LMFDB - Embedded newform 1300.2.bs.b.293.1

Newspace parameters

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.bs (of order $12$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			293.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1300.293
Dual form			1300.2.bs.b.457.1

$q$-expansion

$f(q)$	$=$	$q+(0.500000 - 0.133975i) q^{3} +(2.13397 - 1.23205i) q^{7} +(-2.36603 + 1.36603i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.133975i) q^{3} +(2.13397 - 1.23205i) q^{7} +(-2.36603 + 1.36603i) q^{9} +(-1.13397 - 4.23205i) q^{11} +(2.00000 - 3.00000i) q^{13} +(0.232051 - 0.866025i) q^{17} +(-2.86603 - 0.767949i) q^{19} +(0.901924 - 0.901924i) q^{21} +(0.0358984 + 0.133975i) q^{23} +(-2.09808 + 2.09808i) q^{27} +(-1.50000 - 0.866025i) q^{29} +(-5.19615 - 5.19615i) q^{31} +(-1.13397 - 1.96410i) q^{33} +(-1.33013 - 0.767949i) q^{37} +(0.598076 - 1.76795i) q^{39} +(9.33013 - 2.50000i) q^{41} +(5.96410 + 1.59808i) q^{43} -10.9282i q^{47} +(-0.464102 + 0.803848i) q^{49} -0.464102i q^{51} +(-2.46410 - 2.46410i) q^{53} -1.53590 q^{57} +(2.33013 - 8.69615i) q^{59} +(4.50000 + 7.79423i) q^{61} +(-3.36603 + 5.83013i) q^{63} +(-6.13397 + 10.6244i) q^{67} +(0.0358984 + 0.0621778i) q^{69} +(0.598076 - 2.23205i) q^{71} +14.9282 q^{73} +(-7.63397 - 7.63397i) q^{77} -0.535898i q^{79} +(3.33013 - 5.76795i) q^{81} -2.92820i q^{83} +(-0.866025 - 0.232051i) q^{87} +(-14.7942 + 3.96410i) q^{89} +(0.571797 - 8.86603i) q^{91} +(-3.29423 - 1.90192i) q^{93} +(-3.86603 - 6.69615i) q^{97} +(8.46410 + 8.46410i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9	$ 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 8 q^{19} + 14 q^{21} + 14 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 20 q^{41} + 10 q^{43} + 12 q^{49} + 4 q^{53} - 20 q^{57} - 8 q^{59} + 18 q^{61} - 10 q^{63} - 28 q^{67} + 14 q^{69} - 8 q^{71} + 32 q^{73} - 34 q^{77} - 4 q^{81} - 28 q^{89} + 30 q^{91} + 18 q^{93} - 12 q^{97} + 20 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 8 * q^19 + 14 * q^21 + 14 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 20 * q^41 + 10 * q^43 + 12 * q^49 + 4 * q^53 - 20 * q^57 - 8 * q^59 + 18 * q^61 - 10 * q^63 - 28 * q^67 + 14 * q^69 - 8 * q^71 + 32 * q^73 - 34 * q^77 - 4 * q^81 - 28 * q^89 + 30 * q^91 + 18 * q^93 - 12 * q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$e\left(\frac{11}{12}\right)$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	0.500000	−	0.133975i	0.288675	−	0.0773503i	−0.111576	−	0.993756i	$-0.535590\pi$
$3$	0.500000	−	0.133975i	0.288675	−	0.0773503i	0.400251	+	0.916406i	$0.368923\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.13397	−	1.23205i	0.806567	−	0.465671i	−0.0391956	−	0.999232i	$-0.512480\pi$
$7$	2.13397	−	1.23205i	0.806567	−	0.465671i	0.845762	+	0.533560i	$0.179146\pi$
$8$		0			0
$9$	−2.36603	+	1.36603i	−0.788675	+	0.455342i
$10$		0			0
$11$	−1.13397	−	4.23205i	−0.341906	−	1.27601i	−0.896185	−	0.443680i	$-0.853673\pi$
$11$	−1.13397	−	4.23205i	−0.341906	−	1.27601i	0.554279	−	0.832331i	$-0.312994\pi$
$12$		0			0
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$13$	2.00000	−	3.00000i	0.554700	−	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.232051	−	0.866025i	0.0562806	−	0.210042i	−0.932059	−	0.362306i	$-0.881990\pi$
$17$	0.232051	−	0.866025i	0.0562806	−	0.210042i	0.988340	+	0.152264i	$0.0486563\pi$
$18$		0			0
$19$	−2.86603	−	0.767949i	−0.657511	−	0.176180i	−0.0853887	−	0.996348i	$-0.527213\pi$
$19$	−2.86603	−	0.767949i	−0.657511	−	0.176180i	−0.572123	+	0.820168i	$0.693880\pi$
$20$		0			0
$21$	0.901924	−	0.901924i	0.196816	−	0.196816i
$22$		0			0
$23$	0.0358984	+	0.133975i	0.00748533	+	0.0279356i	0.969567	−	0.244824i	$-0.0787302\pi$
$23$	0.0358984	+	0.133975i	0.00748533	+	0.0279356i	−0.962082	+	0.272760i	$0.912064\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.09808	+	2.09808i	−0.403775	+	0.403775i
$28$		0			0
$29$	−1.50000	−	0.866025i	−0.278543	−	0.160817i	0.354221	−	0.935162i	$-0.384746\pi$
$29$	−1.50000	−	0.866025i	−0.278543	−	0.160817i	−0.632764	+	0.774345i	$0.718080\pi$
$30$		0			0
$31$	−5.19615	−	5.19615i	−0.933257	−	0.933257i	0.0646514	−	0.997908i	$-0.479406\pi$
$31$	−5.19615	−	5.19615i	−0.933257	−	0.933257i	−0.997908	+	0.0646514i	$0.979406\pi$
$32$		0			0
$33$	−1.13397	−	1.96410i	−0.197400	−	0.341906i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.33013	−	0.767949i	−0.218672	−	0.126250i	0.386663	−	0.922221i	$-0.373628\pi$
$37$	−1.33013	−	0.767949i	−0.218672	−	0.126250i	−0.605335	+	0.795971i	$0.706961\pi$
$38$		0			0
$39$	0.598076	−	1.76795i	0.0957688	−	0.283098i
$40$		0			0
$41$	9.33013	−	2.50000i	1.45712	−	0.390434i	0.558627	−	0.829419i	$-0.311329\pi$
$41$	9.33013	−	2.50000i	1.45712	−	0.390434i	0.898494	+	0.438985i	$0.144662\pi$
$42$		0			0
$43$	5.96410	+	1.59808i	0.909517	+	0.243704i	0.683099	−	0.730326i	$-0.260632\pi$
$43$	5.96410	+	1.59808i	0.909517	+	0.243704i	0.226418	+	0.974030i	$0.427298\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.9282i		−	1.59404i	−0.603951	−	0.797021i	$-0.706408\pi$
$47$		−	10.9282i		−	1.59404i	0.603951	−	0.797021i	$-0.293592\pi$
$48$		0			0
$49$	−0.464102	+	0.803848i	−0.0663002	+	0.114835i
$50$		0			0
$51$		−	0.464102i		−	0.0649872i
$52$		0			0
$53$	−2.46410	−	2.46410i	−0.338470	−	0.338470i	0.517321	−	0.855791i	$-0.326929\pi$
$53$	−2.46410	−	2.46410i	−0.338470	−	0.338470i	−0.855791	+	0.517321i	$0.826929\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.53590			−0.203435
$58$		0			0
$59$	2.33013	−	8.69615i	0.303357	−	1.13214i	−0.630994	−	0.775788i	$-0.717353\pi$
$59$	2.33013	−	8.69615i	0.303357	−	1.13214i	0.934351	−	0.356355i	$-0.115981\pi$
$60$		0			0
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	0.995914	+	0.0903080i	$0.0287851\pi$
$61$	4.50000	+	7.79423i	0.576166	+	0.997949i	−0.419748	+	0.907641i	$0.637882\pi$
$62$		0			0
$63$	−3.36603	+	5.83013i	−0.424079	+	0.734527i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.13397	+	10.6244i	−0.749384	+	1.29797i	0.198734	+	0.980053i	$0.436317\pi$
$67$	−6.13397	+	10.6244i	−0.749384	+	1.29797i	−0.948118	+	0.317918i	$0.897016\pi$
$68$		0			0
$69$	0.0358984	+	0.0621778i	0.00432166	+	0.00748533i
$70$		0			0
$71$	0.598076	−	2.23205i	0.0709786	−	0.264896i	−0.921313	−	0.388822i	$-0.872882\pi$
$71$	0.598076	−	2.23205i	0.0709786	−	0.264896i	0.992291	+	0.123927i	$0.0395487\pi$
$72$		0			0
$73$	14.9282			1.74721			0.873607	−	0.486632i	$-0.161775\pi$
$73$	14.9282			1.74721			0.873607	+	0.486632i	$0.161775\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.63397	−	7.63397i	−0.869972	−	0.869972i
$78$		0			0
$79$		−	0.535898i		−	0.0602933i	−0.999545	−	0.0301466i	$-0.990403\pi$
$79$		−	0.535898i		−	0.0602933i	0.999545	−	0.0301466i	$-0.00959743\pi$
$80$		0			0
$81$	3.33013	−	5.76795i	0.370014	−	0.640883i
$82$		0			0
$83$		−	2.92820i		−	0.321412i	−0.987002	−	0.160706i	$-0.948623\pi$
$83$		−	2.92820i		−	0.321412i	0.987002	−	0.160706i	$-0.0513771\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.866025	−	0.232051i	−0.0928477	−	0.0248785i
$88$		0			0
$89$	−14.7942	+	3.96410i	−1.56819	+	0.420194i	−0.935243	−	0.354005i	$-0.884819\pi$
$89$	−14.7942	+	3.96410i	−1.56819	+	0.420194i	−0.632942	+	0.774199i	$0.718153\pi$
$90$		0			0
$91$	0.571797	−	8.86603i	0.0599406	−	0.929412i
$92$		0			0
$93$	−3.29423	−	1.90192i	−0.341596	−	0.197220i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.86603	−	6.69615i	−0.392535	−	0.679891i	0.600248	−	0.799814i	$-0.295069\pi$
$97$	−3.86603	−	6.69615i	−0.392535	−	0.679891i	−0.992783	+	0.119923i	$0.961735\pi$
$98$		0			0
$99$	8.46410	+	8.46410i	0.850674	+	0.850674i
$100$		0			0
$101$	−14.8923	−	8.59808i	−1.48184	−	0.855541i	−0.482052	−	0.876143i	$-0.660108\pi$
$101$	−14.8923	−	8.59808i	−1.48184	−	0.855541i	−0.999788	+	0.0206021i	$0.993442\pi$
$102$		0			0
$103$	11.1962	−	11.1962i	1.10319	−	1.10319i	0.109166	−	0.994024i	$-0.465182\pi$
$103$	11.1962	−	11.1962i	1.10319	−	1.10319i	0.994024	−	0.109166i	$-0.0348181\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.96410	+	7.33013i	0.189877	+	0.708630i	0.993534	+	0.113537i	$0.0362181\pi$
$107$	1.96410	+	7.33013i	0.189877	+	0.708630i	−0.803657	+	0.595093i	$0.797115\pi$
$108$		0			0
$109$	−3.53590	+	3.53590i	−0.338678	+	0.338678i	−0.855869	−	0.517192i	$-0.826977\pi$
$109$	−3.53590	+	3.53590i	−0.338678	+	0.338678i	0.517192	+	0.855869i	$0.326977\pi$
$110$		0			0
$111$	−0.767949	−	0.205771i	−0.0728905	−	0.0195310i
$112$		0			0
$113$	−0.767949	+	2.86603i	−0.0722426	+	0.269613i	−0.992594	−	0.121480i	$-0.961236\pi$
$113$	−0.767949	+	2.86603i	−0.0722426	+	0.269613i	0.920351	+	0.391093i	$0.127903\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.633975	+	9.83013i	−0.0586110	+	0.908796i
$118$		0			0
$119$	−0.571797	−	2.13397i	−0.0524165	−	0.195621i
$120$		0			0
$121$	−7.09808	+	4.09808i	−0.645280	+	0.372552i
$122$		0			0
$123$	4.33013	−	2.50000i	0.390434	−	0.225417i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	19.8923	−	5.33013i	1.76516	−	0.472972i	0.777404	−	0.629001i	$-0.216536\pi$
$127$	19.8923	−	5.33013i	1.76516	−	0.472972i	0.987752	+	0.156029i	$0.0498694\pi$
$128$		0			0
$129$	3.19615			0.281406
$130$		0			0
$131$	5.85641			0.511677			0.255838	−	0.966720i	$-0.417649\pi$
$131$	5.85641			0.511677			0.255838	+	0.966720i	$0.417649\pi$
$132$		0			0
$133$	−7.06218	+	1.89230i	−0.612368	+	0.164084i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.4019	−	7.16025i	1.05957	−	0.611742i	0.134255	−	0.990947i	$-0.457136\pi$
$137$	12.4019	−	7.16025i	1.05957	−	0.611742i	0.925313	+	0.379205i	$0.123802\pi$
$138$		0			0
$139$	−4.50000	+	2.59808i	−0.381685	+	0.220366i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−4.50000	+	2.59808i	−0.381685	+	0.220366i	0.296866	+	0.954919i	$0.404058\pi$
$140$		0			0
$141$	−1.46410	−	5.46410i	−0.123300	−	0.460160i
$142$		0			0
$143$	−14.9641	−	5.06218i	−1.25136	−	0.423321i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.124356	+	0.464102i	−0.0102567	+	0.0382785i
$148$		0			0
$149$	−3.33013	−	0.892305i	−0.272815	−	0.0731005i	0.119818	−	0.992796i	$-0.461769\pi$
$149$	−3.33013	−	0.892305i	−0.272815	−	0.0731005i	−0.392633	+	0.919695i	$0.628436\pi$
$150$		0			0
$151$	−0.267949	+	0.267949i	−0.0218054	+	0.0218054i	−0.717925	−	0.696120i	$-0.754908\pi$
$151$	−0.267949	+	0.267949i	−0.0218054	+	0.0218054i	0.696120	+	0.717925i	$0.254908\pi$
$152$		0			0
$153$	0.633975	+	2.36603i	0.0512538	+	0.191282i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.46410	+	6.46410i	−0.515891	+	0.515891i	−0.916326	−	0.400434i	$-0.868859\pi$
$157$	−6.46410	+	6.46410i	−0.515891	+	0.515891i	0.400434	+	0.916326i	$0.368859\pi$
$158$		0			0
$159$	−1.56218	−	0.901924i	−0.123889	−	0.0715272i
$160$		0			0
$161$	0.241670	+	0.241670i	0.0190462	+	0.0190462i
$162$		0			0
$163$	1.59808	+	2.76795i	0.125171	+	0.216803i	0.921800	−	0.387666i	$-0.126719\pi$
$163$	1.59808	+	2.76795i	0.125171	+	0.216803i	−0.796629	+	0.604469i	$0.793385\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.9904	+	9.23205i	1.23737	+	0.714398i	0.968556	−	0.248794i	$-0.0800342\pi$
$167$	15.9904	+	9.23205i	1.23737	+	0.714398i	0.268816	+	0.963191i	$0.413368\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$	7.83013	−	2.09808i	0.598785	−	0.160444i
$172$		0			0
$173$	2.76795	+	0.741670i	0.210443	+	0.0563881i	0.362500	−	0.931984i	$-0.381923\pi$
$173$	2.76795	+	0.741670i	0.210443	+	0.0563881i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	4.66025i		−	0.350286i
$178$		0			0
$179$	6.96410	−	12.0622i	0.520521	−	0.901570i	−0.479194	−	0.877709i	$-0.659071\pi$
$179$	6.96410	−	12.0622i	0.520521	−	0.901570i	0.999715	−	0.0238604i	$-0.00759573\pi$
$180$		0			0
$181$			22.9282i			1.70424i	0.523347	+	0.852120i	$0.324683\pi$
$181$			22.9282i			1.70424i	−0.523347	+	0.852120i	$0.675317\pi$
$182$		0			0
$183$	3.29423	+	3.29423i	0.243516	+	0.243516i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.92820			−0.287259
$188$		0			0
$189$	−1.89230	+	7.06218i	−0.137645	+	0.513698i
$190$		0			0
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	0.998749	+	0.0500060i	$0.0159241\pi$
$191$	7.50000	+	12.9904i	0.542681	+	0.939951i	−0.456068	+	0.889945i	$0.650743\pi$
$192$		0			0
$193$	−12.7942	+	22.1603i	−0.920949	+	1.59513i	−0.122998	+	0.992407i	$0.539251\pi$
$193$	−12.7942	+	22.1603i	−0.920949	+	1.59513i	−0.797950	+	0.602723i	$0.794082\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.13397	+	3.69615i	−0.152039	+	0.263340i	−0.931977	−	0.362517i	$-0.881917\pi$
$197$	−2.13397	+	3.69615i	−0.152039	+	0.263340i	0.779938	+	0.625857i	$0.215251\pi$
$198$		0			0
$199$	9.42820	+	16.3301i	0.668348	+	1.15761i	0.978366	+	0.206881i	$0.0663314\pi$
$199$	9.42820	+	16.3301i	0.668348	+	1.15761i	−0.310018	+	0.950731i	$0.600335\pi$
$200$		0			0
$201$	−1.64359	+	6.13397i	−0.115930	+	0.432657i
$202$		0			0
$203$	−4.26795			−0.299551
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.267949	−	0.267949i	−0.0186238	−	0.0186238i
$208$		0			0
$209$			13.0000i			0.899229i
$210$		0			0
$211$	−2.96410	+	5.13397i	−0.204057	+	0.353437i	−0.949832	−	0.312761i	$-0.898746\pi$
$211$	−2.96410	+	5.13397i	−0.204057	+	0.353437i	0.745775	+	0.666198i	$0.232080\pi$
$212$		0			0
$213$		−	1.19615i		−	0.0819590i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−17.4904	−	4.68653i	−1.18732	−	0.318143i
$218$		0			0
$219$	7.46410	−	2.00000i	0.504377	−	0.135147i
$220$		0			0
$221$	−2.13397	−	2.42820i	−0.143547	−	0.163339i
$222$		0			0
$223$	1.33013	+	0.767949i	0.0890719	+	0.0514257i	0.543874	−	0.839167i	$-0.316957\pi$
$223$	1.33013	+	0.767949i	0.0890719	+	0.0514257i	−0.454802	+	0.890592i	$0.650290\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.59808	−	2.76795i	−0.106068	−	0.183715i	0.808106	−	0.589037i	$-0.200493\pi$
$227$	−1.59808	−	2.76795i	−0.106068	−	0.183715i	−0.914174	+	0.405322i	$0.867159\pi$
$228$		0			0
$229$	0.0717968	+	0.0717968i	0.00474446	+	0.00474446i	0.709475	−	0.704731i	$-0.248932\pi$
$229$	0.0717968	+	0.0717968i	0.00474446	+	0.00474446i	−0.704731	+	0.709475i	$0.748932\pi$
$230$		0			0
$231$	−4.83975	−	2.79423i	−0.318432	−	0.183847i
$232$		0			0
$233$	−12.8564	+	12.8564i	−0.842251	+	0.842251i	−0.989151	−	0.146900i	$-0.953070\pi$
$233$	−12.8564	+	12.8564i	−0.842251	+	0.842251i	0.146900	+	0.989151i	$0.453070\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.0717968	−	0.267949i	−0.00466370	−	0.0174052i
$238$		0			0
$239$	16.6603	−	16.6603i	1.07766	−	1.07766i	0.0809436	−	0.996719i	$-0.474207\pi$
$239$	16.6603	−	16.6603i	1.07766	−	1.07766i	0.996719	−	0.0809436i	$-0.0257934\pi$
$240$		0			0
$241$	11.3301	+	3.03590i	0.729838	+	0.195559i	0.604557	−	0.796562i	$-0.293350\pi$
$241$	11.3301	+	3.03590i	0.729838	+	0.195559i	0.125281	+	0.992121i	$0.460017\pi$
$242$		0			0
$243$	3.19615	−	11.9282i	0.205033	−	0.765195i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.03590	+	7.06218i	−0.511312	+	0.449356i
$248$		0			0
$249$	−0.392305	−	1.46410i	−0.0248613	−	0.0927837i
$250$		0			0
$251$	−15.3564	+	8.86603i	−0.969288	+	0.559619i	−0.899019	−	0.437910i	$-0.855719\pi$
$251$	−15.3564	+	8.86603i	−0.969288	+	0.559619i	−0.0702687	+	0.997528i	$0.522386\pi$
$252$		0			0
$253$	0.526279	−	0.303848i	0.0330869	−	0.0191027i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−21.6244	+	5.79423i	−1.34889	+	0.361434i	−0.859725	−	0.510757i	$-0.829365\pi$
$257$	−21.6244	+	5.79423i	−1.34889	+	0.361434i	−0.489165	+	0.872191i	$0.662698\pi$
$258$		0			0
$259$	−3.78461			−0.235164
$260$		0			0
$261$	4.73205			0.292907
$262$		0			0
$263$	−13.8923	+	3.72243i	−0.856636	+	0.229535i	−0.660300	−	0.751002i	$-0.729571\pi$
$263$	−13.8923	+	3.72243i	−0.856636	+	0.229535i	−0.196336	+	0.980537i	$0.562904\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−6.86603	+	3.96410i	−0.420194	+	0.242599i
$268$		0			0
$269$	15.8205	−	9.13397i	0.964593	−	0.556908i	0.0670097	−	0.997752i	$-0.478654\pi$
$269$	15.8205	−	9.13397i	0.964593	−	0.556908i	0.897584	+	0.440844i	$0.145321\pi$
$270$		0			0
$271$	4.33013	+	16.1603i	0.263036	+	0.981666i	0.963441	+	0.267920i	$0.0863362\pi$
$271$	4.33013	+	16.1603i	0.263036	+	0.981666i	−0.700405	+	0.713746i	$0.746997\pi$
$272$		0			0
$273$	−0.901924	−	4.50962i	−0.0545869	−	0.272935i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.76795	−	10.3301i	0.166310	−	0.620677i	−0.831560	−	0.555436i	$-0.812552\pi$
$277$	2.76795	−	10.3301i	0.166310	−	0.620677i	0.997869	−	0.0652416i	$-0.0207818\pi$
$278$		0			0
$279$	19.3923	+	5.19615i	1.16099	+	0.311086i
$280$		0			0
$281$	−16.4641	+	16.4641i	−0.982166	+	0.982166i	−0.999844	−	0.0176778i	$-0.994373\pi$
$281$	−16.4641	+	16.4641i	−0.982166	+	0.982166i	0.0176778	+	0.999844i	$0.494373\pi$
$282$		0			0
$283$	−7.96410	−	29.7224i	−0.473417	−	1.76682i	−0.627354	−	0.778735i	$-0.715862\pi$
$283$	−7.96410	−	29.7224i	−0.473417	−	1.76682i	0.153937	−	0.988081i	$-0.450805\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	16.8301	−	16.8301i	0.993451	−	0.993451i
$288$		0			0
$289$	14.0263	+	8.09808i	0.825075	+	0.476357i
$290$		0			0
$291$	−2.83013	−	2.83013i	−0.165905	−	0.165905i
$292$		0			0
$293$	10.2583	+	17.7679i	0.599298	+	1.03801i	0.992925	+	0.118744i	$0.0378869\pi$
$293$	10.2583	+	17.7679i	0.599298	+	1.03801i	−0.393627	+	0.919270i	$0.628780\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	11.2583	+	6.50000i	0.653275	+	0.377168i
$298$		0			0
$299$	0.473721	+	0.160254i	0.0273960	+	0.00926773i
$300$		0			0
$301$	14.6962	−	3.93782i	0.847072	−	0.226972i
$302$		0			0
$303$	−8.59808	−	2.30385i	−0.493947	−	0.132353i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		0			0
$309$	4.09808	−	7.09808i	0.233131	−	0.403795i
$310$		0			0
$311$		−	3.46410i		−	0.196431i	−0.995165	−	0.0982156i	$-0.968687\pi$
$311$		−	3.46410i		−	0.196431i	0.995165	−	0.0982156i	$-0.0313135\pi$
$312$		0			0
$313$	5.53590	+	5.53590i	0.312907	+	0.312907i	0.846035	−	0.533127i	$-0.178983\pi$
$313$	5.53590	+	5.53590i	0.312907	+	0.312907i	−0.533127	+	0.846035i	$0.678983\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	18.9282			1.06311			0.531557	−	0.847023i	$-0.321607\pi$
$317$	18.9282			1.06311			0.531557	+	0.847023i	$0.321607\pi$
$318$		0			0
$319$	−1.96410	+	7.33013i	−0.109969	+	0.410408i
$320$		0			0
$321$	1.96410	+	3.40192i	0.109625	+	0.189877i
$322$		0			0
$323$	−1.33013	+	2.30385i	−0.0740102	+	0.128190i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.29423	+	2.24167i	−0.0715710	+	0.123965i
$328$		0			0
$329$	−13.4641	−	23.3205i	−0.742300	−	1.28570i
$330$		0			0
$331$	4.45448	−	16.6244i	0.244841	−	0.913757i	−0.728623	−	0.684915i	$-0.759839\pi$
$331$	4.45448	−	16.6244i	0.244841	−	0.913757i	0.973464	−	0.228842i	$-0.0734940\pi$
$332$		0			0
$333$	4.19615			0.229948
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.9282	+	15.9282i	0.867665	+	0.867665i	0.992213	−	0.124549i	$-0.0397484\pi$
$337$	15.9282	+	15.9282i	0.867665	+	0.867665i	−0.124549	+	0.992213i	$0.539748\pi$
$338$		0			0
$339$			1.53590i			0.0834185i
$340$		0			0
$341$	−16.0981	+	27.8827i	−0.871760	+	1.50993i
$342$		0			0
$343$			19.5359i			1.05484i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	10.4282	+	2.79423i	0.559815	+	0.150002i	0.527621	−	0.849480i	$-0.323084\pi$
$347$	10.4282	+	2.79423i	0.559815	+	0.150002i	0.0321938	+	0.999482i	$0.489751\pi$
$348$		0			0
$349$	−7.86603	+	2.10770i	−0.421059	+	0.112822i	−0.463126	−	0.886292i	$-0.653272\pi$
$349$	−7.86603	+	2.10770i	−0.421059	+	0.112822i	0.0420673	+	0.999115i	$0.486606\pi$
$350$		0			0
$351$	2.09808	+	10.4904i	0.111987	+	0.559935i
$352$		0			0
$353$	−9.06218	−	5.23205i	−0.482331	−	0.278474i	0.239056	−	0.971006i	$-0.423162\pi$
$353$	−9.06218	−	5.23205i	−0.482331	−	0.278474i	−0.721387	+	0.692532i	$0.756495\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−0.571797	−	0.990381i	−0.0302627	−	0.0524165i
$358$		0			0
$359$	13.1962	+	13.1962i	0.696466	+	0.696466i	0.963647	−	0.267180i	$-0.0860919\pi$
$359$	13.1962	+	13.1962i	0.696466	+	0.696466i	−0.267180	+	0.963647i	$0.586092\pi$
$360$		0			0
$361$	−8.83013	−	5.09808i	−0.464744	−	0.268320i
$362$		0			0
$363$	−3.00000	+	3.00000i	−0.157459	+	0.157459i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.96410	+	14.7942i	0.206924	+	0.772252i	0.988854	+	0.148886i	$0.0475688\pi$
$367$	3.96410	+	14.7942i	0.206924	+	0.772252i	−0.781930	+	0.623366i	$0.785765\pi$
$368$		0			0
$369$	−18.6603	+	18.6603i	−0.971414	+	0.971414i
$370$		0			0
$371$	−8.29423	−	2.22243i	−0.430615	−	0.115383i
$372$		0			0
$373$	−5.30385	+	19.7942i	−0.274623	+	1.02491i	0.681471	+	0.731845i	$0.261341\pi$
$373$	−5.30385	+	19.7942i	−0.274623	+	1.02491i	−0.956094	+	0.293061i	$0.905326\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.59808	+	2.76795i	−0.288316	+	0.142557i
$378$		0			0
$379$	2.59808	+	9.69615i	0.133454	+	0.498058i	0.999999	−	0.00104063i	$-0.000331242\pi$
$379$	2.59808	+	9.69615i	0.133454	+	0.498058i	−0.866545	+	0.499099i	$0.833665\pi$
$380$		0			0
$381$	9.23205	−	5.33013i	0.472972	−	0.273071i
$382$		0			0
$383$	22.1147	−	12.7679i	1.13001	−	0.652412i	0.186073	−	0.982536i	$-0.440424\pi$
$383$	22.1147	−	12.7679i	1.13001	−	0.652412i	0.943937	+	0.330124i	$0.107091\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−16.2942	+	4.36603i	−0.828282	+	0.221938i
$388$		0			0
$389$	−15.0718			−0.764170			−0.382085	−	0.924127i	$-0.624794\pi$
$389$	−15.0718			−0.764170			−0.382085	+	0.924127i	$0.624794\pi$
$390$		0			0
$391$	0.124356			0.00628894
$392$		0			0
$393$	2.92820	−	0.784610i	0.147708	−	0.0395783i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.86603	−	5.69615i	0.495162	−	0.285882i	−0.231552	−	0.972823i	$-0.574380\pi$
$397$	9.86603	−	5.69615i	0.495162	−	0.285882i	0.726713	+	0.686941i	$0.241047\pi$
$398$		0			0
$399$	−3.27757	+	1.89230i	−0.164084	+	0.0947337i
$400$		0			0
$401$	2.66987	+	9.96410i	0.133327	+	0.497583i	0.999999	−	0.00129478i	$-0.000412141\pi$
$401$	2.66987	+	9.96410i	0.133327	+	0.497583i	−0.866672	+	0.498878i	$0.833745\pi$
$402$		0			0
$403$	−25.9808	+	5.19615i	−1.29419	+	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.74167	+	6.50000i	−0.0863314	+	0.322193i
$408$		0			0
$409$	18.5263	+	4.96410i	0.916066	+	0.245459i	0.685903	−	0.727693i	$-0.259407\pi$
$409$	18.5263	+	4.96410i	0.916066	+	0.245459i	0.230163	+	0.973152i	$0.426074\pi$
$410$		0			0
$411$	5.24167	−	5.24167i	0.258553	−	0.258553i
$412$		0			0
$413$	−5.74167	−	21.4282i	−0.282529	−	1.05441i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.90192	+	1.90192i	−0.0931376	+	0.0931376i
$418$		0			0
$419$	−31.5000	−	18.1865i	−1.53888	−	0.888470i	−0.998905	−	0.0467865i	$-0.985102\pi$
$419$	−31.5000	−	18.1865i	−1.53888	−	0.888470i	−0.539971	−	0.841684i	$-0.681565\pi$
$420$		0			0
$421$	6.85641	+	6.85641i	0.334161	+	0.334161i	0.854164	−	0.520003i	$-0.174069\pi$
$421$	6.85641	+	6.85641i	0.334161	+	0.334161i	−0.520003	+	0.854164i	$0.674069\pi$
$422$		0			0
$423$	14.9282	+	25.8564i	0.725834	+	1.25718i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	19.2058	+	11.0885i	0.929432	+	0.536608i
$428$		0			0
$429$	−8.16025	−	0.526279i	−0.393981	−	0.0254090i
$430$		0			0
$431$	−23.2583	+	6.23205i	−1.12031	+	0.300187i	−0.771011	−	0.636822i	$-0.780249\pi$
$431$	−23.2583	+	6.23205i	−1.12031	+	0.300187i	−0.349304	+	0.937010i	$0.613582\pi$
$432$		0			0
$433$	−32.5526	−	8.72243i	−1.56438	−	0.419173i	−0.630330	−	0.776327i	$-0.717081\pi$
$433$	−32.5526	−	8.72243i	−1.56438	−	0.419173i	−0.934046	+	0.357154i	$0.883747\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	0.411543i		−	0.0196868i
$438$		0			0
$439$	0.0358984	−	0.0621778i	0.00171334	−	0.00296759i	−0.865167	−	0.501483i	$-0.832788\pi$
$439$	0.0358984	−	0.0621778i	0.00171334	−	0.00296759i	0.866881	+	0.498515i	$0.166121\pi$
$440$		0			0
$441$		−	2.53590i		−	0.120757i
$442$		0			0
$443$	15.5885	+	15.5885i	0.740630	+	0.740630i	0.972699	−	0.232069i	$-0.0745496\pi$
$443$	15.5885	+	15.5885i	0.740630	+	0.740630i	−0.232069	+	0.972699i	$0.574550\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.78461			−0.0844091
$448$		0			0
$449$	−5.20577	+	19.4282i	−0.245676	+	0.916874i	0.727367	+	0.686249i	$0.240744\pi$
$449$	−5.20577	+	19.4282i	−0.245676	+	0.916874i	−0.973043	+	0.230625i	$0.925923\pi$
$450$		0			0
$451$	−21.1603	−	36.6506i	−0.996397	−	1.72581i
$452$		0			0
$453$	−0.0980762	+	0.169873i	−0.00460802	+	0.00798133i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.79423	−	11.7679i	0.317821	−	0.550481i	−0.662212	−	0.749316i	$-0.730382\pi$
$457$	6.79423	−	11.7679i	0.317821	−	0.550481i	0.980033	+	0.198835i	$0.0637157\pi$
$458$		0			0
$459$	1.33013	+	2.30385i	0.0620850	+	0.107534i
$460$		0			0
$461$	0.813467	−	3.03590i	0.0378869	−	0.141396i	−0.944392	−	0.328823i	$-0.893348\pi$
$461$	0.813467	−	3.03590i	0.0378869	−	0.141396i	0.982279	+	0.187427i	$0.0600148\pi$
$462$		0			0
$463$	21.6077			1.00419			0.502097	−	0.864811i	$-0.332562\pi$
$463$	21.6077			1.00419			0.502097	+	0.864811i	$0.332562\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.6603	−	16.6603i	−0.770945	−	0.770945i	0.207327	−	0.978272i	$-0.433524\pi$
$467$	−16.6603	−	16.6603i	−0.770945	−	0.770945i	−0.978272	+	0.207327i	$0.933524\pi$
$468$		0			0
$469$			30.2295i			1.39587i
$470$		0			0
$471$	−2.36603	+	4.09808i	−0.109021	+	0.188829i
$472$		0			0
$473$		−	27.0526i		−	1.24388i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.19615	+	2.46410i	0.421063	+	0.112823i
$478$		0			0
$479$	−3.13397	+	0.839746i	−0.143195	+	0.0383690i	−0.329705	−	0.944084i	$-0.606949\pi$
$479$	−3.13397	+	0.839746i	−0.143195	+	0.0383690i	0.186510	+	0.982453i	$0.440282\pi$
$480$		0			0
$481$	−4.96410	+	2.45448i	−0.226344	+	0.111915i
$482$		0			0
$483$	0.153212	+	0.0884573i	0.00697141	+	0.00402495i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	0.794229	+	1.37564i	0.0359899	+	0.0623364i	0.883459	−	0.468508i	$-0.155208\pi$
$487$	0.794229	+	1.37564i	0.0359899	+	0.0623364i	−0.847469	+	0.530844i	$0.821875\pi$
$488$		0			0
$489$	1.16987	+	1.16987i	0.0529035	+	0.0529035i
$490$		0			0
$491$	5.89230	+	3.40192i	0.265916	+	0.153527i	0.627030	−	0.778995i	$-0.284270\pi$
$491$	5.89230	+	3.40192i	0.265916	+	0.153527i	−0.361114	+	0.932522i	$0.617604\pi$
$492$		0			0
$493$	−1.09808	+	1.09808i	−0.0494549	+	0.0494549i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.47372	−	5.50000i	−0.0661054	−	0.246709i
$498$		0			0
$499$	20.2679	−	20.2679i	0.907318	−	0.907318i	−0.0887371	−	0.996055i	$-0.528283\pi$
$499$	20.2679	−	20.2679i	0.907318	−	0.907318i	0.996055	+	0.0887371i	$0.0282831\pi$
$500$		0			0
$501$	9.23205	+	2.47372i	0.412458	+	0.110518i
$502$		0			0
$503$	3.35641	−	12.5263i	0.149655	−	0.558519i	−0.849849	−	0.527026i	$-0.823307\pi$
$503$	3.35641	−	12.5263i	0.149655	−	0.558519i	0.999504	−	0.0314933i	$-0.0100263\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.10770	−	5.33013i	−0.182429	−	0.236719i
$508$		0			0
$509$	−7.45448	−	27.8205i	−0.330414	−	1.23312i	−0.908756	−	0.417328i	$-0.862967\pi$
$509$	−7.45448	−	27.8205i	−0.330414	−	1.23312i	0.578342	−	0.815795i	$-0.303700\pi$
$510$		0			0
$511$	31.8564	−	18.3923i	1.40924	−	0.813628i
$512$		0			0
$513$	7.62436	−	4.40192i	0.336624	−	0.194350i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−46.2487	+	12.3923i	−2.03402	+	0.545013i
$518$		0			0
$519$	1.48334			0.0651114
$520$		0			0
$521$	−7.85641			−0.344195			−0.172098	−	0.985080i	$-0.555054\pi$
$521$	−7.85641			−0.344195			−0.172098	+	0.985080i	$0.555054\pi$
$522$		0			0
$523$	−10.9641	+	2.93782i	−0.479427	+	0.128462i	−0.490436	−	0.871477i	$-0.663162\pi$
$523$	−10.9641	+	2.93782i	−0.479427	+	0.128462i	0.0110090	+	0.999939i	$0.496496\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.70577	+	3.29423i	−0.248547	+	0.143499i
$528$		0			0
$529$	19.9019	−	11.4904i	0.865301	−	0.499582i
$530$		0			0
$531$	6.36603	+	23.7583i	0.276262	+	1.03102i
$532$		0			0
$533$	11.1603	−	32.9904i	0.483404	−	1.42897i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.86603	−	6.96410i	0.0805249	−	0.300523i
$538$		0			0
$539$	3.92820	+	1.05256i	0.169200	+	0.0453369i
$540$		0			0
$541$	21.7846	−	21.7846i	0.936594	−	0.936594i	−0.0615128	−	0.998106i	$-0.519592\pi$
$541$	21.7846	−	21.7846i	0.936594	−	0.936594i	0.998106	+	0.0615128i	$0.0195925\pi$
$542$		0			0
$543$	3.07180	+	11.4641i	0.131823	+	0.491972i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.124356	−	0.124356i	0.00531706	−	0.00531706i	−0.704443	−	0.709760i	$-0.748803\pi$
$547$	0.124356	−	0.124356i	0.00531706	−	0.00531706i	0.709760	+	0.704443i	$0.248803\pi$
$548$		0			0
$549$	−21.2942	−	12.2942i	−0.908816	−	0.524705i
$550$		0			0
$551$	3.63397	+	3.63397i	0.154813	+	0.154813i
$552$		0			0
$553$	−0.660254	−	1.14359i	−0.0280769	−	0.0486305i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.3301	−	7.69615i	−0.564816	−	0.326096i	0.190260	−	0.981734i	$-0.439067\pi$
$557$	−13.3301	−	7.69615i	−0.564816	−	0.326096i	−0.755076	+	0.655637i	$0.772400\pi$
$558$		0			0
$559$	16.7224	−	14.6962i	0.707284	−	0.621581i
$560$		0			0
$561$	−1.96410	+	0.526279i	−0.0829244	+	0.0222195i
$562$		0			0
$563$	24.8923	+	6.66987i	1.04909	+	0.281102i	0.741874	−	0.670540i	$-0.233937\pi$
$563$	24.8923	+	6.66987i	1.04909	+	0.281102i	0.307212	+	0.951641i	$0.400604\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	16.4115i		−	0.689220i
$568$		0			0
$569$	18.8205	−	32.5981i	0.788997	−	1.36658i	−0.137585	−	0.990490i	$-0.543934\pi$
$569$	18.8205	−	32.5981i	0.788997	−	1.36658i	0.926582	−	0.376093i	$-0.122733\pi$
$570$		0			0
$571$		−	21.6077i		−	0.904254i	−0.891954	−	0.452127i	$-0.850665\pi$
$571$		−	21.6077i		−	0.904254i	0.891954	−	0.452127i	$-0.149335\pi$
$572$		0			0
$573$	5.49038	+	5.49038i	0.229364	+	0.229364i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−20.7846			−0.865275			−0.432637	−	0.901568i	$-0.642417\pi$
$577$	−20.7846			−0.865275			−0.432637	+	0.901568i	$0.642417\pi$
$578$		0			0
$579$	−3.42820	+	12.7942i	−0.142471	+	0.531710i
$580$		0			0
$581$	−3.60770	−	6.24871i	−0.149672	−	0.259240i
$582$		0			0
$583$	−7.63397	+	13.2224i	−0.316167	+	0.547617i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.7224	−	41.0885i	0.979130	−	1.69590i	0.313556	−	0.949570i	$-0.398480\pi$
$587$	23.7224	−	41.0885i	0.979130	−	1.69590i	0.665573	−	0.746333i	$-0.268187\pi$
$588$		0			0
$589$	10.9019	+	18.8827i	0.449206	+	0.778048i
$590$		0			0
$591$	−0.571797	+	2.13397i	−0.0235206	+	0.0877800i
$592$		0			0
$593$	30.9282			1.27007			0.635035	−	0.772484i	$-0.280986\pi$
$593$	30.9282			1.27007			0.635035	+	0.772484i	$0.280986\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.90192	+	6.90192i	0.282477	+	0.282477i
$598$		0			0
$599$		−	36.2487i		−	1.48108i	−0.672011	−	0.740541i	$-0.734569\pi$
$599$		−	36.2487i		−	1.48108i	0.672011	−	0.740541i	$-0.265431\pi$
$600$		0			0
$601$	7.42820	−	12.8660i	0.303003	−	0.524816i	−0.673812	−	0.738903i	$-0.735344\pi$
$601$	7.42820	−	12.8660i	0.303003	−	0.524816i	0.976815	+	0.214087i	$0.0686775\pi$
$602$		0			0
$603$		−	33.5167i		−	1.36490i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.03590	−	0.813467i	−0.123223	−	0.0330176i	0.196680	−	0.980468i	$-0.436984\pi$
$607$	−3.03590	−	0.813467i	−0.123223	−	0.0330176i	−0.319904	+	0.947450i	$0.603651\pi$
$608$		0			0
$609$	−2.13397	+	0.571797i	−0.0864730	+	0.0231704i
$610$		0			0
$611$	−32.7846	−	21.8564i	−1.32632	−	0.884216i
$612$		0			0
$613$	21.1865	+	12.2321i	0.855716	+	0.494048i	0.862575	−	0.505929i	$-0.168850\pi$
$613$	21.1865	+	12.2321i	0.855716	+	0.494048i	−0.00685934	+	0.999976i	$0.502183\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8.79423	−	15.2321i	−0.354042	−	0.613219i	0.632911	−	0.774224i	$-0.281860\pi$
$617$	−8.79423	−	15.2321i	−0.354042	−	0.613219i	−0.986954	+	0.161005i	$0.948526\pi$
$618$		0			0
$619$	14.6603	+	14.6603i	0.589245	+	0.589245i	0.937427	−	0.348182i	$-0.113201\pi$
$619$	14.6603	+	14.6603i	0.589245	+	0.589245i	−0.348182	+	0.937427i	$0.613201\pi$
$620$		0			0
$621$	−0.356406	−	0.205771i	−0.0143021	−	0.00825732i
$622$		0			0
$623$	−26.6865	+	26.6865i	−1.06917	+	1.06917i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.74167	+	6.50000i	0.0695556	+	0.259585i
$628$		0			0
$629$	−0.973721	+	0.973721i	−0.0388248	+	0.0388248i
$630$		0			0
$631$	41.6506	+	11.1603i	1.65809	+	0.444283i	0.961860	−	0.273541i	$-0.0881948\pi$
$631$	41.6506	+	11.1603i	1.65809	+	0.444283i	0.696225	+	0.717823i	$0.254862\pi$
$632$		0			0
$633$	−0.794229	+	2.96410i	−0.0315678	+	0.117812i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.48334	+	3.00000i	0.0587721	+	0.118864i
$638$		0			0
$639$	1.63397	+	6.09808i	0.0646390	+	0.241236i
$640$		0			0
$641$	5.64359	−	3.25833i	0.222909	−	0.128696i	−0.384388	−	0.923172i	$-0.625587\pi$
$641$	5.64359	−	3.25833i	0.222909	−	0.128696i	0.607296	+	0.794475i	$0.292254\pi$
$642$		0			0
$643$	31.3301	−	18.0885i	1.23554	−	0.713339i	0.267360	−	0.963597i	$-0.413849\pi$
$643$	31.3301	−	18.0885i	1.23554	−	0.713339i	0.968179	+	0.250258i	$0.0805153\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.50000	+	0.669873i	−0.0982851	+	0.0263354i	−0.307626	−	0.951507i	$-0.599535\pi$
$647$	−2.50000	+	0.669873i	−0.0982851	+	0.0263354i	0.209341	+	0.977843i	$0.432868\pi$
$648$		0			0
$649$	−39.4449			−1.54835
$650$		0			0
$651$	−9.37307			−0.367359
$652$		0			0
$653$	−47.9449	+	12.8468i	−1.87623	+	0.502734i	−0.876453	+	0.481487i	$0.840097\pi$
$653$	−47.9449	+	12.8468i	−1.87623	+	0.502734i	−0.999774	+	0.0212467i	$0.993236\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−35.3205	+	20.3923i	−1.37798	+	0.795580i
$658$		0			0
$659$	−35.4282	+	20.4545i	−1.38009	+	0.796794i	−0.992169	−	0.124901i	$-0.960139\pi$
$659$	−35.4282	+	20.4545i	−1.38009	+	0.796794i	−0.387918	+	0.921694i	$0.626805\pi$
$660$		0			0
$661$	−0.794229	−	2.96410i	−0.0308919	−	0.115290i	0.948758	−	0.316003i	$-0.102341\pi$
$661$	−0.794229	−	2.96410i	−0.0308919	−	0.115290i	−0.979650	+	0.200713i	$0.935674\pi$
$662$		0			0
$663$	−1.39230	−	0.928203i	−0.0540726	−	0.0360484i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.0621778	−	0.232051i	0.00240754	−	0.00898504i
$668$		0			0
$669$	0.767949	+	0.205771i	0.0296906	+	0.00795558i
$670$		0			0
$671$	27.8827	−	27.8827i	1.07640	−	1.07640i
$672$		0			0
$673$	−9.62436	−	35.9186i	−0.370992	−	1.38456i	−0.859114	−	0.511784i	$-0.828985\pi$
$673$	−9.62436	−	35.9186i	−0.370992	−	1.38456i	0.488122	−	0.872775i	$-0.337682\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.1769	−	28.1769i	1.08293	−	1.08293i	0.0866916	−	0.996235i	$-0.472371\pi$
$677$	28.1769	−	28.1769i	1.08293	−	1.08293i	0.996235	−	0.0866916i	$-0.0276295\pi$
$678$		0			0
$679$	−16.5000	−	9.52628i	−0.633212	−	0.365585i
$680$		0			0
$681$	−1.16987	−	1.16987i	−0.0448296	−	0.0448296i
$682$		0			0
$683$	−7.33013	−	12.6962i	−0.280480	−	0.485805i	0.691023	−	0.722832i	$-0.257160\pi$
$683$	−7.33013	−	12.6962i	−0.280480	−	0.485805i	−0.971503	+	0.237028i	$0.923827\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	0.0455173	+	0.0262794i	0.00173659	+	0.00100262i
$688$		0			0
$689$	−12.3205	+	2.46410i	−0.469374	+	0.0938748i
$690$		0			0
$691$	−37.6506	+	10.0885i	−1.43230	+	0.383783i	−0.889829	−	0.456293i	$-0.849177\pi$
$691$	−37.6506	+	10.0885i	−1.43230	+	0.383783i	−0.542468	+	0.840076i	$0.682510\pi$
$692$		0			0
$693$	28.4904	+	7.63397i	1.08226	+	0.289991i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	8.66025i		−	0.328031i
$698$		0			0
$699$	−4.70577	+	8.15064i	−0.177989	+	0.308285i
$700$		0			0
$701$		−	21.0718i		−	0.795871i	−0.917413	−	0.397935i	$-0.869727\pi$
$701$		−	21.0718i		−	0.795871i	0.917413	−	0.397935i	$-0.130273\pi$
$702$		0			0
$703$	3.22243	+	3.22243i	0.121536	+	0.121536i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−42.3731			−1.59360
$708$		0			0
$709$	−4.27757	+	15.9641i	−0.160647	+	0.599544i	0.837908	+	0.545812i	$0.183779\pi$
$709$	−4.27757	+	15.9641i	−0.160647	+	0.599544i	−0.998555	+	0.0537328i	$0.982888\pi$
$710$		0			0
$711$	0.732051	+	1.26795i	0.0274541	+	0.0475518i
$712$		0			0
$713$	0.509619	−	0.882686i	0.0190854	−	0.0330568i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.09808	−	10.5622i	0.227737	−	0.394452i
$718$		0			0
$719$	−20.9641	−	36.3109i	−0.781829	−	1.35417i	−0.930875	−	0.365337i	$-0.880954\pi$
$719$	−20.9641	−	36.3109i	−0.781829	−	1.35417i	0.149046	−	0.988830i	$-0.452380\pi$
$720$		0			0
$721$	10.0981	−	37.6865i	0.376072	−	1.40352i
$722$		0			0
$723$	6.07180			0.225813
$724$		0			0
$725$		0			0
$726$		0			0
$727$	33.5885	+	33.5885i	1.24573	+	1.24573i	0.957589	+	0.288138i	$0.0930362\pi$
$727$	33.5885	+	33.5885i	1.24573	+	1.24573i	0.288138	+	0.957589i	$0.406964\pi$
$728$		0			0
$729$			13.5885i			0.503276i
$730$		0			0
$731$	2.76795	−	4.79423i	0.102376	−	0.177321i
$732$		0			0
$733$		−	26.7846i		−	0.989312i	−0.869089	−	0.494656i	$-0.835294\pi$
$733$		−	26.7846i		−	0.989312i	0.869089	−	0.494656i	$-0.164706\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	51.9186	+	13.9115i	1.91245	+	0.512438i
$738$		0			0
$739$	−37.9186	+	10.1603i	−1.39486	+	0.373751i	−0.876495	−	0.481410i	$-0.840125\pi$
$739$	−37.9186	+	10.1603i	−1.39486	+	0.373751i	−0.518362	+	0.855161i	$0.673458\pi$
$740$		0			0
$741$	−3.07180	+	4.60770i	−0.112845	+	0.169268i
$742$		0			0
$743$	19.3301	+	11.1603i	0.709154	+	0.409430i	0.810748	−	0.585396i	$-0.199061\pi$
$743$	19.3301	+	11.1603i	0.709154	+	0.409430i	−0.101594	+	0.994826i	$0.532394\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.00000	+	6.92820i	0.146352	+	0.253490i
$748$		0			0
$749$	13.2224	+	13.2224i	0.483137	+	0.483137i
$750$		0			0
$751$	17.6436	+	10.1865i	0.643824	+	0.371712i	0.786086	−	0.618117i	$-0.212104\pi$
$751$	17.6436	+	10.1865i	0.643824	+	0.371712i	−0.142262	+	0.989829i	$0.545438\pi$
$752$		0			0
$753$	−6.49038	+	6.49038i	−0.236523	+	0.236523i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	4.83975	+	18.0622i	0.175904	+	0.656481i	0.996396	+	0.0848236i	$0.0270327\pi$
$757$	4.83975	+	18.0622i	0.175904	+	0.656481i	−0.820492	+	0.571657i	$0.806301\pi$
$758$		0			0
$759$	0.222432	−	0.222432i	0.00807377	−	0.00807377i
$760$		0			0
$761$	−0.669873	−	0.179492i	−0.0242829	−	0.00650658i	0.246657	−	0.969103i	$-0.420668\pi$
$761$	−0.669873	−	0.179492i	−0.0242829	−	0.00650658i	−0.270940	+	0.962596i	$0.587335\pi$
$762$		0			0
$763$	−3.18911	+	11.9019i	−0.115454	+	0.430879i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−21.4282	−	24.3827i	−0.773728	−	0.880408i
$768$		0			0
$769$	5.47372	+	20.4282i	0.197387	+	0.736660i	0.991636	+	0.129067i	$0.0411982\pi$
$769$	5.47372	+	20.4282i	0.197387	+	0.736660i	−0.794248	+	0.607593i	$0.792135\pi$
$770$		0			0
$771$	−10.0359	+	5.79423i	−0.361434	+	0.208674i
$772$		0			0
$773$	−17.0429	+	9.83975i	−0.612992	+	0.353911i	−0.774136	−	0.633020i	$-0.781815\pi$
$773$	−17.0429	+	9.83975i	−0.612992	+	0.353911i	0.161144	+	0.986931i	$0.448482\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.89230	+	0.507042i	−0.0678861	+	0.0181900i
$778$		0			0
$779$	−28.6603			−1.02686
$780$		0			0
$781$	−10.1244			−0.362278
$782$		0			0
$783$	4.96410	−	1.33013i	0.177403	−	0.0475349i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	37.4545	−	21.6244i	1.33511	−	0.770825i	0.349031	−	0.937111i	$-0.386511\pi$
$787$	37.4545	−	21.6244i	1.33511	−	0.770825i	0.986078	+	0.166286i	$0.0531776\pi$
$788$		0			0
$789$	−6.44744	+	3.72243i	−0.229535	+	0.132522i
$790$		0			0
$791$	1.89230	+	7.06218i	0.0672826	+	0.251102i
$792$		0			0
$793$	32.3827	+	2.08846i	1.14994	+	0.0741633i
$794$		0			0
$795$		0			0
$796$		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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.bs (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.133975i) q^{3} +(2.13397 - 1.23205i) q^{7} +(-2.36603 + 1.36603i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.133975i) q^{3} +(2.13397 - 1.23205i) q^{7} +(-2.36603 + 1.36603i) q^{9} +(-1.13397 - 4.23205i) q^{11} +(2.00000 - 3.00000i) q^{13} +(0.232051 - 0.866025i) q^{17} +(-2.86603 - 0.767949i) q^{19} +(0.901924 - 0.901924i) q^{21} +(0.0358984 + 0.133975i) q^{23} +(-2.09808 + 2.09808i) q^{27} +(-1.50000 - 0.866025i) q^{29} +(-5.19615 - 5.19615i) q^{31} +(-1.13397 - 1.96410i) q^{33} +(-1.33013 - 0.767949i) q^{37} +(0.598076 - 1.76795i) q^{39} +(9.33013 - 2.50000i) q^{41} +(5.96410 + 1.59808i) q^{43} -10.9282i q^{47} +(-0.464102 + 0.803848i) q^{49} -0.464102i q^{51} +(-2.46410 - 2.46410i) q^{53} -1.53590 q^{57} +(2.33013 - 8.69615i) q^{59} +(4.50000 + 7.79423i) q^{61} +(-3.36603 + 5.83013i) q^{63} +(-6.13397 + 10.6244i) q^{67} +(0.0358984 + 0.0621778i) q^{69} +(0.598076 - 2.23205i) q^{71} +14.9282 q^{73} +(-7.63397 - 7.63397i) q^{77} -0.535898i q^{79} +(3.33013 - 5.76795i) q^{81} -2.92820i q^{83} +(-0.866025 - 0.232051i) q^{87} +(-14.7942 + 3.96410i) q^{89} +(0.571797 - 8.86603i) q^{91} +(-3.29423 - 1.90192i) q^{93} +(-3.86603 - 6.69615i) q^{97} +(8.46410 + 8.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9	\( 4 q + 2 q^{3} + 12 q^{7} - 6 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 8 q^{19} + 14 q^{21} + 14 q^{23} + 2 q^{27} - 6 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 20 q^{41} + 10 q^{43} + 12 q^{49} + 4 q^{53} - 20 q^{57} - 8 q^{59} + 18 q^{61} - 10 q^{63} - 28 q^{67} + 14 q^{69} - 8 q^{71} + 32 q^{73} - 34 q^{77} - 4 q^{81} - 28 q^{89} + 30 q^{91} + 18 q^{93} - 12 q^{97} + 20 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 12 * q^7 - 6 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 8 * q^19 + 14 * q^21 + 14 * q^23 + 2 * q^27 - 6 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 20 * q^41 + 10 * q^43 + 12 * q^49 + 4 * q^53 - 20 * q^57 - 8 * q^59 + 18 * q^61 - 10 * q^63 - 28 * q^67 + 14 * q^69 - 8 * q^71 + 32 * q^73 - 34 * q^77 - 4 * q^81 - 28 * q^89 + 30 * q^91 + 18 * q^93 - 12 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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