LMFDB - Embedded newform 1050.3.e.e.701.4

Newspace parameters

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.4
Character	$\chi$	$=$	1050.701
Dual form			1050.3.e.e.701.2

$q$-expansion

$f(q)$	$=$	$q+1.41421i q^{2} +(1.28756 + 2.70965i) q^{3} -2.00000 q^{4} +(-3.83202 + 1.82088i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(-5.68438 + 6.97766i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(1.28756 + 2.70965i) q^{3} -2.00000 q^{4} +(-3.83202 + 1.82088i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(-5.68438 + 6.97766i) q^{9} -10.4698i q^{11} +(-2.57512 - 5.41930i) q^{12} +11.9937 q^{13} +3.74166i q^{14} +4.00000 q^{16} +29.1133i q^{17} +(-9.86791 - 8.03893i) q^{18} -20.3176 q^{19} +(3.40656 + 7.16905i) q^{21} +14.8066 q^{22} +7.71572i q^{23} +(7.66404 - 3.64177i) q^{24} +16.9617i q^{26} +(-26.2260 - 6.41851i) q^{27} -5.29150 q^{28} +47.4046i q^{29} -35.5084 q^{31} +5.65685i q^{32} +(28.3695 - 13.4805i) q^{33} -41.1724 q^{34} +(11.3688 - 13.9553i) q^{36} -58.0907 q^{37} -28.7334i q^{38} +(15.4426 + 32.4988i) q^{39} +52.5850i q^{41} +(-10.1386 + 4.81761i) q^{42} -15.7629 q^{43} +20.9396i q^{44} -10.9117 q^{46} -85.2023i q^{47} +(5.15024 + 10.8386i) q^{48} +7.00000 q^{49} +(-78.8867 + 37.4850i) q^{51} -23.9875 q^{52} +42.7649i q^{53} +(9.07715 - 37.0892i) q^{54} -7.48331i q^{56} +(-26.1601 - 55.0535i) q^{57} -67.0403 q^{58} -37.9769i q^{59} +53.5108 q^{61} -50.2165i q^{62} +(-15.0395 + 18.4612i) q^{63} -8.00000 q^{64} +(19.0643 + 40.1206i) q^{66} +27.6366 q^{67} -58.2265i q^{68} +(-20.9069 + 9.93444i) q^{69} +58.0688i q^{71} +(19.7358 + 16.0779i) q^{72} -67.8080 q^{73} -82.1526i q^{74} +40.6352 q^{76} -27.7005i q^{77} +(-45.9602 + 21.8392i) q^{78} -19.2994 q^{79} +(-16.3756 - 79.3274i) q^{81} -74.3664 q^{82} +33.6998i q^{83} +(-6.81312 - 14.3381i) q^{84} -22.2920i q^{86} +(-128.450 + 61.0363i) q^{87} -29.6131 q^{88} -46.8325i q^{89} +31.7324 q^{91} -15.4314i q^{92} +(-45.7192 - 96.2153i) q^{93} +120.494 q^{94} +(-15.3281 + 7.28354i) q^{96} +65.1337 q^{97} +9.89949i q^{98} +(73.0549 + 59.5145i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} - 44 q^{9}+O(q^{10}) $ 24 * q - 48 * q^4 - 44 * q^9	$ 24 q - 48 q^{4} - 44 q^{9} + 96 q^{16} + 80 q^{19} - 28 q^{21} + 224 q^{31} - 128 q^{34} + 88 q^{36} + 92 q^{39} - 144 q^{46} + 168 q^{49} - 284 q^{51} + 144 q^{54} - 192 q^{64} + 224 q^{66} + 152 q^{69} - 160 q^{76} + 72 q^{79} - 212 q^{81} + 56 q^{84} + 168 q^{91} + 128 q^{94} + 876 q^{99}+O(q^{100}) $ 24 * q - 48 * q^4 - 44 * q^9 + 96 * q^16 + 80 * q^19 - 28 * q^21 + 224 * q^31 - 128 * q^34 + 88 * q^36 + 92 * q^39 - 144 * q^46 + 168 * q^49 - 284 * q^51 + 144 * q^54 - 192 * q^64 + 224 * q^66 + 152 * q^69 - 160 * q^76 + 72 * q^79 - 212 * q^81 + 56 * q^84 + 168 * q^91 + 128 * q^94 + 876 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	1.28756	+	2.70965i	0.429186	+	0.903216i
$3$	1.28756	+	2.70965i	0.429186	+	0.903216i
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	−3.83202	+	1.82088i	−0.638670	+	0.303481i
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		−	2.82843i		−	0.353553i
$9$	−5.68438	+	6.97766i	−0.631598	+	0.775296i
$10$		0			0
$11$		−	10.4698i		−	0.951802i	−0.879499	−	0.475901i	$-0.842122\pi$
$11$		−	10.4698i		−	0.951802i	0.879499	−	0.475901i	$-0.157878\pi$
$12$	−2.57512	−	5.41930i	−0.214593	−	0.451608i
$13$	11.9937			0.922595			0.461297	−	0.887246i	$-0.347384\pi$
$13$	11.9937			0.922595			0.461297	+	0.887246i	$0.347384\pi$
$14$			3.74166i			0.267261i
$15$		0			0
$16$	4.00000			0.250000
$17$			29.1133i			1.71254i	0.516525	+	0.856272i	$0.327225\pi$
$17$			29.1133i			1.71254i	−0.516525	+	0.856272i	$0.672775\pi$
$18$	−9.86791	−	8.03893i	−0.548217	−	0.446607i
$19$	−20.3176			−1.06935			−0.534673	−	0.845059i	$-0.679565\pi$
$19$	−20.3176			−1.06935			−0.534673	+	0.845059i	$0.679565\pi$
$20$		0			0
$21$	3.40656	+	7.16905i	0.162217	+	0.341384i
$22$	14.8066			0.673026
$23$			7.71572i			0.335466i	0.985832	+	0.167733i	$0.0536446\pi$
$23$			7.71572i			0.335466i	−0.985832	+	0.167733i	$0.946355\pi$
$24$	7.66404	−	3.64177i	0.319335	−	0.151740i
$25$		0			0
$26$			16.9617i			0.652373i
$27$	−26.2260	−	6.41851i	−0.971333	−	0.237723i
$28$	−5.29150			−0.188982
$29$			47.4046i			1.63464i	0.576182	+	0.817321i	$0.304542\pi$
$29$			47.4046i			1.63464i	−0.576182	+	0.817321i	$0.695458\pi$
$30$		0			0
$31$	−35.5084			−1.14543			−0.572717	−	0.819753i	$-0.694110\pi$
$31$	−35.5084			−1.14543			−0.572717	+	0.819753i	$0.694110\pi$
$32$			5.65685i			0.176777i
$33$	28.3695	−	13.4805i	0.859683	−	0.408501i
$34$	−41.1724			−1.21095
$35$		0			0
$36$	11.3688	−	13.9553i	0.315799	−	0.387648i
$37$	−58.0907			−1.57002			−0.785009	−	0.619485i	$-0.787342\pi$
$37$	−58.0907			−1.57002			−0.785009	+	0.619485i	$0.787342\pi$
$38$		−	28.7334i		−	0.756142i
$39$	15.4426	+	32.4988i	0.395965	+	0.833302i
$40$		0			0
$41$			52.5850i			1.28256i	0.767307	+	0.641280i	$0.221596\pi$
$41$			52.5850i			1.28256i	−0.767307	+	0.641280i	$0.778404\pi$
$42$	−10.1386	+	4.81761i	−0.241395	+	0.114705i
$43$	−15.7629			−0.366578			−0.183289	−	0.983059i	$-0.558674\pi$
$43$	−15.7629			−0.366578			−0.183289	+	0.983059i	$0.558674\pi$
$44$			20.9396i			0.475901i
$45$		0			0
$46$	−10.9117			−0.237210
$47$		−	85.2023i		−	1.81281i	−0.422405	−	0.906407i	$-0.638814\pi$
$47$		−	85.2023i		−	1.81281i	0.422405	−	0.906407i	$-0.361186\pi$
$48$	5.15024	+	10.8386i	0.107297	+	0.225804i
$49$	7.00000			0.142857
$50$		0			0
$51$	−78.8867	+	37.4850i	−1.54680	+	0.735001i
$52$	−23.9875			−0.461297
$53$			42.7649i			0.806885i	0.915005	+	0.403443i	$0.132187\pi$
$53$			42.7649i			0.806885i	−0.915005	+	0.403443i	$0.867813\pi$
$54$	9.07715	−	37.0892i	0.168095	−	0.686836i
$55$		0			0
$56$		−	7.48331i		−	0.133631i
$57$	−26.1601	−	55.0535i	−0.458949	−	0.965850i
$58$	−67.0403			−1.15587
$59$		−	37.9769i		−	0.643676i	−0.946795	−	0.321838i	$-0.895699\pi$
$59$		−	37.9769i		−	0.643676i	0.946795	−	0.321838i	$-0.104301\pi$
$60$		0			0
$61$	53.5108			0.877226			0.438613	−	0.898676i	$-0.355470\pi$
$61$	53.5108			0.877226			0.438613	+	0.898676i	$0.355470\pi$
$62$		−	50.2165i		−	0.809944i
$63$	−15.0395	+	18.4612i	−0.238722	+	0.293034i
$64$	−8.00000			−0.125000
$65$		0			0
$66$	19.0643	+	40.1206i	0.288854	+	0.607888i
$67$	27.6366			0.412487			0.206243	−	0.978501i	$-0.433876\pi$
$67$	27.6366			0.412487			0.206243	+	0.978501i	$0.433876\pi$
$68$		−	58.2265i		−	0.856272i
$69$	−20.9069	+	9.93444i	−0.302998	+	0.143977i
$70$		0			0
$71$			58.0688i			0.817871i	0.912563	+	0.408935i	$0.134100\pi$
$71$			58.0688i			0.817871i	−0.912563	+	0.408935i	$0.865900\pi$
$72$	19.7358	+	16.0779i	0.274109	+	0.223304i
$73$	−67.8080			−0.928877			−0.464438	−	0.885605i	$-0.653744\pi$
$73$	−67.8080			−0.928877			−0.464438	+	0.885605i	$0.653744\pi$
$74$		−	82.1526i		−	1.11017i
$75$		0			0
$76$	40.6352			0.534673
$77$		−	27.7005i		−	0.359747i
$78$	−45.9602	+	21.8392i	−0.589234	+	0.279990i
$79$	−19.2994			−0.244296			−0.122148	−	0.992512i	$-0.538978\pi$
$79$	−19.2994			−0.244296			−0.122148	+	0.992512i	$0.538978\pi$
$80$		0			0
$81$	−16.3756	−	79.3274i	−0.202168	−	0.979351i
$82$	−74.3664			−0.906907
$83$			33.6998i			0.406021i	0.979177	+	0.203011i	$0.0650726\pi$
$83$			33.6998i			0.406021i	−0.979177	+	0.203011i	$0.934927\pi$
$84$	−6.81312	−	14.3381i	−0.0811086	−	0.170692i
$85$		0			0
$86$		−	22.2920i		−	0.259210i
$87$	−128.450	+	61.0363i	−1.47643	+	0.701566i
$88$	−29.6131			−0.336513
$89$		−	46.8325i		−	0.526208i	−0.964768	−	0.263104i	$-0.915254\pi$
$89$		−	46.8325i		−	0.526208i	0.964768	−	0.263104i	$-0.0847462\pi$
$90$		0			0
$91$	31.7324			0.348708
$92$		−	15.4314i		−	0.167733i
$93$	−45.7192	−	96.2153i	−0.491604	−	1.03457i
$94$	120.494			1.28185
$95$		0			0
$96$	−15.3281	+	7.28354i	−0.159668	+	0.0758702i
$97$	65.1337			0.671482			0.335741	−	0.941954i	$-0.391013\pi$
$97$	65.1337			0.671482			0.335741	+	0.941954i	$0.391013\pi$
$98$			9.89949i			0.101015i
$99$	73.0549	+	59.5145i	0.737928	+	0.601156i
$100$		0			0
$101$			171.169i			1.69474i	0.531000	+	0.847372i	$0.321816\pi$
$101$			171.169i			1.69474i	−0.531000	+	0.847372i	$0.678184\pi$
$102$	−53.0119	−	111.563i	−0.519724	−	1.09375i
$103$	129.879			1.26096			0.630481	−	0.776205i	$-0.282858\pi$
$103$	129.879			1.26096			0.630481	+	0.776205i	$0.282858\pi$
$104$		−	33.9234i		−	0.326186i
$105$		0			0
$106$	−60.4787			−0.570554
$107$			29.1627i			0.272549i	0.990671	+	0.136274i	$0.0435128\pi$
$107$			29.1627i			0.272549i	−0.990671	+	0.136274i	$0.956487\pi$
$108$	52.4520	+	12.8370i	0.485667	+	0.118861i
$109$	−19.3296			−0.177336			−0.0886680	−	0.996061i	$-0.528261\pi$
$109$	−19.3296			−0.177336			−0.0886680	+	0.996061i	$0.528261\pi$
$110$		0			0
$111$	−74.7952	−	157.405i	−0.673830	−	1.41807i
$112$	10.5830			0.0944911
$113$			28.5568i			0.252715i	0.991985	+	0.126358i	$0.0403287\pi$
$113$			28.5568i			0.252715i	−0.991985	+	0.126358i	$0.959671\pi$
$114$	77.8574	−	36.9959i	0.682959	−	0.324526i
$115$		0			0
$116$		−	94.8092i		−	0.817321i
$117$	−68.1769	+	83.6882i	−0.582709	+	0.715284i
$118$	53.7074			0.455148
$119$			77.0264i			0.647281i
$120$		0			0
$121$	11.3828			0.0940727
$122$			75.6757i			0.620292i
$123$	−142.487	+	67.7063i	−1.15843	+	0.550457i
$124$	71.0169			0.572717
$125$		0			0
$126$	−26.1080	−	21.2690i	−0.207207	−	0.168802i
$127$	−162.116			−1.27650			−0.638252	−	0.769828i	$-0.720342\pi$
$127$	−162.116			−1.27650			−0.638252	+	0.769828i	$0.720342\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	−20.2956	−	42.7118i	−0.157330	−	0.331099i
$130$		0			0
$131$		−	49.5756i		−	0.378440i	−0.981935	−	0.189220i	$-0.939404\pi$
$131$		−	49.5756i		−	0.378440i	0.981935	−	0.189220i	$-0.0605959\pi$
$132$	−56.7391	+	26.9610i	−0.429841	+	0.204250i
$133$	−53.7553			−0.404175
$134$			39.0840i			0.291672i
$135$		0			0
$136$	82.3447			0.605476
$137$		−	212.721i		−	1.55271i	−0.630297	−	0.776354i	$-0.717067\pi$
$137$		−	212.721i		−	1.55271i	0.630297	−	0.776354i	$-0.282933\pi$
$138$	−14.0494	−	29.5668i	−0.101807	−	0.214252i
$139$	−74.1650			−0.533561			−0.266780	−	0.963757i	$-0.585960\pi$
$139$	−74.1650			−0.533561			−0.266780	+	0.963757i	$0.585960\pi$
$140$		0			0
$141$	230.868	−	109.703i	1.63736	−	0.778036i
$142$	−82.1217			−0.578322
$143$		−	125.572i		−	0.878128i
$144$	−22.7375	+	27.9107i	−0.157899	+	0.193824i
$145$		0			0
$146$		−	95.8950i		−	0.656815i
$147$	9.01292	+	18.9675i	0.0613124	+	0.129031i
$148$	116.181			0.785009
$149$		−	23.1598i		−	0.155435i	−0.996975	−	0.0777176i	$-0.975237\pi$
$149$		−	23.1598i		−	0.155435i	0.996975	−	0.0777176i	$-0.0247632\pi$
$150$		0			0
$151$	75.1469			0.497661			0.248831	−	0.968547i	$-0.419954\pi$
$151$	75.1469			0.497661			0.248831	+	0.968547i	$0.419954\pi$
$152$			57.4668i			0.378071i
$153$	−203.143	−	165.491i	−1.32773	−	1.08164i
$154$	39.1745			0.254380
$155$		0			0
$156$	−30.8853	−	64.9976i	−0.197983	−	0.416651i
$157$	−66.6968			−0.424820			−0.212410	−	0.977181i	$-0.568131\pi$
$157$	−66.6968			−0.424820			−0.212410	+	0.977181i	$0.568131\pi$
$158$		−	27.2934i		−	0.172743i
$159$	−115.878	+	55.0624i	−0.728792	+	0.346304i
$160$		0			0
$161$			20.4139i			0.126794i
$162$	112.186	−	23.1586i	0.692506	−	0.142954i
$163$	248.817			1.52648			0.763241	−	0.646114i	$-0.223607\pi$
$163$	248.817			1.52648			0.763241	+	0.646114i	$0.223607\pi$
$164$		−	105.170i		−	0.641280i
$165$		0			0
$166$	−47.6587			−0.287100
$167$			92.0222i			0.551031i	0.961297	+	0.275516i	$0.0888485\pi$
$167$			92.0222i			0.551031i	−0.961297	+	0.275516i	$0.911151\pi$
$168$	20.2771	−	9.63521i	0.120697	−	0.0573525i
$169$	−25.1504			−0.148819
$170$		0			0
$171$	115.493	−	141.769i	0.675397	−	0.829060i
$172$	31.5257			0.183289
$173$		−	233.473i		−	1.34956i	−0.738021	−	0.674778i	$-0.764239\pi$
$173$		−	233.473i		−	1.34956i	0.738021	−	0.674778i	$-0.235761\pi$
$174$	−86.3183	−	181.655i	−0.496082	−	1.04400i
$175$		0			0
$176$		−	41.8793i		−	0.237951i
$177$	102.904	−	48.8975i	0.581379	−	0.276257i
$178$	66.2311			0.372085
$179$			185.252i			1.03493i	0.855705	+	0.517465i	$0.173124\pi$
$179$			185.252i			1.03493i	−0.855705	+	0.517465i	$0.826876\pi$
$180$		0			0
$181$	−146.372			−0.808684			−0.404342	−	0.914608i	$-0.632499\pi$
$181$	−146.372			−0.808684			−0.404342	+	0.914608i	$0.632499\pi$
$182$			44.8764i			0.246574i
$183$	68.8983	+	144.995i	0.376493	+	0.792324i
$184$	21.8233			0.118605
$185$		0			0
$186$	136.069	−	64.6567i	0.731554	−	0.347617i
$187$	304.811			1.63000
$188$			170.405i			0.906407i
$189$	−69.3875	−	16.9818i	−0.367129	−	0.0898507i
$190$		0			0
$191$			83.4780i			0.437057i	0.975831	+	0.218529i	$0.0701257\pi$
$191$			83.4780i			0.437057i	−0.975831	+	0.218529i	$0.929874\pi$
$192$	−10.3005	−	21.6772i	−0.0536483	−	0.112902i
$193$	−88.5046			−0.458573			−0.229287	−	0.973359i	$-0.573639\pi$
$193$	−88.5046			−0.458573			−0.229287	+	0.973359i	$0.573639\pi$
$194$			92.1130i			0.474809i
$195$		0			0
$196$	−14.0000			−0.0714286
$197$			36.9343i			0.187484i	0.995597	+	0.0937419i	$0.0298828\pi$
$197$			36.9343i			0.187484i	−0.995597	+	0.0937419i	$0.970117\pi$
$198$	−84.1662	+	103.315i	−0.425082	+	0.521794i
$199$	205.101			1.03066			0.515330	−	0.856992i	$-0.327669\pi$
$199$	205.101			1.03066			0.515330	+	0.856992i	$0.327669\pi$
$200$		0			0
$201$	35.5838	+	74.8854i	0.177034	+	0.372564i
$202$	−242.070			−1.19836
$203$			125.421i			0.617837i
$204$	157.773	−	74.9701i	0.773399	−	0.367500i
$205$		0			0
$206$			183.677i			0.891634i
$207$	−53.8377	−	43.8591i	−0.260085	−	0.211880i
$208$	47.9749			0.230649
$209$			212.721i			1.01781i
$210$		0			0
$211$	49.4829			0.234516			0.117258	−	0.993101i	$-0.462590\pi$
$211$	49.4829			0.234516			0.117258	+	0.993101i	$0.462590\pi$
$212$		−	85.5298i		−	0.403443i
$213$	−157.346	+	74.7670i	−0.738714	+	0.351019i
$214$	−41.2423			−0.192721
$215$		0			0
$216$	−18.1543	+	74.1783i	−0.0840477	+	0.343418i
$217$	−93.9465			−0.432933
$218$		−	27.3362i		−	0.125395i
$219$	−87.3069	−	183.736i	−0.398661	−	0.838976i
$220$		0			0
$221$			349.177i			1.57998i
$222$	222.605	−	105.776i	1.00272	−	0.476470i
$223$	1.69928			0.00762011			0.00381005	−	0.999993i	$-0.498787\pi$
$223$	1.69928			0.00762011			0.00381005	+	0.999993i	$0.498787\pi$
$224$			14.9666i			0.0668153i
$225$		0			0
$226$	−40.3855			−0.178697
$227$			206.970i			0.911762i	0.890041	+	0.455881i	$0.150676\pi$
$227$			206.970i			0.911762i	−0.890041	+	0.455881i	$0.849324\pi$
$228$	52.3202	+	110.107i	0.229474	+	0.482925i
$229$	7.02543			0.0306787			0.0153394	−	0.999882i	$-0.495117\pi$
$229$	7.02543			0.0306787			0.0153394	+	0.999882i	$0.495117\pi$
$230$		0			0
$231$	75.0587	−	35.6661i	0.324930	−	0.154399i
$232$	134.081			0.577933
$233$			116.672i			0.500738i	0.968151	+	0.250369i	$0.0805519\pi$
$233$			116.672i			0.500738i	−0.968151	+	0.250369i	$0.919448\pi$
$234$	−118.353	−	96.4168i	−0.505782	−	0.412037i
$235$		0			0
$236$			75.9538i			0.321838i
$237$	−24.8491	−	52.2945i	−0.104848	−	0.220652i
$238$	−108.932			−0.457697
$239$			157.119i			0.657400i	0.944434	+	0.328700i	$0.106611\pi$
$239$			157.119i			0.657400i	−0.944434	+	0.328700i	$0.893389\pi$
$240$		0			0
$241$	−287.032			−1.19101			−0.595503	−	0.803353i	$-0.703047\pi$
$241$	−287.032			−1.19101			−0.595503	+	0.803353i	$0.703047\pi$
$242$			16.0977i			0.0665194i
$243$	193.865	−	146.511i	0.797797	−	0.602926i
$244$	−107.022			−0.438613
$245$		0			0
$246$	−95.7511	−	201.507i	−0.389232	−	0.819133i
$247$	−243.684			−0.986573
$248$			100.433i			0.404972i
$249$	−91.3145	+	43.3905i	−0.366725	+	0.174259i
$250$		0			0
$251$		−	161.056i		−	0.641658i	−0.947137	−	0.320829i	$-0.896039\pi$
$251$		−	161.056i		−	0.641658i	0.947137	−	0.320829i	$-0.103961\pi$
$252$	30.0789	−	36.9223i	0.119361	−	0.146517i
$253$	80.7822			0.319297
$254$		−	229.267i		−	0.902625i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	356.900i		−	1.38871i	−0.719630	−	0.694357i	$-0.755689\pi$
$257$		−	356.900i		−	1.38871i	0.719630	−	0.694357i	$-0.244311\pi$
$258$	60.4036	−	28.7023i	0.234122	−	0.111249i
$259$	−153.693			−0.593411
$260$		0			0
$261$	−330.774	−	269.466i	−1.26733	−	1.03244i
$262$	70.1105			0.267597
$263$			126.259i			0.480071i	0.970764	+	0.240036i	$0.0771591\pi$
$263$			126.259i			0.480071i	−0.970764	+	0.240036i	$0.922841\pi$
$264$	−38.1287	−	80.2412i	−0.144427	−	0.303944i
$265$		0			0
$266$		−	76.0214i		−	0.285795i
$267$	126.900	−	60.2996i	0.475279	−	0.225841i
$268$	−55.2732			−0.206243
$269$			356.319i			1.32461i	0.749236	+	0.662304i	$0.230421\pi$
$269$			356.319i			1.32461i	−0.749236	+	0.662304i	$0.769579\pi$
$270$		0			0
$271$	−422.588			−1.55937			−0.779683	−	0.626175i	$-0.784620\pi$
$271$	−422.588			−1.55937			−0.779683	+	0.626175i	$0.784620\pi$
$272$			116.453i			0.428136i
$273$	40.8574	+	85.9837i	0.149661	+	0.314959i
$274$	300.833			1.09793
$275$		0			0
$276$	41.8137	−	19.8689i	0.151499	−	0.0719887i
$277$	267.379			0.965266			0.482633	−	0.875823i	$-0.339680\pi$
$277$	267.379			0.965266			0.482633	+	0.875823i	$0.339680\pi$
$278$		−	104.885i		−	0.377284i
$279$	201.843	−	247.766i	0.723453	−	0.888050i
$280$		0			0
$281$			417.470i			1.48566i	0.669480	+	0.742830i	$0.266517\pi$
$281$			417.470i			1.48566i	−0.669480	+	0.742830i	$0.733483\pi$
$282$	155.143	+	326.497i	0.550154	+	1.15779i
$283$	56.1781			0.198509			0.0992546	−	0.995062i	$-0.468354\pi$
$283$	56.1781			0.198509			0.0992546	+	0.995062i	$0.468354\pi$
$284$		−	116.138i		−	0.408935i
$285$		0			0
$286$	177.586			0.620930
$287$			139.127i			0.484762i
$288$	−39.4716	−	32.1557i	−0.137054	−	0.111652i
$289$	−558.582			−1.93281
$290$		0			0
$291$	83.8635	+	176.489i	0.288191	+	0.606493i
$292$	135.616			0.464438
$293$			35.8668i			0.122412i	0.998125	+	0.0612062i	$0.0194947\pi$
$293$			35.8668i			0.122412i	−0.998125	+	0.0612062i	$0.980505\pi$
$294$	−26.8241	+	12.7462i	−0.0912386	+	0.0433544i
$295$		0			0
$296$			164.305i			0.555085i
$297$	−67.2007	+	274.582i	−0.226265	+	0.924517i
$298$	32.7530			0.109909
$299$			92.5402i			0.309499i
$300$		0			0
$301$	−41.7046			−0.138553
$302$			106.274i			0.351900i
$303$	−463.808	+	220.390i	−1.53072	+	0.727361i
$304$	−81.2703			−0.267337
$305$		0			0
$306$	234.039	−	287.287i	0.764835	−	0.938846i
$307$	496.572			1.61750			0.808749	−	0.588154i	$-0.200145\pi$
$307$	496.572			1.61750			0.808749	+	0.588154i	$0.200145\pi$
$308$			55.4011i			0.179874i
$309$	167.227	+	351.926i	0.541188	+	1.13892i
$310$		0			0
$311$		−	49.5016i		−	0.159169i	−0.996828	−	0.0795846i	$-0.974641\pi$
$311$		−	49.5016i		−	0.159169i	0.996828	−	0.0795846i	$-0.0253594\pi$
$312$	91.9204	−	43.6784i	0.294617	−	0.139995i
$313$	219.172			0.700229			0.350114	−	0.936707i	$-0.386143\pi$
$313$	219.172			0.700229			0.350114	+	0.936707i	$0.386143\pi$
$314$		−	94.3235i		−	0.300393i
$315$		0			0
$316$	38.5987			0.122148
$317$		−	96.3016i		−	0.303791i	−0.988397	−	0.151895i	$-0.951462\pi$
$317$		−	96.3016i		−	0.303791i	0.988397	−	0.151895i	$-0.0485377\pi$
$318$	−77.8700	−	163.876i	−0.244874	−	0.515333i
$319$	496.318			1.55586
$320$		0			0
$321$	−79.0206	+	37.5487i	−0.246170	+	0.116974i
$322$	−28.8696			−0.0896570
$323$		−	591.511i		−	1.83130i
$324$	32.7512	+	158.655i	0.101084	+	0.489675i
$325$		0			0
$326$			351.880i			1.07939i
$327$	−24.8880	−	52.3764i	−0.0761102	−	0.160173i
$328$	148.733			0.453453
$329$		−	225.424i		−	0.685180i
$330$		0			0
$331$	538.706			1.62751			0.813755	−	0.581208i	$-0.197420\pi$
$331$	538.706			1.62751			0.813755	+	0.581208i	$0.197420\pi$
$332$		−	67.3995i		−	0.203011i
$333$	330.209	−	405.337i	0.991620	−	1.21723i
$334$	−130.139			−0.389638
$335$		0			0
$336$	13.6262	+	28.6762i	0.0405543	+	0.0853459i
$337$	202.735			0.601589			0.300794	−	0.953689i	$-0.402748\pi$
$337$	202.735			0.601589			0.300794	+	0.953689i	$0.402748\pi$
$338$		−	35.5681i		−	0.105231i
$339$	−77.3790	+	36.7686i	−0.228257	+	0.108462i
$340$		0			0
$341$			371.767i			1.09023i
$342$	200.492	+	163.332i	0.586234	+	0.477578i
$343$	18.5203			0.0539949
$344$			44.5841i			0.129605i
$345$		0			0
$346$	330.181			0.954280
$347$			139.011i			0.400607i	0.979734	+	0.200304i	$0.0641928\pi$
$347$			139.011i			0.400607i	−0.979734	+	0.200304i	$0.935807\pi$
$348$	256.900	−	122.073i	0.738217	−	0.350783i
$349$	−452.984			−1.29795			−0.648974	−	0.760811i	$-0.724801\pi$
$349$	−452.984			−1.29795			−0.648974	+	0.760811i	$0.724801\pi$
$350$		0			0
$351$	−314.547	−	76.9819i	−0.896147	−	0.219322i
$352$	59.2263			0.168256
$353$			453.499i			1.28470i	0.766412	+	0.642349i	$0.222040\pi$
$353$			453.499i			1.28470i	−0.766412	+	0.642349i	$0.777960\pi$
$354$	69.1515	+	145.528i	0.195343	+	0.411097i
$355$		0			0
$356$			93.6650i			0.263104i
$357$	−208.715	+	99.1761i	−0.584635	+	0.277804i
$358$	−261.986			−0.731806
$359$			150.308i			0.418686i	0.977842	+	0.209343i	$0.0671324\pi$
$359$			150.308i			0.418686i	−0.977842	+	0.209343i	$0.932868\pi$
$360$		0			0
$361$	51.8039			0.143501
$362$		−	207.001i		−	0.571826i
$363$	14.6560	+	30.8434i	0.0403747	+	0.0849680i
$364$	−63.4649			−0.174354
$365$		0			0
$366$	−205.054	+	97.4369i	−0.560258	+	0.266221i
$367$	382.992			1.04357			0.521787	−	0.853076i	$-0.325265\pi$
$367$	382.992			1.04357			0.521787	+	0.853076i	$0.325265\pi$
$368$			30.8629i			0.0838665i
$369$	−366.920	−	298.913i	−0.994364	−	0.810062i
$370$		0			0
$371$			113.145i			0.304974i
$372$	91.4384	+	192.431i	0.245802	+	0.517287i
$373$	80.3409			0.215391			0.107696	−	0.994184i	$-0.465653\pi$
$373$	80.3409			0.215391			0.107696	+	0.994184i	$0.465653\pi$
$374$			431.067i			1.15259i
$375$		0			0
$376$	−240.988			−0.640927
$377$			568.558i			1.50811i
$378$	24.0159	−	98.1287i	0.0635341	−	0.259600i
$379$	636.088			1.67833			0.839166	−	0.543875i	$-0.183044\pi$
$379$	636.088			1.67833			0.839166	+	0.543875i	$0.183044\pi$
$380$		0			0
$381$	−208.734	−	439.277i	−0.547858	−	1.15296i
$382$	−118.056			−0.309046
$383$			343.675i			0.897323i	0.893702	+	0.448661i	$0.148099\pi$
$383$			343.675i			0.897323i	−0.893702	+	0.448661i	$0.851901\pi$
$384$	30.6562	−	14.5671i	0.0798338	−	0.0379351i
$385$		0			0
$386$		−	125.164i		−	0.324260i
$387$	89.6021	−	109.988i	0.231530	−	0.284207i
$388$	−130.267			−0.335741
$389$			520.410i			1.33782i	0.743345	+	0.668908i	$0.233238\pi$
$389$			520.410i			1.33782i	−0.743345	+	0.668908i	$0.766762\pi$
$390$		0			0
$391$	−224.630			−0.574500
$392$		−	19.7990i		−	0.0505076i
$393$	134.333	−	63.8316i	0.341813	−	0.162421i
$394$	−52.2330			−0.132571
$395$		0			0
$396$	−146.110	−	119.029i	−0.368964	−	0.300578i
$397$	184.320			0.464281			0.232140	−	0.972682i	$-0.425427\pi$
$397$	184.320			0.464281			0.232140	+	0.972682i	$0.425427\pi$
$398$			290.057i			0.728787i
$399$	−69.2131	−	145.658i	−0.173466	−	0.365057i
$400$		0			0
$401$			10.1322i			0.0252674i	0.999920	+	0.0126337i	$0.00402154\pi$
$401$			10.1322i			0.0252674i	−0.999920	+	0.0126337i	$0.995978\pi$
$402$	−105.904	+	50.3230i	−0.263443	+	0.125182i
$403$	−425.879			−1.05677
$404$		−	342.338i		−	0.847372i
$405$		0			0
$406$	−177.372			−0.436876
$407$			608.199i			1.49435i
$408$	106.024	+	223.125i	0.259862	+	0.546876i
$409$	802.483			1.96206			0.981031	−	0.193853i	$-0.0620985\pi$
$409$	802.483			1.96206			0.981031	+	0.193853i	$0.0620985\pi$
$410$		0			0
$411$	576.399	−	273.891i	1.40243	−	0.666402i
$412$	−259.758			−0.630481
$413$		−	100.477i		−	0.243287i
$414$	62.0261	−	76.1380i	0.149822	−	0.183908i
$415$		0			0
$416$			67.8468i			0.163093i
$417$	−95.4918	−	200.961i	−0.228997	−	0.481921i
$418$	−300.834			−0.719697
$419$		−	27.2524i		−	0.0650414i	−0.999471	−	0.0325207i	$-0.989647\pi$
$419$		−	27.2524i		−	0.0650414i	0.999471	−	0.0325207i	$-0.0103535\pi$
$420$		0			0
$421$	792.513			1.88245			0.941227	−	0.337775i	$-0.109674\pi$
$421$	792.513			1.88245			0.941227	+	0.337775i	$0.109674\pi$
$422$			69.9793i			0.165828i
$423$	594.513	+	484.322i	1.40547	+	1.14497i
$424$	120.957			0.285277
$425$		0			0
$426$	−105.737	−	222.521i	−0.248208	−	0.522349i
$427$	141.576			0.331560
$428$		−	58.3254i		−	0.136274i
$429$	340.257	−	161.682i	0.793139	−	0.376880i
$430$		0			0
$431$		−	483.889i		−	1.12271i	−0.827574	−	0.561357i	$-0.810280\pi$
$431$		−	483.889i		−	1.12271i	0.827574	−	0.561357i	$-0.189720\pi$
$432$	−104.904	−	25.6741i	−0.242833	−	0.0594307i
$433$	567.937			1.31163			0.655816	−	0.754920i	$-0.272325\pi$
$433$	567.937			1.31163			0.655816	+	0.754920i	$0.272325\pi$
$434$		−	132.860i		−	0.306130i
$435$		0			0
$436$	38.6592			0.0886680
$437$		−	156.765i		−	0.358729i
$438$	259.842	−	123.471i	0.593246	−	0.281896i
$439$	−834.645			−1.90124			−0.950621	−	0.310355i	$-0.899552\pi$
$439$	−834.645			−1.90124			−0.950621	+	0.310355i	$0.899552\pi$
$440$		0			0
$441$	−39.7907	+	48.8437i	−0.0902283	+	0.110757i
$442$	−493.810			−1.11722
$443$		−	543.969i		−	1.22792i	−0.789337	−	0.613960i	$-0.789576\pi$
$443$		−	543.969i		−	1.22792i	0.789337	−	0.613960i	$-0.210424\pi$
$444$	149.590	+	314.810i	0.336915	+	0.709033i
$445$		0			0
$446$			2.40315i			0.00538823i
$447$	62.7550	−	29.8197i	0.140392	−	0.0667107i
$448$	−21.1660			−0.0472456
$449$		−	308.306i		−	0.686649i	−0.939217	−	0.343325i	$-0.888447\pi$
$449$		−	308.306i		−	0.686649i	0.939217	−	0.343325i	$-0.111553\pi$
$450$		0			0
$451$	550.555			1.22074
$452$		−	57.1137i		−	0.126358i
$453$	96.7560	+	203.622i	0.213589	+	0.449496i
$454$	−292.700			−0.644713
$455$		0			0
$456$	−155.715	+	73.9919i	−0.341480	+	0.162263i
$457$	475.765			1.04106			0.520530	−	0.853843i	$-0.325734\pi$
$457$	475.765			1.04106			0.520530	+	0.853843i	$0.325734\pi$
$458$			9.93545i			0.0216931i
$459$	186.864	−	763.524i	0.407111	−	1.66345i
$460$		0			0
$461$			764.494i			1.65834i	0.558998	+	0.829169i	$0.311186\pi$
$461$			764.494i			1.65834i	−0.558998	+	0.829169i	$0.688814\pi$
$462$	50.4395	+	106.149i	0.109176	+	0.229760i
$463$	558.506			1.20628			0.603138	−	0.797637i	$-0.293917\pi$
$463$	558.506			1.20628			0.603138	+	0.797637i	$0.293917\pi$
$464$			189.618i			0.408661i
$465$		0			0
$466$	−164.999			−0.354075
$467$			251.321i			0.538160i	0.963118	+	0.269080i	$0.0867196\pi$
$467$			251.321i			0.538160i	−0.963118	+	0.269080i	$0.913280\pi$
$468$	136.354	−	167.376i	0.291354	−	0.357642i
$469$	73.1196			0.155905
$470$		0			0
$471$	−85.8761	−	180.725i	−0.182327	−	0.383705i
$472$	−107.415			−0.227574
$473$			165.034i			0.348910i
$474$	73.9556	−	35.1419i	0.156024	−	0.0741391i
$475$		0			0
$476$		−	154.053i		−	0.323641i
$477$	−298.399	−	243.092i	−0.625575	−	0.509627i
$478$	−222.199			−0.464852
$479$		−	343.617i		−	0.717363i	−0.933460	−	0.358681i	$-0.883226\pi$
$479$		−	343.617i		−	0.717363i	0.933460	−	0.358681i	$-0.116774\pi$
$480$		0			0
$481$	−696.724			−1.44849
$482$		−	405.925i		−	0.842169i
$483$	−55.3144	+	26.2841i	−0.114523	+	0.0544184i
$484$	−22.7656			−0.0470363
$485$		0			0
$486$	207.198	+	274.166i	0.426333	+	0.564128i
$487$	909.858			1.86829			0.934146	−	0.356892i	$-0.116163\pi$
$487$	909.858			1.86829			0.934146	+	0.356892i	$0.116163\pi$
$488$		−	151.351i		−	0.310146i
$489$	320.366	+	674.206i	0.655146	+	1.37874i
$490$		0			0
$491$			584.882i			1.19121i	0.803279	+	0.595603i	$0.203087\pi$
$491$			584.882i			1.19121i	−0.803279	+	0.595603i	$0.796913\pi$
$492$	284.973	−	135.413i	0.579214	−	0.275229i
$493$	−1380.10			−2.79940
$494$		−	344.621i		−	0.697612i
$495$		0			0
$496$	−142.034			−0.286358
$497$			153.636i			0.309126i
$498$	−61.3634	−	129.138i	−0.123220	−	0.259314i
$499$	−903.199			−1.81002			−0.905009	−	0.425392i	$-0.860136\pi$
$499$	−903.199			−1.81002			−0.905009	+	0.425392i	$0.860136\pi$
$500$		0			0
$501$	−249.348	+	118.484i	−0.497700	+	0.236495i
$502$	227.768			0.453720
$503$		−	188.576i		−	0.374903i	−0.982274	−	0.187452i	$-0.939977\pi$
$503$		−	188.576i		−	0.374903i	0.982274	−	0.187452i	$-0.0600228\pi$
$504$	52.2161	+	42.5380i	0.103603	+	0.0844008i
$505$		0			0
$506$			114.243i			0.225777i
$507$	−32.3827	−	68.1488i	−0.0638711	−	0.134416i
$508$	324.232			0.638252
$509$		−	526.061i		−	1.03352i	−0.856131	−	0.516759i	$-0.827138\pi$
$509$		−	526.061i		−	1.03352i	0.856131	−	0.516759i	$-0.172862\pi$
$510$		0			0
$511$	−179.403			−0.351083
$512$			22.6274i			0.0441942i
$513$	532.849	+	130.409i	1.03869	+	0.254208i
$514$	504.732			0.981970
$515$		0			0
$516$	40.5912	+	85.4236i	0.0786652	+	0.165550i
$517$	−892.053			−1.72544
$518$		−	217.355i		−	0.419605i
$519$	632.630	−	300.611i	1.21894	−	0.579211i
$520$		0			0
$521$		−	636.046i		−	1.22082i	−0.792087	−	0.610408i	$-0.791005\pi$
$521$		−	636.046i		−	1.22082i	0.792087	−	0.610408i	$-0.208995\pi$
$522$	381.082	−	467.784i	0.730043	−	0.896139i
$523$	−662.926			−1.26755			−0.633773	−	0.773519i	$-0.718494\pi$
$523$	−662.926			−1.26755			−0.633773	+	0.773519i	$0.718494\pi$
$524$			99.1513i			0.189220i
$525$		0			0
$526$	−178.557			−0.339461
$527$		−	1033.77i		−	1.96161i
$528$	113.478	−	53.9221i	0.214921	−	0.102125i
$529$	469.468			0.887463
$530$		0			0
$531$	264.990	+	215.875i	0.499040	+	0.406545i
$532$	107.511			0.202087
$533$			630.690i			1.18328i
$534$	85.2765	+	179.463i	0.159694	+	0.336073i
$535$		0			0
$536$		−	78.1681i		−	0.145836i
$537$	−501.969	+	238.523i	−0.934765	+	0.444178i
$538$	−503.912			−0.936639
$539$		−	73.2888i		−	0.135972i
$540$		0			0
$541$	−299.602			−0.553793			−0.276897	−	0.960900i	$-0.589306\pi$
$541$	−299.602			−0.553793			−0.276897	+	0.960900i	$0.589306\pi$
$542$		−	597.630i		−	1.10264i
$543$	−188.462	−	396.616i	−0.347076	−	0.730416i
$544$	−164.689			−0.302738
$545$		0			0
$546$	−121.599	+	57.7811i	−0.222709	+	0.105826i
$547$	177.699			0.324862			0.162431	−	0.986720i	$-0.448067\pi$
$547$	177.699			0.324862			0.162431	+	0.986720i	$0.448067\pi$
$548$			425.442i			0.776354i
$549$	−304.176	+	373.380i	−0.554054	+	0.680110i
$550$		0			0
$551$		−	963.147i		−	1.74800i
$552$	28.0989	+	59.1336i	0.0509037	+	0.107126i
$553$	−51.0613			−0.0923351
$554$			378.131i			0.682546i
$555$		0			0
$556$	148.330			0.266780
$557$		−	227.747i		−	0.408882i	−0.978879	−	0.204441i	$-0.934462\pi$
$557$		−	227.747i		−	0.408882i	0.978879	−	0.204441i	$-0.0655377\pi$
$558$	350.394	+	285.450i	0.627946	+	0.511559i
$559$	−189.055			−0.338203
$560$		0			0
$561$	392.462	+	825.930i	0.699575	+	1.47225i
$562$	−590.392			−1.05052
$563$			271.221i			0.481742i	0.970557	+	0.240871i	$0.0774330\pi$
$563$			271.221i			0.481742i	−0.970557	+	0.240871i	$0.922567\pi$
$564$	−461.736	+	219.406i	−0.818682	+	0.389018i
$565$		0			0
$566$			79.4479i			0.140367i
$567$	−43.3258	−	209.881i	−0.0764123	−	0.370160i
$568$	164.243			0.289161
$569$			63.1031i			0.110902i	0.998461	+	0.0554509i	$0.0176596\pi$
$569$			63.1031i			0.110902i	−0.998461	+	0.0554509i	$0.982340\pi$
$570$		0			0
$571$	130.175			0.227977			0.113988	−	0.993482i	$-0.463637\pi$
$571$	130.175			0.227977			0.113988	+	0.993482i	$0.463637\pi$
$572$			251.144i			0.439064i
$573$	−226.196	+	107.483i	−0.394757	+	0.187579i
$574$	−196.755			−0.342779
$575$		0			0
$576$	45.4751	−	55.8213i	0.0789497	−	0.0969120i
$577$	371.832			0.644424			0.322212	−	0.946668i	$-0.395574\pi$
$577$	371.832			0.644424			0.322212	+	0.946668i	$0.395574\pi$
$578$		−	789.954i		−	1.36670i
$579$	−113.955	−	239.816i	−0.196813	−	0.414190i
$580$		0			0
$581$			89.1612i			0.153462i
$582$	−249.594	+	118.601i	−0.428855	+	0.203782i
$583$	447.741			0.767995
$584$			191.790i			0.328408i
$585$		0			0
$586$	−50.7233			−0.0865586
$587$			824.490i			1.40458i	0.711890	+	0.702291i	$0.247840\pi$
$587$			824.490i			1.40458i	−0.711890	+	0.702291i	$0.752160\pi$
$588$	−18.0258	−	37.9351i	−0.0306562	−	0.0645154i
$589$	721.445			1.22486
$590$		0			0
$591$	−100.079	+	47.5551i	−0.169338	+	0.0804655i
$592$	−232.363			−0.392504
$593$			148.957i			0.251192i	0.992081	+	0.125596i	$0.0400843\pi$
$593$			148.957i			0.251192i	−0.992081	+	0.125596i	$0.959916\pi$
$594$	−388.317	−	95.0361i	−0.653732	−	0.159994i
$595$		0			0
$596$			46.3197i			0.0777176i
$597$	264.080	+	555.752i	0.442345	+	0.930909i
$598$	−130.872			−0.218849
$599$		−	98.6062i		−	0.164618i	−0.996607	−	0.0823090i	$-0.973771\pi$
$599$		−	98.6062i		−	0.164618i	0.996607	−	0.0823090i	$-0.0262294\pi$
$600$		0			0
$601$	244.192			0.406310			0.203155	−	0.979147i	$-0.434880\pi$
$601$	244.192			0.406310			0.203155	+	0.979147i	$0.434880\pi$
$602$		−	58.9792i		−	0.0979721i
$603$	−157.097	+	192.839i	−0.260526	+	0.319799i
$604$	−150.294			−0.248831
$605$		0			0
$606$	−311.679	−	655.924i	−0.514322	−	1.08238i
$607$	−504.707			−0.831478			−0.415739	−	0.909484i	$-0.636477\pi$
$607$	−504.707			−0.831478			−0.415739	+	0.909484i	$0.636477\pi$
$608$		−	114.934i		−	0.189035i
$609$	−339.846	+	161.487i	−0.558040	+	0.265167i
$610$		0			0
$611$		−	1021.89i		−	1.67249i
$612$	406.285	+	330.982i	0.663865	+	0.540820i
$613$	416.536			0.679503			0.339752	−	0.940515i	$-0.389657\pi$
$613$	416.536			0.679503			0.339752	+	0.940515i	$0.389657\pi$
$614$			702.259i			1.14374i
$615$		0			0
$616$	−78.3490			−0.127190
$617$			132.241i			0.214330i	0.994241	+	0.107165i	$0.0341773\pi$
$617$			132.241i			0.214330i	−0.994241	+	0.107165i	$0.965823\pi$
$618$	−497.699	+	236.495i	−0.805338	+	0.382677i
$619$	−669.449			−1.08150			−0.540751	−	0.841183i	$-0.681860\pi$
$619$	−669.449			−1.08150			−0.540751	+	0.841183i	$0.681860\pi$
$620$		0			0
$621$	49.5234	−	202.352i	0.0797479	−	0.325849i
$622$	70.0059			0.112550
$623$		−	123.907i		−	0.198888i
$624$	61.7706	+	129.995i	0.0989913	+	0.208326i
$625$		0			0
$626$			309.955i			0.495137i
$627$	−576.400	+	273.891i	−0.919299	+	0.436829i
$628$	133.394			0.212410
$629$		−	1691.21i		−	2.68873i
$630$		0			0
$631$	590.546			0.935889			0.467944	−	0.883758i	$-0.344995\pi$
$631$	590.546			0.935889			0.467944	+	0.883758i	$0.344995\pi$
$632$			54.5869i			0.0863716i
$633$	63.7121	+	134.081i	0.100651	+	0.211819i
$634$	136.191			0.214812
$635$		0			0
$636$	231.756	−	110.125i	0.364396	−	0.173152i
$637$	83.9561			0.131799
$638$			701.900i			1.10016i
$639$	−405.185	−	330.085i	−0.634092	−	0.516565i
$640$		0			0
$641$			542.271i			0.845977i	0.906135	+	0.422989i	$0.139019\pi$
$641$			542.271i			0.845977i	−0.906135	+	0.422989i	$0.860981\pi$
$642$	−53.1019	−	111.752i	−0.0827132	−	0.174069i
$643$	−546.024			−0.849181			−0.424591	−	0.905385i	$-0.639582\pi$
$643$	−546.024			−0.849181			−0.424591	+	0.905385i	$0.639582\pi$
$644$		−	40.8277i		−	0.0633971i
$645$		0			0
$646$	836.523			1.29493
$647$		−	801.011i		−	1.23804i	−0.785376	−	0.619020i	$-0.787530\pi$
$647$		−	801.011i		−	1.23804i	0.785376	−	0.619020i	$-0.212470\pi$
$648$	−224.372	+	46.3172i	−0.346253	+	0.0714772i
$649$	−397.611			−0.612652
$650$		0			0
$651$	−120.962	−	254.562i	−0.185809	−	0.391032i
$652$	−497.633			−0.763241
$653$		−	1193.82i		−	1.82821i	−0.405482	−	0.914103i	$-0.632896\pi$
$653$		−	1193.82i		−	1.82821i	0.405482	−	0.914103i	$-0.367104\pi$
$654$	74.0715	−	35.1970i	0.113259	−	0.0538180i
$655$		0			0
$656$			210.340i			0.320640i
$657$	385.447	−	473.142i	0.586677	−	0.720155i
$658$	318.798			0.484495
$659$			1234.84i			1.87381i	0.349579	+	0.936907i	$0.386325\pi$
$659$			1234.84i			1.87381i	−0.349579	+	0.936907i	$0.613675\pi$
$660$		0			0
$661$	−660.309			−0.998954			−0.499477	−	0.866327i	$-0.666475\pi$
$661$	−660.309			−0.998954			−0.499477	+	0.866327i	$0.666475\pi$
$662$			761.845i			1.15082i
$663$	−946.146	+	449.586i	−1.42707	+	0.678108i
$664$	95.3174			0.143550
$665$		0			0
$666$	573.233	+	466.987i	0.860711	+	0.701181i
$667$	−365.761			−0.548367
$668$		−	184.044i		−	0.275516i
$669$	2.18793	+	4.60446i	0.00327045	+	0.00688260i
$670$		0			0
$671$		−	560.248i		−	0.834945i
$672$	−40.5543	+	19.2704i	−0.0603487	+	0.0286762i
$673$	−372.706			−0.553798			−0.276899	−	0.960899i	$-0.589307\pi$
$673$	−372.706			−0.553798			−0.276899	+	0.960899i	$0.589307\pi$
$674$			286.711i			0.425387i
$675$		0			0
$676$	50.3009			0.0744096
$677$		−	762.928i		−	1.12693i	−0.826142	−	0.563463i	$-0.809469\pi$
$677$		−	762.928i		−	1.12693i	0.826142	−	0.563463i	$-0.190531\pi$
$678$	−51.9987	−	109.430i	−0.0766942	−	0.161402i
$679$	172.328			0.253796
$680$		0			0
$681$	−560.816	+	266.486i	−0.823518	+	0.391316i
$682$	−525.758			−0.770906
$683$			28.2779i			0.0414024i	0.999786	+	0.0207012i	$0.00658987\pi$
$683$			28.2779i			0.0414024i	−0.999786	+	0.0207012i	$0.993410\pi$
$684$	−230.986	+	283.538i	−0.337698	+	0.414530i
$685$		0			0
$686$			26.1916i			0.0381802i
$687$	9.04565	+	19.0364i	0.0131669	+	0.0277095i
$688$	−63.0514			−0.0916445
$689$			512.911i			0.744428i
$690$		0			0
$691$	−587.034			−0.849543			−0.424771	−	0.905301i	$-0.639645\pi$
$691$	−587.034			−0.849543			−0.424771	+	0.905301i	$0.639645\pi$
$692$			466.946i			0.674778i
$693$	193.285	+	157.460i	0.278911	+	0.227216i
$694$	−196.591			−0.283272
$695$		0			0
$696$	172.637	+	363.311i	0.248041	+	0.521999i
$697$	−1530.92			−2.19644
$698$		−	640.616i		−	0.917787i
$699$	−316.140	+	150.222i	−0.452274	+	0.214910i
$700$		0			0
$701$		−	221.461i		−	0.315922i	−0.987445	−	0.157961i	$-0.949508\pi$
$701$		−	221.461i		−	0.315922i	0.987445	−	0.157961i	$-0.0504920\pi$
$702$	108.869	−	444.837i	0.155084	−	0.633671i
$703$	1180.26			1.67889
$704$			83.7586i			0.118975i
$705$		0			0
$706$	−641.344			−0.908419
$707$			452.871i			0.640553i
$708$	−205.808	+	97.7950i	−0.290689	+	0.138129i
$709$	1187.65			1.67511			0.837553	−	0.546356i	$-0.183986\pi$
$709$	1187.65			1.67511			0.837553	+	0.546356i	$0.183986\pi$
$710$		0			0
$711$	109.705	−	134.665i	0.154297	−	0.189402i
$712$	−132.462			−0.186043
$713$		−	273.973i		−	0.384254i
$714$	−140.256	−	295.167i	−0.196437	−	0.413399i
$715$		0			0
$716$		−	370.505i		−	0.517465i
$717$	−425.736	+	202.300i	−0.593774	+	0.282147i
$718$	−212.568			−0.296055
$719$		−	611.937i		−	0.851095i	−0.904936	−	0.425547i	$-0.860082\pi$
$719$		−	611.937i		−	0.851095i	0.904936	−	0.425547i	$-0.139918\pi$
$720$		0			0
$721$	343.628			0.476599
$722$			73.2618i			0.101471i
$723$	−369.571	−	777.757i	−0.511164	−	1.07574i
$724$	292.744			0.404342
$725$		0			0
$726$	−43.6191	+	20.7268i	−0.0600814	+	0.0285492i
$727$	−1136.33			−1.56304			−0.781520	−	0.623881i	$-0.785555\pi$
$727$	−1136.33			−1.56304			−0.781520	+	0.623881i	$0.785555\pi$
$728$		−	89.7529i		−	0.123287i
$729$	646.605	+	336.664i	0.886976	+	0.461816i
$730$		0			0
$731$		−	458.908i		−	0.627781i
$732$	−137.797	−	289.991i	−0.188247	−	0.396162i
$733$	−401.450			−0.547681			−0.273840	−	0.961775i	$-0.588294\pi$
$733$	−401.450			−0.547681			−0.273840	+	0.961775i	$0.588294\pi$
$734$			541.632i			0.737919i
$735$		0			0
$736$	−43.6467			−0.0593026
$737$		−	289.350i		−	0.392606i
$738$	422.727	−	518.904i	0.572801	−	0.703121i
$739$	75.9398			0.102760			0.0513801	−	0.998679i	$-0.483638\pi$
$739$	75.9398			0.102760			0.0513801	+	0.998679i	$0.483638\pi$
$740$		0			0
$741$	−313.757	−	660.297i	−0.423424	−	0.891088i
$742$	−160.012			−0.215649
$743$		−	308.906i		−	0.415755i	−0.978155	−	0.207877i	$-0.933345\pi$
$743$		−	308.906i		−	0.415755i	0.978155	−	0.207877i	$-0.0666555\pi$
$744$	−272.138	+	129.313i	−0.365777	+	0.173808i
$745$		0			0
$746$			113.619i			0.152305i
$747$	−235.146	−	191.562i	−0.314787	−	0.256442i
$748$	−609.621			−0.815002
$749$			77.1572i			0.103014i
$750$		0			0
$751$	629.294			0.837941			0.418971	−	0.908000i	$-0.362391\pi$
$751$	629.294			0.837941			0.418971	+	0.908000i	$0.362391\pi$
$752$		−	340.809i		−	0.453204i
$753$	436.405	−	207.369i	0.579555	−	0.275391i
$754$	−804.063			−1.06640
$755$		0			0
$756$	138.775	+	33.9636i	0.183565	+	0.0449254i
$757$	−526.381			−0.695351			−0.347676	−	0.937615i	$-0.613029\pi$
$757$	−526.381			−0.695351			−0.347676	+	0.937615i	$0.613029\pi$
$758$			899.564i			1.18676i
$759$	104.012	+	218.891i	0.137038	+	0.288394i
$760$		0			0
$761$		−	955.908i		−	1.25612i	−0.778164	−	0.628061i	$-0.783849\pi$
$761$		−	955.908i		−	1.25612i	0.778164	−	0.628061i	$-0.216151\pi$
$762$	621.232	−	295.194i	0.815265	−	0.387394i
$763$	−51.1414			−0.0670267
$764$		−	166.956i		−	0.218529i
$765$		0			0
$766$	−486.029			−0.634503
$767$		−	455.485i		−	0.593852i
$768$	20.6010	+	43.3544i	0.0268242	+	0.0564510i
$769$	−1062.16			−1.38123			−0.690613	−	0.723225i	$-0.742659\pi$
$769$	−1062.16			−1.38123			−0.690613	+	0.723225i	$0.742659\pi$
$770$		0			0
$771$	967.072	−	459.530i	1.25431	−	0.596018i
$772$	177.009			0.229287
$773$		−	548.202i		−	0.709188i	−0.935020	−	0.354594i	$-0.884619\pi$
$773$		−	548.202i		−	0.709188i	0.935020	−	0.354594i	$-0.115381\pi$
$774$	155.546	+	126.717i	0.200964	+	0.163716i
$775$		0			0
$776$		−	184.226i		−	0.237405i
$777$	−197.889	−	416.455i	−0.254684	−	0.535978i
$778$	−735.971			−0.945979
$779$		−	1068.40i		−	1.37150i
$780$		0			0
$781$	607.970			0.778451
$782$		−	317.674i		−	0.406233i
$783$	304.267	−	1243.23i	0.388592	−	1.58778i
$784$	28.0000			0.0357143
$785$		0			0
$786$	90.2715	+	189.975i	0.114849	+	0.241698i
$787$	−590.574			−0.750412			−0.375206	−	0.926942i	$-0.622428\pi$
$787$	−590.574			−0.750412			−0.375206	+	0.926942i	$0.622428\pi$
$788$		−	73.8686i		−	0.0937419i
$789$	−342.117	+	162.566i	−0.433608	+	0.206040i
$790$

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(1.28756 + 2.70965i) q^{3} -2.00000 q^{4} +(-3.83202 + 1.82088i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(-5.68438 + 6.97766i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(1.28756 + 2.70965i) q^{3} -2.00000 q^{4} +(-3.83202 + 1.82088i) q^{6} +2.64575 q^{7} -2.82843i q^{8} +(-5.68438 + 6.97766i) q^{9} -10.4698i q^{11} +(-2.57512 - 5.41930i) q^{12} +11.9937 q^{13} +3.74166i q^{14} +4.00000 q^{16} +29.1133i q^{17} +(-9.86791 - 8.03893i) q^{18} -20.3176 q^{19} +(3.40656 + 7.16905i) q^{21} +14.8066 q^{22} +7.71572i q^{23} +(7.66404 - 3.64177i) q^{24} +16.9617i q^{26} +(-26.2260 - 6.41851i) q^{27} -5.29150 q^{28} +47.4046i q^{29} -35.5084 q^{31} +5.65685i q^{32} +(28.3695 - 13.4805i) q^{33} -41.1724 q^{34} +(11.3688 - 13.9553i) q^{36} -58.0907 q^{37} -28.7334i q^{38} +(15.4426 + 32.4988i) q^{39} +52.5850i q^{41} +(-10.1386 + 4.81761i) q^{42} -15.7629 q^{43} +20.9396i q^{44} -10.9117 q^{46} -85.2023i q^{47} +(5.15024 + 10.8386i) q^{48} +7.00000 q^{49} +(-78.8867 + 37.4850i) q^{51} -23.9875 q^{52} +42.7649i q^{53} +(9.07715 - 37.0892i) q^{54} -7.48331i q^{56} +(-26.1601 - 55.0535i) q^{57} -67.0403 q^{58} -37.9769i q^{59} +53.5108 q^{61} -50.2165i q^{62} +(-15.0395 + 18.4612i) q^{63} -8.00000 q^{64} +(19.0643 + 40.1206i) q^{66} +27.6366 q^{67} -58.2265i q^{68} +(-20.9069 + 9.93444i) q^{69} +58.0688i q^{71} +(19.7358 + 16.0779i) q^{72} -67.8080 q^{73} -82.1526i q^{74} +40.6352 q^{76} -27.7005i q^{77} +(-45.9602 + 21.8392i) q^{78} -19.2994 q^{79} +(-16.3756 - 79.3274i) q^{81} -74.3664 q^{82} +33.6998i q^{83} +(-6.81312 - 14.3381i) q^{84} -22.2920i q^{86} +(-128.450 + 61.0363i) q^{87} -29.6131 q^{88} -46.8325i q^{89} +31.7324 q^{91} -15.4314i q^{92} +(-45.7192 - 96.2153i) q^{93} +120.494 q^{94} +(-15.3281 + 7.28354i) q^{96} +65.1337 q^{97} +9.89949i q^{98} +(73.0549 + 59.5145i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} - 44 q^{9}+O(q^{10}) \) 24 * q - 48 * q^4 - 44 * q^9	\( 24 q - 48 q^{4} - 44 q^{9} + 96 q^{16} + 80 q^{19} - 28 q^{21} + 224 q^{31} - 128 q^{34} + 88 q^{36} + 92 q^{39} - 144 q^{46} + 168 q^{49} - 284 q^{51} + 144 q^{54} - 192 q^{64} + 224 q^{66} + 152 q^{69} - 160 q^{76} + 72 q^{79} - 212 q^{81} + 56 q^{84} + 168 q^{91} + 128 q^{94} + 876 q^{99}+O(q^{100}) \) 24 * q - 48 * q^4 - 44 * q^9 + 96 * q^16 + 80 * q^19 - 28 * q^21 + 224 * q^31 - 128 * q^34 + 88 * q^36 + 92 * q^39 - 144 * q^46 + 168 * q^49 - 284 * q^51 + 144 * q^54 - 192 * q^64 + 224 * q^66 + 152 * q^69 - 160 * q^76 + 72 * q^79 - 212 * q^81 + 56 * q^84 + 168 * q^91 + 128 * q^94 + 876 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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