LMFDB - Embedded newform 1620.3.t.d.269.2

Newspace parameters

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.154550410641.1
Defining polynomial:	$ x^{8} - 15x^{6} + 221x^{4} - 60x^{2} + 16 $ x^8 - 15x^6 + 221x^4 - 60*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.2
Root			$-3.32360 - 1.91888i$ of defining polynomial
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.d.1349.2

$q$-expansion

$f(q)$	$=$	$q+(0.0689865 + 4.99952i) q^{5} +(-2.42096 - 1.39774i) q^{7} +O(q^{10})$	$q+(0.0689865 + 4.99952i) q^{5} +(-2.42096 - 1.39774i) q^{7} +(-15.7154 - 9.07327i) q^{11} +(-19.9416 + 11.5133i) q^{13} +5.72842 q^{17} +23.1852 q^{19} +(-0.135792 - 0.235199i) q^{23} +(-24.9905 + 0.689799i) q^{25} +(34.4674 + 19.8997i) q^{29} +(-23.6852 - 41.0241i) q^{31} +(6.82104 - 12.2001i) q^{35} -34.8712i q^{37} +(-11.4891 + 6.63325i) q^{41} +(40.4571 + 23.3579i) q^{43} +(20.4568 - 35.4323i) q^{47} +(-20.5926 - 35.6675i) q^{49} +91.3705 q^{53} +(44.2779 - 79.1953i) q^{55} +(68.2773 - 39.4199i) q^{59} +(-15.5926 + 27.0072i) q^{61} +(-58.9366 - 98.9042i) q^{65} +(-5.98970 + 3.45816i) q^{67} -81.5870i q^{71} +106.084i q^{73} +(25.3642 + 43.9321i) q^{77} +(-31.7779 + 55.0409i) q^{79} +(0.142081 - 0.246091i) q^{83} +(0.395183 + 28.6394i) q^{85} -28.5210i q^{89} +64.3705 q^{91} +(1.59947 + 115.915i) q^{95} +(80.4657 + 46.4569i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{5}+O(q^{10}) $ 8 * q + 3 * q^5	$ 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95}+O(q^{100}) $ 8 * q + 3 * q^5 - 12 * q^17 + 12 * q^19 - 30 * q^23 - 9 * q^25 - 16 * q^31 - 90 * q^35 + 48 * q^47 - 78 * q^49 + 384 * q^53 + 94 * q^55 - 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 - 288 * q^83 + 100 * q^85 + 168 * q^91 + 318 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0689865	+	4.99952i	0.0137973	+	0.999905i
$5$	0.0689865	+	4.99952i	0.0137973	+	0.999905i
$6$		0			0
$7$	−2.42096	−	1.39774i	−0.345852	−	0.199678i	0.317005	−	0.948424i	$-0.397323\pi$
$7$	−2.42096	−	1.39774i	−0.345852	−	0.199678i	−0.662857	+	0.748746i	$0.730656\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.7154	−	9.07327i	−1.42867	−	0.824843i	−0.431653	−	0.902040i	$-0.642070\pi$
$11$	−15.7154	−	9.07327i	−1.42867	−	0.824843i	−0.997016	+	0.0771971i	$0.975403\pi$
$12$		0			0
$13$	−19.9416	+	11.5133i	−1.53397	+	0.885637i	−0.534796	+	0.844981i	$0.679611\pi$
$13$	−19.9416	+	11.5133i	−1.53397	+	0.885637i	−0.999173	+	0.0406560i	$0.987055\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.72842			0.336966			0.168483	−	0.985705i	$-0.446113\pi$
$17$	5.72842			0.336966			0.168483	+	0.985705i	$0.446113\pi$
$18$		0			0
$19$	23.1852			1.22028			0.610138	−	0.792295i	$-0.291114\pi$
$19$	23.1852			1.22028			0.610138	+	0.792295i	$0.291114\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.135792	−	0.235199i	−0.00590400	−	0.0102260i	0.863058	−	0.505104i	$-0.168546\pi$
$23$	−0.135792	−	0.235199i	−0.00590400	−	0.0102260i	−0.868962	+	0.494878i	$0.835213\pi$
$24$		0			0
$25$	−24.9905	+	0.689799i	−0.999619	+	0.0275920i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	34.4674	+	19.8997i	1.18853	+	0.686198i	0.957972	−	0.286860i	$-0.0926116\pi$
$29$	34.4674	+	19.8997i	1.18853	+	0.686198i	0.230558	+	0.973059i	$0.425945\pi$
$30$		0			0
$31$	−23.6852	−	41.0241i	−0.764040	−	1.32336i	−0.940752	−	0.339094i	$-0.889880\pi$
$31$	−23.6852	−	41.0241i	−0.764040	−	1.32336i	0.176712	−	0.984263i	$-0.443454\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.82104	−	12.2001i	0.194887	−	0.348574i
$36$		0			0
$37$		−	34.8712i		−	0.942465i	−0.882009	−	0.471232i	$-0.843809\pi$
$37$		−	34.8712i		−	0.942465i	0.882009	−	0.471232i	$-0.156191\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.4891	+	6.63325i	−0.280223	+	0.161787i	−0.633524	−	0.773723i	$-0.718392\pi$
$41$	−11.4891	+	6.63325i	−0.280223	+	0.161787i	0.353302	+	0.935509i	$0.385059\pi$
$42$		0			0
$43$	40.4571	+	23.3579i	0.940862	+	0.543207i	0.890231	−	0.455510i	$-0.150543\pi$
$43$	40.4571	+	23.3579i	0.940862	+	0.543207i	0.0506318	+	0.998717i	$0.483877\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	20.4568	−	35.4323i	0.435252	−	0.753878i	−0.562064	−	0.827093i	$-0.689993\pi$
$47$	20.4568	−	35.4323i	0.435252	−	0.753878i	0.997316	+	0.0732153i	$0.0233260\pi$
$48$		0			0
$49$	−20.5926	−	35.6675i	−0.420258	−	0.727908i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	91.3705			1.72397			0.861986	−	0.506932i	$-0.169221\pi$
$53$	91.3705			1.72397			0.861986	+	0.506932i	$0.169221\pi$
$54$		0			0
$55$	44.2779	−	79.1953i	0.805052	−	1.43991i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	68.2773	−	39.4199i	1.15724	−	0.668134i	0.206600	−	0.978425i	$-0.433760\pi$
$59$	68.2773	−	39.4199i	1.15724	−	0.668134i	0.950641	+	0.310292i	$0.100427\pi$
$60$		0			0
$61$	−15.5926	+	27.0072i	−0.255617	+	0.442741i	−0.965063	−	0.262018i	$-0.915612\pi$
$61$	−15.5926	+	27.0072i	−0.255617	+	0.442741i	0.709446	+	0.704760i	$0.248945\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−58.9366	−	98.9042i	−0.906718	−	1.52160i
$66$		0			0
$67$	−5.98970	+	3.45816i	−0.0893986	+	0.0516143i	−0.544033	−	0.839064i	$-0.683103\pi$
$67$	−5.98970	+	3.45816i	−0.0893986	+	0.0516143i	0.454634	+	0.890678i	$0.349770\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	81.5870i		−	1.14911i	−0.818465	−	0.574556i	$-0.805175\pi$
$71$		−	81.5870i		−	1.14911i	0.818465	−	0.574556i	$-0.194825\pi$
$72$		0			0
$73$			106.084i			1.45320i	0.687060	+	0.726601i	$0.258901\pi$
$73$			106.084i			1.45320i	−0.687060	+	0.726601i	$0.741099\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	25.3642	+	43.9321i	0.329405	+	0.570547i
$78$		0			0
$79$	−31.7779	+	55.0409i	−0.402252	+	0.696720i	−0.993997	−	0.109405i	$-0.965106\pi$
$79$	−31.7779	+	55.0409i	−0.402252	+	0.696720i	0.591746	+	0.806125i	$0.298439\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.142081	−	0.246091i	0.00171182	−	0.00296495i	−0.865168	−	0.501482i	$-0.832788\pi$
$83$	0.142081	−	0.246091i	0.00171182	−	0.00296495i	0.866880	+	0.498517i	$0.166122\pi$
$84$		0			0
$85$	0.395183	+	28.6394i	0.00464921	+	0.336934i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	28.5210i		−	0.320461i	−0.987080	−	0.160230i	$-0.948776\pi$
$89$		−	28.5210i		−	0.320461i	0.987080	−	0.160230i	$-0.0512237\pi$
$90$		0			0
$91$	64.3705			0.707368
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.59947	+	115.915i	0.0168365	+	1.22016i
$96$		0			0
$97$	80.4657	+	46.4569i	0.829543	+	0.478937i	0.853696	−	0.520771i	$-0.174356\pi$
$97$	80.4657	+	46.4569i	0.829543	+	0.478937i	−0.0241531	+	0.999708i	$0.507689\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−40.4153	−	23.3338i	−0.400151	−	0.231027i	0.286398	−	0.958111i	$-0.407542\pi$
$101$	−40.4153	−	23.3338i	−0.400151	−	0.231027i	−0.686549	+	0.727083i	$0.740875\pi$
$102$		0			0
$103$	9.68385	−	5.59098i	0.0940180	−	0.0542813i	−0.452254	−	0.891889i	$-0.649380\pi$
$103$	9.68385	−	5.59098i	0.0940180	−	0.0542813i	0.546272	+	0.837608i	$0.316047\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	147.297			1.37661			0.688303	−	0.725424i	$-0.258356\pi$
$107$	147.297			1.37661			0.688303	+	0.725424i	$0.258356\pi$
$108$		0			0
$109$	−121.556			−1.11519			−0.557595	−	0.830113i	$-0.688276\pi$
$109$	−121.556			−1.11519			−0.557595	+	0.830113i	$0.688276\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−48.8148	−	84.5496i	−0.431989	−	0.748227i	0.565056	−	0.825053i	$-0.308855\pi$
$113$	−48.8148	−	84.5496i	−0.431989	−	0.748227i	−0.997045	+	0.0768260i	$0.975521\pi$
$114$		0			0
$115$	1.16651	−	0.695121i	0.0101436	−	0.00604453i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.8683	−	8.00686i	−0.116540	−	0.0672845i
$120$		0			0
$121$	104.148	+	180.390i	0.860730	+	1.49083i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.17267	−	124.893i	−0.0413814	−	0.999143i
$126$		0			0
$127$			88.6481i			0.698017i	0.937120	+	0.349008i	$0.113482\pi$
$127$			88.6481i			0.698017i	−0.937120	+	0.349008i	$0.886518\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	77.9193	−	44.9867i	0.594804	−	0.343410i	−0.172191	−	0.985064i	$-0.555085\pi$
$131$	77.9193	−	44.9867i	0.594804	−	0.343410i	0.766995	+	0.641653i	$0.221751\pi$
$132$		0			0
$133$	−56.1306	−	32.4070i	−0.422035	−	0.243662i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	114.963	−	199.122i	0.839147	−	1.45345i	−0.0514620	−	0.998675i	$-0.516388\pi$
$137$	114.963	−	199.122i	0.839147	−	1.45345i	0.890609	−	0.454770i	$-0.150279\pi$
$138$		0			0
$139$	−28.8148	−	49.9086i	−0.207300	−	0.359055i	0.743563	−	0.668666i	$-0.233134\pi$
$139$	−28.8148	−	49.9086i	−0.207300	−	0.359055i	−0.950863	+	0.309611i	$0.899801\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	417.852			2.92205
$144$		0			0
$145$	−97.1115	+	173.693i	−0.669734	+	1.19788i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.64205	+	5.56684i	−0.0647117	+	0.0373613i	−0.532007	−	0.846740i	$-0.678562\pi$
$149$	−9.64205	+	5.56684i	−0.0647117	+	0.0373613i	0.467295	+	0.884101i	$0.345229\pi$
$150$		0			0
$151$	11.0926	−	19.2130i	0.0734611	−	0.127238i	−0.826955	−	0.562268i	$-0.809929\pi$
$151$	11.0926	−	19.2130i	0.0734611	−	0.127238i	0.900416	+	0.435030i	$0.143262\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	203.467	−	121.245i	1.31269	−	0.782226i
$156$		0			0
$157$	195.148	−	112.669i	1.24298	−	0.717635i	0.273281	−	0.961934i	$-0.411891\pi$
$157$	195.148	−	112.669i	1.24298	−	0.717635i	0.969700	+	0.244299i	$0.0785579\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.759209i			0.00471559i
$162$		0			0
$163$		−	302.948i		−	1.85858i	−0.369352	−	0.929290i	$-0.620420\pi$
$163$		−	302.948i		−	1.85858i	0.369352	−	0.929290i	$-0.379580\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	119.334	+	206.692i	0.714573	+	1.23768i	0.963124	+	0.269057i	$0.0867121\pi$
$167$	119.334	+	206.692i	0.714573	+	1.23768i	−0.248552	+	0.968619i	$0.579955\pi$
$168$		0			0
$169$	180.611	−	312.828i	1.06871	−	1.85105i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−54.4011	+	94.2254i	−0.314457	+	0.544656i	−0.979322	−	0.202308i	$-0.935156\pi$
$173$	−54.4011	+	94.2254i	−0.314457	+	0.544656i	0.664865	+	0.746964i	$0.268489\pi$
$174$		0			0
$175$	61.4652	+	33.2603i	0.351230	+	0.190059i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	168.054i		−	0.938849i	−0.882973	−	0.469425i	$-0.844461\pi$
$179$		−	168.054i		−	0.938849i	0.882973	−	0.469425i	$-0.155539\pi$
$180$		0			0
$181$	−154.297			−0.852468			−0.426234	−	0.904613i	$-0.640160\pi$
$181$	−154.297			−0.852468			−0.426234	+	0.904613i	$0.640160\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	174.339	−	2.40564i	0.942375	−	0.0130035i
$186$		0			0
$187$	−90.0241	−	51.9755i	−0.481412	−	0.277944i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−134.175	−	77.4662i	−0.702489	−	0.405582i	0.105785	−	0.994389i	$-0.466264\pi$
$191$	−134.175	−	77.4662i	−0.702489	−	0.405582i	−0.808274	+	0.588807i	$0.799598\pi$
$192$		0			0
$193$	55.4313	−	32.0033i	0.287209	−	0.165820i	−0.349473	−	0.936946i	$-0.613640\pi$
$193$	55.4313	−	32.0033i	0.287209	−	0.165820i	0.636683	+	0.771126i	$0.280306\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.56832			0.0485702			0.0242851	−	0.999705i	$-0.492269\pi$
$197$	9.56832			0.0485702			0.0242851	+	0.999705i	$0.492269\pi$
$198$		0			0
$199$	−117.815			−0.592034			−0.296017	−	0.955183i	$-0.595658\pi$
$199$	−117.815			−0.592034			−0.296017	+	0.955183i	$0.595658\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−55.6295	−	96.3531i	−0.274037	−	0.474646i
$204$		0			0
$205$	−33.9557	−	56.9826i	−0.165637	−	0.277964i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−364.365	−	210.366i	−1.74337	−	1.00654i
$210$		0			0
$211$	24.4074	+	42.2748i	0.115675	+	0.200355i	0.918049	−	0.396466i	$-0.129764\pi$
$211$	24.4074	+	42.2748i	0.115675	+	0.200355i	−0.802375	+	0.596821i	$0.796430\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−113.987	+	203.878i	−0.530174	+	0.948268i
$216$		0			0
$217$			132.424i			0.610247i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−114.234	+	65.9529i	−0.516895	+	0.298429i
$222$		0			0
$223$	−276.887	−	159.861i	−1.24164	−	0.716864i	−0.272216	−	0.962236i	$-0.587756\pi$
$223$	−276.887	−	159.861i	−1.24164	−	0.716864i	−0.969429	+	0.245372i	$0.921090\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.13579	−	15.8237i	0.0402458	−	0.0697077i	−0.845201	−	0.534449i	$-0.820519\pi$
$227$	9.13579	−	15.8237i	0.0402458	−	0.0697077i	0.885447	+	0.464741i	$0.153853\pi$
$228$		0			0
$229$	−67.0369	−	116.111i	−0.292737	−	0.507036i	0.681719	−	0.731615i	$-0.261233\pi$
$229$	−67.0369	−	116.111i	−0.292737	−	0.507036i	−0.974456	+	0.224578i	$0.927900\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	284.173			1.21963			0.609813	−	0.792546i	$-0.291245\pi$
$233$	284.173			1.21963			0.609813	+	0.792546i	$0.291245\pi$
$234$		0			0
$235$	178.556	+	99.8301i	0.759812	+	0.424809i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−179.068	+	103.385i	−0.749237	+	0.432572i	−0.825418	−	0.564522i	$-0.809061\pi$
$239$	−179.068	+	103.385i	−0.749237	+	0.432572i	0.0761810	+	0.997094i	$0.475727\pi$
$240$		0			0
$241$	81.3336	−	140.874i	0.337484	−	0.584539i	−0.646475	−	0.762935i	$-0.723757\pi$
$241$	81.3336	−	140.874i	0.337484	−	0.584539i	0.983959	+	0.178396i	$0.0570908\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	176.900	−	105.414i	0.722040	−	0.430261i
$246$		0			0
$247$	−462.351	+	266.938i	−1.87187	+	1.08072i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	347.387i		−	1.38401i	−0.721893	−	0.692005i	$-0.756728\pi$
$251$		−	347.387i		−	1.38401i	0.721893	−	0.692005i	$-0.243272\pi$
$252$		0			0
$253$			4.92831i			0.0194795i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.5189	−	18.2192i	−0.0409294	−	0.0708919i	0.844835	−	0.535027i	$-0.179699\pi$
$257$	−10.5189	−	18.2192i	−0.0409294	−	0.0708919i	−0.885764	+	0.464135i	$0.846365\pi$
$258$		0			0
$259$	−48.7410	+	84.4219i	−0.188189	+	0.325953i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−107.728	+	186.591i	−0.409614	+	0.709472i	−0.994846	−	0.101394i	$-0.967670\pi$
$263$	−107.728	+	186.591i	−0.409614	+	0.709472i	0.585233	+	0.810865i	$0.301003\pi$
$264$		0			0
$265$	6.30333	+	456.809i	0.0237861	+	1.72381i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	180.237i		−	0.670024i	−0.942214	−	0.335012i	$-0.891260\pi$
$269$		−	180.237i		−	0.670024i	0.942214	−	0.335012i	$-0.108740\pi$
$270$		0			0
$271$	231.741			0.855133			0.427566	−	0.903984i	$-0.359371\pi$
$271$	231.741			0.855133			0.427566	+	0.903984i	$0.359371\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	398.993	+	215.905i	1.45088	+	0.785109i
$276$		0			0
$277$	−91.4227	−	52.7829i	−0.330046	−	0.190552i	0.325816	−	0.945433i	$-0.394361\pi$
$277$	−91.4227	−	52.7829i	−0.330046	−	0.190552i	−0.655862	+	0.754881i	$0.727694\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	436.993	+	252.298i	1.55514	+	0.897859i	0.997710	+	0.0676323i	$0.0215445\pi$
$281$	436.993	+	252.298i	1.55514	+	0.897859i	0.557426	+	0.830226i	$0.311789\pi$
$282$		0			0
$283$	131.306	−	75.8095i	0.463978	−	0.267878i	−0.249737	−	0.968314i	$-0.580344\pi$
$283$	131.306	−	75.8095i	0.463978	−	0.267878i	0.713716	+	0.700436i	$0.247011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	37.0863			0.129221
$288$		0			0
$289$	−256.185			−0.886454
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−274.519	−	475.481i	−0.936924	−	1.62280i	−0.771166	−	0.636634i	$-0.780326\pi$
$293$	−274.519	−	475.481i	−0.936924	−	1.62280i	−0.165759	−	0.986166i	$-0.553007\pi$
$294$		0			0
$295$	201.791	+	338.634i	0.684037	+	1.14791i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.41582	+	3.12682i	0.0181131	+	0.0104576i
$300$		0			0
$301$	−65.2967	−	113.097i	−0.216933	−	0.375738i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−136.099	−	76.0926i	−0.446226	−	0.249484i
$306$		0			0
$307$		−	146.401i		−	0.476876i	−0.971158	−	0.238438i	$-0.923365\pi$
$307$		−	146.401i		−	0.476876i	0.971158	−	0.238438i	$-0.0766355\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	422.468	−	243.912i	1.35842	−	0.784282i	0.369006	−	0.929427i	$-0.379698\pi$
$311$	422.468	−	243.912i	1.35842	−	0.784282i	0.989410	+	0.145144i	$0.0463647\pi$
$312$		0			0
$313$	94.9915	+	54.8433i	0.303487	+	0.175218i	0.644008	−	0.765018i	$-0.277270\pi$
$313$	94.9915	+	54.8433i	0.303487	+	0.175218i	−0.340521	+	0.940237i	$0.610604\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	181.784	−	314.859i	0.573452	−	0.993247i	−0.422756	−	0.906243i	$-0.638937\pi$
$317$	181.784	−	314.859i	0.573452	−	0.993247i	0.996208	−	0.0870039i	$-0.0277293\pi$
$318$		0			0
$319$	−361.111	−	625.463i	−1.13201	−	1.96070i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	132.815			0.411191
$324$		0			0
$325$	490.408	−	301.478i	1.50895	−	0.927625i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−99.0505	+	57.1868i	−0.301065	+	0.173820i
$330$		0			0
$331$	−150.111	+	260.001i	−0.453509	+	0.785501i	−0.998601	−	0.0528755i	$-0.983161\pi$
$331$	−150.111	+	260.001i	−0.453509	+	0.785501i	0.545092	+	0.838376i	$0.316495\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.7023	−	29.7071i	−0.0528428	−	0.0886779i
$336$		0			0
$337$	323.334	−	186.677i	0.959447	−	0.553937i	0.0634441	−	0.997985i	$-0.479792\pi$
$337$	323.334	−	186.677i	0.959447	−	0.553937i	0.896003	+	0.444049i	$0.146458\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			859.610i			2.52085i
$342$		0			0
$343$			252.112i			0.735020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.72213	−	15.1072i	−0.0251358	−	0.0435365i	0.853184	−	0.521610i	$-0.174668\pi$
$347$	−8.72213	−	15.1072i	−0.0251358	−	0.0435365i	−0.878320	+	0.478074i	$0.841335\pi$
$348$		0			0
$349$	−117.075	+	202.779i	−0.335457	+	0.581029i	−0.983573	−	0.180513i	$-0.942224\pi$
$349$	−117.075	+	202.779i	−0.335457	+	0.581029i	0.648115	+	0.761542i	$0.275558\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−190.642	+	330.202i	−0.540063	+	0.935416i	0.458837	+	0.888520i	$0.348266\pi$
$353$	−190.642	+	330.202i	−0.540063	+	0.935416i	−0.998900	+	0.0468954i	$0.985067\pi$
$354$		0			0
$355$	407.896	−	5.62840i	1.14900	−	0.0158546i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			222.024i			0.618451i	0.950989	+	0.309226i	$0.100070\pi$
$359$			222.024i			0.618451i	−0.950989	+	0.309226i	$0.899930\pi$
$360$		0			0
$361$	176.556			0.489074
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−530.368	+	7.31834i	−1.45306	+	0.0200502i
$366$		0			0
$367$	389.023	+	224.602i	1.06001	+	0.611995i	0.925434	−	0.378908i	$-0.123700\pi$
$367$	389.023	+	224.602i	1.06001	+	0.611995i	0.134573	+	0.990904i	$0.457034\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−221.205	−	127.713i	−0.596239	−	0.344239i
$372$		0			0
$373$	−278.932	+	161.041i	−0.747806	+	0.431746i	−0.824901	−	0.565278i	$-0.808769\pi$
$373$	−278.932	+	161.041i	−0.747806	+	0.431746i	0.0770948	+	0.997024i	$0.475436\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−916.446			−2.43089
$378$		0			0
$379$	−216.074			−0.570115			−0.285058	−	0.958510i	$-0.592013\pi$
$379$	−216.074			−0.570115			−0.285058	+	0.958510i	$0.592013\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	186.988	+	323.873i	0.488220	+	0.845622i	0.999908	−	0.0135493i	$-0.00431302\pi$
$383$	186.988	+	323.873i	0.488220	+	0.845622i	−0.511688	+	0.859171i	$0.670980\pi$
$384$		0			0
$385$	−217.890	+	129.840i	−0.565948	+	0.337246i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	193.217	+	111.554i	0.496702	+	0.286771i	0.727351	−	0.686266i	$-0.240751\pi$
$389$	193.217	+	111.554i	0.496702	+	0.286771i	−0.230648	+	0.973037i	$0.574085\pi$
$390$		0			0
$391$	−0.777873	−	1.34731i	−0.00198944	−	0.00344582i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−277.370	−	155.077i	−0.702204	−	0.392600i
$396$		0			0
$397$		−	610.535i		−	1.53787i	−0.639325	−	0.768936i	$-0.720786\pi$
$397$		−	610.535i		−	1.53787i	0.639325	−	0.768936i	$-0.279214\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−356.319	+	205.721i	−0.888575	+	0.513019i	−0.873476	−	0.486867i	$-0.838140\pi$
$401$	−356.319	+	205.721i	−0.888575	+	0.513019i	−0.0150992	+	0.999886i	$0.504806\pi$
$402$		0			0
$403$	944.643	+	545.390i	2.34403	+	1.35333i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−316.396	+	548.013i	−0.777385	+	1.34647i
$408$		0			0
$409$	−295.315	−	511.500i	−0.722041	−	1.25061i	−0.960180	−	0.279381i	$-0.909871\pi$
$409$	−295.315	−	511.500i	−0.722041	−	1.25061i	0.238139	−	0.971231i	$-0.423462\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−220.396			−0.533646
$414$		0			0
$415$	1.24014	+	0.693359i	0.00298829	+	0.00167074i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	207.462	−	119.778i	0.495135	−	0.285867i	−0.231567	−	0.972819i	$-0.574385\pi$
$419$	207.462	−	119.778i	0.495135	−	0.285867i	0.726702	+	0.686952i	$0.241052\pi$
$420$		0			0
$421$	−296.704	+	513.907i	−0.704760	+	1.22068i	0.262018	+	0.965063i	$0.415612\pi$
$421$	−296.704	+	513.907i	−0.704760	+	1.22068i	−0.966778	+	0.255618i	$0.917721\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−143.156	+	3.95146i	−0.336837	+	0.00929754i
$426$		0			0
$427$	75.4983	−	43.5890i	0.176811	−	0.102082i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	436.566i		−	1.01291i	−0.862265	−	0.506457i	$-0.830955\pi$
$431$		−	436.566i		−	1.01291i	0.862265	−	0.506457i	$-0.169045\pi$
$432$		0			0
$433$			231.736i			0.535187i	0.963532	+	0.267593i	$0.0862284\pi$
$433$			231.736i			0.535187i	−0.963532	+	0.267593i	$0.913772\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.14837	−	5.45314i	−0.00720451	−	0.0124786i
$438$		0			0
$439$	209.611	−	363.058i	0.477475	−	0.827011i	−0.522192	−	0.852828i	$-0.674885\pi$
$439$	209.611	−	363.058i	0.477475	−	0.827011i	0.999667	+	0.0258173i	$0.00821881\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−223.260	+	386.697i	−0.503973	+	0.872906i	0.496017	+	0.868313i	$0.334795\pi$
$443$	−223.260	+	386.697i	−0.503973	+	0.872906i	−0.999989	+	0.00459325i	$0.998538\pi$
$444$		0			0
$445$	142.591	−	1.96756i	0.320430	−	0.00442149i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			495.776i			1.10418i	0.833785	+	0.552089i	$0.186169\pi$
$449$			495.776i			1.10418i	−0.833785	+	0.552089i	$0.813831\pi$
$450$		0			0
$451$	240.741			0.533794
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.44069	+	321.822i	0.00975976	+	0.707301i
$456$		0			0
$457$	−591.810	−	341.681i	−1.29499	−	0.747662i	−0.315454	−	0.948941i	$-0.602157\pi$
$457$	−591.810	−	341.681i	−1.29499	−	0.747662i	−0.979534	+	0.201279i	$0.935490\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−346.521	−	200.064i	−0.751672	−	0.433978i	0.0746257	−	0.997212i	$-0.476224\pi$
$461$	−346.521	−	200.064i	−0.751672	−	0.433978i	−0.826298	+	0.563234i	$0.809557\pi$
$462$		0			0
$463$	−273.892	+	158.132i	−0.591559	+	0.341537i	−0.765714	−	0.643181i	$-0.777614\pi$
$463$	−273.892	+	158.132i	−0.591559	+	0.341537i	0.174155	+	0.984718i	$0.444281\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	419.741			0.898803			0.449401	−	0.893330i	$-0.351637\pi$
$467$	419.741			0.898803			0.449401	+	0.893330i	$0.351637\pi$
$468$		0			0
$469$	19.3345			0.0412249
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−423.865	−	734.156i	−0.896121	−	1.55213i
$474$		0			0
$475$	−579.411	+	15.9932i	−1.21981	+	0.0336698i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	731.766	+	422.485i	1.52769	+	0.882015i	0.999458	+	0.0329177i	$0.0104799\pi$
$479$	731.766	+	422.485i	1.52769	+	0.882015i	0.528237	+	0.849097i	$0.322853\pi$
$480$		0			0
$481$	401.482	+	695.387i	0.834682	+	1.44571i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−226.711	+	405.495i	−0.467446	+	0.836072i
$486$		0			0
$487$			546.384i			1.12194i	0.827837	+	0.560969i	$0.189571\pi$
$487$			546.384i			1.12194i	−0.827837	+	0.560969i	$0.810429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−560.713	+	323.728i	−1.14198	+	0.659324i	−0.946921	−	0.321467i	$-0.895824\pi$
$491$	−560.713	+	323.728i	−1.14198	+	0.659324i	−0.195062	+	0.980791i	$0.562491\pi$
$492$		0			0
$493$	197.443	+	113.994i	0.400494	+	0.231225i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−114.038	+	197.519i	−0.229452	+	0.397423i
$498$		0			0
$499$	−291.407	−	504.732i	−0.583983	−	1.01149i	−0.995001	−	0.0998608i	$-0.968160\pi$
$499$	−291.407	−	504.732i	−0.583983	−	1.01149i	0.411019	−	0.911627i	$-0.365173\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−752.520			−1.49606			−0.748032	−	0.663663i	$-0.769001\pi$
$503$	−752.520			−1.49606			−0.748032	+	0.663663i	$0.769001\pi$
$504$		0			0
$505$	113.870	−	203.667i	0.225484	−	0.403301i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	252.479	−	145.769i	0.496030	−	0.286383i	−0.231042	−	0.972944i	$-0.574214\pi$
$509$	252.479	−	145.769i	0.496030	−	0.286383i	0.727073	+	0.686560i	$0.240880\pi$
$510$		0			0
$511$	148.278	−	256.825i	0.290172	−	0.502593i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	28.6203	+	48.0290i	0.0555733	+	0.0932601i
$516$		0			0
$517$	−642.973	+	371.221i	−1.24366	+	0.718028i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	137.690i		−	0.264280i	−0.991231	−	0.132140i	$-0.957815\pi$
$521$		−	137.690i		−	0.264280i	0.991231	−	0.132140i	$-0.0421848\pi$
$522$		0			0
$523$			330.987i			0.632862i	0.948616	+	0.316431i	$0.102485\pi$
$523$			330.987i			0.632862i	−0.948616	+	0.316431i	$0.897515\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−135.679	−	235.003i	−0.257455	−	0.445926i
$528$		0			0
$529$	264.463	−	458.064i	0.499930	−	0.865905i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	152.741	−	264.555i	0.286568	−	0.496351i
$534$		0			0
$535$	10.1615	+	736.414i	0.0189934	+	1.37647i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			747.370i			1.38659i
$540$		0			0
$541$	556.593			1.02882			0.514412	−	0.857543i	$-0.328010\pi$
$541$	556.593			1.02882			0.514412	+	0.857543i	$0.328010\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.38570	−	607.721i	−0.0153866	−	1.11508i
$546$		0			0
$547$	−402.849	−	232.585i	−0.736470	−	0.425201i	0.0843144	−	0.996439i	$-0.473130\pi$
$547$	−402.849	−	232.585i	−0.736470	−	0.425201i	−0.820784	+	0.571238i	$0.806463\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	799.135	+	461.381i	1.45034	+	0.837351i
$552$		0			0
$553$	153.866	−	88.8347i	0.278239	−	0.160641i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1059.37			−1.90192			−0.950962	−	0.309306i	$-0.899903\pi$
$557$	−1059.37			−1.90192			−0.950962	+	0.309306i	$0.899903\pi$
$558$		0			0
$559$	−1075.70			−1.92434
$560$		0			0
$561$		0			0
$562$		0			0
$563$	193.080	+	334.424i	0.342949	+	0.594004i	0.984979	−	0.172675i	$-0.0552410\pi$
$563$	193.080	+	334.424i	0.342949	+	0.594004i	−0.642030	+	0.766679i	$0.721908\pi$
$564$		0			0
$565$	419.340	−	249.883i	0.742195	−	0.442271i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	594.147	+	343.031i	1.04420	+	0.602866i	0.921019	−	0.389518i	$-0.127359\pi$
$569$	594.147	+	343.031i	1.04420	+	0.602866i	0.123176	+	0.992385i	$0.460692\pi$
$570$		0			0
$571$	410.816	+	711.554i	0.719467	+	1.24615i	0.961211	+	0.275813i	$0.0889471\pi$
$571$	410.816	+	711.554i	0.719467	+	1.24615i	−0.241744	+	0.970340i	$0.577720\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.55575	+	5.78406i	0.00618390	+	0.0100592i
$576$		0			0
$577$		−	183.840i		−	0.318613i	−0.987229	−	0.159306i	$-0.949074\pi$
$577$		−	183.840i		−	0.318613i	0.987229	−	0.159306i	$-0.0509258\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.687945	+	0.397185i	−0.00118407	+	0.000683623i
$582$		0			0
$583$	−1435.92	−	829.029i	−2.46299	−	1.42201i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−160.203	+	277.480i	−0.272919	+	0.472709i	−0.969608	−	0.244664i	$-0.921322\pi$
$587$	−160.203	+	277.480i	−0.272919	+	0.472709i	0.696689	+	0.717373i	$0.254656\pi$
$588$		0			0
$589$	−549.148	−	951.153i	−0.932340	−	1.61486i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1061.34			−1.78977			−0.894887	−	0.446292i	$-0.852744\pi$
$593$	−1061.34			−1.78977			−0.894887	+	0.446292i	$0.852744\pi$
$594$		0			0
$595$	39.0738	−	69.8872i	0.0656702	−	0.117457i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	846.155	−	488.528i	1.41261	−	0.815573i	0.416979	−	0.908916i	$-0.363089\pi$
$599$	846.155	−	488.528i	1.41261	−	0.815573i	0.995634	+	0.0933433i	$0.0297554\pi$
$600$		0			0
$601$	130.908	−	226.740i	0.217817	−	0.377271i	−0.736323	−	0.676630i	$-0.763440\pi$
$601$	130.908	−	226.740i	0.217817	−	0.377271i	0.954140	+	0.299359i	$0.0967730\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−894.681	+	533.137i	−1.47881	+	0.881218i
$606$		0			0
$607$	579.203	−	334.403i	0.954206	−	0.550911i	0.0598212	−	0.998209i	$-0.480947\pi$
$607$	579.203	−	334.403i	0.954206	−	0.550911i	0.894385	+	0.447298i	$0.147614\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			942.101i			1.54190i
$612$		0			0
$613$		−	220.119i		−	0.359086i	−0.983750	−	0.179543i	$-0.942538\pi$
$613$		−	220.119i		−	0.359086i	0.983750	−	0.179543i	$-0.0574618\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−160.531	−	278.049i	−0.260181	−	0.450646i	0.706109	−	0.708103i	$-0.250449\pi$
$617$	−160.531	−	278.049i	−0.260181	−	0.450646i	−0.966290	+	0.257457i	$0.917115\pi$
$618$		0			0
$619$	445.593	−	771.791i	0.719860	−	1.24683i	−0.241195	−	0.970477i	$-0.577539\pi$
$619$	445.593	−	771.791i	0.719860	−	1.24683i	0.961055	−	0.276358i	$-0.0891275\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−39.8651	+	69.0483i	−0.0639889	+	0.110832i
$624$		0			0
$625$	624.048	−	34.4768i	0.998477	−	0.0551629i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	199.757i		−	0.317578i
$630$		0			0
$631$	−583.669			−0.924990			−0.462495	−	0.886622i	$-0.653046\pi$
$631$	−583.669			−0.924990			−0.462495	+	0.886622i	$0.653046\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−443.198	+	6.11552i	−0.697950	+	0.00963074i
$636$		0			0
$637$	821.300	+	474.178i	1.28932	+	0.744392i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−603.006	−	348.146i	−0.940727	−	0.543129i	−0.0505391	−	0.998722i	$-0.516094\pi$
$641$	−603.006	−	348.146i	−0.940727	−	0.543129i	−0.890188	+	0.455593i	$0.849427\pi$
$642$		0			0
$643$	694.262	−	400.832i	1.07972	−	0.623378i	0.148901	−	0.988852i	$-0.452426\pi$
$643$	694.262	−	400.832i	1.07972	−	0.623378i	0.930822	+	0.365474i	$0.119093\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	216.221			0.334191			0.167095	−	0.985941i	$-0.446561\pi$
$647$	216.221			0.334191			0.167095	+	0.985941i	$0.446561\pi$
$648$		0			0
$649$	−1430.67			−2.20442
$650$		0			0
$651$		0			0
$652$		0			0
$653$	219.401	+	380.014i	0.335989	+	0.581951i	0.983674	−	0.179958i	$-0.0575960\pi$
$653$	219.401	+	380.014i	0.335989	+	0.581951i	−0.647685	+	0.761908i	$0.724263\pi$
$654$		0			0
$655$	230.288	+	386.456i	0.351584	+	0.590009i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−317.063	−	183.056i	−0.481127	−	0.277779i	0.239759	−	0.970832i	$-0.422932\pi$
$659$	−317.063	−	183.056i	−0.481127	−	0.277779i	−0.720886	+	0.693054i	$0.756265\pi$
$660$		0			0
$661$	−79.0377	−	136.897i	−0.119573	−	0.207106i	0.800026	−	0.599966i	$-0.204819\pi$
$661$	−79.0377	−	136.897i	−0.119573	−	0.207106i	−0.919599	+	0.392859i	$0.871486\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	158.148	−	282.862i	0.237816	−	0.425357i
$666$		0			0
$667$		−	10.8089i		−	0.0162052i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	490.087	−	282.952i	0.730384	−	0.421687i
$672$		0			0
$673$	−561.683	−	324.288i	−0.834595	−	0.481854i	0.0208282	−	0.999783i	$-0.493370\pi$
$673$	−561.683	−	324.288i	−0.834595	−	0.481854i	−0.855423	+	0.517929i	$0.826703\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	101.173	−	175.236i	0.149443	−	0.258842i	−0.781579	−	0.623806i	$-0.785585\pi$
$677$	101.173	−	175.236i	0.149443	−	0.258842i	0.931022	+	0.364964i	$0.118919\pi$
$678$		0			0
$679$	−129.870	−	224.941i	−0.191266	−	0.331283i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	173.829			0.254508			0.127254	−	0.991870i	$-0.459384\pi$
$683$	173.829			0.254508			0.127254	+	0.991870i	$0.459384\pi$
$684$		0			0
$685$	1003.45	+	561.024i	1.46488	+	0.819013i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1822.07	+	1051.97i	−2.64452	+	1.52681i
$690$		0			0
$691$	423.371	−	733.301i	0.612694	−	1.06122i	−0.378091	−	0.925769i	$-0.623419\pi$
$691$	423.371	−	733.301i	0.612694	−	1.06122i	0.990784	−	0.135448i	$-0.0432474\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	247.531	−	147.503i	0.356160	−	0.212235i
$696$		0			0
$697$	−65.8145	+	37.9980i	−0.0944254	+	0.0545165i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	72.2614i		−	0.103083i	−0.998671	−	0.0515416i	$-0.983587\pi$
$701$		−	72.2614i		−	0.103083i	0.998671	−	0.0515416i	$-0.0164135\pi$
$702$		0			0
$703$		−	808.497i		−	1.15007i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	65.2293	+	112.980i	0.0922621	+	0.159803i
$708$		0			0
$709$	−143.779	+	249.032i	−0.202791	+	0.351244i	−0.949427	−	0.313989i	$-0.898334\pi$
$709$	−143.779	+	249.032i	−0.202791	+	0.351244i	0.746636	+	0.665233i	$0.231668\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.43253	+	11.1415i	−0.00902178	+	0.0156262i
$714$		0			0
$715$	28.8262	+	2089.06i	0.0403163	+	2.92177i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1015.27i		−	1.41205i	−0.708185	−	0.706027i	$-0.750486\pi$
$719$		−	1015.27i		−	1.41205i	0.708185	−	0.706027i	$-0.249514\pi$
$720$		0			0
$721$	−31.2590			−0.0433551
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−875.083	−	473.529i	−1.20701	−	0.653143i
$726$		0			0
$727$	−269.696	−	155.709i	−0.370971	−	0.214180i	0.302911	−	0.953019i	$-0.402041\pi$
$727$	−269.696	−	155.709i	−0.370971	−	0.214180i	−0.673883	+	0.738838i	$0.735375\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	231.755	+	133.804i	0.317038	+	0.183042i
$732$		0			0
$733$	278.035	−	160.523i	0.379310	−	0.218995i	−0.298208	−	0.954501i	$-0.596389\pi$
$733$	278.035	−	160.523i	0.379310	−	0.218995i	0.677518	+	0.735506i	$0.263055\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	125.507			0.170295
$738$		0			0
$739$	−657.185			−0.889290			−0.444645	−	0.895707i	$-0.646670\pi$
$739$	−657.185			−0.889290			−0.444645	+	0.895707i	$0.646670\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	512.482	+	887.645i	0.689747	+	1.19468i	0.971920	+	0.235313i	$0.0756116\pi$
$743$	512.482	+	887.645i	0.689747	+	1.19468i	−0.282173	+	0.959364i	$0.591055\pi$
$744$		0			0
$745$	−28.4967	−	47.8216i	−0.0382506	−	0.0641901i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−356.600	−	205.883i	−0.476101	−	0.274877i
$750$		0			0
$751$	31.5738	+	54.6874i	0.0420423	+	0.0728194i	0.886281	−	0.463148i	$-0.153280\pi$
$751$	31.5738	+	54.6874i	0.0420423	+	0.0728194i	−0.844239	+	0.535968i	$0.819947\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	96.8210	+	54.1324i	0.128240	+	0.0716986i
$756$		0			0
$757$			1452.21i			1.91837i	0.282781	+	0.959185i	$0.408743\pi$
$757$			1452.21i			1.91837i	−0.282781	+	0.959185i	$0.591257\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	422.468	−	243.912i	0.555148	−	0.320515i	−0.196048	−	0.980594i	$-0.562811\pi$
$761$	422.468	−	243.912i	0.555148	−	0.320515i	0.751196	+	0.660079i	$0.229477\pi$
$762$		0			0
$763$	294.282	+	169.904i	0.385691	+	0.222679i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−907.705	+	1572.19i	−1.18345	+	2.04979i
$768$		0			0
$769$	−338.574	−	586.427i	−0.440278	−	0.762584i	0.557432	−	0.830223i	$-0.311787\pi$
$769$	−338.574	−	586.427i	−0.440278	−	0.762584i	−0.997710	+	0.0676389i	$0.978453\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−949.926			−1.22888			−0.614441	−	0.788963i	$-0.710619\pi$
$773$	−949.926			−1.22888			−0.614441	+	0.788963i	$0.710619\pi$
$774$		0			0
$775$	620.204	+	1008.87i	0.800263	+	1.30177i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−266.378	+	153.794i	−0.341949	+	0.197424i
$780$		0			0
$781$	−740.261	+	1282.17i	−0.947837	+	1.64170i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	576.752	+	967.874i	0.734717	+	1.23296i
$786$		0			0
$787$	991.018	−	572.164i	1.25923	−	0.727020i	0.286309	−	0.958137i	$-0.407572\pi$
$787$	991.018	−	572.164i	1.25923	−	0.727020i	0.972926	+	0.231118i	$0.0742383\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			272.922i			0.345034i
$792$		0			0
$793$		−	718.089i		−	0.905535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−325.550	−	563.870i	−0.408470	−	0.707490i	0.586249	−	0.810131i	$-0.300604\pi$
$797$	−325.550	−	563.870i	−0.408470	−	0.707490i	−0.994718	+	0.102641i	$0.967271\pi$
$798$		0			0
$799$	117.185	−	202.971i	0.146665	−	0.254031i
$800$		0			0
$801$		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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0689865 + 4.99952i) q^{5} +(-2.42096 - 1.39774i) q^{7} +O(q^{10})\)	\(q+(0.0689865 + 4.99952i) q^{5} +(-2.42096 - 1.39774i) q^{7} +(-15.7154 - 9.07327i) q^{11} +(-19.9416 + 11.5133i) q^{13} +5.72842 q^{17} +23.1852 q^{19} +(-0.135792 - 0.235199i) q^{23} +(-24.9905 + 0.689799i) q^{25} +(34.4674 + 19.8997i) q^{29} +(-23.6852 - 41.0241i) q^{31} +(6.82104 - 12.2001i) q^{35} -34.8712i q^{37} +(-11.4891 + 6.63325i) q^{41} +(40.4571 + 23.3579i) q^{43} +(20.4568 - 35.4323i) q^{47} +(-20.5926 - 35.6675i) q^{49} +91.3705 q^{53} +(44.2779 - 79.1953i) q^{55} +(68.2773 - 39.4199i) q^{59} +(-15.5926 + 27.0072i) q^{61} +(-58.9366 - 98.9042i) q^{65} +(-5.98970 + 3.45816i) q^{67} -81.5870i q^{71} +106.084i q^{73} +(25.3642 + 43.9321i) q^{77} +(-31.7779 + 55.0409i) q^{79} +(0.142081 - 0.246091i) q^{83} +(0.395183 + 28.6394i) q^{85} -28.5210i q^{89} +64.3705 q^{91} +(1.59947 + 115.915i) q^{95} +(80.4657 + 46.4569i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{5}+O(q^{10}) \) 8 * q + 3 * q^5	\( 8 q + 3 q^{5} - 12 q^{17} + 12 q^{19} - 30 q^{23} - 9 q^{25} - 16 q^{31} - 90 q^{35} + 48 q^{47} - 78 q^{49} + 384 q^{53} + 94 q^{55} - 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} - 288 q^{83} + 100 q^{85} + 168 q^{91} + 318 q^{95}+O(q^{100}) \) 8 * q + 3 * q^5 - 12 * q^17 + 12 * q^19 - 30 * q^23 - 9 * q^25 - 16 * q^31 - 90 * q^35 + 48 * q^47 - 78 * q^49 + 384 * q^53 + 94 * q^55 - 38 * q^61 + 138 * q^65 + 174 * q^77 + 6 * q^79 - 288 * q^83 + 100 * q^85 + 168 * q^91 + 318 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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