LMFDB - Embedded newform 63.4.e.a.46.1

Newspace parameters

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	63.e (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.71712033036$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			46.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	63.46
Dual form			63.4.e.a.37.1

$q$-expansion

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(-3.50000 - 18.1865i) q^{7} -21.0000 q^{8} +O(q^{10})$	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(-3.50000 - 18.1865i) q^{7} -21.0000 q^{8} +(-4.50000 + 7.79423i) q^{10} +(-7.50000 + 12.9904i) q^{11} -64.0000 q^{13} +(-42.0000 + 36.3731i) q^{14} +(35.5000 + 61.4878i) q^{16} +(42.0000 - 72.7461i) q^{17} +(8.00000 + 13.8564i) q^{19} +3.00000 q^{20} +45.0000 q^{22} +(-42.0000 - 72.7461i) q^{23} +(58.0000 - 100.459i) q^{25} +(96.0000 + 166.277i) q^{26} +(17.5000 + 6.06218i) q^{28} +297.000 q^{29} +(126.500 - 219.104i) q^{31} +(22.5000 - 38.9711i) q^{32} -252.000 q^{34} +(-42.0000 + 36.3731i) q^{35} +(158.000 + 273.664i) q^{37} +(24.0000 - 41.5692i) q^{38} +(31.5000 + 54.5596i) q^{40} -360.000 q^{41} +26.0000 q^{43} +(-7.50000 - 12.9904i) q^{44} +(-126.000 + 218.238i) q^{46} +(-15.0000 - 25.9808i) q^{47} +(-318.500 + 127.306i) q^{49} -348.000 q^{50} +(32.0000 - 55.4256i) q^{52} +(181.500 - 314.367i) q^{53} +45.0000 q^{55} +(73.5000 + 381.917i) q^{56} +(-445.500 - 771.629i) q^{58} +(-7.50000 + 12.9904i) q^{59} +(59.0000 + 102.191i) q^{61} -759.000 q^{62} +433.000 q^{64} +(96.0000 + 166.277i) q^{65} +(185.000 - 320.429i) q^{67} +(42.0000 + 72.7461i) q^{68} +(157.500 + 54.5596i) q^{70} +342.000 q^{71} +(-181.000 + 313.501i) q^{73} +(474.000 - 820.992i) q^{74} -16.0000 q^{76} +(262.500 + 90.9327i) q^{77} +(-233.500 - 404.434i) q^{79} +(106.500 - 184.463i) q^{80} +(540.000 + 935.307i) q^{82} -477.000 q^{83} -252.000 q^{85} +(-39.0000 - 67.5500i) q^{86} +(157.500 - 272.798i) q^{88} +(453.000 + 784.619i) q^{89} +(224.000 + 1163.94i) q^{91} +84.0000 q^{92} +(-45.0000 + 77.9423i) q^{94} +(24.0000 - 41.5692i) q^{95} +503.000 q^{97} +(808.500 + 636.529i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} + O(q^{10}) $	$ 2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} - 9q^{10} - 15q^{11} - 128q^{13} - 84q^{14} + 71q^{16} + 84q^{17} + 16q^{19} + 6q^{20} + 90q^{22} - 84q^{23} + 116q^{25} + 192q^{26} + 35q^{28} + 594q^{29} + 253q^{31} + 45q^{32} - 504q^{34} - 84q^{35} + 316q^{37} + 48q^{38} + 63q^{40} - 720q^{41} + 52q^{43} - 15q^{44} - 252q^{46} - 30q^{47} - 637q^{49} - 696q^{50} + 64q^{52} + 363q^{53} + 90q^{55} + 147q^{56} - 891q^{58} - 15q^{59} + 118q^{61} - 1518q^{62} + 866q^{64} + 192q^{65} + 370q^{67} + 84q^{68} + 315q^{70} + 684q^{71} - 362q^{73} + 948q^{74} - 32q^{76} + 525q^{77} - 467q^{79} + 213q^{80} + 1080q^{82} - 954q^{83} - 504q^{85} - 78q^{86} + 315q^{88} + 906q^{89} + 448q^{91} + 168q^{92} - 90q^{94} + 48q^{95} + 1006q^{97} + 1617q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−1.50000	−	2.59808i	−0.530330	−	0.918559i	−0.999374	−	0.0353837i	$-0.988735\pi$
$2$	−1.50000	−	2.59808i	−0.530330	−	0.918559i	0.469044	−	0.883175i	$-0.344599\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.0625000	+	0.108253i
$5$	−1.50000	−	2.59808i	−0.134164	−	0.232379i	0.791114	−	0.611669i	$-0.209502\pi$
$5$	−1.50000	−	2.59808i	−0.134164	−	0.232379i	−0.925278	+	0.379290i	$0.876168\pi$
$6$		0			0
$7$	−3.50000	−	18.1865i	−0.188982	−	0.981981i
$7$	−3.50000	−	18.1865i	−0.188982	−	0.981981i
$8$	−21.0000			−0.928078
$9$		0			0
$10$	−4.50000	+	7.79423i	−0.142302	+	0.246475i
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	−0.950316	−	0.311287i	$-0.899240\pi$
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	0.744740	+	0.667355i	$0.232573\pi$
$12$		0			0
$13$	−64.0000			−1.36542			−0.682708	−	0.730691i	$-0.739198\pi$
$13$	−64.0000			−1.36542			−0.682708	+	0.730691i	$0.739198\pi$
$14$	−42.0000	+	36.3731i	−0.801784	+	0.694365i
$15$		0			0
$16$	35.5000	+	61.4878i	0.554688	+	0.960747i
$17$	42.0000	−	72.7461i	0.599206	−	1.03785i	−0.393733	−	0.919225i	$-0.628817\pi$
$17$	42.0000	−	72.7461i	0.599206	−	1.03785i	0.992939	−	0.118630i	$-0.0378502\pi$
$18$		0			0
$19$	8.00000	+	13.8564i	0.0965961	+	0.167309i	0.910274	−	0.414007i	$-0.135871\pi$
$19$	8.00000	+	13.8564i	0.0965961	+	0.167309i	−0.813678	+	0.581317i	$0.802538\pi$
$20$	3.00000			0.0335410
$21$		0			0
$22$	45.0000			0.436092
$23$	−42.0000	−	72.7461i	−0.380765	−	0.659505i	0.610406	−	0.792088i	$-0.291006\pi$
$23$	−42.0000	−	72.7461i	−0.380765	−	0.659505i	−0.991172	+	0.132583i	$0.957673\pi$
$24$		0			0
$25$	58.0000	−	100.459i	0.464000	−	0.803672i
$26$	96.0000	+	166.277i	0.724121	+	1.25421i
$27$		0			0
$28$	17.5000	+	6.06218i	0.118114	+	0.0409159i
$29$	297.000			1.90178			0.950888	−	0.309535i	$-0.100173\pi$
$29$	297.000			1.90178			0.950888	+	0.309535i	$0.100173\pi$
$30$		0			0
$31$	126.500	−	219.104i	0.732906	−	1.26943i	−0.222731	−	0.974880i	$-0.571497\pi$
$31$	126.500	−	219.104i	0.732906	−	1.26943i	0.955636	−	0.294550i	$-0.0951696\pi$
$32$	22.5000	−	38.9711i	0.124296	−	0.215287i
$33$		0			0
$34$	−252.000			−1.27111
$35$	−42.0000	+	36.3731i	−0.202837	+	0.175662i
$36$		0			0
$37$	158.000	+	273.664i	0.702028	+	1.21595i	0.967753	+	0.251900i	$0.0810553\pi$
$37$	158.000	+	273.664i	0.702028	+	1.21595i	−0.265725	+	0.964049i	$0.585611\pi$
$38$	24.0000	−	41.5692i	0.102456	−	0.177458i
$39$		0			0
$40$	31.5000	+	54.5596i	0.124515	+	0.215666i
$41$	−360.000			−1.37128			−0.685641	−	0.727940i	$-0.740478\pi$
$41$	−360.000			−1.37128			−0.685641	+	0.727940i	$0.740478\pi$
$42$		0			0
$43$	26.0000			0.0922084			0.0461042	−	0.998937i	$-0.485319\pi$
$43$	26.0000			0.0922084			0.0461042	+	0.998937i	$0.485319\pi$
$44$	−7.50000	−	12.9904i	−0.0256970	−	0.0445085i
$45$		0			0
$46$	−126.000	+	218.238i	−0.403863	+	0.699511i
$47$	−15.0000	−	25.9808i	−0.0465527	−	0.0806316i	0.841810	−	0.539774i	$-0.181490\pi$
$47$	−15.0000	−	25.9808i	−0.0465527	−	0.0806316i	−0.888363	+	0.459142i	$0.848157\pi$
$48$		0			0
$49$	−318.500	+	127.306i	−0.928571	+	0.371154i
$50$	−348.000			−0.984293
$51$		0			0
$52$	32.0000	−	55.4256i	0.0853385	−	0.147811i
$53$	181.500	−	314.367i	0.470395	−	0.814748i	−0.529032	−	0.848602i	$-0.677445\pi$
$53$	181.500	−	314.367i	0.470395	−	0.814748i	0.999427	+	0.0338538i	$0.0107781\pi$
$54$		0			0
$55$	45.0000			0.110324
$56$	73.5000	+	381.917i	0.175390	+	0.911354i
$57$		0			0
$58$	−445.500	−	771.629i	−1.00857	−	1.74689i
$59$	−7.50000	+	12.9904i	−0.0165494	+	0.0286645i	−0.874182	−	0.485599i	$-0.838601\pi$
$59$	−7.50000	+	12.9904i	−0.0165494	+	0.0286645i	0.857632	+	0.514264i	$0.171935\pi$
$60$		0			0
$61$	59.0000	+	102.191i	0.123839	+	0.214495i	0.921279	−	0.388903i	$-0.127146\pi$
$61$	59.0000	+	102.191i	0.123839	+	0.214495i	−0.797440	+	0.603399i	$0.793813\pi$
$62$	−759.000			−1.55473
$63$		0			0
$64$	433.000			0.845703
$65$	96.0000	+	166.277i	0.183190	+	0.317294i
$66$		0			0
$67$	185.000	−	320.429i	0.337334	−	0.584279i	−0.646597	−	0.762832i	$-0.723808\pi$
$67$	185.000	−	320.429i	0.337334	−	0.584279i	0.983930	+	0.178553i	$0.0571417\pi$
$68$	42.0000	+	72.7461i	0.0749007	+	0.129732i
$69$		0			0
$70$	157.500	+	54.5596i	0.268926	+	0.0931589i
$71$	342.000			0.571661			0.285831	−	0.958280i	$-0.407731\pi$
$71$	342.000			0.571661			0.285831	+	0.958280i	$0.407731\pi$
$72$		0			0
$73$	−181.000	+	313.501i	−0.290198	+	0.502638i	−0.973856	−	0.227165i	$-0.927054\pi$
$73$	−181.000	+	313.501i	−0.290198	+	0.502638i	0.683658	+	0.729802i	$0.260388\pi$
$74$	474.000	−	820.992i	0.744613	−	1.28971i
$75$		0			0
$76$	−16.0000			−0.0241490
$77$	262.500	+	90.9327i	0.388502	+	0.134581i
$78$		0			0
$79$	−233.500	−	404.434i	−0.332542	−	0.575979i	0.650468	−	0.759534i	$-0.274573\pi$
$79$	−233.500	−	404.434i	−0.332542	−	0.575979i	−0.983010	+	0.183555i	$0.941240\pi$
$80$	106.500	−	184.463i	0.148838	−	0.257795i
$81$		0			0
$82$	540.000	+	935.307i	0.727232	+	1.25960i
$83$	−477.000			−0.630814			−0.315407	−	0.948957i	$-0.602141\pi$
$83$	−477.000			−0.630814			−0.315407	+	0.948957i	$0.602141\pi$
$84$		0			0
$85$	−252.000			−0.321568
$86$	−39.0000	−	67.5500i	−0.0489009	−	0.0846989i
$87$		0			0
$88$	157.500	−	272.798i	0.190790	−	0.330459i
$89$	453.000	+	784.619i	0.539527	+	0.934488i	0.998929	+	0.0462600i	$0.0147303\pi$
$89$	453.000	+	784.619i	0.539527	+	0.934488i	−0.459402	+	0.888228i	$0.651936\pi$
$90$		0			0
$91$	224.000	+	1163.94i	0.258039	+	1.34081i
$92$	84.0000			0.0951914
$93$		0			0
$94$	−45.0000	+	77.9423i	−0.0493765	+	0.0855227i
$95$	24.0000	−	41.5692i	0.0259195	−	0.0448938i
$96$		0			0
$97$	503.000			0.526515			0.263257	−	0.964726i	$-0.415203\pi$
$97$	503.000			0.526515			0.263257	+	0.964726i	$0.415203\pi$
$98$	808.500	+	636.529i	0.833376	+	0.656113i
$99$		0			0
$100$	58.0000	+	100.459i	0.0580000	+	0.100459i
$101$	−543.000	+	940.504i	−0.534956	+	0.926570i	0.464210	+	0.885725i	$0.346338\pi$
$101$	−543.000	+	940.504i	−0.534956	+	0.926570i	−0.999165	+	0.0408451i	$0.986995\pi$
$102$		0			0
$103$	−868.000	−	1503.42i	−0.830355	−	1.43822i	−0.897757	−	0.440491i	$-0.854804\pi$
$103$	−868.000	−	1503.42i	−0.830355	−	1.43822i	0.0674017	−	0.997726i	$-0.478529\pi$
$104$	1344.00			1.26721
$105$		0			0
$106$	−1089.00			−0.997859
$107$	−676.500	−	1171.73i	−0.611212	−	1.05865i	−0.991036	−	0.133592i	$-0.957349\pi$
$107$	−676.500	−	1171.73i	−0.611212	−	1.05865i	0.379824	−	0.925059i	$-0.375985\pi$
$108$		0			0
$109$	185.000	−	320.429i	0.162567	−	0.281574i	−0.773222	−	0.634136i	$-0.781356\pi$
$109$	185.000	−	320.429i	0.162567	−	0.281574i	0.935789	+	0.352562i	$0.114689\pi$
$110$	−67.5000	−	116.913i	−0.0585079	−	0.101339i
$111$		0			0
$112$	994.000	−	860.829i	0.838609	−	0.726256i
$113$	648.000			0.539458			0.269729	−	0.962936i	$-0.413066\pi$
$113$	648.000			0.539458			0.269729	+	0.962936i	$0.413066\pi$
$114$		0			0
$115$	−126.000	+	218.238i	−0.102170	+	0.176964i
$116$	−148.500	+	257.210i	−0.118861	+	0.205873i
$117$		0			0
$118$	45.0000			0.0351067
$119$	−1470.00	−	509.223i	−1.13239	−	0.392272i
$120$		0			0
$121$	553.000	+	957.824i	0.415477	+	0.719627i
$122$	177.000	−	306.573i	0.131351	−	0.227507i
$123$		0			0
$124$	126.500	+	219.104i	0.0916132	+	0.158679i
$125$	−723.000			−0.517337
$126$		0			0
$127$	377.000			0.263412			0.131706	−	0.991289i	$-0.457954\pi$
$127$	377.000			0.263412			0.131706	+	0.991289i	$0.457954\pi$
$128$	−829.500	−	1436.74i	−0.572798	−	0.992115i
$129$		0			0
$130$	288.000	−	498.831i	0.194302	−	0.336541i
$131$	−325.500	−	563.783i	−0.217092	−	0.376015i	0.736826	−	0.676083i	$-0.236324\pi$
$131$	−325.500	−	563.783i	−0.217092	−	0.376015i	−0.953918	+	0.300068i	$0.902991\pi$
$132$		0			0
$133$	224.000	−	193.990i	0.146040	−	0.126474i
$134$	−1110.00			−0.715593
$135$		0			0
$136$	−882.000	+	1527.67i	−0.556109	+	0.963210i
$137$	−885.000	+	1532.86i	−0.551903	+	0.955923i	0.446235	+	0.894916i	$0.352765\pi$
$137$	−885.000	+	1532.86i	−0.551903	+	0.955923i	−0.998137	+	0.0610074i	$0.980569\pi$
$138$		0			0
$139$	−1558.00			−0.950704			−0.475352	−	0.879796i	$-0.657679\pi$
$139$	−1558.00			−0.950704			−0.475352	+	0.879796i	$0.657679\pi$
$140$	−10.5000	−	54.5596i	−0.00633866	−	0.0329366i
$141$		0			0
$142$	−513.000	−	888.542i	−0.303169	−	0.525104i
$143$	480.000	−	831.384i	0.280697	−	0.486181i
$144$		0			0
$145$	−445.500	−	771.629i	−0.255150	−	0.441933i
$146$	1086.00			0.615603
$147$		0			0
$148$	−316.000			−0.175507
$149$	1227.00	+	2125.23i	0.674629	+	1.16849i	0.976577	+	0.215168i	$0.0690298\pi$
$149$	1227.00	+	2125.23i	0.674629	+	1.16849i	−0.301948	+	0.953324i	$0.597637\pi$
$150$		0			0
$151$	−629.500	+	1090.33i	−0.339258	+	0.587612i	−0.984293	−	0.176540i	$-0.943509\pi$
$151$	−629.500	+	1090.33i	−0.339258	+	0.587612i	0.645035	+	0.764153i	$0.276843\pi$
$152$	−168.000	−	290.985i	−0.0896487	−	0.155276i
$153$		0			0
$154$	−157.500	−	818.394i	−0.0824137	−	0.428234i
$155$	−759.000			−0.393318
$156$		0			0
$157$	98.0000	−	169.741i	0.0498169	−	0.0862854i	−0.840042	−	0.542522i	$-0.817470\pi$
$157$	98.0000	−	169.741i	0.0498169	−	0.0862854i	0.889859	+	0.456236i	$0.150803\pi$
$158$	−700.500	+	1213.30i	−0.352714	+	0.610918i
$159$		0			0
$160$	−135.000			−0.0667043
$161$	−1176.00	+	1018.45i	−0.575663	+	0.498539i
$162$		0			0
$163$	626.000	+	1084.26i	0.300810	+	0.521019i	0.976320	−	0.216332i	$-0.0694095\pi$
$163$	626.000	+	1084.26i	0.300810	+	0.521019i	−0.675509	+	0.737351i	$0.736076\pi$
$164$	180.000	−	311.769i	0.0857051	−	0.148446i
$165$		0			0
$166$	715.500	+	1239.28i	0.334540	+	0.579440i
$167$	2646.00			1.22607			0.613035	−	0.790056i	$-0.289949\pi$
$167$	2646.00			1.22607			0.613035	+	0.790056i	$0.289949\pi$
$168$		0			0
$169$	1899.00			0.864360
$170$	378.000	+	654.715i	0.170537	+	0.295379i
$171$		0			0
$172$	−13.0000	+	22.5167i	−0.00576303	+	0.00998186i
$173$	−393.000	−	680.696i	−0.172712	−	0.299147i	0.766655	−	0.642059i	$-0.221920\pi$
$173$	−393.000	−	680.696i	−0.172712	−	0.299147i	−0.939367	+	0.342913i	$0.888586\pi$
$174$		0			0
$175$	−2030.00	−	703.213i	−0.876878	−	0.303759i
$176$	−1065.00			−0.456122
$177$		0			0
$178$	1359.00	−	2353.86i	0.572255	−	0.991174i
$179$	1446.00	−	2504.55i	0.603794	−	1.04580i	−0.388447	−	0.921471i	$-0.626988\pi$
$179$	1446.00	−	2504.55i	0.603794	−	1.04580i	0.992241	−	0.124331i	$-0.0396784\pi$
$180$		0			0
$181$	1352.00			0.555212			0.277606	−	0.960695i	$-0.410459\pi$
$181$	1352.00			0.555212			0.277606	+	0.960695i	$0.410459\pi$
$182$	2688.00	−	2327.88i	1.09477	−	0.948097i
$183$		0			0
$184$	882.000	+	1527.67i	0.353380	+	0.612072i
$185$	474.000	−	820.992i	0.188374	−	0.326273i
$186$		0			0
$187$	630.000	+	1091.19i	0.246365	+	0.426716i
$188$	30.0000			0.0116382
$189$		0			0
$190$	−144.000			−0.0549835
$191$	1956.00	+	3387.89i	0.741001	+	1.28345i	0.952040	+	0.305974i	$0.0989820\pi$
$191$	1956.00	+	3387.89i	0.741001	+	1.28345i	−0.211039	+	0.977478i	$0.567685\pi$
$192$		0			0
$193$	−746.500	+	1292.98i	−0.278416	+	0.482230i	−0.970991	−	0.239115i	$-0.923143\pi$
$193$	−746.500	+	1292.98i	−0.278416	+	0.482230i	0.692575	+	0.721345i	$0.256476\pi$
$194$	−754.500	−	1306.83i	−0.279227	−	0.483635i
$195$		0			0
$196$	49.0000	−	339.482i	0.0178571	−	0.123718i
$197$	4086.00			1.47774			0.738872	−	0.673846i	$-0.235359\pi$
$197$	4086.00			1.47774			0.738872	+	0.673846i	$0.235359\pi$
$198$		0			0
$199$	1778.00	−	3079.59i	0.633362	−	1.09702i	−0.353497	−	0.935436i	$-0.615008\pi$
$199$	1778.00	−	3079.59i	0.633362	−	1.09702i	0.986860	−	0.161580i	$-0.0516590\pi$
$200$	−1218.00	+	2109.64i	−0.430628	+	0.745870i
$201$		0			0
$202$	3258.00			1.13481
$203$	−1039.50	−	5401.40i	−0.359402	−	1.86751i
$204$		0			0
$205$	540.000	+	935.307i	0.183977	+	0.318657i
$206$	−2604.00	+	4510.26i	−0.880725	+	1.52546i
$207$		0			0
$208$	−2272.00	−	3935.22i	−0.757379	−	1.31182i
$209$	−240.000			−0.0794313
$210$		0			0
$211$	1250.00			0.407837			0.203918	−	0.978988i	$-0.434632\pi$
$211$	1250.00			0.407837			0.203918	+	0.978988i	$0.434632\pi$
$212$	181.500	+	314.367i	0.0587994	+	0.101844i
$213$		0			0
$214$	−2029.50	+	3515.20i	−0.648289	+	1.12287i
$215$	−39.0000	−	67.5500i	−0.0123711	−	0.0214273i
$216$		0			0
$217$	−4427.50	−	1533.73i	−1.38506	−	0.479799i
$218$	−1110.00			−0.344856
$219$		0			0
$220$	−22.5000	+	38.9711i	−0.00689523	+	0.0119429i
$221$	−2688.00	+	4655.75i	−0.818165	+	1.41710i
$222$		0			0
$223$	425.000			0.127624			0.0638119	−	0.997962i	$-0.479674\pi$
$223$	425.000			0.127624			0.0638119	+	0.997962i	$0.479674\pi$
$224$	−787.500	−	272.798i	−0.234898	−	0.0813709i
$225$		0			0
$226$	−972.000	−	1683.55i	−0.286091	−	0.495523i
$227$	1927.50	−	3338.53i	0.563580	−	0.976149i	−0.433600	−	0.901105i	$-0.642757\pi$
$227$	1927.50	−	3338.53i	0.563580	−	0.976149i	0.997180	−	0.0750439i	$-0.0239097\pi$
$228$		0			0
$229$	1094.00	+	1894.86i	0.315692	+	0.546795i	0.979584	−	0.201033i	$-0.0644299\pi$
$229$	1094.00	+	1894.86i	0.315692	+	0.546795i	−0.663892	+	0.747828i	$0.731097\pi$
$230$	756.000			0.216735
$231$		0			0
$232$	−6237.00			−1.76500
$233$	426.000	+	737.854i	0.119778	+	0.207461i	0.919679	−	0.392670i	$-0.128449\pi$
$233$	426.000	+	737.854i	0.119778	+	0.207461i	−0.799902	+	0.600131i	$0.795115\pi$
$234$		0			0
$235$	−45.0000	+	77.9423i	−0.0124914	+	0.0216357i
$236$	−7.50000	−	12.9904i	−0.00206868	−	0.00358306i
$237$		0			0
$238$	882.000	+	4583.01i	0.240217	+	1.24820i
$239$	−5508.00			−1.49072			−0.745362	−	0.666660i	$-0.767723\pi$
$239$	−5508.00			−1.49072			−0.745362	+	0.666660i	$0.767723\pi$
$240$		0			0
$241$	−395.500	+	685.026i	−0.105711	+	0.183097i	−0.914029	−	0.405650i	$-0.867045\pi$
$241$	−395.500	+	685.026i	−0.105711	+	0.183097i	0.808317	+	0.588747i	$0.200379\pi$
$242$	1659.00	−	2873.47i	0.440680	−	0.763280i
$243$		0			0
$244$	−118.000			−0.0309597
$245$	808.500	+	636.529i	0.210829	+	0.165985i
$246$		0			0
$247$	−512.000	−	886.810i	−0.131894	−	0.228447i
$248$	−2656.50	+	4601.19i	−0.680193	+	1.17813i
$249$		0			0
$250$	1084.50	+	1878.41i	0.274359	+	0.475204i
$251$	−5265.00			−1.32400			−0.662000	−	0.749504i	$-0.730292\pi$
$251$	−5265.00			−1.32400			−0.662000	+	0.749504i	$0.730292\pi$
$252$		0			0
$253$	1260.00			0.313105
$254$	−565.500	−	979.475i	−0.139695	−	0.241959i
$255$		0			0
$256$	−756.500	+	1310.30i	−0.184692	+	0.319897i
$257$	−3435.00	−	5949.59i	−0.833733	−	1.44407i	−0.895058	−	0.445950i	$-0.852866\pi$
$257$	−3435.00	−	5949.59i	−0.833733	−	1.44407i	0.0613246	−	0.998118i	$-0.480468\pi$
$258$		0			0
$259$	4424.00	−	3831.30i	1.06137	−	0.919171i
$260$	−192.000			−0.0457974
$261$		0			0
$262$	−976.500	+	1691.35i	−0.230261	+	0.398824i
$263$	−111.000	+	192.258i	−0.0260249	+	0.0450765i	−0.878745	−	0.477292i	$-0.841618\pi$
$263$	−111.000	+	192.258i	−0.0260249	+	0.0450765i	0.852720	+	0.522369i	$0.174952\pi$
$264$		0			0
$265$	−1089.00			−0.252441
$266$	−840.000	−	290.985i	−0.193623	−	0.0670730i
$267$		0			0
$268$	185.000	+	320.429i	0.0421667	+	0.0730349i
$269$	3925.50	−	6799.17i	0.889747	−	1.54109i	0.0495729	−	0.998771i	$-0.484214\pi$
$269$	3925.50	−	6799.17i	0.889747	−	1.54109i	0.840174	−	0.542317i	$-0.182453\pi$
$270$		0			0
$271$	−2591.50	−	4488.61i	−0.580895	−	1.00614i	−0.995374	−	0.0960800i	$-0.969370\pi$
$271$	−2591.50	−	4488.61i	−0.580895	−	1.00614i	0.414479	−	0.910059i	$-0.363964\pi$
$272$	5964.00			1.32949
$273$		0			0
$274$	5310.00			1.17076
$275$	870.000	+	1506.88i	0.190774	+	0.330431i
$276$		0			0
$277$	2480.00	−	4295.49i	0.537938	−	0.931736i	−0.461077	−	0.887360i	$-0.652537\pi$
$277$	2480.00	−	4295.49i	0.537938	−	0.931736i	0.999015	−	0.0443755i	$-0.0141298\pi$
$278$	2337.00	+	4047.80i	0.504187	+	0.873277i
$279$		0			0
$280$	882.000	−	763.834i	0.188249	−	0.163028i
$281$	774.000			0.164317			0.0821583	−	0.996619i	$-0.473819\pi$
$281$	774.000			0.164317			0.0821583	+	0.996619i	$0.473819\pi$
$282$		0			0
$283$	−1849.00	+	3202.56i	−0.388380	+	0.672695i	−0.992232	−	0.124402i	$-0.960299\pi$
$283$	−1849.00	+	3202.56i	−0.388380	+	0.672695i	0.603852	+	0.797097i	$0.293632\pi$
$284$	−171.000	+	296.181i	−0.0357288	+	0.0618841i
$285$		0			0
$286$	−2880.00			−0.595447
$287$	1260.00	+	6547.15i	0.259148	+	1.34657i
$288$		0			0
$289$	−1071.50	−	1855.89i	−0.218095	−	0.377751i
$290$	−1336.50	+	2314.89i	−0.270628	+	0.468741i
$291$		0			0
$292$	−181.000	−	313.501i	−0.0362747	−	0.0628297i
$293$	6273.00			1.25076			0.625380	−	0.780321i	$-0.284944\pi$
$293$	6273.00			1.25076			0.625380	+	0.780321i	$0.284944\pi$
$294$		0			0
$295$	45.0000			0.00888136
$296$	−3318.00	−	5746.94i	−0.651537	−	1.12849i
$297$		0			0
$298$	3681.00	−	6375.68i	0.715552	−	1.23937i
$299$	2688.00	+	4655.75i	0.519903	+	0.900499i
$300$		0			0
$301$	−91.0000	−	472.850i	−0.0174258	−	0.0905469i
$302$	3777.00			0.719675
$303$		0			0
$304$	−568.000	+	983.805i	−0.107161	+	0.185609i
$305$	177.000	−	306.573i	0.0332295	−	0.0575551i
$306$		0			0
$307$	−1684.00			−0.313065			−0.156533	−	0.987673i	$-0.550032\pi$
$307$	−1684.00			−0.313065			−0.156533	+	0.987673i	$0.550032\pi$
$308$	−210.000	+	181.865i	−0.0388502	+	0.0336453i
$309$		0			0
$310$	1138.50	+	1971.94i	0.208589	+	0.361286i
$311$	−660.000	+	1143.15i	−0.120338	+	0.208432i	−0.919901	−	0.392151i	$-0.871731\pi$
$311$	−660.000	+	1143.15i	−0.120338	+	0.208432i	0.799563	+	0.600582i	$0.205065\pi$
$312$		0			0
$313$	4251.50	+	7363.81i	0.767760	+	1.32980i	0.938775	+	0.344531i	$0.111962\pi$
$313$	4251.50	+	7363.81i	0.767760	+	1.32980i	−0.171014	+	0.985269i	$0.554704\pi$
$314$	−588.000			−0.105678
$315$		0			0
$316$	467.000			0.0831355
$317$	−1288.50	−	2231.75i	−0.228295	−	0.395418i	0.729008	−	0.684505i	$-0.239982\pi$
$317$	−1288.50	−	2231.75i	−0.228295	−	0.395418i	−0.957303	+	0.289087i	$0.906648\pi$
$318$		0			0
$319$	−2227.50	+	3858.14i	−0.390959	+	0.677162i
$320$	−649.500	−	1124.97i	−0.113463	−	0.196524i
$321$		0			0
$322$	4410.00	+	1527.67i	0.763229	+	0.264390i
$323$	1344.00			0.231524
$324$		0			0
$325$	−3712.00	+	6429.37i	−0.633553	+	1.09735i
$326$	1878.00	−	3252.79i	0.319058	−	0.552624i
$327$		0			0
$328$	7560.00			1.27266
$329$	−420.000	+	363.731i	−0.0703810	+	0.0609517i
$330$		0			0
$331$	242.000	+	419.156i	0.0401859	+	0.0696040i	0.885419	−	0.464794i	$-0.153872\pi$
$331$	242.000	+	419.156i	0.0401859	+	0.0696040i	−0.845233	+	0.534398i	$0.820538\pi$
$332$	238.500	−	413.094i	0.0394259	−	0.0682876i
$333$		0			0
$334$	−3969.00	−	6874.51i	−0.650222	−	1.12622i
$335$	−1110.00			−0.181032
$336$		0			0
$337$	−8359.00			−1.35117			−0.675584	−	0.737283i	$-0.736109\pi$
$337$	−8359.00			−1.35117			−0.675584	+	0.737283i	$0.736109\pi$
$338$	−2848.50	−	4933.75i	−0.458396	−	0.793966i
$339$		0			0
$340$	126.000	−	218.238i	0.0200980	−	0.0348107i
$341$	1897.50	+	3286.57i	0.301335	+	0.521928i
$342$		0			0
$343$	3430.00	+	5346.84i	0.539949	+	0.841698i
$344$	−546.000			−0.0855766
$345$		0			0
$346$	−1179.00	+	2042.09i	−0.183189	+	0.317293i
$347$	−930.000	+	1610.81i	−0.143876	+	0.249201i	−0.928953	−	0.370197i	$-0.879290\pi$
$347$	−930.000	+	1610.81i	−0.143876	+	0.249201i	0.785077	+	0.619398i	$0.212623\pi$
$348$		0			0
$349$	−1918.00			−0.294178			−0.147089	−	0.989123i	$-0.546990\pi$
$349$	−1918.00			−0.294178			−0.147089	+	0.989123i	$0.546990\pi$
$350$	1218.00	+	6328.91i	0.186014	+	0.966556i
$351$		0			0
$352$	337.500	+	584.567i	0.0511046	+	0.0885157i
$353$	−1524.00	+	2639.65i	−0.229786	+	0.398000i	−0.957744	−	0.287620i	$-0.907136\pi$
$353$	−1524.00	+	2639.65i	−0.229786	+	0.398000i	0.727959	+	0.685621i	$0.240469\pi$
$354$		0			0
$355$	−513.000	−	888.542i	−0.0766964	−	0.132842i
$356$	−906.000			−0.134882
$357$		0			0
$358$	−8676.00			−1.28084
$359$	−15.0000	−	25.9808i	−0.00220521	−	0.00381953i	0.864921	−	0.501909i	$-0.167369\pi$
$359$	−15.0000	−	25.9808i	−0.00220521	−	0.00381953i	−0.867126	+	0.498089i	$0.834035\pi$
$360$		0			0
$361$	3301.50	−	5718.37i	0.481338	−	0.833703i
$362$	−2028.00	−	3512.60i	−0.294446	−	0.509995i
$363$		0			0
$364$	−1120.00	−	387.979i	−0.161275	−	0.0558672i
$365$	1086.00			0.155737
$366$		0			0
$367$	5655.50	−	9795.61i	0.804400	−	1.39326i	−0.112296	−	0.993675i	$-0.535820\pi$
$367$	5655.50	−	9795.61i	0.804400	−	1.39326i	0.916696	−	0.399586i	$-0.130846\pi$
$368$	2982.00	−	5164.98i	0.422412	−	0.731638i
$369$		0			0
$370$	−2844.00			−0.399601
$371$	−6352.50	−	2200.57i	−0.888963	−	0.307946i
$372$		0			0
$373$	−604.000	−	1046.16i	−0.0838443	−	0.145223i	0.821054	−	0.570851i	$-0.193387\pi$
$373$	−604.000	−	1046.16i	−0.0838443	−	0.145223i	−0.904898	+	0.425628i	$0.860053\pi$
$374$	1890.00	−	3273.58i	0.261309	−	0.452600i
$375$		0			0
$376$	315.000	+	545.596i	0.0432045	+	0.0748324i
$377$	−19008.0			−2.59672
$378$		0			0
$379$	7640.00			1.03546			0.517731	−	0.855543i	$-0.326777\pi$
$379$	7640.00			1.03546			0.517731	+	0.855543i	$0.326777\pi$
$380$	24.0000	+	41.5692i	0.00323993	+	0.00561173i
$381$		0			0
$382$	5868.00	−	10163.7i	0.785950	−	1.36131i
$383$	6375.00	+	11041.8i	0.850515	+	1.47314i	0.880744	+	0.473592i	$0.157043\pi$
$383$	6375.00	+	11041.8i	0.850515	+	1.47314i	−0.0302291	+	0.999543i	$0.509624\pi$
$384$		0			0
$385$	−157.500	−	818.394i	−0.0208492	−	0.108336i
$386$	4479.00			0.590609
$387$		0			0
$388$	−251.500	+	435.611i	−0.0329072	+	0.0569969i
$389$	1563.00	−	2707.20i	0.203720	−	0.352854i	−0.746004	−	0.665942i	$-0.768030\pi$
$389$	1563.00	−	2707.20i	0.203720	−	0.352854i	0.949724	+	0.313087i	$0.101363\pi$
$390$		0			0
$391$	−7056.00			−0.912627
$392$	6688.50	−	2673.42i	0.861786	−	0.344459i
$393$		0			0
$394$	−6129.00	−	10615.7i	−0.783692	−	1.35739i
$395$	−700.500	+	1213.30i	−0.0892303	+	0.154551i
$396$		0			0
$397$	2966.00	+	5137.26i	0.374960	+	0.649450i	0.990321	−	0.138795i	$-0.0443230\pi$
$397$	2966.00	+	5137.26i	0.374960	+	0.649450i	−0.615361	+	0.788246i	$0.710990\pi$
$398$	−10668.0			−1.34356
$399$		0			0
$400$	8236.00			1.02950
$401$	804.000	+	1392.57i	0.100124	+	0.173420i	0.911736	−	0.410777i	$-0.134743\pi$
$401$	804.000	+	1392.57i	0.100124	+	0.173420i	−0.811611	+	0.584198i	$0.801409\pi$
$402$		0			0
$403$	−8096.00	+	14022.7i	−1.00072	+	1.73330i
$404$	−543.000	−	940.504i	−0.0668695	−	0.115821i
$405$		0			0
$406$	−12474.0	+	10802.8i	−1.52481	+	1.32053i
$407$	−4740.00			−0.577280
$408$		0			0
$409$	2232.50	−	3866.80i	0.269902	−	0.467484i	−0.698934	−	0.715186i	$-0.746342\pi$
$409$	2232.50	−	3866.80i	0.269902	−	0.467484i	0.968836	+	0.247702i	$0.0796753\pi$
$410$	1620.00	−	2805.92i	0.195137	−	0.337987i
$411$		0			0
$412$	1736.00			0.207589
$413$	262.500	+	90.9327i	0.0312755	+	0.0108342i
$414$		0			0
$415$	715.500	+	1239.28i	0.0846326	+	0.146588i
$416$	−1440.00	+	2494.15i	−0.169716	+	0.293957i
$417$		0			0
$418$	360.000	+	623.538i	0.0421248	+	0.0729623i
$419$	1584.00			0.184686			0.0923430	−	0.995727i	$-0.470564\pi$
$419$	1584.00			0.184686			0.0923430	+	0.995727i	$0.470564\pi$
$420$		0			0
$421$	−1330.00			−0.153967			−0.0769837	−	0.997032i	$-0.524529\pi$
$421$	−1330.00			−0.153967			−0.0769837	+	0.997032i	$0.524529\pi$
$422$	−1875.00	−	3247.60i	−0.216288	−	0.374622i
$423$		0			0
$424$	−3811.50	+	6601.71i	−0.436563	+	0.756150i
$425$	−4872.00	−	8438.55i	−0.556063	−	0.963129i
$426$		0			0
$427$	1652.00	−	1430.67i	0.187227	−	0.162143i
$428$	1353.00			0.152803
$429$		0			0
$430$	−117.000	+	202.650i	−0.0131215	+	0.0227271i
$431$	4794.00	−	8303.45i	0.535775	−	0.927989i	−0.463351	−	0.886175i	$-0.653353\pi$
$431$	4794.00	−	8303.45i	0.535775	−	0.927989i	0.999125	−	0.0418139i	$-0.0133137\pi$
$432$		0			0
$433$	494.000			0.0548271			0.0274135	−	0.999624i	$-0.491273\pi$
$433$	494.000			0.0548271			0.0274135	+	0.999624i	$0.491273\pi$
$434$	2656.50	+	13803.6i	0.293816	+	1.52671i
$435$		0			0
$436$	185.000	+	320.429i	0.0203209	+	0.0351968i
$437$	672.000	−	1163.94i	0.0735609	−	0.127411i
$438$		0			0
$439$	8004.50	+	13864.2i	0.870237	+	1.50729i	0.861752	+	0.507330i	$0.169368\pi$
$439$	8004.50	+	13864.2i	0.870237	+	1.50729i	0.00848508	+	0.999964i	$0.497299\pi$
$440$	−945.000			−0.102389
$441$		0			0
$442$	16128.0			1.73559
$443$	3886.50	+	6731.62i	0.416824	+	0.721961i	0.995618	−	0.0935130i	$-0.0298097\pi$
$443$	3886.50	+	6731.62i	0.416824	+	0.721961i	−0.578794	+	0.815474i	$0.696476\pi$
$444$		0			0
$445$	1359.00	−	2353.86i	0.144770	−	0.250749i
$446$	−637.500	−	1104.18i	−0.0676827	−	0.117230i
$447$		0			0
$448$	−1515.50	−	7874.77i	−0.159823	−	0.830464i
$449$	−864.000			−0.0908122			−0.0454061	−	0.998969i	$-0.514458\pi$
$449$	−864.000			−0.0908122			−0.0454061	+	0.998969i	$0.514458\pi$
$450$		0			0
$451$	2700.00	−	4676.54i	0.281903	−	0.488269i
$452$	−324.000	+	561.184i	−0.0337161	+	0.0583980i
$453$		0			0
$454$	−11565.0			−1.19553
$455$	2688.00	−	2327.88i	0.276957	−	0.239852i
$456$		0			0
$457$	−1259.50	−	2181.52i	−0.128921	−	0.223298i	0.794338	−	0.607476i	$-0.207818\pi$
$457$	−1259.50	−	2181.52i	−0.128921	−	0.223298i	−0.923259	+	0.384179i	$0.874485\pi$
$458$	3282.00	−	5684.59i	0.334842	−	0.579964i
$459$		0			0
$460$	−126.000	−	218.238i	−0.0127713	−	0.0221205i
$461$	342.000			0.0345521			0.0172761	−	0.999851i	$-0.494501\pi$
$461$	342.000			0.0345521			0.0172761	+	0.999851i	$0.494501\pi$
$462$		0			0
$463$	−4336.00			−0.435229			−0.217614	−	0.976035i	$-0.569828\pi$
$463$	−4336.00			−0.435229			−0.217614	+	0.976035i	$0.569828\pi$
$464$	10543.5	+	18261.9i	1.05489	+	1.82713i
$465$		0			0
$466$	1278.00	−	2213.56i	0.127043	−	0.220046i
$467$	9318.00	+	16139.2i	0.923310	+	1.59922i	0.794257	+	0.607581i	$0.207860\pi$
$467$	9318.00	+	16139.2i	0.923310	+	1.59922i	0.129052	+	0.991638i	$0.458806\pi$
$468$		0			0
$469$	−6475.00	−	2243.01i	−0.637500	−	0.220837i
$470$	270.000			0.0264982
$471$		0			0
$472$	157.500	−	272.798i	0.0153592	−	0.0266029i
$473$	−195.000	+	337.750i	−0.0189558	+	0.0328325i
$474$		0			0
$475$	1856.00			0.179282
$476$	1176.00	−	1018.45i	0.113239	−	0.0980680i
$477$		0			0
$478$	8262.00	+	14310.2i	0.790575	+	1.36932i
$479$	7539.00	−	13057.9i	0.719135	−	1.24558i	−0.242208	−	0.970224i	$-0.577872\pi$
$479$	7539.00	−	13057.9i	0.719135	−	1.24558i	0.961343	−	0.275354i	$-0.0887951\pi$
$480$		0			0
$481$	−10112.0	−	17514.5i	−0.958560	−	1.66028i
$482$	2373.00			0.224247
$483$		0			0
$484$	−1106.00			−0.103869
$485$	−754.500	−	1306.83i	−0.0706393	−	0.122351i
$486$		0			0
$487$	−3110.50	+	5387.54i	−0.289425	+	0.501300i	−0.973673	−	0.227950i	$-0.926798\pi$
$487$	−3110.50	+	5387.54i	−0.289425	+	0.501300i	0.684247	+	0.729250i	$0.260131\pi$
$488$	−1239.00	−	2146.01i	−0.114932	−	0.199068i
$489$		0			0
$490$	441.000	−	3055.34i	0.0406579	−	0.281686i
$491$	7371.00			0.677492			0.338746	−	0.940878i	$-0.389997\pi$
$491$	7371.00			0.677492			0.338746	+	0.940878i	$0.389997\pi$
$492$		0			0
$493$	12474.0	−	21605.6i	1.13956	−	1.97377i
$494$	−1536.00	+	2660.43i	−0.139895	+	0.242304i
$495$		0			0
$496$	17963.0			1.62613
$497$	−1197.00	−	6219.79i	−0.108034	−	0.561360i
$498$		0			0
$499$	−2137.00	−	3701.39i	−0.191714	−	0.332058i	0.754104	−	0.656755i	$-0.228071\pi$
$499$	−2137.00	−	3701.39i	−0.191714	−	0.332058i	−0.945818	+	0.324696i	$0.894738\pi$
$500$	361.500	−	626.136i	0.0323335	−	0.0560033i
$501$		0			0
$502$	7897.50	+	13678.9i	0.702157	+	1.21617i
$503$	2520.00			0.223382			0.111691	−	0.993743i	$-0.464373\pi$
$503$	2520.00			0.223382			0.111691	+	0.993743i	$0.464373\pi$
$504$		0			0
$505$	3258.00			0.287087
$506$	−1890.00	−	3273.58i	−0.166049	−	0.287605i
$507$		0			0
$508$	−188.500	+	326.492i	−0.0164633	+	0.0285152i
$509$	−7138.50	−	12364.2i	−0.621628	−	1.07669i	−0.989183	−	0.146689i	$-0.953138\pi$
$509$	−7138.50	−	12364.2i	−0.621628	−	1.07669i	0.367555	−	0.930002i	$-0.380195\pi$
$510$		0			0
$511$	6335.00	+	2194.51i	0.548423	+	0.189979i
$512$	−8733.00			−0.753804
$513$		0			0
$514$	−10305.0	+	17848.8i	−0.884308	+	1.53167i
$515$	−2604.00	+	4510.26i	−0.222808	+	0.385914i
$516$		0			0
$517$	450.000			0.0382804
$518$	−16590.0	−	5746.94i	−1.40719	−	0.487464i
$519$		0			0
$520$	−2016.00	−	3491.81i	−0.170014	−	0.294473i
$521$	−3153.00	+	5461.16i	−0.265135	+	0.459228i	−0.967599	−	0.252492i	$-0.918750\pi$
$521$	−3153.00	+	5461.16i	−0.265135	+	0.459228i	0.702464	+	0.711719i	$0.252083\pi$
$522$		0			0
$523$	−4036.00	−	6990.56i	−0.337442	−	0.584466i	0.646509	−	0.762906i	$-0.276228\pi$
$523$	−4036.00	−	6990.56i	−0.337442	−	0.584466i	−0.983951	+	0.178440i	$0.942895\pi$
$524$	651.000			0.0542730
$525$		0			0
$526$	666.000			0.0552072
$527$	−10626.0	−	18404.8i	−0.878322	−	1.52130i
$528$		0			0
$529$	2555.50	−	4426.26i	0.210035	−	0.363792i
$530$	1633.50	+	2829.30i	0.133877	+	0.231881i
$531$		0			0
$532$	56.0000	+	290.985i	0.00456374	+	0.0237139i
$533$	23040.0			1.87237
$534$		0			0
$535$	−2029.50	+	3515.20i	−0.164005	+	0.284066i
$536$	−3885.00	+	6729.02i	−0.313072	+	0.542256i
$537$		0			0
$538$	−23553.0			−1.88744
$539$	735.000	−	5092.23i	0.0587360	−	0.406935i
$540$		0			0
$541$	11429.0	+	19795.6i	0.908264	+	1.57316i	0.816474	+	0.577382i	$0.195926\pi$
$541$	11429.0	+	19795.6i	0.908264	+	1.57316i	0.0917903	+	0.995778i	$0.470741\pi$
$542$	−7774.50	+	13465.8i	−0.616132	+	1.06717i
$543$		0			0
$544$	−1890.00	−	3273.58i	−0.148958	−	0.258003i
$545$	−1110.00			−0.0872425
$546$		0			0
$547$	−24724.0			−1.93258			−0.966291	−	0.257454i	$-0.917116\pi$
$547$	−24724.0			−1.93258			−0.966291	+	0.257454i	$0.917116\pi$
$548$	−885.000	−	1532.86i	−0.0689878	−	0.119490i
$549$		0			0
$550$	2610.00	−	4520.65i	0.202347	−	0.350475i
$551$	2376.00	+	4115.35i	0.183704	+	0.318185i
$552$		0			0
$553$	−6538.00	+	5662.07i	−0.502756	+	0.435399i
$554$	−14880.0			−1.14114
$555$		0			0
$556$	779.000	−	1349.27i	0.0594190	−	0.102917i
$557$	−4921.50	+	8524.29i	−0.374382	+	0.648448i	−0.990234	−	0.139413i	$-0.955478\pi$
$557$	−4921.50	+	8524.29i	−0.374382	+	0.648448i	0.615853	+	0.787861i	$0.288812\pi$
$558$		0			0
$559$	−1664.00			−0.125903
$560$	−3727.50	−	1291.24i	−0.281278	−	0.0974375i
$561$		0			0
$562$	−1161.00	−	2010.91i	−0.0871420	−	0.150934i
$563$	−6685.50	+	11579.6i	−0.500462	+	0.866826i	0.499538	+	0.866292i	$0.333503\pi$
$563$	−6685.50	+	11579.6i	−0.500462	+	0.866826i	−1.00000		0.000533812i	$0.999830\pi$
$564$		0			0
$565$	−972.000	−	1683.55i	−0.0723758	−	0.125359i
$566$	11094.0			0.823879
$567$		0			0
$568$	−7182.00			−0.530546
$569$	−2616.00	−	4531.04i	−0.192739	−	0.333834i	0.753418	−	0.657542i	$-0.228404\pi$
$569$	−2616.00	−	4531.04i	−0.192739	−	0.333834i	−0.946157	+	0.323708i	$0.895070\pi$
$570$		0			0
$571$	7199.00	−	12469.0i	0.527616	−	0.913858i	−0.471866	−	0.881670i	$-0.656419\pi$
$571$	7199.00	−	12469.0i	0.527616	−	0.913858i	0.999482	−	0.0321874i	$-0.0102474\pi$
$572$	480.000	+	831.384i	0.0350871	+	0.0607726i
$573$		0			0
$574$	15120.0	−	13094.3i	1.09947	−	0.952170i
$575$	−9744.00			−0.706701
$576$		0			0
$577$	−9935.50	+	17208.8i	−0.716846	+	1.24161i	0.245397	+	0.969423i	$0.421082\pi$
$577$	−9935.50	+	17208.8i	−0.716846	+	1.24161i	−0.962243	+	0.272191i	$0.912252\pi$
$578$	−3214.50	+	5567.68i	−0.231325	+	0.400666i
$579$		0			0
$580$	891.000			0.0637875
$581$	1669.50	+	8674.98i	0.119213	+	0.619447i
$582$		0			0
$583$	2722.50	+	4715.51i	0.193404	+	0.334985i
$584$	3801.00	−	6583.53i	0.269326	−	0.466487i
$585$		0			0
$586$	−9409.50	−	16297.7i	−0.663315	−	1.14890i
$587$	16137.0			1.13466			0.567330	−	0.823491i	$-0.307976\pi$
$587$	16137.0			1.13466			0.567330	+	0.823491i	$0.307976\pi$
$588$		0			0
$589$	4048.00			0.283183
$590$	−67.5000	−	116.913i	−0.00471005	−	0.00815805i
$591$		0			0
$592$	−11218.0	+	19430.1i	−0.778812	+	1.34894i
$593$	−10662.0	−	18467.1i	−0.738340	−	1.27884i	−0.953242	−	0.302207i	$-0.902276\pi$
$593$	−10662.0	−	18467.1i	−0.738340	−	1.27884i	0.214902	−	0.976636i	$-0.431057\pi$
$594$		0			0
$595$	882.000	+	4583.01i	0.0607705	+	0.315773i
$596$	−2454.00			−0.168657
$597$		0			0
$598$	8064.00	−	13967.3i	0.551441	−	0.955123i
$599$	−4323.00	+	7487.66i	−0.294880	+	0.510747i	−0.974957	−	0.222394i	$-0.928613\pi$
$599$	−4323.00	+	7487.66i	−0.294880	+	0.510747i	0.680077	+	0.733141i	$0.261946\pi$
$600$		0			0
$601$	11195.0			0.759823			0.379911	−	0.925023i	$-0.375954\pi$
$601$	11195.0			0.759823			0.379911	+	0.925023i	$0.375954\pi$
$602$	−1092.00	+	945.700i	−0.0739312	+	0.0640263i
$603$		0			0
$604$	−629.500	−	1090.33i	−0.0424073	−	0.0734515i
$605$	1659.00	−	2873.47i	0.111484	−	0.193096i
$606$		0			0
$607$	4485.50	+	7769.11i	0.299935	+	0.519503i	0.976121	−	0.217228i	$-0.0697015\pi$
$607$	4485.50	+	7769.11i	0.299935	+	0.519503i	−0.676185	+	0.736731i	$0.736368\pi$
$608$	720.000			0.0480261
$609$		0			0
$610$	−1062.00			−0.0704904
$611$	960.000	+	1662.77i	0.0635637	+	0.110096i
$612$		0			0
$613$	6386.00	−	11060.9i	0.420764	−	0.728784i	−0.575251	−	0.817977i	$-0.695096\pi$
$613$	6386.00	−	11060.9i	0.420764	−	0.728784i	0.996014	+	0.0891932i	$0.0284288\pi$
$614$	2526.00	+	4375.16i	0.166028	+	0.287569i
$615$		0			0
$616$	−5512.50	−	1909.59i	−0.360560	−	0.124902i
$617$	−12762.0			−0.832705			−0.416352	−	0.909203i	$-0.636692\pi$
$617$	−12762.0			−0.832705			−0.416352	+	0.909203i	$0.636692\pi$
$618$		0			0
$619$	−6421.00	+	11121.5i	−0.416933	+	0.722150i	−0.995629	−	0.0933936i	$-0.970229\pi$
$619$	−6421.00	+	11121.5i	−0.416933	+	0.722150i	0.578696	+	0.815543i	$0.303562\pi$
$620$	379.500	−	657.313i	0.0245824	−	0.0425780i
$621$		0			0
$622$	3960.00			0.255276
$623$	12684.0	−	10984.7i	0.815688	−	0.706407i
$624$		0			0
$625$	−6165.50	−	10679.0i	−0.394592	−	0.683453i
$626$	12754.5	−	22091.4i	0.814333	−	1.41047i
$627$		0			0
$628$	98.0000	+	169.741i	0.00622711	+	0.0107857i
$629$	26544.0			1.68264
$630$		0			0
$631$	21365.0			1.34790			0.673952	−	0.738775i	$-0.264596\pi$
$631$	21365.0			1.34790			0.673952	+	0.738775i	$0.264596\pi$
$632$	4903.50	+	8493.11i	0.308625	+	0.534554i
$633$		0			0
$634$	−3865.50	+	6695.24i	−0.242143	+	0.419404i
$635$	−565.500	−	979.475i	−0.0353404	−	0.0612114i
$636$		0			0
$637$	20384.0	−	8147.57i	1.26789	−	0.506779i
$638$	13365.0			0.829350
$639$		0			0
$640$	−2488.50	+	4310.21i	−0.153698	+	0.266212i
$641$	4137.00	−	7165.49i	0.254917	−	0.441529i	−0.709956	−	0.704246i	$-0.751285\pi$
$641$	4137.00	−	7165.49i	0.254917	−	0.441529i	0.964873	+	0.262717i	$0.0846186\pi$
$642$		0			0
$643$	27998.0			1.71716			0.858580	−	0.512680i	$-0.171347\pi$
$643$	27998.0			1.71716			0.858580	+	0.512680i	$0.171347\pi$
$644$	−294.000	−	1527.67i	−0.0179895	−	0.0934761i
$645$		0			0
$646$	−2016.00	−	3491.81i	−0.122784	−	0.212668i
$647$	−8733.00	+	15126.0i	−0.530649	+	0.919110i	0.468712	+	0.883351i	$0.344718\pi$
$647$	−8733.00	+	15126.0i	−0.530649	+	0.919110i	−0.999360	+	0.0357592i	$0.988615\pi$
$648$		0			0
$649$	−112.500	−	194.856i	−0.00680433	−	0.0117854i
$650$	22272.0			1.34397
$651$		0			0
$652$	−1252.00			−0.0752026
$653$	1078.50	+	1868.02i	0.0646324	+	0.111947i	0.896531	−	0.442981i	$-0.146079\pi$
$653$	1078.50	+	1868.02i	0.0646324	+	0.111947i	−0.831898	+	0.554928i	$0.812746\pi$
$654$		0			0
$655$	−976.500	+	1691.35i	−0.0582519	+	0.100895i
$656$	−12780.0	−	22135.6i	−0.760633	−	1.31745i
$657$		0			0
$658$	1575.00	+	545.596i	0.0933129	+	0.0323245i
$659$	−19944.0			−1.17892			−0.589460	−	0.807798i	$-0.700659\pi$
$659$	−19944.0			−1.17892			−0.589460	+	0.807798i	$0.700659\pi$
$660$		0			0
$661$	−13753.0	+	23820.9i	−0.809273	+	1.40170i	0.104095	+	0.994567i	$0.466806\pi$
$661$	−13753.0	+	23820.9i	−0.809273	+	1.40170i	−0.913368	+	0.407135i	$0.866528\pi$
$662$	726.000	−	1257.47i	0.0426236	−	0.0738262i
$663$		0			0
$664$	10017.0			0.585444
$665$	−840.000	−	290.985i	−0.0489832	−	0.0169683i
$666$		0			0
$667$	−12474.0	−	21605.6i	−0.724131	−	1.25423i
$668$	−1323.00	+	2291.50i	−0.0766294	+	0.132726i
$669$		0			0
$670$	1665.00	+	2883.86i	0.0960068	+	0.166289i
$671$	−1770.00			−0.101833
$672$		0			0
$673$	−19123.0			−1.09530			−0.547650	−	0.836707i	$-0.684478\pi$
$673$	−19123.0			−1.09530			−0.547650	+	0.836707i	$0.684478\pi$
$674$	12538.5	+	21717.3i	0.716565	+	1.24113i
$675$		0			0
$676$	−949.500	+	1644.58i	−0.0540225	+	0.0935698i
$677$	6928.50	+	12000.5i	0.393329	+	0.681266i	0.992886	−	0.119066i	$-0.0379899\pi$
$677$	6928.50	+	12000.5i	0.393329	+	0.681266i	−0.599557	+	0.800332i	$0.704657\pi$
$678$		0			0
$679$	−1760.50	−	9147.83i	−0.0995019	−	0.517027i
$680$	5292.00			0.298440
$681$		0			0
$682$	5692.50	−	9859.70i	0.319615	−	0.553589i
$683$	−11122.5	+	19264.7i	−0.623120	+	1.07927i	0.365782	+	0.930701i	$0.380802\pi$
$683$	−11122.5	+	19264.7i	−0.623120	+	1.07927i	−0.988901	+	0.148574i	$0.952532\pi$
$684$		0			0
$685$	5310.00			0.296182
$686$	8746.50	−	16931.7i	0.486797	−	0.942353i
$687$		0			0
$688$	923.000	+	1598.68i	0.0511469	+	0.0885890i
$689$	−11616.0	+	20119.5i	−0.642285	+	1.11247i
$690$		0			0
$691$	320.000	+	554.256i	0.0176170	+	0.0305136i	0.874700	−	0.484666i	$-0.161059\pi$
$691$	320.000	+	554.256i	0.0176170	+	0.0305136i	−0.857082	+	0.515179i	$0.827725\pi$
$692$	786.000			0.0431781
$693$		0			0
$694$	5580.00			0.305207
$695$	2337.00	+	4047.80i	0.127550	+	0.220924i
$696$		0			0
$697$	−15120.0	+	26188.6i	−0.821680	+	1.42319i
$698$	2877.00	+	4983.11i	0.156012	+	0.270220i
$699$		0			0
$700$	1624.00	−	1406.43i	0.0876878	−	0.0759398i
$701$	15561.0			0.838418			0.419209	−	0.907890i	$-0.362307\pi$
$701$	15561.0			0.838418			0.419209	+	0.907890i	$0.362307\pi$
$702$		0			0
$703$	−2528.00	+	4378.62i	−0.135626	+	0.234912i
$704$	−3247.50	+	5624.83i	−0.173856	+	0.301128i
$705$		0			0
$706$	9144.00			0.487449
$707$	19005.0	+	6583.53i	1.01097	+	0.350211i
$708$		0			0
$709$	−2767.00	−	4792.58i	−0.146568	−	0.253864i	0.783389	−	0.621532i	$-0.213489\pi$
$709$	−2767.00	−	4792.58i	−0.146568	−	0.253864i	−0.929957	+	0.367668i	$0.880156\pi$
$710$	−1539.00	+	2665.63i	−0.0813488	+	0.140900i
$711$		0			0
$712$	−9513.00	−	16477.0i	−0.500723	−	0.867278i
$713$	−21252.0			−1.11626
$714$		0			0
$715$	−2880.00			−0.150638
$716$	1446.00	+	2504.55i	0.0754742	+	0.130725i
$717$		0			0
$718$	−45.0000	+	77.9423i	−0.00233898	+	0.00405123i
$719$	−10923.0	−	18919.2i	−0.566564	−	0.981317i	−0.996902	−	0.0786494i	$-0.974939\pi$
$719$	−10923.0	−	18919.2i	−0.566564	−	0.981317i	0.430339	−	0.902667i	$-0.358394\pi$
$720$		0			0
$721$	−24304.0	+	21047.9i	−1.25538	+	1.08719i
$722$	−19809.0			−1.02107
$723$		0			0
$724$	−676.000	+	1170.87i	−0.0347007	+	0.0601035i
$725$	17226.0	−	29836.3i	0.882424	−	1.52840i
$726$		0			0
$727$	−11089.0			−0.565706			−0.282853	−	0.959163i	$-0.591281\pi$
$727$	−11089.0			−0.565706			−0.282853	+	0.959163i	$0.591281\pi$
$728$	−4704.00	−	24442.7i	−0.239481	−	1.24438i
$729$		0			0
$730$	−1629.00	−	2821.51i	−0.0825918	−	0.143053i
$731$	1092.00	−	1891.40i	0.0552518	−	0.0956990i
$732$		0			0
$733$	−5881.00	−	10186.2i	−0.296343	−	0.513282i	0.678953	−	0.734182i	$-0.262434\pi$
$733$	−5881.00	−	10186.2i	−0.296343	−	0.513282i	−0.975296	+	0.220900i	$0.929101\pi$
$734$	−33933.0			−1.70639
$735$		0			0
$736$	−3780.00			−0.189311
$737$	2775.00	+	4806.44i	0.138695	+	0.240227i
$738$		0			0
$739$	11363.0	−	19681.3i	0.565622	−	0.979686i	−0.431369	−	0.902175i	$-0.641969\pi$
$739$	11363.0	−	19681.3i	0.565622	−	0.979686i	0.996992	−	0.0775108i	$-0.0246972\pi$
$740$	474.000	+	820.992i	0.0235467	+	0.0407841i
$741$		0			0
$742$	3811.50	+	19805.1i	0.188578	+	0.979878i
$743$	−6678.00			−0.329734			−0.164867	−	0.986316i	$-0.552719\pi$
$743$	−6678.00			−0.329734			−0.164867	+	0.986316i	$0.552719\pi$
$744$		0			0
$745$	3681.00	−	6375.68i	0.181022	−	0.313539i
$746$	−1812.00	+	3138.48i	−0.0889303	+	0.154032i
$747$		0			0
$748$	−1260.00			−0.0615911
$749$	−18942.0	+	16404.3i	−0.924066	+	0.800265i
$750$		0			0
$751$	9993.50	+	17309.2i	0.485577	+	0.841043i	0.999863	−	0.0165754i	$-0.00527637\pi$
$751$	9993.50	+	17309.2i	0.485577	+	0.841043i	−0.514286	+	0.857619i	$0.671943\pi$
$752$	1065.00	−	1844.63i	0.0516444	−	0.0894506i
$753$		0			0
$754$	28512.0	+	49384.2i	1.37712	+	2.38524i
$755$	3777.00			0.182065
$756$		0			0
$757$	314.000			0.0150760			0.00753799	−	0.999972i	$-0.497601\pi$
$757$	314.000			0.0150760			0.00753799	+	0.999972i	$0.497601\pi$
$758$	−11460.0	−	19849.3i	−0.549137	−	0.951133i
$759$		0			0
$760$	−504.000	+	872.954i	−0.0240553	+	0.0416649i
$761$	−5748.00	−	9955.83i	−0.273804	−	0.474242i	0.696029	−	0.718014i	$-0.254949\pi$
$761$	−5748.00	−	9955.83i	−0.273804	−	0.474242i	−0.969833	+	0.243772i	$0.921615\pi$
$762$		0			0
$763$	−6475.00	−	2243.01i	−0.307222	−	0.106425i
$764$	−3912.00			−0.185250
$765$		0			0
$766$	19125.0	−	33125.5i	0.902107	−	1.56250i
$767$	480.000	−	831.384i	0.0225969	−	0.0391389i
$768$		0			0
$769$	2765.00			0.129660			0.0648299	−	0.997896i	$-0.479350\pi$
$769$	2765.00			0.129660			0.0648299	+	0.997896i	$0.479350\pi$
$770$	−1890.00	+	1636.79i	−0.0884557	+	0.0766049i
$771$		0			0
$772$	−746.500	−	1292.98i	−0.0348020	−	0.0602788i
$773$	−7023.00	+	12164.2i	−0.326778	+	0.565997i	−0.981871	−	0.189552i	$-0.939296\pi$
$773$	−7023.00	+	12164.2i	−0.326778	+	0.565997i	0.655092	+	0.755549i	$0.272630\pi$
$774$		0			0
$775$	−14674.0	−	25416.1i	−0.680136	−	1.17803i
$776$	−10563.0			−0.488646
$777$		0			0
$778$	−9378.00			−0.432156
$779$	−2880.00	−	4988.31i	−0.132460	−	0.229428i
$780$		0			0
$781$	−2565.00	+	4442.71i	−0.117520	+	0.203550i
$782$	10584.0	+	18332.0i	0.483994	+	0.838302i
$783$		0			0
$784$	−19134.5	−	15064.5i	−0.871652	−	0.686248i
$785$	−588.000			−0.0267345
$786$		0			0

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(-3.50000 - 18.1865i) q^{7} -21.0000 q^{8} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(-3.50000 - 18.1865i) q^{7} -21.0000 q^{8} +(-4.50000 + 7.79423i) q^{10} +(-7.50000 + 12.9904i) q^{11} -64.0000 q^{13} +(-42.0000 + 36.3731i) q^{14} +(35.5000 + 61.4878i) q^{16} +(42.0000 - 72.7461i) q^{17} +(8.00000 + 13.8564i) q^{19} +3.00000 q^{20} +45.0000 q^{22} +(-42.0000 - 72.7461i) q^{23} +(58.0000 - 100.459i) q^{25} +(96.0000 + 166.277i) q^{26} +(17.5000 + 6.06218i) q^{28} +297.000 q^{29} +(126.500 - 219.104i) q^{31} +(22.5000 - 38.9711i) q^{32} -252.000 q^{34} +(-42.0000 + 36.3731i) q^{35} +(158.000 + 273.664i) q^{37} +(24.0000 - 41.5692i) q^{38} +(31.5000 + 54.5596i) q^{40} -360.000 q^{41} +26.0000 q^{43} +(-7.50000 - 12.9904i) q^{44} +(-126.000 + 218.238i) q^{46} +(-15.0000 - 25.9808i) q^{47} +(-318.500 + 127.306i) q^{49} -348.000 q^{50} +(32.0000 - 55.4256i) q^{52} +(181.500 - 314.367i) q^{53} +45.0000 q^{55} +(73.5000 + 381.917i) q^{56} +(-445.500 - 771.629i) q^{58} +(-7.50000 + 12.9904i) q^{59} +(59.0000 + 102.191i) q^{61} -759.000 q^{62} +433.000 q^{64} +(96.0000 + 166.277i) q^{65} +(185.000 - 320.429i) q^{67} +(42.0000 + 72.7461i) q^{68} +(157.500 + 54.5596i) q^{70} +342.000 q^{71} +(-181.000 + 313.501i) q^{73} +(474.000 - 820.992i) q^{74} -16.0000 q^{76} +(262.500 + 90.9327i) q^{77} +(-233.500 - 404.434i) q^{79} +(106.500 - 184.463i) q^{80} +(540.000 + 935.307i) q^{82} -477.000 q^{83} -252.000 q^{85} +(-39.0000 - 67.5500i) q^{86} +(157.500 - 272.798i) q^{88} +(453.000 + 784.619i) q^{89} +(224.000 + 1163.94i) q^{91} +84.0000 q^{92} +(-45.0000 + 77.9423i) q^{94} +(24.0000 - 41.5692i) q^{95} +503.000 q^{97} +(808.500 + 636.529i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} + O(q^{10}) \)	\( 2q - 3q^{2} - q^{4} - 3q^{5} - 7q^{7} - 42q^{8} - 9q^{10} - 15q^{11} - 128q^{13} - 84q^{14} + 71q^{16} + 84q^{17} + 16q^{19} + 6q^{20} + 90q^{22} - 84q^{23} + 116q^{25} + 192q^{26} + 35q^{28} + 594q^{29} + 253q^{31} + 45q^{32} - 504q^{34} - 84q^{35} + 316q^{37} + 48q^{38} + 63q^{40} - 720q^{41} + 52q^{43} - 15q^{44} - 252q^{46} - 30q^{47} - 637q^{49} - 696q^{50} + 64q^{52} + 363q^{53} + 90q^{55} + 147q^{56} - 891q^{58} - 15q^{59} + 118q^{61} - 1518q^{62} + 866q^{64} + 192q^{65} + 370q^{67} + 84q^{68} + 315q^{70} + 684q^{71} - 362q^{73} + 948q^{74} - 32q^{76} + 525q^{77} - 467q^{79} + 213q^{80} + 1080q^{82} - 954q^{83} - 504q^{85} - 78q^{86} + 315q^{88} + 906q^{89} + 448q^{91} + 168q^{92} - 90q^{94} + 48q^{95} + 1006q^{97} + 1617q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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