LMFDB - Embedded newform 1050.3.e.e.701.13

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			701.13
Character	$\chi$	$=$	1050.701
Dual form			1050.3.e.e.701.15

$q$-expansion

$f(q)$	$=$	$q-1.41421i q^{2} +(-0.554262 + 2.94835i) q^{3} -2.00000 q^{4} +(4.16960 + 0.783844i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-8.38559 - 3.26832i) q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +(-0.554262 + 2.94835i) q^{3} -2.00000 q^{4} +(4.16960 + 0.783844i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-8.38559 - 3.26832i) q^{9} -2.58444i q^{11} +(1.10852 - 5.89671i) q^{12} -0.180526 q^{13} -3.74166i q^{14} +4.00000 q^{16} -7.25119i q^{17} +(-4.62210 + 11.8590i) q^{18} +21.4338 q^{19} +(-1.46644 + 7.80061i) q^{21} -3.65495 q^{22} +11.8858i q^{23} +(-8.33921 - 1.56769i) q^{24} +0.255303i q^{26} +(14.2840 - 22.9122i) q^{27} -5.29150 q^{28} -29.5458i q^{29} -49.5193 q^{31} -5.65685i q^{32} +(7.61985 + 1.43246i) q^{33} -10.2547 q^{34} +(16.7712 + 6.53664i) q^{36} +43.2958 q^{37} -30.3120i q^{38} +(0.100059 - 0.532256i) q^{39} +20.7847i q^{41} +(11.0317 + 2.07386i) q^{42} +74.2183 q^{43} +5.16888i q^{44} +16.8091 q^{46} -7.05106i q^{47} +(-2.21705 + 11.7934i) q^{48} +7.00000 q^{49} +(21.3791 + 4.01906i) q^{51} +0.361053 q^{52} -55.0572i q^{53} +(-32.4027 - 20.2006i) q^{54} +7.48331i q^{56} +(-11.8799 + 63.1944i) q^{57} -41.7841 q^{58} +100.110i q^{59} +8.44678 q^{61} +70.0309i q^{62} +(-22.1862 - 8.64716i) q^{63} -8.00000 q^{64} +(2.02580 - 10.7761i) q^{66} +85.7646 q^{67} +14.5024i q^{68} +(-35.0435 - 6.58784i) q^{69} +97.8086i q^{71} +(9.24420 - 23.7180i) q^{72} +17.8312 q^{73} -61.2295i q^{74} -42.8676 q^{76} -6.83779i q^{77} +(-0.752723 - 0.141505i) q^{78} +110.652 q^{79} +(59.6362 + 54.8136i) q^{81} +29.3940 q^{82} -3.08611i q^{83} +(2.93288 - 15.6012i) q^{84} -104.961i q^{86} +(87.1115 + 16.3761i) q^{87} +7.30991 q^{88} +80.1881i q^{89} -0.477628 q^{91} -23.7716i q^{92} +(27.4467 - 146.001i) q^{93} -9.97171 q^{94} +(16.6784 + 3.13538i) q^{96} +105.794 q^{97} -9.89949i q^{98} +(-8.44678 + 21.6721i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.554262	+	2.94835i	−0.184754	+	0.982785i
$3$	−0.554262	+	2.94835i	−0.184754	+	0.982785i
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	4.16960	+	0.783844i	0.694934	+	0.130641i
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$			2.82843i			0.353553i
$9$	−8.38559	−	3.26832i	−0.931732	−	0.363147i
$10$		0			0
$11$		−	2.58444i		−	0.234949i	−0.993076	−	0.117475i	$-0.962520\pi$
$11$		−	2.58444i		−	0.234949i	0.993076	−	0.117475i	$-0.0374799\pi$
$12$	1.10852	−	5.89671i	0.0923769	−	0.491392i
$13$	−0.180526			−0.0138866			−0.00694332	−	0.999976i	$-0.502210\pi$
$13$	−0.180526			−0.0138866			−0.00694332	+	0.999976i	$0.502210\pi$
$14$		−	3.74166i		−	0.267261i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	7.25119i		−	0.426541i	−0.976993	−	0.213270i	$-0.931589\pi$
$17$		−	7.25119i		−	0.426541i	0.976993	−	0.213270i	$-0.0684115\pi$
$18$	−4.62210	+	11.8590i	−0.256783	+	0.658834i
$19$	21.4338			1.12809			0.564047	−	0.825743i	$-0.309244\pi$
$19$	21.4338			1.12809			0.564047	+	0.825743i	$0.309244\pi$
$20$		0			0
$21$	−1.46644	+	7.80061i	−0.0698304	+	0.371458i
$22$	−3.65495			−0.166134
$23$			11.8858i			0.516774i	0.966042	+	0.258387i	$0.0831909\pi$
$23$			11.8858i			0.516774i	−0.966042	+	0.258387i	$0.916809\pi$
$24$	−8.33921	−	1.56769i	−0.347467	−	0.0653203i
$25$		0			0
$26$			0.255303i			0.00981934i
$27$	14.2840	−	22.9122i	0.529036	−	0.848599i
$28$	−5.29150			−0.188982
$29$		−	29.5458i		−	1.01882i	−0.860524	−	0.509410i	$-0.829864\pi$
$29$		−	29.5458i		−	1.01882i	0.860524	−	0.509410i	$-0.170136\pi$
$30$		0			0
$31$	−49.5193			−1.59740			−0.798699	−	0.601731i	$-0.794478\pi$
$31$	−49.5193			−1.59740			−0.798699	+	0.601731i	$0.794478\pi$
$32$		−	5.65685i		−	0.176777i
$33$	7.61985	+	1.43246i	0.230905	+	0.0434078i
$34$	−10.2547			−0.301610
$35$		0			0
$36$	16.7712	+	6.53664i	0.465866	+	0.181573i
$37$	43.2958			1.17016			0.585078	−	0.810977i	$-0.301064\pi$
$37$	43.2958			1.17016			0.585078	+	0.810977i	$0.301064\pi$
$38$		−	30.3120i		−	0.797683i
$39$	0.100059	−	0.532256i	0.00256561	−	0.0136476i
$40$		0			0
$41$			20.7847i			0.506944i	0.967343	+	0.253472i	$0.0815726\pi$
$41$			20.7847i			0.506944i	−0.967343	+	0.253472i	$0.918427\pi$
$42$	11.0317	+	2.07386i	0.262660	+	0.0493775i
$43$	74.2183			1.72601			0.863004	−	0.505197i	$-0.168580\pi$
$43$	74.2183			1.72601			0.863004	+	0.505197i	$0.168580\pi$
$44$			5.16888i			0.117475i
$45$		0			0
$46$	16.8091			0.365414
$47$		−	7.05106i		−	0.150023i	−0.997183	−	0.0750113i	$-0.976101\pi$
$47$		−	7.05106i		−	0.150023i	0.997183	−	0.0750113i	$-0.0238993\pi$
$48$	−2.21705	+	11.7934i	−0.0461885	+	0.245696i
$49$	7.00000			0.142857
$50$		0			0
$51$	21.3791	+	4.01906i	0.419198	+	0.0788050i
$52$	0.361053			0.00694332
$53$		−	55.0572i		−	1.03882i	−0.854526	−	0.519408i	$-0.826153\pi$
$53$		−	55.0572i		−	1.03882i	0.854526	−	0.519408i	$-0.173847\pi$
$54$	−32.4027	−	20.2006i	−0.600050	−	0.374085i
$55$		0			0
$56$			7.48331i			0.133631i
$57$	−11.8799	+	63.1944i	−0.208420	+	1.10867i
$58$	−41.7841			−0.720415
$59$			100.110i			1.69678i	0.529375	+	0.848388i	$0.322427\pi$
$59$			100.110i			1.69678i	−0.529375	+	0.848388i	$0.677573\pi$
$60$		0			0
$61$	8.44678			0.138472			0.0692359	−	0.997600i	$-0.477944\pi$
$61$	8.44678			0.138472			0.0692359	+	0.997600i	$0.477944\pi$
$62$			70.0309i			1.12953i
$63$	−22.1862	−	8.64716i	−0.352162	−	0.137256i
$64$	−8.00000			−0.125000
$65$		0			0
$66$	2.02580	−	10.7761i	0.0306939	−	0.163274i
$67$	85.7646			1.28007			0.640034	−	0.768347i	$-0.278920\pi$
$67$	85.7646			1.28007			0.640034	+	0.768347i	$0.278920\pi$
$68$			14.5024i			0.213270i
$69$	−35.0435	−	6.58784i	−0.507877	−	0.0954759i
$70$		0			0
$71$			97.8086i			1.37759i	0.724958	+	0.688793i	$0.241859\pi$
$71$			97.8086i			1.37759i	−0.724958	+	0.688793i	$0.758141\pi$
$72$	9.24420	−	23.7180i	0.128392	−	0.329417i
$73$	17.8312			0.244264			0.122132	−	0.992514i	$-0.461027\pi$
$73$	17.8312			0.244264			0.122132	+	0.992514i	$0.461027\pi$
$74$		−	61.2295i		−	0.827426i
$75$		0			0
$76$	−42.8676			−0.564047
$77$		−	6.83779i		−	0.0888025i
$78$	−0.752723	−	0.141505i	−0.00965030	−	0.00181416i
$79$	110.652			1.40066			0.700328	−	0.713821i	$-0.253037\pi$
$79$	110.652			1.40066			0.700328	+	0.713821i	$0.253037\pi$
$80$		0			0
$81$	59.6362	+	54.8136i	0.736249	+	0.676711i
$82$	29.3940			0.358464
$83$		−	3.08611i		−	0.0371820i	−0.999827	−	0.0185910i	$-0.994082\pi$
$83$		−	3.08611i		−	0.0371820i	0.999827	−	0.0185910i	$-0.00591804\pi$
$84$	2.93288	−	15.6012i	0.0349152	−	0.185729i
$85$		0			0
$86$		−	104.961i		−	1.22047i
$87$	87.1115	+	16.3761i	1.00128	+	0.188231i
$88$	7.30991			0.0830671
$89$			80.1881i			0.900990i	0.892779	+	0.450495i	$0.148752\pi$
$89$			80.1881i			0.900990i	−0.892779	+	0.450495i	$0.851248\pi$
$90$		0			0
$91$	−0.477628			−0.00524866
$92$		−	23.7716i		−	0.258387i
$93$	27.4467	−	146.001i	0.295125	−	1.56990i
$94$	−9.97171			−0.106082
$95$		0			0
$96$	16.6784	+	3.13538i	0.173733	+	0.0326602i
$97$	105.794			1.09066			0.545328	−	0.838223i	$-0.316405\pi$
$97$	105.794			1.09066			0.545328	+	0.838223i	$0.316405\pi$
$98$		−	9.89949i		−	0.101015i
$99$	−8.44678	+	21.6721i	−0.0853210	+	0.218910i
$100$		0			0
$101$		−	120.653i		−	1.19459i	−0.802022	−	0.597294i	$-0.796243\pi$
$101$		−	120.653i		−	1.19459i	0.802022	−	0.597294i	$-0.203757\pi$
$102$	5.68380	−	30.2346i	0.0557236	−	0.296417i
$103$	173.031			1.67992			0.839958	−	0.542652i	$-0.182580\pi$
$103$	173.031			1.67992			0.839958	+	0.542652i	$0.182580\pi$
$104$		−	0.510606i		−	0.00490967i
$105$		0			0
$106$	−77.8627			−0.734554
$107$		−	82.1335i		−	0.767603i	−0.923416	−	0.383802i	$-0.874615\pi$
$107$		−	82.1335i		−	0.767603i	0.923416	−	0.383802i	$-0.125385\pi$
$108$	−28.5679	+	45.8244i	−0.264518	+	0.424300i
$109$	−63.4890			−0.582468			−0.291234	−	0.956652i	$-0.594066\pi$
$109$	−63.4890			−0.582468			−0.291234	+	0.956652i	$0.594066\pi$
$110$		0			0
$111$	−23.9972	+	127.651i	−0.216191	+	1.15001i
$112$	10.5830			0.0944911
$113$			145.494i			1.28755i	0.765213	+	0.643777i	$0.222634\pi$
$113$			145.494i			1.28755i	−0.765213	+	0.643777i	$0.777366\pi$
$114$	89.3704	+	16.8008i	0.783951	+	0.147375i
$115$		0			0
$116$			59.0916i			0.509410i
$117$	1.51382	+	0.590018i	0.0129386	+	0.00504289i
$118$	141.577			1.19980
$119$		−	19.1848i		−	0.161217i
$120$		0			0
$121$	114.321			0.944799
$122$		−	11.9456i		−	0.0979144i
$123$	−61.2807	−	11.5202i	−0.498217	−	0.0936599i
$124$	99.0387			0.798699
$125$		0			0
$126$	−12.2289	+	31.3760i	−0.0970550	+	0.249016i
$127$	153.065			1.20523			0.602616	−	0.798031i	$-0.294125\pi$
$127$	153.065			1.20523			0.602616	+	0.798031i	$0.294125\pi$
$128$			11.3137i			0.0883883i
$129$	−41.1364	+	218.822i	−0.318887	+	1.69629i
$130$		0			0
$131$		−	176.646i		−	1.34844i	−0.738529	−	0.674222i	$-0.764479\pi$
$131$		−	176.646i		−	1.34844i	0.738529	−	0.674222i	$-0.235521\pi$
$132$	−15.2397	−	2.86491i	−0.115452	−	0.0217039i
$133$	56.7085			0.426380
$134$		−	121.289i		−	0.905145i
$135$		0			0
$136$	20.5095			0.150805
$137$		−	229.315i		−	1.67383i	−0.547333	−	0.836915i	$-0.684357\pi$
$137$		−	229.315i		−	1.67383i	0.547333	−	0.836915i	$-0.315643\pi$
$138$	−9.31661	+	49.5591i	−0.0675117	+	0.359124i
$139$	−7.93236			−0.0570673			−0.0285337	−	0.999593i	$-0.509084\pi$
$139$	−7.93236			−0.0570673			−0.0285337	+	0.999593i	$0.509084\pi$
$140$		0			0
$141$	20.7890	+	3.90813i	0.147440	+	0.0277173i
$142$	138.322			0.974101
$143$			0.466560i			0.00326266i
$144$	−33.5424	−	13.0733i	−0.232933	−	0.0907866i
$145$		0			0
$146$		−	25.2172i		−	0.172720i
$147$	−3.87983	+	20.6385i	−0.0263934	+	0.140398i
$148$	−86.5916			−0.585078
$149$			111.886i			0.750910i	0.926841	+	0.375455i	$0.122513\pi$
$149$			111.886i			0.750910i	−0.926841	+	0.375455i	$0.877487\pi$
$150$		0			0
$151$	−73.7503			−0.488412			−0.244206	−	0.969723i	$-0.578527\pi$
$151$	−73.7503			−0.488412			−0.244206	+	0.969723i	$0.578527\pi$
$152$			60.6239i			0.398842i
$153$	−23.6992	+	60.8055i	−0.154897	+	0.397421i
$154$	−9.67010			−0.0627928
$155$		0			0
$156$	−0.200118	+	1.06451i	−0.00128281	+	0.00682379i
$157$	52.3958			0.333731			0.166866	−	0.985980i	$-0.446635\pi$
$157$	52.3958			0.333731			0.166866	+	0.985980i	$0.446635\pi$
$158$		−	156.485i		−	0.990414i
$159$	162.328	+	30.5161i	1.02093	+	0.191925i
$160$		0			0
$161$			31.4469i			0.195322i
$162$	77.5181	−	84.3383i	0.478507	−	0.520607i
$163$	−16.9135			−0.103764			−0.0518819	−	0.998653i	$-0.516522\pi$
$163$	−16.9135			−0.103764			−0.0518819	+	0.998653i	$0.516522\pi$
$164$		−	41.5694i		−	0.253472i
$165$		0			0
$166$	−4.36442			−0.0262917
$167$		−	2.03625i		−	0.0121931i	−0.999981	−	0.00609657i	$-0.998059\pi$
$167$		−	2.03625i		−	0.0121931i	0.999981	−	0.00609657i	$-0.00194061\pi$
$168$	−22.0635	−	4.14771i	−0.131330	−	0.0246888i
$169$	−168.967			−0.999807
$170$		0			0
$171$	−179.735	−	70.0525i	−1.05108	−	0.409664i
$172$	−148.437			−0.863004
$173$			258.927i			1.49669i	0.663312	+	0.748343i	$0.269150\pi$
$173$			258.927i			1.49669i	−0.663312	+	0.748343i	$0.730850\pi$
$174$	23.1593	−	123.194i	0.133099	−	0.708013i
$175$		0			0
$176$		−	10.3378i		−	0.0587373i
$177$	−295.159	−	55.4870i	−1.66757	−	0.313486i
$178$	113.403			0.637096
$179$		−	38.3599i		−	0.214301i	−0.994243	−	0.107150i	$-0.965827\pi$
$179$		−	38.3599i		−	0.214301i	0.994243	−	0.107150i	$-0.0341727\pi$
$180$		0			0
$181$	213.719			1.18077			0.590384	−	0.807122i	$-0.298976\pi$
$181$	213.719			1.18077			0.590384	+	0.807122i	$0.298976\pi$
$182$			0.675468i			0.00371136i
$183$	−4.68173	+	24.9041i	−0.0255832	+	0.136088i
$184$	−33.6181			−0.182707
$185$		0			0
$186$	−206.476	−	38.8154i	−1.11009	−	0.208685i
$187$	−18.7403			−0.100215
$188$			14.1021i			0.0750113i
$189$	37.7918	−	60.6199i	0.199957	−	0.320740i
$190$		0			0
$191$		−	245.546i		−	1.28558i	−0.766043	−	0.642790i	$-0.777777\pi$
$191$		−	245.546i		−	1.28558i	0.766043	−	0.642790i	$-0.222223\pi$
$192$	4.43409	−	23.5868i	0.0230942	−	0.122848i
$193$	−32.9007			−0.170470			−0.0852350	−	0.996361i	$-0.527164\pi$
$193$	−32.9007			−0.170470			−0.0852350	+	0.996361i	$0.527164\pi$
$194$		−	149.615i		−	0.771210i
$195$		0			0
$196$	−14.0000			−0.0714286
$197$			219.340i			1.11340i	0.830714	+	0.556700i	$0.187933\pi$
$197$			219.340i			1.11340i	−0.830714	+	0.556700i	$0.812067\pi$
$198$	30.6489	+	11.9456i	0.154793	+	0.0603311i
$199$	−100.635			−0.505703			−0.252852	−	0.967505i	$-0.581368\pi$
$199$	−100.635			−0.505703			−0.252852	+	0.967505i	$0.581368\pi$
$200$		0			0
$201$	−47.5360	+	252.864i	−0.236498	+	1.25803i
$202$	−170.630			−0.844701
$203$		−	78.1708i		−	0.385078i
$204$	−42.7582	−	8.03811i	−0.209599	−	0.0394025i
$205$		0			0
$206$		−	244.703i		−	1.18788i
$207$	38.8466	−	99.6694i	0.187665	−	0.481495i
$208$	−0.722106			−0.00347166
$209$		−	55.3944i		−	0.265045i
$210$		0			0
$211$	119.865			0.568082			0.284041	−	0.958812i	$-0.408325\pi$
$211$	119.865			0.568082			0.284041	+	0.958812i	$0.408325\pi$
$212$			110.114i			0.519408i
$213$	−288.375	−	54.2116i	−1.35387	−	0.254514i
$214$	−116.154			−0.542777
$215$		0			0
$216$	64.8054	+	40.4012i	0.300025	+	0.187042i
$217$	−131.016			−0.603760
$218$			89.7870i			0.411867i
$219$	−9.88317	+	52.5728i	−0.0451286	+	0.240059i
$220$		0			0
$221$			1.30903i			0.00592322i
$222$	180.526	+	33.9372i	0.813182	+	0.152870i
$223$	−315.561			−1.41507			−0.707536	−	0.706677i	$-0.750193\pi$
$223$	−315.561			−1.41507			−0.707536	+	0.706677i	$0.750193\pi$
$224$		−	14.9666i		−	0.0668153i
$225$		0			0
$226$	205.759			0.910438
$227$			315.509i			1.38991i	0.719054	+	0.694954i	$0.244575\pi$
$227$			315.509i			1.38991i	−0.719054	+	0.694954i	$0.755425\pi$
$228$	23.7599	−	126.389i	0.104210	−	0.554337i
$229$	−66.8556			−0.291946			−0.145973	−	0.989289i	$-0.546631\pi$
$229$	−66.8556			−0.291946			−0.145973	+	0.989289i	$0.546631\pi$
$230$		0			0
$231$	20.1602	+	3.78992i	0.0872737	+	0.0164066i
$232$	83.5681			0.360207
$233$		−	301.826i		−	1.29539i	−0.761900	−	0.647695i	$-0.775733\pi$
$233$		−	301.826i		−	1.29539i	0.761900	−	0.647695i	$-0.224267\pi$
$234$	0.834411	−	2.14086i	0.00356586	−	0.00914899i
$235$		0			0
$236$		−	200.220i		−	0.848388i
$237$	−61.3301	+	326.241i	−0.258777	+	1.37654i
$238$	−27.1315			−0.113998
$239$		−	241.028i		−	1.00849i	−0.863562	−	0.504243i	$-0.831771\pi$
$239$		−	241.028i		−	1.00849i	0.863562	−	0.504243i	$-0.168229\pi$
$240$		0			0
$241$	−196.843			−0.816775			−0.408388	−	0.912809i	$-0.633909\pi$
$241$	−196.843			−0.816775			−0.408388	+	0.912809i	$0.633909\pi$
$242$		−	161.674i		−	0.668074i
$243$	−194.664	+	145.448i	−0.801086	+	0.598550i
$244$	−16.8936			−0.0692359
$245$		0			0
$246$	−16.2920	+	86.6640i	−0.0662275	+	0.352293i
$247$	−3.86937			−0.0156654
$248$		−	140.062i		−	0.564766i
$249$	9.09894	+	1.71051i	0.0365419	+	0.00686952i
$250$		0			0
$251$		−	349.301i		−	1.39164i	−0.718217	−	0.695819i	$-0.755041\pi$
$251$		−	349.301i		−	1.39164i	0.718217	−	0.695819i	$-0.244959\pi$
$252$	44.3724	+	17.2943i	0.176081	+	0.0686282i
$253$	30.7182			0.121416
$254$		−	216.466i		−	0.852228i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	4.18732i		−	0.0162931i	−0.999967	−	0.00814653i	$-0.997407\pi$
$257$		−	4.18732i		−	0.0162931i	0.999967	−	0.00814653i	$-0.00259315\pi$
$258$	309.461	+	58.1756i	1.19946	+	0.225487i
$259$	114.550			0.442278
$260$		0			0
$261$	−96.5651	+	247.759i	−0.369981	+	0.949267i
$262$	−249.815			−0.953493
$263$		−	268.485i		−	1.02086i	−0.859921	−	0.510428i	$-0.829487\pi$
$263$		−	268.485i		−	1.02086i	0.859921	−	0.510428i	$-0.170513\pi$
$264$	−4.05160	+	21.5522i	−0.0153470	+	0.0816371i
$265$		0			0
$266$		−	80.1979i		−	0.301496i
$267$	−236.423	−	44.4452i	−0.885479	−	0.166461i
$268$	−171.529			−0.640034
$269$		−	59.1206i		−	0.219779i	−0.993944	−	0.109890i	$-0.964950\pi$
$269$		−	59.1206i		−	0.219779i	0.993944	−	0.109890i	$-0.0350497\pi$
$270$		0			0
$271$	−206.348			−0.761433			−0.380717	−	0.924692i	$-0.624323\pi$
$271$	−206.348			−0.761433			−0.380717	+	0.924692i	$0.624323\pi$
$272$		−	29.0048i		−	0.106635i
$273$	0.264731	−	1.40822i	0.000969710	−	0.00515830i
$274$	−324.300			−1.18358
$275$		0			0
$276$	70.0871	+	13.1757i	0.253939	+	0.0477380i
$277$	−370.482			−1.33748			−0.668740	−	0.743496i	$-0.733166\pi$
$277$	−370.482			−1.33748			−0.668740	+	0.743496i	$0.733166\pi$
$278$			11.2180i			0.0403527i
$279$	415.249	+	161.845i	1.48835	+	0.580090i
$280$		0			0
$281$		−	71.0258i		−	0.252761i	−0.991982	−	0.126380i	$-0.959664\pi$
$281$		−	71.0258i		−	0.252761i	0.991982	−	0.126380i	$-0.0403360\pi$
$282$	5.52693	−	29.4001i	0.0195991	−	0.104256i
$283$	255.816			0.903944			0.451972	−	0.892032i	$-0.350721\pi$
$283$	255.816			0.903944			0.451972	+	0.892032i	$0.350721\pi$
$284$		−	195.617i		−	0.688793i
$285$		0			0
$286$	0.659816			0.00230705
$287$			54.9912i			0.191607i
$288$	−18.4884	+	47.4361i	−0.0641958	+	0.164709i
$289$	236.420			0.818063
$290$		0			0
$291$	−58.6373	+	311.917i	−0.201503	+	1.07188i
$292$	−35.6625			−0.122132
$293$			162.409i			0.554297i	0.960827	+	0.277148i	$0.0893894\pi$
$293$			162.409i			0.554297i	−0.960827	+	0.277148i	$0.910611\pi$
$294$	29.1872	+	5.48691i	0.0992763	+	0.0186630i
$295$		0			0
$296$			122.459i			0.413713i
$297$	−59.2152	−	36.9161i	−0.199378	−	0.124297i
$298$	158.230			0.530973
$299$		−	2.14570i		−	0.00717625i
$300$		0			0
$301$	196.363			0.652370
$302$			104.299i			0.345360i
$303$	355.729	+	66.8735i	1.17402	+	0.220705i
$304$	85.7352			0.282024
$305$		0			0
$306$	85.9919	+	33.5157i	0.281019	+	0.109529i
$307$	532.774			1.73542			0.867709	−	0.497072i	$-0.165592\pi$
$307$	532.774			1.73542			0.867709	+	0.497072i	$0.165592\pi$
$308$			13.6756i			0.0444012i
$309$	−95.9046	+	510.158i	−0.310371	+	1.65100i
$310$		0			0
$311$			353.139i			1.13549i	0.823203	+	0.567747i	$0.192185\pi$
$311$			353.139i			1.13549i	−0.823203	+	0.567747i	$0.807815\pi$
$312$	1.50545	+	0.283009i	0.00482515	+	0.000907081i
$313$	113.835			0.363689			0.181845	−	0.983327i	$-0.441793\pi$
$313$	113.835			0.363689			0.181845	+	0.983327i	$0.441793\pi$
$314$		−	74.0989i		−	0.235984i
$315$		0			0
$316$	−221.304			−0.700328
$317$		−	73.1423i		−	0.230733i	−0.993323	−	0.115366i	$-0.963196\pi$
$317$		−	73.1423i		−	0.230733i	0.993323	−	0.115366i	$-0.0368042\pi$
$318$	43.1563	−	229.567i	0.135712	−	0.721908i
$319$	−76.3594			−0.239371
$320$		0			0
$321$	242.159	+	45.5235i	0.754389	+	0.141818i
$322$	44.4726			0.138114
$323$		−	155.420i		−	0.481178i
$324$	−119.272	−	109.627i	−0.368125	−	0.338355i
$325$		0			0
$326$			23.9193i			0.0733720i
$327$	35.1895	−	187.188i	0.107613	−	0.572441i
$328$	−58.7880			−0.179232
$329$		−	18.6554i		−	0.0567032i
$330$		0			0
$331$	−244.549			−0.738818			−0.369409	−	0.929267i	$-0.620440\pi$
$331$	−244.549			−0.738818			−0.369409	+	0.929267i	$0.620440\pi$
$332$			6.17222i			0.0185910i
$333$	−363.061	−	141.505i	−1.09027	−	0.424938i
$334$	−2.87970			−0.00862185
$335$		0			0
$336$	−5.86575	+	31.2025i	−0.0174576	+	0.0928644i
$337$	−360.620			−1.07009			−0.535045	−	0.844824i	$-0.679705\pi$
$337$	−360.620			−1.07009			−0.535045	+	0.844824i	$0.679705\pi$
$338$			238.956i			0.706970i
$339$	−428.967	−	80.6415i	−1.26539	−	0.237881i
$340$		0			0
$341$			127.980i			0.375308i
$342$	−99.0692	+	254.184i	−0.289676	+	0.743227i
$343$	18.5203			0.0539949
$344$			209.921i			0.610236i
$345$		0			0
$346$	366.178			1.05832
$347$		−	320.079i		−	0.922417i	−0.887292	−	0.461209i	$-0.847416\pi$
$347$		−	320.079i		−	0.922417i	0.887292	−	0.461209i	$-0.152584\pi$
$348$	−174.223	−	32.7522i	−0.500641	−	0.0941155i
$349$	656.250			1.88037			0.940187	−	0.340660i	$-0.110650\pi$
$349$	656.250			1.88037			0.940187	+	0.340660i	$0.110650\pi$
$350$		0			0
$351$	−2.57863	+	4.13625i	−0.00734654	+	0.0117842i
$352$	−14.6198			−0.0415336
$353$			307.348i			0.870673i	0.900268	+	0.435337i	$0.143371\pi$
$353$			307.348i			0.870673i	−0.900268	+	0.435337i	$0.856629\pi$
$354$	−78.4705	+	417.418i	−0.221668	+	1.17915i
$355$		0			0
$356$		−	160.376i		−	0.450495i
$357$	56.5637	+	10.6334i	0.158442	+	0.0297855i
$358$	−54.2490			−0.151534
$359$			149.072i			0.415244i	0.978209	+	0.207622i	$0.0665723\pi$
$359$			149.072i			0.415244i	−0.978209	+	0.207622i	$0.933428\pi$
$360$		0			0
$361$	98.4076			0.272597
$362$		−	302.245i		−	0.834930i
$363$	−63.3635	+	337.058i	−0.174555	+	0.928534i
$364$	0.955256			0.00262433
$365$		0			0
$366$	35.2197	+	6.62096i	0.0962287	+	0.0180901i
$367$	−394.930			−1.07610			−0.538052	−	0.842912i	$-0.680840\pi$
$367$	−394.930			−1.07610			−0.538052	+	0.842912i	$0.680840\pi$
$368$			47.5432i			0.129193i
$369$	67.9311	−	174.292i	0.184095	−	0.472336i
$370$		0			0
$371$		−	145.668i		−	0.392635i
$372$	−54.8933	+	292.001i	−0.147563	+	0.784949i
$373$	−567.874			−1.52245			−0.761225	−	0.648488i	$-0.775402\pi$
$373$	−567.874			−1.52245			−0.761225	+	0.648488i	$0.775402\pi$
$374$			26.5028i			0.0708630i
$375$		0			0
$376$	19.9434			0.0530410
$377$			5.33379i			0.0141480i
$378$	−85.7295	−	53.4457i	−0.226798	−	0.141391i
$379$	−641.443			−1.69246			−0.846230	−	0.532817i	$-0.821133\pi$
$379$	−641.443			−1.69246			−0.846230	+	0.532817i	$0.821133\pi$
$380$		0			0
$381$	−84.8378	+	451.289i	−0.222671	+	1.18448i
$382$	−347.254			−0.909042
$383$			340.090i			0.887963i	0.896036	+	0.443982i	$0.146434\pi$
$383$			340.090i			0.887963i	−0.896036	+	0.443982i	$0.853566\pi$
$384$	−33.3568	−	6.27075i	−0.0868667	−	0.0163301i
$385$		0			0
$386$			46.5286i			0.120541i
$387$	−622.365	−	242.569i	−1.60818	−	0.626794i
$388$	−211.587			−0.545328
$389$			432.582i			1.11204i	0.831170	+	0.556018i	$0.187672\pi$
$389$			432.582i			1.11204i	−0.831170	+	0.556018i	$0.812328\pi$
$390$		0			0
$391$	86.1862			0.220425
$392$			19.7990i			0.0505076i
$393$	520.815	+	97.9081i	1.32523	+	0.249130i
$394$	310.193			0.787292
$395$		0			0
$396$	16.8936	−	43.3441i	0.0426605	−	0.109455i
$397$	−471.352			−1.18728			−0.593642	−	0.804729i	$-0.702310\pi$
$397$	−471.352			−1.18728			−0.593642	+	0.804729i	$0.702310\pi$
$398$			142.319i			0.357586i
$399$	−31.4313	+	167.197i	−0.0787753	+	0.419039i
$400$		0			0
$401$			180.124i			0.449186i	0.974453	+	0.224593i	$0.0721053\pi$
$401$			180.124i			0.449186i	−0.974453	+	0.224593i	$0.927895\pi$
$402$	357.604	+	67.2261i	0.889563	+	0.167229i
$403$	8.93955			0.0221825
$404$			241.307i			0.597294i
$405$		0			0
$406$	−110.550			−0.272291
$407$		−	111.896i		−	0.274928i
$408$	−11.3676	+	60.4692i	−0.0278618	+	0.148209i
$409$	−259.241			−0.633842			−0.316921	−	0.948452i	$-0.602649\pi$
$409$	−259.241			−0.633842			−0.316921	+	0.948452i	$0.602649\pi$
$410$		0			0
$411$	676.101	+	127.100i	1.64501	+	0.309246i
$412$	−346.063			−0.839958
$413$			264.866i			0.641321i
$414$	−140.954	−	54.9374i	−0.340468	−	0.132699i
$415$		0			0
$416$			1.02121i			0.00245484i
$417$	4.39660	−	23.3874i	0.0105434	−	0.0560849i
$418$	−78.3395			−0.187415
$419$		−	146.134i		−	0.348769i	−0.984678	−	0.174384i	$-0.944207\pi$
$419$		−	146.134i		−	0.348769i	0.984678	−	0.174384i	$-0.0557935\pi$
$420$		0			0
$421$	−599.808			−1.42472			−0.712361	−	0.701813i	$-0.752374\pi$
$421$	−599.808			−1.42472			−0.712361	+	0.701813i	$0.752374\pi$
$422$		−	169.515i		−	0.401695i
$423$	−23.0451	+	59.1273i	−0.0544802	+	0.139781i
$424$	155.725			0.367277
$425$		0			0
$426$	−76.6667	+	407.823i	−0.179969	+	0.957331i
$427$	22.3481			0.0523374
$428$			164.267i			0.383802i
$429$	−1.37558	−	0.258596i	−0.00320649	−	0.000602789i
$430$		0			0
$431$		−	825.631i		−	1.91562i	−0.287409	−	0.957808i	$-0.592794\pi$
$431$		−	825.631i		−	1.91562i	0.287409	−	0.957808i	$-0.407206\pi$
$432$	57.1359	−	91.6487i	0.132259	−	0.212150i
$433$	−185.307			−0.427962			−0.213981	−	0.976838i	$-0.568643\pi$
$433$	−185.307			−0.427962			−0.213981	+	0.976838i	$0.568643\pi$
$434$			185.284i			0.426923i
$435$		0			0
$436$	126.978			0.291234
$437$			254.758i			0.582970i
$438$	74.3492	+	13.9769i	0.169747	+	0.0319108i
$439$	809.103			1.84306			0.921529	−	0.388309i	$-0.126941\pi$
$439$	809.103			1.84306			0.921529	+	0.388309i	$0.126941\pi$
$440$		0			0
$441$	−58.6991	−	22.8782i	−0.133105	−	0.0518781i
$442$	1.85125			0.00418835
$443$		−	449.922i		−	1.01563i	−0.861467	−	0.507813i	$-0.830454\pi$
$443$		−	449.922i		−	1.01563i	0.861467	−	0.507813i	$-0.169546\pi$
$444$	47.9944	−	255.303i	0.108095	−	0.575006i
$445$		0			0
$446$			446.271i			1.00061i
$447$	−329.878	−	62.0138i	−0.737983	−	0.138733i
$448$	−21.1660			−0.0472456
$449$			676.008i			1.50559i	0.658257	+	0.752793i	$0.271294\pi$
$449$			676.008i			1.50559i	−0.658257	+	0.752793i	$0.728706\pi$
$450$		0			0
$451$	53.7169			0.119106
$452$		−	290.987i		−	0.643777i
$453$	40.8769	−	217.442i	0.0902361	−	0.480004i
$454$	446.198			0.982814
$455$		0			0
$456$	−178.741	−	33.6015i	−0.391975	−	0.0736875i
$457$	588.327			1.28737			0.643683	−	0.765292i	$-0.277405\pi$
$457$	588.327			1.28737			0.643683	+	0.765292i	$0.277405\pi$
$458$			94.5481i			0.206437i
$459$	−166.141	−	103.576i	−0.361962	−	0.225655i
$460$		0			0
$461$			344.268i			0.746785i	0.927673	+	0.373393i	$0.121806\pi$
$461$			344.268i			0.746785i	−0.927673	+	0.373393i	$0.878194\pi$
$462$	5.35976	−	28.5109i	0.0116012	−	0.0617119i
$463$	−220.223			−0.475643			−0.237821	−	0.971309i	$-0.576433\pi$
$463$	−220.223			−0.475643			−0.237821	+	0.971309i	$0.576433\pi$
$464$		−	118.183i		−	0.254705i
$465$		0			0
$466$	−426.846			−0.915978
$467$			595.502i			1.27516i	0.770382	+	0.637582i	$0.220065\pi$
$467$			595.502i			1.27516i	−0.770382	+	0.637582i	$0.779935\pi$
$468$	−3.02764	−	1.18004i	−0.00646932	−	0.00252144i
$469$	226.912			0.483820
$470$		0			0
$471$	−29.0410	+	154.481i	−0.0616581	+	0.327986i
$472$	−283.153			−0.599901
$473$		−	191.813i		−	0.405524i
$474$	461.374	+	86.7338i	0.973364	+	0.182983i
$475$		0			0
$476$			38.3697i			0.0806086i
$477$	−179.945	+	461.687i	−0.377242	+	0.967898i
$478$	−340.865			−0.713108
$479$		−	710.354i		−	1.48299i	−0.670956	−	0.741497i	$-0.734116\pi$
$479$		−	710.354i		−	1.48299i	0.670956	−	0.741497i	$-0.265884\pi$
$480$		0			0
$481$	−7.81604			−0.0162496
$482$			278.378i			0.577547i
$483$	−92.7165	−	17.4298i	−0.191960	−	0.0360865i
$484$	−228.641			−0.472399
$485$		0			0
$486$	205.694	+	275.296i	0.423239	+	0.566453i
$487$	325.987			0.669378			0.334689	−	0.942329i	$-0.391369\pi$
$487$	325.987			0.669378			0.334689	+	0.942329i	$0.391369\pi$
$488$			23.8911i			0.0489572i
$489$	9.37450	−	49.8670i	0.0191708	−	0.101977i
$490$		0			0
$491$		−	573.407i		−	1.16784i	−0.811813	−	0.583918i	$-0.801519\pi$
$491$		−	573.407i		−	1.16784i	0.811813	−	0.583918i	$-0.198481\pi$
$492$	122.561	+	23.0403i	0.249108	+	0.0468299i
$493$	−214.242			−0.434568
$494$			5.47211i			0.0110771i
$495$		0			0
$496$	−198.077			−0.399350
$497$			258.777i			0.520679i
$498$	2.41903	−	12.8678i	0.00485749	−	0.0258390i
$499$	−410.671			−0.822989			−0.411494	−	0.911412i	$-0.634993\pi$
$499$	−410.671			−0.822989			−0.411494	+	0.911412i	$0.634993\pi$
$500$		0			0
$501$	6.00360	+	1.12862i	0.0119832	+	0.00225273i
$502$	−493.987			−0.984037
$503$		−	903.370i		−	1.79596i	−0.440032	−	0.897982i	$-0.645033\pi$
$503$		−	903.370i		−	1.79596i	0.440032	−	0.897982i	$-0.354967\pi$
$504$	24.4579	−	62.7520i	0.0485275	−	0.124508i
$505$		0			0
$506$		−	43.4420i		−	0.0858538i
$507$	93.6521	−	498.176i	0.184718	−	0.982595i
$508$	−306.129			−0.602616
$509$		−	323.535i		−	0.635628i	−0.948153	−	0.317814i	$-0.897051\pi$
$509$		−	323.535i		−	0.635628i	0.948153	−	0.317814i	$-0.102949\pi$
$510$		0			0
$511$	47.1770			0.0923230
$512$		−	22.6274i		−	0.0441942i
$513$	306.160	−	491.095i	0.596803	−	0.957300i
$514$	−5.92176			−0.0115209
$515$		0			0
$516$	82.2728	−	437.644i	0.159443	−	0.848147i
$517$	−18.2231			−0.0352477
$518$		−	161.998i		−	0.312738i
$519$	−763.408	−	143.513i	−1.47092	−	0.276519i
$520$		0			0
$521$			687.287i			1.31917i	0.751631	+	0.659584i	$0.229268\pi$
$521$			687.287i			1.31917i	−0.751631	+	0.659584i	$0.770732\pi$
$522$	350.384	+	136.564i	0.671233	+	0.261616i
$523$	760.416			1.45395			0.726975	−	0.686664i	$-0.240926\pi$
$523$	760.416			1.45395			0.726975	+	0.686664i	$0.240926\pi$
$524$			353.292i			0.674222i
$525$		0			0
$526$	−379.695			−0.721854
$527$			359.074i			0.681355i
$528$	30.4794	+	5.72983i	0.0577262	+	0.0108519i
$529$	387.728			0.732945
$530$		0			0
$531$	327.191	−	839.480i	0.616179	−	1.58094i
$532$	−113.417			−0.213190
$533$		−	3.75219i		−	0.00703975i
$534$	−62.8550	+	334.353i	−0.117706	+	0.626128i
$535$		0			0
$536$			242.579i			0.452572i
$537$	113.098	+	21.2614i	0.210612	+	0.0395929i
$538$	−83.6091			−0.155407
$539$		−	18.0911i		−	0.0335642i
$540$		0			0
$541$	472.370			0.873142			0.436571	−	0.899670i	$-0.356193\pi$
$541$	472.370			0.873142			0.436571	+	0.899670i	$0.356193\pi$
$542$			291.821i			0.538415i
$543$	−118.456	+	630.120i	−0.218152	+	1.16044i
$544$	−41.0189			−0.0754024
$545$		0			0
$546$	−1.99152	−	0.374386i	−0.00364747	−	0.000685688i
$547$	536.508			0.980818			0.490409	−	0.871492i	$-0.336847\pi$
$547$	536.508			0.980818			0.490409	+	0.871492i	$0.336847\pi$
$548$			458.629i			0.836915i
$549$	−70.8312	−	27.6068i	−0.129019	−	0.0502856i
$550$		0			0
$551$		−	633.278i		−	1.14933i
$552$	18.6332	−	99.1181i	0.0337558	−	0.179562i
$553$	292.757			0.529398
$554$			523.941i			0.945741i
$555$		0			0
$556$	15.8647			0.0285337
$557$		−	288.061i		−	0.517165i	−0.965989	−	0.258583i	$-0.916745\pi$
$557$		−	288.061i		−	0.517165i	0.965989	−	0.258583i	$-0.0832554\pi$
$558$	228.883	−	587.250i	0.410185	−	1.05242i
$559$	−13.3984			−0.0239685
$560$		0			0
$561$	10.3870	−	55.2530i	0.0185152	−	0.0984902i
$562$	−100.446			−0.178729
$563$			16.4938i			0.0292963i	0.999893	+	0.0146482i	$0.00466282\pi$
$563$			16.4938i			0.0292963i	−0.999893	+	0.0146482i	$0.995337\pi$
$564$	−41.5781	−	7.81627i	−0.0737200	−	0.0138586i
$565$		0			0
$566$		−	361.779i		−	0.639185i
$567$	157.783	+	145.023i	0.278276	+	0.255773i
$568$	−276.645			−0.487050
$569$		−	1076.66i		−	1.89219i	−0.323889	−	0.946095i	$-0.604991\pi$
$569$		−	1076.66i		−	1.89219i	0.323889	−	0.946095i	$-0.395009\pi$
$570$		0			0
$571$	1009.39			1.76775			0.883876	−	0.467721i	$-0.154925\pi$
$571$	1009.39			1.76775			0.883876	+	0.467721i	$0.154925\pi$
$572$		−	0.933120i		−	0.00163133i
$573$	723.956	+	136.097i	1.26345	+	0.237516i
$574$	77.7693			0.135487
$575$		0			0
$576$	67.0847	+	26.1466i	0.116467	+	0.0453933i
$577$	901.098			1.56169			0.780847	−	0.624722i	$-0.214788\pi$
$577$	901.098			1.56169			0.780847	+	0.624722i	$0.214788\pi$
$578$		−	334.349i		−	0.578458i
$579$	18.2356	−	97.0030i	0.0314950	−	0.167535i
$580$		0			0
$581$		−	8.16507i		−	0.0140535i
$582$	441.117	+	82.9257i	0.757933	+	0.142484i
$583$	−142.292			−0.244069
$584$			50.4344i			0.0863602i
$585$		0			0
$586$	229.681			0.391947
$587$			265.739i			0.452707i	0.974045	+	0.226354i	$0.0726805\pi$
$587$			265.739i			0.452707i	−0.974045	+	0.226354i	$0.927319\pi$
$588$	7.75966	−	41.2770i	0.0131967	−	0.0701989i
$589$	−1061.39			−1.80202
$590$		0			0
$591$	−646.691	−	121.572i	−1.09423	−	0.205705i
$592$	173.183			0.292539
$593$		−	548.447i		−	0.924869i	−0.886653	−	0.462435i	$-0.846976\pi$
$593$		−	548.447i		−	0.924869i	0.886653	−	0.462435i	$-0.153024\pi$
$594$	−52.2073	+	83.7430i	−0.0878910	+	0.140981i
$595$		0			0
$596$		−	223.771i		−	0.375455i
$597$	55.7781	−	296.708i	0.0934306	−	0.496998i
$598$	−3.03448			−0.00507438
$599$			237.976i			0.397289i	0.980072	+	0.198644i	$0.0636538\pi$
$599$			237.976i			0.397289i	−0.980072	+	0.198644i	$0.936346\pi$
$600$		0			0
$601$	−566.317			−0.942291			−0.471146	−	0.882055i	$-0.656159\pi$
$601$	−566.317			−0.942291			−0.471146	+	0.882055i	$0.656159\pi$
$602$		−	277.700i		−	0.461295i
$603$	−719.186	−	280.306i	−1.19268	−	0.464852i
$604$	147.501			0.244206
$605$		0			0
$606$	94.5734	−	503.077i	0.156062	−	0.830159i
$607$	60.4291			0.0995537			0.0497769	−	0.998760i	$-0.484149\pi$
$607$	60.4291			0.0995537			0.0497769	+	0.998760i	$0.484149\pi$
$608$		−	121.248i		−	0.199421i
$609$	230.475	+	43.3271i	0.378449	+	0.0711446i
$610$		0			0
$611$			1.27290i			0.00208331i
$612$	47.3984	−	121.611i	0.0774484	−	0.198711i
$613$	307.983			0.502419			0.251210	−	0.967933i	$-0.419172\pi$
$613$	307.983			0.502419			0.251210	+	0.967933i	$0.419172\pi$
$614$		−	753.456i		−	1.22713i
$615$		0			0
$616$	19.3402			0.0313964
$617$		−	780.226i		−	1.26455i	−0.774745	−	0.632274i	$-0.782122\pi$
$617$		−	780.226i		−	1.26455i	0.774745	−	0.632274i	$-0.217878\pi$
$618$	721.472	+	135.630i	1.16743	+	0.219465i
$619$	−838.604			−1.35477			−0.677386	−	0.735627i	$-0.736887\pi$
$619$	−838.604			−1.35477			−0.677386	+	0.735627i	$0.736887\pi$
$620$		0			0
$621$	272.330	+	169.776i	0.438534	+	0.273392i
$622$	499.413			0.802916
$623$			212.158i			0.340542i
$624$	0.400235	−	2.12902i	0.000641403	−	0.00341190i
$625$		0			0
$626$		−	160.987i		−	0.257167i
$627$	163.322	+	30.7030i	0.260482	+	0.0489681i
$628$	−104.792			−0.166866
$629$		−	313.946i		−	0.499119i
$630$		0			0
$631$	−433.178			−0.686494			−0.343247	−	0.939245i	$-0.611527\pi$
$631$	−433.178			−0.686494			−0.343247	+	0.939245i	$0.611527\pi$
$632$			312.971i			0.495207i
$633$	−66.4368	+	353.406i	−0.104955	+	0.558303i
$634$	−103.439			−0.163153
$635$		0			0
$636$	−324.656	−	61.0322i	−0.510466	−	0.0959626i
$637$	−1.26368			−0.00198381
$638$			107.988i			0.169261i
$639$	319.670	−	820.183i	0.500266	−	1.28354i
$640$		0			0
$641$		−	571.035i		−	0.890850i	−0.895319	−	0.445425i	$-0.853053\pi$
$641$		−	571.035i		−	0.890850i	0.895319	−	0.445425i	$-0.146947\pi$
$642$	64.3799	−	342.464i	0.100280	−	0.533433i
$643$	194.578			0.302609			0.151304	−	0.988487i	$-0.451653\pi$
$643$	194.578			0.302609			0.151304	+	0.988487i	$0.451653\pi$
$644$		−	62.8937i		−	0.0976611i
$645$		0			0
$646$	−219.798			−0.340244
$647$			432.986i			0.669220i	0.942357	+	0.334610i	$0.108605\pi$
$647$			432.986i			0.669220i	−0.942357	+	0.334610i	$0.891395\pi$
$648$	−155.036	+	168.677i	−0.239253	+	0.260303i
$649$	258.728			0.398656
$650$		0			0
$651$	72.6171	−	386.281i	0.111547	−	0.593366i
$652$	33.8270			0.0518819
$653$			870.907i			1.33370i	0.745192	+	0.666850i	$0.232358\pi$
$653$			870.907i			1.33370i	−0.745192	+	0.666850i	$0.767642\pi$
$654$	−264.724	−	49.7655i	−0.404777	−	0.0760940i
$655$		0			0
$656$			83.1388i			0.126736i
$657$	−149.525	−	58.2782i	−0.227588	−	0.0887035i
$658$	−26.3827			−0.0400952
$659$		−	95.3946i		−	0.144757i	−0.997377	−	0.0723783i	$-0.976941\pi$
$659$		−	95.3946i		−	0.144757i	0.997377	−	0.0723783i	$-0.0230589\pi$
$660$		0			0
$661$	−532.252			−0.805222			−0.402611	−	0.915371i	$-0.631897\pi$
$661$	−532.252			−0.805222			−0.402611	+	0.915371i	$0.631897\pi$
$662$			345.844i			0.522423i
$663$	−3.85949	−	0.725546i	−0.00582125	−	0.00109434i
$664$	8.72883			0.0131458
$665$		0			0
$666$	−200.118	+	513.446i	−0.300477	+	0.770939i
$667$	351.175			0.526500
$668$			4.07251i			0.00609657i
$669$	174.903	−	930.386i	0.261440	−	1.39071i
$670$		0			0
$671$		−	21.8302i		−	0.0325339i
$672$	44.1269	+	8.29543i	0.0656651	+	0.0123444i
$673$	256.746			0.381494			0.190747	−	0.981639i	$-0.438909\pi$
$673$	256.746			0.381494			0.190747	+	0.981639i	$0.438909\pi$
$674$			509.994i			0.756667i
$675$		0			0
$676$	337.935			0.499904
$677$		−	962.182i		−	1.42124i	−0.703574	−	0.710622i	$-0.748414\pi$
$677$		−	962.182i		−	1.42124i	0.703574	−	0.710622i	$-0.251586\pi$
$678$	−114.044	+	606.651i	−0.168207	+	0.894765i
$679$	279.903			0.412229
$680$		0			0
$681$	−930.233	−	174.875i	−1.36598	−	0.256791i
$682$	180.991			0.265383
$683$			981.986i			1.43775i	0.695137	+	0.718877i	$0.255344\pi$
$683$			981.986i			1.43775i	−0.695137	+	0.718877i	$0.744656\pi$
$684$	359.470	+	140.105i	0.525541	+	0.204832i
$685$		0			0
$686$		−	26.1916i		−	0.0381802i
$687$	37.0555	−	197.114i	0.0539381	−	0.286920i
$688$	296.873			0.431502
$689$			9.93928i			0.0144257i
$690$		0			0
$691$	−38.5649			−0.0558103			−0.0279052	−	0.999611i	$-0.508884\pi$
$691$	−38.5649			−0.0558103			−0.0279052	+	0.999611i	$0.508884\pi$
$692$		−	517.853i		−	0.748343i
$693$	−22.3481	+	57.3389i	−0.0322483	+	0.0827401i
$694$	−452.660			−0.652247
$695$		0			0
$696$	−46.3186	+	246.388i	−0.0665497	+	0.354006i
$697$	150.714			0.216232
$698$		−	928.078i		−	1.32962i
$699$	889.889	+	167.290i	1.27309	+	0.239328i
$700$		0			0
$701$			301.635i			0.430293i	0.976582	+	0.215146i	$0.0690228\pi$
$701$			301.635i			0.430293i	−0.976582	+	0.215146i	$0.930977\pi$
$702$	5.84955	+	3.64674i	0.00833269	+	0.00519478i
$703$	927.993			1.32005
$704$			20.6755i			0.0293687i
$705$		0			0
$706$	434.655			0.615659
$707$		−	319.219i		−	0.451512i
$708$	590.318	+	110.974i	0.833783	+	0.156743i
$709$	910.699			1.28448			0.642242	−	0.766502i	$-0.278005\pi$
$709$	910.699			1.28448			0.642242	+	0.766502i	$0.278005\pi$
$710$		0			0
$711$	−927.881	−	361.646i	−1.30504	−	0.508644i
$712$	−226.806			−0.318548
$713$		−	588.577i		−	0.825493i
$714$	15.0379	−	79.9932i	0.0210615	−	0.112035i
$715$		0			0
$716$			76.7197i			0.107150i
$717$	710.637	+	133.593i	0.991125	+	0.186322i
$718$	210.820			0.293622
$719$		−	526.394i		−	0.732119i	−0.930591	−	0.366060i	$-0.880707\pi$
$719$		−	526.394i		−	0.732119i	0.930591	−	0.366060i	$-0.119293\pi$
$720$		0			0
$721$	457.798			0.634948
$722$		−	139.169i		−	0.192755i
$723$	109.102	−	580.362i	0.150902	−	0.802714i
$724$	−427.438			−0.590384
$725$		0			0
$726$	476.672	+	89.6096i	0.656573	+	0.123429i
$727$	348.251			0.479025			0.239513	−	0.970893i	$-0.423012\pi$
$727$	348.251			0.479025			0.239513	+	0.970893i	$0.423012\pi$
$728$		−	1.35094i		−	0.00185568i
$729$	−320.936	−	654.554i	−0.440242	−	0.897879i
$730$		0			0
$731$		−	538.171i		−	0.736212i
$732$	9.36345	−	49.8082i	0.0127916	−	0.0680440i
$733$	−1270.52			−1.73332			−0.866658	−	0.498903i	$-0.833736\pi$
$733$	−1270.52			−1.73332			−0.866658	+	0.498903i	$0.833736\pi$
$734$			558.515i			0.760920i
$735$		0			0
$736$	67.2362			0.0913536
$737$		−	221.654i		−	0.300751i
$738$	−246.486	−	96.0690i	−0.333992	−	0.130175i
$739$	−1072.31			−1.45103			−0.725514	−	0.688207i	$-0.758398\pi$
$739$	−1072.31			−1.45103			−0.725514	+	0.688207i	$0.758398\pi$
$740$		0			0
$741$	2.14464	−	11.4083i	0.00289425	−	0.0153958i
$742$	−206.005			−0.277635
$743$		−	205.210i		−	0.276191i	−0.990419	−	0.138095i	$-0.955902\pi$
$743$		−	205.210i		−	0.276191i	0.990419	−	0.138095i	$-0.0440981\pi$
$744$	412.952	+	77.6309i	0.555043	+	0.104343i
$745$		0			0
$746$			803.094i			1.07653i
$747$	−10.0864	+	25.8788i	−0.0135025	+	0.0346437i
$748$	37.4806			0.0501077
$749$		−	217.305i		−	0.290127i
$750$		0			0
$751$	−167.702			−0.223306			−0.111653	−	0.993747i	$-0.535614\pi$
$751$	−167.702			−0.223306			−0.111653	+	0.993747i	$0.535614\pi$
$752$		−	28.2043i		−	0.0375057i
$753$	1029.86	+	193.604i	1.36768	+	0.257111i
$754$	7.54312			0.0100041
$755$		0			0
$756$	−75.5837	+	121.240i	−0.0999784	+	0.160370i
$757$	468.004			0.618236			0.309118	−	0.951024i	$-0.399966\pi$
$757$	468.004			0.618236			0.309118	+	0.951024i	$0.399966\pi$
$758$			907.137i			1.19675i
$759$	−17.0259	+	90.5680i	−0.0224320	+	0.119325i
$760$		0			0
$761$			252.487i			0.331784i	0.986144	+	0.165892i	$0.0530503\pi$
$761$			252.487i			0.331784i	−0.986144	+	0.165892i	$0.946950\pi$
$762$	638.218	+	119.979i	0.837557	+	0.157452i
$763$	−167.976			−0.220152
$764$			491.091i			0.642790i
$765$		0			0
$766$	480.960			0.627885
$767$		−	18.0725i		−	0.0235625i
$768$	−8.86818	+	47.1737i	−0.0115471	+	0.0614241i
$769$	−146.391			−0.190366			−0.0951828	−	0.995460i	$-0.530344\pi$
$769$	−146.391			−0.190366			−0.0951828	+	0.995460i	$0.530344\pi$
$770$		0			0
$771$	12.3457	+	2.32087i	0.0160126	+	0.00301021i
$772$	65.8014			0.0852350
$773$			1105.53i			1.43018i	0.699032	+	0.715091i	$0.253615\pi$
$773$			1105.53i			1.43018i	−0.699032	+	0.715091i	$0.746385\pi$
$774$	−343.045	+	880.156i	−0.443210	+	1.13715i
$775$		0			0
$776$			299.229i			0.385605i
$777$	−63.4906	+	337.734i	−0.0817125	+	0.434664i
$778$	611.764			0.786328
$779$			445.495i			0.571881i
$780$		0			0
$781$	252.781			0.323663
$782$		−	121.886i		−	0.155864i
$783$	−676.958	−	422.031i	−0.864570	−	0.538993i
$784$	28.0000			0.0357143
$785$		0			0
$786$	138.463	−	736.544i	0.176162	−	0.937079i
$787$	−1176.01			−1.49430			−0.747150	−	0.664655i	$-0.768578\pi$
$787$	−1176.01			−1.49430			−0.747150	+	0.664655i	$0.768578\pi$
$788$		−	438.679i		−	0.556700i
$789$	791.589	+	148.811i	1.00328	+	0.188607i

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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-0.554262 + 2.94835i) q^{3} -2.00000 q^{4} +(4.16960 + 0.783844i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-8.38559 - 3.26832i) q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +(-0.554262 + 2.94835i) q^{3} -2.00000 q^{4} +(4.16960 + 0.783844i) q^{6} +2.64575 q^{7} +2.82843i q^{8} +(-8.38559 - 3.26832i) q^{9} -2.58444i q^{11} +(1.10852 - 5.89671i) q^{12} -0.180526 q^{13} -3.74166i q^{14} +4.00000 q^{16} -7.25119i q^{17} +(-4.62210 + 11.8590i) q^{18} +21.4338 q^{19} +(-1.46644 + 7.80061i) q^{21} -3.65495 q^{22} +11.8858i q^{23} +(-8.33921 - 1.56769i) q^{24} +0.255303i q^{26} +(14.2840 - 22.9122i) q^{27} -5.29150 q^{28} -29.5458i q^{29} -49.5193 q^{31} -5.65685i q^{32} +(7.61985 + 1.43246i) q^{33} -10.2547 q^{34} +(16.7712 + 6.53664i) q^{36} +43.2958 q^{37} -30.3120i q^{38} +(0.100059 - 0.532256i) q^{39} +20.7847i q^{41} +(11.0317 + 2.07386i) q^{42} +74.2183 q^{43} +5.16888i q^{44} +16.8091 q^{46} -7.05106i q^{47} +(-2.21705 + 11.7934i) q^{48} +7.00000 q^{49} +(21.3791 + 4.01906i) q^{51} +0.361053 q^{52} -55.0572i q^{53} +(-32.4027 - 20.2006i) q^{54} +7.48331i q^{56} +(-11.8799 + 63.1944i) q^{57} -41.7841 q^{58} +100.110i q^{59} +8.44678 q^{61} +70.0309i q^{62} +(-22.1862 - 8.64716i) q^{63} -8.00000 q^{64} +(2.02580 - 10.7761i) q^{66} +85.7646 q^{67} +14.5024i q^{68} +(-35.0435 - 6.58784i) q^{69} +97.8086i q^{71} +(9.24420 - 23.7180i) q^{72} +17.8312 q^{73} -61.2295i q^{74} -42.8676 q^{76} -6.83779i q^{77} +(-0.752723 - 0.141505i) q^{78} +110.652 q^{79} +(59.6362 + 54.8136i) q^{81} +29.3940 q^{82} -3.08611i q^{83} +(2.93288 - 15.6012i) q^{84} -104.961i q^{86} +(87.1115 + 16.3761i) q^{87} +7.30991 q^{88} +80.1881i q^{89} -0.477628 q^{91} -23.7716i q^{92} +(27.4467 - 146.001i) q^{93} -9.97171 q^{94} +(16.6784 + 3.13538i) q^{96} +105.794 q^{97} -9.89949i q^{98} +(-8.44678 + 21.6721i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} - 44 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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