+	55.4256i	−0.0769231	+	0.133235i
$417$			84.0000i			0.201439i
$418$	−249.415	+	144.000i	−0.596687	+	0.344498i
$419$			339.482i			0.810219i	0.914268	+	0.405110i	$0.132767\pi$
$419$			339.482i			0.810219i	−0.914268	+	0.405110i	$0.867233\pi$
$420$		0			0
$421$	674.000			1.60095			0.800475	−	0.599366i	$-0.204581\pi$
$421$	674.000			1.60095			0.800475	+	0.599366i	$0.204581\pi$
$422$	242.487	+	420.000i	0.574614	+	0.995261i
$423$	207.846			0.491362
$424$	592.000			1.39623
$425$		0			0
$426$		0			0
$427$	−180.133			−0.421858
$428$	41.5692	−	72.0000i	0.0971243	−	0.168224i
$429$	24.0000			0.0559441
$430$		0			0
$431$			540.400i			1.25383i	0.779088	+	0.626914i	$0.215682\pi$
$431$			540.400i			1.25383i	−0.779088	+	0.626914i	$0.784318\pi$
$432$	41.5692	+	72.0000i	0.0962250	+	0.166667i
$433$		−	334.000i		−	0.771363i	−0.922632	−	0.385681i	$-0.873966\pi$
$433$		−	334.000i		−	0.771363i	0.922632	−	0.385681i	$-0.126034\pi$
$434$	48.0000	+	83.1384i	0.110599	+	0.191563i
$435$		0			0
$436$	92.0000	−	159.349i	0.211009	−	0.365479i
$437$		−	576.000i		−	1.31808i
$438$	−79.6743	−	138.000i	−0.181905	−	0.315068i
$439$		−	117.779i		−	0.268290i	−0.990962	−	0.134145i	$-0.957171\pi$
$439$		−	117.779i		−	0.268290i	0.990962	−	0.134145i	$-0.0428288\pi$
$440$		0			0
$441$	−3.00000			−0.00680272
$442$	−34.6410	+	20.0000i	−0.0783733	+	0.0452489i
$443$	76.2102			0.172032			0.0860161	−	0.996294i	$-0.472586\pi$
$443$	76.2102			0.172032			0.0860161	+	0.996294i	$0.472586\pi$
$444$	156.000	+	90.0666i	0.351351	+	0.202853i
$445$		0			0
$446$	−588.000	+	339.482i	−1.31839	+	0.761170i
$447$	−3.46410			−0.00774967
$448$	443.405			0.989743
$449$	−394.000			−0.877506			−0.438753	−	0.898608i	$-0.644580\pi$
$449$	−394.000			−0.877506			−0.438753	+	0.898608i	$0.644580\pi$
$450$		0			0
$451$			401.836i			0.890988i
$452$	−381.051	−	220.000i	−0.843034	−	0.486726i
$453$			156.000i			0.344371i
$454$	492.000	−	284.056i	1.08370	−	0.625675i
$455$		0			0
$456$	−288.000			−0.631579
$457$			478.000i			1.04595i	0.852347	+	0.522976i	$0.175178\pi$
$457$			478.000i			1.04595i	−0.852347	+	0.522976i	$0.824822\pi$
$458$	−245.951	+	142.000i	−0.537011	+	0.310044i
$459$			51.9615i			0.113206i
$460$		0			0
$461$	142.000			0.308026			0.154013	−	0.988069i	$-0.450780\pi$
$461$	142.000			0.308026			0.154013	+	0.988069i	$0.450780\pi$
$462$	−83.1384	−	144.000i	−0.179953	−	0.311688i
$463$	−630.466			−1.36170			−0.680849	−	0.732423i	$-0.738389\pi$
$463$	−630.466			−1.36170			−0.680849	+	0.732423i	$0.738389\pi$
$464$	−208.000	−	360.267i	−0.448276	−	0.776437i
$465$		0			0
$466$	−82.0000	−	142.028i	−0.175966	−	0.304781i
$467$	20.7846			0.0445067			0.0222533	−	0.999752i	$-0.492916\pi$
$467$	20.7846			0.0445067			0.0222533	+	0.999752i	$0.492916\pi$
$468$	20.7846	+	12.0000i	0.0444116	+	0.0256410i
$469$	−48.0000			−0.102345
$470$		0			0
$471$		−	370.659i		−	0.786962i
$472$	−720.533			−1.52655
$473$			336.000i			0.710359i
$474$	−204.000	−	353.338i	−0.430380	−	0.745440i
$475$		0			0
$476$	240.000	+	138.564i	0.504202	+	0.291101i
$477$		−	222.000i		−	0.465409i
$478$	387.979	+	672.000i	0.811672	+	1.40586i
$479$		−	734.390i		−	1.53317i	−0.642141	−	0.766586i	$-0.721954\pi$
$479$		−	734.390i		−	1.53317i	0.642141	−	0.766586i	$-0.278046\pi$
$480$		0			0
$481$	52.0000			0.108108
$482$	79.6743	−	46.0000i	0.165299	−	0.0954357i
$483$	332.554			0.688517
$484$	146.000	−	252.879i	0.301653	−	0.522478i
$485$		0			0
$486$	27.0000	−	15.5885i	0.0555556	−	0.0320750i
$487$	103.923			0.213394			0.106697	−	0.994292i	$-0.465972\pi$
$487$	103.923			0.213394			0.106697	+	0.994292i	$0.465972\pi$
$488$			208.000i			0.426230i
$489$	36.0000			0.0736196
$490$		0			0
$491$		−	921.451i		−	1.87668i	−0.345711	−	0.938341i	$-0.612362\pi$
$491$		−	921.451i		−	1.87668i	0.345711	−	0.938341i	$-0.387638\pi$
$492$	−200.918	+	348.000i	−0.408370	+	0.707317i
$493$		−	260.000i		−	0.527383i
$494$	−72.0000	+	41.5692i	−0.145749	+	0.0841482i
$495$		0			0
$496$	96.0000	−	55.4256i	0.193548	−	0.111745i
$497$		0			0
$498$	−145.492	+	84.0000i	−0.292153	+	0.168675i
$499$			76.2102i			0.152726i	0.997080	+	0.0763630i	$0.0243308\pi$
$499$			76.2102i			0.152726i	−0.997080	+	0.0763630i	$0.975669\pi$
$500$		0			0
$501$	−168.000			−0.335329
$502$	−145.492	−	252.000i	−0.289825	−	0.501992i
$503$	581.969			1.15700			0.578498	−	0.815684i	$-0.303639\pi$
$503$	581.969			1.15700			0.578498	+	0.815684i	$0.303639\pi$
$504$		−	166.277i		−	0.329914i
$505$		0			0
$506$	−192.000	−	332.554i	−0.379447	−	0.657221i
$507$	−285.788			−0.563685
$508$	−290.985	+	504.000i	−0.572804	+	0.992126i
$509$	842.000			1.65422			0.827112	−	0.562037i	$-0.189982\pi$
$509$	842.000			1.65422			0.827112	+	0.562037i	$0.189982\pi$
$510$		0			0
$511$			318.697i			0.623674i
$512$		−	512.000i		−	1.00000i
$513$			108.000i			0.210526i
$514$	−254.000	−	439.941i	−0.494163	−	0.855916i
$515$		0			0
$516$	−168.000	+	290.985i	−0.325581	+	0.563924i
$517$			480.000i			0.928433i
$518$	−180.133	−	312.000i	−0.347748	−	0.602317i
$519$		−	578.505i		−	1.11465i
$520$		0			0
$521$	−326.000			−0.625720			−0.312860	−	0.949799i	$-0.601287\pi$
$521$	−326.000			−0.625720			−0.312860	+	0.949799i	$0.601287\pi$
$522$	−135.100	+	78.0000i	−0.258812	+	0.149425i
$523$	−311.769			−0.596117			−0.298058	−	0.954548i	$-0.596339\pi$
$523$	−311.769			−0.596117			−0.298058	+	0.954548i	$0.596339\pi$
$524$	408.000	+	235.559i	0.778626	+	0.449540i
$525$		0			0
$526$	264.000	−	152.420i	0.501901	−	0.289773i
$527$	69.2820			0.131465
$528$	−166.277	+	96.0000i	−0.314918	+	0.181818i
$529$	239.000			0.451796
$530$		0			0
$531$			270.200i			0.508851i
$532$	498.831	+	288.000i	0.937652	+	0.541353i
$533$			116.000i			0.217636i
$534$	−246.000	+	142.028i	−0.460674	+	0.265970i
$535$		0			0
$536$			55.4256i			0.103406i
$537$		−	324.000i		−	0.603352i
$538$	453.797	−	262.000i	0.843489	−	0.486989i
$539$		−	6.92820i		−	0.0128538i
$540$		0			0
$541$	530.000			0.979667			0.489834	−	0.871816i	$-0.337058\pi$
$541$	530.000			0.979667			0.489834	+	0.871816i	$0.337058\pi$
$542$	−20.7846	−	36.0000i	−0.0383480	−	0.0664207i
$543$	−3.46410			−0.00637956
$544$	160.000	−	277.128i	0.294118	−	0.509427i
$545$		0			0
$546$	−24.0000	−	41.5692i	−0.0439560	−	0.0761341i
$547$	−339.482			−0.620625			−0.310313	−	0.950635i	$-0.600434\pi$
$547$	−339.482			−0.620625			−0.310313	+	0.950635i	$0.600434\pi$
$548$	−34.6410	−	20.0000i	−0.0632135	−	0.0364964i
$549$	78.0000			0.142077
$550$		0			0
$551$		−	540.400i		−	0.980762i
$552$		−	384.000i		−	0.695652i
$553$			816.000i			1.47559i
$554$	290.000	+	502.295i	0.523466	+	0.906669i
$555$		0			0
$556$	−168.000	−	96.9948i	−0.302158	−	0.174451i
$557$		−	766.000i		−	1.37522i	−0.726078	−	0.687612i	$-0.758659\pi$
$557$		−	766.000i		−	1.37522i	0.726078	−	0.687612i	$-0.241341\pi$
$558$	−20.7846	−	36.0000i	−0.0372484	−	0.0645161i
$559$			96.9948i			0.173515i
$560$		0			0
$561$	−120.000			−0.213904
$562$	−391.443	+	226.000i	−0.696519	+	0.402135i
$563$	−491.902			−0.873717			−0.436858	−	0.899530i	$-0.643909\pi$
$563$	−491.902			−0.873717			−0.436858	+	0.899530i	$0.643909\pi$
$564$	−240.000	+	415.692i	−0.425532	+	0.737043i
$565$		0			0
$566$	516.000	−	297.913i	0.911661	−	0.526348i
$567$	−62.3538			−0.109971
$568$		0			0
$569$	422.000			0.741652			0.370826	−	0.928702i	$-0.379075\pi$
$569$	422.000			0.741652			0.370826	+	0.928702i	$0.379075\pi$
$570$		0			0
$571$		−	284.056i		−	0.497472i	−0.968571	−	0.248736i	$-0.919985\pi$
$571$		−	284.056i		−	0.497472i	0.968571	−	0.248736i	$-0.0800151\pi$
$572$	−27.7128	+	48.0000i	−0.0484490	+	0.0839161i
$573$			384.000i			0.670157i
$574$	696.000	−	401.836i	1.21254	−	0.700062i
$575$		0			0
$576$	−192.000			−0.333333
$577$			46.0000i			0.0797227i	0.999205	+	0.0398614i	$0.0126916\pi$
$577$			46.0000i			0.0797227i	−0.999205	+	0.0398614i	$0.987308\pi$
$578$	−327.358	+	189.000i	−0.566363	+	0.326990i
$579$		−	502.295i		−	0.867521i
$580$		0			0
$581$	336.000			0.578313
$582$	−3.46410	−	6.00000i	−0.00595206	−	0.0103093i
$583$	512.687			0.879395
$584$	368.000			0.630137
$585$		0			0
$586$	362.000	+	627.002i	0.617747	+	1.06997i
$587$	630.466			1.07405			0.537024	−	0.843567i	$-0.319548\pi$
$587$	630.466			1.07405			0.537024	+	0.843567i	$0.319548\pi$
$588$	3.46410	−	6.00000i	0.00589133	−	0.0102041i
$589$	144.000			0.244482
$590$		0			0
$591$		−	45.0333i		−	0.0761985i
$592$	−360.267	+	208.000i	−0.608558	+	0.351351i
$593$			82.0000i			0.138280i	0.997607	+	0.0691400i	$0.0220255\pi$
$593$			82.0000i			0.138280i	−0.997607	+	0.0691400i	$0.977974\pi$
$594$	36.0000	+	62.3538i	0.0606061	+	0.104973i
$595$		0			0
$596$	4.00000	−	6.92820i	0.00671141	−	0.0116245i
$597$			684.000i			1.14573i
$598$	−55.4256	−	96.0000i	−0.0926850	−	0.160535i
$599$		−	55.4256i		−	0.0925303i	−0.998929	−	0.0462651i	$-0.985268\pi$
$599$		−	55.4256i		−	0.0925303i	0.998929	−	0.0462651i	$-0.0147319\pi$
$600$		0			0
$601$	−334.000			−0.555740			−0.277870	−	0.960619i	$-0.589629\pi$
$601$	−334.000			−0.555740			−0.277870	+	0.960619i	$0.589629\pi$
$602$	581.969	−	336.000i	0.966726	−	0.558140i
$603$	20.7846			0.0344687
$604$	−312.000	−	180.133i	−0.516556	−	0.298234i
$605$		0			0
$606$	−222.000	+	128.172i	−0.366337	+	0.211505i
$607$	−367.195			−0.604934			−0.302467	−	0.953160i	$-0.597810\pi$
$607$	−367.195			−0.604934			−0.302467	+	0.953160i	$0.597810\pi$
$608$	332.554	−	576.000i	0.546963	−	0.947368i
$609$	312.000			0.512315
$610$		0			0
$611$			138.564i			0.226782i
$612$	−103.923	−	60.0000i	−0.169809	−	0.0980392i
$613$		−	214.000i		−	0.349103i	−0.984648	−	0.174551i	$-0.944152\pi$
$613$		−	214.000i		−	0.349103i	0.984648	−	0.174551i	$-0.0558475\pi$
$614$	−252.000	+	145.492i	−0.410423	+	0.236958i
$615$		0			0
$616$	384.000			0.623377
$617$			1118.00i			1.81199i	0.423285	+	0.905997i	$0.360877\pi$
$617$			1118.00i			1.81199i	−0.423285	+	0.905997i	$0.639123\pi$
$618$	228.631	−	132.000i	0.369953	−	0.213592i
$619$			672.036i			1.08568i	0.839836	+	0.542840i	$0.182651\pi$
$619$			672.036i			1.08568i	−0.839836	+	0.542840i	$0.817349\pi$
$620$		0			0
$621$	−144.000			−0.231884
$622$	235.559	+	408.000i	0.378712	+	0.655949i
$623$	568.113			0.911898
$624$	−48.0000	+	27.7128i	−0.0769231	+	0.0444116i
$625$		0			0
$626$	478.000	+	827.920i	0.763578	+	1.32256i
$627$	−249.415			−0.397792
$628$	741.318	+	428.000i	1.18044	+	0.681529i
$629$	−260.000			−0.413355
$630$		0			0
$631$		−	145.492i		−	0.230574i	−0.993332	−	0.115287i	$-0.963221\pi$
$631$		−	145.492i		−	0.230574i	0.993332	−	0.115287i	$-0.0367788\pi$
$632$	942.236			1.49088
$633$			420.000i			0.663507i
$634$	−170.000	−	294.449i	−0.268139	−	0.464430i
$635$		0			0
$636$	444.000	+	256.344i	0.698113	+	0.403056i
$637$		−	2.00000i		−	0.00313972i
$638$	−180.133	−	312.000i	−0.282341	−	0.489028i
$639$		0			0
$640$		0			0
$641$	10.0000			0.0156006			0.00780031	−	0.999970i	$-0.497517\pi$
$641$	10.0000			0.0156006			0.00780031	+	0.999970i	$0.497517\pi$
$642$	62.3538	−	36.0000i	0.0971243	−	0.0560748i
$643$	−1212.44			−1.88559			−0.942796	−	0.333370i	$-0.891814\pi$
$643$	−1212.44			−1.88559			−0.942796	+	0.333370i	$0.891814\pi$
$644$	−384.000	+	665.108i	−0.596273	+	1.03278i
$645$		0			0
$646$	360.000	−	207.846i	0.557276	−	0.321743i
$647$	−332.554			−0.513993			−0.256997	−	0.966412i	$-0.582733\pi$
$647$	−332.554			−0.513993			−0.256997	+	0.966412i	$0.582733\pi$
$648$			72.0000i			0.111111i
$649$	−624.000			−0.961479
$650$		0			0
$651$			83.1384i			0.127709i
$652$	−41.5692	+	72.0000i	−0.0637565	+	0.110429i
$653$			670.000i			1.02603i	0.858379	+	0.513017i	$0.171472\pi$
$653$			670.000i			1.02603i	−0.858379	+	0.513017i	$0.828528\pi$
$654$	138.000	−	79.6743i	0.211009	−	0.121826i
$655$		0			0
$656$	−464.000	−	803.672i	−0.707317	−	1.22511i
$657$		−	138.000i		−	0.210046i
$658$	831.384	−	480.000i	1.26350	−	0.729483i
$659$			824.456i			1.25107i	0.780195	+	0.625536i	$0.215120\pi$
$659$			824.456i			1.25107i	−0.780195	+	0.625536i	$0.784880\pi$
$660$		0			0
$661$	−1222.00			−1.84871			−0.924357	−	0.381529i	$-0.875398\pi$
$661$	−1222.00			−1.84871			−0.924357	+	0.381529i	$0.875398\pi$
$662$	−408.764	−	708.000i	−0.617468	−	1.06949i
$663$	−34.6410			−0.0522489
$664$		−	387.979i		−	0.584306i
$665$		0			0
$666$	78.0000	+	135.100i	0.117117	+	0.202853i
$667$	720.533			1.08026
$668$	193.990	−	336.000i	0.290404	−	0.502994i
$669$	−588.000			−0.878924
$670$		0			0
$671$			180.133i			0.268455i
$672$	332.554	+	192.000i	0.494872	+	0.285714i
$673$		−	334.000i		−	0.496285i	−0.968724	−	0.248143i	$-0.920180\pi$
$673$		−	334.000i		−	0.496285i	0.968724	−	0.248143i	$-0.0798202\pi$
$674$	338.000	+	585.433i	0.501484	+	0.868595i
$675$		0			0
$676$	330.000	−	571.577i	0.488166	−	0.845528i
$677$		−	1006.00i		−	1.48597i	−0.669309	−	0.742984i	$-0.733410\pi$
$677$		−	1006.00i		−	1.48597i	0.669309	−	0.742984i	$-0.266590\pi$
$678$	−190.526	−	330.000i	−0.281011	−	0.486726i
$679$			13.8564i			0.0204071i
$680$		0			0
$681$	492.000			0.722467
$682$	83.1384	−	48.0000i	0.121904	−	0.0703812i
$683$	187.061			0.273882			0.136941	−	0.990579i	$-0.456273\pi$
$683$	187.061			0.273882			0.136941	+	0.990579i	$0.456273\pi$
$684$	−216.000	−	124.708i	−0.315789	−	0.182321i
$685$		0			0
$686$	−600.000	+	346.410i	−0.874636	+	0.504971i
$687$	−245.951			−0.358008
$688$	−387.979	−	672.000i	−0.563924	−	0.976744i
$689$	148.000			0.214804
$690$		0			0
$691$			990.733i			1.43377i	0.697193	+	0.716884i	$0.254432\pi$
$691$			990.733i			1.43377i	−0.697193	+	0.716884i	$0.745568\pi$
$692$	1157.01	+	668.000i	1.67198	+	0.965318i
$693$		−	144.000i		−	0.207792i
$694$	−348.000	+	200.918i	−0.501441	+	0.289507i
$695$		0			0
$696$		−	360.267i		−	0.517624i
$697$		−	580.000i		−	0.832138i
$698$	876.418	−	506.000i	1.25561	−	0.724928i
$699$		−	142.028i		−	0.203188i
$700$		0			0
$701$	−1034.00			−1.47504			−0.737518	−	0.675328i	$-0.764002\pi$
$701$	−1034.00			−1.47504			−0.737518	+	0.675328i	$0.764002\pi$
$702$	10.3923	+	18.0000i	0.0148039	+	0.0256410i
$703$	−540.400			−0.768705
$704$		−	443.405i		−	0.629837i
$705$		0			0
$706$	−178.000	−	308.305i	−0.252125	−	0.436693i
$707$	512.687			0.725158
$708$	−540.400	−	312.000i	−0.763277	−	0.440678i
$709$	−530.000			−0.747532			−0.373766	−	0.927523i	$-0.621934\pi$
$709$	−530.000			−0.747532			−0.373766	+	0.927523i	$0.621934\pi$
$710$		0			0
$711$		−	353.338i		−	0.496960i
$712$		−	656.000i		−	0.921348i
$713$			192.000i			0.269285i
$714$	120.000	+	207.846i	0.168067	+	0.291101i
$715$		0			0
$716$	648.000	+	374.123i	0.905028	+	0.522518i
$717$			672.000i			0.937238i
$718$	166.277	+	288.000i	0.231583	+	0.401114i
$719$			706.677i			0.982861i	0.870917	+	0.491430i	$0.163526\pi$
$719$			706.677i			0.982861i	−0.870917	+	0.491430i	$0.836474\pi$
$720$		0			0
$721$	−528.000			−0.732316
$722$	122.976	−	71.0000i	0.170326	−	0.0983380i
$723$	79.6743			0.110200
$724$	4.00000	−	6.92820i	0.00552486	−	0.00956934i
$725$		0			0
$726$	219.000	−	126.440i	0.301653	−	0.174159i
$727$	242.487			0.333545			0.166772	−	0.985995i	$-0.446665\pi$
$727$	242.487			0.333545			0.166772	+	0.985995i	$0.446665\pi$
$728$	110.851			0.152268
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	484.974i		−	0.663439i
$732$	−90.0666	+	156.000i	−0.123042	+	0.213115i
$733$			194.000i			0.264666i	0.991205	+	0.132333i	$0.0422468\pi$
$733$			194.000i			0.264666i	−0.991205	+	0.132333i	$0.957753\pi$
$734$	−348.000	+	200.918i	−0.474114	+	0.273730i
$735$		0			0
$736$	768.000	+	443.405i	1.04348	+	0.602452i
$737$			48.0000i			0.0651289i
$738$	−301.377	+	174.000i	−0.408370	+	0.235772i
$739$			1351.00i			1.82815i	0.405550	+	0.914073i	$0.367080\pi$
$739$			1351.00i			1.82815i	−0.405550	+	0.914073i	$0.632920\pi$
$740$		0			0
$741$	−72.0000			−0.0971660
$742$	−512.687	−	888.000i	−0.690953	−	1.19677i
$743$	−678.964			−0.913814			−0.456907	−	0.889514i	$-0.651043\pi$
$743$	−678.964			−0.913814			−0.456907	+	0.889514i	$0.651043\pi$
$744$	96.0000			0.129032
$745$		0			0
$746$	310.000	+	536.936i	0.415550	+	0.719753i
$747$	−145.492			−0.194769
$748$	138.564	−	240.000i	0.185246	−	0.320856i
$749$	−144.000			−0.192256
$750$		0			0
$751$		−	658.179i		−	0.876404i	−0.898877	−	0.438202i	$-0.855615\pi$
$751$		−	658.179i		−	0.876404i	0.898877	−	0.438202i	$-0.144385\pi$
$752$	−554.256	−	960.000i	−0.737043	−	1.27660i
$753$		−	252.000i		−	0.334661i
$754$	−52.0000	−	90.0666i	−0.0689655	−	0.119452i
$755$		0			0
$756$	72.0000	−	124.708i	0.0952381	−	0.164957i
$757$			1006.00i			1.32893i	0.747319	+	0.664465i	$0.231341\pi$
$757$			1006.00i			1.32893i	−0.747319	+	0.664465i	$0.768659\pi$
$758$	−436.477	−	756.000i	−0.575827	−	0.997361i
$759$		−	332.554i		−	0.438147i
$760$		0			0
$761$	−758.000			−0.996058			−0.498029	−	0.867160i	$-0.665943\pi$
$761$	−758.000			−0.996058			−0.498029	+	0.867160i	$0.665943\pi$
$762$	−436.477	+	252.000i	−0.572804	+	0.330709i
$763$	−318.697			−0.417690
$764$	−768.000	−	443.405i	−1.00524	−	0.580373i
$765$		0			0
$766$	−1056.00	+	609.682i	−1.37859	+	0.795929i
$767$	−180.133			−0.234854
$768$	221.703	−	384.000i	0.288675	−	0.500000i
$769$	−2.00000			−0.00260078			−0.00130039	−	0.999999i	$-0.500414\pi$
$769$	−2.00000			−0.00260078			−0.00130039	+	0.999999i	$0.500414\pi$
$770$		0			0
$771$		−	439.941i		−	0.570611i
$772$	1004.59	+	580.000i	1.30128	+	0.751295i
$773$			262.000i			0.338939i	0.985535	+	0.169470i	$0.0542055\pi$
$773$			262.000i			0.338939i	−0.985535	+	0.169470i	$0.945795\pi$
$774$	−252.000	+	145.492i	−0.325581	+	0.187975i
$775$		0			0
$776$	16.0000			0.0206186
$777$		−	312.000i		−	0.401544i
$778$	−1001.13	+	578.000i	−1.28679	+	0.742931i
$779$		−	1205.51i		−	1.54751i
$780$		0			0
$781$		0			0
$782$	277.128	+	480.000i	0.354384	+	0.613811i
$783$	−135.100			−0.172541
$784$	8.00000	+	13.8564i	0.0102041	+	0.0176740i
$785$		0			0
$786$	204.000	+	353.338i	0.259542	+	0.449540i
$787$	−1447.99			−1.83989			−0.919946	−	0.392046i	$-0.871767\pi$
$787$	−1447.99			−1.83989			−0.919946	+

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)

\(f(q)\)	\(=\)	\(q+(-1.73205 + 1.00000i) q^{2} -1.73205 q^{3} +(2.00000 - 3.46410i) q^{4} +(3.00000 - 1.73205i) q^{6} -6.92820 q^{7} +8.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(-1.73205 + 1.00000i) q^{2} -1.73205 q^{3} +(2.00000 - 3.46410i) q^{4} +(3.00000 - 1.73205i) q^{6} -6.92820 q^{7} +8.00000i q^{8} +3.00000 q^{9} +6.92820i q^{11} +(-3.46410 + 6.00000i) q^{12} +2.00000i q^{13} +(12.0000 - 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} -10.0000i q^{17} +(-5.19615 + 3.00000i) q^{18} -20.7846i q^{19} +12.0000 q^{21} +(-6.92820 - 12.0000i) q^{22} +27.7128 q^{23} -13.8564i q^{24} +(-2.00000 - 3.46410i) q^{26} -5.19615 q^{27} +(-13.8564 + 24.0000i) q^{28} +26.0000 q^{29} +6.92820i q^{31} +(27.7128 + 16.0000i) q^{32} -12.0000i q^{33} +(10.0000 + 17.3205i) q^{34} +(6.00000 - 10.3923i) q^{36} -26.0000i q^{37} +(20.7846 + 36.0000i) q^{38} -3.46410i q^{39} +58.0000 q^{41} +(-20.7846 + 12.0000i) q^{42} +48.4974 q^{43} +(24.0000 + 13.8564i) q^{44} +(-48.0000 + 27.7128i) q^{46} +69.2820 q^{47} +(13.8564 + 24.0000i) q^{48} -1.00000 q^{49} +17.3205i q^{51} +(6.92820 + 4.00000i) q^{52} -74.0000i q^{53} +(9.00000 - 5.19615i) q^{54} -55.4256i q^{56} +36.0000i q^{57} +(-45.0333 + 26.0000i) q^{58} +90.0666i q^{59} +26.0000 q^{61} +(-6.92820 - 12.0000i) q^{62} -20.7846 q^{63} -64.0000 q^{64} +(12.0000 + 20.7846i) q^{66} +6.92820 q^{67} +(-34.6410 - 20.0000i) q^{68} -48.0000 q^{69} +24.0000i q^{72} -46.0000i q^{73} +(26.0000 + 45.0333i) q^{74} +(-72.0000 - 41.5692i) q^{76} -48.0000i q^{77} +(3.46410 + 6.00000i) q^{78} -117.779i q^{79} +9.00000 q^{81} +(-100.459 + 58.0000i) q^{82} -48.4974 q^{83} +(24.0000 - 41.5692i) q^{84} +(-84.0000 + 48.4974i) q^{86} -45.0333 q^{87} -55.4256 q^{88} -82.0000 q^{89} -13.8564i q^{91} +(55.4256 - 96.0000i) q^{92} -12.0000i q^{93} +(-120.000 + 69.2820i) q^{94} +(-48.0000 - 27.7128i) q^{96} -2.00000i q^{97} +(1.73205 - 1.00000i) q^{98} +20.7846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q + 8q^{4} + 12q^{6} + 12q^{9} + O(q^{10}) \)	\( 4q + 8q^{4} + 12q^{6} + 12q^{9} + 48q^{14} - 32q^{16} + 48q^{21} - 8q^{26} + 104q^{29} + 40q^{34} + 24q^{36} + 232q^{41} + 96q^{44} - 192q^{46} - 4q^{49} + 36q^{54} + 104q^{61} - 256q^{64} + 48q^{66} - 192q^{69} + 104q^{74} - 288q^{76} + 36q^{81} + 96q^{84} - 336q^{86} - 328q^{89} - 480q^{94} - 192q^{96} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about